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1993-04-09T17:14:43 | 9303 | alg-geom/9303006 | en | https://arxiv.org/abs/alg-geom/9303006 | [
"alg-geom",
"math.AG"
] | alg-geom/9303006 | Roberto Paoletti | Roberto Paoletti | Seshadri constants, gonality of space curves and restriction of stable
bundles | 36 pages, amslatex | null | null | null | null | We define the Seshadri constant of a space curve and consider ways to
estimate it. We then show that it governs the gonality of the curve. We use an
argument based on Bogomolov's instability theorem on a threefold. The same
methods are then applied to the study of the behaviour of a stable vector
bundle on P^3 under restriction to curves and surfaces.
| [
{
"version": "v1",
"created": "Sun, 28 Mar 1993 23:23:01 GMT"
},
{
"version": "v2",
"created": "Fri, 9 Apr 1993 15:15:04 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Paoletti",
"Roberto",
""
]
] | alg-geom | \section{\bf {Introduction}}
There exist many situations in algebraic geometry
where the extrinsic geometry of a variety is reflected
in clear restrictions in the way that it can map
to projective spaces. For example, it is well-known that
the gonality of a smooth plane curve $C$ of degree $d$ is
$d-1$, and that every minimal pencil has the form
$\cal O_C(H-P)$, where $H$
denotes the hyperplane class and $P\in C$.
In fact, there exist to date various statements of this kind
concerning the existence of morphisms from a divisor to
$\bold P^1$. The first general results in this direction
are due to Sommese (\cite{so:amp}) and Serrano (\cite{se:ext}).
Reider (\cite{re:app}) then showed that at least part of
Serrano's results for surfaces can be obtained by use of
vector bundle methods based on the Bogomolov-Gieseker inequality
for semistable vector bundles on a surface.
In \cite{pa}, a generalization of these methods to higher
dimensional varieties is used to obtain the following statement:
\begin{thm} Let $X$ be a smooth projective $n$-fold,
and let $Y\subset X$ be a reduced irreducible divisor. If
$n\ge 3$ assume that $Y$ is ample, and if $n=2$ assume
that $Y^2>0$ (so that in particular it is at least nef).
Let $\phi :Y@>>>\bold P^1$ be a morphism, and let $F$ denote
the numerical class of a fiber.
\noindent
(i) If
$$F\cdot Y^{n-2}<\sqrt {Y^n}-1,$$
then there exists a morphism $\psi :X@>>>\bold P^1$
extending $\phi$. Furthermore, the restriction
$$H^0(X,\psi ^{*}\cal O_{\bold P^1}(1))@>>>
H^0(Y,\phi ^{*}\cal O_{\bold P^1}(1))$$
is injective. In particular, $\psi$ is linearly normal if
$\phi$ is.
\noindent
(ii) If
$$F\cdot Y^{n-2}=\sqrt {Y^n}-1$$
and $Y^n\neq 4$, then either there exists an
extension $\psi :X@>>>\bold P^1$ of $\phi$, or else
we can find an effective divisor
$D$ on $X$ such that
$(D\cdot Y^{n-1})^2=(D^2\cdot Y^{n-2})Y^n$ and
$D\cdot Y^{n-1}=\sqrt {Y^n}$,
and an inclusion
$$\phi ^{*}\cal O_{\bold P^1}(1)\subset \cal O_Y(D).$$
\end{thm}
\bigskip
However, a much less understood range of situations is the one where
$codim(Y)\ge 2$. In some particular cases there are
rather precise statements. In curve theory, in particular,
one has a clear picture of the gonality of Castelnuovo extremal
curves (\cite{acgh}). In even degree,
for example, if $C\subset \bold P^3$ is a smooth complete intersection
of a smooth quadric and a hypersurface of degree $a\ge 2$, the
gonality is attained by restricting to $C$ the rulings on the quadric.
More generally, unpublished work of Lazarsfeld shows that
if $C\subset \bold P^3$ is a smooth complete intersection of type
$(a,b)$, with $a\ge b$, then $gon(C)\ge a(b-1)$. Lazarsfeld's
argument is also based on Bogomolov's instability theorem.
In a somewhat more general
direction, Ciliberto and Lazarsfeld have studied linear series
of low degree on various classes of space curves (\cite{cl}). Their method
is based on the number of conditions imposed by a linear series
on another.
Naturally enough, one is led to investigate
more general situations. We
shall focus on the gonality
of space curves, and then show how the methods
developped apply to other circumstances as well.
In the codimension $1$ case
we have seen that the self intersection of the divisor
governs the numerical constraint on a free pencil on $Y$.
Loosely speaking, in the
higher codimension case a similar role is played by the
Seshadri costant of the curve.
This is defined as follows.
Consider a smooth curve $C\subset \bold P^3$
and denote by
$$f:X_C@>>>\bold P^3$$
the blow up of $\bold P^3$ along $C$, and by
$$E=f^{-1}C$$
the exceptional divisor.
The {\it Seshadri constant} of $C$ is
\begin{center}
$\epsilon (C)=sup\{\eta \in \bold Q|
f^{*}H-\eta E$ is ample$\}$.
\end{center}
This is a very delicate invariant, and it gathers classical
information such as what secants the curve has and
the minimal degree in which powers of $\cal J_C$ are globally
generated.
For example, if $C\subset \bold P^3$ is a complete intersection
of type $(a,b)$, with $a\ge b$, then $\epsilon (C)=\frac 1a$.
More generally, if $C\subset \bold P^3$ is defined
as the zero locus of a regular section of a rank two vector bundle
$\cal E$,
then we have an estimate
$\epsilon (C)\ge \gamma (\cal E)$, where
$\gamma (\cal E)$ is the Seshadri constant of $\cal E$, defined as
\bigskip
\begin{center}
$\gamma (\cal E)=sup\{\frac nm|S^n\cal E^{*}(m)$ is
globally generated$\}$.
\end{center}
\bigskip
\noindent
It is always true that $\epsilon (C)\ge \frac 1d$.
However, the problem of finding general
optimal estimates $\epsilon (C)$ for an
arbitrary curve seems to be a hard one.
Something can be said,
for example, as soon as $C$ can be expressed
as an irreducible component of a complete intersection of smooth
surfaces.
Interest in Seshadri constants, of course, is not new.
In fact, if $Y$ is a subvariety of any projective variety
$X$, one can define in an obvious way the Seshadri constant
of $Y$ with respect to any polarization $H$ on $X$.
Seshadri constants of points, in particular, have received
increasing attention recently, partly in relation to
the quest for Fujita-type results. A differential geometric
interpretation has been given by DeMailly (\cite{de}).
Seshadri constants of points on a surface have been investigated
by Ein and Lazarsfeld (\cite{el}), who have proved the
surprising fact that
they can be bounded away from zero at all but countably many
points of $S$. However, Seshadri constants of higher dimensional
subvarieties have apparently never been put at use.
What a bound on the gonality of a space curve might look
like is suggested by Lazarsfeld's result. In fact, we may
write $a(b-1)=deg(C)-\frac 1{\epsilon (C)}$, so that for a complete
intersection we have the optimal bound
$$gon(C)\ge d-\frac 1{\epsilon (C)}.$$
Keeping the notation above, let us define
$$H_{\eta}=f^{*}H-\eta E$$
and
$$\delta _{\eta}(C)=\eta \cdot deg(N)-d,$$
where $N$ is the normal bundle of $C$.
For example, for a complete intersection of
type $(a,b)$ with $a\ge b$ we have $\delta _{1/a}(C)=b^2$.
$\delta _{\eta}(C)$ has a simple geometric meaning, that we
explain at the end of Chapter 3.
Our result is
\begin{thm} Let $C\subset \bold P^3$ be a smooth
curve of degree $d$ and Seshadri constant $\epsilon (C)$.
Set $\alpha =min \big \{1,\sqrt d\big (1-\epsilon (C)\sqrt d\big
)\big \}$.
Then
$$gon(C)\ge min \Big \{\frac {\delta _{\epsilon (C)}
(C)}{4\epsilon (C)},
\alpha \Big (d-\frac {\alpha}{\epsilon (C)}\Big )\Big \}.$$
\end{thm}
This reproduces Lazarsfeld's result if $a\ge b+3$. As another
example, it says that if $a\gg b$ and $C$ is residual to a line
in a complete intersection of type $(a,b)$, then
$gon(C)=ab-(a+b-2)$ (consider the pencil of planes through the
line).
In view of the above, one would expect the above bound
to hold with $\alpha =1$ always, but I have been unable to prove it.
The idea of the proof is as follows. If $A$ is a minimal pencil
on $C$, and if $\pi :E@>>>C$ is the induced projection, one
can define a rank two vector bundle on $X_C$ by the exactness
of the sequence
$$0@>>>\cal F@>>>H^0(C,A)\otimes \cal O_{X_C}@>>>\pi ^{*}A@>>>0.$$
The numerical assuptions then force $\cal F$ to be Bogomolov unstable
w.r.t. $H_{\epsilon (C)}$ (see $\S0$) and therefore a
maximal destabilizing line bundle
$$\cal O_{X_C}(-D)\subset \cal F$$
comes into the picture.
$D$ and $A$ are related by the inequalities coming from the instability
of $\cal F$, and from this one can show that $deg(A)$ is forced
to satisfy the above bound.
By its general nature, this argument can be applied to the study
of linear series on arbitrary smooth subvarieties of $\bold P^r$.
We will not detail this generalization here.
\bigskip
In another direction, similar methods have been used by Bogomolov
(\cite{bo:78} and \cite{bo:svb})
to study the behaviour of a stable bundle on a surface under
restriction to a curve $C$ that is linearly equivalent to
a multiple of the polarization at hand.
For example, it follows from Bogomolov's theorem that
if $S$ is a smooth surface with $Pic(S)\simeq \bold Z$
and $\cal E$ is a stable rank two vector bundle on $S$,
then $\cal E|_C$ is also stable, for every irreducible curve
$C\subset S$ such that $C^2>4c_2(\cal E)^2$.
A more complicated statement holds for arbitrary surfaces.
One can see, in fact, that
this result implies a similar one for surfaces in $\bold P^3$.
In the spirit of the above discussion, one is then led to
consider the problem of the behaviour under restriction to
subvarieties of higher codimension. The inspiring idea,
suggested by the divisor case, should be that when some suitable
invariants, describing some form of "positivity" of the
subvariety, become large with respect to the invariants of
the vector bundle, then stability is preserved under restriction.
Furthermore, if in the divisor case one needs the hypothesis
that $\cal E$ be $\cal O_S(C)$-stable, in the higher codimension
case one should still expect some
measure of the relation between the geometry of
the subvariety and the stability of the vector bundle to play
a role in the solution to the problem.
In fact, in the case of space curves
the same kind of argument that proves the theorem
about gonality
can be applied to this question. Before explaining the result,
we need the following definition. Recall
that if $X$ is a smooth projective
threefold, $\cal F$ is a vector bundle on $X$ and $L$ and
$H$ are two
nef line bundles on $X$, $\cal F$ is
said to be $(H,L)$-stable if for every
nontrivial subsheaf $\cal G\subset
\cal F$ we have $(fc_1(\cal G)-gc_1(\cal F))\cdot H\cdot L<
0$,
where $f=rank(\cal F)$ and $g=rank(\cal G)$.
Let then $\cal E$ be a rank two vector bundle on
$\bold P^3$, and consider a curve $C\subset \bold P^3$.
Let us define the
{\it stability constant of $\cal E$ with respect to $C$} as
\bigskip
\begin{center}
$\gamma (C,\cal E)=sup\{\eta \in [0,\epsilon (C)]|
f^{*}\cal E$ is $(H,H_{\eta})$-stable$\}.$
\end{center}
\bigskip
\noindent
For example, if $C$ is a complete intersection of type $(a,b)$
and the restriction of $\cal E$ to one of the two surfaces defining $C$
is stable (with respect to the hyperplane bundle) then
$\gamma (C,\cal E)=\epsilon (C)$.
Then we have
\begin{thm} Let
$\cal E$ be a stable rank two vector bundle on $\bold P^3$
with $c_1(\cal E)=0$. Let $C\subset \bold P^3$ be a smooth curve
of degree $d$ and Seshadri constant $\epsilon (C)$,
and let $\gamma =\gamma (C,\cal E)$ be the stability constant
of $\cal E$ w.r.t. $C$. Suppose that $\cal E|_C$ is not
stable. Then
$$c_2(\cal E)\ge min\Big \{\frac {\delta _{\gamma}(C)}
4, \alpha \gamma \Big (d-\frac {\alpha}{\gamma}\Big ) \Big \},$$
where $\alpha =:min\Big \{1,\sqrt d
\Big (\sqrt {\frac 34}-\gamma \sqrt d
\Big )\Big \}$.
\end{thm}
The problem of the behaviour of stable bundles on $\bold P^r$
under restriction to curves has been studied by many researchers.
In particular, a well-known fundamental theorem of Mehta and
Ramanathan
(\cite{mr:res}) shows that $\cal E|_C$ is stable if $C$ is a {\it general}
complete intersection curve of type $(a_1,a_2,\cdots)$, and all the
$a_i\gg 0$. Flenner (\cite{fl}) has then given an explicit
bound on the $a_i$s in term of the invariants of $\cal E$
which makes the conclusion of Mehta and Ramanathan's Theorem true.
On the other hand, here we give numerical conditions that
imply stability for $\cal E|_C$, with no generality assumption
and without restricting $C$ to be a complete intersection.
We have the following applications:
\begin{cor} Let $\cal E$ be a stable rank two vector bundle
on $\bold P^3$ with $c_1(\cal E)=0$ and $c_2(\cal E)=c_2$.
Suppose that $b\ge c_2+2$. If $V\subset \bold P^3$ is a smooth
surface of degree $b$, then $\cal E|_V$ is
$\cal O_V(H)$-stable.
\end{cor}
\begin{cor} Let $\cal E$ be a stable bundle on $\bold P^3$
with $c_1(\cal E)=0$ and $c_2(\cal E)\neq 1$. Suppose that $C=
V_a\cap V_b\subset \bold P^3$ is an
irreducible smooth complete intersection
curve and that $V_a$ is smooth.
Assume furthermore that $a\ge \frac 43b+\frac {10}3$ that
and that $b\ge c_2(\cal E)+2$. Then $\cal E|_C$
is stable.
\end{cor}
\begin{cor} Let $c_2\ge 0$ be an integer and let
$\cal M(0,c_2)$ denote the moduli space of stable rank two vector
bundles on $\bold P^3$. If $a\gg b\gg c_2$ and
$C\subset \bold P^3$ is an irreducible smooth complete intersection of
type $(a,b)$, then
$\cal
M(0,c_2)$ embeds in the moduli space of stable vector
bundles of degree $0$ on $C$.
\end{cor}
\bigskip
\bigskip
This paper covers part
of the content of my Phd thesis at UCLA.
I want to thank Robert Lazarsfeld, my advisor, for introducing me to
Algebraic Geometry and taking continuous interest in my progress.
I am also endebted to a number of people for valuable comments
and discussions; among them, O. Garcia-Prada, D. Gieseker, M. Green,
J. Li and A. Moriwaki.
\section{\bf {Preliminaries}}\label{section:preliminaries}
In this section we state some results that will be
used in the sequel.
The following fact is well-known:
\begin{lem} Let $X$ be a smooth projective variety
and let $Y\subset X$ be a divisor.
Suppose that we have an exact sequence:
$$0@>>>\cal F@>>>\cal E@>>>A@>>>0,$$
where $A$ is a line bundle on $Y$ and $\cal E$ is a rank
two vector bundle on $X$. Let $[Y]\in A^1(X)$ be the divisor
class of $Y$ and let $[A]\in A^2(X)$ be the image of
the divisor class of $A$ under the push forward
$A^1(Y)@>>>A^2(X)$.
Then $\cal F$ is a rank two vector bundle on $X$,
having Chern classes
$c_1(\cal F)=c_1(\cal E)-[Y]$ and $c_2(\cal F)=c_2(\cal E)+[A]
-Y\cdot c_1(\cal E)$.
\label{lem:eltr}
\end{lem}
{\it Proof.} The first statement follows by considering
local trivializations.
As to
the Chern classes of
$\cal F$, we could prove the statement by directly
computing
$$c_t(\cal F)=c_t(\cal E)\cdot c_t(A)^{-1}.$$
However, the following shorter argument proves
that the above equalities hold numerically,
after multiplying both sides by $n-2$ nef divisor classes
(which is what we need).
First of all, the morphism $\cal F@>>>\cal E$ drops rank
along $Y$, and therefore $c_1(\cal F)=c_1(\cal E)-Y$. Let
us consider the second equality.
If $X$ is a surface, the proof is reduced to
a Riemann-Roch computation. If
$dim(X)=3$, let $H$ be any very
ample divisor on $X$, and let $S\in |H|$ be a general smooth surface.
By generality, we may assume that $C=S\cap Y$ is a smooth irreducible
curve.
Then by restriction we obtain an exact sequence on
$S$:
$0@>>>\cal F|_S@>>>\cal E|_S@>>>A|_C@>>>0$.
By applying the statement for the surface case,
we then obtain $(c_2(\cal F)-c_2(\cal E)-[A]
+Y\cdot c_1(\cal E))\cdot H=0$.
But then the expression between brackets has to be
killed by all ample divisors, and so it is numerically
trivial.
The general case is similar.
$\sharp$
\bigskip
\begin{lem} Let $X$ be a smooth projective threefold, and
let $C\subset X$ be a smooth curve in $X$.
Denote by $f:X_C@>>>X$ the blow up of $X$ along $C$, and
let $E$ be the exceptional divisor.
Then $E^3=-deg(N)$, where $N$ is the normal bundle of
$C$ in $X$.
Furthermore, let $A$ be any line bundle on $X$, and by abuse
of language let $A$ also denote its pull-back to $X_C$.
Then $E^2\cdot A=-C\cdot A$.
\label{lem:segre}
\end{lem}
{\it Proof.} Both statements follow from a simple Segre class
computation (see for example \cite{fu}).
$\sharp$
\bigskip
We now recall some known results about instability of
rank two vector
bundles on projective manifolds, which are one of the main tools
in the following analysis. Recall the following notation.
\begin{defn}
If $S$ is a smooth projective surface, $N(S)$ is the vector
space of the numerical equivalence classes of divisors in
$S$; $K^{+}(S)
\subset N(S)$ is the (positive) cone spanned by those
divisors $D$ such that $D^2>0$ and $D\cdot H>0$ for some
polarization on $S$. In general, if $X$ is a smooth
projective $n$-fold and $H$ is a polarization
on it, we shall denote by $K^{+}(X,H)$ the cone of all
numerical classes $D$ in $N(X)$ such that
$D^2\cdot H^{n-2}>0$ and $D\cdot H^{n-1}>0$ (or,
equivalently, $D\cdot R\cdot H^{n-2}>0$ for
any other polarization $R$ on $X$).
\label{defn:poscone}
\end{defn}
\begin{defn} Let $X$ be a smooth projective $n$-fold, and let
$\cal E$ be a rank two vector bundle on $X$, with Chern
classes $c_1(\cal E)$ and $c_2(\cal E)$.
The {\it discriminant} $\Delta (\cal E)\in A^2(X)$ is
$$\Delta (\cal E)=c_1(\cal E)^2-4c_2(\cal E).$$
\label{defn:discr}
\end{defn}
\begin{lem} Let $X$ be a smooth projective
$n$-fold, and let $\cal E$ be a rank two vector bundle
on $X$.
Fix a polarization $H$ on $X$.
Suppose that $\cal L_1, \cal L_2\subset \cal E$
are two line bundles in $\cal E$. Let us
denote by $l_1$ and $l_2$ their $H$-degrees, respectively
(i.e., $l_i=\cal L_i\cdot H^{n-1}$) and let
$e=deg_H(\cal E)=\wedge ^2\cal E\cdot H^{n-1}$ be the $H$-degree
of $\cal E$. Suppose that
$2l_i>e$ for $i=1$ and $i=2$ (in other words,
$\cal L_1$ and $\cal L_2$ make $\cal E$ $H$-unstable).
If $\cal L_2$ is saturated
in $\cal E$, then $\cal L_1\subseteq \cal L_2$.
\label{lem:contains}
\end{lem}
{\it Proof.} Set
$l=min \{l_1,l_2\}$. By assumption, we have
$$2l-e>0.$$
\begin{claim}
If the statement is false, the morphism of vector
bundles
$$\phi :\cal L_1\oplus \cal L_2@>>>\cal E$$
is generically surjective.
\end{claim}
{\it Proof} Set $\cal Q=\cal E/\cal L_2$. Then
$\cal Q$ is a rank one torsion free sheaf. The morphism
$\cal L_1@>>>\cal Q$ is therefore either identically
zero or generically nonzero. If $\cal L_1 \not \subset
\cal L_2$ the morphism $\cal L_1@>>>\cal Q$ is then
generically nonzero. But this implies that
$\phi$ is generically surjective.
$\sharp$
\bigskip
Therefore,
$\wedge ^2\cal E\otimes \cal L_1^{-1}\otimes \cal L_2
^{-1}$ is an effective line
bundle; it follows that
$$0\le e-(l_1+l_2)\le e-2l,$$
a contradiction.
$\sharp$
\bigskip
\begin{cor} Let $X$ and $\cal E$ be as above, and
let $\cal A\subset \cal E$ be a saturated
$H$-destabilizing line bundle.
Then
$\cal A$ is the maximal $H$-destabilizing line bundle.
\label{cor:max}
\end{cor}
\begin{cor} Let $X$ be a smooth projective $n$-fold,
and fix a very ample
linear series $|V|$ on $X$, with $V\subset
H^0(X,H)$.
Suppose that
$\cal E$ is a rank two vector bundle on $X$ which is
$H$-unstable. Let $C\subset X$ be a general complete intersection
of $n-1$ divisors in $|V|$.
Then the maximal destabilizing line bundle of $\cal E|_C$ is the
restriction to $C$ of the maximal destabilzing line bundle of $\cal
E$.
\label{cor:res}
\end{cor}
{\it Proof.} Let $\cal A$ be the maximal destabilizing line bundle
of $\cal E$. Then the inclusion $
\psi :\cal A@>>>\cal E$ drops rank
in codimension two, because $\cal A$ is saturated in $\cal E$.
Let $Z$ be the locus where $\psi$ drops rank. For a general complete
intersection curve, we have $C\cap Z=\emptyset$.
Hence $\cal A|_C$ is the maximal destabilizing line bundle of
$\cal E|_C$.
$\sharp$
\bigskip
The basic result is the following
\begin{thm} (Bogomolov) Let
$S$ be a smooth projective surface, and let
$\cal E$ be a rank two vector bundle on $S$. Let $c_1(\cal E)$
and $c_2(\cal E)$ be its Chern classes, and suppose that
$$c_1(\cal E)^2-4c_2(\cal E)>0.$$
Then there exists an exact sequence
$$0@>>>A@>>>\cal E@>>>\cal J_Z\otimes B@>>>0,$$
where $A$ and $B$ are line bundles on $S$ and $Z$ is a codimension
two (possibly empty) local complete intersection subscheme,
with the property that
$A-B\in K^{+}(S)$.
\label{thm:bog}
\end{thm}
For a proof, see \cite{bo:st}, \cite{mi:cc}, \cite{re:vbls},
\cite{gi} or \cite{la:svbt}.
\begin{cor} Let $S$ and $\cal E$ be a smooth projective surface
and a rank two vector bundle on it such that
the hypothesis of the theorem are satisfied.
Let $\cal A$ and
$\cal B$ be the line bundles in the above exact sequence.
Then the following inequalities hold:
$$(\cal A-\cal B)\cdot H>0$$
for all polarizations $H$ on $S$, and
$$(\cal A-\cal B)^2\ge c_1(\cal E)^2-4c_2(\cal E).$$
\label{cor:devissage}
\end{cor}
{\it Proof.} The first inequality follows from the condition
$A-B\in K^{+}(S)$.
To obtain the second, just use the above exact sequence to
compute $c_1(\cal E)$ and $c_2(\cal E)$: we obtain
$$c_1(\cal E)^2-4c_2(\cal E)=(A+B)^2-4A\cdot B-4deg[Z]
\le (A-B)^2.$$
$\sharp$
\bigskip
\begin{cor} Let $S$ and $\cal E$ satisfy the hypothesis of
Bogomolov's theorem, and let $H$ be any polarization on $S$.
Then $\cal E$ is $H$-unstable, and $\cal A$ is the
maximal $H$-destabilizing subsheaf of $\cal E$.
\end{cor}
Recall the fundamental theorem
of Mumford-Mehta-Ramanatan (cfr \cite{mi:cc}):
\begin{thm} Let $X$ be a smooth
projective $n$-fold,and let $H$ be a polarization
on $X$. Consider a vector bundle $\cal E$ on $X$.
If $m\gg 0$ and $V\in |mH|$ is general,
then the maximal $H|_V$-destabilizing subsheaf of
$\cal E|_V$ is the restriction of the maximal $H$-destabilizing
subsheaf of $\cal E$.
\label{thm:mumera}
\end{thm}
This theorem is very powerful, because it detects global instability
from instability on the general complete intersection curve.
\begin{thm} Let $X$ be a smooth projective $n$-fold, and let
$H$ be a polarization on $X$. Consider a rank two
vector bundle
$\cal E$ on $X$, and suppose that
$$(c_1(\cal E)^2-4c_2(\cal E))\cdot H^{n-2}>0.$$
Then there exists an exact sequence
$$0@>>>\cal A@>>>\cal E@>>>\cal B\otimes \cal J_Z@>>>0,$$
where $\cal A$ and $\cal B$ are invertible sheaves and
$Z$ is a locally complete intersection of codimension two
(possibly empty) such that
$$\cal A-\cal B\in K^{+}(X,H) $$
and
$$(\cal A-\cal B)^2\cdot H^{n-2}\ge (c_1(\cal E)^2-4c_2(\cal E))
\cdot H^{n-2}.$$
Furthermore, $\cal A$ is the maximal $(H,\cdots ,H,L)$-destabilizing
subsheaf of $\cal E$, for every ample line bundle $L$
on $X$.
\label{thm:main}
\end{thm}
{\it Proof.}
The case $n=2$ is just
the content of Theorem \ref{thm:bog}; for $n\ge 3$, the
statement follows by induction using theorem \ref{thm:mumera}.
$\sharp$
\bigskip
\begin{defn} If $\cal E$ satisfies the hypothesis of the
theorem, we shall say that $\cal E$ is {\it Bogomolov-unstable with
respect to $H$}.
\label{defn:bogunst}
\end{defn}
\bigskip
\begin{lem} Let $f:X@>>>Y$ be a morphism of
projective varieties.
Let $\cal F$
and $A$ be, respectively, a vector bundle and an ample line
bundle on $X$.
For $y\in Y$, let $X_y=f^{-1}y$ and denote by
$\cal J_{X_y}$ the ideal sheaf of $X_y$.
Then there exists $k>0$ such that
$$H^i(X,\cal F\otimes A^n\otimes \cal J_{X_y})=0$$
for all $i>0$, $n\ge k$ and for all $y\in Y$.
\label{lem:fs}
\end{lem}
{\it Proof.} For all $y\in Y$, there is an exact sequence
$$0@>>>\cal F\otimes A^n\otimes \cal J_{X_y}@>>>
\cal F\otimes A^n@>>>\cal F\otimes A^n|_{X_y}@>>>0.$$
Furthermore, there exists a {\it flattening stratification}
of $Y$ w.r.t. $f$, $Y=
\coprod _{l=1}^r Y_l$, with the following property (\cite{mu:cs}).
The $Y_l$ are locally closed subschemes of $Y$, and
if $X_l=:f^{-1}Y_l$, $l=1,\cdots,r$, and $f_l:X_l@>>>Y_l$
is the restriction of $f$, then $f_l$ is a flat morphism.
Let us then start by finding $k_1$ such that
for all $n\ge k_1$ we have
$$H^i(X,\cal F\otimes A^n )=0$$
and
$$H^i(X,\cal F\otimes A^n\otimes \cal J_{X_l})=0$$
for all $i>0$ and for all $l=1,\cdots,r$.
Then
it is easy to see that the statement is equivalent to saying that there is
$k\ge k_1$ such that for all $n\ge k$ the restriction maps
$$H^0(X,\cal F\otimes A^n)@>\phi _y>>H^0(X_y,\cal F\otimes
A^n|_{X_y})$$
are all surjective, and that
$$H^i(X_y,\cal F\otimes A^n|_{X_y})=0,$$
for all $y\in Y$ and for all $i>0$.
If $y\in Y_l$ and $\cal J^{X_l}_{X_y}$ denotes the
ideal sheaf of $X_y$ in $X_l$, then we have an exact sequence
$$0@>>>\cal J_{X_l}@>>>\cal J_{X_y}@>>>\cal J^{X_l}_{X_y}@>>>0.$$
\begin{claim} The lemma is true if there exists $k$ such that
for all $n\ge k$, for $l=1,\cdots,r$ and for all $y\in Y_l$ we
have that $H^i(X_l,\cal F\otimes A^n|_{X_l}\otimes
\cal J^{X_l}_{X_y})=0$ for $i>0$.
\end{claim}
{\it Proof.} It follows from the exact sequences
$$
\CD
H^i(X,\cal F\otimes A^n\otimes \cal J_{X_l})@>>>H^i(X,\cal F\otimes A^n
\otimes \cal J_{X_y})
@>>>H^i(X,\cal F\otimes A^n\otimes \cal J^{X_l}_{X_y})\\
@| @. @| \\
0 @. @. 0
\endCD
$$
for $i>0$.
$\sharp$
\bigskip
This means that we can reduce to the case where $f$ is flat.
For $y_0\in Y$, we can find $k_0$ such that
for $n\ge k_0$ and for $i>0$ we have
$$H^i(X,\cal F\otimes A^n\otimes \cal J_{X_{y_0}})=0.$$
Therefore, the morphism
$$\lim_{y_0\in U}H^0(f^{-1}U,\cal F\otimes A^n)@>\beta _{y_0}>>
H^0(X_y,\cal F\otimes A^n)
$$
is onto, and then so is
$$\psi _{y_0}=:\beta_{y_0} \otimes k(y_0):
f_{*}(\cal F\otimes A^n)(y)@>>>H^0(X_{y_0},\cal F\otimes A^n
|_{X_{y_0}}).$$
By Grauert's theorem (\cite{ha:ag}) we then have that $\psi _{y_0}$
is an isomorphism, and that the same holds for
$\psi _y$, for $y$ in a suitable open neighbourhood $U_0$
of $y_0$.
Therefore the restriction morphism
$$H^0(X,\cal F\otimes A^n)@>>>H^0(X_y,\cal F\otimes A^n|_{X_y})$$
come from a morphism of sheaves, and hence
they are onto for all $y\in V_0$,
for a suitable open set $V_0\subset U_0$.
We can then invoke the quasi-compactness of $Y$ to
conclude that there exists $k$ such that
$H^1(X,\cal F\otimes A^n\otimes \cal J_{X_y})=0$ for all
$y\in Y$.
As to $i\ge 2$, we have isomorphisms
$$H^i(X_y,\cal F\otimes A^n|_{X_y})
\simeq H^{i+1}(X,\cal F\otimes A^n\otimes \cal J_{X_y})=0$$
for all $i>0$, and so we need to show that
$H^i(X_y,\cal F\otimes A^n)=0$ for $n\gg 0$,
$i>0$ and for all $y\in Y$.
But for $n\gg 0$ we have
$$R^if_{*}(\cal F\otimes A^n)=0$$
if $i>0$ and then this implies $h^i(X_y,\cal F\otimes A^n|_{X_y})=0$
for all $y$ (\cite{mu:cs}).
$\sharp$
\bigskip
We record here a trivial numerical lemma that will be handy in
the sequel:
\begin{lem} If $s\ge \alpha$, $a \ge 2s$ and
$b \ge a s-s^2$, then $b \ge a \alpha-\alpha ^2$.
\label{lem:trivial}
\end{lem}
{\it Proof.} $a s-s^2$ is increasing
in $s$ if $a \ge 2s$.
The statement follows.
$\sharp$
\bigskip
\section{\bf {Seshadri Constants of Curves}}\label{section:sc}
Let $C\subset \bold P^3$ be a smooth curve
and
let $H$ denote the hyperplane bundle on $\bold P^3$.
We shall let $f:X_C@>>>\bold P^3$ be the blow up of $\bold P^3$ along $C$,
and $E=f^{-1}C$ be the exceptional divisor.
\begin{defn} The Seshadri constant of $C$ is
\begin{center}
$\epsilon (C)=:
sup \{\eta \in \bold Q|f^{*}H-\eta E$ is ample$\}$.
\end{center}
\label{defn:sc}
\end{defn}
\noindent
In other terms, $\epsilon (C)$ is the supremum of the ratios
$\frac nm$, where $n$ and $m$ are such that
$mH-nE$ is ample (or, equivalently, very ample).
In the sequel we shall use the short hand
$$H_{\eta}=:H-\eta E$$
for $\eta \in \bold Q$; furthermore, we shall generally write
$H$ for $f^{*}H$ (as we just did).
\begin{lem} $H_{\eta}$ is ample if and only if
$0<\eta <\epsilon (C)$. It is nef if and only if
$\eta \in [0,\epsilon (C)]$.
\label{lem:nef}
\end{lem}
{\it Proof.} Since the ample cone of a projective
variety is convex, the line $H-tE\subset N^1(X)$ intersects
$K^{+}(X)$ in a segment $(H-t_1E,H-t_2E)$.
Let $F$ denote the numerical class of a fiber
of $\pi :E@>>>C$. Then $H_{\eta}\cdot F=\eta$, and therefore if
$H_{\eta}$ is ample we must have $\eta >0$.
Hence $t_1\ge 0$. On the other hand, it is well known
that $H-tE$ is ample for $t>0$ sufficiently small,
and therefore $t_1=0$.
By definition,
$t_2=\epsilon _2(C)$.
The remaining part of the statement is clear. $\sharp$
\bigskip
\begin{cor} We have
\begin{center}
$\epsilon (C)=sup\{\eta |H_{\eta}\cdot D\ge 0$ for all curves
$D\subset X_C\}.$
\label{cor:nef}
\end{center}
\end{cor}
\begin{lem} Let $C\subset \bold P^3$ be a smooth curve,
and let $\cal J_C$ be its ideal sheaf.
Let $m$ and $n$ be nonnegative integers.
Then $\cal O_{X_C}(mH-nE)$
is globally generated if $\cal J_C^n(m)$ is.
\label{lem:gg}
\end{lem}
{\it Proof.} Let us suppose that $\cal J_C^n(m)$ is globally generated,
and let
$$F_1,\cdots,F_k\in H^0(\bold P^3,\cal J_C^n(m))$$
be a basis.
Let $P\in C$ and
let $U$ be some open
neighbourhood of $P$.
By assumption, $F_1,\cdots,F_k$ generate $\cal J_C$ in
$U$. By abuse of language, let us write
$F_i$ for the pull-backs $f^{*}F_i$. Then if $e$ is a local
equation for $E$ in a Zariski open set
$V\subset f^{-1}U$, then the ideal generated by the $F_i$s
is $<\{F_i\}>=(e^n)$.
Hence we can write
$$\sum _{i=1}^kP_iF_i=e^n$$
for some $P_i$s regular on $V$. However, by construction we
can write $F_i=\tilde F_ie^n$, and therefore
we have
$$\sum _{i=1}^k\tilde F_iP_i=1$$
in $V$. Hence the $\tilde F_i$ are base point free, and
they can be extended to global sections of
$\cal O_{X_C}(mH-nE)$, which is therefore globally spanned.
$\sharp$
\bigskip
\begin{cor} Let $C\subset \bold P^3$
be a smooth curve.
Then
\begin{center}
$\epsilon (C)\ge sup \{\frac nm|\cal J_Y^n(m)$ is globally
generated$\}$.
\end{center}
\label{cor:gg}
\end{cor}
Let us look at some examples.
\begin{exmp} If $L\subset \bold P^3$ is
a line, then $\cal J_L(1)$ is globally generated.
Therefore $\epsilon (L)\ge 1$.
On the other hand, let $\Lambda \subset \bold P^3$ be a hyperplane
containing $L$ and let $D\subset \Lambda$ be any irreducible curve
distinct from $L$. Then $H_1\cdot \tilde D=deg(D)-L\cdot _{\Lambda}D=0$,
where $\tilde D\subset Bl_L(\bold P^3)$ is the proper transform of $L$.
Hence $\epsilon (L)=1$. As we shall see shortly, this generalizes
to the statement that if $C\subset \bold P^3$ is a smooth complete
intersection of type $(a,b)$ and $a\ge b$, then
$\epsilon (C)=\frac 1a$.
\end{exmp}
\begin{exmp} If $C\subset \bold P^3$ has an $l$-secant line, then
$\epsilon (C)\le \frac 1l$. To see this, let $L$ be the $l$-secant;
denoting by $\tilde L\subset X_C$ the proper transform of $L$
in $Bl_C(\bold P^3)$ we have $H\cdot \tilde L=1$ and $E\cdot \tilde L=l$.
Hence $0\le H_{\epsilon}\cdot \tilde L$ implies $\epsilon \le \frac 1l$.
\end{exmp}
\begin{lem} Let $C\subset \bold P^3$ be a smooth
curve of degree $d$. Then
$$\frac 1{\sqrt d}\ge \epsilon (C)\ge \frac 1d$$
\label{lem:deg}
\end{lem}
{\it Proof.} It is well-known that a
smooth subvariety of degree $d$ of
projective space is cut out by hypersurfaces of degree $d$.
Hence $\cal J_C(d)$ is globally generated, and this proves the
second inequality.
As to the first, we must have $0\le H\cdot H_{\epsilon}^2=1-\epsilon ^2d$,
by a simple Segre class computation.
$\sharp$
\bigskip
The right inequality is sharp if the curve is degenerate; the left
one is sharp for a complete intersection curve of type $(a,a)$.
If the curve is nondegenerate, however, one can say something
more.
\begin{defn} Let $C\subset \bold P^3$ be a smooth
curve, and let $\cal J_C$ be its ideal sheaf.
$C$ is said to be {\it $l$-regular} if
$$H^i(\bold P^3,\cal J_C(l-i))=0$$
for all $i>0$. The {\it regularity} of $C$, denoted by $m(C)$,
is the smallest $l$ such that $C$ is $l$-regular
(\cite{ca}, \cite{mu}, \cite{gr}).
\label{defn:reg}
\end{defn}
\begin{rem} By a celebrated theorem
of Castelnuovo, we
have $m(C)\le d-1$ (\cite{ca}, \cite{glp}).
\label{rem:cast}
\end{rem}
\begin{prop} Let $C
\subset \bold P^3$ be a smooth space curve, and let $m=m(C)$
be its regularity. Then
$$\dfrac 2{m-1}\ge \epsilon (C)\ge \dfrac 1m.$$
\label{prop:reg}
\end{prop}
{\it Proof.} By a classical theorem of
Castelnuovo-Mumford, the homogeneuos ideal of $C$ is saturated in
degree $m(C)$ and therefore $\epsilon (C)\ge \frac 1{m(C)}$.
By definition, to prove the first inequality it is
enough to show that
$H^i(\bold P^3,\cal J_C(k))=0$ for $k\ge \lceil \frac 2{\epsilon (C)} \rceil
-3$
because this implies $m(C)\le \frac 2{\epsilon (C)}+1$
and then the statement.
To prove the above vanishing, observe that
$$\Big \{2/ (\lceil 2/\epsilon (C)\rceil +1)
\Big \} <\epsilon (C)$$
and therefore
$$\Big (\Big \lceil \frac 2{\epsilon (C)}\Big \rceil +1\Big )H-2E$$
is an ample integral divisor in $X_C$.
Since $\omega _{X_C}=\cal O_{X_C}(-4H+E)$,
the Kodaira vanishing theorem gives:
$$H^i(X_C,\cal O_{X_C}((\lceil 2/\epsilon (C) \rceil
-3)H-E))=0$$
for $i>0$, as desired.
$\sharp$
\bigskip
\begin{rem} Using vanishing theorems on the blow up to
obtain bounds on the regularity is a well-known technique:
see \cite{bel} for various results in this direction.
\end{rem}
\begin{rem} It is not possible, in
the above vanishing, to replace the
condition on $k$ by $k\ge \lceil \frac 1{\epsilon}\rceil$. To see this,
suppose that $C$ is a complete intersection of type $(a,b)$ so that we
have a Koszul resolution
$$0@>>>\cal O_{\bold P^3}(-b)@>>> \cal O_{\bold P^3}\oplus
\cal O_{\bold P^3}(a-b)@>>>\cal J_C(a)@>>>0.$$
It follows that $H^2(\bold P^3,\cal J_C(a))\simeq H^3(\bold P^3,
\cal O_{\bold P^3}(-b))\neq 0$, for $b\ge 4$.
\end{rem}
\begin{cor} Let $C\subset \bold P^3$ be a
nondegenerate smooth curve. Then
$\epsilon (C)\ge \frac 1{d-1}$.
\end{cor}
\noindent
Equality is attained in the previous corollary in the case of a twisted
cubic.
\bigskip
It is convenient to introduce the following definition.
\begin{defn} Let $C\subset \bold P^3$ be a smooth curve.
For an irreducible curve $D\subset \bold P^3$ different from $C$
let $\tilde D$ be its proper transform in the blow up of $\bold P^3$
along $C$.
Define
\begin{center}
$\epsilon _1(C)=:sup \{\eta \in \bold Q|
(H-\eta E)|_E$ is ample$\}$
\end{center}
and
\begin{center}
$\epsilon _2(C)=:sup\{\eta \in \bold Q|
H_{\eta}\cdot \tilde D\ge 0 \forall$ irreducible curves $D\neq C$\}.
\end{center}
\label{defn:12}
\end{defn}
\begin{rem} $\epsilon (C)=min\{\epsilon _1(C),
\epsilon _2(C)\}$.
\label{rem:12}
\end{rem}
We are interested in estimating the Seshadri constant of a space curve $C$.
It is convenient to examine $\epsilon _1(C)$ and
$\epsilon _2(C)$ separately. We shall see that $\epsilon _1(C)$ is
determined by the structure of the normal bundle, while $\epsilon _2(C)$
depends on the "linkage" of $C$, and is generally much harder
to estimate.
We start with an analysis of $\epsilon _1(C)$.
\begin{defn} Let $C$ be a smooth projective curve and let
$\cal E$ be a rank two vector bundle on it. For all finite morphisms
$f:\tilde C@>>>C$ and all exact sequences of locally free shaves on $\tilde
C$ of the form $0@>>>L@>>>f^{*}\cal E@>>>M@>>>0$, consider the
ratios $\frac {deg(L)}{deg(f)}$. Let $\Sigma _{\cal E}$ denote the
set of all the numbers obtained in this way. Define
$$s(\cal E)=:sup\Sigma _{\cal E}.$$
\label{defn:wahl}
\end{defn}
\begin{rem}
As in \cite{w}, $s(\cal E)$ can be interpreted as a measure
of the instability of $\cal E$. More precisely, we have
$$s(\cal E)=\frac 12deg(\cal E)$$ if
$\cal E$ is semistable
and
$$s(\cal E)=deg(L)$$
if $\cal E$ is unstable, and
$L\subset \cal E$ is the maximal destabilzing line subundle
of $\cal E$.
In other words, $s(\cal E)-\frac 12deg(\cal E)\ge 0$ always,
and equality holds if and only if $\cal E$ is semistable.
\label{rem:wahl}
\end{rem}
We then have
\begin{prop} Let $C\subset \bold P^3$ be a smooth
curve.
Denote by $N$ the normal bundle of $C$ in $\bold P^3$, and
let $\epsilon _1(C)$ be as above. Then
$$\epsilon _1(C)=\frac {deg(C)}{s(N)}.$$
\label{prop:e1}
\end{prop}
{\it Proof.} Let $X_C=:Bl_C(X)@>f>>X$ be the blow up of $X$ along
$C$ and let $E$ be the exceptional divisor;
recall that $E$ can be identified with the relative projective
space of lines in the vector bundle $N$.
Set $\pi =f|_E$ and
denote by $F$ a fiber of $\pi$.
Let $D\subset E$ be any reduced irreducible curve. If $D$ is a fiber
of $\pi$, then $\eta >0$ ensures that $H_{\eta}\cdot D>0$. Hence we
may assume that $D@>>>C$ is a finite map, whose degree is given
by $a=D\cdot F$.
Let $
\psi :
\tilde D@>>>D\subset X_C$ be the normalization of $D$, and let
$p:\tilde D@>>>C$ be the induced morphism.
Then $\psi$ is equivalent to the assignment of a
sub-line bundle $L\subset p^{*}N$, given by
$L=\psi ^{*}\cal O_{\bold PN}(-1)$.
Since $\cal O_{\bold PN}(-1)\simeq \cal O_E(E)$,
we have $deg(L)=D\cdot E$. Hence
$H_{\eta}\cdot D=aH\cdot C-\eta \cdot deg(L)$; the condition
$\eta \le \epsilon _1(C)$ translates therefore
in the condition $\eta \le inf\{\dfrac {H\cdot C}{deg(L)/a}\}$.
In other words, then, it is equivalent to
$\eta \le \dfrac {H\cdot C}{s(N)}$.
$\sharp$
\bigskip
\begin{exmp} Let $C\subset \bold P^3$ be a smooth
complete intersection curve of type $(a,b)$, with $a\ge b$.
Then we have a Koszul resolution of the ideal sheaf of $C$,
from which it is easy to conclude that
$\epsilon (C)\ge \frac 1a$. On the other hand, $s(N)=a^2b$
and therefore
by Proposition \ref{prop:e1}
$\epsilon _1(C)=\frac 1a$. Hence we have
$\epsilon (C)=\frac 1a$.
\end{exmp}
\begin{exmp} Let $C\subset \bold P^3$ be given as the zero
locus of a regular section of a rank two vector bundle
$\cal E$ on $\bold P^3$. It is well known that this is always
the case provided that the determinant of the normal bundle $N$
extends. The Koszul resolution then is
$$0@>>>det(\cal E)^{-1}@>>> \cal E^{*}@>>>\cal J_C@>>>0.$$
By Corollary \ref{cor:gg}
and Proposition \ref{prop:e1}, we then conclude that
$$\frac {H\cdot C}{s(\cal E|_C)}\ge \epsilon (C)\ge \epsilon (\cal E)$$
where
\begin{center}
$\epsilon (\cal E)=sup \{\frac nm|S^n\cal E^{*}(m)$ is spanned$\}$.
\end{center}
\label{exmp:zl}
\end{exmp}
We shall call $\epsilon (\cal E)$ the Seshadri constant of the
vector bundle $\cal E$. It has the following geometric interpretation.
Let $\bold P\cal E$ be the relative projective
space of lines in $\cal E$. $Pic (\bold P\cal E)$ is generated
by two line bundles $H$ and $\cal O(1)$, where
$H$ is the pull-back of the hyperplane bundle on $\bold P^3$.
Let $R$ be some divisor
associated to the line bundle $\cal O(1)$. It is well known that
the rational divisor $H+\eta R$ is ample, for
sufficiently small $\eta \in \bold Q^{+}$ (\cite{ha:ag}).
\begin{prop} $\epsilon (\cal E)=sup\{\eta \in \bold Q|
H+\eta R\in Div _{\bold Q}(\bold P\cal E)$ is ample$\}$.
\label{prop:scvb}
\end{prop}
{\it Proof.} Provisionally denote by $\gamma (\cal E)$
the right hand side of the statement. Also, for brevity
let us set $X=\bold P\cal E$ and let $X_z$ stand for the
fiber over a point $z\in \bold P^3$.
Let us first prove that $\epsilon (\cal E)\le \gamma (\cal E)$.
Suppose then that $\eta =\frac nm<\epsilon (C)$, where $n$ and
$m$ are such that $S^n\cal E^{*}(m)$ is globally
generated. Since
$$S^n\cal E^{*}(m)=f_{*}\cal O_X(mH+nR),$$
we have the identifications
$$H^0(X,\cal O_X(mH+nR))\simeq H^0(\bold P^3, S^n\cal E^{*}(m))$$
and
$$H^0(X_z,\cal O_{X_z}(mH+nR))\simeq
S^n\cal E^{*}(m)(z).$$
With this in mind, we then have a surjection
$$H^0(X,\cal O_X(mH+nR))@>>>H^0(X_z,\cal O_{X_z}(mH+nR))$$
for all $z\in \bold P^3$, and since $\cal O_X(mH+nR)$ is generated along
the fibers, it is also globally generated.
Let us now prove that $\gamma (\cal E)\le \epsilon (\cal E)$.
Let $\eta =\frac nm <\gamma (\cal E)$, where $n$ and $m$ have been
chosen so that $mH+nR$ is ample. After perhaps multiplying $m$ and $n$
by some large positive integer we may suppose that $mH+nR$ is very ample and
that
$$H^i(X,\cal J_{X_z}(mH+nR))=0$$
for all $i>0$ and all $z\in \bold P^3$ (see Lemma \ref{lem:fs}).
But then we have surjective restriction maps
$$H^0(X,\cal O_X(mH+nR))@>>>H^0(X_z,\cal O_{X_z}(mH+nR))$$
for all $z\in \bold P^3$, and the lemma then follows from the above
identifications.
$\sharp$
\begin{rem}
The inequality $\epsilon (C)\ge \epsilon (\cal E)$
from Example \ref{exmp:zl}
can then be
explained as follows. For each $n\ge 0$ we have surjective
morphisms
$S^n\cal E^{*}@>>>\cal J_C^n$,
and therefore we have a surjection of sheaves of graded algebras
$$\bigoplus _{n\ge 0} S^n\cal E^{*}@>>>
\bigoplus _{n\ge 0}\cal J_C^n,$$
which yields a closed embedding
$$i:X_C\hookrightarrow \bold P\cal E.$$
On the other hand, $i^{*}\cal O_{\bold P\cal E}(R)=\cal O_{X_C}(-E)$
and the above ineqality is just saying that if $H+\eta R$ is
ample, it restricts to an ample divisor on $X_C$.
\end{rem}
\bigskip
We now consider ways to estimate $\epsilon _2(C)$.
$\epsilon _2(C)$ gathers more global information than
$\epsilon _1(C)$, because it relates to how $C$ is
"linked" to the curves in $\bold P^3$.
Recall that our definition was:
\begin{center}
$\epsilon _2(C)=:sup\{\eta \in \bold Q|
H_{\eta}\cdot \tilde D\ge 0$ $\forall$ irreducible curves $D\subset
\bold P^3$, $D\neq C \}$.
\end{center}
As usual, $\tilde D$ denotes the proper transform of
$D$ in the blow up of $C$.
There does not seem
to be much that one can say about $\epsilon _2(C)$
in general; with some extra assumptions, however, we can obtain
an estimate.
Let us make the following definiton:
\begin{defn} Let $D\subset \bold P^3$ be a reduced irreducible curve,
and let $t:D_n@>>>D\subset \bold P^3$ be its normalization.
If the derivative $dt:T_{D_n}@>>>t^{*}T_{\bold P^3}$ never drops rank, we
shall say that $D$ has only ordinary singularities.
\label{defn:os}
\end{defn}
\begin{prop} Let $C\subset \bold P^3$ be a smooth
curve. Suppose that $C$ is contained in the intersection
of two distinct reduced and irreducible hypersurfaces
$V_a$ and $V_b$ of degree $a$ and $b$, respectively.
Suppose
that all the residual curves to $C$ in the complete intersection
$V_a\cap V_b$ are reduced and
that at least one of the two hypersurfaces is smooth.
Then $$ \epsilon _2(C)\ge \frac 1{a+b-2}.$$
If all the residual curves
have ordinary singularities, then
equality holds if and only if the residual curve is a
union of disjoint lines.
\label{prop:e2}
\end{prop}
\begin{exmp} It is well-known that a curve which is
linked to a line
$L$ in a complete intersection of type $(a,b)$
is cut out by the hypersurfaces $V_a$ and $V_b$ and by a third
equation of degree $a+b-2$. Therefore its ideal sheaf is
generated in degree $a+b-2$, so that
$\epsilon (C)\ge \frac 1{a+b-2}$. On the other hand, it is
easy to
check that $\tilde L\cdot E=a+b-2$. Therefore in this
case we find directly that $\epsilon (C)=\frac 1{a+b-2}$.
More generally, the same argument works whenever $C$ is linked
to a union of (reduced) disjoint lines.
\label{exmp:sharp}
\end{exmp}
\begin{exmp} The assumption that the residual curves be
all reduced is necessary. To see this, let $L\subset \bold P^3$
be a line, and let $V$ be a smooth surface of degree $v$ through
$L$.
We have $L\cdot _VL=2-v$. Let $H$ be the hyperplane bundle
restricted to $V$. Then for $s\gg 0$ the linear series
$|sH-2L|$ is very ample. Choose a smooth curve
$C\in |sH-2L|$. Then $C$ is linked to a double line supported
on $L$ in the complete intersection $V\cap W$, where
$W$ is a suitable hypersurface of degree $s$ in $\bold P^3$.
We have
$$\tilde L\cdot E_C=(sH-2L)\cdot _VL=s-2L^2=s+2v-4,$$
and so $\epsilon _2(C)\le \dfrac 1{s+2v-4}$.
\end{exmp}
{\it Proof.}
We need to show that for $\eta \le\frac 1{a+b-2}$ we
have $\tilde D\cdot H_{\eta}\ge 0$, whenever $D\subset \bold P^3$
is some irreducible curve distinct from $C$. Clearly we
may assume that $D$ is reduced.
Let us start with the following simple observation. Let
$C$ and $D$ be reduced curves in $\bold P^3$, and let
$D_n@>t>>D\subset \bold P^3$ be the normalization of
$D$. If $X_C@>f>>\bold P^3$ is the blow up of $C$ and
$E_C$ is the exceptional divisor, clearly $t$ factors
through $f$, i.e. there exists $u:D_n@>>>X_C$ such that
$t=f\circ u$. On the other hand, $t^{-1}C=u^{-1}f^{-1}
C=u^{-1}E_C$ and therefore
\begin{equation}
\tilde D\cdot E_C=D_n\cdot _uE_C=deg\{t^{-1}C\}.
\label{eq:norm}
\end{equation}
Given the geometric situation,
we start testing the
desired positivity condition on the curves that are not
contained in $V_a\cap V_b$.
\begin{lem} Let $C\subset \bold P^3$, $V_a$ and
$V_b$ be as in the statement of the Proposition.
Suppose that $a\ge b$, and let $\eta \le \frac 1a$.
Then for every irreducible curve $D\not\subset V_a\cap V_b$
we have $\tilde D\cdot H_{\eta}\ge 0$.
\label{lem:nonres}
\end{lem}
{\it Proof of the Lemma.} Let $D$
be reduced and have degree $s$, and set
$G=:V_a\cap V_b$.
$G$ is a complete intersection curve, and then we know
from the Koszul resolution of its ideal sheaf that its
Seshadri constant satisfies
$\epsilon (G)\ge \frac 1a$.
Let $X_G@>>>\bold P^3$ be the blow up of $\bold P^3$ along
$G$, and let $E_G$ be the exceptional divisor.
For $\eta \in \bold Q$, let $H^{\prime}_{\eta} =
g^{*}H-\eta E_G$.
By what we have just said, $H^{\prime}_{\frac 1a}$ is
a nef divisor on $X_G$.
Therefore, if we let
$D^{\prime}\subset X_G$ denote the proper transform
of $D$ in $X_G$, we have
$D^{\prime}\cdot H^{\prime}_{\eta}\ge 0$, and this
can be rewritten as $D^{\prime}\cdot E_G\le as$.
Now let as above $t:D_n@>>>D\subset \bold P^3$
be the normalization of $D$, and let
$\tilde D\subset X_C$ denote the proper transform of
$D$ in the blow up of $C$. Then by equation
(\ref{eq:norm})
$$\tilde D\cdot E_C=deg\{t^{-1}C\}\le deg\{t^{-1}G\}
=D^{\prime}\cdot E_G,$$
since $G\supset C$ as schemes. Therefore,
\begin{equation}
H_{\frac 1a}\cdot \tilde D\ge
H^{\prime}_{\frac 1a}\cdot D^{\prime}\ge 0,
\end{equation}
and the statement follows.
$\sharp$
\bigskip
We now need to consider the
condition $H_{\eta}\cdot \tilde D_i\ge 0$,
where the $D_i$s are the irreducible
components of the residual curve
to $C$ in the complete intersection
$V_a\cap V_b$.
Let us drop the index $i$, and let
$D$ be one of the the $D_i$s.
We have to show that $H_{\eta}\cdot \tilde D\ge 0$ for
$\eta \le \frac 1{a+b-2}$.
We shall be using case (b) of the following lemma, but it may
be worthwhile to state it in more generality:
\begin{lem} Let $C$ and $D$ be reduced irreducible space curves,
and suppose that either one of the following
conditions holds:
(a) $C$ and $D$ are both smooth, or
(b) $C$ and $D$ lie in a smooth hypersurface
$S\subset \bold P^3$.
Then $\tilde D\cdot E_C=\tilde C\cdot E_D$, where $\tilde D$
(resp., $\tilde C$) is the proper transform of $D$ in the
blow up of $C$ (resp., the proper transform of $C$ in the blow up of
$D$).
\label{lem:blowups}
\end{lem}
{\it Proof.} Let us first suppose that
$(b)$ holds.
Let $t: C_n@>>>C\subset S$ be the
normalization of $C$. By (\ref{eq:norm}), we know that
$\tilde C\cdot E_D=deg\{t^{-1}D\}=deg\{t^{*}\cal O_S(D)\}
=C\cdot _SD$
and similarly for
$\tilde C\cdot E_D$.
If (a) holds,
the situation is almost the same, because at each intersection
point $P$ of $C$ and $D$ we can still
locally view $C$ and $D$ as lying
in some smooth
open surface in an neighbourhood of $P$, and
the problem is local in $P$.
Explicitly, the argument is the following.
Suppose that $C\cap D$ is supported on $P_1,\cdots,P_k$.
We "measure" the intersection of $C$ and $D$
in the following way (cfr \cite{sev}):
let $\pi:\bold P^3--\to \bold P^2$ be a general
projection, and set
\begin{equation}
C{*}D=:\sum _{i=1}^ki(\pi (P_i),\pi (C),\pi (D)),
\label{eq:int}
\end{equation}
where $i$ denotes the ordinary intersection multiplicity.
Using the projection
formula, one can easily check the following:
\begin{claim} Let $P\in \bold P^3$
be chosen generally, and let $C_P$ be the cone on $C$ with vertex
$P$. Then
$$C{*}D=\sum _{i=1}^k i(P_i,D,C_P).$$
\label{claim:cones}
\end{claim}
Observe that these intersection multiplicities
are generally constant by the principle of
continuity.
Given that $C*D$ is symmetric, Lemma \ref{lem:blowups} will follow once
we establish that
$C*D=\tilde D\cdot E_C$.
Since $C$ is smooth, it is defined scheme-theoretically
by the cones through it (\cite{mu}). Hence for the proof of Lemma 3.5
we are reduced to the following:
\begin{lem} Let $C$ and $D$ be distinct reduced irreducible
curves in $\bold P^3$. Suppose that $C\cap D$ is supported at points
$P_1,\cdots,P_k$.
Let $\cal C\subset H^0(\bold P^3,
\cal J_C(m))$ be an irreducible family of hypersurfaces.
Suppose that the linear series $V=|\cal C|$ spanned by
$\cal C$ globally generates $\cal J_C(m)$ (in other words,
$C$ is cut out scheme-theoretically by the elements of
$\cal C$).
Then for a general $F\in \cal C$ we have
$$\tilde D\cdot E_C=\sum _{i=1}^k i(P_i,D,F).$$
\label{lem:blowups1}
\end{lem}
{\it Proof.} The assumption implies in particular that
$\cal C\not\subset H^0(\bold P^3,\cal J_C^2(m))$, i.e. that
the general $F\in \cal C$ is generically smooth along
$C$.
For such a general $F$, then, if $\tilde F$ denotes the proper
transform in $X_C$ we have
$$\tilde F\in |f^{*}F-E|.$$
Furthermore, the family of all such $\tilde F$ has to be base
point free, so there is
$F\in \cal C$ which is generically smooth along
$C$ and such that
$\tilde F$ does not meet any of the intersection points of
$\tilde D$ and $E_C$.
Let us denote by a subscript $(\cdot ,\cdot )_{P}$
the contribution to a given intersection product on $X_C$ coming from
the points lying over $P\in \bold P^3$.
Then by construction and the projection formula we have
$$(\tilde D\cdot E_C)_{P_i}=
(\tilde D\cdot f^{*}F)_{P_i}=i(P_i,D,F)$$
and this proves the lemma.
$\sharp$
\bigskip
Let then $X_D@>>>\bold P^3$ be the blow up of $\bold P^3$
along $D$, and let $G$ be the complete intersection
$V_a\cap V_b$.
Then $C$ is a component of the effective cycle
$G-D$, and furthermore $G-D$ does not have any component
supported on $D$.
Hence we may consider the proper transform
$\tilde {G-D}\subset X_D$, which is an effective cycle
in $X_D$ containing $\tilde C$ as a component.
Suppose, say, that $V_a$ is smooth. Then
we are in case (b) of lemma 3.5, and
therefore we have
\begin{equation}
\tilde D\cdot E_C=\tilde C\cdot E_D\le (\tilde {G-D})\cdot E_D.
\label{eq:fulton}
\end{equation}
In the hypothesis of the proposition, at a generic point of
$D$ $V_a$ and $V_b$ are both smooth and meet transversally
(for otherwise $D$ would not be reduced).
Therefore $\tilde V_a\equiv f^{*}V_a-E$ and $\tilde V_b\equiv f^{*}V_b
-E$, and no component of $\tilde V_a\cap \tilde V_b$ maps
dominantly to $D$.
Furthermore if, say, $V_a$ is smooth then
$\tilde V_a\simeq V_a$ does not contain any fiber of $\pi$.
Therefore $\tilde {(G-D)}=\tilde V_a\cap \tilde V_b$, and so
$$\tilde {(G-D)}\cdot E_C= (f^{*}V_a-E)\cdot (f^{*}V_b-E)
\cdot E_C.$$
Let $N$ denote the normal bundle to the complete intersection $G$.
{}From intersection theory, the latter term is known to be
\begin{equation}
\{c(N)\cap s(D,\bold P^3)\}_0=s(D,\bold P^3)_0+(a+b)H\cap s(D,\bold P^3)_1
\label{eq:fulton1}
\end{equation}
where $c(N)$ denotes the
total Chern class of $N$, and $s(D,\bold P^3)$
is the Segre class of $D$ in $\bold P^3$ (\cite{fu}, $\S$9 ).
Summing up, we have
\begin{equation}
\tilde D\cdot E_C\le s(D,\bold P^3)_0+(a+b)H\cap s(D,\bold P^3)_1
\label{eq:fulton2}
\end{equation}
and equality holds if and only if
$D$ does not meet any component of $G-C$ different from $C$.
\begin{lem}
We have $s(D,\bold P^3)_1=[D]$ and
$s(D,\bold P^3)_0\le -2deg(D)$; if $C$ only
has ordinary singularities then equality holds
if and only if $D$ is a line.
\end{lem}
{\it Proof.}
If either
(a) or (b) in the statement of Lemma X holds, then
$D$ is a local complete intersection, and therefore it has
a normal bundle $N$ in $\bold P^3$.
Hence $s(D,\bold P^3)=c(N)^{-1}\cap [D]$, and
the statement is then reduced to the inequality
$deg(N)\ge 2deg(D)$. Let $t:D_n@>>>D$ be the normalization
of $D$.
We then have a generically surjective morphism
$t^{*}T_{\bold P^3}@>>>t^{*}N$. On the other hand,
$T_{\bold P^3}(-1)$ is globally generated, and therefore we
must have $deg(N(-1))\ge 0$,
i.e. $deg(N) \ge 2d$.
If furthermore $D$ only has ordinary singularities,
we have an exact sequence $0@>>>T_D@>>>t^{*}T_{\bold P^3}
@>>>N$ and this shows that
equality holds if and only if
$g=0$ and $d=1$.
$\sharp$
\bigskip
We then have
$\tilde D\cdot E_C\le (a+b-2)deg(D)$, and if $D$ has only ordinary
singularities then
equality holds if and only if $D$ is a line not meeting any
component of $G-D$ different from $C$.
The Proposition follows.
$\sharp$
\bigskip
We know define two auxiliary invariants related to the Seshadri
constant that will be useful shortly.
\begin{defn} Let $C\subset \bold P^3$
be a smooth curve of degree $d$ and let $\epsilon (C)$ be its
Seshadri constant.
Let $N$ be the normal bundle of $C$ in $\bold P^3$.
For $0\le \eta \le \epsilon (C)$ a rational number,
define
$$\delta _{\eta}(C)=:\eta \cdot deg(N)-d$$
and
$$\lambda _{\eta}(C)=:\eta ^2d^2-\delta _{\eta}(C).$$
\label{defn:dandl}
\end{defn}
It is easy to check that
\begin{equation}
\delta _{\eta}(C)=:E^2\cdot H_{\eta}.
\label{eq:dint}
\end{equation}
More explicitly,
suppose that $0<\eta <\epsilon (C)$
and let $m$ and $n$ be large positive integers
such that $\eta =\frac nm$ and
$mH-nE$ is very ample.
Then for a general
$S\in |mH-nE|$ the intersection $C^{\prime}=E\cap S$ is
an irreducible smooth curve, and the induced morphism
$C^{\prime}@>>>C$ has degree $n$.
Then
\begin{equation}
\delta _{\eta}(C)=\frac {C^{\prime}\cdot _SC^{\prime}}
{H\cdot _SH}.
\label{eq:dint1}
\end{equation}
Similarly,
\begin{equation}
\lambda _{\eta}(C)=\frac {(H\cdot C^{\prime})^2}{(H\cdot _SH)^2}-
\frac {C^{\prime}\cdot _SC^{\prime}}
{H\cdot _SH}.
\label{eq:lint}
\end{equation}
\begin{rem} If we let $x=\eta d$, we have
$\lambda _{\eta}(C)=f(x)$, where
$$f(x)=x^2-\Big (4+\frac {2g-2}d\Big )x+d.$$
For $C$ subcanonical, $f$ is the polynomial introduced
by Halphen in his celebrated {\it speciality theorem}
(\cite{gp}), given by
$$g(X)=x^2-(4+e)x+d$$
where $e=max\{k|H^1(C,\cal O_C(k))\neq 0\}.$
Observe that $e\le (2g-2)/d$ always.
\end{rem}
\begin{cor}
Suppose that there exists an irreducible surface of degree $m$
through $C$, having multiplicity $n$ along $C$. If $\eta =\frac nm$,
then $\lambda _{\eta}(C)\ge 0$.
In particular, $\lambda _{\eta}(C)\ge 0$
for all $0\le \eta \le \epsilon (C)$.
Equality holds if and only if
$\cal O_S(C^{\prime})$ is numerically equivalent
to a multiple of $\cal O_S(H)$.
In particular, $\lambda _{\epsilon (C)}(C)\ge 0$
and equality holds if
$C$ is a complete intersection.
If $C$ is subcanonical and $\eta d$ is an integer, then
$\lambda _{\eta}(C)=0$
forces $C$ to be a complete intersection.
\label{cor:lpos}
\end{cor}
{\it Proof.} A straightforward application of the Hodge
index theorem.
The last part follows from the corresponding
statement of the speciality theorem (see \cite{gp}).
$\sharp$
\bigskip
\begin{cor} We have:
$$g\le \frac 12 d^2\epsilon (C)+d\Big (\frac 1{2\epsilon (C)}-2
\Big )+1.$$
\end{cor}
The right-hand side of the above inequality is
a decreasing function of $\epsilon$ in the interval
$(1/d, 1/{\sqrt d})$. In other words, higher Seshadri constants
impose tighter conditiond on the genus.
For a Castelnuovo extremal curve of even degree
we have $\epsilon =\frac 2d$ and the right hand side, as
a function of $d$, is asimptotic to $\frac {d^2}4$.
\begin{cor} Let $D$ be a divisor on $X_C$, and
set $s=D\cdot H_{\eta}\cdot H$.
Then for $0\le \eta \le \epsilon (C)$ we have
$$D^2\cdot H_{\eta}-D\cdot H_{\eta}\cdot E\le s^2-s\eta d.$$
\label{cor:sl}
\end{cor}
{\it Proof.} Write
$$D=xH+yE.$$
Then
$$D^2\cdot H_{\eta}=x^2+y^2\delta _{\eta}(C)+2xyd$$
and
$$D\cdot H_{\eta}\cdot E=x\eta d+y\delta _{\eta}(C).$$
{}From this we obtain
$$D^2\cdot H_{\eta}-D\cdot H_{\eta}\cdot E
=s^2-s\eta d-\lambda _{\eta}(C)(y^2-y).$$
Since $y$ is an integer, the statement then follows from
Corollary \ref{cor:lpos}.
$\sharp$
\bigskip
\begin{rem} From the inequality (see Remark \ref{rem:wahl})
$$s(N)\ge \frac 12 deg(N)$$
and the definition of $\delta _{\eta}(C)$, it
is easy to see that
$$d\ge \delta _{\eta}(C).$$
\label{rem:dd}
\end{rem}
\section{\bf {Gonality of space curves
and free pencils on projective
varieties}}\label{section:gon}
We have seen that if $C\subset S$ is a smooth curve with
$C^2>0$, then one can give lower bounds on the
gonality of $C$.
We deal here with the next natural question: if $C\subset \bold P^3$,
what can be said about $gon(C)$
in terms of the invariants of this embedding, and exactly which invariants
should one expect to play a direct role?
A hint to this is given by Lazarsfeld's result, to the
effect that if $C$ is nondegenerate complete intersection
of type $(a,b)$ with $a\ge b$ then
$gon(C)\ge a(b-1)$.
For $C\subset \bold P^r$ a smooth curve,
we let
$$\delta _{\eta}(C)=E^2\cdot H_{\eta}^2.$$
We then have
$\delta _{\eta}(C)=\eta ^{r-3}(\eta deg(N)-deg(C))$.
\begin{thm} Let $C\subset \bold P^r$ be a
smooth curve of degree $d$, $r\ge 3$.
Let $\epsilon (C)$ be the Seshadri constant of $C$, and
set $\alpha =min \Big \{1,\sqrt {\epsilon (C)
^{r-3}d}\Big (1-\epsilon (C)\sqrt {\epsilon (C)^{r-3} d}\Big
)\Big \}$.
Then
$$gon(C)\ge min\Big \{ \frac {\delta _{\epsilon (C)}(C)}{4\epsilon (C)
^{r-2}},
\alpha \Big (deg(C)-\frac {\alpha}{\epsilon (C)^{r-2}}\Big )\Big \}.$$
\end{thm}
Although we state the result for curves in $\bold
P^r$ for the sake of
simplicity, it is easy to see that
the same considerations apply when $\bold P^r$ is replaced by
a general smooth projective manifold $X$ with
$Pic(X)\simeq \bold Z$. Later in this section we shall indicate
how these results generalize to higher dimensional varieties
in $\bold P^r$.
{\it Proof.} To avoid
cumbersome notation, we shall assume that $r=3$. The proof
applies to higher value of $r$, with no significant change.
We want then to show that
\begin{equation}
gon(C)\ge min \Big \{\frac {\delta _{\epsilon (C)}(C)}
{4\epsilon (C)}, \alpha \Big (d-\frac {\alpha}{\epsilon (C)}\Big )
\Big \},
\label{eq:spcv}
\end{equation}
where $\alpha =min \{1,\sqrt d(1-\epsilon (C)\sqrt d)\}$.
Suppose, to the
contrary, that the statement is false: if $k=gon(C)
$, then $k$ is strictly smaller than both terms within the braces in the
last inequality. For $\eta <\epsilon (C)$ sufficiently close
to $\epsilon (C)$ the same inequality holds. More precisely,
if let $\alpha _{\eta}=min \{1,\sqrt d(1-\eta \sqrt d)\}$, we
have:
\begin{equation}
k<\dfrac {\delta _{\eta}(C)}{4\eta}
\label{eq:in1}
\end{equation}
and
\begin{equation}
k< \alpha _{\eta}\Big (d-\dfrac {\alpha _{\eta}}{\eta}\Big ).
\label{eq:in2}
\end{equation}
Pick a minimal pencil $A\in Pic^k(C)$, and set $V=:H^0(C,
A)$. Then $V$ is a two-dimensional vector space.
On $C$ we have an exact sequence of locally free sheaves $0@>>>
-A@>>>V\otimes \cal O_C@>>>A@>>>0$.
Consider the blow up diagram:
\begin{equation}
\CD
E @>>> X_C=Bl_C(X) \\
@V\pi VV @VVfV \\
C@>>> X
\endCD
\label{eq:cd}
\end{equation}
(here $E$ clearly denotes the exceptional divisor).
Define
\begin{equation}
\cal F=: Ker(\psi :V\otimes \cal O_{X_C}@>>>\pi ^{*}A).
\label{eq:F}
\end{equation}
$\pi ^{*}A$ is a line bundle on $E$, and $\psi$ is surjective.
Since $E$ is a Cartier divisor in $X_C$, $\cal F$ is a rank two
vector bundle on $X_C$.
As usual we set $H_{\eta}=H-\eta E$, where $\eta$ is a rational
number.
\begin{claim} Let $\eta$ be a rational number in the interval
$(0,\epsilon (C))$.
If $k<\dfrac {\delta _{\eta}(C)}{4\eta}$, then
$\cal F$ is Bogomolov-unstable with respect to $H_{\eta}$.
\end{claim}
{\it Proof.}
By Lemma \ref{lem:eltr},
the Chern classes of $\cal F$
are $c_1(\cal F)=-E$ and
$c_2(\cal F)=\pi ^{*}[A]$, where $[A]$ denotes the divisor class
in $A^1(C)$ of any element in $|V|$, and we implicitly map
$A^1(E)$ to $A^2(X_C)$.
Then the discriminant of $\cal F$ (definition \ref{defn:discr})
is given by
$$ \Delta (\cal F)=E^2-4[A].$$
Therefore by the assumption we have
\begin{equation}
\Delta (\cal F)\cdot H_{\eta}=
\delta _{\eta}(C)-4\eta k>0,
\label{eq:basin2}
\end{equation}
which implies that $\cal F$ is Bogomolov-unstable
with respect to $H_{\eta}$.
$\sharp$
\bigskip
Therefore, by Theorem \ref{thm:main}, there exists a
unique saturated invertible subsheaf
$\cal L\subset \cal F$ satisfying the following
properties:
(i) $\cal L$ is the maximal destabilizing subsheaf of $\cal F$ with
respect to any pair $(H_{\eta},R)$, with $R$ an arbitrary ample
divisor on $X_C$. In particular, for any such pair we have:
$(2c_1(\cal L)-c_1(\cal F))\cdot H_{\eta}\cdot R>0$.
Incidentally, this implies that $\cal L$ is the same for all
the values of $0< \eta <\epsilon (C)$ which make the hypothesis
of the claim true.
(ii) $(2c_1(\cal L)-c_1(\cal F))^2\cdot H_{\eta}\ge
\Delta (\cal F)\cdot H_{\eta}$.
\bigskip
Given the inclusions $\cal L\subset \cal F\subset \cal O_{X_C}^2$,
we have
\begin{equation}
\cal L=\cal O_{X_C}(-D)
\label{eq:eff}
\end{equation}
for some effective divisor $D$ on
$X_C$.
We can write
$$D=xH+yE,$$
with $x$ and $y$ integers and $x\ge 0$.
Set $$s=:D\cdot H_{\eta}\cdot H=x+y\eta d.$$
Since $\cal F$ has no sections, $D\neq 0$. The same applies for
the restriction to any ample surface. Hence $s\ge 0$
for $0<\eta <\epsilon (C)$.
\begin{lem} Assume that
$s\ge \alpha$. Then $k\ge \alpha(d-\dfrac {\alpha} {\eta})$.
\end{lem}
{\it Proof.} Given
(\ref{eq:eff}), from (ii) and (\ref{eq:basin2}) we get
\begin{equation}
(E-2D)^2\cdot H_{\eta}\ge \delta _{\eta}(C)-4\eta k.
\label{eq:basin3}
\end{equation}
Since $E^2\cdot H_{\eta}=\delta _{\eta}(C)$, this can be rewritten
$$D^2\cdot H_{\eta}-D\cdot H_{\eta}\cdot E\ge
-\eta k.$$
By Corollary \ref{cor:sl}, we then have
\begin{equation}
s^2-s\eta d\ge -\eta k.
\label{eq:basin4}
\end{equation}
On the other hand,
we have the destabilizing condition (i)
\begin{equation}
(E-2D)\cdot H_{\eta}\cdot H\ge 0.
\label{eq:dest}
\end{equation}
Now
$$E\cdot H_{\eta}\cdot H=\eta d$$
and therefore (\ref{eq:dest}) can be written
\begin{equation}
\eta d\ge 2s.
\label{eq:dest1}
\end{equation}
Therefore we can apply Lemma \ref{lem:trivial} with $a=\eta d$ and
$b=\eta k$ to obtain
$$\eta k\ge \eta d \alpha-\alpha ^2.$$
This proves the Lemma.
$\sharp$
\bigskip
The proof of the theorem is then reduced to the following Lemma.
\begin{lem} $s\ge \alpha$.
\end{lem}
{\it Proof.} We shall argue that
$s\ge \alpha _{\eta}$ for all rational $\eta <\epsilon (C)$ such
that the inequalities (\ref{eq:in1}) and
(\ref{eq:in2}) hold.
For all such $\eta$ we are then in the
situation of Claim 4.1.
\begin{claim} $\cal L$ is saturated in $V\otimes \cal O_X$.
\label{claim:sat}
\end{claim}
{\it Proof.} By construction, $\cal L
=\cal O_X(-D)$ is saturated in $\cal F$.
Therefore, if the Claim is false then the inclusion
$\cal L\subset V\otimes \cal O_X$ drops rank along $E$.
Hence there exists an inclusion
$\cal O_X(E-D) \subset \cal O_X^2$.
This implies that $D-E$ is effective, and in particular
$(D-E)\cdot H_{\eta}^2\ge 0$.
Together with
the instability condition $(E-2D)\cdot H_{\eta}^2>0$,
this would imply
$D\cdot H_{\eta}^2<0$,
against the fact that $D$ is effective.
$\sharp$
\bigskip
By Claim \ref{claim:sat}, there is an exact sequence
$$0@>>>\cal O_X(-D)@>>>V\otimes \cal O_X@>>>\cal O_X(D)\otimes
\cal J_Y@>>>0,$$
where $Y$ is a closed subscheme of $X$ of codimension two or empty.
Computing $c_2(\cal O_X^2)=0$ from this sequence, we obtain
$D^2=[Y],$
and therefore
$D^2\cdot H\ge 0$.
On the other hand,
$D^2\cdot H=x^2-y^2d$, and so
$$x\ge |y|\sqrt d.$$
Now,
$$s=x+y\eta d\ge x-|y|\eta d\ge |y|\sqrt d(1-\eta \sqrt d).$$
By construction, $H^0(X,\cal F)=0$, and therefore
$D\neq 0$. Hence, if $y=0$ then $s=x\ge 1$.
If $y\neq 0$, then the above inequality shows
that
$s\ge \sqrt d(1-\eta \sqrt d)$.
$\sharp$
\bigskip
This completes the proof of the Theorem.
$\sharp$
\bigskip
\begin{exmp} Let $C\subset \bold P^3$ be a smooth complete
intersection curve of type $(a,b)$, with $a\ge b+3$, $b\ge 2$.
Then $gon(C)\ge a(b-1)$.
\label{exmp:ci}
\end{exmp}
\begin{rem}
This shows that
the result is generally optimal.
However,
the theorem is void for a complete intersection of type $(a,a)$.
But for complete intersections
one knows more than just the Seshadri constant: not
only $\epsilon (C)=\frac 1a$, but
in fact the linear series $|aH-E|$ is base point free, and the
general element is smooth.
An ad hoc argument proves that $gon(C)\ge a(b-1)$ (\cite{la:unp}).
\label{rem:ci}
\end{rem}
\begin{exmp} Let $C$ be a nondegenerate smooth complete
curve in $\bold P^3$ that is linked to a line in a
complete intersection of type $(a,b)$. Then for
$a\gg b\gg 0$ we obtain
$gon (C)\ge deg(C)-(a+b-2)$.
This is clearly optimal,
because a base point free linear series of that degree
is obtained by considering the pencil of planes through the
line.
The same considerations as in Remark
\ref{rem:ci} apply.
\end{exmp}
\begin{rem} An analysis of "small" linear series on special
classes of space curves is carried out by Ciliberto and
Lazarsfeld in \cite{cl}. It would be interesting to know whether the
present method can be adapted to give a generalization of
their results.
\end{rem}
{}From the Theorem, we immediately get
\begin{cor} Let $X\subset \bold P^r$ be
a smooth projective variety.
Let
$d$ be the degree of $X$ and $\epsilon (X)$ be its Seshadri constant.
Suppose that $A$ is a line bundle on $X$ with a pencil
of sections $V\subset H^0(X,A)$ whose base locus has codimension
at least two. Let $F$ be any divisor in the linear series
$|A|$. Then
$$deg(F)\ge
min \Big \{
\frac 1{4\epsilon (C)^{r-2}} \Big [\epsilon (X) \big (c_1(N)\cdot _XH^{n-1}+
(n-1)d \big )-d\Big ],
\alpha \Big (d-\frac {\alpha}{\epsilon (X)}\Big )\Big \},$$
where
$\alpha =\Big \{1,\sqrt {\eta ^{r-3}d}
\Big (1-\epsilon (X)\sqrt {d\epsilon (C)^{r-3}} \Big )\Big \}$.
\end{cor}
{\it Proof.} Let $C\subset X$ be a curve
of the form $X\cap \Lambda$, where
$\Lambda\subset \bold P^3$ is a linear subspace of dimension
$c+1$, with $c$ the codimension of $X$.
Then $V$ restricts to a base point free pencil on $C$, and
the result follows by applying the theorem.
$\sharp$
\bigskip
Given the general nature of the above arguments,
one clearly expects that they should be applicable to a
wider range of situations. In fact,
we give now the generalization of theorem
3.1 to arbitrary smooth projective varieties in $\bold P^r$.
The proof is exactly the same as the one for theorem 3.1,
the only change consisting in a more involved notation.
\begin{thm} Let $Y\subset \bold P^r$ be a projective manifold
of degree $d$ and codimension $c$. Let $\epsilon (Y)$ be its Seshadri
constant, and suppose $0\le \eta \le \epsilon (Y)$.
If $A$ is base point free pencil on $Y$, then
$$\pi ^{*}[A]\cdot H_{\eta}^{r-2}\ge
min \Big \{\frac {\delta _{\eta}(Y)}4, \frac 1{H^2H_{\eta}^{r-2}}
\alpha \Big (H\cdot E\cdot H_{\eta}^{r-2}- \alpha \Big ) \Big \},$$
where
$\alpha =min \Big \{H^2\cdot H_{\eta}
^{r-2}, \sqrt {\eta ^{c-2} deg(Y)}
\Big (H^2\cdot H_{\eta}^{r-2}-\dfrac {H\cdot E\cdot H_{\eta}^{r-2}}
{\sqrt {\eta ^{c-2} deg(Y)}}\Big ) \Big \}$.
\end{thm}
\section{\bf {Stability of restricted bundles}}\label{section:stability}
We deal here with the following problem:
\begin{prob}
Let $\cal E$ be a rank two vector bundle on $\bold P^3$,
and let $C\subset \bold P^3$ be a smooth curve.
If $\cal E$ is stable, what conditions on $C$ ensure
that $\cal E|_C$ is also stable?
\end{prob}
\begin{rem}
This question has been considered by Bogomolov
(\cite{bo:78} and \cite{bo:svb})
in the case of vector bundles on surfaces.
In particular, he shows that if $S$ is a smooth projective
surface with $Pic(S)\simeq \bold Z$, $\cal E$ is a stable rank
two vector bundle on $S$ with $c_1(\cal E)=0$ and $C\subset S$
is a smooth curve with $C^2>4c_2(\cal E)^2$, then
$\cal E|_C$ is stable.
\label{rem:surfacecase}
\end{rem}
After a suitable twisting, we may also assume that
$\cal E$ is {\it normalized}, i.e.
$c_1(\cal E)=0$ or $-1$. We shall suppose here
that $c_1(\cal E)=0$, the other case being similar.
As usual we adopt the following notation:
$$f:X_C@>>>\bold P^3$$
is the blow up of $\bold P^3$ along $C$,
$$E=f^{-1}C$$
is the exceptional divisor, and
$$\pi :E@>>>C$$
is the induced projection.
Recall that for $\eta \in \bold Q$ we set
$$H_{\eta}=:H-\eta E,$$
where we write $H$ for $f^{*}H$.
If $0<\eta <\epsilon (C)$, $H_{\eta}$ is a polarization
on $X_C$.
\begin{defn} We define the {\it stability constant}
of $\cal E$ w.r.t. $C$ as
\begin{center}
$\gamma (C,\cal E)=
sup \{\eta \in [0,\epsilon (C)]| f^{*}\cal E$ is $(H_{\eta},
H)$-stable$\}$.
\end{center}
\label{defn:gamma}
\end{defn}
\begin{rem}
Recall that $f^{*}\cal E$ is $(H,H_{\eta})$-stable if for all
line bundles $\cal L\subset f^{*}\cal E$ we have
$\cal L\cdot H\cdot H_{\eta}<0$.
\end{rem}
\begin{lem} Suppose $0\le \eta <\epsilon (C)$.
Then $f^{*}\cal E$ is $(H,H_{\eta})$-semistable if and only if
$\eta \le \gamma (C,\cal E)$. If
$\eta <\gamma (C,\cal E)$, $f^{*}\cal E$ is
$(H,H_{\eta})$-stable.
\end{lem}
{\it Proof.} The collection of the numerical classes
of nef divisors $D$ with respect to which $f^{*}\cal E$
is $(H,D)$-semistable (or stable) is convex, hence it
contains
the segment $[H,H_{\gamma (C,\cal E)}]$.
Since $f^{*}\cal E$ is $(H,H)$-stable the second
statement follows.
$\sharp$
\bigskip
\begin{lem} Suppose that $V\subset \bold P^3$ is a smooth
surface of degree $a$ containing $C$, and that
$\cal E|_V$ is $\cal O_V(H)$-stable.
Then
$$\gamma (C,\cal E)\ge min\big \{\frac 1a,\epsilon (C)\big \}.$$
\label{lem:bound}
\end{lem}
{\it Proof.} Let $\tilde V$ be the proper transform
of $V$ in $X_C$.
We have $\tilde V\simeq V$
and
$$\tilde V\in |aH_{\frac 1a}|.$$
The hypothesis implies that for every line-bundle
$$\cal L\subset f^{*}\cal E$$
we have
$$\cal L\cdot H_{\frac 1a}\cdot H<0.$$
Hence the same holds for every
$\eta$ with $0\le \eta \le \frac 1a$.
$\sharp$
\bigskip
\begin{rem} Note that the same argument actually proves the
following stronger statement: let $V\supset C$ be a reduced
irreducible surface
through $C$ having degree $m$ and multiplicity $n$ along $C$,
and such that $f^{*}\cal E|_{\tilde V}$ is $\cal O_{\tilde V}(H)$-stable.
Then $\gamma (C,\cal E)\ge min\{\frac nm,\epsilon (C)\}$.
\end{rem}
\begin{lem} Fix $c_2\ge 0$ an integer.
Then then there exists an integer $k$ with the
following property. If $\cal E$ is a
stable rank two vector bundle
on $\bold P^3$ with $c_1(\cal E)=0$ and $c_2(\cal E)=c_2$,
and if $V\subset \bold P^3$ is a smooth surface of
degree $a\ge k$, then $
\cal E|_V$ is $\cal O_V(H)$-stable.
\label{lem:surfres}
\end{lem}
{\it Proof.} We start by finding $s$ such that for a general
surface $S$ of degree $s$
we have $Pic(S)\simeq \bold ZH$ ($s\ge 4$ will do) and furthermore
the restriction $\cal E|_S$ is $\cal O_S
(H)$-stable.
Bogomolov's theorem
(remark \ref{rem:surfacecase}) then says that for any
curve $C\subset S$ such that $C^2>4c_2(\cal E)^2s^2$ the
restriction $\cal E|_C$ is also stable.
Let now $a>0$ be such that $a^2>4c_2(\cal E)^2s$.
Suppose that $V$ is a smooth surface of degree $a$ and that
$\cal E|_V$ is not stable. Then the same is true for
$C=V\cap S$. For a general choice of $S$, $C$ is a smooth
curve, and since $C\cdot _SC=a^2s>4c_2(\cal E)^2s^2$,
we have a contradiction.
$\sharp$
\bigskip
We can in fact restate the previous lemma as follows:
{\it Let $s$ be the smallest positive integer such that
for a general surface of degree $s$ we have $Pic(S)\simeq \bold
Z$ and $\cal E|_S$ stable.
If $a>2c_2(\cal E)\sqrt s$ and $V\subset \bold P^3$ is any smooth surface
of degree $a$, then $\cal E|_V$ is $\cal O_V(H)$-stable.
}
\begin{cor} Let $\cal E$ be a rank two stable bundle on $\bold P^3$
with $c_1(\cal E)
=0$ but $c_2(\cal E)\neq 1$ (i.e., $\cal E$ is not a null correlation
bundle (\cite{oss}). If $a>2c_2(\cal E)$ and $V\subset \bold P^3$
is a smooth hypersurface of degree $a$, then $\cal E|_V$ is
$\cal O_V(H)$-stable.
\label{cor:surfres}
\end{cor}
{\it Proof.} In fact, a theorem of Barth says that in this case
we can take $s=1$ (\cite{ba}).
$\sharp$
\bigskip
\begin{rem} In light of Barth's restriction theorem,
by induction these statements
generalize to $\bold P^r$ for any $r\ge 2$ (for $r=2$ this is just
Bogomolov's theorem, and the hypothesis $c_2\neq 1$ is not needed).
\end{rem}
\begin{rem}
In the proof of Corollary 5.1, we use stability on the
hyperplane section to deduce stability on the whole surface.
What makes this work is Bogomolov's theorem
(cfr remark \ref{rem:surfacecase}), which gives us
a control of the behaviour of stability under restriction to
plane curves. On the other hand, if we are given an arbitrary
smooth surface $V$, it may well be that $\cal E|_V$ is $H$-stable
while $\cal E|_C$ is not, where $C$ is an hyperplane section of
$V$. In that case, however, $\cal E|_{V\cap W}$ will be stable,
if $W$ is a smooth surface of
very large degree such that $V\cap W$
is a smooth curve. To improve the above result, therefore,
one would need to control the behaviour of stability under restriction
to curves not necessarily lying in a plane.
After proving the restriction theorem \ref{thm:curveres}
we shall strengthen
the above corollary (cfr Corollary \ref{cor:surres1}).
\end{rem}
\bigskip
\begin{defn}
If $X$ is a smooth variety and
$c_i\in A^i(X)$ for $i=1$ and $2$,
let $\cal M_X(c_1,c_2)$ denote the
moduli space of stable rank two vector bundles with Chern classes
$c_1$ and $c_2$.
\end{defn}
\begin{cor} Fix an integer $c_2\ge 0$. Then for
any sufficiently large positive integer $a$ the following
holds: if $V\subset \bold P^r$ is a smooth hypersurface of degree
$a$, then $\cal M_{\bold P^r}(0,c_2)$ embeds as an open subset of
$\cal M_V(0,c_2a)$.
\label{cor:surfmoduli}
\end{cor}
{\it Proof.} $\cal M_{\bold P^r}(0,c_2)$ forms a bounded
family of vector bundles, and therefore so does the collection
of the vector bundles $End (\cal E,\cal F)$, with
$\cal E$, $\cal F\in \cal M_{\bold P^r}(0,c_2)$.
Therefore, if $k\gg 0$, we have
$$H^i(\bold P^r,End (\cal E,\cal F)(-a))=0$$
for all
$i\le 2$,
$a\ge k$ and for all $\cal E$, $\cal F\in \cal M_{\bold
P^r}(0,c_2)$. Furthermore, by the above
lemma we can assume that $\cal E|_V$ is
$\cal O_V(H)$-stable for all
$\cal E\in \cal M_{\bold P^r}(0,c_2)$.
{}From the long exact sequence
in cohomology associated to the exact sequence of sheaves
$$0@>>>End(\cal E,\cal F)(-a)@>>>End(\cal E,\cal F)@>>>
End(\cal E|_V,\cal F|_V)@>>>0$$
we then obtain
isomorphisms
$$H^0(\bold P^r,End(\cal E,\cal F))\simeq H^0(V,
End (\cal E|_V,\cal F|_V))$$
and
$$H^1(\bold P^r,End (\cal E,\cal F))\simeq
H^1(V,End (\cal E|_V,\cal F|_V)).$$
Since there can't be any
nontrivial homomorphism between two nonisomorphic
stable bundles
of the same slope, the first one says that
$$\cal E@>>>\cal E|_V$$
is a one-to-one morphism of $\cal M_{\bold P^r}(0,c_2)$ into
$\cal M_V(0,c_2a)$ and the second says that
the derivative of this morphism is an isomorphism
(\cite{ma}).
$\sharp$
\bigskip
\begin{cor} $\gamma (C,\cal E)>0$.
\label{cor:gammapos}
\end{cor}
{\it Proof.} By Lemma 5.3, for $r\gg 0$
the restiction of $\cal E$ to any smooth surface of degree $r$
is stable with respect to the hyperplane bundle. So we
just need to consider a smooth surface through $C$ of very large
degree and apply Lemma 5.2.
$\sharp$
\bigskip
\begin{exmp} Let
$$C=V_a\cap V_b\subset \bold P^3$$
be
a smooth complete intersection of type $(a,b)$, with
$a\ge b$.
Suppose that $V_a$ is smooth, and that
$\cal E|_{V_a}$ is $\cal O_{V_a}(H)$-stable.
Then
$$\gamma (C,\cal E)=\frac 1a=\epsilon (C).$$
In general, $0<\eta <\gamma (C,\cal E)$ if
and only if for $m$ and $n$ sufficiently large
integers such that $\eta =\frac nm$, and $S\in |mH-nE|$
a smooth surface, we have
that $f^{*}\cal E|_S$ is $\cal O_S(H)$-stable. In other words,
we have a degree $m$ hypersurface with an ordinary singularity
of multiplicity $n$ along $C$, such that the pull-back
of $\cal E$ to the desingularization of $S$ is $H$-stable.
\end{exmp}
\bigskip
Our result is then the following:
\begin{thm} Let $\cal E$ be a stable rank two vector
bundle on $\bold P^3$ with $c_1(\cal E)=0$.
Let $C\subset \bold P^3$ be a smooth curve
of degree $d$ and Seshadri constant $\epsilon (C)$,
and let $\gamma =\gamma (C,\cal E)$ be the stability constant
of $\cal E$ w.r.t $C$.
Let
$\alpha =min\Big \{1,\sqrt d\Big (\sqrt {\frac 34}-\gamma \sqrt d
\Big )\Big \}.$
Suppose that
$\cal E|_C$ is not stable.
Then
$$c_2(\cal E)\ge min\Big \{\frac {\delta _{\gamma}}4,
\alpha \gamma \Big (d-\frac {\alpha}{\gamma}\Big )\Big \}.$$
\label{thm:curveres}
\end{thm}
{\it Proof.} Suppose to the contrary that $c_2(\cal E)$ is strictly
smaller that both quantities on the right hand side. We can
find a rational number $\eta$ with $0<\eta <\gamma $
such that
\begin{equation}
c_2(\cal E)<\dfrac {\delta _{\eta}(C)}4
\label{eq:c2ineq}
\end{equation}
and
\begin{equation}
c_2(\cal E)<\alpha \eta \Big (d-\dfrac {\alpha}{\eta}\Big).
\label{eq:c2ineq1}
\end{equation}
By assumption there exists a line bundle $L$ on $C$ of
degree $l\ge 0$ sitting in an exact sequence
$$0@>>>L@>>>\cal E|_C@>>>L^{-1}@>>>0.$$
Define a sheaf $\cal F$ on $X_C$ by the exactness of the
sequence
\begin{equation}
0@>>>\cal F@>>>f^{*}\cal E@>>>\pi ^{*}L^{-1}@>>>0.
\label{eq:Fi}
\end{equation}
Then $\cal F$ is a rank two vector bundle on $X_C$
having Chern classes
$c_1(\cal F)=-[E]$ and $c_2(\cal F)=f^{*}c_2(\cal E)-\pi
^{*}[L]$ (cfr Lemma \ref{lem:eltr}).
A straightforward computation then gives
\begin{equation}
\Delta (\cal F)\cdot H_{\eta}=
\delta _{\eta}(C)-4c_2(\cal E)+4\eta l\ge \delta _{\eta}(C)-4c_2(\cal E)
\label{eq:DF}
\end{equation}
and this is positive by (\ref{eq:c2ineq}).
Therefore $\cal F$ is Bogomolov-unstable with respect
to $H_{\eta}$ (Theorem \ref{thm:main}). Let
$$\cal L\subset \cal F$$
be the maximal destabilizing line bundle w.r.t. $H_{\eta}$. We
shall write
$$\cal L=\cal O_{X_C}(-D),$$
with
$$D=xH-yE.$$
\begin{claim} $x>0$
\label{claim:xpos}
\end{claim}
{\it Proof.} Pushing forward the inclusion
$\cal L\subset \cal F$ we obtain an inclusion
$\cal O_{\bold P^3}(-x)\subset \cal E$. Therefore the
statement follows from the assumption of stability on
$\cal E$ and the hypothesis $c_1(\cal E)=0$.
$\sharp$
\bigskip
The destabilizing condition says
$(2c_1(\cal L)-c_1(\cal F))\cdot H_{\eta}\cdot R\ge 0$
for all nef divisors on $X_C$, with strict inequality holding
when $R$ is ample. In particular, with $R=H$ we have
\begin{equation}
(E-2D)\cdot H_{\eta}\cdot H\ge 0.
\label{eq:resinst}
\end{equation}
Let us set $s=D\cdot H_{\eta}\cdot H$. Then (\ref{eq:resinst})
reads
\begin{equation}
\eta d\ge 2s.
\label{eq:resinst1}
\end{equation}
On the other hand, since $\cal L$ is saturated in
$\cal F$, we also have
$(E-2D)^2\cdot H_{\eta}\ge \Delta (\cal F)\cdot H_{\eta}$,
and with some algebra this becomes
\begin{equation}
c_2(\cal E)\ge D\cdot E\cdot H_{\eta}-D^2\cdot H_{\eta}+\eta l\ge
D\cdot E\cdot H_{\eta}-D^2\cdot H_{\eta}.
\label{eq:resinst2}
\end{equation}
Invoking Corollary \ref{cor:sl}, we
then have
\begin{equation}
c_2(\cal E)\ge s\eta d-s^2.
\label{eq:resinst3}
\end{equation}
\begin{claim} $\cal L$ is saturated in $f^{*}\cal E$.
\label{claim:ressat}
\end{claim}
{\it Proof.} Suppose not. Then there would be an inclusion
$$\cal L(E)=\cal O_{X_C}(E-D)\subset f^{*}\cal E$$
and therefore the $(H,H_{\eta})$-stability of $f^{*}\cal E$
would force
$$(E-D)\cdot H_{\eta}\cdot H<0.$$
On the other hand by instability we have $E\cdot H_{\eta}\cdot H\ge
2D\cdot H_{\eta}\cdot H$ and from this we would get
$$E\cdot H_{\eta}\cdot H=\eta d<0,$$
absurd.
$\sharp$
\bigskip
Therefore there is an exact sequence
$$0@>>>\cal O_{X_C}(-D)@>>>f^{*}\cal E@>>>\cal O_{X_C}(D)\otimes \cal J_W
@>>>0$$
where $W$ is a local complete intersection subscheme of $X_C$ of codimension
two or empty.
Computing $c_2(f^{*}\cal E)$ from the above
sequence we then get
$f^{*}c_2(\cal E)=W-D^2$, i.e.
$$D^2\cdot H\ge -c_2(\cal E).$$
This can be rewritten
$$x^2\ge y^2d-c_2(\cal E).$$
Recall that we have (remark \ref{rem:dd})
$$d\ge \delta _{\eta}(C),$$
and therefore the assumption $c_2(\cal E)< \delta _{\eta}(C)/4$
implies
\begin{equation}
c_2(\cal E)< \frac d4.
\label{eq:c2d4}
\end{equation}
\begin{lem}
$$s\ge min\Big \{1,\sqrt d\Big (\sqrt {\frac 34}-\eta \sqrt d\Big )
\Big \}.$$
\end{lem}
{\it Proof.}
If $y\le 0$ then $s=x+|y|\eta d\ge 1$.
If $y> 0$,
we have
$s=x-y\eta d\ge
y\sqrt d
\Big (\sqrt {1-\frac {c_2(\cal E)}d} -\eta\sqrt d\Big )$
and therefore using (\ref{eq:c2d4}) we obtain
$$s\ge
\sqrt d\Big (\sqrt {\frac 34}-\eta \sqrt d \Big ).$$
$\sharp$
\bigskip
Hence we can apply lemma
\ref{lem:trivial} with
$a=\eta d$ and $b=c_2(\cal E)$ to deduce
$$c_2(\cal E)\ge \alpha \eta d-\alpha ^2,$$
which contradicts (\ref{eq:c2ineq1}).
This completes the proof of the Theorem.
$\sharp$
\bigskip
\begin{cor} Let $\cal E$ be a stable rank two vector bundle on
$\bold P^3$ with $c_1(\cal E)=0$ and $c_2(\cal E)=c_2$.
If $b\ge c_2+2$ and $V\subset \bold P^3$ is a smooth hypersurface
of degree $b$, then $\cal E|_V$ is $\cal O_V(H)$-stable.
\label{cor:surres1}
\end{cor}
{\it Proof.}
Let $a\gg b$; then we may assume that if $W\subset \bold P^3$ is
a surface of degree $a$ then $\cal E|_W$ is $H$-stable.
If $W$ is chosen generally, we may also assume that $C=W\cap V$
is a smooth curve. Then
by Lemma \ref{lem:bound} we have $\gamma (C,\cal E)=\epsilon (C)=
\frac 1a$. For $a$ large enough, furthermore, we also have
$\alpha =1$. Hence the theorem says that if $\cal E|_C$ is not
stable, then $c_2\ge b-1$. The hypothesis implies therefore that
$\cal E|_C$ is stable, and this forces $\cal E|_V$ to be stable
also.
$\sharp$
\bigskip
\begin{cor}
Let $\cal E$ be as above, and let
$C=V_a\cap V_b$ be a smooth complete intersection
curve of type $(a,b)$, and suppose that $V_a$ is smooth.
Assume that $a\ge \frac 43b+\frac {10}3$ and that $b\ge c_2+2$.
Then $\cal E|_C$ is stable.
\end{cor}
{\it Proof.} By Corollary 5.4, $\cal E|_{V_b}$ is $H$-stable.
Hence by Lemma \ref{lem:bound} $\gamma (C,\cal E)=\frac 1a$.
The first hypothesis implies that $\alpha =1$. Hence if $\cal E|_C$
is not stable the theorem yields $c_2\ge b-1$, a contradiction.
$\sharp$
\bigskip
\begin{cor} Fix a nonnegative integer $c_2$. Then
we can find positive integers $a$ and $b$ such that if $C\subset
\bold P^3$ is any smooth complete intersection curve
of type $(a,b)$ then $\cal M_{\bold P^3}(0,c_2)$ embeds
as a subvariety of $\cal M_C(0)$.
\end{cor}
{\it Proof.} The argument is similar
to the one in the proof
of Corollary 5.2. Here one uses
the Koszul resolution
$$0@>>>\cal O_{\bold P^3}(-a-b)@>>>\cal O_{\bold P^3}(-a)\oplus
\cal O_{\bold P^3}(-b)@>>>\cal J_C@>>>0$$
to show that $H^i(\bold P^3,End(\cal E,\cal F)\otimes
\cal J_C)=0$ for $i\le 1$.
$\sharp$
\bigskip
\begin{rem} Using the above corollary,
we obtain a compactification of
$\cal M_{\bold P^3}(c_1,c_2)$, by simply
taking its closure in the moduli space of semistable bundles on the
curve.
It would be interesting
to know whether these compactifications are intrinsic, i.e. they
are independent of the choice of the curve or, if not,
how they depend on the geometry of the embedding $C\subset \bold P^3$.
\end{rem}
|
1995-08-01T03:37:57 | 9405 | alg-geom/9405014 | en | https://arxiv.org/abs/alg-geom/9405014 | [
"alg-geom",
"math.AG"
] | alg-geom/9405014 | Eckhard Meinrenken | Eckhard Meinrenken | On Riemann-Roch Formulas for Multiplicities | 21 pages, AMS-LaTeX | null | null | null | null | A Theorem due to Guillemin and Sternberg about geometric quantization of
Hamiltonian actions of compact Lie groups $G$ on compact Kaehler manifolds says
that the dimension of the $G$-invariant subspace is equal to the Riemann-Roch
number of the symplectically reduced space. Combined with the shifting-trick,
this gives explicit formulas for the multiplicities of the various irreducible
components. One of the assumptions of the Theorem is that the reduction is
regular, so that the reduced space is a smooth symplectic manifold. In this
paper, we prove a generalization of this result to the case where the reduced
space may have orbifold singularities. Our proof uses localization techniques
from equivariant cohomology, and relies in particular on recent work of
Jeffrey-Kirwan and Guillemin. Since there are no complex geometry arguments
involved, the result also extends to non Kaehlerian settings.
| [
{
"version": "v1",
"created": "Mon, 30 May 1994 16:08:06 GMT"
},
{
"version": "v2",
"created": "Mon, 14 Nov 1994 09:48:48 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Meinrenken",
"Eckhard",
""
]
] | alg-geom | \section{Introduction}
Let $(M,\omega)$ be a compact K\"{a}hler manifold, and
let $\tau:L\rightarrow M$ be a holomorphic line bundle over $M$
with Hermitian fiber metric $h$. $(L,h)$ is said to satisfy
the quantizing condition if $-2\pi i \omega$ is the curvature of
the canonical Hermitian connection on $L$.
Let $H^i(M,{\cal O}(L))$ be the $i$th cohomology group for the
sheaf of germs of holomorphic sections. By the
Riemann-Roch Formula of Hirzebruch and Atiyah-Singer\cite{AS68a},
the Euler number
\begin{equation} \mbox{Eul}(L):=\sum_i (-1)^i \dim H^i(M,{\cal O}(L))\end{equation}
is equal to the characteristic number
\begin{equation} \mbox{Eul}(L)=\int_M Td\,(M)\,Ch\,(L),\label{HRR}\end{equation}
where ${Td}\,(M)$ is the Todd class and ${Ch}\,(L)=e^{[\omega]}$
the Chern character.
Recall that if $L$ is ``sufficiently positive'', in particular
if one replaces $L$ by some sufficiently high tensor power,
all the cohomology groups with $i>0$ are zero by
Kodaira's Theorem \cite{GH78}, so in this case
(\ref{HRR}) gives a formula for the dimension of the space
$H^0(M,{\cal O}(L))=\Gamma_{hol}(M,L)$ of holomorphic sections.
Let $G$ be a compact, connected Lie group that acts on $M$
by K\"ahler diffeomorphisms $\Phi:G\times M\rightarrow M$,
with an equivariant moment map $J:M\rightarrow {\frak g}^*$.
Suppose also that $\Phi$ lifts to Hermitian bundle automorphisms
of $L\rightarrow M$, according to the rules of geometric
quantization \cite{GS82a}.
The corresponding virtual representation of $G$ on
$\sum (-1)^i\,H^i(M,{\cal O}(L))$ may then be regarded as the
``quantization'' of the classical action $\Phi$.
Its character $\chi$ is the element of the representation ring
$R(G)$ defined by
\begin{equation} \chi(g):=\sum_i\, (-1)^i\, \mbox{tr}\big(g|H^i(M,{\cal O}(L))\big).
\label{char}
\end{equation}
{}From the Equivariant Riemann-Roch Formula of Atiyah-Segal-Singer
\cite{AS68b,AS68a}, one has an expression for $\chi(g)$
as the evaluation of certain characteristic classes on
the fixed point set $M^g=\{x\in M|\,g.x=x\}$ (which is a
K\"ahler submanifold of $M$):
Let ${Ch}^g(L|M^g)=c_L(g)\,{Ch}(L|M^g)$, where $c_L(g)\in S^1$ is the
(locally constant) action of $g$ on $L|M^g$. Denote by $N^g$ the
normal bundle of $M^g$ in $M$, by $F(N^g)$ its curvature, and let
\begin{equation} D^g(N^g)=\det(I-(g^\sharp)^{-1}e^{-\frac{i}{2\pi}F(N^g)})\end{equation}
where $g^\sharp$ is the automorphism of $N^g$ defined by $g$.
Then
\begin{equation} \chi(g)=\int_{M^g} \frac{{Td}\,(M^g){Ch}^g(L|{M^g})}{D^g(N^g)}.
\label{character}\end{equation}
By a Theorem of Guillemin and Sternberg \cite{GS82a}, there are also
Riemann-Roch Formulas for the multiplicities of the
irreducible components of the above representation,
at least if certain regularity
assumptions are satisfied. Let $T\subset G$ be a maximal torus, and
${\frak g}={\frak t}\oplus [{\frak t},{\frak g}]$ the corresponding decomposition of the Lie
algebra.
Choose a fundamental Weyl chamber
${\frak t}^*_+\subset{\frak t}^*\subset {\frak g}^*$, let $\Lambda\subset {\frak t}^*$ the
integral lattice, and $\Lambda_+=\Lambda\cap {\frak t}^*_+$ the dominant
weights.
For a given lattice point $\mu\in\Lambda_+$,
let $V_\mu$ denote the corresponding
irreducible representation with highest weight $\mu$,
and define the multiplicity
$N(\mu)$ by the alternating sum
\begin{equation} N(\mu):=\sum_i (-1)^i \dim\big(V_\mu^*\otimes
H^i(M,{\cal O}(L))\big)^G. \label{defmult}\end{equation}
Suppose that $\mu\in\Lambda_+$ is a regular value of $J$, or equivalently
that the action of the isotropy group
$G_\mu$ on $J^{-1}(\mu)$ is locally free.
If the action is in fact free, the reduced space
$M_\mu=J^{-1}(\mu)/G_\mu$ is a smooth symplectic manifold,
and it is well-known that it acquires a natural K\"ahler
structure, together with a quantizing line bundle $L_\mu$.
The main result of \cite{GS82a} is that the multiplicity of $\mu$ in
$\Gamma_{hol}(M,L)$ is equal to the dimension
of the space $\Gamma_{hol}(M_\mu,L_\mu)$, so in particular $N(\mu)$ is
given by the Euler number of $L_\mu$ if $L$ is sufficiently positive:
\begin{theo}[V. Guillemin, S. Sternberg \cite{GS82a}] If the action of
$G_\mu$ on $J^{-1}(\mu)$ is free, and if $L$ is sufficiently
positive,
\begin{equation} N(\mu)=\int_{M_\mu}{Td}\,(M_\mu){Ch}\,(L_\mu). \label{GS} \end{equation}
\end{theo}
The ``physical'' interpretation of this Theorem is that reduction and
quantization commute.
In practice, one is often dealing with situations where the action is
only locally free. In this case, the reduced
space is in general just an
orbifold (or V-manifold) in the sense of Satake \cite{S57},
which means (roughly) that it is locally the quotient of a manifold
by a finite group.
Moreover, the reduced line bundle
is in general just an orbifold-bundle, that is, at some points the
fiber of $L_\mu$ is not ${\Bbb C}$, but its quotient by a
finite group.
Guillemin and Sternberg conjectured that in this case,
the right hand side of
(\ref{GS}) has to be replaced by the expression from T. Kawasaki's
Riemann-Roch Formula for orbifolds \cite{K79}.
It was proved by R. Sjamaar \cite{S93} that this assertion is true if
$L$ is sufficiently positive, and if $L_\mu$ is an honest line bundle.
In fact, his approach also covers the truly singular case where
$\mu$ is not even a regular value, by using Kirwan's partial
desingularization procedure to reduce it to the orbifold case.
On the other hand, the condition that $L_\mu$ be a genuine line
bundle
is rather restrictive. It is the aim of the present paper
to give a different proof of the Guillemin-Sternberg conjecture (for
$\mu$ a regular value), without having to make this assumption.
The method we use is motivated by recent work of V. Guillemin
\cite{G94},
who used localization techniques from equivariant cohomology
to establish the connection between the Multiplicity Formula (\ref{GS})
and a certain formula for counting lattice points in polytopes.
This formula is known to be true in various interesting cases, and
for these gives a new proof of (\ref{GS}) without using any
complex geometry arguments. (In particular, it also works for
{\em almost} K\"ahler polarizations.)
We will adopt this utilization of equivariant cohomology,
but in a slightly different guise.
The main idea is to consider the rescaled problem, where we replace
\begin{equation} \omega\leadsto m\omega,\,L\leadsto\,L^m,\,J\leadsto\, mJ,
\,\mu\leadsto m\mu\end{equation}
for $m\in{\Bbb N}$.
Our starting point will be the Equivariant Riemann-Roch Formula,
but instead of (\ref{character}) we will use it in a
form due to Berline and Vergne \cite {BV85},
involving equivariant characteristic classes.
By a stationary phase
version of the Localization Formula of Jeffrey and Kirwan \cite{JK93},
we pass from equivariant characteristic classes to (ordinary)
characteristic classes on the reduced spaces. This leads to
the desired Multiplicity Formula for $N^{(m)}(m\mu)$,
up to an error term $O(m^{-\infty})$.
Since the multiplicities are integers, one easily finds
that the error term is zero for large $m$.
To investigate the general dependence of $N^{(m)}(m\mu)$ on $m$, we
use a different expression for $N^{(m)}(m\mu)$ via
the number of lattice points in certain polytopes.
If $J(M)$ is contained in the set of regular elements of ${\frak g}^*$,
in particular in the abelian case, this analysis
turns out to be sufficiently
good to show that the above error term is zero for all $m$.
\section{Statement of the result}
In order to state the result, we have to give a closer
description of the reduced space and its singular strata.
Suppose that $\mu\in\Lambda_+$ is a regular value of $J$.
Recall first the shifting trick to express
$M_\mu$ as a reduced space at the zero level set:
Let ${O}=G.\mu$ be the coadjoint orbit through $\mu$,
equipped with its usual Kirillov K\"ahler structure, and let
${O}^-$ denote ${O}$ with the opposite K\"ahler
structure. The action of $G$ on ${O}$ is Hamiltonian, with
moment map $\Psi$ the embedding into ${\frak g}^*$.
Then $\tilde{M}=M\times {O}^-$ is a K\"ahler manifold,
and the diagonal
action of $G$ is Hamiltonian, with moment map $\tilde{J}=J-\Psi$.
There are canonical identifications
\begin{equation} M_\mu=J^{-1}(\mu)/G_\mu\cong J^{-1}({O})/G\cong
\tilde{J}^{-1}(0)/G.\end{equation}
By Kostant's version of the Borel-Weil-Bott Theorem,
one also has a natural quantizing bundle $\Xi\rightarrow {O}$, and
the irreducible representation $V_\mu$ corresponding to $\mu$
gets realized as the space of holomophic sections
of $\Xi$. (The higher order cohomology groups $H^i(O,\O(\Xi))$
vanish.) The tensor product
$\tilde{L}:=L\otimes \Xi^*$ quantizes
$\tilde{M}$, and there is an isomorphism
\begin{equation} H^i(\tilde{M},{\cal O}(\tilde{L}))\cong
V_\mu^*\otimes H^i(M,{\cal O}(L)).\label{qst}\end{equation}
Hence $N(\mu)={\rm Eul}(\tilde{L})$, which is the quantum counterpart
of the shifting-trick.
Using the shifting-trick, it is enough to consider the case $\mu=0$.
The reduced space $M_0$ inherits a natural K\"ahler structure
from $M$ (see \cite{GS82a}), and the reduced bundle $L_0=(L|J^{-1}(0))/G$
renders a quantizing orbifold-line bundle. Note however
that $L_0$ need not be an honest line bundle, not even
over the smooth part of $M_0$.
Sections of an orbifold bundle are defined as coming from
invariant sections for the local orbifold charts,
so all sections of $L_0$ have to vanish at points were the fiber
is not ${\Bbb C}$.
Let us regard $P=J^{-1}(0)$ as an orbifold-principal bundle over
$M_0=J^{-1}(0)/G$. Following \cite{K79,F92}, we introduce
\begin{equation} \tilde{P}=\{(x,g)|\,x\in P,\,g.x=x\}\subset P\times G,\end{equation}
and let $\Sigma=\tilde{P}/G$ be its quotient under the locally free action
$h.(x,g)=(h.x,\,h\,g\,h^{-1})$.
The projection of $\tilde{P}$ to the second factor descends to
a locally constant mapping
\begin{equation} \tau:\Sigma\rightarrow {\rm Conj}(G)\end{equation}
to the set of conjugacy classes. For $g\in G$, let $(g)={\rm Ad}(G).g$
denote the corresponding conjugacy class, and $\Sigma_g$ its preimage under
$\tau$. There is a natural identification $\Sigma_g=P^g/Z_g$, where
$P^g\subset P$ is the fixed point
manifold and $Z_g$ the centralizer of $g$ in $G$.
Since the fixed point set $M^g\subset M$ is a K\"ahler submanifold, and
the action of $Z_g$ on $M^g$ is Hamiltonian with the restriction of $J$
serving as a moment map, this makes clear that $\Sigma$ is a
K\"ahler orbifold (with several components of different dimensions).
Note that this K\"ahler structure does not depend on the choice of the
representative for $(g)$. Observe also that $\Sigma_e\cong M_0$.
The collection of bundles $(L|P^g)/Z_g$ defines a
quantizing orbifold line bundle $L_\Sigma\rightarrow \Sigma$.
As above, let $c_L(g)\in S^1$ be the locally constant action of
$g$ on $L|P^g$, denote by $c_\Sigma:\Sigma\rightarrow S^1$ the
function defined by the $c_L(g)'s$, and let ${Ch}^\Sigma(L_\Sigma)$
be the cohomology class defined by
\begin{equation} Ch^\Sigma(L_\Sigma)=c_\Sigma\,e^{\omega_\Sigma}\end{equation}
where $\omega_\Sigma$ is the K\"ahler form on $\Sigma$.
Consider now the natural mapping $f:\Sigma\rightarrow M_0$, sending
$G.(x,g)\rightarrow G.x$. In a local orbifold chart,
the tangent space to $\Sigma$ at $G.(x,g)$ is isomorphic to
$T_x(M^g\cap J^{-1}(0))/T_x(Z_g.x)$, while the tangent space to
$M_0$ at $G.x$ is $T_x(J^{-1}(0))/T_x(G.x)$. From this,
it is easy to see that $f$ is a Khler immersion.
Let $N_\Sigma\rightarrow \Sigma$ be the normal bundle of this immersion,
and denote by $g^\sharp$ the automorphism of $N_\Sigma|\Sigma_g$
induced by the action of $g$.
Then the collection of differential forms
\begin{equation} \det(I-(g^\sharp)^{-1}e^{-\frac{i}{2\pi}F(N_\Sigma)}),\end{equation}
where $F(N_\Sigma)$ is the curvature of $N_\Sigma$, defines a characteristic
class $D^\Sigma(N_\Sigma)$.
Finally, for each connected component $\Sigma_i$ of $\Sigma$, let
$d_i$ be the number of elements in a generic stabilizer for the
$G$-action on the corresponding component $\tilde{P}_i$, and
$d_\Sigma:\,\Sigma\rightarrow{\Bbb N} $ the function defined by the $d_i$'s.
For general values $\mu\in\Lambda_+$, let $\Sigma_\mu$,
$L_\mu$ etc. be defined by means of the shifting-trick.
The main result of this paper is the following
\begin{theo}[Multiplicity Formula] \label{multf}
If $\mu\in\Lambda_+$ is a
regular value of $J$, the multiplicities $N^{(m)}(m\mu)$
are for $m>>0$ given by the formula
\begin{equation} N^{(m)}(m\mu)=
\int_{\Sigma_\mu} \frac{1}{d_{\Sigma_\mu}}\,
\frac{Td\,({\Sigma_\mu})\,
{Ch}^{\Sigma_\mu}(L^m_{\Sigma_\mu})}
{D^{\Sigma_\mu}(N_{\Sigma_\mu})}.\label{orbifold1}\end{equation}
If the image of the moment map, $J(M)$, is contained in ${\frak g}^*_{reg}$
(the set of regular elements of ${\frak g}^*$), one may set $m=1$ in this
formula:
\begin{equation} N(\mu)= \int_{\Sigma_\mu} \frac{1}{d_{\Sigma_\mu}}\,\frac{{Td}\,
({\Sigma_\mu})\,
{Ch}^{\Sigma_\mu}(L_{\Sigma_\mu})}
{D^{\Sigma_\mu}(N_{\Sigma_\mu})},\label{orbifold}\end{equation}
In particular, this is the case if $G$ is abelian.
\end{theo}
The following sections are aimed at proving this Theorem.
We do not know whether the second part of the Theorem remains true
without the condition $J(M)\subset{\frak g}^*_{reg}$.\\
\noindent{\bf Remarks.}
\begin{enumerate}
\item
Comparing the right hand side of (\ref{orbifold}) to Kawasaki's
Riemann-Roch Formula for orbifolds \cite{K79}, the Theorem says that
$N(\mu)$ is equal to the Euler number of the orbifold-bundle
$L_\mu\rightarrow M_\mu$. In particular, $N(\mu)$ is zero if the fiber of
$L_\mu$ over the smooth stratum of $M_\mu$ is a nontrivial
quotient of ${\Bbb C}$.
\item
Let $\Delta=J(M)\cap{\frak t}_+^*$, which is a convex polytope by
a result of Guillemin-Sternberg and Kirwan,
and $\Delta^*\subset
\Delta$ the set of regular values. By the Duistermaat-Heckman Theorem
\cite{DH82},
the diffeotype of the reduced space $M_\mu$ (and of course also of
$\Sigma_\mu$)
does not change as $\mu$ varies in a connected component of
${\rm int}(\Delta^*)$,
and the cohomology class of the symplectic form $\omega_\mu$
varies linearly.
In particular, the symplectic volume $\mbox{\rm Vol}(M_\mu)$ is
a polynomial on these connected components.
If the action of $G_\mu$ on $J^{-1}(\mu)$ is free, so that
${\Sigma_\mu}=M_\mu$, the right hand side in (\ref{orbifold}) is equal to
a polynomial as well, since all that varies is the Chern character
${Ch}(L_\mu)=e^{\omega_\mu}$.
In the orbifold case, the behaviour is slightly more complicated:
For $\mu\in\Lambda_+$ in any given connected
component of ${\rm int}(\Delta^*)$, and any connected component
$\Sigma_{\mu,j}$ of $\Sigma_\mu$,
\begin{equation} {Ch}^{\Sigma_\mu}(L_{\Sigma_\mu})|\,\Sigma_{\mu,j}
=\rho_\mu(g_j^{-1})\,c_L(g_j)\,e^{\omega_{\Sigma_\mu}},\end{equation}
where $g_j\in G_\mu$ represents $\tau(\Sigma_{\mu,j})$, and
$\rho_\mu:G_\mu\rightarrow S^1$
is defined by $\rho_\mu(\exp(\xi))=e^{2\pi i\l \mu,\xi\rangle}$.
Hence, the right hand side of (\ref{orbifold}) is of the form
\begin{equation} N(\mu)=\sum_j \rho_\mu(g_j^{-1})\, c_L(g_j)\,p_j(\mu),
\end{equation}
where the $p_j$ are polynomials of degree
$\frac{1}{2}\dim(\Sigma_{\mu,j})$.
\item
Since ${Ch}\,((L^m)_{m\mu})=e^{m\omega_\mu}$, the right hand side
of (\ref{orbifold1}) is a polynomial in $m$ if the $G_\mu$-action
on $J^{-1}(\mu)$ is free. In the orbifold-case, this is not true
in general since
\begin{equation} {Ch}^{\Sigma_\mu}(L^m_{\Sigma_\mu})|\,\Sigma_{\mu,j}=
\rho_\mu(g_j^{-1})^m\,c_L(g_j)^m\,e^{m\omega_{\Sigma_\mu}}.\end{equation}
\begin{defi}[Ehrhart \cite{E77}]
A function $f:{\Bbb N}\rightarrow {\Bbb C}$ is called an arithmetic polynomial,
if for some $k\in {\Bbb N}$, all the functions
\begin{equation} q_j(m)=f(km-j),\,\,j=0,\ldots, k-1\end{equation}
are polynomials. $k$ is called the period of $f$.
\end{defi}
Equivalently, $f$ is an arithmetic polynomial
if and only if it can be written in the form
\begin{equation} f(m)=\sum_{l=0}^{k-1}\exp(2\pi i\,\frac{l\,m}{k})\,p_l(m),\end{equation}
where the $p_l$ are polynomials. Taking $k$ such that $g^k=e$
for all $(g)\in \tau({\Sigma_\mu})$, the right hand side of (\ref{orbifold1})
clearly has this property.
\item
Our proof of Theorem \ref{multf} does not really use the
assumption that $M$ is K\"ahler. Everything will be derived
from the Equivariant Riemann-Roch Formula (\ref{char}),
which is of course valid in much more general situations.
Suppose for instance that $M$ is an arbitrary compact symplectic
manifold, equipped with a Hamiltonian $G$-action, and that these data
are quantizable. Then one can always choose a compatible, invariant
almost K\"ahler structure, and replace the virtual space
$\sum (-1)^i \,H^i(M,{\cal O}(L))$ with the index space of
some G-invariant Dirac operator for the Clifford module
$L\otimes \wedge(T^{(0,1)}M)^*$ (see \cite{BGV92,G94}).
As an immediate consequence of the Berline-Vergne Formula for
the character, Theorem \ref{RiemannRoch} below,
the multiplicities $N(\mu)$ defined in this way
do not depend on the choice of the almost K\"ahler structure or
of the quantizing line bundle $L$.
\end{enumerate}
\bigskip
\noindent{\bf Example:}
Let $M={\Bbb C} P(2)$, equipped with the Fubini-Study K\"ahler form
$\omega_{FS}$.
Let $G=S^1$ act by
\[e^{i\phi}.[z_0:z_1:z_2]=[e^{i\phi}z_0:e^{-i\phi}z_1:z_2].\]
This action is Hamiltonian, and has a moment map
\[J([z_0:z_1:z_2])=
\frac{|z_1|^2-|z_0|^2}{|z_0|^2+|z_1|^2+|z_2|^2}.\]
The dual of the tautological line bundle serves as a quantizing
line bundle $L$.
We also consider the tensor powers $L^m$, which are
quantizing line bundles for $(M,m\,\omega_{FS})$.
By Kodaira's Theorem, $H^i(M,\O(L^m))=0$ for all $i>0,\,m\in{\Bbb N}$.
If we identify the
spaces $\Gamma_{hol}(M,L^m)$ with the homogeneous polynomials of
degree $m$ on ${\Bbb C}^3$, the representation of $S^1$ is induced
by the action $e^{i\phi}.(z_0,z_1,z_2)=
(e^{i\phi}z_0,e^{-i\phi}z_1,z_2)$ on ${{\Bbb C}^3}$.
The isotypical subspace of $\Gamma_{hol}(M,L^m)$
corresponding to the weight $l\in{\Bbb Z}$
is, for $l\ge 0$, spanned by
\[ z_0^l\,z_2^{m-l},\,z_0^{l+1}\,z_1\,z_2^{m-l-2},\ldots,
z_0^{l+r}\,z_1^r\,z_2^{m-l-2r}\]
with $r=\left[\frac{m-l}{2}\right]$. For $l \le 0$, the roles
of $z_0$ and $z_1$ are reversed. Thus
\[ N^{(m)}(l)=\left\{\begin{array}{ll}
{1+\left[\frac{m-|l|}{2}\right]}&{\mbox{ if $|l|\le m$}}\\
{0}&{\mbox{ otherwise}}
\end{array} \right.,\]
for all $l\in{\Bbb Z},\,m\in{\Bbb N}$.
On the other hand,
the image of the moment map $J^{(m)}=mJ$ is
the interval $-m\,\le\, \mu\,\le\, m$, with critical values at
$-m,\,0,\,m$. If $0< \, |l|\,< m $, the level set $(mJ)^{-1}(l)$
consists of two orbit type strata: On the set where $z_2\not=0$,
the action is free, and on the set where $z_2=0$, the stabilizer
is ${\Bbb Z}_2$. Writing $S=\{e,g\}$, the reduced space $M^e_\mu=M_\mu$
is an orbifold with a ${\Bbb Z}_2$ singularity (the
``teardrop-orbifold''), and $M^g_\mu$ is the singular point.
Since $c_{L^m}(g)=(-1)^m$ and $\rho_l(g^{-1})=(-1)^l$, we expect
the multiplicities to grow like
$N^{(m)}(l)=p_e(m,l)+(-1)^{(m-l)}p_g(m,l)$ where $p_e$ is a first
order polynomial and $p_g$ a constant. Comparison with the
above formula shows that this is indeed the case, with
$p_e(m,l)=3/4+(m-|l|)/2$ and $p_g(m,l)=1/4$.
Note that for $m-|l|$ even, the fiber of $L^m_l$ at the singular
point is ${\Bbb C}$, whereas form $m-|l|$ odd it is ${\Bbb C}/{\Bbb Z}_2$. This means
that for $m-|l|$ odd, all holomorphic sections of $L^m_l$ have to
vanish at the singular point. Again, this fits with the above
explicit formulas.
\section{Some equivariant cohomology}\label{section3}
We start by reviewing Cartan's model for equivariant cohomology,
following Berline and Vergne \cite{BV85}.
Let $M$ be a compact manifold, $G$ a compact
Lie group, and $\Phi:G\times M\rightarrow M$
a smooth action. Denote by ${\cal A}_G(M)$ the space of $G$-invariant
polynomial mappings $\sigma:{\frak g}\rightarrow {\cal A}(M)$, that is, $\sigma(\xi)$ depends
polynomially on $\xi$ and satisfies the equivariance property
\begin{equation} \sigma(\mbox{Ad}_g(\xi))=\Phi_{g^{-1}}^*(\sigma(\xi)). \end{equation}
The elements of ${\cal A}_G(M)$ are called {\em equivariant differential forms},
and the space ${\cal A}_G(M)$ is preserved under the
{\em equivariant differential}
\begin{equation} {\d_\G}:{\cal A}_G(M)\rightarrow{\cal A}_G(M),\,\,({\d_\G}\sigma)(\xi)=
\d(\sigma(\xi))+2\pi i \big(\iota(\xi_M)\sigma (\xi)\big).\end{equation}
Here,
$\xi_M$ denotes the fundamental vector field, i.e. the generating vector
field of the flow $(t,p)\mapsto \exp(-t\xi).p$.
Equivariance together with Cartan's identity for the Lie derivative,
$\L_Y=\iota_Y\circ \d +\d\circ \iota_Y$, implies $\d_{\frak g}^2=0$.
The cohomology $H_G(M)$ of the complex
$({\cal A}_G(M),{\d_\G})$ is called the {\em equivariant cohomology}.
One can show that if the action of $G$ on $M$
is locally free, the pullback mapping ${\cal A}(M/G)\rightarrow {\cal A}(M)^G_{hor}
\hookrightarrow {\cal A}_G(M)$ gives rise to
an isomorphism $H(M/G)\rightarrow H_G(M)$.
After choosing a principal connection
on $M\rightarrow M/G$, the inverse is induced on the level of forms by the
mapping
\begin{equation} {\cal A}_G(M)\rightarrow {\cal A}(M)^G\rightarrow {\cal A}(M)^G_{hor}\cong {\cal A}
(M/G)\end{equation}
given by substituting $\frac{i}{2\pi}$ times
the curvature in the ${\frak g}$-slot, followed by
projection onto the horizontal part (for the proof, see \cite{DV93}).
For what follows, it will be necessary to relax the polynomial
dependence on $\xi$ to analytic dependence, possibly defined only
on some neighborhood of $0\in{\frak g}$.
We will denote the corresponding space of equivariant forms by
${\cal A}^\omega_G(M)$, and the
cohomology by $H_G^\omega(M)$.
Suppose now that ${\cal V}\rightarrow M$ is a $G$-equivariant Hermitian
vector bundle over $M$, with fiber dimension $N$. Let ${\cal A}(M,{\cal V})$
be the bundle-valued differential forms, and ${\cal A}_G(M,{\cal V})$
their equivariant counterpart.
For each $G$-invariant Hermitian connection $\nabla:{\cal A}(M,{\cal V})\rightarrow
{\cal A}(M,{\cal V})$,
the moment map $\mu\in{\cal A}_G(M,\mbox{End}({\cal V}))$ of Berline
and Vergne is defined by
\begin{equation} \mu(\xi).\sigma :=
\xi.\sigma - \nabla_{\xi_M}\sigma,\end{equation}
where $\sigma\rightarrow \xi.\sigma$ denotes the representation
of ${\frak g}$ on the space of sections.
Geometrically, $\mu(\xi)$ is the vertical part (with respect to the
connection) of the fundamental
vector field $\xi_{\cal V}$ on ${\cal V}$. Let $F({\cal V})\in{\cal A}^2
(M,\mbox{End}({\cal V}))$ denote the curvature of $\nabla$.
The {\em equivariant} curvature is then defined by
\begin{equation} F_{\frak g}({\cal V},\xi)=F({\cal V})+2\pi i \mu(\xi),\end{equation}
and it satisfies the Bianchi identity with respect to the
equivariant covariant derivative $\nabla_{\frak g}=\nabla+2\pi i \iota(\xi_M)$.
Suppose now that $A\rightarrow f(A)$ is the germ of a $U(N)$-invariant
analytic function on ${\frak{u}}(N)$. Then $f(F_{\frak g})\in{\cal A}_G(M)$
is $\d_{\frak g}$-closed, and one can show that choosing a different
connection changes $f(F_{\frak g})$ by a $\d_{\frak g}$-exact form.
The corresponding cohomology classes are called the {\em equivariant
characteristic classes} of ${\cal V}\rightarrow M$.
If the action on $M$ is
locally free, one can choose $\nabla$ in such a way that $\mu=0$,
which shows that
the mapping $H^\omega_G(M)\rightarrow H(M/G)$ sends the
equivariant characteristic classes of ${\cal V}$ to the usual
characteristic classes of the orbifold-bundle ${\cal V}/G$.
The following
characteristic classes will play a role in the sequel:
\begin{itemize}
\item[(a)] The equivariant Chern character, defined by
\begin{equation} {Ch}_{\frak g}({\cal V},\xi)=\mbox{tr}
(e^{\frac{i}{2\pi}F_{\frak g}({\cal V},\xi)}).\end{equation}
In the above geometric quantization setting, ${\cal V}=L$ is a line
bundle, and for the equivariant curvature one has
$\frac{i}{2\pi}F_{\frak g}({\cal V},\xi)=\omega+2\pi i\l J,\xi\rangle $, thus
\begin{equation} Ch_{\frak g}(L,\xi)=e^{\omega+2\pi i\l J,\xi\rangle}.\end{equation}
More generally, if $g\in G$ acts trivially on the base $M$,
one defines
\begin{equation} {Ch}_{\frak g}^g({\cal V},\xi)=\mbox{tr}(g^{\cal V}\,\,e^{
\frac{i}{2\pi}F_{\frak g}({\cal V},\xi)})\end{equation}
where $g^{\cal V}\in\Gamma(M,\mbox{End}({\cal V}))$
is the induced action of $g$.
In the line bundle case,
this is simply $c_L(g){Ch}_{\frak g}(L,\xi)$ where
where $c_L(g)\in S^1$ is the action of $g$ on the fibers.
\item[(b)] The equivariant Todd class,
\begin{equation} {Td}_{\frak g}({\cal V},\xi)=
\det\Big(\frac{\frac{i}{2\pi}F_{\frak g}({\cal V},\xi)}{1-e^{-\frac{i}{2\pi}
F_{\frak g}({\cal V},\xi)}}\Big).\end{equation}
The Todd class of a complex manifold is defined as the Todd class
of its tangent bundle.
\item[(c)] The equivariant Euler class
\begin{equation}\chi_{\frak g}({\cal V},\xi)=\det(\frac{i}{2\pi}F_{\frak g}({\cal V},\xi)).\end{equation}
\item[(d)] The class
\begin{equation} D_{\frak g}^g({\cal V},\xi)=\det (I-(g^{-1})^{\cal V} e^{-\frac{i}{2\pi}
F_{\frak g}({\cal V},\xi)}),\end{equation}
for $g\in G$ acting trivially on $M$.
\end{itemize}
All of this also makes sense for symplectic vector bundles, since the
choice of a compatible complex structure reduces the structure group
to $U(n)$, and any two such choices are homotopic.
The equivariant Euler class occurs in the Localization Formula of
Atiyah-Bott and Berline-Vergne.
\begin{theo}[Atiyah-Bott \cite{AB84}, Berline-Vergne \cite{BV83}] \label{AB84}
Suppose $M$ is an orientable $G$-ma\-ni\-fold, and
$\sigma\in {\cal A}_G^\omega(M)$ is
${\d_\G}$-closed. Assume that $\xi\in{\frak g}$
is in the domain of definition of $\sigma(\xi)$, and
let $F$ be the set of zeroes of $\xi_M$. The connected components of
$F$ are then
smooth submanifolds of even codimension, and the normal
bundle $N_F$ admits a Hermitian structure which is invariant under
the flow of $\xi_M$.
Choose orientations on $F$ and $M$ which are compatible with the
corresponding orientation of $N_F$. Then
\begin{equation} \int_M \sigma(\xi)=\int_{F}
\frac{\iota_F^*\sigma(\xi)}{\chi_{\frak g}(N_F,\xi)},\label{localization} \end{equation}
where $\iota_F:F\rightarrow M$ denotes the embedding.
\end{theo}
We will need this result only in the symplectic
or complex case, where the above orientations are given in a natural way.
Jeffrey and Kirwan \cite{JK93} have proved a different sort
of Localization Formula for the case of {\em Hamiltonian} G-spaces.
We will need a stationary phase version of their result.
Let $(M,\omega)$ be a Hamiltonian $G$-space, with moment map $J:M\rightarrow{\frak g}^*$,
and suppose $0$ is a regular value of $J$. Let
$\sigma\in{\cal A}^\omega_G(M)$ be ${\d_\G}$-closed,
and let $\Delta\in C^\infty_0({\frak g})$ be a cutoff-function, with $\sigma(\xi)$
defined for $\xi\in\mbox{supp}(\Delta)$, and $\Delta=1$ in a neighborhood of
$0$.
Consider the integral
\begin{equation}\int_{\frak g}\int_M \Delta(\xi)\sigma(\xi)e^{m(\omega+2 \pi i\,\l J,\xi\rangle)}
\,\d\xi,\end{equation}
where $\d\xi$ is the measure on ${\frak g}$ corresponding to the normalized measure
on $G$. Notice that the ``equivariant symplectic form''
$\omega+2 \pi i\,\l J,\xi\rangle$ is ${\d_\G}$-closed.
Since $e^{m\omega}$ is simply a polynomial in $m$, the
leading behaviour of this integral for $m\rightarrow\infty$ is determined by the
stationary points of the phase function $e^{2\pi i m\l J,\xi\rangle}$.
Stationarity in ${\frak g}$-direction gives the condition $J=0$, and stationarity
in $M$-direction
the condition $\d \l J,\xi\rangle=0$, or $\xi=0$ since the action on $J^{-1}(0)$
is locally free.
Let $M_0= J^{-1}(0)/G$ be the reduced space, $\pi:J^{-1}(0)\rightarrow
M_0$ the projection and $\iota:J^{-1}(0)\rightarrow M$ the embedding.
Consider the mapping
\begin{equation} \kappa:H^\omega_G(M)\rightarrow H(M_0),\label{kappa}\end{equation}
given by composing pullback to $J^{-1}(0)$ with the mapping
$H^\omega_G(J^{-1}(0))\rightarrow H(M_0)$. On the level of forms
$\sigma\in {\cal A}^\omega_G(M)$, the form $\pi^*\kappa(\sigma)$
is by definition equal to the horizontal part of
$\iota^*\sigma(\frac{i}{2\pi}F^\theta)$, where
$F^\theta\in{\cal A}^2(J^{-1}(0),{\frak g})^G_{hor}$
is the curvature of some connection
$\theta\in{\cal A}^1(J^{-1}(0),{\frak g})^G$.
\begin{theo} \label{jf}
For $m\rightarrow\infty$,
\begin{equation} \int_{\frak g}\int_M
\Delta(\xi)\sigma(\xi)e^{m(\omega+2\pi i\l J,\xi\rangle)}\,\d\xi
=\frac{1}{d}\int_{M_0}\kappa(\sigma) \,
e^{m\omega_0}\,\,\,+O(m^{-\infty}),\label{jklocf} \end{equation}
where $d$ is the number of elements in the generic stabilizer
for the $G$-action on $J^{-1}(0)$.
\end{theo}
{\bf Proof.}\hspace{0.5cm}
This is a variation of Theorem 4.1 in Jeffrey-Kirwan
\cite{JK93}, which deals with polynomial equivariant
cohomology classes, and where one has a Gaussian convergence
factor instead of the cutoff.
Following \cite{JK93}, we will perform the integral in a local
model for $M$ near $J^{-1}(0)$.
By the Coisotropic Embedding Theorem of Gotay, a neighborhood of
$J^{-1}(0)$ in $M$ is equivariantly symplectomorphic to
a neighborhood of the zero section of the trivial bundle
$J^{-1}(0)\times {\frak g}^*$,
with symplectic form $\pi^*\omega_0+\d\l \alpha,\theta\rangle$, where
$\alpha$ is the coordinate function on ${\frak g}^*$. In this model,
$G$ acts by $g.(x,\alpha)=(g.x,\mbox{Ad}_{g^{-1}}^*(\alpha))$ ,
and the moment map is
simply $J(x,\alpha)=\alpha$. Using the model and another
cutoff-function $\Delta'(\alpha)$ on ${\frak g}^*$, equal to $1$ near the origin
and with sufficiently small support, the same computation
as in \cite{JK93} shows that the integral is equal to
\[ m^{\dim G}\int_{J^{-1}(0)}
\int_{{\frak g}^*}\int_{{\frak g}} \Delta(\xi)\Delta'(\alpha)
(\iota^*\sigma)(\xi)e^{m(\pi^*\omega_0+\l \alpha,F^\theta+2\pi i
\xi\rangle)}\, \d\xi\, \d\alpha\,\d g +O(m^{-\infty}).\]
Here, $\d g$ denotes the (vertical) volume form on the fibers of
$J^{-1}(0)\rightarrow M_0$, corresponding to the canonical identification
$T_x(\mbox{fiber})\cong{\frak g}$ by means of the $G$-action.
Now apply the Stationary Phase Theorem (see e.g. \cite{H90}, Theorem 7.7.5)
to the $\alpha,\xi$-integral, the relevant phase function
being $e^{2\pi i m \l \alpha,\xi\rangle}$.
Since $e^{m\l \alpha,F^\theta\rangle}$
is simply a polynomial in $\alpha$, the stationary phase
expansion
terminates after finitely many terms, and the result is
\[ \int_{J^{-1}(0)} e^{m \pi^*\omega_0}
\sum_{r=0}^\infty \frac{1}{r!}\big(\frac{i}{2 \pi m}\big)^r\left.
\Big(\sum_j\frac{\partial}{\partial \xi_j}
\frac{\partial}{\partial \alpha^j}\Big)^r\right|_{\atop\stackrel{\xi=0}{\alpha=0}}
\iota^*\sigma(\xi)\, e^{m\l \alpha,F^\theta\rangle }
\,\,\d g +O(m^{-\infty})\]
\[ =\int_{J^{-1}(0)} \pi^*\,(e^{m \omega_0})\,\iota^*\sigma(\frac{i}{2\pi}
F^\theta)\,\d g +O(m^{-\infty}).\]
Since $\iota^*\sigma(\frac{i}{2\pi}F^\theta)$ gets wedged with $\d g$,
only its horizontal part, which by definition of $\kappa$
is $\pi^*\kappa(\sigma)$,
contributes to
the integral. The result (\ref{jklocf}) now follows by
integration over the fiber; the factor
${1}/{d}$ appears since this is the volume of a generic fiber.
\hspace{0.5cm}\bigskip\mbox{$\Box$}
We now turn to the Equivariant Hirzebruch-Riemann-Roch Theorem, in
the form due to Berline and Vergne \cite{BV85}. Let $M$ be a compact
complex manifold, equipped with a holomorphic action of $G$,
and let $L\rightarrow M$ be a G-equivariant holomorphic line bundle.
Define the character $\chi\in R(G)$ as in (\ref{char}).
\begin{theo}\label{RiemannRoch}
For $\xi$ sufficiently close to zero,
\begin{equation} \chi(e^{\xi})=
\int_M {Td}_{\frak g}(M,\xi)\,{Ch}_{\frak g}(L,\xi).\label{e}\end{equation}
More generally, if $g\in G$, one has for all sufficiently small
$\xi\in {\frak k}$, the Lie algebra of the centralizer $Z_g$ of $g$:
\begin{equation} \chi(g\, e^{\xi})=
\int_{M^g}
\frac{{Td}_{\frak k}(M^g,\xi)\,{Ch}_{\frak k}^g
(L|\,M^g,\xi)}{ D^g_{\frak k}(N^g,\xi)},
\label{g}\end{equation}
where $M^g$ is the fixed point set and $N^g\rightarrow M^g$ its normal bundle.
\end{theo}
To be precise, Berline and Vergne have shown how to rewrite the Equivariant
Atiyah-Segal-Singer Index Theorem for Dirac operators in this style,
with the
equivariant $\hat{A}$-genus appearing
on the right hand side. This formula, however,
implies the above Theorem
in the same way as the usual Atiyah-Singer Index Theorem
leads to the
Hirzebruch-Riemann-Roch Formula; see \cite{BGV92}, p. 152 for the
calculations.
\section{The stationary phase approximation}
In this section, we will prove the first part of Theorem \ref{multf}.
By the shifting trick, it is sufficient to consider
the case $\mu=0$.
The idea is to substitute the expressions from the Equivariant
Hirzebruch-Riemann-Roch Theorem \ref{RiemannRoch} for $\chi^{(m)}$ in
\begin{equation} N^{(m)}(0)=\int_G \chi^{(m)}(h)\d h,\label{4.5}\end{equation}
and apply the Localization Formula, Theorem \ref{jf}.
For this, we need to know what happens to the equivariant
Todd class of $M$ under the mapping (\ref{kappa}):
\begin{lemma}\label{todd} Let
\begin{equation} j_{\frak g}(\xi)=\det\Big({\textstyle\frac{1-e^{-{ad}(\xi)} }
{{ad}(\xi)}}\Big)\end{equation}
be the Jacobian of the exponential mapping $\exp:{\frak g}\rightarrow G$.
Then
\begin{equation} \kappa({Td}_{\frak g}(M) j_{\frak g})={Td}\,(M_0).\end{equation}
\end{lemma}
{\bf Proof.}\hspace{0.5cm}
Identify the vertical subbundle of $TJ^{-1}(0)$ with the trivial
bundle ${\frak g}$, and the symplectic bundle ${\frak g}\oplus I{\frak g}$ (where $I$
is the complex structure of $M$) with ${\frak g}_{\Bbb C}$. Then
\[ \iota^*(TM)=\pi^*(TM_0)\oplus {\frak g}_{\Bbb C}. \]
Since the equivariant Todd class of ${\frak g}_{\Bbb C}$ is just $j_{\frak g}^{-1}$,
this shows $\kappa({Td}_{\frak g}(M))=Td\,(M_0)\,\kappa(j_{\frak g}^{-1})$, q.e.d.
\hspace{0.5cm}\bigskip\mbox{$\Box$}
We will now consider the contribution to (\ref{4.5}) coming from
a small ${\rm Ad}$-invariant
neighborhood of a given orbit $(g)={\rm Ad}(G).g$.
Let $\sigma\in C^\infty(G)$ be an
$\mbox{Ad}$-invariant cutoff-function, supported in a sufficiently
small neighborhood of $(g)$ and equal to 1 near $(g)$.
Consider the integral
\begin{equation} I_g(m)=\int_G \sigma(h)\chi^{(m)}(h)\d h.\end{equation}
Since (\ref{g}) only holds for
$\xi\in{\frak k}$, we want to replace this integral by an integral over $Z_g$.
(Of course, this step is void in the abelian case.)
Let ${\frak r}\subset{\frak g}$
be the orthogonal of ${\frak k}$ with respect
to some invariant inner product (or, more intrinsically, the
annihilator of $({\frak g}^*)^g\cong{\frak k}^*$). For $k\in Z_g$, let
$k^{\frak{r}}$ denote the action of $k$ on ${\frak r}$.
\begin{lemma} \label{lemmaGK}
Let $f\in C^\infty(G)$ be $\mbox{Ad}(G)$-invariant, with support
in a small neighborhood of $\mbox{Ad}(G).g$. Then, for a suitable
$\mbox{Ad}(Z_g)$-invariant cutoff-function
$\tilde{\sigma}\in C^\infty_0(Z_g)$,
supported near $e\in Z_g$ and identically $1$ near $e$,
\begin{equation} \int_G f(h)\d h=\int_{Z_g} f(gk)
\det(I-(g^{-1}k^{-1})^{\frak{r}})\tilde{\sigma}(k)\,\d k.
\label{GK}\end{equation}
\end{lemma}
{\bf Proof.}\hspace{0.5cm}
From the Slice Theorem for actions of compact Lie groups, it
follows that an invariant neighborhood of the orbit
$\mbox{Ad}(G).g$ is equivariantly diffeomorphic to a neighborhood
of the zero section of $G\times_{Z_g}{\frak k}$, where $Z_g$ acts on ${\frak k}$
by the adjoint action. Using the exponential map for $Z_g$,
it follows that the mapping
\begin{equation} \phi:\,G\times_{Z_g}Z_g\rightarrow G,\,\,(h,k)\mapsto \mbox{Ad}_h(gk), \end{equation}
with $Z_g$ acting on itself by $\mbox{Ad}$,
is an equivariant diffeomorphism from a neighborhood of the unit
section to a neighborhood of the orbit.
Let $\d\nu$ be
the canonical measure on the group bundle $G\times_{Z_g}Z_g=:W$,
constructed
from the normalized invariant measures on $Z_g$ and $G/Z_g$.
We have to compute the tangent mapping to $\phi$, but
by equivariance it is sufficient to do this at points
$(e,k)$ in the fiber over $e\,Z_g$, which is canonically
isomorphic to $Z_g$. If we identify $T_{(e,k)}W\cong
\frak{r}\oplus\frak{k}$, and $TG\cong G\times{\frak g}$ using
left trivialization, the tangent mapping is given by
\[ T_{(e,k)}\phi(\xi,\eta)=((1-\mbox{Ad}_{(gk)^{-1}})(\xi),
\eta).\]
This shows that the measure transforms according to
\begin{equation} \phi^*\,\d g = \det(I-(g^{-1}k^{-1})^{\frak r})\,\d \nu.\end{equation}
We can now perform the integral by first pulling $f$ back to
$G\times_{Z_g}Z_g$, multiplying with a suitable cuttoff function
$\tilde{\sigma}$,
integrating over the fibers of $W\rightarrow G/Z_g$, and then
integrate over the base $\mbox{Ad}(G).g\cong G/Z_g$.
\hspace{0.5cm}\bigskip\mbox{$\Box$}
Using the Lemma, we find that
\begin{equation} I_g(m)=
\int_{Z_g} \tilde{\sigma}(k)\chi^{(m)}(gk)
\det(I-(g^{-1}\,k^{-1})^{\frak r})\,
\d k.\end{equation}
Replacing this with an integral over the Lie algebra ${\frak k}$, and using
(\ref{g}) gives
\begin{equation} I_g(m)=\int_{\frak k}\int_{M^g} \Delta(\xi)
\frac{{Td}_{\frak k}(M^g,\xi){Ch}_{\frak k}^g(L^m|M^g,\xi)}{ D^g_{\frak k}(N^g,\xi)}
\det(I-(g^{-1}e^{-\xi})^{\frak r})
j_{\frak k}(\xi)\d \xi,\label{LK}\end{equation}
with $\Delta(\xi)=\tilde{\sigma}(e^{\xi})$.
Let $\kappa_g:\,H_{Z_g}(M^g)\rightarrow H(\Sigma_g)$ be the mapping defined
by (\ref{kappa}), with $M$ replaced by $M^g$ and $G$ by $Z_g$.
By Lemma \ref{todd},
\begin{equation} \kappa_g(Ch^g_{\frak k}(L^m|M^g))\,\kappa_g(j_{\frak k})=
Ch^{\Sigma}(L^m_\Sigma)|\Sigma_g.\end{equation}
For $x\in P^g=M^g\cap J^{-1}(0)$, let
${\frak r}_M(x):=\{\xi_M(x)|\xi\in {\frak r}\}\cong {\frak r}$.
Then
\begin{equation} N^g(x)=N_\Sigma(y)\oplus{\frak r}_M(x)\oplus I{\frak r}_M(x)
=N_\Sigma(y)\oplus {\frak r}_{\Bbb C},\end{equation}
where $y=G.(x,g)$.
But $D^g_{\frak k}({\frak r}_{\Bbb C},\xi)=\det(I-(g^{-1}e^{-\xi}))$, hence
\begin{equation} \kappa_g(D_{\frak k}^g(N^g,\xi))=
\kappa_g(\det(I-(g^{-1}e^{-\xi})^{\frak r}))\,\,
D^\Sigma(N_\Sigma)|\Sigma_g.\end{equation}
With these preperations, we apply Theorem \ref{jf} to the
integral \ref{LK}, and obtain
\begin{equation} I_g(m)={\sum}'_{j} \frac{1}{d_j}
\int_{\Sigma_j} \frac{Td\,(\Sigma_j)
{Ch}^\Sigma(L^m_\Sigma)}
{D^\Sigma(N_\Sigma)}+O(m^{-\infty}),\end{equation}
the sum being over the connected components of $\Sigma_g$.
Summing over all contributions, we get (\ref{orbifold1}) up to an error
term $O(m^{-\infty})$.
As we remarked above, the right hand side of (\ref{orbifold1})
is an arithmetic polynomial as a function of $m$.
But if $f:{\Bbb N}\rightarrow {\Bbb Z}$ is an integer-valued function
with $\lim_{m\rightarrow\infty}(f(m)-p(m))=0$ for some polynomial $p$,
then $f(m)=p(m)$ for large $m$.
This shows that the error term is zero for large $m$, and finishes
the proof of the first part of Theorem \ref{multf}.
\section{Counting lattice points}
To prove the second part of Theorem \ref{multf}, i.e. that we can
set $m=1$, all we have to show is that $m\rightarrow N^{(m)}(m\mu)$ is
an arithmetic polynomial.
\begin{theo} \label{stepw}
Suppose that $J(M)\subset {\frak g}^*_{reg}$. Then the function
$m\mapsto N^{(m)}(m\mu)$ is an arithmetic polynomial for all
$\mu\in\Lambda_+$.
\end{theo}
Before we prove this, we convert the computation of the multiplicities
into a problem of counting lattice points.
The next steps are based on work of
Guillemin-Lerman-Sternberg
\cite{GLS88} and Guillemin-Prato \cite{GP90},
except that we replace their use of the
Atiyah-Bott Lefschetz Formula with the Localization Formula \ref{AB84}
applied to (\ref{e}), since we do not want to assume
isolated fixed points.
Consider the action of the maximal torus $T\subset G$, with
its moment map $J^T$ equal to $J$ followed by projection
to ${\frak t}^*$.
\begin{prop}
\label{prop5.1}
For all generic $\xi\in{\frak t}$,
\begin{equation} \chi(e^{\xi})=\sum_{{\cal F}} \int_{\cal F} \frac{{Td} ({\cal F})e^{\omega
+ 2\pi i \l J_{\cal F},\xi\rangle}}
{\det(I-e^{-\frac{i}{2\pi}F_{\frak t}(N_{\cal F},\xi)})},\label{fixedpoints}\end{equation}
the sum being over the fixed points manifolds of the
$T\subset G$-action, $N_{\cal F}$ the corresponding normal bundles,
and $J_{\cal F}$ the constant value of $J$ on ${\cal F}$.
\end{prop}
{\bf Proof.}\hspace{0.5cm}
By ``generic'' we mean that the zero set of
$\xi_M$ is equal to the fixed point set of the $T$-action.
To get ({\ref{fixedpoints}), apply the
Localization Formula, Theorem \ref{AB84}, to (\ref{e}).
The bundle $TM|_{\cal F}$ splits
into the direct sum $T{\cal F}\oplus N_{\cal F}$. Since $T$ does not act on
$T{\cal F}$,
\[{Td}_{\frak t}(M,\xi)={Td}\,({\cal F})\,{Td}_{\frak t}(N_{\cal F},\xi).\]
The equivariant Euler class of $N_{\cal F}$ in Proposition
\ref{AB84} cancels
the denominator of the equivariant Todd class, which immediately
gives (\ref{fixedpoints}).
\hspace{0.5cm}\bigskip\mbox{$\Box$}
Although the left hand side of (\ref{fixedpoints})
is an analytic function of $\xi$, the
individual summands on the right hand side have poles.
Since they are not in $L^1_{loc}$, they do not a priori
define distributions
on ${\frak t}$.
This problem can be fixed as follows \cite{D93}.
By using the splitting principle (or simply a partition of unity
on ${\cal F}$) if necessary, we can assume
that $N_{\cal F}$ splits into a direct sum of invariant line bundles
$N_{\cal F}^1,\ldots,N_{\cal F}^r$. Let
$\alpha^j_{\cal F}$ be the weight for the
$T$-action on $N_{\cal F}^j$, that is, $e^\xi\in T$ acts by the
character $\exp(2\pi i \l \alpha_{\cal F}^j,\xi\rangle)$.
Each $\alpha^j_{\cal F}$ determines an orthogonal hyperplane in ${\frak t}$,
let $C$ be any fixed connected component in the complement of
the union of all these hyperplanes.
If we replace $\xi$ by $\xi-i\eta$ in
({\ref{fixedpoints}), with $\eta\in C$, the terms on the right hand side
are analytic for all $\xi$. One can therefore regard
(\ref{fixedpoints}) as an equality of distributions, with the
summands on the right hand side defined defined as a
distributional limit for $\eta\rightarrow 0$ in $C$.
Let us now first discuss the abelian case, i.e. assume that
$G=T$ is a torus. Denote by
$F^j(N_{\cal F})$ the components of the curvature.
By expanding
$\det(I-e^{-\frac{i}{2\pi}{F}_{\frak t}(N_{\cal F},\xi-i\eta)})^{-1}$
into its Taylor series w.r.t. ${F}^j(N_{\cal F})$, we can write it
as a finite sum
\begin{equation} \det(I-e^{-\frac{i}{2\pi}{F}_{\frak t}(N,\xi-i\eta)})^{-1}=
\sum_{s\in{\Bbb N}^r} \frac{p_s({F}^1(N_{\cal F}),\ldots,
{F}^r(N_{\cal F}))}
{\prod_j (1-e^{-2\pi i\l {\alpha}^j_{\cal F},\xi-i\eta\rangle})^{s_j}},
\label{5}
\end{equation}
where for all $s=(s_1,\ldots,s_r)$, $p_s$ is a polynomial.
We now invoke the ``polarization trick'' used in ref.
\cite{GLS88,GP90}.
For each $\alpha^j_{\cal F}$,
write
\begin{equation} \check{\alpha}^j_{\cal F}=\left\{\begin{array}{ll}
{\alpha}^j_{\cal F}&\mbox{ if }
\l {\alpha}^j_{\cal F},\eta\rangle>0\\
-{\alpha}^j_{\cal F}&\mbox{ if }
\l {\alpha}^j_{\cal F},\eta\rangle<0\end{array}\right.
\label{signs}\end{equation}
for any, hence all, $\eta\in C$.
Let $l_j^0=0$ if $\check{\alpha}^j_{\cal F}=\alpha^j_{\cal F}$, $1$ otherwise.
Then
\begin{equation} \chi(e^{\xi-i\eta})=\sum_{\cal F}\sum_{s\in {\Bbb N}^r}c_{{\cal F},s}
\frac{e^{2\pi i\l J_{\cal F}-\sum l_j^0 s_j\check{\alpha}^j_{\cal F},\xi-i\eta\rangle}}
{\prod_j (1-e^{-2\pi i\l \check{\alpha}^j_{\cal F},\xi-i\eta\rangle})^{s_j}}
\label{lhs}\end{equation}
with
\begin{equation} c_{{\cal F},s}= (-1)^{k_{{\cal F},s}} \int_{\cal F} {Td}\,({\cal F})\,e^\omega p_s({F}^1
(N_{\cal F}),\ldots,
{F}^r(N_{\cal F})),\end{equation}
where $k_{{\cal F},s}=\sum l_j^0 s_j$ is the number of sign changes.
For given ${\cal F},s$, write $(a^1,\ldots,a^N)$
for the list of $\check{\alpha}^j_{\cal F}$'s, appearing with respective
multiplicities $s_j$. Since
\begin{equation} (1-e^{-2\pi i\l a^j,\xi-i\eta\rangle})^{-1}=\sum_{l_j=0}^\infty
e^{- 2\pi i \l l_j\,a^j,\xi-i\eta \rangle},\end{equation}
we get
\begin{equation} \chi(e^\xi)=\sum_{{\cal F},s} c_{{\cal F},s}\sum_{l\in{\Bbb Z}^N_+}
e^{2\pi i\l J_{\cal F}-\sum (l_j+l_j^0)a^j,\xi\rangle}\end{equation}
(the sum over ${\Bbb Z}^N_+:=\{l\in {\Bbb Z}^N:\,l_j\ge 0\}$
is a well-defined periodic distribution).
Comparing this to
\begin{equation} \chi(e^\xi)=\sum_{\mu\in\Lambda}
N(\mu)e^{2\pi i\l\mu,\xi\rangle}\label{abelian}\end{equation}
yields
\begin{equation} N(\mu) = \sum_{{\cal F},s}c_{{\cal F},s}{\frak P}_{{\cal F},s}(J_{\cal F}-\mu-
\sum l_j^0 a^j)\label{latticepoints}\end{equation}
where the partition function ${\frak P}_{{\cal F},s}(\nu)$
is the number of solutions
$k\in {\Bbb Z}^N$ of $\sum k_j a^j=\nu$, $k_j\ge 0$.
Starting from this expression, we will now show that $N^{(m)}(m\mu)$
is an arithmetic polynomial. We have to replace $\omega$ by
$m\omega$, $\mu$ by $m\mu$ and $J$ by $mJ$.
Since
$c_{{\cal F},s}^{(m)}$ is a polynomial in $m$, it is sufficient to show that
the number of integer solutions of
\begin{equation} m(J_{\cal F}-\mu)=\sum (l_j+l_j^0) a^j,\,\,l_j\ge 0 \label{10}\end{equation}
is an arithmetic polynomial as a function of $m$.
Let us write $\nu=J_{\cal F}-\mu$, and consider $A=(a^1,\ldots,a^N)$ as a
${\Bbb Z}$-linear mapping ${\Bbb Z}^N\rightarrow {\Bbb Z}^p$, where $p=\dim(T)$. We are thus
looking for integer solutions of
\begin{equation} m\nu= A\,l,\,\,l_j\ge l_j^0.\label{11}\end{equation}
We will need the following
\begin{theo}[Ehrhart \cite{E77}]
Let $L$ be a lattice, with underlying vector space $L_{\Bbb R}=L\otimes_{\Bbb Z} {\Bbb R}$,
and $\Delta\subset L_{\Bbb R}$ a lattice polytope,
i.e. a polytope whose vertices are all lattice points. Then,
for all $r\in {\Bbb N}$, the counting function
\begin{equation} f(m)=\#\big(\frac{m}{r}\Delta\cap L\big) \end{equation}
is an arithmetic polynomial, with period $r$.
\end{theo}
Let now $x_0\in{\Bbb R}^N$ be any solution of $Ax=\nu$. The general solution
of $A\,x=m\nu$ is thus given by the affine plane
$E_m=mx_0+\mbox{ker}(A)$. Let $r\in {\Bbb N}$ be the smallest number such
that the vertices of the polytope
$\Delta:=E_r\cap {\Bbb R}^N_+$ are lattice points. If $l^0=0$, the set of
solutions of (\ref{11}) is the intersection $\frac{m}{r}\Delta\cap {\Bbb Z}^N$,
so the number of solutions is an arithmetic polynomial by Ehrhart's Theorem.
If $l^0\not=0$, let $\Delta_j$ be the face of $\Delta$ defined by
$x_j=0$, and let $\Delta'$ be the union of all $\Delta_j$ for which
$l_j^0=1$. Then the solution set of (\ref{11}) is
\[ \frac{m}{r}\Delta\cap {\Bbb Z}^N\,-\,\frac{m}{r}\Delta'\cap {\Bbb Z}^N,\]
and this is again an arithmetic polynomial by Ehrhart's Theorem.
This proves Theorem \ref{stepw} in the abelian case.
Suppose now that $G$ is nonabelian, but that $J(M)$ is contained
in the set of regular elements, ${\frak g}^*_{reg}=G.\mbox{int}({\frak t}^*_+)$.
We will show how this reduces to the abelian case.
By the Symplectic Slice Theorem \cite{GS84}, $Y_+=J^{-1}(\mbox{int}
({\frak t}^*_+))$ is a symplectic (but not necessarily K\"ahler)
submanifold of $M$, and is in fact a
Hamiltonian $T$-space, with the restriction of $J$ serving
as a moment map.
The above assumption implies that $Y_+$ is a {\em closed} submanifold, and
$M=G\times_T Y_+$. The restriction $L_+=L|Y_+$ renders a quantizing
bundle for $Y_+$. Consider the expression
\begin{equation} \chi'(e^\xi):= \int_{Y^+} {Td}_{\frak t}(Y_+,\xi)\,
{Ch}_{\frak t}(L_+,\xi).\end{equation}
We claim that this is of the form
\begin{equation} \chi'(e^\xi)=\sum_{\mu\in\Lambda} N'(\mu) \,e^{2\pi i \l \mu,\xi\rangle},\end{equation}
where $N'(\mu)\not=0$ for only finitely many lattice points, and
$N'(\mu)=0$ unless $\mu\in J(Y_+)\cap\Lambda\subset\Lambda_+$.
Indeed, one can check directly that $\chi(e^\xi)$ comes from
a function on $T$, given near any point $g\in T$
by the formula (\ref{g}), and then repeat
the above analysis. (One can also pick a
$T$-invariant almost K\"ahler structure on $M$, and then realize
$\chi(e^\xi)$ as the equivariant index for some Dirac operator
associated to the Clifford module $L\otimes
\Lambda(T^{(0,1)}Y_+)^*$.)
\begin{lemma} For all $\mu\in\Lambda_+$, $N(\mu)=N'(\mu)$.
\end{lemma}
Since we know that ${N'}^{(m)}(m\mu)$ is an arithmetic polynomial, this
will finish the proof of Theorem \ref{multf}.\\
{\bf Proof.}\hspace{0.5cm}
Let us go back to the formula (\ref{fixedpoints}) for the character.
Notice that the Weyl group $W=N_G(T)/T$ acts on $M^T$ by
permuting the connected components, and that $M^T$ consists of
its portion in $Y_+$ and the $W$-transforms thereof.
Let ${\cal F}\subset Y_+$ be a connected component of $M^T$. The
normal bundle $N_{\cal F}$ of ${\cal F}$ in $M$ splits into into its part in
$Y_+$, $N_{\cal F}':=N_{\cal F}\cap TY_+$, and the symplectic orthogonal of
$TY_+|{\cal F}$, which is canonically isomorphic to the trivial bundle
${\frak g}/{\frak t}$. The weights for the $T$-action on ${\frak g}/{\frak t}$ are of course
simply the positive roots $\beta\in {\frak t}^*$ of $G$. Therefore, by
taking the trivial connection on ${\frak g}/{\frak t}$,
\[ \det(I- e^{-\frac{i}{2\pi}F_{\frak t}(N_{\cal F},\xi)})=\prod_{\beta>0}
(1-e^{-2\pi i \l \beta,\xi\rangle})\det(I-e^{-\frac{i}{2\pi}
F_{\frak t}(N'_{\cal F},\xi)}), \]
hence
\[ \chi(e^\xi)=\sum_{w\in W}\frac{1}{\prod_{\beta>0}
(1-e^{-2\pi i \l \beta,w^{-1}(\xi)\rangle})}\sum_{{\cal F}\subset Y_+}
\int_{\cal F} \frac{{Td}\,({\cal F})e^{\omega
+ 2\pi i \l J_{\cal F},w^{-1}(\xi)\rangle}}
{\det(I-e^{-\frac{i}{2\pi}F_{\frak t}(N'_{\cal F},w^{-1}(\xi))})}.\]
To the sum
\[ \sum_{{\cal F}\subset Y_+}\int_{\cal F} \frac{{Td}\,({\cal F})e^{\omega
+ 2\pi i \l J_{\cal F},\xi\rangle}}
{\det(I-e^{-\frac{i}{2\pi}F_{\frak t}(N'_{\cal F},\xi)})},\]
we can apply the Localization Formula, this time in the opposite
direction, and get that it is equal to the above expression
$\chi'(e^\xi)$. This gives
\[\chi(e^\xi)=\sum_{w\in W} \frac{\chi'(e^{w^{-1}(\xi)})}{ \prod_{\beta>0}
(1-e^{-2\pi i \l \beta,w^{-1}(\xi)\rangle})}.\]
But
\[ \prod_{\beta>0}
(1-e^{-2\pi i \l \beta,w^{-1}(\xi)\rangle})=\det(w)\,
e^{-2\pi i\l w(\delta)-\delta,\xi\rangle}\,\prod_{\beta>0}
(1-e^{-2\pi i \l \beta,\xi\rangle}),\]
where $\delta=\frac{1}{2}\sum_{\beta>0}\beta$ is the magic weight.
Weyl's Character Formula hence shows that
\[ \chi(e^\xi)=\sum_{\mu\in\Lambda_+}
N'(\mu)\sum_{w\in W}
\det(w)\,\frac{e^{2\pi i\l w(\delta+\mu)-\delta,\xi\rangle}}
{\prod_{\beta>0}
(1-e^{-2\pi i \l \beta,\xi\rangle})}=\sum_{\mu\in\Lambda_+}
N'(\mu) \,\chi_\mu(e^\xi),\]
where $\chi_\mu$ is the character of the irreducible representation
corresponding to $\mu$. This proves $N(\mu)=N'(\mu)$.
\bigskip\hspace{0.5cm}\bigskip\mbox{$\Box$}
\noindent{\bf Remarks.}
\begin{enumerate}
\item
If $J(M)\not\subset {\frak g}^*_{reg}$, it is still possible to derive
a formula for $N(\mu)$ similar to (\ref{latticepoints}),
following part II of Guillemin-Prato \cite{GP90}.
However, this formula involves an additional ``shift'', so that
(\ref{11}) gets replaced by an equation of the form
\[ Al=m\nu+\sigma,\,l_j\ge l_j^0.\]
In general,
the number of integer solutions of such an equation is not an
arithmetic polynomial for all $m\in{\Bbb N}$, even though this is
true for large $m$.
\item On the other hand, Theorem \ref{stepw} does
not require that $\mu$ is a regular value of $J$. Even in the
singular case, it is therefore sufficient to prove Multiplicity
Formulas under the assumption $m>>0$.
\end{enumerate}
\noindent{\bf Acknowledgements.}
I would like to thank V. Guillemin, J. Kalkman, E. Lerman,
R. Sjamaar and C. Woodward for useful comments and discussions.
I am very much indebted to Victor Guillemin,
whose recent results \cite{G94} on Multiplicity Formulas of
Riemann-Roch type have been the basic motivation for this work.
The stationary phase
version (\ref{jklocf}) of the Jeffrey-Kirwan Localization Theorem
was worked out jointly with Jaap Kalkman, who has also been
a great help in explaining equivariant cohomology to me.
This work was carried out when I was visiting scholar at the
M.I.T., and I wish to thank the Mathematics Department for its
hospitality.
\bigskip
\noindent{\bf Postscript:} After completing this article, we
learned about independent work of M. Vergne \cite{V94},
who has made a different application of equivariant cohomology
to the multiplicity problem.
|
1994-05-10T19:06:02 | 9405 | alg-geom/9405004 | en | https://arxiv.org/abs/alg-geom/9405004 | [
"alg-geom",
"math.AG"
] | alg-geom/9405004 | Michael Thaddeus | Michael Thaddeus | Geometric invariant theory and flips | 33 pages, LaTeX with AMS fonts | null | null | null | null | We study the dependence of geometric invariant theory quotients on the choice
of a linearization. We show that, in good cases, two such quotients are related
by a flip in the sense of Mori, and explain the relationship with the minimal
model programme. Moreover, we express the flip as the blow-up and blow-down of
specific ideal sheaves, leading, under certain hypotheses, to a quite explicit
description of the flip. We apply these ideas to various familiar moduli
problems, recovering results of Kirwan, Boden-Hu, Bertram-Daskalopoulos-
Wentworth, and the author. Along the way we display a chamber structure,
following Duistermaat-Heckman, on the space of all linearizations. We also give
a new, easy proof of the Bialynicki-Birula decomposition theorem.
| [
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\noindent
{\LARGE \bf Geometric invariant theory and flips}
\medskip \\
{\bf Michael Thaddeus }\\
Mathematical Institute, 24--29 St Giles, Oxford OX1 3LB, England
\medskip
\smallskip
\noindent Ever since the invention of geometric invariant theory, it
has been understood that the quotient it constructs is not entirely
canonical, but depends on a choice: the choice of a {\em
linearization} of the group action. However, the founders of the
subject never made a systematic study of this dependence. In light of
its fundamental and elementary nature, this is a rather surprising
gap, and this paper will attempt to fill it.
In a way, this neglect is understandable, because the different
quotients must be related by birational transformations, whose
structure in higher dimensions is poorly understood. However, it has
been considerably clarified in the last dozen years with the advent of
Mori theory. In particular, the birational transformations that Mori
called {\em flips} are ubiquitous in geometric invariant theory;
indeed, one of our main results \re{1f} describes the mild conditions
under which the transformation between two quotients is given by a
flip. This paper will not use any of the deep results of Mori theory,
but the notion of a flip is central to it.
The definition of a flip does not describe the birational
transformation explicitly, but in the general case there is not much
more to say. So to obtain more concrete results, hypotheses must be
imposed which, though fairly strong, still include many interesting
examples. The heart of the paper, \S\S4 and 5, is devoted to
describing the birational transformations between quotients as
explicitly as possible under these hypotheses. It turns out that
there are fairly explicit smooth loci in two different quotients whose
blow-ups are isomorphic. Thus the two quotients are related by a
blow-up followed by a blow-down. This is somewhat at odds with the
point of view of Mori theory, which views a flip as two contractions,
not two blow-ups; but it facilitates explicit calculations of such
things as topological cohomology or Hilbert polynomials.
The last three sections of the paper put this theory into practice,
using it to study moduli spaces of points on the line, parabolic
bundles on curves, and Bradlow pairs. An important theme is that the
structure of each individual quotient is illuminated by understanding
the structure of the whole family. So even if there is one especially
natural linearization, the problem is still interesting. Indeed, even
if the linearization is unique, useful results can be produced by
enlarging the variety on which the group acts, so as to create more
linearizations.
I believe that this problem is essentially elementary in nature, and I
have striven to solve it using a minimum of technical machinery. For
example, stability and semistability are distinguished as little as
possible. Moreover, transcendental methods, choosing a maximal torus,
and invoking the numerical criterion are completely avoided. The only
technical tool relied on heavily is the marvelous Luna slice theorem
\cite{luna}. Luckily, this is not too difficult itself, and there is
a good exposition in GIT, appendix 1D. This theorem is used, for
example, to give a new, easy proof of the Bialynicki-Birula
decomposition theorem \re{2e}.
Section 1 treats the simplest case: that of an affine variety $X$
acted on by the multiplicative group $\kst$, and linearized on the
trivial bundle. This case has already been treated by Brion and
Procesi \cite{bp}, but the approach here is somewhat different,
utilizing $\Z$-graded rings. The four main results are models for
what comes later. The first result, \re{2i}, asserts that the two
quotients $X \mod \pm$ coming from nontrivial linearizations are
typically related by a flip over the affine quotient $X \mod 0$. The
second, \re{2d}, describes how to blow up ideal sheaves on $X \mod
\pm$ to obtain varieties which are both isomorphic to the same
component of the fibred product ${X \mod -} \times_{X \mod 0} {X \mod
+}$. In other words, it shows how to get from $X \mod +$ to $X \mod
-$ by performing a blow-up followed by a blow-down. The third result,
\re{2j}, asserts that when $X$ is smooth, the blow-up loci are
supported on subvarieties isomorphic to weighted projective fibrations
over the fixed-point set. Finally, the fourth, \re{2k}, identifies
these fibrations, in what gauge theorists would call the quasi-free
case, as the projectivizations of weight subbundles of the normal
bundle to the fixed-point set. Moreover, the blow-ups are just the
familiar blow-ups of smooth varieties along smooth subvarieties.
Sections 3, 4, and 5 are concerned with generalizing these results in
three ways. First, the variety $X$ may be any quasi-projective
variety, projective over an affine. Second, the group acting may be
any reductive algebraic group. Third, the linearization may be
arbitrary. But \S2 assumes $X$ is normal and projective, and is
something of a digression. It starts off by introducing a notion of
$G$-algebraic equivalence, and shows, following Mumford, that
linearizations equivalent in this way give the same quotients. Hence
quotients are really parametrized by the space of equivalence classes,
the $G$-N\'eron-Severi group $\NS^G$. Just as in the
Duistermaat-Heckman theory in symplectic geometry, it turns out
\re{1b} that $\NS^G \otimes \Q$ is divided into chambers, on which the
quotient is constant.
The analogues of the four main results of \S1 then apply to quotients
in adjacent chambers, though they are stated in a somewhat more
general setting. The first two results are readily generalized to
\re{1f} and \re{1g}. The second two, however, require the hypotheses
mentioned above; indeed, there are two analogues of each. The first,
\re{2n} and \re{1x}, make fairly strong hypotheses, and show that the
weighted projective fibrations are locally trivial. The second,
\re{1l} and \re{1n}, relax the hypotheses somewhat, but conclude only
that the fibrations are locally trivial in the \'etale topology.
Counterexamples \re{1r} and \re{2p} show that the hypotheses are
necessary.
The strategy for proving all four of these results is not to imitate
the proofs in the simple case, but rather to reduce to this case by
means of a trick. In fact, given a variety $X$ acted on by a group
$G$, and a family of linearizations parametrized by $t$, we construct
\re{1h} a new variety $Z$, dubbed the ``master space'' by Bertram, acted
on by a torus $T$, and a family of linearizations on $\co(1)$
parametrized by $t$, such that $X \mod G(t) = Z \mod T(t)$ naturally.
This reduces everything to the simple case.
The final sections, \S\S6, 7, and 8, are in a more discursive style;
they explain how to apply the theory of the preceding sections to some
examples. In all cases, the strongest hypotheses are satisfied, so
the best result \re{1x} holds. Perhaps the simplest interesting
moduli problems are those of (ordered or unordered) sets of $n$ points
in $\Pj^1$; these are studied in \S6. The ideas here should have many
applications, but only a very simple one is given: the formula of
Kirwan \cite{k} for the Betti numbers of the moduli spaces when $n$ is
odd. In \S7 the theory is applied to parabolic bundles on a curve,
and the results of Boden and Hu \cite{bhu} are recovered and extended.
Finally, in \S8, the theory is applied to Bradlow pairs on a curve,
recovering the results of the author \cite{t1} and Bertram et al.\
\cite{bdw}.
While carrying out this research, I was aware of the parallel work of
Dolgachev and Hu, and I received their preprint \cite{dh} while this
paper was being written. Their main result is contained in the third
of the four main results I describe, \re{1l}; and of course, some of
the preliminary material, corresponding to my \S2, overlaps. I am
indebted to them for the observation quoted just after \re{1y}, and
for the result \re{1s}, though my proof of the latter is original.
Dolgachev and Hu do not, however, include the results on flips or
blow-ups, study the local triviality of the exceptional loci, or
identify the projective bundles in the quasi-free case. For them,
this is not necessary, since they appear \cite{hu} to be interested
chiefly in computing Betti numbers and intersection Betti numbers of
quotients, and for this, their main result suffices, together with the
deep results of Beilinson, Bernstein, and Deligne \cite{bbd}. I am
more interested in computing algebraic cohomology, as in \cite{t1};
for this, a precise characterization of the birational map between
quotients is necessary, which led me to the present paper. In any
case, even where our results coincide, our methods of proof are quite
different.
A few words on notation and terminology. Many of the statements
involve the symbol $\pm$. This should always be read as two distinct
statements: that is, $X^\pm$ means $X^+$ (resp.\ $X^-$), never $X^+
\cup X^-$ or $X^+ \cap X^-$. Similarly, $X^\mp$ means $X^-$ (resp.\
$X^+$). The quotient of $X$ by $G$ is denoted $X \mod G$, or $X \mod
G(L)$ to emphasize the choice of a linearization $L$. When there is
no possibility of confusion, we indulge in such abuses of notation as
$X \mod \pm$, which are explained in the text. For stable and
semistable sets, the more modern definitions of \cite{n} are followed,
not those of \cite{mf}, which incidentally is often referred to as
GIT. Points are assumed to be closed unless otherwise stated. The
stabilizer in $G$ of a point $x \in X$ is denoted $G_x$. When $E$ and
$F$ are varieties with morphisms to $G$, then $E \times_G F$ denotes
the fibred product; but if $G$ is a group acting on $E$ and $F$, then
$E \times_G F$ denotes the twisted quotient $(E \times F) / G$.
Unfortunately, both notations are completely standard.
All varieties are over a fixed algebraically closed field $\k$. This
may have any characteristic: although we use the Luna slice theorem,
which is usually said \cite{luna,mf} to apply only to characteristic
zero, in fact this hypothesis is used only to show that the stabilizer
must be linearly reductive. Since all the stabilizers we encounter
will be reduced subgroups of the multiplicative group, this will be
true in any characteristic.
By the way, most of the results in \S\S1 and 3 apply not only to
varieties, but to schemes of finite type over $\k$. But for
simplicity everything is stated for varieties.
Finally, since the experts do not entirely agree on the definition of
a flip, here is what we shall use. Let $X_- \to X_0$ be a {\em small
contraction} of varieties over $\k$. This means a small birational
proper morphism; {\em small} means that the exceptional set has
codimension greater than $1$. (This appears to be the prevailing
terminology in Mori theory \cite[(2.1.6)]{ko}, but it is called {\em
semismall\/} in intersection homology, where {\em small} has a
stronger meaning.) Let $D$ be a $\Q$-Cartier divisor class on $X_-$
which is relatively negative over $X_0$; that is, $\co(-D)$ is
relatively ample. Then the {\em $D$-flip} is a variety $X_+$, with a
small contraction $X_+ \to X_0$, such that, if $g: X_- \dasharrow X_+$
is the induced birational map, then the divisor class $g_*D$ is
$\Q$-Cartier, and $\co(D)$ is relatively ample over $X_0$. We
emphasize the shift between ampleness of $\co(-D)$ and that of
$\co(D)$. If a flip exists it is easily seen to be unique. Note that
several authors, including Mori \cite{m}, require that each
contraction reduce the Picard number by exactly $1$. We will not
require this; indeed, it is not generally true of our flips
\cite[4.7]{toric}. For convenience, $D$-flips will be referred to
simply as {\em flips}. However, in the literature, the unmodified
word {\em flip} has traditionally denoted a $K$-flip where $K$ is the
canonical divisor of $X_-$; this is not what we will mean.
\bit{The simplest case}
We begin by examining the simplest case, that of an affine variety
acted on by the multiplicative group of $\k$. This has been studied
before in several papers, that of Brion and Procesi \cite{bp} being
closest to our treatment; but we will clarify, extend, and slightly
correct the existing results.
Let $R$ be a finitely-generated integral algebra over the
algebraically closed field $\k$, so that $X = \Spec R$ is an affine
variety over $\k$. In this section only, $G$ will denote the
multiplicative group of $\k$. An action of $G$ on $\Spec R$ is
equivalent to a $\Z$-grading of $R$ over $\k$, say $R = \bigoplus_{i
\in \Z} R_i$. We will study geometric invariant theory quotients $X
\mod G$, linearized on the trivial bundle.
So choose any $n \in \Z$, and define a $\Z$-grading on $R[z]$ by $R_i
\subset R[z]_i$, $z \in R[z]_{-n}$. Of course $R[z]$ is also
$\N$-graded by the degree in $z$, but this $\Z$-grading is different.
Since $X = \Spec R = \Proj R[z]$, the $\Z$-grading is equivalent to a
linearization on $\co$ of the $G$-action on $X$. The quotient is
$\Proj R[z]^{G} = \Proj R[z]_0 = \Proj \bigoplus_{i \in \N} R_{ni}
z^i$. For $n = 0$, this is just $\Proj R_0[z] = \Spec R_0$, the usual
affine quotient \cite[3.5; GIT Thm.\ 1.1]{n}. For $n > 0$, $\Proj
\bigoplus_{i \in \N} R_{ni} = \Proj \bigoplus_{i \in \N} R_{i}$ by
\cite[II Ex.\ 5.13]{h} (the hypothesis there is not needed for the
first statement); similarly for $n < 0$, $\Proj \bigoplus_{i \in \N}
R_{ni} = \Proj \bigoplus_{i \in \N} R_{-i}$. Hence we need concern
ourselves only with the quotients when $n = 0$, $1$, and $-1$; we
shall refer to them in this section as $X \mod 0$, $X \mod +$ and $X
\mod -$ respectively. Note that $X \mod \pm$ are projective over $X
\mod 0$.
\begin{s}{Proposition}
\label{2h}
If $X \mod + \neq \emptyset} \def\dasharrow{\to \neq X \mod -$, then the natural
morphisms $X \mod \pm \to X \mod 0$ are birational.
\end{s}
Of course, if say $X \mod - = \emptyset} \def\dasharrow{\to$, then $X \mod +$ can be any
$\k$-variety projective over $X \mod 0$.
\pf. For some $d > 0$ $R_{-d}$ contains a nonzero element $t$. The
function field of $X \mod 0$ is $\{ r/s \st r,s \in R_0 \}$, while
that of $X \mod +$ is $\{ r/s \st r,s \in R_{di} \mbox{ \rm for some }
i \geq 0 \}$. But the map $r/s \mapsto (rt^i)/(st^i)$ from the latter
to the former is an isomorphism. \fp \bl
Let $X^\pm \subset X$ be the subvarieties corresponding to the ideals
$\langle R_i \st \mp i > 0 \rangle$ (note the change of sign), and let
$X^0 = X^+ \cap X^-$; then $X^0$ corresponds to the ideal $\langle R_i
\st i \neq 0 \rangle$. Also say $\lim g \cdot x = y$ if the morphism
$G \to X$ given by $g \mapsto g \cdot x$ extends to a morphism $\Aff^1
\to X$ such that $0 \mapsto y$.
\begin{s}{Lemma}
\label{2l}
As sets, $X^\pm = \{ x \in X \st \exists \lim g^{\pm 1} \cdot x \}$,
and $X^0$ is the fixed-point set for the $G$-action.
\end{s}
\pf. A point $x$ is in $X^+$ if and only if for all $n < 0$, $R_n$ is
killed by the homomorphism $R \to \k[x, x^{-1}]$ of graded rings
induced by $g \mapsto g \cdot x$. This in turn holds if the image of
$R$ is contained in $\k[x]$, that is, if $G \to X$ extends to $\Aff^1
\to X$. The proof for $X^-$ is similar.
Hence $x \in X^0$ if and only if $\lim g^{\pm 1} \cdot x$ both exist,
that is, the closure of $G \cdot x$ is a projective variety in $X$.
Since $X$ is affine, this means $x$ is a fixed point of $G$. \fp
\begin{s}{Proposition}
\label{2a}
\mbox{\rm (a)} $X^{ss}(0) = X$; \mbox{\rm (b)} $X^s(0) = X \sans (X^+
\cup X^-)$ ; \mbox{\rm (c)} $X^{ss}(\pm) = X^s(\pm) = X \sans X^\mp$.
\end{s}
\pf. Recall that $x \in X^{ss}(L)$ if for some $n > 0$ there exists
$s \in H^0(L^n)^G$ such that $s(x) \neq 0$, and $x \in X^{s}(L)$ if
the morphism $G \to X^{ss}(L)$ given by $g \mapsto g \cdot x$ is
proper. For $L = 0$, $H^0(L^n)^{G} = R_0$ for all $n$, but this
contains 1, which is nowhere vanishing. That is all there is to (a).
The valuative criterion implies that the morphism $G \to X$ is proper
if and only if the limits do not exist, which together with \re{2l}
implies (b). For $L = \pm$, $H^0(L^n)^{G} = R_{\pm n}$, so
$X^{ss}(\pm) = X \sans X^\mp$ follows immediately from the definition
of $X^\pm$. The additional condition of properness needed for $x \in
X^s(\pm)$ is equivalent, by the valuative criterion, to $\lim g \cdot
x$ and $\lim g^{-1} \cdot x \notin X^{ss}(\pm)$. But one does not
exist, and the other, if it exists, is fixed by $G$, so is certainly
not in $X^{ss}(\pm)$. \fp
\begin{s}{Corollary}
\label{2m}
The morphisms $X \mod \pm \to X \mod 0$ are isomorphisms on the
complements of $X^{\pm} \mod \pm \to X^\pm \mod 0$.
\end{s}
In good cases, $X^{\pm} \mod \pm$ will be exactly the exceptional loci
of the morphisms, but they can be smaller, even empty---for instance
$X \mod - \to X \mod 0$ in \re{2g} below.
\pf. By \re{2a}, the sets $(X \sans X^\pm) \mod \pm$ and $(X \sans
X^\pm) \mod 0$ contain no quotients of strictly semistable points.
They are therefore isomorphic. \fp
\begin{s}{Proposition}
There are canonical dominant morphisms $\pi_\pm: X^\pm \to X^0$ such
that for all $x \in X^\pm$, $\pi_\pm (x) = \lim g^{\pm 1} \cdot x$.
\end{s}
\pf. Note first that $R_0 \cap \langle R_i \st \pm i > 0 \rangle =
R_0 \cap \langle R_i \st i \neq 0 \rangle$. So $R / \langle R_i \st i
\neq 0 \rangle = R_0 / \langle R_i \st i \neq 0 \rangle$ are naturally
included in $R / \langle R_i \st \pm i > 0 \rangle$ as the
$G$-invariant parts. Hence there are natural dominant morphisms
$\pi_\pm: X^\pm \to X^0$.
Because $\pi_\pm$ is induced by the inclusion of the degree 0 part in
$R / \langle R_i \st \mp i > 0 \rangle$, $\pi_\pm (x)$ is the unique
fixed point such that $f(\pi_\pm (x)) = f(x)$ for all $f \in R_0$.
But $f \in R_0$ means it is $G$-invariant, hence constant on orbits,
so the same property is satisfied by $\lim g^{\pm 1} \cdot x$, which
is a fixed point in the closure of $G \cdot x$. \fp \bl
The next two results digress to show what the results so far have to
do with flips.
\begin{s}{Proposition}
\label{2i}
If $X^{\pm} \subset X$ have codimension $\geq 2$, then the birational
map $f: X \mod - \dasharrow X \mod +$ is a flip with respect to
$\co(1)$.
\end{s}
\pf. The hypothesis implies that the open sets $(X \sans X^\pm) \mod
\pm$ in $X \mod \pm$ have complements of codimension $\geq 2$. But by
\re{2m} these open sets are identified by $f$. Hence there is a
well-defined push-forward $f_*$ of divisors.
For some $n > 0$ the twisting sheaves $\co(\pm n) \to X \mod \pm$ are
line bundles. Indeed, they are the descents \cite{dn} from $X$ to $X
\mod \pm$ of the trivial bundle $\co$, with linearization given by $n$
as at the beginning of this section. Consequently, they agree on the
open sets $(X \sans X^\pm) \mod \pm$, so $f_* \co(-n) = \co(n)$. But
$\co(\pm n) \to X \mod \pm$ are relatively ample over $X \mod 0$, so $f$
is a flip. \fp
\begin{s}{Proposition}
\label{2o}
Let $Y_0$ be normal and affine over $\k$, and let $f: Y_- \dasharrow
Y_+$ be a flip of normal varieties over $Y_0$. Then there exists $X$
affine over $\k$ and a $G$-action on $X$ so that $Y_0 = X \mod 0$,
$Y_\pm = X \mod \pm$.
\end{s}
\pf. Let $L = \co(D)$, where $D$ is as in the definition of a flip.
Since $Y_\pm$ are normal, and the exceptional loci of $f$ have
codimension $\geq 2$, $f$ induces an isomorphism $H^0(Y_-, L^n) \cong
H^0(Y_+, f_*L^n)$ for all $n$. The $\N$-graded algebras $\bigoplus_{n
< 0} H^0(Y_-, L^n)$ and $\bigoplus_{n > 0} H^0(Y_+, f_*L^n)$ are the
homogeneous coordinate rings of the quasi-projective varieties $Y_-$
and $Y_+$ respectively, so are finitely-generated over $\k$. Hence
the same is true of the $\Z$-graded algebra $R = \bigoplus_{n \in \Z}
H^0(Y_-, L^n)$. Let $X = \Spec R$ with the $G$-action coming from the
grading. Then $X \mod 0 = Y_0$ and $X \mod \pm = Y_\pm$. I thank
Miles Reid for pointing out this simple proof. \fp \bl
In order to describe the birational map $X \mod - \dasharrow X \mod +$
more explicitly, we will next construct a variety birational to $X
\mod \pm$ which dominates them both. It is admittedly true in general
that any birational map can be factored into a blow-up and blow-down
of some sheaves of ideals. The virtue of the present situation,
however, is that these sheaves can be identified fairly explicitly,
and that the common blow-up is precisely the fibred product.
Choose $d > 0$ such that $\bigoplus_{i \in \Z} R_{di}$ is generated by
$R_0$ and $R_{\pm d}$. Then let $\ci^\pm$ be the sheaves of ideals on
$X$ corresponding to $\langle R_{\mp d} \rangle$. Let $\ci^\pm \mod
\pm$ on $X \mod \pm$ and $\ci^\pm \mod 0$ on $X \mod 0$ be the ideal
sheaves of invariants of $\ci^\pm$, that is, the sheaves of ideals
locally generated by the invariant elements of $\ci^\pm$. Note that
$\ci^\pm$ are supported on $X^\pm$, so that $\ci^\pm \mod \pm$ are
supported on $X^\pm \mod \pm$.
For $i, j \geq 0$, let $R_{i,j} = R_i \cdot R_{-j} \subset
R_{i-j}$.
\begin{s}{Lemma}
\label{2b}
The ideal sheaf $(\ci^+ + \ci^-) \mod 0$ is exactly $\langle R_{d,d}
\rangle$, and its pullbacks by the morphisms $X \mod \pm \to X \mod 0$
are $\ci^\pm \mod \pm$.
\end{s}
\pf. The ideals in $R$ corresponding to $\ci^\pm$ are by definition
$\langle R_{\mp d} \rangle$, and $\langle R_{\mp d} \rangle \cap R_0 =
R_{d,d}$, so $(\ci^+ + \ci^-) \mod 0 = \langle R_{d,d} \rangle$.
Regard $X \mod \pm$ as quotients with respect to the linearizations
$\pm d$. Then $X \mod \pm = \bigoplus_{i \geq 0} R_{\pm di}$, so for
any $\si \in R_{\pm d}$, $\Spec (\si^{-1} \bigoplus_{i \geq 0} R_{\pm
di})_0$ is an affine in $X \mod \pm$. But $\ci^\pm \cap (\si^{-1}
\bigoplus_{i \geq 0} R_{\pm di})_0 = \si^{-1} \langle R_{d,d}
\rangle$, so locally $\ci^\pm \mod \pm$ is the pullback of $\langle
R_{d,d} \rangle$. As $\si$ ranges over $R_{\pm d}$, these affines
cover $X \mod \pm$, so the result holds globally. \fp
\begin{s}{Theorem}
\label{2d}
Suppose $X \mod + \neq \emptyset} \def\dasharrow{\to \neq X \mod -$. Then the blow-ups
of $X \mod \pm$ at $\ci_\pm \mod \pm$, and the blow-up of $X \mod 0$
at $(\ci^+ + \ci^-) \mod 0$, are all naturally isomorphic to the
irreducible component of the fibred product ${X \mod -} \times_{X \mod
0} {X \mod +}$ dominating $X \mod 0$.
\end{s}
\pf. The blow-up of $X \mod +$ at $\ci_+ \mod +$ is $\Proj
\bigoplus_n H^0((\ci_+ \mod +)^n(dn))$ for $d$ sufficiently divisible.
But by \re{2b} $H^0((\ci_+ \mod +)^n(dn)) = R_{(d+1)n, n}$, so the
blow-up is $\Proj \bigoplus_n R_{(d+1)n, n}$. There is a surjection
of $R_0$-modules $R_{(d+1)n} \otimes_{R_0} R_{-n} \to R_{(d+1)n, n}$,
so the blow-up embeds in $\Proj \bigoplus_n R_{(d+1)n} \otimes_{R_0}
R_{-n}$. This is precisely the fibred product ${X \mod -} \times_{X
\mod 0} {X \mod +}$, with polarization $\co(d+1,1)$. By \re{2m} this
naturally contains $(X \sans X^\pm) \mod \pm$ as a nonempty open set,
but so does the blow-up. The blow-up is certainly irreducible, so it
is the component containing $(X \sans X^\pm) \mod \pm$. The proof for
$X \mod -$ is similar. Likewise, the blow-up of $X \mod 0$ at $(\ci^+
+ \ci^-) \mod 0$ is $\Proj \bigoplus_n (\ci^+ + \ci^- \mod 0)^n =
\Proj \bigoplus_n R_{dn,dn} = \Proj \bigoplus_n R_{n,n}$. This embeds
in the fibred product with polarization $\co(1,1)$, but by \re{2m}
contains $(X \sans X^\pm) \mod \pm$ as a nonempty open set. \fp \bl
The ideal sheaves $\ci^\pm$ are supported on $X^\pm$, so $\ci^\pm \mod
\pm$ are supported on $X^\pm \mod \pm$. But they are not just
$\ci_{X^\pm \mod \pm}$, as the following counterexample shows.
\begin{s}{Counterexample}
\label{2g}
To show that the ideal sheaves of $X^\pm \mod \pm$ cannot
generally replace $\ci_\pm \mod \pm$ in \re{2d}.
\end{s}
In other words, the blow-up may be weighted, not just the usual
blow-up of a smooth subvariety. Let $G$ act on $X = \Aff^3$ by
$\la(w,x,y) = (\la^{-1} w, \la x, \la^2 y)$; in other words, $w \in
R_{-1}$, $x \in R_1$, and $y \in R_2$. Then $X \mod 0 = \Spec
\k[wx,w^2y] = \Aff^2$, and
$X \mod - = \Proj \k[wx,w^2y, zw] = \Aff^2$, where the $\N$-grading of
every variable is 0 except $z$, which is graded by 1. However,
\beqas
X \mod + & = & \Proj \k[wx,w^2y,zx,z^2y] \\
& = & \Proj \k[wx,w^2y,z^2(w^2x^2),z^2(w^2y)] \\
& = & \Proj \k[u,v,zu^2,zv],
\eeqas
which is $\Aff^2$ blown up at the ideal $\langle u^2,v \rangle$. This
has a rational double point, so is not the usual blow-up at a point.
\fp
The paper \cite{bp} of Brion and Procesi asserts (in section 2.3) a
result very similar to \re{2d}. They state that the two quotients are
related by blow-ups---``\'eclatements''---of certain subvarieties.
The counterexample above shows that the blow-ups must sometimes be
weighted, that is, must have non-reduced centres. Brion and Procesi
do not state this explicitly, but they are no doubt aware of it.
Another minor contradiction to their result is furnished by the
following counterexample.
\begin{s}{Counterexample}
To show that the fibred product of \re{2d} can be reducible, and the
blow-up one of its irreducible components.
\end{s}
Let $G$ act on the singular variety $X = \Spec \k[a^2,ab,b^2,c,d]/
\langle ad-bc \rangle$ where $a,b,c,d$ are of degree 1, acted on with
weights $1,-1,1,-1$ respectively. Then
\beqas
X \mod 0 & = & \Spec \k[ab,cd,a^2d^2,b^2c^2]/\langle ad-bc \rangle \\
& = & \Spec \k[ab,cd] \\
& = & \Aff^2.
\eeqas
But, using the same $\N$-grading convention as in the previous example,
\beqas
X \mod - & = & \Proj \k[ab,cd,z^2b^2,zd] \\
& = & \Proj \k[u,v,zu,zv] \\
& = & \blowup,
\eeqas
that is, the blow-up of $\Aff^2$ at the origin, and by symmetry $X
\mod + \cong \blowup$ as well. Taking $d =2$ gives ideal sheaves
$\ci_+ = \langle a^2,c^2 \rangle$ and $\ci_- = \langle b^2,d^2
\rangle$; the sheaf of invariants of both is$\langle u^2,v^2 \rangle$,
the ideal sheaf of twice the exceptional divisor. Hence blowing up
$\ci_\pm \mod \pm$ does nothing. The fibred product, however, is
$\blowup \times_{\Aff^2} \blowup$, which is not just $\blowup$: it has
two components, isomorphic to $\blowup$ and $\Pj^1 \times \Pj^1$
respectively and meeting in a $\Pj^1$. \fp \bl
Following Bialynicki-Birula \cite{bb}, define the {\em trivial
$w_i$-fibration} over an affine variety $Y$ to be $\Aff^r \times Y$,
with a $G$-action induced by the action on $\Aff^r$ with weights $w_i$.
A {\em $w_i$-fibration} over $Y$ is a variety over $Y$, with a
$G$-action over the trivial action on $Y$, which is locally the
trivial $w_i$-fibration. As Bialynicki-Birula points out, a
$w_i$-fibration need not be a vector bundle, because the transition
functions need not be linear. But if all the $w_i$ are equal, then it
is a vector bundle.
Suppose now that $G$ acts on an affine variety $X$ which is {\em
smooth}. Then it will be proved in (a) below that $X^0$ is also
smooth. Purely for simplicity, suppose that it is also connected.
(If not, the following theorem is still valid, but the fibrations
involved need have only locally constant rank and weights.) The group
$G$ acts on the normal bundle $N_{X^0 / X}$. Let $N^\pm_X$, or simply
$N^\pm$, be the subbundles of positive and negative weight spaces for
this action, with weights $w^\pm_i \in \Z$.
\begin{s}{Theorem}
\label{2e}
Suppose $X$ is smooth. Then \mbox{\rm (a)} $X^0$ is smooth; \mbox{\rm
(b)} as varieties with $G$-action, $\pi_\pm: X^\pm \to X^0$ are
naturally $w^\pm_i$-fibrations; \mbox{\rm (c)} $N_{X^0 / X}$ has no
zero weights, so equals $N^+ \oplus N^-$; \mbox{\rm (d)} the normal
bundles $N_{X^0 / X^\pm} = N^\pm$; \mbox{\rm (e)} if all $w^\pm_i =
\pm w$ for some $w$, then $\ci^\pm$ and $\ci^+ +\ci^-$ cut out the
$d/w$th infinitesimal neighbourhoods of $X^\pm$ and $X^0$ respectively.
\end{s}
Parts (b), (c), (d) are the {\em Bialynicki-Birula decomposition
theorem} \cite[Thm.\ 4.1]{bb}. Another version of this result,
possibly more familiar, gives a Morse-style decomposition of a
projective variety into a disjoint union of $w_i$-fibrations. It
follows easily from this \cite{bb}.
\pf. First consider the case of a finite-dimensional vector space
$V$, acted on linearly by $G$. Then $V = \Spec S$ for $S$ a
$\Z$-graded polynomial algebra. This decomposes naturally into three
polynomial algebras, $S = S^- \otimes S^0 \otimes S^+$, corresponding
to the subspaces of negative, zero, and positive weight. Then
$\ci^\pm = \langle S^\mp_{\mp d} \rangle$, $V^\pm = \Spec S^\pm
\otimes S^0$, and $V^0 = \Spec S^0$. Parts (a)--(e) all follow
easily. Indeed, the fibrations are naturally trivial.
To return to the general case, note first that if $U \subset X$ is a
$G$-invariant open affine, then $U = \Spec F^{-1}R$ for some $F
\subset R_0$. Hence $(F^{-1}R)_i = F^{-1}(R_i)$ for each $i$, so
$\ci^\pm_U = \ci^\pm_X |_U$, $U^\pm = X^\pm \cap U$, $U^0 = X^0 \cap
U$, and $\pi_\pm$ is compatible with restriction. Consequently, the
whole theorem is local in the sense that it suffices to prove it for a
collection of $G$-invariant open affines in $X$ containing $X^+ \cup
X^-$.
Now for any closed point $x \in X^0$, apply the Luna slice theorem
\cite{luna,mf} to $X$. Since $G_x = G$, the Luna slice is a
$G$-invariant open affine $U \subset X$ containing $x$, and $G
\times_{G_x} N_x = T_x X$. Hence there is a strongly \' etale
$G$-morphism (see \cite{luna,mf}) $\phi: U \to V = T_x X$ such that
$\phi(x) = 0$. In particular, $U = U \mod 0 \times_{V \mod 0} V$.
Any $y \in X^+ \cup X^-$ is contained in some such $U$: indeed, just
take $x = \pi_\pm (y)$. It therefore suffices to prove the theorem
for $U$, so by abuse of notation, say $U = \Spec R$ from now on. The
$G$-morphism $\phi$ then corresponds to a graded homomorphism $S \to
R$, where $S$ is a $\Z$-graded polynomial ring, such that $R = R_0
\otimes_{S_0} S$. In particular, $R_i = R_0 \otimes_{S_0} S_i$ for
each $i$. Hence $R_{\pm d}$ and $S_{\pm d}$ generate the same ideals
in $R$, so $\ci^\pm_U = \phi^{-1}\ci^\pm_V$. Also, $U^\pm = \phi^{-1}
V^\pm = V^\pm \times_{V \mod 0} U \mod 0$, and $U^0 = \phi^{-1}V^0 =
V^0 \times_{V \mod 0} U \mod 0$. This immediately implies (a). Since
$V^\pm \to V^0$ are trivial fibrations, it also gives the local
trivialization of $X^\pm \to X^0$ near $x$ needed to prove (b).
Parts (c) and (d) also follow, since $\phi$, as an \' etale
$G$-morphism, satisfies $\phi^* N_{V^0/V} = N_{U^0/U}$ as bundles with
$G$-action, so in particular $\phi^* N^\pm_V = N^\pm_U$. The
hypotheses of part (e) imply that $d$ is a multiple of $w$; the
conclusion holds if and only if the map $R_{\mp w}^{d/w} \to R_{\mp
d}$ is surjective. This is true for $S$, and follows in general from
$R_i = R_0 \otimes_{S_0} S_i$. \fp
The above methods can be used to describe the local structure of the
non-reduced schemes cut out by $\ci^\pm$ even when not all $w^\pm_i =
\pm w$, but we will not pursue this.
\begin{s}{Corollary}
\label{2j}
Suppose $X$ is smooth. Then $X^\pm \mod \pm$ are locally trivial
fibrations over $X^0$ with fibre the weighted projective space
$\Pj(|w_i^\pm|)$. \fp
\end{s}
If $X \mod + \neq \emptyset} \def\dasharrow{\to \neq X \mod -$, these are the supports
of the blow-up loci of \re{2d}. On the other hand, if $X \mod - =
\emptyset} \def\dasharrow{\to$, then $X^+ = X$ and $X^0 = X \mod 0$, so this says the
natural morphism $X \mod + \to X \mod 0$ is a weighted projective
fibration.
\begin{r}{Remark}
\label{2f}
It follows from the above corollary that, in this smooth case, ${X^-
\mod -} \times_{X \mod 0} {X^+ \mod +}$ is irreducible of codimension
1 in ${X \mod -} \times_{X \mod 0} {X \mod +}$. It must therefore be
exactly the exceptional divisor of each of the two blow-ups of
\re{2d}. In other words, the latter fibred product is irreducible,
and is isomorphic to each of the blow-ups. This implies that, when
$X$ is smooth, the surjection of $R_0$-modules $R_{i} \otimes_{R_0}
R_{-j} \to R_{i,j}$ is an isomorphism for $i,j > 0$ sufficiently
divisible. However, I do not know of a direct algebraic proof of this
fact.
\end{r}
\begin{s}{Theorem}
\label{2k}
Suppose $X$ is smooth, and that all $w_i^\pm = \pm w$ for some $w$.
Then $X^\pm \mod \pm$ are naturally isomorphic to the projective
bundles $\Pj(N^\pm)$ over the fixed-point set $X^0$, their normal
bundles are naturally isomorphic to $\pi_\pm^* N^\mp(-1)$, and the
blow-ups of $X \mod \pm$ at $X^\pm \mod \pm$, and of $X \mod 0$ at
$X^0 \mod 0$, are all naturally isomorphic to the fibred product ${X
\mod -} \times_{X \mod 0} {X \mod +}$.
\end{s}
Note that if each 0-dimensional stabilizer on $X$ is trivial, then all
$w^\pm_i = \pm 1$.
\pf. All the blow-ups and the fibred product are empty if either $X
\mod +$ or $X \mod -$ is empty, so suppose they are not. By the
observation of Bialynicki-Birula quoted above, if all $w^\pm_i = \pm
w$, then the fibrations $X^\pm \to X^0$ are actually vector bundles.
But any vector bundle is naturally isomorphic to the normal bundle of
its zero section, so by \re{2e}(d) $X^\pm \mod \pm = \Pj(X^\pm) =
\Pj(N^\pm)$, and the natural $\co(1)$ bundles correspond. By
\re{2e}(e), $\ci^\pm$ cut out the $d/w$th infinitesimal neighbourhoods
of $X^\pm$. This means that $R_{\mp w}^{d/w} \to R_{\mp d}$ are
surjective and hence that $\ci^\pm \mod \pm$ and $(\ci^+ + \ci^-) \mod
0$ cut out the $d/w$th infinitesimal neighbourhoods of $X^\pm \mod
\pm$ and $X^0 \mod 0$ respectively. Since blowing up a subvariety has
the same result as blowing up any of its infinitesimal neighbourhoods,
the result follows from \re{2d} and \re{2f}, except for the statement
about normal bundles. To prove this, recall first that if $E$ is the
exceptional divisor of the blow-up $\tilde{Y}$ of any affine variety
$Y$ at $Z$, then $N_{E/\tilde{Y}} = \co_E(-1)$, and $N^*_{Z/Y} =
(R^0\pi) N^*_{E/\tilde{Y}}$. Applying this to the case $Y = X \mod 0$
shows that the normal bundle to ${X^- \mod -} \times_{X^0} {X^+ \mod
+}$ is the restriction of $\co(-1,-1) \to {X \mod -} \times_{X \mod 0}
{X \mod +}$, which is exactly the obvious $\co(-1,-1) \to \Pj N^+
\times_{X^0} \Pj N^-$. The normal bundle to $\Pj N^\pm$ is then just
$(R^0\pi_\mp \co(1,1))^* = \pi_\pm^* N^\mp(-1)$. \fp
\begin{r}{Example}
The simplest non-trivial example of these phenomena is also the
best-known; indeed it goes back to an early paper of Atiyah \cite{a}.
Let $X = \Aff^4$, and let $G$ act by $\lambda \cdot(v,w,x,y)=
(\lambda v, \lambda w, \lambda^{-1} x, \lambda^{-1} y)$. Then $X \mod
0 = \Spec \k[vx,vy,wx,wy] = \Spec[a,b,c,d]/\langle ad-bc \rangle$, the
affine cone on a smooth quadric surface in $\Pj^3$. But, using the
$\N$-grading conventions of the previous examples,
\beqas
X \mod + & = & \Proj \k[vx,vy,wx,wy,zv,zw] \\
& = & \Proj \k[a,b,c,d,za,zc] / \langle ad-bc \rangle.
\eeqas
This is the blow-up of $X \mod 0$ at the Weil divisor cut out by $a$
and $c$. But this Weil divisor is generically Cartier, so the
blow-down $X \mod + \to X \mod 0$ is generically an isomorphism even
over the divisor. The exceptional set of the morphism therefore has
codimension 2; indeed, it is the $\Pj^1$ lying over the cone point.
Likewise, $X \mod - = \Proj \k[a,b,c,d,zb,zd]/ \langle ad-bc \rangle$,
and similar remarks apply by symmetry. By \re{2k} the fibred product
${X \mod -} \times_{X \mod 0} {X \mod +}$ is the common blow-up of $X
\mod \pm$ at these $\Pj^1$, and also the blow-up of $X \mod 0$ at the
cone point. This is exactly the proper transform of the quadric cone
in $\Aff^4$ blown up at the origin, so it has fibre $\Pj^1 \times
\Pj^1$ over the cone point, as expected.
\end{r}
\bit{The space of linearizations}
In \S\S3, 4 and 5 we will generalize the results of \S1 in three
directions. First, the group $G$ may now be any reductive group over
$\k$. Second, the variety $X$ may now be any quasi-projective variety
over $\k$, projective over an affine variety. Finally, the
linearization may be arbitrary. Before doing this, though, we will
prove some general facts, in the case where $X$ is normal and
projective, about the structure of the group of all linearizations.
This will show how to apply our general results in this case.
So in this section, suppose $X$ is normal and projective over $\k$,
and that $G$ is a reductive group over $\k$ acting on $X$. We first
recall a few familiar facts about the Picard group.
In the Picard group $\Pic$ of isomorphism classes of line bundles, the
property of ampleness depends only on the algebraic equivalence class
of the bundle. Hence there is a well-defined ample subset $\A$ of the
N\'eron-Severi group $\NS$ of algebraic equivalence classes of line
bundles. This determines the {\em ample cone} $\A_\Q = \A
\otimes_{\N} \Q_{\geq 0} \subset \NS_\Q = \NS \otimes \Q$. The
N\'eron-Severi group is finitely-generated, so $\NS_\Q$ is a
finite-dimensional rational vector space. We will refer to an element
of $\A$ as a {\em polarization}, and an element of $\A_\Q$ as a {\em
fractional polarization}.
Now let $\Pic^G$ be the group of isomorphism classes of linearizations
of the $G$-action (cf.\ 1, \S3 of GIT). There is a forgetful
homomorphism $f: \Pic^G \to \Pic$, whose kernel is the group of
linearizations on $\co$, which is exactly the group $\chi(G)$ of
characters of $G$. There is an equivalence relation on $\Pic^G$
analogous to algebraic equivalence on $\Pic$; it is defined as
follows. Two linearizations $L_1$ and $L_2$ are said to be {\em
$G$-algebraically equivalent} if there is a connected variety $T$
containing points $t_1, t_2$, and a linearization $L$ of the
$G$-action on $T \times X$ induced from the second factor, such that
$L|_{t_1 \times X} \cong L_1$ and $L|_{t_2 \times X} \cong
L_2$.
\begin{s}{Proposition}
\label{1d}
If $L$ is an ample linearization, then $X^{ss}(L)$, and the quotient
$X \mod G(L)$ regarded as a polarized variety, depend only on the
$G$-algebraic equivalence class of $L$.
\end{s}
\pf. The statement about $X^{ss}(L)$ is proved exactly like Cor.\ 1.20
of GIT, except that the Picard group $P$ is replaced by $T$. The
statement about $X \mod G(L)$ as a variety then follows from this,
since $X \mod G(L)$ is a good quotient of $X^{ss}(L)$, hence a
categorical quotient of $X^{ss}(L)$, so is uniquely determined by
$X^{ss}(L)$. As for the polarization, note that, if $L_1$ and $L_2$
are $G$-linearly equivalent ample linearizations, then the
linearization $L$ on $T \times X$ inducing the equivalence can be
assumed ample: just tensor $L$ by the pullback of a sufficiently ample
bundle on $T$. Then $\co(1) \to (T \times X) \mod G (L)$ is a
family of line bundles on $X \mod G$ including $\co(1) \to X \mod G
(L_1)$ and $\co(1) \to X \mod G (L_2)$, so these are algebraically
equivalent. \fp
So define $\NS^G$ to be the group of $G$-algebraic equivalence classes
of linearizations. In light of \re{1d}, by abuse of terminology an
element of $\NS^G$ will frequently be called just a linearization.
The forgetful map $f$ descends to $f: \NS^G \to \NS$.
\begin{s}{Proposition}
\label{2q}
This new $f$ has kernel $\chi(G)$ modulo a torsion subgroup.
\end{s}
\pf. Let $M \to \Pic_0 X \times X$ be the Poincar\'e line bundle, and
let $G$ act on $\Pic_0 X \times X$, trivially on the first factor. By
Cor.\ 1.6 of GIT, some power $M^n$ of $M$ admits a linearization.
Since the $n$th power morphism $\Pic_0 X \to \Pic_0 X$ is surjective,
this shows that any element of $\Pic_0 X$ has a linearization
$G$-algebraically equivalent to a linearization on $\co$. Hence $\ker
f$ is $\chi(G)$ modulo the subgroup of linearizations on $\co$
which are $G$-algebraically equivalent to the trivial linearization.
We will show that this subgroup is torsion.
Suppose there is a linearization $L_1$ on $\co$ which is
$G$-algebraically equivalent to the trivial linearization. Then there
exist $T$ containing $t_1$, $t_2$ and $L$ as in the definition of
$G$-algebraic equivalence. There is an induced morphism $g: T \to
\Pic_0 X$ such that $t_1, t_2 \mapsto \co$. As before, let $M^n$ be
the power of the Poincar\'e bundle admitting a linearization. Then $N
= (1 \times g)^* M^n \otimes L^{-n}$ is a family of linearizations on
$\co \to X$. Since the isomorphism classes of such linearizations
form the discrete group $\chi(G)$, $L_1^n = N_{t_1}^{-1} \otimes
N_{t_2}$ is trivial as a linearization. \fp \bl
Hence $\NS^G$ is finitely-generated and $\NS^G_\Q = \NS^G \otimes
\Q$ is again a finite-dimensional rational vector space. We refer to
an element of $\NS^G_\Q$ as a {\em fractional linearization}.
The map $f: \NS^G \to \NS$ is not necessarily surjective (see 1, \S3
of GIT). But by Cor.\ 1.6 of GIT, $f_\Q: \NS^G_\Q \to \NS_\Q$ is
surjective. By \re{2q}, the kernel is $\chi(G) \otimes \Q$, the group
of {\em fractional characters}. (Not to be confused with $f_\Q$ is
the natural surjective linear map $\NS^G_\Q(X) \to \NS_\Q(X \mod G)$:
this is induced by descent, since divisor classes always descend over
$\Q$.)
An ample linearization $L$ is said to be {\em $G$-effective}
if $L^n$ has a $G$-invariant section for some $n > 0$. This is
equivalent to having a semistable point, so \re{1d} shows that
$G$-effectiveness depends only on the $G$-algebraic equivalence class
of the linearization. Hence there is a well-defined $G$-effective
subset $\E^G \subset f^{-1}\A \subset \NS^G$, and a {\em $G$-effective
cone} $\E^G_\Q = \E^G \otimes_{\N} \Q_{\geq 0} \subset \NS^G_\Q$.
Now a linearization $L$ determines a quotient $X \mod G$ if $L$ is
ample; then $X \mod G \neq \emptyset} \def\dasharrow{\to$ if and only if $L$ is also
$G$-effective. Of course, we can also tensor by $\Q$, allowing
fractional linearizations; the quotient $X \mod G$ will then be
fractionally polarized. Hence any fractional linearization $L \in
f_\Q^{-1}(\A_\Q) \subset \NS^G_\Q$ defines a fractionally
polarized quotient, which will be nonempty if and only if $L \in
\E^G_\Q$. Replacing a fractional linearization $L$ by $L^n$ for
some positive $n \in \Q$ has no effect on the quotient, except to
replace the fractional polarization $\co(1)$ by $\co(n)$. \bl
The first result describing the dependence of the quotient $X \mod
G(L)$ on the choice of $L \in \NS^G_\Q$ is the following, which is
analogous to the Duistermaat-Heckman theory in symplectic geometry.
\begin{s}{Theorem}
\label{1b}
The $G$-effective cone $\E_\Q^G$ is locally polyhedral in the ample
cone $f_\Q^{-1} \A_\Q$. It is divided by homogeneous {\rm walls},
locally polyhedral of codimension 1 in $f_\Q^{-1} \A_\Q$, into convex
{\rm chambers} such that, as $t$ varies within a fixed chamber, the
semistable set $X^{ss}(t)$, and the quotient $X \mod G(t)$, remain
fixed, but $\co(1)$ depends affinely on $t$. If $t_0$ is on a wall or
walls, or on the boundary of $\E_\Q^G$, and $t_+$ is in an adjacent
chamber, then there is an inclusion $X^{ss}(+) \subset X^{ss}(0)$
inducing a canonical projective morphism $X \mod G(+) \to X \mod
G(0)$.
\end{s}
The proposition above could be proved directly, using Kempf's descent
lemma \cite{dn} for the statement about $\co(1)$, and Mumford's
numerical criterion \cite[4.9; GIT Thm.\ 2.1]{n} for the rest. But it
will follow easily from the construction \re{1h} to be introduced in
the next section, so we put off the proof until then. \bl
Theorem \re{1b} asserts that the walls are locally polyhedral, and in
particular, locally finite; but with a little more effort we can prove
a global result.
\begin{s}{Theorem}
\label{1s}
There are only finitely many walls.
\end{s}
\pf. Suppose not. Then there exists an infinite sequence $\{ C_i \}$
of chambers such that, for any $m,n \geq 0$, the convex hull of $C_n
\cup C_{n+m}$ intersects the interior of $C_{n+1}$ nontrivially.
Indeed, let $C_0$ be any chamber; then there exists a wall $W_0$
bounding it such that there are infinitely many chambers on the other
side of $W_0$ (or more properly, the affine hyperplane containing
$W_0$). Let $C_1$ be the other chamber bounded by $W_0$.
Inductively, given $C_0, \dots, C_n$ such that $C_n$ is on the other
side of $W_i$ from $C_i$ for all $i < n$, there is a wall $W_n$
bounding $C_n$ such that there are infinitely many chambers which for
all $i \leq n$ are on the other side of $W_i$ from $C_i$. Let
$C_{n+1}$ be the other chamber bounded by $W_n$. For a sequence
chosen in this manner, $C_{n+m}$ is on the other side of $W_n$ from
$C_n$, so the convex hull of $C_n \cup C_{n+m}$ meets the interior of
$C_{n+1}$.
Choose an $L_i$ in the interior of each $C_i$. For any fixed $x \in
X$, the set $\{ L \in \NS_\Q \st x \in X^{ss}(L) \}$ is convex, since
$s_\pm \in H^0(L_\pm)^G$, $s_\pm(x) \neq 0$ imply $s_+ \cdot s_- \in
H^0(L_+ \otimes L_-)^G$, $(s_+ \cdot s_-)(x) \neq 0$. But by \re{1b}
it is also a union of chambers, so by induction it includes $C_n \cup
C_{n+m}$ if and only if it includes $C_{n+i}$ for all $i \leq m$. Its
intersection with $\{ L_i \}$ is therefore the image of an interval in
$\N$. Hence $X^{ss}(L_{i+1}) \sans X^{ss}(L_i)$ are all disjoint; but
each one is open in the complement of $X^{ss}(L_0)$ and the preceding
ones. Since varieties are noetherian, this implies there exists $i_0$
such that for all $i \geq i_0$, $X^{ss}(L_{i+1}) \sans X^{ss}(L_i) =
\emptyset} \def\dasharrow{\to$, and hence $X^{ss}(L_{i+1}) \subset X^{ss}(L_i)$. There
is therefore an infinite sequence of dominant projective morphisms $$
\cdots \to X \mod G(L_{i_0 + 2}) \to X \mod G(L_{i_0 + 1}) \to X\mod
G(L_{i_0}).$$ Hence the N\'eron-Severi group of $X \mod G(L_i)$ has
arbitrarily large rank for some $i$. But as mentioned before, there
is a natural surjective linear map $\NS^G_\Q(X) \to \NS_\Q(X \mod
G(L_i))$ for all $i$. Since $\NS^G_\Q(X)$ is finite-dimensional, this
is a contradiction. \fp
\bit{The general case}
We now embark on our generalization of the results of \S1. So let $G$
be a reductive group over $\k$, acting on a quasi-projective variety
$X$ over $\k$, projective over an affine variety. This is the largest
category in which geometric invariant theory guarantees that the
semistable set has a good quotient.
All of the arguments in this section use the following trick.
\begin{r}{Construction}
\label{1h}
Let $L_1 , \dots , L_{n+1}$ be ample linearizations. Let $\Delta$ be
the $n$-simplex $\{ (t_i) \in \Q^{n+1} \st \sum t_i = 1 \} $. Then
for any $t = (t_i) \in \Delta$, $L(t) = \bigotimes_i L_i^{t_i}$ is a
fractional linearization on $X$. We refer to the set $\{ L(t) \st t
\in \Delta \}$ as an {\em $n$-simplicial family} of fractional
linearizations.
Put
$$Y = \Pj(\bigoplus_i L_i) = \Proj \sum_{j_i \in \N} H^0(\bigotimes_i
L_i^{j_i}),$$
and let $q: Y \to X$ be the projection. Then $G$ acts naturally on
$\bigoplus_i L_i$, hence on $Y$ with a canonical linearization on the
ample bundle $\co(1)$. Likewise, the $n$-parameter torus $T = \{\la
\in \k^{n+1} \st \prod_i \la_i = 1 \}$ acts on $\bigoplus_i L_i$ by
$\la(u_i) = (\la_i u_i)$, and hence on $Y$. This $T$-action commutes
with the $G$-action. Moreover, since it comes from $\bigoplus_i L_i$,
it too is linearized on $\co(1)$. But this obvious linearization is
not the only one. Indeed, any $t \in \Delta$ determines a fractional
character of $T$ by $t(\la) = \prod_i \la_i^{t_i}$; then $\la(u_i) =
(t^{-1}(\la) \la_i u_i)$ determines a fractional linearization
depending on $t$. This gives an $n$-simplicial family $M(t)$ of
fractional linearizations on $\co(1)$ of the $T$-action on $Y$, each
compatible with the canonical linearization of the $G$-action. In
other words, $M(t)$ is a family of fractional linearizations of the $G
\times T$-action on $Y$. Let $Y^{ss}(t)$ be the semistable set for
this action and linearization, and let $Y^{ss}(G)$ be the semistable
set for the $G$-action alone. For any $t$, $Y^{ss}(t) \subset
Y^{ss}(G)$.
With respect to $M(t)$, $T$ acts trivially on $H^0(\bigotimes_i
L_i^{j_i})$ if and only if $j_i = m t_i$ for some fixed $m$. Hence
the subalgebra of $T$-invariants is $\sum_m H^0 ((\bigotimes_i
L_i^{t_i})^m)$. The quotient $Y \mod T(t)$ is therefore $X$, but with
the residual $G$-action linearized by $L(t)$. Consequently $(Y \mod
T(t)) \mod G = X \mod G(t)$, the original quotients of interest.
Moreover, $X^{ss}(t) = q(Y^{ss}(t))$.
Let $Z$ be the quotient $Y \mod G$ with respect to the canonical
linearization defined above, and let $p: Y^{ss}(G) \to Z$ be the
quotient. Then the $M(t)$ descend to an $n$-simplicial family $N(t)$
of fractional linearizations on $\co(1)$ of the residual $T$-action on
$Z$, and $Y^{ss}(t) = q^{-1}(Z^{ss}(t))$. When two group actions
commute, the order of taking the quotient does not matter, so $(Y \mod
T(t)) \mod G = (Y \mod G) \mod T(t) = Z \mod T(t)$. So we have
constructed a variety $Z$, acted on by a torus $T$, and a simplicial
family $N(t)$ of fractional linearizations on $\co(1)$, such that $Z
\mod T(t) = X \mod G(t)$. Moreover, $X^{ss}(t) = q(p^{-1}(Z^{ss}(t)))$.
\end{r}
As a first application of this construction, let us prove the result
asserted in the last section.
\pf\ of \re{1b}. The result is relatively easy in the case where $X =
\Pj^n$ and $G$ is a torus $T$. Indeed, the $T$-effective cone is
globally polyhedral, as is each chamber; for details see \cite{bp,
toric}. In the general case, choose a locally finite collection of
simplices in $f_\Q^{-1}\A_\Q \subset \NS^G_\Q$ such that every vertex
is in $\NS^G$, and for every $L \in f_\Q^{-1}\A_\Q$, some fractional
power $L^m$ is in one of the simplices. By the homogeneity property
mentioned just before the statement of \re{1b}, it suffices to prove
the statement analogous to \re{1b} where $f_\Q^{-1}\A_\Q$ is replaced
by the simplex parametrizing each of these families. The construction
\re{1h} applies, so there exists $Z \subset \Pj^n$ and a simplicial
family $N(t)$ in $\NS^T(\Pj^n)$ such that $X \mod G(t) = Z \mod T(t)$
for all $t \in \Delta$. The conclusions of the theorem are preserved
by restriction to a $T$-invariant subvariety, so they hold for $Z$ and
$Z \mod T(t)$, and hence for $X$ and $X \mod G(t)$. \fp \bl
The rest of this section and all of \S\S4 and 5 will refer to the
following set-up. Let $X$ and $G$ be as before. Let $L_+$ and $L_-$
be ample linearizations such that, if $L(t) = L_+^t L_-^{1-t}$ for $t
\in [-1,1]$, there exists $t_0 \in (-1,1)$ such that $X^{ss}(t) =
X^{ss}(+)$ for $t > t_0$ and $X^{ss}(t) = X^{ss}(-)$ for $t < t_0$.
For example, \re{1b} implies that this is the case if $X$ is normal
and projective, $L_\pm$ are in adjacent chambers, and the line segment
between them crosses a wall only at $L(t_0)$. Even in the normal
projective case, however, there are other possibilities; for example,
$L_\pm$ could both lie in the same wall, or $L(t_0)$ could lie on the
boundary of $\E^G_\Q$. In future, $L(t_0)$ will be denoted $L_0$. \bl
The following lemma shows how to globalize the results of \S1 within
this set-up. Suppose $T \cong \kst$ acts on $X$, and let $\si
\in H^0(X, L_0^n)^T$ for some $n$, so that $X_\si \subset
X^{ss}(0)$ is a $T$-invariant affine.
\begin{s}{Lemma}
\label{1e}
Suppose $f(L_-) = f(L_+)$. Then \mbox{\rm (a)} $X^{ss}(\pm) \subset
X^{ss}(0)$; \mbox{\rm (b)} $X_\si^{ss}(0) = X_\si \cap X^{ss}(0)$; and
\mbox{\rm (c)} $X_\si^{ss}(\pm) = X_\si \cap X^{ss}(\pm)$; so there is
a natural commutative diagram
$$\begin{array}{ccc}
X_\si \mod \pm & \emb & X \mod \pm \vspace{.7ex} \\
\down{} & & \down{} \vspace{.7ex} \\
X_\si \mod 0 & \emb & X \mod 0
\end{array} $$
whose rows are embeddings.
\end{s}
\pf. Put $R_m = H^0(X, L_0^m)$, so that $X = \Proj \bigoplus_{m \in
\N}R_m$, and let $R_m = \bigoplus_{n \in \Z} R_{m,n}$ be the weight
decomposition for the $\kst$-action. Suppose $x \in X^{ss}(+) \sans
X^{ss}(0)$. Then for $m>0$, every element of $R_{m,0}$ vanishes at
$x$. Since $\bigoplus_m R_m$ is finitely-generated, this implies
that, for $m/n$ large enough, every element of $R_{m,n}$ vanishes at
$x$. But then there exists $t>t_0$ such that $x \notin X^{ss}(t)$,
contradicting the set-up. The proof for $X^{ss}(-)$ is similar. This
proves (a).
Without loss of generality suppose $\si \in H^0(X, L_0)$. Then $\si
\in R_{1,0}$ and $X_\si = \Spec (\si^{-1}R)_0$. Since $\bigoplus_m
R_{m,0}$ is finitely-generated, for $m$ large the map $R_{m,0} \to
(\si^{-1} R)_{0,0}$ given by dividing by $\si^m$ is surjective. But
$R_{m,0} = H^0(X, L_0^m)^T$ and $(\si^{-1} R)_{0,0} = H^0(X_\si,
L_0^m)^T$ (the latter since $L_0^m$ is trivial on $X_\si$), so this
implies that $X_\si^{ss}(0) = X_\si \cap X^{ss}(0)$, hence that $X_\si
\mod 0$ embeds in $X \mod 0$. This proves (b).
Without loss of generality take $L_+$ to be $L_0$ twisted by the
fractional character $\la \mapsto \la^{1/p}$ for $p$ large. Since $R$ is
finitely-generated, $R_{m,n} \to (\si^{-1} R)_{0,n}$ is
surjective for $m/n$ large. But for $m = np$, $R_{m,n} = H^0(X,
L_+^m)^T$ and $(\si^{-1} R)_{0,n} = H^0(X_\si, L_+^m)^T$,
so this implies that $X_\si^{ss}(+) = X_\si \cap X^{ss}(+)$,
hence that $X_\si \mod +$ embeds in $X \mod +$. The case
of $L_-$ is similar. This proves (c). \fp
Hence, in studying $\kst$-quotients where $L_+ \cong L_-$ as bundles,
we may work locally, using the methods of \S1.
\begin{s}{Theorem}
\label{1f}
If $X \mod G(+)$ and $X \mod G(-)$ are both nonempty, then the
morphisms $X \mod G(\pm) \to X \mod G(0)$ are proper and birational.
If they are both small, then the rational map $X \mod G(-) \dasharrow X
\mod G(+)$ is a flip with respect to $\co(1) \to X \mod G(+)$.
\end{s}
Again, this could be proved directly, by first examining the stable
sets to show birationality, then applying Kempf's descent lemma to the
linearization $L_+$. But again, we will use the trick.
\pf\ of \re{1f}. Perform the construction of \re{1h} on $L_+$ and
$L_-$. This gives a variety $Z$ with an action of $T \cong \kst$ and
a family $N(t)$ of fractional linearizations with $f_\Q(N(t))$ constant
such that $Z \mod T(t) = X \mod G (t)$. The whole statement is local
over $X \mod G(0)$, so by \re{1e} it suffices to prove it for affines
of the form $Z_\si$, with the $T$-action and fractional
linearizations $N(t)$. But this is the case considered in \S1, so
\re{2h} and \re{2i} complete the proof. \fp
There is a converse to \re{1f} analogous to \re{2o}, which we leave to
the reader.
\begin{r}{Application}
For an application, suppose that $X$ is normal and projective. Choose
any nonzero $M \in \NS_\Q X$, let $L(t) = L \otimes M^t$ and consider
the ray $\{ L(t) \st t \geq 0 \} \subset \NS^G_\Q(X)$. By \re{1b},
the quotient $X \mod G(t)$ is empty except for $t$ in some bounded
interval $[0,\omega]$, and this interval is partitioned into finitely
many subintervals in whose interior $X \mod G(t)$ is fixed. But when
a critical value $t_0$ separating two intervals is crossed, there are
morphisms $X \mod G(t_\pm) \to X \mod G(t_0)$, which by \re{1f} are
birational except possibly at the last critical value $\omega$. Since
the fractional polarization on $X \mod G(t)$ is the image of $L
\otimes M^{-t}$ in the natural descent map $\NS^G_\Q(X) \to \NS_\Q(X
\mod G(t))$, the descents of $M$ to $\Q$-Cartier divisor classes on $X
\mod G(\pm)$ are relatively ample for each morphism $X \mod G(t_+) \to
X \mod G(t_0)$, and relatively negative for each morphism $X \mod
G(t_-) \to X \mod G(t_0)$. So suppose that each $X \mod G(t_+) \to X
\mod G(t_0)$ is small when $X \mod G(t_-) \to X \mod G(t_0)$ is small,
and that each $X \mod G(t_+) \to X \mod G(t_0)$ is an isomorphism when
$X \mod G(t_-) \to X \mod G(t_0)$ is divisorial. It then follows that
the finite sequence of quotients $X \mod G(t)$ runs the $M$-minimal
model programme \cite[(2.26)]{ko} on $X \mod G(L)$, where by abuse of
notation $M$ denotes its image in the descent map.
\end{r}
For some $d > 0$, the ideal sheaves $\langle H^0(X, L_\pm^{nd})^G
\rangle$ and $\langle H^0(X, L_\pm^{d})^G\rangle^n$ on $X$ are equal
for all $n \in \N$. For such a $d$, let $\ci^\pm = \langle H^0(X,
L_\mp^{d})^G \rangle$ (note the reversal of sign), and let $\ci^\pm
\mod G(\pm)$ be the corresponding sheaves of invariants on $X \mod G
(\pm)$. Also let $(\ci^+ + \ci^-) \mod G(0)$ be the sheaf of
invariants of the ideal sheaf $\ci^+ + \ci^-$ on $X \mod G(0)$.
\begin{s}{Theorem}
\label{1g}
Suppose $X \mod G(+)$ and $X \mod G(-)$ are both nonempty. Then the
pullbacks of $(\ci^+ + \ci^-) \mod G(0)$ by the morphisms $X \mod
G(\pm) \to X \mod G(0)$ are exactly $\ci^\pm \mod G(\pm)$, and the
blow-ups of $X \mod G(\pm)$ at $\ci^\pm \mod G(\pm)$, and of $X \mod
G(0)$ at $(\ci^+ + \ci^-) \mod G(0)$, are all naturally isomorphic to
the irreducible component of the fibred product $X \mod G(-) \times_{X
\mod G(0)} X \mod G(+)$ dominating $X \mod G(0)$.
\end{s}
\pf. Construct a variety $Z$ as in the proof of \re{1f}.
Notice that for $d$ large, since $\ci^\pm_X = \langle H^0(X,
L_\pm^d)^G \rangle$ on $X$ and $\ci^\pm_Z = \langle H^0(Z, N_\pm^d)^T
\rangle$ on $Z$, the pullbacks of both $\ci_X^\pm$ and $\ci_Z^\pm$ to
$Y$ are $\ci_Y^\pm = \langle H^0(Y, M_\pm^d)^{G \times T} \rangle$.
Hence $\ci_X^\pm$ and $\ci_Z^\pm$ have the same sheaves of invariants
on the quotients $Z \mod T(t) = X \mod G (t)$. It therefore suffices
to prove the statement for $Z$ and $N_\pm$. All statements are local
over $Z \mod T(0)$, so by \re{1e} it suffices to prove them for
affines of the form $Z_\si$. But this is the case considered in
\S1, so \re{2b} and \re{2d} complete the proof. \fp \bl
\bit{The smooth case: strong results}
In the next two sections we seek to generalize the other two main
results of \S1, \re{2j} and \re{2k}. Indeed, we will give two
different generalizations of each. The generalizations in \S4 make
fairly strong hypotheses, and prove that, as in \S1, $X^\pm \mod
G(\pm)$ are locally trivial over $X^0 \mod G(0)$. Moreover, the
proofs are quite easy using the tools already at hand. Those in \S5
relax the hypotheses somewhat, but conclude only that $X^\pm \mod
G(\pm)$ are locally trivial in the \'etale topology. The proofs
therefore require \'etale covers and are more difficult; in fact we
confine ourselves to a sketch of the \'etale generalization of
\re{2k}.
Let $X$, $G$, and $L_\pm$ be as in \S3. As in \re{1h}, let $Y =
\Pj(L_+ \oplus L_-)$, let $T$ be the torus acting on $Y$, let $Z = Y
\mod G$, and let $p: Y^{ss}(G) \to Z$ be the quotient morphism. Fix
the isomorphism $T \cong \kst$ given by projection on the first
factor. Define $Y^\pm$, $Y^0$, $Z^\pm$, and $Z^0$ similarly to
$X^\pm$ and $X^0$. Also let $i_\pm : X \to Y$ be the embeddings given
by the sections at 0 and $\infty$. Write $q : Y \to X$ for the
projection as before, but let $\pi$ denote the restriction of $q$ to
$Y \sans (i_+(X) \cup i_-(X))$. So in particular $\pi^{-1}(X)$ denotes
$Y \sans (i_+(X) \cup i_-(X))$ itself.
\begin{s}{Lemma}
\label{2r}
$X^{ss}(\pm) \subset X^{ss}(0)$.
\end{s}
\pf. This is true for $Z$ by \re{1e}(a), but $X^{ss}(\pm) =
q(p^{-1}(Z^{ss}(\pm)))$ and $X^{ss}(0) = q(p^{-1}(Z^{ss}(0)))$. \fp
\begin{s}{Lemma}
\label{1j}
\mbox{\rm (a)} $i_\pm(X) \cap Y^{ss}(0) = \emptyset} \def\dasharrow{\to$; \mbox{\rm (b)}
$i_\pm(X) \cap Y^{ss}(G) = i_\pm(X^{ss}(\pm))$; \mbox{\rm (c)}
$\pi^{-1}(X) \cap Y^{ss}(0) = \pi^{-1}(X) \cap Y^{ss}(G) =
\pi^{-1}(X^{ss}(0))$.
\end{s}
\pf. For $i_\pm(x)$ to be in $Y^{ss}(0)$, it must certainly be
semistable for the $T$-action on the fibre $q^{-1}(x) = \Pj^1$. But
in the fractional linearization $M_0$, $T$ acts with nontrivial weight
on both homogeneous coordinates of $\Pj^1$, so any invariant section
of $\co(n)$ for $n > 0$ must vanish both at $0$ and $\infty$. Hence
$i_\pm(x)$ are unstable, which proves (a). However, for $i_\pm(x)$ to
be in $Y^{ss}(G)$ requires only that the section of $\co(n)$ which is
nonzero at $x$ be $G$-invariant. Pushing down by $q$ shows that
$$H^0(Y, \co(n)) = H^0(X, \bigoplus_{j=0}^n L_+^j \otimes L_-^{n-j}) =
\bigoplus_{j=0}^n H^0(X, L_+^j \otimes L_-^{n-j}),$$
and a section of $\co(n)$ is nonzero at $i_\pm(x)$ if and only if its
projection on $H^0(X, L_\pm^n)$ is nonzero at $x$. Hence $i_\pm(x)
\in X^{ss}(G)$ if and only if $x \in X^{ss}(\pm)$, which proves (b).
With respect to the fractional linearization $M(t)$, the $T$-invariant
subspace in the above decomposition consists of that $H^0(X, L_+^j
\otimes L_-^{n-j})$ such that $L_+^j \otimes L_-^{n-j}$ is a power of $L(t)$,
and an invariant section is nonzero on $\pi^{-1}(x)$ if and only if
the corresponding element of $H^0(X, L_+^j \otimes L_-^{n-j})$ is nonzero at
$x$.
Hence there is a $G$-invariant section of some $\co(n)$ non-vanishing
on $\pi^{-1}(x)$ if and only if there is a $G$-invariant section of
some $L(t)^n$ non-vanishing at $x$; this implies $\pi^{-1}(x) \subset
Y^{ss}(G)$ if and only if $x \in \cup_t X^{ss}(t)$, which equals
$X^{ss}(0)$ by \re{2r}. On the other hand, $x \in X^{ss}(0)$ if and
only if there is a $G$-invariant section of some $L_0^n$ non-vanishing
at $x$, and hence a $G \times T$-invariant section of some $M_0^n$
non-vanishing on $\pi^{-1}(x)$, that is, $\pi^{-1}(x) \subset
Y^{ss}(0)$. This proves (c). \fp \bl
Let $X^\pm$ and $X^0$ be the intersections with $X^{ss}(0)$ of the
supports of the sheaves $\ci^\pm$ and $\ci^+ + \ci^-$, defined as in
\S3. Note that this generalizes the definitions of \S1. Indeed,
\beqas
X^\pm & = & X^{ss}(0) \sans X^{ss}(\mp); \\
X^0 & = & X^{ss}(0) \sans (X^{ss}(+) \cup X^{ss}(-)).
\eeqas
\begin{s}{Lemma}
\label{1u}
\mbox{\rm (a)} $\pi^{-1}X^\pm = Y^\pm = p^{-1}Z^\pm$; \mbox{\rm (b)}
$\pi^{-1}X^0 = Y^0 = p^{-1}Z^0$.
\end{s}
\pf. These follow immediately from $\pi^* \ci^\pm_X = \ci^\pm_Y = p^*
\ci^\pm_Z$. \fp \bl
Choose $x \in X^0$; throughout this section, we will assume the
following.
\begin{r}{Hypothesis}
\label{1y}
Suppose that $X$ is smooth at $x \in X^0$, that $G \cdot x$ is closed
in $X^{ss}(0)$, and that $G_x \cong \kst$.
\end{r}
Note that if $G_x \cong \kst$ for {\em all} $x \in X^0$, then an orbit
in $X^0$ cannot specialize in $X^0$, so it is closed in $X^0$ and
hence in $X^{ss}(0)$. So the second part of the hypothesis is
redundant in this case. The third part is necessary, as the
counterexample \re{1r} will show. But it is always true when $G$ is a
torus or when $G$ acts diagonally on the product of its flag variety
with another variety \cite{dh}. \bl
Since $x \in X^{ss}(0)$, $G_x$ acts trivially on $(L_0)_x$. If it
acts nontrivially on $(L_+)_x$, requiring it to act with some negative
weight $v_+ < 0$ fixes an isomorphism $G_x \cong \kst$. It then acts
on $(L_-)_x$ with some positive weight $v_- > 0$. To obtain the
first, stronger generalizations, assume that these two weights are
coprime: $(v_+, v_-) = 1$. When $X$ is normal and projective and
$L_\pm$ are in adjacent chambers, this additional hypothesis can be
interpreted as follows. The weight of the $G_x$-action defines a
homomorphism $\rho: \NS^G \to \Z$, and $L_\pm$ can be chosen within
their chambers to satisfy this hypothesis, and the conditions of the
set-up, if and only if $\rho$ is surjective. Again, this is always
true when $G$ is a torus or when $G$ acts diagonally on the product of
its flag variety with another variety.
If $(v_+, v_-) = 1$, then $\pi^{-1}(x)$ is contained in a
$G$-orbit, so $p(\pi^{-1}(x))$ is a single point in $Z$.
\begin{s}{Lemma}
\label{1v}
If \re{1y} holds and $(v_+, v_-) = 1$, then $L_\pm$ can be chosen so
that $G$ acts freely on $Y$ at $\pi^{-1}(x)$ and $Z$ is smooth at
$p(\pi^{-1}(x))$.
\end{s}
\pf. Since $G_x$ acts on $(L_\pm)_x$ with weights $v_\pm$, there
exist positive powers of $L_+$ and $L_-$ whose weights add to 1.
Replace $L_\pm$ by these powers. Then $G_x$ acts freely on $(L_+
\otimes L_-^{-1})_x \sans 0$. But this is exactly $\pi^{-1}(x)$, so
$G$ acts freely on $Y$ at $\pi^{-1}(x)$. To show that $Z$ is smooth
at $p(\pi^{-1}(x))$, it therefore suffices to show that $\pi^{-1}(x)
\subset Y^s(G)$, that is, that the $G$-orbit of $\pi^{-1}(x)$ is
closed in $Y^{ss}(G)$. But if $y \in Y^{ss}(G)$ is in the closure of
$G \cdot \pi^{-1}(x)$, then $y \notin i_\pm(X)$ by \re{1j}(b) and
\re{1u}(b), so $y \in \pi^{-1}(x')$ for some $x' \in X^0$ by
\re{1j}(c). Then $x'$ is in the closure of $G \cdot x \subset X^0$,
so by \re{1y}, $x' \in G \cdot x$ and hence $y \in G \cdot
\pi^{-1}(x)$. \fp
\begin{s}{Proposition}
\label{1w}
If \re{1y} holds and $(v_+, v_-) = 1$, then \mbox{\rm (a)} $X^0$ is
smooth at $x$; \mbox{\rm (b)} on a neighbourhood of $x$ in $X^0$,
there exists a vector bundle $N$ with $\kst$-action, whose fibre at
$x$ is naturally isomorphic to $N_{G \cdot x / X}$; \mbox{\rm (c)} the
bundle $N^0$ of zero weight spaces of $N$ is exactly the image of
$TX^0$ in $N$; \mbox{\rm (d)} the bundles $N^\pm$ of positive and
negative weight spaces of $N$ are naturally isomorphic to $N_{X^0 /
X^\pm}$.
\end{s}
\pf. By \re{2e}(a), \re{1u}(b) and \re{1v}, $Z^0$ is smooth at
$p(\pi^{-1}(x))$, and $Y^0$ is locally a principal $G$-bundle over
$Z^0$. Hence $Y^0$ is smooth at $\pi^{-1}(x)$, so $X^0$ is smooth at
$x$. The bundle $N_Z$ is just $TZ|_{Z^0}$, so define $N_Y = p^* N_Z$.
This is acted upon by $\kst$, so by Kempf's descent lemma \cite{dn}
descends to a bundle $N_X$ which has the desired property. This
proves (b); the proofs of (c) and (d) are similar, using \re{2e}(c)
and (d). \fp \bl
As in \S1, let $w_i^\pm \in \Z$ be the weights of the $\kst$-actions
on $N^\pm$.
\begin{s}{Theorem}
\label{2n}
If \re{1y} holds and $(v_+, v_-) = 1$, then over a neighbourhood of
$x$ in $X^0 \mod G(0)$, $X^\pm \mod G(\pm)$ are locally trivial
fibrations with fibre the weighted projective space $\Pj(|w_i^\pm|)$.
\end{s}
\pf. This now follows immediately from \re{2j} and \re{1v}. \fp \bl
If $X \mod G(\pm)$ are both nonempty, then $X^\pm \mod G(\pm)$ are the
supports of the blow-up loci of \re{1g}. But if $X \mod G(-) =
\emptyset} \def\dasharrow{\to$, then $X^+ \mod G(+) = X \mod G(+)$ and $X^0 \mod G(0) =
X \mod G(0)$, so \re{2n} says the natural morphism $X \mod G(+) \to X
\mod G(0)$ is a locally trivial weighted projective fibration. \bl
If moreover all $w_i^\pm = \pm w$ for some $w$, then for any
linearization $L$ such that $L_x$ is acted on by $G_x$ with weight
$-1$, the bundles $N^\pm \otimes L^{\pm w}$ are acted upon trivially
by all stabilizers. So by Kempf's descent lemma \cite{dn} they
descend to vector bundles $W^\pm$ over a neighbourhood of $x$ in $X^0
\mod G(0)$.
\begin{s}{Theorem}
\label{1x}
Suppose that \re{1y} holds, that $(v_+, v_-) = 1$, and that all
$w_i^\pm = \pm w$ for some $w$. Then over a neighbourhood of $x$ in
$X^0 \mod G(0)$, $X^\pm \mod G(\pm)$ are naturally isomorphic to the
projective bundles $\Pj W^\pm$, their normal bundles are naturally
isomorphic to $\pi_\pm^* W^\mp(-1)$, and the blow-ups of $X \mod
G(\pm)$ at $X^\pm \mod G(\pm)$, and of $X \mod G(0)$ at $X^0 \mod
G(0)$, are all naturally isomorphic to the fibred product $X \mod G(-)
\times_{X \mod G(0)} X \mod G(+)$.
\end{s}
\pf. First notice that, although $W^\pm$ depend on the choice of $L$,
the projectivizations $\Pj W^\pm$, and even the line bundle $\co(1,1)
\to \Pj W^+ \times_{X^0 \mod G(0)} \Pj W^-$, are independent of $L$.
Now on $Z$, taking $L \cong \co$ yields $W^\pm = N^\pm$. But $N^\pm_Y
= p^* N^\pm_Z$, so taking $L = p^* \co$ on $Y$, with the induced
linearization, yields $W^\pm_Y = W^\pm_Z$. On the other hand,
$N^\pm_Y = \pi^* N^\pm_X$ also, so for another choice of $L$ on $Y$,
$W^\pm_Y = W^\pm_X$. Hence $\Pj W^\pm_X \cong \Pj W^\pm_Z$, and the
line bundles $\co(1,1) \to \Pj W^+_X \times_{X^0 \mod G(0)} \Pj W^-_X$
and $\co(1,1) \to \Pj W^+_Z \times_{Z^0} \Pj W^-_Z$ correspond under
this isomorphism; pushing down and taking duals, the bundles
$\pi_\pm^* W^\mp_X (-1) \to \Pj W^\pm_X$ and $\pi_\pm^* W^\mp_Z (-1)
\to \Pj W^\pm_Z$ also correspond. On the other hand, by \re{1u}(a)
$X^\pm \mod G(\pm) = Z^\pm \mod T(\pm)$. The theorem then follows
from \re{2k} together with \re{1e}. \fp \bl
The hypothesis on $w_i^\pm$ is most easily verified as follows.
\begin{s}{Proposition}
\label{1z}
If every 0-dimensional stabilizer is trivial near $x$, then all
$w_i^\pm = \pm 1$.
\end{s}
\pf. If not all $w_i^\pm = \pm 1$, then by \re{2e}(b) there is a point
$z \in Z$ with proper nontrivial stabilizer $T_z$ such that
$p(\pi^{-1}(x))$ is in the closure of $T \cdot z$. Then any $y \in
p^{-1}(z)$ has nontrivial 0-dimensional stabilizer $(G \times T)_y$,
and the closure of $(G \times T) \cdot y$ contains $\pi^{-1}(x)$. But
then $G_{\pi(y)} \cong (G \times T)_y$, and the closure of $G \cdot
\pi(y)$ contains $x$. \fp \bl
For most applications, the hypothesis \re{1y} will hold for all $x \in
X^0$. Then the conclusions of \re{2n} and \re{1x} hold globally,
because they are natural. Notice, however, that if $X^0$ is not
connected, then the $w_i^\pm$ need be only locally constant.
\bit{The smooth case: \'etale results}
If $(v_+, v_-) \neq 1$, however, then the proof of \re{1v} fails, and
$X^\pm \mod G(\pm)$ need not be locally trivial over $X^0 \mod G(0)$,
even if \re{1y} holds: see \re{2p} for a counterexample. But they
will be locally trivial in the \'etale topology. The proof uses the
Luna slice theorem. The first step, however, is to check that $v_\pm$
are always nonzero. As always, let $L_\pm$ and $L_0$ be as in the
set-up of \S3.
\begin{s}{Lemma}
\label{1t}
If \re{1y} holds, then $G_x$ acts nontrivially on $(L_\pm)_x$.
\end{s}
\pf. If $G_x$ acts trivially on $L_+$ (and hence on $L_-$), then the
embedding $\kst = \pi^{-1}(x) \subset Y$ descends to an embedding
$\kst \subset Z$. This is completed by two points, which must come
from two equivalence classes of $G$-orbits in $Y^{ss}(G)$. These
semistable orbits cannot be $G \cdot (i_\pm(x))$, since $i_\pm(x)
\notin Y^{ss}(G)$ by \re{1j}(a) and the definition of $X^0$. Hence
our two classes of semistable $G$-orbits must be contained in
$\pi^{-1}(\overline{G \cdot x} \sans G \cdot x)$. By \re{1j}(b) and
(c) their images in $\pi$ are in $X^{ss}(0)$. But they are also in
the closure of $G \cdot x$, which contradicts \re{1y}. \fp
Again, requiring $G_x$ to act on $(L_+)_x$ with negative weight $v_+ <
0$ fixes an isomorphism $G_x \cong \kst$. It then acts on $(L_-)_x$
with positive weight $v_- > 0$. We no longer require $(v_+, v_-) =
1$, but assume instead the following.
\begin{r}{Hypothesis}
\label{2s}
Suppose that either $\chr \k = 0$ or $(v_+, v_-)$ is coprime to
$\chr \k$.
\end{r}
Now choose $y \in \pi^{-1}(x)$, and let $S = (G \times T)_y$.
\begin{s}{Lemma}
\label{1o}
If \re{1y} holds, then there is a fixed isomorphism $S \cong
\kst$, and $G_y = S \cap G$ is a proper subgroup such that $S / G_y
\cong T$ naturally. Moreover, if \re{2s} holds, then $L_\pm$ may be
chosen so that $G_y$ is reduced.
\end{s}
\pf. Since $\pi$ is the quotient morphism for the $T$-action, $S
\subset G_x \times T = \kst \times T$; indeed, it is the subgroup
acting trivially on $\pi^{-1}(x)$. Since by \re{1t} $\kst$ acts
nontrivially, and $T$ obviously acts with weight 1, this has a fixed
isomorphism to $\kst$, and its intersection with $\kst \times 1$ is a
proper subgroup having the desired property.
If $\chr \k = 0$, then $G_y$ is certainly reduced. Otherwise, let $v
= (v_+, v_-)$, and replace $L_+$ and $L_-$ with positive powers so
that $G_x \cong \kst$ acts on $\pi^{-1}(x) = (L_+ \otimes L_-^{-1})_x$
with weight $v$. Then $G_y = \Spec \k[z]/ \langle z^v - 1 \rangle$,
which is reduced if $v$ is coprime to $\chr \k$. \fp \bl
Notice that for any $y \in \pi^{-1}(x)$, $(G \times T) \cdot y = G
\cdot y$. Together with lemma \re{1o}, this implies that any
$S$-invariant complement to $T_y (G \times T) \cdot y$ in $T_y Y$ is
also a $G_y$-invariant complement to $T_y (G \cdot y)$ in $T_y Y$. It
follows from the definition of the Luna slice \cite{luna,mf} that a
slice for the $(G \times T)$-action at $y$ is also a slice for the
$G$-action at $y$. Luna's theorem then implies that there exists a
smooth affine $U \subset Y$ containing $y$ and preserved by $S$, and a
natural diagram
$$\begin{array}{ccccc}
G \times U & \lrow & G
\times_{G_y} U & \lrow & Y^{ss}(G) \vspace{.7ex} \\
\down{} & & \down{} & & \down{} \vspace{.7ex} \\
U & \lrow & U / G_y & \lrow & Z,
\end{array} $$
such that the two horizontal arrows on the right are strongly \'etale
with respect to the actions of
$$\begin{array}{ccccc}
G \times S &\lrow & G \times T & \lrow & G \times T \vspace{.7ex} \\
\down{} & & \down{} & & \down{} \vspace{.7ex} \\
S &\lrow & T & \lrow & T.
\end{array} $$
These actions are obvious in every case except perhaps on $G \times
U$; there the $G \times S$-action is given by $(g, s)\cdot(h,u)
= (gh\hat{s}^{-1}, s u)$, where $\hat{s}$ is the
image of $s$ in the projection $S \to G$. Each of these actions
has a 1-parameter family of fractional linearizations, pulled back from the
right-hand column. For any object $V$ in the diagram, define $V^0$
and $V^\pm$ with respect to these linearizations.
\begin{s}{Lemma}
\label{1k}
For every arrow $f: V \to W$ in the diagram, $f^{-1}(W^\pm) = V^\pm$
and $f^{-1}(W^0) = V^0$.
\end{s}
\pf. This is straightforward for the vertical arrows, and for the
morphism $U \to U / G_y$, because they are all quotients by subgroups
of the groups which act. The result for $G \times U \lrow G
\times_{G_y} U$ follows from the commutativity of the diagram. As for
the strongly \'etale morphisms, these are treated as in the proof of
\re{2e}(b). \fp \bl
With this construction, \re{1w} can now be strengthened.
\begin{s}{Proposition}
If \re{1y} and \re{2s} hold, then the conclusions of \re{1w} follow
even if $(v_+, v_-) \neq 1$.
\end{s}
\pf. By \re{1o} and \re{1k}, in a neighbourhood of $\pi^{-1}(x)$ in
$Y$, a point is in $Y^0$ if and only if it has stabilizer conjugate to
$S$. Hence in a neighbourhood of $x$ in $X$, a point is in $X^0$ if
and only if it has stabilizer conjugate to $G_x$. So if $W \subset
X^0$ is the fixed-point set for the $G_x$-action on a neighbourhood of
$x \in X$, then a neighbourhood of $x$ in $X^0$ is precisely the
affine quotient of $W \times G$ by the diagonal action of the
normalizer of $G_x$ in $G$. In particular, this is smooth as claimed
in (a), because $W$ is smooth by \re{2e}(a) and the normalizer acts
freely. Also, its tangent bundle has a natural subbundle consisting
of the tangent spaces to the $G$-orbits. Let $N_X$ be the quotient of
$TX|_{X^0}$ by this subbundle. Then $N_X$ certainly satisfies (b),
and $N_Y = \pi^* N_X$, so by it suffices to prove (c) and (d) for $Y$
with its $G \times T$-action. But (c) and (d) hold for $U$ with its
$S$-action by \re{2e}(c) and (d), hence for $G \times U$ with its $G
\times S$-action since $N_U$ pulls back to $N_{G \times U}$. But the
morphism $G \times_{G_y} U \to Y$ is \'etale, and so is the morphism
$G \times U \to G \times_{G_y} U$, since $G_y$ is reduced. The result
for $Y$ then follows from \re{1k}, since \'etale morphisms are
isomorphisms on tangent spaces. \fp \bl
Let $w_i^\pm \in \Z$ be the weights of the $\kst$-action on $N^\pm$.
\begin{s}{Theorem}
\label{1l}
If \re{1y} and \re{2s} hold, then over a neighbourhood of $x$ in $X^0
\mod G(0)$, $X^\pm \mod G(\pm)$ are fibrations, locally trivial in the
\'etale topology, with fibre the weighted projective space
$\Pj(|w_i^\pm|)$.
\end{s}
As before, if $X \mod G(\pm)$ are both nonempty, then $X^\pm \mod
G(\pm)$ are the supports of the blow-up loci of \re{1g}; but if $X
\mod G(-) = \emptyset} \def\dasharrow{\to$, then \re{1l} says the natural morphism $X
\mod G(+) \to X \mod G(0)$ is a weighted projective fibration, locally
trivial in the \'etale topology.
The proof requires the following lemma.
\begin{s}{Lemma}
\label{1m}
If $\phi: V \to W$ is a strongly \'etale morphism of affine varieties
with $\kst$-action, then $\phi \mod \pm : V \mod \pm \to W \mod \pm$
are \'etale, and $V \mod \pm = {W \mod \pm} \times_{W \mod 0} {V \mod
0}$.
\end{s}
\pf. Say $V = \Spec R$, $W = \Spec S$. The $\kst$-actions induce
$\Z$-gradings on $R$ and $S$, and $V = W \times_{W \mod 0} V \mod 0$
implies $R = S \otimes_{S_0} R_0$. Hence $\bigoplus_{i \in \N} R_{\pm
i} = \bigoplus_{i \in \N} S_{\pm i} \otimes_{S_0} R_0$, which implies
the second statement. Then $\phi \mod \pm$ are certainly \'etale,
since being \'etale is preserved by base change. \fp
\pf\ of \re{1l}. The Luna slice $U$ associated to any $y \in
\pi^{-1}(x)$ is smooth, and for $S = (G \times T)_y$, $U \mod S(t) =
(U / G_y) \mod T(t)$ since $T = S / G_y$ by \re{1o}. But as stated
when $U$ was constructed, $U / G_y$ is strongly \'etale over $Z$, so
by \re{1m} and \re{1e} $U \mod S(t)$ is \'etale over $Z \mod T(t) = X
\mod G(t)$, and $U \mod S(\pm) = X \mod G(\pm) \times_{X \mod G(0)} U
\mod S(0)$. In particular, $U^0 \mod S(\pm) = X^0 \mod G(\pm)
\times_{X^0 \mod G(0)} U^0 \mod S(0)$, which is exactly the pullback
of $X^0 \mod G(\pm)$ by the \'etale morphism $U^0 \mod S(0) \to X^0
\mod G(0)$. The theorem therefore follows from the analogous result
\re{2j} for quotients of smooth affines by $\kst$. \fp
\begin{s}{Counterexample}
\label{1r}
To show that the hypothesis $G_x \cong \kst$ is necessary in \re{1l}.
\end{s}
Let $G$ be any semisimple reductive group, and let $V_+$ and $V_-$ be
representations of $G$. Let $X = \Pj (V_+ \oplus V_- \oplus \k)$, and
let $G \times \kst$ act on $X$, $G$ in the obvious way, and $\kst$
with weights $1,-1,0$. Then $\NS_\Q^{G \times \kst} \cong \Q$, with
two chambers separated by a wall at 0. Moreover $X^\pm = \Pj(V_\pm
\oplus \k)$, so $X^0 = \{ (0,0,1) \}$. But $G_{(0,0,1)} = G \times
\kst$, so the hypothesis is violated. Now $X^\pm \mod \kst (\pm) =
\Pj (V_\pm)$, so $X^\pm \mod (G \times \kst)(\pm) = \Pj (V_\pm) \mod
G$. This certainly need not be a projective space, as the theorem
would predict; see for example the discussion of the case $G = {\rm
PSL(2)}$ in \S6. \fp \bl
Since $G$ acts on $N^\pm$, there are quotients $N^\pm \mod G(\pm)$,
which are fibrations with fibre $\Pj(|w_i^\pm|)$ over a neighbourhood
of $x$ in $X^0 \mod G(0)$, locally trivial in the \'etale topology.
Notice that by \re{1k}, since $N^\pm_V = N_{V^0 / V^\pm}$ for $V = X$,
$Y$, $G \times_{G_y} U$, and $U$,
\beqas
N^\pm_U \mod S(\pm) & = & N^\pm_{G \times U} \mod (G \times S)(\pm) \\
& = & N^\pm_Y \mod (G \times T)(\pm) \times_{Y^0 \mod (G \times
T)(0)} (G \times U)^0 \mod (G \times S)(0) \\
& = & N^\pm_X \mod G(\pm) \times_{X^0 \mod G(0)} U^0 \mod
S(0),
\eeqas
which is the pullback of $N^\pm_X \mod G(\pm)$ by the \'etale morphism
$U^0 \mod S(0) \to X^0 \mod G(0)$. The following result then ought to
be true, but proving it conclusively is rather cumbersome, so we content
ourselves with a sketch.
\begin{s}{Theorem}
\label{1n}
Suppose that \re{1y} and \re{2s} hold, and that all $w_i^\pm = \pm w$
for some $w$. Then $X^\pm \mod G(\pm)$ are naturally isomorphic to
$N^\pm \mod G(\pm)$, and the blow-ups of $X \mod G(\pm)$ at $X^\pm
\mod G(\pm)$, and of $X \mod G(0)$ at $X^0 \mod G(0)$, are all
naturally isomorphic to the fibred product $X \mod G(-) \times_{X \mod
G(0)} X \mod G(+)$.
\end{s}
{\em Sketch of proof}. All the blow-ups and the fibred product are
empty if either $X \mod G(+)$ or $X \mod G(-)$ is empty, so suppose
they are not. Now $X^\pm \mod G(\pm) \to X^0 \mod G(0)$ are covered
in the \'etale topology by $U^\pm \mod S(\pm) \to U^0 \mod S(0)$ by
\re{1m}, and $N^\pm_X \mod G(\pm) \to X^0 \mod G(0)$ are covered in
the \'etale topology by $N^\pm_U \mod S(\pm) \to U^0 \mod S(0)$ by the
remarks above. But the analogous result for $U$ holds by \re{2k}.
The theorem would therefore follow if we could display a morphism
$N^\pm_X \mod G(\pm) \to X^\pm \mod G(\pm)$ compatible with the
\'etale morphisms and the isomorphisms of \re{2k}. Unfortunately,
this is somewhat awkward to construct. One way to do it is to imitate
the argument of \re{1x}, using a bundle of tangent cones with
$\kst$-action over $Z^0$, which is typically in the singular locus of
$Z$. This requires generalizing the Bialynicki-Birula decomposition
theorem to the mildly singular space $Z$, which can still be
accomplished using the Luna slice theorem. \fp \bl
The hypothesis on $w_i^\pm$ is again most easily verified as follows.
\begin{s}{Proposition}
If every 0-dimensional stabilizer is trivial near $x$, then all
$w_i^\pm = \pm 1$.
\end{s}
\pf. If not all $w_i^\pm = \pm 1$, then for each Luna slice $U$ there
is a point $u \in U$ with nontrivial proper stabilizer $S_u$. Then
any $(g,u) \in G \times U$ satisfies $(G \times S)_{(g,u)} \cong S_u$.
Since the morphism $G \times U \to Y$ is \'etale, this implies that
there exists $y \in Y$ with a nontrivial 0-dimensional stabilizer $(G
\times T)_y$. But then $G_{\pi(y)} = (G \times T)_y$. \fp \bl
Again, for most applications, the hypothesis \re{1y}, and hence the
conclusions of \re{1l} and \re{1n}, will hold globally.
\bit{The first example}
In this section we turn to a simple application of our main results,
the much-studied diagonal action of $\PSL{2}$ on the $n$-fold product
$(\Pj^1)^n$. This has $n$ independent line bundles, so it is tempting
to study the quotient with respect to an arbitrary $\co(t_1, \dots,
t_n)$. We will take a different approach, however: to add an $n+1$th
copy of $\Pj^1$, and consider only fractional linearizations on
$(\Pj^1)^{n+1}$ of the form $\co(t,1,1,\dots,1)$. This has the
advantage that it does not break the symmetry among the $n$ factors.
In other words, the symmetric group $S_n$ acts compatibly on
everything, so in addition to $(\Pj^1)^n$, we learn about quotients by
$\PSL{2}$ of the symmetric product $(\Pj^1)^n / S_n = \Pj^n$.
So for any $n>2$, let $(\Pj^1)^n$ be acted on diagonally by $G =
\PSL{2}$, fractionally linearized on $\co(1,1,\dots,1)$. We wish to
study the quotient $(\Pj^1)^n \mod G$. The stability condition for
this action is worked out in \cite[4.16; GIT Ch.\ 3]{n}, using the
numerical criterion. This is readily generalized to an arbitrary
linearization on $X = \Pj^1 \times (\Pj^1)^n = (\Pj^1)^{n+1}$; indeed
for the fractional linearization $\co(t_0, \dots, t_n)$, it turns out
that $(x_j) \in X^{ss}(t_j)$ if and only if, for all $x \in \Pj^1$,
$$\sum_{j=0}^n t_j \, \delta(x,x_j) \leq \sum_{j=0}^n t_j / 2.$$
Moreover, $(x_j) \in X^s(t_j)$ if and only if the inequality is always
strict. We will study the case where $t_0$ is arbitrary, but $t_j =
1$ for $j>0$.
For $t_0 < 1$, it is easy to see that $\Pj^1 \times ((\Pj^1)^n)^s
\subset X^{ss} \subset \Pj^1 \times ((\Pj^1)^n)^{ss}$. So the
projection $X \to (\Pj^1)^n$ induces a morphism $X \mod G(t_0) \to
(\Pj^1)^n \mod G$ whose fibre over each stable point is $\Pj^1$.
Indeed, each diagonal in $X = \Pj^1 \times (\Pj^1)^n$ is fixed by $G$,
so descends to $X \mod G(t_0)$. Hence, over the stable set in
$(\Pj^1)^n \mod G$, $X \mod G(t_0)$ is exactly the total space of the
universal family.
Now because $G = \PSL{2}$, not $\SL{2}$, the bundle
$\co(1,0,0,\dots,0)$ has no bona fide linearization, only a fractional
one. However, $\co(1,1,0,\dots,0)$ does admit a bona fide
linearization, as does $\co(1,1,\dots,1)$ if $n$ is odd. So these
bundles descend to $X \mod G(t_0)$ for $t_0 < 1$, yielding line
bundles whose restriction to each $\Pj^1$ fibre is $\co(1)$. This
implies that, over the stable set in $(\Pj^1)^n \mod G$, $X \mod
G(t_0)$ is a locally trivial fibration. In particular, if $n$ is odd,
it is a fibration everywhere. However, if $n$ is even, there is no
$S_n$-invariant line bundle having the desired property. Hence the
quotient $(\Pj^1 \times \Pj^n) \mod G(t_0) = (X \mod G(t_0)) / S_n$,
though it has generic fibre $\Pj^1$ over $\Pj^n \mod G$, and is
generically trivial in the \'etale topology, is not even locally
trivial anywhere. It is (generically) what is sometimes called a {\em
conic bundle}.
To apply our results to this situation, note first that for numerical
reasons the stability condition only changes when equality can occur
in the inequality above, that is, when $t_0 = n - 2m$ for some integer
$m \leq n/2$. These will be our walls. A point $(x_j) \in X$ is in
$X^0$ for one of these walls if it is semistable for $t = t_0$, but
unstable otherwise. This means there exist points $x, x' \in \Pj^1$
such that
$$t \, \delta(x, x_0) + \sum_{j=1}^n \delta(x,x_j) \geq \frac{t+n}{2} $$
for $t \leq t_0$, and
$$t \, \delta(x', x_0) + \sum_{j=1}^n \delta(x',x_j) \geq \frac{t+n}{2} $$
for $t \geq t_0$, with equality in both if and only if $t = t_0$.
This requires that all $x_j$ be either $x$ or $x'$; indeed, $x_0$ and
exactly $m$ other $x_j$ must be $x$, and the $n-m$ remaining $x_j$
must be $x'$. In particular, this implies $m \geq 0$, so there are
only finitely many walls, as expected. On the other hand, any $(x_j)
\in X$ of this form will belong to $X^{ss}(t_0)$, provided that $x
\neq x'$. So $X^0(t_0)$ consists of ${n \choose m}$ copies of $(\Pj^1
\times \Pj^1) \sans \Delta$.
Hence every point $(x_j) \in X^0(t_0)$ is stabilized by the subgroup
of $G$ fixing $x$ and $x'$, which is isomorphic to $\kst$. So the
hypothesis \re{1y} is satisfied. Moreover, the bundle
$\co(2,0,0,\dots,0)$ is acted on by this $\kst$ with weight 1, so the
strong results of \S4 will apply. Finally, we claim that, even though
$X^0$ is disconnected, the weights $w_i^\pm$ are globally constant,
and are all $\pm 1$. Indeed, it is easy to see that the $w_i^\pm$ are
independent of the component, because the action of the symmetric
group $S_n$ on $X$ commutes with the $G$-action, and acts transitively
on the components of $X^0$. To evaluate $w_i^\pm$, note that each
component is a single orbit, and that setting $x=0$, $x' = \infty$
determines an unique point in this orbit with stabilizer $\{ \diag
(\lambda^{-1}, \lambda) \st \lambda \in \kst \} / \pm 1$. This acts
on $T\Pj^1$ with weight $-1$ at 0, $1$ at $\infty$; so it acts on
$T_{(x_j)}X$ with $m+1$ weights equal to $-1$ and $n-m$ weights equal
to $1$. But it acts on the $G$-orbit $G / \kst$ with one weight equal
to 1 and one equal to $-1$, so $N$ is acted on with $m$ weights $-1$
and $n-m-1$ weights $1$. So the very strongest result \re{1x}
applies. Hence $X^\pm \mod G(\pm)$ are bundles with fibre
$\Pj^{n-m-2}$ and $\Pj^{m-1}$, respectively, over $X^0 \mod G(0)$.
Since this is just ${n \choose m}$ points, $X^\pm \mod G(\pm)$ are
disjoint unions of ${n \choose m}$ projective spaces. Moreover, the
blow-ups of $X \mod G(\pm)$ at $X^\pm \mod G(\pm)$ are both isomorphic
to $X \mod G(-) \times_{X \mod G(0)} X \mod G(+)$.
This does not seem to say much about $(\Pj^1)^n \mod G$ itself, only
about $X \mod G(t)$, which for $t$ small is (at least generically) a
$\Pj^1$-bundle over it. But this is enough to compute quite a lot
(cf.\ \cite{t1}). We content ourselves with just one calculation, of
the Betti numbers of $(\Pj^1)^n \mod G$ and $\Pj^n \mod G$ for $n$
odd, in the case where the ground field is the complex numbers $\C$.
These formulas are originally due to Kirwan \cite{k}.
\begin{s}{Proposition}
For $n$ odd,
$$P_t ((\Pj^1)^n \mod G) = \sum_{m=0}^{(n-1)/2} {n \choose m} \frac{t^{2m} -
t^{2(n-m-1)}}{1-t^4} $$
and
$$P_t (\Pj^n \mod G) = \sum_{m=0}^{(n-1)/2} \frac{t^{2m} -
t^{2(n-m-1)}}{1-t^4}. $$
\end{s}
\pf. Let $t_0 = n - 2m$, and $t_\pm = t_0 \mp 1$. Then the blow-ups
of $X \mod G(\pm)$ at $X^\pm \mod G(\pm)$ are equal. So by the
standard formula for Poincar\'e polynomials of blow-ups,
$$ P_t(X \mod G(-)) - P_t(X^- \mod G(-)) + P_t(E)
= P_t(X \mod G(+)) - P_t(X^+ \mod G(+)) + P_t(E), $$
where $E$ is the exceptional divisor. Cancelling and rearranging
yields
$$ P_t(X \mod G(+)) - P_t(X \mod G(-))
= P_t(X^+ \mod G(+)) - P_t(X^- \mod G(-)). $$
But $X^\pm \mod G(\pm)$ are ${n \choose m}$ copies of $\Pj^{n-m-2}$
and $\Pj^{m-1}$ respectively, so
\beqas
P_t(X \mod G(+)) - P_t(X \mod G(-))
& = & {n \choose m}
\left( \frac{1-t^{2(n-m-1)}}{1-t^2}-\frac{1-t^{2m}}{1-t^2} \right) \\
& = & {n \choose m} \frac{t^{2m}-t^{2(n-m-1)}}{1-t^2}.
\eeqas
Summed over $m$, the left-hand side telescopes, so for $t<1$
$$P_t(X \mod G(t)) = \sum_{m=0}^{(n-1)/2} {n \choose m}
\frac{t^{2m}-t^{2(n-m-1)}}{1-t^2}. $$
But for $n$ odd, this is a $\Pj^1$-bundle over $(\Pj^1)^n \mod G$, and the
Poincar\'e polynomial of any projective bundle splits, so
$$P_t((\Pj^1)^n \mod G) = \sum_{m=0}^{(n-1)/2} {n \choose m}
\frac{t^{2m}-t^{2(n-m-1)}}{1-t^4}, $$
as desired.
As for $\Pj^n \mod G$, it is the quotient of $(\Pj^1)^n \mod G$ by the
action of the symmetric group $S_n$. A result from Grothendieck's
T\^ohoku paper then implies \cite{gr,mac} that $H^*(\Pj^n \mod G, \C)$
is the $S_n$-invariant part of $H^*((\Pj^1)^n \mod G, \C)$. But since
$S_n$ acts on $X \mod G(t)$ for all $t$, the calculation above
actually decomposes $H^*((\Pj^1)^n \mod G, \C)$ as a representation of
$S_n$: the term with coefficient ${n \choose m}$ gives the
multiplicity of the permutation representation induced by the natural
action of $S_n$ on subsets of $\{ 1, \dots, n\}$ of size $m$. The
trivial summand of this representation is exactly one-dimensional, so
the cohomology of $\Pj^n \mod G$ is as stated. \fp \bl
We round off this section by using some of the ideas discussed above
to give the counterexample promised in \S5.
\begin{s}{Counterexample}
\label{2p}
To show that the hypothesis $(v_+, v_-) = 1$ is necessary in \re{2n}.
\end{s}
Let $V$ be the standard representation of $\GL{2}$, and let $W = S^n V
\otimes (\Lambda^2 V)^{-n/2}$, where $S^n V$ is the $n$th symmetric
power for some even $n>2$. Let $X = \Pj (V \oplus V^* \oplus W)$, and
let $\GL{2}$ act on $X$. Then $\NS_\Q^{\GL{2}} \cong \Q$, with two
chambers separated by a wall at 0. The central $\kst \subset \GL{2}$
acts on $V$, $V^*$, and $W$ with weight $1$, $-1$, and $0$
respectively, so $X^+$ is open in $\Pj(V \oplus W)$, $X^-$ is open in
$\Pj(V^* \oplus W)$, and $X^0$ is open in $\Pj W$. By construction,
the scalars $\kst \subset \GL{2}$ act trivially on $\Pj W$ with the
linearization $L_0$, so the action reduces to the action of $\PSL{2}$
on $\Pj^n$ considered above. A generic $x \in \Pj^n$ is stable and is
acted on freely by $\PSL{2}$, so $\GL{2}_x = \kst$. Moreover, it is
stable, so $\GL{2} \cdot x = \PSL{2} \cdot x$ is closed in
$(\Pj^n)^{ss}$ and hence in $X^{ss}(0)$. Therefore \re{1y} holds for
the $\GL{2}$-action at $x$. On the other hand, the hypothesis $(v_+,
v_-) = 1$ cannot be satisfied: the tautological linearization $L_0$ on
$\co(1)$ is acted on with weight 0, and the linearization $L_+$
obtained by tensoring $L_0$ with the character $\det: \GL{2} \to \kst$
is acted on with weight $2$, but together these generate $\Pic^\GL{2}
X$.
Now $X^+ \mod \kst (+) = \Pj V \times \Pj W$. As a variety with
$\PSL{2}$-action, this is exactly $\Pj^1 \times \Pj^n$ as considered
above. The $+$ linearization on $\Pj V \times \Pj W$ corresponds to
the linearization given by $t<1$ on $\Pj^1 \times \Pj^n$, so $X^\pm
\mod \GL{2}(+) = (\Pj V \times \Pj W) \mod \PSL{2}(+) = (\Pj^1 \times
\Pj^n) \mod \PSL{2}(t)$. As mentioned above, this is a conic bundle,
so it is not even locally trivial over $X^0 \mod \GL{2}(0) = \Pj W
\mod \PSL{2}(0) = \Pj^n \mod \PSL{2}$ at $x$. \fp
\bit{Parabolic bundles}
In the last two sections we apply our main results to moduli problems
of vector bundles with additional structure over a curve. Throughout
these sections, $C$ will denote a smooth projective curve over $k$, of
genus $g>0$.
Fix a point $p \in C$. In this section we will study parabolic
bundles of rank $r$ and degree $d$ over $C$, with parabolic structure
at $p$. We refer to \cite{ms,s} for basic definitions and results on
parabolic bundles. However, we insist for simplicity on full flags at
$p$, so the weights $\ell_j \in [0,1)$ are strictly increasing. The
space of all possible weights is therefore parametrized by $$W = \{
(\ell_j) \in \Q^r \st 0 \leq \ell_1 < \ell_2 < \cdots < \ell_r < 1
\}.$$
There are several constructions of the moduli space $M(\ell_j)$ of
parabolic bundles semistable with respect to $(\ell_j)$. The one best
suited for our purposes is due to Bhosle \cite{bho}, following
Gieseker \cite{g}; so we first review his construction, then hers.
Suppose without loss of generality that $d > > 0$, and let $\chi = d +
r(1-g)$. Let $\Quot$ be the Grothendieck Quot scheme \cite{quot}
parametrizing quotients $\phi: \co_C^\chi \to E$, where $E$ has
Hilbert polynomial $\chi +ri$ in $i$, and let $\co^\chi \to \be$ be
the universal quotient over $\Quot \times C$. Let $R \subset \Quot$
be the smooth open subvariety consisting of locally free sheaves $E$
such that $H^0(\co^\chi) \to H^0(E)$ is an isomorphism, and let
$R^{ss}$ be the subset corresponding to semistable bundles. For $d$
large, every semistable bundle of rank $r$ and degree $d$ is
represented by a point in $R$. Let $Z$ be the bundle over $\Pic^d C$,
constructed as a direct image, with fibre $\Pj \Hom (H^0(\La^r
\co^\chi), H^0(M))$ at $M$. The group $G = \SL{\chi}$ acts on $R$ and
$Z$, and there is a natural $G$-morphism $T: R \to Z$, and a
linearization $L$ on $Z$, such that $T^{-1}Z^{ss}(L) = R^{ss}$.
Moreover, the restriction $T: R^{ss} \to Z^{ss}(L)$ is finite. The
existence of a good quotient of $R^{ss}$ by $G$ then follows from a
lemma \cite[Lemma 4.6]{g} which states that if a set has a good
$G$-quotient, then so does its preimage by a finite $G$-morphism.
This quotient is the moduli space of semistable bundles on $C$.
To construct the moduli space of semistable parabolic bundles in an
analogous way, let $\tilde{R}$ be the bundle $\Fl \be|_{R \times \{ p
\} }$ of full flags in $\be_p$. This parametrizes a family of
quasi-parabolic bundles; for $d$ large, any bundle which is semistable
for some weights $(\ell_j)$ is represented by a point in $\tilde{R}$.
Let $\tilde{R}^{ss}(\ell_j)$ be the subset corresponding to parabolic
bundles semistable with respect to $(\ell_j)$. Also let $G_r$ be the
product of Grassmannians $\prod_{j=1}^r \Gr(\chi - j,\chi)$. Then $G$
acts on $\tilde{R}$, $Z$, and $G_r$, and there is a $G$-morphism
$\tilde{T}: \tilde{R} \to Z \times G_r$, and a family $L(\ell_j)$ of
fractional linearizations on $Z \times G_r$ depending affinely on
$(\ell_j)$, such that $\tilde{T}^{-1}(Z \times G_r)^{ss}(\ell_j) =
\tilde{R}^{ss}(\ell_j)$. Moreover, the restriction $\tilde{T}:
\tilde{R}^{ss}(\ell_j) \to (Z \times G_r)^{ss}(\ell_j)$ is finite.
The existence of a good quotient $M(\ell_j)$ again follows from the
lemma.
In fact, we can say more.
\begin{s}{Proposition}
\label{2u}
$T$ is an embedding.
\end{s}
\pf. Since $T$ is injective \cite[4.3]{g}, it suffices to show its
derivative is everywhere injective. At a quotient $\phi: \co^\chi \to
E$, the tangent space to $\Quot$ is given by the hypercohomology group
$\Hyp^1(\End E \stackrel{\phi}{\to} E \otimes \co^\chi)$ (cf.\
\cite{bdw,t1}). Since $H^1(E \otimes \co^\chi) = \k^\chi \otimes
H^1(E) = 0$, this surjects onto $H^1(\End E)$, and hence onto
$H^1(\co)$, which is the tangent space to $\Pic^d C$. So it suffices
to show the kernel of this surjection injects into the tangent space
to $\Pj \Hom(H^0(\La^r \co^\chi), H^0(\La^r E))$. The kernel is
isomorphic to the quotient of $\Hyp^1(\End_0 E \stackrel{\phi}{\to} E
\otimes \co^\chi)$ by the 1-dimensional subspace generated by $\phi$
(cf.\ \cite[2.1]{t1}), where $\End_0$ denotes trace-free
endomorphisms. So the kernel injects as desired if and only if the
natural map $\Hyp^1(\End E \stackrel{\phi}{\to} E \otimes \co^\chi)
\to \Hom(H^0(\La^r \co^\chi), H^0(\La^r E))$ is injective as well.
What is this natural map? It is obtained from the derivative of $T$;
since $T$ is essentially $\phi \mapsto \La^r \phi$, the derivative of
$T$ at $\phi$ is essentially $\psi \mapsto (\La^{r-1} \phi) \wedge
\psi$. More precisely, an element of the hypercohomology group above
is determined by \v Cech cochains $g \in C^1(\End_0 E)$ and $\psi
\in C^0(E \otimes \co^\chi)$ such that $g \phi = d\psi$. Since $\phi$
is surjective, the hypercohomology class of the pair is uniquely
determined by $\psi$; on the other hand, a cochain $\psi$ determines
the trivial hypercohomology class if and only if $\psi = f \phi$ for
some $f \in C^0(\End_0 E)$. The natural map to $\Hom(H^0(\La^r
\co^\chi), H^0(\La^r E))$ is then indeed given by $\psi \mapsto
(\La^{r-1} \phi) \wedge \psi$; its injectivity follows from the lemma
below. \fp
\begin{s}{Lemma}
If $\phi: \k^\chi \to \k^r$ is a linear surjection of vector spaces,
and $\psi: \k^\chi \to \k^r$ is a linear map, then $(\La^{r-1} \phi)
\wedge \psi = 0$ if and only if $\psi = f \phi$ for some $f \in \End_0
\k^r$.
\end{s}
\pf. Suppose first that $\chi = r$. Then $\phi$ is invertible, and
$(\La^{r-1} \phi) \wedge \psi$ is a homomorphism of 1-dimensional
vector spaces. Indeed, if $e_1, \dots, e_r$ is the standard basis for
$\k^r$, then $\La^r \k^\chi$ is spanned by $\phi^{-1}e_1 \wedge \cdots
\wedge \phi^{-1}e_r$. But
\beqas
((\La^{r-1} \phi) \wedge \psi)(\phi^{-1}e_1 \wedge \cdots \wedge
\phi^{-1}e_r) & \!\!\!\!\!\!\!= & \!\!\!\!\!\!\!1/r \sum_i e_1 \wedge \cdots
\wedge e_{i-1} \wedge
\psi\phi^{-1}(e_i) \wedge e_{i+1} \wedge \cdots \wedge e_r \\
& \!\!\!\!\!\!\! = & \!\!\!\!\!\!\! 1/r \, (\tr \psi\phi^{-1})(e_1 \wedge
\cdots \wedge e_r),
\eeqas
so the lemma is true when $\chi = r$. In the general case,
$(\La^{r-1} \phi) \wedge \psi = 0$ implies $\ker \psi \supset \ker
\phi$, for if not, let $\phi^{-1}$ be a right inverse for $\phi$, and
let $u \in \ker \phi \sans \ker \psi$. For some $i$, the coefficient
of $e_i$ in $\psi(u)$ is nonzero. Then
\beqas
((\La^{r-1} \phi) \wedge \psi) (\phi^{-1}e_1 \wedge \cdots \wedge
\phi^{-1}e_{i-1} \wedge u \wedge \phi^{-1}e_{i+1} \wedge \cdots \wedge
\phi^{-1}e_r) & & \\
& \hspace{-60ex} = & \hspace{-31ex} e_i \wedge \cdots \wedge e_{i-1} \wedge
\psi(u) \wedge
e_{i+1} \wedge \cdots \wedge e_r \neq 0.
\eeqas
So $\psi$ descends to $\k^\chi / \ker \phi$; this has dimension $r$,
so the case above applies. \fp
\begin{s}{Corollary}
\label{2t}
$\tilde{T}$ is an embedding.
\end{s}
\pf. $\tilde{R}$ is a bundle of flag varieties over $R$, and each
fibre clearly embeds in $G_r$. \fp
Let $X$ be the Zariski closure in $Z \times G_r$ of $\tilde{T}
(\tilde{R})$.
\begin{s}{Corollary}
The moduli space $M(\ell_j)$ of semistable parabolic bundles is $X
\mod G(\ell_j)$.
\end{s}
\pf. Since $\tilde{T}^{-1}(Z \times G_r)^{ss}(\ell_j) =
\tilde{R}^{ss}(\ell_j)$, this is automatic provided there are no
semistable points in $X \sans \tilde{T}(\tilde{R})$. Since
$M(\ell_j)$ is already projective, any such points would be in the
orbit closures of semistable points in $\tilde{T}(\tilde{R})$. Hence
there would be $x \in \tilde{R}^{ss}(\ell_j)$, and a 1-parameter
subgroup $\la(t) \subset G$, such that $\lim \la(t) \cdot x \notin
\tilde{T}(\tilde{R})$, but $\mu^{\ell_j} (\tilde{T}(x), \la) = 0$,
where $\mu^{\ell_j}$ is the valuation used in the numerical criterion
\cite[4.8; GIT Defn.\ 2.2]{n}. But all the destabilizing subgroups of
points in $\tilde{T}(\tilde{R})$ correspond to destabilizing
subbundles, and their limits are points corresponding to the
associated graded subbundles; in particular, they are in
$\tilde{T}(\tilde{R})$. \fp
We are therefore in a position to apply our main results. Let us
first look for walls and chambers. Notice that the stability
condition only changes at values where there can exist subbundles
whose parabolic slope equals that of $E$. If such a subbundle has
rank $r^+$, degree $d^+$, and weights $\ell_{j^+_i}$ for some $ \{
j^+_i \} \subset \{ 1, \dots, r \}$, then the slope condition is $$
\frac{d^+ + \sum_{i = 1}^{r^+} \ell_{j^+_i}}{r^+} = \frac{d + \sum_{j
= 1}^r \ell_j}{r}. $$ This determines a codimension 1 affine subset of
$W$, which is one of our walls. The complementary numbers $r^- = r -
r^+$, $d^- = d - d^+$, and $\{ \ell_{j^-_i} \} = \{ \ell_j \} \sans \{
\ell_{j^+_i} \}$ of course determine the same wall, but no other
numbers do. Also, there are only finitely many walls, since for a
given $r^+$ and $\{ \ell_{j^+_i} \}$, the affine hyperplane defined by
the above equation only intersects $W$ for finitely many $d^+$. The
connected components of the complement of the walls are the chambers;
for purely numerical reasons, the semistability condition is constant
in the interior of a chamber.
Now, as in the set-up of \S3, suppose $(\ell_j)$ lies on a single
wall in $W$, and choose $(\ell_j^+)$ and $(\ell_j^-)$ in the adjacent
chambers such that the line segment connecting them crosses a wall
only at $(\ell_j)$. Then $x \in X$ belongs to $X^0$ if and only if it
represents a parabolic bundle which splits as $E_+ \oplus E_-$, where
$E_\pm$ are $(\ell_{j^\pm_i})$-stable parabolic bundles. This is
because, to be in $X^{ss}(0) \sans X^{ss}(+)$, a parabolic bundle must
have a semistable parabolic subbundle $E_+$ of rank $r^+$, degree
$d^+$, and weights $\ell_{j^+_i}$. Indeed, $E_+$ must be stable, for
since $(\ell_j)$ lies on only one wall, $E_+$ can have no parabolic
subbundle of the same slope. For the same reason $E / E_+$ must be
stable. On the other hand, to be in $X^{ss}(0) \sans X^{ss}(-)$, $E$
must have an stable parabolic subbundle $E_-$ of rank $r^-$, degree
$d^-$, and weights $\ell_{j^-_i}$. Since all the weights are
distinct, $E_-$ cannot be isomorphic to $E_+$; so there is a nonzero
map $E_- \to E / E_+$. By \cite[III Prop.\ 9(c)]{s}, this map must be
an isomorphism, so $E$ splits as $E_+ \oplus E_-$.
On the other hand, if $E_+$ and $E_-$ are any stable parabolic bundles
with rank, degree, and weights as above, then $E_+ \oplus E_-$ is
certainly represented in $X^0$. Hence $X^0 \mod G(0) = M(\ell_{j_i^+})
\times M(\ell_{j_i^-})$, the product of two smaller moduli spaces.
It is now easy to verify the hypotheses of our strongest result
\re{1x}. First, $X$ is smooth on $X^{ss}(\ell_j)$, hence at $X^0$.
Second, for any $x \in X^0$, the stabilizer $G_x$ is the subgroup
isomorphic to $\kst$ acting on $H^0(E_+)$ with weight $\chi^- / c$ and
on $H^0(E_-)$ with weight $-\chi^+ / c$, where $c$ is the greatest
common divisor $(\chi^+, \chi_-)$. This is because any $g \in
\GL{\chi}$ stabilizing a point in $X^0$ induces an automorphism of
$\be_x$, and vice versa, so for $x \in X^0$ there is an isomorphism
$\GL{\chi}_x \cong \Par \Aut (E_+ \oplus E_-) = \kst \times \kst$; but
only the automorphisms acting with the weights above correspond to
special linear $g$. Third, if $L_j$ is the ample generator of $\Pic
\Gr(\chi - j, \chi)$, then for $x \in X^0$, $G_x \cong \kst$ acts on
$(L_j)_x$ with weight $(n_j^+ \chi^- - n_j^- \chi^+) / c$, where
$n_j^\pm$ are the number of $j_i^\pm$ less than or equal to $j$. But
since $\chi^+ / c$ and $\chi^- / c$ are coprime, so are these weights
for some two values of $j$. Bhosle gives a formula for the
linearization on $Z \times \Gr$ determined by $(\ell_j)$ in terms of
the $L_i$; an easy argument using this formula shows that
$(\ell_j^\pm)$ can be chosen within their chambers so that $G_x$ acts
with coprime weights on the corresponding linearizations.
Because all semistable parabolic bundles are represented by points in
$X$, and because semistability is an open condition, the universal
family of parabolic bundles is a versal family near any point $x \in
X^0$. Moreover, two points in $X$ represent the same parabolic bundle
if and only if they are in the same orbit. The normal bundle $N_{G
\cdot x / X}$ to an orbit is therefore exactly the deformation space
of the parabolic bundle. For $\be_x = E_+ \oplus E_-$ as above this
is $H^1(X, \Par \End (E_+ \oplus E_-))$. The stabilizer $G_x$ acts
with weight $r^\pm / (r^+, r^-)$ on $E_\pm$, so $N^0 = H^1(X, \Par
\End E_+) \oplus H^1(X, \Par \End E_-)$, and $N^\pm = H^1(X, \Par \Hom
(E_\mp, E_\pm))$. Moreover, every element in $N^\pm$ is acted on with
weight exactly $\pm(\chi^+ + \chi_-)/ c$, so $N^\pm$ descend to vector
bundles $W^\pm$ over $M(\ell_{j_i^+}) \times M(\ell_{j_i^-})$.
Indeed, if $\be_\pm \to M(\ell_{j_i^\pm}) \times C$ are universal
bundles, then $W^\pm = (R^1 \pi) \Par \Hom(\be_\mp, \be_\pm)$.
Theorem \re{1x} then states that $\Pj W^\pm$ are the exceptional loci
of the morphisms $M(\ell_j^\pm) \to M(\ell_j)$. This is the result of
Boden and Hu \cite{bhu}. Moreover, \re{1x} asserts that the blow-ups
of $M(\ell_j^\pm)$ at $\Pj W^\pm$, and of $M(\ell_j)$ at
$M(\ell_{j_i^+}) \times M(\ell_{j_i^-})$, are all naturally isomorphic
to the fibred product $M(\ell_j^-) \times_{M(\ell_j)} M(\ell_j^+)$.
With the obvious modifications, the same techniques and results go
through for bundles with parabolic structure at several marked points,
or with degenerate flags.
\bit{Bradlow pairs}
The moduli spaces of Bradlow pairs on our curve $C$ can be studied in
the same way. The role of the weights will be played by a positive
parameter $\si \in \Q$.
A {\em Bradlow pair} is a pair $(E, \phi)$ consisting of a vector
bundle $E$ over $C$ and a nonzero section $\phi \in H^0(X,E)$. We
refer to \cite{brad,bd,t1} for basic definitions and results on Bradlow
pairs. As in \cite{t1}, we confine ourselves to the study of rank 2
pairs. In this case a Bradlow pair of degree $d$ is $\si$-{\em
semistable} if for all line bundles $L \subset E$,
$$\begin{array}{cl}
\deg L \leq d/2 - \si & \mbox{if $\phi \in H^{0}(L)$ and} \\
\deg L \leq d/2 + \si & \mbox{if $\phi \not\in H^{0}(L)$.}
\end{array}$$
It is $\si$-{\em stable} if both inequalities are strict.
The moduli spaces $B_d(\si)$ of $\si$-semistable rank 2 pairs were
constructed in \cite{t1}. In that paper, the determinant was fixed,
but to parallel the discussion of parabolic bundles above we shall now
allow arbitrary determinant. With that modification, the construction
goes as follows.
It suffices to construct $B_d(\si)$ for $d$ sufficiently large. This
is because, for any effective divisor $D$, $B_d(\si)$ will be embedded
in $B_{d+ 2|D|}(\si)$ as the locus where $\phi$ vanishes on $D$. So
assume that $d/2 - \si > 2g-2$, and let $\chi = d + 2(1-g)$. Let
$\Quot$, $R$ and $Z$ be as in \S7 above, and let $G = \SL{\chi}$ act
diagonally on $R \times \Pj^{\chi-1}$. The hypothesis $d/2 - \si >
2g-2$ implies that every $\si$-semistable pair is represented by a
point in $R \times \Pj^{\chi-1}$. Let $(Z \times
\Pj^{\chi-1})^{ss}(\si)$ denote the semistable set with respect to the
fractional linearization $\co(\chi + 2 \si, 4 \si)$, and let $(R
\times \Pj^{\chi-1})^{ss}(\si)$ denote the $\si$-semistable set in the
sense of the definition above. Then the natural $G$-morphism $T
\times 1: R \times \Pj^{\chi-1} \to Z \times \Pj^{\chi-1}$ satisfies
$T^{-1} (Z \times \Pj^{\chi-1})^{ss}(\si) = (R \times
\Pj^{\chi-1})^{ss}(\si)$. Moreover, by \re{2u}, it is an embedding.
So if $X$ denotes the Zariski closure of its image, then for reasons
like those given in \S7, the moduli space $B_d(\si)$ is the geometric
invariant theory quotient $X \mod G(\si)$, where $\si$ denotes the
fractional linearization $T^* \co(\chi + 2 \si, 4 \si)$.
So again our main results apply. The stability condition only changes
for $\si \in d/2 + \Z$, so these points are the walls. Fix one such
$\si$. Then $x \in X$ belongs to $X^0$ if and only if it represents a
pair which splits as $L \oplus M$, where $\deg L = d/2 - \si$ and
$\phi \in H^0(L)$. Indeed, a subbundle $L$ of degree $d/2 - \si$ is
needed to violate the first semistability condition to the right of
the wall, and a subbundle $M$ of degree $d/2 + \si$ is needed to
violate the second semistability condition to the left. But since
$\deg M = \deg E/L$, the map $M \to E/L$ is an isomorphism, so $E$ is
split. On the other hand, for any pair $(L, \phi)$ with $\deg L = d/2
- \si$ and $\phi \in H^0(L) \sans 0$, and for any line bundle $M$ with
$\deg M = d/2 + \si$, certainly $(L \oplus M, \phi \oplus 0)$ is
represented in $X^0$. Hence $X^0 \mod G(0) = S^iC \times \Pic^{d-i}
C$, where $i = d/2 - \si$ and $S^iC$ is the $i$th symmetric product.
It is now easy to verify the hypotheses of our strongest result
\re{1x}. First, $X$ is smooth at $X^{ss}(0)$, hence at $X^0$.
Second, for any $x \in X^0$, the stabilizer $G_x$ is the subgroup
isomorphic to $\kst$ acting on $H^0(L)$ with weight $\chi(M)/c$ and on
$H^0(M)$ with weight $-\chi(L)/c$, where $c$ is the greatest common
divisor $(\chi(L), \chi(M))$. This is because any $g \in \GL{\chi}$
stabilizing a point in $X^0$ induces an automorphism of the
corresponding pair, and vice versa, but only the automorphisms acting
with the weights above correspond to special linear $g$. Third, this
stabilizer $\kst$ acts on $\co(1,0)_x$ with weight $(\chi(M) -
\chi(L))/c$, and on $\co(0,1)_x$ with weight $\chi(M)/c$. These are
coprime, so linearizations with coprime weights can be chosen within
the chambers adjacent to $\si$.
As in \S7, the normal bundle to an orbit is exactly the deformation
space of the Bradlow pair. For any pair $(E, \phi)$, this is the
hypercohomology group $\Hyp^1(\End E \stackrel{\phi}{\to} E)$. (See
\cite{bd} or \cite[(2.1)]{t1}; the slightly different formula in
\cite{t1} arises because the determinant is fixed.) More naturally,
the term $E$ in the complex is actually $E \otimes \co$, where $\co$
is the dual of the subsheaf of $E$ generated by $\phi$; it is acted on
accordingly by $G$. For a pair $(L \oplus M, \phi \oplus 0)$
represented in $X^0$, this splits as
$$\Hyp^1(\co \oplus \co \stackrel{\phi \oplus 0}{\lrow} L)
\oplus \Hyp^1(LM^{-1} \to 0)
\oplus \Hyp^1(ML^{-1} \stackrel{\phi}{\to} M).$$
These are acted on by the stabilizer $\kst$ with weights $0$ and
$\pm(\chi(M) + \chi(L))/c$ respectively, so they are exactly $N^0$,
$N^+$, and $N^-$. The expressions for $N^\pm$ can be simplified:
$\Hyp^1(LM^{-1} \to 0)$ is just $H^1(LM^{-1})$, and if $D$ is the
divisor of zeroes of $\phi$, the long exact sequence of $$ 0 \lrow
ML^{-1} \lrow M \lrow \co_D \otimes M \lrow 0 $$ implies that
$\Hyp^1(ML^{-1} \stackrel{\phi}{\to} M)$ is just $H^0(\co_D \otimes
M)$. Since every element in $N^\pm$ is acted on with weight exactly
$\pm (\chi(M) + \chi(L))/c$, $N^\pm$ descend to vector bundles $W^\pm$
over $S^iC \times \Pic^{d-i}C$. Indeed, if ${\bf M} \to \Pic^{d-i}C
\times C$ is a Poincar\'e bundle and $\Delta \subset S^iC \times C$ is
the universal divisor, then $W^+ = (R^1 \pi) {\bf M}^{-1}(\Delta)$ and
$W^- = (R^0 \pi) \co_\Delta \otimes {\bf M}$.
Theorem \re{1x} then states that $\Pj W^\pm$ are the exceptional loci
of the morphisms $B_d(\si \pm \half) \to B_d(\si)$, and that the
blow-ups of $B_d(\si \pm \half)$ at $\Pj W^\pm$, and of $B_d(\si)$ at
$S^iC \times \Pic^{d-i} C$, are all isomorphic to the fibred product
$B_d(\si - \half) \times_{B_d(\si)} B_d(\si + \half)$. This includes
the main result (3.18) of \cite{t1}; to recover the $W^\pm$ obtained
there for a fixed determinant line bundle $\La$, just substitute ${\bf
M} = \La(-\Delta)$.
Notice that since the construction only works for $d$ large, the
result has so far only been proved in that case. For general $d$,
choose disjoint divisors $D$, $D'$ of degree $|D| = |D'|$ such that $d
+ 2|D|$ is large enough. Then $D$ and $D'$ determine two different
embeddings $B_{d+2|D|}(\si) \to B_{d+4|D|}(\si)$ whose images
intersect in $B_d(\si)$. The result for $B_d(\si)$ then follows
readily from the result for $B_{d+2|D|}(\si)$ and $B_{d+4|D|}(\si)$. \bl
A similar argument proves the analogous result of Bertram,
Daskalopoulos and Wentworth \cite{bdw} on Bradlow $n$-pairs. These
are pairs $(E, \phi)$, where $E$ is as before, but $\phi$ is now a
nonzero element of $H^0(E \otimes \co^n)$. The stability condition is
just like that for ordinary Bradlow pairs, except that the two cases
are $\phi \in H^0(L \otimes \co^n)$ and $\phi \notin H^0(L \otimes
\co^n)$. There is no geometric invariant theory construction in the
literature of moduli spaces of $n$-pairs, but the construction of
\cite{t1} for 1-pairs generalizes in the obvious way; for example,
$\Pj^{\chi-1}$ gets replaced by $\Pj^{n\chi-1}$.
Again the stability condition only changes for $\si \in d/2 + \Z$, so
fix one such $\si$. Assume that $d/2 - \si > 2g-2$, so that the
moduli space can be constructed directly as a geometric invariant
theory quotient. Then $x \in X^0$ if and only if it splits as $L
\oplus M$, where $\deg L = i = d/2 - \si$ and $\phi \in H^0(L \otimes
\co^n)$. Hence $X^0 \mod G(0) = \times^n_{\Pic^i C} S^i C
\times \Pic^{d-i}C$, where $\times^n_{\Pic^i C}$ denotes the $n$-fold
fibred product over $\Pic^i C$. The hypotheses of \re{1x} are
verified exactly as before. The normal bundle to an orbit is again
the deformation space, but this is now $\Hyp^1(\End E
\stackrel{\phi}{\to} E \otimes \co^n)$; for a pair $(L \oplus M, \phi
\oplus 0)$ represented in $X^0$, this splits as
$$\Hyp^1(\co\oplus \co \stackrel{\phi\oplus 0}{\lrow} L \otimes \co^n)
\oplus \Hyp^1(LM^{-1} \to 0)
\oplus \Hyp^1(ML^{-1} \stackrel{\phi}{\to} M \otimes \co^n).$$
Again these are exactly $N^0$, $N^+$, and $N^-$. The expression for
$N^+$ is simply $H^1(LM^{-1})$, but the expression for $N^-$
cannot be simplified very much. If $F$ is defined to be the cokernel
of the sheaf injection $ML^{-1} \to M \otimes \co^n$ induced by
$\phi$, then $N^- = H^0(F)$, but this is not very helpful as $F$ may
not be locally free. In any case, $N^\pm$ descend as before. If
$\Delta_j \subset \times^n_{\Pic^i C} S^i C \times C$ is the pullback
from the $j$th factor of the universal divisor $\Delta \subset S^iC
\times C$, then there is a universal map $\co \to \oplus_j
\co(\Delta_j)$ of bundles on $\times^n_{\Pic^i C} S^i C \times C$, and
hence a sheaf injection ${\bf ML}^{-1} \to {\bf ML}^{-1} \otimes
\oplus_j \co(\Delta_j)$ of bundles on $\times^n_{\Pic^i C} S^i C
\times \Pic^{d-i} C \times C$, where ${\bf M}$ and ${\bf L}$ are
Poincar\'e bundles on $\Pic^i C \times C$ and $\Pic^{d-i}C \times C$
respectively. Let ${\bf F}$ be the cokernel of this injection; then
$W^- = (R^0\pi) {\bf F}$. As before, $W^+ = (R^1 \pi){\bf LM}^{-1}$.
Theorem \re{1x} then gives a result precisely analogous to the one
stated above for 1-pairs. However, the argument passing from large
$d$ to general $d$ no longer works, so this result is only valid for
$i = d/2 - \si > 2g-2$. Indeed, for $i$ smaller than this, the fibred
product $\times^n_{\Pic^i C} S^i C$, and hence $X^0 \mod G(0)$, have
bad singularities. Nevertheless, Bertram et al.\ succeed in using
this result to compute certain Gromov invariants of Grassmannians.
{\em Acknowledgements.} I am grateful to Aaron Bertram, Igor
Dolgachev, David Eisenbud, Jack Evans, Antonella Grassi, Stuart
Jarvis, J\'anos Koll\'ar, Kenji Matsuki, Rahul Pandharipande, David
Reed, Eve Simms, and especially Frances Kirwan, Peter Kronheimer, and
Miles Reid for very helpful conversations and advice.
|
1994-05-10T21:41:38 | 9405 | alg-geom/9405003 | en | https://arxiv.org/abs/alg-geom/9405003 | [
"alg-geom",
"math.AG"
] | alg-geom/9405003 | Alan Hyungju Park | H. Park and C. Woodburn | An Algorithmic Proof of Suslin's Stability Theorem over Polynomial Rings | 23 pages, LaTex | null | null | null | null | Let $k$ be a field. Then Gaussian elimination over $k$ and the Euclidean
division algorithm for the univariate polynomial ring $k[x]$ allow us to write
any matrix in $SL_n(k)$ or $SL_n(k[x])$, $n\geq 2$, as a product of elementary
matrices. Suslin's stability theorem states that the same is true for the
multivariate polynomial ring $SL_n(k[x_1,\ldots ,x_m])$ with $n\geq 3$. As
Gaussian elimination gives us an algorithmic way of finding an explicit
factorization of the given matrix into elementary matrices over a field, we
develop a similar algorithm over polynomial rings.
| [
{
"version": "v1",
"created": "Tue, 10 May 1994 19:40:16 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Park",
"H.",
""
],
[
"Woodburn",
"C.",
""
]
] | alg-geom | \section{Introduction}
Immediately after proving the famous {\em Serre's Conjecture}
(the {\em Quillen-Suslin theorem}, nowadays) in 1976 \cite{suslin:serre},
A. Suslin went on \cite{suslin} to prove the following $K_1$-analogue of
{\em Serre's Conjecture} which is now known as
{\em Suslin's stability theorem}:
\begin{quote}
Let $R$ be a commutative Noetherian ring and $n\geq \max(3,\dim(R)+2)$.
Then, any $n\times n$ matrix $A=(f_{ij})$ of determinant $1$, with
$f_{ij}$ being elements of the polynomial ring $R[x_1,\ldots ,x_m]$, can be
written as a product of elementary matrices over $R[x_1,\ldots ,x_m]$.
\end{quote}
\begin{definition}
For any ring $R$, an $n\times n$ elementary matrix $E_{ij}(a)$ over $R$ is
a matrix of the form
$I+a\cdot e_{ij}$ where $i\neq j,a\in R$ and $e_{ij}$ is the
$n\times n$ matrix whose $(i,j)$ component is $1$ and all other
components are zero.
\end{definition}
For a ring $R$, let $SL_n(R)$ be be the group of all the $n\times n$
matrices of determinant $1$ whose entries are elements of $R$, and let
$E_n(R)$ be the subgroup of $SL_n(R)$ generated by the elementary matrices.
Then {\em Suslin's stability theorem} can be expressed as
\begin{eqnarray}
SL_n(R[x_1,\ldots ,x_m]) = E_n(R[x_1,\ldots ,x_m])
\quad {\rm for\ all}\ n\geq \max(3,\dim(R)+2).
\end{eqnarray}
In this paper, we develop an algorithmic proof of the above assertion
over a field $k$. By implementing this algorithm, for a given
$A\in SL_n(k[x_1,\ldots ,x_m])$
with $n\geq 3$, we are able to find those elementary
matrices $E_1,\ldots ,E_t\in E_n(k[x_1,\ldots ,x_m])$ such that
$A=E_1\cdots E_t$.
\begin{remark}
If a matrix $A$ can be written as a product of elementary matrices, we
will say $A$ is {\em realizable}.
\end{remark}
\bigskip
\begin{itemize}
\item
In section 2, an algorithmic proof of the normality of
$E_n(k[x_1,\ldots ,x_m])$ in $SL_n(k[x_1,\ldots ,x_m])$ for $n\geq 3$
is given, which will be used in the rest of paper.
\item
In section 3, we develop an algorithm for the {\em Quillen Induction Process},
a standard way of reducing a given problem over a ring to an
easier problem over a local ring. Using this {\em Quillen Induction Algorithm},
we reduce our realization problem over the polynomial
ring $R[X]$ to one over $R_M[X]$'s, where $R=k[x_1,\ldots, x_{m-1}]$
and $M$ is a maximal ideal of $R$.
\item
In section 4, an algorithmic proof of the {\em Elementary Column Property},
a stronger version of the {\em Unimodular Column Property}, is given,
and we note that this algorithm gives another constructive proof
of the {\em Quillen-Suslin theorem}.
Using the {\em Elementary Column Property}, we show that a
realization algorithm for
$SL_n(k[x_1,\ldots ,x_m])$ is obtained from
a realization algorithm for the matrices of the following
special form:
$$\left(\! \begin{array}{ccc} p & q & 0 \\ r & s & 0 \\ 0 & 0 & 1
\end{array}\!\right) \in SL_3(k[x_1,\ldots ,x_m]),$$
where $p$ is monic in the last variable $x_m$.
\item
In section 5, in view of the results in the preceding two sections, we note
that a realization algorithm over $k[x_1,\ldots ,x_m]$ can be obtained
from a realization algorithm for the matrices of the special
form $\left(\!\begin{array}{ccc} p & q & 0 \\ r & s & 0 \\ 0 & 0 & 1
\end{array}\!\right)$
over $R[X]$, where $R$ is now a local ring and $p$ is monic in $X$.
A realization algorithm
for this case was already found by M.P. Murthy in \cite{murthy}. We
reproduce {\em Murthy's Algorithm} in this section.
\item
In section 6, we suggest using the {\em Steinberg relations} from algebraic
$K$-theory to lower the number of elementary matrix factors in a
factorization produced by our algorithm.
We also mention an ongoing effort of using our algorithm in {\em Signal
Processing}.
\end{itemize}
\section{Normality of $E_n(k[x_1,\ldots ,x_m])$ in $SL_n(k[x_1,\ldots ,x_m])$}
\begin{lemma}
The Cohn matrix $A=\left(\! \begin{array}{cc} 1+xy & x^2 \\ -y^2 & 1-xy
\end{array}\!\right)$ is not realizable, but $\left(\! \begin{array}{cc}
A & 0 \\ 0 & 1 \end{array}\!\right) \in SL_3(k[x,y])$ is.
\end{lemma}
\noindent {\bf Proof:\ }
The nonrealizability of $A$ is proved in \cite{cohn}, and a complete
algorithmic criterion for the realizability of
matrices in $SL_2(k[x_1,\ldots ,x_m])$
is developed in \cite{thk}. Now consider
\begin{eqnarray}
\left(\! \begin{array}{cc} A & 0 \\ 0 & 1 \end{array}
\!\right) =\left(\! \begin{array}{ccc} 1+xy & x^2 & 0 \\ -y^2 & 1-xy & 0
\\ 0 & 0 & 1 \end{array}\!\right).
\end{eqnarray}
Noting that $\left(\!\begin{array}{ccc} 1+xy & x^2 & 0 \\ -y^2 & 1-xy & 0
\\ 0 & 0 & 1 \end{array}\!\right)=I+\left(\! \begin{array}{c} x\\ -y \\ 0
\end{array}\!\right)\cdot (y,x,0)$, we see that
the realizability of this matrix is
a special case of the following {\bf Lemma 3}.
\hspace*{\fill}{\bf $\Box$} \bigskip \\
\begin{definition}
Let $n\geq 2$. A {\bf Cohn-type matrix} is a matrix of the form
$$ I+a{\bf v} \cdot (v_j{\bf e_i}-v_i{\bf e_j}) $$
where ${\bf v}=\left(\!\begin{array}{c}v_1\\ \vdots\\ v_n\end{array}\!
\right)\in (k[x_1,\ldots ,x_m])^n$, $i<j\in\{ 1,\ldots ,n\}$,
$a\in k[x_1,\ldots ,x_m]$, and ${\bf e_i}=(0,\ldots ,0,1,0,\ldots,0)$
with $1$ occurring only at the $i$-th position.
\end{definition}
\begin{lemma} \label{lem;Cohn}
Any Cohn-type matrix for $n\geq 3$ is realizable.
\end{lemma}
\noindent {\bf Proof:\ }
First, let's consider the case $i=1,j=2$.
In this case,
\begin{eqnarray}
B & = & I+a\left(\!\begin{array}{c} v_1\\ \vdots\\ v_n \end{array}\!\right)
\cdot (v_2, -v_1,0,\ldots , 0) \nonumber\\
& = & \left(\!\begin{array}{ccccc}
1+av_1v_2 & -av_1^2 & 0 & \cdots & 0 \\
av_2^2 & 1-av_1v_2 & 0 & \cdots & 0 \\
av_3v_2 & -av_3v_1 & & & \\
\vdots & \vdots & & I_{n-2} & \\
av_nv_2 & -av_nv_1 & & & \end{array}\!\right) \nonumber\\
& = & \left(\!\begin{array}{ccccc}
1+av_1v_2 & -av_1^2 & 0 & \cdots & 0 \\
av_2^2 & 1-av_1v_2 & 0 & \cdots & 0 \\
0 & 0 & & & \\
\vdots & \vdots & & I_{n-2} & \\
0 & 0 & & & \end{array}\!\right)
\prod_{l=3}^nE_{l1}(av_lv_2)E_{l2}(-av_lv_1),
\end{eqnarray}
So, it's enough to show that
\begin{eqnarray}
A=\left( \!\begin{array}{ccc} 1+av_1v_2 & -av_1^2 & 0 \\
av_2^2 & 1-av_1v_2 & 0 \\
0 & 0 & 1 \end{array}\!\right)
\end{eqnarray}
is realizable for any $a,v_1,v_2\in k[x_1,\ldots ,x_m]$.
Let ``$\rightarrow$'' indicate that we are applying elementary operations,
and consider the following:
\begin{eqnarray}
A & = & \left(\! \begin{array}{ccc} 1+av_1v_2 & -av_1^2 & 0 \\
av_2^2 & 1-av_1v_2 & 0 \\ 0 & 0 & 1\end{array}\!\right)
\rightarrow
\left(\!\begin{array}{ccc} 1+av_1v_2 & -av_1^2 & v_1 \\
av_2^2 & 1-av_1v_2 & v_2\\ 0 & 0 & 1\end{array}\!\right)\nonumber\\
& \rightarrow &
\left( \!\begin{array}{ccc} 1 & -av_1^2 & v_1\\
0 & 1-av_1v_2 & v_2 \\ -av_2 & 0 & 1\end{array}\!\right)
\rightarrow
\left(\! \begin{array}{ccc} 1 & 0 & v_1 \\ 0 & 1 & v_2 \\
-av_2 & av_1 & 1 \end{array} \!\right)
\rightarrow
\left( \!\begin{array}{ccc} 1 & 0 & v_1 \\ 0 & 1 & v_2 \\
0 & av_1 & 1+av_1v_2 \end{array}\!\right)\nonumber\\
& \rightarrow &
\left(\! \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & v_2 \\
0 & av_1 & 1+av_1v_2 \end{array}\!\right)
\rightarrow
\left(\! \begin{array}{ccc} 1 & 0 &0 \\ 0 & 1 & v_2 \\
0 & 0 & 1 \end{array}\! \right)
\rightarrow
\left(\!\begin{array}{ccc}
1 & 0 &0 \\ 0 & 1 &0 \\ 0 & 0 & 1\end{array}\! \right).
\end{eqnarray}
Keeping track of all the elementary operations involved, we get
\begin{eqnarray}
A=E_{13}(-v_1)E_{23}(-v_2)E_{31}(-av_2)E_{32}(av_1)E_{13}(v_1)
E_{23}(v_2)E_{31}(av_2)E_{32}(-av_1).
\end{eqnarray}
In general (i.e., for arbitrary $i<j$),
\begin{eqnarray}
B & = & I+a\left(\!\begin{array}{c} v_1 \\ \vdots \\v_n \end{array}\!\right)
\cdot (0,\ldots ,0,v_j,0,\ldots ,0,-v_i,0,\ldots ,0) \nonumber\\
& & \mbox{(Here,\ $v_j$\ occurs\ at\ the\ $i$-th\ position\
and\ $-v_i$\ occurs\ at\ the}\nonumber\\
& & j\mbox{-th\ position.)}\nonumber\\
& = & \left(\!\begin{array}{ccccccc}
1 & \cdots & av_1v_j & \cdots & -av_1v_i & \cdots & 0 \\
& \ddots & \vdots & & \vdots & & 0 \\
& & 1+av_iv_j & & -av_i^2 & & \\
& & \vdots & & \vdots & & \\
& & av_j^2 & & 1-av_iv_j & & \\
& & \vdots & & \vdots & & \\
& & v_nv_j & & -v_nv_i & & 1 \end{array}\!\right)\nonumber\\
& = & \left(\!\begin{array}{ccccccc}
1 & \cdots & 0 & \cdots & 0 & \cdots & 0 \\
& \ddots & \vdots & & \vdots & & 0 \\
& & 1+av_iv_j & & -av_i^2 & & \\
& & \vdots & & \vdots & & \\
& & av_j^2 & & 1-av_iv_j & & \\
& & \vdots & & \vdots & & \\
& & 0 & & 0 & & 1 \end{array}\!\right)\nonumber\\
& & \cdot\prod_{1\leq l\leq n, l\neq i,j}
E_{li}(av_lv_j)E_{lj}(-av_lv_i)
\nonumber\\
& = & E_{it}(-v_i)E_{jt}(-v_j)E_{ti}(-av_j)E_{tj}(av_i)E_{it}(v_i)
E_{jt}(v_j)E_{ti}(av_j)E_{tj}(-av_i) \nonumber\\
& & \cdot\prod_{1\leq l\leq n, l\neq i,j}
E_{li}(av_lv_j)E_{lj}(-av_lv_i).
\end{eqnarray}
In the above, $t\in \{1,\ldots, n\}$ can be chosen to be any number other
than $i$ and $j$.
\hspace*{\fill}{\bf $\Box$} \bigskip \\
Since a {\em Cohn-type matrix} is realizable, any product of
{\em Cohn-type matrices} is also realizable.
This observation motivates the following generalization of the above lemma.
\begin{definition}
Let $R$ be a ring and ${\bf v}=(v_1,\ldots ,v_n)^t
\in R^n$ for some $n\in I\!\!N$. Then ${\bf v}$ is called a {\em unimodular
column vector} if its components generate $R$, i.e.
if there exist $g_1,\ldots ,g_n\in R$ such that
$v_1g_1+\cdots +v_ng_n=1$.
\end{definition}
\begin{cor}
Suppose that $A \in SL_n(k[x_1, \ldots ,x_m])$ with $n\geq 3$
can be written in the
form $A=I+{\bf v}\cdot {\bf w}$ for a unimodular column vector $\ {\bf v}$
and a row vector $\ {\bf w}$ over $\ k[x_1,\ldots ,x_m]\ $
such that ${\bf w}\cdot {\bf v}=0$. Then $A$ is realizable.
\end{cor}
\noindent {\bf Proof:\ }
Since ${\bf v}=(v_1,\ldots ,v_n)^t$ is unimodular, we
can find $g_1,\ldots ,g_n \in k[x_1,\ldots ,x_m]$ such that
$v_1g_1+\cdots +v_ng_n=1$. We can use the {\em effective Nullstellensatz}
to explicitly find these $g_i$'s (See \cite{fitchas:galligo}).
This combined with ${\bf w}\cdot {\bf v}=w_1v_1+\cdots +w_nv_n=0$
yields a new expression for ${\bf w}$:
\begin{eqnarray}
{\bf w}=\sum_{i<j}a_{ij}(v_j{\bf e_i}-v_i{\bf e_j})
\end{eqnarray}
where $a_{ij}=w_ig_j-w_jg_i$. Now,
\begin{eqnarray}
A=\prod_{i<j}\left(I+{\bf v}\cdot a_{ij}(v_j{\bf e_i}-v_i{\bf e_j})
\right).
\end{eqnarray}
Each component on the right hand side of this equation is a {\em Cohn-type
matrix} and thus realizable, so $A$ is also realizable.
\hspace*{\fill}{\bf $\Box$} \bigskip \\
\begin{cor}
\quad $BE_{ij}(a)B^{-1}$ is realizable for any $\ B\in GL_n(k[x_1, \ldots
,x_m])$ with $n\geq 3$ and $\ a\in k[x_1, \ldots ,x_m]$.
\end{cor}
\noindent {\bf Proof:\ }
Note that $i\neq j$, and
$$BE_{ij}(a)B^{-1}=I+(i\mbox{-th\ column\ vector\ of}\ B)\cdot a
\cdot (j\mbox{-th\ row\ vector\ of}\ B^{-1}).$$
Let ${\bf v}$ be the $i$-th column vector of $B$ and ${\bf w}$ be
$a$ times the $j$-th row vector of $B^{-1}$. Then $(i$-th row vector of
$B^{-1})\cdot {\bf
v}=1$ implies ${\bf v}$ is unimodular, and ${\bf w}\cdot {\bf v}$ is
clearly zero since $i\neq j$. Therefore, $BE_{ij}(a)B^{-1}=I+{\bf v}\cdot
{\bf w}$ satisfies the condition of the above corollary, and is thus
realizable.
\hspace*{\fill}{\bf $\Box$} \bigskip \\
\begin{remark}
One important consequence of this corollary is that for $n\geq 3$,
$E_n(k[x_1,\ldots ,x_m])$ is a normal subgroup of $SL_n(k[x_1,\ldots ,x_m])$,
i.e. if $A\in SL_n(k[x_1,\ldots ,x_m])$ and $E\in E_n(k[x_1,\ldots ,x_m])$,
then the above corollary gives us an algorithm for finding elementary matrices
$E_1,\ldots ,E_t$ such that $A^{-1}EA=E_1\cdots E_t$.
\end{remark}
\section{Glueing of Local Realizability}
Let $R=k[x_1,\ldots , x_{m-1}],\ X=x_m$ and $M\in$ Max($R$)
=\{ maximal ideals of $R$\}.
For $A\in SL_n(R[X])$, we let $A_M\in SL_n(R_M[X])$ be
its image under the canonical mapping $SL_n(R[X])\rightarrow SL_n(R_M[X])$.
Also, by induction, we may assume $SL_n(R)=E_n(R)$ for $n\geq 3$.
Now consider the following analogue of Quillen's
theorem for elementary matrices;
\begin{quote}
Suppose $n\geq 3$ and $A\in SL_n(R[X])$. Then $A$ is realizable
over $R[X]$ if and only if $A_M\in SL_n(R_M[X])$ is realizable
over $R_M[X]$ for every $M\in$ Max($R$).
\end{quote}
While a non-constructive proof of this assertion is given in
\cite{suslin} and a more general functorial treatment of this
{\em Quillen Induction Process} can be found in \cite{knus}, we will attempt
to give a constructive proof for it here.
Since the necessity of the condition is clear, we have to prove
the following;
\begin{thm}\label{thm;glueing}
(Quillen Induction Algorithm)
For any given $A\in SL_n(R[X])$, if $A_M\in E_n(R_M[X])$
for every $M\in {\rm Max}(R)$, then $A\in E_n(R[X])$.
\end{thm}
\begin{remark}
In view of this theorem, for any given $A\in SL_n(R[X])$,
now it's enough to have a realization
algorithm for each $A_M$ over $R_M[X]$.
\end{remark}
\noindent {\bf Proof:\ }
Let ${\bf a_1}=(0,\ldots ,0)\in k^{m-1}$, and
$M_1=\{ g\in k[x_1,\ldots ,x_{m-1}]\mid g({\bf a_1})=0\}$
be the corresponding maximal ideal. Then by the condition of the
theorem, $A_{M_1}$ is realizable over $R_{M_1}[X]$. Hence, we can
write
\begin{eqnarray}
A_{M_1}=\prod_jE_{s_jt_j}\left(\frac{c_j}{d_j}\right)
\end{eqnarray}
where $c_j,d_j\in R, d_j\not\in M_1$.
Letting $r_1=\prod_jd_j\notin M_1$, we can rewrite this as
\begin{eqnarray}
A_{M_1}=\prod_jE_{s_jt_j}\left(\frac{c_j\prod_{k\neq j}d_k}{r_1}\right)
\in E_n(R_{r_1})\subset E_n(R_{M_1}).
\end{eqnarray}
Denote an algebraic closure of $k$ by $\bar{k}$.
Inductively, let ${\bf a_j} \in {\bar{k}}^{m-1}$ be a common zero of
$r_1,\ldots ,r_{j-1}$
and $M_j=\{ g\in k[x_1,\ldots ,x_{m-1}]\mid g({\bf a_j})=0\}$
be the corresponding maximal ideal of $R$ for
each $j\geq 2$. Define $r_j\notin M_j$ in the same way
as in the above so that
\begin{eqnarray}
A_{M_j}\in E_n(R_{r_j}[X]).
\end{eqnarray}
Since ${\bf a_j}$ is a common zero of
$r_1,\ldots ,r_{j-1}$ in this construction, we immediately see
$r_1,\ldots ,r_{j-1}\in M_j=\{g\in R\mid g({\bf a_j})=0\}$.
But noting $r_j\notin M_j$,
we conclude that $r_j\notin r_1R+\cdots +r_{j-1}R$.
Now, since the Noetherian condition on $R$ guarantees that we will
get to some $L$ after a finite number of steps such that
$r_1R+\cdots +r_LR=R$,
we can use the usual {\em Ideal Membership Algorithm} to determine
when $1_R$ is in the ideal $r_1R+\cdots +r_LR$.
Let $l$ be a {\em large} natural number (It will soon be clear what {\em
large} means). Then since $r_1^lR+\cdots +r_L^lR=R$, we can
use the {\em effective Nullstellensatz} to find
$g_1,\ldots ,g_L\in R$ such that $r_1^lg_1+\cdots +r_L^lg_L=1$.
Now, we express $A(X)\in SL_n(R[X])$ in the following way:
\begin{eqnarray}
A(X) & = & A(X-Xr_1^lg_1) \cdot [ A^{-1}(X-Xr_1^lg_1) A(X) ]\nonumber\\
& = & A(X-Xr_1^lg_1-Xr_2^lg_2) \cdot
[ A^{-1}(X-Xr_1^lg_1-Xr_2^lg_2) A(X-Xr_1^lg_1) ] \nonumber\\
& & \cdot [ A^{-1}(X-Xr_1^lg_1) A(X) ]\nonumber\\
& = & \cdots \nonumber\\
& = & A(X-\sum_{i=1}^LXr_i^lg_i) \cdot [ A^{-1}(X-\sum_{i=1}^LXr_i^lg_i)
A(X-\sum_{i=1}^{L-1}Xr_i^lg_i) ] \cdots \nonumber \\
& & \cdots [ A^{-1}(X-Xr_1^lg_1) A(X) ].
\end{eqnarray}
Note here that the first matrix $A(X-\sum_{i=1}^LXr_i^lg_i)=A(0)$
on the right hand side is
in $SL_n(R)=E_n(R)$ by the induction hypothesis. What will be shown
now is that for a sufficiently large $l$, each expression in the
brackets in the above equation for $A$ is actually in $E_n(R[X])$,
so that $A$ itself is in $E_n(R[X])$.
To this end, by letting $A_{M_i}=A_i$ and identifying
$A\in SL_n(R[X])$ with $A_i\in SL_n(R_{M_i}[X])$, note that each
expression in the brackets is in the following form:
\begin{eqnarray}
A_i^{-1}(cX)A_i((c+r_i^lg)X).
\end{eqnarray}
\newline {\bf *Claim:} For any $c,g\in R$, we can find a sufficiently large
$l$ such that $A_i^{-1}(cX)A_i((c+r_i^lg)X)\in E_n(R[X])$ for all
$i=1,\ldots ,L$.
\newline Let
\begin{eqnarray}
D_i(X,Y,Z)=A_i^{-1}(Y\cdot X)A_i((Y+Z)\cdot X)\in E_n(R_{r_i}[X,Y,Z])
\end{eqnarray}
and write $D_i$ in the form
\begin{eqnarray}
D_i=\prod_{j=1}^hE_{s_jt_j}(b_j+Zf_j)
\end{eqnarray}
where $b_j\in R_{r_i}[X,Y]$ and $f_j\in R_{r_i}[X,Y,Z]$.
{}From now on, the elementary matrix $E_{s_jt_j}(a)$ will be simply
denoted as $E^j(a)$ for notational convenience.
Now define $C_p$ by
\begin{eqnarray}
C_p=\prod_{j=1}^pE^j(b_j)\in E_n(R_{r_i}[X,Y]).
\end{eqnarray}
Then the $C_p$'s satisfy the following recursive relations;
\begin{eqnarray}
E^1(b_1) & = & C_1 \nonumber \\
E^p(b_p) & = & C_{p-1}^{-1}C_p\quad (2\leq p\leq h)\nonumber \\
C_h & = & I.
\end{eqnarray}
Hence, using $E_{ij}(a+b)=E_{ij}(a)E_{ij}(b)$,
\begin{eqnarray}
D_i & = & \prod_{j=1}^hE^j(b_j+Zf_j) \nonumber \\
& = & \prod_{j=1}^hE^j(b_j)E^j(Zf_j) \nonumber \\
& = & [E^1(b_1)E^1(Zf_1)][E^2(b_2)E^2(Zf_2)]
\ \cdots\ [E^h(b_h)E^h(Zf_h)] \nonumber \\
& = & [C_1E^1(Zf_1)][C_1^{-1}C_2E^2(Zf_2)]\ \cdots\
[C_{h-1}^{-1}C_hE^h(Zf_h)] \nonumber \\
& = & \prod_{j=1}^hC_jE^j(Zf_j)C_j^{-1}.
\end{eqnarray}
Now in the same way as in the proof of {\bf Corollary 1} and
{\bf Corollary 2} of section 2, we can write
$C_jE^j(Zf_j)C_j^{-1}$ as a product of Cohn-type matrices, i.e.
for any given $j\in \{1,\ldots ,h\}$, let
${\bf v}=\left(\!\begin{array}{c} v_1 \\ \vdots \\ v_n \end{array}\!\right)$
be the $s_j$-th column vector of $C_j$. Then
\begin{eqnarray}
C_jE_{s_jt_j}(Zf_j)C_j^{-1}=\prod_{1\leq \gamma < \delta \leq n}
[I+{\bf v}\cdot Zf_j\cdot a_{\gamma \delta}(v_{\gamma}{\bf e_{\delta}}-v_{\delta}{\bf e_{\gamma}})]
\end{eqnarray}
for some $a_{\gamma \delta}\in R_{r_i}[X,Y]$.
Also we can find a natural number $l$ such that
\begin{eqnarray}
v_{\gamma}=\frac{v_{\gamma}'}{r_i^l},\quad a_{\gamma \delta}=\frac{a_{\gamma \delta}'}{r_i^l},\quad
f_j=\frac{f_j'}{r_i^l}
\end{eqnarray}
for some $v_{\gamma}',a_{\gamma \delta}'\in R[X,Y],\ f_j'\in
R[X,Y,Z]$. Now, replacing $Z$ by $r_i^{4l}g$, we see that
all the Cohn-type matrices in the above expression for $C_jE^j(Zf_j)C_j^{-1}$
have denominator-free entries. Therefore,
\begin{eqnarray}
C_jE^j(r_i^{4l}gf_j)C_j^{-1}\in E_n(R[X,Y]).
\end{eqnarray}
Since this is true for each $j$, we conclude that for a sufficiently
large $l$,
\begin{eqnarray}
D_i(X,Y,r_i^{l}g)=\prod_{j=1}^hC_jE^j(r_i^{l}gf_j)
C_j^{-1}\in E_n(R[X,Y]).
\end{eqnarray}
Now, letting $Y=c$ proves the claim.
\hspace*{\fill}{\bf $\Box$} \bigskip \\
\section{Reduction to $SL_3(k[x_1,\ldots ,x_m])$}
Let $A\in SL_n(k[x_1,\ldots ,x_m])$ with $n\geq 3$, and ${\bf v}$
be its last column vector. Then ${\bf v}$ is unimodular. (Recall that
the cofactor expansion along the last column gives a required relation.)
Now, if we can reduce ${\bf v}$ to ${\bf e_n}=(0,0,\ldots ,0,1)^t$
by applying elementary operations, i.e. if we can
find $B\in E_n(k[x_1,\ldots ,x_m])$ such that $B{\bf v}={\bf e_n}$,
then
\begin{eqnarray}
BA=\left(\! \begin{array}{cccc} & & & 0 \\
& \tilde{A} & & \vdots \\
& & & 0 \\ p_1 &\ldots & p_{n-1} & 1 \end{array}\! \right)
\end{eqnarray}
for some $\tilde{A} \in SL_{n-1}(k[x_1,\ldots ,x_m])$ and $p_i \in
k[x_1,\ldots ,x_m]$ for
$i=1,\ldots ,n-1$. Hence,
\begin{eqnarray}
BAE_{n1}(-p_1)\cdots E_{n(n-1)}(-p_{n-1})=
\left(\!\begin{array}{cc} \tilde{A} & 0 \\ 0 & 1 \end{array}\!\right).
\end{eqnarray}
Therefore our problem of expressing $A\in SL_n(k[x_1,\ldots ,x_m])$
as a product of elementary matrices is now reduced to the same problem
for $\tilde{A}\in SL_{n-1}(k[x_1,\ldots ,x_m])$. By repeating this
process, we get to the problem of expressing $A=
\left(\! \begin{array}{ccc} p & q & 0 \\ r & s & 0 \\ 0 & 0 & 1
\end{array}\! \right) \in SL_3(k[x_1,\ldots ,x_m])$ as a product of
elementary matrices, which is the subject of the next section.
In this section, we will develop an algorithm for finding
elementary operations that reduce a given unimodular column vector
${\bf v} \in (k[x_1,\ldots ,x_m])^n$ to ${\bf e_n}$.
Also, as a corollary to this {\em Elementary Column Property}, we
give an algorithmic proof of
the {\em Unimodular Column Property} which states that
for any given unimodular column vector ${\bf v} \in
(k[x_1,\ldots ,x_m])^n$, there exists a unimodular matrix $B$,
i.e. a matrix of constant determinant, over $k[x_1,\ldots ,x_m]$
such that $B{\bf v}={\bf e_n}$. Lately, {\sl A. Logar, B. Sturmfels}
in \cite{logar:sturmfels} and {\sl N. Fitchas, A. Galligo} in
\cite{fitchas:galligo}, \cite{fitchas} have given different
algorithmic proofs of this {\em Unimodular Column Property}, thereby
giving algorithmic proofs of the {\em Quillen-Suslin theorem}.
Therefore, our algorithm gives another constructive proof of the
{\em Quillen-Suslin theorem}. The second author has given a different
algorithmic proof of the {\em Elementary Column Property} based
on a localization and patching process in \cite{cynthia}.
\begin{definition}
For a ring $R$, ${\rm Um}_n(R)=\{ n$-dimensional unimodular
column vectors over $R\}$.
\end{definition}
\begin{remark}
Note that the groups $GL_n(k[x_1,\ldots ,x_m])$
and $E_n(k[x_1,\ldots ,x_m])$ act on the set
${\rm Um}_n(k[x_1,\ldots ,x_m])$ by matrix multiplication.
\end{remark}
\begin{thm}\label{thm;reduction}
(Elementary Column Property)
For $n\geq 3$, the group $E_n(k[x_1,\ldots ,x_m])$ acts transitively
on the set Um$_n(k[x_1,\ldots ,x_m])$.
\end{thm}
\begin{remark}
According to this theorem, if ${\bf v,v'}$ are $n$-dimensional
unimodular column vectors over $k[x_1,\ldots ,x_m]$, then we can find $B\in
E_n(k[x_1,\ldots ,x_m])$
such that $B{\bf v}={\bf v'}$. Letting ${\bf v'}={\bf e_n}$
gives a desired algorithm.
\end{remark}
\begin{cor}
(Unimodular Column Property)
For $n\geq 2$, the group $GL_n(k[x_1,\ldots ,x_m])$ acts transitively
on the set Um$_n(k[x_1,\ldots ,x_m])$.
\end{cor}
\noindent {\bf Proof:\ }
For $n\geq 3$, the {\em Elementary Column Property} cleary implies
the {\em Unimodular Column Property} since a product of elementary
matrices is always unimodular, i.e. has a constant determinant.
If $n=2$, for any ${\bf v}=(v_1,v_2)^t\in
\mbox{Um}_2(k[x_1,\ldots ,x_m])$,
find $g_1,g_2\in k[x_1,\ldots ,x_m]$ such that $v_1g_1+v_2g_2=1$.
Then the unimodular matrix
$U_{\bf v}= \left(\!\begin{array}{cc} v_2 & -v_1 \\ g_1 & g_2
\end{array}\!\right)$ satisfies
$U_{\bf v}\cdot {\bf v}={\bf e_2}$. Therefore we see that, for any
${\bf v}, {\bf w}\in \mbox{Um}_2(k[x_1,\ldots ,x_m])$,
$U_{\bf w}^{-1}U_{\bf v}\cdot {\bf v}={\bf w}$ where
$U_{\bf w}^{-1}U_{\bf v}\in GL_2(k[x_1,\ldots ,x_m])$.
\hspace*{\fill}{\bf $\Box$} \bigskip \\
Let $R=k[x_1,\ldots ,x_{m-1}]$ and $X=x_m$. Then $k[x_1,\ldots ,x_m]=R[X]$.
By identifying $A\in SL_2(R[X])$ with
$\left(\! \begin{array}{cc} A & 0 \\ 0 & I_{n-2}
\end{array}\!\right)\in SL_n(R[X])$,
we can regard $SL_2(R[X])$ as a subgroup of
$SL_n(R[X])$. Now consider the following theorem.
\begin{thm}\label{thm;link}
Suppose ${\bf v}(X)=
\left(\!\begin{array}{c} v_1(X) \\ \vdots \\ v_n(X) \end{array}\!\right)
\in {\rm Um}_n(R[X])$, and $v_1(X)$ is monic in $X$.
Then there exists $B_1\in SL_2(R[X])$ and $B_2\in E_n(R[X])$ such that
$B_1B_2\cdot {\bf v}(X)={\bf v}(0)$.
\end{thm}
\noindent {\bf Proof:\ }
Later
\hspace*{\fill}{\bf $\Box$} \bigskip \\
We will use this theorem to prove the {\bf Theorem~\ref{thm;reduction}}, now.
\medskip
\noindent {\bf Proof of Theorem~\ref{thm;reduction}:}
Since the {\em Euclidean division algorithm} for $k[x_1]$ proves the
theorem for $m=1$ case,
by induction, we may assume the statement of the theorem for $R=
k[x_1,\ldots ,x_{m-1}]$. Let $X=x_m$ and
${\bf v}=\left(\!\begin{array}{c} v_1\\ \vdots\\ v_n\end{array}\!\right)
\in {\rm Um}_n(R[X])$. We may also assume that $v_1$ is monic
by applying a change of variables
(as in the well-known proof of the {\em Noether Normalization Lemma}).
Now by the above {\bf Theorem~\ref{thm;link}}, we can find
$B_1\in SL_2(R[X])$ and $B_2\in E_n(R[X])$ such that
\begin{eqnarray}
B_1B_2\cdot {\bf v}(X)={\bf v}(0)\in R.
\end{eqnarray}
And then by the inductive hypothesis,
we can find $B'\in E_n(R)$ such that
\begin{eqnarray}
B'\cdot {\bf v}(0)={\bf e_n}.
\end{eqnarray}
Therefore, we get
\begin{eqnarray}
{\bf v}=B_2^{-1}B_1^{-1}B'^{-1}{\bf e_n}.
\end{eqnarray}
By the normality of $E_n(R[X])$ in $SL_n(R[X])$ ({\bf Corollary 2}),
we can write $B_1^{-1}B'^{-1}=B''B_1^{-1}$ for some $B''\in E_n(R[X])$.
Since
\begin{eqnarray}
B_1^{-1}=\left(\! \begin{array}{ccccc}
p & q& 0 &\ldots & 0 \\ r & s & 0 & \ldots &0\\
0 & 0 &&& \\\vdots & \vdots & & I_{n-2} & \\ 0 & 0 &&&
\end{array}\!\right)
\end{eqnarray}
for some $p,q,r,s \in R[X]$, we have
\begin{eqnarray}
{\bf v} & = & B_2^{-1}B_1^{-1}B'^{-1}{\bf e_n}\nonumber\\
& = & (B_2^{-1}B'')B_1^{-1}{\bf e_n}\nonumber\\
& = & (B_2^{-1}B'')\left(\! \begin{array}{ccccc}
p & q& 0 &\ldots & 0 \\ r & s & 0 & \ldots &0\\
0 & 0 &&& \\\vdots & \vdots & & I_{n-2} & \\ 0 & 0 &&&
\end{array}\!\right)\left(\! \begin{array}{c} 0 \\ 0 \\ \vdots
\\ 0 \\ 1 \end{array} \!\right)\nonumber\\
& = & (B_2^{-1}B''){\bf e_n}
\end{eqnarray}
where $B_2^{-1}B''\in E_n(R[X])$.
Since we have this relationship for any ${\bf v}\in {\rm Um}_n(R[X])$,
we get the desired transitivity.
\hspace*{\fill}{\bf $\Box$} \bigskip \\
Now, we need one lemma to construct an algorithm for
the {\bf Theorem~\ref{thm;link}}.
\begin{lemma}\label{lem;link}
Let $f_1,f_2,b,d\in R[X]$ and $r$ be the resultant of $f_1$ and $f_2$.
Then there exists $B\in SL_2(R[X])$ such that
\begin{eqnarray}
B\left(\! \begin{array}{c} f_1(b) \\f_2(b) \end{array}\!\right)
=\left( \!\begin{array}{c} f_1(b+rd) \\f_2(b+rd) \end{array}\!\right).
\end{eqnarray}
\end{lemma}
\noindent {\bf Proof:\ }
By the property of the resultant of two polynomials, we can find
$g_1,g_2\in R[X]$ such that $f_1g_1+f_2g_2=r$. Also let
$s_1,s_2,t_1,t_2\in R[X,Y,Z]$ be the polynomials defined by
\begin{eqnarray}
f_1(X+YZ) & = & f_1(X)+Ys_1(X,Y,Z)\nonumber\\
f_2(X+YZ) & = & f_2(X)+ Ys_2(X,Y,Z)\nonumber\\
g_1(X+YZ) & = & g_1(X)+Yt_1(X,Y,Z) \nonumber\\
g_2(X+YZ) & = & g_2(X)+ Yt_2(X,Y,Z).
\end{eqnarray}
Now, let
\begin{eqnarray}
B_{11} & = & 1+s_1(b,r,d)\cdot g_1(b)+t_2(b,r,d)\cdot f_2(b) \nonumber\\
B_{12} & = & s_1(b,r,d)\cdot g_2(b)-t_2(b,r,d)\cdot f_1(b) \nonumber \\
B_{21} & = & s_2(b,r,d)\cdot g_1(b)-t_1(b,r,d)\cdot f_2(b) \nonumber \\
B_{22} & = & 1+s_2(b,r,d)\cdot g_2(b)+t_1(b,r,d)\cdot f_1(b).
\end{eqnarray}
Then one checks easily that
$B=\left(\! \begin{array}{cc} B_{11} & B_{12} \\ B_{21} & B_{22}
\end{array}\!\right)$ satisfies the desired property and that
$B\in SL_2(R[X])$.
\hspace*{\fill}{\bf $\Box$} \bigskip \\
{\bf Proof of Theorem~\ref{thm;link}:}
Let ${\bf a_1}=(0,\ldots ,0)\in k^{m-1}$. Define
$M_1=\{ g\in k[x_1,\ldots ,x_{m-1}]\mid g({\bf a_1})=0\} $
and $k_1=R/M_1$ as the corresponding maximal ideal and residue
field, respectively.
Since ${\bf v}\in (R[X])^n$ is a unimodular column vector, its image
${\bf \bar{v}}$ in $(k_1[X])^n=((R/M_1)[X])^n$ is also unimodular.
Since $k_1[X]$ is a principal ideal ring, the minimal Gr\"{o}bner basis of
its ideal $<\bar{v}_2,\ldots ,\bar{v}_n>$ consists of
a single element, $G_1$.
Then $\bar{v}_1$ and $G_1$ generate the unit ideal in $k_1[X]$ since
$\bar{v}_1,\bar{v}_2,\ldots ,\bar{v}_n$ generate the unit ideal.
Using the Euclidean division algorithm for $k_1[X]$, we can find $E_1\in
E_{n-1}(k_1[X])$ such that
\begin{eqnarray}
E_1\left( \!\begin{array}{c} \bar{v}_2 \\ \vdots \\ \bar{v}_n
\end{array} \right)=
\left( \begin{array}{c} G_1 \\ 0 \\ \vdots \\ 0
\end{array}\! \right).
\end{eqnarray}
By identifying $k_1$ with a subring of $R$, we may regard
$E_1$ to be an element of $E_n(R[X])$
and $G_1$ to be an element of $R[X]$.
Then,
\begin{eqnarray}
\left( \!\begin{array}{cc} 1 & 0 \\ 0 & E_1 \end{array} \right)
{\bf v}=\left( \begin{array}{c} v_1
\\ G_1+q_{12} \\ q_{13}
\\ \vdots \\ q_{1n}
\end{array} \!\right)
\end{eqnarray}
for some $q_{12},\ldots, q_{1n}\in M_1[X]$.
Now, define $r_1\in R$ by
\begin{eqnarray}
r_1 & = & {\rm Res}(v_1, G_1+q_{12}) \nonumber\\
& = & {\rm the\ resultant\ of}\
v_1\ {\rm and}\ G_1+q_{12}
\end{eqnarray}
and find $f_1,h_1\in R[X]$ such that
\begin{eqnarray}
f_1\cdot v_1+h_1\cdot (G_1+q_{12})=r_1.
\end{eqnarray}
Since $v_1$ is monic, and $\bar{v}_1$ and $G_1\in k_1[X]$ generate the unit
ideal, we have
\begin{eqnarray}
\bar{r}_1 & = & \overline{{\rm Res}(v_1, G_1+q_{12})}\nonumber\\
& = & {\rm Res}(\bar{v}_1, G_1)\nonumber\\
& \neq & 0.
\end{eqnarray}
Therefore, $r_1\notin M_1$. Denote an algebraic closure of $k$ by
$\bar{k}$.
Inductively, let ${\bf a_j}\in {\bar{k}}^{m-1}$ be a common zero of
$r_1,\ldots ,r_{j-1}$
and $M_j$ be the corresponding maximal ideal of $R$ for
each $j\geq 2$. Define $r_j\notin M_j$ in the same way
as in the above. Define also, $E_j\in E_{n-1}(k_j[X]), G_j\in k_j[X],
f_j,h_j\in R[X]$,
and $q_{j2}, \ldots , q_{jn}\in M_j[X]$ in an analogous way.
Since we let ${\bf a_j}$ be a common zero of
$r_1,\ldots ,r_{j-1}$ in this construction, we see
$r_1,\ldots ,r_{j-1}\in M_j=\{g\in R\mid g({\bf a_j})=0\}$.
But noting $r_j\notin M_j$,
we conclude that $r_j\notin r_1R+\cdots +r_{j-1}R$.
Now, since $R$ is Noetherian, after a
finite number of steps, we will get to some $L$ such that
$r_1R+\cdots +r_LR=R$.
We can use the {\em effective Nullstellensatz} to explicitly find
those $g_i$'s in
$R$ such that $r_1g_1+\cdots +r_Lg_L= 1$.
Define, now, $b_0,b_1,\ldots ,b_L\in R[X]$ in the following way:
\begin{eqnarray}
b_0 & = & 0 \nonumber\\
b_1 & = & r_1g_1X \nonumber\\
b_2 & = & r_1g_1X+r_2g_2X \nonumber\\
& \vdots & \nonumber\\
b_L & = & r_1g_1X+r_2g_2X+\cdots +r_Lg_LX=X.
\end{eqnarray}
Then these $b_i$'s satisfy the recursive relations:
\begin{eqnarray}
b_0 & = & 0 \nonumber\\
b_i & = & b_{i-1} + r_ig_iX \quad {\rm for}\ i=1,\ldots ,L.
\end{eqnarray}
{\bf *Claim:}
For each $i\in \{ 1,\ldots ,L\}$, there exists $B_i\in SL_2(R[X])$
and $B_i'\in E_n(R[X])$ such that
${\bf v}(b_i)=B_iB_i'{\bf v}(b_{i-1})$.
If this claim is true, then using $E_n(R[X])\cdot SL_2(R[X])\subseteq
SL_2(R[X])\cdot E_n(R[X])$ (Normality of $E_n(R[X])$; {\bf Corollary 2}),
we inductively get
\begin{eqnarray}
{\bf v}(X) & = & {\bf v}(b_L)\nonumber\\
& = & B_LB_L'{\bf v}(b_{L-1})\nonumber\\
& \vdots & \nonumber\\
& = & BB'{\bf v}(b_0)\nonumber\\
& = & BB'{\bf v}(0)
\end{eqnarray}
for some $B\in SL_2(R[X])$
and $B'\in E_n(R[X])$. Therefore it's enough to prove the above
claim.
For this purpose, let $\tilde{G_i} =G_i+q_{i2}$. Then
\begin{eqnarray}
\left(\! \begin{array}{cc} 1 & 0 \\ 0 & E(X) \end{array} \!\right)
{\bf v}(X)=\left(\! \begin{array}{c} v_1(X) \\ \tilde{G_i}(X) \\
q_{i3}(X) \\ \vdots \\ q_{in}(X) \end{array} \!\right).
\end{eqnarray}
For $3\leq l \leq n$,
we have
\begin{eqnarray}
q_{il}(b_i)-q_{il}(b_{i-1}) & \in & (b_i - b_{i-1})\cdot R[X] \nonumber\\
& = & r_ig_iX \cdot R[X].
\end{eqnarray}
Since $r_i\in R$ doesn't depend on $X$, we have
\begin{eqnarray}
r_i & = & f_i(X)v_1(X)+h_i(X)\tilde{G_i}(X)\nonumber\\
& = & f_i(b_{i-1})v_1(b_{i-1})+h_i(b_{i-1})\tilde{G_i}
(b_{i-1})\nonumber\\
& = & {\rm a\ linear\ combination\ of}\ v_1(b_{i-1})\
{\rm and}\ \tilde{G_i}(b_{i-1})\ {\rm over}\ R[X].
\end{eqnarray}
Therefore, we see that for $3\leq l \leq n$,
\begin{eqnarray}
q_{il}(b_i)=q_{il}(b_{i-1})+{\rm a\ linear\ combination\ of}\ v_1(b_{i-1})\
{\rm and}\ \tilde{G_i}(b_{i-1})\ {\rm over} R[X].\nonumber
\end{eqnarray}
Hence we can find $C\in E_n(R[X])$ such that
\begin{eqnarray}
C \left(\! \begin{array}{cc} 1 & 0 \\ 0 & E(b_{i-1}) \end{array} \!\right)
{\bf v}(b_{i-1}) & = &
C \left(\! \begin{array}{c} v_1(b_{i-1}) \\ \tilde{G}_i(b_{i-1}) \\
q_{i3}(b_{i-1}) \\ \vdots \\ q_{in}(b_{i-1}) \end{array} \!\right)
\nonumber\\
& = & \left(\! \begin{array}{c} v_1(b_{i-1})\\ \tilde{G}_i(b_{i-1}) \\
q_{i3}(b_{i}) \\ \vdots \\ q_{in}(b_{i}) \end{array} \!\right).
\end{eqnarray}
Now, by the {\bf Lemma~\ref{lem;link}}, we can find
$\tilde{B} \in SL_2(R[X])$ such that
\begin{eqnarray}
\tilde{B} \left(\! \begin{array}{c} v_1(b_{i-1}) \\ \tilde{G_i}(b_{i-1})
\end{array} \!\right)=\left(\! \begin{array}{c} v_1(b_i) \\ \tilde{G_i}(b_i)
\end{array} \!\right).
\end{eqnarray}
Finally, define $B\in SL_n(R[X])$ as follows:
\begin{eqnarray}
B=\left(\! \begin{array}{cc} 1 & 0 \\ 0 & E(b_i)^{-1} \end{array} \!\right)
\left(\! \begin{array}{cc} \tilde{B} & 0 \\ 0 & I_{n-2} \end{array} \!\right)
\cdot C\cdot \left(\!\begin{array}{cc} 1 &0 \\ 0& E(b_i)\end{array}\!\right).
\end{eqnarray}
Then this $B$ satisfies
\begin{eqnarray}
B{\bf v}(b_{i-1})={\bf v}(b_i),
\end{eqnarray}
and by using the normality of $E_n(R[X])$ again, we see that
\begin{eqnarray}
B\in SL_2(R[X])E_n(R[X])
\end{eqnarray}
and this proves the claim.
\hspace*{\fill}{\bf $\Box$} \bigskip \\
\section{Realization Algorithm for $SL_3(R[X])$}
Now, we want to find a realization
algorithm for the matrices of the special type in $SL_3(k[x_1,\ldots ,x_m])$,
i.e.
matrices of the form $\left(\!\begin{array}{ccc} p & q & 0 \\ r & s & 0 \\ 0 &
0 & 1
\end{array}\!\right) \in SL_3(k[x_1,\ldots ,x_m])$. Again, by applying
a change of variables, we may assume
that $p\in k[x_1,\ldots ,x_m]$ is a monic polynomial in the
last variable $x_m$. In view of the
{\em Quillen Induction Algorithm} developed in the section 3, we see that
it's enough to develop a realization algorithm for the matrices of the
form $\left(\!\begin{array}{ccc} p & q & 0 \\ r & s & 0 \\ 0 & 0 & 1
\end{array}\! \right) \in SL_3(R[X])$, where $R$ is now a commutative local
ring and $p\in R[X]$ is a monic polynomial. A realization algorithm for this
case
was obtained by M.P. Murthy, and we present in the below a slightly
modified version of the {\bf Lemma 3.6}
in \cite{murthy} {\sl Suslin's Work on Linear Groups over Polynomial Rings
and Serre Problem} by S.K. Gupta and M.P. Murthy.
\begin{lemma}\label{lemma:split}
Let $L$ be a commutative ring, and $a,a',b\in L$. Then, the followings are
true.
\begin{enumerate}
\item $(a,b)$ and $(a',b)$ are unimodular over $L$ if and only if
$(aa',b)$ is unimodular over $L$.
\item For any $c,d\in L$ such that $aa'd-bc=1$,
there exist $c_1,c_2,d_1,d_2\in L$ such that $ad_1-bc_1=1,\ a'd_2-bc_2=1$, and
$$\left(\!\begin{array}{ccc} aa' & b & 0 \\ c & d & 0 \\ 0 & 0 & 1
\end{array}\!\right)\equiv \left(\!\begin{array}{ccc}a&b&0\\ c_1&d_1&0\\ 0&0&1
\end{array}\!\right)\cdot \left(\!\begin{array}{ccc}a'&b&0\\ c_2&d_2&0\\ 0&0&1
\end{array}\!\right) \pmod {E_3(L)}.$$
\end{enumerate}
\end{lemma}
\noindent {\bf Proof:\ }
(1) If $(aa',b)$ is unimodular over $L$, there exist $h_1,h_2\in L$ such that
$h_1\cdot (aa')+h_2\cdot b=1$. Now $(h_1a')\cdot a+h_2\cdot b=1$ implies
$(a,b)$ is unimodular, and $(h_1a)\cdot a'+h_2\cdot b=1$ implies $(a',b)$
is unimodular.
Suppose, now, that $(a,b)$ and $(a',b)$ are unimodular over $L$. Then, we can
find $h_1,h_2,h_1',h_2'\in L$ such that
$h_1a+h_2b=1,\ h_1'a'+h_2'b=1$.
Now, let $g_1=h_1h_1',\ g_2=h_2'+a'h_2h_1'$, and consider
\begin{eqnarray}
g_1aa'+g_2b & = & h_1h_1'aa'+(h_2'+a'h_2h_1')b \nonumber\\
& = & h_1'a'(h_1a+h_2b)+h_2'b\nonumber\\
& = & h_1'a'+h_2'b\nonumber\\
& = & 1.
\end{eqnarray}
So we have a desired unimodular relation.
\medskip
\noindent (2) If $c,d\in L$ satisfy $aa'd-bc=1$, then $(aa',b)$ is
unimodular, which in turn implies that $(a,b)$ and $(a',b)$ are unimodular.
Therefore, we can find
$c_1,d_1,d_1,d_2\in L$ such that $ad_1-bc_1=1$ and $a'd_2-bc_2=1$.
For example, we can let
\begin{eqnarray}
c_1=c_2=c,\quad d_1=a'd, \quad d_2=ad.
\end{eqnarray}
Now, consider
\begin{eqnarray}
\left(\!\begin{array}{ccc} aa' & b & 0 \\ c & d & 0 \\ 0 & 0 & 1
\end{array}\!\right) & = & E_{21}(cd_1d_2-d(c_2+a'c_1d_2))
\left(\!\begin{array}{ccc}aa'&b& 0\\ c_2+a'c_1d_2&d_1d_2 &0 \\ 0&0&1
\end{array}\!\right)\nonumber\\
& = & E_{21}(cd_1d_2-d(c_2+a'c_1d_2))E_{23}(d_2-1)E_{32}(1)E_{23}(-1)
\nonumber\\
& & \left(\!\begin{array}{ccc} a & b & 0\\ c_1 & d_1 & 0\\ 0 & 0 & 1
\end{array}\!\right)E_{23}(1)E_{32}(-1)E_{23}(1)
\left(\!\begin{array}{ccc}a'&b&0\\ c_2&d_2&0 \\ 0&0&1\end{array}
\!\right)\nonumber\\
& & E_{23}(-1)E_{32}(1)E_{23}(a-1)E_{31}(-a'c_1)E_{32}(-d_1).
\end{eqnarray}
This explicit expression tells us that
\begin{eqnarray}
\left(\!\begin{array}{ccc} aa' & b & 0 \\ c & d & 0 \\ 0 & 0 & 1
\end{array}\!\right) & \equiv & \left(\!\begin{array}{ccc} a&b&0\\ c_1&d_1&0
\\ 0&0&1\end{array}\!\right) \cdot \left(\!\begin{array}{ccc}a'&b&0
\\ c_2&d_2&0\\ 0&0&1\end{array}\!\right)\pmod {E_3(L)}.
\end{eqnarray}
\hspace*{\fill}{\bf $\Box$} \bigskip \\
\begin{thm}
Suppose $(R,M)$ is a commutative local ring, and
$A=\left(\!\begin{array}{ccc} p & q & 0 \\ r & s & 0 \\ 0 & 0 & 1
\end{array}\!\right) \in SL_3(R[X])$ where $p$ is monic.
Then $A$ is realizable over $R[X]$.
\end{thm}
\noindent {\bf Proof:\ }
By induction on $\deg (p)$. If $\deg (p)=0$, then $p=0\ \mbox{or}\ 1$, and
$A$ is clearly
realizable. Now, suppose $\deg(p)=d>0$ and $\deg(q)=l$. Since $p\in R[X]$
is monic, we can find $f,g\in R[X]$ such that
\begin{eqnarray}
q & = & fp+g,\quad \deg(g)<d.
\end{eqnarray}
Then,
\begin{eqnarray}
AE_{12}(-f) & = & \left(\!\begin{array}{ccc}p&q-fp&0\\ r&s-fr&0\\ 0&0 & 1
\end{array}\!\right)=\left(\!\begin{array}{ccc}p&g&0\\ r&s-fr&0\\ 0&0 & 1
\end{array}\!\right).
\end{eqnarray}
Hence we may assume $\deg(q)<d$. Now, we note that either $p(0)$ or
$q(0)$ is a unit in $R$, otherwise, we would have $p(0)s(0)-q(0)r(0)\in M$
that contradicts to $ps-qr=p(0)s(0)-q(0)r(0)=1$. Let's consider these
two cases, separately.
\medskip
\noindent Case 1: When $q(0)$ is a unit.
\newline Using the invertibility of $q(0)$, we have
\begin{eqnarray}
AE_{21}(-q(0)^{-1}p(0)) & = & \left(\!\begin{array}{ccc}
p-q(0)^{-1}p(0)q & q&0\\ r-q(0)^{-1}p(0)s& s &0\\ 0&0&1
\end{array}\!\right).
\end{eqnarray}
So, we may assume $p(0)=0$. Now, write $p=Xp'$. Then, by the above
{\bf Lemma}~\ref{lemma:split}, we can find $c_1,d_1,c_2,d_2\in R[X]$
such that $Xd_1-qc_1=1,\ p'd_2-qc_2=1$ and
\begin{eqnarray}
\left(\!\begin{array}{ccc} p & q & 0 \\ r & s & 0 \\ 0 & 0 & 1
\end{array}\!\right) & \equiv & \left(\!\begin{array}{ccc}X&q&0\\ c_1&d_1&0
\\ 0&0&1\end{array}\!\right) \cdot \left(\!\begin{array}{ccc}p'&q&0
\\ c_2&d_2&0\\ 0&0&1\end{array}\!\right)\pmod {E_3(R[X])}
\end{eqnarray}
Since $\deg(p')<d$, the second matrix on the right hand side is
realizable by the induction hypothesis. As for the first one, we may
assume that $q$ is a unit of $R$ since we can assume $\deg(q)<\deg(X)=1$
and $q(0)$ is a unit. And then invertibility of $q$ leads easily to
an explicit factorization of $\left(\!\begin{array}{ccc}X&q&0\\ c_1&d_1&0
\\ 0&0&1\end{array}\!\right)$ into elementary matrices.
\medskip
\noindent Case 2: When $q(0)$ is not a unit.
\newline First we claim the following; there exist $p',q'\in R[X]$
such that $\deg(p')<l,\deg(q')<d$ and $p'p-q'q=1$. To prove this
claim, we let $r\in R$ be the resultant of $p$ and $q$. Then, there
exist $f,g\in R[X]$ with $\deg(f)<l,\deg(g)<d$ such that
$fp+gq=r$. Since $p$ is monic and $p,q\in R[X]$ generate the unit ideal,
we see that $r\notin M$, i.e. $r\in A^{*}$. Now, letting
$p'=f/r,\ q'=-g/r$ shows the claim. Also note that the two relations,
$p'(0)p(0)-q'(0)q(0)=1$ and $q(0)\in M$, imply $p'(0)\notin M$. This means
$q(0)+p'(0)$ is a unit. Now, consider the following.
\begin{eqnarray}
\left(\!\begin{array}{ccc} p & q & 0 \\ r & s & 0 \\ 0 & 0 & 1
\end{array}\!\right) & = & E_{21}(rp'-sq')
\left(\!\begin{array}{ccc} p & q & 0 \\ q' & p' & 0 \\ 0 & 0 & 1
\end{array}\!\right)\nonumber\\
& = & E_{21}(rp'-sq')E_{12}(-1)
\left(\!\begin{array}{ccc} p+q'& q+p'&0\\ q' & p' & 0 \\ 0 & 0 & 1
\end{array}\!\right).
\end{eqnarray}
Noting that the last matrix on the right hand side is realizable
by the Case~1 since $q(0)+p'(0)$ is a unit and $\deg(p+q')=d$, we see
that $\left(\!\begin{array}{ccc} p & q & 0 \\ r & s & 0 \\ 0 & 0 & 1
\end{array}\!\right)$ is also realizable.
\hspace*{\fill}{\bf $\Box$} \bigskip \\
\section{Eliminating Redundancies}
When applied to a specific matrix, the algorithm presented in this
paper will produce a factorization into elementary matrices, but
this factorization may contain more factors than is necessary.
The {\em Steinberg relations} \cite{milnor}
from algebraic $K$--theory provide a method for
improving a given factorization by eliminating some of the
unnecessary factors.
The {\em Steinberg relations} that elementary matrices satisfy are
\begin{enumerate}
\item
$E_{ij}(0) = I$
\item
$E_{ij}(a)E_{ij}(b) = E_{ij}(a+b)$
\item
For $i\neq l$, $[E_{ij}(a),E_{jl}(b)]=E_{ij}(a)E_{jl}(b)E_{ij}(-a)E_{jl}(-b)
= E_{il}(ab)$
\item
For $j\neq l$, $[E_{ij}(a),E_{li}(b)]= E_{ij}(a)E_{li}(b)E_{ij}(-a)E_{li}(-b)
= E_{lj}(-ab)$
\item
For $i\neq p$, $j\neq l$, $[E_{ij}(a),E_{lp}(b)]= E_{ij}(a)E_{lp}(b)E_{ij}(-a)
E_{lp}(-b)=I.$
\end{enumerate}
The first author is in the process of implementing the realization algorithm
of this paper, together with a {\em Redundancy Elimination Algorithm}
based on the above set of relations, using existing computer algebra systems.
As suggested in \cite{thk}, an algorithm of this kind has application
in {\em Signal Processing} since it gives a way of expressing a given
multidimensional filter bank as a cascade of simpler filter banks.
\section{Acknowledgement}
The authors wish to thank A. Kalker, T.Y. Lam, R. Laubenbacher, B. Sturmfels
and M. Vetterli for all the valuable support, insightful discussions
and encouragement.
|
1996-09-15T19:18:28 | 9405 | alg-geom/9405013 | en | https://arxiv.org/abs/alg-geom/9405013 | [
"alg-geom",
"hep-th",
"math.AG"
] | alg-geom/9405013 | Vadim Schechtman | Vladimir Hinich and Vadim Schechtman | Deformation theory and Lie algebra homology | amslatex (Replacement of the previous version. Minor corrections are
made) | null | null | null | null | A description of a ring of functions on the base of a universal formal
deformation for several moduli problems is given. The answer is given in terms
of a homology group of a certain dg Lie algebra canonically (up to an
essentially unique quasi-isomorphism) associated with a problem.
| [
{
"version": "v1",
"created": "Wed, 25 May 1994 18:36:20 GMT"
},
{
"version": "v2",
"created": "Wed, 14 Sep 1994 15:32:45 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Hinich",
"Vladimir",
""
],
[
"Schechtman",
"Vadim",
""
]
] | alg-geom | \section{Introduction}
\subsection{}
\label{pose} Let $X$ be a smooth proper scheme $X$ over a field $k$
of characteristic $0$, $G$ an algebraic group over $k$ and $p: P\lra X$
a $G$-torsor over $X$. Consider the following deformation
problems.
{\bf Problem 1.} Flat deformations of $X$.
{\bf Problem 2.} Flat deformations of the pair $(X,P)$.
{\bf Problem 3.} Deformations of $P$ ($X$ being fixed).
According to Grothendieck, one can assign to a deformation problem
of the above type a sheaf of Lie algebras
over $X$ which can be defined as "a sheaf of infinitesimal automorphisms"
of the corresponding deformation functor (cf. Section ~\ref{univers}).
Let us describe the sheaves corresponding to our problems. For Problem 1 it is
a tangent sheaf $\CA_1=\CT_X$. For Problem 2 it is a sheaf $\CA_2=\CA_P$
defined as follows. For a Zariski open $U\subset X$, $\Gamma(U,\CA_P)$
is the space
of $G$-invariant vector fields on $p^{-1}(U)$. The map $p$ induces
(epimorhic) map $\epsilon:\CA_P\lra\CT_X$. For Problem 3, $\CA_3=\fg_P:=
\ker(\epsilon)$. The last sheaf may be also defined as a sheaf of sections
of a vector bundle associated with $P$ and the adjoint representation
of $G$ on its Lie algebra. Note that the sheaves $\CA_i$ are locally free
$\CO_X$-modules of finite type (but the bracket is not $\CO_X$-linear for
$i=1,2$).
Suppose that $H^0(X,\CA_i)=0$. Then it is known that there exists
a universal formal deformation space $\fS_i=\Spf(R_i)$ for Problem $i$
(see Section ~\ref{univers}). Here $R_i$ is a complete local $k$-algebra
with residue field $k$. Let $\bm_{R_i}$ denote its maximal ideal.
We have the Kodaira-Spencer isomorphism
\begin{equation}
\label{ksg}
\kappa^1: T_{\fS_i,s}=(\bm_{R_i}/\bm_{R_i}^2)^*\iso H^1(X,\CA_i)
\end{equation}
describing the tangent space of $\fS_i$ at the closed point $s$.
The main goal of the present paper is to describe, in case when $\fS_i$ is
smooth, the whole ring $R_i$ in terms of the sheaf $\CA_i$.
\subsection{}
\label{sullivan} To formulate the answer, we need a certain cohomological
construction. Let $\fg$ be a sheaf of $k$-Lie algebras over $X$ which is
also a quasicoherent
$\CO_X$-module. Consider an affine open covering $\CU$
of $X$ and the corresponding complex of \v{C}ech cochains $\CHC(\CU,\fg)$.
Using a generalization of Thom-Sullivan construction
\footnote{introduced in ~\cite{hlha}} used in Rational
homotopy theory one can construct (see Section ~\ref{direct})
a certain differential graded Lie algebra
$R\Gamma^{Lie}(X,\fg)$ canonically quasi-isomorphic to $\CHC(\CU,\fg)$.
This dg Lie algebra does not depend, up to (essentially) unique
quasi-isomorphism, on a covering, hence we omit it from the notation.
Now one can apply to $R\Gamma^{Lie}(X,\fg)$ the Quillen functor $C$
(which is a generalization to dg Lie algebras of the Chevalley complex
computing the homology of a Lie algebra with trivial coefficients) and
get the complex $C(R\Gamma^{Lie}(X,\fg))$. This complex carries the canonical
increasing filtration $\{ F_nC(R\Gamma^{Lie}(X,\fg))\}$ with
graded quotients isomorphic to symmetric powers
$S^n(R\Gamma^{Lie}(X,\fg)[1])$ (for details, see \ref{quillen}).
Let us define homology spaces
$$
H^{Lie}_i(R\Gamma^{Lie}(X,\fg)):=H^{-i}(C(R\Gamma^{Lie}(X,\fg)));\
F_nH^{Lie}_i(R\Gamma^{Lie}(X,\fg)):=H^{-i}(F_nC(R\Gamma^{Lie}(X,\fg))).
$$
These spaces depend only on $X$ and on the sheaf $\fg$.
We have evident maps
$$
\ldots\lra F_{n-1}H^{Lie}_i(R\Gamma^{Lie}(X,\fg))\lra
F_nH^{Lie}_i(R\Gamma^{Lie}(X,\fg))\lra\ldots\lra
H^{Lie}_i(R\Gamma^{Lie}(X,\fg))
$$
and
$$
F_nH^{Lie}_i(R\Gamma^{Lie}(X,\fg))\lra
H^{n-i}(\Lambda^iR\Gamma^{Lie}(X,\fg))
$$
($\Lambda^i$ denotes the exterior power).
\subsection{} Return to the assumptions ~\ref{pose}. Pick $i=1,2$ or $3$.
Recall that we suppose that $H^0(X,\CA_i)=0$. Suppose also that
$\fS_i$ is {\em smooth} i.e. isomorphic to a formal power series ring
over $k$.
Let $R^*_i$ denote the space of continuous $k$-linear maps
$R_i\lra k$ ($k$ is equipped with the discrete topology). The main result
of this paper is (see Thm.~\ref{complet}).
\subsubsection{}
\label{thm-intro}
\begin{thm}{} One has compatible canonical isomorphisms
\begin{equation}
\label{isomor}
\kappa: R^*_i\iso H^{Lie}_0(R\Gamma^{Lie}(X,\CA_i));\
\end{equation}
\begin{equation}
\kappa^{\leq n}: (R_i/\bm^{n+1}_{R_i})^*\iso
F_nH^{Lie}_0(R\Gamma^{Lie}(X,\CA_i))
\end{equation}
After passing to the graded quotients, isomorphisms $\kappa^{\leq n}$ induce
isomorphisms
$$
S^n(T_{\fS_i,s})\cong (\bm^{n}/\bm^{n+1})^*\iso
H^{n}(\Lambda^nR\Gamma^{Lie}(X,\CA_i))\cong S^nH^1(X,\CA_i)
$$
which coincide with $(-1)^n\kappa^1$.
\end{thm}
\subsection{} Let us describe the contents of the paper in a more detail.
Our construction of the isomorphism $\kappa$ is based on the construction of
{\bf higher Kodaira-Spencer morphisms} --- they are relative versions
of ~(\ref{isomor}). Similarly to a usual KS map they are not necessarily
isomorphisms and are defined for arbitrary --- not necessarily universal ---
deformation\footnote{this idea is used in ~\cite{bs}}.
In the same way as the usual KS map comes from a certain boundary homomorphism
of a short exact sequence of complexes, our higher KS map comes from
a map, which we call {\bf connecting morhism}. It arises from
an extension of {\em dg Lie algebras}
$$
0\lra\fg\lra\fa\lra\ft\lra 0
$$
and maps the enveloping algebra of the cone of the map $\fg\ra\fa$
(which is quasi-isomorphic to $U(\ft)$) to the standard complex
$C(\fg)$.
The construction is given in Section ~\ref{envstand}, cf. ~\ref{constr-thm}.
For applications we need a generalization of this construction to
{\em dg Lie algebroids}. The simplest example of a sheaf of Lie algebroids is
the tangent sheaf $\CT_S$ over a variety $S$. We borrow
their definition from ~\cite{bb} and ~\cite{bfm}. Following
{\em loc.cit.}, one defines an
analogue of enveloping algebra for Lie algebroids --- we call them
{\em twisted enveloping algebras}. For example, the twisted enveloping algebra
of $\CT_S$ will be the sheaf $\Diff_S$ of differential operators
provided $S$ is smooth.
These sheaves are sources of higher KS maps. Section ~\ref{twisted}
generalizes the principal construction of Section ~\ref{envstand} to
(sheaves of) dg Lie algebroids.
The construction of connecting morphisms
is our first point. (They may be of an independent interest.)
Section ~\ref{formal} is technical; we discuss there some basic
properties of differential operators on formal schemes.
\subsection{} In Sections~\ref{direct} and~\ref{thoms} we develop
certain formalism of {\bf Homotopy Lie algebras}. We define there for
(locally noetherian)
schemes $X$ categories $\Holie^{qc}(X)$ which contain sheaves
of dg Lie algebras with bounded cohomology and flat quasicoherent components,
and have good functorial properties. For example, the functor $R\Gamma^{Lie}$
mentioned in ~\ref{sullivan} is a particular case of the {\bf direct image}
functor --- its construction is the main result of these Sections.
Although it is a technical tool, we regard the contents of Sect.~\ref{direct}
and ~\ref{thoms} as a {\bf second main outcome} of the present paper.
We believe that unlike usual types of algebras,
{\em Homotopy Lie algebras} are correctly defined only as objects of
appropriate {\em Homotopy categories}. The main property one needs
from them is good functorial behaviour. This is what Thom-Sullivan
functor does. The success of the construction is due to the fact that
Thom-Sullivan functor has excellent exactness, base change, etc.
properties. They are proved in ~\ref{thoms}.
Another approach to the definition and functoriality of Homotopy Lie algebras
has been developed in ~\cite{hla}. In {\em loc.cit.} a HLA was a
"dg Lie algebra up to higher homotopies". There are strong indications that
both definitions give rise to equivalent Homotopy categories.
It seems that one can use also cosimplicial Lie algebras
for the construction of the same Homotopy category.
From this point of view, different versions of HLA are nothing but
different "models" of these Homotopy categories. We believe that
every "model" might be useful in applications.
Of course these remarks apply also to other types of algebras.
\subsection{} In Section ~\ref{kodaira} we take all
the results together and start the machine of Section ~\ref{twisted} which
cooks up higher KS maps for us. In Section ~\ref{univers} we discuss
universal formal deformations.
\subsection{} One can generalize the above approach and get a description
of sheaves of differential operators on moduli spaces acting on natural
vector bundles (for example, determinant bundles) in terms of Lie algebra
homology (with non-trivial coefficients).
For example, for operators of order $\leq 1$ on determinant
bundles one gets a result equivalent to ~\cite{bs}.
Also there are strong indications that combining our description with
Serre duality it is possible to get a "global counterpart" of the results
of ~\cite{bd}.
We will return to these subjects in future publications.
\subsection{} The general main idea that
{\bf the completion of a local ring of a moduli
space at a given point $X$ is isomorphic to the dual of the $0$-th
homology group
of the "Lie algebra of infinitesimal automorphisms of $X$"}
was spelled out very clearly a few years ago by Drinfeld,
Deligne, Feigin (cf. ~\cite{d}, ~\cite{del}, \cite{f}). We knew this
idea from Drinfeld.
For deformations of complex structures a result analogous
to Theorem ~\ref{thm-intro} was proven in ~\cite{gm} (by different argument).
The present paper develops further the results of ~\cite{hdt}.
Note also some recent related work: ~\cite{ev}, \cite{r}.
This paper owes much to various ideas of A.Beilinson and V.Drinfeld.
We express to them our deep gratitude. We thank H.~Esnault, V.~Ginzburg
and E.~Viehweg for useful discussions.
We are especially grateful to Professor Han Sah who made possible several
visits of V.H. to Stony Brook.
\section{Enveloping algebras and standard complexes}
\label{envstand}
\subsection{Preliminaries}
\subsubsection{} Let us fix some notations and sign conventions
(cf. ~\cite{de}, 1.1).
Throughout this Section we fix a commutative ground ring $k$ of containing
$\Bbb Q$.
$\Mod (k),\ \Gr (k)$ will denote a category of $k$-modules and that of
$\Bbb Z$-graded $k$-modules respectively. $\CC (k)$
denotes the category of complexes of $k$-modules (all differentials have
degree $+1$). We have an obvious forgetful functor $\CC (k)\lra \Gr (k)$.
If $X\in \CC (k)$ (or $\Gr (k)$), $x\in X^p$, we refer to $p$ as to degree of
$x$, and denote it $|x|$.
We identify $k$-modules with complexes concentrated in degree $0$.
If $n\in \Bbb Z$, $X\in \Gr (k)$, $X[n]$ will denote the shifted module
$X[n]^p=X^{p+n}$. If $X\in \CC(k)$, we define $X[n]\in \CC(k)$ which is as
above as a graded module, the differential being $d_{X[n]}=(-1)^nd_X$.
{\em A map of degree $n$}, $f: X\lra Y$ in $\CC(k)$ is by definition a
morphism $f: X\lra Y[n]$ in $\Gr (k)$. For such an $f$ we set $df=d_Y\circ f+
(-1)^nf\circ d_X: X\lra Y[n+1]$.
The category $\CC (k)$ has a tensor structure --- for $X,Y\in \CC (k)$,
$X\otimes Y$ is the usual tensor product of complexes over $k$. We have
natural associativity and commutativity isomorphisms
\begin{equation}
\label{assoc}
a_{X,Y,Z}:(X\otimes Y)\otimes Z\iso X\otimes (Y\otimes Z),\
(x\otimes y)\otimes z\mapsto x\otimes (y\otimes z)
\end{equation}
and
\begin{equation}
\label{r}
R_{X,Y}: X\otimes Y\iso Y\otimes X
\end{equation}
defined by the formula $R_{X,Y}(x\otimes y)=(-1)^{|x||y|}y\otimes x$.
This formula, as well as other formulas in the "dg world" are obtained by
implementing {\em the Quillen sign rule}:
"whenever something of degree
$p$ is moved past something of degree $q$ the sign $(-1)^{pq}$ accrues",
{}~\cite{q}, p. 209.
Isomorphisms ~(\ref{assoc}) and ~(\ref{r}) endow $\CC(k)$ with a structure
of a
{\em symmetric monoidal category} in the sense of MacLane, ~\cite{mac}.
We have canonical {\em shifting} isomorphisms
\begin{equation}
\label{shift}
X[n]\otimes Y[m]\iso (X\otimes Y)[n+m]
\end{equation}
sending $x\otimes y$, $x\in X^i,\ y\in Y^j$ to $(-1)^{im}x\otimes y$.
(They may be obtained by identifying (following ~\cite{de}, 1.1) $X[n]$ with
$k[n]\otimes X$ , and applying $R_{X,k[m]}$).
{\em Algebras.} The structure of a symmetric monoidal category allows one
to define usual types of algebras in $\CC (k)$.
We shall refer to them by adding "dg" to their original name.
More specifically, we shall use the following algebras (cf. ~\cite{q}).
--- A {\em dg Lie algebra} is a complex $X$ together with a bracket
$[,]: X\otimes X\lra X$ which is skew symmetric, i.e.
$[,]\circ R_{X,X}=-[,]$, and satisfies the Jacobi identity
$$
[x,[y,z]]+(-1)^{|x|(|y|+|z|)}[y,[z,x]]+(-1)^{|z|(|x|+|y|)}[z,[x,y]]=0
$$
We shall denote $\Dglie=\Dglie_k$ the category of dg Lie algebras
over $k$.
--- A {\em dg coalgebra} is a complex $X$ together with an coassociative
comultiplication $\Delta: X\lra X\otimes X$ and a counit $\epsilon: X\lra k$.
$X$ is called {\em cocommutative} if $\Delta= R_{X,X}\circ \Delta$.
We shall denote $\Dgcoalg=\Dgcoalg_k$ the category of cocommutative
dg coalgebras over $k$.
--- A {\em dg algebra} is a complex $X$ together with an associative
multiplication $\mu: X\otimes X\lra X$ and the unit $1\in X^0$ (which
may be considered as a map $k\lra X$). $X$ is called {\em commutative}
is $\mu=\mu\circ R_{X,X}$.
--- A {\em dg Hopf algebra} is a complex $X$ together with a multiplication
$\mu$, comultiplication $\Delta$, a unit and a counit, making it
a dg algebra and a dg coalgebra, and such that these two structures are
compatible in the standard way. In particular, $\Delta$ is
a map of dg algebras, where the multiplication in $X\otimes X$ is defined
by the rule $(x\otimes y)(x'\otimes y')=(-1)^{|y||x'|}xx'\otimes yy'$.
$X$ is called (co)commutative if the underlying (co)algebra is.
An element $x\in X$ is called {\em primitive} if
$\Delta (x)=x\otimes 1+1\otimes x$.
\subsubsection{}
\label{connected}
An element $u$ of a dg coalgebra $C$ is called {\em group-like}
if the following conditions are fulfilled:
(i) $du=0$ (ii) $\Delta(u)=u\otimes u$ (iii) $\epsilon(u)=1\in k$
A group-like element $u\in C$ defines a decomposition $C=k\cdot u\oplus
C^+$ with $C^+=\ker(\epsilon)$.
Let $\pi_u:C\ra C^+$ be the projection onto the second summand.
For a positive integer $n$ define the map
$$
\Delta^{(n)}: C\lra C^{\otimes n}
$$
by induction on $n$: set $\Delta^{(1)}=\id_C,
\ \Delta^{(n)}=(\Delta\otimes \id_{C^{\otimes(n-2)}})\circ \Delta^{(n-1)}$.
The choice of a group-like element $u$ defines an increasing filtration
$F^u=\{F^u_n\}$ of $C$ by the formula
$$ F^u_n=\ker\left(C\overset{\Delta^{(n+1)}}{\lra}C^{\otimes n+1}
\overset{\pi_u^{\otimes n+1}}{\lra}(C^+)^{\otimes n+1}\right).$$
\begin{defn}{unital}
1. A group-like element $u\in C$ is called {\em a unit} if the corresponding
filtration $F^u$ is exhaustive: $C=\cup_{i=0}^{\infty} F_i$.
2. {\em A unital coalgebra} is a pair $(C,1_C)$ where $C$ is a coalgebra and
$1_C\in C$ is a unit it in.
\end{defn}
The category $\Dgcu(k)$ has as objects unital cocommutative dg $k$-coalgebras;
a morphism in $\Dgcu(k)$ is a coalgebra morphism preserving the units.
\subsubsection{} Let $X\in \CC (k)$. For an integer $n\geq 1$ denote
$T^n(X)$ its $n$-th tensor power $X\otimes\ldots \otimes X$. Set $T^0X=k$.
The direct sum $T(X)=\oplus_{n\geq 0} T^n(X)$ has a natural structure
of a dg algebra --- the {\em tensor algebra} of $X$.
The commutativity isomorphisms ~(\ref{r}) define the action of the symmetric
group on $n$ letters $\Sigma_n$ on $T^n(X)$. We denote $S^n(X)$, $\Lambda^n(X)$
the complexes of coinvariants (resp., coantiinvariants) of this action
and $\pi_{S,n}: T^n(X)\lra S^n(X)$,
$\pi_{\wedge,n}: T^n(X)\lra \Lambda^n(X)$
the canonical projections.
For $x_1,\ldots, x_n\in X$ we set
$$
x_1\cdot\ldots\cdot x_n=\pi_{S,n}(x_1\otimes\ldots \otimes x_n)\in S^n(X);
$$
$$
x_1\wedge\ldots\wedge x_n=\pi_{\wedge,n}(x_1\otimes\ldots \otimes x_n)\in
\Lambda^n(X)
$$
The projections $\pi_{S,n},\ \pi_{\wedge,n}$ have
canonical sections
\begin{equation}
\label{sym}
i_{S,n}: S^n(X)\lra T^n(X),\ i_{S,n}(x_1\cdot\ldots\cdot x_n)=
\frac{1}{n!}\sum_{\sigma\in \Sigma_n}\sigma(x_1\otimes\ldots\otimes x_n)
\end{equation}
\begin{equation}
\label{antisym}
i_{\wedge,n}: \Lambda^n(X)\lra T^n(X),\ i_{\wedge,n}(x_1\cdot\ldots\cdot x_n)=
\frac{1}{n!}\sum_{\sigma\in \Sigma_n}(-1)^{|\sigma|}
\sigma(x_1\otimes\ldots\otimes x_n)
\end{equation}
The isomorphisms ~(\ref{shift}) induce canonical isomorphisms
$$
a_n: T^n(X[1])\cong T^n(X)[n]
$$
such that for $\sigma\in \Sigma_n$, $a_n\circ \sigma=(-1)^{|\sigma|}\sigma
\circ a_n$. After passing to coinvariants, we get canonical shifting
({\em "d\'{e}calage"}) isomorphisms
\begin{equation}
\label{dec}
\dec_n:S^n(X[1])\cong \Lambda^n(X)[n]
\end{equation}
Explicit formula: for $x_i\in X^{p_i},\ i=1,\ldots , n$,
\begin{equation}
\label{decalage}
\dec_n(x_1\cdot\ldots\cdot x_n)=(-1)^{\sum_{i=1}^n (n-i)p_i}
(x_1\wedge\ldots\wedge x_n)
\end{equation}
\subsubsection{}
The direct sum $S(X)=\sum_{n\geq 0} S^n(X)$ is naturally a commutative
dg algebra. Let us define the counit $S(X)\lra k=S^0(X)$ as a canonical
projection, and a map $\Delta: S(X)\lra S(X)\otimes S(X)$ by two conditions
which characterize it uniquely:
(i) $\Delta(x)=x\otimes 1 + 1\otimes x$ for $x\in X=S^1(X)$
(ii) $\Delta$ is a map of dg algebras.
This makes $S(X)$ a commutative and cocommutative dg Hopf algebra, cf.
{}~\cite{q},
App B, 3.3.
The argument of {\em loc.cit.} shows that $S(X)$ is a unital dg coalgebra
with
$1_{S(X)}=1\in k=S^0(X)$. The corresponding filtration is
$F_iS(X)=\oplus_{p=0}^iS^p(X)$.
\subsubsection{Universal property of $S(X)$}
\label{coalg-s}
Let $C\in\Dgcu$ and suppose we are given a map of unital dg coalgebras
$f: C\lra S(X)$. Set $f_n=p_n\circ f: C\lra S^n(X)$ where $p_n: S(X)\lra
S^n(X)$
is the projection. We have $f_0=\epsilon_C$.
\begin{lem}{} For $n\geq 1$ we have
$$
f_n=\frac{1}{n!}\pi_{S,n}\circ f_1^{\otimes n}\circ \Delta^{(n)}
$$
\end{lem}
\begin{pf}
This follows from the compatibility of $f$ with the comultiplication.
\end{pf}
As a consequence, we get
\begin{prop}{} (\cite{q}, App. B, 4.4) The assignment $f\mapsto f_1$
establishes a bijection between the
set of all unital dg coalgebra maps $f: C\lra S(X)$ and the set of maps
$f_1: C\lra X$
in $\CC (k)$ such that $f_1(1_C)=0$.
\end{prop}
This follows from the previous lemma and the remark that for
$x\in F_iC$ we have $f_n(x)=0$ for $n>i$. $\Box$
\subsubsection{Universal enveloping algebra}
\label{envel} If $\fg$ is a dg Lie algebra,
its {\em universal enveloping algebra $U(\fg)$} is a dg algebra which
is a quotient of the tensor algebra $T(\fg)$ by the two-sided dg ideal
generated by all elements
$$
xy-(-1)^{|x||y|}yx -[x,y],
$$
$x,y\in \fg=T^1(\fg)$ homogeneous.
The composition $\fg=T^1(\fg)\hra T(\fg)\lra U(\fg)$ is injective;
one identifies $\fg$ with the subcomplex of $U(\fg)$. Evidently,
$U(\fg)$ is generated by $\fg$ as a dg algebra. We shall denote
$F_iU(\fg)\subset U(\fg)$ the subspace spanned by all products of $\leq i$
elements of $\fg$. We have $F_0U(\fg)=k\cdot 1$.
$U(\fg)$ has a canonical structure of a unital cocommutative dg Hopf
algebra, the comultiplication being defined uniquely by the
requirement that $\fg$ consists of primitive elements. (See ~\cite{q} for
details.) $U(\fg)^+\subset U(\fg)$ will denote the kernel of the counit
({\em augmentation ideal}); one has
a canonical decomposition $U(\fg)\cong k\cdot 1\oplus U(\fg)^+$.
\subsection{Quillen standard complex}
\label{quillen}
\subsubsection{} Let $X\in \CC (k)$. Consider the dg coalgebra $C(X):=S(X[1])$.
Using the d\'{e}calage isomorphisms we identify $C(X)$ with $\oplus_{n\geq 0}
\Lambda^n(X)[n]$.
Let us write down the precise formula for the comultiplication in $C(X)$.
Let $x_i\in X^{\alpha_i},\ i=1,\ldots , n$. For a finite subset
$I=\{ p_1,\ldots , p_i\}\subseteq\{ 1,\ldots , n\};\ p_1<\ldots <p_i$,
set
$$
x_{I}=x_{p_1}\wedge x_{p_2}\wedge\ldots \wedge x_{p_i}\in \Lambda^i(X),
\ x_{\emptyset}=1,
$$
$\bar I:=\{1,\ldots,n\}-I$. Then
\begin{equation}
\label{comult-in-c}
\Delta(x_1\wedge\ldots\wedge x_n)=\sum_I(-1)^{s(I)}x_I\otimes x_{\bar I}
\end{equation}
where the summation is over all subsets $I\subseteq \{1,\ldots,n\}$, and
signs $s(I)$
are defined by the rule
$$
x_1\wedge\ldots\wedge x_n=(-1)^{s(I)}x_I\wedge x_{\bar I}
$$
For future, we shall denote this sign $s(I;\alpha_1,\ldots \alpha_n)$.
Set
$$
C(X)^{pq}=(\Lambda^{-p}(X))^q
$$
(we agree that $\Lambda^p=0$ for $p<0$). It is clear that
$C(X)^n=\oplus_{p+q=n} C(X)^{pq}$. The differential in $C(X)$ has bidegree
$(0,1)$. For future, let us denote it $d_{II}$.
Note that $d_{II}^{-p,*}:\Lambda^p(X)^*\lra \Lambda^p(X)^{*+1}$ is equal to
$(-1)^p$ times the differential on $\Lambda^p(X)$ induced by that on $X$.
\subsubsection{} Now suppose that $\fg$ is a dg Lie algebra. Let us define
maps
$$
d_n: \Lambda^n(\fg)\lra \Lambda^{n-1}(\fg)
$$
by the formula
$$
d_n(x_1\wedge\ldots \wedge x_n)=\sum_{1\leq i<j\leq n}(-1)^{s(\{ i,j\};
\alpha_1,\ldots, \alpha_n)} [x_i,x_j]\wedge x_{\ol{\{ i,j\}}}
$$
where $x_i\in X^{\alpha_i}$.
In particular,
(i) {\em the composition $d_2\circ \pi_{\wedge, 2}$ coincides with the bracket
on $\fg$.}
Set
$$
d_I^{pq}=(d_{-p})^q: C(\fg)^{pq}\lra C(\fg)^{p+1,q};\ d_I=\sum_{p,q} d_I^{pq}
$$
Set $d=d_I+d_{II}$; it is an endomorphism of degree $1$ of the graded
$k$-module $C(\fg)$.
\begin{prop}{} (a) The map $d$ has the following properties.
(ii) $d^2=0$. In particular, $d_I^2=0$ and $d_Id_{II}+d_{II}d_I=0$.
(iii) The comultiplication $\Delta: C(\fg)\lra C(\fg)\otimes C(\fg)$ is
compatible with $d$.
(b) Given $d_{II}$, the differential $d_I$ is uniquely determined by the
properties (i), (ii), and (iii).
(c) Conversely, given $d=d_I+d_{II}$ satisfying (ii) and (iii),
define the bracket $\fg\otimes \fg\lra \fg$ as the composition
$(d_I)^{-2,*}\circ \pi_{\wedge, 2}$. This endows $\fg$ with the structure of a
dg Lie algebra.
\end{prop} $\Box$
So, for a dg Lie algebra $\fg$ we get a unital cocommutative dg coalgebra
$C(\fg)$ which is called
{\em the standard complex} of $\fg$. It was introduced by
Quillen, \cite{q}, App. B, no. 6.
\subsubsection{Maurer-Cartan condition} Let $A$ be a unital cocommutative dg
coalgebra,
and $f: A\lra C(\fg)$ be a map in $\Dgcu$. Let us denote
$f_i: A\lra S^i(\fg[1])$ the composition $p_i\circ f$ where $ p_i$ is
the projection $p_i: C(\fg)\lra S^i(\fg[1])$.
Define the map $[f_1,f_1]: A\lra \fg[2]$ in $\Gr (k)$ as follows:
if $x\in A,\ \Delta (x)=\sum y_i\otimes z_i$, set
$[f_1,f_1](x)=\sum (-1)^{|y_i|} [f_1(y_i),f_1(z_i)]$.
On the other hand, consider
$df_1=d_{\fg}\circ f_1+f_1\circ d_A: A\lra \fg[2]$.
For $x\in A$ we have $f(dx)=f_0(dx)+f_1(dx)+\ldots$. Since
$f(dx)=d\circ f(x)=(d_I+d_{II})\circ f(x)$, we have
$$
f_1(dx)=d_I\circ f_2(x)+d_{II}\circ f_1(x)= d_I\circ f_2(x)-d_{\fg}\circ f_1(x)
$$
On the other hand, if $\Delta (x)=\sum y_i\otimes z_i$, we have by
{}~\ref{coalg-s}
$$
f_2(x)=\frac{1}{2}\sum f_1(y_i)\cdot f_2(z_i)=\frac{1}{2}\sum (-1)^{|y_i|+1}
f_1(y_i)\wedge f_1(z_i)
$$
whence
$$
d_I\circ f_2 =-\frac{1}{2}[f_1,f_1]
$$
Hence we get
\begin{equation}
\label{mc}
df_1+\frac{1}{2}[f_1,f_1]=0
\end{equation}
--- the {\em Maurer-Cartan} equation.
Let us denote $MC(A,\fg)$ the set of all maps $f_1: A\lra \fg[1]$ in $\Gr (k)$
satisfying ~(\ref{mc}) and such that $f_1(1_A)=0$.
\subsubsection{}
\begin{prop}{} (~\cite{q}, App. B, 5.3) The
assignment $f\mapsto f_1$ yields a bijection
$$
\Hom_{\Dgcu}(A,C(\fg))\cong MC(A,\fg)
$$
\end{prop} $\Box$
\subsubsection{}
\label{formula} Suppose that $A$ is a dg Hopf algebra, $a_1,\ldots a_n\in A^0$
primitive elements.
For a subset $I\subset \{1,\ldots,n\}$ set
$a_I=a_{i_1}\cdot a_{i_2}\cdot\ldots\cdot a_{i_s}$ where
$I=\{ i_1,\ldots i_s\}$; $i_1<i_2<\ldots < i_s$; $a_{\emptyset}=1$.
Let us call a {\em $p$-partition} of $\{1,\ldots,n\}$ a sequence
$P=(I_1,\ldots ,I_p)$ of
subsets $I_j\subset \{1,\ldots,n\}$ such that $\{1,\ldots,n\}$ is the
disjoint union $I_1\cup\ldots\cup I_p$. Denote $\CP_p(n)$ the set of all
$p$-partitions.
One computes without difficulty that
$$
\Delta^{(p)}(a_1\cdot\ldots\cdot a_n)=\sum_{P=(I_1,\ldots ,I_p)\in \CP_p(n)}
a_{I_1}\otimes\ldots\otimes a_{I_p}
$$
Suppose we are given a dg coalgebra map $f: A\lra C(\fg)$. It follows form
{}~\ref{coalg-s} that
$$
f_p(a_1\cdot\ldots\cdot a_n)=\frac{1}{p!}
\sum_{P=(I_1,\ldots ,I_p)\in \CP_p(n)}
f_1(a_{I_1})\cdot\ldots\cdot f_1(a_{I_p}) \in S^p(\fg^1)
$$
In particular,
\begin{equation}
f_n(a_1\cdot\ldots\cdot a_n)=f_1(a_1)\cdot\ldots\cdot f_1(a_n)
\end{equation}
\subsection{Connecting morphism}
\subsubsection{Conic dg Lie algebras}
\label{conic} Let $\fg$ be a dg Lie algebra;
$\fh \subset \fg$ a dg Lie ideal. Denote by $i: \fh \lra \fg$ the embedding.
Define a dg Lie algebra $\fX$ as follows. Set $\fX^n=\fh^{n+1}\oplus \fg^n$.
The differential $d:\fX^n\lra \fX^{n+1}$ sends $(h,g)$ to $(-dh,i(h)+dg)$.
So, as a complex, $\fX$ is the usual cone of $i$.
The bracket in $\fX$ is defined as follows. We have $\fX=\fh [1]\oplus \fg$
(as graded modules). The bracket $\fX\otimes \fX\lra \fX$ has components:
$\fg\otimes \fg\lra \fg$ is the bracket in $\fg$;
$\fh [1]\otimes\fg\cong (\fh\otimes\fg)[1]\lra \fh[1]$ and
$\fg\otimes\fh [1]\cong (\fg\otimes\fh)[1]\lra \fh[1]$ are compositions of the
shifting isomorphisms ~(\ref{shift}) and the adjoint action of $\fg$ on $\fh$.
Explicitely:
\begin{equation}
\label{bracket}
[(h,g),(h',g')]=((-1)^a[g,h']+[h,g'],[g,g'])
\end{equation}
for $g\in \fg^a$.
Define maps in $\Gr (k):\ \phi: \fX\lra \fh[1],\ \phi (h,g)=h;\
\theta: \fX\lra \fg,\ \theta (h,g)=g.$ Note that $\phi$ is a map
of complexes. On the other hand, $\theta$ is a morphism
of graded (not dg) Lie algebras. We have
\begin{equation}
\label{dtheta}
d\theta=-i\circ \phi
\end{equation}
Consequently, $\theta$ induces the map $\Theta: U(\fX)\lra U(\fg)$ of
enveloping algebras which is a morphism of graded (not dg) Hopf algebras.
\subsubsection{Construction of the connecting morphism}
\label{constr} Define the map
$\tilde c_1: T(\fX) \lra \fh[1]$ in $\Gr (k)$ as follows.
Set
$$
\tilde c_1|_{T^0(\fX)}=0,\ \tilde c_1|_{T^1(\fX)}=\phi
$$
Suppose we have defined $\tilde c_1$ on $T^n(\fX)$ for $n\geq 1$.
For $u\in T^n(\fX),\ x\in \fX$ set
\begin{equation}
\label{tc}
\tilde c_1(xu)= (-1)^{|x|}\ad_{\fg}(\theta (x))(\tilde c_1(u))
\end{equation}
This defines $\tilde c_1$ on $T^{n+1}(\fX)$.
Here $\ad_{\fg}$ denotes the adjoint action of $\fg$ on $\fh$:
$\ad_{\fg}(g)(h)=[g,h]$.
Let us denote $\ad_{U(\fg)}$ the induced action of $U(\fg)$ on $\fh$.
{}~(\ref{tc}) implies the important equality:
\begin{equation}
\label{imp}
\tilde c_1(uv)=(-1)^{|u|}\ad_{U(\fg)}(\Theta(u))(\tilde c_1(v))
\end{equation}
for all $u\in T(\fX), v\in T^+(\fX):=\sum_{n>0} T^n(\fX)$.
\subsubsection{}
\begin{thm}{constr-thm} (i) The map $\tilde c_1$ vanishes on the kernel
of the projection $T(\fX)\lra U(\fX)$, and hence it induces the map
\begin{equation}
\label{c1}
c_1: U(\fX)\lra \fh[1]
\end{equation}
(ii) $c_1$ satisfies the Maurer-Cartan equation ~(\ref{mc}).
Consequently, $c_1$ defines the map of unital dg coalgebras
\begin{equation}
\label{conn}
c: U(\fX)\lra C(\fh)
\end{equation}
\end{thm}
We will call $c$ {\bf connecting morphism}.
\begin{pf} (i) We have to prove that
\begin{equation}
\label{toprove}
\tilde c_1(uxyv-(-1)^{ab}uyxv-u[x,u]v)=0
\end{equation}
for all $u,v\in T(\fX),\ x\in \fX^a,\ y\in \fX^b$. If $v\in T^+(\fX)$
this follows from ~(\ref{imp}); so we can suppose $v=1$.
Again, from ~(\ref{imp}) follows that it suffices to prove ~(\ref{toprove})
for $u=1$. This reduces to proving that
$$
(-1)^a[\theta(x),\phi(y)]-(-1)^{(a+1)b}[\theta(y),\phi(x)]=\phi([x,y])
$$
which is equivalent to ~(\ref{bracket}).
(ii) Recall the canonical filtration $F_nU(\fX)$ from ~\ref{envel}.
Let us prove by induction on $n$ that
\begin{equation}
\label{mcprove}
(dc_1+\frac{1}{2}[c_1,c_1])(u)=0
\end{equation}
for all $u\in F_nU(\fX)$. If $n\leq 1$ then both summands in ~(\ref{mcprove})
are $0$.
Suppose we have $x\in \fX^a,\ u\in U(\fX)^b$ such that $\epsilon(u)=0$
where $\epsilon: U(\fX)\lra k$ is the counit. We have
\begin{multline}
(dc_1)(xu)=d(c_1(xu))+c_1(d(xu))=(-1)^ad(\theta(x),c_1(u)])+ c_1(dx\cdot u)+
(-1)^ac_1(x\cdot du)= \\
=(-1)^ad([\theta(x),c_1(u)])+(-1)^{a+1}[\theta(dx),c_1(u)]+[\theta(x),c_1(du)]
\\
\label{eq1}
\end{multline}
Note that
\begin{eqnarray}
[\theta(x),c_1(du)]=[\theta(x),(dc_1)(u)]-[\theta(x),d(c_1(u))]= \nonumber \\
=[\theta(x),(dc_1)(u)]+(-1)^{a+1}d([\theta(x),c_1(u)])+
(-1)^a[d(\theta(x)),c_1(u)] \nonumber
\end{eqnarray}
(we have used that $d([g,h])=[dg,h]+(-1)^a[g,dh]$ for $g\in \fg^a$).
Substituting this in ~(\ref{eq1}) we get
\begin{equation}
\label{dcone}
(dc_1)(xu)=(-1)^a[d(\theta(x))-\theta(dx),c_1(u)]=(-1)^{a+1}[\phi(x),c_1(u)]
\end{equation}
(we have used ~(\ref{dtheta})).
Suppose that $\Delta(u)=u\otimes 1+1\otimes u+\sum_iu'_i\otimes u''_i$.
Let $b,b'_i,b''_i$ be the degrees of $u,u'_i,u''_i$ respectively.
Then
$$
\Delta(xu)=xu\otimes 1+1\otimes xu+x\otimes u+(-1)^{ab}u\otimes x+
\sum_ixu'_i\otimes u''_i+\sum_i(-1)^{ab'_i}u'_i\otimes xu''_i
$$
so that
\begin{multline}
\label{conecone}
[c_1,c_1](xu)=2(-1)^a[\phi(x),c_1(u)]+\sum_i(-1)^{a+b'_i}[c_1(xu'_i),c_1(u''_i)]
+ \\
+\sum_i(-1)^{(a+1)b'_i}[c_1(u'_i),c_1(xu''_i)]=2(-1)^a[\phi(x),c_1(u)]+
\ad_{\fg}(\theta(x))([c_1,c_1](u))
\end{multline}
Adding up ~(\ref{dcone}) and ~(\ref{conecone}) we get
$$
(dc_1+\frac{1}{2}[c_1,c_1])(xu)=\ad_{\fg}(\theta(x))
((dc_1+\frac{1}{2}[c_1,c_1])(u))
$$
Now ~(\ref{mcprove}) follows by induction on $n$.
This proves the theorem.
\end{pf}
\subsubsection{} Let $\fa$ be a graded Lie algebra, $M$ a graded $\fa$-module,
$\phi: \fa\lra M$ a $1$-cocycle of $\fa$ with values in $M$, i.e.
$$
\phi([a,b])=a\phi(b)-(-1)^{|a||b|}b\phi(a)
$$
for $a,b\in \fa$. Then there exists a unique map of graded modules
$$
c_1^+: U(\fa)^+\lra M
$$
such that
(i) $c_1^+|_{\fa}=\phi$; (ii) $c_1^+$ commutes with the action of $\fa$, where
the $\fa$ acts on $U(\fa)^+$ by the left multiplication. Taking the
composition with the projection $U(\fa)\lra U(\fa)^+$ we get a map
$$
c_1: U(\fg)\lra M
$$
The map ~(\ref{c1}) is obtained by applying this remark to
$\fa=\fX,\ M=\fh[1]$,
the action of $\fX$ on $\fh[1]$ being induced from the adjoint action of
$\fg$ on $\fh[1]$ through the graded Lie algebra morphism $\theta:\fX\lra \fg$.
\section{Twisted enveloping algebras and connecting morphism}
\label{twisted}
\subsection{}
\label{dif-smo}
From now on until the end of the paper we fix
a ground field $k$ of characteristic $0$.
Let $(X,\CO_X)$ be a topological space equipped with a sheaf of
commutative $k$-algebras $\CO_X$. Define the {\em tangent sheaf} $\CT_X$
as the sheaf of $\CO_X$-modules associated with the presheaf
$$
U\mapsto \operatorname{Der}_k(\Gamma(U,\CO_X),\Gamma(U,\CO_X))
$$
(the space of $k$-derivations). $\CT_X$ is a sheaf of $k$-Lie algebras.
We will say that $X$ is {\em differentially smooth} if there exists an open
covering $X=\bigcup U_i$ such that for each $U_i$ the restriction
$\CT_X|_{U_i}$ is a free $\CO_X|_{U_i}$-module admitting a finite basis
of {\em commuting} sections $\dpar_1,\ldots,\dpar_n\in\Gamma(U_i,\CT_X)$.
\subsection{} If $\CF$ a sheaf on $X$, the notation $t\in \CF$ will mean
that $t$ is a local
section of $\CF$. We will use below the straightforward "sheaf" versions
of the definitions and results from Sect.~\ref{envstand}. In particular,
we will use the notion of a {\em dg $\CO_X$-Lie algebra} (the
bracket is supposed to be $\CO_X$-linear).
If $\CF^{\cdot}$ is a complex of sheaves, $\CZ^i(\CF^{\cdot})$ will denote
sheaves of $i$-cocycles, $\CH^i(\CF^{\cdot})$ cohomology sheaves
(not to be confused with cohomology {\em spaces} $H^i(X,\CF)$).
If $\fg$ is a dg $\CO_X$-Lie algebra, we will consider its standard
complex $C(\fg)=C_{\CO_X}(\fg)$ (this is a complex of sheaves of dg
$\CO_X$-coalgebras) over $X$) and its canonical filtration $F_iC(\fg)$;
we set $\gr_iC(\fg)=F_iC(\fg)/F_{i-1}C(\fg)$. We will use notations
\begin{equation}
\label{homology}
\CH_i^{Lie}(\fg):=\CH^{-i}(C(\fg));\ \CF_j\CH_i^{Lie}(\fg):=\CH^{-i}(F_jC(\fg))
\end{equation}
\subsection{Dg Lie algebroids} Below we will use some definitions and
constructions from ~\cite{bb}, 1.2 (see also ~\cite{bfm}, 3.2) and their
generalization to a dg-situation.
\begin{defn}{} A {\em dg Lie algebroid} over $X$ is
a sheaf of dg $k$-Lie algebras $\CA$ on $X$ together with a structure
of a left $\CO_X$-module on it and a map
$$
\pi: \CA\lra \CT_X
$$
of dg $k$-Lie algebras and $\CO_X$-modules such that
$$
[a,fb]=f[a,b]+\pi(a)(f)b
$$
for all $a,b\in \CA,\ f\in \CO_X$. We denote $\CA_{(0)}:=\ker \pi$.
It is an $\CO_X$-Lie algebra and a dg Lie ideal in $\CA$.
$\CA$ is called {\em transitive} if $\pi$ is epimorphic (i.e.
$\pi^0: \CA^0\lra \CT_X$ is epimorphic).
A Lie algebroid is a dg Lie algebroid concentrated in degree $0$.
\end{defn}
Dg Lie algebroids over $X$ form a category in an obvious way, with
the final object $(\CT_X, \id_{\CT_X})$.
A dg Lie algebroid with $\pi=0$ is the same as a dg $\CO_X$-Lie algebra.
\subsection{Modules} Let $\CA$ be a dg Lie algebroid over $X$.
A {\em dg $\CA$-module} is a complex of left $\CO_X$ modules $M$ which is
(as a complex of $k$-modules) equipped with the action of the
dg $k$-Lie algebra $\CA$ such that
\begin{equation}
\label{eq1-mod}
f(am)=(fa)m
\end{equation}
\begin{equation}
\label{eq2-mod}
a(fm)=f(am)+\pi(a)(f)m
\end{equation}
for all $f\in \CO_X,\ a\in \CA,\ m\in M$. We will say in this situation
that we have an {\em action} of a dg Lie algebroid $\CA$ on $M$.
\subsection{Twisted enveloping algebras}
\label{twist-env} Let $(\CA, \pi)$ be a dg Lie algebroid
over $X$. Let $U_k(\CA)$ denote the enveloping algebra of $\CA$ considered
as a dg $k$-Lie algebra, $U_k(\CA)^+\subseteq U_k(\CA)$ the augmentation ideal.
Let us consider $U_k(\CA)^+$ as a sheaf of dg algebras without unit. Denote by
$U_{\CO_X}(\CA)^+$ its quotient by the two-sided dg ideal generated by all
elements
$$
a_1\cdot fa_2-fa_1\cdot a_2-\pi(a_1)(f)a_2,
$$
$a_1,a_2\in \CA,\ f\in \CO_X$.
Set $U_{\CO_X}(\CA)=U_{\CO_X}(\CA)^+\oplus \CO_X$ and define the structure
of a dg algebra with unit by the rule
$$
f\cdot a=fa,\ a\cdot f=fa+\pi(a)(f),\ f\in \CO_X,a\in \CA
$$
We have a canonical algebra map
$$
\CO_X\lra U_{\CO_X}(\CA)
$$
providing $U_{\CO_X}(\CA)$ with a structure of an $\CO_X$-bimodule, and
a map of left $\CO_X$-modules and dg $k$-Lie algebras (with the structure of
a dg Lie algebra on $U_{\CO_X}(\CA)$ given by the commutator)
$$
i: \CA\lra U_{\CO_X}(\CA)
$$
which is a composition of evident maps $\CA\lra U_k(\CA)^+\lra U_{\CO_X}(\CA)$.
The map $i$ induces an equivalence of the category of dg modules over
$U_{\CO_X}(\CA)$ (as an associative dg algebra) and that of $\CA$-modules.
We define the canonical filtration $F_iU_{\CO_X}(\CA)\subset U_{\CO_X}(\CA)$
as the image of $F_iU_k(\CA)^+\oplus \CO_X$.
\subsection{Coalgebra structure} Let $\CA$ be a dg Lie algebroid
over $X$. Consider the comultiplication
$\Delta: U_k(\CA)\lra U_k(\CA)\otimes_k U_k(\CA)$. For $x\in U_k(\CA)^+$,
$\Delta(x)=x\otimes 1+1\otimes x+\Delta^+(x)$ where
$\Delta^+(x)\in U_k(\CA)^+\otimes_k U_k(\CA)^+$. Consider the composition
\begin{equation}
\label{comp}
U_k(\CA)^+\overset{\Delta^+}{\lra}U_k(\CA)^+\otimes_k U_k(\CA)^+
\lra U_{\CO_X}(\CA)^+\otimes_{\CO_X} U_{\CO_X}(\CA)^+
\end{equation}
(we consider the tensor square of $U_{\CO_X}(\CA)^+$ as a {\em left}
$\CO_X$-module).
One checks directly that $\ker (U_k(\CA)^+\lra U_{\CO_X}(\CA))$ is contained
in the kernel of ~(\ref{comp}). Hence, ~(\ref{comp}) induces the map
$$
\Delta^+:U_{\CO_X}(\CA)^+\lra U_{\CO_X}(\CA)^+\otimes_{\CO_X} U_{\CO_X}(\CA)^+
$$
Define
$$
\Delta:U_{\CO_X}(\CA)\lra U_{\CO_X}(\CA)\otimes_{\CO_X} U_{\CO_X}(\CA)
$$
as follows: for $f\in \CO_X,\ \Delta(f)=f\otimes 1$; for
$x\in U_{\CO_X}(\CA)^+,\ \Delta(x)=x\otimes 1+1\otimes x+\Delta^+(x)$.
Define the counit $U_{\CO_X}(\CA)\lra \CO_X$ to be the canonical projection.
This defines a structure of a unital cocommutative dg $\CO_X$-coalgebra
on $U_{\CO_X}(\CA)$.
\subsection{Differential operators}
Let $(X,\CO_X)$ be a differentially smooth $k$-ringed space.
\begin{defn}{} The sheaf of differential operators
(resp., that of operators of order $\leq n$) on $X$ is
$$
\Diff_X=U_{\CO_X}(\CT_X);\ \Diff^{\leq n}_X=F_nU_{\CO_X}(\CT_X)\ .
$$
\end{defn}
\subsection{Poincare-Birkhoff-Witt condition} The associated graded
algebra
$$
\gr U_{\CO_X}(\CA):=\oplus_{i\geq 0} F_iU_{\CO_X}(\CA)/F_{i-1}U_{\CO_X}(\CA)
$$
is a commutative dg $\CO_X$-algebra. Hence, we have a canonical surjective
morphism
\begin{equation}
\label{pbw-map}
j: S_{\CO_X}{\CA}\lra \gr U_{\CO_X}(\CA)
\end{equation}
We will say that {\em $\CA$ satisfies the Poincare-Birkhoff-Witt (PBW)
condition} if ~(\ref{pbw-map}) is isomorphism.
\subsection{}
\begin{thm}{pbw-algebr}
Let $(X,\CO_X)$ be differentially smooth and let $\CA$ be a transitive dg Lie
algebroid over $X$. Then $\CA$ satisfies PBW.
\end{thm}
{\em Proof} will be done in several steps.
\subsubsection{}
\label{pbw-lie}
\begin{lem}{} Let $\fg$ be a dg $\CO_X$-Lie algebra. Then one has a canonical
isomorphism of unital dg $\CO_X$-coalgebras
$$
e: S_{\CO_X}(\fg)\lra U_{\CO_X}(\fg)
$$
defined by the formula
$$
e(x_1\cdot\ldots\cdot x_n)=\frac{1}{n!}\sum_{\sigma\in \Sigma_n}\pm i(x_1)\cdot
\ldots i(x_n)
$$
where $i:\fg\lra U(\fg)$ is the canonical map, the sign $\pm$ is inserted
according to the Quillen rule.
\end{lem}
\begin{pf} The argument of the proof of ~\cite{q}, App. B, Thm. 3.2
works in our situation.
\end{pf}
Let us continue the proof of ~\ref{pbw-algebr}. We can forget about
differentials. The question is local on $X$, so we can suppose that
$\CT_X$ is freely generated over $\CO_X$ by $n$ commuting global vector
fields $\dpar_1,\ldots, \dpar_n\in \CT_X(X)$ which can be lifted to
global sections $\gamma_1,\ldots,\gamma_n\in \CA^0(X)$. Consider the map
$$
\mu: U_{\CO_X}(\CA_{(0)})(X)\otimes_k k[\gamma_1,\ldots,\gamma_n]\lra
U_{\CO_X}(\CA)(X)
$$
given by the multiplication. Here $k[\gamma_1,\ldots,\gamma_n]$ denotes the
free left $k$-module on the basis $\gamma_1^{d_1},\ldots,\gamma_n^{d_n}$,
$d_i$ being nonnegative integers.
\subsubsection{}
\begin{lem}{} $\mu$ is an isomorphism of filtered graded $k$-modules.
\end{lem}
Here the filtration on $U_{\CO_X}(\CA_{(0)})(X)$ is the canonical one;
on $k[\gamma_1,\ldots,\gamma_n]$ the filtration by the total degree,
and the filtration on their tensor product is the tensor product
of filtrations:
$F_i(A\otimes B)=\sum_{p+q=i}\Ima(F_pA\otimes F_qB\lra A\otimes B)$.
\begin{pf} We apply the idea of Serre's proof of the PBW theorem, see
{}~\cite{se}, proof of Thm. 3, p. I, ch.
III.4. First, it is clear that $\mu$ is surjective.
To prove the injectivity,
consider the free left $U_{\CO_X}(\CA_{(0)})(X)$-module $F$ with the basis
$\{ x_M\}$ indexed by finite non-decreasing sequences $M=(i_1,\ldots,i_d)$
with $i_j\in \{1,\ldots,n\}$.
One can introduce on $M$ the structure of an $\CA(X)$-module as follows.
For $M$ as above, call $d$ the length of $M$, and denote it $l(M)$. For
$i\in \{1,\ldots,n\}$ say that $i\leq M$ if $i\leq i_1$, and denote $iM$ the
concatenation
$(i,i_1,\ldots,i_d)$. Note that $\CA(X)$ is generated as
an abelian group by elements $a\gamma_i,\ a\in \CA_{(0)}(X)$, so it
suffices to define $\gamma_i\cdot ux_M,\ u\in U_{\CO_X}(\CA_{(0)})(X)$.
Let us do this by induction. We set
$$
\gamma_i\cdot ux_M=u\gamma_i\cdot x_M-\ad_{\gamma_i}(u)\cdot x_M
$$
so it suffices to define $\gamma_i\cdot x_M$.
Suppose we have defined $\gamma_i\cdot ux_N$ for all $N:\ l(N)<l(M)$ and
$\gamma_j\cdot ux_N$ for all $j<i,\ l(N)=l(M)$; we suppose that these
elements are linear combinations over $U_{\CO_X}(\CA_{(0)})(X)$ of $x_{N'}$
with $l(N')\leq l(N)+1$. Set
$$
\gamma_i\cdot x_M=\left\{ \begin{array}{ll}
x_{iM} & \mbox{if $i\leq M$} \\
\gamma_j\cdot (\gamma_i\cdot x_N)+
[\gamma_i,\gamma_j]\cdot x_N & \mbox{if $M=jN,\
i>j$}
\end{array}
\right.
$$
the right hand side being defined by induction (note that
$[\gamma_i,\gamma_j]\in \CA_{(0)}(X)$). One checks that this definition is
correct.
Using this, one proves, as in {\em loc. cit.} that all $x_M$ form the
$U_{\CO_X}(\CA_{(0)})(X)$-basis of $U_{\CO_X}(\CA)(X)$, which implies the
injectivity of $\mu$.
\end{pf}
To finish the proof of ~\ref{pbw-algebr}, it remains to note that
$$
\gr (\mu): S_{\CO_X}(\CA_{(0)})(X)\otimes_k k[\gamma_1\ldots,\gamma_n]\lra
\gr U_{\CO_X}(\CA)(X)
$$
coincides with $j(X)$. Theorem ~\ref{pbw-algebr} is proven. $\Box$
\subsection{}
\label{quasi-iso}
\begin{cor}{}
Let $(X,\CO_X)$ be differentially smooth and $f: \CA\lra \CB$ be a map of
transitive dg Lie algebroids
satisfying one of the two assumprions:
(i) $f$ locally $\CO_X$-homotopy equivalence;
(ii) $f$ is quasiisomorphism, all components $\CA^i,\ \CB^i$ are flat over
$\CO_X$, $\CH^i(\CA)=\CH^i(\CB)=0$ for big $i$.
Then the induced map $U_{\CO_X}(f): U_{\CO_X}(\CA)\lra U_{\CO_X}(\CB)$
is a filtered quasi-isomorphism.
\end{cor}
\begin{pf} Each of our hypotheses implies that
$T_{\CO_X}(f): T_{\CO_X}(\CA)\lra T_{\CO_X}(\CB)$ is a quasi-isomorphism;
hence this is true for $S_{\CO_X}(f)$. Now by ~\ref{pbw-algebr} the same
is true for $U_{\CO_X}(f)$.
\end{pf}
\subsection{Pushout} Let $\CA$ be a dg Lie algebroid over $X$ and
$\fg$ be a dg $\CO_X$-Lie algebra which is also an $\CA$-module.
An {\em $\CA$-morphism} $\psi: \CA_{(0)}\lra \fg$
is a morphism of $\CO_X$-Lie algebras which commutes with the $\CA$-action,
where the action of $\CA$ on $\CA_{(0)}$ is the adjoint one, and such that
$$
a\cdot x=[\psi(a),x]
$$
for all $a\in \CA_{(0)},\ x\in \fg$.
Given such a morphism, the dg Lie algebroid $\CA_{\psi}$ is defined as follows.
Consider the dg Lie algebra semi-direct product $\CA \semid \fg$ (so, the
bracket is
$[(a,x),(b,y)]=([a,b],-(-1)^{|b||x|}b\cdot x+a\cdot y+[x,y]),\ a,b\in \CA,\
x,y\in fg$). By definition, $\CA_{\psi}$ is the quotient of $\CA \semid \fg$
by the dg Lie ideal $\CA_{(0)}\hra \CA \semid \fg,\ a\mapsto (a, -\psi(a))$.
The map $\pi_{\CA_{\psi}}:\CA_{\psi}\lra \CT_X$ maps $(a,x)$ to $\pi_{\CA}(a)$.
If $\CA$ is transitive then so is $\CA_{\psi}$, and $\CA_{\psi (0)}=\fg$.
\subsection{Boundary morphism}
\subsubsection{Conic Lie algebroids}
Let $(\CA,\pi)$ be a dg Lie algebroid over $X$, $\fh$ a dg Lie algebra and a
left $\CO_X$-module, $i: \fh\hra \CA$ an embedding of dg Lie algebras and
of $\CO_X$-modules such that $i(\fh)$ is a dg Lie ideal in $\CA$, and
$\pi\circ i=0$. This implies that $\fh$ is a dg $\CO_X$-Lie algebra.
Let us consider the complex $\fA=\Cone (i)$. According to ~\ref{conic},
$\fA$ has a canonical structure of a dg $k$-Lie algebra. Together with the
evident structure of an $\CO_X$-module and
$\pi_{\fA}:\fA\overset{\theta}{\lra} \CA\overset{\pi}{\lra} \CT_X$,
$\fA$ becomes a dg Lie algebroid.
Let us consider $\fA$ as a dg $k$-Lie algebra, and apply to it (the
sheaf version of) the construction ~\ref{constr}. We get a map of
sheaves of graded $k$-modules
\begin{equation}
\label{conn-prep}
c_{1/k}^+: U_k(\fA)^+\lra \fh[1]
\end{equation}
satisfying the Maurer-Cartan equation.
\subsubsection{}
\begin{thm}{}
\label{conn-thm} The map $c_{1/k}^+$ factors through the
canonical map $U_k(\fA)^+\lra U_{\CO_X}(\fA)^+$ and hence induces the map
$$
c_1^+:U_{\CO_X}(\fA)^+\lra \fh[1].
$$
Taking its composition with the projection
$U_{\CO_X}(\fA)\lra U_{\CO_X}(\fA)^+$, we get
$$
c_1:U_{\CO_X}(\fA)\lra \fh[1].
$$
This map is a morphism of left graded $\CO_X$-modules and it satisfies
the Maurer-Cartan equation.
Consequently, it induces the map of filtered dg $\CO_X$-coalgebras
\begin{equation}
c:U_{\CO_X}(\fA)\lra C_{\CO_X}(\fh)
\end{equation}
\end{thm}
The map $c$ will be called {\bf boundary morphism} associated to $\fA$.
\begin{pf} We have only to check that
\begin{equation}
\label{check}
ux(fy)v-u(fx)yv-u(\pi_{\fA}(x)(f)y)v\in \ker(c_{1/k}^+)
\end{equation}
for all $x,y\in \fA,\ u,v\in U_k(\fA),\ f\in \CO_X$.
Note that $\fh$ is an $\CA$-module with respect to the adjoint action
(NB! this is not true for $\CA$: the Axiom ~(\ref{eq1-mod}) does not hold).
It follows that the action of $U_k(\CA)$ on $\fh[1]$ factors through
$U_{\CO_X}(\CA)$, hence ~(\ref{check}) holds true for $v\in U_k(\fA)^+$;
so we can suppose $v=1$; obviously, it is enough to check ~(\ref{check})
for $u=1$.In that case it is a direct check (note that the maps $\phi$ and
$\theta$ are $\CO_X$-linear):
$$
c_{1/k}^+(x(fy)-(fx)y-\pi_{\fA}(x)(f)y)=[\theta(x),f\phi(y)]-
[f\theta(x),\phi(y)]-\pi_{\CA}(\theta(x))(f)\phi(y)=0
$$
\end{pf}
\subsection{Connecting morphisms}
\label{abstr-ksmaps}
Here $X$ is supposed to be differentially smooth.
\subsubsection{} Let $\CA$ be a transitive dg Lie algebroid
over $X$. Set $\fh:=\CA_{(0)}$, so we have an exact sequence
\begin{equation}
\label{fund-abstr}
0\lra \fh\overset{i}{\lra} \CA\overset{\pi}{\lra} \CT_X\lra 0\ .
\end{equation}
It induces the map
\begin{equation}
\label{ks1-abstr}
\kappa^1: \CT_X\lra \CH^1(\fh)\ .
\end{equation}
Set $\fA=\Cone (i)$. By Thm.~\ref{quasi-iso} $\pi$ induces a
filtered quasi-isomorphism $U_{\CO_X}(\fA)\lra U_{\CO_X}(\CT_X)=\Diff_X$,
whence isomorphisms
$$
\CH^0(U_{\CO_X}(\fA))\cong \Diff_X;\ \CH^0(F_nU_{\CO_X}(\fA))\cong
\Diff_X^{\leq n}\ .
$$
On the other hand, Thm.~\ref{conn-thm} gives the filtered map
$$
c: U_{\CO_X}(\fA)\lra C_{\CO_X}(\fh)\ .
$$
By taking $\CH^0(c)$, we get maps
\begin{equation}
\label{abstr-ks}
\kappa:\Diff_X\lra \CH_0^{Lie}(\fh)
\end{equation}
as well as
\begin{equation}
\label{abstr-ks-i}
\kappa^{\leq n}: \Diff^{\leq n}_X\lra \CF_n\CH_0^{Lie}(\fh)
\end{equation}
which are called {\bf connecting morphisms}.
\subsubsection{} We have by definition
$$
\gr_nC_{\CO_X}(\fh)=S^n_{\CO_X}(\fh[1])
$$
hence the maps
\begin{equation}
\label{proj-lie}
\CF_n\CH_0^{Lie}(\fh)\lra \CH_0(S^n_{\CO_X}(\fh[1]))\ .
\end{equation}
On the other hand, the embeddings $S^n(\fh^1)\subset S^n(\fh[1])$ induce
embedding of cocycles $S^n\CZ^1(\fh)\hra \CZ^0(S^n(\fh[1]))$ which pass to
cohomology and give the maps
\begin{equation}
\label{emb}
S^n_{\CO_X}(\CH^1(\fh))\lra \CH^0(S^n(\fh[1]))\ .
\end{equation}
\subsubsection{}
\label{main-thm}
\begin{thm}{} The connecting morphisms ~(\ref{abstr-ks}) and
{}~(\ref{abstr-ks-i}) have the following properties.
(i) The squares
$$\begin{array}{ccc}
\Diff^{\leq n-1}_X & \overset{\kappa^{\leq n-1}}{\lra} &
\CF_{n-1}\CH_0^{Lie}(\fh) \\
\downarrow & \; & \downarrow \\
\Diff^{\leq n}_X & \overset{\kappa^{\leq n}}{\lra} & \CF_{n}\CH_0^{Lie}(\fh) \\
\end{array}$$
commute. We have
$$
\kappa=\lim_{\ra}\kappa^{\leq n}\ .
$$
(ii) The squares
$$\begin{array}{ccccc}
\Diff^{\leq n}_X &\; & \overset{\kappa^{\leq n}}{\lra} &\;
& \CF_{n}\CH_0^{Lie}(\fh) \\
\downarrow & \; & \; &\; & \downarrow \\
S^n_{\CO_X}(\CT_X) & \overset{(-1)^nS^n(\kappa^1)}{\lra} & S^n_{\CO_X}
(\CH^1(\fh)) & \overset{(\ref{emb})}{\lra} & \CH^0(S^n(\fh[1])) \\
\end{array}$$
commute. Here the left vertical arrow is the symbol map,
and the right one is ~(\ref{proj-lie}).
\end{thm}
The property (i) is obvious. (ii) will be proven in the next Subsection.
\subsection{Explicit formulas}
\subsubsection{}
\label{expl1} In the previous assumptions, suppose we have local
sections $\dpar_1,\ldots ,\dpar_n\in \CT_X$.
Let us pick $0$-cocycles in
$\fA$ lifting them: $a_p=(-\alpha_p,\gamma_p)\in \fA^0,\ \gamma_p\in
\CA^0,\alpha_p\in\fh^1,\ \pi(\gamma_p)=\dpar_p;\
d_{\CA}(\gamma_p)=i(\alpha_p)$.
Consider the map $c_{1/k}^+$, ~(\ref{conn-prep}). For $I$ as in ~\ref{formula}
define $a_I$ as there, and set
$\dpar_I=\dpar_{i_1}\cdot\ldots\cdot\dpar_{i_s}$,
$$
\alpha(\dpar_I)=c_{1/k}^+(a_I)=\ad (\gamma_{i_1})\circ\ldots\circ
\ad(\gamma_{i_{s-1}})(-\alpha_{i_s})
$$
Define elements (we use the notations of ~\ref{formula})
$$
\kappa^{(p)}_n=\frac{1}{p!}
\sum_{P=(I_1,\ldots,I_p)\in\CP_p(n)}\alpha(\dpar_{I_1})\cdots
\alpha(\dpar_{I_p})\in S^p_{\CO_X}(\fh^1),
$$
$p=1,\ldots,n$. In particular,
\begin{equation}
\label{kappa-nn}
\kappa^{(n)}_n=(-1)^n\alpha_1\cdots
\alpha_n.
\end{equation}
\subsubsection{}
\label{expl2}
\begin{prop}{} The class $\kappa^{\leq n}(\dpar_1\cdots\dpar_n)$
is represented by the cocycle
$$
(0,\kappa^{(1)}_n,\ldots,\kappa^{(n)}_n)\in\oplus_{p=0}^nS^p_{\CO_X}(\fh^1)
\subset F_nC_{\CO_X}(\fh)^0.
$$
\end{prop}
\begin{pf} This follows from ~\ref{formula} applied to $c_{1/k}$ (note
that $\fA$ is a Hopf algebra).
\end{pf}
As a corollary, we get the claim (ii) of ~\ref{main-thm} which follows
from ~(\ref{kappa-nn}).
\subsubsection{Schur polynomials} Let us define polynomials
$P_n(\alpha_1,\alpha_2,\ldots)$ by means of the generating function
$$
\exp (\sum_{p=1}^{\infty}\alpha_p\frac{t^p}{p!})=
\sum_{n=0}^{\infty}P_n(\alpha_1,\ldots)\frac{t^n}{n!}
$$
It is easy to see that $P_n(\alpha_1,\ldots)=P_n(\alpha_1,\ldots,\alpha_n)$
and
\begin{equation}
\label{schur}
P_n(\alpha_1,\ldots,\alpha_n)=\sum_{(n_1,\ldots,n_s):\sum_j jn_j=n}
\frac{n!}{(1!)^{n_1}\cdot\ldots\cdot (s!)^{n_s}
n_1!\cdot\ldots\cdot n_s!} \alpha_1^{n_1}\cdot\ldots\cdot\alpha_s^{n_s}
\end{equation}
The first polynomials are: $P_0=1,\ P_1=\alpha_1,\ P_2=\frac{\alpha_1^2}{2}+
\alpha_2$.
\subsubsection{} Now suppose we are given a local section $\dpar\in \CT_X$
together with a lifting $a=(-\alpha,\gamma)\in \fA^0$ as in ~\ref{expl1}.
For $i\geq 1$ set
$$
\alpha_i=(\ad (\gamma))^{i-1}(-\alpha)\in \fh^1
$$
\subsubsection{}
\label{explicit}
\begin{prop}{} The class $\kappa^{\leq n}(\dpar^n)$ is represented by the
cocycle
$$
P_n(\alpha_1,\ldots,\alpha_n)\in F_nS_{\CO_X}(\fh^1)\subseteq
F_nC_{\CO_X}(\fh)^0.
$$
\end{prop}
\begin{pf} From ~\ref{expl2} follows that $\kappa^{\leq n}(\dpar^n)$ can be
represented by the polynomial $Q(\alpha_1,\ldots,\alpha_n)$ where
the coefficient of $Q$ at $\alpha_1^{n_1}\cdots\alpha_s^{n_s}$
is equal to the number of partitions of the set $\{1,\ldots,n\}$ containing
$n_1$ of $1$-element subsets, $n_2$ of $2$-element subsets, ...,
$n_s$ of $s$-element subsets. From ~(\ref{schur}) follows that
$Q=P_n$.
\end{pf}
\subsubsection{} Summing up the expressions ~\ref{explicit} over $n$, we can
rewrite our formulas as
$$
\kappa(\exp(t\dpar))=\exp (\frac{\exp(\ad(\gamma))-\Id}{\ad(\gamma)}(\alpha)).
$$
This was pointed out to us by I.T.Leong. Cf. Deligne's formula
{}~\cite{gmd}, p. 51, (1-1).
\section{Differential calculus on formal schemes}
\label{formal}
The aim of this Section is to extend (a part of) the classical Grothendieck's
language of differential calculus, \cite{ega} IV, \S\S 16, 17, to formal
schemes.
\subsection{} Recall that we have fixed a ground field $k$ of
characteristic $0$.
We will work in the category $\Fsch$ of separated locally noetherian formal
schemes over $k$. All necessary definitions and facts about them are contained
in \cite{ega} I, \S10. Objects of $\Fsch$ will be called simply
{\em formal schemes}.
Inside $\Fsch$, we will consider a full subcategory $\Sch$ of separated
locally noetherian schemes of over $k$ whose objects will be called simply
(usual) {\em schemes}.
By definition, a formal scheme is a topological space $\fX$ equipped with
a sheaf of topological rings $\CO_{\fX}$. $\fX$ is a union of
affine formal schemes $\Spf(A)$ where $A$ is a noetherian $k$-algebra
complete in the $I$-adic topology for some ideal $I\subset A$. As a topological
space, $\Spf(A)=\Spec(A/I)$, and $\Gamma(\Spf(A),\CO_{\Spf(A)})=A$.
There exists a sheaf of ideals $\fI\subset\CO_{\fX}$ such that for sufficiently
small affine $U\subset \fX$ the ideals $\Gamma(U,\fI)^n,\ n\geq 1$, form
a base for the topology of $\Gamma(U,\CO_{\fX})$. Such a sheaf is called
the {\em ideal of definition} of $\fX$. All ringed spaces
$$
X_n:=(\fX,\CO_{\fX}/\fI^{n+1})
$$
are (usual) schemes, and we have
\begin{equation}
\label{indlim}
\fX=\dirlim\ X_n
\end{equation}
cf. {\em loc.cit.}, 10.11. Among ideals of definition there exists a unique
maximal one.
Let $f:\fX\lra \fS$ be a morphism of formal schemes, $\fK\subseteq\CO_{\fS}$
an ideal of definition of $\fS$ and $\fI$ the maximal ideal of definition
of $\fX$. Then $f^*(\fK)\subseteq\fI$, so $f$ induces maps of schemes
$f_n:X_n\lra S_n$ such that
\begin{equation}
\label{adic}
f=\dirlim\ f_n
\end{equation}
cf. {\em loc.cit.}, 10.6.10.
\subsubsection{} The category $\Fsch$ has fibered products, {\em loc.cit.},
10.7.
\subsection{}
\label{aff} Let $\fX$ be affine, $\fX=\Spf(A)$. We have a canonical
functor
$$
\Delta: \Mod(A)\lra \Mod(\CO_{\fX}),\ M\mapsto M^{\Delta}
$$
from the category of $A$-modules to the category of sheaves
$\CO_{\fX}$-modules.
If $A$ is noetherian then $\Delta$ establishes an equivalence between
the category of $A$-modules of finite type and that of coherent $\CO_{\fX}$-
modules, {\em loc.cit.}, 10.10.2.
\subsection{} Let $\fX$ be a formal scheme, $\CM$ a coherent
$\CO_{\fX}$-module.
If $\fX$ is represented as in ~(\ref{indlim}) then
\begin{equation}
\label{indlim-m}
\CM\cong \invlim\ M_n
\end{equation}
for a suitable inverse system of coherent $X_n$-modules $M_n$, {\em loc.cit.},
10.11.3.
We will consider $\CM$ as a sheaf of topological $\CO_{\fX}$ modules equipped
with the topology defined in {\em loc.cit.}, 10.11.6. Note that
for every affine open $U\subseteq\fX$ the module $\Gamma(U,\CM)$ is
complete.
If $\CN$ is another coherent $\CO_{\fX}$-module, we have isomorphisms
$$
\CM\otimes_{\CO_{\fX}}\CN\cong\invlim\ (M_n\otimes_{\CO_{X_n}}N_n)
$$
and
$$
\CHom_{\CO_{\fX}}(\CM,\CN)\cong \invlim\ \CHom_{\CO_{X_n}}(M_n,N_n),
$$
cf. {\em loc.cit.}, 10.11.7.
\subsection{Jets} Let $f: \fX\lra \fS$ be a morphism of formal schemes,
$f=\dirlim\ f_i$ a representation as in ~(\ref{adic}).
Let us consider the diagonal
$$
\Delta_f: \fX\hra \fX\times_{\fS}\fX
$$
We have
$$
\fX\times_{\fS}\fX=\dirlim\ X_i\times_{S_i}X_i
$$
and $\Delta_f=\dirlim\ \Delta_{f,i}$ where
$$
\Delta_{f,i}: X_i\hra X_i\times_{S_i}X_i
$$
are diagonal mappings. If
$\CI_i\subseteq\CO_{X_i\times_{S_i}X_i}$ is the ideal of $\Delta_{f,i}$
then
$$
\fI=\invlim\CI_i\subseteq\invlim\ \CO_{X_i\times_{S_i}X_i}=
\CO_{\fX\times_{\fS}\fX}
$$
is the ideal of $\Delta_f$.
Let $p_j:\fX\times_{\fS}\fX\lra \fX,\ j=1,2$ be projections.
For any integer $n\geq 0$ set
$$
\fX^{(n)}_f=(\Delta_f(\fX),\CO_{\fX\times_{\fS}\fX}/\CI^{n+1})
$$
--- it is a closed formal subscheme of $\fX\times_{\fS}\fX$.
Consider canonical projections $p_i^{(n)}:\fX^{(n)}\lra \fX,\ i=1,2$.
\subsubsection{}
\label{defn-jets}
\begin{defn}{} (Cf. \cite{ega} IV 16.3.1, 16.7.1.) We define a sheaf of
rings over $\fX$ which is called the {\em sheaf of $n$-jets} as
$$
\CP^n_f=\CP^n_{\fX/\fS}:=p_{1*}(\CO_{\fX\times_{\fS}\fX}/\fI^{n+1})=
p_{1*}^{(n)}p^{(n)*}_2(\CO_{\fX})
$$
We have two canonical morphisms of sheaves of topological rings
\begin{equation}
\label{first}
\CO_{\fX}\lra \CP_f^n,\ x\mapsto x\otimes 1
\end{equation}
and
\begin{equation}
\label{second}
d^n_{\fX/\fS}:\CO_{\fX}\lra \CP_f^n,\ x\mapsto 1\otimes x
\end{equation}
by means of which one introduces a structure of left (resp., right)
$\CO_{\fX}$-module on $\CP^n_f$.
More generally, for a coherent sheaf $\CM$ over $\fX$ set
$$
\CP^n_{\fX/\fS}(\CM):=p_{1*}^{(n)}p^{(n)*}_2(\CM)
$$
\end{defn}
We have
$$
\CP^n_{\fX/\fS}(\CM)=\CP^n_{\fX/\fS}\otimes_{\CO_{\fX}}\CM
$$
where the right $\CO_{\fX}$-module structure on $\CP^n_{\fX/\fS}$
is used on the right-hand side, cf. {\em loc.cit.}, 16.7.2.1.
The $\CO_{\fX}$-bimodule structure on $\CP^n_f$ induces an
$\CO_{\fX}$-bimodule structure on $\CP^n_{\fX/\fS}(\CM)$.
Following {\em loc.cit.}, if we do not specify the structure of an
$\CO_{\fX}$-module on $\CP^n_{\fX/\fS}(\CM)$, we mean that
of a {\em left} module.
\subsubsection{} If $\CM=\invlim\ M_i$ as in ~(\ref{indlim-m}) then
\begin{equation}
\label{invlim-p}
\CP^n_{\fX/\fS}(\CM)=\invlim\ \CP^n_{X_i/S_i}(M_i)\ .
\end{equation}
\subsubsection{} Consider affine open formal subschemes
$U=\Spf(B)\subseteq\fX,\ V=\Spf(A)\subseteq \fS$
such that $f(U)\subseteq V$, so that $A,B$ are adic rings, and $B$ is a
topological $A$-algebra. Let $I=\ker(B\otimes_AB\lra B)$ be the kernel
of the multiplication, and
$$
P_{B/A}=(B\otimes_AB)/I^{n+1}
$$
considered as a $B$-module by means of a map $x\mapsto x\otimes 1$.
It is naturally a topological $B$-module, and we can consider its completion,
$\hP_{B/A}$. We have
\begin{equation}
\CP^n_U\cong (\hP_{B/A})^{\Delta}
\end{equation}
\subsubsection{}
\begin{claim}{} $d^n_{\fX/\fS}(\CO_{\fX})$ topologically generates
$\CO_{\fX}$-module $\CP^n_f$.
\end{claim}
(cf. \cite{ega} IV, 16.3.8) $\Box$
\subsubsection{} We have evident projections
\begin{equation}
\label{cofilt}
\CP^n_f\lra\CP^{n-1}_f\ .
\end{equation}
Consider a projection
\begin{equation}
\label{proj}
\CP^n_f\lra \CP^0_f=\CO_{\fX}
\end{equation}
and let $\bCP^n_f$ denote its kernel. A map
$$
d_1: \CO_{\fX}\lra \CP^n_f,\ x\mapsto x\otimes 1,
$$
is left inverse to ~(\ref{proj}), so we have a splitting
$$
\CP^n_f\cong\bCP^n_f\oplus\CO_{\fX}.
$$
\subsection{Differentials} We define
$$
\Omega^1_f=\Omega^1_{\fX/\fS}:=\bCP^1_f\ .
$$
If $\fX=\Spf(B),\ \fS=\Spf(B)$ then we will use the notation $\Omega^1_{B/A}$
for $\Gamma(\fX,\Omega^1_{\fX/\fS})$.
We have natural isomorphisms
$$
\Omega^1_f\cong p_{1*}(\CI/\CI^2)\cong p_{2*}(\CI/\CI^2)\ .
$$
We have a canonical continuous $\CO_{\fS}$-linear map
\begin{equation}
\label{difl}
d:\CO_{\fX}\lra \Omega^1_{\fX/\fS}
\end{equation}
induced by the map $x\mapsto x\otimes 1-1\otimes x$.
\subsubsection{}
From ~(\ref{invlim-p}) follows that we have a natural isomorphism
\begin{equation}
\Omega^1_{\fX/\fS}\cong\invlim\ \Omega^1_{X_i/S_i}
\end{equation}
\subsection{Example}
\label{power} Suppose $\fS=\Spf(A)$, and $\fX=\Spf(A_n)$ where
$A_n=A\{ T_1,\ldots,T_n\}$ is the completion of the polynomial
ring $A[T_1,\ldots,T_n]$ in the $J$-adic topology $J$, being an ideal
generated by an ideal
of definition of $A$ and $T_1,\ldots, T_n$. Then $\Omega^1_{A_n/A}$ is a
free $A_n$-module with the basis $dT_1,\ldots,dT_n$.
\subsection{Derivations} Let $\CM$ be a coherent $\CO_{\fX}$-module. We
define {\em the sheaf of derivations}
$$
\Der_{\fS}(\CO_{\fX},\CM):=\CHom_{\CO_{\fX}}(\Omega^1_{\fX/\fS},\CM).
$$
The map ~(\ref{difl}) induces a canonical embedding
$$
\Der_{\fS}(\CO_{\fX},\CM)\hra\CHom_{\CO_{\fS}}(\CO_{\fX},\CM)
$$
which identifies $\Der_{\fS}(\CO_{\fX},\CM)$ with the sheaf of local
$\CO_{\fS}$-homomorphisms $\alpha:\CO_{\fX}\lra\CM$ such that
$$
\alpha(xy)=x\alpha(y)+y\alpha(x).
$$
We set
$$
\CT_{\fX/\fS}:=\Der_{\fS}(\CO_{\fX},\CO_{\fX})
$$
and call this sheaf the {\em tangent sheaf of $\fX$ with respect to $\fS$},
cf. {\em loc.cit.}, 16.5.7. It is a sheaf of $\CO_{\fS}$-Lie algebras.
\subsection{Differential operators} Let $\CM,\ \CN$ be coherent
$\CO_{\fX}$-modules.
We define {\em the sheaf of differential operators of order $\leq n$ from
$\CM$ to $\CN$}:
$$
\Diff^{\leq n}_{\fX/\fS}(\CM,\CN)=\CHom_{\CO_{\fX}}(\CP^n_{\fX/\fS}(\CM),\CN).
$$
The structure of $\CO_{\fX}$-bimodule on $\CP^n({\CM})$ induces the structure
of $\CO_{\fX}$-bimodule on $\Diff^{\leq n}_{\fX/\fS}(\CM,\CN)$.
The projections ~(\ref{cofilt}) induce embeddings
$$
\Diff^{\leq n-1}_{\fX/\fS}(\CM,\CN)\hra\Diff^{\leq n}_{\fX/\fS}(\CM,\CN)\ .
$$
We set
$$
\Diff_{\fX/\fS}(\CM,\CN)=\dirlim\ \Diff^{\leq n}_{\fX/\fS}(\CM,\CN)\ .
$$
We denote
$$
\Diff^{\leq n}_{\fX/\fS}:=\Diff^{\leq n}_{\fS}(\CO_{\fX},\CO_{\fX});\
\Diff_{\fX/\fS}:=\Diff_{\CO_{\fS}}(\CO_X,\CM)\ .
$$
As in the case of schemes, we have compositions
$$
\Diff^{\leq n}_{\fX/\fS}\otimes_{\CO_{\fS}}\Diff^{\leq m}_{\fX/\fS}\lra
\Diff^{\leq n+m}_{\fX/\fS}
$$
(where in the tensor product the 1st (resp., 2nd) factor has a structure
of a right (resp., left) $\CO_{\fX}$-module).
\subsection{} Note that we have canonically
$$
\Diff^{\leq 1}_{\fX/\fS}\cong\CO_{\fX}\oplus\CT_{\fX/\fS}\ .
$$
Thus, the multiplication induces a canonical map
\begin{equation}
\label{twist-dif}
U_{\CO_{\fX}}(\CT_{\fX/\fS})\lra \Diff_{\fX/\fS}
\end{equation}
--- cf. ~\ref{twist-env}.
\subsection{}
\label{functor} Suppose we have a commutative square of formal schemes
$$\begin{array}{ccc}
\fX & \overset{u}{\lra} & \fX' \\
\downarrow & \; & \downarrow \\
\fS & \lra & \fS'
\end{array}$$
It induces natural maps
$$
\nu_n: u^*\CP^n_{\fX'/\fS'}\lra \CP^n_{\fX/\fS}
$$
If the square is cartesian, these maps are isomorphisms, cf. {\em loc.cit.},
16.4.5.
\subsection{}
\label{point} Let $\fX$ be a formal scheme, $x:\Spec(k)\lra \fX$ a point.
Then one has a canonical isomorphism of $k(x)$-algebras
$$
(\CP^n_{\fX/k})_x\otimes_{\CO_x}k(x)\cong \CO_x/\fm_x^{n+1}
$$
--- cf. {\em loc.cit.}, 16.4.12.
\subsection{Formally smooth morphisms} Let us call a morphism $f:\fX\lra \fS$
of formal schemes {\em formally smooth} if for every commutative
square
$$\begin{array}{ccc}
Y & \lra & \fX \\
\downarrow & \; & \downarrow \\
T & \lra & \fS
\end{array}$$
where $Y=\Spec(B)\lra T=\Spec(A)$ is a closed embedding of affine
(usual) schemes
corresponding to an epimorphism $A\lra B$ whose kernel $I$ satisfies $I^2=0$,
there exists a lifting $T\lra\fX$.
\subsubsection{} Suppose that $f=\dirlim\ f_n$, where all
$f_n:X_n\lra S_n$ are smooth morphisms of (usual) schemes. Then $f$ is
formally smooth.
This follows from ~\cite{ega} 0$_{\mbox{IV}}$, 19.4.1.
\subsection{} Let $\fX\overset{f}{\lra}\fY\overset{g}{\lra}\fS$ be
morphisms of formal schemes.
By ~\ref{functor} they induce maps
$f^*\Omega^1_{\fY/\fS}\lra\Omega^1_{\fX/\fS}$ and
$\Omega^1_{\fX/\fS}\lra\Omega^1_{\fX/\fY}$.
They in turn induce the maps in the sequence
\begin{equation}
\label{seq}
0\lra\CT_{\fX/\fY}\overset{g'}{\lra}\CT_{\fX/\fS}
\overset{f'}{\lra} f^*\CT_{\fY/\fS}\ .
\end{equation}
\subsubsection{}
\label{seq-thm}
\begin{thm}{} The sequence ~(\ref{seq}) is exact. If $f$ is formally
smooth then $f'$ is surjective.
\end{thm}
\begin{pf} Follows from ~\cite{ega} 0$_{\mbox{IV}}$, 20.7.18.
\end{pf}
\subsection{Smooth morphisms}
\label{smooth}
\begin{defn}{} A morphism $f:\fX\lra\fS$ is called {\em smooth} if it is
formally smooth and locally of finite type (\cite{ega} I, 10.13).
\end{defn}
\subsubsection{}
\begin{lem}{} If $f$ is smooth then $\Omega^1_f$ is locally free of finite
type.
\end{lem}
\begin{pf}
Everything is reduced to the case of affine formal schemes,
$\fX=\Spf(B),\ \fS=\Spf(A)$, and $B$ is a noetherian topological $A$-algebra
topologically of finite type over $A$. First, $\Omega^1_{B/A}$ is of finite
type by ~\ref{power} and ~\ref{seq-thm} since $B$ is a quotient of some
$A\{ T_1,\ldots, T_n\}$.
Let us prove that $\Omega^1_{B/A}$ is projective $B$-module. Suppose we have
a diagram
$$\begin{array}{ccc}
\; & \; & \Omega^1_{B/A}\\
\; & \; & \downarrow\\
M & \overset{\psi}{\lra}& N\\
\end{array}$$
with epimorhic $\psi$ and finitely generated $M$ . Let $I$ be an ideal of
definition in $B$.
By \\ \cite{ega}~0$_{\mbox{IV}}$, 20.4.9,
we can find liftings $\phi_n:\Omega^1_{B/A}\lra M/I^nM$ for each $n$.
We claim that we can choose $\phi_n$'s in such a way that
for all $n$ the composition $\Omega^1_{B/A}\overset{\phi_{n+1}}{\lra}
M/I^{n+1}M\lra M/I^nM$ is equal to $\phi_n$. Indeed, we do it step by step,
using again the lifting property and the fact that canonical maps
$$
M/I^{n+1}M\lra M/I^nM\times_{N/I^nN}N/I^{n+1}N
$$
are surjective.
Since $B$ is complete, $M=\invlim\ M/I^nM$, hence there exists a lifting
$\phi:\Omega^1_{B/A}\lra M$.
Now, since $\Omega^1_{B/A}$ is of finite type, we can find an epimorphic
map $F\lra \Omega^1_{B/A}$ from a finitely generated free $B$-module $F$;
using the lifting property proved above, we conclude that $\Omega^1_{B/A}$
is a direct summand of $F$.
\end{pf}
\subsection{Differentially smooth morphisms}
\begin{defn}{} A morphism $f:\fX\lra\fS$ of formal schemes is called
{\em differentially smooth} if $\Omega^1_f$ is locally free of finite rank.
\end{defn}
For an arbitrary $f$, define the graded ring
$$
\CGr.(\CP_f)=\oplus_{n=0}^{\infty}\CGr_n(\CP_f);\
\CGr_n(\CP_f)=\fI^n/\fI^{n+1},
$$
$\fI$ being as in ~\ref{defn-jets}. Evidently $\CGr_1(\CP_f)=\Omega^1_f$ and
we have the canonical surjective morphism from the symmetric algebra
\begin{equation}
\label{symm}
S^._{\CO_{\fX}}(\Omega^1_f)\lra\CGr_.(\CP_f)\ .
\end{equation}
\subsubsection{}
\begin{lem}{} If $f$ is differentially smooth then ~(\ref{symm}) is
isomorphism.
\end{lem}
\begin{pf} The same as in ~\cite{ega} IV 16.12.2 (cf. {\em loc.cit.},
16.10).
\end{pf}
\subsubsection{}
\label{basis}
\begin{thm}{} Let $f$ be differentially smooth, $U\subseteq\fX$ an open,
$\{ t_i\}_{i\in I}\in\Gamma(U,\CO_{\fX})$ a set of sections such that
$\{ dt_i\}_{i\in I}$ is the basis of $\Omega^1_{U/\fS}$. Let
$\{\dpar_i\}\in\Gamma(U,\CT_{\fX/\fS})$ be the dual basis. Then
(i) all $\dpar_i$ commute with each other;
(ii) $\Diff^{\leq n}_{\fX/\fS}$ is freely generated by all monomials on
$\dpar_i$ of degrees $\leq n$.
\end{thm}
\begin{pf} The same as in {\em loc.cit.}, 16.11.2.
\end{pf}
Thus, if $\fS=\Spec(k)$, if $f$ is differentially smooth then
$\fX$ is differentially smooth in the sense of ~\ref{dif-smo}.
\subsection{}
\begin{cor}{} If $f$ is differentially smooth then the map ~(\ref{twist-dif})
is isomorphism.
\end{cor}
\begin{pf}
Follows immediately from \Thm{basis} and \Thm{pbw-algebr}: the
map~(\ref{twist-dif}) preserves filtrations and induces an isomorphism on
the associated graded rings.
\end{pf}
Hence, if $\fS=\Spec(k)$, our sheaf $\Diff_{\fX/\fS}$ coincides with
$\Diff_{\fX}$ defined in Section ~\ref{twisted}.
\subsection{Examples of differentially smooth morphisms}
\subsubsection{} Smooth morphisms. This follows from {\em loc.cit.}, 16.10.2.
\subsubsection{} The structure morphism
$\Spf(A\{ T_1,\ldots,T_n\})\lra\Spf(A)$ (see ~\ref{power}).
\section{Homotopy Lie algebras and direct image functor}
\label{direct}
In this Section we develop a formalism of homotopy Lie algebras which
is sufficiently good for our needs.
In~\ref{holie} we define the category $\Holie$ of homotopy Lie algebras
and a functor from it to the filtered derived category given by the Quillen
standard complex (see~\ref{quillen}). In order to define higher direct
images for $\Holie$ in~\ref{dihla}, we provide in~\ref{s-derham} a construction
of Thom-Sullivan functor from cosimplicial modules to complexes, and some
homotopical properties of it in~\ref{cob}. The proof of the properties of
the Thom-Sullivan functor is given in Section~\ref{thoms}.
The main result of this Section is~\Thm{main-5}.
\subsection{Homotopy Lie algebras}
\label{holie}
\subsubsection{}
\label{cat}
Let $X$ be a formal scheme. We define $\Dglie(X)$
(resp., $\Dglie^{qc}(X),\ \Dglie^{c}(X)$) as a category
of dg $\CO_X$-Lie algebras $\fg$ such that ---
(i) all components $\fg^i$ are $\CO_X$-flat
(and quasicoherent or coherent over $\CO_X$ respectively);
(ii) one has $\CH^i(\fg)=0$ for sufficiently big $i$.
A morphism $f: \fg\lra \fh$ in this category
is a map of complexes of $\CO_X$-modules compatible with
brackets. Let us call $f$ {\em a quasi-isomorphism} if it induces an
isomorphism of all cohomology sheaves $\CH^i(f):\CH^i(\fg)\lra \CH^i(\fh)$.
By definition, the category $\Holie^{(*)}(X)$, is
the localization of $\Dglie^{(*)}(X)$ with respect to the
class of all quasi-isomorphisms. Objects of $\Holie(X)$ are called
{\em homotopy Lie algebras}. So, each dg $\CO_X$-Lie algebra defines
a homotopy Lie algebra.
\subsubsection{} If $f: X\lra Y$ is a flat morphism, the inverse
image functor for $\CO_X$-modules induces the functor
\begin{equation}
\label{inv-im-dglie}
f^{*Lie}:\Dglie^{(*)}(Y)\lra \Dglie^{(*)}(X)
\end{equation}
It takes quasi-isomorphisms to quasi-isomorphisms, and hence induces
the functor (to be denoted by the same letter)
\begin{equation}
\label{inv-im-holie}
f^{*Lie}:\Holie^{(*)}(Y)\lra \Holie^{(*)}(X)
\end{equation}
\subsubsection{Filtered derived categories} (For more details, see ~\cite{i},
ch. V, no.1 where the case of finite filtrations is considered.)
Let $\CC(X)$ denote the category of complexes of $\CO_X$-modules,
and $\CCF(X)$ the category
whose objects are complexes of $\CO_X$-modules $A$ together with a
filtration by $\CO_X$-subcomplexes $\ldots\subseteq F_iA\subseteq
F_{i+1}A\subseteq\ldots,\ i\in \Bbb Z$ such that $F_iA=0$ for sufficiently
small $i$ and
$A=\cup_i F_iA$; the morphisms being morphisms of complexes
compatible with filtrations.
We set $\gr_i(A)=F_iA/F_{i-1}A$. For $a,b\in \Bbb Z$, let $\CCF_{[a,b]}(X)$
be the full subcategory consisting of complexes
with $F_{a-1}A=0,F_{b}A=A$.
A morphism $f:A\lra B$ in $\CCF(X)$ is called {\em a filtered
quasi-isomorphism} if the induced maps $\gr_i(f): \gr_i(A)\lra \gr_i(B)$ are
quasi-isomorphisms. From our assumptions on filtrations follows that
a filtered quasi-isomorphism is a quasi-isomorphism.
We denote by $\CD(X)$ the localization of $\CC(X)$ with respect
to quasi-isomorphisms, and $\CDF(X)$, $\CDF_{[a,b]}(X)$ the localization
of $\CCF(X)$ (resp., $\CCF_{[a,b]}(X)$) with respect to filtered
quasi-isomorphisms.
\subsubsection{}
\begin{lem}{} Suppose that $f:\fg\lra \fh$ is a quasi-isomorphism in
$\Dglie(X)$. Then the induced morphism $C(f): C(\fg)\lra C(\fh)$ is a filtered
quasi-isomorphism.
\end{lem}
\begin{pf} It suffices to prove that all
$\gr_i(f): S^i(\fg[1])\lra S^i(\fh[1])$ are quasi-isomorphisms.
Repeated application of the K\"{u}nneth spectral sequence ~\cite{g},
Ch.~I, 5.5.1,
shows that $f^{\otimes i}:\fg^{\otimes i}\lra \fh^{\otimes i}$ are
quasi-isomorphisms (we use the assumptions ~\ref{cat} (i)-(ii)).
After passing to $\Sigma_n$-invariants, we get the desired claim.
\end{pf}
\subsubsection{} It follows that the functors $\fg\mapsto C(\fg)$,
$\fg\mapsto F_nC(\fg)$ induce functors between homotopy categories
$$
C: \Holie(X)\lra \CDF(X)
$$
$$
F_nC:\Holie(X)\lra \CDF_{[0,n]}(X)
$$
For $\fg\in \Holie(X)$ we define homology sheaves
$$
\CH^{Lie}_n(\fg)=\CH^{-n}(C(\fg));\ \CF_m\CH_n^{Lie}(\fg)=H^{-n}(F_mC(\fg))
$$
--- these are sheaves of $\CO_X$-modules. If $X=\Spec (k)$, we denote them
$H^{Lie}_n(\fg),\ F_mH_n^{Lie}(\fg)$ respectively --- these are
$k$-vector spaces.
\subsubsection{}
\label{sec-spectr-seq} Let $\fg\in \Holie(X)$. We have a spectral sequence
\begin{equation}
\label{spectr-seq}
E_1^{pq}=\CH^q(\Lambda^{-p}_{\CO_X}(\fg)) \Lra \CH^{Lie}_{-p-q}(\fg)
\end{equation}
If all sheaves $\CH^i(\fg)$ are $\CO_X$-flat then by K\"{u}nneth formula,
we have
\begin{equation}
\label{kunneth}
\CH^q(\Lambda^{n}_{\CO_X}(\fg))\cong
\left(\sum_{q_1+\ldots+q_n=q}\CH^{q_1}(\fg)\otimes_{\CO_X}\ldots\otimes_{\CO_X}
\CH^{q_n}(\fg)\right)^{\Sigma_n,-}
\end{equation}
where $(\cdot)^{\Sigma_n,-}$ denotes the subspace of anti-invariants of
the symmetric group $\Sigma_n$.
\subsection{Thom-Sullivan complex}
\label{s-derham}
\subsubsection{} In this subsection $\Delta$ will denote the category
of totally ordered finite sets $[n]=\{ 0,\ldots, n\}$,
$n\geq 0,$
and non-decreasing maps. For any category $C$, we denote $\Delta^0C,\
\Delta C$ the categories of simplicial and cosimplicial objects
in $C$ respectively. For $A\in \Delta^0 C$ (resp., $B\in \Delta C$) we
denote $\alpha^*: A_m\lra A_n$ (resp., $\alpha_*: A^n\lra A^m$) the
map in $C$ corresponding to a morphism $\alpha: [n]\lra[m]$
in $\Delta$.
$\Ens$ will denote the category of sets, $\Delta[n]\in \Delta^0\Ens$ the
standard $n$-simplex.
\subsubsection{} (For details, see \cite{bug}).
Denote by $R_n$ the commutative $k$-algebra
{}~{$k[t_0,\ldots,t_n]/(\sum_{i=0}^nt_i-1)$} ($t_i$ being independent
variables).
Set $\Bbb A[n]=\Spec R_n$.
Together with the standard face and degeneracy maps, algebras $R_n,\ n\geq 0,$
form a simplicial algebra $R$.
Let $\Omega_n$ denote the algebraic de Rham complex of $R_n$ over $k$,
$$
\Omega_n=\Gamma(\Bbb A[n],\Omega_{\Bbb A[n]/k})
$$
It is a commutative dg $k$-algebra which may be identified with
$$
R_n[dt_0,\ldots,dt_n]/(\sum dt_i),
$$
with $\deg(dt_i)=1$ and the differential given by the formula $d(t_i)=dt_i$.
The algebras $\Omega_n,\ n\geq 0$ together with coface and codegeneracy
maps induced from $R$, form a simplicial commutative dg algebra
$\Omega=\{\Omega_n\}$. So, for a fixed $p,\ \Omega^p$ is a
simplicial vector space, and $\Omega_p$ is a complex.
\subsubsection{}
\label{invtensor} (Cf. ~\cite{hlha}, 3.1.3) Let $C$ be a small category.
Let us denote $\Mor(C)$
the category whose objects are morphisms $f: x\lra y$ in $C$,
a map $f\lra g$ is given by a commutative diagram
$$\begin{array}{ccc}
\cdot & \overset{f}{\lra} & \cdot \\
\uparrow & \; & \downarrow \\
\cdot & \overset{g}{\lra} & \cdot \\
\end{array}$$
The composition is defined in the evident way. For $f$ as above set $s(f)=x,\
t(f)=y$.
Let $A$ be a commutative $k$-algebra, $X: C^0\lra \Mod(k),\
Y:C\lra \Mod(A)$ two functors ($C^0$ denotes the opposite
category). Denote by
$$
X\otimes Y: \Mor(C)\lra \Mod(A)
$$
the functor given by $X\otimes Y(f)=X(s(f))\otimes_kY(t(f))$ and defined
on morphisms in the evident way. Set
$$
X\invtimes Y=\invlim (X\otimes Y)\in \Mod(A)
$$
In other words, $X\invtimes Y$ is an $A$-submodule of
$\prod_{c\in Ob(C)}X(c)\otimes Y(c)$ consisting of all $\{a(c)\}_{c\in Ob(C)},$
$ a(c)\in X(c)\otimes Y(c)$ such that for every
$f:b\lra c\in \Mor(C)$
$$
(f^*\otimes 1)(a(c))=(1\otimes f_*)(a(b)).
$$
\subsubsection{Thom-Sullivan complex} (Cf. ~\cite{hlha}, \S 4) Applying
the previous construction to $C=\Delta,\ X=\Omega^p\in \Delta^0\Mod(k),
\ Y\in\Delta\Mod(A)$, we get $A$-modules
$$
\Omega^p(Y)=\Omega^p\invtimes Y
$$
When $p$ varies, they form a complex $\Omega(Y)\in \CC(A)$ which is called
{\em Thom-Sullivan complex} of $Y$.
If $Y$ is a constant cosimplicial object, then
\begin{equation}
\label{const}
\Omega(Y)\cong Y
\end{equation}
canonically.
For example, if $Y$ is the cosimplicial space of $k$-valued cochains of a
simplicial set $X$, $\Omega(Y)$ is the Thom-Sullivan complex of
$X$ described in ~\cite{bug}.
If $Y$ is a {\em complex} in $\Delta\Mod(A)$ or, what is the same,
a cosimplicial complex of $A$-modules, then, applying the previous
construction componentwise we get a {\em bicomplex} of $A$-modules;
we will denote the corresponding simple complex again $\Omega(Y)$.
This way we get a functor
\begin{equation}
\label{thom}
\Omega: \Delta\Mod(A)\lra \CC(A).
\end{equation}
\subsubsection{}
\begin{lem}{} The functor $\Omega$ is exact.
\end{lem}
For a proof, see \ref{exact-6}.
\subsubsection{Normalization} For $Y\in \Delta\Mod(A)$ denote by
$N(Y)\in \CC(A)$ its
normalization, i.e. set $N(Y)^i\subset Y^i$ to be the intersection
of kernels of all codegeneracies $Y^i\lra Y^{i-1}$, the differential
$N(Y)^i\lra N(Y)^{i+1}$ being the alternating sum of the cofaces. ($N(Y)^i=0$
for $i<0$). We say that $Y$ is {\em finite dimensional} if $N(Y)^i=0$
for $i>>0$.
This way we get a functor
$$
N:\Delta\Mod(A)\lra \CC(A)
$$
For each $n\geq 0$ denote by $\CZ^{\cdot}_n=C^{\cdot}(\Delta[n],k)$ the
complex of
normalized $k$-valued cochains of the standard simplex. When $n$ varies,
the complexes $\CZ^{\cdot}_n$ form a simplicial object
$\CZ=\CZ^{\cdot}_{\cdot}\in \Delta^0\CC(k)$.
Given $Y\in \Delta\Mod(A)$, we can apply the construction
{}~\ref{invtensor} to each $\CZ^n_{\cdot}$ and $Y$, and obtain
$\CZ\invtimes Y\in \CC(A)$. It follows from the definitions that
we have a natural isomorphism
$$
\CZ\invtimes Y\cong N(Y)
$$
(cf. ~\cite{hlha}, 2.4).
\subsubsection{} For each $n$ we have the {\em integration}
map
$$
\int: \Omega_n\lra C^*(\Delta[n],k)
$$
in $\CC(k)$, ~\cite{bug},\S 2. Taken together, they give rise to the morphism
$$
\int: \Omega\lra \CZ
$$
in $\Delta^0C(k)$. It induces natural maps
\begin{equation}
\label{int-y}
\int_Y:\Omega(Y)\lra N(Y)
\end{equation}
for every $Y\in \Delta\Mod(A)$.
\subsubsection{"De Rham theorem"}
\label{derham-thm}
\begin{lem}{} For every $Y\in \Mod(A)$ the map $\int_Y$ is a
quasi-isomorphism.
\end{lem}
\begin{pf}
See ~\cite{hlha}, 4.4.1.
\end{pf}
\subsubsection{Base change} Let $A'$ be a commutative $A$-algebra,
$Y\in \Delta\Mod(A)$, whence $Y\otimes_AA'\in \Delta\Mod(A')$. We have an
evident base change morphism
\begin{equation}
\label{bchange-map}
\Omega(Y)\otimes_AA'\lra \Omega(Y\otimes_AA').
\end{equation}
\subsubsection{}
\label{base}
\begin{lem}{} If $Y$ is finite dimensional then ~(\ref{bchange-map}) is
isomorphism.
\end{lem}
\subsubsection{}
\label{flat}
\begin{lem}{} Let $A$ be noetherian and $Y\in \Delta\Mod(A)$;
suppose that all $Y^n$ are flat over $A$.
Then for every $p,\ \Omega^p(Y)$ is flat over $A$.
\end{lem}
The proof of ~\ref{base} and ~\ref{flat} will be given in the next Section,
see ~\ref{base-6},~\ref{flat-6}.
\subsubsection{}
\label{cosimpl-lie} Suppose that $B$ is a commutative dg algebra and
$\fg$ is a dg Lie algebra. Then $B\otimes \fg$ is naturally
a dg Lie algebra, the bracket being defined as
$$
[a\otimes x,b\otimes y]=(-1)^{|x||b|} ab\otimes [x,y].
$$
Let $\fg$ be a cosimplicial dg $A$-Lie algebra. Then all
$\Omega_n\otimes \fg^n$ are dg $A$-Lie algebras; hence their inverse limit
$\Omega(\fg)$ is. This way we get a functor $\Omega$ from the
category of cosimplicial dg $A$-Lie algebras to dg $A$-Lie algebras.
\subsubsection{} Let $X$ be a (formal) scheme. We can sheafify Thom-Sullivan
construction. Denote by $\Mod(\CO_X)$ the category of sheaves
of $\CO_X$-modules. For $\CF\in \Delta\Mod(\CO_X)$ we get
$N(\CF), \Omega(\CF)\in \CC(X)$. We call $\CF$ {\em finite dimensional}
if $N(\CF)^i=0$
for $i>>0$. We denote $\Delta(\Mod(\CO_X))^f\subseteq \Delta(\Mod(\CO_X))$ the
full subcategory consisting of finite dimensional cosimplicial sheaves.
By~\ref{qc} if $\CF$ is finite dimensional, and all $\CF^i$ are quasicoherent
then all $\Omega^i(\CF)$ are quasicoherent.
By~\ref{flat} if all $\CF^i$ are $\CO_X$-flat then all $\Omega^i(\CF)$
are $\CO_X$-flat.
If $\fg$ is a cosimplicial dg $\CO_X$-Lie algebra, then, applying
{}~\ref{cosimpl-lie} we get a dg $\CO_X$-Lie algebra $\Omega(\fg)$.
\subsubsection{}
\label{derham-dir-im} Let $f:X\lra Y$ be a map of schemes,
$\CF\in\Delta\Mod(\CO_X)$. Then we have an evident equality
$$
f_*\Omega(\CF)=\Omega(f_*\CF)
$$
\subsection{Cosimplicial homotopies}
\label{cob}
Let $\CA$ be a category with finite products.
\subsubsection{}
\begin{defn}{path}
Define the {\em path
functor} $X\mapsto X^I$ from $\Delta{\CA}$ to itself as follows:
$$ (X^I)^n=\prod_{s:[n]\ra[1]}X^n,$$
the map $f_*:(X^I)^m\ra(X^I)^n$ for any $f:[m]\ra[n]$ being defined
by the formula
$$ f_*(\{x_s\})_t=f_*(x_{tf}),\ t:[n]\ra[1].$$
\end{defn}
The path functor is endowed with natural transformations
$$i_X:X\ra X^I,\ \pr_{i,X}:X^I\ra X (i=0,1)$$
given by the formulas
$$ i_X(x)_s=x;\ \pr_{i,X}(\{x_s\})=x_{s_i}$$
where $s_i:[n]\ra[1]$ denotes the constant map with value $i$.
\Defn{path} is a special case of constructions given in~\cite{q2}, ch.~2,
see Prop. 2 for the dual statement. In particular, one can define in the same
way a functor $X\mapsto X^S$ for any finite simplicial set $S$, this
construction is functorial on $S\in\Delta^0\Ens$ and the natural
transformations $i_X,\pr_{i,X}$ are induced by the corresponding maps
$\pi:I\ra *\text{ and }\iota_i:*\ra I (i=0,1)$ in $\Delta^0\Ens$.
\subsubsection{}
\begin{defn}{strict-homo}
Let $X,Y\in\Delta{\CA}$. Maps $f_i:X\ra Y \ (i=0,1)$ are said to be
{\em strictly homotopic} if there exists a map $F:X\ra Y^I$ such that
$f_i=\pr_i\circ F$.
\end{defn}
\subsubsection{}
\begin{exa}{=}
Let $X\in\Delta{\CA}$. The maps $\id,\ i\circ\pr_0:X^I\ra X^I$ are
strictly homotopic.
In fact, the maps $\id_I$ and $\iota_0\circ\pi$ are strictly homotopic
in $\Delta^0\Ens$.
\end{exa}
\subsubsection{} Let now $\CA$ be additive. Recall that
for $X\in\Delta{\CA}$ the total complex $\Tot(X)\in \CC({\CA})$
is defined by the
properties
$$ \Tot(X)^n=X^n,$$
$$d=\sum (-1)^i\delta^i:\Tot(X)^n\ra\Tot(X)^{n+1}.$$
This defines a functor $\Tot:\Delta{\CA}\ra \CC({\CA})$.
\begin{lem}{homo-homo}
A strict homotopy $H:X\ra Y^I$ between $f$ and $g$ induces a (chain)
homotopy $h:\Tot(X)\ra\Tot(Y)[-1]$ between $\Tot(f)$ and $\Tot(g)$.
\end{lem}
\begin{pf}
For $x\in X^n$ define
$$ h(x)=\sum_{i=0}^{n-1}(-1)^i\sigma^i(y_{\alpha_i})$$
where $y_{\alpha}$ are defined by $H(x)=\{y_{\alpha}\}$ and
\begin{equation}
\alpha_i(t)=\begin{cases} 0& \text{ if } t\leq i\\
1& \text{ if } t>i.
\end{cases}
\end{equation}
A direct calculation shows that $h$ is the chain homotopy we need.
\end{pf}
\subsubsection{} We apply the above constructions to homotopy Lie algebras.
Fix a commutative ring $A\supseteq{\Bbb Q}$ and put ${\CA}=\Dglie(A)$.
\begin{cor}{glueh}
Let $X,Y\in\Delta\Dglie(A)$. Let $f,g:X\ra Y$ be strictly homotopic. Then
the maps $\Omega(f), \Omega(g):\Omega(X)\ra\Omega(Y)$ induce equal maps
in the homotopy category $\Holie(A)$.
\end{cor}
\begin{pf}
It suffices to check that $\Omega(\pr_{0,Y})=\Omega(\pr_{1,Y})$ in the homotopy
category. Since the both maps split $\Omega(i_Y)$, it sufficies to check that
the latter one is a quasi-isomorphism. For this we can substitute the
functor $\Omega$ with $\Tot$ (since they are naturally quasi-isomorphic).
We can also substitute the category $\Dglie(A)$ with the category ${\CC}(A)$
since the forgetful functor $\#:\Dglie(A)\ra{\CC}(A)$ commutes with direct
products. Then~\Exa{=} and~\Lem{homo-homo} prove even more that we actually
need.
\end{pf}
\subsection{Direct image of homotopy Lie algebras}
\label{dihla}
\subsubsection{\v{C}ech resolutions} Let $X$ be a topological space, $\CF$
an abelian sheaf over $X$, $\CU=\{ U_i\}_{i\in I}$ an open covering of $X$.
For each $n\geq 0$ set
$$
\CHC^n(\CU,\CF)=\prod_{(i_0,\ldots,i_n)\in I^{n+1}}j_{i_0\ldots i_n*}
j_{i_0\ldots i_n}^*\CF
$$
where $j_{i_0\ldots i_n}:U_{i_0}\cap\ldots\cap U_{i_n}\hra X$.
Together with the standard cofaces and codegeneracies, the sheaves
$\CHC^n(\CU,\CF),\ n\geq 0,$ form a cosimplicial sheaf $\CHC(\CU,\CF)$.
It is finite dimensional if the covering $\CU$ is finite.
We have an embedding $\CF\lra \CHC^0(\CU,\CF)$; it induces the augmentation
map
\begin{equation}
\label{augm}
\CF\lra\CHC(\CU,\CF)
\end{equation}
where $\CF$ is considered as a constant cosimplicial sheaf. The induced
map
\begin{equation}
\label{augm-n}
\CF\lra N(\CHC(\CU,\CF))
\end{equation}
is a quasi-isomorphism, \cite{g}, ch.~II, 5.2.1.
\subsubsection{} Suppose that
--- either $X$ is a scheme and $\CF$ is a quasicoherent sheaf of
$\CO_X$-modules,
--- or $X$ is a formal scheme and $\CF$ is a coherent sheaf.
Choose an affine covering $\CU$. Then the complex
$\Gamma(X,N(\CHC(\CU,\CF)))$ represents $R\Gamma(X,\CF)$, ~\cite{h},
III, 4.5; ~\ref{aff}.
If $f:X\lra Y$ is a morphism of (formal) schemes then $f_*N(\CHC(\CU,\CF))$
represents $Rf_*(\CF)$.
\subsubsection{} Applying to ~(\ref{augm}) the functor $\Omega$, and using
{}~(\ref{const}), we get a canonical map of complexes
\begin{equation}
\label{augm-omega}
\CF\lra \Omega(\CHC(\CU,\CF))
\end{equation}
which is a quasi-isomorphism by the above and ~\ref{derham-thm}.
\subsubsection{} Suppose we are in one of the following situations:
{\bf Case 1.} $f: X\lra Y$ is a flat morphism of schemes,
$\fg\in \Dglie^{qc}(X)$.
{\bf Case 2.} $f: X\lra Y$ is a flat morphism of formal schemes,
$\fg\in \Dglie^{c}(X)$.
Choose an open affine covering $\CU$ of $X$,
and consider the cosimplicial complex of $\CO_Y$-modules
$f_*\CHC(\CU,\fg)$; it has an evident structure of a cosimplicial
dg $\CO_Y$-Lie algebra. Applying the Thom-Sullivan functor,
we get a dg $\CO_Y$-Lie algebra $\Omega(f_*\CHC(\CU,\fg))$.
Note that by ~\ref{derham-dir-im} this is the same as
$f_*\Omega(\CHC(\CU,\fg))$. Let us denote it $f_{*,\CU}^{Lie}(\fg)$.
\subsubsection{} Let now $\CU=\{U_i\}_{i\in I}$ and
$\CV=\{V_j\}_{j\in J}$ be two open coverings of $X$ so that cosimplicial
dg Lie algebras $\CHC(\CU,\fg),\ \CHC(\CV,\fg)$ are defined.
Let maps $f,g:I\ra J$
satisfy the conditions $U_i\subseteq V_{f(i)},\ U_i\subseteq V_{g(i)}.$
\begin{lem}{to5.}
The maps from $\CHC(\CV,\fg)$ to $\CHC(\CU,\fg)$ induced by $f$ and $g$
are strictly homotopic.
\end{lem}
\begin{pf} Immediate.
\end{pf}
\begin{cor}{} In the notations above the maps from $f_{*,\CV}^{Lie}(\fg)$
to $f_{*,\CU}^{Lie}(\fg)$ induced by $f$ and by $g$, coincide in $\Holie(Y)$.
\end{cor}
\begin{pf}
Compare~\ref{to5.} with~\ref{glueh}.
\end{pf}
\subsubsection{} If $\fg=f^{*\Lie}\fh$ for some $\fh\in\Dglie(Y)$ then the
augmentation ~(\ref{augm-omega})
$$
f^{*\Lie}\fh\lra \Omega(\CHC(\CU,f^{*\Lie}\fh))
$$
and the adjunction map $\fh\lra f_*f^*\fh$ induce the map
of dg $\CO_X$-Lie algebras
\begin{equation}
\label{adj}
\fh\lra f_{*,\CU}^{Lie}(f^{*\Lie}\fh).
\end{equation}
On the other hand, for any $\fg\in\Holie(X)$ a map
\begin{equation}
f^{*\Lie}f_*^{\Lie}(\fg)\lra\fg
\end{equation}
in $\Holie(X)$ is defined as the composition
\begin{equation}
f^{*\Lie}f_*^{\Lie}(\fg)=f^{*\Lie}f_*\Omega(\CHC(\CU,\fg))\ra
\Omega(\CHC(\CU,\fg))\ra\fg
\end{equation}
the last map being inverse to the composition of the
quasi-isomorphisms~(\ref{augm-n}) and~(\ref{int-y}).
Putting together the above considerations, we get the following
\subsubsection{}
\begin{thm}{main-5} (i) In Case 1 (resp., Case 2) $f_{*,\CU}^{Lie}(\fg)$
belongs to
$\Dglie^{qc}(Y)$ (resp., $\Dglie(Y)$).
(ii) The class of $f_{*,\CU}^{Lie}(\fg)$ in $\Holie^{qc}(Y)$ (resp.,
$\Holie(Y)$) does not depend, up to a unique isomorphism, on $\CU$.
(iii) The functor $f_{*,\CU}^{Lie}$ takes quasi-isomorphisms to
quasi-isomorphisms; thus it induces the functor
$$
f_*^{Lie}:\Holie^{qc}(X)\lra \Holie^{qc}(Y)
$$
in Case 1, and
$$
f_*^{Lie}:\Holie^{c}(X)\lra \Holie(Y)
$$
in Case 2 respectively, such that the square
$$\begin{array}{ccc}
\Holie^{(*)}(X) & \overset{f_*^{Lie}}{\lra} & \Holie^{(*')}(Y) \\
\downarrow & \; & \downarrow \\
\CD(X) & \overset{f_*}{\lra} & \CD(Y) \\
\end{array}$$
the vertical arrows being forgetful functors, 2-commutes in both cases.
This means that there is a natural isomorphism between the two functors
from $\Holie^{(*)}(X)$ to $\CD(Y)$.
(iv) In Case 1, maps ~(\ref{adj}) induce the natural transformation
$\Id_{\Holie(Y)}\lra f_*^{Lie}f^{*Lie}$ which makes the functor $f_*^{Lie}$
right adjoint to $f^{*Lie}$. $\Box$
\end{thm}
\subsubsection{} In case $Y=\Spec(k),\ f:X\lra Y$ the structure
morphism, we will denote $f_*^{Lie}(\fg)$ also by $\Gamma^{Lie}(X,\fg)$.
\section{Thom-Sullivan functor}
\label{thoms}
In this Section we will compute more explicitely the Thom-Sullivan
functor and prove some fundamental properties of it.
\subsection{} We keep the assumptions and notations from ~\ref{s-derham}.
In particular, $k$ is a base field of characteristic zero and
$A$ denotes a commutative $k$-algebra.
Expressing $t_0$ as $t_0=1-\sum_{i=1}^nt_i$ we identify $k$-algebras
$R_n$ with $k[t_1,\ldots,t_n]$ and commutative
dg $k$-algebras $\Omega_n$ with $R_n[dt_1,\ldots,dt_n]$.
The standard simplicial morphisms of dg $k$-algebras
$$
d_i:\Omega_n\lra\Omega_{n-1},\ s_i:\Omega_n\lra
\Omega_{n+1},
$$
$i=0,\ldots, n$ are defined by the formulas
\begin{equation}
d_i(t_j)=
\begin{cases}
t_j & \text{ if } j<i,\\
0 & \text{ if } j=i,\\
t_{j-1}& \text{ if } j>i
\end{cases}
\label{d_i}
\end{equation}
for $i>0$ and
\begin{equation}
d_0(t_j)=
\begin{cases}
t_{j-1} & \text{ if } j>1,\\
1-\sum t_i& \text{ if } j=1.
\end{cases}
\label{d_0}
\end{equation}
\begin{equation}
s_i(t_j)=
\begin{cases}
t_j & \text{ if } j<i,\\
t_i+t_{i+1} & \text{ if } j=i,\\
t_{j+1}& \text{ if } j>i
\end{cases}
\label{s_i}
\end{equation}
for $i>0$ and
\begin{equation}
s_0(t_j)=t_{j+1}.
\label{s_0}
\end{equation}
These maps satisfy standard simplicial identities.
\subsection{} Let $X\in\Delta\Mod(A)$. So $X$ is a set of $A$-modules
$X^p,\ p\geq 0$, together with maps
$$
\delta^i:X^{p-1}\lra X^{p},\ \sigma^i:X^{p+1}\lra X^{p},
$$
$i=0,\ldots,p$, satisfying the standard cosimplicial identities.
By definition, $\Omega(X)$ is a complex of $A$-modules
$$
0\lra\Omega^0(X)\overset{d}{\lra}\ldots\overset{d}{\lra}\Omega^n(X)
\overset{d}{\lra}\ldots
$$
where $\Omega^n(X)$ is the space of all collections
$\{x_p\in\Omega_p^n\otimes X^p\}_{p\geq 0}$ satisfying the following conditions
\begin{equation}
(1\otimes\delta^i)(x_p)=(d_i\otimes 1)(x_{p+1})
\label{d-eq}
\end{equation}
\begin{equation}
(s_i\otimes 1)(x_p)=(1\otimes\sigma^i)(x_{p+1})
\label{s-eq}
\end{equation}
for all $p,i$ --- see the diagram below.
\begin{center}
\begin{picture}(14,6)
\put(5,0){\makebox(4,2){$\Omega^n_{p+1}\otimes X^{p+1}$}}
\put(5,4){\makebox(4,2){$\Omega^n_p\otimes X^{p}$}}
\put(0,2){\makebox(4,2){$\Omega^n_{p+1}\otimes X^{p}$}}
\put(10,2){\makebox(4,2){$\Omega^n_p\otimes X^{p+1}$}}
\put(5.5,4.5){\vector(-2,-1){2}}
\put(5.5,1.5){\vector(-2,1){2}}
\put(8.5,1.5){\vector(2,1){2}}
\put(8.5,4.5){\vector(2,-1){2}}
\put(3.5,4){\makebox(1,0.5){$\scriptsize s_i\otimes 1$}}
\put(3.5,1.5){\makebox(1,0.5){$\scriptsize 1\otimes\sigma^i$}}
\put(9.5,1.5){\makebox(1,0.5){$\scriptsize d_i\otimes 1$}}
\put(9.5,4){\makebox(1,0.5){$\scriptsize 1\otimes\delta^i$}}
\end{picture}
\end{center}
Until the end of this Section, let us fix $n\geq 0$. Our aim will be
an explicit computation of $\Omega^n(X)$.
\subsection{} In what follows ${\Bbb N}=\{0,1,\ldots\}$.
Let $p\in\Bbb N$. For $a\in{\Bbb N}^p,\alpha\in\{0,1\}^p$ denote
$$
\omega_{a,\alpha}=t_1^{a_1}\cdots t_p^{a_p}dt_1^{\alpha_1}\wedge
\cdots\wedge dt_p^{\alpha_p}\in\Omega_p.
$$
We have $\deg\omega_{a,\alpha}=\sum\alpha_i$.
Let $I^p\subset {\Bbb N}^p\times\{0,1\}^p$ consist of all pairs
$(a,\alpha)$ such that $\sum\alpha_i=n$.
Evidently, forms $\omega_{a,\alpha},\ (a,\alpha)\in I^p$, make up a basis of
$\Omega_p^n$. Set $I=\coprod_p I^p$.
An arbitrary element
$x_p\in\Omega_p^n\otimes X^p$ takes form
$$
x_p=\sum_{(a,\alpha)\in I^p}\omega_{a,\alpha}\otimes x_{a,\alpha}
$$
with $x_{a,\alpha}\in X^p$. This way we get a mapping
\begin{equation}
\label{mapping}
x\mapsto\{ x_{a,\alpha}\}_{(a,\alpha)\in I}
\end{equation}
from $\Omega^n(X)$ to the set of collections
\begin{equation}
\{ x_{a,\alpha}\}_{(a,\alpha)\in I},\ x_{a,\alpha}\in X^p\
\mbox{for}\ (a,\alpha)\in I^p
\label{collect}
\end{equation}
\subsubsection{} Let us introduce two operations on $I$.
For $(a,\alpha)\in I^p$ denote by
$\eta_i(a,\alpha)\in I^{p+1},\
i=1,\ldots,p+1$ the element
obtained from $(a,\alpha)$ by inserting $(0,0)$ on the place $i$.
Further, denote by $\zeta_i(a,\alpha)\in I^{p-1},\ i=1,\ldots,p-1$,
the pair $(a',\alpha')$ with $a'=(\ldots,a_i+a_{i+1},\ldots)$ and
$\alpha'=(\ldots,\alpha_i+\alpha_{i+1},\ldots)$--- this operation is defined
only if $\alpha_i+\alpha_{i+1}\leq 1$.
\subsubsection{} Let
$ x_p=\sum_{(a,\alpha)\in I^p}\omega_{a,\alpha}\otimes x_{a,\alpha}\in
\Omega^{n}_p\otimes X^p$
and
$ x_{p+1}=\sum_{(b,\beta)\in I^{p+1}}\omega_{b,\beta}\otimes x_{b,\beta}\in
\Omega^{n}_{p+1}\otimes X^{p+1}$.
The condition~(\ref{d-eq}) is equivalent to the following
formulas~(\ref{d_i-eq}) and~(\ref{d_0-eq}):
\begin{equation}
x_{\eta_i(a,\alpha)}=\delta^ix_{a,\alpha},\ \mbox{for }i\geq 1
\label{d_i-eq}
\end{equation}
\begin{equation}
\delta^0(x_{a,\alpha})=
\sum\begin{Sb}
e_0;\ e_i\leq a_i\\
\epsilon_i\leq\alpha_i\\
\beta_1:=\sum\epsilon_i\leq 1
\end{Sb}
(-1)^{e+\beta_1+\sum_{i\geq j\geq 2}\epsilon_i\beta_j}{b_1!\over
{e_0!\cdots e_p!}} x_{b,\beta}
\label{d_0-eq}
\end{equation}
where the (big) sum is taken over non-negative $e_i,\epsilon_j$ satisfying
the conditions written and
$b_1=\sum_{i=0}^pe_i,
\ e=\sum_{i=1}^pe_i,\ b_{i+1}=a_i-e_i,\
\beta_1=\sum\epsilon_i,
\beta_{i+1}=\alpha_i-\epsilon_i.$
The condition~(\ref{s-eq}) is equivalent to the following
formulas~(\ref{s_i-eq}) and~(\ref{s_0-eq}):
\begin{equation}
\sigma^i(x_{b,\beta})=\begin{cases}
0 &\text{ if } \beta(i)=\beta(i+1)=1\\
\binom{b_i+b_{i+1}}{b_i}x_{\zeta_i(b,\beta)}& \text{ otherwise }
\end{cases}
\label{s_i-eq}
\end{equation}
for $i\geq 1$;
\begin{equation}
\sigma^0(x_{b,\beta})=\begin{cases}
x_{a,\alpha}& \text{ if } (b,\beta)=\eta_1(a,\alpha)\\
0 & \text{ otherwise.}
\end{cases}
\label{s_0-eq}
\end{equation}
Let us denote $\bT(X)$ the set of all collections ~(\ref{collect}) satisfying
{}~(\ref{d_i-eq})---~(\ref{s_0-eq}).
Let us call a collection ~(\ref{collect}) {\em locally finite} if
for every $p$ the set $\{ (a,\alpha)\in I^p|x_{a,\alpha}\neq 0\}$ is finite.
Let us denote by $\bT^{lf}(X)\subset\bT(X)$ the subset consisting of all
locally finite collections.
The above argument proves
\subsubsection{}
\begin{lem}{} The mapping ~(\ref{mapping}) defines an isomorphism
$$
\rho:\Omega^n(X)\iso\bT^{lf}(X)
$$
\end{lem} $\Box$
Elements $x_{a,\alpha},\ a,\alpha\in I$ are coordinates of $x\in\Omega^n(X)$.
Now our strategy will be: using relations ~(\ref{d_i-eq})---~(\ref{s_0-eq})
to express these coordinates in terms of smaller subsets of coordinates.
\subsection{} An element $(a,\alpha)\in I^p$ is called {\em reduced} if
none of $(a_i,\alpha_i),\ i=1,\ldots, p,$ is equal to $(0,0)$.
An element $(a,\alpha)\in I^p$ is called {\em special} if it
is reduced and $(a_1,\alpha_1)=(1,0)$.
An element $(a,\alpha)\in I^p$ is called {\em d-free} if it is reduced
and not special.
The set of all d-free elements in $I^p$ will be denoted by ${\cal F}^p$.
We set ${\cal F}=\cup{\cal F}^p\subset I$.
Let us denote by $\bT_{\CF}(X)$ the set of all collections
\begin{equation}
\label{coll-f}
\{ x_{b,\beta}\}_{(b,\beta)\in\CF}, \mbox{ where } x_{b,\beta}\in X^p
\mbox{ for } (b,\beta)\in\CF^p
\end{equation}
satisfying following conditions:
\begin{align}
\sigma_0(x_{b,\beta})&=0; \nonumber\\
\sigma_i(x_{b,\beta})&=0 \text{ if } \beta(i)=\beta(i+1)=1\\
\sigma_i(x_{b,\beta})&=\binom{b_i+b_{i+1}}{b_i}x_{\zeta_i(b,\beta)}
\text{ otherwise } \nonumber
\label{props}
\end{align}
\subsubsection{}
\begin{lem}{} The natural projection defines an isomorphism
$$
\pi_1: \bT(X)\iso\bT_{\CF}(X)
$$
\end{lem}
{\bf Proof.} The formula~(\ref{d_i-eq}) allows one to express $x_{b,\beta}$ for
non-reduced $(b,\beta)\in I^{p+1}$ through
$x_{a,\alpha},(a,\alpha)\in I^p$. Furthermore, the formula~(\ref{d_0-eq})
allows one to calculate $x_{b,\beta}$ for special $(b,\beta)\in I^{p+1}$
by induction as follows. Endow $I^p$ with the partial order in which
$(a,\alpha)\geq (a',\alpha')$ iff $a_i\geq a_i'$ and $\alpha_i=\alpha'_i$
for all $i$. We determine the value of $x_{b,\beta}$ for special
$(b,\beta)\in I^{p+1}$
by induction on $(a,\alpha)$ such that $b=a\cup 1_1,\ \beta=\alpha
\cup 0_1$ in the obvious notation. For this one should consider the
equation~(\ref{d_0-eq}) and
see that all special summands in the right hand side except of $x_{b,\beta}$
correspond to smaller values of $(a,\alpha)$.
This immediately implies that
the map which takes $x=\{x_{a,\alpha}\}_{(a,\alpha)\in I^{\cdot}}\in\Th(X)^n$
to the collection
$\{x_{b,\beta}\}_{(b,\beta)\in {\cal F}^{\cdot}}$, is injective.
To prove bijectivity, we proceed by induction on $p$. Suppose that, apart
from $x_{b,\beta}$ with d-free $(b,\beta)$, all elements $x_{a,\alpha}$ with
$({a,\alpha})\in I^i,\ i\leq p$, are constructed and satisfy the
equations~(\ref{d_i-eq})--(\ref{s_0-eq}). In order to make the next
induction step, we have to check the following claims:
1) If $(b,\beta)=\eta_i(a,\alpha)=\eta_j(a',\alpha')$
then $\delta^ix_{a,\alpha}=\delta^jx_{a',\alpha'}.$
2) The condition~(\ref{d_0-eq}) is satisfied for any $(a,\alpha)\in I^p$
(and not only for reduced $(a,\alpha)$).
3) The conditions~(\ref{s_i-eq}) and~(\ref{s_0-eq}) are satisfied
for any $(b,\beta)\in I^{p+1}$ (and not only for d-free $(b,\beta)$).
These claims can be checked by a direct calculation using standard
identities for the morphisms in the category $\Delta$. $\Box$
\subsection{} An element $(a,\alpha)\in I$ is called {\em basic} if
--- $(a_1,\alpha_1)=$ either $(2,0)$ or $(0,1)$;
--- $(a_i,\alpha_i)=$ either $(1,0)$ or $(0,1)$ for $i>1$.
If $(a_1,\alpha_1)=(2,0)$ then $(a,\alpha)$ is called {\em basic
element of the first kind}, otherwise --- {\em basic element of the
second kind}.
We denote $\CB$ the set of all basic elements; we define
$\CB^p=\CB\cap I^p$. Evidently $\CB^p\subset\CF^p\subset I^p$.
\subsubsection{}
\begin{lem}{} Suppose we are given $x=(x_{a,\alpha})_{(a,\alpha)\in I}\in
\bT(X)$. The elements $x_{a,\alpha},\ (a,\alpha)\in\CB$ satisfy the
following relations:
\begin{equation}
\label{rel-1}
\sigma^0(x_{a,\alpha})=0
\end{equation}
\begin{equation}
\sigma^i(x_{a,\alpha})=0 \text{ if } \alpha_i=\alpha_{i+1}=1
\end{equation}
\begin{equation}
\sigma^i(x_{a,\alpha})=\sigma^i(x_{a',\alpha'})
\end{equation}
for $i>1$ where
$$ a=(\ldots,0,1,\ldots), a'=(\ldots,1,0,\ldots)$$
$$ \alpha=(\ldots,1,0,\ldots), \alpha'=(\ldots,0,1,\ldots).$$
(switching the places $i,i+1$);
\begin{equation}
\sigma^1(x_{a,\alpha})=(\sigma^1)^2(x_{a',\alpha'})
\label{rel-3}
\end{equation}
where
$$ a=(2,0,\ldots),\ a'=(0,1,1,\ldots)$$
$$ \alpha=(0,1,\ldots),\ \alpha'=(1,0,0,\ldots).$$
\end{lem}
{\bf Proof.} Direct. $\Box$
\subsubsection{} We denote by $\bT_{\CB}(X)$
the set of all collections $\{ x_{a,\alpha}\}_{(a,\alpha)\in \CB}$
satisfying the relations ~(\ref{rel-1}) -- ~(\ref{rel-3}).
Thus we get a map
\begin{equation}
\label{map-pi2}
\pi_2:\bT_{\CF}(X)\lra \bT_{\CB}(X)
\end{equation}
\subsubsection{}
\begin{lem}{} The map ~(\ref{map-pi2}) is an isomorphism.
\end{lem}
{\bf Proof.} One can easily see that any $d$-free element may be obtained
from an element from $\CB$ by applying operations $\zeta_i$. This proves
injectivity of ~(\ref{map-pi2}). The proof of surjectivity is standard.
$\Box$
Summing up, we get a sequence of natural maps
\begin{equation}
\label{map-pi}
\Omega^n(X)\iso \bT^{lf}(X)\hra\bT(X)\iso\bT_{\CF}(X)\iso\bT_{\CB}(X)
\end{equation}
\subsection{} Let us call a collection
$\{x_{a,\alpha}\}_{(a,\alpha)\in{\cal B}^{\cdot}}$ {\em locally finite} if
$$
\forall p\in{\Bbb N}\ \exists m\in{\Bbb N}\ \forall(a,\alpha)\in
{\cal B}^q_{m'}
\text{ with }m'\geq m\ \forall f:[q]\ra[p]\in\Delta\text{ one has }
f(x_{a,\alpha})=0.
$$
Let us denote by $\bT^{lf}_{\CB}(X)\subset\bT_{\CB}(X)$ the subspace of all
locally finite collections.
\subsubsection{}
\begin{lem}{} The map ~(\ref{map-pi}) induces an isomorphism
$$
\pi:\Omega^n(X)\iso \bT^{lf}_{\CB}(X)
$$
\end{lem}
{\bf Proof.} Direct check. $\Box$
\subsection{} Set $\CB^p_m=\{ (a,\alpha)\in\CB^p|\sum a_i=m\}$;
$\CB_m=\cup_p\ \CB_m^p$. By definition,
$$
\CB_m=\CB_m^{m+n-1}\cup\CB_m^{m+n},
$$
the first (resp., second) subset consisting of all elements of the
first (resp., second) kind.
Let us introduce the following total order on $\CB_m$:
$(a,\alpha)>(a',\alpha')$ iff $a>a'$ in the lexicographic order.
Let us denote by $\bT_{\CB,m}(X)$ the set of all
collections $\{ x_{a,\alpha}\}_{(a,\alpha)\in \CB_m}$ satisfying the
relations ~(\ref{rel-1}) -- ~(\ref{rel-3}).
Given $(b,\beta)\in\CB_m$, denote by $\bT_{\leq (b,\beta)}(X)$ the
space
of all collections $\{ x_{a,\alpha}\}_{(a,\alpha)\in \CB_m,\ (a,\alpha)
\leq (b,\beta)}$.
We have obvious maps
$$
\bT_{\leq (b,\beta)}(X)\lra\bT_{\leq (a,\alpha)}(X)
$$
for $(a,\alpha)\leq (b,\beta)$.
\subsubsection{}
\label{inverse}
\begin{lem}{} We have
$$
\bT_{\CB,m}(X)=\invlim\ \bT_{\leq (b,\beta)}(X)
$$
the inverse limit over $\CB_m$.
\end{lem}
{\bf Proof.} Obvious. $\Box$
\subsection{}
\label{step}
\begin{lem}{}
Let $(a,\alpha)\in{\cal B}_m$. Let $(b,\beta)\in{\cal B}^l_m$ be the first
element such that $(b,\beta)>(a,\alpha)$.
Then the map
$$\bT_{\leq(b,\beta)}(X)\ra\bT_{\leq(a,\alpha)}(X)$$
is surjective and its kernel is isomorphic to a direct summand of $X^l$.
\end{lem}
{\bf Proof}
In order to lift an element of $\bT_{a,\alpha}(X)$, we have to find an
element $x_{b,\beta}$ having prescribed values for some of
$\sigma^i(x_{b,\beta})$.
1. Existence of the lifting:
One has the following conditions on $\sigma^i(x_{b,\beta})$:
(0) $\sigma^0(x_{b,\beta})=0$ --- always.
(a) if $\beta_i=\beta_{i+1}=1$. Then one has $\sigma^i(x_{b,\beta})=0$.
(b) if $(\beta_i,\beta_{i+1})=(0,1)$ and $i>1$. Then the condition is
$\sigma^i(x_{b,\beta})=\sigma^i(x_{b',\beta'})$ where the pair $(b',\beta')$
is obtained from $(b,\beta)$ by switching the places $(i,i+1)$.
(c) if $(\beta_1,\beta_2)=(0,1)$. Then the condition is
$\sigma^1(x_{b,\beta})=(\sigma^1)^2(x_{a,\alpha})$ where
$(\alpha_1,\alpha_2,\alpha_3)=(1,0,0),\alpha_i=\beta_{i-1}\text{ for }
i>3$ and $a_i$
are defined uniquely by $\alpha_i$.
Let $I\subseteq[0,l-1]$ be the set of indices $i$ such that
$\sigma^i(x_{b,\beta})$ is defined by the conditions (0)---(c) above.
Let $y^i\in A^{l-1},\ i\in I,$ be the right hand sides of the
conditions (0)---(c) so that they take form
$$ \sigma^i(x_{b,\beta})=y^i,\ i\in I.$$
The first observation is that for any couple $i<j$ in $I$ one has
\begin{equation}
\sigma^{j-1}y^i=\sigma^iy^j.
\label{compatibility}
\end{equation}
This can be checked easily by an
explicit calculation.
Now consider two different cases:
(1st case) $I$ does not coincide with $[0,l-1]$.
In this case \Lem{kan} below asserts the existence of a solution
for the system of equations for $x_{b,\beta}$.
(2nd case) $I=[0,l-1]$. This is possible only in two cases:
--- $\beta_i=1$ for all $i$. Then $y^i=0$ for all $i$.
--- $\beta_1=0;\beta_i=1$ for $i>1$. Then $y^i=0$ for $i\not=1$ and
the condition~(\ref{compatibility}) gives that $y^1\in N^{l-1}X$.
In both cases \Lem{more} assures the existence of a solution.
2. The kernel of the map.
The kernel of the map in question has always form
$$ N^p_I(X):=\{a\in X^p| \sigma^i(a)=0\text{ for all }i\in I\}$$
for some subset $I\subseteq[0,p]$. According to~\Lem{kan2} below, this is
a direct summand of $X^p$. $\Box$
\subsection{} Let $\cal M$ be the set of all non-decreasing functions
$f:{\Bbb N}\ra{\Bbb N}$ equipped with a partial order:
$$
f\geq g\text{ iff }f(n)\geq g(n)\text{ for each }n.
$$
\subsubsection{}
\begin{defn}{growth}
Given $f\in{\cal M}$, an element $x\in\bT(X)$ {\em has growth $\leq f$}
if
$\pi_1(x)=\{x_{b,\beta}\}_{(b,\beta)\in{\cal F}}$
satisfies the property
$$
x_{b,\beta}=0 \text{ if } (b,\beta)\in{\cal F}^p\text{ and }
\sum b_i>f(p).
$$
Denote by $\bT^f(X)\subset \bT(X)$ the subspace of all elements of growth
$\leq f$.
\end{defn}
We have $\bT^f(X)\subseteq \bT(X)$. Moreover,
\subsubsection{}
\label{directlim}
\begin{lem}{} We have
$$
\bT^{lf}(X)=\dirlim\ \bT^f(X)
$$
the limit taken over $f\in\CM$.
\end{lem}
{\bf Proof.} Obvious.
\subsection{}
For $X\in\Delta\Mod(A)$ and $d\in{\Bbb N}\cup\{ -1,\infty\}$ define
$X_{>d}\in\Delta\Mod(A)$ as follows. Set $X_{>-1}=X;\ X_{>\infty}=0$.
For $d\in \Bbb N$ set
\begin{equation}
X_{>d}^i=\{x\in X^i|f(x)=0\ \forall f:[i]\ra[d]\in\Delta\}
\end{equation}
We have obviously $X_{>d}^i=0\text{ for }i\leq d$, and $X_{>d}^{d+1}=N^{d+1}X$
where $N^iX$ denotes the normalization (see ~\ref{normal} below).
\subsubsection{} Given $f\in\CM$, define a function
$f^{\circ}: \Bbb N\lra \Bbb N\cup\{ -1,\infty\}$ by a formula
$$
f^{\circ}(m)=p\text{ iff } f(p)<m\leq f(p+1).
$$
For $f\in\CM$ let us denote
$$
\bT_{\CB}^f(X)=\pi_2\pi_1(\bT^f(X))
$$
\subsubsection{}
\begin{lem}{} We have
$$
\bT_{\CB}(X)=\prod_{m=0}^{\infty}\bT_{\CB,m}(X)
$$
\end{lem}
{\bf Proof.} Evident: the equations~(\ref{rel-1}--\ref{rel-3})
are homogeneous on $m=\sum a_i$.
$\Box$
\subsubsection{}
\begin{lem}{product} For any $f\in{\cal M}$ one has
\begin{equation}
\label{eq-prod}
\bT^f_{\CB}(X)=\prod_{m=0}^{\infty}\bT_{\CB,m}(X_{>f^{\circ}(m)})
\end{equation}
\end{lem}
\begin{pf} An element $x\in\bT(X)$ has growth $\leq f$ iff
$\pi_1(x)=\{x_{b,\beta}\}_{(b,\beta)\in{\cal F}}$ satisfies the property
$$
x_{b,\beta}=0 \text{ if } (b,\beta)\in{\cal F}^p\text{ and }
p\leq f^{\circ}(\sum b_i).
$$
Then the formulas~(\ref{props}) give immediately the result.
\end{pf}
Recall that $X$ is called {\em finite dimensional} if there exists
$d\in\Bbb N$
such that $N^i(X)=0$ for $i>d$ (we say that $\dim(X)\leq d$.
If this is the case then $X_{>d}=0$, hence the product in ~(\ref{eq-prod})
is finite.
Combining all the previous results together, we get
\subsection{}
\begin{thm}{} For every $n\in\Bbb N$ the $A$-module $\Omega^n(X)$ may be
obtained from $X$ applying the following operations (naturally in $X$):
(1) taking modules $\bT_{\CB,m}(X)$ which have a natural finite filtration
with graded factors isomorphic to direct summands of modules
$X^l,\ l\in\Bbb N$;
(2) taking direct products over $\Bbb N$; if $\dim(X)<\infty$ then
products are finite;
(3) passing to a filtered direct limit.
\end{thm}
{\bf Proof.} Follows immediately from ~\ref{directlim}, ~\ref{product},
{}~\ref{step} and ~\ref{inverse}. $\Box$
We have the following easy corollaries which are the main properties
of the functor $\Omega$.
\subsection{}
\label{exact-6}
\begin{cor}{} The functor $X\mapsto\Omega(X)$ is exact. $\Box$
\end{cor}
\subsection{}
\label{base-6}
\begin{cor}{} Suppose $X\in\Delta\Mod(A)$ is finite dimensional. Let $A'$ be
a (commutative) $A$-algebra. Then the natural base change map
$$
\Omega(X)\otimes_AA'\lra\Omega(X\otimes_AA')
$$
is an isomorphism. $\Box$
\end{cor}
\subsection{}
\label{flat-6}
\begin{cor}{}
Let $X\in\Delta\Mod(A)$ and suppose that either $A$ is noetherian
or $X$ is finite dimensional.
If all
$X^i$ are flat $A$-modules so are all $\Omega^n(X)$. $\Box$
\end{cor}
\subsection{}
\label{qc}
\begin{cor}{} Let $S$ be topological space endowed with a sheaf $\CO_S$
of commutative $\Bbb Q$-algebras. Let $X\in\Delta\Mod({\CO}_S)$ be
finite. If all
$X^i$ are quasi-coherent ${\CO}_S$-modules so are all $\Omega^n(X)$.
$\Box$
\end{cor}
In the remaining part of this Section we will prove some facts about
cosimplicial modules needed above. Most of them are more or less standard.
In fact, mostly we will discuss an explicit form of Dold-Puppe correspondence.
\subsection{}
\label{normal}
Let $X$ be a cosimplicial $A$-module. For any $i\geq 0$ define
$$ N^i(X)=\{x\in X^i|\sigma^j(x)=0\text{ for all }j\}.$$
Define a subcategory $\Lambda$ in $\Delta$ as follows. $\Lambda$ has
the same objects as $\Delta$; The set of morphisms of $\Lambda$ is generated
by the faces $\delta^i:[n]\ra[n+1]$ with $i=0,\ldots, n$
(that is: (a) only faces appear; (b) the last face $\delta^{n+1}:[n]\ra[n+1]$
disappear).
Define {\em a shift functor} $ s:\Delta\ra\Delta$
by the formulas
$$s[n]=[n+1],\ s(\delta^i)=\delta^{i+1},\ s(\sigma^i)=\sigma^{i+1}.$$
\subsubsection{}
\begin{prop}{decomposition}
For any cosimplicial $A$-module $X$ one has
$$
X^n=\bigoplus_{m\geq 0}\bigoplus_{f:[m]\ra[n]\in\Lambda}f(N^m(X)).
$$
\end{prop}
\begin{pf}
We will prove the claim by induction.
For $n=0$ the claim is trivial. Suppose it is true for degrees $< n$
and for all cosimplicial modules $X$. Apply this to the shift $Y=Xs$
of $X$. We have by the inductive hypothesis
$$ X^n=Y^{n-1}=\bigoplus_{m\geq 0}
\bigoplus_{f:[m]\ra[n-1]\in{\Lambda}}f(M^m)$$
where $M^i=N^i(Y)$ is the normalization of $Y=Xs$.
An element $x\in X^{m+1}$ belongs to $M^m$ iff $\sigma^i(x)=0$ for
$i>0$. Write $x=y+\delta^0\sigma^0(x)$ where $y=x-\delta^0\sigma^0(x)$.
One checks that $y\in N^{m+1}(X)$ and (of course) $\sigma^0(x)\in X^m$.
Thus by induction (note that $m\leq n-1$)
$$\sigma^0(x)=\sum_k\sum_{g:[k]\ra[m]\in\Lambda}g(z_g)$$
with $z_g\in N^k(X)$.
If $x'\in M^m$ is the element corresponding to $x\in X^{m+1}$ then
$f(x')=(sf)(x)$ and therefore
$$ f(x')=(sf)(x)=(sf)(y)+(sf)\delta^0\sum g(z_g)=(sf)(y)+\sum\delta^0
fg(z_g)$$
which proves that
$$ X^n=\sum_m\sum_{f:[m]\ra[n]\in\Lambda}f(N^m(X)).$$
Let us finally prove the uniqueness of the presentation
of an element $x\in X^n$ into sum
\begin{equation}
x=\sum_m\sum_{f:[m]\ra[n]\in\Lambda}f(x_f)
\label{presentation}
\end{equation}
with $x_f\in N^m(X)$.
For any $f:[m]\ra[n]\in\Lambda$ define a left-inverse $f^l:[n]\ra[m]$
as follows: if $f=\delta^{i_1}\cdots\delta^{i_{n-m}}$ with
$i_1>\ldots>i_{n-m}$ then
$$ f^l=\sigma^{i_{n-m}}\cdots\sigma^{i_1}.$$
We will check the uniqueness of the elements $x_f$ in the
presentation~(\ref{presentation}) by induction on $m$.
Suppose that $x_f$ are defined uniquely for all $f:[m']\ra[n]$ with $m'<m$
(this is true, say, for $m=0$).
Define an order on the set $\Hom_{\Lambda}([m],[n])$ saying that
$\delta^{i_1}\cdots\delta^{i_{n-m}}\geq\delta^{j_1}\cdots\delta^{j_{n-m}}$
iff $(i_1,\ldots,i_{n-m})\geq(j_1,\ldots,j_{n-m})$ in the lexicographic order.
One immediately checks the following
\begin{lem}{}
If $f>g$ then $f^lg\not=\id_{[m]}$.
\end{lem}
By the inductive hypothesis we can suppose that $x_f=0$ for $f:[m']\ra[n]$
with $m'<m$. Then
\begin{equation}
f^l(g(x_g))=\begin{cases}
x_g&\text{ if }f=g\\
0& \text{ if } f>g
\end{cases}
\end{equation}
Thus, the transition matrix expressing the values of $f^l(x)$ for different $f$
through $x_g$ is upper-triangular and hence invertible.
The proposition is proven.
\end{pf}
It is very convenient to rewrite the statement of~\Prop{decomposition} as
follows.
Let ${\Bbb Q}\Lambda_{mn}$ be the rational vector space spanned by
the set $\Hom_{\Lambda}([m],[n])$. Then one has
\begin{cor}{}
One has a canonical on $X\in\Delta\Mod(A)$ isomorphism
$$ X^n=\bigoplus_{m\geq 0}{\Bbb Q}\Lambda_{mn}\otimes_{\Bbb Q}N^m(X).$$
\end{cor}
\subsection{}
\begin{lem}{kan}
Let $I\subset[0,n]$ be a proper subset and let $y_i\in X^{n-1},\ i\in I$
be given.
Then the system of equations
$$ \sigma^i(x)=y^i,\ i\in I,$$
has a solution if (and only if) $y^i$ satisfy the compatibilities
$$ \sigma^{j-1}y^i=\sigma^iy^j\text{ for }i<j\in I.$$
\end{lem}
\begin{pf}
Consider the functor
$$ i:\Delta^0\ra\Delta $$
defined by the formulas
$$ i([n])=[n+1],\ i(\delta^i)=\sigma^i,\ i(\sigma^i)=\delta^{i+1}$$
--- see Gabriel-Zisman's functor II, ~\cite{gz}, 3.1.1).
If $X$ is a cosimplicial module then $Xi$ is a simplicial module.
The property of $X$ we have to prove just means that $Xi$ is a
Kan simplicial set. This is well-known to be always true
(see, e.g., ~\cite{q2}, Prop. II.3.1).
\end{pf}
\subsection{}
\begin{lem}{more}For any system $y^i\in N^{n-1}X,\ i\in[0,n-1]$
there exists an element $x\in X^n$ such that $\sigma^ix=y^i$ for
all $i\in[0,n-1]$.
\end{lem}
\begin{pf}
We will look for $x=\sum\delta^i(z^i)$ with $z^i\in N^{n-1}X$.
Then the conditions on $x$
are expressed by the equations $y^i=\sigma^i(x)=z^i+z^{i+1}$
which are clearly solvable.
\end{pf}
\subsection{}
{\em Notation.} For $I\subseteq[0,n]$ denote
$$ N^i_I(X)=\{x\in X^i|\sigma^j(x)=0\text{ for all }j\in I\}.$$
\subsubsection{}
\begin{lem}{kan2}
Let $I\subseteq[0,n]$. There exist a collection of vector subspaces
${\Bbb Q}\Lambda^I_{m,n}\subseteq{\Bbb Q}\Lambda_{m,n}$
such that
$$
N^n_I(X)=\bigoplus_m{\Bbb Q}\Lambda^I_{m,n}\otimes_{\Bbb Q}N^m(X)
$$
\end{lem}
Here is a more general statement.
\subsubsection{}
\begin{lem}{kan3}
Fix $d\leq n\in{\Bbb N}$ and a subset $\Phi\subseteq\Hom_{\Delta}([n],[d])$.
Define
$$X^n_{\Phi}=\{x\in X^n|f(a)=0\text{ for all }f\in\Phi\}.$$
There exists a collection of vector subspaces
${\Bbb Q}\Lambda^{\Phi}_{m,n}\subseteq{\Bbb Q}\Lambda_{m,n}$
such that
$$ X^n_{\Phi}=\bigoplus_m{\Bbb Q}\Lambda^{\Phi}_{m,n}\otimes_{\Bbb Q}N^m(X).$$
\end{lem}
\begin{pf} of~\ref{kan3}.
Let $t:[n]\ra[k]\in\Delta$.
The condition $tx=0$ for $x=\sum_ff(x_f)$ takes form
$$0=tx=\sum_ftf(x_f)=\sum_gg(\sum_{f: tf=g}x_f)$$
which is equivalent to the system of equations
$$ \sum_{f: tf=g}x_f=0$$
numbered by $g\in\Mor\Delta$.
The lemma immediately follows from this observation.
\end{pf}
\section{Higher Kodaira-Spencer morphisms}
\label{kodaira}
\subsection{}
\label{laawm}
(Cf. \cite{bs}, 1.2.) Let $S$ be a differentially smooth formal
scheme (for example, a smooth scheme). Let $\pi: X\lra S$
be a smooth separated map of formal schemes, ~\ref{smooth}, for example
a smooth morphism of usual schemes.
We have the exact sequence ~(\ref{seq}):
$$
0\lra \CT_{X/S}\lra \CT_X\overset{\epsilon_{\CT}}{\lra} \pi^*\CT_S\lra 0
$$
The first embedding makes $\CT_{X/S}$ a Lie algebroid over $X$. Note that
the sheaf $\pi^*\CT_S$ is not a sheaf of Lie algebras.
Let $\pi^{-1}$ denotes the functor of set-theoretical inverse image, so that
$\pi^*=\CO_X\otimes_{\pi^{-1}\CO_S}\pi^{-1}$. The subsheaf
$\pi^{-1}\CT_S\subset \pi^*\CT_S$ is a $\pi^{-1}\CO_S$-Lie algebra.
Set $\CT_{\pi}:=\epsilon_{\CT}^{-1}(\pi^{-1}\CT_S)$ - this is a sheaf
of $k$-Lie algebras and $\pi^{-1}\CO_S$-modules (consisting of vector fields
with the constant projection to $S$ along fibers of $\pi$).
We have an exact sequence of $k$-Lie algebras and $\pi^{-1}\CO_S$-modules
$$
0\lra\CT_{X/S}\lra \CT_{\pi}\lra \pi^{-1}\CT_S\lra 0
$$
Let $\epsilon: \CA\lra \CT_X$ be a transitive Lie algebroid over $X$.
We set $\CA_{/S}:=\epsilon^{-1}(\CT_{X/S})\subset\CA_{\pi}:=
\epsilon^{-1}(\CT_{\pi})\lra \CA$. These are subsheaves of Lie
algebras. $\CA_{/S}$ is a subsheaf of $\CO_X$-modules, and a Lie algebroid
over $X$ included into an exact sequence
$$
0\lra \CA_{(0)}\lra \CA_{/S}\lra \CT_{X/S}\lra 0
$$
$\CA_{\pi}$ is a subsheaf of $\pi^{-1}\CO_S$-modules. We have an exact
sequence of $k$-Lie algebras and $\pi^{-1}\CO_S$-modules
\begin{equation}
\label{a}
0\lra\CA_{/S}\lra\CA_{\pi}\lra \pi^{-1}\CT_S\lra 0
\end{equation}
\subsection{}
\label{dglie-fund} Pick a finite affine open covering $\CU$ of $X$.
We will suppose that $\CA$ is a locally free $\CO_X$-module of finite
type, hence, so is $\CA_{/S}$.
Let us apply the functor $\pi_*\CHC(\CU,\cdot)$ to the exact sequence
{}~(\ref{a}). We have $R^1\pi_*\CHC^i(\CU,\CA_{/S})=0$; so we get
an exact sequence of cosimplicial $\CO_S$-modules
$$
0\lra \pi_*\CHC(\CU,\CA_{/S})\lra \pi_*\CHC(\CU,\CA_{\pi})
\lra \pi_*\CHC(\CU,\pi^{-1}\CT_S)\lra 0
$$
Next, applying the Thom-Sullivan functor $\Omega$ we get an exact sequence
of complexes
\begin{equation}
\label{ex-derham}
0\lra \pi^{Lie}_{*,\CU}(\CA_{/S})\lra \Omega(\pi_*\CHC(\CU,\CA_{\pi}))
\lra \Omega(\pi_*\CHC(\CU,\pi^{-1}\CT_S))\lra 0
\end{equation}
Note that $\Omega(\pi_*\CHC(\CU,\CA))$ is naturally a dg $k$-Lie algebra.
We have a canonical adjunction map
$$
\CT_S\lra \Omega(\pi_*\CHC(\CU,\pi^{-1}\CT_S))
$$
so, taking the pullback of ~(\ref{ex-derham}) we get an exact sequence
\begin{equation}
\label{fund}
0\lra \pi^{Lie}_{*,\CU}(\CA_{/S})\lra \CA^{\pi}_{\CU}\lra \CT_S\lra 0
\end{equation}
By definition,
$$
\CA^{\pi}_{\CU}=\Omega(\pi_*\CHC(\CU,\CA_{\pi}))
\times_{\Omega(\pi_*\CHC(\CU,\pi^{-1}\CT_S))}\CT_S,
$$
and this sheaf inherits the structure of a dg $k$-Lie algebra and
$\CO_S$-module from $\Omega(\pi_*\CHC(\CU,\CA_{\pi}))$ and $\CT_S$.
One sees that this way we get a structure of a transitive dg Lie algebroid on
$\CA^{\pi}_{\CU}$.
\subsection{}
\label{higher} Now we can apply to ~(\ref{fund}) the construction
{}~\ref{abstr-ksmaps}. We get the maps:
\begin{equation}
\label{ks-class}
\kappa^1:\CT_S\lra R^1\pi_*(\CA_{/S})
\end{equation}
--- the "classical" KS map;
\begin{equation}
\label{ks}
\kappa:\Diff_S\lra \CH_0^{Lie}(\pi_*^{Lie}(\CA_{/S}))
\end{equation}
and
\begin{equation}
\label{ks-leq-n}
\kappa^{\leq n}:\Diff^{\leq n}_S\lra \CF_n\CH^{Lie}_0(\pi_*^{Lie}(\CA_{/S}))
\end{equation}
satisfying the compatibilities ~\ref{main-thm}. These maps are called
{\bf higher Kodaira-Spencer morphisms.}
\subsection{Split case}
\label{dglie-fund-const} Suppose that $X=Y\times S$, and $\pi$ is a projection
to the second factor. In this case we have canonical embeddings
$$
\pi^{-1}\CT_S\hra\pi^*\CT_S\hra \CT_X
$$
By taking the pull-back of the exact sequence
$$
0\lra\CA_{(0)}\lra \CA\lra\CT_X\lra 0,
$$
we get an exact sequence
\begin{equation}
\label{a-const}
0\lra \CA_{(0)}\lra \bar{\CA}\lra\pi^{-1}\CT_S\lra 0
\end{equation}
Now we can repeat the construction of ~\ref{dglie-fund}, replacing
the sequence ~(\ref{a}) by ~(\ref{a-const}). This will
provide a dg Lie algebroid
\begin{equation}
\label{fund-const}
0\lra\CA_{(0)}\lra\bar{\CA}^{\pi}_{\CU}\lra\CT_S\lra 0
\end{equation}
Again we can apply the construction ~\ref{abstr-ksmaps},
and get the KS maps analogous to ~(\ref{ks-class}) -~(\ref{ks-leq-n}):
\begin{equation}
\label{ks-class0}
\kappa^1_{(0)}:\CT_S\lra R^1\pi_*(\CA_{(0)})
\end{equation}
--- the "classical" KS map;
\begin{equation}
\label{ks0}
\kappa_{(0)}:\Diff_S\lra \CH_0^{Lie}(\pi_*^{Lie}(\CA_{(0)}))
\end{equation}
and
\begin{equation}
\label{ks-leq-n0}
\kappa^{\leq n}_{(0)}:\Diff^{\leq n}_S\lra \CF_n\CH^{Lie}_0(\pi_*^{Lie}
(\CA_{(0)}))
\end{equation}
satisfying the same compatibilities.
\subsection{}
\label{isom}
\begin{thm}{} Suppose that
--- either we are in the situation ~\ref{higher}, $\pi_*\CA_{/S}=0$,
and $\kappa^1$ is an isomorphism,
--- or we are in the situation ~\ref{dglie-fund-const}, $\pi_*\CA_{(0)}=0$
and $\kappa^1_{(0)}$ is an isomorphism.
Then all $\kappa^{\leq n}$ (resp., $\kappa^{\leq n}_{(0)}$) are isomorphisms.
\end{thm}
{\bf Proof.} Suppose for definiteness that we are in the first situation.
The claim is proved by induction on $n$, using commutative diagrams
\begin{center}
\begin{picture}(20,4)
\put(0,0){\makebox(1,1){$0$}}
\put(2.5,0){\makebox(4,1){$\CF_{n-1}\CH^{Lie}_0(\pi^{Lie}_*(\CA_{/S}))$}}
\put(8,0){\makebox(4,1){$\CF_{n}\CH^{Lie}_0(\pi^{Lie}_*(\CA_{/S}))$}}
\put(13.5,0){\makebox(4,1){$S^n(R^1\pi_*(\CA_{/S}))$}}
\put(19,0){\makebox(1,1){$0$}}
\put(0,3){\makebox(1,1){$0$}}
\put(2.5,3){\makebox(4,1){$\Diff^{\leq n-1}_S$}}
\put(8,3){\makebox(4,1){$\Diff_S^{\leq n}$}}
\put(13.5,3){\makebox(4,1){$S^n(\CT_S)$}}
\put(19,3){\makebox(1,1){$0$}}
\put(4.5,3){\vector(0,-1){2}}
\put(10,3){\vector(0,-1){2}}
\put(15.5,3){\vector(0,-1){2}}
\put(1,0.5){\vector(1,0){1}}
\put(7,0.5){\vector(1,0){1}}
\put(12.5,0.5){\vector(1,0){1}}
\put(17.8,0.5){\vector(1,0){1}}
\put(1,3.5){\vector(1,0){2}}
\put(5.5,3.5){\vector(1,0){3}}
\put(11,3.5){\vector(1,0){3}}
\put(17,3.5){\vector(1,0){2}}
\put(2.5,1){\makebox(2,2){$\kappa^{\leq n-1}$}}
\put(8,1){\makebox(2,2){$\kappa^{\leq n}$}}
\put(16,1){\makebox(2,2){$(-1)^nS^n(\kappa^1)$}}
\end{picture}
\end{center}
Note that our assumptions imply that
$$H^{-1}(F_nC(\pi^{Lie}_*(\CA_{/S}))/F_{n-1}C(\pi^{Lie}_*(\CA_{/S})))=0$$
and
$$H^{0}(F_nC(\pi^{Lie}_*(\CA_{/S}))/F_{n-1}C(\pi^{Lie}_*(\CA_{/S})))=
S^n(R^1\pi_*(\CA_{/S})).$$
$\Box$
\subsection{Deformations of schemes.}
\label{def-schemes} Set
$\CA=\CT_X$. Applying the previous construction, we get the maps
\begin{equation}
\label{ks-class-1}
\kappa^1: \CT_S\lra R^1\pi_*(\CT_{X/S})
\end{equation}
--- the classical Kodaira-Spencer map;
\begin{equation}
\label{ks-1}
\kappa: \Diff_S\lra \CH_0^{Lie}(\pi_*^{Lie}(\CT_{X/S}))
\end{equation}
and
\begin{equation}
\label{ks-leq-n-1}
\kappa^{\leq n}: \Diff_S^{\leq n}\lra \CF_n\CH_0^{Lie}(\pi_*^{Lie}(\CT_{X/S}))
\end{equation}
satisfying the compatibilities ~\ref{main-thm}.
\subsection{Deformations of $G$-torsors}
\label{def-tors} (a) Let $G$ be an algebraic group over $k$, $\fg=\Lie(G)$,
and $p:P\lra X$ a $G$-torsor over $X$. We define sheaves of Lie algebras
$\CA_P$ and $\fg_P$ as in ~\ref{pose}. $\CA_P$ is naturally a transitive Lie
algebroid over $X$, with $\CA_{P(0)}=\fg_P$.
Applying the previous construction, we get the maps analogous to
{}~(\ref{ks-class-1}) - ~(\ref{ks-leq-n-1}), with $\CT_{X/S}$
replaced by $\CA_{P/S}$, subject to the same compatibilities.
(b) Suppose in addition that $X=Y\times S$ as in ~\ref{dglie-fund-const}.
Then we can apply the construction of {\em loc. cit.} to $\CA=\CA_P$, and get
higher KS maps taking value in $\CF_n\CH_0^{Lie}(\fg_P)$.
\section{Universal deformations}
\label{univers}
\subsection{} Let us fix a smooth scheme $X$, an algebraic group $G$ and
a $G$-torsor $P$ over $X$. Consider the following deformation
problems. To each problem we assign a sheaf of $k$-Lie algebras
$\CA_i,\ i=1,2,3$ over $X$.
{\bf Problem 1.} Flat deformations of $X$; $\CA_1=\CT_X$.
{\bf Problem 2.} Flat deformations of the pair $(X,P)$; $\CA_2=\CA_P$,
cf. \ref{def-tors}.
{\bf Problem 3.} Deformations of $P$ ($X$ being fixed); $\CA_3=\fg_P$.
Accordig to Grothendieck, to each problem corresponds a (2-)functor
of infinitesimal deformations
$$
F_i:\Artin_k\lra\Groupoids
$$
from the category of local artinian $k$-algebras with the residue
field $k$ to the (2-)category of groupoids.
In each case, $\CA_i$ is "a sheaf of infinitesimal automorphisms"
corresponding to $F_i$ (in the sense of ~\cite{sga1}, Exp.III, 5,
especially Cor. 5.2 for Problem 1; for the other problems the meaning is
analogous).
In particular, we have the Kodaira-Spencer-Grothendieck isomorphisms
$$
\ft_{\bF_i}\cong H^1(X,\CA_i).
$$
where
$$
\bF_i:\Artin_k\lra \Ens
$$
is the composition of $F_i$ and the connected components functor
$\pi_0:\Groupoids\lra \Ens$. Here for a functor
$$
F:\Artin_k\lra\Ens
$$
$\ft_F$ denotes the tangent space to $F$:
$$
\ft_F=F(\Spec(k[\epsilon]/(\epsilon^2)),
$$
cf. ~\cite{gr}, Exp. 195.
\subsection{} One can verify that in each case the conditions of
Schlessinger's Theorem 2.11, ~\cite{sch}, are fullfilled, and there exists
a versal formal deformation $\fS_i$.
\subsubsection{}
\begin{lem}{} Suppose that $H^0(X,\CA_i)=0$. Then $\fS_i$ is a universal
deformation, i.e. $\bF_i$ is prorepresentable.
\end{lem}
\begin{pf} This fact is presumably classic. We give a sketch of a proof for
completeness. We have $\fS_i=\Spf(R)$, and we have a canonical morphism
$x: h_R\lra F$ (we use notations of ~\cite{sch}). Consider the functor
$$
G:\Artin_R\lra\Ens
$$
from the category of local artinian $R$-algebras with the residue field $k$,
defined as $G(\alpha:R\lra A))=\Aut_{F_i(A)}(\alpha_*(x))$.
{\bf Claim 1.} $G$ is prorepresentable.
Indeed, one can check that the hypotheses of {\em loc.cit.}, 2.11 hold true
for $G$.
Let $\Spf(T)$ prorepresents $G$, where $T$ is a complete local $R$-algebra.
{\bf Claim 2.} The structure morphism $\phi: R\lra T$ is injective.
In fact, this morphism has a section since groups of automorphisms have
identity. Now, we have
$$
\bm_T/\bm^2_T+\bm_R=H^0(X,\CA_i)=0
$$
hence $R=T$, whence $\bF_i$ is prorepresentable by {\em loc.cit.} 3.12.
\end{pf}
\subsection{}
\label{complet}
\begin{thm}{} Suppose that $H^0(X,\CA_i)=0$; let $\fS=\Spf(R)$ be the base
of the universal formal deformation for Problem $i$.
Suppose that $\fS$ is formally smooth. Then we have a canonical
isomorphism
$$
R^*=H^{Lie}_0(R\Gamma^{Lie}(X,\CA_i))
$$
where $R^*$ denotes the space of continuous $k$-linear maps $R\lra k$
($k$ considered in the discrete topology).
\end{thm}
\begin{pf} Let us treat Problem 2 for definiteness (for other problems
the proof is the same). Since $\fS$ is formally
smooth, $R$ is isomorphic to a power series algebra $k[[T_1,\ldots,T_n]]$,
hence $\fS$ is differentially smooth. Let
$\pi: \fX\lra\fS$ be the universal deformation and $\fP$ the
universal $G$-torseur over $\fX$. Applying ~\ref{def-tors}, we get the map
$$
\kappa:\Diff_{\fS}\lra \CH^{Lie}_0(\pi_*^{Lie}(\CA_{\fP/\fS}))
$$
Since $\fS$ is universal, the usual KS map
$\kappa^1:\CT_{\fS}\lra R^1\pi_*(\CA_{\fX/\fS})$ is isomorpism.
Note that $\CA_{\fP/\fS}|_X\cong \CA_i$.
Since $H^0(X,\CA_i)$, we have $\pi_*(\CA_{\fX/\fS})=0$; hence by
{}~\ref{isom} $\kappa$ is isomorphism.
Now let us take the geometric fiber of $\kappa$ at the closed point $s:\Spec(k)
\hra \fS$. We have $\Diff(\fS)_{k(s)}\cong R^*$ by ~\ref{point} and
$$
\CH^{Lie}_0(\pi_*^{Lie}(\CA_{\fP/\fS}))_{k(s)}\cong
H^{Lie}_0(R\Gamma^{Lie}(X,\CA_i))
$$
by ~\ref{base}. The theorem follows.
\end{pf}
|
1994-05-12T21:20:27 | 9405 | alg-geom/9405005 | en | https://arxiv.org/abs/alg-geom/9405005 | [
"alg-geom",
"math.AG"
] | alg-geom/9405005 | Dr. Yakov Karpishpan | Yakov Karpishpan | Higher-order differentials of the period map and higher Kodaira-Spencer
classes | 18 pages, LaTeX | null | null | null | null | In \cite{K} we introduced two variants of higher-order differentials of the
period map and showed how to compute them for a variation of Hodge structure
that comes from a deformation of a compact K\"ahler manifold. More recently
there appeared several works (\cite{BG}, \cite{EV}, \cite{R}) defining higher
tangent spaces to the moduli and the corresponding higher Kodaira-Spencer
classes of a deformation. The $n^{th}$ such class $\kappa_n$ captures all
essential information about the deformation up to $n^{th}$ order. A well-known
result of Griffiths states that the (first) differential of the period map
depends only on the (first) Kodaira-Spencer class of the deformation. In this
paper we show that the second differential of the Archimedean period map
associated to a deformation is determined by $\kappa_2$ taken modulo the image
of $\kappa_1$, whereas the second differential of the usual period map, as well
as the second fundamental form of the VHS, depend only on $\kappa_1$ (Theorems
2, 5, and 6 in Section~3). Presumably, similar statements are valid in
higher-order cases (see Section~4).
| [
{
"version": "v1",
"created": "Thu, 12 May 1994 19:19:29 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Karpishpan",
"Yakov",
""
]
] | alg-geom | \section{Constructing linear maps out of connections}
We start by reviewing the definitions of higher-order
differentials of the period map from \cite{K}, using a slightly
different approach. Let $S$ be a polydisc in ${\bf C}^s$
centered at 0. Consider a free ${\cal O}_S$-module $\cal V$ with
a decreasing filtration by ${\cal O}_S$-submodules
$\ldots\subseteq{\cal F}^p\subseteq{\cal
F}^{p-1}\subseteq\ldots$ and an integrable connection
$$\nabla:{\cal V}\rightarrow{\cal V}\otimes\Omega_S^1$$
satisfying the Griffiths transversality condition
$\nabla({\cal F}^p)\subseteq{\cal F}^{p-1}\otimes\Omega_S^1$.
\begin{lemma}
(a) $\nabla$ induces an \ ${\cal O}_S$-linear map
\begin{eqnarray*}
\Theta_S & \longrightarrow & {\cal H}om_{{\cal O}_S} ({\cal
F}^p,{\cal F}^{p-1}/{\cal F}^p)\\
\xi & \longmapsto & \nabla_{\xi} \bmod {\cal F}^p
\end{eqnarray*}
(b) Analogously, we also have an \ ${\cal O}_S$-linear
symmetric map
\begin{eqnarray*}
\Theta_S^{\otimes 2} & \longrightarrow & {\cal H}om_{{\cal
O}_S}
({\cal F}^p,{\cal F}^{p-2}/{\cal F}^p+span\,\{\nabla_{\eta}
({\cal F}^p)\,|\ all\ \eta\in\Gamma(S,\Theta_S)\}) \\
\zeta\otimes\xi & \mapsto &
\nabla_{\zeta}\nabla_{\xi} \bmod {\cal F}^p +
span\,\{\nabla_{\eta}({\cal F}^p)\}\ .
\end{eqnarray*}
\end{lemma}
\ \\ \noindent {\bf Proof.\ \ } Both (a) and (b) are proved by straightforward computations;
the fact that the map in (b) is symmetric follows from the
integrability of $\nabla$:
$$\nabla_{\zeta}\nabla_{\xi}-
\nabla_{\xi}\nabla_{\zeta}=\nabla_{[\zeta,\xi]}\ ,$$
and so
$$\nabla_{\zeta}\nabla_{\xi}\equiv\nabla_{\xi}\nabla_{\zeta}
\bmod span\,\{\nabla_{\eta}({\cal F}^p)\,|\ \eta\in\Theta_S\}\
.$$
\ $\Box$\\ \ \par
We will apply the Lemma to two connections arising from a
deformation of a compact K\"{a}hler manifold $X$,
\begin{eqnarray}
\nonumber {\cal X} & \supset & X\\
\pi\ \downarrow & & \downarrow\\
\nonumber S & \ni & 0
\end{eqnarray}
\noindent 1. The usual Gauss-Manin connection $\nabla$ on
${\cal H}=R^m\pi_{\ast}{\bf C}_{\cal X}$\ . In this case we
denote the map in (a)
$$d\Phi: \Theta_S\rightarrow
{\cal H}om_{{\cal O}_S} ({\cal F}^p,{\cal F}^{p-1}/{\cal F}^p)\
. $$
This is {\em the (first) differential of the (usual) period
map}. The same notation and terminology will be applied to the
induced map $$\Theta_S\rightarrow\bigoplus_p{\cal H}om_{{\cal
O}_S}
({\cal F}^p/{\cal F}^{p+1},{\cal F}^{p-1}/{\cal F}^p)\ .$$
The map given by part (b) of the Lemma is called {\em the
second differential of the (usual) period map} and denoted
$$d^2\Phi: \Theta_S^2\longrightarrow{\cal H}om_{{\cal O}_S}
({\cal F}^p,{\cal F}^{p-2}/{\cal F}^p+span\,\{\nabla_{\eta}
({\cal F}^p)\,|\ \eta\in\Theta_S\})\ .$$
\noindent 2. The Archimedean Gauss-Manin connection
$\nabla=\nabla_{ar}$ on
$${\cal H}\otimes B_{ar}= R^m\pi_{\ast}{\bf
C}_{\cal X}[T,T^{-1}]$$
(see Appendix). The corresponding map from (a) is called
{\em the (first) differential of the {\em Archimedean}
period map}, denoted $$d\Psi:\Theta_S\longrightarrow
{\cal H}om_{{\cal O}_S}({\cal H}_{ar},
{\cal F}_{ar}^{-1}/{\cal H}_{ar})\ .$$
Again, we will abuse notation and write $d\Psi$ to denote the
induced map
\begin{equation}
\Theta_S\rightarrow {\cal H}om_{{\cal O}_S}
(Gr_{{\cal F}_{ar}}^0, Gr_{{\cal F}_{ar}}^{-1})\ .
\end{equation}
Finally, the map in part (b) of the Lemma, for
$\nabla=\nabla_{ar}$, is {\em the second differential of the
Archimedean period map} and will be denoted
$$d^2\Psi:\Theta_S^2\longrightarrow {\cal H}om_{{\cal O}_S}
({\cal H}_{ar},{\cal F}_{ar}^{-1}/{\cal H}_{ar}+
span\,\{\nabla_{\eta}({\cal H}_{ar})\,|\ \eta\in\Theta_S\})\ .$$
We have an identification of ${\cal O}_S$-modules
$$Gr_{{\cal F}_{ar}}^0({\cal H}\otimes B_{ar})\cong
{\bf R}^m\pi_{\ast}(Gr_F^0(\Omega^{\textstyle\cdot}_{{\cal X}/S}\otimes
B_{ar}))\cong{\bf R}^m\pi_{\ast}\Omega^{\textstyle\cdot}_{{\cal X}/S}=
{\cal H}\ ,$$
obtained from the obvious isomorphism of sheaf complexes
$$\begin{array}{ccccc}
\longrightarrow & \Omega^p_{{\cal X}/S}.T^p &
\stackrel{\bf d}{\longrightarrow} &
\Omega^{p+1}_{{\cal X}/S}.T^{p+1} & \longrightarrow\\
& \simeq\downarrow & & \downarrow\simeq & \\
\longrightarrow & \Omega^p_{{\cal X}/S} &
\stackrel{d}{\longrightarrow} &
\Omega^{p+1}_{{\cal X}/S} & \longrightarrow
\end{array}$$
\noindent (``dropping the T's").
Similarly, $Gr_{{\cal F}_{ar}}^{-1}\cong {\cal H}$.
We use these identifications to obtain a version of (2)
``without
T's," $$\overline{d\Psi}:\Theta_S\longrightarrow{\cal E}nd_
{{\cal O}_S}({\cal H})\ .$$
Analogously, we also define
$$\overline{d^2\Psi}:\Theta_S^{\otimes 2}\longrightarrow{\cal
E}nd_ {{\cal O}_S}({\cal H})\ ,$$
with $\overline{d^2\Psi}(\zeta,\xi)$ being the composition
$${\cal H}\cong Gr_{{\cal F}_{ar}}^0
\stackrel{d^2\Psi(\zeta,\xi)}{\longrightarrow}{\cal
F}_{ar}^{-2}/{\cal H}_{ar}+span\,\{\nabla_{\eta}({\cal
F}_{ar}^{-1})\,|\ \eta\in\Theta_S\}\,\longrightarrow\hspace{-12pt
Gr_{{\cal F}_{ar}}^{-2}\cong{\cal H}\ .$$
\ \\ \noindent {\bf Remark.\ \ } Let ${\bf t}=(t_1,\ldots,t_s)$ be a coordinate system on
$S$ centered at 0. Then, in the
notation of \cite{K}, $d\Psi(\partial/\partial t_i)|_{t=0}$
is $\widetilde{L}^{(i)}$,
$\overline{d\Psi}(\partial/\partial t_i)|_{t=0}$ is
$\overline{L}^{(i)}$, $d^2\Psi(\partial/\partial t_i\otimes
\partial/\partial t_j)|_{t=0}$ is $\widetilde{L}^{(ij)}$, and
$\overline{d^2\Psi}(\partial/\partial t_i\otimes
\partial/\partial t_j)|_{t=0}$ is $\overline{L}^{(ij)}$.
\ ${\cal O}_S$-linearity of $d\Psi$, $d^2\Psi$, etc. is essential
for the ability to restrict to 0 in $S$.
To formulate the next Lemma, we bring out the natural
$C_S^{\infty}$-linear identification
$$h:{\cal H}=\bigoplus_{p+q=m}{\cal H}^{p,q}
\stackrel{\simeq}{\longrightarrow}
\bigoplus_p Gr_{\cal F}^p{\cal H}=:
Gr_{\cal F}^{\textstyle\cdot}{\cal H}\ .$$
\begin{lemma}
(a) For any \ $\xi\in\Theta_S$ and all \ $p$ we have
$$\overline{d\Psi}(\xi)({\cal F}^p{\cal H})\subset
{\cal F}^{p-1}{\cal H}\ ,$$
the induced endomorphism of degree \
$-1$ of \ $Gr_{\cal F}^{\textstyle\cdot}{\cal H}$ coincides with \ $d\Phi(\xi)$
and, in fact,
$$\overline{d\Psi}(\xi)=h^{-1}\circ d\Phi(\xi)\circ h\ .$$
(b) For any \ $\zeta,\xi\in\Theta_S$ and all \ $p$ we have
$$\overline{d^2\Psi}(\zeta,\xi)({\cal F}^p{\cal H})\subset
{\cal F}^{p-2}{\cal H}\ ,$$ and
$$d^2\Phi(\zeta,\xi):{\cal F}^p\longrightarrow{\cal F}^{p-2}/
{\cal F}^p+span\,\{\nabla_{\eta}({\cal F}^p)\,|\
\eta\in\Theta_S\}$$
factors through \ $\overline{d^2\Psi}(\zeta,\xi):
{\cal F}^p\longrightarrow{\cal F}^{p-2}$.
\end{lemma}
\ \\ \noindent {\bf Proof.\ \ } (a) For any $\xi\in\Theta_S$
$$\nabla_{\xi}{\cal H}^{p,q}\subset
{\cal H}^{p,q}\oplus{\cal H}^{p-1,q+1}$$
and, correspondingly,
$$\nabla_{\xi}^{ar}{\cal H}^{p,q}.T^p\subset
{\cal H}^{p,q}.T^p\oplus{\cal H}^{p-1,q+1}.T^p\ .$$
Therefore, $d\Psi(\xi)$ maps ${\cal H}^{p,q}.T^p$ to its image
under $\nabla_{\xi}^{ar}$ modulo ${\cal H}_{ar}={\cal
F}_{ar}^0$, i.e. into ${\cal H}^{p-1,q+1}.T^p$. Hence
$$\overline{d\Psi}(\xi)({\cal H}^{p,q})\subset
{\cal H}^{p-1,q+1}\ ,$$
which implies every statement in part (a) of the Lemma.
Part (b) is established by similar reasoning.\ $\Box$\\ \ \par
The connection $\nabla$ on $\cal H$ naturally induces a connection
$\nabla$ on ${\cal E}nd_{{\cal O}_S}({\cal H})$ subject to the
rule $$\nabla(Ax)=(\nabla A)x+A\nabla x$$
for any $A\in{\cal E}nd_{{\cal O}_S}({\cal H})$ and $x\in{\cal H}$.
In accordance with Lemma 1, for any $\zeta\in\Theta_S$
the covariant derivative $\nabla_{\zeta}$ on ${\cal E}nd({\cal
H})$ determines an ${\cal O}_S$-linear map
$${\cal E}_{\zeta}:
im\,(\overline{d\Psi})\longrightarrow
{\cal E}nd({\cal H})/im\,(\overline{d\Psi})\ .$$
\begin{lemma}
For any \ $\zeta,\xi\in\Theta_S$
(a) \ $\nabla_{\zeta}(\overline{d\Psi}(\xi))({\cal F}^p)\subset
{\cal F}^{p-1}$ for all \ $p$.
(b) \ ${\cal E}_{\zeta}(\overline{d\Psi}(\xi))$ determines an
element of \ ${\cal E}nd_{{\cal O}_S}^{(-1)}(Gr^{\textstyle\cdot}
_{\cal F}{\cal H})/im\,(d\Phi)$.
\end{lemma}
\ \\ \noindent {\bf Proof.\ \ } (a) $\nabla_{\zeta}(\overline{d\Psi}(\xi))\omega=
\nabla_{\zeta}(\overline{d\Psi}(\xi)\omega)-
\overline{d\Psi}(\xi)\nabla_{\zeta}\omega$ for any
$\omega\in{\cal H}$. Assume $\omega\in{\cal H}^{p,q}$. We want to
show that the $(p-2,q+2)$-component of the right-hand side is 0.
But this component is
$$(\nabla_{\zeta}\nabla_{\xi}\omega)_{(p-2,q+2)}-
(\nabla_{\xi}\nabla_{\zeta}\omega)_{(p-2,q+2)}=
(\nabla_{[\xi,\zeta]}\omega)_{(p-2,q+2)}=0\ !$$
(b) follows from (a) and the relation between
$\overline{d\Psi}(\xi)$ and $d\Phi(\xi)$ established in part (a)
of the previous Lemma.\ $\Box$\\ \ \par
\begin{dfntn}
The {\em second fundamental form of the VHS}
$${\rm II}:\Theta_S^{\otimes 2}\longrightarrow
{\cal E}nd_{{\cal O}_S}(Gr^{\textstyle\cdot}_{\cal F})/im\,(d\Phi)$$
is defined by
$${\rm II}(\zeta,\xi):=h\circ\nabla_{\zeta}(h^{-1}d\Phi(\xi)h)\circ
h^{-1}\bmod im\,(d\Phi)\ .$$
\end{dfntn}
\ \\ \noindent {\bf Remark.\ \ } ${\rm II}|_{t=0}$ was denoted $d^2\Phi$ in \cite{K}.
\
We will omit the identification $h$ in what follows.
\begin{prop}[(2.4) in \cite{K}]
$${\rm II}(\zeta,\xi)\equiv\overline{d^2\Psi}(\zeta,\xi)-
\overline{d\Psi}(\zeta) \circ \overline{d\Psi}(\xi)
\bmod im\,(d\Phi)\ .$$
\end{prop}
\ \\ \noindent {\bf Proof.\ \ }
$\nabla_{\zeta}(\overline{d\Psi}(\xi))\omega=
\nabla_{\zeta}(\overline{d\Psi}(\xi)\omega)-
\overline{d\Psi}(\xi)\nabla_{\zeta}\omega$ for any
$\omega\in{\cal H}$. Now, let $\mbox{\boldmath $\omega$}$ be
the element of $Gr_{{\cal F}_{ar}}^0({\cal H}\otimes B_{ar})$
corresponding to $\omega$ under the
isomorphism
${\cal H}\cong Gr_{{\cal F}_{ar}}^0({\cal H}\otimes B_{ar})$.
Using a similar identification of ${\cal H}$ with
$Gr_{{\cal F}_{ar}}^{-2}({\cal H}\otimes B_{ar})$, we have the
following correspondences:
$$\nabla_{\zeta}(\overline{d\Psi}(\xi)\omega)\longleftrightarrow
(\nabla_{\zeta}(d\Psi(\xi)\mbox{\boldmath$\omega$})
\bmod {\cal F}_{ar}^{-1})=(d^2\Psi(\zeta,\xi)
\mbox{\boldmath$\omega$}\bmod {\cal F}_{ar}^{-1})
\in Gr_{{\cal F}_{ar}}^{-2}\ ,$$
and
$$\overline{d\Psi}(\xi)\nabla_{\zeta}\omega\longleftrightarrow
(d\Psi(\zeta)\circ d\Psi(\xi)\mbox{\boldmath$\omega$}
\bmod {\cal F}_{ar}^{-1})\in Gr_{{\cal F}_{ar}}^{-2}\ .$$
It remains to pass to ${\cal H}$ on the right-hand side, i.e.
put bars on $d\Psi$ and $d^2\Psi$.\ $\Box$\\ \ \par
\section{The second Kodaira-Spencer class}
First, let us recall the construction of the (first)
Kodaira-Spencer map $\kappa_1=\kappa$ of the deformation (1): it
is the connecting morphism in the higher-direct-image sequence
\begin{equation}
\longrightarrow\pi_{*}\Theta_{\cal
X}\longrightarrow\Theta_S
\stackrel{\kappa}{\longrightarrow}R^1\pi_{*}\Theta_{{\cal
X}/S} \longrightarrow
\end{equation}
associated with the short exact sequence
\begin{equation}
0\longrightarrow\Theta_{{\cal X}/S}\longrightarrow
\Theta_{\cal X}\longrightarrow\pi^{*}\Theta_S\longrightarrow
0\ .
\end{equation}
Given $\xi\in\Theta_S$, the corresponding covariant derivative of
the Gauss-Manin connection
$$\nabla_{\xi}:{\bf R}^m\pi_{*}\Omega^{\textstyle\cdot}_{{\cal X}/S}
\longrightarrow{\bf R}^m\pi_{*}\Omega^{\textstyle\cdot}_{{\cal X}/S}$$
is computed as follows (see \cite{Del}, \cite{KO}, or
\cite{K}). Choose a Stein covering ${\cal U}=\{U_i\}$ of $X$.
Then $\{W_i=U_i\times S\}$ constitute a Stein covering $\cal W$
of $\cal X$. Consider a class in $\Gamma(S,{\bf
R}^m\pi_{*}\Omega^{\textstyle\cdot}_{{\cal X}/S})$ represented by the \v{C}ech
cocycle $\omega=\{\omega_Q\in \Gamma({\cal W}_Q,\Omega^p_{{\cal
X}/S})\}$, where $Q=(i_1<\ldots<i_q)$ and $p+q+1=m$. Let the same
symbol $\omega_Q$ denote a pull-back of $\omega_Q\in
\Gamma({\cal W}_Q,\Omega^p_{{\cal X}/S})$ to an element of
$\Gamma({\cal W}_Q,\Omega^p_{\cal X})$.
Let $v=\{v_i\}$ denote liftings of $\xi\in\Theta_S$, or rather,
in $\pi^{*}\Theta_S$, to $\Gamma(W_i,\Theta_{\cal X})$. Then
$$\nabla_{\xi}[\omega]=[\check{\pounds}_v\omega]$$
where the brackets denote cohomology classes in
${\bf R}^m\pi_{*}\Omega^{\textstyle\cdot}_{{\cal X}/S}$ (more precisely, in
$\Gamma(S,{\bf R}^m\pi_{*}\Omega^{\textstyle\cdot}_{{\cal X}/S})$), and
$\check{\pounds}_v$ is the Lie derivative on
$\check{C}^m({\cal W},\Omega^{\textstyle\cdot}_{\cal X}) $ with respect to
$v=\{v_i\}\in \check{C}^0({\cal W},\Theta_{\cal X})$:
\begin{equation}
\check{\pounds}_v\omega=
\{\check{D}v_{i_1}\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\omega_{i_1,\ldots,i_q}+
v_{i_1}\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\check{D}\omega_{i_1,\ldots,i_q}\}
\end{equation}
with $\check{D}=d\pm\delta$, $\delta=\check{\delta}$ being the
\v{C}ech differential, as usual.
Now, when $\xi\in\Theta_S$ lifts to all of $\cal X$, i.e. $\xi$
lies in the image of $\pi_*\Theta_{\cal X}\rightarrow\Theta_S$
($= ker\,(\kappa)$ !), the cochain
$v\in\check{C}^0({\cal W},\Theta_{\cal X})$ lifting $\xi$ is a
cocycle, i,e, $\delta v=0$. But then formula (5) reduces to
$$\check{\pounds}_v\omega=
\{dv_{i_1}\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\omega_{i_1,\ldots,i_q}+
v_{i_1}\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\, d\omega_{i_1,\ldots,i_q}\}=
\{\pounds_{v_{i_1}}\omega_{i_1,\ldots,i_q}\} ,$$
where $\pounds_{v_{i_1}}$ now denotes the usual Lie derivative
with respect to the vector field $v_{i_1}$. Evidently, in this
case $\nabla_{\xi}{\cal F}^p\subset{\cal F}^p$. Consequently,
the (first) differential of the period map
$$d\Phi:\Theta_S\longrightarrow
{\cal H}om({\cal F}^p,{\cal H}/{\cal F}^p)$$
factors through
$\Theta_S/\,im\,\{\pi_*\Theta_{\cal X}\rightarrow\Theta_S\}=
\Theta_S/\,ker\,(\kappa)$, and thus we arrive at
\begin{thm}[Griffiths]
There is a commutative
diagram
$$\begin{array}{ccccc}
\Theta_S & \stackrel{d\Phi}{\longrightarrow} &
\bigoplus_p{\cal H}om({\cal F}^p,{\cal H}/{\cal F}^p) &
\,\longrightarrow\hspace{-12pt & {\cal E}nd^{(-1)}(Gr^{\textstyle\cdot}_{\cal F}{\cal H}) \\
& & & & \\
& \searrow\kappa & \uparrow & \smile\nearrow & \\
& & & & \\
& & {\bf T}^1_{{\cal X}/S} & &
\end{array}$$
where \ ${\bf T}^1_{{\cal X}/S}:=R^1\pi_*\Theta_{{\cal X}/S}$ and
the northeast arrow sends \ $x$ to the map \ $x\smile$ \ \ (the
cup product with \ $x$).
\end{thm}
Analogous results hold for $d\Psi$ and $\overline{d\Psi}$
(see \cite{K}). We seek a similar statement for $d^2\Phi$ and
$d^2\Psi$.
First we need to review the construction of the ``second-order
tangent space to the moduli." Here we are following (a
relativized version of) the presentation in \cite{R}. We will
make the simplifying assumption that $X$ has no global
holomorphic vector fields, i.e. $\pi_*\Theta_{{\cal X}/S}=0$.
Consider the diagram
\begin{eqnarray}
\nonumber{\cal X}\times_S{\cal X} &
\stackrel{g}{\longrightarrow} & {\cal X}_2/S \\
f\searrow & & \swarrow p \\
\nonumber & S &
\end{eqnarray}
where ${\cal X}_2/S$ denotes the symmetric
product of $\cal X$ with itself over $S$ (fiberwise).
Let ${\cal K}^{\textstyle\cdot}$ denote the complex of sheaves on ${\cal X}_2/S$
$$\begin{array}{ccc}
{\scriptstyle -1} & & {\scriptstyle 0} \\
(g_*(\Theta_{{\cal X}/S}^{\makebox[0in][l]{$\scriptstyle\times$ 2}))^{-} &
\stackrel{[\, ,\,]}{\longrightarrow} & \Theta_{{\cal X}/S}
\end{array}$$
where $\makebox[0in][l]{$\times$}\raisebox{-1pt}{$\Box$}$ stands for the exterior tensor product on
${\cal X}\times_S{\cal X}$, \ \ $(\ \ )^{-}$ denotes the
anti-invariants of the
${\bf Z}/2{\bf Z}$-action, and the differential is the
restriction to the diagonal
$\Delta\subset{\cal X}\times_{S}{\cal X}$,
followed by the Lie bracket of vector fields.
\begin{dfntn}
${\bf T}^{(2)}_{{\cal X}/S}:={\bf R}^1p_*{\cal K}^{\textstyle\cdot}$ is
the sheaf on $S$ whose fiber over each $t\in S$ is the
{\em second-order (Zariski) tangent space to the base $V_t$ of
the miniversal deformation of} $X_t$, i.e.
${\bf T}^{(2)}_{X_t}\cong({\bf m}_{V_t,0}/{\bf m}_{V_t,0}^3)^*$.
\end{dfntn}
This should not be confused with the sheaf
${\bf T}^{2}_{{\cal X}/S}=R^2\pi_*\Theta_{{\cal X}/S}$ whose
fiber over each $t\in S$ is the {\em obstruction space}\ \
${\bf T}^{2}_{X_t}$ for deformations of $X_t$.
In fact, we have this exact sequence:
\begin{equation}
0\longrightarrow{\bf T}^1_{{\cal X}/S}\longrightarrow
{\bf T}^{(2)}_{{\cal X}/S}\longrightarrow
Sym^2{\bf T}^1_{{\cal X}/S}\stackrel{o}{\longrightarrow}
{\bf T}^{2}_{{\cal X}/S}\ ,
\end{equation}
where $o$ is the first obstruction map, given by the
${\cal O}_S$-linear graded Lie bracket:
$$Sym^2R^1\pi_*\Theta_{{\cal
X}/S}\stackrel{[\, ,\,]}{\longrightarrow}
R^2\pi_*\Theta_{{\cal X}/S}\ .$$
We will find it easier to deal with an ``unsymmetrized version"
of ${\bf T}^{(2)}_{{\cal X}/S}$.
\begin{dfntn}
$\widetilde{\bf T}^{(2)}_{{\cal X}/S}:={\bf R}^1f_*
\widetilde{\cal K}^{\textstyle\cdot}$,
where $\widetilde{\cal K}^{\textstyle\cdot}$ is the complex on
${\cal X}\times_S{\cal X}$,
$$\begin{array}{ccc}
{\scriptstyle -1} & & {\scriptstyle 0} \\
\Theta_{{\cal X}/S}^{\makebox[0in][l]{$\scriptstyle\times$ 2} &
\stackrel{[\, ,\,]}{\longrightarrow} & \Theta_{{\cal X}/S}\ .
\end{array}$$
\end{dfntn}
$\widetilde{\bf T}^{(2)}_{{\cal X}/S}$ fits in the commutative diagram
with exact rows:
$$\begin{array}{cccccccc}
0\rightarrow & {\bf T}^1_{{\cal X}/S} & \longrightarrow &
\widetilde{\bf T}^{(2)}_{{\cal X}/S} & \longrightarrow &
({\bf T}^1_{{\cal X}/S})^{\otimes 2} &
\stackrel{\widetilde{o}}{\longrightarrow} &
{\bf T}^2_{{\cal X}/S} \\
& \| & & \downsurj & & \downsurj & & \| \\
0\rightarrow & {\bf T}^1_{{\cal X}/S} & \longrightarrow &
{\bf T}^{(2)}_{{\cal X}/S} & \longrightarrow &
Sym^2{\bf T}^1_{{\cal X}/S} &
\stackrel{o}{\longrightarrow} &
{\bf T}^2_{{\cal X}/S}
\end{array}$$
\begin{dfntn}
$T_S^{(2)}:={\cal D}_S^{(2)}/{\cal O}_S$ will denote the sheaf of
{\em second-order tangent vectors}\ \ on $S$.
\end{dfntn}
As part of a more general construction in \cite{EV}, there is
{\em the second Kodaira-Spencer map} associated to every
deformation as in (1):
$$\kappa_2:T_S^{(2)}\longrightarrow{\bf T}^{(2)}_{{\cal X}/S}\ .$$
We will work with a natural lifting $\widetilde{\kappa}_2$
of $\kappa_2$:
\begin{equation}
\begin{array}{ccc}
\Theta_S\oplus\Theta_S^{\otimes 2} &
\stackrel{\widetilde{\kappa}_2}{\longrightarrow} &
\widetilde{\bf T}^{(2)}_{{\cal X}/S} \\
\downsurj & & \downsurj \\
T_S^{(2)} & \stackrel{\kappa_2}{\longrightarrow} &
{\bf T}^{(2)}_{{\cal X}/S}
\end{array}
\end{equation}
or, rather, with the restriction of $\widetilde{\kappa}_2$ to
$\Theta_S^{\otimes 2}$.
It is easy to describe $\widetilde{\kappa}_2$ explicitly.
Let
$$\kappa:\Theta_S\longrightarrow {\bf T}^1_{{\cal X}/S}=
R^1\pi_*\Theta_{{\cal X}/S}$$
be the ({\em relative, first}\ ) Kodaira-Spencer map of the
family $\pi$ as in (1).
It is equivalent to the datum of a section of
$\Gamma(S,\Omega_S^1\otimes R^1\pi_*\Theta_{{\cal X}/S})$. This
section can be represented by a
$\check{C}^1({\cal U},\Theta_X)$-valued one-form on $S$,
\begin{equation}
\theta({\bf t})d{\bf t}:=
\sum_{\ell=1}^s\theta({\bf t})_{\ell}dt_{\ell}=
\sum_{\ell=1}^s\sum_{I\in{\bf Z}_+^s}^s
\theta_{\ell}^{(I)}{\bf t}^Idt_{\ell}\ \ \
({\bf Z}_+:=\{0\}\cup{\bf N})\ .
\end{equation}
Here each $\theta_{\ell}^{(I)}=\{\theta_{ij,\ell}^{(I)}\}_{ij}$ is
a cochain in $\check{C}^1({\cal U},\Theta_X)$ and each
$\theta({\bf t})_{\ell}$ is a cocycle on $X_t$ for every
value of $\bf t$, but only the leading coefficients
$\theta_{\ell}^{(0)}\ \ (\ell=1,\ldots,s)$ are \v{C}ech {\em
cocycles}\ on $X$\ \ $(t=0)$. The rest satisfy the ``deformation
equation"
\begin{equation}
\delta\left(\frac{\partial\theta({\bf t})_{\ell}}
{\partial_{t_k}}\right)=
[\theta({\bf t})_{\ell},\theta({\bf t})_k]
\end{equation}
When $s=1$, this
reduces to $\delta\dot{\theta}(t)=[\theta(t),\theta(t)]$.
\noindent Thus, it is natural to make the following
\begin{dfntn}
$\widetilde{\kappa}_2:\Theta_S^{\otimes 2}\longrightarrow
\widetilde{\bf T}^{(2)}_{{\cal X}/S}$
sends
$\frac{\partial}{\partial t_k}\otimes
\frac{\partial}{\partial t_{\ell}}$ to the cohomology class of
the cocycle
\begin{equation}
(\theta({\bf t})_k\times
\theta({\bf t})_{\ell},
\frac{\partial\theta({\bf t})_{\ell}}
{\partial_{t_k}})
\in \check{C}^1({\cal W}\times_S{\cal W},
\widetilde{\cal K}^{\textstyle\cdot})\ .
\end{equation}
\end{dfntn}
For example, if
\begin{equation}
\theta({\bf t})d{\bf t}=\sum_{\ell=1}^s(\theta_{\ell}^{(0)}+
\sum_{k=1}^s\theta_{\ell}^{(k)}t_k)dt_{\ell}+O({\bf t}^2)
\end{equation}
is the expansion of $\theta({\bf t})d{\bf t}$ at 0 up to order
two, then
$\widetilde{\kappa}_2|_{0}:\Theta_S^{\otimes 2}|_{0}\rightarrow
\widetilde{\bf T}_X^{(2)}$ sends
$\frac{\partial}{\partial t_k}\otimes
\frac{\partial}{\partial t_{\ell}}$ to the cohomology class of
the cocycle
\begin{equation}
(\theta_k^{(0)}\times\theta_{\ell}^{(0)},
\theta_{\ell}^{(k)})\in \check{C}^1({\cal U}\times{\cal U},
\widetilde{\cal K}^{\textstyle\cdot}|_{0})\ .
\end{equation}
Indeed, for the definition of $\widetilde{\kappa}_2$ to make any
sense, we must have the following commutative diagram with exact
rows:
\begin{equation}
\begin{array}{cccccccc}
0\rightarrow & \Theta_S & \rightarrow &
\Theta_S\oplus\Theta_S^{\otimes 2} &
\rightarrow & \Theta_S^{\otimes 2} & \rightarrow & 0 \\
& \kappa_1\ \downarrow & & \downarrow \widetilde{\kappa}_2 &
& \downarrow \kappa_1^2 & & \\
0\rightarrow & {\bf T}_{{\cal X}/S}^1 & \rightarrow &
\widetilde{\bf T}_{{\cal X}/S}^{(2)} & \rightarrow &
({\bf T}_{{\cal X}/S}^1)^{\otimes 2} &
\stackrel{o}{\rightarrow} & {\bf T}_{{\cal X}/S}^2\ .
\end{array}
\end{equation}
The square on the right induces a commutative triangle
\begin{equation}
\begin{array}{ccc}
\Theta_S^{\otimes 2} &
\stackrel{\widetilde{\kappa}_2}{\longrightarrow} &
\widetilde{\bf T}_{{\cal X}/S}^{(2)} \\
& \kappa_1^2\ \searrow & \downarrow \\
& & ({\bf T}_{{\cal X}/S}^1)^{\otimes 2}\ .
\end{array}
\end{equation}
Therefore,
$\widetilde{\kappa}_2(\frac{\partial}{\partial t_k}\otimes
\frac{\partial}{\partial t_\ell})$ must project onto
$$\kappa_1(\frac{\partial}{\partial t_k})\otimes
\kappa_1(\frac{\partial}{\partial t_\ell})=
[\theta({\bf t})_k]\otimes[\theta({\bf t})_\ell]\ .$$
Since
$$\check{C}^1({\cal W}\times_S{\cal W},\widetilde{\cal K}^{\textstyle\cdot})=
\check{C}^2({\cal W}\times_S{\cal W},\widetilde{\cal K}^{-1})
\oplus\check{C}^1(\widetilde{\cal K}^0)\ ,$$
and
$$\check{C}^2({\cal W}\times_S{\cal W},\widetilde{\cal K}^{-1})
\simeq\check{C}^1({\cal U},\Theta_{{\cal X}/S})^{\otimes 2}\ ,$$
this means that the $\check{C}^2(\widetilde{\cal
K}^{-1})$-component of a representative of
$\widetilde{\kappa}_2(\frac{\partial}{\partial t_k}\otimes
\frac{\partial}{\partial t_\ell})$ in
$\check{C}^1(\widetilde{\cal K}^{\textstyle\cdot})$ may be taken to be
$$\theta({\bf t})_k\times\theta({\bf t})_{\ell}\ .$$
And, in view of (10), the cochain (11) is indeed a {\em cocycle}
\ in $\check{C}^1(\widetilde{\cal K}^{\textstyle\cdot})$.
We still need to check that $\widetilde{\kappa}_2$ is well-defined.
\begin{prop}\ \ \
$\widetilde{\kappa}_2:\Theta_S^{\otimes 2}\longrightarrow
\widetilde{\bf T}^{(2)}_{{\cal X}/S}$ can
be presented as a connecting morphism in the higher-direct-image
sequence of a short exact sequence.
\end{prop}
\ \\ \noindent {\bf Proof.\ \ } The starting point is the sequence of
${\cal O}_{\cal X}$-modules
(4), whose direct-image sequence (3) gives the first
Kodaira-Spencer mapping $\kappa_1$ as a connecting morphism.
Now, (4) contains an exact subsequence
\begin{equation}
0\longrightarrow\Theta_{{\cal X}/S}
\stackrel{\alpha}{\longrightarrow}\widetilde{\Theta}_{\cal X}
\stackrel{\beta}{\longrightarrow}\pi^{-1}\Theta_S
\longrightarrow 0
\end{equation}
of $\pi^{-1}{\cal O}_S$-modules, whose direct-image sequence also
has $\kappa_1$ as a connecting morphism. From now on we will
work with (16) in place of (4).
We can splice two sequences produced from (16) by exterior
tensor products with $\Theta_{{\cal X}/S}$ and with
$\pi^{-1}\Theta_S$, respectively:
$$\begin{array}{cccc}
& & 0 & \\
& & \uparrow & \\
& & \pi^{-1}\Theta_S\displayboxtimes\pi^{-1}\Theta_S & \\
& & \uparrow & \\
& & \pi^{-1}\Theta_S\displayboxtimes
\widetilde{\Theta}_{\cal X} & \\
& \nearrow & \uparrow & \\
0\longrightarrow\Theta_{{\cal X}/S}\displayboxtimes
\Theta_{{\cal X}/S} \longrightarrow
\widetilde{\Theta}_{\cal X}\displayboxtimes
\Theta_{{\cal X}/S} & \rightarrow &
\pi^{-1}\Theta_S\displayboxtimes\Theta_{{\cal X}/S} &
\longrightarrow 0 \\
& & \uparrow & \\
& & 0 &
\end{array}$$
The resulting four-term exact sequence
\begin{equation}
0\rightarrow\Theta_{{\cal X}/S}^{\makebox[0in][l]{$\scriptstyle\times$ 2}
\rightarrow\widetilde{\Theta}_{\cal X}
\displayboxtimes\Theta_{{\cal X}/S}
\stackrel{\beta\makebox[0in][l]{$\scriptstyle\times$\alpha}{\longrightarrow}
\pi^{-1}\Theta_S\displayboxtimes\widetilde{\Theta}_{\cal X}
\rightarrow(\pi^{-1}\Theta_S)^{\makebox[0in][l]{$\scriptstyle\times$ 2}
\rightarrow 0
\end{equation}
can be extended to a commutative diagram
\begin{equation}
\begin{array}{ccccc}
0\rightarrow & \Theta_{{\cal X}/S}^{\makebox[0in][l]{$\scriptstyle\times$ 2} &
\rightarrow & \widetilde{\Theta}_{\cal X}
\displayboxtimes\Theta_{{\cal X}/S} &
\rightarrow
\pi^{-1}\Theta_S\displayboxtimes\widetilde{\Theta}_{\cal X}
\rightarrow(\pi^{-1}\Theta_S)^{\makebox[0in][l]{$\scriptstyle\times$ 2}
\rightarrow 0 \\
& \downarrow & & \downarrow & \\
& \Theta_{{\cal X}/S} & = & \Theta_{{\cal X}/S} &
\end{array}
\end{equation}
where the vertical maps are composed of the restriction to
the diagonal $\Delta\subset{\cal X}\times_S{\cal X}$ followed
by Lie brackets.
\ \\ \noindent {\bf Remark.\ \ } Here we use the fact that the restriction of
the Lie bracket
$$[\ ,\ ]:\Theta_{\cal X}^{\makebox[0in][l]{$\scriptstyle\times$ 2}
\longrightarrow\Theta_{\cal X}$$
to
$\widetilde{\Theta}_{\cal X}\makebox[0in][l]{$\times$}\raisebox{-1pt}{$\Box$}\Theta_{{\cal X}/S}$ takes
values in $\Theta_{{\cal X}/S}$ (see \cite{BS}, and
also \cite{EV}).
We note that the first column of the diagram (18) constitutes
the complex $\widetilde{\cal K}^{\textstyle\cdot}$ computing
$\widetilde{\bf T}^{(2)}_{{\cal X}/S}$. Let ${\cal L}^{\textstyle\cdot}$
denote the complex
$$\begin{array}{ccc}
{\scriptstyle -1} & & {\scriptstyle 0} \\
\widetilde{\Theta}_{\cal X}\displayboxtimes
\Theta_{{\cal X}/S} & \stackrel{\ell}{\longrightarrow} &
(\pi^{-1}\Theta_S\displayboxtimes
\widetilde{\Theta}_{\cal X})\oplus\Theta_{{\cal X}/S}
\end{array}$$
with $\ell=(\beta\makebox[0in][l]{$\times$}\raisebox{-1pt}{$\Box$}\alpha,[\ ,\ ])$\ .
Then we can rewrite (18) as a short exact sequence
of {\em complexes}\ \ on ${\cal X}\times_S{\cal X}$:
\begin{equation}
0\longrightarrow\widetilde{\cal K}^{\textstyle\cdot}\longrightarrow
{\cal L}^{\textstyle\cdot}\longrightarrow
(\pi^{-1}\Theta_S)^{\makebox[0in][l]{$\scriptstyle\times$ 2}\longrightarrow 0
\end{equation}
The associated direct-image sequence yields
\begin{equation}
\longrightarrow{\bf R}^0f_*{\cal L}^{\textstyle\cdot}
\longrightarrow\Theta_S^{\otimes 2}
\stackrel{\widetilde{\kappa}_2}{\longrightarrow}
\widetilde{\bf T}^{(2)}_{{\cal X}/S}\longrightarrow\ \ .
\end{equation}
Tracing out the definition of a connecting
morphism (bearing in mind that if $\zeta\in\Theta_{\cal X}$ is
a local lifting of $\partial/\partial t_k$, and $\theta({\bf t})$ is
any element of $\Theta_{{\cal X}/S}$, then
$[\zeta,\theta({\bf t})]=
\frac{\partial\theta({\bf t})}{\partial t_k}$)
shows that it is indeed the same as $\widetilde{\kappa}_2$ given by
the explicit Definition in coordinates given above. \ $\Box$\\ \ \par
\ \\ \noindent {\bf Remark.\ \ } The explicit construction above shows how the data, up to
second order, of the (first) Kodaira-Spencer mapping
$$\kappa:\Theta_{S,0}/{\bf m}_{S,0}^2\Theta_{S,0}\longrightarrow
{\bf T}^1_{{\cal X}/S,0}/{\bf m}_{S,0}^2{\bf T}^1_{{\cal X}/S,0}$$
determines the second Kodaira-Spencer class
$$\widetilde{\kappa}_2:\Theta_S^{\otimes 2}|_0=
\Theta_{S,0}^{\otimes 2}/{\bf m}_{S,0}\Theta_{S,0}^{\otimes 2}
\longrightarrow
\widetilde{\bf T}^{(2)}_X$$
(see (12)). Conversely, if $\zeta,\xi\in\Theta_S|_0$, and
$\widetilde{\kappa}_2(\zeta\otimes\xi)$ is represented by a
cocycle $(\widehat{\zeta}\times\widehat{\xi},\theta)\in
\check{C}^1(\widetilde{\cal K}^{\textstyle\cdot}|_0)$, then we can choose
coordinates $\bf t$ on $S$ so that $\zeta=\partial/\partial t_k$,
$\xi=\partial/\partial t_{\ell}$, and the Kodaira-Spencer mapping
of the deformation in question is represented
in
${\bf T}^1_{{\cal X}/S,0}\otimes\Omega^1_{S,0}/
{\bf m}_{S,0}^2{\bf
T}^1_{{\cal X}/S,0}\otimes\Omega^1_{S,0}$
by the form
$$\sum_{\mu=1}^s(\theta_{\mu}^{(0)}+\sum_{\nu=1}^s
\theta_{\mu}^{(\nu)}t_k)dt_{\ell}
$$
with
$\theta_k^{(0)}=\widehat{\zeta}$,
$\theta_{\ell}^{(0)}=\widehat{\xi}$, and
$\theta_{\ell}^{(k)}=\theta$.
\section{Main results}
There is a natural composition map
\begin{equation}
\Theta_S^{\otimes 2}\hookrightarrow
\Theta_S\oplus\Theta_S^{\otimes 2}\,\longrightarrow\hspace{-12pt T_S^{(2)}\ .
\end{equation}
However, this map is not
${\cal O}_S$-linear. For example, $x\otimes y-y\otimes x$ is
mapped to $[x,y]$, whereas for any $f\in{\cal O}_S$
$$f.(x\otimes y-y\otimes x)=(f.x)\otimes y- y\otimes(f.x) $$
is sent to $f.[x,y]-y(f).x$ \ . Nevertheless, (21) induces an
${\cal O}_S$-linear map
$$\Theta_S^{\otimes 2}\,\longrightarrow\hspace{-12pt T_S^{(2)}/\Theta_S\ \
(\simeq Sym^2\Theta_S)\ .$$
The latter fits in a commutative square of ${\cal O}_S$-linear
maps obtained from (8),
\begin{equation}
\begin{array}{ccc}
\Theta_S^{\otimes 2} &
\stackrel{\overline{\widetilde{\kappa}}_2}{\longrightarrow} &
\widetilde{\bf T}^{(2)}_{{\cal X}/S}/
\, im\, (\kappa_1) \\
\downsurj & & \downsurj \\
T_S^{(2)}/\Theta_S &
\stackrel{\overline{\kappa}_2}{\longrightarrow} &
{\bf T}^{(2)}_{{\cal X}/S}/\,im\,(\kappa_1)
\end{array}
\end{equation}
\begin{thm}
The second differential of the Archimedean period map
$d^2\Psi$
factors through the
diagonal of (22),
$$\overline{\widetilde{\kappa}}_2:
\Theta_S^{\otimes 2}\longrightarrow
\widetilde{\bf T}_{{\cal X}/S}^{(2)}/\,im\,(\kappa_1)\ . $$
\end{thm}
\ \\ \noindent {\bf Proof.\ \ } Since the statement deals with ${\cal O}_S$-linear
maps, it is enough to prove it pointwise, for each $t\in S$. It
suffices to restrict to $0\in S$. We need to show that
$d^2\Psi(y)=0$ for any $y\in \Theta_S^{\otimes 2}|_0$ with
$\widetilde{\kappa}_2(y)\in im\,(\kappa_1)$. The condition on $y$
implies that
$\widetilde{\kappa}_2(y)\in \widetilde{\bf T}^{(2)}_X$
can be represented by a cocycle of the form
\begin{equation}
(0,\theta)\in \check{C}^2(\Theta_X^{\makebox[0in][l]{$\scriptstyle\times$ 2}[1])\oplus
\check{C}^1(\Theta_X)=\check{C}^1(\widetilde{\cal K}^{\textstyle\cdot}|_0)\ ,
\end{equation}
where $\theta$ is a {\em cocycle} \ in
$\check{C}^1(\Theta_X)$ representing $\kappa_1(\eta)$ for some
$\eta\in \Theta_S|_0$.
At this point we ``recall" two theorems from \cite{K}.
\begin{thm}[(5.4) in \cite{K}]
If \
$\kappa\in {\bf T}^1_{{\cal X}/S,0}\otimes\Omega_{S,0}^1/
{\bf m}_{S,0}^2{\bf T}^1_{{\cal X}/S,0}\otimes\Omega_{S,0}^1$
is represented by the form
$$\sum_{\mu=1}^s(\theta_{\mu}^{(0)}+\sum_{\nu=1}^s
\theta_{\mu}^{(\nu)}t_k)dt_{\ell}
$$
with
$\theta_k^{(0)}=\widehat{\zeta}$,
$\theta_{\ell}^{(0)}=\widehat{\xi}$, and
$\theta_{\ell}^{(k)}=\theta$,
then the second differential of the Archimedean period map
$$d^2\Psi(\frac{\partial}{\partial t_k}\otimes
\frac{\partial}{\partial t_{\ell}}):
H_{ar}\longrightarrow F_{ar}^{-2}/
H_{ar}+span\,\{\nabla_{\eta}|_0({\cal H}_{ar})\ |\
\eta\in\Theta_S\}$$
is induced by the map
\begin{eqnarray*}
\lefteqn{H_{ar}=H_{ar}^m\longrightarrow} \\
& & {\bf H}^m(\Omega^{\textstyle\cdot}_X\otimes B_{ar}/
F_{ar}^0(\Omega^{\textstyle\cdot}_X\otimes B_{ar})+
span\,\{ \check{\boldpounds}_{\eta}|_0{\cal
F}_{ar}^0\ |\ \eta\in\Theta_S\})
\end{eqnarray*}
given on the cochain level by
\begin{eqnarray}
\nonumber
\lefteqn{\omega_{i_1,\ldots,i_q}.T^p=\omega_Q.T^p\mapsto}\\
& &
\widehat{\zeta}_{i_{-1}i_0}\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\widehat{\xi}_{i_0i_1}\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,
\omega_Q.T^p-\widehat{\xi}_{i_0i_1}\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,
\check{\varpounds}_{\widehat{\zeta}_{i_0i_1}}
\omega_Q.T^{p+1}+
\theta_{i_0i_1}\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\omega_Q.T^p\ .
\end{eqnarray}
\end{thm}
\
\begin{thm}[(5.7) in \cite{K}]
$d^2\Psi$ on $\Theta_S^{\otimes 2}|_0$ is determined by
$$\kappa\in {\bf T}^1_{{\cal X}/S,0}\otimes\Omega_{S,0}^1/
{\bf m}_{S,0}^2{\bf T}^1_{{\cal X}/S,0}\otimes\Omega_{S,0}^1\
.$$
\end{thm}
Reading the two theorems in light of the Remark at the end of
Section~2, Theorem 4 shows that $d^2\Psi(y)$ is determined by
$\widetilde{\kappa}_2(y)=[(0,\theta)]\in \widetilde{\bf
T}^{(2)}_X$,
and Theorem 3 says that $d^2\Psi(y)$ is induced on the
cochain level by the contraction with the {\em cocycle} \
$\theta$ representing $\kappa_1(\eta)$. This contraction is
equivalent to $\nabla_{\eta}|_0$ modulo $H_{ar}$ (in fact, it is
none other than $d\Psi(\eta)$), and so we have proved that
$d^2\Psi(y)=0$. \ $\Box$\\ \ \par
\begin{thm}
The graded version of the second differential of the
Archimedean period map \ $\overline{d^2\Psi}$, as well as the
second differential of the usual period map \ $d^2\Phi$ and the
second fundamental form of the VHS, \ {\rm II}, \ all factor through
$$\kappa_1^2:\Theta_S^{\otimes 2}\longrightarrow
({\bf T}^1_{{\cal X}/S})^{\otimes 2}\ ,$$
and thus depend on \ $\kappa_1$ only.
\end{thm}
\ \\ \noindent {\bf Proof.\ \ } Again it suffices to restrict to $0\in S$.
Suppose
$$\widetilde{\kappa}_2(y)=[(\widehat{\zeta}\times
\widehat{\xi},\theta)]\in \widetilde{\bf T}^{(2)}_X$$
for some $y=\zeta\otimes\xi\in\Theta_S^{\otimes 2}|_0$.
Examining formula (24), we observe that the term involving
$\theta$ lies in $F_{ar}^{-1}$. Therefore,
$\overline{d^2\Psi}(y)$ depends only on
$\kappa_1^2(y):=\kappa_1(\zeta)\otimes\kappa_1(\xi)$.
This proves the Theorem for $\overline{d^2\Psi}$.
The statements for $d^2\Phi$ and II follow from this by Lemma 2
(b) and Proposition 1, respectively. \ $\Box$\\ \ \par
Finally, all the maps in question are symmetric, and so we may
pass from $\kappa_1^2$ to $Sym^2\kappa_1$ and from
$\overline{\widetilde{\kappa}}_2$ to $\overline{\kappa}_2$
(see (8)). Referring to the following symmetrized version of
(14),
\begin{equation}
\begin{array}{cccccccc}
0\rightarrow & \Theta_S & \rightarrow &
{\bf T}^{(2)}_S &
\rightarrow & Sym^2\,\Theta_S & \rightarrow & 0 \\
& \kappa_1 \downarrow & & \downarrow \kappa_2 &
& \downarrow Sym^2\kappa_1 & & \\
0\rightarrow & {\bf T}_{{\cal X}/S}^1 & \rightarrow &
{\bf T}_{{\cal X}/S}^{(2)} & \rightarrow &
Sym^2{\bf T}_{{\cal X}/S}^1 &
\stackrel{o}{\rightarrow} & {\bf T}_{{\cal X}/S}^2\ ,
\end{array}
\end{equation}
we conclude with
\begin{thm} $d^2\Psi$ factors through
$$\overline{\kappa}_2:Sym^2\Theta_S\longrightarrow
{\bf T}_{{\cal X}/S}^{(2)}/\,im\,(\kappa_1),$$
whereas \ $\overline{d^2\Psi}$, $d^2\Phi$ and \ ${\rm II}$ \ factor
through
$$Sym^2\kappa_1:Sym^2\Theta_S\longrightarrow
Sym^2{\bf T}_{{\cal X}/S}^1\ .$$
\end{thm}
\ \\ \noindent {\bf Remark.\ \ } When the deformation is versal, i.e.
$im\,(\kappa_1)$ is all of ${\bf T}_{{\cal X}/S}^1$, there is
no difference between $\overline{\kappa}_2$ and $Sym^2\kappa_1$.
\section{The higher-order cases}
The definition of the second differential of the period map in
Section 1 easily generalizes to higher-order cases (cf.
\cite{K}).
All three papers mentioned in the introduction define ``tangent
spaces to the moduli" ${\bf T}_{{\cal X}/S}^{(n)}$ of all
orders $n$. However, these definitions seem more complicated
than in the case $n=2$.
Still, we should have a diagram analogous to (25),
\begin{equation}
\begin{array}{cccccccc}
0\rightarrow & {\bf T}^{(n-1)}_S & \rightarrow &
{\bf T}^{(n)}_S &
\rightarrow & Sym^n\,\Theta_S & \rightarrow & 0 \\
& \kappa_{n-1} \downarrow & & \downarrow \kappa_n &
& \downarrow Sym^n\kappa_1 & & \\
0\rightarrow & {\bf T}_{{\cal X}/S}^{(n-1)} & \rightarrow &
{\bf T}_{{\cal X}/S}^{(n)} & \rightarrow &
Sym^n{\bf T}_{{\cal X}/S}^1 &
\stackrel{o_n}{\rightarrow} & {\bf T}_{{\cal X}/S}^2\ ,
\end{array}
\end{equation}
where $o_n$ is the $n^{th}$ obstruction map, and we expect that
the $n^{th}$ differential of the Archimedean period map
$d^n\Psi$ factors through the $n^{th}$ Kodaira-Spencer map
$\kappa_n$ modulo the image of $\kappa_{n-1}$, whereas the
$n^{th}$ differential of the usual period map $d^n\Phi$ and the
$n^{th}$ fundamental form of the VHS I$n$I factor through
$Sym^n\kappa_1$.
\section{Appendix: Archimedean cohomology}
In this section we summarize what we need about Archimedean
cohomology. For more information on this subject we refer to
\cite{Den}.
\begin{dfntn}
$B_{ar}={\bf C}[T,T^{-1}], \ \ {\bf L}={\bf C}[T^{-1}]$. \
$B_{ar}$ is filtered by the ${\bf L}$-submodules
$F^p=T^{-p}.{\bf L}$.
\end{dfntn}
Thus, if $X$ is a compact K\"{a}hler manifold,
$H^m(X)\otimes_{\bf C}B_{ar}$ receives the filtration
$F^{\textstyle\cdot}_{ar}$ obtained as the tensor product of the Hodge
filtration on $H^m(X,{\bf C})$ and the filtration $F^{\textstyle\cdot}$ on
$B_{ar}$. $F^{\textstyle\cdot}_{ar}$ is a decreasing filtration with
infinitely many levels, and
$$Gr_{F_{ar}}^k\cong\bigoplus_{p+q=m}H^{p,q}.T^{p-k}\ .$$
\begin{dfntn}
The {\em Archimedean cohomology}\ of $X$ is
$$H^m_{ar}(X):=F_{ar}^0(H^m(X,{\bf C})\otimes B_{ar})\ .$$
\end{dfntn}
Consider the complex of sheaves
$\Omega_X^{\textstyle\cdot}\otimes_{\bf C}B_{ar}$ with the differential
$${\bf d}(\omega.T^k):=d\omega.T^{k+1}\ .$$
This complex is also filtered by the tensor product of the
stupid filtration on $\Omega^{\textstyle\cdot}_X$ and $F^{\textstyle\cdot}$ on $B_{ar}$,
and we have
\begin{eqnarray*}
\lefteqn{H^m_{ar}(X)=F^0{\bf H}^m(X,\Omega^{\textstyle\cdot}_X\otimes
B_{ar}) \cong} \\
& & {\bf H}^m(X,F^0(\Omega^{\textstyle\cdot}_X\otimes B_{ar}))
\cong
{\bf H}^m(X,\Omega^{\textstyle\cdot}_X)\otimes{\bf L}
\cong H^m(X,{\bf C})\otimes{\bf L}\ .
\end{eqnarray*}
Note that ${\bf H}^m(X,\Omega^{\textstyle\cdot}_X\otimes B_{ar})$ is a
complex infinite-dimensional Hodge structure (of weight $m$),
and $(\Omega^{\textstyle\cdot}_X\otimes B_{ar},{\bf d})$ is a Hodge complex.
Hence
$$Gr_{F_{ar}}^k{\bf H}^m(X,\Omega^{\textstyle\cdot}_X\otimes B_{ar})\cong
{\bf H}^m(X,Gr_{F_{ar}}^k(\Omega^{\textstyle\cdot}_X\otimes B_{ar}))\ .$$
We will write boldface $\check{\bf D}$ for the differential in
the \v{C}ech cochain complex computing
${\bf H}^m(X,\Omega^{\textstyle\cdot}_X\otimes B_{ar})$, and
$$\check{\boldpounds}_v:=\check{\bf D}v\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\, +
v\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\check{\bf D}$$ for the corresponding Lie derivative
with respect to a vector field $v$ on $X$.
These constructions extend without any difficulty to the
relative situation. In particular, given a flat family
$\pi:{\cal X}\longrightarrow S$ of compact K\"{a}hler
manifolds, the bundle
$${\cal H}\otimes B_{ar}={\bf R}^m\pi_*(\Omega^{\textstyle\cdot}_{{\cal
X}/S}\otimes B_{ar})$$
is filtered by ${\cal F}^{\textstyle\cdot}_{ar}$, and the Gauss-Manin
connection $\nabla$ extends to
$$\nabla_{ar}:{\cal H}\otimes B_{ar}\longrightarrow
{\cal H}\otimes B_{ar}\otimes\Omega_S^1\ ,$$
with the usual Griffiths' transversality property
$$\nabla_{ar}({\cal F}^p_{ar})\subset{\cal F}^{p-1}_{ar}\otimes
\Omega_S^1\ .$$
Specifically, if $x$ is a section of $\cal H$, then
$$\nabla_{ar}(x.T^p)=\nabla x.T^p\ .$$
The real difference arises when one examines the definition of
$\nabla_{ar}$ on the cochain level, due to the fact that $\bf
d$ increases the exponent at $T$.
|
1994-05-16T19:17:14 | 9405 | alg-geom/9405008 | en | https://arxiv.org/abs/alg-geom/9405008 | [
"alg-geom",
"math.AG"
] | alg-geom/9405008 | Klaus Altmann | Klaus Altmann | Infinitesimal Deformations and Obstructions for Toric Singularities | 26 pages, LaTeX (uses diagram.sty) | null | null | null | null | The obstruction space T^2 and the cup product T^1 x T^1 -> T^2 are computed
for toric singularities.
| [
{
"version": "v1",
"created": "Mon, 16 May 1994 17:16:49 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Altmann",
"Klaus",
""
]
] | alg-geom | \section{#1}
\protect\setcounter{secnum}{\value{section}}
\protect\setcounter{equation}{0}
\protect\renewcommand{\theequation}{\mbox{\arabic{secnum}.\arabic{equation}}}}
\setcounter{tocdepth}{1}
\begin{document}
\title{Infinitesimal Deformations and Obstructions for Toric Singularities}
\author{Klaus Altmann\footnotemark[1]\\
\small Dept.~of Mathematics, M.I.T., Cambridge, MA 02139, U.S.A.
\vspace{-0.7ex}\\ \small E-mail: [email protected]}
\footnotetext[1]{Die Arbeit wurde mit einem Stipendium des DAAD unterst\"utzt.}
\date{}
\maketitle
\begin{abstract}
The obstruction space $T^2$ and the cup product
$T^1\times T^1\to T^2$ are computed for toric singularities.
\end{abstract}
\tableofcontents
\sect{Introduction}\label{s1}
\neu{11}
For an affine scheme $\,Y= \mbox{Spec}\; A$, there are two important $A$-modules,
$T^1_Y$ and $T^2_Y$, carrying information about its deformation theory:
$T^1_Y$ describes the infinitesimal deformations, and $T^2_Y$ contains the
obstructions for extending deformations of $Y$ to larger base spaces.\\
\par
In case $Y$ admits a versal deformation, $T^1_Y$ is the tangent space of the
versal base space $S$. Moreover, if $J$ denotes the ideal defining $S$ as a
closed
subscheme of the affine space $T^1_Y$, the module
$\left( ^{\displaystyle J}\! / \! _{\displaystyle m_{T^1} \,J} \right) ^\ast$
can be canonically embedded into $T^2_Y$, i.e. $(T_Y^2)^\ast$-elements induce
the
equations defining $S$ in $T^1_Y$.\\
\par
The vector spaces $T^i_Y$ come with a cup product
$T_Y^1 \times T^1_Y \rightarrow T^2_Y$.
The associated quadratic form $T^1_Y \rightarrow T^2_Y$ describes the
quadratic part of the elements of $J$, i.e. it can be used to get a better
approximation of the versal base space $S$ as regarding its tangent space
only.\\
\par
\neu{12}
In \cite{T1} we have determined the vector space $T^1_Y$ for affine toric
varieties.
The present paper can be regarded as its continuation - we will compute $T^2_Y$
and
the cup product. \\
These modules $T^i_Y$ are canonically graded (induced from the character group
of
the torus). We will describe their homogeneous pieces as cohomology groups of
certain complexes, that are directly induced from the combinatorial structure
of the
rational, polyhedral cone defining our variety $Y$. The results
can be found in \S \ref{s3}.\\
\par
Switching to another, quasiisomorphic complex provides a second formula for the
vector spaces $T^i_Y$ (cf. \S \ref{s6}). We will use this particular version
for
describing these spaces and the cup product in the special case of
three-dimensional toric Gorenstein singularities (cf. \S \ref{s7}).\\
\par
\sect{$T^1$, $T^2$, and the cup product (in general)}\label{s2}
In this section we will give a brief reminder to the well known
definitions of these objects. Moreover, we will use this opportunity to fix
some
notations.\\
\par
\neu{21}
Let $Y \subseteq \,I\!\!\!\!C^{w+1}$ be given by equations $f_1,\dots,f_m$, i.e.
its ring of regular functions equals
\[
A=\;^{\displaystyle P}\!\! / \! _{\displaystyle I} \quad
\mbox{ with }
\begin{array}[t]{l}
P = \,I\!\!\!\!C[z_0,\dots, z_w]\\
I = (f_1,\dots,f_m)\, .
\end{array}
\]
Then, using $d:^{\displaystyle I}\! / \! _{\displaystyle I^2}
\rightarrow A^{w+1}\;$ ($d(f_i):= (\frac{\partial f_i}{\partial z_0},\dots
\frac{\partial f_i}{\partial z_w})$),
the vector space $T^1_Y$ equals
\[
T^1_Y = \;^{\displaystyle \mbox{Hom}_A(^{\displaystyle I}\!\! / \!
_{\displaystyle I^2}, A)} \! \left/ \!
_{\displaystyle \mbox{Hom}_A(A^{w+1},A)} \right.\; .
\vspace{1ex}
\]
\par
\neu{22}
Let ${\cal R}\subseteq P^m$ denote the $P$-module of relations between the equations
$f_1,\dots,f_m$. It contains the so-called Koszul relations
${\cal R}_0:= \langle f_i\,e^j - f_j \,e^i \rangle$ as a submodule.\\
Now, $T^2_Y$ can be obtained as
\[
T^2_Y = \;^{\displaystyle \mbox{Hom}_P(^{\displaystyle {\cal R}}\! / \!
_{\displaystyle {\cal R}_0}, A)} \! \left/ \!
_{\displaystyle \mbox{Hom}_P(P^m,A)} \right.\; .
\vspace{1ex}
\]
\par
\neu{23}
Finally, the cup product $T^1\times T^1 \rightarrow T^2$ can be defined in the
following way:
\begin{itemize}
\item[(i)]
Starting with an $\varphi\in \mbox{Hom}_A(^{\displaystyle I}\! / \!
_{\displaystyle I^2}, A)$, we lift the images of the $f_i$ obtaining
elements $\tilde{\varphi}(f_i)\in P$.
\item[(ii)]
Given a relation $r\in {\cal R}$, the linear combination
$\sum_ir_i\,\tilde{\varphi}(f_i)$ vanishes in $A$, i.e. it is contained in the
ideal $I\subseteq P$. Denote by $\lambda(\varphi)\in P^m$ any set of
coefficients such that
\[
\sum_i r_i \, \tilde{\varphi}(f_i) + \sum_i \lambda_i(\varphi)\, f_i =0\quad
\mbox{ in } P.
\]
(Of course, $\lambda$ depends on $r$ also.)
\item[(iii)]
If $\varphi, \psi \in \mbox{Hom}_A(^{\displaystyle I}\! / \!
_{\displaystyle I^2}, A)$ represent two elements of $T^1_Y$, then we define for
each relation $r\in {\cal R}$
\[
(\varphi \cup \psi)(r) := \sum_i \lambda_i (\varphi)\, \psi(f_i) +
\sum_i \varphi(f_i)\, \lambda_i(\psi)\; \in A\, .
\vspace{1ex}
\]
\end{itemize}
{\bf Remark:}
The definition of the cup product does not depend on the choices we made:
\begin{itemize}
\item[(a)]
Choosing a $\lambda'(\varphi)$ instead of $\lambda(\varphi)$ yields
$\lambda'(\varphi) - \lambda(\varphi) \in {\cal R}$, i.e. in $A$ we obtain the same
result.
\item[(b)]
Let $\tilde{\varphi}'(f_i)$ be different liftings to $P$. Then, the difference
$\tilde{\varphi}'(f_i) - \tilde{\varphi}(f_i)$ is contained in $I$, i.e. it can
be written as some linear combination
\[
\tilde{\varphi}'(f_i) - \tilde{\varphi}(f_i) = \sum_j t_{ij}\, f_j\, .
\]
Hence,
\[
\sum_i r_i \,\tilde{\varphi}'(f_i) = \sum_i r_i \,\tilde{\varphi}(f_i) +
\sum_{i,j} t_{ij}\, r_i\, f_j\,,
\]
and we can define $\lambda'_j(\varphi):= \lambda_j(\varphi) -
\sum_it_{ij}\,r_i$
(corresponding to $\tilde{\varphi}'$ instead of $\tilde{\varphi}$). Then, we
obtain
for the cup product
\[
(\varphi\cup\psi)'(r) - (\varphi\cup\psi)(r) = -\sum_ir_i\cdot
\left( \sum_j t_{ij}\, \psi(f_j)\right)\, ,
\]
but this expression comes from some map $P^m\rightarrow A$.
\vspace{3ex}
\end{itemize}
\sect{$T^1$, $T^2$, and the cup product (for toric varieties)}\label{s3}
\neu{31}
We start with fixing the usual notations when dealing with affine toric
varieties (cf. \cite{Ke} or
\cite{Oda}):
\begin{itemize}
\item
Let $M,N$ be mutually dual free Abelian groups, we denote by $M_{I\!\!R}, N_{I\!\!R}$
the associated real
vector spaces obtained by base change with $I\!\!R$.
\item
Let $\sigma=\langle a^1,\dots,a^N\rangle \subseteq N_{I\!\!R}$ be a rational,
polyhedral
cone with apex - given by its fundamental generators. \\
$\sigma^{\scriptscriptstyle\vee}:= \{ r\in M_{I\!\!R}\,|\; \langle \sigma,\,r\rangle \geq 0\}
\subseteq M_{I\!\!R}$
is called the dual cone. It induces a partial order on the lattice $M$ via
$[\,a\geq b$ iff
$a-b \in \sigma^{\scriptscriptstyle\vee}\,]$.
\item
$A:= \,I\!\!\!\!C[\sigma^{\scriptscriptstyle\vee}\cap M]$ denotes the semigroup algebra. It is the ring of
regular
functions of the toric variety $Y= \mbox{Spec}\; A$ associated to $\sigma$.
\item
Denote by $E\subset \sigma^{\scriptscriptstyle\vee}\cap M$ the minimal set of generators of this
semigroup
("the Hilbert basis"). $E$ equals the set of all primitive (i.e.
non-splittable) elements
of $\sigma^{\scriptscriptstyle\vee}\cap M$.
In particular, there is a surjection of semigroups $\pi:I\!\!N^E \longrightarrow\hspace{-1.5em}\longrightarrow
\sigma^{\scriptscriptstyle\vee}\cap M$, and
this fact translates into a closed embedding $Y\hookrightarrow \,I\!\!\!\!C^E$.\\
To make the notations
coherent with \S \ref{s2}, assume that $E=\{r^0,\dots,r^w\}$ consists of $w+1$
elements.
\vspace{2ex}
\end{itemize}
\neu{32}
To a fixed degree $R\in M$ we associate ``thick facets'' $K_i^R$ of the dual
cone
\[
K_i^R := \{r\in \sigma^{\scriptscriptstyle\vee}\cap M \, | \; \langle a^i, r \rangle <
\langle a^i, R \rangle \}\quad (i=1,\dots,N) .
\vspace{2ex}
\]
\par
{\bf Lemma:}{\em
\begin{itemize}
\item[(1)]
$\cup_i K_i^R = (\sigma^{\scriptscriptstyle\vee}\cap M) \setminus (R+ \sigma^{\scriptscriptstyle\vee})$.
\item[(2)]
For each $r,s\in K_i^R$ there exists an $\ell\in K_i^R$ such that $\ell\geq
r,s$.
Moreover, if $Y$ is smooth in codimension 2, the intersections $K^R_i\cap
K^R_j$
(for 2-faces $\langle a^i,a^j\rangle <\sigma$) have the same property.
\vspace{1ex}
\end{itemize}
}
\par
{\bf Proof:}
Part (i) is trivial; for (ii) cf. (3.7) of \cite{T1}.
\hfill$\Box$\\
\par
Intersecting these sets with $E\subseteq \sigma^{\scriptscriptstyle\vee}\cap M$, we obtain the
basic objects for describing the modules $T^i_Y$:
\begin{eqnarray*}
E_i^R &:=& K_i^R \cap E = \{r\in E\, | \; \langle a^i,r \rangle <
\langle a^i, R \rangle \}\, ,\\
E_0^R &:=& \bigcup_{i=1}^N E_i^R\; ,\mbox{ and}\\
E^R_{\tau} &:=& \bigcap_{a^i\in \tau} E^R_i \; \mbox{ for faces }
\tau < \sigma\,.
\end{eqnarray*}
We obtain a complex
$L(E^R)_{\bullet}$ of free Abelian groups via
\[
L(E^R)_{-k} := \bigoplus_{\begin{array}{c}
\tau<\sigma\\ \mbox{dim}\, \tau=k \end{array}} \!\!L(E^R_{\tau})
\]
with the usual differentials.
($L(\dots)$ denotes the free Abelian group of integral, linear dependencies.)
\\
The most interesting part ($k\leq 2$) can be written explicitely as
\[
L(E^R)_{\bullet}:\quad \cdots
\rightarrow
\oplus_{\langle a^i,a^j\rangle<\sigma} L(E^R_i\cap E^R_j)
\longrightarrow
\oplus_i L(E_i^R) \longrightarrow L(E_0^R) \rightarrow 0\,.
\vspace{1ex}
\]
\par
\neu{33}
{\bf Theorem:}
{\em
\begin{itemize}
\item[(1)]
$T^1_Y(-R) = H^0 \left( L(E^R)_{\bullet}^\ast \otimes_{Z\!\!\!Z}\,I\!\!\!\!C\right)$
\item[(2)]
$T^2_Y(-R) \supseteq H^1 \left( L(E^R)_{\bullet}^\ast \otimes_{Z\!\!\!Z}\,I\!\!\!\!C\right)$
\item[(3)]
Moreover, if $Y$ is smooth in codimension 2
(i.e.\ if the 2-faces $\langle a^i, a^j \rangle < \sigma$ are spanned
by a part of a $Z\!\!\!Z$-basis of the lattice $N$), then
\[
T^2_Y(-R) = H^1 \left( L(E^R)_{\bullet}^\ast \otimes_{Z\!\!\!Z}\,I\!\!\!\!C\right)\, .
\]
\item[(4)]
Module structure: If $x^s\in A$ (i.e. $s\in \sigma^{\scriptscriptstyle\vee}\cap M$), then
$E_i^{R-s}\subseteq R_i^R$, hence $L(E^R)_{\bullet}^\ast \subseteq
L(E^{R-s})^\ast_{\bullet}$. The induced map in cohomology corresponds to the
multiplication with $x^s$ in the $A$-modules $T^1_Y$ and $T^2_Y$.
\vspace{2ex}
\end{itemize}
}
\par
The first part was shown in \cite{T1}; the formula for $T^2$ will be proved
in \S \ref{s4}. Then, the claim concerning the module structure will become
clear
automatically.\\
\par
{\bf Remark:}
The assumption made in (3) can not be dropped: \\
Taking for $Y$ a 2-dimensional cyclic quotient
singularity given by some 2-dimensional cone $\sigma$, there are only two
different sets
$E_1^R$ and $E_2^R$ (for each $R\in M$). In particular,
$H^1 \left( L(E^R)_{\bullet}^\ast \otimes_{Z\!\!\!Z}\,I\!\!\!\!C\right)=0$.\\
\par
\neu{34}
We want to describe the isomorphisms connecting the general $T^i$-formulas of
\zitat{2}{1} and \zitat{2}{2} with the toric ones given in \zitat{3}{3}.\\
\par
$Y\hookrightarrow\,I\!\!\!\!C^{w+1}$ is given by the equations
\[
f_{ab}:= \underline{z}^a-\underline{z}^b\quad (a,b\in I\!\!N^{w+1} \mbox{ with } \pi(a)=\pi(b)
\mbox{ in } \sigma^{\scriptscriptstyle\vee}
\cap M),
\]
and it is easier to deal with this infinite set of equations
(which generates the ideal $I$ as a $\,I\!\!\!\!C$-vector
space) instead of selecting a finite number of them in some non-canonical way.
In particular, for
$m$ of \zitat{2}{1} and \zitat{2}{2} we take
\[
m:= \{ (a,b)\in I\!\!N^{w+1}\timesI\!\!N^{w+1}\,|\;\pi(a)=\pi(b)\}\,.
\]
The general $T^i$-formulas mentioned in \zitat{2}{1} and \zitat{2}{2} remain
true.\\
\par
{\bf Theorem:}
{\em
For a fixed element $R\in M$ let $\varphi: L(E)_{\,I\!\!\!\!C}\rightarrow \,I\!\!\!\!C$ induce some
element of
\[
\left(\left. ^{\displaystyle L(E_0^R)}\!\!\right/
\!_{\displaystyle \sum_i L(E_i^R)} \right)^\ast \otimes_{Z\!\!\!Z} \,I\!\!\!\!C
\cong T^1_Y(-R)\quad \mbox{(cf. Theorem \zitat{3}{3}(1)).}
\]
Then, the $A$-linear map
\begin{eqnarray*}
^{\displaystyle I}\!\!/\!_{\displaystyle I^2} &\longrightarrow& A\\
\underline{z}^a-\underline{z}^b & \mapsto & \varphi(a-b)\cdot x^{\pi(a)-R}
\end{eqnarray*}
provides the same element via the formula \zitat{2}{1}.
}\\
\par
Again, this Theorem follows from the paper \cite{T1} - accompanied by the
commutative diagram
of \zitat{4}{3} in the present paper. (Cf. Remark \zitat{4}{4}.)\\
\par
{\bf Remark:}
A simple, but nevertheless important check shows that the map
$(\underline{z}^a-\underline{z}^b) \mapsto
\varphi(a-b)\cdot x^{\pi(a)-R}$ goes into $A$, indeed:\\
Assume $\pi(a)-R \notin \sigma^{\scriptscriptstyle\vee}$. Then, there exists an index $i$ such
that
$\langle a^i, \pi(a)-R \rangle <0$.
Denoting by "supp $q$" (for a $q\in I\!\!R^E$) the set of those $r\in E$ providing
a non-vanishing entry
$q_r$, we obtain
\[
\mbox{supp}\,(a-b) \subseteq \mbox{supp}\,a \cup \mbox{supp}\, b \subseteq
E^R_i\, ,
\]
i.e. $\varphi(a-b)=0$.\\
\par
\neu{35}
The $P$-module ${\cal R}\subseteq P^m$ is generated by relations of two different
types:
\begin{eqnarray*}
r(a,b;c) &:=& e^{a+c,\,b+c}- \underline{z}^c\, e^{a,b}\quad
(a,b,c\in I\!\!N^{w+1};\, \pi(a)=\pi(b))\quad \mbox{ and}\\
s(a,b,c) &:=& e^{b,c} - e^{a,c} + e^{a,b}\quad
(a,b,c\in I\!\!N^{w+1};\, \pi(a)=\pi(b)=\pi(c))\,.\\
&&\qquad\qquad(e^{\bullet,\bullet} \mbox{ denote the standard basis vectors of
} P^m.)
\vspace{1ex}
\end{eqnarray*}
\par
{\bf Theorem:}
{\em
For a fixed element $R\in M$ let $\psi_i: L(E_i^R)_{\,I\!\!\!\!C}\rightarrow \,I\!\!\!\!C$ induce
some
element of
\[
\left( \frac{\displaystyle
\mbox{Ker}\,\left( \oplus_i L(E_i^R) \longrightarrow
L(E'^R)\right)}{\displaystyle
\mbox{Im}\, \left( \oplus_{\langle a^i,a^j\rangle<\sigma} L(E_i^R\cap E_j^R)
\rightarrow \oplus_i L(E_i^R)\right)} \right)^\ast
\otimes_{Z\!\!\!Z}\,I\!\!\!\!C \subseteq T^2_Y(-R)
\quad \mbox{(cf. \zitat{3}{3}(2)).}
\]
Then, the $P$-linear map
\begin{eqnarray*}
^{\displaystyle {\cal R}}\!\!/\!_{\displaystyle {\cal R}_0} &\longrightarrow & A\\
r(a,b;c) & \mapsto &
\left\{ \begin{array}{ll}
\psi_i(a-b)\, x^{\pi(a+c)-R} & \mbox{for } \pi(a)\in K_i^R;\; \pi(a+c)\geq R\\
0 & \mbox{for }\pi(a)\geq R \mbox{ or } \pi(a+c)\in\bigcup_i K_i^R
\end{array}\right.\\
s(a,b,c) &\mapsto & 0
\end{eqnarray*}
is correct defined, and via the formula \zitat{2}{2} it induces the same
element of
$T^2_Y$.
}
\vspace{2ex}
\\
\par
For the prove we refer to \S \ref{s4}. Nevertheless, we check the {\em
correctness of
the definition} of the $P$-linear map
$^{\displaystyle {\cal R}}\!/\!_{\displaystyle {\cal R}_0} \rightarrow A$ instantly:
\begin{itemize}
\item[(i)]
If $\pi(a)$ is contained in two different sets $K_i^R$ and $K_j^R$, then the
two
fundamental generators $a^i$ and $a^j$ can be connected by a sequence
$a^{i_0},\dots,a^{i_p}$, such that
\begin{itemize}
\item[$\bullet$]
$a^{i_0}=a^i,\, a^{i_p}=a^j,$
\item[$\bullet$]
$a^{i_{v-1}}$ and $a^{i_v}$ are the edges of some 2-face of $\sigma$
($v=1,\dots,p$),
and
\item[$\bullet$]
$\pi(a)\in K^R_{i_v}$ for $v=0,\dots,p$.
\end{itemize}
Hence, $\mbox{supp}\,(a-b)\subseteq E^R_{i_{v-1}}\cap E^R_{i_v}$
($v=1,\dots,p$),
and we obtain
\[
\psi_i(a-b)=\psi_{i_1}(a-b)=\dots=\psi_{i_{p-1}}(a-b)=\psi_j(a-b)\,.
\]
\item[(ii)]
There are three types of $P$-linear relations between the generators $r(\dots)$
and
$s(\dots)$ of ${\cal R}$:
\begin{eqnarray*}
0 &=& \underline{z}^d\,r(a,b;c) -r(a,b;c+d) + r(a+c,b+c;d)\,,\\
0 &=& r(b,c;d) - r(a,c;d) + r(a,b;d) - s(a+d,b+d,c+d) + \underline{z}^d\,
s(a,b,c)\,,\\
0 &=& s(b,c,d) - s(a,c,d) + s(a,b,d) - s(a,b,c)\,.
\end{eqnarray*}
Our map respects them all.
\item[(iii)]
Finally, the typical element
$(\underline{z}^a-\underline{z}^b)e^{cd} - (\underline{z}^c-\underline{z}^d)e^{ab} \in {\cal R}_0$
equals
\[
-r(c,d;a)+r(c,d;b)+r(a,b;c)-r(a,b;d) - s(a+c,b+c,a+d) - s(a+d,b+c,b+d)\,.
\]
It will be sent to 0, too.
\vspace{2ex}
\end{itemize}
\par
\neu{36}
The cup product $T^1_Y\times T^1_Y\rightarrow T^2_Y$ respects the grading, i.e.
it splits
into pieces
\[
T^1_Y(-R)\times T^1_Y(-S) \longrightarrow T^2_Y(-R-S)\quad (R,S\in M)\,.
\]
To describe these maps in our combinatorial language, we choose some
set-theoretical
section $\Phi:M\rightarrowZ\!\!\!Z^{w+1}$ of the $Z\!\!\!Z$-linear map
\begin{eqnarray*}
\pi: Z\!\!\!Z^{w+1} &\longrightarrow& M\\
a&\mapsto&\sum_v a_v\,r^v
\end{eqnarray*}
with the additional property $\Phi(\sigma^{\scriptscriptstyle\vee}\cap M)\subseteq I\!\!N^{w+1}$.\\
\par
Let $q\in L(E)\subseteqZ\!\!\!Z^{w+1}$ be an integral relation between the generators
of
the semigroup $\sigma^{\scriptscriptstyle\vee}\cap M$. We introduce the following notations:
\begin{itemize}
\item
$q^+,q^-\inI\!\!N^{w+1}$ denote the positive and the negative part of $q$,
respectively.
(With other words: $q=q^+-q^-$ and $\sum_v q^-_v\,q^+_v=0$.)
\item
$\bar{q}:=\pi(q^+)=\sum_v q_v^+\,r^v = \pi(q^-)=\sum_v q_v^-\,r^v \in M$.
\item
If $\varphi,\psi: L(E)\rightarrowZ\!\!\!Z$ are linear maps and $R,S\in M$, then we
define
\[
t_{\varphi,\psi,R,S}(q):=
\varphi(q)\cdot \psi \left( \Phi(\bar{q}-R)+\Phi(R)-q^-\right) +
\psi(q)\cdot \varphi\left( \Phi(\bar{q}-S)+\Phi(S)-q^+\right)\,.
\vspace{2ex}
\]
\end{itemize}
\par
{\bf Theorem:}
{\em
Assume that $Y$ is smooth in codimension 2.\\
Let $R,S\in M$, and let $\varphi,\psi: L(E)_{\,I\!\!\!\!C}\rightarrow\,I\!\!\!\!C$ be linear maps
vanishing on $\sum_i L(E_i^R)_{\,I\!\!\!\!C}$ and $\sum_i L(E_i^S)_{\,I\!\!\!\!C}$, respectively.
In particular,
they define elements $\varphi\in T^1_Y(-R),\,\psi\in T^1_Y(-S)$ (which involves
a slight
abuse of notations).\\
Then, the cup product $\varphi\cup\psi\in T^2_Y(-R-S)$ is given (via
\zitat{3}{3}(3))
by the linear maps $(\varphi\cup\psi)_i: L(E_i^{R+S})_{\,I\!\!\!\!C}\rightarrow\,I\!\!\!\!C$
defined as follows:
\begin{itemize}
\item[(i)]
If $q\in L(E_i^{R+S})$ (i.e. $\langle a^i,\mbox{supp}\,q\rangle < \langle
a^i,R+S\rangle$) is an integral relation,
then there exists a decomposition $q=\sum_k q^k$ such that
\begin{itemize}
\item
$q^k\in L(E_i^{R+S})$, and moreover
\item
$\langle a^i, \bar{q}^k\rangle < \langle a^i,R+S\rangle$.
\end{itemize}
\item[(ii)]
$(\varphi\cup\psi)_i\left( q\in L(E_i^{R+S})\right):= \sum_k
t_{\varphi,\psi,R,S}(q^k)$.
\vspace{2ex}
\end{itemize}
}
\par
It is even not obvious that the map $q\mapsto \sum_k t(q^k)$
\begin{itemize}
\item
does not depend on the representation of $q$ as a particular sum of $q_k$'s
(which
would instantly imply linearity on $L(E_i^{R+S})$), and
\item
yields the same result on $L(E_i^{R+S}\cap E_j^{R+S})$ for $i,j$ corresponding
to edges
$a^i, a^j$ of some 2-face of $\sigma$.
\end{itemize}
The proof of these facts (cf.\ \zitat{5}{4})and of the entire theorem is
contained
in \S \ref{s5}.\\
\par
{\bf Remark 1:}
Replacing all the terms $\Phi(\bullet)$ in the $t$'s of the previous formula
for
$(\varphi\cup\psi)_i(q)$ by arbitrary liftings from $M$ to $Z\!\!\!Z^{w+1}$,
the result in $T^2_Y(-R-S)$ will be unchanged as long as we obey the following
two
rules:
\begin{itemize}
\item[(i)]
Use always (for all $q$, $q^k$, and $i$)
the {\em same liftings} of $R$ and $S$ to $Z\!\!\!Z^{w+1}$ (at the places of
$\Phi(R)$ and $\Phi(S)$, respectively).
\item[(ii)]
Elements of $\sigma^{\scriptscriptstyle\vee}\cap M$ always have to be lifted to $I\!\!N^{w+1}$.
\vspace{2ex}
\end{itemize}
{\bf Proof:}
Replacing $\Phi(R)$ by $\Phi(R)+d$ ($d\in L(E)$) at each occurence changes all
maps $(\varphi\cup\psi)_i$ by the summand $\psi(d)\cdot\varphi(\bullet)$.
However,
this additional linear map comes from $L(E)^\ast$, hence it is trivial on
$\mbox{Ker}\left(\oplus_iL(E_i^{R+S})\rightarrow L(E_0^{R+S})\subseteq
L(E)\right)$.\\
\par
Let us look at the terms $\Phi(\bar{q}-R)$ in $t(q)$ now:
Unless $\bar{q}\geq R$, the factor $\varphi(q)$ vanishes (cf. Remark
\zitat{3}{4}). On
the other hand, the expression $t(q)$ is never used for those relations $q$
satisfying
$\bar{q}\geq R+S$ (cf. conditions for the $q^k$'s). Hence, we may assume that
\[
(\bar{q}-R)\geq 0\; \mbox{ and, moreover, } (\bar{q}-R)\in \bigcup_i K_i^S\,.
\]
Now, each two liftings of $(\bar{q}-R)$ to $I\!\!N^{w+1}$ differ by an element of
$\mbox{Ker}\,\psi$ only (apply the method of Remark \zitat{3}{4} again), in
particular,
they cause the same result for $t(q)$.
\hfill$\Box$\\
\par
{\bf Remark 2:}
In the special case of $R\geq S\kgeq0$ we can choose liftings $\Phi(R)\geq
\Phi(S)
\geq 0$ in $I\!\!N^{w+1}$. Then,
there exists an easier description for $t(q)$:
\begin{itemize}
\item[(i)]
Unless $\bar{q}\geq R$, we have $t(q)=0$.
\item[(ii)]
In case of $\bar{q}\geq R$ we may assume that $q^+\geq\Phi(R)$ is true
in $I\!\!N^{w+1}$.
(The general $q$'s are differences of those ones.) Then, $t$ can be computed as
the
product $t(q)=\varphi(q)\,\psi(q)$.
\vspace{2ex}
\end{itemize}
\par
{\bf Proof:}
(i) As used many times, the property $\bar{q}\in\bigcup_iE_i^R$ implies
$\varphi(q)=0$.
Now, we can distinguish between two cases:\\
{\em Case 1: $\bar{q}\in\bigcup_iE_i^S$.} We obtain $\psi(q)=0$, in particular,
both
summands of $t(q)$ vanish.\\
{\em Case 2: $\bar{q}\geq S$.} Then, $\bar{q}-S,\,S\in \sigma^{\scriptscriptstyle\vee}\cap M$,
and $\Phi$ lifts
these elements to $I\!\!N^{w+1}$. Now, the condition $\bar{q}\in\bigcup_iE_i^R$
implies
that $\varphi\left( \Phi(\bar{q}-S)+\Phi(S)-q^+\right)=0$.\\
\par
(ii)
We can choose $\Phi(\bar{q}-R):=q^+-\Phi(R)$ and
$\Phi(\bar{q}-S):=q^+-\Phi(S)$. Then,
the claim follows straight forward.
\hfill$\Box$\\
\par
\sect{Proof of the $T^2$-formula}\label{s4}
\neu{41}
We will use the sheaf $\Omega^1_Y=\Omega^1_{A|\,I\!\!\!\!C}$ of K\"ahler
differentials for computing the modules $T^i_Y$. The maps
\[
\alpha_i: \mbox{Ext}^i_A\left(
\;^{\displaystyle\Omega_Y^1}\!\!\left/\!_{\displaystyle
\mbox{tors}\,(\Omega_Y^1)}\right. ,
\, A \right)
\hookrightarrow
\mbox{Ext}^i_A\left( \Omega^1_Y,\,A\right) \cong T^i_Y\quad
(i=1,2)
\]
are injective. Moreover, they are isomorphisms for
\begin{itemize}
\item
$i=1$, since $Y$ is normal, and for
\item
$i=2$, if $Y$ is smooth in codimension 2.
\vspace{2ex}
\end{itemize}
\par
\neu{42}
As in \cite{T1}, we build a special $A$-free resolution (one step further now)
\[
{\cal E}\stackrel{d_E}{\longrightarrow}{\cal D}\stackrel{d_D}{\longrightarrow}
{\cal C}\stackrel{d_C}{\longrightarrow}{\cal B} \stackrel{d_B}{\longrightarrow}
\;^{\displaystyle\Omega_Y^1}\!\!\left/\!_{\displaystyle
\mbox{tors}\,(\Omega_Y^1)}\right.
\rightarrow 0\,.
\vspace{2ex}
\]
With $L^2(E):=L(L(E))$, $L^3(E):=L(L^2(E))$, and
\[
\mbox{supp}^2\xi:= \bigcup_{q\in supp\,\xi} \mbox{supp}\,q\quad (\xi\in
L^2(E)),\quad
\mbox{supp}^3\omega:= \bigcup_{\xi\in supp\,\omega}\mbox{supp}^2\xi\quad
(\omega\in L^3(E)),
\]
the $A$-modules involved in this resolution are defined as follows:
\[
\begin{array}{rcl}
{\cal B}&:=&\oplus_{r\in E} \,A\cdot B(r),\qquad
{\cal C}\,:=\,\oplus_{\!\!\!\!\!\begin{array}[b]{c}\scriptstyle q\in L(E) \vspace{-1ex}\\
\scriptstyle\ell\geq supp\, q\end{array}}
\!\!\!A\cdot C(q;\ell),\\
{\cal D}&:=&\left(
\oplus_{\!\!\!\!\!\!\!\begin{array}{c}\scriptstyle q\in L(E)\vspace{-1ex}\\
\scriptstyle\eta\geq\ell\geq supp\, q\end{array}}
\!\!\!\!A\cdot D(q;\ell,\eta) \right)
\oplus \left(
\oplus_{\!\!\!\!\begin{array}{c}\scriptstyle\xi\in L^2(E)\vspace{-1ex}\\
\scriptstyle\eta\geq supp^2 \xi\end{array}}
\!\!\!A\cdot D(\xi;\eta) \right),\;\mbox{ and}\\
{\cal E}&:=&
\begin{array}[t]{r} \left(
\oplus_{\!\!\!\!\!\!\!\!\begin{array}{c}\scriptstyle q\in L(E)\vspace{-1ex}\\
\scriptstyle\mu\geq\eta\geq\ell\geq supp\, q\end{array}}
\!\!\!\!\!A\cdot E(q;\ell,\eta,\mu) \right)
\oplus \left(
\oplus_{\!\!\!\!\!\!\!\begin{array}{c}\scriptstyle\xi\in L^2(E)\vspace{-1ex}\\
\scriptstyle\mu\geq\eta\geq supp^2 \xi\end{array}}
\!\!\!A\cdot E(\xi;\eta,\mu) \right) \oplus \qquad \\
\oplus \left(
\oplus_{\!\!\!\!\begin{array}{c}\scriptstyle\omega\in L^3(E)\vspace{-1ex}\\
\scriptstyle\omega\geq supp^3 \omega\end{array}}
\!\!\!\! A\cdot E(\omega;\mu)\right)
\end{array}
\end{array}
\]
($B,C,D,$ and $E$ are just symbols).
The differentials equal
\[
\begin{array}{cccl}
d_B: &B(r)&\mapsto &d\,x^r\vspace{1ex}\\
d_C: &C(q;\ell)&\mapsto &\sum_{r\in E} q_r\,x^{\ell-r}\cdot B(r)\vspace{1ex}\\
d_D: &D(q;\ell,\eta)&\mapsto &C(q;\eta) - x^{\eta-\ell}\cdot C(q,\ell)\\
d_D: &D(\xi;\eta)&\mapsto& \sum_{q\in L(E)}\xi_q\cdot C(q,\eta)\vspace{1ex}\\
d_E: &E(q;\ell,\eta,\mu)&\mapsto& D(q;\eta,\mu)-D(q;\ell,\mu)+
x^{\mu-\eta}\cdot D(q;\ell,\eta) \\
d_E: &E(\xi;\eta,\mu)&\mapsto &D(\xi;\mu) - x^{\mu-\eta}\cdot D(\xi;\eta) -
\sum_{q\in L(E)} \xi_q\cdot D(q;\eta,\mu)\\
d_E: &E(\omega;\mu)&\mapsto &\sum_{\xi\in L^2(E)} \omega_{\xi}\cdot
D(\xi;\mu)\, .
\vspace{2ex}
\end{array}
\]
\par
Looking at these maps, we see that the complex is $M$-graded: The degree of
each of
the elements $B$, $C$, $D$, or $E$ can be obtained by taking the last of its
parameters
($r$, $\ell$, $\eta$, or $\mu$, respectively).\\
\par
{\bf Remark:} If one prefered a resolution with free $A$-modules of finite rank
(as it was
used in
\cite{T1}), the following replacements would be necessary:
\begin{itemize}
\item[(i)]
Define succesively $F\subseteq L(E)$, $G\subseteq L(F) \subseteq L^2(E)$, and
$H\subseteq L(G)\subseteq L^2(F) \subseteq L^3(E)$ as the finite
sets of normalized, minimal relations between elements of $E$, $F$, or
$G$, respectively. Then, use them instead of $L^i(E)$ ($i=1,2,3$).
\item[(ii)]
Let $\ell$, $\eta$, and $\mu$ run through finite generating
(under $(\sigma^{\scriptscriptstyle\vee}\cap M)$-action)
systems of all possible elements meeting the desired inequalities.
\end{itemize}
The disadvantages of those treatment are a more comlplicated description of
the resolution, on the one hand, and difficulties to obtain the
commutative diagram \zitat{4}{3},
on the other hand.\\
\par
\neu{43}
Combining the two exact sequences
\[
^{\displaystyle {\cal R}}\!/\!_{\displaystyle I\,{\cal R}} \longrightarrow
A^m \longrightarrow
^{\displaystyle I}\!\!/\!_{\displaystyle I^2}\rightarrow 0\quad
\mbox{and}\quad
^{\displaystyle I}\!\!/\!_{\displaystyle I^2}\longrightarrow
\Omega^1_{\,I\!\!\!\!C^{w+1}}\otimes A \longrightarrow
\Omega_Y^1 \rightarrow 0\,,
\]
we get a complex (not exact at the place of $A^m$) involving $\Omega_Y^1$. We
will compare
in the following commutative diagram this complex with the previous resolution
of
$^{\displaystyle\Omega_Y^1}\!\!\left/\!_{\displaystyle
\mbox{tors}\,(\Omega_Y^1)}\right.$:
\vspace{-5ex}\\
\[
\dgARROWLENGTH=0.8em
\begin{diagram}
\node[5]{^{\displaystyle I}\!\!/\!_{\displaystyle I^2}}
\arrow{se,t}{d}\\
\node[2]{^{\displaystyle {\cal R}}\!/\!_{\displaystyle I\,{\cal R}}}
\arrow[2]{e}
\arrow{se,t}{p_D}
\node[2]{A^m}
\arrow{ne}
\arrow[2]{e}
\arrow[2]{s,l}{p_C}
\node[2]{\Omega_{\,I\!\!\!\!C^{w+1}}\!\otimes \!A}
\arrow{e}
\arrow[2]{s,lr}{p_B}{\sim}
\node{\Omega_Y}
\arrow[2]{s}
\arrow{e}
\node{0}\\
\node[3]{\mbox{Im}\,d_D}
\arrow{se}\\
\node{{\cal E}}
\arrow{e,t}{d_E}
\node{{\cal D}}
\arrow[2]{e,t}{d_D}
\arrow{ne}
\node[2]{{\cal C}}
\arrow[2]{e,t}{d_C}
\node[2]{{\cal B}}
\arrow{e}
\node{^{\displaystyle\Omega_Y^1}\!\!\!\left/\!\!_{\displaystyle
\mbox{tors}\,(\Omega_Y^1)}\right.}
\arrow{e}
\node{0}
\end{diagram}
\]
\par
Let us explain the three labeled vertical maps:
\begin{itemize}
\item[(B)]
$p_B: dz_r \mapsto B(r)$ is an isomorphism between two free $A$-modules of rank
$w+1$.
\item[(C)]
$p_C: e^{ab} \mapsto C(a-b;\pi(a))$. In particular, the image of this map is
spanned by
those $C(q,\ell)$ meeting $\ell\geq \bar{q}$ (which is stronger than just
$\ell\geq\mbox{supp}\,q$).
\item[(D)]
Finally, $p_D$ arises as pull back of $p_C$ to $^{\displaystyle {\cal R}}\!/\!_{\displaystyle I\,{\cal R}}$.
It can
be described by
$r(a,b;c)\mapsto D(a-b; \pi(a),\pi(a+c))$ and $s(a,b,c)\mapsto D(\xi;\pi(a))$
($\xi$ denotes
the relation $\xi=[(b-c)-(a-c)+(a-b)=0]$).
\vspace{2ex}
\end{itemize}
\par
{\bf Remark:}
Starting with the typical ${\cal R}_0$-element mentioned in \zitat{3}{5}(iii), the
previous
description of the map $p_D$ yields 0 (even in ${\cal D}$).\\
\par
\neu{44}
By \zitat{4}{1} we get the $A$-modules $T^i_Y$ by computing the cohomology
of the complex dual to those of \zitat{4}{2}.\\
\par
As in \cite{T1}, denote by $G$ one of the capital letters $B$, $C$, $D$, or
$E$. Then, an element $\psi$ of the dual free module $(\bigoplus\limits_G
\,I\!\!\!\!C[\check{\sigma}\cap M]\cdot G)^\ast$ can be described by giving elements
$g(x)\in\,I\!\!\!\!C[\check{\sigma}\cap M]$ to be the images of the generators $G$
($g$ stands for $b$, $c$, $d$, or $e$, respectively).\\
\par
For $\psi$ to be homogeneous of degree $-R\in M$, $g(x)$ has to be
a monomial of degree
\[
\deg g(x)=-R+\deg G.
\]
In particular, the corresponding complex coefficient $g\in \,I\!\!\!\!C$ (i.e.
$g(x)=g\cdot x^{-R+\deg G}$) admits the property that
\[
g\neq 0\quad\mbox{implies}\quad -R+\deg G\ge 0\quad (\mbox{i.e.}\;
-R+\deg G\in\check{\sigma}).
\vspace{2ex}
\]
\par
{\bf Remark:}
Using these notations,
Theorem \zitat{3}{3}(1) was proved in \cite{T1} by showing that
\begin{eqnarray*}
\left(\left. ^{\displaystyle L(E_0^R)}\!\!\right/
\!_{\displaystyle \sum_i L(E_i^R)} \right)^\ast \otimes_{Z\!\!\!Z} \,I\!\!\!\!C
&\longrightarrow&
^{\displaystyle \mbox{Ker}({\cal C}^\ast_{-R}\rightarrow {\cal D}^\ast_{-R})}\!\!\left/
\!_{\displaystyle \mbox{Im}({\cal B}^\ast_{-R}\rightarrow{\cal C}^\ast_{-R})}\right.\\
\varphi &\mapsto&
[\dots,\, c(q;\ell):=\varphi(q),\dots]
\end{eqnarray*}
is an isomorphism of vector spaces.\\
Moreover, looking at the diagram of
\zitat{4}{3}, $e^{ab}\in A^m$ maps to both $\underline{z}^a-\underline{z}^b\in ^{\displaystyle
I}\!\!/\!
_{\displaystyle I^2}$ and $C(a-b;\pi(a))\in {\cal C}$. In particular, we can verify Theorem
\zitat{3}{4}:
Each $\varphi:L(E)_{\,I\!\!\!\!C} \rightarrow\,I\!\!\!\!C$, on the one hand,
and its associated $A$-linear map
\begin{eqnarray*}
^{\displaystyle I}\!\!/\!_{\displaystyle I^2} &\longrightarrow& A\\
\underline{z}^a-\underline{z}^b & \mapsto & \varphi(a-b)\cdot x^{\pi(a)-R},
\end{eqnarray*}
on the other hand, induce the same element of $T^1_Y(-R)$. \\
\par
\neu{45}
For computing $T_Y^2(-R)$,
the interesting part of the dualized complex
$\zitat{4}{2}^\ast$ in degree $-R$ equals the complex
of $\,I\!\!\!\!C$-vector spaces
\[
{\cal C}^{\ast}_{-R} \stackrel{d_D^{\ast}}{\longrightarrow} {\cal D}^{\ast}_{-R}
\stackrel{d_E^{\ast}}{\longrightarrow} {\cal E}^{\ast}_{-R}
\]
with coordinates $\underline{c}$, $\underline{d}$, and $\underline{e}$,
respectively:
\begin{eqnarray*}
{\cal C}^{\ast}_{-R} &=&
\{\underline{c(q;\ell)}\, |\;
c(q;\ell)=0 \;\mbox{for}\;\ell-R\notin\check{\sigma}\}\\
{\cal D}^{\ast}_{-R} &=&
\{[\underline{d(q;\ell,\eta)},\underline{d(\xi;\eta)}]\;|\;
\begin{array}[t]{ccccl}
d(q;\ell,\eta)&=&0& \mbox{for} &\eta-R\notin\check{\sigma}\mbox{, and}\\
d(\xi;\eta)&=&0& \mbox{for} &\eta-R\notin\check{\sigma} \}
\end{array}\\
{\cal E}^{\ast}_{-R} &=&
\{ [\underline{e(q;\ell,\eta,\mu)},
\underline{e(\xi;\eta,\mu)},
\underline{e(\omega ;\mu)}]\,|\;
\mbox{each coordinate vanishes for } \mu - R \notin \check{\sigma} \}.
\vspace{1ex}
\end{eqnarray*}
\par
The differentials
$d_D^{\ast}$ and $d_E^{\ast}$ can be described by
\[
\begin{array}{lcll}
d(q;\ell,\eta)&=&c(q;\eta)-c(q;\ell),&\\
d(\xi;\eta)&=&\sum\limits_{q\in F}\xi_q\cdot c(q;\eta),&
\mbox{and}\\
e(q;\ell,\eta,\mu) &=&
d(q;\eta,\mu) - d(q;\ell,\mu) + d(q;\ell,\eta),\\
e(\xi;\eta,\mu) &=&
d(\xi;\mu) - d(\xi;\eta) - \sum_{q\in F} \xi_q\cdot
d(q;\eta,\mu),\\
e(\omega ;\mu) &=&
\sum\limits_{\xi\in G} \omega_{\xi}\cdot d(\xi;\mu).
\end{array}
\vspace{1ex}
\]
\par
Denote $V:= \mbox{Ker}\,d^{\ast}_E \subseteq {\cal D}_{-R}^{\ast}\,$ and
$\,W:= \mbox{Im}\,d_D^{\ast}\subseteq V$, i.e.
\begin{eqnarray*}
V&=& \{ [\underline{d(q;\ell,\eta)};\,\underline{d(\xi;\eta)}]\,|\;
\begin{array}[t]{l}
q\in L(E), \;\eta\geq\ell\geq\mbox{supp}\,q\mbox{ in }M;\\
\xi\in L^2(E), \;\eta\geq\mbox{supp}^2\xi;
\vspace{0.5ex}\\
d(q;\ell,\eta) = d(\xi;\eta) = 0 \mbox{ for } \eta -R \notin \check{\sigma},\\
d(q;\ell,\mu) = d(q;\ell,\eta) + d(q;\eta,\mu) \; (\mu\geq\eta\geq\ell\geq
\mbox{supp}\, q),\\
d(\xi;\mu)= d(\xi;\eta) + \sum_q \xi_q \cdot d(q;\eta,\mu)\;
(\mu\geq\eta\geq \mbox{supp}^2 \xi),\\
\sum_{\xi\in G}\omega_{\xi} \,d(\xi;\mu) =0 \mbox{ for }\omega \in L^3(E)
\mbox{ with } \mu \geq \mbox{supp}^3\omega\,\},
\end{array}\\
W&=& \{ [\underline{d(q;\ell,\eta)};\,\underline{d(\xi;\eta)}]\,|\;
\mbox{there are $c(q;\ell)$'s with}
\begin{array}[t]{l}
c(q,\ell)=0 \mbox{ for } \ell-R\notin\check{\sigma},\\
d(q;\ell,\eta) = c(q;\eta)-c(q;\ell),\\
d(\xi;\eta)= \sum_q\xi_q\cdot c(q;\eta)\,\}.
\end{array}
\end{eqnarray*}
By construction, we obtain
\[
V\!\left/\!_{\displaystyle W}\right. =
\mbox{Ext}^i_A\left(
\;^{\displaystyle\Omega_Y^1}\!\!\left/\!_{\displaystyle
\mbox{tors}\,(\Omega_Y^1)}\right.
, \, A \right)(-R)
\subseteq T^2_Y(-R)
\]
(which is an isomorphism, if $Y$ is smooth in codimension 2).\\
\par
\neu{46}
Let us define the much easier vector spaces
\begin{eqnarray*}
V_1&:=& \{[\underline{x_i(q)}_{(q\in L(E_i^R))}]\,|\;
\begin{array}[t]{l}
x_i(q)=x_j(q) \mbox{ for }
\begin{array}[t]{l}
\bullet\,
\langle a^i, a^j \rangle < \sigma \mbox{ is a 2-face and}\\
\bullet\,
q\in L(E_i^R\cap E_j^R)\,,
\end{array}\\
\xi\in L^2(E_i^R) \mbox{ implies } \sum_q\xi_q\cdot
x_i(q)=0
\,\}\;\mbox{ and}
\end{array}
\vspace{1ex}
\\
W_1&:=& \{[\underline{x(q)}_{(q\in \cup_i L(E_i^R))}]
\,|\;
\begin{array}[t]{l}
\xi\in L(\bigcup_i L(E_i^R)) \mbox{ implies } \sum_q\xi_q\cdot
x(q)=0 \,\}.
\vspace{2ex}
\end{array}
\end{eqnarray*}
\par
{\bf Lemma:}
{\em
The linear map $V_1\rightarrow V$ defined by
\begin{eqnarray*}
d(q;\ell,\eta) &:=& \left\{
\begin{array}{lll}
x_i(q) &\mbox{ for }& \ell\in K_i^R,\;\; \eta \geq R\\
0 &\mbox{ for }& \ell \geq R \;\mbox{ or } \;\eta \in
\bigcup_i K_i^R\,;
\end{array} \right. \\
d(\xi;\eta) &:=&
0\,
\end{eqnarray*}
induces an injective map
\[
V_1\!\left/\!_{\displaystyle W_1}\right.
\hookrightarrow
V\!\left/\!_{\displaystyle W}\right.\,.
\]
If $Y$ is smooth in codimension 2, it will be an isomorphism.
}
\\
\par
{\bf Proof:}
1) The map $V_1 \rightarrow V$ is {\em correct defined}:
On the one hand, an argument as used in \zitat{3}{5}(i) shows that $\ell\in
K_i^R\cap K_j^R$ would imply $x_i(q)=x_j(q)$. On the other hand,
the image of
$[x_i(q)]_{q\in L(E_i^R)}$ meets all conditions in the definition of $V$.
\vspace{1ex}
\\
2) $W_1$ maps to $W$ (take $c(q,\ell):=x(q)$ for $\ell\geq R$ and
$c(q,\ell):=0$ otherwise).
\vspace{1ex}
\\
3) The map between the two factor spaces is {\em injective}: Assume for
$[x_i(q)]_{q\in L(E_i^R)}$ that there exist elements $c(q,\ell)$, such that
\begin{eqnarray*}
c(q;\ell) &=& 0 \; \mbox{ for } \ell \in \bigcup_i K^R_i\, ,\\
x_i(q) &=& c(q;\eta) - c(q;\ell) \; \mbox{ for }
\eta \geq \ell,\, \ell\in K_i^R,\, \eta\geq R\,,\\
0 &=&
c(q;\eta) - c(q;\ell)\; \mbox{ for } \eta\geq \ell \mbox{ and }
[\ell\geq R\mbox{ or } \eta\in \bigcup_iK_i^R]\, , \mbox{ and}\\
0 &=&
\sum_q \xi_q \cdot c(q;\eta) \; \mbox{ for } \eta \geq
\mbox{supp}^2\xi\, .
\end{eqnarray*}
In particular, $x_i(q)$ do not depend on $i$, and these elements
meet the property
\[
\sum_q \xi_q \cdot x_{\bullet}(q) = 0 \; \mbox{ for } \xi\in L(\bigcup_i
L(E_i^R)).
\]
4) If $Y$ is smooth in codimension 2, the map is {\em surjective} :\\
Given an element $[d(q;\ell,\eta),\,d(\xi;\eta)]\in V$, there exist
complex numbers $c(q;\eta)$ such that:
\begin{itemize}
\item[(i)]
$d(\xi;\eta) = \sum_q\xi_q\cdot c(q;\eta)\,$ ,
\item[(ii)]
$c(q;\eta)=0 \mbox{ for } \eta\notin R+\sigma^{\scriptscriptstyle\vee}\,
(\mbox{i.e. }\eta\in \bigcup_iK_i^R)\,$.
\end{itemize}
(Do this separately for each $\eta$ and distinguish between the cases
$\eta\in R +\sigma^{\scriptscriptstyle\vee}$ and $\eta\notin R+\sigma^{\scriptscriptstyle\vee}$.)\\
In particular, $[c(q;\eta) - c(q;\ell),\, d(\xi;\eta)]\in W$. Hence, we
have seen that we may assume $d(\xi;\eta)=0$.\\
\par
Let us choose some sufficiently high degree $\ell^\ast\geq E$.
Then,
\[
x_i(q):= d(q;\ell,\eta) - d(q;\ell^\ast\!,\eta)
\]
(with $\ell\in K_i^R$, $\ell\geq \mbox{supp}\,q$
(cf.\ Lemma \zitat{3}{2}(2)), and $\eta\geq\ell,\ell^\ast\!,R$)
defines some preimage:
\begin{itemize}
\item[(i)]
It is independent from the choice of $\eta$: Using a different $\eta'$
generates
the difference $d(q;\eta,\eta')-d(q;\eta,\eta')$.
\item[(ii)]
It is independent from $\ell\in K_i^R$: Choosing another $\ell'\in K_i^R$
with $\ell'\geq\ell$ would add the summand $d(q;\ell,\ell')$, which is 0;
for the general case use Lemma \zitat{3}{2}(2).
\item[(iii)]
If $\langle a^i,a^j\rangle < \sigma$ is a 2-face with $\mbox{supp}\,q
\subseteq L(E^R_i)\cap L(E_j^R)$, then by Lemma \zitat{3}{2}(2) we can choose
an
$\ell\in K_i^R\cap K_j^R$ achieving $x_i(q)=x_j(q)$.
\item[(iv)]
For $\xi\in L^2(E_i^R)$ we have
\[
\sum_q \xi_q\cdot d(q;\ell,\eta) = \sum_q \xi_q\cdot d(q;\ell^\ast\!,\eta) =
0\,,
\]
and this gives the corresponding relation for the $x_i(q)'$s.
\item[(v)]
Finally, if we apply to
$[\underline{x_i(q)}]\in V_1$
the linear map $V_1\rightarrow V$, the result differs from
$[d(q;\ell,\eta),0]\in V$ by
the $W$-element built from
\[
c(q;\ell) := \left\{ \begin{array}{ll}
d(q;\ell,\eta) - d(q;\ell^\ast\!,\eta) & \mbox{ if } \ell\geq R \\
0 & \mbox{ otherwise }.
\end{array} \right.
\vspace{-2ex}
\]
\end{itemize}
\hfill$\Box$\\
\par
\neu{47}
Now, it is easy to complete the proofs for Theorem \zitat{3}{3} (part 2 and 3)
and
Theorem \zitat{3}{5}:\\
\par
First, for a tuple $[\underline{x_i(q)}]_{q\in L(E_i^R)}$, the condition
\[
\xi\in L^2(E_i^R) \mbox{ implies } \sum_q\xi_q\cdot x_i(q)=0
\]
is equivalent to the fact the components $x_i(q)$ are induced by elements
$x_i\in L(E_i^R)_{\,I\!\!\!\!C}^\ast$.\\
The other condition for elements of $V_1$ just says that for 2-faces
$\langle a^i,a^j\rangle<\sigma$ there is $x_i=x_j$ on
$L(E_i^R\cap E_j^R)_{\,I\!\!\!\!C}=L(E_i^R)_{\,I\!\!\!\!C}\cap L(E_j^R)_{\,I\!\!\!\!C}$. In particular, we
obtain
\[
V_1= \mbox{Ker}\left( \oplus_i L(E_i^R)_{\,I\!\!\!\!C}^\ast \rightarrow
\oplus_{\langle a^i,a^j\rangle <\sigma} L(E_i^R\cap E_j^R)_{\,I\!\!\!\!C}^\ast \right)\,.
\]
In the same way we get
\[
W_1 = \left( \sum_i L(E^R_i)_{\,I\!\!\!\!C}\right)^\ast\,,
\]
and our $T^2$-formula is proven.\\
\par
Finally, if $\psi_i:L(E_i^R)_{\,I\!\!\!\!C}\rightarrow \,I\!\!\!\!C$ are linear maps defining an
element of
$V_1$, they induce the following $A$-linear map on ${\cal D}$ (even on
$\mbox{Im}\,d_D$):
\begin{eqnarray*}
D(q;\ell,\eta) &\mapsto& \left\{
\begin{array}{lll}
\psi_i(q)\cdot x^{\eta-R} &\mbox{ for }& \ell\in K_i^R,\;\; \eta \geq R\\
0 &\mbox{ for }& \ell \geq R \;\mbox{ or } \;\eta \in
\bigcup_i K_i^R
\end{array} \right.\\
D(\xi;\eta &\mapsto& 0\,.
\end{eqnarray*}
Now, looking at the diagram of \zitat{4}{3}, this translates exactly into the
claim of
Theorem \zitat{3}{5}.\\
\par
\sect{Proof of the cup product formula}\label{s5}
\neu{51}
Fix an $R\in M$, and let $\varphi\in L(E)^\ast_{\,I\!\!\!\!C}$ induce some element
(also denoted by $\varphi$)
of $T^1_Y(-R)$. Using the notations of \zitat{2}{3}, \zitat{3}{4},
and \zitat{3}{6} we can take
\[
\tilde{\varphi}(f_{\alpha\beta}):=
\varphi(\alpha-\beta)\cdot \underline{z}^{\Phi(\pi(\alpha)-R)}
\]
for the auxiliary $P$-elements needed to compute the $\lambda(\varphi)$'s
(cf. Theorem \zitat{3}{4}).\\
\par
Now, we have to distinguish between the two several types of relations
generating the $P$-module ${\cal R}\subseteq P^m$:
\begin{itemize}
\item[(r)]
Regarding the relation $r(a,b;c)$ we obtain
\begin{eqnarray*}
\sum_{(\alpha,\beta)\in m} r(a,b;c)_{\alpha\beta}\cdot
\tilde{\varphi}(f_{\alpha\beta}) &=&
\tilde{\varphi}(f_{a+c,b+c}) - \underline{z}^c\,\tilde{\varphi}(f_{ab})
\\
&=&
\varphi(a-b)\cdot \left(
\underline{z}^{\Phi(\pi(a+c)-R)} - \underline{z}^{c+\Phi(\pi(a)-R)} \right)
\\
&=&
\varphi(a-b)\cdot
f_{\Phi(\pi(a+c)-R),\,c+\Phi(\pi(a)-R)}\,.
\end{eqnarray*}
In particular,
\[
\lambda_{\alpha\beta}^{r(a,b;c)}(\varphi) =
\left\{\begin{array}{ll}
\varphi(a-b) & \mbox{ for } [\alpha,\beta] = [c+\Phi(\pi(a)-R),\,
\Phi(\pi(a+c)-R)]\\
0 & \mbox{ otherwise}\,.
\end{array} \right.
\]
\item[(s)]
The corresponding result for the relation $s(a,b,c)$ is much nicer:
\begin{eqnarray*}
\sum_{(\alpha,\beta)\in m} s(a,b,c)_{\alpha\beta}\cdot
\tilde{\varphi}(f_{\alpha\beta}) &=&
\tilde{\varphi}(f_{bc})-
\tilde{\varphi}(f_{ac})+
\tilde{\varphi}(f_{ab})\\
&=&
[\varphi(b-c)-\varphi(a-c)+\varphi(a-b)]\cdot
\underline{z}^{\Phi(\pi(a)-R)}\\
&=& 0\,.
\end{eqnarray*}
In particular, $\lambda^{s(a,b,c)}(\varphi)=0$.
\vspace{2ex}
\end{itemize}
\par
\neu{52}
Now, let $R,S,\varphi$, and $\psi$ as in the assumption of Theorem
\zitat{3}{6}. Using formula \zitat{2}{3}(iii), our previous computations
yield $(\varphi\cup\psi)(s(a,b,c))=0$ and
\[
\begin{array}{l}
(\varphi\cup\psi)(r(a,b;c))=
\sum_{\alpha,\beta}\lambda^{r(a,b;c)}_{\alpha\beta}(\varphi)
\cdot \psi(f_{\alpha\beta}) +
\sum_{\alpha,\beta} \varphi(f_{\alpha\beta})\cdot
\lambda^{r(a,b;c)}_{\alpha\beta}(\psi)
\vspace{2ex}\\
\qquad=
\begin{array}[t]{r}
\varphi(a-b)\cdot \psi\left(
c^{}+\Phi(\pi(a)-R)-\Phi(\pi(a+c)-R)\right)\cdot
x^{\pi(c+\Phi(\pi(a)-R))-S} +\qquad\\
+\psi(a-b)\cdot \varphi\left(
c+\Phi(\pi(a)-S)-\Phi(\pi(a+c)-S)\right)\cdot
x^{\pi(c+\Phi(\pi(a)-S))-R}
\end{array}
\vspace{2ex}\\
\qquad=
\begin{array}[t]{r}
\left[ \varphi(a-b)\cdot \psi\left(
c+\Phi(\pi(a)-R)-\Phi(\pi(a+c)-R)\right) +\right.
\qquad\qquad\qquad\qquad\qquad\\
\left. + \psi(a-b)\cdot \varphi\left(
c+\Phi(\pi(a)-S)-\Phi(\pi(a+c)-S)\right)
\right]
\cdot x^{\pi(a+c)-R-S}\,.
\vspace{1ex}
\end{array}
\end{array}
\]
\par
{\bf Remark:}
Unless $\pi(a+c)\geq R+S$, both summand in the brackets will vanish. For
instance,
on the one hand, $\pi(a)\in\bigcup_iK_i^R$ would cause $\varphi(a-b)=0$, and,
on the
other hand, $\pi(a)-R\geq 0$ and $\pi(c+\Phi(\pi(a)-R))\in \bigcup_iK_i^S$
imply $\psi(c+\Phi(\pi(a)-R)-\Phi(\pi(a+c)-R))=0$.\\
\par
To apply Theorem \zitat{3}{5} we would like to remove the argument $c$ from
the big coefficient. This will be done by adding a suitable
coboundary $T$.\\
\par
\neu{53}
Let us start with defining for $(\alpha,\beta)\in m$
\[
t(\alpha,\beta):= \begin{array}[t]{r}
\varphi(\alpha-\beta)\cdot
\psi \left( \Phi(\pi(\alpha)-R)+\Phi(R)-\beta\right)+\qquad\qquad\qquad\\
+ \psi(\alpha-\beta) \cdot
\varphi \left( \Phi(\pi(\alpha)-S)+\Phi(S)-\alpha\right)\,.
\end{array}
\]
(This expression is related to $t_{\varphi,\psi,R,S}$ from
\zitat{3}{6} by $t(q)=t(q^+,q^-)$.) \\
\par
{\bf Lemma:}{\em
Let $\alpha,\beta,\gamma\inI\!\!N^E$ with $\pi(\alpha)=\pi(\beta)=\pi(\gamma)$.
\begin{itemize}
\item[(1)]
$t(\alpha,\beta)=t(\alpha-\beta)$
as long as $\pi(\alpha)\in\bigcup_i K_i^{R+S}$.
\item[(2)]
$t(\beta,\gamma)-t(\alpha,\gamma)+t(\alpha,\beta)=0$.
\vspace{2ex}
\end{itemize}
}
\par
{\bf Proof:}
(1) It is enough to show that $t(\alpha+r,\beta+r)=t(\alpha,\beta)$ for
$r\in I\!\!N^E$, $\pi(\alpha+r)\in\bigcup_iK_i^{R+S}$. But the difference of these
two terms has exactly the shape of the coefficient of $x^{\pi(a+c)-R-S}$ in
\zitat{5}{2}. In particular, the argument given in the previous remark
applies again.\\
\par
(2) By extending $\varphi$ and $\psi$ to linear maps $\,I\!\!\!\!C^E\rightarrow\,I\!\!\!\!C$,
we obtain
\[
t(\alpha,\beta) = \begin{array}[t]{r}
[\varphi(\alpha-\beta)\,\psi\left(
\Phi(\pi(\alpha)-R) + \Phi(R)\right) + \psi(\alpha-\beta) \,
\varphi\left( \Phi(\pi(\alpha)-S)+\Phi(S)\right)]+\,\\
+[\varphi(\beta)\,\psi(\beta)-\varphi(\alpha)\,\psi(\alpha)].
\end{array}
\]
Now, since $\pi(\alpha)=\pi(\beta)=\pi(\gamma)$, both types of summands add
up to 0 separately in
$t(\beta,\gamma)-t(\alpha,\gamma)+t(\alpha,\beta)$.
\hfill$\Box$\\
\par
{\bf Remark:} The previous lemma does not imply that $t(q)$ is
$Z\!\!\!Z$-linear in $q$. The
assumption for $\pi(\alpha)$ made in (1) is really essential.\\
\par
Now, we obtain a
$P$-linear map $T\in \mbox{Hom}(P^m,A)$ by
\[
T: e^{\alpha\beta}\mapsto
\left\{ \begin{array}{ll}
t(\alpha,\beta)\,x^{\pi(\alpha)-R-S} & \mbox{ for } \pi(\alpha)\geq R+S\\
0 & \mbox{ otherwise}\,.
\end{array} \right.
\]
Pulling back $T$ to ${\cal R}\subseteq P^m$ yields (in case of $\pi(a+c)\geq R+S$)
\begin{eqnarray*}
T(r(a,b;c)) &=& \left\{ \begin{array}{ll}
[t(a+c,b+c)-t(a,b)]\cdot x^{\pi(a+c)-R-S} & \mbox{ for } \pi(a)\geq R+S\\
t(a+c,b+c)\cdot x^{\pi(a+c)-R-S} & \mbox{ otherwise}
\end{array} \right.\\
&=&
\left\{ \begin{array}{ll}
-(\varphi\cup\psi)(r(a,b;c)) & \mbox{ for } \pi(a)\geq R+S\\
t(a,b)\,x^{\pi(a+c)-R-S} -(\varphi\cup\psi)(r(a,b;c)) & \mbox{ otherwise}\,
\end{array} \right.
\end{eqnarray*}
and $T(s(a,b,c))=0$ (by (2) of the previous lemma).\\
\par
On the other hand, $T$ yields a trivial element of $T^2_Y(-R-S)$,
i.e. inside this group we may replace
$\varphi\cup\psi$ by $(\varphi\cup\psi)+T$ to obtain
\begin{eqnarray*}
(\varphi\cup\psi)(r(a,b;c)) &=&
\left\{ \begin{array}{ll}
t(a,b)\cdot x^{\pi(a+c)-R-S}&\mbox{ for } \pi(a)\in \bigcup_i K_i^{R+S};\;
\pi(a+c)\geq R+S\\
0 & \mbox{ otherwise}\,,
\end{array} \right.
\vspace{1ex}\\
(\varphi\cup\psi)(s(a,b,c)) &=& 0\,.
\vspace{1ex}
\end{eqnarray*}
\par
Having Theorem \zitat{3}{5} in mind, this formula for $\varphi\cup\psi$ is
exactly what
we were looking for:\\
Given an $r(a,b;c)$ with $\pi(a)\in K_i^{R+S}$,
let us compute $(\varphi\cup\psi)_i(q:=a-b)$ following the recipe of (i), (ii)
of Theorem
\zitat{3}{6}. We do not need to split
$q=a-b$ into a sum $q=\sum_k q^k$ - the element $q$ itself already satisfies
the condition
\[
\langle a^i,\bar{q}\rangle \leq \langle a^i, \pi(a) \rangle < \langle
a^i,R+S\rangle.
\]
In particular, with $(\varphi\cup\psi)_i(a-b)=t(a-b)=t(a,b)$ we
will obtain the right result - if the recipe is assumed to be correct. \\
\par
\neu{54}
We will fill those remaining gap now, i.e. we will show that
\begin{itemize}
\item[(a)]
each $q\in L(E_i^{R+S})$
admits a decomposition $q=\sum_k q^k$ with the desired properties,
\item[(b)]
$\sum_k q^k=0$ (with $\bar{q}^k\in K_i^{R+S}$) implies $\sum_k t(q^k)=0$, and
\item[(c)]
for adjacent $a^i,a^j$ the relations $q\in L(E_i^{R+S}\cap E_j^{R+S})$ admit
a decomposition $q=\sum_kq^k$ that works for both $i$ and $j$.
\end{itemize}
(In particular, this answers the questions arised right after stating the
theorem in
\zitat{3}{6}.)\\
\par
Let us fix an element $i\in \{1,\dots,N\}$. Since $\sigma^{\scriptscriptstyle\vee}\cap M$
contains elements $r$ with $\langle a^i,r\rangle =1$, some of them must be
contained in the generating set $E$, too. We choose one of these elements
and call it $r(i)$.\\
Now, to each $r\in E$ we associate some relation $p(r)\in L(E)$ by
\[
p(r):= e^r - \langle a^i, r \rangle\cdot e^{r(i)} +
[\mbox{suitable element of } Z\!\!\!Z^{E\cap (a^i)^\bot}]\,.
\]
The two essential properties of these special relations are
\begin{itemize}
\item[(i)]
$\langle a^i, \bar{p}(r)\rangle = \langle a^i, r\rangle$, and
\item[(ii)]
if $q\in L(E)$ is any relation, then $q$ and $\sum_{r\in E}q_r\cdot p(r)$
differ
by some element of $L(E\cap (a^i)^\bot)$ only.
\vspace{1ex}
\end{itemize}
\par
In particular, this proves (a). For (b) we start with the following\\
\par
{\em Claim:}
Let $q^k\in L(E)$ be relations such that
$\sum_k \langle a^i,\bar{q}^k\rangle < \langle a^i, R+S\rangle$.
Then, $\sum_k t(q^k)=t(\sum_k q^k)$.\\
\par
{\em Proof:} We can restrict ourselves to the case of two summands, $q^1$ and
$q^2$. Then,
by Lemma \zitat{5}{3},
\begin{eqnarray*}
t(q^1)+t(q^2) &=&
t\left((q^1)^+,(q^1)^-\right) + t\left((q^2)^+,(q^2)^-\right)\\
&=&
t\left((q^1)^++(q^2)^+,(q^1)^-+(q^2)^+\right) +
t\left((q^2)^++(q^1)^-,(q^2)^-+(q^1)^-\right)\\
&=&
t\left((q^1)^++(q^2)^+,(q^2)^-+(q^1)^-\right)\\
&=& t(q^1+q^2)\,.
\hspace{9cm} \Box
\end{eqnarray*}
\par
In particular, if $\sum_kq^k=0$ (with $\bar{q}^k\in K_i^{R+S}$), then this
applies for
the special decompositions
\[
q^k=\sum_r q^k_r\cdot p(r) + q^{0,k} \quad (q^{0,k}\in L(E\cap(a^i)^\bot))
\]
of the summands $q^k$ themselves. We obtain
\[
\sum_{q^k_r>0}q^k_r\cdot t\left(p(r)\right) + t(q^{0,k}) = t\left(
\sum_{q^k_r>0}q^k_r\,p(r)+q^{0,k}\right) =: t(q^{1,k})
\]
and
\[
\sum_{q^k_r<0}q^k_r\cdot
t\left(p(r)\right)= t\left( \sum_{q^k_r<0}q^k_r\,p(r)\right)=:t(q^{2,k})\,.
\]
Up to elements of $E\cap (a^i)^\bot$, the relations $q^{1,k}$ and $q^{2,k}$ are
connected by
the common
\[
(q^{1,k})^-=-q^{1,k}_{r(i)}\cdot e^{r(i)}=\langle a^i,\bar{q}^k\rangle
\cdot e^{r(i)}=q^{2,k}_{r(i)}\cdot e^{r(i)}=(q^{2,k})^+\,.
\]
Hence, Lemma \zitat{5}{3} yields
\[
\sum_r q^k_r\cdot t\left(p(r)\right) + t(q^{0,k}) = t(q^{1,k}) + t(q^{2,k}) =
t\left(
q^{1,k}+q^{2,k}\right) = t(q^k)\,,
\]
and we conclude
\begin{eqnarray*}
\sum_k t(q^k) &=&
\sum_k \left(\sum_r q^k_r\cdot t\left(p(r)\right) + t(q^{0,k})\right)\\
&=&
\sum_r \left( \sum_k q^k_r \right) t\left(p(r)\right) + t\left(\sum_k
q^{0,k}\right)
\quad (\mbox{cf. previous claim})\\
&=&
0+ t\left( \sum_k q^k - \sum_{k,r} q^k_r\,p(r) \right)\\
&=& 0\,.
\vspace{2ex}
\end{eqnarray*}
\par
Finally, only (c) is left. Let $a^i$, $a^j$ be two adjacent edges of $\sigma$.
We adapt the construction of the elementary relations
$p(r)$. Instead of the $r(i)$'s, we will use elements $r(i,j)\in E$
characterized by the
property
\[
\langle a^i, r(i,j)\rangle = 1\,,\; \langle a^j, r(i,j)\rangle = 0\,.
\]
(Those elements exist, since $Y$ is assumed to be smooth in codimension 2.)\\
Now, we define
\[
p(r):= e^r - \langle a^i,r\rangle \cdot e^{r(i,j)} - \langle a^j,r \rangle
\cdot e^{r(j,i)}
+ [\mbox{suitable element of }Z\!\!\!Z^{E\cap(a^i)^\bot\cap(a^j)^\bot}]\,.
\]
These special $p(r)$'s meet the usual properties (i) and (ii) - but for the two
different
indices $i$ and $j$ at the same time. In particular, if $q\in L(E)$ is any
relation, then
$q$ and $\sum_{r\in E}q_r\cdot p(r)$ differ by some element of
$L(E\cap(a^i)^\bot\cap(a^j)^\bot)$ only.\\
\par
\sect{An alternative to the complex $L(E^R)_{\bullet}$}\label{s6}
\neu{61}
Let $R\in M$ be fixed for the whole \S \ref{s6}. The complex $L(E^R)_{\bullet}$
introduced in \zitat{3}{2} fits naturally into the exact sequence
\[
0\rightarrow L(E^R)_{\bullet} \longrightarrow (Z\!\!\!Z^{E^R})_{\bullet}
\longrightarrow \mbox{span}(E^R)_{\bullet}\rightarrow 0
\]
of complexes built in the same way as $L(E^R)_{\bullet}$, i.e.
\[
(Z\!\!\!Z^{E^R})_{-k} := \oplus\!\!\!\!\!\!_{\begin{array}{c}
\scriptstyle\tau<\sigma\vspace{-1ex} \\ \scriptstyle dim\, \tau=k \end{array}}
\!\!\!\!Z\!\!\!Z^{E^R_{\tau}}
\qquad \mbox{and}\qquad
\mbox{span}(E^R)_{-k} := \oplus\!\!\!\!\!\!_{\begin{array}{c}
\scriptstyle\tau<\sigma\vspace{-1ex} \\ \scriptstyle dim\, \tau=k \end{array}}
\!\!\!\!\mbox{span}(E^R_{\tau})\,.
\]
\par
{\bf Lemma:}{\em
The complex $(Z\!\!\!Z^{E^R})_{\bullet}$ is exact.\\
}
\par
{\bf Proof:}
The complex $(Z\!\!\!Z^{E^R})_{\bullet}$ can be decomposed into a direct sum
\[
(Z\!\!\!Z^{E^R})_{\bullet} = \bigoplus_{r\in M} (Z\!\!\!Z^{E^R})(r)_{\bullet}
\]
showing the contribution of each $r\in M$. The complexes occuring as summands
are
defined as
\begin{eqnarray*}
(Z\!\!\!Z^{E^R})(r)_{-k} &:=&
\oplus\!\!\!\!\!\!_{\begin{array}{c}
\scriptstyle\tau<\sigma\vspace{-1ex}\\ \scriptstyle dim\, \tau=k \end{array}} \!\!\!\!
\left\{ \begin{array}{ll}
Z\!\!\!Z=Z\!\!\!Z^{\{r\}} & \mbox{ for } r\in E^R_{\tau}\\
0 & \mbox{ otherwise}
\end{array} \right\}\\
&=&
Z\!\!\!Z^{\#\{\tau\,|\; dim\,\tau=k; \, r\in E^R_{\tau}\}}\,.
\end{eqnarray*}
Denote by $H^+$ the halfspace
\[
H^+ := \{ a\in N_{I\!\!R}\,|\; \langle a,r\rangle < \langle a, R\rangle\} \subseteq
N_{I\!\!R}.
\]
Then, for $\tau \neq 0$, the fact that $r\in E^R_{\tau}$ is equivalent to
$\tau \setminus \{0\} \subseteq H^+$. On the other hand, $r\in E^R_0$
corresponds to
the condition $\sigma \cap H^+ \neq \emptyset$.\\
In particular, $(Z\!\!\!Z^{E^R})(r)_{\bullet}$,
shifted by one place, equals the complex for computing the reduced homolgy of
the
topological space $\cup \{\tau\,|\;\tau \setminus \{0\} \subseteq H^+\}
\subseteq \sigma$ cut
by some affine hyperplane. Since this space is contractable, the complex is
exact.
\hfill$\Box$\\
\par
{\bf Corollary:}{\em
The complexes $L(E^R)_{\bullet}^\ast$ and $\mbox{span}(E^R)_{\bullet}^\ast[1]$
are
quasiisomorphic. In particular, under the usual assumptions (cf. Theorem
\zitat{3}{3}), we obtain
\[
T^i_Y(-R) = H^i\left( \mbox{span}(E^R)_{\bullet}^\ast\otimes _{Z\!\!\!Z}\,I\!\!\!\!C\right)\,.
\vspace{2ex}
\]
}
\par
\neu{62}
We define the $I\!\!R$-vector spaces
\begin{eqnarray*}
V^R_i &:= &\mbox{span}_{I\!\!R}(E_i^R)=\left\{
\begin{array}{l@{\quad\mbox{for}\;\:}l}
0 &\langle a^i,R\rangle\le 0\\
\left[ a^i=0\right] \subseteq M_{I\!\!R} & \langle a^i,R\rangle =1\\
M_{I\!\!R}=I\!\!R^n & \langle a^i,R\rangle\ge2
\end{array}
\right.\\
&& \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad (i=1,\ldots,N),\;
\mbox{ and}\\
V^R_{\tau} &:=& \cap_{a^i\in \tau} V^R_i
\supseteq \mbox{span}_{I\!\!R}(E^R_{\tau})
\quad (\mbox{for faces } \tau<\sigma)\,.
\end{eqnarray*}
$\,$
\vspace{-2ex}\\
\par
{\bf Proposition:}{\em
With ${\cal V}^R_{-k}:= \oplus\!\!\!\!\!\!_{\begin{array}{c}
\scriptstyle\tau<\sigma\vspace{-1.5ex} \\ \scriptstyle dim\, \tau=k \end{array}}
\!\!\!\!V_{\tau}^R$ we obtain a complex
${\cal V}^R_{\bullet}
\supseteq \mbox{span}_{I\!\!R}(E^R)_{\bullet}$.
Moreover, if $Y$ is smooth in
codimension $k$, then both complexes are equal at $\geq\!(-k)$.
}
\\
\par
{\bf Proof:}
$V^R_{\tau} = \mbox{span}_{I\!\!R}(E^R_{\tau})$ is true
for smooth cones $\tau<\sigma$ (cf.(3.7) of \cite{T1}).
\hfill$\Box$\\
\par
{\bf Corollary:}{\em
\begin{itemize}
\item[(1)]
If $Y$ is smooth in codimension 2, then $T^1_Y(-R)=
H^1\left(({\cal V}^R_{\bullet})^\ast \otimes_{I\!\!R} \,I\!\!\!\!C \right)$.
\item[(2)]
If $Y$ is smooth in codimension 3, then $T^2_Y(-R)=
H^2\left(({\cal V}^R_{\bullet})^\ast \otimes_{I\!\!R} \,I\!\!\!\!C \right)$.
\vspace{1ex}
\end{itemize}
}
\par
The formula (1) for $T^1_Y$ (with a more boring proof) was already obtained
in (4.4) of \cite{T1}.\\
\par
\sect{3-dimensional Gorenstein singularities}\label{s7}
\neu{71}
We want to apply the previous results for the special case of an isolated,
3-dimensional,
toric Gorenstein singularity. We start with fixing the notations.\\
\par
Let $Q=\mbox{conv}(a^1,\dots,a^N)\subseteq I\!\!R^2$ be a lattice polygon with
primitive
edges
\[
d^i:= a^{i+1}-a^i\in Z\!\!\!Z^2\,.
\]
Embedding $I\!\!R^2$ as the affine hyperplane $[a_3=1]$ into $N_{I\!\!R}:=I\!\!R^3$,
we can define the cone
\[
\sigma:= \mbox{Cone}(Q) \subseteq N_{I\!\!R}\,.
\]
The fundamental generators of $\sigma$ equal the vectors
$(a^1,1),\dots,(a^N,1)$, which we
will also denote by $a^1,\dots,a^N$, respectively. \\
\par
The vector space $M_{I\!\!R}$ contains a special element $R^\ast:=[0,0;1]$:
\begin{itemize}
\item
$\langle \bullet,R^\ast\rangle = 1$ defines the affine hyperplane containing
$Q$,
\item
$\langle \bullet,R^\ast\rangle = 0$ describes the vectorspace containing the
edges
$d^i$ of $Q$.
\end{itemize}
The structure of the dual cone $\sigma^{\scriptscriptstyle\vee}$ can be described as follows:
\begin{itemize}
\item
$[c;\eta]\in M_{I\!\!R}$ is contained in $\sigma^{\scriptscriptstyle\vee}$, iff $\langle Q,-c\rangle
\leq \eta$.
\item
$[c;\eta]\in \partial \sigma^{\scriptscriptstyle\vee}$ iff there exists some $i$ with
$\langle a^i,-c\rangle = \eta$.
\item
The set $E$ contains $R^\ast$. However, $E\setminus \{R^\ast\}\subseteq
\partial
\sigma^{\scriptscriptstyle\vee}$.
\vspace{1ex}
\end{itemize}
\par
{\bf Remark:}
The toric variety $Y$ built by the cone $\sigma$ is 3-dimensional, Gorenstein,
and
regular outside its 0-dimensional orbit. Moreover, all those singularities can
be
obtained in this way.\\
\par
\neu{72}
Let $V$ denote the $(N-2)$-dimensional $I\!\!R$-vector space
\[
V:=\{(t_1,\dots,t_N)\,|\; \sum_i t_i\,d^i=0\}\subseteq I\!\!R^N\,.
\]
The non-negative tuples among the $\underline{t}\in V$ describe the set of Minkowski
summands
$Q_{\underline{t}}$
of positive multiples of the polygon $Q$. ($t_i$ is the scalar by which $d^i$
has to be
multiplied to get the $i$-th edge of $Q_{\underline{t}}$.)\\
\par
We consider the bilinear map
\[
\begin{array}{cclcl}
V&\times& I\!\!R^E & \stackrel{\Psi}{\longrightarrow}& I\!\!R\\
\underline{t}&,&[c;\eta]\in E &\mapsto&
\left\{ \begin{array}{ll}
0& \mbox{ if } c=0\quad(\mbox{i.e. } [c;\eta]=R^\ast)\\
\sum_{v=1}^{i-1} t_v\cdot
\langle d^v,-c\rangle & \mbox{ if }\langle a^i,-c\rangle =\eta\,.
\end{array} \right.
\end{array}
\]
Assuming both $a^1$ and the associated vertices of all Minkowski sumands
$Q_{\underline{t}}$
to coincide with $0\inI\!\!R^2$, the map $\Psi$ detects the maximal values of the
linear
functions $c$ on these summands
\[
\Psi(\underline{t},[c;\eta]) = \mbox{Max}\,(\langle a,-c\rangle\,|\; a\in
Q_{\underline{t}})\,.
\]
In particular, $\Psi(\underline{1},[c;\eta])=\eta$, i.e. $\Psi$ induces a map
\[
\Psi: \quad^{\displaystyle V}\!\!/\!_{\displaystyle I\!\!R\cdot\underline{1}} \times L_{I\!\!R}(E) \longrightarrow
I\!\!R\,.
\]
The results of \cite{Gor} and \cite{Sm} imply that $\Psi$ provides an
isomorphism
\[
^{\displaystyle V_{\,I\!\!\!\!C}}\!\!/\!_{\displaystyle \,I\!\!\!\!C\cdot\underline{1}}\stackrel{\sim}{\longrightarrow}
\left(\left. ^{\displaystyle L(E_0^{R^\ast})}\!\!\right/
\!_{\displaystyle \sum_i L(E_i^{R^\ast})} \right)^\ast \otimes_{Z\!\!\!Z} \,I\!\!\!\!C
\cong T^1_Y(-R^\ast)= T^1_Y\,.
\]
In particular, $\mbox{dim}\,T^1_Y = N-3$.\\
\par
\neu{73}
Let $R\in M$. Combining the general results of \S \ref{s6} with the fact
\[
\bigcap_i V_i^R = \mbox{Ker}\left[ \oplus_i(V_i^R\cap V^R_{i+1})\longrightarrow
\oplus_i V^R_i\right]
\]
coming from the special situation we are in, we obtain the handsome formula
\[
T^2_Y(-R)=
\left[ \left.^{\displaystyle \bigcap_i (\mbox{span}_{\,I\!\!\!\!C} E_i^R)}\!\!\! \right/
\!\!\! _{\displaystyle \mbox{span}_{\,I\!\!\!\!C} (\bigcap_i E_i^R)} \right] ^\ast\,.
\]
$T^1_Y$ is concentrated in the degree $-R^\ast$. Hence, for computing $T^2_Y$,
the
degrees $-kR^\ast$ ($k\geq 2$) are the most interesting (but not only) ones. In
this
special case, the vector spaces $V^{kR^\ast}_i$ equal $M_{I\!\!R}$, i.e.
\[
T^2_Y(-kR^\ast)= \left[ \left. ^{\displaystyle M_{\,I\!\!\!\!C}}\!\!\! \right/ \!\!\!
_{\displaystyle \mbox{span}_{\,I\!\!\!\!C} (\bigcap_i E_i^{kR^\ast})} \right] ^\ast
\subseteq
\left[ \left. ^{\displaystyle M_{\,I\!\!\!\!C}}\!\!\! \right/ \!\!\! _{\displaystyle \,I\!\!\!\!C\cdot R^\ast}\right]
^\ast =
\mbox{span}_{\,I\!\!\!\!C}(d^1,\dots,d^N) \subseteq N_{\,I\!\!\!\!C}\,.
\vspace{2ex}
\]
\par
{\bf Proposition:}
{\em
For $c\in I\!\!R^2$ denote by
\[
d(c):= \mbox{Max}\,(\langle a^i,c\rangle\,|\; i=1,\dots,N) -
\mbox{Min}\,(\langle a^i,c\rangle\,|\; i=1,\dots,N)
\]
the diameter of $Q$ in $c$-direction. If
\[
k_1:= \!\!\begin{array}[t]{c}
\mbox{Min}
\vspace{-1ex}\\
\scriptstyle c\inZ\!\!\!Z^2\setminus 0
\end{array} \!\!d(c) \quad \mbox{ and } \quad
k_2:= \!\!\!\begin{array}[t]{c}
\mbox{Min}
\vspace{-1ex}\\ \scriptstyle c,c'\inZ\!\!\!Z^2
\vspace{-1ex}\\ \scriptstyle lin.\, indept.
\end{array} \!\!\!\mbox{Max}\,[ d(c), d(c')]\,,
\]
then
$\quad\begin{array}[t]{lll}
\dim T^2_Y(-kR^\ast) = 2 & \mbox{ for } & 2\leq k \leq k_1\,,\\
\dim T^2_Y(-kR^\ast) = 1 & \mbox{ for } & k_1+1\leq k \leq k_2\,,\mbox{ and}\\
\dim T^2_Y(-kR^\ast) = 0 & \mbox{ for } & k_2+1\leq k \,.
\end{array}
\vspace{2ex}
$
}
\par
{\bf Proof:}
We have to determine the dimension of $\;\mbox{span}_{\,I\!\!\!\!C}\left( \bigcap_i
E_i^{kR^\ast}\right)\!\!\left/\!\!_{\displaystyle \,I\!\!\!\!C\cdot R^\ast}\right.$. Computing
modulo $R^\ast$
simply means to forget the $\eta$ in $[c;\eta]\in M$. Hence, we are done by the
following
observation for each $c\in Z\!\!\!Z^2\setminus0$:
\[
\begin{array}{rcl}
\exists \eta\inZ\!\!\!Z: \;[c,\eta]\in \bigcap_iK_i^{kR^\ast}
& \Longleftrightarrow &
\exists \eta\inZ\!\!\!Z: \;(k-1)R^{\ast}\geq [c;\eta] \geq 0\\
& \Longleftrightarrow &
d(c) \leq k-1\,.
\end{array}
\vspace{-3ex}
\]
\hfill$\Box$\\
\par
{\bf Corollary:}
{\em
Unless $Y=\,I\!\!\!\!C^3$ or $Y=\mbox{cone over }I\!\!P^1\timesI\!\!P^1$, we have
\[
T^2_Y(-2R^\ast)= \mbox{span}_{\,I\!\!\!\!C}(d^1,\dots,d^N),
\]
i.e. $\dim T^2_Y(-2R^\ast)=2$.}
\\
\par
\neu{74}
{\bf Proposition:}
{\em
Using both the isomorphism $\;V_{\,I\!\!\!\!C}\!\!\left/\!\!_{\displaystyle \,I\!\!\!\!C\cdot\underline{1}}\right.
\stackrel{\sim}{\rightarrow} T^1_Y$ and the injection
$T^2_Y(-2R^\ast)\hookrightarrow \mbox{span}_{\,I\!\!\!\!C}(d^1,\dots,d^N)$,
the cup product $T^1_Y\times T^1_Y \rightarrow T^2_Y$ equals the bilinear map
\[
\begin{array}{ccccc}
V_{\,I\!\!\!\!C}\!\!\left/\!\!_{\displaystyle \,I\!\!\!\!C\cdot\underline{1}}\right. &
\times &
V_{\,I\!\!\!\!C}\!\!\left/\!\!_{\displaystyle \,I\!\!\!\!C\cdot\underline{1}}\right. &
\longrightarrow &
\mbox{span}_{\,I\!\!\!\!C}(d^1,\dots,d^N)\\
\underline{s} &
,
&
\underline{t} &
\mapsto &
\sum_i s_i\,t_i\,d^i\,.
\end{array}
\vspace{1ex}
\]
}
\par
{\bf Proof:}
{\em Step 1:} To apply Theorem \zitat{3}{6} we will combine the isomorphisms
for $T^2_Y$ presented in \S \ref{s6} and \zitat{7}{3}. Actually, we will
describe the dual map
by associating to each $r\in M$ an element
$[q^1(r),\dots,q^N(r)]\in\oplus_iL(E_i^{2R^\ast})$.\\
First, for every $i=1,\dots,N$, we have to write
$r\in M = (\mbox{span}\, E_i^{2R^\ast})\cap (\mbox{span}\, E_{i+1}^{2R^\ast})$
as a linear
combination of elements from $E_i^{2R^\ast}\cap E_{i+1}^{2R^\ast}$.
This set contains a $Z\!\!\!Z$-basis for $M$ consisting of
\begin{itemize}
\item
$r^i:=$ primitive element of $\sigma^{\scriptscriptstyle\vee}\cap (a^i)^\bot \cap
(a^{i+1})^\bot$,
\item
$R^\ast$, and
\item
$r(i):= r(i,i+1)$ (cf. notation at the end of \zitat{5}{4}), i.e.
$\begin{array}[t]{l}
\langle a^i, r(i)\rangle = 1 \mbox{ and}\\
\langle a^{i+1}, r(i) \rangle = 0\,.
\end{array} $
\end{itemize}
In particular, we can write
\[
r= g^i(r)\cdot r^i + \langle a^{i+1},r\rangle\cdot R^\ast + \left(
\langle a^i,r \rangle - \langle a^{i+1},r\rangle \right) \cdot r(i)
\]
with some integer $g^i(r)\in Z\!\!\!Z$.\\
\par
Now, we have to apply the differential in the complex
$(Z\!\!\!Z^{E^{2R^\ast}})_{\bullet}$,
i.e. we map the previous expression via the map
\[
\oplus_i Z\!\!\!Z^{E_i^{2R^\ast}\cap E_{i+1}^{2R^\ast}} \longrightarrow
\oplus_i Z\!\!\!Z^{E_i^{2R^\ast}}\, .
\]
The result is (for every $i$) the element of $L(E_i^{2R^\ast})$
\[
\begin{array}{l}
g^i(r)\, e^{r^i} - g^{i-1}(r)\, e^{r^{i-1}} + \langle a^i-a^{i+1},r\rangle\cdot
e^{r(i)} -
\langle a^{i-1} - a^i,r\rangle \cdot e^{r(i-1)} +
\langle a^{i+1}-a^i, r \rangle\cdot e^{R^\ast}
\vspace{1ex}\\
\qquad = \langle d^i,r\rangle \cdot \left( e^{R^\ast} -
e^{r(i)}\right) + [(a^i)^\bot \mbox{-summands}] =: q^i(r)\,.
\end{array}
\vspace{2ex}
\]
\par
{\em Step 2:}
Defining
\[
q^i:= e^{R^\ast}-e^{r(i)} + [(a^i)^\bot \mbox{-summands}] \in
L(E_i^{2R^\ast})\quad
(i=1,\dots,N)\,,
\]
we use Theorem \zitat{3}{6} and the second remark of \zitat{3}{6} to obtain
\[
(\underline{s}\cup\underline{t})_i \left( q^i(r) \right) =
\langle d^i,r \rangle \cdot t_{\Psi(\underline{s},\bullet), \Psi(\underline{t},\bullet),
R^\ast,R^\ast}
(q^i) = \Psi(\underline{s},q^i)\cdot \Psi(\underline{t},q^i)\,.
\]
To compute those two factors, we take a closer look at the $q^i$'s. Let
\[
q^i= e^{R^\ast}-e^{r(i)} + \sum_v \lambda^i_v \,e^{[c^v; \eta^v]}\,,
\vspace{-1ex}
\]
and the sum is taken over those $v$'s meeting the property
$\langle a^i,-c^v\rangle = \eta^v$. Then, by definition of $\Psi$ in
\zitat{7}{2},
\[
\Psi(\underline{s},q^i)= \sum_{j=1}^{(i+1)-1}s_j\,\langle d^j, r(i)\rangle -
\sum_v \lambda^i_v \cdot \left( \sum_{j=1}^{i-1}s_j\, \langle d^j,c^v\rangle
\right)\,.
\]
On the other hand, we know that $q^i$ is a relation, i.e. the equation
\[
R^\ast - r(i) + \sum_v \lambda^i_v [c^v; \eta^v] =0
\vspace{-1ex}
\]
is true in $M$. Hence,
\[
\begin{array}{rcl}
\Psi(\underline{s},q^i)&=& \sum_{j=1}^i s_j\,\langle d^j, r(i)\rangle -
\sum_{j=1}^{i-1} s_j \langle d^j, r(i)\rangle
\vspace{0.5ex}\\
&=& s_i\cdot \langle d^i, r(i)\rangle
\vspace{0.5ex}\\
&=& -s_i\,.
\end{array}
\vspace{-3ex}
\]
\hfill$\Box$
\vspace{2ex}\\
\par
$T^1_Y\subseteq \,I\!\!\!\!C^N$ is the tangent space of the versal base space $S$ of our
singularity
$Y$. It is given by the linear equation $\sum_i t_i\cdot d^i=0$.\\
On the other hand,
the cup product $T^1_Y\times T^1_Y\rightarrow T^2_Y$ shows the quadratic part
of the
equations defining $S\subseteq\,I\!\!\!\!C^N$. By the previous proposition, it equals
$\sum_i t_i^2 \cdot d^i$.\\
\par
These facts suggest an idea how the equations of $S\subseteq\,I\!\!\!\!C^N$ could look
like. In
\cite{Vers} we have proved this conjecture; $S$ is indeed given by the
equations
\[
\sum_{i=1}^N t_i^k \cdot d^i =0\quad (k\geq 1)\,.
\vspace{3ex}
\]
\par
|
1994-07-07T21:26:29 | 9405 | alg-geom/9405002 | en | https://arxiv.org/abs/alg-geom/9405002 | [
"alg-geom",
"math.AG"
] | alg-geom/9405002 | Ron Stern | Ronald Fintushel and Ronald Stern | The blowup formula for Donaldson invariants | 16 pages, AMS-LaTeX | null | null | null | null | In this paper we present a formula which relates the Donaldson invariants of
a 4-manifold X with the Donaldson invariants of its blowup X#-CP(2). This
blow-up formula is independent of X and involves sigma-functions associated to
a naturally arising elliptic function. This, the final version, corrects some
earlier misconceptions regarding signs.
| [
{
"version": "v1",
"created": "Mon, 9 May 1994 17:19:18 GMT"
},
{
"version": "v2",
"created": "Thu, 7 Jul 1994 19:26:10 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Fintushel",
"Ronald",
""
],
[
"Stern",
"Ronald",
""
]
] | alg-geom | \section{Introduction}
Since their introduction in 1984 \cite{Donpoly}, the Donaldson invariants of
smooth $4$-manifolds
have remained as mysterious as they have been useful. However, in the past year
there has been a surge
of activity pointed at comprehension of the structure of these invariants
\cite{KM,FS}. One key to
these advances and to future insights lies in understanding the relation of the
Donaldson invariants
of a $4$-manifold $X$ and those of its blowup $\hat{X}=X\#\overline{\bold{CP}}^{\,2}$. It is the
purpose of this paper to
present such a blowup formula. This formula is independent of $X$ and is given
in terms of an
infinite series
\[ B(x,t) = \sum\limits_{k=0}^{\infty}B_k(x){t^k\over k!} \]
which is calculated in \S4 below.
This formula has been the target of much recent work.
The abstract fact that there exists such a formula which is
independent of $X$ was first proved by C. Taubes using techniques of
\cite{Reds}. J. Bryan \cite{B}
and P. Ozsvath \cite{Ozs} have independently calculated the coefficients
through $B_{10}(x)$. Quite
recently, J. Morgan and Ozsvath have announced a scheme which can recursively
compute all of the
$B_k(x)$. The special case of the blowup formula for manifolds of ``simple
type'' (see \S5 below) was
first given by P. Kronheimer and T. Mrowka. However, none of the techniques in
these cases approach
the simplicity of that offered here.
Before presenting the formula, we shall first establish notation for the
Donaldson invariants of a
simply connected $4$-manifold $X$ with $b^+>1$ and odd. (The hypothesis of
simple
connectivity is not necessary, but makes the exposition easier.) An orientation
of $X$, together with
an orientation of $H^2_+(X;\bold{R})$ is called a {\em homology orientation} of $X$.
Such a homology
orientation determines the ($SU(2)$) Donaldson invariant, a linear function
\[ D=D_X:\bold{A}(X)=\text{Sym}_*(H_0(X)\oplus H_2(X))\to \bold{R} \]
which is a homology orientation-preserving diffeomorphism invariant.
Here
\[\bold{A}(X)=\text{Sym}_*(H_0(X)\oplus H_2(X))\]
is viewed as a graded algebra where $H_i(X)$ has degree $\frac12(4-i)$.
We let $x\in H_0(X)$ be the generator $[1]$ corresponding to
the orientation. Then as usual, if $a+2b=d>\frac34(1+b^+_X)$ and $\alpha\in
H_2(X)$,
\[ D(\alpha^ax^b)=\langle\mu(\alpha)^a\nu^b,[\cal M_X^{2d}]\rangle \]
where $[\cal M_X^{2d}]$ is the fundamental class of the (compactified)
$2d$-dimensional moduli space of
anti-self-dual connections on an $SU(2)$ bundle over $X$, $\mu:H_i(X)\to H^{4-i}(\cal
B^*_X)$ is the
canonical map to the cohomology of the space of irreducible connections on that
bundle
\cite{Donpoly}, and $\nu=\mu(x)$. The extension of the definition to smaller
$d$ is given in
\cite{MMblowup} (and is accomplished, in fact, from the knowledge of the lowest
coefficient in the
$SO(3)$ blowup formula). Since an $SU(2)$ bundle $P$ over $X$ has a moduli
space of dimension
\[\dim\cal M_X(P) = 8c_2(P)-3(1+b^+_X) \]
it follows that such moduli spaces $\cal M_X^{2d}$ can exist only for
$d\equiv\frac12(1+b^+_X)\pmod4$. Thus the Donaldson invariant $D$ is defined
only on elements of
$\bold{A}(X)$ whose total degree is congruent to $\frac12(1+b^+_X)\pmod4$.
By definition, $D$ is $0$ on all elements of other degrees.
We can now state the blowup formula. Let $\hat{X}=X\#\overline{\bold{CP}}^{\,2}$ and let $e\in
H_2(X)$ denote the homology
class of the exceptional divisor. Since $b^+_X=b^+_{\hat{X}}$, the
corresponding Donaldson invariants $D=D_X$ and $\hat{D}=D_{\hat{X}}$ have their
(possible) nonzero
values in the same degrees $\pmod4$. We first show that there are polynomials
$B_k(x)$ satisfying
\[ \hat{D}(e^k\,z) = D(B_k(x)\,z) \]
for all $\,z\in \bold{A}(X)$ and then define the formal power series $B(x,t)$ as
above. Our result is that
\[ B(x,t)=e^{-{t^2x\over6}}\sigma_3(t) \]
where $\sigma_3$ is a particular quasi-periodic Weierstrass
sigma-function \cite{Ak} associated to the $\wp$-function which
satisfies the differential equation
\[(y')^2=4y^3-g_2y-g_3 \]
where
\[ g_2=4\,(\frac{x^2}{3}-1)\, , \hspace{.25in} g_3={8x^3-36x\over 27} \,. \]
There are also Donaldson invariants associated to $SO(3)$ bundles $V$ over $X$.
To define these
invariants one needs, along with a homology orientation of $X$, an integral
lift of $w_2(V)$. If
$c\in H_2(X;\bold{Z})$ is the Poincar\'e dual of the lift, the invariant is denoted
$D_c$ or $D_{X,c}$ if
the manifold $X$ is in doubt. $D_c$ is nonzero only in degrees congruent to
$-c\cdot c+\frac12(1+b^+)\pmod4$. If $c'\equiv c\pmod2$ then
\[ D_{c'}=(-1)^{({c'-c\over2})^2}D_c\,. \]
The $SO(3)$ blowup formula states that there are polynomials $S_k(x)$ such
that
\[ \hat{D}_e(e^k) = D(S_k(x)) \]
and if
\[ S(x,t) = \sum\limits_{k=0}^{\infty}S_k(x){t^k\over k!} \]
then
\[ S(x,t)=e^{-{t^2x\over6}}\sigma(t) \] where $\sigma(t)$ is the standard
Weierstrass
sigma-function \cite{Ak} associated to $\wp$.
The coefficients $S_k(x)$ for $k\le 7$ were earlier computed by T. Leness
\cite{Leness}.
The discriminant of the cubic equation
$4y^3-g_2y-g_3=0$ turns out to be ${x^2-4\over 4}$. Thus, when (viewed as a
function on
$\bold{A}(X)$) $D(x^2-4)=0$, the Weierstrass sigma-functions degenerate to elementary
functions, and the
blowup formula can be restated in terms of these functions. This is done in the
final section. It is
interesting to note that the condition $D(x^2z)=4\,D(z)$ is the {\em simple
type} condition introduced
by Kronheimer and Mrowka \cite{KM}.
Our formulas are proved by means of a simple relation satisfied by $D(\tau^4z)$
where $\tau\in
H_2(X;\bold{Z})$ is represented by an embedded $2$-sphere of self-intersection $-2$.
When this
relation is applied to $\tau=e_1-e_2$, the difference of the two exceptional
classes of the double
blowup $X\#2\overline{\bold{CP}}^{\,2}$, one obtains a differential equation for $B(x,t)$. Solving
this equation gives our
formulas.
\section{Some Relations among Donaldson Invariants}
The key to the blowup formula lies in a few simple relations which are useful
for evaluating
Donaldson invariants on classes represented by embedded spheres of
self-intersection $-2$ and $-3$.
We begin by studying the behavior of the Donaldson invariant of a $4$-manifold
with a homology class $\tau$ represented by an embedded $2$-sphere $S$ of
self-intersection
$\tau\cdot \tau=-2$. Let $\langle \tau\rangle^\perp$ denote $\{\alpha\in H_2(X)|\tau\cdot\alpha=0\}$
and let
\[\bold{A}(\tau^\perp)=\bold{A}_X(\tau^\perp)=\text{Sym}_*(H_0(X)\oplus \langle \tau\rangle^\perp)\,.\]
\begin{thm}{\em (Ruberman\; \cite{R})} Suppose that $\tau\in H_2(X;\bold{Z})$ with
$\tau\cdot \tau=-2$ is represented by an embedded sphere $S$. Then for $\,z \in
\bold{A}(\tau^\perp)$, we have
$D(\tau^2\,z)=2\,D_{\tau}(\,z)$. \label{Ruber}\end{thm}
\begin{pf} Write $X=X_0 \cup N$ where $N$ is a tubular
neighborhood of $S$, and note that $\partial N$ is the lens space $L(2,-1)$. Since
$b^+_{X_0}>0$, generically there are no reducible anti-self-dual connections on $X_0$.
However, since $b^+_{N}=0$, there are nontrivial reducible anti-self-dual connections
arising
from complex line bundles $\lambda^m$, $m \in\bold{Z}$, where $\langle c_1(\lambda),\tau\rangle=-1$.
The corresponding moduli spaces $\cal M_N(\lambda^m\oplus\bar{\lambda}^m)$ have
dimensions
$4m^2-3$ and have boundary values $\zeta^m$ where $\zeta$ generates the character
variety of
$SU(2)$ representations of $\pi_1(\partial N)=\bold{Z}_2$ mod conjugacy. (Of course,
$\zeta^{2m}$
is trivial, and $\zeta^{2m+1}=\zeta$.)
Since $\langle \tau\rangle^\perp=H_2(X_0)$, we need to evaluate the Donaldson
invariant on two copies of $\tau$ and classes in $H_2(X_0)$. After cutting down
moduli spaces by
intersecting with transverse divisors representing the images under $\mu$ of
these classes in
$H_2(X_0)$ and using the given homology orientation, we may assume without loss
of generality that
there are no such classes and that we are working with a $4$-dimensional
moduli space $\cal M_X$. Let
$V_1$ and $V_2$ be divisors representing $\mu(\tau)$, coming from general
positioned surfaces in $N$.
The Donaldson invariant is the signed intersection number
\[ D(\tau^2)=\#(\cal M_X\cap V_1\cap V_2). \]
A standard dimension counting argument (cf. \cite{Donpoly}) shows that if we
choose a metric on $X$
with long enough neck length, $\partial N\times [0,T]$, then
all the intersections take place in a neighborhood $\cal{U}$ of the grafted
moduli
space $\cal M_{X_0}[\zeta]\# \{A_{\lambda}\}$ where $A_{\lambda}$ is the reducible anti-self-dual
connection on
$\lambda\oplus \bar{\lambda}$, and $\cal M_{X_0}[\zeta]$ is the $0$-dimensional cylindrical
end moduli space on
$X_0$ consisting of anti-self-dual connections which decay exponentially to the boundary
value $\zeta$. Let
$m_{X_0}$ be the signed count of points in $\cal M_{X_0}[\zeta]$. A neighborhood of
$A_{\lambda}$ in the
moduli space $\cal M_N(\lambda\oplus\bar{\lambda})$ is diffeomorphic to $(\bold C
\times_{S^1} SO(3))/SO(3)
\cong\bold C/S^1\cong[0,\infty)$. Here $S^1$ acts on $SO(3)$ so that
$SO(3)/S^1=S^2$ and on $\bold C$
with weight $-2$. Thus the neighborhood $\cal{U}$ is
\[(\tilde{\cal M}_{X_0}[\zeta]\times (\bold C \times_{S^1} SO(3)))/SO(3)\] where
``$\tilde{\cal M}_{X_0}[\zeta]$''
denotes the based moduli space.
Now $\tilde{V}_1\cap (\bold C \times_{S^1} SO(3))=\{0\} \times_{S^1} SO(3)$,
and the
intersection of $V_1$ with all of $\cal M_X$ is
\[(\cal M_{X_0}[\zeta] \times(\{0\} \times_{S^1} SO(3)))/SO(3)=\Delta .\]
Fix a point $p\in\cal M_{X_0}[\zeta]$, let $SO(3)\cdot p$ denote its orbit in
$\tilde{\cal M}_{X_0}[\zeta]$,
and let
\[\Delta_p=SO(3)\cdot p\times (\{0\} \times_{S^1} SO(3)))/SO(3)\cong S^2.\]
Identify $\Delta_p$ with a transversal in $\tilde{\Delta}_p$ and compute the
intersection number
$\tilde{V}_2\cdot \Delta_p=\iota_p$. Since $\iota_p$ is independent of $p\in
\cal M_{X_0}[\zeta]$, we
have $D(\tau^2)=\iota_p\cdot m_{X_0}$. The constant $\iota_p$ is
computed in \cite{FMbook} as follows. Note that $\Delta_p=\{0\}\times_{S^1}
SO(3)
\subset \bold C\times_{S^1} SO(3)$ is a \,zero-section of the $c_1=-2$ complex
line
bundle over $S^2$ and $\tilde{V}_2$ is another section. Thus
$\tilde{V}_2\cdot \Delta_p=-2$; and so $D(\tau^2)=-2\,m_{X_0}$.
To identify the relative invariant $m_{X_0}$, view $\cal M_{X_0}[\zeta]$ as
$\cal M_{X_0,0}[\text{ad}(\zeta)]$, an $SO(3)$ moduli space. Since $\text{ad}(\zeta)$ is
the
trivial $SO(3)$-representation, we may graft connections in
$\cal M_{X_0,0}[\text{ad}(\zeta)]$ to the trivial $SO(3)$ connection over $N$, and
since
$b_N^+=0$, there is no obstruction to doing this. We obtain an $SO(3)$ moduli
space
over $X$ corresponding to an $SO(3)$ bundle over $X$ with $w_2$
Poincar\'e dual to $\tau$. (This is the unique nonzero class in $H^2(X;\bold{Z}_2)$
which
restricts trivially to both $N$ and $X_0$.) Thus for $\,z \in \bold{A}(\tau^\perp)$,
we have
$D(\tau^2\,z)=\pm2\,D_{\tau}(\,z)$. (Note that since $\tau\cdot\tau=-2$, we have
$D_{-\tau}=D_\tau$.)
To determine the sign in this equation, we need to compare orientations on the
moduli spaces which
are involved. Let $A_0\in \cal M_{X_0}[\zeta]$. The way that a sign is attached
to this point is described in \cite{Donor,K}. By addition and subtraction of
instantons, $A_0$ is
related to a connection $B_0$ in a reducible bundle $E$ over $X_0$, and $B_0$
can be connected by a
path to a reducible connection $R$ which comes from a splitting $E\cong
L_0\oplus\bar{L}_0$.
There is a standard orientation for the determinant line of
the operator $d_R^+\oplus d_R^*$, and this can be followed back to give an
orientation
for the determinant line at $A_0$. This determinant line is canonically
oriented because the
cohomology $H^*_{A_0}$ vanishes. Comparing the two orientations gives a sign,
$\varepsilon$.
To determine the sign at the grafted connection $A_0\# A_{\lambda}$, note that the
same sequence of
instanton additions and subtractions as above relates $A_0\# A_{\lambda}$ to
$B_0\# A_{\lambda}$ which can
be connected to $R\# A_{\lambda}$, a reducible connection on the bundle
$L\oplus\bold{R}$ over $X$, where
the Mayer-Vietoris map $H^2(X)\to H^2(X_0)\oplus H^2(N)$ carries $c=c_1(L)$ to
$c_1(L_0)+c_1(\lambda)$.
Since $R\# A_{\lambda}$ is reducible, there is an orientation of the determinant
line, and it relates to the orientation which can be pulled back from the
trivial connection by
$(-1)^{c\cdot c}$. Thus pulling the orientation back over $A_0\# A_{\lambda}$
gives the
sign $\varepsilon\cdot (-1)^{c\cdot c}$.
To get the sign for $A_0\#\Theta$ we first pass to $SO(3)$, and then
$\text{ad}(A_0)$ is related as
above to the reducible connection $\text{ad}(R)$ which lives in the line bundle
$L_0^2$.
Grafting to the trivial connection $\Theta_N$, we get
$\text{ad}(A_0)\#\Theta_N$ which is
connected to the reducible connection $\text{ad}(R)\#\Theta_N$. This lives in
the grafted line bundle
$L_0^2\#\bold{R}$ which has $c_1=2c_1(L_0)$. (Note that although $c_1(L_0)$ is not a
global class,
$2c_1(L_0)$ is.) The class $2c_1(L_0)$ restricts trivially to $X_0$ and to $N$
(mod $2$); so its
mod $2$ reduction is the same as that of $\tau$. (We are here identifying $\tau$
and its Poincar\'e
dual.) Since $\tau=2c_1(\lambda)$, the difference in these reductions is
$2c_1(L_0)-\tau=2(c-\tau)$. The
corresponding orientations compare via the parity of $(c-\tau)\cdot(c-\tau)\equiv
c\cdot c\pmod2$. Thus
the sign which is attached to $A_0\#\Theta$ is $\varepsilon\cdot (-1)^{c\cdot c}$, the
same as for $A_0\#
A_{\lambda}$, and the sign in the formula above is `${}\,+\,{}$'. \end{pf}
For the case of the $SO(3)$ invariants the proof of Theorem~\ref{Ruber} can be
easily adapted to show:
\begin{thm} Suppose that $\tau\in H_2(X;\bold{Z})$ with $\tau\cdot\tau=-2$ is represented
by an
embedded sphere $S$. Let $c\in H_2(X;\bold{Z})$ satisfy $c\cdot\tau\equiv0\pmod2$.
Then for
$\,z \in \bold{A}(\tau^\perp)$ we have
$D_c(\tau^2\,z)=2\,D_{c+\tau}(z)$. \ \ \ \qed
\label{RuberSO3}\end{thm}
We next need to review some elementary facts concerning the Donaldson
invariants of blowups. These
can be found, for example in \cite{FMbook,Ko,Leness}. Let $X$ have the
Donaldson invariant $D$, and
let $\hat{X}=X\#\overline{\bold{CP}}^{\,2}$ have the invariant $\hat{D}$.
\begin{lem} Let $e\in H_2(\overline{\bold{CP}}^{\,2};\bold{Z})\subset H_2(\hat{X};\bold{Z})$ be the exceptional
class, and let
$c\in H_2(X;\bold{Z})$. Then for all $\,z\in \bold{A}(X)$:
\begin{enumerate}
\item $\hat{D}_c(e^{2k+1}\,z)=0$ for all $k\ge 0$.
\item $\hat{D}_c(\,z)=D_c(\,z)$.
\item $\hat{D}_c(e^2\,z)=0$.
\item $\hat{D}_c(e^4\,z)=-2\,D_c(\,z)$.
\item $\hat{D}_{c+e}(e^{2k}\,z)=0$ for all $k\ge 0$.
\item $\hat{D}_{c+e}(e\,z)=D_c(\,z)$.
\item $\hat{D}_{c+e}(e^3\,z)=-D_c(x\,z)$.
\end{enumerate} \label{blowuplow} \end{lem}
\begin{pf} Items (1)--(5) are standard and are explained in \cite{FMbook}. Both
(1) and (5)
follow because the automorphism of $H_2(X\#\overline{\bold{CP}}^{\,2};\bold{Z})$ given by reflection in $e$
is realized by a
diffeomorphism. Items (2) and (3) follow from counting arguments, and (4)
follows
from simple arguments as in the proof of Theorem~\ref{Ruber} above
\cite{FMbook}. Item (6) is due to
D. Kotschick \cite{Ko}.
A proof of (7) is given in \cite{Leness}. (However, the sign there differs from
ours since item
(6) is stated in \cite{Leness} with an incorrect sign.) We sketch a proof here.
Consider a
neighborhood $N$ of the exceptional curve, and let $X_0=\hat{X}\setminus N
\cong X\setminus B^4$. As
in the proof of Ruberman's theorem we lose no generality by assuming that we
are evaluating $\hat{D}$
only on $e^3$. A dimension counting argument shows that if we stretch the neck
between $X_0$ and $N$
to have infinite length by taking a sequence of generic metrics $\{ g_n\}$, and
if $V_i$ are
transverse divisors representing $\mu(e)$, then any sequence of connections \[
A_n\in
\cal M_{\hat{X},c+e}(g_n)\cap V_1\cap V_2\cap V_3\] must converge to the sum of a
connection in the
$4$-manifold $\cal M_{X_0,c}$ and the unique reducible connection on $N$
corresponding to the line bundle $\lambda$ over $N$ whose Euler class is
Poincar\'e dual to $e$. The
based moduli space $\tilde{\cal M}_N(\lambda)$ is the orbit of this reducible
connection, a $2$-sphere,
$S^2_e$. Let $v$ denote the (positive) generator of the equivariant cohomology
$H^*_{SO(3)}(S^2_e)\cong H^*(\bold{CP}^{\infty})$ in dimension $2$. The class
$\mu(e)$ lifts to the
equivariant class $-\frac12\langle c_1(\lambda),e\rangle\,v=\frac12 v\in
H^2_{SO(3)}(S^2_e)$. The connections
in $S^2_e$ are asymptotically trivial and this induces an $SO(3)$ equivariant
push-forward map
\[ \partial_*(N): H^*_{SO(3)}(S^2_e)\to H^*_{SO(3)}({1})=H^*(BSO(3)).\]
If $u\in H^*_{SO(3)}({1})$ is the generator in dimension $4$ then
$\partial_*(N)(v^{2k+1})=2\,u^k$. So $\partial_*(N)(e^3)=\frac14 u$.
Since each connection in $\cal M_{X_0,c}$ is also asymptotically trivial, there is
an induced map
$\partial^*(X_0):H^*_{SO(3)}({1})\to H^*_{SO(3)}(\tilde{\cal M}_{X_0,c})$.
It follows from \cite{Reds,AB} that $\hat{D}_{c+e}(e^3)$ is obtained by
evaluating
\[\langle \partial^*(X_0)\,\partial_*(N)(v^3),[\tilde{\cal M}_{X_0,c}]\rangle =
\frac14\,\langle \partial^*(X_0)(u),[\tilde{\cal M}_{X_0,c}]\rangle
= \frac14\,\langle \pi_*\partial^*(X_0)(u), [\cal M_{X_0,c}] \rangle \]
where basepoint fibration $\beta$ over $X_0$ is
\[\pi:\tilde{\cal M}_{X_0,c}\to\cal M_{X_0,c}\, ,\]
the last equality because the $SO(3)$ action on $\tilde{\cal M}_{X_0,c}$ is free.
But Austin and Braam \cite{AB}, for example, show that
$\pi_*\partial^*(X_0)(u)=p_1(\beta)$. Since
$\nu=-\frac14 p_1(\beta)$, we get $\langle\nu,[\cal M_{X_0,c}]\rangle = -D_c(x)$.
\end{pf}
We next consider embedded $2$-spheres of self-intersection $-3$.
\begin{thm} Suppose that $\tau\in H_2(X;\bold{Z})$ is represented by an
embedded 2-sphere $S$ with self-intersection $-3$. Let $\omega\in H_2(X;\bold{Z})$
satisfy
$\omega\cdot\tau\equiv0\pmod2$. Then for all $\,z\in\bold{A}(\tau^\perp)$ we have
\[D_\omega(\tau\,z) = -D_{\omega+\tau}(\,z).\]
\label{3curve}\end{thm}
\begin{pf} The proof is similar in structure to that of
Theorem~\ref{Ruber}. Write $X=X_0\cup N$ where $N$ is a tubular neighborhood of
$S$.
Then $\partial N=L(3,-1)$. Let $\eta$ generate the character variety of $SO(3)$
representations of $\pi_1(\partial N)$. Reducible anti-self-dual $SO(3)$ connections on $N$
arise
from complex line bundles $\lambda^m$, $m\in\bold{Z}$, where $\langle c_1(\lambda),\tau\rangle=-1$.
The corresponding moduli spaces $\cal M_N(\lambda^m\oplus\bold{R})$ have boundary values
$\eta^m$
and dimensions $\frac{2}{3}m^2-3$ if $m\equiv 0\pmod3$ and ${2m^2+1\over3}-2$
if
$m\not\equiv0\pmod3$.
Since it is easiest to work with an $\omega$ which satisfies $\partial\omega_{X_0}=0\in
H_1(\partial X;\bold{Z})=\bold{Z}_3$, we simply work with $\rho=3\,\omega$ rather than $\omega$. This is
no
problem, since $D_{3\omega}=(-1)^{\omega\cdot\omega}D_{\omega}$. Thus we may write
$\rho=\rho_0+\rho_N\in H_2(X_0;\bold{Z})\oplus H_2(N;\bold{Z})$.
As in our previous arguments, we assume that we are evaluating $D_{\rho}$ only
on $\tau$.
A dimension counting argument
shows that $D_{\rho}(\tau)$ is the product of relative invariants
$D_{X_0}[\eta^m]$
coming from a $0$-dimensional cylindrical end moduli space over $X_0$
with terms coming from nontrivial reducible
connections on $N$. These reducible connections must live in moduli spaces of
dimension $\le 1$, and the corresponding line bundles must have $c_1\equiv
\rho_N\pmod2$. Our hypothesis, $\omega\cdot\tau\equiv0\pmod2$ implies that
$\rho_N\cdot\tau\equiv0\pmod2$; so the line bundle in question must be an even
power of
$\lambda$. Recalling the constraint that the dimension of the corresponding moduli
space be $\le1$, the only possibility is $\cal M_N(\lambda^2\oplus\bold{R})$.
Consider an anti-self-dual connection $A_0$ lying in the finite $0$-dimensional moduli
space $\cal M_{X_0}[\eta^2]$, and let $A_{\lambda^2}$ be the reducible anti-self-dual
connection on $N$. A neighborhood of the $SO(3)$ orbit of $A_{\lambda^2}$ in the
based
moduli space $\tilde{\cal M}_N(\lambda^2\oplus\bold{R})$ is modelled by
$SO(3)\times_{S^1}\bold{C}$ and the (based) divisor for $\tau$ is
$-\frac12 \langle c_1(\lambda^2),\tau\rangle (SO(3)\times_{S^1}\{0\})=SO(3)\times_{S^1}\{0\}$.
The based connections obtained from grafting the orbit $SO(3)_{A_0}$ of $A_0$
to the orbit of
$A_{\lambda^2}$ are given by the fibered product of these orbits over the 2-sphere
in
$SO(3)$ consisting of representations of $\pi_1(\partial N)$ which are in the
conjugacy
class $\eta^2$. By cutting this down by the divisor for $\tau$ we obtain (up to
sign) the fibered product of $SO(3)_{A_0}$ with $S^2$ over $S^2$; i.e. simply
$SO(3)_{A_0}$. Taking the quotient by $SO(3)$,
\[ D_{\rho}(\tau)=\pm D_{X_0}[\eta^2].\]
Since $\eta^2=\eta$ in the character variety (a copy of $\bold{Z}_3$), we can
graft anti-self-dual connections $A_0$ to the unique (reducible) connection $A_{\lambda}$
lying in the
moduli space $\cal M_N(\lambda\oplus\bold{R})$ of formal dimension $-1$. As the
glued-together
bundle has $w_2$ which is Poincar\'e dual to $\rho+\tau\pmod2$, we have
\[ D_{\rho_0}[\eta^2]=\pm D_{\rho+\tau},\]
so our result is proved up to a sign.
To get this sign, we need to compare signs induced at $A_0\# A_{\lambda^2}$ and
$A_0\# A_{\lambda}$ using
a fixed homology orientation of $X$ and the integral lifts $\rho$ and $\rho+\tau$
of the
corresponding Stiefel-Whitney classes. By an excision argument \cite{Donor},
the difference in signs
depends only on the part of the connections over the neighborhood $N$. Thus the
sign is universal,
and may be determined by an example. For this, let $X$ be the $K3$ surface and
$\hat{X}=X\#\overline{\bold{CP}}^{\,2}$.
Let $s$ be any class in $H_2(X)$ of square $-2$ represented by an embedded
$2$-sphere (e.g. a
section), and let $\tau=s+e$. Note that $s-2e\in\bold{A}(\tau^{\perp})$.
Then using Theorems \ref{Ruber} and \ref{blowuplow},
\begin{eqnarray*}
\hat{D}((s-2e)\tau)&=&D(s^2)=2\,D_s\\
\hat{D}_{\tau}(s-2e)&=&-2\,D_s
\end{eqnarray*}
so the overall sign is `${}\,-\,{}$'.
\end{pf}
Next we combine our two relations to obtain a relation which is crucial in
obtaining
the general blowup formula. This relation was first proved by Wojciech
Wieczorek
using different methods. His proof will appear in his thesis \cite {Wiecz}.
\begin{cor} Suppose that $\tau\in H_2(X;\bold{Z})$ is represented by an embedded
$2$-sphere with
self-intersection $-2$, and let $c\in H_2(X;\bold{Z})$ with $c\cdot\tau\equiv0\pmod2$.
Then for all
$\,z\in\bold{A}(\tau^\perp)$
\[ D_c(\tau^4\,z)=-4\,D_c(\tau^2 \, x\,z)-4\,D_c(\,z). \]
\label{4ofem}\end{cor}
\begin{pf} In $\hat{X}=X\#\overline{\bold{CP}}^{\,2}$ the class $\tau+e$ is represented by a $2$-sphere
of self-intersection
$-3$, and $(\tau-2e)\cdot(\tau+e)=0$. From Lemma~\ref{blowuplow} we get
\[ \hat{D}_c((\tau-2e)^3 \,(\tau+e)\,z) = D_c(\tau^4\,z)-8\,\hat{D}_c(e^4\,z) =
D_c(\tau^4\,z)+16\,D_c(\,z).
\] On the other hand, by Theorems~\ref{RuberSO3} and \ref{3curve} and by
Lemma~\ref{blowuplow},
\begin{eqnarray*}
\hat{D}_c((\tau-2e)^3 \,(\tau+e)\,z)&=&-\hat{D}_{c+\tau+e}((\tau-2e)^3\,z)
=6\,\hat{D}_{c+\tau+e}(\tau^2\, e\,z)+8\,\hat{D}_{c+\tau+e}(e^3\,z)\\
&=&6\,D_{c+\tau}(\tau^2\,z)-8\,D_{c+\tau}(x\,z) = 12\,D_c(\,z)-4\,D_c(\tau^2\, x\,z)
\end{eqnarray*}
and the result follows.
\end{pf}
\bigskip
\section{The blowup equation}
Let $X$ be a simply connected oriented $4$-manifold and let $\hat{X}=X\#\overline{\bold{CP}}^{\,2}$.
Let $c\in H_2(X;\bold{Z})$. Of course $H_2(\hat{X};\bold{Z})=H_2(X;\bold{Z})\oplus\bold{Z} e$ with $e$
the exceptional class.
It follows from Lemma~\ref{blowuplow}(1),(5) that we can write
\[\hat{D}_c=\sum\beta_{c,k}\,E^{2k}\]
where $E$ denotes the $1$-form given by $E(y)=e\cdot y$ and
$\beta_{c,k}({\alpha^d})=\hat{D}_c(\alpha^d \, e^{2k})$ for any $\alpha\in H_2(X)$. Similarly
\[\hat{D}_{c+e}=\sum\gamma_{c,k}\,E^{2k+1}.\]
Consider $\bar{X}=X\#2\overline{\bold{CP}}^{\,2}$ with exceptional classes $e_1,\, e_2\in
H_2(\bar{X};\bold{Z})$, and let
$\bar{D}$ denote its Donaldson invariant. Then $e_1+e_2$ has self-intersection
$-2$ and is
represented by an embedded $2$-sphere. Furthermore, the intersection
$(e_1-e_2)\cdot (e_1+e_2)=0$;
so we can apply Corollary~\ref{4ofem} to get
\begin{equation}
\bar{D}_c((e_1-e_2)^r \, (e_1+e_2)^4\,z)=
-4\,\bar{D}_c((e_1-e_2)^r \, (e_1+e_2)^2\,x\,z)-4\,\bar{D}_c((e_1-e_2)^r\,z)
\label{blowuprecursion}
\end{equation}
for all $\,z\in \bold{A}(X)$.
\begin{lem} There are polynomials, $B_{k}(x)$, independent of $X$, so that for
any
$c\in H_2(X;\bold{Z})$ and $\,z\in \bold{A}(X)$ we have $\hat{D}_c(e^k\,z)=
D_c(B_k(x)\,z)$.\end{lem}
\begin{pf} Lemma~\ref{blowuplow}(1) implies that $\beta_{c,0}=D_c$. Thus we have
$B_0=1$.
Assume inductively that for $j\le k$, $\hat{D}_c(e^{j}z)= D_c(B_{j}(x)z)$.
Expanding
\eqref{blowuprecursion} via the induction hypothesis we have
\begin{multline*}
\bar{D}_c((e_1-e_2)^{k-3}\,(e_1+e_2)^4\,z) =
-4\,D_c(\,z\,\sum_{i=0}^{k-3}\binom{k-3}{i}\{ xB_{i+2}(x)B_{k-3-i}(x) \\
-2x B_{i+1}(x)B_{k-2-i}(x)
+ x B_i(x)B_{k-1}(x) + B_i(x)B_{k-3-i}(x)\}\,) = D_c(P(x)\,z)
\end{multline*}
for some polynomial $P$.
On the other hand, expanding the argument of
\[ \bar{D}_c((e_1-e_2)^{k-3}(e_1+e_2)^4\,z) \]
and using the induction hypothesis in a similar fashion, we get
\[\bar{D}_c((e_1-e_2)^{k-3}(e_1+e_2)^4\,z) = 2\,\hat{D}_c(e^{k+1}\,z) +
D_c(R(x)\,z) \]
for another polynomial $R$. The lemma follows.\end{pf}
\begin{lem} There are polynomials, $S_{k}(x)$, independent of $X$, so that for
any
$c\in H_2(X;\bold{Z})$ and $\,z\in \bold{A}(X)$ we have $\hat{D}_{c+e}(e^k\,z)=
D_c(S_k(x)\,z)$.
\end{lem}
\begin{pf} By Theorem~\ref{Ruber} we have for any even $k>0$,
\[
\bar{D}_c((e_1+e_2)^k(e_1-e_2)^2)=2\,D_{c+e_1-e_2}((e_1+e_2)^k)=-2\,D_{c+e_1+e_2}((e_1+e_2)^k) \]
This formula can then be used as above to inductively calculate $S_{k-1}(x)$ in
terms of
$S_1(x),\dots,S_{k-3}(x)$ and $B_0(x),\dots,B_{k+2}(x)$.
\end{pf}
We now explicitly determine the polynomials $B_k(x)$ and $S_k(x)$.
Set \[ B(x,t)=\sum_{t=0}^\infty
B_k(x)\frac{t^k}{k!}\hspace{.25in}\text{and}\hspace{.25in}
S(x,t)=\sum_{t=0}^\infty S_k(x)\frac{t^k}{k!}\]
Note that
\begin{equation}
\begin{split}\frac{d^n}{dt^n}\,\hat{D}(\exp(te)\,z) & =\hat{D}(\,z\,\sum
e^{k+n}\frac{t^k}{k!})=
D(B_{k+n}(x)\frac{t^k}{k!}\,z) \\ &=\frac{d^n}{dt^n}\,D(B(x,t)\,z) =
D(B^{(n)}(x,t)\,z)
\end{split}
\label{diff}
\end{equation}
where the last differentiation is with respect to $t$.
On $\bar{X} = X\#2\overline{\bold{CP}}^{\,2}$, we get
$\bar{D}(\exp(t_1e_1+t_2e_2)z)=D(B(x,t_1)\,B(x,t_2)z)$.
Now apply Corollary \ref{4ofem} to $e_1-e_2\in H_2(\bar{X};\bold{Z})$. Since for any
$t\in\bold{R}$ the class
$te_1+te_2\in\langle e_1-e_2\rangle^\perp$, we have the equation
\begin{equation}
\begin{split}\bar{D}(\exp(te_1+te_2)\,(e_1-e_2)^4\,z) & + 4\,
\bar{D}(x\,\exp(te_1+te_2)\,(e_1-e_2)^2\,z)
\\ &+4\,\bar{D}(\exp(te_1+te_2)\,z) = 0
\end{split}
\label{pre}
\end{equation}
But, for example,
\[ e_1^4\,\exp(te_1+te_2) = \left(\sum e_1^{k+4}\frac{t^k}{k!}\right)
\left(\sum e_2^k\frac{t^k}{k!}\right) =
\frac{d^4}{dt^4}(\exp(te_1))\,\exp(te_2) \]
Arguing similarly and using \eqref{diff}
\begin{equation*}
\begin{split}
& \bar{D}(\exp(te_1+te_2)\,(e_1-e_2)^4\,z) \\
&=D((2\,B^{(4)}(x,t)\,B(x,t)-8\,B'''(x,t)\,B'(x,t)
+6\,(B''(x,t))^2)\,z)\\
&= 2\,D((B^{(4)}\,B-4\,B'''\,B'+3\,(B'')^2 )\,z)
\end{split}
\end{equation*}
where $B=B(x,t)$. Completing the expansion of \eqref{pre} we get
\[2\,D((B^{(4)}\,B-4\,B'''\,B'+3\,(B'')^2 +4x\,(B''B-(B')^2)+2\,B^2)\,z)=0 \]
for all $\,z\in \bold{A}(X)$. This means that the expression
\[ B^{(4)}\,B-4\,B'''\,B'+3\,(B'')^2 +4x\,(B''B-(B')^2)+2\,B^2 \]
lies in the kernel of $D:\bold{A}(X)\to\bold{R}$.
Thus the ``blowup function'' $B(x,t)$ satisfies the differential equation
\[ B^{(4)}\,B-4\,B'''\,B'+3\,(B'')^2 +4x\,(B''B-(B')^2)+2\,B^2 =0 \]
modulo the kernel of $D$. Of course, the fact that this equation
holds only modulo the kernel of $D$ is really no constraint, since our interest
in
$B(x,t)$ comes from the equation $\hat{D}(\exp(te)z)=D(B(x,t)z)$.
Now let $B=\exp(f(t))$.
\begin{prop} Modulo the kernel of $D$, the logarithm $f(t)$ of $B(x,t)$
satisfies the differential
equation \[ f^{(4)}+6\,(f'')^2+4xf''+2=0\]
with the initial conditions $f=f'=f''=f'''=0$. \ \ \qed \label{DE}\end{prop}
The initial conditions follow from Lemma~\ref{blowuplow}.
\bigskip
\section{The blowup formula}
In order to solve the differential equation of Proposition~\ref{DE}, we set
$u=f''$. Then
the differential equation becomes
\[ u''+6u^2+4xu+2=0 \] with initial conditions $u(0)=u'(0)=0$.
This is equivalent to the equation
\begin{equation}
(u')^2=-4u^3-4xu^2-4u
\label{uDE}\end{equation}
as can be seen by differentiating both sides of the last equation with respect
to $t$.
Replacing $u$ by $-v$ and completing the cube yields
\[ (v')^2=4(v-\frac{x}{3})^3-\frac43 vx^2 +\frac{4x^3}{27}+4v \]
Finally, letting $y=v-\frac{x}{3}$ we get
\begin{equation}
(y')^2=4y^3-g_2y-g_3 \hspace{.2in} \text{where} \hspace{.2in}
g_2=4\,(\frac{x^2}{3}-1)
\hspace{.1in} \text{and} \hspace{.1in} g_3={8x^3-36x\over 27} .
\label{yDE}\end{equation}
This is the equation which defines the Weierstrass $\wp$-function. In fact, if
we rewrite \eqref{yDE}
as
\[ \frac{dt}{dy}={1\over\sqrt{4y^3-g_2y-g_3}}\]
then
\[t=\int_y^\infty{ds\over\sqrt{4s^3-g_2s-g_3}}=\wp^{-1}(y)\]
and we see that for arbitrary constants $c$, $y=\wp(t+c)$ gives all solutions
to \eqref{yDE}, and
so $f''=u=-(\wp(t+c)+{x\over3})$ is the general solution of \eqref{uDE}.
The roots of the cubic equation
\[ 4s^3-g_2s-g_3=0 \]
are
\begin{equation}
e_1=\frac{x}{6}+{\sqrt{x^2-4}\over2}, \hspace{.15in}
e_2=\frac{x}{6}-{\sqrt{x^2-4}\over2},
\hspace{.15in} e_3=-\frac{x}{3} \label{roots}
\end{equation}
where we have followed standard notation (cf. \cite{Ak}). These correspond to
the half-periods
$\omega_i=\wp^{-1}(e_i)$ of the $\wp$-function. The initial condition $f''(0)=0$
implies
that $\wp(c)=-\frac{x}{3}=e_3$; so $c=\omega_3+2\varpi$, where
$2\varpi=2m_1\omega_1+2m_3\omega_3$, with $m_1$,
$m_3\in\bold{Z}$, is an arbitrary period. (Note that the initial condition
$f'''(0)=0$ follows
because the half-periods are \,zeros of $\wp'$.) The Weierstrass
\,zeta-function satisfies
$\zeta'=-\wp$; thus $f'(t)=\zeta(t+\omega_3+2\varpi)-{tx\over3}+ a$. The constant $a$ is
determined by the
initial condition $f'(0)=0$;\; $a=-\zeta(\omega_3+2\varpi)$. Since the logarithmic
derivative of the
Weierstrass sigma-function is $\zeta$, integrating one more time gives
$f(t)=\log\sigma(t+\omega_3+2\varpi)-t\zeta(\omega_3+2\varpi)-{t^2x\over6}+b$,
and the initial condition $f(0)=0$ shows that $b=-\log\sigma(\omega_3+2\varpi)$.
Thus
\[
B(x,t)=e^{f(t)}=e^{-{t^2x\over6}}e^{-t\zeta(\omega_3+2\varpi)}\,{\sigma(t+\omega_3+2\varpi)\over\sigma(\omega_3+2\varpi)}.
\]
For $\omega=\omega_1$ or $\omega_3$ and $\eta=\zeta(\omega)$ we have the formulas
\[\zeta(u+2m\omega)=2m\eta+\zeta(u)\, ,\hspace{.2in}
\sigma(u+2m\omega)=(-1)^me^{2\eta(mu+m^2\omega)}\sigma(u)\]
(which follow easily from \cite[p.199]{Ak}). Using them, our formula for
$B(x,t)$ becomes
\[ B(x,t)=e^{-{t^2x\over6}}e^{-\eta_3 t}\,{\sigma(t+\omega_3)\over\sigma(\omega_3)}.\]
The above addition formula for the sigma-function implies that
\[\sigma(t+\omega_3)=\sigma((t-\omega_3)+2\omega_3)=-e^{2\eta_3t}\sigma(t-\omega_3).\]
Thus
\[ B(x,t)=-e^{-{t^2x\over6}}e^{\eta_3
t}\,{\sigma(t-\omega_3)\over\sigma(\omega_3)}=e^{-{t^2x\over6}}\sigma_3(t), \]
the last equality by the definition of the quasi-periodic function $\sigma_3$. In
conclusion,
\begin{thm} Modulo the kernel of $D$, the blowup function $B(x,t)$ is given by
the formula
\[ B(x,t)=e^{-{t^2x\over6}}\sigma_3(t).\ \ \ \qed\] \label{blowup}\end{thm}
\noindent The indexing of the Weierstrass functions $\sigma_i$ depends on the
ordering of the roots
$e_i$ of the equation $4s^3-g_2s-g_3=0$. This can be confusing. The important
point is that the
sigma-function we are using corresponds to the root $-{x\over3}$.
One can now obtain the individual blowup polynomials from the formula for
$B(x,t)$. For example,
$B_{12}=-512\,x^4-960\,x^2-408$ and (for fun),
\begin{multline*}
\!B_{30}(x)\!=\!134,217,728\,x^{13}+4,630,511,616\,x^{11}+
68,167,925,760\,x^9-34,608,135,536,640\,x^7\\
-39,641,047,695,360\,x^5-9,886,101,110,784\,x^3+543,185,367,552\,x
\end{multline*}
(We thank Alex Selby for help with some computer calculations.)
We also have
\begin{thm} Modulo the kernel of $D$, the blowup function $S(x,t)$ is given by
the formula
\[ S(x,t)=e^{-{t^2x\over6}}\sigma(t).\] \label{SO3blowup}\end{thm}
\begin{pf} As usual we let $\bar{X}=X\#2\overline{\bold{CP}}^{\,2}$ with exceptional classes $\varepsilon_1$
and
$\varepsilon_2$. (We have temporarily changed notation to avoid confusion with the
roots $e_i$ of
$4s^3-g_2s-g_3=0$.) Consider the class $\varepsilon_1-\varepsilon_2$ which is
represented by a sphere of self-intersection $-2$. By Theorem~\ref{Ruber} we
have
\[\bar{D}(\exp(t\varepsilon_1+t\varepsilon_2)\,(\varepsilon_1-\varepsilon_2)^2\,z)=2\,D_{\varepsilon_1-\varepsilon_2}(\exp(t\varepsilon_1+t\varepsilon_2)\,z)=
-2\,D_{\varepsilon_1+\varepsilon_2}(\exp(t\varepsilon_1+t\varepsilon_2)\,z)\] for all $\,z\in \bold{A}(X)$.
Equivalently we get $D((2\,B''B-2\,(B')^2)\,z)=-2\,D(S^2\,z)$. In other words,
\[ S^2=e^{-{t^2x\over3}}({x\over3}\sigma_3^2+(\sigma_3')^2-\sigma_3\sigma_3'')\]
Write $\sigma_3(t)=\exp(h(t))$. Then
\[ S^2= e^{-{t^2x\over3}}e^{2h}({x\over3}-h'') \]
i.e.
\[ S=\pm e^{-{t^2x\over6}}\sigma_3(t)({x\over3}-h'')^{\frac12}. \]
Since $\exp(h)=\sigma_3(t)=\sigma(t)(\wp(t)-e_3)^{\frac12}$, it follows that
$h=\log\sigma(t)+\frac12\log(\wp(t)-e_3)$. Then
\[h'=\zeta(t)+\frac12{\wp'(t)\over\wp(t)-e_3}=\zeta(t)+\frac12(\zeta(t+\omega_3)+\zeta(t-\omega_3)-2\zeta(t))\]
by \cite[p.41]{Ak}. Thus $h'= \frac12(\zeta(t+\omega_3)+\zeta(t-\omega_3))$, and
\[ h''=\frac12(-\wp(t+\omega_3)-\wp(t-\omega_3))=-\wp(t+\omega_3).\] Thus
\[ S=\pm e^{-{t^2x\over6}}\sigma_3(t)(\wp(t+\omega_3)-e_3)^{\frac12}=
\pm e^{-{t^2x\over6}}\sigma_3(t)\left({(e_3-e_1)(e_3-e_2)\over
\wp(t)-e_3}\right)^{\frac12} \]
(\cite[p.200]{Ak}). However, $(e_3-e_1)(e_3-e_2)=1$ (see\eqref{roots}); so
\[ S =\pm e^{-{t^2x\over6}}{\sigma_3(t)\over\sqrt{\wp(t)-e_3}}=\pm
e^{-{t^2x\over6}}\sigma(t)\, . \]
To determine the sign, note that (fixing $x$) $S'(0)=S_1=1$ by
Theorem~\ref{blowuplow}(6).
But from our formula $S'(0)=\pm\sigma'(0)$, and $\sigma'(0)=1$.
\end{pf}
\bigskip
\section{The blowup formula for manifolds of simple type}
A $4$-manifold is said to be of {\em simple type} \cite{KM} if for all $z\in
\bold{A}(X)$ the relation
$D(x^2z)=4\,D(z)$ is satisfied by its Donaldson invariant. It is clear that if
$X$ has simple type,
then $\hat{X}=X\#\overline{\bold{CP}}^{\,2}$ does as well. In this case, following \cite{KM}, one
considers the invariant
$\bold{D}$ defined by
\[\bold{D}(\alpha)=D((1+{x\over2})\exp(\alpha))\]
for all $\alpha\in H_2(X)$. $\bold{D}$ is called the {\em Donaldson series} of $X$.
Note that the simple type condition implies that for any $z\in\bold{A}(X)$,
\[D((1+{x\over2})\,z\,x)=2\,D((1+{x\over2})\,z)\, ,\]
i.e. $x$ acts as multiplication by $2$ on $D(1+{x\over2})$.
The blowup formula in this case has been determined previously by Kronheimer
and Mrowka.
In this section, we derive that formula by setting $x=2$ in
Theorems~\ref{blowup} and \ref{SO3blowup}. This gives a degenerate case of the
associated
Weierstrass functions. All the formulas below involving elliptic functions can
be found in
\cite{Ak}. The squares $k^2$, ${k'}^2$ of the modulus and complementary modulus
of our Weierstrass
functions are given by
\[ k^2={x-\sqrt{x^2-4}\over x+\sqrt{x^2-4}} \hspace{.35in}
{k'}^2={2\sqrt{x^2-4}\over x+\sqrt{x^2-4}}. \]
Thus $k^2=1$ and ${k'}^2= 0$ when $x=2$. The corresponding complete elliptic
integrals of
the first kind are
\begin{eqnarray*}
K&=&\int_0^1{ds\over\sqrt{(1-s^2)(1-k^2s^2)}}= \int_0^1{ds\over1-s^2}\\
K'&=& \int_0^1{ds\over\sqrt{(1-s^2)(1-{k'}^2s^2)}}=
\int_0^1{ds\over\sqrt{1-s^2}}
\end{eqnarray*}
Thus $K=\infty$ and $K'={\pi\over2}$ when $x= 2$. Also, when $x=2$ we have
$g_2=\frac43$
and $g_3=-\frac{8}{27}$; so the roots of $4s^3-g_2s-g_3=0$ are
$e_1=e_2=\frac13$ and $e_3=-\frac23$.
This means that when $x=2$ the basic periods are
\[ \omega_1={K\over\sqrt{e_1-e_3}}= K=\infty \hspace{.35in}
\omega_3={iK'\over\sqrt{e_1-e_3}}= iK' ={i\pi\over2}\,.\]
In this situation,
\[\sigma(t)={2\omega_3\over\pi}e^{{1\over6}({\pi t\over2\omega_3})^2}\sin{\pi t\over2\omega_3}=
e^{-{t^2\over6}}\sinh t \]
and
\[\wp(t)=-{1\over3}({\pi\over2\omega_3})^2+({\pi\over2\omega_3})^2{1\over\sin^2({\pi
t\over2\omega_3})}
={1\over3}+{1\over\sinh^2t} \, .\]
So
\[ \sigma_3(t)=\sigma(t)\sqrt{\wp(t)-e_3}=e^{-{t^2\over6}}\sinh
t\sqrt{1+{1\over\sinh^2t}}
= e^{-{t^2\over6}}\cosh t. \]
\begin{thm} If $X$ has simple type, the Donaldson series of $\hat{X}=X\#\overline{\bold{CP}}^{\,2}$
is
\[ \hat{\bold{D}}=\bold{D}\cdot e^{-{E^2\over2}}\cosh E \]
where $E$ is the form dual to the exceptional class $e$, i.e. $E(\,z)=e\cdot
\,z$ for all $\,z\in H_2(\hat{X})$. Also
\[ \hat{\bold{D}}_e=-\bold{D}\cdot e^{-{E^2\over2}}\sinh E\,. \]
\end{thm}
\begin{pf} For $\alpha\in H_2(X)$ we calculate
\begin{eqnarray*}
\hat{\bold{D}}(\alpha+te) &=&
\hat{D}((1+{x\over2})\exp(\alpha)\exp(te))=D((1+{x\over2})\exp(\alpha)B(x,t))\\
&=& D((1+{x\over2})\exp(\alpha)e^{-{t^2x\over6}} e^{-{t^2\over6}}\cosh t)
\end{eqnarray*}
The simple type condition implies that $D((1+{x\over2})e^{-{t^2x\over6}})=
D((1+{x\over2})e^{-{t^2\over3}})$.
Hence
\[ \hat{\bold{D}}(\alpha+te)=\bold{D}(\alpha)e^{-{t^2\over2}}\cosh t=\bold{D}(\alpha)(e^{-{E^2\over2}}\cosh
E)(te)\]
as desired. The formula for $\hat{\bold{D}}_e$ follows similarly since
$\sinh(E)(te)=-\sinh(t)$.
\end{pf}
A $4$-manifold $X$ is said to have $c$-{\em simple type} if, for $c\in
H_2(X;\bold{Z})$,
$D_c(x^2\,z)=4\,D_c(z)$ for all $z\in \bold{A}(X)$.
It is shown in \cite{FS}, and also by Kronheimer and Mrowka, that if $X$ has
simple type, then it has
$c$-simple type for all $c\in H_2(X;\bold{Z})$. As above we have,
\begin{thm}If $X$ has $c$-simple type,
\begin{eqnarray*}
\hat{\bold{D}}_c&=&\bold{D}_c\cdot e^{-{E^2\over2}}\cosh E\,. \\
\hat{\bold{D}}_{c+e}&=&-\bold{D}_c\cdot e^{-{E^2\over2}}\sinh E\,.\ \ \ \qed
\end{eqnarray*}
\end{thm}
\newpage
|
1995-10-11T05:20:16 | 9510 | alg-geom/9510006 | en | https://arxiv.org/abs/alg-geom/9510006 | [
"alg-geom",
"math.AG"
] | alg-geom/9510006 | Yekutieli Amnon | Amnon Yekutieli | Some Remarks on Beilinson Adeles | AMSLaTeX 1.1, 6 pages, to appear in: Proc. AMS (replaced only to fix
a latex problem) | null | null | null | null | This paper contains two remarks on Beilinson's adeles with values in the De
Rham complex of a scheme. The first is an interpretation, in terms of adeles,
of the decomposition of the De Rham complex on a scheme defined modulo $p^{2}$
(the result of Deligne-Illusie). The second remark is about the possible
relation between adeles and Hodge decomposition. We work out a counter example.
| [
{
"version": "v1",
"created": "Thu, 5 Oct 1995 16:57:07 GMT"
},
{
"version": "v2",
"created": "Tue, 10 Oct 1995 18:01:53 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Yekutieli",
"Amnon",
""
]
] | alg-geom | \subsection{Introduction}
In this note we consider two aspects of Beilinson adeles on schemes.
Let $X$ be a scheme of finite type over a field $k$. Given a quasi-coherent
sheaf $\cal{M}$ let
$\underline{\Bbb{A}}^{q}_{\mathrm{red}}(\cal{M})$
be the sheaf of reduced Beilinson adeles of degree $q$ (see \cite{Be},
\cite{Hr}, \cite{HY1}). It is known that
$\underline{\Bbb{A}}^{q}_{\mathrm{red}}(\cal{M}) \cong
\underline{\Bbb{A}}^{q}_{\mathrm{red}}(\cal{O}_{X})
\otimes_{\cal{O}_{X}} \cal{M}$.
For any open set $U \subset X$
\begin{equation} \label{eqn0.1}
\Gamma(U, \underline{\Bbb{A}}^{q}_{\mathrm{red}}(\cal{M})) \subset
\prod_{\xi \in S(U)^{\mathrm{red}}_{q}} \cal{M}_{\xi}
\end{equation}
where $S(U)^{\mathrm{red}}_{q}$ is the set of reduced chains of points
in $U$ of length $q$, and $\cal{M}_{\xi}$ is the Beilinson completion
of $\cal{M}$ along the chain $\xi$ (cf.\ \cite{Ye1}). For $q=0$ and $\cal{M}$
coherent one has
$\cal{M}_{(x)} = \widehat{\cal{M}}_{x}$, the $\frak{m}_{x}$-adic completion,
and (\ref{eqn0.1}) is an equality.
Let $\Omega^{{\textstyle \cdot}}_{X/k}$ be the De Rham complex on $X$, relative to $k$.
As shown in \cite{HY1}, setting
$\cal{A}_{X}^{p, q} := \underline{\Bbb{A}}^{q}_{\mathrm{red}}(\Omega^{p}_{X/k})$
and
$\cal{A}_{X}^{i} := \bigoplus_{p+q=i} \cal{A}_{X}^{p, q}$
we get a differential graded algebra (DGA), which is quasi-isomorphic
to $\Omega^{{\textstyle \cdot}}_{X/k}$ and is flasque.
Thus
$\mathrm{H}^{{\textstyle \cdot}}(X, \Omega^{{\textstyle \cdot}}_{X/k}) =
\mathrm{H}^{{\textstyle \cdot}} \Gamma(X, \cal{A}_{X}^{{\textstyle \cdot}})$. In particular if $X$
is smooth, we get the De Rham cohomology
$\mathrm{H}^{{\textstyle \cdot}}_{\mathrm{DR}}(X/k)$.
More generally, let $\frak{X}$ be a formal scheme, of formally finite
type (f.f.t.) over $k$ (see \cite{Ye2}). Then applying the adelic
construction to the complete De Rham complex
$\widehat{\Omega}^{{\textstyle \cdot}}_{\frak{X}/k}$ we get a DGA
$\cal{A}_{\frak{X}}^{{\textstyle \cdot}}$. If $X \subset \frak{X}$ is a smooth formal
embedding (op.\ cit.) and $\operatorname{char} k = 0$ then
$\mathrm{H}^{{\textstyle \cdot}} \Gamma(X, \cal{A}_{\frak{X}}^{{\textstyle \cdot}})=
\mathrm{H}^{{\textstyle \cdot}}_{\mathrm{DR}}(X/k)$.
There is an analogy between the sheaf $\cal{A}_{X}^{p,q}$
on a smooth $n$-dimensional variety $X$ and the sheaf of smooth
$(p,q)$-forms on a complex manifold. The coboundary operator $\mathrm{D}$ of
$\cal{A}_{X}^{{\textstyle \cdot}}$ is defined as a sum
$\mathrm{D} := \mathrm{D}' + \mathrm{D}''$,
and
$\mathrm{D}'' : \cal{A}_{X}^{p, q} \rightarrow \cal{A}_{X}^{p, q+1}$
plays the role of the anti-holomorphic derivative. The
map
$\int_{X} = \sum_{\xi} \operatorname{Res}_{\xi} : \Gamma(X, \cal{A}_{X}^{2n})
\rightarrow k$
is the counterpart of the integral ($\operatorname{Res}_{\xi}$ is
the Parshin-Lomadze residue along the maximal chain $\xi$ in $X$, see
\cite{Ye1}).
This analogy to the complex manifold picture is quite
solid; for example, in \cite{HY2} there is an algebraic
proof of the Bott residue formula, which in some parts is just a
translation of the original proof of Bott to the setting of adeles.
The main purpose of this note is to examine the potential applicability of
adeles for the study of algebraic De Rham cohomology.
In \S 1 the construction of Deligne-Illusie \cite{DI} is
rewritten in terms of adeles.
In \S 2 we consider a possibility to relate adeles to Hodge theory, and
show by example its failure.
\subsection{Lifting Modulo $p^{2}$}
We interpret, in terms of adeles, the result of Deligne-Illusie
on the decomposition of the De Rham complex in characteristic $p$.
In this part we shall follow closely the ideas and notation of \cite{DI}.
Let $k$ be a perfect field of characteristic $p$.
Write $\tilde{k} := W_{2}(k)$.
Let $F_{k} : \operatorname{Spec} k \rightarrow \operatorname{Spec} k$ be the Frobenius morphism,
i.e.\ $F^{*}_{k}(a) = a^{p}$ for $a \in k$. By pullback along $F_{k}$ we
get a scheme
$X' := X \times_{k} k$ and a finite, bijective $k$-morphism
$F = F_{X/k} : X \rightarrow X'$.
Assume we are given some lifting $\tilde{X}$ of $X$ to $\tilde{k}$. By this
we mean a smooth scheme $\tilde{X}$ over $\tilde{k}$ s.t.\
$X \cong \tilde{X} \times_{\tilde{k}} k$. Using the Frobenius
$F_{\tilde{k}}$ we also define a scheme
$\tilde{X}'$, and a $\tilde{k}$-morphism
$F_{\tilde{X}} : \tilde{X} \rightarrow \tilde{X}'$.
For any point $x \in X$ the relative Frobenius homomorphism
$F^{*}_{x} : \cal{O}_{X', F(x)} \rightarrow \cal{O}_{X, x}$
can be lifted to a $\tilde{k}$-algebra homomorphism
$\tilde{F}^{*}_{x} : \cal{O}_{\tilde{X}', F(x)} \rightarrow
\cal{O}_{\tilde{X}, x}$
(cf.\ \cite{DI}). In view of (\ref{eqn0.1}), the collection
$\{ \tilde{F}^{*}_{x} \}_{x \in X}$
induces a homomorphism of sheaves of DG $\tilde{k}$-algebras
\[ \tilde{F}^{*} :
\underline{\Bbb{A}}^{0}_{\mathrm{red}}(\Omega^{{\textstyle \cdot}}_{\tilde{X}' / \tilde{k}})
\rightarrow F_{*}
\underline{\Bbb{A}}^{0}_{\mathrm{red}}(\Omega^{{\textstyle \cdot}}_{\tilde{X} / \tilde{k}}) . \]
\begin{lem} \label{lem1}
The liftings $\{ \tilde{F}^{*}_{x} \}_{x \in X}$
determine $\cal{O}_{X'}$-linear homomorphisms
\[ f : \Omega^{1}_{X' / k} \rightarrow F_{*} \cal{A}^{1, 0}_{X} \]
\[ h : \Omega^{1}_{X' / k} \rightarrow F_{*} \cal{A}^{0, 1}_{X} \]
such that
\[ \mathrm{D} (f + h) = 0 . \]
\end{lem}
\begin{pf}
Let
$\nmbf{p} : \Omega^{{\textstyle \cdot}}_{X/k} \stackrel{\simeq}{\rightarrow}
p \Omega^{{\textstyle \cdot}}_{\tilde{X}/\tilde{k}}$
be multiplication by $p$. This extends to an
$\underline{\Bbb{A}}^{0}_{\mathrm{red}}(\cal{O}_{X})$-linear isomorphism
\[ \nmbf{p} : \cal{A}^{{\textstyle \cdot}, 0}_{X} =
\underline{\Bbb{A}}^{0}_{\mathrm{red}}(\Omega^{{\textstyle \cdot}}_{X/k}) \stackrel{\simeq}{\rightarrow}
p \underline{\Bbb{A}}^{0}_{\mathrm{red}}(\Omega^{{\textstyle \cdot}}_{\tilde{X}/\tilde{k}}) . \]
Just as in \cite{DI} we get a homomorphism $f$ making the diagram
\[ \setlength{\unitlength}{0.20mm}
\begin{array}{ccc}
\Omega^{1}_{\tilde{X}'/\tilde{k}} & \lrar{\tilde{F}^{*}} &
p F_{*}
\underline{\Bbb{A}}^{0}_{\mathrm{red}}(\Omega^{1}_{\tilde{X}/\tilde{k}}) \\
\ldar{} & & \luar{\nmbf{p}} \\
\Omega^{1}_{X'/k} & \lrar{f} &
F_{*} \underline{\Bbb{A}}^{0}_{\mathrm{red}}(\Omega^{1}_{X/k})
\end{array} \]
commutative.
Next, for any chain of points $(x_{0}, x_{1})$ in $X$ and a local section
$a \in \cal{O}_{\tilde{X}'}$ we have
\[ \mathrm{D}'' \tilde{F}^{*}(a) =
\tilde{F}^{*}_{x_{0}}(a) - \tilde{F}^{*}_{x_{1}}(a) \in
p \cal{O}_{\tilde{X}, (x_{0}, x_{1})} . \]
Therefore
\[ \mathrm{D}'' \tilde{F}^{*} : \cal{O}_{\tilde{X}'} \rightarrow
p F_{*} \underline{\Bbb{A}}^{1}_{\mathrm{red}}(\cal{O}_{\tilde{X}}) \]
is a derivation which kills $p \cal{O}_{\tilde{X}'}$, and we get an
$\cal{O}_{X'}$-linear homomorphism $h$ s.t.\ the diagram
\[ \setlength{\unitlength}{0.20mm}
\begin{array}{ccc}
\cal{O}_{\tilde{X}'} & \lrar{\mathrm{D}'' \tilde{F}^{*}} &
F_{*} p
\underline{\Bbb{A}}^{1}_{\mathrm{red}}(\cal{O}_{\tilde{X}}) \\
\ldar{\mathrm{d}} & & \luar{\nmbf{p}} \\
\Omega^{1}_{X'/k} & \lrar{h} &
F_{*} \underline{\Bbb{A}}^{1}_{\mathrm{red}}(\cal{O}_{X})
\end{array} \]
commutes.
Reinterpreting the calculations of \cite{DI} in terms of adeles we see that
the following hold: for each point $x_{0} \in X$,
$\mathrm{D}' f = 0$ in $\Omega^{1}_{X/k, (x_{0})}$;
for each chain $(x_{0}, x_{1}, x_{2})$ in $X$,
$\mathrm{D}'' h = 0$ in $\cal{O}_{X, (x_{0}, x_{1}, x_{2})}$;
lastly, for each chain $(x_{0}, x_{1})$,
$\mathrm{D}'' f = - \mathrm{D}' h$ in $\Omega^{1}_{X/k, (x_{0}, x_{1})}$.
This implies that on the level of sheaves $\mathrm{D}(f + h) = 0$.
\end{pf}
\begin{prop} \label{prop1}
The liftings $\{ \tilde{F}^{*}_{x} \}_{x \in X}$
determine an $\cal{O}_{X'}$-linear homomorphism of complexes
\[ \psi_{\tilde{X}} : \bigoplus_{i = 0}^{n}
\underline{\Bbb{A}}^{{\textstyle \cdot}}_{\mathrm{red}}(\Omega^{i}_{X' / k})[-i] \rightarrow
F_{*} \cal{A}^{{\textstyle \cdot}}_{X} \]
making the diagram
\begin{equation} \label{eqn6}
\setlength{\unitlength}{0.25mm}
\begin{array}{ccc}
\bigoplus_{i} \Omega^{i}_{X' / k} & \lrar{C^{-1}} &
F_{*} \mathrm{H}^{{\textstyle \cdot}} \Omega^{{\textstyle \cdot}}_{X / k} \\
\ldar{} & & \ldar{} \\
\bigoplus_{i} \mathrm{H}^{{\textstyle \cdot}}
\underline{\Bbb{A}}^{{\textstyle \cdot}}_{\mathrm{red}}(\Omega^{i}_{X' / k})[-i] &
\lrar{\mathrm{H}^{{\textstyle \cdot}}(\psi_{\tilde{X}})} &
F_{*} \mathrm{H}^{{\textstyle \cdot}} \cal{A}^{{\textstyle \cdot}}_{X}
\end{array}
\end{equation}
commute. Here $C^{-1}$ is the Cartier operation, and the vertical arrows
are the canonical isomorphisms. Therefore $\psi_{\tilde{X}}$ is a
quasi-isomorphism.
\end{prop}
\begin{pf}
Since
\[ \underline{\Bbb{A}}^{j}_{\mathrm{red}}(\Omega^{i}_{X' / k}) \cong
\underline{\Bbb{A}}^{j}_{\mathrm{red}}(\cal{O}_{X'}) \otimes_{\cal{O}_{X'}}
\Omega^{i}_{X' / k} , \]
and since
$F^{*} : \underline{\Bbb{A}}^{{\textstyle \cdot}}_{\mathrm{red}}(\cal{O}_{X'}) \rightarrow
F_{*} \underline{\Bbb{A}}^{{\textstyle \cdot}}_{\mathrm{red}}(\cal{O}_{X})$
commutes with $\mathrm{D}''$ and is killed by $\mathrm{D}'$
(i.e.\ $\mathrm{D}' F^{*} = 0$)
it suffices to define $\cal{O}_{X'}$-linear homomorphisms
$\psi_{\tilde{X}}^{i} : \Omega^{i}_{X' / k} \rightarrow F_{*} \cal{A}^{i}_{X}$
s.t.\ $\mathrm{D} \psi_{\tilde{X}}^{i} = 0$.
Define
$\psi_{\tilde{X}}^{0} := F^{*}$, and
$\psi_{\tilde{X}}^{1} := f+h$ as in Lemma \ref{lem1}. For
$1 \leq i \leq n$ let
$\nmbf{a} : \Omega^{i}_{X' / k} \rightarrow (\Omega^{1}_{X' / k})^{\otimes i}$
be the anti-symmetrizing operator (this makes sense since $n < p$;
cf.\ \cite{DI}), and define $\psi_{\tilde{X}}^{i}$ by
\[ \setlength{\unitlength}{0.30mm}
\begin{array}{ccc}
(\Omega^{1}_{X' / k})^{\otimes i} &
\lrar{(\psi_{\tilde{X}}^{1})^{\otimes i}} &
(F_{*} \cal{A}^{1}_{X})^{\otimes i} \\
\luar{\nmbf{a}} & & \ldar{\mathrm{product}} \\
\Omega^{i}_{X' / k} & \lrar{\psi_{\tilde{X}}^{i}} &
F_{*} \cal{A}^{i}_{X}
\end{array} \]
Let $a \in \cal{O}_{\tilde{X}}$
be a local section, with corresponding pullback
$a \otimes 1 \in \cal{O}_{\tilde{X}'}$, and with image
$a_{0} \in \cal{O}_{X}$. Then according to the calculations in \cite{DI},
we have
$\tilde{F}^{*}(a \otimes 1) = a^{p} + \nmbf{p} u$
for some local section
$u \in \underline{\Bbb{A}}^{0}_{\mathrm{red}}(\cal{O}_{X})$. Therefore
$f (\mathrm{d} a_{0} \otimes 1) = a_{0}^{p-1} \mathrm{d} a_{0} +
\mathrm{D}' u$ and
$h (\mathrm{d} a_{0} \otimes 1) = \mathrm{D}'' u$,
so
\[ \psi_{\tilde{X}} (\mathrm{d} a_{0} \otimes 1) =
a_{0}^{p-1} \mathrm{d} a_{0} + \mathrm{D} u . \]
This means that
\[ \mathrm{H}^{1} (\psi_{\tilde{X}}) = C^{-1} : \Omega^{1}_{X' / k} \stackrel{\simeq}{\rightarrow}
F_{*} \mathrm{H}^{1} \Omega^{{\textstyle \cdot}}_{X / k} \cong
F_{*} \mathrm{H}^{1} \cal{A}^{{\textstyle \cdot}}_{X} . \]
Clearly in degree $0$,
$\mathrm{H}^{0} (\psi_{\tilde{X}}) = F^{*} = C^{-1}$.
Since the vertical arrows in diagram (\ref{eqn6}) are isomorphisms of
(sheaves of) graded algebras, it follows that
$\mathrm{H}^{{\textstyle \cdot}} \cal{A}^{{\textstyle \cdot}}_{X}$ is a graded-commutative algebra,
and therefore
\[ \mathrm{H}^{{\textstyle \cdot}}(\psi_{\tilde{X}}) :
\bigoplus_{i} \Omega^{i}_{X' / k} \rightarrow F_{*} \mathrm{H}^{{\textstyle \cdot}}
\cal{A}^{{\textstyle \cdot}}_{X} \]
is a homomorphism of graded algebras. But then
$\mathrm{H}^{{\textstyle \cdot}}(\psi_{\tilde{X}}) = C^{-1}$ in all degrees, and it's
an isomorphism.
\end{pf}
Of course in the derived category the map
$\psi_{\tilde{X}}$ is independent of the choices of Frobenius liftings.
\subsection{A Hodge-type Decomposition?}
The second aspect is a naive attempt to use adeles for a Hodge-type
decomposition of De Rham cohomology. Suppose $\operatorname{char} k = 0$ and
$X$ is smooth over $k$, of dimension $n$. For any $0 \leq p, q \leq n$
define a canonical subspace
\begin{equation}
\mathrm{H}^{p,q} :=
\frac{ \Gamma(X, \cal{A}^{p,q}_{X}) \cap \operatorname{Ker} D }
{ \Gamma(X, \cal{A}^{p,q}_{X}) \cap \operatorname{Im} D }
\subset \mathrm{H}^{p+q}_{\mathrm{DR}}(X/k)
\end{equation}
(cf.\ \cite{GH} p.\ 116).
Since the sheaves $\cal{A}^{p, q}_{X}$ imitate the Dolbeault sheaves on
a complex manifold so nicely, one can imagine that
\[ \mathrm{H}^{i}_{\mathrm{DR}}(X/k) =
\bigoplus_{p+q=i} \mathrm{H}^{p, q} \]
if $X$ is proper.
Yet this is {\em false}, as can be seen from the example below.
What we get is a serious breakdown in the analogy to smooth forms on
a complex manifold. I should mention that even in \cite{HY2} there
was a breakdown in this analogy; there it was not possible to define
a connection on the adelic sections of a vector bundle, and hence
an auxiliary algebraic device, the sheaf
$\tilde{\cal{A}}_{X}^{{\textstyle \cdot}}$ of Thom-Sullivan adeles,
had to be introduced.
\begin{prob}
Is it true that for $X$ smooth, the filtration on
$\cal{A}_{X}^{{\textstyle \cdot}}$ by the subcomplexes
$\cal{A}_{X}^{{\textstyle \cdot}, \geq q}$
induces the coniveau filtration on
$\mathrm{H}^{{\textstyle \cdot}}_{\mathrm{DR}}(X/k)$?
\end{prob}
\begin{exa} \label{exa1}
Suppose $k$ is algebraically closed and $X$ is an elliptic
curve. Then
$\operatorname{dim} \mathrm{H}^{1}_{\mathrm{DR}}(X/k) = 2$.
Consider the nondegenerate pairing on $\mathrm{H}^{1}_{\mathrm{DR}}(X/k)$
given by
\[ \langle [\alpha] , [\beta] \rangle =
\int_{X} [\alpha] \smile [\beta] =
\sum_{\xi} \operatorname{Res}_{\xi} (\alpha \cdot \beta) \]
for adeles $\alpha, \beta \in \cal{A}^{1}_{X}$. We see that
$\langle \mathrm{H}^{1, 0} , \mathrm{H}^{1, 0} \rangle =
\langle \mathrm{H}^{0, 1} , \mathrm{H}^{0, 1} \rangle = 0$.
Therefore if
$\mathrm{H}^{1}_{\mathrm{DR}}(X) = \mathrm{H}^{1, 0} + \mathrm{H}^{0, 1}$,
then
$\operatorname{dim} \mathrm{H}^{1, 0} = \operatorname{dim} \mathrm{H}^{0, 1} = 1$.
It is easy to find
$0 \neq [\alpha] \in \mathrm{H}^{1, 0}$;
take any $0 \neq [\alpha] \in \Gamma(X, \Omega^{1}_{X/k})$.
On the other hand an adele
\[ \beta = (b_{(\mathrm{gen}, x)}) \in \Gamma(X, \cal{A}^{0, 1}_{X}) =
\Bbb{A}^{1}_{\mathrm{red}}(X, \cal{O}_{X}) \]
(where $x$ runs over the set $X_{0}$ of closed points, and $\mathrm{gen}$
is the generic point) satisfies $\mathrm{D} \beta = 0$ iff
$\mathrm{d} b_{(\mathrm{gen}, x)} = 0$ for every $x$. This forces
$b_{(\mathrm{gen}, x)} \in k$. But taking
$b = (b_{(\mathrm{gen})}, b_{(x)}) \in
\Bbb{A}^{0}_{\mathrm{red}}(X, \cal{O}_{X})$, with
$b_{(\mathrm{gen})} = 0$, $b_{(x)} = b_{(\mathrm{gen}, x)}$
we get $\beta = \mathrm{D} b$. Hence
$\mathrm{H}^{0, 1} = 0$.
\end{exa}
\begin{prob} \label{prob2}
For $\alpha$ as above find explicitly a cocycle
$\beta \in \Gamma(X, \cal{A}^{1}_{X})$ s.t.\
$\langle \alpha , \beta \rangle = 1$.
\end{prob}
The best I can do is:
\begin{prop} \label{prop2}
Suppose $X$ is a smooth proper curve and $k$ is algebraic\-ally closed.
Let $\alpha_{(\mathrm{gen})} \in \Omega^{1}_{k(X) / k}$ be a differential
of the $2$-nd kind, namely
$\operatorname{Res}_{(\mathrm{gen}, x)} \alpha_{(\mathrm{gen})} = 0$
for every $x \in X_{0}$.
Then it defines a cocycle
$\alpha \in \Gamma(X, \cal{A}^{1}_{X})$
whose component at $(\mathrm{gen})$ is $\alpha_{(\mathrm{gen})}$.
Every cohomology class in $\mathrm{H}^{1}_{\mathrm{DR}}(X / k)$
is gotten in this way. The Hodge filtration is induced by the differentials
of the $1$-st kind.
\end{prop}
\begin{pf}
The adele $\alpha$ will be given by its bihomogeneous components,
$\alpha = \alpha^{1,0} + \alpha^{0,1}$. We set
$\alpha^{1,0} := (\alpha_{(\mathrm{gen})}, \alpha_{(x)})$
where for $x \in X_{0}$, $\alpha_{(x)} = 0$.
Since
$\operatorname{Res}_{(\mathrm{gen}, x)} \alpha_{(\mathrm{gen})} = 0$
there is some
$a_{(\mathrm{gen}, x)} \in k(X)_{(\mathrm{gen}, x)}$
(unique up to adding a constant) s.t.\
$\mathrm{d} a_{(\mathrm{gen}, x)} = \alpha_{(\mathrm{gen})}$.
Set
$\alpha^{0,1} := ( a_{(\mathrm{gen}, x)} )$.
Then $\alpha$ is evidently a cocycle.
If $\alpha_{(\mathrm{gen})}$ is of the $1$-st kind then actually we get
$a_{(\mathrm{gen}, x)} \in \cal{O}_{X, (x)}$; call this element also
$a_{(x)}$.
So we can define an adele
$\tilde{\alpha} = \tilde{\alpha}^{1,0} + \tilde{\alpha}^{0,1}$
with
$\tilde{\alpha}^{1,0} := (\alpha_{(\mathrm{gen})}, \mathrm{d} a_{(x)} )$
and
$\tilde{\alpha}^{0,1} := 0$. We get a cocycle (cohomologous to $\alpha$),
and conversely any
cocycle in $\Gamma(X, \cal{A}^{1,0}_{X})$ looks like this.
Consider the niveau spectral sequence of De Rham homology (cf.\ \cite{Ye3}).
A comparison of dimensions shows that this degenerates at the $E_{2}$
term. Also the niveau filtration on $\mathrm{H}_{1}^{\mathrm{DR}}(X / k)$
is trivial. Hence we get
\[ \begin{array}{rcl}
\mathrm{H}_{1}^{\mathrm{DR}}(X / k) & = &
\operatorname{Ker} \left( \mathrm{H}^{1} \Omega^{{\textstyle \cdot}}_{k(X) / k} \rightarrow
\bigoplus_{x \in X_{0}} k \right) \\
& \cong & (\text{forms of the $2$-nd kind}) / (\text{exact forms}) .
\end{array} \]
Now the map
$\mathrm{H}^{1}_{\mathrm{DR}}(X / k) \rightarrow
\mathrm{H}_{1}^{\mathrm{DR}}(X / k)$,
$[\alpha] \mapsto [\alpha] \frown [X] = \pm [\mathrm{C}_{X} \cdot \alpha]$
is bijective. A direct inspection reveals that the adele
$\alpha = \alpha^{1,0} + \alpha^{0,1}$
is sent to the differential of the second kind
$\alpha_{(\mathrm{gen})} \in \Omega^{1}_{k(X)/k}$.
\end{pf}
|
1995-10-03T05:20:28 | 9510 | alg-geom/9510003 | en | https://arxiv.org/abs/alg-geom/9510003 | [
"alg-geom",
"hep-th",
"math.AG",
"math.QA",
"q-alg"
] | alg-geom/9510003 | Nakajima Hiraku | Hiraku Nakajima | Instantons and affine Lie algebras | 12 pages, AMSLaTeX v 1.1 | Nucl.Phys.Proc.Suppl.46:154-161,1996 | 10.1016/0920-5632(96)00017-5 | null | null | Various constructions of the affine Lie algebra action on the homology group
of moduli spaces of instantons on 4-manifolds are discussed. The analogy
between the local-global principle and the role of mass is also explained. The
detailed proofs are given in separated papers \cite{Na-algebra,Na-Hilbert}.
| [
{
"version": "v1",
"created": "Thu, 2 Nov 1995 18:36:05 GMT"
}
] | 2011-07-19T00:00:00 | [
[
"Nakajima",
"Hiraku",
""
]
] | alg-geom | \section{Introduction}
Vafa and Witten \cite{VW} introduced topological invariants\footnote{
The author does not know how to define their invariants in
a mathematically rigorous way. The difficulty lies in the lack of the
compactness of relevant moduli spaces.}
for $4$-manifolds using $N = 4$ topological supersymmetric Yang-Mills
theory.
Then the $S$-duality conjecture implies that
the generating function of those invariants is a
modular form of certain weight, where the summation runs over
all $\operatorname{SU}(2)$ or $\operatorname{SO}(3)$-principal bundles
of any topological types.
(In general, it has the modular invariance
only for $\Gamma_0(4)$, a subgroup of $\operatorname{SL}(2,\Bbb Z)$.)
For some $4$-manifolds, they identify those invariants with the
Euler numbers of the instanton moduli spaces.
Then they can check the modular invariance
for various $4$-manifolds, using mathematical results,
Yoshioka's formulae \cite{Yo} for ${\Bbb P}^2$ and the blow up,
G\"ottsche and Huybrechts's result \cite{GotHu} for the K3 surface,
and the author's result for ALE spaces.
On the other hand, the author's motivation of the study
\cite{Na-quiver,Na-gauge,Na-algebra} of the homology groups of the
instanton moduli spaces on ALE spaces is totally different.
The author's motivation was trying to understand Ringel \cite{Ri} and
Lusztig's \cite{Lu1,Lu2} constructions of the lower triangular part
${\bold U}_q^-$ of the quantized enveloping algebra.
They used the moduli spaces of representations of quivers, and their
cotangent bundle\footnote{They are not cotangent bundles rigorously.
The situation is very much similar to the relation between the moduli
space of vector bundles over a curve and Hitchin's moduli space of
Higgs bundles.}
can be identified with the instanton moduli spaces on ALE spaces,
via the ADHM description \cite{KN}.
The author showed that the generating function of the Euler numbers of
the instanton moduli spaces on ALE spaces
becomes the character of the affine Lie algebra, which has been
known to have modular transformation property by
Kac-Peterson (see \cite{Kac}).
The definition of the affine Lie algebra representation on the
homology group of the instanton moduli spaces is very geometric, and
seems to be generalized, at least, to projective surfaces.
The results of similar direction are announced recently by
Ginzburg-Kapranov-Vasserot \cite{GKV} and Grojnowski \cite{Gj}.
Unfortunately, our construction depends heavily on the complex
structure of the base manifold.
It is a challenging problem to generalize the construction to more
general $4$-manifolds.
One may need to reformulate the homology group of the moduli spaces, etc.....
\subsection*{Acknowledgements}
The author's understanding of the analogy between the local-global
principle and the theory of mass came from lectures given by the
seminar on Seiberg-Witten theory organized by K.~Ueno.
He would like to speakers, especially S.-K.Yang.
He also thank to G.~Moore and K.~Yoshioka for valuable discussions.
\section{The Hilbert scheme of points and the Heisenberg algebra:
Twist around points}
In this section, we study the relationship between the Hilbert scheme
of points and the Heisenberg algebra.
The reasons why we study the Hilbert scheme are (a) it is a toy model
for moduli spaces of instantons, (b) it appears in the boundary of
the compactification of the instanton moduli spaces over projective
surfaces, and (c) its homology group is isomorphic to that of a moduli
space for some special cases \cite{GotHu}.
We explain the reason~(b) a little bit more. Since the instanton
moduli spaces are usually noncompact, one must compactify them to
consider their Euler numbers. When the base manifold is a projective
surface, the results of Donaldson and Uhlenbeck-Yau enable us to
identify the instanton moduli space with the moduli space of
$\mu$-stable holomorphic vector bundles (Hitchin-Kobayashi
correspondence). Then the one of the most
natural compactifications seems to be Gieseker-Maruyama's
compactifications $\overline{\frak M}$\nobreak
{}~\footnote{The Gieseker-Maruyama's compactifications are not smooth in
general. In fact, Vafa-Witten's formula for the K3 surface gives the
fractional Euler number. This may be the contribution of the
singularities.},
namely moduli spaces of semi-stable torsion free sheaves.
If $\cal E$ is a torsion free sheaf which is not locally free, its
double dual ${\cal E}^{\vee\vee}$ is a locally free sheaf and we have
an exact sequence
\begin{equation*}
0 @>>> {\cal E} @>>> {\cal E}^{\vee\vee} @>>>
{\cal E}^{\vee\vee}/{\cal E} @>>> 0.
\end{equation*}
Thus ${\cal E}$ can be determined by (a) ${\cal E}^{\vee\vee}$ and
(b) ${\cal E}^{\vee\vee}\to{\cal E}^{\vee\vee}/{\cal E}$.
The double dual ${\cal E}^{\vee\vee}$ is contained in the interior of
$\overline{\frak M}$, but in the different component with lower second
Chern number.
Thus it is natural to expect that those studies
can be decomposed into two parts, the interior~(a) and
the quotient map~(b).
And the variety of the quotient map~(b),
which depends only on the rank of $\cal E$ and
the length of ${\cal E}^{\vee\vee}/{\cal E}$, looks very much like the
Hilbert scheme of points.
In fact, the Hilbert scheme is the special case
${\cal E}^{\vee\vee} = {\cal O}$.
The Betti numbers of the variety was computed by
Yoshioka~\cite[0.4]{Yo}.
Let $X$ be a projective surface defined over $\Bbb C$.
Let $\Hilbn{X}$ be the component of the
Hilbert scheme of $X$ parameterizing the ideals of ${\cal O}_X$ of
colength $n$.
It is smooth and irreducible.
Let $S^n X$ denotes the $n$-th symmetric product of $X$.
It parameterizes formal linear combinations $\sum n_i [x_i]$ of points
$x_i$ in $X$ with coefficients $n_i\in{\Bbb Z}_{> 0}$ with $\sum n_i = n$.
There is a canonical morphism $\pi\colon \Hilbn{X}\to S^n X$
defined by
\begin{equation*}
\pi({\cal J}) = \sum_{x\in X}
\operatorname{length}({\cal O}_X/{\cal J})_x [x].
\end{equation*}
It is known that $\pi$ is a resolution of singularities.
G\"ottsche \cite{Got} computed the generating function of the
Poincar\'e polynomials
\begin{equation*}
\begin{split}
& \sum_{n=0}^\infty q^n P_t(\Hilbn{X}) \\
= &\prod_{m=1}^\infty
\prod_{i=0}^4 \, (1 - (-t)^{2m - 2 + i}q^m)^{(-1)^{i+1}b_i(X)}\, ,
\end{split}
\end{equation*}
where $b_i(X)$ is the Betti number of $X$.
It was shown that the Euler number of $\Hilbn{X}$ is equal to
the orbifold Euler number of $S^n X$ by Hirzebruch-Hofer \cite{HH}.
It was also pointed out by Vafa and Witten that this is equal to the
character of the Fock space.
We shall construct the representation of the Heisenberg and
Clifford algebras in a geometric way. The key point is to introduce
appropriate ``Hecke correspondence'' which give the generators of the
Heisenberg/Clifford algebra.
Take a basis of $H_*(X)$ and
assume that each element is represented by a (real) compact submanifold
$C^a$. ($a$ runs over $1, 2, \dots, \dim H_*(X)$.)
Take a dual basis for $H_*(X)$, and assume
that each element is also represented by a submanifold $D^a$.
(Those assumptions are only for the brevity. The modification to
the case of cycles is clear.)
For each $a = 1,2,\dots,\dim H_*(X)$, $n = 1,2,\dots$ and
$i=1,2,\dots$, we introduce cycles of products of the Hilbert
schemes by
\begin{equation*}
\begin{split}
&E_i^a(n) = \{\, ({\cal J}_1,{\cal J}_2)
\in\HilbX{n-i}\times\HilbX{n} \mid {\cal J}_1\supset {\cal J}_2 \\
& \quad\text{and
$\Supp({\cal J}_1/{\cal J}_2) = \{ p\}$ for some $p\in D^a$}\;
\}, \\
&F_i^a(n) = \{\, ({\cal J}_1,{\cal J}_2)\in\HilbX{n+i}\times\HilbX{n}
\mid {\cal J}_1\subset {\cal J}_2 \\
& \quad\text{and
$\Supp ({\cal J}_2/{\cal J}_1) = \{ p \}$ for some $p\in C^a$}\;\}.
\end{split}
\end{equation*}
Then we define an endomorphism $H_*(\HilbX{n}) \to H_*(\HilbX{n-i})$ by
\begin{equation*}
c\mapsto (p_1)_* ([E_i^a(n)]\cap p_2^* c),
\end{equation*}
where $p_1$, $p_2$ are projections of the first and second factor of
$\HilbX{n-i}\times\HilbX{n}$ and $p_2^* c = [\HilbX{n-i}]\times c$ and
$(p_1)_*$ is a push-forward.
Similarly, we have an endomorphism
$H_*(\HilbX{n}) \to H_*(\HilbX{n+i})$ using $F_i^a(n)$.
Collecting the operators with respect to $n$, we have operators
$[E_i^a]$, $[F_i^a]$ acting on the direct sum
$\bigoplus_n H_*(\HilbX{n})$.
Then
\begin{Theorem}. The following relations hold as operators on
$\bigoplus_{n} H_*(\HilbX{n})$.
\begin{gather*}
[E_i^a] [E_j^b] =
(-1)^{\dim D^a\dim D^b}[E_j^b] [E_i^a]\\
[F_i^a] [F_j^b] =
(-1)^{\dim C^a\dim C^b}[F_j^b] [F_i^a]\\
[E_i^a] [F_j^b] =
(-1)^{\dim D^a\dim C^b}[F_j^b] [E_i^a]\\
\qquad\qquad+ \delta_{ab}\delta_{ij}c_i \operatorname{Id},
\end{gather*}
where $c_i$ is a
nonzero integer depending only on $i$ \rom(independent of $X$\rom).
\label{th:main}\end{Theorem}
In particular, for each fixed $a$, $[E_i^a]$, $[F_i^a]$ ($i =
1,\dots$) define the action of the Heisenberg or Clifford algebra
according to the parity of $\dim C^a$.
Moreover, comparing with G\"ottsche's formula, we can conclude our
representation is irreducible.
The definition of the correspondence $E_i^a$, $F_i^a$ can be naturally
generalized to the case of moduli spaces of torsion free sheaves (see
\cite{Gj}).
However, the author has no idea to generalize to more general
$4$-manifolds.
\section{Elementary transformation: Twist along embedded submanifolds}
The opearator of the previous section twists sheaves around points.
There is another kind of operator which twists sheaves along an
embedded $2$-dimensional submanifold.
This operation is called the elementary transformation in the
literature.
Suppose $C$ is a holomorphic curve embedded in a projective surface
$X$.
Let $i$ denote the inclusion map.
Let $\frak M$ be the moduli space of $\operatorname{U}(r)$-instantons,
namely Einstein-Hermitian connections.
We identify it with the moduli space of holomorhic vector bundles over
$X$ by the Hitchin-Kobayashi correspondence.
It decomposes by the first and second Chern classes.
For each integer $d$, we also consider the following
moduli space ${\frak P}$ of parabolic bundles
$({\cal E_1},{\cal E_2},\varphi)$ where
$\cal E_i$ is a holomorphic vector bundle over $X$, and
$\varphi\colon{\cal E_1} \to {\cal E_2}$ is an injection which is an
isomorphism outside $C$.
In order to define the moduli space, we need to introduce the notion
of the stability to parabolic bundles (see \cite{MY}),
for this we need to choose an ample line bundle $L$ or the K\"ahler
metric which is a curvature of $L$.
Moreover, it is necessary to consider the Gieseker-Maruyama
compactification of moduli spaces, as explained in the previous
section.
But we do not go in detail.
There is a morphism $f\colon {\frak P}\to {\frak M}\times {\frak M}$.
Then we can define two operators on the homology group exactly as in the
previous section:
\begin{equation*}
\begin{split}
c &\mapsto (p_1)_* (f_*[{\frak P}]\cap p_2^* c),\\
c &\mapsto (p_2)_* (p_1^* c\cap f_*[{\frak P}])
\end{split}
\end{equation*}
Note that the first and second Chern classes are not preserved this
operator.
Strictly speaking, we do not have a globally defined morphism
since the stability conditions for the parabolic bundles
and their underlying vector bundles are not equivalent in general.
But it is enough for our purpose to have $f_*[{\frak P}]$ as an
element of homology group of ${\frak M}\times {\frak M}$.
For example, $f$ could be a meromorphic map.
Since we do not know what is the right setting for general
projective surfaces\footnote{If one could define the Hecke operators using
Kronheimer-Mrowka's singular anti-self-dual connections, they might be
the right setting.}, we focus on particular examples, namely ALE
spaces.
The ALE spaces are the minimal resolution of simple singularities
${\Bbb C}^2/\Gamma$, where $\Gamma$ is a finite subgroup of
$\operatorname{SU}(2)$.
The second homology group $H_2$ of the ALE space is spanned by
the irreducible components $\Sigma_1,\dots, \Sigma_n$ of the exceptional set,
which are the projective line.
The intersection matrix is the negative of the Cartan matrix of type
ADE.
The classification of simple singularities are given by the Dynkin
graphs in this way.
In particular, there is a bijection between simple singularities and
simple Lie algebra of type ADE.
The rank is equal to the number of the number of the irreducible
components, namely $n$.
By the work of Kronheimer, it is known that
they have hyper-K\"ahler metrics.
There are a variant of the ADHM description, which identifies the
framed moduli spaces of instantons, or more precisely, torsion-free
sheaves with the cotangent bundles of the moduli space
of representations of quivers of affine Dynkin graphs.
Since we shall work on non-compact spaces, we have extra discrete
parameters which parameterizes the boundary condition.
We consider instantons which converge to a flat connection at the end
of the ALE space $X$.
The flat connection on the end can be classified by its monodromy,
namely a representation $\rho$ of the finite group $\Gamma$.
Let $\rho_0$, $\rho_1$, \dots, $\rho_n$ be the irreducible
representations of $\Gamma$ with $\rho_0$ the trivial representation.
By the McKay correspondence, there is a bijection between the vertices
of the affine Dynkin graph and the irreducible representations
(see \cite{Na-gauge} for more detail).
The monodromy representation $\rho$ is decomposed as
$\rho = \bigoplus_{k=0}^n \rho_k^{\oplus w_k}$, where $w_k$ is the
multiplicity. This datum will be preserved under the Hecke operator.
Corresponding to each irreducible component $\Sigma_k$,
we take a component of the moduli space of parabolic bundles where
${\cal E_2}/\varphi({\cal E_1})$ is rank $1$ and degree $-1$.
We then define operators
$e_k$ and $f_k$ on the homology group of the moduli space as above.
We also have an operator $\alpha_k^\vee$ which is the multiplication
by $-\langle c_1, [\Sigma_k]\rangle$ on the homology class belonging
to the component with the first Chern class $c_1$.
For $k = 0$, we can define similar operators $e_0$, $f_0$ by replacing
${\cal O}_{\Sigma_k}(-1)$ by a sheaf ${\cal O}_{\bigcup \Sigma_k}$.
The operator $\alpha_0^\vee$ is defined so that
\begin{equation*}
\sum_{k=0}^n \dim\rho_k \alpha_k^\vee
= \operatorname{rank} E\, \operatorname{Id}.
\end{equation*}
Finally define the operator $d$ to detect the instanton number.
Namely the mulitiplication by
\begin{equation*}
-\int_X \operatorname{ch}({\cal E}).
\end{equation*}
on the homology group of the each component of the moduli spaces.
\begin{Theorem}.
Operators $\alpha_k^\vee$, $e_k$, $f_k$ \rom($k = 0, \dots, n$\rom), $d$
satisfy the relation of the affine Lie algebra corresponding to the
extended Dynkin graph.
Moreover, the representation on $H_*({\frak M})$ is integrable.
\end{Theorem}
The irreducible decomposition of the representation is complicated,
but we have one irreducible factor whose geometric meaning is clear.
\begin{Theorem}.
If we take the middle degree part of the homology group
$H_*({\frak M})$ \rom(since the dimension of $\frak M$ are changing on
components, the middle degree also changes\rom), it is preserved by
the affine Lie algebra action.
Moreover, it is the integrable highest weight representation with the
highest weight vector $\,{}^{t}(w_0,\dots,w_n)$.
The level is equal to the rank of the vector bundle.
\end{Theorem}
The highest weight vector lies in the paticularly chosen moduli space
which consists of a single point.
\section{Local-Global Principle and the Mass}
Historically the Hecke operators were originally introduced in the
theory of modular forms. There are also analogoues operators in the
theory of the moduli spaces vector bundles over curves \cite{NR},
which are used in the geometric Langlands program.
Our operators can be considered as natural complex $2$-dimensional
analogue of these operators.
The importance of the Hecke operators comes from the fact
that they lie in the heart of the ``local-global principle''.
We shall explain it only very briefly.
The interested reader should consult to good literatures about the
Hecke operators and the modular forms (see e.g., \cite{Langlands}).
The local-global principle roughly says that
a global problem could be studied as a collection of local
problems.
The basic example is the Hasse principle: A quadratic from with
integer coefficients has a nontrivial integer solution if and only if it
has real solution and a $p$-adic solution.
In this case the global problem is to find an integer solution and the
local problems are find solutions in $\Bbb R$ and $\Bbb Q_p$.
Thus the base manifold, which is the parameter space of the local
places, is the set of prime numbers plus infinity $\Bbb R$.
The theory of modular forms are also examples of the global-local
principle.
Consider the space of modular forms of weight $k$, which can be
considered as functions of lattices in
$\Bbb C$ with homogeneous degree $-k$, i.e., $F(\lambda L) =
\lambda^{-k}F(L)$.
Then for each prime number $p$, we define the Hecke operator $T(p)$ by
\begin{equation*}
(T(p) F)(L) = p^{k-1} \sum_{[ L' : L] = p} F(L'),
\end{equation*}
where the summation runs over the set of sublattices of $L$ with index
$p$.
These operators commute each other.
If a modular form is a simultaneous eigenfunction, its $L$-function
has an Euler product expansion. The analogy between the modular forms
and $4$-dimensional gauge theory our theory are given in the table.
\begin{table}[hbt]
\begin{tabular}{l|l}
\hline
prime numbers & points in a $4$-manifold \\
& \ and submanifolds $C$\\
\hline
the space of & the homology gruop \\
\ modular forms & \ of moduli spaces \\
\hline
Hecke operators & our Hecke operators\\
\hline
\end{tabular}
\end{table}
In physics, there is a good explanation why the local-global principle
holds in some topological field theories\footnote{Topological field
theories of cohomological type according to the terminology in
\cite{CMR}}. In these theories, topological invariants, like
Donaldson's invariants, are expressed as correlation functions. It is
not by no means obvious that the results are topological invariants,
since one needs to introduce a Riemannian metric for the definition of
the Lagrangian. However, by a clever choice of the Lagrangian, the
resulted correlation functions are independent of the choice of the
metric. The mechanism is just like the fact that the euler class,
which is defined as the pfaffian of the curvature, is a topological
invariant. Thus one can take a family of Riemannian metrics $g_t =
tg$ with $t > 0$, and consider the limiting behaviour for $t\to
\infty$. In the limit, the distance of two different points goes to
infinity. Hence if the ``mass'' of all particles is not zero, there
are no interaction between two points. Then one can compute the
correlation functions by integrating local contributions over the base
manifold. In this sense, the local-global principle holds in this
theory.
In $N = 1$ topological supersymmetric Yang-Mills theory, it is
believed that all particles have non-trivial mass. However in $N = 2$
topological supersymmetric Yang-Mills theory, which is relevant to
Donaldson's invariants, it is no longer true. Hence there may be
massless particles which make interaction even when the distance
between two points are very large. Anyway, if all particles would
have non-trivial mass,
Donaldson's invariants would depend only on homology classes of the
underlying manifolds. On a K\"ahler manifold $X$
with a non-trivial holomorphic $2$-form $\omega$, Witten
\cite{Wi-SUSY} used a
perturabation from the $N = 2$ theory to the $N = 1$ theory adding a
term depending on $\omega$. The remarkable observation was that there
remain particles which have zero mass where $\omega$ vanishes.
Unless the manifold $X$ is a K3 surface or a
torus, $\omega$ vanishes along a divisor $C$.
Thus the local-global principle holds in the $N = 2$ theory with only
one modification; there are non-trivial contribution from $C$.
In other words, the $2$-dimensional submanifold $C$ cannot be divided
any more, and should be considered as a point.
Similarly, in \cite{VW}, again for the same class of K\"ahler
manifolds, the $N = 4$ theory was perturbed to the $N = 1$ theory, and
the correlation function was calculated in a similar way.
And finally, Witten conjectured \cite{Wi-monopole}
that the Kronheimer-Mrowka's basic classes \cite{KrMr}
coincide with homology classes whose Seiberg-Witten invariants are
nonzero.
It means that the local-global principle fails exactly along basic
classes.
Now it becomes clear why we must introduce two kinds of Hecke
correspondences, twist along a point and twist along a $2$-dimensional
submanifold.
For the Hilbert scheme on a projective surface, it is apparent that
the local-global principle holds without the introduction of $C$.
This should be the basic reason why the homology group of the Hilbert
scheme is generated only by the first kind of the Hecke correspondence.
For higher rank case, we mighty need the second type of the
Hecke correspondence in order to get all homology classes in the
moduli spaces.
However, the relation between two kinds of Hecke operators is not clear, at
this moment.
Moreover, there might be other types of correspondences which is
useful to describe the homology group of the moduli space.
For example, G\"ottsche and Huybrechts
\cite{GotHu} used an interesting correspondence in order to relate moduli
spaces of rank $2$ bundles and Hilbert schemes.
Finally, we would like to point out the difference between our
situation and the classical one (i.e., the Hecke operators on
modular forms).
The first kind of the Hecke
operator is independent of the choice of the representative $C^a$ of
the homology class.
This is because we are studing the homology group of the moduli space.
It is not clear that the second kind of the Hecke operator depends
only on the homology class of $C$. But it depends only on the rational
equivalence class.
|
1995-12-06T06:20:13 | 9510 | alg-geom/9510010 | en | https://arxiv.org/abs/alg-geom/9510010 | [
"alg-geom",
"math.AG"
] | alg-geom/9510010 | Zhenbo Qin | Zhenbo Qin and Yongbin Ruan | Quantum cohomology of projective bundles over $\Pee^n$ | AMS-TEX Version 2.1 | null | null | null | null | Results in the preliminary version have been strengthed. In addition,
Batyrev's conjectural formula for quantum cohomology of projective bundles
associated to direct sum of line bundles over $\Pee^n$ is partially verified.
| [
{
"version": "v1",
"created": "Sat, 14 Oct 1995 02:03:31 GMT"
},
{
"version": "v2",
"created": "Tue, 5 Dec 1995 18:48:12 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Qin",
"Zhenbo",
""
],
[
"Ruan",
"Yongbin",
""
]
] | alg-geom | \section{1. Introduction}
Quantum cohomology, proposed by Witten's study \cite{16} of
two dimensional nonlinear sigma models, plays a fundamental role
in understanding the phenomenon of mirror symmetry for Calabi-Yau manifolds.
This phenomenon was first observed by physicists motivated by
topological field theory. A topological field theory starts
with correlation functions. The correlation functions of sigma model
are linked with the intersection numbers of cycles in the moduli space
of holomorphic maps from Riemann surfaces to manifolds.
For some years, the mathematical construction of these correlation functions
remained to be a difficult problem because the moduli spaces
of holomorphic maps usually are not compact and may have wrong dimension.
The quantum cohomology theory was first
put on a firm mathematical footing by \cite{12,13} for semi-positive
symplectic manifolds (including Fano and Calabi-Yau manifolds),
using the method of symplectic topology. Recently, an algebro-geometric
approach has been taken by \cite{8,9}. The results of \cite{12,13} have been
redone in the algebraic geometric setting for the case of homogeneous spaces.
The advantage of homogeneous spaces is that the moduli spaces of
holomorphic maps always have expected dimension and
their compactifications are nice. Beyond the homogeneous spaces,
one can not expect such nice properties for the moduli spaces.
The projective bundles are perhaps the simplest examples.
However, by developing sophisticated excessive intersection theory,
it is possible that the algebro-geometric method can work for
any projective manifolds.
In turn, it may shed new light to removing the semi-positive condition
in the symplectic setting.
Although we have a solid foundation for quantum cohomology theory at least for
semi-positive symplectic manifolds, the
calculation remains to be a difficult task.
So far, there are only a few examples
which have been computed, e.g., Grassmannian \cite{14},
some rational surfaces \cite{6},
flag varieties \cite{4}, some complete intersections \cite{3},
and the moduli space of stable bundles over Riemann surfaces \cite{15}.
One of the common feature for these examples is
that the relevant moduli spaces of rational curves have expected dimension.
Then, one can use the intersection theory. We should mention that there are
many predications based on mathematically unjustified mirror symmetry (for
Calabi-Yau 3-folds) and linear sigma model (for toric varieties). In this
paper, we attempt to determine the quantum cohomology of
projective bundles over the projective space $\Pee^n$. In contrast to the
previous examples, the relevant moduli spaces in our case frequently do not
have expected dimensions. It makes the calculation more difficult. We
overcome this difficulty by using excessive intersection theory.
There are two main ingredients in our arguments. The first one is a result of
Siebert and Tian (the Theorem 2.2 in \cite{14}),
which says that if the ordinary cohomology $H^*(X; \Zee)$ of
a symplectic manifold $X$ with the symplectic form $\omega$
is the ring generated by $\alpha_1, \ldots, \alpha_s$
with the relations $f^1, \ldots, f^t$, then the quantum cohomology
$H^*_\omega(X; \Zee)$ of $X$ is the ring generated
by $\alpha_1, \ldots, \alpha_s$ with $t$ new relations
$f_\omega^1, \ldots, f_\omega^t$
where each new relation $f_\omega^i$ is just
the relation $f^i$ evaluated in the quantum cohomology ring structure.
It was known that the quantum product $\alpha \cdot \beta$ is
the deformation of ordinary cup product by the lower order terms
called quantum corrections. The second ingredient is that
under certain numerical conditions, most of the quantum corrections vanishes.
Moreover, the nontrivial quantum corrections seem to come from
Mori's extremal rays.
Let $V$ be a rank-$r$ bundle over $\Pee^n$,
and $\Pee(V)$ be the corresponding projective bundle.
Let $h$ and $\xi$ be the cohomology classes of a hyperplane in $\Pee^n$
and the tautological line bundle in $\Pee(V)$ respectively.
For simplicity, we make no distinction between $h$ and $\pi^*h$
where $\pi: \Pee(V) \to \Pee^n$ is the natural projection.
Denote the product of $i$ copies of $h$ and $j$ copies of $\xi$
in the ordinary cohomology ring by $h_i \xi_j$,
and the product of $i$ copies of $h$ and $j$ copies of $\xi$
in the quantum cohomology ring by $h^i \cdot \xi^j$.
For $i = 0, \ldots, r$, put $c_i(V) = c_i \cdot h_i$ for some integer $c_i$.
It is well known that $-K_{\Pee(V)} = (n + 1 - c_1)h + r\xi$
and the ordinary cohomology ring $H^*(\Pee(V); \Zee)$ is
the ring generated by $h$ and $\xi$ with the two relations:
$$h_{n+1} = 0 \qquad \text{and} \qquad \sum_{i=0}^r (-1)^i c_i \cdot h_i
\xi_{r-i} = 0. \eqno (1.1)$$
In particular, $H^{2(n+r-2)}(\Pee(V); \Zee)$ is generated by
$h_{n-1}\xi_{r-1}$ and $h_{n}\xi_{r-2}$,
and its Poincar\'e dual $H_2(\Pee(V); \Zee)$ is generated by
$(h_{n-1}\xi_{r-1})_*$ and $(h_{n}\xi_{r-2})_*$
where for $\alpha \in H^*(\Pee(V); \Zee)$,
$\alpha_*$ stands for its Poincar\'e dual. We have
$$-K_{\Pee(V)}(A) = a(n + 1 - c_1) + r \cdot \xi(A)
= a(n + 1 - c_1) + r(ac_1 + b) \eqno (1.2)$$
for $A = (a h_{n-1}\xi_{r-1}+ b h_{n}\xi_{r-2})_* \in H_2(\Pee(V); \Zee)$.
By definition, $V$ is an ample (respectively, nef) bundle if and only if
the tautological class $\xi$ is an ample (respectively, nef) divisor
on $\Pee(V)$. Assume that $V$ is ample such that either $c_1 \le (n+1)$
or $c_1 \le (n + r)$ and $V\otimes {\Cal O}_{\Pee^n}(-1)$ is nef.
Then both $\xi$ and $-K_{\Pee(V)}$ are ample divisors.
Thus, $\Pee(V)$ is a Fano variety, and its quantum cohomology ring
is well-defined \cite{13}. Here we choose the symplectic form $\omega$ on
$\Pee(V)$ to be the Kahler form $\omega$ such that $[\omega] = -K_{\Pee(V)}$.
Let $f_\omega^1$ and $f_\omega^2$ be the two relations in (1.1)
evaluated in the quantum cohomology ring
$H^*_{\omega}(\Pee(V); \Zee)$. Then by the Theorem 2.2 in \cite{14},
the quantum cohomology $H^*_{\omega}(\Pee(V); \Zee)$ is
the ring generated by $h$ and $\xi$ with the two relations
$f_\omega^1$ and $f_\omega^2$:
$$H_\omega^*(\Pee(V); \Zee) = \Zee [h, \xi]/(f_\omega^1, f_\omega^2)
\eqno (1.3)$$
By Mori's Cone Theorem \cite{5}, $\Pee(V)$ has exactly
two extremal rays $R_1$ and $R_2$. Up to an order of $R_1$ and $R_2$,
the integral generator $A_1$ of $R_1$ is represented by lines
in the fibers of the projection $\pi$.
We shall show that under certain numerical conditions,
the nontrivial homology classes
$A \in H_2(\Pee(V); \Zee)$ which give nontrivial quantum corrections
are $A_1$ and $A_2$, where $A_2$ is represented by
some smooth rational curves in $\Pee(V)$ which are isomorphic
to lines in $\Pee^n$ via $\pi$. In general, it is unclear whether
$A_2$ generates the second extremal ray $R_2$. However, we shall prove
that under further restrictions on $V$, $A_2$ generates
the extremal ray $R_2$. These analyses enable us to determine
the quantum cohomology ring $H^*_{\omega}(\Pee(V); \Zee)$.
The simplest ample bundle over $\Pee^n$ is perhaps the direct sum of
line bundles $V=\oplus^r_{i=1} {\Cal O}(m_i)$ where $m_i>0$ for every $i$.
Since we can twist $V$ by ${\Cal O}(-1)$ without changing $\Pee(V)$,
we can assume that $\hbox{min}\{ m_1, \ldots, m_r \}=1$.
In this case, $\Pee(V)$ is a special case of toric variety.
Batyrev \cite{2} conjectured a general formula for quantum cohomology of
toric varieties. Furthermore, he computed the contributions
from certain moduli spaces of holomorphic maps
which have expected dimensions. In our case,
the contributions Batyrev computed are only part of the data
to compute the quantum cohomology. As we explained earlier,
the difficulty in our case lies precisely in computing the contributions
from the moduli spaces with wrong dimensions. Nevertheless, in our case,
Batyrev's formula (see also \cite{1}) reads as follows.
\vskip 0.1in
\noindent
{\bf Batyrev's Conjecture:}
{\it Let $V=\oplus^r_{i=1} {\Cal O}(m_i)$ where $m_i > 0$ for every $i$.
Then the quantum cohomology ring $H^*_{\omega}(\Pee(V); \Zee)$ is
generated by $h$ and $\xi$ with two relations}
$$h^{n+1}=\prod^r_{i=1}(\xi-m_ih)^{m_i-1} \cdot e^{-t(n+1+r-\sum_{i=1}^r m_i)}
\qquad {and} \qquad
\prod^r_{i=1}(\xi-m_ih) = e^{-tr}.$$
\vskip 0.1in
Our first result partially verifies Batyrev's conjecture.
\theorem{A} Batyrev's conjecture holds if
$$\sum_{i=1}^r m_i < \text{min}(2r, (n+1+2r)/2, (2n+2+r)/2).$$
\endproclaim
Note that under the numerical condition of Theorem A,
only extremal rational curves with fundamental classes $A_1$ and $A_2$
give the contributions to the two relations in the quantum cohomology.
The moduli space of rational curves $\frak M(A_2, 0)$ with fundamental
class $A_2$ does not have expected dimension in general. But it is compact.
This fact simplifies a great deal of the excessive intersection theory
involved.
To remove the numerical condition, we have to consider other moduli spaces
(for example $\frak M(kA_2, 0)$ with $k > 1$ and
its excessive intersection theory).
These moduli spaces are not compact in general.
Then, we have an extra difficulty of the compactification and
the appropriate excessive intersection theory with it.
It seems to be a difficult problem and we shall not pursue here.
In general, ample bundles over $\Pee^n$ are not direct sums of line bundles.
We can say much less about its quantum cohomology. However,
we obtain some result about its general form and
compute the leading coefficient.
\theorem{B} {\rm (i)} Let $V$ be a rank-$r$ ample bundle over $\Pee^n$.
Assume either $c_1 \le n$ or $c_1 \le (n + r)$ and
$V\otimes {\Cal O}_{\Pee^n}(-1)$ is nef so that $\Pee(V)$ is Fano.
Then the quantum cohomology $H^*_{\omega}(\Pee(V); \Zee)$ is
the ring generated by $h$ and $\xi$ with two relations
$$h^{n+1} = \sum_{i+j \le (c_1 - r)} a_{i,j} \cdot h^i \cdot \xi^j
\cdot e^{-t(n+1-i-j)}$$
$$\sum_{i=0}^r (-1)^i c_i \cdot h^i \cdot \xi^{r-i} = e^{-tr}
+ \sum_{i+j \le (c_1-n-1)} b_{i,j} \cdot h^i \cdot \xi^{j}
\cdot e^{-t(r-i-j)}$$
where the coefficients $a_{i, j}$ and $b_{i, j}$ are integers depending on $V$;
{\rm (ii)} If we further assume that $c_1<2r$,
then the leading coefficient $a_{0,c_1-r}=1$.
\endproclaim
It is understood that when $c_1 \le n$, then the summation
$\sum_{i+j \le (c_1-n-1)}$
in the second relation in Theorem B (i) does not exist.
In general, it is not easy to determine all the integers $a_{i, j}$ and
$b_{i, j}$ in Theorem B (i). However, it is possible to
compute these numbers when $(c_1 - r)$ is relatively small. For instance,
when $(c_1 - r) = 0$, then necessarily $V = \Cal O_{\Pee^n}(1)^{\oplus r}$
and it is well-known that the quantum cohomology
$H^*_{\omega}(\Pee(V); \Zee)$ is the ring generated by $h$ and $\xi$
with the two relations $h^{n+1} = e^{-t(n+1)}$ and
$\sum_{i=0}^r (-1)^i c_i \cdot h^i \cdot \xi^{r-i} = e^{-tr}$.
When $(c_1 - r) = 1$ and $r < n$,
then necessarily $V = \Cal O_{\Pee^n}(1)^{\oplus (r - 1)}
\oplus \Cal O_{\Pee^n}(2)$. When $(c_1 - r) = 1$ and $r = n$,
then $V = \Cal O_{\Pee^n}(1)^{\oplus (r - 1)} \oplus \Cal O_{\Pee^n}(2)$
or $V = T_{\Pee^n}$ the tangent bundle of $\Pee^n$.
In these cases, $V\otimes {\Cal O}_{\Pee^n}(-1)$ is nef. In particular, the
direct sum cases have been computed by Theorem A.
We shall prove the following.
\proposition{C} The quantum cohomology ring
$H^*_{\omega}(\Pee(T_{\Pee^n}); \Zee)$ with $n \ge 2$ is the ring
generated by $h$ and $\xi$ with the two relations:
$$h^{n+1} = \xi \cdot e^{-tn} \qquad \text{and}
\qquad \sum_{i=0}^n (-1)^i c_i \cdot h^i \cdot \xi^{n-i}
= (1 + (-1)^n) \cdot e^{-tn}.$$
\endproclaim
Recall that for an arbitrary projective bundle over a general manifold,
its cohomology ring is a module over the cohomology ring of the base
with the generator $\xi$ and the second relation of (1.1).
Naively. one may think that the quantum cohomology of projective bundle
is a module over the quantum cohomology of base with the generator $\xi$
and the quantanized second relation. Our calculation shows that
one can not expect such simplicity for its quantum cohomology ring.
We hope that our results could shed some light
on the quantum cohomology for general projective bundles,
which we shall leave for future research.
Our paper is organized as follows.
In section 2, we discuss the extremal rays and extremal rational curves.
In section 3, we review the definition of quantum product
and compute some Gromov-Witten invariants.
In the remaining three sections, we prove Theorem B, Theorem A,
and Proposition C respectively.
\medskip\noindent
{\bf Acknowledgements:} We would like to thank Sheldon Katz, Yungang Ye,
and Qi Zhang for valuable helps and stimulating discussions.
In particular, we are grateful to Sheldon Katz for bringing us
the attention of Batyrev's conjecture.
\section{2. Extremal rational curves}
Assume that $V$ is ample such that either $c_1 \le (n+1)$
or $c_1 \le (n + r)$ and $V\otimes {\Cal O}_{\Pee^n}(-1)$ is nef.
In this section, we study the extremal rays and extremal rational curves
in the Fano variety $\Pee(V)$. By Mori's Cone Theorem (p.25 in \cite{5}),
$\Pee(V)$ has precisely two extremal rays $R_1 = \Bbb R_{\ge 0} \cdot A_1$
and $R_2 = \Bbb R_{\ge 0} \cdot A_2$ such that the cone
$\hbox{NE}(\Pee(V))$ of curves in $\Pee(V)$ is equal to $R_1 + R_2$ and
that $A_1$ and $A_2$ are the homology classes of two rational curves
$E_1$ and $E_2$ in $\Pee(V)$ with
$0 < -K_{\Pee(V)}(A_i) \le \hbox{dim}(\Pee(V)) + 1$.
Up to orders of $A_1$ and $A_2$, we have $A_1 = (h_{n}\xi_{r-2})_*$,
that is, $A_1$ is represented by lines in the fibers of $\pi$.
It is also well-known that if $V = \oplus_{i=1}^r \Cal O_{\Pee^n}(m_i)$
with $m_1 \le \ldots \le m_r$,
then $A_2 = [h_{n-1}\xi_{r-1}+ (m_1 - c_1) h_{n}\xi_{r-2}]_*$
which is represented by a smooth rational curve in $\Pee(V)$
isomorphic to a line in $\Pee^n$ via $\pi$. However, in general,
it is not easy to determine the homology class $A_2$ and
the extremal rational curves representing $A_2$. Assume that
$$V|_\ell = \oplus_{i = 1}^r \Cal O_\ell(m_i) \eqno (2.1)$$
for generic lines $\ell \subset \Pee^n$ where we let $m_1 \le \ldots \le m_r$.
Since $V$ is ample, $m_1 \ge 1$.
\lemma{2.2} Let $A = [h_{n-1}\xi_{r-1} + (m_1 - c_1) h_{n}\xi_{r-2}]_*$. Then,
\roster
\item"{(i)}" $A$ is represented by a smooth rational curve isomorphic to
a line in $\Pee^n$;
\item"{(ii)}" $A_2 = A$ if and only if $(\xi- m_1 h)$ is nef;
\item"{(iii)}" $A_2 = A$ if $2c_1 \le (n + 1)$;
\item"{(iv)}" $A$ can not be represented by reducible or nonreduced curves
if $m_1 = 1$.
\endroster
\endproclaim
\proof (i) Let $\ell \subset \Pee^n$ be a generic line.
Then we have a natural projection $V|_\ell = \oplus_{i = 1}^r \Cal O_\ell(m_i)
\to \Cal O_\ell(m_1)$. By the Proposition 7.12 in Chapter II of \cite{7},
this surjective map $V|_\ell \to \Cal O_\ell(m_1) \to 0$ induces
a morphism $g: \ell \to \Pee(V)$. Then $g(\ell)$ is isomorphic to $\ell$
via the projection $\pi$. Since $h([g(\ell)]) = 1$ and $\xi([g(\ell)]) = m_1$,
we have
$$[g(\ell)] = [h_{n-1}\xi_{r-1}+ (m_1 - c_1) h_{n}\xi_{r-2}]_* = A.$$
(ii) First of all, if
$A_2 = [h_{n-1}\xi_{r-1}+ (m_1 - c_1) h_{n}\xi_{r-2}]_*$,
then for any curve $E$, $[E] = a(h_{n}\xi_{r-2})_* +
b[h_{n-1}\xi_{r-1}+ (m_1 - c_1) h_{n}\xi_{r-2}]_*$ for some nonnegative numbers
$a$ and $b$; so $(\xi- m_1h)([E]) = a \ge 0$; therefore $(\xi- m_1h)$ is nef.
Conversely, if $(\xi- m_1h)$ is nef, then $0 \le (\xi- m_1h)([E])
= ac_1 + b - am_1$ where $[E] = (ah_{n-1}\xi_{r-1}+ b h_{n}\xi_{r-2})_*$
for some curve $E$; thus $[E] = (ac_1 + b - am_1)(h_{n}\xi_{r-2})_* +
a[h_{n-1}\xi_{r-1}+ (m_1 - c_1) h_{n}\xi_{r-2}]_*$;
it follows that $A_2 = [h_{n-1}\xi_{r-1}+ (m_1 - c_1) h_{n}\xi_{r-2}]_* = A$.
(iii) Let $A_2 = (ah_{n-1}\xi_{r-1} + bh_{n}\xi_{r-2})_*$.
Since $A_1 = (h_{n}\xi_{r-2})_*$ and $a = h(A_2) \ge 0$, $a \ge 1$.
If $a > 1$, then since $2c_1 \le (n + 1)$, we see that
$$\align
-K_{\Pee(V)}(A_2) &= (n+1-c_1)a +r \cdot \xi(A_2) \ge 2(n+1-c_1) + r \\
&> n + r = \text{dim}(\Pee(V)) + 1; \\
\endalign$$
but this contradicts with $-K_{\Pee(V)}(A_2) \le \hbox{dim}(\Pee(V)) + 1$.
Thus $a = 1$ and $A_2 = (h_{n-1}\xi_{r-1} + bh_{n}\xi_{r-2})_*$.
Now $[\pi(E_2)] = \pi_*(A_2) = (h_{n-1})_*$.
So $\pi(E_2)$ is a line in $\Pee^n$.
Since $V|_\ell = \oplus_{i = 1}^r \Cal O_\ell(m_i)$ for a generic line
$\ell \subset \Pee^n$,
$V|_{\pi(E_2)} = \oplus_{i = 1}^r \Cal O_{\pi(E_2)}(m_i')$
where $m_i' \ge m_1$ for every $i$. Thus, $\xi(A_2) \ge m_1$,
and so $c_1 + b \ge m_1$. It follows that
$$A_2 = [h_{n-1}\xi_{r-1} + (m_1- c_1) h_{n}\xi_{r-2}]_*
+ (c_1 + b - m_1) \cdot (h_{n}\xi_{r-2})_*.$$
Therefore, $A_2 = [h_{n-1}\xi_{r-1}+ (m_1 - c_1) h_{n}\xi_{r-2}]_* = A$.
(iv) Since $\xi(A) = m_1 = 1$ and $\xi$ is ample, the conclusion follows.
\endproof
Next, let $\frak M(A,0)$ be the moduli space of morphisms
$f: \Pee^1 \to \Pee(V)$ with $[\hbox{Im}(f)] = A$.
In the lemma below, we study the morphisms in $\frak M(A,0)$
when $A = [h_{n-1}\xi_{r-1}+ (m - c_1) h_{n}\xi_{r-2}]_*$.
Note that $\xi(A) = m$.
\lemma{2.3} Let $A = [h_{n-1}\xi_{r-1}+ (m - c_1) h_{n}\xi_{r-2}]_*$.
\roster
\item"{(i)}" If $\frak M(A, 0) \ne \emptyset$, then $m \ge m_1$ and
$\frak M(A, 0)$ consists of embeddings $f: \ell \to \Pee(V)$
induced by surjective maps $V|_\ell \to \Cal O_\ell(m) \to 0$
where $\ell$ are lines in $\Pee^n$;
\item"{(ii)}" If $m = m_1$ and $m_1 = \ldots = m_k < m_{k+1} \le
\ldots \le m_r$, then the moduli space $\frak M(A, 0)$
has (complex) dimension $(2n +k)$;
\item"{(iii)}" If $m \ge m_r$, then $\frak M(A, 0)$ has dimension
$(2n+r+rm-c_1)$.
\endroster
\endproclaim
\proof
(i) Let $f: \Pee^1 \to \Pee(V)$ be a morphism in $\frak M(A, 0)$.
Then $[\hbox{Im}(f)] = A = [h_{n-1}\xi_{r-1}+ (m - c_1) h_{n}\xi_{r-2}]_*$.
Since $h(A) = 1$, $\pi^*H \cap f(\Pee^1)$ consists of a single point
for any hyperplane $H$ in $\Pee^n$. Thus,
$\pi|_{f(\Pee^1)}: f(\Pee^1) \to (\pi \circ f)(\Pee^1)$ is an isomorphism
and $\ell = (\pi \circ f)(\Pee^1)$ is a line in $\Pee^n$.
Since $h([\ell]) = 1$, $(\pi \circ f): \Pee^1 \to \ell = (\pi \circ f)(\Pee^1)$
is also an isomorphism, and so is $f: \Pee^1 \to f(\Pee^1)$.
Replacing $f: \Pee^1 \to \Pee(V)$ by
$f \circ (\pi \circ f)^{-1}: \ell \to \Pee(V)$,
we conclude that $\frak M(A, 0)$ consists of embeddings $f: \ell \to \Pee(V)$
such that $[\hbox{Im}(f)] = A$, $\ell$ are lines in $\Pee^n$,
and $\pi|_{f(\ell)}: f(\ell) \to \ell$ are isomorphisms.
In particular, these embeddings $f: \ell \to \Pee(V)$ are
sections to the natural projection
$\pi|_{\Pee(V|_\ell)}: \Pee(V|_\ell) \to \ell$.
Thus, by the Proposition 7.12 in Chapter II of \cite{7},
these embeddings are induced by surjective maps
$V|_\ell \to \Cal O_\ell(m) \to 0$.
By (2.1), the splitting type of the restrictions of $V$ to generic lines
in $\Pee^n$ is $(m_1, \ldots, m_r)$ with $m_1 \le \ldots \le m_r$;
thus we must have $V|_\ell = \oplus_{i = 1}^r \Cal O_\ell(m_i')$ where
$m_i' \ge m_1$ for every $i$. It follows that
$m \ge \hbox{min} \{ m_1', \ldots, m_r' \} \ge m_1$.
(ii) Note that all the lines in $\Pee^n$ are parameterized by
the Grassmannian $G(2, n+1)$ which has dimension $2(n -1)$.
For a fixed generic line $\ell \subset \Pee^n$, the surjective maps
$V|_\ell \to \Cal O_\ell(m_1) \to 0$ are parameterized by
$$\Pee(\hbox{Hom}(V|_\ell, \Cal O_\ell(m_1))) \cong
\Pee(\oplus_{i=1}^r H^0(\ell, \Cal O_\ell(m_1-m_i))) \cong \Pee^{k - 1};$$
It follows from (i) that as the generic line $\ell$ varies,
the morphisms $f: \ell \to \Pee(V)$ induced by these surjective maps
$V|_\ell \to \Cal O_\ell(m_1) \to 0$ form an open dense subset of
$\frak M(A, 0)$. Thus, $\hbox{dim}(\frak M(A, 0)) = 3+ 2(n -1) + (k - 1)
= 2n +k$.
(iii) As in the proof of (ii), for a fixed generic line $\ell \subset \Pee^n$,
the surjective maps $V|_\ell \to \Cal O_\ell(m) \to 0$ are
parameterized by a nonempty open subset of
$$\Pee(\hbox{Hom}(V|_\ell, \Cal O_\ell(m))) \cong
\Pee(\oplus_{i=1}^r H^0(\ell, \Cal O_\ell(m-m_i)))
\cong \Pee^{(rm - c_1 +r) - 1}.$$
As the generic line $\ell$ varies, the morphisms $f: \ell \to \Pee(V)$
induced by these surjective maps $V|_\ell \to \Cal O_\ell(m) \to 0$
form an open dense subset of $\frak M(A, 0)$. It follows that
$\frak M(A, 0)$ has dimension $(2n+r+rm-c_1)$.
\endproof
\section{3. Calculation of Gromov-Witten invariants}
In this section, we shall compute some Gromov-Witten invariants of $\Pee(V)$.
First of all, we recall that for two homogeneous elements
$\alpha$ and $\beta$ in $H^*(\Pee(V); \Zee)$, the quantum product
$\alpha \cdot \beta \in H^*(\Pee(V); \Zee)$ can be written as
$$\alpha \cdot \beta = \sum_{A \in H_2(\Pee(V); \Zee)}
(\alpha \cdot \beta)_A \cdot e^{t \cdot K_{\Pee(V)}(A)} \eqno (3.1)$$
where $(\alpha \cdot \beta)_A$ has degree
$\hbox{deg}(\alpha) + \hbox{deg}(\beta) + 2K_{\Pee(V)}(A)$
and is defined by
$$(\alpha \cdot \beta)_A(\gamma_*) = \Phi_{(A, 0)}(\alpha, \beta, \gamma)$$
for a homogeneous cohomology class $\gamma \in H^*(\Pee(V); \Zee)$ with
$$\hbox{deg}(\gamma) = -2K_{\Pee(V)}(A) + 2(n+r-1) - \hbox{deg}(\alpha)
- \hbox{deg}(\beta). \eqno (3.2)$$
Furthermore, for higher quantum products, we have
$$\alpha_1 \cdot \alpha_2 \cdot \ldots \cdot \alpha_k =
\sum_{A \in H_2(\Pee(V); \Zee)} (\alpha_1 \cdot \alpha_2 \cdot
\ldots \cdot \alpha_k)_A \cdot e^{t \cdot K_{\Pee(V)}(A)} \eqno (3.3)$$
where $(\alpha_1 \cdot \alpha_2 \cdot \ldots \cdot \alpha_k)_A$ is defined as
$(\alpha_1 \cdot \alpha_2 \cdot \ldots \cdot \alpha_k)_A(\gamma_*)
= \Phi_{(A,0)}(\alpha_1, \alpha_2, \ldots, \alpha_k, \gamma)$. Thus,
$\alpha_1 \cdot \alpha_2 \cdot \cdots \cdot \alpha_k
= \alpha_1\alpha_2\dots\alpha_k + \text{(lower\ order\ terms)}$,
where $\alpha_1\alpha_2\dots\alpha_k$
stands for the ordinary cohomology product of
$\alpha_1, \alpha_2, \ldots, \alpha_k$,
and the degree of a lower order term is dropped by $2K_{\Pee(V)}(A)$
for some $A \in H_2(\Pee(V); \Zee)$ which is represented by
a nonconstant effective rational curve.
There are two explanations for the Gromov-Witten invariant
$\Phi_{(A, 0)}(\alpha, \beta, \gamma)$ defined by the second author \cite{12}.
Recall that the Gromov-Witten invariant is only defined for
a generic almost complex structure and that $\frak M(A,0)$ is
the moduli space of morphisms $f: \Pee^1 \to \Pee(V)$
with $[\hbox{Im}(f)] = A$. Assume the genericity conditions:
\roster
\item"{(i)}" $\frak M(A, 0)/PSL(2; \Cee)$ is smooth
in the sense that $h^1(N_f) = 0$ for every $f \in \frak M(A, 0)$ where
$N_f$ is the normal bundle, and
\item"{(ii)}" the homology class $A$ is only
represented by irreducible and reduced curves.
\endroster
\noindent
Then the complex structure is already generic and one can use
algebraic geometry to calculate the Gromov-Witten invariants.
Moreover, $\frak M(A, 0)/PSL(2; \Cee)$ is
compact with the expected complex dimension
$$-K_{\Pee(V)}(A) + (n+r-1) - 3. \eqno (3.4)$$
The first explanation for $\Phi_{(A, 0)}(\alpha, \beta, \gamma)$ is that
when $\alpha, \beta, \gamma$ are classes of subvarieties $B, C, D$
of $\Pee(V)$ in general position, $\Phi_{(A, 0)}(\alpha, \beta, \gamma)$
is the number of rational curves $E$ in $\Pee(V)$ such that
$[E] = A$ and that $E$ intersects with $B, C, D$
(counted with suitable multiplicity). The second explanation for
$\Phi_{(A, 0)}(\alpha, \beta, \gamma)$ is that
$$\Phi_{(A, 0)}(\alpha, \beta, \gamma) = \int_{\frak M(A, 0)} e_0^*(\alpha)
\cdot e_1^*(\beta) \cdot e_2^*(\gamma) $$
where the evaluation map $e_i: \frak M(A, 0) \to \Pee(V)$ is defined by
$e_i(f) = f(i)$.
Assume that the genericity condition (i) is not satisfied but
$h^1(N_f)$ is independent of $f \in \frak M(A, 0)$ and
$\frak M(A, 0)/PSL(2; \Cee)$ is smooth with dimension
$$-K_{\Pee(V)}(A) + (n+r-1) - 3+h^1(N_f).$$
Then one can form an obstruction bundle $COB$ of rank $h^1(N_f)$
over the moduli space $\frak M(A, 0)$. Moreover,
if the genericity condition (ii) is satisfied,
then by the Proposition 5.7 in \cite{11}, we have
$$\Phi_{(A, 0)}(\alpha, \beta, \gamma) = \int_{\frak M(A, 0)} e_0^*(\alpha)
\cdot e_1^*(\beta) \cdot e_2^*(\gamma) \cdot e(COB) \eqno (3.5)$$
where $e(COB)$ stands for the Euler class of the bundle $COB$.
We remark that in general, the cohomology class $h_i \xi_j$ may not
be able to be represented by a subvariety of $\Pee(V)$.
However, since $\xi$ is ample, $s \xi$ is very ample for $s \gg 0$.
Thus, the multiple $t h_i \xi_j$ with $t \gg 0$ can be represented
by a subvariety of $\Pee(V)$ whose image in $\Pee^n$ is
a linear subspace of codimension $i$.
Since $\Phi_{(A, 0)}(\alpha, \beta, h_i \xi_j) =
{1/t} \cdot \Phi_{(A, 0)}(\alpha, \beta, t \cdot h_i \xi_j)$
for $\alpha$ and $\beta$ in $H^*(\Pee(V); \Zee)$,
it follows that to compute $\Phi_{(A, 0)}(\alpha, \beta, h_i \xi_j)$,
it suffices to compute $\Phi_{(A, 0)}(\alpha, \beta, t \cdot h_i \xi_j)$.
In the proofs below, we shall assume implicitly that $t = 1$ for simplicity.
Now we compute the Gromov-Witten invariant
$\Phi_{((h_{n}\xi_{r-2})_*, 0)}(\xi, \xi_{r-1}, h_{n}\xi_{r-1})$.
\lemma{3.6}
$\Phi_{((h_{n}\xi_{r-2})_*, 0)}(\xi, \xi_{r-1}, h_{n}\xi_{r-1}) = 1$.
\endproclaim
\proof
First of all, we notice that $A = (h_{n} \xi_{r-2})_*$ can only be
represented by lines $\ell$ in the fibers of $\pi$. In particular,
there is no reducible or nonreduced effective curves representing $A$.
Thus, $\frak M(A, 0)/PSL(2; \Cee)$ is compact and has dimension:
$$\text{dim}(\Pee^n) + \text{dim} G(2, r)= n + 2(r-2) = n + 2r - 4$$
which is the expected dimension by (3.4) (here we use $G(2, r)$ to stand for
the Grassmannian of lines in $\Pee^{r-1}$). Next, we want to show that
$\frak M(A, 0)/PSL(2; \Cee)$ is smooth. Let $p = \pi(\ell)$.
Then from the two inclusions $\ell \subset \pi^{-1}(p) \subset \Pee(V)$,
we obtain an exact sequence relating normal bundles:
$$0 \to N_{\ell|\pi^{-1}(p)} \to N_{\ell|\Pee(V)} \to
(N_{\pi^{-1}(p)|\Pee(V)})|_\ell \to 0.$$
Since $N_{\ell|\pi^{-1}(p)} = N_{\ell|\Pee^{r-1}}
= \Cal O_{\ell}(1)^{\oplus (r-2)}$
and $N_{\pi^{-1}(p)|\Pee(V)} = (\pi|_{\pi^{-1}(p)})^*T_{p, \Pee^n}$,
the previous exact sequence is simplified into the exact sequence
$$0 \to \Cal O_{\ell}(1)^{\oplus (r-2)} \to N_{\ell|\Pee(V)} \to
(\pi|_\ell)^*T_{p, \Pee^n} \to 0.$$
It follows that $H^1(\ell, N_{\ell|\Pee(V)}) = 0$. Thus,
$\frak M(A, 0)/PSL(2; \Cee)$ is smooth.
Finally, the Poincar\'e dual of $h_n \xi_{r-1}$ is represented by a point
$q_0 \in \Pee(V)$. If a line $\ell \in \frak M(A,0)$ intersects $q_0$,
then $\ell \subset \pi^{-1}(\pi(q_0))$. Since the restriction of $\xi$ to
the fiber $\pi^{-1}(\pi(q_0)) \cong \Pee^{r-1}$ is
the cohomology class of a hyperplane in $\Pee^{r-1}$, we conclude that
$\Phi_{((h_{n}\xi_{r-2})_*, 0)}(\xi, \xi_{r-1}, h_{n}\xi_{r-1}) = 1$.
\endproof
Next we show the vanishing of some Gromov-Witten invariant.
\lemma{3.7} Let $A = b(h_n \xi_{r-2})_*$ with $b \ge 1$
and $\alpha \in H^*(\Pee(V); \Zee)$. Then,
$$\Phi_{(A, 0)}(h_{p_1}\xi_{q_1}, h_{p_2}\xi_{q_2}, \alpha) = 0$$
if $p_1, q_1, p_2, q_2$ are nonnegative integers with $(q_1+q_2) < r$.
\endproclaim
\proof
We may assume that $\alpha$ is a homogeneous class in
$H^*(\Pee(V); \Zee)$. By (3.2),
$$\align
{1 \over 2} \cdot \hbox{deg}(\alpha)
&= (n+r-1) - K_{\Pee(V)}(A) - (p_1+p_2+q_1+q_2) \\
&= (n+r+br-1)-(p_1+p_2+q_1+q_2). \\
\endalign$$
Let $\alpha = h_{(n+r+br-1)-(p_1+p_2+q_1+q_2+q_3)} \xi_{q_3}$ with
$0 \le q_3 \le (r - 1)$.
Let $B, C, D$ be the subvarieties of $\Pee(V)$ in general position,
whose homology classes are Poincar\'e dual to
$h_{p_1}\xi_{q_1}, h_{p_2}\xi_{q_2}, \alpha$ respectively.
Then the homology classes of $\pi(B), \pi(C), \pi(D)$ in $\Pee^n$ are
Poincar\'e dual to $h_{p_1}, h_{p_2},
h_{(n+r+br-1)-(p_1+p_2+q_1+q_2+q_3)}$ respectively.
Since $(q_1+q_2+q_3) < (2r -1)$, we have
$p_1 + p_2 + [(n+r+br-1)-(p_1+p_2+q_1+q_2+q_3)]
= (n+r+br-1)-(q_1+q_2+q_3) > n$.
Thus, $\pi(B) \cap \pi(C) \cap \pi(D) = \emptyset$.
Notice that the genericity conditions (i) and (ii)
mentioned earlier in this section are not satisfied for $b\ge 2$.
However, we observe that these conditions can be relaxed by assuming:
\roster
\item"{(i$'$)}" $h^1(N_f)=0$ for every $f\in \frak M(A, 0)$ such that
$\hbox{Im}(f)$ intersects $B, C, D$, and
\item"{(ii$'$)}" there is no reducible or nonreduced effective
(connected) curve $E$ such that $[E]=A$ and $E$ intersects $B, C, D$.
\endroster
\noindent
In fact, we will show that there is no effective connected curve $E$ at all
representing $A$ and intersecting $B, C, D$.
It obviously implies (i$'$), (ii$'$) and
$$\Phi_{(A, 0)}(h_{p_1}\xi_{q_1}, h_{p_2}\xi_{q_2}, \alpha) = 0.$$
Suppose that $E=\sum a_i E_i$ is such an effective connected curve
where $a_i > 0$ and $E_i$ is irreducible and reduced.
Then, $\sum a_i [E_i] = [E] = A$.
Since $(h_n \xi_{r-2})_*$ generates an extremal ray for $\Pee(V)$,
$[E_i]=b_i (h_n \xi_{r-2})_*$ for $0<b_i\le b$.
Thus the curves $E_i$ are contained in the fibers of $\pi$.
Since $E$ is connected, all the curves $E_i$ must be contained
in the same fiber of $\pi$. So $\pi(E)$ is a single point.
Since $E$ intersects $B, C, D$, $\pi(E)$ intersects with
$\pi(B), \pi(C), \pi(D)$.
It follows that $\pi(B) \cap \pi(C) \cap \pi(D)$ contains $\pi(E)$
and is nonempty. Therefore we obtain a contradiction.
\endproof
Finally, we show that if $c_1 < 2r$ and
$A = [h_{n-1}\xi_{r-1} + (1 - c_1) h_{n}\xi_{r-2}]_*$,
then $\Phi_{(A, 0)}(h, h_n, h_n \xi_{2r-c_1-1}) = 1$.
Since $c_1 < 2r$, we see that for a generic line $\ell \subset \Pee^n$,
$$V|_{\ell} = \Cal O_{\ell}(1)^{\oplus k}
\oplus \Cal O_{\ell}(m_{k+1}) \oplus \ldots \oplus \Cal O_{\ell}(m_r)$$
where $k \ge 1$ and $2 \le m_{k+1} \le \ldots \le m_r$.
We remark that even though the moduli space $\frak M(A, 0)/PSL(2; \Cee)$
is compact by Lemma 2.2 (iv),
it may not have the correct dimension by Lemma 2.3 (ii).
The proof is lengthy, but the basic idea is that
we shall determine the obstruction bundle and
use the formula (3.5).
\lemma{3.8} Let $V$ be a rank-$r$ ample vector bundle over
$\Pee^n$ satisfying $c_1<2r$ and the assumption of Theorem B (i).
If $A = [h_{n-1}\xi_{r-1} + (1 - c_1) h_{n}\xi_{r-2}]_*$, then
$$\Phi_{(A, 0)}(h, h_n, h_n \xi_{2r-c_1-1}) = 1.$$
\endproclaim
\noindent
{\it Proof.} Note that by Lemma 2.2 (iv),
the moduli space $\frak M(A, 0)/PSL(2; \Cee)$ is compact.
Let $B, C, D$ be the subvarieties of $\Pee(V)$ in general position,
whose homology classes are Poincar\'e dual to
$h, h_n, h_n \xi_{2r-c_1-1}$ respectively. Then the homology classes of
$\pi(B), \pi(C)$, $\pi(D)$ in $\Pee^n$ are Poincar\'e dual to
$h, h_n, h_n$ respectively. Thus $\pi(C)$ and $\pi(D)$ are
two different points in $\Pee^n$. Let $\ell_0$ be the unique line passing
$\pi(C)$ and $\pi(D)$. Let $V|_{\ell_0} = \Cal O_{\ell_0}(1)^{\oplus k}
\oplus \Cal O_{\ell_0}(m_{k+1}) \oplus \ldots \oplus \Cal O_{\ell_0}(m_r)$
where $2 \le m_{k+1} \le \ldots \le m_r$. Since $c_1 < 2r$, $k \ge 1$.
Let $f: \ell \to \Pee(V)$ be
a morphism in $\frak M(A, 0)$ for some line $\ell \in \Pee^n$.
If $\hbox{Im}(f)$ intersects with $B, C$, and $D$, then $\ell = \ell_0$.
As in the proof of Lemma 2.3 (ii), the morphisms $f: \ell_0 \to \Pee(V)$
in $\frak M(A, 0)$ are parameterized by
$\Pee(\hbox{Hom}(V|_{\ell_0}, \Cal O_{\ell_0}(1))) \cong \Pee^{k - 1}$;
moreover, $\hbox{Im}(f)$ are of the form:
$$\ell_0 \times \{ q \} \subset \ell_0 \times \Pee^{k - 1}
= \Pee(\Cal O_{\ell_0}(1)^{\oplus k})
\subset \Pee(V|_{\ell_0}) \subset \Pee(V) \eqno (3.9)$$
where $q$ stands for points in $\Pee^{k - 1} \subset \Pee^{r - 1} \cong
\pi^{-1}(\pi(D))$. Note that $\ell_0 \times \{ q \}$ always intersects
with $B$ and $C$, and that $D$ is a dimension-$(c_1 - r)$ linear subspace
in $\Pee^{r - 1} \cong \pi^{-1}(\pi(D))$. Thus, $\ell_0 \times \{ q \}$
intersects with $B, C, D$ simultaneously if and only if
$\ell_0 \times \{ q \}$ intersects with $D$, and if only only if
$$q \in \Pee^{c_1+k-2r} \overset \hbox{def} \to =
\Pee^{k - 1} \cap D \subset \Pee^{r - 1} \cong \pi^{-1}(\pi(D)).
\eqno (3.10)$$
It follows that $\frak M/PSL(2; \Cee) \cong \Pee^{c_1+k-2r}$ where
$\frak M$ consists of morphisms $f \in \frak M(A, 0)$ such that
$\hbox{Im}(f)$ intersects with $B, C, D$ simultaneously.
If $c_1+k-2r = 0$, then $a_0 = \Phi_{(A, 0)}(h, h_n, h_n \xi_{2r-c_1-1}) = 1$.
But in general, we have $c_1+k-2r \ge 0$. We shall use (3.5) to compute
$a_0 = \Phi_{(A, 0)}(h, h_n, h_n \xi_{2r-c_1-1})$.
Let $N_f = N_{\ell_0 \times \{ q \}|\Pee(V)}$ be the normal bundle
of $\hbox{Im}(f) = \ell_0 \times \{ q \}$ in $\Pee(V)$.
If $h^1(N_f)$ is constant for every $f \in \frak M$,
then by (3.5), $\Phi_{(A, 0)}(h, h_n, h_n \xi_{2r-c_1-1})$
is the Euler number $e(COB)$ of the rank-$(c_1+k-2r)$ obstruction bundle
$COB$ over
$$\frak M/PSL(2; \Cee) \cong \Pee^{c_1+k-2r}.$$
Thus we need to show that $h^1(N_f)$ is constant for every $f \in \frak M$.
First, we study the normal bundle
$N_{\ell_0 \times \Pee^{c_1+k-2r}|\Pee(V)}$. The three inclusions
$$\ell_0 \times \Pee^{c_1+k-2r}
\subset \ell_0 \times \Pee^{k - 1} = \Pee(\Cal O_{\ell_0}(1)^{\oplus k})
\subset \Pee(V|_{\ell_0}) \subset \Pee(V) \eqno (3.11)$$
give rise to two exact sequences relating normal bundles:
$$0 \to N_{\ell_0 \times \Pee^{c_1+k-2r}|\Pee(V|_{\ell_0})}
\to N_{\ell_0 \times \Pee^{c_1+k-2r}|\Pee(V)} \to
N_{\Pee(V|_{\ell_0})|\Pee(V)} \to 0$$
$$0 \to N_{\ell_0 \times \Pee^{c_1+k-2r}|\Pee(\Cal O_{\ell_0}(1)^{\oplus k})}
\to N_{\ell_0 \times \Pee^{c_1+k-2r}|\Pee(V|_{\ell_0})} \to
N_{\Pee(\Cal O_{\ell_0}(1)^{\oplus k})|\Pee(V|_{\ell_0})} \to 0$$
Notice that $N_{\Pee(V|_{\ell_0}) | \Pee(V)} =
(\pi|_{\Pee(V|_{\ell_0})})^*(N_{\ell_0 | \Pee^n})
= \Cal O_{\ell_0}(1)^{\oplus (n-1)}$ and that
$$N_{\ell_0 \times \Pee^{c_1+k-2r}|\Pee(\Cal O_{\ell_0}(1)^{\oplus k})} =
N_{\ell_0 \times \Pee^{c_1+k-2r}|\ell_0 \times \Pee^{k-1}}
= \Cal O_{\Pee^{c_1+k-2r}}(1)^{\oplus (2r-c_1-1)}.$$
Since $V|_{\ell_0} = \Cal O_{\ell_0}(1)^{\oplus k}
\oplus \oplus_{i=k+1}^r \Cal O_{\ell_0}(m_i)$,
$\xi|_{\ell_0 \times \Pee^{k-1}} =
\Cal O_{\ell_0}(1) \otimes \Cal O_{\Pee^{k-1}}(1)$ and
$$\align
N_{\Pee(\Cal O_{\ell_0}(1)^{\oplus k}) | \Pee(V|_{\ell_0})}
&= \oplus_{i=k+1}^r \Cal O_{\ell_0}(-m_i) \otimes
\xi|_{\ell_0 \times \Pee^{k-1}} \\
&= \oplus_{i=k+1}^r \Cal O_{\ell_0}(1-m_i) \otimes \Cal O_{\Pee^{k-1}}(1).\\
\endalign$$
Thus the previous two exact sequences are simplified to:
$$0 \to N_{\ell_0 \times \Pee^{c_1+k-2r}|\Pee(V|_{\ell_0})}
\to N_{\ell_0 \times \Pee^{c_1+k-2r}|\Pee(V)} \to
\Cal O_{\ell_0}(1)^{\oplus (n-1)} \to 0 \eqno (3.12)$$
$$0 \to \Cal O_{\Pee^{c_1+k-2r}}(1)^{\oplus (2r-c_1-1)}
\to N_{\ell_0 \times \Pee^{c_1+k-2r}|\Pee(V|_{\ell_0})} \to$$
$$\oplus_{i=k+1}^r \Cal O_{\ell_0}(1-m_i) \otimes
\Cal O_{\Pee^{c_1+k-2r}}(1) \to 0 \eqno (3.13)$$
Now (3.13) splits since for $k+1 \le i \le r$, we have $m_i \ge 2$ and
$$\align
&\quad \text{Ext}^1(\Cal O_{\ell_0}(1-m_i)
\otimes \Cal O_{\Pee^{c_1+k-2r}}(1), \Cal O_{\Pee^{c_1+k-2r}}(1)) \\
&= H^1(\ell_0 \times \Pee^{c_1+k-2r}, \Cal O_{\ell_0}(m_i - 1)) = 0.\\
\endalign$$
Thus, the normal bundle $N_{\ell_0 \times \Pee^{c_1+k-2r}|\Pee(V|_{\ell_0})}$
is isomorphic to
$$\oplus_{i=k+1}^r \Cal O_{\ell_0}(1-m_i) \otimes
\Cal O_{\Pee^{c_1+k-2r}}(1) \oplus
\Cal O_{\Pee^{c_1+k-2r}}(1)^{\oplus (2r-c_1-1)},$$
and the exact sequence (3.12) becomes to the exact sequence:
$$0 \to \oplus_{i=k+1}^r \Cal O_{\ell_0}(1-m_i) \otimes
\Cal O_{\Pee^{c_1+k-2r}}(1) \oplus
\Cal O_{\Pee^{c_1+k-2r}}(1)^{\oplus (2r-c_1-1)} \to$$
$$N_{\ell_0 \times \Pee^{c_1+k-2r}|\Pee(V)} \to
\Cal O_{\ell_0}(1)^{\oplus (n-1)} \to 0 \eqno (3.14)$$
Restricting (3.14) to $\ell_0 \times \{ q \}$ and taking long exact
cohomology sequence result
$$\oplus_{i=k+1}^r H^1(\Cal O_{\ell_0}(1-m_i))
\otimes \Cal O_{\Pee^{c_1+k-2r}}(1)|_{q} \to$$
$$H^1((N_{\ell_0 \times \Pee^{c_1+k-2r}|\Pee(V)})|_{\ell_0 \times \{ q \}})
\to 0. \eqno (3.15)$$
Next, we determine $N_f$ and show that $h^1(N_f) \le c_1+k-2r$.
The two inclusions $\ell_0 \times \{ q \} \subset
\ell_0 \times \Pee^{c_1+k-2r} \subset \Pee(V)$ give an exact sequence
$$0 \to N_{\ell_0 \times \{ q \} | \ell_0 \times \Pee^{c_1+k-2r}}
\to N_{\ell_0 \times \{ q \}|\Pee(V)} \to
(N_{\ell_0 \times \Pee^{c_1+k-2r}|\Pee(V)})|_{\ell_0 \times \{ q \}} \to 0.$$
Since $N_{\ell_0 \times \{ q \} | \ell_0 \times
\Pee^{c_1+k-2r}} = T_{q, \Pee^{c_1+k-2r}}$, the above exact sequence becomes
$$0 \to T_{q, \Pee^{c_1+k-2r}} \to N_f \to
(N_{\ell_0 \times \Pee^{c_1+k-2r}|\Pee(V)})|_{\ell_0 \times \{ q \}} \to 0.
\eqno (3.16)$$
Thus, $h^1(N_f) =
h^1((N_{\ell_0 \times \Pee^{c_1+k-2r}|\Pee(V)})|_{\ell_0 \times \{ q \}})$.
By (3.15), we obtain
$$\align
h^1(N_f) &= h^1((N_{\ell_0 \times \Pee^{c_1+k-2r}|\Pee(V)})|_{\ell_0 \times
\{ q \}}) \le \sum_{i=k+1}^r h^1(\Cal O_{\ell_0}(1-m_i)) \\
&= \sum_{i=k+1}^r (m_i - 2) = c_1 + k - 2r.\\
\endalign$$
Finally, we show that $h^1(N_f) = c_1 + k - 2r$. It suffices to prove that
$h^1(N_f) \ge c_1 + k - 2r$. Since $\ell_0$ is a generic line in $\Pee^n$
and $V|_{\ell_0} = \Cal O_{\ell_0}(1)^{\oplus k}
\oplus \oplus_{i = k+1}^r \Cal O_{\ell_0}(m_i)$,
$\hbox{dim} \frak M(A, 0) = (2n + k)$ by Lemma 2.3 (ii).
Since $h^0(N_f)$ is the dimension of the Zariski tangent space of
$\frak M(A, 0)/PSL(2; \Cee)$ at $f$, $h^0(N_f) \ge (2n + k-3)$. Thus,
$$h^1(N_f) = h^0(N_f) - \chi(N_f) \ge (2n + k-3) - (2n + 2r-c_1-3)
= k + c_1 -2r.$$
Therefore, $h^1(N_f) = c_1 + k - 2r$. In particular, $h^1(N_f)$ is
independent of $f \in \frak M$. To obtain the obstruction bundle $COB$
over $\Pee^{c_1+k-2r}$, we notice that (3.15) gives
$$\oplus_{i=k+1}^r H^1(\Cal O_{\ell_0}(1-m_i)) \otimes
\Cal O_{\Pee^{c_1+k-2r}}(1)|_{q} \cong
H^1((N_{\ell_0 \times \Pee^{c_1+k-2r}|\Pee(V)})|_{\ell_0 \times \{ q \}}).$$
Thus by the exact sequence (3.16), we conclude that
$$\align
H^1(N_f)
&\cong H^1((N_{\ell_0 \times
\Pee^{c_1+k-2r}|\Pee(V)})|_{\ell_0 \times \{ q \}}) \\
&\cong \oplus_{i=k+1}^r H^1(\Cal O_{\ell_0}(1-m_i))
\otimes \Cal O_{\Pee^{c_1+k-2r}}(1)|_{q}. \tag 3.17
\endalign
$$
It follows that $COB = \Cal O_{\Pee^{c_1+k-2r}}(1)^{\oplus (c_1+k-2r)}$.
By (3.5), we obtain
$$a_0 = \Phi_{(A, 0)}(h, h_n, h_n \xi_{2r-c_1-1})
= e(COB) = 1. \qed$$
\section{4. Proof of Theorem B}
In this section, we prove Theorem B which we restate below.
\theorem{4.1} {\rm (i)} Let $V$ be a rank-$r$ ample bundle over $\Pee^n$.
Assume either $c_1 \le n$ or $c_1 \le (n + r)$ and
$V\otimes {\Cal O}_{\Pee^n}(-1)$ is nef so that $\Pee(V)$ is Fano.
Then the quantum cohomology $H^*_{\omega}(\Pee(V); \Zee)$ is
the ring generated by $h$ and $\xi$ with two relations
$$h^{n+1} = \sum_{i+j \le (c_1 - r)} a_{i,j} \cdot h^i \cdot \xi^j
\cdot e^{-t(n+1-i-j)} \eqno (4.2)$$
$$\sum_{i=0}^r (-1)^i c_i \cdot h^i \cdot \xi^{r-i} = e^{-tr}
+ \sum_{i+j \le (c_1-n-1)} b_{i,j} \cdot h^i \cdot \xi^{j}
\cdot e^{-t(r-i-j)} \eqno (4.3)$$
where the coefficients $a_{i, j}$ and $b_{i, j}$ are integers depending on $V$;
{\rm (ii)} If we further assume that $c_1<2r$,
then the leading coefficient $a_{0,c_1-r}=1$.
\endproclaim
\noindent
{\it Proof.} (i) First, we determine the first relation $f_\omega^1$
in (1.3). By Lemma 3.7,
$$h \cdot h_p = h_{p +1} + \sum_{A \in H_2'} (h \cdot h_p)_A \cdot
e^{t K_{\Pee(V)}(A)} \eqno (4.4)$$
where $p \ge 1$ and $H_2'$ stands for $H_2(\Pee(V); \Zee) -
\Zee \cdot (h_n\xi_{r-2})_*$. Thus,
$$h^{n - p} \cdot h_{p +1} = h^{n - p + 1} \cdot h_{p} - \sum_{A \in H_2'}
h^{n - p} \cdot (h \cdot h_p)_A \cdot e^{t K_{\Pee(V)}(A)}.$$
If $(h \cdot h_p)_A \ne 0$, then $A = [E]$ for some effective curve $E$.
So $a = h(A) \ge 0$. Since $A \in H_2'$, $a \ge 1$.
We claim that $-K_{\Pee(V)}(A) \ge (n+1-c_1 + r)$ with equality
if and only if $A = [h_{n-1}\xi_{r-1} + (1 - c_1) h_{n}\xi_{r-2}]_*
\overset \hbox{def} \to = A_2$. Indeed, if $c_1 \le n$,
then $-K_{\Pee(V)}(A) = (n+1-c_1)a + r \cdot \xi(A) \ge (n+1-c_1 + r)$
with equality if and only if $a = \xi(A) = 1$, that is,
if and only if $A = A_2$; if $c_1 \le (n + r)$ and $(\xi -h)$ is nef,
then again $-K_{\Pee(V)}(A) = (n+1+r-c_1)a + r \cdot (\xi - h)(A)
\ge (n+1-c_1 + r)$ with equality if and only if $a = 1$ and
$(\xi - h)(A) = 0$, that is, if and only if $A = A_2$.
Thus, $\hbox{deg}((h \cdot h_p)_A) = 1 + p + K_{\Pee(V)}(A)
\le (p - n +c_1 -r)$,
and $\hbox{deg}(h^{n - p} \cdot (h \cdot h_p)_A) \le (c_1 - r)$.
Using induction on $p$ and keeping track of the exponential
$e^{t K_{\Pee(V)}(A)}$, we obtain
$$0 = h_{n+1} = h^{n+1} - \sum_{i+j \le (c_1 - r)} a_{i,j} \cdot
h^i \cdot \xi^j \cdot e^{-t(n+1-i-j)}.$$
Therefore, the first relation $f_\omega^1$ for the quantum cohomology ring is:
$$h^{n+1} = \sum_{i+j \le (c_1 - r)} a_{i,j} \cdot h^i \cdot \xi^j \cdot
e^{-t(n+1-i-j)}.$$
Next, we determine the second relation $f_\omega^2$ in (1.3).
We need to compute the quantum product $h^i \cdot \xi^{r-i}$
for $0 \le i \le r$. First, we calculate the quantum product $\xi^r$.
Note that if $A = (bh_{n}\xi_{r-2})_*$ with $b \ge 1$,
then $-K_{\Pee(V)}(A) = br \ge r$ with
$-K_{\Pee(V)}(A) = r$ if and only if
$A = (h_{n}\xi_{r-2})_* \overset \hbox{def} \to = A_1$.
Thus for $p \ge 1$,
$$\xi \cdot \xi_p =
\cases
\xi_{p+1} + \sum_{A \in H_2'} (\xi \cdot \xi_p)_A \cdot e^{t K_{\Pee(V)}(A)},
&\text{if $p < r -1$}\\
\xi_{r} + (\xi \cdot \xi_{r-1})_{A_1} \cdot e^{-tr} +
\sum_{A \in H_2'} (\xi \cdot \xi_{r-1})_A \cdot e^{t K_{\Pee(V)}(A)},
&\text{if $p = r -1$.}\\
\endcases
$$
Note that $(\xi \cdot \xi_{r-1})_{A_1}$ is of
degree zero; by Lemma 3.6, we obtain $(\xi \cdot \xi_{r-1})_{A_1} =
\Phi_{(A_1, 0)}(\xi, \xi_{r-1}, h_{n}\xi_{r-1}) = 1$. Therefore for $p \ge 1$,
$$\xi \cdot \xi_p =
\cases
\xi_{p+1} + \sum_{A \in H_2'} (\xi \cdot \xi_p)_A \cdot e^{t K_{\Pee(V)}(A)},
&\text{if $p < r -1$}\\
\xi_{r} + e^{-tr} +
\sum_{A \in H_2'} (\xi \cdot \xi_{r-1})_A \cdot e^{t K_{\Pee(V)}(A)},
&\text{if $p = r -1$.}\\
\endcases
\eqno (4.5)$$
Now, for $i \ge 1$ and $j \ge 1$ with $i + j \le r$, we have
$$h_i \cdot \xi_j =
\cases
h_i \xi_{j} + \sum_{A \in H_2'} (h_i \cdot \xi_j)_A \cdot e^{t K_{\Pee(V)}(A)},
&\text{if $i + j < r$}\\
h_i \xi_{j} + (h_i \xi_{j})_{A_1} \cdot e^{-tr} +
\sum_{A \in H_2'} (h_i \cdot \xi_j)_A \cdot e^{t K_{\Pee(V)}(A)},
&\text{if $i + j = r$;}\\
\endcases
$$
when $i + j = r$, $(h_i \xi_{j})_{A_1}$ is of degree zero; by Lemma 3.7,
we have $(h_i \cdot \xi_j)_{A_1} =
\Phi_{(A_1, 0)}(h_i, \xi_{r-i}, h_{n}\xi_{r-1}) = 0$.
Therefore for $i \ge 1$ and $j \ge 1$ with $i + j \le r$,
$$h_i \cdot \xi_j = h_i \xi_{j} + \sum_{A \in H_2'}
(h_i \cdot \xi_j)_A \cdot e^{t K_{\Pee(V)}(A)}. \eqno (4.6)$$
From the proof of the first relation $f_\omega^1$, we see that
if $\alpha$ and $\beta$ are homogeneous elements in
$H^*(\Pee(V); \Zee)$ with $\hbox{deg}(\alpha) + \hbox{deg}(\beta) = m \le r$,
then $\hbox{deg}((\alpha \cdot \beta)_A) \le m - (n+1-c_1 + r)$
for $A \in H_2'$. Thus if $\gamma$ is a homogeneous element in
$H^*(\Pee(V); \Zee)$ with $\hbox{deg}(\gamma) = r-m$,
then $\hbox{deg}(\gamma \cdot (\xi \cdot \xi_p)_A) \le (c_1 - n - 1)$.
Since $\sum_{i=0}^r (-1)^i c_i \cdot h_i \xi_{r-i} = 0$,
it follows from (4.4), (4.5), and (4.6) that
the second relation $f_\omega^2$ is
$$\sum_{i=0}^r (-1)^i c_i \cdot h^i \cdot \xi^{r-i} = e^{-tr}
+ \sum_{i+j \le (c_1-n-1)} b_{i,j} \cdot h^i \cdot \xi^{j}
\cdot e^{-t(r-i-j)}.$$
(ii) From the proof of the first relation in (i), we see that
$-K_{\Pee(V)}(A) \ge (n+1-c_1 + r)$ with equality
if and only if $A = A_2$;
moreover, the term $\xi^{c_1-r}$ can only come from the quantum correction
$(h \cdot h_n)_{A_2}$. Now
$$(h \cdot h_n)_{A_2} = (\sum_{i = 0}^{c_1 - r} a_i' h_i \xi_{c_1 - r - i})
\cdot e^{-t(n+1-c_1 + r)}$$
where $a_0' = \Phi_{(A_2, 0)}(h, h_n, h_n \xi_{2r-c_1-1})$.
Since $c_1 < 2r$, $(c_1 - r)< r$. By (4.4), (4.5), and (4.6), we conclude that
$h_i \xi_{c_1 - r - i} = h^i \cdot \xi^{c_1 - r - i} +
\text{(lower degree terms)}$.
Thus $a_{0,c_1-r} = a_0' = \Phi_{(A_2, 0)}(h, h_n, h_n \xi_{2r-c_1-1})$.
By Lemma 3.8, $a_{0,c_1-r} = 1.\qed$
It is understood that when $c_1 \le n$, then the summations
on the right-hand-sides of the second relations (4.3) and (4.9)
below do not exist.
Next, we shall sharpen the results in Theorem 4.1 by imposing additional
conditions on $V$. Let $V$ be a rank-$r$ ample vector bundle over $\Pee^n$.
Then $c_1 \ge r$. Thus if $c_1<2r$ and if
either $2c_1 \le (n + r)$ or $2c_1 \le (n + 2r)$ and
$V\otimes {\Cal O}_{\Pee^n}(-1)$ is nef, then the conditions in Theorem 4.1
are satisfied.
\corollary{4.7} {\rm (i)} Let $V$ be a rank-$r$ ample vector bundle over
$\Pee^n$ with $c_1<2r$. Assume that either $2c_1 \le (n + r)$ or
$2c_1 \le (n + 2r)$ and $V\otimes {\Cal O}_{\Pee^n}(-1)$ is nef so that
$\Pee(V)$ is a Fano variety. Then the first relation (4.2) is
$$h^{n+1}= \left ( \sum_{i=0}^{c_1-r} a_i \cdot h^i \cdot \xi^{c_1-r-i}
\right ) \cdot e^{-t(n+1+r-c_1)} \eqno (4.8)$$
where the integers $a_i$ depend on $V$. Moreover, $a_0 = 1$.
{\rm (ii)} Let $V$ be a rank-$r$ ample vector bundle over $\Pee^n$.
Assume that $2c_1 \le (2n + r+1)$ and $V\otimes {\Cal O}_{\Pee^n}(-1)$ is nef
so that $\Pee(V)$ is Fano. Then the second relation (4.3) is
$$\sum_{i=0}^r (-1)^i c_i \cdot h^i \cdot \xi^{r-i} = e^{-tr} +
\sum_{i=0}^{c_1-n-1} b_{i} \cdot h^i \cdot \xi^{c_1-n-1-i}
\cdot e^{-t(n+1+r-c_1)} \eqno (4.9)$$
where the integers $b_i$ depend on $V$.
\endproclaim
\proof (i) From the proof of Theorem 4.1 (i),
we notice that it suffices to show that the only homology class
$A \in H_2' = H_2(\Pee(V); \Zee) - \Zee \cdot (h_n\xi_{r-2})_*$
which has nonzero contributions to the quantum corrections in (4.4) is
$A = [h_{n-1}\xi_{r-1} + (1 - c_1) h_{n}\xi_{r-2}]_*
\overset \hbox{def} \to = A_2$. In other words,
if $A = (ah_{n-1}\xi_{r-1}+ b h_{n}\xi_{r-2})_*$ with $a \ne 0$
and if $\Phi_{(A, 0)}(h, h_p, \alpha) \ne 0$ for $1 \le p \le n$ and
$\alpha \in H^*(\Pee(V); \Zee)$, then $A = A_2$. First of all,
we show that $a = 1$. Suppose $a \ne 1$. Then $a \ge 2$. By (3.2),
$$\align
{1 \over 2} \cdot \hbox{deg}(\alpha)
&= (n+r-1) - K_{\Pee(V)}(A) - 1 - p\cr
&= (n+r-1) + [(n + 1-c_1)a + r \cdot \xi(A)] - 1-p\cr
&\ge \hbox{dim}(\Pee(V)) + [(n + 1-c_1)a + r \cdot \xi(A)] - 1-n.\cr
\endalign$$
If $2c_1 \le (n + r)$, then $c_1 \le n$,
and $[(n + 1-c_1)a + r \cdot \xi(A)] - 1-n \ge 2(n + 1-c_1) + r - 1-n > 0$.
If $2c_1 \le (n + 2r)$ and $(\xi - h)$ is nef,
then $c_1 \le n + r$, and $[(n + 1-c_1)a + r \cdot \xi(A)] - 1-n =
[(n + 1 + r-c_1)a + r \cdot (\xi-h)(A)] - 1-n \ge 2(n + 1 + r-c_1)-1 -n > 0$.
Thus, $[(n + 1-c_1)a + r \cdot \xi(A)] - 1-n > 0$,
and so $\hbox{deg}(\alpha)/2 > \hbox{dim}(\Pee(V))$. But this is absurd.
Next, we prove that $b = (1 - c_1)$, or equivalently, $\xi(A) = 1$.
Suppose $\xi(A) \ne 1$. Then $\xi(A) \ge 2$. By (3.2),
$$\align
{1 \over 2} \cdot \hbox{deg}(\alpha)
&= (n+r-1) + [(n + 1-c_1) + r \cdot \xi(A)] - 1-p\cr
&\ge \hbox{dim}(\Pee(V)) + [(n + 1-c_1) + 2r] - 1-n\cr
&> \hbox{dim}(\Pee(V))\cr
\endalign$$
since $c_1 < 2r$. But once again this is absurd.
(ii) We follow the previous arguments for (i). Again it suffices to show that
if $A = (ah_{n-1}\xi_{r-1}+ b h_{n}\xi_{r-2})_*$ with $a \ne 0$
and if $\Phi_{(A, 0)}(\alpha_1, \alpha_2, \alpha) \ne 0$ for
some $\alpha_1, \alpha_2, \alpha \in H^*(\Pee(V); \Zee)$
with $\hbox{deg}(\alpha_1) + \hbox{deg}(\alpha_2) \le r$, then $A = A_2$.
Indeed, if $a \ne 1$ or if $a = 1$ but $\xi(A) \ne 1$,
then we must have $\hbox{deg}(\alpha)/2 > \hbox{dim}(\Pee(V))$.
But this is impossible. Therefore, $a = 1$ and $\xi(A) = 1$. So $A = A_2$.
\endproof
Now we discuss the relation between the quantum corrections and
the extremal rays of the Fano variety $\Pee(V)$.
Let $V$ be a rank-r ample vector bundle over $\Pee^n$
with $c_1<2r$ and $2c_1 \le (n + r)$. By (4.8) and (4.3),
the quantum cohomology ring $H^*_{\omega}(\Pee(V); \Zee)$ is
the ring generated by $h$ and $\xi$ with two relations
$$h^{n+1}= \left ( \sum_{i=0}^{c_1-r} a_i \cdot h^i \cdot \xi^{c_1-r-i}
\right ) \cdot e^{-t(n+1+r-c_1)} \eqno (4.10)$$
$$\sum_{i=0}^r (-1)^i c_i \cdot h^i \cdot \xi^{r-i} = e^{-tr}.
\eqno (4.11)$$
From the proof of Theorem 4.1 (i), we notice that the quantum correction to
the second relation (4.11) comes from the homology class
$A_1 = (h_n\xi_{r-2})_*$ which is represented by the lines in
the fibers of $\pi: \Pee(V) \to \Pee^n$. Also,
we notice from the proof of Corollary 4.7 (i) that the quantum correction
to the first relation (4.10) comes from the homology class
$A_2 = [h_{n-1}\xi_{r-1} + (1 - c_1) h_{n}\xi_{r-2}]_*$;
from the proof of Lemma 3.8, $A_2$ can be represented by
a smooth rational curve isomorphic to lines in $\Pee^n$ via $\pi$.
Now $A_1$ generates one of the two extremal rays of
$\Pee(V)$. It is unclear whether $A_2$ generates the other extremal ray.
By Lemma 2.2 (iii), if we further assume that $2c_1 \le (n + 1)$,
then indeed $A_2$ generates the other extremal ray of $\Pee(V)$.
By Lemma 2.2 (ii), $A_2$ generates the other extremal ray of
$\Pee(V)$ if and only if $(\xi - h)$ is nef, that is,
$V \otimes \Cal O_{\Pee^n}(-1)$ is a nef vector bundle over $\Pee^n$.
\section{5. Direct sum of line bundles over $\Pee^n$}
In this section, we partially verify Batyrev's conjecture on
the quantum cohomology of projective bundles associated to
direct sum of line bundles over $\Pee^n$.
We shall use (3.5) to compute the necessary Gromov-Witten invariants.
Our first step is to recall some standard materials for
the Grassmannian $G(2, n+1)$ from \cite{3}.
Then we determine certain obstruction bundle and its Euler class.
Finally we proceed to determine the first and second relations
for the quantum cohomology.
On the Grassmannian $G(2, n+1)$, there exists a tautological exact sequence
$$0 \to S \to (\Cal O_{G(2, n+1)})^{\oplus (n + 1)} \to Q \to 0 \eqno (5.1)$$
where the sub- and quotient bundles $S$ and $Q$ are of
rank $2$ and $(n -1)$ respectively. Let $\alpha$ and $\beta$
be the virtual classes such that $\alpha + \beta = -c_1(S)$
and $\alpha \beta = c_2(S)$. Then
$$\hbox{cl}(\{ \ell \in G(2, n+1)| \ell \cap h_p \ne \emptyset \})
= {\alpha^p - \beta^p \over \alpha - \beta} \eqno (5.2)$$
where $\hbox{cl}(\cdot)$ denotes the fundamental class and
$h_p$ stands for a fixed linear subspace of $\Pee^n$ of codimention $p$.
If $P(\alpha, \beta)$ is a symmetric homogeneous polynomial of
degree $(2n - 2)$ (so that $P(\alpha, \beta)$ can be written as
a polynomial of maximal degree in the Chern classes of the bundle $S$),
then we have
$$\int_{G(2, n+1)} P(\alpha, \beta) =
\left ( \text{the coefficient of } \alpha^n \beta^n \text{ in}
-{1 \over 2}(\alpha - \beta)^2 P(\alpha, \beta) \right ). \eqno (5.3)$$
Let $F_n = \{ (x, \ell) \in \Pee^n \times G(2, n+1)| x \in \ell \}$,
and $\pi_1$ and $\pi_2$ are the two natural projections from
$F_n$ to $\Pee^n$ and $G(2, n+1)$ respectively.
Then $F_n = \Pee(S^*)$ where $S^*$ is the dual bundle of $S$,
and $(\pi_1^*\Cal O_{\Pee^n}(1))|_{F_n}$ is
the tautological line bundle over $F_n$.
Let $\hbox{Sym}^m(S^*)$ be the $m$-th symmetric product of $S^*$.
Then for $m \ge 0$,
$$\pi_{2*} (\pi_1^*\Cal O_{\Pee^n}(m)|_{F_n})
\cong \hbox{Sym}^m(S^*). \eqno (5.4)$$
By the duality theorem for higher direct image sheaves
(see p.253 in \cite{7}),
$$\align
R^1 \pi_{2*} (\pi_1^*\Cal O_{\Pee^n}(-m)|_{F_n})
&\cong (\pi_{2*} (\pi_1^*\Cal O_{\Pee^n}(m - 2)|_{F_n}))^*
\otimes (\hbox{det}S^*)^* \\
&\cong \hbox{Sym}^{m-2}(S) \otimes (\hbox{det}S) \tag 5.5 \\
\endalign$$
Now, let $V = \oplus_{i = 1}^r \Cal O_{\Pee^n}(m_i)$ where
$1 = m_1 = \ldots = m_k < m_{k+1} \le \ldots \le m_r$.
Assume that $k \ge 1$ and $\Pee(V)$ is Fano.
Then the two extremal rays of $\Pee(V)$ are generated by the two classes
$A_1 = (h_{n}\xi_{r-2})_*$ and
$A_2 = [h_{n-1}\xi_{r-1}+ (1 - c_1) h_{n}\xi_{r-2}]_*$. From the proof
of Lemma 2.3 (ii), we see that
$$\frak M(A_2, 0)/PSL(2; \Cee) = G(2, n+1) \times \Pee^{k -1}. \eqno (5.6)$$
Let a morphism $f \in \frak M(A_2, 0)$ be induced by some surjective map
$V|_\ell \to \Cal O_\ell(1) \to 0$ such that
the image $\hbox{Im}(f)$ of $f$ is of the form
$$\hbox{Im}(f) = \ell \times \{q\} \subset \ell \times \Pee^{k-1}
\subset \Pee^n \times \Pee^{k-1}. $$
Then by arguments similar to the proof of (3.17), we have
$$H^1(N_f) \cong \oplus_{u=k+1}^r H^1(\Cal O_{\ell}(1-m_u))
\otimes \Cal O_{\Pee^{k-1}}(1)|_{q}. \eqno (5.7)$$
It follows that the obstruction bundle $COB$ over
$\frak M(A_2, 0)/PSL(2; \Cee)$ is
$$COB \cong \oplus_{u=k+1}^r R^1 \pi_{2*} (\pi_1^*\Cal
O_{\Pee^n}(1-m_u)|_{F_n})
\otimes \Cal O_{\Pee^{k-1}}(1). \eqno (5.8)$$
Since $c_1(S) = -(\alpha + \beta)$ and $c_2(S) = \alpha \beta$,
we obtain from (5.5) the following.
\lemma{5.9} The Euler class of the obstruction bundle $COB$ is
$$e(COB) = \prod_{u = k +1}^r \prod_{v=0}^{m_u - 3}
[(1+v)(-\alpha) + (m_u - 2 -v)(-\beta) + \tilde h] \eqno (5.10)$$
where $\tilde h$ stands for the hyperplane class in $\Pee^{k-1}$. \qed
\endproclaim
Next assuming $c_1 < 2r$, we shall compute the Gromov-Witten invariant
$$W_i \overset \hbox{def} \to = \Phi_{(A_2, 0)}(h_{\tilde n},
h_{n+1 - \tilde n}, h_{n-i}\xi_{2r-c_1 - 1+i}) \eqno (5.11)$$
where $0 \le i \le (c_1 - r)$ and $\tilde n = \left [{n+1 \over 2} \right ]$
is the largest integer $\le (n+1)/2$.
\lemma{5.12} Assume $c_1 < \text{min}(2r, (n+1+2r)/2)$ and
$0 \le i \le (c_1 - r)$. Then $W_i$ is the coefficient of $t^i$
in the power series expansion of
$$\prod_{u = 1}^r (1 - m_u t)^{m_u-2}.$$
\endproclaim
\noindent
{\it Proof.} Note that the restriction of $\xi$ to $\Pee^n \times \Pee^{k-1}
= \Pee(\Cal O_{\Pee^n}(1)^{\oplus k})$ is $(h + \tilde h)$. Thus,
$$\align
h_{n-i}\xi_{2r-c_1 - 1+i}|_{\Pee^n \times \Pee^{k-1}}
&= \sum_{j = 0}^{2r-c_1-1+i} {2r-c_1 - 1+i \choose j}
h_{n-i+j} \tilde h_{2r-c_1-1+i-j}\\
&= \sum_{j = 0}^{i} {2r-c_1 - 1+i \choose j}
h_{n-i+j} \tilde h_{2r-c_1-1+i-j}. \\
\endalign$$
So by (3.5) (replacing $\frak M(A_2, 0)$ by
$\frak M(A_2, 0)/PSL(2; \Cee)$), (5.2), and Lemma 5.9,
$$W_i = \int_{G(2, n+1) \times \Pee^{k-1}} \tilde P(\alpha, \beta)
\eqno (5.13)$$
where $\tilde P(\alpha, \beta)$ is the symmetric homogeneous polynomial of
degree $(2n - 2) + (k-1)$:
$$\align
\tilde P(\alpha, \beta) &= {\alpha^{\tilde n} - \beta^{\tilde n}
\over \alpha - \beta} \cdot {\alpha^{n + 1- \tilde n} -
\beta^{n + 1- \tilde n} \over \alpha - \beta} \\
&\qquad \cdot \sum_{j = 0}^{i} {2r-c_1 - 1+i \choose j}
{\alpha^{n-i+j} - \beta^{n-i+j} \over \alpha - \beta}
\cdot \tilde h_{2r-c_1-1+i-j} \\
&\qquad \cdot \prod_{u = k +1}^r \prod_{v=0}^{m_u - 3}
[(1+v)(-\alpha) + (m_u - 2 -v)(-\beta) + \tilde h] \\
&= \sum_{j = 0}^{i} {2r-c_1 - 1+i \choose j}
{\alpha^{n+1} - \alpha^{n + 1- \tilde n}\beta^{\tilde n} - \alpha^{\tilde n}
\beta^{n + 1- \tilde n} + \beta^{n+1} \over (\alpha - \beta)^2}\\
&\qquad \cdot \sum_{t= 0}^{n-i+j-1} \alpha^{t} \beta^{n-i+j-1-t} \cdot
\tilde h_{2r-c_1-1+i-j}\\
&\qquad \cdot \prod_{u = k +1}^r \prod_{v=0}^{m_u - 3}
[(1+v)(-\alpha) + (m_u - 2 -v)(-\beta) + \tilde h]. \\
\endalign$$
By (5.3) and (5.13), we conclude from straightforward manipulations that:
$$\align
W_i &= \sum_{j = 0}^{i} {2r-c_1 - 1+i \choose j} \cdot (-1)^{i-j} \\
&\qquad \cdot \sum_{j_{k+1} + \ldots + j_r = i -j} \quad
\prod_{u = k+1}^r {m_u-2 \choose j_u} (m_u -1)^{j_u} \\
&= \sum_{j = 0}^{i} {2r-c_1 - 1+i \choose i-j} \cdot (-1)^{j} \\
&\qquad \cdot \sum_{j_{k+1} + \ldots + j_r = j} \quad
\prod_{u = k+1}^r {m_u-2 \choose j_u} (m_u -1)^{j_u}. \\
\endalign$$
Thus $W_i$ is the coefficient of $t^i$ in the polynomial
$$\align
&\qquad (1+t)^{2r-c_1 - 1+i} \cdot \prod_{u = k+1}^r [1 - (m_u -1) t]^{m_u-2}\\
&= (1+t)^{2r-c_1 - 1+i} \cdot \prod_{u = k+1}^r [(1+t) - m_u t]^{m_u-2}\\
&= (1+t)^{2r-c_1 - 1+i} \cdot \sum_{j=0}^{c_1-2r+k}
\sum_{j_{k+1} + \ldots + j_r = j} \\
&\qquad \cdot \prod_{u = k+1}^r {m_u-2 \choose j_u} (-m_ut)^{j_u}
\cdot (1+t)^{m_u-2-j_u}\\
&= \sum_{j=0}^{c_1-2r+k} \sum_{j_{k+1} + \ldots + j_r = j} \quad
\prod_{u = k+1}^r {m_u-2 \choose j_u} (-m_ut)^{j_u} \cdot (1+t)^{i+k-1-j}\\
\endalign$$
since $\sum_{u = k+1}^r (m_u - 2 - j_u) = c_1 -2r+k - j$.
So $W_i$ is the coefficient of $t^i$ in
$$\align
\prod_{u = k+1}^r (1-m_u t)^{m_u-2} \cdot \sum_{j=0}^{+ \infty}
{j +k-1 \choose k-1} t^j
&= \prod_{u = k+1}^r (1-m_u t)^{m_u-2} \cdot {1 \over (1 - t)^k}\\
&= \prod_{u = 1}^r (1 - m_u t)^{m_u-2}. \qed\\
\endalign$$
\proposition{5.14} Let $V = \oplus_{i = 1}^r \Cal O_{\Pee^n}(m_i)$ where
$m_i \ge 1$ for each $i$ and
$$\sum_{i=1}^r m_i < \text{min}(2r, (n+1+2r)/2).$$
Then the first relation $f^1_\omega$ for the quantum cohomology ring
$H^*_{\omega}(\Pee(V); \Zee)$ is
$$h^{n+1} = \prod_{u = 1}^r (\xi - m_u h)^{m_u-1} \cdot
e^{-t(n+1+r-\sum_{i=1}^r m_i)}. \eqno (5.15)$$
\endproclaim
\noindent
{\it Proof.} We may assume that
$1 = m_1 = \ldots = m_k < m_{k+1} \le \ldots \le m_r$.
Since the conclusion clearly holds when $k = r$, we also assume that $k < r$.
Let $c_1 = \sum_{i=1}^r m_i$.
Notice that the conditions in Corollary 4.7 (i) are satisfied. Thus,
$$h^{n+1}= \left ( \sum_{i=0}^{c_1-r} a_i \cdot h^i \cdot \xi^{c_1-r-i}
\right ) \cdot e^{-t(n+1+r-c_1)}.$$
More directly, putting $\tilde n = \left [{n+1 \over 2} \right ]$,
then $\tilde n < -K_{\Pee(V)}(A_2) = (n+1+r -c_1)$,
and $(n+1-\tilde n) < -K_{\Pee(V)}(A_2)$
unless $n$ is even and $c_1 = (n + 2r)/2$. From the proofs in Theorem 4.1
and Corollary 4.7 (i) for the first relation $f^1_\omega$,
we have $h^{\tilde n} = h_{\tilde n}$,
and $h^{n+1-\tilde n} = h_{n+1-\tilde n}$ unless $n$ is even and
$c_1 = (n + 2r)/2$. Moreover, if $n$ is even and $c_1 = (n + 2r)/2$,
then $h^{n+1-\tilde n} = h \cdot h^{n-\tilde n} = h \cdot h_{n-\tilde n}
= h_{n+1-\tilde n} + (h \cdot h_{n-\tilde n})_{A_2} \cdot e^{-t(n+1+r-c_1)}$.
Since $(h \cdot h_{n-\tilde n})_{A_2}$ is of degree zero,
$(h \cdot h_{n-\tilde n})_{A_2} =
\Phi_{(A_2, 0)}(h, h_{n-\tilde n}, h_n\xi_{r-1})$. Since $1 \le k <r$,
we can choose a point $q_0$ in $\Pee(V)$ representing the homology class
$(h_n\xi_{r-1})_*$ such that the point $q_0$ is not contained in
the $(k-1)$-dimensional linear subspace
$$\Pee^{k -1} = \Pee((\Cal O_{\Pee^n}(1)^{\oplus k})|_{\pi(q_0)})
\subset \Pee(V|_{\pi(q_0)}) \cong \Pee^{r-1}.$$
Note that for every $f \in \frak M(A_2, 0)$,
$\hbox{Im}(f) = \ell \times \{ q \}$ for some line $\ell \subset \Pee^n$
and some point $q \in \Pee^{k -1}$. Thus $\hbox{Im}(f)$ can not pass $q_0$.
As in the proof of Lemma 3.7, we conclude that
$\Phi_{(A_2, 0)}(h, h_{n-\tilde n}, h_n\xi_{r-1}) = 0$.
Therefore, $h^{n+1-\tilde n} = h_{n+1-\tilde n}$. So
$$h^{n+1} = h^{\tilde n} \cdot h^{n+1-\tilde n}
= h_{\tilde n} \cdot h_{n+1-\tilde n}.$$
By similar arguments in the proofs of Theorem 4.1 and Corollary 4.7 (i)
for the first relation $f^1_\omega$,
we see that if $(h_{\tilde n} \cdot h_{n+1-\tilde n})_A \ne 0$,
then $A = 0, A_2$. Thus
$$h^{n+1} = h_{n+1} + (h_{\tilde n} \cdot h_{n+1-\tilde n})_{A_2} \cdot
e^{-t(n+1+r-c_1)} = (h_{\tilde n} \cdot h_{n+1-\tilde n})_{A_2}
\cdot e^{-t(n+1+r-c_1)}.$$
So it suffices to show that $(h_{\tilde n} \cdot h_{n+1-\tilde n})_{A_2}
= \prod_{u = 1}^r (\xi - m_u h)^{m_u-1}$. Note that
$$\prod_{u = 1}^r (\xi - m_u h)^{m_u-1} =
\prod_{u = 1}^r (\xi - m_u h)_{m_u-1}$$
where the right-hand-side stands for the product in the ordinary cohomology.
Thus we need to show that $(h_{\tilde n} \cdot h_{n+1-\tilde n})_{A_2}
= \prod_{u = 1}^r (\xi - m_u h)_{m_u-1}$, or equivalently,
$$\Phi_{(A_2, 0)}(h_{\tilde n}, h_{n+1 - \tilde n}, h_{n-i}\xi_{2r-c_1 - 1+i})
= \prod_{u = 1}^r (\xi - m_u h)_{m_u-1} h_{n-i}\xi_{2r-c_1 - 1+i}
\eqno (5.16)$$
for $0 \le i \le (c_1 - i)$. The left-hand-side of (5.16) is computed
in Lemma 5.12.
Denote the right-hand-side of (5.16) by $\tilde W_i$.
Let $s_i$ be the $i$-th Segre class of $V$. Then
we have $s_i = (-1)^i \cdot \sum_{j_1 + \ldots + j_r = i}
\prod_{u=1}^r m_u^{j_u}$ and
$$\sum_{i=0}^{+ \infty} (-1)^{i} s_{i} t^{i} =
\prod_{u=1}^r {1 \over 1-m_u t}. \eqno (5.17)$$
Moreover from the second relation in (1.1), we obtain for $i \ge r$,
$$\xi_i = (-1)^{i - (r-1)} s_{i-(r-1)} \xi_{r-1} +
\text{ (terms with exponentials of } \xi \text{ less than } (r-1)).$$
It follows from the right-hand-side of (5.16) that
$\tilde W_i$ is equal to
$$\align
&\quad \sum_{j=0}^{c_1-r} \quad \sum_{j_1 + \ldots + j_r = j} \quad
\prod_{u=1}^r {m_u-1 \choose j_u} \xi_{m_u-1-j_u} (-m_uh)_{j_u}
h_{n-i}\xi_{2r-c_1 - 1+i} \\
&= \sum_{j=0}^{i} \quad \sum_{j_1 + \ldots + j_r = j} \quad \prod_{u=1}^r
{m_u-1 \choose j_u} (-m_u)^{j_u} h_{n-i+j}\xi_{r - 1+i-j} \\
&= \sum_{j=0}^{i} (-1)^{i-j} s_{i-j} \quad \sum_{j_1 + \ldots + j_r = j}
\quad \prod_{u=1}^r {m_u-1 \choose j_u} (-m_u)^{j_u}. \\
\endalign$$
Therefore, the formal power series $\sum_{i=0}^{+ \infty} \tilde W_i t^i$
is equal to
$$\align
&\quad \sum_{i=0}^{+ \infty} \sum_{j=0}^{i} (-1)^{i-j} s_{i-j} t^{i-j}
\quad \sum_{j_1 + \ldots + j_r = j} \quad
\prod_{u=1}^r {m_u-1 \choose j_u} (-m_u t)^{j_u} \\
&= \sum_{j=0}^{+ \infty} \sum_{i=j}^{+ \infty} (-1)^{i-j} s_{i-j} t^{i-j}
\quad \sum_{j_1 + \ldots + j_r = j} \quad
\prod_{u=1}^r {m_u-1 \choose j_u} (-m_u t)^{j_u} \\
&= \sum_{j=0}^{+ \infty} \sum_{i=0}^{+ \infty} (-1)^{i} s_{i} t^{i}
\quad \sum_{j_1 + \ldots + j_r = j} \quad
\prod_{u=1}^r {m_u-1 \choose j_u} (-m_u t)^{j_u} \\
&= \sum_{j=0}^{+ \infty} \prod_{u=1}^r {1 \over 1-m_u t}
\quad \sum_{j_1 + \ldots + j_r = j} \quad
\prod_{u=1}^r {m_u-1 \choose j_u} (-m_u t)^{j_u} \\
&= \prod_{u=1}^r {1 \over 1-m_u t} \sum_{j=0}^{+ \infty}
\quad \sum_{j_1 + \ldots + j_r = j} \quad
\prod_{u=1}^r {m_u-1 \choose j_u} (-m_u t)^{j_u} \\
&= \prod_{u=1}^r {1 \over 1-m_u t} \quad \prod_{u=1}^r (1-m_u t)^{m_u-1}\\
&= \prod_{u=1}^r (1-m_u t)^{m_u-2}\\
\endalign$$
where we have applied (5.17) in the third equality.
By Lemma 5.12, $\tilde W_i = W_i$ for $0 \le i \le (c_1 - r)$.
Hence the formule (5.16) and (5.15) hold.
\qed
It turns out that under certain conditions on the integers $m_i$,
the second relation $f^2_\omega$ for the quantum cohomology ring
$H^*_{\omega}(\Pee(V); \Zee)$ is much easier to be determined.
Note that the second relation $f^2$ in (1.1) can be rewritten as
$$\prod_{i=1}^r (\xi-m_i h) = 0 \eqno (5.18)$$
where the left-hand-side stands for the product in the ordinary cohomology
ring.
\proposition{5.19} Let $V = \oplus_{i = 1}^r \Cal O_{\Pee^n}(m_i)$ where
$m_i \ge 1$ for each $i$, $m_i = 1$ for some $i$, and
$\sum_{i=1}^r m_i < (2n+2+r)/2$.
Then the second relation $f^2_\omega$ for the quantum cohomology ring
$H^*_{\omega}(\Pee(V); \Zee)$ is
$$\prod_{i=1}^r (\xi-m_i h) = e^{-tr} \eqno (5.20)$$
where the left-hand-side stands for the product in the quantum cohomology ring.
\endproclaim
\noindent
{\it Proof.} We may assume that
$1 = m_1 = \ldots = m_k < m_{k+1} \le \ldots \le m_r$. So $k \ge 1$.
We notice that the conditions in Corollary 4.7 (ii)
are satisfied. From the proofs of Theorem 4.1 (i) and Corollary 4.7 (ii),
we see that the quantum corrections to the second relation (5.18)
can only come from the classes $A_1, A_2$; moreover,
the quantum correction from $A_1$ is $e^{-tr}$.
Thus it suffices to show that the quantum correction from $A_2$
is zero. In view of (3.3), it suffices to show that
$$\Phi_{(A_2, 0)}(\xi-m_1 h, \ldots, \xi-m_r h, \alpha) = 0$$
for every $\alpha \in H^*(\Pee(V); \Zee)$. For $1 \le i \le r$,
let $V_i$ be the subbundle of $V$:
$$V_i = \Cal O_{\Pee^n}(m_1) \oplus \ldots \oplus
\Cal O_{\Pee^n}(m_{i-1}) \oplus \Cal O_{\Pee^n}(m_{i+1})
\oplus \ldots \oplus \Cal O_{\Pee^n}(m_r),$$
and let $B_i = \Pee(V_i)$ be
the codimension-$1$ subvariety of $\Pee(V)$ induced by the projection
$V \to V_i \to 0$. Then the fundamental class of $B_i$ is $(\xi-m_i h)$.
As in the proof of Lemma 3.7, we need only to show that if
$f \in \frak M(A_2, 0)$, then the image $\hbox{Im}(f)$ can not intersect
with $B_1, \ldots, B_r$ simultaneously. In fact, we will show that
$\hbox{Im}(f)$ can not intersect with $B_1, \ldots, B_k$ simultaneously.
Indeed, $\hbox{Im}(f)$ is of the form
$$\hbox{Im}(f) = \ell \times \{q\} \subset \ell \times \Pee^{k-1}
\subset \Pee^n \times \Pee^{k-1} = \Pee(\Cal O_{\Pee^n}(1)^{\oplus k})$$
for some line $\ell \subset \Pee^n$,
and $B_i|_{\pi^{-1}(\ell)} = \Pee(V_i|_\ell)$.
Put $p = \pi(q) \in \Pee^n$, and
$$V|_p = \oplus_{i=1}^k \Cee \cdot e_i \oplus
(\oplus_{i=k+1}^r \Cal O_{\Pee^n}(m_i)|_p)$$
where $e_i$ is a global section of
$\Cal O_{\Pee^n}(m_i) = \Cal O_{\Pee^n}(1)$ for $i \le k$.
Now the point $q$ is identified with $\Cee \cdot v$ for some nonzero vector
$v \in \oplus_{i=1}^k \Cee \cdot e_i$. Let $v = \sum_{i=1}^k a_i e_i$.
Since $\ell \times \{q\}$ and $B_i$ ($1 \le i \le k$) intersect,
the one-dimensional vector space $\Cee \cdot v$ is also contained
in $(V_i)|_p$. It follows that $a_i = 0$ for every $i$ with
$1 \le i \le k$. But this is impossible since $v$ is a nonzero vector.
\qed
In summary, we partially verify Batyrev's conjecture.
\theorem{5.21} Let $V = \oplus_{i = 1}^r \Cal O_{\Pee^n}(m_i)$ where
$m_i \ge 1$ for each $i$ and
$$\sum_{i=1}^r m_i < \text{min}(2r, (n+1+2r)/2, (2n+2+r)/2).$$
Then the quantum cohomology $H^*_{\omega}(\Pee(V); \Zee)$ is
generated by $h$ and $\xi$ with relations
$$h^{n+1}=\prod^r_{i=1}(\xi-m_ih)^{m_i-1} \cdot
e^{-t(n+1+r-\sum_{i=1}^r m_i)} \qquad {and} \qquad
\prod^r_{i=1}(\xi-m_ih) = e^{-tr}.$$
\endproclaim
\proof Follows immediately from Propositions 5.14 and 5.19.
\endproof
\section{6. Examples}
In this section, we shall determine the quantum cohomology of $\Pee(V)$
for ample bundles $V$ over $\Pee^n$ with $2 \le r \le n$ and $c_1 = r + 1$.
In these cases, $V|_\ell = \Cal O_\ell(1)^{\oplus (r - 1)} \oplus
\Cal O_\ell(2)$ for every line $\ell \subset \Pee^n$. In particular,
$V$ is a uniform bundle. If $r < n$, then by the Theorem 3.2.3 in \cite{10},
$V = \Cal O_{\Pee^n}(1)^{\oplus (r - 1)} \oplus \Cal O_{\Pee^n}(2)$;
if $r = n$, then by the results on pp.71-72 in \cite{10},
$V = \Cal O_{\Pee^n}(1)^{\oplus (n - 1)} \oplus \Cal O_{\Pee^n}(2)$
or $V = T_{\Pee^n}$ the tangent bundle of $\Pee^n$.
When $V = \Cal O_{\Pee^n}(1)^{\oplus (r - 1)} \oplus \Cal O_{\Pee^n}(2)$
with $r \le n$, the conditions in Theorem 5.21 are satisfied,
so the quantum cohomology ring $H^*_{\omega}(\Pee(V); \Zee)$ is
the ring generated by $h$ and $\xi$ with two relations
$$h^{n+1}= (\xi-2h) \cdot e^{-t(n+1+r-c_1)}
\qquad \hbox{and} \qquad (\xi-h)^{r-1} (\xi-2h) = e^{-tr}.$$
In the rest of this section, we compute the quantum cohomology of
$\Pee(T_{\Pee^n})$. It is well-known that $(\xi-h)$ is a nef divisor
on $\Pee(T_{\Pee^n})$, and the two extremal rays of $\Pee(T_{\Pee^n})$
are generated by $A_1 = (h_{n}\xi_{n-2})_*$ and
$A_2 = (h_{n-1}\xi_{n-1} - n h_{n}\xi_{n-2})_*$.
Moreover, $A_2$ is represented by smooth rational curves in $\Pee(T_{\Pee^n})$
induced by the surjective maps $T_{\Pee^n}|_\ell \to \Cal O_\ell(1) \to 0$
for lines $\ell \subset \Pee^n$. Since $c_1 = n+1$ and $n \ge 2$,
the assumptions in Corollary 4.7 are satisfied,
so the quantum cohomology ring $H^*_{\omega}(\Pee(T_{\Pee^n}); \Zee)$
is the ring generated by $h$ and $\xi$ with two relations
$$h^{n+1}= (a_1 h + \xi) \cdot e^{-tn} \quad \text{and} \quad \sum_{i=0}^n
(-1)^i c_i \cdot h^i \cdot \xi^{n-i} = (1 + b_0) \cdot e^{-tn}. \eqno (6.1)$$
More precisely, putting $H_2' = H_2(\Pee(V); \Zee) -
\Zee \cdot (h_n\xi_{n-2})_*$, then we see from the proof of Corollary 4.7 (i)
that the only homology class $A \in H_2'$ which has nonzero contributions to
the quantum corrections in (4.4) is $A = A_2$. Thus by (4.4),
$$h \cdot h_p = \cases h_{p+1}, &\text{if $p \le n -2$}\\
h_n + a_1' \cdot e^{-tn}, &\text{if $p = n -1$}\\
h_{n+1} + (a_2'h+a_3'\xi) \cdot e^{-tn}, &\text{if $p = n$}.\\
\endcases
\eqno (6.2)$$
where $a_1' = \Phi_{(A_2, 0)}(h, h_{n-1}, h_n \xi_{n-1})$,
$a_3' = \Phi_{(A_2, 0)}(h, h_{n}, h_n \xi_{n-2})$, and
$$a_2' = \Phi_{(A_2, 0)}(h, h_{n}, h_{n-1} \xi_{n-1})-c_1 a_3'.$$
By Lemma 3.8, $a_3' = 1$. Thus $a_1 = (a_1'+a_2')$ and the first relation
$f_\omega^1$ in (6.1) is
$$h^{n+1}= ((a_1'+a_2')h + \xi) \cdot e^{-tn} \eqno (6.3)$$
Similarly, from the proof of Corollary 4.7 (ii), we see that
the only homology class $A \in H_2'$ which has nonzero contributions to
the quantum corrections in (4.5) and (4.6) is also $A = A_2$.
By (4.5), $\xi \cdot \xi_p = \xi_{p+1}$
if $p < n -1$, and $\xi \cdot \xi_{n-1} = \xi_{n} + e^{-tn} +
b_2^{(n)} \cdot e^{-tn}$ where
$b_2^{(n)} = \Phi_{(A_2, 0)}(\xi, \xi_{n-1}, h_n \xi_{n-1})$. Thus,
$$\xi^p =
\cases
\xi_{p}, &\text{if $p < n$}\\
\xi_{n} + (1 + b_2^{(n)}) \cdot e^{-tn}, &\text{if $p = n$}\\
\endcases
\eqno (6.4)$$
By (6.2), we have $h \cdot h_p = h_{p+1}$ if $p < n -1$,
and $h \cdot h_{n-1} = h_{n} + b_2^{(0)} \cdot e^{-tn}$ where
$b_2^{(0)} = a_1' = \Phi_{(A_2, 0)}(h_{n-1}, h, h_n \xi_{n-1})$. Thus, we
obtain
$$h^p =
\cases
h_{p}, &\text{if $p < n$}\\
h_{n} + b_2^{(0)} \cdot e^{-tn}, &\text{if $p = n$}\\
\endcases
\eqno (6.5)$$
By (4.6), for $1 \le i \le (n-1)$, $h_{n-i} \cdot \xi_i
= h_{n-i}\xi_i + b_2^{(i)} \cdot e^{-tn}$ where
$b_2^{(i)} = \Phi_{(A_2, 0)}(h_{n-i}, \xi_i, h_n \xi_{n-1})$.
Thus by (6.4) and (6.5), we have
$$h^{n-i} \cdot \xi^i = h_{n-i} \cdot \xi_i =
h_{n-i}\xi_i + b_2^{(i)} \cdot e^{-tn}. \eqno (6.6)$$
Since $\sum_{i=0}^n (-1)^i c_i \cdot h_i \xi_{n-i} = 0$,
it follows from (6.4), (6.5), (6.6) that
$$\sum_{i=0}^n (-1)^i c_i \cdot h^i \cdot \xi^{n-i} =
(1 + \sum_{i=0}^n (-1)^i c_i b_2^{(n-i)}) \cdot e^{-tn}. \eqno (6.7)$$
Next, we compute the above integers $a_1', a_2'$, and $b_2^{(i)}$
where $0 \le i \le n$.
\lemma{6.8} Let $V = T_{\Pee^n}$ with $n \ge 2$ and
$A_2 = (h_{n-1}\xi_{n-1} - nh_{n}\xi_{n-2})_*$.
\roster
\item"{(i)}" $\Phi_{(A_2, 0)}(h, h_n, h_{n-1} \xi_{n-1}) = n$;
\item"{(ii)}" Let $\alpha = h_j\xi_k$ and $\beta = h_s\xi_t$ where
$j, k, s, t$ are nonnegative integers such that
{\rm max}$(j, k) > 0$, {\rm max}$(s, t) > 0$, and $(j+k+s+t) = n$. Then,
$$\Phi_{(A_2, 0)}(\alpha, \beta, h_n \xi_{n-1}) = 1.$$
\endroster
\endproclaim
\noindent
{\it Proof.} (i) By Lemma 2.2 (iv),
$\frak M({A_2}, 0)/PSL(2; \Cee)$ is compact.
By (3.17), we have $h^1(N_f) = 0$ for every $f \in \frak M({A_2}, 0)$.
Thus, $\frak M({A_2}, 0)/PSL(2; \Cee)$ is also smooth.
Fix a line $\ell_0$ in $\Pee^n$. Let $g: \ell_0 \to \Pee(T_{\Pee^n}|_{\ell_0})
\subset \Pee(T_{\Pee^n})$ be the embedding induced by the natural projection
$T_{\Pee^n}|_{\ell_0} = \Cal O_{\ell_0}(1)^{\oplus (n - 1)} \oplus
\Cal O_{\ell_0}(2) \to \Cal O_{\ell_0}(2) \to 0.$
Since $h([g(\ell_0)]) = 1$ and $\xi([g(\ell_0)]) = 2$,
we have $[g(\ell_0)] = [h_{n-1}\xi_{n-1} - (n - 1) h_{n}\xi_{n-2}]_*$.
So $h_{n-1}\xi_{n-1} = [g(\ell_0)]_* + (n - 1) h_{n}\xi_{n-2}$, and
$$\Phi_{({A_2}, 0)}(h, h_{n}, h_{n-1} \xi_{n-1}) =
\Phi_{({A_2}, 0)}(h, h_{n}, [g(\ell_0)]_*)
+ (n - 1) \Phi_{({A_2}, 0)}(h, h_{n}, h_{n}\xi_{n-2}).$$
By Lemma 3.8, it suffices to show that
$\Phi_{(A_2, 0)}(h, h_{n}, [g(\ell_0)]_*) = 1$.
Let $B$ and $C$ be the subvarieties of $\Pee(T_{\Pee^n})$ in general position,
whose homology classes are Poincar\'e dual to $h$ and $h_{n}$ respectively.
Then the homology classes of $\pi(B)$ and $\pi(C)$ in $\Pee^n$ are
Poincar\'e dual to $h$ and $h_n$ respectively.
Let $f: \ell \to \Pee(T_{\Pee^n})$ be a morphism in $\frak M(A_2, 0)$
induced by a surjective map $T_{\Pee^n}|_\ell \to \Cal O_{\ell}(1) \to 0$
for some line $\ell \subset \Pee^n$. If the image $\hbox{Im}(f)$
intersects with $B, C$, and $g(\ell_0)$, then $\ell$ intersects with
$\pi(B)$, $\pi(C)$, and $\pi(g(\ell_0)) = \ell_0$. In other words,
$\ell$ passes through the point $\pi(C)$ and intersects with $\ell_0$.
Moreover, putting $p = \ell \cap \ell_0$ and noticing that
every surjective map $T_{\Pee^n}|_\ell \to \Cal O_{\ell}(1) \to 0$
factors through the natural projection
$T_{\Pee^n}|_\ell = \Cal O_{\ell}(1)^{(n-1)} \oplus \Cal O_{\ell}(2)
\to \Cal O_{\ell}(1)^{(n-1)}$, we conclude that the $(n-1)$-dimensional
subspace $(\Cal O_{\ell}(1)^{(n-1)})|_p$ in
$(T_{\Pee^n}|_\ell)|_p = T_{p, \Pee^n}$ must contain
the $1$-dimensional subspace $(\Cal O_{\ell_0}(2))|_p$ in
$(T_{\Pee^n}|_{\ell_0})|_p = T_{p, \Pee^n}$.
Conversely, let $p \in \ell_0$ and let $\ell_p$ be the unique line
connecting the two points $\pi(C)$ and $p$. If the $(n-1)$-dimensional
subspace $(\Cal O_{\ell_p}(1)^{(n-1)})|_p$ in
$(T_{\Pee^n}|_{\ell_p})|_p = T_{p, \Pee^n}$ contains
the $1$-dimensional subspace $(\Cal O_{\ell_0}(2))|_p$ in
$(T_{\Pee^n}|_{\ell_0})|_p = T_{p, \Pee^n}$,
then there exists a unique surjective map
$T_{\Pee^n}|_{\ell_p} \to \Cal O_{\ell_p}(1) \to 0$
such that the image of the induced morphism $f: \ell_p \to \Pee(T_{\Pee^n})$
intersects $g(\ell_0)$ at the point $g(p)$.
Since there exists a unique point $p \in \ell_0$ such that
the $(n-1)$-dimensional subspace $(\Cal O_{\ell_p}(1)^{(n-1)})|_p$ in
$(T_{\Pee^n}|_{\ell_p})|_p = T_{p, \Pee^n}$ contains
the $1$-dimensional subspace $(\Cal O_{\ell_0}(2))|_p$ in
$(T_{\Pee^n}|_{\ell_0})|_p = T_{p, \Pee^n}$, it follows that
$$\Phi_{(A_2, 0)}(h, h_{n}, [g(\ell_0)]_*) = 1.$$
(ii) It is well-known (see p.176 of \cite{7}) that there is an exact sequence
$$0 \to \Cal O_{\Pee^n} \to \Cal O_{\Pee^n}(1)^{\oplus (n+1)}
\to T_{\Pee^n} \to 0. \eqno (6.9)$$
The surjective map $\Cal O_{\Pee^n}(1)^{\oplus (n+1)} \to T_{\Pee^n} \to 0$
induces the inclusion $\phi: \Pee(T_{\Pee^n}) \subset \Pee^n \times \Pee^n$
such that $\xi$ is the restriction of the $(1, 1)$ class in
$\Pee^n \times \Pee^n$. Let $B, C, q_0$ be the subvarieties of
$\Pee(T_{\Pee^n})$ in general position, whose homology classes are
Poincar\'e dual to $\alpha, \beta, h_n \xi_{n-1}$ respectively.
Then $q_0$ is a point. Put $p_0 = \pi(q_0) \in \Pee^n$.
Now the morphisms in $\frak M(A_2, 0)$ are of the forms
$f: \ell \to \Pee(T_{\Pee^n})$ induced by surjective maps
$T_{\Pee^n}|_\ell \to \Cal O_{\ell}(1) \to 0$
for lines $\ell \subset \Pee^n$. If the image $\hbox{Im}(f)$ passes $q_0$,
then the line $\ell$ passes $p_0$ and $q_0$ is contained in the hyperplane
$$\Pee^{n-2} = \Pee((\Cal O_\ell(1)^{\oplus (n-1)})|_{p_0})
\subset \Pee((T_{\Pee^n}|_\ell)|_{p_0}) = \pi^{-1}(p_0) = \Pee^{n-1}.$$
Conversely, if $\ell$ passes $p_0$ and $q_0$ is contained in the hyperplane
$$\Pee^{n-2} = \Pee((\Cal O_\ell(1)^{\oplus (n-1)})|_{p_0})
\subset \Pee((T_{\Pee^n}|_\ell)|_{p_0}) = \pi^{-1}(p_0) = \Pee^{n-1},
\eqno (6.10)$$
then there exists a unique $f \in \frak M(A_2, 0)$ of the form
$f: \ell \to \Pee(T_{\Pee^n})$ such that $\hbox{Im}(f)$ passes $q_0$;
moreover, putting $q_0 = (p_0, p_0') \in \Pee^n \times \Pee^n$ such that
$\pi$ is the first projection of $\Pee^n \times \Pee^n$,
then $\hbox{Im}(f) = \ell \times \{ p_0' \} \subset \Pee^n \times \Pee^n$.
The set of all lines $\ell$ passing $p_0$ such that $q_0$ is contained
in the hyperplane (6.10) is parameterized by an $(n - 2)$-dimensional
linear subspace $\Pee^{n - 2}$ in $\Pee^n$ (the first factor in
$\Pee^n \times \Pee^n$). It follows that the images
$\hbox{Im}(f) \subset \Pee(T_{\Pee^n})$ sweep a hyperplane
$$H \overset \hbox{def} \to = \Pee^{n - 1} \times \{ p_0' \} \subset
\Pee^n \times \{ p_0' \}. \eqno (6.11)$$
Since $\xi$ is the restriction of the $(1, 1)$ class in $\Pee^n \times \Pee^n$,
$\xi|_H$ is the hyperplane class $\tilde h$ in
$H = \Pee^{n - 1} \times \{ p_0' \} \cong \Pee^{n - 1}$.
Thus $\alpha|_H = \tilde h_{j + k}$ and $\beta|_H = \tilde h_{s+t}$.
Since $(j+k+s+t) = n$ and $B$ and $C$ are in general position,
there is a unique line in $H$ passing $q_0 = (p_0, p_0')$ and
intersecting with $B$ and $C$. Therefore,
$$\Phi_{(A_2, 0)}(\alpha, \beta, h_n \xi_{n-1}) = 1. \qed$$
Finally, we summarize the above computations and prove the following.
\proposition{6.12} The quantum cohomology ring
$H^*_{\omega}(\Pee(T_{\Pee^n}); \Zee)$ with $n \ge 2$ is the ring
generated by $h$ and $\xi$ with the two relations:
$$h^{n+1} = \xi \cdot e^{-tn} \qquad \text{and}
\qquad \sum_{i=0}^n (-1)^i c_i \cdot h^i \cdot \xi^{n-i}
= (1 + (-1)^n) \cdot e^{-tn}.$$
\endproclaim
\proof
By Lemma 6.8 (ii), $a_1' = 1$. By Lemma 3.8, $a_3' = 1$.
By Lemma 6.8 (i),
$$a_2' = \Phi_{(A_2, 0)}(h, h_{n}, h_{n-1} \xi_{n-1}) -c_1 a_3'= -1.$$
Thus by (6.3), the first relation $f_\omega^1$ is
$h^{n+1}= \xi \cdot e^{-tn}$. By Lemma 6.8 (ii),
$b_2^{(i)} = 1$ for $0 \le i \le n$. By (6.7),
the second relation $f_\omega^2$ is
$\sum_{i=0}^n (-1)^i c_i \cdot h^i \cdot \xi^{n-i} =
(1 + \sum_{i=0}^n (-1)^i c_i) \cdot e^{-tn}$. From the exact sequence (6.9),
$c_i = {n+1 \choose i}$ for $0 \le i \le n$.
Therefore, the relation $f_\omega^2$ is
$\sum_{i=0}^n (-1)^i c_i \cdot h^i \cdot \xi^{n-i}
= (1 + (-1)^n) \cdot e^{-tn}$.
\endproof
\Refs
\ref \no {1} \by A. Astashkevich, V. Sadov \paper Quantum cohomology of partial
flag manifolds $F_{n_1 \ldots n_k}$ \jour Preprint
\endref
\ref \no {2} \by V.V. Batyrev \paper Quantum cohomology rings of toric
manifolds \jour Preprint
\endref
\ref \no {3} \by A. Beauville \paper Quantum cohomology of complete
intersections \jour Preprint
\endref
\ref \no {4} \by I. Ciocan-Fontanine \paper Quantum cohomology of flag
varieties \jour Preprint
\endref
\ref \no {5} \by H. Clemens, J. Kolla\'r, S. Mori \book Higher dimensional
complex geometry. {\rm (Asterisque, Vol. 166) Paris: Soc. Math. Fr.} \yr 1988
\endref
\ref \no {6} \by B. Crauder, R. Miranda \paper Quantum cohomology of rational
surfaces \inbook The moduli space of curves \eds R. Dijkgraaf, C. Faber, G.
van der Geer \bookinfo Progress in Mathematics {\bf 129} \publ Birkh\" auser
\publaddr Boster Basel Berlin \yr 1995
\endref
\ref \no {7} \by R. Hartshorne \book Algebraic Geometry
\publ Springer \publaddr Berlin-Heidelberg-New York\yr 1978
\endref
\ref \no {8} \by M. Kontsevich, Y. Manin \paper Gromov-Witten classes, quantum
cohomology, and enumerative geometry \jour Preprint
\endref
\ref \no {9} \by J. Li, G. Tian \paper Quantum cohomology of homogeneous
varieties \jour Preprint
\endref
\ref \no {10} \by C. Okonek, M. Schneider, H. Spindler \book Vector bundles on
complex projective spaces, {\rm Progress in Math.} \publ Birkh{\" a}user \yr
1980
\endref
\ref \no {11} \by Y. Ruan \paper Symplectic topology and extremal rays \jour
Geom. Func. Anal. \vol 3 \pages 395-430 \yr 1993
\endref
\ref \no {12} \bysame \paper Topological sigma model and Donaldson type
invariants in Gromov theory \jour To appear in Duke Math. J.
\endref
\ref \no {13} \by Y. Ruan, G. Tian \paper A mathematical theory of quantum
cohomology \jour To appear in J. Diffeo. Geom. \yr 1995
\endref
\ref \no {14} \by B. Siebert, G. Tian \paper On quantum cohomology rings of
Fano manifolds and a formula of Vafa and Intriligator \jour Preprint
\endref
\ref \no {15} \bysame \paper Quantum cohomology of moduli space of stable
bundles \jour In preparation
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\ref \no{16} \by E. Witten \paper Topological sigma models \jour
Commun. Math. Phys. \vol 118 \pages 411-449 \yr 1988
\endref
\endRefs
\enddocument
|
1995-10-27T05:20:14 | 9510 | alg-geom/9510014 | en | https://arxiv.org/abs/alg-geom/9510014 | [
"alg-geom",
"math.AG"
] | alg-geom/9510014 | V. Batyrev | Victor V. Batyrev and Yuri Tschinkel | Manin's conjecture for toric varieties | 45 pages, LaTeX | null | null | null | null | We prove an asymptotic formula conjectured by Manin for the number of
$K$-rational points of bounded height with respect to the anticanonical line
bundle for arbitrary smooth projective toric varieties over a number field $K$.
| [
{
"version": "v1",
"created": "Thu, 26 Oct 1995 20:51:33 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Batyrev",
"Victor V.",
""
],
[
"Tschinkel",
"Yuri",
""
]
] | alg-geom | \section{Algebraic tori and toric varieties}
Let $X_K$ be an algebraic variety defined over a number
field $K$ and $E/K$ a finite extension of number fields.
We will denote the set of $E$-rational points of $X_K$
by $X(E)$ and by $X_E$ the $E$-variety obtained from
$X_K$ by base change. We sometimes omit the subscript in
$X_E$ if the respective field of definition
is clear from the context. Let
${\bf G}_{m,E}= {\rm Spec}( E[x,x^{-1}])$ be
the multiplicative group scheme over $E$.
\begin{dfn}
{\rm A linear algebraic group $T_K$ is
called a {\em $d$-dimen\-sio\-nal algebraic torus} if
there exists a finite extension $E/K$ such that
$T_{E}$ is isomorphic to
$({\bf G}_{m,E})^d$.
The field $E$ is called the
{\em splitting field } of $T$.
For any field $E$ we denote by $\hat{T}_E =
{\rm Hom}\,( T, E^*)$
the group of regular $E$-rational
characters of $T$.
}
\label{opr.tori}
\end{dfn}
\begin{theo} {\rm \cite{grothendieck,ono1,vosk}}
There is a contravariant equivalence
between the category of algebraic tori
defined over a number field $K$ and the category of
torsion free
${\rm Gal}(E/K)$-modules of finite rank over
${\bf Z}$.
The functors are given by
$$
M \rightarrow T = {\rm Spec}(K\lbrack M \rbrack); \;\;
T \rightarrow \hat{T}_E.
$$
The above contravariant equivalence is
functorial under field extensions of
$K$.
\label{represent}
\end{theo}
Let ${\rm Val}(K)$ be the set of all
valuations of a global field $K$. Denote by $S_{\infty}$
the set of archimedian valuations of $K$.
For any $v \in {\rm Val}(K)$, we
denote by $K_v$ the completion of
$K$ with respect to $v$.
Let $E$ be a
finite Galois extension of $K$.
Let ${\cal V}$ be an extension of $v$ to $E$,
$E_{\cal V}$ the completion
of $E$ with respect to ${\cal V}$. Then
\[ {\rm Gal}(E_{\cal V}/ K_v ) \cong G_v
\subset G, \]
where $G_{v}$ is the decomposition subgroup of
$G$ and $ K_v \otimes_K E \cong \prod_{{\cal V} \mid v} E_{\cal V}. $
Let $T$ be an algebraic torus over $K$
with the splitting field $E$.
Denote by
$T(K_v)=$ the $v$-adic
completion of $T(K)$ and by $T({\cal O}_v)\subset T(K_v)$
its maximal compact subgroup.
\begin{dfn}
{\rm Denote by $T({\bf A}_K)$ the adele group of $T$.
Define
\[T^1({\bf A}_K) = \{ {\bf t} \in T({\bf A}_K) \, : \,
\prod_{v \in {\rm Val}(K)}
\mid m(t_v) \mid_v = 1, \; {\rm for \; all}\; m \in \hat{T}_K \subset M \}.
\]
Let
\[ {\bf K}_T = \prod_{v \in {\rm Val}(K)} T({\cal O}_v), \]
be the maximal compact subgroup of $T({\bf A}_K)$.
}
\end{dfn}
\begin{prop} {\rm \cite{ono1}}
The groups $T({\bf A}_K)$, $T^1({\bf A}_K)$, $T(K)$,
${\bf K}_T$ have the following properties:
{\rm (i)} $T({\bf A}_K)/T^1({\bf A}_K) \cong {\bf R}^t$, where $t$
is the rank
of $\hat{T}_K$;
{\rm (ii)} $T^1({\bf A}_K)/T(K)$ is compact;
{\rm (iii)} $T^1({\bf A}_K)/ T(K)\cdot {\bf K}_T $
is isomorphic to the direct product of a finite group
${\bf cl}(T_K)$ and
a connected compact abelian topological group which dimension
equals the rank $r'$ of the group
of ${\cal O}_K$-units in $T(K)$;
{\rm (iv)} $W(T) = {\bf K}_T \cap T(K)$ is a finite
group of all torsion elements in $T(K)$.
\label{subgroups}
\end{prop}
\begin{dfn}
{\rm
We define the following
cohomological invariants of the algebraic
torus $T$:
$$
h(T)={\rm Card}[H^1(G,M)],
$$
$$
{\rm III}(T)={\rm Ker}\, \lbrack
H^1(G, T(K)) \rightarrow \prod_{v\in {\rm Val}(K)}
H^1(G_v, T(K_v)) \rbrack,
$$
$$
i(T)= {\rm Card}[{\rm III}(T)].
$$
\label{coh.inv}
}
\end{dfn}
\begin{dfn}
{\rm
Let $\overline{T(K)}$ be the closure
of $T(K)$ in $T({\bf A}_K)$ in the
{\em direct product topology}.
Define the {\em
obstruction group to weak approximation} as
$$
A(T)= T({\bf A}_K)/\overline{T(K)}.
$$
\label{weak0}
}
\end{dfn}
\begin{rem}
{\rm
It is known that over the splitting field $E$ one has
$A(T_E)=0$.
}
\end{rem}
Let us recall standard facts about toric varieties over
arbitrary fields
\cite{danilov,demasur,fulton,oda,BaTschi}.
\begin{dfn}
{\rm A finite set $\Sigma$ consisting of convex rational polyhedral
cones in $N_{\bf R} = N \otimes {\bf R}$ is called a {\em
$d$-dimensional fan} if the following conditions are satisfied:
(i) every cone $\sigma \in \Sigma$
contains $0 \in N_{\bf R}$;
(ii) every face $\sigma'$ of
a cone $\sigma \in \Sigma$ belongs to $\Sigma$;
(iii) the intersection of any
two cones in $\Sigma$ is a face of
both cones. }
\end{dfn}
\begin{dfn}
{\rm A $d$-dimensional fan $ \Sigma $ is called
{\em complete and regular} if the
following additional conditions are satisfied:
(i) $N_{\bf R}$ is the union of cones from $\Sigma$;
(ii) every
cone $\sigma \in \Sigma$ is generated by a
part of a ${\bf Z}$-basis of
$N$.\\
We denote by $\Sigma(j)$ the set of all
$j$-dimensional cones in
$\Sigma$. For each cone $\sigma \in \Sigma$
we denote by
$N_{{\sigma}, \bf R}$ the minimal
linear subspace containing $\sigma$. }
\label{def.fan}
\end{dfn}
\noindent
Let $T_K$ be a $d$-dimensional algebraic torus
over $K$
with splitting field $E$ and
$G = {\rm Gal}\, (E/K)$. Denote
by $M$ the lattice $\hat{T}_E$ and by
$N ={\rm Hom}\, (M, {\bf Z})$ the dual abelian group.
\begin{theo}
A complete regular $d$-dimensional fan $ \Sigma $ defines
a smooth
equivariant compactification ${\bf P}_{ \Sigma ,E}$
of the $E$-split algebraic torus $T_E$. The {\em
toric variety} ${\bf P}_{ \Sigma ,E}$ has the following
properties:
(i) There is a
$T_E$-invariant open covering by affine subsets $U_{\sigma,E}$:
\[ {\bf P}_{\Sigma,E} = \bigcup_{ \sigma \in \Sigma} U_{\sigma,E}. \]
The affine subsets are defined as
$U_{\sigma,E} = {\rm Spec}(E \lbrack M \cap \check{\sigma}
\rbrack$), where $\check{\sigma}$
is the cone in $M_{\bf R}$ which is
dual to $\sigma$.
(ii) There is a representation of
${\bf P}_{\Sigma,E}$ as a disjoint
union of split algebraic
tori $T_{\sigma,E}$ of dimension
${\rm dim}\, T_{\sigma,E} = d - {\rm dim}\, \sigma $:
\[ {\bf P}_{\Sigma,E} = \bigcup_{ \sigma \in \Sigma } T_{\sigma,E}. \]
For each $j$-dimensional cone $\sigma \in \Sigma{(j)}$
we denote by
$T_{\sigma,E}$ the kernel of a
homomorphism $T_E \rightarrow
({\bf G}_{m,E})^j$ defined by a ${\bf Z}$-basis of
the sublattice $N \cap N_{{\sigma},{\bf R}} \subset N$.
\end{theo}
\noindent
To construct compactifications of non-split tori
$T_K$ over $K$,
we need a complete fan $\Sigma$ of cones
having an additional combinatorial structure: an {\em
action of the Galois group }
$G={\rm Gal}(E/K)$ \cite{vosk1}.
The lattice $M=\hat{T}_E$ is a $G$-module and
we have a representation $\rho: G \rightarrow {\rm Aut}(M)$.
Denote by $\rho^*$ the induced dual
representation of $G$ in ${\rm Aut}(N)
\cong {\rm GL}(d,{\bf Z})$.
\begin{dfn}
{\rm A complete fan $\Sigma \subset
N_{\bf R}$ is called
{\em $G$-invariant} if for any $g
\in G$ and for any $\sigma \in \Sigma$, one
has $\rho^*(g) (\sigma) \in \Sigma$.
Let $N^G$ (resp. $M^G$, $N_{\bf R}^G$,
$M_{\bf R}^G$ and $ \Sigma ^G$) be the subset
of $G$-invariant elements
in $N$ (resp. in $M$, $N\otimes {\bf R}$,
$M\otimes {\bf R}$ and $ \Sigma $).
Denote by $\Sigma_G \subset N_{\bf R}^G$
the fan consisting of all possible
intersections $\sigma \cap N_{\bf R}^G$
where $\sigma$ runs over all cones in $\Sigma$.
}
\label{opr.invar}
\end{dfn}
\noindent
The following theorem is due to Voskresenski\^i \cite{vosk1}:
\begin{theo}
Let $\Sigma$ be a complete regular $G$-invariant
fan in $N_{\bf R}$. Assume that the complete toric variety
${\bf P}_{\Sigma,E}$ defined over the splitting field
$E$ by the $G$-invariant fan $\Sigma$ is projective.
Then there exists a unique complete
algebraic variety ${\bf P}_{\Sigma,K}$ over $K$
such that its base extension
${\bf P}_{\Sigma,K} \otimes_{{\rm Spec} (K)}
{\rm Spec}(E)$ is isomorphic to
the toric variety ${\bf P}_{\Sigma,E}$.
The above isomorphism respects
the natural $G$-actions on ${\bf P}_{\Sigma,K} \otimes_{{\rm Spec}(K)}
{\rm Spec}(E)$ and ${\bf P}_{\Sigma,E}$.
\end{theo}
\begin{rem}
{\rm
Our definition of heights
and the proof of the analytic properties of height zeta
functions do not use the projectivity of respective
toric varieties. We note that there exist non-projective compactifications of
split algebraic tori. We omit
the technical question of existence of non-projective
compactifications of non-split tori.
}
\end{rem}
We proceed to describe the algebraic geometric structure
of the variety ${\bf P}_{ \Sigma ,K}$ in terms of the fan with
Galois-action. Let ${\rm Pic}({\bf P}_{ \Sigma ,K}) $ be the
Picard group and
$\Lambda_{\rm eff}$ the cone in
${\rm Pic}({\bf P}_{ \Sigma ,K})$ generated by
classes of effective divisors.
Let ${\cal K}$ be the canonical line
bundle of ${\bf P}_{ \Sigma ,K}$.
\begin{dfn}
{\rm A continuous function $\varphi\; : \;
N_{\bf R} \rightarrow {\bf R}$ is called {\em $\Sigma$-piecewise linear}
if the restriction of $\varphi$ to every
cone $\sigma \in \Sigma$ is a linear function. It is
called {\em integral} if $\varphi(N) \subset {\bf Z}$.
Denote the group of $ \Sigma $-piecewise linear integral functions by $PL( \Sigma )$.
}
\end{dfn}
We see that the $G$-action on $M$ (and $N$)
induces a $G$-action on the free
abelian group $PL( \Sigma )$.
Denote by $e_1, \ldots, e_n$ the primitive
integral generators
of all $1$-dimensional cones in $\Sigma$.
A function $\varphi\in PL( \Sigma )$ is determined by
its values on $e_i,\, (i=1,...,n)$.
Let $T_{i}$
be the $(d-1)$-dimensional torus orbit
corresponding to the cone
${\bf R}_{\geq 0}e_i \in \Sigma(1)$ and
$\overline{T}_i$ the Zariski closure of $T_i$ in
${\bf P}_{\Sigma,E}$.
\begin{prop}
Let ${\bf P}_{ \Sigma ,K}$ be a smooth toric variety
over $K$ which is an equivariant compactification
of an algebraic torus $T_K$
with splitting field $E$ and $ \Sigma $
the corresponding complete regular fan
with $G={\rm Gal}(E/K)$-action. Then:
(i) There is an exact sequence
$$
0 \rightarrow M^G \rightarrow PL( \Sigma )^G \rightarrow {\rm Pic}({\bf P}_{ \Sigma ,K}) \rightarrow
H^1(G,M) \rightarrow 0.
$$
(ii) Let
$$
\Sigma (1)= \Sigma _1(1)\cup ...\cup \Sigma _r(1)
$$
be the decomposition of $ \Sigma (1)$ into a union
of $G$-orbits.
The cone of effective divisors
$ \Lambda _{\rm eff}$ is generated by classes
of $G$-invariant divisors
$$
D_j = \sum_{{\bf R}_{\geq 0}e_i \in \Sigma _j(1)}
\overline{T}_i \,\,\, (j=1,...,r).
$$
(iii)
The class of the anticanonical line
bundle ${\cal K}^{-1}\in {\rm Pic}({\bf P}_{ \Sigma ,K})$
is the class of the $G$-invariant piecewise linear
function $\varphi_{ \Sigma }\in PL( \Sigma )^G$ given by
$\varphi_{ \Sigma }(e_j)=1$ for all $j=1,...,n$.
\label{nonsplit.geom}
\end{prop}
\begin{theo} {\rm \cite{vosk,ct}} Let $T$ be an
algebraic torus over $K$ with splitting field $E$.
Let ${\bf P}_{ \Sigma ,K}$ be a complete
smooth equivariant compactification of $T$.
There is an exact sequence:
\[ 0 \rightarrow A(T) \rightarrow
Hom (H^1(G,{\rm Pic}({\bf P}_{ \Sigma ,E})),{\bf Q}/{\bf Z})
\rightarrow {\rm III}(T) \rightarrow
0.
\]
\label{weak}
\end{theo}
\begin{rem}
{\rm
The group $H^1(G,{\rm Pic}({\bf P}_{ \Sigma ,E}))$
is canonically isomorphic to the non-trivial
part of the Brauer group
${\rm Br}({\bf P}_{\Sigma,K})/{\rm Br}(K)$, where
${\rm Br}({\bf P}_{\Sigma,K}) = H^2_{\rm et}({\bf P}_{\Sigma,K}, {\bf G}_m)$.
This group appears
as the obstruction group to the Hasse principle
and weak approximation in \cite{manin,ct}.
}
\end{rem}
\begin{coro}
{\rm
Let $\beta({\bf P}_{\Sigma})$ be the cardinality of
$H^1(G,{\rm Pic}({\bf P}_{ \Sigma ,E}))$. Then
\[ {\rm Card} \lbrack A(T) \rbrack =
\frac{\beta({\bf P}_{\Sigma})}{i(T)}. \]
\label{weak1}
}
\end{coro}
\section{Tamagawa numbers}
In this section we recall the definitions of Tamagawa
numbers of tori following A. Weil \cite{weil}
and of algebraic varieties with a
metrized canonical line bundle following E. Peyre \cite{peyre}.
The constructions of Tamagawa numbers depend on
a choice of a finite set of valuations
$S\subset {\rm Val}(K)$ containing archimedian
valuations and places of bad reduction, but the Tamagawa numbers
themselves do not depend on $S$.
Let $X$ be a smooth algebraic variety over $K$, $X(K_v)$ the
set of $K_v$-rational points of $X$.
Then a choice of local analytic coordinates $x_1, \ldots, x_d$ on $X(K_v)$
defines a homeomorphism $\phi\,: \, U \rightarrow K_v^d$
in $v$-adic topology between an
open subset $U \subset X(K_v)$ and $\phi(U) \subset K_v^d$.
Let $dx_1 \cdots dx_d$ be the Haar measure on $K_v^d$ normalized
by the condition
\[ \int_{{\cal O}_v^d} dx_1 \cdots dx_d = \frac{1}{(\sqrt{\delta_v})^d} \]
where $\delta_v$ is the absolute different of $K_v$.
Denote by $dx_1 \wedge \cdots \wedge dx_d$ the standard
differential form on $K_v^{d}$. Then
$f = \phi^*(dx_1 \wedge \cdots \wedge dx_d)$ is a local analytic section of
the canonical sheaf ${\cal K}$. If $\| \cdot \|$ is a $v$-adic
metric on ${\cal K}$, then we obtain the $v$-adic measure on $U$ by
the formula
\[ \int_{U'} \omega_{{\cal K},v} =
\int_{\phi(U')} \| f(\phi^{-1}(x))
\|_v dx_1 \cdots dx_d,
\]
where $U'$ is arbitrary open subset in $U$.
The measure $\omega_{{\cal K},v}$ does not
depend on the
choice of local coordinates
and extends to a global
measure on $X(K_v)$ \cite{peyre}.
\begin{dfn}
{\rm
\cite{ono1} Let $T$ be an algebraic torus
defined over a number field $K$ with
splitting field $E$. Denote by
\[ L_S(s, T;E/K) =
\prod_{v \in {\rm Val}(K)} L_v(s, T ;E/K) \]
the Artin $L$-function corresponding to
the representation
\[ \rho \; :\; G= {\rm Gal}(E/K)
\rightarrow {\rm Aut}(\hat{T}_E)
\]
and a finite set $S \subset {\rm Val}(K)$
containing all
archimedian valuations
and all non-archimedian valuations
of $K$ which are
ramified in $E$.
By definition,
$L_v(s,T;E/K) \equiv 1$ if $v \in S$, $L_v(s,T;E/K)=
{\rm det}(Id - q^{-s}_v F_v)^{-1}$ if $v \not\in S$, where $F_v \in
{\rm Aut}(\hat{T}_E)$ is a representative of the Frobenius automorphism.
}
\end{dfn}
Let $T$ be an algebraic torus of dimension $d$ and
$\Omega$ a $T$-invariant algebraic $K$-rational differential
$d$-form. The form $\Omega$ defines an isomorphism
of the canonical sheaf on $T$ with the structure sheaf on
$T$. Since the structure sheaf has a canonical metrization,
using the above construction, we obtain a $v$-adic measure
$\omega_{\Omega,v}$ on $T(K_v)$. Moreover, according to A. Weil \cite{weil},
we have
\[
\int_{T({\cal O}_v)} \omega_{\Omega,v} =
\frac{{\rm Card} \lbrack T(k_v) \rbrack}{q^d_v} = L_v(1, T; E/K)^{-1} \]
for all $v\not\in S$.
We put $d\mu_v = L_v(1, T; E/K) \omega_{\Omega,v}$
for all $v\in {\rm Val}(K)$.
Then the local measures $d\mu_v$ satisfy
$$
\int_{T({\cal O}_v)} d\mu_v = 1
$$
for all $v\not\in S$.
\begin{dfn}
{\rm
We define the {\em canonical measure}
on the adele group $T({\bf A}_K)$
$$
\omega_{\Omega,S} = \prod_{v \in {\rm Val}(K)}
L_v(1, T; E/K) \omega_{\Omega,v} =
\prod_{v \in {\rm Val}(K)} d\mu_v.
$$
\label{can.meas}
}
\end{dfn}
\noindent
By the product formula, $\omega_{\Omega,S} $
does not depend on the choice of $\Omega$.
Let ${\bf dx}$ be the standard Lebesgue measure
on $T({\bf
A}_K)/T^1({\bf A}_K)$. There exists
a unique Haar measure
$\omega^1_{\Omega,S}$ on $T^1({\bf A}_K)$ such that $\omega^1_{\Omega,S}
{\bf dx} =
\omega_{\Omega,S}$.
\bigskip
We proceed to define {\em Tamagawa measures}
on algebraic varieties following E. Peyre \cite{peyre}.
Let $X$ be a smooth projective
algebraic variety over $K$ with a metrized
canonical sheaf ${\cal K}$. We assume that $X$ satisfies
the conditions
$h^1(X, {\cal O}_X) = h^2(X, {\cal O}_X) = 0$.
Under these
assumptions, the N{\'e}ron-Severi group $NS(X)$ (or, equivalently,
the Picard group ${\rm Pic}(X)$ modulo torsion)
over the algebraic closure
$\overline{K}$ is a discrete continuous ${\rm Gal}(\overline{K}/K)$-module
of finite rank over ${\bf Z}$.
Denote by $T_{NS}$ the corresponding
torus under the duality from \ref{represent}
and by $E_{NS}$ a splitting field.
\begin{dfn}
{\rm \cite{peyre}
The {\em adelic Tamagawa measure}
$\omega_{{\cal K},S}$ on $X({\bf A}_K)$ is defined by
$$
\omega_{{\cal K},S} = \prod_{v \in {\rm Val}(K)}
L_v(1, T_{NS}; E_{NS}/K)^{-1}\omega_{{\cal K},v}. $$
}
\end{dfn}
\begin{dfn}
{\rm
Let $t$ be the rank of the group of
$K$-rational characters $\hat{T}_K$ of $T$.
Then the {\em Tamagawa number of } $T$ is defined as
\[ \tau(T) = \frac{b_S(T)}{l_S(T)} \]
where
\[ b_S(T) = \int_{T^1({\bf A}_K)/T(K)} \omega^1_{\Omega,S} , \]
\[ l_S(T) = \lim_{s \rightarrow 1} (s-1)^t L_S(s, T; E/K). \]}
\label{tamagawa1}
\end{dfn}
\begin{dfn}
{\rm \cite{peyre}
Let $k$ be the rank of the N{\'e}ron-Severi group of
$X$ over $K$, and $\overline{X(K)}$
the closure of $X(K) \subset X({\bf A}_K)$
in the
direct product topology.
Then the {\em Tamagawa number} of $X$ is defined by
\[ \tau_{\cal K}(X) = \frac{b_S(X)}{l_S(X)} \]
where
\[ b_S(X) =
\int_{\overline{X(K)}} \omega_{{\cal K},S} \]
whenever the adelic integral converges, and
\[ l_S^{-1}(X) =
\lim_{s \rightarrow 1} (s-1)^k L_S(s, T_{NS};
E_{NS}/K). \]
}
\label{tamagawa2}
\end{dfn}
\begin{rem}
{\rm
Notice that there is a difference in the choice
of convergence factors for the
Tamagawa measure on
an algebraic variety $X$ and for
the Tamagawa measure on an algebraic torus
$T$. In the first case, we choose $L_v^{-1}(1, T_{NS};E_{NS}/K)$
whereas in the second case one uses $L_v(1, T; E/K)$.
This explains the difference in the
definitions of
$l_S(X)$ and $l_S(T)$.
}
\end{rem}
\begin{rem}
{\rm For a toric variety ${\bf P}_{ \Sigma }\supset T$
one can take $E_{NS}=E$, where $E$ is a splitting field
of $T$.
\label{ENS}
}
\end{rem}
\begin{rem}
{\rm
It is clear that in both definitions the Tamagawa numbers do not
depend on the choice of the finite subset
$S\subset {\rm Val}(K)$.
E. Peyre ($\cite{peyre}$) proves the existence
of the Tamagawa number for
Fano varieties by using
the Weil conjectures. The same method shows the existence
of the Tamagawa number for smooth complete varieties $X$ satisfying
the conditions $h^1(X, {\cal O}_X) = h^2(X, {\cal O}_X) = 0$.
}
\end{rem}
\begin{theo} {\rm \cite{ono2}} Let $T$ be an algebraic torus
defined over $K$.
The Tamagawa number $\tau (T)$ doesn't depend on
the choice of a splitting field $E/K$. We have
$$
\tau (T)=h(T)/i(T).
$$
The constants $h(T),i(T)$ were defined in \ref{coh.inv}.
\label{tamagawa}
\end{theo}
We see that the Tamagawa number of an algebraic torus
is a rational number. We have $\tau({\bf G}_m(K)) =1$.
The Tamagawa number of a Fano variety with a metrized
canonical line bundle is certainly not rational
in general. For ${\bf P}^1_{{\bf Q}}$ with our metrization
we have $\tau_{\cal K}({\bf P}^1_{{\bf Q}}) =1/\zeta_{{\bf Q}}(2)$.
\begin{prop} {\rm \cite{BaTschi}}
One has
$$
\int_{\overline{T(K)}} \omega_{{\cal K},S} =
\int_{\overline{{\bf P}_{ \Sigma }(K)}} \omega_{{\cal K},S}.
$$
\label{two-integrals}
\end{prop}
\section{Heights and their Fourier transforms}
Let $\varphi \in PL(\Sigma)^G_{\bf C}$.
Using the decomposition of $ \Sigma (1)$ into a union
of $G$-orbits
$$
\Sigma (1)= \Sigma _1(1)\cup ...\cup \Sigma _r(1),
$$
we can identify $\varphi$ with a $T$-invariant divisor with
complex coefficients
$$
D_{\varphi} = s_1 D_1 + \cdots + s_r D_r
$$
where $s_j = \varphi(e_j) \in {{\bf C}}$ and $e_j$ is a primitive lattice
generator of some cone $ \sigma \in \Sigma _j(1)$ $(j =1, \ldots, r)$.
It will be convenient to identify an element
$\varphi =\varphi_{\bf s}\in PL(\Sigma)^G_{\bf C}$ with the
vector ${\bf s} = (s_1, \ldots, s_r)$ of its complex
coordinates.
Let us recall the definition of heights
on toric varieties from \cite{BaTschi}. For our purposes
it will be sufficient to describe the restrictions of heights to
the Zariski
open subset $T\subset {\bf P}_{ \Sigma ,K}$.
\begin{prop} Let $v\in {\rm Val} (K) $ be a valuation
and $G_v\subset G$ the decomposition group of $v$.
There is an injective homomorphism
$$
\pi_v: T(K_v)/T({\cal O}_v)\hookrightarrow N_v,
$$
which is an isomorphism for all
but finitely many $v\in {\rm Val}(K)$.
Here $N_v=N^{G_v}\subset N$ for
non-archimedian $v$ and $N_v=N_{{\bf R}}^{G_v}$ for
archimedian valuations $v$.
For every non-archimedian valuation we can identify
the image of $\pi_v$ with a sublattice of finite
index in $N_v$.
\label{pi-image}
\end{prop}
\begin{dfn}
{\rm Let ${\bf s } \in {{\bf C}}^r$ be a complex vector defining
a complex piecewise linear $G$-invariant function $\varphi
\in PL(\Sigma)^G_{\bf C}$.
For any point $x_v \in T(K_v) \subset {\bf P}_{\Sigma}(K_v)$,
denote by $\overline{x}_v$ the image of
$x_v$ in $N_v$, where $N_v$ is considered
as a canonical lattice in the real
space $N_{\bf R}^{G_v}$ for non-archimedian
valuations (resp. as the real Lie-algebra
$N_{{\bf R},v}$ of $T(K_v)$ for archimedian valuations).
Define the {\em complexified
local Weil function}
$H_{ \Sigma ,v}(x_v, {\bf s})$ by the formula
\[H_{ \Sigma ,v}(x_v, {\bf s}) =
e^{\varphi(\overline{x}_v)\log q_v }\]
where $q_v$ is the cardinality of the residue field
$k_v$ of $K_v$ if $v$ is non-archimedian
and $\log q_v = 1$ if
$v$ is archimedian.
}
\end{dfn}
\begin{theo} {\rm \cite{BaTschi}}
The complexified local Weil function $H_{ \Sigma ,v}(x_v, {\bf s}
)$ satisfies the
following properties:
{\rm (i)} $H_{ \Sigma ,v}(x_v,{\bf s})$ is $T({\cal O}_v)$-invariant.
{\rm (ii)} If ${\bf s} = 0$, then $H_{ \Sigma ,v}(x_v,{\bf s}) = 1$
for all $x_v \in T(K_v)$.
{\rm (iii)} $H_{ \Sigma ,v}(x_v, {\bf s} + {\bf s}') =
H_{ \Sigma ,v}(x_v,{\bf s}) H_{ \Sigma ,v}(x_v,{\bf s}')$.
{\rm (iv)} If ${\bf s}=(s_1,...,s_r)\in {\bf Z}^r$,
then $H_{ \Sigma ,v}(x_v, {\bf s})$ is a classical local Weil
function corresponding to
a Cartier divisor $D_{\bf s} =
s_1 D_1 + \cdots + s_r D_r$ on ${\bf P}_{ \Sigma ,K}$.
\label{local.f}
\end{theo}
\begin{dfn}
{\rm For a piecewise
linear function $\varphi_{\bf s} \in PL(\Sigma)^G_{\bf C}$
we define the {\em complexified
height function on $T(K)\subset {\bf P}_{ \Sigma ,K}(K)$} by
\[ H_{\Sigma}(x, {\bf s}) =
\prod_{v \in {\rm Val}(K)} H_{ \Sigma ,v}(x_v, {\bf s}). \]}
\end{dfn}
\begin{rem}
{\rm
Although
the local heights are defined only
as functions on $PL( \Sigma )_{{\bf C}}^G \cong {{\bf C}}^r$, the
product formula implies that
for $x\in T(K)$ the global
complexified height function descends to
the Picard group ${\rm Pic}({\bf P}_{ \Sigma ,K})_{{\bf C}}$.
Moreover, since $H_{\Sigma}(x, {\bf s})$
is the product of local complex
Weil functions $H_{ \Sigma ,v}(x, {\bf s})$ and
since for all $x_v \in T({\cal O}_v)$ we have
$H_{ \Sigma ,v}(x_v, {\bf s}) = 1$ for all
$v$,
we can immediately
extend $H_{\Sigma}(x,{\bf s})$ to a
function on
$T({\bf A}_K)\times PL( \Sigma )^G_{{\bf C}}$.
}
\end{rem}
\begin{dfn}
{\rm Let
$ \Sigma (1) = \Sigma _1(1) \cup \cdots \cup \Sigma _l(1) $
be the decomposition of $ \Sigma (1)$ into a disjoint union of $G_v$-orbits.
Denote by $d_j$ the length of the $G_v$-orbit $ \Sigma _j(1)$
$(d_1 + \cdots + d_l = n)$.
We establish a
1-to-1 correspondence $ \Sigma _j(1) \leftrightarrow u_j$ between
the $G_v$-orbits $ \Sigma _1(1), \ldots, \Sigma _l(1)$ and independent variables
$u_1, \ldots, u_l$.
Let $\sigma \in \Sigma ^{G_v}$ be any
$G_v$-invariant cone and
$ \Sigma _{j_1}(1) \cup \cdots \cup \Sigma _{j_k}(1)$ the set of all
$1$-dimensional faces of $\sigma$.
We define the rational function
$R_{\sigma}(u_1, \ldots, u_l)$ corresponding to $\sigma$ as follows:
\[ R_{\sigma}(u_1, \ldots, u_l) : =
\frac{u_{j_1}^{d_{j_1}} \cdots u_{j_k}^{d_{j_k}}
}{(1 - u_{j_1}^{d_{j_1}}) \cdots (1 - u_{j_k}^{d_{j_k}}) }. \]
Define the polynomial $Q_{ \Sigma }(u_1, \ldots, u_l)$ by the
formula
\[\sum_{\sigma \in \Sigma ^{G_v}} R_{\sigma}(u_1, \ldots, u_l) =
\frac{Q_{ \Sigma }(u_1, \ldots, u_l)}
{(1 - u_1^{d_1}) \cdots (1- u_l^{d_l}) }. \]
}
\end{dfn}
\begin{prop} {\rm \cite{BaTschi}}
Let $\Sigma$ be a complete regular $G_v$-invariant
fan. Then the polynomial
\[ Q_{ \Sigma } (u_1, \ldots, u_l) - 1 \]
contains only monomials of degree $\geq 2$.
\label{p-function}
\end{prop}
Let $\chi$ be a topological character
of $T({\bf A}_K)$ such that
its $v$-component $\chi_v\, : \, T(K_v) \rightarrow S^1 \subset
{\bf C}^*$ is trivial on $T({\cal O}_v)$.
For each $ j \in \{ 1, \ldots, l\}$, we denote by
$n_j$ one of $d_j$ generators of all $1$-dimensional
cones of the $G_v$-orbit $ \Sigma _j(1)$; i.e., $G_vn_j$ is the set of
generators of $1$-dimensional cones in $ \Sigma _j(1)$.
Recall ($\ref{pi-image}$) that for
non-archimedian valuations,
$n_j$ represents an element
of $T(K_v)$ modulo $T({\cal O}_v)$. Therefore, $\chi_v(n_j)$ is
well-defined.
By ($\ref{pi-image}$) we know that the homomorphism
$$
\pi_v : T(K_v)/T({\cal O}_v) \rightarrow N_v
$$
is an isomorphism for almost all $v$.
We call these valuations {\em good}.
\begin{dfn}
{\rm Denote by
$\hat{H}_{\Sigma,v} (\chi_v, -{\bf s})$ the value
at $\chi_v$ of the
{\it multiplicative}
Fourier transform of the local Weil function
$H_{ \Sigma ,v}(x_v,-{\bf s})$ with
respect to the $v$-adic Haar measure
$d\mu_v$ on $T(K_v)$ normalized by
$\int_{T({\cal O}_v)} d\mu_v = 1$.}
\end{dfn}
\begin{prop} {\rm \cite{BaTschi}}
Let $v$ be a good non-archimedian
valuation of $K$ .
For any topological character
$\chi_v$ of $T(K_v)$ which is trivial on the
subgroup $T({\cal O}_v)$
and a piecewise linear function $\varphi = \varphi_{\bf s}
\in PL( \Sigma )^G_{{\bf C}}$
one has
\[ \hat{H}_{\Sigma,v} (\chi_v, -{\bf s}) =
\int_{T(K_v)} H_{ \Sigma ,v}(x_v,-{\bf s}) \chi_v(x_v)
d\mu_v = \]
\[ =
\frac{Q_{ \Sigma }\left( \frac{\chi_{v}(n_1)}{{q_v}^{\varphi(n_1)}},
\ldots, \frac{\chi_{v}(n_l)}{{q_v}^{\varphi(n_l)}} \right)}
{(1- \frac{\chi_{v}(n_1)}{{q_v}^{\varphi(n_1)}} )
\cdots (1 -
\frac{\chi_{v}(n_l)}{{q_v}^{\varphi(n_l)}} ) }. \]
\label{integral.1}
\end{prop}
\begin{coro} {\rm \cite{BaTschi}}
{\rm
Let $v$ be a good non-archimedian valuation of $K$.
The restriction of
\[ \int_{T(K_v)} H_{ \Sigma ,v}(x_v,-{\bf s}) d \mu_v \]
to the line $s_1 = \cdots = s_r = s$ is equal to
\[ L_v( s,T; E/K)\cdot L_v(s, T_{NS}, E/K) \cdot
Q_{ \Sigma } (q_v^{-s}, \ldots, q_v^{-s}). \]
}
\label{loc-int}
\end{coro}
\begin{rem}
{\rm
It is difficult to calculate the
Fourier transforms of local heights for
the finitely many
"bad" non-archimedian
valuations $v$, because there is only
an embedding of finite index
$$
T(K_v)/T({\cal O}_v)\hookrightarrow N_v.
$$
However, for our purposes it will be sufficient
to use upper estimates for these local
Fourier transforms. One immediately sees that
for all non-archimedian valuations $v$
the local Fourier transforms
of $H_{ \Sigma ,v}(x_v,-{\bf s})$ can be bounded
absolutely and uniformly in
all characters by a finite combination
of multidimensional geometric series in $q_v^{-1/2}$ in the domain
${\rm Re}({\bf s})\in {\bf R}_{>1/2}$.
\label{badreduction}
}
\end{rem}
Now we assume that $v$ is an archimedian valuation.
By ($\ref{pi-image}$), we have
$T(K_v)/T({\cal O}_v) = N_{\bf R}^{G_v} \subset
N_{\bf R}$
where $G_v$ is the trivial group for the case
$K_v = {\bf C}$,
and $G_v = {\rm Gal}({\bf C}/ {\bf R}) \cong
{\bf Z}/2{\bf Z}$
for the case $K_v = {\bf R}$.
Let $\langle \cdot,\cdot\rangle $ be the pairing
between $N_{{\bf R}}$ and $M_{{\bf R}}$ induced from the
duality between $N$ and $M$.
Let $y$ be an arbitrary element of
the dual ${\bf R}$-space
$M_{\bf R}^{G_v} = Hom(T(K_v)/T({\cal O}_v), {\bf R})$.
Then $\chi_y(x_v) = e^{- i \langle
\overline{x}_v,y \rangle}$
is a topological character
of $T(K_v)$ which is trivial on
$T({\cal O}_v)$. We choose the Haar measure $d\mu_v$ on $T(K_v)$
as the product of the Haar measure $d\mu_v^0$
on $T({\cal O}_v)$ and
the Haar measure $d\overline{x}_v$ on
$T(K_v)/T({\cal O}_v)$.
We normalize the measures such that
the $d\mu_v^0$-volume of $T({\cal O}_v)$ equals $1$ and
$d\overline{x}_v$ is
the standard Lebesgue measure on $N_{\bf
R}^{G_v}$ normalized by the full sublattice $N^{G_v}$.
\begin{prop} {\rm \cite{BaTschi}}
Let $v$ be an archimedian
valuation of $K$.
The Fourier transform
$\hat{H}_{ \Sigma ,v}(\chi_y,-{\bf s})$ of a
local archimedian Weil function
$$
H_{ \Sigma ,v} (x_v,-{\bf s}) = e^{-\varphi_{\bf s}(\overline{x}_v)}
$$
is a rational function in
variables $s_j = \varphi_{\bf s}(e_j)$ for
${\rm Re}({\bf s}) \in {\bf R}_{>0}$.
\label{archim.tr}
\end{prop}
{\it Proof.}
Let us consider the case $K_v = {\bf C}$.
One has a decomposition of the space $N_{\bf R}$
into a union of $d$-dimensional cones $N_{\bf R} =
\bigcup_{\sigma \in \Sigma (d)} \sigma$.
We calculate the Fourier transform as follows:
\[
\hat{H}_{ \Sigma ,v}(\chi_y,-{\bf s}) =
\int_{N_{\bf R}} e^{-\varphi_{\bf s}(\overline{x}_v) -
i \langle \overline{x}_v,y \rangle}
d\overline{x}_v=
\]
\[
= \sum_{\sigma \in \Sigma(d)} \int_{\sigma}
e^{-\varphi_{\bf s}(\overline{x}_v) -
i \langle \overline{x}_v,y \rangle}
d\overline{x}_v=
\sum_{\sigma \in \Sigma(d)}
\frac{1}{\prod_{e_j \in \sigma}
(s_j + i \langle e_j,y \rangle)}.
\]
The case $K_v = {\bf R}$ can be
reduced to the above situation.
\hfill $\Box$
\medskip
\section{Poisson formula}
Let ${\bf P}_{ \Sigma } $ be a toric variety and
$H_{ \Sigma }(x,{\bf s})$ the height function constructed
above.
\begin{dfn}
{\rm We define the zeta-function of the complex
height-function $H_{\Sigma}(x, {\bf s})$ as
\[ Z_{\Sigma}({\bf s}) = \sum_{x \in T(K)} H_{\Sigma}(x,{\bf -s}). \]}
\end{dfn}
\begin{theo}
The series $Z_{\Sigma}({\bf s})$
converges absolutely and uniformly for
${\bf s}$ contained in any compact in the
domain ${\rm Re}({\bf s})\in {\bf R}^r_{>1}$.
\label{convergence}
\end{theo}
\noindent
{\em Proof.}
It was proved in \cite{ono1} that we can always choose a finite set $S$
such that the natural map
$$
\pi_{S}\; : \; T(K) \rightarrow \bigoplus_{v \not\in S} T(K_v)/T({\cal O}_v) =
\bigoplus_{v \not\in S} N_v
$$
is surjective. Denote by $T({\cal O}_S)$ the kernel of $\pi_S$ consisting
of all $S$-units in $T(K)$. Let $W(T) \subset T({\cal O}_S)$ the subgroup of
torsion elements in $T({\cal O}_S)$. Then $T({\cal O}_S)/W(T)$ has a natural
embedding into the finite-dimensional logarithmic space
$$
N_{{\bf R},S} = \bigoplus_{v \in S} T(K_v)/T({\cal O}_v) \otimes {{\bf R}}
$$
as a sublattice of codimension $t$. Let $\Gamma$ be a full sublattice in
$N_{{\bf R},S}$ containing the image of $T({\cal O}_S)/W(T)$. Denote by
$\Delta$ a bounded fundamental domain of $\Gamma$ in $N_{{\bf R},S}$.
For any $x \in T(K)$ we denote by $\overline{x}_S$ the image of $x$ in
$N_{{\bf R},S}$. Define $\phi(x)$ to be the element of $\Gamma$ such that
$\overline{x}_S - \phi(x) \in \Delta$. Thus, we have obtained
the mapping
$$
\phi\; : \; T(K) \rightarrow \Gamma.
$$
Define a new function $\tilde{H}_{ \Sigma }(x, {\bf s})$ on $T(K)$ by
$$
\tilde{H}_{ \Sigma }(x, {\bf s}) = \prod_{v \in S} H_{ \Sigma ,v}(\phi(x)_v,
{\bf s}) \prod_{v \not\in S} H_{ \Sigma ,v}(x_v, {\bf s}).
$$
If ${\bf K}\subset {\bf C}^r$ is a compact in the domain
${\rm Re}({\bf s}) \in {\bf R}^r_{>1}$, then there exist two positive constants
$C_1({\bf K}) < C_2({\bf K})$ such that
$$
0 < C_1({\bf K})
< \frac{\tilde{H}_{ \Sigma }(x, {\bf s})}{H_{ \Sigma }(x, {\bf s})} <
C_2({\bf K}) \;\; \mbox{\rm for ${\bf s} \in {\bf K}, \,x \in T(K)$},
$$
since $\overline{x}_v - \phi(x)_v$ belongs to some bounded subset $\Delta_v$
in $N_{{\bf R},v}$ for any $x \in T(K)$ and $v \in S$.
Therefore, it is sufficient to prove that the series
$$
\tilde{Z}_{ \Sigma }({\bf s}) = \sum_{x \in T(K)} \tilde{H}_{ \Sigma }(x, - {\bf s})
$$
is absolutely converent for ${\bf s} \in {\bf K}$.
Notice that $\tilde{Z}_{ \Sigma }({\bf s})$ can be estimated from above by the
the following Euler product
$$
\left( \sum_{ \gamma \in \Gamma} \prod_{v \in S}H_{ \Sigma ,v}( \gamma_v, -{\bf s}) \right)
\prod_{v \not\in S} \left( \sum_{z \in N_v} H_{ \Sigma ,v}(z, - {\bf s}) \right).
$$
The sum
$$
\sum_{ \gamma \in \Gamma} \prod_{v \in S}H_{ \Sigma ,v}( \gamma_v, -{\bf s})
$$
is an absolutely convergent geometric series for
${\rm Re}({\bf s}) \in {\bf R}^r_{>1}$.
On the other hand, the Euler product
$$
\prod_{v \not\in S} \left( \sum_{z \in N_v} H_{ \Sigma ,v}(z, - {\bf s}) \right)
$$
can be estimated from above by the product of zeta-functions
$$ C_3({\bf K}) \prod_{j =1}^r \zeta_{K_j}(s_j),$$
where $C_3({\bf K})$ is some constant depending on ${\bf K}$.
Since each $ \zeta_{K_j}(s_j)$ is absolutely convergent for
${\rm Re}(s_j) > 1$, we obtain the statement.
\hfill
$\Box$
We need the Poisson formula in the following form:
\begin{theo}
Let ${\cal G}$ be a locally compact abelian group with
Haar measure $dg, {\cal H}\subset {\cal G} $ a closed
subgroup with Haar measure $dh$.
The factor group ${\cal G}/{\cal H}$ has a unique Haar measure $dx$
normalized by the condition $dg=dx\cdot dh$.
Let $F\,:\, {\cal G} \rightarrow {\bf R} $ be an ${L}^1$-function on
${\cal G}$ and $\hat{F}$ its Fourier transform with respect
to $dg$. Suppose that $\hat{F}$ is also an ${L}^1$-function
on ${\cal H}^{\perp}$, where ${\cal H}^{\perp}$ is the group
of topological characters $\chi \,: \, {\cal G} \rightarrow S^1$
which are trivial on ${\cal H}$.
Then
$$
\int_{\cal H} F(x)dh=\int_{{\cal H}^{\perp}}\hat{F}(\chi) d\chi,
$$
where $d\chi$ is the orthogonal Haar measure on ${\cal H}^{\perp}$
with respect to the Haar measure $dx$ on ${\cal G}/{\cal H}$.
\label{poi}
\end{theo}
We will apply this theorem in the case when ${\cal G}=T({\bf A}_K)$
and ${\cal H}=T(K)$, $dg = \omega_{\Omega,S}$, and $dh$ is the
discrete measure on $T(K)$.
\begin{theo} (Poisson formula)
For all ${\bf s}$ with ${\rm Re}({\bf s})\in {\bf R}^r_{>1}$
we have the following formula:
$$
Z_{\Sigma}({\bf s})=\frac{1}{(2\pi )^t b_S(T)}
\int_{(T({\bf A}_K)/T(K))^*}
\left( \int_{T(A_F)}H_{ \Sigma }(x,-{\bf s})\chi(x)\omega_{\Omega,S}
\right) d\chi,
$$
where $\chi \in (T({\bf A}_K)/T(K))^* $
is a topological character of $T({\bf A}_K)$, trivial on
the closed subgroup $T(K)$ and
$d\chi$ is the orthogonal Haar measure on $(T({\bf A}_K)/T(K))^*$.
The integral converges
absolutely and uniformly to a holomorphic function in ${\bf s}$
in any compact in the domain ${\rm Re}({\bf s})\in {\bf R}^r_{>1}$.
\label{poiss}
\end{theo}
{\em Proof.} Because of \ref{convergence} we only
need to show that the Fourier transform
$\hat{H}_{ \Sigma }(\chi,-{\bf s})$ of the height function
is an $ L^1$-function on $(T({\bf A}_K)/T(K))^*$.
By \ref{integral.1} and uniform estimates at places of
bad reduction \ref{badreduction},
we know that the Euler product
$$
\prod_{v\not\in S_{\infty}}\hat{H}_{ \Sigma ,v}(\chi_v,-{\bf s})
$$
converges absolutely and is
uniformly bounded by a constant $c({\bf K})$ for all characters
$\chi$ and all ${\bf s}\in {\bf K}$, where ${\bf K}$ is some
compact in the domain ${\rm Re}({\bf s})\in {\bf R}_{>1}^r$.
Since the height function
$H_{ \Sigma ,v}(x,-{\bf s})$ is invariant under
$T({\cal O}_v)$ for all $v$, the Fourier transform
of $\hat{H}_{ \Sigma }(\chi,-{\bf s})$ equals
zero for characters $\chi$ which are non-trivial on the maximal
compact subgroup ${\bf K}_T$.
Denote by ${\cal P}$ the set of all such characters $\chi\in
(T({\bf A}_K)/T(K))^* $.
We have a non-canonical splitting
of characters $\chi =\chi_l\cdot \chi_y$,
where $\chi_l\in (T^1({\bf A}_K)/T(K))^*$
and $\chi_y\in (T({\bf A}_K)/T^1({\bf A}_K))^*$.
Let us consider the logarithmic space
$$
N_{{\bf R},\infty}=\bigoplus_{v\in S_{\infty}}T(K_v)/T({\cal O}_v)=
\bigoplus_{v\in S_{\infty}}N_{{\bf R},v}.
$$
It contains the lattice $T({\cal O}_K)/W(T)$
of ${\cal O}_K$-integral points of
$T(K)$ modulo torsion.
Denote by
$
M_{{\bf R},\infty}=\bigoplus_{v\in S_{\infty}}M_{{\bf R},v}
$
the dual space.
It has a decomposition as a direct sum of vector spaces
$M_{{\bf R},\infty}=M_L\oplus M_Y$,
such that the space $M_L$ contains
the dual lattice $L:=(T({\cal O}_K)/W(T))^*$
as a full sublattice and the space $M_Y$ is isomorphic
to $(T({\bf A}_K)/T^1({\bf A}_K))^*= \hat{T}_K\otimes {\bf R}$.
By \ref{subgroups}, we have an exact sequence
$$
0 \rightarrow {\bf cl}^*(T) \rightarrow {\cal P} \rightarrow {\cal M} \rightarrow 0,
$$
where ${\cal M}$ is the image of the projection of ${\cal P}$ to
$M_{{\bf R},\infty}$ and ${\bf cl}^*(T)$ is a finite group.
We see that the character
$\chi\in {\cal P}$ is determined
by its archimedian component up to a finite choice.
Denote by $y(\chi)\in {\cal M}\subset M_{{\bf R},\infty}$ the image
of $\chi\in {\cal P}$.
The following lemmas
will provide the necessary estimates of the
Fourier transform of local heights at archimedian places.
This allows to apply the Poisson formula \ref{poi}. \hfill $\Box $
\begin{lem} {\rm \cite{BaTschi}}
{\rm Let $\Sigma\subset N_{{\bf R}}$ be a
complete fan
in a real vector space of dimension $d$. Denote by $M_{{\bf R}}$ the
dual space. For all $m\in M_{{\bf R}}$
we have the following estimate
$$
|\sum_{ \sigma \in \Sigma (d)}
\frac{1}{\prod_{e_j\in \sigma }(s_j+i<e_j,m>)}|\le
\frac{1}{(1 + \|m\|)^{1+1/d}}.
$$
}
\end{lem}
\begin{coro}
{\rm
Consider
\[ \hat{H}_{ \Sigma ,\infty}(\chi, -{\bf s}) =
\prod_{v \in S_{\infty}}
\hat{H}_{ \Sigma ,v}(\chi, -{\bf s}) \]
as a function on
\[ {\cal M}\subset M_{{\bf R}, \infty}
= \bigoplus_{v \in S_{\infty}} M_{{\bf R},v}. \]
Let $d'$ be the dimension of $M_{{\bf R}, \infty}$.
We have a direct sum decomposition
$M_{{\bf R}, \infty}=M_L\oplus M_Y$
of real vector spaces.
Let $M'_Y\subset M_Y$ be any affine
subspace, $dy'$ the Lebesgue measure on $M'_Y$
and $L'\subset M_L$ any lattice.
Let $g(y,-{\bf s})$ be a function on $M_{{\bf R}, \infty}$
satisfying the inequality $|g(y,-{\bf s})|\le c \|y\|^{\delta } $ for all
$y\in M_{{\bf R}, \infty}$, all ${\bf s}$ in some compact domain in
${\rm Re}({\bf s})\in {\bf R}_{>1/2}$, some $0<\delta < 1/d'$
and some constant $c>0$. Then the series
\[
\sum_{y(\chi) \in L} \int_{M'_Y}
g(y,-{\bf s})\hat{H}_{ \Sigma ,\infty}(y(\chi), -{\bf s}) dy'
\]
is absolutely and uniformly convergent to a holomorphic
function in ${\bf s}$ in this domain.
\label{infconver}
}
\end{coro}
{\em Proof.} We apply \ref{archim.tr} and observe that
on the space $N_{{\bf R}, \infty}$ we have a fan $ \Sigma _{\infty}$
obtained as the direct product of fans $ \Sigma ^{G_v}$ for $v\in S_{\infty}$
(i.e., every cone in $ \Sigma _{\infty}$ is a direct product of cones in
$ \Sigma ^{G_v}$).
\hfill $\Box $
\section{${\cal X}$-functions of convex cones}
Let $(A, A_{\bf R}, \Lambda ) $ be a triple consisting of
a free abelian group
$A$ of rank $k$, a $k$-dimensional real vector space
$A_{\bf R} = A \otimes {\bf R}$ containing $A$ as a sublattice of
maximal rank, and a convex $k$-dimensional cone
$\Lambda \subset A_{{\bf R}}$ such that $\Lambda \cap - \Lambda = 0
\in A_{{\bf R}}$. Denote by $ \Lambda ^{\circ}$ the interior of $ \Lambda $ and
by ${ \Lambda }_{\bf C}^{\circ} = { \Lambda }^{\circ} + iA_{{\bf R}}$
the complex tube domain over ${ \Lambda }^{\circ}$.
Let $( A^*, A^*_{{\bf R}}, \Lambda ^*) $ be the triple
consisting of the dual abelian group
$A^* = {\rm Hom}(A, {\bf Z})$, the dual real vector space
$A^*_{{\bf R}} = {\rm Hom}(A_{{\bf R}}, {\bf R})$, and the dual cone
$ \Lambda ^* \subset A^*_{{\bf R}}$.
We normalize the Haar measure $ {\bf d}{\bf y}$ on $A_{{\bf R}}^*$
by the condition:
${\rm vol}(A^*_{{\bf R}}/A^*)=1$.
\begin{dfn}
{\rm The {\em ${\cal X}$-function of}
${ \Lambda }$ is defined as
the integral
\[ {\cal X}_{ \Lambda }({\bf s}) =
\int_{{ \Lambda }^*} e^{- \langle {\bf s}, {\bf y}
\rangle} {\bf d}{\bf y}, \]
where ${\bf s} \in { \Lambda }_{\bf C}^{\circ}$. }
\end{dfn}
\begin{rem}
{\rm ${\cal X}$-functions of convex cones
have been investigated
in the theory of homogeneous cones
by M. K\"ocher, O.S. Rothaus, and
E.B. Vinberg \cite{koecher,rothaus,vinberg}. In these papers
${\cal X}$-functions were called {\em characteristic functions of
cones}, but we find such a notion rather misleading in view of the
fact that ${\cal X}_{ \Lambda }({\bf s})$ is the Fourier-Laplace transform
of the standard set-theoretic characteristic function of the
dual cone $ \Lambda ^*$. }
\end{rem}
\noindent
The function ${\cal X}_{ \Lambda }({\bf s})$
has the following
properties \cite{rothaus,vinberg}:
\begin{prop}
{\rm (i)} If ${\cal A}$ is any invertible
linear operator on ${\bf C}^k$, then
\[ {\cal X}_{ \Lambda } ({\cal A}{\bf s}) = \frac{{\cal X}_{ \Lambda }({\bf s})}
{{\rm det}{\cal A}}; \]
{\rm (ii)} If ${ \Lambda } = {\bf R}^k_{\geq 0}$, then
\[ {\cal X}_{ \Lambda }({\bf s}) = (s_1 \cdots s_k)^{-1}, \;{\rm for }
\;{\rm Re}(s_i) > 0 ; \]
{\rm (iii)} If ${\bf s} \in { \Lambda }^{\circ}$, then
\[ \lim_{{\bf s} \rightarrow \partial { \Lambda }}
{\cal X}_{ \Lambda }({\bf s}) = \infty; \]
{\rm (iv)} ${\cal X}_{ \Lambda }({\bf s}) \neq 0$ for all
${\bf s} \in { \Lambda }_{\bf C}^{\circ}$.
\label{zeta.cone}
\end{prop}
\begin{prop}
If ${ \Lambda }$ is a $k$-dimensional
finitely generated polyhedral cone,
then ${\cal X}_{ \Lambda }({\bf s})$ is a rational function
$$
{\cal X}_{ \Lambda }({\bf s}) = \frac{P({\bf s})}{Q({\bf s})},
$$
where $P$ is a homogeneous polynomial,
$Q$ is a product of all linear homogeneous forms defining
the codimension $1$ faces of
$ \Lambda $, and ${\rm deg}\, P -
{\rm deg}\, Q = -k$.
\label{merom}
\end{prop}
\noindent
{\em Proof.} We subdivide the dual cone ${ \Lambda }^*$ into a finite
union of simplicial
subcones $ \Lambda _j^*$ $(j \in J)$.
Let $ \Lambda _j \subset A_{{\bf R}}$ be the dual cone
to $ \Lambda _j^*$. Then
$$
{\cal X}_{ \Lambda }({\bf s}) = \sum_{j \in J} {\cal X}_{ \Lambda _j}({\bf s}).
$$
By \ref{zeta.cone}(i) and (ii),
$${\cal X}_{ \Lambda _j}({\bf s}) = \frac{P_j({\bf s})}{Q_j({\bf s})} \;\;
( j \in J),$$
where $P_j$ is a homogeneous polynomial of degree $0$ and $Q_j$ is
the product of $k$ homogeneous linear forms defining the
codimension $1$ faces of $ \Lambda _j$.
Therefore, ${\cal X}_{ \Lambda }({\bf s})$ can be uniquely
represented up to constants
as a ratio of two homogeneous polynomials $P({\bf s})/Q({\bf s})$
with $g.c.d.(P,Q)=1$ where $Q$ is a product of linear homogeneous
forms defining some faces of $ \Lambda _j$ of codimension $1$. Since
$Q$ does not depend on a choice of a subdivision of $ \Lambda ^*$ into a finite
union of simplicial cones $ \Lambda ^*_j$, only linear homogeneous forms
which vanish on codimension $1$ faces of $ \Lambda $ can be factors of
$Q$. On the other hand, by \ref{zeta.cone}(iii), every linear homogeneous
form vanishing on a face of $ \Lambda $ of codimension $1$ must divide $Q$.
\hfill $\Box$
\begin{theo} Let $(A, A_{{\bf R}}, \Lambda )$ and $(\tilde{A}, \tilde{A}_{{\bf R}},
\tilde{ \Lambda })$ be two triples as above, $k = {\rm rk}\, A$ and
$\tilde{k} = {\rm rk}\, \tilde{A}$, and $\psi\;:\; A \rightarrow \tilde{A}$
a homomorphism of free abelian groups with a finite cokernel
$A'$ (i.e., the corresponding
linear mapping of real vector spaces $\psi \;:\; A_{{\bf R}} \rightarrow
\tilde{A}_{{\bf R}}$ is surjective), and $\psi( \Lambda ) = \tilde{ \Lambda }$.
Let $B= {\rm Ker}\, \psi \subset A$, ${\bf d}{\bf b}$ the Haar measure
on $B_{{\bf R}} = B \otimes {{\bf R}}$ normalized by the condition
${\rm vol}(B_{{\bf R}}/B)=1$.
Then for all ${\bf s}$ with
${\rm Re}({\bf s}) \in \Lambda^{\circ}$
the following formula holds:
$$
{\cal X}_{\tilde{ \Lambda }}(\psi({\bf s}))
= \frac{1}{(2\pi)^{k-\tilde{k}}|A'|}
\int_{B_{{\bf R}}} {\cal X}_{{ \Lambda }}
({\bf s} + i {\bf b}) {\bf db},
$$
where $|A'|$ is the order of the finite abelian group $A'$.
\label{char0}
\end{theo}
{\em Proof.} We have the dual injective homomorphisms
of free abelian groups
$\psi^*\;:\; \tilde{A}^* \rightarrow A^*$ and of the corresponding
real vector spaces $\psi^*\;:\; \tilde{A}^*_{{\bf R}} \rightarrow A^*_{{\bf R}}$.
Moreover, $\tilde{ \Lambda }^* = \Lambda ^* \cap \tilde{A}^*_{{\bf R}}$. Let
$C_{ \Lambda ^*}({\bf y})$ be the set-theoretic characteristic function
of the cone $ \Lambda ^* \subset A^*_{{\bf R}}$ and $C_{ \Lambda ^*}(\tilde{\bf {y}})$ the
restriction of $C_{ \Lambda ^*}({\bf y})$ to $\tilde{A}_{{\bf R}}^*$ which is
the set-theoretic characteristic function of $\tilde{ \Lambda }^* \subset
\tilde{A}_{{\bf R}}^*$. Then
$$
{\cal X}_{\tilde{ \Lambda }}(\psi({\bf s})) =
\int_{\tilde{A}^*_{{\bf R}}} C_{ \Lambda ^*}(\tilde{\bf {y}}) e^{- \langle
\psi({\bf s}), \tilde{\bf y}
\rangle} {\bf d}\tilde{\bf y}.
$$
Now we apply the Poisson formula to the last integral.
For this purpose we notice that any additive topological character of
${A}^*_{{\bf R}}$ which vanishes on the subgroup $\tilde{A}^*_{{\bf R}} \subset
{A}^*_{{\bf R}}$ has the form
$$ e^{- i \langle {\bf b}, {\bf y} \rangle}, \;\;\; \mbox{ \rm where
${\bf b} \in B_{{\bf R}}$}.
$$
Moreover,
$$
\frac{{\bf db}}{(2\pi)^{k-\tilde{k}}|A'|}
$$
is the orthogonal Haar measure on $B_{{\bf R}}$ with respect to
the Haar measures
${\bf d}\tilde{\bf y}$ and ${\bf d}{\bf y}$ on $\tilde{A}^*_{{\bf R}}$ and
$A^*_{{\bf R}}$ respectively. It remains to notice that
$${\cal X}_{{ \Lambda }}
({\bf s} + i {\bf b}) =
\int_{{A}^*_{{\bf R}}} C_{ \Lambda ^*}({\bf {y}}) e^{- \langle
{\bf s} + i{\bf b}, {\bf y}
\rangle} {\bf d}{\bf y}
$$
is the value of the Fourier transform of
$C_{ \Lambda ^*}({\bf {y}}) e^{- \langle {\bf s} , {\bf y} \rangle}$ on
the topological character of $A_{{\bf R}}^*/\tilde{A}_{{\bf R}}^*$ corresponding
to an element ${\bf b} \in B_{{\bf R}} \subset A_{{\bf R}}$.
\hfill $\Box$
\begin{coro}
{\rm
Assume that in the above situation ${\rm rk}\,= k - \tilde{k} =1$ and
$\tilde{A} = A/B$. Denote by $ \gamma$ a generator of $B$. Then
$$
{\cal X}_{\tilde{ \Lambda }}({\psi}({\bf s})) =
\frac{1}{2\pi i}\int_{{\rm Re}(z) = 0}
{\cal X}_{ \Lambda }({\bf s} + z \cdot \gamma) dz,
$$
where $z = x + iy \in {{\bf C}}$.
\label{char1}
}
\end{coro}
\begin{coro}{\rm
Assume that a $\tilde{k}$-dimensional rational finite polyhedral cone
$\tilde{\Lambda} \subset \tilde{A}_{{\bf R}}$ contains exactly $r$ one-dimensional
faces with primitive lattice generators $a_1, \ldots, a_r \in \tilde{A}$.
We set $k := r$, $A := {{\bf Z}}^r$ and denote by $\psi$
the natural homomorphism of lattices ${\bf Z}^r \rightarrow \tilde{A}$
which sends the standard basis of ${{\bf Z}}^r$ into
$a_1, \ldots, a_r \in \tilde{A}$, so that $\tilde{ \Lambda }$ is the image
of the simplicial cone ${\bf R}^r_{\ge 0}\subset {\bf R}^r$
under the surjective map of vector spaces $\psi\; : \; {\bf R}^r
\rightarrow A_{{\bf R}}$.
Denote by $M_{{\bf R}}$ the kernel of $\psi$ and set $M := {{\bf Z}}^r \cap M_{{\bf R}}$.
Let ${\bf s}=(s_1,...,s_r)$ be the standard
coordinates in ${\bf C}^r$. Then
$$
{\cal X}_{ \Lambda }(\psi({\bf s}))=\frac{1}{(2\pi )^{r-k}|A'|}
\int_{M_{{\bf R}}}\frac{1}{\prod_{j=1,n}(s_j+iy_j)} {\bf d}{\bf y}
$$
where ${\bf dy}$ is the Haar measure on the additive
group $M_{{\bf R}}$ normalized
by the lattice $M$,
$y_j$ are the coordinates of ${\bf y}$ in ${\bf R}^r$, and
$|A'|$ is the index of the sublattice in $\tilde{A}$ generated by
$a_1, \ldots, a_r$.
\label{int.formula}
}
\end{coro}
\begin{exam}
{\rm Consider an example of a non-simplicial
convex cone which
appears as the cone of effective divisors of the
split toric
Del Pezzo surface $X$ of anticanonical degree 6.
The cone ${\Lambda}_{{\rm eff}}$ has
6 generators corresponding to exceptional curves
of the first kind on $X$. We can construct
$X$ as the blow up of
3 points $p_1, p_2, p_3$ in general position
in ${\bf P}^2$.
Denote the exceptional curves by $C_1, C_2, C_3,
C_{12}, C_{13}, C_{23}$, where $C_{ij}$ is
the proper pullback of the line
joining $p_i$ and $p_j$. Let
${\bf s} = s_1 [C_1] + s_2 [C_2] + s_3 [C_3]
+ s_{12}[C_{12}] +
s_{13}[C_{13}] + s_{23} [C_{23}] \in
\Lambda _{\rm eff}^{\circ}$
be an element in the
interior of the cone of effective divisors.
The sublattice $M \subset {\bf Z}^6$ of rank $2$ consisting of
principal divisors is generated by
$ \gamma_1 = C_1 + C_{13} - C_2 - C_{23}$ and $ \gamma_2 = C_1 + C_{12} - C_3 -
C_{23} = 0$.
In our case, the integral formula in \ref{int.formula} is a 2-dimensional
integral ($r =6$) which can be computed by applying twice the residue
theorem to two $1$-dimensional integrals like the one in \ref{char1}.
We obtain the
following formula for the characteristic
function of ${\Lambda}_{\rm eff}$: }
\[ {\cal X}_{\Lambda}(\psi({\bf s})) =
\frac{ s_1 + s_2 + s_3 + s_{12} + s_{13} + s_{23} }
{(s_1 + s_{23}) (s_2 + s_{13})(s_3 +
s_{12})(s_1 + s_2 + s_3 )
(s_{12} + s_{13} + s_{23})}. \]
\end{exam}
\begin{dfn}
{\rm Let $X$ be a smooth proper algebraic variety.
Consider the triple $({\rm Pic}(X), {\rm Pic}(X) \otimes{\bf R},
\Lambda _{\rm eff})$ where
$ \Lambda _{\rm eff} \subset {\rm Pic}(X)\otimes {\bf R}$ is
the cone generated by classes of effective
divisors on $X$.
Assume that the anticanonical
class $ \lbrack {\cal K}^{-1}
\rbrack \in {\rm Pic}(X)_{\bf R}$
is contained in the interior of $ \Lambda _{\rm eff}$. We define
the constant $\alpha(X)$ by
\[ \alpha(X) = {\cal X}_{ \Lambda _{\rm eff}}( \lbrack {\cal K}^{-1}
\rbrack). \]
}
\end{dfn}
\begin{coro}{\rm
If ${ \Lambda }_{\rm eff}$ is a finitely generated
polyhedral cone,
then $\alpha(X)$ is a rational number.
}
\end{coro}
\section{Some technical statements}
Let $E$ be a number field and
$\chi$ an unramified Hecke character
on ${\bf G}_m(A_E)$.
Its local components $\chi_v$ for all
valuations $v$ are given by:
$$
\chi_v:
{\bf G}_m(E_v)/{\bf G}_m({\cal O}_{v}) \rightarrow S^1
$$
$$
\chi_v(x_v)=|x_v|_v^{it_v}.
$$
\begin{dfn}
{\rm Let $\chi$ be an unramified Hecke character. We set
$$
y(\chi) : = \{ t_v \}_{v \in S_{\infty}(E)} \in
{{\bf R}}^{r_1 + r_2},
$$
where $r_1$ (resp. $ r_2$) is the number of real (resp. pairs of
complex) valuations of $E$. We also set
$$
\| y(\chi) \| := \max_{v \in S_{\infty}(E)} |t_v|.
$$
\label{y-comp}
}
\end{dfn}
We will need uniform estimates for
Hecke $L$-functions in vertical strips.
They can be deduced using the Phragmen-Lindel\"of
principle \cite{rademacher}.
\begin{theo}
For any $\varepsilon > 0$
there exists a $\delta>0 $ such that for any
$0<\delta_1<\delta $ there exists a constant $c(\varepsilon,\delta_1) > 0$
such that the inequality
$$
| L_E(s, \chi) |
\leq c(\varepsilon) ( 1 + |{\rm Im}(s)| + \|y(\chi)\| )^{\varepsilon}
$$
holds for all
$s$ with
$ \delta_1<|{\rm Re}(s) -1|< \delta$ and every Hecke L-function
$L_E(s, \chi)$ corresponding to an unramified
Hecke character $\chi$.
\label{estim}
\end{theo}
\begin{coro}{\rm
For any $\varepsilon >0 $
there exists a $ \delta>0$ such
that for any compact
${\bf K}$ in the domain
$ 0<| {\rm Re}\,( s) - 1| <\delta$
there exists a
constant $C({\bf K},\varepsilon)$ depending only on
${\bf K}$ and $\epsilon$
such that
\[ | L_E(s,\chi) | \leq C({\bf K},\varepsilon)
(1 + \|y(\chi)\|)^{\varepsilon}
\]
for $s \in {\bf K}$ and every unramified
character $\chi$.
\label{m.estim}
}
\end{coro}
Let $ \Sigma $ be the Galois-invariant fan defining ${\bf P}_{ \Sigma }$ and
$ \Sigma (1)= \Sigma _1(1)\cup ...\cup \Sigma _r(1)$ the decomposition
of the set of one-dimensional generators of $ \Sigma $ into $G$-orbits.
Let $e_j$ be a primitive integral generator of $\sigma_j$,
$G_j \subset G$ the stabilizer of $e_j$. Denote by
$K_j \subset E$ the subfield of
$G_j$-fixed elements.
Consider the $n$-dimensional torus
$$
T'=\prod_{j=1}^r R_{K_j/K}({\bf G}_m).
$$
Let us recall the exact sequence
of Galois-modules from Proposition \ref{nonsplit.geom}:
$$
0 \rightarrow M^G \rightarrow PL( \Sigma )^G \rightarrow {\rm Pic}({\bf P}_{ \Sigma }) \rightarrow H^1(G,M) \rightarrow 0.
$$
It induces a map of tori $T' \rightarrow T$ and a homomorphism
$$
a\,:\,\,T'({\bf A}_K)/T'(K) \rightarrow T({\bf A}_K)/T(K).
$$
So we get a dual homomorphism for characters
$$
a^*:\, (T({\bf A}_{K})/T(K))^* \rightarrow \prod_{j =1}^r
({\bf G}_{m}({\bf A}_{K_j})/{\rm G}_m(K_j))^*.
$$
\begin{prop}{\rm \cite{drax1}}
The kernel of $a^*$ is dual
to the obstruction group to weak approximation $A(T) $
defined in \ref{weak0}.
\label{kera}
\end{prop}
Let $\chi\in (T({\bf A}_{K})/T(K))^*$ be a character. Then
$\chi\circ a$ defines $r$ Hecke characters of the idele groups
\[ \chi_j \; :\;
{\bf G}_m({\bf A}_{K_j}) \rightarrow S^1 \subset {\bf C}^*, \;
j =1, \ldots, r. \]
If $\chi$ is trivial on ${\bf K}_T$, then all characters $\chi_j$
$(j =1, \ldots, r)$ are trivial on the maximal
compact subgroups in ${\bf G}_m({\bf A}_{K_j})$.
We denote by $L_{K_j}(s,\chi_j)$ the
Hecke $L$-function corresponding to the unramified
character $\chi_j$.
\begin{prop}
Let $\chi=(\chi_v)$ be a character and
$\hat{H}_{ \Sigma ,v}(\chi_v,-{\bf s})$
the local Fourier transform
of the complex local height
function $H_{ \Sigma ,v}(x_v,-{\bf s})$.
For any compact ${\bf K}$ contained
in the domain ${\rm Re}({\bf s})\in {\bf R}^r_{>1/2} $ there exists
a constant $c({\bf K})$ such that
\[ \prod_{v\not\in S}\hat{H}_{ \Sigma ,v}(\chi_v,-{\bf s})\cdot
\prod_{i=1}^r L^{-1}_{K_j}(s_j,\chi_j) \le c({\bf K})\]
for all characters $\chi\in (T({\bf A}_{K})/T(K))^*$.
\label{dmethod}
\end{prop}
The proof follows from explicit computations
of local Fourier transforms \ref{integral.1} and
is almost identical with the proof of
Proposition 3.1.3 in \cite{BaTschi}.
\begin{prop} There exists an $\varepsilon >0$ such that
for any open $U\subset {\bf C}^r$
contained in the domain
$ 0< |{\rm Re}(s_j)-1|<\varepsilon $ for $j=1,...,r$
the integral
$$
\int_{(T({\bf A}_{K})/T(K))^*}\hat{H}_{ \Sigma }(\chi, -{\bf s})d\chi
$$
converges absolutely and uniformly to a holomorphic function
for ${\rm Re}({\bf s})\in U$.
\label{analytic}
\end{prop}
{\em Proof.} Using uniform estimates of Fourier transforms for
non-archimedian places of bad reduction (\ref{badreduction}) and
the proposition above we need only to consider the following
integral
$$
\int_{(T({\bf A}_{K})/T(K))^*}\hat{H}_{ \Sigma ,\infty}(\chi,-{\bf s})
\prod_{j=1}^r L_{K_j}(s_j,\chi_j)d\chi.
$$
Observe that there exist constants $c_1>0$ and $c_2>0$
such that we have the following inequalities:
$$
c_1\|y(\chi)\|\le \sum_{j=1}^r\|y(\chi_j)\|\le c_2\|y(\chi)\|.
$$
Here we denoted by $\|y(\chi)\|$ the norm of $y(\chi)\in M_{{\bf R},\infty}$.
Recall that since we only consider $\chi$ which are trivial on the
maximal compact subgroup ${\bf K}_T$, all characters $\chi_j$ are
unramified.
To conclude, we apply uniform estimates
of Hecke L-functions from Corollary \ref{m.estim}
and the Corollary \ref{infconver}.
\hfill $\Box $
The rest of this section is devoted to
the proof of our main technical result.
Let ${{\bf R}}\lbrack {\bf s} \rbrack$
(resp. ${{\bf C}}\lbrack {\bf s} \rbrack$)
be the ring of polynomials in $s_1, \ldots, s_r$
with coefficients in
${{\bf R}}$ (resp. in ${{\bf C}}$), ${{\bf C}}\lbrack \lbrack {\bf s}
\rbrack \rbrack$ the ring of formal power series in $s_1, \ldots, s_r$
with complex coefficients.
\begin{dfn}
{\rm Two elements $f({\bf s}),\, g({\bf s})\in {{\bf C}}\lbrack \lbrack {\bf s}
\rbrack \rbrack$ will be called {\em coprime}, if
$g.c.d.(f({\bf s}),\, g({\bf s})) =1$.
}
\end{dfn}
\begin{dfn}
{\rm Let $f({\bf s})$ be an element of ${{\bf C}}\lbrack \lbrack {\bf s}
\rbrack \rbrack$. By the {\em order} of a monomial
$s_1^{ \alpha _1}...s_r^{ \alpha _r}$ we mean the sum
of the exponents $ \alpha _1+...+ \alpha _r$.
By {\em multiplicity $\mu(f({\bf s}))$
of $f({\bf s})$ at
${\bf 0} = (0, \ldots, 0)$} we always mean
the minimal order of non-zero monomials appearing in the
Taylor
expansion of $f({\bf s})$ at ${\bf 0}$ . }
\label{mult1}
\end{dfn}
\begin{dfn}
{\rm
Let $f({\bf s})$ be a meromorphic at ${\bf 0}$ function.
Define the {\em multiplicity $\mu(f({\bf s}))$ }
of $f({\bf s})$ at
${\bf 0}$ as
\[ \mu(f({\bf s})) = \mu(g_1({\bf s})) - \mu(g_2({\bf s})) \]
where $g_1({\bf s})$ and $g_2({\bf s})$ are two coprime
elements in ${{\bf C}}\lbrack \lbrack {\bf s} \rbrack \rbrack$
such that $f = g_1/g_2$.
}
\label{mult2}
\end{dfn}
\begin{rem}
{\rm It is easy to show that for any two
meromorphic at ${\bf 0}$ functions $f_1({\bf s})$ and $f_2({\bf s})$,
one has
(i) $\mu(f_1 \cdot f_2) = \mu(f_1) \cdot \mu(f_2)$ (in
particular, one can omit "coprime" in Definition \ref{mult2});
(ii) $\mu(f_1 + f_2) \geq \min \{ \mu(f_1), \mu(f_2) \}$;
(iii) $\mu(f_1 + f_2) = \mu(f_1)$ if $\mu(f_2) > \mu(f_1)$. }
\label{mult3}
\end{rem}
Using the properties \ref{mult3}(i)-(ii), one immediately
obtains from Definition \ref{mult1} the following:
\begin{prop}
Let $f_1({\bf s})$ and $f_2({\bf s})$ be two analytic at ${\bf
0}$ functions, $l({\bf s})$ a homogeneous linear function,
$ \gamma = ( \gamma_1, \ldots, \gamma_r) \in {{\bf C}}^r$ an arbitrary complex
vector with $l( \gamma) \neq 0$,
and $g({\bf s}) := f_1({\bf s})/f_2({\bf s})$. Then
the multiplicity of
$$
\tilde{g}({\bf s}): = \left(\frac{\partial}{\partial
z}\right)^k
g({\bf s} + z \cdot \gamma) |_{z = - l({\bf s})/l( \gamma)}
$$
at ${\bf 0}$ is at least $\mu(g) - k$.
\label{mult4}
\end{prop}
Let $\Gamma \subset {{\bf Z}}^r$ be a
sublattice, $\Gamma_{{\bf R}} \subset {{\bf R}}^r$ (resp.
$\Gamma_{{\bf C}} \subset {{\bf C}}^r$) the scalar extension of
$\Gamma$ to a {{\bf R}}-subspace
(resp. to a {{\bf C}}-subspace). We always assume that $\Gamma_{{\bf R}}
\cap {{\bf R}}_{\geq 0}^r = 0$. We set $V_{{\bf R}}: = {{\bf R}}^r/\Gamma_{{\bf R}}$ and
$V_{{\bf C}}: = {{\bf C}}^r/\Gamma_{{\bf C}}$. Denote by $\psi$ the
canonical ${{\bf C}}$-linear projection ${\bf C}^r \rightarrow V_{{\bf C}}$.
\begin{dfn}
{\rm A complex analytic function $f({\bf s})=
f(s_1, \ldots, s_r): U \rightarrow {\bf C}$ defined
on an open subset $U \subset {\bf C}^r$
is said to {\em descend to $V_{{\bf C}}$} if for any
vector $ \alpha \in \Gamma_{{\bf C}}$ and any ${\bf u}= (u_1, \ldots, u_r) \in U$
one has
\[ f({\bf u}+ z \cdot \alpha ) = f({\bf u}) \;\; \mbox{\rm for
all $\{ z \in {\bf C}\, :\,{\bf u}+ z \cdot \alpha \in U\}$}.\]
}
\end{dfn}
\begin{rem}
{\rm By definition, if $f({\bf s})$ descends to $V_{{\bf C}}$, then there exists
an analytic function $g$ on $\psi(U) \subset V_{{\bf C}}$ such that
$f = g \circ \psi$. Using Cauchy-Riemann equations, one
immediatelly obtains that $f$ descends to $V_{{\bf C}}$ if and only
if for any vector $ \alpha \in \Gamma_{{\bf R}}$ and
any ${\bf u}= (u_1, \ldots, u_r) \in U$, one has
\[ f({\bf u}+ iy \cdot \alpha ) = f({\bf u})\; \;
\mbox{\rm for all $\{ y \in {\bf R}\, :\,{\bf u}+ iy \cdot \alpha \in
U\}$}. \]}
\label{desc}
\end{rem}
\begin{dfn}
{\rm An analytic function $W({\bf s})$ in the domain
${\rm Re}({\bf s}) \in {{\bf R}}_{>0}^r$ is called
{\em good with respect to $\Gamma$} if it satisfies the following conditions:
{(i)} $W({\bf s})$ descends to $V_{{\bf C}}$;
{(ii)} There exist pairwise coprime linear
homogeneous polynomials
$$
l_1({\bf s}), \ldots,
l_p({\bf s}) \in {{\bf R}}\lbrack {\bf s} \rbrack$$
and positive integers
$k_1, \ldots, k_p$ such that for every $j \in \{1, \ldots, p \}$
the
linear form $l_j({\bf s})$ descends to $V_{{\bf C}}$, $l_j({\bf s})$
does not vanish for ${\bf s} \in {{\bf R}}_{>0}^n$, and
$$
P({\bf s}) = W({\bf s}) \cdot \prod_{j =1}^p l_j^{k_j}({\bf s})
$$
is analytic at ${\bf 0}$.
(iii) There exist a non-zero complex number $C(W)$ and
a decomposition of $P({\bf s})$ into the sum
$$
P({\bf s}) = P_0({\bf s}) + P_1({\bf s})
$$
so that $P_0({\bf s})$ is a homogeneous polynomial of degree $\mu(P)$,
$P_1({\bf s})$ is an analytic function at ${\bf 0}$ with
$\mu(P_1) > \mu(P_0)$, both
functions $P_0$, $P_1$ descend to $V_{{\bf C}}$, and
$$
\frac{P_0({\bf s})}{\prod_{j =1}^p l_j^{k_j}({\bf s})} =
C(W) \cdot {\cal X}_{ \Lambda }(\psi({\bf s})),
$$
where ${\cal X}_{ \Lambda }$ is the ${\cal X}$-function of the cone $ \Lambda =
\psi({{\bf R}}^r_{\geq 0}) \subset V_{{\bf C}}$;
}
\end{dfn}
\begin{dfn}
{\rm If $W({\bf s})$ is a good with respect to $\Gamma$ as above, then the
meromorphic function
$$
\frac{P_0({\bf s})}{\prod_{j =1}^p l_j^{k_j}({\bf s})}
$$
will be called the {\em principal part of $W({\bf s})$ at ${\bf 0}$}
and the non-zero constant $C(W)$ the {\em principal coefficient
of $W({\bf s})$ at ${\bf 0}$}. }
\end{dfn}
Suppose that $\Gamma \neq {{\bf Z}}^r$.
Let $ \gamma \in {{\bf Z}}^r$ be an element which is
not contained in $\Gamma$, $\tilde{\Gamma}: = \Gamma \oplus {\bf Z} < \gamma >$,
$\tilde{\Gamma}_{{\bf R}} := \Gamma_{{\bf R}} \oplus {\bf R} < \gamma >$, $\tilde{V}_{{\bf R}} :=
{{\bf R}}^r /\tilde{\Gamma}_{{\bf R}}$ and $\tilde{V}_{{\bf C}} :=
{{\bf R}}^r /\tilde{\Gamma}_{{\bf C}}$.
The following easy statement will be helpful in the sequel:
\begin{prop}
Let $f({\bf s})$ be an analytic at ${\bf 0}$ function,
$l({\bf s})$ a homogeneous linear function such that $l( \gamma) \neq
0$. Assume
that $f({\bf s})$ and $l({\bf s})$ descend to $V_{{\bf C}}$. Then
$$
\tilde{f}({\bf s}) : = f\left({\bf s} - \frac{l({\bf s})}{l( \gamma)}
\cdot \gamma\right)
$$
descends to $\tilde{V}_{{\bf C}}$.
\label{desc2}
\end{prop}
\begin{theo}
Let $W({\bf s})$ be a good function with respect to $\Gamma$ as above,
$$
\Phi({\bf s}) = \prod_{j\;:\; l_j( \gamma)=0} l_j^{k_j}({\bf s})
$$
the product of those linear
forms $l_j({\bf s})$ $j \in \{ 1, \ldots, p\}$ which vanish on $ \gamma$.
Assume that $\tilde{\Gamma}_{{\bf R}} \cap {{\bf R}}_{\geq 0}^r = 0$
and the following statements hold:
{\rm (i)} The integral
$$
\tilde{W}({\bf s}) : = \int_{{\rm Re}(z) = 0} W({\bf s} + z \cdot \gamma) dz
, \;\; z \in {\bf C}
$$
converges absolutely and uniformly
on any compact in the domain ${\rm Re}({\bf s}) \in {{\bf R}}_{>0}^r$;
{\rm (ii)}
There exists $ \delta > 0$ such that the integral
$$
\int_{{\rm Re}(z) = \delta }
\Phi({\bf s}) \cdot W({\bf s} + z \cdot \gamma) dz
$$
converges absolutely and uniformly
in an open neighborhood of ${\bf 0}$. Moreover, the
multiplicity of the meromorphic function
$$
\tilde{W}_{ \delta }({\bf s}): = \int_{{\rm Re}(z) = \delta }
W({\bf s} + z \cdot \gamma) dz
$$
at ${\bf 0}$ is at least $1 + {\rm rk}\, \tilde{\Gamma} - r$;
{\rm (iii)} For any ${\rm Re}({\bf s}) \in {{\bf R}}_{>0}^r$, the function
$$
\phi(t,{\bf s}) =
\sup_{0\leq {\rm Re}(z) \leq \delta ,\,{\rm Im}(z)=t }|W({\bf s}+ z\cdot \gamma)|
$$
tends to $0$ as $|t| \rightarrow + \infty$.
Then $\tilde{W}({\bf s})$ is a good function with
respect to $\tilde{\Gamma}$, and $C(\tilde{W}) = 2 \pi i \cdot
C(W)$.
\label{desc3}
\end{theo}
\noindent
{\em Proof.} Assume that $l_j( \gamma) < 0$ for $j=1, \ldots, p_1$,
$l_j( \gamma) = 0$ for $j=p_1 +1, \ldots, p_2$, and
$l_j( \gamma) > 0$ for $j=p_2 +1, \ldots, p$. In particular, one has
\[ \Phi({\bf s}) = \prod_{j = p_1 + 1}^{p_2} l_j^{k_j}({\bf s}). \]
Denote by $z_j$ the solution of
the equation
\[
l_j({\bf s}) + z l_j( \gamma) = 0,\;\;j =1, \ldots, p_1.
\]
Let $U$ be the intersection of ${{\bf R}}^r_{>0}$ with an open
neighborhood of ${\bf 0}$ where
$$ \Phi({\bf s}) \cdot
\tilde{W}_{ \delta }({\bf s})
$$
is analytic. Then both functions
$\tilde{W}_{ \delta }({\bf s})$ and $\tilde{W}({\bf s})$
are analytic in $U$. Moreover, the
integral formulas for $\tilde{W}_{ \delta }({\bf s})$ and $\tilde{W}({\bf s})$
show that the equalities
$\tilde{W}_{ \delta }({\bf u}+ iy \cdot \gamma) =\tilde{W}_{ \delta }({\bf u})$
and $\tilde{W}({\bf u}+ iy \cdot \gamma) =\tilde{W}({\bf u})$ hold
for any $y \in {\bf R}$ and ${\bf u},{\bf u}+ iy \cdot \gamma \in U$. Therefore,
both functions
$\tilde{W}_{ \delta }({\bf s})$ and $\tilde{W}({\bf s})$
descend to $\tilde{V}_{\bf C}$ (see Remark \ref{desc}).
Using assumptions (i)-(iii) of Theorem, we
can apply the residue theorem and obtain
\[ \tilde{W}({\bf s}) - \tilde{W}_{ \delta }({\bf s}) = 2 \pi i\cdot
\sum_{j=1}^{p_1} {\rm Res}_{z = z_j} W({\bf s} + z \cdot \gamma)\]
for ${\bf s} \in U$.
We denote by $U( \gamma)$ the open subset of $U$ defined by the
inequalities
$$
\frac{l_j({\bf s})}{l_j( \gamma)} \neq \frac{l_{m}({\bf s})}{l_{m}( \gamma)}\;\;
\mbox{\rm for all $j \neq m$, $\;\;j,m \in \{ 1, \ldots, p\}$.}
$$
The open set $U( \gamma)$ is non-empty, since we assume that
$g.c.d.(l_j, l_{m})=1$ for $j \neq m$.
For ${\bf s} \in U( \gamma)$, we have
$$
{\rm Res}_{z = z_j} W({\bf s} + z \cdot \gamma) =
\frac{1}{(k_j-1)!}
\left( \frac{\partial}{\partial z} \right)^{k_j-1}
\frac{l_{j}({\bf s} + z \cdot \gamma)^{k_j}
P({\bf s} + z \cdot \gamma)}{l_j^{k_j}
( \gamma) \cdot \prod_{m =1}^p
l_{m}^{k_m}({\bf s} + z \cdot \gamma) }|_{z = z_j},
$$
where
$$
z_j = - \frac{l_j({\bf s})}{l_j( \gamma)}.
$$
Let
$$
W({\bf s}) \cdot \prod_{j =1}^p l_j^{k_j}({\bf s})
= {P}({\bf s}) = {P}_0({\bf s}) + {P}_1({\bf s}),
$$
where ${P}_0({\bf s})$ is a uniquely determined homogeneous polynomial
and ${P}_0({\bf s})$ is an analytic at ${\bf 0}$ function
such that $\mu({P}) = \mu({P}_0) < \mu({P}_1)$ and
$$
\frac{P_0({\bf s})}{\prod_{j =1}^p l_j^{k_j}({\bf s})} =
C(W) \cdot {\cal X}_{ \Lambda }({\bf s})
$$
(${\cal X}_{ \Lambda }({\bf s})$ is the ${\cal X}$-function of the cone $ \Lambda =
\psi({{\bf R}}^r_{\geq 0})$). We set
$$
R_0({\bf s}) : = \frac{P_0({\bf s})}{\prod_{j =1}^p {l}^{k_j}_j({\bf
s})}, \;\;
R_1({\bf s}) : = \frac{P_1({\bf s})}{\prod_{j =1}^p {l}_j^{k_j}({\bf
s})}.
$$
Then $\mu(W)= \mu (R_0) < \mu (R_1)$. Moreover,
$\mu(W) = - {\rm dim} V_{{\bf R}} = r - {\rm rk}\, \Gamma$.
Define
$$
\tilde{R}_0({\bf s}):= 2\pi i \cdot
\sum_{j=1}^{p_1} {\rm Res}_{z = z_j}
R_0({\bf s}+ z\cdot \gamma)
$$
and
$$
\tilde{R}_1({\bf s}):=
2\pi i \cdot \sum_{j=1}^{p_1} {\rm Res}_{z = z_j}
R_1({\bf s}+ z\cdot \gamma).
$$
By Proposition \ref{mult4}, we
have
$\mu (\tilde{R}_1) \geq 1+ \mu(R_1) \geq 2 + \mu(R_0)= 1+ {\rm
rk}\, \tilde{\Gamma} - r $.
We claim
$$
\tilde{R}_0({\bf s}) = 2 \pi i \cdot C(W) {\cal X}_{\tilde{ \Lambda }}
(\tilde{\psi}({\bf s}))
$$
in particular $\mu (\tilde{R}_0) = \mu(R_0) + 1 = {\rm
rk}\, \tilde{\Gamma}$. Indeed, repeating for ${\cal X}_{ \Lambda }(\psi({\bf s}))$
the same arguments as for $W({\bf s})$ we obtain
$$
\int_{{\rm Re}(z) = 0} {\cal X}_{ \Lambda }(\psi({\bf s} + z \cdot \gamma)) dz
- \int_{{\rm Re}(z) = \delta } {\cal X}_{ \Lambda }(\psi({\bf s} + z \cdot \gamma)) dz
$$
$$
= 2\pi i \cdot \sum_{j=1}^{k_1} {\rm Res}_{z = z_j}
{\cal X}_{ \Lambda }(\psi({\bf s} + z_j \cdot \gamma)).
$$
Moving the contour of integration
${\rm Re}(z) = \delta $, by residue theorem,
we obtain
$$
\int_{{\rm Re}(z) = \delta } {\cal X}_{ \Lambda }(\psi({\bf s} + z \cdot \gamma)) dz =0.
$$
On the other hand,
$$
{\cal X}_{\tilde{ \Lambda }}(\tilde{\psi}({\bf s})) =
\frac{1}{2\pi i}\int_{{\rm Re}(z) = 0}
{\cal X}_{ \Lambda }(\psi({\bf s} + z \cdot \gamma)) dz
$$
(see Theorem \ref{char1}).
By \ref{mult2}(iii), using the decomposition
$$
\tilde{W}({\bf s}) = \tilde{W}_{ \delta }({\bf s}) + \tilde{R}_0({\bf
s}) + \tilde{R}_1({\bf s})
$$
and our assumption $\mu(\tilde{W}_{ \delta }) \geq
1 + {\rm rk}\, \tilde{\Gamma} - r$,
we obtain that $\mu (\tilde{W}) = \mu (\tilde{R}_0) = {\rm
rk}\, \tilde{\Gamma} - r$.
By \ref{desc2}, the linear forms
\[ h_{m,j}({\bf s}):= l_{m}({\bf s} + z_j \cdot \gamma)
= l_{m}({\bf s}) -
\frac{l_j({\bf s})}{l_{j}( \gamma)} l_{m}( \gamma) \]
and the analytic in the domain $U( \gamma)$ functions
\[ {\rm Res}_{z = z_j} W({\bf s} + z \cdot \gamma), \;\;
{\rm Res}_{z = z_j} R_0({\bf s} + z \cdot \gamma) \]
descend to $\tilde{V}_{{\bf C}}$.
For any $j \in \{ 1, \ldots, p_1\}$, let us denote
$$
Q_j ({\bf s}) = \prod_{m \neq j, m=1}^p h_{m,j}^{k_m}({\bf s}).
$$
It is clear that
$$
Q_j^{k_j}({\bf s}) \cdot {\rm Res}_{z = z_j} W({\bf s} + z \cdot \gamma)\;
\;\mbox{\rm and}\;\;
Q_j^{k_j}({\bf s}) \cdot {\rm Res}_{z = z_j} R_0({\bf s} + z \cdot \gamma)
$$
are analytic at ${\bf 0}$ and $\Phi({\bf s})$ divides
each $Q_j ({\bf s})$. So we obtain that
$$
\tilde{W}({\bf s}) \prod_{j=1}^{p_1} Q_j^{k_j}({\bf s})
= \left( \tilde{W}_{ \delta }({\bf s}) +
2 \pi i\cdot
\sum_{j=1}^{p_1} {\rm Res}_{z = z_j} W({\bf s} + z \cdot \gamma) \right)
\prod_{j=1}^{p_1} Q_j^{k_j}({\bf s})
$$
and
$$
\tilde{R}_0({\bf s}) \prod_{j=1}^{p_1} Q_j^{k_j}({\bf s})
= \left( 2 \pi i\cdot
\sum_{j=1}^{p_1} {\rm Res}_{z = z_j} R_0({\bf s} + z \cdot \gamma) \right)
\prod_{j=1}^{p_1} Q_j^{k_j}({\bf s})
$$
are analytic at ${\bf 0}$.
Define the set $\{ \tilde{l}_1({\bf s}), \ldots, \tilde{l}_q({\bf s}) \}$ as
a subset of pairwise coprime elements
in the set of homogeneous linear forms $\{ h_{m,j}({\bf s}) \}$ $(m
\in \{1, \ldots, p\}, \; j \in \{1, \ldots, p_1\})$ such that
there exist positive integers $n_1, \ldots, n_q$ and a
representation of the meromorphic functions $\tilde{W}({\bf s})$
and $\tilde{R}_0({\bf s})$
as quotients
\[ \tilde{W}({\bf s}) =
\frac{\tilde{P}({\bf s})}{\prod_{j =1}^q \tilde{l}^{n_j}_j({\bf s})},\;\;
\tilde{R}_0({\bf s}) =
\frac{\tilde{P}_0({\bf s})}{\prod_{j =1}^q \tilde{l}^{n_j}_j({\bf s})},\]
where $\tilde{P}({\bf s})$ is analytic at ${\bf 0}$,
$\tilde{P}_0({\bf s})$ is a homogeneous polynomial and
none of the forms
$\tilde{l}_1({\bf s}), \ldots,
\tilde{l}_q({\bf s})$ vanishes for $ {\bf s} \in {{\bf R}}_{>0}^r$
(the last property can be achieved, because both functions
$\tilde{W}({\bf s})$ and $\tilde{R}_0({\bf s})$ are analytic in $U$).
Define
$$
\tilde{P}_1({\bf s}) = \left( \tilde{W}_{ \delta }({\bf s}) +
\tilde{R}_1({\bf s}) \right)
\cdot \prod_{j =1}^q \tilde{l}^{n_j}_j({\bf s}).
$$
Then
$$
\tilde{P}({\bf s}) = \tilde{P}_0({\bf s}) +
\tilde{P}_1({\bf s})
$$
where $\tilde{P}_0({\bf s})$ is a homogeneous polynomial
and $\tilde{P}_1({\bf s})$ is an analytic at ${\bf 0}$ function
such that $\mu(\tilde{P}) = \mu(\tilde{P}_0) < \mu(\tilde{P}_1)$ and
$$
\frac{\tilde{P}_0({\bf s})}{\prod_{j =1}^q
\tilde{l}_j^{n_j}({\bf s})} =
2\pi i \cdot C(W) \cdot
{\cal X}_{\tilde{ \Lambda }}(\tilde{\psi}({\bf s})).
$$
\section{Main theorem}
Let us set
$$W_{ \Sigma }({\bf s}) :=
Z_{ \Sigma }(\varphi_{\bf s} + \varphi _{ \Sigma }) = Z_{ \Sigma }(s_1 +1, \ldots, s_r +1).$$
By Theorem \ref{convergence}, $W_{ \Sigma }({\bf s})$ is
an analytic function in the
domain ${\rm Re}({\bf s}) \in {\bf R}^r_{>0}$.
\begin{theo}
The analytic function $W_{ \Sigma }({\bf s})$
is good with respect to the lattice
$M^G \subset PL( \Sigma )^G = {\bf Z}^r$.
\label{analytic.cont}
\end{theo}
\noindent
{\em Proof.}
By Theorem \ref{poiss}, we have the following integral
representation for $Z_{ \Sigma }({\bf s}) $ in the domain
${\rm Re}({\bf s}) \in {\bf R}^r_{>1}$
$$
Z_{\Sigma}({\bf s})=\frac{1}{(2\pi )^t b_S(T)}
\int_{(T({\bf A}_K)/T(K))^*}\hat{H}_{ \Sigma }(\chi, -{\bf s})d\chi
$$
We need only to consider characters $\chi$
which are trivial
on the maximal compact subgroup
${\bf K}_T\subset T^1({\bf A}_K)$, because
for all other characters the Fourier transform
$\hat{H}_{ \Sigma }(\chi, -{\bf s})$ vanishes.
Choosing a non-canonical splitting of characters
corresponding to some
splitting of the sequence
$$
0 \rightarrow T^1({\bf A}_K) \rightarrow T({\bf A}_K) \rightarrow T({\bf A}_K)/T^1({\bf A}_K) \rightarrow 0
$$
we obtain
$$
Z_{\Sigma}({\bf s})=\frac{1}{(2\pi )^t b_S(T)}
\int_{(T({\bf A}_K)/T^1({\bf A}_K))^*}d\chi_y
\int_{(T^1({\bf A}_K)/T(K))^*}\hat{H}_{ \Sigma }(\chi, -{\bf s})d\chi_l
$$
We have an isomorphism
$M^G_{{\bf R}}\simeq(T({\bf A}_K)/T^1({\bf A}_K))^*$ and the
measure $d\chi_y $ coincides with the usual Lebesgue measure
on $M^G_{{\bf R}}$.
Recall that a character $\chi\in (T({\bf A}_K)/T(K))^*$
defines $r$ Hecke characters $\chi_1,...,\chi_r$
of the idele groups ${\bf G}_m({\bf A}_{K_j})$.
In particular, we get $r$ characters $\chi_{1,y},...,\chi_{r,y}$.
We have an embedding $M^G\subset PL( \Sigma )^G$,
which together with explicit formulas for Fourier transforms
of local heights shows that the integral
$$
A_{ \Sigma }({\bf s},\chi_y):=\frac{1}{b_S(T)}
\int_{(T^1({\bf A}_K)/T(K))^*}
\hat{H}_{ \Sigma }(\chi, -({\bf s}+{\bf 1}))d\chi_l
$$
is a function on $PL( \Sigma )^G_{{\bf C}}$ and we have
$$
A_{ \Sigma }({\bf s},\chi_y)=A_{ \Sigma }({\bf s}+i{\bf y})= A_{ \Sigma }(s_1+iy_1,...,s_r+iy_r).
$$
Denote by $\Gamma:=M^G$ the lattice of $K$-rational characters
of $T$. Let $t$ be the rank of $\Gamma$.
The case $t =0$ corresponds to
an anisotropic torus $T$. It has been considered already in
\cite{BaTschi}. So we assume $t >0$.
For any element $ \gamma \in \Gamma\subset {{\bf Z}}^r$ we
denote by $l( \gamma)$ the number of its
coordinates which are zero $(0 \leq l( \gamma) \leq r)$.
Let $l(\Gamma)$ be the minimum of $l( \gamma)$ among $ \gamma \in \Gamma$.
Notice that
$l(\Gamma) \leq r - t - 1$. Indeed, if we had
$l(\Gamma) \geq r -t$, then $M^G$ would be contained in
the intersection of $r - t$ linear coordinate hyperplanes
$s_j = 0$ (the latter contradicts the condition $M^G_{{\bf R}} \cap
{\bf R}^r_{\geq 0} = 0$).
We can always choose a
${{\bf Z}}$-basis $ \gamma^1, \ldots , \gamma^t$ of $\Gamma$ in such a
way that $l(\Gamma) = l( \gamma^u)$ $(u =1, \ldots, t)$. Without loss
of generality we assume
that $\Gamma$ is contained in the intersection of coordinate
hyperplanes $s_j = 0$, $j \in \{1, \ldots, l(\Gamma) \}$.
We set
$$\Phi({\bf s}) := \prod_{j =1}^{l(\Gamma)} s_j. $$
For any $u \leq t$ we define a subgroup $\Gamma^{(u)} \subset
\Gamma$ of rank $u$ as
$$\Gamma^{(u)}:= \bigoplus_{j =1}^u {{\bf Z}}< \gamma_j>.$$
We introduce
some auxiliary functions
$$
W^{(u)}_{ \Sigma }({\bf s}) =
\int_{\Gamma^{(u)}_{{\bf R}}}
A_{ \Sigma }({\bf s}+i{\bf y}^{(u)}){\bf dy}^{(u)}
$$
where $ {\bf dy}^{(u)}$ is the induced measure
on $\Gamma^{(u)}_{{\bf R}}\subset PL( \Sigma )^G_{{\bf R}}$.
Denote $V^{(u)}_{{\bf C}} = {{\bf C}}^r/\Gamma_{{\bf C}}^{(u)}$.
We prove by induction
that $W^{(u)}_{ \Sigma }({\bf s})$ is good with respect to
$\Gamma^{(u)} \subset {{\bf Z}}^r$.
By \ref{infconver}, $W^{(u)}_{ \Sigma }({\bf s})$ is an analytic function
in the domain ${\rm Re}({\bf s}) \in {{\bf R}}^r_{>0}$.
There exist $ \delta _1,..., \delta _t > 0$ such that the
integral
$$
\int_{{\rm Re}(z) = \delta _u}
\Phi({\bf s}) \cdot W_{ \Sigma }^{(u-1)}({\bf s} + z \cdot \gamma^u) dz
$$
converges absolutely and uniformly
in an open neighborhood of ${\bf 0}$.
This can be seen as follows:
For any $ \varepsilon $ with $0< \varepsilon <1/rd'$,
where $d'=\dim M_{{\bf R},\infty}$, we can
choose a
ball $B_{e_1}\subset {\bf R}$ of radius $e_1$ around ${\bf 0}$ such that
for any ball
$B_{e_2}\subset B_{e_1} $ of radius $e_2$ ($0<e_2 <e_1$)
around $0$ we can uniformly bound the Hecke $L$-functions
$L_{K_j}(s_j+1,\chi_j)$
appearing in $\hat{H}_{ \Sigma }(\chi,{\bf s})$ by
$$
c_j(e_2 )(\|y(\chi_j)\|+ |{\rm Im}(s_j)|+1)^{ \varepsilon }
$$
with some constants $c_j(e_2 )$ for all
${\bf s}$ in the domain ${\rm Re}(s_j)\in B_{e_1}\backslash B_{e_2}$
for $j=1,...,r$ (see \ref{estim}).
By \ref{infconver}, this assures the
absolute and uniform convergence
of the integral
$$
\int_{\Gamma^{(u)}_{{\bf R}}}
A_{ \Sigma }({\bf s}+i{\bf y}^{(u)}){\bf dy}^{(u)}
$$
for all ${\bf s}$ contained
in a compact in ${\bf C}^r$ such that
${\rm Re}(s_j)\in B_{ \varepsilon _1}\backslash B_{ \varepsilon _2}$
for $j=1,...r$.
We know that the coordinates $ \gamma_j^u$ of
the vectors $ \gamma^u=( \gamma_1^u,..., \gamma^u_r)\in {\bf Z}^r$
are not equal to zero for $l(\Gamma )<j \le r$.
Therefore, we can now choose such real
$ \delta _u>0$ that $ \delta _u \gamma^u_j$
are all contained in the {\em open} ball $B_{e_1}$.
So there must exist some $e_2>0$ such that
$ \delta _u \gamma^u_j\not\in B_{e_2}$ for all $u=1,...,t$ and all
$l(\Gamma )<j\le r$. It follows that there
exists an open neighborhood of
${\bf 0}$, such that for all
${\bf s}$ contained in this neighborhood
we have ${\rm Re}(s_j+ \delta _u \gamma^u_j)\in
B_{e_1}\backslash B_{e_2}$ for all
$l(\Gamma )<j\le r$. Since we remove the remaining poles
by multiplying with
$\Phi({\bf s})$ we obtain the absolute
and uniform convergence of $W^{(u)}_{ \Sigma }({\bf s})$ to a
holomorphic function in ${\bf s}$ in this neighborhood.
Moreover, the
multiplicity of the meromorphic function
$$
\tilde{W}_{ \delta _u}^{(u)}({\bf s}): = \int_{{\rm Re}(z) = \delta _u}
W^{(u-1)}_{ \Sigma }({\bf s} + z \cdot \gamma_u) dz
$$
at ${\bf 0}$ is at least $1 + {\rm rk}\, {\Gamma} - r \geq 1 +
{\rm rk}\, {\Gamma}^{(u)} - r$.
We apply Theorem \ref{desc3}. The property (iii) follows from
estimates \ref{m.estim} and \ref{infconver}.
This concludes
the proof.
\hfill $\Box$
\begin{theo} Denote by $\hat{H}_{ \Sigma ,S}(\chi,-{\bf s})$
the multiplicative Fourier transform of the height function
with respect to the measure $\omega_{\Omega, S}$ (see \ref{can.meas}).
The principal coefficient $C( \Sigma )$
of
$$
A_{ \Sigma }({\bf s})=\frac{1}{b_S(T)} \int_{(T^1({\bf A}_K/T(K){\bf K}_T)^*}
\hat{H}_{ \Sigma ,S}(\chi_l,-{\bf s})d\chi_l
$$
at $s_1=...=s_r=1$ is equal to
$\beta({\bf P}_{ \Sigma }) \tau_{\cal K}({\bf P}_{ \Sigma })$.
\label{beta.tau}
\end{theo}
{\em Proof.} We follow closely the exposition
of the proof of theorem 3.4.6 in \cite{BaTschi}.
Since $M^G\hookrightarrow PL( \Sigma )^G$ we have an embedding
of characters
$$
(T({\bf A}_K)/T^1({\bf A}_K))^* = M^G_{{\bf R}}\hookrightarrow
\prod_{j=1}^r ({\bf G}_m({\bf A}_{K_j})/{\bf G}_m^1({\bf A}_{K_j}))^*.
$$
Recall that the kernel of
$$
a^*\,:\, (T({\bf A}_K)/T({\bf A}_K))^* \rightarrow
\prod_{j=1}^r ({\bf G}_m({\bf A}_{K_j})/{\bf G}_m({\bf A}_{K_j}))^*
$$
is dual to the obstruction group to weak approximation
$A(T)=T({\bf A}_K)/\overline{T(K)}$.
We have a splitting
$$
\overline{T(K)} = \overline{T(K)}_S \times T(A_{K,S}).
$$
Here we denoted by
$\overline{T(K)}_S$ the image of $\overline{T(K)}$ in
$\prod_{v \in S} T(K_v)$ and
$T(A_{K,S})=T({\bf A}_K)\cap \prod_{v\not\in S}T(K_v)$.
The pole of the highest order
$r$ of $\hat{H}_{ \Sigma ,S}(\chi_l,-{\bf s}) $ at $s_1=...=s_r=1$ appears
from characters $\chi_l$ such that the corresponding
$\chi_1,...,\chi_r$
are trivial characters of the groups
${\bf G}_m({\bf A}_{K_j})/{\bf G}_m({K_j})$,
i.e., $\chi_l$ is a character of the finite group
$A(T)=\prod_{v\in S}T(K_v)/\overline{T(K)}_S$,
and is trivial on the group
$T({\bf A}_{K,S})$.
For ${\bf s}\in {\bf R}_{>1}^r$ we can again apply the Poisson
formula to $A(T)$. By \ref{weak1}, the order of $A(T)$ equals
$\beta({\bf P}_{ \Sigma })/i(T)$. We obtain
$$
\frac{1}{b_S(T)}
\sum_{\chi \in (A(T))^* }
\hat{H}_{ \Sigma ,S}(\chi_l, - {\bf s})=
\frac{\beta({\bf P}_{\Sigma})}{i(T) b_S(T)}
\int_{\overline{T(K)}} H_{ \Sigma }(x,-{\bf s})
\omega_{\Omega,S} \]
(see \ref{weak1}).
We restrict to the line $s_1=...=s_r=s$ and we want to compute the
limit
$$
\lim_{s \rightarrow 1} (s-1)^r \int_{\overline{T(K)}} H_{ \Sigma }(x,-{\bf s})
\omega_{\Omega,S}.
$$
We have
\begin{equation}
\int_{\overline{T(K)}} H_{ \Sigma }(x,-{\bf s})
\omega_{\Omega,S} =
\label{const1}
\end{equation}
$$
= \int_{\overline{T(K)}_S}
\prod_{v \in S} H_{ \Sigma ,v}(x_v,-{\bf s}) \omega_{\Omega,v}
\cdot
\prod_{v \not\in S} \int_{T(K_v)} H_{ \Sigma ,v}(x_v,-{\bf s}) d\mu_v
$$
(recall that $\omega_{\Omega,v}=\prod_{v\in {\rm Val}(K)} d\mu_v$
and
$d\mu_v = L_v(1,T;E/K) \omega_{\Omega,v}$ for all
$v$ and $L_v(1,T;E/K) =1$ for $v \in S$).
{}From our calculations of the Fourier transform of local
height functions for $v \not\in S$ (\ref{loc-int}),
we have
\begin{equation}
\prod_{v \not\in S} \int_{T(K_v)}
H_{ \Sigma ,v}(x_v,-{\bf s}) d\mu_v
=
\label{const2}
\end{equation}
\[ = L_S(s, T;E/K)
\cdot L_S(s, T_{NS}; E/K) \prod_{v \not\in S}
Q_{ \Sigma }(q_v^{-s}, \ldots, q_v^{-s}). \]
By \ref{p-function},
\[ \prod_{v \not\in S}
Q_{ \Sigma }(q_v^{-s}, \ldots, q_v^{-s}) \]
is an absolutely convergent Euler product for $s =1$.
Moreover, the limits
$$
\lim_{s \rightarrow 1} (s-1)^t L_S(s, T;E/K)
$$
$$
\lim_{s \rightarrow 1} (s-1)^{(r-t)} L_S(s, T_{NS};E/K)
$$
exist and equal the non-zero constants
$l_S(T)$ and $l^{-1}_S({\bf P}_{ \Sigma })$ ($r=t+k$).
By \ref{badreduction},
\[ \int_{\overline{T(K)}_S}
\prod_{v \in S} H_{ \Sigma ,v}(x_v,-{\bf s}) \omega_{\Omega,v} \]
is absolutely convergent for $s_1=,,,=s_r=1$.
Using (\ref{const1}) and (\ref{const2}), we obtain:
\begin{equation}
\lim_{s \rightarrow 1} (s-1)^r
\int_{\overline{T(K)}} H_{ \Sigma }(x,-{\bf s})
\omega_{\Omega,S} =
\label{const3}
\end{equation}
\[ = \frac{l_S(T)}{l_S({\bf P}_{ \Sigma })}
\int_{\overline{T(K)}_S}
\prod_{v \in S} H_{ \Sigma ,v}(x_v,-{\bf s}) \omega_{\Omega,v} \cdot
\prod_{v \not\in S}
Q_{ \Sigma }(q_v^{-1}, \ldots, q_v^{-1}).\]
Now recall (\ref{loc-int}),
that for $v\not\in S$ we have
$$
Q_{ \Sigma }(q_v^{-1}, \ldots, q_v^{-1})=
\int_{T(K_v)}L_v^{-1}(1,T_{NS};E/K)
H_{ \Sigma ,v}(x_v,-{\bf 1})\omega_{\Omega,v}.
$$
It was
proved in \cite{BaTschi} Proposition 3.4.4
that the restriction of the $v$-adic
measure $\omega_{{\cal K},v}$ to $T(K_v) \subset
{\bf P}_{\Sigma}(K_v)$ coincides with the measure
\[ H_{ \Sigma ,v}(x, -{\bf 1}) \omega_{\Omega,v}. \]
Here ${\cal K}$ is the
canonical sheaf on the toric variety ${\bf P}_{\Sigma}$ metrized
as above.
We also have
\begin{equation}
\int_{\overline{T(K)}_S}
\prod_{v \in S} H_{ \Sigma ,v}(x_v,-{\bf 1}) \omega_{\Omega,v} =
\int_{\overline{T(K)}_S}
\prod_{v \in S} \omega_{{\cal K},v}.
\label{const6}
\end{equation}
Using the splitting
$\overline{T(K)} = \overline{T(K)}_S \times T(A_{K,S})$
and multiplying the above equations we get
$$
\int_{\overline{T(K)}} \omega_{{\cal
K},S} = \int_{\overline{T(K)}_S} \prod_{v \in S} \omega_{{\cal K},v}
\cdot \prod_{v \not\in S} \int_{{T(K_v)}}
L_v^{-1}(1,T_{NS};E/K) \omega_{{\cal K},v}.
$$
On the other hand, it was proved in \cite{BaTschi} Proposition
3.4.5 that we have
\[ \int_{\overline{T(K)}} \omega_{{\cal
K},S} = \int_{\overline{{\bf P}_{ \Sigma }(K)}}
\omega_{{\cal K},S} = b_S({\bf P}_{ \Sigma }). \]
Therefore,
\[ b_S({\bf P}_{ \Sigma }) =
\int_{\overline{T(K)}_S}
\prod_{v \in S} H_{ \Sigma }(x,-\varphi_{\Sigma}) \omega_{\Omega,v}
\cdot \prod_{v \not\in S}
Q_{ \Sigma }(q_v^{-1}, \ldots, q_v^{-1})
.\]
Collecting terms, we obtain
$$
C( \Sigma )= \frac{\beta({\bf P}_{ \Sigma })}{i(T) b_S(T)} \cdot
\frac{l_S(T)}{l_S({\bf P}_{ \Sigma })} \cdot b_S({\bf P}_{ \Sigma }).
$$
By \ref{tamagawa1} and \ref{tamagawa}, we have
the following equality
$$
i(T) b_S(T) = h(T) l_S(T).
$$
It remains to notice that
we have an exact sequence of lattices
$$
0 \rightarrow M^G \rightarrow PL( \Sigma )^G \rightarrow {\rm Pic}({\bf P}_{ \Sigma }) \rightarrow H^1(G,M) \rightarrow 0
$$
and that the number $h(T)= |H^1(G,M)|$
appears in the integral formula
for the ${\cal X}$-function of the
cone ${ \Lambda }_{\rm eff}\subset {\rm Pic}({\bf P}_{ \Sigma })$.
We apply Theorem \ref{char0} and obtain that
$$
W_{ \Sigma }({\bf s})=\frac{1}{(2\pi)^t b_S(T)}
\int_{M^G_{{\bf R}}}A_{ \Sigma }({\bf s}+i{\bf y}){\bf dy}
$$
is good with respect to the lattice $M^G$ and that
$$
C( \Sigma )= \beta( {\bf P}_{ \Sigma })\tau_{\cal K}({\bf P}_{ \Sigma })
$$
is the principal coefficient of $W_{ \Sigma }({\bf s})$ at ${\bf 0}$.
\hfill $\Box $
\begin{theo} There exists a $\delta >0$ such that the
height zeta-function $\zeta_{ \Sigma }(s)$ obtained by restiction of
the zeta-function
$Z_{ \Sigma }({\bf s})$ to the complex line $s_j = \varphi (e_j)=s$ for
all $j=1,...,r$ has a representation of the form
$$
\zeta_{ \Sigma }(s)= \frac{\Theta( \Sigma )}{(s-1)^k} + \frac{g(s)}{(s-1)^{k-1}}
$$
with $k= r-t = {\rm rk}\, {\rm Pic}({\bf P}_{ \Sigma })$ and some
holomorphic function $g(s)$ in the domain
${\rm Re}(s)>1-\delta$. Moreover, we have
$$
\Theta(\Sigma) = \alpha({\bf P}_{\Sigma})\beta({\bf P}_{\Sigma})
\tau_{\cal K}({\bf P}_{\Sigma}).
$$
\end{theo}
{\it Proof.} Since $W_{ \Sigma }({\bf s})$ is good with respect to the
lattice $M^G \subset {{\bf Z}}^r$, we have the following representation
of $W_{ \Sigma }({\bf s})$ in a small open neighborhood of ${\bf 0}$:
$$
W_{ \Sigma }({\bf s}) = \frac{ P({\bf s})}{\prod_{j =1}^p l_j^{k_j}({\bf s})}
$$
where $P({\bf s}) = P_0({\bf s}) + P_1({\bf s})$, $\mu(P_1) >
\mu(P_0)$ and
$$
\frac{P_0({\bf s})}{\prod_{j =1}^p l_j^{k_j}({\bf s})} =
\beta({\bf P}_{ \Sigma }) \tau_{\cal K}({\bf P}_{ \Sigma })
\cdot {\cal X}_{ \Lambda _{\rm eff}}(\psi({\bf s})),
$$
where ${\cal X}_{ \Lambda _{\rm eff}}$ is the ${\cal X}$-function of the cone
$ \Lambda _{\rm eff} = \psi({{\bf R}}^r_{\geq 0}) \subset {\rm Pic}({\bf P}_{ \Sigma })_{{\bf R}}$.
If we restrict
$$
\frac{P_0({\bf s})}{\prod_{j =1}^p l_j^{k_j}({\bf s})}
$$
to the line
$s_j = s - 1$ $(j =1, \ldots, r)$, then
we get the meromorphic function $\Theta (s-1)^{-k}$ with
$\Theta = \alpha({\bf P}_{\Sigma})\beta({\bf P}_{\Sigma})
\tau_{\cal K}({\bf P}_{\Sigma})$.
Moreover, the order of the pole at $s =1$
of the restriction of
$$
\frac{P_1({\bf s})}{\prod_{j =1}^p l_j^{k_j}({\bf s})}
$$
to the line
$s_j = s - 1$ $(j =1, \ldots, r)$ is less than $k$.
Therefore, this restriction
can be written as $g(s)/(s-1)^{k-1}$
for some analytic at $s =1 $ function
$g(s)$.
\begin{coro}
{\rm
Let $T$ be an algebraic
torus and ${\bf P}_{\Sigma}$ its smooth
projective compactification.
Let $k$ be
the rank of ${\rm Pic}({\bf P}_{\Sigma})$.
Then the number
of $K$-rational points $x \in T(K)$ having the anticanonical
height $H_{{\cal K}^{-1}}(x) \leq B$ has the asymptotic
\[ N(T,{\cal K}^{-1}, B) = \frac{\Theta(\Sigma)}{(k-1)!}
\cdot B (\log B)^{k-1}(1+o(1)), \hskip 0,3cm B \rightarrow \infty.\]
}
\end{coro}
\noindent
{\em Proof.}
We apply a Tauberian theorem to $\zeta_{ \Sigma }(s)$.
\hfill $\Box$
|
1995-10-23T00:11:40 | 9510 | alg-geom/9510012 | en | https://arxiv.org/abs/alg-geom/9510012 | [
"alg-geom",
"math.AG"
] | alg-geom/9510012 | Selman Akbulut | Selman Akbulut (Michigan State University) | Lectures on Seiberg-Witten Invariants | AMSLaTeX, 26 pages with 1 figure | null | null | null | null | These are yet another lecture notes on Seiberg-Witten invariants, where no
claim of originality is made, they contain a discussion of some related results
from the recent literature.
| [
{
"version": "v1",
"created": "Sun, 15 Oct 1995 18:09:27 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Akbulut",
"Selman",
"",
"Michigan State University"
]
] | alg-geom | \section{Introduction}
Every compact oriented smooth $4$-manifold has a $Spin_{c}$ structure, i.e. the
second Steifel-Whitney $w_{2}(X)\in H^{2}(X;\Z_{2})\;$ has an integral
lifting. This is because:
$w_{2}(X)$ can be represented by an imbedded surface $F\subset X$. If $F$ is
orientable then clearly the homology class
$[F]$ comes from an integral class; if not then it suffices to show the circle
$S\subset F$ representing $w_{1}(F)$ is null homologous in $H_{1}(X;\Z)$,
because the Bockstein $\delta[F]=[S]$ in the coefficient exact sequence:
$$ ..\to H_{2}(X;\Z)\stackrel{\times 2}{\longrightarrow}
H_{2}(X;\Z)\stackrel{\rho}{\longrightarrow}
H_{2}(X;\Z_{2})\stackrel{\delta}{\to}
H_{1}(X;\Z)\to..$$ where $\rho $ is the reduction map. Now if
$\delta[F]\neq 0$, we
can choose an imbedded oriented $3$-manifold $\Sigma \subset X$ representing
the
Poincare dual of
$\delta[F]$,
which is transverse to $F$. Then $T=F\cap \Sigma \subset F$ has a trivial
normal
bundle $\nu (T,F)$ since
$$\nu (T,X) = \nu (T,F)\oplus \nu (T,\Sigma)$$ and the two other normal
bundles in
the above equality are trivial. This gives a contradiction, since in $F$ the
$1$-manifold $T$ meets $S$ transversally at one point and $[S]=w_{1}(F)$
implies
$\nu (T,F)$ must necessarily be nontrivial $\;\;\;\Box$.
\vspace{.05in}
\begin{eqnarray*}\mbox{Recal:}\;\;\;\;\;\;\; Spin(4)&=&SU (2)\times SU (2)\\
Spin_{c}(4)&=&(\;SU (2)\times SU(2)\times S^{1}\;)/\Z_{2} =(\;Spin(4)\times
S^{1})/\Z_{2}\\ SO(4)&=&(\;SU (2)\times SU (2)\;)/\Z_{2}\\ U(2)&=&(\;SU
(2)\times S^{1}\;)/\Z_{2}\end{eqnarray*}
\noindent We have fibrations:
$$S^{1}\longrightarrow Spin_{c}(4)\to SO(4)$$
$$\Z_{2}\longrightarrow Spin_{c}(4)\to SO(4)\times S^{1}$$
\vspace{.01in}
\noindent We can also identify
$Spin_{c}(4)=\{ (A,B)\in U_{2}\times U_{2}\; |\; det(A)=det(B)\;\}$ by
$$(A,B)\leadsto (A.(det A)^{-1/2}\; I\;,\; B.(det B)^{-1/2}\; I\;,\; (det
A)^{1/2}
)$$
We also have $2$ fold cover $Spin_{c}(4)\to SO(4)\times S^{1}$ .The
fibrations above
extend to fibrations:
$$S^{1}\to Spin_{c}(4)\to SO(4)\to K(\Z,2)\to BSpin_{c}(4)\to BSO(4)\to
K(\Z,3)$$ The last map in the sequence is given by the Bokstein of the second
Steifel-Whitney class
$\delta (w_{2})$ which explains why lifting of $w_{2}$ to an integral class
corresponds to a $Spin_{c}(4)$-structure. We also have the fibration:
$$\Z_{2}\to Spin_{c}(4)\to SO(4)\times S^{1}\to K(\Z_{2},1)\to
BSpin_{c}(4)\to BSO(4)\times BS^{1}\to K(\Z_{2},2) $$ The last map in this
sequence is given by
$\;w_{2}\times 1 + 1\times
\rho(c_{1})\;$ which clearly vanishes exactly when $\delta (w_{2})=0$ .
Finally we
have the fibration:
$$\Z_{2}\to Spin(4)\times S^{1}\to Spin_{c}(4) \to K(\Z_{2},1)\to
BSpin(4)\times BS^{1}\to BSpin_{c}(4)\to K(\Z_{2},2) $$
\noindent The last map is given by $w_{2}$. This sequence says that locally a
$\;Spin_{c}(4)$ bundle consists a pair of a $Spin(4)$ bundle and a complex line
bundle. Also recall
$\;\;H^{2}(X;\Z)=[X,K(\Z,2)]=[X,BS^{1}]= \{\mbox{complex line bundles on
X}\}$
\vspace{.15in}
\noindent {\bf Definition}: Let $ L\longrightarrow X$ be a complex line
bundle over
a smooth oriented $4$-manifold with
$c_{1}(L)=w_{2}(TX)$ (i.e. $L$ is a characteristic line bundle). A
$Spin_{c}(4)\;$ structure on $X$, corresponding $L$, is a principal
$\;Spin_{c}(4)$-bundle
$\;P\longrightarrow X\;$ such that the associated framed bundles of
$TX$ and $L$ satisfy:
$$P_{SO(4)}(TX)=P\times_{{\rho}_{0}} SO(4) $$
$$P_{S^{1}}(L)=P\times_{{\rho}_{1}} S^{1} $$
\noindent where $(\rho_{0}, \rho_{1}): Spin_{c}(4)\to SO(4)\times S^{1}$ are
the
obvious projections
$$\begin{array}{ccccc} & & Spin(4)\times S^{1} & & \\ &&&&\\ & &
\downarrow \pi & &\\ &&&&\\ SO(4) & \stackrel{\rho_{0}} {\longleftarrow }
&Spin_{c}(4)&
\stackrel{\rho_{1}} { \longrightarrow} & S^{1}\\ &&&&\\ &&&&\\
& \swarrow \rho_{+} && \rho_{-}\searrow & \\ &&&&\\
U(2) && \downarrow
\tilde{\rho}_{+}\;\;\;\;\;\;\tilde{\rho}_{-}\downarrow & & U(2)\\ &&&&\\
& Ad \searrow&& \swarrow Ad& \\ &&&&\\ & &SO(3) & & \\
\end{array}$$
So $\;\tilde{\rho}_{\pm}= Ad\circ \rho _{\pm}\;$, also call
$\;\bar{\rho}_{\pm}=
\rho_{\pm}\circ \pi\;$.
For $x\in \H = \R^{4}$ we have
\begin{eqnarray*}
\rho_{1}[\;q_{+},q_{-},\lambda \;]&=&\lambda^{2} \\
\rho_{0}[\;q_{+},q_{-},\lambda
\;]&=&[\;q_{+},q_{-}\;]\;\;\;\;\;,\;\mbox{i.e.}
\;\;\;x\longmapsto q_{+}xq_{-}^{-1}\\
\rho_{\pm}[\;q_{+},q_{-},\lambda \;]&=&[\;q_{\pm},\lambda \;]
\;\;\;\;\;\;\;,\;\mbox{i.e.}
\;\;\;x\longmapsto q_{\pm} x \lambda^{-1} \\
\tilde{\rho}_{\pm}[\;q_{+},q_{-},\lambda \;]&=& Ad\circ q_{\pm}
\;\;\;\;\;\;\;,\;
\mbox{i.e.}\;\;\;x\longmapsto q_{\pm}xq_{\pm}^{-1}\\
\bar{\rho}_{\pm}(\;q_{+},q_{-},\lambda \;) &=& \lambda q_{\pm}
\end{eqnarray*}
\noindent Apart from $TX$ and $L$, $Spin_{c}(4)$ bundle $P\to X$ induces a
pair of
$U(2)$ bundles:
$$W^{\pm}=P\times _{\rho_{\pm}}\C^{2}\longrightarrow X$$ Let
$\Lambda^{p}(X)=\Lambda^{p}T^{*}(X) $ be the bundle of exterior $p$ forms.
If $X$ is
a Riemanian manifold (i.e. with metric), we can construct the bundle of
self(antiself)-dual 2-forms
$\;\Lambda_{\pm}^{2}(X)$ which we abbreviate by $ \Lambda^{\pm}(X)
\;$. We can identify $ \Lambda ^{2}(X) \;$ by the Lie algebra
$so(4)$-bundle
$$\Lambda ^{2}(X)=P(T^{*}X)\times _{ad}so(4)\;\;\;\;\;\;\mbox{by}\;\;\;
\Sigma \;a_{ij}\;dx^{i}\wedge dx^{j}\;\longleftrightarrow \; (a_{ij})$$ where
$ad:SO(4)\to so(4)\;$ is the adjoint representation. The adjoint action
preserves
the two summands of
$so(4)=spin(4)=so(3)\times so(3)=\R^{3}\oplus \R^{3}$. By above
identification it is easy to see that the $\pm 1$ eigenspaces $
\Lambda^{\pm}(X) \;$
of the star operator
$*:\Lambda (X)\to \Lambda (X)$ corresponds to these two ${\R}^{3}$-bundles;
this
gives:
$$\Lambda^{\pm}(X)= P\times _{\tilde{\rho}_{\pm}} \R^{3} $$
If the $Spin_{c}(4)$ bundle $ P\to X$ lifts to $Spin(4)$ bundle $
\bar{P}\to X$ (i.e. when $w_{2}(X)=0$), corresponding to the obvious
projections $
p_{\pm}: Spin(4)\to SU(2) $,
$p_{\pm}(q_{-},q_{+})=q_{\pm}$ we get a pair of $SU(2)$ bundles:
$$V^{\pm}= P\times_{p_{\pm}} \C^{2} $$ Clearly since
$\;x\longmapsto q_{\pm} x \lambda ^{-1}=
q_{\pm} x \;(\lambda^{2})^{-1/2}\; $ in this case we have:
$$\;W^{\pm}= V^{\pm}\otimes L^{-1/2}\;$$
\subsection {Action of $\;\Lambda^*(X) \;\;$on$\;\; W_\pm$ }
From the definition of $Spin_{c}(4)$ structure above we see that
$$T^{*}(X)=P\times \H/ (p,v)\sim
(\tilde{p},q_{+}v\;q_{-}^{-1})\;\;,\;\mbox{where}\;\;\;
\tilde{p}=p[\;q_{+},q_{-},\lambda\;]$$ We define left actions (Clifford
multiplications), which is well defined by
$$ T^{*}(X) \otimes W^{+}\longrightarrow W^{-}\;\;,\;\mbox{by}\;\;\;\;
[\;p,v\;]\otimes[\;p,x\;]\longmapsto [\;p,-\bar{v}x\;]$$
$$T^{*}(X) \otimes W^{-} \longrightarrow W^{+}\;\;,\;\mbox{by}\;\;\;\;
[\;p,v\;]\otimes[\;p,x\;]
\longmapsto [\;p,vx\;]$$ From identifications, we can check the well
definededness
of these actions, e.g.:
$$[p,v\;]\otimes [\;p,x\;]\sim [\;\tilde{p},q_{+}v\;q_{-}^{-1}\;]
\otimes [\;\tilde{p}, q_{+}x\lambda^{-1}\;]
\longmapsto [\;\tilde{p}, \; q_{-}(-\bar{v}x)\lambda^{-1}\;]
\sim[\;p,-\bar{v}x\;]$$
\vspace{.05in}
\noindent By dimension reason complexification of these representation give
$$ \rho:T^{*}(X)_\C\stackrel{\cong }{\longrightarrow} Hom(W^{\pm},
W^{\mp})\equiv W^{\pm}\otimes W^{\mp}$$
\noindent We can put them together as a single representation (which we
still call
$\rho $)
$$ \rho: T^{*}(X)\longrightarrow Hom(W^{+}\oplus W^{-})\;\;,\;\;\mbox{by}
\;\;\;v\;\longmapsto \;
\rho (v)=\left(\begin{array}{cc} 0& v \\
-\bar{v} & 0\end{array} \right) $$ We have $\;\rho (v)\circ\rho
(v)=-|v|^{2}I\;$.
By universal property of the Clifford algebra this representation extends
to the
Clifford algebra
$C(X)=\Lambda^{*}(X)$ (exterior algebra)
$$\begin{array}{ccc}
\Lambda^{*}(X)& & \\ & & \\
\downarrow &\searrow & \\ & & \\ T^{*}(X) & \longrightarrow & Hom\;(W^{+}\oplus
W^{-})
\end{array}$$
\noindent One can construct this extension without the aid of the universal
property of the Clifford algebra, for example since
$$\Lambda^{2}(X)=\left \{\; v_{1}\wedge v_{2}=\frac{1}{2}(v_{1}\otimes
v_{2}-v_{2}\otimes v_{1})\;|\; v_1,v_{2}\in T^{*}(X)\;\right \}$$ The action of
$T^{*}(X)$ on
$W^{\pm}$ determines the action of
$\;\Lambda^{2}(X)=\Lambda^{+}(X)\otimes\Lambda^{-}(X) $, and since
$\;2Im\;(v_{2}\bar{v}_{1})=-v_{1}\bar{v}_{2} + v_{2}\bar{v}_{1} $ we have
the action
$\rho$ with property:
$$\; \Lambda^{+}(X)\otimes W^{+} \longrightarrow W^{+}\;\;\;\;\mbox{to
be}\;\;\;
[\;p,v_{1}\wedge v_{2}\;]\otimes [\;p,x\;]
\longrightarrow [\;p,Im\;(v_{2}\bar{v}_{1})x\;] $$
$$\rho: \Lambda^{+} \longrightarrow Hom(W^{+}, W^{+})$$
\begin{eqnarray}\rho (v_{1}\wedge v_{2})&=&
\;\frac{1}{2}\;[\;\rho(v_{1}),\rho (v_{2})\;]
\end{eqnarray}
\vspace{.1in} Let us write the local descriptions of these representations:
We first
pick a
local orthonormal basis $\;\{e^{1},e^{2},e^{3},e^{4}\}\;$ for
$T^{*}(X)$, then we can take
$$\;\{\;f_{1}=\frac{1}{2}( e^{1}\wedge e^{2} \pm e^{3}\wedge e^{4}),\;
f_{2}=\frac{1}{2}( e^{1}\wedge e^{3} \pm e^{4}\wedge e^{2}),\;
f_{3}=\frac{1}{2}(
e^{1}\wedge e^{4} \pm e^{2}\wedge e^{3})\;\}$$ to be a basis for
$\Lambda^{\pm}(X)$. After the local identification $T^{*}(X)=\H$ we can take
$e^{1}=1,\;e^{2}=i,\;e^{3}=j,\; e^{4}=k$. Let us identify
$W^{\pm}=\C^{2}=\{z+jw\;|\; z,w\in \C\;\}$, then the multiplication by
$1,i,j,k$ (action on $\C^{2}$ as multiplication on left) induce the
representations
$\rho (e^{i})\;,\;i=1,2,3,4$. From this we see that
$\Lambda^{+}(X)$ acts trivially on $W^{-}$; and the basis
$f_{1},f_{2},f_{3}$ of
$\Lambda^{+}(X)$ acts on $W^{+}$ as multiplication by $i,j,k$, respectively
(these
are called Pauli matrices).
$$\begin{array}{cc}
\rho(e^{1})=\left( \begin{array} {cccc} & &1 & 0\\ & & 0 & 1
\\-1 & 0 & & \\ 0 & -1 & & \end{array}
\right) & \;\;\;\;
\rho(e^{2})=\left( \begin{array}{cccc} &&i&0\\ &&0&-i \\i&0&&\\ 0&-i&&
\end{array} \right) \\ &\\ &\\
\rho(e^{3})=\left( \begin{array}{cccc} &&0&-1\\ &&1&0 \\0&-1&&\\1&0&&
\end{array} \right) &\;\;\;
\rho(e^{4})=\left( \begin{array}{cccc} &&0&-i\\ &&-i&0
\\0&-i&&\\-i&0&&
\end{array} \right) \\
\end{array}$$
\vspace{.15in}
$$\begin{array}{ccc}
\rho(f_1)=\left( \begin{array}{cc} i&0\\ 0&-i \end{array} \right)&
\rho(f_2)=\left( \begin{array}{cc} 0&-1\\ 1&0 \end{array} \right)&
\rho(f_3)=\left( \begin{array}{cc} 0&-i\\ -i&0 \end{array} \right)
\\
\end{array}$$
\vspace{.15in}
In particular we get an isomorphism $\Lambda^{+}(X)\longrightarrow su\;(W^{+})$
(traceless skew adjoint endemorphism of $W^{+}$); which after complexifying
extends
to an isomorphism
$\rho:\Lambda^{+}(X)_\C\cong sl\;(W^{+})$ (traceless endemorphism of $W^{+}$)
$$\begin{array}{ccc}
\Lambda^{+}(X)&\stackrel{\cong}{\longrightarrow}& su\;(W^{+})\\ &&\\
\bigcap & & \bigcap\\ &&\\
\Lambda^{+}(X)_\C&\stackrel{\rho}{\longrightarrow} & sl\;(W^{+})
\end{array}$$
\vspace{.1in}
Recall $\;Hom(W^{+}, W^{+})\cong W^{+}\otimes (W^{+})^{*}\;$; we identify
the dual
space $(W^{+})^{*}$ naturally with $\bar{W}^{+}$ (= $W^{+}$ with scalar
multiplication $\;c.v=\bar {c}v$) by the pairing
$$W^{+}\otimes \bar{W}{+}\longrightarrow \C $$ given by
$\;z\otimes w\to z\bar{w}$. Usually
$sl\;(W^{+})$ is denoted by $(W^{+}\otimes \bar{W}^{+})_0\;$ and the trace map
gives the identification:
$$W^{+}\otimes \bar{W}^{+}=(W^{+}\otimes \bar{W}^{+})_0\oplus \C=
\Lambda^{+}(X)_\C\oplus \C$$ Let $\sigma
:W^{+}\longrightarrow\Lambda^{+}(X)\; $ be the map
$\;[\;p,x\;]\longmapsto [\;p,\;\frac{1}{2}(x i \;\bar{x})\;] $. By local
identification as above $ W^{+}=\C^{2}$ and
$\Lambda^{+}(X)=\R\oplus \C$, we see $\sigma$ corresponds to
$$\;(z,w)\longmapsto i\;\left(\frac{|z|^{2}-|w|^{2}}{2}\right) -k\;
Re(z\bar{w})+
j\;Im(z\bar{w}) =
\left(\frac{|w|^{2}-|z|^{2}}{2}\right) + z\bar{w} $$
\noindent We identify this by the element $i\sigma(z,w)$ of
$su\;(W^{+})$ (by Pauli matrices) where:
\begin{eqnarray}\;(z,w)\; \longmapsto \sigma(z,w)\;=\left(
\begin{array}{cc} (\;|z|^{2}-|w|^{2})/2& {z}\bar{w}\\
\bar{z}{w}& (\;|w|^{2}-|z|^{2})/2 \end{array} \right)\end{eqnarray}
$\sigma$ is the projection of the diagonal elements of $W^{+}\otimes
\bar{W}^{+}$ onto
$(W^{+}\otimes \bar{W}^{+})_0$
\vspace{.05in}
\noindent We can check:
\begin{eqnarray} i\;\sigma(z,w)=\rho
\;[\;\frac{|z|^{2}-|w|^{2}}{2}\; f_{1} + Im(z\bar{w})
\;f_{2} - Re(z\bar{w})\; f_{3}\;]
\end{eqnarray}
From these identifications we see:
\begin{eqnarray} |\;\sigma(\psi) \;|^{2}&=&\frac{1}{4} |\;\psi
\;|^{4} \\ <\;\sigma(\psi)\;\psi,\psi\;> &=& \frac{1}{2} |\;\psi
\;|^{4}\\
<\rho(\omega)\;\psi\;,\;\psi>&=&2i\;<\rho (\omega)\;,i\;
\sigma(\psi)>
\end{eqnarray} Here the norm in $\;su(2)\;$ is induced by the inner product
$<A,B>=\frac{1}{2}trace(AB)\;$. \\
By calling $\sigma(\psi, \psi)=\sigma (\psi)$ we extend the definition of
$\sigma $ to $ W^{+}\otimes \bar{W}^{+} $ by
$$<\rho(\omega)\;\psi\;,\;\varphi>=2i\;<\rho (\omega)\;,i\;
\sigma(\psi,
\varphi)> $$
\vspace{.02in}
$$\begin{array}{cccc}
\Lambda^{+}(X)\;\;\;\;\;=&su (W^{+})&\;\;\stackrel{i\;\sigma}{\longleftarrow}&
W^{+}\\ &&&\\ &\bigcap & & \bigcap\\ &&&\\ (W^{+}\otimes \bar{W}^{+})_0\;\;
=&sl\;(W^{+})&\stackrel{i\;\sigma}{\longleftarrow} & W^{+}\otimes
\bar{W}^{+}
\end{array}$$
\vspace{.15in}
\noindent{\bf Remark}: A $Spin_{c}(4)$ structure can also be defined as a
pair of
$U(2)$ bundles:
$$W^{\pm}\longrightarrow
X\;\;\mbox{with}\;\;det(W^{+})=det(W^{-})\longrightarrow X
\;\;\mbox{(a complex line bundle), }$$
$$\mbox{ and an action}\;\; c_{\pm}:T^{*}(X)\longrightarrow
Hom(W^{\pm},W^{\mp})\;\;
\mbox{with}\;
\;c_{\pm}(v)c_{\mp}(v)=-|v|^{2}I $$
\vspace{.1in}
\noindent The first definition clearly implies this, and conversely we can
obtain
the first definition by letting the principal
$Spin_{c}(4)$ bundle to be:
$$P=\{\;(p_{+},p_{-})\in P(W^{+})\times P(W^{-})\;|\;
det(p_{+})=det(p_{-})\;\} $$
Clearly, $\;Spin_{c}(4)=\{ (A,B)\in U_{2}\times U_{2}\; |\;
det(A)=det(B)\;\}$ acts
on
$P$ freely.
\vspace{.15in}
\noindent This definition generalizes and gives way to the following
definition:
\vspace{.1in}
\noindent{\bf Definition}: A Dirac bundle $W\longrightarrow X$ is a Riemanian
vector bundle with an action $
\rho:T^{*}(X)\longrightarrow Hom(W,W)\;$ satisfying
$\;\rho (v)\circ \rho (v)=-|v|^{2}I $. $W$ is also equipped with a
connection $D$
satisfying:
$$<\rho (v)x,\rho (v)y>=<x,y>$$
$$D_{Y}(\rho (v)s)=\rho (\nabla_{X} v) s + \rho (v) D_{Y}(s)$$
\noindent where $\nabla $ is the Levi-Civita connection on
$T^{*}(X)$, and $Y$ is a vector filed on $X$
\vspace{.1in}
An example of a Dirac bundle is $\;W=W^{+}\oplus W^{-}\longrightarrow X\;$ and
$\;D=d+d^{*}\;$ with
$W^{+}=\oplus\Lambda^{2k}(X)\; $ and
$W^{-}=\oplus\Lambda^{2k+1}(X)\; $ where
$\rho (v)= v\wedge . + v \;\bot ..$ (exterior $+$ interior product with
$v$). In this case
$\rho : W^{\pm}\to W^{\mp} $. In the next section we will discuss the
natural
connections
$D$ for $Spin_{c}$ structures $W^{\pm}$ .
\section{ Dirac Operator}
Let $\cal{A}(L)$ denote the space of connections on a $\;U(1)$ bundle
$L\longrightarrow X$. Any $A\in \cal{A}(L)$ and the Levi-Civita connection
$A_{0}$
on the tangent bundle coming from Riemanian metric of
$X$ defines a product connection on $P_{SO(4)}\times P_{S^{1}}$. Since
$Spin_{c}(4)$ is the two fold covering of
$SO(4)\times S^{1}$, they have the same Lie algebras
$spin_{c}(4)=so(4)\oplus i\;\R$. Hence we get a connection
$\tilde{A}$ on the $Spin_{c}(4)$ principle bundle
$P\longrightarrow X$. In particular the connection
$\tilde{A}$ defines connections to all the associated bundles of P, giving back
$A,\;A_{0}$ on $L, \;T(X)$ respectively, and two new connections $A^{\pm}$ on
bundles $W^{\pm}$. We denote the corresponding covariant derivatives by
$\nabla_{A}$.
$$\nabla _{A} :\Gamma(W^{+})\to \Gamma(\;T^{*} X\otimes W^{+})$$ Composing
this with
the Clifford multiplication $\Gamma(\;T^{*} X\otimes W^{+} )\to \Gamma(W^{-}) $
gives the Dirac operator
$$D\!\!\!\!/\, _{A} :\Gamma(W^{+})\to \Gamma(W^{-})$$ Locally, by choosing orthonormal
tangent vector field
$e=\{e_{i}\;\}_{i=1}^{4}$ and the dual basis of $1$-forms
$\{e^{i}\;\}_{i=1}^{4}$ in a neighborhood
$U$ of a point $x\in X$ we can write
$$D\!\!\!\!/\, _{A}=\sum \rho (e^{i}) \nabla_{e_{i}}$$ where $\nabla_{e_{i}}:
\Gamma(W^{+})\to \Gamma(W^{+})$ is the covariant derivative
$\nabla _{A}$ along $e_{i}$. Also locally $W^{\pm}=V^{\pm}\otimes L^{1/2}
$, hence by Leibnitz rule, the connection $A$ and the untwisted Dirac operator
$$\partial\!\!\!/\, : \Gamma(V^{+})\to \Gamma(V^{-})$$ determines $D\!\!\!\!/\,_{A}$. Notice
that as
in $W^{\pm}$, forms $\Lambda^{*}(X)$ act on $V^{\pm}$. Now let
$\omega=(\omega_{ij})$ be the Levi-Civita connection $1$-form, i.e.
$so(4)$-valued ``equivariant"
$1$-form on $P_{SO(4\;)}(X)$ and
$\tilde{\omega } =(\tilde{\omega }_{ij})=e^{*}(\omega) $ be the pull-back
$1$-form on $U$. Since
$P_{SO(4\;)}(U)=P_{Spin(4\;)}(U)$ the orthonormal basis $e\in P_{SO(4)}(U)$
determines an orthonormal basis
$\sigma=\{\sigma^{k}\}\in P_{SU_{2}}(V^{+})$, then (e.g. [{\bf LM}])
$$\partial\!\!\!/\,(\sigma^{k})=\frac{1}{2}\sum_{i<j}
\rho( \tilde{\omega}_{ji})\;\rho( e^{i}) \rho ( e^{j})\;\sigma^{k} $$
Metrics on $T(X)$ and $L$ give metrics on $W^{\pm}$ and
$T^{*}(X) \otimes W^{\pm} $, hence we can define the adjoint $\nabla
_{A}^{*}:\Gamma(T^{*} X\otimes W^{-})\to
\Gamma(W^{+})$. Similarly we can define $D\!\!\!\!/\, _{A} :\Gamma(W^{-})\to
\Gamma(W^{+}) $ which turns out to be the adjoint of the previous
$D\!\!\!\!/\,_{A}$ and makes the following commute (vertical maps are Clifford
multiplications):
$$\begin{array}{ccccc}
\Gamma(W^{+})&\stackrel{ \nabla _{A}}{\longrightarrow}&\Gamma(\;T^{*} X
\otimes W^{+})&
\stackrel{ \nabla _{A}}{\longrightarrow}&\Gamma(\;T^{*} X \otimes T^{*}
X\otimes
W^{+})\\
\parallel & & \downarrow & &\downarrow \\
\Gamma(W^{+}) & \stackrel{ D\!\!\!\!/\, _{A}}{\longrightarrow} &\Gamma(W^{-})&
\stackrel{ D\!\!\!\!/\, _{A}}{\longrightarrow}&\Gamma(W^{+})\\
\end{array}$$
\noindent Let $F_{A}\in \Lambda^{2}(X)$ be the curvature of the connection
$A$ on $L$, and
$F^{+}_{A}\in \Lambda^{+}(X)$ be the self dual part of this curvature, and
$s$ be
the scalar curvature of $X$. Weitzenbock formula says that:
\begin{eqnarray}
D\!\!\!\!/\,_{A}^{2}(\psi)&=& \nabla _{A}^{*}\nabla _{A}\psi
+\frac{s}{4}\psi + \frac{1}{4}\rho(F_{A}^{+})\psi
\end{eqnarray} To see this we we can assume
$\nabla_{e_{i}}(e^{j})=0$ at the point $x$
\begin{eqnarray*}
D\!\!\!\!/\,_{A}^{2}\psi&=& \sum \rho( e^{i}).\;\nabla_{e_{i}}\;[\;\sum \rho
(e^{j}).\;\nabla_{e_{j}}\psi\;]\;\\ &=&
\nabla^{*}_{A}\nabla_{A}\psi+\frac{1}{2}\sum_{i,j}\rho (e^{i})\;\rho
(e^{j})\;(\nabla_{e_{i}}\nabla_{e_{j}} -
\nabla_{e_{j}}\nabla_{e_{i}})\;\psi\\ &=&
\nabla^{*}_{A}\nabla_{A}\psi+\frac{1}{2}
\sum_{i,j}\rho (e^{i})\;\rho (e^{j})\; \Omega_{ij}^{A} \;\psi
\end{eqnarray*}
$\Omega_{ij}^{A}=R_{ij}+ \frac{1}{2} F_{ij}\;$ is curvature on
$V^{+}\otimes L^{1/2}$, i.e. $R_{ij}$ is the Riemanian curvature and the
imaginary
valued $2$-form
$F_{ij}$ is the curvature of $A$ for the line bundle $L$ (endemorphisims of
$W^{+}$). So if $\psi=\sigma\otimes \alpha \in
\Gamma(V^{+}\otimes L^{1/2})\;$, then
\begin{eqnarray*}\frac{1}{2}\sum_{i,j}\rho (e^{i})\;\rho (e^{j})\;
\Omega_{ij}^{A}\;(\sigma\otimes \alpha)&=&\frac{1}{2}
(\sum \rho (e^{i})\;\rho (e^{j})\;R_{ij} \;\sigma )\otimes \alpha
\\ && + \;\frac{1}{4}\sum \rho (e^{i})\;\rho (e^{j}) \sigma \otimes ( F_{ij}
\alpha)\\ &&\\ &=& \frac{1}{8} \;\sum \rho (e^{i})\;\rho (e^{j})\;
\rho (e^{k})\;
\rho (e^{l})\;R_{ijkl} \;(\psi) \\ && + \;\frac{1}{4}\;\rho
\;(\sum F_{ij} \; e^{i}\wedge e^{j}) \;(\psi )
\end{eqnarray*} The last identity follows from (1). It is a standard
calculation
that the first term is $s/4$ ( e.g.[{\bf LM}], pp. 161), and since $\Lambda
^{-}
(X)$ act as zero on $W^{+}$, the second term can be replaced by
$$\frac{1}{4}\;\rho(F_{A}^{+})\;\psi=\frac{1}{4}\;\rho \;(\sum F_{ij}^{+}
\; e^{i}\wedge e^{j}) \;\psi $$
\subsection {A Special Calculation}
In Section 4 we need some a special case (7). For this, suppose
$$V^{+}=L^{1/2}\oplus L^{-1/2}$$ where $L^{1/2}\longrightarrow X$ is some
complex
line bundle with
$L^{1/2}\otimes L^{1/2}=L$.
Hence $W^{+}=(L^{1/2}\oplus L^{-1/2})\otimes L^{-1/2}=L^{-1}\oplus \C$. In
this case there is a unique connection
$\frac{1}{2}A_{0}$ in $L^{-1/2}\to X$ such that the induced Dirac operator
$D_{A_{0}}$ on $W^{+}$ restricted to the trivial summand $\underline \C\to X$
is the exterior derivative $d$. This is because for
$\sigma _{\pm}\in \Gamma (L^{\pm 1/2})$, the following determines
$\nabla_{\frac{A_{0}}{2}}(\sigma_{-})$ :
\begin{eqnarray*}
\nabla_{A_{0}} (\sigma_{+}+0)\otimes \sigma_{-}&=&
\partial\!\!\!/\, (\sigma_{+}+0 )\otimes \sigma_{-} + (\sigma_{+}+ 0 )\otimes
\nabla_{\frac{A_{0}}{2}}(\sigma_{-})
\\ =\nabla_{A_{0}}(\sigma_{+}\otimes \sigma_{-})&=&
d(\sigma_{+}\otimes \sigma_{-})
\end{eqnarray*}
The following is essentially the Leibnitz formula for Laplacian applied to
Weitzenbock formula (7)
\vspace{.12in}
\noindent{\bf Proposition}:
Let $A, A_{0} \in \cal{A}(L^{-1}) $ and $i\;a=A-A_{0}$. Let
$\nabla _{a}=d +i\;a $ be the covariant derivative of the trivial bundle
$\underline{\C}\longrightarrow X$, and $\alpha :X\to \C$. Let
$u_{0}$ be a section of $W^{+}=L^{-1}\oplus{\C}$ with a constant
$\C $ component and $D\!\!\!\!/\,_{A_{0}}(u_{0})=0$ then:
\begin{eqnarray}
D\!\!\!\!/\,_{A}^{2}(\alpha u_{0})= (\nabla _{a}^{*}\nabla _{a} \alpha )u_{0} +
\frac{1}{2} \rho (F_{a} )\;\alpha \;u_{0} -2<\nabla _{a}\alpha \;,
\nabla_{A_{0}}(u_{0})> \end{eqnarray}
Proof: By writing $\nabla_{A}=\nabla ^{A}$ for the sake of not cluttering
notations,
and abbreviating
$\;\nabla_{e_{j}}=\nabla_{j}\;$ and $\;\nabla^{a}_{j}(\alpha)=
\nabla_{j}(\alpha) + i\; a_{j} \alpha \;$, and leaving out summation signs for
repeated indices (Einstein convention) we calculate:
\begin{eqnarray}
\nabla^{A}(\alpha u_{0})&=&\nabla^{A}(\alpha)u_{0}+\alpha
\nabla^{A}(u_{0})\nonumber \\ &=& e^{j}\otimes\nabla_{j}(\alpha)u_{0}+
\alpha (\nabla^{A_{0}}(u_{0}) +i\; e^{j}\otimes a_{j}\;u_{0}) \nonumber
\nonumber \\ &=&
e^{j}\otimes(\nabla_{j}(\alpha) +i\; a_{j} \alpha\;) u_{0} + \alpha
\nabla^{A_{0}}(u_{0}) \nonumber \\
D\!\!\!\!/\,_{A}(\alpha u_{0} )&=&\rho ( e^{j}) \;\nabla ^{a}_{j}(\alpha)\; u_{0} +
\alpha \;D\!\!\!\!/\,_{A_{0}}(u_{0}) = \rho ( e^{j}) \;\nabla ^{a}_{j}(\alpha)\; u_{0}
\end{eqnarray}
\vspace{.1in}
\noindent By abbreviating $\;\mu=\nabla ^{a}_{j}(\alpha)\; $ we calculate:
\begin{eqnarray}
\nabla^{A} ( \rho ( e^{j}) \;\mu\; u_{0} )&=& e^{k}\otimes \rho ( e^{j})
\;\nabla_{k}(\mu ) u_{0} + e^{k}\otimes\; \rho(e^{j})\;\mu
\; (\nabla^{A_{0}}_{k}(u_{0}) +i\; a_{k}\;u_{0}) \nonumber \\ &=& e^{k}\otimes
\rho(e^{j}) \;\nabla^{a}_{k}(\mu)\; u_{0}\; + e^{k}\otimes
\rho(e^{j})\;\mu\;\nabla^{A_{0}}_{k}(u_{0}) \nonumber\\
D\!\!\!\!/\,_{A}( \rho ( e^{j}) \;\mu\; u_{0})&=& \rho (e^{k}
)\rho(e^{j})\;\nabla^{a}_{k}(\mu)\; u_{0}\; +
\rho (e^{k} )\rho(e^{j})\;\mu\;\nabla^{A_{0}}_{k}(u_{0}) \nonumber \\ & = &
-\nabla^{a}_{j}(\mu)\; u_{0}\; + \frac{1}{2}
\sum_{k,j} \rho (e^{k} )\rho(e^{j}) (\nabla^{a}_{k}(\mu)-
\nabla^{a}_{j}(\mu) )u_{0} \nonumber\\ & & - \mu\;\nabla^{A_{0}}_{j}(u_{0}) -
\mu\;\rho(e^{j})
\sum_{k\neq j} \rho (e^{k} ) \nabla^{A_{0}}_{k}(u_{0})
\end{eqnarray}
Since $0=D\!\!\!\!/\,_{A_{0}}(u_{0})=\sum \rho (e^{k} )
\nabla^{A_{0}}_{k}(u_{0}) $ the last term of (3) is
$-\mu\;\nabla^{A_{0}}_{j}(u_{0}) $.
\vspace{.1in}
\noindent By plugging $\mu=\nabla ^{a}_{j}(\alpha)\; $ in (10) and summing over
$j$, from (2) we see
\begin{eqnarray*}
D\!\!\!\!/\,_{A}^{2}(\alpha u_{0})=-\nabla ^{a}_{j}\nabla ^{a}_{j}(\alpha) u_{0}
+\frac{1}{2}
\rho(\sum F^{a}_{k,j}\; e^{k}\wedge e^{j})\;\alpha \;u_{0}-2\sum
\nabla ^{a}_{j}(\alpha) \nabla^{A_{0}}_{j}(u_{0})\;\;\;\;\Box
\end{eqnarray*}
\noindent{\bf Remark}: Notice that since $u_{0}$ has a constant
$\C$ component and
$\nabla_{A_{0}}$ restricts to the usual $d$ the $\C$ component, the term
$<\nabla _{a}\alpha \;, \nabla_{A_{0}}(u_{0})>$ lies entirely in $L$
component of
$W^{+}$
\section{Seiberg-Witten invariants} Let $X$ be a closed oriented Riemanian
manifold, and $L\longrightarrow X$ a characteristic line bundle. Seiberg
-Witten
equations are defined for
$(A,\psi)\in {\cal A}(L)\times \Gamma(W^{+})$,
\begin{eqnarray}
D\!\!\!\!/\,_{A}(\psi)&=&0\\
\rho(F_{A}^{+})&=&\sigma(\psi)
\end{eqnarray}
Gauge group $\;{\cal G}(L)=Map(X,S^{1})\;$ acts on
$\;\tilde{\cal B}(L)={\cal A}(L)\times \Gamma(W^{+})\;$ as follows: for
$\;s=e^{if}\in {\cal G}(L) $
$$s^{*}(A, \psi)= (s^{*} A, s^{-1}\psi)=(A+s^{-1}ds\;,\; s^{-1}
\psi)= (A+i\; df \; ,\;s^{-1}\psi) $$
\noindent By locally writing $W^{\pm}= V^{\pm}\otimes L^{1/2} $, and
$\psi= \varphi\otimes \lambda\in \Gamma ( V^{\pm} \otimes L^{1/2})$ and from:
\begin{eqnarray*}
D\!\!\!\!/\, _{s^{*}A}( \varphi\otimes \lambda)=\; \partial\!\!\!/\, (\varphi)\otimes
\lambda + [\; \varphi\otimes D_{A}(\lambda) + i\;df\; ( \varphi
\otimes \lambda)\;]\;
\end{eqnarray*} we see that
$D\!\!\!\!/\, _{s^{*}A}\; (s^{-1}\psi) =s^{-1} D\!\!\!\!/\,_{A} (\psi) $, and from definitions
$$\rho(F_{s^{*}A}^{+})=s^{-1}\rho(F_{A}^{+})\;s =
\rho(F_{A}^{+})=\sigma(\psi) =\sigma(s^{-1}\psi) $$
\noindent Hence the solution set $\;\tilde{\cal M}(L)\subset
\tilde{\cal B}(L)\;$ of Seiberg-Witten equations is preserved by the action
$\;(A,\psi)\longmapsto s^{*}( A,\psi )\;$ of ${\cal G }(L)$ on
$\tilde{\cal M}(L)$. Define
$$ {\cal M}(L)=\tilde{\cal M}(L)/{\cal G }(L) \;\;\subset\;\; {\cal B}(L)=
\tilde{\cal B}(L)/{\cal G} (L)$$
We call a solution $(A,\psi)$ of (11) and (12) an irreducible solution if
$\psi\neq 0 $. $\;{\cal G}(L)$ acts on the subset $\tilde{\cal M}^{*}(L)$
of the
irreducible solutions freely, we denote
$$ {\cal M}^{*}(L)=\tilde{\cal M}^{*}(L)/{\cal G }(L) $$
\vspace{.005in}
Any solution $(A,\psi) $ of Seiberg -Witten equations satisfies the
$C^{0}$ bound
\begin{eqnarray} |\psi|^{2}\leq \mbox{max}(0,-2s)
\end{eqnarray} where $s$ is the scalar curvature function of $X$. This
follows by
plugging (12) in the Weitzenbock formula (7).
\begin{eqnarray}
D\!\!\!\!/\,_{A}^{2}(\psi)&=& \nabla _{A}^{*}\nabla _{A}\psi
+\frac{s}{4}\psi + \frac{1}{4}\sigma (\psi)\psi
\end{eqnarray} Then at the points where where $|\psi|^{2}$ is maximum, we
calculate
\begin{eqnarray*} 0\leq \frac{1}{2}\Delta |\psi |^{2}&=& \frac{1}{2}
d^{*}d<\;\psi,\psi\;> =
\frac{1}{2} d^{*} (\; <\nabla_{A} \psi,\psi>+<\psi,\nabla_{A} \psi>
\;) \\ &=& \frac{1}{2} d^{*} ( \;\bar{<\psi, \nabla _{A} \psi>}+
<\psi,\nabla_{A}
\psi>\; )= d^{*} < \psi\;, \nabla_{A} \psi>_{\;\R}\\ &=& <\psi, \nabla_{A}
^{*}\nabla_{A} \psi>-|\nabla_{A}
\psi|^{2} \;\leq \; <\psi, \nabla_{A} ^{*}\nabla _{A} \psi>\\ &\leq & - \;
\frac{s}{4} |\; \psi \; |^{2} - \frac{1}{8} |\;\psi \; |^{4}
\end{eqnarray*} The last step follows from (14), (11) and (5), and the last
inequality gives (13)
\vspace{.15in}
\noindent{\bf Proposition 3.1} $\;{\cal M}(L)$ is compact
\vspace{.12in}
Proof: Given a sequence $[\;A_{n},\psi_{n}\;] \in {\cal M}(L) $ we claim
that there
is a convergent subsequence (which we will denote by the same index), i.e.
there is
a sequence of gauge transformations $g_{n}\in {\cal G}(L)$ such that
$g_{n}^{*}(A_{n},\psi_{n})$ converges in $C^{\infty}$. Let $A_{0}$ be a base
connection. By Hodge theory of the elliptic complex:
$$\Omega^{0}(X)\stackrel {d^{0}}{\longrightarrow} \Omega^{1}(X)
\stackrel{d^{+}}{\longrightarrow} \Omega^{2}_{+}(X) $$
$$A-A_{0}=h_{n}+a_{n}+b_{n}\in {\cal H}\oplus im (d^{+})^{*}\oplus im (d)
$$ where
$\cal{H}$ are the harmonic $1$-forms. After applying gauge transformation
$g_{n}$
we can assume that $b_{n}=0$, i.e. if
$b_{n}=i\;d f _{n}$ we can let
$g_{n}=e^{if}$. Also $$h_{n}\in {\cal H}=H^{1}(X;\R)\;\;\;\mbox{and a
component of}\;\;{\cal G}(L)\;
\;\mbox{is}\;\; H^{1}(X;\Z)$$ Hence after a gauge transformation we can assume
$ h_{n}\in H^{1}(X;\R)/ H^{1}(X;\Z)$ so $h_{n}$ has convergent
subsequence. Consider the first order elliptic operator:
$$D =d^{*} \oplus d^{+} :
\Omega^{1}(X)_{L^{p}_{k}}\longrightarrow \Omega^{0}(X)_{L^{p}_{k-1}}
\oplus \Omega^{2}_{+}(X)_{L^{p}_{k-1}}$$
\noindent The kernel of $D$ consists of harmonic $1$-forms, hence by Poincare
inequality
if $a$ is a $1$-form orthogonal to the harmonic forms, then for some
constant $C$
$$ ||a ||_{L^{p}_{k}}\leq C ||D(a)||_{L^{p}_{k-1}} $$
\noindent Now $a_{n}=(d^{+})^{*} \alpha_{n}$ implies
$d^{*}(a_{n})=0 $. Since $\alpha_{n}$ is orthogonal to harmonic forms, and
by calling
$A_{n}=A_{0}+a_{n}$ we see :
$$ || a_{n}||_{L^{p}_{1}}\leq C\; ||D(a_{n}) ||_{L^{p}}
\leq C || d^{+} a_{n} ||_{L^{p}}= C \;|| F_{A_{n}}^{+}-F_{A_{0}}^{+}
||_{L^{p}}$$
Here we use C for a generic constant. By (12), (4) and (13) there is a $C$
depending only on the scalar curvature $s$ with
\begin{eqnarray}|| a_{n}||_{L^{p}_{1}}\leq C \end{eqnarray} By iterating this
process we get $ || a_{n}||_{L^{p}_{k}}\leq C $ for all
$k$ , hence
$|| a_{n}||_{\infty}\leq C$. From the elliptic estimate and
$D\!\!\!\!/\,_{A_{n}}(\psi_{n})=0$ :
\begin{eqnarray} ||\psi_{n}||_{L^{p}_{1}}&\leq & C(\; ||D\!\!\!\!/\,
_{A_{0}}\psi_{n}\;||_{L^{p}} + || \psi _{n} ||_{L^{p}})= C(\;
||a_{n}\psi_{n}\;||_{L^{p}}+|| \psi _{n} ||_{L^{p}}) \nonumber \\
||\psi_{n}||_{L^{p}_{1}}
&\leq & C(\; ||a_{n}||_{\infty} ||\psi _{n} ||_{L^{p}} + ||\psi _{n}
||_{L^{p}} )
\leq C
\end{eqnarray}
By repeating this (boothstrapping) process we get
$||\psi_{n}||_{L^{p}_{k}}\leq C $, for all $k$, where C depends only on
the scalar
curvature $s$ and $A_{0}$. By Rallich theorem we get convergent subsequence of
$\;(a_{n},\psi_{n})\;$ in
${L^{p}_{k-1}} $ norm for all $k$. So we get this convergence to be
$C^{\infty}$ convergence.
\hspace{2in}$\Box$
\vspace{.15in} It is not clear that the solution set of Seiberg-Witten
equations is
a smooth manifold. However we can perturb the Seiberg-Witten equations
(11), (12)
by any self dual
$2$-form $\delta \in \Omega^{+}(X)$, in a gauge invariant way, to obtain a
new set
of equations whose solutions set is a smooth manifold:
\begin{eqnarray}
D\!\!\!\!/\,_{A}(\psi)&=&0\\
\rho(F_{A}^{+} + i\;\delta )&=&\sigma(\psi)
\end{eqnarray}
\vspace{.15in} Denote this solution space by $\tilde{\cal M}_{\delta}(L)$, and
parametrized solution space by
$$\tilde{\bf{\cal M}}=\bigcup_{\delta \in \Omega^{+}}
\tilde{\cal M}_{\delta} (L)\times \{\;\delta\;\}\subset {\cal A}(L)\times
\Gamma(W^{+})
\times \Omega ^{+}(X)$$
$${\cal M}_{\delta}(L) = \tilde{\cal M}_{\delta}(L)\;/{\cal G}(L)
\;\;\subset\;\; {\bf{\cal M}} =\tilde{\bf{\cal M}}\;/{\cal G}(L) $$
\vspace{.08in}
\noindent Let $\tilde{\cal M}_{\delta}(L)^{*}\subset\tilde{\bf{\cal
M}}^{*}$ be the
corresponding irreducible solutions, and also let
$ {\cal M}_{\delta}(L)^{*} \subset {\bf{\cal M}}^{*}$ be their quotients by
Gauge
group. The following theorem says that for a generic choice of $\delta $
the set
${\cal M}_{\delta}(L)^{*}$ is a closed smooth manifold.
\vspace{.15in}
\noindent {\bf Proposition 3.2} $\;{\cal M}^{*}$ is a smooth manifold.
Projection
$\pi:{\cal M}^{*}\longrightarrow \Omega^{+}(X) $ is a proper surjection of
Fredholm
index:
$$d(L)=\frac{1}{4}[\;c_{1}(L)^{2}-(2\chi +3\sigma)\;]$$ where $\chi$ and
$\sigma$
are Euler characteristic and the signature of $X$.
\vspace{.12in}
Proof: The linearization of the map
$\;(A,\psi,\delta)\longmapsto (\rho(F_{A}^{+} + i\;\delta )-
\sigma(\psi) ,D\!\!\!\!/\,_{A}(\psi)\;) $ at $(A_{0},\psi_{0},\delta_{0})$ is given by:
$$ P: \Omega^{1}(X)\oplus \Gamma(W^{+})\oplus
\Omega^{+}(X)\longrightarrow su(W^{+})\oplus \Gamma(W^{-})$$
$$ P(a,\psi, \epsilon)=(\rho (d^{+}a + i\; \epsilon ) -2
\;\sigma(\psi,\psi_{0})\;,\;
D\!\!\!\!/\,_{A_{0}}\psi +\rho(a)\psi_{0}) $$ To see that this is onto we pick $
(\kappa,\theta)\in su(W^{+})\oplus
\Gamma(W^{-})$, by varying $\epsilon$ we can see that $(\kappa,0) $ is in
the image
of $P$. To see
$(0,\theta)$ is in the image of $P$, we prove that if it is in the orthogonal
complement to
$\;image(P)\;$ then it is $(0,0)$; i.e. assume
$$<D\!\!\!\!/\,_{A_{0}}\psi, \theta> +<\rho(a)\psi_{0}, \theta >=0$$ for all
$a$ and $\psi$, then by choosing $\psi=0$ we see $<\rho(a)\psi_{0},
\theta >=0$ for all $a$ which implies $\theta=0$
\vspace{.05in}
By implicit function theorem $\;\tilde {\cal M}\;$ is a smooth manifold, and by
Sard's theorem $\;\tilde{\cal M}_{\delta}(L)\;$ are smooth manifolds, for
generic
choice of $\delta $'s. Hence their free quotients
$\;{\bf{\cal M}}^{*\;}$ and $\;{\cal M}_{\delta}(L)^{*}\;$ are smooth
manifolds.
After taking ``gauge fixig" account, the dimension of $ {\cal M}_{\delta}(L)$
is
given by the index of
$\;P+ d^{*}$ (c.f. [{\bf DK}]). $\;P+ d^{*}$ is the compact perturbation of
$$ S: \Omega^{1}(X)\oplus
\Gamma(W^{+})\longrightarrow [\;\Omega^{0}(X)\oplus
\Omega_{+}^{2}(X)\;]\oplus \Gamma(W^{-}) $$
$$S=\left( \begin{array}{cc} d^{*}\oplus d^{+} & 0 \\ 0 & D\!\!\!\!/\,_{A_{0}}
\end{array} \right)$$
By Atiyah-Singer index theorem
\begin{eqnarray}
\mbox{ dim }{\cal M}_{\delta} (L)=\mbox{ind} (S)&=&
\mbox{index} (d^{*}\oplus d^{+}) + \mbox{index}
_{\R}\;D\!\!\!\!/\,_{A_{0}}\nonumber\\ &=&
-\frac{1}{2} (\chi + \sigma )+\frac{1}{4}( c_{1}(L)^{2}-\sigma ) \nonumber
\\ &=&
\frac{1}{4}\;[\;c_{1}(L)^{2}-(2\chi + 3 \sigma )\;]\nonumber \\ &=&
\frac{c_{1}(L)^{2}-\sigma
}{4}\;-\;(1+b^{+})
\end{eqnarray} where $b^{+}$ is the dimension of positive define part
$\;H^{2}_{+}\;$ of
$\;H^{2}(X;\Z)$. Notice that when $b^{+}$ is odd this expression is even,
since $L$ being a characteristic line bundle we have
$c_{1}(L)^{2}=\sigma\; \mbox{mod}\; 8 $
\hspace{3in} $\Box$
\vspace{.15in}
Now assume that $H^{1}(X)=0$, then ${\cal G}(L)=K(\Z,1)$. Than being a free
quotient of a contractible space by ${\cal G}(L)$ we have
$${\cal B}^{*}(L)=K(\Z,2)=\C\P^{\infty}$$ The orientation of
$H^{2}_{+}$ gives an orientation to
${\cal M}_{\delta}(L)$. Now By (19) if $ b^{+} $ is odd
${\cal M}_{\delta}(L)\subset{\cal B}^{*}(L) $ is an even dimensional $2d$
smooth closed oriented submanifold, then we can define Seiberg-Witten
invariants as:
$$SW_{L}(X)=<{\cal M}_{\delta}(L)\;,\; [\;\C\P^{d}\;]> $$
As in the case of Donaldson invariants ([{\bf DK}]), even though
${\cal M}_{\delta}(L)$ depends on metric (and on the perturbation
$\delta$) the invariant $SW_{L}(X)$ is independent of these choices, provided
$\;b^{+}\ge 2\;$, i.e. there is a generic metric theorem.
Also by (13) if
$X$ has
nonnegative scalar curvature then all the solutions are reducible, i.e.
$\psi=0$.
This implies that $A$ is anti-self-dual, i.e. $F_{A}^{+}=0$; but just as in
[{\bf
DK}] , If
$b^{+}\geq 2$ and $L$ nontrivial, for a generic metric $L$ can not admit such
connections. Hence $\tilde{\cal M}=\emptyset$ which implies $SW_{L}(X)=0$.
Similar to Donaldson invariants there is a ``connected sum theorem" for
Seiberg-Witten invariants: If $X_{i}\;i=1,2\;$ are oriented compact smooth
manifolds
with common boundary, which is a
$3$-manifold with a positive scalar curvature; then gluing these manifolds
together
along their boundaries produces a manifold
$X=X_{1}\smile X_{2}$ with vanishing Seiberg-Witten invariants (cf [{\bf
F}],[{\bf
FS}]). There is also conjecture that only
$0$-dimensional moduli spaces ${\cal M}_{\delta}(L)$ give nonzero invariants
$SW_{L}(X)$.
\section{Almost Complex and Symplectic Structures}
Now assume that $X$ has an almost complex structure. This means that there is a
principal
$GL(2,\C)$-bundle $Q\longrightarrow X$ such that
$$T(X)\cong Q\times_{GL(2,\C)} \C^{2}$$ By choosing Hermitian metric on
$T(X)$ we can assume $Q\longrightarrow X$ is a
$U(2)$ bundle, and the tangent frame bundle $P_{SO(4)}(TX)$ comes from $Q$
by the
reduction map
$$U(2)=(\;S^{1}\times SU (2) \;)/\Z_{2}\hookrightarrow (\;SU (2)\times SU
(2)\;)/\Z_{2} =SO(4)$$ Equivalently there is an endemorphism $I\in
\Gamma(End(TX))$ with $I^{2}=-Id$
$$\begin{array}{ccc} T(X)& \stackrel{I}{\longrightarrow} & T(X)\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\searrow & &
\swarrow\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\\ & X &
\end{array}$$ The $\;\pm i \;$ eigenspaces of $I$ splits the complexified
tangent
space
$T(X)_{\C}$
$$T(X)_{\C}\cong T^{1,0}(X)\oplus T^{0,1}(X)=\Lambda^{1,0}(X)\oplus
\Lambda^{0,1}(X)$$ This gives us a complex line bundle which is called the
canonical
line bundle:
$$K=K_{X}=\Lambda^{2,0}(X)=\Lambda^{2}(T^{1,0})\longrightarrow X$$
Both $K^{\pm}$ are characteristic; corresponding to line bundle $K
\longrightarrow X$ there is a canonical $Spin_{c}(4)$ structure on
$X$, given by the lifting of $f[\lambda , A ]=([\lambda ,A],
\lambda^{2})$
$$\begin{array}{ccc} && Spin_{c}(4)\\ & & \\ & F \nearrow & l\downarrow\\ & &
\\
U(2) &\stackrel{f}{\longrightarrow} & SO(4)\times S^{1}
\end{array}$$
$\;F[\lambda ,A]=[\lambda , A,\lambda]$. Transition function
$\lambda^{2}$ gives $K$, and the corresponding
${\C}^{2}$-bundles are given by:
\begin{eqnarray*} W^{+}&= & \Lambda ^{0,2}(X)\oplus\Lambda^{0,0}(X) =
K^{-1}\oplus{\C}\\ W^{-}&=&\Lambda^{0,1}(X)
\end{eqnarray*} We can check this from the transition functions, e.g. for
$W^{+}$,
$x=z+jw\in{\H}$
$$x\longmapsto \lambda x \lambda^{-1} =\lambda (z+jw) \bar{\lambda } = z + jw
\bar{\lambda} \bar{ \lambda} = z +jw \lambda ^{-2}$$
Since we can identify $\bar{\Lambda}^{0,1}(X)\cong
\Lambda^{1,0}(X)$, and
$\Lambda^{0,2}(X)\otimes \Lambda^{1,0}(X) \cong \Lambda^{0,1}(X)$ we
readily see the
decomposition
$T(X)_{\C}\cong W^{+}\otimes \bar{W}^{-}$. As real bundles we have
$$ \Lambda^{+}(X)\cong K\oplus {\R}$$
\noindent We can verify this by taking
$\;\{e^{1},e^{2}=I(e^{1}),e^{3},e^{4}=I(e^{3} )\}\;$ to be a local orthonormal
basis for $T^{*}(X)$, then
$$\Lambda^{1,0}(X)=\;<e^{1}-ie^{2},e^{3}-ie^{4}>\;\;\;,and\;\;\;\;
\Lambda^{0,1}(X)=\;<e^{1}+ie^{2},e^{3}+ie^{4}>$$
$$K =\;< f =(e^{1}-ie^{2})\wedge (e^{3}-ie^{4})>$$
$$\Lambda^{+}(X)=\;<\omega=\frac{1}{2}( e^{1}\wedge e^{2} + e^{3}\wedge
e^{4}),\;f_{2}=
\frac{1}{2}( e^{1}\wedge e^{3} + e^{4}\wedge e^{2}),\; f_{3}=
\frac{1}{2}( e^{1}\wedge e^{4} + e^{2}\wedge e^{3})> $$
$\omega $ is the global form $ \omega(X,Y)=g(X,IY) $ where $ g $ is the
hermitian
metric (which makes the basis
$\{e^{1},e^{2},e^{3},e^{4}\} $ orthogonal). Also since
$f= 2(f_{2} -i f_{3}) $, we see as ${\R}^{3}$-bundles
$\Lambda^{+}(X)\cong K\oplus {\R}(\omega)$. We can check:
$$W^{+}\otimes \bar{W}^{+}\cong{\C}\oplus{\C}\oplus K\oplus
\bar{K}= (K\oplus{\R})_{\C}\oplus {\C}$$
As before by writing the sections of $W^{+}$ by $z+jw \in \Gamma ({\C}\oplus
K^{-1})$
we see that $\omega, f_{2},f_{3}$ act as Pauli matrices; in particular
\begin{eqnarray*}
\omega & \longmapsto & {\left(\begin{array}{cc} i &0 \\ 0 &- i
\end{array}\right)}\\ f &\longmapsto &{2\left(\begin{array}{cc} 0 & -1 \\ 1 & 0
\end{array}\right)- 2i\left(\begin{array}{cc} 0 & -i
\\- i & 0 \end{array}\right)=
\left(\begin{array}{cc} 0 & -4 \\ 0 & 0 \end{array}\right)}\\
\bar{f} &\longmapsto &{2\left(\begin{array}{cc} 0 & -1 \\ 1 & 0
\end{array}\right)+ 2i\left(\begin{array}{cc} 0 &- i \\ -i & 0
\end{array}\right)=
\left(\begin{array}{cc} 0 & 0 \\ 4 & 0 \end{array}\right)}\\
\end{eqnarray*}
So in particular, if we write $\psi\in
\Gamma (W^{+})=\Gamma({\C}\oplus K^{-1})$ by $\psi=\alpha u_{0} + {\beta} \;$,
where $\beta$ is a section of $K^{-1}$, and
$\alpha: X\to {\C} $ and $u_{0}$ is a fixed section of the trivial bundle
$\underline{\C}\to X$ with $||u_{0}||=1$, then
$$\rho(\omega )\; u_{0}= iu_{0} \;\;\;\; \rho(\omega
)\;\beta=-i\beta\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$$
$$\rho(\beta)\; u_{0}=4\beta\;\;\;\; \rho(\beta
)\;\beta=0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(*)$$
$$\;\;\;\;\;\;\rho(\bar{\beta})\; u_{0}=0
\;\;\;\;\;\;\;\;\;\rho(\bar{\beta} )
\;\beta =-4\;|\beta |^{2}u_{0}
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$$
\noindent We see these by locally writing $\psi$ in terms of basis
$\;\psi = \alpha u_{0} + \lambda \bar{f} $, where $\beta= \lambda
\bar{f}$ with
$\;||\bar{f}||=1 $. Writing Formula (3) in terms of the basis $\{
\omega, f, \bar{f}\}$ we get:
\begin{eqnarray}i\; \sigma ( \alpha, \lambda)& =&
\rho\;[\;\frac{|\alpha |^{2}-|\lambda |^{2}}{2}\;\omega -
\frac{i}{4}\;\alpha \bar {\lambda} f +\frac{i}{4}
\;\bar{\alpha}{\lambda} \bar{f}\;] \nonumber\\
\sigma (\psi)&=& \rho\;[\;\frac{|\beta |^{2}-|\alpha |^{2}}{2}\;i\;
\omega - \frac{1}{4}\;\alpha \; \bar {\beta} +\frac{1}{4}
\;\bar{\alpha}\; \beta \;]
\end{eqnarray}
If we consider the decomposition $\;F_{A}^{+}=F^{2,0}_{A} +F^{0,2}_{A}
+F^{1,1}_{A}\; $
the equation $\rho(F_{A})=\sigma(\psi)$ gives Witten's formulas:
\begin{eqnarray} F^{2,0}_{A}&=& -\frac{1}{4}\alpha \;\bar{\beta}\\
F^{0,2}_{A}&=&
\;\frac{1}{4}\bar{\alpha} \;\beta\\ F^{1,1}_{A}&=&
\frac{|\beta |^{2}-|\alpha |^{2}}{2}\;i\;\omega
\end{eqnarray}
\vspace{.18in}
In case $\;X\;$ is a Kahler surface the Dirac operator is given by
(c.f.[{\bf LM}])
$$D\!\!\!\!/\,_{A}=\bar{\partial\!\!\!/\,}^{*}_{A} + \bar{\partial\!\!\!/\,}_{A} :\Gamma (W^{+})\to
\Gamma (W^{-})$$ Hence from the Dirac part of the Seiberg-Witten equation
(17) we
have
\begin{eqnarray}
\bar{\partial\!\!\!/\,}_{A}^{*}(\beta) + \bar{\partial\!\!\!/\,}_{A}(\alpha u_{0})&=&0
\nonumber \\
\bar{\partial\!\!\!/\,}_{A} \bar{\partial\!\!\!/\,}^{*}_{A}(\beta) +
\bar{\partial\!\!\!/\,}_{A}\bar{\partial\!\!\!/\,}_{A}(\alpha u_{0})&=&0
\end{eqnarray}
\noindent The second term is $\;
\bar{\partial\!\!\!/\,}_{A}\bar{\partial\!\!\!/\,}_{A}(\alpha u_{0})= F^{0,2}_{A}
\alpha u_{0}= \frac{1}{4} |\alpha|^{2}\beta $. By taking inner product
both sides
of (24) by $\;\beta \;$ and integrating over $X$ we get the
$L^{2}$ norms satisfy
\begin{eqnarray}||\alpha||^{2} ||\beta||^{2}=0\;\;& \Longrightarrow
\;\;\alpha= 0\; \mbox{ or }\;\beta=0
\end{eqnarray} This argument eventually calculates
$\;SW_{K}(X)=1\;$ ([{\bf W}]). We will not repeat this argument here, instead
we
will review a stronger result of C.Taubes for symplectic manifolds below, which
implies this result.
\vspace{.18in}
We call an almost complex manifold with Hermitian metric $\{X, I,g\}$
syplectic if
$d\omega=0$. Clearly a nondegenerate closed form $\omega$ and a hermitian
metric
determines the almost complex structure $I$. Given
$\omega$ then $I$ is called an almost complex structure taming the
symplectic form
$\omega$
\vspace{.1in}
By Section 2.1 there is a unique connection $A_{0}$ in $\; K\longrightarrow
X \;$
such that the induced Dirac operator
$D_{A_{0}}$ on $W^{+}$ restricted to the trivial summand $\underline {\C}\to X$
is the exterior derivative $d$. Let $u_{0}$ be the section of $W^{+}$ with
constant
${\C}$ component and
$||u_{0}||=1$. Taubs's first fundamental observation is
$$D\!\!\!\!/\,_{A} (u_{0})=0 \;\;\;\; \mbox{ if and only if}\;\;\;\;\; d\omega=0 $$
This can
be seen by applying the Dirac operator to both sides of $ i u_{0} =
\rho(\omega).
u_{0}$, and observing that by the choice of $u_{0}$ the term
$\nabla_{A_{0}}(u_{0})$
lies entirely in
$K^{-1}$ component:
\begin{eqnarray*} i D\!\!\!\!/\, _{A_{0}}(u_{0})&=&\sum
\rho(e^{i})\nabla_{i}\;(\rho (\omega )\;u_{0}) \\ &=&\sum \rho (e^{i}) \;[\
\nabla_{i}\;(\rho(\omega) )\;u_{0} +
\rho(\omega)\;\nabla_{i}\;(u_{0})\;]\\ &=& \sum \rho (e^{i})
\nabla_{i}\;(\rho(\omega) ) \;u_{0} -i\;\sum\rho (e^{i}\;)\nabla
_{i}\;(u_{0})\\
2i\;D\!\!\!\!/\, _{A_{0}}(u_{0}) &=& \sum \rho (e^{i})
\nabla_{i}\;(\rho(\omega) )
\;u_{0}=
\rho ( (d +d^{*}) \omega)\;u_{0}=\rho ((d-*d)\omega)\;u_{0}
\end{eqnarray*} Last equality holds since $\omega \in
\Lambda^{+}(X)_{\C}\oplus {\C}$, and by naturality, the Dirac operator on
$ \Lambda^{*}(X)_{\C} $ is
$\;d+d^{*}$, and since $d=-*d*$ on $2$ forms and $\omega$ is self dual
$$2i\;D\!\!\!\!/\, _{A_{0}}(u_{0})=-\rho (*d \omega ) u_{0}$$
\vspace{.15in}
\noindent{\bf Theorem (Taubes) }: Let $(X,\omega)$ be a closed symplectic
manifold
such that $b_{2}(X)^{+}\geq2$, then
$SW_{K}(X)=\pm1 $.
\vspace{.1in}
Proof: $\mbox{Write}\;\;\;\psi=\alpha u_{0} + {\beta} \in \Gamma( W^{+})=
\Gamma ({\C}\oplus K^{-1}) $ where $\alpha: X\to {\C} $, and
$u_{0}$ is the section as above. Consider the perturbed Seiberg-Witten
equations :
For
$(A,\psi)\in {\cal A}(L)\times \Gamma (W^{+})$ :
\begin{eqnarray} D\!\!\!\!/\, _{A}(\psi)&=&0\\
\rho (F_{A}^{+})&=&\rho (F_{A_{0}}^{+})+r\;[\;
\sigma(\psi)+i\;\rho(\omega)\;]
\end{eqnarray} By (20) the second equation is equivalent to:
\begin{eqnarray}
F_{A}^{+}-F_{A_{0}}^{+} &=&r\;\ [\; ( \;\frac{|\beta
|^{2}-|\alpha|^{2}}{2} +1 )\;
i \omega - \frac{1}{4}\alpha
\bar{\beta} +\frac{1}{4} \bar{\alpha} \beta \; ]
\end{eqnarray}
We will show that up to gauge equivalence there is a unique solution to these
equations.
Write $A=A_{0}+a$, after a gauge transformation we can assume that
$a$ is coclosed, i.e.
$d^{*}(a)=0$. Clearly $ (A,\psi)=(A_{0},u_{0})$, and $r=0$ satisfy these
equations.
It suffices to show that for $r\longmapsto \infty $ these equations admit only
$(A_{0},u_{0})$ as a solution. From Weitzenbock formulas (7), (8) and
abbreviating
$\nabla_{A_{0}}(u_{0})=b$ we get
\begin{eqnarray}
D\!\!\!\!/\,_{A}^{2}(\psi)= D\!\!\!\!/\,_{A}^{2}(\beta)+(\nabla _{a}^{*}\nabla_{a}\alpha)u_{0}
-2<\nabla_{a} \alpha,b> + \frac{1}{2}\alpha\;
\rho(F_{A}^{+}-F_{A_{0}}^{+})\;u_{0}\\
D\!\!\!\!/\,_{A}^{2}(\beta) =(\nabla _{A}^{*}\nabla_{A}\;\beta) +\frac{s}{4}\;\beta
+\frac{1}{4} \rho ( F_{A_{0}}^{+})\;\beta +\frac{1}{4} \rho
(F_{A}^{+}-F_{A_{0}}^{+})
\;\beta
\hspace{.3in}
\end{eqnarray}
\noindent From (28) and (*) we see that
\begin{eqnarray}\frac{1}{2}\alpha
\;\rho(F_{A}^{+}-F_{A_{0}}^{+})u_{0}&=&
\frac {r}{4}\alpha \;(\;|\alpha|^{2}-|\beta|^{2}-2)\;u_{0} +
\frac{r}{2} \;|\alpha |^{2}\beta \\
\frac{1}{4}\;\rho(F_{A}^{+}-F_{A_{0}}^{+})\;\beta &= & -
\;\frac{r}{8}\;(\;|\alpha|^{2}-|\beta|^{2}-2)
\;\beta +\frac{r}{4}\alpha |\beta|^{2} u_{0}
\end{eqnarray}
\noindent By substituting (31) in (29) we get
\begin{eqnarray}D\!\!\!\!/\,_{A}^{2}(\psi -\beta)&=& [\;\nabla _{a}^{*}\nabla_{a}\alpha
+
\frac{r}{4}\alpha
\;(|\alpha|^{2}-|\beta|^{2}-2)\;]\;u_{0}\nonumber\\ && -2 <\nabla_{a}
\alpha,b> +\frac{r}{2}\;
|\alpha\;|^{2} \beta
\end{eqnarray}
\noindent By substituting (32) in (30), then substituting (30) in (33) we
obtain:
\begin{eqnarray}0=D\!\!\!\!/\,_{A}^{2}(\psi)&=& [\;\nabla _{a}^{*}\nabla_{a}\alpha +
\frac{r}{4}\alpha
\;(|\alpha|^{2}-2)\;]
\;u_{0}-2 <\nabla_{a} \alpha,b> \nonumber \\ & &
+[\;\nabla _{A}^{*}\nabla_{A}+\frac{s}{4} +
\frac{1}{4}\;\rho(F_{A_{0}}^{+})+\frac{r}{8} (\;3|\alpha|^{2}+|\beta
|^{2}+2)\;]\;\beta
\end{eqnarray}
\noindent By recalling that $\beta$ and $u_{0}$ are orthogonal sections of
$W^{+}$, we take inner product of both sides of (8) with $\beta$ and
integrate over
$X$ and obtain:
\begin{eqnarray*}
\int_{X}(\;|\nabla_{A}\beta\;|^{2} +\frac{r}{8}\;|\beta|^{4} +
\frac{r}{4}|\beta|^{2}+\frac{3r}{8}|\alpha|^{2}|\beta|^{2}\;)&=& \\
2\int_{X}(<<\nabla_{a} \alpha \;,\;b>\;,\;\beta> -\;\frac{s}{4}\;|\beta|^{2}
-\;\frac{1}{4}<\rho(F_{A_{0}}^{+})\beta
\;,\; \beta> & &
\end{eqnarray*}
$$\mbox{Hence}\;\; \int_{X} \;|\nabla_{A}\beta\;|^{2}
+\frac{r}{8}\;|\beta|^{4} +
\frac{r}{4}|\beta|^{2}+\frac{3r}{8}|\alpha|^{2}|\beta|^{2}\;
\leq \int_{X}c_{1}|\beta|^{2}+ c_{2}|\beta | |\nabla_{a} \alpha| $$ where
$c_1$
and $c_2$ are positive constants depending on the Riemanian metric and the base
connection $A_{0}$. Choose $r\gg 1$, by calling $c_{2}=2c_{3}$ we get :
$$ \int_{X}(\;|\nabla_{A}\beta\;|^{2} +\frac{r}{8}\;|\beta|^{4} +
\frac{r}{8}|\beta|^{2}+\frac{3r}{8}|\alpha|^{2}|\beta|^{2}\;)
\leq \int_{X}(c_{1}-\frac{r}{8})\;|\beta\;|^{2}+ 2c_{3}| \beta\; | |\nabla_{a}
\alpha| $$
$$=- \left[ \;(r/8-c_{1})^{1/2}\;|\beta\;|-c_{3}(r/8-c_{1})^{-1/2}\;
|\nabla_{a}
\alpha|\;\right]^{2}+ \frac{c_{3}^{2}}{(r/8-c_{1})} |\nabla_{a} \alpha|^{2}
\leq \frac{C}{r}|\nabla_{a} \alpha|^{2}$$ For some $C$ depending on the
metric and
$A_{0}$. In particular we have
\begin{eqnarray*}
\int_{X}r \;|\beta|^{2} -\frac{8C}{r}\;|\nabla_{a} \alpha|^{2}&\leq &0 \\
8c_{2}\;|\beta \;|\;|\nabla_{a} \alpha|- \frac{8C}{r}|\nabla_{a}
\alpha|^{2}&\leq&
\int_{X}({r}-8c_{1})\;|\beta\;|^{2}
\end{eqnarray*}
\begin{eqnarray}\mbox{Hence}\;\;\;\;\;\;\;\; c_{2}\;|\beta
\;|\;|\nabla_{a} \alpha|- \frac{2C}{r}|\nabla_{a}
\alpha|^{2}&\leq &0
\end{eqnarray}
\vspace{.1in}
Now by self adjointness of the Dirac operator, and by $\alpha
u_{0}=\psi-\beta $ we
get:
\begin{eqnarray} <D\!\!\!\!/\,_{A}^{2}(\psi)\;,\;\alpha u_{0}>&=&<D\!\!\!\!/\,_{A}^{2}(\psi
-\beta)\;,\;\alpha u_{0}>+ <D\!\!\!\!/\,_{A}^{2}\;(\beta)\;,\;\alpha u_{0}>\nonumber\\
&=&<D\!\!\!\!/\,_{A}^{2}(\psi -\beta)\;,\;\alpha u_{0}>+
<\beta\;,\;D\!\!\!\!/\,_{A}^{2}(\psi-\beta)>
\end{eqnarray}
\noindent We can calculate (36) by using (33) and obtain the inequalities:
\begin{eqnarray*} 0=<D\!\!\!\!/\,_{A}^{2}(\psi)\;,\;\alpha
u_{0}>&=&|\nabla_{a}\alpha|^{2} +
\frac{r}{4}|\alpha|^{4}-
\frac{r}{4}|\alpha|^{2}|\beta|^{2}-
\frac{r}{2}|\alpha|^{2}\\ && +
\;\frac{r}{2}\;|\alpha|^{2}|\beta|^{2}-2 <<\nabla_{a} \alpha,b>,
\beta>
\end{eqnarray*}
\begin{eqnarray*}
\int_{X}|\nabla_{a}\alpha|^{2} +\frac{r}{4}|\alpha|^{4}-\frac{r}{2}|\alpha|^{2}
&\leq &\int_{X}2 <<\nabla_{a} \alpha,b>, \beta> -\frac{r}{4}
|\alpha|^{2}|\beta|^{2}\\ &\leq& \int_{X}2 <<\nabla_{a} \alpha,b>, \beta>\leq
\int_{X}c_{2} |\nabla_{a} \alpha|\; |\beta\; |
\end{eqnarray*} By choosing $c_{4}=1-{2C}/{r}$ and by (35), we see
\begin{eqnarray}
\int_{X}c_{4}\;|\nabla_{a}\alpha|^{2}
+\frac{r}{4}|\alpha|^{4}-\frac{r}{2}|\alpha|^{2} &\leq &0
\end{eqnarray} Since for a connection $A$ in $K\longrightarrow X$ the class
$({i}/{2\pi}) F_{A}$ represents the Chern class
$c_{1}(K)$, and since $\omega$ is a self dual two form we can write:
$$\int_{X}\omega \wedge F_{A}=-2\pi\;i\; \omega c_{1}(K) \;\;\;\;\;\;
\int_{X}\omega \wedge F_{A}=\int_{X}\omega \wedge F_{A}^{+}$$
$$\int_{X} \omega \wedge (F_{A}^{+}-F_{A_{0}}^{+})=0$$ By (28) this implies:
\begin{eqnarray}
\frac{r}{2} \int_{X} (2-|\alpha|^{2}+|\beta|^{2})=0
\end{eqnarray}
By adding (38) to (37) we get
\begin{eqnarray}
\int_{X}c_{4}\;|\nabla_{a}\alpha|^{2} + \frac{r}{2}|\beta|^{2} +r (1 -
\frac{1}{2}|\alpha|^{2})^{2}\leq 0
\end{eqnarray}
Assume $r\gg 1$, then $c_{4}\geq 0$ and hence $\nabla_{a}\alpha=0$ and
$\beta=0$ and $|\alpha|=\sqrt{2}$, hence:
$$\beta =0\;\;\mbox{and}\;\;\alpha =\sqrt{2}e^{i\;\theta}\;\;\mbox{ and}\;\;
\nabla_{a}(\alpha)=d(e^{i\;\theta})+i\;a\;e^{i\;\theta}=0$$ Hence
$a= d(-\theta)$, recall that we also have $d^{*}(a)=0$ which gives
$$0 = <d^{*}d(\theta),\theta> = <d(\theta),d(\theta>)>=||d(\theta)||^{2}
$$ Hence
$\;a=0\;$ and
$\;\alpha \;$=constant. So up to a gauge equivalence
$(A,\psi)=(A_{0},u_{0})$
\hspace{4.5in} $\Box$
\section{Applications}
Let $X$ be a simply connected closed smooth $4$-manifold. By
J.H.C.Whitehead the
intersection form
$$ q_{X}:H_{2}(X;\Z)\otimes H_{2}(X;\Z) \longrightarrow \Z$$
determines the homotopy type of $X$. By C.T.C Wall in fact $q_{X}$
determines the
$h$-cobordism class of $X$. Donaldson (c.f. [{\bf DK}]) showed that if
$q_{X}$ is
definite then it is dioganalizable, i.e.
$$q_{X}=p<1>\oplus q<-1>$$ We call $q_{X}$ is even if q(a,a) is even for
all $a$,
otherwise we call
$q_{X}$ odd. Since integral liftings $c$ of the second Steifel Whitney calass
$w_{2}$ of $X$ are characterized by $\;c.a=a.a\;$ for all $a\in H_{2}(X;\Z)$,
the condition of $q_{X}$ being even is equivalent to $X$ being spin. From
classification of unimodular even integral quadratic forms and the Rohlin
theorem it
follows that the intersection form of a closed smooth spin manifold is in
the form:
\begin{eqnarray}q_{X}=2k E_{8}\oplus lH \end{eqnarray} where
$E_{8}$ is the $8\times 8$ intersection matrix given by the Dynikin diagram
\begin{figure}[htb]
\vspace{.8in}
\special{picture 1 scaled 700}
\caption{}
\end{figure}
\noindent and $H$ is the form
$\; H=\left(\begin{array}{cc}0&1\\1&0
\end{array} \right)\;$. The intersection form of the manifold
$S^{2}\times S^{2}$ realizes the form
$H$, and the K3 surface (quadric in ${\C}{\P}^{3}$) realizes
$ 2E_{8}\oplus 3H$. Donaldson had shown that if $k=1$, then $l\geq 3$
([{\bf D}]).
Clearly connected sums of K3 surface realizes
$\;2kE_{8}\oplus 3k H$. In general it is a conjecture that in (40) we must
necessarily have
$l\geq 3k$ (sometimes this is called $11/8$ conjecture). Recently by using
Seiberg-Witten theory M.Furuta has shown that
\vspace{.15in}
\noindent{\bf Theorem (Furuta) }: Let $X$ be a simply connected closed
smooth spin \\
$4$-manifold with the intersection form $q_{X}=2k E_{8}\oplus lH $, then
$l\geq 2k+1$
\vspace{.15in}
Proof: We will only sketch the proof of $\;l\geq 2k$. We pick
$L\longrightarrow X$ to be the trivial bundle (it is characteristic since $X$
is
spin). Notice that the spinor bundles
$$V^{\pm}=P\times _{\rho_{\pm}}{\C}^{2}\longrightarrow X$$
$\rho_{\pm}: x\longmapsto q_{\pm} x $ , are quoternionic vector bundles.
That is,
there is an action $j: V^{\pm}\to V^{\pm}$ defined by $[p,x]\to [p,xj]$,
which is
clearly well defined. This action commutes with
$$\partial\!\!\!/\,: \Gamma(V^{+})\longrightarrow \Gamma(V^{-}) $$ Let
$A_{0}$ be the trivial connection, and write $\;\pm A=A_{0}\pm i\;a\in
\cal{A}(L)$
\begin{eqnarray*}
\partial\!\!\!/\, _{A}(\psi j)&=&\sum \rho (e^{k}) \;[\;\nabla _{k} +i\ a\;]\; (\psi
j)=
\sum \rho (e^{k}) \;[\;\nabla _{k}(\psi)j+ \psi j ia\;] \\ &=& \sum
\rho (e^{k}) \;[\;\nabla _{k}(\psi)j- ia \psi j\;]=
\partial\!\!\!/\,_{-A}(\psi)j
\end{eqnarray*}
$\Z_{4}\;$ action $\;(A , \psi)\longmapsto (-A , \psi j)\;$ on
$\;\Omega^{1}(X)\times \Gamma(V^{+})\;$ preserves the compact set
$${\cal M}_{0}=\tilde {\cal M}\cap \; ker( d^{*}) \oplus \Gamma (V^{+})$$
\vspace{.05in}
\noindent where $\;\;\tilde {\cal M}=\{ (a, \psi)\in
\Omega^{1}(X)\oplus \Gamma(V^{+})\;|\;
\partial\!\!\!/\,_{A}(\psi)=0 \;,\;\;\;\rho(F_{A}^{+})=\sigma(\psi) \;\} $ \\ For
example
from the local description of $\sigma $ in (2) we can check
$$\sigma (\psi j)=\sigma (z+jw)j = \sigma(-\bar{w} + j\bar{z})=-\sigma(\psi)=
-\;F_{A}^{+}=F_{-A}^{+}$$ This $\Z_{4}$ in fact extends to an action of the
subgroup $G$ of
$SU(2)$ which is generated by $<S^{1}\;,\;j>$, where $S^{1}$ acts trivially on
$\Omega^{*}$ and by complex multiplication on $\Gamma(V^{+})$, and
$j$ acts by $-1$ on $\Omega^{*}$ and by quaternionic multiplication on
$\Gamma(V^{+})$ In particular we get a
$G$-equivariant map
$\;\varphi = L + \theta : {\cal V}\to {\cal W} \;$ where:
$$ \varphi : {\cal V}=ker(d^{*})\oplus
\Gamma(V^{+})\longrightarrow {\cal W}= \Omega_{+}^{2}\oplus
\Gamma(V^{-}) $$
$$L=\left( \begin{array}{cc} d^{+} & 0 \\ 0 & \partial\!\!\!/\, \end{array}
\right)\;\;\;
\mbox{and}\;\;\;
\theta (a,\psi)=(\;\sigma (\psi)\;,\; a \psi\;)$$ with
$\varphi^{-1}(0)= {\cal M}_{0}$ and $\varphi (v)=L(v) + \theta (v)$ with
$L$ linear
Fredholm and $\theta $ quadratic. We apply the ``usual" Kuranishi technique (cf
[{\bf L}]) to obtain a finite dimensional local model
$\;V\longmapsto W\;$ for $\varphi$.
We let ${\cal V}=\oplus V_{\lambda }$ and ${\cal W}=\oplus W_{\lambda }$,
where
$V_{\lambda }$ and $W_{\lambda }$ be $\lambda $ eigenspaces of
$L^{*}L :V\to V$ and $LL^{*}:W\to W$
repectively. Since $L L^{*}$ is a multiplication by $\lambda $ on
$V_{\lambda}$, for $\lambda > 0$ we have isomorphisms $L
:V_{\lambda}\stackrel{\cong
}{\longrightarrow} W_{\lambda }$. Now pick
$\Lambda >0$ and consider projections:
$$\oplus_{\lambda\leq
\Lambda}W_{\lambda}\;\;\stackrel{p_{\Lambda}}{\longleftarrow}
\;\; W \;\;
\stackrel{1-p_{\Lambda}}{\longrightarrow}\;\;
\oplus_{\Lambda > \Lambda} W_{\lambda}$$ Consider the local diffeomorphism
$f_{\Lambda}: V \stackrel{ }{\longrightarrow} V$ given by:
$$u=f_{\Lambda}(v)=v+L^{-1}(1-p_{\Lambda})\theta (v)
\;\;\Longleftrightarrow
\;\;L(u)=L(v)+(1-p_{\Lambda})\theta (v)$$ The condition
$\varphi(v)=0$ is equivalent to $\;p_{\Lambda}\;\varphi (v)=0 $ and
$ (1-p_{\Lambda})\;\varphi (v)=0 $, but
\begin{eqnarray*} (1-p_{\Lambda})\;\varphi (v)=0\;\Longleftrightarrow \;
(1-p_{\Lambda})\;L(v) + (1-p_{\Lambda})\;\theta(v)=0
\;\Longleftrightarrow\;\;\;\;\;\;
\;\;\;\;\;\;\;\;\;\;\; \\ (1-p_{\Lambda})\;L(v) + L(u)-L(v) =0
\; \Longleftrightarrow\;L(u)=p_{\Lambda}\;L(v) \;
\Longleftrightarrow\; u\in
\oplus_{\lambda\leq \Lambda}V_{\lambda}
\end{eqnarray*} Hence $\;\varphi
(v)=0\;\Longleftrightarrow\;p_{\Lambda}\;\varphi
(v)=0\; $ and
$\;u\in \oplus_{\lambda\leq \Lambda}V_{\lambda}\;$, let
$$ \varphi _{\Lambda}:V=\oplus_{\lambda\leq \Lambda}
V_{\lambda}\longrightarrow W=
\oplus_{\lambda\leq
\Lambda}W_{\lambda}\hspace{.15in}
\mbox{where}\hspace{.15in}
\varphi _{\Lambda}(u)=p_{\Lambda}\;\varphi\; f_{\Lambda}^{-1}(u)$$
Hence in the local diffeomorphism
$f_{\Lambda}: {\cal O}\stackrel{ \approx}{\longrightarrow} {\cal O}
\subset {\cal V }$ takes the piece of the compact set
$f_{\lambda}({\cal O}\cap {\cal M}_{0} )$ into the finite dimensional subspace
$V\subset {\cal V}$, where ${\cal O}$ is a neighborhood of $(0,0)$.
As a side fact note that near $(0,0)$ we have
$${\cal M}(L) \approx{\cal M}_{0}(L)/S^{1} $$ We claim that for
$\lambda \gg 1 \;$, the local diffeomorphism
$f_{\Lambda}: {\cal O}\stackrel{ \approx}{\longrightarrow} {\cal O}
\subset V $
extends to a ball $B_{R}$ of large radius $R$ containig the compact set
${\cal M}_{0}(L) $, i.e. we can make the zero set
$\varphi_{\Lambda}^{-1}(0)$ a compact set.
We see this by applying the Banach contraction principle. For example for
a given
$u\in B_{R}$, showing that there is $v\in V$ such that
$f_{\Lambda}(v)=u$ is equivalent of showing that the map
$\;T_{u}(v)=u- L^{-1}(1-p_{\Lambda})\theta (v)\;$ has a fixed point. Since
$L^{-1}(1-p_{\Lambda})$ has eigenvalues
$1/\lambda$ on each
$W_{\lambda}$ in appropriate Sobolev norm we can write
$$|| T_{u}(v_{1})- T_{u}(v_{2}) ||\leq \frac{C}{\Lambda }||\theta(v_{1})-
\theta (v_{2})||
\leq \frac{C}{\Lambda }||v_{1}-v_{2}||$$
Vector subspaces $V_{\lambda}$ and $W_{\lambda}$ are either quaternionic or
real
depending on whether they are subspaces of
$\Gamma (V^{\pm})$ or
$\Omega^{*}(X)$. For a generic metric we can make the cokernel of
$\partial\!\!\!/\,$ zero hence the dimension of the kernel (as a complex vector
space) is
$\;ind(\partial\!\!\!/\,)=-\sigma /8=2k\;$, and since $H^{1}(X)=0$ the dimension of
the
cokernel of $d^{+}$ (as a real vector space) is
$b^{+}=l$.
Hence $\;\varphi _{\Lambda}\;$ gives a $G$-equivariant map
$$\varphi: {\H}^{k+y}\oplus {\R}^{x}\longrightarrow {\H}^{y}\oplus
{\R}^{l+x}$$ with
compact zero set. From this Furuta shows that $l \geq 2l+1$. Here we give
an easier
argument of D.Freed which gives a slightly weaker result of $l\geq 2k$. Let
$E_{0}$ and $E_{1}$ be the complexifications of the domain an the range of
$\varphi
$; consider $E_{0}$ and $E_{1}$ as bundles over a point
$x_{0}$ with projections $\pi_{i}:E_{i}\to x_{0}$, and with
$0$-sections
$s_{i}:x_{0}\to E_{i}\;, i=0,1$. Recall $K_{G}(x_{0})=R(G)$, and we have Bott
isomorphisms
$\beta (\rho)=\pi _{i}^{*}(\rho)\;\lambda_{E_{i}}\;$, for $\;i=0,1$ where
$\;\lambda_{E_{i}}\;$ are the Bott classes. By compactness we get an
induced map
$\varphi ^{*}$:
$$\begin{array} {ccc} K_{G}(B(E_{1}),S(E_{1}))&\stackrel{\varphi ^{*}}
\longrightarrow & K_{G}(B(E_{0}),S(E_{0}))\\ &&\\
\approx \;\uparrow \beta & & \approx \;\uparrow \beta \\ R(G) && R(G)
\end{array}$$ Consider $s_{i}^{*}(\lambda_{E_{i}})=
\sum(-1)^{k}\Lambda^{k}(E_{i})=\Lambda_{-1}(E_{i})\in R(G)$, then by some
$\rho$ we have
$$\Lambda_{-1}(E_{1})=s_{1}^{*}(\lambda_{E_{1}})=s_{0}^{*}
\varphi^{*}(\lambda_{E_{1}}) =s_{0}^{*} (\pi
_{0}^{*}(\rho)\;\lambda_{E_{0}})=\rho
\;\Lambda_{-1}(E_{0})$$ So in particular $tr_{j}(\Lambda_{-1}(E_{0}))$ divides
$tr_{j}(\Lambda_{-1}(E_{1}))$. By recalling $\;j:E_{i}\to E_{i}\;$
$$tr_{j}(\Lambda_{-1}(E_{i}))=det (I-j)\;\;\;\;\;\mbox{for}\;\;i=0,1$$ Since
$(z,w)j=(z+jw)j=-\bar{w}+j\bar{z}=(-\bar{w},\bar{z})\;$ $j$ acts on
${\H}\otimes{\C}$ by matrix
$$ A=\left(\begin{array}{cccc}0 &0 &1 &0\\ 0&0&0&-1\\-1&0&0&0\\0&1&0&0
\end{array}\right) $$ so $\;det (I-A)=4$, and $j$ acts on $\;{\R}\otimes
{\C}\;$ by
$j(x)=-x$
so $det (I-(-I))=2\;$. Hence $\;4^{k+y}\;2^{x}\;$ divides
$\;4^{y}\;2^{l+x}\;$ which implies
$\;l\geq 2k\;$ \hspace{1.4in} $\Box$
\vspace{.15in}
There is another nice application of Seiberg-Witten invariants: It is an
old problem
whether the quotient of a simply connected smooth complex surface by an
antiholomorphic involution
$\sigma:\tilde{X}\to \tilde{X}\;$ (an involution which anticommutes with
the almost
complex homomorphism
$\sigma _{*}J=-J\sigma_{*}$) is a "standard" manifold (i.e. connected sums of
$S^{2}\times S^{2}$ and $\;\pm \C\P^{2}\;$). A common example of a
antiholomorphic involution is the complex conjugation on a complex projective
algebraic surface with real coefficients. It is known that the quotient of
$\C\P^{2}$ by complex conjugation is $S^{4}$ (Arnold, Massey, Kuiper); and
for every
$d$ there is a curve of degree $d$ in $\C\P^{2}$ whose two fold
branched cover has a standard quotient ([{\bf A}]). This problem makes
sense only if
the antiholomorphic involution has a fixed point, otherwise the quotient
space has
fundamental group
$\Z_{2}$ and hence it can not be standard. By ``connected sum" theorem,
Seiberg-Witten invariants of ``standard" manifolds vanish, so it is natural
question
to ask whether
Seiberg-Witten invariants of the quotients vanish. Shugang Wang has shown
that this
is the case for free antiholomorphic involutions.
\vspace{.15in}
\noindent {\bf Theorem} ({\bf S.Wang}) Let $\tilde{X}$ be a minimal Kahler
surface
of general type, and $\sigma: \tilde{X}\to \tilde{X}$ be a free antiholomorphic
involution, then the quotient
$X=\tilde{X}/\sigma$ has all Seiberg-Witten invariants zero
\vspace{.12in} Proof: Let $h$ be the Kahler metric on $\tilde{X}$, i.e.
$\omega(X,Y)=h(X,JY)$ is the Kahler form. Then
$\;\tilde{g}=h+\sigma^{*}h\;$ is an invariant metric on $\tilde {X}$ with the
Kahler form
$\;\tilde{\omega}=\omega - \sigma^{*}\omega \;$. Let $g$ be the
``push-down" metric
on $X$. Now we claim that all $SW_{L}(X)=0$ for all $L\to X$, in fact we
show that
there are no solutions to Seiberg-Witten equations for $X$: Otherwise if
$L\longrightarrow (X,g)$ is the characteristic line bundle supporting a
solution
$(A,\psi)$, then the pull-back pair
$(\tilde{A},\tilde{\psi})$ is a solution for the pull-back line bundle
$\tilde{L}\longrightarrow \tilde{X}$ with the pull-back $Spin_{c}$
structure, hence
$$0\leq dim {\cal M_{\tilde{L}}}(\tilde{X})=\frac{1}{4}c_{1}^{2}(\tilde{L})-
\frac{1}{4} (3\sigma(\tilde{X})+2 \chi (\tilde{X}))$$ But
$\tilde{X}$ being a minimal Kahler suface of general type
$3\sigma(\tilde{X})+2 \chi (\tilde{X})=K^{2}_{\tilde{X}}>0\;$, hence
$\;c_{1}^{2}(\tilde{L})>0 $. This implies that
$(\tilde{A},\tilde{\psi})\;
$ must be an irreducible solution (i.e. $\psi \neq 0 $), otherwise
$F^{+}_{\tilde{A}}=0$ would imply
$\;c_{1}^{2}(\tilde{L})<0\;$. Now by (25) the nonzero solution
$\;\psi=
\alpha u_{0} + \beta$ must have either one of ${\;\alpha\;}$ or
$\;\beta\;$ is zero (so the other one is nonzero), and since
$\tilde{\omega}\wedge \tilde{\omega}$ is the volume element:
$$\tilde{\omega}.c_{1}(\tilde{L})=\frac{i}{2\pi}
\int \tilde{\omega}\wedge F^{+}_{\tilde{A}}=
\frac{i}{2\pi}\int \tilde{\omega}\wedge(\frac{|\beta |^{2}- |\alpha
|^{2}}{2})\;i\;\tilde{\omega}
\neq 0$$ But since $\;\sigma^{*} (\tilde{\omega})=-
\tilde{\omega}\;$,
$\;\sigma^{*} c_{1}(\tilde{L})= c_{1}(\tilde{L})\;$, and $\;\sigma
\;$ is an orientation preserving map we get a contradiction \\
$$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\tilde{\omega}.c_{1}(\tilde{L})=\sigma^{*}(\tilde{\omega}.c_{1}(\tilde{L}))=
-\tilde{\omega}.c_{1}(\tilde{L})
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\Box$$
|
1995-10-17T05:20:25 | 9510 | alg-geom/9510011 | en | https://arxiv.org/abs/alg-geom/9510011 | [
"alg-geom",
"hep-th",
"math.AG"
] | alg-geom/9510011 | Pablo A. Gastesi | Indranil Biswas, Pablo Gastesi and Suresh Govindarajan | Parabolic Higgs bundles and Teichm\"uller spaces for punctured surfaces | AMSLaTeX v 2.09, 13 pages, DVI file available at
http://www.imsc.ernet.in/~pablo/ | Trans.Am.Math.Soc. 349 (1997) 1551-1560 | null | TIFR/TH/95-50 | null | In this paper we study the relation between parabolic Higgs bundles and
irreducible representations of the fundamental group of punctured Riemann
surfaces established by Simpson. We generalize a result of Hitchin, identifying
those parabolic Higgs bundles that correspond to Fuchsian representations. We
also study the Higgs bundles that give representations whose image is
contained, after conjugation, in SL($k,\Bbb R$). We compute the real dimension
of one of the components of this space of representations, which in the absence
of punctures is the generalized Teichm\"uller space introduced by Hitchin, and
which in the case of $k=2$ is the Teichm\"uller space of the Riemann surface.
| [
{
"version": "v1",
"created": "Sat, 14 Oct 1995 18:35:25 GMT"
}
] | 2007-07-31T00:00:00 | [
[
"Biswas",
"Indranil",
""
],
[
"Gastesi",
"Pablo",
""
],
[
"Govindarajan",
"Suresh",
""
]
] | alg-geom | \section{Introduction}
In the well-known paper \cite{H1}, Hitchin introduced Higgs
bundles, and established a one-to-one correspondence between
equivalence classes of irreducible GL($2,\Bbb{C}$) representations of
the fundamental group of a compact Riemann surface and
isomorphism classes of rank two stable Higgs of degree zero.
In \cite{S2}, Simpson defined parabolic Higgs bundles, which
generalized Hitchin's correspondence to the case of
open Riemann surfaces (see also \cite{S1}).
Here, by an open Riemann surface we mean the complement of
finitely many points in a compact surface. More
precisely, Simpson identified what he calls filtered local
systems with parabolic Higgs bundles.
In \cite{H2}, Hitchin identified the Higgs bundles
corresponding to the Fuchsian representations.
Our main aim here is to generalize his results to the case of open
Riemann surfaces.
Before giving more details, we describe the result of Hitchin on
Fuchsian representations.
Let $X$ be a compact Riemann surface of genus $g \geq 2$, and let
$L$ be a line bundle on $X$ such that $L^2 = K_X$, that is $L$
is a square root of the canonical bundle of $X$. Define $$ E
\, := \, L^* \oplus L$$ which is a rank $2$ bundle on $X$. For
$a\in H^0(X,K^2)$, let $${\theta}(a) \, :=\, \left(\begin{matrix}0 &
1\\ a & 0\\ \end{matrix}\right) \, \in \,H^0\big({\bar X},
End(E)\otimes K\big)$$ be the Higgs field. Hitchin proved that
the conjugacy classes of Fuchsian representations of $\pi_1(X)$
(homomorphisms of ${\pi}_1(X)$ into PSL$(2,\Bbb R$) such that the
quotient of the action on the upper half plane is a compact
Riemann surface of genus $g$) correspond to the Higgs bundles of
the form $(E,{\theta}(a))$ defined above. Moreover, the Higgs
bundle $(E,{\theta}(0))$ corresponds to the Fuchsian representation
for the Riemann surface $X$ itself.
Consider now an open Riemann surface $X = {\bar X} - \{ p_1, \ldots, p_n
\}$, where $\bar X$ is a compact surface of genus $g$, and $p_1, \ldots,
p_n$ are $n>0$ distinct points of $\bar X$. Let $D$ denote the divisor
given by these points, {\it i.e.\/}\ $D=\{p_1,\dots,p_n\}$.
We will further assume that
$2g-2+n>0$, which is equivalent to the condition that the
universal covering space of $X$ is (conformally equivalent to) the upper
half plane.
Consider the bundle $$E\,\, :=\,\, (L\otimes {\cal O}_X(D))^*\,
\oplus \,L $$ and give parabolic weight $1/2$ to the fiber
$E_{p_i}$, $1\leq i \leq n$. For $a\in H^0({\bar X},K^2\otimes
{\cal O}_{\bar X}(D))$ let $${\theta}(a) \, :=\, \left(\begin{matrix}0 & 1\\ a
& 0\\ \end{matrix}\right) \, \in \,H^0\big({\bar X}, End(E)\otimes
K\otimes {\cal O}_X(D)\big) $$ be a parabolic Higgs field on the
parabolic bundle $E$.
We prove that under the identification between filtered local
systems and parabolic Higgs bundles, Fuchsian representations of
$n$-punctured Riemann surfaces are in one-one correspondence
with parabolic Higgs bundles of the type $(E,{\theta}(a))$ defined above.
Moreover, the parabolic Higgs bundle $(E,{\theta}(0))$ corresponds to the
Fuchsian representation of the punctured surface $X$ itself.
Thus this is a direct generalization of the result of Hitchin on
Fuchsian representations of compact Riemann surfaces to the
punctured case.
In section $3$, we generalize the above results to the case of
representations of the fundamental group of the surface $X$ into
PSL($k,\Bbb R$), for $k>2$. More precisely, we consider a parabolic
bundle $W_k$,
obtained by tensoring the $(k-1)$th symmetric product of the bundle
$E$ defined above with an appropriate power of ${\cal O}_{\bar X}(D)$.
The Higgs fields we consider are generalizations of the $2$-dimensional
case, namely they are of the form
$$\theta(a_2,\ldots,a_{k-1}):=\left(\begin{matrix}0 & 1 & \cdots & 0 \\
0 & 0 & 1 & \vdots \\
\vdots & & & 0 \\
a_k & \cdots & a_2 & 0 \end{matrix}\right) ,$$
$a_j$ is a section of the line bundle
$K^j \otimes ({\cal O}_{{\bar X}(D)})^{j-1}$.
As in section $2$, we have that
the pair $(W_k,\theta(a_2, \ldots, a_k))$ is a stable parabolic bundle
of parabolic degree $0$. It is not difficult to see that the parabolic
dual of $W_k$ is naturally isomorphic to
the parabolic bundle $W_k$ itself. This
implies that the holonomy of the flat connection corresponding to these
bundles is contained in a split real form of SL($k,\Bbb C$),
which is isomorphic to SL($k,\Bbb R$).
We prove that one of the components of the space of
representations of $\pi_1(X)$ into SL$(k,\Bbb R)$, with fixed
conjugacy class of monodromy
around the punctures, has real dimension equal to
$2(k^2-1)(g-1)+k(k-1)n$. Observe that for $k=2$, this dimension is
$2(3g-3+n)$, which is precisely the real dimension of ${\cal T}_g^n$,
the Teichm\"uller space of compact surfaces of genus $g$ with $n$
punctures. It is therefore natural, following \cite{H2}, to call the
above component the {\it Teichm\"uller component} of the corresponding
space of representations. Further study of this set is worthwhile.
\section{Higgs bundles for Fuchsian representations}
Let $\bar X$ be a compact Riemann surface of genus $g$, and let $$D \,\,
:= \,\, \{p_1,p_2,\ldots ,p_n\}$$ be $n$ distinct points on $\bar X$.
Define $X := {\bar X} - D$ to be the punctured Riemann surface given by
the complement of the divisor $D$.
We will assume that $2g-2+n > 0$, that is, the surface $X$ supports a
metric of constant curvature $(-4)$.
The degree of the holomorphic cotangent bundle $K$, of $\bar X$ is
$2g-2$. Therefore, there is a line bundle $L$ on $\bar X$ such that
$L^2 = K$. Fix such a line bundle $L$. Note that any two of the
$4^g$ possible choices of $L$ differ by a line bundle of order
$2$.
Using $L$ we will construct a parabolic Higgs bundle on $\bar X$, as
follows.
Let $\xi = {\cal O}_{\bar X}(D)$ denote the line bundle on $\bar X$ given by
the divisor $D$. Define
\begin{equation} E\,\, :=\,\, (L\otimes \xi)^*\,
\oplus \,L \label{eq:bundle}\end{equation}
to be the rank 2 vector bundle on
$\bar X$. To define a parabolic structure on $E$
(we will follow the definition of parabolic Higgs bundle in \cite{S2}),
on each point
$p_i\in D$, $1\leq i \leq n$, we consider the trivial flag
$$ E_{p_i} \, \supset \, 0,$$
and give parabolic weight $1/2$ to
$E_{p_i}$. This gives a parabolic structure on $E$.
Note that
\begin{equation} Hom(L , L^{*}\otimes {\xi}^*)\otimes K\otimes\xi
\, =\, {\cal O} \, \subset \, End(E)\otimes K\otimes \xi
\label{eq:hom}\end{equation}
Let $1$ denote the section of $\cal O$ given by the
constant function $1$. So from (\ref{eq:hom}) we have
\begin{equation} \theta \, :=\,
\left(\begin{matrix}0 & 1\\ 0 & 0\\ \end{matrix}\right) \, \in
\,H^0\big({\bar X}, End(E)\otimes K\otimes\xi\big) \label{eq:theta}
\end{equation}
\begin{lemma} The parabolic Higgs bundle
$(E, \theta)$ is a parabolic stable Higgs bundle of parabolic
degree zero.\label{lemma:stable}\end{lemma}
\begin{pf} From the definition of parabolic degree
(see \cite[definition 1.11]{MS} or \cite{S2}) we immediately
conclude that the parabolic degree of $E$ is zero.
To see that $(E, \theta)$ is stable, first note that there is only
one sub-bundle of $E$ which is invariant under $\theta$, namely the summand
$(L\otimes \xi)^*$ in (2.1). (A sub-bundle $F \subset E$ is
called {\it invariant} under $\theta$ if $\theta (F) \subset F\otimes
K\otimes \xi$.) The degree of $(L\otimes \xi)^*$ is $1-g -n$. So
the parabolic degree of $(L\otimes \xi)^*$, for
the induced parabolic structure, is $1-g-n/2$.
But, from our assumption that $2g-2+n > 0$ we have $1-g-n/2 <
0$. So $(E,\theta)$ is stable.
\end{pf}
{}From the proof of the \lemref{lemma:stable} it follows that $(E,\theta)$
constructed above is stable if and only if $2g-2+n > 0$. We will
show later that this corresponds to the fact that $X$ admits a
complete metric of constant negative curvature if and only if
$2g-2+n > 0$.
{}From the main theorem of \cite[pg. 755]{S2} we know that there
is a tame harmonic metric on the bundle $E$. (See the Synopsis
of that paper for the definition of tame harmonic metric.)
It is well-known that there is an unique complete K\"ahler metric
on $X$, known as the {\it Poincar\'e} {\it metric},
such that its curvature is $(-4)$.
Both the bundles $L$ and $(L\otimes \xi)^*$ are equipped with
metrics induced by the tame harmonic metric on $E$. So $$
Hom(L,(L\otimes \xi)^*) \, =\, L^2\otimes{\xi}^* \, =\, T\otimes
\xi$$ is equipped with a metric. The restriction to $X$ of the line
bundle $\xi$, and hence ${\xi}^*$, on $\bar X$
has a canonical trivialization. Therefore we have a hermitian
metric on $T_X$ the tangent bundle of $X$. We will denote
this hermitian metric on $T_X$ by $H$. Note that $H$ is
singular at $D$, {\it i.e.\/}\ does not induce a hermitian metric on
$T_{\bar X}$.
\begin{lemma} The hermitian metric $H$ on
the holomorphic tangent bundle on $X$ obtained above is the
Poincar\'e metric.
\label{lemma:metric}\end{lemma}
\begin{pf} We recall the Hermitian-Yang-Mills
equation which gives the harmonic metric on $E$ \cite{S2}. This
equation was first introduced in \cite{H1}.
Let $\nabla$ denote the holomorphic hermitian connection on the
restriction of $E$ to $X$ for the harmonic metric. Then the
Hermitian-Yang-Mills equation of the curvature of $\nabla$ is the following:
\begin{equation} K(\nabla) \, :=\, {\nabla}^2 \, = \, -\, [\theta,
{\theta}^*]\label{eq:yang}\end{equation}
If the decomposition (\ref{eq:bundle}) is orthogonal
with respect to the metric, then $[\theta, {\theta}^*]$ is a 2-form
with values in the diagonal endomorphisms of $E$ (diagonal for
the decomposition (\ref{eq:bundle})). Using this, the equation
(\ref{eq:yang})
reduces to the following equation on $X$
\begin{equation} F_H \,=\, -2{\bar H},
\label{eq:curv}\end{equation} where $H$ is a hermitian metric on $T_X$ and
$\bar H$ is the $(1,1)$-form on $X$ given by $H$. Observe that
$\bar H$ also denotes the K\"ahler $2$-form for the metric $H$.
A metric $H'$ on $T_X$ induces a metric on $L$. Since the bundle
$\xi$ has a natural trivialization over $X$, the metric $H'$
also induces a metric on $(L\otimes\xi)^*$, and therefore also
on $E$. If $H'$ satisfies the equation (\ref{eq:curv}) then the
metric on $E$ obtained this way satisfies (\ref{eq:yang}). Now from the
uniqueness of the solution of (\ref{eq:yang}) (\cite{S2}), we have that
such metric is obtained from the solution of (\ref{eq:curv}) in the above
fashion.
{}From the computation in Example (1.5) of \cite[pg. 66]{H1}, we
conclude that the K\"ahler metric $H$ on $X$ has Gaussian
curvature $(-4)$.
So in order to complete the proof of the lemma we must show that
the K\"ahler metric on $X$ is complete.
Recall the asymptotic behavior of the harmonic metric near the
punctures given in Section 7 of \cite{S2}. First of all, observe
that the fiber of $K\otimes \xi$ at any
$p_i \in D$ is canonically isomorphic to $\Bbb C$. So the fiber
$(End(E)\otimes K\otimes \xi)_{p_i}$ is $End(E_{p_i})$. The
evaluation of the section ${\theta}$ at $p_i$ as an element of
$End(E_{p_i})$ is defined to be the residue of $\theta$ at $p_i$.
For the Higgs field $\theta$, we have that the residue
at each $p_i$ is $$N \, := \, \left(\begin{matrix}0 & 1\\ 0 &
0\\ \end{matrix}\right).$$
In \cite[pg. 755]{S2}, Simpson studies parabolic Higgs
bundles with residue $N$ as above. Consider the displayed
equation in page 758 of \cite{S2}, which describes the asymptotic
behavior of the harmonic metric. Using the fact that the
parabolic weight of $E_{p_i}$ is $1/2$ we conclude that for the
metric on $L$ induced by the tame harmonic metric on
$E$, both $a_i$ and $n_i$ in the equation in page 758 of
\cite{S2} are $1/2$. (We also use the fact that, in the notation
of \cite[pg. 755]{S2}, $L \subset W_1$ and $L$ is not contained in
$W_0$.) In other words, in a suitable trivialization of $L$
on an open set containing a puncture $p_i \in D$, and with
holomorphic coordinate $z$ around $p_i$, the hermitian metric on
$L$ obtained by restricting the harmonic metric on $E$ is
$$r^{1/2}|{\mathrm{log}}(r)|^{1/2},$$ where $r = |z|$.
Similarly, for $(L\otimes\xi)^*$, the $a_i$ and $n_i$ in
the equation \cite[pg. 758]{S2} are $1/2$ and $-1/2$
respectively.
So the metric on $Hom(L,(L\otimes\xi)^*)$ is $({\mathrm{
log}}|(r)|)^{-1}$. Recall the earlier remark that ${\xi}^*$ has a
natural trivialization on $X$. The section of ${\xi}^*$ on
$X$ has a pole of order $1$ at the points of $D$,
when it is considered as a
meromorphic section of ${\xi}^*$ on $\bar X$. This implies that the
hermitian metric on $T = L^{-2}$ is
\begin{equation} r^{-1}|{\mathrm{log}}(r)|^{-1}.
\label{eq:metric}\end{equation}
But this is the expression the Poincar\'e metric
of the punctured disk in $\Bbb C$. This proves that the K\"ahler
metric on $X$ induced by $H$ is indeed complete. This completes
the proof of the lemma.
\end{pf}
{}From the decomposition (\ref{eq:bundle}) it follows that
\begin{equation} Hom(L^{*}\otimes
{\xi}^* ,L)\otimes K\otimes\xi \, =\, K^2\otimes {\xi}^2 \,
\subset \, End(E)\otimes K\otimes \xi
\label{eq:hom2}\end{equation}
Note that the bundle $\xi$ has a natural section, which we will denote
by $1_{\xi}$. We may imbedd $H^0({\bar X} ,K^2\otimes \xi)$ into
$H^0({\bar X} ,K^2\otimes {\xi}^2)$ by the homomorphism $s \longmapsto
s\otimes 1_{\xi}$. So using (\ref{eq:hom2}) we have a natural homomorphism
\begin{equation} \rho \, :\, H^0({\bar X}, K^2\otimes \xi) \,
\longrightarrow \, H^0({\bar X} ,End(E)\otimes K\otimes \xi)
\label{eq:rho} \end{equation}
Note that the image
of $\rho$ is contained in the image of the inclusion $$H^0({\bar X},
End(E)\otimes K) \, \longrightarrow \,H^0({\bar X},
End(E)\otimes K\otimes \xi)$$ With a slight abuse of notation,
for any $a\in H^0({\bar X},K^2\otimes \xi)$, the corresponding element in
$H^0({\bar X} ,End(E)\otimes K)$ will also be denoted by $\rho (a)$.
The following theorem is a generalization of theorem (11.2) of
\cite{H1} to the case of open Riemann surfaces.
\begin{thm} For any $a \in
H^0({\bar X} ,K^2\otimes \xi)$, the Higgs structure $${\theta}_a \,
:=\,{\theta}
+ {\rho}(a) \, =\, \left(\begin{matrix}0 & 1\\ 0 & 0\\
\end{matrix}\right) + {\rho}(a)$$ on the parabolic bundles $E$
(defined in (\ref{eq:bundle})) makes $(E,{\theta}_a)$ a parabolic stable Higgs
bundle of parabolic degree zero.
Let $H_a$ denote the harmonic metric (given by the main theorem of
\cite{S2}) on the restriction of $E$ to $X$, and let $h$ denote
the K\"ahler metric on $X$ induced by the tame harmonic metric
${H}_{a}$ as in \lemref{lemma:metric}. Then the following holds :
\begin{enumerate}
\item{} The section of the $2$-nd symmetric power of the complex
tangent bundle $$h_a \, :=\, a+h+{\bar a}+a{\bar a}/h \, \in
\, {\Omega}^0(X, S^2T^*\otimes\Bbb C)$$ is a Riemannian metric on $X$.
\item{} The metric $h_a$ is a complete Riemannian metric of
constant Gaussian curvature $(-4)$. The Riemann surface
structure on $X$ given by metric $h_a$ is a Riemann surface with
punctures, {\it i.e.\/}\ there are no holes. (A Riemann surface with a hole
is a complement of a disk in a compact Riemann surface.)
\item{} Associating to $a \in H^0({\bar X} ,K^2\otimes \xi)$ the complex
structure on the $C^{\infty}$ surface $X$ given by the metric
$h_a$, the map obtained from $H^0({\bar X} ,K^2\otimes \xi)$ to the
Teichm\"uller space ${\cal T}^n_g$ of surfaces of genus $g$ and $n$
punctures
is a bijection.
\end{enumerate}
\label{thm:main}\end{thm}
\begin{pf} To prove that $(E,{\theta}_a)$ is stable we
use a trick of \cite{H2}. For $\mu >0$, define an
automorphism of $E$ by $$T\, := \, \left(
\begin{matrix}1 & 0\\ 0 & \mu\\ \end{matrix}\right) .$$
The parabolic Higgs bundle $(E,{\theta}_a)$ is
isomorphic to $(E,T^{-1}\circ {\theta}_a\circ T)$, and hence
$(E,T^{-1}\circ {\theta}_a\circ T)$ is parabolic stable if and
only if $(E,{\theta}_a)$ is so. Since $\mu \neq 0$, we have
$(E,T^{-1}\circ {\theta}_a\circ T)$ is parabolic stable if and only
if $(E,\frac{1}{\mu} T^{-1}\circ {\theta}_a\circ T)$ is parabolic stable.
Now $$1/\mu T^{-1}\circ {\theta}_a\circ T\,=\,\left(\begin{matrix}
0 & 1 \\ 0 & 0\\ \end{matrix}\right) +{\rho}(a) /\mu \, = \,
{\theta}_{a/\mu}.$$ But from the openness of the stability
condition we have that since $(E, \theta)$ is stable [\lemref{lemma:stable}],
there is a non-empty open set $U$ in $H^0({\bar X} ,K^2\otimes \xi)$
containing the origin such that for any $a\in U$, the bundle
$(E,{\theta}_a)$ is parabolic stable. Taking $\mu$ to be sufficiently
large so that ${\theta}_{a/\mu} \in U$, we conclude that any
$(E,{\theta}_a)$ is parabolic stable.
The bundle $E$ is equipped with the harmonic metric $H_a$, and
$K$ has a metric induced by $h_a$. Using these metrics we
construct a hermitian metric on $End(E)\otimes K$. Since $\rho(a)
\in H^0({\bar X} ,End(E)\otimes K)$, we may take its pointwise norm.
To prove the statement ($1$) we first want to calculate the
behavior of $||\rho(a)||$ near the punctures. Since $\rho(a)
\in H^0({\bar X} ,End(E)\otimes K)$, we have $$
{\mathrm{residue}}\, ({\theta}_a)\, =\, {\mathrm{residue}}\,
({\theta}) \, =\, N.$$ So the
two hermitian metrics $H_0$ (corresponding to $a=0$) and $H_a$
on $E$ are mutually bounded, {\it i.e.\/}\ $C_1.H_0 \leq H_a \leq C_2.H_0$
for some constants $C_1$ and $C_2$. (Recall that the metric in
\lemref{lemma:metric} was induced by $H_0$.) From this it is easy to check
that around any puncture $p_i$, the norm $||\rho(a)||$ is bounded
by $r|{\mathrm{log}}(r)|^{3/2}$. This implies that $||\rho(a)||$
converges to zero as we approach a puncture.
Arguing as in
($11.2$) of \cite{H1}, if $h_a$ is not a metric then $$1\, - \,
||\rho(a)|| \, \leq\, 0$$ at some point $x\in X$. Since $||\rho(a)||$
converges to zero as we approach a puncture, the infimum of the
function $1 - ||\rho(a)||$ on $X$ must be attained somewhere, say
at $x_0\in X$.
Let $\Delta$ denote the Laplacian operator acting on smooth
functions on
$X$. Since the operator ${\cal L} := - \Delta -4.||\rho(a)||^2$ is
uniformly elliptic on $X$, we may apply \cite[Section VI.3.,
Proposition 3.3]{JT} for the operator ${\cal L}$ and the point
$x_0$.
We conclude that either $1 - ||\rho(a)|| >0$ or $1 - ||\rho(a)||$
is a constant
function. This proves that $h_a$ is a Riemannian metric on $X$.
{}From the computation in the proof of Theorem (11.3)(ii) of
\cite[pg. 120]{H1}, we conclude that
$h_a$ is a metric of curvature $(-4)$.
To complete the proof of the statement ($2$) we must show that
$h_a$ is complete and it has finite volume. (If the volume of
the Poincar\'e metric on a Riemann surface is finite then the
Riemann surface is a complement of finite number of points in a
compact Riemann surface. In particular, the Riemann surface can
not have any holes.)
The above established fact that the metrics $H_0$ and $H_a$
on $E$ are mutually bounded, together with \lemref{lemma:metric} imply
that the Riemannian metric $h_a$ and the Poincar\'e metric on
$X$ are mutually bounded. Since the Poincar\'e metric is
complete and of finite volume, the same must hold for $h_a$.
To prove the statement ($3$) we have to show that map from $H^0({\bar X}
,K^2\otimes \xi)$ to the Teichm\"uller space ${\cal T}^n_g$ obtained
in ($2$) is surjective. This will follow from Section $3$ where we
will prove that the image is both open and closed, and hence it
must be surjective as ${\cal T}^n_g$ is connected.
However we may also use the argument in \cite[Theorem
(11.2)(iii)]{H1} to prove statement (3). Let $h_0$ denote the
Poincar\'e metric on $X$. Indeed, to make the argument work all
we need to show is the following generalization of the
Eells-Sampson theorem to punctured Riemann surfaces:
given a complete Riemannian
metric $h$ of constant curvature $(-4)$ and finite volume on the
$C^{\infty}$ surface $X$, there is a unique diffeomorphism $f$,
of $X$ homotopic to the identity map, such that $f$ is a harmonic
map from $(X,h_0)$ to $(X,h)$.
This follows from the generalization of the theorem of Corlette,
\cite{C}, to the non-compact case as mentioned in \cite[pg. 754]{S2}.
Let $(V,\nabla)$ be the flat rank two bundle given by the Fuchsian
representation for the Riemann surface $(X,g)$. Let $H$ be the
harmonic metric on $V$ given by the main theorem of \cite{S2}
(pg. $755$) for the flat bundle $(V,\nabla)$ on the Riemann surface
$(X,h_0)$. In other words, $H$ gives a section, denoted by $s$,
of the associated bundle with fiber SL$(2,{\Bbb R})/$SO($2$) = $\Bbb H$,
where $\Bbb H$ is the upper half plane. This section $s$ gives the
harmonic map $f$ mentioned above. This completes the proof of
the theorem.
\end{pf}
The vector space $H^0({\bar X} ,K^2\otimes \xi)$ has a natural complex
structure. So does the Teichm\"uller space ${\cal T}_g^n$.
The identification of $H^0({\bar X} ,K^2\otimes \xi)$
with ${\cal T}^n_g$ given by Theorem 2.11 does not preserve the
complex structures. Indeed, ${\cal T}^n_g$ is known to be
biholomorphic to a bounded domain in ${\Bbb C}^{3g-3+n}$. Since any
bounded holomorphic function on an affine space must be
constant, the identification in Theorem 2.11 is never
holomorphic.
\noindent {\bf Remark}\, The parabolic dual of the
parabolic bundle $E$ is $E^*\otimes {\xi}^*$ with trivial
parabolic flag and parabolic weight $1/2$ at the parabolic
points $p_i$, $1\leq i \leq n$. So the parabolic dual of $E$ is
$E$ itself. Any parabolic Higgs bundle $(E, {\theta}_a)$ (as in
\thmref{thm:main}) is naturally isomorphic to the parabolic Higgs
bundle $(E^*, {\theta}^*_a)$, where $E^*$ is the parabolic dual of
$E$. This implies that the holonomy of the flat connection on
$X$ corresponding to the Higgs bundle $(E,{\theta}_a)$ is contained
(after conjugation) in SL$(2,\Bbb R$). This of course is also
implied by \thmref{thm:main} since the image of a Fuchsian
representation is contained in PSL$(2,\Bbb R$).
\section{Higgs bundles for
SL(${\load{\normalsize}{\it}k},\Bbb R$) representations}
Recall the bundle $E$ of section 2, which was defined by $E = (L \otimes
\xi)^* \oplus L$, where $L$ is a (fixed) square root of the canonical
bundle $K$, and $\xi={\cal O}_{\bar X}(D)$. The ($k-1$)-th
symmetric product of ${\Bbb C}^2$ produces an embedding of
SL($2,\Bbb R$) into SL($k,\Bbb R$), via action on homogeneous polynomials of
degree $k$. Let $V_k$ denote the bundle given by the ($k-1$)-th
symmetric product of $E$, that is $V_k:=S^{k-1}(E)$.
At each point $p_i\in D$ we have the trivial flag
$(V_k)_{p_i}\supset 0$, $1\leq p_i \leq n$, with weight
equal to $\frac{k-1}{2}$. In order to construct a parabolic bundle, we
need to reduce the weight to a number in the interval $[0,1)$. We do
this by tensoring $V_k$ with $\xi^{m(k)}$, where $m(k)$ is equal to
$\frac{k}{2}-1$, if $k$ is even, or $\frac{k-1}{2}$, if $k$ is odd. We
will denote the bundle $V_k\otimes\xi^{m(k)}$ by $W_k$. At each point
$p_i\in D$, we take the trivial flag $(W_k)_{p_i} \supset 0$
of $W_k$, with weight equal to
$\frac{1}{2}$, if $k$ is even, or $0$, if $k$ is odd.
Considering $1$ as the section of $\cal O$ given by the
constant function $1$, we can define
\begin{equation}
\theta(0,\ldots,0):=\left(\begin{matrix}0 & 1 & \cdots & 0 \\
0 & 0 & 1 & \vdots \\
\vdots & & & 1 \\
0 & 0 & \cdots & 0 \end{matrix}\right) ,\label{eqn:theta0}\end{equation}
which represents an element of $H^0\big( {\bar X}, End(W) \otimes K \otimes
\xi)$.
\begin{lemma}The bundle $(W_k,\theta(0,\ldots,0))$ is a parabolic stable
Higgs bundle of para-\newline bolic degree zero.\end{lemma}
\begin{pf}
If $k$ is even, we have that the parabolic degree of $W_k$ is
equal to $\frac{k(k+1)}{2}n + \frac{k}{2}(k+1)n=0$.
In the case of odd $k$,
it is easy to see that the degree (as a bundle)
of $W_k$ is $0$, and since the weight is equal to $0$,
we get that the parabolic degree of $W_k$ is zero.
The invariant proper sub-bundles of \ref{eqn:theta0} are
$L^{1-k}\otimes \xi^{-k/2},\ldots,
L^{1-k}\otimes \xi^{-k/2}\oplus\cdots\oplus
L^{(k/2)-1}\otimes\xi^{(k-4)/2}$, if
$k$ is even; or
$L^{1-k}\otimes \xi^{(1-k)/2},\ldots,
L^{1-k}\otimes \xi^{(1-k)/2}\oplus\cdots\oplus
L^{k-3}\otimes\xi^{(k-3)/2}$,
if $k$ is odd.
It is not difficult to see that all these sub-bundles have negative
parabolic degree.
\end{pf}
Using the natural section $1_\xi$ of $\xi$, we embed the spaces
$H^0({\bar X},K^j \otimes \xi^{j-1})$, $j=2,\ldots,k$, into $H^0({\bar
X},End(W_k)\otimes K \otimes \xi)$.
By an abuse of notation, if $a_j \in H^0({\bar X},K^j \otimes
\xi^{j-1})$, we understand the above embedding as producing an element
\begin{equation}
\theta(a_2,\ldots,a_{k-1}):=\left(\begin{matrix}0 & 1 &
\cdots & 0 \\
0 & 0 & 1 & \vdots \\
\vdots & & & 1 \\
a_k & \cdots & a_2 & 0 \end{matrix}\right)\label{eqn:thetaa}\end{equation}
of $H^0({\bar X},End(W_k) \otimes K \otimes \xi)$.
Now, by the arguments of Hitchin, based on the
openness of the stability of bundles, we get that the pair $(W_k,
\theta (a_2,\ldots,a_k))$ is a stable parabolic Higgs bundle of
parabolic degree $0$.
Using these special Higgs bundles, one can obtain some information about
the space of representations of the fundamental group of $X$ into
SL($k,\Bbb R$). More precisely, our result is as follows.
\begin{prop}
The space of representations of the fundamental group of $X$ in
SL($k,\Bbb R$), with fixed conjugacy class of monodromy around the punctures,
has a component of real dimension
$2(k^2-1)(g-1)+k(k-1)n$.
\end{prop}
\begin{pf}
By the work of Simpson \cite{S2} and Balaji Srinivasan
\cite{B}, we have a one-to-one continuous
correspondence
between the space $M$ of stable parabolic Higgs bundles of degree zero,
and the space of representations of the fundamental group of $X$ into
SL($k,\Bbb C$).
Consider the parabolic dual of $W_k$, which is constructed as follows.
First, take the dual bundle $W_k^*$ of $W_k$.
If $k$ is odd, since the weight of
the flag is $0$, we have that the parabolic dual of $W_k$ is $W_k^*$,
with trivial flag at the points $p_i\in D$, and weight equal to zero.
If $k$ is even, we have a weight of $-\frac{1}{2}$ associated to the
trivial flag of $W_k^*$.
Tensor $W_k^*$ with $\xi$ to obtain that the parabolic dual
of $W_k$ is $W_k^* \otimes \xi$. So we always have that the parabolic
dual of the bundle $W_k$ is $W_k$ itself. This implies that the image of
the fundamental group under the representation induced by $(W_k,\theta)$
lies in SL($k,\Bbb R$).
Since $a_j$ is a section of $K^j \otimes \xi^{j-1}$, we have that the residue
of the Higgs field is invariant, {\it i.e.\/}\
$${\mathrm{residue}}\,
(\theta(a_2,\ldots,a_{k-1})) =
{\mathrm{residue}}\, (\theta(0,\ldots,0)) =
\left(\begin{matrix}0 & 1 & \cdots & 0 \\
0 & 0 & 1 & \vdots \\
\vdots & & & 1 \\
0 & 0 & \cdots & 0 \end{matrix}\right) .$$ This implies that in the above
representation, the conjugacy class of the elements corresponding to
small loops around the punctures of $X$
is invariant. By the embedding of SL($2,\Bbb R$) into SL($k,\Bbb R$),
we have that this is the class of the element
\begin{equation}
\left(\begin{matrix}1 & 1 & \cdots & 0 \\
0 & 1 & 1 & \vdots \\
\vdots & & & 1 \\
0 & 0 & \cdots & 1 \end{matrix}\right) .\end{equation}
Using a bases, $\{p_1,\ldots,p_{k-1}\}$,
for the set of invariant polynomials of the
Lie algebra of SL($k,\Bbb C$), we can construct a continuous mapping
$p:M\rightarrow \bigoplus_{j=2}^{k}H^0(\bar{X},K^j\otimes\xi^j)$, given by
assigning to the Higgs field $(W_k,\Phi)$ the elements
$(p_1(\Phi),\ldots,p_{k-1}(\Phi))$. The Higgs fields of the form
(\ref{eqn:thetaa})
produce a section $s$ of $p$, defined over the closed subspace
$\bigoplus_{j=2}^{k}H^0(\bar{X},K^j\otimes\xi^{j-1})$.
Therefore, we have that the image of $s$ is closed.
One can easily compute that the dimension
(over $\Bbb R$) of the space of sections
$\bigoplus_{j=2}^{k}H^0(\bar{X},K^j\otimes\xi^{j-1})$ is equal to
$$\sum_{j=2}^k 2(2j-1)(g-1)+2\sum_{j=2}^k(j-1)n=2(k^2-1)(g-1)+k(k-1)n.$$
On the
other hand, the dimension of the space of representations of the
fundamental group of $X$ into SL($k,\Bbb R$), with the condition that the
monodromy around the punctures lies in the above conjugacy class, can be
computed as follows. The fundamental group of $X$ can be identified
with a group of M\"obius transformations (or elements of SL($2,\Bbb
R$)), generated by elements
$\{c_1,d_1,\ldots,c_g,d_g,e_1,\ldots,e_n\}$, and with one
relation of the form $\prod_{j=1}^g[c_j,d_j]\prod_{j=1}^n e_j=id$, where
$[c,d]=cdc^{-1}d^{-1}$ denotes the commutator of the elements $c$ and $d$.
In classical terms, the transformations $c_j$'s and $d_j$'s are hyperbolic,
that is conjugate to dilatations, while the $e_j$'s are parabolic, or
conjugate to translations. In terms of loops on $X$, we have that the
$c_j$'s and $d_j$'s can be identified with paths around the handles of
$X$, while the $e_j$'s are simple loops around the punctures.
The image of the elements
$c_j$ and $d_j$ depends on dim(SL($k,\Bbb R$))=$k^2-1$ parameters.
In order to compute the number of parameters of the elements $e_j$,
first observe that these transformations belong to the conjugacy classes
of elements of $U=\{$regular unipotent elements of PSL($2, \Bbb
R$)$\}$.
Any matrix of SL($k,\Bbb R$) can be written as $ldu$, where $l$ is
unipotent and lower triangular, $d$ is diagonal, and $u$ is unipotent
upper triangular. We therefore have $(ldu)U(ldu)^{-1}=lUl^{-1}$. So the
conjugacy class of $U$ depends on $k(k-1)$ parameters.
Therefore, we have that the real dimensions of
$\bigoplus_{j=2}^{k}H^0(\bar{X},K^j\otimes\xi^{j-1})$ and the space of
representations of $\pi_1(X)$, with fixed conjugacy class for the
monodromy elements around the punctures, are equal.
Standard arguments using the invariance of domain theorem complete the
proof.\end{pf}
\ifx\undefined\leavevmode\hbox to3em{\hrulefill}\,
\newcommand{\leavevmode\hbox to3em{\hrulefill}\,}{\leavevmode\hbox to3em{\hrulefill}\,}
\fi
|
1995-10-05T05:20:31 | 9510 | alg-geom/9510005 | en | https://arxiv.org/abs/alg-geom/9510005 | [
"alg-geom",
"math.AG"
] | alg-geom/9510005 | Richard Wentworth | Georgios Daskalopoulos, and Richard Wentworth | On the Brill-Noether Problem for Vector Bundles | LaTeX 2e (amsart) | null | null | null | null | On an arbitrary compact Riemann surface, necessary and sufficient conditions
are found for the existence of semistable vector bundles with slope between
zero and one and a prescribed number of linearly independent holomorphic
sections. Existence is achieved by minimizing the Yang-Mills-Higgs functional.
| [
{
"version": "v1",
"created": "Thu, 5 Oct 1995 01:15:47 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Daskalopoulos",
"Georgios",
""
],
[
"Wentworth",
"Richard",
""
]
] | alg-geom | \section{Introduction}
In this note we exhibit the existence of semistable vector bundles on compact
Riemann
surfaces with a prescribed number of linearly independent holomorphic sections.
This may be
regarded as a higher rank version of the classical Brill-Noether problem for
line bundles.
Fix a compact Riemann surface $\Sigma$ of genus $g\geq 2$ and integers $r$ and
$d$ satisfying
\begin{equation}
0\leq d \leq r\ , \quad r\geq 2\ . \label{d-range}
\end{equation}
Then the main result may be stated as follows:
\begin{Main}
Let $k$ be a positive integer and suppose that $r$ and $d$ satisfy
\eqref{d-range}. Then
the necessary and sufficient conditions for the existence of a semistable
bundle on
$\Sigma$ with at least $k$ linearly independent holomorphic sections are $k\leq
r$ and if
$d\neq 0$,
$r\leq d+(r-k)g$.
\end{Main}
By analogy with the classical situation of special divisors (cf.\
\cite{ACGH,N}) one can
define the higher rank version of the Brill-Noether number:
\begin{equation}
\rho^{k-1}_{r,d}= r^2(g-1)+1-k(k-d+r(g-1))\ .
\end{equation}
Then $\rho^{k-1}_{r,d}$ is the formal dimension of the locus $W^{k-1}_{r,d}$ in the moduli space
of
semistable bundles of rank $r$ and degree $d$. $W^{k-1}_{r,d}$ is defined as the
closure of the
set of stable bundles with at least $k$ linearly independent sections. Note
that the
condition in the Main Theorem implies that $\rho^{k-1}_{r,d}\geq 1$, except in the trivial
case $d=0$
where $W^{k-1}_{r,d}$ is necessarily empty. The converse, in general, is not true.
Thus, unlike
the case of divisors, there are situations where
$\rho^{k-1}_{r,d}\geq 0$ and
$W^{k-1}_{r,d}=\emptyset$.
It would be interesting to improve the Main Theorem to a statement concerning
stable
bundles; however, our method does not immediately imply such a result except in
special
cases. We do have the following:
\begin{MainCor} {\rm (i) (see \cite[Thm III.2.4]{S})} For $d>0$ and any rank
$r$, there
exists a stable bundle of rank $r$ and degree $d$ with a nontrivial holomorphic
section.
{\rm (ii)} If $0<d<r$ and $r\leq d+g$, then there exists a stable bundle of
rank
$r$ and degree $d$ with precisely $r-1$ linearly independent sections.
\end{MainCor}
Instead of the constructive approach to theorems of this type taken in
references
\cite{S,T}, we use a variational method. More precisely, we study the Morse
theory of the
Yang-Mills-Higgs functional (cf.\ \cite{B}). The idea is simply the following:
Let
$(A^i,\vec\varphi^i)$ be a minimizing sequence with respect to the
Yang-Mill-Higgs
functional~\eqref{YMH}. Here,
$\vec\varphi_i=(\varphi^i_1,\ldots,\varphi^i_k)$ is a $k$-tuple of linearly
independent
holomorphic sections with respect to $A^i$. The minimal critical values
correspond to
solutions to the $k$-$\tau$-vortex equations, which for an appropriate choice
of $\tau$ imply that the limiting holomorphic structure is semistable (cf.\
\cite{BDW}).
If the sequence is assumed to converge to a nonminimal critical value, then we
show that
under the assumptions of the Main Theorem there exist ``negative directions"
which contradict the fact that the sequence is minimizing.
The energy estimates used closely follow \cite{D}. However, an extra
combinatorial
argument is needed to ensure that the bundles constructed have the correct
number of
holomorphic sections, and this is where the assumption $r\leq d+(r-k)g$ is
needed.
We have been informed that the Main Theorem stated above has been proven using
somewhat different methods in \cite{BGN}.
\medskip
\noindent {\it Acknowledgements.} The authors would like to thank L. Brambila
Paz for
introducing them to this problem and for several useful discussions during the
preparation of
this manuscript. They are also grateful for the warm hospitality of UAM,
Mexico and the
Max-Planck Institute in Bonn, where a portion of this work was completed.
\section{The Yang-Mills-Higgs Functional} \label{S:functional}
Let $\Sigma$, $d$, and $r$ be as in the Introduction, and let $k$ be a positive
integer.
Let $E$ be a fixed hermitian vector bundle on $\Sigma$ of rank $r$ and degree
$d$. Let $\mathcal{A}$
denote the space of hermitian connections on $E$, $\Omega^0(E)$ the space of
smooth sections
of $E$, and $\mathcal{H}\subset \mathcal{A}\times \Omega^0(E)^{\oplus k}$ the subspace
consisting of
holomorphic $k$-pairs. Thus,
$$
\mathcal{H}=\left\{ \left(A, \vec\varphi=(\varphi_1,\ldots,\varphi_k)\right) :
D''_A\varphi_i=0 , \
i=1,\ldots,k\right\}\ .
$$
Given a real parameter $\tau$, we define the Yang-Mills-Higgs functional:
\begin{align}
f_\tau &: \mathcal{A}\times\Omega^0(E)^{\oplus k}\longrightarrow \Bbb R \notag \\
f_\tau(A,\vec\varphi)&=
\Vert F_A\Vert^2 + \sum_{i=1}^k\Vert D_A\varphi_i\Vert^2 +\frac{1}{4}\left\Vert
\sum_{i=1}^k \varphi_i\otimes\varphi_i^\ast-\tau {\bf I}\, \right\Vert^2
-2\pi\tau d
\label{YMH}
\end{align}
In the above, the $\Vert\cdot\Vert$ denotes $L^2$ norms. Using a Weitzenb\"ock
formula
we obtain (cf.\ \cite[Theorem 4.2]{B})
$$
f_\tau(A,\vec\varphi)=2\sum_{i=1}^k\Vert D''_A
\varphi_i\Vert^2+\left\Vert\sqrt{-1}\Lambda
F_A+\frac{1}{2}\sum_{i=1}^k\varphi_i\otimes\varphi_i^\ast-\frac{\tau}{2}{\bf
I}\,
\right\Vert^2\ ,
$$
and therefore the absolute minimum of $f_\tau$ consists of holomorphic
$k$-pairs satisfying
the $k$-$\tau$-vortex equations discussed in \cite{BDW}.
\begin{Prop} \label{P:gradient}
{\rm (i)} The $L^2$-gradient of $f_\tau$ is given by
\begin{align*}
\left(\nabla_{(A,\vec\varphi)} f_\tau\right)_1 &=
D_A^\ast F_A+\frac{1}{2}\sum_{j=1}^k\left(
D_A\varphi_j\otimes\varphi_j^\ast-\varphi_j\otimes D_A\varphi_j^\ast\right) \\
\left(\nabla_{(A,\vec\varphi)} f_\tau\right)_{2,i} &=
\Delta_A\varphi_i-\frac{\tau}{2}\varphi_i+\frac{1}{2}\sum_{j=1}^k\langle
\varphi_i,\varphi_j\rangle\varphi_j
\end{align*}
{\rm (ii)} If $(A,\vec\varphi)\in\mathcal{H}$, then under the usual identification
$\Omega^0(\Sigma,{\rm ad}\, E)\simeq$ \break $\Omega^{0,1}(\Sigma,{\rm End}\, E)$,
we have
\begin{align*}
\left(\nabla_{(A,\vec\varphi)} f_\tau\right)_1 &=
-D''_A\left(\sqrt{-1}\Lambda
F_A+\frac{1}{2}\sum_{j=1}^k\varphi_j\otimes\varphi_j^\ast\right) \\
\left(\nabla_{(A,\vec\varphi)} f_\tau\right)_{2,i} &=
\sqrt{-1}\Lambda
F_A(\varphi_i)-\frac{\tau}{2}\varphi_i+\frac{1}{2}\sum_{j=1}^k\langle
\varphi_i,\varphi_j\rangle\varphi_j
\end{align*}
{\rm (iii)} If $(A,\vec\varphi)\in\mathcal{H}$ is a critical point of $f_\tau$, then
either {\rm (I)}
$\vec\varphi\equiv 0$ and $A$ is a direct sum of Hermitian-Yang-Mills
connections (not
necessarily of the same slope), or {\rm (II)} $A$ splits as $A=A'\oplus A_Q$ on
$E=E'\oplus E_Q$, where $(A',\vec\varphi)$ solves the $k$-$\tau$-vortex
equations and $A_Q$
is a direct sum of Hermitian-Yang-Mills connections (not necessarily of the
same slope).
\end{Prop}
\begin{proof}
(i) is a standard calculation, and (ii) follows from (i) via the K\"ahler
identities. We
are going to prove (iii). If $(A,\vec\varphi)$ is critical, then since
$
\sqrt{-1}\Lambda F_A+\frac{1}{2}\sum_{j=1}^k\varphi_j\otimes\varphi_j^\ast
$
is a self-adjoint holomorphic endomorphism, it must give a splitting
$A=A_0\oplus\cdots\oplus A_\ell$ according to its distinct (constant)
eigenvalues
$\sigma_0,\ldots,\sigma_\ell$. Write
$$
\sqrt{-1}\Lambda F_A=
\begin{pmatrix}
-\frac{1}{2}\sum_{j=1}^k\varphi_j\otimes\varphi_j^\ast +\sigma_0\ {\bf I}
& 0 & \cdots & 0 \\
0& \sigma_1\ {\bf I} & & \vdots \\
\vdots && \ddots & \\
0 & \cdots && \sigma_\ell\ {\bf I}
\end{pmatrix}
\quad .
$$
Thus,
\begin{align*}
0 &=
\sqrt{-1}\Lambda
F_A(\varphi_i)-\frac{\tau}{2}\varphi_i+\frac{1}{2}\sum_{j=1}^k\langle
\varphi_i,\varphi_j\rangle\varphi_j\\
&= -\frac{1}{2}\sum_{j=1}^k\langle\varphi_i,
\varphi_j\rangle\varphi_j+\sigma_0\varphi_i
-\frac{\tau}{2}\varphi_i+\frac{1}{2}\sum_{j=1}^k\langle
\varphi_i,\varphi_j\rangle\varphi_j\\
&= \left(\sigma_0-\frac{\tau}{2}\right)\varphi_i\ ,
\end{align*}
from which we obtain either Case I or Case II, depending upon whether
$\vec\varphi\equiv 0$.
\end{proof}
Next, recall that $\mathcal{H}$ is an infinite dimensional complex analytic
variety whose tangent space is given by the kernel of a certain differential
defined in
\cite[3.15]{BDW}. Moreover, $\mathcal{H}$ is preserved by the action of the complex
gauge
group ${\G}^{\CBbb}$. We have the following:
\begin{Prop}
\label{P:tangent}
If $(A,\vec\varphi)\in \mathcal{H}$, then $\nabla_{(A,\vec\varphi)}f_\tau$ is tangent
to the
orbits of ${\G}^{\CBbb}$. In particular, $\nabla_{(A,\vec\varphi)}f_\tau$ is tangent to
$\mathcal{H}$
itself.
\end{Prop}
\begin{proof}
Set
$
u=\sqrt{-1}\Lambda
F_A+\frac{1}{2}\sum_{j=1}^k\varphi_j\otimes\varphi_j^\ast-\frac{\tau}{2}{\bf I}
$.
By Proposition \ref{P:gradient} (ii) we have that
$\nabla_{(A,\vec\varphi)}f_\tau=d_1(u)$,
where
$d_1$ is the differential defined in \cite[3.15]{BDW}. The Proposition
follows.
\end{proof}
Because of Proposition \ref{P:tangent}, the critical points of the functional
$f_\tau$
restricted to $\mathcal{H}$ are characterized by Proposition \ref{P:gradient} (iii).
A solution $(A(t),\vec\varphi(t))$, $t\in [0,t_0)$ to the initial value problem
\begin{equation}
\label{flow}
\left(\frac{\partial A}{\partial t}, \frac{\partial\vec\varphi}{\partial
t}\right)
=
-\nabla_{(A,\vec\varphi)}f_\tau\ , \quad
\left( A(0),\vec\varphi(0)\right)
=
(A_0,\vec\varphi_0)\ ,
\end{equation}
is called the $L^2$-gradient flow of $f_\tau$ starting at
$(A_0,\vec\varphi_0)$. Notice
that
\begin{equation}
\label{E:decay}
\frac{d}{dt}f_\tau(A(t),\vec\varphi(t))
=-\left\Vert\nabla_{(A(t),\vec\varphi(t))}
f_\tau\right\Vert^2\ ,
\end{equation}
and so the energy decreases along the $L^2$-gradient flow.
\begin{Prop}
\label{P:shorttime}
Given $(A_0,\vec\varphi_0)\in\mathcal{H}$, there is a $t_0>0$ such that the
$L^2$-gradient flow
exists for $0\leq t< t_0$.
\end{Prop}
\begin{proof}
The proof is an application of the implicit function theorem as in \cite{R}.
\end{proof}
\section{Technical Lemmas}
\label{S:technical}
In this section we collect several results needed for the proof of the Main
Theorem.
Throughout,
$E$ will denote a holomorphic bundle of rank $r$ and degree $d$ on the compact
Riemann
surface $\Sigma$.
\begin{Lem}
\label{L:bounds}
Let $E$ be as above with $0\leq d\leq r$ and $h^0(E)=k$. If either {\rm (i)}
$E$ is semistable, or {\rm (ii)} $E$ satisfies the k-$\tau$-vortex equation
for some $0<\tau<1$ and
$E$ does not contain the trivial bundle as a split factor; then $k\leq r$ and
if $d\neq 0$, $r\leq d+(r-k)g$.
\end{Lem}
\begin{proof} We first show that $k\leq r$. Suppose $k\geq r$. Thus, $E$ has
at least $r$
linearly independent holomorphic sections. If the sections generate
$E$ at every point, then $E\simeq\mathcal{O}^{\oplus r}$; in which case $d=0$ and
$k=r$.
Suppose the sections fail to generate at every point. Then we
can find a point
$p\in\Sigma$ and a section of $E$ vanishing at $p$. Thus $E$ contains
$\mathcal{O}(p)$ as a
subsheaf, which is a contradiction to (ii). If (i) is assumed, then $E$ is
strictly
semistable with $d=r$, and the bound $k\leq r$ follows from induction on the
rank. Note
that the second inequality is also satisfied in this case.
Assume $0< d< r$. In both cases (i) and (ii) we obtain
$0\to \mathcal{O}^{\oplus k} \xrightarrow{\pi} E\to F\to 0$,
where $F$ is locally
free. By dualizing and taking the resulting long exact sequence in cohomology,
we find
$$
0\longrightarrow H^0(F^\ast)\longrightarrow H^0(E^\ast)
\xrightarrow{\delta} H^0(\mathcal{O}^{\oplus k}) \longrightarrow H^1(F^\ast)
\ .
$$
We are going to show that $H^0(E^\ast)=0$. The result then follows by the
Riemann-Roch
formula. For
(i), $H^0(E^\ast)=0$ by semistability. For (ii), note first that $\delta=0$.
For if not,
there would be a section $s:\mathcal{O}\to E^\ast$ with $\pi^\ast\circ s=\sigma\neq
0$. But
$\sigma$ could not have any zeros, and so $\mathcal{O}$ would be a split factor in
$E^\ast$; hence,
also in $E$. Secondly, we show that $H^0(F^\ast)=0$. Let
$L\subset F^\ast$ be a subbundle. Then $\tau$-stability immediately implies
$c_1(L^\ast) >
\tau >0$. Thus, in particular, $F^\ast$ cannot contain $\mathcal{O}$ as a subsheaf.
This
completes the proof.
\end{proof}
\begin{Lem}
\label{L:extension}
Let $E_1$, $E_2$ be holomorphic bundles of rank $r_1, r_2$ and degree $d_1,
d_2$,
satisfying $0\leq \mu_1=d_1/r_1 < d_2/r_2=\mu_2\leq 1$. Suppose
$h^0(E_1)=k_1 \leq r_1$, $h^0(E_2)=k_2\leq r_2$, and
$$
d_2+(r_2-k_2-1)g < r_2\leq d_2+(r_2-k_2)g\ .
$$
Furthermore,
\begin{itemize}
\item If $d_1\neq 0$ assume $r_1\leq d_1+(r_1-k_1)g$.
\item If $d_1=0$ and $k_1=r_1$, assume $r_2 < d_2+(r_2-k_2)g$.
\end{itemize}
Then there exists a nontrivial extension $0\to E_1\to E\to E_2\to 0$ such that
$h^0(E)=
k_1+k_2$.
\end{Lem}
\begin{proof} If $k_2=0$, the result follows from Riemann-Roch. Suppose
$k_2\geq 1$. The
condition that the
$k_2$ sections of
$E_2$ lift for some nontrivial extension is
$k_2 h^1(E_1) < h^1(E_1\otimes E_2^\ast)$. Notice that
\begin{align*}
h^1(E_1) &= h^0(E_1)-d_1+r_1(g-1)=k_1-d_1+r_1(g-1) \\
h^1(E_1\otimes E_2^\ast) &= h^0(E_1\otimes E_2^\ast)+r_1 r_2(\mu_2-\mu_1+g-1)
\\
&\geq r_1 r_2(\mu_2-\mu_1+g-1)\ ,
\end{align*}
hence, it suffices to show that
$$
k_2(k_1-d_1+r_1(g-1)) < r_1 r_2(\mu_2-\mu_1+g-1)\ ,
$$
or equivalently, that
\begin{equation}
\label{E:one}
r_1(d_2-r_2+(r_2-k_2)g)-r_2 d_1 + k_2 d_1 -k_1 k_2 + k_2 r_1 > 0\ .
\end{equation}
Now if $k_2=r_2=d_2$, then \eqref{E:one} is trivially satisfied by the
hypotheses.
Similarly for $d_1=0$. So assume
$k_2\leq r_2-1$, $d_1\neq 0$. Write $d_2=r_2-(r_2-k_2)g+p$, where $0\leq p < g$
by
assumption. On the other hand,
$$
d_1 < r_1\frac{d_2}{r_2}\leq
\left(d_1+(r_1-k_1)g\right)\frac{r_2-(r_2-k_2)g+p}{r_2}\ ,
$$
which is equivalent to
$$
-\frac{d_1 p}{g}+k_1 p + (r_1-k_1)(r_2-k_2)(g-1) < r_1 p-r_2d_2+k_2 d_1-k_1
k_2+k_2 r_1
\ .
$$
Therefore, \eqref{E:one} will follow from
\begin{equation}
\label{E:two}
-\frac{d_1 p}{g}+k_1 p + (r_1-k_1)(r_2-k_2)(g-1)\geq 0\ .
\end{equation}
Now if $p=0$ then \eqref{E:two} is trivially satisfied. Assume that $1\leq
p\leq g-1$.
Then
\begin{align*}
-\frac{d_1 p}{g} &+k_1 p + (r_1-k_1)(r_2-k_2)(g-1) \\
&\geq -d_1+r_1
p-(r_1-k_1)p+(r_1-k_1)(r_2-k_2)(g-1) \\
&\geq (r_1-d_1)+(r_1-k_1)(r_2-k_2-1)(g-1) \\
&\geq 0 \ ,
\end{align*}
which proves \eqref{E:two}, \eqref{E:one}, and hence the Lemma.
\end{proof}
In order to get an upper bound on the infimum of the Yang-Mills-Higgs
functional in the next
section, we shall need the following construction and energy estimate:
\begin{Lem}
\label{L:special}
Assume $0< d < r$, $k\geq 1$, and $r\leq d+(r-k)g$. Let $F$ be a holomorphic
bundle of
degree
$d$ and rank
$r-1$ with $h^0(F)=k-1$. Then there exists a non-split extension $0\to\mathcal{O}\to
E\to F\to 0$
with $h^0(E)=k$.
\end{Lem}
\begin{proof}
The condition for all of the sections of $F$ to lift is
\begin{align*}
(k-1)h^1(\mathcal{O}) < h^1(F^\ast)\
&\iff\quad g(k-1) < d+(r-1)(g-1) \\
&\iff\quad r< d+(r-k)g+1\ ,
\end{align*}
and hence the result.
\end{proof}
\begin{Prop}[{cf.\ \cite[Prop. 3.5]{D}}]
\label{P:energyestimate}
Let $E_1, E_2$ be hermitian bundles with slope $\mu_1, \mu_2$. Let $A_1, A_2$
be hermitian
connections on $E_1, E_2$, and $\vec\varphi^1, \vec\varphi^2$ be $k_1$ and
$k_2$ tuples of
holomorphic sections. Set $k=k_1+k_2$. Let $\tau_1, \tau_2$ and $\tau$ be
real numbers
satisfying $\mu_1\leq\tau_1\leq \tau < \mu_2 \leq\tau_2$, and assume that
$(A_1,\vec\varphi^1)$ and
$(A_2,\vec\varphi^2)$ satisfy the $\tau_1$ and $\tau_2$ vortex equations,
respectively.
Set $E=E_1\oplus E_2$, $\varphi_i=(\varphi^1_i,0)$ for $i=1,\ldots, k_1$, and
$\varphi_{k_1+i}=(0,\varphi^2_i)$ for $i=1,\ldots, k_2$. Then there exist
constants
$\varepsilon_1, \varepsilon_2 , \eta >0$ such that for all
$$
\beta\in H^{0,1}\left(\Sigma, {\rm Hom}(E_2,E_1)\right)\ ,\quad
\vec\psi\in\Omega^0(E)^{\oplus k}\ ,
$$
with $\Vert\beta\Vert =\varepsilon_1$, $\Vert\vec\psi\Vert\leq \varepsilon_2$,
and
$$
\left( A_\beta=
\begin{pmatrix}
A_1 & \beta \\ 0 & A_2
\end{pmatrix},
\vec\varphi+\vec\psi\right)\in\mathcal{H}\ ,
$$
it follows that
$
f_\tau(A_\beta,\vec\varphi+\vec\psi) < f_\tau(A_1\oplus A_2,\vec\varphi)-\eta
$.
\end{Prop}
\begin{proof}
By assumption,
$$
\sqrt{-1}\Lambda
F_{A_\ell}+\frac{1}{2}\sum_{j=1}^{k_\ell}
\varphi^{\ell}_j\otimes(\varphi^{\ell}_j)^\ast
=
\frac{\tau}{2}{\bf I}_{\ell}\ ,\quad \ell=1,2\ .
$$
It follows that
$$
\sqrt{-1}\Lambda
F_{A_1\oplus
A_2}+\frac{1}{2}\sum_{j=1}^{k_1}\varphi^1_j\otimes(\varphi^1_j)^\ast
+\frac{1}{2}\sum_{j=1}^{k_2}\varphi^2_j
\otimes(\varphi^2_j)^\ast-\frac{\tau}{2}{\bf I}=
\begin{pmatrix}
\frac{\tau_1-\tau}{2}{\bf I}_1 & 0 \\
0 & \frac{\tau_2-\tau}{2}{\bf I}_2
\end{pmatrix}
$$
The argument of \cite[pp.\ 715-716]{D} shows that there is a constant
$\varepsilon_1$ such
that for $\beta$ and $A_\beta$ as in the statement,
$$
f_\tau\left(
A_\beta,\varphi^1_1,\ldots,\varphi^1_{k_1},
\varphi^2_1,\ldots,\varphi^2_{k_2}\right)
<
f_\tau\left(
A_1\oplus
A_2,\varphi^1_1,\ldots,\varphi^1_{k_1},
\varphi^2_1,\ldots,\varphi^2_{k_2}\right)
\ .
$$
Now if we take $\varepsilon_2$ sufficiently small the Proposition follows (note
that which
norms we use is irrelevent, since $\beta$ and $\vec\varphi+\vec\psi$ satisfy
elliptic
equations, and hence the $L^2$ norm is equivalent to any other).
\end{proof}
\section{Proof of the Main Theorem}
\label{S:proof}
Necessity of the conditions follows from Lemma \ref{L:bounds}, and sufficiency
for $d=0$ or
$d=r$ is clear as well. To prove sufficiency in general, we shall proceed by
induction on the
rank. The case
$r=2$,
$d=1$ is clear from a direct construction. Assume the Main Theorem holds for
bundles of
rank $< r$. We show that it holds for $r$ as well.
Let $\mathcal{H}^\ast\subset\mathcal{H}$ denote the subspace of $k$-pairs
$\left(A,\vec\varphi=(\varphi_1,\ldots,\varphi_k)\right)$
such that the sections $\varphi_1,\ldots,\varphi_k$ are linearly independent.
Fix $\tau$ as
in Assumption 1 of \cite{BD}, i.e. $\mu(E)<\tau=\mu(E)+\gamma < \mu_+$, where
$\mu_+$ is
the smallest possible slope greater that $\mu=\mu(E)$ of a subbundle of $E$
(note that
$0<\tau<1$ and that we also normalize the volume of $\Sigma$ to be $4\pi$).
\begin{Lem}
\label{L:inf}
Let $m=\inf_{\mathcal{H}^\ast} f_\tau$. Then $0\leq m<\pi/(r-1)$.
\end{Lem}
\begin{proof}
Let $F$ be a vector bundle of degree $d$ and rank $r-1$. Then by the
inductive hypothesis, we may assume there exist hermitian connections $A_1$ and
$A_2$ on
$\mathcal{O}$ and
$F$, respectively, and holomorphic sections $\varphi_1\neq 0$ in
$H^0(\Sigma,\mathcal{O})$, and
$\varphi_2,\ldots,\varphi_k$ linearly independent sections in $H^0(\Sigma,F)$,
such that
$(A_1,\varphi_1)$ and $(A_2,\varphi_2,\ldots,\varphi_k)$ satisfy the $\tau_1$
and
$\tau_2$ vortex equations, respectively, for $\tau_1=\tau$,
$\tau_2=d/(r-1)+\gamma$.
It follows from Lemma \ref{L:special} and Proposition \ref{P:energyestimate}
that there is
a nontrivial extension $\beta: 0\to\mathcal{O}\to E\to F\to 0$, and $\vec\psi$ such
that
$(A_\beta,\vec\varphi+\vec\psi)\in\mathcal{H}^\ast$ and
\begin{align*}
f_\tau(A_\beta,\vec\varphi+\vec\psi)&<f_\tau(A_1\oplus
A_2,\varphi_1,\ldots,\varphi_k)-\eta \\
&=
\left\Vert\frac{1}{2}\left(\frac{d}{r-1}-\frac{d}{r}\right){\bf I_F}\,
\right\Vert^2-\eta <
\frac{\pi}{r-1}\ .
\end{align*}
\end{proof}
Let $(A^i,\vec\varphi^i)$ be a minimizing sequence in $\mathcal{H}^\ast$. Thus,
$f_\tau(A^i,\vec\varphi^i)\to m$. By weak compactness (more precisely, see the
argument
in \cite[Lemma 5]{BD}) there is a subsequence converging to
$(A^\infty,\vec\varphi^\infty)$ in the $C^\infty$ topology. By the continuity
of
$f_\tau$ with respect to the $C^\infty$ topology, Propositions
\ref{P:shorttime} and
\ref{P:tangent}, and equation \eqref{E:decay}, it follows that
$(A^\infty,\vec\varphi^\infty)$ is a critical point of
$f_\tau$. If the holomorphic structure $E^\infty$ defined by $A^\infty$ is
semistable, then
by semicontinuity of cohomology we are finished. We therefore
assume
$E^\infty$ is unstable and derive a contradiction. According to Proposition
\ref{P:gradient} (iii) we must consider the following cases:
\begin{align*}
\vec\varphi^\infty &= 0\ , \quad E^\infty=E_1\oplus\cdots\oplus E_\ell
\tag{I} \\
\vec\varphi^\infty &\neq 0\ , \quad E^\infty=E_{\varphi}\oplus
E_1\oplus\cdots\oplus E_\ell
\tag{II}
\end{align*}
Set $\mu_j=\mu(E_j)$, and assume $\mu_1 < \cdots < \mu_\ell$. If $\mu_\ell >
1$ (or
similarly,
$\mu_1 < 0$), then
$$
f_\tau\left(A^\infty,\vec\varphi^\infty\right)\geq \pi(\mu_\ell-\tau)^2 r_\ell
\geq \pi(\mu_\ell-1)^2 r_\ell
\geq\frac{\pi}{r_\ell}\geq\frac{\pi}{r-1} > m\ ,
$$
contradicting Lemma \ref{L:inf}.
We therefore rule out this possibility.
We will consider Cases I and II separately.
\medskip
\noindent \emph{Case I}\@. Let $k_i=h^0(E_i)$. By semicontinuity of
cohomology,
$\sum_{i=1}^\ell k_i\geq k$. If $\mu_\ell =1$, then we may replace $E_\ell$ by
a
Hermitian-Yang-Mills bundle $\widehat E_\ell$ with exactly $\hat k_\ell=r_\ell$
sections.
Hence, we may assume that
$$
d_\ell + (r_\ell -\hat k_\ell-1)g < r_\ell \leq d_\ell +(r_\ell-\hat k_\ell)g\
{}.
$$
For $1<i<\ell$, the inductive hypothesis implies that we may replace $E_i$ by a
Hermitian-Yang-Mills bundle $\widehat E_i$ with
$$
h^0(\widehat E_i)=\hat k_i=\left[ \frac{d_i+r_i(g-1)}{g}\right]\ ,
$$
the maximal number of sections allowed for $d_i, r_i$, and $g$.
Note that
\begin{equation}
\label{E:max}
d_i + (r_i -\hat k_i-1)g < r_i \leq d_i +(r_i-\hat k_i)g\ .
\end{equation}
If $\mu_1\neq 0$, then we
can replace $E_1$ by $\widehat E_1$ as above. If $\mu_1=0$, we may replace
$E_1$ with
$\mathcal{O}^{\oplus r_1}$, with $\hat k_1=r_1\geq k_1$ sections.
By our choices of $\hat
k_i$, $\sum_{i=1}^\ell \hat k_i\geq \sum_{i=1}^\ell k_i\geq k$.
Let $0\leq\mu_1<\cdots<\mu_s\leq \mu <\mu_{s+1} <\cdots <\mu_\ell\leq 1$.
Suppose first that $\mu_s\neq 0$.
By Lemma \ref{L:extension} there is a
nontrivial extension $0\to\widehat E_s\to G\to \widehat E_{s+1}\to 0$, with
$h^0(G)=\hat
k_s+\hat k_{s+1}$. Thus,
$$
h^0\left(\widehat E_1\oplus\cdots\oplus\widehat E_{s-1}\oplus G\oplus \widehat
E_{s+1}\oplus \cdots\oplus \widehat E_\ell\right)=\sum_{i=1}^\ell \hat k_i\geq
k\ .
$$
On the other hand, by Proposition \ref{P:energyestimate} there is a hermitian
connection
on $\widehat E_1\oplus\cdots\oplus\widehat E_{s-1}\oplus G\oplus \widehat
E_{s+1}\oplus \cdots\oplus \widehat E_\ell$ and linearly independent sections
$\varphi_1,\ldots, \varphi_k$ such that $f_\tau(A,\vec\varphi)<
f_\tau(A_\infty, 0)=m$,
contradicting the minimality of $(A_\infty, 0)$.
Now suppose $\mu_s=\mu_1=0$, $\mu < \mu_i$ for $2\leq i\leq \ell$. If for any
$2\leq i\leq
\ell$ we have $r_i < d_i+(r_i-\hat k_i)g$, then by Lemma \ref{L:extension}
there is a
nontrivial extension $0\to\widehat E_1\to G\to \widehat E_i\to 0$, with
$h^0(G)=\hat
k_1+\hat k_i$, and Proposition \ref{P:energyestimate} yields a contradiction as
before.
Suppose that for all $2\leq i\leq \ell$, $r_i=d_i+(r_i-\hat k_i)g$. We claim
that
$\sum_{i=1}^\ell \hat k_i > k$. For if $\sum_{i=1}^\ell \hat k_i = k$, then
$\sum_{i=2}^\ell (r_i-\hat k_i)= r-k$, and hence
$$
r > \sum_{i=2}^\ell r_i = \sum_{i=2}^\ell d_i+ (r_i-\hat k_i)g = d+ (r-k)g\ ;
$$
a contradiction. Thus, we may replace $\widehat E_1$ by a bundle $\widehat
E_1^\prime$
having $\hat k_1^\prime=\hat k_1-1$ sections. According to Lemma
\ref{L:extension}
there is a
nontrivial extension $0\to\widehat E_1^\prime \to G\to \widehat E_2\to 0$, with
$h^0(G)=\hat
k_1^\prime +\hat k_2$, $\hat k_1^\prime +\sum_{i=2}^\ell\hat k_i\geq k$, and
Proposition
\ref{P:energyestimate} yields a contradiction as before.
\medskip
\noindent \emph{Case II}\@. First notice that by the invariance of the
Yang-Mills-Higgs equations under the natural action by U($k$), we may assume
that
$\varphi_1^\infty, \ldots, \varphi_k^\infty$ form an $L^2$-orthogonal set of
sections. In
particular, we may assume that there exists $s\leq k$ such that
$\varphi_1^\infty,\ldots,\varphi_s^\infty$ are linearly independent and
$\varphi_{s+1}^\infty,\ldots,\varphi_k^\infty\equiv 0$. Write
$E_\varphi=E_\varphi^\prime\oplus\mathcal{O}^{\oplus t}$, where $E_\varphi^\prime $
contains no
split factor of
$\mathcal{O}$. Set $k_i=h^0(E_i)$,
$k_\varphi=h^0(E_\varphi)$, $k_\varphi^\prime
=h^0(E_\varphi^\prime)=k_\varphi-t$. By
semicontinuity of cohomology, $k_\varphi+\sum_{i=1}^\ell k_i\geq k$. As in
Case I, we may
replace each $E_i$ by a Hermitian-Yang-Mills bundle $\widehat E_i$ such that
$h^0(\widehat
E_i)=\hat k_i\geq k_i$, and \eqref{E:max} is satisfied for $i=1,\ldots,\ell$.
On the other
hand, since $E_\varphi$ satisfies the $k$-$\tau$-vortex equation for
$\tau=\mu+\gamma$ as
above, it follows that $E_\varphi^\prime$ is $\tau$-stable. Therefore,
$0\neq \mu(E_\varphi^\prime)\leq \mu=\mu(E)$; and since $\tau <
1$, we obtain from Lemma \ref{L:bounds} that $r_\varphi^\prime \leq
d_\varphi+(r_\varphi^\prime-k_\varphi^\prime)g$. Finally, notice that since
$E^\infty$ is
unstable, $\mu_\ell > \mu$. We may now apply Lemma \ref{L:extension} to
$E_\varphi^\prime$
and $\widehat E_\ell$ to obtain a nontrivial extension $0\to
E_\varphi^\prime\to G\to
\widehat E_\ell\to 0$, with $h^0(G)=k_\varphi^\prime+\hat k_\ell$. It follows
that
$$
h^0\left(
G\oplus\mathcal{O}^{\oplus t}\oplus \widehat E_1\oplus\cdots\oplus\widehat E_{\ell-1}
\right) = k_\varphi +\sum_{i=1}^\ell \hat k_i \geq k\ .
$$
By Proposition \ref{P:energyestimate} there is a hermitian connection $A$ on
$G\oplus\mathcal{O}^{\oplus t}\oplus \widehat E_1\oplus\cdots\oplus\widehat
E_{\ell-1}$
and linearly independent sections $\varphi_1,\ldots,\varphi_k$ extending
$\varphi_1^\infty,\ldots,\varphi_s^\infty$ such that $f_\tau(A,\vec\varphi)<
f_\tau(A_\infty,\vec\varphi^\infty)=m$, again contradicting the minimality of
$m$.
This completes the proof of the Main Theorem.
\section{Stable Bundles}
We conclude by proving the Corollary stated in the Introduction. Consider
first part
(ii). The upper bound follows from Lemma \ref{L:bounds}. By the Main Theorem,
it suffices
to show that if $E$ is semistable with $0<\mu < 1$ and $h^0(E)=r-1$, then $E$
is stable.
Suppose to the contrary. Then we can find a semistable subbundle $E'$ with
$Q=E/E'$ stable
and $\mu(E')=\mu(Q)=\mu$. By Lemma \ref{L:bounds}, $h^0(E')\leq r'-1$, and
$h^0(Q)\leq r_Q-1$; contradiction.
Now consider part (i). Tensoring by ample line bundles allows us to restrict to
the case
$0<d\leq r$. Let
$\mathfrak{B}_\tau$ be the set of gauge equivalence classes of solutions to the (one section)
$\tau$-vortex
equation for bundles of rank
$r$ and degree $d$. By the proof of the Main Theorem, $\mathfrak{B}_\tau\neq\emptyset$. One
can therefore
show as in
\cite{BDW,BD} that
$\mathfrak{B}_\tau$ is a smooth projective variety of dimension $(r^2-r)(g-1)+d$ with a
morphism $\psi :
\mathfrak{B}_\tau\to\mathfrak{M}(r,d)$, where
$\mathfrak{M}(r,d)$ is the moduli space of semistable bundles of rank $r$ and degree $d$. The
image of
$\psi$ is precisely the set of isomorphism classes of
semistable bundles $E$ with $h^0(E)\geq 1$.
\begin{Lem}
Suppose that there exists a semistable (resp.\ stable) bundle $E_0$ of rank
$r$, degree
$d$, $0< d\leq r$, and $h^0(E_0)=k\geq 1$. Then there exists a semistable
(resp.\ stable)
bundle
$E$ of the same rank and degree with $h^0(E)=k-1$.
\end{Lem}
\begin{proof} By Lemma \ref{L:bounds}, $k\leq r$. The case where $d=r$ and
$E_0$ is
strictly semistable is trivial. In the other cases, $k<r$, and we may write
$$
\beta_0 : 0\to\mathcal{O}^{\oplus k}\to E_0\to F\to 0\ ,
$$
where by assumption the connecting homomorphism $\delta_0: H^0(F)\to
H^1(\mathcal{O}^{\oplus k})$
is injective. Consider $\{L_t : t\in D\}$ a smooth local family of line
bundles
parametrized by the open unit disk $D\subset \Bbb C$ and satisfying
$L_0=\mathcal{O}$ and $H^0(L_t)=0$, $t\neq 0$. Set $G_t=\mathcal{O}^{k-1}\oplus L_t$. The
semistability
of $E_0$ implies that $ H^0(F^\ast\otimes G_t)=0$. Hence, $\{ H^1(F^\ast\otimes
G_t) : t\in
D\}$ defines a smooth vector bundle $V$ over $D$. Let $\beta=\{\beta(t) :
t\in
D\}$ be a nowhere vanishing section of $V$ with $\beta(0)=\beta_0$. Then
$\beta$ defines a
smooth family of nonsplit extensions $0\to G_t\to E_t\to F\to 0$ and a smooth
family of
connecting homomorphisms
$$
\delta_t : H^0(F)\longrightarrow H^1(G_t)\subset\Omega^{0,1}(U)\ ,
$$
where $U$ is the trivial rank $k$, $C^\infty$ vector bundle on $\Sigma$. By
assumption,
$\delta_0$ is injective; hence, $\delta_t$ is injective for small $t$. It
follows that
$h^0(E_t)=k-1$ for small $t$. Furthermore, since $E_0$ is semistable (resp.\
stable) then
$E_t$ is also semistable (resp.\ stable) for small $t$.
\end{proof}
\noindent Since the condition $h^0(E)=1$ is open in $\mathfrak{B}_\tau$, by the Lemma there
exists some
component
$\mathfrak{B}_\tau^\prime$ of $\mathfrak{B}_\tau$ containing an open dense set $\mathfrak{B}_\tau^\ast$ consisting of pairs
$[E,\varphi]$
with
$h^0(E)=1$. Let $W^\ast=\psi(\mathfrak{B}_\tau^\ast)$. We will assume that $W^\ast$ is
contained in the
strictly semistable locus of $\mathfrak{M}(r,d)$ and derive a contradiction. We may assume
that each
irreducible component of $W^\ast$ is contained a subvariety $S$ parametrized by
bundles of
the form
$E_1\oplus E_2$, where $E_1$ is stable with $h^0(E_1)\geq 1$, $E_2$ is
semistable, and
$\mu(E_1)=\mu(E_2)=\mu(E)$. It follows that
\begin{equation} \label{E:dim1}
\begin{aligned}
\dim S&=(r_1^2-r_1)(g-1)+d_1+r_2^2(g-1)+1 \\
&=r^2(g-1)-2r_1r_2(g-1)-r_1(g-1)+d_1+1\ .
\end{aligned}
\end{equation}
For $[E]\in W^\ast\subset S$, we have
\begin{equation}
\label{E:dim2}
\dim_{[E,\varphi]}\mathfrak{B}_\tau\leq \dim_{[E]} S +\dim \psi^{-1}([E]) \ .
\end{equation}
The dimension of the fiber of $\psi$ is given by
$h^1(E_2\otimes E_1^\ast)$. Assume first that a generic $E_1$ is not
isomorphic to any
factor of
$E_2$. Then the fiber dimension is $r_1 r_2(g-1)$. Thus, we obtain from
\eqref{E:dim1} and \eqref{E:dim2} that
$$
(r^2-r)(g-1)+d \leq r^2(g-1)-r_1r_2(g-1)-r_1(g-1)+d_1+1\ ,
$$
or,
$$
r_2(r_1-1)(g-1)+ d_2-1 \leq 0\ .
$$
This yields a contradiction in all cases other than
$r=d=2$. The latter situation is covered by the following construction which
is verified
by straightforward dimension counting:
\begin{Prop}
For generic line bundles $L$ of degree 2 and generic extensions $0\to\mathcal{O}\to
E\to L\to 0$,
$E$ is a stable rank 2 bundle of degree 2 with a nontrivial holomorphic
section.
\end{Prop}
\noindent In case $E_1$ is isomorphic to some factor of $E_2$, the fiber
dimension
increases by 1. On the other hand, in this case $W^\ast$ is contained in a
strict
subvariety of $S$, so by \eqref{E:dim2} the same argument applies.
Isomorphisms with more
factors are handled similarly. This completes the proof of the Corollary.
|
1995-10-13T05:20:11 | 9510 | alg-geom/9510009 | en | https://arxiv.org/abs/alg-geom/9510009 | [
"alg-geom",
"math.AG"
] | alg-geom/9510009 | null | Ludmil Katzarkov | Nilpotent groups and universal coverings of smooth projective varieties | LaTeX 2.09, 15 pages | null | null | null | null | In this paper we prove that the universal cover of a smooth projective
variety with nilpotent fundamental group is holomorphically convex.
| [
{
"version": "v1",
"created": "Fri, 13 Oct 1995 03:32:37 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Katzarkov",
"Ludmil",
""
]
] | alg-geom | \section{Introduction}
Characterizing the universal coverings of smooth projective
varieties is an old and hard question. Central to the subject is a conjecture
of Shafarevich according to which
the universal cover $\widetilde{X}$ of a smooth projective variety is
holomorphically convex, meaning that for every infinite sequence of points
without limit points in $\widetilde{X}$ there exists a holomorphic function
unbounded on this sequence.
\medskip
In this paper we try to study the universal covering of a smooth projective
variety $X$ whose fundamental group $\pi_{1}(X)$ admits an infinite
image homomorphism
\[ \rho : \pi_{1}(X) \longrightarrow L \]
into a complex linear algebraic group $L$. We will say that a nonramified
Galois covering $X' \rightarrow X$ corresponds to a representation
$\rho : \pi_{1}(X) \rightarrow L$ if its group of deck transformations is
${\rm im}(\rho)$.
\begin{defi}
We call a representation $\rho : \pi_{1}(X) \rightarrow L$ linear,
reductive, solvable or nilpotent if the Zariski closure of its image is a
linear, reductive, solvable or nilpotent algebraic subgroup in $L$. We call
the corresponding covering linear, reductive, solvable or nilpotent
respectively.
The natural homomorphism $\pi_{1}(X,x) \rightarrow \fgc{{\rm uni}}{X,x}$ to
Malcev's pro-uni\-po\-tent completion will be called the Malcev representation
and the corresponding covering the Malcev covering.
\end{defi}
One may ask not only if the universal covering of $X$ is
holomorphically convex but also if some special intermediate coverings that
correspond to representations $\rho : \pi_{1}(X) \longrightarrow L$
are holomorphically convex.
\bigskip
In case $X$ is an algebraic surface and $\rho : \pi_{1}(X) \longrightarrow L$
is a reductive representation this question has been answered in \cite{KR}.
The author and M. Ramachandran proved there that if $X' \longrightarrow X$ is
a Galois covering of a smooth projective surface corresponding to a reductive
representation of $\pi_{1}(X)$ and such that ${\rm Deck}(X'/X)$ does not have two
ends, then $X'$ is holomorphically convex. The proof is based on two major
developments in K\"{a}hler geometry that occured in the last decade. The first
is a correspondence, established through the work of Hitchin \cite{HI},
Corlette \cite{C} and Simpson \cite{SC},
between Higgs bundles, representations of the fundamental group of a
smooth projective variety $\rho : \pi_{1}(X) \rightarrow G$
(here $G$ is a linear algebraic group over ${\Bbb C}$) and $\rho$ equivariant
harmonic maps from the universal covering of $X$ to the corresponding
symmetric space for $G$. This correspondence is called now - non-abelian
Hodge theory. The second is the theory of harmonic maps to buildings
developed by Gromov and Schoen
\cite{GS}. This theory gives the $p$-adic version of the theory of Higgs
bundles developed by Corlette, Hitchin and Simpson and can be thought as of
a $p$-adic non-abelian Hodge theory.
\medskip
These two ideas are used simultaneously in \cite{KR} in order to get more
information about $\pi_{1}(X)$. The proof in \cite{KR} uses also some very
powerful ideas of Lasell, Ramachandran \cite{BR} and Napier \cite{N1}, which
can be
interpreted as a non-abelian strictness property. These ideas provide a bridge
and make the Nonabelian Hodge theory suitable for questions related to the
Shafarevich conjecture.
\medskip
In this paper we elaborate further on the idea that the answer
to certain uniformization questions depends heavily on the fundamental
group of the variety. We study the question if solvable or nilpotent coverings
$X' \rightarrow X$ are holomorphically convex for $X$ smooth projective
variety.
First we prove the following:
\begin{theo}
The Malcev covering of any smooth projective $X$ is holomorphically
convex.
\end{theo}
As an immediate consequence of this statement we get:
\begin{theo} Let $X$ be a smooth projective variety with a virtually
nilpotent fundamental group. Then the Shafarevich conjecture is true for $X$.
\end{theo}
(Recall that a finitely generated group is nilpotent if its lower
central series has finitely many terms. A group is virtually nilpotent if it
has a finite index subgroup which is nilpotent.)
\bigskip
The proof of Theorem 1.1 uses the functorial Mixed Hodge
Structure (MHS) on $\pi_{1}(X)$ combined with some new ideas of J\'anos
Koll\'ar from \cite{K1} and \cite{K2}.
At the end of section 2 we give a different proof of Theorem 1.2, which
combined with the strictness property for the nonabelian Hodge theory seems
to be a very promising idea ( see \cite{LM}).
Observe that what helps us prove Theorem 1.1 is the use of all nilpotent
representations of $\pi_{1}(X)$ at the same time.
We can ask even more basic question than the Shafarevich conjecture:
\bigskip
\noindent
{\bf Question 1.} Are there any nonconstant holomorphic functions on the
universal covering $X$ of any smooth projective variety?
\bigskip
Clearly it is enough to restrict ourselves to the case when $\pi_{1}(X)$ is an
infinite group.
To study this question in bigger generality we add some more Hodge theoretic
tools - the results of Arapura \cite{A}, Beauvile \cite{BE}, Green,
Lazarsfeld \cite{GL} and Simpson \cite{SA} about characterizing the
absolute sets in the moduli space of rank one local systems. We also need
the following variant of the result of Arapura and Nori \cite{AN} saying
that linear solvable K\"{a}hler groups are nilpotent.
\begin{theo} Let $\Gamma$ be a quotient of a K\"{a}hler group $\pi_{1}(X)$
so that $\Gamma$ is a ${\Bbb Q}$-linear solvable group, then there are two
possibilities - either $\Gamma$ is
virtually nilpotent or $\pi_{1}(X)$ surjects onto the fundamental group of a
curve of genus bigger than zero.
\end{theo}
The above theorem gives a way of constructing new examples of non-K\"{a}hler
groups.
In particular any group $\Gamma$ with infinite $H^{1}([\Gamma,\Gamma],
{\Bbb Q})$ possesing a solvable linear quotient defined over ${\Bbb
Q}$ that is not virtually nilpotent cannot be K\"{a}hler.
Unfortunately we could not prove a solvable variant of the theorem 1.1. The
maximum we were able to say is how much the solvable coverings differ from the
nilpotent ones. We show the following.
\begin{theo} If $\Gamma$ is a quotient of a K\"{a}hler group $\pi_{1}(X)$
so that $\Gamma$ is a complex linear solvable group, then there are two
possibilities - either $\Gamma$ is deformable to a
virtually nilpotent representation of $\pi_{1}(X)$ or $\pi_{1}(X)$ surjects
onto the fundamental group of a curve of genus bigger than zero.
\end{theo}
This theorem gives a way of constructing new examples of non-K\"{a}hler
groups.
In particular any group $\Gamma$ with infinite $H^{1}([\Gamma,\Gamma],
{\Bbb Q})$ possessing a solvable linear quotient defined over ${\Bbb
Q}$ that is not virtually nilpotent cannot be K\"{a}hler. In \cite{LM} it is
proved that the linear covering are holomorphically convex for $X$ an algebraic
surface. Of course this implies a solvable variant of the theorem 1.1 for
algebraic surfaces. The above theorem implies immediately:
\begin{corr} Let $\rho : \pi_{1}(X) \longrightarrow S({\Bbb C})$ be a Zariski
dense representation of the fundamental group of a smooth projective
variety $X$ to the complex points of
an affine solvable group defined over ${\Bbb Q}$. Then the image of
$\pi_{1}(X)$ in the Malcev completion of $\pi_{1}(X)$ is infinite.
\end{corr}
In particular this implies that the first Betti number of $X$ is nonzero so the
universal covering of $X$ $\widetilde{X}$ admits nonconstant holomorphic
functions. The above corollary can be proved of course in a different way too.
If we restrict ourselves to the case when $X$ is an algebraic surface we get a
stronger statement.
\begin{theo} Let $ X $ be a smooth projective surface with an infinite
complex linear representation of its fundamental group. Then there exist
non-constant holomorphic functions on $\widetilde{X}$.
\end{theo}
In some sense the above theorem says that the universal coverings are
different from arbitrary coverings. The well known example of Cousin (see
e.g. \cite{N1}) gives a ${\Bbb Z}$-covering of the two dimensional torus
which does not admit holomorphic functions. The Theorem 1.5 raises a natural
question:
\bigskip
\noindent
{\bf Question 2.} Are there examples of infinite $\pi_{1}(X)$ without any
infinite linear representation?
\bigskip
There are known examples of groups with this properties, e.g. Higman's
four generator group. The question is if they can be fundamental groups.
Even more interesting question was asked by J. Koll\'ar and C. Simpson.
\bigskip
\noindent
{\bf Question 3.} Are there examples of infinite residually finite
$\pi_{1}(X)$ without any infinite linear representation?
\bigskip
As it was pointed out to me by S. Gersten the answer of this question is
positive if we are looking for an arbitrary group not for $\pi_{1}(X)$.
There are the groups of Grigorchuk and Gupta-Sidki which are finitely
generated infinite torsion groups. These groups
are known to be residually finite ( see e.g. \cite{BU}).
\medskip
A negative answer to this question could have a great impact on the answer to
Shafarevich conjecture for residually finite groups (see \cite{LM},
\cite{LP}). From another side a recent paper of Bogomolov and the author
\cite{BL} shows that things can get quite exotic even for $\pi_{1}(X)$. In
some sense the examples constructed in \cite{BL} indicate that if the answer
of {\bf Question 2.} is negative then the statement of Theorem 1.5
could be the best statement in such a generality.
Theorem 1.5 and Corollary 1.1 suggest the following:
\begin{con} Let $ X $ be a smooth projective variety with an infinite
linear representation of its fundamental group. Then there exist non-constant
holomorphic functions on $\widetilde{X}$.
\end{con}
All of this strongly suggests that Hodge theory has a lot to offer
in studying uniformization questions. We stop at the border line, before we
introduce the next level of Hodge theoretic considerations, the theory of
Nonabelian Mixed Hodge Structures- a theory that is giving us a way of
working with all linear representations at the same time to get maximal
information about $\pi_{1}(X)$. The first steps in this theory are done in
\cite{SL}, \cite{SIM1}, \cite{SIM2}, \cite{SIM3} and \cite{LP} and it is far
from being sufficiently developed. In
any case it has fast consequences even on a very primitive level. Using
these very first steps we prove in \cite{LM} the Shafarevich conjecture for
surfaces with linear fundamental groups. The same method implies that the
coverings that correspond to any linear representation are holomorphically
convex. The proof uses basically only the mixed Hodge structure on the
relative completion of $\pi_{1}(X)$ with respect to some complex variation of
mixed Hodge structures. Our feeling is that this is just the beginning.
\bigskip
\noindent
{\bf Acknowledgements:} I would like to thank A. Beilinson F. Bogomolov, J.
Carlson, K. Corlette, R. Donagi, M. Gromov, S. Gersten, M. Larsen, M. Nori,
T. Pantev, C. Simpson, D. Toledo and S. Weinberger for the useful
conversations and H. Clemens, P. Deligne, R. Hain, J. Koll\'ar and M.
Ramachandran for teaching me all ingredients of the technique used in this
paper. Special thanks to Professor J. Koll\'ar for inviting me to visit
University of Utah, where most of this work was done.
\section{The Malcev covering}
In this section we prove Theorem 1.1. and give some applications.
We start with some ideas of J\'anos
Koll\'ar from \cite{K1} and \cite{K2}.
In \cite{K1} Koll\'ar observed that the Shafarevich conjecture
is equivalent to:
1) There exists a normal variety ${\bf Sh}(X)$ and a proper
map with connected fibers ${\bf Sh} : X
\longrightarrow {\bf Sh}(X) $, which contracts precisely the subvarieties
$Z$
in $X$ with the property that $ {\rm im} [\pi_{1}(Z')\longrightarrow
\pi_{1}(X)]$ is finite. Here $Z'$ denotes a desingularization of $Z$.
2) ${\bf Sh}(\widetilde{X} )$ is a Stein space. Here we denote by ${\bf
Sh}(\widetilde{X} )$ the Grauert- Remmert reduction of ${\bf Sh}(\widetilde{X}
)$. In our notations ${\bf Sh}(X)={\bf Sh}(\widetilde{X} ) / \pi_{1}(X)$. The
action of $\pi_{1}(X)$ may have fixed points on ${\bf Sh}(\widetilde{X} )$ but
we can still take a quotient.
One can consider also a relative version of condition 1). Let $H
\triangleleft \pi_{1}(X)$ be a normal subgroup. We will say that
a subgroup $R \subset \pi_{1}(X)$ is almost contained in $H$ if
the intersection $R \cap H$ has finite index in $R$ and we will
write $R \lesssim H$.
We have the following condition.
\medskip
\begin{enumerate}
\item There exists a normal variety ${\bf Sh}^{H}(X)$ and a proper
map with connected fibers ${\bf Sh}^{H}: X \longrightarrow {\bf Sh}^{H}(X),$
which contracts exactly the subvarieties $Z$
in $X$ having the property that ${\rm im}[\pi_{1}(Z')\longrightarrow \pi_{1}(X)]
\lesssim H$. Again $Z'$ denotes a desingularization of $Z$.
The relative version of 2) is the following:
\item ${\bf Sh}^{H}(\widetilde{X} )$ is a Stein space. Here we denote by
${\bf Sh}^{H}(\widetilde{X} )$ the Grauert- Remmert reduction of
${\bf Sh}(\widetilde{X} )$. In our notations ${\bf Sh}^{H}(X)
= {\bf Sh}^{H}(\widetilde{X} ) / (\pi_{1}(X)/H)$.
\end{enumerate}
This was also independently observed by F. Campana in \cite{CM}.
\bigskip
Our approach is that if there is a natural candidate for ${\bf Sh}(X) $ it is
enough to check condition 1) only for $Z$ - an algebraic curve. This certainly
is the case when $\pi_{1}(X)$ is a nilpotent group. In the simplest case when
$\pi_{1}(X)$ is virtually abelian one uses for ${\bf Sh}(X) $ the Albanese
variety ${\rm Alb} (X)$.
It is clear (see e,g, \cite{CT}) that for a smooth projective variety $X$
with $\pi_{1}(X)$ an infinite nilpotent group the Albanese map:
\[ {\rm Alb} : X \longrightarrow {\rm Alb}(X) \]
has nontrivial image. In other words $\dim_{\Bbb{C}}({\rm im}({\rm Alb}))>0.$
Moreover if we denote by $S$ the Stein factorization of the Albanese map,
then this is a natural candidate for ${\bf Sh}(X) $ in case $\pi_{1}(X)$ is a
nilpotent group. Observe that the map
\[ X \longrightarrow S \]
contracts all subvarieties $Z$ with the property that $ {\rm im} [H_{1}(Z,
{{\Bbb Q}})\longrightarrow H_{1}(X, {{\Bbb Q}})]$ is trivial.
Now using that $\pi_{1}(X)$ is a nilpotent group and the theory of Mixed Hodge
Structures on its Malcev completion we show that the fact that $ {\rm im}
[H_{1}(Z, {{\Bbb Q}})
\longrightarrow H_{1}(X, {{\Bbb Q}})]$ is trivial is equivalent to the fact
that $ {\rm im} [\pi_{1}(Z)\longrightarrow \pi_{1}(X)]$ is finite for $Z$ an
algebraic curve. We finish the proof by reducing the argument for $Z$ of
arbitrary dimension to $Z$ an algebraic curve.
To prove Theorem 1.1 we need to show again that there is natural candidate for
${\bf Sh}^{H}(X) $, where $H={\rm ker}(\rho : \pi_{1}(X) \longrightarrow
\fgc{{\rm uni}}{X,x}$ of the Malcev representation.
Again this candidate is $S$ the Stein factorization of the Albanese map.
At the end of section we give a different proof of Theorem 1.1, which is
basically spelling of the proof we have given already in the language of
equivariant harmonic maps.
\subsection{Mixed Hodge Structure considerations}
In this subsection we explain why if $\pi_{1}(X)$ is a nilpotent group the
theory of Mixed Hodge Structures on it implies that ${\rm im} [H_{1}(Z, {{\Bbb Q}})
\longrightarrow H_{1}(X, {{\Bbb Q}})]$ is trivial is equivalent to the fact
that $ {\rm im} [\pi_{1}(Z)\longrightarrow \pi_{1}(X)]$ is finite for $Z$ an
algebraic curve. For some background one can look at \cite{D}, \cite{DG} or
\cite{H}.
For the proof of Theorem 1.1 we need to work with $X$ smooth but for
completeness in this section we will require only the MHS on $H^{1}(X)$ is
of weights $> 0$.
\begin{lemma} If Z is a compact nodal curve and
$f:Z \longrightarrow X$ is a map to a variety such that MHS on $H^{1}(X)$
is of weights $> 0$ then the map
\[ f_{* }: L(Z,x) \longrightarrow L(X,f(x)) \]
is trivial if and only if the map
\[f^{*} : H^{1}(X,{\Bbb Q}) \longrightarrow H^{1}(Z,{\Bbb Q})\]
is trivial. Here $L(Z,x)$ and $L(X,f(x))$ are the corresponding Lie algebras
of the unipotent completions $\fgc{{\rm uni}}{Z,x}$ and $\fgc{{\rm uni}}{
X,f(x)}$ of
the fundamental groups $\pi_{1}(Z,x)$ and $\pi_{1}(X,f(x))$ respectively and
$x$ is a point in $Z$.
\end{lemma}
{\bf Proof.} Observe that the map in unipotent completions determines and is
determined by a map on the corresponding Lie algebras:
\[L(Z,x) \longrightarrow L(X,f(x)).\]
First let us consider the case where $H_1(Z)$ is pure of weight $-1$. This
is the case when the dual graph of $Z$ is a tree. By a standard strictness
argument the weight filtration on $L(Z,x)$ is its lower central
series, and the associated graded Lie algebra is generated by
$Gr_{-1} L(Z,x) = H_{1}(Z, {\Bbb Q})$.
Since
\[L(Z,x) \longrightarrow L(X,f(x))\]
is a morphism of MHS, it is non-zero if and only if the
map
\[Gr L(Z,x) \longrightarrow Gr L(X,f(x))\]
on weight graded quotients is. Since
\[Gr_{-1} L(X,f(x)) = H_{1}(X,{\Bbb Q})/W_{-2},\]
and since $H_{1}(Z,{\Bbb Q}) \longrightarrow H_{1}(X,{\Bbb Q})$ is trivial, it
follows that $L(Z,x) \longrightarrow L(X,f(x))$ is trivial.
\medskip
To prove the general case, we take a partial normalization
\[Z' \longrightarrow Z\]
with the property that $Z'$ is connected and such that $H^{1}(Z')$ is a pure
MHS of weight 1.
This can be done as follows.
Take a maximal tree $T$ in the dual graph
of $Z$ and normalize only those double points corresponding to
edges not in $ T$. Then $H_{1}(Z) $ is pure MHS of weight -1. The previous
argument implies that
\[L(Z',x) \longrightarrow L (X,f(x))\]
is trivial.
To complete the proof, note that we have an exact sequence
\[1 \longrightarrow N \longrightarrow \pi_{1}(Z,x) \longrightarrow
\pi_{1}(\Gamma,*) \longrightarrow 1,\]
where $\Gamma$ denotes the dual graph of $Z $ and $N$ is the normal subgroup
of $\pi_{1}(Z)$ generated by $\pi_{1}(Z',x)$. After passing to unipotent
completions, we obtain an exact sequence
\[0 \longrightarrow (L(Z'))\longrightarrow L(Z,x) \longrightarrow
L(\Gamma,*) \longrightarrow 0.\]
This is an exact sequence in the category of Malcev Lie
algebras with MHS. The ideal $(L(Z'))$ generated by $L(Z')$
is exactly $W_{-1} L(Z)$, so the MHS induced
on $L(\Gamma,*)$ is pure of weight 0.
It follows that the homomorphism $L(Z,x) \longrightarrow L(X,f(x))$
induces a homomorphism
\[L(\Gamma,*) \longrightarrow L(X,f(x)).\]
This is a morphism of MHS of (0,0) type. It is injective
if and only if the map
\[L(\Gamma,*) = Gr L(\Gamma,*) \longrightarrow Gr L(X,f(x))\]
is also injective. Since $H_{1}(X)$ has weights $< 0$ and $L(\Gamma,*)$
has weight zero, it follows that
\[L(\Gamma,*) = Gr L(\Gamma,*) \longrightarrow Gr L(X,f(x))\]
is zero. This proves the statement in general.
Namely, we have that for any nodal curve (singular, reducible) the map
\[ f_{* }: L(Z,x) \longrightarrow L(X,f(x)) \]
is trivial if and only if the map
\[f^{*} : H^{1}(X,{\Bbb Q}) \longrightarrow H^{1}(Z,{\Bbb Q})\]
is trivial. \hfill $\Box$
\begin{lemma} Let $X$ be a smooth projective variety with a nilpotent
fundamental group $\pi_{1}(X)$. Then for any algebraic curve $Z \subset X $
the fact $ {\rm im} [H_{1}(Z, {{\Bbb Q}})\longrightarrow
H_{1}(X, {{\Bbb Q}})]$ is trivial is equivalent to the fact that $ {\rm im}
[\pi_{1}(Z)\longrightarrow \pi_{1}(X)]$ is finite.
\end{lemma}
{\bf Proof.} Since we
can always find a partial normalization $\widetilde{Z} \rightarrow Z$ with
$\widetilde{Z}$-nodal and $\pi_{1}(\widetilde{Z},\tilde{x}) \rightarrow
\pi_{1}(Z,x)$ surjective it follows from the previous lemma that the map
\[ f_{* }: L(Z,x) \longrightarrow L(X,f(x)) \]
is the zero map.
Furthermore, if $\pi_{1}(X)$ is a torsion free nilpotent group then by
definition it embeds in $\pi_{un}(X, f(x))$. It is easy to see that torsion
elements of a nilpotent group generate a finite group and hence
\[\pi_{1}(X,f(x)) \longrightarrow \fgc{{\rm uni}}{X,f(x)}\]
is an embedding up to torsion which proves the lemma.
\hfill
$\Box$
We have actually proved more:
\begin{lemma} Let $X$ be a smooth projective variety and $\rho : \pi_{1}(X)
\longrightarrow L(X,f(x))$ be the Malcev representation of $\pi_{1}(X)$.
Then for any algebraic curve $Z \subset X $ the fact $ {\rm im} [H_{1}(Z,
{{\Bbb Q}})\longrightarrow
H_{1}(X, {{\Bbb Q}})]$ is trivial is equivalent to the fact that $ {\rm im}
[\pi_{1}(Z) \longrightarrow \pi_{1}(X) / H ]$ is finite. Here $H$ is the
kernel of the Malcev representation.
\end{lemma}
\subsection{A reduction to the case of an algebraic curve}
In this section we show how to reduce the argument for $Z$ of arbitrary
dimension to $Z$ an algebraic curve.
\begin{lemma} Let $F$ be a connected subvariety in $X$ then we can find a
curve $Z
\subset F$ such that $\pi_{1}(Z) $ surjects on $\pi_{1}(F) $.
\end{lemma}
{\bf Proof.} If $F$ is smooth variety the above lemma is just the Lefschetz
hyperplane section theorem. Let $F= F_{1}+ \ldots + F_{i}$ be singular and
with many components of different dimension. Denote by $n$ the normalization
$n:F' \longrightarrow F $ of $F$. In every component of $F'$ after
additional desingularization we can find finitely many points $x_{k}, y_{k}$
such that $n(x_{k})=n(y_{k})$ and $\pi_{1}(F'/ x_{k}=y_{k})$ surjects onto
$\pi_{1}(F)$. The way to do that is to take the Whitney stratification of
$F$ and put the points $x_{k},y_{k}$ in every stratum in a way that all loops
that come from singularities pass through these points. Now following
\cite{GM}(ii, 1.1) we take hypesurfaces with big degrees that pass through
the points $x_{k}, y_{k}$ and intersect every component of $F'$, $F'_{l}$
in a curve $Z_{l}$ such that $Z'= \cup Z_{l}$ and $\pi_{1}(Z')$ surjects
on $\pi_{1}(F') $. We make $Z=n(Z')$. Observe that $Z$ might be singular
and have many components but it will be connected.
\hfill $\Box$
Now we are ready to finish the proof of Theorem 1.1. We start with the
Stein factorization of the Albanese map for $X$
\[ {\rm Alb} : X \longrightarrow S \longrightarrow {\rm im}({\rm Alb}) \subset {\rm Alb}(X).\]
Denote by $S'$ the fiber product of the universal covering $\widetilde{{\rm Alb}(
X)}$ of ${\rm Alb}(X)$ and $S$ over ${\rm Alb}(X)$. By definition the map
\[ S' \longrightarrow \widetilde{{\rm Alb}(X)} \]
is a covering map and since $\widetilde{{\rm Alb}(X)}$ is a Stein manifold $S'$
is a Stein manifold as well. It follows from the definition of the Albanese
morphism that the fibers of the map
\[ {\rm Alb} : X \longrightarrow S \]
are all subvarieties $F$ in $X$ for which the map
$H_{1}(F, {{\Bbb Q}})\longrightarrow H_{1}(X, {{\Bbb Q}})$ is trivial. We
willshow that if the fact that $H_{1}(F, {{\Bbb Q}})
\longrightarrow H_{1}(X, {{\Bbb Q}})$ is trivial implies that $ {\rm im}
[\pi_{1}(F)\longrightarrow \pi_{1}(X) / H ]$ is finite.
We have shown this in Lemma 2.3 when $F$ is an algebraic curve.
Now if $dim_{\Bbb{C}}(F)>1$ we apply Lemma 2.4 to find a curve $Z$ in $F$
such that $\pi_{1}(Z) $ surjects on $\pi_{1}(F) $. The argument of
Lemma 2.1.2 implies that $\pi_{1}(F) $ goes to a finite group in
$\pi_{1}(X)/H $ since
$\pi_{1}(Z) $ goes to afinite group in $\pi_{1}(X)/H $. Observe that the
curve $Z$ is also contained in the fiber $F$ of the map
\[ {\rm Alb} : X \longrightarrow S .\]
Therefore the map
$H_{1}(Z, {{\Bbb Q}})\longrightarrow H_{1}(X, {{\Bbb Q}})$ is also trivial.
To finish the proof of Theorem 1.1 we need to observe that $S$ satisfies
the conditions for being the Shafarevich variety of $X$, $S= {\bf Sh}^{H}(X)$.
Namely
1) There exists a holomorphic map with connected fibers $ X
\longrightarrow S $, which contracts only the subvarieties $Z$
in $X$ with the property that $ {\rm im} [\pi_{1}(Z)\longrightarrow
\pi_{1}(X)/H]$ is finite.
2) $ {\bf Sh}^{H}(\widetilde{X})=S'$ is a Stein space.
\hfill $\Box$
To prove Theorem 1.2 we use the same argument as above but $H$ is a
finite
group.
Actually we have shown more:
\begin{corr} Let $X$ be a smooth projective variety with a virtually
residually nilpotent
fundamental group. Then the Shafarevich conjecture is true for $X$.
\end{corr}
\subsection{Some examples}
In this subsections we give some examples and geometric applications of our
method. We start with the following result that was also proved by Campana in
\cite{CM1}.
\begin{corr} Let $X$ be a smooth projective surface and $\Gamma$ is the
image of $\pi_{1}(X)$ in $L(X,f(x))$. Let as before $S$ be the Stein
factorization of the map $X \longrightarrow {\rm im}({\rm Alb}(X))$. After taking an
etale finite covering $X'' \longrightarrow X$ the homomorphism $\pi_{1}(X'')
\longrightarrow \Gamma$ factors through the map $\pi_{1}(S) \longrightarrow
\Gamma$.
\end{corr}
{\bf Proof.} According to \cite{K2} 4.8 after taking some etale finite
covering $X'' \longrightarrow X$, $\pi_{1}(X'')$ is the same as the
fundamental group of $\pi_{1}(S)$. This follows from the fact that residually
nilpotent groups are residually finite.
\hfill $\Box$
Nilpotent K\"{a}hler groups were constructed by Sommese and Van de Ven
\cite{SV}, and Campana \cite{CM1}. The construction goes as follows:
Start with a finite morphism from an abelian variety $A$ to ${\Bbb P}^{n}$.
Now take the preimage $X$ in $A$ of any abelian d-fold in ${\Bbb P}^{n}$. A
double cover of $X$ has as fundamental group a nonsplit central extension of
an abelian group by ${\Bbb Z}$.
Let us following \cite{SV} give more explicit example. We start with a
four dimensional abelian variety $A$ and a finite morphism $f$ to
${\Bbb P}^{4}$. Take the Mumford-Horrocks abelian surface $Z$ in ${\Bbb
P}^{4}$
and pull it back to $A$. Let us call the new surface
$f^{-1}(Z)$. The following exact sequence was established in \cite{SV}
\[\pi_{2}(A)\oplus \pi_{2}(Z) \longrightarrow \pi_{2}({\Bbb P}^{4})
\longrightarrow \pi_{1}(f^{-1}(Z)) \longrightarrow \pi_{1}(A)\oplus
\pi_{1}(Z) \longrightarrow 0.\]
In our case this sequence reads as:
\[0 \longrightarrow {\Bbb Z} \longrightarrow \pi_{1}(f^{-1}(Z)) \longrightarrow
{\Bbb Z}^{12} \longrightarrow 0\]
and shows that $f^{-1}(Z)$ has a two steps nilpotent fundamental group.
Actually we know more. By theorem of Arapura
and Nori \cite{AN} all K\"{a}hler linear solvable groups are
virtually nilpotent. So we have the following:
\begin{corr} Let $X$ be a smooth projective variety with a linear solvable
fundamental group. Then the universal covering $\widetilde{X}$ is
holomorphically convex.
\end{corr}
Now we will use the technique from Lemmas 2.3 and 2.4 to show that the
theorem of Arapura
and Nori \cite{AN} is the marginal statement meaning that there exists a
residually solvable linear group which does not embed in its Malcev
completion. By residually solvable we mean a group that embeds in its
completion with respect to all finitely generated solvable representations.
The following example came out from a discussion with D. Arapura, J\'anos
Koll\'ar, M. Nori, T. Pantev, M. Ramachandran and D. Toledo.
Consider nontrivial smooth family of smooth abelian varieties of dimension
$N$ over curve $C$. Let us denote this family by $X$. The fundamental group
of $X$ is given by the following exact sequence
\[ 0 \longrightarrow {\Bbb Z}^{2N} \longrightarrow \pi_{1}(X) \longrightarrow
\pi_{1}(C) \longrightarrow 0.\]
The group $\pi_{1}(X) $ is a semidirect product of the groups ${\Bbb Z}^{2N}$
and $\pi_{1}(C)$. For generic enough family we can make the monodromy action
\[M: \pi_{1}(C) \longrightarrow Sp(2N, {\Bbb Z}) \]
to be irreducible and from here one can get that the image of ${\Bbb Z}^{2N}$
in $H^{1}(X,{\Bbb Z})$ is trivial.
The the group $\pi_{1}(X)$ is linear. To see that we consider the morphism
\[l: \pi_{1}(X) \longrightarrow SL(2,{\Bbb C}) \times [ GL(V) \ltimes V ].\]
Here $V$ is a vector space over $\Bbb{C}$ of dimension $N$ on which
${\Bbb Z}^{2N}$ acts discretely. It is easy to see that $L$ is an injection.
The group $\pi_{1}(X)$ is also virtually residually solvable. This can be
seen as follows:
Choose a prime number $p$. Since the group $\pi_{1}(X)$ is linear, namely it
embeds in $GL(T)$ for some vector space $T$ we can embed it up to a finite
index in a series of finite solvable groups $GL(T_{p^{q}})$ for $q=1,2,
\ldots$.
{}From another point $\pi_{1}(X)$ does not embed in its Malcev completion up
to a finite index.
Assume that $\pi_{1}(X)$ does embed in its Malcev completion. Then
lemma 2.3 and lemma 2.4 imply that if the image of ${\Bbb Z}^{2N}$ in
$H^{1}(X,{\Bbb Z})$ is trivial then the image of ${\Bbb Z}^{2N}$ in the Malcev
completion of $\pi_{1}(X)$ is trivial which is not the case in our situation.
Therefore our technique does not answer the question if the universal
coverings of $X$ or of generic hyperplane sections of it is holomorphically
convex. Of course this is true and can be seen as follows:
\begin{prop} The universal covering of any smooth family of Abelian varieties
or of any generic hyperplane section of them is holomorphically convex.
\end{prop}
{\bf Proof.} It follows from \cite{K1}, Theorem 6.3 that every smooth
family of abelian varieties over a curve has a linear fundamental group since
according
to 6.3 \cite{K1} after a finite etale covering it is birational to a family
of a smooth abelian varieties. But the universal covering of a family of
smooth abelian varieties or a generic hyperplane section of it is
holomorphically convex since it is a Stein space since. It embeds in $(SIEG
\times C^{N})$, where $SIEG $ is the Siegel upperhalf plane.
\hfill $\Box$
It also follows from \cite{LM} where more
powerful technique, the theory of Nonabelian Mixed Hodge Structures, is used.
We formulate:
\begin{corr} Let $X$ be a smooth projective variety with an infinite
virtually nilpotent fundamental group and such that ${\rm rank}\,
{\rm Pic}(X) = 1$ (or better ${\rm rank}\, NS(X)=1$). Then for every
subvariety $Z$
in $X$ we have that $ {\rm im} [\pi_{1}(Z)\longrightarrow \pi_{1}(X)]$ is infinite.
\end{corr}
The proof is an easy consequence of \cite{K2} (Chapter 1).
We demonstrate a quick application of the above corollary. Denote by $A$ a
four
dimensional abelian variety. Let us say that $X$ is a hypersurface in it with
an isolated singular point $s$ of the following type $xy=zt$. Denote by $X'$
the blow of $X$ in $s$.
We glue in $X$ ${\Bbb P}^{1} \times {\Bbb P}^{1}$ instead of $s$.
It is easy to see that $X'$ is smooth and that ${\Bbb P}^{1}$ can be blown
down. The new space $X''$, obtained after blowing down one of the above
${\Bbb P}^{1}$, is smooth, ${\rm rank} NS(X'')=1$ and
$\pi_{1}(X)={\Bbb Z}^{4}$. Therefore the Shafarevich conjecture should be true
for $X''$. But as we can see $X''$ contains ${\Bbb P}^{1}$. This contradicts
the
above corollary and we conclude that $X''$ is not projective. Of course all
this can be seen in many different ways. This is another spelling of the fact
that ${\Bbb P}^{1}$, that remains in $X''$, should be homologically nontrivial
if $X''$ is projective.
We give now an idea of an alternative proof of Theorem 1.1 which came from
conversations with M. Ramachandran. It is based on the use of $\pi_{1}(X)$
equivariant harmonic maps to the universal coverings to Higher Albanese
varieties defined in \cite{HZ}. Combined with the strictness property for the
nonabelian Hodge theory this seems to be a very promising idea ( see
\cite{LM}).
Denote by $G_{s}$ the complex simply connected group $\pi_{1}(X) /
\Gamma^{s+1}$, where $\Gamma^{i}$ are the groups from the lower central series
for $\pi_{1}(X)$ and $\Gamma^{s}$ is the smallest nontrivial one. The
corresponding Lie algebra $g_{s}$ has MHS. Denote by $F^{0}G_{s}$ the closed
subgroup in $G_{s}$ group that corresponds to $F^{0}g_{s}$. Since the group
$\pi_{1}(X) / \Gamma^{s+1}$ is unipotent then as it is easy to see we have a
free action of the corresponding to $G_{s}$ lattice $G_{s}({\Bbb Z})$ on
$G_{s}/F^{0}G_{s}$.
Therefore in the same way as in \cite{KR} we obtain a $\pi_{1}(X) $
equivariant proper horizontal holomorphic map ( see \cite{HZ})
\[ \widetilde{X} \longrightarrow G_{s}/F^{0}G_{s}.\]
According to \cite{H1} $G_{s}/F^{0}G_{s}$ is biholomorphic to $\Bbb{C}^{N}$.
Therefore $\widetilde{X}$ is holomorphically convex.
\begin{rem}{\rm The above argument is weaker then the argument we have used in
the first proof. It cannot be generalized to the case of residually nilpotent
groups since in this case $G_{s}/F^{0}G_{s}$ will not be a manifold.}
\end{rem}
\section{Solvable coverings}
We would like to obtain the analog of Theorem 1.1 for solvable
coverings. The analog of Theorem 1.2 for solvable groups - Corollary 2.3
was proved in the previous section as a consequence of the result of Arapura
and Nori. We cannot prove solvable analog of theorem 1.1. The maximum we can do
is to realize how close the solvable representations come to nilpotent ones. To
be able to do so we need to generalize slightly the result of Arapura and
Nori.
First we prove Theorem 1.3.
\noindent
{\bf Proof.} (The idea of the proof was suggested to me by T. Pantev.) Denote
by $\Gamma$ the image of the solvable representation $\rho : \pi_{1}(X)
\rightarrow L$. We need to show that either $\Gamma$ is virtually nilpotent
or there exists a holomorphic map with connected fibers $f : X \rightarrow C$
to a smooth curve $C$ of genus $\geq 1$.
First we introduce some notations. For a
finitely generated group $\Gamma$ denote by $\Sigma(\Gamma)$ the set of all
special characters of $\Gamma$. That is
\[
\Sigma(\Gamma) := \left\{ \alpha : \Gamma \rightarrow {\Bbb C}^{\times}
\left| H^{1}(\Gamma, {\Bbb C}_{\alpha}) \neq 0 \right. \right\},
\]
where ${\Bbb C}_{\alpha}$ is the one dimensional $\Gamma$-module associated to
$\alpha$. Now we recall the following:
\begin{prop} [Arapura-Nori \cite{AN}] Let $\Gamma$ be a finitely generated
${\Bbb Q}$-linear solvable group. Then the following are equivalent
\begin{enumerate}
\item $\Gamma$ is virtually nilpotent.
\item $\Sigma(\Gamma)$ consists of finitely many torsion characters.
\end{enumerate}
\end{prop}
Due to this proposition it is enough to show that either $\Sigma(\Gamma)$
consists of finitely many torsion characters or $X$ has a non-trivial map to
a curve of genus bigger than zero. Now, since $\pi_{1}(X)$ surjects on
$\Gamma$ it follows that $\Sigma(\Gamma) \subset \Sigma(\pi_{1}(X))$ and hence
it suffices to show that either $\Sigma(\pi_{1}(X))$ consists of finitely
many torsion characters or $X$ has an irrational pencil.
\medskip
For a smooth projective variety $X$ denote by $M(X)$ the moduli space of
homomorphisms from $\pi_{1}(X)$ to ${\Bbb C}^{\times}$. The locus of special
characters is a jump locus in $M(X)$ and hence it is a subscheme in a natural
way.
It turns out that $\Sigma(\pi_{1}(X))$ is actually a smoooth subvariety having
very special geometric properties which we are going to exploit. Since the
subvariety $\Sigma(\pi_{1}(X)) \subset M(X)$ is completely canonical one
expects it to have an intrinsic description. One way to construct natural
subvarieties in $M(X)$ is via pullbacks. Namely, given any surjective morphism
$\varphi : X \rightarrow Y$ we can pullback the moduli space of characters of
$\pi_{1}(Y)$ to get a subvariety $\varphi^{*}M(Y) \subset M(X)$. According to
\cite{SA}, Lemma 2.1 and Theorem 6.1 every connected component $\Sigma$ of the
subvariety $\Sigma(\pi_{1}(X)) \subset M(X)$ is of this kind. More specifically
for every such $\Sigma$ there exists a torsion character $\sigma \in \Sigma$
and a connected abelian subvariety $P \subset {\rm Alb}(X)$ so that $\Sigma$ is
the translation of $\varphi^{*}M({\rm Alb}(X)/P) \subset M(X)$ by $\sigma$. Here
$\varphi : X
\rightarrow {\rm Alb}(X) \rightarrow {\rm Alb}(X)/P$ is the composition of the Albanese
map and the natural quotient morphism. In particular, $\Sigma(\pi_{1}(X))$
has a positive dimensional component if and only if its intersection with the
set of all unitary characters has a positive dimensional component. Now the
Hodge decomposition of the cohomology of a unitary local system implies that
unless $\Sigma(\pi_{1}(X))$ consists of finitely many torsion characters
the subvariety of all special line bundles in ${\rm Pic}^{\tau}(X)$ has a
positive dimensional component. Indeed, for a unitary character $\alpha$
denote by ${\Bbb L}_{\alpha}$ the corresponding rank one local system and
by $L_{\alpha} = {\Bbb L}_{\alpha}\otimes_{\Bbb C} {\cal O}_{X}$ the
corresponding holomorphic line bundle. Now by the Hodge theorem
\[h^{1}(\pi_{1}(X),{\Bbb C}_{\alpha})
= h^{1}(X, {\Bbb L}_{\alpha}) = h^{1}(X,L_{\alpha})+h^{0}(X,\Omega^{1}_{X}
\otimes L_{\alpha}) = 2h^{1}(X,L_{\alpha}),\]
i.e. $\alpha$ is a special character iff the line bundle $L_{\alpha}$ is
special.
Furthermore a theorem of Beauville (\cite{BE}, Proposition 1) asserts that
the subvariety of ${\rm Pic}^{0}(X)$ consisting of special line bundles is a
union of a finite set and the subvarieties of the form $f^{*}{\rm Pic}^{0}(B)$
where $f : X \rightarrow B$ is a morphism with connected fibers to a curve $B$
of genus $\geq 1$. Thus $X$ posseses irrational pencils which finishes the
proof of Theorem 1.3 \hfill $\Box$
\medskip
The above theorem can be seen as the solvable analog of the theorem of
Simpson's that $SL(n,{\Bbb Z})$ is not a K\"{a}hler group, $n>2$. This
theorem gives a way of constructing new examples of non-K\"{a}hler groups.
In particular any group $\Gamma$ with infinite $H^{1}([\Gamma,\Gamma],
{\Bbb Q})$ possessing a solvable linear quotient defined over ${\Bbb
Q}$ that is not virtually nilpotent cannot be K\"{a}hler.
\medskip
Now we prove theorem 1.4.
\noindent
{\bf Proof.} We would like to use theorem 3.1. Therefore we need an infinite
solvable representation defined over ${\Bbb
Q}$. First we show
\begin{lemma} Let $S$ be a connected affine solvable group defined over ${\Bbb
Q}$. If $\rho : \pi_{1}(X) \rightarrow S({\Bbb C})$ is a Zariski dense
represenattion into the group of complex points of $S$, then $\rho$ can be
deformed to a representation $\nu : \pi_{1}(X) \rightarrow S(\overline{{\Bbb
Q}})$ having an infinite image.
\end{lemma}
{\bf Proof.}
Denote by $\Lambda$ the image of $\rho:\pi_{1}(X) \to S$. Let $\phi:\Lambda
\to S(\Bbb{C})$ be a homomorphism from $\Lambda$
to the group of complex points of $S$ with an infinite image.
We want to find a subgroup $\Lambda'$ of $ \Lambda$ of finite index and a
homomorphism $\phi':\Lambda'\to S (\overline{{\Bbb Q}})$ with infinite image,
such that $\phi '$ is arbitrarily close to $\phi$.
Let $S_{0} := S$, and consider the (upper) derived series $S_{i+1} :=
[S_{i},S_{i}]$ for $S$.
We choose the maximal $i$ such that $\Lambda\cap S_{i(\Bbb{C})}$ is of finite
index of $\Lambda$.
We replace $G$ by $\Lambda' = \Lambda\cap S_i(\Bbb{C})$. The image of
$\Lambda'$ in $S_i(\Bbb{C})/S_{i+1}(\Bbb{C})$
is infinite. The group $A = S_{i}/S_{i+1}$ is either a torus or a vector
space group. Let $X$ be the affine variety of homomorphisms $\Lambda' \to A$,
$Y$ the affine variety $Hom(\Lambda',S_{i}), X' $ the image of $Y$ in $X$.
Thus $X'$ is the affine subvariety of $X$ consisting of homomorphisms which
factor through $S_{i}$. We want to find points on $X'(\overline{{\Bbb Q}})$
arbitarily close in $X'(\Bbb{C})$ to the point defined by $\phi$. If the
original point is defined
over $\overline{{\Bbb Q}}$, we are done. If not, the original point cannot
be isolated,
since $X'$ is defined over $\overline{{\Bbb Q}}$. Thus we have arbitrarily
close points
defined over $\overline{{\Bbb Q}}$. To find a $\overline{{\Bbb Q}}$ point
with an infinite image we consider two cases:
1) $A$ is a vector space group. Let $g$ be an element of
$\Lambda'$ such that $\phi(g) $ maps to a non-trivial element of $A$. Then for
every nearby representation $\phi'$ the element $\phi'(g)$ is non-trivial
in $A$, therefore it is of infinite order in $A$.
2) $A$ is a torus $T$.
We fix an element $g$ in $\Lambda'$ such that $\phi(g)$ maps to a point of
infinite order on $T$. Let $Z \subset T$ denote the image of $X'$ in $T$
under the map which takes each homomorphism $\Lambda'\to T$ to the
image of $g$ in $T$. By definition $X' \to Z$ is a surjective map and it is
defined over
$\overline{{\Bbb Q}}$, so every $\overline{{\Bbb Q}}$ -point of $Z$ comes from
a $\overline{{\Bbb Q}}$-point of $X'$
and therefore from a $\overline{{\Bbb Q}}$ -point of $Y$, i.e. an actual $\bar
Q$ -homomorphism
from $\Lambda'$ to $S_{i}$. So it is enough to find a $\overline{{\Bbb Q}}$
-point of $Z$ which is
close to the image of the original homomorphism $\phi$ but which is also
of infinite order.
We prove the following:
\begin{claim} Let $T$ be a torus, $Z$ a $\overline{{\Bbb Q}}$-affine subvariety
of $T$, $p$ a point in
$Z(C) $ of infinite order. Then $p$ is in the closure of the subset of
$Z(\overline{{\Bbb Q}}) $ consisting of points of infinite order.
\end{claim}
{\bf Proof.} If $p$ is in $Z(\overline{{\Bbb Q}})$, we are done. If not,
there exists a character
$\chi: Z \to GL(1,\Bbb{C})$ such that $\chi(p)$ is not in $ \overline{{\Bbb
Q}}$ . As $Z, T$ and $\chi$
are defined over $ \overline{{\Bbb Q}}$ , and $GL(1,\Bbb{C})$ is
1-dimensional, it follows
that $\chi(Z)$ is an open subset of $GL(1,\Bbb{C})$. In particular, every
neighborhood of $p$ in $Z$ maps to a neighborhood of $\chi(p)$ containing
non-torsion elements. If $ q \in Z(\overline{{\Bbb Q}})$ maps to a non-torsion
element,
then of course $q$ is a point of infinite order in $T(\Bbb{C})$. This finishes
the proof of the claim and the lemma.
\hfill $\Box$
\medskip
To get an infinite solvable representation defined over ${\Bbb Q}$
consider the affine solvable group $\widetilde{S}$ obtained from $S$ by
restriction of scalars, i.e. $\widetilde{S} := {\rm res}_{\overline{{\Bbb Q}}/
{\Bbb Q}}S$. The representation $\nu$ induces a representation $\tilde{\nu}:
\pi_{1}(X) \rightarrow \widetilde{S}({\Bbb Q})$ which has an image isomorphic
to the image of $\nu$. \hfill $\Box$
\hfill $\Box$
Now to prove Corollary~1.1 we have to consider the following two alternatives
1) The group $\tilde{\nu}(\pi_{1}(X)$
is virtually nilpotent so it has a subgroup of
finite index which is nilpotent.
2) There exists an holomorphic map with connected fibers $f : X
\rightarrow C$ to a curve of genus one or higher. But then we know that $
\pi_{1}(C)$ embeds in its Malcev completion.
In both cases there exists a finite \'{e}tale cover of $X$ which has a
non-trivial Albanese variety, which is what we need.
Theorem 1.5 follows easily from Theorem 1.1 and the result from \cite{KR}.
{\bf Proof.} Let us start with a complex linear representation
${\rm im}[\pi_{1}(X) \rightarrow L]$. Then we have the following three
possibilities.
(a) The image of $\pi_{1}(X)$ in $L/R^{u}L$ does not have zero or two ends.
Then we can apply \cite{KR} to conclude that $\widetilde{X}$ has
a non-constant holomorphic function.
(b) The image ${\rm im}[\pi_{1}(X) \rightarrow L/R^{u}L]$ has two ends. Then
by the theorem of Hopf and Freudenthal it follows that ${\rm im}[\pi_{1}(X)
\rightarrow L/R^{u}L]$ has a subgroup of finite index that is isomorphic to
${\Bbb Z}$. Therefore the abealianization of $\pi_{1}(X)$ is not finite. This
implies that the Malcev representation is not trivial and we apply
Theorem 1.1 to finish the proof.
(c) The image of $\pi_{1}(X)$ in $L/R^{u}L$ has zero ends. Then $L/R^{u}L$ is
a finite group. So the Malcev representation is not trivial and we are taken
applying Theorem 1.1.
\hfill $\Box$
\medskip
Theorem 1.5 follows from \cite{LM} as well. To be able to attack conjecture
1.1 we should be able to analyze the real issue, the semisimple
representations. Some initial steps in this direction are done in \cite{LM}.
\medskip
What should we do if the answer of {\bf Question 3} is positive? We hope
using \cite{LP} to be able to handle the case when the image of $ \pi_{1}(X)$
in its proalgebraic completion is infinite.
\medskip
What should we do if the answer of {\bf Question 2} is positive and the
$ \pi_{1}(X)$ in question has finite image in its proalgebraic completion.
At the moment this case seems to be out of reach.
|
1998-02-16T01:32:57 | 9510 | alg-geom/9510007 | en | https://arxiv.org/abs/alg-geom/9510007 | [
"alg-geom",
"math.AG"
] | alg-geom/9510007 | Yekutieli Amnon | Amnon Yekutieli | Smooth Formal Embeddings and the Residue Complex | 33 pages, AMSLaTeX, final version (some corrections, section on
D-modules omitted), to appear in Canadian Math. J | null | null | null | null | Let \pi : X -> S be a finite type morphism of noetherian schemes. A smooth
formal embedding of X (over S) is a bijective closed immersion X -> \frak{X},
where \frak{X} is a noetherian formal scheme, formally smooth over S. An
example of such an embedding is the formal completion \frak{X} = Y_{/X} where X
\subset Y is an algebraic embedding. Smooth formal embeddings can be used to
calculate algebraic De Rham (co)homology. Our main application is an explicit
construction of the Grothendieck residue complex when S is a regular scheme. By
definition the residue complex is the Cousin complex of \pi^{!} \cal{O}_{S}. We
start with Huang's theory of pseudofunctors on modules with 0-dimensional
support, which provides a graded sheaf \cal{K}^{.}_{X/S}. We then use smooth
formal embeddings to obtain the coboundary operator on \cal{K}^{.}_{X / S}. We
exhibit a canonical isomorphism between the complex (\cal{K}^{.}_{X/S}, \delta)
and the residue complex of Grothendieck. When \pi is equidimensional of
dimension n and generically smooth we show that H^{-n} \cal{K}^{.}_{X/S} is
canonically isomorphic to the sheaf of regular differentials of Kunz-Waldi.
Another issue we discuss is Grothendieck Duality on a noetherian formal scheme
\frak{X}. Our results on duality are used in the construction of
\cal{K}^{.}_{X/S}.
| [
{
"version": "v1",
"created": "Thu, 5 Oct 1995 17:21:46 GMT"
},
{
"version": "v2",
"created": "Tue, 17 Oct 1995 15:41:33 GMT"
},
{
"version": "v3",
"created": "Wed, 14 Aug 1996 11:28:24 GMT"
},
{
"version": "v4",
"created": "Mon, 16 Feb 1998 00:32:56 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Yekutieli",
"Amnon",
""
]
] | alg-geom | \section{Introduction}
It is sometimes the case in algebraic geometry, that in order to define
an object associated to a singular variety $X$, one first embeds $X$
into a nonsingular variety $Y$. One such instance is algebraic
De Rham cohomology
$\mrm{H}^{{\textstyle \cdot}}_{\mrm{DR}}(X) =
\mrm{H}^{{\textstyle \cdot}}(Y, \widehat{\Omega}^{{\textstyle \cdot}})$,
where $\widehat{\Omega}^{{\textstyle \cdot}}$ is the completion along $X$ of
the De Rham complex $\Omega_{Y / k}^{{\textstyle \cdot}}$ (relative to a base field
$k$ of characteristic $0$; cf.\ \cite{Ha}).
Now $\widehat{\Omega}^{{\textstyle \cdot}}$ coincides with the complete De Rham
complex
$\widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{X} / k}$,
where $\mfrak{X}$ is the formal scheme $Y_{/ X}$.
It is therefore reasonable to ask what sort of embedding
$X \subset \mfrak{X}$ into a formal scheme
would give rise to the same cohomology.
The answer we provide in this paper is that any
{\em smooth formal embedding} works. Let us define this notion.
Suppose $S$ is a noetherian base scheme $S$ and
$\pi : X \rightarrow S$ is a finite type morphism.
A smooth formal embedding of $X$ consists of morphisms
$X \rightarrow \mfrak{X} \rightarrow S$, where
$X \rightarrow \mfrak{X}$ is a closed immersion of $X$ into a noetherian
formal scheme $\mfrak{X}$, which is a homeomorphism
of the underlying topological spaces;
and $\mfrak{X} \rightarrow S$ is a {\em formally smooth} morphism.
A smooth formal embedding $X \subset \mfrak{X} = Y_{/ X}$ like
in the previous paragraph is said to be algebraizable. But in general
$X \subset \mfrak{X}$ will not be algebraizable.
Smooth formal embeddings enjoy a few advantages over
algebraic embeddings. First consider an embedding $X \subset \mfrak{X}$
and an \'{e}tale morphism $U \rightarrow X$. Then it is pretty clear
(cf.\ Proposition \ref{prop2.4}) that there is an \'{e}tale
morphism of formal schemes $\mfrak{U} \rightarrow \mfrak{X}$ and a
smooth formal embedding $U \subset \mfrak{U}$, s.t.\
$U \cong \mfrak{U} \times_{\mfrak{X}} X$.
Next suppose $X \subset \mfrak{X}, \mfrak{Y}$ are two smooth formal
embeddings, and we are given either a closed immersion
$\mfrak{X} \rightarrow \mfrak{Y}$ or a formally smooth morphism
$\mfrak{Y} \rightarrow \mfrak{X}$, which restrict to the identity on $X$.
Then locally on $X$,
\begin{equation} \label{eqn0.1}
\mfrak{Y} \cong \mfrak{X} \times \operatorname{Spf}
\mbb{Z} [\sqbr{t_{1}, \ldots, t_{n}}]
\end{equation}
(Theorem \ref{thm2.2}).
As mentioned above, De Rham cohomology can be calculated by
smooth formal embeddings. Indeed, when $\operatorname{char} S = 0$,
$\mrm{H}^{q}_{\mrm{DR}}(X / S) =
\mrm{R} \pi_{*} \widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{X} / S}$,
where $X \subset \mfrak{X}$ is any smooth formal embedding
(Corollary \ref{cor2.1}). Moreover, in
\cite{Ye3} it is proved that De Rham homology
$\mrm{H}_{{\textstyle \cdot}}^{\mrm{DR}}(X)$
can also be calculated by smooth formal embeddings,
when $S = \operatorname{Spec} k$, $k$ a field.
According to the preceding paragraph, given an \'{e}tale morphism
$g : U \rightarrow X$ there is a homomorphism
$g^{*} : \mrm{H}_{{\textstyle \cdot}}^{\mrm{DR}}(X) \rightarrow
\mrm{H}_{{\textstyle \cdot}}^{\mrm{DR}}(U)$,
and we conclude that homology is contravariant w.r.t.\
\'{e}tale morphisms.
See Remark \ref{rem2.4} for an application to $\mcal{D}$-modules
on singular varieties.
The main application of smooth formal embeddings in the present paper
is for an {\em explicit
construction of the Grothendieck residue complex}
$\mcal{K}^{{\textstyle \cdot}}_{X / S}$, when $S$ is any regular scheme.
By definition $\mcal{K}^{{\textstyle \cdot}}_{X / S}$ is the Cousin complex
$\mrm{E} \pi^{!} \mcal{O}_{S}$, in the notation of \cite{RD}
Sections IV.3 and VII.3.
Recall that Grothendieck Duality, as developed by
Hartshorne in \cite{RD},
is an abstract theory, stated in the language of derived categories.
Even though this abstraction is suitable for many important
applications, often one wants more explicit information.
In particular a significant
amount of work was directed at finding an explicit presentation of
duality in terms of differential forms and residues.
Mostly the focus was on the dualizing sheaf $\omega_{X}$,
in various circumstances. The structure of $\omega_{X}$ as a
coherent $\mcal{O}_{X}$-module and its variance properties are
thoroughly understood by now, thanks to an extended effort
including \cite{KW}, \cite{Li}, \cite{HK1}, \cite{HK2},
\cite{LS1} and \cite{HS}.
Regarding an explicit presentation of the full duality theory of
dualizing complexes, there have been some advances in recent years,
notably in the papers \cite{Ye1}, \cite{SY}, \cite{Hu}, \cite{Hg1}
\cite{Sa} and \cite{Ye3}.
The later papers \cite{Hg2}, \cite{Hg3} and \cite{LS2}
somewhat overlap our present paper in their results, but their methods
are quite distinct; specifically, they do not use formal schemes.
We base our construction of $\mcal{K}^{{\textstyle \cdot}}_{X / S}$
on I-C.\ Huang's theory of pseudofunctors on modules with
zero dimensional support (see \cite{Hg1}). Suppose $\phi : A \rightarrow B$
is a residually finitely generated homomorphism between complete
noetherian local rings, and $M$ is a discrete $A$-module
(i.e.\ $\operatorname{dim} \operatorname{supp} M = 0$). Then according
to \cite{Hg1} there is a discrete $B$-module $\phi_{\#} M$,
equipped with certain variance properties (cf.\ Theorem \ref{thm6.1}).
In particular
when $\phi$ is residually finite there is a map
$\operatorname{Tr}_{\phi} : \phi_{\#} M \rightarrow M$. Huang's theory is developed
using only methods of commutative algebra.
Now given a point $x \in X$ with $s := \pi(x) \in S$, consider
the local homomorphism
$\phi : \widehat{\mcal{O}}_{S, s} \rightarrow \widehat{\mcal{O}}_{X, x}$.
Define
$\mcal{K}_{X / S}(x) :=
\phi_{\#} \mrm{H}^{d}_{\mfrak{m}_{s}} \widehat{\mcal{O}}_{S, s}$,
where $d := \operatorname{dim} \widehat{\mcal{O}}_{S, s}$,
$\mfrak{m}_{s}$ is the maximal ideal and
$\mrm{H}^{d}_{\mfrak{m}_{s}}$ is local cohomology.
Then $\mcal{K}_{X / S}(x)$ is an injective hull of $k(x)$ as
$\mcal{O}_{X, x}$-module.
As a graded $\mcal{O}_{X}$-module we take
$\mcal{K}^{{\textstyle \cdot}}_{X / S} := \bigoplus_{x \in X}
\mcal{K}_{X / S}(x)$, with the obvious grading. Then
for any scheme morphism $f : X \rightarrow Y$, we deduce from Huang's theory
a homomorphism of graded sheaves
$\operatorname{Tr}_{f} : f_{*} \mcal{K}^{{\textstyle \cdot}}_{X / S}
\rightarrow \mcal{K}^{{\textstyle \cdot}}_{Y / S}$.
The problem is to exhibit a coboundary operator
$\delta : \mcal{K}^{q}_{X / S} \rightarrow \mcal{K}^{q + 1}_{X / S}$,
and to determine that the complex we obtain is indeed isomorphic
to $\mrm{E} \pi^{!} \mcal{O}_{S}$. For this we use smooth formal
embeddings, as explained below.
In Section 5 we discuss Grothendieck Duality on formal schemes,
extending the theory of \cite{RD}.
We propose a definition of dualizing complex $\mcal{R}^{{\textstyle \cdot}}$ on
a noetherian formal scheme (Definition \ref{dfn5.1}), and prove its
uniqueness (Theorem \ref{thm5.1}).
It is important to note that the cohomology sheaves
$\mrm{H}^{q} \mcal{R}^{{\textstyle \cdot}}$ are discrete quasi-coherent
$\mcal{O}_{\mfrak{X}}$-modules, and in general {\em not coherent}.
We define the Cousin functor
$\mrm{E}$ associated to $\mcal{R}^{{\textstyle \cdot}}$,
and show that
$\mrm{E} \mcal{R}^{{\textstyle \cdot}} \cong \mcal{R}^{{\textstyle \cdot}}$
in the derived category, and
$\mrm{E} \mcal{R}^{{\textstyle \cdot}}$ is a residual complex.
On a regular formal scheme $\mfrak{X}$ the (surprising) fact is that
$\mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{O}_{\mfrak{X}}$
is a dualizing complex, and not $\mcal{O}_{\mfrak{X}}$
(Theorem \ref{thm5.3}).
Now let $U \subset X$ be an affine open set and suppose
$U \subset \mfrak{U}$ is a smooth formal embedding.
Say $n := \operatorname{rank} \widehat{\Omega}^{1}_{\mfrak{U} / S}$,
so $\widehat{\Omega}^{n}_{\mfrak{U} / S}$ is a locally free
$\mcal{O}_{\mfrak{U}}$-module of rank $1$,
and
$\mrm{R} \underline{\Gamma}_{\mrm{disc}} \widehat{\Omega}^{n}_{\mfrak{U} / S}[n]$
is a dualizing complex.
Since the Cousin complex is a sum of local cohomology modules,
there is a natural identification of graded
$\mcal{O}_{\mfrak{U}}$-modules
$\mrm{E} \mrm{R} \underline{\Gamma}_{\mrm{disc}}
\widehat{\Omega}^{n}_{\mfrak{U} / S}[n] \cong
\mcal{K}^{{\textstyle \cdot}}_{\mfrak{U} / S}$.
This makes
$\mcal{K}^{{\textstyle \cdot}}_{\mfrak{U} / S}$ into a complex.
Since
$\mcal{K}^{{\textstyle \cdot}}_{U / S} \cong
\mcal{H}om_{\mfrak{U}} \left( \mcal{O}_{U},
\mcal{K}^{{\textstyle \cdot}}_{\mfrak{U} / S} \right)$
we come up with an operator $\delta$ on
$\mcal{K}^{{\textstyle \cdot}}_{U / S} = \mcal{K}^{{\textstyle \cdot}}_{X / S}|_{U}$.
Given another smooth formal embedding $U \subset \mfrak{V}$
we have to compare the complexes $\mcal{K}^{{\textstyle \cdot}}_{\mfrak{U} / S}$
and $\mcal{K}^{{\textstyle \cdot}}_{\mfrak{V} / S}$. This is rather easy to do using
the following trick.
Choosing a sequence $\underline{a}$ of generators of some defining ideal of
$\mfrak{U}$, and letting $\mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a})$ be
the associated Koszul complex, we obtain an explicit presentation of
the dualizing complex, namely
\[ \mrm{R} \underline{\Gamma}_{\mrm{disc}}
\widehat{\Omega}^{n}_{\mfrak{U} / S}[n] \cong
\mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}) \otimes
\widehat{\Omega}^{n}_{\mfrak{U} / S}[n] \]
(cf.\ Lemma \ref{lem4.3}).
By the structure of smooth formal embeddings
we may assume there is a morphism $f : \mfrak{U} \rightarrow \mfrak{V}$ which is
either formally smooth or a closed immersion. Then choosing relative
coordinates (cf.\ formula \ref{eqn0.1})
and using Koszul complexes we produce a morphism
$\mrm{R} \underline{\Gamma}_{\mrm{disc}} \widehat{\Omega}^{n}_{\mfrak{U} / S}[n]
\rightarrow
\mrm{R} \underline{\Gamma}_{\mrm{disc}} \widehat{\Omega}^{m}_{\mfrak{V} / S}[m]$.
Applying the Cousin functor $\mrm{E}$ we recover
$\operatorname{Tr}_{f} : \mcal{K}^{{\textstyle \cdot}}_{\mfrak{U} / S} \rightarrow
\mcal{K}^{{\textstyle \cdot}}_{\mfrak{V} / S}$
as a map of complexes! We conclude that $\delta$ is independent of
$\mfrak{U}$ and hence it glues to a global operator
(Theorem \ref{thm6.2}).
If $f : X \rightarrow Y$ is a finite morphism, then the trace map
$\operatorname{Tr}_{f} : f_{*} \mcal{K}^{{\textstyle \cdot}}_{X / S} \rightarrow
\mcal{K}^{{\textstyle \cdot}}_{Y / S}$,
which is provided by Huang's theory, is actually a homomorphism of
complexes (Theorem \ref{thm7.6}).
We show this by employing the same trick as above of going from
Koszul complexes to Cousin complexes, this time inserting a
``Tate residue map'' into the picture.
We use Theorem \ref{thm7.6} to prove directly that if
$\pi : X \rightarrow S$ is equidimensional of dimension $n$ and generically
smooth, then
$\mrm{H}^{-n} \mcal{K}^{{\textstyle \cdot}}_{X / S}$
coincides with the sheaf of regular differentials
$\tilde{\omega}^{n}_{X / S}$ of Kunz-Waldi \cite{KW}
(Theorem \ref{thm7.4}).
Finally in Theorem \ref{thm8.10} we exhibit a canonical isomorphism
$\zeta_{X}$
between the complex $\mcal{K}^{{\textstyle \cdot}}_{X / S}$ constructed here and
the complex
$\pi^{\triangle} \mcal{O}_{S} = \mrm{E} \pi^{!} \mcal{O}_{S}$
of \cite{RD}.
Given a morphism of schemes $f : X \rightarrow Y$ the isomorphisms
$\zeta_{X}$ and $\zeta_{Y}$ send Huang's trace map
$\operatorname{Tr}_{f} : f_{*} \mcal{K}^{{\textstyle \cdot}}_{X / S} \rightarrow
\mcal{K}^{{\textstyle \cdot}}_{Y / S}$
to the trace
$\operatorname{Tr}^{\mrm{RD}}_{f} : f_{*} \mrm{E} \pi^{!}_{X} \mcal{O}_{S}
\rightarrow \mrm{E} \pi^{!}_{Y} \mcal{O}_{S}$
of \cite{RD} Section VI.4.
In particular it follows that for $f$ proper, $\operatorname{Tr}_{f}$
is a homomorphism of complexes (Corollary \ref{cor8.1}).
Sections 1 and 3 of the paper contain the necessary supplements
to \cite{EGA}. Perhaps the most noteworthy result there is
Theorem \ref{thm1.10}, which states that formally finite type
morphisms are stable under base change.
This was also proved in \cite{AJL2}.
\medskip \noindent
{\em Acknowledgments.}\
The author wishes to thank L.\ Alonso, I-C.\ Huang, R.\ H\"{u}bl,
A.\ Jerem\'{\i}as, J.\ Lipman and P.\ Sastry for helpful discussions,
some of which took place during a meeting in Oberwolfach in May 1996.
\tableofcontents
\section{Formally Finite Type Morphisms}
In this section we define formally finite type morphisms between
noetherian formal schemes. This mild generalization of the finite
type morphism of \cite{EGA} I \S 10 has the advantage that it
includes the completion morphism
$\mfrak{X} \rightarrow \mfrak{X}_{/ Z}$ (cf.\ Proposition \ref{prop1.12}), and
still is preserved under base change (Theorem \ref{thm1.10}).
We follow the conventions of \cite{EGA} $0_{\mrm{I}}$ \S 7 on {\em adic}
rings. Thus an adic ring is a commutative ring $A$ which is
complete and separated in the $\mfrak{a}$-adic topology, for some ideal
$\mfrak{a} \subset A$.
As for formal schemes, we follow the conventions of \cite{EGA} I
\S 10. Throughout the paper all formal schemes are by default noetherian
(adic) formal schemes.
We write $A \sqbr{\underline{t}} = A \sqbr{t_{1}, \ldots, t_{n}}$
for the polynomial algebra with variables
$t_{1}, \ldots,$ \linebreak
$t_{n}$ over a ring $A$.
The easy lemma below is taken from \cite{AJL2}.
\begin{lem} \label{lem1.11}
Let $A \rightarrow B$ be a continuous homomorphism between noetherian adic rings,
and let $\mfrak{b} \subset B$ be a defining ideal. Then the following
are equivalent:
\begin{enumerate}
\rmitem{i} $A \rightarrow B / \mfrak{b}$ is a finite type homomorphism.
\rmitem{ii} For some homomorphism
$f : A \sqbr{\underline{t}} \rightarrow B$ extending $A \rightarrow B$ one has
$\mfrak{b} = B \cdot f^{-1}(\mfrak{b})$ and
$A \sqbr{\underline{t}} \rightarrow B / \mfrak{b}$ is surjective.
\end{enumerate}
\end{lem}
\begin{proof}
(i) $\Rightarrow$ (ii): Say $b_{1}, \ldots, b_{m}$ generate $\mfrak{b}$
as a $B$-module, and the images of $b_{m+1}, \ldots, b_{n}$ generate
$B / \mfrak{b}$ as an $A$-algebra. Then the homomorphism
$A \sqbr{\underline{t}} \rightarrow B$, $t_{i} \rightarrow b_{i}$ has the required properties.
\noindent (ii) $\Rightarrow$ (i): Trivial.
\end{proof}
\begin{dfn} \label{dfn1.10}
Let $A \rightarrow B$ be a continuous homomorphism between adic noetherian
rings.
We say that $A \rightarrow B$ is of {\em formally finite type} (f.f.t.)
if the equivalent conditions of Lemma \ref{lem1.11} hold.
We shall also say that $B$ is a formally finite type $A$-algebra.
\end{dfn}
\begin{exa} \label{exa1.9}
Let $I \subset A$ be any open ideal, and let
$B := \lim_{\leftarrow i} A / I^{i}$. Then $A \rightarrow B$ is f.f.t.
\end{exa}
Recall that if $A'$ and $B$ are adic $A$-algebras, with defining ideals
$\mfrak{a}'$ and $\mfrak{b}$, the complete tensor product
$A' \widehat{\otimes}_{A} B$ is the completion of
$A' \otimes_{A} B$ w.r.t.\ the topology defined by the image of
$(\mfrak{a}' \otimes_{A} B) \oplus (A' \otimes_{A} \mfrak{b})$.
\begin{prop} \label{prop1.11}
Let $A, A'$ and $B$ be noetherian adic rings, $A \rightarrow B$ a f.f.t.\
homomorphism, and $A \rightarrow A'$ any continuous homomorphism.
Then
$B' := A' \widehat{\otimes}_{A} B$
is a noetherian adic ring, and $A' \rightarrow B'$ is a f.f.t.\ homomorphism.
\end{prop}
\begin{proof}
Choose a homomorphism
$f : A \sqbr{\underline{t}} \rightarrow B$
satisfying condition (ii) of Lemma \ref{lem1.11}. Let
$\mfrak{b} \subset B$ and $\mfrak{a}' \subset A'$ be defining ideals.
Write
$C := A' \otimes_{A} B$ and
$\mfrak{c} := \mfrak{a}' \cdot C + C \cdot \mfrak{b}$,
so
$B' = \lim_{\leftarrow i} C / \mfrak{c}^{i}$.
Consider the homomorphism
$f' : A' \sqbr{\underline{t}} \rightarrow C$, and let
$\mfrak{c}' := {f'}^{-1}(\mfrak{c})$
and
$\widehat{A' \sqbr{\underline{t}}} :=
\lim_{\leftarrow i} A' \sqbr{\underline{t}} / {\mfrak{c}'}^{i}$.
Since $\mfrak{c} = C \cdot \mfrak{c}'$, it follows from
\cite{CA} Section III.2.11 Proposition 14 that
$\widehat{A' \sqbr{\underline{t}}} \rightarrow B'$ is surjective. Hence $B'$ is a
noetherian adic ring with the $\mfrak{b}'$-adic topology, where
$\mfrak{b}' = B' \cdot \mfrak{c}$.
Furthermore
$A' \sqbr{\underline{t}} \rightarrow B' / \mfrak{b}'$
is surjective, and we conclude that $A' \rightarrow B'$ is f.f.t.
\end{proof}
In the next three examples $A$ is an adic ring with defining ideal
$\mfrak{a}$.
\begin{exa} \label{exa1.2}
Recall that for $a \in A$, the complete ring of fractions
$A_{\{a\}}$ is the completion of the localized ring $A_{a}$
w.r.t.\ the $\mfrak{a}_{a}$-adic topology. Then
$A_{\{a\}} \cong A \widehat{\otimes}_{\mbb{Z} \sqbr{t}}
\mbb{Z} \sqbr{t, t^{-1}}$,
which proves that $A \rightarrow A_{\{a\}}$ is f.f.t.
\end{exa}
\begin{exa} \label{exa1.6}
Given indeterminates $t_{1}, \ldots, t_{n}$, the ring of restricted
formal power series
$A\{ \underline{t} \} = A\{ t_{1}, \ldots, t_{n} \}$ is the completion of the
polynomial ring $A \sqbr{ \underline{t} }$ w.r.t.\ the
$(A \sqbr{ \underline{t} } \cdot \mfrak{a})$-adic topology. Hence
$A \{ \underline{t} \} \cong A \widehat{\otimes}_{\mbb{Z}}
\mbb{Z} \sqbr{ \underline{t} }$,
which demonstrates that $A \rightarrow A \{ \underline{t} \}$ is f.f.t.
\end{exa}
\begin{exa} \label{exa1.1}
Consider the adic ring
$A \widehat{\otimes}_{\mbb{Z}} \mbb{Z} [\sqbr{ \underline{t} }]$,
where
$\mbb{Z} [\sqbr{ \underline{t} }] = \mbb{Z} [\sqbr{ t_{1}, \ldots, t_{n} }]$
is the ring of formal power series, with the $(\underline{t})$-adic topology.
Since inverse limits commute, we see that
$A \widehat{\otimes}_{\mbb{Z}} \mbb{Z} [\sqbr{ \underline{t} }]
\cong A [\sqbr{ \underline{t} }]$,
the ring of formal power series over $A$, endowed with the
$(A [\sqbr{ \underline{t} }] \cdot (\mfrak{a} + \underline{t}))$-adic topology.
By Prop.\ \ref{prop1.11},
$A \rightarrow A [\sqbr{ \underline{t} }]$ is f.f.t.
\end{exa}
Let $A \rightarrow B$ be a f.f.t\ homomorphism between adic rings. Choose
a defining ideal $\mfrak{b} \subset B$, and set
$B_{i} := B / \mfrak{b}^{i+1}$. For $n \geq 0$ define
\[ \widehat{\Omega}^{n}_{B / A} :=
\lim_{\leftarrow i} \Omega^{n}_{B_{i} / A} \cong
\lim_{\leftarrow i} B_{i} \otimes_{B} \Omega^{n}_{B / A} \]
(cf.\ \cite{EGA} $0_{\mrm{IV}}$ 20.7.14).
Let
$\widehat{\Omega}^{{\textstyle \cdot}}_{B / A} := \bigoplus_{n \geq 0}
\widehat{\Omega}^{n}_{B / A}$, which is a topological DGA
(differential graded algebra), with
$\widehat{\Omega}^{0}_{B / A} = B$.
This definition is independent of the ideal $\mfrak{b}$.
Since $\Omega^{n}_{B_{i} / A}$ is finite over $B_{i}$ it follows that
$\widehat{\Omega}^{n}_{B / A}$ is finite over $B$.
\begin{rem}
If $A \rightarrow B$ is f.f.t.\ then
$\widehat{\Omega}^{{\textstyle \cdot}}_{B/A} \cong \Omega^{{\textstyle \cdot}, \mrm{sep}}_{B/A}$,
where $\Omega^{{\textstyle \cdot}, \mrm{sep}}_{B/A}$ is the separated algebra
of differentials defined in
\cite{Ye1} \S 1.5 for semi-topo\-logical algebras.
Also $\widehat{\Omega}^{{\textstyle \cdot}}_{B/A}$ is the universally finite
differential algebra in the sense of \cite{Ku}.
\end{rem}
\begin{prop} \label{prop1.1}
Let $L \rightarrow A \rightarrow B$ be f.f.t.\ homomorphisms between adic noetherian
rings.
\begin{enumerate}
\item $A \rightarrow B$ is formally smooth relative to $L$ iff the sequence
\[ 0 \rightarrow B \otimes_{A} \widehat{\Omega}^{1}_{A / L} \xrightarrow{v}
\widehat{\Omega}^{1}_{B / L} \xrightarrow{u}
\widehat{\Omega}^{1}_{B / A} \rightarrow 0 \]
is split exact.
\item $A \rightarrow B$ is formally \'{e}tale relative to $L$ iff
$B \otimes_{A} \widehat{\Omega}^{1}_{A / L} \rightarrow
\widehat{\Omega}^{1}_{B / L}$
is bijective.
\end{enumerate}
\end{prop}
\begin{proof}
Use the results of \cite{EGA} $0_{\mrm{IV}}$ Section 20.7, together
the fact that these are finite $B$-modules.
\end{proof}
\begin{prop} \label{prop1.3}
Let $f: A \rightarrow B$ be a formally smooth, f.f.t.\ homomorphism
between noetherian adic rings. Then $B$ is flat over $A$ and
$\widehat{\Omega}^{1}_{B/A}$ is a projective finitely generated
$B$-module.
\end{prop}
\begin{proof}
For flatness it suffices to show that if
$\mfrak{n}$ is a maximal ideal of $B$ and
$\mfrak{m} := f^{-1}(\mfrak{n})$,
then
$\widehat{A}_{\mfrak{m}} \rightarrow \widehat{B}_{\mfrak{n}}$
is flat ($\widehat{B}_{\mfrak{n}}$ is the completion of
$B_{\mfrak{n}}$ with the $\mfrak{n}$-adic topology).
Now $\mfrak{n}$ is open,
and hence so is $\mfrak{m}$. Both
$A \rightarrow \widehat{A}_{\mfrak{m}}$ and
$B \rightarrow \widehat{B}_{\mfrak{n}}$ are formally \'{e}tale,
therefore $\widehat{A}_{\mfrak{m}} \rightarrow \widehat{B}_{\mfrak{n}}$
is formally smooth. Because $A \rightarrow B$ is f.f.t.\ it follows that
$A / \mfrak{m} \rightarrow B / \mfrak{n}$
is finite type, and hence finite (and $\mfrak{m}$ is a maximal ideal).
By \cite{EGA} $0_{\mrm{IV}}$ Thm.\ 19.7.1,
$\widehat{B}_{\mfrak{n}}$ is flat over $\widehat{A}_{\mfrak{m}}$.
The second assertion follows from
\cite{EGA} $0_{\mrm{IV}}$ Thm.\ 20.4.9.
\end{proof}
\begin{prop} \label{prop1.4}
Let $f : A \rightarrow B$ be a f.f.t., formally smooth homomorphism of
noetherian adic rings, and let $\mfrak{q} \in \operatorname{Spf} B$.
Suppose
$\operatorname{rank} \widehat{\Omega}^{1}_{\widehat{B}_{\mfrak{q}} / A} = n$.
Then:
\begin{enumerate}
\item For some $b \in B - \mfrak{q}$ there is a formally \'{e}tale
homomorphism
$\tilde{f} : A\sqbr{\underline{t}} = A\sqbr{t_{1}, \ldots, t_{n}}
\rightarrow B_{ \{b\} }$ extending $f$.
\item For any
$\mfrak{q}' \in \operatorname{Spf} B_{ \{b\} }$
let
$\mfrak{r} := \tilde{f}^{-1}(\mfrak{q}')$.
Then
$\widehat{A\sqbr{\underline{t}}}_{\mfrak{r}} \rightarrow \widehat{B}_{\mfrak{q}'}$
is finite \'{e}tale.
\item Let $\mfrak{p} := f^{-1}(\mfrak{q})$.
Assume $\widehat{A}_{\mfrak{p}}$ is regular of dimension $m$,
and
$\operatorname{tr.deg}_{k(\mfrak{p})} k(\mfrak{q}) = l$. Then
$\widehat{B}_{\mfrak{q}}$ is regular of dimension $n + m - l$.
\end{enumerate}
\end{prop}
\begin{proof}
1.\ By Prop.\ \ref{prop1.3} we can find $b$ s.t.\
$\widehat{\Omega}^{1}_{B_{ \{b\} } / A} \cong B_{ \{b\} } \otimes_{B}
\widehat{\Omega}^{1}_{B / A}$
is free, say with basis $\mrm{d} b_{1}, \ldots, \mrm{d} b_{n}$.
Then we get a homomorphism
$A \sqbr{ \underline{t} } \rightarrow B_{ \{b\} }$, $t_{i} \mapsto b_{i}$.
In order to stay inside the category of adic rings we may replace
$A \sqbr{\underline{t}}$ with its completion $A \{ \underline{t} \}$
(cf.\ Examples \ref{exa1.2} - \ref{exa1.1} for the notation).
According to Proposition \ref{prop1.1} we see that
$A \sqbr{\underline{t}} \rightarrow B_{ \{b\} }$ is formally \'{e}tale relative to $A$.
But since $A \rightarrow B_{ \{b\} }$ is formally smooth, this implies that
$A \sqbr{\underline{t}} \rightarrow B_{ \{b\} }$ is actually (absolutely)
formally \'{e}tale.
\noindent 2.\
Consider the formally \'{e}tale homomorphism
$k(\mfrak{r}) \rightarrow
\widehat{B}_{\mfrak{q}'} / \mfrak{r} \widehat{B}_{\mfrak{q}'}$.
Since $\mfrak{q}'$ is an open prime ideal it follows that
$A \rightarrow B / \mfrak{q}'$ is a finite type homomorphism, so the field
extension $k(\mfrak{r}) \rightarrow k(\mfrak{q}')$ is finitely generated.
By \cite{Hg1} Lemma 3.9 we see that in fact
$\widehat{B}_{\mfrak{q}'} / \mfrak{r} \widehat{B}_{\mfrak{q}'} =
k(\mfrak{q}')$, so
$k(\mfrak{r}) \rightarrow k(\mfrak{q}')$ is finite separable.
Hence
$\widehat{A\sqbr{\underline{t}}}_{\mfrak{r}} \rightarrow \widehat{B}_{\mfrak{q}'}$
is finite \'{e}tale.
\noindent 3.\
Take $\mfrak{q}' := \mfrak{q}$.
Under the assumption the ring
$\widehat{A\sqbr{\underline{t}}}_{\mfrak{r}}$ is regular, and
according to \cite{Ma} \S 14.c Thm.\ 23,
$\operatorname{dim} \widehat{A\sqbr{\underline{t}}}_{\mfrak{r}} = m + n - l$.
By part 2, $\widehat{B}_{\mfrak{q}}$ is also regular, and
$\operatorname{dim} \widehat{B}_{\mfrak{q}} =
\operatorname{dim} \widehat{A\sqbr{\underline{t}}}_{\mfrak{r}}$.
\end{proof}
Let us now pass to formal schemes.
Given a noetherian formal scheme $\mfrak{X}$,
choose a defining ideal
$\mcal{I} \subset \mcal{O}_{\mfrak{X}}$, and set
\begin{equation} \label{eqn1.1}
X_{n} := (\mfrak{X}, \mcal{O}_{\mfrak{X}}/\mcal{I}^{n+1}) .
\end{equation}
$X_{n}$ is a noetherian (usual) scheme, and
$\mfrak{X} \cong \lim_{n \rightarrow} X_{n}$
in the category of formal schemes.
One possible choice for $\mcal{I}$ is the largest defining ideal,
in which case one has
$X_{0} = \mfrak{X}_{\mrm{red}}$,
the reduced closed subscheme (cf.\ \cite{EGA} I \S 10.5).
\begin{lem} \label{lem1.12}
Suppose $f : \mfrak{X} \rightarrow \mfrak{Y}$ is a morphism between noetherian
formal schemes. There are defining ideals
$\mcal{I} \subset \mcal{O}_{\mfrak{X}}$ and
$\mcal{J} \subset \mcal{O}_{\mfrak{Y}}$ s.t.\
$f^{-1} \mcal{J} \cdot \mcal{O}_{\mfrak{X}} \subset \mcal{I}$.
Letting $X_{n}$ and $Y_{n}$ be the corresponding schemes
\textup{(}cf.\ \textup{(\ref{eqn1.1})} above\textup{)}, we get morphisms
of schemes
$f_{n} : X_{n} \rightarrow Y_{n}$, with $f = \lim_{n \rightarrow} f_{n}$.
\end{lem}
\begin{proof}
See \cite{EGA} I \S 10.6. For instance, one could take $\mcal{I}$
to be the largest defining ideal and $\mcal{J}$ arbitrary.
\end{proof}
\begin{dfn} \label{dfn1.2}
Let $f: \mfrak{X} \rightarrow \mfrak{Y}$ be a morphism of noetherian (adic)
formal schemes. We say
that $f$ is of {\em formally finite type} (or that $\mfrak{X}$ is a
formally finite type formal scheme over $\mfrak{Y}$) if the morphism
$f_{0} : X_{0} \rightarrow Y_{0}$ in Lemma \ref{lem1.12} is finite type, for some
choice of defining ideals of $\mfrak{X}$ and $\mfrak{Y}$.
\end{dfn}
Observe that if the morphism $f_{0}$ is finite type then so are all
the $f_{n}$, and the definition doesn't depend on the defining ideals
chosen.
\begin{rem} \label{rem1.10}
The definition of f.f.t.\ morphism we gave in an earlier version of the
paper was more cumbersome, though equivalent.
The present Definition \ref{dfn1.2} is taken from \cite{AJL2},
where the name is ``pseudo-finite type morphism'', and I wish to thank
A.\ Jerem\'{\i}as for bringing it to my attention.
\end{rem}
Here are a couple of examples of f.f.t.\ morphisms:
\begin{exa}
A finite type morphism $\mfrak{X} \rightarrow \mfrak{Y}$ (in the sense of
\cite{EGA} I \S 10.13) is f.f.t.
\end{exa}
\begin{exa} \label{exa1.7}
Let $X$ be a scheme of finite type over a noetherian scheme $S$, and let
$X_{0} \subset X$
be a locally closed subset. Then the completion $\mfrak{X} = X_{/X_{0}}$
(see \cite{EGA} I \S 10.8) is of f.f.t.\
over $S$. Such a formal scheme is called {\em algebraizable}.
\end{exa}
\begin{dfn} \label{dfn1.1}
A f.f.t.\ morphism $f : \mfrak{X} \rightarrow \mfrak{Y}$ is called
{\em formally finite} (resp.\ {\em formally proper}) if the morphism
$f_{0} : X_{0} \rightarrow Y_{0}$ in Lemma \ref{lem1.12} is finite
(resp.\ proper), for some choice of defining ideals.
\end{dfn}
\begin{exa} \label{exa1.8}
If in Example \ref{exa1.7} the subset $X_{0} \subset X$ is closed, then
$\mfrak{X} \rightarrow X$ is formally finite.
If $X_{0} \rightarrow S$ is proper, then $\mfrak{X} \rightarrow S$ is
formally proper.
\end{exa}
\begin{prop} \label{prop1.2}
\begin{enumerate}
\item An immersion $\mfrak{X} \rightarrow \mfrak{Y}$ is f.f.t.
\item If $\mfrak{X} \rightarrow \mfrak{Y}$ and $\mfrak{Y} \rightarrow \mfrak{Z}$ are
f.f.t., then so is $\mfrak{X} \rightarrow \mfrak{Z}$.
\item Let $\mfrak{U} = \operatorname{Spf} B$ and
$\mfrak{V} = \operatorname{Spf} A$. Then a morphism
$\mfrak{U} \rightarrow \mfrak{V}$ is f.f.t.\ iff the ring homomorphism
$A \rightarrow B$ is f.f.t.
\end{enumerate}
\end{prop}
\begin{proof}
Consider morphisms of schemes $X_{0} \rightarrow Y_{0}$ etc.\
as in Lemma \ref{lem1.12}.
For part 3 use condition (i) of Lemma \ref{lem1.11}.
\end{proof}
\begin{prop} \label{prop1.12}
Let $\mfrak{X}$ be a noetherian formal scheme and $Z \subset \mfrak{X}$
a locally closed subset. Then there is a noetherian formal scheme
$\mfrak{X}_{/Z}$, with underlying topological space $Z$, and the
natural morphism $\mfrak{X}_{/Z} \rightarrow \mfrak{X}$
is f.f.t.
\end{prop}
\begin{proof}
Pick an open subset $\mfrak{U} \subset \mfrak{X}$ s.t.\
$Z \subset \mfrak{U}$ is closed, and choose a defining ideal
$\mcal{I}$ of $Z$. Let
$\mcal{O}_{\mfrak{Z}} := \lim_{\leftarrow i}
\mcal{O}_{\mfrak{U}} / \mcal{I}^{i}$.
According to \cite{EGA} I Section 10.6,
$\mfrak{X}_{/Z} := (Z, \mcal{O}_{\mfrak{Z}})$
is a noetherian formal scheme. Clearly
$\mfrak{X}_{/Z} \rightarrow \mfrak{X}$ is f.f.t.
\end{proof}
In \cite{EGA} I \S 10.3 it is shown that finite type morphisms
between noetherian formal schemes are preserved by base change.
This is true also for f.f.t.\ morphisms:
\begin{thm} \label{thm1.10}
Suppose $\mfrak{X}$, $\mfrak{Y}$ and $\mfrak{Y}'$ are noetherian
formal schemes,
$\mfrak{X} \rightarrow \mfrak{Y}$ is a f.f.t.\ morphism, and
$\mfrak{Y}' \rightarrow \mfrak{Y}$ is an arbitrary morphism. Then
$\mfrak{X}' := \mfrak{X} \times_{\mfrak{Y}} \mfrak{Y}'$
is also noetherian, and the morphism $\mfrak{X}' \rightarrow \mfrak{Y}'$
is f.f.t.
\end{thm}
\begin{proof}
First note that the formal scheme
$\mfrak{X}' = \mfrak{X} \times_{\mfrak{Y}} \mfrak{Y}'$
exists (\cite{EGA} I \S 10.7).
For any affine open sets
$\mfrak{U} = \operatorname{Spf} B \subset \mfrak{X}$,
$\mfrak{V}' = \operatorname{Spf} A' \subset \mfrak{Y}'$ and
$\mfrak{V} = \operatorname{Spf} A \subset \mfrak{Y}$ such that
$\mfrak{U} \rightarrow \mfrak{V}$ and $\mfrak{V}' \rightarrow \mfrak{V}$,
one has
$\mfrak{U}' = \mfrak{U} \times_{\mfrak{V}} \mfrak{V}' =
\operatorname{Spf} B \widehat{\otimes}_{A} A'$,
and $\mfrak{U}' \subset \mfrak{X}'$ is open.
By Propositions \ref{prop1.11} and \ref{prop1.2}, $\mfrak{U}'$ is a
noetherian formal scheme, and
$\mfrak{U}' \rightarrow \mfrak{V}'$ is f.f.t. But finitely many such
$\mfrak{U}'$ cover $\mfrak{X}'$.
\end{proof}
\begin{cor} \label{cor1.10}
If $\mfrak{X}_{1}$, $\mfrak{X}_{2}$ and $\mfrak{Y}$ are
noetherian and $\mfrak{X}_{i} \rightarrow \mfrak{Y}$ are f.f.t.\ morphisms,
then
$\mfrak{X}_{3} := \mfrak{X}_{1} \times_{\mfrak{Y}} \mfrak{X}_{2}$
is also noetherian, and
$\mfrak{X}_{3} \rightarrow \mfrak{Y}$ is f.f.t.
\end{cor}
\begin{rem} \label{rem1.4}
I do not know an example of a f.f.t.\ formal scheme $\mfrak{X}$ over a
scheme $S$ which is not locally algebraizable. (Locally algebraizable
means there is an open
covering $\mfrak{X} = \bigcup \mfrak{U}_{i}$, with $\mfrak{U}_{i} \rightarrow S$
algebraizable, in the sense of Example \ref{exa1.7}.)
\end{rem}
\begin{dfn} \label{dfn1.3}
A morphism of formal schemes $\mfrak{X} \rightarrow \mfrak{Y}$ is said to be {\em
formally smooth} (resp.\ {\em formally \'{e}tale}) if, given
a (usual) affine scheme $Z$ and a closed subscheme $Z_{0} \subset Z$
defined by a nilpotent ideal, the map
$\operatorname{Hom}_{\mfrak{Y}}(Z, \mfrak{X}) \rightarrow
\operatorname{Hom}_{\mfrak{Y}}(Z_{0}, \mfrak{X})$
is surjective (resp.\ bijective).
\end{dfn}
This is the definition of formal smoothness used in \cite{EGA} IV
Section 17.1. We shall also require the next notion.
\begin{dfn} \label{dfn1.4}
A morphism $g: \mfrak{X} \rightarrow \mfrak{Y}$ between noetherian formal schemes
is called {\em \'{e}tale} if it is of finite
type (see \cite{EGA} I \S 10.13) and formally \'{e}tale.
\end{dfn}
Note that if $\mfrak{Y}$ is a usual scheme, then so is
$\mfrak{X}$, and $g$ is an \'{e}tale morphism of schemes.
According to \cite{EGA} I Prop.\ 10.13.5 and by the obvious properties
of formally \'{e}tale morphisms, if $\mfrak{U} \rightarrow \mfrak{X}$ and
$\mfrak{V} \rightarrow \mfrak{X}$ are \'{e}tale, then so is
$\mfrak{U} \times_{\mfrak{X}} \mfrak{V} \rightarrow \mfrak{X}$.
Hence for fixed $\mfrak{X}$, the
category of all \'{e}tale morphisms $\mfrak{U} \rightarrow \mfrak{X}$ forms a site
(cf.\ \cite{Mi} Ch.\ II \S 1). We call this site the small \'{e}tale site
on $\mfrak{X}$, and denote it by $\mfrak{X}_{\mrm{et}}$.
\section{Smooth Formal Embeddings and De Rham Cohomology}
Fix a noetherian base scheme $S$ and a finite type $S$-scheme $X$.
\begin{dfn} \label{dfn2.1}
A {\em smooth formal embedding} (s.f.e.) of $X$ (over $S$)
is the following data:
\begin{enumerate}
\rmitem{i} A noetherian formal scheme $\mfrak{X}$.
\rmitem{ii} A formally finite type, formally smooth morphism
$\mfrak{X} \rightarrow S$.
\rmitem{iii} An $S$-morphism $X \rightarrow \mfrak{X}$, which is a
closed immersion and a homeomorphism between the underlying
topological spaces.
\end{enumerate}
We shall refer to this by writing ``$X \subset \mfrak{X}$ is a s.f.e.''
\end{dfn}
\begin{exa} \label{exa2.1}
Suppose $Y$ is a smooth $S$-scheme, $X \subset Y$ a locally closed subset,
and $\mfrak{X} = Y_{/X}$ the completion. Then $X \subset \mfrak{X}$ is a
smooth formal embedding. Such an embedding is called an
{\em algebraizable embedding} (cf.\ Remark \ref{rem1.4}).
\end{exa}
The smooth formal embeddings of $X$ form a category, in which a morphism
of embeddings is an $S$-morphism of formal schemes
$f : \mfrak{X} \rightarrow \mfrak{Y}$
inducing the identity on $X$.
Note that any morphism of embeddings $f: \mfrak{X} \rightarrow \mfrak{Y}$
is affine (cf.\ \cite{EGA} I Prop.\ 10.6.12), and the functor
$f_{*} : \mathsf{Mod}(\mfrak{X}) \rightarrow \mathsf{Mod}(\mfrak{Y})$
is exact. Let
$\mfrak{X}$ and $\mfrak{Y}$ be two smooth formal embeddings of $X$.
Consider the formal scheme
$\mfrak{X} \times_{S} \mfrak{Y}$. Then the diagonal
$\Delta : X \rightarrow \mfrak{X} \times_{S} \mfrak{Y}$
is an immersion (we do not assume our formal schemes are separated!).
\begin{prop} \label{prop2.1}
The completion
$(\mfrak{X} \times_{S} \mfrak{Y})_{/ X}$
of $\mfrak{X} \times_{S} \mfrak{Y}$ along $\Delta(X)$ is a smooth formal
embedding of $X$, and moreover it is a product of $\mfrak{X}$ and
$\mfrak{Y}$ in the category of smooth formal embeddings.
\end{prop}
\begin{proof}
By Theorem \ref{thm1.10} and Proposition \ref{prop1.12} it follows that
$(\mfrak{X} \times_{S} \mfrak{Y})_{/ X}$ is
formally finite type over $S$, so in particular it is noetherian.
Clearly
$(\mfrak{X} \times_{S} \mfrak{Y})_{/ X} \rightarrow S$
is formally smooth.
\end{proof}
The benefit of using formal rather than algebraic embeddings is in:
\begin{prop} \label{prop2.4}
Let $X \subset \mfrak{X}$ be a smooth formal embedding \textup{(}over
$S$\textup{)}
and $g : U \rightarrow X$ an \'{e}tale morphism. Then there exists a noetherian
formal scheme $\mfrak{U}$ and an \'{e}tale morphism
$\widehat{g} : \mfrak{U} \rightarrow \mfrak{X}$ s.t.\
$U \cong \mfrak{U} \times_{\mfrak{X}} X$.
$\widehat{g} : \mfrak{U} \rightarrow \mfrak{X}$
is unique \textup{(}up to a unique isomorphism\textup{)}, and moreover
$U \rightarrow \mfrak{U}$ is a smooth formal embedding.
\end{prop}
\begin{proof}
This is essentially the ``topological invariance of \'{e}tale morphisms'',
cf.\ \cite{EGA} IV \S 18.1 (or \cite{Mi} Ch.\ I Thm.\ 3.23).
Let
$\mcal{I} := \operatorname{Ker}(\mcal{O}_{\mfrak{X}} \rightarrow \mcal{O}_{X})$ and
$X_{i} := (\mfrak{X}, \mcal{O}_{\mfrak{X}} / \mcal{I}^{i+1})$;
so $X = X_{0}$. For every $i$ there is a unique \'{e}tale morphism
$g_{i} : U_{i} \rightarrow X_{i}$ s.t.\
$U \cong U_{i} \times_{X_{i}} X$. Identifying the underlying topological
spaces of $U_{i}$ and $U$ we get an inverse system of sheaves
$\{ \mcal{O}_{U_{i}} \}$ on $U$. Setting
$\mcal{O}_{\mfrak{U}} := \lim_{\leftarrow i} \mcal{O}_{U_{i}}$ we get a
noetherian formal scheme $\mfrak{U}$ having the proclaimed properties
(cf.\ \cite{EGA} I \S 10.6).
\end{proof}
Thus we can consider $\mfrak{X}_{\mrm{et}}$ as a ``smooth formal
embedding'' of $X_{\mrm{et}}$.
If $\mcal{M}$ is a sheaf on $X_{\mrm{et}}$ and $U \rightarrow X$ is an \'{e}tale
morphism, we denote by $\mcal{M}|_{U}$ the restriction of $\mcal{M}$ to
$U_{\mrm{Zar}}$.
\begin{cor} \label{cor2.3}
Let $X \subset \mfrak{X}$ be a smooth formal embedding over $S$.
Then there is a sheaf of DGAs
$\widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{X}_{\mrm{et}} / S}$
on $X_{\mrm{et}}$, with the property that for each
$g: U \rightarrow X$ in $X_{\mrm{et}}$ and corresponding
$\widehat{g}: \mfrak{U} \rightarrow \mfrak{X}$ in $\mfrak{X}_{\mrm{et}}$, one has
$\widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{X}_{\mrm{et}} / S} |_{U} \cong
\widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{U} / S}$.
\end{cor}
\begin{proof}
By Prop.\ \ref{prop1.1},
$\widehat{\Omega}^{p}_{\mfrak{U} / S} \cong
\widehat{g}^{*} \widehat{\Omega}^{p}_{\mfrak{X} / S}$.
Now $\widehat{\Omega}^{p}_{\mfrak{X} / S}$ is coherent, so we can use
\cite{Mi} Ch.\ II Cor.\ 1.6 (which applies to our \'{e}tale site
$\mfrak{X}_{\mrm{et}}$).
\end{proof}
For smooth formal embeddings, closed immersions and smooth morphisms
are locally trivial, in the following sense. Recall that for an
adic algebra $A$, the ring of formal power series
$A [\sqbr{ \underline{t} }] = A [\sqbr{ t_{1}, \ldots, t_{n} }]$
is adic (cf.\ Example \ref{exa1.1}).
\begin{thm} \label{thm2.2}
Let $f : \mfrak{X} \rightarrow \mfrak{Y}$ be a morphism of smooth formal
embeddings of $X$ over $S$. Assume $f$ is a closed immersion
\textup{(}resp.\ formally smooth\textup{)}.
Then, given a point $x \in X$, there are affine open sets $U \subset X$
and $W \subset S$, with $x \in U$ and $U \rightarrow W$, satisfying condition
\textup{($*$)} below.
\begin{enumerate}
\item[\textup{($*$)}]
Let $W = \operatorname{Spec} L$, and let
$\operatorname{Spf} A \subset \mfrak{Y}$ and
$\operatorname{Spf} B \subset \mfrak{X}$
be the affine formal schemes supported on $U$. Then
there is an isomorphism of topological $L$-algebras
$A \cong B [\sqbr{\underline{t}}]$
\textup{(}resp.\ $B \cong A [\sqbr{\underline{t}}]$\textup{)}
such that $f^{*} : A \rightarrow B$ is projection modulo $(\underline{t})$
\textup{(}resp.\ the inclusion\textup{)}.
\end{enumerate}
\end{thm}
\begin{proof}
1.\ Assume $f$ is a closed immersion.
According to \cite{EGA} $0_{\mrm{IV}}$ Thm.\ 19.5.3 and
Cor.\ 20.7.9, by choosing $U = \operatorname{Spec} C$ small enough, and setting
$I := \operatorname{Ker}(f^{*} : A \rightarrow B)$, we obtain
an exact sequence
\[ 0 \rightarrow I / I^{2} \rightarrow B \otimes_{A} \widehat{\Omega}^{1}_{A/L} \rightarrow
\widehat{\Omega}^{1}_{B/L} \rightarrow 0 \]
of free $B$-modules. Choose
$a_{1}, \ldots, a_{n}, b_{1}, \ldots, b_{m} \in A$
s.t.\ $\{ a_{i} \}$ is a $B$-basis of $I / I^{2}$, and
$\{ \mrm{d} b_{i} \}$
is a $B$-basis of $\widehat{\Omega}^{1}_{B/L}$.
By the proof of Prop.\ \ref{prop1.4} the homomorphisms
$L \sqbr{\underline{s}} \rightarrow B$,
$L \sqbr{\underline{s}, \underline{t}} \rightarrow A$ and
$L \sqbr{\underline{s}, \underline{t}} \rightarrow B [\sqbr{\underline{t}}]$,
sending
$s_{i} \mapsto b_{i}$ and $t_{i} \mapsto a_{i}$,
are all formally \'{e}tale.
Take $\mfrak{a} := \operatorname{Ker}(A \rightarrow C)$, which is a defining ideal of
$A$, containing $A \cdot (\underline{t}) = I$.
Let $\mfrak{b} := \mfrak{a} \cdot B$, which is a
defining ideal of $B$. Hence the ideal
$\mfrak{c} = B [\sqbr{\underline{t}}] \cdot (\mfrak{b}, \underline{t})$
is a defining ideal of $B [\sqbr{\underline{t}}]$. By formal \'{e}taleness
of
$L \sqbr{\underline{s}, \underline{t}} \rightarrow A$ and
$L \sqbr{\underline{s}, \underline{t}} \rightarrow B [\sqbr{\underline{t}}]$,
the isomorphism
$A / \mfrak{a} \cong B [\sqbr{\underline{t}}] / \mfrak{c} \cong C$
lifts uniquely to an isomorphism
$A \cong B [\sqbr{\underline{t}}]$.
\noindent 2.\
Now assume $f$ is formally smooth. Let
$\mfrak{b} := \operatorname{Ker}(B \rightarrow C)$,
which is a defining ideal of $B$.
Since $A \rightarrow B / \mfrak{b}$ is surjective it follows that
$(B / \mfrak{b}) \otimes_{B} \widehat{\Omega}^{1}_{B/A}$
is generated by $\mrm{d}(\mfrak{b})$. By Nakayama's Lemma we see that
$\widehat{\Omega}^{1}_{B/A} = B \cdot \mrm{d}(\mfrak{b})$.
After shrinking $U$ sufficiently we get
$\widehat{\Omega}^{1}_{B/A} =
\bigoplus_{i = 1}^{n} B \cdot \mrm{d} b_{i}$
with $b_{i} \in \mfrak{b}$, and the homomorphism
$A [\sqbr{\underline{t}}] \rightarrow B$, $t_{i} \mapsto b_{i}$,
is formally \'{e}tale. Continuing like in part 1 of the proof we
conclude that this is actually an isomorphism.
\end{proof}
\begin{thm} \label{thm2.1}
Suppose $S$ is a noetherian scheme of characteristic $0$, and $X$ is a
finite type $S$-scheme.
Let $f : \mfrak{X} \rightarrow \mfrak{Y}$ be a morphism of smooth formal
embeddings of $X$. Then the DGA homomorphism
$f^{*} : \widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{Y} / S} \rightarrow
\widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{X} / S}$
is a quasi-isomorphism. Moreover, if $g : \mfrak{X} \rightarrow \mfrak{Y}$ is any
other morphism, then
$\mrm{H}(f^{*}) = \mrm{H}(g^{*})$.
\end{thm}
\begin{proof}
The assertions of the theorem are both local, and they will be proved in
three steps.
\noindent Step 1.\
Assume $f$ is a closed immersion. By Thm.\ \ref{thm2.2}
it suffices to check the case
$f : \operatorname{Spf} B = \mfrak{U} \rightarrow \operatorname{Spf} A = \mfrak{V}$
with $A \cong B [\sqbr{\underline{t}}]$ as topological $L$-algebras.
We must show that
$\widehat{\Omega}^{{\textstyle \cdot}}_{A/L} \rightarrow \widehat{\Omega}^{{\textstyle \cdot}}_{B/L}$
is a quasi-isomorphism. But since $\mbb{Q} \subset L$,
this is the well known Poincar\'{e} Lemma for
formal power series (cf.\ \cite{Ha} Ch.\ II Prop.\ 1.1, or \cite{Ye3}
Lemma 7.5).
\noindent Step 2.\
Suppose $f_{1}, f_{2}: \mfrak{X} \rightarrow \mfrak{Y}$ are two morphisms.
We wish to show that $\mrm{H}(f_{1}^{*}) = \mrm{H}(f_{2}^{*})$.
First consider
\[ \mfrak{Y} \xrightarrow{\mrm{diag}} (\mfrak{Y} \times_{k} \mfrak{Y})_{/X}
\xrightarrow{p_{i}} \mfrak{Y} . \]
Since the diagonal immersion is closed, we can apply the result of the
previous paragraph to it. We conclude that
$\mrm{H}(p_{1}^{*}) = \mrm{H}(p_{2}^{*})$,
and that these are isomorphisms.
But looking at
\[ \mfrak{X} \xrightarrow{\mrm{diag}} (\mfrak{X} \times_{k}
\mfrak{X})_{/X}
\xrightarrow{f_{1} \times f_{2}} (\mfrak{Y} \times_{k} \mfrak{Y})_{/X}
\xrightarrow{p_{i}} \mfrak{Y} \]
we see that our claim is proved.
\noindent Step 3.\
Consider an arbitrary morphism
$f: \mfrak{X} \rightarrow \mfrak{Y}$. Take any affine open
set $U \subset X$, with corresponding affine formal schemes
$\operatorname{Spf} B = \mfrak{U} \subset \mfrak{X}$ and
$\operatorname{Spf} A = \mfrak{V} \subset \mfrak{Y}$.
The definition of formal smoothness implies
there is some morphism of embeddings $g : \mfrak{V} \rightarrow \mfrak{U}$. This
morphism will not necessarily be an inverse of $f|_{\mfrak{U}}$, but
nonetheless, according to Step 2, $\mrm{H}(g^{*})$ and
$\mrm{H}(f|_{\mfrak{U}}^{*})$ will be isomorphisms between
$\mrm{H} \widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{U} / S}$
and
$\mrm{H} \widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{V} / S}$, inverse to each
other.
\end{proof}
In \cite{Ha} the relative De Rham cohomology
$\mrm{H}^{{\textstyle \cdot}}_{\mrm{DR}} (X / S)$
was defined. In the situation of Example \ref{exa2.1},
where $X \subset Y$ is a smooth algebraic embedding of $S$-schemes,
$\mfrak{X} = Y_{/X}$ and $\pi : \mfrak{X} \rightarrow S$ is the structural
morphism, the definition is
$\mrm{H}^{{\textstyle \cdot}}_{\mrm{DR}} (X / S) =
\mrm{H}^{{\textstyle \cdot}} \mrm{R} \pi_{*}
\widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{X} / S}$.
Even if $X$ is not globally embeddable,
$\mrm{H}^{{\textstyle \cdot}}_{\mrm{DR}} (X / S)$ can still be defined, by taking
a system of local embeddings
$\{ U_{i} \subset V_{i} \}$, $X = \bigcup U_{i}$,
and putting together a ``\v{C}ech-De Rham'' complex (cf.\ \cite{Ha} pp.\
28-29; it seems one should also assume $X$ separated and the
$U_{i}$ are affine).
\begin{cor} \label{cor2.1}
Suppose $S$ has characteristic $0$.
Let $X \subset \mfrak{X}$ be any smooth formal embedding \textup{(}not
necessarily algebraizable\textup{)}. Then
$\mrm{H}^{{\textstyle \cdot}}_{\mrm{DR}} (X / S) = \mrm{H}^{{\textstyle \cdot}} \mrm{R} \pi_{*}
\widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{X} / S}$
as graded $\mcal{O}_{S}$-algebras.
\end{cor}
\begin{proof}
Assume for simplicity that a global smooth algebraic embedding exists.
The general case, involving a system of embeddings,
only requires more bookkeeping.
Say $X \subset Y$ is the given algebraic embedding, and let
$\mfrak{Y} := Y_{/X}$.
Now the two formal embeddings $\mfrak{X}$ and $\mfrak{Y}$
are comparable: their product $(\mfrak{X} \times_{S} \mfrak{Y})_{/X}$
maps to both. By the theorem we get quasi-isomorphic DGAs on $X$.
\end{proof}
\begin{rem} \label{rem2.1}
From Corollaries \ref{cor2.3} and \ref{cor2.1} we see that there is a
sheaf of DGAs
$\widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{X}_{\mrm{et}} / S}$ on
$X_{\mrm{et}}$, with the property that for any $U \rightarrow X$ \'{e}tale,
$\mrm{H}^{{\textstyle \cdot}}_{\mrm{DR}}(U/S) =$ \linebreak
$\mrm{H}^{{\textstyle \cdot}} \Gamma(U,
\widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{X}_{\mrm{et}} / S})$.
As will be shown in \cite{Ye4}, the DGA
$\widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{X} / S}$ has an adelic resolution
$\mcal{A}^{{\textstyle \cdot}}_{\mfrak{X} / S}$,
where
$\mcal{A}^{p,q}_{\mfrak{X} / S} = \underline{\mbb{A}}^{q}_{\mrm{red}}(
\widehat{\Omega}^{p}_{\mfrak{X} / S})$, Beilinson's sheaf of adeles.
The adeles calculate cohomology:
$\mrm{H}^{{\textstyle \cdot}}_{\mrm{DR}}(X/S) =
\mrm{H}^{{\textstyle \cdot}} \Gamma(X, \mcal{A}^{{\textstyle \cdot}}_{\mfrak{X} / S})$.
Furthermore the adeles extend to an \'{e}tale sheaf
$\mcal{A}^{{\textstyle \cdot}}_{\mfrak{X}_{\mrm{et}} / S}$.
\end{rem}
\begin{rem} \label{rem2.3}
Suppose $S = \operatorname{Spec} k$, a field of characteristic $0$.
In \cite{Ye3} a complex
$\mcal{F}^{{\textstyle \cdot}}_{\mfrak{X}}$, called the De Rham-residue complex, is
defined. One has
$\mrm{H}^{i}(X, \mcal{F}^{{\textstyle \cdot}}_{\mfrak{X}}) =
\mrm{H}_{-i}^{\mrm{DR}}(X)$, the De Rham homology.
Moreover there is a sheaf
$\mcal{F}^{{\textstyle \cdot}}_{\mfrak{X}_{\mrm{et}}}$
on $X_{\mrm{et}}$, which directly implies that the De Rham homology
is contravariant for \'{e}tale morphisms.
Furthermore $\mcal{F}^{{\textstyle \cdot}}_{\mfrak{X}}$
is naturally a DG $\mcal{A}^{{\textstyle \cdot}}_{\mfrak{X}}$-module,
\end{rem}
\begin{rem} \label{rem2.4}
Smooth formal embeddings can be also used to define the category
of $\mcal{D}$-modules on a singular scheme $X$ (in characteristic $0$).
Say $X \subset \mfrak{X}$ is such an embedding. Then a formal
version of Kashiwara's Theorem (cf.\ \cite{Bo} Theorem VI.7.11)
implies that
$\msf{Mod}_{\mrm{disc}}(\mcal{D}_{\mfrak{X}})$,
the category of discrete modules over the ring of differential operators
$\mcal{D}_{\mfrak{X}}$ is, as an abelian category, independent of
$\mfrak{X}$.
\end{rem}
\section{Quasi-Coherent Sheaves on Formal Schemes}
Let $\mfrak{X}$ be a noetherian (adic) formal scheme.
By definition, a quasi-coherent sheaf on $\mfrak{X}$ is an
$\mcal{O}_{\mfrak{X}}$-module $\mcal{M}$, such that on sufficiently small
open sets $\mfrak{U} \subset \mfrak{X}$ there are exact sequences
$\mcal{O}_{\mfrak{U}}^{(J)} \rightarrow \mcal{O}_{\mfrak{U}}^{(I)} \rightarrow
\mcal{M}|_{\mfrak{U}} \rightarrow 0$,
for some indexing sets $I,J$ (cf.\ \cite{EGA} $0_{\mrm{I}}$ \S 5.1).
We shall denote by
$\mathsf{Mod}(\mfrak{X})$ (resp.\ $\mathsf{Coh}(\mfrak{X})$, resp.\
$\mathsf{QCo}(\mfrak{X})$)
the category of $\mcal{O}_{\mfrak{X}}$-modules (resp.\ the full subcategory
of coherent, resp.\ quasi-coherent, modules).
It seems that the only important quasi-coherent sheaves are the coherent
and the discrete ones (Def.\ \ref{dfn3.1}). Nevertheless we shall
consider all quasi-coherent sheaves, at the price of a little extra
effort.
\begin{rem}
There is some overlap between results in this section and
\cite{AJL2}.
\end{rem}
Let $A$ be a noetherian adic ring, and let
$\mfrak{U} := \operatorname{Spf} A$ be the affine formal scheme.
Then there is an exact functor $M \mapsto M^{\triangle}$ from the category
$\mathsf{Mod}_{\mrm{f}}(A)$ of
finitely generated $A$-modules to $\mathsf{Mod}(\mfrak{U})$.
It is an equivalence between $\mathsf{Mod}_{\mrm{f}}(A)$ and
$\mathsf{Coh}(\mfrak{U})$ (see \cite{EGA} I \S 10.10).
\begin{prop} \label{prop3.2}
The functor $M \mapsto M^{\triangle}$ extends uniquely to a functor
$\mathsf{Mod}(A) \rightarrow \mathsf{Mod}(\mfrak{U})$,
which is exact and commutes with direct limits.
The $\mcal{O}_{\mfrak{U}}$-module $M^{\triangle}$ is
quasi-coherent. For any $\mcal{O}_{\mfrak{U}}$-module $\mcal{M}$ the
following are equivalent:
\begin{enumerate}
\rmitem{i} $\mcal{M} \cong M^{\triangle}$ for some $A$-module $M$.
\rmitem{ii} $\mcal{M} \cong \lim_{\alpha \rightarrow} \mcal{M}_{\alpha}$
for some directed system $\{ \mcal{M}_{\alpha} \}$ of coherent
$\mcal{O}_{\mfrak{U}}$-modules.
\rmitem{iii} For every affine open set
$\mfrak{V} = \operatorname{Spf} B \subset \mfrak{U}$, one has
$\Gamma(\mfrak{V}, \mcal{M}) \cong
B \otimes_{A} \Gamma(\mfrak{U}, \mcal{M})$.
\end{enumerate}
\end{prop}
\begin{proof}
Take any $A$-module $M$ and write it as
$M = \lim_{\alpha \rightarrow} M_{\alpha}$ with finitely generated modules
$M_{\alpha}$. Define a presheaf $M^{\triangle}$ on $\mfrak{U}$ by
$\Gamma(\mfrak{V}, M^{\triangle}) := \lim_{\alpha \rightarrow}
\Gamma(\mfrak{V}, M^{\triangle}_{\alpha})$, for
$\mfrak{V} \subset \mfrak{U}$
open. Since $\mfrak{U}$ is a noetherian topological space it follows that
$M^{\triangle}$ is actually a sheaf. By construction
$M \mapsto M^{\triangle}$ commutes with direct limits. Since the
functor is exact on $\mathsf{Mod}_{\mrm{f}}(A)$, it's also exact on
$\mathsf{Mod}(A)$.
The implication (i) $\Rightarrow$ (ii) is because $M^{\triangle}_{\alpha}$
is coherent.
(ii) $\Rightarrow$ (iii): for such $B$ one has
$\Gamma(\mfrak{V}, \mcal{M}_{\alpha}) \cong
B \otimes_{A} \Gamma(\mfrak{U}, \mcal{M}_{\alpha})$;
now apply $\lim_{\alpha \rightarrow}$.
(iii) $\Rightarrow$ (i): set
$M := \Gamma(\mfrak{U}, \mcal{M})$. Then for every affine $\mfrak{V}$ we
have
$\Gamma(\mfrak{V}, \mcal{M}) = B \otimes_{A} M =
\Gamma(\mfrak{V}, M^{\triangle})$,
so $\mcal{M} = M^{\triangle}$.
Finally the module $M$ has a presentation
$A^{(I)} \rightarrow A^{(J)} \rightarrow M \rightarrow 0$. By exactness we get a presentation for
$M^{\triangle}$.
\end{proof}
It will be convenient to write $\mcal{O}_{\mfrak{U}} \otimes_{A} M$
instead of $M^{\triangle}$.
\begin{rem}
I do not know whether Serre's Theorem holds,
namely whe\-ther {\em every} quasi-coherent $\mcal{O}_{\mfrak{U}}$-module
$\mcal{M}$ is of the form
$\mcal{M} \cong \mcal{O}_{\mfrak{U}} \otimes_{A} M$.
Thus it may be that $\mathsf{QCo}(\mfrak{U})$ is not closed under direct
limits in $\mathsf{Mod}(\mfrak{U})$ (cf.\ Lemma \ref{lem4.1}).
\end{rem}
\begin{cor} \label{cor3.1}
Let $\mcal{M}$ be a quasi-coherent $\mcal{O}_{\mfrak{X}}$-module and
$x \in \mfrak{X}$ a point. Then there is an open neighborhood
$\mfrak{U} = \operatorname{Spf} A$ of $x$ s.t.\
$\mcal{M}|_{\mfrak{U}} \cong \mcal{O}_{\mfrak{U}} \otimes_{A}
\Gamma(\mfrak{U}, \mcal{M})$.
For such $\mfrak{U}$ one has
$\mrm{H}^{1}(\mfrak{U}, \mcal{M}) = 0$.
\end{cor}
\begin{proof}
Choose $\mfrak{U}$ affine such that $\mcal{M}|_{\mfrak{U}}$ has a
presentation
$\mcal{O}_{\mfrak{U}}^{(J)} \xrightarrow{\phi} \mcal{O}_{\mfrak{U}}^{(I)}
\xrightarrow{\psi} \mcal{M}|_{\mfrak{U}} \rightarrow 0$. Define
$M := \operatorname{Coker}(\phi: A^{(I)} \rightarrow A^{(J)})$.
Applying the exact functor $\mcal{O}_{\mfrak{U}} \otimes_{A}$ to
$A^{(I)} \xrightarrow{\phi} A^{(J)} \rightarrow M \rightarrow 0$
we get
$\mcal{M}|_{\mfrak{U}} \cong \mcal{O}_{\mfrak{U}} \otimes_{A} M$.
By the Proposition
$M \cong \Gamma(\mfrak{U}, \mcal{M})$.
As for $\mrm{H}^{1}(\mfrak{U}, - )$, use the fact that
it vanishes on coherent sheaves.
\end{proof}
\begin{prop} \label{prop3.3}
Let $\mcal{M}$ be coherent and $\mcal{N}$ quasi-coherent \textup{(}resp.\
coherent\textup{)}. Then
$\mcal{H}om_{\mcal{O}_{\mfrak{X}}}(\mcal{M}, \mcal{N})$
is quasi-coherent \textup{(}resp.\ coherent\textup{)}.
\end{prop}
\begin{proof}
For small enough $\mfrak{U} = \operatorname{Spf} A$ we get
$\mcal{M}|_{\mfrak{U}} \cong \mcal{O}_{\mfrak{U}} \otimes_{A} M$ and
$\mcal{N}|_{\mfrak{U}} \cong \mcal{O}_{\mfrak{U}} \otimes_{A} N$.
Now for any $\mfrak{V} = \operatorname{Spf} B \subset \mfrak{U}$, $A \rightarrow B$ is flat;
so
\[ \operatorname{Hom}_{B}(B \otimes_{A} M, B \otimes_{A} N) \cong
B \otimes_{A} \operatorname{Hom}_{A}(M, N) . \]
Hence
\[ \mcal{H}om_{\mcal{O}_{\mfrak{X}}}(\mcal{M}, \mcal{N})|_{\mfrak{U}} \cong
\mcal{O}_{\mfrak{U}} \otimes_{A} \operatorname{Hom}_{A}(M,N) . \]
\end{proof}
Recall that a subcategory $\mathsf{B}$ of an abelian category $\mathsf{A}$
is called a thick abelian subcategory if for any exact sequence
$M_{1} \rightarrow M_{2} \rightarrow N \rightarrow M_{3} \rightarrow M_{4}$
in $\mathsf{A}$ with $M_{i} \in \mathsf{B}$, also $N \in \mathsf{B}$.
\begin{prop} \label{prop3.1}
The category $\mathsf{QCo}(\mfrak{X})$ is a thick abelian subcategory of
$\mathsf{Mod}(\mfrak{X})$.
\end{prop}
\begin{proof}
First observe that the kernel and cokernel of a homomorphism
$\mcal{M} \rightarrow \mcal{N}$ between quasi-coherent sheaves is also
quasi-coherent.
This is immediate from Cor.\ \ref{cor3.1} and Prop.\ \ref{prop3.2}.
So it suffices to prove:
$0 \rightarrow \mcal{M}' \rightarrow \mcal{M} \rightarrow \mcal{M}'' \rightarrow 0$ exact,
$\mcal{M}', \mcal{M}''$ quasi-coherent $\Rightarrow$ $\mcal{M}$
quasi-coherent. For a sufficiently small affine open formal subscheme
$\mfrak{U} = \operatorname{Spf} A$ we will get, by Cor.\ \ref{cor3.1}, that
$\mrm{H}^{1}(\mfrak{U}, \mcal{M}') = 0$. Hence the
sequence
\[ 0 \rightarrow \Gamma(\mfrak{U}, \mcal{M}') \rightarrow
M= \Gamma(\mfrak{U}, \mcal{M}) \rightarrow
\Gamma(\mfrak{U}, \mcal{M}'') \rightarrow 0 \]
is exact. This implies that
$\mcal{M}|_{\mfrak{U}} \cong \mcal{O}_{\mfrak{U}} \otimes_{A} M$.
\end{proof}
\begin{dfn} \label{dfn3.1}
Let $\mcal{M}$ be an $\mcal{O}_{\mfrak{X}}$-module. Define
\[ \underline{\Gamma}_{\mrm{disc}} \mcal{M} := \lim_{n \rightarrow}
\mcal{H}om_{\mcal{O}_{\mfrak{X}}}(\mcal{O}_{\mfrak{X}} / \mcal{I}^{n},
\mcal{M}) \subset \mcal{M} \]
where $\mcal{I} \subset \mcal{O}_{\mfrak{X}}$ is any defining ideal.
$\mcal{M}$ is called {\em discrete} if
$\underline{\Gamma}_{\mrm{disc}} \mcal{M} = \mcal{M}$.
\end{dfn}
\begin{prop} \label{prop3.4}
Let $\mcal{M}$ be a quasi-coherent
$\mcal{O}_{\mfrak{X}}$-module. Then
$\underline{\Gamma}_{\mrm{disc}} \mcal{M}$ is quasi-coherent, and in fact
is a direct limit of discrete coherent $\mcal{O}_{\mfrak{X}}$-modules.
\end{prop}
\begin{proof}
Let $X_{n}$ be as in formula (\ref{eqn1.1}) and
$\mcal{M}_{n} := \mcal{H}om_{\mcal{O}_{\mfrak{X}}}(\mcal{O}_{X_{n}},
\mcal{M})$, so
$\underline{\Gamma}_{\mrm{disc}} \mcal{M} = \lim_{n \rightarrow} \mcal{M}_{n}$.
If $\mcal{M}$ is quasi-coherent, then $\mcal{M}_{n}$ is a quasi-coherent
$\mcal{O}_{X_{n}}$-module (by Prop.\ \ref{prop3.3}), and hence is a
direct limit of coherent modules.
\end{proof}
\section{Some Derived Functors of $\mcal{O}_{\mfrak{X}}$-Modules}
Denote by $\mathsf{Mod}_{\mrm{disc}}(\mfrak{X})$
(resp.\ $\mathsf{QCo}_{\mrm{disc}}(\mfrak{X})$)
the full subcategory of $\mathsf{Mod}(\mfrak{X})$ consisting
of discrete modules (resp.\ discrete quasi-coherent modules).
These are thick abelian subcategories.
In this section we study injective objects in the category
$\msf{QCo}_{\mrm{disc}}(\mfrak{X})$, and introduce the discrete Cousin
functor $\mrm{E} \mrm{R} \underline{\Gamma}_{\mrm{disc}}$.
\begin{lem} \label{lem4.1}
$\msf{Mod}_{\mrm{disc}}(\mfrak{X})$
is a locally noetherian category, with enough injectives.
\end{lem}
\begin{proof}
A family of noetherian generators consists of the sheaves
$\mcal{O}_{U}$, where $X \subset \mfrak{X}$ is a closed subscheme,
$U \subset X$ is an open set, and $\mcal{O}_{U}$ is extended by $0$
to all of $X$ (cf.\ \cite{RD} Theorem II.7.8).
If $\mcal{J} \in \msf{Mod}(\mfrak{X})$ is injective then
$\underline{\Gamma}_{\mrm{disc}} \mcal{J}$ is injective in
$\msf{Mod}_{\mrm{disc}}(\mfrak{X})$.
\end{proof}
Given a point $x \in \mfrak{X}$ let $J(x)$ be an injective hull
of the residue field $k(x)$ over the local ring
$\mcal{O}_{\mfrak{X},x}$,
and let $\mcal{J}(x)$ be the corresponding
$\mcal{O}_{\mfrak{X}}$-module. Then $\mcal{J}(x)$ is a
discrete quasi-coherent sheaf, constant on $\overline{\{ x \}}$,
and it is injective in $\msf{Mod}(\mfrak{X})$.
\begin{prop} \label{prop4.1}
\begin{enumerate}
\item
$\msf{QCo}_{\mrm{disc}}(\mfrak{X})$ is a locally noetherian
category with enough injectives.
\item Let $\mcal{J} \in \msf{QCo}_{\mrm{disc}}(\mfrak{X})$
be an injective object. Then $\mcal{J}$ is injective in
$\msf{Mod}_{\mrm{disc}}(\mfrak{X})$ and injective on
$\msf{Coh}(\mfrak{X})$.
For any
$\mcal{M} \in \msf{Mod}_{\mrm{disc}}(\mfrak{X})$
or
$\mcal{M} \in \msf{Coh}(\mfrak{X})$ the sheaf
$\mcal{H}om_{\mfrak{X}}(\mcal{M}, \mcal{J})$ is flasque.
\end{enumerate}
\end{prop}
\begin{proof}
1.\
Let $\mcal{N} \in \msf{QCo}_{\mrm{disc}}(\mfrak{X})$.
Choose a defining ideal $\mcal{I}$ of $\mfrak{X}$ and let
$X_{0}$ be the scheme $(\mfrak{X},
\mcal{O}_{\mfrak{X}} / \mcal{I})$.
Define $\mcal{N}_{0} :=
\mcal{H}om_{\mfrak{X}}(\mcal{O}_{X_{0}}, \mcal{N})$,
which is a quasi-coherent $\mcal{O}_{X_{0}}$-module.
Then the injective hull of $\mcal{N}_{0}$ in $\msf{Mod}(X_{0})$
is isomorphic to
$\bigoplus_{\alpha} \mcal{J}_{0}(x_{\alpha})$
for some $x_{\alpha} \in X_{0}$.
According to Proposition \ref{prop3.4},
$\msf{QCo}_{\mrm{disc}}(\mfrak{X})$
is locally noetherian, and this implies that
$\bigoplus_{\alpha} \mcal{J}(x_{\alpha})$
is an injective object in it. Now
$\mcal{N}_{0} \subset \mcal{N}$
and
$\mcal{N}_{0} \subset \bigoplus_{\alpha} \mcal{J}(x_{\alpha})$
are essential submodules, so there is some homomorphism
$\mcal{N} \rightarrow \bigoplus_{\alpha} \mcal{J}(x_{\alpha})$,
which is necessarily injective and essential.
\noindent 2.\
If $\mcal{N} = \mcal{J}$ is injective in
$\msf{QCo}_{\mrm{disc}}(\mfrak{X})$, it follows that
$\mcal{J} \rightarrow \bigoplus_{\alpha} \mcal{J}(x_{\alpha})$
is an isomorphism.
Since $\msf{Mod}_{\mrm{disc}}(\mfrak{X})$ is locally noetherian it
follows that $\mcal{J}$ is injective in it.
Given $\mcal{M} \in \msf{Mod}_{\mrm{disc}}(\mfrak{X})$
and open sets
$\mfrak{V} \subset \mfrak{U} \subset \mfrak{X}$
consider the sheaves
$\mcal{M}|_{\mfrak{V}} \subset \mcal{M}|_{\mfrak{U}} \subset
\mcal{M}$
(extension by $0$). Then
$\mcal{H}om_{\mfrak{X}}(\mcal{M}|_{\mfrak{U}}, \mcal{J}) \rightarrow
\mcal{H}om_{\mfrak{X}}(\mcal{M}|_{\mfrak{V}}, \mcal{J})$
is surjective.
The category
$\msf{Coh}(\mfrak{X})$ is noetherian, and therefore the functor
$\operatorname{Hom}_{\mfrak{X}}(-, \mcal{J})$ is exact on it.
Given $\mcal{M} \in \msf{Coh}(\mfrak{X})$ we have
$\mcal{H}om_{\mfrak{X}}(\mcal{M}, \mcal{J}) \cong
\bigoplus \mcal{H}om_{\mfrak{X}}(\mcal{M}, \mcal{J}(x_{\alpha}))$
which is clearly flasque.
\end{proof}
\begin{cor} \label{cor4.1}
Let
$\mcal{J}^{{\textstyle \cdot}} \in \msf{D}^{+}(\msf{QCo}_{\mrm{disc}}(\mfrak{X}))$
be a complex of injectives. Then for any
$\mcal{M}^{{\textstyle \cdot}} \in \msf{Mod}_{\mrm{disc}}(\mfrak{X})$
or
$\mcal{M}^{{\textstyle \cdot}} \in \msf{Coh}(\mfrak{X})$ one has
\[ \begin{aligned}
\mrm{R} \mcal{H}om_{\mfrak{X}}(\mcal{M}^{{\textstyle \cdot}}, \mcal{J}^{{\textstyle \cdot}})
& \cong \mcal{H}om_{\mfrak{X}}(\mcal{M}^{{\textstyle \cdot}}, \mcal{J}^{{\textstyle \cdot}}) \\
\mrm{R} \operatorname{Hom}_{\mfrak{X}}(\mcal{M}^{{\textstyle \cdot}}, \mcal{J}^{{\textstyle \cdot}})
& \cong \operatorname{Hom}_{\mfrak{X}}(\mcal{M}^{{\textstyle \cdot}}, \mcal{J}^{{\textstyle \cdot}})
\cong \Gamma(\mfrak{X},
\mcal{H}om_{\mfrak{X}}(\mcal{M}^{{\textstyle \cdot}}, \mcal{J}^{{\textstyle \cdot}})) .
\end{aligned} \]
\end{cor}
\begin{proof}
The first equality follows from Proposition \ref{prop4.1}
(cf.\ \cite{RD} Section I.6). Since each sheaf
$\mcal{H}om_{\mfrak{X}}(\mcal{M}^{p}, \mcal{J}^{q})$
is flasque we obtain the second equality.
\end{proof}
The functor
$\underline{\Gamma}_{\mrm{disc}} : \msf{Mod}(\mfrak{X}) \rightarrow
\msf{Mod}_{\mrm{disc}}(\mfrak{X})$
has a derived functor
\[ \mrm{R} \underline{\Gamma}_{\mrm{disc}} : \msf{D}^{+}(\msf{Mod}(\mfrak{X}))
\rightarrow \msf{D}^{+}(\msf{Mod}_{\mrm{disc}}(\mfrak{X})) , \]
which is calculated by injective resolutions.
There is another way to compute cohomology with supports.
Let $t$ be an indeterminate. Define
$\mbf{K}^{{\textstyle \cdot}}(t)$ to be the Koszul complex
$\mbb{Z}\sqbr{t} \xrightarrow{\cdot t} \mbb{Z}\sqbr{t}$, in dimensions $0$ and $1$,
and let
$\mbf{K}^{{\textstyle \cdot}}_{\infty}(t) := \lim_{i \rightarrow} \mbf{K}^{{\textstyle \cdot}}(t^{i})$.
Given a sequence $\underline{t} = (t_{1}, \ldots, t_{n})$
define
$\mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{t}) := \mbf{K}^{{\textstyle \cdot}}_{\infty}(t_{1})
\otimes \cdots \otimes \mbf{K}^{{\textstyle \cdot}}_{\infty}(t_{n})$,
a complex of flat $\mbb{Z} \sqbr{\underline{t}}$-modules
(in fact it's a commutative DGA).
If $A$ is a noetherian commutative ring and
$\underline{a} = (a_{1}, \ldots, a_{n}) \in A^{n}$, then we write
$\mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a})$ instead of
$\mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{t}) \otimes_{\mbb{Z} \sqbr{\underline{t}}} A$.
Now suppose $\mfrak{a} \subset A$ is an ideal, and $\underline{a}$ are
generators of $\mfrak{a}$. Then for any
$M^{{\textstyle \cdot}} \in \msf{D}^{+}(\msf{Mod}(A))$
there is a natural isomorphism
\begin{equation} \label{eqn4.1}
\mrm{R} \Gamma_{\mfrak{a}} M^{{\textstyle \cdot}} \cong
\mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}) \otimes M^{{\textstyle \cdot}}
\end{equation}
in $\msf{D}(\msf{Mod}(A))$.
We refer to \cite{LS1}, \cite{Hg1} and \cite{AJL1} for full details and
proofs. For sheaves one has:
\begin{lem} \label{lem4.3}
Suppose $\underline{a} \in \Gamma(\mfrak{U}, \mcal{O}_{\mfrak{U}})^{n}$
generates a defining ideal of the formal sche\-me $\mfrak{U}$.
Then for any $\mcal{M}^{{\textstyle \cdot}} \in \msf{D}^{+}(\msf{Mod}(\mfrak{U}))$
there is a natural isomorphism
\[ \mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{M}^{{\textstyle \cdot}} \cong
\mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}) \otimes \mcal{M}^{{\textstyle \cdot}} . \]
\end{lem}
\begin{proof}
Let $\mcal{I} := \mcal{O}_{\mfrak{U}} \cdot \underline{a}$. Then
$\underline{\Gamma}_{\mrm{disc}} = \underline{\Gamma}_{\mcal{I}}$, and we may use
\cite{AJL1} Lemma 3.1.1.
\end{proof}
\begin{prop} \label{prop4.2}
Let $X$ be a noetherian scheme, $X_{0} \subset X$ a closed subset,
$\mfrak{X} = X_{/ X_{0}}$ and
$g : \mfrak{X} \rightarrow X$ the completion morphism. Then for any
$\mcal{M}^{{\textstyle \cdot}} \in \msf{D}^{+}_{\mrm{qc}}(\msf{Mod}(X))$
there is a natural isomorphism
$g^{*} \mrm{R} \underline{\Gamma}_{X_{0}} \mcal{M}^{{\textstyle \cdot}} \cong
\mrm{R} \underline{\Gamma}_{\mrm{disc}} g^{*} \mcal{M}^{{\textstyle \cdot}}$.
In particular for a single quasi-coherent sheaf $\mcal{M}$ one has
$g^{*} \underline{\Gamma}_{X_{0}} \mcal{M} \cong
\underline{\Gamma}_{\mrm{disc}} g^{*} \mcal{M}$.
\end{prop}
\begin{proof}
Let
$\mcal{M}^{{\textstyle \cdot}} \rightarrow \mcal{J}^{{\textstyle \cdot}}$ be a resolution by quasi-coherent
injectives. Since $g$ is flat we get
\[ \phi :
g^{*} \mrm{R} \underline{\Gamma}_{X_{0}} \mcal{M}^{{\textstyle \cdot}} =
g^{*} \underline{\Gamma}_{X_{0}} \mcal{J}^{{\textstyle \cdot}} \rightarrow
\underline{\Gamma}_{\mrm{disc}} g^{*} \mcal{J}^{{\textstyle \cdot}} \rightarrow
\mrm{R} \underline{\Gamma}_{\mrm{disc}} g^{*} \mcal{J}^{{\textstyle \cdot}}
= \mrm{R} \underline{\Gamma}_{\mrm{disc}} g^{*} \mcal{M}^{{\textstyle \cdot}} . \]
Locally on any affine open $U \subset X$, with $U_{0} = U \cap X_{0}$
and $\mfrak{U} = U_{/U_{0}}$, we can find $\underline{a}$ in
$\Gamma(U, \mcal{O}_{U})$ which define $U_{0}$. It's known that
$\underline{\Gamma}_{U_{0}} (\mcal{J}^{{\textstyle \cdot}}|_{U}) \rightarrow
\mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}) \otimes (\mcal{J}^{{\textstyle \cdot}}|_{U})$
is a quasi-isomorphism. Since $g$ is flat we obtain quasi-isomorphisms
\begin{multline*}
\hspace{10mm}
\phi|_{\mfrak{U}} : g^{*} \underline{\Gamma}_{U_{0}} (\mcal{J}^{{\textstyle \cdot}}|_{U})
\rightarrow g^{*} \left( \mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}) \otimes
(\mcal{J}^{{\textstyle \cdot}}|_{U}) \right) \\
\cong \mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}) \otimes
g^{*} (\mcal{J}^{{\textstyle \cdot}}|_{U}) =
\mrm{R} \underline{\Gamma}_{\mrm{disc}} g^{*} (\mcal{J}^{{\textstyle \cdot}}|_{U}) .
\hspace{10mm}
\end{multline*}
It follows that $\phi$ is an isomorphism.
\end{proof}
Denote by $\msf{D}^{+}_{\mrm{d}}(\msf{Mod}(\mfrak{X}))$ the
subcategory of complexes with discrete cohomologies.
\begin{lem} \label{lem4.2}
\begin{enumerate}
\item If
$\mcal{M}^{{\textstyle \cdot}} \in \msf{D}^{+}_{\mrm{d}}(\msf{Mod}(\mfrak{X}))$
then
$\mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{M} \rightarrow \mcal{M}$
is an isomorphism.
\item If
$\mcal{M}^{{\textstyle \cdot}} \in \msf{D}^{+}_{\mrm{qc}}(\msf{Mod}(\mfrak{X}))$
then
$\mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{M} \in
\msf{D}^{+}_{\mrm{qc}}(\msf{Mod}_{\mrm{disc}}(\mfrak{X}))$.
\end{enumerate}
\end{lem}
\begin{proof}
From Lemma \ref{lem4.3} we see that the functor
$\mrm{R} \underline{\Gamma}_{\mrm{disc}}$ has finite cohomological dimension.
By way-out reasons (cf.\ \cite{RD} Section I.7) we may assume
$\mcal{M}^{{\textstyle \cdot}}$ is a single discrete (resp.\ quasi-coherent) sheaf.
Then the claims are obvious (use Proposition \ref{prop3.4} for 2).
\end{proof}
\begin{thm} \label{thm4.1}
The identity functor
$\msf{D}^{+}(\msf{QCo}_{\mrm{disc}}(\mfrak{X})) \rightarrow
\msf{D}^{+}_{\mrm{dqc}}(\msf{Mod}(\mfrak{X}))$
is an equivalence of categories.
In particular any
$\mcal{M}^{{\textstyle \cdot}} \in \msf{D}^{+}_{\mrm{dqc}}(\msf{Mod}(\mfrak{X}))$
is isomorphic to a complex of injectives
$\mcal{J}^{{\textstyle \cdot}} \in \msf{D}^{+}(\msf{QCo}_{\mrm{disc}}(\mfrak{X}))$.
\end{thm}
\begin{proof}
According to Lemma \ref{lem4.2} we see that
$\msf{D}^{+}_{\mrm{qc}}(\msf{Mod}_{\mrm{disc}}(\mfrak{X})) \rightarrow
\msf{D}^{+}_{\mrm{dqc}}(\msf{Mod}(\mfrak{X}))$
is an equivalence with quasi-inverse $\mrm{R} \underline{\Gamma}_{\mrm{disc}}$.
Next, by Proposition \ref{prop4.1} and by \cite{RD} Proposition I.4.8,
the functor
$\msf{D}^{+}(\msf{QCo}_{\mrm{disc}}(\mfrak{X})) \rightarrow
\msf{D}^{+}_{\mrm{qc}}(\msf{Mod}_{\mrm{disc}}(\mfrak{X}))$
is an equivalence.
\end{proof}
\begin{rem} \label{rem6.10}
In \cite{AJL2} it is proved that
$\msf{D}(\msf{QCo}_{\mrm{disc}}(\mfrak{X}))
\rightarrow \msf{D}_{\mrm{dqc}}(\msf{Mod}(\mfrak{X}))$
is an equivalence, using the quasi-coherator functor.
\end{rem}
Suppose there is a codimension function
$d : \mfrak{X} \rightarrow \mbb{Z}$,
i.e.\ a function satisfying $d(y) = d(x) + 1$ whenever $(x, y)$ is
an immediate specialization pair. Then there is a filtration
$\cdots \supset Z^{p} \supset Z^{p+1} \supset \cdots$
of $\mfrak{X}$, with
$Z^{p} := \{ F \subset \mfrak{X} \mid F \text{ closed}, d(F) \geq p \}$.
Here $d(F) := \operatorname{min} \{ d(x) \mid x \in F \}$.
This filtration determines a Cousin functor
\begin{equation}
\mrm{E} : \msf{D}^{+}(\msf{Ab}(\mfrak{X})) \rightarrow
\msf{C}^{+}(\msf{Ab}(\mfrak{X}))
\end{equation}
where $\msf{C}^{+}$ denotes the abelian category of bounded below
complexes (cf.\ \cite{RD} \S IV.1).
Given a point $x \in \mfrak{X}$ and a sheaf
$\mcal{M} \in \msf{Ab}(\mfrak{X})$ we let
$\Gamma_{x} \mcal{M} := (\underline{\Gamma}_{\, \overline{\{x\}}\, }
\mcal{M})_{x}$ $\subset \mcal{M}_{x}$.
The derived functor
$\mrm{R} \Gamma_{x} : \msf{D}^{+}(\msf{Ab}(\mfrak{X})) \rightarrow
\msf{D}(\msf{Ab})$
is calculated by flasque sheaves. Let us write
$\mrm{H}_{x}^{q} \mcal{M} :=
\mrm{H}^{q} \mrm{R} \Gamma_{x} \mcal{M}$,
the local cohomology, and let $i_{x} : \{x\} \rightarrow \mfrak{X}$
be the inclusion
According to \cite{RD} \S IV.1 Motif F one has a natural isomorphism
\begin{equation} \label{eqn4.3}
\mrm{E}^{p} \mcal{M}^{{\textstyle \cdot}} =
\mcal{H}_{Z^{p} / Z^{p+1}}^{p} \mcal{M}^{{\textstyle \cdot}} \cong
\bigoplus_{d(x) = p} i_{x *} \mrm{H}_{x}^{p} \mcal{M}^{{\textstyle \cdot}} .
\end{equation}
Observe that if
$\mcal{M} \in \msf{D}^{+}(\msf{Mod}(\mfrak{X}))$
then
$\mrm{E} \mcal{M}^{{\textstyle \cdot}} \in
\msf{C}^{+}(\msf{Mod}(\mfrak{X}))$
and
$\mrm{R} \Gamma_{x} \mcal{M} \in$ \newline
$\msf{D}^{+}(\msf{Mod}(\mcal{O}_{\mfrak{X}, x}))$.
Unlike an ordinary scheme, on a formal scheme the topological
support of a quasi-coherent sheaf does not coincide with its
algebraic support. But for discrete sheaves these two notions of support
do coincide. This suggests:
\begin{dfn}
Given $\mcal{M} \in \msf{D}^{+}(\msf{Mod}(\mfrak{X}))$
its {\em discrete Cousin complex} is \newline
$\mrm{E} \mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{M}^{{\textstyle \cdot}}$.
\end{dfn}
\begin{thm} \label{thm4.2}
For any
$\mcal{M}^{{\textstyle \cdot}} \in \msf{D}^{+}_{\mrm{qc}}(\msf{Mod}(\mfrak{X}))$
the complex
$\mrm{E} \mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{M}^{{\textstyle \cdot}}$
consists of discrete quasi-coherent sheaves. So we get a functor
\[ \mrm{E} \mrm{R} \underline{\Gamma}_{\mrm{disc}} :
\msf{D}^{+}_{\mrm{qc}}(\msf{Mod}(\mfrak{X})) \rightarrow
\msf{C}^{+}(\msf{QCo}_{\mrm{disc}}(\mfrak{X})). \]
\end{thm}
\begin{proof}
According to Theorem \ref{thm4.1} we may assume
$\mcal{N}^{{\textstyle \cdot}} = \mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{M}^{{\textstyle \cdot}}$
is in \newline
$\msf{D}^{+}(\msf{QCo}_{\mrm{disc}}(\mfrak{X}))$.
On any open formal subscheme $\mfrak{U} = \operatorname{Spf} A$ we get
$\mcal{N}^{{\textstyle \cdot}} = \mcal{O}_{\mfrak{U}} \otimes_{A} N^{{\textstyle \cdot}}$,
where
$N^{q} = \Gamma(\mfrak{U}, N^{q})$
(cf.\ Propositions \ref{prop3.4} and \ref{prop3.2})
Then for $x \in \mfrak{U}$,
\[ \mrm{R} \Gamma_{x} \mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{M}^{{\textstyle \cdot}}
= \mrm{R} \Gamma_{x} \mcal{N}^{{\textstyle \cdot}} =
\mrm{R} \Gamma_{\mfrak{p}} N^{{\textstyle \cdot}}_{\mfrak{p}} \]
where $\mfrak{p} \subset A$ is the prime ideal of $x$. Hence
$\mrm{H}^{q}_{x} \mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{M}^{{\textstyle \cdot}}
= \mrm{H}^{q}_{\mfrak{p}} N^{{\textstyle \cdot}}_{\mfrak{p}}$
is $\mfrak{p}$-torsion. So the sheaf corresponding to $x$ in
(\ref{eqn4.3}) is quasi-coherent and discrete.
\end{proof}
\section{Dualizing Complexes on Formal Schemes}
In this section we propose a theory of duality on noetherian formal
sche\-mes. There is a fundamental difference between this theory and the
duality theory on schemes, as developed in \cite{RD}. A dualizing
complex $\mcal{R}^{{\textstyle \cdot}}$ on a scheme $X$ has coherent cohomology
sheaves; this will not be true on a general formal scheme $\mfrak{X}$,
where $\mrm{H}^{q} \mcal{R}^{{\textstyle \cdot}}$ are discrete quasi-coherent sheaves
(Def.\ \ref{dfn5.1}).
We prove uniqueness of dualizing complexes (Thm.\ \ref{thm5.1}),
and existence in some cases (Prop.\ \ref{prop5.8} and Thm.\ \ref{thm5.3}).
Before we begin here is an instructive example due to J.\ Lipman.
\begin{exa} \label{exa5.2}
Consider the ring $A = k[\sqbr{t}]$ of formal power series over a field
$k$. Let $\mfrak{X} := \operatorname{Spf} A$, which has a single point. The modules
$A$ and $J = \mrm{H}^{1}_{(t)} A$ both have finite injective dimension
and satisfy $\operatorname{Hom}_{A}(A, A) = \operatorname{Hom}_{A}(J, J) = A$. Which
one is a dualizing complex on $\mfrak{X}$? We will see that $J$ is
the correct answer (Def.\ \ref{dfn5.1}), and $A$ is a ``fake'' dualizing
complex (Thm.\ \ref{thm5.3}). The relevant relation between them is:
$J = \mrm{R} \Gamma_{\mrm{disc}} A [1]$.
\end{exa}
Suppose
$\mcal{N}^{{\textstyle \cdot}} \in
\msf{D}^{+}(\msf{Mod}_{\mrm{disc}}(\mfrak{X}))$.
We say $\mcal{N}^{{\textstyle \cdot}}$ has finite injective dimension on
$\msf{QCo}_{\mrm{disc}}(\mfrak{X})$
if there is an integer $q_{0}$ s.t.\ for all $q > q_{0}$ and
$\mcal{M} \in \msf{QCo}_{\mrm{disc}}(\mfrak{X})$,
$\mrm{H}^{q} \mrm{R} \operatorname{Hom}_{\mfrak{X}}
(\mcal{M}, \mcal{N}^{{\textstyle \cdot}}) = 0$.
\begin{dfn} \label{dfn5.1}
A {\em dualizing complex} on $\mfrak{X}$ is a complex
$\mcal{R}^{{\textstyle \cdot}} \in
\msf{D}^{\mrm{b}}_{\mrm{dqc}}(\msf{Mod}(\mfrak{X}))$
satisfying:
\begin{enumerate}
\rmitem{i} $\mcal{R}^{{\textstyle \cdot}}$ has finite injective dimension
on $\msf{QCo}_{\mrm{disc}}(\mfrak{X})$.
\rmitem{ii} The adjunction morphism
$\mcal{O}_{\mfrak{X}} \rightarrow
\mrm{R} \mcal{H}om_{\mfrak{X}}
(\mcal{R}^{{\textstyle \cdot}}, \mcal{R}^{{\textstyle \cdot}})$
is an isomorphism.
\rmitem{iii} For some defining ideal $\mcal{I}$ of $\mfrak{X}$,
$\mrm{R} \mcal{H}om_{\mfrak{X}}(\mcal{O}_{\mfrak{X}} / \mcal{I},
\mcal{R}^{{\textstyle \cdot}})$
has coherent cohomology sheaves.
\end{enumerate}
\end{dfn}
\begin{lem} \label{lem5.2}
Let
$\mcal{N}^{{\textstyle \cdot}} \in
\msf{D}^{+}_{\mrm{dqc}}(\msf{Mod}(\mfrak{X}))$.
Then $\mcal{N}^{{\textstyle \cdot}}$ has finite injective dimension on
$\msf{QCo}_{\mrm{disc}}(\mfrak{X})$ iff it is isomorphic to a bounded
complex of injectives in $\msf{QCo}_{\mrm{disc}}(\mfrak{X})$.
\end{lem}
\begin{proof}
Because of Theorem \ref{thm4.1}, the proof is just like
\cite{RD} Prop.\ I.7.6.
\end{proof}
In light of this, we can, when convenient, assume the dualizing complex
$\mcal{R}^{{\textstyle \cdot}}$ is a bounded complex of discrete quasi-coherent
injectives.
\begin{prop} \label{prop5.2}
Let $\mcal{R}^{{\textstyle \cdot}}$ be a dualizing complex on $\mfrak{X}$.
Then for any
$\mcal{M}^{{\textstyle \cdot}} \in \msf{D}^{\mrm{b}}_{\mrm{c}}(\msf{Mod}(\mfrak{X}))$
the morphism of adjunction
\[ \mcal{M}^{{\textstyle \cdot}} \rightarrow
\mrm{R} \mcal{H}om_{\mfrak{X}} (
\mrm{R} \mcal{H}om_{\mfrak{X}} (\mcal{M}^{{\textstyle \cdot}},
\mcal{R}^{{\textstyle \cdot}}), \mcal{R}^{{\textstyle \cdot}}) \]
is an isomorphism.
\end{prop}
\begin{proof}
We can assume $\mfrak{X}$ is affine, and so replace $\mcal{M}^{{\textstyle \cdot}}$
with a complex of coherent sheaves.
By ``way-out'' arguments (cf.\ \cite{RD} Section I.7)
we reduce to the case $\mcal{M}^{{\textstyle \cdot}} = \mcal{O}_{\mfrak{X}}$,
which property (ii) applies.
\end{proof}
\begin{lem} \label{lem5.4}
Suppose $\mcal{R}^{{\textstyle \cdot}}$ is a dualizing complex on $\mfrak{X}$.
Let $\mcal{I}$ be any defining ideal of $\mfrak{X},$ and let $X_{0}$ be
the scheme
$(\mfrak{X}, \mcal{O}_{\mfrak{X}} / \mcal{I})$.
Then
$\mrm{R} \mcal{H}om_{\mfrak{X}}
(\mcal{O}_{X_{0}}, \mcal{R}^{{\textstyle \cdot}})$
is a dualizing complex on $X_{0}$.
\end{lem}
\begin{proof}
We can assume $\mcal{R}^{{\textstyle \cdot}}$ is a bounded complex of injectives
in $\msf{QCo}_{\mrm{disc}}(\mfrak{X})$, so
$\mcal{R}^{{\textstyle \cdot}}_{0} := \mcal{H}om_{\mfrak{X}}
(\mcal{O}_{X_{0}}, \mcal{R}^{{\textstyle \cdot}})$
is a complex of injectives on $X_{0}$. Property (iii) implies that
$\mcal{R}^{{\textstyle \cdot}}_{0}$ has coherent cohomology sheaves. Now
\[ \mcal{H}om_{X_{0}}(\mcal{R}^{{\textstyle \cdot}}_{0}, \mcal{R}^{{\textstyle \cdot}}_{0})
\cong \mcal{H}om_{\mfrak{X}} (
\mcal{H}om_{\mfrak{X}} (\mcal{O}_{X_{0}},
\mcal{R}^{{\textstyle \cdot}}), \mcal{R}^{{\textstyle \cdot}})
\cong \mcal{O}_{X_{0}} , \]
so $\mcal{R}^{{\textstyle \cdot}}_{0}$ is dualizing.
\end{proof}
\begin{thm} \label{thm5.1} \textup{(Uniqueness)}\
Suppose $\mcal{R}^{{\textstyle \cdot}}$ and
$\tilde{\mcal{R}}^{{\textstyle \cdot}}$ are dualizing complexes and $\mfrak{X}$
is connected. Then
$\tilde{\mcal{R}}^{{\textstyle \cdot}} \cong \mcal{R}^{{\textstyle \cdot}} \otimes
\mcal{L}[n]$
in
$\msf{D}(\msf{Mod}(\mfrak{X}))$,
for some invertible sheaf $\mcal{L}$ and integer $n$.
\end{thm}
\begin{proof}
We can assume both $\mcal{R}^{{\textstyle \cdot}}$ and $\tilde{\mcal{R}}^{{\textstyle \cdot}}$
are bounded complexes of injectives in
$\msf{QCo}_{\mrm{disc}}(\mfrak{X})$.
Choose a defining ideal $\mcal{I}$ and let
$X_{m}$ be the scheme
$(\mfrak{X}, \mcal{O}_{\mfrak{X}} / \mcal{I}^{m+1})$.
Define a complex
$\mcal{R}^{{\textstyle \cdot}}_{m} := \mcal{H}om_{\mfrak{X}}
(\mcal{O}_{X_{m}}, \mcal{R}^{{\textstyle \cdot}})$
and likewise $\tilde{\mcal{R}}^{{\textstyle \cdot}}_{m}$.
These are dualizing complexes on $X_{m}$, so by \cite{RD} Thm.\ IV.3.1
there is an isomorphism
\[ \phi_{m} : \mcal{R}^{{\textstyle \cdot}}_{m} \otimes \mcal{L}_{m}[n_{m}]
\rightarrow \tilde{\mcal{R}}^{{\textstyle \cdot}}_{m} \]
in $\msf{D}(\msf{Mod}(X_{m}))$, for some
invertible sheaf $\mcal{L}_{m}$ and integer $n_{m}$.
Writing
$\mcal{M}_{m}^{{\textstyle \cdot}} := \mcal{H}om_{X_{m}}(
\mcal{R}^{{\textstyle \cdot}}_{m}, \tilde{\mcal{R}}^{{\textstyle \cdot}}_{m})$
we have
$\mcal{M}_{m}^{{\textstyle \cdot}} \cong \mcal{L}_{m}[n_{m}]$
in $\msf{D}(\msf{Mod}(X_{m}))$.
Now
\[ \mcal{M}_{m}^{{\textstyle \cdot}} \cong
\mcal{H}om_{X_{m+1}}(
\mcal{H}om_{X_{m+1}}(\mcal{O}_{X_{m}},
\mcal{R}^{{\textstyle \cdot}}_{m+1}), \mcal{R}^{{\textstyle \cdot}}_{m+1})) \otimes
\mcal{L}_{m+1}[n_{m+1}] \]
as complexes of $\mcal{O}_{X_{m+1}}$-modules, so by the dualizing
property of $\mcal{R}^{{\textstyle \cdot}}_{m+1}$ we deduce an isomorphism
$\mcal{M}_{m}^{{\textstyle \cdot}} \cong \mcal{O}_{X_{m}}
\otimes \mcal{L}_{m+1}[n_{m+1}]$
in $\msf{D}(\msf{Mod}(X_{m+1}))$.
We conclude that
$n_{m} = n_{m+1}$ and
$\mcal{L}_{m} \cong \mcal{O}_{X_{m}} \otimes \mcal{L}_{m+1}$.
Set $n := n_{m}$ and
$\mcal{L} := \lim_{\leftarrow m} \mcal{L}_{m}$.
Next, since
$\mcal{R}^{q}_{m} \subset \mcal{R}^{q}_{m+1}$
and $\tilde{\mcal{R}}^{q}_{m+1}$
is injective in $\msf{Mod}(X_{m+1})$, we see that
$\mcal{M}_{m+1}^{q} \rightarrow \mcal{M}_{m}^{q}$
is surjective for all $q,m$. Furthermore,
$\mrm{H}^{q} \mcal{M}^{{\textstyle \cdot}}_{m+1} \rightarrow
\mrm{H}^{q} \mcal{M}^{{\textstyle \cdot}}_{m}$
is also surjective, since
$\mrm{H}^{q} \mcal{M}^{{\textstyle \cdot}}_{m} = \mcal{L}_{m}$ or $0$.
Define
\[ \mcal{M}^{{\textstyle \cdot}} := \mcal{H}om_{\mfrak{X}}(
\mcal{R}^{{\textstyle \cdot}}, \tilde{\mcal{R}}^{{\textstyle \cdot}})
\cong \lim_{\leftarrow m} \mcal{M}^{{\textstyle \cdot}}_{m} . \]
According to \cite{Ha} Cor.\ I.4.3 and Prop.\ I.4.4 it follows that
$\mrm{H}^{q} \mcal{M}^{{\textstyle \cdot}} = \lim_{\leftarrow m}
\mrm{H}^{q} \mcal{M}^{{\textstyle \cdot}}_{m}$.
This implies that
$\mcal{H}om_{\mfrak{X}}(
\mcal{R}^{{\textstyle \cdot}} \otimes \mcal{L}[n], \tilde{\mcal{R}}^{{\textstyle \cdot}}))
\cong \mcal{O}_{\mfrak{X}}$
in $\msf{D}(\msf{Mod}(\mfrak{X}))$, so by Corollary \ref{cor4.1}
\[ \mrm{H}^{0} \operatorname{Hom}_{\mfrak{X}}
(\mcal{R}^{{\textstyle \cdot}}\otimes \mcal{L}[n], \tilde{\mcal{R}}^{{\textstyle \cdot}})
\cong \Gamma(\mfrak{X}, \mcal{O}_{\mfrak{X}}) .
\]
Choose a homomorphism of complexes
$\phi : \mcal{R}^{{\textstyle \cdot}} \otimes \mcal{L}[n] \rightarrow
\tilde{\mcal{R}}^{{\textstyle \cdot}}$
corresponding to
$1 \in \Gamma(\mfrak{X}, \mcal{O}_{\mfrak{X}})$.
Backtracking we see that for every $m$, $\phi$ induces a homomorphism
$\mcal{R}^{{\textstyle \cdot}}_{m} \otimes \mcal{L}[n] \rightarrow
\tilde{\mcal{R}}^{{\textstyle \cdot}}_{m}$
which represents $\phi_{m}$ in
$\msf{D}(\msf{Mod}(X_{m}))$. So
$\phi = \lim_{m \rightarrow} \phi_{m}$ is a quasi-isomorphism.
\end{proof}
\begin{prob}
Let $\mcal{R}^{{\textstyle \cdot}}$ be a dualizing complex. Is it true that the following
conditions on
$\mcal{N}^{{\textstyle \cdot}} \in \msf{D}^{\mrm{b}}_{\mrm{dqc}}(\msf{Mod}(\mfrak{X}))$
are equivalent?
\begin{enumerate}
\rmitem{i}
$\mcal{N}^{{\textstyle \cdot}} \cong \mrm{R} \mcal{H}om_{\mfrak{X}}^{{\textstyle \cdot}}(
\mcal{M}^{{\textstyle \cdot}}, \mcal{R}^{{\textstyle \cdot}})$
for some
$\mcal{M}^{{\textstyle \cdot}} \in \msf{D}^{\mrm{b}}_{\mrm{c}}(\msf{Mod}(\mfrak{X}))$.
\rmitem{ii} For any $\mcal{M}$ discrete coherent,
$\mrm{R} \mcal{H}om_{\mfrak{X}}^{{\textstyle \cdot}}(
\mcal{M}, \mcal{N}^{{\textstyle \cdot}}) \in
\msf{D}^{\mrm{b}}_{\mrm{c}}(\msf{Mod}(\mfrak{X}))$.
\end{enumerate}
\end{prob}
Recall that for a point $x \in \mfrak{X}$ we denote by $J(x)$ an
injective hull of $k(x)$ over $\mcal{O}_{\mfrak{X}, x}$,
and $\mcal{J}(x)$ is the corresponding quasi-coherent sheaf.
\begin{lem} \label{lem5.6}
Suppose $\mcal{R}^{{\textstyle \cdot}}$ is a dualizing complex on $\mfrak{X}$.
For any $x \in \mfrak{X}$ there is a unique integer $d(x)$ s.t.\
\[ \mrm{H}^{q}_{x} \mcal{R}^{{\textstyle \cdot}} \cong
\begin{cases}
J(x) & \text{ if } q = d(x)\\
0 & \text{ otherwise}.
\end{cases} \]
Furthermore $d$ is a codimension function.
\end{lem}
\begin{proof}
We can assume $\mcal{R}^{{\textstyle \cdot}}$ is a bounded complex of injectives
in $\msf{QCo}_{\mrm{disc}}(\mfrak{X})$. Then as seen before
$\mrm{H}^{q}_{x} \mcal{R}^{{\textstyle \cdot}} =
\mrm{H}^{q} \Gamma_{x} \mcal{R}^{{\textstyle \cdot}}$. Define schemes $X_{m}$
and complexes $\mcal{R}^{{\textstyle \cdot}}_{m}$ like in the proof of Thm.\
\ref{thm5.1}. Since $\mcal{R}^{{\textstyle \cdot}}_{m}$ is dualizing it determines
a codimension function $d_{m}$ on $X_{m}$ (cf.\ \cite{RD} Ch.\ V \S 7).
But the arguments used before show that $d_{m} = d_{m+1}$. Finally
$\mrm{H}^{q} \Gamma_{x} \mcal{R}^{{\textstyle \cdot}} =
\lim_{m \rightarrow} \mrm{H}^{q} \Gamma_{x} \mcal{R}^{{\textstyle \cdot}}_{m}$,
and
$\mrm{H}^{q} \Gamma_{x} \mcal{R}^{{\textstyle \cdot}}_{m} \cong J_{m}(x)$,
an injective hull of $k(x)$ over $\mcal{O}_{X_{m}, x}$.
\end{proof}
\begin{dfn} \label{dfn5.3}
A residual complex on the noetherian formal scheme $\mfrak{X}$
is a dualizing complex $\mcal{K}^{{\textstyle \cdot}}$ which
is isomorphic, as $\mcal{O}_{\mfrak{X}}$-module, to
$\bigoplus_{x \in \mfrak{X}} \mcal{J}(x)$.
\end{dfn}
\begin{prop} \label{prop5.7}
Say $\mcal{R}^{{\textstyle \cdot}}$ is a dualizing complex on $\mfrak{X}$.
Let $d$ be the codimension function above, and let $\mrm{E}$ be the
associated Cousin functor. Then
$\mcal{R}^{{\textstyle \cdot}} \cong \mrm{E} \mcal{R}^{{\textstyle \cdot}}$ in
$\msf{D}(\msf{Mod}(\mfrak{X}))$, and
$\mrm{E} \mcal{R}^{{\textstyle \cdot}}$ is a residual complex.
\end{prop}
\begin{proof}
By Lemma \ref{lem5.6} $\mcal{R}^{{\textstyle \cdot}}$ is a Cohen-Macaulay complex,
in the sense of \cite{RD} p.\ 247, Definition. So there exists some
isomorphism
$\mcal{R}^{{\textstyle \cdot}} \rightarrow \mrm{E} \mcal{R}^{{\textstyle \cdot}}$
in $\msf{D}^{\mrm{b}}(\msf{Mod}(\mfrak{X}))$.
\end{proof}
To conclude this section we consider some situations where a dualizing
complex exists. If $f : \mfrak{X} \rightarrow \mfrak{Y}$ is a morphism
then $(\mfrak{Y}, f_{*} \mcal{O}_{\mfrak{X}})$ is a ringed space,
and
$\bar{f} : \mfrak{X} \rightarrow (\mfrak{Y}, f_{*} \mcal{O}_{\mfrak{X}})$
is a morphism of ringed spaces.
\begin{prop} \label{prop5.8}
Let $f : \mfrak{X} \rightarrow \mfrak{Y}$ be a formally finite morphism,
and assume $\mcal{K}^{{\textstyle \cdot}}$ is a residual complex on $\mfrak{Y}$.
Then
$\bar{f}^{*} \mcal{H}om_{\mfrak{Y}}(f_{*} \mcal{O}_{\mfrak{X}},
\mcal{K}^{{\textstyle \cdot}})$
is a residual complex on $\mfrak{X}$.
\end{prop}
\begin{proof}
Let $f_{n} : X_{n} \rightarrow Y_{n}$ be morphisms as in Lemma \ref{lem1.12},
and let
$\mcal{K}_{n}^{{\textstyle \cdot}}$ \linebreak
$:= \mcal{H}om_{\mfrak{Y}}(\mcal{O}_{Y_{n}},
\mcal{K}^{{\textstyle \cdot}})$.
Since $f_{n}$ is a finite morphism,
$\bar{f}_{n}^{*} \mcal{H}om_{Y_{n}}(f_{n *} \mcal{O}_{X_{n}},
\mcal{K}_{n}^{{\textstyle \cdot}})$
is a residual complex on $X_{n}$. As in the proof of Thm.\ \ref{thm5.1},
\[ \bar{f}^{*} \mcal{H}om_{\mfrak{Y}}(f_{*} \mcal{O}_{\mfrak{X}},
\mcal{K}^{{\textstyle \cdot}}) \cong \lim_{n \rightarrow}
\bar{f}_{n}^{*} \mcal{H}om_{Y_{n}}(f_{n *} \mcal{O}_{X_{n}},
\mcal{K}_{n}^{{\textstyle \cdot}}) \]
is residual.
\end{proof}
\begin{exa} \label{exa5.1}
Suppose $X_{0} \subset X$ is closed, $\mfrak{X} = X_{/ X_{0}}$
and $g : \mfrak{X} \rightarrow X$ is the completion morphism.
Let $\mcal{K}^{{\textstyle \cdot}}$ be a residual complex on $X$. In this case
$g = \bar{g}$, and by Proposition \ref{prop4.2}
\[ g^{*} \mcal{H}om_{X}(g_{*} \mcal{O}_{\mfrak{X}},
\mcal{K}^{{\textstyle \cdot}}) \cong
\lim_{n \rightarrow} g^{*} \mcal{K}_{n}^{{\textstyle \cdot}} \cong
g^{*} \underline{\Gamma}_{X_{0}} \mcal{K}^{{\textstyle \cdot}} \cong
\underline{\Gamma}_{\mrm{disc}} g^{*} \mcal{K}^{{\textstyle \cdot}} \]
is a residual complex.
We see that if $\mcal{R}^{{\textstyle \cdot}}$ is
any dualizing complex on $X$ then
$\mrm{E} \mrm{R} \underline{\Gamma}_{\mrm{disc}} g^{*} \mcal{R}^{{\textstyle \cdot}}$
is dualizing on $\mfrak{X}$.
\end{exa}
We call a formal scheme $\mfrak{X}$ {\em regular} of all its local rings
$\mcal{O}_{\mfrak{X}, x}$ are regular.
\begin{lem} \label{lem5.7}
Suppose $\mfrak{X}$ is a regular formal scheme. Then
$d(x) := \operatorname{dim} \mcal{O}_{\mfrak{X},x}$
is a bounded codimension function on $\mfrak{X}$.
\end{lem}
\begin{proof}
Let
$\mfrak{U} = \operatorname{Spf} A \subset \mfrak{X}$ be a connected affine open
set. If $x \in \mfrak{U}$ is the point corresponding to an open prime
ideal $\mfrak{p}$, then
$\widehat{A}_{\mfrak{p}} \cong \widehat{\mcal{O}}_{\mfrak{X}, x}$.
Therefore $A_{\mfrak{p}}$ is a regular local ring.
Now in the adic noetherian ring $A$ any maximal ideal $\mfrak{m}$ is open.
Hence, by \cite{Ma} \S 18 Lemma 5 (III), $A$ is a regular ring,
of finite global dimension equal to its Krull dimension.
Now let
$U := \operatorname{Spec} A$, so as a topological space, $\mfrak{U} \subset U$
is the closed set defined by any defining ideal $I \subset A$.
Since $U$ is a regular scheme, $\mcal{O}_{U}$ is a dualizing complex
on it. The codimension function $d'$ corresponding to $\mcal{O}_{U}$
satisfies $d'(y) = \operatorname{dim} \mcal{O}_{U, y}$. Thus
$0 \leq d'(y) \leq \operatorname{dim} U$. But clearly
$d|_{\mfrak{U}} = d'|_{\mfrak{U}}$. By covering $\mfrak{X}$ with
finitely many such $\mfrak{U}$ this implies
that $d$ is a bounded codimension function.
\end{proof}
\begin{thm} \label{thm5.3}
Suppose $\mfrak{X}$ is a regular formal scheme. Then
$\mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{O}_{\mfrak{X}}$
is a dualizing complex on $\mfrak{X}$.
\end{thm}
\begin{proof}
By the proof of Theorem \ref{thm4.2} and known properties of regular
local rings, for any $x \in \mfrak{X}$
\[ \mrm{H}^{q}_{x} \mrm{R} \underline{\Gamma}_{\mrm{disc}}
\mcal{O}_{\mfrak{X}} \cong
\mrm{H}^{q}_{\mfrak{m}_{x}} \widehat{\mcal{O}}_{\mfrak{X}, x} \cong
\begin{cases}
J(x) & \text{ if } q = d(x)\\
0 & \text{ otherwise}
\end{cases} \]
where $\mfrak{m}_{x} \subset \widehat{\mcal{O}}_{\mfrak{X}, x}$
is the maximal ideal, and $J(x)$ is an injective hull of $k(x)$.
Since $d$ is bounded it follows that
$\mcal{K}^{{\textstyle \cdot}} :=
\mrm{E} \mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{O}_{\mfrak{X}}$
is a bounded complex of injectives in $\msf{QCo}_{\mrm{disc}}(\mfrak{X})$.
Like in the proof of Proposition \ref{prop5.7},
$\mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{O}_{\mfrak{X}} \cong
\mcal{K}^{{\textstyle \cdot}}$
in $\msf{D}(\msf{Mod}(\mfrak{X}))$.
To complete the proof it suffices to show that for any affine open set
$\mfrak{U} = \operatorname{Spf} A \subset \mfrak{X}$ the complex
$\mcal{K}^{{\textstyle \cdot}}|_{\mfrak{U}}$ is residual on $\mfrak{U}$.
Let $U := \operatorname{Spec} A$ and let $g : \mfrak{U} \rightarrow U$ be the canonical
morphism Let $U_{0} \subset U$ be the closed set $g(\mfrak{U})$,
so that $\mfrak{U} \cong U_{/ U_{0}}$.
Define
$\mcal{K}^{{\textstyle \cdot}}_{U} := \mrm{E} \mcal{O}_{U}$,
which is a residual complex on $U$. Then according to
Proposition \ref{prop4.2}
\[ \mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{O}_{\mfrak{U}} \cong
g^{*} \mrm{R} \underline{\Gamma}_{U_{0}} \mcal{O}_{U} \cong
g^{*} \underline{\Gamma}_{U_{0}} \mcal{K}^{{\textstyle \cdot}}_{U} . \]
As in Example \ref{exa5.1} this is a dualizing complex, so
$\mcal{K}^{{\textstyle \cdot}}|_{\mfrak{U}} \cong
\mrm{E} \mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{O}_{\mfrak{U}}$
is a residual complex.
\end{proof}
\begin{rem}
According to \cite{RD} Thm.\ VI.3.1,
if $f : X \rightarrow Y$ is a finite type morphism
between finite dimensional noetherian schemes, and if $\mcal{K}^{{\textstyle \cdot}}$
is a residual complex on $Y$, then there is a residual complex
$f^{\triangle} \mcal{K}^{{\textstyle \cdot}}$ on $X$. Now suppose
$f : \mfrak{X} \rightarrow \mfrak{Y}$ is a f.f.t.\ morphism and
$f_{n} : X_{n} \rightarrow Y_{n}$ are like in Lemma \ref{lem1.12}.
In the same fashion as in Prop.\ \ref{prop5.8} we set
$f^{\triangle} \mcal{K}^{{\textstyle \cdot}} := \lim_{n \rightarrow} f_{n}^{\triangle}
\mcal{K}^{{\textstyle \cdot}}_{n}$. This is a residual complex on $\mfrak{X}$.
If $f$ is formally proper then
$\operatorname{Tr}_{f} = \lim_{n \rightarrow} \operatorname{Tr}_{f_{n}}$ induces a duality
\[ \mrm{R} f_{*} \mcal{M}^{{\textstyle \cdot}} \rightarrow
\mrm{R} \mcal{H}om_{\mfrak{Y}}(\mrm{R} f_{*}
\mrm{R} \mcal{H}om_{\mfrak{X}}(\mcal{M}^{{\textstyle \cdot}},
f^{\triangle} \mcal{K}^{{\textstyle \cdot}}), \mcal{K}^{{\textstyle \cdot}}) \]
for every
$\mcal{M}^{{\textstyle \cdot}} \in \msf{D}^{\mrm{b}}(\msf{Coh}(\mfrak{X}))$.
The proofs are standard, given the results of this section.
\end{rem}
\section{Construction of the Complex $\mcal{K}^{{\textstyle \cdot}}_{X/S}$}
In this section we work over a regular noetherian base scheme $S$.
We construct the relative residue complex
$\mcal{K}^{{\textstyle \cdot}}_{X/S}$ on any finite type $S$-scheme $X$.
The construction is explicit and does not rely on \cite{RD}.
Let $A, B$ be complete local rings, with maximal ideals
$\mfrak{m}, \mfrak{n}$.
Recall that a local homomorphism $\phi : A \rightarrow B$ is called residually
finitely generated if the field extension
$A / \mfrak{m} \rightarrow B / \mfrak{n}$ is finitely generated.
Denote by $\msf{Mod}_{\mrm{disc}}(A)$ the category of $\mfrak{m}$-torsion
$A$-modules (equivalently, modules with $0$-dimensional support).
Suppose
$A \sqbr{\underline{t}} = A \sqbr{t_{1}, \ldots, t_{n}}$
is a polynomial algebra and
$\mfrak{p} \subset A \sqbr{\underline{t}}$ is some maximal ideal.
Then
$A \rightarrow B = \widehat{A \sqbr{\underline{t}}}_{\mfrak{p}}$
is formally smooth of relative dimension $n$ and residually finite.
Let $b_{i} \in B / \mfrak{n}$ be the image of $t_{i}$ and
$\bar{q}_{i} \in
(A / \mfrak{m}) \sqbr{b_{1}, \ldots, b_{i-1}}\sqbr{t_{i}}$
the monic irreducible polynomial of $b_{i}$, of degree $d_{i}$.
Choose a monic lifting
$q_{i} \in A \sqbr{t_{1}, \ldots, t_{i}}$.
Then for a discrete $A$-module $M$ one has
\[ \mrm{H}^{n}_{\mfrak{p}}
\left( \widehat{\Omega}^{n}_{B / A} \otimes_{A} M \right)
\cong \bigoplus_{1 \leq i_{l}} \ \bigoplus_{0 \leq j_{l} < d_{l}}
\gfrac{ t_{1}^{j_{1}} \cdots t_{n}^{j_{n}}
\mrm{d} t_{1} \cdots \mrm{d} t_{n} }
{ q_{1}^{i_{1}} \cdots q_{n}^{i_{n}} } \otimes M . \]
As in \cite{Hg1} Section 7 define the Tate residue
\begin{equation} \label{eqn6.6}
\operatorname{res}_{t_{1}, \ldots, t_{n}; A, B} :
\mrm{H}^{n}_{\mfrak{p}}
\left( \widehat{\Omega}^{n}_{B / A} \otimes_{A} M \right) \rightarrow M
\end{equation}
by the rule
\[ \gfrac{ t_{1}^{j_{1}} \cdots t_{n}^{j_{n}}
\mrm{d} t_{1} \cdots \mrm{d} t_{n} }
{ q_{1}^{i_{1}} \cdots q_{n}^{i_{n}} } \otimes m \mapsto
\begin{cases}
m & \text{ if } i_{l} = 1, j_{l} = d_{l} - 1 \\
0 & \text{ otherwise}
\end{cases} \]
(cf.\ \cite{Ta}).
Observe that any residually finite homomorphism $A \rightarrow C$ factors into some
$A \rightarrow B = \widehat{A \sqbr{\underline{t}}}_{\mfrak{p}} \rightarrow C$.
\begin{thm} \label{thm6.1} \textup{(Huang)}\
Consider the category $\msf{Loc}$ of complete noetherian local rings and
residually finitely generated local homomorphisms. Then:
\begin{enumerate}
\item For any morphism
$\phi : A \rightarrow B$ in $\msf{Loc}$ there is a functor
\[ \phi_{\#} : \msf{Mod}_{\mrm{disc}}(A)
\rightarrow \msf{Mod}_{\mrm{disc}}(B) . \]
For composable morphisms
$A \xrightarrow{\phi} B \xrightarrow{\psi} C$
there is an isomorphism
$(\psi \phi)_{\#} \cong \psi_{\#} \phi_{\#}$,
and
$(1_{A})_{\#} \cong 1_{\msf{Mod}_{\mrm{disc}}(A)}$.
These data form a pseudofunctor on $\msf{Loc}$ \textup{(}cf.\ \cite{Hg1}
Def.\ \textup{4.1)}.
\item If $\phi : A \rightarrow B$ is formally smooth of relative dimension $q$,
and
$n = \operatorname{rank} \widehat{\Omega}^{1}_{B / A}$,
then there is an isomorphism, functorial in
$M \in \msf{Mod}_{\mrm{disc}}(A)$,
\[ \phi_{\#} M \cong \mrm{H}^{q}_{\mfrak{n}}(
\widehat{\Omega}^{n}_{B/A} \otimes_{A} M) . \]
\item If $\phi : A \rightarrow B$ is residually finite then
there is an $A$-linear homomorphism, functorial in
$M \in \msf{Mod}_{\mrm{disc}}(A)$,
\[ \operatorname{Tr}_{\phi} : \phi_{\#} M \rightarrow M , \]
which induces an isomorphism
$\phi_{\#} M \cong \mrm{Hom}^{\mrm{cont}}_{A}(B, M)$.
For composable homomorphisms
$A \xrightarrow{\phi} B \xrightarrow{\psi} C$
one has
$\operatorname{Tr}_{\psi \phi} = \operatorname{Tr}_{\phi} \operatorname{Tr}_{\psi}$
under the isomorphism of part \textup{1}.
\item If $B = \widehat{A \sqbr{\underline{t}}}_{\mfrak{p}}$
then
$\operatorname{Tr}_{\phi} = \operatorname{res}_{t_{1}, \ldots, t_{n}; A, B}$
under the isomorphism of part \textup{2}.
\end{enumerate}
\end{thm}
\begin{proof}
Parts 1 and 2 are \cite{Hg1} Thm.\ 6.12. Parts 3 and 4
follow from \cite{Hg1} Section 7.
\end{proof}
\begin{dfn} \label{dfn6.3}
Suppose $L$ is a regular local ring of dimension $q$, with maximal
ideal $\mfrak{r}$. Given a homomorphism $\phi : L \rightarrow A$
in $\msf{Loc}$, define
\[ \mcal{K}(A / L) := \phi_{\#} \mrm{H}^{q}_{\mfrak{r}} L , \]
the {\em dual module of $A$ relative to $L$}.
\end{dfn}
Since $\mrm{H}^{q}_{\mfrak{r}} L$ is an injective hull of the field
$L / \mfrak{r}$, it follows that $\mcal{K}(A / L)$ is an injective hull
of $A / \mfrak{m}$ (cf.\ \cite{Hg1} Corollary 3.10).
\begin{cor} \label{cor6.1}
If $\psi : A \rightarrow B$ is a residually finite homomorphism, then there
is an $A$-linear homomorphism
\[ \operatorname{Tr}_{\psi} = \operatorname{Tr}_{B / A} : \mcal{K}(B / L) \rightarrow
\mcal{K}(A / L) . \]
Given another such homomorphism $B \rightarrow C$, one has
$ \operatorname{Tr}_{C / A} = \operatorname{Tr}_{B / A} \operatorname{Tr}_{C / B}$.
\end{cor}
\begin{rem} \label{rem6.1}
One can show that when $L$ is a perfect field,
there is a functorial isomorphism between
$\mcal{K}(A / L) = \phi_{\#} L$ above and the dual module
$\mcal{K}(A)$ of \cite{Ye2}, which was defined via Beilinson completion
algebras.
\end{rem}
Suppose $\pi : \mfrak{X} \rightarrow S$ is a formally finite type (f.f.t.)
formally smooth morphism. According to Proposition \ref{prop1.4},
$\mfrak{X}$ is a regular formal scheme. When we write
$n = \operatorname{rank} \widehat{\Omega}^{1}_{\mfrak{X}/S}$
we mean that $n$ is a locally constant function
$n : \mfrak{X} \rightarrow \mbb{N}$.
\begin{lem} \label{lem6.1}
Given a f.f.t.\ morphism $\pi : \mfrak{X} \rightarrow S$ and a point
$x \in \mfrak{X}$, let $s := \pi(x)$, and define
\[ d_{S}(x) := \operatorname{dim} \widehat{\mcal{O}}_{S, s} -
\operatorname{tr.deg}_{k(s)} k(x) . \]
Then:
\begin{enumerate}
\item $d_{S}$ is a codimension function.
\item If $\pi$ is formally smooth then
\[ d_{S}(x) = \operatorname{dim} \widehat{\mcal{O}}_{\mfrak{X}, x} -
\operatorname{rank} \widehat{\Omega}^{1}_{\mfrak{X}/S} . \]
\end{enumerate}
\end{lem}
\begin{proof}
We shall prove 2 first.
Let $L := \widehat{\mcal{O}}_{S, s}$
and
$A := \widehat{\mcal{O}}_{\mfrak{X}, x}$.
By Prop.\ \ref{prop1.4},
\[ \operatorname{rank} \widehat{\Omega}^{1}_{A/L} =
\operatorname{dim} A - \operatorname{dim} L + \operatorname{tr.deg}_{L / \mfrak{r}} A / \mfrak{m} . \]
We see that $d_{S}$ is the codimension function associated with the
dualizing complex
$\mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{O}_{\mfrak{X}}[n]$
(see Theorem \ref{thm5.3}).
As for 1, the property of being a codimension function is local.
But locally there is always a closed immersion
$\mfrak{X} \subset \mfrak{Y}$ with $\mfrak{Y} \rightarrow S$ formally smooth.
\end{proof}
We shall use the codimension function $d_{S}$ by default.
\begin{dfn} \label{dfn6.1}
Let $\pi : \mfrak{X} \rightarrow S$ be a formally finite type morphism. Given a
point $x \in \mfrak{X}$, consider
$\phi : L = \widehat{\mcal{O}}_{S ,\pi(x)} \rightarrow
A = \widehat{\mcal{O}}_{\mfrak{X}, x}$,
which is a morphism in $\msf{Loc}$.
Since $L$ is a regular local ring, the dual module $\mcal{K}(A / L)$
is defined.
Let $\mcal{K}_{\mfrak{X} / S}(x)$ be the quasi-coherent sheaf
which is constant on
$\overline{\{x\}}$ with group of sections $\mcal{K}(A / L)$,
and define
\[ \mcal{K}_{\mfrak{X} / S}^{q} :=
\bigoplus_{d_{S}(x) = q} \mcal{K}_{\mfrak{X} / S}(x) . \]
\end{dfn}
In Theorem \ref{thm6.2} we are going to prove that on
the graded sheaf $\mcal{K}_{X / S}^{{\textstyle \cdot}}$
there is a canonical coboundary operator $\delta$ which makes it into
residual complex.
\begin{dfn} \label{dfn6.2}
Let $f : \mfrak{X} \rightarrow \mfrak{Y}$ be a morphism of formal schemes over $S$.
Define a homomorphism of graded $\mcal{O}_{\mfrak{Y}}$-modules
$\operatorname{Tr}_{f} : f_{*} \mcal{K}_{\mfrak{X} / S}^{{\textstyle \cdot}} \rightarrow
\mcal{K}_{\mfrak{Y} / S}^{{\textstyle \cdot}}$
as follows. If $x \in \mfrak{X}$ is closed in its fiber and $y = f(x)$,
then
$A = \widehat{\mcal{O}}_{\mfrak{Y}, y} \rightarrow
B = \widehat{\mcal{O}}_{\mfrak{X}, x}$
is a residually finite $L$-algebra homomorphism.
The homomorphism
$\operatorname{Tr}_{B / A} : \mcal{K}(B / L) \rightarrow \mcal{K}(A / L)$
of Cor.\ \ref{cor6.1} gives a map of sheaves
\[ \operatorname{Tr}_{f} : f_{*} \mcal{K}_{\mfrak{X} / S}(x) \rightarrow
\mcal{K}_{\mfrak{Y} / S}(y) . \]
If $x$ is not closed in its fiber, we let $\operatorname{Tr}_{f}$ vanish on
$f_{*} \mcal{K}_{\mfrak{X} / S}(x)$.
\end{dfn}
\begin{prop} \label{prop6.3}
\begin{enumerate}
\item $\operatorname{Tr}_{f}$ is functorial: if $g : \mfrak{Y} \rightarrow \mfrak{Z}$
is another morphism, then
$\operatorname{Tr}_{gf} = \operatorname{Tr}_{g} \operatorname{Tr}_{f}$.
\item If $f$ is formally finite \textup{(}see Def.\
\textup{\ref{dfn1.1})},
then $\operatorname{Tr}_{f}$ induces an isomorphism
of graded sheaves
\[ f_{*} \mcal{K}_{\mfrak{X} / S}^{{\textstyle \cdot}} \cong
\mcal{H}om_{\mfrak{Y}}(f_{*} \mcal{O}_{\mfrak{X}},
\mcal{K}_{\mfrak{Y} / S}^{{\textstyle \cdot}}) . \]
\item If $g : \mfrak{U} \rightarrow \mfrak{X}$ is an open immersion, then
there is a natural isomorphism
$\mcal{K}_{\mfrak{U} / S}^{{\textstyle \cdot}} \cong
g^{*} \mcal{K}_{\mfrak{X} / S}^{{\textstyle \cdot}}$.
\end{enumerate}
\end{prop}
\begin{proof}
Part 3 is trivial. Part 1 is a consequence of Cor.\ \ref{cor6.1}.
As for part 2, $f$ is an affine morphism, and fibers of $f$ are all
finite, so all points of $X$ are closed in their fibers.
\end{proof}
Suppose $\underline{a} = (a_{1}, \ldots, a_{n})$ is a sequence of elements
in the noetherian ring $A$.
Let us write $\tilde{\mbf{K}}^{{\textstyle \cdot}}_{\infty}(\underline{a})$
for the subcomplex $\mbf{K}^{\geq 1}_{\infty}(\underline{a})$, so we get an
exact sequence
\begin{equation}
0 \rightarrow \tilde{\mbf{K}}^{{\textstyle \cdot}}_{\infty}(\underline{a}) \rightarrow
\mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}) \rightarrow A \rightarrow 0 .
\end{equation}
For any
$M^{{\textstyle \cdot}} \in \msf{D}^{+}(\msf{Mod}(A))$
let $\mcal{M}^{{\textstyle \cdot}}$ be the complex of
sheaves $\mcal{O}_{X} \otimes M^{{\textstyle \cdot}}$ on
$X := \operatorname{Spec} A$, and let $U \subset X$ be the open set
$\bigcup \{ a_{i} \neq 0 \}$. Then
\[ \mrm{R} \Gamma(U, \mcal{M}^{{\textstyle \cdot}}) \cong
\tilde{\mbf{K}}^{{\textstyle \cdot}}_{\infty}(\underline{a})[1] \otimes M^{{\textstyle \cdot}} \]
in $\msf{D}(\msf{Mod}(A))$. In fact
$\tilde{\mbf{K}}^{{\textstyle \cdot}}_{\infty}(\underline{a}) \otimes \mcal{O}_{X}$
is a shift by $1$ of the \v{C}ech complex corresponding to the open
cover of $U$.
\begin{lem} \label{lem6.2}
Let $A$ be an adic noetherian ring and
$M^{{\textstyle \cdot}} \in \msf{D}^{+}(\msf{Mod}(A))$.
Define $\mfrak{U} := \operatorname{Spf} A$ and
$\mcal{M}^{{\textstyle \cdot}} := \mcal{O}_{\mfrak{U}} \otimes M^{{\textstyle \cdot}}$.
\begin{enumerate}
\item Let $x \in \mfrak{U}$ with corresponding open prime ideal
$\mfrak{p} \subset A$. Suppose the sequence
$\underline{a}$ generates $\mfrak{p}$. Then
\[ \mrm{R} \Gamma_{x} \mrm{R} \underline{\Gamma}_{\mrm{disc}}
\mcal{M}^{{\textstyle \cdot}} \cong
\mrm{R} \Gamma_{\mfrak{p}} M^{{\textstyle \cdot}}_{\mfrak{p}} \cong
\mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}) \otimes M^{{\textstyle \cdot}}_{\mfrak{p}} \]
in $\msf{D}^{+}(\msf{Mod}(A_{\mfrak{p}})).$
\item Suppose $y \in \mfrak{U}$ is an immediate specialization of $x$,
and its ideal $\mfrak{q}$ has generators $\underline{a}, \underline{b}$. Then
\[ \mrm{R} \Gamma_{x} \mrm{R} \underline{\Gamma}_{\mrm{disc}}
\mcal{M}^{{\textstyle \cdot}} \cong
\mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}) \otimes
\tilde{\mbf{K}}^{{\textstyle \cdot}}_{\infty}(\underline{b})[1] \otimes
M^{{\textstyle \cdot}}_{\mfrak{q}} \]
in $\msf{D}^{+}(\msf{Mod}(A_{\mfrak{q}})).$
\item Assume $d$ is a codimension function on $\mfrak{U}$.
Then in the Cousin complex
$\mrm{E} \mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{M}^{{\textstyle \cdot}}$
the map
\[ \mrm{H}^{d(x)}_{x} \mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{M}^{{\textstyle \cdot}}
\rightarrow \mrm{H}^{d(y)}_{y} \mrm{R} \underline{\Gamma}_{\mrm{disc}} \mcal{M}^{{\textstyle \cdot}} \]
is given by applying $\mrm{H}^{d(y)}$ to
\[ \left( \mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}) \otimes
\tilde{\mbf{K}}^{{\textstyle \cdot}}_{\infty}(\underline{b}) \rightarrow
\mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}, \underline{b}) \right) \otimes
M^{{\textstyle \cdot}}_{\mfrak{q}} . \]
\end{enumerate}
\end{lem}
\begin{proof}
Part 1 follows immediately from formula (\ref{eqn4.1}).
Parts 2 and 3 are true because
$\operatorname{Spec} (A / \mfrak{p})_{\mfrak{q}} =
\{ \mfrak{p}, \mfrak{q} \}$.
\end{proof}
As a warm up for Thm.\ \ref{thm6.2}, here is:
\begin{prop} \label{prop6.4}
If $\pi : \mfrak{X} \rightarrow S$ is formally smooth, with
$n = \operatorname{rank} \widehat{\Omega}^{1}_{\mfrak{X} / S}$,
then there is a canonical isomorphism of graded sheaves
\[ \mcal{K}_{\mfrak{X} / S}^{{\textstyle \cdot}} \cong
\mrm{E} \mrm{R} \underline{\Gamma}_{\mrm{disc}}
\widehat{\Omega}^{n}_{\mfrak{X} / S}[n] . \]
This makes $\mcal{K}_{\mfrak{X} / S}^{{\textstyle \cdot}}$ into a residual complex.
\end{prop}
\begin{proof}
Take any point $x$, and with the notation of Def.\
\ref{dfn6.1} let $p := \operatorname{dim} L$ and $q := \operatorname{dim} A$.
Then by Lemma \ref{lem6.2} part 1 and \cite{Hg1} Proposition 2.6
we have a canonical isomorphism
\[ \mrm{H}^{d(x)}_{x}
\mrm{R} \underline{\Gamma}_{\mrm{disc}} \widehat{\Omega}^{n}_{\mfrak{X} / S}[n]
\cong
\mrm{H}^{q}_{\mfrak{m}} \widehat{\Omega}^{n}_{A / L} \cong
\mrm{H}^{q-p}_{\mfrak{m}} \left( \widehat{\Omega}^{n}_{A / L}
\otimes_{L} \mrm{H}^{p}_{\mfrak{r}} L \right)
\cong \mcal{K}(A / L) . \]
According to Theorem \ref{thm5.3} and Proposition \ref{prop5.7},
$\mrm{E} \mrm{R} \underline{\Gamma}_{\mrm{disc}}
\widehat{\Omega}^{n}_{\mfrak{X} / S}[n]$
is a residual complex.
\end{proof}
In particular taking $\mfrak{X} = S$ we get
$\mcal{K}_{S / S}^{{\textstyle \cdot}} = \mrm{E} \mcal{O}_{S}$.
\begin{lem} \label{lem6.3}
Suppose $X \subset \mfrak{X}$ and $X \subset \mfrak{Y}$ are
s.f.e.'s and $f : \mfrak{X} \rightarrow \mfrak{Y}$ is a morphism of
embeddings.
Then
$\operatorname{Tr}_{f} : \mcal{K}^{{\textstyle \cdot}}_{\mfrak{X}} \rightarrow
\mcal{K}^{{\textstyle \cdot}}_{\mfrak{Y}}$
is a homomorphism of complexes.
\end{lem}
\begin{proof}
Factoring $f$ through $(\mfrak{X} \times_{S} \mfrak{Y})_{/ X}$
we can assume that $f$ is either a closed immersion, or that it is
formally smooth. At any rate $f$ is an affine morphism, so we can take
$\mfrak{X} = \operatorname{Spf} B$, $\mfrak{Y} = \operatorname{Spf} A$ and
$S = \operatorname{Spec} L$.
By Theorem \ref{thm2.2} we can suppose one of the following holds:
(i) $B \cong A [\sqbr{\underline{t}}]$ for a sequence of indeterminates
$\underline{t} = (t_{1}, \ldots, t_{l})$, and $A \rightarrow B$ is the inclusion;
or
(ii) $A \cong B [\sqbr{\underline{t}}]$ and $A \rightarrow B$ is the projection modulo
$\underline{t}$. We shall treat each case separately.
\noindent (i)\
Choose generators $\underline{a}$ for a defining ideal of $A$. Let
$m := \operatorname{rank} \widehat{\Omega}^{1}_{A / L}$
and
$n := \operatorname{rank} \widehat{\Omega}^{1}_{B / L}$,
so $n = m + l$. Define an $A$-linear map
$\rho : \mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{t}) \otimes
\widehat{\Omega}^{l}_{B / A}[l] \rightarrow A$
by
$\rho(\underline{t}^{(-1, \ldots, -1)} \mrm{d} \underline{t}) = 1$
and
$\rho(\underline{t}^{\underline{i}}\, \mrm{d} \underline{t}) = 0$ if
$\underline{i} \neq (-1, \ldots, -1)$.
Extend $\rho$ linearly to
\[ \rho : \mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}, \underline{t}) \otimes
\widehat{\Omega}^{n}_{B / L}[n] \rightarrow
\mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}) \otimes
\widehat{\Omega}^{m}_{A / L}[m] . \]
This $\rho$ sheafifies to give a map of complexes in $\msf{Ab}(X)$
\[ \tilde{\rho} :
\mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}, \underline{t}) \otimes
\widehat{\Omega}^{n}_{\mfrak{X} / S}[n] \rightarrow
\mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}) \otimes
\widehat{\Omega}^{m}_{\mfrak{Y} / S}[m] . \]
By Lemma \ref{lem6.2} and \cite{Hg1} \S 5, for any point $x \in X$,
$\mrm{H}^{d(x)}_{x}(\tilde{\rho})$ recovers
$\operatorname{Tr}_{f} : \mcal{K}_{\mfrak{X} / S}(x) \rightarrow
\mcal{K}_{\mfrak{Y} / S}(x)$.
Thus $\operatorname{Tr}_{f} = \mrm{E}(\tilde{\rho})$ is a homomorphism of complexes.
\noindent (ii)\
Now $l = m - n$. Take $\underline{a}$ to be generators of a defining ideal of
$B$. Define a $B$-linear map
$\rho' : B \rightarrow \mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{t}) \otimes
\widehat{\Omega}^{l}_{A / L}[l]$
by
$\rho'(1) = \underline{t}^{(-1, \ldots, -1)} \mrm{d} \underline{t}$.
Extend $\rho$ linearly to
\[ \rho' : \mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}) \otimes
\widehat{\Omega}^{n}_{B / L}[n] \rightarrow
\mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}, \underline{t}) \otimes
\widehat{\Omega}^{m}_{A / L}[m] . \]
Again this extends to a map of complexes of sheaves
$\tilde{\rho}'$ in $\msf{Ab}(X)$,
and checking punctually we see that
$\operatorname{Tr}_{f} = \mrm{E}(\tilde{\rho}')$.
\end{proof}
\begin{thm} \label{thm6.2}
Suppose $X \rightarrow S$ is a finite type morphism.
There is a unique operator
$\delta : \mcal{K}_{X / S}^{q} \rightarrow \mcal{K}_{X / S}^{q+1}$,
satisfying the following local condition:\\[2mm]
\textup{\bf (LE)}\ \blnk{4mm}
\begin{minipage}{11cm}
Suppose $U \subset X$ is
an open subset, and $U \subset \mfrak{U}$ is a smooth formal embedding.
By Proposition \textup{\ref{prop6.3}} there is an inclusion of graded
$\mcal{O}_{U}$-modules
$\mcal{K}_{X / S}^{{\textstyle \cdot}}|_{U} \subset
\mcal{K}_{\mfrak{U} / S}^{{\textstyle \cdot}}$.
Then $\delta|_{U}$ is compatible with the coboundary operator on
$\mcal{K}_{\mfrak{U} / S}^{{\textstyle \cdot}}$ coming from Proposition
\textup{\ref{prop6.4}}.
\end{minipage}\\[2mm]
Moreover $(\mcal{K}_{X / S}^{{\textstyle \cdot}}, \delta)$
is a residual complex on $X$.
\end{thm}
\begin{proof}
Define $\delta|_{U}$ using {\bf LE}. According to Lemma \ref{lem6.3},
$\delta|_{U}$ is independent of $\mfrak{U}$, so it glues.
We get a bounded complex of quasi-coherent injectives on $X$.
By Proposition \ref{prop6.4} it follows that it is residual.
\end{proof}
\begin{rem}
This construction of $\mcal{K}_{X / S}^{{\textstyle \cdot}}$ actually allows
a computation of the operator $\delta$, given the data of a local
embedding. The formula is in part 3 of Lemma \ref{lem6.2},
with $M^{{\textstyle \cdot}} = \widehat{\Omega}^{n}_{A / L}[n]$.
The formula for changing the embedding can be extracted from the
proof of Lemma \ref{lem6.3}. Of course when
$\operatorname{rank} \widehat{\Omega}^{1}_{\mfrak{X} / S}$ is high
these computations can be nasty.
\end{rem}
\begin{rem} \label{rem6.6}
The recent papers \cite{Hg2}, \cite{Hg3} and \cite{LS2} also
use the local theory of \cite{Hg1} as a starting point for explicit
constructions of Grothendieck Duality. Their constructions are more
general than ours: Huang constructs $f^{!} \mcal{M}^{{\textstyle \cdot}}$ for a finite
type morphism $f : X \rightarrow Y$ and a residual complex complex
$\mcal{M}^{{\textstyle \cdot}}$; and Lipman-Sastry even allow $\mcal{M}^{{\textstyle \cdot}}$
to be any Cousin complex.
\end{rem}
\section{The Trace for Finite Morphisms}
In this section we prove that
$\operatorname{Tr}_{f}$ is a homomorphism of complexes when $f$ is a finite
morphism. The proof is by a self contained calculation involving
Koszul complexes and a comparison of global and local Tate residue maps.
In Theorem \ref{thm7.4} we compare the complex
$\mcal{K}_{X / S}^{{\textstyle \cdot}}$ to the
sheaf of regular differentials of Kunz-Waldi.
Throughout $S$ is a regular noetherian scheme.
\begin{thm} \label{thm7.6}
Suppose $f : X \rightarrow Y$ is finite. Then
$\operatorname{Tr}_{f} : f_{*} \mcal{K}_{X / S}^{{\textstyle \cdot}} \rightarrow
\mcal{K}_{Y / S}^{{\textstyle \cdot}}$
is a homomorphism of complexes.
\end{thm}
The proof appears after some preparatory work, based on and
inspired by \cite{Hg1} \S7.
\begin{rem} \label{rem7.1}
In Section 8 we prove a much stronger result, namely Corollay \ref{cor8.1},
but its proof is indirect and relies on the Residue Theorem of \cite{RD}
Chapter VII. We have decided to include Theorem \ref{thm7.6} because of
its direct algebraic proof.
\end{rem}
Let $A$ be an adic noetherian ring with defining ideal $\mfrak{a}$.
Suppose $p \in A \sqbr{t}$ is a monic polynomial of degree $e > 0$. Define
an $A$-algebra
\begin{equation} \label{eqn7.3}
B := \lim_{\leftarrow i} A \sqbr{t} / A \sqbr{t} \cdot p^{i} .
\end{equation}
Let
$\mfrak{b} :=B \mfrak{a} + B p$; then
$B \cong \lim_{\leftarrow i} B / \mfrak{b}^{i}$, so that $B$ is an adic
ring with the $\mfrak{b}$-adic topology.
The homomorphism $\phi : A \rightarrow B$ is f.f.t.\ and formally smooth, and
$\widehat{\Omega}^{1}_{B / A} = B \cdot \mrm{d} t$.
Furthermore $p \in B$ is a non-zero-divisor,
and by long division we obtain an isomorphism
\begin{equation} \label{eqn7.2}
\mrm{H}^{1}_{(p)} B =
\mrm{H}^{1} \left(\mbf{K}^{{\textstyle \cdot}}_{\infty}(p) \otimes B \right) \cong
\bigoplus_{1 \leq i}\ \bigoplus_{0 \leq j < e} A \cdot
\gfrac{t^{j}}{p^{i}} .
\end{equation}
Define an $A$-linear homomorphism
$\operatorname{Res}_{B / A} : \mrm{H}^{1}_{(p)} \widehat{\Omega}^{1}_{B / A}
\rightarrow A$
by
\[ \operatorname{Res}_{B / A} \left( \gfrac{t^{j} \mrm{d}t}{p^{i}} \right) :=
\begin{cases}
1 & \text{ if } i=1, j=e-1 \\
0 & \text{ otherwise} .
\end{cases} \]
We call $\operatorname{Res}_{B / A}$ the {\em global Tate residue}. It
gives rise to a map of complexes in $\msf{Mod}(A)$:
\begin{equation} \label{eqn7.9}
\operatorname{Res}_{B / A} : \mbf{K}^{{\textstyle \cdot}}_{\infty}(p)[1] \otimes
\widehat{\Omega}^{1}_{B / A} \rightarrow A .
\end{equation}
Note that both the algebra $B$ and the map $\operatorname{Res}_{B / A}$
depend on $t$ and $p$.
Suppose $\mfrak{q} \subset B$ is an open prime ideal and
$\mfrak{p} = \phi^{-1}(\mfrak{q}) \subset A$. Then the local homomorphism
$\phi_{\mfrak{q}} : \widehat{A}_{\mfrak{p}} \rightarrow
\widehat{B}_{\mfrak{q}}$
is formally smooth of relative dimension $1$ and residually finite.
Let
$\tilde{\mfrak{q}} := \mfrak{q} \cap \widehat{A}_{\mfrak{p}} \sqbr{t}$,
and denote by $\bar{\mfrak{q}}$ the image of $\tilde{\mfrak{q}}$
in $k(\mfrak{p}) \sqbr{t}$, so
$k(\mfrak{p}) \sqbr{t} / \bar{\mfrak{q}} = k(\mfrak{q})$.
For a polynomial $q \in \widehat{A}_{\mfrak{p}} \sqbr{t}$
let $\bar{q}$ be its image in $k(\mfrak{p}) \sqbr{t}$.
Suppose $q$ satisfies:
\begin{equation} \label{eqn7.5}
q \text{ is monic, and the ideal }
(\bar{q}) \subset k(\mfrak{p}) \sqbr{t} \text{ is }
\bar{\mfrak{q}}\text{-primary.}
\end{equation}
Then
$\widehat{B}_{\mfrak{q}} \cdot \mfrak{q} =
\sqrt{\widehat{B}_{\mfrak{q}} \cdot (\mfrak{p}, q)} \subset
\widehat{B}_{\mfrak{q}}$,
and
\[ \widehat{B}_{\mfrak{q}} \cong
\lim_{\leftarrow i} \widehat{A}_{\mfrak{p}} \sqbr{t} / \tilde{\mfrak{q}}^{i}
\cong \lim_{\leftarrow i}
\widehat{A}_{\mfrak{p}} \sqbr{t} / \widehat{A}_{\mfrak{p}} \sqbr{t}
\cdot q^{i} . \]
Hence $q$ is a non-zero-divisor in $\widehat{B}_{\mfrak{q}}$ and
$\widehat{B}_{\mfrak{q}} / \widehat{B}_{\mfrak{q}} \cdot q$ is a free
$\widehat{A}_{\mfrak{p}}$-module with basis
$1, t, \ldots, t^{d-1}$, where $d = \operatorname{deg} q$. We see that
a decomposition like (\ref{eqn7.2}) exists for
$\mrm{H}^{1}_{(q)} \widehat{B}_{\mfrak{q}}$.
Suppose we are given a discrete $\widehat{A}_{\mfrak{p}}$-module $M$.
Then one gets
\[ \mrm{H}^{1}_{\mfrak{q}} \left(
\widehat{\Omega}^{1}_{\widehat{B}_{\mfrak{q}} / \widehat{A}_{\mfrak{p}}}
\otimes_{\widehat{A}_{\mfrak{p}}} M \right) \cong
\left( \mrm{H}^{1}_{(q)}
\widehat{\Omega}^{1}_{\widehat{B}_{\mfrak{q}} / \widehat{A}_{\mfrak{p}}}
\right) \otimes_{\widehat{A}_{\mfrak{p}}} M
\cong
\bigoplus_{1 \leq i}\ \bigoplus_{0 \leq j < d}
\gfrac{t^{j} \mrm{d} t}{q^{i}} \otimes M \]
(cf.\ \cite{Hg1} pp.\ 41-42). Define the {\em local Tate residue map}
\[ \operatorname{Res}_{\widehat{B}_{\mfrak{q}} / \widehat{A}_{\mfrak{p}}} :
\mrm{H}^{1}_{\mfrak{q}} \left(
\widehat{\Omega}^{1}_{\widehat{B}_{\mfrak{q}} / \widehat{A}_{\mfrak{p}}}
\otimes_{\widehat{A}_{\mfrak{p}}} M \right) \rightarrow M \]
by
\[ \operatorname{Res}_{\widehat{B}_{\mfrak{q}} / \widehat{A}_{\mfrak{p}}}
\left( \gfrac{t^{j} \mrm{d}t \otimes m}{q^{i}}
\right) :=
\begin{cases}
m & \text{ if } i=1, j=d-1 \\
0 & \text{ otherwise} .
\end{cases} \]
Clearly $\operatorname{Res}_{\widehat{B}_{\mfrak{q}} / \widehat{A}_{\mfrak{p}}}$
is functorial in $M$, and it depends on $t$.
\begin{lem} \label{lem7.7}
$\operatorname{Res}_{\widehat{B}_{\mfrak{q}} / \widehat{A}_{\mfrak{p}}}$
is independent of $q$. It coincides with the residue map
$\operatorname{res}_{t; \widehat{B}_{\mfrak{q}} / \widehat{A}_{\mfrak{p}}}$
of \textup{(\ref{eqn6.6})},
i.e.\ of \cite{Hg1} Definition \textup{8.1}.
\end{lem}
\begin{proof}
Suppose the polynomials
$q_{1}, q_{2} \in \widehat{A}_{\mfrak{p}} \sqbr{t}$
satisfy (\ref{eqn7.5}). Then so does $q_{3} := q_{1} q_{2}$.
Let $\operatorname{deg} q_{h} = d_{h}$, and let
$\operatorname{Res}_{\widehat{B}_{\mfrak{q}} / \widehat{A}_{\mfrak{p}}; q_{h}}$
be the residue map determined by $q_{h}$.
Pick any $1 \leq i$ and $0 \leq j < d_{1}$, and write
$q_{2}^{i} = \sum_{l = 0}^{i d_{2}} a_{l} t^{l}$, so $a_{i d_{2}} = 1$.
By the rules for manipulating generalized fractions (cf.\ \cite{Hg1}
\S 1) we have
\begin{equation} \label{eqn7.1}
\operatorname{Res}_{\widehat{B}_{\mfrak{q}} / \widehat{A}_{\mfrak{p}}; q_{3}}
\left( \gfrac{t^{j} \mrm{d} t \otimes m}{q_{1}^{i}} \right) =
\sum_{l = 0}^{i d_{2}}
\operatorname{Res}_{\widehat{B}_{\mfrak{q}} / \widehat{A}_{\mfrak{p}}; q_{3}}
\left(
\gfrac{t^{l + j} \mrm{d} t \otimes a_{l} m}{q_{3}^{i}} \right) .
\end{equation}
If $i \geq 2$ or $j \leq d_{1} - 2$ one has
$l + j \leq i d_{3} -2$,
and therefore each summand of the right side of (\ref{eqn7.1}) is $0$.
When $i = 1$ and $j = d_{1} - 1$ the only possible nonzero
residue there is for $l = d_{2}$, and this residue is $m$.
We conclude that
$\operatorname{Res}_{\widehat{B}_{\mfrak{q}} / \widehat{A}_{\mfrak{p}}; q_{3}} =
\operatorname{Res}_{\widehat{B}_{\mfrak{q}} / \widehat{A}_{\mfrak{p}}; q_{1}}$.
Clearly also
$\operatorname{Res}_{\widehat{B}_{\mfrak{q}} / \widehat{A}_{\mfrak{p}}; q_{3}} =
\operatorname{Res}_{\widehat{B}_{\mfrak{q}} / \widehat{A}_{\mfrak{p}}; q_{2}}$.
If we take $q$ such that $(\bar{q}) = \bar{\mfrak{q}}$, this is by
definition the residue map of (\ref{eqn6.6}).
\end{proof}
\begin{lem} \label{lem7.8}
Let $F$ be the set of prime ideals in $B / (p)$ lying over
$\mfrak{p}$. Then for any
$M \in \msf{Mod}_{\mrm{disc}}(\widehat{A}_{\mfrak{p}})$ one has
\[ \left( \mrm{H}^{1}_{(p)} \widehat{\Omega}^{1}_{B / A} \right)
\otimes_{A} M
\cong
\bigoplus_{\mfrak{q}' \in F}
\mrm{H}^{1}_{\mfrak{q}'} \left(
\widehat{\Omega}^{1}_{\widehat{B}_{\mfrak{q}'} / \widehat{A}_{\mfrak{p}}}
\otimes_{\widehat{A}_{\mfrak{p}}} M \right) , \]
and w.r.t.\ this isomorphism,
\[ \operatorname{Res}_{B / A} \otimes 1 = \sum_{\mfrak{q}' \in F}
\operatorname{Res}_{\widehat{B}_{\mfrak{q}'} / \widehat{A}_{\mfrak{p}}} . \]
\end{lem}
\begin{proof}
The isomorphism of modules is not hard to see. Let
$\bar{p} = \prod_{\mfrak{q}' \in F} \bar{p}_{\mfrak{q}'}$
be the primary decomposition in $k(\mfrak{p}) \sqbr{t}$
(all the $\bar{p}_{\mfrak{q}'}$ monic). By Hensel's Lemma this
decomposition lifts to
$p = \prod_{\mfrak{q}' \in F} p_{\mfrak{q}'}$
in $\widehat{A}_{\mfrak{p}} \sqbr{t}$.
Since each polynomial $p_{\mfrak{q}'}$ satisfies condition (\ref{eqn7.5})
for the prime ideal $\mfrak{q}'$, we can use it to calculate
$\operatorname{Res}_{\widehat{B}_{\mfrak{q}'} / \widehat{A}_{\mfrak{p}}}$.
\end{proof}
\begin{proof} (of Thm.\ \ref{thm7.6})\
This claim is local on $Y$, so we may assume $X$, $Y$ and $S$ are affine,
say $X = \operatorname{Spec} \bar{B}$, $Y = \operatorname{Spec} \bar{A}$ and
$S = \operatorname{Spec} L$.
By the functoriality of $\operatorname{Tr}$ we can assume
$\bar{B} = \bar{A} \sqbr{b}$
for some element $b \in \bar{B}$. It will suffice to find suitable
s.f.e.'s $X \subset \mfrak{X}$ and $Y \subset \mfrak{Y}$
with a morphism
$\widehat{f} : \mfrak{X} \rightarrow \mfrak{Y}$
extending $f$, and to check that
$\operatorname{Tr}_{\widehat{f}} : \widehat{f}_{*}
\mcal{K}^{{\textstyle \cdot}}_{\mfrak{X} / S} \rightarrow \mcal{K}^{{\textstyle \cdot}}_{\mfrak{Y} / S}$
commutes with $\delta$.
Pick any s.f.e.\
$Y \subset \mfrak{Y} = \operatorname{Spf} A$, so
$\mfrak{a} := \operatorname{Ker}(A \rightarrow \bar{A})$ is a defining ideal.
Let $A \sqbr{t} \rightarrow \bar{B}$ be the homomorphism $t \mapsto b$.
Choose any monic polynomial $p(t) \in A \sqbr{t}$ s.t.\ $p(b) = 0$, and
define the adic ring $B$ as in formula (\ref{eqn7.3}). So
$\mfrak{X} := \operatorname{Spf} B$ is the s.f.e.\ of $X$ we want.
Let $(y_{0}, y_{1})$ be an immediate specialization pair in $Y$, and let
$F_{i} := f^{-1}(y_{i}) \subset X$.
Let $\mfrak{p}_{0} \subset \mfrak{p}_{1} \subset A$ be the prime ideals
corresponding to $(y_{0}, y_{1})$.
Pick a sequence of generators $\underline{a}$ for $\mfrak{p}_{0}$, and
generators $(\underline{a}, \underline{a}')$ for $\mfrak{p}_{1}$.
Let $m := \operatorname{rank} \widehat{\Omega}^{1}_{A / L}$.
Consider the commutative diagram of complexes
\[ \begin{CD}
\mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}, p)[1] \otimes
\tilde{\mbf{K}}^{{\textstyle \cdot}}_{\infty}(\underline{a}') \otimes
(\widehat{\Omega}^{m + 1}_{B / L})_{\mfrak{p}_{1}}
@> \operatorname{Res}_{B / A} \otimes 1 >>
\mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}) \otimes
\tilde{\mbf{K}}^{{\textstyle \cdot}}_{\infty}(\underline{a}') \otimes
(\widehat{\Omega}^{m}_{A / L})_{\mfrak{p}_{1}} \\
@VVV @VVV \\
\mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}, \underline{a}', p)[1] \otimes
(\widehat{\Omega}^{m + 1}_{B / L})_{\mfrak{p}_{1}}
@> \operatorname{Res}_{B / A} \otimes 1 >>
\mbf{K}^{{\textstyle \cdot}}_{\infty}(\underline{a}, \underline{a}') \otimes
(\widehat{\Omega}^{m}_{A / L})_{\mfrak{p}_{1}}
\end{CD} \]
gotten from tensoring the map $\operatorname{Res}_{B / A}$ of (\ref{eqn7.9})
with
$A_{\mfrak{p}_{1}} \otimes \widehat{\Omega}^{m}_{A / L}$
and the various $\mbf{K}^{{\textstyle \cdot}}_{\infty}$.
Applying $\mrm{H}^{i}$ to this diagram, where
$i := \operatorname{dim} \widehat{A}_{\mfrak{p}_{1}}$,
and using Lemmas \ref{lem6.2} and \ref{lem7.8} we obtain a
commutative diagram
\[ \begin{CD}
\bigoplus_{\mfrak{q}_{0} \in F_{0}}
\mrm{H}^{1}_{\mfrak{q}_{0}} \left( \widehat{\Omega}^{1}_{
\widehat{B}_{\mfrak{q}_{0}} / \widehat{A}_{\mfrak{p}_{0}}}
\otimes
\mrm{H}^{i - 1}_{\mfrak{p}_{0}} \widehat{\Omega}^{m}_{
\widehat{A}_{\mfrak{p}_{0}} / L} \right)
@> \sum \operatorname{Res} >>
\mrm{H}^{i - 1}_{\mfrak{p}_{0}} \widehat{\Omega}^{m}_{
\widehat{A}_{\mfrak{p}_{0}} / L} \\
@VVV @VVV \\
\bigoplus_{\mfrak{q}_{1} \in F_{1}}
\mrm{H}^{1}_{\mfrak{q}_{1}} \left( \widehat{\Omega}^{1}_{
\widehat{B}_{\mfrak{q}_{1}} / \widehat{A}_{\mfrak{p}_{1}}}
\otimes
\mrm{H}^{i}_{\mfrak{p}_{1}} \widehat{\Omega}^{m}_{
\widehat{A}_{\mfrak{p}_{1}} / L} \right)
@> \sum \operatorname{Res} >>
\mrm{H}^{i}_{\mfrak{p}_{1}} \widehat{\Omega}^{m}_{
\widehat{A}_{\mfrak{p}_{1}} / L} .
\end{CD} \]
In this diagram
$\operatorname{Res} = \operatorname{Res}_{
\widehat{B}_{\mfrak{q}_{0}} / \widehat{A}_{\mfrak{p}_{0}}}$
etc. Using the definitions this is the same as
\[ \begin{CD}
\bigoplus_{x_{0} \in F_{0}}
f_{*} \mcal{K}_{\mfrak{X} / S}(x_{0})
@> \operatorname{Tr}_{f} >>
\mcal{K}_{\mfrak{Y} / S}(y_{0}) \\
@V \delta VV @V \delta VV \\
\bigoplus_{x_{1} \in F_{1}}
f_{*} \mcal{K}_{\mfrak{X} / S}(x_{1})
@> \operatorname{Tr}_{f} >>
\mcal{K}_{\mfrak{Y} / S}(y_{1}) .
\end{CD} \]
\end{proof}
According to \cite{KW}, if $\pi : X \rightarrow S$ is equidimensional of
dimension $n$ and generically smooth, and $X$ is integral,
then the {\em sheaf of regular
differentials} $\tilde{\omega}^{n}_{X/S}$ (relative to the DGA
$\mcal{O}_{S}$) exists. It is a coherent subsheaf of
$\Omega^{n}_{k(X)/k(S)}$.
\begin{thm} \label{thm7.4}
Suppose $\pi : X \rightarrow S$ is equidimensional of
dimension $n$ and generically smooth, and $X$ is integral. Then
$\mcal{K}^{-n}_{X/S} = \Omega^{n}_{k(X)/k(S)}$, and
\[ \tilde{\omega}^{n}_{X/S} = \mrm{H}^{-n} \mcal{K}^{{\textstyle \cdot}}_{X/S} . \]
\end{thm}
First we need:
\begin{lem} \label{lem7.1}
Suppose $L_{0} \rightarrow A_{0} \rightarrow B_{0}$ are finitely generated field
extensions, with
$L_{0} \rightarrow A_{0}$ and $L_{0} \rightarrow B_{0}$ separable, $A_{0} \rightarrow B_{0}$
finite, and
$\operatorname{tr.deg}_{L_{0}} A_{0} = n$. Then
$\mcal{K}(A_{0} / L_{0}) = \Omega^{n}_{A_{0} / L_{0}}$,
$\mcal{K}(B_{0} / L_{0}) = \Omega^{n}_{B_{0} / L_{0}}$,
and
$\operatorname{Tr}_{B_{0} / A_{0}} : \mcal{K}(B_{0} / L_{0}) \rightarrow
\mcal{K}(A_{0} / L_{0})$
coincides with
$\sigma^{L_{0}}_{B_{0} / A_{0}} : \Omega^{n}_{B_{0} / L_{0}}
\rightarrow \Omega^{n}_{A_{0} / L_{0}}$
of \cite{Ku} \S \textup{16}.
\end{lem}
\begin{proof}
Since $L_{0} \rightarrow A_{0}$ is formally smooth, we get
$\mcal{K}(A_{0} / L_{0}) = \Omega^{n}_{A_{0} / L_{0}}$. The same for
$B_{0}$.
Consider the trivial DGA $L_{0}$. Then the universal
$B_{0}$-extension of $\Omega^{{\textstyle \cdot}}_{A_{0} / L_{0}}$ is
$\Omega^{{\textstyle \cdot}}_{B_{0} / L_{0}}$, so
$\sigma^{L_{0}}_{B_{0} / A_{0}}$ makes sense. To check that
$\sigma^{L_{0}}_{B_{0} / A_{0}} = \operatorname{Tr}_{B_{0} / A_{0}}$
we may reduce to the cases $A_{0} \rightarrow B_{0}$ separable, or purely
inseparable of prime degree, and then use the properties of the trace.
\end{proof}
\begin{proof} (of the Theorem)\
Given any point $x \in X$ there is an open
neighborhood $U$ of $x$ which admits a factorization $\pi|_{U} = h g f$,
with $f : U \rightarrow Y$ an open immersion; $g : Y \rightarrow Z$ finite;
and $h : Z \rightarrow S$ smooth of relative dimension $n$
(in fact one can take $Z$ open in $\mbf{A}^{n} \times S$).
This follows from quasi-normalization (\cite{Ku} Thm.\ B20)
and Zariski's Main Theorem (\cite{EGA} IV 8.12.3; cf.\
\cite{Ku} Thm.\ B16).
We can also assume $Y, Z, S$ are affine, say
$Y = \operatorname{Spec} B$,
$Z = \operatorname{Spec} A$ and $S = \operatorname{Spec} L$.
Let us write
$\tilde{\omega}^{n}_{B / L} := \Gamma(Y, \tilde{\omega}^{n}_{Y/S})$
and
$\mcal{K}^{{\textstyle \cdot}}_{B / L} := \Gamma(Y, \mcal{K}^{{\textstyle \cdot}}_{Y/S})$.
Also let us write
$B_{0} := k(Y)$, $A_{0} := k(Z)$ and $L_{0} := k(S)$.
By \cite{KW} \S 4,
\[ \tilde{\omega}^{n}_{B / L} =
\{ \beta \in \Omega^{n}_{B_{0} / L_{0}} \mid
\sigma^{L_{0}}_{B_{0} / A_{0}} (b \beta) \in \Omega^{n}_{A / L}
\text{ for all } b \in B \} . \]
One has
\[ \mcal{K}^{-n}_{B / L} = \mcal{K}(B_{0} / L_{0}) =
\Omega^{n}_{A_{0} / L_{0}} \]
and the same for $A$.
According to Prop.\ \ref{prop6.4} there is a quasi-isomorphism
$\Omega^{n}_{A / L}[n]$ \linebreak
$ \rightarrow \mcal{K}^{{\textstyle \cdot}}_{A / L}$.
From the commutative diagram
\[ \begin{CD}
0 @> >> \mrm{H}^{-n} \mcal{K}^{{\textstyle \cdot}}_{B / L} @> >>
\mcal{K}^{-n}_{B / L} @>{\delta}>>
\mcal{K}^{-n+1}_{B / L} \\
& & @VVV @V{\operatorname{Tr}_{g}}VV @V{\operatorname{Tr}_{g}}VV \\
0 @>>> \Omega^{n}_{A / L} @>>>
\mcal{K}^{-n}_{A / L} @>{\delta}>>
\mcal{K}^{-n+1}_{A / L}
\end{CD} \]
and the isomorphism
\[ \mcal{K}^{-n+1}_{B / L} \cong
\operatorname{Hom}_{A} ( B, \mcal{K}^{-n+1}_{A / L}) \]
induced by $\operatorname{Tr}_{g}$
we conclude that
$\tilde{\omega}^{n}_{B / L} = \mrm{H}^{-n} \mcal{K}^{{\textstyle \cdot}}_{B / L}$.
Since $\tilde{\omega}^{n}_{Y / S}$ and
$\mrm{H}^{-n} \mcal{K}^{{\textstyle \cdot}}_{Y / S}$
are coherent sheaves and $f : U \rightarrow Y$ is an open immersion, this shows
that
$\tilde{\omega}_{U/S} = \mrm{H}^{-n} \mcal{K}^{{\textstyle \cdot}}_{U/S}$.
\end{proof}
\begin{cor}
If $X$ is a Cohen-Macaulay scheme then the sequence
\[ 0 \rightarrow \tilde{\omega}^{n}_{X / S} \rightarrow
\mcal{K}^{-n}_{X / S} \rightarrow \cdots \rightarrow
\mcal{K}^{m}_{X / S} \rightarrow 0 \]
\textup{(}$m = \operatorname{dim} S$\textup{)} is exact.
\end{cor}
\begin{proof}
$X$ is Cohen-Macaulay iff any dualizing complex has a single nonzero
cohomology sheaf.
\end{proof}
\begin{exa} \label{exa7.1}
Suppose $X$ is an $(n+1)$-dimensional integral scheme and
$\pi : X \rightarrow \operatorname{Spec} \mbb{Z}$ is a finite type
dominant morphism (i.e.\ $X$ has mixed characteristics). Then
$\pi$ is flat, equidimensional of dimension $n$ and
generically smooth. So
\[ \tilde{\omega}^{n}_{X / \mbb{Z}} =
\mrm{H}^{-n} \mcal{K}^{{\textstyle \cdot}}_{X / \mbb{Z}} \subset
\Omega^{n}_{k(X) / \mbb{Q}} . \]
\end{exa}
\begin{rem} \label{rem7.7}
In the situation of Thm.\ \ref{thm7.4} there is a homomorphism
\[ \mrm{C}_{X} : \Omega^{n}_{X / S} \rightarrow \mcal{K}^{-n}_{X / S} \]
called the {\em fundamental class of} $X / S$. According to \cite{KW},
when $\pi$ is flat one has
$\mrm{C}_{X}(\Omega^{n}_{X / S}) \subset \tilde{\omega}^{n}_{X/S}$; so
$\mrm{C}_{X} : \Omega^{n}_{X / S}[n] \rightarrow \mcal{K}^{{\textstyle \cdot}}_{X / S}$
is a homomorphism of complexes.
\end{rem}
\begin{rem}
In \cite{LS2} Theorem 11.2 we find a stronger statement
than our Theorem \ref{thm7.4}: $S$ is only
required to be an excellent equidimensional scheme without embedded
points, satisfying Serre's condition $\mrm{S}_{2}$; and $\pi$ is
finite type, equidimensional and generically smooth. Moreover,
for $\pi$ proper, the trace is compared to the integral
of \cite{HS} (cf.\ Remark \ref{rem8.1}).
The price of this generality is that the proofs in
\cite{LS2} are not self-contained but rely on rather complicated
results from other papers.
\end{rem}
\section{The Isomorphism $\mcal{K}^{{\textstyle \cdot}}_{X / S}
\protect \cong \pi^{!} \mcal{O}_{S}$}
In this section we describe the canonical isomorphism between the
complex $\mcal{K}^{{\textstyle \cdot}}_{X / S}$ constructed in
Section 6, and the twisted inverse image $\pi^{!} \mcal{O}_{S}$
of \cite{RD}.
Recall that for residual complexes there is an inverse image
$\pi^{\triangle}$, and
$\pi^{\triangle} \mcal{K}^{{\textstyle \cdot}}_{S / S} =
\mrm{E} \pi^{!} \mcal{O}_{S}$,
where $\mrm{E}$ is the Cousin functor corresponding to the dualizing
complex $\pi^{!} \mcal{O}_{S}$.
For an $S$-morphism $f : X \rightarrow Y$ denote by
$\operatorname{Tr}^{\mrm{RD}}_{f}$ the homomorphism of graded sheaves
\[ \operatorname{Tr}^{\mrm{RD}}_{f} :
f_{*} \pi^{\triangle}_{X} \mcal{K}^{{\textstyle \cdot}}_{S / S} \cong
f_{*} f^{\triangle} \pi^{\triangle}_{Y} \mcal{K}^{{\textstyle \cdot}}_{S / S} \rightarrow
\pi^{\triangle}_{Y} \mcal{K}^{{\textstyle \cdot}}_{S / S} \]
of \cite{RD} Section VI.4.
\begin{thm} \label{thm8.10}
Let $\pi : X \rightarrow S$ be a finite type morphism. Then there exists a
unique isomorphism of complexes
\[ \zeta_{X} : \mcal{K}^{{\textstyle \cdot}}_{X / S} \rightarrow
\pi^{\triangle} \mcal{K}^{{\textstyle \cdot}}_{S / S} \]
such that for every morphism $f : X \rightarrow Y$ the diagram
\begin{equation} \label{eqn8.4}
\begin{CD}
f_{*} \mcal{K}^{{\textstyle \cdot}}_{X / S} @>{\operatorname{Tr}_{f}}>>
\mcal{K}^{{\textstyle \cdot}}_{Y / S} \\
@V{f_{*} (\zeta_{X})}VV @V{\zeta_{Y}}VV \\
f_{*} \pi^{\triangle}_{X} \mcal{K}^{{\textstyle \cdot}}_{S / S}
@>{\operatorname{Tr}^{\mrm{RD}}_{f}}>>
\pi^{\triangle}_{Y} \mcal{K}^{{\textstyle \cdot}}_{S / S}
\end{CD}
\end{equation}
is commutative.
\end{thm}
The proof of Thm.\ \ref{thm8.10} is given later in this section,
after some preparation. Here is one corollary:
\begin{cor} \label{cor8.1}
If $f : X \rightarrow Y$ is proper then $\operatorname{Tr}_{f}$ is a homomorphism of
complexes, and for any
$\mcal{M}^{{\textstyle \cdot}} \in \msf{D}^{-}_{\mrm{qc}}(\msf{Mod}(X))$
the induced morphism
\[ f_{*} \mcal{H}om_{X}(\mcal{M}^{{\textstyle \cdot}}, \mcal{K}^{{\textstyle \cdot}}_{X / S}) \rightarrow
\mcal{H}om_{X}(\mrm{R} f_{*} \mcal{M}^{{\textstyle \cdot}}, \mcal{K}^{{\textstyle \cdot}}_{Y / S}) \]
is an isomorphism.
\end{cor}
\begin{proof}
Use \cite{RD} Theorem VII.2.1 and Corollary VII.3.4.
\end{proof}
\begin{rem} \label{rem8.1}
In \cite{Hg3} and \cite{LS2} the authors prove that in their
respective constructions the trace
$\operatorname{Tr}_{f} : f_{*} f^{!} \mcal{N}^{{\textstyle \cdot}} \rightarrow \mcal{N}^{{\textstyle \cdot}}$
is a homomorphism of complexes for any proper morphism $f$
and residual (resp.\ Cousin) complex $\mcal{N}^{{\textstyle \cdot}}$
(cf.\ Remark \ref{rem6.6}).
\end{rem}
Let $Y = \operatorname{Spec} A$ be an affine noetherian scheme,
$X := \mbf{A}^{n} \times Y =$ \newline
$\operatorname{Spec} A \sqbr{t_{1}, \ldots, t_{n}}$
and $f : X \rightarrow Y$ the projection.
Fix a point $x \in X$, and let $y := f(x)$,
$Z_{0} := \overline{\{x\}}_{\mrm{red}}$. Assume $Z_{0} \rightarrow Y$ is finite.
\begin{lem} \label{lem8.10}
There exists an open set $U \subset Y$ containing $y$ and
a flat finite morphism $g : Y' \rightarrow U$ s.t.:
\begin{enumerate}
\rmitem{i} $g^{-1}(y)$ is one point, say $y'$.
\rmitem{ii} Define $X' := \mbf{A}^{n} \times Y'$, and let
$f' : X' \rightarrow Y'$, $h : X' \rightarrow X$. Then for every point $x' \in h^{-1}(x)$
there is some section $\sigma_{x'} : Y' \rightarrow X'$ of $f'$ with
$x' \in \sigma_{x'}(Y')$.
\end{enumerate}
\end{lem}
\begin{proof}
Choose any finite normal field extension $K$ of $k(y)$ containing $k(x)$.
Define recursively open sets
$U_{i} = \operatorname{Spec} A_{i} \subset Y$
and finite flat morphisms
$g_{i} : Y_{i} = \operatorname{Spec} A'_{i} \rightarrow U_{i}$
s.t.\
$g_{i}^{-1}(y) = \{y_{i}\}$ and
$k(y_{i}) \subset K$,
as follows. Start with
$U_{0} = Y_{0} := Y$ and
$A'_{0} = A_{0} := A$.
If $k(y_{i}) \neq K$ take some $\bar{b} \in K - k(y_{i})$ and let
$\bar{p} \in k(y_{i}) \sqbr{t}$ be the monic irreducible polynomial of
$\bar{b}$. Choose a monic polynomial
$p \in \mcal{O}_{Y_{i}, y_{i}} \sqbr{t}$
lifting $\bar{p}$. There is some open set
$U_{i + 1} = \operatorname{Spec} A_{i + 1} \subset U_{i}$
s.t.\
$p \in (A'_{i} \otimes_{A_{i}} A_{i + 1}) \sqbr{t}$.
Define
$A'_{i + 1} := (A'_{i} \otimes_{A_{i}} A_{i + 1}) \sqbr{t} / (p)$
and $Y_{i + 1} = \operatorname{Spec} A'_{i + 1}$.
For $i = r$ this stops, and $k(y_{r}) = K$.
For every point
$x' \in \operatorname{Spec} (K \otimes_{k(y)} k(x))$
and $1 \leq i \leq n$ let $\bar{a}_{i, x'} \in k(x') \cong k(y_{r})$
be the image of $t_{i}$, and let
$a_{i, x'} \in \mcal{O}_{Y_{r}, y_{r}}$
be a lifting. Take an open set
$U = \operatorname{Spec} A_{r + 1} \subset U_{r}$
s.t.\ each
$a_{i, x'} \in A' = (A'_{r} \otimes_{A_{r}} A_{r + 1})$,
and define
$Y' := \operatorname{Spec} A'$.
So for each $x'$ the homomorphism
$B' = A' \sqbr{t} \rightarrow A'$, $t_{i} \mapsto a_{i, x'}$ gives the desired
section $\sigma_{x'} : Y' \rightarrow X'$.
\end{proof}
Let $Z_{i}$ be the $i$-th infinitesimal neighborhood of
$Z_{0}$ in $X$, so $f_{i} : Z_{i} \rightarrow Y$ is a finite morphism.
Suppose we are given a quasi-coherent $\mcal{O}_{Y}$-module
$\mcal{M}$ which is supported on $\overline{\{ y \}}$. One has
\[ \mcal{H}^{n}_{Z_{0}} \left( \Omega^{n}_{X / Y} \otimes f^{*} \mcal{M}
\right) \cong
\lim_{i \rightarrow} \mcal{E}xt^{n}_{X} \left( \mcal{O}_{Z_{i}},
\Omega^{n}_{X / Y} \otimes f^{*} \mcal{M} \right) \]
and by \cite{RD} Thm.\ VI.3.1
\[ \mcal{E}xt^{n}_{X} \left( \mcal{O}_{Z_{i}},
\Omega^{n}_{X / Y} \otimes f^{*} \mcal{M} \right) =
\mcal{H}^{0} f_{i}^{!} \mcal{M} . \]
Note that we can also factor $f_{i}$ through $\mbf{P}^{n} \times Y$,
so $f_{i}$ is projectively embeddable, and by \cite{RD} Thm.\ III.10.5
we have a map
\begin{equation} \label{eqn8.10}
\operatorname{Tr}_{f}^{\mrm{RD}} :
f_{*} \mcal{H}^{n}_{Z_{0}} \left( \Omega^{n}_{X / Y} \otimes
f^{*} \mcal{M} \right) \rightarrow \mcal{M} .
\end{equation}
Now define
$\widehat{A} := \widehat{\mcal{O}}_{Y , y}$ and
$\widehat{B} := \widehat{\mcal{O}}_{X , x}$, with
$\mfrak{n} \subset \widehat{B}$ the maximal ideal and
$\phi = f^{*} : \widehat{A} \rightarrow \widehat{B}$.
Set $M := \mcal{M}_{y}$, which is a discrete $\widehat{A}$-module.
We then have a natural isomorphism of $\widehat{A}$-modules
\begin{equation} \label{eqn8.1}
\left( f_{*} \mcal{H}^{n}_{Z_{0}} \left( \Omega^{n}_{X / Y} \otimes
f^{*} \mcal{M} \right) \right)_{y} \cong
\mrm{H}^{n}_{\mfrak{n}} \left(
\widehat{\Omega}^{n}_{\widehat{B} / \widehat{A}} \otimes_{\widehat{A}}
M \right) \cong \phi_{\#} M .
\end{equation}
\begin{lem} \label{lem8.1}
Under the isomorphism \textup{(\ref{eqn8.1})},
\[ \operatorname{Tr}_{f}^{\mrm{RD}} = \operatorname{Tr}_{\phi} : \phi_{\#} M \rightarrow M . \]
\end{lem}
\begin{proof}
The proof is in two steps.\\
Step 1.\ Assume there is a section $\sigma : Y \rightarrow X$ to $f$ with
$x \in W_{0} = \sigma(Y)$. The homomorphism
$\sigma^{*} : B = A \sqbr{\underline{t}} \rightarrow A$ chooses
$a_{i} = \sigma^{*}(t_{i}) \in A$,
so after the linear change of variables $t_{i} \mapsto t_{i} - a_{i}$
we may assume that $\sigma$ is the $0$-section (i.e.\
$\mcal{O}_{W_{0}} = \mcal{O}_{X} / \mcal{O}_{X} \cdot \underline{t}$).
Let $W_{i}$ be the $i$-th infinitesimal neighborhood of $W_{0}$.
Since $f : W_{i} \rightarrow Y$ is projectively embeddable, there is a trace
map
\[ \operatorname{Tr}_{f}^{\mrm{RD}} :
f_{*} \mcal{H}^{n}_{W_{0}} \Omega^{n}_{X / Y} \rightarrow \mcal{O}_{Y} . \]
For any $a \in A$ one has
\begin{equation} \label{eqn8.11}
\operatorname{Tr}_{f}^{\mrm{RD}} \left(
\gfrac{a \mrm{d} t_{1} \wedge \cdots \wedge \mrm{d} t_{n}}
{t_{1}^{i_{1}} \cdots t_{n}^{i_{n}}} \right) =
\begin{cases}
a & \text{ if } \underline{i} = (1, \ldots, 1) \\
0 & \text{ otherwise} .
\end{cases}
\end{equation}
This follows from properties R6 (normalization) and
R7 (intersection) of the residue symbol (\cite{RD} Section III.9).
Alternatively this can be checked as follows. Note that
$\operatorname{Tr}_{f}^{\mrm{RD}}$ factors through
$\mrm{R} f_{*} \Omega^{n}_{\mbf{P}^{n}_{Y} / Y}$.
For the case $\underline{i} = (1, \ldots, 1)$ use \cite{RD}
Proposition III.10.1.
For $\underline{i} \neq (1, \ldots, 1)$ consider a change of coordinates
$t_{i} \mapsto \lambda_{i} t_{i}$, $\lambda_{i} \in A$. By
\cite{RD} Corollary III.10.2, $\operatorname{Tr}_{f}^{\mrm{RD}}$ is independent
of homogeneous coordinates, so it must be $0$.
Now since $W_{0} \cap f^{-1}(y) = Z_{0}$ we have
\[ \mcal{H}^{n}_{Z_{0}} \left( \Omega^{n}_{X / Y} \otimes f^{*} \mcal{M}
\right) \cong
\mcal{H}^{n}_{W_{0}} \left( \Omega^{n}_{X / Y} \otimes f^{*} \mcal{M}
\right) \]
and so the formula for $\operatorname{Tr}_{f}^{\mrm{RD}}$ in (\ref{eqn8.10})
is given by (\ref{eqn8.11}).
But the same formula is used in \cite{Hg1} to define
$\operatorname{Tr}_{\phi}$.\\
Step 2.\
The general situation: take $g : Y' \rightarrow Y$ as in Lemma \ref{lem8.10},
and set $Z_{0}' := Z_{0} \times_{Y} Y'$.
The flatness of $g$ implies there is a natural
isomorphism of $\mcal{O}_{Y'}$-modules
\[ g^{*} f_{*} \mcal{H}^{n}_{Z_{0}} \left( \Omega^{n}_{X / Y} \otimes
f^{*} \mcal{M} \right) \cong
f_{*}' \mcal{H}^{n}_{Z_{0}'} \left( \Omega^{n}_{X' / Y'} \otimes
{f'}^{*} \mcal{M}' \right) \]
(where $\mcal{M}' := g^{*} \mcal{M}$) and by \cite{RD} Thm.\ III.10.5
property TRA4 we have
\begin{equation} \label{eqn8.14}
g^{*}(\operatorname{Tr}_{f}^{\mrm{RD}}) = \operatorname{Tr}_{f'}^{\mrm{RD}} .
\end{equation}
Let $\widehat{A}' := \widehat{\mcal{O}}_{Y', y'} \cong A' \otimes_{A}
\widehat{A}$,
so $\widehat{A} \rightarrow \widehat{A}'$ is finite flat. Therefore
\begin{equation} \label{eqn8.2}
\widehat{A}' \otimes_{\widehat{A}} \mrm{H}^{n}_{\mfrak{n}}
\left( \widehat{\Omega}^{n}_{\widehat{B} / \widehat{A}}
\otimes_{\widehat{A}} M \right)
\cong
\bigoplus_{\mfrak{n}' \in Z_{0}'}
\mrm{H}^{n}_{\mfrak{n}'}
\left( \widehat{\Omega}^{n}_{\widehat{B}_{\mfrak{n}'} / \widehat{A}'}
\otimes_{\widehat{A}'} M' \right) .
\end{equation}
Here
$M' := \mcal{M}'_{y'} \cong \widehat{A}' \otimes_{\widehat{A}} M$
and
$\prod_{\mfrak{n}' \in Z_{0}'} \widehat{B}_{\mfrak{n}'}$
is the decomposition of $A' \otimes_{A} \widehat{B}$
to local rings. Write
$\phi'_{\mfrak{n}'} : \widehat{A}' \rightarrow \widehat{B}_{\mfrak{n}'}$.
Direct verification shows that under the isomorphism (\ref{eqn8.2}),
\begin{equation} \label{eqn8.15}
1 \otimes \operatorname{Tr}_{\phi} = \sum_{\mfrak{n}' \in Z_{0}'}
\operatorname{Tr}_{\phi'_{\mfrak{n}'}} .
\end{equation}
Since $\widehat{A} \rightarrow \widehat{A}'$ is faithfully flat it follows that
$M \rightarrow M'$ is injective. In view of
the equalities (\ref{eqn8.14}) and (\ref{eqn8.15}), we conclude
that it suffices to check for each $\mfrak{n}' = x' \in Z_{0}$ that
$\operatorname{Tr}_{\phi'_{\mfrak{n}'}} =
\operatorname{Tr}^{\mrm{RD}}_{f'}$
on
$\mrm{H}^{n}_{\mfrak{n}'}
\left( \widehat{\Omega}^{n}_{\widehat{B}_{\mfrak{n}'} / \widehat{A}'}
\otimes_{\widehat{A}'} M' \right)$.
But there is a section $\sigma_{x'} : Y' \rightarrow X'$, so we can apply
step 1.
\end{proof}
\begin{proof} (of Thm.\ \ref{thm8.10}.)\\
Step 1. (Uniqueness)\
Suppose
$\zeta_{X}' : \mcal{K}^{{\textstyle \cdot}}_{X / S} \rightarrow
\pi^{\triangle} \mcal{K}^{{\textstyle \cdot}}_{S / S}$
is another isomorphism satisfying
$\operatorname{Tr}_{\pi} = \operatorname{Tr}^{\mrm{RD}}_{\pi} \pi_{*}(\zeta_{X}')$.
Then $\zeta_{X}' = a \zeta_{X}$ for some
$a \in \Gamma(X, \mcal{O}_{X}^{*})$, and by
assumption for any closed point $x \in X$ and
$\alpha \in \mcal{K}_{X / S}(x)$ there is equality
$\operatorname{Tr}_{\pi}(\alpha) = \operatorname{Tr}_{\pi} (a \alpha)$.
Now writing $s := \pi(x)$, it's known that
\[ \operatorname{Hom}_{\mcal{O}_{S, s}} \left(
\mcal{K}_{X / S}(x), \mcal{K}_{S / S}(s) \right) \]
is a free $\widehat{\mcal{O}}_{X, x}$-module with basis
$\operatorname{Tr}_{\pi}$. Therefore $a = 1$ in
$\widehat{\mcal{O}}_{X, x}$. Because this is true for all closed points
we see that $a = 1$.\\
Step 2.\
Assume $X = \mbf{A}^{n} \times S$ and $f = \pi$.
In this case there is a canonical isomorphism of complexes
\[ \mcal{K}_{X / S}^{{\textstyle \cdot}} \cong \mrm{E} \Omega^{n}_{X / S}[n] \cong
\mrm{E} \pi^{!} \mcal{O}_{S} \cong
\pi^{\triangle} \mcal{K}_{S / S}^{{\textstyle \cdot}} \]
(cf.\ \cite{RD} Thm.\ VI.3.1 and our Prop.\ \ref{prop6.4}),
which we use to define
$\zeta_{X} : \mcal{K}^{{\textstyle \cdot}}_{X / S} \rightarrow
\pi^{\triangle} \mcal{K}^{{\textstyle \cdot}}_{S / S}$.
Consider $x \in X$, $Z := \overline{\{x\}}_{\mrm{red}}$,
$s := \pi(x)$ and assume $x$ is closed in $\pi^{-1}(s)$.
By replacing $S$ with a suitable open neighborhood of $s$ we can
assume $Z \rightarrow S$ is finite. Then we are allowed to apply Lemma
\ref{lem8.1} with $Y = S$, $\mcal{M} = \mcal{K}_{S / S}(s)$.
It follows that (\ref{eqn8.4}) commutes on
$\pi_{*} \mcal{K}_{X / S}(x) \subset \pi_{*} \mcal{K}^{{\textstyle \cdot}}_{X / S}$.\\
Step 3.\ Let $X$ be any finite type $S$-scheme. For every affine open
subscheme $U \subset X$ we can find a closed immersion
$h : U \rightarrow \mbf{A}^{n}_{S}$. Write $Y := \mbf{A}^{n}_{S}$
and let
$\pi_{U}$ and $\pi_{Y}$ be the structural morphisms.
Now
$\operatorname{Tr}_{h}$ induces an isomorphism
\[ \mcal{K}^{{\textstyle \cdot}}_{U / S} \cong
\mcal{H}om_{Y}(\mcal{O}_{U}, \mcal{K}^{{\textstyle \cdot}}_{Y / S}) , \]
and
$\operatorname{Tr}^{\mrm{RD}}_{h}$ induces an isomorphism
\[ \pi_{U}^{\triangle} \mcal{K}^{{\textstyle \cdot}}_{S / S} \cong
\mcal{H}om_{Y}(\mcal{O}_{U},
\pi_{Y}^{\triangle} \mcal{K}^{{\textstyle \cdot}}_{S / S}) . \]
So the isomorphism
$\zeta_{Y}$ of Step 2 induces an isomorphism
$\zeta_{U} : \mcal{K}^{{\textstyle \cdot}}_{U / S} \rightarrow
\pi_{U}^{\triangle} \mcal{K}^{{\textstyle \cdot}}_{S / S}$, which satisfies
$\operatorname{Tr}_{\pi_{U}} = \operatorname{Tr}^{\mrm{RD}}_{\pi_{U}}
\pi_{U *}(\zeta_{U})$.
According to Step 1 the local isomorphisms $\zeta_{U}$ can be glued
to a global isomorphism $\zeta_{X}$.\\
Step 4.\ Let $f : X \rightarrow Y$ be any $S$-morphism. To check (\ref{eqn8.4})
we may assume $X$ and $Y$ are affine, and in view of step 3
we may in fact assume
$Y = \mbf{A}^{m} \times S$ and
$X = \mbf{A}^{n} \times Y \cong \mbf{A}^{n + m} \times S$.
Now apply Lemma \ref{lem8.1} with $x \in X$ closed in its fiber
and
$\mcal{M} := \mcal{K}_{Y / S}(y)$.
\end{proof}
|
1997-08-15T22:41:35 | 9510 | alg-geom/9510018 | en | https://arxiv.org/abs/alg-geom/9510018 | [
"alg-geom",
"math.AG"
] | alg-geom/9510018 | Ezra Getzler | Ezra Getzler | Mixed Hodge structures of configuration spaces | AmSLaTeX 1.v2, 18 pages, revised version | null | null | Preprint 96-61, Max-Planck-Institut f. Mathematik, Bonn | null | The symmetric group S_n acts freely on the configuration space of n distinct
points in a quasi-projective variety. In this paper, we study the induced
action of the symmetric group S_n on the de Rham cohomology of this space,
using mixed Hodge theory, combined with methods from the theory of symmetric
functions. (We prove a motivic version of this as well.) As an application of
our results, we calculate the S_n-equivariant Hodge polynomial of the
Fulton-MacPherson compactification X[n] of the configuration space.
| [
{
"version": "v1",
"created": "Wed, 1 Nov 1995 04:59:25 GMT"
},
{
"version": "v2",
"created": "Thu, 9 Nov 1995 21:01:26 GMT"
},
{
"version": "v3",
"created": "Sun, 19 May 1996 19:56:04 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Getzler",
"Ezra",
""
]
] | alg-geom | \subsection*{Acknowledgments} The author wishes to thanks the Department of
Mathematics at the Universit\'e de Paris-VII for its hospitality during the
inception of this paper. He is partially supported by a research grant of
the NSF, a fellowship of the A.P. Sloan Foundation, and the
Max-Planck-Institut f\"ur Mathematik in Bonn.
\section{Symmetric functions and $\lambda$-rings}
\subsection{Symmetric functions} In this section, we recall some results on
symmetric functions and representations of $\SS_n$ which we need later. For
the proofs of these results, we refer to Macdonald \cite{Macdonald}.
The ring of symmetric functions is the inverse limit
$$
\Lambda = \varprojlim \mathbb{Z}[x_1,\dots,x_k]^{\SS_k} .
$$
It is is a polynomial ring in the complete symmetric functions
$$
h_n = \sum_{i_1\le\dots\le i_n} x_{i_1}\dots x_{i_n} .
$$
The power sums (also known as Newton polynomials)
$$
p_n = \sum_i x_i^n
$$
form a set of generators of the polynomial ring $\Lambda_\mathbb{Q}=\Lambda\o\mathbb{Q}$.
This is shown by means of the elementary formula
\begin{equation} \label{P-H}
P_t = t \frac{d}{dt} \log H_t ,
\end{equation}
where
$$
H_t = \sum_{n=0}^\infty t^n h_n = \prod_i (1-tx_i)^{-1}
\quad\text{and}\quad
P_t = \sum_{n=0}^\infty t^n p_n = \sum_i (1-tx_i)^{-1} .
$$
Written out explicitly, we obtain Newton's formula relating the two sets of
generators:
$$
nh_n = p_n + h_1p_{n-1} + \dots + h_{n-1}p_1 .
$$
We may also invert \eqref{P-H}, obtaining the formula
\begin{equation} \label{H-P}
H_t = \exp \Bigl( \sum_{n=1}^\infty \frac{t^np_n}{n} \Bigr) .
\end{equation}
A partition $\lambda$ is a decreasing sequence
$(\lambda_1\ge\dots\ge\lambda_\ell)$ of positive integers; we write
$\lambda\vdash n$, where $n=\lambda_1+\dots+\lambda_\ell$, and denote the
length of $\lambda$ by $\ell(\lambda)$. Identifying $\Lambda$ with the ring
of characters of the Lie algebra $\gl_\infty = \varprojlim \gl_k$, we see
that partitions correspond to dominant weights, and thus $\Lambda$ has a
basis of consisting of the characters of the irreducible representations of
$\gl_\infty$. These characters, given by the Weyl character formula
$$
s_\lambda = \lim_{k\to\infty}
\frac{\det(x_i^{\lambda_j+k-j})_{1\le i,j\le k}}
{\det(x_i^{k-j})_{1\le i,j\le k}} ,
$$
are known as the Schur functions. In terms of the polynomial generators
$h_n$, they are given by the Jacobi-Trudy formula
$s_\lambda=\det\bigl(h_{\lambda_i-i+j}\bigr)_{1\le i,j\le\ell(\lambda)}$.
There is a non-degenerate integral bilinear form on $\Lambda$, denoted
$\<f,g\>$, for which the Schur functions $s_\lambda$ form an orthonormal
basis. (This is sometimes called the Hall inner product.) The adjoint of
multiplication by $f\in\Lambda$ with respect to this inner product is
denoted $D(f)$. Written in terms of the power sums $p_n$, the operator
$D(f)$ has the formula $D(p_n)=n\partial/\partial p_n$, while the inner product
$\<f,g\>$ has the formula
$$
\< f , g \> = D(f)g \Big|_{p_n=0,n\ge1} .
$$
\subsection{Pre-$\lambda$-rings} A pre-$\lambda$-ring is a commutative ring
$R$, together with a morphism of commutative rings $\sigma_t:R\to R\[t\]$
such that $\sigma_t(a)=1+ta+O(t^2)$. Expanding $\sigma_t$ in a power series
$$
\sigma_t(a) = \sum_{n=0}^\infty t^n \sigma_n(a) ,
$$
we obtain endomorphisms $\sigma_n$ of $R$ such that $\sigma_0(a)=1$,
$\sigma_1(a)=a$, and
$$
\sigma_n(a+b) = \sum_{k=0}^n \sigma_{n-k}(a) \sigma_k(b) .
$$
There are also operations $\lambda_k(a)=(-1)^k\sigma_k(-a)$, with
generating function
\begin{equation} \label{invert}
\lambda_t(a) = \sum_{n=0}^\infty t^n \lambda_n(a) = \sigma_{-t}(a)^{-1} .
\end{equation}
The $\lambda$-operations are polynomials in the $\sigma$-operations with
integral coefficients, and vice versa. In this paper, we take the
$\sigma$-operations to be more fundamental; nevertheless, following custom,
the structure they define is called a pre-$\lambda$-ring.
Given a pre-$\lambda$-ring $R$, there is a bilinear map $\Lambda\o R\to R$,
which we denote $f\circ a$, defined by the formula
$$
(h_{n_1}\dots h_{n_k})\circ a = \sigma_{n_1}(a)\dots\sigma_{n_k}(a) .
$$
The image of the power sum $p_n$ under this map is the operation on $R$
known as the Adams operation $\psi_n$. We denote the operation
corresponding to the Schur function $s_\lambda$ by $\sigma_\lambda$. Note
that \eqref{H-P} implies the relation
$$
\sigma_t(a) = \exp \Bigl( \sum_{n=1}^\infty \frac{t^n\psi_n(a)}{n} \Bigr) .
$$
The following formula (I.4.2 of \cite{Macdonald}) is known as Cauchy's
formula:
\begin{equation} \label{Cauchy}
H_t(...,x_iy_j,...) = \prod_{i,j} (1-tx_iy_j)^{-1}
= \sum_{\lambda\vdash n} s_\lambda(x) \o s_\lambda(y)
= \exp\Bigl( \sum_{k=1}^\infty \frac{p_k(x) \o p_k(y)}{k} \Bigr) .
\end{equation}
From it, the following result is immediate.
\begin{proposition}
If $R$ and $S$ are pre-$\lambda$-rings, their tensor product $R\o S$ is a
pre-$\lambda$-ring, with $\sigma$-operations
$$
\sigma_n(a\o b) = \sum_{\lambda\vdash n} \sigma_\lambda(a) \o
\sigma_\lambda(b) ,
$$
and Adams operations $\psi_n(a\o b) = \psi_n(a) \o \psi_n(b)$.
\end{proposition}
For example, $\sigma_2(a\o b) = \sigma_2(a)\o\sigma_2(b) +
\lambda_2(a)\o\lambda_2(b)$.
\subsection{$\lambda$-rings} The polynomial ring $\mathbb{Z}[x]$ is a
pre-$\lambda$-ring, with $\sigma$-operations characterized by the formula
$\sigma_n(x^i)=x^{ni}$. Taking tensor powers of this pre-$\lambda$-ring
with itself, we see that the polynomial ring $\mathbb{Z}[x_1,\dots,x_k]$ is a
pre-$\lambda$-ring. The $\lambda$-operations on this ring are equivariant
with respect to the permutation action of the symmetric group $\SS_k$ on
the generators, hence the ring of symmetric functions
$\mathbb{Z}[x_1,\dots,x_k]^{\SS_k}$ is a pre-$\lambda$-ring. Taking the limit
$k\to\infty$, we obtain a pre-$\lambda$-ring structure on $\Lambda$.
\begin{definition}
A $\lambda$-ring is pre-$\lambda$-ring such that if $f,g\in\Lambda$ and
$x\in R$,
\begin{equation} \label{lambda-ring}
f\circ(g\circ x)=(f\circ g)\circ x .
\end{equation}
\end{definition}
By definition, the pre-$\lambda$-ring $\Lambda$ is a $\lambda$-ring; in
particular, the operation $f\circ g$, called plethysm, is associative.
The following result (see Knutson, \cite{Knutson}) is the chief result in
the theory of $\lambda$-rings.
\begin{theorem} \label{universal}
$\Lambda$ is the universal $\lambda$-ring on a single generator $h_1$.
\end{theorem}
This theorem makes it straighforward to verify identities in
$\lambda$-rings: it suffices to verify them in $\Lambda$. As an
application, we have the following corollary.
\begin{corollary}
The tensor product of two $\lambda$-rings is a $\lambda$-ring.
\end{corollary}
\begin{proof}
We need only verify this for $R=\Lambda$. A torsion-free pre-$\lambda$-ring
whose Adams operations are ring homomorphisms which satisfy
$\psi_m(\psi_n(a))=\psi_{mn}(a)$ is a $\lambda$-ring. It is easy to verify
these conditions for $\Lambda\o\Lambda$, since $\psi_n(a\o b) = \psi_n(a)
\o \psi_n(b)$.
\end{proof}
In the definition of a $\lambda$-ring, it is usual to adjoin the axiom
$$
\sigma_n(xy) = \sum_{\lambda\vdash n} \sigma_\lambda(a) \o
\sigma_\lambda(y) .
$$
However, this formula follows from our definition of a $\lambda$-ring: by
universality, it suffices to check it for $R=\Lambda\o\Lambda$, $x=h_1\o1$
and $y=1\o h_1$, for which it is evident.
\section{Complete $\lambda$-rings}
A filtered $\lambda$-ring $R$ is a $\lambda$-ring with decreasing
filtration
$$
R = F^0R \supset F^1R \supset \dots ,
$$
such that
\begin{enumerate}
\item $\bigcap_k F^kR = 0$ (the filtration is discrete);
\item $F^mR\*F^nR\subset F^{m+n}R$ (the filtration is compatible with the
product);
\item $\sigma_m(F^nR)\subset F^{mn}R$ (the filtration is compatible with
the $\lambda$-ring structure).
\end{enumerate}
The completion of a filtered $\lambda$-ring is again a $\lambda$-ring;
define a complete $\lambda$-ring to be a $\lambda$-ring equal to its
completion. For example, the universal $\lambda$-ring $\Lambda$ is filtered
by the subspaces $F^n\Lambda$ of polynomials vanishing to order $n-1$, and
its completion is the $\lambda$-ring of symmetric power series, whose
underlying ring is the power series ring $\mathbb{Z}\[h_1,h_2,h_3,\dots\]$.
The tensor product of two filtered $\lambda$-rings is again a filtered
$\lambda$-ring, when furnished with the filtration
$$
F^n(R\o S) = \sum_{k=0}^n F^{n-k}R \o F^kS .
$$
If $R$ and $S$ are filtered $\lambda$-rings, denote by $R\Hat{\otimes} S$ the
completion of $R\o S$.
Let $\mathcal{R}$ be a Karoubian rring\xspace over a field of characteristic zero, and
consider the complete $\lambda$-ring $\Lambda\Hat{\otimes} K_0(\mathcal{R})$, where $K_0(\mathcal{R})$
has the discrete filtration. This $\lambda$-ring has a natural realization,
as the Grothendieck group of the Karoubian rring\xspace
$$
\hom{\mathcal{R}} = \prod_{n=0}^\infty [\SS_n,\mathcal{R}] ,
$$
whose objects are the $\SS$-modules in $\mathcal{R}$. In this rring\xspace, the sum and
product are given by the same formulas as in the rring\xspace $[\SS,\mathcal{R}]$ of
bounded $\SS$-modules.
Without assuming the existence of infinite sums in $\mathcal{R}$, plethysm does not
extend to a monoidal structure on $\hom{\mathcal{R}}$. However, $\mathcal{X}\circ\mathcal{Y}$ is
well-defined in $\[\SS,\mathcal{R}\]$ under either of the following two hypotheses:
$$
\text{i) $\mathcal{X}$ is bounded, or ii) $\mathcal{Y}(0)=0$.}
$$
The first of these situations allows us to construct a $\lambda$-ring
structure on the Grothendieck group of $\hom{\mathcal{R}}$, by the same method as
for $[\SS,\mathcal{R}]$, while the second will be needed in the proof of our main
theorem. Introducing the notation $\hom[k]{\mathcal{R}}$ for the subcategory of
$\hom{\mathcal{R}}$ consisting of $\SS$-modules $\mathcal{X}$ such that $\mathcal{X}(n)=0$ for $n<k$,
we see that plethysm extends to a symmetric monoidal structure on
$\hom[1]{\mathcal{R}}$.
Denote the Grothendieck group of the full subcategory
$\hom[1]{\mathcal{R}}\subset\hom{\mathcal{R}}$ by $\Check{K}^\SS_0(\mathcal{R})$. Since
$\Check{K}^\SS_0(\mathcal{R})$ is a (non-unital) $\lambda$-ring, we may define a
bilinear operation
$$
\circ : \Hat{K}^\SS_0({\mathsf{Proj}}) \o \Check{K}^\SS_0(\mathcal{R}) \to \check{K}_\SS(\mathcal{R}) ,
$$
satisfying \eqref{lambda-ring}. This operation may be extended to a
bilinear operation (which we denote by the same symbol),
$$
\circ : \Hat{K}^\SS_0(\mathcal{R}) \o \Check{K}^\SS_0(\mathcal{R}) \to \check{K}_\SS(\mathcal{R}) ,
$$
using the Peter-Weyl Theorem: to define $x\circ y$, we expand $x$ in a
series $x=\sum_\lambda x_\lambda\*s_\lambda$, and define
$$
x\circ y = \sum_\lambda x_\lambda \* \sigma_\lambda(y) .
$$
The interest of this operation lies in the following lemma, which is a
simple consequence of the definition of $x\circ y$.
\begin{lemma} \label{plethysm}
If $\mathcal{X}$ and $\mathcal{Y}$ are objects of $\hom{\mathcal{R}}$ and $\hom[1]{\mathcal{R}}$
respectively, $[\mathcal{X}\circ\mathcal{Y}]=[\mathcal{X}]\circ[\mathcal{Y}]$.
\end{lemma}
If $R$ is a complete $\lambda$-ring, the operation
$$
\Exp(a) = \sum_{n=0}^\infty \sigma_n(a) : R \to 1+F_1R
$$
is an analogue of exponentiation, whose logarithm is given by a formula of
Cadogan \cite{Cadogan}.
\begin{proposition} \label{Cadogan}
On a complete filtered $\lambda$-ring $R$, the operation $\Exp:R\to1+F_1R$
has inverse
$$
\Log(1+a) = \sum_{n=1}^\infty \frac{\mu(n)}{n} \log(1+\psi_n(a)) .
$$
\end{proposition}
\begin{proof}
Expanding $\Log(1+a)$, we obtain
$$
\Log(1+a) = - \sum_{n=1}^\infty \frac{1}{n} \sum_{d|n} \mu(d)
\psi_d(-a)^{n/d} = \sum_{n=1}^\infty \Log_n(a) .
$$
Let $\chi_n$ be the character of the cyclic group $C_n$ equalling $e^{2\pi
i/n}$ on the generator of $C_n$. The characteristic of the $\SS_n$-module
$\Ind^{\SS_n}_{C_n}\chi_n$ equals
$$
\frac{1}{n} \sum_{k=0}^{n-1} e^{2\pi ik/n} p_{(k,n)}^{n/(k,n)}
= \frac{1}{n} \sum_{d|n} \mu(d) p_d^{n/d} ,
$$
while the characteristic of the $\SS_n$-module
$\Ind^{\SS_n}_{C_n}\chi_n\o\varepsilon_n$, where $\varepsilon_n$ is the sign
representation of $\SS_n$, equals
$$
\frac{1}{n} \sum_{d|n} \mu(d) \bigl( (-1)^{d-1}p_d \bigr)^{n/d}
= \frac{(-1)^n}{n} \sum_{d|n} \mu(d) (-p_d)^{n/d} .
$$
It follows that $(-1)^{n-1}\Log_n$ is the operation associated to the
$\SS_n$-module $\Ind^{\SS_n}_{C_n}\chi_n\o\varepsilon_n$, and hence defines a map
from $F_1R$ to $F_nR$.
To prove that $\Log$ is the inverse of $\Exp$, it suffices to check this
for $R=\Lambda$ and $x=h_1$. We must prove that
$$
\Exp\left( \sum_{n=1}^\infty \frac{\mu(n)}{n} \log(1+p_n) \right) = 1+h_1 .
$$
The logarithm of the expression on the left-hand side equals
$$
\exp \Bigl( \sum_{k=1}^\infty \frac{p_k}{k} \Bigr) \circ
\Bigl( \sum_{n=1}^\infty \frac{\mu(n)}{n} \log(1+p_n) \Bigr)
= \sum_{n=1}^\infty \sum_{d|n} \mu(d) \frac{\log(1+p_n)}{n}
= \log(1+p_1) ,
$$
and the formula follows.
\end{proof}
\begin{example} \label{Log(t)}
If $a\in F^1R$ is a line bundle in the complete $\lambda$-ring $R$ (that
is, $\sigma_n(a)=a^n$ for all $n\ge0$), we see that
$$
\Exp(a) = \frac{1}{1-a} .
$$
In particular, this shows that $\Exp(t^n)=(1-t^n)^{-1}$, and that
$$
\Exp(t-t^2) = \frac{\Exp(t)}{\Exp(t^2)} = \frac{1-t^2}{1-t} = 1 + t .
$$
It follows that $\Log(1-t)=t$ and that $\Log(1+t)=t-t^2$.
\end{example}
We now introduce the operations on $\lambda$-rings which will arise in the
calculation of the Serre polynomials of the local systems
$\mathsf{F}(X,n)\times_{\SS_n}V_\lambda$. We start by considering the case $X=\mathbb{C}$.
\begin{proposition}
$$
\sum_\lambda s_\lambda \o \Serre(\mathsf{F}(\mathbb{C},n),V_\lambda)
= \prod_{k=1}^\infty (1+p_k)^{\frac{1}{k}\sum_{d|k}\mu(k/d)\mathsf{L}^d}
\in \Lambda \Hat{\otimes} \mathbb{Z}[\mathsf{L}]
$$
\end{proposition}
\begin{proof}
It is proved in Lehrer-Solomon \cite{LS} that
\begin{equation} \label{Lehrer-Solomon}
\sum_{n=0}^\infty \sum_{i=0}^\infty (-x)^i \ch_n(H^i(\mathsf{F}(\mathbb{C},n),\mathbb{C})) =
\prod_{k=1}^\infty (1+x^kp_k)^{\frac{1}{k}\sum_{d|k}\mu(k/d)x^{-d}} ,
\end{equation}
where $H^i(\mathsf{F}(\mathbb{C},n),\mathbb{C})$ is the $\SS_n$-module associated to the de Rham
cohomology of degree $i$. By Poincar\'e duality, we see that
$$
\sum_{n=0}^\infty \sum_{i=0}^\infty (-x)^i \ch_n(H_c^i(\mathsf{F}(\mathbb{C},n),\mathbb{C})) =
\prod_{k=1}^\infty (1+p_k)^{\frac{1}{k}\sum_{d|k}\mu(k/d)x^d} .
$$
But the mixed Hodge structure on the cohomology group $H^i_c(\mathsf{F}(\mathbb{C},n),\mathbb{C})$
is pure of weight $2i$, and indeed
$H^i_c(\mathsf{F}(\mathbb{C},n),\mathbb{C})=H^i_c(\mathsf{F}(\mathbb{C},n),\mathbb{C})^{i,i}$, proving the result.
\end{proof}
Motivated by this proposition, we define operations $\Phi_\lambda$ in a
$\lambda$-ring $R$, parametrized by partitions $\lambda$, by means of the
generating function
\begin{equation} \label{Phi}
\Phi(x) \equiv \sum_\lambda s_\lambda\o\Phi_\lambda(x)
= \prod_{k=1}^\infty (1+p_k)^{\frac{1}{k}\sum_{d|k}\mu(k/d)\psi_d(x)}
\in \Lambda \Hat{\otimes} R .
\end{equation}
\begin{theorem} \label{explicit}
We have the formula $\Phi(x) = \Exp(\Log(1+p_1)x)$. In particular, the
operations $\Phi_\lambda$ are defined on any $\lambda$-ring.
\end{theorem}
\begin{proof}
Applying $\Log$ to the definition of $\Phi(x)$, we obtain
\begin{align*}
\Log(\Phi(x)) &= \sum_{n=1}^\infty \frac{\mu(n)}{n} \psi_n \log(\Phi(x)) \\
&= \sum_{n=1}^\infty \frac{\mu(n)}{n} \psi_n
\sum_{k=1}^\infty \frac{1}{k}\sum_{d|k}\mu(k/d) \log(1+p_k) \psi_d(x) \\
&= \sum_{n,d,e=1}^\infty \frac{\mu(n)\mu(e)}{nde}
\psi_{nd}(x) \log(1+p_{nde}) \\
&= \sum_{e=1}^\infty \frac{\mu(e)}{e} \log(1+p_e) x ,
\end{align*}
by M\"obius inversion. On applying $\Exp$, we obtain the desired formula.
\end{proof}
Using this theorem, we can prove more explicit formulas for $\Phi_n$ and
$\Phi_{1^n}$.
\begin{corollary} \label{braid}
$$
\sum_{n=0}^\infty t^n \Phi_n(y) =
\frac{\sigma_t(y)}{\sigma_{t^2}(y)} \text{ and }
\Phi_{1^n}(y) = \lambda_n(y)
$$
\end{corollary}
\begin{proof}
We obtain $\sum_{n=0}^\infty t^n \Phi_n(y)$ from $\Phi(x)$ by replacing
$p_n$ by $t^n$. By Theorem \ref{explicit}, it follows that
$$
\sum_{n=0}^\infty t^n \Phi_n(y) = \Exp(\Log(1+t)x) = \Exp((t-t^2)x) =
\frac{\sigma_t(x)}{\sigma_{t^2}(x)} ,
$$
since $\Log(1+t)=t-t^2$ by Example \ref{Log(t)}. The proof of the second
formula is similar, except that we replace $p_n$ by $(-t)^n$, and apply the
formula $\Log(1-t)=-t$.
\end{proof}
\section{Representations of finite groups in Karoubian rrings\xspace}
Let $(\mathcal{R},\otimes,{1\!\!1})$ be a symmetric monoidal category with coproducts,
denoted $X\oplus Y$. We say that $\mathcal{R}$ is a \textbf{rring\xspace} (this is our
abbreviation for the usual term \emph{ring category}) if there are natural
isomorphisms
$$
(X\oplus Y)\o Z \cong (X\o Z)\oplus(Y\o Z) \quad\text{and}\quad X\o 0 \cong 0
$$
which describe the distributivity of the tensor product over the sum,
satisfying the coherence axioms of Laplaza \cite{Laplaza}. If $\o$ is the
categorical product, we say that $\mathcal{R}$ is a Cartesian rring\xspace.
The Grothendieck group $K_0(-)$ is a functor from rrings\xspace to commutative
rings. Given an object $X$ of a rring\xspace $\mathcal{R}$, denote by $[X]$ its
isomorphism class; then $K_0(\mathcal{R})$ is generated as an abelian group by the
isomorphism classes of objects, with the relation
$$
[X] + [Y] = [X\oplus Y] .
$$
The product on $K_0(\mathcal{R})$ is given by the formula $[X]\*[Y]=[X\o
Y]$. (Here, we suppose that the isomorphism classes of objects of $\mathcal{R}$
form a set; this hypothesis will always be fulfilled in this paper.)
If $\mathcal{R}$ and $\mathcal{S}$ are two rrings\xspace, $\mathcal{R}\o\mathcal{S}$ is a rring\xspace whose objects
are formal sums of tensor products $X\o Y$, where $X$ and $Y$ are objects
of $\mathcal{R}$ and $\mathcal{S}$ respectively; note that $K_0(\mathcal{R}\o\mathcal{S})\cong K_0(\mathcal{R})\o
K_0(\mathcal{S})$.
Recall that an additive category over a commutative ring $R$ is a category
$\mathcal{R}$ such that the set of morphisms $\mathcal{R}(X,Y)$ is a $R$-module for all
objects $X$ and $Y$, the composition maps $\mathcal{R}(Y,Z)\o_K\mathcal{R}(X,Y)\to\mathcal{R}(X,Z)$
are $R$-linear, and every finite set of objects has a direct sum. A
\textbf{Karoubian category} over a ring $R$ is an additive category over
$R$ such that every idempotent has an image, denoted $\im(p)$. (Karoubian
categories are also sometimes known as pseudo-abelian categories.)
\begin{definition}
A \textbf{Karoubian rring\xspace} $\mathcal{R}$ is a rring\xspace which is a Karoubian category,
and whose sum $X\oplus Y$ is the direct sum.
\end{definition}
An example of a Karoubian rring\xspace is the category ${\mathsf{Proj}}$ of finitely
generated projective $R$-modules.
If $\mathcal{R}$ is a Karoubian rring\xspace and $G$ is a group, let $[G,\mathcal{R}]$ be the
Karoubian rring\xspace of $G$-modules in $\mathcal{R}$, that is, functors from $G$ to
$\mathcal{R}$. If $X$ and $Y$ are objects of $[G,\mathcal{R}]$, the $R$-module of morphisms
$\mathcal{R}(X,Y)$ carries a natural $R[G]$-module structure, given by the formula
$f^g=g^{-1}\*f\*g$.
There is a natural bifunctor $V\boxtimes X$, the external tensor product,
from $[G,{\mathsf{Proj}}]\times[G,\mathcal{R}]$ to $[G,\mathcal{R}]$, characterized by the identity
of $R[G]$-modules
$$
\mathcal{R}(V\boxtimes X,Y) \cong V \o \mathcal{R}(X,Y) .
$$
For the finitely generated free module $R[G]^n$, we have
$$
R[G]^n\boxtimes X = \bigoplus_{g\in G} X^{\oplus n} .
$$
For general $V$, we realize $V$ as the image of an idempotent $p$ in a free
module $R[G]^n$, and define $V\boxtimes X$ to be the image of the
corresponding idempotent in $R[G]^n\boxtimes X$. Using the external tensor
product, we may embed $[G,{\mathsf{Proj}}]$ into $[G,\mathcal{R}]$ by the functor $V\DOTSB\mapstochar\to
V\boxtimes{1\!\!1}$.
If $G$ is a group whose order is invertible in $R$, the functor $(-)^G$ of
$G$-invariants from $[G,\mathcal{R}]$ to $\mathcal{R}$ is defined by taking the image of
the idempotent automorphism of $\mathcal{R}$
$$
p = \frac{1}{|G|} \sum_{g\in G} g .
$$
From now on, we restrict attention to groups satisfying this condition.
If $H$ is a subgroup of $G$, the induction functor
$\Ind^G_H:[H,\mathcal{R}]\to[G,\mathcal{R}]$ is defined by the formula
$$
\Ind^G_H X = (R[G]\boxtimes X)^H .
$$
Here, we use the $G\times H$-module structure of $R[G]$, where $G$ acts on
the left and $H$ acts on the right.
The following is a generalization of the Peter-Weyl theorem to Karoubian
categories.
\begin{theorem}[Peter-Weyl] \label{Peter-Weyl}
If $\mathcal{R}$ is a Karoubian rring\xspace over a commutative ring $R$ and $G$ is a
group whose order is invertible in $R$, the composition
$$
[G,{\mathsf{Proj}}] \o \mathcal{R} \hookrightarrow [G,{\mathsf{Proj}}] \o [G,\mathcal{R}]
\xrightarrow{\boxtimes} [G,\mathcal{R}]
$$
is an equivalence of categories.
\end{theorem}
\begin{proof}
Since the order of $G$ is invertible in $R$, the group algebra $R[G]$ is
semi-simple, and may be written
$$
R[G] \cong \bigoplus_a \End(V_a) \cong \bigoplus_a V_a \o V_a^* ,
$$
where we sum over the isomorphism classes of irreducible representations
$\{V_a\}$ of $G$. This permits us to rewrite the induction functor as
$$
\Ind_H^GX = (R[G]\o X)^H \cong \bigoplus_a V_a \o (V_a^*\boxtimes X)^H .
$$
Taking $H=G$, and recalling that $\Ind_G^G$ is equivalent to the identity
functor, we obtain the desired equivalence between $[G,{\mathsf{Proj}}]\o\mathcal{R}$ and
$[G,\mathcal{R}]$.
\end{proof}
\section{$\SS$-modules in Karoubian rrings\xspace}
Let $\SS$ be the category of permutations $\coprod_{n=0}^\infty\SS_n$ and
let $\mathcal{R}$ be a rring\xspace. A bounded $\SS$-module in $\mathcal{R}$ is an object $\mathcal{X}$ of
$$
[\SS,\mathcal{R}] = \bigoplus_{n=0}^\infty \, [\SS_n,\mathcal{R}] ,
$$
in other words, a sequence $\{\mathcal{X}(n)\mid n\ge0\}$ of $\SS_n$-modules in
$\mathcal{R}$ such that $\mathcal{X}(n)=0$ for $n\gg0$. Let ${1\!\!1}_n$ denote the $\SS$-module
such that ${1\!\!1}_n(n)$ is the trivial $\SS_n$-module, while ${1\!\!1}_n(k)=0$ for
$k\ne n$.
The category $[\SS,\mathcal{R}]$ is itself a rring\xspace:
\begin{enumerate}
\item the sum of two $\SS$-modules is $(\mathcal{X}\oplus\mathcal{Y})(n)=\mathcal{X}(n)\oplus\mathcal{Y}(n)$;
\item the product of two $\SS$-modules is defined using induction:
$$
(\mathcal{X}\o\mathcal{Y})(n) = \bigoplus_{j+k=n} \Ind_{\SS_j\times\SS_k}^{\SS_n} \mathcal{X}\o\mathcal{Y} ;
$$
\item the unit of the product is ${1\!\!1}_0$.
\end{enumerate}
Denote the Grothendieck group of the rring\xspace $[\SS,\mathcal{R}]$ by $K_0^\SS(\mathcal{R})$.
There is another monoidal structure $\mathcal{X}\circ\mathcal{Y}$ on $[\SS,\mathcal{R}]$, called
plethysm. If $\lambda$ is a partition of $n$, let $\SS_\lambda =
\SS_{\lambda_1}\times\dots\times\SS_{\lambda_{\ell(\lambda)}}\subset\SS_n$,
and let $N(\SS_\lambda)$ be the normalizer of $\SS_\lambda$ in $\SS_n$. The
quotient $W(\SS_\lambda)=N(\SS_\lambda)/\SS_\lambda$ may be identified with
$$
\{ \sigma\in\SS_{\ell(\lambda)} \mid \text{$\lambda_{\sigma(i)}=\lambda_i$
for $1\le i\le\ell(\lambda)$} \} \subset \SS_{\ell(\lambda)} .
$$
Given bounded $\SS$-modules $\mathcal{X}$ and $\mathcal{Y}$, we obtain an action of
$N(\SS_\lambda)$ on the tensor product
$$
\mathcal{X}(\ell(\lambda)) \o \bigotimes_{1\le i\le\ell(\lambda)} \mathcal{Y}(\lambda_i) .
$$
\textbf{Plethysm} is the monoidal structure (not symmetric) defined by the
formula
$$
(\mathcal{X}\circ\mathcal{Y})(n) = \bigoplus_{\lambda\vdash n} \bigoplus_{k=0}^\infty
\Ind^{\SS_n}_{N(\SS_\lambda)}
\biggl( \mathcal{X}(\ell(\lambda)+k) \o \bigotimes_{1\le i\le\ell(\lambda)}
\mathcal{Y}(\lambda_i) \o \mathcal{Y}(0)^{\o k} \biggr)^{\SS_k} ,
$$
and with unit ${1\!\!1}_1$.
\begin{lemma}
Let $\mathcal{R}$ be a Karoubian rring\xspace over a field of characteristic zero. The
Grothendieck group $K_0^\SS(\mathcal{R})$ is a pre-$\lambda$-ring, with
$\sigma$-operations characterized by the formula
$$
\sigma_n([\mathcal{X}]) = \bigl[{1\!\!1}_n\circ\mathcal{X}\bigr] ,
$$
where $\mathcal{X}$ is a bounded $\SS$-module.
\end{lemma}
\begin{proof}
We must prove that for bounded $\SS$-modules $\mathcal{X}$ and $\mathcal{Y}$,
\begin{equation} \label{pre}
\sigma_n([\mathcal{X}]+[\mathcal{Y}]) = \sum_{i=0}^n \sigma_i([\mathcal{X}]) \* \sigma_{n-i}([\mathcal{Y}]) .
\end{equation}
Observe that ${1\!\!1}_n\circ(\mathcal{X}\oplus\mathcal{Y})$ equals
$$
\bigoplus_{i=0}^n \bigoplus_{\lambda\vdash i} \bigoplus_{\mu\vdash n-i}
\bigoplus_{j,k=0}^\infty \Ind^{\SS_n}_{N(\SS_\lambda)\times N(\SS_\mu)}
\biggl( \bigotimes_{1\le i\le\ell(\lambda)} \mathcal{X}(\lambda_i) \o \mathcal{X}(0)^{\o j}
\o \bigotimes_{1\le i\le\ell(\mu)} \mathcal{Y}(\mu_i) \o \mathcal{Y}(0)^{\o k}
\biggr)^{\SS_j\times\SS_k}
$$
Since
$$
\Ind^{\SS_n}_{N(\SS_\lambda)\times N(\SS_\mu)}V\o W
= \Ind^{\SS_n}_{\SS_i\times\SS_{n-i}} \Bigl(
\Ind^{\SS_i}_{N(\SS_\lambda)} V \o \Ind^{\SS_{n-i}}_{N(\SS_\mu)} W \Bigr) ,
$$
it follows that
$$
{1\!\!1}_n\circ(\mathcal{X}\oplus\mathcal{Y}) \cong \bigoplus_{i=0}^n
({1\!\!1}_i\circ\mathcal{X})\o({1\!\!1}_{n-i}\circ\mathcal{Y}) ,
$$
proving \eqref{pre} for elements of $K_0^\SS(\mathcal{R})$ of the form $[\mathcal{X}]$ and
$[\mathcal{Y}]$. The definition of the sigma operations on virtual elements
$[\mathcal{X}_0]-[\mathcal{X}_1]$ is now forced by \eqref{invert}:
$$
\sigma_n([\mathcal{X}_0]-[\mathcal{X}_1]) = \sum_{k=0}^\infty
\sum_{\substack{j_\ell>0 \\ i+j_1+\dots+j_\ell=n}}
(-1)^k \sigma_i([\mathcal{X}_0]) \sigma_{j_1}([\mathcal{X}_1]) \dots \sigma_{j_k}([\mathcal{X}_1]) .
$$
\def{}
\end{proof}
\begin{lemma} \label{atiyah}
There is an isomorphism of $\lambda$-rings $K_0^\SS({\mathsf{Proj}})\cong\Lambda$.
\end{lemma}
\begin{proof}
The pre-$\lambda$-ring $K_0^\SS({\mathsf{Proj}})$ is the sum of abelian groups
$
K_0^\SS(\mathcal{R}) = \bigoplus_{n=0}^\infty R(\SS_n) ,
$
where $R(\SS_n)=K_0([\SS_n,{\mathsf{Proj}})$ is the abelian group underlying the
virtual representation ring of $\SS_n$. The identification of
$K_0^\SS({\mathsf{Proj}})$ with $\Lambda$ is via the Frobenius characteristic
$\ch:R(\SS)\to\Lambda$, which sends the irreducible representation
$V_\lambda$ associated to the partition $\lambda$ to the Schur function
$s_\lambda$. The Frobenius characteristic is given by the explicit formula
$$
\ch_n(V) = \frac{1}{n!} \sum_{\sigma\in\SS_n} \Tr_V(\sigma) p_\sigma ,
$$
where $p_\sigma$ is the monomial in the power sums obtained by taking one
factor $p_k$ for each cycle of $\sigma$ of length $k$. For the proof that
$\ch(\dots)$ is a map of $\lambda$-rings, see Knutson \cite{Knutson} or
Appendix~A of Macdonald \cite{Macdonald}.
\end{proof}
Using these lemmas and the Peter-Weyl Theorem, we will show that
$K_0^\SS(\mathcal{R})$ is a $\lambda$-ring for any Karoubian rring\xspace over a field of
characteristic zero. First, we prove some simple lemmas which are of
interest in their own right.
Plethysm is distributive on the left with respect to sum.
\begin{lemma} \label{additive}
$
(\mathcal{X}_1\oplus\mathcal{X}_2)\circ\mathcal{Y} \cong (\mathcal{X}_1\circ\mathcal{Y}) \oplus (\mathcal{X}_2\circ\mathcal{Y})
$
\end{lemma}
\begin{proof}
Clear.
\end{proof}
It is also distributive on the left with respect to product.
\begin{lemma} \label{multiplicative}
$
(\mathcal{X}_1\o\mathcal{X}_2)\circ\mathcal{Y} \cong (\mathcal{X}_1\circ\mathcal{Y}) \o (\mathcal{X}_2\circ\mathcal{Y})
$
\end{lemma}
\begin{proof}
By Lemma \ref{additive}, it suffices to check this formula when
$\mathcal{X}_1(j)=X_1$, $\mathcal{X}_2(k)=X_2$, $\mathcal{X}_1(i)=0$, $i\ne j$ and $\mathcal{X}_1(i)=0$, $i\ne
k$. We have
\begin{multline*}
\bigl((\mathcal{X}_1\o\mathcal{X}_2) \circ \mathcal{Y}\bigr)(n) \\
= \bigoplus_{q=0}^n
\bigoplus_{\substack{\lambda\vdash n \\ \ell(\lambda)+q=j+k}}
\Ind^{\SS_n}_{N(\SS_\lambda)} \biggl( \Ind^{\SS_{j+k}}_{\SS_j\times\SS_k}
\bigl( X_1\o X_2 \bigr) \o \bigotimes_{1\le i\le\ell(\lambda)}
\mathcal{Y}(\lambda_i) \o \mathcal{Y}(0)^{\o q} \biggr)^{\SS_q} .
\end{multline*}
But we have
\begin{multline*}
\bigoplus_{\substack{\lambda\vdash n \\ \ell(\lambda)+q=j+k}} \biggl(
\Ind^{\SS_{j+k}}_{\SS_j\times\SS_k} \bigl( X_1\o X_2 \bigr) \o
\bigotimes_{1\le i\le\ell(\lambda)} \mathcal{Y}(\lambda_i) \o \mathcal{Y}(0)^{\o q}
\biggr)^{\SS_q} \cong \bigoplus_{p=0}^q \bigoplus_{i=0}^n \\
\bigoplus_{\substack{\mu\vdash i \\ \ell(\mu)+p=j}}
\biggl( X_1 \o \bigotimes_{1\le i\le\ell(\mu)} \mathcal{Y}(\lambda_i) \o
\mathcal{Y}(0)^{\o p} \biggr)^{\SS_q}
\o \bigoplus_{\substack{\lambda\vdash n-i \\ \ell(\lambda)+q-p=k}}
\biggl( X_2 \o \bigotimes_{1\le i\le\ell(\lambda)} \mathcal{Y}(\lambda_i) \o
\mathcal{Y}(0)^{\o q} \biggr)^{\SS_q} ,
\end{multline*}
from which the lemma follows.
\end{proof}
\begin{lemma} \label{associative}
If $\mathcal{V}$ is a bounded $\SS$-module in ${\mathsf{Proj}}$ and $\mathcal{X}$ is a bounded
$\SS$-module in $\mathcal{R}$,
$$
\ch(\mathcal{V}) \circ [\mathcal{X}] = [\mathcal{V}\circ\mathcal{X}] .
$$
\end{lemma}
\begin{proof}
By Lemma \ref{additive}, we may assume that $\mathcal{V}$ is an irreducible
$\SS_n$-module $V_\lambda$. It remains to show that
$\sigma_\lambda([\mathcal{X}])=[V_\lambda\circ\mathcal{X}]$ for all partitions $\lambda$.
By Lemma \ref{multiplicative}, we see that for any partition $\mu$ with
$\ell=\ell(\mu)$, we have
$$
\bigl( {1\!\!1}_{\mu_1} \o \dots \o {1\!\!1}_{\mu_\ell} \bigr) \circ \mathcal{X}
\cong \bigl( {1\!\!1}_{\mu_1} \o \mathcal{X} \bigr) \o \dots \o
\bigl( {1\!\!1}_{\mu_\ell} \circ \mathcal{X} \bigr) .
$$
Taking the class in $K_0^\SS(\mathcal{R})$ of both sides, we see that
$$
\bigl[ \bigl( {1\!\!1}_{\mu_1} \o \dots \o {1\!\!1}_{\mu_\ell} \bigr) \circ \mathcal{X} \bigr] =
\sigma_{\mu_1}([\mathcal{X}]) \dots \sigma_{\mu_\ell}([\mathcal{X}]) .
$$
The irreducible representation $V_\lambda$ is a linear combination of
representations ${1\!\!1}_{\mu_1}\o\dots\o{1\!\!1}_{\mu_\ell}$ with integral
coefficients, and by Lemma \ref{atiyah}, the Schur function $s_\lambda$ is
a linear combination of symmetric functions $h_{\mu_1}\o\dots\o
h_{\mu_\ell}$ with the same coefficients; the proof is completed by
application of Lemma \ref{additive}.
\end{proof}
\begin{theorem}
The Grothendieck group $K_0^\SS(\mathcal{R})$ of a Karoubian rring\xspace $\mathcal{R}$ over a field
of characteristic zero is a $\lambda$-ring.
\end{theorem}
\begin{proof}
If $f=\ch(\mathcal{V})$ and $g=\ch(\mathcal{W})$, where $\mathcal{V}$ and $\mathcal{W}$ are bounded
$\SS$-modules in ${\mathsf{Proj}}$, and $x=[\mathcal{X}]$, where $\mathcal{X}$ is a bounded
$\SS$-module in $\mathcal{R}$, it follows from Lemma \ref{associative} that
$$
f \circ \bigl( g \circ x \bigr) = \ch\bigl( \mathcal{V} \circ (\mathcal{W}\circ\mathcal{X}) \bigr)
= \ch\bigl( (\mathcal{V}\circ\mathcal{W}) \circ \mathcal{X} \bigr) = \ch(\mathcal{V}\circ\mathcal{W}) \circ x .
$$
Since $\ch$ is a morphism of $\lambda$-rings, we see that
$\ch(\mathcal{V}\circ\mathcal{W})=f\circ g$, and from which we obtain the formula
\eqref{lambda-ring} characterizing $\lambda$-rings in this case:
$$
f\circ(g\circ x) = (f\circ g) \circ x .
$$
It only remains to extend \eqref{lambda-ring} to virtual elements
$g=\ch(\mathcal{W}_0)-\ch(\mathcal{W}_1)$ and $x=[\mathcal{X}_0]-[\mathcal{X}_1]$. Both sides of
\eqref{lambda-ring} are polynomial functions of $g\in\Lambda$ and $x\in
K_0^\SS(\mathcal{R})$ and hence must coincide, since they are equal on a cone with
non-empty interior.
\end{proof}
It follows that the Grothendieck group $K_0(\mathcal{R})$ is a $\lambda$-ring,
namely the sub-$\lambda$-ring of $K_0^\SS(\mathcal{R})$ consisting of virtual
objects such that $X(n)=0$, $n>0$. The Peter-Weyl Theorem now has the
following consequence.
\begin{theorem}
If $\mathcal{R}$ is a Karoubian rring\xspace over a field of characteristic zero, there is
an isomorphism $K_0^\SS(\mathcal{R})\cong\Lambda\o K_0(\mathcal{R})$ of $\lambda$-rings.
\end{theorem}
\begin{proof}
The Peter-Weyl Theorem gives isomorphisms of rings
$$
\Lambda \o K_0(\mathcal{R}) \xleftarrow{\ch\o1} K_0^\SS({\mathsf{Proj}}) \o K_0(\mathcal{R})
\xrightarrow{\boxtimes} K_0^\SS(\mathcal{R}) .
$$
The first of these arrows is an isomorphism of $\lambda$-rings by Lemma
\ref{atiyah}. As rings, both $K_0^\SS({\mathsf{Proj}})\o K_0(\mathcal{R})$ and $K_0^\SS(\mathcal{R})$
are generated by $K_0(\mathcal{R})$ and $[{1\!\!1}_n]$, $n\ge1$, and $\boxtimes$ respects
the $\sigma$-operations of these elements, proving that it is a map of
$\lambda$-rings.
\end{proof}
\section{The main result}
If $\mathcal{R}$ is a Karoubian rring\xspace, denote by $\mathcal{R}[\mathbb{N}]$ the Karoubian ring of
bounded sequences
$$
(A^0,A^1,A^2,\dots\mid\text{$A^n=0$ for $n\gg0$}) .
$$
The sum on $\mathcal{R}[\mathbb{N}]$ is defined by $(A\oplus\mathcal{Y})^n=A^n\oplus\mathcal{Y}^n$, while the
product is defined by
$$
(A\o B)^n = \bigoplus_{i+j=n} A^i \o B^j .
$$
\begin{definition}
A \textbf{K\"unneth functor} with values in the Karoubian rring\xspace $\mathcal{R}$ is a
rring\xspace functor $\GS$ from the Cartesian {rring\xspace} ${\mathsf{Var}}$ of quasi-projective
varieties and open embeddings to $\mathcal{R}[\mathbb{N}]$.
\end{definition}
In other words, a functor $\GS:{\mathsf{Var}}\to\mathcal{R}[\mathbb{N}]$ is a K\"unneth functor if
there are natural isomorphisms
\begin{gather*}
\textstyle
\GS^i(X\coprod Y)\cong\GS^i(X)\oplus\GS^i(Y) , \\
\GS^n(X\times Y) \cong \bigoplus_{n=i+j} \GS^i(X)\o\GS^j(Y) .
\end{gather*}
If $\GS=\{\GS^n\}$ is a K\"unneth functor, denote by $\Serre(X)$ the
associated Euler characteristic
$$
\Serre(X) = \sum_{n=0}^\infty (-1)^n [\GS^n(X)]
$$
in the Grothendieck group $K_0(\mathcal{R})$.
\begin{definition}
A \textbf{Serre functor} with values in $\mathcal{R}$ is a K\"unneth functor $\GS$
such that for any closed sub-variety $Z$ of $X$,
$$
\Serre(X) = \Serre(X\setminus Z) + \Serre(Z) .
$$
\end{definition}
If $\GS$ is a Serre functor and $X=X^0\subset X^1\subset X^2\subset\dots$
is a filtered quasi-projective variety such that $X^n=\emptyset$ for
$n\gg0$, we have
\begin{equation} \label{gr}
\Serre(\gr X) \equiv \sum_n \Serre(\gr^nX) = \Serre(X) .
\end{equation}
Here are two examples of Serre functors:
\begin{enumerate}
\item The category of mixed Hodge structures over $\mathbb{C}$ is a rring\xspace, whose
Grothendieck group may be identified with the polynomial ring $\mathbb{Z}[u,v]$ by
means of the Serre polynomial \eqref{Serre}. The functor $\GS^n(X)$ which
takes a quasi-projective variety $X$ to the mixed Hodge structure
$(H^n_c(X,\mathbb{C}),F,W)$ over $\mathbb{C}$ is a Serre functor. The associated
characteristic $\Serre(X)$ may be identified with the Serre polynomial.
\item Gillet and Soul\'e \cite{GS} have constructed a functor to the
homotopy category of chain complexes of (pure effective rational) Chow
motives; let $\GS^n(X)$ be the $n$th cohomology of this complex.
\end{enumerate}
If $\mathcal{R}$ is a rring\xspace, let $\mathsf{T}:\mathcal{R}\to\[\SS,\mathcal{R}\]$ be the rring\xspace functor with
$\mathsf{T}(X,n)=X^n$. (More precisely, $\mathsf{T}(X,n)$ is defined by induction:
$\mathsf{T}(X,0)={1\!\!1}$, and $\mathsf{T}(X,n)=\mathsf{T}(X,n-1)\o X$.) The following result is a
generalization of Macdonald's formula \cite{Macdonald-symmetric} for the
Poincar\'e polynomial of the symmetric power $S^nX=X^n/\SS_n$.
\begin{proposition} \label{Macdonald}
If $X$ is a quasi-projective variety,
$$
\Serre(\mathsf{T}(X)) = \Exp\bigl(p_1\Serre(X)\bigr) \in \Hat{K}^\SS_0(\mathcal{R}) .
$$
Here, $\Serre(\mathsf{T}(X))$ denotes the class $n\DOTSB\mapstochar\to\Serre(\mathsf{T}(X,n))$ in the
Grothendieck group $\Hat{K}^\SS_0(\mathcal{R})$.
\end{proposition}
\begin{proof}
Since $\GS$ is a rring\xspace-functor, $\GS\*\mathsf{T}=\mathsf{T}\*\GS$. By the Peter-Weyl
Theorem,
$$
\GS(\mathsf{T}(X,n)) = \mathsf{T}[\GS(X)](n)
= \bigoplus_{\lambda\vdash n} V_\lambda \boxtimes \bigl( V_\lambda^*\o
\GS(X)^{\o n} \bigr)^{\SS_n} .
$$
Descending to the Grothendieck group, we see that
$$
\Serre(\mathsf{T}(X,n)) = \bigoplus_{\lambda\vdash n} s_\lambda \o
\sigma_\lambda(\Serre(X)) \in \Lambda_n \o K_0(\mathcal{R}) \subset K_0^\SS(\mathcal{R}) .
$$
Summing over $n\ge0$, and applying Cauchy's formula \eqref{Cauchy}, we see
that
$$
\Serre(\mathsf{T}(X)) = \exp \Bigl( \sum_{k=1}^\infty
\frac{p_k\o\psi_k\Serre(X)}{k} \Bigr) \in \Hat{K}^\SS_0(\mathcal{R}) .
$$
The proposition now follows by the definition of $\Exp(\dots)$.
\end{proof}
Consider the following decreasing filtration on the $\SS$-module $\mathsf{T}(X)$,
where $X$ is a quasi-projective variety:
$$
\mathsf{T}^i(X)(n) = \{ (z_1,\dots,z_n) \in X^n \mid \text{$\{z_1,\dots,z_n\}$
has cardinality at most $n-i$} \} .
$$
Let $\gr^i\mathsf{T}(X)=\mathsf{T}^i(X)\setminus\mathsf{T}^{i+1}(X)$ be the associated graded
$\SS$-module.
\begin{lemma} \label{gr-T}
Let ${\mathsf{Z}}$ be the object of $\hom[1]{{\mathsf{Var}}}$
$$
{\mathsf{Z}}(n) = \begin{cases} \mathbb{A}^0 , & n>0 , \\ \emptyset , & n=0 .
\end{cases}$$
Then $\gr\mathsf{T}(X)=\mathsf{F}(X)\circ{\mathsf{Z}}$; in particular, $\gr^0\mathsf{T}(X)=\mathsf{F}(X)$.
\end{lemma}
\begin{proof}
This lemma reflects the fact that an element of $\gr^i\mathsf{T}(X,n)$ determines,
and is determined by, a partition of the set $\{1,\dots,n\}$ into $n-i$
disjoint subsets, together a point in $\mathsf{F}(X,n-i)$.
\end{proof}
We now arrive at the main theorem of this paper.
\begin{theorem} \label{MAIN}
Let $X$ be a quasi-projective variety over $\mathbb{C}$. If $\GS$ is a Serre
functor and $V_\lambda$ is an irreducible representation of $\SS_n$,
$$
\Serre(\mathsf{F}(X,n),V_\lambda) = \Phi_\lambda(\Serre(X)) .
$$
\end{theorem}
\begin{proof}
If $\GS$ is Serre functor, \eqref{gr} and Lemma \ref{gr-T} show that
$$
\Serre(\mathsf{T}(X)) = \Serre(\gr\mathsf{T}(X)) = \Serre(\mathsf{F}(X)) \circ \Serre({\mathsf{Z}}) .
$$
To calculate $\Serre(\mathsf{F}(X))$, we invert the operation $-\circ\Serre({\mathsf{Z}})$
on $\Hat{K}^\SS_0(\mathcal{R})$. Indeed, $\Serre({\mathsf{Z}})=\Exp(p_1)-1$ and by Lemma
\ref{Cadogan},
\begin{align*}
\Serre(\mathsf{F}(X)) &= \Serre(\mathsf{F}(X)) \circ \bigl( \Exp(p_1) - 1 \bigr)
\circ \Bigl( \sum_{n=1}^\infty \frac{\mu(n)}{n} \log(1+p_n) \Bigr) \\
&= \Serre(\mathsf{T}(X)) \circ \Bigl( \sum_{n=1}^\infty \frac{\mu(n)}{n}
\log(1+p_n) \Bigr) .
\end{align*}
By Proposition \ref{Macdonald}, this equals
\begin{multline*}
\exp \Bigl( \sum_{k=1}^\infty \frac{p_k\*\psi_k\Serre(X)}{k} \Bigr) \circ
\Bigl( \sum_{\ell=1}^\infty \frac{\mu(\ell)}{\ell} \log(1+p_\ell) \Bigr)
= \exp \Bigl( \sum_{k=1}^\infty \sum_{\ell=1}^\infty \frac{\mu(\ell)}{k\ell}
\log(1+p_{k\ell}) \psi_k\Serre(X)\Bigr) \\
= \exp \Bigl( \sum_{n=1}^\infty \sum_{d|n} \frac{\mu(n/d)}{n}
\log(1+p_n) \psi_d\Serre(X) \Bigr) ,
\end{multline*}
from which the theorem follows by extracting the coefficient of the Schur
function $s_\lambda$ on both sides.
\end{proof}
The concise formulation
$$
\Serre(\mathsf{F}(X)) = \Exp(\Log(1+p_1)\Serre(X))
$$
of this result makes the resemblance with the formula
$\Serre(\mathsf{T}(X))=\Exp(p_1\Serre(X))$ clearer.
In the special cases $\lambda=(n)$ or $\lambda=(1^n)$, when $\Phi_\lambda$
is given by the explicit formula of Corollary \ref{braid}, we obtain the
following corollary.
\begin{corollary}
If $\Serre(X)=\sum_{p,q}h_{pq}u^pv^q$ is the Serre polynomial of $X$, then
\begin{gather*}
\sum_{n=0}^\infty t^n \Serre(\mathsf{F}(X,n)/\SS_n) =
\frac{\sigma_t(X)}{\sigma_{t^2}(X)} = \prod_{p,q=0}^\infty
\left( \frac{1-t^2u^pv^q}{1-tu^pv^q} \right)^{h_{pq}} , \\
\sum_{n=0}^\infty t^n \Serre(\mathsf{F}(X,n),\varepsilon) = \sigma_{-t}(X)^{-1} =
\prod_{p,q=0}^\infty (1+tu^pv^q)^{h_{pq}} .
\end{gather*}
\end{corollary}
For example, if $X=\mathbb{C}$, $\mathsf{F}(\mathbb{C},n)/SS_n$ is the classifying space
$K(B_n,1)$ of the braid group $B_n$ on $n$ strands. Our formula becomes in
this case
$$
\sum_{n=0}^\infty t^n \Serre(\mathsf{F}(\mathbb{C},n)/\SS_n) = \frac{1-t^2uv}{1-tuv}
= 1 + t\mathsf{L} + t^2(\mathsf{L}^2-\mathsf{L}) + t^3(\mathsf{L}^3-\mathsf{L}^2) + \dots ,
$$
reflecting the isomorphism of rational cohomology groups
$H^\bullet(B_n,\mathbb{Q})\cong H^\bullet(\mathbb{G}_m,\mathbb{Q})$ as mixed Hodge structures.
\section{The Fulton-MacPherson compactification}
Fulton and MacPherson \cite{FM} have introduced a sequence of functors
$X\DOTSB\mapstochar\to X[n]$ from ${\mathsf{Var}}$ to $[\SS_n,{\mathsf{Var}}]$, with the following
properties.
\begin{enumerate}
\item If $X$ is projective, then so is $X[n]$.
\item There is natural transformation of functors $\mathsf{F}(X,n)\hookrightarrow
X[n]$, which is an embedding.
\item The complement $X[n]\setminus\mathsf{F}(X)$ is a divisor with normal
crossings.
\end{enumerate}
In this section, we calculate the equivariant Serre polynomial
$\Serre(X[n])$. Denote by $\FM(X)$ the functor $X\DOTSB\mapstochar\to(n\DOTSB\mapstochar\to X[n])$
from ${\mathsf{Var}}$ to $\[\SS,{\mathsf{Var}}\]$,
\subsection{Trees and $\SS$-modules}
Let $\Gamma(n)$, $n\ge2$, be the set of isomorphism classes of labelled
rooted trees with $n$ leaves, such that each vertex has at least two
branches. It is easily seen that $\Gamma(n)$
is finite: in fact, the generating function
\begin{equation} \label{enumerate}
x + \sum_{n=2}^\infty \frac{x^n |\Gamma(n)|}{n!}
\end{equation}
is the inverse under composition of $x-x^2-x^3-x^4-\dots$.
Given a tree $T\in\Gamma(n)$, denote by $\VERT(T)$ the set of vertices of
$T$; given a vertex $v\in\VERT(T)$, denote by $n(v)$ the valence of $v$
(its number of branches). Given a tree $T\in\Gamma(n)$ and an $\SS$-module
$\mathcal{V}$ in the rring\xspace $\mathcal{R}$, let $\mathcal{V}(T)$ be the object
\begin{equation} \label{V(T)}
\mathcal{V}(T) = \bigotimes_{v\in\VERT(T)} \mathcal{V}(n(v)) ,
\end{equation}
and let ${\mathbb{T}}\mathcal{V}(n)$ be the $\SS_n$-module
$$
{\mathbb{T}}\mathcal{V}(n) = \bigoplus_{T\in\Gamma(n)} \mathcal{V}(T) .
$$
Thus, ${\mathbb{T}}$ is a functor from $\hom[2]{\mathcal{R}}$ to itself. (Recall that
$\hom[2]{\mathcal{R}}$ is the full subcategory of $\SS$-modules such that
$X(0)=X(1)=0$.)
A proof of the following formula for $\mathcal{R}={\mathsf{Proj}}$ may be found in
\cite{modular}; however, the same proof works in general. Observe that this
theorem may be used to prove \eqref{enumerate}.
\begin{theorem} \label{revert}
The elements
$$
f = h_1 - \sum_{n=2}^\infty [\mathcal{V}] \quad\text{and}\quad
g = h_1 + \sum_{n=2}^\infty [{\mathbb{T}}\mathcal{V}]
$$
of $\Check{K}^\SS_0(\mathcal{R})$ satisfy the formula $f\circ g = g\circ f=h_1$.
\end{theorem}
\subsection{The varieties $\protect\overset{\circ}{\mathsf{P}}_k(n)$}
The algebraic groups $\mathbb{C}^k$ and ${\mathbb{G}}_m$ act on the affine space $\mathbb{C}^k$ by
translation and dilatation respectively; by functoriality, these actions
extend to $\mathsf{F}(\mathbb{C}^k,n)$. Denote by $G_k=\mathbb{C}^k\rtimes{\mathbb{G}}_m$ the semidirect
product of these groups, and by $\overset{\circ}{\mathsf{P}}_k(n)$, $n>1$, the quotient of the
configuration space $\mathsf{F}(\mathbb{C}^k,n)$ by the free $G_k$-action. This action is
$\SS_n$-equivariant, and $\overset{\circ}{\mathsf{P}}_k(n)$ is a smooth $\SS_n$-variety of
dimension $nk-k-1$. For example, $\overset{\circ}{\mathsf{P}}_k(2)$ is naturally isomorphic to the
projective space $\mathbb{CP}^{k-1}$, with trivial $\SS_2$-action.
\begin{proposition}
$$
\Serre\bigl( \overset{\circ}{\mathsf{P}}_k(n),\SS_n \bigr) =
\frac{\Serre(\mathsf{F}(\mathbb{C}^k,n),\SS_n)}{\Serre(\mathbb{C}^k)\Serre({\mathbb{G}}_m)} =
\frac{\Serre(\mathsf{F}(\mathbb{C}^k,n),\SS_n)}{\mathsf{L}^k(\mathsf{L}-1)}
$$
\end{proposition}
\begin{proof}
We start with a lemma.
\begin{lemma}
Let $G$ be an algebraic group and $P$ be a $G$-torsor with base $X=P/G$.
If the projection $P\to X$ is locally trivial in the Zariski topology,
$\Serre(P) = \Serre(G) \Serre(X)$.
\end{lemma}
\begin{proof}
We stratify $X$ by locally closed subvarieties $X_i$ of codimension $i$
over which the torsor $P$ is trivial. The strata are chosen inductively:
$X_{-1}$ is empty, while $X_i$ is a Zariski-open subset of $X\setminus
X_{i-1}$ over which $P$ is trivial. The formula follows, since
$$
\Serre(P) = \sum_i \Serre(P_i) = \sum_i \Serre(G) \Serre(X_i) .
$$
\def{}
\end{proof}
The action of $\mathbb{C}^k$ on $\mathsf{F}(\mathbb{C}^k,n)$ is not just locally, but globally,
trivial: a global section is given by $(z_1,\dots,z_n) \DOTSB\mapstochar\to
(z_1-{\bar{z}},\dots,z_n-{\bar{z}})$, where ${\bar{z}} = \frac{1}{n} \sum_{i=1}^n z_i$.
On the other hand, any free action of ${\mathbb{G}}_m$ on a variety is locally
trivial in the Zariski topology: free actions with quotient $X$ are
classified by $H^1(X_{\text{fl}},{\mathbb{G}}_m)$, locally trivial free actions with
quotient $X$ are classified by $H^1(X,{\mathbb{G}}_m)$, and these two groups are
isomorphic by Hilbert's Theorem 90 (see Proposition XI.5.1 of Grothendieck
\cite{SGA1}).
\end{proof}
\subsection{Stratification of $\FM(X)$}
The $\SS$-variety $\overset{\circ}{\mathsf{P}}_k$ has a natural compactification to a smooth
projective $\SS$-variety $\P_k$, which has a natural stratification. The
strata are labelled by trees $T\in\Gamma(n)$, and the stratum associated to
$T$ is isomorphic to $\overset{\circ}{\mathsf{P}}_k(T)$, in the notation of \eqref{V(T)}. It
follows from Theorem \ref{revert} that $\Serre(\P_k)$ is the inverse of
\begin{align*}
h_1 - \Serre\bigl( \overset{\circ}{\mathsf{P}}_k \bigr) &= p_1 - \frac{\prod_{n=1}^\infty
(1+p_n)^{\frac{1}{n}\sum_{d|n}\mu(n/d)\mathsf{L}^{kd}} - 1 - \mathsf{L}^kp_1}{\mathsf{L}^k(\mathsf{L}-1)} \\
& = \frac{\mathsf{L}^{k+1}p_1 + 1 - \prod_{n=1}^\infty
(1+p_n)^{\frac{1}{n}\sum_{d|n}\mu(n/d)\mathsf{L}^{kd}}}{\mathsf{L}^k(\mathsf{L}-1)} .
\end{align*}
under plethysm.
The importance of the spaces $\P^k(n)$ comes from the following result of
Fulton and MacPherson.
\begin{proposition}
The $\SS$-module $\FM(X)$ has a filtration such that
$$
\gr\FM(X) \cong \mathsf{F}(X)\circ\P_k .
$$
\end{proposition}
Since $X[n]$ is a projective $Q$-variety (it has singularities which are
quotients of affine space by a finite group), $\Serre(\FM(X))(n)$ equals
the $\SS_n$-equivariant Hodge polynomial of $X[n]$. The above proposition
shows that $\Serre(\FM(X))=\Serre(\mathsf{F}(X))\circ\Serre(\P_k)$, and leads to a
practical algorithm for the calculation of the $\SS_n$-equivariant Hodge
numbers of $X[n]$.
On forgetting the action of the symmetric groups $\SS_n$, we recover the
formula of Fulton and Macpherson for the Poincar\'e polynomials of
$\FM(X,n)$, in a form stated by Manin \cite{Manin:1}. On replacing $h_n$
by $x^n/n!$, we obtain
$$
1 + \sum_{n=1}^\infty x^n \Serre(X[n]) = (1+x)^{\Serre(X)} \circ
\biggl( \frac{\mathsf{L}^{k+1}x+1-(1+x)^{\mathsf{L}^k}}{\mathsf{L}^k(\mathsf{L}-1)} \biggr)^{-1}
$$
In this formula, we may take the limit $\mathsf{L}\to1$ using L'H\^opital's rule,
obtaining a formula for the Euler characteristic of $\FM(X,n)$:
$$
1 + \sum_{n=1}^\infty x^n \chi(X[n])
= (1+x)^{\chi(X)} \circ \bigl( (k+1)x - k(1+x)\log(1+x) \bigr)^{-1} .
$$
The one dimensional case has special interest, since the spaces $\overset{\circ}{\mathsf{P}}_1(n)$
and $\P_1(n)$ are naturally isomorphic to the moduli spaces $\mathcal{M}_{0,n+1}$
and $\overline{\mathcal{M}}_{0,n+1}$; this isomorphism comes about because the translations
and dilatations in one dimension generate the isotropy group of the point
$\infty\in\mathbb{CP}^1$ with respect to the action of the group $\PSL(2,\mathbb{C})$. This
identification means that the action of $\SS_n$ on these spaces is the
restriction of an action of $\SS_{n+1}$. We have calculated the
$\SS_{n+1}$-equivariant Serre polynomials of these spaces in
\cite{gravity}; in a sequel to this paper, we calculate the
$\SS_n$-equivariant Serre polynomial of $\overline{\mathcal{M}}_{1,n}$.
|
1995-11-01T06:20:17 | 9510 | alg-geom/9510017 | en | https://arxiv.org/abs/alg-geom/9510017 | [
"alg-geom",
"math.AG"
] | alg-geom/9510017 | Frank Sottile | Frank Sottile | Explicit Enumerative Geometry for the Real Grassmannian of Lines in
Projective Space | LaTeX 2e, 18 pages with three figures | Duke Math. J., 87 (1997) 59-85 | null | null | null | We extend the classical Schubert calculus of enumerative geometry for the
Grassmann variety of lines in projective space from the complex realm to the
real. Specifically, given any collection of Schubert conditions on lines in
projective space which generically determine a finite number of lines, we show
there exist real generic conditions determining the expected number of real
lines. Our main tool is an explicit description of rational equivalences which
also constitutes a novel determination of the Chow rings of these Grassmann
varieties.
| [
{
"version": "v1",
"created": "Tue, 31 Oct 1995 18:25:06 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Sottile",
"Frank",
""
]
] | alg-geom | \section{Introduction}
A basic problem in algebraic geometry is to describe the common
zeroes of a set of polynomials.
This is more difficult over non-algebraically closed fields.
For systems of polynomials with few monomials on a complex torus,
Khovanskii~\cite{Khovanskii_fewnomials} showed
that the real zeroes are at most a small fraction of
the complex zeroes.
Fulton (\cite{Fulton_introduction_intersection}, \S7.2) asked
how many solutions to a problem of
enumerative geometry can be real; in particular, how many of the
3264 conics tangent to five general real conics can be real.
He later showed that all, in fact, can be real.
Recently, this was independently rediscovered by Ronga, Tognoli, and
Vust~\cite{Ronga_Tognoli_Vust}.
Robert Spesier suggested the classical Schubert calculus of enumerative
geometry would be a
good testing ground for these questions.
For any problem of enumerating lines in ${\bf P}^n$ incident on
real linear subspaces in general position, we show
that all solutions can be real.
\medskip
A flag and a partition $\lambda = (\alpha,\beta)$ determine a Schubert
subvariety of the Grassmannian of lines in ${\bf P}^n$
of type $\lambda$, which has codimension
$|\lambda| = \alpha+\beta$.
Any generically transverse intersection of
Schubert varieties is rationally equivalent
to a sum of Schubert varieties.
The classical Schubert calculus gives algorithms for determining
how many of each type.
For partitions $\lambda^1,\ldots,\lambda^m$,
we describe a cycle $\Omega({\cal T})$ (depending upon $\lambda^1,
\ldots,\lambda^m$) which is
a sum of distinct Schubert varieties.
Let ${\cal G}= {\cal G}(\lambda^1,\ldots,\lambda^m)$
be the set of points of the Chow variety
representing generically transverse intersections
of Schubert varieties of types $\lambda^1,\ldots,\lambda^m$.
In \S 4, we show that ${\cal G}$ is unirational.
Cycles represented by the points of a rational curve on the Chow
variety are rationally equivalent.
In \S 3, we show
\medskip
\noindent{\bf Theorem A.} \ {\em
Let $X\in {\cal G}$.
Then there is a chain of rational curves between
$X$ and the cycle $\Omega({\cal T})$.
Furthermore, these curves may be explicitly
described and each lies in the
Zariski closure of ${\cal G}$.
In particular, the point representing
$\Omega({\cal T})$ is in the
Zariski closure of $\,{\cal G}$.}
\medskip
The proof of Theorem A constitutes an explicitly geometric
determination of the Schubert
calculus of enumerative geometry for lines in ${\bf P}^n$.
In fact, it shows these `Schubert-type' enumerative problems
may be solved {\em without} reference to the Chow ring, a traditional tool
in enumerative geometry.
We use it to compute products in the Chow ring.
Let $\sigma_\lambda$ be the rational equivalence class
of a Schubert variety of type $\lambda$.
\medskip
\noindent{\bf Theorem B. }
{\em
Let $c^\lambda$ be the number of components of $
\,\Omega({\cal T})$ of type
$\lambda$.
Then
$$
\prod_{i=1}^m \sigma_{\lambda^i} = \sum_\lambda c^\lambda \sigma_\lambda.
$$
}
Thus we derive the structure of these
Chow rings in a strong sense:
All products among classes from the Schubert basis
are expressed as linear combinations of basis elements
and these expressions are obtained by exhibiting
rational equivalences between a generically transverse intersection
of Schubert varieties and the cycle $\Omega({\cal T})$.
We believe this is the first non-trivial explicit description of
rational equivalences giving all products among a set
generators of the Chow group for any variety.
When $k= \mbox{\bf R}$, we show
\medskip
\noindent{\bf Theorem C.} \ {\em
Let $\lambda^1,\ldots,\lambda^m$ be partitions
with $|\lambda^1|+\cdots+|\lambda^m|$ equal to the dimension
of the Grassmannian of lines in $\mbox{\bf P}^n$.
Then there exists a nonempty classically open
subset in the product of $m$ real flag manifolds whose
corresponding Schubert varieties meet transversally,
with all points of intersection real.
}
\medskip
To the best of our knowledge, this is the first result
showing that a large class of non-trivial enumerative problems can have all
of their solutions real.
The construction of the cycles $\Omega({\cal T})$ and
rational curves of Theorem~A use
a `calculus of tableaux' outlined
in \S\ref{sec:calculus_of_tableaux} and extended in \S6,
where we define a non-commutative associative algebra with additive basis
the set of Young tableaux.
This algebra has surjections to the Chow rings of Grassmann
varieties and the algebra of symmetric functions.
However, it differs fundamentally from the plactic algebra
of Lascoux and
Sch\"utzenberger~\cite{Lascoux_Schutzenberger_monoid_plactic}.
In \S7, we ask which enumerative
problems may be solved over which (finite) fields
and give the answer for two classes of Schubert-type
enumerative problems.
We also show how some of our constructions may be carried out over
finite fields.
The rational equivalences we construct are a modification of the
classical method of degeneration.
This method may fail when applied to more than
a few conditions; an intersection typically
becomes improper before the conditions become special enough to
completely determine the intersection.
Considering deformations of intersection cycles, rather
than of conditions, an idea of
Chaivacci and
Escamilla-Castillo~\cite{Chiavacci_Escamilla-Castillo},
enables
us to deform generically transverse
intersections into sums of distinct Schubert varieties.
Theorem C is from our 1994
Ph.D. Thesis from the University of Chicago,
written under the direction of William Fulton.
We would like to thank William Fulton for suggesting these problems,
for his thoughtful advice, and above all for introducing us
to algebraic geometry.
\section{Preliminaries}
Let $k$ be an infinite field.
Varieties will be closed, reduced, projective
(not necessarily irreducible), and defined over $k$.
When $k={\mbox{\bf R}}$, let $X(\mbox{\bf R})$ be the points of $X$
with residue field $\bf R$.
We use the classical topology on $X(\mbox{\bf R})$.
A subset $Y$ (not necessarily algebraic) of a variety
$X$ is {\em unirational} if $Y$ contains the image $U$ of a dense
open subset of affine space under an algebraic morphism and
$Y\subset \overline{U}$.
A subset $Y\subset X(\mbox{\bf R})$ is {\em real unirational} if
$Y$ contains the image $U$ of a dense open subset of $\mbox{\bf R}^n$ under a
real algebraic map and $Y\subset \overline{U}$.
Let $X$ be a smooth variety,
$U$ and $W$ subvarieties of $X$, and set $Z = U\cap W$.
Then $U$ and $W$ meet
{\em generically transversally}
if $U$ and $V$ meet transversally at the generic point of each component of
$Z$.
Then $Z$ is generically reduced,
the fundamental cycle $[Z]$ of $Z$ is multiplicity free,
and in the Chow ring $A^*X$ of $X$:
$$
[U] \cdot [W] = [U\cap W] = [Z] = \sum_{i=1}^r\, [Z_i],
$$
where $Z_1,\ldots,Z_r$ are the irreducible components of $Z$.
\vspace{.5in}
\subsection{Chow Varieties}\label{sec:Chow}
Let $X$ be a projective variety.
The Chow variety $\mbox{\it Chow}\, X$ is a projective variety parameterizing
positive cycles
on $X$.
Let $U$ be a smooth variety and $W$ a subvariety
of $X\times U$ with equidimensional fibres over $U$.
Then there is an dense open subset $U'$ of $U$ such that
the association of a point $u$ of $U$ to the fundamental cycle of the
fibre $W_u$ determines a morphism $U'\rightarrow \mbox{\it Chow}\, X$.
If $U$ is a smooth curve, then $U' = U$.
Moreover, if $X$, $U$, and $W$ are defined over $k$, then so
are $\mbox{\it Chow}\, X$, $U'$, and the map $U' \rightarrow\mbox{\it Chow}\, X$
(\cite{Samuel}, \S I.9).
Cycles represented all points on a rational curve
in $\mbox{\it Chow}\, X$ are rationally equivalent.
We will use the same notation for a subscheme
of $X$, its fundamental cycle and the point representing that cycle
in $\mbox{\it Chow}\, X$.
\vspace{.5in}
\subsection{Grassmannians and Schubert Subvarieties}
For $S\subset {\mbox{\bf P}^n}$, let $\Span{S}$ be its linear span.
For a vector space $V$, let ${\bf P} V$ be
the projective space of all one dimensional subspaces of $V$.
Suppose $K = {\mbox{\bf P}}U$ and $M={\mbox{\bf P}}W$.
Set
$\mbox{Hom}(K,M) = \mbox{Hom}(U,W)$, the space of linear maps from
$U$ to $W$.
If $K\subset M$, set $M/K = {\mbox{\bf P}}(W/U)$.
A complete flag ${F\!_{\DOT}}$ is a collection of subspaces
$F_n\subset\cdots\subset F_1\subset F_0 = {\mbox{\bf P}^n}$,
where $\dim F_i = n-i$.
If $p>n$, set $F_p = \emptyset$.
For $0\leq s\leq n$, let $\mbox{\bf G}_s{\mbox{\bf P}^n}$ be the Grassmannian of $s$-dimensional
subspaces of ${\mbox{\bf P}^n}$.
We sometimes write
$\mbox{\bf G}^{n-s}{\mbox{\bf P}^n}$ for this variety.
Its dimension is $(n-s)(s+1)$.
For a partition $\lambda=(\alpha,\beta)$,
let $\mbox{\bf Fl}(\lambda)$
denote the variety of partial flags of {\em type} $\lambda$; those
$K\subset M$ with $K$ a $(n-\alpha-1)$-plane
and $M$ a $(n-\beta)$-plane.
A partial flag $K\subset M$
determines a {\em Schubert variety} $\Omega(K,M)$;
those lines contained in $M$ which also meet $K$.
The {\em type} of $\Omega(K,M)$ is the type, $\lambda=(\alpha,\beta)$,
of $K\subset M$ and its codimension is $|\lambda| =\alpha+\beta$.
If $\alpha=\beta$, then $\Omega(K,M) = \mbox{\bf G}_1M$, the Grassmannian of lines
in $M$.
If $\beta=0$, so $M={\mbox{\bf P}^n}$, then we write $\Omega_K$ for
this Schubert variety.
The tangent space to $\ell \in {\mbox{\bf G}_1{\mbox{\bf P}^n}}$ is naturally
identified with the linear space $\mbox{Hom}(\ell,{\mbox{\bf P}^n}/\ell)$.
It is not hard to verify the following Lemma, whose proof we omit.
\subsection{Lemma.}\label{lemma:one}
{\em
\begin{enumerate}
\item The smooth locus of $\Omega(K,M)$ consists of those
$\ell$ with $\ell \not\subset K$.
For such $\ell$,
$$
T_{\ell}\Omega(K,M) = \{ \phi\in \mbox{Hom}(\ell,{\mbox{\bf P}^n}/\ell)\,|\,
\phi(\ell)\subset M/\ell\mbox{ and }
\phi(\ell\cap K)\in (K+\ell)/\ell\}.
$$
\item Let $K,M\subset {\bf P}^n$.
Then $\Omega_K\bigcap\mbox{\bf G}_1M =
\Omega(K\cap M,\, M)$,
and this is transverse at the
smooth points of $\,\Omega(K\cap M,\, M)$
if and only if $K$ and $M$ meet properly in ${\mbox{\bf P}^n}$.
\item Let $K_i\subset M_i$, for $i=1,2$.
Then the intersection $\Omega(K_1,M_1)\bigcap
\Omega(K_2,M_2)$ is improper unless
$M_i$ meets both $K_j$ and $M_j$ properly for $i\neq j$.
\end{enumerate}
}
\medskip
An intersection of two Schubert varieties
may be generically transverse and reducible:
\subsection{ Lemma.}\label{lemma:component_calculation}
{\em
Let $F,P,N,H$ be linear subspaces of ${\bf P}^n$ and suppose
$H$ is a hyperplane not
containing $P$ or $N$,
$F\subsetneq P\cap H$, and $N$ meets $F$ properly.
Set $L=N\cap H$.
Then $\Omega(F,P)$ and $\Omega_L$ meet generically transversally,
$$
\Omega(F,P)\bigcap\Omega_L \ =\
\Omega(N\cap F,P) +\Omega(F,P\cap H)\bigcap\Omega_N,
$$
and the second component is itself a generically transverse intersection.
}
\medskip
\noindent{\bf Proof:}
The right hand side is a subset of the left, we show the other inclusion.
Let $\ell \in\Omega(F,P)\bigcap\Omega_L$.
If $\ell$ meets $L\cap F= N\cap F$, then
$\ell \in \Omega(N\cap F,P)$.
Otherwise, $\ell$ is spanned by its intersections with $F$ and $L$,
hence $\ell\subset \Span{F,L}\cap P \subset P\cap H$.
Since $N\cap P\cap H = L\cap P\cap H$, we see that
$\ell \in \Omega(F,P\cap H)\bigcap\Omega_N$.
Verifying these intersections are generically
transverse is left to the reader.
\QED
\subsection{A Calculus of Tableaux}\label{sec:calculus_of_tableaux}
The {\em Young diagram} of a partition $\lambda = (\alpha,\beta)$
is a two rowed array of boxes with
$\alpha$ boxes in the first row and $\beta$ in the second.
Note that $\alpha\geq \hfl{|\lambda|+1}\geq \hfl{|\lambda|}
\geq \beta$.
We make no distinction between a partition and its Young diagram.
A {\em Young tableau} $T$ of {\em shape} $\lambda$ is a
filling of the boxes of $\lambda$ with the integers $1,2,\ldots,|\lambda|$.
These integers increase left to right across each row and down each column.
Thus the $j$th entry in the second row of $T$ must be at
least $2j$.
Call $|\lambda|$ the {\em degree} of $T$, denoted $|T|$.
If $\alpha = \beta$, then $T$ is said to be {\em rectangular}.
Let $T$ be a tableau and $\alpha$ a positive integer.
Define $T* \alpha$ to be the set of all tableaux of degree
$|T|+\alpha$ whose first $|T|$ entries comprise $T$
and last $\alpha$ entries occur in distinct columns,
increasing from left to right.
For example:
$$
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\ *\ 4 \ =\
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\put(0,0){\usebox{\Shading}}
\put(32,11.8){\bf 5}
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\,,\
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\,,\
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\put(42,11.3){\bf 8}
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\right\}.
$$
Let $T(\alpha)\in T*\alpha$ be the tableau whose last
$\alpha$ entries lie in the first row.
Let $T^{+\alpha}\in T*\alpha$ be the tableau whose last $\alpha$ entries
lie in the second row, if that is possible.
Write $T^+$ for $T^{+1}$ and
define $T^{+\alpha}(\beta)$ to be $(T^{+\alpha})(\beta)$.
Then
$$
T*\alpha = \{T(\alpha),\, T^{+}(\alpha-1),\ldots,T^{+\alpha}\}.
$$
Given a set ${\cal T}$ of tableaux, define ${\cal T}*\alpha$ to be
$\bigcup_{T\in {\cal T}} T*\alpha$,
a disjoint union.
Similarly define ${\cal T}(\alpha)$, ${\cal T}^{+\alpha}$,
and ${\cal T}^{+\alpha}(\beta)$.
For $0\leq s\leq \alpha$, set
${\cal T}_{s,\alpha} = {\cal T}(\alpha) \bigcup {\cal T}^{+}(\alpha -1)
\bigcup\cdots\bigcup {\cal T}^{+s}(\alpha - s)$.
It follows that
${\cal T}_{s,\alpha} = {\cal T}_{s-1,\alpha} \bigcup
\left( {\cal T}^{+s} (\alpha-s)\right)$
and ${\cal T}*\alpha = {\cal T}_{\alpha,\alpha}$.
Finally, for positive integers $\alpha_1,\ldots,\alpha_m$, define
$\alpha_1*\cdots*\alpha_m$ to be
$\emptyset*\alpha_1*\cdots*\alpha_m$, where $\emptyset$
is the unique tableau of shape $(0,0)$.
\subsection{Arrangements}\label{sec:arrangements}
An {\em arrangement} ${\cal F}$ is a collection of $2n-3$ hyperplanes
$H_2,\ldots,H_{2n-2}$ and a complete flag
${F\!_{\DOT}}$ in ${\mbox{\bf P}^n}$,
which satisfy some additional assumtions.
For a tableau $T$ of degree at most $2n-2$, let $H_T$ be
$\bigcap \{H_j\,|\, j \in\mbox{ the second row of $T$}\}$.
For $p=2,3,\ldots,2n-2$, the linear spaces in an arrangement
are required to
satisfy:
\begin{enumerate}
\item $H_p \cap F_{\hfl{p}} = F_{\hfl{p}+1}$,
\item For tableaux $S, T$ of degrees at most $p-1$,
$H_T\cap H_S\subset H_p \Rightarrow H_T\cap H_S=F_{\hfl{p}+1}$.
\end{enumerate}
\noindent
The only obstruction to constructing arrangements over $k$ is that
$k$ have enough elements to find $H_p$ satisfying condition 2.
In particular, there are arrangements for any infinite field.
In \S\ref{thm:arrangement_finite_field}, we give an
estimate over which finite
fields it is possible to construct arrangements.
Let ${\cal F}$ be an arrangement and $T$ a tableau of shape
$(\alpha,\beta)$ with $\alpha + \beta \leq 2n-2$.
Define $\Omega(T) = \Omega(F_{\alpha+1}, H_T)$.
If $\alpha\geq n$, then $F_{\alpha+1} = \emptyset$, so
$\Omega(T) = \emptyset$.
\subsection{Lemma}\label{lemma:arrangements}
{\em Let ${\cal F}$ be an arrangement,
$S,T$ tableaux of degree $l\leq 2n{-}2$ and suppose $T$ has shape
$(\alpha,\beta)$ with $\alpha\leq n-1$.
Then
\begin{enumerate}
\item $F_{\alpha+1} \subset F_{\hfl{l+1}+1} \subset H_T$
and $F_{\alpha+1}\subset H_T$ is a partial flag of type $(\alpha,\beta)$.
\item If $\beta>0$, then $H_T\neq F_\beta$.
\item If either $\Omega(T) =\Omega(S)$ or
$H_T=H_S$, then $T=S$.
\end{enumerate}
}
\medskip
\noindent{\bf Proof:}
Since $\alpha\geq \hfl{l{+}1}$ and for $p\leq l$,
$F_{\hfl{l{+}1}}\subset H_p$, we see
$F_{\alpha+1}\subset F_{\hfl{l{+}1}{+}1}\subset H_T$,
by the definition of arrangement.
We show the codimension of $H_T$ is $\beta$
by induction on $\beta$.
When $\beta = 0$ or $1$, this is clear.
Removing, if necessary, extra entries from the first row of $T$,
we may assume $l$ is in the second row of $T$.
Let $S$ be the tableau obtained from $T$ by removing $l$.
By induction, $H_S$ has codimension $\beta-1$.
By the definition of arrangement, $H_S \not\subset H_l$.
Noting $H_T = H_S\cap H_l$ proves (1).
For (2), if $H_T=F_\beta$, then $F_\beta\subset H_{p}$, where $p$ is
the $\beta$th entry in the second row of $T$.
Thus $F_\beta \subset F_{\hfl{p}+1}$ so
$\beta\geq\hfl{p}{+}1$.
But this contradicts $p \geq 2\beta$.
For (3), since $\Omega(T)\neq \emptyset$,
$H_T$ is the union of all lines in $\Omega(T)$.
Thus either assumption implies $H_T=H_S$.
Suppose $|S|+|T|$ is minimal subject to $S\neq T$
but $H_S = H_T$.
Let $s = |S|$ and $t=|T|$, then $s$ and $t$ are necessarily in
the second rows of $S$ and $T$, respectively.
If $s\neq t$, suppose $s>t$.
Then $H_T = H_S\cap H_s$, which contradicts
$H_T \not\subset H_s$.
Thus $s=t$.
Let $S'$ and $T'$ be the tableaux obtained by removing $s$ from
each of $S$ and $T$.
Then $H_{S'}\neq H_{T'}$, but $H_{S'}\cap H_{T'} = H_S \subset H_s$.
Thus $H_S = F_{\hfl{s}+1}$ a contradiction, as
the codimension of $H_S$ is at most $\hfl{s}$.
\QED
\subsection{The Cycles $\Omega({\cal T})$ and
$\Omega({\cal T}_{s,\alpha};L)$}
Let ${\cal F}$ be an arrangement.
For a set ${\cal T}$ of tableaux with common degree $l\leq 2n{-}2$,
define $\Omega({\cal T})$ to be the sum over $T\in {\cal T}$ of
the Schubert varieties $\Omega(T)$.
By Lemma~\ref{lemma:arrangements}(3), these are
distinct, so $\Omega({\cal T})$ is a multiplicity
free cycle.
Let $0\leq s\leq \alpha$ be integers and $L$
a subspace of codimension $\alpha-s+1$
which meets the subspaces $F_{\hfl{l+s}+1},\ldots,F_{l+1}$
properly with $L\cap F_{l+1} = F_{l+\alpha-s+2}$ and
if $|T| = l$, then $L$ meets $H_{T^{+(s-1)}}$ and $H_{T^{+s}}$
properly.
When this occurs, we shall say that $L$
{\em meets ${\cal F}_{l,s}$ properly}.
Let $T\in {\cal T}_{s-1,\alpha}$.
Then $T$ is a tableau whose first row has length
$b\geq\hfl{l+s}+1+\alpha-s$ and no entry in its second row
exceeds $l+s-1$.
Define
$$
\Omega(T;L) = \left\{\begin{array}{ll}
\Omega(F_{b+1},H_T) & \mbox{ if } b\geq l+1\\
\Omega(F_{b-\alpha+s}\cap L, H_T) & \mbox{ otherwise }
\end{array}
\right. .
$$
Then we define $\Omega({\cal T}_{s-1,\alpha};L)$
to be the sum over $T\in {\cal T}_{s-1,\alpha}$
of the varieties $\Omega(T;L)$.
By Lemma~\ref{lemma:arrangements}(3), these are
distinct, so $\Omega({\cal T}_{s-1,\alpha};L)$ is a multiplicity
free cycle.
\subsection{The sets $U_{i,s}$ and ${\cal G}_{i,s}$.}%
\label{sec:sets_of_cycles}
Let $\alpha_1,\ldots,\alpha_m$ be positive integers and
${\cal F}$ an arrangement.
Fix $1\leq i\leq m$.
Let ${\cal T}=\alpha_1*\cdots*\alpha_{i-1}$ and
$l$ be the common degree of tableaux in
${\cal T}$.
Let $\Sigma_{i,0}\subset \mbox{\bf G}_1{\mbox{\bf P}^n}\times\prod_{j=i}^m\mbox{\bf G}^{\alpha_j+1}{\mbox{\bf P}^n}$
be the subvariety whose fibre over $(K_i,\ldots,K_m)$ is
$$
\Omega({\cal T})
\bigcap \Omega_{K_i}\bigcap\cdots\bigcap\Omega_{K_m}.
$$
Let $U_{i,0} \subset \prod_{j=i}^m\mbox{\bf G}^{\alpha_j+1}{\mbox{\bf P}^n}$
be those points for which this intersection is generically
transverse.
Let ${\cal G}_{i,0}\subset \mbox{\it Chow}\, \mbox{\bf G}_1{\mbox{\bf P}^n}$ be
the fundamental cycles of fibres of $\Sigma_{i,0}$ over $U_{i,0}$.
For $1\leq s\leq \alpha_i$, let
$\Sigma_{i,s}\subset \mbox{\bf G}_1{\mbox{\bf P}^n}\times
\mbox{\bf G}^{\alpha_i-s+1}({\mbox{\bf P}^n}/F_{l+\alpha_i-s+2})
\times \prod_{j=i+1}^m\mbox{\bf G}^{\alpha_j+1}{\mbox{\bf P}^n}$
be the subvariety where $L$ meets ${\cal F}_{l,s}$ properly and
whose fibre over $(L,K_{i+1},\ldots,K_m)$
is
$$
\left[\Omega({\cal T}_{s-1,\alpha_i};L)+
\Omega({\cal T}^{+s})\bigcap\Omega_L\right]
\bigcap \Omega_{K_{i+1}}
\bigcap\cdots\bigcap\Omega_{K_m}.
$$
Define $U_{i,s}$
and ${\cal G}_{i,s}$ analogously to $U_{i,0}$ and ${\cal G}_{i,0}$.
Set ${\cal G}_{m+1,0}$ to be the singleton
$\{ \Omega(\alpha_1*\cdots*\alpha_m)\}$.
When $k = \mbox{\bf R}$,
let ${\cal G}_{i,s;\mbox{\scriptsize\bf R}}$ be the fundamental cycles of fibres of
$\Sigma_{i,s}$ over $U_{i,s}(\mbox{\bf R})$.
\section{Main Results}
In \S4, we prove
\medskip
\noindent{\bf Theorem D.}\ {\em
Let $\alpha_1,\ldots,\alpha_m$ be positive
integers and ${\cal F}$ any arrangement.
Then
\begin{enumerate}
\item For all $1\leq i\leq m$, and
$0\leq s\leq \alpha_i$, $U_{i,s}$ is a dense
open subset of the corresponding product of Grassmannians.
\item For all $1\leq i\leq m$, $0\leq s\leq \alpha_i$,
${\cal G}_{i,s}$ is a unirational subset of $\mbox{\it Chow}\,{\mbox{\bf G}_1{\mbox{\bf P}^n}}$.
When $k=\mbox{\bf R}$, ${\cal G}_{i,s;\mbox{\scriptsize\bf R}}$ is a real unirational
subset of $\mbox{\it Chow}\,{\mbox{\bf G}_1{\mbox{\bf P}^n}}(\mbox{\bf R})$.
\end{enumerate}
}
\medskip
Let $U$ be an open subset of ${\mbox{\bf P}^1}$ and $\phi:U \rightarrow U_{i,s}$.
Then $\phi^*\Sigma_{i,s}$ has equidimensional fibres over $U$.
As in \S\ref{sec:Chow}, the association of a point $u$ of $U$ to the
fundamental cycle of the fibre $(\phi^*\Sigma_{i,s})_u$ is
an algebraic morphism,
which we denote $\phi_* : U\rightarrow \mbox{\it Chow}\, {\mbox{\bf G}_1{\mbox{\bf P}^n}}$.
Let $\phi_*$ be the unique extension of $\phi_*$ to
${\mbox{\bf P}^1}$, as well.
In \S5, we prove:
\medskip
\noindent{\bf Theorem E.}\ {\em
Let $\alpha_1,\ldots,\alpha_m$ be positive
integers and ${\cal F}$ any arrangement.
Then
\begin{enumerate}
\item For all $1\leq i\leq m$, if $X$ is a closed point of
${\cal G}_{i+1,0}$, then there is an open subset $U$ of
${\mbox{\bf P}^1}- \{0\}$ and a map $\phi: U \rightarrow U_{i,\alpha_i}$
such that $X= \phi_*(0)$.
\item For all $1\leq i\leq m$ and $0\leq s\leq \alpha_i$,
if $X$ is a closed point of
${\cal G}_{i,s+1}$, then there is an open subset $U$ of
$\,{\mbox{\bf P}^1}- \{0\}$ and a map $\phi: U \rightarrow U_{i,s}$
such that $X= \phi_*(0)$.
\end{enumerate}
}\medskip
\subsection{Proof of Theorem A}
In the situation of Theorem E, let ${\cal T} =\alpha_1*\cdots*\alpha_m$.
Then ${\cal G}_{1,0}$ is the set ${\cal G}$
and ${\cal G}_{m+1,0}$ is the cycle $\Omega({\cal T})$.
By Theorem~D part 2, any two points of ${\cal G}_{i,s}$ are connected
by a chain of rational curves, each lying within the closure of
${\cal G}_{i,s}$.
Downward induction in the lexicographic order on pairs $(i,s)$
gives a chain of rational curves between
$\Omega({\cal T})$ and a cycle $X\in {\cal G}$.
Thus Theorem E implies Theorem A
when $\beta_i = 0$ for $1\leq i \leq m$.
Suppose $\lambda^i = (\alpha_i+\beta_i,\beta_i)$ for $1\leq i \leq m$.
Let $M_0\subset {\mbox{\bf P}^n}$ have codimension
$\beta=\beta_1+\cdots+\beta_m$ and ${\cal F}$ be any arrangement
in $M_0$.
Define $U_{i,s}$ and ${\cal G}_{i,s}$ as in \S\ref{sec:sets_of_cycles},
with $M_0$ replacing ${\mbox{\bf P}^n}$.
Let $X\in {\cal G}$, so $X$ is a generically transverse intersection
$$
\Omega(K_1,M_1)\bigcap\cdots\bigcap\Omega(K_m,M_m),
$$
where $K_i\subset M_i$ has type $\lambda^i$ for $1\leq i\leq m$.
Set $M = M_1\cap\cdots\cap M_m$.
Iteration of Lemma~\ref{lemma:one} shows that $M$ has codimension
$\beta$ and $L_i = M \cap K_i$ has codimension $\alpha_i +1$ in $M$.
Thus
$$
X = \Omega_{L_1}\bigcap\cdots\bigcap \Omega_{L_m}
$$
is a generically transverse intersection in ${\mbox{\bf G}_1} M$.
Let $\gamma$ be any automorphism of ${\mbox{\bf P}^n}$ with $\gamma M = M_0$
and $\Gamma$ a one parameter subgroup containing $\gamma$.
The orbit $\Gamma \cdot X$ is a rational curve
(or a point) in $\mbox{\it Chow}\, {\mbox{\bf G}_1{\mbox{\bf P}^n}}$ containing $\gamma(X)$.
Since $\gamma(X)$ is in ${\cal G}_{i,0}$,
previous arguments have shown there exists
a chain of rational curves between $\gamma(X)$ and $\Omega({\cal T})$,
each contained within the closure of ${\cal G}_{i,0}$.
\QED
\subsection{The Schubert Calculus.}
Let $\lambda^1,\ldots,\lambda^m$ be partitions with
$\lambda^i = (\alpha_i+\beta_i,\beta_i)$.
Set $\beta = \beta_1+\cdots+\beta_m$ and
$s_i = \alpha_i+\cdots+\alpha_{i-1}$.
For a partition $\lambda$ with $|\lambda| = s_m$, define
${\cal C}^\lambda = {\cal C}^\lambda(\alpha_1,\ldots,\alpha_m)$
to be those tableaux of shape $\lambda$
such that for $1\leq i\leq m$ the integers
$s_i+1,\ldots, s_i+\alpha_i$ occur in distinct columns
increasing from left to right.
Then $\alpha_1*\cdots*\alpha_m =
\bigcup_{|\lambda| = s_m} {\cal C}^\lambda$.
Let $c^\lambda_{\alpha_1,\ldots,\alpha_m} = \#{\cal C}^\lambda$.
Interpreting Theorems A and E in terms of products in the \medskip
Chow ring of ${\mbox{\bf G}_1{\mbox{\bf P}^n}}$, we have:
\noindent{\bf Theorem B$'$.}\
{\em
\begin{enumerate}
\item ${\displaystyle
\sigma_{\alpha_1}\cdots\sigma_{\alpha_m} =
\sum_{|\lambda| = s_m} c^\lambda_{\alpha_1,\ldots,\alpha_m} \sigma_\lambda}$.
\item ${\displaystyle
\sigma_{\lambda^1}\cdots\sigma_{\lambda^m} =
\sum_{\stackrel{\mbox{\scriptsize $\lambda = (\beta+a,\beta+b)$}}{a+b = s_m}}
c^{(a,b)}_{\alpha_1,\ldots,\alpha_m} \sigma_\lambda}$.
\end{enumerate}
}
\medskip
In particular, if $\alpha_1+\cdots +\alpha_m = 2n-2$, then
the only non-zero term on the right hand side of (1) is
$c^{(n-1,n-1)}_{\alpha_1,\ldots,\alpha_m}\sigma_{(n-1,n-1)}$, or
$c^{(n-1,n-1)}_{\alpha_1,\ldots,\alpha_m}$ times the class of a point (line).
Thus
\subsection{ Corollary}
{\em The number of lines meeting general
$(n-\alpha_i - 1)$-planes for $1\leq i\leq m$ is equal to the number of
tableaux of shape $(n-1,n-1)$ such that for $1\leq i\leq m$, the integers
$s_i+1,\ldots, s_i+\alpha_i$ occur in distinct columns increasing from
left to right.
}
\medskip
This number, $c^{(n-1,n-1)}_{\alpha_1,\ldots,\alpha_m}$,
is also known as a Kostka number~\cite{Sagan}.
\subsection{Enumerative Geometry of the Real Grassmannian.}
Let ${\cal G}_{\mbox{\scriptsize\bf R}}$ consist of the fundemental cycles
of generically transverse intersections of Schubert varieties
of types $\lambda^1,\ldots,\lambda^m$ defined by real flags.
\medskip
\noindent{\bf Theorem C$'$.} \ {\em
Let $\lambda^1,\ldots,\lambda^m$ be
partitions, suppose $\lambda^i = (\alpha_i+\beta_i,\beta_i)$,
and set $\beta=\sum_{i=1}^m \beta_i$.
Let $M \subset {\mbox{\bf P}^n}$ be a real $(n-\beta)$-plane and
${\cal F}$ an arrangement in $M$.
\begin{enumerate}
\item
$\Omega(\alpha_1*\cdots*\alpha_m)$ is in the closure of
$\,{\cal G}_{\mbox{\scriptsize\bf R}}$.
\item
If $\,|\lambda^1|+\cdots+|\lambda^m|=2n-2$,
then there is a nonempty classically open
subset in the product of $m$ real flag manifolds whose
corresponding Schubert varieties meet transversally,
with all points of intersection real.
\end{enumerate}
}
\medskip
\noindent{\bf Proof:}
Suppose that $\lambda^i = (\alpha_i+\beta_i,\beta_i)$.
Define $U_{i,s}$ and ${\cal G}_{i,s}$ as in \S\ref{sec:sets_of_cycles}
for the arrangement ${\cal F}$ in $M$ and the integers
$\alpha_1,\ldots,\alpha_m$.
Arguing as in the proof of Theorem~A
shows ${\cal G}_{1,0;\mbox{\scriptsize\bf R}}\subset {\cal G}_{\mbox{\scriptsize\bf R}}$.
Restricting to the real points of the varieties in Theorem E
shows ${\cal G}_{i,s;\mbox{\scriptsize\bf R}}\subset \overline{{\cal G}_{1,0;\mbox{\scriptsize\bf R}}}$.
The case $(i,s) = (m+1,0)$ is part 1.
For 2, let $d = c^{(n-1,n-1)}_{\alpha_1,\ldots,\alpha_m}$.
Then $\Omega(\alpha_1*\cdots*\alpha_m)$
is $d$ distinct real lines.
Hence ${\cal G}_{i,s} \subset S^d{\mbox{\bf G}_1{\mbox{\bf P}^n}}$, the Chow variety of
effective degree $d$ zero cycles on ${\mbox{\bf G}_1{\mbox{\bf P}^n}}$.
The real points $S^d{\mbox{\bf G}_1{\mbox{\bf P}^n}}(\mbox{\bf R})$ of $S^d{\mbox{\bf G}_1{\mbox{\bf P}^n}}$ are
effective degree $d$ zero cycles
stable under complex conjugation.
The dense subset of $S^d{\mbox{\bf G}_1{\mbox{\bf P}^n}}(\mbox{\bf R})$ of
multiplicity free cycles
has a component ${\cal M}$ parameterizing sets of $d$ distinct real lines
and $\Omega(\alpha_1*\cdots*\alpha_m)\in {\cal M}$.
By part 1,
$\Omega(\alpha_1*\cdots*\alpha_m)\in
\overline{{\cal G}_{\mbox{\scriptsize\bf R}}}$, which shows
${\cal G}_{\mbox{\scriptsize\bf R}}\bigcap{\cal M}\neq \emptyset$, a restatement
of 2.
\QED
\section{Generically Transverse Intersections}
\subsection{Lemma.}\label{lemma:gic}
{\em
Let $\lambda^1,\ldots,\lambda^m$ be partitions.
Then the set $U$ of partial flags
$K_1\subset M_1,\,\ldots,$
$\,K_m\subset M_m$ for which the intersection
$$
\Omega(K_1,M_1)\bigcap\cdots\bigcap\Omega(K_m,M_m)
$$
is generically transverse is a dense open subset of
$\,\prod_{i=1}^m \mbox{\bf Fl}(\lambda^i)$.
}
\medskip
\noindent{\bf Proof:}
For $1\leq i\leq m$, let $K_i\subset M_i$ be a partial flag of
type $\lambda^i= (\alpha_i+\beta_i,\beta_i)$ and suppose the
corresponding Schubert varieties meet generically transversely.
By Lemma~\ref{lemma:one},
$$
\Omega(K_1,M_1)\bigcap\cdots\bigcap\Omega(K_m,M_m)
\ =\ \mbox{\bf G}_1M\bigcap \Omega_{L_1}\bigcap\cdots\bigcap\Omega_{L_m},
$$
where $M = M_1\cap\cdots\cap M_m$, $K_i = L_i\cap M_i$
where $L_i$ meets $M_i$ properly, and $M$ has codimension
$\beta = \beta_1+\cdots+\beta_m$.
\medskip
Fix a codimension $\beta$ subspace $M$ of ${\mbox{\bf P}^n}$.
As $U$ is stable under the diagonal action of $Gl_{n+1}$, it is the
union of the translates of $V = U\cap X$, where
$X$ consists of those $m$-tuples of
flags with $M \subset M_i, 1\leq i\leq m$.
Moreover, $U$ is open if and only if $V$ is open in $X$.
Let $Y\subset X$ be those flags where
$M = M_1\cap\cdots\cap M_m$ and $K_i$ meets $M$ properly.
The product of maps defined by $(K_i,M_i) \mapsto K_i\cap M=L_i$
exhibits $Y$ as a fibre bundle over the product
$\prod_{i=1}^m \mbox{\bf G}^{\alpha_i+1}M$, and $V$ is the inverse image of the set
$W$ consisting of those $(L_1,\ldots,L_m)$ for which
$\Omega_{L_1}\bigcap\cdots\bigcap\Omega_{L_m}$ is generically transverse.
Thus we may assume $\beta_i = 0$.
\medskip
Let $\Sigma\subset({\mbox{\bf P}^n})^m\times\mbox{\bf G}_1{\mbox{\bf P}^n}\times\prod_{i=1}^m\mbox{\bf G}^{\alpha_i+1}{\mbox{\bf P}^n}$
consist of those
$(p_1,\ldots,p_m,\ell,L_1,\ldots,L_m)$ such that
$p_i\in \ell \cap L_i$ for $1\leq i\leq m$.
The projection of $\Sigma$ to $({\mbox{\bf P}^n})^m\times \mbox{\bf G}_1{\mbox{\bf P}^n}$
exhibits $\Sigma$ as a fibre bundle with fibre
$\prod_{i=1}^m\mbox{\bf G}^{\alpha_i+1}({\mbox{\bf P}^n}/p_i)$
and image those $(p_1,\ldots,p_m,\ell)$ with each
$p_i\in \ell$.
This image has dimension $m + 2n-2$.
Thus $\Sigma$ is irreducible of dimension
$$
m+2n-2+\sum_{i=1}^m(n-\alpha_i-1)(\alpha_i+1)\ = \
2n-2 -\sum_{i=1}^m \alpha_i+ \sum_{i=1}^m (n-\alpha_i)(\alpha_i+1).
$$
The image of $\Sigma$ in $\prod_{i=1}^m\mbox{\bf G}^{\alpha_i+1}{\mbox{\bf P}^n}$ consists of those
$(L_1,\ldots,L_m)$ whose corresponding Schubert varieties have
nonempty intersection.
This image is a proper subvariety if
$2n-2 < \sum_{i=1}^m \alpha_i$.
In this case, $U$ is the complement of this image.
Suppose $2n-2 \geq \sum_{i=1}^m \alpha_i$.
Let $W\subset \Sigma$ consist of those points where
$\Omega_{L_1},\ldots,\Omega_{L_m}$ meet transversally at $\ell$.
By Lemma~\ref{lemma:one}, $W$ consists of those points such that
\begin{enumerate}
\item $\ell \not\subset L_i$ for $1\leq i\leq m$,
thus $p_i = \ell\cap L_i$ and $\ell$ is a smooth point of
$\Omega_{L_i}$.
\item The tangent spaces
$T_{\ell}\Omega_{L_i}$
meet transversally.
\end{enumerate}
Thus $W$ is an open subset of $\Sigma$.
We show $W\neq \emptyset$.
\smallskip
Fix $\ell \in \mbox{\bf G}_1{\mbox{\bf P}^n}$ and distinct points
$p_1,\ldots,p_m$ of $\ell$.
Define $f : \mbox{Hom}(\ell,{\mbox{\bf P}^n}/\ell)-\{0\} \rightarrow ({\mbox{\bf P}^n}/\ell)^m$
by $\phi \mapsto (\phi(p_1),\ldots,\phi(p_m))$.
Let $G\subset Gl_{n+1}$ fix $\ell$ pointwise.
Then $G^m$ acts transitively on $({\mbox{\bf P}^n}/\ell)^m$.
Choose $(L_1,\ldots,L_m)\in
\prod_{i=1}^m\mbox{\bf G}^{\alpha_i+1}{\mbox{\bf P}^n}$
with $p_i = \ell \cap L_i$.
By Theorem 2 (i) of~\cite{Kleiman}, there is a dense open subset $V$ of
$G^m$ consisting of those {\boldmath $g$} such that
either
$f^{-1}(\mbox{\boldmath $g$}
((L_1+\ell)/\ell \times\cdots\times (L_m+\ell)/\ell))$
is empty or its codimension equals that of
$(L_1+\ell)/\ell \times\cdots\times (L_m+\ell)/\ell$ in ${\mbox{\bf P}^n}$,
which is $\sum_{i=1}^m \alpha_i$.
Let {\boldmath $g$} $ = (g_1,\ldots,g_m)\in V$ and set
$L_i' = g_i L_i$.
Then
$f^{-1}((L_1'+\ell)/\ell \times\cdots\times
(L_m'+\ell)/\ell)\cup \{0\}$
is the intersection of the tangent spaces
$T_\ell\Omega_{L_i'}$ for $1\leq i\leq m$.
Since $\alpha_i$ is the codimension of $T_\ell\Omega_{K_i'}$ in
$T_{\ell}{\mbox{\bf G}_1{\mbox{\bf P}^n}}$ and
$\sum_{i=1}^m \alpha_i \leq 2n-2$,
we see that
$\Omega_{L_1'},\ldots,\Omega_{L_m'}$ meet transversally at $\ell$.
Thus $W\neq \emptyset$.
\medskip
Let $Z = \Sigma - W$ and $\pi$ be the projection
$\Sigma\rightarrow \prod_{i=1}^m \mbox{\bf G}^{\alpha_i+1}{\mbox{\bf P}^n}$.
The desired set $U$
consists of those $(L_1,\ldots,L_m)$ with
$\dim(\pi^{-1}(L_1,\ldots,L_m) \bigcap Z)< 2n-2-\sum_{i=1}^m \alpha_i$.
$U$ is open and non-empty, for otherwise $\dim Z = \dim \Sigma$, which
implies $Z = \Sigma$ and contradicts $W\neq\emptyset$.
\QED
\subsection{Lemma.}\label{lemma:good_dimension}
{\em Let $d, \alpha_1,\ldots,\alpha_m$ be positive integers
and $Z$ a subscheme of ${\mbox{\bf G}_1{\mbox{\bf P}^n}}$ with $\dim(Z)<d$.
Then the set $W \subset \prod_{i=1}^m\mbox{\bf G}^{\alpha_i+1}{\mbox{\bf P}^n}$
consisting of those $(K_1,\ldots,K_m)$
for which
$\dim(Z\bigcap \Omega_{K_1}\bigcap\cdots\bigcap\Omega_{K_m})<
d-\sum_{i=1}^m\alpha_i$ is open and dense.
}
\medskip
\noindent{\bf Proof:}
Let $\Sigma\subset Z\times \prod_{i=1}^m\mbox{\bf G}^{\alpha_i+1}{\mbox{\bf P}^n}$
be the subscheme whose fibre over $(K_1,\ldots,K_m)$
is
$Z\bigcap \Omega_{K_1}\bigcap\cdots\bigcap\Omega_{K_m}$.
By the upper semicontinuity of fibre dimension,
$W$ is open.
If $W$ were empty, then all fibres of the projection to
$\prod_{i=1}^m\mbox{\bf G}^{\alpha_i+1}{\mbox{\bf P}^n}$
would have dimension at least
$d-\sum_{i=1}^m\alpha_i$ and so
$\dim\Sigma \geq d-\sum_{i=1}^m\alpha_i +\sum_{i=1}^m
(n-\alpha_i)(\alpha_i+1)$.
Projecting to $Z$ exhibits $\Sigma$ as a fibre bundle with fibre
over a point $\ell$ of $Z$ equal to $X_1(\ell)\times\cdots \times
X_m(\ell)$, where
$X_i(\ell)\subset \mbox{\bf G}^{\alpha_i+1}{\mbox{\bf P}^n}$ is the set of those
$K_i\subset\mbox{\bf G}^{\alpha_i+1}{\mbox{\bf P}^n}$ which meet $\ell$,
which has codimension $\alpha_i$.
Thus $\Sigma$ has dimension
$$
\dim Z - \sum_{i=1}^m \alpha_i + \sum_{i=1}^m
(n-\alpha_i)(\alpha_i+1).
$$
Since $d> \dim Z$, $W$ must be non-empty.
\QED
\subsection{Proof of Theorem D, part 1}\label{sec:Proof_D}
We show that for each $1\leq i\leq m$ and
$0\leq s\leq \alpha_i$, the sets $U_{i,s}$ are open dense
subsets of the corresponding products of Grassmannians.
Let ${\cal T} = \alpha_1*\cdots*\alpha_{i-1}$.
Recall that $U_{i,0}$
consists of those $(K_i,\ldots,K_m)$ such that
the intersection
$$
\Omega({\cal T})
\bigcap \Omega_{K_i}\bigcap\cdots\bigcap\Omega_{K_m}
$$
is generically transverse.
Such a cycle has dimension $d = 2n-2 -\sum_{i=1}^m \alpha_i$.
Let $Z$ be the singular locus
of $\Omega({\cal T})$.
The above intersection is generically transverse
if $\dim (Z \bigcap \Omega_{K_i}\bigcap\cdots\bigcap\Omega_{K_m}) <d$
and if, for every component $\Omega(T)$ of $\Omega({\cal T})$,
the intersection
$\Omega(T)\bigcap \Omega_{K_i}\bigcap\cdots\bigcap\Omega_{K_m}$
is generically transverse.
By Lemma~\ref{lemma:arrangements},
$\Omega(T) = \Omega(S)\neq \emptyset$ implies that
$T=S$, thus $Z$ is a union of intersections of components and
the singular loci of components, and hence
$\dim(Z) < \dim(\Omega({\cal T}))$.
By Lemma~\ref{lemma:good_dimension},
there is an open subset $W$ of $\prod_{j=i}^m \mbox{\bf G}^{\alpha_j+1}{\mbox{\bf P}^n}$
consisting those $(K_1,\ldots,K_m)$ for which
$\dim(Z\bigcap \Omega_{K_i}\bigcap\cdots\bigcap\Omega_{K_m})<d$.
For $T\in {\cal T}$, let $U_T\subset \prod_{j=i}^m\mbox{\bf G}^{\alpha_j+1}{\mbox{\bf P}^n}$
be those $(K_i,\ldots,K_m)$ where the intersection
$$
\Omega(T)\bigcap \Omega_{K_i}\bigcap\cdots\bigcap\Omega_{K_m}
$$
is generically transverse.
It suffices to show that for each $T\in {\cal T}$,
$U_T$ is a dense open subset of $\prod_{j=i}^m\mbox{\bf G}^{\alpha_j+1}{\mbox{\bf P}^n}$,
since $U_{i,0} =
W\cap\bigcap_{T\in {\cal T}} U_T$.
Suppose $T$ has shape $\lambda = (\alpha,\beta)$ and let
$V\subset \mbox{\bf Fl}(\lambda)\times\prod_{j=i}^m\mbox{\bf G}^{\alpha_j+1}{\mbox{\bf P}^n}$
be those flags for which the intersection
$$
\Omega(F,H)\bigcap \Omega_{K_i}\bigcap\cdots\bigcap\Omega_{K_m}
$$
is generically transverse.
By Lemma~\ref{lemma:gic}, $V$ is dense and open.
Note that
$$
\{F_{\alpha+1}\subset H_T\}\times U_T =
V\bigcap\left( \{F_{\alpha+1}\subset H_T\}\times
\prod_{j=i}^m\mbox{\bf G}^{\alpha_j+1}{\mbox{\bf P}^n}\right),
$$
so $U_T$ is open.
Since $\mbox{\bf Fl}(\lambda) = Gl_{n+1} \cdot \{F_{\alpha+1}\subset H_T\}$,
and $V$ is stable under the diagonal
action of $Gl_{n+1}$,
$V = Gl_{n+1} \cdot (\{F_{\alpha+1}\subset H_T\}\times U_T)$.
Thus $U_T$ is non-empty.
\medskip
The case of $U_{i,s}$ for $s>0$ follows by similar arguments.
\QED
\subsection{Unirationality of ${\cal G}_{i,s}$}
\mbox{ }\medskip
\noindent{\bf Lemma.}\label{lemma:unirationality}
{\em
Let $X$ be a projective variety, $U$ a dense open subset of a
variety $Y$ which is covered by affine spaces, and suppose that
$\Sigma \subset X\times U$ is closed and the projection to $U$
has generically reduced fibres of pure dimension.
Then the set ${\cal G}\subset \mbox{\it Chow}\, X$ of
fundamental cycles of the fibres of $\,\Sigma$ is unirational.
When $k=\mbox{\bf R}$, let ${\cal G}_{\mbox{\scriptsize\bf R}}$ be those cycles
lying over $U(\mbox{\bf R})$, then ${\cal G}_{\mbox{\scriptsize\bf R}}$ is
real unirational.
}
\medskip
Part 2 of Theorem D is a consequence of this Lemma.
\noindent{\bf Proof:}
Let $\pi$ be the projection $\Sigma \rightarrow U$.
As in \S 2.1, let $U'\subset U$ be the open set where the map $\phi$
which associates a point of $U$ to the fundamental cycle of the fibre
of $\pi$ at $x$ is an algebraic morphism.
Then $\phi(U')\subset {\cal G}$.
We show ${\cal G}\subset \overline{\phi(U')}$.
Let $x\in U$ and
choose a map $f:{\bf A}^1 \rightarrow Y$ with $f(0) = x$ and
$f^{-1}(U')\neq \emptyset$.
This is possible as $Y$ has a covering by affine spaces.
Then $f^{-1}\Sigma \rightarrow f^{-1}(U)$ is a family over a smooth curve
with generically reduced fibres of pure dimension.
The association of a point $u$ of $f^{-1}(U)$ to the
fundamental cycle of the fibre $\pi^{-1}(f(u))$ gives a map
$\psi: f^{-1}(U) \rightarrow \mbox{\it Chow}\, X$ agreeing with
$\phi\circ f$ on $f^{-1}(U')$.
Thus the fundamental cycle of $\pi^{-1}(x)$
is in $\phi(U')$.
If $k = \mbox{\bf R}$, these maps show
$\phi(U'(\mbox{\bf R}))\subset{\cal G}_{\mbox{\scriptsize\bf R}}
\subset \overline{\phi(U'(\mbox{\bf R}))}$,
thus ${\cal G}_{\mbox{\scriptsize\bf R}}$ is real unirational.
\QED
\section{Construction of Explicit Rational Equivalences}
We use the following to parameterize the explicit rational
equivalences we construct.
\subsection{Lemma.}\label{lemma:limits_are_good}
{\em
Let ${F\!_{\DOT}}$ be a complete flag in ${\mbox{\bf P}^n}$.
Suppose $L_{\infty}$ is a hyperplane not containing $F_n$.
Then there exists a pencil of hyperplanes $L_t$, for
$t\in \mbox{\bf P}^1 = {\bf A}^1\bigcup \{\infty\}$,
such that if $t\neq 0$, then $L_t$ meets the subspaces
in ${F\!_{\DOT}}$ properly,
and, for each $i\leq n-1$, the family of codimension $i+1$ planes
induced by $L_t\bigcap F_i$, for $t\neq 0$ has fibre $F_{i+1}$ over $0$.
}
\medskip
\noindent{\bf Proof:}
Let $x_0,\ldots,x_n$ be coordinates for ${\mbox{\bf P}^n}$ such that
$L_{\infty}$ is given by $x_n=0$ and
$F_i$ by $x_0=\cdots=x_{i-1}=0$.
Let $e_0,\ldots,e_n$ be a basis for ${\mbox{\bf P}^n}$ dual to these coordinates
and define
$$
L_t = \Span{te_j + e_{j+1}\,|\, 0\leq j\leq n-1}.
$$
For $t\neq 0$,
$L_t \bigcap F_i = \Span{te_j + e_{j+1}\,|\, i\leq j\leq n-1}$
and so has codimension $i+1$.
The fibre of this family at $0$ is
$\Span{e_{j+1}\,|\, i\leq j\leq n-1} = F_{i+1}$.
\QED
In the situation of Lemma~\ref{lemma:limits_are_good}, we write
$\lim_{t\rightarrow 0} L_t\cap F_i = F_{i+1}$.
\medskip
For the remainder of this section, fix an arrangement ${\cal F}$.
Set ${\cal T} = \alpha_1*\cdots*\alpha_{i-1}$ and let
$l$ be the common degree of tableaux in
${\cal T}$.
\subsection{Proof of Theorem E, part 1.}\label{sec:proof_E_1}
Let $1\leq i\leq m$,
and suppose $X_0$ is a cycle in ${\cal G}_{i+1,0}$:
$$
X_0 = \Omega({\cal T}*\alpha_i)
\bigcap\Omega_{K_{i+1}}\bigcap\cdots\bigcap\Omega_{K_m}.
$$
Let $L_\infty$ be any hyperplane which meets ${\cal F}_{l,\alpha_i}$ properly.
By Lemma~\ref{lemma:limits_are_good} applied to the flag
induced by ${F\!_{\DOT}}$ in ${\mbox{\bf P}^n}/F_{l+2}$ and the hyperplane $L_\infty/F_{l+2}$,
there is a pencil $L_t$ of hyperplanes such that if $t\neq 0$
and $i\leq l+1$, then $L_t$ meets $F_i$ properly and
$\lim_{t\rightarrow 0} L_t\cap F_i = F_{i+1}$.
Let ${\cal X}\subset \mbox{\bf P}^1\times {\mbox{\bf G}_1{\mbox{\bf P}^n}}$ be the subscheme whose fibre
at $t\neq 0$ is
$$
X_t = \left[ \Omega({\cal T}_{\alpha_i-1,\alpha_i};L_t) + \rule{0pt}{13pt}
\Omega({\cal T}^{+\alpha})\right] \bigcap
\Omega_{K_{i+1}}\bigcap\cdots\bigcap\Omega_{K_m}.
$$
Since $L_\infty$ meets ${\cal F}_{l,\alpha_i}$ properly,
the set $U'\subset {\mbox{\bf P}^1}$ of $t$ where $L_t$ meets
${\cal F}_{l,\alpha_i}$ properly is open and dense.
We claim that $X_0$ is the fibre of ${\cal X}$ over $0$.
In that case, let $U''\subset \mbox{\bf P}^1$ be the open subset of those
$t$ for which
$X_t$ is generically reduced.
Since $X_0$ is generically reduced,
$0 \in U''$ so $U'' \neq \emptyset$.
For $t\in U'\cap U''$, the fibre
$X_t\in {\cal G}_{i-1,\alpha_i}$,
as $\Omega_{L_t} = {\mbox{\bf G}_1{\mbox{\bf P}^n}}$.
The restriction of ${\cal X}$ to $U\cup \{0\}$ gives a family
over a smooth curve with generically reduced equidimensional fibres.
Thus the association of a point of $U\cup \{0\}$ to its fibre
gives a map $\phi: U\cup \{0\}\rightarrow \mbox{\it Chow}\,{\mbox{\bf G}_1{\mbox{\bf P}^n}}$ with
$\phi(U)\subset {\cal G}_{i-1,\alpha_i}$ and $\phi(0) = X_0$,
proving part 1.
For $T\in {\cal T}_{\alpha_i-1,\alpha_i}$, let
${\cal X}_T\subset \mbox{\bf P}^1\times {\mbox{\bf G}_1{\mbox{\bf P}^n}}$ be the subscheme whose fibre
at $t\neq 0$ is
$$
(X_T)_t = \Omega(T;L_t)
\bigcap\Omega_{K_{i+1}}\bigcap\cdots\bigcap\Omega_{K_m}.
$$
Since ${\cal X}
= \sum_{T\in {\cal T}_{\alpha_i-1,\alpha_i}} {\cal X}_T \ +\
\mbox{\bf P}^1\times \Omega({\cal T}^{+\alpha_i})
\bigcap\Omega_{K_{i+1}}\bigcap\cdots\bigcap\Omega_{K_m}$,
to show that $X_0$ is the fibre of ${\cal X}$ over $0$
it suffices to show that for each $T\in {\cal T}_{\alpha_i-1,\alpha_i}$,
the fibre of ${\cal X}_T$ at 0 is
$\Omega(T)\bigcap\Omega_{K_{i+1}}\bigcap\cdots\bigcap\Omega_{K_m}$.
Let $T\in {\cal T}_{\alpha_i-1,\alpha_i}$.
If the first row of $T$ has length exceeding $l+1$, then
$\Omega(T;L_t) = \Omega(T)$, so ${\cal X}_T$ is the constant family
$\mbox{\bf P}^1\times\Omega(T)\bigcap\Omega_{K_{i+1}}\bigcap\cdots\bigcap\Omega_{K_m}$.
Now suppose the first row of $T$ has length $b\leq l+1$.
Then, for $t\neq 0$, $\Omega(T;L_t) = \Omega(F_b\cap L_t,H_T)$.
Since $\lim_{t\rightarrow 0}F_b\bigcap L_t = F_{b+1}$, we see that
$\Omega(T)$ is the fibre over 0 of the family over ${\bf P}^1$
whose fibre over $t\neq 0$ is $\Omega(T;L_t)$.
Since
$\Omega(T)\bigcap\Omega_{K_{i+1}}\bigcap\cdots\bigcap\Omega_{K_m}$
is generically transverse, there is an open subset $U_T\subset {\bf P}^1$
such that for $t\in U_T-\{0\}$,
$\Omega(T;L_t)\bigcap\Omega_{K_{i+1}}\bigcap\cdots\bigcap\Omega_{K_m}$
is generically transverse.
This shows that the fibre over 0
of ${\cal X}_T$ is
$\Omega(T)\bigcap\Omega_{K_{i+1}}\bigcap\cdots\bigcap\Omega_{K_m}$.
\medskip
\subsection{Proof of Theorem E, part 2}
Let $0\leq s\leq \alpha_i-1$
and suppose $X_0\in {\cal G}_{i,s+1}$
$$
X_0 = \left[\Omega({\cal T}_{s,\alpha_i};N)+
\Omega({\cal T}^{+s+1})\bigcap\Omega_N
\right]\bigcap\Omega_{K_{i+1}}\bigcap\cdots\bigcap\Omega_{K_m}.
$$
Then $N$ has codimension $\alpha_i-s$ in ${\mbox{\bf P}^n}$
and meets ${\cal F}_{l,s+1}$ properly,
and the above intersection is generically transverse.
We make a useful calculation.
Let $L_0 = N\cap H_{l+s+1}$.
\medskip
\noindent{\bf Lemma. }
$$
\Omega({\cal T}^{+s}(\alpha_i-s);N)+\Omega({\cal T}^{+s+1})\bigcap\Omega_N
= \Omega({\cal T}^{+s})\bigcap \Omega_{L_0}.
$$
\medskip
Since ${\cal T}_{s,\alpha_i} = {\cal T}_{s-1,\alpha_i}
\bigcup {\cal T}^{+s}(\alpha_i-s)$, we see that
$$
X_0 = \left[\Omega({\cal T}_{s-1,\alpha_i};N)+
\Omega({\cal T}^{+s})\bigcap \Omega_{L_0}
\right]\bigcap \Omega_{K_{i+1}}\bigcap\cdots\bigcap\Omega_{K_m}.
$$
\noindent{\bf Proof:}
Let $T\in {\cal T}^{+s}$
and suppose that $b$ is the length of the first row of $T$.
Then $\hfl{l+s+1}\leq b\leq l$.
The degree of $T$ is $l+s$, so
$L_0 = N\cap H_{l+s+1}$ meets $H_T$ properly,
because $H_T\cap H_{l+s+1}$ equals either $H_{T^+}$ or $F_{\hfl{l+s+1}+1}$,
each of which meets $N$ properly.
If $T$ is rectangular, $b = \frac{l+s}{2}$ and
$\Omega(T) = \mbox{\bf G}_1 H_T$.
Thus,
$\Omega(T)\bigcap \Omega_{L_0} =
\Omega( H_T\cap L_0, H_T)$.
Since $L_0$ meets $H_T$ properly, this is generically
transverse, by Lemma~\ref{lemma:one}.
We calculate $ H_T\cap L_0$.
First note that
$ H_T\cap H_{l+s+1}=F_{\hfl{l+s+1}+1} = F_{\hfl{l+s}+1}$.
So
$L_0\cap H_T = N\cap H_{l+s+1}\cap H_T
= N\cap F_{\hfl{l+s}+1}$.
As $H_T = H_{T(\alpha_i-s)}$,
$$
\Omega(F_{b+1},H_T)\bigcap \Omega_{L_0} =
\Omega(N\cap F_{\hfl{l+s}+1},H_{T(\alpha_i-s)}) = \Omega(T(\alpha_i-s);N).
$$
Suppose $T$ is not rectangular.
Since $F_{b+1}\subset F_{\hfl{l+s+1}+1}\subset H_{l+s+1}$,
Lemma~\ref{lemma:component_calculation} implies
$$
\Omega(F_{b+1},H_T)\bigcap \Omega_{L_0} =
\Omega(F_{b+1}\cap N,H_T) + \Omega(F_{b+1},H_T\cap H_{l+s+1})
\bigcap\Omega_N.
$$
But this is
$\Omega(T(\alpha_i-s);N) + \Omega(T^+)\bigcap \Omega_N$.
Summing over $T\in {\cal T}^{+s}$ completes the proof.
\QED
Let ${N_{\DOT}}$ be any complete flag in
$N/F_{l+\alpha_i-s+2}$ refining the images of
$$
F_{l+\alpha_i-s+1}\subset N\cap F_l\subset \cdots\subset
N\cap F_{\hfl{l+s}+1}\subset L_0.
$$
Let $L_\infty$ be any hyperplane of $N$ which meets ${\cal F}_{l,s}$
properly.
By Lemma~\ref{lemma:limits_are_good}
applied to ${N_{\DOT}}$ in $N/F_{l+\alpha_i-s+2}$, there is a
pencil $L_t$ of hyperplanes
of $N$ each containing $F_{l+\alpha_i-s+2}$,
and for $\hfl{l+s}+1\leq j\leq l$ and $t\neq 0$,
$L_t$ meets $N\cap F_j$ properly, with
$\lim_{t\rightarrow 0} L_t\cap N\cap F_j
= N\cap F_{j+1}$.
Since $L_t\cap N\cap F_j = L_t\cap F_j$ for $j$ in this
range, $L_t$ meets $F_j$ properly.
Let ${\cal X}\subset {\mbox{\bf P}^1}\times{\mbox{\bf G}_1{\mbox{\bf P}^n}}$ be the subscheme whose fibre
over $t\in {\mbox{\bf P}^1}$ is
$$
X_t = \left[\Omega({\cal T}_{s-1,\alpha_i};L_t) +
\Omega({\cal T}^{+s})\bigcap \Omega_{L_t}\right] \bigcap
\Omega_{K_{i+1}}\bigcap\cdots\bigcap\Omega_{K_m}.
$$
Since $L_\infty$ meets ${\cal F}_{l,\alpha_i}$ properly,
The set $U'\subset {\mbox{\bf P}^1}$ of $t$ where $L_t$ meets
${\cal F}_{l,\alpha_i}$ properly is open and dense.
We claim that $X_0$ is the fibre of ${\cal X}$ over $0$.
In that case, let $U''\subset \mbox{\bf P}^1$ be the open subset of those
$t$ for which
$X_t$ is generically reduced.
Since $X_0$ is generically reduced,
$0 \in U''$ so $U'' \neq \emptyset$.
For $t\in U'\cap U''$, the fibre
$X_t\in {\cal G}_{i,s}$.
The restriction of ${\cal X}$ to $U\cup \{0\}$ gives a family
over a smooth curve with generically reduced equidimensional fibres.
Thus the association of a point of $U\cup \{0\}$ to its fibre
gives a map $\phi: U\cup \{0\}\rightarrow \mbox{\it Chow}\,{\mbox{\bf G}_1{\mbox{\bf P}^n}}$ with
$\phi(U)\subset {\cal G}_{i,s}$ and $\phi(0) = X_0$,
proving part 1.
For $T\in {\cal T}_{s-1,\alpha_i}$
let ${\cal X}_T$ be the subscheme of ${\mbox{\bf P}^1}\times {\mbox{\bf G}_1{\mbox{\bf P}^n}}$ whose fibre over
$t\neq 0$ is
$$
({\cal X}_T)_t =
\Omega(T;L_t)\bigcap \Omega_{K_{i+1}}\bigcap\cdots\bigcap\Omega_{K_m}.
$$
Arguing as at the end of \S\ref{sec:proof_E_1},
we may conclude that
$\Omega(T;N)\bigcap \Omega_{K_{i+1}}\bigcap\cdots\bigcap\Omega_{K_m}$
is the fibre of ${\cal X}_T$ at 0.
For $S\in {\cal T}^{+s}$, let ${\cal X}_S$ be the subscheme of
${\mbox{\bf P}^1}\times {\mbox{\bf G}_1{\mbox{\bf P}^n}}$ whose fibre over $t$ is
$$
({\cal X}_S)_t =
\Omega(S)\bigcap \Omega_{L_t} \bigcap
\Omega_{K_{i+1}}\bigcap\cdots\bigcap\Omega_{K_m}.
$$
Arguing as at the end of \S\ref{sec:proof_E_1},
we may conclude that
$\Omega(S)\bigcap \Omega_{L_0} \bigcap
\Omega_{K_{i+1}}\bigcap\cdots\bigcap\Omega_{K_m}$
is the fibre of ${\cal X}_S$ at 0.
Since ${\cal X} = \sum_{T\in {\cal T}_{s-1,\alpha_i}}
{\cal X_T} +\sum_{S\in {\cal T}^{+s}}{\cal X}_S$,
we conclude that the fibre of ${\cal X}$ at 0 is $X_0$.
\QED
\section{An Algebra of Tableaux}
The Schubert classes, $\sigma_{\lambda}$, form an integral basis for
the Chow ring of any Grassmann variety.
Thus there exist integral constants
$c^{\lambda}_{\mu\,\nu}$ defined by the identity:
$$
\sigma_{\mu}\cdot\sigma_{\nu} =
\sum_\lambda c^{\lambda}_{\mu\,\nu}\sigma_{\lambda}.
$$
In 1934,
Littlewood and Richardson~\cite{Littlewood_Richardson} gave a conjectural
formula for these constants,
which was proven in 1978 by Thomas~\cite{Thomas_schensted_construction}.
Lascoux and Sch\"utzenberger~\cite{Lascoux_Schutzenberger_monoid_plactic}
constructed the ring of symmetric functions as a subalgebra of a
non-commutative
associative ring called the plactic algebra whose additive group is the
free abelian group $\Lambda$ with basis the set of Young tableaux.
For that, each tableau $T$ of shape $\lambda$
determines a monomial summand of the
Schur function, $s_{\lambda}$.
Evaluating $s_{\lambda}$ at Chern roots of the dual to
the tautological bundle of the Grassmannian gives the Schubert class
$\sigma_\lambda$.
Non-symmetric monomials in these Chern roots are not defined,
so individual Young tableaux are not expected to appear in the
geometry of Grassmannians.
In this context, the crucial use we made of the Schubert varieties
$\Omega(T)$ is surprising.
A feature of our methods is the correspondence
between an iterative construction of the set
$\alpha_1*\cdots*\alpha_m$ and the rational
curves in the proof of Theorem~E.
This suggests an alternate non-commutative
associative product $\circ$ on $\Lambda$.
The resulting algebra has surjections
to the ring of symmetric functions and to Chow rings of
Grassmannians.
Additional combinatorial preliminaries for this section
may be found in~\cite{Sagan}.
Here, partitions $\lambda$, $\mu$, and $\nu$ may have any
number of rows.
Suppose $T$ and $U$ are,
respectively, a tableau of shape $\mu$ and a skew tableau
of shape $\lambda/\mu$.
Let $T\bigcup U$ be the tableau of shape $\lambda$ whose
first $|\mu|$ entries comprise $T$, and remaining entries
comprise $U$, with each increased by $|\mu|$.
For tableaux $S$ and $T$ where the shape of $S$ is $\lambda$, define
$$
S \circ T = \sum S\bigcup U,
$$
the sum over all $\nu$ and all skew tableaux $U$
of shape $\nu/\lambda$ Knuth equivalent to $T$.
For example:
\begin{eqnarray*}
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\put(30,10){\line(0,1){10}}
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\put(12,11.8){\bf 2}
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\put( 2, 1.8){\bf 4}
\end{picture}
\ \circ\
\begin{picture}(40,10)(0,2)
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\put( 2, 1.8){\bf 1}
\put(12, 1.8){\bf 2}
\put(22, 1.8){\bf 3}
\put(32, 1.8){\bf 4}
\end{picture}
&=&
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\thicklines
\put(0, 0){\line(1,0){10}}
\put(0,10){\line(1,0){70}}
\put(0,20){\line(1,0){70}}
\put( 0, 0){\line(0,1){20}}
\put(10, 0){\line(0,1){20}}
\put(20,10){\line(0,1){10}}
\put(30,10){\line(0,1){10}}
\put(40,10){\line(0,1){10}}
\put(50,10){\line(0,1){10}}
\put(60,10){\line(0,1){10}}
\put(70,10){\line(0,1){10}}
\put(0,0){\usebox{\Shading}}
\put(32,11.8){\bf 5}
\put(42,11.3){\bf 6}
\put(52,11.3){\bf 7}
\put(62,11.3){\bf 8}
\end{picture}
\ +\
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\begin{picture}(60,10)(0,12)
\thicklines
\put(0, 0){\line(1,0){20}}
\put(0,10){\line(1,0){60}}
\put(0,20){\line(1,0){60}}
\put( 0, 0){\line(0,1){20}}
\put(10, 0){\line(0,1){20}}
\put(20, 0){\line(0,1){20}}
\put(30,10){\line(0,1){10}}
\put(40,10){\line(0,1){10}}
\put(50,10){\line(0,1){10}}
\put(60,10){\line(0,1){10}}
\put(0,0){\usebox{\Shading}}
\put(12, 1.6){\bf 5}
\put(32,11.8){\bf 6}
\put(42,11.3){\bf 7}
\put(52,11.3){\bf 8}
\end{picture}
\ +\
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\begin{picture}(50,10)(0,12)
\thicklines
\put(0, 0){\line(1,0){30}}
\put(0,10){\line(1,0){50}}
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\put( 0, 0){\line(0,1){20}}
\put(10, 0){\line(0,1){20}}
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\put(30, 0){\line(0,1){20}}
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\put(50,10){\line(0,1){10}}
\put(0,0){\usebox{\Shading}}
\put(12, 1.6){\bf 5}
\put(22, 1.3){\bf 6}
\put(32,11.8){\bf 7}
\put(42,11.3){\bf 8}
\end{picture}
\ + \raisebox{-12pt}{\rule{0pt}{5pt}}
\\
& &
\setlength{\unitlength}{1.3pt}%
\begin{picture}(70,20)(0,12)
\thicklines
\put(0,-10){\line(1,0){10}}
\put(0, 0){\line(1,0){10}}
\put(0,10){\line(1,0){60}}
\put(0,20){\line(1,0){60}}
\put( 0,-10){\line(0,1){30}}
\put(10,-10){\line(0,1){30}}
\put(20,10){\line(0,1){10}}
\put(30,10){\line(0,1){10}}
\put(40,10){\line(0,1){10}}
\put(50,10){\line(0,1){10}}
\put(60,10){\line(0,1){10}}
\put(0,0){\usebox{\Shading}}
\put( 2,-8.2){\bf 5}
\put(32,11.3){\bf 6}
\put(42,11.3){\bf 7}
\put(52,11.3){\bf 8}
\end{picture}
\ +\
\setlength{\unitlength}{1.3pt}%
\begin{picture}(60,20)(0,12)
\thicklines
\put(0,-10){\line(1,0){10}}
\put(0, 0){\line(1,0){20}}
\put(0,10){\line(1,0){50}}
\put(0,20){\line(1,0){50}}
\put( 0,-10){\line(0,1){30}}
\put(10,-10){\line(0,1){30}}
\put(20, 0){\line(0,1){20}}
\put(30,10){\line(0,1){10}}
\put(40,10){\line(0,1){10}}
\put(50,10){\line(0,1){10}}
\put(0,0){\usebox{\Shading}}
\put( 2,-8.2){\bf 5}
\put(12, 1.8){\bf 6}
\put(32,11.3){\bf 7}
\put(42,11.3){\bf 8}
\end{picture}
\ +\
\setlength{\unitlength}{1.3pt}%
\begin{picture}(50,20)(0,12)
\thicklines
\put(0,-10){\line(1,0){10}}
\put(0, 0){\line(1,0){30}}
\put(0,10){\line(1,0){40}}
\put(0,20){\line(1,0){40}}
\put( 0,-10){\line(0,1){30}}
\put(10,-10){\line(0,1){30}}
\put(20, 0){\line(0,1){20}}
\put(30, 0){\line(0,1){20}}
\put(40,10){\line(0,1){10}}
\put(0,0){\usebox{\Shading}}
\put( 2,-8.2){\bf 5}
\put(12, 1.3){\bf 6}
\put(22, 1.8){\bf 7}
\put(32,11.3){\bf 8}
\end{picture}
\ .
\raisebox{-35pt}{\rule{0pt}{5pt}}
\end{eqnarray*}
This product is related to the composition $*$ of
\S\ref{sec:calculus_of_tableaux}:
Let $Y_\alpha$ be the unique standard
tableau of shape $(\alpha,0)$.
Then $T*\alpha$ consists of the summands of
$T\circ Y_{\alpha}$ with at most two rows.
\medskip
\noindent{\bf Theorem F.}\ {\em
The product $\circ $ determines an associative non-commutative
$\bf Z$-algebra structure on $\Lambda$ with unit the empty tableau
$\emptyset$.
Moreover, $\circ$ is not the plactic product.
}
\medskip
\noindent{\bf Proof:}
In the plactic algebra, the product of two tableaux is always a third,
showing $\circ$ is not the plactic product.
For any tableau $T$, $\emptyset \circ T = T\circ \emptyset = T$.
Note
$$
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\put(10, 0){\line(0,1){10}}
\put( 2, 1.8){\bf 1}
\end{picture}
\ \circ\
\begin{picture}(20,8)(0,3)
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\put(0, 0){\line(1,0){20}}
\put(0,10){\line(1,0){20}}
\put( 0, 0){\line(0,1){10}}
\put(10, 0){\line(0,1){10}}
\put(20, 0){\line(0,1){10}}
\put( 2, 1.8){\bf 1}
\put(12, 1.8){\bf 2}
\end{picture}
\ = \
\begin{picture}(30,10)(0,3)
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\put( 2, 1.8){\bf 1}
\put(12, 1.8){\bf 2}
\put(22, 1.8){\bf 3}
\end{picture}
\ +\
\setlength{\unitlength}{1.3pt}%
\begin{picture}(20,20)(0,8)
\thicklines
\put(0, 0){\line(1,0){10}}
\put(0,10){\line(1,0){20}}
\put(0,20){\line(1,0){20}}
\put( 0, 0){\line(0,1){20}}
\put(10, 0){\line(0,1){20}}
\put(20,10){\line(0,1){10}}
\put( 2,11.8){\bf 1}
\put(12,11.8){\bf 3}
\put( 2, 1.8){\bf 2}
\end{picture}
\ \ \ \neq\ \ \
\begin{picture}(30,10)(0,3)
\thicklines
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\put(0,10){\line(1,0){30}}
\put( 0, 0){\line(0,1){10}}
\put(10, 0){\line(0,1){10}}
\put(20, 0){\line(0,1){10}}
\put(30, 0){\line(0,1){10}}
\put( 2, 1.8){\bf 1}
\put(12, 1.8){\bf 2}
\put(22, 1.8){\bf 3}
\end{picture}
\ +\
\setlength{\unitlength}{1.3pt}%
\begin{picture}(20,20)(0,8)
\thicklines
\put(0, 0){\line(1,0){10}}
\put(0,10){\line(1,0){20}}
\put(0,20){\line(1,0){20}}
\put( 0, 0){\line(0,1){20}}
\put(10, 0){\line(0,1){20}}
\put(20,10){\line(0,1){10}}
\put( 2,11.8){\bf 1}
\put(12,11.8){\bf 2}
\put( 2, 1.8){\bf 3}
\end{picture}
\ =\
\begin{picture}(20,10)(0,3)
\thicklines
\put(0, 0){\line(1,0){20}}
\put(0,10){\line(1,0){20}}
\put( 0, 0){\line(0,1){10}}
\put(10, 0){\line(0,1){10}}
\put(20, 0){\line(0,1){10}}
\put( 2, 1.8){\bf 1}
\put(12, 1.8){\bf 2}
\end{picture}
\circ
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\put(0,10){\line(1,0){10}}
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\put(10, 0){\line(0,1){10}}
\put( 2, 1.8){\bf 1}
\end{picture}
\ ,\raisebox{-20pt}{\rule{0pt}{5pt}}
$$
so $\circ$ is non-commutative.
To show associativity,
let $R$, $S$, and $T$ be tableaux.
Then
$$
R\circ (S\circ T) = \sum R \bigcup W,
$$
the sum over $W$ Knuth equivalent to $S\bigcup V$,
where $V$ is Knuth equivalent to $T$.
Let $U'$ be the first $|S|$ entries in $W$, and
$V'$ the last $|T|$ entries, each decreased by $|S|$,
thus,
$$
R\circ (S\circ T) = \sum R \bigcup U' \bigcup V',
$$
the sum over $U'$ Knuth equivalent
to $S$ and $V'$ to $T$, which is $(R\circ S)\circ T$.
\QED
Let $m<n$.
For a tableau $T$ of shape $\lambda$, let
$\phi(T)$ be the Schur function $s_\lambda$.
Define $\phi_{m,\,n}(T)$ to be 0 if $\lambda_1+m \geq n$
or $\lambda_{m+1}\neq 0$ and $\sigma_\lambda$ otherwise.
Then $\phi$ and $\phi_{m,n}$ are, respectively, additive surjections
from $\Lambda$
to the algebra of symmetric functions and to $A^*\mbox{\bf G}_m{\bf P}^n$.
\medskip
\noindent{\bf Theorem G.} {\em
The maps $\phi$ and $\phi_{m,\,n}$ are $\bf Z$-algebra
homomorphisms.
}
\medskip
\noindent{\bf Proof:}
For any tableaux $S$ and $T$ of shape $\nu$ and partitions $\lambda$
and $\mu$, there is a natural bijection
(given by dual equivalence of Haiman~\cite{Haiman_dual_equivalence})
between the set of
tableaux with shape $\lambda/\mu$ Knuth equivalent to $S$ and those
Knuth equivalent to $T$, and this common number is
$c^\lambda_{\mu\,\nu}$.
This shows that $\phi$ is an algebra homomorphism.
It follows that
$\phi_{m,\,n}$ is as well.
\QED
\section{Enumerative Geometry and Arrangements Over Finite Fields}
A main result of this paper, Theorem~C, shows that any Schubert-type
enumerative problem concerning lines in projective space may be
solved over $\bf R$.
By `solved' over a field $k$, we mean
there are flags in ${\bf P}^n_k$ determining Schubert varieties which meet
transversally in finitely many points, all of which are
defined over $k$.
Given an enumerative problem, we feel it is legitimate
to inquire over which (finite) fields it may be solved.
We present two families of enumerative problems
for which this question may be resolved, and consider the problem
of finding arrangements over finite fields.
\subsection{The $n$ lines meeting four $(n-1)$-planes in ${\bf P}^{2n-1}$.}
Given three non-intersecting $(n-1)$-planes $L_1,L_2,$ and $L_3$
in ${\bf P}^{2n-1}$,
there are coordinates $x_1,\ldots,x_{2n}$
such that
\begin{eqnarray*}
L_1 &:& x_1 = x_2 = \cdots = x_n = 0\\
L_2 &:& x_{n+1} = \cdots = x_{2n} = 0\\
L_3 &:& x_1 - x_{n+1} = \cdots = x_n - x_{2n}= 0
\end{eqnarray*}
One may check that
$ \Omega_{L_1}\bigcap\Omega_{L_2}\bigcap\Omega_{L_3}$
is a transverse intersection, and if $\Sigma_{1,n-1}\subset {\bf P}^{2n-1}$
is the union of the lines meeting each of
$L_1,L_2,$ and $L_3$, then $\Sigma_{1,n-1}$ is the image of the standard
Segre embedding of
${\bf P}^1\times{\bf P}^{n-1}$ into ${\bf P}^{2n-1}$
(cf.~\cite{Harris_geometry}):
$$
\psi:[a,b]\times[y_1,\ldots,y_n] \longmapsto
[ay_1,\ldots,ay_n,by_1,\ldots,by_n].
$$
The lines meeting $L_1$, $L_2$, and $L_3$ are
the images of ${\bf P}^1\times \{p\}$, for $p\in{\bf P}^{n-1}$.
$\Sigma_{1,n-1}$ has degree $n$, so a general $(n-1)$-plane
$L_4$ meets $\Sigma_{1,n-1}$ in $n$ distinct points, each determining
a line meeting $L_1,\ldots,L_4$.
These lines, $\ell_1,\ldots,\ell_n$, meet $L_1$ in distinct points
which span $L_1$.
Changing coordinates if necessary, we may
assume $\ell_j$ is the span of $x_j$ and $x_{n+j}$.
For $1\leq j\leq n$, let $p_j = [\alpha_j,\beta_j]\in {\bf P}^1$ be the
first coordinate of $\psi^{-1}(\ell_j\cap L_4)$.
Then
$$
L_4 \quad :\quad \beta_1x_1 -\alpha_1 x_{n+1}=\cdots=
\beta_nx_n - \alpha_n x_{2n}=0.
$$
Also, $p_1,\ldots,p_n$ are distinct; otherwise
$L_4\cap \Sigma_{1,n-1}$ contains a line.
Thus, if $k$ has at least $n-1$ elements, this enumerative problem
may be solved over $k$.
\subsection{The $n$ lines meeting a fixed line and $n+1$
$(n-1)$-planes in ${\bf P}^{n+1}$.}
A line $\ell$ and $(n-1)$-planes
$K_1,\ldots,K_n$ in ${\bf P}^{n+1}$
are independent if for every $p\in \ell$, the hyperplanes
$\Gamma_i(p) = \Span{p,K_i}$, for $1\leq i\leq n$, meet in a line.
In this case, the union
$$
S_{1,n-1} = \bigcup_{p\in \ell}
\Gamma_1(p)\cap \cdots\cap \Gamma_n(p)
$$
is a rational normal surface scroll.
Moreover, the lines meeting each of $\ell$, $K_1,\ldots,K_n$
are precisely those lines
$\lambda(p) = \Gamma_1(p)\cap \cdots\cap \Gamma_n(p)$
for $p\in \ell$.
Since $S_{1,n-1}$ has degree $n$, a general $(n-1)$-plane
$K_{n+1}$ meets $S_{1,n-1}$ in $n$ distinct points, each determining a
line $\lambda(p)$ which meets
$\ell,K_1,\ldots,K_{i+1}$.
If $k$ is finite with $q$ elements,
there are only $q+1$ lines $\lambda(p)$ defined over $k$.
Thus it is necessary that $q\geq n-1$ to solve this problem over $k$.
We show this condition suffices.
All rational normal surface scrolls are projectively equivalent,
(cf.~\cite{Harris_geometry}, \S9),
thus we may assume that $S_{1,n-1}$ has the following standard form.
Let $x_1,x_2,y_1,\ldots,y_n$ be coordinates for ${\bf P}^{n+1}$
where $\ell$ has equation $y_1 = \cdots = y_n = 0$.
Then for $p = [a,b,0,\ldots,0] \in \ell$,
$\lambda(p)$ is the linear span of $p$ and the point
$[0,0,a^{n-1},a^{n-2}b,\ldots,ab^{n-2},b^{n-1}]$.
Let $\alpha_1,\ldots,\alpha_n\in {\bf P}^1$ be distinct points.
Let $F = \sum_{i=0}^n A_i b^i a^{n-i}$
be a form on ${\bf P}^1$ vanishing at $\alpha_1,\ldots,\alpha_n$.
Define $K_{n+1}$ by the vanishing of the two linear forms
$$
\Lambda_1 \ :\ x_2 - y_1\ \ \ \ \ \
\Lambda_2 \ :\ A_0 x_1 + A_1y_1 + \cdots + A_n y_n.
$$
The intersection of $S_{1,n-1}$ and the hyperplane defined by
$\Lambda_1$ is the rational normal curve
$$
\psi : [a,b] \longmapsto [a^n,a^{n-1}b,a^{n-1}b,
a^{n-2}b^2,\ldots,a b^{n-1},b^n].
$$
Since $\psi^* (\Lambda_2) = F$, the lines meeting each of
$\ell,K_1,\ldots,K_{n+1}$ are
$\lambda(\alpha_1),\ldots,\lambda(\alpha_n)$.
Thus, if $k$ has at least $n-1$ elements, this enumerative problem
may be solved over $k$.
\medskip
These two families
are the only non-trivial examples of Schubert-type enumerative
problems for which we know an explicit description
of their solutions.
Each of these problems can be solved over any field $k$
where $\#{\bf P}^1_k$ exceeds the number of solutions.
It would be interesting to find explicit
solutions to other enumerative problems to test whether this holds more
generally.
\subsection{Arrangements over Finite Fields}
In \S\ref{sec:arrangements} we remarked it is possible to construct
arrangements over some
finite fields.
Here we show how.
Recall that an arrangement is complete flag ${F\!_{\DOT}}$
and $2n-3$ hyperplanes $H_2,\ldots,H_{2n-2}$
such that for any $p$,
\begin{enumerate}
\item $H_p \cap F_{\hfl{p}} = F_{\hfl{p}+1}$,
\item For tableaux $S, T$ of degrees at most $p-1$, if
$H_T\cap H_S \subset H_p$, then $H_T\cap H_S = F_{\hfl{p}+1}$.
\end{enumerate}
We give an equivalent set of conditions.
For any subset $A\neq \emptyset$ of $\{2,3,\ldots,2n-2\}$, let $H_A$ be
$\bigcap\{H_i\,|\, i\in A\}$.
Set $H_{\emptyset} = {\mbox{\bf P}^n}$.
\medskip
\noindent{\bf Lemma.}\label{lemma:alt_def_arrangement}
{\em A complete flag ${F\!_{\DOT}}$ and hyperplanes $\,H_2,\ldots,H_{2n}$
constitute an arrangement if and only if
for each $m = 1,\ldots,n-1$, they satisfy
\begin{enumerate}
\item [1$'$.] $H_{2m}\cap F_m = F_{m+1}$ and $F_{m+1}\subset H_{2m+1}$.
\item [2$'$.] For any $A\subset \{2,3,\ldots,2m\}$ where
$H_A$ has codimension $\#A \leq m$, $H_A\not\subset H_{2m+1}$.
\end{enumerate}
}
\medskip
\noindent{\bf Proof:}
Let $H_2,\ldots,H_{2n-2}$ and ${F\!_{\DOT}}$ satisfy 1$'$ and 2$'$.
We show they constitute an arrangement by induction on $m$.
Suppose that for $p<2m$ conditions 1 and 2 for arrangements are satisfied.
We show that 1 and 2 are satisfied for $p = 2m$ and $2m+1$.
For $p=2m$, 1 and 1$'$ are equivalent.
Moreover, if $S,T$ are tableaux of degree less than $2m$,
then $F_m \subset H_T\cap H_S$, so $H_T\cap H_S\not\subset H_{2m}$,
so 2 is satisfied.
Thus $H_2,\ldots,H_{2m}$, $F_1,\ldots,F_{m+1}$
constitute an arrangement in ${\bf P}^n/F_{m+2}\simeq {\bf P}^{m+1}$.
Then by Lemma~\ref{lemma:arrangements}, if $T$ has shape $(\alpha,\beta)$
with $\alpha+\beta < 2m$, $H_T$ has codimension $\beta$ and is not equal to
$F_\beta$.
Let $S,T$ be tableaux of degree at most $2m$
and suppose $F_{m+1}\neq H_S\cap H_T$.
Let $B$ be the union of the second rows of $S$ and $T$.
Since $H_B = H_S\cap H_T$ has codimension $s \leq m$, there is a
set $A\subset B$ of order $s$ with $H_A = H_B$.
By 2$'$, $ H_S\cap H_T = H_A \not\subset H_{2m+1}$.
So $H_{2m+1}$ satisfies 1 and 2.
Conversely, suppose $H_2,\ldots,H_{2n-2}$ and ${F\!_{\DOT}}$
constitute an arrangement.
These satisfy 1$'$.
To show they satisfy 2$'$,
let $A\subset \{2,\ldots,2m\}$ where $H_A$ has
codimension $s = \# A$.
Suppose $A = a_1<\cdots<a_s$ and let $j$ be the largest index such that
$a_j<2j$.
If $j=0$, then $A$ is the second row of a tableau $T$,
so $H_A = H_T \not\subset H_{2m+1}$.
If $j\neq 0$, then $j\geq 2$.
Since $p<2j$ implies $F_j\subset H_p$ and $H_A$ has codimension $s$, we
have $F_j = H_{a_1}\cap \cdots\cap H_{a_j}$.
An induction using condition 1$'$ shows that
$F_j = H_3\cap H_2\cap H_4\cap \cdots\cap H_{2j-2}$.
Thus $H_A = H_{A'}$ where
$$
A' = 3,2<\cdots<2j-2<a_{j+1}<\cdots<a_s,
$$
and if $i>j$, $a_i\geq 2i$.
Let $B: 2<\cdots<2j-2<a_{j+1}<\cdots<a_s$.
then we see that $B$ is the second row a tableau $T$ of degree at most $2s$.
Let $S$ be the tableau of degree 3 whose second row consists of $3$.
Since $s\leq m$, $F_{m+1} \neq H_S\cap H_T$,
so $H_A = H_S\cap H_T \not\subset H_{2m+1}$.
\QED
\subsection{Corollary.}\label{sec:refined_condition} {\em
Condition $2'$ may be replaced by
\begin{enumerate}
\item[2$''$.] If $A: 3,2<\cdots<2j-2<a_{j+1}<\cdots<a_m\leq 2m$
satisfies
$i>j$ implies $a_i\geq 2i$, then $H_A\not\subset H_{2m+1}$.
\end{enumerate}}
\medskip
\noindent{\bf Proof:}
Suppose $H_2,\ldots,H_{2n-2}$ and ${F\!_{\DOT}}$ satisfy 1$'$ and 2$'$
and $A : 3,2<\cdots<2j-2<a_{j+1}<\cdots<a_m\leq 2m$ satisfies
$i>j$ implies $a_i \geq 2i$.
Then $2<\cdots<2j-2<a_{j+1}<\cdots<a_m$ is the second row of
a tableau $T$.
By Lemma~\ref{lemma:arrangements}, $H_{T}$ has codimension $m-1$ and
does not equal $F_{m-1}$.
Thus $H_T\not\subset H_3$ so $H_A = H_T\cap H_3$ has codimension $m$.
Conversely, suppose
$H_2,\ldots,H_{2n-2}$ and ${F\!_{\DOT}}$ satisfy 1$'$ and 2$''$.
{}From the proof of the previous Lemma, it suffices to know
2$'$ for those subsets $A$ of the form
$3,2<\cdots<2j-2<a_{j+1}<\cdots<2s\leq 2m$, where $i>j$ implies $a_i\geq 2i$.
If $B = A\cup\{2s+2,\ldots,2m\}$, then $B$ also has the form
in 2$''$.
So $H_B \not\subset H_{2m+1}$.
Since $H_B\subset H_A$, $H_A\not\subset H_{2m+1}$,
showing 2$'$ holds for $H_2,\ldots,H_{2n-2},{F\!_{\DOT}}$.
\QED
We estimate the size of a field $k$ necessary to
construct an arrangement.
\subsection{Theorem.}\label{thm:arrangement_finite_field}
{\em
There exists an arrangement in ${\bf P}^n_k$ if the order of $k$
is at least
$$
\frac{(2n-4)!}{(n-2)!(n-1)!}\ +\ \sum_{i=1}^{n-4} \frac{(2i)!}{i! (i+1)!}.
$$
}
\medskip
\noindent{\bf Proof:}
Consider the problem of inductively constructing an arrangement
in ${\bf P}^n_k$ satisfying 1$'$ and 2$''$ of
\S\S\ref{lemma:alt_def_arrangement} and~\ref{sec:refined_condition}.
Since it is always possible to find a hyperplane not
containing any particular proper linear subspace of ${\mbox{\bf P}^n}$, the only possible
obstruction is the selection of hyperplanes $H_{2m+1}\supset F_{m+1}$
satisfying 2$''$ for $m=0,1,\ldots,n-2$.
Let $\check{\bf P}^m$ be the set of hyperplanes defined over $k$
containing $F_{m+1}$.
Every codimension $m$ subspace $H_A$ containing $F_{m+1}$
determines a hyperplane $\check{H}_A$ in $\check{\bf P}^m$
consisting of those hyperplanes $H$ of ${\bf P}^n$ containing
$H_A$.
Thus there exists a hyperplane $H_{2m+1}$ satisfying 2$'$ if,
as sets of $k$-points,
$$
X = \check{\bf P}^m - \bigcup_{A\in {\cal S}} \check{H}_A\neq \emptyset,
$$
Where ${\cal S}$ is the set of those sequences
$3,2<\cdots<2j-2<a_{j+1}<\cdots<a_m\leq 2m$ such that
if $i>j$, then $a_i\geq 2i$.
We claim
$\#{\cal S} = s_m = \frac{(2m)!}{m!(m+1)!} +
\sum_{i=1}^{m-2} \frac{(2i)!}{i! (i+1)!}$.
Suppose $k$ has $q\geq s_{n-2}$ elements.
Since $\check{\bf P}^m$ has $(q^{m+1}-1)/(q-1)$ elements and
each $\check{H}_A$ has $(q^m-1)/(q-1)$ elements, $X$
is non-empty if
$q^{m+1}-1 > (q^m-1)s_m$.
This holds as
$$
\left\lfloor\frac{q^{m+1}-1}{q^m-1}\right\rfloor \geq q \geq s_{n-2}\geq s_m.
$$
To enumerate ${\cal S}$, let
$\{3,2<\cdots<2j-2<a_{j+1}<\cdots<a_m\} \in {\cal S}$.
If $b_j = a_{j+i} - 2j$, then
$b_1,\ldots, b_{m-j}$ is the second row of a tableau of shape
$(m-j,\,m-j)$.
Conversely, if $b_1,\ldots, b_{m-j}$ is the second row of a
tableau of shape $(m-j,\,m-j)$, then
$$
\{3, 2<\cdots<2j-2<b_1+2j<\cdots<b_{m-j}+2j\} \in {\cal S}.
$$
Let ${\cal T}_s$ be the set of tableaux of shape $(s,s)$.
These considerations show there is a bijection
$$
{\cal S} \longleftrightarrow
{\cal T}_m\cup{\cal T}_{m-2}\cup{\cal T}_{m-3}\cup\ldots\cup{\cal T}_0.
$$
Noting that $\#{\cal T}_s = \frac{(2s)!}{s!(s+1)!}$,
by the hook length formula of Frame, Robinson, and Thrall~\cite{FRT},
shows that the order of ${\cal S}$ is
$\frac{(2m)!}{m!(m+1)!}+\sum_{i=1}^{m-2} \frac{(2i)!}{i! (i+1)!}$.
\QED
This result is not the best possible:
For ${\bf P}^4$, this gives $q\geq 5$, but
arrangements in ${\bf P}^4$ may be constructed over
the field with three elements.
|
1998-04-01T03:07:34 | 9510 | alg-geom/9510002 | en | https://arxiv.org/abs/alg-geom/9510002 | [
"alg-geom",
"math.AG"
] | alg-geom/9510002 | Lev Borisov | Lev A. Borisov | Finiteness Theorem for Sp(4,Z) | null | null | null | null | null | We consider Siegel upper half space of rank two ${\cal H}^2$ and different
subgroups $H\subseteq {\bf Sp(4,Z)}$ of finite index. The purpose of this paper
is to prove that the field of rational functions of ${\cal H}^2/H$ has general
type for all but the finite number of $H$.
| [
{
"version": "v1",
"created": "Mon, 2 Oct 1995 17:30:45 GMT"
},
{
"version": "v2",
"created": "Wed, 1 Apr 1998 01:07:33 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Borisov",
"Lev A.",
""
]
] | alg-geom | \section{Introduction}
The Siegel upper half space of rank two consists of complex symmetric two by two
matrices whose imaginary part is positive definite. It will be denoted by
${\cal H}$ throughout the paper. It is the moduli space of principally
polarized marked abelian surfaces.
The group ${\rm Sp(4,{\bf Z})}$ acts on ${\cal H}$ by the automorphisms of the
marking. This group consists of four by four integer matrices of the form
$\pmatrix{A&B \cr C&D\cr}$ where $A,B,C,$ and $D$ are two by two matrices that
obey $A{}^tB=B{}^tA,C{}^tD=D{}^tC,A{}^tD-B{}^tC={\bf 1}.$
Written in coordinates, this action becomes
$$\pmatrix{A&B\cr C&D\cr}\cdot M=(AM+B)(CM+D)^{-1}.$$
It is a natural generalization of the usual upper half plane with the action
of ${\rm Sl(2,{\bf Z})}$. It is related to various moduli spaces of
abelian surfaces in the same way the usual upper half plane is related
to moduli spaces of elliptic curves.
We shall be concerned mostly with quotients of ${\cal H}$ by
the action of subgroups $H$ of finite index in ${\rm Sp(4,{\bf Z})}.$
These quotients are known to be algebraic varieties of dimension $3$.
They have been studied extensively since the end of last century.
Some of these varieties have extremely rich
and beautiful geometry, see for instance \cite{Geer},\cite{Lee} and \cite{GeerII}.
The goal of this paper is to prove the following statement, see
proposition \ref{fintheorem}.
{\bf Finiteness Theorem.} There are only finitely many subgroups
$H\subseteq {\rm Sp(4,{\bf Z})}$ of finite index such that ${\cal H}/H$
is not of general type.
The important corollary of this result is that there are only finitely many
subgroups $H$ such that the quotient ${\cal H}/H$ is rational.
Varieties of general type can be viewed as the generalization to higher
dimension of curves of genus two or more.
It is reasonable to expect that they do not have any special geometric properties,
and thus all interesting quotients ${\cal H}/H$ can be in
principle listed. This theorem is analogous to the result of J.G. Thompson
(see \cite{Thompson}) for the usual upper half plane.
More accurate estimates for certain classes of subgroups of ${\rm Sp(4,{\bf Z})}$
have been proved in \cite{Grady,Gritsenko,Hulek}.
The method of the proof is roughly the following. It is known that $H$
contains a principal congruence subgroup $\Gamma(n)$ of some level $n$.
The quotient ${\cal H}/\Gamma(n)$ admits a well understood smooth
compactification, constructed in the paper of Igusa \cite{Igusa}.
Our aim is to construct global sections of the multicanonical
line bundle on the desingularization of the compactification of
$({\cal H}/\Gamma(n))/(H/\Gamma(n))$
from the sections of certain line bundles on the Igusa compactification of
${\cal H}/\Gamma(n)$.
We will use standard facts about singular algebraic varieties, which are
collected in Section 7. The results of Sections 2 and 4
are probably known to specialists in the field, although there
are not many convenient references. Section 3 and 5 are the key
sections of the paper. The former is a purely combinatorial calculation, and
the latter is an algebra-geometrical one. In both sections we assume
that $n$ is a power of a prime, and Section 6 allows
us to drop this restriction.
This paper is essentially my University of Michigan thesis.
Major part of it was done when I was still in Moscow.
It is influenced a lot by my advisor Vasilii Iskovskikh
who taught me the basics of algebraic geometry as well as some
singularity theory which comes in very handy in the paper.
I would like to thank
Osip Shvartsman for many stimulating discussions on the subject of this paper.
My thesis advisor Igor Dolgachev has been a constant source of inspiration for my
studies of algebraic geometry at the University of Michigan. I also wish
to thank
Gopal Prasad for several valuable conversations and Melvin Hochster for providing
a useful reference.
\section{Algebraic cycles on Satake and Igusa compactifications}
The purpose of this section is to recall the basic facts about some
special algebraic cycles on the Satake and Igusa compactifications of
${\cal H}/\Gamma(n)$ and to find a nice combinatorial description of
their components.
We consider the principal congruence subgroup $\Gamma(n)$ of level
$n$ inside ${\rm Sp(4,{\bf Z})}$. For the rest of the section $n$ is fixed
and is greater than two. The group $\Gamma(n)$ acts on the Siegel upper
half space of rank two ${\cal H}$ according to the formula
$$\pmatrix{A&B\cr C&D\cr}\cdot\tau=(A\tau+B)(C\tau+D)^{-1}.$$
The quotient ${\cal H}/\Gamma(n)$ is a nonsingular algebraic variety.
It is a Zariski open subset of the compact singular algebraic
variety called the Satake compactification of ${\cal H}/\Gamma(n)$.
The exact references can be found in \cite{Igusa}. The monoidal
transformation of the Satake compactification along the singular
locus is nonsingular. This
variety was first considered by Igusa in \cite{Igusa}, and is called
the Igusa compactification. We denote it by $X_n$. Points of
${\cal H}/\Gamma(n)$ are referred to as the {\it finite}\/ part of the
compactification and the rest is the part {\it at infinity.}
The part at infinity of the Satake compactification consists of a finite
number of curves that intersect in a finite number of {\it cusp}\/
points. The part at infinity of the Igusa compactification is a divisor
$D=\sum_iD_i$, which has simple normal crossings. Its components are elliptic
fibrations over the curves at infinity of the Satake compactification.
The group $G=\Gamma(1)/\Gamma(n)$ acts on both compactifications,
and the map between them is equivariant. The group $G$ is isomorphic
to ${\rm Sp(4,{\bf Z}/n{\bf Z})}$, and $\pm{\bf 1}$ act as the identity.
There are two more types of divisors on the Igusa compactification
that will be important to our discussion. First of all, there are
divisors $E_i$ that are conjugates of the closure of the image
in ${\cal H}/\Gamma(n)$ of the set of diagonal matrices in
${\cal H}$. They are disjoint and are isomorphic to the product of
two modular curves (see \cite{Yamazaki}). We denote their sum by $E$.
We also consider divisors that are conjugates of
the closure of the image of the set of matrices $\pmatrix{x&y\cr y&x\cr}$
in ${\cal H}$. Geometrically, these matrices correspond to Jacobians
of genus two curves with an extra involution, see \cite{Bolza}.
We denote them by $F_i$ and their sum by $F$. They do intersect with
each other and their geometry is somewhat more complicated. We prove
the necessary statements regarding these at the end of this section.
We abuse notation somewhat to denote ${\rm Sp(4,{\bf Z})}$-conjugates
of the sets $\pmatrix{x&0\cr 0&z\cr}$ and $\pmatrix{x&y\cr y&x\cr}$ by
$E_i$ and $F_j$ as well.
Let us introduce the abelian group $V$ of column vectors of
height four with coefficients in ${\bf Z}/n{\bf Z}$
provided with the skew form $\langle ~,~\rangle $ defined by the formula
$\langle {}^t(x^1,...,x^4),{}^t(y^1,...,y^4)\rangle =x^1y^3+x^2y^4-x^3y^1-x^4y^2.$
The group $G$ acts naturally on $V$ by left multiplication.
Our goal here is to construct $G$-equivariant
correspondences between components of cycles on the Satake
and Igusa compactifications mentioned above and some objects defined
in terms of the group $V$.
\begin{prop}
{ The infinity divisors of the Igusa compactification (or equivalently,
the curves at infinity of the Satake compactification) are in one-to-one $G$-
equivariant correspondence with the primitive $\pm$vectors $\pm v$ in $V$.
Here we call a vector $v$ primitive iff its order is exactly $n$.
The $\pm$ means that we identify opposite vectors.}
\label{indexD}
\end{prop}
{\em Proof.} It is known (see \cite{Igusa}) that all components of $D$ are
$G$-conjugate. It can be shown that the group $G$ also acts transitively on
the set of primitive $\pm$vectors. It remains to notice that the stabilizer
of the $\pm$vector ${}^t(0,1,0,0)$ coincides with the stabilizer of $D_0$,
where $D_0$ is the {\it standard}\/ divisor that corresponds to the basis
of open subsets $\{\pmatrix{x&y\cr y&z\cr},~{\rm Im}(z)\to+\infty\}$ of ${\cal H}.$
The description of the stabilizer of $D_0$ can be derived from \cite{Igusa}.
It consists of matrices of the form
$$\pm\pmatrix{a&0&b&m_3\cr m_1&1&m_2&m_4\cr
c&0&d&m_5\cr 0&0&0&1\cr}({\rm mod}n),$$$$~ad-bc=1({\rm mod}n),
~bm_1+m_3=am_2({\rm mod}n),~dm_1+m_5=cm_2({\rm mod}n).$$
This allows us to construct a bijective correspondence between infinity
divisors on the Igusa compactification and $\pm$vectors in $V.$
We will use the notation $\pm v_\alpha$ for the $\pm$vector that corresponds
to the divisor $D_\alpha$ and vice versa.\hfill$\Box$
\begin{prop}
{ Cusp points $Q_i$ of the Satake compactification
are in one-to-one $G$-equivariant correspondence with the following pairs
$(W,\pm f)$. We consider all possible $W\subset V$ and $f:W\times W\to
{{\bf Z}/n{\bf Z}}$ such that
(1) $W$ is a subgroup of $\,V$ isomorphic to $({{\bf Z}/n{\bf Z})}^2$,
(2) $\langle ,\rangle |_W=0$,
(3) $f$ is a non-degenerate skew form on $W$ with values in
${{\bf Z}/n{\bf Z}}$, where non-degeneracy means $f(W\times W)\ni 1(n)$.}
\label{indexptsSatake}
\end{prop}
{\em Proof.} All cusp points are conjugates of the one described
by the basis of open sets $$\{\pmatrix{x&y\cr y&z\cr},~{\rm Im}(\pmatrix{x&y\cr
y&z\cr})\to+\infty\}$$
(see \cite{Igusa}). We call this point {\it standard}\/.
The stabilizer of the standard point consists of matrices
of the form
$$\{\pmatrix{A&B\cr {\bf 0}&{}^tA^{-1}\cr}, A{}^tB=B{}^tA, det(A)=\pm1(n)\}$$
However, this is exactly the stabilizer of the {\it standard}\/
pair $$(W,\pm f)=({}^t(*,*,0,0),f({}^t(1,0,0,0),{}^t(0,1,0,0))=1(n)).$$
It can be shown that any pair $(W,f)$ is a $G$-conjugate of the standard pair.
As a result, we can define the required $G$-equivariant correspondence.
We will use the notation $(W_\alpha,\pm f_\alpha)$ for the pair that
corresponds to the point $Q_\alpha$ and vice versa. \hfill$\Box$
\begin{prop}
{ The curve at infinity of the Satake compactification that corresponds
to the divisor $D_\alpha$ contains the cusp point $Q_\beta$ iff
$v_\alpha\in W_\beta$.}
\label{indexcurvethroughpointonSatake}
\end{prop}
{\em Proof.} Consider the action of the group that stabilizes the standard
curve. It acts transitively on the set of cusp points of this curve, which
are exactly the $Q_i$'s. Therefore, all inclusion pairs are acted upon
transitively. The standard curve passes through the standard point,
and ${}^t(0,1,0,0)\in {{}^t(}*,*,0,0)$, which proves the only if part of the
statement. On the other hand, the stabilizer of the standard point acts
transitively on the $\pm$vectors in ${}^t(*,*,0,0)$, which proves the if
part. \hfill$\Box$
\begin{prop}
{ Two infinity divisors $D_\alpha$ and $D_\beta$ intersect over
the point $Q_\delta$ iff $v_\alpha,v_\beta \in W_\delta$ and
$f_\delta(v_\alpha,v_\beta)=\pm 1(n)$. In this case the intersection is
isomorphic to ${\bf P}^1$.}
\label{indexDD}
\end{prop}
{\em Proof.} Because of transitivity of the action, the point $Q_\delta$
may be considered standard. We follow the argument of \cite{Igusa} for the
case where $g_0=0$ and $g_1=2$. Curves of the intersection
of the two infinity divisors are conjugate to one of the curves obtained by
taking the limits of the points $\pmatrix{x&y\cr y&z\cr}$,
with imaginary parts of two out of three normal coordinates
$y,(-x-y),(-z-y)$ going to $-\infty$ and the remaining one being bounded.
They are pairwise intersections of the divisors that correspond to
the limits where exactly one of the imaginary parts goes to $-\infty$
and the other two are bounded. The divisor that corresponds to
$Im(z+y)\to\infty$ is exactly the standard divisor, because ${\rm Im}(y)$
is bounded. The other two divisors are obtained from the standard one
by the action of $\{\pmatrix{A&{\bf 0}\cr {\bf 0}&{}^tA^{-1}\cr}\}$ with
$A=\pmatrix{1&-1\cr 0&1\cr}$ and $A=\pmatrix{0&1\cr 1&0\cr}$
respectively. Therefore, these divisors correspond to the $\pm$vectors
$\pm {}^t(0,1,0,0)$, $\pm {}^t(-1,1,0,0)$, and $\pm {}^t(1,0,0,0).$
This proves the "only if" part. The "if" part follows from the transitivity
of the action of $G$ on the combinatorial data on the right hand side of
the statement. The fact that each irreducible component of the intersection
is isomorphic to ${\bf P}^1$ is proven in \cite{Igusa}, and the uniqueness of
the irreducible component can be derived easily from the description of
the divisors in terms of the above limits.
\hfill$\Box$
\begin{prop}
{ Three infinity divisors $D_\alpha, D_\beta, D_\gamma$ intersect over
the point $Q_\delta$ iff $v_\alpha,v_\beta,v_\gamma \in W_\delta$,
the set $\{\pm v_\alpha \pm v_\beta \pm v_\gamma\}$ contains $0$, and
$f_\delta(v_\alpha,v_\beta)=\pm 1(n)$. In this case the intersection point
is unique.}
\label{indextDDD}
\end{prop}
{\em Proof.} As in the previous proposition, we prove that all points
of triple intersection are conjugates of the intersection point of the
divisors that correspond to $\pm {}^t(0,1,0,0)$, $\pm {}^t(-1,1,0,0)$,
and $\pm {}^t(1,0,0,0).$ Then we notice that any triple of $\pm$vectors
with the above properties can be transformed to the triple
$(\pm {}^t(0,1,0,0)$, $\pm {}^t(-1,1,0,0)$, $\pm {}^t(1,0,0,0)).$
\hfill$\Box$
\begin{prop}
{ Divisors $E_i$ are in one-to-one $G$-equivariant correspondence
with unordered pairs $(W_1,W_2)$ such that
(1) $W_1$ and $W_2$ are subgroups of $V$ isomorphic to
$({{\bf Z}/n{\bf Z}})^2$ each,
(2) $W_1 \perp W_2=V$.}
\label{indexE}
\end{prop}
{\em Proof.} All divisors $E_i$ are conjugates of the {\it standard}\/
one defined as the closure of the image of the set of diagonal matrices.
The stabilizer of this standard divisor is described in \cite{Yamazaki}.
It is exactly the stabilizer of the {\it standard}\/ pair
$({}^t(*,0,*,0),{}^t(0,*,0,*)).$ It is easy to show that every pair $(W_1,W_2)$
with above properties is conjugate to this standard one, which
completes the proof. For a given $E_\alpha$ the corresponding pair
will be denoted by $(W_{\alpha1},W_{\alpha2})$ and vice versa.
\hfill$\Box$
\begin{prop}
{ The divisor $E_\alpha$ intersects the divisor $D_\beta$ iff
$v_\beta$ lies in one of the subgroups $W_{\alpha1},W_{\alpha2}$.
In this case the intersection is isomorphic to the modular curve
of principal level $n$.}
\label{indexED}
\end{prop}
{\em Proof.} We assume that the divisor $E_\alpha$ is a standard one.
Then the statement of the proposition follows from the description of
the action of the group $\Gamma(n)$ in a neighborhood of the set of
diagonal matrices (see \cite{Yamazaki}).
\hfill$\Box$
There is an alternative way to describe divisors $E_i$.
\begin{prop}
{ Divisors $E_i$ are in one-to-one $G$-equivariant correspondence
with conjugates of the involution
$$\pm\pmatrix{1&0&0&0\cr 0&-1&0&0\cr 0&0&1&0\cr 0&0&0&-1\cr}$$
in the group ${\rm Sp(4,{\bf Z}/n{\bf Z})/\{\pm 1\}}.$}
\label{indexinvE}
\end{prop}
{\em Proof.} The action of
$$\pm\pmatrix{1&0&0&0\cr 0&-1&0&0\cr 0&0&1&0\cr 0&0&0&-1\cr}$$
on ${\cal H}$ is defined by $\pmatrix{x&y\cr y&z\cr}\to \pmatrix{x&-y\cr
-y&z\cr}$, so it fixes exactly the points of the standard divisor
$E_0.$ This gives a one-to-one correspondence between $\Gamma(1)$
conjugates of this involution and $\Gamma(1)$ conjugates of the
diagonal in ${\cal H}.$ This correspondence survives when we mod out
by $\Gamma(n)$, and then we use surjectivity of
$\Gamma(1)/\Gamma(n)\to {\rm Sp(4,{\bf Z}/n{\bf Z})}.$
\hfill$\Box$
This alternative description is related to the original one as follows.
\begin{prop}
{ The involution $\varphi_\alpha$ that fixes all points of the
divisor $E_\alpha$ is defined by
(1) $\varphi_\alpha |_{W_{\alpha1}}={\rm id} |_{W_{\alpha1}}$,
(2) $\varphi_\alpha |_{W_{\alpha2}}=-{\rm id} |_{W_{\alpha2}}$.
This definition makes sense, because the switch of the order of two
subgroups $W_{\alpha1},W_{\alpha2}$ results in the change of sign
of the involution $\varphi_\alpha$.}
\label{twodescrE}
\end{prop}
{\em Proof.} It is true for the standard divisor, and the rest follows
from the transitivity of the action of the group $G$. \hfill$\Box$
We can describe divisors $F_i$ in the same fashion, because there is
also an involution in ${\rm Sp(4,{\bf Z})}$,
whose fixed points on ${\cal H}$ are exactly the matrices $\pmatrix{x&y\cr
y&x\cr}$ that form the standard divisor $F_0.$
\begin{prop}
{ Divisors $F_i$ are in one-to-one $G$-equivariant correspondence
with conjugates in ${\rm Sp(4,{\bf Z}/n{\bf Z})/\{\pm 1\}}$ of the involution
$$\pm\pmatrix{0&1&0&0\cr 1&0&0&0\cr 0&0&0&1\cr 0&0&1&0\cr}.$$}
\label{indexinvF}
\end{prop}
{\em Proof.} It is completely analogous to the proof of \ref{indexinvE}.
\hfill$\Box$
Now we are going to discuss the geometry of the divisor $F$.
\begin{prop}
{ Divisors $F_i$ are smooth surfaces of general type if $n$ is
sufficiently big. Moreover, ${\rm dim}H^0(F_i,K_{F_i}) > 0$.}
\label{gentypeF}
\end{prop}
{\em Proof.} Because $F_\alpha$ is an irreducible component
of the set of fixed points of an involution on $X$, it is a smooth surface.
The finite part of $F_\alpha$ is isomorphic to
the quotient of ${\cal H}^1/\Gamma_1(2n)\times{\cal H}^1/\Gamma_1(2n)$
by the diagonal action of the group $\Gamma_1(n)/\Gamma_1(2n)$, where
${\cal H}^1$ is the usual upper half plane, and $\Gamma_1(n)$ is the principal
congruence subgroup of ${\rm Sl(2,{\bf Z})}.$ This can be shown by direct calculation,
using an element of ${\rm Sp(4,{\bf R})}$ that maps a matrix
$\pmatrix{x&y\cr y&x\cr}$ to the matrix $\pmatrix{x-y&0\cr 0&x+y\cr}$.
As a result, $F_\alpha$ admits a finite morphism to
$({\cal H}^1/\Gamma_1(n))^2$, which is of general type and has global
$2$-forms, if $n$ is sufficiently big. \hfill$\Box$
The divisor $F+D$ does not have simple normal crossings.
\begin{prop}
{ There are exactly $n$ divisors $F_\gamma$ on $X_n$ that contain any
given curve $l_{\alpha\beta}=D_\alpha\cap D_\beta$.}
\label{nFl}
\end{prop}
{\em Proof.} We assume that the line $l_{\alpha\beta}$ is standard. Let us
consider the involution that fixes all points of the divisor $F_i$.
It fixes all points of the line $l_{\alpha\beta}$. This implies that
the matrix of this involution equals
$$\pm\pmatrix{0&1&0&b\cr 1&0&-b&0\cr 0&0&0&1\cr 0&0&1&0\cr}.$$
We can lift these involutions to $\Gamma(1)$ so that they map
$\pmatrix{x&y\cr y&z\cr}$ to $\pmatrix{z+b&y\cr y&x-b\cr}$. The corresponding
divisors $F_i$ are $\pmatrix{x&y\cr y&x-b\cr}.$ The number of
$\Gamma(n)$-inequivalent divisors of this form is equal to $n$.
\hfill$\Box$
\begin{prop}
{ If a divisor $F_\gamma$ contains a line $l_{\alpha\beta}$,
then $c_1(F_\gamma)l_{\alpha\beta}=-2$.}
\label{Fintl}
\end{prop}
{\em Proof.} The line $l_{\alpha\beta}$ may be assumed to be standard.
Calculations in the local coordinates show that the normal bindle to
$l_{\alpha\beta}$ inside $X_n$ is isomorphic to ${\cal O}(2)\oplus{\cal O}(2)$,
and the normal bundle to $l_{\alpha\beta}$ inside $F_\gamma$ is the subbundle
of the form $(x,e^{2\pi i b/n}x)$.\hfill{$\Box$}
\section{Upper bounds on the indices of subgroups of
${\rm Sp(4,{\bf Z}/p^t{\bf Z})}$}
This is the key section of the paper. Its purpose is to estimate
the index of the subgroup $H\subseteq {\rm Sp(4,{\bf Z}/n{\bf Z})}$ if $H$
contains sufficiently many elements of a special type. We additionally
assume that $n=p^t$ for some prime number $p$ and integer $t$.
We fix $H$ and assume that $H\ni\pm{\bf 1}$ throughout the rest
of the section. We use the notation $[x]_p$ with $x\in {\bf R}_{\geq 1}$
for the maximum number of form $p^t,t\in {\bf N}$ that does not exceed $x$.
We first discuss subgroups that contain many elements that fix $D_i$
pointwise.
\begin{dfn}
{ For any primitive vector $v$ we consider
the subgroup ${\rm Ram}_G(v)$ of $G={\rm Sp(4,{\bf Z}/n{\bf Z})}$ that consists of
transvections, which are operators of the form
$$r_{v,\alpha}: w \to w+\alpha\langle w,v\rangle v,~\alpha\in{{\bf Z}/n{\bf Z}}.$$
Because $v$ is primitive, ${\rm Ram}_G(v)\simeq {{\bf Z}/n{\bf Z}}$. We denote
$${\rm Ram}_H(v)=H\cap {\rm Ram}_G(v),~{\rm ram}_H(v)=|{\rm Ram}_H(v)|/n.$$
Clearly, ${\rm ram}_H(-v)={\rm ram}_H(v)$.}
\label{dfnramD}
\end{dfn}
\begin{rem}
{ We shall see later in Proposition \ref{ramdiv} that ${\rm Ram}_G(v_\alpha)$
is exactly the group that fixes
all points of the divisor $D_\alpha$.}
\end{rem}
\begin{prop}
{ If
$\sum_{\pm v}{\rm ram}_H(v)\geq\epsilon \cdot \sharp (\pm v),$
then $|G:H| < 2^5\epsilon^{-2}[2^{72}\epsilon^{-42}]_p.$}
\label{boundD}
\end{prop}
{\em Proof.} We can forget about $\pm$ signs in the above proposition.
For any set $I$ of primitive vectors we define the {\it ramification
mean}\/ of $I$ to be equal to $$(\sum_{v\in I}{\rm ram}_H(v))/|I|.$$ Clearly,
the ramification mean never exceeds $1$.
Among the subgroups of $V$ that are isomorphic to $({{\bf Z}/n{\bf Z}})^3$,
we choose a subgroup $V_3$, such that the ramification mean of the set
of primitive vectors that lie in it is maximum among all such subgroups.
Any two primitive vectors are contained in the same number of subgroups
that are isomorphic to $({{\bf Z}/n{\bf Z}})^3$, so the sum of the ramification means
among these subgroups is at least $\epsilon$ times the number of subgroups.
Hence, the ramification mean of $V_3$ is at least $\epsilon$. Analogously,
we can choose the subgroup $V_2$ that has the maximum ramification mean
among the subgroups with the properties
(1) $V_2\simeq ({{\bf Z}/n{\bf Z}})^2$,
(2) $V_2\subseteq V_3$,
(3) $\langle,\rangle |_{V_2}=0$.
Any two primitive vectors in $V_3$ are conjugates with respect to the
stabilizer of $V_3$ and therefore are contained in the same number of subgroups
$V_2$ that satisfy the above three properties. As a result, the ramification
mean of the set of the primitive vectors that lie in $V_2$ is also
at least $\epsilon$. The total number of primitive vectors $v$ in $V_2$ is
$n^2(1-p^{-2}).$ One can easily show that
at least $(\epsilon/2)n^2(1-p^{-2})$ of them have ${\rm ram}_H(v)$ bigger than
$\epsilon/2$, because otherwise the ramification mean of $V_2$ would
be less than $\epsilon.$ We call these vectors {\it good.}
We may additionally assume without loss of generality that
$V_2={}^t(*,*,0,0)$ and $V_3={}^t(*,*,*,0).$
If $v={}^t(x,y,0,0)$, then $r_{v,1}$ has the matrix $\pmatrix{{\bf 1}&B\cr
{\bf 0}&{\bf 1}\cr}$, where $B=\pmatrix{x^2&xy\cr xy&y^2\cr}.$
Denote the group that consists of matrices $\pmatrix{{\bf 1}&B\cr
{\bf 0}&{\bf 1}\cr}$ by $G_{V_2}.$ We can prove the following
statement.
\begin{lem}
{ $|G_{V_2}:(G_{V_2}\cap H)|\leq[\epsilon^{-9}2^{14}]_p.$}
\label{V_2}
\end{lem}
{\em Proof of the lemma.}
We assume that ${\rm ram}_H({}^t(1,0,0,0))\geq \epsilon/2.$ We can do it,
because there is a primitive vector in $V_2$ with this property and
we may transform it to ${}^t(1,0,0,0)$ by an element of $G$ that stabilizes
$V_2.$ This transformation may not stabilize $V_3$, so we can not use
this assumption in the proof of Proposition \ref{boundD}.
At least $\epsilon n^2(1-p^{-2})/4$ good vectors ${}^t(x,y,0,0)$
satisfy ${\rm g.c.d.}(y,n)\leq [4/(\epsilon(1-p^{-2})]_p.$ Really, the number of
vectors in $V_2$ that do not satisfy this condition is at most $\epsilon
n^2(1-p^{-2})/4.$ We pick one such vector and call it ${}^t(x_1,y_1,0,0).$
Consider the set of vectors $v={}^t(x,y,0,0)$ that
have the following properties
(1) $v$ is good,
(2) ${\rm g.c.d.}(y,n)\leq [4/(\epsilon(1-p^{-2}))]_p$,
(3) ${\rm g.c.d.}(x_1y-y_1x,n)\leq [4/(\epsilon(1-p^{-2}))]_p$.
There are at least $\epsilon n^2(1-p^{-2})/4$ vectors that satisfy
the first two conditions and there are less than $\epsilon n^2(1-p^{-2})/4$
vectors that do not satisfy the third one. As a result, such a vector exists,
and we denote it by $v={}^t(x_2,y_2,0,0)$.
So $H$ contains three elements of $G_{V_2}$ with the matrices
$$B=\pmatrix{\alpha^2&0\cr 0&0\cr},\pmatrix{\alpha^2x_1^2&\alpha^2x_1y_1\cr
\alpha^2x_1y_1&\alpha^2y_1^2\cr},\pmatrix{\alpha^2x_2^2&\alpha^2x_2y_2\cr
\alpha^2x_2y_2&\alpha^2y_2^2\cr},$$
where ${\rm g.c.d.}(\alpha,n)\leq [2/\epsilon]_p.$ They generate a subgroup of $G_{V_2}$
of index equal to the greatest common divisor of $n$ and the determinant
of the corresponding three by three matrix. This is equal to
$${\rm g.c.d.}(n,\alpha^6y_1y_2(x_1y_2-x_2y_1))\leq
[(2/\epsilon)]_p^6[4/(\epsilon(1-p^{-2}))]^3
\leq [\epsilon^{-9}2^{14}]_p.$$
This proves the lemma. \hfill{$\Box$}
We now recall that the ramification mean of the set of vectors
$v=~{}^t(x,y,z,0)$ is at least $\epsilon$. It implies that there are at least
$\epsilon n^3(1-p^{-3})/2$ of them that have ${\rm ram}_H(v)\geq \epsilon/2.$
There are at least $\epsilon n^3(1-p^{-3})/4$ of them that additionally
satisfy ${\rm g.c.d.}(z,n)\leq 4/(\epsilon(1-p^{-3})).$
We abuse the notations and also call such vectors good. The operator
$r_{v,\alpha}$ that
corresponds to a vector $v\in V_3$ and a number $\alpha$ has the matrix
$$\pmatrix{1+\alpha xz&0&-\alpha x^2&-\alpha xy\cr \alpha yz&1&-
\alpha xy&-\alpha y^2\cr \alpha z^2&0&1-\alpha xz&-\alpha yz\cr 0&0&0&1\cr}.$$
All the elements we have described so far lie inside the group
$$G_{V_3}=\pmatrix{a&0&b&m_3\cr m_1&1&m_2&m_4\cr
c&0&d&m_5\cr 0&0&0&1\cr}({\rm mod}n),$$$$~ad-bc=1({\rm mod}n),
~bm_1+m_3=am_2({\rm mod}n),~dm_1+m_5=cm_2({\rm mod}n)\}.$$
This group has a natural projection $\lambda$ to the ${\rm Sl(2,{\bf Z}/n{\bf Z})}$
defined by the entries $a,b,c,d.$ Our next step is to show that
the images of elements of $H$ generate a subgroup of ${\rm Sl(2,{\bf Z}/n{\bf Z})}$
of bounded index.
We have at our disposal the matrices $\pmatrix{1+\alpha xz&-\alpha x^2\cr
\alpha z^2&1-\alpha xz\cr}$, as well as $\pmatrix{1&\beta\cr 0&1\cr}$
with ${\rm g.c.d.}(\beta,n)\leq[\epsilon^{-9}2^{14}]_p.$
Here we use the estimate of $\beta$ that comes from the
result of lemma \ref{V_2}.
We fix $\alpha_0$ that satisfies
${\rm g.c.d.}(\alpha_0,n)=[2/\epsilon]_p.$ Notice that if $(1+\alpha_0 x_1z_1,
\alpha_0 z_1^2)\neq (1+\alpha_0 x_2z_2,\alpha_0 z_2^2)$, then the matrices
$$\pmatrix{1+\alpha_0 x_1z_1&-\alpha_0 x_1^2\cr \alpha_0 z_1^2&
1-\alpha_0x_1z_1\cr},\pmatrix{1+\alpha_0 x_2z_2&-\alpha_0 x_2^2\cr
\alpha_0 z_2^2&1-\alpha_0 x_2z_2\cr}$$
lie in different cosets of the subgroup $\pmatrix{1&*\cr 0&1\cr}.$
Therefore, we can estimate the order of the subgroup generated by the
elements that lie in $H$ simply by multiplying
$n/[\epsilon^{-9}2^{14}]_p$ by the number of different pairs
$(1+\alpha_0 xz, \alpha_0 z^2)$ that we are guaranteed to have.
We have at least $\epsilon n^2(1-p^{-3})/4$ pairs $(x,z)$ that correspond
to at least one good vector ${}^t(x,y,z,0)$ and thus give rise to an
element in $H$ of the above form. We now need to estimate the number of
pairs $(x,z)$ that can give the same $(1+\alpha_0 xz, \alpha_0 z^2).$
The number of different $z$ that have the same $\alpha_0 z^2$
is at most $4\cdot {\rm g.c.d.}(\alpha_0 z^2,n)$, which does not exceed
$4\cdot [2/\epsilon]_p[4/(\epsilon(1-p^{-3}))]_p^2$. Once we know $z$,
the number of $x$ that give the same $1+\alpha_0 xz$ is at most
${\rm g.c.d.}(\alpha_0 z,n)$, which is at most
$[2/\epsilon]_p[4/\epsilon(1-p^{-3})]_p.$
So the total number of pairs $(1+\alpha_0 xz, \alpha_0 z^2)$ is at least
$$(\epsilon n^2(1-p^{-3})/4)/(4[2/\epsilon]_p^2[4/(\epsilon(1-p^{-3}))]_p^3)
\geq \epsilon n^2/(2^5[2^8\epsilon^{-5}]_p).$$
This implies that the images of elements that lie in $H$ generate a
subgroup of ${\rm Sl(2,{\bf Z}/n{\bf Z})}$ of index at most
$${{n^3(1-p^{-2})(1-p^{-1})} \over {(\epsilon n^2/(2^5[2^8\epsilon^{-5}]_p))\cdot
(n/[\epsilon^{-9}2^{14}]_p)}}~\leq~[2^{22}\epsilon^{-14}]_p\epsilon^{-1}2^5.$$
On the other hand, let us estimate the index of $H\cap {\rm Ker}(\lambda)$ in
${\rm Ker}(\lambda)$. We use the formula
$$
\pmatrix{1+\alpha xz&0&-\alpha x^2&-\alpha xy\cr \alpha yz&1&-\alpha
xy&-\alpha y^2\cr \alpha z^2&0&1-\alpha xz
&-\alpha yz\cr 0&0&0&1\cr}\cdot
\pmatrix{1&0&0&b\cr 0&1&b&0\cr 0&0&1&0\cr 0&0&0&1\cr}\cdot
$$$$
\pmatrix{1+\alpha xz&0&-\alpha x^2&-\alpha xy\cr \alpha yz&1&-\alpha
xy&-\alpha y^2\cr \alpha z^2&0&1-\alpha xz&-\alpha yz\cr 0&0&0&1\cr}^{
\hspace{-2pt}-1}
\hspace{-4pt}\cdot
\pmatrix{1&0&0&-b-b\alpha xz\cr 0&1&-b-b\alpha xz&b\alpha z(-2y+
bz+b\alpha xz^2)\cr 0&0&1&0\cr 0&0&0&1\cr}
$$
$$
=\pmatrix{1&0&0&0\cr -b\alpha z^2&1&0&0\cr 0&0&1&b\alpha z^2\cr 0&0&0&1}
$$
to generate the subgroup of $\pmatrix{1&0&0&0\cr *&1&0&0\cr 0&0&1&*\cr 0&0&0&1}$
of index at most ${\rm g.c.d.}(\beta\alpha z^2,n),$ which we can estimate.
$${\rm g.c.d.}(\beta\alpha z^2,n)\leq [\epsilon^{-9}2^{14}]_p[2\epsilon^{-1}]_p
[4\epsilon^{-1}(1-p^{-3})]^2_p\leq [\epsilon^{-12}2^{20}]_p.$$
Because the kernel of $\lambda$ is a semidirect product of the above group
and a subgroup of $G_{V_2}$, we have
$$|{\rm Ker}\lambda:({\rm Ker}\lambda\cap H)|\leq [\epsilon^{-12}2^{20}]_p
[\epsilon^{-9}2^{14}]_p\leq [\epsilon^{-21}2^{34}]_p$$
and
$$|G_{V_3}:(G_{V_3}\cap H)|\leq [\epsilon^{-21}2^{34}]_p[\epsilon^{-14}
2^{22}]_p\epsilon^{-1}2^5 \leq [\epsilon^{-35}2^{56}]_p\epsilon^{-1}2^5.$$
There is only one more step necessary to prove this proposition. Because the
ramification mean of $V$ is at least $\epsilon$, there are at least
$\epsilon n^4(1-p^{-4})/4$ primitive vectors $v={}^t(x,y,z,t)$ that satisfy
(1) ${\rm Ram}_H(v)\geq \epsilon/2$,
(2) ${\rm g.c.d.}(t,n)\leq [4\epsilon^{-1}(1-p^{-4})^{-1}]_p.$
We continue to abuse the notations and call these vectors good.
We use the number $\alpha_0$ defined earlier and consider elements
$r_{v,\alpha_0}$ for all good vectors. They all lie in $H$, and the claim
is that they lie in $\sim n^4$ different cosets of $G:G_{V_3}.$
Indeed, all elements of the group $G_{V_3}$ fix ${}^t(0,1,0,0)$, and
$r_{v,\alpha_0}$ pushes ${}^t(0,1,0,0)$ to ${}^t(x\alpha_0 t, 1+y\alpha_0 t,
z\alpha_0 t,\alpha_0 t^2).$ So if
$${}^t(x\alpha_0 t, 1+y\alpha_0 t,
z\alpha_0 t,\alpha_0 t^2)\neq {}^t(x_1\alpha_0 t_1, 1+y_1\alpha_0 t_1,
z_1\alpha_0 t_1,\alpha_0 t_1^2),$$
then $r_{v,\alpha_0}$ and $r_{v_1,\alpha_0}$ lie in different cosets.
We can estimate the number of vectors that can give the same fourtuple
as follows. If we know $\alpha_0 t^2$, it leaves us with at most
$$4\cdot {\rm g.c.d.}(\alpha_0 t^2,n)\leq 4[2\epsilon^{-1}]_p
[4\epsilon^{-1}(1-p^{-4})^{-1}]_p^2\leq 4[2^6\epsilon^{-3}]_p$$
options for $t.$ Once we know $t$, we have at most
$({\rm g.c.d.}(\alpha_0 t,n))^3$ choices for $(x,y,z).$
This gives us a total of at most
$$ 4[2^6\epsilon^{-3}]_p \cdot ([2\epsilon^{-1}]_p
[4\epsilon^{-1}(1-p^{-4})^{-1}]_p)^3\leq 4[2^{16}\epsilon^{-7}]_p$$
different good vectors $v$ that give $r_{v,\alpha_0}$ from the same
coset. Therefore, we can estimate the number of different cosets
that have representatives in $H$ by
$$(\epsilon n^4(1-p^{-4})/4)/(4[2^{16}\epsilon^{-7}]_p)
\geq \epsilon n^4/(2^5[2^{16}\epsilon^{-7}]_p).$$
Hence, we can estimate the order of $H$ by multiplying the
estimate on the order of its intersection with $G_{V_3}$ by
the number of cosets that it has representatives in, which gives
$$|H|\geq {n^6(1-p^{-2})(1-p^{-1})\over 2^5\epsilon^{-1}[2^{56}\epsilon^{-35}]
_p}\cdot{\epsilon n^4\over (2^5[2^{16}\epsilon^{-7}]_p)}\geq
n^{10}\cdot{\epsilon^2 2^{-5}(1-p^{-2})(1-p^{-1})\over
[2^{72}\epsilon^{-42}]_p}.$$
Therefore,
$$|G:H|\leq n^{10}(1-p^{-4})(1-p^{-3})(1-p^{-2})(1-p^{-1}):
(n^{10}\cdot{\epsilon^2 2^{-5}(1-p^{-2})(1-p^{-1})\over
[2^{72}\epsilon^{-42}]_p})
$$$$
< 2^5\epsilon^{-2}[2^{72}\epsilon^{-42}]_p.
$$
\hfill{$\Box$}
\begin{rem}
{ The estimate of Proposition \ref{boundD} is probably far from
optimum.}
\label{remboundD}
\end{rem}
Now we consider subgroups that contain many elements that fix $E_i$
pointwise.
\begin{dfn}
{ Let $(W_{\alpha1},W_{\alpha2})$ be a pair of complementary isotropic
subgroups that corresponds to the divisor $E_\alpha$, as described
in \ref{indexE}, and $\varphi_\alpha$ be the corresponding involution
described in \ref{twodescrE}. We define ${\rm ram}_H(E_\alpha)$ to equal $1$
if $H\ni \varphi_\alpha$, and to equal $0$ otherwise. This definition makes
sense because $H\ni\pm{\bf 1}$.}
\end{dfn}
\begin{rem}
{ We have shown already that $\varphi_\alpha$ fixes all points of
$E_\alpha$.}
\end{rem}
\begin{prop}
{ If
$\sum_\alpha {\rm ram}_H(E_\alpha) \geq \epsilon\sharp(\alpha),$
then $|G:H| < 2^7\epsilon^{-2}[2^{246}\epsilon^{-130}]_p$.}
\label{boundE}
\end{prop}
{\em Proof.} For every set of indices $I$ we define the ramification mean
of $I$ to be $\sum_{\alpha\in I}{\rm ram}_H(E_\alpha)/|I|$.
For every primitive vector $v$ we consider the set $I_v$ of indices
$\alpha$ such that $v$ is an eigenvector of $\varphi_\alpha$.
Each index $\alpha$ belongs to the same number of sets $I_v$, therefore
$$\sum_v {\rm ramif.mean}(I_v)\geq \epsilon \sharp(v).$$
Hence there are at least $(\epsilon/2)\cdot\sharp(v)$ vectors $v$ such that
the ramification mean of $I_v$ is at least $\epsilon/2$. So now we
try to estimate ${\rm ram}_H(v)$ for a vector $v$ with this property, and then
use \ref{boundD}.
We assume that $v={}^t(0,1,0,0).$
\begin{lem}
{ Involutions $\varphi_\alpha, \alpha \in I_v$ have matrices of the
form
$$\pmatrix{1&0&0&-2x\cr -2z&-1&2x&0\cr 0&0&1&-2z\cr 0&0&0&-1}.$$
The sign is chosen to satisfy $\varphi_\alpha v=-v$.}
\end{lem}
{\em Proof of the lemma.} Any involution of this kind is defined uniquely
by the choice of $W_{\alpha_2}$. Because of $\langle
W_{\alpha1},W_{\alpha2}\rangle =0$,
the form $\langle,\rangle$ is unimodular on $W_{\alpha2}$. It implies that
there is a basis of $W_{\alpha2}$ that consists of $v$ and
${}^t(x,0,z,1).$ The rest is just a calculation. \hfill$\Box$
We denote the involution with the matrix
$$\pmatrix{1&0&0&-2x\cr -2z&-1&2x&0\cr 0&0&1&-2z\cr 0&0&0&-1}$$
by $\varphi_{x,z}$. We may assume without loss of generality that
$\varphi_{0,0}\in H$. There are at least $\epsilon n^2/2$ pairs
$(x,z)$ such that $\varphi_{x,z}\in H$. We call these pairs good.
There are at least $\epsilon n^2/4$ good pairs that satisfy
${\rm g.c.d.}(z,n) \leq [4/\epsilon]_p$. We choose one of them and denote it by
$(x_1,z_1)$. There is at least one good pair $(x,z)$ such that
${\rm g.c.d.}(xz_1-zx_1,n)\leq[2/\epsilon]\cdot {\rm g.c.d.}(z,n)$.
Then $ {\rm g.c.d.}(xz_1-zx_1,n)\leq [8\epsilon^{-2}]_p.$ We denote
this pair by $(x_2,z_2).$
It is a matter of calculation to check that
$$(\varphi_{x_1,z_1}\varphi_{0,0}\varphi_{x_2,z_2})^2
= \pmatrix{1&0&0&0\cr 0&1&0&8(x_1z_2-x_2z_1)\cr
0&0&1&0\cr 0&0&0&1\cr}.$$
This element lies in $H$, therefore ${\rm ram}_H(v)\geq 1/[8\epsilon^{-2}]_p.$
Because we can prove the same estimate for every vector $v$ for
which the ramification mean of $I_v$ is at least $\epsilon/2$, we get
$$\sum_v{{\rm ram}_H(v)}\geq (8\epsilon^{-2})_p^{-1}\cdot(\epsilon/2).$$
Now we use Proposition \ref{boundD} to get
$$|G:H| < 2^7\epsilon^{-2}[2^{246}\epsilon^{-130}]_p.$$
\hfill{$\Box$}
Now let us consider subgroups that contain many elements that fix lines
$D_i\cap D_j$ pointwise.
\begin{dfn}
{ Every line $l_{\alpha\beta}=D_\alpha\cap D_\beta$ is a conjugate
of the {\it standard}\/ line $l_0$, which is the intersection of the
divisors that correspond to the $\pm$vectors $\pm{}^t(1,0,0,0)$,
$\pm{}^t(0,1,0,0)$. We define ${\rm Ram}_G(l_0)$ to consist of matrices
$$\pmatrix{1&0&*&0\cr 0&1&0&*\cr 0&0&1&0\cr 0&0&0&1\cr} ({\rm mod}n).$$
We then define ${\rm Ram}_G(g\cdot l_0)=g\cdot {\rm Ram}_G(l_0)\cdot g^{-1}.$
It can be defined invariantly as a subgroup of all matrices that fix
both $v_\alpha$ and $v_\beta$, and also fix a pair of the isotropic
subgroups $W_1\ni v_\alpha,W_2\ni v_\beta$ that correspond to a divisor
$E_i$ that intersects $l_{\alpha\beta}$. It does not matter which $E_i$
we consider.}
\end{dfn}
\begin{rem}
{ We shall see later in Proposition \ref{jumpDD}
that ${\rm Ram}_G(l_{\alpha\beta})$ consists of
transformations that fix all points of the line $l_{\alpha\beta}$ and
do not switch the divisors $D_\alpha$ and $D_\beta$.}
\end{rem}
\begin{dfn}
{ We define ${\rm Ram}_H(l_{\alpha\beta})=H\cap {\rm Ram}_G(l_{\alpha\beta}).$
We define ${\rm ram}_H(l_{\alpha\beta})$ to be the maximum order of the element
of ${\rm Ram}_H(l_{\alpha\beta})$ divided by $n.$}
\end{dfn}
\begin{prop}
{ If $\sum_{\alpha\beta}{\rm ram}_H(l_{\alpha\beta})\geq \epsilon\sharp
(\alpha\beta)$, then
$$|G:H| \leq 2^{11}\epsilon^{-2}[2^{1020}\epsilon^{-350}]_p.$$}
\label{boundDD}
\end{prop}
{\em Proof.} We will eventually use Proposition \ref{boundD}.
We need another definition.
\begin{lem}
{ Let $l_{\alpha\beta}$ be the line of the intersection of the divisors
$D_\alpha$ and $D_\beta.$
Then ${\rm Ram}_G(l_{\alpha\beta})={\rm Ram}_G(v_\alpha)\oplus {\rm Ram}_G(v_\beta).$}
\label{ramDD}
\end{lem}
{\em Proof of the lemma.} It is enough to consider the standard line,
for which the statement follows from the explicit matrix representations of
the three groups in question. \hfill$\Box$
\begin{dfn}
{ We define
$${\rm ram}_H(l_{\alpha\beta}\subset D_\alpha)=
{|{\rm Ram}_H(l_{\alpha\beta})|\over |{\rm Ram}_H(l_{\alpha\beta})\cap {\rm Ram}_G(v_\alpha)|
\cdot n }$$
If $\alpha,\beta$ are standard, then ${\rm ram}_H(l_{\alpha\beta}\subset
D_\alpha)$ is the inverse of the minimum ${\rm g.c.d.}(a,n)$ for
$$\pmatrix{1&0&a&0\cr 0&1&0&c\cr 0&0&1&0\cr 0&0&0&1\cr}\in H.$$}
\end{dfn}
We notice that
$${\rm ram}_H(l_{\alpha\beta})\leq {\rm max}\{{\rm ram}_H(l_{\alpha\beta}\subset
D_\alpha),{\rm ram}_H(l_{\alpha\beta}\subset D_\beta)\}.$$
The usual argument shows that at least $(\epsilon/6)\cdot\sharp|D_\alpha|$
of divisors $D_\alpha$ obey the following property
(1) at least $(\epsilon/6)\cdot\sharp (l_{\alpha\beta}\subset
D_\alpha)$ of lines $l_{\alpha\beta}$ that are contained in it
have ${\rm ram}_H(l_{\alpha\beta}\subset D_\alpha)\geq \epsilon/6.$
We call these divisors good. Now our goal is to prove that
every good divisor $D_\alpha$ has sufficiently big ${\rm ram}_H(v_\alpha).$
We may assume without loss of generality that $D_\alpha=D_0$ is standard.
We may also assume that the arrangement of lines in $D_0$ over the standard point on
the Satake compactification contains at least $(\epsilon/6)\cdot n$
of the lines with ${\rm ram}_H(l_{0\beta}\subset D_0)\geq
\epsilon/6.$ Divisors $D_\beta$ that intersect $D_0$ over
the standard point of the Satake compactification correspond to
$\pm$vectors of the form $\pm{}^t(1,b,0,0)$, see \ref{indexDD}.
It implies, that there are at least $(\epsilon/6)\cdot n$ numbers $b$
such that
$$ H\ni\pmatrix{1&0&a_0&ba_0\cr 0&1&ba_0&*\cr 0&0&1&0\cr
0&0&0&1\cr}$$
where ${\rm g.c.d.}(a_0,n) \leq [(\epsilon/6)^{-1}]_p$, and $*$ is an unknown
number.
We can choose $b_1$ and $b_2$ that give us the above elements in $H$
and additionally satisfy ${\rm g.c.d.}(b_1-b_2,n)\leq [6/\epsilon]_p.$
Then we can divide one such element by another to get
$$H\ni\pmatrix{1&0&0&(b_1-b_2)a_0\cr
0&1&(b_1-b_2)a_0&*\cr 0&0&1&0\cr 0&0&0&1\cr}=\pmatrix{1&0&0&x\cr
0&1&x&*\cr 0&0&1&0\cr 0&0&0&1\cr}.$$
We can estimate ${\rm g.c.d.}(x,n)\leq [6^2\epsilon^{-2}]_p.$
We denote the above element by $\rho.$
Now we wander away from the standard point on the Satake
compactification. All other divisors $D_\beta$ that intersect $D_0$
correspond to the $\pm$vectors $\pm{}^t(d,e,f,0)$ with $(d,f)\neq(0,0)(p)$.
This also follows from Proposition \ref{indexDD}. At least $(\epsilon/6) \cdot
n^3(1-p^{-2})$ of lines $l_{0\beta}$ satisfy ${\rm ram}_H(l_{0\beta}\subset
D_0)\geq \epsilon/6.$ Therefore, at least one of them satisfies
additionally ${\rm g.c.d.}(f,n)\leq [6\epsilon^{-1}(1-p^{-2})^{-1}]_p.$
It implies, that $H$ contains an element $\rho_1$ of the form
$$\pmatrix{1+
dfa_0&0&-d^2a_0&-dea_0\cr efa_0&1&-dea_0&-ea_0^2+c\cr
f^2a_0&0&1-dfa_0&-efa_0\cr 0&0&0&1\cr}$$
with ${\rm g.c.d.}(f,n)\leq [6\epsilon^{-1}(1-p^{-2})^{-1}]_p.$
One can calculate that
$$\rho_1\rho\rho_1^{-1}\rho^{-1}\rho_1\rho^{-1}\rho_1^{-1}\rho=
\pmatrix{1&0&0&0\cr 0&1&0&-2x^2f^2a\cr 0&0&1&0\cr 0&0&0&1},$$
which implies
$${\rm ram}_H(v_0)\geq [2\cdot(6^2\epsilon^{-2})^2
\cdot(6\epsilon^{-1}(1-p^{-2})^{-1})^2\cdot(\epsilon/6)^{-1}]_p^{-1}\geq
[2^{19}\epsilon^{-7}]_p^{-1}.$$
As a result,
$\sum_{\pm v}{\rm ram}_H(v)\geq [2^{19}\epsilon^{-7}]_p^{-1}
(\epsilon/6) \cdot \sharp (\pm v).$
By \ref{boundD}, it implies
$$|G:H| \leq 2^{11}\epsilon^{-2}
[2^{1020}\epsilon^{-350}]_p.$$
\hfill$\Box$
We also need to deal with subgroups that contain many elements that fix
$F_i$ pointwise.
\begin{dfn}
{ Let $\psi_\alpha$ be the involution that corresponds to the
divisor $F_\alpha$ as described in \ref{indexinvF}.
We define ${\rm ram}_H(F_\alpha)$ to equal $1$ if $H\ni \psi_\alpha$,
and to equal $0$ otherwise.}
\end{dfn}
\begin{rem}
{ We have shown already that $\psi_\alpha$ fixes all points of
$F_\alpha$.}
\end{rem}
\begin{prop}
{ If
$\sum_\alpha {\rm ram}_H(F_\alpha) \geq \epsilon\sharp(\alpha),$
then $|G:H| \leq 2^{13}\epsilon^{-2}[2^{1722}\epsilon^{-702}]_p$.}
\label{boundF}
\end{prop}
{\em Proof.} There are at least $(\epsilon/2)\sharp(\alpha\beta)$ lines
$l_{\alpha\beta}$ such that at least $\epsilon n/2$ of divisors
$F_\gamma$ that contain $l_{\alpha\beta}$ are ramification divisors.
We call these lines good. Our goal is to estimate ${\rm ram}_H(l_{\alpha\beta})$
for a good line $l_{\alpha\beta}$.
We may assume that $l_{\alpha\beta}$ is the standard line.
If it is good, then the group $H$ contains at least $\epsilon n/2$ elements
of the form
$$\varphi_b=\pmatrix{0&1&0&b\cr 1&0&-b&0\cr 0&0&0&1\cr 0&0&1&0\cr}.$$
There are two elements $\varphi_{b_1}$ and $\varphi_{b_2}$ in $H$ such that
${\rm g.c.d.}(n,b_1-b_2)\leq [2\epsilon^{-1}]_p$. The matrix of the element
$\varphi_{b_1}\varphi_{b_2}$ is equal to
$$\pmatrix{1&0&b_1-b_2&0\cr 0&1&0&b_2-b_1\cr 0&0&1&0\cr 0&0&0&1}.$$
Therefore, ${\rm ram}_H(l_{\alpha\beta})\geq [2\epsilon^{-1}]_p^{-1}$.
As a result,
$$\sum_{\alpha\beta}{\rm ram}_H(l_{\alpha\beta})\geq
\epsilon2^{-1}[2\epsilon^{-1}]_p^{-1}\sharp(\alpha\beta).$$
Proposition \ref{boundDD} gives $|G:H| \leq 2^{13}\epsilon^{-2}
[2^{1722}\epsilon^{-702}]_p$. \hfill$\Box$
Finally, we will get an index estimate for subgroups such that their quotient
varieties have bad singularities at the images of $D_i\cap D_j\cap D_k$.
This is the most delicate calculation of the whole paper. We need some
preliminary definitions.
\begin{dfn}
{ Let $P_{\alpha\beta\gamma}$ be the point of the intersection of
three infinity divisors $D_\alpha,D_\beta$, and $D_\gamma$.
Define $${\rm Ram}_G(P_{\alpha\beta\gamma})={\rm Ram}_G(v_\alpha)\oplus
{\rm Ram}_G(v_\beta)\oplus {\rm Ram}_G(v_\gamma).$$
If $P$ is the standard point, that is the one that corresponds to
$v_\alpha={}^t(1,-1,0,0),v_\beta={}^t(1,0,0,0),v_\gamma={}^t(0,1,0,0),$
then this group consists of matrices
$$\pmatrix{1&0&a&b\cr 0&1&b&c\cr 0&0&1&0\cr 0&0&0&1\cr}({\rm mod}n).$$
As usual, we define
${\rm Ram}_H(P_{\alpha\beta\gamma})=H\cap {\rm Ram}_G(P_{\alpha\beta\gamma})$.}
\label{ramDDD}
\end{dfn}
\begin{dfn}
{ Consider the singularity at the image of $P_{\alpha\beta\gamma}$
in the quotient of a neighborhood of $P_{\alpha\beta\gamma}$
by the group ${\rm Ram}_H(P_{\alpha\beta\gamma})$.
We define ${\rm mult}_H(P_{\alpha\beta\gamma})$ to be the multiplicity
of this singular point.}
\label{multDDD}
\end{dfn}
\begin{prop}
{ If $\,\sum\nolimits^* {\rm mult}_H{P_i}\geq \epsilon \sharp(i)$,
where $\sum\nolimits^*$ means taking one point $P_{\alpha\beta\gamma}$ per
orbit
of the action of the group $H$, then
$|G:H|\leq 2^{69}\epsilon^{-34}[2^{11170}\epsilon^{-5950}]_p$.}
\label{boundDDD}
\end{prop}
{\em Proof.} For each point $P_{\alpha\beta\gamma}$ we define
$\delta(H,P_{\alpha\beta\gamma})$ as a number $\delta$ defined in \ref{appdelta}
for the group ${\rm Ram}_H(P_{\alpha\beta\gamma})$ acting in the tangent space at
$P_{\alpha\beta\gamma}$. Notice that there is a natural choice of
coordinates $(x_1,x_2,x_3)$ in a neighbourhood of $P_{\alpha\beta\gamma}$, such that the
weights of an element $h\in {\rm Ram}_H(P_{\alpha\beta\gamma})$ are determined
using ${\rm Ram}_G(P_{\alpha\beta\gamma})={\rm Ram}_G(v_\alpha)\oplus
{\rm Ram}_G(v_\beta)\oplus {\rm Ram}_G(v_\gamma)$.
Then $\delta(H)$ is defined as $(1/n) {\rm min}_{l\neq 0}(l_1+l_2+l_3)$, where minimum
is taken over all $H$-invariant monomials
$x_1^{l_1}x_2^{l_2}x_3^{l_3}$.
First of all, we rewrite the condition of the proposition in terms of
$\delta(H,P_{\alpha\beta\gamma})$. By \ref{appmult},
${\rm mult}_H P_{\alpha\beta\gamma} \leq n^3\delta(H,P_{\alpha\beta\gamma})
/|{\rm Ram}_H(P_{\alpha\beta\gamma})|$. Therefore,
$$\sum_{P_{\alpha\beta\gamma}}\delta(H,P_{\alpha\beta\gamma})\geq
\sum_{P_{\alpha\beta\gamma}}n^{-3}|{\rm Ram}_H(P_{\alpha\beta\gamma})|{\rm mult}_H
P_{\alpha\beta\gamma}$$
$$\geq \sum\nolimits^*(6n^3)^{-1}|H|{\rm mult}_H P_{\alpha\beta\gamma} \geq
(6n^3)^{-1}\epsilon|H|\cdot|G:H|=\epsilon\sharp(P_{\alpha\beta\gamma}).$$
For every isotropic subgroup $V_2\simeq({{\bf Z}/n{\bf Z}})^2$ in $V$ we consider
the set of the points $P_{\alpha\beta\gamma}$ with $v_\alpha,v_\beta,
v_\gamma\in V_2$. Geometrically, these are the points that lie over certain
cusp points of the Satake compactification, see \ref{indexptsSatake}.
There are at least $(\epsilon/2)\sharp(V_2)$ of these subgroups that have
$$\sum_{v_\alpha,v_\beta,v_\gamma\in V_2}\delta(H,P_{\alpha\beta\gamma})
\geq (\epsilon/2)\sharp(v_\alpha,v_\beta,v_\gamma\in V_2).$$
We call these subgroups good. We are going to prove that if $V_2$ is a good
isotropic subgroup, then
$$\sum_{v_\alpha,v_\beta\in V_2}{\rm ram}_H(l_{\alpha\beta})\geq
\epsilon_1(\epsilon)\sharp(v_\alpha,v_\beta\in V_2),$$
and then use Proposition \ref{boundDD}.
We assume without loss of generality that $V_2={}^t(*,*,0,0)$, and
$\delta(H,P_0)\geq (\epsilon/2)$, where $P_0$ is the standard point.
Notice that ${\rm Ram}_G(P_{\alpha\beta\gamma})$ and ${\rm Ram}_H(P_{\alpha\beta\gamma})$
do not depend on the point $P_{\alpha\beta\gamma}$, provided
$v_\alpha,v_\beta,v_\gamma\in V_2$. We denote these groups by $G_1$ and
$H_1$ respectively. The group $G_1$ is described in Definition \ref{ramDDD}.
We are dealing with points $P_{\alpha\beta\gamma}$ obtained from
the standard one by the action of elements of type $\pmatrix{A&0\cr 0&A\cr}$,
where $A\in {\rm Gl(2,{\bf Z}/n{\bf Z})}$. Although the group $H_1$ is the same for all
$P_{\alpha\beta\gamma}$, its action in the tangent spaces depends on
$P_{\alpha\beta\gamma}$. It is the same as the action in the tangent space
to the standard point $P_0$ of the group $AH{}^tA,~A\in{\rm Gl(2,{\bf Z}/n{\bf Z})}$,
if we think of $G_1$ as the group of symmetric $2\times 2$ matrices.
We define $\epsilon_1$ by the formula
$$
\sum_{v_\alpha,v_\beta\in V_2}{\rm ram}_H(l_{\alpha\beta})=
\epsilon_1\sharp(v_\alpha,v_\beta\in V_2).$$
There is a line $l_{\alpha\beta}$ such that
${\rm ram}_Hl_{\alpha\beta}\leq \epsilon_1$. It implies that the group
$H_2=(H_1+[\epsilon_1^{-1}]_pG_1)/[\epsilon_1^{-1}]_pG_1$ is cyclic.
When we pass from $H_1$ to $H_1+[\epsilon_1^{-1}]_pG_1$, the numbers
$\delta$ do not decrease. Hence,
$$\sum_{v_\alpha,v_\beta,v_\gamma\in V_2}\delta(H_1+[\epsilon_1^{-1}]_pG_1,
P_{\alpha\beta\gamma})\geq (\epsilon/2)\sharp(v_\alpha,v_\beta,v_\gamma\in
V_2).$$
This is equivalent to
$$\sum_{A\in {\rm Gl(2,{\bf Z}/n{\bf Z})}}\delta(AH_1{}^tA+[\epsilon_1^{-1}]_pG_1,P_0)
\geq (\epsilon/2)\sharp(A).$$
Let $H_2$ be generated by $B=\pmatrix{a&b\cr b&c\cr}$. One can show that
$\delta(AH{}^tA+[\epsilon_1^{-1}]_pG_1,P_0)$ equals $\delta(({{\bf Z}/n{\bf Z}})
{\bar A}{\bar B}{}^t{\bar A},{\bar P}_0)$, where $n$ is replaced by
$[\epsilon_1^{-1}]_p$ and bars means reduction
${\rm mod}[\epsilon_1^{-1}]_p$. Therefore,
$$\sum_{C\in {\rm Gl(2,{\bf Z}/}[\epsilon_1^{-1}]_p{\bf Z})}
\delta(({\bf Z}/[\epsilon_1^{-1}]_p{\bf Z})CB{}^tC,{\bar P_0})\geq
(\epsilon/2)\sharp(C).$$
Because of the result of Proposition \ref{appfinmany},
there are at most $2^{10}\epsilon^{-8}[2^{12}\epsilon^{-5}]_p$ different
matrices $CB{}^tC$ up to
proportionality that give $\delta(({\bf Z}/[\epsilon_1^{-1}]_p{\bf Z})
CB{}^tC,{\bar P_0})\geq (\epsilon/4)\sharp(C)$. This implies that
the orbit $CB{}^tC({\rm mod~proportionality})$ of the action of the group
${\rm Gl(2,{\bf Z}/}[\epsilon_1^{-1}]_p{\bf Z})$ has length at most
$2^{12}\epsilon^{-9}[2^{12}\epsilon^{-5}]_p$.
However, we can estimate this length by looking at matrices $C=
\pmatrix{t&0\cr 0&1\cr}$. They give $CB{}^tC=\pmatrix{t^2a&tb\cr tb&c\cr}$,
and so length of the orbit is at least $[\epsilon_1^{-1}]_p(1-p^{-1})
/{\rm g.c.d.}(bc,[\epsilon_1^{-1}]_p).$ Because we have assumed that $\delta
({\bf Z}/[\epsilon_1^{-1}]_p{\bf Z})B,{\bar P_0})\geq (\epsilon/2)$,
we have ${\rm g.c.d.}(b,n)\leq[2\epsilon^{-1}]_p$ and ${\rm g.c.d.}(c,n)\leq
[4\epsilon^{-1}]_p$. Really, the weights of $B$ are $(-b,a+b,c+b)$,
and if ${\rm g.c.d.}[c,n]$ is greater than $[4\epsilon^{-1}]_p$, then
we get $\delta\geq (\epsilon/2)$ because of the invariant monomial of the form
$(x_1x_3)^{[\epsilon_1^{-1}]_p/[4\epsilon^{-1}]_p}$.
As a result, the length of the orbit is at least $[\epsilon_1^{-1}]_p
(1-p^{-1})/[8\epsilon^{-2}]_p$, and we have
$[\epsilon_1^{-1}]_p(1-p^{-1})/[8\epsilon^{-2}]_p\leq 2^{12}\epsilon^{-9}[2^{12}
\epsilon^{-5}]_p$ and $\epsilon_1\geq 2^{-28}\epsilon^{16}$.
We now may use the result of Proposition \ref{boundDD} with
$(2^{-28}\epsilon^{16})(\epsilon/2)$ in place of $\epsilon$.
Thus, $|G:H|\leq 2^{69}\epsilon^{-34}[2^{11170}\epsilon^{-5950}]_p$.
\hfill$\Box$
\section{Singularities of ${\cal H}/H$}
It is easy to describe all elements of finite order in $\Gamma(2)$
by means of the following proposition.
\begin{prop}
{ Any nonidentity element of finite order in $\Gamma(2)/\{\pm 1\}$
is conjugate in $\Gamma(1)/\{\pm 1\}$ to the element with the matrix
$$\varphi_0=\pmatrix{1&0&0&0\cr0&-1&0&0\cr0&0&1&0\cr0&0&0&-1\cr}.$$}
\label{einv}
\end{prop}
{\em Proof.} Denote the matrix of this element by $\varphi=\pmatrix
{A&B\cr C&D\cr}.$ Because $\Gamma(4)$ is torsion free and
$\varphi^2\in\Gamma(4)$, we obtain $\varphi^2=1$. Hence the following
equalities hold $$A~{}^tB=B~{}^tA,~~C~{}^tD=D~{}^tC,~~A~{}^tD-B~{}^tC=1$$
$$A=~{}^tD,~~B=-{}^tB,~~C=-{}^tC.$$Really, the first three equalities hold
for all symplectic matrices, and they imply $\varphi^{-1}=\pmatrix
{{}^tD&-{}^tB\cr -{}^tC&{}^tA\cr}$, so $\varphi^{-1}=\varphi$ gives the
last three ones. Six equalities together show that
$$\varphi=\pmatrix{a_1&a_2&0&b\cr a_3&a_4&-b&0\cr0&c&a_1&a_3\cr
-c&0&a_2&a_4\cr}$$ with $(a_1+a_4)b=(a_1+a_4)c=(a_1+a_4)a_2=
(a_1+a_4)a_3=0,~a_1^2+a_2a_3-bc=a_4^2+a_2a_3-bc=1.$ Hence if
$\varphi\neq1$, then $a_1+a_4=0$, so
$$\varphi=\pmatrix{a_1&a_2&0&b\cr
a_3&-a_1&-b&0\cr0&c&a_1&a_3\cr -c&0&a_2&-a_1\cr}$$
with $a_1^2+a_2a_3
-bc=1,~(a_1-1),a_2,a_3,b,c\equiv 0{\rm mod}(2)$.
We need to prove that any matrix with these properties is conjugate to
$\varphi_0$. The vector spaces ${\rm Ker}(\varphi-1)$
and ${\rm Ker}(\varphi+1)$ are orthogonal, so we should simply find
four integer vectors $e_1,...,e_4$ that obey $\varphi(e_i)=(-1)^{i+1}
e_i$ and $\langle e_2,e_4\rangle =\langle e_1,e_3\rangle =1$.
Because of symmetry, it is enough
to find $e_1$ and $e_3$. Let us denote $d={\rm g.c.d.}(b/2,a_3/2,(a_1-1)/2).$
There holds $\alpha b/2+\beta(a_1-1)/2+\gamma(-a_3/2)=d$ for some
integers $\alpha,\beta,\gamma.$ Now we simply put
$$e_1=\pmatrix{b/2d\cr 0\cr a_3/2d\cr(1-a_1)/2d\cr},~
e_3=\alpha\pmatrix{0\cr-b/2\cr(a_1+1)/2\cr a_2/2\cr}+\beta
\pmatrix{a_2/2\cr(1-a_1)/2\cr c/2\cr 0\cr}+\gamma\pmatrix{(a_1+1)
/2\cr a_3/2\cr0\cr-c/2\cr}$$
and check the required conditions by direct calculation. \hfill$\Box$
\begin{dfn}
{ Let $H$ be a subgroup of finite index in $\Gamma(1)$. We call
$E_i$ or $F_j$ a {\em ramification divisor} iff $H$ contains the
involution that fixes all points of the divisor. Because of
the results of \ref{indexinvE} and \ref{indexinvF}, $E_\alpha$ is a
ramification divisor iff ${\rm ram}_H(E_\alpha)=1$, and similarly
for $F_\beta$.}
\end{dfn}
We are interested in singularities of ${\cal H}/H$. They occur
at the images of the points of ${\cal H}$ that have nontrivial
stabilizers in $H$. The goal of the rest of this section is to prove
the following statement.
\begin{prop}
{ Singularities of the images of the points $\xi\in{\cal H}$ that
do not lie in ramification divisors $E_i$ or $F_j$ are
canonical. Points that do lie in ramification divisors have
solvable stabilizers of order at most $72$. We refer to
\ref{defcan} for the definitions of canonical and terminal
singularities. }
\label{finsing}
\end{prop}
{\em Proof. }There are two possibilities: $\xi\in\cup E_i$ and
$\xi\notin\cup E_i$.
{\em Case 1. } $\xi\notin\cup E_i$. The stabilizer of $\xi$ in $\Gamma(2)$
equals $\{\pm 1\}$ because of Proposition
\ref{einv} and the definition of $E_i$. We
consider the quotient of ${\cal H}$ by the action of $\Gamma(2)$.
It is the smooth part of the singular quartic $V$ defined by the equation
$(\sum x_i^2)^2=4\sum x_i^4$ in coordinates $(x_1:...:x_6,~\sum x_i=0)$
of ${\bf P}^4$, see \cite{Geer}. The group $\Gamma(1)/\Gamma(2)\simeq
\Sigma_6$ acts on $V$ by the permutations of the coordinates $x_i.$ The
stabilizer $\xi$ in $\Gamma(1)$ equals that of the image of $\xi$ in $V$ in
the group $\Sigma_6.$ Moreover, locally their actions are the same, so
the resulting quotient singularities are isomorphic. Therefore, we need to
study fixed points of $\Sigma_6$-action on $V$.
\begin{lem}
{ A point $\xi\notin\cup E_i$ with a nontrivial stabilizer in $\Gamma(1)$
either lies in $\cup F_j$ or has the image in $V$ of type
$\sigma(0:\theta:\theta^2:\theta^3:\theta^4:1),~\theta=\exp(2\pi i/5),
~\sigma\in\Sigma_6.$}
\label{fixedsigma}
\end{lem}
{\em Proof of the lemma.} Denote by $x=(x_1:...:x_6)$ the image
of $\xi$ in $V$. We may assume that the stabilizer of $x$ contains one
of the permutations
$$(1,2);(1,2)(3,4);(1,2)(3,4)(5,6);(1,2,3);(1,2,3)(4,5,6);(1,2,3,4,5).$$
Let us calculate the sets of fixed points of these permutations that
lie in $V$.
Case (1,2). We have $(x_1,x_2,...,x_6)=\lambda(x_2,x_1,...,x_6).$
If $\lambda=-1$, then $x=(-1:1:0:0:0:0)$, but this point does not lie in $V$.
Hence $\lambda=1$. The set defined by "$x_1=x_2$" constitutes an irreducible divisor on
$V$, so it is the closure of the image of some submanifold of dimension two
in ${\cal H}$.
Case (1,2)(3,4). We have $(x_1,x_2,x_3,x_4,x_5,x_6)=\lambda(x_2,x_1,x_4,
x_3,x_5,x_6).$ If $\lambda=-1$, then $x_1=-x_2,~x_3=-x_4,~x_5=x_6=0$.
The equality $(\sum x_i^2)^2=4\sum x_i^4$ implies that
$x_1=x_3$ or $x_1=x_4$, so $x\in {\rm Sing}(V)$, see \cite{Geer}. If $\lambda=1$,
then $x$ lies in the divisor "$x_1=x_2$".
Case (1,2),(3,4),(5,6). We have $(x_1,x_2,x_3,x_4,x_5,x_6)=\lambda
(x_2,x_1,x_4,x_3,x_6,x_5).$ If $\lambda=-1$, then
$x_1=-x_2,~x_3=-x_4,~x_5=-x_6.$ Equality $(\sum x_i^2)^2=4\sum x_i^4$
leads to $(x_1+x_3+x_5)\cdot(x_1+x_3-x_5)\cdot(x_1-x_3+x_5)\cdot(
-x_1+x_3+x_5)=0.$ Each of these linear equations implies that $x$
lies in the image of $\cup E_i$, see \cite{Geer}.
If $\lambda=1$, then $x_1=x_2,~x_3=x_4,~x_5=x_6$ so $x\in {\rm Sing}(V)$.
Case (1,2,3). We have $(x_1,x_2,x_3,x_4,x_5,x_6)=\lambda(x_2,x_3,x_1
,x_4,x_5,x_6).$ If $\lambda\neq1$, then $x_1+x_2+x_3=0$, so $x$ lies in
the image of $\cup E_i$. If $\lambda=1$, then $x_1=x_2=x_3$, and $x$ lies
in the divisor "$x_1=x_2$".
Case (1,2,3)(4,5,6). We have $(x_1,x_2,x_3,x_4,x_5,x_6)=\lambda(x_2,
x_3,x_1,x_5,x_6,x_4).$ If $\lambda=1$, then
$x=(1:1:1:-1:-1:-1)\notin V$. Otherwise, $x_1+x_2+x_3=0$, and $\xi\in\cup E_i$.
Case (1,2,3,4,5). We have $(x_1,x_2,x_3,x_4,x_5,x_6)=\lambda(x_2,x_3
,x_4,x_5,x_1,x_6).$ If $\lambda=1$, then
$x=(1:1:1:1:1:-5)\notin V$. Otherwise $x=\sigma(0:\theta:
\theta^2:\theta^3:\theta^4:1),~\theta=\exp(2\pi i/5),~\sigma\in\Sigma_6.$
The above calculation shows that there is only one up to $\Sigma_6$-action
divisor on $V$ with a nontrivial stabilizer of a generic closed point.
On the other hand, the images of $F_j$ on $V$ obey this condition.
Therefore the images of $F_j$ are the conjugates of the divisor "$x_1=x_2$",
which proves the lemma. \hfill$\Box$
\begin{rem}
{ As a corollary of this lemma, codimension one components
of the ramification locus of the map from ${\cal H}/\Gamma_n$
to ${\cal H}/H$ can only be divisors $E_\alpha$ and $F_\beta$.
Moreover, ramification occurs
iff $E_\alpha(F_\beta)$ is a ramification divisor as defined
above, and in this case the only nontrivial element that preserves all points
of the divisor is the corresponding involution. Of course, when we consider
the Igusa compactifications, we may have ramification at infinity divisors.}
\label{ramfin}
\end{rem}
Let us come back to the proof of \ref{finsing}. We try to estimate
the singularity
at the image of the point $\xi\notin \cup E_i$ under the quotient map ${\cal
H}\to{\cal H}/H.$ The group $\Gamma(2)/\{\pm 1\}$ acts freely on
${\cal H}-\cup E_i$, so we can work in terms of the image point
$x\in V-{\rm Sing}(V)$ and the group ${\rm Stab}^H\xi\cdot\Gamma(2)/\Gamma(2)$,
because these quotient singularities are isomorphic.
There exists a useful criterion that enables one to find out
whether the quotient singularity is canonical, see \cite{Reid}.
In particular, it is always canonical, if the image of the group in ${\rm
Gl}(T_x)$ lies in ${\rm Sl}(T_x)$. We use these facts
extensively.
First of all we consider the case $x=\sigma(0:\theta:...:1)$. Then
either ${\rm Stab}^H\xi\cdot\Gamma(2)/\Gamma(2)=1$ or ${\rm Stab}^H\xi\cdot
\Gamma(2)/\Gamma(2)={{\bf Z}/5{\bf Z}}.$ A direct calculation of the weights of
the generator in the tangent space and the criterion of \cite{Reid}
show that the quotient singularity is terminal, hence canonical.
Now let us consider other points $x=(x_1:x_2:x_3:x_4:x_5:x_6)\in V-{\rm Sing}(V)$.
The group $S={\rm Stab}^H\xi\cdot\Gamma(2)/\Gamma(2)$ contains no transpositions,
because $\xi$ does not belong to any ramification divisors $F_i$.
The proof of \ref{fixedsigma} shows that $S$ does not contain
permutations of types $(*,*)(*,*)(*,*),(*,*,*)(*,*,*)$, and
$(*,*,*,*,*)$. As a result, $S$ consists of permutations of types $(*,*)
(*,*),(*,*,*),(*,*,*,*)$, and $(*,*)(*,*,*,*)$ only.
Calculations similar to those of \ref{fixedsigma} show that if the group $S$
contains $(*,*)(*,*,*,*)$, then $\xi\in \cup E_i$. Moreover, if it
contains a permutation of type $(*,*,*,*)$ and the proportionality
coefficient $\lambda$ does not equal $1$, then $\xi\in \cup E_i$.
Notice (see the proof of \ref{fixedsigma}) that the proportionality
coefficients of elements of the group $S$ of types $(*,*)(*,*)$ and $(*,*,*)$
must also equal $1$. All these restrictions on the group $S$ imply that it
consists of even permutations, and all proportionality coefficients are
equal to $1$. Therefore, the group $S$ acts in the tangent space
of $x$ by matrices from ${\rm Sl}$. The criterion of M. Reid
shows that the quotient singularity is canonical.
{\em Case 2.} $\xi\in \cup E_i$. all divisors $E_i$ are conjugates,
so we may assume that $\xi$ is represented by a diagonal matrix. Different
$E_i$ do not intersect, so the stabilizer $S$ of $\xi$ in $\Gamma(1)$
is a subgroup of $${\rm Stab}(\Delta)=\{\pmatrix{
a&0&b&0\cr0&a_1&0&b_1\cr c&0&d&0\cr0&c_1&0&d_1\cr}\cup\pmatrix{
0&a_1&0&b_1\cr a&0&b&0\cr 0&c_1&0&d_1\cr c&0&d&0\cr},~ad-bc=a_1
b_1-c_1d_1=1\}.$$
Point $\xi$ may be transformed by the group ${\rm Stab}(\Delta)$ to the point
$\pmatrix{x_0&0\cr0&z_0\cr}$ with $|{\rm Re}(x_0)|\leq 1/2,~|x_0|\geq
1,~|{\rm Re}(z_0)|\leq 1/2,~|z_0|\geq 1$. Without any loss of generality
one may consider points of this type only. The stabilizer of
the general such point in $\Gamma(1)/\{\pm 1\}$ equals ${{\bf Z}/2{\bf Z}}.$
It is generated by the involution of Proposition \ref{einv}.
If this element is in $H$, then $\Delta$ is a ramification divisor
by our definition. The order of the stabilizer can increase in the
following curves and points (we have used the symmetry between $x$
and $z$)
$$\pmatrix{x&0\cr0&x\cr},\pmatrix{i&0\cr0&x\cr},\pmatrix{\rho&0\cr0&x\cr}
,\pmatrix{i&0\cr0&\rho\cr},\pmatrix{i&0\cr0&i\cr},
\pmatrix{\rho&0\cr0&\rho\cr}.$$
Let us check all these cases.
Case $\pmatrix{x&0\cr0&x\cr}$. Because $\Delta$ is not a ramification
divisor, the order of ${\rm Stab}^H\xi$ is at most two, so the quotient
singularity is canonical.
Case $\pmatrix{i&0\cr0&x\cr}$. We get $| {\rm Stab}^H\xi|=1$ by the same
argument.
Case $\pmatrix{\rho&0\cr0&x\cr}$. In this case either $| {\rm Stab}^H\xi|=1$
or $ {\rm Stab}^H\xi$ is generated by the element of order $3$ whose action
in the tangent space of $\xi$ has determinant $1$.
Case $\pmatrix{i&0\cr0&\rho\cr}$. The argument is the same as in the
previous case.
Case $\pmatrix{i&0\cr0&i\cr}$. The stabilizer of $\xi$ in $\Gamma(1)
/\{1,-1\}$ is generated by the images of elements of $\Gamma(1)$
with matrices
$$\varphi=\pmatrix{0&1&0&0\cr 1&0&0&0\cr 0&0&0&1\cr 0&0&1&0\cr},
~\alpha=\pmatrix{1&0&0&0\cr 0&0&0&1\cr 0&0&1&0\cr 0&-1&0&0\cr},
~\beta=\pmatrix{0&0&1&0\cr 0&1&0&0\cr -1&0&0&0\cr 0&0&0&1\cr}.$$
The relations are $\alpha\beta=\beta\alpha,~\alpha^2=\beta^2,~ \varphi\alpha
=\beta\varphi,~\varphi^2=\alpha^4=\beta^4=1.$ The order of the group
is $16$.
The point $\xi$ does not lie in the ramification divisors, so ${\rm Stab}^H\xi$
does not contain any conjugates of $\varphi$. As in the above cases,
${\rm Stab}^H\xi$ does not contain $\alpha^2$ either. We can also employ the
following simple statement: if $s^2=1$ for all $s\in {\rm Stab}^H\xi$, then
the quotient singularity is canonical. In our case it implies that if the
quotient singularity is not canonical, then the group ${\rm Stab}^H\xi$ contains an
element of order $4$. All these conditions on ${\rm Stab}^H\xi$ together
hold iff this group is generated by a conjugate of $\varphi\beta.$
A direct calculation of the weights in the tangent space completes the
argument.
Case $\pmatrix{\rho&0\cr0&\rho\cr}.$ In this case it is possible to check
that the condition "$\xi$ does not belong to any ramification divisors"
implies that ${\rm Stab}^H\xi$ acts in the tangent space of
$\xi$ by matrices with determinant $1$.
To finish the proof of Theorem \ref{finsing}, we only need to check that
stabilizers of all points of ${\cal H}$ are solvable groups whose
orders are at most $72$. It can be done using the description of
${\cal H}/\Gamma(2)$ as the smooth part of the singular quartic. I skip
the details, because this number is clearly bounded and only slightly
affects the constant in the final result.
\hfill$\Box$
\section{Finiteness Theorem for subgroups $H\supseteq \Gamma(p^t)$}
We assume that $n=p^t$ throughout this section. We denote the subgroup
$H\supseteq \Gamma(n)$ and the quotient $H/\Gamma(n)$ by the same letter, which
should not lead to a confusion. The Igusa compactifications of ${\cal H}
/\Gamma(n)$ and ${\cal H}/H$ are denoted by $X$ and $Y$. The quotient map
$X\to X/H=Y$ is denoted by $\mu$.
We start by pulling the problem from $Y$ to $X$.
\begin{dfn}
{ Let $\pi:Z\to Y$ be a desingularization of $Y$. Denote by $-1+\delta$
the minimum discrepancy of $Y$, see \ref{discrep}. Because of
\ref{qulog}, $\delta$ is a positive rational number.}
\end{dfn}
\begin{dfn}
{ Let $m$ be a sufficiently divisible number, so that $mK_Y$ is
a Cartier divisor on $Y$. The vector space $H^0(Y,mK_Y-{\rm mlt})$ is
defined as the space of global sections $s$ of the coherent subsheaf of
${\cal O}_Y(mK_Y)$ that consists of sections that lie in
$m_y^{m(1-\delta)}({\cal O}_Y(mK_Y))_y$ for all noncanonical singular points
$y\in Y$.}
\end{dfn}
\begin{rem}
{ We assume $m$ to be sufficiently divisible whenever it
is necessary. We also omit ${\cal O}$ in the notations of the
space of global sections, unless it can lead to a misunderstanding.}
\end{rem}
\begin{prop}
{ ${\rm dim}H^0(Z,mK_Z)\geq {\rm dim}H^0(Y,mK_Y-{\rm mlt}).$}
\label{ZtoY}
\end{prop}
{\em Proof.} The pullbacks $\pi^*s$ vanish with the multiplicity at least
$m(1-\delta)$ along exceptional divisors with negative discrepancies.
Hence we can define an injective linear map from $H^0(Y,mK_Y-{\rm mlt})$
to $H^0(Z,mK_Z)$. \hfill$\Box$
\begin{dfn}
{ Denote by $H^0(Y,mK_Y-{\rm mlt}^0)$ the space of global
sections $s\in H^0(Y,mK_Y)$ that satisfy $s\in m_y^{m(1-\delta)}({\cal
O}_Y(mK_Y))_y$ for all noncanonical singular points of $Y$ except for
the images of points $P_{\alpha\beta\gamma}$ that are triple intersections
of infinity divisors on $X$.}
\end{dfn}
Clearly, $H^0(Y,mK_Y-{\rm mlt}^0))\supseteq H^0(Y,mK_Y-{\rm mlt})$.
\begin{prop}
{ If $|G:H| > 2^{953}[2^{165870}]_p$, then
${\rm dim} H^0(Y,mK_Y-{\rm mlt}^0)-{\rm dim}H^0(Y,mK_Y-{\rm mlt})
\preceq_{m\to\infty} 2^{-8}3^{-6}5^{-1}m^3|G:H|.$}
\label{excludeDDD}
\end{prop}
{\em Proof.} When $m\to\infty$, the codimension we are trying
to estimate grows no faster than
$(\sum_{Q\in Y}{\rm mult}_Q)(m^3/6)$, where $\sum_{Q\in Y}{\rm mult}_Q$ is the
sum over all points $Q$ in the image of $\cup P_{\alpha\beta\gamma}$,
and ${\rm mult}_Q$ is the multiplicity of the local ring of $Y$ at $Q$.
We want to relate it to the statement of Proposition \ref{boundDDD}.
We need an easy lemma.
\begin{lem}
{ Let $P_{\alpha\beta\gamma}=D_\alpha\cap D_\beta \cap D_\gamma$
be a point on $X$, such that $\mu (P_{\alpha\beta\gamma})=Q$.
Then ${\rm mult}_Q\leq 6^3 {\rm mult}_H(P_{\alpha\beta\gamma})$ with
${\rm mult}_H(P_{\alpha\beta\gamma})$ defined in \ref{multDDD}.}
\end{lem}
{\em Proof of the lemma.} Every element of $G$ that fixes
$P_{\alpha\beta\gamma}$ permutes the triple of the $\pm$vectors
$(\pm v_\alpha, \pm v_\beta, \pm v_\gamma)$. Hence the subgroup in ${\rm Stab}^H(P)$
of the elements that induce trivial permutations is a normal subgroup
of order at most $6$. One can show that this subgroup coincides with
${\rm Ram}_H(P_{\alpha\beta\gamma})$ by the explicit matrix calculation for the
standard triple $v_\alpha={}^t(0,1,0,0),v_\beta={}^t(-1,1,0,0), v_\gamma=
{}^t(1,0,0,0)$.
Therefore, the singularity of $Y$ at $Q$ can be obtained as the quotient
of the singularity of $X/{\rm Ram}_H(P_{\alpha\beta\gamma})$ by the group of
order at most $6$. Its multiplicity can be estimated by means of
Proposition \ref{klem}. \hfill$\Box$
As a result of this lemma, the codimension we are trying to estimate grows
no faster than $m^36^2\sum\nolimits^*{\rm mult}_H(P_{\alpha\beta\gamma})$, where
one takes
one point $P_{\alpha\beta\gamma}$ per orbit of $H$. By \ref{boundDDD} with
$\epsilon=2^{-26}$, it grows no faster than $2^{-8}
3^{-6}5^{-1}m^3|G:H|$, if $|G:H| > 2^{953}[2^{165870}]_p$.
\hfill$\Box$
\begin{prop}
{ Let $L_Y$ be a divisor of the modular form of weight $1$ on $Y$.
Then ${\rm dim}H^0(Y,mL_Y)$ grows as $2^{-7}3^{-6}5^{-1}m^3|G:H|$.}
\label{growthL}
\end{prop}
{\em Proof.} It can be derived, for instance, from the formula for
${\rm dim}H^0(X,mL_X)$ and $\oplus_mH^0(Y,mL_Y)=(\oplus_mH^0(X,mL_X))^H$.
\hfill$\Box$
\begin{prop}
{ If ${\rm dim} H^0(Y,m(K_Y-L_Y)-{\rm mlt}^0)\neq 0$
for sufficiently big $m$ and $|G:H| >
2^{953}[2^{165870}]_p $, then the variety $Y$ is of general type.}
\label{getridofDDD}
\end{prop}
{\em Proof.} We get
$${\rm dim}H^0(Z,mK_Z)\geq {\rm dim}H^0(Y,mK_Y-{\rm mlt})$$
$$\geq {\rm dim}H^0(Y,mK_Y-{\rm mlt}^0)-2^{-8}3^{-6}5^{-1}|G:H|m^3$$
$$\succeq
{\rm dim}H^0(Y,mL_Y)-2^{-8}3^{-6}5^{-1}|G:H|m^3\sim
2^{-8}3^{-6}5^{-1}|G:H|m^3.$$
\hfill$\Box$
We shall eventually prove that if $|G:H|$ is big, then
${\rm dim}H^0(Y,m(K_Y-L_Y)-{\rm mlt}^0)\neq 0$ for big $m$.
\begin{dfn}
{ Let $R$ be the ramification divisor of the morphism $\mu$.
We define $H^0(X,mK-mR-mL-{\rm mlt}^0)$ to be the space
of global sections of ${\cal O}_X(m(K_X-R-L_X))$ that satisfy
certain vanishing conditions. Namely, we require their germs to lie
in $m_x^{m\cdot k({\rm Stab}^H(x))} {\cal O}_X(m(K_X-R-L_X))_x$ for all
points $x\in X$ whose images in $Y$ have noncanonical singularities,
except for $x=P_{\alpha\beta\gamma}.$ Here $k({\rm Stab}^H(x))$ is defined
according to remark \ref{k}.}
\end{dfn}
\begin{prop}
{ If $|G:H|> 2^{953}[2^{165870}]_p$ and
${\rm dim}H^0(X,mK-mR-mL-{\rm mlt}^0)\neq 0$ for some $m>0$,
then the variety $Y$ is of general type.}
\label{alreadyonX}
\end{prop}
{\em Proof.} Because of \ref{klem}, all $H$-invariant elements
of $H^0(X,mK-mR-mL-{\rm mlt}^0)$ can be pushed down to
elements of $H^0(Y,m(K_Y-L_Y)-{\rm mlt}^0).$ Notice, that
$\mu^*(m(K_Y-L_Y))=m(K_X-R-L_X)$, and $m\delta$ is dropped from the
vanishing conditions to compensate for the constant $N$ from \ref{klem}.
One can multiply the $H$-conjugates of any section to get an $H$-invariant
one, so if ${\rm dim}H^0(X,mK-mR-mL-{\rm mlt}^0)\neq 0$ for some $m>0$,
then ${\rm dim}H^0(Y,m(K_Y-L_Y)-{\rm mlt}^0)\neq 0$ for big $m$,
and Proposition \ref{getridofDDD} finishes the proof.\hfill$\Box$
Let us now describe ramification divisors and points with noncanonical
images.
\begin{prop}
{ The ramification divisor $R$ equals
$\sum_\alpha(n\cdot {\rm ram}_H(v_\alpha)-1)D_\alpha +
\sum_\alpha {\rm ram}_H(E_\alpha)E_\alpha +\sum_\alpha {\rm ram}_H(F_\alpha)F_\alpha.$}
\label{ramdiv}
\end{prop}
{\em Proof.} We know by \ref{ramfin} that the ramification divisor in
the finite part is equal to $\sum_\alpha {\rm ram}_H(E_\alpha)E_\alpha+\sum_\alpha
{\rm ram}_H(F_\alpha)F_\alpha$. We only need to show that the group of
elements of $G$ that fix all points of the divisor $D_\alpha$ is
exactly $\pm {\rm Ram}_G(v_\alpha)$. It can be done explicitly in coordinates
for the standard divisor $D_0.$ \hfill$\Box$
\begin{prop}
{ If $x\in D_\alpha$, but $x\notin \cup (D_\alpha\cap D_\beta)$,
then ${\rm Stab}^H(x)/(\pm {\rm Ram}_H(v_\alpha))$ is a group of order at most $6$.}
\label{jumpD}
\end{prop}
{\em Proof.} We only need to consider the standard divisor $D_0$.
It is the universal elliptic curve with level $n$ structure.
It can be shown that the group ${\rm Stab}^G_{D_0}$ acts on it by
a combination of modular transformations of the base, additions
of points of order $n$ in the fibers, and the involution
$a\to -a$ of the fibers. The order $6$ can be reached for the point
$x$ on the curve with complex multiplication, such that $x$ satisfies
$2n\cdot x=0$, and all other stabilizers are even smaller. I skip
the details, because a different bound here would only slightly
affect the final estimate. \hfill$\Box$
\begin{prop}
{ If $x\in l_{\alpha\beta}= D_\alpha\cap D_\beta$,
but $x\notin \cup P_{\alpha\beta\gamma}$,
then the order of the group
${\rm Stab}^H(x)/\pm {\rm Ram}_H(l_{\alpha\beta})$ is at most
$4$.}
\label{jumpDD}
\end{prop}
{\em Proof.} We may assume that $l_{\alpha,\beta}=l_0$ is the standard line.
The group ${\rm Stab}^G(l_0)$ contains a subgroup of index $2$ of elements
that preserve both $\pm{}^t(1,0,0,0)$ and $\pm{}^t(0,1,0,0).$
It in turn contains a subgroup of
index $2$ that consists of matrices $\pm \pmatrix{{\bf 1}&B\cr
{\bf 0}&{\bf 1}\cr}, ~B=\pmatrix{a&b\cr b&c\cr}.$
One can show using the explicit coordinate on $l_0$, that
if $b$ is nonzero, then the action of this element has no fixed points
on $l_0$, except for the points of triple intersection of the
infinity divisors, which finishes the proof. \hfill$\Box$
\begin{prop}
{ If $\,|G:H|> 2^{953}[2^{165870}]_p$ and
$${\rm dim}H^0(m(K-L)-m\sum_\alpha n\cdot {\rm ram}_H(v_\alpha) 7 D_\alpha
-m\sum_\alpha {\rm ram}_H(E_\alpha) 73 E_\alpha$$
$$-m\sum_\alpha {\rm ram}_H(F_\alpha) 73 F_\alpha
-m\sum_{\alpha\beta}n\cdot {\rm ram}_H(l_{\alpha\beta}) 4 l_{\alpha\beta})
\neq 0$$
for some $m>0$, then the variety $Y$ is of general type.}
\label{73}
\end{prop}
{\em Proof.} We know from Proposition \ref{finsing} that the
points in the finite part, that do not
lie in the ramification divisors $E_\alpha$ or $F_\beta$,
do not contribute to ${\rm mlt}^0.$ Therefore,
$$m\sum_\alpha {\rm ram}_H(E_\alpha) 73 E_\alpha+m\sum_\alpha {\rm ram}_H(F_\alpha)
73 F_\alpha \geq {\rm mlt}+mR$$
in the finite part. This inequality, strictly speaking, is the inclusion of
the sheaves of ideals. Analogously, Propositions \ref{jumpD} and
\ref{jumpDD} show that
$$m\sum_\alpha n\cdot {\rm ram}_H(v_\alpha) 7 D_\alpha\geq mR_D+{\rm mlt}^0$$
on $D$ away from $\cup(D_\alpha\cap D_\beta)$, and
$$m\sum_{\alpha\beta}n\cdot {\rm ram}_H(l_{\alpha\beta}) 4 l_{\alpha\beta}
\geq {\rm mlt}^0$$
on $\cup(D_\alpha\cap D_\beta)$ away from points $P_{\alpha\beta\gamma}$.
Then it remains to use Proposition \ref{alreadyonX}.
\hfill$\Box$
\begin{prop}
{ If the variety $Y$ is not of general type, then at least
one of the following inequalities holds true.
(1) $$|G:H|\leq 2^{953}[2^{165870}]_p$$
(2) $$ {\rm dim}H^0(m(K-L))-{\rm dim}H^0(m(K-L)-m\sum_\alpha n\cdot {\rm ram}_H(v_\alpha)
7 D_\alpha)$$
$$\geq ((1/6)c_1(K-L)^3m^3)/5$$
(3) $$ {\rm dim}H^0(m(K-L))-{\rm dim}H^0(m(K-L)-m\sum_\alpha {\rm ram}_H(E_\alpha)
73 E_\alpha)$$
$$\geq ((1/6)c_1(K-L)^3m^3)/5$$
(4) $$ {\rm dim}H^0(m(K-L))-{\rm dim}H^0(m(K-L)-m\sum_\alpha {\rm ram}_H(F_\alpha)
73 F_\alpha)$$
$$\geq ((1/6)c_1(K-L)^3m^3)/5$$
(5) $$ {\rm dim}H^0(m(K-L)-m\sum_\alpha n\cdot {\rm ram}_H(v_\alpha) 7 D_\alpha
-m\sum_\alpha {\rm ram}_H(E_\alpha) 73 E_\alpha$$
$$-m\sum_\alpha {\rm ram}_H(F_\alpha) 73 F_\alpha)
-{\rm dim}H^0(m(K-L)-m\sum_\alpha n\cdot {\rm ram}_H(v_\alpha) 7 D_\alpha$$
$$-m\sum_\alpha {\rm ram}_H(E_\alpha) 73 E_\alpha
-m\sum_\alpha {\rm ram}_H(F_\alpha) 73 F_\alpha$$
$$ -m\sum_{\alpha\beta}n\cdot {\rm ram}_H(l_{\alpha\beta}) 4 l_{\alpha\beta})
\geq ((1/6)c_1(K-L)^3m^3)/5$$}
\label{splitcases}
\end{prop}
{\em Proof.} If (2),(3), and (4) are all false, then
$${\rm dim}H^0(m(K-L)-m\sum_\alpha n\cdot {\rm ram}_H(v_\alpha) 7 D_\alpha
-\sum_\alpha {\rm ram}_H(E_\alpha) 73 E_\alpha$$
$$-\sum_\alpha {\rm ram}_H(F_\alpha) 73 F_\alpha)\succeq (2/5)(1/6)c_1(K-L)^3m^3.$$
Really, ${\rm dim}H^0(m(K-L)$ grows like $(1/6)c_1(K-L)^3m^3$, because $K-L$ is
ample for big $n$, and $E_\alpha, F_\beta, D_\gamma$ are different divisors.
Hence, if (1) and (5) are also false, then Proposition
\ref{73} proves that the variety $Y$ is of general type. \hfill$\Box$
Our next goal is to show that each of the statements (2)-(5)
implies that $|G:H|$ is less than some constant.
We use results of Yamazaki \cite{Yamazaki} and statements of
Section 3.
\begin{prop}
{ If $$ {\rm dim}H^0(m(K-L))-{\rm dim}H^0(m(K-L)-m\sum_\alpha n\cdot {\rm ram}_H(v_\alpha)
7 D_\alpha)$$
$$\geq ((1/6)c_1(K-L)^3m^3)/5,$$
then $|G:H|< 2^{41}[2^{828}]_p$.}
\label{5.D}
\end{prop}
{\em Proof.} First of all, we get
$$ {\rm dim}H^0(m(K-L))-{\rm dim}H^0(m(K-L)-m\sum_\alpha n\cdot {\rm ram}_H(v_\alpha)
7 D_\alpha)$$
$$\leq \sum_\alpha({\rm dim}H^0(m(K-L))-{\rm dim}H^0(m(K-L)-7mn\cdot {\rm ram}_H(v_\alpha)
D_\alpha)).$$
The standard exact sequences associated to $D_\alpha\subset X$ allow us
to estimate that
$${\rm dim}H^0(m(K-L))-{\rm dim}H^0(m(K-L)-7mn\cdot {\rm ram}_H(v_\alpha)
D_\alpha)$$
$$\leq \sum_{j=0}^{7mn\cdot {\rm ram}_H(v_\alpha)-1}
{\rm dim}H^0(D_\alpha,m(K-L)-jD_\alpha)$$
$$=\sum_{j=0}^{7mn\cdot {\rm ram}_H(v_\alpha)-1}
{\rm dim}H^0(D_\alpha,m(K-L)+(2j/n)(L+E)).$$
The divisor $L+E$ is nef on $X$, because $L$ is nef,
divisors $E_i$ are disjoint, and $(L+E)E_i=0$. The divisor $K-L$
is ample on $X$, if $n$ is sufficiently big. Therefore, we may use the
Riemann-Roch
formula to calculate ${\rm dim}H^0(D_\alpha,m(K-L)+(2j/n)(L+E))$. Because we
are only interested in the coefficient of $m^3$, as $m\to \infty$,
we only need to take into account the term
$(1/2)c_1(m(K-L)+(2j/n)(L+E))^2c_1(D_\alpha)$.
When $j$ grows, this intersection number grows, therefore
$${\rm dim}H^0(m(K-L))-{\rm dim}H^0(m(K-L)-7mn\cdot {\rm ram}_H(v_\alpha)
D_\alpha)$$
$$\leq 7mn\cdot {\rm ram}_H(v_\alpha) (1/2) m^2c_1(K-L+14{\rm ram}_H(v_\alpha)(L+E))^2
c_1(D_\alpha)$$
$$\leq m^3 \sharp(v_\alpha)^{-1} {\rm ram}_H(v_\alpha)
(7n/2)c_1(K-L+14 (L+E))^2c_1(D).$$
Hence, if the condition of the proposition is true, then
$$\sharp(v_\alpha)^{-1}\sum_\alpha {\rm ram}_H(v_\alpha)\geq
105^{-1}c_1(K-L)^3/(c_1(K-L+14(L+E))^2c_1(nD)).$$
The right hand side can be calculated using the formulas of Yamazaki for
the intersection numbers of the divisors $D,L,K$, and $E$.
It is bigger than $2^{-18}$ if $n$ is sufficiently big, which we may assume
without loss of generality. Therefore, by the result of Proposition
\ref{boundD}, $|G:H|< 2^{41}[2^{828}]_p$.\hfill$\Box$
\begin{prop}
{ If $$ {\rm dim}H^0(m(K-L))-{\rm dim}H^0(m(K-L)-m\sum_\alpha {\rm ram}_H(E_\alpha)
73 E_\alpha)$$
$$\geq ((1/6)c_1(K-L)^3m^3)/5,$$
then $|G:H|< 2^{53}[2^{3236}]_p$.}
\label{5.E}
\end{prop}
{\em Proof.} Analogously to the proof of \ref{5.D}, we estimate
$$ {\rm dim}H^0(m(K-L))-{\rm dim}H^0(m(K-L)-m\sum_\alpha {\rm ram}_H(E_\alpha)
73 E_\alpha)$$
$$\leq \sum_\alpha {\rm ram}_H(E_\alpha) \sum_{j=0}^{73m-1}
{\rm dim}H^0(E_\alpha,m(K-L)-jE_\alpha)$$
$$=\sum_\alpha {\rm ram}_H(E_\alpha)
\sum_{j=0}^{73m-1} {\rm dim}H^0(E_\alpha,m(K-L)+jL)$$
$$\preceq
\sharp(E_\alpha)^{-1}\sum_\alpha {\rm ram}_H(E_\alpha) (73/2)m^3
c_1(K+72L)^2c_1(E).$$
Therefore,
$$\sharp(E_\alpha)^{-1}\sum_\alpha {\rm ram}_H(E_\alpha)
\geq 73^{-1}15^{-1}c_1(K-L)^3/(c_1(K+72L)^2c_1(E))>2^{-23}.$$
Then Proposition \ref{boundE} tells us that
$|G:H|< 2^{53}[2^{3236}]_p$.\hfill$\Box$
\begin{prop}
{ If $$ {\rm dim}H^0(m(K-L))-{\rm dim}H^0(m(K-L)-m\sum_\alpha {\rm ram}_H(F_\alpha)
73 F_\alpha)$$
$$\geq ((1/6)c_1(K-L)^3m^3)/5,$$
then $|G:H|< 2^{73}[2^{22782}]_p$.}
\label{5.F}
\end{prop}
{\em Proof.} As in the proof of \ref{5.E}, we estimate
$$ {\rm dim}H^0(m(K-L))-{\rm dim}H^0(m(K-L)-m\sum_\alpha {\rm ram}_H(F_\alpha)
73 F_\alpha)$$
$$\leq \sum_\alpha {\rm ram}_H(F_\alpha) \sum_{j=0}^{73m-1}
{\rm dim}H^0(F_\alpha,m(K-L)-jF_\alpha).$$
Unfortunately, the geometry of $F$ is more complicated than that of $E$,
and we do not have a nice formula like $(L+E_\alpha)E_\alpha=0$. We can
get away with it by using the adjunction formula together with the Proposition
\ref{gentypeF}.
We can estimate
$$\sum_\alpha {\rm ram}_H(F_\alpha) \sum_{j=0}^{73m-1}
{\rm dim}H^0(F_\alpha,m(K-L)-jF_\alpha)$$
$$\leq \sum_\alpha {\rm ram}_H(F_\alpha) \sum_{j=0}^{73m-1}
{\rm dim}H^0(F_\alpha,m(K-L)+jK-jK_{F_\alpha})$$
$$\leq \sum_\alpha {\rm ram}_H(F_\alpha) \sum_{j=0}^{73m-1}
{\rm dim}H^0(F_\alpha,m(K-L)+jK)$$
$$\leq \sharp(F_\alpha)^{-1}\sum_\alpha {\rm ram}_H(F_\alpha)
(73/2)m^3c_1(74K-L)^2c_1(F).$$
Therefore,
$$\sharp(F_\alpha)^{-1}\sum_\alpha {\rm ram}_H(F_\alpha)\geq
73^{-1}15^{-1}c_1(K-L)^3/(c_1(74K-L)^2c_1(F)).$$
We need to have some upper bound on $c_1(74K-L)^2c_1(F).$
To do this, we recall the proof of Proposition \ref{finsing},
where we have shown that the images of the divisors $F_\alpha$ on the singular
quartic $V$ have form $x_i=x_j$. The product $\prod_{i\neq j}(x_i-x_j)^2$
is invariant under the permutations of the coordinates, so it defines a
modular form of weight $60$, that vanishes on $F$. Here we use the fact
that the coordinates of ${\bf P}^4$ are given by the modular forms of weight $2$,
see \cite{Geer}.
As a result, $c_1(74K-L)^2c_1(F)\leq 60c_1(74K-L)^2c_1(L)$,
and we can estimate
$\sharp(F_\alpha)^{-1}\sum_\alpha {\rm ram}_H(F_\alpha)>2^{-30}$.
Now Proposition \ref{boundF} implies that $|G:H|< 2^{73}[2^{22782}]_p$.
\hfill$\Box$
\begin{prop}
{ If
$$ {\rm dim}H^0(m(K-L)-m\sum_\alpha n\cdot {\rm ram}_H(v_\alpha) 7 D_\alpha
-m\sum_\alpha {\rm ram}_H(E_\alpha) 73 E_\alpha$$
$$-m\sum_\alpha {\rm ram}_H(F_\alpha) 73 F_\alpha)
-{\rm dim}H^0(m(K-L)-m\sum_\alpha n\cdot {\rm ram}_H(v_\alpha) 7 D_\alpha$$
$$-m\sum_\alpha {\rm ram}_H(E_\alpha) 73 E_\alpha
-m\sum_\alpha {\rm ram}_H(F_\alpha) 73 F_\alpha$$
$$ -m\sum_{\alpha\beta}n\cdot {\rm ram}_H(l_{\alpha\beta}) 4 l_{\alpha\beta})
\geq ((1/6)c_1(K-L)^3m^3)/5,$$
then $|G:H|<2^{65}[2^{10470}]_p$.}
\label{5.DD}
\end{prop}
{\em Proof.} Denote
$$L_1=K-L-7\sum_\alpha n\cdot {\rm ram}(v_\alpha)D_\alpha - 73\sum_\alpha
{\rm ram}_H(E_\alpha)E_\alpha-73\sum_\alpha {\rm ram}_H(F_\alpha)F_\alpha.$$
Then the left hand side of the proposition does not exceed the sum over
all $l_{\alpha\beta}$ of
$${\rm dim}H^0(mL_1)-{\rm dim}H^0(mL_1-4mn\cdot {\rm ram}_H(l_{\alpha\beta})l_{\alpha\beta}).$$
To estimate this codimension, we consider the blow-up of the variety $X$
along the line $l_{\alpha\beta}$, which we denote by $\pi:X_1\to X$.
The normal bundle to $l_{\alpha\beta}$ is isomorphic to
${\cal O}(2)\oplus{\cal O}(2)$. This can be checked by direct calculation.
Therefore, the exceptional divisor of $\pi$ is isomorphic to ${\bf P}^1\times
{\bf P}^1$. We get
$${\rm dim}H^0(mL_1)-{\rm dim}H^0(mL_1-4mn\cdot {\rm ram}_H(l_{\alpha\beta})l_{\alpha\beta})$$
$$={\rm dim}H^0(m\pi^*L_1)-{\rm dim}H^0(m\pi^*L_1-4mn\cdot {\rm ram}_H(l_{\alpha\beta})S)$$
$$\leq\sum_{j=0}^{4mn\cdot {\rm ram}(l_{\alpha\beta})-1}{\rm dim}H^0(S,m\pi^*L_1-jS).$$
We denote the fiber and the section of $S\to l_{\alpha\beta}$ by $f$ and
$s$ respectively and get $(m\pi^*L_1-jS)S=m\cdot c_1(L_1)l_{\alpha\beta}
\cdot f +j(2f+s)$. Hence, $H^0(S,m\pi^*L_1-jS)$ grows when $j$ grows, and
we have
$${\rm dim}H^0(mL_1)-{\rm dim}H^0(mL_1-4mn\cdot {\rm ram}_H(l_{\alpha\beta})l_{\alpha\beta})$$
$$\leq 4mn\cdot {\rm ram}_H(l_{\alpha\beta}){\rm dim}H^0(S,mc_1(L_1)l_{\alpha\beta}\cdot
2f+4mn\cdot {\rm ram}_H(l_{\alpha\beta})\cdot j )$$
$$\leq m^3{\rm ram}_H(l_{\alpha\beta})\cdot (8n\cdot {\rm ram}_H(l_{\alpha\beta})
+c_1(L_1)l_{\alpha\beta})\cdot4n\cdot {\rm ram}_H(l_{\alpha\beta})$$
$$\leq m^3{\rm ram}_H(l_{\alpha\beta})(128n^3+16n^2c_1(L_1)l_{\alpha\beta})$$
$$\preceq_{n\to\infty}~m^3{\rm ram}_H(l_{\alpha\beta})(128n^3+
16n^2\cdot (7n\cdot 2\cdot 2+73\cdot 2\cdot n)
$$
$$=m^3{\rm ram}_H(l_{\alpha\beta})\cdot 2912n^3.$$
The number of $l_{\alpha\beta}$ is equal to $2^{-3}n^7(1-p^{-4})(1-p^{-2})$,
see \cite{Yamazaki}. Therefore, if the condition of the proposition is true,
then
$$(\sharp(l_{\alpha\beta}))^{-1}\sum_{\alpha\beta}{\rm ram}_H(l_{\alpha\beta})
\geq {{c_1(K-L)^3
}\over{
(30\cdot 2912\cdot 2^{-3}n^{10}(1-p^{-4})(1-p^{-2}))}}>2^{-27}.$$
Now the result of Proposition \ref{boundDD} gives
$|G:H|<2^{65}[2^{10470}]_p$. \hfill$\Box$
We are now ready to prove the finiteness theorem for $H\supseteq
\Gamma(p^t)$.
\begin{prop}
{ If $|G:H|>2^{953}[2^{165870}]_p$,
then the variety $Y$ is of general type.}
\end{prop}
{\em Proof.} We simply combine the results of Propositions
\ref{5.D}, \ref{5.E}. \ref{5.F}, \ref{5.DD}, and \ref{splitcases}.
\hfill$\Box$
\begin{prop}
{ {\bf Finiteness theorem for $H\supseteq \Gamma(p^t).$}
There are only finitely many subgroups $H\subseteq {\rm Sp(4,{\bf Z})}$
of finite index that contain $\Gamma(p^t)$ for some $p$ and $t$,
such that the variety ${\cal H}/H$ is not of general type.}
\label{finthmprimary}
\end{prop}
{\em Proof.} It follows from the fact that $|G:H|$ is bounded.
\hfill$\Box$
In particular, if $p$ is sufficiently big, then for any $H$,
${\rm Sp(4,{\bf Z})}\supset H\supseteq \Gamma(p^t)$ the variety $Y$
is of general type.
\section{Finiteness Theorem, the general case}
Now we no longer assume that $n$ is a power of a prime number. Our goal
is to prove that the condition $n=p^t$ can be dropped from the statement
of Proposition \ref{finthmprimary}. Our proof is the direct
generalization of the argument of \cite{Thompson}.
We first estimate prime factors of $n$.
\begin{prop}
{ If $p>3$, and
$$H\cdot\Gamma(p)=\Gamma(1),~
H\supseteq\Gamma(mp^\alpha),~{\rm g.c.d.}(m,p)=1,$$
then
$H\supseteq\Gamma(m)$.}
\label{bigprime}
\end{prop}
{\em Proof. } For any group $G$ we denote its image modulo
$\Gamma(mp^\alpha)$ by $\hat G$. We have isomorphisms
$$\hat\Gamma(1)\simeq
\hat\Gamma(m)\times\hat\Gamma(p^\alpha),~\hat\Gamma(m)\simeq{\rm
Sp(4,{\bf Z}/p^\alpha {\bf Z})},~\hat\Gamma(p^\alpha)\simeq{\rm Sp(4,{\bf Z}/m
{\bf Z})}.$$
The group ${\rm PSp(4,{\bf Z}/p {\bf Z})}$ is simple for all $p\geq 3$.
Because of $\hat H\cdot\hat\Gamma(p)/\hat\Gamma(p)\simeq
{\rm Sp(4,{\bf Z}/p {\bf Z})}$, the group $\hat H$ has a section isomorphic to
${\rm PSp(4,{\bf Z}/p {\bf Z})}$. Consider the following normal subgroups of
$\hat\Gamma(1)$.
$$\hat\Gamma(1)\supset\hat\Gamma(m)\supset\hat\Gamma(mp)\supseteq\{e\}.$$
We easily get that $\hat H\cap\hat\Gamma(m)/\hat H\cap\hat\Gamma(mp)$
has a section isomorphic to ${\rm PSp(4,{\bf Z}/p{\bf Z})}$, so there holds
$$(\hat H\cap\hat\Gamma(m))\cdot\hat\Gamma(mp)=\hat\Gamma(m)$$
Now it will suffice to prove that the last equality implies $\hat H\supseteq
\hat\Gamma(m)$. Note that $\hat\Gamma(m)\simeq {\rm Sp(4,{\bf Z}/p^\alpha
{\bf Z})}$ and $\hat\Gamma(mp)\simeq {\rm Ker}({\rm Sp(4,{\bf Z}/p^\alpha {\bf Z})}
\to{\rm Sp(4,{\bf Z}/p {\bf Z}}))$. We denote by $K_i$ the kernels of ${\rm Sp(4
,{\bf Z}/p^\alpha {\bf Z})}\to{\rm Sp(4,{\bf Z}/p^i {\bf Z})}$
for $i=1,...,\alpha$ and prove that $\hat H\supseteq K_i$ by the decreasing
induction on $i$.
For $i=\alpha$ there is nothing to prove. Besides we already have the last
step of the induction. Suppose that $\hat H\supseteq K_i,~i>1$. To prove
that $\hat H\supseteq K_{i-1}$ consider $h\in\hat H\cap\hat\Gamma(m)$
such that $$h\equiv \pmatrix{
1&0&1&0\cr0&1&0&0\cr0&0&1&0\cr0&0&0&1}({\rm mod}~p).$$
Clearly, $h^{p^i}\in K_i$. Besides, a simple calculation shows that for
$p\geq 5$ $$h^{p^{i-1}}\equiv\pmatrix{1&0&p^{i-1}&0\cr0&1&0&0\cr0&0&1&0
\cr0&0&0&1}({\rm mod}~p^i).$$ When the group $\hat\Gamma(m)$ acts on
$K_{i-1}/K_i$ by conjugation, its subgroup $\hat\Gamma(mp)$ acts as
identity. We have already known that $(\hat H\cap\hat\Gamma(m))\cdot
\hat\Gamma(mp)=\hat\Gamma(m)$, so it is enough to show that
conjugates of the element $h^{p^{i-1}}$ generate the whole group
$K_{i-1}$ modulo $K_i$. This can be done by a direct calculation in the
abelian group $K_{i-1}/K_i$. \hfill $\Box$
\begin{prop}
{ There exists a natural number $N$ such that if ${\cal H}/H$ is not
of general type, then $$H\supseteq \Gamma(\prod_{p_i\leq N}p_i^{n_i})$$
for some natural numbers $n_i$.}
\label{boundprime}
\end{prop}
{\em Proof.} Let $n$ be the minimum number such that $H\supseteq \Gamma(n)$.
Because of the result of \ref{bigprime}, $H\cdot \Gamma(p)\neq \Gamma(1)$
for all prime factors of $p$ of $n$ bigger than $3$. If ${\cal H}/H$ is not of
general type, then ${\cal H}/(H\cdot\Gamma_p)$ is not of general type
either, see \ref{fincov}. Because of Proposition \ref{finthmprimary},
there are only finitely many choices for $p.$ \hfill$\Box$
We now prove the Finiteness Theorem in full generality.
Define for any $H\subseteq\Gamma(1)$ and any prime $p$ the
$p$-projection of $H$ as $H_p=\cap_1^\infty H\cdot\Gamma(p^j)$. Note
that $H_p\supseteq H$ and $H_p\supseteq\Gamma(p^j)$ for some $j$. The
following proposition allows us to work with $p$-projections only,
after we have got an estimate on the primes.
\begin{prop}
{ For any given set of subgroups $G_i\supseteq\Gamma(p_i^{n_i}),
~i=1,...,k$, there are only finitely many subgroups $H\supseteq\Gamma
(p_1^{\alpha_1}\cdot...\cdot p_k^{\alpha_k})$ with $H_{p_i}=G_i$.}
\label{splitprime}
\end{prop}
{\em Proof.} We can simply estimate the index of $H$ if we employ
the fact that $\Gamma(p_i)$ are pro-$p_i$-groups.\hfill $\Box$
Now we can easily prove the Finiteness Theorem.
\begin{prop}
{ {\bf Finiteness Theorem.} There are only finitely many subgroups
$H\subseteq {\rm Sp(4,{\bf Z})}$ of finite index, such that ${\cal H}/H$
is not of general type.}
\label{fintheorem}
\end{prop}
{\em Proof.} If ${\cal H}/H$ is not of general type, then
${\cal H}/H_p$ is not of general type either. Therefore, Proposition
\ref{finthmprimary} tells us that there are only finitely many choices
for $H_p$. By \ref{boundprime}, all prime factors of $n$ are bounded,
so Proposition \ref{splitprime} finishes the proof.
\hfill $\Box$
\section{Varieties of general type and singularities}
We first recall some standard facts about varieties of general type and
singularities.
\begin{dfn}
{ A smooth compact algebraic variety X over ${\bf C}$ is called a variety of
general type if there exists some constant $c>0$ such that ${\rm dim}H^0(X,{\cal
O}_X(mK_X))>cm^{{\rm dim}X}$ for all sufficiently big
(equivalent condition -- divisible by some integer $d$) positive integers $m$.
Here $K_X$ is the canonical divisor of $X$.}
\end{dfn}
\begin{rem}
{ If $X$ and $Y$ are birational smooth compact algebraic varieties, then
${\rm dim}H^0(X,{\cal O}_X(mK_X))= {\rm dim}H^0(Y,{\cal O}_Y(mK_Y))$ for $m\geq0$. }
\end{rem}
\begin{dfn}
{ A field ${\cal K}\supset{\bf C}$ is called a field of general type if it
is a field of the rational functions
of a smooth compact algebraic variety of general type.}
\end{dfn}
\begin{dfn}
{ An algebraic variety over ${\bf C}$ is called a variety of general type if
its field of functions is a field of general type.}
\end {dfn}
\begin{dfn}
{ A canonical divisor $K_X$ of a normal variety $X$ is a Weil divisor
on $X$ that coincides with a canonical divisor on $X-{\rm Sing}(X)$. The variety $X$
is called ${\bf Q}$-$Gorenstein$ if $mK_X$ is a Cartier divisor for some integer
$m$.}
\end{dfn}
\begin{rem}
{ If the variety $Y$ is normal ${\bf Q}$-Gorenstein but has singularities,
then the condition "${\rm dim}H^0(Y,{\cal O}_Y(mK_Y))>cm^{{\rm dim}Y}$ for $m\to+\infty$"
does not imply by itself that $Y$ is of general type. Really, if
$\pi:Z\to Y$ is some desingularization, then there holds
$$K_Z=\pi^*(K_Y)+\sum_i\alpha_iF_i,~~\alpha_i\in{\bf Q}$$
in the sense of equivalence of ${\bf Q}$-Cartier divisors, where $F_i$ are
exceptional divisors of morphism $\pi$ and $\alpha_i$ are some rational numbers
called {\it discrepancies}\/. If some $\alpha_i$ are negative,
then ${\rm dim}H^0(Z,{\cal O}_Z(mK_Z))$ may be less than ${\rm dim}H^0(Y,{\cal
O}_Y(mK_Y))$}.
\label{discrep}
\end{rem}
\begin{dfn}
{ A normal ${\bf Q}$-Gorenstein variety $Y$ is said to have $log-terminal$
singularities if for some desingularization $\pi:Z\to Y$, such that the
exceptional divisor $\sum F_i$ has simple normal crossings, all
discrepancies are greater than $(-1)$. A singular point $y\in Y$ is called
{\it canonical}\/ (resp. {\it terminal}\/) if the discrepancies $\alpha_i$ are
nonnegative (resp. positive) for all $i$ such that $\pi(F_i)\ni y$.
Once satisfied for some desingularization, whose exceptional locus is a divisor
with simple normal crossings, these conditions are satisfied for any
desingularization (see \cite{CKM}).}
\label{defcan}
\end{dfn}
\begin{prop}
{ If $\mu:X\to Y$ is a finite morphism of algebraic varieties and $Y$ is of
general type, then $X$ is also of general type.}
\label{fincov}
\end{prop}
{\em Proof.} We find a surjective morphism $\hat\mu:\hat X\to\hat Y$,
where $\hat X,\hat Y$ are smooth projective birational models of $X,Y$,
and then pull back multicanonical forms.
\hfill$\Box$
The following statement is well-known.
\begin{prop}
{ {\rm (see \cite{CKM})}
Let $X$ be a smooth projective algebraic variety over ${\bf C}$ with an
action of a finite group $G$.
Then the quotient variety $Y=X/G$ has log-terminal singularities.}
\label{qulog}
\end{prop}
Now we shall prove a simple but important technical result about quotient
singularities. Let $X$ be a projective algebraic variety with an action of a
finite solvable group $H$. Let $x$ be a (closed) point of $X$, such that $hx=x$
for all $h\in H$. Suppose we have $\{e\}=H_0\subset H_1\subset ...\subset
H_t=H$, where $H_{i-1}$ are normal subgroups of $H_i$ and $H_i/H_{i-1}$ are
abelian groups with exponents $k_i$. Denote $k=k_1\cdot...\cdot k_t$.
Denote the local ring of $x$ in $X$ by $(A,{\bf m})$. Then $(B,{\bf n})=(A^H,{\bf m}^H)$ is the
local ring of the image of $x$ under the quotient morphism.
\begin{prop}
{ In the above setup there exists a constant $N$, which depends only on $X$
and $H$ but not on $x$, such that there holds ${\bf m}^{kl+N}\cap B\subseteq {\bf n}^l$
for all $l\geq0$.}
\label{klem}
\end{prop}
{\em Proof.} We do not suppose $X$ to be smooth, so it is enough to consider
just the case of an abelian group $H$ with $kH=0$. There exists a linearized
$H$-invariant very ample invertible sheaf ${\cal L}$ on $X$. Consider the
corresponding closed embedding $X\to {\bf P}^{N_0}$. Because
$H$ is abelian, there exists an open $H$-invariant affine neighborhood of
$x$ with the ring $R$ equal to ${\bf C}[1,l_1/l_0,...,l_{N_0}/l_0]/I$
where
$l_i\in H^0(X,{\cal L}),~
h(l_i)=\mu_i(h)\cdot l_i,~\forall h\in H$ and $I$ is some ideal. Moreover,
we may assume
that $f_i=l_i/l_0$ vanish at $x$, because of $Hx=x$. Hence the local ring
$(A,{\bf m})$ is the localization of $R$ in $p=(f_1,...,f_{N_0})$.
Because $H$ is finite, one can assume that all denominators are
$H$-invariant. Therefore, the statement of the proposition
is equivalent to $p^{kl+N}\cap R^H\subseteq (p^H)^l$.
Each element of $p$ can be represented as a polynomial in $f_i$ with zero
constant term. Therefore, each element of $p^{ kl+N}$ can be represented as a
polynomial in $f_i$ with monomials of degree no less than $kl+N$. For any
given $f\in p^{kl+N}\cap R^H$ consider such a representation with the minimum
possible number of monomials. Then if for some monomial $g$ of this
representation and some element $h\in H$ there holds $h(g)=w\cdot g,~w\neq1$,
then the formula $f=f\cdot w/(w-1)-h(f)/(w-1)$ allows us to reduce the number
of monomials. Hence every element $f\in p^{kl+N}\cap R^H$ is a sum of
$H$-invariant monomials of degree at least $kl+N$.
Now we only need to prove that any $H$-invariant monomial $g=f_1^{\alpha_1}
\cdot...\cdot f_{N_0}^{\alpha_{N_0}}$ of degree at least $kl+N$ is a product
of at least $l$ $H$-invariant monomials of positive degree. It is time to
choose $N$, namely $N=k\cdot N_0$. Denote by $\gamma_i$ the maximum integers
that do not exceed $\alpha_i/k$. Then $g=f_1^{k\gamma_1}\cdot...\cdot
f_{N_0}^{k\gamma_{N_0}}\cdot g_1$ gives the required decomposition,
because $\sum\gamma_i>\sum\alpha_i/k-N_0\geq l$.\hfill $\Box$
\begin{rem}
{ Due to the result of \cite{Hochster}, the above proposition holds
for scheme points which correspond to the subvarieties that are pointwise
$H$-invariant. I wish to thank Melvin Hochster for pointing out this
reference.}
\label{afterklem}
\end{rem}
\begin{rem}
{ In the rest of the paper $k(H)$ for a finite solvable group $H$ denotes
the least possible value of $k$ that could be obtained in the above way.}
\label{k}
\end{rem}
The rest of the section is devoted to multiplicities of certain toric
singularities. Somewhat unnatural choice of notation is motivated by
the notation of Section 3.
\begin{dfn}
{ Let $G_1\simeq ({{\bf Z}/n{\bf Z}})^3$ act on ${\bf C}^3$ according to the formula
$$(\xi_1,\xi_2,\xi_3)(x_1,x_2,x_3)=(e^{2\pi i\xi_1/n}x_1,
e^{2\pi i\xi_2/n}x_2,e^{2\pi i\xi_3/n}x_3).$$
Let $H_1$ be a subgroup of $G_1$. Define $\delta(H_1)=(1/n)
{\rm min}_{l\neq 0}(l_1+l_2+l_3)$, where the minimum is taken
among all $H_1$-invariant monomials
$x_1^{l_1}x_2^{l_2}x_3^{l_3}$.}
\label{appdelta}
\end{dfn}
\begin{prop}
{ The multiplicity of the local ring of $C^3/H_1$ at zero is at most
$n^3\delta(H_1)/|H_1|.$}
\label{appmult}
\end{prop}
{\em Proof.} The exponents of the $H_1$-invariant monomials form a semigroup,
which we denote by $K$. One can show that the multiplicity is equal to
$vol({\bf R}^n_{>0}-conv(K-\{0\}))/|H_1|$, where the volume is normalized
to be equal one on the basic tetrahedron. This result does not seem to
be stated explicitly anywhere in the literature, but its proof is completely
analogous to the calculation of \cite{Teissier} of multiplicities
of the ideals in the polynomail ring that are generated by monomials.
On the other hand, this set is contained in the set
$$conv((l_1,l_2,l_3),(0,0,n),(0,n,0),(0,0,0))\cup...$$
$$...\cup
conv((l_1,l_2,l_3),(0,n,0),(n,0,0),(0,0,0)),$$
which has volume $n^3\delta(H_1)$. \hfill$\Box$
\begin{rem}
{ Our results on the multiplicities of certain toric singularities
can be generalized to arbitrary dimension, but we only need the case
of dimension three.}
\end{rem}
Now we consider in detail the case when $n$ is a power of a prime number,
and the group $H_1$ is cyclic.
\begin{prop}
{ Let $K=K_{uvw}$ be a semigroup, defined by the conditions $\alpha u+
\beta v +\gamma w =0({\rm mod}{\em p^s})$ and $\alpha,\beta,\gamma\in
{\bf Z}_{\geq 0},$ where $u$, $v$, and $w$ are some natural numbers.
The number $\delta$ defined in \ref{appdelta} equals
$p^{-s}{\rm min}_{K-\{0\}}(\alpha+\beta+\gamma).$ Then the number of homogeneous
triples $(u:v:w)$ such that $\delta(u,v,w)\geq \epsilon$ is at most
$2^2\epsilon^{-8}[4\epsilon^{-5}]_p$.}
\label{appfinmany}
\end{prop}
{\em Proof.} Consider the intersection of $K$ and the coordinate plane
$\alpha = 0$. It is the semigroup $K_1$ defined by the conditions
$\beta,\gamma\in {\bf Z}_{\geq 0},~\beta v+\gamma w = 0({\rm mod}{\em p^s}).$
If $\delta(u,v,w)\geq \epsilon$, then $\beta+\gamma\geq \epsilon p^s$
for all nonzero $(\beta,\gamma)\in K_1$. Therefore, the area of
${\bf R}^2_{>0}-conv(K_1-\{0\})$ is at least $\epsilon^2p^{2s}$, if
the area of the basic triangle in ${\bf Z}^2$ is equal to one. Because
any triangle in ${\bf Z}^2$ with no lattice points inside and on the
edges is basic, the number of points of $K_1$ that lie inside the
positive quadrant and on the boundary of $conv(K_1-\{0\})$ is at least
$-1+\epsilon^2p^{2s}/|{\bf Z}^2:span(K_1)|\geq -1+\epsilon^2p^s$.
The function $\beta-\gamma$ is monotone on the boundary of
$conv(K_1-\{0\})$, and changes by at most $2p^s$ inside the positive
quadrant. Hence, there is a segment of this boundary, that is represented
by the vector $(\beta_1,-\gamma_1)$ with $0<\beta_1,\gamma_1,~\beta_1
+\gamma_1\leq 2\epsilon^{-2}$. Hence there holds
$v\beta_1=w\gamma_1({\rm mod}{\em p^s})$ with $0<\beta_1,\gamma_1,~\beta_1
+\gamma_1\leq 2\epsilon^{-2}$.
Analogously, we have $u\alpha_2=w\gamma_2({\rm mod}{\em p^s})$ with
$0<\alpha_2,\gamma_2,~\alpha_2+\gamma_2\leq 2\epsilon^{-2}$.
Besides, ${\rm g.c.d.}(w,p^s)\leq[\epsilon^{-1}]_p$, because otherwise
$(0,0,p^s/{\rm g.c.d.}(w,p^s))$ lies in $K$ and gives $\delta<\epsilon$.
There are at most $[\epsilon^{-1}]_p$ choices of $w({\rm mod}{\em p^s})$
up to multiplication by $({\bf Z}/p^s{\bf Z})^*$. There are at most
$2^2\epsilon^{-8}$ choices for the fourtuple $(\beta_1,\gamma_1,
\alpha_2,\gamma_2)$. Once we know $(w,\beta_1,\gamma_1,
\alpha_2,\gamma_2)$, there are at most $[2\epsilon^{-2}]_p$
for each of the numbers $u,v({\rm mod}{\em p^s})$. This proves the proposition.
\hfill$\Box$
\bigskip
|
1995-05-22T06:20:24 | 9505 | alg-geom/9505020 | en | https://arxiv.org/abs/alg-geom/9505020 | [
"alg-geom",
"math.AG"
] | alg-geom/9505020 | Dan Abramovich | Dan Abramovich | Lang's conjectures, Conjecture H, and uniformity | 7 pages. AMSLaTeX, dvi file available at
http://math.bu.edu/INDIVIDUAL/abrmovic/conjh.dvi | null | null | null | null | The purpose of this note is to wish a happy birthday to Professor Lucia
Caporaso.*
We prove that Conjecture H of Caporaso et. al. ([CHarM], sec. 6) together
with Lang's conjecture implies the uniformity of rational points on varieties
of general type, as predicted in [CHarM]; a few applications in arithmetic and
geometry are stated.
Let X be a variety of general type defined over a number field K. It was
conjectured by S. Lang that the set of rational points X(K) is not Zariski
dense in X. In the paper [CHarM] of L. Caporaso, J. Harris and B. Mazur it is
shown that the above conjecture of Lang implies the existence of a uniform
bound on the number of K-rational points of all curves of fixed genus g over K.
The paper [CHarM] has immediately created a chasm among arithmetic geometers.
This chasm, which often runs right in the middle of the personalities involved,
divides between loyal believers of Lang's conjecture, who marvel in this
powerful implication, and the disbelievers, who try (so far in vain) to use
this implication to derive counterexamples to the conjecture. In this paper we
will attempt to deepen this chasm, using the techniques of [CHarM] and
continuing [aleph], by proving more implications, some of which very strong, of
various conjectures of Lang. Along the way we will often use a conjecture
donned by Caporaso et. al. Conjecture H (see again [CHarM], sec. 6) about
Higher dimensional varieties, which is regarded very plausible among experts of
higher dimensional algebraic geometry. In particular, we will show
| [
{
"version": "v1",
"created": "Sun, 21 May 1995 17:49:36 GMT"
}
] | 2015-06-30T00:00:00 | [
[
"Abramovich",
"Dan",
""
]
] | alg-geom | \section{Introduction}
Let $X$ be a variety of general type defined over a number field $K$. It was
conjectured by S. Lang that the set of rational points $X(K)$ is not Zariski
dense in $X$.
In the paper \cite{chm} of L. Caporaso, J. Harris and B. Mazur
it is shown that the above conjecture of Lang implies
the existence of a uniform bound on the number of $K$-rational points of all
curves of fixed genus $g$ over $K$.
The paper \cite{chm} has immediately created a chasm
among arithmetic geometers.
This chasm, which often runs right in the
middle of the personalities involved, divides between loyal believers of Lang's
conjecture, who marvel in this powerful implication, and the disbelievers, who
try to use this implication to derive counterexamples to the conjecture.
In this paper we will attempt to deepen this chasm, using the
techniques of \cite{chm} and continuing \cite{abr}, by proving more
implications, some of which very strong, of various conjectures of Lang. Along
the way we will often use a conjecture donned by Caporaso et al. {\em
Conjecture H} (see again \cite{chm}, \S 6) about {\em H}igher dimensional
varieties,
which is regarded very plausible among experts of higher dimensional algebraic
geometry.
Before we state any results, we need to specify various conjectures which we
will apply.
\subsection{A few conjectures of Lang} Let $X$ be a variety of general type
over a field $K$ of characteristic 0. In view of Faltings's proof of Mordell's
conjecture, Lang has
stated the following conjectures:
\begin{conj} \begin{enumerate} \item (Weak Lang conjecture) If $K$ is finitely
generated over $\Bbb{Q}$ then the set of rational points $X(K)$ is not Zariski
dense in $X$.
\item (Weak Lang conjecture for function fields) If $k\subset K$
is a finitely generated regular extension in characteristic 0, and if $X(K)$ is
Zariski dense in $X$, then $X$ is birational to a variety $X_0$ defined over
$k$ and the {\em ``non-constant points''} $X(K)\setminus X_0(k)$ are not
Zariski dense in $X$.
\item (Geometric Lang's conjecture) Assuming only
$Char(K) = 0$, there is a proper Zariski closed subset $Z(X) \subset X$, called
in \cite{chm} the {\em Langian exceptional set}, which is the union of all
positive dimensional subvarieties which are not of general type.
\item (Strong Lang conjecture) If $K$ is finitely generated over $\Bbb{Q}$
then there is a Zariski closed subset $Z\subset X$ such that for any finitely
generated field $L\supset K$ we have that $X(L)\setminus Z(L)$ is finite.
\end{enumerate}
\end{conj}
These conjectures and the relationship between them are studied in
\cite{langbul}, \cite{lang3} and in the introduction of \cite{chm}. For
instance, it should be
noted that the weak Lang conjecture together with the geometric conjecture
imply the strong Lang conjecture.
It should also be remarked that the analogous situation over fields of positive
characteristic is subtle and interesting. See a recent survey by Voloch
\cite{voloch}.
\subsection{Conjecture H} An important tool used by Caporaso et al. in
\cite{chm} is that of fibered powers. Let $X\rightarrow B$ be a morphism of varieties
in characteristic 0, where the general fiber is a variety of general type. We
denote by $X^n_B$ the
$n$-th fibered power of $X$ over $B$.
\begin{conj} (Conjecture H of \cite{chm}) For sufficiently large $n$, there
exists a dominant rational map $h_n:X^n_B \dashrightarrow W_n$ where $W_n$ is a
variety of general type, and where the restriction of $h_n$ to the general
fiber $(X_b)^n$ is generically finite.
\end{conj}
This conjecture is known for curves and surfaces:
\begin{thh} (Correlation theorem of \cite{chm}) Conjecture H holds when $X\rightarrow
B$ is a family of curves of genus $>1$. \end{thh}
\begin{thh} (Correlation theorem of \cite{hassett}) Conjecture H holds when
$X\rightarrow B$ is a family of surfaces of general type . \end{thh}
Using their Theorem H 1, and Lemma 1.1 of \cite{chm}, Caporaso
et al. have shown that the weak Lang conjecture implies a uniform bound on the
number of rational points on curves (Uniform bound theorem, \cite{chm} Theorem
1.1).
It should be noted that the proofs of theorems H 1 and H 2 give a bit more:
they describe a natural dominant rational map $X^n_B\rightarrow W$.
For
the case of curves, if $B_0$ is the image of $B$ in the moduli space, $
\mbox{ \bf M}_g$, then for sufficiently large $n$ the inverse image $B_n\subset
{\mbox{ \bf M}}_{g,n}$ in the moduli space of $n$-pointed curves is a variety of general
type. Therefore the moduli map $X^n_B \dashrightarrow B_n\subset
\mbox{ \bf M}_{g,n}$ satisfies the requirements. A similar construction works for
surfaces of general type, and one may ask whether this should hold in general.
It is convenient to make the following definitions when discussing Lang's
conjectures: \\
{\bf Definition:} {\em 1. A variety $X/K$ is said to be a {\bf Lang variety} if
there is a
dominant rational map $X_{\overline{K}} \dashrightarrow W$, where $W$ is a
positive
dimensional variety of general type.
2. A positive dimensional variety $X$ is said to be {\bf geometrically
mordellic} (In short GeM) if $X_{\overline{K}}$ does not
contain subvarieties which are not of general type.}
In \cite{lang3}, in the course of stating even more far reaching conjectures,
Lang defined by a notion of {\em algebraically hyperbolic}
varieties which is very similar, and conjecturally the same as that of GeM
varieties. I chose to use a different terminology here, to avoid confusion.
Note that the weak Lang conjecture directly implies that the rational points on
a Lang variety over a number field are not Zariski dense, and that there are
only finitely many rational points over a number field on a GeM variety.
\subsection{Summary of results}
An indicated in \cite{chm} \S 6, Conjecture H together with Lang's conjectures
should have very strong implications for counting rational points on varieties
of general type, similar to the uniform bound theorem of \cite{chm} . Here we will
prove the following basic result:
\begin{th}\label{unif} Assume that the weak Lang conjecture as well as
conjecture H hold.
Let $X \rightarrow B$ be a family of GeM varieties over a number field $K$ (or any
finitely generated field over $Q$). Then there is a uniform bound on $\sharp
X_b(K)$.
\end{th}
One may refine this theorem for arbitrary families of varieties of
general type, obtaining a bound on the number of points which do not lie in
lang exceptional sets of fibers. If one assumes Lang's geometric conjecture,
one obtains a closed subset $Z(X_b)$ for every $b\in B$. A natural question
which arises in such a refinement is: how do these subsets fit together? An
answer was given in \cite{chm} , Theorem 6.1, assuming conjecture H as well: the
varieties $Z(X)$ are uniformly bounded. We will
show that, using results of Viehweg, one does not need to assume conjecture H:
\begin{th}\label{Z(X)}(Compare \cite{chm} , Theorem 6.1) Assume that the geometric
Lang conjecture holds. Let $X\rightarrow B$ be a family of varieties of general type.
Then there is a proper closed subvariety $\tilde{Z}\subset X$ such that for any
$b\in B$ we have $Z(X_b)\subset \tilde{Z}$.
\end{th}
Using theorem \ref{Z(X)}, we can apply theorem \ref{unif} to any family $X\rightarrow
B$ of varieties of general type, assuming that the geometric Lang conjecture
holds: we can bound the rational points in the complement of $\tilde{Z}$.
We will apply our theorem \ref{unif} in various natural cases. An immediate but
rather surprising application is the following theorem:
\begin{th}\label{unideg} Assume that the weak Lang conjecture as well as
conjecture H hold.
Let $X \rightarrow B$ be a family of GeM varieties over a field $K$ finitely generated
over $Q$. Fix a number $d$. Then there is a uniform bound $N_d$ such that for
any field extension $L$ of $K$ of degree $d$ and every $b\in B(L)$ we have
$\sharp X_b(L)<N_d$.
\end{th}
As a corollary, we see that Lang's conjecture together with conjecture H
imply the existence of a bound on the number of points on curves of fixed
genus $g$ over a number field $K$ which depends only on the degree of the
number field.
These results have natural analogues for function fields. We will state a few
of these, notably:
\begin{th}\label{unigon} Assume that Lang's conjecture for function fields
holds. Fix an integer $g>1$.
Then there is an integer $N(g)$ such that for any generically smooth
fibration of
curves $C\rightarrow D$ where the fiber has genus $g$ and the base is hyperelliptic
curve, there are at most $N$ non-constant sections $s:D\rightarrow C$.
\end{th}
We remind the reader that the {\em gonality } of a curve $D$ is the minimal
degree of a nonconstant rational function on $D$ (so a curve of gonality 2 is
hyperelliptic). One expects the above theorem to be generalized to the
situation where ``hyperelliptic curve'' is replaced by ``curve of gonality
$\leq d$'' for fixed $d$.
\section{Proof of theorem \ref{unif}}
\subsection{Preliminaries}
Throughout this subsection {\bf we assume that conjecture H holds}, and the
base field is algebraically closed.
Observe that a positive dimensional subvariety of an GeM variety is GeM; and
the normalization of an GeM variety is GeM. Note also that a variety dominating
a Lang variety is a Lang variety as well.
\begin{prp} Let $X \rightarrow B$ be a family of GeM varieties. Let $F \subset X$ be a
reduced subscheme such that every component of $F$ dominating $B$ has positive
fiber dimension. Then for $n$
sufficiently large, every component of the fibered power $F^n_B$ which
dominates $B$ is a Lang variety.
\end{prp}
The proof will use the following lemmas:
\begin{lem} Let $X\rightarrow B$ and $F$ be as above, and assume that the general
fiber of $F\rightarrow B$ is irreducible. Then for $n$ sufficiently large, the
dominant component of $F^n_B$ is a Lang variety.
\end{lem}
{\bf Proof.} Apply conjecture $H$ to $F \rightarrow B$, using the fact that the fibers
of $F$ are of general type.
\begin{lem} Let $X\rightarrow B$ and $F$ be as in the proposition, with $F$
irreducible. Then for $n$ sufficiently large, every component of the fibered
power $F^n_B$ which dominates $B$ is a Lang variety.
\end{lem}
{\bf Proof.} Let $\tilde{F}$ be the normalization of $F$, and let $\tilde{F}
\rightarrow \tilde{B} \rightarrow B$ be the Stein factorization. Denote by $c$ the degree of
$\tilde{B}$ over $B$. Let $G \subset \tilde{F}^n_B$ be a dominant component.
Then $G$ parametrizes $n$-tuples of points in the fibers of $\tilde{F}$ over
$B$, and since $G$ is irreducible, there is a decomposition $\displaystyle
\{1,\ldots,n\} = \cup_{i=1}^c J_i$ and $G$ surjects onto the dominant component
of $\tilde{F}^{J_i}_{\tilde{B}}$. At least one of $J_i$ has at least $n/c$
elements. Using lemma 1 applied to $\tilde{F}\rightarrow \tilde{B}$, we see
that for $n/c$ large enough $G$ is a Lang variety.
{\bf Proof of proposition. } Let $F= F_1 \cup\ldots \cup F_m$ be the
decomposition into irreducible components. Let $G$ be a dominant component of
$F^n_B$. Then $G$ dominates $(F_1)^{n_1}_B \times_B
\cdots\times_B(F_m)^{n_m}_B$. For at least one $i$ we have $n_i > n/m$, so
applying the previous
lemma we obtain that $G$ is a Lang variety.
\subsection{Prolongable points}
We return to the setup in theorem \ref{unif}.
{\bf Definition.} 1. A point $x_n = (P_1,\ldots,P_n)\in X^n_B(K) $ is said to
be off diagonal if for any $1\leq i< j\leq n$ we have $P_i\neq P_j$. We extend
this for $n=0$ trivially by agreeing that any point of $B(K)$ is off diagonal.
2. Let $m>n$. An off diagonal point $x_n$ is said to be $m$-prolongable if
there is an off-diagonal $x_m\in X^m_B(K)$ whose first $n$ coordinates give
$x_n$.
Let $E_n^{(m)}$ be the set of $m$-prolongable points on $X^n_B$, and let
$F_n^{(m)}$ be the Zariski closure. Let $ F_n = \displaystyle\cap_{m>n}
F_n^{(m)}$. By the Noetherian property of the Zariski topology we have $F_n =
F_n^{(m)}$ for some $m$.
All we need to show is $F_n = \emptyset$ for some $n$.
\begin{lem} We have a surjection $F_{n+1} \rightarrow F_n$.
\end{lem}
{\bf Proof.} The set $E_{n+1}^{(m)}$ surjects to $E_n^{(m)}$ for any $m>n+1$.
\begin{lem} Every fiber of $F_{n+1} \rightarrow F_n$ is positive dimensional.
\end{lem}
{\bf Proof.} Suppose that over an open set in $F_n$ the degree of the map is
$d$. Then $E_n^{(n+d+1)}$ cannot be dense in $F_n$: if $(y_1,\ldots,y_{n+d+1})$
is an off diagonal prolongation of $(y_1,\ldots,y_{n+d+1})\in E_n^{(n+d+1)}$,
then for $n+1\leq j\leq n+d+1$ we have that the points $(y_1,\ldots,y_{n},
y_j)\in E_{n+1}^{(n+d+1)}$ are distinct, therefore the degree of the map is at
least $d+1$.
\subsection{Proof of theorem.} We show by induction on $i$ that for any $n$ and
$i$ the dimension of any fiber of $F_{n+1}\rightarrow F_n$ is at least $i+1$.
Lemma 4 shows this for $i=0$. Assume it holds true for $i-1$, let $n\geq 0$ and
let $G$ be a component of $F_{n}$, such that the fiber dimension of $F_{n+1}$
over $G$ is $i$. Applying the inductive assumption to each $F_{n+j+1}\rightarrow
F_{n+j}$, we have that the dimension of every fiber of $F_{n+k}$ over
$F_n$ is at least $ik$. On the other hand, $F_{n+k}$ is a subscheme of the
fibered power $(F_{n+1})^k_{F_n}$, so over $G$ it has fiber dimension precisely
$ik$. Therefore
there exists a component $H_k$ of $F_{n+k}$ dominant over $G$ of fiber
dimension
$ik$, which is therefore identified as a dominant component of the fibered
power $(F_{n+1})^k_{F_n}$. By proposition 1, for $k$ sufficiently large we
have that $H_k$ is a
Lang variety. Lang's conjecture implies that $H_k(K)$ is not dense in $K$,
contradicting the definition of $F_{n+k}$. \qed
\section{A few refinements and applications in arithmetic and geometry}
\subsection{Proof of Theorem \ref{Z(X)}}
Assume that $X\rightarrow B$ is a family of varieties of general type. By Hironaka's
desingularization theorem, we may assume that $B$ is a smooth projective
variety. Let $L$ be an
ample line bundle on $B$, let $n>>0$ be a sufficiently large integer and let
$H$ be a smooth
divisor of $L^{\otimes n}$. Let $B_1\rightarrow B$ be the cyclic cover ramified to
order $n$ along $H$. Then by adjunction, $B_1$ is a variety of general type.
Let $X_1\rightarrow X$ be the pullback of $X$ to $B_1$. By the main theorem (Satz III)
of \cite{viehweg}, the variety $X_1$ is of general type. Assuming the geometric
Lang conjecture, Let $Z_1(X_1)$ be the Langian exceptional set. Let
$\tilde{Z}$ be the image of $Z_1(X_1)$ in $X$. Then for any $b\in B$, we have
by definition that $Z(X_b)\subset \tilde{Z}$. \qed
It has been noted in \cite{chm} that Viehweg's work goes a long way towards proving
conjecture H. It is therefore not surprising that it may be used on occasion to
replace the assumption of conjecture H.
\subsection{Uniformity in terms of the degree of an extension}
Let $X \rightarrow B$ be a family of GeM varieties over $K$. Assuming the conjectures,
theorem 1 gave us a uniform bound on the number of rational points over finite
extension fields in the fibers. We will now see that this in fact implies a
much stronger result, namely our theorem \ref{unideg}: the uniform bound only
depends on the degree of the field extension.
{\bf Proof of theorem \ref{unideg}:}
for $n=1$ or $2$, Let $Y_n = \mbox{ \rm Sym}^d(X^n_B)$, and $Y_0 =\mbox{ \rm Sym}^d(B)$.
Then we have natural maps $p_n:Y_n \rightarrow Y_{n-1}$. Let $\Gamma$
be the branch locus of the quotient map $X^d \rightarrow Y_1$, namely the set of
points which are fixed by some permutation. If
$P\not\in \Gamma$ then $p_2^{-1}(P)$ is a GeM variety, isomorphic over
$\overline{K}$ to the product of $d$ fibers of $X$. Denote $Y_1' = Y_1\setminus
\Gamma_1$, and $Y_2'=p_2^{-1}Y_1'$. Then $Y_2'\rightarrow Y_1'$ is a family of GeM
varieties, and by Theorem 1 we have a bound on the cardinality of
$(Y_2')_y(K)$ uniform over $y\in Y_1'(K)$.
By induction, it suffices to bound the number of points in
$X_b(L)$ over any field $L$ of degree $d$ over $K$, which are
defined over $L$ but not over any intermediate field. If
$\sigma_1,\ldots,\sigma_d$ are the distinct embeddings of $L$ in
$\overline{K}$,
and $P\in X_b(L)$ not defined over an intermediate field, then the points
$\sigma_i(P)\in X_{\sigma_i(b)}(\sigma_i(K))\subset
X(\overline{K})$ are distinct. If $(P_1,P_2)\in X_B^2(L)$ is a
pair of such points, then the Galois orbit $\{\sigma_i(P_1,P_2),
i=1,\ldots,d\}$ is Galois stable, therefore it gives rise to a point in
$Y_2(K)$. This point has the further property that its image in $Y_1$ does not
lie in $\Gamma_1$, so it gives rise to a point in $Y'_2(K)$. The
previous paragraph shows that the number of points on a fiber is bounded. \qed
Applying theorem 3 where $X\rightarrow B$ is the universal family over the Hilbert
scheme of 3-canonical
curves of genus $g$ (as in \cite{chm} , \S\S 1.2), we obtain the following:
\begin{cor} Assume that the weak Lang conjecture as well as conjecture H hold.
Fix integers $d, g>1$ and a number field $K$. Then there is a uniform bound
$N_d$ such that for
any field extension $L$ of $K$ of degree $d$ and every curve $C$ of genus $g$
over $L$ we have $\sharp C(L)<N_d$.
\end{cor}
We remark that in the cases of degrees $d\leq 3$ one does not need to assume
conjecture H: this was proven in \cite{abr}, using the fact that conjecture H
holds for families of curves or surfaces. A similar result is being
worked out by P. Pacelli for arbitrary $d$.
Here is a special case: let $f(x)\in
{\Bbb Q}(x)$ be a polynomial of degree $>4$ with distinct complex roots. Then,
assuming the weak Lang conjecture,
the number of rational points over any quadratic field on the curve $C: y^2 =
f(x)$ is bounded uniformly. We remark that, if $\deg f>6$, this in
fact may be
deduced using a combination of \cite{chm} and a theorem of Vojta \cite{vojta}
which says that all but finitely many quadratic points on $C$ have rational $x$
coordinate. One then applies \cite{chm} which gives a uniform bound on the
rational points on the twists $ty^2 = f(x)$.
Following the suggestion of \cite{chm}, \S 6 one can apply Theorem 1 to
symmetric powers of curves. Since conjecture H is known for surfaces, one
obtains the following (stated without proof in \cite{chm}, Theorem 6.2):
\begin{cor}(Compare \cite{chm} , Theorem 6.2) Assume that the weak Lang conjecture
holds. Fix a number field $K$.
Then there is a uniform bound $N$ for the number of quadratic
points on any nonhyperelliptic, non-bielliptic curve $C$ of genus $g$
over $K$.
\end{cor}
Similarly, it was shown in \cite{ah}, lemma 1 that if the gonality of a curve
$C$ is $>2d$ then $\mbox{ \rm Sym}^d(C)$ is GeM. Recall that a closed point $P$ on $C$ is
said to be
of degree $d$ over $K$ if $[K(P):K]=d$. We deduce the following:
\begin{cor} Assume that the weak Lang conjecture holds. Fix a number field $K$
and an integer $d$.
Then there is a uniform bound $N$ for the number of
points of degree $d$ over $K$ on any curve $C$ of genus $g$ and gonality $>2d$
over $K$.
\end{cor}
\subsection{The geometric case}
One can use the same methods using Lang's conjecture for function fields of
characteristic 0, say over $\Bbb C$. Given a fibration
$X\rightarrow B$ where the generic fiber is a variety of general type, a rational
point
$s\in X(K_B)$ over the function field of $B$
is called {\em constant} if $X$ is birational to a product $X_0\times B$ and
$s$ corresponds to a point on $X_0$. Lang's conjecture for function fields
says
that the non-constant points are not Zariski dense.
In this section we will restrict attention to the case where the base is the
projective line ${\Bbb{P}^1}$. We will only assume the following statement: if $X$ is
a variety of general type, then the rational curves in $X$ are not Zariski
dense. It is easy to see that this statement in fact follows from the
geometric Lang conjecture, as well as from Lang's conjecture for function
fields.
We would like to apply this conjecture to obtain geometric uniformity results.
One has to be careful here, since the conjecture does not
apply to Lang varieties, and one has to use a variety of general type directly.
As stated in the introduction, if $X\rightarrow B$ is a family of curves of genus $>1$
the appropriate variety $W$ of general type dominated by $X^n_B$ is identified
in \cite{chm} as the image
$B_n\subset{\mbox{ \bf M}}_{g,n}$ of $X^n_B$ by the moduli map. We use this in the proof
of the following proposition:
\begin{prp}\label{unigeom} Assume that Lang's conjecture for function fields
holds. Fix
an integer $g>1$.
Then there is a bound $N$ such that for any generically smooth family of
curves $C\rightarrow \Bbb{P}^1$ of genus $g$ there are at most $N$ non-constant
sections $s:\Bbb{P}^1\rightarrow C$.
\end{prp}
{\bf Proof.} First note that if $s:\Bbb{P}^1\rightarrow C$ is a nonconstant section
whose image in $ \mbox{ \bf M}_{g,1}$ is a point, then $s$ becomes a
constant section after
a finite base change $D\rightarrow \Bbb{P}^1$. This implies that $s$ is fixed by
a nontrivial automorphism of $C$, and the number of such points is bounded
in terms of $g$.
Therefore it suffices to bound the number of sections whose image in
$\mbox{ \bf M}_{g,1}$
is non-constant. We will call such sections {\em strictly non-constant}.
Let $B_0\subset \mbox{ \bf M}_g$ be a closed subvariety, and choose $n$ such
that $B_n\subset \mbox{ \bf M}_{g,n}$ is of general type. If a family $C\rightarrow {\Bbb{P}^1}$ has
moduli in $B_0$, then for any $n$-tuple of strictly
non-constant
sections $s_i:{\Bbb{P}^1}\rightarrow C$, we obtain a non-constant rational map ${\Bbb{P}^1} \rightarrow
B_n$. Let ${F} \subset
B_n$ be the Zariski closure of the images of the collection of non-constant
rational maps obtained this way.
Since $B_n$ is of general type, Lang's
conjecture implies that $F\neq B_n$. Applying
lemma 1.1 of \cite{chm} we obtain that there is an closed subset set $F_0\subset B_0$
and an
integer $N$ such that, given a family of curve $C\rightarrow{\Bbb{P}^1}$ such that the
rational image
of ${\Bbb{P}^1}$ in $\mbox{ \bf M}_g$ lies in $B_0$ but not in $F_0$, there are at most $N$
non-constant sections of $C$. Noetherian induction gives the theorem. \qed
Choosing a coordinate $t$ on $ {\Bbb P}^1$ we can pull back the curve $C$
along the map ${\Bbb{P}^1}\rightarrow {\Bbb{P}^1}$ obtained by
taking $n$-th roots of $t$. Let ${\Bbb C}(t^{1/\infty}) = {\Bbb C}(\{t^{1/n},
n\geq 1\})$, the field obtained by adjoining all roots of $t$. If one restricts
attention to non-isotrivial curves, one obtains the following amusing result
(suggested to the author by Felipe Voloch):
\begin{cor} Assume that the Lang conjecture for function fields holds. Fix an
integer $g>1$.
Then there is a bound $N$ such that for any smooth
nonisotrivial
curve $C$ over ${\Bbb C}(t)$ of genus $g$ there are at most $N$ points in
$C({\Bbb C}(t^{1/\infty}))$.
\end{cor}
One can also try to prove uniformity results analogous to theorem 3. Using the
results in \cite{abr} we can refine proposition \ref{unigeom} and obtain
theorem \ref{unigon}.
{\bf Proof of theorem \ref{unigon}.} The proof is a slight modification of the
theorem of \cite{abr}, keeping track of the dominant map to a variety of
general type.
As in the proof of theorem 3, it suffices
to look at sections $s:D\rightarrow C$ which are not pullbacks of sections of a
family
over ${\Bbb{P}^1}$.
In an analogous way to the
proof of theorem \ref{unif}, we say that an
$n$-tuple of distinct, strictly non-constant sections is $m$-prolongable if
it may be prolonged to an $m$-tuple of distinct, strictly non-constant
sections, none of which being the pullback from a family over ${\Bbb{P}^1}$. Any
$n$-tuple of distinct
sections $s_i:D\rightarrow C$ over a hyperelliptic curve $D$ gives rise to a rational
map
${\Bbb{P}^1}\rightarrow \mbox{ \rm Sym}^2(\mbox{ \bf M}_{g,n})$. We define $F_n^{(m)}$ to be the
closure in $\mbox{ \rm Sym}^2(\mbox{ \bf M}_{g,n})$ of the images of $m$-prolongable sections, and
$F_n = \cap_{m>n} F_n^{(m)}$.
As in Lemma 1, we have that the relative
dimension of
any fiber of $F_{n+1}\rightarrow F_n$ is positive. We have two cases to consider:
either for high $n$ there is a component of $F_{n+1}$ having fiber dimension 1
over $F_n$, or for all $n$ the fiber dimension is everywhere 2.
In case the fiber dimension is 1,
we will see that there is a
component of $F_{n+k}$ which is a variety of general type.
Assuming Lang's conjecture for function fields this contradicts the fact that
the images of non-constant sections are dense. Fix a
general fiber $f$ of $F_{n+1}$ over $F_{n}$. The curve $f$ lies inside a
surface isomorphic to the product of two curves
$C_{b_1} \times C_{b_2}$. By the definition of $m$-prolongable sections, and
analogously to lemma 1, we obtain that there is a component $f'$ of $f$ which
maps surjectively to both $C_{b_1}$ and $C_{b_2}$. Therefore as either $b_1$ or
$b_2$ moves in $B_0$, the moduli of $f'$ move as well.
Let $F'$ be a component
of
$F_{n+1}$ whose fibers have this property. If we follow the proof of
proposition 1 and use the moduli description of the dominant map to a variety
of general type $m:(F')^k_{F_n}\rightarrow W$, we see that if $E$ is a general curve
in
$(F')^k_{F_n}$ lying in a fiber of $m$, then $E$ projects to a point
in $B_0$; moreover, by the definition of prolongable points, $E$ projects to an
off diagonal point in some $(F')^l_{F_n}$. But the fibers over off-diagonal
points are GeM varieties, therefore the general fiber of the map $m$ is of
general type. By the main theorem of \cite{viehweg},
$(F')^k_{F_n}$ is itself a variety of general type, and therefore $F_{n+k}$ has
a component of general type, contradicting Lang's conjecture.
In case of fiber dimension 2, we
use
proposition 1 of \cite{abr}: let $B\subset \mbox{ \rm Sym}^2(\mbox{ \bf M}_g)$. Then for high $n$,
the inverse image
$B_n\subset \mbox{ \rm Sym}^2(\mbox{ \bf M}_{g,n})$ of $B$ is a variety of general type. Since the
images of non-constant sections are dense in $F_n$, this again
contradicts Lang's conjecture. \qed
If one restricts attentions to trivial fibrations, one obtains as an immediate
corollary:
\begin{cor} Assume that the Lang conjecture for function fields holds. Fix an
integer $g>1$.
Then there is an integer $N$ such that for any
curve $C$ of genus $g$ and any hyperelliptic curve $D$
there are at most $N$ non-trivial morphisms $f:D\rightarrow C$.
\end{cor}
It should be noted that the theory of Hilbert schemes gives the existence of a
bound depending on the genus of $D$, which is clearly not as strong. As in the
arithmetic case, I expect that work in progress of Pacelli should generalize
these results to the case where $D$ is $d$-gonal, for fixed $d$.
|
1996-02-27T06:25:21 | 9505 | alg-geom/9505029 | en | https://arxiv.org/abs/alg-geom/9505029 | [
"alg-geom",
"hep-th",
"math.AG"
] | alg-geom/9505029 | Teleman | Ch. Okonek and A. Teleman | Quaternionic Monopoles | LaTeX, 35 pages | null | 10.1007/BF02099718 | null | null | We present the simplest non-abelian version of Seiberg-Witten theory:
Quaternionic monopoles. These monopoles are associated with
Spin^h(4)-structures on 4-manifolds and form finite-dimensional moduli spaces.
On a Kahler surface the quaternionic monopole equations decouple and lead to
the projective vortex equation for holomorphic pairs. This vortex equation
comes from a moment map and gives rise to a new complex-geometric stability
concept. The moduli spaces of quaternionic monopoles on Kahler surfaces have
two closed subspaces, both naturally isomorphic with moduli spaces of
canonically stable holomorphic pairs. These components intersect along
Donaldsons instanton space and can be compactified with Seiberg-Witten moduli
spaces. This should provide a link between the two corresponding theories.
Notes: To appear in CMP The revised version contains more details concerning
the Uhlenbeck compactfication of the moduli space of quaternionic monopoles,
and possible applications are discussed. Attention ! Due to an ununderstandable
mistake, the duke server had replaced all the symbols "=" by "=3D" in the
tex-file of the revised version we sent on February, the 2-nd. The command
"\def{\ad}" had also been damaged !
| [
{
"version": "v1",
"created": "Sat, 27 May 1995 19:21:11 GMT"
},
{
"version": "v2",
"created": "Mon, 10 Jul 1995 15:18:41 GMT"
},
{
"version": "v3",
"created": "Fri, 2 Feb 1996 10:32:57 GMT"
},
{
"version": "v4",
"created": "Mon, 5 Feb 1996 13:50:55 GMT"
}
] | 2009-10-28T00:00:00 | [
[
"Okonek",
"Ch.",
""
],
[
"Teleman",
"A.",
""
]
] | alg-geom | \section{Introduction}
Recently, Seiberg and Witten [W] introduced new 4-manifold invariants,
essentially by
counting solutions of the monopole equations. The new invariants have
already found nice
applications, like e.g. in the proof of the Thom conjecture [KM] or in a
short proof of the Van
de Ven conjecture [OT2]. In this paper we introduce and study the simplest
and the most
natural non-abelian version of the Seiberg-Witten monopoles, the
quaternionic monopoles.
Let $(X,g)$ be an oriented Riemannian manifold of dimension 4. The
structure group $SO(4)$
has as natural extension the quaternionic spinor group
$Spin^h(4):=Spin(4)\times_{{\Bbb Z}_2}Sp(1)$:
$$1\longrightarrow Sp(1)\longrightarrow Spin^h(4)\longrightarrow SO(4)\longrightarrow 1 \ .$$
The projection onto the second factor $Sp(1)=SU(2)$ induces a "determinant map"
$\delta:Spin^h(4)\longrightarrow PU(2)$.
A $Spin^h(4)$-structure on $(X,g)$ consists of a $Spin^h(4)$-bundle over
$X$ and an
isomorphism of its $Sp(1)$-quotient with the (oriented) orthonormal frame
bundle of
$(X,g)$. Given a $Spin^h(4)$-structure on $X$, one has a one-one
correspondence between
$Spin^h$-connections projecting onto the Levi-Civita connection and
$PU(2)$-connections in
the associated "determinant" $PU(2)$-bundle. The quaternionic monopole
equations are:
$$\left\{\begin{array}{ccc}\hskip 4pt{\not}{D}_{A}\Psi&=&0\ \ \ \ \\
\Gamma(F_{A}^+)&=&(\Psi\bar\Psi)_0 \ \ ,
\end{array}\right.\eqno{ }$$
where $A$ is a $PU(2)$-connection in the "determinant" of the
$Spin^h(4)$-structure and
$\hskip 4pt{\not}{D}_A$ the induced Dirac operator; $\Psi$ is a positive quaternionic
half-spinor. The Dirac
operator satisfies the crucial Weitzenb\"ock formula :
$$\hskip 4pt{\not}{D}_A^2=\nabla_{\hat A}^*\nabla_{\hat A}+\Gamma(F_A)+\frac{s}{4}{\rm id}$$
It can be used to show that the solutions of the quaternionic monopole
equations are
the absolute minima of a certain functional, just like in the
$Spin^c(4)$-case [JPW].
The moduli space of quaternionic monopoles associated with a fixed
$Spin^h(4)$-structure
${\fam\meuffam\tenmeuf h}$ is a real analytic space of virtual dimension
$$m_{\fam\meuffam\tenmeuf h}=-\frac{1}{2}(3p_1+3e+4\sigma)\ .$$
Here $p_1$ is the first Pontrjagin class of
the determinant, $e$ and $\sigma$ denote the Euler characteristic and the
signature of $X$.
Note that $m_{\fam\meuffam\tenmeuf h}$ is an even integer iff $X$ admits an almost complex
structure.
The moduli spaces of quaternionic monopoles contain the Donaldson
instanton moduli spaces as
well as the classical Seiberg-Witten moduli spaces, which suggests that
they could provide a
method of comparing the two theories. We study the analytic structure
around the Donaldson
moduli space.
Much more can be said if the holonomy of $(X,g)$ reduces to $U(2)$, i.e. if
$(X,g)$ is a
K\"ahler surface. In this case we use the canonical $Spin^c(4)$-structure with
$\Sigma^+=\Lambda^{00}\oplus\Lambda^{02}$ and $\Sigma^-=\Lambda^{01}$ as spinor
bundles. The data of a $Spin^h(4)$-structure ${{\germ h}}$ in $(X,g)$ is then
equivalent to the data
of a Hermitian 2-bundle $E$ with $\det E=\Lambda^{02}$. The determinant
$\delta({\germ h})$ coincides with the $PU(2)$-bundle $P(E)$ associated with $E$.
A positive spinor
can be written as
$\Psi=\varphi+\alpha$, where
$\varphi\in A^0(E^{\vee})$ and $\alpha\in A^{02}(E^{\vee})$ are
$E^{\vee}$-valued forms.
To give a $PU(2)$-connection in $P(E)$ means to give a $U(2)$-connection
in $E$ inducing the
Chern connection in $\Lambda^{02}$, or equivalently, a $U(2)$-connection $C$ in
$E^{\vee}$
inducing the Chern connection in $K_X=\Lambda^{20}$. A pair
$(C,\varphi+\alpha)$ solves the
quaternionic monopole equation iff $C$ is a connection of type $(1,1)$, one
of $\alpha$ or
$\varphi$ vanishes while the other is $\bar\partial_C$-holomorphic, and a
certain projective
vortex equation is satisfied. This shows that in the K\"ahler case the
moduli space decomposes as
a union of two Zariski closed subspaces intersecting along the Donaldson
locus. The two subspaces
are interchanged by a natural real analytic involution, whose fixed point
set is precisely the
Donaldson moduli space.
The projective vortex equation comes from a moment map which corresponds to
a new stability
concept for pairs
$({\cal E},\varphi)$ consisting of a holomorphic bundle ${\cal E}$ with
canonical determinant
$\det{\cal E}={\cal K}_X$ and a holomorphic section $\varphi$. We call such
a pair canonically
stable iff either ${\cal E}$ is stable, or $\varphi\ne 0$ and the
divisorial component $D_\varphi$
of the zero locus satisfies the inequality
$$c_1\left({\cal O}_X(D_\varphi )^{\otimes2}\otimes K_X^{\vee}\right) \cup
[\omega_g]<0 \ \ .$$
Our main result identifies the moduli spaces of irreducible quaternionic
monopoles on a
K\"ahler surface with the algebro-geometric moduli space of canonically
stable pairs.
In the algebraic case, moduli spaces of quaternionic monopoles can easily
be computed using our
main result (Theorem 7.3) and Lemma 5.5. The moduli spaces may have
several components:
Every component contains a Zariski open subset which is a holomorphic
${\Bbb C}^*$-bundle. For some components, this ${\Bbb C}^*$-bundle consists only of
pairs $({\cal E},\varphi)$ with
${\cal E}$ stable as a bundle; components of this type can be obtained by
compactifying the
corresponding ${\Bbb C}^*$-bundle with a Donaldson moduli space at infinity. In
the other direction,
the component is not compact, but has a {\sl natural compactification}
obtained by adding spaces
associated with Seiberg-Witten moduli spaces. The other components can
also be
naturally compactified by using Seiberg-Witten moduli spaces in
both directions.
This compactification process, as well as the corresponding differential
geometric
interpretation will be the subject of a later paper. \footnote{ After
having completed our
results we received a manuscript by Labastida and Marino [LM] in which
related ideas are
proposed from a physical point of view, and physical implications are
discussed }
\section{$Spin^h$-structures}
The quaternionic spinor group is defined as
$$Spin^h:=Spin\times_{{\Bbb Z}/2}Sp(1)=Spin\times_{{\Bbb Z}/2}SU(2)\ ,$$
and fits in the exact
sequences
$$\begin{array}{c}
1\longrightarrow Sp(1)\longrightarrow Spin^h\stackrel{\pi}{\longrightarrow}SO\longrightarrow 1\\
1\longrightarrow Spin\longrightarrow Spin^h\stackrel{\delta}{\longrightarrow}\ PU(2)\longrightarrow 1\end{array}
\eqno{(1)}$$
These can be combined in the sequence
$$1\longrightarrow{\Bbb Z}/2\longrightarrow Spin^h\textmap{(\pi,\delta)} SO\times PU(2)\longrightarrow 1\eqno{(2)}$$
In dimension 4, $Spin^h(4)$ has a simple description, coming from the splitting
$Spin(4)=SU(2)\times SU(2)$:
$$Spin^h(4)=\qmod{SU(2)\times SU(2) \times SU(2)}{{\Bbb Z}/2}$$
with ${\Bbb Z}/2=\langle(-{\rm id},-{\rm id},-{\rm id})\rangle$. There is another useful way to
think of $Spin^h(4)$:
let $G$ be the group
$$G:=\{(a,b,c)\in U(2)\times U(2)\times U(2)|\ \det a=\det b= \det c\}\ .$$
One has an obvious isomorphism
$Spin^h(4)=\qmod{G}{S^1}\ $ , and a commutative diagram with exact rows
$$\matrix{1\rightarrow&{\Bbb Z}_2&\longrightarrow &SU(2)\times SU(2)\times SU(2)&\longrightarrow
&Spin^h(4)&\rightarrow 1\cr
&\downarrow& &\downarrow&&\parallel&\cr
1\rightarrow &S^1&\longrightarrow &G&\longrightarrow &Spin^h(4)&\rightarrow 1\cr
}\eqno{(3)}$$
\begin{dt} Let $P$ be a principal $SO$-bundle over a space $X$. A
$Spin^h$-structure in $P$ is
a pair consisting of a $Spin^h$ bundle $P^h$ and an isomorphism $P\simeq
P^h\times_\pi SO$.
The $PU(2)$-bundle associated with a $Spin^h$-structure is the bundle
$P^h\times_\delta
PU(2)$.
\end{dt}
\begin{lm} A principal $SO$-bundle admits a $Spin^h$-structure iff there exists
a
$PU(2)$-bundle with the same second Stiefel-Whitney class.
\end{lm}
{\bf Proof: } This follows from the cohomology sequence
$$\longrightarrow H^1(X,\underline{Spin^h})\longrightarrow
H^1(X,\underline{{\phantom(}SO{\phantom)}}\times\underline{PU(2)})\textmap{\
beta}
H^2(X,{\Bbb Z}/2)$$
associated to (2), since the connecting homomorphism $\beta$ is given by
taking the sum of
the second Stiefel-Whitney classes of the two factors.
\hfill\vrule height6pt width6pt depth0pt \bigskip
In this paper we will only use $Spin^h$-structures in $SO(4)$-bundles
whose second Stiefel
Whitney class admit {\sl integral} lifts. Then we have:
\begin{lm}
Let $P$ be a principal $SO(4)$-bundle whose second Stiefel-Whitney class
$w_2(P)$ is the
reduction of an integral class.
Isomorphism classes of $Spin^h(4)$-structures in $P$ are in 1-1
correspondence with
equivalence classes of triples consisting of a $Spin^c(4)$-structure
$\qmod{P^c}{S^1}\simeq P$ in
$P$, a
$U(2)$-bundle $E$, and an isomorphism $\det P^c\simeq\det E$, where two
triples are
equivalent if they can be obtained from each other by tensoring with an
$S^1$-bundle.
\end{lm}
{\bf Proof: } The cohomology sequence associated with the second row in (3) shows that
$Spin^h$-structures in bundles whose second Stiefel-Whitney classes admit
integral lifts are
given by $G$-structures modulo tensoring with $S^1$-bundles. On the other
hand, to give a
$G$-structure in $P$ simply means to give a triple $(\Sigma^+,\Sigma^-,E)$
of $U(2)$-bundles
together with isomorphisms
$$\det\Sigma^+\simeq\det\Sigma^-\simeq\det E \ .$$
This is equivalent to giving a triple consisiting of a
$Spin^c(4)$-structure $\qmod{P^c}{S^1}\simeq
P$ in $P$, a $U(2)$-bundle, and an isomorphism $\det P^c\simeq\det E$.
\hfill\vrule height6pt width6pt depth0pt \bigskip
In the situation of this lemma, we get well defined vector bundles
$${\cal H}^{\pm}:=\Sigma^{\pm}\otimes E^{\vee}\ $$
depending only on the $Spin^h$-structure and
not on the chosen $G$-lifting. These spinor bundles have the following
intrinsic interpretation:
identify
$SU(2)\times_{{\Bbb Z}/2} SU(2)$ with $SO(4)$, and denote by
$$\pi_{ij}:Spin^h\longrightarrow SO(4)$$
the projections of $Spin^h(4)=\qmod{SU(2)\times SU(2)\times SU(2)}{{\Bbb Z}/2}$
onto the indicated
factors ($\pi=\pi_{12}$). Using the inclusion $SO(4)\subset SU(4)$, we can
form three
$SU(4)$-vector bundles $P^h\times_{\pi_{ij}}{\Bbb C}^4$,
$(i,j)\in\{(1,2),(1,3),(2,3)\}$.
Under the conditions of the previous lemma we have
$${\cal H}^+=P^h\times_{\pi_{13}}{\Bbb C}^4\ ,\ \ {\cal
H}^-=P^h\times_{\pi_{23}}{\Bbb C}^4\ ,\ \
\Sigma^+\otimes(\Sigma^-)^{\vee}=P^h\times_{\pi}{\Bbb C}^4 \ .$$
The $PU(2)$-bundle $P^h\times_\delta PU(2)$ associated with the
$Spin^h$-structure
$\qmod{P^c}{S^1}\simeq P$ has in this case a very simple description: it is
the projectivization
$P(E)$ of the
$U(2)$-bundle $E$.
\section{ The quaternionic monopole equations}
Let $(X,g)$ be an oriented Riemannian 4-manifold with orthonormal frame
bundle $P$. The exact
sequence (2) in the previous section shows two things: first, isomorphism
classes of
$PU(2)$-bundles with second Stiefel-Whitney class equal to $w_2(P)$ are in
1-1 correspondence with orbits of $Spin^h(4)$-structures in $P$ under the
action of
$H^1(X,{\Bbb Z} /2)$; second, $Spin^h(4)$-connections in a $Spin^h(4)$-bundle
$P^h$ which induce the
Levi-Civita connection in $P$ correspond bijectively to connections in the
associated
$PU(2)$-bundle $P^h\times_\delta PU(2)$.
Now it is well known that $w_2(P)=w_2(X)$ is always the reduction of an
integral class [HH], so
that we can think of a $Spin^h$-structure in $P$ as a triple
$(\Sigma^+,\Sigma^-,E)$ of
$U(2)$-bundles with isomorphisms $\det\Sigma^+\simeq\det\Sigma^-\simeq\det
E$ modulo
tensoring with unitary line bundles. We denote the $Spin^h(4)$-connection
corresponding to a
connection $A\in{\cal A}(P(E))$ in the associated $PU(2)$-bundle by $\hat A$.
\begin{re} Given a fixed $U(1)$-connection $c$ in $\det E$, the elements in
${\cal A}(P(E))$ can
be identified with those $U(2)$-connections in $E$, which induce the fixed
connection $c$.
\end{re}
Now view a $Spin^h(4)$-structure in $P$ as a $Spin^c(4)$-structure
$\qmod{P^c}{S^1}\simeq P$
together with a $U(2)$-bundle $E$ and an isomorphism $\det P^c\simeq\det E$
. Recall
that the choice of $\qmod{P^c}{S^1}\simeq P$ induces an isomorphism
$$\gamma:\Lambda^1\otimes{\Bbb C}\longrightarrow(\Sigma^+)^{\vee}\otimes\Sigma^-$$
which extends to a homomorphism
$$\Lambda^1\otimes{\Bbb C}\longrightarrow{\rm End}_0(\Sigma^+\oplus\Sigma^-)\ ,$$
mapping the bundle $\Lambda^1$ of real 1-forms into the bundle of
trace-free skew-Hermitian
endomorphisms. The induced homomorphism
$$\Gamma:\Lambda^2\otimes{\Bbb C}\longrightarrow {\rm End}_0(\Sigma^+\oplus\Sigma^-)$$
maps the subbundles $\Lambda^2_{\pm}\otimes{\Bbb C}$ isomorphically onto the bundles
${\rm End}_0(\Sigma^{\pm})$, and identifies $\Lambda_{\pm}$ with the trace-free,
skew-Hermitian
endomorphisms ([H], [OT1]).
\begin{dt}
Let $P^h\times_{\pi}SO(4)\simeq P$ be a ${\rm Spin}^h(4)$-structure in $P$ with
spinor bundle ${\cal
H}:={\cal H}^+\oplus{\cal H}^-$ and associated $PU(2)$-bundle $P(E)$.
Choose a connection
$A\in{\cal A}(P(E))$, and let $\hat A$ be the corresponding
$Spin^h(4)$-connection in $P^h$.
The associated Dirac opearor is defined as the composition
$$\hskip 4pt{\not}{D}_A:A^0({\cal H})\textmap{\nabla_{\hat A}} A^1({\cal
H})\textmap{\gamma} A^0({\cal
H})\ ,$$
where $\nabla_{\hat A}$ is the covariant derivative of $\hat A$ and
$\gamma$ the Clifford
multiplication.
\end{dt}
Note that the restricted operators
$$\hskip 4pt{\not}{D}_A : A^0({\cal H}^{\pm})\longrightarrow A^0({\cal H}^{\mp})$$
interchange the positive and negative half-spinors.
Let $s$ be the scalar curvature of $(X,g)$.
\begin{pr} The Dirac operator $\hskip 4pt{\not}{D}_A:A^0({\cal H})\longrightarrow A^0({\cal H})$ is an
elliptic, selfadjoint
operator whose Laplacian satisfies the Weitzenb\"ock formula
$$\hskip 4pt{\not}{D}_A^2=\nabla_{\hat A}^*\nabla_{\hat A}+\Gamma(F_A)+\frac{s}{4}id_{\cal
H}\eqno{(4)}$$
\end{pr}
{\bf Proof: } Choose a $Spin^c(4)$-structure $\qmod{P^c}{S^1}\simeq P$ and a
$S^1$-connection $c$ in the
unitary line bundle $\det P^c$. The connection $A\in {\cal A}(P(E))$lifts
to a unique
$U(2)$-connection $C$ in the bundle $E^{\vee}$ which induces the dual
connection of $c$ in $\det
E^{\vee}=(\det P^c)^{\vee}$. In [OT1] we introduced the Dirac operator
$$\hskip 4pt{\not}{D}_{C,c}:A^0(\Sigma \otimes E^{\vee}) \longrightarrow A^0(\Sigma \otimes E^{\vee})\
;$$
by construction it coincides with the operator $\hskip 4pt{\not}{D}_A:A^0({\cal H})\longrightarrow
A^0({\cal H})$, and its
Weitzenb\"ock formula reads
$$\hskip 4pt{\not}{D}_{C,c}^2=\nabla_{\hat A}^*\nabla_{\hat
A}+\Gamma(F_{C,c})+\frac{s}{4}id_{\cal H}\ ,$$
where $F_{C,c}=F_C+\frac{1}{2}F_c{\rm id}_{E^{\vee}}\in A^2( {\rm End} E^{\vee})$.
Substituting $$F_C
=\frac{1}{2}{\rm Tr}(F_C){\rm id}_{E^{\vee}}+F_A$$ and using
$\frac{1}{2}{\rm Tr}(F_C)=-\frac{1}{2}F_c$ we
get the Weitzenb\"ock formula (4) for $ \hskip 4pt{\not}{D}_A$.
\hfill\vrule height6pt width6pt depth0pt \bigskip
Consider now a section $ \Psi\in A^0({\cal H}^{\pm})$. We denote by
$$(\Psi\bar\Psi)_0\in A^0({\rm End}_0 \Sigma^{\pm}\otimes{\rm End}_0 E^{\vee})$$
the projection of $\Psi\otimes\bar\Psi\in A^0({\rm End}{\cal H}^{\pm})$ onto the
fourth summand in
the decomposition
$${\rm End}({\cal H}^{\pm})={\Bbb C}{\rm id}\oplus{\rm End}_0\Sigma^{\pm}\otimes{\rm End}_0
E^{\vee}\otimes({\rm End}_0\Sigma^{\pm}\otimes{\rm End}_0 E^{\vee})\ .$$
$(\Psi\bar\Psi)_0$ is a Hermitian endomorphism which is trace-free in both
factors.
\begin{dt}Choose a $Spin^h(4)$-structure in $P$ with spinor bundle ${\cal
H}$ and associated
$PU(2)$-bundle $P(E)$. The quaternionic monopole equations for the pair
$(A,\Psi)\in{\cal
A}(P(E))\times A^0({\cal H})$ are the following equations:
$$\left\{\begin{array}{ccc}\hskip 4pt{\not}{D}_{A}\Psi&=&0\ \ \ \ \\
\Gamma(F_{A}^+)&=&(\Psi\bar\Psi)_0 \ .\ \
\end{array}\right.\eqno{(SW^h) }$$
\end{dt}
The following result is the analog of Witten's formula in the
quaternionic case (see [W], \S 3 ):
\begin{pr} Let $\Psi\in A^0({\cal H}^+)$ be a positive half-spinor, $A\in
{\cal A}(P(E))$ a connection in $P(E)$. Then we have
$$\parallel\hskip 4pt{\not}{D}_{A}\Psi\parallel^2+
\frac{1}{2}\parallel\Gamma(F_{A}^+)-(\Psi\bar\Psi)_0\parallel^2=$$ $$=
\parallel\nabla_{\hat A}\Psi\parallel^2+
\frac{1}{2}\parallel
F_{A}^+\parallel^2+\frac{1}{2}\parallel(\Psi\bar\Psi)_0\parallel^2+
\frac{1}{4}\int\limits_X s|\Psi|^2. \eqno{(5)}$$
\end{pr}
{\bf Proof: } The pointwise inner product $(\Gamma(F_A)\Psi,\Psi)$ for a positive
half-spinor
$\Psi$ simplifies:
$(\Gamma(F_A)\Psi,\Psi)= (\Gamma(F_A^+)\Psi,\Psi)=
(\Gamma(F_A^+),(\Psi\bar\Psi)_0)$, since $\Gamma(F_A^-)$ vanishes on
$A^0({\cal H}^+)$,
and since $ \Gamma(F_A^+)$ is trace-free in both arguments.
Using the Weitzenb\"ock formula (5), we find
$$(\hskip 4pt{\not}{D}_A\Psi,\Psi)=(\nabla_{\hat A}^*\nabla_{\hat
A}\Psi,\Psi)+(\Gamma(F_A^+),(\Psi\bar\Psi)_0)+\frac{s}{4}|\Psi|^2\
,\eqno{(6)}$$
which shows that
$$(\hskip 4pt{\not}{D}_A^2\Psi,\Psi)+\frac{1}{2}|\Gamma(F_{A}^+)-(\Psi\bar\Psi)_0|^2=
(\nabla_{\hat A}^*\nabla_{\hat A}\Psi,\Psi)+
\frac{1}{2}|F_{A}^+|^2+\frac{1}{2}|(\Psi\bar\Psi)_0|^2+\frac{s}{4}|\Psi|^2
$$
The identity (5) follows by integration over $X$.
\section{Moduli spaces of quaternionic monopoles}
Let $E$ be $U(2)$-bundle with $w_2(P)\equiv \overline{c_1(E)} $ (mod 2),
and let $c$ be a
fixed $S^1$-connection in $\det E^{\vee}$. We identify ${\cal A}(P(E))$
with the space
${\cal A}_c(E^{\vee})$ of $U(2)$-connections in $E^{\vee}$ which induce the
fixed
connection in $\det E^{\vee}$, and we set:
$${\cal A}:={\cal A}_c(E^{\vee})\times A^0({\cal H}^+)$$
The natural gauge group is the group ${\cal G}$ consisting of unitary
automorphisms in
$E^{\vee}$ which induce the identity in $\det E^{\vee}$. ${\cal G}$ acts on
${\cal A}$ from
the right in a natural way. Let ${\cal A}^*\subset {\cal A}$ be the open
subset of ${\cal
A}$ consisting of pairs $(C,\Psi)$ whose stabilizer ${\cal G}_{(C,\Psi)}$
is contained in the
center ${\Bbb Z}/2=\{\pm{\rm id}_E\}$ of the gauge group.
\begin{re} A pair $(C,\Psi)$ does not belong to ${\cal A}^*$ iff $\Psi=0$
and $C$ is a
reducible connection.
\end{re}
Indeed, the isotropy group of ${\cal G}$ acting only on the first factor
${\cal A}_c(E^{\vee})$ is
the centralizer of the holonomy of $C$ in $SU(2)$. The latter is $S^1$ if
$C$ is reducible,
and ${\Bbb Z}/2$ in the irreducible case.
\hfill\vrule height6pt width6pt depth0pt \bigskip
A pair belonging to ${\cal A}^*$ will be called irreducible. Note that the
stabilizer of
\underbar{any} pair with vanishing second componenent
$\Psi$ contains
${\Bbb Z}/2$.
{}From now on we also assume that ${\cal A}$ and ${\cal G}$ are completed
with respect to
suitable Sobolev norms $L^2_k$, such that ${\cal G}$ becomes a Hilbert Lie
group acting
smoothly on ${\cal A}$. Let ${\cal B}:=\qmod{{\cal A}}{{\cal G}}$ be the
quotient, ${\cal
B}^*:=\qmod{{\cal A}^*}{{\cal G}}$, and denote the orbit-map $[\ ]:{\cal
A}\longrightarrow{\cal B}$
by $\pi$.
An element in ${\cal A}^{*}$ will be called {\sl strongly irreducible} if
its stabilizer is
trivial. Let ${\cal A}^{**}\subset{\cal A}^*$ be the subset of strongly
irreducible pairs,
and put ${\cal B}^{**}:=\qmod{{\cal A}^{**}}{{\cal G}}$.
\begin{pr} ${\cal B}$ is a Hausdorff space. ${\cal B}^{**}\subset{\cal B}$
is open and has
the structure of a differentiable Hilbert manifold. The map ${\cal
A}^{**}\longrightarrow{\cal
B}^{**}$ is a differentiable principal ${\cal G}$-bundle.
\end{pr}
{\bf Proof: } Standard, cf. [DK], [FU]. \\
Fix a point $p=(C,\Psi)\in{\cal A}$. The differential of the map ${\cal
G}\longrightarrow{\cal A}$
given by the action of ${\cal G}$ on $p$ is the map
$$
\begin{array}{cccc}D^0_{p}&:A^0(su(E^{\vee}))&\longrightarrow& A^1(su(E^{\vee}))\oplus
A^0(\Sigma^+\otimes E^{\vee})\\ &f&\longmapsto&(D_{C}(f),-f\Psi)
\end{array}$$
Setting
$$N_{p}(\varepsilon):=\{\beta\in A^1(su(E^{\vee}))\oplus
A^0(\Sigma^+\otimes E^{\vee})|\ {D^0_{p}}^*\beta=0,\
||\beta||<\varepsilon\}\ ,$$
for $\varepsilon>0$ sufficiently small, one obtains local slices for the
action of ${\cal G}$
on ${\cal A}^{**}$ and charts
$\pi|_{N_p(\varepsilon)}:N_p(\varepsilon)\longrightarrow {\cal B}^{**}$ for
${\cal B}^{**}$.
\hfill\vrule height6pt width6pt depth0pt \bigskip
Note that the curvature $F_A$ of a connection in $P(E)$ equals the
trace-free part $F_C^0$ of the
curvature of the corresponding connection $C\in{\cal A}_c(E^{\vee})$.
Using the identification ${\cal A}(P(E))={\cal A}_c(E^{\vee})$, we can
rewrite the
quaternionic monopole equations in terms of pairs $(C,\Psi)\in{\cal A}$.
Let ${\cal
A}^{SW^h}\subset {\cal A}$ be the space of solutions.
\begin{dt} Fix a $Spin^h$-structure in $P$. The moduli space of quaternionic
monopoles is
the quotient ${\cal M}:=\qmod{{\cal A}^{SW^h}}{{\cal G}}$. We denote by
${\cal
M}^{**}:=\qmod{({\cal A}^{SW^h}\cap{\cal A}^{**})}{{\cal G}}$, ${\cal
M}^{*}:=\qmod{({\cal A}^{SW^h}\cap{\cal A}^{*})}{{\cal G}}$ the subspaces of
(strongly) irreducible monopoles.
\end{dt}
The tangent space to ${\cal A}^{SW^h}$ at $p=(C,\Psi)\in {\cal A}$ is the
kernel of the
operator
$$
\begin{array}{c}D^1_{p} :A^1(su(E^{\vee}))\oplus A^0(\Sigma^+\otimes
E^{\vee}) \longrightarrow
A^0(su(\Sigma^+)\otimes su(E^{\vee}))\oplus A^0(\Sigma^-\otimes
E^{\vee})\end{array}$$
defined by
$$D^1_{p}((\alpha,\psi))= \left(\Gamma(D_C^+(\alpha))-[(\psi\bar\Psi)_0+
(\Psi\bar\psi)_0],\hskip 4pt{\not}{D}_{C,c}\psi +\gamma(\alpha)\Psi\right) \ ,
$$
where we consider $\gamma(\alpha)$ as map $\gamma(\alpha):\Sigma^+\longrightarrow
\Sigma^-\otimes su(E^{\vee})$. Clearly $D^1_p\circ D_p^0=0$, since the
monopole equations
are gauge invariant.
Using the isomorphism $\Gamma^{-1}: A^0(su(\Sigma^+))\longrightarrow A^2_+$, we can
consider
$D^1_p$ as an operator $D^1_p:A^1(su(E^{\vee}))\oplus A^0(\Sigma^+\otimes
E^{\vee}) \longrightarrow
A^2_+(su(E^{\vee}))\oplus A^0(\Sigma^-\otimes E^{\vee})$.
Let $\sigma(X)$ and $e(X)$ be the signature and the topological Euler
characteristic of the
oriented manifold $X$.
\begin{pr} For a solution $p=(C,\Psi)\in{\cal A}^{SW^h}$, the complex
$$0\rightarrow A^0 su(E^{\vee}) \textmap{{D}^0_p}
A^1 su(E^{\vee}) \oplus A^0 {\cal H}^+ \textmap{{D}^1_p}A^2_+ su(E^{\vee})
\oplus A^0 {\cal
H}^- \rightarrow 0\eqno{({\cal C}_p)}$$
is elliptic and its index is
$$\frac{3}{2}(4c_2(E^{\vee})-c_1(E^{\vee})^2)-\frac{1}{2}(3e(X)+4\sigma(X))
\ .\eqno{(7)}$$
\end{pr}
{\bf Proof: }
The complex ${\cal C}_p$ has the same symbol sequence as
$$0\rightarrow A^0 su(E^{\vee}) \stackrel{(D_C,0)}{\longrightarrow}
A^1 su(E^{\vee}) \oplus A^0 {\cal
H}^+ \stackrel{(D_C^+,\hskip 4pt{\not}{D}_{C,c})}{\longrightarrow}A^2_+ su(E^{\vee}) \oplus A^0 {\cal
H}^- \rightarrow 0 $$
which is an elliptic complex with index
$$2(4c_2(E^{\vee})-c_1(E^{\vee})^2)-\frac{3}{2}(\sigma(X)+e(X))+index
\hskip 4pt{\not}{D}_{C,c}\ .$$
The latter term is
$$index\hskip 4pt{\not}{D}_{C,c}=[ch(E^{\vee})e^{\frac{1}{2} c_1(E^{\vee})}\hat
A(X)]_4=-2c_2(E^{\vee})+\frac{1}{2}c_1(E^{\vee})^2-\frac{1}{2}\sigma(X)\ .$$
\hfill\vrule height6pt width6pt depth0pt \bigskip
\begin{re} The integer in (7) is always an even number if $X$ admits almost
complex
structures.
\end{re}
Our next step is to endow the spaces ${\cal M}^{**}$ (${\cal M}^{*}$) with
the structure of
a real analytic space (orbifold).
In the first case (compare with [FU], [DK], [OT1], [LT]), we have an
analytic map
$\sigma:{\cal A}\longrightarrow A^2_+(su(E^{\vee}))\oplus A^0( {\cal H}^-)$ defined by
$$\sigma(C,\Psi)=
\left((F_C^0)^+-\Gamma^{-1}(\Psi\bar\Psi)_0,\hskip 4pt{\not}{D}_{C,c}\Psi\right)$$
which gives rise to a section $\tilde\sigma$ in the bundle ${\cal
A}^{**}\times_{{\cal G}}
\left(A^2_+(su(E^{\vee}))\oplus A^0( {\cal H}^-) \right)$. We endow ${\cal
M}^{**}$ with a real
analytic structure by identifying it with the vanishing locus
$Z(\tilde\sigma)$ of
$\tilde\sigma$, regarded as a subspace of the Hilbert manifold ${\cal
B}^{**}$ (in Douady's sense) ([M], [LT]).
Now fix a point $p=(C,\Psi)\in{\cal A}^*$. We put
$$S_p(\varepsilon):=\{p+\beta|\ \beta\in A^1su(E^{\vee}) \oplus A^0 {\cal
H}^+ ,\
D^0_p{D^0_p}^* \beta +{D^1}^*_p\sigma(p+\beta)=0,\ ||\beta||<\varepsilon\}\ .$$
\begin{cl} For sufficiently small $\varepsilon>0$, $S_p(\varepsilon)$ is a
finite
dimensional submanifold of ${\cal A}$ which is contained in the slice
$N_p(\varepsilon)$
and whose tangent space at $p$ is the first harmonic space ${\Bbb H}^1_p$ of the
deformation
complex ${\cal C}_p$.
\end{cl}
To prove this claim, we consider the map
$$s_p:A^1(su(E^{\vee}))\oplus A^0({\cal H}^+)\longrightarrow {\rm im}( D^0_p)\oplus{\rm im}
(D^1_p)^*$$
given by the left hand terms in the equations defining $S_p(\varepsilon)$.
The derivative of
$s_p$ at 0 is the first Laplacian
$$\Delta^1_p:A^1(su(E^{\vee}))\oplus A^0({\cal H}^+)\longrightarrow {\rm im}(
D^0_p)\oplus{\rm im} (D^1_p)^*$$
associated with the elliptic complex ${\cal C}_p$, hence $s_p$ is a
submersion in 0. This
proves the claim.
\hfill\vrule height6pt width6pt depth0pt \bigskip
The intersection ${\cal A}^{SW^h}\cap N_p(\varepsilon)=Z(\sigma)\cap
N_p(\varepsilon)$
of the space of solutions with the standard slice through $p$ is contained in
$S_p(\varepsilon)$ and can be identified with the finite dimensional model
$$Z(\sigma)\cap N_p(\varepsilon)=Z(\sigma|_{S_p(\varepsilon)})\ .$$
If $p\in {\cal A}^{**}$ is strongly irreducible, then the map
$$\pi|_{Z(\sigma|_{S_p(\varepsilon)})}:Z(\sigma|_{S_p(\varepsilon)})\longrightarrow
{\cal M}^{**}$$
is a local parametrization of ${\cal M}^*$ at $p$, hence
$Z(\sigma|_{S_p(\varepsilon)})$ is
a local model for the moduli space around $p$.
If $p\in{\cal A}^*\setminus{\cal A}^{**}$ is irreducible but not strongly
irreducible, then
necessarily $\Psi=0$, and the isotropy group ${\cal G}_p={\Bbb Z}/2$ acts on
$S_p(\varepsilon)$. Since
$\sigma$ is ${\Bbb Z}/2$-equivariant, we obtain an induced action on
$Z(\sigma|_{S_p(\varepsilon)})$. In
this case $\pi|_{Z(\sigma|_{S_p(\varepsilon)})}$ induces a homeomorphism of
the quotient
$\qmod{Z(\sigma|_{S_p(\varepsilon)})}{{\Bbb Z}/2}$ with an open neighbourhood of
$p$ in ${\cal
M}^*$, and ${\cal M}^*$ becomes an orbifold at $p$, if we use the map
$$\pi|_{Z(\sigma|_{S_p})}:Z(\sigma|_{S_p(\varepsilon)})\longrightarrow {\cal M}^*$$
as an orbifold chart.
\begin{re} Using a real analytic isomorphism which identifies the germ of
$S_p(\varepsilon)$ at $p$ with the germ of ${\Bbb H}_p^1=T_p(S_p(\varepsilon))$
at 0, we
obtain a local model of Kuranishi-type for ${\cal M}^*$ at $p$.
\end{re}
\begin{re} The points in ${\cal D}^*:={\cal M}^*\setminus{\cal M}^{**}$
have the form
$[(C,0)]$, where
$C$ is projectively anti-self-dual, i.e $(F_C^0)^+=0$. There is a natural
finite map
$${\cal D}^*\longrightarrow{\cal M}(P(E^{\vee}))$$
into the Donaldson moduli space of $PU(2)$-instantons in
$P(E^{\vee})$, which maps ${\cal D}^*$ isomorphically onto ${\cal
M}(P(E^{\vee})^*$ if
$H^1(X,{\Bbb Z}/2)=0$. In general ${\cal D}^*$ and ${\cal M}(P(E^{\vee})^*$
cannot be identified.
The difference comes from the fact that our gauge group is $SU(E^{\vee})$,
whereas the
$PU(2)$-instantons are classified modulo $PU(E^{\vee})$.
\end{re}
For simplicity we shall however refer to ${\cal D}^*$ as Donaldson
instanton moduli space.
Concluding, we get
\begin{pr} ${\cal M}^{**}$ is a real analytic space. ${\cal M}^*$ is a real
analytic orbi\-fold,
and the points in ${\cal M}^*\setminus{\cal M}^{**}$ have neighbourhoods
modeled on
${\Bbb Z}/2$-quotients. ${\cal M}^*\setminus{\cal M}^{**}$ can be identified as a
set with the
Donaldson moduli space ${\cal D}^*$ of irreducible projectively
anti-self-dual connections in
$E^{\vee}$ with fixed determinant $c$.
\end{pr}
The local structure of the moduli space ${\cal M}$ in reducible points, which
correspond to pairs formed by a reducible instanton and a trivial spinor,
can also be
described using the method above (compare with [DK]).
Let ${\cal M}^{SW}\subset{\cal M}$ be the subspace of ${\cal M}$ consisting of
all orbits of the
form
$(C,\Psi)\cdot SU(E^{\vee})$, where $C$ is a reducible connection and
$\Psi$ belongs to one
of the summands. Let $L:=\det\Sigma^{\pm}=\det E$. It is easy to see that
$${\cal M}^{SW}\simeq\mathop{\bigcup}\limits_{\matrix{^{S {\rm \ summand}}\cr ^{{\rm of} \
E^{\vee}}\cr}}{\cal M}^{SW}_{L \otimes S^{\otimes 2}} \ ,$$
where ${\cal M}^{SW}_M$ denotes the rank-1 Seiberg-Witten moduli space
associated to a $Spin^c(4)$-structure of determinant $M$.
The fact that the moduli spaces of quaternionic monopoles contain
Donaldson moduli spaces as well of Seiberg-Witten moduli spaces suggests
that they could
provide a method for comparing the invariants given by the two theories.
\section{Quaternionic monopoles on K\"ahler surfaces}
Let $(X,g)$ be a K\"ahler surface with canonical $Spin^c(4)$-structure; in
this case
$\Sigma^+=\Lambda^{00}\oplus\Lambda^{02}$, and $\Sigma^-=\Lambda^{01}$. A
$Spin^h(4)$-structure in the frame bundle is given by a unitary vector
bundle $E$
together with an isomorphism $\det E\simeq \Lambda^{02}$. A
$Spin^h(4)$-connection
$\hat A$ corresponds to a $PU(2)$-connection $A$ in the associated bundle
$P(E)$, or
alternatively, to a unitary connection $C$ in $E^{\vee}$ which induces a fixed
$S^1$-connection $c$ in $\Lambda^{20}$. Recall that the curvature $F_A$ of
$A$ equals the
trace-free component $F_C^0$ of $F_C$.
If we choose
$c$ to be the Chern connection in the canonical bundle $\Lambda^{20}=K_X$,
then the
$Spin^h(4)$-connection in
${\cal H}=\Sigma \otimes E^{\vee}$ is simply the tensor product of the
canonical
connection in $ \Sigma=\Sigma^{+}\oplus\Sigma^-$ and the connection $C$.
A positive quaternionic spinor $\Psi\in A^0 ( {\cal H}^+) $ can be written as
$\Psi=\varphi+\alpha$, with $\varphi\in A^0( E^{\vee}) $, and $\alpha\in
A^{02}( E^{\vee})$.
\begin{pr} Let $C$ be a unitary connection in $E^{\vee}$ inducing the Chern
connection $c$ in
$\det E^{\vee}=K_X$. A pair $(C,\varphi+\alpha)$ solves the quaternionic
monopole
equations if and only if $F_C$ is of type $(1,1)$
and one of the following conditions holds
$$\matrix{1.\ \alpha=0\ ,\ \bar\partial_C\varphi=0\ and\ i\Lambda_g
F_{C}^0 +\frac{1}{2} (\varphi\otimes\bar\varphi)_0\ =\ 0\ , \cr
2. \ \varphi=0\ ,\ \partial_C\alpha=0\ and\ i\Lambda_g
F_{C}^0 -\frac{1}{2}* (\alpha\otimes\bar\alpha)_0=0\ .\cr}\eqno{(8)}$$
\end{pr}
{\bf Proof: } Using the notation in the proof of the Weitzenb\"ock formula, we have
$F_{C,c}=\frac{1}{2}({\rm Tr} F_C+F_c){\rm id}_{E^{\vee}}+ F_A=F_A=F_C^0\in A^2(su
(E^{\vee}))$. By
Proposition 2.6 of [OT1] the quaternionic Seiberg-Witten equations become
$$\left\{
\begin{array}{lll}
F_{A}^{20}&=&-\frac{1}{2}(\varphi\otimes\bar\alpha)_0\\
F_{A}^{02}&=&\frac{1}{2}(\alpha\otimes\bar\varphi)_0\\
i\Lambda_g F_{A}&=&-\frac{1}{2}\left[(\varphi\otimes\bar\varphi)_0-
*(\alpha\otimes\bar\alpha)_0\right]\\
\bar\partial_C\varphi&=&i\Lambda_g\partial_C\alpha\ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \
.\end{array}\right.$$
Note that the right-hand side of formula (5) is invariant under Witten's
transformation
$(C,\varphi+\alpha)\longmapsto (C, \varphi-\alpha)$. Therefore, every
solution satisfies
$F_A^{20}= F_A^{02}=0$,
and $(\varphi\otimes\bar\alpha)_0=(\alpha\otimes\bar\varphi)_0=0$. Elementary
computations show that this can happen only if $\varphi=0$ or $\alpha=0$.
On the other
hand, since the Chern connection in $K_X$ is integrable, we also get
$F_C^{20}=F_C^{02}=0$.
\hfill\vrule height6pt width6pt depth0pt \bigskip
\begin{re} The second case in this proposition reduces to the first: in fact,
if
$\varphi=0$ and $\alpha\in A^{02}(E^{\vee})$ satisfies
$i\Lambda_g\partial\alpha=0$, we set
$\psi:=\bar\alpha\in A^{20}(\bar E^{\vee})=A^0(\Lambda^{20}\otimes
E)=A^0(E^{\vee})$, and
we obtain
$\bar\partial_C\psi=\overline{\partial_C\bar\psi}=\overline{\partial_C\alpha
}=0$. Here
we used the fact that $\Lambda_g:\Lambda^{12}\longrightarrow \Lambda^{01}$ is an
isomorphism, the
adjoint of the Lefschetz isomorphism $\cdot\wedge\omega_g$ [LT]. A simple
calculation
in coordinates gives
$-*(\alpha\otimes\bar\alpha)_0=(\bar\alpha\otimes\overline{\bar\alpha})_0=
(\psi\otimes\bar\psi)_0$.
\end{re}
\section{Stability}
Let $(X,g)$ be a compact K\"ahler manifold of arbitrary dimension, $E$ a
differentiable
vector bundle, and let ${\cal L}$ be a fixed holomorphic line
bundle, whose underlying differentiable line bundle is $L:=\det E$.
\begin{dt} A holomorphic pair of type $(E,{\cal L})$ is a pair $({\cal
E},\varphi)$
consisting of a holomorphic bundle ${\cal E}$ and a section $\varphi\in
H^0(X,{\cal E})$
such that the underlying differentiable bundle of ${\cal E}$ is $E$ and
$\det{\cal E}={\cal
L}$.
\end{dt}
Note that the determinant of the holomorphic bundle ${\cal E}$ is fixed,
not only its
isomorphism type.
Two pairs $({\cal E}_i,\varphi_i)$, $i=1, 2$ of the same type are
isomorphic if there
exists an isomorphism $f:{\cal E}_1\longrightarrow{\cal E}_2$ with
$f^*(\varphi_2)=\varphi_1$ and $\det f={\rm id}_{\cal L}$.
In other words, $({\cal E}_i,\varphi_i)$ are isomorphic iff there exists a
complex gauge
transformation $f\in SL(E)$ with $f^*(\varphi_2)=\varphi_1$ such that $f$ is
holomorphic as a map $f:{\cal E}_1\longrightarrow{\cal E}_2$.
%
\begin{dt}
A holomorphic pair $({\cal E},\varphi)$ is simple if any automorphism of
it is
of the form $f=\varepsilon{\rm id}_{\cal E}$, where $\varepsilon^ {{\rm rk}{\cal E}}=1$.
A pair $({\cal E},\varphi)$ is strongly simple if its only automorphism
is ${\rm id}_{\cal
E}$.
\end{dt}
Note that a simple pair $({\cal E},\varphi)$ with $\varphi\ne 0$ is stongly
simple,
whereas a pair $({\cal E},0)$ is simple iff ${\cal E}$ is a simple bundle.
Note also that $({\cal E},\varphi)$ is simple iff any trace-free
holomorphic endomorphism
$f$ of ${\cal E}$ with $f(\varphi)=0$ vanishes.
For a nontrivial torsion free sheaf ${\cal F}$ on $X$, we denote by
$\mu_g({\cal F})$ its slope
with respect to the K\"ahler metric $g$. Given a holomorphic bundle ${\cal
E}$ over $X$ and
a holomorphic section $\varphi\in H^0(X,{\cal E})$, we let ${\cal S}({\cal
E})$ be the set of
reflexive subsheaves ${\cal F}\subset{\cal E}$ with $0<{\rm rk}({\cal
F})<{\rm rk}({\cal E})$, and
we define
$${\cal S}_\varphi({\cal E}):=\{{\cal F}\in{\cal S}({\cal E})|\ \varphi\in
H^0(X,{\cal F})\} \
.$$
Recall the following stability concepts [B2]:
\begin{dt}\hfill{\break}
1. ${\cal E}$ is $\varphi$-stable if
$$\max\left(\mu_g({\cal E}),\sup\limits_{{\cal F}'\in{\cal S}({\cal E})}
\mu_g({\cal
F}')\right)<
\inf\limits_{{\cal F}\in {\cal S}_\varphi({\cal E})}\mu_g(\qmod{{\cal
E}}{{\cal F} })\ .$$
2. Let $\lambda\in{\Bbb R}$ be a real parameter. The pair $({\cal E},\varphi)$ is
$\lambda$-stable iff
$$\max\left(\mu_g({\cal E}),\sup\limits_{{\cal F}'\in{\cal S}({\cal E})}
\mu_g({\cal
F}')\right)<\lambda<
\inf\limits_{{\cal F}\in {\cal S}_\varphi({\cal E})}\mu_g(\qmod{{\cal
E}}{{\cal F}})\ .$$
3. $({\cal E},\varphi)$ is called $\lambda$-polystable if ${\cal E}$
splits holomorphically
as ${\cal E}={\cal E}'\oplus{\cal E}''$, such that $\varphi\in H^0(X,{\cal
E}')$, $({\cal
E}',\varphi)$ is a $\lambda$-stable pair, and ${\cal E}''$ is a polystable
vector bundle of
slope $\lambda$.
\end{dt}
{}From now on we restrict ourselves to the case ${\rm rk}({\cal E})=2$.
\begin{dt} \hfill{\break}
1. A holomorphic pair $({\cal E},\varphi)$ of type $({ E},{\cal L})$
is called stable if one of the following conditions is satisfied:\\
i) ${\cal E}$ is $\varphi$-stable.\\
ii) $\varphi\ne 0$ and ${\cal E}$ splits in direct sum of line bundle
${\cal E}={\cal
E}'\oplus{\cal E}''$, such that $\varphi\in H^0({\cal E}')$ and the pair
$({\cal
E}',\varphi)$ is $\mu_g({ E})$-stable.\\
2. A holomorphic pair $({\cal E},\varphi)$ of type $({ E},{\cal L})$
is called polystable if it is stable, or $\varphi=0$ and ${\cal E}$ is
a polystable
bundle.
\end{dt}
Note that there is \underbar{no} parameter $\lambda$ in the stability
concept for
holomorphic pairs of a fixed type. The conditions depend only on the metric
$g$ and on the
slope $\mu_g(E)$ of the underlying differentiable bundle $E$.
\begin{lm} Let $({\cal E},\varphi)$ be a holomorphic pair of type $(E,{\cal
L})$ with
$\varphi\ne 0$. There exists a uniquely determined effective divisor
$D=D_\varphi$ and a
commutative diagram
$$\begin{array}{cccc}
0\longrightarrow&{\cal O}_X(D)&\textmap{\hat\varphi}&{\cal E}\longrightarrow {\cal L}(-D)\otimes
J_Z\longrightarrow 0\
,\\
&{\scriptstyle D\cdot}\uparrow{\phantom i}&{\scriptstyle\varphi}\nearrow&\\
&\ {\cal O}_X&\end{array}\eqno{(9)}$$
with a local complete intersection $Z\subset X$ of codimension 2. The pair
$({\cal
E},\varphi)$ is stable if and only if $\mu_g({\cal O}_X(D))<\mu_g(E)$.
\end{lm}
{\bf Proof: } $D=D_\varphi$ is the divisorial component of the zero locus
$Z(\varphi)$ of ${\cal E}$
which is defined by the ideal ${\rm im}(\varphi^{\vee}:{\cal E}^{\vee}\longrightarrow{\cal
O}_X)$, and
$\hat\varphi$ is the induced map. The set ${\cal S}_\varphi({\cal E})$
consists precisely
of the line bundles ${\cal F}\subset{\cal O}_X({D})$, so that
$$\inf\limits_{{\cal F}\in
{\cal S}_\varphi({\cal E})}\mu_g(\qmod{{\cal E}}{{\cal
F}})=2\mu_g(E)-\mu_g({\cal
O}_X(D))\ .$$
Suppose $({\cal E},\varphi)$ is stable. If ${\cal E}$ is $\varphi$-stable,
we have
$\mu_g(E)<2\mu_g(E)-\mu_g({\cal O}_X(D))$, which gives the required
inequality. If ${\cal
E}$ is not $\varphi$-stable, then $Z=\emptyset$, the extension (9) splits,
and the
pair $({\cal O}_X(D),\varphi)$ is $\mu_g(E)$-stable, i.e. $\mu_g({\cal
O}_X(D))<\mu_g(E)$.
Conversely, suppose $\mu_g({\cal O}_X(D))<\mu_g(E)$, and assume first that
the extension
(9) does not split. In this case ${\cal E}$ is $\varphi$-stable: in fact,
if ${\cal
F}'\subset {\cal E}$ is an arbitrary line bundle, either ${\cal
F}'\subset{\cal O}_X(D)$, or
the induced map ${\cal F}'\subset{\cal E}\longrightarrow {\cal J}_Z\otimes{\cal L}(-D)$ is
non-trivial. But then ${\cal F}'\simeq {\cal L}\otimes{\cal
O}_X(-D-\Delta)$ for an
effective divisor $\Delta$ containing $Z$, and we find
$$\mu_g({\cal
F}')=2\mu_g(E)-\mu_g(D)-\mu_g(\Delta)\leq 2\mu_g(E)-\mu_g({\cal
O}_X(D)) \ . $$
Furthermore, strict inequality holds, unless
$Z=\emptyset$ and the extension (9) splits, which it does not by assumption.
In the case of a split extension, we only have to notice that a pair
$({\cal E}',\varphi)$
is $\lambda$-stable for any parameter
$\lambda>\mu_g({\cal E}') $ [B1].
\hfill\vrule height6pt width6pt depth0pt \bigskip
\begin{re} Consider a pair $({\cal E},\varphi)$ of type $(E,{\cal L})$ with
$\varphi\ne 0$
and associated extension (9). ${\cal E}$ is $\varphi$-stable iff $\mu_g({\cal
O}_X(D))<\mu_g(E)$, and the extension does not split.
\end{re}
Indeed, if the extension splits, then ${\cal E}$ is not $\varphi$-stable, since
$$\mu_g({\cal L}(-D))=\inf\limits_{{\cal F}\in
{\cal S}_\varphi({\cal E})}\mu_g(\qmod{{\cal E}}{{\cal F}}) \ .$$
\section{The projective vortex equation}
Let $E$ be a differentiable vector bundle over a compact K\"ahler manifold
$(X,g)$. We
fix a holomorphic line bundle ${\cal L}$ and a Hermitian metric $l$ in
${\cal L}$. Let
$({\cal E},\varphi)$ be a holomorphic pair of type $(E,{\cal L})$.
\begin{dt} A Hermitian metric in ${\cal E}$ with $\det h=l$ is a solution
of the projective
vortex equation iff the trace free part $F^0_h$ of the curvature $F_h$
satisfies the
equation
$$i\Lambda_g F_h^0 +\frac{1}{2}(\varphi\bar\varphi^h)_0=0\ .\eqno{(V)}$$
\end{dt}
\begin{th} Let $({\cal E},\varphi)$ be a holomorphic pair of type $(E,{\cal
L})$ with
${\rm rk}({\cal E})=2$. Fix a Hermitian metric $l$ in ${\cal L}$.
The pair $({\cal E},\varphi)$ is polystable iff ${\cal E}$ admits a
Hermitian metric $h$
with $\det h=l$ which is a solution of the projective vortex equation. If
$({\cal
E},\varphi)$ is stable, then the metric $h$ is unique.
\end{th}
{\bf Proof: }
Suppose first that $h$ is a solution of the projective vortex equation
$(V)$. Then we have
$$i\Lambda F_h+\frac{1}{2}(\varphi\bar\varphi^h)=\frac{1}{2}(i\Lambda{\rm Tr}
F_h+\frac{1}{2}|\varphi|^2){\rm id}_E \ ,
$$
i.e. $h$ satisfies the weak vortex equation $(V_t)$ associated to the real
function
$t:=\frac{1}{2}(2i\Lambda{\rm Tr} F_h+|\varphi|^2)$. Therefore, by [OT1], the
pair $({\cal
E},\varphi)$ is $\lambda$-polystable for the parameter
$\lambda=\frac{(n-1)!}{4\pi}\int\limits_X t vol_g=\mu_g({\cal
E})+\frac{(n-1)!}{8\pi}||\varphi||^2$.
Let $A$ be the Chern connection of $h$, and denote by ${\cal E}'$ the minimal
$A$-invariant
subbundle which contains $\varphi$. If ${\cal E}'={\cal E}$, then ${\cal E}$
is
$\varphi$-stable and the pair $({\cal E},\varphi)$ is stable.
If ${\cal E}'=0$, hence $\varphi=0$, then $h$ is a weak Hermitian-Einstein
metric,
${\cal E}$ is a polystable bundle, and the pair $({\cal E},\varphi)$ is
polystable by
definition.
In the remaining case ${\cal E}'$ is a line bundle and $\varphi\ne 0$.
Let ${\cal E}'':={\cal E}'^{\bot}$ be the orthogonal complement of ${\cal
E}'$, and let $h'$
and $h''$ be the induced metrics in ${\cal E}'$ and ${\cal E}''$. We put
$s:=i\Lambda_g{\rm Tr}
F_h$. Then, since $h=h'\oplus h''$, the projective vortex equation can be
rewritten as:
$$
\left\{\begin{array}{ll}
i\Lambda F_{h'}+\frac{1}{2}(\varphi\bar\varphi^{h'})=&\frac{1}{2}(s+
\frac{1}{2}|\varphi|_{h'}^2){\rm id}_{{\cal E}'}\\
i\Lambda F_{h''}=&\frac{1}{2}(s+\frac{1}{2}|\varphi|_{h'}^2)
{\rm id}_{{\cal E}''} \ .
\end{array}\right. $$
The first of these equations is equivalent to
$$i\Lambda
F_{h'}+\frac{1}{4}(\varphi\bar\varphi^{h'})=\frac{s}{2}{\rm id}_{{\cal E}'}\ ,$$
which implies that $({\cal E}',\frac{\varphi}{\sqrt 2})$ is $\mu_g({\cal
E})$-stable by
[OT1].
Conversely, suppose first that $({\cal E},\varphi)$ is stable. We have to
consider two
cases:\\
\underbar{Case 1}: ${\cal E}$ is $\varphi$ stable.
Using Bradlow's existence theorem, we obtain Hermitian metrics in ${\cal
E}$ satisfying
the usual vortex equations associated with suitable chosen $\lambda$ , and,
of course
these metrics all satisfy the equation $(V)$. The problem
is, however, to find a solution with an a priori given determinant
$l$.
In order to achieve this stronger result, Bradlow's proof has to modified
slightly at some
points:
One starts by fixing a background metric $k$ such that $\det k=l$. Denote
by $S_0(k)$
the space of {trace-free} $k$-Hermitian endomorphisms of $E$, and let
${\cal M} et(l)$ be the space of Hermitian metrics in $E$ with $\det h=l$. On
$$ {\cal M} et(l)^p_2:=\{ke^s |\ s\in L^p_2(S_0(k))\}$$
we define the functional $M_\varphi:{\cal M} et(l)^p_2\longrightarrow{\Bbb R}$ by
$$M_\varphi(h):=M_D(k,h)+||\varphi||^2_h-||\varphi||^2_k \ .$$
Here $M_D$ is the Donaldson functional, which is known to satisfy the
identity
$ \frac{d}{dt}M_D(k,h(t))=2\int\limits_X{\rm Tr}[ h^{-1}(t)\dot h(t) i\Lambda_g
F_h ]\ $
for any smooth path of metrics $h(t)$ [Do], [Ko]. Since $h^{-1}(t)\dot
h(t)$
is trace-free for a path in ${\cal M} et(l)$, we obtain
$$\frac{d}{dt}|M_D(k,h(t))= 2\int\limits_X{\rm Tr}[ h^{-1}\dot h(t) i\Lambda_g
F_h^0]\ .$$
Similarly, for a path of the form $h(t)=he^{ts}$, with $s \in S_0(h)$, we get
$$\frac{d}{dt}_{|_{t=0}}||\varphi||^2_{h_t}=\frac{d}{dt}_{|_{t=0}}\langle
e^{ts}\varphi,\varphi\rangle_h=\left\langle
\frac{d}{dt}_{|_{t=0}}e^{ts}\varphi,\varphi\right\rangle_h=
\left\langle s ,\varphi\bar\varphi^h\right\rangle_h=\int\limits_X{\rm Tr}[s
(\varphi\bar\varphi^h)_0] \ .$$
This means that, putting $m_\varphi(h):=i\Lambda
F_h^0+\frac{1}{2}(\varphi\bar\varphi^h)_0$, we always have
$$\frac{d}{dt}_{|_{t=0}}M_\varphi(he^{ts})= 2\int\limits_X{\rm Tr}[ s\
m_\varphi(he^{ts})]\ ,\ $$
so that solving the projective vortex equation is equivalent to finding
a critical
point of the functional $M_\varphi$ (compare with Lemma 3.3 [B2]).
\begin{cl} Suppose $({\cal E},\varphi)$ is simple. Choose $B>0$ and put
$$ {\cal M} et(l)^p_2(B):=\{ h\in {\cal M} et(l)^p_2|\ ||m_\varphi(h)||_{L^p}\leq B\}
\ .$$
Then any
$h\in{\cal M} et(l)^p_2(B)$ which minimizes $M_\varphi$ on ${\cal M} et(l)^p_2(B)$ is
a weak
solution of the projective vortex equation.
\end{cl}
The essential point is the injectivity
of the operator $s\longmapsto\Delta_h'(s)+\frac{1}{2}[(\varphi\bar\varphi)
s]_0$ acting on
$L^p_2 S_0(h)$. But from
%
$$\left\langle\Delta_h's+\frac{1}{2}[\varphi\bar\varphi^h) s]_0,\
s\right\rangle_h=||\bar\partial_h(s)||_{h}^2+||s\varphi||^2_{h} \ $$
we see that this operator is injective on trace-free
endomorphisms if $({\cal E},\varphi)$ is simple.
\hfill\vrule height6pt width6pt depth0pt \bigskip
Now we can follow again Bradlow's proof : if ${\cal E}$ is
$\varphi$-stable, then there exist positive constants $C_1$, $C_2$ such
that for all $s\in
L^p_2 S_0(k) $ with $ke^s\in{\cal M} et(l)^p_2(B)$ the following "main estimate"
holds:
$$\sup|s|\leq C_1 M_\varphi(ke^{s})+C_2\ .$$
This follows by applying Proposition 3.2 of [B2] to an arbitrary
$\tau\in{\Bbb R}$ with
$$\max\left(\mu_g({\cal E}),\sup\limits_{{\cal F}'\in{\cal S}({\cal E})}\mu_g({\cal
F'})\right)<\frac{(n-1)!\tau
Vol_g(X)}{4\pi}<\inf\limits_{{\cal F}\in{\cal S}_\varphi({\cal E})}
\mu_g(\qmod{\cal E}{\cal F})\ ,
$$
since Bradlow's functional ${\cal
M}_{\varphi,\tau}$ coincides on ${\cal M} et(l)$ with $M_{\varphi}$ .
It remains to be shown that the existence of this main
estimate implies the existence of a solution of the projective vortex
equation.
The main estimate implies that for any $c>0$, the set
$$\{s\in\ L^p_2S_0(k) |\ ke^s\in {\cal M} et(l)^p_2(B) ,\ \ M_\varphi(ke^{s})<c\}$$
is bounded in $L^p_2$. Let $(s_i) $ be a sequence in $L^p_2 S_0(k) $ such that
$ke^{s_i}\in {\cal M} et(l)^p_2(B)$ is a minimizing sequence for $M_\varphi$,
and let $s$ be weak
limit. Then $h:=ke^s$ is a weak solution of the projective vortex equation,
which is smooth by
elliptic regularity [B2].
Finally, we have to treat
\underbar{Case 2}: $\varphi\ne 0$, ${\cal E}={\cal E}'\oplus{\cal E}''$,
with $\varphi\in
H^0({\cal E}')$, and the pair $({\cal E}',\varphi)$ is $\mu_g(E)$-stable.\\
We wish to find
metrics $h'$ and $h''$ in ${\cal E}'$ and ${\cal E}''$, such that for
$s:=i\Lambda F_l$ the
following equations are satisfied:
$$\left\{ \begin{array}{lll}h'\cdot h''&=&l\\
i\Lambda
F_{h'}+\frac{1}{4}(\varphi\bar\varphi^{h'})&=&\frac{1}{2}s{\rm id}_{{\cal E}'}
\\
i\Lambda F_{h''}&=&\frac{1}{2}(s+\frac{1}{2}|\varphi|_{h'}^2)
{\rm id}_{{\cal E}''} \ .
\end{array}\right.\ $$
Since the pair $({\cal E}',\frac{1}{\sqrt 2}\varphi)$ is
$\mu_g(E)$-stable, there exists by
[OT1] a unique Hermitian metric
$h'$ in ${\cal E}'$ solving the second of these equations. With this
solution the third
equation can be rewritten as
$$i\Lambda_g F_{h''}=s-i\Lambda_g F_{h'}\ . $$
Since $\int\limits_X(s-i\Lambda_g F_{h'})={\rm deg}({\cal E}'')$, we can solve
this weak
Hermitian-Einstein equation by a metric $h''$, which is unique up to
constant
rescaling. The pro\-duct $h'\cdot h''$ is a metric in
${\cal E}'\otimes{\cal E}''={\cal L}$ which has the same mean curvature
$s$ as $l$, and therefore
differs from $l$ by a constant factor. We can now simply rescale $h''$ by
the inverse of this
constant, and we get a pair of metrics satisfying the three equations above.
\hfill\vrule height6pt width6pt depth0pt \bigskip
\section{Moduli spaces of pairs}
Let $E$ be a differentiable vector bundle of rank $r$ over a K\"ahler
manifold $(X,g)$, and let
${\cal L}$ be a holomorphic line bundle whose underlying differentiable
bundle is $L:=\det E$.
\begin{pr} There exists a possibly non-Hausdorff complex analytic
orbifold ${\cal M}^s(E,{\cal
L} )$ parametrizing isomorphism classes of simple holomorphic pairs of type
$(E,{\cal L})$. The open subset ${\cal M}^{ss}(E,{\cal L})\subset {\cal
M}^{s}(E,{\cal L})$ consisting
of strongly simple pairs is a complex analytic space, and the points in
${\cal M}^s(E,{\cal
L})\setminus{\cal M}^{ss}(E,{\cal L})$ have neighbourhoods modeled on
${\Bbb Z}/r$-quotients.
\end{pr}
{\bf Proof: } Since we use the same method as in the proof of Proposition 3.9, we
only sketch the main
ideas.
Let $\bar\lambda$ be the semiconnection defining the holomorphic structure
of ${\cal
L}$, and put $\bar{\cal A}:=\bar{\cal A}_{\bar\lambda}(E)\times A^0(E)$,
where $\bar{\cal
A}_{\bar\lambda}(E)$ denotes the affine space of semiconnections in $E$
inducing $\bar\lambda$
in
$L=\det E$. The complex gauge group $SL(E)$ acts on $\bar{\cal A}$, and
we write $\bar{\cal A}^s$ ($\bar{\cal A}^{ss})$ for the open subset of
pairs whose stabilizer is
contained in the center ${\Bbb Z}/r$ of $SL(E)$ ( is trivial). After suitable
Sobolev
completions,
$\bar{\cal A}^{ss}$ becomes the total space of a holomorphic Hilbert principal
$SL(E)$-bundle over $\bar{\cal B}^{ss}:=\qmod{\bar{\cal A}^{ss}}{SL(E)}$.
A point $(\bar\delta,\varphi)\in \bar{\cal A}$ defines a pair of type
$(E,{\cal L})$ iff it is
integrable, i.e. iff it satisfies the following equations:
$$\left\{\begin{array}{lll}F^{02}_{\bar\delta}&=&0\\ \bar\delta\varphi&=&0\ .
\end{array}\right.\eqno{(10)}$$
Here $F^{02}_{\bar\delta}:=\bar\delta^2$ is a $(0,2)$-form with values in
the bundle ${\rm End}_0(E)$
of trace-free endomorphisms. Moreover, isomorphy of pairs of type
$(E,{\cal L})$ corresponds
to equivalence modulo the action of the complex gauge group $SL(E)$.
Let $\bar\sigma$ be the map ${\cal A}\longrightarrow A^{02}({\rm End}_0(E))\oplus
A^{01}(E)$ sending a pair
$(\bar\delta,\varphi)$ to the left hand sides of (10). We endow the sets
${\cal
M}^{ss}_X(E,{\cal L})=\qmod{Z(\sigma)\cap\bar{\cal A}^{ss}}{SL(E)}$ (
${\cal M}^{s }_X(E,{\cal
L})=\qmod{Z(\sigma)\cap\bar{\cal A}^{s }}{SL(E)}$ ) with the structure of
a complex
analytic space (orbifold) as follows:
${\cal M}^{ss}_X(E,{\cal L})$ is defined to be the vanishing locus of the
section
$\tilde{\bar\sigma}$ in the Hilbert vector bundle
$\bar{\cal A}^{ss}\times_{SL(E)}\left(A^{02}{\rm End}_0(E)\oplus A^{01}E\right)$
over $\bar{\cal B}^{ss}$ which is defined by $\bar\sigma$.
To define the orbifold structure in ${\cal M}^{s }_X(E,{\cal L})$, we use
local models derived from a deformation complex:
Let $\bar p=(\bar\delta,\varphi)\in\bar{\cal A}$ an integrable point. The
associated
{deformation} {complex} $\bar{\cal D}_{\bar p}$ is the cone over the
evaluation map
$ev^*_\varphi$:
$$ ev^q_\varphi:A^{0q}({\rm End}_0(E))\longrightarrow A^{0q}(E)\ ,$$
and has the form
$$\matrix{0\rightarrow A^0 ({\rm End}_0(E))\textmap{\bar D^0_{\bar p}}
A^{01}({\rm End}_0(E))\oplus
A^0(E)\textmap{\bar D^1_{\bar p}}\cr\ \ \ \ \ \ \textmap{\bar D^1_{\bar p}}
A^{02}({\rm End}_0(E))\oplus A^{01}(E)\textmap{\bar D^2_{\bar
p}}\dots\cr}\eqno{(\bar{\cal D}_{\bar
p})}$$
(compare with [OT1] \S 4).
We define
$$\bar S_{\bar p}(\varepsilon):=\{{\bar p}+\beta|\beta\in
A^{01}{\rm End}_0(E)\oplus A^0E, \bar
D^0_{\bar p}{\bar {D_{\bar p}^0}}^*(\beta)+{\bar {D_{\bar
p}^1}}^*(\bar\sigma({\bar p}+\beta))=0,
||\beta||<\varepsilon\}.$$
The same arguments as in the proof of Proposition 3.9 show that for
sufficiently small
$\varepsilon>0$,
$\bar S_{\bar p}(\varepsilon)$ is a submanifold of $\bar{\cal A}$, whose
tangent space in $\bar p$
coincides with the first harmonic space $\bar{\Bbb H}^1_{\bar p}$ of the elliptic
complex $(\bar{\cal
D}_{\bar p})$. Therefore, we get a local finite dimensional model
$Z(\bar\sigma|_{\bar
S_{\bar p}(\varepsilon)}) $ for the intersection $Z(\bar\sigma)\cap \bar
N_{\bar p}(\varepsilon)$
of the integrable locus with the standard slice
$$ \bar N_{\bar p}(\varepsilon):=\{{\bar p} +\beta|\beta\in
A^{01}({\rm End}_0(E))\oplus A^0(E), {\bar{ D_{\bar p}^0}}^*(\beta)=0,\
||\beta||<\varepsilon\}$$
through $\bar p$. The restriction
$$\bar\pi|_{Z(\bar\sigma|_{\bar S_{\bar p}(\varepsilon)})} :
Z(\bar\sigma|_{\bar
S_{\bar p}(\varepsilon)})\longrightarrow {\cal M}^{s}_X(E,{\cal L})$$
of the orbit map is \'etale if $[{\bar p}]\in {\cal M}^{ss}_X(E,{\cal
L})$, and induces an open
injection
$$\qmod{ Z(\bar\sigma|_{\bar S_{\bar p}(\varepsilon)})}{ {\Bbb Z}/r }\longrightarrow {\cal
M}^{s}_X(E,{\cal L})$$
if $[{\bar p}]\in {\cal M}^{s}_X(E,{\cal L})\setminus {\cal
M}^{ss}_X(E,{\cal L})$. We define the
orbifold structure of ${\cal M}^{s}_X(E,{\cal L})$ by taking the maps
$\bar\pi|_{Z(\bar\sigma|_{\bar S_{\bar p}(\varepsilon)})}$ as orbifold-charts.
\hfill\vrule height6pt width6pt depth0pt \bigskip
Our next purpose is to compare the two types of moduli spaces constructed
in this paper.
Let $(X,g)$ be a K\"ahler surface endowed with the canonical
$Spin^c$-structure ${\germ c} $. Let
$E$ be a $U(2)$ bundle with $\det E=K_X$, and denote by ${\cal M}^*(E)$ the
moduli space
of irreducible quaternionic monopoles associated to the
$Spin^h(4)$-structure defined by
$({\germ c},E^{\vee})$ (Lemma 1.3)
It follows from Proposition 4.1 that ${\cal M}^*(E)$ has a
decomposition
$${\cal M}^*(E)= {\cal M}^*(E)_{\alpha=0}\cup {\cal M}^*(E)_{\varphi=0}\ ,$$
where ${\cal M}^*(E)_{\alpha=0}$ ( ${\cal M}^*(E)_{\varphi=0}$ ) is the
Zariski closed
subspace of ${\cal M}^*(E)$ cut out by the equation $\alpha=0$ (
${\varphi=0}$ ). The
intersection
$${\cal M}^*(E)_{\alpha=0}\cap {\cal M}^*(E)_{\varphi=0}$$
is the Donaldson instanton moduli space ${\cal D}^*$ of irreducible
projectively anti-self-dual
connections in $E$, inducing the Chern connection in ${\cal K}_X$.
\begin{pr} The affine isomorphism ${\cal
A}\ni(C,\varphi)\longmapsto(\bar\partial_C,\varphi)\in\bar{\cal A}$ induces
a natural real
analytic open embedding
$$J:{\cal M}^*(E)_{\alpha=0}\hookrightarrow {\cal M}^s(E,{\cal K}_X)$$
whose image is the suborbifold of stable pairs of type $(E,{\cal K}_X)$.
\end{pr}
{\bf Proof: } Standard arguments (cf. [OT1]) show that $J$ is an \'etale map which
induces natural
identifications of the local models.
A point $[(\bar\delta,\varphi)]$ lies in the image of $J$ iff the
$SL(E)$-orbit of
$(\bar\delta,\varphi)$ intersects the zero locus of the map
$$m:\bar{\cal A}\longrightarrow A^0(su(E)), \ \ (\bar\partial_C,\varphi)\longmapsto
\Lambda_g
F_C^0-\frac{1}{2}(\varphi\bar\varphi)_0 \ .$$
Let $({\cal E},\varphi)$ be the holomorphic pair of type $(E,{\cal K}_X)$
defined by
$(\bar\delta,\varphi)$. We can reformulate the condition above in the
following way:
$[({\cal E},\varphi)]$ lies in the image of $J$ iff there exists a
Hermitian metric $h$ in
${\cal E}$ inducing the K\"ahler metric in ${\cal K}_X=\det{\cal E}$ which
satisfies the
projective vortex equation $(V)$. But we know already that this holds iff
$({\cal
E},\varphi)$ is stable. Moreover, the unicity of the solution of the
projective vortex
equation is equivalent to the injectivity of $J$.
\hfill\vrule height6pt width6pt depth0pt \bigskip
Using the remark after Proposition 4.1, we can now state the main result
of this
paper:
\begin{th}
Let $(X,g)$ be a K\"ahler surface with canonical bundle ${\cal K}_X$, and
let $E$ be a
$U(2)$-bundle with $\det E=K_X$. Consider the
$Spin^h$-structure associated with the canonical $Spin^c(4)$-structure and the
$U(2)$-bundle $E^{\vee}$. The corresponding moduli space of irreducible
quaternionic
monopoles is a union of two Zariski closed subspaces. Each of these
subspaces is
naturally isomorphic as a real analytic orbifold to the moduli space of
stable pairs of type
$(E,{\cal K}_X)$. There exists a real analytic involution on the
quaternionic moduli space
which interchanges these two closed subspaces. The fixed point set of this
involution is
the Donaldson moduli space of instantons in $E$ with fixed determinant, modulo
the gauge group $SU(E)$. The closure of the complement of the Donaldson
moduli space
intersects the moduli space of instantons in the Brill-Noether locus.
The union ${\cal M}^{SW}$ of all rank 1-Seiberg-Witten moduli spaces
associated with
splittings $E=E' \oplus E''$ corresponds to the subspace of stable
pairs of type ii).
\end{th}
\newpage
\centerline{\large{\bf References}}
\vspace{10 mm}
\parindent 0 cm
[B1] Bradlow, S. B.: {\it Vortices in holomorphic line bundles over closed
K\"ahler manifolds}, Comm. Math. Phys. 135, 1-17 (1990)
[B2] Bradlow, S. B.: {\it Special metrics and stability for holomorphic
bundles with global sections}, J. Diff. Geom. 33, 169-214 (1991)
[D] Donaldson, S.: {\it Anti-self-dual Yang-Mills connections over complex
algebraic surfaces and stable vector bundles}, Proc. London Math. Soc. 3,
1-26 (1985)
[DK] Donaldson, S.; Kronheimer, P.B.: {\it The Geometry of four-manifolds},
Oxford Science Publications (1990)
[FU] Freed D. S. ; Uhlenbeck, K.:
{\it Instantons and Four-Manifolds.}
Springer-Verlag 1984.
[HH] Hirzebruch, F.; Hopf H.: {\it Felder von Fl\"achenelementen in
4-dimensionalen 4-Mannigfaltigkeiten}, Math. Ann. 136 (1958)
[H] Hitchin, N.: {\it Harmonic spinors}, Adv. in Math. 14, 1-55 (1974)
[JPW] Jost, J.; Peng, X.; Wang, G. :{\it Variational aspects of the
Seiberg-Witten functional},
Preprint, dg-ga/9504003, April (1995)
[K] Kobayashi, S.: {\it Differential geometry of complex vector bundles},
Princeton University Press (1987)
[KM] Kronheimer, P.; Mrowka, T.: {\it The genus of embedded surfaces in the
projective plane}, Preprint (1994)
[LM] Labastida, J. M. F.; Marino M.: {\it Non-abelian monopoles on four
manifolds}, Preprint,
Departamento de Fisica de Particulas, Santiago de Compostela, April
(1995)
[LT] L\"ubke, M.; Teleman, A.: {\it The Kobayashi-Hitchin correspondence},
World Scientific Publishing Co, to appear.
[M] Miyajima K.: {\it Kuranishi families of
vector bundles and algebraic description of
the moduli space of Einstein-Hermitian
connections}, Publ. R.I.M.S. Kyoto Univ. 25,
301-320 (1989)
[OSS] Okonek, Ch.; Schneider, M.; Spindler, H: {\it Vector bundles on complex
projective spaces}, Progress in Math. 3, Birkh\"auser, Boston (1980)
[OT1] Okonek, Ch.; Teleman A.: {\it The Coupled Seiberg-Witten Equations,
Vortices, and Moduli Spaces of Stable Pairs}, Preprint, Z\"urich, Jan. 13-th,
(1995)
[OT2] Okonek, Ch.; Teleman A.: {\it Les invariants de Seiberg-Witten et la
conjecture de Van De Ven}, to appear in Comptes Rendus
[OT3] Okonek, Ch.; Teleman A.: {\it Seiberg-Witten invariants and rationality
of complex surfaces}, Preprint, Z\"urich, March (1995)
[W] Witten, E.: {\it Monopoles and four-manifolds}, Mathematical Research
Letters 1, 769-796 (1994)
\vspace{0.4cm}\\
Authors addresses:\\
\\
Mathematisches Institut, Universit\"at Z\"urich,\\
Winterthurerstrasse 190, CH-8057 Z\"urich\\
e-mail: [email protected] \ ; \
[email protected]
\end{document}
--========================_16111448==_
Content-Type: text/plain; charset="us-ascii"
Dr. Andrei Teleman (e-mail: [email protected])
Mathematisches Institut der Universitaet Zuerich
Winterthurer Strasse 190, CH-8057 Zuerich-Irchel
Tel.: (+411) 257 58 65; Fax 2575706
--========================_16111448==_--
|
1996-03-05T06:19:25 | 9505 | alg-geom/9505014 | en | https://arxiv.org/abs/alg-geom/9505014 | [
"alg-geom",
"math.AG"
] | alg-geom/9505014 | Teleman | Andrei Teleman and Christian Okonek | Seiberg-Witten Invariants and Rationality of Complex Surfaces | Duke preprint, LaTeX | null | null | null | null | The purpose of this paper is: 1) to explain the Seiberg-Witten invariants, 2)
to show that - on a K\"ahler surface - the solutions of the monopole equations
can be interpreted as algebraic objects, namely effective divisors, 3) to give
- as an application - a short selfcontained proof for the fact that rationality
of complex surfaces is a ${\cal C}^{\infty}$-property.
| [
{
"version": "v1",
"created": "Mon, 8 May 1995 19:55:24 GMT"
},
{
"version": "v2",
"created": "Wed, 10 May 1995 15:53:39 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Teleman",
"Andrei",
""
],
[
"Okonek",
"Christian",
""
]
] | alg-geom | \section{Introduction}
Recently, Seiberg and Witten introduced new differential invariants for
4-manifolds, which are defined by counting solutions of the so called {\sl
monopole
equations}, a system of non-linear differential equations of Yang-Mills-Higgs
type [18].
The new invariants are expected to be equivalent to the Donaldson polynomial
invariants, and they have already found important applications [15].
The purpose of this paper is:\\
- to explain the Seiberg-Witten invariants\\
- to show that --- on a K\"ahler surface --- the solutions of the monopole
equations can be interpreted as algebraic objects, namely effective divisors\\
- to give --- as an application --- a short selfcontained proof for the
fact that
rationality of complex surfaces is a ${\cal C}^{\infty}$-property.
\section{${\rm Spin}^c$-structures and the monopole equation}
\begin{dt} {\rm [1], [11]} The group ${\rm Spin}^c(n):={\rm Spin}(n)\times_{{\Bbb Z}_2}S^1$ is
called the complex spinor group.
\end{dt}
For the case $n=4$, there is a natural identification
$${\rm Spin}^c(4)=\{(A,B)\in\U(2)\times\U(2)|\ {\rm det} A={\rm det} B\}\ .$$
The following diagram summarizes some of the basic relations of
${\rm Spin}^c(4)$ to other
groups:
$$\begin{array}{rrclcrl}
&&&&\U(2)&&\\
&&&^l\swarrow& &\searrow^i\\
\ \ \ S^1&\longrightarrow&\ \ {\rm Spin}^c(4)\ &&\longrightarrow&&{\rm SO}(4)\\
(\cdot)^2\downarrow\ \ &{\scriptstyle{\rm det}}\swarrow&\ \downarrow&&
\phantom{{\scriptstyle\lambda^+}}&&\ \ \ \ \ \downarrow
{\scriptstyle(\lambda^+,\lambda^-)}\\
\ \ \ S^1&&\ \U(2)\times U(2)&&\textmap{{\rm ad}}&&{\rm SO}(3)\times SO(3) \ \ \ \
\end{array}$$
Here \ $l:\U(2)\longrightarrow{\rm Spin}^c(4)$ \ is the canonical lifting of the
homomorphism \
$i\times{\rm det}:\U(2)\longrightarrow{\rm SO}(4)\times S^1$ [11], and acts by the formula\linebreak
$\U(2)\ni a\longmapsto\left(\left(\matrix{{\rm id}&0\cr0&{\rm det}
a\cr}\right),a\right)\in{\rm Spin}^c(4)$.
$\lambda^{\pm}:{\rm SO}(4)\longrightarrow{\rm SO}(3)$ are the maps induced by the two projections of
${\rm Spin}(4)={\rm SU}(2)^+\times{\rm SU}(2)^-$ onto the factors.
Let $X$ be a closed, oriented {\sl simply connected} 4-manifold,
$\Lambda^p$ the bundle of $p$-forms on $X$, and $A^p:=A^0(X,\Lambda^p)$ the
space of
sections in this bundle. Let $g$ be a Riemannian metric on $X$, denote by
$P$ the
associated principal
${\rm SO}(4)$-bundle, and by $P^{\pm}$ the ${\rm SO}(3)$-bundles induced via the
morphisms
$\lambda^{\pm}$. The real 3-vector bundles
$\Lambda^2_{\pm}:=P^{\pm}\times_{{\rm SO}(3)}{\Bbb R}^3$ can be identified with the
bundles of
(anti)self-dual 2-forms, hence there is an orthogonal splitting
$\Lambda^2=\Lambda^2_+\oplus\Lambda^2_-$.
\begin{lm}{\rm [10]} Given $c\in H^2(X,{\Bbb Z})$ with $w_2(X)\equiv\bar c$
(mod 2) there exists a unique ${\rm Spin}^c(4)$-bundle $\hat{P_c}$ with
$P\simeq\qmod{\hat{P_c}}{S^1}$, and $c_1({\rm det}(\hat{P_c}))=c$.
\end{lm}
We denote by $\Sigma_c^{\pm}$ the induced $\U(2)$-vector bundles, and we put
$\Sigma_c:=\Sigma^+_c\oplus\Sigma^-_c$.
\begin{lm} {\rm [1], [11]} The choice of a ${\rm Spin}^c(4)$-lift $\hat{P_c}$ of $P$
induces an isomorphism
$$\gamma_+:\Lambda^1\otimes{\Bbb C}\longrightarrow{\rm Hom}_{{\Bbb C}}(\Sigma^+_c,\Sigma^-_c)$$
satisfying the identity
$\gamma_+(u)^*\gamma_+(v)+\gamma_+(v)^*\gamma_+(u)=2g(u,v){\rm id}_{\Sigma^+_c}$
for {\sl
real} cotangent vectors $u,\ v\in\Lambda^1.$
\end{lm}
We define the homomorphisms $\gamma:\Lambda^1\longrightarrow{\rm End}_0(\Sigma_c)$,
$\Gamma:\Lambda^2\longrightarrow{\rm End}_0(\Sigma_c)$ by
$$\gamma(u):=\left(\matrix{0&-\gamma_+(u)^*\cr\gamma_+(u)&0\cr}\right)$$
$$\Gamma(u\wedge v):=\frac{1}{2}[\gamma(u),\gamma(v)]\ ,$$
and we denote by the same symbols
also their ${\Bbb C}$-linear extensions
\linebreak$\Lambda^1\otimes{\Bbb C}\longrightarrow{\rm End}_0(\Sigma_c)$,
and $\Lambda^2\otimes{\Bbb C}\longrightarrow{\rm End}_0(\Sigma_c)$. The homomorphism $\gamma$
defines a map
$\Lambda^1\otimes\Sigma_c\longrightarrow\Sigma_c$\ , called the {\sl Clifford
multiplication}.
The map
$\Gamma$ identifies the bundles $\Lambda^2_{\pm}$ with the bundles of {\sl
trace free
skew-Hermitian} endomorphisms of $\Sigma^{\pm}_c$.
Fix a ${\rm Spin}^c(4)$-bundle $\hat{P_c}$ with $P\simeq\qmod{\hat{P_c}}{S^1}$,
and let $L_c:={\rm det}(\hat{P_c})$ be the associated $S^1$-vector bundle. $L_c$
is the
unique unitary line bundle with Chern class $c$.
\begin{re} {\rm [11]} The choice of a $S^1$-connection $a$ in $L_c$ is
equivalent to the
choice of a ${\rm Spin}^c(4)$-connection $A$ in $\hat{P_c}$ projecting onto the
Levi-Civita connection.
\end{re}
\begin{dt} The composition
$\hskip 4pt{\not}{D}_a:A^0(\Sigma_c)\textmap{\nabla_A}A^1(\Sigma_c)\stackrel{\gamma}{\longrightarrow}A^
0(\Sigma_c)$
is called the Dirac operator associated to the connection $a\in{\cal A}(L_c)$.
\end{dt}
{\bf Notation:} Let ${\cal A}(L_c)$ be the affine space of $S^1$-connections
in $L_c$. For a connection $a\in{\cal A}(L_c)$, we denote by $F_a\in
A^2({\rm ad}(L_c))=
iA^2$ its curvature, and by $F_a^{\pm}\in i A^2_{\pm}$ the components of $F_a$
with respect to the orthogonal splitting $A^2=A^2_+\oplus A^2_-$.
Every spinor $\Psi\in A^0(\Sigma^+_c)$ has a conjugate $\bar\Psi\in
A^0(\bar\Sigma^+_c)$, and we can interpret $\Psi\otimes\bar\Psi$ as a Hermitian
endomorphism of $\Sigma^+_c$. Let $(\Psi\otimes\bar\Psi)_0\in
A^0({\rm End}_0(\Sigma^+_c))$ denote the trace-free component of it.
\vspace{4mm}\\
The {\sl monopole equations} for a pair $(a,\Psi)\in
{\cal A}(L_c)\times A^0(\Sigma^+_c)$ are the equations [18]:
$$\left\{\begin{array}{ll} \hskip 4pt{\not}{D}_a\Psi=&0\\
\Gamma(F_a^+)=&2(\Psi\otimes\bar\Psi)_0
\end{array}\right.\eqno{(SW)}$$
\begin{pr} {\rm (The Weitzenb\"ock formula [11])}. Let $s$ be the scalar
curvature of
$(X,g)$. Fix a
${\rm Spin}^c(4)$-structure on $X$, and choose a $S^1$-connection $a\in{\cal
A}(L_c)$.
Then the following identity holds on $A^0(\Sigma_c)$ :
$$\hskip 4pt{\not}{D}_a^2=\nabla_A^*\nabla_A+\frac{1}{2}\Gamma(F_a)+\frac{s}{4}{\rm id}_{\Sigma_c
}\ .$$
\end{pr}
\begin{co} Let $\Psi\in A^0(\Sigma^+_c)$. Then
$$\parallel\hskip 4pt{\not}{D}_a\Psi\parallel^2+\frac{1}{2}\parallel\frac{1}{2}\Gamma(F_a^+)-
(\Psi\bar\Psi)_0)\parallel^2=\parallel\nabla_A\Psi\parallel^2+
\frac{1}{8}\parallel F_a^+\parallel^2+\frac{1}{4}\parallel\Psi\parallel^4+
\frac{1}{4}\int\limits_X s|\Psi|^2.$$
\end{co}
{\bf Proof: } By the Weitzenb\"ock formula we have
$$(\hskip 4pt{\not}{D}^2_a\Psi,\Psi)=(\nabla_A^*\nabla_A\Psi,\Psi)+
\frac{1}{2}(\Gamma(F_a^+)(\Psi),\Psi)+\frac{s}{4}(\Psi,\Psi)\ ,$$
since $\Gamma(F_a^-)$ vanishes on $\Sigma_c^+$; integration over $X$ yields:
$$\matrix{\parallel\hskip 4pt{\not}{D}_a\Psi\parallel^2+\frac{1}{2}\parallel\frac{1}{2}
\Gamma(F_a^+)-
(\Psi\bar\Psi)_0)\parallel^2=\int\limits_X
(\hskip 4pt{\not}{D}^2_a\Psi,\Psi)+\frac{1}{2}\int\limits_X|\frac{1}{2}\Gamma(F_a^+)-
(\Psi\bar\Psi)_0|^2=\cr
=\parallel\nabla_A\Psi\parallel^2+
\frac{1}{2}\int\limits_X(\Gamma(F_a^+),(\Psi\bar\Psi)_0)+\frac{1}{4}\int\lim
its_X
s|\Psi|^2+\cr+\frac{1}{2}\int\limits_X\frac{1}{4}|\Gamma(F_a^+)|^2-
\frac{1}{2}\int\limits_X(\Gamma(F_a^+),(\Psi\bar\Psi)_0)+
\frac{1}{4}\parallel\Psi\parallel^4\ .} $$
\begin{re} {\rm [18]} If $s\geq 0$ on $X$, then the only solutions $(a,\Psi)$
of
$(SW)$ are pairs $(a,0)$ with $F_a^+=0$.
\end{re}
\section{Seiberg-Witten Invariants}
The {\sl gauge group} ${\cal G}:={\cal C}^{\infty}(X,S^1)$ in the
Seiberg-Witten
theory {\sl is abelian} and acts on ${\cal A}(L_c)\times A^0(\Sigma^+_c)$ by
$(a,\Psi)\cdot f:=(a+f^{-1}df,f^{-1}\Psi)$, letting invariant the set of
solutions
of the equations $(SW)$. We denote by ${\cal W}^g_X(c)$ the moduli space of
solutions of the Seiberg-Witten equations, modulo gauge equivalence. A standard
technique provides a natural structure of finite dimensional {\sl real
analytic}
space in ${\cal W}^g_X(c)$ [6], [5], [16]. The {\sl expected dimension} of this
moduli space is
$$w_c=\frac{1}{4}(c^2-2e(X)-3\sigma(X))\ ,$$
where $e(X)$ and $\sigma(X)$ stand for the Euler characteristic and the
signature
of the oriented manifold $X$. A solution $(a,\Psi)$ is reducible (has
nontrivial
stabilizer) if and only if $\Psi=0$, and then the connection $a$ must be
anti-selfdual. We say that the metric $g$ is $c$-{\sl good} if the $g$-harmonic
representative of the de Rham cohomology class $c_{\rm DR}$ is {\sl not}
anti-selfdual.
If $g$ is $c$-good, then ${\cal W}^g_X(c)$ consists only of irreducible orbits.
Using the same technique as in Yang-Mills theory
([6], [5]), one defines a {\sl gauge invariant perturbation} of the
Seiberg-Witten
equations in order to get smooth moduli spaces of the expected dimension.
For a selfdual
form $\mu\in A^2_+$ we denote by ${\cal W}^{g,\mu}_X(c)$ the moduli space
of solutions
of the perturbed Seiberg-Witten equations
$$\left\{\begin{array}{lll} \hskip 4pt{\not}{D}_a\Psi&=&0\\
\Gamma(F_a^+ +i\mu)&=&2(\Psi\otimes\bar\Psi)_0
\end{array}\right.\eqno{(SW_{\mu})}$$
We refer to [15] for the following
\begin{lm}\hfill{\break}
1. For every $\mu\in A^2_+$, the moduli space ${\cal W}^{g,\mu}_X(c)$ is
compact.\hfill{\break}
2. There is a dense, second category set of perturbations $\mu\in A^2_+$, for
which the
irreducible part
${\cal W}^{g,\mu}_X(c)^*$ of ${\cal W}^{g,\mu}_X(c)$ is smooth and has the
expected
dimension.\hfill{\break}
3. If $g$ is $c$-good, and $\mu$ is small enough in the $L^2$ topology,
then ${\cal
W}^{g,\mu}_X(c)$ consists only of irreducible orbits, i.e. ${\cal
W}^{g,\mu}_X(c)$=${\cal W}^{g,\mu}_X(c)^*$ .\hfill{\break}
4. Let $g_0$ and $g_1$ be $c$-good metrics which can be connected
by a smooth path of $c$-good metrics, and let $\varepsilon_i>0$ be small
enough such
that ${\cal W}^{g_i,\mu_i}_X(c)={\cal W}^{g_i,\mu_i}_X(c)^*$ for all
perturbations
$\mu_i$ with $\parallel\mu_i\parallel<\varepsilon_i$. Then any two
moduli spaces
${\cal W}^{g_i,\mu_i}_X(c)\ ,\ \ i=0, 1$ , with
$\parallel\mu_i\parallel<\varepsilon_i$, which are smooth and have the expected
dimension, are cobordant.
\end{lm}
The first assertion is a simple consequence of the Weitzenb\"ock formula
and of the
Maximum Principle. The other three assertions follow as in Donaldson theory
by the
Sard theorem for smooth Fredholm maps, and by transversality arguments [5].
Note that
in 4. we mean cobordism between {\sl non-oriented} compact smooth
manifolds. A more
delicate analysis of the monopole equations [18] shows that, in fact, the
moduli
spaces ${\cal W}^{g,\mu}_X(c)^*$ come with
natural orientations, as soon as they are smooth and have the expected
dimension, and
that the conclusion in 4. holds for the oriented moduli spaces.
\vspace{3mm}\\
The Seiberg-Witten theory provides strong differentiable
invariants using only moduli spaces of dimension 0. Let $c$ be an integral
lift of
$w_2(X)$, with $w_c=0$, i.e.
$$c^2=2e(X)+3\sigma(X)\ .$$
Such a lift is called an {\sl almost canonical class}, since the condition
\hbox{$w_c=0$} is equivalent to the existence of an almost complex structure on
$X$ with first Chern class $c$ [10], [16].
Now fix an almost canonical class $c$, choose a $c$-good metric $g$, and a
small,
sufficiently general perturbation $\mu$. Then ${\cal W}^{g,\mu}_X(c)={\cal
W}^{g,\mu}_X(c)^*$ is compact, smooth of the expected dimension 0, and its
bordism
class is independent of $\mu$. Let
$n_c^g:=|{\cal W}^{g,\mu}_X(c)|$ mod 2 be the number of points modulo 2 of this
moduli space. Lemma 2.1 implies that
$n_c^g$ is also independent of $g$ if any two $c$-good metrics can be
connected by a
smooth 1-parameter family of $c$-good metrics.
The numbers $n_c:=n_c^g$ associated to such almost canonical
classes are called the mod 2-Seiberg-Witten invariants, and the classes
$c$ with
$n_c\ne 0$ are then called mod 2-Seiberg-Witten classes of index 0. By
definition
they are differentiable invariants, in the following sense: If $f:X'\longrightarrow
X$ is an
orientation-preserving diffeomorphism, and for an almost canonical class $c$ of
$X$ the Seiberg-Witten invariant $n_c$ is well defined, then $f^*(c)$ has the
same property, and $n_{f^*(c)}=n_{c}$.
\begin{re}
Let $c$ be an almost canonical class of $X$. \hfill{\break}
1. If $c^2\geq 0$ and $c_{\rm DR}\ne 0$ , then {\sl any} Riemannian metric
on $X$ is
$c$-good.\hfill{\break} 2. If $b_2^+\geq 2$, then any two $c$-good metrics
can be
connected by a smooth path of $c$-good metrics.
\end{re}
Therefore, if one of the two conditions above is satisfied, then the
\linebreak\hbox{mod
2-Seiberg-Witten} invariant $n_c$ is well-defined.
\vspace{3mm}
In the case $b_2^+=1$, invariants can still be defined, but the dependence of
$n_c^g$ on the metric $g$ must be taken into account: In the real vector space
$H^2_{\rm DR}(X)$ , consider the positive cone
$${\cal K}=\{u\in H^2_{\rm DR}(X) \ | u^2>0\}\ .$$
Fix a {\sl non-vanishing} cohomology class $k\in H^2_{\rm DR}(X)$ with
$k^2\geq 0$. The
cone
${\cal K}$ splits as the disjoint union of its connected components ${\cal
K}_{\pm}$,
where
$${\cal K}_{\pm}:=\{u\in{\cal K} \ |\ \pm u\cdot k>0\}\ .$$
If $c$ is
an almost canonical class, let $c^{\bot}$ be the hyperplane
$$c^{\bot}:=\{u\in H^2_{\rm DR}(X)\ |\ c\cdot u=0\}$$
If $c^{\bot}$ meets ${\cal K}_+$, then the intersection $c^{\bot}\cap{\cal
K}_+$ is
called the {\sl wall} of type $c$, and the two components of ${\cal
K}_+\setminus
c^{\bot}$ are called {\sl chambers} of type $c$.
For every Riemannian metric $g$ on $X$, let $\omega_g$ be a generator of
the real line
of $g$-harmonic selfdual 2-forms, such that $[\omega_g]\in{\cal K}_+$ .
Then the ray
${\Bbb R}_{>0}[\omega_g]\subset {\cal K}_+$ depends smoothly on the metric $g$.
The property of a
metric to be $c$-good has the following simple geometric interpretation:
\begin{re}\hfill{\break} Suppose $b_2^+(X)=1$. Then:\hfill{\break}
1. The metric $g$ is $c$-good iff the ray ${\Bbb R}_{>0}[\omega_g]$ does not lie in
the wall $c^{\bot}\cap{\cal K}_+$. \hfill{\break}
2. If $g_0$, and $g_1$ are $c$-good metrics, then $n_c^{g_0}=n_c^{g_1}$ iff
the two
rays ${\Bbb R}_{>0}[\omega_{g_i}]$ belong to the same chamber of type $c$.
\end{re}
The first assertion follows immediately from the definition. The second needs a
careful analysis of a 1-parameter family of 0-dimensional smooth
moduli spaces ${\cal W}^{g_t}_X(c)$ around the value of the parameter $t$ for
which the ray
${\Bbb R}_{>0}[\omega_{g_t}]$ crosses the wall $c^{\bot}\cap{\cal K}_+$ (see
[18], [15]).
\section{Monopoles on K\"ahler surfaces}
Let $(X,J,g)$ be an almost complex 4-manifold endowed with a Hermitian
metric $g$. The
almost complex structure $J$ defines a reduction of the structure group of the
tangent bundle $T_X$ of $X$ from ${\rm SO}(4)$ to $\U(2)$. In particular, we get
a {\sl
canonical}
${\rm Spin}^c(4)$-structure on \ $X$ \ via \ the \ canonical \ lifting \
\ $l:\U(2)\longrightarrow{\rm Spin}^c(4)$ [11]. Let $\omega_g$ be the K\"ahler form of $g$.
\begin{lm} {\rm [11]} The canonical ${\rm Spin}^c$-structure of an almost
complex Hermitian
4-manifold has the following properties:\hfill{\break}
1. There are canonical identifications
$\Sigma^+=\Lambda^{00}\oplus\Lambda^{02}$,
$\Sigma^-=\Lambda^{01}$.\hfill{\break}
2. Via these identifications, the map
$\Gamma:\Lambda^2_+\otimes{\Bbb C}\longrightarrow{\rm End}_0(\Sigma^+)$ is given by:
$$\Lambda^{20}\oplus\Lambda^{02}\oplus\Lambda^{00}\omega_g\ni(\lambda^{20},\
lambda^{02},
f\omega_g)\textmap{\Gamma}2\left[\matrix{-if&-*(\lambda^{20}\wedge\cdot)\cr
\lambda^{02}\wedge\cdot&if\cr}\right]\in{\rm End}_0(\Lambda^{00}\oplus\Lambda^{02
})\ .$$
\end{lm}
Suppose now that $(X,J,g)$ is a K\"ahler surface. This means that $J$ is
integrable,
and $\omega_g$ is closed (or equivalently, $J$ is Levi-Civita parallel). In
particular
the holonomy group of the Levi-Civita connection also reduces to $\U(2)$,
and the
splittings $\Lambda^p\otimes{\Bbb C}=\bigoplus\limits_{i+j=p}\Lambda^{ij}$ are
Levi-Civita
parallel. We get a $\U(2)$-connection in the holomorphic tangent bundle ${\cal
T}_X=T^{10}_X\simeq\Lambda^{01}$, which coincides with the {\sl Chern
connection} of
this bundle, i.e. with the unique connection compatible with the
holomorphic structure
and the Hermitian metric. The induced connection $c_0$ in the line bundle
$K_X^{\vee}={\rm det}({\cal T}_X)\simeq\Lambda^{02}$ also coincides with the Chern
connection of this Hermitian holomorphic line bundle.
Every other ${\rm Spin}^c(4)$ structure $\hat{P_c}\longrightarrow P$ on $(X,g)$ has as
spinor bundle
$$\Sigma_c=\Sigma\otimes M\ ,$$
where $M$ is a differentiable $S^1$-bundle with $2c_1(M)+c_1(K_X^{\vee})=c$.
(For a simply connected manifold $X$, $M$ is well defined up to
isomorphy by this condition.) $S^1$-connections in
${\rm det}(\Sigma^{\pm}_c)=K_X^{\vee}\otimes M^{\otimes 2}$ correspond to
$S^1$-connections
in $M$. Given $b\in{\cal A}(M)$, the curvature of the corresponding connection
$a\in{\cal A}(K_X^{\vee}\otimes M^{\otimes 2})$ is $F_a=F_{c_0}+2F_b$.
A half-spinor $\Psi\in A^0(\Sigma^+\otimes M)$ can be written as
$$\Psi=\varphi+\alpha\ ,\ \ \varphi\in A^0(M)\ ,\ \ \alpha\in
A^{02}(M)\ .$$
We put $J(M):=c_1(\Sigma^+\otimes M)\cup[\omega_g]$.
\begin{pr} Let $(X,g)$ be a K\"ahler surface with Chern connection $c_0$ in
$K_X^{\vee}$, $M$ a differentiable $S^1$-bundle with $J(M)<0$. A pair
$(b,\varphi+\alpha)\in {\cal A}(M)\times\left(A^0(M)\oplus A^{02}(M)\right)$
solves the
monopole equations iff:
$$
\begin{array}{l}F_b^{20}=F_b^{02}=0\ \\
\alpha=0\ ,\ \ \bar\partial_b(\varphi)=0 \ \\
i\Lambda F_b+\frac{1}{2}\varphi\bar\varphi+\frac{s}{2}=0\ .\end{array}\
\eqno{(*)}$$
\end{pr}
{\bf Proof: } The pair $(b,\varphi+\alpha)$ solves the equations $(SW)$ iff the
corresponding pair $(a,\varphi+\alpha)$ satisfies
$$\begin{array}{ll}F_a^{20}&=-\varphi\otimes\bar\alpha\\
F_a^{02}&=\ \alpha\otimes\bar\varphi\\
\bar\partial_b(\varphi)&=\ i\Lambda\partial_b(\alpha) \\
i\Lambda
F_a&=-\left(\varphi\bar\varphi-*(\alpha\wedge\bar\alpha)\right).\end{array}\
$$
By Corollary 1.7 it follows that $(b,\varphi+\alpha)$ solves $(SW)$ iff
$(b,\varphi-\alpha)$ does (Witten's trick).
Therefore
$\varphi\otimes\bar\alpha=\alpha\otimes\bar\varphi=0$, hence
$F_a^{20}=F_a^{02}=0$, and
$\varphi$ or $\alpha$ must vanish. Integrating the equation $i\Lambda
F_a=-\left(\varphi\bar\varphi-
*(\alpha\wedge\bar\alpha)\right)$ over $X$, we find:
$$J(M)=(2c_1(M)-c_1(K_X))\cup[\omega_g]=\int\limits_X\frac{i}{2\pi}
F_a\wedge\omega_g=
\frac{1}{8\pi}\int\limits_X(-|\varphi|^2+|\alpha|^2)\ ,$$
hence $\alpha=0$ if $J(M)<0$.
\hfill\vrule height6pt width6pt depth0pt \bigskip
The above proposition must be interpreted as follows: If $J(M)<0$, then the
solutions
of the monopole equations $(SW)$ are the pairs $(b,\varphi)\in{\cal A}(M)\times
A^0(M)$, such that
$b$ is the Chern connection of a holomorphic structure in $M$, $\varphi$ is a
holomorphic section, and the mean curvature $i\Lambda F_b$ of $b$ satisfies
the {\sl
generalized vortex equation} [16], [3], [4], [9]
$$i\Lambda F_b+\frac{1}{2}\varphi\bar\varphi+\frac{s}{2}=0\ .\eqno{(V_s)}$$
Moreover, every {\sl infinitesimal deformation} of a solution of the form
$(b,\varphi)$, $\varphi\ne 0$ of the monopole equation still vanishes in the
$\alpha$-direction. Therefore ${\cal W}_X^g(c)$ can be identified (as {\sl real
analytic space}) with the moduli space of pairs
$(b,\varphi)$ satisfying the above conditions, modulo the gauge group ${\cal
C}^{\infty}(X,S^1)$ of unitary automorphisms of $M$. Under the assumption
$J(M)<0$,
the action of the gauge group is free on the space of solutions, because
any solution
$(b,\varphi)$ has a non-vanishing section $\varphi$.
Alternatively, let ${\cal M}$ be a holomorphic line bundle with differentiable
support $M$, and $\varphi$ a holomorphic section of ${\cal M}$. For a
Hermitian metric
$h$ in ${\cal M}$, we denote by $F_h$ the curvature of the associated Chern
connection,
and we consider the following equation for $h$:
$$i\Lambda F_h+\frac{1}{2}\varphi\bar\varphi^h+\frac{s}{2}=0\ . \eqno{(V'_s)}$$
Standard arguments (see for instance [16], [9]) show that the problem of
classifying
the solutions $(b,\varphi)$ of $(*)$ modulo {\sl unitary automorphisms of
$M$} is
equivalent to the problem of classifying those pairs $({\cal M},\varphi)$
modulo
{\sl holomorphic isomorphisms}, for which the equation $(V'_s)$ has a solution.
\begin{pr}
Let $(X,g)$ be a compact K\"ahler surface, $({\cal M},\varphi)$ a
holomorphic line
bundle with a {\sl non-vanishing} holomorphic section $\varphi\in
H^0(X,{\cal M})$.
${\cal M}$ admits a metric
$h$ satisfying the equation $(V'_s)$ iff
$$c_1({\cal M})\cup[\omega_g]<\frac{1}{2}c_1(K_X)\cup[\omega_g]\ .$$
\end{pr}
{\bf Proof: } (cf. [3]) Fix a background metric $h_0$; any other metric $h$ has the form
$h=e^{2u}h_0$, with $u\in A^0$ a smooth function.The vortex equation $(V'_s)$
translates into
$$\Delta u+\frac{1}{2}|\varphi|^2_{h_0} e^{2u}+(i\Lambda
F_{h_0}+\frac{s}{2})=0\ .\eqno{(1)}$$
Set $q:=\int\limits_X(i\Lambda F_{h_0}+\frac{s}{2})=2\pi(c_1({\cal
M})-\frac{1}{2}c_1(K_X)\cup[\omega_g]$, and choose
$v\in A^0$ with
$$-\Delta v=(i\Lambda F_{h_0}+\frac{s}{2})-q \ .$$
Define $w:=2(u-v)$. Then $(1)$ is equivalent to the following equation in $w$:
$$\Delta w+(|\varphi|^2_{h_0} e^{2v})e^w +2q=0\ .\eqno{(2)}$$
Integrating over $X$, we see that if (2) has solutions, then
$q$ must be negative. On the other hand, by a well known result of Kazdan
and Warner
[3], (2) has a unique solution if $q<0$.
\hfill\vrule height6pt width6pt depth0pt \bigskip
\begin{th} {\rm [18], [16]}
Let $(X,g)$ be a simply connected K\"ahler surface, $c\in H^2(X,{\Bbb Z})$ with
$c\equiv c_1(K_X)$ mod 2, and $\pm c\cup[\omega_g]<0$.
\hfill{\break}
1. \ If $c\ \not\in\ {\rm NS}(X)$, then ${\cal W}_X^g(c)=\emptyset$ .\hfill{\break}
2. Suppose $c\in{\rm NS}(X)$. Then there is a natural real analytic isomorphism
\linebreak ${\cal W}_X^g(c)\simeq{\Bbb P}(H^0(X,{\cal M}))$, where ${\cal M}$ is
the (unique,
up to isomorphy) holomorphic line bundle with
$c_1(K_X^{\vee}\otimes{\cal M}^{\otimes 2})=\pm c$. \hfill{\break}
3. ${\cal W}^g_X(c)$ is always smooth.\ Let $D$ be the divisor of a
nontrivial section in
${\cal M}$. Then ${\cal W}^g_X(c)$ has the expected dimension iff
\hbox{$h^1({\cal O}_X(D)|_D)=0$}.
\end{th}
\section{Rationality of complex surfaces}
A compact complex surface is {\sl rational} iff its field of meromorphic
functions is
isomorphic to ${\Bbb C}(u,v)$. Such a surface is always simply connected and has
$b_2^+=1$
[2]. The following result has been
has been announced by R. Friedman and Z. Qin [8]. Whereas their proof uses
Donaldson theory and vector bundles techniques, our proof uses the new
Seiberg-Witten invariants, and our interpretation of these invariants in
terms of linear systems.
\begin{th} {\rm [17]}
A complex surface $X$ which is diffeomorphic to a rational surface is rational.
\end{th}
{\bf Proof: } The proof consists of the following three steps:
1.Any rational surface $X_0$ admits a
{\sl Hitchin metric} [12], i.e. a K\"ahler metric
$g_0$ with positive {\sl total scalar curvature}. This condition can be
written as
$c_1(K_{X_0})\cup[\omega_{g_0}]<0$.
Let $c$ be any integral lift of $w_2(X_0)$, such that $g_0$ is $c$-good,
i.e. such that
the moduli space ${\cal W}^{g_0}_{X_0}(c)$ contains no reducible solutions.
Since $p_g(X_0)=0$, $c$ has always type (1,1), and $g_0$ is $c$-good iff
$c\cup[{\omega_{g_0}}]\ne 0$.
We assert
that ${\cal W}^{g_0}_{X_0}(c)$ is then empty, and in particular, all
Seiberg-Witten
invariants $n_c^{g_0}$ computed with respect to this metric vanish.
Indeed, let ${\cal M}$ be the holomorphic line bundle defined in Theorem
3.4. If the
moduli space ${\Bbb P}(H^0(X_0, {\cal M}))$ was not empty, then
$$c_1({\cal M})\cup[\omega_{g_0}]\geq 0 \ .\eqno{(1)}$$
But we have
$$0>\pm c\cup[\omega_{g_0}]=(2c_1({\cal M})-c_1(K_{X_0}))\cup[\omega_{g_0}]\
,$$
hence, by (1)
$$0\leq 2c_1({\cal M})\cup[\omega_{g_0}]<c_1(K_{X_0})\cup[\omega_{g_0}]\ ,$$
which contradicts the assumption on the total scalar curvature of $g_0$.
\vspace{2mm}\\
2. Let now $X$ be a simply connected projective surface with ${\rm kod}(X)>
0$. We may
suppose that $X$ is the blow up in $k$ {\sl distinct} points of its minimal
model
$X_{\min}$. Denote by
$\sigma:X\longrightarrow X_{\min}$ the contraction to the minimal model, and by
$E=\sum\limits_{i=1}^k E_i$ the exceptional divisor. Fix an ample divisor
$H_{\min}$ on
$X_{\min}$, set
$H_n:=\sigma^*(n H_{\min})-E$, and for $n\gg 0$ choose a K\"ahler metric
$g_n$ on
$X$ with $[\omega_{g_n}]=c_1(H_n)$. Given $I\subset\{1,\dots,k\}$, define
$$\begin{array}{l}E_I:=\sum\limits_{i\in I} E_i\\ c_I:=2c_1(E_I)-c_1(K_X)\\
\bar I:=\{1,\dots,k\}\setminus I \ .\end{array}$$
Since $c_I$ is an almost canonical class, the expected dimension of the
corresponding Seiberg-Witten moduli space is 0. For
$n\gg 0$ we get
$c_I\cup [\omega_{g_n}]<0$, and Theorem 3.4 gives
$${\cal W}^{g_n}_X(c_I)\simeq\{E_I\}\ .$$
Therefore ${\cal W}^{g_n}_X(c_I)$ consists of a single smooth point, and
$$n_{c_I}^{g_n}= 1\ {\rm mod}\ 2\ .\eqno{(2)}$$
3. Suppose now that there is an orientation-preserving diffeomorphism
$f:X\longrightarrow X_0$, where
$X$ is projective surface with ${\rm kod} X\geq 0$. Since $X$ must have
$p_g(X)=0$, and
$\pi_1(X)=\{1\}$, it follows that, in fact, ${\rm kod} X>0$. Let
$g=f^*(g_0)$ denote the pull-back of a Hitchin metric to
$X$; clearly
$$n_{c_I}^g=0 \eqno{(3)}$$
for all $I\subset\{1,\dots,k\}$ such that $g$ is $c_I$-good.
We will now derive a contradiction in the following way: Using the
Enriques-Kodaira
classification of surfaces, it easy to see that the de Rham cohomology class
$ k_{\min}:=\sigma^*(c_{1,{\rm DR}}(K_{\min}))$ is non-trivial and satisfies
the condition
$k^2_{\min}\geq 0$. Therefore we can consider the upper positive cone
$${\cal K}_+:=\{u\in H^2_{\rm DR}(X)\ |\ u^2>0, \ u\cdot k_{\min}>0\}\ .$$
Clearly $[\omega_{g_n}]$ belongs to ${\cal K}_+$. We choose a harmonic
$g$-selfdual form
$\omega_g$, with
$[\omega_g]\in{\cal K}_+$.
\vspace{3mm}\\
{\bf Claim:} {\sl The rays ${\Bbb R}_{>0}[\omega_g]$ and ${\Bbb R}_{>0}[\omega_{g_n}]$
belong
either to the same chamber of type $c_I$ or to the same chamber of type
$c_{\bar I}$.}
\vspace{1mm}\\
{\bf Proof: } If not, then, since $c_I\cup [\omega_{g_n}]<0$, we get
$[\omega_g]\cdot c_I\geq0$
and
$[\omega_g]\cdot c_{\bar I}\geq 0$. Write
$$[\omega_g]=\sum\limits_{i=1}^k\lambda_i E_i+\sigma^*[\omega]\ ,$$
with $[\omega]\in
H^2_{\rm DR}(X_{\min})$. Then
$$\begin{array}{l} -\sum\limits_{i\in I}\lambda_i+\sum\limits_{j\in\bar
I}\lambda_j-[\omega]\cdot [K_{\min}]\geq 0\\
-\sum\limits_{j\in\bar I}\lambda_j+\sum\limits_{i\in
I}\lambda_i-[\omega]\cdot [K_{\min}]\geq 0\ .\end{array}$$
Adding these inequalities we find $[\omega]\cdot[K_{\min}]\leq 0$. But
$[\omega]\cdot [K_{\min}]=[\omega_g]\cdot k_{\min}>0$, because
$[\omega_g]\in{\cal K}_+$. This contradiction proves the claim.
\vspace{3mm}
It follows that either $g$ and $g_n$ are both $c_I$-good and
$n_{c_I}^g=n_{c_I}^{g_n}$, or $g$ and $g_n$ are both $c_{\bar I}$-good and
$n_{c_{\bar I}}^g=n_{c_{\bar I}}^{g_n}$. This gives now a contradiction
with (2) and
(3).
\hfill\vrule height6pt width6pt depth0pt \bigskip
Together with the results of Friedman and Morgan [7], we have:
\begin{th}{\rm (The Van de Ven conjecture [19])} The Kodaira dimension of
complex
surfaces is a
${\cal C}^{\infty}$-invariant.
\end{th}
{\bf Remark:} It is possible to couple the Seiberg-Witten equations to
connections
in unitary bundles. The solutions of these coupled Seiberg-Witten equations
over
K\"ahler surfaces again have a purely complex-geometric interpretation
[16]: The moduli
space of solutions can be identified---via generalized vortex equations--- with
moduli spaces of stable pairs [13], [4]. This construction could lead to
new invariants
which might be nontrivial for K\"ahler surfaces wit $p_g=0$.
\vspace{0.8cm}\\
\parindent0cm
\centerline {\Large {\bf Bibliography}}
\vspace{0.5cm}
1. Atiyah M., Hitchin N. J., Singer I. M.: {\it Selfduality in
four-dimensional Riemannian geometry}, Proc. R. Lond. A. 362, 425-461 (1978)
2. Barth, W., Peters, C., Van de Ven, A.: {\it Compact complex surfaces},
Springer Verlag (1984)
3. Bradlow, S. B.: {\it Vortices in holomorphic line bundles over closed
K\"ahler manifolds}, Comm. Math. Phys. 135, 1-17 (1990)
4. Bradlow, S. B.: {\it Special metrics and stability for holomorphic
bundles with global sections}, J. Diff. Geom. 33, 169-214 (1991)
5. Donaldson, S.; Kronheimer, P. B.: {\it The Geometry of four-manifolds},
Oxford Science Publications (1990)
6. Freed, D. S.; Uhlenbeck, K. K.: {\it Instantons and Four-Manifolds},
Springer Verlag (1984)
7. Friedman, R., Morgan, J.W.: {\it Smooth 4-manifolds and Complex Surfaces},
Springer Verlag 3. Folge, Band 27 (1994)
8. Friedman, R., Qin, Z.: {\it On complex surfaces diffeomorphic to
rational surfaces}, Preprint (1994)
9. Garcia-Prada, O.: {\it Dimensional reduction of stable bundles, vortices
and stable pairs}, Int. J. of Math. Vol. 5, No 1, 1-52 (1994)
10. Hirzebruch, F., Hopf H.: {\it Felder von Fl\"achenelementen in
4-dimensionalen 4-Mannigfaltigkeiten}, Math. Ann. 136, (1958)
11. Hitchin, N.: {\it Harmonic spinors}, Adv. in Math. 14, 1-55 (1974)
12. Hitchin, N.: {\it On the curvature of rational surfaces}, Proc. of Symp.
in Pure Math., Stanford, Vol. 27 (1975)
13. Huybrechts, D.; Lehn, M.: {\it Stable pairs on curves and surfaces},
J. Alg. Geometry, (1995)
14. Kobayashi, S.: {\it Differential geometry of complex vector bundles},
Princeton University Press, (1987)
15. Kronheimer, P., Mrowka, T.: {\it The genus of embedded surfaces in the
projective plane}, Preprint (1994)
16. Okonek, Ch.; Teleman A.: {\it The Coupled Seiberg-Witten Equations,
Vortices, and Moduli Spaces of Stable Pairs}, Preprint, January, 13-th 1995
17. Okonek, Ch.; Teleman A.: {\it Seiberg-Witten invariants and the Van de
Ven conjecture}, Preprint, February, 8-th 1995
18. Witten, E.: {\it Monopoles and four-manifolds}, Mathematical Research
Letters 1, 769-796 (1994)
19. Van de Ven, A,: {\it On the differentiable structure of certain algebraic
surfaces}, S\'em. Bourbaki ${\rm n}^o$ 667, Juin (1986)
\vspace{0.5cm}\\
Authors addresses:\\
Mathematisches Institut, Universit\"at Z\"urich,\\
Winterthurerstrasse 190, CH-8057 Z\"urich\\
e-mail:[email protected]
\ \ \ \ \ \ \ \ \ [email protected]
\end{document}
--========================_18689640==_
Content-Type: text/plain; charset="us-ascii"
Dr. Andrei Teleman (e-mail: [email protected])
Mathematisches Institut der Universitaet Zuerich
Winterthurer Strasse 190, CH-8057 Zuerich-Irchel
Tel.: (+411) 257 58 65; Fax 2575706
--========================_18689640==_--
|
1995-06-01T06:20:23 | 9505 | alg-geom/9505038 | en | https://arxiv.org/abs/alg-geom/9505038 | [
"alg-geom",
"math.AG"
] | alg-geom/9505038 | Dan Abramovich | Dan Abramovich | Uniformity of stably integral points on elliptic curves | 10 pages. Postscript file available at
http://math.bu.edu/INDIVIDUAL/abrmovic/integral.ps, AMSLaTeX | null | null | null | null | A common practice in arithmetic geometry is that of generalizing rational
points on projective varieties to integral points on quasi-projective
varieties.
Following this practice, we demonstrate an analogue of a result of L.
Caporaso, J. Harris and B. Mazur, showing that the Lang - Vojta conjecture
implies a uniform bound on the number of stably integral points on an elliptic
curve over a number field, as well as the uniform boundedness conjecture
(Merel's theorem).
| [
{
"version": "v1",
"created": "Thu, 1 Jun 1995 01:02:11 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Abramovich",
"Dan",
""
]
] | alg-geom | \section{Introduction}
Let $X$ be a variety of logarithmic general type, defined over a number field
$K$. Let $S$ be a finite set of places in $K$ and let ${{\cal O}_{K,S}}$ be the ring of
$S$-integers. Suppose that ${\cal{X}}$ is a model of $X$ over ${\mbox{Spec }}{{\cal O}_{K,S}}$.
As a natural generalizasion of theorems of Siegel and Faltings, It was
conjectured by S. Lang and P. Vojta (\cite{vojta}, conjecture 4.4) that the
set of $S$-integral points ${\cal{X}}({{\cal O}_{K,S}})$ is not
Zariski dense in ${\cal{X}}$. In case $X$ is projective, one may chose an arbitrary
projective model ${\cal{X}}$ and then ${\cal{X}}({{\cal O}_{K,S}})$ is identified with $X(K)$. In
such a case, one often refers to this conjecture of Lang and Vojta as just
Lang's conjecture.
L. Caporaso, J. Harris
and B. Mazur \cite{chm} apply Lang's conjecture in the following way: Let
$X\rightarrow B$
be a smooth family of curves of genus $g>1$. Let $X^n_B\rightarrow B$ be the $n$-th
fibered
power of $X$ over $B$. In \cite{chm} it is shown that for high enough $n$,
the variety $X^n_B$ dominates a variety of general type. Assuming Lang's
conjecture, they deduce the following remarkable result: the number of rational
points on a curve of genus $g$ over a fixed number field is uniformly bounded.
In this note we study an analogous implication for elliptic curves. Let $E/K$
be an elliptic curve over a number field, and let $P\in E(K)$. We say that $P$
is stably $S$-integral, denoted $P\in E(K,S)$, if $P$ is $S$-integral after
semistable reduction (see \S\ref{stably}). Our main theorem states (see
\S\ref{main}):
\begin{th}(Main theorem in terms of points) Assume that the Lang - Vojta
conjecture
holds. Then for any number field $K$ and a finite set of places $S$, there is
an
integer $N$ such that for any elliptic curve $E/K$ we have $\#E(K,S)<N$.
\end{th}
Since the moduli space of elliptic curves is only one-dimensional, the
computations and the proofs are a bit simpler than the higher genus cases.
One can view the results in this paper as a simple application of
the methods of \cite{chm}.
\subsection{Overview} In section \ref{correlation} we prove a basic lemma
analogous to lemma 1.1 \cite{chm} on uniformity of correlated points. In
section \ref{level3} we study a particular pencil of elliptic
curves which is the main building block for proving theorem 1. In section
\ref{twist} we look at quadratic twists of an elliptic curve, motivating the
study in section
\ref{stably} of stably integral points. Section \ref{main} gives a proof of the
main theorem.
In section \ref{ubc} we will refine our methods and show that the Lang - Vojta
conjecture implies the uniform boundedness conjecture for torsion on elliptic
curves, thus giving a conditional (and therefore obsolete) proof of the
following theorem of Merel:
\begin{th}(Merel, \cite{merel}) For and integer $d$ there is an integer $N(d)$
such that given a number field $K$ with $[K:{\Bbb{Q}}]=d$, and given an elliptic
curve $E/K$, then $\#E(K)_{tors}<N(d)$.
\end{th}
It should be noted that the methods introduced in section \ref{ubc} were
essential for the developments in \cite{abr},\cite{abr1}.
It would be interesting if one could apply the method to the study
points on abelian varieties in general.
\subsection{Acknowledgements} I am indebted to the ideas in the
work \cite{chm} of Lucia Caporaso, Joe Harris and Barry Mazur and to
conversations with
them. Thanks to Joe Silverman and Sinnou David for discussions of points on
elliptic curves and much encouragement. Much of this text was written during a
pleasant
visit with the group ``Problemes Diophantiens'' in Paris. It is also a pleasure
to thank Henri Darmon, Gerhard Frey Brendan Hassett and Felipe Voloch for
their suggestions. Special thanks are due to Frans Oort who read an
earlier version of this paper and sent me many helpful comments\footnote{I
take the opportunity to wish Professor Oort a happy 60th birthday.}.
\subsection{The Lang - Vojta conjecture}\label{lv}
A common practice in arithmetic geometry is that of generalizing rational
points on projective varieties to integral points on quasi-projective
varieties. We can summarize this in the following table, which will be
explained
below:\\
\begin{tabular}{l|l}\hline\hline
Number field $K$ & Ring of $S$-integers $O_{K,S}$ \\[2mm] \hline
Projective variety $X$ over $K$ & \parbox{3in}{\vspace*{1mm} Quasi projective
variety $X$
and a model
${\cal{X}}$ over $O_{K,S}$ } \\ \hline \vspace*{1mm}
Rational point $P\in X(K)$ & Integral point $P\in {\cal{X}}({{\cal O}_{K,S}})$ \\[2mm] \hline
$X$ of general type & $X$ of log-general type \\
e.g.: $C$ a curve of genus $>1$ & \parbox{3in}{ e.g.: $E$ an elliptic curve
with the origin
removed } \\ \hline
Faltings' theorem: $C(K)$ finite & Siegel's theorem: $E(O_{K,S})$ finite
\\
\hline \vspace*{1mm}
\parbox{3in}{\vspace*{1mm} Lang's conjecture: If $X$ is of general type then
$X(K)$ not
Zariski
dense\\ } &
\parbox{3in}{\vspace*{1mm} Lang-Vojta conjecture: If $X$ is of logarithmic
general type
then
${\cal{X}}({{\cal O}_{K,S}})$ not
Zariski dense } \\ \hline
\parbox{3in}{\cite{chm}: Lang's conjecture implies uniformity of $\#C(K)$} &
\parbox{3in}{
\vspace*{1mm} ?? }\\ \hline\hline
\end{tabular}
\vspace*{2mm}
We remind the reader of the definition of a variety of log general type:
\begin{dfn} Let $X$ be a quasi-projective variety over ${\Bbb{C}}$. Let $f:Y \rightarrow X$
be a resolution of singularities, that is, a proper, birational morphism where
$Y$ is a smooth variety. Let $Y\subset Y_1$ be a projective compactification,
such that $Y_1$ is smooth and such that
$D = Y_1\setminus Y$ is a divisor of normal crossings. Then $X$ is said to be
of logarithmic general type if for some positive integer $m$, the rational map
defined by the complete linear system $|m(K_{Y_1}+D)|$ is birational to the
image.
\end{dfn}
Let $X$ be a quasi-projective variety
of logarithmic general type, defined over a field $K$ which is finitely
generated over ${\Bbb{Q}}$ (e.g., a number field). Let $R$ be a ring, finitely
generated over ${\Bbb{Z}}$, whose
fraction field is $K$ (e.g., the ring of $S$-integers in a number field).
Choose a model ${\cal{X}}$ of $X$ over $R$. The following is
a well-known
conjecture of Lang and Vojta (\cite{vojta}, conjecture 4.4).
{\bf Conjecture.} The set of integral points ${\cal{X}}(R)$ is not Zariski dense in
${\cal{X}}$.
In case $X$ in the conjecture above is projective, then logarithmic general
type means just general type; and integral points are just rational points.
\subsection{What should the last entry in the table read?}
We would like to fill in the question mark in the last entry in the table. One
is tempted to ask:
{\em Does the Lang - Vojta conjecture imply the uniformity of
$\#E(O_{K,S})$?}\\
but one sees immediately that this cannot be true without some restrictions.
Most importantly, one has to restrict the choice of the model $E$, as can be
seen in the following example:
Let $E$ be an elliptic curve over a number field $K$ such that $E(K) $ is
infinite. Fix $P_1,\ldots ,P_n\in E(K)$. Choose an equation $$y^2 = x^3 +
Ax+B$$ for $E$, where $A,B\in O_{K,S}$ for some finite $S$. Choose $c\in {{\cal O}_{K,S}}$
such that for each $P_i$ one has $c^2x(P_i),c^3 y(P_i)\in {{\cal O}_{K,S}}$. By changing
coordinates $x_1 = c^2 x, y_1 = c^3y$, one obtains a new model $E_1$ given by
the equation $y_1^2 = x_1^3 + c^4 A x_1 + c^6B$, on which all the points $P_i$
are integral.
The problem with this new model arises because when one changes coordinates,
one blows up the closed point corresponding to the origin at primes dividing
$c$, so the resulting model has ``extraneous'' components over these primes. We
are led to modify the statement:
\begin{th}(Main theorem in terms of models, see \S\ref{stably}) Assume that the
Lang-Vojta
conjecture holds. Then for any number field $K$ and a finite set of places $S$
there is an integer $N$ such that for any stably
minimal elliptic curve ${\cal{E}}$ over ${{\cal O}_{K,S}}$ we have $\#{\cal{E}}({{\cal O}_{K,S}})<N$.
\end{th}
It turns out that stably minimal models are very minimal indeed. In
particular we will see that N\'eron models, the canonical models of
elliptic curves over rings of integers, are not necessarily sufficiently
minimal for the purpose of
our methods. On the other hand
we will see that semistable models are stably minimal. A precise
definition of a stablly minimal model, and how to obtain a canonical one
from the N\'eron model, will be given in \S\ref{stably}.
In the case of semistable elliptic curves,
it is worthwhile to state an immediate corollary of the theorem:
\begin{cor} The Lang-Vojta conjecture implies that the number of
integral points on semistable elliptic curves over ${\Bbb{Q}}$ is bounded.\end{cor}
{\bf Remark.} Another conjecture of Lang (see \cite{lang1}) predicts that the
number of all
$S$-integral points on so called {\em quasi-minimal} elliptic curves should be
bounded in
terms of the rank and the number of elements in $S$. In view of the corollary,
one is tempted to ask whether the rank of an elliptic curve can be bounded in
terms of the places of additive reduction of the elliptic curve.
\section{Boundedness of correlated points}\label{correlation}
One of the main ideas in \cite{chm} is, that in order to bound the number of
points on curves it is enough to show that they are {\em correlated}, that is,
there is
an algebraic relation between all $n$-tuples of these points. This is the
content of the lemma below. First,
some notation.
Let $\pi:X\rightarrow B$ be a family of smooth irreducile curves over a field $K$. We
denote by
$\pi_n:X_B^n\rightarrow B$ the n-th
fibered power of $X$ over $B$. Given a point $b\in B$ we denote by $X_b$ the
fiber of $X$ over $b$; Similarly, given $Q=(P_1,\ldots,P_n)\in X^n_B$ we
denote by $X_Q\subset X_B^{n+1}$ the fiber of $X_B^{n+1}$ over $Q$. Note that
if $\pi_n(Q) = b$ then $X_Q \simeq X_b$. Denote by $p_n:X_n\rightarrow X_{n-1}$ the
projection onto the first $n-1$ factors.
Assume that we are given a
subset ${\cal{P}}\subset X(K)$ (typical examples would be rational points, or
integral
points on some model of $X$). Again, we denote by ${\cal{P}}_B^n\subset
X_B^n$ the fibered power of ${\cal{P}}$ over $B$ (namely the union of the
$n$-tuples of points in ${\cal{P}}$ consisting of points in the same fiber), and by
${\cal{P}}_b$ the points of ${\cal{P}}$ lying over $b$.
\begin{dfn} Assume
that for some $n$ there is a proper closed subset $F_n\subset
X^n_B$ such that ${\cal{P}}_B^n\subset F_n$. In such a case we say that the
subset ${\cal{P}}$ is {\em $n$-correlated}.
\end{dfn}
For instance, a subset ${\cal{P}}$ is
1-correlated if and only if it is not Zariski dense; in which case it is easy
to see that, over some open set in $B$, the number of points of ${\cal{P}}$ in each
fiber is bounded. This is generalized by the following lemma:
\begin{lem}(compare \cite{chm}, lemma 1.1) Let $X\rightarrow B$ be a family of smooth
irreducible curves, and let
${\cal{P}}\subset
X(K)$ be an $n$-correlated subset.
Then there is a dense open
set $U\subset B$ and an integer $N$ such that for every $b\in U$, we have
$\# {\cal{P}}_b\leq N$.
\end{lem}
{\bf Proof:} Let $F_n = \overline{{\cal{P}}_B^n}$ be the Zariski closure, and
$U_n= X_B^n\setminus F_n$ the complement. We now define be descending
induction: $U_{i-1} =
p_i (U_i)$ and $F_{i-1}=X^{i-1}_B\setminus U_{i-1} $ the complement. Notice
that
over $U_{i-1}$, the map $p_i$ restricts to a finite map on $F_i$: by definition
if $x\in U_{i-1}$ then $p_i^{-1}(x)\not\subset F_i$, and $p_i^{-1}(x) $ is
an irreducible curve. Therefore
the number of points in the fibers of this map is bounded: if $x\in U_{i-1}$
then we can write $\#(p_i^{-1}(x)\cap F )\leq d_i$.
Let $U=U_0\subset B$. We claim that over $U$, the number of points of ${\cal{P}}$
in each fiber is bounded. Consider a point $b\in U$.
Case 1: ${\cal{P}}_b \subset F_1$. In this case, the number of points on
${\cal{P}}_b$ is bounded by $d_1$.
Case 2: there is some $P\in {\cal{P}}_b, P\not\in F_1$, but $X_P\cap{\cal{P}}_b^2
\subset F_2$.
In this case the number of points is bounded by $d_2$.
Case $i$: $Q=(P_1,\ldots,P_{i-1})\in {\cal{P}}_b^{i-1}\setminus F_{i-1}$ but
$X_Q\cap {\cal{P}}_b^i\subset
F_i$. Here the number of points is bounded
by $d_i$.
Notice that in the case $i=n$ we have by definition $X_Q\cap {\cal{P}}_B^n\in
F_n$, and the process stops. Therefore $N=\displaystyle \max_i d_i$ is a bound
for the number of ${\cal{P}}$ points in each fiber over $U$.
{\bf Example} (\cite{chm}): Let $X\rightarrow B$ be a family of smooth, irreducible
curves of
genus $>1$ over a number field $K$. Assume that Lang's conjecture holds true.
Then in \cite{chm} it is shown that $X(K)$ is $n$-correlated, and the lemma
above, with noetherian induction, is used to obtain the existence of a uniform
bound on the number of rational points on such curves.
{\bf Example:} Assume that $X_K\rightarrow B_K$ is a semistable family of curves of
genus 1,
together
with a section $s:B_K\rightarrow X_K$, and assume
that over an open set $B_0\subset B_K$ the restricted family $X_0\rightarrow B_0$ is
smooth. Assume that the Lang - Vojta conjecture holds true. Given a semistable
model $X$ of $X_K\setminus s(B_K)$ over ${{\cal O}_{K,S}}$, we
will later show that $X({{\cal O}_{K,S}})\cap X_0$ is $n$-correlated for some integer $n$.
We will deduce the existence of a uniform bound on the number of integral
points on curves in this family.
{\bf Example:} Let $S$ be a finite set of places in $K$. Assuming that the
Lang - Vojta
conjecture holds true, we will show that the set of
stably $S$-integral points on any family of elliptic curves over a number field
$K$ is $n$-correlated for some $n$. We will deduce the existence of a uniform
bound on the number of stably $S$-integral points on an elliptic curve.
\section{Moduli of elliptic curves with level 3 structure}\label{level3} We
introduce here a building block in the proof of the main theorem.
\subsection{The geometry} Let $E_1$ be the universal family of elliptic
curves over ${\Bbb{C}}$ with full symplectic level 3 structure. The surface $E_1$
can be identified with the total space of the elliptic
pencil written in bi-homogeneous coordinates as: $$(*)\quad
\lambda(X^3+Y^3+Z^3)
- 3\mu XYZ=0,$$ mapping to the moduli space ${\Bbb{P}}^1$ via
$[\lambda:\mu]$. This equation gives a smooth model, which by abuse of notation
we will also call $E_1$, of this space over ${\mbox{Spec }}{\Bbb{Z}}[1/3]$. We
may choose the section $\Theta$ over the point $[X:Y:Z]=[1:-1:0]$ as the origin
of
the elliptic surface. Over ${\mbox{Spec }} {\Bbb{Z}}[1/3]$, the fibers of the elliptic pencil
possess level 3 structure of type $\mu_3 \times {\Bbb{Z}}/3{\Bbb{Z}}$, in a way which is
described precisely by Rubin and Silverberg in \cite{rs}; however in this
section we will work over ${\Bbb{C}}$.
The pencil $E_1\rightarrow {\Bbb{P}}^1$ is semistable, possessing four singular fibers,
having 3 nodes each, over $\Sigma_0=\{0,1,\zeta_3,\zeta_3^2\}\subset {\Bbb{P}}^1$
where $\zeta_3$ is a primitive third root of 1.
Let $L$ be the pullback of a line from the plane, and
let $S_1,\ldots,S_9$ be the exceptional curves over the nine base points of the
pencil fixing $S_1=\Theta$ to be the origin of the elliptic surface. Let
$F$ be a fiber of the elliptic surface. We have the linear equivalence $-F\sim
-3L +S_1+\cdots+S_9$.
As a pencil of cubics with smooth total space, one easily calculates the
relative dualizing sheaf, as follows:
We know that $\omega_{{\Bbb{P}}^2} =
{\cal{O}}_{{\Bbb{P}}^2}(-3)$. The canonical sheaf of the blown up surface is therefore
${\cal{O}}( -3L +S_1+\cdots+S_9)$. Therefore we have
$\omega_{E_1} \simeq {\cal{O}}(-F)$, and $\omega_{E_1/{\Bbb{P}}^1} \simeq {\cal{O}}(F)$.
Let $\pi_n:E_n\rightarrow {\Bbb{P}}^1$ be
the $n$-th fibered power of $E_1$ over ${\Bbb{P}}^1$. Denote by $\pi_{n,i}:E_n\rightarrow
E_1$ the projection onto the $i$-th factor. We have that
$\omega_{E_n/{\Bbb{P}}^1} \simeq {\cal{O}}(nF)$, and therefore $\omega_{E_n}
\simeq {\cal{O}}((n-2)F)$. We denote by
$\Theta_n=\sum_{i=1}^n\pi_{n,i}^*\Theta$, the theta divisor. We denote by
$\Sigma_n=\pi_n^{-1}\Sigma_0\subset E_n$
the locus of singular
fibers, the inverse image of $\Sigma_0\subset {\Bbb{P}}^1$.
It should be noted that $E_n$ is singular, but not too singular:
\begin{lem}
There is a desingularization $f_n:\tilde{E_n}\rightarrow{E_n}$ such that
$\omega_{\tilde E_n} \simeq f_n^*\omega_{ E_n}(D)$, for some effective divisor
$D$ such that $f_n(D)\subset \Sigma_n$, and such that $f_n^*\Theta_n$ is a
reduced divisor of normal crossings.
\end{lem}
{\bf Proof:} The existence of a desingularization with $\omega_{\tilde E_n}
\simeq f_n^*\omega_{ E_n}(D)$ follows from \cite{chm}, lemma 3.3, or lemma
3.6 of \cite{viehweg}. The desingularization is given by a succession of
blowups along smooth centers. Since the
singular locus of $E_n$ meets $\Theta_n$ transversally, the centers of the
blowups can be taken to be transversal to $\Theta_n$, and therefore its inverse
image is a divisor of normal crossings.
This lemma shows that $E_n\setminus\Theta_n$ has log canonical
singularities. This means that sections of powers of $\omega_{E_n}(\Theta_n)$
give regular sections of the logarithmic pluricanonical sheaves of
$\tilde{E_n}$; therefore, in order to prove that $E_n$ is of logarithmic
general type, there is no need to pass to a resolution of singularities - it
suffices to show that $\omega^k_{E_n}(k\Theta_n)$ has many sections.
\begin{lem} \begin{enumerate}
\item The line bundle $\omega_{E_1/{\Bbb{P}}^1}(\Theta_1)$ is the pullback of
an ample bundle along a birational morphism.
\item Fix $n>2$. Then
${E_n}\setminus\Theta_n$ is of logarithmic general type. Moreover, the base
locus of the logarithmic pluricanonical linear series is contained in
$\Theta_n$.
\end{enumerate}
\end{lem}
{\bf Proof: } Let $Y$ be the blowup of ${\Bbb{P}}^2$ at all the base points of our
pencil except
$[1,-1,0]$. The surface $Y$ is the same as $E_1$ blown down along $\Theta$.
The line
bundle $\omega_{E_1/{\Bbb{P}}^1}(\Theta_1)$ is the pullback of
$M={\cal{O}}(3)\otimes{\cal{O}}(-(S_2+\cdots+S_9))$ from $Y$. On $Y$, $M$ is
represented by the strict transform of one of the cubics of the pencil, hence
it is a nef line bundle; it
has self intersection number 1, therefore it is nef and big. In fact, it is
easy to see by a dimension count that the complete linear system of sections of
$M^{\otimes 3}$ gives a birational morphism of $Y$ to a surface in projective
space,
which blows down only the fibral components of $E_1$ which do not meet $S_1$.
Part (2) follows by taking the products of sections pulled back along the
projections:
On $E_3$, let $p_{ij},p_k$ be the projections to the $i$-th and $j$-th
factors,
respectively $k$-th factor. We have the inclusion $p_{12}^*\omega_{E_2}
\otimes p_3^*\omega_{E_1/{\Bbb{P}}^1}(\Theta_1) \subset\omega_{E_3}(\Theta_3)$.
The sections of a power of this subsheaf give a map which generically
separates between
points whose third factors are different. Part 1 of this lemma implies
that the base locus of these sections is
contained in the theta divisors. By repeating this
for the other two
projections, we find that sections of powers of $\omega_{E_3}(\Theta_3)$
generically separate points; in particular $E_3\setminus\Theta_3$ is of
logarithmic general type. Similarly, $E_n$ is of logarithmic general type for
$n\geq 3$.
\subsection{Boundedness of integral points on elliptic curves with level 3
structure}
Let $K$ be a number field containing ${\Bbb{Q}}(\zeta_3)$, and let $E$ be an
elliptic curve with full symplectic level 3
structure over $K$. Let $R={\cal O}_K[1/3]$. The curve $E$ occurs as a fiber in
the
surface over a point in ${\Bbb{P}}^1(K)$, which automatically has semistable
reduction over $R$. Let ${\cal{E}}$ be the semistable model. Given any three
$R$-integral
points $P_i$ on ${\cal{E}}\setminus 0$, the point $(P_{1}, P_{2}, P_{3})$ gives
rise to
an $R$-integral point of the scheme $E_3\setminus\Theta_3$ (where by abuse of
notation, we use the model of $E_3$ over $R$ which is the fibered cube of the
given model $(*)$ of $E_1$).
Assume that the Lang - Vojta conjecture holds for the variety $E_3\setminus
\Theta_3$. Thus the Zariski closure $F$ of the set of integral
points $(E_3\setminus\Theta_3)(R)$ is
a proper subvariety of $E_3$; in other words, the set ${\cal{P}}=(E\setminus
\Theta)(R)$ is 3-correlated. By lemma 1,
there is a dense open set $U
\in {\Bbb{P}}^1$ such that the number of integral points of fibers over $U(K)$ is
bounded. The complement of
$U$ is a finite number of points, therefore by Siegel's theorem there is a
bound on the number of integral points on these curves as well.
{\bf Open problem:} Show that the $S$-integral points on $E_3\setminus
\Theta_3$
are not Zariski dense.
\section{Quadratic twists of an elliptic curve}\label{twist}
As a ``complementary case'' to the last section we will discuss here a typical
case of isotrivial families of elliptic curves. This is in direct analogy with
the exposition in \cite{chm}, \S\S 2.2. It will give us a good hint about the
type of
models of elliptic curves we need in order to obtain boundedness. A slightly
more general version of the example here will be used in the proof of the main
theorem.
Let $E: y^2 = x^3 + Ax + B$ be a fixed elliptic curve. We denote $f(x) =
x^3 + Ax + B$. We assume that $A$ and
$B$ are relatively prime $S$-integers in a number field $L$, where $S$ is a
finite set of
places.
All the quadratic twists of the curve over $L$ can be written in the form:
$$E_t: ty^2 = f(x)$$
where $t$ may be chosen $S$-integral.
We may form the family of Kummer surfaces associated to $E_t$:
$$K_t: t^2 z^2 = f(x_1) f(x_2). $$ We have a morphism of $K_t$ to
$K_1$ via $(x_1,x_2,z,t) \mapsto (x_1,x_2,tz)$. It can be easily verified that
the affine surface $K_1$ is of
logarithmic general type.
Assume that the
Lang - Vojta conjecture
holds for $K_1$. It now follows from lemma 1, that there is a uniform bound on
the number of integral points
on $E_t$: the integral points on $K_1$ are not Zariski dense, therefore the
integral points on $K_t$ are not Zariski dense, since they map to integral
points on $K_1$. Lemma 1 says that there is an open set $U\subset {\Bbb A}_1$
such that there is a bound on the number of points on $E_t$ for an
$S$-integer
$t$ in $U$; for the remaining finitely many integers $t$ we can use Siegel's
theorem. The same result can be obtained using any of the higher Kummer
varieties $(E\times\cdots\times E)/(\pm 1)$.
Note that the integral points on $E_t$ are not the same as the
integral points on the N\'eron model. Suppose that $t$ is square free. Then a
N\'eron integral point $P$ is
integral on $E_t$, away from characteristic 2 and 3, if at a prime of additive
reduction $P$ does not reduce to the
component of the origin on the N\'eron model. In other words, even after
semistable reduction (obtained by taking $y' = \sqrt{t} y$), $P$ remains
integral on the N\'eron model. We call such
points {\em stably integral}.
{\bf Open problem:} Show that the $S$-integral points on $K_1$
are not Zariski dense. As a first step, describe the images of nontrivial
morphisms ${\Bbb
A}_1\setminus 0 \rightarrow K_1$.
\section{Stably integral points}\label{stably}
\begin{dfn}
Let $E$ be an elliptic curve over a number field $K$, let $S$ be a finite set
of
places in $K$,
and $P$ be a $K$-rational point on $E$. We say that $P$ is {\em stably
$S$-integral}, written $P\in E(K,S)$ if the following holds: let $L$ be a
finite extension of $K$,
and let $T$ be the set of places above $S$, and assume that $E$ has semistable
reduction ${\cal{E}}$ over ${\cal{O}}_{L,T}$; then $P\in ({\cal{E}}\setminus 0)({\cal{O}}_{L,T})$.
In
other words, $P$ is
integral on the semistable model of $E\setminus 0$ over some
finite
field extension $L$ of $K$, where $T$ is the set of all places over $S$.
\end{dfn}
Stably integral points should be thought of as the rational points which are
integral over the algebraic closure of the field. In this sense, they are a
good
analogue on elliptic curves, for rational points on curves of higher genus.
It is important to note that stably integral points can be described as the
integral points on a certain type of model of the curve.
\begin{dfn} Let $E$ be an elliptic curve over a number field $K$, let $S$ be a
set of places containing all places dividing 2 and 3, and let ${\cal{E}}$
be the N\'eron model over ${{\cal O}_{K,S}}$. Let $D_0$ be the zero section of ${\cal{E}}$. Let
$S_a$ be the set
of places of additive reduction, and for a place $v$ let ${\cal{E}}^0_v$ be the zero
component. Let $D= D_0\cup \bigcup_{v\in S_a} {\cal{E}}^0_v$ and let ${\cal{E}}_0 =
{\cal{E}}\setminus
D$. We call ${\cal{E}}_0$ the {\em Stably minimal model} of $E$.
\end{dfn}
\begin{prp} Let $S$ be a set of places containing all places dividing 2 and
3. Then the $S$-integral points on the stably minimal model are
precisely the stably $S$-integral points.
\end{prp}
{\bf Proof:} One can prove this proposition using the explicit list of
possible reduction of the
N\'eron model and their semistable reduction (Tate's algorithm). If one
goes through this list, one
sees that the kernel of the semistable reduction map away from characteristic 2
and 3 is
precisely the additive components of the identity on the N\'eron model. A
much more appealing proof follows directly from \cite{edix}, section 5
(especially remark 5.4.1): assume given
a field extension $L\supset K$ which is tamely and totally ramified at a given
prime $p$ (this can be assumed for a local field of semistable reduction of an
elliptic curves once
one avoids the primes dividing 2 and 3). Let ${\cal{E}}_K, {\cal{E}}_L$ be the N\'eron
models of $E$ over $K$ and $L$ respectively. In \cite{edix} one obtains a
description of
the map induced on N\'eron models ${\cal{E}}_K\times{\mbox{Spec }}{\cal O}_L \rightarrow {\cal{E}}_L$,
and one there sees that the group of components of the reduction $({\cal{E}}_K)_p$
maps isomorphically to the group of components of the fixed locus under
the Galois action of $({\cal{E}}_L)_p$. Therefore the kernel of the map of N\'eron
models is connected.
{\bf Remark:} in order to include primes over 2 and 3 one simply needs to
remove all
the additive components which are in the kernel of the semistable reduction
map. Since we are allowed to remove a finite number of places anyway, we can
ignore this problem altogether.
\section{The main theorem}\label{main}
{\bf Proof of the main theorem:} We may assume that $S$ contains all places
above 2 and 3, and that
$K$ contains $\zeta_3$.
Let $X \rightarrow U$ be the family of all elliptic curves given by the equation
$$y^2 = x^3 + ax + b,$$
over an open set $U$ in $\Bbb{A}^2$, with parameters $a,b$. Let $B_0$ be
any
irreducible closed subset in $U$ and $B$ a compactification of $B_0$. We
can add level 3 structure to produce a semistable family $X_1\rightarrow B_1$, over a
Galois, generically finite cover $B_1$ of
$B$. The Galois group of the cover is some $G_1\subset Sl_2({\Bbb{F}}_3) =G$. We
have a natural map $X_1\rightarrow E_1$,
coming from the moduli interpretation of $E_1$, which is $G_1$ equivariant.
We
have that $X^n_{B_0}$ maps to $((X_1)^n_{B_1})/G_1,$ which maps down to
$E_n/G$.
Since the family $E_1$ is semistable, an $n$-tuple of stably integral
points on an elliptic curve gives rise to an integral point on
$(E_n\setminus\Theta_n)/G$.
Note that if $X_{B_0}\rightarrow B_0$ is not isotrivial, then $X_{B_0}^n$ dominates
$E_n/G$.
Otherwise, its image is isomorphic to $(E\setminus 0)^n/Aut\,E$ for some fixed
elliptic curve
$E$, where $Aut\, E$ acts diagonally.
The following lemmas show that
$(E_n\setminus\Theta_n)/G$ is of logarithmic general type for large $n$, and
that for any
fixed elliptic curve $E$, $(E\setminus 0)^n/Aut\,E$ is of logarithmic general
type.
Assuming the
Lang-Vojta
conjecture, the integral points on the variety $((X_1)^n_{B_1}\setminus
\Theta)/G_1$ are not Zariski dense. By lemma 1, we obtain a uniform bound on
the
number of stably integral
points on all elliptic curves away from a closed subset $B'$ of $B$. By
Noetherian
induction, we have a bound on all elliptic curves. This gives the theorem.
\begin{lem}\label{flem}(Compare \cite{chm}, lemma 4.1) Let $X_0\subset X$ be an
open inclusion of an
irreducible
variety $X_0$ in a smooth complex projective irreducible $X$ of dimension $n$,
such that the complement
$D = X \setminus X_0$ is a divisor of normal crossings. Let $G$ be a finite
group acting on $ X,X_0,D$ compatibly. Let $\omega$ be a $G$- equivariant
logarithmic $k-$canonical form on $X_0$. If at any point $x$ of $X_0$ which is
fixed
by some element in $G$, the form $\omega$ vanishes to order at least
$C=k(|G|-1)$, then
$\omega$ descends to a regular logarithmic $k$-canonical form on any
desingularization of $X_0/G$
\end{lem}
{\bf Proof:} let $Y_0$ be a desingularization of $X_0/G$ and let $Y$ be
a regular compactification, mapping to $X/G$. Let $Z'$ be the graph of the
rational map $X\rightarrow Y$. Let $Z$ be a $G$-equivariant desingularization of $Z'$,
and $Z_0$ the inverse image of $X_0$. Let $S$ be the branch locus of $Z$ over
$Y$. By a theorem of Hironaka, such
desingularizations may be chosen such that $(Z\setminus Z_0)\cup S$ is a
divisor
of normal crossings. Let $F_1$ be the closed set in $Y$ where the fibers in $Z$
are positive dimensional. Let
$D_Z$ be the inverse image of $D$ in $Z$, $D_Y$ its image in $Y$. Let
$F_2\subset Y$ be the singular locus of $D_Y\cup S$. Clearly $F=F_1\cup F_2$
is of codimension at least 2 in $Y$. Note that away from $F_1$ the branch
locus $S$ is of codimension 1, since $Y$ is smooth.
Clearly $\omega$ descends to a logarithmic form on $Y\setminus S$. It is
enough to show that it extends over $Y\setminus F$, since $F$ has codimension
at least 2.
Given a point $y\in S\setminus F$ let $z\in Z$ be a point mapping to
it. We can choose formal coordinates $(z_1,z_2,\ldots,z_n)$ on $Z$ such that
$(y_1,z_2,\ldots,z_n)$ are coordinates on $Y$, with $y_1 = z_1^m$. Since we
removed the intersections of components of $D\cup S$, there are
only two cases to consider:
Case 1: $z\not\in D_Z$. We can write
$\omega = f(z_1\ldots,z_m)z_1^C (dz_1\wedge\cdots\wedge dz_m)^k$. We have
$dy_1 = m z_1^{m-1} dz_1$. Since $m<|G|$, we have that
$\omega = f z_1^{C-k(m-1)} (dy_1\wedge dz_2\wedge\cdots\wedge dz_m)^k$, is
regular, and since it is invariant it descends.
Case 2: $z\in D_Z$ and $z_1=0$ is the equation of $D_Z$. We can write
$\omega = f(z_1\ldots,z_m) (dz_1\wedge\cdots\wedge dz_m)^k/z_1^k$. Since
$m dz_1/z_1 = dy_1/y_1,$ the invariance of $\omega$ means that $f$
descends to $Y$, and
therefore
$\omega$ descends.
\begin{lem}\label{flem1}(Compare \cite{chm}, theorem 1.3)
\begin{enumerate} \item There exists a positive integer $n$ such that
$(E_n\setminus \Theta)/G$ ($G$ acting diagonally) is of
logarithmic general type.
\item For a fixed elliptic curve $E$, there is $n$ so that $(E\setminus
0)_n/Aut\,E$
($Aut\,E$ acting diagonally) is of logarithmic
general type.
\end{enumerate}
\end{lem}
{\bf Proof:} Let $S\subset E_1$ be any divisor containing the locus of fixed
points of elements of $G$, and let $F$ be a fiber. Then the fixed
points in $E_n$ are contained in $S^n_{{\Bbb{P}}^1}$, the fibered product of $S$
with itself $n$ times. Recall that we have shown that
$L=\omega_{E/{\Bbb{P}}^1}(\Theta) $ is big; therefore for some large $k$, the
${\Bbb{Q}}$-line bundle $L(-(S+2F)/k)$ is big. This means that for $n=k|G|$, on
$E_n$ there are
many sections of $\omega^m_{E_n}$ vanishing to order $m|G|$ on the fixed
points in $E_n$. As in \cite{chm}, lemma 2.1, it follows that there are also
many {\em invariant} sections
vanishing to such order. The proof of part (2) is identical.
\section{The uniform boundedness conjecture}\label{ubc}
It is well known that torsion points of high order on an elliptic curve are
integral; we will use this to study torsion points in terms
of integral points. As quoted in the introduction, a long standing conjecture
which was recently
proved by Merel says that the order of a torsion point on an elliptic curve
over a number field is bounded in terms of the degree of the field of
definition only. We now indicate how Merel's theorem follows from the
Lang - Vojta conjecture.
We start with the basic proposition which makes things work (see the case of
elliptic curves in \cite{oester}):
\begin{prp}
Let $P$ be a torsion point on an abelian variety $A$,
both defined over a field $K$
of degree $d$. Denote by $n$ the order of $P$. Let $g$ be the dimension of
$A$ and let $C$ be the order of the group $Sp_g(Z/5Z)$. Assume that either $n$
is not a prime power, or $p^k = n$, such that $p^k-p^{k-1} > Cd$. Then
$P$ is stably integral on $A$, that is, its reduction at any place on the
N\'eron model after semistable reduction is not the origin.
\end{prp}
{\bf Proof:} by adding level 3 structure we have (by a theorem of Raynaud) that
there is a field of degree at most $Cd$ where $A$ has semistable reductions
over all $p\neq 3$. Theorem IV.6.1 in \cite{silv} says that if $P$ is not
integral, then it is not integral at a place $\frak{p}$ above some prime $p$
where $n=p^k$;
and
the valuation satisfies $v(p)>p^k-p^{k-1}$. Here by definition,
$p=u\pi^{v(p)}$, where $u$ is a unit and $\pi$ a uniformizer of the valuation
ring. But $v(p)$ is
at most the degree of the field. We can similarly deal with primes over 3 by
adding level
5 structure instead. \qed
This following corollary is probably well known: torsion on abelian varieties
is bounded in terms of the degree of the field, the dimension and a prime of
potentially good reduction.
\begin{cor}
For any triple $(d,g,p)$ there is an (explicit) integer $N$ such that if $K$
is a
number field of degree $d$, $A$ an abelian variety of dimension $g$ over $K$,
and $p$ is a rational prime over which there is a place $\frak{p}$ of $K$
where $A$ has potentially good reduction, then $A(K)_{tors}<N$.
\end{cor}
{\bf Proof:} Let $L$ be a field of degree $\leq Cd$ over which $A$ has good
reduction at some prime $\frak{p}$ over $p$. Let $A(L)_{tors}\rightarrow A_{\frak{p}}$
be the reduction map. By Weil's theorem, the image has cardinality $\leq
(1+p^{Cd/2})^{2g}$. But by the proposition, any point in the kernel is of
order $p^k$ satisfying $p^k-p^{k-1} < Cd$, which can be bounded as well.\qed
We would like to apply the correlation method to torsion points of high order
on elliptic curves, defined over all fields of degree $d$. By the proposition,
we may
use the fact that when $p$ is large these points are stably integral.
Since we want to show that there is a bound for torsion over number fields
depending only on the degree, we might as well assume that $E$ has level 3
structure: this has the effect of increasing the degree $d$ by a factor of 24.
In \cite{kama}, Kamienny
and Mazur show that it is enough to
bound the order of prime torsion points. We will show the existence of a bound
on prime order torsion points, assuming the Lang - Vojta conjecture.
Let $P$ be a torsion point of large prime order $p$ on an elliptic curve $E$
which has
level 3 structure, defined over some number field of degree $d$. The point $P$
gives a point on the surface $E_1$ introduced in \S\ref{level3}, defined over
the
same number field, and is in fact integral on $E_1\setminus \Theta$. The Galois
orbit of $P$ gives a ${\Bbb{Q}}$-rational point on
the $d-$th symmetric power of $E_1$. We can do a bit better: fix an integer
$n$. Given $n$ torsion
points on an elliptic curve defined over the same number field (e.g. multiples
of a given torsion point) we in fact get a rational point on
$Y_n={\mbox{Sym}}^d(E_n)$. In $Y_n$ there is a
divisor $\Theta_{Y_n}$ which consists of those tuples of points such that
at least one point is the origin, and the points thus obtained are in fact
integral on the scheme $Y_n\setminus \Theta_{Y_n}$.
Given an auxiliary integer $k$, let $F_{n,k}$ be the Zariski closure in $Y_n$
of the set of all points
corresponding to Galois orbits of $n$-tuples of distinct torsion points of
prime order {\em larger than $k$} defined over fields of degree exactly $d$.
By definition, if
$l>k$ then $F_{n,l}$ is contained in $F_{n,k}$. Let $F_n$ be the intersection
of
$F_{n,k}$ over all integers $k$. By the noetherian property of algebraic
varieties, $F_n = F_{n,k}$ for some $k$. What we want to show is that $F_n$
is empty. We will assume the contrary and derive a contradiction.
We have the natural symmetrization map $(E_n)^d \rightarrow Y_n$. Let $G_n$ be the
inverse image of $F_n$ under this map.
We denote by $\pi^d_i:(E_n)^d \rightarrow (E_1)^d$ the map induced from
$\pi_i:E_n\rightarrow
E_1$.
The varieties $(E_n)^d$ can be viewed as compactified
semiabelian schemes over the space ${\Bbb{P}}=({\Bbb{P}}^1)^d$.
\begin{lem}
Let $G$ be a component of $G_n$.
There exists a closed subscheme $B\subset {\Bbb{P}}$, and subvarieties $A_i \subset
E_1^d, 0< i<n+1$ mapping onto $B$, such that the general fiber of $A_i$ over
$B$ is a
finite union of abelian subvarieties, and $G$ is a component of
the fibered product of $A_i$ over $B$. The varieties $A_i$ are not contained
in any diagonal in $E_1^d$ or in the theta divisor, nor in the locus of
singular
fibers.
\end{lem}
Proof: if $P$ is a torsion point of some prime order $p$ defined over a number
field, then
any multiple of it $kP$, for $k$ prime to $p$, is also torsion defined over the
same field. Fix an integer $1\leq i\leq n$. We look at the projection of
$q_i:G\rightarrow E_{n-1}^d$ forgetting the $i$-th factor. It follows that each
fiber of
$G_n$ over $E_{n-1}^d$ is stable
under multiplication by $k$ for any integer $k$. We now use the trick of Neeman
and Hindry (see \cite{neeman} or \cite{hindry}), which tells us that a
subvariety of an abelian
variety which is stable under multiplication by all integers, is a union of
abelian subvarieties.
Let $G'\subset G_{n-1}$ be the image of $G$ under $q_i$. For each point $P\in
E_{n-1}^d$ in $G'$ we have that
$q_i^{-1}(P)\cap F_n$ is a union of finitely many
abelian
subvarieties of $E_1^d$. Since a subvariety of a constant
abelian scheme is constant (say, by looking at torsion points), these abelian
subvarieties depend only on the image of $P$ in ${\Bbb{P}}$. Therefore there is a
subvariety $A_i$ as in the lemma such that $G$ is a component of
$(\pi^d_i)^{-1}A_i\cap q_i^{-1}G'$. By induction we obtain the product
structure.
Since the variety $G$ was obtained from the closure of Galois orbits of points
over fields of degree exactly $d$, none of them is fixed by any permutation,
and none is in the theta divisor. Similarly, we see that they are not contained
in the singular fibers of $E_n^d$.
\qed
We will now show that for high enough $n$, any candidate for a component of $F$
is of logarithmic general type. First note that, by noetherian induction, one
may assume that the base $B$ of $A_i$ remains constant as $n$ grows. We will
now see that if one of the $A_i$ appears many times in the product, then the
image $F$ of $G$ is of logarithmic general type.
\begin{lem} Let
$B\subset
{\Bbb{P}}$ be an irreducible closed subvariety.
Let $A_i\subset (E_1)^d,\quad 1\leq i\leq m$ and $A_{m+1}$ be subschemes
mapping to
$B$ satisfying the conclusions in the previous lemma.
There is an integer $k_0$ such that for any $k>k_0$ and any $l_i\geq 0$ the
following
holds:
Let $G$ be a component of the scheme
$$(A_1)^{l_1}_{B}\times_B\cdots\times_B (A_m)^{l_m}_B\times_B
(A_{m+1})^k_B.$$ Let $F$ be
the image of
$G$ in $Y_n$, where $n=l_1+\cdots +l_n+k$. Let $F' = F\setminus\Theta_{Y_n}$.
Then $F'$ is of logarithmic general type.
\end{lem}
{\bf Proof:} Let $M_i$ be the dimension of the fibers of $A_i$ over $B$. For
each
choice of a subset $J_i$ of $\{1,\ldots,d\}$ of size $M_i$, we have a variety
$E_{J_i,B}$,
the
pullback of
$(E_1)^{M_i}$ to $B$ along the projection $\pi_{J_i}$ to the factors in $J_i$.
Since
$A_i$ is
not contained in the theta divisor of $E_1^d$, we have that $A_i$ surjects
generically finitely onto
$E_{J_i,B}$, whenever $J_i$ has size $M_i$. We treat $A_{m+1}$ a bit
differently: using $d$ different generically finite surjections $A_{m+1}\rightarrow
E_{J'_i,B}$ where $J'_i=\{i,\ldots,(i+M_{m+1} \mod d)\}\subset \{1,\ldots,d\}$,
we can cook up a special generically finite surjection: write $k=qd+r$, then
we map map $(A_{m+1})^k_B\rightarrow (E_{J_{m+1},B})^r_B \times_B (E_{qm})^d|_B $.
In order to deal with the singularities, we desingularize the base: $B'\rightarrow B$.
Now the pullback of the product of $E_{J_i,B}$ to $B'$ has semistable
fibers, therefore has log canonical singularities as in lemma 2. Choose a
canonical divisor
$K_{B'}$. Choose an effective divisor
$H\subset B$ such that the pull-back of $H$ to $B'$ is bigger than $-K_{B'}$.
If $G'$ is a
desingularization of $G$, it admits a generically finite surjection
$$p:G\rightarrow V= (E_{J_1,B})^{l_1}_B \times_B\cdots\times_B
(E_{J_m,B})^{l_m}_B\times_B
(E_{J_{m+1},B})^r_B \times (E_{qm})^d|_B.$$
Notice that $V$ is the restriction to $B$ of a variety of the form
$E_{r_1}\times \cdots \times E_{r_d}$, and if $k$ is large then {\em each of
the $r_i$} is large as well.
We can choose $G'$ so that it maps to $B'$. Let $V'$ be the pullback of $V$ to
$B'$.
We wish to use the sections of powers of the logarithmic relative dualizing
sheaf of the product variety $V$ to construct
differential forms on $F$. In view
of lemmas \ref{flem},\ref{flem1} we need the sections to vanish sufficiently
along the preimage of $H$, and their pullback to $G$ should vanish along the
fixed points $\Delta$ of the symmetric group action to sufficiently high order.
Since the relative dualizing sheaf $\omega_{E_1/{\Bbb{P}}^1}(\Theta)$ is nef and
big, then the sheaf $\omega_{E_{J,B}/B}(\Theta)$ is nef, and the sheaf
$\omega_{(E_{1})^d|_B/B}(\Theta)$ is
nef and big. We have
an injection $p^*(\omega^m_{V'/B'})(-mH+m\Theta) \rightarrow
\omega^m_{G'}(m\Theta)$, since $V'\setminus \Theta$ has log
canonical singularities. Since each of the $r_i$ in the description of $V$ can
be made as large as we wish, the argument of lemma \ref{flem1} shows that $F$
is of logarithmic general type. \qed
We can now show by induction on $M$ that the relative dimension of $G_{n+1}$
over
$G_{n}$ is at least $M+1$, thus obtaining a contradiction. Clearly the
relative dimension is at least 1. If for some $n$ the relative dimension is
precisely $M$, then by induction, using the embedding $G_{n+k}\subset
(G_{n+1})^k_{G_n}$, the relative
dimension of $G_{n+k}$ over $G_n$ is $Mk$, and therefore there is a component
$G$ of $G_{n+k}$ of relative dimension $Mk$. From lemma 6 it follows that $G$
is a component of a product variety of the form described in lemma 7, and
therefore $F\setminus \Theta$ is of logarithmic general type. The Lang - Vojta
conjecture implies that the integral points on $F$ are not dense, contradicting
the definition of $F$.
We arrived at a contradiction, therefore $F_n$ must be empty, and we conclude
that there is a bound for torsion points of prime order. \qed
|
1995-11-21T06:02:20 | 9505 | alg-geom/9505023 | en | https://arxiv.org/abs/alg-geom/9505023 | [
"alg-geom",
"math.AG"
] | alg-geom/9505023 | Rahul Pandharipande | R. Pandharipande | A Note On Elliptic Plane Curves With Fixed j-Invariant | 10 pages, AMSLatex | null | null | null | null | Let N_d be the number of degree d, nodal, rational plane curves through 3d-1
points in the complex projective plane. The number of degree d>=3, nodal,
elliptic plane curves with a fixed (general) j-invariant through 3d-1 points is
found to be {d-1 \choose 2}*N_d.
| [
{
"version": "v1",
"created": "Wed, 24 May 1995 01:31:06 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Pandharipande",
"R.",
""
]
] | alg-geom | \section{Summary}
Let $N_d$ be the number of irreducible, reduced, nodal, degree
$d$ {\em rational plane curves} passing through $3d-1$ general points in the
complex projective plane $\bold P^2$. The numbers $N_d$ satisfy a
beautiful recursion relation ([K-M], [R-T]):
$$N_1=1$$
$$\forall d>1, \ \ \ N_d= \sum_{i+j=d, \ i,j>0}
N_iN_j \bigg( i^2j^2 {3d-4 \choose 3i-2} - i^3j {3d-4 \choose 3i-1} \bigg).$$
Let $E_{d,j}$ be the number of irreducible, reduced, nodal,
degree $d$, {\em elliptic plane curves with fixed j-invariant j}
passing through $3d-1$ general points $\bold P^2$.
$E_{d,j}$ is defined for $d\geq 3$ and $\infty \neq j \in \overline{M}_{1,1}$.
In this note, the following relations are established:
\begin{eqnarray*}
\forall j\neq 0,1728,\infty, & E_{d,j}= {d-1\choose 2}N_d, \\
j=0, & E_{d,0}= {1\over 3} {d-1 \choose 2} N_d, \\
j=1728, & \ E_{d,1728}= {1\over 2} {d-1\choose 2} N_d.
\end{eqnarray*}
If $d\equiv 0 \ mod \ 3$, then $ 3 \not\mid {d-1 \choose 2}$.
Since $E_{3\hat{d},0}$ is an integer,
$N_{3\hat{d}} \equiv 0 \ mod \ 3$ for $\hat{d}\geq 1$.
In fact, a check of values in [D-I] shows
$N_d \equiv 0 \ mod \ 3$ if and only if $d \equiv 0 \ mod \ 3$
for $3\leq d \leq 12$. P. Aluffi has calculated $E_{3,j}$ for
$j<\infty$ in [A]. Aluffi's results agree with the above formulas.
Thanks are due to Y. Ruan for discussions on
Gromov-Witten invariants and quantum cohomology. The question
of determining the numbers $E_{d,j}$ was first considered
by the author in a conversation with Y. Ruan.
\section{Kontsevich's Space of Stable Maps}
\subsection{ The Quasi-Projective Subvarieties $U_C(\Gamma,c,\overline{w})$
, $U_{j=\infty}(\Lambda,\overline{w})$}
\label{qpd} Fix $d\geq 3$ for the entire paper.
Let $C$ be a nonsingular elliptic curve or an irreducible, nodal
rational curve of arithmetic genus 1.
Consider the coarse moduli space of $3d-1$-pointed stable maps
from $C$ to $\bold P^2$ of degree $d\geq 3$,
$\overline{M}_{C, 3d-1}(\bold P^2, d)$. For convenience, the
notation $\barr{M}_C(d) = \overline{M}_{C,3d-1}(\bold P^2,d)$ will be used.
Let $S_d= \{1,2,\ldots,3d-1\}$ be the marking set. Constructions of
$\barr{M}_C(d)$ can be found in [Al], [K], [P].
Let $\Gamma$ be a tree consisting of
a distinguished vertex $c$, $k\geq 0$ {\em other} vertices
$v_1, \ldots, v_k$, and $3d-1$ marked legs.
Let $0\leq e \leq d$. Weight the vertex $c$ by $e$.
Let $w_1, \ldots, w_k$ be non-negative integral weights
of the vertices $v_1, \ldots, v_k$ satisfying
$$e+ w_1 + \cdots + w_k =d.$$
Denote the weighting by $\overline{w}=(e,w_1,\ldots, w_k)$.
The marked, weighted tree with distinguished vertex
$(\Gamma, c, \overline{w})$ is {stable} if the following
implication holds for all $1\leq i\leq k$:
$$w_i=0 \ \ \Rightarrow \ \ valence(v_i)\geq 3.$$
Two marked, weighted trees
with distinguished vertex $(\Gamma, c,\overline{w})$ and $(\Gamma', c',\overline{w}')$
are isomorphic if there is an isomorphism of marked trees $\Gamma\rightarrow \Gamma'$
sending $c$ to $c'$ and respecting the weights.
A quasi-projective subvariety $U_C(\Gamma,c, \overline{w})$ of $\barr{M}_C(d)$
is associated to each isomorphism class
of stable, marked, weighted graph with distinguished vertex $(\Gamma,
c,\overline{w})$.
$U_C(\Gamma,c,\overline{w})$ consists of stable maps
$\mu: (D, p_1,\ldots, p_{3d-1})
\rightarrow \bold P^2$ satisfying the following conditions. The domain $D$ is
equal to a union:
$$D=C \cup \bold P^1_1 \cup \cdots \cup \bold P^1_k.$$ The
marked, weighted dual graph with distinguished vertex of the map $\mu$ is
isomorphic to $(\Gamma, c, \overline{w})$. The distinguished vertex of the
dual graph of $\mu$ corresponds to the (unique) component of $D$ isomorphic to
$C$.
Weights of the
dual graph of $\mu$ are obtained by the degree of $\mu$ on the components.
Note $U_C(\Gamma,c,\overline{w})=\emptyset$ if and only if $e=1$.
Let $(\Gamma,c,\overline{w})$ be a stable, marked, weighted tree
with distinguished vertex. Assume $e\neq 1$. The dimension of
$U_{C}(\Gamma,c,\overline{w})$ is determined as follows.
If $e\geq 2$, then
$$dim \ U_{C}(\Gamma,c,\overline{w})= 6d-2-k.$$
If $e=0$, then
$$dim \ U_{C}(\Gamma,c,\overline{w})= 6d-k$$
(where $k$ is the number of non-distinguished vertices of $\Gamma$).
These calculations are straightforward.
Let $C$ be a nonsingular elliptic curve. Every stable
map in $\barr{M}_C(d)$
has domain obtained by attaching a finite number of marked trees to $C$.
By the definition of tree and map stability:
$$\bigcup_{(\Gamma,c, \overline{w})} U_C(\Gamma,c, \overline{w}) \ = \ \barr{M}_C(d).$$
Let $C$ be an irreducible, 1-nodal rational curve.
The quasi-projective varieties $U_C(\Gamma,c,\overline{w})$
do not cover $\barr{M}_C(d)$. The curve $C$ can degenerate
into a simple circuit of $\bold P^1$'s.
Let $\Lambda$ be a graph with 1 circuit ($1^{st}$ Betti number equal to $1$,
no self edges),
$k\geq 1$ vertices $v_0,\ldots, v_k$, and $3d-1$ marked legs.
Note the different vertex numbering convention. At least $2$ vertices
are required to make a circuit, so $k\geq 1$.
Let $w_0, w_1,\ldots,w_k$ be non-negative, integral weights summing to $d$.
The marked, weighted graph with $1$ circuit $(\Lambda, \overline{w})$ is
stable if each zero weighted vertex has valence at least 3.
A quasi-projective subvariety $U_C(\Lambda, \overline{w})$
of $\barr{M}_C(d)$ is associated to each
isomorphism class of stable, marked, weighted graph with $1$ circuit
$(\Lambda, \overline{w})$. $U_C(\Lambda, \overline{w})$ consists of
stable maps with marked, weighted dual graphs isomorphic to
$(\Lambda, \overline{w})$. The union
$$\bigcup_{(\Gamma,c, \overline{w})} U_C(\Gamma,c, \overline{w}) \
\ \cup \ \ \bigcup_{(\Lambda, \overline{w})} U_C(\Lambda, \overline{w})\ \ = \ \barr{M}_C(d)$$
holds by the definition of stability.
Finally, the dimensions of the loci $U_C(\Lambda,\overline{w})$
will be required. Let $(\Lambda, \overline{w})$ be a stable, marked,
weighted graph with $1$ circuit. Let $c_1,\ldots, c_l$ be the
unique circuit of vertices of $\Lambda$.
Let $e$ be the sum of the weights of the circuit vertices.
$U_C(\Lambda, \overline{w})=\emptyset$ if and only if
$e=1$.
If $e\geq 2$, then
$$dim\ U_C(\Lambda,\overline{w})= 6d-2-k.$$
If $e=0$, then
$$dim\ U_C(\Lambda, \overline{w})= 6d-k$$
(where $k+1$ is the total number of vertices of $\Lambda$).
Again, these results are straightforward.
\subsection{The Component $\overline{W}_{1}(d)$}
Let $\barr{M}_1(d) = \overline{M}_{1,3d-1}(\bold P^2,d)$ be
Kontsevich's space of $3d-1$-pointed stable maps
from genus $1$ curves to $\bold P^2$. There is canonical morphism
$$\pi: \barr{M}_1(d) \rightarrow \overline{M}_{1,1}$$
obtained by forgetting the map and all the markings except $1\in S_d$ (the
$3d-1$ possible choices of marking in $S_d$ all yield the
same morphism $\pi$).
Let $j\in \overline{M}_{1,1}$. By the
universal properties of the moduli spaces, there is a canonical bijection
$$\overline{M}_{C_j}(d) \rightarrow \pi^{-1}(j)$$
where $C_j$ is the elliptic curve (possibly nodal
rational) with $j$-invariant $j$.
When $j\in \overline{M}_{1,1}$ is automorphism-free, this bijection
is an isomorphism. For $j=0,1728$, the scheme theoretic
fiber $\pi^{-1}(j)$ is nonreduced.
Define an open locus $W_{1}(d) \subset \barr{M}_1(d)$ by
$[\mu: (D, p_1, \ldots, p_{3d-1}) \rightarrow \bold P^2]\in W_{1}(d)$ if and only if
$D$ is irreducible.
By considering the natural tautological spaces over the
universal Picard variety of degree $d$ {\em line bundles} over
$M_{1,3d-1}$, it is easily seen that $W_{1}(d)$ is a
reduced, irreducible open set of dimension
$6d-1$. Let $\overline{W}_1(d)$ be the
closure of $W_{1}(d)$ in $\barr{M}_1(d)$.
\section{A Deformation Result}
Let $\Phi$ be a stable, marked, weighted tree with
distinguished vertex $c$ determined by the data: $k=1$, $(e,w_1)=(0,d)$.
There are $2^{3d-1}$ isomorphism classes of such $\Phi$
determined by the marking distribution.
Let $j\in \overline{M}_{1,1}$. The dimension of
$U_j(\Phi)$ is $6d-1$.
A point $[\mu]\in U_j(\Phi)$ has domain $C_j \cup \bold P^1$.
There are $3d-1$ dimensions of the map $\mu|_{\bold P^1}:\bold P^1\rightarrow \bold P^2$.
The incidence point $p= C_j\cap \bold P^1$ moves in a $1$-dimensional family on
$\bold P^1$. The remaining $3d-1$ markings move in $3d-1$ dimensions on $C_j$ and
$\bold P^1$ (specified by the marking distribution).
$6d-1=3d-1+1+3d-1$.
A technical result is
needed in the computation of the numbers $E_{d,j}$.
\begin{lm}
\label{aa}
Let $I(\Phi,j)= \overline{W}_1(d) \cap U_j(\Phi) \subset \barr{M}_1(d)$.
The dimension of $I(\Phi,j)$ is bounded by $dim\ I(\Phi,j) \leq 6d-3$.
\end{lm}
\begin{pf}
Let $[\mu]\in I(\Phi,j)$ be a point. Let $D=C_j \cup \bold P^1$ be
the domain of $\mu$ as above. The following condition will be shown to hold:
the linear series on $\bold P^1$ determined
by $\mu|_{\bold P^1}$ has vanishing sequence $\{0, \geq 2,*\}$ at the
incidence point $p=C_j \cap \bold P^1$. The existence of
a point with vanishing sequence $\{0,\geq 2,*\}$ is a $1$-dimensional
condition on the linear series. The condition that the incidence
point $p$ has this vanishing sequence is an additional
$1$-dimensional constraint on $p$. Therefore, the dimension of $I(\Phi,j)$
is at most $6d-1-1-1=6d-3$. The vanishing sequence $\{0, \geq 2, *\}$
is equivalent to $d(\mu|_{\bold P^1})=0$ at $p$.
It remains to establish the vanishing sequence $\{0,\geq 2,*\}$ at
$p$. This result is easily seen in explicit holomorphic coordinates.
Let $\bigtriangleup_t$ be a disk at the origin in $\Bbb{C}$ with coordinate $t$.
Let $\eta: \cal{E} \rightarrow \bigtriangleup_t$ be a flat family of
curves of arithmetic genus $1$ satisfying:
\begin{enumerate}
\item[(i.)] $\eta^{-1}(0)\stackrel{\sim}{=} C_j$.
\item[(ii.)] $\eta^{-1}(t\neq 0)$ is irreducible, reduced, and (at worst)
nodal.
\end{enumerate}
For each $1\leq i \leq d$, let $\cal{G}_i=\cal{H}_i\subset \cal{E}$ be the
open subset of $\cal{E}$ on which the morphism $\eta$ is {\em smooth}.
Consider the fiber product:
$$X= \cal{G}_1 \times_{\bigtriangleup_t} \cdots \times_{\bigtriangleup_t} \cal{G}_d
\times_{\bigtriangleup_t} \cal{H}_1 \times_{\bigtriangleup_t} \cdots \times_{\bigtriangleup_t}
\cal{H}_d.$$
$X$ is a nonsingular open set of the $2d$-fold fiber product
of $\cal{E}$ over $\bigtriangleup_t$. Let $Y\subset X$ be the
subset of points $y=(g_1,\ldots, g_d,h_1, \ldots, h_d)$ where
the two divisors $\sum g_i$ and $\sum h_i$ are linearly
equivalent on the curve $\eta^{-1}(\eta(y))$.
$Y$ is a nonsingular divisor in $X$.
Let $p\in C_j=\eta^{-1}(0)$ be a nonsingular point of $C_j$.
Certainly $p\in \cal{G}_i, \cal{H}_i$ for all $i$.
Let $\gamma: \bigtriangleup_t \rightarrow \cal{E}$ be any local
holomorphic section of $\eta$ such that $\gamma(0)=p$.
Let $V$ be a local holomorphic field of vertical tangent vectors
to $\cal{E}$ on an open set containing $p$. The section $\gamma$ and the
vertical vector field $V$ together determine
local holomorphic coordinates $(t,v)$ on $\cal{E}$ at $p$.
Let $\phi_{V}:\cal{E}\times \Bbb{C} \rightarrow \cal{E}$ be the
holomorphic flow of $V$ defined locally near $(p,0)\in \cal{E}\times \Bbb{C}$.
The coordinate map $$\psi:(t,v) \rightarrow \cal{E}$$
is determined by $\psi(t,v)= \phi_V(\gamma(t),v)$.
Local coordinates on $X$ near the point $x_p=(p,\ldots, p,p,\ldots, p)\in X$
are given by $$(t,v_1,\ldots, v_d, w_1, \ldots w_d).$$
The coordinate map is determined by:
$$\psi_X(t,v_1,\ldots, v_d, w_1, \ldots w_d)=
(\psi(t,v_1),\ldots, \psi(t,v_d), \psi(t,w_1), \ldots, \psi(t,w_d))\in X.$$
Note $x_p\in Y$. Let $f(t,v_1,\ldots, v_d, w_1, \ldots w_d)$ be a
local equation of $Y$ at $x_p$.
Since $f$ is identically $0$ on the line $(t,0,\ldots,0,0,\ldots,0)$,
\begin{equation}
\label{tvan}
\forall k\geq 0, \ \ {\partial ^k f\over \partial t^k} |_{x_p} =0.
\end{equation}
The tangent directions in the plane $t=0$ correspond to
divisors on the fixed curve $C_j$. Here, it is well know (up to $\Bbb{C}^*$-
factor)
\begin{equation}
\label{lion}
{\partial f\over \partial v_i}|_{x_p}=+1, \ \
{\partial f\over \partial w_i}|_{x_p}=-1.
\end{equation}
Equations (\ref{tvan}) and (\ref{lion})
are the only properties of $f$ that will be used.
Let $\hat{\eta}:\cal{\hat{E}} \rightarrow \bigtriangleup_t$ be the family obtained
by blowing-up $\cal{E}$ at $p$ and adding $3d-1$-marking.
Let $\mu:\cal{\hat{E}}\rightarrow \bold P^2$ be a morphism.
Let $\hat{\eta}^{-1}(0)=D= C_j \cup \bold P^1$.
Assume the following conditions are satisfied:
\begin{enumerate}
\item[(i.)] $\mu$, $\hat{\eta}$, and the $3d-1$ markings determine
a family of Kontsevich stable pointed maps to $\bold P^2$.
\item[(ii.)] The markings of $D$ are distributed according to $\Phi$.
\item[(iii.)] $deg(\mu|_{C_j})=0$, $deg(\mu|_{\bold P^1})=d$.
\end{enumerate}
Let $L_1$, $L_2$ be general divisors of $\mu^*({\cal{O}}_{\bold P^2}(1))$
that each intersect $\bold P^1$ transversely at $d$ distinct points.
For $1\leq \alpha \leq 2$,
$L_{\alpha}$ breaks into holomorphic sections
$s_{\alpha,1}+\ldots+s_{\alpha,d}$ of $\hat{\eta}$ over a holomorphic disk
at $0\in \bigtriangleup_t$. These sections $s_{\alpha,i}$ ($1\leq \alpha \leq 2$,
$1\leq i \leq d$) determine a map $\lambda: \bigtriangleup_t \rightarrow Y$ locally at $0\in
\bigtriangleup_t$.
Let an affine coordinate on $\bold P^1$ be given by $\xi$ corresponding
to the normal direction
\begin{equation}
\label{cord}
{d\gamma\over dt}|_{t=0} + \xi \cdot V(p).
\end{equation}
Let $s_{1,i}(0)=\nu_i \in \Bbb{C} \subset \bold P^1$,
$s_{2,i}(0)=\omega_i \in \Bbb{C} \subset \bold P^1$ be given
in terms of the affine coordinate $\xi$.
The map $\lambda$ has the form
$$\lambda(t)= (t, \nu_1 t,\ldots, \nu_d t, \omega_1 t, \ldots, \omega_d t)$$
to first order in $t$ (written in the coordinates determined by $\psi_X$).
Equations (\ref{tvan}), (\ref{lion}), and the condition $f(\lambda(t))=0$
implies
\begin{equation}
\label{cony}
\sum_{i=1}^{d} \nu_i = \sum_{1}^{d} \omega_i.
\end{equation}
$L_1 \cap \bold P^1$ is a degree $d$ polynomial with roots at
$\nu_i$. Condition (\ref{cony}) implies that the
sums of the roots (in the coordinates (\ref{cord}))
of general elements of the linear
series $\mu|_{\bold P^1}$ are the same. Therefore, a constant
$K$ exists with the following property. If
$$\beta_0 + \beta_1 \xi+ \ldots + \beta_{d-1} \xi^{d-1} +\beta_{d} \xi^{d}$$
is an element of the linear series $\mu|_{\bold P^1}$, then
$\beta_{d-1}+ K\cdot \beta_d=0$. The vanishing sequence at $\xi=\infty$
is therefore $\{0, \geq 2,*\}$. The point $\xi=\infty$ is the intersection
$C_j \cap \bold P^1$.
Suppose $\tilde{\eta}:\cal{\tilde{E}} \rightarrow \bigtriangleup_t$ is obtained
from $\cal{E}$ by a sequence of $n$ blow-ups over $p$. The fiber
$\tilde{\eta}^{-1}(0)$ is assumed to be $C_j$ union
a chain of $\bold P^1$'s of length $n$. Each blow-up occurs in the
exceptional divisor of the previous blow-up. Let $\bold P$ denote the
extreme exceptional divisor. Let $\mu: \cal{\tilde{E}} \rightarrow \bold P^2$
be of degree $d$ on $\bold P$ and degree $0$ on the other components
of the special fiber $\tilde{\eta}^{-1}(0)$. Let there be $3d-1$ markings
as before. It must be again concluded that the linear series on
$\bold P$ has vanishing sequence $\{0, \geq 2,*\}$ at the node.
Let $\gamma$ be section of $\eta$ such that the lift of $\gamma$ to
$\tilde{\eta}$ meets $P$. Let the coordinates $(t,v)$ on $\cal{E}$
be determined by this $\gamma$ (and any $V$). An affine coordinate
$\xi$ is obtained on $\bold P$ in the follow manner.
Let $\gamma_{\xi}$ be
the section of $\eta$ determined in $(t,v)$ coordinates by
$$\gamma_{\xi}(t)=(t,\xi t^n).$$
Let $\tilde{\gamma}_{\xi}$ be the lift of $\gamma_{\xi}$ to
a section of $\tilde{\eta}$.
The association $$\Bbb{C} \ni \xi \mapsto \tilde{\eta}(0)\in \bold P$$
is an affine coordinate on $\bold P$.
Let $L_1, L_2$ be divisors in the linear series
$\mu$ intersecting $\bold P$ transversely. As before,
$L_{\alpha}$ breaks into
holomorphic sections $s_{\alpha,1}$. Let
$s_{1,i}=\nu_i \in \Bbb{C}\subset \bold P$, $s_{2,i}=\omega_i \in \Bbb{C} \subset
\bold P$.
As before, a map $\lambda:\bigtriangleup_t \rightarrow Y$ is obtained from the
sections $s_{\alpha,i}$. In the coordinates determined by $\psi_X$,
$$\lambda(t)=(t, \nu_1 t^n+O(t^{n+1}), \ldots, \nu_d t^n+ O(t^{n+1}),
\omega_1 t^n+O(t^{n+1}), \ldots, \omega_d t^n+O(t^{n+1})).$$
As before $f(\lambda(t))=0$. The term of leading order in $t$ of
$f(\lambda(t))$ is
$$ (\sum_{i=1}^{d} \nu_i- \sum_{i=1}^{d}\omega_i) \cdot t^n.$$
This follows from equations (\ref{tvan}) and (\ref{lion}).
The vanishing sequence $\{0,\geq 2,*\}$ is obtained as before.
By definition,
an element $[\mu]\in I(\Phi,j)$ can be obtained as the special fiber
of family of Kontsevich stable maps where
the domain is a smoothing of the node $p$.
After resolving the singularity in the total space at the node
$p$ by blowing-up, a family $\cal{\tilde{E}}$ is obtained.
The above results show the linear series on $\bold P^1$ has
vanishing sequence $\{0,\geq 2,*\}$ at $p$.
\end{pf}
The markings play no role in the preceding proof. An identical
argument establishes the following:
\begin{lm}
\label{bb}
Let $\Phi$ be a stable, marked, weighted
tree with distinguished vertex satisfying $e=0$ and $w_i=d$ for some $i$.
Let $k$ be the number of non-distinguished vertices of $\Phi$.
Let $j\in \overline{M}_{1,1}$. Let $I(\Phi,j)= \overline{W}_1(d) \cap U_j(\Phi)$.
The dimension of $I(\Phi,j)$ is bounded by
$dim\ I(\Phi,j)\leq 6d-k-2$
\end{lm}
\begin{lm}
\label{cc}
Let $\Omega$ be a stable, marked, weighted graph with $1$ circuit.
Let $v_i$ be a non-circuit vertex with
weight $w_i=d$ (this implies $e=0$).
Let $k+1$ be the total number of vertices of $\Omega$.
Let $I(\Omega,\infty)=\overline{W}_1(d) \cap U_{\infty}(\Omega)$.
The dimension of $I(\Omega,\infty)$ is bounded by
$dim \ I(\Omega,\infty)\leq 6d-k-2$.
\end{lm}
\noindent
The vanishing sequence $\{0,\geq 2, *\}$ condition reduces
the dimensions of $U_C(\Phi)$, $U_{\infty}(\Omega)$ by $2$.
\section{The Numbers $E_{d,j}$}
The space of maps $\barr{M}_1(d)$ is equipped with $3d-1$ evaluation
maps corresponding to the marked points.
For $i\in S_d$, let $e_i:\overline{W}_1(d)\rightarrow \bold P^2$ be the
restriction of the $i^{th}$ evaluation map to $\overline{W}_1(d)$.
let $\cal{L}_i= e_i^*({\cal{O}}_{\bold P^2})$
Let
$$Z=c_1(\cal{L}_1)^2 \cap \ldots \cap c_1(\cal{L}_{3d-1})^2$$
Let $\pi_{\overline{W}}:
\overline{W}_{1}(d) \rightarrow \overline{M}_{1,1}\cong \bold P^1$ be the
restriction of $\pi$ to $\overline{W}_1(d)$.
Let $$T=c_1(\pi_{\overline{W}}^*({\cal{O}}_{\bold P^1}(1))).$$
Note $\overline{W}_{1}(d)$ is an irreducible, projective scheme of
dimension $6d-1$. The top intersection of line bundles on $\overline{W}_1(d)$,
$Z\cap T$,
is an integer.
\begin{lm}
\label{pal}
\begin{eqnarray*}
\forall j\neq 0,1728,\infty, & Z\cap T=E_{d,j}\ , \\
j=0, & \ Z\cap T = 3\cdot E_{d,0}\ , \\
j=1728, & \ \ Z\cap T= 2\cdot E_{d,1728}\ .
\end{eqnarray*}
\end{lm}
\begin{pf}
Via pull-back, lines in $\bold P^2$ yield representative classes of
$c_1(\cal{L}_i)$. Therefore $3d-1$ general points in $\bold P^2$,
$\overline{x}=(x_1, \ldots, x_{3d-1})$,
determine a representative cycle $Z_{\overline{x}}$ of the the class $Z$.
Let $\infty > j \in \overline{M}_{1,1}$.
Let $\pi_W$ be the restriction of $\pi$ to
$W_1(d)$.
It is first established for a general representative $Z_{\overline{x}}$,
\begin{equation}
\label{erst}
Z_{\overline{x}} \cap \pi_{\overline{W}}^{-1}(j) \subset \pi_W^{-1}(j).
\end{equation}
The statement (\ref{erst}) is proven by considering
the quasi-projective strata of $\overline{M}_{C_j}(d)$.
Note $\pi_W^{-1}(j)$ is the strata $U_{C_j}(\Gamma,c,\overline{w})$ where
$(\Gamma,c,\overline{w})$ is the trivial, stable, marked, weighted tree
with distinguished vertex. Assume now $(\Gamma,c,\overline{w})$ is
not the trivial tree.
By the equations for the dimension of $(\Gamma,c,\overline{w})$
$$dim U_{C_j}(\Gamma,c,\overline{w}) \leq 6d-3$$
unless $e=0$ and $k=1,2$.
Since the linear series determined by the evaluation maps
are base point free, the general intersection
(\ref{erst}) will miss all loci of dimension less than
$6d-2$.
It remains to consider the trees $(\Gamma,c,\overline{w})$ where
$e=0$ and $k=1,2$. If $k=1$, $(\Gamma,c,\overline{w})=\Phi$
satisfies the conditions of Lemma (\ref{aa}).
By Lemma (\ref{aa}),
$$dim\ I(\Phi,j) \leq 6d-3.$$ Hence, the general intersection
(\ref{erst}) will miss all the loci $U_C(\Phi,c,(0,d))$.
If $k=2$, there are two cases to consider. If there exists
a vertex of weight $d$, then $(\Gamma,c,\overline{w})=\Phi$
satisfies the conditions of Lemma (\ref{bb}).
By Lemma (\ref{bb}),
$$dim\ I(\Phi,j) \leq 6d-4.$$
If $w_1+w_2=d$ is a positive partition, then the
image of every map $[\mu]\in U_C(\Gamma,c,\overline{w})$ is
the union of two rational curves of degrees $w_1$ and $w_2$.
No such unions pass through $3d-1$ general points.
The proof of claim (\ref{erst}) is complete.
For $\infty>j\neq 0,1728$, $\pi_W^{-1}(j)$ is a
reduced, irreducible divisor of $W_{1}(d)$. Since
the linear series determined by the evaluation maps are
base point free, the general intersection cycle
\begin{equation}
\label{al}
Z_{\overline{x}} \cap \pi_W^{-1}(j)
\end{equation}
is a reduced collection of $Z\cap T$ points. The general intersection
cycle $(\ref{al})$ also consists exactly of the reduced, nodal, degree $d$
elliptic plane curves with $j$-invariant $j$ passing through the
points $\overline{x}$.
The argument for $j=0,1728$ is identical except that
$\pi_W^{-1}(0)$, and $\pi_W^{-1}(1728)$ are divisors in $W_1(d)$
with multiplicity $3$, $2$ respectively.
These multiplicities arise from the
extra automorphisms for $j=0,1728$.
Therefore the cycle (\ref{al}) is a collection of
$${1\over 3}\cdot Z\cap T,$$ $${1\over 2}\cdot Z\cap T$$ triple and double
points respectively.
\end{pf}
It remains to evaluate $Z\cap T$.
\begin{lm}
$Z\cap T= {d-1\choose 2} N_d$.
\end{lm}
\begin{pf}
It is first established for a general representative $Z_{\overline{x}}$,
\begin{equation}
\label{lrst}
Z_{\overline{x}} \cap \pi_{\overline{W}}^{-1}(\infty) \subset \pi_W^{-1}(\infty).
\end{equation}
The statement (\ref{lrst}) is proven by considering
the quasi-projective strata of $\overline{M}_{\infty}(d)$.
By arguments of Lemma (\ref{pal}),
all the loci $U_{\infty}(\Gamma,c,\overline{w})$ where
$(\Gamma,c,\overline{w})$ is not the trivial tree are
avoided in the general intersection (\ref{lrst}). Only the
strata $U_{\infty}(\Lambda,\overline{w})$ remain to be considered.
Let $k+1\geq 2$ be the total number of vertices of $\Lambda$.
By the equations for the dimensions of $U_{\infty}(\Lambda,\overline{w})$,
$$dim \ U_{\infty}(\Lambda, \overline{w}) \leq 6d-2-k\leq 6d-3$$
unless all the circuit vertices have weight zero.
If all circuit vertices have weight zero, $k+1\geq 3$.
Now $$dim \ U_{\infty}(\Lambda, \overline{w}) \leq 6d-k \leq 6d-3$$
unless $k=2$.
Only one stable, marked, weighted, graph with $1$-circuit
$(\Lambda, \overline{w})$
need be considered. Vertices $c_1, c_2$ form a weightless circuit.
Vertex $v_3$ is connected to $c_2$ and $w_3=d$. $(\Lambda,\overline{w})=
\Omega$ satisfies the conditions of Lemma (\ref{cc}). Therefore,
$$dim\ I(\Omega, \infty) \leq 6d-4.$$
Claim (\ref{lrst}) is now proven.
The divisor $\pi_W^{-1}(\infty)$ is
reduced and irreducible in $W_{1}(d)$. As above,
\begin{equation}
\label{tal}
Z_{\overline{x}} \cap \pi_W^{-1}(\infty)
\end{equation}
is a reduced collection of $Z\cap T$ points. The general intersection
cycle $(\ref{tal})$ also consists exactly of degree $d$ {\em maps} of the
$1$-nodal rational curve $C_{\infty}$ passing through $\overline{x}$.
The image of such a map must be
one of the $N_d$ degree $d$, nodal, rational plane curves passing
through $\overline{x}$. The number of distinct birational maps (up to isomorphism)
from
$C_{\infty}$ to a ${d-1 \choose 2}$-nodal plane curve is exactly
${d-1 \choose 2}$. Therefore, $Z\cap T= {d-1\choose 2} N_d$.
\end{pf}
|
1995-05-25T06:20:26 | 9505 | alg-geom/9505025 | en | https://arxiv.org/abs/alg-geom/9505025 | [
"alg-geom",
"math.AC",
"math.AG"
] | alg-geom/9505025 | Tim Ford | Timothy J. Ford | Topological invariants of a fan associated to a toric variety | 16 pages with 2 figures, Author-supplied DVI file available at
ftp://ftp.math.fau.edu/pub/Ford/itv.dvi, Author-supplied PostScript file
available at ftp://ftp.math.fau.edu/pub/Ford/itv.ps, AMSLaTeX v 1.2 | null | null | null | null | Associated to a toric variety $X$ of dimension $r$ over a field $k$ is a fan
$\Delta$ on $\Bbb R^r$. The fan $\Delta$ is a finite set of cones which are in
one-to-one correspondence with the orbits of the torus action on $X$. The fan
$\Delta$ inherits the Zariski topology from $X$. In this article some
cohomological invariants of $X$ are studied in terms of whether or not they
depend only on $\Delta$ and not $k$. Secondly some numerical invariants of $X$
are studied in terms of whether or not they are topological invariants of the
fan $\Delta$. That is, whether or not they depend only on the finite
topological space defined on $\Delta$. The invariants with which we are mostly
concerned are the class group of Weil divisors, the Picard group, the Brauer
group and the dimensions of the torsion free part of the \'etale cohomology
groups with coefficients in the sheaf of units. The notion of an open
neighborhood of a fan is introduced and examples are given for which the above
invariants are sufficiently fine to give nontrivial stratifications of an open
neighborhood of a fan all of whose maximal cones are nonsimplicial.
| [
{
"version": "v1",
"created": "Wed, 24 May 1995 18:59:56 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Ford",
"Timothy J.",
""
]
] | alg-geom | \section{Introduction} \label{sec1}
Let $k$ be a field.
Let $N = \Bbb Z^r$ and denote by $T_N$ the $k$-torus on $N$. Let $\Delta$
be a finite fan on $N \otimes \Bbb R$ and $X = T_N\emb(\Delta,k)$ the toric
variety over $k$ associated to $\Delta$ \cite{D:Gtv}, \cite{F:ITV},
\cite{O:CBA}.
This defines a functor $T_N\emb$ on the product category
\begin{equation} \label{eq33}
\begin{array}{ccc}
(\text{finite fans on }N \otimes \Bbb R) \times (\text{fields}) &
\stackrel{T_N\emb}{\longrightarrow} & (\text{toric varieties}) \\
(\Delta,k) & \mapsto & T_N\emb(\Delta,k)
\end{array}
\text{.}
\end{equation}
We define the topology on $\Delta$
as follows (cf. \cite[pp. 137--138]{DFM:CBg}).
The orbit space $\tilde X$ of $X$ under the action of the torus $T_N$ is in
one-to-one correspondence with the finite set of cones that belong to
$\Delta$. There is a topology on $\tilde X$ inherited from $X$ by the
continuous function $X \to \tilde{X}$. Identifying a cone $\sigma \in
\Delta$ with the orbit $\operatorname{orb}{\sigma}$ in $\tilde X$, we see that the
topology on $\tilde X$ corresponds to the topology on $\Delta$ under which
the open sets are the subfans of $\Delta$. The fan $\Delta$ is now a
two-faced beast. On the one hand $\Delta$ is an object in the category of
fans on $N \otimes \Bbb R$. At the same time $\Delta$ is an object of the
category of finite topological spaces. To distinguish between these roles
played by $\Delta$, we denote by $\Delta_{fan}$ the object in the category
of fans on $N \otimes \Bbb R$ and by $\Delta_{top}$ the object in the
category of finite topological spaces. This defines a functor $\frak T$
(which
factors via $T_N\emb$ through the category of toric varieties)
\begin{equation} \label{eq26}
\begin{array}{ccc}
(\text{finite fans on }N \otimes \Bbb R) \times (\text{fields}) &
\xrightarrow{\frak T} &
(\text{finite top. spaces}) \\
(\Delta_{fan},k) & {\mapsto} & \Delta_{top}
\end{array} \text{.}
\end{equation}
In Section~\ref{sec2} we consider some invariants of $\Delta_{fan}$ that
are constant for all $k$.
Suppose $\gamma(\Delta_{fan},k)$
is an invariant
that is defined for any pair $(\Delta_{fan},k)$ (in this article $\gamma$
is usually an abelian group). We call $\gamma$ a {\em fan invariant} in
case $\gamma(\Delta_{fan},k)$ depends only on $\Delta_{fan}$ and not on
$k$ --- that is, given a fan $\Delta_{fan}$, $\gamma(\Delta_{fan},k_1)
\cong \gamma(\Delta_{fan},k_2)$ for every pair of fields $k_1$, $k_2$.
We show that the Brauer group
$\operatorname{B}(~)$ is not a fan invariant for nonsingular fans. This is an observation
based on a theorem of Hoobler and \cite[Theorem~1.1]{DF:Bgt}.
In \cite[Theorem~1.1]{DF:Bgt} a complete computation of the Brauer group of
a nonsingular toric variety $X=T_N\emb(\Delta)$ over an algebraically closed
field $k$ was given in terms of the
so-called invariant factors of the fan $\Delta$. In Theorem~\ref{th5} we
give the Brauer group of $T_N\emb(\Delta,k)$ for any field $k$ in terms of
the Brauer group and Galois group of $k$.
The main result of
Section~\ref{sec2} is Theorem~\ref{th6} in which it is stated
that the class group, $\operatorname{Cl}(~)$, the Picard group, $\operatorname{Pic}(~)$, and the
relative cohomology group, $H^2(K(~)/X(~)_{\operatorname{\acute{e}t}},\Bbb G_m)$ (where $K(X)$
is the function field of $X$), are fan
invariants. The proof of Theorem~\ref{th6} follows from that of
\cite[Theorem~1]{DFM:CBg} and is omitted.
In Section~\ref{sec3} we consider some invariants of $\Delta_{fan}$ that are
constant on fibers of the map $\frak T$ in \eqref{eq26}, hence
depend only
on $\Delta_{top}$. That is, suppose we have an invariant
$\beta(\Delta_{fan})$ (usually a numerical invariant) associated to any fan
$\Delta_{fan}$. If two fans $\Delta_1$, $\Delta_2$ have the same
$\beta$-invariant whenever $(\Delta_1)_{top} \cong (\Delta_2)_{top}$, then
we say $\beta$ is a {\em topological invariant of $\Delta_{fan}$}.
We consider several invariants, all being cohomologically defined. The
first sequence is defined by
\[
\rho_0 = \operatorname{dim}_{\Bbb Q} \left[ H^0(X_{\operatorname{\acute{e}t}},\Bbb G_m) / k^* \otimes \Bbb Q
\right] \text{,}
\]
and for $i \ge 1$,
\[
\rho_i = \operatorname{dim}_{\Bbb Q} \left[ H^i(X_{\operatorname{\acute{e}t}},\Bbb G_m) \otimes \Bbb Q \right]
\text{.}
\]
Also set
\[
\rho_1' = \operatorname{dim}_{\Bbb Q} \left[ \operatorname{Cl}(X) \otimes \Bbb Q \right] \text{.}
\]
For $0 \le i \le 2$ these numbers are finite and are fan invariants.
The first main result of Section~\ref{sec3} lists some facts about $\rho_0$
and $\rho_1'$.
\medskip\noindent
{\bf Theorem~\ref{th1}.}
{\em
Let $N = \Bbb Z^r$, $\Delta$ a fan on $N \otimes \Bbb R$,
$X = T_N\emb(\Delta)$, and
$s =
\operatorname{dim}_{\Bbb R} \Bbb R|\Delta_{fan}|$ (that is, $s$ is the dimension of the $\Bbb
R$-vector space spanned by the vectors in the support $|\Delta_{fan}|$). Then
\begin{itemize}
\item[(a)]
$\rho_0 = r-s$, hence is a fan invariant, but not a topological invariant.
\item[(b)]
Suppose $\Delta_{fan}$ contains a cone $\sigma$ such that $\operatorname{dim}{\sigma} =
r$. This is true for example if $\Delta_{fan}$ is a complete fan on $N
\otimes \Bbb R$. Then $\rho_0 = 0$ and $\rho_0$ is a topological invariant
of $\Delta_{fan}$.
\item[(c)]
$\rho_1'=\operatorname{dim}_{\Bbb Q}(\operatorname{Cl}(X) \otimes \Bbb Q) = \#(\Delta(1)) -s$.
If $\operatorname{dim}{\Delta_{top}} =r$, then $\rho_1'$ is a topological invariant of
$\Delta_{fan}$.
\item[(d)]
The number $\rho_0-\rho_1'$ is a topological
invariant of $\Delta_{fan}$.
\end{itemize}
}
The second main result of Section~\ref{sec3} gives some results on
$\rho_0$, $\rho_1$ and $\rho_2$ for simplicial fans.
\medskip\noindent
{\bf Theorem~\ref{th2}.~}{\em
Let $N = \Bbb Z^r$.
Let $\Delta$ be a simplicial fan on $N \otimes \Bbb R$
and $s = \operatorname{dim}_{\Bbb R}\Bbb R |\Delta_{fan}|$.
Then
\begin{itemize}
\item[(a)]
$\rho_1 =
\#(\Delta(1))-s$.
\item[(b)]
$\rho_2 = 0$ hence is a topological invariant of
$\Delta_{fan}$.
\item[(c)]
If $\operatorname{dim}{\Delta_{top}}=r$,
then $\rho_1$ is a topological invariant of $\Delta_{fan}$.
\item[(d)]
$\rho_0-\rho_1+\rho_2$ is a topological
invariant of $\Delta_{fan}$.
\end{itemize}}
The third main result of Section~\ref{sec3} gives some results on
$\rho_0$, $\rho_1$ and $\rho_2$ for 3-dimensional fans.
\medskip\noindent
{\bf Theorem~\ref{th3}.~}{\em
Let $\Delta$ be a fan on $N \otimes \Bbb R$.
Let $\sigma_0, \dots, \sigma_w$ be the maximal cones in $\Delta$.
Assume
$\sigma_i \cap
\sigma_j$ is simplicial for each $i \not = j$. These assumptions are
satisfied for example if $\operatorname{dim}{\Delta_{top}} \le 3$.
Then
\begin{itemize}
\item[(a)]
\[
\rho_1 +s + \sum_{i=0}^w(\#(\Delta(\sigma_i)(1))-s_i) =
\rho_2 + \#( \Delta(1)) \text{,}
\]
where we set $s_i = \operatorname{dim}{\sigma_i}$ for each $i = 0, \dots, w$ and $s =
\operatorname{dim}_{\Bbb R} \Bbb R |\Delta_{fan}|$.
\item[(b)]
$\rho_0-\rho_1+\rho_2$ is a topological invariant of $\Delta_{fan}$.
\end{itemize}}
In Section~\ref{sec4} we introduce the notion of an {\em open neighborhood
$B$ of a fan $\Delta$}. This is a subset of the fiber $\frak
T^{-1}(\Delta_{top})$ that is parametrized by a dense subset of a real
manifold.
Let $\scr{S}\scr{F}$ denote the sheaf of $\Delta$-linear support functions on the
topological space $\Delta_{top}$. It was shown in
\cite{DFM:CBg} that the numbers $\rho_i$, $1\le i\le 2$, can be
determined by the cohomology of the sheaf $\scr{S}\scr{F}$ on the finite
topological space $\Delta_{top}$. Therefore we define another sequence of
invariants by
\[
\kappa_i = \operatorname{dim}_{\Bbb Q}\left[ H^i(\Delta_{top},\scr{S}\scr{F}) \otimes \Bbb Q
\right]
\]
for $i \ge 0$.
We consider the stratification of $B$ by the numerical invariant
$\kappa_0$.
Several examples are given for which the stratification
of $B$ is nontrivial.
We conjecture that $\kappa_0 = 3$ on a nonempty open subset of $B$ if
$\Delta$ is a complete fan on $\Bbb R^3$ such that every maximal cone of
$\Delta$ is nonsimplicial. Algorithm~\ref{alg1} is presented which
computes an upper bound for $\kappa_0$. For complete 3-dimensional fans,
this algorithm can be used to compute an upper bound for $\rho_1$ and
$\rho_2$.
For the benefit of the reader the following notation will be fixed
throughout the rest of the paper.
\begin{table}[htp]
\label{tab1}
\end{table}
\begin{center}
\begin{tabular}{|lp{14pc}|ll|}\hline
$k$ & a field & $r$ & a positive integer \\
$N$ & $=\Bbb Z^r$ & $M $ & $=\operatorname{Hom}_{\Bbb Z}(N,\Bbb Z)$
\\
$\Delta$ & \raggedright a finite rational fan on $N\otimes \Bbb R$
& $X$ & $ =T_N\emb(\Delta,k)$ toric variety
\\
$\Delta_{fan}$ & object in the category of fans
& $\Delta_{top}$ & finite topological
space \\
$\left|\Delta_{fan}\right|$ & support of the fan $\Delta$
& $\frak T$ & functor that maps $(\Delta_{fan},k)$ to
$\Delta_{top}$ \\
$\operatorname{Cl}(X)$ &\raggedright class group of Weil divisor classes & $\operatorname{Pic}{X}$ & Picard
group of
invertible modules \\
$\operatorname{B}(X)$ & \raggedright Brauer group of Azumaya algebra classes
& $\Bbb G_m$ & \'etale sheaf of units \\
$\rho_0 $ & $ = \operatorname{dim}_{\Bbb Q}\left[ H^0(X_{\operatorname{\acute{e}t}},\Bbb G_m)\otimes\Bbb Q\right]$
&
$\rho_i $ & $ = \operatorname{dim}_{\Bbb Q}\left[ H^i(X_{\operatorname{\acute{e}t}},\Bbb G_m)\otimes\Bbb
Q\right]$ (for $i>0$) \\
$\rho'_1$ & $ = \operatorname{dim}_{\Bbb Q}\left[ \operatorname{Cl}(X)\otimes\Bbb Q\right]$ &
$\kappa_i$ & $=\operatorname{dim}_{\Bbb Q}\left[
H^i(\Delta_{top},\scr{S}\scr{F})\otimes\Bbb Q\right]$
(for $i\ge 0$) \\
$s $ & $=\operatorname{dim}_{\Bbb R}\Bbb R \left|\Delta_{fan}\right|$ & $\Delta(i)$ &
$=\{\sigma\in\Delta | \operatorname{dim}{\sigma}=i\}$ \\
$K$ & $=K(X)$ the function field of $X$ & $\scr{S}\scr{F}$ & sheaf of $\Delta$-linear
support functions \\
$\scr{W}$ & sheaf of Weil divisors & $\scr{P}$ & quotient sheaf $\scr{W} / \scr{S}\scr{F}$ \\
\hline
\end{tabular}\end{center}
\section{Fan Invariants} \label{sec2}
In Theorem~\ref{th5} we determine the Brauer group of a nonsingular toric
variety over $k$. This invariant depends on $k$. We then show in
Theorem~\ref{th6} that $\operatorname{Cl}(X)$, $\operatorname{Pic}{X}$ and the relative cohomology
group $H^2(K/X_{\operatorname{\acute{e}t}},\Bbb G_m)$ depend only on $\Delta_{fan}$, not on $k$.
In order to determine the Brauer group of a nonsingular toric variety over
$k$, we use the following theorem of Hoobler.
\begin{theorem}
\label{th4}
Let $R = A[x_1,x_1^{-1}, \dots, x_r,x_r^{-1}]$,
where $A$ is a connected,
normal integral domain. Suppose $\nu$ is an integer relatively prime to the
residue characteristics of $A$. Then
\begin{equation}
\label{eq8}
H^1(R,\Bbb Z/\nu) = H^1(A,\Bbb Z/\nu) \oplus \left( \bigoplus^r \mu_\nu^{-1}
\right) \text{ ,}
\end{equation}
and
\begin{equation}
\label{eq27}
_\nu\operatorname{B}(R) = {_\nu\operatorname{B}(A)} \oplus \left( \bigoplus^r H^1(A, \Bbb Z/\nu)
\right) \oplus \left( \bigoplus^{r(r-1)/2} \mu_\nu^{-1} \right) \text{ .}
\end{equation}
\end{theorem}
\begin{pf} See \cite[Cor. 2.6]{H:Fgr}. \end{pf}
Therefore the $\nu$-torsion of the Brauer group of $R=A[x_1,x_1^{-1},
\dots, x_r,x_r^{-1}]$ is
generated by the Azumaya $A$-algebras and the classes of cyclic crossed
product algebras of 2 types. For each cyclic Galois extension $C/A$ of
degree $\nu$ with group $\langle \sigma \rangle$ and for each $1 \le i \le
r$, there is the cyclic crossed
product $(C/R,\langle \sigma \rangle, x_i)$ which is an Azumaya algebra
over $R$. If there exists a primitive $\nu$-th root of unity $\zeta$ over
$A$, then the symbol algebras $(x_i,x_j)_\nu$ are Azumaya algebras over $R$.
\begin{example}
\label{ex13}
Let $R= \Bbb R [x_1,x_1^{-1},\dots, x_r,x_r^{-1}]$. Then by
Theorem~\ref{th4}
$\operatorname{B}(R)$ is an elementary 2-group and
\begin{equation}
\label{eq28}
\begin{split}
\operatorname{B}(R) & \cong {\operatorname{B}(\Bbb R)} \oplus \left( \bigoplus^r H^1(\Bbb R, \Bbb Z/2)
\right) \oplus \left( \bigoplus^{r(r-1)/2} \mu_2^{-1} \right) \\
& \cong \left( \Bbb Z/2 \right)^{1+r+
r(r-1)/2}
\end{split}
\end{equation}
\end{example}
Define a sheaf ${\scr{S}\scr{F}}$ on $\Delta_{top} $
by assigning to each open set $\Delta ' \subseteq \Delta_{top}$
the abelian group $\operatorname{SF}(\Delta ')$ of support functions on $\Delta '$.
Let $M = \operatorname{Hom}(N,{\Bbb Z})$ be the dual of $N$.
There is a natural map $M \rightarrow \operatorname{SF}(\Delta ')$
which is locally surjective.
If ${\scr{M}}$ denotes the constant sheaf of $M$ on $\Delta_{top}$,
then there is an exact sequence of sheaves on $\Delta_{top}$:
\begin{equation}
\label{eq32}
0 \to {\scr{U}} \to {\scr{M}} \to {\scr{S}\scr{F}} \to 0
\end{equation}
\noindent
where ${\scr{U}}$ is defined by the sequence \eqref{eq32}.
On any open $\Delta ' \subseteq \Delta_{top}$,
${\scr{U}} (\Delta ') = |\Delta '|^\perp \cap M =
\{m \in M | \langle m,y\rangle = 0$ for all $y \in |\Delta '|\}$.
Because ${\scr{M}}$ is flasque,
$H^p(\Delta_{top} ,{\scr{M}} ) = 0$ for all $p \geq 1$, so
$H^p(\Delta_{top} ,{\scr{S}\scr{F}} ) \cong H^{p+1}(\Delta_{top} ,{\scr{U}} )$
for all $p \geq 1.$
Let $k$ be a field and $X = T_N\emb(\Delta,k)$ a nonsingular toric variety
over $k$.
Let
$N' = \langle \bigcup_{\sigma \in \Delta} \sigma \cap N \rangle$, let $\nu
\ge 2$ be relatively prime to $\operatorname{char}{k}$,
and let $M_\nu = \{ m \in M | \langle m,n' \rangle \equiv 0 \pmod \nu
\text{ for all } n' \in N' \}$.
The basis theorem for finitely generated abelian groups gives a basis $n_1,
\dots, n_r$ of $N$ such that
$N' = \Bbb Z a_1 n_1 \oplus \Bbb Z a_2 n_2 \oplus \dots \oplus \Bbb Z a_r n_r$
where the $a_i$ are nonnegative integers and
$a_i | a_{i+1}$ for $1 \le i \le r-1$.
As in \cite{DF:Bgt} call $a_1, \dots, a_r$ the set of invariant factors
of $X$.
\begin{theorem} \label{th5} In the above terminology,
if $(\nu, a_i)$ is the greatest common
divisor of $\nu$ and $a_i$, then
\begin{equation}
\label{eq29}
\begin{split}
H^1(X,\Bbb Z/\nu) &\cong
H^1(k,\Bbb Z/\nu) \oplus
\left(
M_\nu/\nu M \otimes \mu_\nu ^{-1}
\right) \\
&\cong
H^1(k,\Bbb Z/\nu) \oplus \left(
\bigoplus_{i=1}^r \Bbb Z/(\nu,a_i) \otimes \mu_\nu ^{-1}
\right) \text{ .}
\end{split}
\end{equation}
\begin{equation}
\label{eq30}
\begin{split}
{_\nu \operatorname{B}(X)} & = {_\nu \operatorname{B}'(X)} \cong \\ &
{_\nu \operatorname{B}(k)} \oplus
\left( \bigoplus_{i=1}^r H^1(k,\Bbb Z/\nu) \otimes \Bbb Z/(\nu,a_i)
\right) \oplus
\left(
\bigoplus_{i=1}^r \operatorname{Hom}(\Bbb Z/a_i \otimes \mu_\nu, \Bbb Q/ \Bbb Z )^{r-i}
\right)
\end{split}
\end{equation}
\end{theorem}
\begin{pf} Follows from the proof of \cite[Theorem~1.1]{DF:Bgt} and
Theorem~\ref{th4}. \end{pf}
Therefore the $\nu$-torsion of the Brauer group of the nonsingular toric
variety $X$ is
generated by the classes of algebras from $k$ and
cyclic crossed
product algebras of 2 types. For each cyclic Galois extension $C/k$ of
degree $\nu$ with group $\langle \sigma \rangle$ and for each $1 \le i \le
r$, there is the cyclic crossed
product $(C/k,\langle \sigma \rangle, x_i)$ which is an Azumaya algebra
over the torus $T_N$. This algebra is unramified on $X$ if and only if the
function $x_i$ corresponds to an element of $M_\nu$.
If there exists a $\nu$-th root of unity $\zeta$ over
$k$, then the symbol algebras $(x_i,x_j)_\nu$ are Azumaya algebras over
$T_N$. Those symbols which are unramified on $X$ correspond to the
last summand of \eqref{eq30}.
\begin{example}
\label{ex14}
Let $k= \Bbb R$ and $X = T_N\emb(\Delta)$ a nonsingular toric variety over
$\Bbb R$.
Then by
Theorem~\ref{th5}
$\operatorname{B}(X)$ is an elementary 2-group. If $t = \left| \{ a_i | (2,a_i) \not = 1
\} \right|$, then
\begin{equation}
\label{eq31}
\begin{split}
\operatorname{B}(X) & \cong
\operatorname{B}(\Bbb R) \oplus
\left( \bigoplus_{i=1}^r \Bbb Z/(2,a_i)
\right) \oplus
\left(
\bigoplus_{i=1}^r \operatorname{Hom}(\Bbb Z/a_i \otimes \mu_2, \Bbb Q/ \Bbb Z )^{r-i}
\right) \\
& \cong \Bbb Z/2 \oplus (\Bbb Z/2)^t \oplus (\Bbb Z/2)^{t(t-1)/2}
\end{split}
\end{equation}
\end{example}
\begin{theorem} \label{th6}
Let $k$ be a field and $X = T_N\emb(\Delta)$ a toric variety
over $k$ with function field $K$.
Then
\begin{enumerate}
\item
$H^p(\Delta_{top},\scr{U}) \cong H^p(X_{\operatorname{Zar}},\scr{O}^*)$ for all $p \ge 1$ hence
$H^p(X_{\operatorname{Zar}},\scr{O}^*)$ depends only on $\Delta_{fan}$, not $k$. In particular
$\operatorname{Cl}(X)$ and $\operatorname{Pic}{X}$ depend only on $\Delta_{fan}$.
\item
$H^1(\Delta_{top},\scr{S}\scr{F}) \cong H^2(X_{\operatorname{Zar}},\scr{O}^*) \cong H^2(K/X_{\operatorname{\acute{e}t}},\Bbb
G_m)$ hence $H^2(K/X_{\operatorname{\acute{e}t}},\Bbb G_m)$ depends only on $\Delta_{fan}$, not $k$.
\item
If $\tilde \Delta $ is a nonsingular subdivision of $\Delta $
and $\tilde X = T_N\emb(\tilde \Delta )$, then the sequence
\[
0 \to H^2(K/X_{\operatorname{\acute{e}t}},\Bbb G_m) \to
H^2(X_{\operatorname{\acute{e}t}},\Bbb G_m) \to
H^2(\tilde X_{\operatorname{\acute{e}t}},\Bbb G_m) \to 0
\]
(with natural maps) is split-exact.
\end{enumerate}
\end{theorem}
\begin{pf}
The theorem follows from
\cite{DFM:CBg},
noting that the proof of \cite[Theorem~1]{DFM:CBg} did not assume
that $k$ is algebraically closed until the proof of Lemma~7 where it was
not necessary anyway.
\end{pf}
\section{Topological Invariants} \label{sec3}
The first invariants to be considered as candidates for topological
invariants are the following. Let $\Delta$ and $X$ be as in the
Introduction. For each $i \ge 0$ we define a positive
integer $\rho_i$.
Set
\[
\rho_0 = \operatorname{dim}_{\Bbb Q} \left[ H^0(X_{\operatorname{\acute{e}t}},\Bbb G_m) / k^* \otimes \Bbb Q
\right] \text{,}
\]
\[
\rho_1 = \operatorname{dim}_{\Bbb Q} \left[ H^1(X_{\operatorname{\acute{e}t}},\Bbb G_m) \otimes \Bbb Q \right]
= \operatorname{dim}_{\Bbb Q} \left( \operatorname{Pic}{X} \otimes \Bbb Q \right)
\]
and for $i \ge 2$,
\[
\rho_i = \operatorname{dim}_{\Bbb Q} \left[ H^i(X_{\operatorname{\acute{e}t}},\Bbb G_m) \otimes \Bbb Q \right]
\text{.}
\]
The number $\rho_1$ is the traditional Picard number $\rho$ associated to
$X$. Also set
\[
\rho_1' = \operatorname{dim}_{\Bbb Q} \left[ \operatorname{Cl}(X) \otimes \Bbb Q \right] \text{.}
\]
It follows from Theorem~\ref{th1} below that $\rho_0$ is a fan invariant
and from Theorem~\ref{th6} above that $\rho_1$, $\rho_1'$, and $\rho_2$ are
fan invariants.
Since $\Delta$ is finite, $\rho_0$, $\rho_1$, $\rho_1'$ and $\rho_2$ are
finite. For $\rho_0$, $\rho_1$ and $\rho_1'$ see \cite{O:CBA} or
\cite{F:ITV}. For $\rho_2$ this follows from \cite{DFM:CBg}.
Examples where the number $\rho_2$ is computed seem to be somewhat scarce.
Grothendieck \cite[II]{G:GB} and Childs \cite{C:Bgn} each give an example
of a local ring $\scr{O}_x$ on a normal surface where
$H^2((\scr{O}_x)_{\operatorname{\acute{e}t}},\Bbb G_m)$
is torsion free, but in each case $H^2((\scr{O}_x)_{\operatorname{\acute{e}t}},\Bbb G_m)$ is not
finitely generated.
\begin{remark}\label{re6}
The dimension of the topological space $\Delta_{top}$ is defined to be the
length of a maximal chain of irreducible closed subsets. One can check that
this is equal to $\max{ \{ \operatorname{dim}{\sigma} |}$ ${ \sigma \in \Delta \} }$.
Therefore $\operatorname{dim}{\Delta_{top}}$ is a topological invariant of
$\Delta_{fan}$.
\end{remark}
\begin{remark}\label{re7}
Define another sequence of invariants by
\[
\kappa_i = \operatorname{dim}_{\Bbb Q}\left[ H^i(\Delta_{top},\scr{S}\scr{F}) \otimes \Bbb Q
\right]
\]
for $i \ge 0$. It follows from Theorem~\ref{th6}~(2) that $\kappa_1 =
\rho_2$.
Let $\sigma_0, \dots, \sigma_m$ be the maximal cones in
$\Delta$. From \cite[Lemma 8]{DFM:CBg} $\kappa_i$ can be computed from the
$\operatorname{\check Cech}$
complex
\begin{equation}
\label{eq12}
0 \to \underset{i}{\oplus}
\scr{S}\scr{F}(\sigma_i) \stackrel{\delta^0}{\rightarrow} \underset{i<j}{\oplus}
\scr{S}\scr{F}(\sigma_{ij}) \stackrel{\delta^1}{\rightarrow} \underset{i<j<k}{\oplus}
\scr{S}\scr{F}(\sigma_{ijk}) \to \dots
\end{equation}
For any cone $\tau \in \Delta$, $\operatorname{dim}_{\Bbb Q}(\scr{S}\scr{F}(\Delta(\tau)) \otimes
\Bbb Q) = \operatorname{dim}{\tau}$. Therefore, if $C^i$ denotes the $i$-th group of $\operatorname{\check Cech}$
cochains in \eqref{eq12} and $c_i = \operatorname{dim}_{\Bbb Q}(C^i \otimes \Bbb Q)$,
then the integer
\begin{equation}
\label{eq24}
c_0 - c_1 + c_2 - \dots
\end{equation}
is a topological invariant of $\Delta_{fan}$. Note that there exists an
integer $M$ such that $C^j = 0$ for all $j > M$. If $\operatorname{dim}(\Delta_{top}) =
t$, then $\kappa_j = 0$ for all $j>t$. So the left hand side of
\begin{equation}
\label{eq25}
\kappa_0 - \kappa_1 + \dots (-1)^t \kappa_t =
c_0 - c_1 + c_2 - \dots (-1)^M c_M
\end{equation}
is a topological invariant of $\Delta_{fan}$.
\end{remark}
\begin{remark}\label{re5}
Let $\Delta$
be a finite fan on $N \otimes \Bbb R$ where $N = \Bbb Z^r$.
Setting $s = \operatorname{dim}_{\Bbb R} \Bbb R|\Delta_{fan}| $, we see
that $s$ is not a topological invariant of $\Delta_{fan}$. Since $s \ge
\operatorname{dim}{\Delta_{top}}$, if $\operatorname{dim}{\Delta_{top}} = r$, then $s=r$ so if $\Delta$
contains a cone $\sigma$ such that $\operatorname{dim}{\sigma}=r$, then $s=r$ and $s $ is
a topological invariant of $\Delta_{fan}$. This condition is satisfied, for
instance, if $\Delta$ is a complete fan on $N \otimes \Bbb R$.
\end{remark}
\begin{remark}\label{re2}
Let $\sigma \in \Delta$ and let $\Delta(\sigma)$ denote the subfan of
$\Delta$ consisting of the cone $\sigma$ and all of its faces. Then
\[
\operatorname{dim}{\sigma} = \operatorname{dim}{\Delta(\sigma)_{top}} \text{,}
\]
so the dimensions of the cones in $\Delta$ depend only on $\Delta_{top}$.
In particular, the number of 1-dimensional cones in $\Delta_{fan}$ is a
topological invariant.
\end{remark}
\begin{remark}\label{re3}
The fan $\Delta_{fan}$ is complete by definition if $|\Delta_{fan}| = \Bbb
R^r$. This is true if and only if
\begin{itemize}
\item[(i)] $\Delta(r) \not = \emptyset$ and
\item[(ii)]
for each cone $\sigma \in \Delta(r)$ and every $r-1$-dimensional face
$\tau$ of $\sigma$ there is a cone $\sigma_1 \in \Delta(r)$ such that $\tau
= \sigma \cap \sigma_1$.
\end{itemize}
But these two conditions depend only on
$\Delta_{top}$. That is, completeness can be thought of as a topological
property of $\Delta_{fan}$.
\end{remark}
{F}rom the next theorem, which combines some results on $\rho_0$ and
$\rho_1'$, we see that $\rho_0$ depends only on the dimension of
the subspace spanned by $|\Delta_{fan}|$.
\begin{theorem}
\label{th1}
Let $N = \Bbb Z^r$, $\Delta$ a fan on $N \otimes \Bbb R$,
$X = T_N\emb(\Delta)$, and
$s =
\operatorname{dim}_{\Bbb R} \Bbb R|\Delta_{fan}|$ (that is, $s$ is the dimension of the $\Bbb
R$-vector space spanned by the vectors in the support $|\Delta_{fan}|$). Then
\begin{itemize}
\item[(a)]
$\rho_0 = r-s$, hence is a fan invariant, but not a topological invariant.
\item[(b)]
Suppose $\Delta_{fan}$ contains a cone $\sigma$ such that $\operatorname{dim}{\sigma} =
r$. This is true for example if $\Delta_{fan}$ is a complete fan on $N
\otimes \Bbb R$. Then $\rho_0 = 0$ and $\rho_0$ is a topological invariant
of $\Delta_{fan}$.
\item[(c)]
$\rho_1'=\operatorname{dim}_{\Bbb Q}(\operatorname{Cl}(X) \otimes \Bbb Q) = \#(\Delta(1)) -s$.
If $\operatorname{dim}{\Delta_{top}} =r$, then $\rho_1'$ is a topological invariant of
$\Delta_{fan}$.
\item[(d)]
The number $\rho_0-\rho_1'$ is a topological
invariant of $\Delta_{fan}$.
\end{itemize}
\end{theorem}
\begin{pf} (a)
Let $N_1 = N \cap \Bbb R |\Delta_{fan}|$, $M_1 = \operatorname{Hom}_{\Bbb Z}(N_1,\Bbb
Z)$, $M_2 = N_1^{\perp}$, $N_2 = \operatorname{Hom}_{\Bbb Z}(M_2,\Bbb Z)$. Then $M = M_1
\oplus M_2$ and $N = N_1 \oplus N_2$. Viewing $\Delta$ as a fan on the
$s$-dimensional vector space $N_1 \otimes \Bbb R$, $X = T_{N_1}\operatorname{emb}(\Delta)
\times T_{N_2}$ and $H^0(X,\Bbb G_m) / k^* \cong H^0(T_{N_2},\Bbb G_m) /
k^* \cong \Bbb Z^{r-s}$. So $\rho_0 = r-s$.
(b)
In this case $s = \operatorname{dim}_{\Bbb R} \Bbb R|\Delta_{fan}|
= \operatorname{dim}_{\Bbb R} (\Bbb R \sigma) = \operatorname{dim} \sigma = r$.
(c)
Let $N_0 = N \cap \Bbb R |\Delta_{fan}|$ be the set of lattice points in the
subspace $\Bbb R |\Delta_{fan}|$
and $M_0 = \operatorname{Hom}_{\Bbb Z}(N_0,\Bbb Z)$. Then
$\operatorname{dim}_{\Bbb Q}(M_0 \otimes \Bbb Q) =
\operatorname{dim}_{\Bbb Q}(N_0 \otimes \Bbb Q) = \operatorname{dim}_{\Bbb R} \Bbb R |\Delta_{fan}| =
s$. From \cite[Corollary~2.5]{O:CBA} there is a presentation
of $\operatorname{Cl}(X)$
\begin{equation}
\label{eq0}
0 \to M_0 \to \bigoplus_{\rho \in \Delta(1)} \Bbb Z \rho \to
\operatorname{Cl}(X) \to 0 \text{.}
\end{equation}
So
$\operatorname{dim}_{\Bbb Q}(\operatorname{Cl}(X) \otimes \Bbb Q)
= \#(\Delta(1)) -s$. In
particular, if $\operatorname{dim}{\Delta_{top}} =r$, then $r=s$ so $\operatorname{dim}_{\Bbb Q}(\operatorname{Cl}(X)
\otimes \Bbb Q)$ is a topological invariant.
(d)
{F}rom (a) and (c),
\[
\rho_0-\rho_1' = (r - s) - (\#(\Delta(1)) -s) = r - \#(\Delta(1))
\]
which depends only on $\Delta_{top}$.
\end{pf}
\begin{remark}
\label{re1}
Let $N = \Bbb Z^r$, $\Delta$ a fan on $N \otimes \Bbb R$, $s = \operatorname{dim}_{\Bbb
R} \Bbb R |\Delta_{fan}|$, $t = \underset{\sigma \in \Delta}{\max} \{
\operatorname{dim}{\sigma} \}$.
It follows from \cite[Theorem~2.3]{DF:Bgt} that if
$t \le 2$, then $\rho_1 = \#(\Delta(1)) - s$ and $\rho_2 = 0$.
In this case $\Delta$ is a simplicial fan, so this is a
special case of the following theorem.
\end{remark}
\begin{theorem}
\label{th2}
Let $N = \Bbb Z^r$.
Let $\Delta$ be a simplicial fan on $N \otimes \Bbb R$
and $s = \operatorname{dim}_{\Bbb R}\Bbb R |\Delta_{fan}|$.
Then
\begin{itemize}
\item[(a)]
$\rho_1 =
\#(\Delta(1))-s$.
\item[(b)]
$\rho_2 = 0$ hence is a topological invariant of
$\Delta_{fan}$.
\item[(c)]
If $\operatorname{dim}{\Delta_{top}}=r$,
then $\rho_1$ is a topological invariant of $\Delta_{fan}$.
\item[(d)]
$\rho_0-\rho_1+\rho_2$ is a topological
invariant of $\Delta_{fan}$.
\end{itemize}
\end{theorem}
\begin{pf}
Suppose $\sigma \in \Delta$ is a simplicial cone and
$\operatorname{dim}{\sigma} = d$. For any support function $h \in \scr{S}\scr{F}(\Delta)$,
$h|_\sigma$ is linear and completely determined by its values on a
spanning set $\{\eta_1, \dots, \eta_d \} \subseteq N$ for $\sigma$. Since
$\operatorname{dim}{\sigma}=d$, $\sigma$ is spanned by $d$ lattice points. So
$\scr{S}\scr{F}(\Delta(\sigma)) \otimes \Bbb Q \cong \Bbb Q^d$.
If $\tau_0, \dots, \tau_n$ are the cones in $\Delta(1)$, and $\Gamma =
\{0,\tau_0, \dots, \tau_n \}$, then $\Gamma_{top}$ is an open subset of the
topological space $\Delta_{top}$. Define the sheaf $\scr{W}$ on
$\Delta_{top}$ to be the direct image $i_*(\scr{S}\scr{F} |_{\Gamma_{top}})$.
Since $\Gamma_{fan}$ is a nonsingular fan, $\scr{S}\scr{F} |_{\Gamma_{top}}$ is the
sheaf defined by $\Xi \mapsto \Bbb Z^{\#(\Xi(1))}$ for each open $\Xi
\subseteq \Delta_{top}$.
It follows that $\scr{W}(\Xi) = \Bbb Z^{\# (\Xi(1))}$, hence $\scr{W}$ is a
flasque sheaf. So there is an embedding $\scr{S}\scr{F} \to \scr{W}$ of sheaves on
$\Delta_{top}$ and we define $\scr{P}$ by the exact sequence of sheaves
\cite[(13), p. 149]{DFM:CBg}
\begin{equation}
\label{eq1}
0 \to \scr{S}\scr{F} \to \scr{W} \to \scr{P} \to 0 \text{.}
\end{equation}
Since $\Delta$ is simplicial,
$ \scr{S}\scr{F}(\Delta(\sigma))$ and $\scr{W}(\Delta(\sigma))$
are free of the same rank $\operatorname{dim}{\sigma}$. Therefore, $\scr{P}$ is
locally torsion, hence torsion.
Because $\scr{W}$ is flasque, $H^1(\Delta_{top},\scr{W}) = 0$ and the long exact
sequence associated to \eqref{eq1} becomes
\begin{equation}
\label{eq2}
0 \to H^0(\Delta_{top},\scr{S}\scr{F}) \to H^0(\Delta_{top},\scr{W}) \to
H^0(\Delta_{top},\scr{P})
\to H^1(\Delta_{top}, \scr{S}\scr{F}) \to 0 \text{.}
\end{equation}
Because $\scr{P}$ is torsion, $H^0(\Delta_{top},\scr{P})$ is torsion. So
$H^1(\Delta_{top}, \scr{S}\scr{F}) \otimes \Bbb Q = 0$. By
\cite[Theorem~1]{DFM:CBg} $\rho_2 = \operatorname{dim}(H^1(\Delta_{top}, \scr{S}\scr{F}) \otimes
\Bbb Q) = 0$. This proves (b).
It also follows from \eqref{eq2} that we obtain the isomorphism of
\cite[Proposition~2.1(v), p. 69]{O:CBA}
\begin{equation}
\label{eq10}
H^0(\Delta_{top},\scr{S}\scr{F}) \otimes \Bbb Q \cong H^0(\Delta_{top},\scr{W})
\otimes \Bbb Q \text{.}
\end{equation}
By \cite[Lemma~8]{DFM:CBg} there is an exact sequence
\begin{equation}
\label{eq6}
0 \to M_0 \to \scr{S}\scr{F}(\Delta_{top}) \to \operatorname{Pic}{X} \to 0 \text{.}
\end{equation}
\noindent
Combining \eqref{eq0} and \eqref{eq6},
we have a commutative diagram with exact rows and columns.
\begin{equation}
\label{eq9}
\begin{CD}
0 @. 0 @. 0 \\
@VVV @VVV @VVV \\
M @>>> \scr{S}\scr{F}(\Delta_{top}) @>>> \operatorname{Pic}{X} @>>> 0 \\
@VV=V @VVV @VVV \\
M @>>> \scr{W}(\Delta_{top}) @>>> \operatorname{Cl}(X) @>>> 0
\end{CD}
\end{equation}
Because the center vertical arrow in \eqref{eq9} tensored with $\Bbb Q$ is
the isomorphism \eqref{eq10}, from \eqref{eq9} it follows that
\begin{equation}
\label{eq11}
\operatorname{Pic}(X) \otimes \Bbb Q \cong \operatorname{Cl}(X) \otimes \Bbb Q \text{.}
\end{equation}
It follows from Theorem~\ref{th1} that $\rho_1 = \#(\Delta(1))-s$.
This proves (a). In case (c), $s=r$ so $\rho_1$ is a topological invariant.
(d)
{F}rom Theorem~\ref{th1} and parts (a) and (b),
\[
\rho_0-\rho_1+\rho_2 =
(r - s) - (\#(\Delta(1)) -s) + 0 = r - \#(\Delta(1))
\]
which only depends on $\Delta_{top}$.
\end{pf}
\begin{lemma}
\label{lem2}
For any cone $\sigma \in \Delta$, let $\Delta(\sigma)$ denote the subfan of
$\Delta$ consisting of $\sigma$ and all of its faces and $U_\sigma =
T_N\emb(\Delta(\sigma))$.
Then $H^0(\Delta(\sigma)_{top},\scr{P}) = \operatorname{Cl}(U_\sigma)$.
\end{lemma}
\begin{pf}
For each $\sigma \in \Delta$ we have $H^1(\Delta(\sigma)_{top}, \scr{S}\scr{F})
= 0$ \cite[Lemma 2.a, p. 139]{DFM:CBg} so from \eqref{eq2}
$$H^0(\Delta(\sigma)_{top},\scr{P}) = \scr{W}(\Delta(\sigma)_{top}) /
\scr{S}\scr{F}(\Delta(\sigma)_{top}) \text{.}$$
Now $\scr{W}(\Delta(\sigma)_{top}) = \Bbb Z^{\#(\Delta(\sigma)(1))}$ and
support functions are linear on a cone $\sigma$, so
$$\scr{W}(\Delta(\sigma)_{top}) / \scr{S}\scr{F}(\Delta(\sigma)_{top})
\cong \Bbb Z^{\#(\Delta(\sigma)(1))}/ \operatorname{im}(M)
= \operatorname{Cl}(U_\sigma) \text{.}$$
\end{pf}
\begin{lemma}
\label{lem3}
Let $\sigma$ be a cone in $N \otimes \Bbb R$ and $s = \operatorname{dim}{\sigma}$. Then
$\operatorname{dim}_{\Bbb Q}(\operatorname{Cl}(U_\sigma) \otimes \Bbb Q) = \#(\Delta(\sigma)(1))-s$.
Also $\sigma$ is simplicial if and only if $\operatorname{Cl}(U_\sigma) $ is torsion.
\end{lemma}
\begin{pf}
Follows from Theorem~\ref{th1}.
\end{pf}
The following can be considered a theorem for 3-dimensional fans.
\begin{theorem}
\label{th3}
Let $\Delta$ be a fan on $N \otimes \Bbb R$.
Let $\sigma_0, \dots, \sigma_w$ be the maximal cones in $\Delta$.
Assume
$\sigma_i \cap
\sigma_j$ is simplicial for each $i \not = j$. These assumptions are
satisfied for example if $\operatorname{dim}{\Delta_{top}} \le 3$.
Then
\begin{itemize}
\item[(a)]
\[
\rho_1 +s + \sum_{i=0}^w(\#(\Delta(\sigma_i)(1))-s_i) =
\rho_2 + \#( \Delta(1)) \text{,}
\]
where we set $s_i = \operatorname{dim}{\sigma_i}$ for each $i = 0, \dots, w$ and $s =
\operatorname{dim}_{\Bbb R} \Bbb R |\Delta_{fan}|$.
\item[(b)]
$\rho_0-\rho_1+\rho_2$ is a topological invariant of $\Delta_{fan}$.
\end{itemize}
\end{theorem}
\begin{pf} (a)
The set $\{ \Delta(\sigma_i)_{top}\}_{i=0}^w$ is an open cover of
$\Delta_{top}$ and the sequence
\begin{equation}
\label{eq3}
0 \to H^0(\Delta_{top},\scr{P}) \to \bigoplus_{i=0}^w
H^0(\Delta(\sigma_i)_{top},\EuScript P) \to
\bigoplus_{i=1}^w\bigoplus_{j=0}^{i-1}H^0(\Delta(\sigma_i \cap
\sigma_j)_{top},\scr{P})
\end{equation}
is exact since $\scr{P}$ is a sheaf. Applying Lemma~\ref{lem2}, the sequence
\eqref{eq3} can be written
\begin{equation}
\label{eq4}
0 \to H^0(\Delta_{top},\scr{P}) \to \bigoplus_{i=0}^w \operatorname{Cl}(U_{\sigma_i}) \to
\bigoplus_{i=1}^w\bigoplus_{j=0}^{i-1} \operatorname{Cl}(U_{\sigma_i \cap \sigma_j}) \text{.}
\end{equation}
By our assumption $\sigma_i \cap \sigma_j$ is
simplicial. By Lemma~\ref{lem3}, $\operatorname{Cl}(U_{\sigma_i \cap \sigma_j})$ is
torsion. By \cite[Theorem 1]{DFM:CBg} $H^1(\Delta_{top}, \scr{S}\scr{F}) \cong
H^2(K/X_{\operatorname{\acute{e}t}},\Bbb G_m)$
and the torsion-free part of $H^2(X_{\operatorname{\acute{e}t}},\Bbb G_m)$ is equal
to the torsion-free part of $H^2(K/X_{\operatorname{\acute{e}t}},\Bbb G_m)$. We compute the rank of
the
torsion-free part of $H^2(K/X_{\operatorname{\acute{e}t}},\Bbb G_m)$ from \eqref{eq2} tensored with
$\Bbb Q$:
\begin{equation}
\label{eq5}
0 \to \scr{S}\scr{F}(\Delta_{top}) \otimes \Bbb Q \to \scr{W}(\Delta_{top}) \otimes
\Bbb Q \to
\scr{P}(\Delta_{top}) \otimes \Bbb Q \to H^2(K/X_{\operatorname{\acute{e}t}}, \Bbb G_m) \otimes
\Bbb Q \to 0 \text{.}
\end{equation}
Tensoring \eqref{eq6} with $\Bbb Q$ and counting dimensions we find
$\operatorname{dim}_{\Bbb Q}(\scr{S}\scr{F}(\Delta_{top}) \otimes \Bbb Q) =$ $ \operatorname{dim}_{\Bbb
Q}(\operatorname{Pic}{X}$ $\otimes \Bbb Q) + s$ $ = \rho_1 +s$. By definition
$\scr{W}(\Delta_{top}) \otimes \Bbb Q
= \Bbb Q^{\#(\Delta(1))}$.
{}From \eqref{eq4} $\scr{P}(\Delta_{top}) \otimes \Bbb Q
= \bigoplus_{i=0}^w \operatorname{Cl}(U_{\sigma_i}) \otimes \Bbb Q$.
{F}rom
Lemma~\ref{lem3}, $\operatorname{dim}_{\Bbb Q}(\operatorname{Cl}(U_{\sigma_i}) \otimes \Bbb Q) =
\#(\Delta(\sigma_i)(1))-s_i$.
Counting dimensions in
\eqref{eq5}, we have the equation
\begin{equation}
\label{eq7}
\rho_2 = \rho_1 +s + \sum_{i=0}^w(\#(\Delta(\sigma_i)(1))-s_i) - \#(\Delta(1))
\text{ .}
\end{equation}
(b)
{F}rom (a) and Theorem~\ref{th1} we have
\begin{align*}
\rho_0-\rho_1+\rho_2
= & (r-s) +s +
\sum_{i=0}^w(\#(\Delta(\sigma_i)(1))-s_i) - \# (\Delta(1)) \\
\mbox{} = & r+ \sum_{i=0}^w(\#(\Delta(\sigma_i)(1))-s_i) - \# (\Delta(1))
\end{align*}
which depends only on $\Delta_{top}$.
\end{pf}
As the next example shows, $\rho_1$ and $\rho_2$ are not topological
invariants of $\Delta_{fan}$ when $r \ge 3$ and $\Delta$ is not simplicial.
\begin{example}
\label{ex1}
Let $\Delta$ be a fan on $\Bbb R^3$ and suppose $\Delta$ consists of
three cones of dimension 3 and 6 cones of dimension 1 such that for each
$\sigma_i \in \Delta(3)$, $\#(\Delta(\sigma_i)(1)) = 4$. Assume that the
intersection of the fan $\Delta$ with the unit sphere traces a graph that
looks like that shown in Figure~\ref{fig1}.
\setlength{\unitlength}{.005in}
\begin{figure}
\center{
\begin{picture}(400,400)
\thinlines
\put(50,50){\line(1,0){300}}
\put(50,50){\line(5,6){250}}
\put(300,350){\line(1,-6){50}}
\put(50,50){\line(2,1){100}}
\put(150,100){\line(1,0){150}}
\put(150,100){\line(1,1){100}}
\put(300,100){\line(1,-1){50}}
\put(250,200){\line(1,-2){50}}
\put(250,200){\line(1,3){50}}
\put(50,50){\circle*{5}}
\put(150,100){\circle*{5}}
\put(250,200){\circle*{5}}
\put(300,100){\circle*{5}}
\put(350,50){\circle*{5}}
\put(300,350){\circle*{5}}
\put(45,50){\makebox(0,0)[r]{\small{5}}}
\put(355,50){\makebox(0,0)[l]{\small{4}}}
\put(145,100){\makebox(0,0)[br]{\small{2}}}
\put(305,100){\makebox(0,0)[bl]{\small{1}}}
\put(245,200){\makebox(0,0)[br]{\small{0}}}
\put(300,355){\makebox(0,0)[b]{\small{3}}}
\put(180,180){\makebox(0,0)[l]{\small{$\sigma_0$}}}
\put(240,75){\makebox(0,0)[r]{\small{$\sigma_2$}}}
\put(310,180){\makebox(0,0)[r]{\small{$\sigma_1$}}}
\end{picture} }
\vspace{-0.35in}
\caption{}
\label{fig1}
\end{figure}
For any such fan $\Delta$, $\Delta_{top}$ is unique up to homeomorphism. We
consider 2 such fans $\Delta$ and $\Delta'$ such that $\rho_1(\Delta) \not
= \rho_1(\Delta')$ and $\rho_2(\Delta) \not
= \rho_2(\Delta')$.
For $\Delta$, take $\Delta(1)$ to be $\{ \Bbb R_{\ge} \eta_i | i=0..5\}$
where $\{\eta_0, \dots, \eta_5\}$ $ =$
\begin{equation} \label{eq23.5}
\left\{
\begin{pmatrix} 1 \\ 0 \\ -2 \end{pmatrix},
\begin{pmatrix} -1 \\ 2 \\ -2 \end{pmatrix},
\begin{pmatrix} -1 \\ -2 \\ -2 \end{pmatrix},
\begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix},
\begin{pmatrix} -1 \\ 2 \\ 2 \end{pmatrix},
\begin{pmatrix} -1 \\ -2 \\ 2 \end{pmatrix} \right\}
\text{.}
\end{equation}
Using the methods of \cite[Section 4]{F:Elt} we find that
$\rho_1(\Delta) =1$ and $\rho_2(\Delta) =1$.
For $\Delta'$, take $\Delta'(1)$ to be $\{ \Bbb R_{\ge} \eta_i | i=0..5\}$
where $\{ \eta_0, \dots, \eta_5\}$ $ =$
\[ \left\{
\begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix},
\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix},
\begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix},
\begin{pmatrix} -1 \\ 3 \\ 1\end{pmatrix},
\begin{pmatrix} -2 \\ -1 \\ 1\end{pmatrix},
\begin{pmatrix} 3 \\ -1 \\ 1 \end{pmatrix} \right\} \text{.} \]
Using the methods of \cite[Section 4]{F:Elt} we find that
$\rho_1(\Delta') =0$ and $\rho_2(\Delta') =0$.
\end{example}
\section{A Stratification of the fibers of $\frak T$}
\label{sec4}
Let $\Delta$ be a fan on $N \otimes \Bbb R= \Bbb R^r$ with $\Delta(1) =
\{r_0, \dotsc, r_n \}$. The intersection of $\Delta(1)$ with the unit
sphere $S$ in $\Bbb R^r$ is a finite set of points, say $\{ p_0, \dotsc,
p_n\}$.
About each $p_i$ we can find an open ball $B_i$ on $S$ such that if
$p_i$ is parametrized by $B_i$, then each choice of $\vec{p} = (p_0, p_1,
\dotsc,
p_n)$ in $B_0 \times B_1 \times \dotsm \times B_n$ defines a fan $\Phi =
\Phi(\vec{p})$ such that $\Phi_{top} \cong \Delta_{top}$. The manifold
$\displaystyle{B = \prod_{i=0}^n B_i}$
parametrizes a subset of fans in the fiber
$\frak T^{-1}(\Delta_{top})$. Call $B$ an {\em open neighborhood of
$\Delta$}.
If $\vec{p} \in B$, then the fan $\Phi = \Phi(\vec{p})$ is not necessarily
rational.
Sometimes it will be necessary to refer to
points in $B$ that give rise to rational fans. In this case let
\begin{multline}
\label{eq35}
B_{rat} = \{ (p_0, \dotsc, p_n) | \text{ for each } i \text{, } \\
p_i \text{ is the
intersection of a rational 1-dimensional cone } r_i \text{ with } B_i \}
\text{.}
\end{multline}
For the present section only
we define the set of support functions on a fan to be a real vector space.
If $\sigma$ is a cone, define $\scr{S}\scr{F}(\sigma) $ to be $\operatorname{Hom}_{\Bbb R}(\Bbb R
\sigma, \Bbb R)$. Define $\scr{S}\scr{F}(\Delta)$ to be the kernel of $\delta^0$ in
the $\operatorname{\check Cech}$ complex
\begin{equation}
\label{eq34}
0 \to \underset{i}{\oplus}
\scr{S}\scr{F}(\sigma_i) \stackrel{\delta^0}{\rightarrow} \underset{i<j}{\oplus}
\scr{S}\scr{F}(\sigma_{ij}) \stackrel{\delta^1}{\rightarrow} \underset{i<j<k}{\oplus}
\scr{S}\scr{F}(\sigma_{ijk}) \to \dots
\end{equation}
where $\{ \sigma_0, \dotsc, \sigma_w \}$ is the set of maximal cones of
$\Delta$. Define $\kappa_0(\Delta) = \operatorname{dim}_{\Bbb R} \scr{S}\scr{F}(\Delta)$. If
$\Delta$ is a rational fan, then this definition of $\kappa_0$ agrees with
the definition given in Remark~\ref{re7} of Section~\ref{sec3}.
In this section we consider the stratification of the manifold $B$ by the
invariant $\kappa_0$.
\begin{example}
\label{ex15} If $\Delta$ is simplicial, then $\kappa_0 = \#(\Delta(1)) =
n+1$ so $B$ has only 1 stratum. As was suggested in Example~\ref{ex1}, we
expect the stratification to be more interesting when $\Delta$ is
nonsimplicial.
\end{example}
\begin{example}
\label{ex16} Let $\Delta$ be the fan on $\Bbb R^3$ given in
equation \eqref{eq23.5} of Example~\ref{ex1}.
Let $\displaystyle{B = \prod_{i=0}^5 B_i}$ be an open neighborhood of $\Delta$.
One can check that any support function $h \in \scr{S}\scr{F}(\Delta)$ is
completely determined by its values on $r_0, r_1, r_2, r_3$ so
$\kappa_0(\Delta) \le 4$. {F}rom Example~\ref{ex1} we know that
$\kappa_0(\Delta) = 4$.
It is possible to vary any one of the $r_i$ to
achieve a fan $\Phi$ in $B$ with $\kappa = 3$.
So $B$ has exactly 2 strata. We will see later that the stratum where
$\kappa_0 = 4$ is a Zariski closed subset of $B$.
\end{example}
\begin{conjecture}
\label{conj1}
Let $\Delta$ be a complete fan on $\Bbb R^3$ such that
for each cone $\sigma \in \Delta(3)$,
$\sigma$ is nonsimplicial. Let $B$ be an open neighborhood of $\Delta$ as
described above.
Then for a general choice of $\vec{p} \in B$, if
$\Phi = \Phi(\vec{p})$, then every $\Phi$-linear support
function is linear. In particular for a general choice of $\vec{p} \in
B_{rat}$,
$\kappa_0(\Phi) = 3$ hence $\rho_1(\Phi) = 0$ and
$\rho_2(\Phi)$ is a topological invariant.
\end{conjecture}
In Conjecture~\ref{conj1} by ``general choice'' of
$\vec{p}$ we mean that there is a dense open subset $G \subseteq B$ and
each fan in the
set $\{ \Phi(\vec p) | \vec{p} \in G \}$ satisfies
the conjecture. That is, if Conjecture~\ref{conj1} is true, a sufficiently
general fan $\Delta'$ with $\Delta'_{top} \cong \Delta_{top}$
should satisfy $\kappa_0(\Delta') = 3$.
As motivation for Conjecture~\ref{conj1}, consider the case where each
$\sigma \in \Delta(3)$ has exactly 4 1-dimensional faces. Let $\Delta(3) =
\{\sigma_0, \dots, \sigma_w \}$, $\Delta(2) =
\{\tau_0, \dots, \tau_e \}$, $\Delta(1) =
\{r_0, \dots, r_n \}$. The intersections of the cones in $\Delta(2)$ with
the unit sphere $S$ in $\Bbb R^3$ trace out the edges of a graph on $S$.
This graph has $e+1$ edges, $n+1$ vertices and $w+1$ regions. So $w+1 =
(e+1)-(n+1)+2$. Each $\sigma_j$ has exactly 4 $\tau_i$'s and each $\tau_i$
is in exactly 2 $\sigma_j$'s, so
$2(e+1)=4(w+1)$ or $e+1 = 2(w+1)$. Hence $w+1=n-1$.
{F}rom Theorem~\ref{th1}~(c) $\rho'_1 = \operatorname{dim}_{\Bbb Q}(\operatorname{Cl}(X) \otimes
\Bbb Q) = (n+1)-3 = n-2$ and $\rho'_1(\Delta(\sigma_i)) = \operatorname{dim}_{\Bbb
Q}(\operatorname{Cl}(U_{\sigma_i}) \otimes \Bbb Q) = 4-3 = 1$. From \eqref{eq5} we have
an exact sequence
\begin{equation}
\label{eq13}
0 \to \scr{S}\scr{F}(\Delta_{top}) \otimes \Bbb Q \to \scr{W}(\Delta_{top}) \otimes
\Bbb Q \to \bigoplus_{i=0}^w (\operatorname{Cl}(U_{\sigma_i}) \otimes \Bbb Q)
\end{equation}
\noindent
Since linear Weil divisors correspond to linear Cartier divisors
\eqref{eq13} gives rise to
\begin{equation}
\label{eq14}
0 \to \operatorname{Pic}(X) \otimes \Bbb Q \to \operatorname{Cl}(X) \otimes
\Bbb Q \to \bigoplus_{i=0}^w (\operatorname{Cl}(U_{\sigma_i}) \otimes \Bbb Q)
\end{equation}
\noindent
In \eqref{eq14} the middle term has dimension $n-2$ and the third term
dimension $n-1$. For each $i$ the map $\operatorname{Cl}(X) \to \operatorname{Cl}(U_{\sigma_i})$ is
surjective. So we can view $ \operatorname{Pic}(X) \otimes \Bbb Q$ as the intersection of
$n-1$ hyperplanes through $(0)$ in $\Bbb Q^{n-2}$. In general, this
intersection should be $(0)$.
If the conclusion of Conjecture~\ref{conj1} is satisfied, then from
\eqref{eq7} it follows that $\rho_2 = 3 + (w+1)-(n+1)$ (for a general
choice of $\Delta_{fan}$).
Next we give an algorithm for computing an upper bound for $\kappa_0$ for a
fan of arbitrary dimension.
The algorithm can also be used to obtain an upper bound for $\rho_1$ and
$\rho_2$ for complete rational 3-dimensional fans.
If $\Delta$ is a fan on $\Bbb R^3$ which contains at least 1 cone of
dimension 3, then $\rho_0 = 0$ and by Theorem~\ref{th3} we have
$\rho_1 = \rho_2 + (\text{topological invariant})$.
In this setting $\rho_1 = \kappa_0-3$ .
\begin{algorithm}
\label{alg1}
Let $\Delta$ be a fan on $N \otimes \Bbb R$. The following is an
algorithm for computing an upper bound for $\kappa_0$.
\end{algorithm}
The algorithm is based on the fact that the map $\scr{S}\scr{F}(\Delta) \to
\Bbb Z^{\# (\Delta(1))}$ is injective.
The algorithm finds a subset $G$ of
$\Delta(1)$ such that any support function $h$ in
$\scr{S}\scr{F}(\Delta) \otimes \Bbb Q$ is completely determined by its values on
the 1-dimensional cones in $G$.
If $\sigma$ is a maximal cone in $\Delta$ of dimension $d$, then a support
function $h$ is determined by its values on any $d$ 1-dimensional faces of
$\sigma$ that span a $d$-dimensional subspace of $N\otimes \Bbb R$. Pick
$d$ such elements of $\sigma(1)$ and place them in a set called $G$. Place
all other elements of $\sigma(1)$ in a set called $R$. Initially, $G$ and
$R$ are both empty, and the starting cone $\sigma$ is chosen somewhat
arbitrarily. The algorithm proceeds to branch from the starting cone
$\sigma$ outward until all maximal cones of $\Delta$ have been visited and
$\Delta(1)$ has been partitioned into $\Delta(1)=G\cup R$. The order in
which the maximal cones are traversed is somewhat arbitrary and may affect
both the resulting set $G$ and the resulting cardinality of $G$.
\begin{itemize}
\item[Step 0.]
Set $B= \{ \sigma \in \Delta | \sigma \text{ is a maximal cone in } \Delta \}$.
Set
$G= \emptyset$ and $R= \emptyset$. Go to Step 3.
\item[Step 1.]
If there is a maximal cone $\sigma \in B$ such that
$\sigma(1) \cap (G \cup R)$
contains a spanning set for $\Bbb R \sigma$,
then add the remaining cones in $\sigma(1)-G-R$ to $R$. Remove $\sigma$
from $B$. repeat Step~1 until the condition is false.
\item[Step 2.]
If there is a maximal cone $\sigma \in B$ such that
$\sigma(1) \cap (G \cup R) \not = \emptyset$, then pick $\sigma \in B$ such
that
\begin{itemize}
\item[(i)] $e = \operatorname{dim}_{\Bbb R} \langle \sigma(1) \cap (G \cup R) \rangle$ is
maximal and
\item[(ii)] $d = \operatorname{dim}{\sigma}$ is maximal among all $\sigma \in B$
satisfying (i).
\end{itemize}
For any $\sigma$ satisfying (i) and (ii), pick $\tau_1, \dots, \tau_e$ in
$\sigma(1) \cap (G \cup R)$ such that $\tau_1+ \dots+ \tau_e$ has dimension
$e$. Choose
$\tau_{e+1}, \dots, \tau_d$ in $\sigma(1)$ such that $\tau_1+ \dots+
\tau_d$ has dimension $d$. Add $\tau_{e+1}, \dots, \tau_d$ to $G$ and add
the remaining elements $\sigma(1)-G-R-\{ \tau_{e+1}, \dots, \tau_d \}$ to $R$.
Delete $\sigma$ from $B$. Go to Step 1.
\item[Step 3.]
If $B \not = \emptyset$, then pick $\sigma \in B$ such that $d =
\operatorname{dim}{\sigma}$ is maximal. Pick
$\tau_{1}, \dots, \tau_d$
in $\sigma(1)$ such that $\tau_1+ \dots+ \tau_d$
has dimension $d$. Add $\tau_{1}, \dots, \tau_d$ to $G$ and add the
remaining cones in $\sigma(1)-\{ \tau_{1}, \dots, \tau_d \}$ to $R$.
Delete $\sigma$ from $B$. Go to Step 1.
\item[Step 4.] This point is reached only if $B = \emptyset$. Now
$\Delta(1)$ is partitioned into 2 sets: $\Delta(1) = G \cup R$. Any support
function $h$ in $\scr{S}\scr{F}(\Delta) \otimes \Bbb Q$ is determined completely by
its values on $G$. So $\scr{S}\scr{F}(\Delta) \to \Bbb Z^{\#(G)}$ is injective.
Therefore $\#(G)$ is an upper bound for $\kappa_0$.
\end{itemize}
\begin{example}
\label{ex5}
Let $\Delta$ be a fan on $\Bbb R^3$ that
consists of three 3-dimensional cones and assume that the
intersection of the fan $\Delta$ with the unit sphere $S$ traces a graph
as shown in Figure~\ref{fig2}(a).
In this example, we step through Algorithm~\ref{alg1} to see that
$\kappa_0(\Delta) \le 4$.
It is shown later in Example~\ref{ex2} that for this fan, $\kappa_0=4$.
Initially, $B=\{\sigma_0,\sigma_1,\sigma_2\}$ and $G=R=\emptyset$.
The algorithm proceeds to Step~3. Place $r_1$, $r_4$,
$r_2$ from $\sigma_0(1)$ in $G$ and $r_0$ in $R$. Delete $\sigma_0$ from
$B$. The condition in Step~1 is still false, so the algorithm goes to
Step~2. For $\sigma_1$, $r_0$ and $r_2$ are both in $G\cup R$ and $r_0+r_2$
has dimension $e=2$. Place $r_5$ in $G$ and $r_3$ in $R$. Delete $\sigma_1$
from $B$. Go to Step~1. This time the set $G\cup R$ contains $\{ r_0, r_1,
r_3\}$ which is a spanning set for $\Bbb R \sigma_2$. Therefore,
remove $\sigma_2$ from $B$ and
place $r_6$ in $R$. Any support function $h$ is completely determined by
its values on $r_1$, $r_2$, $r_4$ and $r_5$, so $\kappa_0 \le 4$.
\end{example}
\setlength{\unitlength}{.005in}
\begin{figure}
\center{
\hfill (a)
\begin{picture}(400,400)
\thinlines
\put(200,200){\line(0,1){150}}
\put(75,125){\line(5,3){125}}
\put(200,200){\line(5,-3){125}}
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\put(75,125){\line(5,-3){125}}
\put(75,275){\line(5,3){125}}
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\put(200,50){\circle*{5}}
\put(75,275){\circle*{5}}
\put(195,205){\makebox(0,0)[br]{\small{$r_0$}}}
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\put(10,200){\makebox(0,0)[c]{\small{$\sigma_5$}}}
\end{picture} \hfill
}
\caption{}
\label{fig2}
\end{figure}
\begin{example}
\label{ex2}
Let $\Delta$ be a complete fan on $\Bbb R^3$ and assume that the
intersection of the fan $\Delta$ with the unit sphere $S$ traces a graph
that corresponds to the edges of a cube as shown in Figure~\ref{fig2}(b).
Applying Algorithm~\ref{alg1} to $\Delta$, we see that $\rho_1(\Delta) \le
1$ and $\rho_2(\Delta) \le 1$.
Take $\Delta(1)$ to be $\{ \Bbb R_{\ge} \eta_i | i=0..7\}$
where $\{ \eta_0, \dots, \eta_7\}$ $ =$
\[ \left\{
\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix},
\begin{pmatrix} 1 \\ -1 \\ 1\end{pmatrix},
\begin{pmatrix} -1 \\ -1 \\ 1 \end{pmatrix},
\begin{pmatrix} -1 \\ 1 \\ 1 \end{pmatrix},
\begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix},
\begin{pmatrix} 1 \\ -1 \\ -1\end{pmatrix},
\begin{pmatrix} -1 \\ -1 \\ -1\end{pmatrix},
\begin{pmatrix} -1 \\ 1 \\ -1 \end{pmatrix}
\right\} \text{.} \]
Using the methods of \cite[Section 4]{F:Elt} we find that the upper bounds
predicted by Algorithm~\ref{alg1} are reached:
$\rho_1(\Delta) =1$ and $\rho_2(\Delta) =2$.
Now change the fan so that $\Delta'(1)$ is no longer symmetrical about the
origin. For example,
take $\Delta'(1)$ to be $\{ \Bbb R_{\ge} \eta_i | i=0..7\}$
where $\{ \eta_0, \dots, \eta_7\}$ $ =$
\[ \left\{
\begin{pmatrix} 2 \\ 1 \\ 1 \end{pmatrix},
\begin{pmatrix} 1 \\ -1 \\ 1\end{pmatrix},
\begin{pmatrix} -1 \\ -1 \\ 1\end{pmatrix},
\begin{pmatrix} -1 \\ 1 \\ 1 \end{pmatrix},
\begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix},
\begin{pmatrix} 1 \\ -1 \\ -1\end{pmatrix},
\begin{pmatrix} -1 \\ -1 \\ -1\end{pmatrix},
\begin{pmatrix} -1 \\ 1 \\ -1 \end{pmatrix}
\right\} \text{.} \]
Using the methods of \cite[Section 4]{F:Elt} we find that the lower bounds
predicted by Conjecture~\ref{conj1} are attained:
$\rho_1(\Delta') =0$ and $\rho_2(\Delta') =1$.
Now $\operatorname{Pic}{X'}$ is torsion-free for the complete toric variety $X' =
T_N\emb{\Delta'}$. Since $\rho_1(\Delta')=0$ we see that $\operatorname{Pic}{X'}=0$. This
proves that $X'$ is nonprojective (see Remark~\ref{re4}).
If $B$ is an open neighborhood of $\Delta$, then the strata of $B$ are
$\kappa_0 = 4$ and $\kappa_0 = 3$. We will show later that $\kappa_0 = 4$
corresponds to a Zariski closed subset of $B$.
\end{example}
\begin{remark}
\label{re4}
In general any toric variety satisfying
Conjecture~\ref{conj1} is nonprojective. This is because a projective
normal
variety $X$ will always have a nonprincipal Cartier divisor corresponding
to a hyperplane section.
\samepage{
This follows from commutative
diagram \eqref{eq15}. See \cite[Ex. 6.2, p. 146]{H:AG}.
}
\begin{equation}
\label{eq15}
\begin{CD}
\operatorname{Pic}{ \Bbb P^N} = \operatorname{Cl}(\Bbb P^N) @>>> \operatorname{Pic}{X} @>>> \operatorname{Cl}(X) \\
@V{\operatorname{deg}}V{\cong}V @. @VV{\operatorname{deg}}V \\
\Bbb Z @>{\cdot(\operatorname{deg}{X})}>> \Bbb Z @= \Bbb Z \\
\end{CD}
\end{equation}
\end{remark}
\begin{example}
\label{ex3}
We give an example to illustrate how \eqref{eq13} can be used to compute
$\kappa_0$. Say $\Delta$ consists of three 3-dimensional cones as shown in
Figure~\ref{fig2}(a).
Then $\scr{W}(\Delta_{top}) = \Bbb Zr_0 \oplus \dots \oplus \Bbb Zr_6$ and
$\scr{W}(\Delta(\sigma_0)) = \Bbb Zr_0 \oplus \Bbb Zr_1 \oplus \Bbb Zr_2
\oplus \Bbb Zr_4$.
Let $\eta_i$ be a primitive lattice point on $r_i$, so that $r_i = \Bbb
R_{\ge} \eta_i$ for $i=0..6$.
The kernel of the surjection $\phi_0 : \scr{W}(\Delta)
\to \operatorname{Cl}(U_{\sigma_0})$ is spanned by the vectors
$ \begin{pmatrix} 0 & 0 & 0 & 1 & 0 & 0 & 0 \end{pmatrix} ^\top$,
$ \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 1 & 0 \end{pmatrix} ^\top$,
$ \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix} ^\top$,
and the columns of
\(
\begin{pmatrix} \eta_0 & \eta_1 & \eta_2 & 0 & \eta_4 & 0 & 0 \end{pmatrix}
^\top \).
So $\operatorname{ker}{\phi_0}$ is a subspace of codimension 1. Consider the matrix
equation
\begin{equation}
\label{eq16}
\begin{pmatrix}
\eta_0 & \eta_1 & \eta_2 & \eta_4
\end{pmatrix}
\overrightarrow{v_0} = 0 \text{.}
\end{equation}
Set $A = \begin{pmatrix}
\eta_0 & \eta_1 & \eta_2
\end{pmatrix}$. Then \eqref{eq16}
becomes
\begin{equation}
\label{eq16.6}
\begin{pmatrix}
I & A^{-1} \eta_4
\end{pmatrix}
\overrightarrow{v_0} = 0 \text{.}
\end{equation}
Set $ A^{-1} \eta_4 = \begin{pmatrix}
a_0 & b_0 & c_0 \end{pmatrix}^ \top $.
So $\overrightarrow{v_0} = \begin{pmatrix} -a_0 z & -b_0 z & -c_0 z & z
\end{pmatrix} ^ \top $.
Since any 3 columns of the matrix in \eqref{eq16} are linearly independent,
$\overrightarrow{v_0}$ has 4
nonzero entries, or $\overrightarrow{v_0}=0$.
Normalize $\overrightarrow{v_0}$ by taking $z = -1$.
Then $\operatorname{ker}{\phi_0}$ is the set of solutions to
\begin{equation}
\label{eq17}
\begin{pmatrix}
a_{0} & b_{0} & c_{0} & 0 & -1 & 0 & 0
\end{pmatrix}
\vec{x} = 0 \text{.}
\end{equation}
Hence if $\phi : \scr{W}(\Delta) \to \operatorname{Cl}(U_{\sigma_0}) \oplus
\operatorname{Cl}(U_{\sigma_1}) \oplus \operatorname{Cl}(U_{\sigma_2})$, then $\operatorname{ker}{\phi}$ is the set of
solutions to
\begin{equation}
\label{eq18}
\begin{pmatrix}
a_{0} & b_{0} & c_{0} & 0 & -1 & 0 & 0 \\
a_{1} & 0 & b_{1} & c_{1} & 0 & -1 & 0 \\
a_{2} & b_{2} & 0 & c_{2} & 0 & 0 & -1
\end{pmatrix} \vec{x} = 0 \text{.}
\end{equation}
This coefficient matrix has rank 3 so $\operatorname{ker}{\phi}$ has rank 4. Therefore
$\kappa_0 =4$ and for any open neighborhood $B$ of $\Delta$, $B$ has only 1
stratum. In this case, we see that $\kappa_0$ and hence $\rho_1$ and
$\rho_2$ are topological invariants of the set of all (rational) fans that
look like the one shown in Figure~\ref{fig2}(a). We could assume
$\Delta$ has more than three (say $w+1$) 3-dimensional cones each with four
1-dimensional faces meeting around the common
1-dimensional face $r_0$. By a similar argument we see that $\kappa_0 = w+2$.
\end{example}
\begin{example}
\label{ex4}
Let $\Delta$ be a fan on $\Bbb R^3$ such that $\Delta_{top}$ is
homeomorphic to the fan in Example~\ref{ex1}.
Following the procedure of Example~\ref{ex3}, set up equations analogous to
\eqref{eq16} \eqref{eq17} and \eqref{eq18}. Then $\operatorname{ker}{\phi}$ is the set of
solutions to
\begin{equation}
\label{eq19}
\begin{pmatrix}
a_{0} & 0 & b_{0} & c_{0} & 0 & -1 \\
a_{1} & b_{1} & 0 & c_{1} & -1 & 0 \\
0 & a_{2} & b_{2} & 0 & c_{2} & -1
\end{pmatrix}
\vec{x} = 0 \text{.}
\end{equation}
The coefficient matrix in \eqref{eq19} clearly has rank 2 or more. This
agrees with the upper bound 4 predicted for $\kappa_0$ by
Algorithm~\ref{alg1}. The third, fourth and sixth columns of \eqref{eq19}
are independent if and only if
\begin{equation}
\label{eq20}
(b_2-b_0) c_1 \not = 0 \text{ .}
\end{equation}
This shows that on the complement of a Zariski open subset of $B$,
$\kappa_0 = 3$. We check that \eqref{eq20} is satisfied on a nonempty
subset of $B$.
Note that \eqref{eq20} is satisfied if
\begin{equation}
\label{eq20.5}
\text{the second row of }
\begin{pmatrix}
\eta_0 & \eta_2 & \eta_3
\end{pmatrix}^{-1} \eta_5
\not =
\text{the second row of }
\begin{pmatrix}
\eta_1 & \eta_2 & \eta_4
\end{pmatrix}^{-1} \eta_5
\end{equation}
which will be true for a sufficiently general choice of the fan. To see
this, consider letting $p_0$ vary in $B_0$. Then in \eqref{eq20.5} the
matrix
$\begin{pmatrix}
\eta_0 & \eta_2 & \eta_3
\end{pmatrix}^{-1} $ varies but
the matrix
$\begin{pmatrix}
\eta_1 & \eta_2 & \eta_4
\end{pmatrix}^{-1}$ remains constant.
\begin{comment}
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\begin{figure}
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\end{picture} }
\vspace{-0.75in}
\caption{}
\label{fig4}
\end{figure}
\end{comment}
\end{example}
Consider \eqref{eq13} once again. Let $\Delta$ be a complete fan on $\Bbb
R^3$. Let $B$ be an open neighborhood of $\Delta$.
Proceed as in Examples \ref{ex3} and \ref{ex4}.
Set up the matrix equation $\Phi \vec{x} = 0$ for $\operatorname{ker}{\phi}$.
Since $M
\to \operatorname{SF}(\Delta'_{top})$ is injective, $\operatorname{ker}{\phi}$ has rank at least 3.
Consider an arbitrary
$(n-2)- \text{by}-(n-2)$ submatrix
$\Phi_0$ of $\Phi$. Then $\Phi_0$ has rank $n-2$
exactly when $\det(\Phi_0) \not = 0$. As in \eqref{eq19} and \eqref{eq20},
we can show that $\det(\Phi_0)=0$ is an equation in no more than $3(n-2)$
variables which are parametrized by points in $B$.
The equation $\det(\Phi_0) =0$
defines a Zariski closed subset of $B$. On the complement of this closed
set $\det(\Phi_0) \not = 0$, $\operatorname{rank}{\Phi} = n-2$ and $\operatorname{ker}{\phi}$ has rank
3. If there is at least one choice of $\Delta_{fan}$ for which
$\det(\Phi_0) \not =0$, then the open set making up the complement of the
determinant equations will be nonempty, hence the conclusion of
Conjecture~\ref{conj1} will
be satisfied. This shows for example that the general fan which is
topologically homeomorphic to that of Figure~\ref{fig2}(b) satisfies the
conclusion to Conjecture~\ref{conj1}, because in Example~\ref{ex2} an
example is given which shows the
determinants are nonzero on a nonempty Zariski open in $B$.
|
1995-05-18T06:20:15 | 9505 | alg-geom/9505018 | en | https://arxiv.org/abs/alg-geom/9505018 | [
"alg-geom",
"math.AG"
] | alg-geom/9505018 | Ron Stern | Ronald Fintushel and Ronald Stern | Rational Blowdowns of Smooth 4-Manifolds | 34 pages with 14 figures (author-supplied), AMSLaTeX | null | null | null | null | In this paper we introduce a surgical procedure, called a rational blowdown,
for a smooth 4-manifold X and determine how this procedure affects both the
Donaldson and Seiberg-Witten invariants of X.
| [
{
"version": "v1",
"created": "Wed, 17 May 1995 17:57:08 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Fintushel",
"Ronald",
""
],
[
"Stern",
"Ronald",
""
]
] | alg-geom | \section{Introduction\label{Intro}}
The invariants of Donaldson and of Seiberg and Witten are powerful tools for
studying
smooth $4$-manifolds. A fundamental problem is to determine procedures which
relate smooth $4$-manifolds in such a fashion that their effect on both the
Donaldson
and Seiberg-Witten invariants can be computed. The purpose of this paper is to
initiate
this study by introducing a surgical procedure, called rational blowdown,
and to
determine how this procedure affects these two sets of invariants. The
technique of
rationally blowing down and its effect on the the Donaldson invariant
were first announced at the 1993 Georgia International Topology Conference and
represents the bulk of the mathematics in this paper. We fell upon this
surgical
procedure while we were investigating the behavior of the Donaldson invariant
in the
presence of embedded spheres and while investigating methods for producing
a topological logarithmic transform. As it turns out, this rational blowdown
procedure
allows for the full computation of the Donaldson series (and Seiberg-Witten
invariants) of all elliptic surfaces with $p_g \ge 1$ with the only input being
the
Donaldson invariants of the Kummer surface; in particular this computation
shows that
the Donaldson series of elliptic surfaces is that conjectured by Kronheimer and
Mrowka
in \cite{KM}:
\begin{th} Let $E(n;p,q)$ be the simply connected elliptic surface with
$p_g=n-1$ and with multiple fibers of relatively prime orders $p,q\ge1$. Then
\[{\bold{D}}_{E(n;p,q)}=\exp(Q/2){\sinh^n(f)\over\sinh(f_p)\sinh(f_q)}.\]
\end{th}
\noindent This theorem gives another, more topological, proof of the
diffeomorphism classification of elliptic surfaces
(\cite{Bauer,MorganMrowka,MorganOGrady,Fried1}). This procedure also goes
further and
routinely computes the Donaldson series (and Seiberg-Witten invariants) for
many
$4$-manifolds, some of which are complex surfaces, and for most of the
currently known
examples which are not even homotopy equivalent to complex surfaces.
The ideas presented in this paper have led to rather easy proofs of the blowup
formulas
for the Donaldson invariants for arbitrary smooth $4$-manifolds
\cite{FSblowup} and
alternate proofs and generalizations
\cite{FSstructure} of some of the results announced by Kronheimer and Mrowka
(\cite{KM},\cite{KMbigpaper}). While we chose to first write up these later
results,
another major delay in the appearance of this paper was the introduction of the
Seiberg-Witten invariants.
{ }From the beginning, Witten has conjectured how the Seiberg-Witten
invariants and the
Donaldson invariants determine each other (cf. \cite{Witten}). Some progress in
proving this relationship has been announced by V. Pidstrigach and A. Tyurin.
Our
techniques verify Witten's conjecture for elliptic surfaces and for a large
class of
manifolds obtained from them by rational blowdowns. (See \S 8.)
Here is an outline of the paper: In \S 2 we introduce the
concept of a rational blowdown and discuss relevant topological issues. Our
main analytical result, Theorem~\ref{basic}, gives a universal formula which
relates
the Donaldson invariants of a manifold with those of its rational blowdown.
Three
examples of the effect of a rational blowdown are given in \S 3 and these
examples are
used in subsequent sections to compute the universal quantities given in
Theorem~\ref{basic}. In \S 4 we give the fundamental definitions of the
Donaldson
series, and \S 5 presents our key analytical results. Here we shall take
advantage of
our later results and techniques (\cite{FSblowup},\cite{FSstructure}) to
streamline
our earlier arguments. In particular, we will utilize the \lq\lq \ pullback
--- pushforward "
point of view introduced and developed by Cliff Taubes in
\cite{Sxl,Reds,Circle,Holo}
(or, alternatively the thesis of Wieczorek \cite{W}) to prove our basic
universal formula
(Theorem~\ref{basic}). Under the assumption of simple type, this universal
formula
takes on a particularly simple form (Theorem~\ref{BASIC}). Starting with the
computations of the Donaldson series for elliptic surfaces without multiple
fibers
given in \cite{KM},\cite{FSstructure} and \cite{L}, we apply
Theorem~\ref{BASIC} and
some of the examples presented in \S 3 to compute the Donaldson series of the
elliptic
surfaces with multiple fibers in \S 6. Under the assumption of simple type and
the additional assumption that the configuration of curves that is blown down
is
`taut', Theorem~\ref{BASIC} yields a very simple formula relating the basic
classes of $X$ with those of its rational blowdowns (cf.
Theorem~\ref{tautcalc}). This,
as well as applications to the computations of the Donaldson series of other
manifolds,
is discussed in \S 7. Theorem~\ref{BASIC} has a straightforward analogue
relating the
Seiberg-Witten invariants of $X$ and those of its rational blowdowns. We
conclude this
paper with a statement and proof of this relationship in \S 8.
\bigskip
\section{The Topology of Rational Blowdowns\label{topology}}
In this section we define what is meant by a rational blowdown. Let $C_p$
denote the
simply-connected smooth
$4$-manifold obtained by plumbing the $(p-1)$ disk bundles over the $2$-sphere
according to the linear diagram
\centerline{\unitlength 1cm
\begin{picture}(5,2)
\put(.9,.7){$\bullet$}
\put(1,.8){\line(1,0){1.3}}
\put(2.2,.7){$\bullet$}
\put(2.3,.8){\line(1,0){.75}}
\put(3.3,.8){.}
\put(3.5,.8){.}
\put(3.7,.8){.}
\put(4,.8){\line(1,0){.75}}
\put(4.65,.7){$\bullet$}
\put(.35,1.1){$-(p+2)$}
\put(2.1,1.1){$-2$}
\put(4.55,1.1){$-2$}
\put(.45,.4){$u_{p-1}$}
\put(2.1,.4){$u_{p-2}$}
\put(4.55,.4){$u_1$}
\end{picture}}
\noindent Here, each node denotes a disk bundle over $S^2$ with Euler class
indicated
by the label; an interval indicates that the endpoint disk bundles are
plumbed, i.e
identified fiber to base over the upper hemisphere of each $S^2$. Label the
homology
classes represented by the spheres in $C_p$ by $u_1,\dots,u_{p-1}$ so that the
self-intersections are
$u_{p-1}^2=-(p+2)$ and, for $j=1,\dots,p-2$, $u_j^2=-2$. Further, orient the
spheres
so that $u_j\cdot u_{j+1}=+1$. Then $C_p$ is a $4$-manifold with negative
definite
intersection form and with boundary the lens space $L(p^2,p-1)$.
\begin{lem}\label{ratball} The lens space $L(p^2,p-1)=\partial C_p$ bounds a
rational
ball
$B_p$ with \
$\pi_1(B_p)={\bold{Z}}_p$ and a surjective inclusion induced homomorphism \
$\pi_1(L(p^2,p-1)={{\bold{Z}}}_{p^2}\to \pi_1(B_p)$.
\end{lem}
\begin{pf} There are several constructions of $B_p$; we present three here.
The first construction is perhaps amenable to showing that if the
configuration of
spheres $C_p$ are symplectically embedded in a symplectic
$4$-manifold $X$, then the rational blowdown $X_p$ is also symplectic (cf.
\cite{Gompf}). For this construction let
${\bold{F}}_{p-1}$, $p\ge 2$, be the simply connected ruled surface whose
negative
section $s_-$ has square $-(p-1)$. Let $s_+$ be a positive section (with square
$(p-1)$) and $f$ a fiber. Then the homology classes
$s_++f$ and $s_-$ are represented by embedded $2$-spheres which intersect each
other
once and have intersection matrix
\[ \begin{pmatrix} p+1& 1\\ 1 & -(p-1) \end{pmatrix} \] It follows that the
regular
neighborhood of this pair of $2$-spheres has boundary
$L(p^2,p-1)$. Its complement in ${\bold{F}}_{p-1}$ is the rational ball $B_p$.
The second construction begins with the configuration of $(p-1)$ $2$-spheres
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\put(3.3,.8){.}
\put(3.5,.8){.}
\put(3.7,.8){.}
\put(4,.8){\line(1,0){.75}}
\put(4.65,.7){$\bullet$}
\put(.6,1.1){p+2}
\put(2.2,1.1){2}
\put(4.65,1.1){2}
\end{picture}}
\noindent in $\#(p-1){\bold{CP}}^{\,2}$ where the spheres (from left to right)
represent
\[ 2h_1-h_2+\cdots-h_{p-1}, \ h_1+h_2, \ h_2+h_3, \dots , h_{p-2}+h_{p-1}\]
where $h_i$
is the hyperplane class in the $i$\,th copy of ${\bold{CP}}^{\,2}$. The
boundary of the
regular neighborhood of the configuration is $L(p^2,p-1)$ and the classes of
the
configuration span $H_2({\bold{CP}}^{\,2};{\bold{Q}})$. The complement is the
rational ball
$B_p$.
The third construction is due to Casson and Harer \cite{CH}. It utilizes the
fact that any
lens space is the double cover of $S^3$ branched over a 2-bridge knot. The
2-bridge knot
$K((1-p)/p^2)$ corresponding to $L(p^2,1-p)$ is slice, and $B_p$ is the double
cover of the
$4$-ball branched over the slice disk. \end{pf}
That all these constructions produce the same rational ball $B_p$ is an
exercise in Kirby calculus.
However, for the purposes of this paper, it is the third construction that is
the
most useful, since it allows us to quickly prove:
\begin{cor} Each diffeomorphism of $L(p^2,1-p)$ extends over the rational ball
$B_p$.
\label{lensdiff}\end{cor}
\begin{pf} It is a theorem of Bonahon \cite{Bonahon} that
$\pi_0(\text{Diff}(L(p^2,1-p))=\bold{Z}_2$, and is generated by the deck
transformation $\tau$ of
the double branched cover of $K((1-p)/p^2)$. The extension of $\tau$ to $B_p$ is
given by the
deck transformation of the double cover of $B^4$ branched over the slice
disk.\end{pf}
Suppose that $C_p$ embeds in a closed smooth $4$-manifold $X$. Then let $X_p$
be the
smooth $4$-manifold obtained by removing the interior of $C_p$ and replacing it
with
$B_p$. Corollary~\ref{lensdiff} implies that this construction is well-defined.
We call this
procedure a {\bf rational blowdown} and say that
$X_p$ is obtained by {\bf rationally blowing down} $X$. Note that
$b^+(X)=b^+(X_p)$ so that
rationally blowing down increases the signature while keeping $b^+$ fixed. An
algebro-geometric analogue of rationally blowing down is discussed in
\cite{KSB}.
With respect to the basis $\{u_1,\dots,u_{p-1}\}$ for $H_2(C_p)$, the plumbing
matrix
for $C_p$ is given by the symmetric $(p-1)\times(p-1)$ matrix
\[ P= \begin{pmatrix} -2 & 1& & & & & \\ 1 & -2& 1& & &0 & \\ 0 & 1&-2 &1 & &
& \\
& & & &\ddots & & \\
& 0& & & & -2& 1 \\ & & & & &1 & -(p+2)
\end{pmatrix}\] with inverse given by $(P^{-1})_{i,j} =-j+{(ij)(p+1)\over
p^2}$ for
$j\le i$.
Let $Q:H_2(C_p,\partial C_p;{{\bold{Z}}})\times H_2(C_p;{\bold{Z}}) \to {\bold{Z}}$ be the (relative)
intersection form of $C_p$ and let $\{\gamma_1,\dots,\gamma_{p-1}\}$ be the basis of
$H_2(C_p,\partial C_p;\bold{Z})$ dual to the basis $\{u_1,\dots, u_{p-1}\}$ of
$H_2(C_p;{\bold{Z}})$
with respect to $Q$. I.e.
$\gamma_k\cdot u_\ell=\delta_{k\ell}$. Let $i_\ast:H_2(C_p;{{\bold{Z}}}) \to
H_2(C_p,\partial
C_p;{\bold{Z}})$ be the inclusion induced homomorphism. Then the intersection form
of
$H_2(C_p,\partial C_p;{\bold{Q}})$ is defined by
\[ \gamma_k\cdot\gamma_\ell={1\over p^2}\gamma_k \cdot {\gamma}'_\ell \] where ${\gamma}'_\ell\in
H_2(C_p;{\bold{Z}})$ is chosen such that
$i_\ast({\gamma}'_\ell)=p^2\gamma_\ell$. Since ${\gamma}'_\ell=p^2P^{-1}({\gamma}_\ell)$, the
intersection matrix
for $H_2(C_p,\partial C_p;{\bold{Q}})$ is $(\gamma_k\cdot\gamma_\ell)=P^{-1}$. Note also that
using
the sequence
\[\begin{CD} 0 \to H_2(C_p;{\bold{Z}}) @>P>> H_2(C_p,\partial C_p;{\bold{Z}}) @>\partial>>
H_1(L(p^2,1-p;{\bold{Z}})
\to 0 \end{CD}\] we may identify $H_1(L(p^2,1-p;{\bold{Z}})$ with ${{\bold{Z}}}_{p^2}$ so
that
$\partial$ is given by $\partial(\gamma_j)=j$.
There is an alternative choice of dual bases for $H_2(C_p;{\bold{Z}})$ and
$H_2(C_p,\partial C_p;{\bold{Z}})$ that we shall find useful because of its symmetry.
Define
the basis $\{v_i\}$ of $H_2(C_p;{\bold{Z}})$ by
\[ v_i= u_{p-1}+\cdots + u_i, \hspace{.25in} u_j=v_j-v_{j+1} \] so
$v_i^2=-(p+2)$ for
each $i$, and if $i\ne j$ then $v_i\cdot v_j=-(p+1)$. The dual basis
$\{\delta_i\}$ of
$H_2(C_p,\partial C_p;{\bold{Z}})$ is given in terms of $\{\gamma_i\}$ by
\begin{eqnarray*} \delta_i&=&\gamma_i-\gamma_{i-1}, \ \ i\ne 1\\
\delta_1&=&\gamma_1 \end{eqnarray*} Then
\begin{eqnarray*} \delta_i\cdot\delta_j&=&{(p+1)\over p^2}, \ \ i\ne j\\
\delta_i^2&=&-{(p^2-p-1)\over p^2} \end{eqnarray*} and
\[ \partial(\sum a_i\delta_i) = \sum a_i. \]
Let the character variety of $SO(3)$ representations of
$\pi_1(L(p^2,1-p))$ mod conjugacy be denoted by $\chi_{SO(3)}(L(p^2,1-p))$, and
identify $\pi_1(L(p^2,1-p))$ with ${\bold{Z}}_{p^2}$ as above. Then we have an
identification
\[ \chi_{SO(3)}(L(p^2,1-p))\cong {{\bold{Z}}}_{p^2}/\{\pm1\}\cong
H_1(L(p^2,1-p);{{\bold{Z}}})/\{\pm1\}. \] Let $\eta$ be the generator of
$\chi_{SO(3)}(L(p^2,1-p))$ satisfying
\[ \eta(1) =\begin{pmatrix} \cos({2\pi i/p^2}) &\sin({2\pi i/p^2})&0\\
-\sin({2\pi i/p^2})& \cos({2\pi i/p^2})&0\\ 0&0&1 \end{pmatrix} \] Let $e\in
H_2(C_p,\partial C_p;{\bold{Z}})$; so $\partial e$ is some $n_e\in {\bold{Z}}_{p^2}$. Since
$b^+(C_p)=0$, $e$ defines an anti-self-dual connection $A_e$ on the complex line bundle
$L_e$ over
$C_p$ whose first chern class is the Poincar\'e dual of $e$. Throughout this
paper we
shall identify $H_2(C_p,\partial C_p;{\bold{Z}})\equiv H^2(C_p;{\bold{Z}})$; so we may write
$c_1(L_e)=e$. Consider $C_p$ with a metric which gives a collar $L(p^2,1-p)\times
[
0,\infty)$. The connection $A_e$ has an asymptotic value as $t\to\infty$, and
this is a
flat connection on $L(p^2,1-p)$. Dividing out by gauge equivalence, we obtain
the
element
$\partial A_e=\eta^{n_e}\in\chi_{SO(3)}(L(p^2,1-p))$. For later use, we define
\[ \partial':H_2(C_p,\partial C_p;{{\bold{Z}}})\to\chi_{SO(3)}(L(p^2,p-1))
={{\bold{Z}}}_{p^2}/\{\pm1\}=\{0,1,\dots,[p/2]\} \] by $\partial'(e)=\bar{n}_e$, the
equivalence
class of $\partial e$.
\bigskip
\section{Examples of Rational Blowdowns\label{examples}}
In this section we present four examples of the effect of rational blowdowns.
These are
essential for our later computations.
\noindent {\bf Example 1.} Logarithmic transform as a rational blowdown
This first example, whose discovery motivated our interest in this procedure,
shows
that a logarithmic transform of order $p$ can be obtained by a sequence of
$(p-1)$
blowups (i.e. connect sum with $(p-1)$ copies of $\overline{\bold{CP}}^{\,2}$) and one rational
blowdown of
a natural embedding of the configuration $C_p$.
First, some terminology. Recall that simply connected elliptic surfaces
without
multiple fibers are classified up to diffeomorphism by their holomorphic Euler
characteristic $n=e(X)/12=p_g(X)+1$. The underlying smooth $4$-manifold is
denoted
$E(n)$. The tubular neighborhood of a torus fiber is a copy of $T^2\times
D^2=S^1\times(S^1\times D^2)$. By a {\it log transform} on $E(n)$ we mean the
result
of removing this $T^2\times D^2$ from $E(n)$ and regluing it by a
diffeomorphism
$$\varphi: T^2\times\partial D^2\to T^2\times\partial D^2.$$ The {\it order}
of the
log transform is the absolute value of the degree of
$${\text {pr}}_{\partial D^2}\circ\varphi:{\text {pt}}\times \partial D^2\to
\partial
D^2.$$ Let $E(n)_{\varphi}$ denote the result of this operation on $E(n)$. Note
that
multiplicity 0 is a possibility. It follows from Moishezon
\cite{Moish} that if $\varphi$ and $\varphi'$ have the same order, there is a
diffeomorphism, fixing the boundary, from $E(n)_{\varphi}$ to
$E(n)_{\varphi'}$. What
is needed here is the existence of a cusp neighborhood (cf. \cite{FScusp}). Let
$E(n;p)$ denote any $E(n)_{\varphi}$ where the multiplicity of $\varphi$ is
$p$.
In $E(n;p)$ there is again a copy of the fiber $F$, but there is also a new
torus
fiber, the {\em multiple fiber}. Denote its homology class by $f_p$; so in
$H_2(E(n;p);{\bold{Z}})$ we have $f=p\,f_p$. We can continue this process on
other torus
fibers; to insure that the resulting manifold is simply connected we can take
at most
two log-transforms with orders that are pairwise relatively prime.
Let the orders be $p$ and $q$ and denote the result by $E(n;p,q)$. We
sometimes
write $E(n;p,q)$ in general, letting $p$ or $q$ equal $1$ if there are fewer
than $2$
multiple fibers. Of course one can take arbitrarily many log transforms (which
we
shall sometimes do) and we denote the result of taking $r$ log transforms of
orders
$p_1,\dots,p_r$ by $E(n;p_1,\dots,p_r)$.
The homology class $f$ of the fiber of $E(n)$ can be represented by an
immersed
sphere with one positive double point (a nodal fiber). Figure 1 represents a
handlebody
(Kirby calculus) picture for a cusp neighborhood $N$ which contains this nodal
fiber.
(See \cite{Kirby} for an explaination of such pictures and how to manipulate
them.)
Blow up this double point (i.e. take the proper transform of $f$) so that the
class
$f-2e_1$ (where $e_1$ is the homology class of the exceptional divisor) is
represented by an embedded sphere with square $-4$ (cf. Figure 2). This is just
the
configuration $C_2$.
Now the exceptional divisor intersects this sphere in two positive points.
Blow up
one of these points, i.e. again take a proper transform. One obtains the
homology
classes $u_2=f-2e_1-e_2$ and $u_1=e_1-e_2$ which form the configuration
$C_3$. Continuing in this fashion, $C_p$ naturally embeds in
$N\#_{p-1}{\overline{\bold{CP}}^{\,2}}\subset E(n)\#_{p-1}{\overline{\bold{CP}}^{\,2}}$ as in Figure 3.
Our first important example of a rational blown down is:
\begin{thm}\label{lgtr} The rational blowdown of the above configuration
$C_p\subset
E(n)\#(p-1)\overline{\bold{CP}}^{\,2}$ is diffeomorphic $E(n;p)$.
\end{thm}
\begin{pf} As proof, we offer a sequence of Kirby calculus moves in Figures 4
through
8. In Figure 4 we add to Figure 3 the handle (with framing $-1$) which has
the property that when added to $\partial C_p$ one obtains $S^2\times S^1$ (so
that when a
further $3$ and $4$-handle are attached $B_p$ is obtained). Then we
blow down the added handle, keeping track of the dual 2-handle (which is
labelled in
Figure 4 with 0-framing). In Figure 5 we blow down this added handle with
framing
$-1$ and rearrange to obtain Figure 6. Now slide $e_1$ over the handle with
framing
$+1$ and rearrange to obtain Figure 7. Blow down the $-1$ curve in Figure 7; so
the $-2$ curve becomes a $-1$ curve. Continue this process $p-2$ times to
obtain Figure
8. If in this final picture one replaces the handle with a dot on it by a
1-handle,
there results the handlebody picture given by Gompf in \cite{nuc} for $N_p$,
the order
$p$ log-transformed cusp neighborhood. \end{pf}
\noindent For the case $p=2$, this theorem was first observed by Gompf
\cite{Gompf}.
Here is a useful observation: To perform a log transform of order $pq$, first
perform
a log transform of order $p$ and then perform a log transform of order $q$ on
the
resulting multiple fiber $f_p$. This can also be obtained via a rational
blowdown
procedure. Figure 9 is a handlebody decomposition $N_p\#_{q-1}{\overline{\bold{CP}}^{\,2}}$ with an
easily
identified copy of $C_q$. The proof that the result of blowing down $C_q$
results in
$E(n;pq)$ is to again follow through the steps of the proof of
Theorem~\ref{lgtr}.
\begin{prop}\label{ponq} Let $f_p$ be the multiple fiber in $E(n;p)$. Then
there is an
immersed (nodal) 2-sphere $S\subset E(n;p)$ representing the homology class of
$f_q$. Let $q$ be a positive integer relatively prime to $p$. If the process of
Theorem~\ref{lgtr} is applied to $S$, i.e. if $Y$ is the rational blowdown of
the
configuration $C_q$ in $E(n;p)\#(q-1)\overline{\bold{CP}}^{\,2}$ obtained from blowing up $S$, then
$Y\cong E(n;pq)$, the result of a multiplicity $pq$ log transform on
$E$.\end{prop}
\noindent {\bf Example 2.} In $E(2)$ there is an embedded sphere with
self-intersection $-4$ such that its blowdown is diffeomorphic to
$3{\bold{CP}}^2\#18\overline{\bold{CP}}^{\,2}$.
For this, any $-4$ curve suffices; however to verify that the rational blowdown
decomposes requires more Kirby calculus manipulations. The Milnor fiber
$M(2,3,5)$ for the Poincar\'e homology $3$-sphere $P=\Sigma(2,3,5)$ embeds in
$E(2)$ so
that $E(2)=M(2,3,5)\cup W$ for some $4$-manifold $W$ (cf. \cite{FScusp}). Now
$\partial M(2,3,5)=P$ also bounds another negative definite $4$-manifold $S$
which is the
trace of $-1$ surgery on the left handed trefoil. It is known that $S\cup W$ is
diffeomorphic to $3{\bold{CP}}^2\#11\overline{\bold{CP}}^{\,2}$. Thus, to construct the example, it
suffices find a $-4$ curve in $M(2,3,5)$ whose rational blowdown produces
$S\#7\overline{\bold{CP}}^{\,2}$.
Recall that
$M(2,3,5)$ is just the $E_8$ plumbing manifold given in Figure 10. Slide the
handle
labeled $h$ over the handle labeled $k$ to obtain the $-4$ curve $h+k$ in
Figure 11.
Blow down this $-4$ curve to obtain Figure 12. Now slide the handle labeled
$h'$ over
the handle labeled $k'$ to obtain Figure 13. Now succesively blow down the $-1$
curves
to obtain Figure 14. Cancelling the $1-$ handle with the $2-$handle with
framing $-2$
yields $S\#7\overline{\bold{CP}}^{\,2}$.
\noindent{\bf Example 3.} Given any smooth $4$-manifold $X$, there is an
embedding of
the configuration
$C_p\subset X\#(p-1)\overline{\bold{CP}}^{\,2}=Y$ with $u_i=e_{p-(i+1)}-e_{p-i}$ for $i=1,\dots,p-2$,
and
$u_{p-1}=-2e_1-e_2-\cdots-e_{p-1}$ such that the rational blowdown $Y_p$ of $Y$
is
diffeomorphic to
$X\#H_p$ where $H_p$ is the homology $4$-sphere with $\pi_1={\bold{Z}}_p$ which is
the
double of the rational ball $B_p$.
In fact $C_p\subset \#(p-1){\overline{\bold{CP}}^{\,2}}=Y$, and, from the proof of
Lemma~\ref{ratball}, the
result of blowing down this configuration is just the double of $B_p$.
Note that Example 3 points out that although a smooth $4$-manifold $Y$ may have
a
symplectic structure, it need not be the case that a rational blowdown $Y_p$
of $Y$
also have a symplectic structure. For in this example $X\#H_p$ will never have
a
symplectic structure since its $p-$fold cover can be written as a connected sum
of two
$4$-manifolds with positive $b_+$ so has vanishing Seiberg-Witten invariants
and hence,
by Taubes \cite{TSymplectic1}, is not symplectic. Of course, in this example
the
configuration $C_p$ is not symplectically embedded. This brings up the
possibility
that any smooth $4$-manifold can be rationally blown up to a symplectic
$4$-manifold.
\bigskip
\section{The Donaldson Series\label{def}}
In this section we outline the definition of the Donaldson invariant. We refer
the
reader to
\cite{Donpoly} and \cite{DK} for a more complete treatment. Given an oriented
simply
connected $4$-manifold with a generic Riemannian metric and an $SU(2)$ or
$SO(3)$ bundle
$P$ over $X$, the moduli space of gauge equivalence classes of anti-self-dual connections
on $P$
is a manifold ${\cal M}_X(P)$ of dimension \[8\,c_2(P)-3\,(1+b_X^+)\] if $P$ is an
$SU(2)$
bundle, and
\[-2p_1(P)-3\,(1+b_X^+)\] if $P$ is an $SO(3)$ bundle. It will often be
convenient to
treat these two cases together by identifying ${\cal M}_X(P)$ and
${\cal M}_X(\text{ad}(P))$
for an $SU(2)$ bundle $P$. Over the product ${\cal M}_X(P) \times X$ there is a
universal
$SO(3)$ bundle
${\bold{P}}$ which gives rise to a homomorphism $\mu:H_i(X;\bold{R})\to
H^{4-i}({\cal M}_X(P);\bold{R})$
obtained by decomposing the class
$-{1\over 4}p_1({\bold{P}})\in H^4({\cal M}_X \times X)$.
When either $w_2(P)\ne 0$ or when $w_2(P)= 0$, $d>\frac34(1+b_X^+)$, the
Uhlenbeck
compactification $\overline{{\cal M}}_X(P)$ carries a fundamental class. In
practice, one is
able to get around this latter restriction by blowing up
$X$ and considering bundles over $X\#\overline{\bold{CP}}^{\,2}$ which are nontrivial when restricted
to the
exceptional divisor \cite{MMblowup}. In
\cite{FMbook} it is shown that for $\alpha\in H_2(X;{{\bold{Z}}})$ the classes
$\mu(\alpha)\in
H^2({{\cal M}}_X(P))$ extend over $\overline{{\cal M}}_X(P)$. When $b_X^+$ is odd,
$\dim
{\cal M}_X(P)$ is even, say equal to $2d$. In fact, a class $c\in H_2(X;{{\bold{Z}}})$
and a
nonnegative integer
$d\equiv -c^2+\frac12(1+b^+)$ determine an $SO(3)$ bundle $P_{c,d}$ over $X$
with
$w_2(P_{c,d})\equiv c$ (mod $2$) and formal dimension $\dim {\cal M}_X(P_{c,d}) =
2d$. For
$\bar{\alpha}=(\alpha_1,\dots,\alpha_d)\in H_2(X;{{\bold{Z}}})^d$, write
$\mu(\bar{\alpha})=\mu(\alpha_1)\cup\cdots\cup
\mu(\alpha_d)$. Then one has
\[ \langle
\mu(\bar{\alpha}),[\overline{{\cal M}}_X(P_{c,d})]\rangle=\int_{\overline{{\cal M}}_X(P_{c,d})}\mu(\bar{\alpha}) \]
when $\mu(\bar{\alpha})$ is viewed as a $2d$-form.
If $[1]\in H_0(X;{{\bold{Z}}})$ is the generator, then
$\nu=\mu([1])=-\frac14p_1(\beta)\in
H^4({\cal M}_X(P))$ where $\beta$ is the basepoint fibration
$\tilde{{\cal M}}_X(P)\to{\cal M}_X(P)$ with
$\tilde{{\cal M}}_X(P)$ the manifold of anti-self-dual connections on $P$ modulo based gauge
transformations, i.e. those that are the identity on the fiber over a fixed
basepoint.
The class $\nu$ extends over the Uhlenbeck compactification
$\overline{{\cal M}}_X(P)$ if
$w_2(P)\ne0$, and in case $P$ is an $SU(2)$ bundle, the class will extend under
certain
dimension restrictions. Once again, these restrictions can be done away with
via the
tricks mentioned above \cite{MMblowup}.
Consider the graded algebra
\[{\bold{A}}(X)=\text{Sym}_*(H_0(X)\oplus H_2(X))\] where $H_i(X)$ has degree
$\frac12(4-i)$.
The Donaldson invariant $D_c=D_{X,c}$ is then an element of the dual algebra
${\bold{A}}^*(X)$, i.e. a linear function \[ D_c:{\bold{A}}(X)\to {\bold{R}}. \] This is a
homology
orientation-preserving diffeomorphism invariant for manifolds $X$ satisfying
$b_X^+\ge3$. Throughout this paper we assume $b_X^+\ge3$ and odd.
We let $x\in H_0(X)$ be the generator $[1]$ corresponding to the orientation.
In case
$a+2b=d>\frac34(1+b^+_X)$ and $\alpha\in H_2(X)$, \[
D_c(\alpha^ax^b)=\langle\mu(\alpha)^a\nu^b,[{\overline{{\cal M}}_X(P_{c,d})}]\rangle\, . \]
We may extend
$\mu$ over ${\bold{A}}(X)$, and write for $z\in{\bold{A}}(X)$ of degree $d$,
$D_c(z)=\langle\mu(z),[{\overline{{\cal M}}_X(P_{c,d})}]\rangle$. Since such moduli spaces
${\cal M}_X(P_{c,d})$ exist only for $d\equiv -c^2+\frac12(1+b^+_X)$ (mod 4), the
Donaldson
invariant $D_c$ is defined only on elements of ${\bold{A}}(X)$ whose total degree is
congruent to
$-c^2+\frac12(1+b^+_X)$ (mod $4$). By definition, $D_c$ is $0$ on all elements
of other
degrees. When $P$ is an $SU(2)$ bundle one
simply writes $D$ or $D_X$.
If $Y$ is a simply connected $4$-manifold with boundary, one can similarly
construct
relative Donaldson invariants. A good reference for this is \cite{MMR}. When
the
boundary is a lens space, the theory simplifies considerably, and we get
relative Donaldson invariants
\[D_{Y,c}[\lambda_i]:{\bold{A}}(Y)\to {\bold{R}}.\]
Following \cite{KM}, one considers the invariant \[
\hat{D}_{X,c}:\text{Sym}_*(H_2(X))\to
{\bold{R}} \] defined by
$\hat{D}_{X,c}(u)=D_{X,c}((1+\frac{x}{2})u)$. Whereas $D_{X,c}$ can be nonzero only
in
degrees congruent to $-c^2+\frac12(1+b^+)$ (mod $4$), $\hat{D}_{X,c}$ can be
nonzero in degrees
congruent to $-c^2+\frac12(1+b^+)$ (mod 2). The {\em Donaldson series}
${\bold{D}}_c={\bold{D}}_{X,c}$ is defined by
\[{\bold{D}}_{X,c}(\alpha)=\hat{D}_{X,c}(\exp(\alpha))=\sum_{d=0}^{\infty}{{\hat{D}_{X,c}}(\alpha^d)\over
d!}\]
for all $\alpha\in H_2(X)$. This is a formal power series on $H_2(X)$.
A simply connected $4$-manifold $X$ is said to have {\em simple type} if the
relation
$D_{X,c}(x^2\,z)=4\,D_{X,c}(z)$ is satisfied by its Donaldson invariant for all
$z \in
{\bold{A}}(X)$ and for all $c\in H_2(X;{\bold{Z}})$. This important definition is due to
Kronheimer
and Mrowka \cite{KM} and was observed to hold for many $4$-manifolds
\cite{KMbigpaper,FSstructure}. In terms of
$\hat{D}_{X,c}$, the simple type condition is that $\hat{D}_{X,c}(zx)=2\hat{D}_{X,c}(z)$
for all
$z\in {\bold{A}}(X)$ and for all $c\in H_2(X;{\bold{Z}})$. The assumption of simple type
assures
that for each $c$, the complete Donaldson invariant $D_{X,c}$ is determined by
the
Donaldson series ${\bold{D}}_{X,c}$. It is still an open question whether all
$4$-manifolds are of simple type.
The structure theorem is:
\begin{thm}[Kronheimer and Mrowka
\cite{KMbigpaper,FSstructure}]\label{KMstruct}
Let $X$ be a simply connected 4-manifold of simple type. Then, there exist
finitely
many `basic' classes $\kappa_1$, \dots,
$\kappa_p\in H_2(X,{\bold{Z}})$ and nonzero rational numbers
$a_1$, \dots, $a_p$ such that \[{\bold{D}}_X\ =\ \exp(Q/2)\,\sum_{s=1}^p
a_se^{\kappa_s}\] as
analytic functions on $H_2(X)$. Each of the `basic classes' $\kappa_s$ is
characteristic,
i.e.
$\kappa_s\cdot x \equiv x\cdot x$ (mod $2$)for all $x\in H_2(X;{\bold{Z}})$.
Further, suppose $c\in H_2(X;{\bold{Z}})$. Then
\[ {\bold{D}}_{X,c}\ =\ \exp(Q/2)\,\sum_{s=1}^p(-1)^{{c^2+\kappa_s\cdot
c\over2}}a_se^{\kappa_s}\]
\end{thm}
\noindent Here the homology class $\kappa_s$ acts on an arbitrary homology class
by
intersection, i.e. $\kappa_s(u)=\kappa_s\cdot u$. The basic classes $\kappa_s$ satisfy
certain inequalities analogous to the adjuction formula in a complex surface
\cite{KMbigpaper,FSstructure}. We shall need
\begin{thm}[\cite{FSstructure}]\label{FSadj}
Let $X$ be a simply connected 4-manifold of simple type and let $\{\kappa_s\}$ be
the set
of basic classes as above. If $u\in H_2(X;{\bold{Z}})$ is represented by an immersed
$2$-sphere with $p\ge 1$ positive double points, then for each $s$
\begin{equation}
2p-2\
\ge u^2 + |\kappa_s\cdot u|. \label{adjintro} \end{equation}\end{thm}
\begin{thm}[\cite{FSstructure}]\label{FSadjspecial} Let $X$ be a simply
connected
4-manifold of simple type with basic classes
$\{\kappa_s\}$ as above. If the nontrivial class $u\in H_2(X;{\bold{Z}})$ is represented
by an
immersed
$2$-sphere with no positive double points, then let \[ \{\kappa_s|\,
s=1,\dots,2m\}\] be the
collection of basic classes which violate the inequality (\ref{adjintro}). Then
$\kappa_s\cdot u=\pm u^2$ for each such $\kappa_s$. Order these classes so that
$\kappa_s\cdot
u=-u^2\,(>0)$ for
$s=1,\dots,m$. Then
\[\sum_{s=1}^ma_se^{\kappa_s+u}-(-1)^{1+b_X^+\over2}\sum_{s=1}^ma_se^{-\kappa_s-u}=0.\]
\end{thm}
\bigskip
\section{The Basic Computational Theorem\label{gauge}}
Recall that for
$y\in H_2(X)$ and $F\in {\bold{A}}(X)$, interior product
\[ \iota_uF(v)= (\deg(v)+1) F(uv) \] defines a derivation which we denote by
$\partial_u$
and call `partial derivation'. Our basic theorem is:
\begin{thm}\label{basic} Let $X$ be a simply connected $4$-manifold of simple
type
containing the configuration $C_p$, and let $X_p$ be the result of rationally
blowing
down $C_p$. Then, restricted to $X^*=X_p \setminus B_p= X\setminus C_p$:
\[{\bold{D}}_{X_p}=\sum_{i=1}^{m(p)}\alpha_i(p)\partial^{n_i(p)}{\bold{D}}_{X,c_i(p)} \] where
$\alpha_i(p)\in
{\bold{Q}}$, $c_i(p)\in H_2(C_p;{\bold{Z}})$, $\partial^{n_i(p)}$ is an $n_i$th order partial
derivative
with respect to classes in $H_2(C_p;{\bold{Z}})$, and these quantities depend only
on
$p$, not on $X$. \end{thm}
As motivation, and for use in the next section, we begin with a `by hand'
calculation.
\begin{lem}\label{C2} Let $X$ be a simply connected $4$-manifold containing an
embedded
2-sphere $\Sigma$ of square $-4$ representing the homology class $\sigma$. Let $X_2$
be the
result of rationally blowing down $\Sigma$. Then
\[ {{\bold{D}}_{X_2}|}_{X^*}={\bold{D}}_X-{\bold{D}}_{X,\sigma}. \]\end{lem}
\begin{pf} Here we work with $SU(2)$ connections over $X_2$ and $X$. The
conjugacy
classes of $SU(2)$ representations of $L(4,-1)$ are $\{\pm1,i\}$. Since a
multiple of
any class in $H_2(X_2;{\bold{Z}})$ lives in $H_2(X^*;{\bold{Z}})$, it suffices to evaluate
$D_{X_2}(z)$ for $z\in{\bold{A}}(X^*)$. The lemma is proved by a standard counting
argument
obtained by stretching the neck $\partial X^*\times {\bold{R}}$ in $X_2$. Doing this with
nonempty
moduli spaces leads to a sequence of
anti-self-dual connections (with respect to a sequence of generic metrics on $X_2$) which
limit
to anti-self-dual connections $A^*$ over $X^*$, and
$A_B$ over $B_2$ together perhaps with instantons on $X^*$ and $B_2$. Dimension
counting shows that $A^*$ is irreducible, $A_B$ is reducible (hence flat), and
that no
instantons occur. (The key fact is that each representation of $L(4,-1)$ has a
positive
dimensional isotropy group.) The flat $SU(2)$ connections on $B_2$ are
$\pm1$. Thus we have
\[ D_{X_2}(z)=\pm D_{X^*}[1](z)\pm D_{X^*}[-1](z). \] The invariants
$D_{X^*}[\pm1](z)$
are relative Donaldson invariants of $X^*$ with the given boundary values.
We first claim that $D_{X^*}[1](z)=\pm D_X(z)$. This is almost obvious by
applying an
argument like the one above. We need to know that there are no nontrivial
reducible
connections on the neighborhood $C_2$ of $\Sigma$ with boundary value $1$ and in
a moduli
space of negative dimension. This follows simply from the fact that if $\lambda$
is the
complex line bundle whose first chern class generates $H^2(C_2;{\bold{Z}})$, then the
moduli
space of anti-self-dual connections on $\lambda^m+\lambda^{-m}$ has dimension $4m-3$ (see
\cite{FSstructure}). To compute $D_{X^*}[-1](z)$, note that the Poincar\'e dual
of
$\sigma$ in $H^2(X;{\bold{Z}}_2)$ is the unique nonzero class whose restrictions to $X^*$
and
$C_2$ are both $0$. When passing to structure group $SO(3)$, the representation
$-1$
becomes trivial, and thus extends over $C_2$ as the trivial $SO(3)$ connection.
Now one
can see that $D_{X^*}[-1](z)=\pm D_{X,\sigma}(z)$.
Finally, we need to determine signs. A key point following from our discussion
is that
they are independent of $X$. Recall from Example 2 that there is a sphere
$\Sigma$ of square $-4$ in the $K3$-surface $X$ which has a rational blowdown
$X_2$ with ${\bold{D}}_{X_2}=0$. Since ${\bold{D}}_{X,\sigma}=\exp(Q/2)={\bold{D}}_X$, our formula
must read
\[ D_{X_2}(z)=\pm(D_X(z)-D_{X,\sigma}(z)). \] To compute the overall sign, we must
compare
the way that signs are attached to
$A_0\#\Theta_{B_2}$, and $A_0\#\Theta_{C_2}$ where $A_0$ is an anti-self-dual connection
on $X^*$
with boundary value $1$ and $\Theta_{B_2}$ and $\Theta_{C_2}$ are the trivial
connections on $B_2$ and $C_2$. This is done in a way similar to the proof of
\cite[Theorem 2.1]{FSblowup}, and the sign is easily seen to be `$+$'.
\end{pf}
We now proceed toward the proof of Theorem~\ref{basic}. The first step is to
understand
reducible connections over $C_p$. It will be convenient here to use the
symmetric dual
bases $\{v_i\}$ and $\{\delta_i\}$ of \S\ref{topology}. Using these coordinates,
we express
elements of $H_2(C_p,\partial C_p;{\bold{Z}})$ as
\[ \beta=\sum t_i\delta_i = \langle t_1,\dots,t_{p-1}\rangle. \] Classes of the form $\langle
t,\dots,t,s,\dots,s\rangle $ will play a special role. We shall use the
abbreviation
\[ \langle t,\dots,t,s,\dots,s \rangle =\langle t,s;b\rangle \] if the number of $s$'s is $1\le
b\le
p-1$. If $e\in H_2(C_p,\partial C_p;{\bold{Z}})$, write
${\cal M}_e$ for the $SO(3)$-moduli space of anti-self-dual connections on
$C_p$ which contains the reducible connection in the bundle $L_e\oplus {\bold{R}}$
where
$c_1(L_e)=e$, and which are asymptotically flat with boundary value $\partial' e\in
\chi_{SO(3)}(L(p^2,1-p))$. Note that $\partial\langle t,t+1;b\rangle =(p-1)t+b$.
\begin{lem}\label{dim} Let $e=\langle t,t+1;b\rangle$ with $0\le t\le p$. Then \
$\dim{{\cal M}}_e=2t-1$.
\end{lem}
\begin{pf} With respect to the basis $\{\delta_i\}$, the intersection form of
$H_2(C_p,\partial)$ is
\begin{equation}\label{Q} Q= -{(p^2-p-1)\over p^2}\sum x_i^2 + 2\ {p+1\over
p^2}\sum_{i<j}x_ix_j\end{equation} and
\begin{multline*} e^2=(b(t+1)^2+(p-b-1)t^2)(-{(p^2-p-1)\over p^2})\\
+2((p-b-1)bt(t+1)+
\binom{p-b-1}{2}t^2+\binom{b}{2}(t+1)^2){p+1\over p^2}\end{multline*} Hence
\begin{equation}\label{esquare} e^2={1\over p^2}(b^2+b^2p-bp^2-2bt+t^2-pt^2).
\end{equation} By hypothesis, $\partial e=(p-1)t+b\ne 0$. From \cite{Lawson} we
have
\[ {\rho\over 2}(\partial e)=-{1\over
p^2}(-2b^2-2b^2p-p^2+2bp^2+4bt-2p^2t-2t^2+2pt^2) \]
and by the index theorem \cite{APS}:
\[ \dim {\cal M}_e= -2e^2-\frac32-\frac12(h+\rho)(\partial e)=-2e^2-2-{\rho\over 2}(\partial
e)=2t-1.\]
\end{pf}
\begin{lem}\label{bvlem1} Let $e=\langle t,t+1;b\rangle $ with $t\ge 0$ and
$(p-1)t+b\le p^2/2$.
Suppose also that
$e'=\langle\alpha_1,\dots,\alpha_{p-1}\rangle$ with $\sum\alpha_i=(p-1)t+b+rp^2$, $r\ne 0,-1$. Then
\
$\dim{\cal M}_{e'}>\dim{\cal M}_e$.\end{lem}
\begin{pf} Using \eqref{Q}, it follows from symmetry that for fixed $s=\sum
x_i$, the
minimum absolute value of
$Q\langle x_1,\dots,x_{p-1}\rangle$ occurs at $\mu(s)=\langle s/(p+1),\dots,s/(p+1)\rangle$,
and
\[ \mu(s)^2= -{(p^2-p-1)\over p^2}(p-1){s^2\over (p-1)^2} + 2\
{p+1\over p^2}\binom{p-1}{2} {s^2\over (p-1)^2} ={s^2\over p^2-p^3}.\] On the
other
hand by \eqref{esquare}, $e^2={1\over p^2}(b^2+b^2p-bp^2-2bt+t^2-pt^2)$.
Set $s= (p-1)t+b+rp^2$. Then
\[ \mu(s)^2-e^2=-{1\over p-1}(b+b^2+2br+p^2r^2+2rt(p-1)-bp).\] Since $1\le b\le
p-1$,
we have $bp\le p^2-p\le p^2r^2$. So
\[ \mu(s)^2-e^2\le -{1\over p-1}(b+b^2+2br+2rt(p-1))\] and if we assume $r\ge
1$, \
$\mu(s)^2< e^2 \text{(}<0\text{)}$. By the index theorem,
\[\dim {\cal M}_{e'}= -2{e'}^2-\frac32-\frac12(h+\rho)(\partial e')\ge -2\mu(s)^2-
\frac32-\frac12(h+\rho)(\partial e)\ge \dim {\cal M}_e \]
since $(h+\rho)(\partial e')=(h+\rho)(\partial e)$. Notice that we have not yet used the
hypothesis that
$(p-1)t+b\le p^2/2$.
If $r<-1$, set $\bar{e}=\langle t',t'+1;c\rangle$ with $t',c$ chosen such that
\[ (p-1)t'+c=p^2-((p-1)t+b)\ge p^2/2.\] By Lemma \ref{dim},
$\dim{\cal M}_{\bar{e}}\ge\dim{\cal M}_e$ with equality only if $t'=t$. Note that
$\dim{\cal M}_{-e'}=\dim{\cal M}_{e'}$, and $-\sum\alpha_i=(p-1)t'+c-(r+1)p^2$. Since
$-(r+1)\ge 1$, the case we have already handled shows that
$\dim{\cal M}_{-e'}\ge\dim{\cal M}_{\bar{e}}$.
\end{pf}
\begin{lem}\label{bvlem2} Let $e=\langle t,t+1;b\rangle $ with $t\ge 0$. Suppose that
$e'=\langle\alpha_1,\dots,\alpha_{p-1}\rangle \ne e$ but $\sum\alpha_i=(p-1)t+b$. Then
$\dim{\cal M}_{e'}>\dim{\cal M}_e$ unless $e'$ is a permutation of $e$.\end{lem}
\begin{pf} It suffices to show that ${e'}^2<e^2$. Write $e'=e+\nu$ where
\[\nu=\langle n_1,\dots,n_{p-b-1},n_{p-b},\dots,n_{p-1}\rangle.\] Since the sum of the
coordinates of $e$ and $e'$ is the same, $\sum n_i=0$. Let
\[ N_L=\sum_{i=1}^{p-b-1}n_i\hspace{.5in} N_R=\sum_{i=p-b}^{p-1}n_i.\]
\begin{eqnarray*} {e'}^2&=&e^2+2(N_L((p-2)t+b)+N_R((p-2)t+b-1))({p+1\over
p^2})\\
&& \hspace{2in} - 2(N_Lt+N_R(t+1))({p^2-p-1\over p^2}) +\nu^2\\
&=&e^2-2N_R+\nu^2
\end{eqnarray*} since $N_L+N_R=0$. Hence
$\frac12(\dim{\cal M}_{e'}-\dim{\cal M}_e)=e^2-{e'}^2=-\nu^2+2N_R$. However, if $y$
is the
result of adding $+1$ to $x_{i_0}$ and $-1$ to $x_{i_1}$ in
$x=\langle x_1,\dots,x_{p-1}\rangle$, then $y^2-x^2= 2(x_{i_1}-x_{i_0}-1)$. Starting
with
$x=\la0,\dots,0\rangle$ and making these $\pm1$ moves with constant sign in each
coordinate
until reaching $\nu$, we see that the minimum change in the square is $-2$.
This is
achieved only if each coordinate operated on is originally $0$. Thus, if $N_+$
is the
sum of the positive coordinates $n_i$, we have $-\nu^2\ge 2N_+$. Equality
occurs only
if each $n_i$ is $\pm1$ or $0$. In this case there are $N_+$ such $-1$'s. If
$|N_R|<N_+$
then $-\nu^2+2N_R\ge 2(N_+-|N_R|)>0$. If $|N_R|=N_+$ then each $-1$ occurs in a
coordinate $n_i$, $i= p-b,\dots,p-1$, and so $e'$ is a permutation of $e$. If
$-\nu^2>2N_+$ then since $|N_R|\le N_+$, we have $-\nu^2+2N_R>0$.
\end{pf}
\begin{prop}\label{bv} Let $e=\langle t,t+1;b\rangle $ with $t\ge 0$ and $(p-1)t+b\le
p^2/2$.
If $e'=\langle\alpha_1,\dots,\alpha_{p-1}\rangle$ with $e'\equiv e$ (mod $2$) and
$\dim{\cal M}_{e'}\le\dim{\cal M}_e$, then $\partial e' \le \partial e$ as elements of
${\bold{Z}}_{p^2}$.
\end{prop}
\begin{pf} Let $\bar{e}=\langle s,s+1;c\rangle$ with $s\ge 0$, be the unique class of
this form
with $0\le\partial\bar{e}\le p^2/2$ satisfying $\partial e'=\partial\bar{e}$. Lemmas
\ref{bvlem1} and \ref{bvlem2} imply that unless $-p^2/2\le\sum a_i<0$, we have
$\dim{\cal M}_{\bar{e}}\le \dim{\cal M}_{e'}$; so $s\le t$. This holds in any case,
since we
can always work with $-e'$. If $s=t$ then
$\bar{e}=e$ since no class $\langle t,t+1;b'\rangle$ with $b'\ne b$ is congruent to
$e\pmod2$.
This means that $\partial e' \le \partial e$.
\end{pf}
\begin{cor}\label{bddim} Let $e=\langle t,t+1;b\rangle $ with $t\ge 0$ and $(p-1)t+b\le
p^2/2$.
Suppose that $e'=\langle\alpha_1,\dots,\alpha_{p-1}\rangle$ with $\partial' e'=\partial'
e\in\chi_{SO(3)}(L(p^2,1-p))$ and
$e'\equiv e$ (mod $2$). Then $\dim{\cal M}_{e'}=\dim{\cal M}_e+4k$, $k\ge
0$.\end{cor}
\begin{pf} As above, $\dim{\cal M}_e\le \dim{\cal M}_{e'}$. But $e'\equiv e$ (mod
$2$) implies
that ${e'}^2=e^2$ (mod $4$); so the corollary follows from the index
theorem.\end{pf}
We need one more simple fact. Let $\iota :(C_p,\emptyset)\to (C_p,\partial)$ be the
inclusion.
\begin{lem}\label{getc2} Let $e\in H_2(C_p,\partial;{\bold{Z}})$, and suppose that
$\partial e\equiv 0$ (mod $2$) in case $p$ is even. Then there is a \ $c\in
H_2(C_p;{\bold{Z}})$ such
that $\iota_*(c)\equiv e$ (mod $2$).\end{lem}
\begin{pf} This follows directly from the exact sequence
\[ 0\to H_2(C_p;{\bold{Z}})\to H_2(C_p,\partial;{\bold{Z}})\to {\bold{Z}}_{p^2}\to 0\, .\]
\end{pf}
We now proceed toward the proof of Theorem \ref{basic}. We shall work always
with
structure group $SO(3)$ and identify $SU(2)$ connections with $SO(3)$
connections on
$w_2=0$ bundles. We wish to calculate $D_{X_p}(z)$ for $z\in{\bold{A}}(X^*)$. If we
blow up
$X^*$ and evaluate $D_{X_p\#\overline{\bold{CP}}^{\,2},e}(ze)=D_{X_p}(z)$ where $e$ is the
exceptional class
\cite{MMblowup}, we can work under the assumption that there are no flat
connections on
the complement of $B_p$ with the same $w_2$ as our given bundle. Keeping this
in mind,
we may simplify notation without loss by making the same assumption for our
given
situation, $X_p=X^*\cup B_p$. Consider a sequence of generic metrics on $X_p$
which
stretch a collar on $L(p^2,1-p)=\partial B_p$ to infinite length, giving the
disjoint union
of $X^*$ and $B_p$ with cylindrical ends as the limit. A sequence of
anti-self-dual connections $\{ A_n\}$ with respect to these metrics, each of which also
lies in the
divisor $V_z$ corresponding to $z\in{\bold{A}}(X^*)$, must limit to $A_{X^*} \amalg
A_{B_p}$.
These are anti-self-dual connections over $X^*$ and $B_p$, and a counting argument shows
that
$A_{X^*}\in V_z$ and $A_{B_p}$ is reducible. (Our above assumption is helpful
here.)
Since the only reducible connections on the rational ball $B_p$ are flat, we
get
\begin{equation}\label{bareqn} D_{X_p}(z)=\sum_{n=0}^{[p/2]}\pm
D_{X^*}[\eta^{np}](z).
\end{equation} The notation $D_{X^*}[\eta^{np}]$ stands for the relative
Donaldson
invariant on
$X^*$ constructed from the moduli space of anti-self-dual connections over $X^*$ (with a
cylindrical end) which decay exponentially to a flat connection whose gauge
equivalence
class corresponds to the conjugacy class of the representation $\eta^{np}$.
We need to calculate the summands of \eqref{bareqn}. We begin with $n=0$, i.e.
$D_{X^*}[1](z)$. Consider $D_X(z)$. To calculate this, we use a neck-stretching
argument
as above. We see that on $C_p$ we must get a reducible anti-self-dual connection
corresponding to
chern class $e$ with $\dim{\cal M}_e<0$ and
$e\equiv 0\pmod2$. This last condition means that $e$ cannot have the form
$\langle 0,1;b\rangle$ (recall $1\le b\le p-1$); so by Lemma \ref{dim}, $e\ne \langle
t,t+1;b\rangle$,
$t\ge 0$. Now Proposition \ref{bv} implies that $e=0$. Thus
\[ D_X(z)=\pm D_{X^*}[1](z) \]
and the sign is independent of $X$.
To calculate the other terms, we must utilize techniques of Taubes
\cite{Sxl,Reds,Circle,Holo} or Wieczorek \cite{W} as in
\cite[\S4]{FSstructure}. We
shall quickly review the methods involved and refer the reader to
\cite{FSstructure} and
the references given there for more details. Our plan is to evaluate all the
$D_{X^*}[\eta^m](z)$ inductively. (In case $p$ is even, we only need to
calculate this
for $m$ even.) We do this by computing
$D_{X,c_m}(z\,w_m)$ where $c_m\in H_2(X;{\bold{Z}})$ is supported in
$C_p$, $m=(p-1)t+b$, and $w_m\in \text{Sym}_t(H_2(C_p;{\bold{Z}}))$ depending only on
$m$ and
$p$. First we obtain $c_m$. Let
$e_m\in H^2(C_p;{\bold{Z}})$ be the Poincar\'e dual of $\langle t,t+1;b\rangle$. By Lemma
\ref{getc2} we can find $c_m\in H_2(C_p;{\bold{Z}})\subset H_2(X;{\bold{Z}})$ such that
$\iota_*(c_m)\equiv \langle t,t+1;b\rangle\pmod2$. Thus the Poincar\'e dual of $c_m$ in
$H^2(X;{\bold{Z}})$ restricts to $C_p$ congruent to $e_m\pmod2$ and restricts
trivially to
$X^*$. A dimension counting argument shows that in the formalism of Taubes
\cite{Sxl},
$D_{X,c_m}(z\,w_m)$ is the sum of terms of the form
\begin{equation}\label{terms}
\int_{\tilde{{\cal M}}_{X^*}[\eta^j]\times_j\tilde{{\cal M}}_{C_p,\epsilon,\ell}}
\tau\wedge\tilde{\mu}(z)\wedge\tilde{\mu}(w_m). \end{equation} In this
formula,
$\tilde{{\cal M}}_{C_p,\epsilon,\ell}$ is the based moduli space of exponentially decaying
asymptotically flat anti-self-dual connections on the $SO(3)$ bundle
$E_{\epsilon,\ell}$ which is obtained from the reducible bundle
$L_\epsilon\oplus{\bold{R}}$ by grafting in $\ell$ instanton bundles. (The euler class of
$L_\epsilon$ is $\epsilon$, $\partial\epsilon = j$, $\epsilon\equiv e_m\pmod2$, and
$\dim{\cal M}_\epsilon+8\ell\le 2t-1$.) The notation `$\times_j$' in the formula denotes the
fiber
product with respect to the $SO(3)$-equivariant boundary value maps
\[ \partial_{C_p,\epsilon,\ell}:\tilde{{\cal M}}_{C_p,\epsilon,\ell}\to G[j], \hspace{.25in}
\partial_{X^*}[j]:\tilde{{\cal M}}_{X^*}[\eta^j]\to G[j] \] where $G[j]\subset SO(3)$
is the
conjugacy class $\eta^j$ of representations of
$\pi_1(L(p^2,1-p))$ to $SO(3)$. If $j\ne 0,p^2/2$ then $G[j]$ is a 2-sphere,
$G[0]=\{ I\}$, and, in case $p$ is even, $G[p^2/2]\cong{\bold{RP}}^2$. Also,
$\tau$
denotes a 3-form which integrates to 1 over the fibers of the basepoint
fibration
$\beta_{X^*,j}$ i.e.
$\tilde{{\cal M}}_{X^*}[\eta^j]\to {\cal M}_{X^*}[\eta^j]$. The form
$\tilde{\mu}(w_m)$ is
supported near the orbit of the reducible connection corresponding to $\epsilon$. (If
$\ell>0$, this reducible connection lies in the Uhlenbeck compactification of
${\cal M}_{C_p,\epsilon,\ell}$.) The principal $SO(3)$ bundle $\beta_{X^*,j}$ has a reduction
to a
bundle with structure group $S^1$. As in \cite[\S4]{FSstructure}, we let
$\varepsilon\in H^2({\cal M}_{X^*}[\eta^j])$ denote the euler class of this $S^1$ bundle.
The upshot of Taubes' work cited above is that there is a form
$\tilde{\mu}(w_m)$ representing a class $\mu_{SO(3)}(w_m)$ in the
$SO(3)$-equivariant
cohomology of an enlargement of $\tilde{{\cal M}}_{C_p,\epsilon,\ell}$. The lift
$\tilde{\mu}(z)$ defines an element of the equivariant cohomology
$H^{2d}_{SO(3)}(\tilde{{\cal M}}_{X^*}[\eta^j])$. Furthermore, Taubes has shown
that the
push-forward $(\partial_{C_p,\epsilon,\ell})_*$ is well-defined, and
\[ \int_{\tilde{{\cal M}}_{X^*}[\eta^j]\times_j\tilde{{\cal M}}_{C_p,\epsilon,\ell}}
\tau\wedge\tilde{\mu}(z)\wedge\tilde{\mu}(w_m)
= \int_{\tilde{{\cal M}}_{X^*}[\eta^j]}\tau\wedge\tilde{\mu}(z)\wedge
(\partial_{X^*}[j])^*(\partial_{C_p,\epsilon,\ell})_*(\tilde{\mu}(w_m)) \] where
$(\partial_{X^*}[j])^*$
denotes pullback in equivariant cohomology.
For $j=0$, $\partial_{C_p,\epsilon,\ell}:\tilde{{\cal M}}_{C_p,\epsilon,\ell}\to \{1\}$, has fiber
dimension
equal to $\dim\tilde{{\cal M}}_{C_p,\epsilon,\ell}= 4k+8\ell$ for some
$k\ge 0$. The cohomology class of $(\partial_{C_p,\epsilon,\ell})_*(\tilde{\mu}(w_m))$ lies
in
$H^{2t-4k-8\ell}_{SO(3)}(\{1\};{\bold{R}})=H^{2t-4k-8\ell}(BSO(3);{\bold{R}})$ which is a
polynomial
algebra on the 4-dimensional class $\wp$, which pulls back over
$\tilde{{\cal M}}_{X^*}[\eta^j]$ as $p_1(\beta_{X^*,j})$. For $j\ne 0,p^2/2$, let
$j=t_j(p-1)+b_j$ where $1\le b_j\le p-1$. Then
$\partial_{C_p,\epsilon,\ell}:\tilde{{\cal M}}_{C_p,\epsilon,\ell}\to G[j]$ has fiber dimension
$2t_j+2+8\ell+4k-2$ for some $k\ge 0$; so the cohomology class of
$(\partial_{C_p,\epsilon,\ell})_*(\tilde{\mu}(w_m))$ lies in
$H^{2(t-t_j)-8\ell-4k}_{SO(3)}(S^2;{\bold{R}})=
H^{2(t-t_j)-8\ell-4k}({\bold{CP}}^\infty;{\bold{R}})$. Let $v$ be the 2-dimensional
generator of
$H^*({\bold{CP}}^\infty;{\bold{R}})$. The pullback
$(\partial_{X^*}[j])^*(v)=\varepsilon$. Using the fact that $\varepsilon^2=p_1(\beta_{X^*,j})$, and
arguing as
in \cite[Prop.4.5,4.6]{FSstructure} we get
\begin{equation}\label{expand}
D_{X,c_m}(z\,w_m)=\sum_{t_j\equiv t\,(2)}\sum_q r_{m,j,q}D_{X^*}[\eta^j](zx^q)
+\sum\begin{Sb} t_j\not\equiv t\,(2)\\j\ne 0\end{Sb}
\sum_q r'_{m,j,q}D_{X^*}[\eta^j](zx^q\varepsilon).
\end{equation} The notation $D_{X^*}[\eta^j](zx^q\varepsilon)$ is not standard, but its
meaning
is clear. It follows from Proposition \ref{bv} that the $\eta^j$ in
\eqref{expand} have
$j\le m$; so this bounds $j$ in both terms. We emphasize that in order to
obtain
$r_{m,j,q}$ or $r_{m,j,q}'\ne 0$ we must have an $\epsilon\in H^2(C_p;{\bold{Z}})$
satisfying
$\partial'\epsilon = j$, $\epsilon\equiv e_m\pmod2$, and $\dim{\cal M}_\epsilon+8q\le 2t-1$.
Assume inductively that:
\begin{itemize}
\item[a)]For each $j< m$ ($j\equiv 0\pmod2$ if $p$ is even) there are classes
$w_{j,i}\in\text{Sym}_*(H_2(C_p;{\bold{Z}}))$ and rational numbers $a_{j,i}$
satisfying
\begin{equation}\label{induct}
D_{X^*}[\eta^j](z)=\sum_{i=1}^j a_{j,i}D_{X,c_i}(zw_{j,i})
\end{equation}
\item[b)] For each $j$ with $t_j< t-1$ (and $j\equiv 0\pmod2$ if $p$ is even)
there are
classes $w'_{j,i}\in\text{Sym}_*(H_2(C_p;{\bold{Z}}))$ and rational numbers
$a'_{j,i}$
satisfying
\begin{equation}\label{inductepsilon}
D_{X^*}[\eta^j](z\varepsilon)=\sum_{i=1}^j a_{j,i}D_{X,c_i}(zw'_{j,i})
\end{equation}
\end{itemize} for all $z\in{\bold{A}}(X^*)$, and the coefficients $a_{j,i},a_{j,i}'$
are
independent of $z$ and $X$.
Recall that we are writing $m=(t-1)p+b$ with $1\le b\le p-1$, and let $e_m$ be
the
Poincar\'e dual of $\langle t,t+1;b\rangle=(t+1)\gamma_{p-1}-\gamma_{p-1-b}$. Also, we suppose
that $m$
is even if $p$ is even. We set
\[ w_m=(u_{p-1}-(t-1)u_{p-1-b})\cdot (u_{p-1})^{t-1}\in{\bold{A}}(C_p). \] We wish to
calculate $D_{X,c_m}(z\,w_m)$ using
\eqref{expand}. For $j=m$ in this formula, we need to compute
$(\partial_{C_p,e_m,0})_*(\tilde{\mu}(w_m))\in H^0_{SO(3)}(G[m];{\bold{R}})={\bold{R}}$ since
$t_m=t$.
In fact,
\begin{multline*} (\partial_{C_p,e_m,0})_*(\tilde{\mu}(w_m))=r_{m,m,0}\\ =
-\frac12\langle
u_{p-1}-(t-1)u_{p-1-b},e_m\rangle\,(-\frac12\langle u_{p-1},e_m\rangle)^{t-1} =
(-\frac12)^t(2t)(t+1)^{t-1}\ne 0\end{multline*} (cf. \cite[p.187]{DK}). In
\eqref{expand}, $ r_{m,m,0}D_{X^*}[\eta^m](z)$ is the only term which involves
the
boundary value
$\eta^m$. If $j$ is the boundary value of an $\epsilon$ with $\epsilon\equiv e_m\pmod2$,
and
$\dim{\cal M}_\epsilon+8q\le 2t-1$, and if $t_j=t-1$, then by Corollary~\ref{bddim} and
Lemma~\ref{bvlem2}, $\epsilon$ must be a permutation of $\langle t-1,t;p-1-b\rangle=
t\gamma_{p-1}-\gamma_b$.
In fact $\epsilon\equiv e_m\pmod2$ implies that $\epsilon = \langle
t,t-1;b\rangle=(t-1)\gamma_{p-1}+\gamma_{p-1-b}$. So
$j=(t-1)(p-1)+(p-1-b)$. Hence
$\langle u_{p-1}-(t-1)u_{p-1-b},\epsilon\rangle =0$. Thus, no such $j$ occurs in the second
sum of
the expansion \eqref{expand} for $D_{X,c_m}(z\,w_m)$. (I.e. for such
$j$, necessarily $q=0$ and $r'_{m,j,q}=0$.) Finally, if $p$ is even, then we
are
assuming that $m$ is also even. If $r_{m,i,q}$ or $r_{m,i,q}'\ne 0$ then as
above there
is an
$\epsilon$ with $\partial\epsilon = i$ and $\epsilon\equiv e_m\pmod2$; so for
\[ \partial_2:H_2(C_p,\partial;{\bold{Z}}_2)\to H_1(L(p^2,1-p);{\bold{Z}}_2)={\bold{Z}}_2 \]
$j\equiv\partial_2(\epsilon)\equiv\partial_2 e_m\equiv m\pmod2$. Accordingly, all the other
terms in
\eqref{expand} are given inductively by \eqref{induct} and
\eqref{inductepsilon}, and
the powers of $x$ can be removed using the hypothesis that $X$ has simple type.
Since the
coefficient of
$D_{X^*}[\eta^m](z)$ is nonzero, we may solve for it, completing the induction
step for
\eqref{induct}.
For \eqref{inductepsilon}, we show how to compute $D_{X^*}[\eta^{m'}](z\varepsilon)$
for
$m'=(t-1)(p-1)+(p-1-b)$ as required. Thus after completing the inductive step
for each
$t(p-1)+c$, $1\le c\le p-1$, we will have completed the calculation of
$D_{X^*}[\eta^j](z\varepsilon)$ for all $j=(t-1)p+a$, $1\le a\le p-1$. So to calculate
$D_{X^*}[\eta^{m'}](z\varepsilon)$ and thus complete the induction, we calculate
$D_{X,c_m}(z\,w'_{m'})$ where
$w'_{m'}=(u_{p-1}+(t+1)u_{p-1-b})\cdot(u_{p-1}+(t-1)u_{p-1-b})\cdot
(u_{p-1})^{t-2}$.
Using \eqref{expand}
\begin{equation}\label{another} D_{X,c_m}(z\,w'_{m'})=\sum_{t_j\equiv
t\,(2)}\sum_q
s_{{m'},j,q}D_{X^*}[\eta^j](zx^q)
+\sum\begin{Sb} t_j\not\equiv t\,(2)\\j\ne 0\end{Sb}
\sum_q s'_{{m'},j,q}D_{X^*}[\eta^j](zx^q\varepsilon).\end{equation} Computing as
above, we
see that $s_{m',m',0}=0$. What we need to see is that
$s'_{m',m',0}\ne 0$. By the argument of the above paragraph, $m'$ is the only
possible
boundary value not covered by the induction step. Let
$\epsilon=(t-1)\gamma_{p-1}+\gamma_{p-1-b}$. This is the only euler class that can give
boundary
value $m'$ in \eqref{another}. Then
\[ (\partial_{C_p,\epsilon,0})_*(\tilde{\mu}(w'_m))\in H^2_{SO(3)}(S^2;{\bold{R}})\cong
H^2_{SO(3)}(G[m'];{\bold{R}})={\bold{R}} \] and $(\partial_{C_p,\epsilon,0})_*(\tilde{\mu}(w'_m))=
(-\frac12)^{t-1}2t(2t-2)(t-1)^{t-2}v$ which pulls back over
$\tilde{{\cal M}}_{X^*}[\eta^j]$
as
$(-\frac12)^{t-3}t(t-1)(t-1)^{t-2}\varepsilon$. This means that we can solve
\eqref{another} for
$D_{X^*}[\eta^{m'}](z\varepsilon)$, completing the induction and the proof of Theorem
\ref{basic}.
The argument above shows that all of the relative invariants
$D_{X^*}[\eta^{np}]$ can be
expressed in terms of absolute invariants of $X$. Since we are assuming that
$X$ has
simple type, it follows that each of the relative invariants satisfies the
formula
\[ D_{X^*}[\eta^{np}](z\,x^2) = 4\,D_{X^*}[\eta^{np}](z). \] Hence it follows
from
\eqref{bareqn} that:
\begin{cor}\label{st} Let $X_p$ be the result of rationally blowing down
$C_p\subset X$.
If $X$ has simple type, then so does $X_p$. \ \ \qed \end{cor}
Now we shall make stronger use of the hypothesis that $X$ has simple type. By
\cite{KM,FSstructure} we can write
\begin{eqnarray*} {\bold{D}}_X&=&\exp(Q_X/2)\sum_{s=1}^na_se^{\kappa_s} \\
{\bold{D}}_{X,c}&=&\exp(Q_X/2)\sum_{s=1}^n(-1)^{\frac12(c^2+c\cdot\kappa_s)}a_se^{\kappa_s}
\end{eqnarray*}
for nonzero rational numbers $a_s$ and basic classes $\kappa_1,\dots,\kappa_n\in
H_2(X;{\bold{Z}})$. Here $Q_X$ is the intersection form of $X$. Now
\[ \partial_u(\exp(Q_X/2)e^{\kappa})=\exp(Q_X/2)(\tilde{u}+\kappa\cdot u)e^{\kappa} \]
where $\tilde{u}: H_2(X)\to\bold{R}$ is $\tilde{u}(\alpha)=u\cdot\alpha$ and
$\partial_v\tilde{u}=v\cdot u$.
Apply Theorem \ref{basic}: since all derivatives are taken with respect to
classes $u\in
H_2(C_p;\bold{Z})$, after all derivatives are taken, the remaining $\tilde{u}$'s
restricted to
$X^*$ vanish. Hence,
\begin{equation}\label{barseries0}
{{\bold{D}}_{X_p}|_{X^*}}=\exp(Q_{X^*}/2)\sum_{s=1}^na_sb_se^{\kappa_s}|_{X^*}=
\exp(Q_{X^*}/2)\sum_{s=1}^na_sb_se^{\kappa'_s}
\end{equation}
where $\kappa'_s=\kappa_s|_{X^*}=\text{PD}(i^*(\text{PD}(\kappa_s)))\in
H_2(X^*,\partial;{\bold{Z}})$, where $\text{PD}$ denotes Poincar\'e duality, $i$ is the
inclusion $X^*\subset X$, and $b_s$ depends only on the intersection numbers of
$\kappa_s$
with the generators $u_i$ of $H_2(C_p;\bold{Z})$.
\begin{lem}\label{theyextend} If $b_s\ne0$ in \eqref{barseries0} then
\[ \partial\kappa'_s\in p{\bold{Z}}_{p^2}\subset H_1(L(p^2,1-p);{\bold{Z}})={\bold{Z}}_{p^2}. \] \end{lem}
\begin{pf} Corollary~\ref{st} implies that $X_p$ has simple type. We thus have
\begin{equation}\label{barseries}
{\bold{D}}_{X_p}=\exp(Q_{X_p}/2)\sum_{r=1}^mc_re^{\lambda_r}
\end{equation} where the basic classes of $X_p$ are $\lambda_1,\dots,\lambda_m$.
Restrict
${\bold{D}}_{X_p}$ to
$X^*$ and compare the restrictions of $\exp(Q_{X_p}/2)^{-1}{\bold{D}}_{X_p}$ in
\eqref{barseries0} and \eqref{barseries}. Since for distinct $\alpha\in
H_2(X^*,\partial;{\bold{Z}})$ the
functions $e^\alpha:H_2(X^*)\to{\bold{R}}$ are linearly independent, it follows that if
$b_s\ne 0$,
then $\kappa'_s=\lambda_i|_{X^*}$ for some $i$. Thus $\kappa'_s$ extends over $B_p$, and
hence $\partial
\kappa'_s\in p{\bold{Z}}_{p^2}$. \end{pf} As a result, we have the following restatement
of
Theorem \ref{basic}.
\begin{thm}\label{BASIC} Suppose that $X$ has simple type and
\[{\bold{D}}_X=\exp(Q_X/2)\sum_{s=1}^na_se^{\kappa_s}.\] Let $C_p\subset X$ and let
$X_p$ be its
rational blowdown. Let $\{\kappa_t|t=1,\dots,m\}$ be the basic classes of $X$ which
satisfy
$\partial\kappa'_t\in p{\bold{Z}}_{p^2}$, and for each $t$, let $\bar{\k}_t$ be the unique
extension of $\kappa'_t$.
Then
\[{\bold{D}}_{X_p}=\exp(Q_{X_p}/2)\sum_{t=1}^ma_tb_te^{\bar{\k}_t}\] where the $b_t$
depend only
on the intersection numbers $u_i\cdot\kappa_t$,
$i=1,\dots,p-1$.
\ \ \qed \end{thm}
\bigskip
\section{The Donaldson Invariant of Elliptic Surfaces\label{ellipticcompute}}
In this section we shall compute the result on the Donaldson series of
performing log
transforms. The Donaldson invariants of the elliptic surfaces
$E(n), n\ge2$ without multiple fibers have been known for some time. There is a
complete calculation in \cite{FSstructure}, for example. For $n\ge2$:
\[ {\bold{D}}_{E(n)} = \exp(Q/2)\sinh^{n-2}(f) \] where $f$ is the class of a fiber.
In this
notation, the $K3$ surface is $E(2)$. As in Theorem~\ref{lgtr}, let
$X=E(2)\#(p-1)\overline{\bold{CP}}^{\,2}$, and let $X_p$ be the rational blowdown of $C_p\subset X$,
so that
$X_p\cong E(2;p)$. Since ${\bold{D}}_{E(2)}=\exp(Q/2)$, the blowup formula
\cite{FSblowup}
yields
\begin{equation}\label{blowup} {\bold{D}}_X={1\over
2^{p-1}}\exp(Q/2)\sum_J\exp(\sum_{i=1}^{p-1}\epsilon_{J,i}e_i)
\end{equation} where the outer sum is taken over all
$J=(\epsilon_{J,1},\dots,\epsilon_{J,p-1})\in\{\pm1\}^{p-1}$. The basic classes of $X$ are
$\{\kappa_J=\sum\epsilon_{J,i}e_i\}$, and applying Theorem~\ref{BASIC} we get
\begin{equation}\label{DXbar}
{\bold{D}}_{X_p}={1\over2^{p-1}}\exp(Q_{X_p}/2)\sum_Jb_Je^{\bar{\k}_J}
\end{equation}
where $\bar{\k}_J\in H_2(X_p;{\bold{Z}})$ is the unique extension of
${\kappa_J|}_{X^*}$. Recall that the spheres of the configuration $C_p$ represent
homology
classes
$u_i=e_{p-(i+1)}-e_{p-i}$ for $1\le i\le p-2$, and
$u_{p-1}=f-2e_1-e_2-\cdots-e_{p-1}$.
In $X_p$ we have the multiple fiber $f_p=f/p$.
\begin{prop}\label{J} $\bar{\k}_J= |J|\cdot f_p$ where
$|J|=\sum_{i=1}^{p-1}\epsilon_{J,i}$.\end{prop}
\begin{pf} First we find a class $\zeta\in H_2(C_p;{\bold{Q}})$ so that $(\kappa_J+\zeta)\cdot
u_i=0$
for each $i$. This means that $\kappa_J+\zeta\in H_2(X^*;{\bold{Q}})$, and as dual forms:
$H_2(X^*;{\bold{Z}})\to{\bold{Z}}$, ${\kappa_J|}_{X^*}=\kappa_J+\zeta$. To find $\zeta$ we need to solve
the
linear system
\[ (\kappa_J+\sum x_iu_i)\cdot u_j = 0,\ \ \ j=1,\dots,p-1. \] We begin by
rewriting
these equations. Let $\{\omega_i\}$ be a standard basis for
${\bold{Q}}^{p-1}$, and let $A$ be the $(p-1)\times(p-1)$ matrix whose $i$th row vector
is
\begin{eqnarray*} A_i&=& \omega_{p-(i+1)}-\omega_{p-i}, \ \ i=1,\dots,p-2\\
A_{p-1}&=&-2\omega_1-\omega_2-\dots-\omega_{p-1}. \end{eqnarray*} We have
$u_i=A^t(\omega_i)\cdot{{\bold{e}}}$ and $u_{p-1}=f+A^t(\omega_{p-1})\cdot{\bold{e}}$ ,
where
${\bold{e}}=(e_1,\dots,e_{p-1})$. Our linear system is equivalent to
\[ P{{\bold{x}}}=A\pmb{\epsilon}_J\] where ${{\bold{x}}}=(x_1,\dots,x_{p-1})$ and
$\pmb{\epsilon}_J=(\epsilon_{J,1},\dots,\epsilon_{J,p-1})$. (The matrix $P$ is the plumbing
matrix for
$C_p$.) Hence ${\bold{x}}=P^{-1}A\pmb{\epsilon}_J$.
We claim that $P(A^t)^{-1}=-A$. This can be checked on the basis
\[ \{\omega_2-\omega_1,\dots,\omega_{p-1}-\omega_{p-2},\omega_{p-1} \}\] using
\begin{eqnarray*} A(\omega_i)&=&-\omega_{p-1}-\omega_{p-(i+1)}+\omega_{p-i},\ 2\le i\le p-1 \ \
(\omega_0=0),\\
A(\omega_1)&=&-2\omega_{p-1}+\omega_{p-2},\\ P(\omega_i)&=&\omega_{i+1}-2\omega_i+\omega_{i-1},\ i\ne
p-1,\\
P(\omega_{p-1})&=&-(p+2)\omega_{p-1}+\omega_{p-2}.
\end{eqnarray*}
It follows that $A^tP^{-1}A=-I$. Thus
\[\kappa_J+\zeta=\kappa_J+\sum
x_iu_i=({\bold{\epsilon}}_J+A^t{{\bold{x}}})\cdot{\bold{e}}+x_{p-1}f
=(\pmb{\epsilon}_J-\pmb{\epsilon}_J)\cdot{\bold{e}}+x_{p-1}f=x_{p-1}f.\] To compute
$x_{p-1}$ note
that
\[
A\pmb{\epsilon}_J=(\epsilon_{J,p-2}-\epsilon_{J,p-1},\epsilon_{J,p-3}-\epsilon_{J,p-2},\dots,\epsilon_{J,1}-\epsilon_{J,2},
-2\epsilon_{J,1}-\epsilon_{J,2}-\cdots-\epsilon_{J,p-1}) \] so that if $(P^{-1})_{p-1}$ denotes
the
bottom row of $P^{-1}$:
\[x_{p-1}=(P^{-1})_{p-1}(A\pmb{\epsilon}_J)=-{1\over
p^2}(1,2,\dots,p-1)\cdot(A{\bold{\epsilon}}_J)
={1\over p}\sum\epsilon_{J,i}={1\over p}|J|.\] Thus ${\kappa_J|}_{X^*}=\kappa_J+\zeta={1\over
p}|J|f$ as
forms: $H_2(X^*;{\bold{Z}})\to{\bold{Z}}$. The homology class $\kappa_J+\zeta$ is in fact an
integral
class $\bar{\kappa}_J=|J|f_p\in H_2(X_p;{\bold{Z}})$ which is the unique extension of
${\kappa_J|}_{X^*}$ \end{pf}
In an arbitrary smooth $4$-manifold $X$, define a {\em nodal fiber} to be an
immersed
2-sphere $S$ with one singularity, a positive double point, such that the
regular
neighborhood of $S$ is diffeomorphic to the regular neighborhood of a nodal
fiber in an
elliptic surface. (There need not be any associated ambient fibration of $X$.)
Given
such a nodal fiber $S$, one can perform a `log transform' of multiplicity $p$
by
blowing up to get $C_p\subset X\#(p-1)\overline{\bold{CP}}^{\,2}$ with
$u_{p-1}=S-2e_1-e_2-\cdots-e_{p-1}$,
and then blowing down $C_p$. We denote the result of this process by $X_p$.
Throughout, we use the following notation. If $X$ has simple type, and
$${\bold{D}}_X=\exp(Q/2)\sum a_se^{\kappa_s},$$ then we write ${\bold{K}}_X=\sum a_se^{\kappa_s}$.
\begin{prop}\label{formallog} Let $S$ be a nodal fiber which satisfies
$S\cdot\lambda_j=0$
for each basic class $\lambda_j$ of $X$. Then
\[{\bold{D}}_{X_p}=\begin{cases}
\exp(Q_{X_p}/2){\bold{K}}_X\cdot(b_{p,0}+\sum\limits_{i=1}^{p-1\over2}b_{p,2i}(e^{2iS/p}+e^{-2iS/p})),\
&p\ \text{odd}\\
\exp(Q_{X_p}/2){\bold{K}}_X\cdot(\sum\limits_{i=1}^{p\over2}b_{p,2i-1}(e^{(2i-1)S/p}+e^{-(2i-1)S/p})),
&p\ \text{even}\end{cases}\] where the coefficients $b_{p,j}$ depend only on
$p$, not on
$X$.
\end{prop}
\begin{pf} The Donaldson series of $X\#(p-1)\overline{\bold{CP}}^{\,2}$ is
\[
{1\over2^{p-1}}{\bold{D}}_X\cdot\exp(Q_{(p-1)\overline{\bold{CP}}^{\,2}}/2)\sum_J\exp(\sum_{i=1}^{p-1}\epsilon_{J,i}e_i).
\]
Theorem~\ref{basic} states that ${\bold{D}}_{X_p}$ is obtained from this by applying
a
differential operator which by hypothesis evaluates trivially on ${\bold{D}}_X$. The
proposition now follows from \eqref{DXbar} and Propostion~\ref{J} by the
Leibniz rule.
(That the coefficients of $e^{mp}$ and $e^{-mp}$ are equal follows from the
fact that
${\bold{D}}_{E(2;p)}$ is an even function.) \end{pf}
\begin{prop}\label{log2} The Donaldson series of the simply connected elliptic
surface
$E(n;2)$ with $p_g=n-1$ ($>0$) and one multiple fiber of multiplicity $2$ is
\[ {\bold{D}}_{E(n;2)}=\exp(Q/2){\sinh^{n-1}(f)\over\sinh(f_2)}. \] \end{prop}
\begin{pf} According to Theorem~\ref{lgtr}, we obtain $E(n;2)$ from
$E(n)\#\overline{\bold{CP}}^{\,2}$ by
blowing down the sphere of square $-4$ representing $f-2e$. We have
${\bold{D}}_{E(n)\#\overline{\bold{CP}}^{\,2}}=\exp(Q/2)\sinh^{n-2}(f)\cosh(e)$. Lemma~\ref{C2} gives
\[ {{\bold{D}}_{E(n;2)}|}_{X^*} = ({\bold{D}}_{E(n)\#\overline{\bold{CP}}^{\,2}}-{\bold{D}}_{E(n)\#\overline{\bold{CP}}^{\,2},f-2e})|_{X^*}=
2\exp(Q/2)\sinh^{n-2}(f)\cosh(e)|_{X^*} \] (cf.\cite{KMbigpaper},
\cite[Thm.5.13]{FSstructure}). By Proposition~\ref{J}
\[
{\bold{D}}_{E(n;2)}=2\exp(Q/2)\sinh^{n-2}(f)\cosh(f_2)=\exp(Q/2){\sinh^{n-1}(f)\over\sinh(f_2)}.\]
\end{pf}
Proposition~\ref{formallog} now implies:
\begin{cor} If $S$ is a nodal fiber in $X$ orthogonal to all basic classes and
$X_2$ is
the multiplicity $2$ log transform of $X$ formed from $S$, then
\[{\bold{D}}_{X_2}=\exp(Q_{X_2}/2){\bold{K}}_X\cdot(e^{S/2}+e^{-S/2}). \ \ \qed\]
\end{cor}
\begin{lem}\label{sum} Let $X$ contain a nodal fiber $S$ orthogonal to all
basic
classes. Then the sum of the coefficients $b_{p,j}$ in the expression for
${\bold{D}}_{X_p}$ in
Proposition~\ref{formallog} is equal to $p$.\end{lem}
\begin{pf} In Example 3 we showed that there is a configuration
$C'_p\subset X\#(p-1)\overline{\bold{CP}}^{\,2}=Y$ where $u'_i=e_{p-(i+1)}-e_{p-i}$ for
$i=1,\dots,p-2$, and
$u'_{p-1}=-2e_1-e_2-\cdots-e_{p-1}$ such that the rational blowdown
$Y_p=X\#H_p$ where $H_p$ is a homology $4$-sphere with $\pi_1={\bold{Z}}_p$. It
follows
easily that ${\bold{D}}_{Y_p}=p\cdot {\bold{D}}_X$.
As above, we let $\kappa_J=\sum\epsilon_{J,i}e_i$, $J\in\{\pm1\}^{p-1}$; so
\[ {\bold{D}}_Y={1\over2^{p-1}}{\bold{D}}_X\cdot\exp(Q_{(p-1)\overline{\bold{CP}}^{\,2}}/2)\sum_J e^{\kappa_J}. \]
All
partial derivatives of ${\bold{D}}_X$ with respect to classes in $H_2(C'_p)$ are
trivial; so
\[ p{\bold{D}}_X={\bold{D}}_{{Y}_p}={\bold{D}}_X\cdot\sum_Jb_Je^{\bar{\kappa}_J}. \]
The proof of Proposition~\ref{J} shows that each $\bar{\kappa}_J=0$; so
$\sum_Jb_J=p$.
We can also form the configuration $C_p\subset Y$ whose blowdown is the $p$-log
transform of the nodal fiber $S\subset X$. The configurations $C_p$, $C'_p$
agree,
$u_i=u'_i$, except that $u_{p-1}=u'_{p-1}+S$. However, since $S$ is orthogonal
to all
the basic classes of $X$, for all $i$, all intersections of $u_i$ and $u'_i$
with all
basic classes of $Y=X\#(p-1)\overline{\bold{CP}}^{\,2}$ agree. Thus, according to
Theorem~\ref{BASIC}, the
coefficients $b_J$ are the same coefficients that arise in the formula
\[ {\bold{D}}_{X_p}=\exp(Q_{X_p}/2){\bold{K}}_X\sum_Jc_Je^{|J|S/p}. \] This means that
the sum of the coefficients of the expression for ${\bold{D}}_{X_p}$ in
Proposition~\ref{formallog} is $\sum_Jb_J=p$. \end{pf}
We next invoke Proposition ~\ref{ponq} to see that if $p$ is any positive odd
integer,
then a multiplicity $2p$ log transform can be obtained as the result of either
a
multiplicity $p$ log transform on a nodal fiber of multiplicity 2, or by a
multiplicity 2 log transform on a nodal fiber of multiplicity $p$. Thus
\begin{eqnarray*}
{\bold{D}}_{E(n;2p)}&=&\exp(Q/2)(e^{f_2}+e^{-f_2})(b_{p,0}+\sum_{i=1}^{(p-1)/2}
b_{p,2i}(e^{2if_2/p}+e^{-2if_2/p}))\\
&=&\exp(Q/2)(b_{p,0}+\sum_{i=1}^{(p-1)/2}b_{p,2i}(e^{2if_p}+e^{-2if_p}))
(e^{f_p/2}+e^{-f_p/2})
\end{eqnarray*} since we already know the formula for a log transform of
multiplicity 2. We compare coefficients using $f_2=pf_{2p}$ and
$f_p=2f_{2p}$.
Assume for the sake of definiteness that $p\equiv 1\pmod4$ and let $r=(p-1)/4$.
In the
top expansion, the coefficient of $e^{\pm pf_{2p}}$ is $b_{p,0}$ and
$b_{p,2j}$ is the coefficient of $e^{\pm (p+2j)f_{2p}}$ and $e^{\pm
(p-2j)f_{2p}}$.
In the second expansion, the coefficient of $e^{\pm f_{2p}}$ is $b_{p,0}$, and
$b_{p,2j}$ is the coefficient of $e^{\pm (4j-1)f_{2p}}$ and $e^{\pm
(4j+1)f_{2p}}$. To
simplify notation, let $(m)_1$ be the coefficient of $e^{mf_{2p}}$ in the top
expansion
and $(m)_2$ its coefficient in the bottom expansion. Then,
\begin{eqnarray*}
b_{p,0}=(p)_1&=&(p)_2=b_{p,2r}=(p-2)_2=(p-2)_1=b_{p,2}=(p+2)_1=(p+2)_2
\\&=&b_{p,2(r+1)}=
(p+4)_2=(p+4)_1=b_{p,4}=(p-4)_1=(p-4)_2\\&=&b_{p,2(r-1)}=(p-6)_2=(p-6)_1=b_{p,6}=\cdots
\end{eqnarray*} and we see inductively that when $p$ is odd, all the $b_{p,2i}$
are
equal. But by Lemma~\ref{sum}
\[ b_{p,0}+2\sum_{i=1}^{(p-1)/2}b_{p,2i}=p.\] It follows that each
$b_{p,2i}=1$,
$i=0,\dots,(p-1)/2$.
Similarly, if $p$ is even, let $q=p-1$. Expanding ${\bold{D}}_{E(n;pq)}$ we see that
all
$b_{p,2i-1}$, $i=1,\dots,p/2$ are equal; and so again each $b_{p,2i-1}=1$.
\begin{thm}\label{toplt} Let $X$ be a $4$-manifold of simple type and suppose
that $X$
contains a nodal fiber $S$ orthogonal to all its basic classes. Then
\[ {\bold{D}}_{X_p}=\exp(Q_{X_p}/2){\bold{K}}_X\cdot{\sinh(S)\over\sinh(S/p)}. \] \end{thm}
\begin{pf} If, e.g., $p$ is odd,
\begin{eqnarray*} {\bold{D}}_{X_p}&=&
\exp(Q_{X_p}/2){\bold{K}}_X\cdot(1+2\cosh(2S/p)+2\cosh(4S/p)+\cdots+2\cosh((p-1)S/p))\\&=&
\exp(Q_{X_p}/2){\bold{K}}_X\cdot{\sinh(S)\over \sinh(S/p)}. \end{eqnarray*} \end{pf}
As a result we have the calculation of the Donaldson series for all simply
connected
elliptic surfaces with $p_g\ge1$.
\begin{thm}\label{ellformula} If $n\ge2$ and $p,q\ge1$ are relatively prime,
\[{\bold{D}}_{E(n;p,q)}=\exp(Q/2){\sinh^n(f)\over\sinh(f_p)\sinh(f_q)}.\ \ \
\qed\]\end{thm}
\noindent This formula was originally conjectured by Kronheimer and Mrowka
\cite{KM}.
As an example of Theorem~\ref{toplt} consider $E(n)$. It follows from
\cite{GM1} and
\cite{FScusp} that in $E(n)$ there are 3 pairs of disjoint nodal fibers such
that the
nodal fibers in each pair are homologous, but give three linearly independent
homology
classes. Form $E(n;p_1,q_1;p_2,q_2;p_3,q_3)$ by performing log transforms with
each
pair $\{ p_i,q_i\}$ relatively prime. The resulting manifold is simply
connected and,
\begin{prop}\label{noncomplex}\hspace{.1in}$\displaystyle
{\bold{D}}_{E(n;p_1,q_1;p_2,q_2;p_3,q_3)}=
\exp(Q/2){\sinh^{n+4}(f)\over\prod\limits_{i=1}^3\sinh(f_{p_i})\sinh(f_{q_i})}.\ \
\qed $\end{prop}
\noindent Applying Theorem~\ref{ellformula} and
Proposition~\ref{noncomplex} to the manifolds $E(n; p_1,q_1;p_2,q_2; p_3,q_3)$,
we see
that they do not admit complex structures with either orientation
(cf.\cite{GM1},\cite[Theorem 8.3]{FScusp}). The manifolds
$E(2;p_1,q_1;p_2,q_2;p_3,q_3)$ are the Gompf-Mrowka fake K3-surfaces
\cite{GM1}.
\bigskip
\section{Tautly Embedded Configurations\label{taut}}
Consider a $4$-manifold $X$ of simple type containing the configuration $C_p$.
By
Theorem~\ref{FSadj} for each 2-sphere $u_i$ in $C_p$ and each basic class
$\kappa$ of $X$, we have
\begin{equation}\label{tautly} -2\ge u_i^2+|u_i\cdot\kappa|
\end{equation} except in the special case described in
Theorem~\ref{FSadjspecial} where
$0\ge u_i^2+|u_i\cdot\kappa|$. The only examples known where the special case
arises are in
blowups. This was the situation in the previous section where we studied log
transforms. In this section, we assume that we are not in the special case. We
say
that a configuration is {\em tautly embedded} if \eqref{tautly} is satisfied
for each
$u_i$ of the configuration and each basic class $\kappa$ of $X$. Thus, if $C_p$ is
tautly
embedded, then for every basic class $\kappa$, $u_i\cdot\kappa=0$ for $i=1,\dots,p-2$
and
$|u_{p-1}\cdot\kappa|\le p$.
\begin{thm}\label{tautcalc} Suppose that $X$ is of simple type and contains the
tautly
embedded configuration $C_p$. If
\[ {\bold{D}}_X=\exp(Q_X/2)\sum a_se^{\kappa_s} \] then the rational blowdown $X_p$
satisfies
\[ {\bold{D}}_{{X_p}}=\exp(Q_{X_p}/2)\sum \bar{a}_se^{\bar{\kappa}_s} \] where
\[\bar{a}_s=\begin{cases} 2^{p-1}a_s, \ \ \ &|u_{p-1}\cdot\kappa_s|=p\\
0, &|u_{p-1}\cdot\kappa_s|<p \end{cases}\] Furthermore,
if
$|u_{p-1}\cdot\kappa_s|= p$, then $\bar{\kappa}_s^2={\kappa_s}^2+(p-1)$.
\end{thm}
\begin{pf} If $\kappa_s\cdot u_{p-1}\ne 0,\pm p$ then $\bar{a}_s=0$ by
Lemma~\ref{theyextend}. For $\kappa_s\cdot u_{p-1}=0$, $\kappa_s\ne0$, note that since
the $\kappa_s$ are
characteristic, $p$ must be even. But then $\bar{\kappa}_s$ cannot even be
characteristic in $X_p$, since $\bar{\kappa}_s^2=\kappa^2$ is not mod $4$ congruent to
$(3\text{sign}+2e)(X_p)$.
Thus, Theorem~\ref{BASIC} implies
that $\bar{a}_s=0$.
In case $\kappa_s\cdot u_{p-1}=\pm p$, we compare with the model for the order $p$
log
transforms of $E(2)$; $C''_p\subset Y=E(2)\#(p-1)\overline{\bold{CP}}^{\,2}$ which is blown down to
obtain
$Y_p=E(2;p)$. Again let
$\lambda_0=\pm(e_1+\cdots+e_{p-1})$; so by Lemma~\ref{J}, $\pm\lambda_0$ are the
unique basic
classes of $Y_p$ satisfying $\pm\bar{\lambda}_0=\pm(p-1)f_p\in H_2(Y_p;{\bold{Z}})$. Now
\begin{eqnarray*} {\bold{D}}_{Y}&=&{1\over2^{p-1}}\exp(Q/2)\sum\exp(\pm
e_1\pm\cdots\pm
e_{p-1})=
\exp(Q/2)\sum_J{1\over2^{p-1}}e^{\lambda_J}\\
{\bold{D}}_{Y_p}&=&\exp(Q/2)\sum\begin{Sb}|\ell|\le p-1\\\ell\equiv p\pmod2\end{Sb}e^{\ell
f_p}=
\exp(Q/2)\sum{1\over2^{p-1}}b_Je^{\lambda_J}
\end{eqnarray*} Since $\pm\lambda_0$ are the unique $\lambda_J$ with
$\bar{\lambda}_0=\pm
(p-1)f_p$, the corresponding coefficient is
$b_0=2^{p-1}$. We may now apply Theorem~\ref{BASIC} to obtain our result since
$\kappa_s\cdot u_i=\lambda_0\cdot u''_i$ for each $i$.
In order to compute $\bar{\kappa}_s^2$, we find $x_i\in{\bold{Q}}$, $i=1,\dots,p-1$, such
that
\[ \kappa_s+\zeta=\kappa_s+\sum_{i=1}^{p-1}x_iu_i \in H_2(X^*;{\bold{Q}}) \] as in the proof of
Proposition~\ref{J}. We can solve for the $x_i$ using the model
$C''_p\subset Y''$, and $\pmb{\epsilon}_J=\pm(1,\dots,1)$ in the proof of
Proposition~\ref{J}. Referring there, we get
\[ {\bold{x}}=P^{-1}A\pmb{\epsilon}_J=-(A^t)^{-1}\pmb{\epsilon}_J=\pm {1\over
p}(1,2,\dots,p-1). \]
So $\zeta=\pm\sum{i\over p}u_i$, and $\zeta^2={\bold{x}}\cdot P{\bold{x}}=1-p$.
Hence
\[ \bar{\kappa}_s^2 = (\kappa_s+\zeta)^2 = \kappa_s^2 + 2\kappa_s\cdot\zeta +\zeta^2= \kappa_s^2 +(p-1). \]
\end{pf}
Now consider the elliptic surface $E(1)$. It can be constructed by blowing up
${\bold{CP}}^2$ at the nine intersection points of a generic pencil of cubic
curves. The
fiber class of $E(1)$ is $f=3h-e_1-\cdots e_9$ where $3h$ is the class of the
cubic in
$H_2({\bold{CP}}^2;{\bold{Z}})$. The nine exceptional curves are disjoint sections of
the
elliptic fibration. The elliptic surface $E(n)$ can be obtained as the fiber
sum of $n$
copies of $E(1)$, and these sums can be made so that the sections glue together
to give
nine disjoint sections of $E(n)$, each of square $-n$. In particular, consider
$E(4)$ with 9 disjoint sections of square $-4$. The basic classes of $E(4)$ are
$0$ and
$2f$; so we see that each of the 9 sections gives us a tautly embedded
configuration
$C_2$. Let $W_n$ be the rational blowdown of $n$ of these sections, $1\le n\le
9$. For
$n\le 8$, $W_n$ is simply connected. Gompf has shown that all these manifolds
admit
symplectic structures, and it is not hard to see that $W_2$ is the 2-fold
branched
cover of ${\bold{CP}}^2$ branched over the octic curve \cite[\S5.2]{Gompf}.
\begin{prop} \hspace{.1in}$\displaystyle
{\bold{D}}_{W_n}=2^{n-1}\exp(Q/2)\cosh(\kappa_n)$ \
where \ $\kappa_n^2=n$.\end{prop}
\begin{pf} We have
\[ {\bold{D}}_{E(4)}=\exp(Q/2)\sinh^2(f)=\exp(Q/2)(\frac12\cosh(2f)+\frac12). \] The
basic
classes $\pm 2f$ intersect each section twice; so Theorem~\ref{tautcalc}
implies that
each $X_n$ has only the basic classes, $\pm\kappa_n$, and that each blowdown
multiplies its
coefficient by 2 and increases its square by $1$. (We start with coefficient
$\frac12$ and square $0$.)
\end{pf}
To further illustrate the utility of Theorem~\ref{tautcalc} we compute the
Donaldson
invariants of a family of Horikawa surfaces $\{ H(n)\}$ with
$c_1(H(n))^2=2n-6$. To obtain $H(n)$, start with the simply connected ruled
surface
${\bold{F}}_{n-3}$ whose negative section $s_-$ has square $-(n-3)$. We have
seen in
the proof of Lemma~\ref{ratball} that the classes $s_++f$ and $s_-$ form a
configuration in ${\bold{F}}_{n-3}$ whose regular neighborhood $D_{n-2}$ has
complement
the rational ball $B_{n-2}$. The Horikawa surface $H(n)$ is defined to be the
$2$-fold branched cover of ${\bold{F}}_{n-3}$ branched over a smoothing of
$4(s_++f)+2s_-$. (Equivalently this is a smooth surface representing (6,$n+1$)
in
$S^2\times S^2$.)
\begin{lem} For $n\ge 4$, the elliptic surface $E(n)$ contains a pair of
disjoint
configurations $C_{n-2}$ in which the spheres $u_{n-1}$ are sections of $E(n)$
and for
$1\le j\le n-2$, $u_j\cdot f=0$. Furthermore, the rational blowdown of this
pair of
configurations is the Horikawa surface $H(n)$.
\end{lem}
\begin{pf} It follows from our description of $H(n)$ that there is a
decomposition
\[ H(n)= B_{n-2}\cup \tilde{D}_{n-2}\cup B_{n-2} \] where $\tilde{D}_{n-2}$ is
the
branched cover of $D_{n-2}$. Rationally blow up each
$B_{n-2}$; this is then the $2-$fold branched cover of ${\bold{F}}_{n-3}$ with
$B_{n-2}$ blown up. The result is the complex surface $ C_{n-2}\cup
\tilde{D}_{n-2}\cup C_{n-2}$ which, by computing characteristic numbers, is
just $E(n)$.
\end{pf}
\noindent The first case, $n=4$, gives the example $H(4)=W_2$ above. The
Horikawa
surfaces $H(n)$ lie on the Noether line $5c_1^2-c_2+36=0$, and of course the
elliptic
surfaces $E(n)$ lie on the line $c_1^2=0$ in the plane of coordinates
$(c_1^2,c_2)$.
Let $Y(n)$ be the simply connected $4$-manifold obtained from $E(n)$ by blowing
down
just one of the configurations $C_{n-2}$. Then $c_1(Y(n))^2=n-3$ and
$c_2(Y(n))=11n+3$; so $Y(n)$ lies on the bisecting line $11c_1^2-c_2+36=0$.
The
calculation of Donaldson invariants of $Y(n)$ and $H(n)$ follows directly from
Theorem~\ref{tautcalc}.
\begin{prop} The Donaldson invariants of $Y(n)$ and $H(n)$ are:
\begin{eqnarray*} {\bold{D}}_{Y(n)}&=&\begin{cases}\exp(Q/2)\sinh(\lambda_n),\ \ n\
\text{odd}\\
\exp(Q/2)\cosh(\lambda_n),\ \ n\ \text{even}\end{cases}\\
{\bold{D}}_{H(n)}&=&\begin{cases}2^{n-3}\exp(Q/2)\sinh(\kappa_n),\ \ n\ \text{odd}\\
2^{n-3}\exp(Q/2)\cosh(\kappa_n),\ \ n\ \text{even}\end{cases}
\end{eqnarray*} where $\lambda_n^2=n-3$ and $\kappa_n^2=2n-6$.\ \ \qed\end{prop}
\begin{cor} The simply connected $4$-manifolds $Y(n)$ are not homotopy
equivalent to
any complex surface.\end{cor}
\begin{pf} If $Y(n)$ were homeomorphic to a complex surface, this computation
shows
that it would have to be minimal, since the formula for ${\bold{D}}_{Y(n)}$ does not
contain
a factor $\cosh(e)$ where $e^2=-1$. Certainly the surface in question could not
be
elliptic since $c_1(Y(n))^2\ne 0$. Bt neither could the surface be of general
type
since $Y(n)$ violates the Noether inequality. Thus $Y(n)$ is not homeomorphic
to any
complex surface.\end{pf}
D. Gomprecht \cite{Gomprecht} has computed the value of the Donaldson invariant
$D_X(F^k)$ for any Horikawa surface $X$ and $k$ large, where $F$ is the
branched cover
of the fiber $f$ of $F_{n-3}$.
\bigskip
\section{Seiberg-Witten Invariants of Rational Blowdowns\label{SW}}
Suppose we are given a spin$^{\text{c}}$ structure on an oriented closed
Riemannian $4$-manifold $X$. Let $W^+$ and $W^-$ be the associated
spin$^{\text{c}}$ bundles with
$L=\det W^+=\det W^-$ the associated determinant line bundle. Since $c_1(L)\in
H^2(X;{\bold{Z}})$ is a characteristic cohomology class, i.e. has mod 2 reduction
equal to
$w_2(X)\in H^2(X;{\bold{Z}}_2)$, we refer to $L$ as a characteristic line bundle.
We will confuse a characteristic line bundle $L$ with its first Chern class
$L \in H^2(X;{\bold{Z}})$. For simplicity we assume that
$H^2(X;{\bold{Z}})$ has no $2$-torsion so that the set $Spin^{\text{c}}(X)$ of
spin$^{\text{c}}$ structures on $X$ is precisely the set of characteristic
line bundles on $X$.
Clifford multiplication, $c$, maps $T^\ast X$ into the skew adjoint
endomorphisms of
$W^+\oplus W^-$ and is determined by the requirement that
$c(v)^2$ is multiplication by $-|v|^2$. Thus $c$ induces a map
$$c: T^\ast X \to {\text{Hom}}(W^+, W^-).$$
The $2$-forms $\Lambda^2=\Lambda^+\oplus\Lambda^-$ of $X$ then act on
$W^+$ leading to a map $\rho:\Lambda^+\to {\text{su}}(W^+)$. A connection
$A$ on $L$ together with the Levi-Civita connection on the tangent bundle of
$X$ induces a connection
$\nabla_A:\Gamma(W^+)\to \Gamma(T^\ast X\otimes W^+)$ on $W^+$. This
connection, followed by Clifford multiplication, induces the Dirac operator
$D_A:\Gamma(W^+)\to\Gamma(W^-)$. (Thus $D_A$ depends both on the connection
$A$ and the Riemannian metric on $X$.) Given a pair $(A,\psi) \in
{\cal{A}}_X(L)\times
\Gamma(W^+)$, i.e. $A$ a connection in $L$ and $\psi$ a section of $W^+$, the
monopole equations of Seiberg and Witten \cite{Witten} are
\begin{eqnarray}\label{monopole} D_A\psi&=&0\\\rho(F_A^+)\notag &=
&(\psi\otimes\psi^\ast)_o \end{eqnarray}
where $(\psi\otimes\psi^\ast)_o$ is the trace-free part of the endomorphism
$\psi\otimes\psi^\ast$.
The gauge group
$\text{Aut}(L)=\text{Map}(X,S^1)$ acts on the space of solutions, and its
orbit space is the moduli space $M_X(L)$ whose formal dimension is
\begin{equation} \dim M_X(L) = \frac14(c_1(L)^2-(3\,\text{sign}(X)+2\,e(X)).
\label{dims} \end{equation} If this formal dimension is nonnegative and if
$b^+>0$, then for a generic metric on $X$ the moduli space
$M_X(L)$ contains no reducible solutions (solutions of the form $(A,0)$
where $A$ is an anti-self-dual connection on $L$), and for a generic perturbation of the
second equation of \eqref{monopole} by the addition of a self-dual 2-form of
$X$, the moduli space $M_X(L)$ is a compact manifold of the given dimension
(\cite{Witten}).
The {\em Seiberg-Witten invariant} for $X$ is the function
$SW_X:Spin^{\text{c}}(X)\to {\bold{Z}}$ defined as follows. Let $L$ be a
characteristic line bundle.
If $\dim M_X(L)<0$ or is odd, then
$SW_X(L)$ is defined to be $0$. If $\dim M_X(L)= 0$, the moduli space
$M_X(L)$ consists of a finite collection of points and
$SW_X(L)$ is defined to be the number of these points counted with signs.
These signs are determined by an orientation on $M_X(L)$, which in turn is
determined by an orientation on the determinant line
$\det(H^0(X;{\bold{R}}))\otimes\det(H^1(X;{\bold{R}}))\otimes
\det(H^2_+(X;{\bold{R}}))$. If $\dim M_X(L)>0$ then we consider the basepoint
map
$$
\tilde{M}_X(L)=\{\text{solutions}\, (A,\psi)\}/\text{Aut}^0(L)\to M_X(L)
$$ where $\text{Aut}^0(L)$ consists of gauge transformations which are the
identity on the fiber of $L$ over a fixed basepoint in $X$. If there are no
reducible solutions, the basepoint map is an $S^1$ fibration, and we denote
its euler class by $\beta\in H^2(M_X(L);\bold{Z})$. The moduli space $M_X(L)$
represents an integral cycle in the configuration space
$B_X(L) =({\cal{A}}_X(L)\times \Gamma(W^+))/\text{Aut}(L)$, and if $\dim M_X(L)=2n$,
the Seiberg-Witten invariant is defined to be the integer
$$
SW_X(L)=\langle\beta^n,[M_X(L)]\rangle.
$$
A fundamental result is that if $b^+(X)\ge2$, the map
$$ SW_X: Spin^{\text{c}}(X) \to {\bold{Z}}
$$ is a diffeomorphism invariant ({\cite{Witten}); i.e.
$SW_X(L)$ does not depend on the (generic) choice of Riemannian metric on
$X$ nor the choice of generic perturbation of the second equation of
\eqref{monopole}.
It is often convenient to observe that the space ${\cal{A}}_X(L)\times \Gamma(W^+)$ is
contractible and $\text{Aut}(L)\cong\text{Map}(X,S^1)$ acts freely on
${\cal{A}}_X(L)\times (\Gamma(W^+)\setminus\{ 0 \})$. Since $S^1$ is a $K({\bold{Z}},1)$, if
we
further assume that
$H^1(X;\bold{R})=0$, then the quotient
$$ B^*_X(L)=\left({\cal{A}}_X(L)\times(\Gamma(W^+)\setminus\{ 0 \})\right)/S^1
$$ of this action is homotopy equivalent to ${\bold{CP}}^{\infty}$. So if
there are no reducible solutions, we may view $M_X(L)\subset
{\bold{CP}}^{\infty}$. Under these identifications, the class
$\beta$ becomes the standard generator of $H^2({\bold{CP}}^{\infty};{\bold{Z}})$.
Call a characteristic line bundle with nontrivial Seiberg-Witen invariant a
{\it Seiberg-Witten class}. The assumption in Seiberg-Witten theory which is
analogous
to the assumption of simple type in Donaldson theory is
\begin{enumerate}\item[(*)] For each Seiberg-Witten class $L$, $\dim M_L(X)=0$.
\end{enumerate}
If this condition is satisfied, $X$ is said to have {\it Seiberg-Witten simple
type}.
\begin{lem}\label{char} Let $C_p\subset X$ and let $X_p$ be its rational
blowdown.
Assume that $X_p$ is simply connected.
\begin{enumerate}
\item[(a)] A line bundle $L^*$ on $X^*$ extends over $X_p$ if and only if
$c_1(L^*|_{L(p^2,1-p)})\in p{\bold{Z}}_{p^2}$.
\item[(b)] If $\bar{L}$ is a characteristic line bundle on $X_p$, then there is a
characteristic line bundle $L$ on $X$ such that $L|_{X^*}=\bar{L}|_{X^*}$.
\item[(c)] If $L$ is a characteristic line bundle on $X$, then an extension
$\bar{L}$ of
$L|_{X^*}$ is characteristic on $X^*$ if and only if $\bar{L}|_{B_p}$ is
characteristic.
\end{enumerate}\end{lem}
\begin{pf} (a) is obvious.
For any simply connected manifold $Y=V\cup W$ where $\partial V=\partial W$ is a rational
homology sphere, a class $c\in H^2(Y;\bold{Z})$ will be characteristic provided
$\langle c,\alpha\rangle\equiv\alpha\cdot\alpha$ (mod 2) for all
$\alpha\in H^2(Y;\bold{Z})$. Thus we need not worry about torsion classes; so a class is
characteristic if and only if its restrictions to $V$ and $W$ are both
characteristic. Applying this observation to $X_p=X^*\cup B_p$ proves (c).
To prove (b), let $\bar{L}$ be a characteristic line bundle on $X_p$ and
let $L^*=\bar{L}|_{X^*}$. By (a), $\delta c_1(L^*)=mp$ for some
integer $m$. Suppose that $p$ is odd, then since $mp=(p+m)p\in\bold{Z}_{p^2}$, we may
assume
that $m$ is even. Let $L'$ be the line bundle on $C_p$ such that the Poincar\'e
dual
of $c_1(L')$ is $(m+1)\gamma_{p-1}+(m-p+1)\gamma_1$. Then $L'$ is characteristic on
$C_p$
and $\delta c_1(L')=\delta c_1(L^*)$. It follows that $L^*$
extends to a characteristic line bundle on $X$ by our observation above. If
$p$ is
even, we may take $c_1(L')$ to be the Poincar\'e dual of $mp\gamma_1$ and get the
extension of $L^*$ to a global line bundle $L$ on $X$ whose restriction to both
$X^*$
and $C_p$ is characteristic.
\end{pf}
If $\bar{L}$ is a line bundle on $X_p$ and $L$ is a line bundle on $X$ satisfying
$L|_{X^*}=\bar{L}|_{X^*}$, we say that $L$ is a {\em lift} of $\bar{L}$.
\begin{thm}\label{swgen} Let $C_p\subset X$ and let $X_p$ be its rational
blowdown. Let $\bar{L}$ be
a characteristic line bundle on $X_p$ and let $L$ be any lift of $\bar{L}$ which is
characteristic on
$X$. Suppose that $\dim M_X(L)\equiv\dim M_{X_p}(\bar{L})\pmod 2$. Then
\[ SW_{X_p}(\bar{L})=SW_X(L).\] \end{thm}
\begin{pf} Since the rational ball $B_p$ embeds in the ruled surface
${\bold{F}}_{p-1}$ (see Lemma~\ref{ratball}), it admits a metric of positive
scalar curvature.
The gluing theory for solutions of the Seiberg-Witten equations follows the
same pattern as for
solutions of the anti-self-duality equations. Thus we study the solutions on $X_p$ for $\bar{L}$
by stretching the
neck between $X^*$ and $B_p$. We may assume that there are positive scalar
curvature metrics on
both the neck $L(p^2,1-p)\times\bold{R}^+$ and on $B_p$. This means that the only
solution to the
Seiberg-Witten equations on $B_p$ with a cylindrical end is the reducible
solution $(A',0)$,
where $A'$ is an anti-self-dual connection on $L'=\bar{L}|_{B_p}$. Possible global solutions
are constructed
from asymptotically reducible solutions on $X^*$ glued to $(A',0)$. The formal
dimension of
$M_{B_p}(L')$ is odd and negative, and there is one gluing parameter (since the
asymptotic value
is reducible); so
\[ \dim M_{X^*}(L^*)+1+\dim M_{B_p}(L') = \dim M_{X_p}(\bar{L})=2d_{\bar{L}}, \] say.
(If $\dim M_{X_p}(\bar{L})$ is odd, there is nothing to prove.) Thus $\dim
M_{X^*}(L^*)=2d_{L^*}$
where $d_{L^*}\ge d_{\bar{L}}$. This means that there is an obstruction to
perturbing a glued-up
$(A^*,\psi^*)\# (A',0)$ to a solution. As in Donaldson theory, there is an
obstruction bundle
$\xi$ over $M_{X^*}(L^*)$, and it is the complex vector bundle of rank
$d_{L^*}-d_{\bar{L}}$
associated to the basepoint fibration. The zero set of a generic section of
$\xi$ is homologous
to $M_{X_p}$ in $B_{X_p}(\bar{L})$. Thus
\[ SW_{X_p}(\bar{L})=\langle\beta^{d_{\bar{L}}},[M_{X_p}(\bar{L})]\rangle=\langle\beta^{d_{\bar{L}}},
\beta^{d_{L^*}-d_{\bar{L}}}\cap [M_{X^*}(L^*)]\rangle=\langle\beta^{d_{L^*}},[M_{X^*}(L^*)]\rangle.
\]
Let $L$ be a characteristic line bundle on $X$ which is a lift of $\bar{L}$, and
let
$\dim M_X(L)=2d_L$. The second construction of
Lemma~\ref{ratball} shows that $C_p$ has a metric of positive scalar curvature.
So the discussion
of the last paragraph applies to show that
\[ SW_X(L)=\langle\beta^{d_{L^*}},[M_{X^*}(L^*)]\rangle, \]
completing the proof of the theorem.
\end{pf}
\begin{lem}\label{swdim} Suppose that $C_p\subset X$ with rational blowdown
$X_p$.
Let $L$ be a characteristic line bundle on $X$ such that $\langle c_1(L), u_i\rangle=0$
for
$i=1,\dots,p-2$ and $\langle c_1(L), u_{p-1}\rangle=mp$ for some $m\in\bold{Z}$. Let $\bar{L}$ be
a
characteristic extension of $L|_{X^*}$ to all of $X_p$. Then $m$ is odd, and
$\dim M_{\bar{L}}(X_p)=\dim M_L(X)+{m^2-1\over4}(p-1)$.\end{lem}
\begin{pf} The proof of Theorem~\ref{tautcalc} shows that $m$ must be odd if
$\bar{L}$ is to
be characteristic and that $c_1(\bar{L})^2=c_1(L)^2+m^2(p-1)$. Since
$3\,\text{sign}(X_p)+2\,e(X_p)=3\,\text{sign}(X)+2\,e(X)+(p-1)$, the lemma
follows.
\end{pf}
\noindent Note that this shows that, unless
$m=\pm1$, the dimensions of the moduli spaces will increase.
We shall consider the two situations analogous to those studied in the previous
sections:
\smallskip
\begin{itemize}
\item[(i)] $C_p$ is embedded in $X=Y\#(p-1)\overline{\bold{CP}}^{\,2}$ so that $X_p$ is the result of
an
order $p$ log transform performed on a nodal fiber of $X$.\\
\vspace{-.1in}\item[(ii)] $C_p$ is tautly embedded in $X$ with respect to $L$,
i.e.
$\langle c_1(L),u_i\rangle=0$ for $i=1,\dots,p-2$, and $\langle c_1(L),u_{p-1}\rangle\le p$.
\end{itemize}
The next theorem follows directly from Theorem~\ref{swgen} and
Lemma~\ref{swdim}.
\begin{thm}\label{tautagain} Suppose that $X$ has Seiberg-Witten simple type
and that
$C_p\subset X$ with $X_p$ its rational blowdown. Assume that $X_p$ is simply
connected and that $\bar{L}$ is a characteristic line bundle on $X_p$. Suppose
further that
$L$ is a characteristic lift of $\bar{L}$ and that $C_p$ is tautly embedded with
respect to $L$. Then
\[ SW_{X_p}({\bar{L}})=SW_X(L)\]
and $c_1(\bar{L})^2=c_1(L)^2+(p-1)$. \ \ \qed
\end{thm}
Say that the configuration $C_p$ is {\it SW-tautly embedded} in $X$ if it is
tautly
embedded with respect to each Seiberg-Witten class.
\begin{cor} Suppose that $X$ has Seiberg-Witten simple type and contains the
SW-tautly
embedded configuration $C_p$. Assume that the rational blowdown $X_p$ is
simply
connected. Then the Seiberg-Witten classes of $X_p$ are the
characteristic line bundles $\bar{L}$ which have a lift to a Seiberg-Witten class
$L$ of
$X$, and $SW_{X_p}({\bar{L}})=SW_X(L)$. \ \ \qed\end{cor}
In a fashion similar to the proof of Theorem~\ref{swgen}, one can prove a
blowup formula
for Seiberg-Witten invariants. The characteristic line bundles of $X\#\overline{\bold{CP}}^{\,2}$ are
those of
the form $L\otimes E^{2k+1}$ where
$L$ is characteristic on $X$ and $c_1(E)=e$, and
$\dim M_{L\otimes E^{2k+1}}(X\#\overline{\bold{CP}}^{\,2})=\dim M_L(X)-k(k+1)$. It is shown in
\cite{Turkey}
that $SW_{X\#\overline{\bold{CP}}^{\,2}}(L\otimes E^{2k+1})=SW_X(L)$ provided $\dim
M_L(X)-k(k+1)\ge0$. It
follows that if $X$ satisfies the Seiberg-Witten simple-type condition (*),
then so
does $X\#\overline{\bold{CP}}^{\,2}$.
Suppose that $X$ contains the nodal fiber $S$, and $X_p$ is the result of
performing
an order $p$ log transform on $S$. The characteristic line bundles on $X_p$ are
obtained from characteristic bundles
$L\otimes E_1^{2k_1+1}\otimes\cdots\otimes E_{p-1}^{2k_{p-1}+1}$ on
$Y=X\#(p-1)\overline{\bold{CP}}^{\,2}$ by
restricting to $Y^*=Y\setminus C_p$ and then extending over $B_p$. If we assume
that
$\langle c_1(L),S\rangle=0$, then for each $L\otimes E_1^{\pm1}\otimes\cdots\otimes
E_{p-1}^{\pm1}=L({\pmb\epsilon}_J)$ with $c_1(L({\pmb\epsilon}_J))=
c_1(L)+\sum_J\epsilon_{J,i}e_i$, it
follows from Proposition~\ref{J} that the unique extension $\bar{L}_J$ over $X_p$
has
$c_1(\bar{L}_J)=c_1(L)+|J|\sigma_p$, where $\sigma_p$ is the Poincar\'e dual of $S/p$.
(Note
that when $p$ is even, $|J|$ must be odd; so the extension $\bar{L}_J$ is
characteristic.) Hence
\[ \dim M_{\bar{L}_J}(X_p) =\dim M_L(X),\]
and Theorem~\ref{swgen} implies:
\begin{thm}\label{SWlog} Suppose that $X$ has Seiberg-Witten simple type and
contains the
nodal fiber $S$. Let $L$ be a characteristic line bundle on $X$ with $\langle
c_1(L),S\rangle=0$.
Let $X_p$ be the result of performing an order $p$ log transform on $S$. For
each $J\in
\{\pm1\}^{p-1}$, we have $SW_{X_p}({\bar{L}_J})=SW_X(L)$. Suppose furthermore that
$\langle
c_1(L),S\rangle=0$ for each characteristic $L$ on $X$ with $SW_X(L)\ne0$. Then
$X_p$ also
has Seiberg-Witten simple type and each line bundle $\Lambda$ on $X_p$ with
$SW_{X_p}(\Lambda)\ne 0$ is of the form $\Lambda=\bar{L}_J$.\ \ \qed
\end{thm}
By a the {\em nodal configuration} we shall mean a configuration
$C_p\subset X\#(p-1)\overline{\bold{CP}}^{\,2}$ as above, obtained from a nodal fiber $S$ satisfying
the
condition $\langle c_1(L),S\rangle=0$ for each characteristic $L$ on $X$ with
$SW_X(L)\ne0$.
Witten \cite{Witten} has conjectured that (for manifolds with
$b^+>1$) the Seiberg-Witten simple type condition is equivalent to the
simple type condition of Kronheimer and Mrowka for Donaldson theory. Further,
under this
hypothess of simple type, Witten gives a precise conjecture for relating the
Seiberg-Witten invariants and the Donaldson series, namely:
\begin{conj}[Witten]
The set of basic classes in the two theories are the same, and
\[ {\bold{D}}_X= 2^{3\text{sign}+2e-({b^+-3\over 2})}\exp(Q/2)\sum
SW_X(\kappa_s)e^{\kappa_s}.\]
\end{conj}
\begin{thm} Witten's conjecture is true for simply connected elliptic
surfaces.\end{thm}
\begin{pf} Witten has given a recipe for calculating $SW_X$ for all Kahler
manifolds
$X$. So one could prove this theorem simply by comparing the answer obtained
with that
of Theorem \ref{ellformula}. Alternatively, note that Witten's recipe gives the
result
that the nonzero Seiberg-Witten invariants of $E(n)$ are:
\begin{equation}\label{SWEn}
SW_{E(n)}((n-2-2r)f)= (-1)^r\binom{n-2}{r},\ \ r=0,\dots,n-2
\end{equation}
(where $f$ is the fiber class). Suppose we define
\[{\bold{W}}_X=2^{3\text{sign}+2e-({b^+-3\over 2})}\sum SW_X(\kappa_s)e^{\kappa_s}, \ \
{\bold{SW}}_X=\exp(Q_X/2){\bold{W}}_X\]
Then \eqref{SWEn} shows that ${\bold{D}}_{E(n)}={\bold{SW}}_{E(n)}$. Suppose
that $X_p$ is the result of an order $p$ log transform on a nodal fiber which
is
orthogonal to all classes in $H_2(X)$ with nontrivial Seiberg-Witten
invariants. Then
Theorem \ref{SWlog} implies that
${\bold{W}}_{X_p}={\bold{W}}_X\cdot(\sinh(f_p)/\sinh(f))$.
It follows that ${\bold{SW}}_{E(n;p,q)}={\bold{D}}_{E(n;p,q)}.$\end{pf}
Furthermore, we have
\begin{thm} If $X$ satisfies the Witten conjecture, then so do all blowups and
blowdowns and any rational blowdown $X_p$ of a nodal or taut configuration.
\end{thm}
\bigskip
|
1995-06-15T02:01:10 | 9505 | alg-geom/9505013 | en | https://arxiv.org/abs/alg-geom/9505013 | [
"alg-geom",
"math.AG"
] | alg-geom/9505013 | Teleman | Andrei Teleman and Christian Okonek | Seiberg-Witten Invariants and the Van De Ven Conjecture | Duke preprint, LATEX | null | null | null | null | The purpose of this note is to give a short, selfcontained proof of the
following result: A complex surface which is diffeomeorphic to a rational
surface is rational.
| [
{
"version": "v1",
"created": "Mon, 8 May 1995 19:46:17 GMT"
},
{
"version": "v2",
"created": "Wed, 10 May 1995 15:48:24 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Teleman",
"Andrei",
""
],
[
"Okonek",
"Christian",
""
]
] | alg-geom | \section{Introduction}
The purpose of this note is to give a short, selfcontained proof of the
following
result:
\begin{th}
A complex surface which is diffeomeorphic to a rational surface is rational.
\end{th}
This result has been announced by R. Friedman and Z. Qin [FQ].
Whereas their proof uses Donaldson theory and vector bundles techniques, our
proof uses the new Seiberg-Witten invariants [W], and the interpretation of
these invariants in terms of stable pairs [OT].
Combining the theorem above with the results of [FM], one obtains a proof of
the
Van de Ven conjecture [V]:
\begin{co}
The Kodaira dimension of a complex surface is a differential invariant.
\end{co}
{\bf Proof: } (of the Theorem) It suffices to prove the theorem for algebraic
surfaces [BPV].
Let $X$ be an algebraic surface of non-negative Kodaira dimension, with
$\pi_1(X)=\{1\}$ and $p_g(X)=0$. We may suppose that $X$ is the blow up in $k$
{\sl distinct} points of its minimal model $X_{\min}$. Denote the
contraction to the
minimal model by
$\sigma:X\longrightarrow X_{\min}$, and the exceptional divisor by $E=\sum\limits_{i=1}^k
E_i$.
Fix an ample divisor $H_{\min}$ on $X_{\min}$, a sufficiently large integer
$n$, and
let $H_n:=\sigma^*(n H_{\min})-E$ be the associated polarization of $X$.
For every subset $I\subset\{1,\dots,k\}$ we put
$E_I:=\sum\limits_{i\in I} E_i$, and
$L_I:=2[E_I]-[K_X]$, where $K_X$ is
a canonical divisor. Clearly $L_I=[E_I]-[E_{\bar I}]-\sigma^*([K_{\min}])$,
where
$\bar I$ denotes the complement of $I$ in $\{1,\dots,k\}$. The cohomology
classes
$L_I$ are almost canonical classes in the sense of [OT]. Now choose
a K\"ahler metric $g_n$ on $X$ with K\"ahler class
$[\omega_{g_n}]=c_1({\cal O}_X(H_n))$. Since $[\omega_{g_n}]\cdot L_I<0$ for
sufficiently large
$n$, the main result of [OT] identifies the Seiberg-Witten moduli space
${\cal W}_X^{g_n}(L_I)$ with the union of all complete linear systems $|D|$
corresponding to effective divisors $D$ on $X$ with
$c_1({\cal O}_X(2D-K_X))=L_I$.
Since $H^2(X,{\Bbb Z})$ has no 2-torsion, and $q(X)=0$, there is only one such
divisor, $D=E_I$. Furthermore, from
$h^1({\cal O}_X(E_I)|_{E_I})=0$, and the smoothness criterion in [OT], we obtain:
$${\cal W}_X^{g_n}(L_I)=\{E_I\}, $$
i.e. ${\cal W}_X^{g_n}(L_I)$ consists of a single smooth point.
The corresponding Seiberg-Witten invariants are therefore odd:
$n_{L_I}^{g_n}=\pm 1$.
Consider now the positive cone ${\cal K}:=\{\eta\in H^2_{\rm DR}(X)|\ \eta^2>0\}$;
using the Hodge index theorem, the fact that $K_{\min}$ is cohomologically
nontrivial, and $K_{\min}^2\geq 0$, we see that
${\cal K}$ splits as a disjoint union of two components ${\cal K}_{\pm}:=\{\eta\in{\cal K}|\
\pm\eta\cdot\sigma^*(K_{\min})>0\}$. Clearly $[\omega_{g_n}]$ belongs to
${\cal K}_+$.
Let $g$ be an {\sl arbitrary} Riemannian metric on $X$, and let $\omega_g$ be
a $g$-selfdual closed 2-form on $X$ such that $[\omega_g]\in{\cal K}_+$.
For a fixed $I\subset\{1,\dots,k\}$, we denote by $L_I^{\bot}\subset{\cal K}_+$ the
wall associated with $L_I$, i.e. the subset of classes $\eta$ with $\eta\cdot
L_I=0$.\\ \\
{\bf Claim:} The rays ${\Bbb R}_{>0}[\omega_g]$, ${\Bbb R}_{>0}[\omega_{g_n}]$ belong
either to the same component of ${\cal K}_+\setminus L_I^{\bot}$ or to the same
component of ${\cal K}_+\setminus L_{\bar I}^{\bot}$. \\
Indeed, since $[\omega_{g_n}]\cdot L_I<0$ and $[\omega_{g_n}]\cdot L_{\bar
I}<0$,
we just have to exclude that
$$[\omega_{g}]\cdot L_I\geq 0 \ \ \ \ {\rm and}\ \ \ \ [\omega_{g}]\cdot
L_{\bar
I}\geq 0 .\eqno{(*)}$$
Write $[\omega_g]=\sum\limits_{i=1}^{k}\lambda_i[E_i]+\sigma^*[\omega]$,
for some
class
$[\omega]\in H^2_{\rm DR}(X_{\min})$; then
$[\omega]^2>\sum\limits_{i=1}^k\lambda_i^2$, and $[\omega]\cdot
K_{\min}>0$, since
$\omega_g$ was chosen such that its cohomology class belongs to ${\cal K}_+$. The
inequalities $(*)$ can now be written as
$$-\sum\limits_{i\in I}\lambda_i+\sum\limits_{j\in\bar I}\lambda_j-
[\omega]\cdot K_{\min}\geq 0 \ \ \ {\rm and}\ \ \
-\sum\limits_{j\in\bar I}\lambda_j+\sum\limits_{i\in I}\lambda_i-
[\omega]\cdot K_{\min}\geq 0,$$
and we obtain the contradiction $[\omega]\cdot K_{\min}\leq 0$. This proves the
claim.
\dfigure 80mm by 165mm (kegel scaled 700 offset 1mm:)
We know already that the mod 2 Seiberg-Witten invariants $n^{g_n}_{L_I}$(mod 2)
and $n^{g_n}_{L_{\bar I}}$(mod 2) are nontrivial for the special metric $g_n$.
Since the invariants $n^{g}_{L_{I}}$(mod 2) and $n^{g}_{L_{\bar I}}$(mod 2)
depend only on the chamber of the ray ${\Bbb R}_{>0}[\omega_g]$ with repect to the
wall $L_{I}^{\bot}$, respectively $L_{\bar I}^{\bot}$ (see [W], [KM]), at
least
one of the invariants associated with the metric $g$ must be non-zero, too.
But any rational surface admits a Hodge metric with positive total scalar
curvature [H], and with respect to such a metric {\sl all} Seiberg-Witten
invariants
are trivial [OT].
\hfill\vrule height6pt width6pt depth0pt \bigskip
\vspace{0.5cm}\\
\parindent0cm
\centerline {\Large {\bf Bibliography}}
\vspace{0.5cm}
[BPV] Barth, W., Peters, C., Van de Ven, A.: {\it Compact complex surfaces},
Springer Verlag (1984)
[FM] Friedman, R., Morgan, J.W.: {\it Smooth 4-manifolds and Complex Surfaces},
Springer Verlag 3. Folge, Band 27, (1994)
[FQ] Friedman, R., Qin, Z.: {\it On complex surfaces diffeomorphic to
rational surfaces}, Preprint (1994)
[H] Hitchin, N.: {\it On the curvature of rational surfaces}, Proc. of Symp.
in Pure Math., Stanford, Vol. 27 (1975)
[KM] Kronheimer, P., Mrowka, T.: {\it The genus of embedded surfaces in the
projective plane}, Preprint (1994)
[OT] Okonek, Ch.; Teleman A.: {\it The Coupled Seiberg-Witten Equations,
Vortices, and Moduli Spaces of Stable Pairs}, Preprint, January, 13-th 1995
[W] Witten, E.: {\it Monopoles and four-manifolds}, Mathematical Research
Letters 1, 769-796 (1994)
[V] Van de Ven, A,: {\it On the differentiable structure of certain algebraic
surfaces}, S\'em. Bourbaki ${\rm n}^o$ 667, Juin (1986)
\vspace{1cm}\\
Authors addresses:\\
\\
Mathematisches Institut, Universit\"at Z\"urich,\\
Winterthurerstrasse 190, CH-8057 Z\"urich\\
e-mail:[email protected]
\ \ \ \ \ \ \ \ \ [email protected]
\end{document}
|
1995-07-01T06:17:36 | 9505 | alg-geom/9505022 | en | https://arxiv.org/abs/alg-geom/9505022 | [
"alg-geom",
"math.AG"
] | alg-geom/9505022 | Rahul Pandharipande | R. Pandharipande | A Geometric Invariant Theory Compactification of M_{g,n} Via the
Fulton-MacPherson Configuration Space | 14 pages. AMSLatex | null | null | null | null | A compactification over $\overline{M}_g$ of $M_{g,n}$ is obtained by
considering the relative Fulton-MacPherson configuration space of the universal
curve. The resulting compactification differs from the Deligne-Mumford space
$\overline{M}_{g,n}$. In case $n=2$, the compactification constructed here and
the Deligne-Mumford compactification are essentially the distinct minimal
resolutions of the fiber product over $\overline{M}_g$ of the universal curve
with itself.
| [
{
"version": "v1",
"created": "Tue, 23 May 1995 17:55:29 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Pandharipande",
"R.",
""
]
] | alg-geom | \section{Introduction}
\label{int}
In [F-M], a compactification of the
configuration space of $n$ marked points on an algebraic variety
is defined. For a nonsingular curve $X$ of genus $g \geq 2$,
the Fulton-MacPherson
configuration space of $X$
(quotiented by the automorphism group of $X$) is isomorphic to
the (reduced)
fiber of $\gamma:\overline{M}_{g,n} \rightarrow \overline{M}_{g}$ over $[X]\in M_g$.
Since the Fulton-MacPherson configuration
space is defined for singular varieties, it is natural to ask
whether a compactification of $\gamma^{-1}(M_g)$ can be obtained over
$\overline{M}_{g}$.
First, we consider the Fulton-MacPherson configuration
space for families of varieties. This relative construction is then
applied to the universal curve over the Hilbert scheme of
$10$-canonical, genus $g\geq 2$ curves.
Following results of Gieseker, it is shown there
exist linearizations of the natural ${SL}$-action on the relative
configuration space of the universal curve
that yield G.I.T. quotients compactifying
$\gamma^{-1}(M_g)$. These new compactifications, $M_{g,n}^{c}$, are described.
For $n=1$, $M_{g,1}^c$ and
the Deligne-Mumford compactification $\overline{M}_{g,1}$ coincide. For $n=2$,
$M_{g,2}^c$ and $\overline{M}_{g,2}$ are isomorphic on open
sets with codimension $2$ complements. $M_{g,2}^c$ and $\overline{M}_{g,2}$
differ essentially by the birational modification corresponding
to the two minimal resolutions of an ordinary threefold double point.
For higher $n$, the compactifications
$M_{g,n}^c$ and $\overline{M}_{g,n}$ differ more substantially.
Thanks are due to
J. Harris for mathematical guidance.
The author has benefited from many discussions with him.
\section{Relative Fulton-MacPherson Configuration Spaces}
\label{rfm}
\subsection{Terminology}
\label{rmft}
Let $\Bbb{C}$ be the ground field of complex numbers. A
morphism $\mu: X \rightarrow Y$ is an {\it immersion} if
$\mu$ is an isomorphism of $X$ onto an open subset of
a closed subvariety of $Y$. A morphism $\gamma$ is {\it quasi-projective}
if it factors as $\gamma =\rho \circ \mu$ where $\mu$ is an open immersion
and $\rho$ is projective. The only smooth morphisms considered will be
smooth morphisms of relative dimension $k$ between nonsingular
varieties.
\subsection{Definition}
\label{rfmd}
We carry out the construction of Fulton and MacPherson
in the relative context.
Suppose $\pi:{ F}\rightarrow{ B}$ is a (separated) morphism
of algebraic varieties. Let $n$ be a positive integer.
${ N}= \{ 1,\ldots, n \} $. Wherever possible,
products will be taken in the category of varieties over $B$.
Define:
$$ F_B^N=\prod_{N}F=
\underbrace { F \times {_B} F \times {_B} \ldots \times {_B} F}_{n}\;\;. $$
And define:
$$(F_B^N)_0= F_B^N \;\;\setminus \;\; (\bigcup \bigtriangleup_{\{a,b\}})$$
Where $\bigtriangleup_{\{a,b\}}$ denotes the large diagonal
corresponding to the indices $a,b \in N$ and the union is over
all pairs $\{ a,b \}$ of distinct element of $N$.
For each subset $S$ of $N$ define
$F_B^S=\prod_{S}F$. Following the notation of [F-M],
let $Bl_\bigtriangleup (F_B^S)$ denote the blow-up of $F_B^S$ along
the small diagonal. There exists a natural immersion:
\begin{equation}
(F_B^N)_0 \subset F_B^N \times \prod_{|S|\geq 2}{Bl_\bigtriangleup
(F_B^S)}\;\; .\label{mouse}
\end{equation}
The relative Fulton-MacPherson configuration space of
$n$ marked points of $F$ over $B$, $F_B[n]$, is defined
to be the closure of $(F_B^N)_0$ in the above product. When $B$ is
a point, this definition coincides with that of [F-M].
Consider the composition:
\begin{equation}
F_B[n] \subset F_B^N \times \prod_{|S|\geq 2}{Bl_\bigtriangleup
(F_B^S)}
\stackrel{\mu}{\rightarrow}
F_B^N \times \prod_{|S|\geq 2}{(F_B^S)}
\stackrel{\beta}{\rightarrow}
F_B^N
\end{equation}
where $\mu$ is the natural blow-down morphism and $\beta$ is
the projection on the first factor. Since $\mu$ is a projective
morphism and $F_B[n]$ is a closed subvariety,
$$\mu: F_B[n] \rightarrow \mu(F_B[n])$$
is also projective. Since $\beta: \mu(F_B[n]) \stackrel{\sim}
{\rightarrow} F_B^N$ is an isomorphism, the morphism $\rho=\beta
\circ \mu$
$$\rho: F_B[n] \rightarrow F_B^N$$
is projective.
For our
purposes, we shall only consider the case where
$\pi:{F}\rightarrow{ B}$ is a
quasi-projective morphism. Also, we will
be mainly interested in the case where
$F$ and $F^N_B$ are irreducible varieties.
\subsection{The Blow-Up Construction}
\label{rfmb}
Consider again the birational projective morphism
$$\rho :F_B[n]\rightarrow F_B^N$$
It is natural to inquire
whether $\rho$ can be expressed as a composition of explicit
blow-ups along canonical subvarieties. In [F-M], such a blow-up
construction is given for the configuration space in case
$B$ is a point. The blow-ups in [F-M] are canonical in the following
sense: if $Y \rightarrow X$ is an immersion, the sequence of
blow-ups resolving $Y[n] \rightarrow Y^N$ is the sequence of
strict transformations of $Y^N$ under the blow-ups resolving
$X[n] \rightarrow X^N$.
The blow-up construction of Fulton and MacPherson is valid in the
relative context.
We now assume that $\pi: F\rightarrow B$ is a quasi-projective morphism.
In this case, there exists a factorization:
\begin{equation*}
\begin{CD}
F @>i>> {\bold P}^r \times B \\
@VV{\pi}V @VVV \\
B @= B
\end{CD}
\end{equation*}
where $i$ is an immersion.
We use the notation $\bold P^r \times B = \bold P_B^r$ and drop
extra $B$ subscripts when the meaning is clear. For example,
$(\bold P_B^r)^N$ instead of $(\bold P_B^r)_B^N$ . We have
the following commutative diagram:
\begin{equation}
\begin{CD}
(F_B^N)_0 @>>> F_B^N \times \prod_{|S|\geq 2}{Bl_\bigtriangleup (F_B^S)}\\
@VV{i^N}V @VV{i^{Bl}}V \\
(\bold P_B^r)^N_0 @>>> (\bold P_B^r)^N \times \prod_{|S|\geq 2}{Bl_\bigtriangleup
((\bold P_B^r)^S)}
\label{snake}
\end{CD}
\end{equation}
where $i^N$, $i^{Bl}$ are immersions. We conclude from
diagram (\ref{snake}) that $F_B[n]$ is immersed in $\bold P_B^r[n]$.
Hence:
\begin{equation}
\begin{CD}
F_B[n] @>j>> \bold P_B^r[n] \\\
@VV{\rho}V @VV{\eta}V \\
F_B^N @>{i^N}>> (\bold P_B^r)^N
\label{eagle}
\end{CD}
\end{equation}
where $i^N$, $\;j$ are immersions. Since $\rho$ is
a projective morphism, $j(F_B[n])$ is closed
in $\eta^{-1}(i^N(F_B^N))$.
$F_B[n]$ is therefore the strict transformation of $F_B^N$ under $\eta$.
It is clear the following diagram holds:
\begin{equation*}
\begin{CD}
\bold P_B^r[n] @= \bold P^r[n] \times B \\
@VV{\eta}V @VV{\gamma \times id}V \\
(\bold P_B^r)^N @=( \bold P^r)^N \times B
\end{CD}
\end{equation*}
In [F-M], an explicit and canonical blow-up construction of $\gamma$
is given. By extending each exceptional locus over the base
$B$, a blow-up construction of $\eta$ is obtained. We see from
diagram (\ref{eagle}) that a blow-up construction of $F_B[n]$ exists by
taking the strict transformation of $F_B^N$ at each blow-up of
$(\bold P_B^r)^N$.
\subsection{Comparing $F_b[n]$ and $F_B[n]_b$}
\label{rfmc}
For a given $b \in B$ let $F_b$ denote the fiber of $\pi$ over $b$.
{}From equation (\ref{mouse}) and the definitions, it is clear there exists
a natural closed immersion:
$$F_b[n] \stackrel{i_b}{\hookrightarrow} F_B[n]_b .$$
It is possible for $i_b$ to be a proper inclusion. Examples of this
behavior will be seen in section (\ref{disc}).
We have the following:
\begin{pr}
\label{pfiber}
If $B$ is irreducible and $\pi:F \rightarrow B$ is a smooth,
quasi-projective morphism of nonsingular varieties, then
for every $b \in B$, $i_b$ is an isomorphism.
\end{pr}
\begin{pf}
Suppose $X$ is a fixed nonsingular algebraic variety. In [F-M],
the canonical construction of $X[n]$ is given by
a sequence of explicit blow-ups
of $X^N$ along {\it nonsingular} centers. In the previous section, it was
shown how the construction of [F-M] could be lifted to the relative
context. Let $m$ be the number of blow-ups needed in the Fulton-
MacPherson construction resolving
$\rho: F_B[n] \rightarrow F_B^N$. Let $F_{B,j}^N$ for
$0 \leq j \leq m$ denote the $j^{th}$ stage. $F_{B,0}^N= F_B^N$
and $F_{B,m}^N= F_B[n]$. Since the blow-up construction in [F-M] is
canonical, for any variety $X$ similar definitions can be made. We
show inductively, for each $b \in B$, the natural inclusion:
\begin{equation}
\label{duck}
F_{b,j}^N \hookrightarrow (F_{B,j}^N)_b
\end{equation}
is an isomorphism. For $j=0$ the assertion is clear. The induction
step rests on a simple {\bf Claim}:
Suppose $S$ is an irreducible nonsingular variety,
$ R \rightarrow S$ is a smooth morphism,
and $T \hookrightarrow R$ is a closed immersion smooth over $S$ . Then, for
any $s \in S$, the blow-up of $R_s$ along $T_s$ is naturally isomorphic to
the fiber over $s$ of the blow-up of $R$ along $T$. Since all spaces are
nonsingular, the assertion follows from examining normal directions of $T$
in $R$; the various smoothnesses imply all normal directions are
represented in the fiber.
Assume equation (\ref{duck}) is an isomorphism for all $b \in B$.
Since $\pi$ is smooth of relative dimension $k$,
$F_b$ and $F_{b,j}^N$ are nonsingular of pure dimensions $k$, $nk$. Hence,
$(F_{B,j}^N)_b$ is nonsingular of pure dimension $nk$.
Also, every irreducible
component of $F_B^N$ (and hence $F_{B,j}^N$) is of relative
dimension $nk$. The last two facts
imply the morphism: $$\pi_j^N:F_{B,j}^N
\rightarrow B$$ is smooth. Examination of the $(j+1)^{th}$ center is
straightforward. Because of the assumed isomorphism (\ref{duck}) and the
knowledge that the [F-M] construction of the configuration space of a
nonsingular variety over a point only involves nonsingular centers, we see
that the $(j+1)^{th}$ center is smooth over B. The above claim now proves
the induction step.
\end{pf}
\subsection{Universal Families}
\label{rfmu}
Let $X$ be a nonsingular algebraic variety. Let
$\overline{x}=(x_1, \ldots, x_n)$ be $n$ ordered points of $X$.
A subset $S\subset N$ is said to be coincident
at $z\in X$ if $|S| \geq 2$ and
for all $i \in S$, $x_i=z$. Following [F-M], for every $S$ coincident
at $z$, we define a {\it screen} of $S$ at $z$ to be an
equivalence class of the data $(t_i)_{i\in S}$ where:
\begin{enumerate}
\item $t_i \in T_z$, the tangent space of $X$ at $z$.
\item $\exists i,j \in S$ such that $t_i \neq t_j$.
\end{enumerate}
Two data sets $(t_i)_{i\in S}$ and $(t'_i)_{i\in S}$
are equivalent if there exists $\lambda \in C^*$ and
$v \in T_z$ so that for all $i \in S$,
$\lambda \cdot t_i + v= t'_i$.
A screen shows the tangential separation of infinitely
near points.
An $n$-tuple $\overline{x}=(x_1, \ldots, x_n)$ together
with a screen $Q_S$ for each coincident subset $S\subset N$
constitute an {\it n-pointed stable class} in $X$ if the
screens are compatible in the following sense.
Suppose $S_1\subset S_2$ are two subsets coincident at
$z$ where $Q_{S_2}$ is represented by the data $(t_j)_{j \in S_2}$.
If there exist
$k,\hat{k} \in S_1$ so that $t_k\neq t_{\hat k}$,
then $(t_j)_{j\in S_1}$ defines a screen for $S_1$.
The compatibility condition requires that when
this restriction of $Q_{S_2}$ is defined, it equals $Q_{S_1}$.
For a nonsingular space $X$, $X[n]$ is the parameter
space of $n$-pointed stable classes in $X$.
Given an $n$-pointed stable class in $X$, an
{\it $n$-pointed stable degeneration } of $X$ can be constructed
(up to isomorphism)
as follows. Let $z\in X$ occur with multiplicity in $\overline{x}$.
Blow-up $X$ at $z$ and attach a
$\bold P(T_z \oplus {\bold 1})$ in the natural way along
the exceptional divisor at $z$. Note that
$\bold P(T_z \oplus {\bold 1})$ minus the hyperplane at infinity,
$\bold P(T_z)$, is naturally isomorphic to the affine space
$T_z$. Let $S_z \subset N$ be the maximal subset coincident
at $z$. The screen $Q_{S_z}$ associates (up to equivalence)
points of $T_z$ to the indices that lie in $S_z$. Condition
(2) of the screen data implies some separation of the
marked points has occurred. The further screens specify
in a natural way (up to equivalence of screens)
the further blow-ups and markings required to separate the marked points.
The final space obtained along with $n$ distinct marked points is
the $n$-pointed stable degeneration associated to the given
$n$-pointed stable class.
See [F-M] for further details.
It is shown in [F-M] that if $X$ is an nonsingular
variety, there exists a universal family of $n$-pointed
stable degenerations of $X$ over $X[n]$. Let $X[n]^+$ denote
this universal family. $X[n]^+$ is equipped with the following
maps:
\begin{equation*}
\begin{CD}
X[n]^+ @>{\mu}>> X[n] \times X \\
@VV{\mu_p}V @VVV \\
X[n] @= X[n]
\end{CD}
\end{equation*}
There are $n$ sections of ${\mu_p}$, $\{ \sigma_i \}_{i \in N}$ :
$$X[n] \stackrel{\sigma_i}{\rightarrow} X[n]^+ . $$
For any $d\in X[n]$, the fiber $\mu^{-1}_p(d)$ along with
the $n$-tuple $(\sigma_1(d), \ldots,\sigma_n(d))$ is
the $n$-pointed stable degeneration of $X$ associated to
the $n$-pointed stable class corresponding to $d$.
We note here that if $C$ is
a nonsingular automorphism-free curve, $n$-pointed stable classes
in $C$ correspond bijectively to isomorphism classes of
$n$-pointed Deligne-Mumford
stable curves over $C$ . Moreover, the universal family over $C[n]$ defines
a map to the reduced fiber $$\phi:C[n] \rightarrow \gamma^{-1}([C])$$ where
$\gamma :\overline{M}_{g,n} \rightarrow \overline{M}_g$.
Since $\phi$ is proper bijective
and both spaces are normal, $\phi$ is an isomorphism. If $C$
has a finite automorphism group, $A$, we see $A$ acts on $C[n]$
and $\phi$ is $A$-invariant. Therefore $\phi$ descends to the quotient
$$\phi (C[n]/A) \rightarrow \gamma^{-1}([C]).$$
It is not hard to see that this map is proper bijective and hence
an isomorphism by normality.
The map $\mu$ is a birational morphism and can be expressed as an explicit
sequence of blow-ups of $X[n] \times X$ along canonical, nonsingular loci.
Canonical here has the same meaning as in section (\ref{rfmb}) :
if $Y \rightarrow X$ is an immersion of nonsingular varieties, the
blow-up sequence resolving $Y[n]^+ \rightarrow Y[n]\times Y$ is
the strict transform of of $Y[n] \times Y$ under the blow-up
sequence resolving $X[n]^+ \rightarrow X[n] \times X$. Moreover,
the sections of $Y[n]^+ \rightarrow Y[n]$ are restrictions
of the sections of $X[n]^+ \rightarrow X[n]$. This
canonical blow-up construction is given in [F-M].
\subsection{Relative Universal Families}
\label{rfmru}
Suppose $\pi:F \rightarrow B$ is a smooth,
quasi-projective morphism of nonsingular varieties. In this case,
the construction of the
universal family that appears in [F-M] can be lifted to the relative
context. Using the notation of section (\ref{rfmb}), we have an immersion:
$$F_B[n] \times_B F \rightarrow \bold P_B^r[n] \times_B \bold P_B^r \;.$$
Consider the diagram:
\begin{equation*}
\begin{CD}
\bold P_B^r[n]^+ @= \bold P^r[n]^+ \times B \\
@VV{\omega}V @VV{\mu \times id}V \\
\bold P_B^r[n] \times_B \bold P_B^r @= \bold P^r[n]\times \bold P^r
\times B
\end{CD}
\end{equation*}
For $\omega=\mu \times id$, the Fulton-MacPherson construction of the
universal family can be carried out uniformly over the base by
extending the centers of the blow-ups resolving $\mu$
trivially over $B$. Define
$F_B[n]^+$ to be the
proper transform of $ F_B[n] \times_B F $ under $\omega$. We have:
$$\upsilon: F_B[n]^+ \rightarrow F_B[n] \times_B F$$
To show the space defined above, $F_B[n]^+$, has the
desired geometrical properties, we argue as in the proof
of Proposition 1. Let $ (F_B[n] \times_B F)_j $ denote the
$j^{th}$ stage of the canonical sequence of blow-ups resolving $\upsilon$.
Inductively, it is shown that for each $b \in B$ there is an isomorphism:
\begin{equation}
\label{cow}
(F_b[n] \times F_b)_j \rightarrow
(F_B[n] \times_B F)_{j,b}.
\end{equation}
The $j=0$ case is established by Proposition 1. The induction step follows
from the the claim made in the proof of Proposition 1 and the fact
that for a nonsingular variety $X$, the canonical Fulton-MacPherson resolution
of $X[n]^+ \rightarrow X[n]\times X$ involves only nonsingular centers.
We conclude that fiber $F_B[n]^+_b$ over $b \in B$ is
naturally isomorphic
to $F_b[n]^+$.
It is clear that $n$ sections ${\sigma_i}$ exist for
$$\omega_p: \bold P_B^r[n]^+ \rightarrow \bold P_B^r[n].$$
For each $b \in B$, these sections
${\sigma_i}$ are compatible with the $n$ natural sections of
$F_b[n]^+ \rightarrow F_b[n].$
Therefore, via restriction, the ${\sigma_i}$ yield $n$ sections of
$$\upsilon_p: F_B[n]^+ \rightarrow F_B[n].$$
The fiber of $F_B[n]^+_\xi$ over
$\xi \in F_B[n]$ is a $n$-pointed stable degeneration of
$F_{\pi(\xi)}$. We have:
\begin{pr}
Suppose $B$ is irreducible and $\pi: F \rightarrow B$ is a smooth,
quasi-projective morphism of nonsingular varieties, then
$F_B[n]^+$ along with $\upsilon$ and
$\{\sigma_i\}_{i \in N}$ is a universal family of n-pointed stable
degenerations of $F_B$ over $F_B[n]$.
\end{pr}
\subsection{Final Note}
\label{rfmfn}
Suppose $\pi:G \rightarrow B$ is a projective morphism,
$G$ is nonsingular, irreducible, $B$ is nonsingular, $\pi$ is flat, and
for every $b \in B$ the fiber $G_b$ is reduced.
In this case, let $F \subset G$ be the open set where
$\pi$ is smooth. Using flatness and a tangent space
calculation, we see:
$$ F=\{\xi \in G|\xi \hbox{ is a nonsingular point of } G_{\pi(\xi)} \} $$
and $\pi: F \rightarrow B$ is a smooth, surjective
morphism of nonsingular varieties.
We know the space $F_B[n]$ is equipped with a
universal family $F_B[n]^+$ obtained from $F_B[n]\times_B F$
by a sequence of canonical blow-ups. The problem with this
universal family is that its fibers over $F_B[n]$ are $n$-pointed
stable degenerations of $F_B$ not $G_B$. This problem can easily
be fixed. Note there is an open inclusion:
$$F_B[n]\times_B F \subset F_B[n]\times_B G .$$
It is the case that the centers of the blow-ups resolving
$$\upsilon:F_B[n]^+ \rightarrow F_B[n] \times_B F$$
are closed in $F_B[n]\times_B G$.
Using the isomorphism (\ref{cow}) and the explicit
description of the centers of blow-ups in [F-M], this
closure is not hard to check.
Hence, if the sequence of
blow-ups is carried out over $F_B[n]\times_B G$ the desired
family of $n$-pointed stable degenerations of $G_B$ is obtained
over $F_B[n]$. An $n$-pointed stable degeneration of a fiber
$G_b$ is as before with the additional condition
that the marked points must lie
over the smooth locus of $G_b$.
\section{The Geometric Invariant Theory Set-Up}
\label{git}
\subsection{Notation}
\label{gitn}
Let ${\overline{ M}_g}$
denote the Deligne-Mumford compactification of the moduli space of
nonsingular, genus $g$, projective curves, $M_g$. Let ${\overline{M}_{g,n}}$
denote the Deligne-Mumford compactification of the moduli space of genus
$g$ curves with $n$ marked points. There exists a natural projective
morphism
$$\gamma : {\overline {M}_{g,n}} \rightarrow {\overline {M}_g}.$$
All these spaces are normal.
\subsection{Gieseker's construction of $\overline{M}_g$}
\label{gitg}
Fix an integer $g \geq 2$. Define: $$d=10 \cdot (2g-2)$$
$$R=d-g.$$ Define the polynomial:
$$f(m)= d \cdot m -g +1.$$
Note $f(m)$ is the Hilbert polynomial of a complete, genus $g$,
$10$-canonical curve in $\bold P ^R$. Let $H_{f,R}$ denote
the Hilbert scheme of the polynomial $f$ in $\bold P ^R$.
If $X$ is a closed subscheme of $\bold P ^R$ with Hilbert polynomial
$f$, we denote the point of $H_{f,R}$ corresponding to $X$ by $[X]$.
It is well known that there exists an integer ${\widehat{m}}$ such
that, for any $m \geq \widehat{m}$ and any closed subscheme
$X \subset \bold P ^R$ corresponding to a point $ [X] \in H_{f,R}$,
\begin{equation}
\label{hippo}
h^1(I_X(m),\bold P ^R) = 0
\end{equation}
\begin{equation}
\label{bear}
h^0({\cal O}_X(m), X)= f(m) \; .
\end{equation}
Therefore, for any $m \geq \widehat{m}$, there is a natural map:
$$i_m: H_{f,R} \rightarrow
{\bold P}(\bigwedge ^{f(m)} H^0( {\cal O}_{{\bold P}^R}(m), \bold P ^R)^*).$$
Where $i_m$ is defined for each $[X] \in H_{f,R}$ as follows:
by (\ref{hippo}), there is a natural surjection
$$ H^0({\cal O}_{\bold P ^R}(m), \bold P ^R) \rightarrow
H^0({\cal O}_X(m),X) $$
which yields, by (\ref{bear}), a surjection
\begin{equation}
\label{whale}
\bigwedge ^{f(m)} H^0({\cal O}_{\bold P ^R}(m), \bold P ^R) \rightarrow
\bigwedge ^{f(m)} H^0({\cal O}_X(m),X) \cong {\bold C}.
\end{equation}
The last surjection (\ref{whale}) is an element of
${\bold P}(\bigwedge ^{f(m)} H^0( {\cal O}_{{\bold P}^R}(m), {\bold P}^R)^*)$.
The map $i_m$ is now defined on sets. That $i_m$ is an algebraic morphism
of schemes can be seen by constructing (\ref{whale}) uniformly
over $ H_{f,R}$ and using the universal property of
${\bold P}(\bigwedge ^{f(m)} H^0({\cal O}_{{\bold P}^R}(m), {\bold P}^R)^*)$.
In fact, it can be shown there
exists an integer $\overline{m}$ such that for every
$m \geq \overline{m}$, $i_m$ is a closed immersion.
{}From the universal property of the Hilbert scheme, we obtain
a natural $SL_{R+1}$-action on $H_{f,R}$.
For each $m \geq \overline{m}$,
the closed immersion $i_m$ defines a linearization of the natural
$SL_{R+1}$-action on $H_{f,R}$.
Define the following locus $\overline{K}_g\subset H_{f,R}$:
$[X] \in \overline{K}_g$ if and only if $X$ is a nondegenerate, 10-canonical,
genus g, Deligne-Mumford stable curve in $\bold P^R$.
$\overline{K}_g$ is a quasi-projective, $SL_{R+1}$-invariant subset. In [G],
Gieseker shows a linearization $i_m$ can be chosen satisfying:
\begin{enumerate}
\item[(i)] The stable locus of the corresponding G.I.T. quotient contains
$\overline{K}_g$.
\item[(ii)] $\overline{K}_g$ is closed in the semistable locus.
\end{enumerate}
{}From (i), we see $\overline{K}_g/SL_{R+1}$ is a geometric quotient.
By (ii), $\overline{K}_g/SL_{R+1}$ is a projective variety. Since
$\overline{K}_g$ is a nonsingular variety ([G]), it follows that
$\overline{K}_g/SL_{R+1}$ is normal.
{}From the definition
of $\overline{K}_g$, the universal family over $H_{f,R}$
restricted to $\overline{K}_g$ is a
family of Deligne-Mumford stable curves. Therefore there exists a
natural map $\mu: \overline{K}_g \rightarrow \overline{M}_g$. Since $\mu$ is
$SL$-invariant, $\mu$ descends to a projective morphism
from the quotient $\overline{K}_g/SL_{R+1}$ to $\overline{M}_g$. Since $\mu$
is one to one and $\overline{M}_g$ is normal, $\mu$ is an
isomorphism. Note that since $\overline{M}_g$ is irreducible,
$\overline{K}_g$ is also irreducible.
\subsection{The Relative $n$-pointed Fulton-MacPherson Configuration
Space of the Universal Curve}
\label{gitrf}
Let $\pi: U_{H} \rightarrow H_{f,R}$ be the universal
family over the Hilbert scheme defined in section (\ref{gitg}) where
$\pi$ is a flat, projective morphism.
Let $\overline{K}_g \subset H_{f,R}$
be defined as above. Let $U_{\overline{K}_g}$ be the
restriction of $U_H$ to $\overline{K}_g$. Following the notation of
section (\ref{rfmd}), we define $U_{\overline {K}_g}[n]$ to be the relative Fulton-
MacPherson space of $n$-marked points on $U_{\overline {K}_g}$ over
${\overline {K}_g}$.
{}From section (\ref{rfmb}), we see the immersion $\zeta$:
\begin{equation*}
\begin{CD}
U_{\overline {K}_g} @>{\zeta}>> \bold P ^R \times H_{f,R} \\
@VV{\pi}V @VV{\rho}V \\
{\overline {K}_g} @>>> H_{f,R}
\end{CD}
\end{equation*}
yields another immersion $\zeta[n]$:
\begin{equation*}
\begin{CD}
U_{\overline{K}_g}[n] @>{\zeta[n]}>> \bold P ^R[n] \times H_{f,R} \\
@VV{\pi[n]}V @VV{\rho[n]}V \\
{\overline{K}_g} @>>> H_{f,R}
\end{CD}
\end{equation*}
There exists a natural $SL_{R+1}$-action on $ \bold P ^R[n]$
and therefore on $ \bold P ^R[n] \times H_{f,R}$. Since $U_{\overline {K}_g}$ is
invariant under the natural $SL_{R+1}$-action, we see $U_{\overline{K}_g}[n]$
is also $SL_{R+1}$-invariant.
Since $\pi$ is projective,
$U_{\overline{K}_g}[n] \subset \rho[n]^{-1}(\overline {K}_g)$ is
a closed subset. It follows from (i) and (ii) of section (\ref{gitg})
and Propositions (7.1.1) and (7.1.2) of [P] that there exist linearizations
of the natural $SL_{R+1}$-action on $ \bold P ^R[n] \times H_{f,R}$ satisfying:
\begin{enumerate}
\item[(i)] $U_{\overline{K}_g}[n]$ is contained in the stable locus of the
corresponding G.I.T. quotient.
\item[(ii)] $(\rho[n]^{-1}(\overline{K}_g))^{SS}$
is closed in the semistable locus.
\end{enumerate}
{}From (i), (ii), and the fact that $U_{\overline{K}_g}[n]$ is closed in
$\rho[n]^{-1}(\overline {K}_g)$,
we see that $U_{\overline {K}_g}[n]/SL_{R+1}$ is a geometric quotient
and a projective variety. Define:
$$M_{g,n}^{c} = U_{\overline {K}_g}[n]/SL_{R+1} .$$
Note there is a natural projective morphism
$$\rho : M_{g,n}^{c} \rightarrow {\overline {M}_{g}}$$
descending from the $SL_{R+1}$-invariant maps:
$$U_{\overline {K}_g}[n] \rightarrow {\overline {K}_g}
\rightarrow {\overline {M}_g}.$$
It follows easily that $M_{g,n}^{c}$ is a compactification
of $\gamma^{-1}(M_g)$. To see this first make the definition:
$${K_g} =\{[X] \in H_{f,R}|X \hbox{ is a nondegenerate, 10-canonical,
nonsingular, genus g curve} \}. $$
$U_{K_g}[n]$ is a dense open $SL_{R+1}$-invariant subset of
$U_{\overline{K}_g}[n]$.
Since the morphism $\pi :U_{K_g} \rightarrow K_g$ is smooth, we see from
section (\ref{rfmru}) that there exists a universal family of
Deligne-Mumford stable $n$-pointed genus $g$ curves over $ U_{K_g}[n]$.
This universal family yields a canonical morphism
$$\mu: U_{K_g}[n] \rightarrow \gamma^{-1}(M_g).$$
It is easily checked that $\mu$ is $SL_{R+1}$-invariant. Therefore,
$\mu$ descends to the open set, $\rho^{-1}(M_g)$, of $M_{g,n}^{c}$. One
sees $$\mu_d:\rho^{-1}(M_g) \rightarrow \gamma^{-1}(M_g)$$ is
a bijection by Proposition (\ref{pfiber})
and the fact that, for a smooth curve C,
$$ (C[n]/\hbox{automorphisms}) \cong \gamma^{-1}([C])\subset
\gamma^{-1}(M_g). $$
(See section (\ref{rfmu})).
Since $\rho: \rho^{-1}(M_g) \rightarrow M_g$ is projective,
$\gamma: \gamma^{-1}(M_{g}) \rightarrow M_g$ is separated,
and $\rho= \gamma \circ \mu_d$, we conclude $\mu_d$
is projective. A bijective projective morphism onto a normal
variety is an isomorphism. Since $\gamma^{-1}(M_g)$ is normal, $\mu_d$ is
an isomorphism.
\section{A Description Of $M_{g,n}^c$}
\label{disc}
\subsection{}
\label{dis1}
Let $\pi: U_{\overline{K}_g} \rightarrow \overline{K}_g$ be as above.
Following section (\ref{rfmfn}), we define $F\subset U_{\overline{K}_g}$
to be the locus where $\pi$ is smooth. $F_{\overline{K}_g}[n]
\subset U_{\overline{K}_g}[n]$ is an open $SL$-invariant subset.
The points of $F_{\overline{K}_g}[n]$ parameterized $n$-pointed stable
classes on the nonsingular locus of the fibers of $\pi$. There
exists a universal family over $F_{\overline{K}_g}[n]$ which defines an
$SL$-invariant morphism :
\begin{equation*}
\mu :F_{\overline{K}_g}[n] \rightarrow \overline{ M}_{g,n}^{s}.
\end{equation*}
Where $\overline{ M}_{g,n}^{s}$
parameterizes $n$-pointed, genus g,
Deligne-Mumford stable curves with marked points lying over
nonsingular points of the contracted stable model. Let
$$F_{\overline{K}_g}[n]/SL_{R+1} = (M_{g,n}^c)^{s}.$$
$SL$-invariance implies $\mu$ descends to:
\begin{equation}
\label{quail}
\mu_d: (M_{g,n}^c)^{s} \rightarrow \overline{ M}_{g,n}^{s}.
\end{equation}
{}From the arguments of section (\ref{rfmu}), we see
$\mu_d$ is bijective. From the valuative criterion,
it follows $\mu_d$ is proper.
As before, by normality, it follows that $\mu_d$ is
an isomorphism.
\subsection{Points Of $M_{g,n}^c$ Over A Singular Point}
\label{dissing}
{}From section (\ref{dis1}), it is clear only the behavior
of $U_{\overline {K}_g}[n]$ over a singular point of
$U_{\overline{K}_g}$ remains to be investigated. Since this is
a local question about the the smooth deformation
of a node, it suffices to investigate the family:
\begin{equation*}
\begin{CD}
G @>>> Spec(C[x,y]) \times Spec(C[t]) \\
@VV{\pi}V @VVV \\
Spec(C[t]) @= Spec(C[t])
\end{CD}
\end{equation*}
Where $G$ is defined by the equation $xy-t$.
In the Fulton-MacPherson configuration space
$Spec(C[x,y])[n]$, there is a closed subset $\cal {T}_n$
corresponding to the points lying over $(0,0)$.
In the notation of section (\ref{rfmb}),
$$ \cal{T}_n = \rho^{-1}(\underbrace{(0,0),(0,0), \ldots,(0,0)}_{n}).$$
Recall the notation of section (\ref{rfmd}).
Let $B=Spec(C[t])$, $B^*=Spec(C[t])-(0)$, and $G^*=\pi^{-1}(B^*)$.
We want to investigate the subset $\cal{W}_n \subset (\cal{T}_n,0)$ that
lies in the closure of $G_{B^*}^{*}[n]$ in
$Spec(C[x,y])[n]\times B$.
Suppose $\kappa$ is a family in $(G_{B^*}^{*N})_0$ where
all the marked points specialize to the node $\zeta$ of $G_0$.
After a base change, $t \rightarrow t^r$,
$\kappa$ can be defined by $n$ sections,
$(\kappa_1, \ldots, \kappa_n)$,
of $\pi$ in a neighborhood of $0 \in B$.
The equation of $G$ after base change is now
$G_r=xy-t^r$.
Let us take $r=2$. The blow-up
of $G_2$ at $\zeta$ is nonsingular and is
defined in an open set
by the equation $ab-1$ in $Spec(C[a,b]) \times Spec(C[t])$.
The
blow-down morphism is defined by the equations:
$$x=at$$
$$y=bt.$$
Now assume that the strict transforms of the
sections, $(\kappa_1, \ldots, \kappa_n)$, meet
the exceptional curve $(ab=1,t=0)$ in distinct points
$((a_1,a_1^{-1}), \ldots, (a_n,a_n^{-1}))$,
$\forall i$ $a_i\neq 0$. Then it is clear that the
$n$-pointed stable class in $\cal{T}_n$ that is the limit of $\kappa$
is the class in the tangent space of $C[x,y]$ at $(0,0)$
defined by the pairs of vectors:
$$((a_1,a_1^{-1}), \ldots, (a_n,a_n^{-1}))$$
in the basis $(\frac{\partial}{\partial x}, \frac{\partial}{\partial y})$.
We now define a map:
\begin{equation*}
\theta_n: (C^{*N})_0 \rightarrow \cal{T}_n
\end{equation*}
Where $\theta_n((a_1,\ldots,a_n))$ is the $n$-pointed
stable class defined by the tangent vectors
$((a_1,a_1^{-1}), \ldots, (a_n,a_n^{-1}))$.
The preceding paragraph shows that $Image(\theta_n)\subset \cal{W}_n$.
In fact, it is not hard to see that $Image(\theta_n)$
is dense in a component of $\cal{W}_n$.
For $n=2$, $\cal{W}_2= \cal{T}_2$ where $\cal{T}_2$ is just
the $\bold P^1$ of normal directions.
Suppose $n \geq 3$. Let $\hat{a}=(a_1,\ldots,a_n)$ and
$\hat{b}=(b_1,\ldots,b_n)$ be distinct points of $(C^{*N})_0$.
Then, $\theta_n(\hat{a})=\theta_n(\hat{b})$ if and only if
there exists a tangent vector $(v_1,v_2)$
and an element $\lambda \in C^*$ such
that :
$$ \forall i, \;\;\; \lambda \cdot a_i+v_1=b_i \hbox{ and }
\lambda \cdot a_i^{-1}+v_2=b_i^{-1} .$$
These equations imply
\begin{equation}
\label{elk}
\forall i,j,\; \;\;\; \lambda \cdot (a_i-a_j)=(b_i-b_j)
\end{equation}
\begin{equation}
\label{rhino}
\forall i,j,\; \;\;\; \lambda \cdot(a_i^{-1}-a_j^{-1})=(b_i^{-1}-b_j^{-1})
\end{equation}
Dividing (\ref{elk}) by (\ref{rhino}) yields $a_i\cdot a_j=b_i\cdot b_j$.
For $n\geq 3$, we easily obtain $\hat{a}=\pm\hat{b}$.
Therefore, a component of
$\cal{W}_n$ can be viewed as a compactification of $(C^{*N})_0/(\pm)$.
We note that the dimension of $\cal{W}_n$ is $n$ for
$n\geq 3$.
\section{Comparison with $\overline {M}_{g,n}$ for $n=1,2$}
\subsection{$n=1$}
{}From the definitions, $M_{g,1}^c$ equals $U_{\overline {K}_g}/SL_{R+1}$.
$\pi^*(U_{\overline{K}_g})$ is a family of 1-pointed Deligne-Mumford
genus $g$ curves over $ U_{\overline{K}_g}$ via the natural diagonal section.
This tautological family yields an $SL$-invariant morphism:
$$\mu:U_{\overline{K}_g} \rightarrow \overline{M}_{g,1} $$
that descends to
$$\mu_d: M_{g,1}^c \rightarrow \overline{M}_{g,1}.$$
Since $\mu_d$ is proper bijective and $\overline{M}_{g,1}$ is normal,
$\mu_d$ is an isomorphism.
\subsection{$n=2$}
Consider the family:
\begin{equation}
U_{\overline {K}_g}^2=U_{\overline {K}_g} \times_{\overline {K}_g} U_{\overline {K}_g}\;.
\label{mongoose}
\end{equation}
The singular locus of $U_{\overline {K}_g}^2$, $S$, is nonsingular
of pure codimension $3$ and $SL_{R+1}$-invariant. The singular points are pairs
$(\zeta,\zeta)$ where $\zeta \in U_{\overline {K}_g}$ is a node
of a fiber. Moreover, the singularities of $ U_{\overline {K}_g}^2$
are \'etale-locally ordinary threefold double point
singularities. That is, the
singularities are of the form
\begin{equation}
\label{possum}
W\times Spec(C[a,b,c,d]/(ab-cd)) \subset W\times Spec(C[a,b,c,d])
\end{equation}
Where $W$ is nonsingular. These assertions about the
singular locus follow from the deformation theory of a Deligne-Mumford
stable curve and [G].
There are three standard resolutions of
the ordinary double point singularity $Spec(C[a,b,c,d]/(ab-cd))$:
\begin{enumerate}
\item The blow-up along $(a,b,c,d)$.
\item For any $\lambda \in C$, the blow-up along
$(a-\lambda \cdot c, \lambda \cdot b-d)$.
\item For any $\lambda \in C$, the blow-up along
$(a-\lambda \cdot d, \lambda \cdot b-c)$.
\end{enumerate}
Methods (2) and (3) yield the distinct small resolutions.
The local description (\ref{possum}) implies that the blow-up
of $U_{\overline {K}_g}^2$ along $S$ is nonsingular with
an exceptional divisor $E$ that is a $\bold P^1\times \bold P^1$-
bundle over $S$. Using the techniques of
section (\ref{gitrf}), it can be shown that the
natural $SL_{R+1}$-action on the blow-up
$Bl_{(S)}(U_{\overline {K}_g}^2)$ can be linearized so
that all the points in question are stable and the
quotient is projective.
The diagonal embedding
$$D:U_{\overline {K}_g} \hookrightarrow U_{\overline {K}_g}^2$$
is divisorial except along $S$ where it of the form
of (2) and (3) in the local description (\ref{possum}).
By definition,
$$M_{g,2}^c= Bl_{(D)}(U_{\overline {K}_g}^2)\;\;/SL_{R+1}.$$
There is a natural blow-down map:
$$\rho: Bl_{(S)}(U_{\overline {K}_g}^2)\rightarrow
Bl_{(D)}(U_{\overline {K}_g}^2).$$
Another $SL_{R+1}$-invariant small resolution of
$U_{\overline {K}_g}^2$ can be obtain by blowing-down uniformly along
the opposite ruling of $E$ blown-down by $\rho$. Let $Y$ denote
this other small resolution and let
$$\overline{\rho}: Bl_{(S)}(U_{\overline {K}_g}^2) \rightarrow Y$$
be the blow-down.
Linearizations can be chosen so that
$$Y/SL_{R+1} \cong \overline{M}_{g,2}.$$
There are birational morphisms
\begin{equation}
\label{birad}
M_{g,2}^c \ \leftarrow \ \ Bl_{(S)}(U_{\overline {K}_g}^2)\;\;/SL_{R+1}
\ \ \rightarrow\
\overline{M}_{g,2}.
\end{equation}
Consider the open loci of $M^{c}_{g,2}$ and $\overline{M}_{g,2}$
where the underlying curve has no (nontrivial) automorphism.
On the automorphism free loci the birational modification
(\ref{birad}) is easy to describe. Let $F_1\subset M_{g,2}^c$
be the locus of
of $2$-pointed stable classes that lie over a
node in a Deligne-Mumford stable curve of genus g. Similarly, let
$F_2\subset \overline{M}_{g,2}$ be the locus of $2$-pointed, genus g,
Deligne-Mumford stable curves such that the
marked points are coincident at a node in
the stable contraction. On the
automorphism free loci,
$M_{g,2}^c$ and $\overline{M}_{g,2}$ are the distinct
small resolution of the fiber product of the universal
curve with itself. Hence, on the automorphism free loci,
the blow-up of $M_{g,2}^2$ along $F_1$
is isomorphic to the blow-up of $\overline{M}_{g,2}$ along
$F_2$.
The modification (\ref{birad}) obtained by this
isomorphism.
|
1995-05-08T06:20:47 | 9505 | alg-geom/9505009 | en | https://arxiv.org/abs/alg-geom/9505009 | [
"alg-geom",
"math.AG"
] | alg-geom/9505009 | Ron Donagi | Ron Donagi | Spectral Covers | null | null | null | null | null | This is a survey of various results about spectral covers and their
relationship to Higgs bundles. To a G-principal Higgs bundle on a variety S
corresponds a cameral cover \widetilde{S} of S (a W-Galois cover, where W is
the Weyl group of G) together with a sheaf on \widetilde{S} which in simple
cases is a line bundle, and is W-equivariant up to certain twists and shifts.
Various other types of spectral covers, depending on the choice of a
representation or weight of G, arise as associated objects of \widetilde{S}. We
focus on the decomposition of the Picards of these spectral covers into Pryms
(this includes various well-known Prym identities as special cases) and on the
interpretation, in the spirit of Hitchin's abelianization program, of a
distinguished Prym component as parameter space for higgs bundles.
| [
{
"version": "v1",
"created": "Sun, 7 May 1995 17:56:05 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Donagi",
"Ron",
""
]
] | alg-geom | \section{Introduction} \label{intro}\
\indent Spectral curves arose historically out of the study of differential
equations of Lax type. Following Hitchin's work \cite{H1}, they have acquired a
central role in understanding the moduli spaces of vector bundles and Higgs
bundles on a curve. Simpson's work \cite{S} suggests a similar role for
spectral covers $\widetilde{S}$ of higher dimensional varieties $S$ in moduli
questions for bundles on $S$.
The purpose of these notes is to combine and review various results about
spectral covers, focusing on the decomposition of their Picards (and the
resulting Prym identities) and the interpretation of a distinguished Prym
component as parameter space for Higgs bundles. Much of this is modeled on
Hitchin's system, which we recall in section \ref{Hitchin}, and on several
other systems based on moduli of Higgs bundles, or vector bundles with twisted
endomorphisms,
on curves.
By peeling off several layers of data which are not essential for our purpose,
we arrive at the notions of an {\em abstract principal Higgs bundle} and a {\em
cameral} (roughly, a principal spectral) cover. Following \cite{D3}, this
leads to the statement of the main result, theorem \ref{main}, as an
equivalence between these somewhat abstract `Higgs' and `spectral' data, valid
over an arbitrary complex variety and for a reductive Lie group $G$. Several
more familiar forms of the equivalence can then be derived in special cases by
adding choices of representation, value bundle and twisted endomorphism. This
endomorphism is required to be {\em regular}, but not semesimple. Thus the
theory works well even for Higgs bundles which are everywhere nilpotent. After
touching briefly on the symplectic side of the story In section
\ref{symplectic}, we discuss some of the issues involved in removing the
regularity assumption, as well as some applications and open problems, in sect!
ion \ref{apps}.
This survey is based on talks at the Vector Bundle Workshop at UCLA (October
92) and the Orsay meeting (July 92), and earlier talks at Penn, UCLA and MSRI.
I would like to express my thanks to Rob Lazarsfeld and Arnaud Beauville for
the invitations, and to them and Ching Li Chai, Phillip Griffiths, Nigel
Hitchin, Vasil Kanev, Ludmil Katzarkov, Eyal Markman, Tony Pantev, Emma
Previato and Ed Witten for stimulating and helpful conversations.
We work throughout over $\bf C$ . The total space of a vector bundle (=locally
free sheaf) $K$ is denoted $\Bbb{K}$. Some more notation:
\begin{tabbing}
bundles of algebras: \= \kill
Groups: \>$G $ \=$B$ \=$T$
\=$N $ \=$C$\\
algebras: \>$\frak g$ \>$\frak b$ \>$\frak t$
\>$\frak n $ \>$\frak c $\\
Principal bundles: \>$\cal G$ \>$\cal B$ \>$\cal T$ \>$\cal N $
\>$\cal C$\\
bundles of algebras: \>\bdl{g} \>\bdl{b} \>\bdl{t} \>\bdl{n}
\>\bdl{c}
\end{tabbing}
\section{Hitchin's system}\
\indent Let ${\cal M} := {\cal M} _{C}(n,d)$ be the moduli space of stable
vector bundles of rank n and degree d on a smooth projective complex curve $C$.
It is smooth and quasiprojective of dimension
\begin{equation}
\tilde{g} := n^{2} (g-1)+1.
\end{equation}
\noindent Its cotangent space at a point $ E \in {\cal M} $ is given by
\begin{equation}
T^*_{E}{\cal M} := H^{0} ( \End{E} \otimes \omega_{C} )
\end{equation}
\noindent where $ \omega_{C} $ is the canonical bundle of $C$. Our starting
point is:
\begin{thm}[Hitchin\cite{H1}]
\label{Hitchin}
The cotangent bundle $T^*{\cal M}$ is an algebraically completely integrable
Hamiltonian system.
\end{thm}
{\em Complete integrability} means that there is a map
\[ h:T^*{\cal M}\longrightarrow B \]
to a $\tilde{g}$-dimensional vector space $B$ which is Lagrangian with
respect to the natural symplectic structure on $ T^*{\cal M}$ (i.e. the
tangent spaces to a general fiber $h^{-1}(a)$ over $a \in B$ are maximal
isotropic subspaces with respect to the symplectic form). In this situation one
gets, by contraction with the symplectic form, a trivialization of the tangent
bundle:
\begin{equation}
T_{h^{-1}(a)} \stackrel{ \approx}{ \longrightarrow} {\cal O}_{h^{-1}(a)}
\otimes T^{*}_{a} B. \end{equation}
In particular, this produces a family of ({\em `Hamiltonian'}) vector fields on
$h^{-1}(a)$ which is parametrized by $T^{*}_{a} B$ , and the flows generated
by these on $h^{-1}(a)$ all commute.
{\em Algebraic complete integrability} means additionally that the fibers
$h^{-1}(a)$ are Zariski open subsets of abelian varieties on which the
Hamiltonian flows are linear, i.e. the vector fields are constant.
We describe the idea of the proof in a slightly more general setting, following
\cite{BNR}. Let $K$ be a line bundle on $C$, with total space $ \Bbb K$ . (In
Hitchin's situation, $K$ is $ \omega_{C} $ and $ \Bbb K$ is $T^*C$ .) A {\em
$K$-valued Higgs bundle} is a pair
\[ (E \quad,\quad \phi :E \longrightarrow E \otimes K) \]
consisting of a vector bundle $E$ on $C$ and a $K$-valued endomorphism. One
imposes an appropriate stability condition, and obtains a good moduli space
${\cal M}_K$ parametrizing equivalence classes of $K$-valued semistable Higgs
bundles, with an open subset ${\cal M}_K^s$ parametrizing isomorphism classes
of stable ones, cf. \cite{S}.
Let $B:=B_K$ be the vector space parametrizing polynomial maps
\[ p_a : \Bbb K \longrightarrow \Bbb K^{n} \]
of the form
\[ p_{a} (x) = x^{n} + a_{1} x^{n-1} + \cdots + a_{n}, \qquad\qquad a_{i}
\in H^{0}(K ^{\otimes i}). \]
in other words,
\begin{equation}
B := \bigoplus_{i=1}^{n} H^{0}(K ^{\otimes i}).
\end{equation}
\noindent The assignment
\begin{equation}
(E, \phi) \longmapsto char(\phi):= \det{(xI-\phi})
\end{equation}
\noindent gives a morphism
\begin{equation}
h_K:{\cal M}_{K}\longrightarrow B_{K}.
\end{equation}
In Hitchin's case, the desired map $h$ is the restriction of $h_{\omega _{C}
}$ to $T^{*}{\cal M}$, which is an open subset of ${\cal M}_{\omega_C}^s$.
Note that in this case $\dim{B}$ is, in Hitchin's words, `somewhat
miraculously' equal to $\tilde{g} = \dim{{\cal M}}$.
The {\em spectral curve} $\widetilde{C}:= \widetilde{C}_{a}$ defined by $a \in
B_{K}$ is the inverse image in $\Bbb K$ of the $0$-section of $\Bbb K
^{\otimes n}$ under $ p_a : \Bbb K \longrightarrow \Bbb K^{n} $. It is
finite over $C$ of degree $n$. The general fiber of $h_K$ is given by:
\begin{prop} \cite{BNR} \label{prop:BNR}
For $a \in B$ with {\em integral} spectral curve $\widetilde{C}_{a}$, there is
a natural equivalence between isomorphism classes of:
\begin{enumerate}
\item Rank-$1$, torsion-free sheaves on $\widetilde{C}_{a}$.
\item Pairs $(E \ , \ \phi:E \rightarrow E \otimes K)$ with
$char(\phi)=a$.
\end{enumerate}
\end{prop}
When $\widetilde{C}_{a}$ is non-singular, the fiber is thus
$Jac(\widetilde{C}_{a})$, an abelian variety.
In $T^*\cal M$ the fiber is an open subset of this abelian variety. One checks
that the missing part has codimension $ \geq 2$, so the symplectic form, which
is exact, must restrict to $0$ on the fibers, completing the proof.
\section {Some related systems}
\noindent \underline{\bf Polynomial matrices} \nopagebreak
\noindent One of the earliest appearances of an ACIHS (algebraically
completely integrable Hamiltonian system) was in Jacobi's work on the geodesic
flow on an ellipsoid (or more generally, on a nonsingular quadric in ${\bf
R}^k$). Jacobi discovered that this differential equation, taking place on the
tangent (=cotangent!) bundle of the ellipsoid, can be integrated explicitly in
terms of hyperelliptic theta functions. In our language, the total space of the
flow is an ACIHS, fibered by (Zariski open subsets of) hyperelliptic Jacobians.
We are essentially in the special case of
Proposition \ref{prop:BNR} where
\[ C={\bf P}^1, \quad n=2, \quad K= {\cal O}_{{\bf P}^1} (k).
\]
A variant of this system appeared in Mumford's solution \cite{Mu} of the
Schottky problem for hyperelliptic curves.
The extension to all values of n is studied in \cite{B} and, somewhat more
analytically, in \cite{AHP} and \cite{AHH}. Beauville considers, for fixed $n$
and $k$, the space $B$ of polynomials:
\begin{equation}
p=y^n + a_1(x)y^{n-1}+\cdots +a_n(x), \quad, \deg{(a_i)}\leq ki
\end{equation}
\noindent
in variables $x$ and $y$. Each $p$ determines an $n$-sheeted branched cover
$$\widetilde{C}_p \rightarrow {\bf P}^1.$$
The total space is the space of polynomial matrices
\begin{equation}
M := H^0 ({\bf P}^1 , \End{{\cal O}^{\oplus n}} \otimes {\cal O}(d) ),
\end{equation}
\noindent
the map $h:M \rightarrow B$ is the characteristic polynomial,
and $M_p$ is the fiber over a given $p \in B$.
The result is that for smooth spectral curves $\widetilde{C}_p$,
$ {\bf P}GL(n)$ acts freely and properly on $M_p$;
the quotient is isomorphic to
$J(\widetilde{C}_p) \smallsetminus \Theta. $
(In order to obtain the entire $J(\widetilde{C}_p) ,$ one must allow all pairs
$(E,\phi)$ with $E$ of given degree, say $0$. Among those, the ones with
$E\approx {{\cal O}_{{\bf P}^1}}^{\oplus n}$
correspond to the open set
$J(\widetilde{C}_p) \smallsetminus \Theta. $ )
This system is an ACIHS, in a slightly weaker sense than before: instead of a
symplectic structure, it has a {\em Poisson structure}, i.e. a section $\beta$
of $\wedge^2 T$, such that the $\bf C$-linear sheaf map given by contraction
with $\beta$
$$\begin{array}{ccc}
{\cal O} & \rightarrow & {\cal T} \\
f & \mapsto & df \rfloor \beta
\end{array}$$
sends the Poisson bracket of functions to the bracket of vector fields. Any
Poisson manifold is naturally foliated, with (locally analytic) symplectic
leaves. For a Poisson ACIHS, we want each leaf to inherit a (symplectic)
ACIHS, so the symplectic foliation should be pulled back via $h$ from a
foliation of the base $B$.
The result of \cite{BNR} suggests that analogous systems should exist when
${\bf P}^1$ is replaced by an arbitrary base curve $C$. The main point is to
construct the Poisson structure. This was achieved by Bottacin \cite{Bn} and
Markman \cite{M1}, cf. section \ref{symplectic}.
In the case of the polynomial matrices though, everything (the commuting vector
fields, the Poisson structure, etc.) can be written very explicitly. What makes
these explicit results possible is that every vector bundle over ${\bf P}^1$
splits. This of course fails in genus $>1$, but for elliptic curves the moduli
space of vector bundles is still completely understood, so here too the system
can be described explicitly:
For simplicity, consider vector bundles with fixed determinant. When the
degree is $0$, the moduli space is a projective space ${\bf P}^{n-1}$ (or more
canonically, the fiber over $0$ of the Abel-Jacobi map
$$C^{[n]} \longrightarrow J(C) = C.$$
The ACIHS which arises is essentially the Treibich-Verdier theory \cite{TV} of
elliptic solitons. When, on the other hand, the degree is $1$ (or more
generally, relatively prime to $n$), the moduli space is a single point; the
corresponding system was studied explicitly in \cite{RS}. \\
\noindent \underline{\bf Reductive groups} \nopagebreak
\noindent In another direction, one can replace the vector bundles by
principal $G$-bundles ${\cal G}$ for any reductive group $G$. Again, there is
a moduli space ${\cal M}_{G,K}$ parametrizing equivalence classes of semistable
$K$-valued $G$-Higgs bundles, i.e. pairs
$({\cal G}, \phi)$ with $\phi \in K \otimes \bdl{ad}(\cal G)$. The Hitchin map
goes to $$B:=\oplus_{i} H^0(K^{\otimes d_i}),$$ where the $d_i$ are the degrees
of the $f_i$, a basis for the $G$-invariant polynomials on the Lie algebra
$\frak g$:
\[ h: ({\cal G}, \phi) \longrightarrow (f_i (\phi))_{i}.
\]
When $K=\omega_C$, Hitchin showed \cite{H1} that one still gets a completely
integrable system, and that it is algebraically completely integrable for the
classical groups $GL(n), SL(n), SP(n), SO(n).$ The generic fibers are in each
case (not quite canonically; one must choose various square roots! cf.
sections \ref{reg.ss} and \ref{reg}) isomorphic to abelian varieties given in
terms of the spectral curves $\widetilde{C}$:
\begin{center}
\begin{equation}
\begin{array}{cl} \label{Pryms for groups}
GL(n)& \widetilde{C}
\mbox{ has degree n over C, the AV is Jac(}
\widetilde{C}). \\
SL(n)& \widetilde{C}
\mbox{ has degree n over C, the AV is Prym(}
\widetilde{C} / C). \\
SP(n)& \widetilde{C}
\mbox{ has degree 2n over C and an involution }
x \mapsto -x. \\
& \mbox{ The map factors through the quotient }
\overline{C}. \nonumber \\
& \mbox{ The AV is }
Prym( \widetilde{C} / \overline{C}). \nonumber \\
SO(n)& \widetilde{C} \mbox{ has degree n and an involution , with: } \\
& \bullet \mbox{ a fixed component, when n is odd.} \\
& \bullet \mbox{ some fixed double points, when n is even.} \\
& \mbox{ One must desingularize }
\widetilde{C}
\mbox{ and the quotient }
\overline{C}, \\
& \mbox{and ends up with the Prym of the} \\
& \mbox{desingularized double cover.} \
\end{array}
\end{equation}
\end{center}
The algebraic complete integrability was verified in \cite{KP1} for the
exceptional group $G_2$.
A sketch of the argument for any reductive $G$ is in \cite{BK}, and a complete
proof was given in \cite{F}. We will outline a proof in section
\ref{abelianization} below. \\
\noindent \underline{\bf Higher dimensions} \nopagebreak
\noindent Finally, a sweeping extension of the notion of Higgs bundle is
suggested by the work of Simpson \cite{S}. To him, a Higgs bundle on a
projective variety S is a vector bundle (or principal $G$-bundle \ldots) $E$
with a {\em symmetric}, $\Omega^1_S$-valued endomorphism
\[ \phi : E \longrightarrow E \otimes \Omega^1_S.
\]
Here {\em symmetric} means the vanishing of:
\[ \phi\wedge\phi : E \longrightarrow E \otimes \Omega^2_S,
\]
a condition which is obviously vacuous on curves. He constructs a moduli space
for such Higgs bundles (satisfying appropriate stability conditions), and
establishes diffeomorphisms to corresponding moduli spaces of representations
of $\pi_1(S)$ and of connections.
\section {Decomposition of spectral Picards}
\subsection{The question}\
\indent Let $({\cal G},\phi)$ be a $K$-valued principal Higgs bundle on a
complex variety $S$. Each representation
\[ \rho : G \longrightarrow Aut(V)
\]
determines an associated $K$-valued Higgs bundle
\[ ( {\cal V} := {\cal G} \times^{G} V, \qquad{\rho}(\phi)\ ),
\]
which in turn determines a spectral cover $\widetilde{S}_V \longrightarrow S$.
The question, raised first in \cite{AvM} when $S={\bf P}^1$, is to relate the
Picard varieties of the
$\widetilde{S}_V$ as $V$ varies, and in particular to find pieces common to all
of them. For Adler and van Moerbeke, the motivation was that many evolution
DEs (of Lax type) can be {\em linearized} on the Jacobians of spectral curves.
This means that the "Liouville tori", which live naturally in the complexified
domain of the DE (and hence are independent of the representation $V$) are
mapped isogenously to their image in $\mbox{Pic}(\widetilde{S}_V)$ for each
nontrivial $V$ ; so one should be able to locate these tori among the pieces
which occur in an isogeny decomposition of each of the
$\mbox{Pic}(\widetilde{S}_V)$. There are many specific examples where a pair of
abelian varieties constructed from related covers of curves are known to be
isomorphic or isogenous, and some of these lead to important identities among
theta functions.
\begin{eg}
\begin{em}
Take $G=SL(4)$ . The standard representation $V$ gives a branched cover
$\widetilde{S}_V \longrightarrow S$ of degree 4. On the other hand, the
6-dimensional representation $\wedge ^2 V$ (=the standard representation of the
isogenous group $SO(6)$) gives a cover
$ \stackrel{\approx}{S} \longrightarrow S$ of degree 6, which factors through
an involution:
\[ \stackrel{\approx}{S} \longrightarrow \overline{S} \longrightarrow S.
\]
One has the isogeny decompositions:
\[ Pic \, (\widetilde{S}) \sim Prym(\widetilde{S} / S) \oplus Pic \,(S)
\]
\[ Pic \,(\stackrel{\approx}{S}) \sim
Prym(\stackrel{\approx}{S} / \overline{S}) \oplus
Prym(\overline{S} / S) \oplus Pic \,(S).
\]
It turns out that
\[ Prym(\widetilde{S} / S) \sim Prym(\stackrel{\approx}{S} /
\overline{S}) .
\]
For $S={\bf P}^1$, this is Recillas' {\em trigonal construction} \cite{R}. It
says that every Jacobian of a trigonal curve is the Prym of a double cover of a
tetragonal curve, and vice versa.
\end{em}
\end{eg}
\begin{eg}
\begin{em}
Take $G=SO(8)$ with its standard 8-dimensional representation $V$. The
spectral cover has degree 8 and factors through an involution,
$ \stackrel{\approx}{S} \longrightarrow \overline{S} \longrightarrow S.$
The two half-spin representations $V_1, V_2$ yield similar covers
\[ \stackrel{\approx}{S} _i \longrightarrow \overline{S} _i \longrightarrow S,
\qquad i=1,2.
\]
The {\em tetragonal construction} \cite{D1} says that the three Pryms of the
double covers are isomorphic. (These examples, as well as Pantazis' {\em
bigonal construction} and constructions based on some exceptional groups, are
discussed in the context of spectral covers in \cite{K} and \cite{D2}.)
\end{em}
\end{eg}
It turns out in general that there is indeed a distinguished, Prym-like isogeny
component common to all the spectral Picards, on which the solutions to
Lax-type DEs evolve linearly. This was noticed in some cases already in
\cite{AvM}, and was greatly extended by Kanev's construction of Prym-Tyurin
varieties. (He still needs $S$ to be ${\bf P}^1$ and the spectral cover to have
generic ramification; some of his results apply only to {\em minuscule
representations}.)
Various parts of the general story have been worked out recently by a number of
authors, based on either of two approaches: one, pursued in \cite{D2,Me,MS},
is to decompose everything according to the action of the Weyl group $W$ and to
look for common pieces; the other, used in \cite{BK,D3,F,Sc}, relies on the
correspondence of spectral data and Higgs bundles . The group-theoretic
approach is described in the rest of this section. We take up the second
method, known as {\em abelianization}, in section~\ref{abelianization}.
\subsection{Decomposition of spectral covers} \label{decomp
covers}\
\indent The decomposition of spectral Picards arises from three sources.
First, the spectral cover for a sum of representations is the union of the
individual covers $\widetilde{S}_V$. Next, the cover $\widetilde{S}_V$ for an
irreducible representation is still the union of subcovers
$\widetilde{S}_{\lambda}$ indexed by weight orbits. And finally, the Picard
of $\widetilde{S}_{\lambda}$ decomposes into Pryms.
We start with a few observations about the dependence of the covers themselves
on the representation. The decomposition of the Picards is taken up in the
next subsection. \\
\noindent \underline{\bf Spectral covers} \nopagebreak
\noindent There is an {\em infinite} collection (of irreducible
representations $V := V_{\mu}$, hence) of spectral covers $\widetilde{S}_V$,
which can be parametrized by their highest weights $\mu$ in the dominant Weyl
chamber $\overline{C}$ , or equivalently by the $W$-orbit of extremal weights,
in $\Lambda / W$. Here $T$ is a maximal torus in $G$, $\Lambda := Hom(T, {\bf
C}^*)$ is the {\em weight lattice } (also called {\em character lattice }) for
$G$, and $W$ is the Weyl group. Each of these $\widetilde{S}_V$ decomposes as
the union of its subcovers $\widetilde{S}_{\lambda}$, parametrizing eigenvalues
in a given $W$-orbit $W{\lambda}$ . ($\lambda$ runs over the weight-orbits in
$V_{\mu}$.) \\
\noindent \underline{\bf Parabolic covers} \nopagebreak
\noindent There is a {\em finite} collection of covers $\widetilde{S}_P$,
parametrized by the conjugacy classes in $G$ of parabolic subgroups (or
equivalently by arbitrary dimensional faces $F_P$ of the chamber
$\overline{C}$) such that (for general $S$) each eigenvalue cover
$\widetilde{S}_{\lambda}$ is birational to some parabolic cover
$\widetilde{S}_{P}$, the one whose open face $F_P$ contains ${\lambda}$. \\
\noindent \underline{\bf The cameral cover} \nopagebreak
\noindent There is a $W$-Galois cover $\widetilde{S} \longrightarrow S$ such
that each
$\widetilde{S}_{P}$ is isomorphic to $\widetilde{S} / W_P$, where $W_P$ is
the Weyl subgroup of $W$ which stabilizes $F_P$. We call $\widetilde{S}$ the
{\em cameral cover} , since, at least generically, it parametrizes the
chambers determined by $\phi$ (in the duals of the Cartans), or equivalently
the Borel subalgebras containing $\phi$. This is constructed as follows: There
is a morphism
${\frak g}\longrightarrow {\frak t}/W$ sending $g \in {\frak g}$ to the
conjugacy class of its semisimple part $g_{ss}$. (More precisely, this is
$Spec$ of the composed ring homomorphism
${\bf C} [ {\frak t} ] ^{W}
{ \stackrel{\simeq}{\leftarrow}}
{\bf C}[{\frak g}]^{G} \label{t/W}
\hookrightarrow
{\bf C}[{\frak g}]$.)
Taking fiber product with the quotient map ${\frak t}\longrightarrow {\frak
t}/W$, we get the cameral cover ${\tilde{\frak g}}$ of ${\frak g}$. The
cameral cover $\widetilde{S} \longrightarrow S$ of a $K$-valued principal Higgs
bundle on $S$ is glued from covers of open subsets in $S$ (on which $K$ and
$\cal G$ are trivialized) which in turn are pullbacks by $\phi$ of
${\tilde{\frak g}} \longrightarrow {\frak g} $.
\subsection{Decomposition of spectral Picards}\
\indent The decomposition of the Picard varieties of spectral covers can be
described as follows:\\
\noindent \underline{\bf The cameral Picard} \nopagebreak
\noindent From each isomorphism class of irreducible $W$-representations,
choose an integral representative $\Lambda _i$. (This can always be done, for
Weyl groups.) The group ring
${\bf Z} [W]$ which acts on $Pic(\widetilde{S}) $ has an isogeny
decomposition:
\begin{equation}\label{regular rep}
{\bf Z} [W] \sim \oplus _i \Lambda _i \otimes_{\bf Z} \Lambda _i^{*},
\end{equation}
\noindent
which is just the decomposition of the regular representation. There is a
corresponding isotypic decomposition:
\begin{equation}\label{cameral Pic decomposition}
Pic(\widetilde{S}) \sim \oplus _i \Lambda _i \otimes_{\bf Z} Prym_{\Lambda
_i}(\widetilde{S}),
\end{equation}
\noindent
where
\begin{equation}\label{def of Prym_lambda}
Prym_{\Lambda _i}(\widetilde{S} ):= Hom_W (\Lambda _i , Pic(\widetilde{S})).
\end{equation}\\
\noindent \underline{\bf Parabolic Picards} \nopagebreak
\noindent There are at least three reasonable ways of obtaining an isogeny
decomposition of $Pic(\widetilde{S}_P) $, for a parabolic subgroup $P \subset
G$:
\begin{itemize}
\item The `Hecke' ring $Corr_P$ of correspondences on $\widetilde{S}_P$ over
$S$ acts on $Pic(\widetilde{S}_P) $, so every irreducible integral
representation $M$ of $Corr_P$ determines a generalized Prym
$$ Hom_{Corr_P} (M, Pic(\widetilde{S}_P)), $$
and we obtain an isotypic decomposition of $Pic(\widetilde{S}_P)$ as before.
\item $Pic(\widetilde{S}_P)$ maps, with torsion kernel, to
$Pic(\widetilde{S})$, so we obtain a decomposition of the former by
intersecting its image with the isotypic components
$\Lambda _i \otimes_{\bf Z} Prym_{\Lambda _i}(\widetilde{S})$ of the latter.
\item Since $\widetilde{S}_P$ is the cover of $S$ {\em associated} to the
$W$-cover $\widetilde{S}$ via the permutation representation ${\bf Z} [W_P
\backslash W]$ of $W$, we get an isogeny decomposition of
$Pic(\widetilde{S}_P)$ indexed by the irreducible representations in
${\bf Z} [W_P \backslash W]$.
\end{itemize}
It turns out (\cite{D2},section 6) that all three decompositions agree
and can be given explicitly as
\begin{equation}
\label{multiplicity spaces}
\oplus _i M _i \otimes Prym_{\Lambda _i}(\widetilde{S}) \subset
\oplus _i \Lambda _i \otimes Prym_{\Lambda _i}(\widetilde{S}),\qquad
M_i := (\Lambda_i)^{W_P}.
\end{equation}
\noindent \underline{\bf Spectral Picards} \nopagebreak
\noindent To obtain the decomposition of the Picards of the original covers
$\widetilde{S}_V$ or
$\widetilde{S}_{\lambda}$, we need, in addition to the decomposition of
$Pic(\widetilde{S}_P)$, some information on the singularities. These can arise
from two separate sources:
\begin{description}
\item[Accidental singularities of the $\widetilde{S}_{\lambda}$. ]
For a sufficiently general Higgs bundle, and for a weight $\lambda$ in the
interior of the face $F_P$ of the Weyl chamber $\overline{C}$, the natural
map:
$$ i_{\lambda}: \widetilde{S}_P\longrightarrow \widetilde{S}_{\lambda} $$
is birational. For the {\em standard} representations of the classical groups
of types
$A_n, B_n$ or $C_n$, this {\em is} an isomorphism. But for general ${\lambda}$
it is {\em not}: In order for $i_{\lambda}$ to be an isomorphism, ${\lambda}$
must be a multiple of a fundamental weight, cf. \cite{D2}, lemma 4.2. In fact,
the list of fundamental weights for which this happens is quite short; for the
classical groups we have only: $\omega_1$ for $A_n, B_n$ and $C_n$,
$\omega_n$ (the dual representation) for $A_n$, and $\omega_2$ for $B_2$. Note
that for $D_n$ the list is {\em empty}. In particular, the covers produced by
the standard representation of $SO(2n)$ are singular; this fact, noticed by
Hitchin In \cite{H1}, explains the need for desingularization in his
result~(\ref{Pryms for groups}).
\item[Gluing the $\widetilde{S}_{V}$. ]
In addition to the singularities of each $i_{\lambda}$, there are the
singularities created by the gluing map $\amalg_{\lambda}
\widetilde{S}_{\lambda} \longrightarrow \widetilde{S}_V$. This makes explicit
formulas somewhat simpler in the case, studied by Kanev \cite{K}, of {\em
minuscule} representations, i.e. representations whose weights form a single
$W$-orbit. These singularities account, for instance, for the
desingularization required in the $SO(2n+1)$ case in
(\ref{Pryms for groups}).
\end{description}
\subsection{The distinguished Prym} \label{distinguished}\
\indent Combining much of the above, the Adler--van Moerbeke problem of finding
a component common to the $Pic(\widetilde{S}_V)$ for all non-trivial $V$
translates into: \\
\begin{em}
Find the irreducible representations
$\Lambda_i $ of $W$ which occur in ${\bf Z} [W_P \backslash W] $
with positive multiplicity for all proper Weyl subgroups
$W_P \subsetneqq W.$
\end{em} \\
By Frobenius reciprocity, or (\ref{multiplicity spaces}), this is equivalent to
\\
\begin{em}
Find the irreducible representations
$\Lambda_i $
of W such that for every proper Weyl subgroup
$W_P \subsetneqq W, $
the space of invariants
$M_i := (\Lambda_i)^{W_P} $
is non-zero.
\end{em} \\
One solution is now obvious: the {\em{reflection representation}} of $W$ acting
on the weight lattice $\Lambda$ has this property. In fact,
$\Lambda^{W_P}$ in this case is just the face $F_P$ of $\overline{C}$. The
corresponding component $Prym_{\Lambda }(\widetilde{S})$ , is called {\em{the
distinguished Prym}.} We will see in section \ref{abelianization} that its
points correspond, modulo some corrections, to Higgs bundles.
For the classical groups, this turns out to be the only common component. For
$G_2$ and $E_6$ it turns out (\cite{D2}, section 6) that a second common
component exists. The geometric significance of points in these components is
not known. As far as I know, the only component other than the distinguished
Prym which has arisen `in nature' is the one associated to the 1-dimensional
sign representation of $W$, cf. section \ref{apps} and \cite{KP2}.
\section {Abelianization}\label{abelianization}
\subsection{Abstract vs. $K$-valued objects}\
\indent We want to describe the abelianization procedure in a somewhat abstract
setting, as an equivalence between {\em{principal Higgs bundles}} and certain
{\em spectral data}.
Once we fix a {\em{values}} vector bundle $K$, we obtain an equivalence between
{\em $K$-valued principal Higgs bundles} and {\em K-valued spectral data}.
Similarly,
the choice of a representation $V$ of $G$ will determine an equivalence of
{\em $K$-valued Higgs bundles} (of a given representation type) with $K$-valued
spectral data.
As our model of a $W$-cover we take the natural quotient map
$$G/T \longrightarrow G/N
$$
and its partial compactification
\begin{equation}
\overline{G/T} \longrightarrow \overline{G/N}. \label{partial
compactification}
\end{equation}
Here $T \subset G$ is a maximal torus, and $N$ is its normalizer in $G$.
The quotient $G/N$ parametrizes maximal tori (=Cartan subalgebras) $\frak{t}$
in $\frak{g}$,
while $G/T$ parametrizes pairs ${\frak t \subset \frak b}$
with ${\frak b \subset \frak g}$ a Borel subalgebra.
An element $x \in {\frak g}$ is {\em regular} if the dimension of its
centralizer
${\frak c \subset \frak g}$ equals $\dim{T}$ (=the rank of $\frak{g}$). The
partial compactifications
$ \overline{G/N}$ and $ \overline{G/T}$ parametrize regular centralizers
${\frak c }$ and pairs ${\frak c \subset \frak b}$, respectively.
In constructing the cameral cover in section \ref{t/W}, we used the $W$-cover
$\frak t \longrightarrow \frak t / W$ and its pullback cover ${
\widetilde{\frak g} \longrightarrow \frak g}$.
Over the open subset $\frak g_{reg}$ of regular elements, the same cover is
obtained by pulling back (\ref{partial compactification}) via the map
$\alpha : \frak g_{reg} \longrightarrow \overline{G/N}$ sending an element to
its centralizer:
\begin{equation}
\label{commutes}
\begin{array}{lccccc}
\frak t & \longleftarrow & \widetilde{\frak g}_{{reg}} & \longrightarrow &
\overline{G/T} & \\
\downarrow & &\downarrow & & \downarrow & \\
\frak t /W & \longleftarrow & {\frak g}_{{reg}} &
\stackrel{\alpha}{\longrightarrow} & \overline{G/N} &.
\end{array}
\end{equation}
When working with $K$-valued objects, it is usually more convenient to work
with the left hand side of (\ref{commutes}), i.e. with eigen{\em values}. When
working with the abstract objects, this is unavailable, so we are forced to
work with the eigen{\em vectors},
or the right hand side of (\ref{commutes}). Thus:
\begin{defn}
An abstract {\em cameral cover} of $S$ is a finite morphism $\widetilde{S}
\longrightarrow S$
with $W$-action, which locally (etale) in $S$ is a pullback of (\ref{partial
compactification}). \\
\end{defn}
\begin{defn}
A {\em $K$-valued cameral cover} ($K$ is a vector bundle on $S$) consists of a
cameral cover $\pi : \widetilde{S} \longrightarrow S$ together with an
$S$-morphism
\begin{equation}
\widetilde{S} \times \Lambda \longrightarrow \Bbb{K} \label{K-values}
\end{equation}
which is $W$-invariant ($W$ acts on $ \widetilde{S} , \Lambda,$ hence
diagonally on
$\widetilde{S} \times \Lambda $ ) and linear in $\Lambda$. \\
\end{defn}
We note that a cameral cover $\widetilde{S}$ determines quotients
$\widetilde{S}_P$ for parabolic subgroups $P \subset G$. A $K$-valued cameral
cover determines additionally the $\widetilde{S}_{\lambda}$ for $\lambda \in
\Lambda$, as images in $\Bbb{K}$ of
$\widetilde{S} \times \{ \lambda \}$. The data of (\ref{K-values}) is
equivalent to a $W$-equivariant map $\widetilde{S} \longrightarrow
\frak{t}\otimes_{\bf C} K.$
\begin{defn} \label{princHiggs}
A $G$-principal Higgs bundle on $S$ is a pair ($\cal{G}, \bdl{c})$ with
$\cal{G}$ a principal $G$-bundle and $\bdl{c} \subset ad(\cal{G})$ a subbundle
of regular centralizers.
\\
\end{defn}
\begin{defn}
A $K$-valued $G$-principal Higgs bundle consists of $( \cal{G}, \bdl{c} )$
as above together with a section $\varphi$ of $\bdl{c} \otimes K$.
\end{defn}
A principal Higgs bundle $(\cal{G}, \bdl{c})$ determines a cameral cover
$\widetilde{S}\longrightarrow S$ and a homomorphism $\Lambda \longrightarrow
\mbox{Pic}(\widetilde{S}).$ Let $F$ be a parameter space for Higgs bundles with
a given $\widetilde{S}$. Each non-zero $\lambda \in \Lambda$ gives a
non-trivial map
$F\longrightarrow \mbox{Pic}(\widetilde{S})$. For $\lambda$ in a face $F_P$ of
$\overline{C}$, this factors through $\mbox{Pic}(\widetilde{S}_P)$. The
discussion in section \ref{distinguished} now suggests that $F$ should be given
roughly by the distinguished Prym,
$$ Hom_W (\Lambda , \mbox{Pic}(\widetilde{S})).
$$
It turns out that this guess needs two corrections. The first correction
involves restricting to a coset of a subgroup; the need for this is visible
even in the simplest case where
$\widetilde{S}$
is etale over
$S$,
so
$(\cal{G}, \bdl{c})$
is everywhere regular and semisimple
(i.e.
$ \bdl{c}$
is a bundle of Cartans.)
The second correction involves a twist along the ramification of
$\widetilde{S}$
over
$S$.
We explain these in the next two subsections.
\subsection{The regular semisimple case: the shift} \label{reg.ss}
\begin{eg} \label{unramified}
\begin{em}
Fix a smooth projective curve
$C$
and a line bundle
$K \in \mbox{Pic}(C)$
such that
$K^{\otimes 2} \approx \cal{O}_C.$
This determines an etale double cover
$\pi : \widetilde{C} \longrightarrow C$
with involution
$i$,
and homomorphisms
\begin{center}
$\begin{array}{cccccc}
\pi^{*} &:& \mbox{Pic}(C) &\longrightarrow
&\mbox{Pic}(\widetilde{C}) &, \\
\mbox{Nm} &:& \mbox{Pic}(\widetilde{C}) &\longrightarrow &\mbox{Pic}(C)
&, \\
i^{*} &:& \mbox{Pic}(\widetilde{C}) &\longrightarrow
&\mbox{Pic}(\widetilde{C}) &,
\end{array}$
\end{center}
satisfying
$$ 1+i^{*} = \pi^* \circ \mbox{Nm}.
$$
\begin{itemize}
\item For
$G = GL(2)$
we have
$\Lambda = \bf{Z} \oplus \bf{Z}$,
and
$W = {\cal{S}}_{2}$
permutes the summands, so
$$ Hom_W (\Lambda , \mbox{Pic}(\widetilde{C})) \approx
\mbox{Pic}(\widetilde{C}).
$$
And indeed, the Higgs bundles corresponding to
$\widetilde{C}$
are parametrized by
$\mbox{Pic}(\widetilde{C})$:
send
$L \in \mbox{Pic}(\widetilde{C})$
to
$(\cal{G}, \bdl{c})$,
where
$\cal{G}$
has associated rank-2 vector bundle
${\cal V} := \pi_* L$,
and
$ \bdl{c} \subset \End{{\cal{V}}}$
is
$\pi_* {\cal O}_{\widetilde{C}}.$
\item On the other hand, for
$G=SL(2)$
we have
$\Lambda=\bf{Z}$
and
$W={\cal{S}}_2$
acts by
$\pm 1$,
so
$$ Hom_W (\Lambda , \mbox{Pic}(\widetilde{S})) \approx
\{L \in \mbox{Pic}(\widetilde{C})\ | \ i^*L \approx L^{-1} \}
= \mbox{ker}(1+i^*).
$$
This group has 4 connected components. The subgroup
$\mbox{ker(Nm)}$
consists of 2 of these. The connected component of 0 is the classical Prym
variety, cf. \cite{MuPrym}. Now the Higgs bundles correspond, via the above
bijection
$L\mapsto \pi_*L$,
to
$$\{L \in \mbox{Pic}(\widetilde{C}) \ |\ \det (\pi_*L) \approx {\cal O}_C \} =
{\mbox{Nm}}^{-1}(K).
$$
Thus they form the {\em non-zero} coset of the subgroup
$\mbox{ker(Nm)}$.
(If we return to a higher dimensional
$S$, it is possible for $K$ not to be in the image of
$\mbox{Nm}$,
so there may be {\em no}
$SL(2)$-Higgs bundles corresponding to such a cover.)
\end{itemize}
\end{em}
\end{eg}
This example generalizes to all
$G$,
as follows. The equivalence classes of extensions
$$1 \longrightarrow T \longrightarrow N' \longrightarrow W \longrightarrow
1
$$
(in which the action of $W$ on $T$ is the standard one) are parametrized by the
group cohomology
$H^2(W,T)$.
Here the 0 element corresponds to the semidirect product . The class
$[N] \in H^2(W,T)$
of the normalizer $N$ of $T$ in $G$ may be 0, as it is for
$G=GL(n) , {\bf P}GL(n) , SL(2n+1) $;
or not, as for
$G=SL(2n)$.
Assume first, for simplicity, that
$S,\widetilde{S}$
are connected and projective. There is then a natural group homomorphism
\begin{equation}
\label{c}
c: Hom_W (\Lambda , \mbox{Pic}(\widetilde{S}))\longrightarrow H^2(W,T).
\end{equation}
Algebraically, this is an edge homomorphism for the Grothendieck spectral
sequence of equivariant cohomology, which gives the exact sequence
\begin{equation}
\label{c-edge}\qquad
0 \longrightarrow H^1(W,T)
\longrightarrow H^1(S,{\cal{C}})
\longrightarrow Hom_W (\Lambda ,
\mbox{Pic}(\widetilde{S}))
\stackrel{c}{\longrightarrow} H^2(W,T).
\end{equation}
where
${\cal{C}} := \widetilde{S} \times _W T.$
Geometrically, this expresses a {\em Mumford group} construction: giving
${\cal{L}} \in \mbox{Hom}(\Lambda,\mbox{Pic}(\widetilde{S}))$
is equivalent to giving a principal $T$-bundle
$\cal T$
over
$\widetilde{S}$;
for
${\cal{L}} \in \mbox{Hom}_W(\Lambda,\mbox{Pic}(\widetilde{S}))$,
$c({\cal{L}})$
is the class in
$H^2(W,T)$
of the group
$N'$
of automorphisms of
$\cal T$
which commute with the action on
$\widetilde{S}$
of some
$w \in W$.
To remove the restriction on
$S, \widetilde{S}$,
we need to replace each occurrence of $T$ in (\ref{c}, \ref{c-edge}) by
$\Gamma (\widetilde{S}, T)$,
the global sections of the trivial bundle on
$\widetilde{S}$
with fiber $T$. The natural map
$H^2(W,T) \longrightarrow H^2(W,\Gamma (\widetilde{S}, T))$
allows us to think of
$[N]$
as an element of
$H^2(W,\Gamma (\widetilde{S}, T))$.
\begin{prop} \cite{D3}
Fix an etale $W$-cover
$\pi: \widetilde{S}\longrightarrow S$.
The following data are equivalent:
\begin{enumerate}
\item Principal $G$-Higgs bundles
$(\cal{G}, \bdl{c})$
with cameral cover
$\widetilde{S}$.
\item Principal $N$-bundles
$\cal N$
over $S$ whose quotient by $T$ is
$\widetilde{S}.$
\item $W$-equivariant homomorphisms
${\cal{L}} : \Lambda \longrightarrow \mbox{Pic}(\widetilde{S})$
with
$c({\cal L}) = [N] \in H^2(W,\Gamma (\widetilde{S}, T))$.
\end{enumerate}
\end{prop}
We observe that while the shifted objects correspond to Higgs bundles,
the unshifted objects
$$
{\cal{L}} \in \mbox{Hom}_W(\Lambda,\mbox{Pic}(\widetilde{S})), \qquad c({\cal
L})=0
$$
\noindent
come from the $\cal C$-torsers in $H^1(S, {\cal C} ).$
\subsection{The regular case: the twist along the ramification}
\label{reg}
\begin{eg} \label{ramified}
\begin{em}
Modify example \ref{unramified} by letting
$K \in \mbox{Pic}(C) $
be arbitrary, and choose a section $b$ of
$K ^{\otimes 2}$
which vanishes on a simple divisor
$B \subset C$.
We get a double cover
$\pi : \widetilde{C} \longrightarrow C$
branched along $B$, ramified along a divisor
$$
R \subset \widetilde{C}, \quad \pi(R)=B.
$$
Via
$L\mapsto \pi_*L$,
the Higgs bundles still correspond to
$$\{L \in \mbox{Pic}(\widetilde{C}) \ |\ \det (\pi_*L) \approx {\cal O}_C \} =
{\mbox{Nm}}^{-1}(K).
$$
But this is no longer in
$ Hom_W (\Lambda , \mbox{Pic}(\widetilde{S}))$;
rather, the line bundles in question satisfy
\begin{equation}
\label{SL(2) twist}
i^*L \approx L^{-1}(R).
\end{equation}
\end{em}
\end{eg}
For arbitrary $G$, let
$\Phi$
denote the root system and
$\Phi^+$
the set of positive roots. There is a decomposition
$$ \overline{G/T} \ \smallsetminus \ G/T = \bigcup _{\alpha \in
\Phi^+}R_{\alpha}
$$
of the boundary into components, with
$R_{\alpha}$
the fixed locus of the reflection
$\sigma_{\alpha}$
in
$\alpha$.
(Via (\ref{commutes}), these correspond to the complexified walls in
$\frak t$.)
Thus each cameral cover
$\widetilde{S} \longrightarrow S$
comes with a natural set of (Cartier) {\em ramification divisors}, which we
still denote
$R_{\alpha}, \quad \alpha \in \Phi^+.$
For
$w \in W$,
set
$$ F_w := \left\{ \alpha \in \Phi^+ \ | \ w^{-1} \alpha \in \Phi^- \right\}
= \Phi^+ \cap w \Phi^-,
$$
and choose a $W$-invariant form
$\langle , \rangle$
on
$\Lambda$.
We consider the variety
$$ Hom_{W,R} (\Lambda , \mbox{Pic}(\widetilde{S}))
$$
of $R$-twisted $W$-equivariant homomorphisms, i.e. homomorphisms
$\cal L$
satisfying
\begin{equation} \qquad
\label{G twist}
w^*{\cal L}(\lambda) \approx
{\cal L}(w\lambda)\left( \sum_{\alpha \in F_w}{
{\langle-2\alpha,w\lambda \rangle \over \langle \alpha ,\alpha \rangle}
R_{\alpha}
} \right) , \qquad \lambda \in \Lambda, \quad w \in W.
\end{equation}
This turns out to be the correct analogue of (\ref{SL(2) twist}). (E.g. for a
reflection
$w=\sigma_{\alpha}$,
\quad $F_w$
is
$\left\{ \alpha \right\}$,
so this gives
$ w^*{\cal L}(\lambda) \approx
{\cal L}(w\lambda)\left(
{{\langle\alpha,2\lambda \rangle \over \langle \alpha,\alpha \rangle}
R_{\alpha}}
\right),$
which specializes to (\ref{SL(2) twist}).) As before, there is a class map
\begin{equation}
\label{c,R}
c: Hom_{W,R} (\Lambda , \mbox{Pic}(\widetilde{S}))\longrightarrow
H^2(W,\ \Gamma (\widetilde{S}, T))
\end{equation}
\noindent
which can be described via a Mumford-group construction.
To understand this twist, consider the formal object
\begin{center}
$\begin{array}{cccc}
{1 \over 2} \mbox{Ram}: & \Lambda & \longrightarrow & {\bf Q}\otimes
\mbox{Pic}\widetilde{S}, \\
& \lambda & \longmapsto &
\sum_{ ( \alpha \in {\Phi^+} ) }{{\langle\alpha,\lambda \rangle \over \langle
\alpha,\alpha \rangle} R_{\alpha}}.
\end{array}$
\end{center}
In an obvious sense, a principal $T$-bundle
$\cal T$
on
$\widetilde{S}$
(or a homomorphism
${\cal L}: \Lambda \longrightarrow \mbox{Pic}(\widetilde{S})$)
is $R$-twisted $W$-equivariant if and only if
${\cal T} (-{1 \over 2} Ram)$
is $W$-equivariant, i.e. if
${\cal T}$
and
${1 \over 2} Ram$
transform the same way under $W$.
The problem with this is that
${1 \over 2} Ram$
itself does not make sense as a $T$-bundle, because the coefficients
${\langle\alpha,\lambda\rangle \over \langle\alpha,\alpha\rangle} $
are not integers. (This argument shows that if
$Hom_{W,R} (\Lambda , \mbox{Pic}(\widetilde{S}))$
is non-empty, it is a torser over the untwisted
$Hom_{W} (\Lambda , \mbox{Pic}(\widetilde{S}))$.)
\begin{thm} \cite{D3}
\label{main}
For a cameral cover
$\widetilde{S} \longrightarrow S$,
the following data are equivalent: \\
(1) $G$-principal Higgs bundles with cameral cover
$\widetilde{S}$. \\
(2) $R$-twisted $W$-equivariant homomorphisms
${\cal L} \in c^{-1}([N]).$
\end{thm}
The theorem has an essentially local nature, as there is no requirement that
$S$ be, say, projective. We also do not need the condition of generic behavior
near the ramification, which appears in \cite{F, Me, Sc}. Thus we may consider
an extreme case, where
$\widetilde{S}$
is `everywhere ramified':
\begin{eg}\begin{em}
In example \ref{ramified}, take the section
$b=0$.
The resulting cover
$\widetilde{C}$
is a `ribbon', or length-2 non-reduced structure on $C$: it is the length-2
neighborhood of $C$ in
$\Bbb{K}$.
The SL(2)-Higgs bundles
$({\cal G},\bdl{c})$
for this
$\widetilde{C}$
have an everywhere nilpotent
$\bdl{c}$,
so the vector bundle
${\cal V} := {\cal G} \times^{SL(2)} V \approx \pi_* L$
(where $V$ is the standard 2-dimensional representation) fits in an exact
sequence
$$ 0 \longrightarrow {\cal S} \longrightarrow {\cal V} \longrightarrow
{\cal Q} \longrightarrow 0
$$
with
${\cal S} \otimes K \approx {\cal Q}.$
Such data are specified by the line bundle
${\cal Q}$,
satisfying
${\cal Q}^{\otimes 2} \approx K$,
and an extension class in
$\mbox{Ext}^1({\cal Q}, {\cal S}) \approx H^1(K^{-1})$.
The kernel of the restriction map
$ \mbox{Pic}(\widetilde{C}) \longrightarrow \mbox{Pic}(C) $
is also given by
$H^1(K^{-1})$
(use the exact sequence
$0 \longrightarrow K^{-1} \longrightarrow \pi_*{\cal
O}_{\widetilde{C}}^{\times}
\longrightarrow {\cal O}_C^{\times} \longrightarrow 0$),
and the $R$-twist produces the required square roots of $K$. (For more details
on the nilpotent locus, cf. \cite{L} and \cite{DEL}.)
\end{em}\end{eg}
\subsection{Adding values and representations}\
\indent Fix a vector bundle $K$, and consider the moduli space $ {\cal
M}_{S,G,K} $ of $K$-valued $G$-principal Higgs bundles on $S$. (It can be
constructed as in Simpson's \cite{S}, even though the objects we need to
parametrize are slightly different than his. In this subsection we outline a
direct construction.)
It comes with a Hitchin map:
\begin{equation}
\label{BigHitchin}
h: {\cal M}_{S,G,K} \longrightarrow B_K
\end{equation}
\noindent where $B := B_K$ parametrizes all possible Hitchin data. Theorem
\ref{main} gives a precise description of the fibers of this map, independent
of the values bundle $K$. This leaves us with the relatively minor task of
describing, for each $K$, the corresponding base, i.e. the closed subvariety
$B_s$ of $B$ parametrizing {\em split} Hitchin data, or $K$-valued cameral
covers. The point is that Higgs bundles satisfy a symmetry condition, which
in Simpson's setup is
$$ \varphi \wedge \varphi = 0,
$$
and is built into our definition \ref{princHiggs} through the assumption that
\bdl{c} is regular, hence abelian. Since commuting operators have common
eigenvectors, this gives a splitness condition on the Hitchin data, which we
describe below. (When $K$ is a line bundle, the condition is vacuous, $B_s =
B$.) The upshot is:
\begin{lem}
\label{parametrization}
The following data are equivalent: \\
(a) A $K$-valued cameral cover of $S$. \\
(b) A split, graded homomorphism
$R{\bf \dot{\ }} \longrightarrow {Sym}{\bf \dot{\ }}K.$ \\
(c) A split Hitchin datum
$b \in B_s$.
\end{lem}
Here $R{\bf \dot{\ }}$ is the graded ring of $W$-invariant polynomials on
$\frak t$:
\begin{equation}
R{\bf \dot{\ }} := (\mbox{Sym}{\bf \dot{\ }} {\frak t}^*)^W
\approx {\bf C}[\sigma_1,\ldots,\sigma_l], \qquad \deg (\sigma_i) = d_i
\end{equation}
\noindent
where
$l := \mbox{Rank}({\frak g})$
and the
$\sigma_i$
form a basis for the $W$-invariant polynomials. The Hitchin base is the vector
space
$$ B := B_K := \oplus _{i=1}^l H^0(S, {Sym}^{d_i}K)
\approx \mbox{Hom}(R{\bf \dot{\ }},\mbox{Sym}{\bf \dot{\ }}K).
$$
\noindent For each
$\lambda \in \Lambda$
(or
$\lambda \in {\frak t}^*$,
for that matter), the expression in an indeterminate $x$:
\begin{equation}
\label{rep poly}
q_{\lambda}(x,t) := \prod_{w \in W}{(x-w\lambda(t))}, \qquad t \in {\frak t},
\end{equation}
is $W$-invariant (as a function of $t$), so it defines an element
$q_{\lambda}(x) \in R{\bf \dot{\ }}[x].$
A Hitchin datum
$b \in B \approx \mbox{Hom}(R{\bf \dot{\ }},\mbox{Sym}{\bf \dot{\ }}K)$
sends this to
$$ q_{\lambda,b}(x) \in \mbox{Sym}\dot{\ }(K)[x].
$$
We say that $b$ is {\em split} if, at each point of $S$ and for each
$\lambda$,
the polynomial
$q_{\lambda,b}(x)$
factors completely, into terms linear in $x$.
We note that, for $\lambda$ in the interior of $C$ (the positive Weyl chamber),
$q_{\lambda,b}$
gives the equation in
$\Bbb K$
of the spectral cover
$\widetilde{S}_{\lambda}$
of section (\ref{decomp covers}):
$q_{\lambda,b}$ gives a morphism
$\Bbb K \longrightarrow \mbox{Sym}^N \Bbb K$,
where $N:=\#W$, and $\widetilde{S}_{\lambda}$ is the invere image of the
zero-section.
(When
$\lambda$
is in a face
$F_P$
of
$\overline{C}$,
we define analogous polynomials
$q_{\lambda}^P(x,t)$
and
$q_{\lambda,b}^P(x)$
by taking the product in (\ref{rep poly}) to be over
$w \in W_P \backslash W.$
These give the reduced equations in this case, and
$q_{\lambda}$
is an appropriate power.)
Over $B_s$ there is a universal $K$-valued cameral cover
$$ \widetilde{\cal S} \longrightarrow B_s
$$
with ramification divisor $R \subset \widetilde{\cal S}$. From the relative
Picard,
$$ \mbox{Pic}( \widetilde{\cal S} / B_s)
$$
we concoct the relative $N$-shifted, $R$-twisted Prym
$$ \mbox{Prym}_{\Lambda ,R}( \widetilde{\cal S} / B_s).
$$
By Theorem \ref{main}, this can then be considered as a parameter space
$ {\cal M}_{S,G,K} $
for all $K$-valued $G$-principal Higgs bundles on $S$. (Recall that our objects
are assumed to be everywhere {\em regular}!) It comes with a `Hitchin map',
namely the projection to $B_s$, and the fibers corresponding to smooth
projective $\widetilde{S}$ are abelian varieties. When $S$ is a smooth,
projective curve, we recover this way the algebraic complete integrability of
Hitchin's system and its generalizations.
\section {Symplectic and Poisson structures}
\label{symplectic}\
\indent The total space of Hitchin's original system is a cotangent bundle,
hence has a natural symplectic structure. For the polynomial matrix systems of
\cite{B} and \cite{AHH} there is a natural Poisson structure which one writes
down explicitly.
In \cite{Bn} and \cite{M1}, this result is extended to the systems
${\cal M}_{C,K}$
of $K$-valued GL(n) Higgs bundles on $C$, when
$K \approx \omega_C(D)$
for an effective divisor $D$ on $C$. There is a general-nonsense pairing on the
cotangent spaces, so the point is to check that this pairing is `closed', i.e.
satisfies the identity required for a Poisson structure. Bottacin does this by
an explicit computation along the lines of \cite{B}. Markman's idea is to
consider the moduli space
${\cal M}_D$
of stable vector bundles on $C$ with level-$D$ structure. He realizes an open
subset
${\cal M}^0_{C,K}$
of
${\cal M}_{C,K}$,
parametrizing Higgs bundles whose covers are nice, as a quotient (by an action
of the level group) of
$T^*{\cal M}_D$,
so the natural symplectic form on
$T^*{\cal M}_D$
descends to a Poisson structure on
${\cal M}^0_{C,K}$. This is identified with the general-nonsense form (wherever
both exist),
proving its closedness.
In \cite{Muk}, Mukai constructs a symplectic structure on the moduli space of
simple sheaves on a $K3$ surface $S$. Given a curve
$C \subset S$,
one can consider the moduli of sheaves having the numerical invariants of a
line bundle on a curve in the linear system
$ |nC| $
on $S$. This has a support map to the projective space
$ |nC| $,which turns it into an ACIHS. This system specializes, by a
`degeneration to the normal cone' argument, cf. \cite{DEL}, to Hitchin's,
allowing translation of various results about Hitchin's system (such as
Laumon's description of the nilpotent cone, cf. \cite{L} ) to Mukai's.
In higher dimensions, the moduli space $\cal M$ of $\Omega^1$-valued Higgs
bundles carries a natural symplectic structure \cite{S}. (Corlette points out
in \cite{C} that certain components of an open subet in $\cal M$ can be
described as cotangent bundles.) It is not clear at the moment exactly when one
should expect to have an ACIHS, with symplectic, Poisson or quasi symplectic
structure, on the moduli spaces of $K$-valued Higgs bundles for higher
dimensional $S$, arbitrary $G$, and arbitrary vector bundle $K$. A beautiful
new idea \cite{M2} is that Mukai's results extend to the moduli of those
sheaves on a (symplectic, Poisson or quasi symplectic) variety $X$ whose
support in $X$ is {\em Lagrangian.}
Again, there is a general-nonsense pairing. At points where the support is
non-singular projective, this can be identified with another, more geometric
pairing, constructed using the {\em cubic condition} of \cite{DM1}, which is
known to satisfy the closedness requirement. This approach is quite powerful,
as it includes many non-linear examples such as Mukai's, in addition to the
line-bundle valued spectral systems of \cite{Bn,M1} and also Simpson's
$\Omega^1$-valued GL(n)-Higgs bundles: just take $X := T^*S
\stackrel{\pi}{\rightarrow} S$, with its natural symplectic form, and the
support in $X$ to be proper over $S$ of degree n; such sheaves correspond to
Higgs bundles by $\pi_*$.
The structure group $GL(n)$ can of course be replaced by an arbitrary reductive
group $G$. Using Theorem \ref{main}, this yields (in the analogous cases) a
Poisson structure on the Higgs moduli space ${\cal M}_{S,G,K}$ described at
the end of the previous section. The fibers of the generalized Hitchin map are
Lagrangian with respect to this structure. Along the lines of our general
approach, the necessary modifications are clear: $\pi_*$ is replaced by the
equivalence of Theorem \ref{main}. One thus considers only Lagrangian supports
which retain a $W$-action, and only {\em equivariant} sheaves on them (with the
numerical invariants of a line bundle). These two restrictions are symplecticly
dual, so the moduli space of Lagrangian sheaves with these invariance
properties is a symplectic (respectvely, Poisson) subspace of the total moduli
space, and the fibers of the Hitchin map are Lagrangian as expected.
A more detailed review of the ACIHS aspects of Higgs bundles will appear in
\cite{DM2}.
\section {Some applications and problems}
\label{apps}
\noindent \underline{\bf Some applications} \nopagebreak
\noindent In \cite{H1}, Hitchin used his integrable system to compute several
cohomology groups of the moduli space ${\cal SM}$ (of rank 2, fixed odd
determinant vector bundles on a curve $C$) with coefficients in symmetric
powers of its tangent sheaf ${\cal T}$. The point is that the symmetric algebra
$Sym{\bf \dot{\ }} {\cal T}$
is the direct image of
$ {\cal O}_{T^*{\cal SM}}$,
and sections of the latter all pull back via the Hitchin map $h$ from functions
on the base $B$, since the fibers of $h$ are open subsets in abelian varieties,
and the missing locus has codimension $\geq 2$. Hitchin's system is used in
\cite{BNR} to compute a couple of "Verlinde numbers" for GL(n), namely the
dimensions
$h^0({\cal M}, \Theta) = 1, \qquad h^0({\cal SM}, \Theta) = n^g$.
These results are now subsumed in the general Verlinde formulas, cf.
\cite{F2}, \cite{BL}, and other references therein.
A pretty application of spectral covers was obtained by Katzarkov and Pantev
\cite{KP2}. Let $S$ be a smooth, projective, complex variety, and
$\rho : \pi_1(S)\longrightarrow G$
a Zariski dense representation into a simple $G$ (over $\bf{C}$). Assume That
the $\Omega^1$-valued Higgs bundle $ ( {\cal V}, \phi) $ associated to $\rho$
by Simpson is (regular and) generically semisimple, so the cameral cover is
reduced. Among other things, they show that $\rho$ factors through a
representation of an orbicurve if and only if the non-standard component
$Prym_{\epsilon}(\widetilde{S})$
is non zero, where
$\epsilon $
is the one-dimensional sign representation of $W$.
(In a sense, this is the opposite of
$Prym_{\Lambda}(\widetilde{S})$:
while
$Prym_{\Lambda}(\widetilde{S})$
is common to
$\mbox{Pic}(\widetilde{S}_P)$
for all proper Weyl subgroups,
$Prym_{\epsilon}(\widetilde{S})$
occurs in none except for the full cameral Picard.)
Another application is in \cite{KoP}: the moduli spaces of SL(n)- or
GL(n)-stable bundles on a curve have certain obvious automorphisms, coming from
tensoring with line bundles on the curve, from inversion, or from automorphisms
of the curve. Kouvidakis and Pantev use the dominant direct-image maps from
spectral Picards and Pryms to the moduli spaces to show that there are no
further, unexpected automorphisms. This then leads to a `non-abelian Torelli
theorem', stating that a curve is determined by the isomorphism class of the
moduli space of bundles on it. \\ \mbox{}\\
\noindent \underline{\bf Compatibility?} \nopagebreak
\noindent Hitchin's construction \cite {H2} of the projectively flat
connection on the vector bundle of non-abelian theta functions over the moduli
space of curves does not really use much about spectral covers. Nor do other
constructions of Faltings \cite{F} and Witten et al \cite{APW}. Hitchin's work
suggests that the `right' approach should be based on comparison of the
non-abelian connection near a curve $C$ with the abelian connection for
standard theta functions on spectral covers $\widetilde{C}$ of $C$. One
conjecture concerning the possible relationship between these connections
appears in \cite{A}, and some related versions have been attempted by several
people, so far in vain. What's missing is a compatibility statement between the
actions of the two connections on pulled-back sections. If the expected
compatibility turns out to hold, it would give another proof of the projective
flatness. It should also imply projective finiteness and projective unitarity
of mo!
nodromy for the non-abelian thetas
, and may or may not bring us closer to a `finite-dimensional' proof of
Faltings' theorem (=the former Verlinde conjecture).\\ \mbox{}\\
\noindent \underline{\bf {Irregulars?} } \nopagebreak
\noindent The Higgs bundles we consider in this survey are assumed to be
everywhere regular. This is a reasonable assumption for line-bundle valued
Higgs bundles on a curve or surface, but {\em not} in $\dim \geq 3$. This is
because the complement of ${\frak g}_{{reg}}$ has codimension 3 in ${\frak g}$.
The source of the difficulty is that the analogue of (\ref{commutes}) fails
over
${\frak g}$. There are two candidates for the universal cameral cover:
$\widetilde{\frak g}$, defined by the left hand side of (\ref{commutes}), is
finite over ${\frak g}$ with $W$ action, but does not have a family of line
bundles parametrized by $\Lambda$.
These live on $\stackrel{\approx}{\frak g}$, the object defined by the right
hand side, which parametrizes pairs
$(x,{\frak b}), \qquad x \in {\frak b} \subset {\frak g}$ .
This suggests that the right way to analyze irregular Higgs bundles may involve
spectral data consisting of a tower
$$ \stackrel{\approx}{S} \stackrel{\sigma}{\longrightarrow} \widetilde{S}
\longrightarrow S
$$
together with a homomorphism
$ {\cal L} : \Lambda \longrightarrow \mbox{Pic}(\stackrel{\approx}{S})$
such that the collection of sheaves
$$ \sigma_*({\cal L}(\lambda)), \qquad \lambda \in \Lambda
$$
on
$\widetilde{S}$
is
$R$-twisted $W$-equivariant in an appropriate sense. As a first step, one may
wish to understand the direct images
$ R^i \sigma_*({\cal L}(\lambda)) $
and in particular the cohomologies
$H^i(F, {\cal L}(\lambda))$
where $F$, usually called a {\em Springer fiber}, is a fiber of $\sigma$. For
regular $x$, this fiber is a single point. For $x=0$, the fiber is all of
$G/B$, so the fiber cohomology is given by the Borel-Weil-Bott theorem. The
question may thus be considered as a desired extension of BWB to general
Springer fibers.
|
1995-08-01T03:39:16 | 9505 | alg-geom/9505012 | en | https://arxiv.org/abs/alg-geom/9505012 | [
"alg-geom",
"math.AG"
] | alg-geom/9505012 | Teleman | Ch. Okonek, and A. Teleman | The Coupled Seiberg-Witten Equations, vortices, and Moduli spaces of
stable pairs | latex | null | null | null | null | We introduce coupled Seiberg-Witten equations, and we prove, using a
generalized vortex equation, that, for Kaehler surfaces, the moduli space of
solutions of these equations can be identified with a moduli space of
holomorphic stable pairs. In the rank 1 case, one recovers Witten's result
identifying the space of irreducible monopoles with a moduli space of divisors.
As application, we give a short proof of the fact that a rational surface
cannot be diffeomorphic to a minimal surface of general type.
| [
{
"version": "v1",
"created": "Mon, 8 May 1995 19:30:12 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Okonek",
"Ch.",
""
],
[
"Teleman",
"A.",
""
]
] | alg-geom | \section{Introduction}
Recently, Seiberg and Witten [W] introduced new invariants of 4-manifolds,
which are defined by counting solutions of a certain non-linear differential
equation.
The new invariants are expected to be equivalent to Donaldson's
polyno\-mial-invariants---at least for manifolds of simple type [KM
1]---and they
have already found important applications, like e.g. in the proof of the Thom
conjecture by Kronheimer and Mrowka [KM 2].
Nevertheless, the equations themselves remain somewhat mysterious, especially
from a mathematical point of view.
The present paper contains our attempt to understand and to generalize the
Seiberg-Witten equations by coupling them to connections in unitary vector
bundles,
and to relate their solutions to more familiar objects, namely stable pairs.
Fix a ${\rm Spin}^c$-structure on a Riemannian 4-manifold $X$, and denote by
$\Sigma^{\pm}$ the associated spinor bundles. The equations which we will study
are:
$$
\left\{\begin{array}{lcc}
\hskip 4pt{\not}{D}_{A,b}\Psi &=&0\\
\Gamma(F_{A,b}^+)&=&(\Psi\bar\Psi)_0\end{array}\right.$$
This is a system of equations for a pair $(A,\Psi)$ consisting of a unitary
connection in a unitary bundle $E$ over $X$, and a
positive spinor $\Psi\in A^0(\Sigma^+\otimes E)$. The symbol $b$ denotes a
connection in the determinant line bundle of the spinor bundles
$\Sigma^{\pm}$ and
\hbox{$\hskip 4pt{\not}{D}_{A,b}:\Sigma^+\otimes E\longrightarrow\Sigma^-\otimes E$} is the Dirac operator
obtained by coupling the connection in $\Sigma^+$ defined by $b$ (and by the
Levi-Civita connection in the tangent bundle) with the variable connection
$A$ in $E$.
These equations specialize to the original Seiberg-Witten equations
if
$E$ is a line bundle. We show that the coupled equations can be interpreted
as a
differential version of the generalized vortex equations [JT].
Vortex equations over K\"ahler manifolds have been investigated by Bradlow
[B1], [B2] and by Garcia-Prada [G1], [G2]: Given a pair $({\cal E},\varphi)$
consisting of a holomorphic vector bundle with a section, the vortex
equations ask
for a Hermitian metric $h$ in ${\cal E}$ with prescribed mean curvature: more
precisely, the equations---which depend on a real parameter
$\tau$---are
$$i\Lambda F_h=\frac{1}{2}(\tau{\rm id}_{\cal E}-\varphi\otimes\varphi^*).$$
A solution exists if and only if the pair $({\cal E},\varphi)$ satisfies a
certain stability condition ($\tau$-stability), and the moduli space of
vortices can be identified with the moduli space of $\tau$-stable pairs. A GIT
construction of the latter space has been given by Thaddeus [T] and
Bertram [B] if the base manifold is a projective curve, and by Huybrechts
and Lehn
[HL1], [HL2] in the case of a projective variety. Other constructions have been
given by Bradlow and Daskalopoulus [BD1], [BD2] in the case of a Riemann
surface,
and by Garcia-Prada for compact K\"ahler manifolds [G2]. In this connection
also
[BD2] is relevant.
In this note we prove the following result:
\begin{th} Let $(X,g)$ be a K\"ahler surface of total scalar curvature
$\sigma_g$, and let $\Sigma$ be the canonical ${\rm Spin}^c$-structure with
associated Chern connection $c$. Fix a unitary vector bundle $E$ of rank $r$
over $X$, and define $\mu_g(\Sigma^+\otimes E):=\frac{{\rm deg}_g(E)}{r} +\sigma_g$.
Then for $\mu_g<0$, the space of solutions of the coupled
Seiberg-Witten equation is isomorphic to the moduli space of stable pairs of
topological type
$E$, with parameter $\sigma_g$.
\end{th}
If the constant $\mu_g(\Sigma^+\otimes E)$ is positive, one simply replaces the
bundle
$E$ with $E^{\vee}\otimes K_X$, where $K_X$ denotes the canonical line bundle
of
$X$ (cf. Lemma 3.1).
Note that the above theorem gives a complex geometric interpretation of the
moduli space of solutions of the coupled Seiberg-Witten equation associated
to \underbar{all} ${\rm Spin}^c$-structures on $X$: The change of the
${\rm Spin}^c$-structure is equivalent to tensoring $E$ with a line bundle.
Notice also
that in the special case
$r=1$ one recovers Witten's result identifying the space of irreducible
monopoles
on a K\"ahler surface with the set of all divisors associated to
line bundles of a fixed topological type; the stability condition which he
mentions is the rank-1 version of the stable pair-condition.
Having established this correspondence, we describe some of the basic
properties of the moduli spaces, and give a first application: We show that
minimal surfaces of general type cannot be diffeomorphic to rational ones. This
provides a short proof of one of the essential steps in Friedman and Qin's
proof
of the Van de Ven conjecture [FQ]. More detailed investigations and
applications
will appear in a later paper.
We like to thank A. Van de Ven for very helpful questions and remarks.
\section{${\rm Spin}^c$-structures and almost canonical classes}
The complex spinor group is defined as ${\rm Spin}^c:={\rm Spin}\times_{{\Bbb Z}_2}S^1$, and
there are two non-split exact sequences
$$\begin{array}{c}
1\longrightarrow S^1\longrightarrow{\rm Spin}^c\longrightarrow{\rm SO}\longrightarrow 1\\
1\longrightarrow{\rm Spin}\longrightarrow{\rm Spin}^c\longrightarrow\ S^1\longrightarrow 1\end{array}$$
In dimension 4, ${\rm Spin}^c(4)$ can be identified with the subgroup of
${\rm U}(2)\times{\rm U}(2)$ consisting of pairs of unitary matrices with the same
determinant, and one has two commutative diagrams:
$$\begin{array}{ccllclll}
& & 1 & & 1 & & & \\
& &\downarrow & &\downarrow& & & \\
1&\longrightarrow& {\Bbb Z}_2 &\longrightarrow&{\rm Spin}(4) &\longrightarrow &{\rm SO}(4)\ \ \
&\longrightarrow
1\\
& &\downarrow & &\downarrow& & \ \
\parallel&
\\
1&\longrightarrow&S^1 &\longrightarrow&{\rm Spin}^c(4) &\longrightarrow &{\rm SO}(4) \ \ \
&\longrightarrow 1\\
& &\downarrow(\cdot)^2&
&{\scriptstyle{\det}}\downarrow\ \ \ \
&\ \nwarrow{\scriptstyle\Delta}&\ \ \uparrow&
\\
& &S^1
&= & S^1 &\longleftarrow&{\rm U}(2)\ \ \ & \\
& &\downarrow & &\downarrow&{\ }^{\det}&&\\
& &1 & &1 &&&
\end{array} \eqno{(1)}$$
where $\Delta:{\rm U}(2)\longrightarrow{\rm Spin}^c(4)\subset{\rm U}(2)\times{\rm U}(2)$ acts by
$a\longrightarrow\left(\left(\matrix{{\rm id}&0\cr 0&\det a\cr}\right),a\right)$, and
$$
\matrix{
& & & & & & & & 1\ \ \ & &\cr
& & & & & & & &\downarrow\ \ \ & &\cr
& & & & 1 & & 1 & & {\Bbb Z}_2\ \ \ & &\cr
& & & &\downarrow& &\downarrow& &\downarrow\ \ \ & &\cr
& &1&\rightarrow&S^1 &\rightarrow&{\rm Spin}^c(4) &\rightarrow&{\rm SO}(4)
&\rightarrow &1\cr
& & & &\downarrow& &\downarrow& &\ \
\ \ \ \ \ \downarrow{\scriptstyle(\lambda^+,\lambda^-)}& &\cr
& &1&\rightarrow&S^1\times S^1
&\rightarrow&{\rm U}(2)\times{\rm U}(2)&\stackrel{\rm
ad}{\rightarrow}&{\rm SO}(3)\times{\rm SO}(3)\ &\rightarrow&1\cr
& & & &\downarrow& &\downarrow& &\downarrow\ \ & &\cr
1&\rightarrow&{\Bbb Z}_2&\rightarrow&S^1&\stackrel{(\cdot)^2}{\rightarrow}&S^1&
\rightarrow&1\ \ &\cr
& & & &\downarrow& &\downarrow& & \ \ \ \ & &\cr
& & & & 1 & & 1 & & & &\cr}$$
where $\lambda^{\pm}:{\rm SO}(4)\longrightarrow{\rm SO}(3)$ are induced by the
two projections of ${\rm Spin}(4)={\rm SU}(2)^+\times{\rm SU}(2)^-$ [HH]. $\lambda^{\pm}$ can
be also be seen as the representations of ${\rm SO}(4)$ in
$\Lambda^2_{\pm}({\Bbb R}^4)\simeq{\Bbb R}^3$ induced by the canonical representation in
${\Bbb R}^4$.
Let $X$ be a closed, oriented 4-manifold. Given any principal ${\rm SO}(4)$-bundle
$P$ over $X$, we denote by $P^{\pm}$ the induced principal ${\rm SO}(3)$-bundles.
If
$\hat{P}$ is a ${\rm Spin}^c(4)$-bundle, we let $\Sigma^{\pm}$ be the
associated ${\rm U}(2)$-vector bundles, and we set (via the vertical determinant-map
in (1)) $\det(\hat{P})=L$, so that $\det(\Sigma^{\pm})=L$.
\begin{lm}
Let $P$ be a principal $SO(4)$-bundle over $X$ with characteristic classes
$w_2(P)\in H^2(X,{\Bbb Z}_2)$, and $p_1(P), e(P)\in H^4(X,{\Bbb Z})$. Then\hfill{\break}
i) $P$ lifts to a principal ${\rm Spin}^c(4)$-bundle $\hat{P}$ iff $w_2(P)$ lifts to
an integral cohomology class.\hfill{\break}
ii) Given a class $L\in H^2(X,{\Bbb Z})$ with $w_2(P)\equiv\bar L$(mod 2), the
${\rm Spin}^c(4)$-lifts $\hat{P}$ of $P$ with $\det\hat{P}=L$ are in 1-1
correspondence with the 2-torsion elements in $H^2(X,{\Bbb Z})$.\hfill{\break}
iii) Let $\hat P$ be a ${\rm Spin}^c(4)$-principal bundle with $P\simeq\hat{P}/S^1$,
and let $L=\det\hat{P}$. Then the Chern classes of $\Sigma^{\pm}$ are:
$$\begin{array}{rl}c_1(\Sigma^{\pm})&=L\\
c_2(\Sigma^{\pm})&=\frac{1}{4}\left(L^2 -p_1(P)\mp 2e(P)\right)\end{array}$$
\end{lm}
{\bf Proof: } [HH] and the diagrams above.
\hfill\vrule height6pt width6pt depth0pt \bigskip
Consider now a Riemannian metric $g$ on $X$, and let $P$ be the associated
principal ${\rm SO}(4)$-bundle. In this case the real vector bundles associated to
$P^{\pm}$ via the standard representations are the bundles $\Lambda^2_{\pm}$
of (anti-) self-dual 2-forms on $X$.
The integral characteristic classes of
$P$ are given by $p_1(P)=3\sigma$ and $e(P)=e$, where $\sigma$ and $e$
denote the
signature and the Euler number of the oriented manifold $X$. Furthermore,
$w_2(P)$ always lifts to an integral class, the lifts are precisely the
characteristic elements in $H^2(X,{\Bbb Z})$, i.e. the classes $L$ with $x^2\equiv
x\cdot L$ for every $x\in H^2(X,{\Bbb Z})$ [HH].
Let $T_X$ be the tangent bundle of $X$, and denote by $\Lambda^p$ the bundle
of $p$-forms on $X$. The choice of a ${\rm Spin}^c(4)$-lift $\hat{P}$ of $P$ with
associated ${\rm U}(2)$-vector bundles $\Sigma^{\pm}$ defines a vector bundle
isomorphism
$\gamma:\Lambda^1\otimes{\Bbb C}\longrightarrow{\rm Hom}_{{\Bbb C}}(\Sigma^+,\Sigma^-)$. There is also a
${\Bbb C}$-linear isomorphism $(\cdot)^{\#}:{\rm Hom}_{{\Bbb C}}(\Sigma^+,\Sigma^-)\longrightarrow
{\rm Hom}_{{\Bbb C}}(\Sigma^-,\Sigma^+)$ which satisfies the identity:
$$\gamma(u)^{\#}\gamma(v)+\gamma(v)^{\#}\gamma(u)=2g^{{\Bbb C}}(u,v){\rm id}_{\Sigma^+},$$
and $\gamma(u)^{\#}=\gamma(u)^*=g(u,u)\gamma(u)^{-1}$ for real non-vanishing
cotangent vectors $u$.
It is convenient to extend the homomorphisms $\gamma(u)$ to endomorphisms of
the direct sum $\Sigma:=\Sigma^+\oplus\Sigma^-$. Putting
$\gamma(u)|_{\Sigma^-}:=-\gamma(u)^{\#}$, we obtain a vector-bundle
homomorphism $\gamma:\Lambda^1\otimes{{\Bbb C}}\longrightarrow{\rm End}_0(\Sigma)$, which maps the
bundle $\Lambda^1$ of real 1-forms into the bundle of trace-free
skew-Hermitian endomorphisms of $\Sigma$. With this convention, we get:
$$\gamma(u)\circ\gamma(v)+\gamma(v)\circ\gamma(u)=-2g^{{\Bbb C}}(u,v){\rm id}_{\Sigma}.$$
Consider the induced homomorphism
$$\Gamma:\Lambda^2\otimes{\Bbb C} \longrightarrow{\rm End}_0(\Sigma)$$
defined on decomposable elements by
$$\Gamma(u\wedge v):=\frac{1}{2}[\gamma(u),\gamma(v)].$$
The restriction $\Gamma|_{\Lambda^2}$ identifies the bundle $\Lambda^2$
with the bundle ${\rm ad}_0(\hat{P})\simeq{\rm ad}(P)$ of skew-symmetric endomorphisms of
the tangent bundle of $X$.
$\Lambda^2$ splits as an orthogonal sum
$\Lambda^2=\Lambda^2_+\oplus\Lambda^2_-$ and $\Gamma$ maps
the bundle $\Lambda^2_{\pm}\otimes{\Bbb C}$ (respectively $\Lambda^2_{\pm}$)
isomorphically onto the bundle
${\rm End}_0(\Sigma^{\pm})\subset{\rm End}(\Sigma)$
($su(\Sigma^{\pm})\subset su(\Sigma)$) of trace-free (trace free
skew-Hermitian)
endomorphisms of
$\Sigma^{\pm}$.
We give now an explicit description of the two spinor bundles $\Sigma^{\pm}$
and of the map $\Gamma$ in the case of a ${\rm Spin}^c(4)$-structure coming from an
almost Hermitian structure.
\begin{dt}
A characteristic element $K\in H^2(X,{\Bbb Z})$ is an almost canonical class if
$K^2=3\sigma+2e$.
\end{dt}
Such classes exist on $X$ if and only if $X$ admits an almost complex
structure.
More precisely:
\begin{pr} ({\rm Wu}) $K\in H^2(X,{\Bbb Z})$ is an almost canonical class if and only
if there exists an almost complex structure $J$ on $X$ which is compatible with
the orientation, such that $K=c_1(\Lambda^{10}_J)$.
\end{pr}
{\bf Proof: } [HH]
\hfill\vrule height6pt width6pt depth0pt \bigskip
Here we denote, as usual, by $\Lambda^{pq}_J$ the bundle of $(p,q)$-forms
defined by the almost complex structure $J$.
Notice that any almost complex structure $J$ on $X$ can be deformed into a
$g$-orthogonal one, and that $J$ is $g$-orthogonal iff $g$ is $J$-Hermitian.
The choice of a $g$-orthogonal almost complex structure $J$ on $X$ corresponds
to to a reduction of the ${\rm SO}(4)$-bundle $P$ of $X$ to a $U(2)$-bundle via the
inclusion ${\rm U}(2)\subset{\rm SO}(4)$; since the inclusion factors through the
embedding $\Delta:{\rm U}(2)\longrightarrow{\rm Spin}^c(4)$ (see diagram (1)), this
reduction defines a unique ${\rm Spin}^c(4)$-bundle $\hat{P_J}$ over $X$. By
construction we have $\hat{P}_J/{S^1}\simeq P$, and $\det\hat{P}_J=-K$.
\begin{pr} Let $J$ be a $g$-orthogonal almost complex structure on $X$,
compatible with the orientation.\hfill{\break}
i) The spinor bundles $\Sigma^{\pm}_J$ of $\hat{P}_J$ are:
$$\Sigma^+_J\simeq\Lambda^{00}\oplus\Lambda^{02}_J,\ \
\Sigma^-_J\simeq\Lambda^{01}_J.$$
ii) The map $\Gamma:\Lambda_+^2\otimes{\Bbb C}\longrightarrow{\rm End}_0(\Sigma^+_J)$ is given by
$$\Lambda^{20}_J\oplus\Lambda^{02}_J\oplus\Lambda^{00}\omega_g
\ni(\lambda^{20},\lambda^{02},\omega_g)\stackrel{\Gamma}{\longmapsto}
2\left[\matrix{-i&-*(\lambda^{20}\wedge\cdot)\cr
\lambda^{02}\wedge\cdot&i\cr}\right]\in{\rm End}_0(\Lambda^{00}\oplus\Lambda^{02}).$$
\end{pr}
{\bf Proof: } i) $c_1(\Sigma^+_J)=c_1(\Sigma^-_J)=-K$,
$c_2(\Sigma^+_J)=\frac{1}{4}[K^2-3\sigma-2e]$,
$c_2(\Sigma^-_J)=\frac{1}{4}[K^2-3\sigma+2e]=c_2(\Sigma^+)+e$,
and ${\rm U}(2)$-bundles on a 4-manifold are classified by their Chern
classes.\hfill{\break}
ii) With respect to a suitable choice of the isomorphisms i), the Clifford map
$\gamma$ acts by
$$\gamma(u)(\varphi+\alpha)=\sqrt{2}\left(\varphi u^{01}-i\Lambda_g
u^{10}\wedge\alpha\right),$$
$$\gamma(u)^{\#}(\theta)=\sqrt{2}\left(i\Lambda_g(u^{10}\wedge\theta)-u^{01}
\wedge
\theta\right),
\eqno{(3)}$$
where $\Lambda_g:\Lambda^{pq}_J\longrightarrow\Lambda^{p-1,q-1}_J$ is the adjoint of
the map
$\cdot\wedge\omega_g$ [H1].
\hfill\vrule height6pt width6pt depth0pt \bigskip
\section{The coupled Seiberg-Witten equations}
Let $P$ be the principal ${\rm SO}(4)$-bundle associated with the tangent bundle
of the oriented, closed Riemannian 4-manifold $(X,g)$, and fix a ${\rm Spin}^c(4)$
structure $\hat{P}$ over $P$ with $L=\det(\hat{P})$. The choice of a
${\rm Spin}^c(4)$-connection in $\hat{P}$ projecting onto the Levi-Civita
connection in $P$ is equivalent to the choice of a connection $b$ in the
unitary line bundle $L$ [H1]. We denote by $B(b)$ the ${\rm Spin}^c(4)$-connection
in
$\hat{P}$ corresponding to $b$, and also the induced
connection in the vector bundle $\Sigma=\Sigma^+\oplus\Sigma^-$.
The curvature $F_{B(b)}$ of the connection $B(b)$ in $\Sigma$ has the form
$$F_{B(b)}=\frac{1}{2}F_b{\rm id}_{\Sigma}+F_g=
\left[\matrix{\frac{1}{2}F_b{\rm id}_{\Sigma^+}+F_g^+&0\cr
0&\frac{1}{2}F_b{\rm id}_{\Sigma^-}+F_g^-\cr}\right],$$
where $F_g$, and $F_g^{\pm}$ denote the Riemannian curvature operator, and its
components with respect to the splitting ${\rm ad}(P)=\Lambda^2_+\oplus\Lambda^2_-$.
Let now $E$ be an arbitrary Hermitian bundle of rank $r$ over $X$, and $A$ a
connection in $E$. We denote by $A_b$ the induced connection in the tensor
product $\Sigma\otimes E$, and by $\hskip 4pt{\not}{D}_{A,b}:A^0(\Sigma\otimes E)\longrightarrow
A^0(\Sigma\otimes E)$ the associated Dirac operator. $\hskip 4pt{\not}{D}_{A,b}$ is defined as
the composition:
$$A^0(\Sigma\otimes E)\stackrel{\nabla_{A_b}}{\longrightarrow}A^1(\Sigma\otimes
E)\stackrel{m}{\longrightarrow} A^0(\Sigma\otimes E)$$
where $m$ is the Clifford multiplication $m(u,\sigma\otimes
e):=\gamma(u)(\sigma)\otimes e$. $\hskip 4pt{\not}{D}_{A,b}$ is an elliptic, self-adjoint
operator and its square $\hskip 4pt{\not}{D}_{A,b}^2$ is related to the usual Laplacian
$\nabla_{A_b}^*\nabla_{A_b}$ by the Weitzenb\"ock formula
$$\hskip 4pt{\not}{D}_{A,b}^2=\nabla_{A_b}^*\nabla_{A_b}+\Gamma(F_{A_b}).$$
Here $\Gamma(F_{A_b})\in A^0({\rm End}(\Sigma\otimes E))$ is the Hermitian
endomorphism defined as the composition
$$A^0(\Sigma\otimes E)\textmap{F_{A_b}}A^0(\Lambda^2\otimes\Sigma\otimes E)
\textmap{\Gamma} A^0({\rm End}_0(\Sigma)\otimes\Sigma\otimes
E)\textmap{ev}A^0(\Sigma\otimes E).$$
We set $F_{A,b}:=F_A+\frac{1}{2}F_b{\rm id}_E\in A^0(\Lambda^2\otimes{\rm End}(E))$.
\begin{pr}
Let $s$ be the scalar curvature of the Riemannian 4-manifold $(X,g)$. Fix a
${\rm Spin}^c(4)$-structure on $X$ and choose connections $b$ and $A$ in $L$ and $E$
respectively. Then
$$\hskip 4pt{\not}{D}_{A,b}^2=\nabla_{A_b}^*\nabla_{A_b}+\Gamma(F_{A,b})+\frac{s}{4}
{\rm id}_{\Sigma\otimes E}.$$
\end{pr}
{\bf Proof: } Since $\Gamma(F_g)=\frac{s}{4}{\rm id}_{\Sigma}$ [H1], and
$F_{A_b}=F_{B(b)}\otimes
{\rm id}_E+{\rm id}_{\Sigma}\otimes
F_A=\frac{1}{2}F_b{\rm id}_{\Sigma}\otimes{\rm id}_E+F_g\otimes{\rm id}_E+{\rm id}_{\Sigma}\otimes
F_A={\rm id}_{\Sigma}\otimes(F_A+\frac{1}{2}F_b{\rm id}_E)+F_g{\rm id}_E$, we find
$\Gamma(F_{A_b})=\Gamma(F_{A,b})+\frac{s}{4}{\rm id}_{\Sigma\otimes E}$.
\hfill\vrule height6pt width6pt depth0pt \bigskip
\begin{re} One has a Bochner-type result for spinors $\Psi$ on which\linebreak
\hbox{$\Gamma(F_{A,b})+\frac{s}{4}{\rm id}_{\Sigma\otimes E}$} is positive: Such a
spinor is harmonic if and only if it is parallel [H1].
\end{re}
Let $(\ ,\ )$ be the pointwise inner product on $\Sigma\otimes E$, $|\ |$
the associated pointwise norm, and $\parallel\ \parallel$ the corresponding
$L^2$-norm. For a spinor
$\Psi\in A^0(\Sigma^{\pm}\otimes E)$ we define
$(\Psi\bar\Psi)_0\in A^0({\rm End}_0(\Sigma^{\pm}\otimes E))$ as the image of the
Hermitian endomorphism $\Psi\otimes\bar\Psi\in A^0({\rm End}(\Sigma^{\pm}\otimes
E))$
under the projection ${\rm End}(\Sigma^{\pm}\otimes
E)\longrightarrow{\rm End}_0(\Sigma^{\pm})\otimes{\rm End}(E)$.
\begin{co}
With the notations above, we have
$$(\hskip 4pt{\not}{D}_{A,b}^2\Psi,\Psi)=(\nabla_{A_b}^*\nabla_{A_b}\Psi,\Psi)+(\Gamma(F_{A,
b}^+),
(\Psi_+\bar\Psi_+)_0)+(\Gamma(F_{A,b}^-),
(\Psi_-\bar\Psi_-)_0)+\frac{s}{4}|\Psi|^2,$$
where ($F_{A,b}^-$) $F_{A,b}^{+}$ is the (anti-)self-dual component of
$F_{A,b}$.
\end{co}
{\bf Proof: } Indeed, since $\Gamma(F_{A,b}^{\pm})$ vanishes on $\Sigma^{\mp}$ and is
trace
free with respect to
$\Sigma^{\pm}$, the inner product
$(\Gamma(F_{A,b}),(\Psi\bar\Psi))$ in the Weitzenb\"ock formula simplifies for
a
spinor
$\Psi\in A^0(\Sigma^{\pm}\otimes E) $:
$$(\Gamma(F_{A,b}),(\Psi\bar\Psi))=(\Gamma(F_{A,b}^{\pm}),(\Psi\bar\Psi)_0)$$
\hfill\vrule height6pt width6pt depth0pt \bigskip
For a positive spinor $\Psi\in A^0(E\otimes\Sigma^+)$, the following
important identity follows immediately:
$$(\hskip 4pt{\not}{D}_{A,b}^2\Psi,\Psi)+\frac{1}{2}|\Gamma(F_{A,b}^+)-(\Psi\bar\Psi)_0|^2=
(\nabla_{A_b}^*\nabla_{A_b}\Psi,\Psi)+
\frac{1}{2}|F_{A,b}^+|^2+\frac{1}{2}|(\Psi\bar\Psi)_0|^2+\frac{s}{4}|\Psi|^2
\eqno{(4)}$$
If we integrate both sides of (4) over $X$, we get:
\begin{pr} Let $(X,g)$ be an oriented, closed Riemannian 4-manifold with scalar
curvature $s$, $E$ a Hermitian bundle over $X$. Choose a ${\rm Spin}^c(4)$-structure
on $X$ and a connection $b$ in the determinant line bundle
$L=\det(\Sigma^+)=\det(\Sigma^-)$. Let $A$ be a connection in $E$. For any
$\Psi\in A^0(\Sigma^+\otimes E)$ we have:
$$\parallel\hskip 4pt{\not}{D}_{A,b}\Psi\parallel^2+
\frac{1}{2}\parallel\Gamma(F_{A,b}^+)-(\Psi\bar\Psi)_0\parallel^2=$$ $$=
\parallel\nabla_{A_b}\Psi\parallel^2+
\frac{1}{2}\parallel
F_{A,b}^+\parallel^2+\frac{1}{2}\parallel(\Psi\bar\Psi)_0\parallel^2+
\frac{1}{4}\int\limits_X s|\Psi|^2.$$
\end{pr}
We introduce now our coupled variant of the Seiberg-Witten equations. The
unknown is a pair $(A,\Psi)$ consisting of a connection in the Hermitian bundle
$E$ and a section $\Psi\in A^0(\Sigma^+\otimes E)$. The equations ask for the
vanishing of the left-hand side in the above formula.
$$\left\{\begin{array}{ccc}\hskip 4pt{\not}{D}_{A,b}\Psi&=&0\\
\Gamma(F_{A,b}^+)&=&(\Psi\bar\Psi)_0
\end{array}\right.\eqno{(SW)}$$
Proposition 2.4 and the inequality
$|(\Psi\bar\Psi)_0|^2\geq\frac{1}{2}|\Psi|^4$
give immediately:
\begin{re} If the scalar curvature $s$ is nonnegative on $X$, then the only
solutions of the equations are the pairs $(A,0)$, with $F_{A,b}^+=0$.
\end{re}
If $L$ is the square of a line bundle $L^{\frac{1}{2}}$, and if we choose a
connection $b^{\frac{1}{2}}$ in $L^{\frac{1}{2}}$ with square $b$, then
$F_{A,b}$ is simply the curvature of the connection $A_{b^{\frac{1}{2}}}$ in
$E\otimes L^{\frac{1}{2}}$. The solution of the coupled Seiberg-Witten
equations
on a manifold with $s\geq 0$ are in this case just ${\rm U}(r)$-instantons on
$E\otimes L^{\frac{1}{2}}$.
In the case of a K\"ahler surface $(X,g)$, the coupled Seiberg-Witten equation
can be reformulated in terms of complex geometry. The point is that if we
consider the canonical ${\rm Spin}^c(4)$-structure associated to the K\"ahler
structure, the Dirac operator has a very simple form [H1]. The determinant of
this ${\rm Spin}^c(4)$-structure is the anti-canonical bundle $K_X^{\vee}$ of the
surface, which comes with a holomorphic structure and a natural metric
inherited from the holomorphic tangent bundle.
Let $c$ be the Chern connection in $K_X^{\vee}$. With this choice, the induced
connection $B(c)$ in
$\Sigma=\Lambda^{00}\oplus\Lambda^{02}\oplus\Lambda^{01}$ coincides with the
connection defined by the Levi-Civita connection. Recall that on a K\"ahler
manifold, the almost complex structure is parallel with respect to the
Levi-Civita connection, so that the splitting of the exterior algebra
$\bigoplus\limits_{p}\Lambda^p\otimes{\Bbb C}$ becomes parallel, too.
\begin{pr}
Let $(X,g)$ be a K\"ahler surface with Chern connection $c$ in $K_X^{\vee}$.
Choose a connection $A$ in a Hermitian vector bundle
$E$ over $X$ and a section $\Psi=\varphi+\alpha\in
A^0(E)+A^0(\Lambda^{02}\otimes E)$.
The pair $(A,\Psi)$ satisfies the Seiberg-Witten equations iff the following
identities hold:
$$
\begin{array}{lll}
F_{A,c}^{20}&=&-\frac{1}{2}\varphi\otimes\bar\alpha\\
F_{A,c}^{02}&=&\frac{1}{2}\alpha\otimes\bar\varphi\\
i\Lambda_g F_{A,c}&=&-\frac{1}{2}\left(\varphi\otimes\bar\varphi-
*(\alpha\otimes\bar\alpha)\right)\\
\bar\partial_A\varphi&=&i\Lambda_g\partial_A\alpha\end{array}$$
\end{pr}
{\bf Proof: } The Dirac operator is in this case
$\hskip 4pt{\not}{D}_{A,c}=\sqrt{2}(\bar\partial_A-i\Lambda_g\partial_A)$, and the endomorphism
$(\Psi\bar\Psi)_0$ has the form:
$$\left[\matrix{\frac{1}{2}(\varphi\otimes\bar\varphi-
*(\alpha\otimes\bar\alpha))&*(\varphi\otimes\bar\alpha\wedge\cdot)\cr
\alpha\otimes\bar\varphi&-\frac{1}{2}(\varphi\otimes\bar\varphi-
*(\alpha\otimes\bar\alpha))\cr}\right].$$
Since $\Gamma(F_{A,c}^+)=\Gamma(F_{A,c}^{20}+F_{A,c}^{02}+\frac{1}{2}\Lambda_g
F_{A,c}\cdot\omega_g)$ equals
$$2\left[\matrix{-\frac{i}{2}\Lambda_gF_{A,c}&-*(F_{A,c}^{20}\wedge\cdot)\cr
F_{A,c}^{20}\wedge\cdot&\frac{i}{2}\Lambda_gF_{A,c}\cr}\right],$$
the equivalence of the two systems of equations follows.
\hfill\vrule height6pt width6pt depth0pt \bigskip
\section{Monopoles on K\"ahler surfaces and the generalized vortex equation}
Let $(X,g)$ be a K\"ahler surface with canonical ${\rm Spin}^c(4)$-structure, and
Chern connection $c$ in the anti-canonical bundle $K_X^{\vee}$.
We fix a unitary vector bundle $E$ of rank $r$ over $X$, and define
$J(E):={\rm deg}_g(\Sigma^+\otimes E)$, i.e.
$J(E)=2r(\mu_g(E)-\frac{1}{2}\mu_g(K_X))$, where $\mu_g$ denotes the slope with
respect to $\omega_g$.
Every spinor $\Psi\in A^0(\Sigma^+\otimes E)$ has the form
$\Psi=\varphi+\alpha$ with $\varphi\in A^0(E)$ and $\alpha\in
A^{0}(\Lambda^{02}\otimes E)$.
We have seen that the coupled Seiberg-Witten equations are equivalent to the
system:
$$\left\{
\begin{array}{lll}
F_{A,c}^{20}&=&-\frac{1}{2}\varphi\otimes\bar\alpha\\
F_{A,c}^{02}&=&\frac{1}{2}\alpha\otimes\bar\varphi\\
i\Lambda_g F_{A,c}&=&-\frac{1}{2}\left(\varphi\otimes\bar\varphi-
*(\alpha\otimes\bar\alpha)\right)\\
\bar\partial_A\varphi&=&i\Lambda_g\partial_A\alpha\end{array}\right.
\eqno{(SW^*)}$$
\begin{lm} \hfill{\break}
A. Suppose $J(E)<0$: \hfill{\break}
A pair $(A,\varphi+\alpha)$ is a solution of the system $(SW^*)$ if and only
if \hfill{\break}
i) $F_A^{20}=F_A^{02}=0$\hfill{\break}
ii) $\alpha=0$, $\bar\partial_A\varphi=0$ \hfill{\break}
iii) $i\Lambda_g F_A+\frac{1}{2}\varphi\otimes\bar\varphi+\frac{1}{2}s{\rm id}_E=0$.
\hfill{\break}
B. Suppose $J(E)>0$, and put $a:=\bar\alpha\in A^{20}(\bar E)=A^0(E^{\vee}
\otimes K_X)$:\hfill{\break}
A pair $(A,\varphi+\bar a)$ is a
solution of the system
$(SW^*)$ if and only if\hfill{\break}
i) $F_A^{20}=F_A^{02}=0$\hfill{\break}
ii) $\varphi=0$, $\bar\partial_A a=0$ \hfill{\break}
iii) $i\Lambda_g F_A-\frac{1}{2}*(a\otimes\bar a)+\frac{1}{2}s{\rm id}_E=0$.
\end{lm}
{\bf Proof: } (cf. [W]) The splitting $\Sigma^+\otimes E=\Lambda^{00}\otimes
E\oplus\Lambda^{02}\otimes E$ is parallel with respect to $\nabla_{A,c}$, so
that, by Proposition 2.4
$$\parallel\hskip 4pt{\not}{D}_{A,c}\Psi\parallel^2+
\frac{1}{2}\parallel\Gamma(F_{A,c}^+)-(\Psi\bar\Psi)_0\parallel^2=$$ $$=
\parallel\nabla_{A_c}\varphi\parallel^2+
\parallel\nabla_{A_c}\alpha\parallel^2+
\frac{1}{2}\parallel
F_{A,c}^+\parallel^2+\frac{1}{2}\parallel(\Psi\bar\Psi)_0\parallel^2+
\frac{1}{4}\int\limits_X s(|\varphi|^2+|\alpha|^2).$$
The right-hand side is invariant under the transformation
$(A,\varphi,\alpha)\longmapsto (A,\varphi,-\alpha)$, hence any solution
$(A,\varphi+\alpha)$ must have $F_A^{20}=F_A^{02}=0$ and
$\varphi\otimes\bar\alpha=\alpha\otimes\bar\varphi=0$; the latter implies
obviously $\alpha=0$ or $\varphi=0$. Integrating the trace of the equation
$i\Lambda F_{A,c}=-\frac{1}{2}\left(\varphi\otimes\bar\varphi-
*(\alpha\otimes\bar\alpha)\right)$, we find:
$$J(E)=c_1(\Sigma^+\otimes
E)\cup[\omega_g]=(2c_1(E)-rc_1(K_X))\cup[\omega_g]=$$ $$=
2\int\limits_X\frac{i}{2\pi}{\rm Tr}(F_{A,c})\wedge\omega_g=
\frac{1}{4\pi}\int\limits_X{\rm Tr}(i\Lambda
F_{A,c})\omega_g^2=\frac{1}{8\pi}\int\limits_X{\rm Tr}(-\varphi\otimes\bar\varphi)
+*(\alpha\otimes\bar\alpha))\omega_g^2$$
This equation shows that we must have $\alpha=0$, if $J(E)<0$, and
$\varphi=0$, if $J(E)>0$.
Notice
that, replacing $E$ by $E^{\vee}\otimes K_X$, the second case reduces to the
first one.
The assertion follows now immediately from the identity
$i\Lambda_g F_c=s$.
\hfill\vrule height6pt width6pt depth0pt \bigskip
Notice that the last equation
$$i\Lambda_g F_A+\frac{1}{2}\varphi\otimes\bar\varphi+\frac{1}{2}s{\rm id}_E=0$$
has the form of a generalized vortex equation as studied by Bradlow [B1], [B2]
and by Garcia-Prada [G2]; it is precisely the vortex equation with constant
$\tau=-{s}$, if $(X,g)$ has constant scalar curvature.
Let $s_m$ be the mean scalar curvature defined by
$\int\limits_Xs\omega_g^2=s_m\int\limits_X\omega^2=2s_m{\rm Vol}_g(X)$.
We are going to prove that the system
$$\left\{\begin{array}{cl}\bar\partial_A^2&=0\\ \bar\partial_A\varphi&=0\\
i\Lambda_g
F_A+\frac{1}{2}\varphi\otimes\bar\varphi+\frac{1}{2}s{\rm id}_E&=0
\end{array}\right.$$
for the pair $(A,\varphi)$ consisting of a unitary connection in $E$, and a
section in $E$, is always equivalent to the vortex system with parameter
$\tau=-s_m$, i.e. to the system obtained by replacing the third equation with
$$i\Lambda_g F_A+\frac{1}{2}\varphi\otimes\bar\varphi+\frac{1}{2}s_m{\rm id}_E=0.$$
"Equivalent" means here that the corresponding moduli spaces of solutions are
naturally isomorphic.
Let generally $t$ be a smooth real function on $X$ with mean value $t_m$, and
consider the following system of equations:
$$\left\{\begin{array}{cl}\bar\partial_A^2&=0\\ \bar\partial_A\varphi&=0\\
i\Lambda_g F_A+\frac{1}{2}\varphi\otimes\bar\varphi-\frac{1}{2}t{\rm id}_E&=0
\end{array}\right.\eqno(V_t)$$
$(V_t)$ is defined on the space ${\cal A}(E)\times A^0(E)$, where ${\cal A}(E)$ is the
space of unitary connections in $E$. The product ${\cal A}(E)\times A^0(E)$
(completed with respect to sufficiently large Sobolev indices) carries a
natural
$L^2$ K\"ahler metric $\tilde g$ and a natural right action of the gauge group
$U(E)$: $(A,\varphi)^f:=(A^f,f^{-1}\varphi)$, where
$d_{A^f}:=f^{-1}\circ d_A\circ f$.
For every real function $t$ let
$$m_t:{\cal A}(E)\times A^0(E)\longrightarrow A^0({\rm ad}(E))$$
be the map given by $m_t:=\Lambda_g
F_A-\frac{i}{2}\varphi\otimes\bar\varphi+\frac{i}{2}t{\rm id}_E.$
\begin{pr}
$m_t$ is a moment map for the action of $U(E)$ on \linebreak ${\cal A}(E)\times
A^0(E)$.
\end{pr}
{\bf Proof: } Let $a^{\#}$ be the vector field on ${\cal A}(E)\times A^0(E)$ associated
with the infinitesimal transformation $a\in A^0({\rm ad}(E))={\rm Lie }(U(E))$, and
define the real function $m^a_t:{\cal A}(E)\times A^0(E)\longrightarrow{\Bbb R}$ by:
$$ m^a_t(x)=\langle m_t(x),a\rangle_{L^2} .$$
We
have to show that
$m_t$ satisfies the identities:
$$\iota_{a^{\#}}\omega_{\tilde g}=dm_t^a \ ,\ \ \ m_t^a\circ
f=m^{{\rm ad}_f(a)}\ \ \ {\rm for\ all}\
\ a\in A^0({\rm ad}(E)), \ \ f\in U(E).$$
It is well known that, in general, a moment map for a group action in a
symplectic
manifold is well defined up to a constant central element in the Lie algebra
of the group. In our case, the center of the Lie algebra
$A^0({\rm ad}(E))$ of the gauge group is just $iA^0{\rm id}_E$, hence it suffices to show
that $m_0$ is a moment map. This has already been noticed by Garcia-Prada [G1],
[G2].
\hfill\vrule height6pt width6pt depth0pt \bigskip
Note also that in our case every moment map has the form $m_t$ for some
function
$t$, which shows that from the point of view of symplectic geometry, the
natural
equations are the generalized vortex equations
$(V_t)$.
In order to show that Bradlow's main result [B2] also holds for the
generalized system $(V_t)$, we have to recall some definitions.
Let ${\cal E}$ be a holomorphic vector bundle of topological type $E$, and let
$\varphi\in H^0({\cal E})$ be a holomorphic section. The pair $({\cal E},\varphi)$ is
$\lambda$-\underbar{stable} with respect to a constant $\lambda\in{\Bbb R}$ iff the
following conditions hold:\\
(1) $\mu_g({\cal E})<\lambda$ and $\mu_g({\cal F})<\lambda$ for all reflexive subsheaves
${\cal F}\subset{\cal E}$ with $0<{\rm rk}({\cal F})<r$.\\
(2) $\mu_g({\cal E}/{\cal F})>\lambda$ for all reflexive subsheaves
${\cal F}\subset{\cal E}$ with $0<{\rm rk}({\cal F})<r$ and $\varphi\in H^0({\cal F})$.
\begin{th} Let $(X,g)$ be a closed K\"ahler manifold, $t\in A^0$ a real
function, and $({\cal E},\varphi)$ a holomorphic pair over $X$. Set
$\lambda:=\frac{1}{4\pi} t_m{\rm Vol}_g(X)$. ${\cal E}$ admits a Hermitian metric $h$
such that the associated Chern connection $A_h$ satisfies the vortex equation
$$i\Lambda_g F_A+\frac{1}{2}\varphi\otimes\bar\varphi-\frac{1}{2}t{\rm id}_E=0$$
iff one of the following conditions holds:\\
(i) $({\cal E},\varphi)$ is $\lambda$-stable\\
(ii) ${\cal E}$ admits a splitting ${\cal E}={\cal E}'\oplus{\cal E}''$ with $\varphi\in
H^0({\cal E}')$ such that $({\cal E}',\varphi)$ is $\lambda$-stable, and ${\cal E}''$ admits
a weak Hermitian-Einstein metric with factor $\frac{t}{2}$. In particular
${\cal E}''$ is polystable of slope $\lambda$.
\end{th}
{\bf Proof: } In the case of a constant function $t=\tau$, the theorem was proved by
Bradlow [B2], and his arguments work in the general context, too: The fact
that the existence of a solution of the vortex equation implies $(i)$ or
$(ii)$ follows by replacing the constant $\tau$ in [B2] everywhere with the
function $t$. The difficult part consists in showing that every
$\lambda$-stable pair $({\cal E},\varphi)$ admits a metric $h$ such that
$(A_h,\varphi)$ satisfies the vortex equation $(V_t)$. To this end let
$Met(E)$ be the space of Hermitian metrics in $E$, and fix a background
metric $k\in Met(E)$. Bradlow constructs a functional
$M_{\varphi,\tau}(\cdot,\cdot):Met(E)\times Met(E)\longrightarrow{\Bbb R}$, which is convex in
the second argument, such that any critical point of
$M_{\varphi,\tau}(k,\cdot)$ is a solution of the vortex equation; the point
is then to find an absolute minimum of $M_{\varphi,\tau}(k,\cdot)$. The
existence of an absolute minimum follows from the following basic
${\cal C}^0$ estimate:
\begin{lm}{\rm (Bradlow)} Let $Met^p_2(E,B):=\{h=ke^a| a\in L^2_p({\rm End}(E)),
a^{*k}=a, \parallel\mu_t(A_h,\varphi)\parallel_{L^p}\leq B\}$. If
$({\cal E},\varphi)$ is $\frac{\tau}{4\pi}{\rm Vol}_g(X)$-stable, then there exist
positive constants $C_1$, $C_2$ such that
$$\sup|a|\leq C_1M_{\varphi,\tau}(k,ke^a)+ C_2,$$
for all $k$-Hermitian endomorphisms $a\in L^2_p({\rm End}(E))$. Moreover, any
absolute minimum of $M_{\varphi,t}(k,\cdot)$ on $Met^p_2(E,B)$ is a critical
point of $M_{\varphi,t}(k,\cdot)$, and gives a solution of the vortex
equation $V_{\tau}$.
\end{lm}
Let now $t$ be a real function on $X$, and choose a solution $v$ of the
Laplace equation $i\Lambda_g\bar\partial\partial v=\frac{1}{2}(t-t_m)$. If we
make the substituion $h=h'e^v$, then $h$ solves the vortex equation $(V_t)$
iff $h'$ is a solution of
$$i\Lambda_g
F_{h'}+\frac{1}{2}e^v\varphi\otimes\bar\varphi^{h'}-\frac{1}{2}t_m{\rm id}_E=0.$$
Define $\mu_{t_m,v}(h'):=i\Lambda_g
F_{h'}+\frac{1}{2}e^v\varphi\otimes\bar\varphi^{h'}-\frac{1}{2}t_m{\rm id}_E=0$, and
$$M_{\varphi,t_m,v}(k,h):=M_D(k,h)+\parallel
e^{\frac{v}{2}}\varphi\parallel^2_h-\parallel
e^{\frac{v}{2}}\varphi\parallel^2_k-t_m\int\limits_X{\rm Tr}(\log(k^{-1}h)),$$
where $M_D$ is the Donaldson functional [D]. Then it is not difficult to show
that all arguments of Bradlow remain correct after replacing $\mu_{t_m}$ and
$M_{\varphi,t_m}$ with $\mu_{t_m,v}$ and $M_{\varphi,t_m,v}$ respectively.
Indeed, let $l$ be a positive bound from below for the map $e^v$. Then
$$\begin{array}{ll}M_{\varphi,t_m}(k,ke^{a+\log l})&\leq
M_D(k,ke^{a})+M_D(ke^{a},lke^{a})+\parallel
l\varphi\parallel^2_h-t_m\int\limits_X{\rm Tr}\log(lk^{-1}h)\cr
&\leq M_{\varphi,t_m,v}(k,ke^{a})+\parallel
e^{\frac{v}{2}}\varphi\parallel^2_k+2\log l{\rm deg}_g(E)-rt_m\log l{\rm Vol}_g(X)\cr
&\leq M_{\varphi,t_m,v}(k,ke^{a})+C'(k,\varphi,v,l).\end{array}$$
Similarly, we get constants $n>0$, $C''$ and an inequality
$$M_{\varphi,t_m,v}(k,ke^{a+\log n})\leq M_{\varphi,t_m}(k,ke^{a})+C'',$$
which shows that the basic ${\cal C}^0$ estimate in the Lemma above holds for
$M_{\varphi,t_m,v}$ iff it holds for Bradlow's functional
$M_{\varphi,t_m}$.
\hfill\vrule height6pt width6pt depth0pt \bigskip
\begin{re}
In the special case of a rank-1 bundle $E$, a much more elementary proof
based on [B1] is possible.
\end{re}
\section{Moduli spaces of monopoles, vortices, and stable pairs}
Let $(X,g)$ be a closed K\"ahler manifold of arbitrary dimension, and fix a
unitary vector bundle $E$ of rank $r$ over $X$. We denote by $\bar{{\cal A}}(E)$
the affine space of semiconnection of type $(0,1)$ in $E$. the complex gauge
group $GL(E)$ acts on $\bar{{\cal A}}(E)\times A^0(E)$ from the right by
$(\bar\partial_A,\varphi)^g:=(g^{-1}\circ\bar\partial_A\circ g,
g^{-1}\varphi)$;
this action becomes complex analytic after suitable Sobolev completions. We
denote
by
$\bar{\cal S}(E)$ the set of pairs $(\bar\partial_A,\varphi)$ with trivial isotropy
group. Notice that $\varphi\ne 0$ when
$(\bar\partial_A,\varphi)\in\bar{\cal S}(E)$,
and that $\bar{\cal S}(E)$ is an open subset of $\bar{{\cal A}}(E)\times A^0(E)$, by
elliptic semi-continuity [K].
The action of $GL(E)$ on $\bar{\cal S}(E)$ is free, by definition, and we denote
the Hilbert manifold $\qmod{\bar{\cal S}(E)}{GL(E)}$ by $\bar{{\cal B}}^s(E)$.
The map $p:\bar{{\cal A}}(E)\times A^0(E)\longrightarrow A^{02}({\rm End}(E)\oplus A^{01}(E)$
defined by $p(\bar\partial_A,\varphi)=(F_A^{02},\bar\partial_A\varphi)$ is
equivariant with respect to the natural actions of $GL(E)$, hence it gives
rise to a section $\hat p$ in the associated vector bundle
$\bar{\cal S}(E)\times_{GL(E)}\left(A^{02}({\rm End}(E)\oplus A^{01}(E)\right)$ over
$\bar{{\cal B}}^s(E)$. We define the moduli space of \underbar{simple pairs} of type
$E$ to be the zero-locus $Z(\hat{p})$ of this section. $Z(\hat{p})$ can be
identified with the set of isomorphism classes consisting of a holomorphic
bundle ${\cal E}$ of differentiable type $E$, and a holomorphic section
$\varphi\ne 0$, such that the kernel of the evaluation map
$ev(\varphi):H^0({\rm End}({\cal E}))\longrightarrow H^0({\cal E})$ is trivial.
In a similar way we define the moduli space
${\cal V}^g_t$ of gauge-equivalence classes of irreducible solutions of the
generalized vortex equation $V_t$:
Let $B^+$ denote as usual the subbundle
$\left((\Lambda^{02}+\Lambda^{20})\cap \Lambda^2\right)\oplus
\Lambda^0\omega$ of
the bundle $\Lambda^2$ of real 2-forms on $X$. We denote by ${\cal D}^*$ the
open subset of the product ${\cal D}:={\cal A}(E)\times A^0(E)\simeq\bar{{\cal A}}(E)\times
A^0(E)$ consisting of pairs with trivial isotropy group with respect to the
action of the gauge group $U(E)$. The quotient
${\cal B}^*(E):=\qmod{{\cal D}^*(E)}{U(E)}$ comes with the structure of a real-analytic
manifold.
Let $v:{\cal D}(E)\longrightarrow A^0(B^+\otimes{\rm ad}(E))\oplus A^{01}(E)$ be the map given by:
$$v(A,\varphi)=(F^{20}+F^{02},m_t(A,\varphi)\omega_g{\rm id}_E,\bar\partial_A\varphi).$$
Again $v$ is $U(E)$-equivariant, and the moduli space
${\cal V}^g_t$ of $t$-vortices is defined to be the zero-locus $Z(\hat{v})$ of the
induced section $\hat v$ of \linebreak
\hbox{${\cal D}^*(E)\times_{U(E)}A^0(B^+\otimes{\rm ad}(E))\oplus A^{01}(E)$} over
${\cal B}^*(E)$.
Notice now that by Proposition 3.2, the second component $v^2$ of $v$ is a
moment map for the $U(E)$ action. It is easy to see that (at least
in a neighbourhood of $Z(v)\cap{\cal D}^*$) it has the general
property of a moment map in the finite dimensional K\"ahler geometry: Its zero
locus $Z(v^2)$ is smooth and intersects every $GL(E)$ orbit along a $U(E)$
orbit,
and the intersection is transversal. This means that the natural map
$A\longrightarrow\bar\partial_A$ defines in a neighbourhood of $Z(\hat{v})\cap{\cal B}^*(E)$ an
open embedding $i:Z({\hat{v}^2})\longrightarrow\bar{\cal B}^s$ of smooth Hilbert manifolds.
Regard now ${\cal V}^g_t$ as the subspace of $Z({\hat{v}^2})\subset{\cal B}^*(E)$
defined by the equation $(\hat{v}^1,\hat{v}^3)=0$. On the other hand, the
pullback of the equation
$\hat p=0$, cutting out the moduli space $Z(\hat{p})$ of simple holomorphic
pairs,
via the open embedding $i$ is precisely the equation
$(\hat{v}^1,\hat{v}^3)=0$, cutting out ${\cal V}^g_t$. We get therefore an open
embedding $i_0:{\cal V}^g_t\longrightarrow Z(\hat{p})$ of real analytic spaces induced by
$i$, and by Theorem 3.3 the image of $i_0$ consists of the set of
$\lambda$-stable pairs, with $\lambda:=\frac{1}{4\pi}t_m{\rm Vol}_g(X)$.
Finally we denote by ${\cal M}_X^g(E,\lambda)\subset Z(\hat p)$ the open subspace of
$\lambda$-stable pairs, with the induced complex space-structure. Putting
everything together, we have:
\begin{th}
Let $(X,g)$ be a closed K\"ahler manifold, $t\in A^0$ a real function, and
$\lambda: =\frac{1}{4\pi}t_m{\rm Vol}_g(X)$. Fix a unitary vector bundle $E$ of rank
$r$ over $X$. There are natural real-analytic isomorphisms of moduli spaces
$${\cal V}^g_t(E)\simeq{\cal V}^g_{t_m}(E)\simeq{\cal M}_X^g(E,\lambda).$$
\end{th}
Let us come back now to the monopole equation $(SW^*)$ on a K\"ahler surface.
In
this case the function $t$ is the negative of the scalar curvature $s$, so that
the corresponding constant $\lambda$ becomes:
$$\lambda=\frac{-s_m}{4\pi}{\rm Vol}_g(X)=-\frac{1}{8\pi}\int\limits_Xs\omega^2=
-\frac{1}{8\pi}\int\limits_X(i\Lambda
F_c)\omega^2=-\frac{1}{4\pi}\int\limits_X i
F_c\wedge\omega=\frac{1}{2}\mu_g(K).$$
This yields our main result:
\begin{th}
Let $(X,g)$ be a K\"ahler surface with canonical ${\rm Spin}^c(4)$-structure, and
Chern connection $c$ in $K_X^{\vee}$. Fix a unitary vector bundle $E$ of
rank $r$
over $X$, and suppose $J(E)={\rm deg}_g(\Sigma^+\otimes E)<0$. The moduli space of
solutions of the coupled Seiberg-Witten equations is isomorphic to the moduli
space ${\cal M}_X^g(E,\frac{1}{2}\mu_g(K))$ of $\frac{1}{2}\mu_g(K)$-stable pairs of
topological type $E$.
\end{th}
At this point it is natural to study the properties of the moduli spaces
${\cal M}^g_X(E,\lambda)$. We do not want to go into details here, and we content
ourselves by describing some of the basic steps.
The infinitesimal structure of the moduli space around a point
$[(A,\varphi)]$ is
given by a deformation complex
$(C_{\bar\partial_A,\varphi}^*, d_{A,\varphi}^*)$ which is the cone over the
evaluation map $ev^*$, $ev^q(\varphi):A^{0q}({\rm End}(E))\longrightarrow A^{0q}(E)$. More
precisely $C_{\bar\partial_A,\varphi}^q=A^{0q}({\rm End}(E))\oplus A^{0,q-1}(E)$ and
the differential $d_{A,\varphi}^q$ is given by the matrix
$$d_{A,\varphi}^q=\left[\matrix{-\bar
D_A&0\cr ev(\varphi)&\bar\partial_A\cr}\right],$$
where $\bar\partial_A$ and $\bar D_A$ are the operators of the Dolbeault
complexes $(A^{0*}(E),\bar\partial_A)$ and $(A^{0*}{\rm End}(E),\bar D_A)$
respectively.
Associated to the morphism $ev^*(\varphi)$ is an exact sequence
$$\dots\longrightarrow H^q({\rm End}({\cal E}_A))\textmap{ev^q(\varphi)}H^q({\cal E}_A)\longrightarrow
H_{\bar\partial_A,\varphi}^{q+1}\longrightarrow H^{q+1}({\rm End}({\cal E}_A))\longrightarrow\dots $$
with finite dimensional vector spaces
$$H^q_{\bar\partial_A,\varphi}=
\ker(ev^q(\varphi))\oplus{\rm coker}(ev^{q-1}(\varphi)).$$
$H^0_{\bar\partial_A,\varphi}$ vanishes for a simple pair
$(\bar\partial_A,\varphi)$, and $H^1_{\bar\partial_A,\varphi}$ is the Zariski
tangent space of ${\cal M}^g_X(E,\lambda)$ at $[\bar\partial_A,\varphi]$.
A Kuranishi type argument yields local models of the moduli space, which can be
locally described as the zero loci of holomorphic map germs
$$K_{[\bar\partial_A,\varphi]}:H^1_{\bar\partial_A,\varphi}\longrightarrow
H^2_{\bar\partial_A,\varphi}$$
at the origin.
One finds that $H^2_{\bar\partial_A,\varphi}=0$ is a sufficient smoothness
criterion in the point $[\bar\partial_A,\varphi]$ of the moduli space, and that
the expected dimension is \linebreak\hbox{$\chi(E)-\chi({\rm End}(E))$}. The
necessary
arguments are very similar to the ones in [BD1], [BD2].
The moduli spaces ${\cal M}^g(E,\lambda)$ will be quasi-projective varieties if the
underlying manifold $(X,g)$ is Hodge, i.e. if $X$ admits a projective embedding
such that a multiple of the K\"ahler class is a polarisation [G1].
A GIT construction for projective varieties of any dimension has been given in
[HL2]. The spaces ${\cal M}^g_X(E,\lambda)$ vary with the
parameter
$\lambda$, and flip-phenomena occur just like in the case of curves [T].
\section{Applications}
The equations considered by Seiberg and Witten are associated to a
${\rm Spin}^c(4)$-structure, and correspond to the case when (in our notations) the
unitary bundle
$E$ is the trivial line bundle. Alternatively, we can fix a ${\rm Spin}^c(4)$
structure ${\germ s}_0$ on $X$ , and regard the Seiberg-Witten equations
corresponding
to the other ${\rm Spin}^c(4)$-structures as {\sl coupled} Seiberg-Witten
equations associated to ${\germ s}_0$ and to a unitary line bundle $E$. The
${\rm Spin}^c(4)$-structure we fix will always be the canonical structure
defined by a
K\"ahler metric. In the most interesting case of rank-1 bundles
$E$ over K\"ahler surfaces the central result is:
\begin{pr}
Let $(X,g)$ be a K\"ahler surface with canonical class $K$, and let $L$ be a
complex line bundle over $X$ with $L\equiv K$ (mod 2). Denote by
${\cal W}_X^g(L)$ the
moduli space of solutions of the Seiberg-Witten equation for all
${\rm Spin}^c(4)$-structures with determinant $L$. Then\hfill{\break}
i) If $\mu(L)<0$, ${\cal W}_X^g(L)$ is isomorphic to the space of all linear systems
$|D|$, where $D$ is a divisor with $c_1({\cal O}_X(2D-K))=L$.\hfill{\break}
ii) If $\mu(L)>0$, ${\cal W}_X^g(L)$ is isomorphic to the space of all linear
systems
$|D|$, where $D$ is a divisor with $c_1({\cal O}_X(2D-K))=-L$.
\end{pr}
{\bf Proof: } Use Theorem 4.2 and Bradlow's description of the moduli spaces of stable
pairs in the case of line bundles [B1].
\hfill\vrule height6pt width6pt depth0pt \bigskip
We have already noticed (Remark 2.5) that in the case of a Riemannian
4-manifold
with nonnegative scalar curvature $s_g$, the Seiberg-Witten equations have only
reducible solutions. In the K\"ahler case, the same result can be obtained
under
the weaker assumption $\sigma_g \geq 0$ on the total scalar curvature.
\begin{co}
Let $(X,g)$ be a K\"ahler surface with nonnegative total scalar curvature
$\sigma_g$. Then all solutions of the Seiberg-Witten equations in rank 1 are
reducible. If moreover the surface has $K^2>0$, then for every almost
canonical class
$L$, the corresponding Seiberg-Witten equations are incompatible.
\end{co}
{\bf Proof: } The first assertion follows directly from the theorem, since the condition
$\sigma_g \geq 0$ is equivalent to $K\cup[\omega_g]\leq 0$. For the second
assertion, note that if $L$ is an almost canonical class, then $L^2=K^2>0$,
hence (regarded as line bundle) it cannot admit anti-selfdual connections.
\hfill\vrule height6pt width6pt depth0pt \bigskip
\begin{re}
The Seiberg-Witten invariants associated to almost canonical classes are
well-defined for oriented, closed 4-manifolds $X$ satisfying $3\sigma+2e>0$.
\end{re}
{\bf Proof: } Recall that if $L$ is an almost canonical class, then the expected
dimension of the moduli space of solutions of the perturbed Seiberg-Witten
equations [W, KM]corresponding to a ${\rm Spin}^c(4)$-structure of determinant $L$
is 0. Seiberg and Witten associate to every such class $L$ the number $n_L$
of points (counted with the correct signs [W]) of such a moduli space chosen to
be smooth and of the expected dimension. In the case $b_+\geq 2$, using
the same cobordism argument as in Donaldson theory, it follows that that
these numbers are well-defined, i.e. independent of the metric, provided the
moduli space has the expected dimension [KM]. The point is that the space of
$L$-good metrics [KM] (i.e. metrics with the property that the space of
harmonic anti-selfdual forms does not contain the harmonic representative of
$c_1^{{\Bbb R}}(L)$) is in this case path-connected. On the other hand, under the
assumption $3\sigma +2e>0$, it follows that
$L^2>0$ for any almost canonical class $L$, hence all metrics are $L$-good.
\hfill\vrule height6pt width6pt depth0pt \bigskip
\begin{pr}
Let $(X,H_0)$ be a polarised surface with $K$ nef and big, and choose a
K\"ahler
metric $g$ with K\"ahler class $[\omega_g]=H_0+nK=:H$ for some $n\geq
KH_0$. Then
${\cal W}_X^g(L)$ is empty for all almost canonical classes, except for $L=\pm K$,
when it consists of a simple point.
\end{pr}
{\bf Proof: } Let $L$ be an almost canonical class with $L H<0$.
Suppose $D$ is an effective divisor with $c_1({\cal O}_X(2D-K))=L$, so that
$D(D-K)=0$. Then $D^2=DK\geq 0$ since $K$ is nef. If $D^2$ were strictly
positive, the Hodge index theorem would give $(D-K)^2\leq 0$, i.e. $K^2\leq
D^2$.
But from $LH<0$ we get $0>(2D-K)(H_0+nK)=(2D-K)H_0+n(2D^2-K^2)\geq
(2D-K)H_0+n$,
which leads to the contradiction $n<(K-2D)H_0\leq KH_0$. Therefore
$D^2=DK=0$, so
that, again by the Hodge index theorem, $D$ must be numerically zero. Since $D$
is effective, it must be empty, and $L=-K$.
Replacing $L$ by $-L$ if $L$ is an almost
canonical class with $LH>0$, we find $L=K$ in this case. The corresponding
Seiberg-Witten moduli spaces are simple points in both cases, since
$H^2_{\bar\partial_A,\varphi}=H^1({\cal O}(D)|_D)=0$.
\hfill\vrule height6pt width6pt depth0pt \bigskip
\begin{co}
There exists no orientation-preserving diffeomorphism between a rational
surface
and a minimal surface of general type.
\end{co}
{\bf Proof: }
Indeed, any rational surface $X$ admits a Hodge metric with
positive total scalar curvature [H2]. If $X$ was
orientation-preservingly diffeomorphic to a minimal surface of general
type, then
$K^2>0$, hence the Seiberg-Witten invariants are well defined (Remark 5.3),
and vanish by Corollary 5.2. Proposition 5.4 shows, however, that the
Seiberg-Witten invariants of a minimal surface of general type are non-trivial
for two almost canonical classes.
\hfill\vrule height6pt width6pt depth0pt \bigskip
\vspace{3mm}\\
Witten has already proved [W] that for a
minimal surface of general type with $p_g>0$ ($b_+\geq 2$), the only almost
canonical classes which give non-trivial invariants are $K$ and $-K$. Their
proof
uses the moduli space of solutions of the perturbation of the Seiberg-Witten
equation with a holomorphic form. Proposition 5.4 shows that a stronger result
can be obtained with the non-perturbed equations by choosing the Hodge metric
$H=H_0+nK,\ n\gg 0$.
For the proof of Corollary 5.5, we need in fact only the mod. 2 version of the
Seiberg-Witten invariants [KM2].
\newpage
\parindent0cm
\centerline {\Large {\bf Bibliography}}
\vspace{1cm}
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[BD2] Bradlow, S. B.; Daskalopoulos, G.: {\it Moduli of stable pairs for
holomorphic bundles over Riemann surfaces II}, Intern. J. Math. 4, 903-925
(1993)
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a compact Riemann surface}, Bull. London Math. Soc., 26, 88-96 (1994)
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modules and their moduli.} Int. J. Math.
6, 297-324 (1995)
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(1980)
[K] Kobayashi, S.: {\it Differential geometry of complex vector bundles},
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[KM2] Kronheimer, P.; Mrowka, T.: {\it The genus of embedded surfaces in the
projective plane}, Preprint (1994)
[OSS] Okonek, Ch.; Schneider, M.; Spindler, H: {\it Vector bundles on complex
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Invent. math. 117, 181-205 (1994)
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\vspace{2cm}\\
Authors addresses:\\
\\
Mathematisches Institut, Universit\"at Z\"urich,\\
Winterthurerstrasse 190, CH-8057 Z\"urich\\
e-mail:[email protected]
\ \ \ \ \ \ \ \ \ [email protected]
\end{document}
|
1995-05-22T06:20:19 | 9505 | alg-geom/9505019 | en | https://arxiv.org/abs/alg-geom/9505019 | [
"alg-geom",
"math.AG"
] | alg-geom/9505019 | Carmen Schuhmann | Carmen Schuhmann | Mapping threefolds onto three-quadrics | 13 pages, LaTeX v. 2.09 | null | null | null | null | We prove that the degree of a nonconstant morphism from a smooth projective
3-fold $X$ with N\'{e}ron-Severi group ${\bf Z}$ to a smooth 3-dimensional
quadric is bounded in terms of numerical invariants of $X$. In the special case
where $X$ is a 3-dimensional cubic we show that there are no such morphisms.
The main tool in the proof is Miyaoka's bound on the number of double points of
a surface.
| [
{
"version": "v1",
"created": "Fri, 19 May 1995 11:46:49 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Schuhmann",
"Carmen",
""
]
] | alg-geom | \section{Proof of Theorem 1}
The proof of Theorem \ref{stelling} is based upon the following result of
Miyaoka (see \cite{Mi}):
\begin{theorem} \label{Miyaoka}
Let $S$ be a complex projective surface with only ordinary double points and
numerically effective dualizing sheaf $K_S$. Let $\tilde{S}$ be the minimal
resolution of $S$. Then
$$\#\mbox{\{double points\}} \leq \frac{2}{3}(c_2(\tilde{S}) - \frac{1}{3}
K_S^2).$$
\end{theorem}
Another important ingredient of the proof is the following lemma,
which will be proven later on. Here the tangent hyperplane to a smooth
quadric $Q$ of dimension 3 at a point
$p \in Q$ is denoted by $T_pQ$.
\begin{lemma} \label{lemma}
Let $X$ be a smooth, projective variety of dimension 3 and
$f$ a finite morphism from $X$ to $Q$. Then there is
a dense open subset $U$ in $Q$ such that $f^*(T_pQ \cap Q)$ has no
singularities away from $f^{-1}(p)$ for all $p\in U$.
\end{lemma}
Before starting the proof of Theorem \ref{stelling}, let us state the
following lemma, which has an elementary proof.
\begin{lemma} \label{NS}
Let $X$, $Y$ be smooth projective threefolds and $f:X \rightarrow Y$ a
morphism. Assume that $X$ has N\'{e}ron-Severi group ${\bf Z}$. Then the
following
two statements are equivalent:
$i$) $f$ is nonconstant;
$ii$) $f$ is finite.
\end{lemma}
{\it Proof of Theorem \ref{stelling}:} Suppose $f:X \rightarrow Q$ is
a nonconstant morphism of generator degree d. By Lemma \ref{NS}, $f$ is finite.
The degree of $f$ is equal to
$H_X^3d^3/2$, where $H_X$ is the ample generator of the N\'{e}ron-Severi
group of $X$.
For every point $p$ on $Q$, the hyperplane section $H_p=T_pQ \cap Q$ is a
quadric with one ordinary double point, namely $p$. So, if $p$ is not
contained in the branch divisor of $f$,
then the surface $f^*(H_p)$ contains $H_X^3d^3/2$ ordinary double
points, which map to $p$ under $f$.
{}From Lemma \ref{lemma} it follows that $f^*(H_p)$ has no other
singularities for general $p$.
Now fix a point $p$ such that $f^*(H_p)$ has exactly $H_X^3d^3/2$
ordinary double points and no other singularities and denote $f^*(H_p)$ by
$S_p$. It will be shown that, if $d$ is large enough, Theorem \ref{Miyaoka} can
be applied to $S_p$ and provides an upper bound on the number of ordinary
double points of $S_p$ which is smaller than $H_X^3d^3/2$. In order to
compute this bound, we have to compute $c_2(\tilde{S_p})$ and $K_{S_p}^2$,
where $\tilde{S_p}$ is the minimal resolution of $S_p$.
By Bertini's Theorem (see \cite{Jo}, Th\'{e}or\`{e}me 6.10), there is
a hyperplane section $H$ of $Q$ such that the surface $f^*(H)$ is
nonsingular. Denote this surface by $S$.
The surfaces $S$ and $\tilde{S_p}$ are homeomorphic (see \cite{A}, Theorem 3),
so $c_2(\tilde{S_p})=c_2(S)$ and $c_1^2(\tilde{S_p})=c_1^2(S)$. As $S_p$
has only ordinary double points, it follows that $K_{S_p}^2=K_{S}^2$. Using
$K_X \equiv kH_X$, where k is
the numerical index of $X$, the adjunction formula gives:
$$K_{S} \cong (K_X + S) | _{S} \equiv (k+d)H_X | _{S},$$
so
$$ K_{S}^2 = (k+d)^2H_X^2 | _{S} = (k+d)^2dH_X^3. $$
The second Chernclass of $S$ can easily be computed by means of adjunction:
$$ c_2(S)=dc_2(X)H_X+d^2(d+k)H_X^3.$$
Thus the expression $\frac{2}{3}(c_2(\tilde{S_p}) - \frac{1}{3}
K_{S_p}^2)$, which equals $\frac{2}{3}(c_2(S) - \frac{1}{3}K_S^2)$ by
previous remarks, becomes the following polynomial expression in $d$:
\begin{equation}
\frac{4}{9}H_X^3d^3 + \frac{2}{9}kH_X^3d^2 + \frac{2}{3}(c_2(X)H_X -
\frac{1}{3}k^2H_X^3)d. \label{eq:bound}
\end{equation}
We can apply Theorem \ref{Miyaoka} to $S_p$ if $S_p$ has only double points and
$K_{S_p}$ is nef. As we remarked before, the first condition is certainly
satisfied because of Lemma \ref{lemma}. As for the second one, using
the adjunction formula, it follows that $K_{S_p}$ is linearly equivalent to
$(K_{X}+S_p) |_{S_p}$. So $K_{S_p}$ is nef if and only if $(K_{X}+S_p)
|_{S_p}$ is nef. As $(K_{X}+S_p) |_{S_p} \equiv (k+d)H_X |_{S_p}$,
this is certainly true if $d \geq -k$.
Notice that this condition is empty if $k \geq -1$. If $k=-2$, then the
condition becomes $d \geq2$, which is also an empty condition as there are no
morphisms of degree 1 between a variety of numerical index -2 and the
quadric $Q$, which has numerical index -3. The only smooth threefolds
with numerical index less than -2 are ${\bf P}^3$,
which has numerical index -4 and $Q$ (see \cite{K-O}). So,
the only cases in which
we cannot apply Theorem \ref{Miyaoka} to $S_p$ are when $X$ is ${\bf P}^3$
and $d \leq 3$ or $X$ is $Q$ and $d \leq 2$. In the other cases, it
tells us that the
number of ordinary double points on $S_p$ is restricted by
the expression (\ref{eq:bound}).
However, we remarked that $S_p$ contains exactly $H_X^3d^3/2$
double points. As the leading term of (\ref{eq:bound}), $4H_X^3d^3/9$,
is smaller than $H_X^3d^3/2$, the expression (\ref{eq:bound})
becomes smaller than $H_X^3d^3/2$ for large $d$. So we obtain a
contradiction if $d$ is larger than the largest positive zero of the polynomial
$H_X^3d^3/2-$(\ref{eq:bound}) (if $X$ is ${\bf P}^3$ resp. $Q$, then we
moreover have to require $d \geq 4$ resp. $d \geq 3$). We conclude
that the generator degree $d$ of $f$ and thus also the degree of $f$
is bounded in terms of the coefficients of this polynomial.
Since $H_X=-c_1(X)/k$, the statement of the theorem follows.\\
{\it Proof of Lemma \ref{lemma}:} Denote the 4-dimensional dual projective
space by ${{\bf P}^4} ^{\vee}$. Let ${Q}^{\vee}$ be the dual variety of $Q$.
It is isomorphic to $Q$ via the following isomorphism:
\[ \begin{array}{rcl}
Q & \longrightarrow & {Q}^{\vee} \\
p & \longmapsto & T_p Q.
\end{array} \]
Given $T_p Q \in {Q}^{\vee}$, denote the surface $f^*(T_p Q \cap Q)
\subset Y$ by $S_p$. We will study the singularities of $S_p$, using
the following criterion:
$$
{}~~~~~~~~~~~~~~~~~~~ x \in S_p \mbox{ is a singularity of } S_p
\Leftrightarrow T_p Q \supset f_* T_x X .~~~~~~~~~~~~~~~~~~~~~~(*)
$$
In order to describe $f_* T_x X$, we introduce some notation. For $i \in \{0,
1,2,3\}$, the set
$$ X_i := \{x \in X \mid f \mbox{ has rank at most } i \mbox{ at }x\} $$
is an algebraic subset of $X$ of dimension $i$. So its image under the
finite map $f$ is an algebraic subset of $Q$ of dimension $i$. Denote
the surface $f(X_2)$ by $B$ (this is just the branch locus of $f$), the
curve $f(X_1)$ by $C$ and the finite set $f(X_0)$ by $R$. Furthermore,
denote the union of all irreducible curves along which $B$ is singular
by $\Gamma$ and the set of isolated singularities of $B$ by $\Sigma$.
Finally, the singular locus of the curve $C$ respectively $\Gamma$ is denoted
by
$Sing(C)$ respectively $Sing(\Gamma)$. Let us now examine when a point $x \in
X$
is a singularity of $S_p$.
If $x \in X_i \backslash X_{i+1}~(i \in \{1,2,3\})$ and $f(x_i)$ is a smooth
point of $f(X_i)$, then $f_*T_xX=T_{f(x)}f(X_i)$. So, according to
criterion $(*)$, $x$ is a singularity of $S_p$ if and only if $T_pQ$
contains $T_{f(x)}X_i$, in other words if and only if $T_pQ$ is tangent
to $f(X_i)$ at the point $f(x)$. Especially, taking $i=3$, it follows that
$x \in X_3 \backslash X_2$ is a singularity of $S_p$ if and only if
$f(x)=p$.
If $x \in X_2 \backslash X_1$ and $f(x) \in \Gamma \backslash Sing(\Gamma)$,
then $f_*T_xX$ contains $T_{f(x)}\Gamma$. So in this case we see from $(*)$
that
if $x$ is a singularity of $S_p$ then $T_pQ$ contains $T_{f(x)}\Gamma$, which
means that $T_pQ$ is tangent to $\Gamma$ at $f(x)$.
Finally, if $x \in X_0$, then $f_*T_xX=0$, so by $(*)$ the point $x$ is
certainly a singularity of $S_p$.
Combining these observations, it follows that $S_p$ has no singularities
away from $f^{-1}(p)$ if the hyperplane $T_pQ$ lies in the intersection
of the following 4 subsets of ${Q}^{\vee}$:
\[
\begin{array}{lll}
U_1 & := & \{H \in {Q}^{\vee} \mid H \cap (R \cup Sing(C) \cup
Sing(\Gamma) \cup \Sigma) = \emptyset \}, \\
U_2 & := & \{H \in {Q}^{\vee} \mid H \mbox{ intersects }
C \mbox{ transversally} \}, \\
U_3 & := & \{H \in {Q}^{\vee} \mid H \mbox{ intersects } \Gamma
\mbox{ transversally} \}, \\
U_4 & := & \{H \in {Q}^{\vee} \mid H \cap B \backslash (\Gamma \cup
\Sigma) \mbox{ is nonsingular} \}.
\end{array}
\]
Thus, in order to prove the lemma, it is sufficient to show that the sets
$U_i$ ($i \in \{1,2,3,4\}$) are Zariski-open in ${Q}^{\vee}$ and nonempty.
As $R \cup Sing(C) \cup
Sing(\Gamma) \cup \Sigma$ is a finite set, this is certainly true for $U_1$.
As for $U_2$, notice that in order to show that a general
hyperplane $H \in {Q}^{\vee}$ intersects $C$
transversally, it is sufficient to show that this is true for every irreducible
component of $C$. So we can without loss of generality assume that $C$ is
irreducible.
Let $Z$ denote the following closed subscheme of
$Q \times {Q}^{\vee}$:
$$ Q \times {Q}^{\vee} \supset Z := \{(x,H) \in Q \times
{Q}^{\vee} \mid x \in H \}. $$
Consider the scheme-theoretic intersection
$(C \times {Q}^{\vee}) \cap Z$ and the restriction of
the projections from $Q \times {Q}^{\vee}$ onto $Q$ respectively ${Q}^{\vee}$
to this subscheme of $Q \times {Q}^{\vee}$:
\[
\begin{array}{l}
q: (C \times {Q}^{\vee}) \cap Z \longrightarrow C,\\
r: (C \times {Q}^{\vee}) \cap Z \longrightarrow {Q}^{\vee}.
\end{array}
\]
The fiber of $r$ over $H \in {Q}^{\vee}$ is the scheme-theoretic
intersection of $C$ and $H$. So in order to show that $U_2$ is open and dense
in ${Q}^{\vee}$,
we have to prove that the general fiber of $r$ is reduced.
Notice that all fibers of $q$ are singular quadric surfaces, so they
have constant Hilbertpolynomial. It follows that
$q$ is flat (see \cite{Ha}, Chapter III, Theorem 9.9). As all fibers of $q$ are
reduced and $q$ itself is flat, we conclude that
$U := q^{-1}(C \setminus Sing(C))$ is reduced.
Now $U$ is a dense open subscheme of $(C \times {Q}^{\vee}) \cap Z$, so
the general fiber of $r$, which is finite, is contained in $U$.
Restricting to those fibers which are contained in the smooth part
of $U$ and have empty intersection with the ramification locus
of $r$, we see that the general fiber of $r$ is reduced.
It follows that $U_2$ is Zariski-open in ${Q}^{\vee}$ and nonempty.
Replacing $C$ by $\Gamma$ and
repeating the above reasoning proves that $U_3$ is also Zariski-open
in ${Q}^{\vee}$ and nonempty.
In order to prove that $U_4$ is open and dense in ${Q}^{\vee}$, we will show
that, for a general element $H \in
{Q}^{\vee}$, the intersection of $H$ and $B$ is smooth away from the
singular locus $\Gamma \cup \Sigma $ of $B$. As it is sufficient to show that
this holds for every irreducible component of $B$, we may without
loss of generality assume that $B$ is irreducible.
Assume first that $B$ is a hyperplane section of $Q$, corresponding to
an element $H_B$ in ${{\bf P}^4} ^{\vee}$. Then the hyperplane sections of
$B$ correspond to the lines in ${{\bf P}^4} ^{\vee}$ through $H_B$. If
$H_B$ is not contained in ${Q}^{\vee}$, then every line through $H_B$
intersects ${Q}^{\vee}$ in 2 (not necessarily distinct) points. So
every hyperplane section of $B$ can be written as the intersection
of $B$ and an element of ${Q}^{\vee}$. If $H_B$ is an element of
${Q}^{\vee}$, then every line through $H_B$ which is not contained in the
tangent space to ${Q}^{\vee}$ at $H_B$ intersects ${Q}^{\vee}$ in exactly
one point different from $H_B$. So in this case the elements of a dense open
subset of the space of hyperplane sections of $B$ can uniquely be written as
the intersection of $B$ and an element of ${Q}^{\vee}$. In both
cases we conclude from Bertini's Theorem (see \cite{G-H}, page 137),
applied to $B$, that the intersection of $B$ and a general element of
${Q}^{\vee}$ is smooth away from $\Gamma \cup \Sigma $. So
$U_4$ is Zariski-open in ${Q}^{\vee}$ and nonempty.
{}From now on, assume that $B$ is not a hyperplane section of $Q$ and
interprete ${Q}^{\vee}$ as a quadric system contained in the space
${\bf P}(H^0(B,{\cal O}_B(1)))$ of all hyperplane sections of $B$.
In order to prove that the intersection of $H$ and $B$ is smooth away from
$\Gamma \cup \Sigma $ for general $H$ in ${Q}^{\vee}$, we will use the
following special case of Bertini's Theorem (see \cite{G-H},
page 137). Here a one-dimensional linear system is called a pencil.
\begin{lemma} \label{Bertini}
Let $X \subset {\bf P}^N$ be a projective variety and $\Lambda \subset {\bf P}(
H^0(X,{\cal O}_X(1)))$ a pencil. Then the general element of $\Lambda$
is smooth away from the base locus of $\Lambda$ and the singular locus of $X$.
\end{lemma}
We will show that ${Q}^{\vee}$ contains enough pencils to globalise
Lemma \ref{Bertini}, which holds for any of these pencils, to a
Bertini type theorem for the whole space ${Q}^{\vee}$.
For every element $H$ of ${Q}^{\vee}$ there is a one-dimensional family
of pencils in ${Q}^{\vee}$ containing $H$, parametrised by a
plane quadric curve. By Lemma \ref{Bertini}, the general element of a
pencil is smooth away from the base locus of the pencil and
$\Gamma \cup \Sigma $. We will call such an element general for that pencil.
Denote the base locus in $B$ of a pencil $\Lambda$ in ${Q}^{\vee}$
by $B_{\Lambda}$.
Denote the isomorphism from $Q$ to ${Q}^{\vee}$, mapping $p \in Q$ to the
hyperplane $T_p Q$, by $T$. An easy computation shows that
\begin{equation}
B_{\Lambda} = T^{-1}(\Lambda) \cap B. \label{eq:base locus}
\end{equation}
By a dimension argument, there is a Zariski-open subset $V$ of ${Q}^{\vee}$
such that every element $H$ of $V$ is general for almost all
pencils containing $H$. We will show that $V \backslash (V \cap T(B))$ is
contained in $U_4$.
Let $T_p Q$ be an element of $V$. By (\ref{eq:base locus}) the intersection
$B_{\Lambda} \cap B_{\Lambda'}$ of the base loci of any 2 pencils $\Lambda$
and $\Lambda'$ in ${Q}^{\vee}$ is empty if $p \not \in B$ and equals $p$
if $p \in B$. As $T_p Q$ is an element of $V$, it follows that the hyperplane
section $T_p Q \cap B$ is smooth away from $\Gamma \cup \Sigma $ if
$p \not \in B$. So the nonempty Zariski-open subset $V \backslash
(V \cap T(B))$ of ${Q}^{\vee}$ is contained in $U_4$. We conclude that $U_4$
is open and dense in ${Q}^{\vee}$.
\begin{remark} {\rm Notice that, if $X$ is ${\bf P}^3$ respectively a smooth
threedimensional quadric $Q$, then the statement of Theorem \ref{stelling}
follows also from Theorem \ref{Lazarsfeld} respectively the following theorem
(see \cite{P-S}):
\begin{theorem} \label{Pasri}
Let $Y$ be a smooth quadric hypersurface of dimension at least 3, $X$ a smooth
projective variety and $f:Y \rightarrow X$ a surjective
morphism. Then either $f$ is an isomorphism or $X$ is isomorphic to a
projective space.
\end{theorem}
Conversely, we conclude from Theorem \ref{stelling} that the degree of a
nonconstant morphism from $Q$ to itself is bounded. So every such morphism
must have degree 1, as otherwise we could produce nonconstant selfmaps of $Q$
of arbitrarily high degree by composition.
More generally, let $X$ be a smooth threefold with N\'{e}ron-Severi
group ${\bf Z}$. From
Theorem \ref{stelling} it follows that, if $X$ does allow some nonconstant
morphism to the quadric (for example if $X$ is a Fermat hypersurface of
even degree in ${\bf P}^4$), then every nonconstant morphism from $X$ to itself
has degree 1, so is an isomorphism.}
\end{remark}
\begin{remark}{\rm Replacing $Q$ by another smooth threefold with
N\'{e}ron-Severi group
${\bf Z}$, I did not manage to prove an analogue of Theorem \ref{stelling}. For
instance, let $Y$ be a smooth complete intersection of $N-3$ hyperplanes of
degrees $m_1, \dots,m_{N-3}$ in ${\bf P}^N$. Let $X$ be as in Theorem
\ref{stelling}
and $f:X \rightarrow Y$ a finite morphism of generator degree $d$. Then the
degree of $f$ equals $H_X^3d^3/(\prod_{i=1}^{N-3} m_i)$.
So, if $H$ is a hyperplane section of $Y$ with $n$ ordinary double points,
none of which lie in the branch locus of $f$, then $f^*(H)$ has
$nH_X^3d^3/(\prod_{i=1}^{N-3} m_i)$ ordinary double points. As the
leading term of the Miyaoka bound for $f^*(H)$ in Theorem \ref{Miyaoka}
is equal to $\frac{4}{9}H_X^3$, it is clear that the idea of the proof of
Theorem \ref{stelling} can only work in this case if
\begin{equation}
n \geq \frac{4}{9}\prod_{i=1}^{N-3} m_i. \label{eq:ci}
\end{equation}
The point is that, in order to apply Theorem \ref{Miyaoka}, it still has
to be checked that for at least one such hyperplane
section $H$ the surface $f^*(H)$ has only ordinary double
points. If $Y$ is a cubic in
${\bf P}^4$ or the intersection of two quadrics in ${\bf P}^5$, it follows from
(\ref{eq:ci}) that we have to study hyperplane sections of $Y$ with at
least 2 ordinary double points. In both cases I did not succeed in proving
an analogue of Lemma \ref{lemma}. If $Y$ is a cubic, the system of
hyperplane sections with 2 ordinary double points has only dimension 2 and
I couldn't apply any Bertini type argument; if $Y$ is an intersection of
2 quadrics, this system has dimension 3, but I could not get any grip
on it.
Another possibility is to try to prove something weaker than Lemma \ref{lemma}.
If one can prove that $f^*(H)$ has only mild singularities apart from the
ordinary double points lying over the double points of $H$, then one may try to
apply Theorem \ref{Miyaoka} after blowing up these singularities.
Of course this is only possible if one knows these singularities well.
If one can prove that $f^*(H)$ contains only quotient singularities, then
one may try to apply a more general version of Theorem \ref{Miyaoka}, valid
for surfaces with only quotient singularities (see \cite{Mi}).
However, these approaches seem quite hard and I did not try them
seriously up to now.}
\end{remark}
\section{Proof of Theorem 2}
Look at the special case where $X$ is a smooth hypersurface of degree
$m$ in ${\bf P}^4$ and $f:X \rightarrow Q$ a nonconstant morphism of
generator degree $d$.
As $X$ has N\'{e}ron-Severi group ${\bf Z}$, we can apply Theorem
\ref{stelling}
which gives that
the generator degree of $f$ is bounded. To estimate this bound,
consider expression (\ref{eq:bound}). As
\[ \begin{array}{lll}
H_X^3 & = & m, \\
H_Xc_2(X) & = & m^3 - 5m^2 + 10m
\end{array} \]
and the index of $X$ is equal to $-5+m$, this expression becomes:
\begin{equation}
\frac{4}{9}md^3+(\frac{2}{9}m^2-\frac{10}{9}m)d^2+
(\frac{4}{9}m^3-\frac{10}{9}m^2+\frac{10}{9}m)d. \label{eq:boun}
\end{equation}
The degree of $f$ is equal to $md^3/2$. Thus, in this
case, the upper bound on the generator degree of $f$ we obtained in the proof
of
Theorem \ref{stelling} is given by the maximal positive integer $d$
for which (\ref{eq:boun}) $ \geq md^3/2$. Denote this
integer by $d_m$. A calculation shows that, for $m \gg 0$, the bound $d_m$
we obtain in this way on the generator degree of $f$ grows approximately
linearly with $m$: $d_m \sim (2+2\surd 3)m+constant$.
However, the generator degree of $f$ is also bounded from below.
Choose coordinates $(x_0: \dots :x_4)$ on the projective space ${\bf P}^4$
containing $X$ and coordinates $(y_0: \dots :y_4)$ on the projective
space ${\bf P}^4$ containing $Q$ such that $Q$ is given by the equation
$\sum_{i=0}^{4} y_i^2 =0$. Then $f$ is given by
\[ \begin{array}{rcl}
f:X & \longrightarrow & Q, \\
(x_0: \dots :x_4) & \longmapsto & ({\phi}_0 (x_0: \dots :x_4): \dots :
{\phi}_4 (x_0: \dots :x_4))
\end{array} \]
where the ${\phi}_i$ are homogeneous polynomials of degree $d$, defined on
$X$. As the natural map $H^0({\bf P}^4, {\cal O}_{{\bf P}^4}(d))
\rightarrow H^0(X, {\cal O}_X(d))$ is surjective,
these polynomials can be extended to polynomials of degree $d$, defined on
${\bf P}^4$ (but not necessarily in a unique way). The
extensions will also be denoted by ${\phi}_i$.
Let $X \subset {\bf P}^4$ be given by the equation $F_X=0$, where
$F_X$ is a homogeneous polynomial of degree $m$. As $\sum_{i=0}^{4}
{\phi}_i^2 =0$ on $X$, it follows that
\begin{equation}
\sum_{i=0}^{4} {\phi}_i^2 = F_XG, \label{eq:vgl}
\end{equation}
where $G$ is a homogeneous polynomial of degree $2d-m$. Thus, the
generator degree of $f$ must be larger than or equal to $m/2$.
In fact, if $m$ is even, then there exist hypersurfaces $X$ of degree $m$ and
morphisms of generator degree $m/2$ from $X$ to $Q$, for instance if $X$
is the Fermat hypersurface of degree $m$ in ${\bf P}^4$. From (\ref{eq:vgl}) it
follows that, if $m$ is even, there is a morphism of generator degree $m/2$
from
$X$ to $Q$ if and only if $F_X$ can be written as the sum of 5 squares
of homogeneous polynomials of degree $m/2$, having no common zeroes
on $X$.
Now consider the case $m=3$, where $X$ is a smooth cubic. In order to prove
Theorem \ref{kubiek}, we will use the following theorem of Lazarsfeld (see
\cite{La}):
\begin{theorem} \label{Lazarsfeld}
Let $X$ be a smooth projective variety of dimension at least 1 and let
$f:{\bf P}^n \rightarrow X$ be a surjective morphism. Then $X \cong {\bf P}^n$.
\end{theorem}
{\it Proof of Theorem \ref{kubiek}}: Let $f:X \rightarrow Q$ be a
morphism of generator degree $d$. A computation shows that the
upper bound $d_3$ on the generator degree of $f$, which was introduced
above, is equal to 3. Morphisms of generator degree 3 cannot occur as
the expression $3d^3/2$ which should be equal to the degree of
such a morphism is not integer for $d=3$. So, all that remains to
be proven is that there are no morphisms of generator degree 2
between cubics and quadrics.
Assume $f$ has generator degree 2. As above, choose coordinates
$(x_0: \dots :x_4)$ and $(y_0: \dots :y_4)$ on ${\bf P}^4$, and let
${\phi}_0, \dots, {\phi}_4$ be homogeneous polynomials of degree $2$,
defining $f$. In this case we get, as in (\ref{eq:vgl}):
\begin{equation}
\sum_{i=0}^{4} {\phi}_i^2 = F_XL, \label{eq:kvgl}
\end{equation}
where $L$ is a homogeneous linear polynomial, defining a hyperplane in ${\bf
P}^4$.
This hyperplane will also be denoted by $L$, for convenience.
We claim that the ${\phi}_i$ do not have any common zeroes on the hyperplane
$L$. As the ${\phi}_i$ do not have any common zeroes on $X$, the claim
follows for points in $X \cap L$. Now let $p$ be a point in $L \backslash
(X \cap L)$. If ${\phi}_i(p)=0$ for all $i \in \{0, \dots,4 \}$, equation
(\ref{eq:kvgl}) implies that
$$ \frac{\partial F_XL}{\partial x_i}(p)=0,\mbox{ for all } i \in
\{0, \dots,4 \}. $$
As $L(p)=0$ and $F_X(p)\not = 0$ by assumption, we get:
$$ \frac{\partial L}{\partial x_i}(p)=0 \mbox{ for all } i \in \{0, \dots,4 \}.
$$
But this is impossible because $L$, being a hyperplane, is nonsingular. This
proves the claim.
Thus, the ${\phi}_i$ define a morphism from $L$ to $Q$. Restricted to the
surface $L \cap X$, this morphism equals $f | _{L \cap X}$, so it is
not constant. This is a contradiction by Theorem \ref{Lazarsfeld}.
\begin{remark}
{\rm Notice that the argument in the proof of Theorem \ref{kubiek} works for
every hypersurface $X$ of degree $m$ in ${\bf P}^4$ with a morphism
$f:X \rightarrow Q$ of generator degree $d$ such that $2d-m=1$. So, if $m$ is
odd, there are no morphisms of generator degree $(m+1)/2$ from $X$ to $Q$
(if $m \equiv 1mod4$, this is also clear from the fact that
the expression $md^3/2$ which should be equal to the degree
of such a morphism is not integer in this case).}
\end{remark}
|
1996-03-20T06:20:17 | 9505 | alg-geom/9505024 | en | https://arxiv.org/abs/alg-geom/9505024 | [
"alg-geom",
"hep-th",
"math.AG"
] | alg-geom/9505024 | Subhashis Nag | Indranil Biswas, Subhashis Nag, and Dennis Sullivan | Determinant Bundles, Quillen Metrics, and Mumford Isomorphisms Over the
Universal Commensurability Teichm\"uller Space | ACTA MATHEMATICA (to appear); finalised version with a note of
clarification regarding the connection of the commensurability modular group
with the virtual automorphism group of the fundamental group of a closed
Riemann surface; 25 pages. LATEX | null | null | IHES/M/95/43 (Paris) | null | There exists on each Teichm\"uller space $T_g$ (comprising compact Riemann
surfaces of genus $g$), a natural sequence of determinant (of cohomology) line
bundles, $DET_n$, related to each other via certain ``Mumford isomorphisms''.
There is a remarkable connection, (Belavin-Knizhnik), between the Mumford
isomorphisms and the existence of the Polyakov string measure on the
Teichm\"uller space. This suggests the question of finding a genus-independent
formulation of these bundles and their isomorphisms. In this paper we combine a
Grothendieck-Riemann-Roch lemma with a new concept of $C^{*} \otimes Q$ bundles
to construct such an universal version. Our universal objects exist over the
universal space, $T_\infty$, which is the direct limit of the $T_g$ as the
genus varies over the tower of all unbranched coverings of any base surface.
The bundles and the connecting isomorphisms are equivariant with respect to the
natural action of the universal commensurability modular group.
| [
{
"version": "v1",
"created": "Wed, 24 May 1995 12:33:07 GMT"
},
{
"version": "v2",
"created": "Tue, 19 Mar 1996 14:06:57 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Biswas",
"Indranil",
""
],
[
"Nag",
"Subhashis",
""
],
[
"Sullivan",
"Dennis",
""
]
] | alg-geom | \section{Introduction}
Let ${\cal T}_g$ denote the Teichm\"uller space comprising compact marked
Riemann surfaces of genus $g$. Let $DET_n \longrightarrow {\cal T}_g$ be
the determinant (of cohomology) line bundle on ${\cal T}_g$ arising from
the $n$-th tensor power of the relative cotangent bundle on the
universal family ${\cal C}_g$ over ${\cal T}_g$. The bundle $DET_0$ is called the Hodge line
bundle. The bundle $DET_n$ is equipped with a hermitian structure which is
obtained from the construction of Quillen of metrics on
determinant bundles using the Poincar\'e metric on the relative tangent
bundle of ${\cal C}_g$, [Q].
These natural line bundles over ${\cal T}_g$ carry liftings of the standard
action of the mapping class group, $MC_g$, on ${\cal T}_g$. We shall think
of them as $MC_g$-equivariant line bundles, and the isomorphisms
we talk about will be $MC_g$-equivariant isomorphisms.
By applying the Grothendieck-Riemann-Roch theorem,
Mumford [Mum] had shown that $DET_n$ is a certain fixed
(genus-independent) tensor power of the Hodge bundle. More precisely,
$$
DET_{n} ~=~ DET^{\otimes (6n^2-6n+1)}_{0}
\eqno{(1.1)}
$$
this isomorphism of equivariant line bundles being ambiguous only
up to multiplication by a non-zero scalar. (Any choice of such an
isomorphism will be called a Mumford isomorphism in what follows.)
There is a remarkable connection, discovered by Belavin and Knizhnik [BK],
between the Mumford isomorphism above for the case $n=2$, [i.e., that
$DET_{2}$ is the $13$-th tensor power of Hodge], and the existence of the
Polyakov string measure on the moduli space ${\cal M}_g$. (See the discussion after
Theorem 5.5 for more details.) This suggests the question of finding a
genus-independent formulation of the Mumford isomorphisms over some
``universal'' parameter space of Riemann surfaces (of varying topology).
In this paper we combine a Grothendieck-Riemann-Roch lemma (Theorem 2.9)
with a new concept of ${{\CC}^{*}} \otimes {\QQ}$ bundles (Section 5), to construct a
universal version of the determinant bundles and Mumford's isomorphism.
Our objects exist over a universal base space ${\cal T}_{\infty} = {\cal T}_{\infty}(X)$, which is the
infinite directed union of the complex manifolds that are the Teichm\"uller
spaces
of higher genus surfaces that are unbranched coverings of any (pointed)
reference surface $X$. The bundles and the relating isomorphisms
are equivariant with respect to the natural action of the
universal commensurability group ${CM_{\infty}}$ -- which is defined (up to isomorphism)
as the group of isotopy classes of unbranched self correspondences of the
surface $X$ arising from pairs of non-isotopic pointed covering maps
$X' {\rightarrow \atop \rightarrow } X$, (see below and in Section 5).
In more detail, our universal objects are obtained by taking the direct
limits using the following category ${\cal S}$: for each integer $g \geq 2$,
there is one object in ${\cal S}$, an oriented closed pointed surface $X_g$
of genus $g$, and one morphism $X_{\tilde g} \rightarrow X_g$ for each based
isotopy class of finite unbranched pointed covering map. For each morphism
of ${\cal S}$ (say of degree $d$) we have an induced holomorphic injection of
Teichm\"uller spaces arising from pullback of complex structure:
$$
{\cal T}(\pi):~{\cal T}_g \rightarrow {\cal T}_{\tilde g}
\eqno(1.2)
$$
The GRR lemma provides a natural isomorphism of the line bundle
$DET_{n,g}^{\otimes 12d}$ on ${\cal T}_g$ with the pullback line bundle
${{\cal T}(\pi)}^*DET_{n,{\tilde g}}^{\otimes 12}$. We may view this isomorphism,
equivalently, as a degree $d$ homomorphism covering the injection ${\cal T}(\pi)$
between the principal $\C*$ bundles associated to $DET_{n,g}$ and
$DET_{n,{\tilde g}}$, respectively. Each commutative triangle in ${\cal S}$ yields
a commutative triangular prism whose top face is the following triangle
of total spaces of principal $\C*$ bundles:
$$
\matrix{
{DET}_{n,g}^{\otimes {12}}
&
&\mapright{}
&
&{DET}_{n,\tilde g}^{\otimes {12}}
\cr
&
\searrow
&
&\swarrow
&
\cr
&&
{DET}_{n,\bar g}^{\otimes {12}}
&&
\cr}
$$
and whose bottom face is the commuting triangle of base spaces for these
bundles:
$$
\matrix{
{\cal T}_g~~~~~
&
&\mapright{}
&
&~~~~~{\cal T}_{\tilde g}
\cr
&
\searrow
&
&\swarrow
&
\cr
&&
{\cal T}_{\bar g}
&&
\cr}
\eqno(1.3)
$$
Moreover, the canonical mappings above relating these $DET_n$ bundles
over the various Teichm\"uller spaces preserve the Quillen
hermitian structure of these bundles in the sense that unit circles are
carried to unit circles.
We explain in brief the commensurability Teichm\"uller space ${\cal T}_{\infty}$ and the
large mapping class group ${CM_{\infty}}$ acting thereon. For
each object $X$ in ${\cal S}$, consider the directed set
$\{\alpha \}$
of all morphisms in ${\cal S}$ with range $X$. Then we form
$$
{\cal T}_{\infty}(X) := dir. lim. {\cal T}_{g(\alpha )}
\eqno(1.4)
$$
where the limit is taken over $\{\alpha \}$, and $g(\alpha )$ is the genus for
the domain of morphism $\alpha $. Each morphism
$X_{g'} \rightarrow X_g$ induces a holomorphic {\it bijection} of the
corresponding direct limits, and we denote any of these isomorphic
``ind-spaces'' (inductive limit of finite-dimensional complex spaces
-- see [Sha]) by ${\cal T}_{\infty}$ -- the universal commensurability
Teichm\"uller space. (Compare Section 2 and Example 4 on p. 547 of [S].)
Notice that a pair of morphisms $X' {\rightarrow \atop \rightarrow } X$
determines an automorphism of ${\cal T}_{\infty}$; we call the group of automorphisms
of ${\cal T}_{\infty}$ obtained this way the {\it commensurability modular group} ${CM_{\infty}}$.
We now take the direct limit of the $\C*$ principal bundles associated
to ${DET}_{n,g}^{\otimes 12}$ over ${\cal T}_g$ to obtain a new object -- a ${{\CC}^{*}} \otimes {\QQ}$ bundle
over ${\cal T}_{\infty}$ -- denoted ${DET}(n,{\QQ})$. As sets the total space with action of
the group ${{\CC}^{*}} \otimes {\QQ}$ is defined by the direct limit construction. Continuity
and complex analyticity for maps into these sets are defined by the
corresponding properties for factorings through the strata of
the direct system. (Section 5.)
There are the natural isomorphisms of Mumford, as stated in (1.1),
at the finite-dimensional stratum levels. By our construction
these isomorphisms are {\it rigidified} to be natural over the
category ${\cal S}$. Therefore we have natural Mumford isomorphisms
between the following ${{\CC}^{*}} \otimes {\QQ}$ bundles over the universal commensurability
Teichm\"uller space ${\cal T}_{\infty}$:
$$
{DET}(n,{\QQ})~~ {\rm and}~~ {{DET}(0,{\QQ})}^{\otimes (6n^2-6n+1)}
\eqno(1.5)
$$
We also show that the natural Quillen metrics of the $DET$ bundles
fit together to define a natural analogue of Hermitian structure on
these ${{\CC}^{*}} \otimes {\QQ}$ bundles; in fact, for all our canonical mappings in
the direct system the unit circles are preserved. Note Theorem 5.5.
In fact, the existence of the canonical relating morphism between
determinant bundles (fixed $n$) in the fixed covering
$\pi:X_{\tilde {g}} \rightarrow X_g$
situation was first conjectured and deduced by us utilizing the
differential geometry of these Quillen metrics. Recall that the
Teichm\"uller spaces ${\cal T}_g$ and ${\cal T}_{\tilde g}$ carry natural symplectic
forms (defined using the Poincare metrics on the Riemann surfaces)
-- the Weil-Petersson K\"ahler forms -- which are in fact
the curvature forms of the natural Quillen metrics of these $DET$
bundles ([Wol], [ZT], [BGS]). If the covering $\pi$ is unbranched of
degree $d$, a direct calculation shows that this natural WP
form on ${\cal T}_{\tilde g}$ (appropriately renormalized by the degree)
pulls back to the WP form of ${\cal T}_g$ by ${\cal T}(\pi)$ (the embedding of
Teichm\"uller spaces induced by $\pi$).
One expects therefore that if one raises the $DET_n$ bundle on ${\cal T}_g$
by the tensor power $d$, then it extends over the larger Teichm\"uller
space ${\cal T}_{\tilde g}$ as the $DET_n$ bundle thereon. This intuition is, of course,
what is fundamentally behind our direct limit constructions. Since it
turns out to be technically somewhat difficult to actually prove that the
relevant bundles are isomorphic using this differential geometric
method, we have separated that aspect of our work into a different
article [BN].
Can objects on ${\cal T}_{\infty}$ that are equivariant by the commensurability modular
group ${CM_{\infty}}$ be viewed as objects on the quotient ${{\cal T}_{\infty}}/{{CM_{\infty}}}$ ?
This quotient is problematical and interesting, so we work with the
equivariant statement.
We end the Introduction by mentioning some problems.
The universal commensurability Teichm\"uller space,
${\cal T}_{\infty}$, is made up from embeddings ${\cal T}(\pi)$ that are isometric
with respect to the natural
Teichm\"uller metrics, so it carries a natural Teichm\"uller metric.
Our theorems give us genus independent
determinant line bundles ${DET}(n,{\QQ})$, Quillen metrics and Mumford
isomorphisms over ${\cal T}_{\infty}$, all compatible with
each other and the commensurability group ${CM_{\infty}}$.
{\it Are the above structures uniformly continuous for this metric?}
Then they would pass to the completion $\widetilde{T}_{\infty}$ of ${\cal T}_{\infty}$ for the
Teichm\"uller metric. One knows that $\widetilde{T}_{\infty}$ is a separable complex
Banach manifold which is the Teichm\"uller space of complex structures
on the universal solenoidal surface $H_{\infty}=\limproj {\tilde X}$,
where ${\tilde X}$ ranges, as above, over all finite covering
surfaces of $X$. (See [S], [NS], for the Teichm\"uller theory of
$H_{\infty}$.) We
would conjecture that this continuity is true and that the ${{\CC}^{*}} \otimes {\QQ}$
bundles ${DET}(n,{\QQ})$, Quillen metrics, and Mumford isomorphisms can be
defined over ${\cal T}(H_{\infty})=\widetilde{T}_{\infty}$ directly by looking
at compact solenoidal Riemann surfaces themselves. Now we may consider
square integrable holomorphic forms on the leaves of ${H}_{\infty}$ (which are
uniformly distributed copies of the hyperbolic plane in ${H}_{\infty}$)
regarded as modules over the $\CC*$ algebra of ${H}_{\infty}$ with chosen
transversal. The measure of this transversal would become a real
parameter extending the genus above. One expects that A. Connes'
version of Grothendieck-Riemann-Roch would replace Deligne's functorial
version which we are using here.
Finally one would hope that
the Polyakov measure (Section 5) on Teichm\"uller space, when viewed as a
metric on the canonical bundle, would also make sense at infinity in
the direct limit because this measure can be constructed by applying
the $13$-th power Mumford isomorphism ((1.1) for $n=2$)
to the $L^2$ inner product on the Hodge line bundle.
That issue remains open.
\noindent {\it Acknowledgements:} We would like express our gratitude
to several mathematicians
for their interest and discussions. In particular, we thank
M.S.Narasimhan and E.Looijenga for helpful discussions in early
stages of this work. Laurent Moret-Bailly deserves a very special
place in our acknowledgements. In fact, he brought to our attention
the Deligne pairing version of the Grothendieck-Riemann-Roch theorem
that we use crucially here, and showed us Lemma 2.9,
after seeing an earlier (less strong version) of the main
theorem based on topology and the curvature calculations mentioned above.
\section{A lemma on determinant bundles}
Let $X$ be a compact Riemann surface, equivalently, an
irreducible smooth projective curve over $\CC$. Let $L$ be a
holomorphic line bundle on $X$. The determinant of $L$ is the
defined to be the $1$-dimensional complex vector space
$(\ext{top}H^0(X,L))\bigotimes (\ext{top}H^1(X,L)^*)$, and will
be denoted by $det(L)$. Take a Riemannian metric $g$ on $X$
compatible with the conformal structure of $X$. Fix a hermitian
metric $h$ on $L$. Using $g$ and $h$, a hermitian structure can
be constructed on ${\Omega }^i(X,L)$, the space of $i$-forms on $X$
with values in $L$. Moreover the vector space $H^1(X,L)$ is
isomorphic, in a natural way, with the space of harmonic
$1$-forms with values in $L$. Consequently the vector spaces
$H^0(X,L)$ and $H^1(X,L)$ are equipped with hermitian structures
which in turn induce a hermitian structure on $det(L)$ -- this
metric on $det(L)$ is usually called the $L^2$ metric. Let
$\Delta := {\overline\partial}^*{\overline\partial}$ be the
laplacian acting on the space of smooth sections of $L$. Let
$\{{\lambda }_i\}_{i\geq 1}$ be the set of non-zero eigenvalues of
$\Delta$; let $\zeta$ denote the analytic continuation of the
function $s \longmapsto \sum_{i}1/{\lambda }^s_i$. The {\it Quillen
metric} on $det(L)$ is defined to be the hermitian structure on
$det(L)$ obtained by multiplying the $L^2$ metric with ${\rm
exp}(-{\zeta}'(0))$, [Q].
To better suit our purposes, we will modify the above (usual)
definition of the Quillen metric by a certain factor.
Consider the real number $a(X)$ appearing in Th\'eor\`eme 11.4 of [D].
This number $a(X)$ depends only on the genus of $X$. The statement in
Remark 11.5 of [D] -- to the effect that there is a constant $c$ such
that $a(X) = c.\chi (X)$, where $\chi (X)$ is the Euler characteristic
of $X$ -- has been established in [GS]. (The constant $c$ is related
to the derivative at $(-1)$ of the zeta function for the trivial
hermitian line bundle on $\CC P^1$ (4.1.7 of [GS]).) Let $H_Q(L; g,h)$
denote the Quillen metric on $det(L)$ defined above. Henceforth, by
{\it Quillen metric} on $det(L)$ we will mean the hermitian metric
$$
{\rm exp}(a(X)/12) H_Q(L; g, h)
\eqno{(2.1)}
$$
Next we will describe briefly some key properties of the
determinant line and the Quillen metric.
Let $\pi:{\cal X} \longrightarrow S$ be a family of compact Riemann
surfaces parametrized by a base $S$. We can work with either
holomorphic (Kodaira-Spencer) families over a complex-analytic
variety $S$, or with algebraic families over complex algebraic
varieties (or, more generally, over a scheme) $S$. In the
algebraic category one means that $\pi$ is a proper smooth
morphism of relative dimension one with geometrically connected
fibers. In the analytic category, $\pi$ is a holomorphic
submersion again with compact and connected fibers. Take a
hermitian line bundle $L_S \longrightarrow {\cal X}$ with hermitian
metric $h_S$. Fix a hermitian metric $g_S$ on the relative
tangent bundle $T_{{\cal X}/S}$.
For any point $s \in S$, the above construction gives a
hermitian line $det(L_s)$ (the hermitian structure is given by
the Quillen metric). The basic fact is that these lines fit
together to give a line bundle on $S$ [KM], which is called the
{\it determinant bundle} of $L_S$, and is denoted by $det(L_S)$.
Moreover the function on the total space of $det(L_S)$ given by
the norm with respect to the Quillen metric on each fiber is a
$C^{\infty}$ function, and hence it induces a hermitian metric
on $det(L_S)$ [Q]. This bundle will be denoted by $det(L_S)$.
We shall make clear in Remark 2.13 below that this ``determinant
of cohomology'' line bundle is also an algebraic or analytic
bundle -- according to the category within which we work.
The determinant bundle $det(L_S)$ is functorial with respect to
base change. We describe what this means. For a morphism $\gamma :
S' \longrightarrow S$ consider the bundle, $p^*_2L_S
\longrightarrow S'\times_S{\cal X}$, on the fiber product, where $p_2
: S'\times_S{\cal X} \longrightarrow {\cal X}$ is the projection onto the
second factor. The hermitian structure $h_S$ pulls back to a
hermitian structure on $L_{S'} := p^*_2L_S$; and, similarly, the
metric $g_S$ induces a hermitian structure on the relative
tangent bundle of $S'\times_S{\cal X}$. ``Functorial with respect to
base change'' now means that in the above situation there is a
{\it canonical isometric isomorphism} $$ {\rho}_{S',S} ~:~
det(L_{S'})~\longrightarrow ~ {\gamma }^*det(L_S)$$ such that if
$$S''~\mapright{\gamma '}~S'~\mapright{{\gamma }}~S$$ are two morphisms
then the following diagram is commutative $$
\matrix{det(L_{S''})&\mapright{{\rho}_{S'',S'}}&
{\gamma '}^*(det(L_{S'}))&\cr \mapdown{{\rho}_{S'',S}}&&
\mapdown{{\gamma '}^*{\rho}_{S',S}}&\cr (\gamma \circ{\gamma }')^*det(L_S)
&\mapright{id}& (\gamma \circ{\gamma }')^*det(L_S)&\cr}
\eqno(2.2)
$$
The determinant of cohomology construction $det(L_S)$ produces a
bundle over the parameter space $S$ induced by the bundle over
the total space ${\cal X}$; now, the Grothendieck-Riemann-Roch (GRR)
theorem gives a canonical isomorphism of $det(L_S)$ with a
combination of certain bundles obtained (on $S$) from the direct
images of the bundle $L_S$ and the relative tangent bundle
$T_{{\cal X}/S}$. In order to relate canonically the determinant
bundle obtained from a given family ${\cal X} \rightarrow S$ (fibers
of genus $g$, say) with the determinant arising from a covering
family $\mbox{${\tilde{\cal X}}$}$ (having fibers of some higher genus ${\tilde g}$), we
shall utilize the GRR theorem in a formulation due to Deligne,
[D, Theorem 9.9(iii)].
In fact, Deligne introduces a ``bilinear pairing'' that
associates a line bundle, denoted by $<L_S,M_S>$, over $S$ from
any pair of line bundles $L_S$ and $M_S$ over the total space of
the fibration ${\cal X} \rightarrow S$. If $L_S$ and $M_S$ carry
hermitian metrics then a canonically determined hermitian
structure gets induced on the Deligne pairing bundle $<L_S,M_S>$
as well. Denoting by ${\cal L} = L_S$ the given line bundle over ${\cal X}$,
the GRR theorem in Deligne's formulation reads:
$$ det({\cal L})^{\otimes 12}~=~ <T^*_{{\cal X} /S},
T^*_{{\cal X} /S}> \bigotimes <{{\cal L}} , {\cal L} {\otimes} T_{{\cal X} /S}>^{\otimes
6}
\eqno(2.3)
$$
Here $T^*_{{\cal X} /S}$ denotes the
relative cotangent bundle over ${\cal X}$, and the equality asserts
that there is a {\it canonical isomorphism, functorial with
respect to base change,} between the bundles on the two sides.
Furthermore, Th\'eor\`eme 11.4 of [D] says that the canonical
identification in (2.3) is actually an {\it isometry} with the
Quillen metric on the left side and the Deligne pairing metrics
on the right. (The constant ${\rm exp}(a(X))$ in the statement
of Th\'eor\`eme 11.4 of [D] has been absorbed in the definition
(2.1).) We proceed to explain the Deligne pairing and the metric
thereon in brief; details are to be found in sections 1.4 and
1.5 of [D].
Let $L$ and $M$ be two line bundles on a compact Riemann surface
$X$. For a pair of meromorphic sections $l$ and $m$ of $L$ and
$M$ respectively, with the divisor of $l$ being disjoint from
the divisor of $m$, let $\CC <l,m>$ be the one dimensional
vector space with the symbol $<l,m>$ as the generator. For two
meromorphic functions $f$ and $g$ on $X$ such that $div(f)$ is
disjoint from $div(m)$ and $div(g)$ is disjoint from $div(l)$,
the following identifications of complex lines are to be made
$$
\matrix{<fl,m> &= & f(div(m))<l,m>&\cr
<l,gm>&=&g(div(l))<l,m>&}
\eqno{(2.4)}
$$ The Weil reciprocity law says that for any two meromorphic
functions $f_1$ and $f_2$ on $X$ with disjoint divisors,
$f_1(div(f_2)) = f_2(div(f_1))$ [GH, page 242]. So we have
$$<fl,gm>~=~f(div(gm)).g(div(l))<l,m>~=~g(div(fl)).f(div(m))<l,m>.$$
{}From the above equality it follows that the identifications in
(2.4) produce a complex one dimensional vector space, denoted by
$<L,M>$, from the pair of line bundles $L$ and $M$. If $L$ and
$M$ are both equipped with hermitian metrics then the hermitian
metric on $\CC <l,m>$ defined by
$$
log||<l,m>|| := {1\over{2{\pi}i}}\int_X\partial
{\overline\partial}(log||l||.log||m||)~+\, log||l||(div(m))
\,+\,log||m||(div(l))
\eqno{(2.5)}
$$
is compatible with the relations in (2.4) -- hence it gives
a hermitian structure on $<L,M>$, see [D, 1.5.1].
Consider now a family of Riemann surfaces ${\cal X} \longrightarrow
S$; let $L_S$ and $M_S$ be two line bundles on ${\cal X}$, equipped
with hermitian structures. Over an open subset $U\subset S$, let
$l_U$, $m_U$ be two meromorphic sections of $L_S$ and $M_S$
respectively, with finite supports over $U$ such that the
support of $l_U$ is disjoint from the support of $m_U$. (Support
of a section is the divisor of the section.) For another open
set $V$ and two such sections $l_V$ and $m_V$, the relations in
(2.4) give a function $$C_{U,V}~\in ~{{\cal O}}^*_{U\cap V}\,.$$ Using
the Weil reciprocity law it can be shown that $\{C_{U,V}\}$
forms a 1-cocycle on $S$. In other words, we get a line bundle
on $S$, which we will denote by $<L_S,M_S>$. The hermitian
structure on $<L,M>$, described earlier, makes $<L_S,M_S>$ into
a hermitian bundle.
Given a meromorphic section $m$ of $M_S$, let $m^{\otimes n}$ be
the meromorphic section of $M^n_S$ obtained by taking the $n$-th
tensor power of $m$. Note that $div(m^{\otimes n}) = n.div(m)$.
The map $<l,m^{\otimes n}> \longrightarrow <l,m>^{\otimes n}$
can be checked to be compatible with the relations (2.4) and
hence it induces an isomorphism
$$
<L_S,M^n_S> \longrightarrow <L_S,M_S>^n
\eqno{(2.6)}
$$ From the definition (2.5) we see that (2.6) is an isometry
for the metric on $M^n_S$ induced by the metric on $M_S$.
We shall now see how the critical formula (2.3) follows from the
general GRR theorem of [D]. Indeed, let ${\cal L}$ denote any rank $n$
vector bundle on the total space of the family ${\cal X}$; we
reproduce below the statement of Theorem 9.9(iii) of [D]:
$$
det({\cal L} )^{\otimes 12}~=~ <T^*_{{\cal X} /S}, T^*_{{\cal X} /S}> \bigotimes
<{\Lambda^{n}}({\cal L} ),{\Lambda^{n}}({\cal L} ) {\otimes} T_{{\cal X}
/S}>^{\otimes 6} \bigotimes I_{{\cal X} /S}C^2({\cal L} )^{-12}
\eqno{(GRR-D)}
$$
Now, from the definition of $I_{{\cal X} /S}C^2$ in [D, 9.7.2] it
follows that for a line bundle ${\cal L}$, the bundle
$I_{{\cal X} /S}C^2({\cal L})$ is the
trivial bundle on $S$, and the metric on it is the constant
metric [D, Theorem 10.2(i)]. From Th\'eor\`eme 11.4 of [D] we
conclude that that the canonical identification in the statement
above is actually an isometric identification. (The factor
${\rm exp}(a(X))$ in Th\'eor\`eme 11.4 of [D]
is taken care of by the definition (2.1).) Thus we have obtained
the isometric isomorphism stated in (2.3).
With this background behind us we can formulate our main lemma.
Let ${\cal X}$ and $\mbox{${\tilde{\cal X}}$}$ be two families of Riemann surfaces over $S$
(say with fibers of genus $g$ and ${\tilde g}$, respectively), and $p:
\mbox{${\tilde{\cal X}}$} \longrightarrow {\cal X}$ be an \'etale ({\it i.e.\/}\ unramified) covering
of degree $d$, commuting with the projections onto $S$. In other
words, the map $p$ fits into the following commutative diagram
$$
\matrix{\mbox{${\tilde{\cal X}}$}& &\mapright{p}& & {\cal X}
\cr &\searrow& & \swarrow &
\cr & &S& & \cr}
\eqno(2.7)
$$ The situation implies that each fiber of the family $\mbox{${\tilde{\cal X}}$}$ is
a degree $d=({\tilde g} - 1)/(g - 1)$ holomorphic covering over the
corresponding fiber of the family ${\cal X}$. Fix also a hermitian
metric $g$ on $T_{{\cal X} /S}$. Since $p$ is \'etale, $p^*T_{{\cal X} /S}
= T_{\mbox{${\tilde{\cal X}}$} /S}$, and hence $g$ induces a hermitian metric $p^*g$
on $T_{{\mbox{${\tilde{\cal X}}$}}/S}$. Let ${{\cal X}}' \longrightarrow S$ be a third
family of Riemann surface which is again an \'etale cover of
$\mbox{${\tilde{\cal X}}$}$ and fits into the following commutative diagram $$
\matrix
{{{\cal X}}' &\mapright{q} &\mbox{${\tilde{\cal X}}$} &\mapright{p}& {\cal X}
\cr &\searrow& \mapdown{} &\swarrow &
\cr & &S && \cr}
\eqno(2.8)
$$ We want to prove the following:
\medskip
\noindent{\bf Lemma 2.9.}\,\ {\it (i)~Let ${\cal L}$ be a hermitian
line bundle on ${\cal X}$ and let $p^*{\cal L} \longrightarrow \mbox{${\tilde{\cal X}}$}$ be the
pullback of ${\cal L}$ equipped with the pullback metric. Then there
is a canonical isometric isomorphism $$det(p^*({\cal L} ))^{\otimes
12}~\cong ~ det({\cal L} )^{\otimes 12.{\rm deg}(p)}$$ of bundles on
$S$. This isomorphism is functorial with respect to base change.
\noindent (ii)~ Denoting the isometric isomorphism obtained in
(i) by ${\Gamma }(p)$, and similarly defining ${\Gamma }(q)$ and
${\Gamma }(p\circ q)$, the following diagram commutes:
$$\matrix{det((p\circ q)^*({\cal L} ))^{\otimes
12}&\mapright{{\Gamma }(q)}& det(p^*({\cal L} ))^{\otimes 12.{\rm
deg}(q)}\cr \mapdown{{\Gamma }(p\circ q)}&&
\mapdown{{\Gamma }(p)^{\otimes {\rm deg}(q) }}\cr det({\cal L} )^{\otimes
12.{\rm deg}(p\circ q)}&\mapright{id}& det({\cal L} )^{\otimes 12.{\rm
deg}(p\circ q)}\cr} $$ where ${\Gamma }(p)^{\otimes {\rm deg}(q) }$
is the isomorphism on appropriate bundles, obtained by taking
the ${\rm deg }(q)$-th tensor product of the isomorphism
${\Gamma }(p)$.}
\medskip
(The terminology ``functorial with respect to base change'' was
explained earlier. We will use ``canonical'' to mean functorial
with respect to base change.)
\medskip
\noindent {\bf Proof of Lemma 2.9.} The idea of the proof is to
relate -- utilizing GRR in form (2.3) -- the determinant
bundles, which are difficult to understand, with the more
tractable ``Deligne pairings''.
Let ${\cal M}$ be any line bundle on $\mbox{${\tilde{\cal X}}$}$ equipped with a hermitian
structure. First we want to show that there is a canonical
isometric isomorphism
$$<p^*{\cal L} ,{\cal M} >~\longrightarrow ~<{\cal L} ,N({\cal M} )>\, , \eqno{(2.10)}$$
where $N({\cal M} ) \longrightarrow {\cal X}$ is the norm of ${\cal M}$. We recall
the definition of $N({\cal M})$. The direct image $R^0p_*({\cal M} )$ is
locally free on ${\cal X}$, and the bundle $R^0p_*({\cal M} )$ admits a
natural reduction of structure group to the {\it monomial group}
$G \subset GL({\rm deg}(p), \CC)$. (The group $G$ is the
semi-direct product of permutation group, $P_{{\rm deg}(p)}$,
with the invertible diagonal matrices defined using the
permutation action of $P_{{\rm deg}(p)}$.) Mapping $g \in G$
to the permanent of $g$ (on $G$ it is simply the product
of all non-zero entries) we get a homomorphism to ${\CC}^*$,
which is denoted by $\mu$. Using this homomorphism $\mu$
we have a holomorphic line bundle on ${\cal X}$, associated to
$R^0p_*({\cal M} )$, which is defined to be $N({\cal M})$. Clearly the
fiber of $N({\cal M})$ over a point $x\in {\cal X}$ is the tensor product
$$ N({\cal M} )_x ~ = ~
\bigotimes_{y\in p^{-1}(x)}{{\cal M}}_y\,,
\eqno{(2.11)}
$$
The hermitian metric on ${\cal M}$ gives a reduction of
the structure group of $R^0p_*({\cal M} )$ to the maximal compact
subgroup $G_U \subset G$. Since $\mu (G_U) = U(1)$, we have
a hermitian metric on $N({\cal M})$. Note that the hermitian metric on
$N({\cal M} )$ is such that the above equality (2.11) is actually an
isometry.
For a meromorphic section $m$ of $M$, the above identification
of fibers gives a meromorphic section of $N(M)$ which is denoted
by $N(m)$. Given sections $l$ and $m$ of ${\cal L}$ and ${\cal M}$
respectively, with finite support over $U\subset S$ (the support
of $p^*l$ and $m$ being assumed disjoint) we map $<p^*l,m>$ to
$<l,N(m)>$. It can be checked that this map is compatible with
the relations in (2.4). Hence we get a homomorphism from the
bundle $<p^*{\cal L} ,{\cal M}>$ to $<{\cal L} ,N({\cal M} )>$; this is our candidate
for (2.10). To check that it is an isometry, we evaluate the
(logarithms of) norms of the sections $<p^*l,m>$ and $<l,N(m)>$
given by definition (2.5). It is easy enough to see from (2.5) that
the norms of these two sections coincide.
Therefore for a hermitian line bundle ${{\cal L}}'$ on ${\cal X}$, the
isomorphism (2.10) implies that $$<p^*{\cal L} ,p^*{{\cal L}}'> ~=~ <{\cal L}
,N(p^*{{\cal L}}')>$$ But $N(p^*{{\cal L}}') = {{\cal L}'}^d$, where $d :={\rm
deg}(p)$, and moreover the hermitian metric on $N(p^*{{\cal L}}')$
coincides with that of ${{\cal L}'}^d$. Hence from the isometric
isomorphism obtained in (2.6) we get the following
identification of hermitian line bundles (the isomorphism so
created being again functorial with respect to change of base
space): $$ <p^*{\cal L},p^*{{\cal L}}'> ~=~<{\cal L} ,{{\cal L}}'>^d
\eqno{(2.12)}
$$
To prove part $(i)$ of the Lemma we apply the GRR isomorphism
(2.3) to both ${\cal L}$ and $p^*{\cal L}$, and compare the Deligne pairing
bundles appearing on the right hand sides using the result
(2.12). To simplify notation set $\omega = T^*_{{\cal X} /S}$. By
applying (2.3) to $p^*{\cal L}$, and noting that since the map $p$ is
\'etale, the relative tangent bundle $T_{\mbox{${\tilde{\cal X}}$} /S} = p^*T_{{\cal X}
/S}$, we deduce that $det(p^*{\cal L} )^{\otimes 12}$ is canonically
isometrically isomorphic to $<p^*{\cal L} ,p^*({\cal L}\bigotimes{\omega }^{-1}
)>^{\otimes 6}
\bigotimes <p^*\omega ,p^*{\omega }>$.
Taking ${{\cal L}}'$ to be ${{\cal L}}\bigotimes\omega $ in (2.12) we have
$<{\cal L},{\cal L}\bigotimes{\omega }>^d=<p^*{\cal L},(p^*{\cal L}\bigotimes \omega )>$.
Substituting $\omega $ in place on ${\cal L}$ and ${{\cal L}}'$ in (2.12) we
have $<\omega ,\omega >^d=<p^*\omega ,p^*\omega >$. Therefore the bundle
$<p^*{\cal L} ,p^*({\cal L}\bigotimes{\omega }^{-1})>^{\otimes 6}\bigotimes
<p^*\omega ,p^*{\omega }>$ is isometrically isomorphic to
$<{\cal L},{\cal L}\bigotimes{\omega }^{-1}>^{6d}\bigotimes <\omega ,\omega >^d$. But
now applying (2.3) to ${\cal L}$ itself we see that this last bundle
is isometrically isomorphic to $det({\cal L} )^{\otimes 12d}$. That
completes the proof. Notice that since all isomorphisms used in
the above proof were canonical (functorial with base change),
the final isomorphism asserted in part $(i)$ is also canonical
in the same sense.
In order to prove part $(ii)$ of the Lemma, we first note that
the isometric isomorphisms in (2.10) and (2.12) actually fit
into the following commutative diagram $$
\matrix{<(p\circ q)^*{\cal L} ,{\cal M} >&\mapright{}&<p^*{\cal L} ,N({\cal M} )_q>
\cr \mapdown{}&& \mapdown{}\cr <{\cal L} ,N({{\cal M}})>& = & <{\cal L}
,N({{\cal M}})>\cr} $$
where ${\cal L}$ is a hermitian line bundle on ${\cal X}$ and ${\cal M}$ is a
hermitian line bundle on ${{\cal X}}'$, $N({\cal M} ) \longrightarrow {\cal X}$ is
the norm of ${\cal M}$ for the covering $p\circ q$, and $N({\cal M} )_q
\longrightarrow \mbox{${\tilde{\cal X}}$}$ is the norm of ${\cal M}$ for the covering $q$.
Indeed, the commutativity of the above diagram is
straightforward to deduce from the fact that the following two
bundles on ${\cal X}$: namely, $N({\cal M} )$ and the norm of $N({\cal M} )_q$,
are isometrically isomorphic. The isomorphism can be defined,
for example, using (2.11). Now using (2.3), and repeatedly using
the above commutative diagram, we obtain part $(ii)$.
$\hfill{\Box}$
\medskip
We will have occasion to use this general lemma in concrete
situations.
\medskip
\noindent{\bf Remark 2.13.} In [KM] and in [D] the basic context
is the algebraic families category, and the determinant of
cohomology bundle as well as the Deligne pairing bundles are
constructed in this category. However, since the constructions
of the determinant bundles and of the Deligne pairing are {\it
canonical} and {\it local}, they work equally well for
holomorphic families of Riemann surfaces also. The point is that
if ${\cal X} \rightarrow S$ is a holomorphic family of Riemann
surfaces parametrized by a complex manifold $S$, and ${\cal L}
\rightarrow {\cal X}$ is a holomorphic line bundle, then $det({\cal L})
\rightarrow S$ is a holomorphic line bundle which is functorial
with respect to holomorphic base changes. And if ${\cal L}$ and ${\cal M}$
are two holomorphic line bundles on ${\cal X}$ then $<{\cal L},{\cal M}>$ is a
holomorphic line bundle on $S$ -- again functorial with respect
to holomorphic base changes. In fact, an analytic construction of
the determinant bundle and the Quillen metric is to be found in [BGS].
Since the constructions of the Quillen metric and the metric on
the Deligne pairing, (using (2.5)), also hold true for
holomorphic families, consequently, Lemma 2.9 holds in the
holomorphic category as well as in the algebraic category.
\noindent{\bf Remark 2.14.} The statement that
$det(p^*({\cal L}))^{\otimes 12}~\cong ~ det({\cal L} )^{\otimes 12{\rm
deg}(p)}$ as line bundles actually holds for curves over any
field. The statement about isometry makes sense only when we
have Riemann surfaces.
\section{Determinant bundles over Teichm\"uller spaces}
Our aim in this section is to apply the Lemma 2.9 to the universal
family of marked Riemann surfaces of genus $g$ over the Teichm\"uller
space ${\cal T}_g$. The situation of Lemma 2.9 is precipitated by choosing
any finite covering space over a topological surface of genus $g$.
Let $\pi:{\tilde X} \longrightarrow X$ be an unramified covering map between
two compact connected oriented two manifolds ${\tilde X}$ and $X$ of genera
${\tilde g}$ and $g$, respectively. Assume that $g \geq 2$. The degree of the
covering $\pi$, which will play an important role, is the ratio
of the respective Euler characteristics;
namely, $deg(\pi)=({\tilde g}-1)/(g-1)$.
We recall the basic deformation spaces of complex (conformal)
structures on smooth closed oriented surfaces -- the Teichm\"uller spaces.
Let $Conf(X)$ (respectively, $Conf({\tilde X})$) denote the space of
all smooth conformal structures on $X$ (respectively, ${\tilde X}$). Define
${\rm Diff}^{+}(X)$ (respectively, ${\rm Diff}^+({\tilde X})$)
to be the group of all orientation preserving diffeomorphisms of
$X$ (respectively, ${\tilde X}$), and denote by
${\rm Diff}^{+}_0(X)$ (respectively, ${\rm Diff}^+_0({\tilde X})$)
the subgroup of those that are homotopic to the identity.
The group ${\rm Diff}^+(X)$ acts naturally on $Conf(X)$ by pullback
of conformal structure. We define
$$
{\cal T}(X)={\cal T}_g~:=~Conf(X)/{\rm Diff}^{+}_0(X)
\eqno{(3.1)}
$$
to be the Teichm\"uller space of genus $g$ (marked) Riemann surfaces.
Similarly obtain ${\cal T}_{{\tilde g}} := Conf({\tilde X} )/{\rm Diff}^+_0({\tilde X} )$
-- the Teichm\"uller space for genus ${\tilde g}$. The Teichm\"uller space
${\cal T}_g$ carries naturally the structure of a $(3g-3)$ dimensional
complex manifold which is embeddable as a contractible domain
of holomorphy in the affine space $\CC^{3g-3}$. The mapping class
group of the genus $g$ surface, namely the discrete group
$MC_g := {\rm Diff}^+(X)/{\rm Diff}^+_0(X)$, acts properly
discontinuously on ${\cal T}_g$ by holomorphic automorphisms, the quotient
being the moduli space ${\cal M}_g$. For these basic facts see, for example,
[N].
The Teichm\"uller spaces are fine moduli spaces. In fact,
the total space $X\times {\cal T}_g$ admits a natural complex
structure such that the projection to the second factor
$$
{\psi}_g:{\cal C}_g:=X\times{\cal T}_g \longrightarrow {\cal T}_g
\eqno{(3.2)}
$$
gives the universal Riemann surface over ${\cal T}_g$. This means that for
any $\eta \in {\cal T}_g$, the submanifold $X\times\eta$ is a complex
submanifold of ${\cal C}_g$, and the complex structure on $X$ induced by
this embedding is represented by $\eta$. As is well-known, (Chapter 5
in [N]), the family ${\cal C}_g \rightarrow {\cal T}_g$ is the {\it universal} object
in the category of holomorphic families of genus $g$ marked
Riemann surfaces.
Given a complex structure on $X$, using $\pi$ we may pull back this to a
complex structure on ${\tilde X}$. This gives an injective map
$Conf(X)\longrightarrow Conf({{\tilde X}})$. Given an element of $f\in {\rm
Diff}^+_0(X)$, from the homotopy lifting property, there is a unique
diffeomorphism ${\tilde f}\in{\rm Diff}^{+}_0({{\tilde X}})$ such that $\tilde f$
is a lift of $f$. Mapping $f$ to $\tilde f$ defines an injective
homomorphism of ${\rm Diff}^{+}_0(X)$ into ${\rm Diff}^{+}_0({{\tilde X}})$.
We therefore obtain an injection
$$
{\cal T}(\pi ):{\cal T}_g~\longrightarrow ~{\cal T}_{{\tilde g}}
\eqno(3.3)
$$
It is known that this map ${\cal T}(\pi )$ is a {\it proper holomorphic
embedding} between these finite dimensional complex manifolds; ${\cal T}(\pi)$
respects the quasiconformal-distortion (=Teichm\"uller) metrics.
{}From the definitions it is evident that this embedding between the
Teichm\"uller spaces depends only on the (unbased) isotopy class of
the covering $\pi$.
\noindent{\bf Remark 3.4.} In fact, we see that
$\cal T$ is thus a contravariant functor from the category of
closed oriented topological surfaces, morphisms being
covering maps, to the category of finite dimensional complex
manifolds and holomorphic embeddings. We shall have more to say
about this in Section 5.
Over each genus Teichm\"uller space we have a sequence of natural
determinant bundles arising from the powers of the relative
(co-)tangent bundles along the fibers of the universal curve. Indeed,
let ${\omega }_g \longrightarrow {\cal C}_g$ be the relative cotangent bundle
for the projection ${\psi }_g$ in $(3.2)$. The determinant line bundle
over ${\cal T}_g$ arising from its $n$-th tensor power is fundamental,
and we shall denote it by:
$$
DET_{n,g}:= det({\omega }^n_{g}) \longrightarrow {\cal T}_g, ~~n \in {\bf Z}
\eqno(3.5)
$$
Applying Serre duality shows that there is a canonical isomorphism
$DET_{n,g} = DET_{1-n,g}$, for all $n$. $DET_{0,g} = DET_{1,g}$ is
called the {\it Hodge} line bundle over ${\cal T}_g$.
These holomorphic line bundles carry natural {\it Quillen hermitian
structure} arising from the Poincar\'e metrics on
the fibers of the universal curve.
Recall that any Riemann surface $Y$ of genus $g\geq 2$ admits a unique
conformal Riemannian metric of constant curvature $-1$, called the
Poincar\'e metric of $Y$. This metric depends smoothly on the conformal
structure, (because of the uniformization theorem with moduli parameters),
and hence, for a family of Riemann surfaces of genus at least
two, the Poincar\'e metric induces a hermitian metric on the relative
tangent/cotangent bundle. We thus obtain Quillen metrics on each
$DET_{n,g}$. The metric functorially assigned by the Quillen metric
on any tensor power of $DET_{n,g}$ will also be referred to as the Quillen
metric on that tensor power.
Observe that by the naturality of the above constructions it follows
that the action of $MC_g$ on ${\cal T}_g$ has a natural lifting as unitary
automorphisms of these $DET$ bundles.
We invoke back into play the unramified finite covering
$\pi: {\tilde X} \rightarrow X$. Let
$$
{\cal T}(\pi )^*{\cal C}_{{\tilde g}}~\longrightarrow ~{\cal T}_g
\eqno{(3.6)}
$$
be the pull-back to ${\cal T}_g$ of the universal family ${\cal C}_{{\tilde g}}
\longrightarrow {\cal T}_{{\tilde g}}$ using the map ${\cal T}(\pi)$.
Given the topological covering space $\pi$ we therefore
obtain the following \'etale covering map between families of
Riemann surfaces parametrized by ${\cal T}_g$:
$$
{\pi}\times id ~:~ {\cal T}(\pi )^*{\cal C}_{{\tilde g}}~
\longrightarrow ~{\cal C}_g~:=~X\times {\cal T}_g
$$
This is clearly a holomorphic map. In fact, we have the following
commutative diagram
$$
\matrix{{\cal T}(\pi )^*{\cal C}_{{\tilde g}}&
&\mapright{\pi\times id}& & {\cal C}_g\cr &\searrow& &
\swarrow &\cr & &{\cal T}_g& & \cr}
\eqno{(3.7)}
$$
exactly as in the general situation (2.7) above Lemma 2.9.
Now let
$$
id\times {\cal T}(\pi )~:~ {\cal T}(\pi )^*{\cal C}_{{\tilde g}}
{}~\longrightarrow ~{\cal C}_{{\tilde g}}
$$
denote the tautological lift of the map ${\cal T}(\pi)$.
{}From the definition of the Poincar\'e metric it is clear that
for an unramified covering of Riemann surfaces, ${\tilde Y} \longrightarrow Y$,
the Poincar\'e metric on ${\tilde Y}$ is the pull-back of the Poincar\'e
metric on $Y$. If ${\omega }_{{\tilde g}}$ is the relative cotangent bundle
on ${\cal C}_{{\tilde g}}$ then this
compatibility between Poincar\'e metrics implies that the two hermitian
line bundles on ${\cal T}(\pi )^*{\cal C}_{{\tilde g}}$ namely,
$(\pi\times id)^*{\omega }_g$ and $(id\times{\cal T}(\pi ))^*{\omega }_{{\tilde g}}$
are canonically isometric.
But since the determinant bundle of a pullback family is the
pullback of the determinant bundle, the holomorphic hermitian bundle
${\cal T}(\pi)^*(det({\omega }^n_{{\tilde g}})) \longrightarrow {\cal T}_g$ is
canonically isometrically isomorphic to the determinant bundle of
$(id\times {\cal T}(\pi))^*{\omega }^n_{{\tilde g}} \longrightarrow
{\cal T}(\pi )^*{\cal C}_{{\tilde g}}$. Using this and simply applying
Lemma 2.9 to the commutative diagram (3.7) we obtain the following
theorem. (All the Quillen metrics are with respect to the Poincar\'e
metric on fibers.)
\medskip
\noindent{\bf Theorem 3.8a.} {\it The two holomorphic hermitian line
bundles $det({\omega }^n_g)^{12.{\rm deg}(\pi )}$ and\\ ${\cal T}(\pi
)^*(det({\omega }^n_{{\tilde g}}))^{12}$ on ${\cal T}_g$ are canonically isometrically
isomorphic for every integer $n$. In other words, there is a canonical
isometrical line bundle morphism $\Gamma (\pi)$ lifting ${\cal T}(\pi)$
and making the following diagram commute:
$$
\matrix{
{{DET}_{n,g}}^{\otimes {12.deg(\pi)}}
&\mapright{{\Gamma }(\pi)}
&{{DET}_{n,\tilde g}^{\otimes 12}}
\cr
\mapdown{}
&
&\mapdown{}
\cr
{\cal T}_g
&\mapright{{\cal T}(\pi)}
&{\cal T}_{\tilde g}
\cr}
$$
}
\medskip
\noindent
{\bf Remark 3.9.} The bundle morphism $\Gamma (\pi)$ has been obtained from
Riemann-Roch isomorphisms -- as evinced by the proof of Lemma 2.9. We shall
therefore, in the sequel, refer to these canonical mappings as GRR morphisms.
Tensor powers of the GRR morphisms will also be referred to as GRR
morphisms. The functoriality of these morphisms is explained below
in Theorem 3.8b.
Let ${\bar X}~\mapright{\rho }~{\tilde X}~\mapright{\pi}~X$ be two unramified
coverings between closed surfaces of respective genera
${{\bar g}}$, ${\tilde g}$ and $g$. By applying the Teichm\"uller functor we
have the corresponding commuting triangle of embeddings between
the Teichm\"uller spaces:
$$
\matrix{
{\cal T}_g
&
&\mapright{{\cal T}(\pi)}
&
&{\cal T}_{\tilde g}
\cr
&
\searrow
&
&\swarrow
&
\cr
&&
{\cal T}_{\bar g}
&&
\cr}
\eqno(3.10)
$$
Here the two slanting embeedings are, of course,
${\cal T}(\pi\circ\rho )$ and ${\cal T}(\rho )$.
Applying Lemma 2.9(ii) we have
\medskip
\noindent{\bf Theorem 3.8b.}{\it The following triangle of
GRR line bundle morphisms commutes:
$$
\matrix{
{DET}_{n,g}^{\otimes {12.deg(\pi\circ\rho )}}
&
&\mapright{}
&
&{DET}_{n,\tilde g}^{\otimes {12.deg(\rho )}}
\cr
&
\searrow
&
&\swarrow
&
\cr
&&
{DET}_{n,\bar g}^{\otimes {12}}
&&
\cr}
\eqno(3.11)
$$
All three maps in the diagram are obtained by applications of Theorem
3.8a, and raising to the appropriate tensor powers.
The triangle above sits over the triangle of Teichm\"uller spaces (3.10),
and the entire triangular prism is a commutative diagram.}
\medskip
\noindent {\bf Remark 3.12.}\,\ The nagging factor of $12$ in Theorems
3.8a and 3.8b can be dealt with as follows. The Teichm\"uller space being a
contractible Stein domain, any two line bundles on ${\cal T}_g$ are
isomorphic. Choose an isomorphism between ${\delta } : det({\omega }^n_g)^{{\rm
deg}(\pi )} \longrightarrow {\cal T}(\pi )^*(det({\omega }^n_{{\tilde g}}))$. Hence
$${\delta }^{\otimes 12} : det({\omega }^n_g)^{{12.\rm deg}(\pi )}
\longrightarrow {\cal T}(\pi )^*(det({\omega }^n_{{\tilde g}}))^{12}$$ is an
isomorphism. Let $${\tau} : det({\omega }^n_g)^{{12.\rm deg}(\pi )}
\longrightarrow {\cal T}(\pi )^*(det({\omega }^n_{{\tilde g}}))^{12}$$ be the
isomorphism given by the Theorem 3.8a. So $f := {\tau}\circ ({\delta }^{\otimes
12})^{-1}$ is a nowhere zero function on ${\cal T}_g$. Since ${\cal T}_g$
is simply connected, there is a function $h$ on ${\cal T}_g$ such that
$h^{12} = f$. Any two such choices of $h$ will differ by a $12$-th root of
unity. Consider the homomorphism ${\bar \tau} := h.\delta $. Clearly ${\bar
\tau}^{\otimes 12} = \tau$. It is easy to see that
for two different choices of the isomorphism $\delta $, the two ${\bar\tau}'$s
differ by multiplication with a $12$-th root of unity. Moreover, if we
consider a similar diagram to that in Theorem 3.8b with the factor 12
removed and all the homomorphisms being replaced by the corresponding
analogues of $\bar\tau$, then the diagram commutes up to multiplication
with a $12$-th root of unity.
\noindent{\bf Remark 3.13.}
Recall from above that the action of $MC_g$ in ${\cal T}_g$
lifts to the total space of $det({\omega }^n_g)$ as bundle
automorphisms preserving the Quillen metric. There is no action,
a priori, of $MC_g$ on the total space of the the pullback bundle
${\cal T}(\pi )^*(det({\omega }^n_{{\tilde g}}))$. However, from Theorem 3.8a
the bundle ${\cal T}(\pi )^*(det({\omega }^n_{{\tilde g}}))^{12}$ gets an
action of $MC_g$ which preserves the pulled back Quillen metric.
Theorem 3.8b ensures the identity between the $MC_g$ actions
obtained by different pullbacks.
\medskip
In [BN] we will consider two special classes of coverings,
namely characteristic covers and cyclic covers. In such situations the map
between Teichm\"uller spaces, induced by the covering, actually descends
to a map between moduli spaces (possibly with level structure).
As mentioned in the Introduction, in that
context we were able to give a proof of the existence of the
GRR morphism of Theorem 3.8a using Weil-Petersson geometry and
topology.
\section{Power law (Principal) bundle morphisms over Teichm\"uller spaces}
We desire to obtain certain canonical geometric objects over the
inductive limits of the finite dimensional Teichm\"uller spaces
by coherently fitting together the determinant line bundles
$DET_{n,g}$ thereon; the limit is taken as $g$ increases by
running through a universal tower of covering maps.
To this end it is necessary to find canonical mappings relating
${DET}_{n,g}$ to ${DET}_{n,\tilde g}$ where genus ${\tilde g}$ covers genus $g$.
Now, given any complex line bundle $\lambda \rightarrow T$ over any base $T$,
there is a certain canonical mapping of $\lambda $ to any positive integral
($d$-th) tensor power of itself, given by:
$$
\omega _d: \lambda \longrightarrow {\lambda }^{\otimes d}
\eqno(4.1)
$$
where $\omega _d$ on any fiber of $\lambda $ is the map $z \mapsto z^{d}$.
Observe that $\omega _d$ maps $\lambda $ minus its zero section to
${\lambda }^{\otimes d}$ minus its zero section by a map which
is of degree $d$ on the $\CC^{*}$ fibers.
We record the following properties of these maps:
\noindent
{\bf 4.1a.}
The map $\omega _d$ is defined independent of any choices of basis,
and it is evidently compatible with base change. [Namely, if we pull
back both $\lambda $ and $\lambda ^{d}$ over some base $T_1 \rightarrow T$,
then the connecting map $\omega _d$ (over $T$) also pulls back to
the corresponding $\omega _d$ over $T_1$.]
\noindent
{\bf 4.1b.}
The map $\omega _d$ is a homomorphism of the corresponding $\CC^{*}$
principal bundles. When $T$ is a complex manifold, and $\lambda $ is
a line bundle in that category, then the map $\omega _d$ is a holomorphic
morphism between the total spaces of the source and target bundles.
\noindent
{\bf 4.1c.}
If $\lambda $ is equipped with a hermitian fiber metric, and its tensor
powers are assigned the corresponding hermitian structures, the map
$\omega _d$ carries the unit circles to unit circles. (The choice of
a unit circle amongst the natural family of zero-centered
circles in any complex line is clearly equivalent to specifying a
hermitian norm. In this section we will think of hermitian
structure on a line bundle as the choice of a smoothly varying
family of unit circles in the fibers.)
Thus, given a topological covering $\pi:{\tilde X} \rightarrow X$,
as in the situation of Theorem 3.8a, we may define a canonical map
$$
\Omega (\pi) := \Gamma (\pi) \circ \omega _{deg(\pi)}:{DET}_{n,g}^{\otimes 12}
\longrightarrow {DET}_{n,\tilde g}^{\otimes 12}
\eqno(4.2)
$$
where $\Gamma (\pi)$ is the canonical GRR line bundle morphism
found in Theorem 3.8a. Translating Theorems 3.8a and 3.8b in
terms of these holomorphic maps $\Omega (\pi)$ of positive
integral fiber degree, we get:
\medskip
\noindent{\bf Theorem 4.3a.} {\it For each integer $n$, there is
a canonical isometrical holomorphic bundle morphism $\Omega (\pi)$
lifting ${\cal T}(\pi)$ and making the following diagram commute:
$$
\matrix{
{{DET}_{n,g}^{\otimes 12}}
&\mapright{{\Omega }(\pi)}
&{{DET}_{n,\tilde g}^{\otimes 12}}
\cr
\mapdown{}
&
&\mapdown{}
\cr
{\cal T}_g
&\mapright{{\cal T}(\pi)}
&{\cal T}_{\tilde g}
\cr}
$$
By ``isometrical'' we mean that the unit circles of the Quillen
hermitian structures are preserved by the $\Omega (\pi)$.}
\smallskip
\noindent{\bf 4.3b.} {\it Let $\pi$ and $\rho $ denote two composable
covering spaces between surfaces, as in the situation of Theorem 3.8b.
The following triangle of non-linear isometrical
holomorphic bundle morphisms commutes:
$$
\matrix{
{DET}_{n,g}^{\otimes {12}}
&
&\mapright{}
&
&{DET}_{n,\tilde g}^{\otimes {12}}
\cr
&
\searrow
&
&\swarrow
&
\cr
&&
{DET}_{n,\bar g}^{\otimes {12}}
&&
\cr}
$$
The horizontal map is $\Omega (\pi)$, and the two slanting maps are,
(reading from left to right), $\Omega (\pi \circ \rho )$ and $\Omega (\rho )$.
The triangle above sits over the triangle of Teichm\"uller spaces
(3.10), and the entire triangular prism is a commutative diagram.}
The canonical and functorial nature of these connecting maps,
$\Omega (\pi)$, will now allow us to produce direct systems of
line/principal bundles over direct systems of Teichm\"uller spaces.
\section{Commensurability Teichm\"uller space and its Automorphism group}
We construct a category ${\cal S}$ of certain topological objects and
morphisms: the objects, $Ob({\cal S})$, are a set of compact oriented topological
surfaces each equipped with a base point ($\star$), there being exactly
one surface of each genus $g \geq 0$; let the object of genus $g$ be
denoted by $X_g$. The morphisms are based isotopy classes of pointed
covering mappings
$$
\pi: (X_{\tilde g}, \star) \rightarrow (X_g, \star)
$$
there being one arrow for each such isotopy class. Note that the
monomorphism of fundamental groups induced by (any representative of
the based isotopy class) $\pi$, is unambiguously defined.
Fix a genus $g$ and let $X = X_g$.
Observe that all the morphisms with the fixed target $X_g$:
$$
M_g = \{\alpha \in Mor({\cal S}): Range(\alpha )=X_g \}
$$
constitute a {\it directed set} under the partial ordering given by
factorisation of covering maps. Thus if $\alpha $ and $\beta $ are two morphisms from
the above set, then $\beta \succ \alpha $ if and only if the image of the
monomorphism $\pi_1(\beta )$ is contained within the image of $\pi_1(\alpha )$;
that happens if and only if there is a commuting triangle of morphisms of
${\cal S}$ as follows:
$$
\matrix{
X_{g(\beta )}
&
&\mapright{\theta}
&
&X_{g(\alpha )}
\cr
&
\searrow
&
&\swarrow
&
\cr
&&
X_g
&&
\cr}
$$
Here $X_{g(\alpha )}$ denotes the domain surface for $\alpha $ (similarly
$X_{g(\beta )}$), and the two slanting arrows are (reading from left to
right), $\beta $ and $\alpha $. It is important to note that the factoring
morphism $\theta$ is {\it uniquely} determined because we are working
with base points. The directed property of $M_g$ follows by a simple
fiber-product argument.
[Remark: Notice that the object of genus $1$ in ${\cal S}$ only has
morphisms to itself -- so that this object together with all its
morphisms (to and from) form a subcategory.]
Recall from Section 3 that each morphism of ${\cal S}$ induces a
proper, holomorphic, Teichm\"uller-metric preserving
embedding between the corresponding finite-dimensional Teichm\"uller
spaces. We can thus create the natural {\it direct system of Teichm\"uller
spaces} over the above directed set $M_g$, by associating to each $\alpha
\in M_g$ the Teichm\"uller space ${\cal T}(X_{g(\alpha )})$, and for each $\beta
\succ \alpha $ the corresponding holomorphic embedding ${\cal T}(\theta)$ (with
$\theta$ as in the diagram above).
Consequently, we may form the {\it direct limit Teichm\"uller space over
$X=X_g$}:
$$
{\cal T}_{\infty}(X_g) = {\cal T}_{\infty}(X) := ind. lim. {\cal T}(X_{g(\alpha )})
\eqno(5.1)
$$
the inductive limit being taken over all $\alpha $ in the directed set
$M_g$. This is our {\it commensurability Teichm\"uller space}.
\noindent
{\bf Remark:} Over the same directed set $M_g$ we may also define a
natural {\it inverse system of surfaces}, by asscoiating to $\alpha \in
M_g$ a certain copy, $S_{\alpha }$ of the pointed surface $X_{g(\alpha )}$.
[Fix a universal covering over $X=X_g$. $S_{\alpha }$ can be taken to be the
universal covering quotiented by the action of the subgroup
$Im(\pi_1(\alpha )) \subset {\pi_1}(X,\star)$.] If $g \ge 2$, then
the inverse limit of this system is the {\it universal solenoidal surface}
$H_{\infty}$ whose Teichm\"uller theory was studied in [S], [NS].
The completion of ${\cal T}_{\infty}(X)$ in the Teichm\"uller metric is
${\cal T}(H_{\infty})$.
A remarkable but obvious fact about this construction is that every
morphism $\pi:Y \rightarrow X$ of ${\cal S}$ induces a natural Teichm\"uller
metric preserving {\it homeomorphism}
$$
{\cal T}_{\infty}(\pi): {\cal T}_{\infty}(Y) \longrightarrow {\cal T}_{\infty}(X)
\eqno(5.2)
$$
${\cal T}_{\infty}(\pi)$ is invertible simply because the morphisms
of ${\cal S}$ with target $Y$ are cofinal with those having target $X$.
If we consider objects and maps to be
continuous/holomorphic on the inductive limit spaces when they are
continuous/holomorphic when restricted to the finite-dimensional
strata, then it is clear that ${\cal T}_{\infty}(\pi)$ is a biholomorphic
identification. (Note that ${\cal T}_{\infty}$ acts covariantly, since it is defined by
a morphism of direct systems, although the Teichm\"uller functor ${\cal T}$
of (3.3) was contravariant.)
It follows that each ${\cal T}_{\infty}(X)$, (and its completion ${\cal T}(H_{\infty})$),
is equipped with a large {\it automorphism group} -- one from each
(undirected) cycle of morphisms of ${\cal S}$ starting from $X$ and
returning to $X$. By repeatedly using pull-back diagrams (i.e., by
choosing the appropriate connected component of the fiber product of
covering maps), it is easy to see that the automorphism arising from
any (many arrows) cycle can be obtained simply from a two-arrow cycle
${\tilde X} {\rightarrow \atop \rightarrow } X$. Namely, whenever we have
(the isotopy class of) a ``self-correspondence'' of $X$ given by
two non-isotopic coverings, say $\alpha $ and $\beta $,
$$
{\tilde X} {\rightarrow \atop \rightarrow } X
\eqno(5.3)
$$
we can create an automorphism of ${\cal T}_{\infty}(X)$ defined as the composition:
${{\cal T}_{\infty}(\beta )}\circ{({\cal T}_{\infty}(\alpha ))^{-1}}$. Therefore each of these automorphisms
-- arising from any arbitrarily complicated cycle of coverings (starting
and ending at $X$) -- is obtained as one of these simple
``two-arrow'' compositions. These automorphisms form a group that we
shall call the {\it commensurability modular group}, $CM_{\infty}(X)$,
acting on the universal commensurability Teichm\"uller space ${\cal T}_{\infty}(X)$.
We make some further remarks regarding this large new mapping class
group. Consider the abstract graph ($1$-complex), $\Gamma ({\cal S})$,
obtained from the category ${\cal S}$ by looking at the objects
as vertices and the (undirected) arrows as edges. It is
clear from the definition above that the fundamental group
of this graph, viz. $\pi_{1}(\Gamma ({\cal S}),X)$, is acting on ${\cal T}_{\infty}(X)$
as these automorphisms. In fact, we may fill in all the ``commuting
triangles'' -- i.e., fill in the $2$-cells in this abstract
graph whenever two morphisms (edges) compose to give a third edge;
the thereby-reduced fundamental group of this $2$-complex produces
on ${\cal T}_{\infty}(X)$ the action of $CM_{\infty}(X)$.
It is interesting to observe that this new modular group $CM_{\infty}(X)$
of automorphisms on ${\cal T}_{\infty}(X)$ corresponds exactly to ``virtual automorphisms''
of the fundamental group $\pi_{1}(X)$, -- generalizing the
classical situation where the usual automorphism group $Aut(\pi_{1}(X))$
appears as the action via modular automorphisms on ${\cal T}(X)$.
Indeed, given any group $G$, one may define its associated group of
``virtual'' automorphisms; as opposed to usual automorphisms,
for virtual automorphisms we demand that they be defined only
on some finite index subgroup of $G$. To be precise,
consider isomorphisms $\rho :H \rightarrow K$ where $H$ and $K$ are
subgroups of finite index in $G$. Two such isomorphisms (say $\rho _1$
and $\rho _2$) are considered equivalent if there is a finite index subgroup
(sitting in the intersection of the two domain groups) on which they
coincide. The equivalence class $[\rho ]$ -- which is like the {\it
germ} of the isomorphism $\rho $ -- is called a {\it virtual automorphism}
of $G$; clearly the virtual automorphisms of $G$ constitute a group,
$Vaut(G)$, under the obvious law of composition, (namely, compose
after passing to deeper finite index subgroups, if necessary).
We shall apply this concept to the fundamental group of a surface of
genus $g$, ($g>1$). It is clear from definition that the group
$Vaut(\pi_{1}(X_g))$ {\it is genus independent}, as is to be expected
in our constructions.
In fact, $Vaut$ presents us a neat way of formalizing the ``two-arrow
cycles'' which we introduced to represent elements of ${CM_{\infty}}$.
Letting $G = \pi_{1}(X)$, (recall that $X$ is already equipped with a
base point), the two-arrow diagram (5.3) above corresponds to the
following well-defined virtual automorphism of $G$:
$$
[\rho ] = [{\beta }_{*}\circ{\alpha }_{*}^{-1}:{\alpha }_{*}(\pi_{1}({\tilde X})) \rightarrow
{\beta }_{*}(\pi_{1}({\tilde X}))]
$$
Here ${\alpha }_{*}$ denotes the monomorphism of the fundamental group
$\pi_{1}({\tilde X})$ into $\pi_{1}(X) = G$, and similarly ${\beta }_{*}$.
We let $Vaut^{+}({\pi}_{1}(X))$ denote the subgroup of $Vaut$ arising
from pairs of orientation preserving coverings. The final upshot is
that ${CM_{\infty}(X)}$ is {\it isomorphic} to $Vaut^{+}(\pi_{1}(X))$ and there
is a natural surjective homomorphism:
${\pi}_{1}(\Gamma ({\cal S}),X) \rightarrow Vaut^{+}({\pi}_{1}(X))$ whose kernel is
obtained by filling in all commuting triangles in $\Gamma ({\cal S})$.
\noindent
{\it Acknowledgement:} The concept of $Vaut$ has arisen in group
theory papers -- for example [Ma],[MT]. We are grateful to Chris Odden for
pointing out these references.
\noindent {\bf Remark 5.4.} For the genus one object $X_1$ in ${\cal S}$,
we know that the Teichm\"uller spaces for all unramified coverings
are each a copy of the upper half-plane $H$. The maps ${\cal T}(\pi)$ are
M\"obius identifications of copies of the half-plane with itself,
and we easily see that the pair $({\cal T}_{\infty}(X_1),CM_{\infty}(X_1))$ is
identifiable as $(H,PGL(2,\rlap {\raise 0.4ex \hbox{$\scriptscriptstyle |$}))$. In fact, $GL(2,\rlap {\raise 0.4ex \hbox{$\scriptscriptstyle |$}) \cong Vaut({\bf Z} \oplus
{\bf Z})$,
and $Vaut^{+}$ is precisely the subgroup of index $2$ therein, as expected.
Notice that the action has dense orbits in the genus one case.
On the other hand, if $X \in Ob({\cal S})$ is of any genus $g \geq 2$,
then we get an infinite dimensional ``ind-space'' as ${\cal T}_{\infty}(X)$ with
the action of ${\cal G}(X)$ on it as described. Since the tower of coverings
over $X$ and $Y$ (both of genus higher than $1$) eventually become
cofinal, it is clear that {\it for any choice of genus higher than one we get
one isomorphism class of pairs} $({\cal T}_{\infty}, CM_{\infty})$. (It is not known
whether the action has dense orbits in this situation; this matter is
related to some old queries on coverings of Riemann surfaces.)
We work now over the direct system of the higher genus example
$(T_{\infty}, CM_{\infty})$ and obtain the main theorem. We will first
explain some preliminary material on direct limits of holomorphic
line bundles over a direct system of complex manifolds.
Given a direct system $T_{\alpha }$ of complex manifolds, and line
bundles $\xi_{\alpha }$ over these, suppose that there are power
law maps as the $\Omega (\pi)$ above, between the corresponding principal
$\C*$ bundles covering the mappings in the direct system of
base manifolds.
Let $N$ denote the directed set of positive integers ordered by
divisibility. For each $\lambda \in N$ take a copy of $\C*$, call it
$(\C*,\lambda )$ and form the direct system $\{ (\C* ,\lambda )\}$ where
$(\C*,\lambda ) \rightarrow (\C* ,{\lambda }')$ is given by the power law map:
$z \rightarrow z^d$ when ${\lambda }'=d{\lambda }$. These maps are homomorphisms
of groups, and the direct limit over $N$ is canonically isomorphic to
the group ${{\CC}^{*}} \otimes {\QQ} := \CC\otimes_{{\bf Z}}\rlap {\raise 0.4ex \hbox{$\scriptscriptstyle |$}$. [The isomorphism maps the
equivalence class of the element $(z,\lambda ) \in (\C*,\lambda )$
to $z \otimes {1/\lambda } \in {{\CC}^{*}} \otimes {\QQ}$.]
The direct limit object obtained from the power law connecting maps
between the principal bundles associated to the ${DET}_{n}^{12}$ system
over the Teichm\"uller spaces will give us a ${{\CC}^{*}} \otimes {\QQ}$ principal bundle
over the universal commensurability Teichm\"uller space ${\cal T}_{\infty}$, at least at
the level of sets. The topological and holomorphic structure on these
sets is defined for maps into these objects which factor through
the direct system by imposing these properties on the factorizations.
Let us consider the direct limit bundles obtained from a family of such
bundles $\xi_{\alpha }$, and from the family obtained by raising each
$\xi_{\alpha }$ to the tensor power $d$. These are two ${{\CC}^{*}} \otimes {\QQ}$ bundles over
the direct limit of the bases which may be thought to have the
same total spaces (as sets) but the ${{\CC}^{*}} \otimes {\QQ}$ action on the second one is
obtained by precomposing the original action by the automorphism
of ${{\CC}^{*}} \otimes {\QQ}$ obtained from the homomorphism $z \mapsto z^d$ on $\C*$.
\noindent
{\bf Theorem 5.5.} {\it Fix any integer $n$. Starting from any
base surface $X \in Ob({\cal S})$, we obtain a direct system of principal
$\CC^*$ bundles ${\cal L}_n(Y) := DET_{n,g(Y)}^{\otimes 12}$ over the
Teichm\"uller spaces ${\cal T}(Y)$ with holomorphic homomorphisms $\Omega (\pi)$
(see Theorem 4.3) between the total spaces; here $Y \mapright{\pi} X$
is an arbitrary morphism of ${\cal S}$ with target $X$.
By passing to the direct limit, one
therefore obtains over the universal commensurability Teichm\"uller space,
${\cal T}_{\infty}(X)$, a principal ${{\CC}^{*}} \otimes {\QQ}$ bundle:
$$
{\cal L}_{n,\infty}(X) = ind. lim. {\cal L}_n(Y)
$$
Since the maps $\Omega (\pi)$ preserved the Quillen unit circles, the limit
object also inherits such a Quillen ``hermitian'' structure.
The construction is functorial with respect to change of the base $X$
in the obvious sense that the directed systems and their limits are
compatible with the biholomorphic identifications ${\cal T}_{\infty}(\pi)$ of
equation (5.2). In particular, the commensurability modular group
action $CM_{\infty}(X)$ on ${\cal T}_{\infty}(X)$ has a natural lifting to
${\cal L}_{n,\infty}(X)$ -- acting by unitary automorphisms.
Finally, the Mumford isomorphisms persist:
$$
{\cal L}_{n,\infty}(X) = {\cal L}_{0,\infty}(X)^{\otimes (6n^{2} - 6n + 1)}
$$
Namely, if we change the action of ${{\CC}^{*}} \otimes {\QQ}$ on the ``Hodge'' bundle
${\cal L}_{0,\infty}$ by the ``raising to the $(6n^2-6n+1)$ power''
automorphism of ${{\CC}^{*}} \otimes {\QQ}$, then the principal ${{\CC}^{*}} \otimes {\QQ}$ bundles are
canonically isomorphic.
}
\noindent {\bf Remark 5.6.}
In other words,
the Mumford isomorphism in the above theorem means that
${\cal L}_{n,\infty}$ and ${\cal L}_{0,\infty}$ are equivariantly isomorphic
relative to the automorphism of ${{\CC}^{*}} \otimes {\QQ}$ induced by the homomorphism
of $\C*$ that raises to the power exhibited.
Also, we could have used the Quillen hermitian structure to reduce the
structure group from $\CC^{*}$ to $U(1)$, and thus obtain direct systems
of $U(1)$ bundles over the Teichm\"uller spaces. Passing to the direct
limit would then produce $U(1) \otimes \rlap {\raise 0.4ex \hbox{$\scriptscriptstyle |$} :=$ {\it ``tiny circle''}
bundles over ${\cal T}_{\infty}$, which can be tested for maps into these objects
as above.
\smallskip
\noindent
{\bf Rational line bundles on ind-spaces:} A line bundle on the inductive
limit of an inductive system of varieties or spaces, is, by definition
([Sha]), a collection of line bundles on each stratum (i.e., each member
of the inductive system of spaces) together with compatible line bundle
(linear on fibers) morphisms. Such a direct system of line bundles
determines an element of the inverse limit of the Picard groups of
the stratifying spaces. See [KNR], [Sha].
(Recall: For any complex space $T$, $Pic(T)$ := the group (under $\otimes$)
of isomorphism classes of line bundles on $T$. In the case of the
Teichm\"uller spaces, we refer to the modular-group invariant
bundles as constituting the relevant Picard group -- see [BN].)
Now, utilising the GRR morphisms $\Gamma (\pi)$ themselves, (without
involving the power law maps), we know from Section 3 that the
``$d$-th root'' of the bundle ${DET}_{n,\tilde g}$ fits together with the bundle
${DET}_{n,g}$ ($d=({\tilde g}-1)/(g-1)$).
A ``rational'' line bundle over the inductive limit is defined to be
an element of the inverse limit of the $Pic_{\rlap {\raise 0.4ex \hbox{$\scriptscriptstyle |$}} = Pic \otimes {\rlap {\raise 0.4ex \hbox{$\scriptscriptstyle |$}}$.
Therefore we may also state a result about the existence of
canonical elements of the inverse limit,
$\lim_{\leftarrow}Pic({\cal T}_{g_i})_{\rlap {\raise 0.4ex \hbox{$\scriptscriptstyle |$}}$, by our construction.
Indeed, in the notation of Section 3, by using the morphisms
${\Gamma (\pi)}\otimes {1/{deg(\pi)}}$ between ${DET}_{n,g}$ and ${{DET}_{n,\tilde g}}\otimes {1/{deg(\pi)}}$
to create a directed system, we obtain canonical elements representing
the Hodge and higher $DET_n$ bundles, with respective Quillen metrics:
$$ {\Lambda }_{m} \in
\lim_{\leftarrow}Pic({\cal T}_{g_i})_{\rlap {\raise 0.4ex \hbox{$\scriptscriptstyle |$}}, ~~m=0,1,2,..
\eqno(5.7)
$$
The pullback (i.e., restriction) of $\Lambda _m$ to each of the stratifying
Teichm\"uller spaces ${\cal T}_{g_i}$ is $(n_i)^{-1}$ times the
corresponding $DET_{m}$ bundle, (with $(n_i)^{-1}$ times its Quillen metric),
over ${\cal T}_{g_i}$. Here $n_i$ is the degree of the covering of the
surface of genus $g_i$ over the base surface. As rational hermitian line
bundles the Mumford isomorphisms persist:
$$
\Lambda _{m} = {\Lambda _{0}}^{\otimes (6m^{2} - 6m +1)}
\eqno(5.8)
$$
as desired. This statement is different from that of the Theorem.
For further details see [BN].
\smallskip
\noindent{\bf Polyakov measure on ${\cal M}_g$ and our constructions:} In his
study of bosonic string theory, Polyakov constructed a measure
on the moduli space ${\cal M}_g$ of curves of genus $g(\geq 2)$.
Details can found, for example, in
[Alv], [Nel]. Subsequently, Belavin and Knizhnik, [BK],
showed that the Polyakov measure has the following elegant mathematical
description. First note that a hermitian metric on the canonical bundle of
a complex space gives a measure on that space. Fixing a volume form
(up to scale) on a space therefore amounts to fixing a fiber
metric (up to scale) on the canonical line bundle, $K$, over
that space. But the Hodge bundle $\lambda $ has its natural Hodge metric
(arising from the $L^2$ pairing of holomorphic
1-forms on Riemann surfaces). Therefore we may transport the corresponding
metric on ${\lambda }^{13}$ to $K$ by Mumford's isomorphism, (as we know the
choice of this isomorphism is unique up to scalar) -- thereby obtaining a
volume form on ${\cal{M}}_g$. [BK] showed that this is none other than the
Polyakov volume. Therefore, the presence of Mumford isomorphisms over the
moduli space of genus $g$ Riemann surfaces describes the Polyakov measure
structure thereon.
Above we have succeeded in fitting together the Hodge and higher $DET$
bundles over the ind space ${\cal T}_{\infty}$, together with the relating Mumford
isomorphisms. We thus have from our results a structure on ${\cal T}_{\infty}$ that
suggests a genus-independent, universal, version of the Polyakov
structure.
We remark that since the genus is considered the
perturbation parameter in the above formulation of the standard
perturbative bosonic Polyakov string theory, our work can be considered as
a contribution towards a {\it non-perturbative} formulation of that theory.
\newpage
|
1995-09-14T06:20:08 | 9505 | alg-geom/9505037 | en | https://arxiv.org/abs/alg-geom/9505037 | [
"alg-geom",
"math.AG"
] | alg-geom/9505037 | Alice Silverberg | A. Silverberg and Yu. G. Zarhin | Hodge groups of abelian varieties with purely multiplicative reduction | This is an updated version of the paper. LaTeX2e or LaTeX2.09 or
AMSLaTeX. Contact: [email protected] | null | 10.1070/IM1996v060n02ABEH000074 | null | null | The main result of the paper is that if $A$ is an abelian variety over a
subfield $F$ of ${\bold C}$, and $A$ has purely multiplicative reduction at a
discrete valuation of $F$, then the Hodge group of $A$ is semisimple. Further,
we give necessary and sufficient conditions for the Hodge group to be
semisimple. We obtain bounds on certain torsion subgroups for abelian varieties
which do not have purely multiplicative reduction at a given discrete
valuation, and therefore obtain bounds on torsion for abelian varieties,
defined over number fields, whose Hodge groups are not semisimple.
| [
{
"version": "v1",
"created": "Wed, 31 May 1995 13:02:07 GMT"
},
{
"version": "v2",
"created": "Wed, 13 Sep 1995 08:57:45 GMT"
}
] | 2015-06-24T00:00:00 | [
[
"Silverberg",
"A.",
""
],
[
"Zarhin",
"Yu. G.",
""
]
] | alg-geom | \section{Introduction}
We show that if $A$ is an abelian variety over a subfield
$F$ of ${\bold C}$, and $A$ has purely multiplicative reduction at a discrete
valuation of $F$, then the Hodge group of $A$ is semisimple (Theorem
\ref{hodsemi}). Since the non-semisimplicity of the Hodge group of an
abelian variety can be translated into a condition on the endomorphism
algebra and its action on the tangent space (see Theorem \ref{notsemi}),
this gives a useful criterion for determining when an abelian variety
does not have purely multiplicative reduction.
For abelian varieties over number fields, a result analogous to
Theorem \ref{hodsemi} holds
where the Hodge group is replaced by a certain linear algebraic
group $H_\ell$ over ${\bold Q}_\ell$ arising from
the image of the $\ell$-adic representation associated to $A$ (see
Theorem \ref{notsemiGal}).
The Mumford-Tate conjecture predicts that $H_\ell$ is the
extension of scalars to ${\bold Q}_\ell$ of the Hodge group.
Our result generalizes a result of Mustafin (Corollary after
Theroem 3.2 of \cite{Mustafin}), which says that for a Hodge
family of abelian varieties (as in \cite{MumP}) admitting a
``strong degeneration'', generically the fibers have
semisimple Hodge group.
The problem of describing the Hodge group of an
abelian variety with
purely multiplication reduction was posed by V.\ G.\ Drinfeld,
in a conversation with Zarhin in the 1980's.
In \S\ref{boundssect} we provide bounds on torsion for
abelian varieties
which do not have purely multiplicative reduction at a
given discrete valuation. We apply this and Theorem \ref{hodsemi}
to obtain bounds on torsion for abelian varieties whose Hodge
groups are not
semisimple.
Silverberg would like to thank MSRI and IHES for their generous
hospitality, and NSF for financial support.
Zarhin would like to thank the Institute f\"ur Experimentelle
Mathematik for its hospitality,
and Gerhard Frey for his interest in the paper and useful
discussions.
\section{Definitions, notation, and lemmas}
Suppose $A$ is an abelian variety defined over a field $F$ of
characteristic zero, and $L$ is an algebraically
closed field containing $F$.
Write $\mbox{End}_F(A)$
for the set of endomorphisms of $A$ which are defined over $F$, let
$\mbox{End}(A) = \mbox{End}_L(A)$, let $\mbox{End}^0(A) = \mbox{End}(A) \otimes_{\bold Z} {\bold Q}$,
and let $\mbox{End}^0_F(A) = \mbox{End}_F(A) \otimes_{\bold Z} {\bold Q}$.
Suppose $K$ is a field and $\iota : K \hookrightarrow \mbox{End}_F^0(A)$
is an embedding such that $\iota(1) = 1$. Let $\mbox{Lie}_F(A)$ be
the tangent
space of $A$ at the origin,
an $F$-vector space. If $\sigma$ is an embedding of $K$ into $L$, let
$$n_\sigma = \mbox{dim}_L\{t \in \mbox{Lie}_L(A) :
\iota(\alpha)t = \sigma(\alpha)t {\text{ for all }} \alpha \in K\}.$$
Note that $n_\sigma$ is independent of the choice of an algebraically
closed field $L$ containing $F$.
Write ${\bar \sigma}$ for the composition of
$\sigma$ with the involution complex conjugation of $K$.
\begin{defn}
If $A$ is an abelian variety over an algebraically
closed field $L$ of
characteristic zero, $K$ is a CM-field, and
$\iota : K \hookrightarrow \mbox{End}^0(A)$
is an embedding such that $\iota(1) = 1$,
we say $(A,K,\iota)$ is {\em of Weil type}
if $n_\sigma = n_{\bar \sigma}$ for all embeddings $\sigma$
of $K$ into
$L$.
\end{defn}
\begin{lem}
\label{freeof}
If $A$ is an abelian variety defined over a field $F$ of characteristic
zero, $L$ is an algebraically closed field containing $F$,
$K$ is a CM-field,
and
$\iota : K \hookrightarrow \mbox{End}_F^0(A) \subseteq \mbox{End}^0(A)$
is an embedding such that $\iota(1) = 1$,
then the following statements are equivalent:
\begin{enumerate}
\item[\normalshape{(i)}]
$(A,K,\iota)$ is of Weil type,
\item[\normalshape{(ii)}]
$\iota$ makes $\mbox{Lie}_{L}(A)$ into a free
$(K \otimes_{\bold Q} L)$-module,
\item[\normalshape{(iii)}]
$\iota$ makes $\mbox{Lie}_F(A)$ into a free $(K \otimes_{\bold Q} F)$-module.
\end{enumerate}
\end{lem}
\begin{pf}
Let
$\Sigma$ be the set of embeddings of $K$ into $L$, let
$\psi_F : K \otimes_{\bold Q} F \to \mbox{End}(\mbox{Lie}_F(A))$
be the homomorphism induced by $\iota$,
let $\psi : K \otimes_{\bold Q} L \to \mbox{End}(\mbox{Lie}_L(A))$ be the extension
of scalars of $\psi_F$ to $K \otimes_{\bold Q} L$, let
$m = 2\mbox{dim}(A)/[K : {\bold Q}]$, let $M_F = (K \otimes_{\bold Q} F)^m$, and let
$M = (K \otimes_{\bold Q} L)^m = M_F \otimes_F L$. Let
$\psi_F^\prime : K \otimes_{\bold Q} F \to \mbox{End}(M_F)$
and
$\psi^\prime : K \otimes_{\bold Q} L \to \mbox{End}(M)$ be the natural homomorphisms.
By \S 2.1 of \cite{Shimura}, for every $\sigma \in \Sigma$ we have
$n_\sigma + n_{\bar \sigma} = m$.
For $\alpha \in K$, taking the trace of $\psi(\alpha)$ gives
$$\mbox{tr}(\psi(\alpha)) = \sum_{\sigma \in \Sigma}n_{\sigma}\sigma(\alpha).$$
The traces of $\psi$ and of $\psi^\prime$ coincide on $K$ if and only if
$n_\sigma = n_{\bar \sigma} = m/2$ for every $\sigma \in \Sigma$.
Since $K \otimes_{\bold Q} L$ is a semisimple ring,
$\mbox{Lie}_L(A)$ and $M$ are semisimple $(K \otimes_{\bold Q} L)$-modules.
Therefore, $\mbox{Lie}_L(A)$ is a free $(K \otimes_{\bold Q} L)$-module if and only if
the traces of $\psi$ and of $\psi^\prime$ coincide on $K$.
Therefore, $\mbox{Lie}_L(A)$ is a free $(K \otimes_{\bold Q} L)$-module if and only if
$(A,K,\iota)$ is of Weil type.
If $\mbox{Lie}_F(A)$ is a free
$(K \otimes_{\bold Q} F)$-module, then clearly
$\mbox{Lie}_L(A)$ ( $= \mbox{Lie}_F(A) \otimes_F L$) is a free
$(K \otimes_{\bold Q} L)$-module.
Conversely, if $\mbox{Lie}_L(A)$ is a free
$(K \otimes_{\bold Q} L)$-module, then the traces of $\psi$ and of
$\psi^\prime$ (and therefore of $\psi_F$ and of
$\psi_F^\prime$) coincide on $K$. Since
$K \otimes_{\bold Q} F$ is a semisimple ring, $\mbox{Lie}_F(A)$ and $M_F$ are
semisimple $(K \otimes_{\bold Q} F)$-modules. Therefore $\mbox{Lie}_F(A)$ and $M_F$
are isomorphic as $(K \otimes_{\bold Q} F)$-modules, i.e., $\mbox{Lie}_F(A)$
is a free $(K \otimes_{\bold Q} F)$-module.
\end{pf}
See also p.~525 of \cite{Ribet} for
the case where $K$ is an imaginary quadratic field.
\begin{rem}
If $A$ is an abelian variety defined over a field $F$ of characteristic
zero, $K$ is a totally real number field, and $\iota : K \hookrightarrow
\mbox{End}_F^0(A)$ is an embedding such that $\iota(1) = 1$, then
$\iota$ makes $\mbox{Lie}_F(A)$ into a free $(K \otimes_{\bold Q} F)$-module.
To see this, let $L$ be an algebraically closed field containing $F$, let
$\Sigma$ be the set of embeddings of $K$ into $L$, and let
$\psi : K \to \mbox{End}(\mbox{Lie}_L(A))$
be the homomorphism induced by $\iota$.
Let $m^{\prime} = \mbox{dim}(A)/[K : {\bold Q}]$. We have
$$\mbox{tr}(\psi(\alpha)) = m^{\prime}\sum_{\sigma \in \Sigma}\sigma(\alpha)$$
for every $\alpha \in K$, by \S 2.1 of \cite{Shimura}. Therefore,
$\mbox{Lie}_L(A)$ is a free $(K \otimes_{\bold Q} L)$-module. As in the
proof of Lemma \ref{freeof}, it follows that $\mbox{Lie}_F(A)$ is a free
$(K \otimes_{\bold Q} F)$-module.
\end{rem}
Suppose $A$ is a complex abelian variety.
Let $V = H_1(A,{\bold Q})$ and let ${\bold S} = \mbox{Res}_{{\bold C}/{\bold R}}{\bold G}_m$.
The complex structure on $A$ gives rise to a rational Hodge structure
on $V$ of weight $-1$, i.e., a homomorphism of algebraic groups
$h : {\bold S} \to \mbox{GL}(V)_{\bold R}$.
Let ${\bold T}$ be the kernel of the norm map ${\bold N} : {\bold S} \to {\bold G}_m$.
Then ${\bold T}({\bold R}) = \{x \in {\bold C} : |x| = 1\}$.
\begin{defn}
If $A$ is an abelian variety over ${\bold C}$ and $V = H_1(A,{\bold Q})$, then the
{\em Hodge group} $H$ is the smallest
algebraic subgroup of $\mbox{GL}(V)$ defined over ${\bold Q}$ such that $H({\bold R})$
contains $h({\bold T}({\bold R}))$.
Equivalently, $H$ is the largest algebraic subgroup of $\mbox{GL}(V)$
defined over ${\bold Q}$ such that all Hodge classes in
$V^{\otimes p} \otimes (V^\ast)^{\otimes q}$, for all non-negative
integers $p$ and $q$, are tensor invariants of $H$. I.e.,
$H$ is the largest algebraic subgroup of $\mbox{GL}(V)$ defined over ${\bold Q}$
which fixes all Hodge classes of all powers of $A$.
\end{defn}
It follows from the definition of $H$ that $\mbox{End}^0(A) = \mbox{End}_H(V)$.
If now $F$ is a number field and $\ell$ is a prime number, let
$T_\ell(A) = {\displaystyle \lim_\leftarrow A_{\ell^r}}$
(the Tate module), let
$V_\ell(A) = T_\ell(A) \otimes_{{\bold Z}_\ell}{\bold Q}_\ell$, and let $\rho_{A,\ell}$
denote the $\ell$-adic representation
$$\rho_{A,\ell} : \mbox{Gal}({\bar F}/F) \to \mbox{GL}(T_\ell(A))
\subseteq \mbox{GL}(V_\ell(A)).$$
Let $G_\ell$
denote the algebraic envelope of the image of $\rho_{A,\ell}$, i.e., the
Zariski closure in $\mbox{GL}(V_\ell(A))$ of the image of $\rho_{A,\ell}$.
By \cite{Faltings}, $G_\ell$ is a reductive algebraic group, and
$\mbox{End}^0_F(A) \otimes_{\bold Q} {\bold Q}_\ell = \mbox{End}_{G_\ell}(V_\ell(A))$.
Let $H_\ell$ be the identity connected component of
$G_\ell \cap \mbox{SL}(V_\ell(A))$.
Then $H_\ell$ is
a connected reductive group and
$\mbox{End}^0(A) \otimes_{\bold Q} {\bold Q}_\ell = \mbox{End}_{H_\ell}(V_\ell(A))$.
We will repeatedly use the fact (see the first Theorem on p.~220
of \cite{Hum})
that if $G$ is a connected linear algebraic group over a field $F$
of characteristic zero, then $G(F)$ is Zariski-dense in $G$.
\begin{lem}
\label{semfin}
If $G$ is a reductive linear algebraic group over a
field $F$ of characteristic zero, and $Z$ is the center of $G$,
then $G$ is semisimple if and only if $Z(F)$ is finite.
\end{lem}
\begin{pf}
Let $Z^0$ denote the identity connected component of $Z$.
Since $G$ is reductive, $G$ is semisimple if and only if $Z^0 = 1$
(see the lemma on p.~125 of \cite{Hum}). Since $Z^0(F)$ is Zariski-dense
in $Z^0$, $Z^0 = 1$ if and only if $Z(F)$ is finite.
\end{pf}
\section{Semisimplicity criteria for the groups $H$ and $H_\ell$}
If the center of $\mbox{End}^0(A)$
is a direct sum of totally real number fields, then
it is well-known that
the groups $H$ and $H_\ell$ are semisimple (see, for instance,
Corollary 1 in \S 1.3.1 of
\cite{Zarhintor} and Lemma 1.4 of \cite{Tankeev}).
The following result follows easily from a result in \cite{moonzar}, and
characterizes the endomorphism algebras of abelian varieties whose
Hodge groups are not semisimple.
\begin{thm}
\label{notsemi}
Suppose $A$ is an
abelian variety defined over ${\bold C}$.
Then the Hodge group of $A$ is not semisimple if and only if
for some simple component $B$ of $A$, the
center of $\mbox{End}^0(B)$ is a CM-field $K$ such that
$(B,K,\mathrm{id})$ is not of Weil type, with $\mathrm{id}$ the identity embedding
of $K$ in $\mbox{End}^0(B)$.
\end{thm}
\begin{pf}
Let $V = H_1(A,{\bold Q})$.
Fix a polarization on $A$. The polarization induces a
non-degenerate alternating bilinear form
$\varphi : V \times V \to {\bold Q}$ such that
$H \subseteq \mbox{Sp}(V,\varphi)$ (see \cite{MumMatAnn}). Then
$$H({\bold Q}) \subseteq \mbox{Sp}(V,\varphi)({\bold Q}) = \{ g \in \mbox{End}(V) : gg^\prime = 1 \},$$
where $g \mapsto g^\prime$ is the involution on $\mbox{End}(V)$ defined
by
$$\varphi(g(x),y) = \varphi(x,g^\prime (y)) \mbox{ for $x, y \in V$}.$$
The restriction of the involution ${}^\prime$ to $\mbox{End}^0(A)$
is the Rosati involution.
Let $Z$ denote the center of $H$ and let $Z_{\mathrm{End}}$ denote the center of
$\mbox{End}^0(A)$.
If $\alpha \in Z({\bold Q})$, then $\alpha$ commutes with all elements of
$H({\bold Q})$, so $\alpha \in \mbox{End}^0(A)$. Further, since $\alpha \in H({\bold Q})$,
$\alpha$ commutes with all elements of $\mbox{End}^0(A)$, and therefore
$\alpha \in Z_{\mathrm{End}}$. Therefore,
\begin{equation}
\label{Zs}
Z({\bold Q}) \subseteq \{\alpha \in Z_{\mathrm{End}} : \alpha\alpha^{\prime} = 1\}.
\end{equation}
If $A$ is isogenous to a product of two abelian varieties, then
the Hodge group $H$ of $A$ is a
subgroup of the product of the Hodge groups $H_1$ and $H_2$ of
the factors, in such a way that for $i = 1$ and $2$ the restriction to $H$
of the projection map from $H_1 \times H_2$ onto $H_i$ induces a
surjective homomorphism from $H$ onto $H_i$ (see Proposition 1.6 of
\cite{Hazama}). It follows easily that
$H$ is semisimple if and only if both $H_1$ and $H_2$ are semisimple.
We may therefore reduce to the case where $A$ is a simple abelian
variety.
Then the center $Z_{\mathrm{End}}$ of $\mbox{End}^0(A)$ is either a totally real number
field or a CM-field.
Suppose $Z_{\mathrm{End}}$ is totally real. Then all Rosati
involutions are the identity when restricted to $Z_{\mathrm{End}}$.
By (\ref{Zs}), $Z({\bold Q}) \subseteq \{\pm 1\}$. Therefore,
$Z({\bold Q})$ is finite, so $H$ is semisimple by Lemma \ref{semfin}.
Suppose $Z_{\mathrm{End}}$ is a CM-field $K$.
Then every Rosati involution induces complex conjugation on $K$.
Choose $\alpha \in K^\times$ such that ${\bar \alpha} = -\alpha$.
Then there exists a unique $K$-Hermitian form $\psi : V \times V \to K$
such that $\varphi(x,y) = Tr_{K/{\bold Q}}(\alpha \psi(x,y))$ (see \cite{Shimura}).
The unitary group $\mbox{U}(V,\psi)$ is an algebraic group over $K_0$, the
maximal totally real subfield of $K$. Let
$U = \mbox{Res}_{K_0/{\bold Q}} \mbox{U}(V,\psi)$,
let $\mbox{SU}$ denote the kernel of the determinant homomorphism
$\mbox{det}_K : \mbox{U} \to \mbox{Res}_{K/{\bold Q}} {\bold G}_m$, and let
$\mbox{End}_K(V)$ denote the ring of $K$-linear endomorphisms of $V$.
Then
$$\mbox{U}({\bold Q}) = \{g \in \mbox{End}_K(V) :
\psi(g(x),g(y)) = \psi(x,y) {\text{ for all }} x, y \in V\}$$
and $H \subseteq \mbox{U} \subseteq \mbox{Sp}(V,\varphi)$.
By Lemma 2.8 of \cite{moonzar}, $H \subseteq \mbox{SU}$
if and only if $(A,K,\mathrm{id})$ is of Weil type.
If $H$ is semisimple, then all homomorphisms from $H$ to
commutative groups are trivial. Therefore
$\mbox{det}_K(H) = 1$, so $H \subseteq \mbox{SU}$.
Conversely,
if $H \subseteq \mbox{SU}$, then
$Z({\bold Q}) \subseteq \mbox{SU}({\bold Q}) \cap K$,
the group of $(\mbox{dim}(V)/[K:{\bold Q}])$-th
roots of unity in $K$. Therefore $Z({\bold Q})$ is
finite and $H$ is semisimple.
\end{pf}
\begin{thm}
\label{notsemiGal}
Suppose $A$ is an abelian variety defined over a number field $F$. Then
the following are equivalent:
\begin{enumerate}
\item[\normalshape{(i)}] $H$ is semisimple,
\item[\normalshape{(ii)}] $H_\ell$ is semisimple, for one prime $\ell$,
\item[\normalshape{(iii)}] $H_\ell$ is semisimple, for every prime $\ell$.
\end{enumerate}
\end{thm}
\begin{pf}
Let $\ell$ be a prime number and let $V_\ell = V_\ell(A)$.
By Theorem \ref{notsemi}, it suffices to show that
$H_\ell$ is not
semisimple if and only if for some simple component $B$ of $A$, the
center of $\mbox{End}^0(B)$ is a CM-field $K$ such that
$(B,K,\mathrm{id})$ is not of Weil type, with $\mathrm{id}$ the identity embedding
of $K$ in $\mbox{End}^0(B)$.
Since $H_\ell$ is connected, it is invariant under
finite extensions of the number field $F$.
By replacing $F$ by a finite extension, we may suppose
that $\mbox{End}^0(A) = \mbox{End}_F^0(A)$.
We parallel the proof of Theorem \ref{notsemi}. Fix a polarization
on $A$ defined over $F$.
Let $V$ and $\varphi$ be as in the proof of Theorem \ref{notsemi}.
Then $V_\ell = V \otimes_{\bold Q} {\bold Q}_\ell$.
Let
$\varphi_\ell : V_\ell \times V_\ell \to {\bold Q}_\ell$ be the ${\bold Q}_\ell$-linear
extension of $\varphi$.
It follows
immediately from p.~516 of \cite{zarhin} and the definition of $H_\ell$ that
$H_\ell \subseteq \mbox{Sp}(V_\ell,\varphi_\ell)$.
Let $Z_\ell$ denote the center of $H_\ell$, let $Z_{\mathrm{End}}$
denote the center of $\mbox{End}^0(A)$, and let
${}^\prime$ denote the involution on $\mbox{End}(V_\ell)$ induced by
$\varphi_\ell$. Following the proof of
Theorem \ref{notsemi}, we conclude that
$$Z_\ell({\bold Q}_\ell) \subseteq \{\alpha \in Z_{\mathrm{End}} \otimes_{\bold Q} {\bold Q}_\ell :
\alpha\alpha^{\prime} = 1\}.$$
If $A$ is $F$-isogenous to a product of two abelian varieties, then
$H_\ell$ is a
subgroup of the product of the corresponding groups $H_{1,\ell}$ and
$H_{2,\ell}$ for
the factors, in such a way that for $i = 1$ and $2$ the restriction to $H_\ell$
of the projection map from $H_{1,\ell} \times H_{2,\ell}$ onto
$H_{i,\ell}$ induces a
surjective homomorphism from $H_\ell$ onto $H_{i,\ell}$. It follows that
we may reduce to the case where $A$ is $F$-simple.
If $Z_{\mathrm{End}}$ is totally real, we conclude that $H_\ell$ is semisimple
as in the proof of Theorem \ref{notsemi}.
Suppose $Z_{\mathrm{End}}$ is a CM-field $K$ and
let $K_\ell = K \otimes_{\bold Q} {\bold Q}_\ell$. Let $\psi$ and $U$ be as in
the proof of Theorem \ref{notsemi},
let $\psi_\ell : V_\ell \times V_\ell \to K_\ell$
denote the $K_\ell$-Hermitian form which extends the pairing $\psi$,
let $\mbox{U}_\ell = \mbox{U} \times {\bold Q}_\ell$,
let $\mbox{SU}_\ell$ denote the kernel of the determinant homomorphism
$\mbox{det}_{K_\ell} : \mbox{U}_\ell \to \mbox{Res}_{K/{\bold Q}}{\bold G}_m \times {\bold Q}_\ell$, and let
$\mbox{End}_{K_\ell}(V_\ell)$ denote the ring of
$K_\ell$-linear endomorphisms of $V_\ell$. Then
$$\mbox{U}_\ell({\bold Q}_\ell) =
\{g \in \mbox{End}_{K_\ell}(V_\ell) :
\psi_\ell(g(x),g(y)) = \psi_\ell(x,y) {\text{ for all }} x, y \in V_\ell\}$$
and
$H_\ell \subseteq \mbox{U}_\ell \subseteq \mbox{Sp}(V_\ell,\varphi_\ell)$.
By Lemma 2.8 of \cite{moonzar}, $H_\ell \subseteq \mbox{SU}_\ell$
if and only if $(A,K,\mathrm{id})$ is of Weil type.
The group $\mbox{SU}_\ell({\bold Q}_\ell) \cap K_\ell$ is
the finite group of
$(\mbox{dim}_{{\bold Q}_\ell}(V_\ell)/\mbox{dim}_{{\bold Q}_\ell}(K_\ell))$-th
roots of unity in the ring $K_\ell$.
Paralleling the proof of Theorem \ref{notsemi},
$H_\ell \subseteq \mbox{SU}_\ell$
if and only if $H_\ell$ is semisimple.
\end{pf}
\begin{ex}
If $A$ is odd-dimensional and the center of
$\mbox{End}^0(A)$ is a CM-field $K$, then $H$ is not semisimple.
To show this, note that $A$ is isogenous to a power of a simple
odd-dimensional abelian variety $B$ such that $K$ is the center
of $\mbox{End}^0(B)$. Then the Hodge groups of $A$ and of $B$ coincide,
so we may reduce to the case where $A$ is simple. Let
$d = \mbox{dim}(A)$ and use the
notation of the proof of Lemma \ref{freeof}. Then
$n_\sigma + n_{\bar \sigma} = 2d/[K : {\bold Q}]$.
If $H$ were semisimple, then by Theorem \ref{notsemi},
$(A,K,\mathrm{id})$ would be of Weil type. We would therefore have
$n_\sigma = n_{\bar \sigma}$, and so $2d/[K : {\bold Q}]$ would be even.
However, $d$ is odd and $[K : {\bold Q}]$ is even, so this cannot happen.
\end{ex}
\section{Abelian varieties having purely multiplicative reduction}
\begin{thm}
\label{hodsemi}
Suppose $A$ is an abelian variety over a subfield $F$ of ${\bold C}$,
$v$ is a discrete valuation on $F$, and $A$ has purely
multiplicative reduction
at $v$. Then the Hodge group $H$ of $A$ is semisimple.
\end{thm}
\begin{pf}
Since $H$ is semisimple if the Hodge groups of each of its
$F$-simple components are,
we may reduce to the case where $A$ is an $F$-simple abelian
variety with purely multiplicative reduction at $v$.
Since the properties of having semisimple Hodge group and having
purely multiplicative reduction are invariant under finite
extensions of the ground field, we may assume
$\mbox{End}_F^0(A) = \mbox{End}^0(A)$.
Suppose that $H$ is not semisimple.
By Theorem \ref{notsemi}, the
center of $\mbox{End}^0(A)$ is a CM-field $K$ such that $(A,K,\mathrm{id})$ is not
of Weil type, with $\mathrm{id}$ the identity embedding
of $K$ in $\mbox{End}^0(A)$.
Let $L$ be a fixed algebraic closure of the completion $F_v$ of
$F$ at $v$. Since $A$ has purely multiplicative reduction at $v$,
$A$ admits a non-archimedean uniformation; i.e.,
(see \cite{Mumford} and \cite{Raynaud}) there are a discrete
subgroup $\Gamma$ of ${\bold G}_m^d(F_v) = (F_v^{\times})^d$,
isomorphic to ${\bold Z}^d$, and a $\mbox{Gal}(L/F_v)$-equivariant $v$-adically
continuous isomorphism $(L^{\times})^d/\Gamma \cong A(L)$ which for
some finite extension $M$ of $F_v$ induces
an isomorphism $(M^{\times})^d/\Gamma \cong A(M)$
as $M$-Lie groups.
Let ${\cal O}$ be the center of $\mbox{End}(A)$. Then ${\cal O}$ is an order in $K$.
By Satz 6 of \cite{Gerritzen}, there is a homomorphism ${\cal O} \hookrightarrow
\mbox{End}({\bold G}_m^d)$ which induces the inclusion ${\cal O} \subseteq \mbox{End}(A)$.
Composing with the natural homomorphism
$$\mbox{End}({\bold G}_m^d) \hookrightarrow \mbox{End}(\mbox{Hom}({\bold G}_m,{\bold G}_m^d))$$
and tensoring with ${\bold Q}$, we have
$$K \hookrightarrow \mbox{End}(\mbox{Hom}({\bold G}_m,{\bold G}_m^d) \otimes {\bold Q}).$$
Therefore, the inclusion of $K$ in $\mbox{End}^0(A)$ induces a $K$-vector space
structure on $\mbox{Hom}({\bold G}_m,{\bold G}_m^d) \otimes {\bold Q}$.
Tensoring with $M$ makes $\mbox{Hom}({\bold G}_m,{\bold G}_m^d) \otimes_{\bold Z} M$ ( $ = M^d$) into
a free $(K \otimes_{\bold Q} M)$-module. We can view $(M^\times)^d$
as a (non-archimedean) analytic variety over $M$. The tangent
space to
$(M^{\times})^d$ at $1$ is isomorphic to $M^d$.
By \cite{Morik1} (see also Chapter 2 of \cite{Manin}),
$(L^{\times})^d/\Gamma$ can be embedded, via
theta functions, as an analytic subvariety of a projective space
${\bold P}^n(L)$, so that the image of $(M^{\times})^d/\Gamma$ is $A(M)$.
Let $T$ denote the analytic tangent space at the origin of the
analytic variety $A(M)$. The tangent map is an isomorphism
$M^d \cong T$.
The algebraic tangent space at the origin to the algebraic variety
$A$ over $M$ is $\mbox{Lie}_M(A) = \mbox{Lie}_F(A) \otimes_F M$, and there is
a canonical isomorphism between the analytic and algebraic
tangent spaces to $A(M)$
(see subsection 3 of \S 2 of Chapter II of \cite{shaf}). Therefore,
the identity embedding of $K$ into $\mbox{End}^0(A)$ makes
$\mbox{Lie}_M(A)$ into a free $(K \otimes_{\bold Q} M)$-module.
By Lemma \ref{freeof}, $(A,K,\mathrm{id})$
is of Weil type, contradicting our assumptions.
\end{pf}
Theorem \ref{hodsemi} remains true if we replace the assumption
that $v$ is a discrete valuation by the assumption that $v$ is a
valuation of rank $1$ and $A$ admits non-archimedean
uniformization (Gerritzen's theorem remains true under these assumptions).
\section
{Bounds on torsion of abelian varieties which do not have purely
multiplicative reduction}
\label{boundssect}
It is easy to find uniform bounds on orders of torsion points over
number fields for
abelian varieties with potential good reduction, or for elliptic
curves which do not have multiplicative reduction, at a given
discrete valuation (see \cite{contemp}, \cite{CM}, \cite{flexoest}).
In this section we extend these results by finding bounds on torsion
subgroups of abelian varieties
which do not have purely multiplicative reduction at a given
discrete valuation.
Suppose $A$ is a $d$-dimensional abelian variety over a field $F$,
$v$ is a discrete valuation on $F$ with finite residue field $k$
of order $q$, $n$ is a positive integer
relatively prime to $q$, and $J$ is a non-zero subgroup of the group
$A_n(F)$ of points in $A(F)$ of order dividing $n$.
Let $A_v^0$ denote the connected component of the identity of the
special fiber $A_v$ of the N\'eron minimal model of $A$ at $v$.
Let $a$, $u$, and $t$ denote respectively the abelian, unipotent,
and toric ranks of $A_v^0$. Then $d = a + t + u$.
If $\lambda$ is a polarization on $A$
defined over an extension of $F$ which is unramified over $v$,
define a skew-symmetric Galois-equivariant pairing
$e_{\lambda,n}$ on $A_n$ by
$e_{\lambda,n}(x,y) = e_n(x,\lambda(y))$, where $e_n$ is the
Weil pairing.
If $J$ is not isotropic
with respect to $e_{\lambda,n}$, and $n$ is a prime number,
then $\zeta_n \in F$, so the prime $n$ can be bounded independent
of $A$ (with a bound depending on $F$).
Therefore, the more interesting case is when $J$ is an isotropic
subgroup of $A_n(F)$.
If $J$ is a maximal isotropic subgroup of $A_n(F)$, and
$A$ does not have semistable reduction at $v$, then $n \le 4$,
by Theorem 6.2 of \cite{semistab}.
The remaining case to consider
is the case where $A$ has semistable reduction at $v$. Theorem \ref{bounds}
below implies that in
this case we can bound $n$ in terms of $q$ and $d$, as long
as $A$ does not have purely multiplicative reduction at $v$.
Note that if $P$ is a point of $A(F)$ of order $n$ which reduces to a point of
$A_v^0$, then $n$ is bounded above by $\#A_v^0(k)$. Therefore
even in the case of abelian varieties with purely
multiplicative reduction one can easily bound, by
a constant depending only on $d$ and $q$,
the orders of torsion points
whose reductions lie in $A_v^0$. As was the case for
elliptic curves, the most difficult case is the case when the
reduction is purely multiplicative and the reductions of the torsion
points do not lie in the identity connected component of the
special fiber of the N\'eron minimal model.
\begin{lem}[Lemma 1 on pp.~494--495 of \cite{serretate}]
\label{serretatelem}
If $A$ is an abelian variety over a field $F$,
$v$ is a discrete valuation on $F$, and $n$ is a positive
integer relatively prime to the residue characteristic of $v$,
then $(A_v^0)_n$ is a free ${\bold Z}/n{\bold Z}$-module of rank $2a + t$.
\end{lem}
\begin{prop}
\label{lessthan}
Suppose $A$ is an abelian variety over a field $F$,
$v$ is a discrete valuation on $F$ with finite residue field $k$
of order $q$, $n$ is a positive integer relatively prime to $q$,
and $J$ is a subgroup of $A_n(F)$.
Suppose there is a positive constant $\epsilon$ such that
$|J| \ge n^{t+2u+\epsilon}$.
Then
$n \le (\#(A_v^0)_n(k))^{1/\epsilon} \le (\#A_v^0(k))^{1/\epsilon}$.
\end{prop}
\begin{pf}
Let $d = \mbox{dim}(A)$.
Via the reduction map we may view $A_n(F)$, and therefore $J$,
as a subgroup of $(A_v)_n$ (see \cite{serretate}).
Therefore, $\#J \#(A_v^0)_n$ divides $n^{2d} \#(J \cap (A_v^0)_n)$.
Thus by Lemma \ref{serretatelem},
$\#J$ divides $n^{t+2u}\#(J \cap (A_v^0)_n)$.
Therefore,
$$n^\epsilon \le \#(J \cap (A_v^0)_n) \le
\#(A_v^0)_n(k) \le \#A_v^0(k).$$
\end{pf}
\begin{thm}
\label{bounds}
Suppose $A$ is a $d$-dimensional abelian variety over a field $F$,
$v$ is a discrete valuation on $F$ with finite residue field $k$
of order $q$, $n$ is a positive integer relatively prime to
$q$, and $A_n(F)$ has a subgroup of
order $n^d$. Suppose the reduction of $A$ at $v$ is semistable but
not purely multiplicative.
Then
$$n \le (1 + \sqrt{q})^{2a}(1 + q)^t \le (1 + \sqrt{q})^{2d}.$$
\end{thm}
\begin{pf}
Since $A$ has semistable reduction at $v$, $u = 0$.
Since the reduction of $A$ at $v$
is not purely multiplicative, $a \ge 1$.
Applying Proposition \ref{lessthan} with
$\epsilon = 1$, we have $n \le \#A_v^0(k)$.
Since $A$ has semistable reduction at $v$, $A_v^0$ is an extension
of an abelian variety $B$ by a torus $T$. We have the Weil bound
$\# B(k) \le (1 + \sqrt{q})^{2a}$. Similarly, we have the bound
$\# T(k) \le (1 + q)^t$, as follows.
Let $X$ be the group of characters of $T \otimes {\bar k}$.
The Frobenius element of $\mbox{Gal}({\bar k}/k)$ acts on $X$, say by
$\varphi_0$. Since the torus $T$
splits over some finite extension of $k$, $\mbox{Gal}({\bar k}/k)$ acts on
$X$ through a finite quotient, so
all the eigenvalues of $\varphi_0$ have absolute value $1$.
Therefore all eigenvalues of $q - \varphi_0$ are non-zero and
have absolute value at most $1 + q$. We have (see Theorem 6.2
in \S 1 of Chapter VI of \cite{Vo})
$$\# T(k) = |\mbox{det}(q - \varphi_0)| \le (1 + q)^t.$$
Therefore,
$$\#A_v^0(k) \le (1 + \sqrt{q})^{2a}(1 + q)^t =
(1 + \sqrt{q})^{2a}(1 + q)^{d-a}
\le (1 + \sqrt{q})^{2d}.$$
\end{pf}
If $c$ and $d$ are positive integers, let $f(c,d)$ be the maximum of
the orders of the elements of $GL_{2d}({\bold Z}/c{\bold Z})$.
\begin{thm}
\label{nrbounds}
Suppose $A$ is a $d$-dimensional abelian variety over a number field $F$
of degree $m$, $v$ is a discrete valuation on $F$ at which $A$ does not
have purely multiplicative reduction, $p$ is the residue
characteristic of $v$, $n$ and $r$ are positive integers not divisible
by $p$, $r \ge 3$, and $A_n(F)$ has a subgroup of order $n^d$. Then
$$n \le (1 + p^{mf(r,d)/2})^{2d}.$$
\end{thm}
\begin{pf}
By a theorem of Raynaud (Proposition 4.7 of \cite{SGA}), $A$ has semistable
reduction at the discrete valuations on the field $F(A_{r})$ of residue
characteristic not dividing $r$. Let $v^\prime$ be a valuation
on $F(A_r)$ extending $v$, and let $k$ be the corresponding residue
field. Then $\# k$ divides $p^{mf}$, where $f$ is the order of
Frobenius at $v^\prime$ in $\mbox{Gal}(F(A_r)/F)$. Since
$\mbox{Gal}(F(A_r)/F)$ injects into $GL_{2d}({\bold Z}/r{\bold Z})$,
$\# k$ divides $p^{mf(r,d)}$. The result now follows from Theorem
\ref{bounds}.
\end{pf}
\begin{cor}
\label{Hsemibds}
Suppose $A$ is a $d$-dimensional abelian variety over a number field $F$
of degree $m$, and suppose the Hodge group of $A$ is not semisimple.
Suppose $n$ is a positive integer and $A_n(F)$ has a subgroup of
order $n^d$. Then
$$n \le [(1 + 2^{mf(3,d)/2})(1 + 3^{mf(4,d)/2})]^{2d}
< (1+10^{-11})\cdot (2^{3^{4d^2}}\cdot 3^{4^{4d^2}})^{md}.$$
\end{cor}
\begin{pf}
The result follows from Theorem \ref{hodsemi}, by applying
Theorem \ref{nrbounds} with
$p = 2$, $r = 3$ to bound the prime-to-two part of $n$,
and with $p = 3$, $r = 4$ to bound the prime-to-three part of $n$.
The final inequality follows from the bound
$f(c,d) \le \# GL_{2d}({\bold Z}/c{\bold Z}) < c^{4d^2}$.
\end{pf}
The bounds on $n$ given in Corollary \ref{Hsemibds} were shown
in Theorem 3.3 and Remark 2 of \cite{contemp} to be bounds
on the orders of torsion subgroups of
abelian varieties with potential good reduction at discrete valuations
of residue characteristics $2$ and $3$.
If we assume the existence of a polarization on $A$ of degree prime
to $n$ (for example, a principal polarization) we obtain stronger
bounds. The following results give bounds on torsion subgroups of
order prime to the degree of a given polarization.
\begin{thm}
\label{lambdabounds}
Suppose $A$ is a $d$-dimensional abelian variety over a number field $F$
of degree $m$, $v$ is a discrete valuation on $F$ at which $A$ does not
have purely multiplicative reduction, $p$ is the residue
characteristic of $v$, $\ell$ is a prime number, $\ell \ne p$, $J$ is
a subgroup of $A_\ell(F)$ of order $\ell^d$, $\lambda$ is a polarization
on $A$ defined over an extension of $F$ unramified at $v$, and $\ell$
does not divide the degree of $\lambda$. Then
$$\ell \le (1 + p^{m/2})^{2d}.$$
\end{thm}
\begin{pf}
Since $\ell$ does not divide the degree of $\lambda$,
the pairing $e_{\lambda,\ell}$ is nondegenerate.
If $J$ is not isotropic with respect to $e_{\lambda,\ell}$, then $F$
contains a primitive $\ell$-th root of unity. Therefore $\ell - 1$
divides $[F : {\bold Q}]$, so $\ell \le 1 + m$.
Suppose $J$ is isotropic with respect to $e_{\lambda,\ell}$.
Then $J$ is a maximal isotropic subgroup of $A_\ell$ (since
$\# J = \ell^d$ and $e_{\lambda,\ell}$ is nondegenerate).
If $A$ does not have semistable reduction at $v$, then $\ell \le 3$
by Theorem 6.2 of \cite{semistab}. If $A$ has semistable reduction
at $v$, then $\ell \le (1 + p^{m/2})^{2d}$ by Theorem \ref{bounds}.
The result now follows since $(1 + p^{m/2})^{2d}$ is greater than
$3$ and than $1 + m$.
\end{pf}
\begin{cor}
\label{Hsemilambdabds}
Suppose $(A,\lambda)$ is a $d$-dimensional polarized abelian variety
over a number field $F$ of degree $m$, and suppose the Hodge group of
$A$ is not semisimple.
Suppose $\ell$ is a prime number which does not divide the degree of
$\lambda$, and $J$ is a subgroup of $A_\ell(F)$ of order $\ell^d$.
Then
$$\ell \le (1 + 2^{m/2})^{2d}.$$
\end{cor}
\begin{pf}
By Theorem \ref{hodsemi}, $A$ does not have
purely multiplicative reduction at any discrete valuations.
Since $2 < (1 + 2^{m/2})^{2d}$, we may assume $\ell$ is an odd prime, and
we obtain the result by applying Theorem \ref{lambdabounds} with $p = 2$.
\end{pf}
The proof of Theorem \ref{lambdabounds} shows that
$\ell \le \max\{ 1+m, (1+\sqrt{q})^{2d}\}$, where $q$ is the order
of the residue field of $v$. Therefore in Corollary \ref{Hsemilambdabds}
we can conclude that $\ell \le \max\{ 1+m, (1+\sqrt{f})^{2d}\}$, where $f$
is the minimal order of the residue fields of the valuations on $F$ of
residue characteristic $2$.
\begin{cor}
Suppose $(A,\lambda)$ is a $d$-dimensional polarized abelian variety
over a number field $F$ of degree $m$, and suppose the Hodge group of
$A$ is not semisimple.
Suppose $n$ is a positive integer relatively prime to the degree of
$\lambda$, and $J$ is a subgroup of $A_n(F)$ of order $n^d$
which is a maximal isotropic subgroup with
respect to $e_{\lambda,n}$.
Then
$$n \le (1 + 2^{m/2})^{2d}(1 + 3^{m/2})^{2d}.$$
\end{cor}
\begin{pf}
By Theorem \ref{hodsemi}, $A$ does not have
purely multiplicative reduction at any discrete valuations.
The prime-to-$p$ part of $n$ is
bounded above by $4$ if there is a valuation on $F$ of
residue characteristic
$p$ at which $A$ does not have semistable reduction (by Theorem 6.2 of
\cite{semistab}), and otherwise is bounded above by $(1 + p^{m/2})^{2d}$
(by Theorem \ref{bounds}). Note that $4 < (1 + p^{m/2})^{2d}$.
The result follows by letting $p = 2, 3$.
\end{pf}
|
1995-05-15T06:20:17 | 9505 | alg-geom/9505016 | en | https://arxiv.org/abs/alg-geom/9505016 | [
"alg-geom",
"math.AG"
] | alg-geom/9505016 | Gerd Dethloff | Gerd Dethloff | Iitaka-Severi's Conjecture for Complex Threefolds | This paper has been withdrawn by the author. Withdrawn since its
content had been subsumed (with improvement) in arXiv:alg-geom/9608033 | null | null | null | null | We prove the following generalization of Severi's Theorem: Let $X$ be a fixed
complex variety. Then there exist, up to birational equivalence, only finitely
many complex varieties $Y$ of general type of dimension at most three which
admit a dominant rational map $f$ from $X$ to Y$.
| [
{
"version": "v1",
"created": "Fri, 12 May 1995 08:45:12 GMT"
},
{
"version": "v2",
"created": "Sat, 29 Nov 2014 14:32:28 GMT"
}
] | 2014-12-02T00:00:00 | [
[
"Dethloff",
"Gerd",
""
]
] | alg-geom | \section{Introduction}
\noindent Let $X$ and $Y$ be algebraic varieties, i.e. complete integral schemes
over a field of characteristic zero, and
denote by $R(X,Y)$ the set of all dominant rational maps $f:X \rightarrow Y$.
Moreover denote by ${\cal F} = {\cal F}(X)$ the set
$\{ f: X \rightarrow Y|$ $f$ is a dominant rational map onto an algebraic
variety $Y$ of general type$\}$ and by
${\cal F}_m = {\cal F}_m(X)$ the set
$\{ f: X \rightarrow Y|$ $f$ is a dominant rational map and $Y$ is birationally
equivalent to a nonsingular algebraic variety for which the $m$-th
pluricanonical
mapping is birational onto its image $\}$. We introduce an equivalence relation
$\sim$ on the sets ${\cal F}$ and ${\cal F}_m$ as follows: $(f:X \rightarrow Y) \sim
(f_1:X \rightarrow Y_1)$
iff there exists a birational map $b:Y \rightarrow Y_1$ such that $b \circ f = f_1$.\\
The classical theorem
of Severi can be stated as follows (cf.\cite{Sa}):
\begin{theo} \label{1.1}
For a fixed algebraic variety $X$ there exist only finitely many hyperbolic
Riemann surfaces $Y$
such that $R(X,Y)$ is nonempty.
\end{theo}
\noindent We may ask if a finiteness theorem of this kind also can be true
in higher dimensions. This leads to the following:
\begin{conj} \label{1.2}
For a fixed variety $X$ there exist, up to birational
equivalence, only finitely many varieties $Y$ of
general type such that $R(X,Y)$ is nonempty. \\
Moreover, the set ${\cal F}/ \sim$ is a finite set.
\end{conj}
\noindent Maehara calls this conjecture Iitaka's Conjecture based on Severi's
theorem (cf. \cite{Ma3}), and we abbreviate this as Iitaka-Severi's Conjecture.
In \cite{Ma3} Maehara states the Conjecture more
generally for algebraic varieties (over any field) and separable dominant
rational maps.
He also mentioned that K. Ueno proposed that a variety of general type
could be replaced by a polarized non uniruled variety in this Conjecture.\\
Maehara proved in Proposition 6.5. in \cite{Ma2} that in characteristic zero
the Conjecture is true if one restricts the
image varieties $Y$ to such varieties that can be birationally embedded
by the $m$-th pluricanonical map for any given $m$, i.e. ${\cal F}_m / \sim$
is finite for all $m$. This especially proves
the Conjecture for surfaces $Y$ (take $m=5$). Furthermore Maehara shows
that one can find a fixed $m$ such that for all {\bf smooth} varieties $Y$
which have nef and big canonical bundle the m-th pluricanonical map is a
birational
embedding, which proves the Conjecture also in this case.
Earlier Deschamps and Menegaux \cite{DM2}, \cite{DM3} proved, in characteristic
zero,
the cases where the varieties
$Y$ are surfaces which satisfy $q >0$ and $P_g \geq 2$, or where the maps
$f:X \rightarrow Y$ are morphisms. In this direction Maehara \cite{Ma1} also showed
finiteness of isomorphism classes of smooth varieties with ample canonical
bundles
which are dominated by surjective morphisms from a fixed variety.\\
There is a related classical result due to de Franchis \cite{Fr} which states
that for any Riemann surface $X$ and any fixed hyperbolic Riemann surface $Y$
the set $R(X,Y)$ is finite. At the same time he gives an upper bound
for $\#R(X,Y)$ only in terms of $X$. The generalization of this theorem
to higher dimensions is not a conjecture any more:
Kobayashi and Ochiai \cite{KO} proved that if $X$ is a Moisheson space and $Y$
a compact complex space of general type, then the set of surjective meromorphic
maps from $X$ to $Y$ is finite. Deschamps and Menegaux \cite{DM1} proved that
if $X$ and $Y$ are smooth projective varieties over a field of arbitrary
characteristic, and $Y$ is of general type, then $\#R(X,Y)$ is finite (where
one has additionally to assume that the dominant rational maps $f:X \rightarrow Y$
are separable). \\
{}From these results it follows that the second part of Conjecture \ref{1.2} is
a consequence of the first part, hence we only have to deal with the first
part.\\
Bandman \cite{Ba1}, \cite{Ba2} and Bandman and Markushevich \cite{BM}
also generalized the second part of de Franchis' theorem, proving that
for projective varieties $X$ and $Y$ with only canonical singularities
and nef and big canonical line bundles $K_X$ and $K_Y$ the number
$\#R(X,Y)$ can be bounded in terms of invariants of $X$ and the index of
$Y$.\\
Another generalization of the (first part of) de Franchis' theorem was
given by Noguchi \cite{No}, who proved that there are only finitely many
surjective meromorphic mappings from a Zariski open subset $X$ of an
irreducible
compact complex space onto an irreducible
compact hyperbolic complex space $Y$. Suzuki \cite{Su} generalized this result
to the case where $X$ and $Y$ are Zariski open subsets of irreducible
compact complex spaces $\overline{X}$ and $\overline{Y}$ and $Y$ is
hyperbolically embedded in $\overline{Y}$. These results can be
generalized to finiteness results for nontrivial sections in hyperbolic
fiber spaces. But since a more precise discussion would lead us too
far from the proper theme of this paper, we refer the interested reader
to Noguchi \cite{No} and Suzuki \cite{Su}, or to the survey \cite{ZL}
of Zaidenberg-Lin, where he also can find an overview for earlier results
which generalized de Franchis' theorem.\\
It is a natural question if Conjecture \ref{1.2} can also be stated in terms
of complex spaces. In \cite{No} Noguchi proposed the following:
\begin{conj} \label{1.3}
Let $X$ be a Zariski open subset of an irreducible compact complex
space. Then the set of compact irreducible hyperbolic complex spaces $Y$
which admit a dominant meromorphic map $f:X \rightarrow Y$ is finite.
\end{conj}
Let us now return to Conjecture \ref{1.2}. In this paper we are only interested
in the case of complex varieties.
Since we want to prove finiteness only up to birational equivalence,
we may assume without loss of generality that $X$ and all $Y$ in the
Conjecture are nonsingular projective complex varieties, by virtue
of Hironaka's resolution theorem \cite{Hi}, cf. also \cite{Ue}, p.73.
Now fix a complex projective variety $X$.
We define ${\cal G}_m:=\{$ a nonsingular complex projective
variety $Y$ : the $m$-th pluricanonical map $\Phi_m: Y \rightarrow \Phi_m(Y)$ is
birational onto its image and there exists a dominant rational map $f:X \rightarrow Y \}
$.
In order to show Conjecture \ref{1.2}
it is sufficient, by Proposition 6.5. of Maehara \cite{Ma2},
to show the following
\begin{conj} \label{1.4}
There exists a natural number $m$ only depending on $X$ such that
all smooth complex projective varieties $Y$ of general
type which admit a dominant rational map $f:X \rightarrow Y$ belong to
${\cal G}_m$.
\end{conj}
\noindent We will prove that Conjecture \ref{1.4} is true for varieties $Y$
which are of
dimension three, thus we prove Iitaka-Severi's Conjecture for complex 3-folds.
Since for varieties $Y$ of dimension one resp. two
we can take $m=3$ resp. $m=5$, our main theorem is:
\begin{theo} \label{1.5}
Let $X$ be a fixed complex variety. Then there exist, up to birational
equivalence,
only finitely many
complex varieties $Y$ of general type of dimension at most three
which admit a dominant rational
map $f:X \rightarrow Y$.\\
Moreover the set ${\cal F}/ \sim$ is a finite set if one resticts
to complex varieties $Y$ of dimension at most three .
\end{theo}
\noindent As Maehara \cite{Ma3}, p.167 pointed out already, in order to prove
Conjecture \ref{1.4}
it is enough to show that for all varieties $Y$
there exists a minimal model and the index of these minimal models
can be uniformly bounded from above by a constant only depending on $X$.
Since in dimension three minimal models and even canonical models do
exist, the problem is reduces to the question how to bound the index.\\
But it turns out that one is running into problems if one directly
tries to bound the indices of the canonical models $Y_c$ of threefolds
$Y$, only using that they are all dominated by dominant rational maps
from a fixed variety $X$. So we will proceed in a different way:\\
The
first step of the proof is to
show that the Euler characteristic $\chi (Y, {\cal O}_{Y})$ is
uniformly bounded by an entire constant $C$ depending only on $X$ (Proposition
3.2),
that is how we use the fact that
all threefolds $Y$ are dominated by a fixed variety $X$. \\
In the second step of the proof, we show that we can choose another
entire constant $R$, also only depending on $X$, such that for any
threefold $Y$ of general type for which the Euler characteristic is bounded by
$C$ the
following holds (Proposition 3.3):
Either the index of the canonical model $Y_c$ of $Y$ divides $R$ (first case)
or the
pluricanonical sheaf ${\cal O}_{Y_c}((13C)K_{Y_c})$
has two linearly independant
sections on $Y_c$ (second case). In order to prove this Proposition,
we use the Plurigenus Formula due to Barlow, Fletcher and Reid and
estimates of some terms in this formula due to Fletcher.
In the first case the index is bounded, and we are done (Proposition
\ref{2.8}).\\
The third step of the proof deals with the second case.
Here we remark
that the two linearly independant sections on $Y_c$ can be lifted to sections
in
$H^0 (Y,{\cal O}_{Y}(mK_{Y}))$, and then we can apply a theorem of Kollar
\cite{Ko}
which states that now the $(11m+5)$-th pluricanonical map gives a
birational embedding (Proposition 3.4), and we are also done in the second
case. \\
Hence we do not prove directly that under our
assumptions the index is uniformly bounded, we prove that if it is not,
then there is some other way to show that some fixed pluricanonical
map gives a birational embedding. The fact that the index actually has
such a uniform bound then follows as a result of Theorem \ref{1.5}.\\
It finally might be worth while to point out that the second and the
third step of our proof actually yield:
\begin{theo}
Let $C$ be a positive entire constant. Define $R={\rm lcm}(2,3,...$ $,26C-1)$
and
$m={\rm lcm}(18R+1,143C+5)$. Then for all smooth projective 3-folds of
general type for
which the Euler characteristic is bounded above by $C$ the $m$-th
pluricanonical map is birational onto its image.
\end{theo}
\noindent Despite the fact that our $m=m(C)$ is explicit, it is so huge that it
is only of theoretical interest. For example for $C=1$ it is known
by Fletcher \cite{Fl} that one can choose $m=269$, but for $C=1$ our $m$ is
already for of the size $10^{12}$. Moreover J.P. Demailly
recently told me that he conjectures that for 3-folds of general type
any $m \geq 7$ should work, independantly of the size of the Euler
characteristic.\\
The paper is organized as follows: In section 2 we collect, for the
convenience of the reader and also for fixing the notations, the basic
facts from canonical threefolds which we need. We try to give precise
references to all these facts, but do not try to trace these facts back
to the original papers. Where we could not find such references we
give short proofs. However we expect that all these facts should be
standard to specialists on threefolds.
In section 3 we give the proof of Theorem \ref{1.5}.\\
The author would like to thank S.Kosarew (Grenoble) for pointing out
Noguchi's Conjecture \ref{1.3} to him. This was his starting point for
working on problems of this kind. He would also like to thank
F.Catanese (Pisa) for pointing out Fletcher's paper \cite{Fl} to him, since
this paper later gave him the motivation for the key step in the proof
of Theorem \ref{1.5}. He finally would like to thank the Institut Fourier in
Grenoble, the University of Pisa and the organizers of the conference
Geometric Complex Analysis in Hayama for inviting him, since this gave him
the possibility to discuss with many specialists.
\section{Some Tools from the Theory of 3-folds}
Let $Y$ be a normal complex variety of dimension $n$, $Y_{reg}$ the subspace of
regular points
of $Y$ and $j: Y_{reg} \hookrightarrow Y$ the inclusion map. Then the sheaves
${\cal O}_Y(mK_Y)$ are defined as
$$ {\cal O}_Y(mK_Y) := j_*((\Omega^n_{Y_{reg}})^{\otimes m})$$
Equivalently ${\cal O}_Y(mK_Y)$ can be defined as the sheaf of $m$-fold tensor
products of rational canonical differentials on $Y$ which are regular
on $Y_{reg}$. The $mK_Y$ can be considered as Weil divisors.
For this and the following definitions, cf. \cite{Re2} and \cite{Mo1}.
\begin{defi} \label{2.1}
$Y$ has only {\bf canonical} singularities if it satisfies the following
two conditions:\\
i) for some integer $r \ge 1$, the Weil divisor $rK_Y$ is a Cartier divisor.\\
ii) if $f: \tilde{Y} \rightarrow Y $ is a resolution of $Y$ and $\{ E_i \}$ the family
of all exceptional prime divisors of $f$, then
$$ rK_{\tilde{Y}} = f^*(rK_Y) + \sum a_iE_i $$
with $a_i \geq 0$. \\
If $a_i >0$ for every exceptional divisor $E_i$, then $Y$ has only
{\bf terminal} singularities.\\
The smallest integer $r$ for which the Weil divisor $rK_Y$ is Cartier
is called the {\bf index} of $Y$.
\end{defi}
\begin{defi} \label{2.2}
A complex projective algebraic variety $Y$ with only canonical (resp.
terminal) singularities is called a {\bf canonical} (resp. {\bf minimal})
model if $K_Y$ is an ample (resp. a nef) $I \!\!\! Q$-divisor.\\
We say that a variety $Z$ has a {\bf canonical (resp. minimal) model}
if there exists a canonical (resp. minimal) model which is birational to
$Z$.
\end{defi}
Later on we will need the following theorem due to Elkik \cite{El} and Flenner
\cite{Fle}, 1.3 (cf. \cite{Re2}, p.363):
\begin{theo} \label{2.3}
Canonical singularities are rational singularities.
\end{theo}
The first part of the following theorem, which is of high importance for
the theory of 3-folds, was proved by Mori \cite{Mo2}, the second
part follows from the first part by works of Fujita \cite{Fu}, Benveniste
\cite{Be}
and Kawamata \cite{Ka}:
\begin{theo} \label{2.4} Let $Y$ be a non singular projective 3-fold of general
type.\\
i) There exists a minimal model of $Y$.\\
ii) There exists a unique canonical model of $Y$, the canonical ring
$R(Y,K_Y)$ of
$Y$ is finitely generated, and the canonical model is just
${\rm Proj} R(Y,K_Y)$.
\end{theo}
We have the following Plurigenus Formula due to Barlow, Fletcher and Reid
(cf. \cite{Fl}, \cite{Re2}, see also \cite{KM}, p.666 for the last part):
\begin{theo} \label{2.5} Let $Y$ be a projective 3-fold with only canonical
singularities.
Then we have
$$ \chi (Y, {\cal O}_Y(mK_Y)) = \frac{1}{12}(2m-1)m(m-1)K_Y^3 - (2m-1) \chi (Y, {\cal O}_Y)
+ \sum_Q l(Q,m)$$
with
$$ l(Q,m) = \sum_{k=1}^{m-1} \frac{\overline{bk}(r- \overline{bk})}{2r}$$
Here the summation takes place over a basket of singularities $Q$ of type
$\frac{1}{r}(a,-a,1)$ (see below for these notations). $\overline{j}$
denotes the smallest nonnegative residue of $j$ modulo $r$, and $b$ is chosen
such that
$\overline{ab}=1$.\\
Furthermore we have
$$ {\rm index}(Y) = {\rm lcm}\{r=r(Q): \: Q\in {\rm basket} \} $$
\end{theo}
\noindent A singularity of type $\frac{1}{r}(a,-a,1)$ is a cyclic quotient
singularity $ I \!\!\!\! C^3 / \mu_r$, where $\mu_r$ denotes the cyclic group
of $r$th roots of unity in $ I \!\!\!\! C$, and $\mu_r$ acts on $ I \!\!\!\! C^3$ via
$$\mu_r \ni \epsilon: (z_1,z_2,z_3) \rightarrow (\epsilon^a z_1, \epsilon^{-a}z_2,
\epsilon z_3)$$
Reid introduced the term `basket of singularities' in order to point out
that the singularities $Q$ of the basket are not
necessarily singularities of $Y$, but only `fictitious singularities'.
However the singularities of $Y$ make
the same contribution to $\chi (Y, {\cal O}_Y(mK_Y))$ as if they were those of the
basket, hence we also can work with the singularities of the basket,
which have the advantage that their contributions are usually easier than
those of the original singularities. More precisely, one can pass from
$Y$ to a variety where the singularities of the basket
actually occur by a crepant partial resolution of singularities and then by a
flat
deformation. For the details cf. \cite{Re2}, p.404, 412. \\
\noindent For estimating from below the terms $l(Q,m)$ in the Plurigenus
Formula, we will need two Propositions due to Fletcher \cite{Fl}. In those
Propositions
$[s]$ denotes
the integral part of $s \in I \!\! R$.
\begin{prop} \label{2.6}
$$l( \frac{1}{r}(1,-1,1), m) = \frac{\overline{m} (\overline{m} -1)(
3r+1-2 \overline{m})}{12r} + \frac{r^2-1}{12} [\frac{m}{r}]$$
\end{prop}
\begin{prop} \label{2.7} For $\alpha, \beta \in Z \!\!\! Z$ with $0 \leq \beta \leq
\alpha$
and for all $m \leq [(\alpha +1)/2]$, we have:
$$ l(\frac{1}{\alpha} (a,-a,1), m) \geq l(\frac{1}{\beta} (1,-1,1),m)$$
\end{prop}
At last, we want to give a proof for Maehara's remark \cite{Ma3}, p.167 which
we already mentioned,
namely that it is enough to show that the index of the canonical models of the
varieties $Y$ can be uniformly bounded from above by a constant only depending
on $X$.
We prove more precisely:
\begin{prop} \label{2.8}
Let $Y$ be a smooth projective 3-fold of general type
and $l$ a natural number such that $l$ is an integer multiple of the
index $r$ of the canonical model $Y_c$ of $Y$. Then
the $(18l+1)$-th pluricanonical map is birational onto its image.
\end{prop}
{\bf Proof:} We could pass from the canonical model $Y_c$ of $Y$ to a minimal
model $Y_m$ and then apply Corollary 4.6 of the preprint \cite{EKL} of Ein,
K\"uchle
and Lazarsfeld. But since it might even be easier we want to pass directly
from $Y_c$ to $Y$, and then apply the corresponding result of Ein, K\"uchle
and Lazarsfeld for smooth projective 3-folds, namely Corollary 3. of
\cite{EKL}.\\
Since $l$ is a multiple of the index of $Y_c$, $lK_{Y_c}$ is an ample line
bundle. Since we only are interested in $Y$ up to birational equivalence
we may assume that $\pi : Y \rightarrow Y_c$ is a desingularization. Since the
bundle $lK_{Y_c}$ on $Y_c$ is ample, the pulled back bundle $\pi^*(lK_{Y_c})$
on $Y$ is still nef and big. Hence we can apply Corollary 3. of \cite{EKL}
to this bundle and get that the map obtained by the sections of
the bundle $K_Y + 18\pi^*(lK_{Y_c})$ maps
$Y$ birationally onto its image. But since by the Definition 2.1 of
canonical singularities every section of the bundle $K_Y + 18\pi^*(lK_{Y_c})$
is also a section of $K_Y + 18lK_Y = (1+18l)K_Y$, the claim follows. \qed
{\bf Remark:}
Notice that for Proposition 2.8 we do not need the assumption
that the smooth projective 3-folds $Y$ are dominated by a fixed complex
variety $X$. This only will be needed to bound the indices of the
canonical models $Y_c$ of the $Y$.
\section{Bounding the Index of a Dominated Canonical 3-fold}
In order to prove Theorem \ref{1.5}, it is enough to prove the following
\begin{theo} \label{3.1}
Let $X$ be a fixed smooth complex variety. Then there exists a natural number
$m$ depending only on $X$ such that all smooth complex projective 3-folds
$Y$ of general type which admit a dominant rational map $f:X \rightarrow Y$ belong to
${\cal G}_m$.
\end{theo}
\noindent Here, as in the introduction, ${\cal G}_m$ is defined as
${\cal G}_m:=\{$ a nonsingular complex projective
variety $Y$ : the $m$-th pluricanonical map $\Phi_m: Y \rightarrow \Phi_m(Y)$ is
birational onto its image and there exists a dominant rational map $f:X \rightarrow Y \}
$. \\
The rest of this chapter is devoted to the proof of Theorem \ref{3.1}.
We denote by $Y_c$ the canonical
model of $Y$. Furthermore we may assume without loss of generality that
$\pi : Y \rightarrow Y_c$ is a desingularization (since we only need to look at
those smooth projective varieties $Y$ up to birational equivalence).\\
In the first step of the proof we show:
\begin{prop} \label{3.2}
Under the assumptions of Theorem \ref{3.1} there exists an entire constant $C
\geq 1$
only depending on $X$, such that
for all $Y$ we have $ \chi (Y, {\cal O}_Y) = \chi (Y_c, {\cal O}_{Y_c}) \leq C$.
\end{prop}
{\bf Proof:}
First we get
by Hodge theory on compact K\"ahler manifolds (cf. \cite{GH}, or \cite{Ii},
p.199):
$$ h^i(Y, {\cal O}_Y) = h^0(Y, \Omega_Y^i), \:\: i=0,1,2,3$$
Now by Theorem 5.3. in Iitaka's book \cite{Ii}, p.198 we get that
$$ h^0(Y, \Omega_Y^i) \leq h^0(X, \Omega_X^i), \:\: i=0,1,2,3$$
Hence by the triangle inequality we get a constant $C$, only depending
on $X$, such that
$$| \chi (Y, {\cal O}_Y) | \leq C$$
Now by the theorem of Elkik and Flenner (Theorem \ref{2.3}) $Y_c$ has only
rational singularities, hence by degeneration of the Leray spectral sequence
we get that
$$ \chi (Y, {\cal O}_Y) = \chi (Y_c, {\cal O}_{Y_c})$$
This finishes the proof of Proposition \vspace{.5cm} \ref{3.2}. \qed
In the second step of the proof of Theorem \ref{3.1} we show:
\begin{prop} \label{3.3}
Let $C\geq 1$ be an entire constant, $R := {\rm lcm}(2,3,...,
26C-1)$ and $m_1=18R+1$. Then for all smooth projective complex 3-folds $Y$
of general type with $\chi (Y, {\cal O}_Y) \leq C$ we have\\
either $Y \in {\cal G}_{m_1}$ or $h^0(Y_c, {\cal O}_{Y_c}((13C)K_{Y_c})) \geq 2$.
\end{prop}
{\bf Proof:} We distinguish between two cases: The first case is that the
index of $Y_c$ divides $R$. Then applying Proposition \ref{2.8} we get that
$Y \in {\cal G}_{m_1}$ and we are done. The second case is that the index
does not divide $R$. Then in the Plurigenus Formula Theorem \ref{2.5} of
Barlow, Fletcher and Reid
we necessarily have at least one singularity $\tilde{Q}$ in the basket of
singularities
which is of the type $\frac{1}{r}(a,-a,1)$ with $r \geq 26C$.
Now applying first a vanishing theorem for ample sheaves (cf. Theorem 4.1 in
\cite{Fl}), the fact that $K^3_{Y_c} >0$ (since $K_{Y_c}$ is an ample
$I \!\!\! Q$-divisor)
and then the Propositions \ref{2.6} and \ref{2.7} due to Fletcher, we get:
$$ h^0(Y_c, {\cal O}_{Y_c}((13C)K_{Y_c}))$$
$$ = \chi (Y_c, {\cal O}_{Y_c}((13C)K_{Y_c}))$$
$$ \geq (1-26C)\chi (Y_c, {\cal O}_{Y_c}) + \sum_{ Q \in {\rm basket}} l(Q,13C)$$
$$ \geq (1-26C)C + l(\tilde{Q},13C)$$
$$ \geq (1-26C)C + l(\frac{1}{26C}(1,-1,1),13C)$$
$$ = (1-26C)C + \frac{13C(13C-1)(78C+1-26C)}{312C}$$
$$ = \frac{312C^2 - 8112C^3 + 8788C^3 - 507C^2 -13C}{312C}$$
$$ = \frac{52C^2 - 15C - 1}{24}$$
$$ \geq \frac{36}{24} = 1.5 $$
\noindent The last inequality is true since $C\geq 1$.
Since $ h^0(Y_c, {\cal O}_{Y_c}((13C)K_{Y_c}))$ is an entire,
this finishes the proof of Proposition \vspace{.5cm} \ref{3.3}. \qed
In the third step of the proof of Theorem \ref{3.1} we show:
\begin{prop} \label{3.4}
Assume that for a smooth projective complex 3-fold $Y$
of general type we have $h^0(Y_c, {\cal O}_{Y_c}((13C)K_{Y_c})) \geq 2$. Then
$Y \in {\cal G}_{m_2}$ with $m_2=143C+5$.
\end{prop}
{\bf Proof:} Kollar proved that if $h^0(Y, {\cal O}_Y(lK_Y
)) \geq 2$ then
the $(11l+5)$-th pluricanonical map is birational onto its image
(Corollary 4.8 in \cite{Ko}). So the only thing which remains to prove is
that from $h^0(Y_c, {\cal O}_{Y_c}((13C)K_{Y_c})) \geq 2$
we get $h^0(Y, {\cal O}_{Y}((13C)K_{Y})) \geq 2$. This fact is standard
for experts (cf. e.g. \cite{Re1}, p.277, or
\cite{Fl}, p.225), but since we remarked in talks about this paper that
this fact doesn't seem to be generally well known, we want to indicate how one
can prove it:\\
What we have to prove is that taking linearly independant sections $s_1$, $s_2$
from $H^0(Y_c, {\cal O}_{Y_c}(lK_{Y_c}))$ we can get from them
linearly independant sections $t_1$, $t_2$ from $H^0(Y, {\cal O}_Y(lK_Y))$. We
mentioned
at the beginning of section 1 that
${\cal O}_{Y_c}(lK_{Y_c})$ can also be defined as the sheaf of $l$-fold tensor
products of rational canonical differentials on $Y_c$ which are regular
on $(Y_c)_{reg}$. But since $Y$ and $Y_c$ are birationally equivalent,
from this definition it is immediate that any
linearly independant sections $s_1$, $s_2$ from $H^0(Y_c, {\cal O}_{Y_c}(lK_{Y_c}))$
can be lifted, namely as pull backs of (tensor products of rational) canonical
differentials with the holomorphic map $\pi$, to linearly independant {\bf
rational} sections $t_1$, $t_2$
of the bundle
${\cal O}_Y(lK_Y)$. These lifted sections are regular outside the family of the
exceptional
prime divisors $\{ E_i \}$ of the resolution $\pi : Y \rightarrow Y_c$. We have to show
that $t_1$ and $t_2$ are regular everywhere. Since $Y$ is a manifold, by the
First Riemann Extension
Theorem it is sufficient to show that these sections are bounded near points
of the $\{ E_i \}$. In order to show this, choose a natural number $p$,
which now may depend on $Y$, such that index($Y_c$) divides $pl$.
Then by the definition of canonical singularities (Definition \ref{2.1}) the
sections $s_1^p$ and $s_2^p$ lift to {\bf regular} sections $t_1^p$ and
$t_2^p$. Hence $t_1$ and $t_2$ have to be bounded near points of the $\{ E_i
\}$,
and we are \vspace{.5cm} done. \qed
Now the proof of Theorem \ref{3.1} is immediate: If we take $m_0 := {\rm
lcm}(m_1,m_2)$,
then by Proposition \ref{3.3} and Proposition \ref{3.4} we have $Y \in {\cal
G}_{m_0}$
for all $Y$ which occur in Theorem \vspace{0.5cm} \ref{3.1}. \qed
|
1995-05-30T06:20:45 | 9505 | alg-geom/9505033 | en | https://arxiv.org/abs/alg-geom/9505033 | [
"alg-geom",
"math.AG"
] | alg-geom/9505033 | Bruce Hunt | Bruce Hunt | Symmetric subgroups of rational groups of hermitian type | 29 pages (11 pt), ps-file also available at the home page
http://www.mathematik.uni-kl.de/~wwwagag, preprints. LaTeX v2.09 | null | null | null | null | A rational group of hermitian type is an algebraic group over the rational
numbers whose symmetric space is a hermitian symmetric space. We assume such a
group $G$ to be given, which we assume is isotropic. Then, for any rational
parabolic $P$ in the group $G$, we find a reductive rational subgroup $N$
closely related with $P$ by a relation we call incidence. This has implications
to the geometry of arithmetic quotients of the symmetric space by arithmetic
subgroups of $G$, in the sense that $N$ defines a subvariety on such an
arithmetic quotient which has special behaviour at the cusp corresponding to
the parabolic with which $N$ is incident.
| [
{
"version": "v1",
"created": "Mon, 29 May 1995 09:17:24 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Hunt",
"Bruce",
""
]
] | alg-geom | \section{Real parabolics of hermitian type}
\subsection{Notations}\label{section1.1}
In this paper we will basically adhere to the notations of \cite{BB}. In
the first two paragraphs $G$ will denote a real Lie group; later $G$ will
be a $\fQ$-group of hermitian type. We assume $G$ is reductive, connected
and with compact center; $K\subset} \def\nni{\supset} \def\und{\underline G$ will denote a maximal compact subgroup,
${\cal D} =G/K$ the corresponding symmetric space. Throughout this paper we will
assume $G$ is of {\it hermitian type}, meaning that ${\cal D} $ is a hermitian
symmetric space, hence a product ${\cal D} ={\cal D} _1\times \cdots \times {\cal D} _d$ of
irreducible factors, each of which we assume is {\it non-compact}. Let
$\Gg=\kk+\pp$ denote a Cartan decomposition of the Lie algebra of $G$,
$\Gg_{\fC}=\kk_{\fC}+\pp^++\pp^-$ the decomposition of the complexified Lie
algebra, with $\pp^{\pm}$ abelian subalgebras (and $\pp_{\fC}=\pp^+\oplus
\pp^-$). Chooosing a Cartan subalgebra $\hh\subset} \def\nni{\supset} \def\und{\underline \Gg$, the set of roots of
$\Gg_{\fC}$ with respect to $\hh_{\fC}$ is denoted
$\Phi=\Phi(\hh_{\fC},\Gg_{\fC})$. As usual, we choose root vectors
$E_{\ga}\in \Gg^{\ga}$ such that the relations
$$[E_{\ga},E_{-\ga}]=H_{\ga}\in \hh_{\fC},\quad
\ga(H_{\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda})=2{<\ga,\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda>\over <\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda,\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda>},\ \ \ga,\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda\in \Phi,$$
hold.
Complex conjugation maps $\pp^+$ to $\pp^-$, in fact permuting $E_{\ga}$
and $E_{-\ga}$ for $E_{\ga}\in \pp^{\pm}$. Moreover, if
$\gS^{\pm}:=\{\ga|E_{\ga} \in \pp^{\pm}\}$, then
$$\pp^{\pm}=\span_{\fC}(E_{\ga}),\ \ga\in \gS^{\pm};\quad
\pp=\span_{{\Bbb R}} \def\fH{{\Bbb H}}(X_{\ga},Y_{\ga}),\ \ga\in \gS^+,$$
where $X_{\ga}=E_{\ga}+E_{-\ga},\ Y_{\ga}=i(E_{\ga}-E_{-\ga})$ (twice the
real and the
(negative of the) imaginary parts, respectively). Let
$\mu_1,\ldots,\mu_t$ denote a maximal
set of strongly orthogonal roots, determined
as in \cite{Helg}: $\mu_1$ is the smallest root in $\gS^+$, and $\mu_j$ is
the smallest root in $\gS^+$ which is strongly orthogonal to
$\mu_1,\ldots,\mu_{j-1}$. This set will be fixed once and for all.
Once this set of strongly orthogonal roots has been chosen, a maximal
${\Bbb R}} \def\fH{{\Bbb H}$-split torus $A$ is uniquely determined by
$Lie(A)=\aa=\span_{{\Bbb R}} \def\fH{{\Bbb H}}(X_{\mu_1},\ldots,X_{\mu_t})$. Then
$\Phi_{{\Bbb R}} \def\fH{{\Bbb H}}=\Phi(\aa,\Gg)$ will denote the set of ${\Bbb R}} \def\fH{{\Bbb H}$-roots, and $\Gg$
has a decomposition
$$\Gg=\frak z} \def\qq{\frak q(\aa)\oplus \sum_{\eta\in\Phi_{{\Bbb R}} \def\fH{{\Bbb H}}}\Gg^{\eta},$$
where $\Gg^{\eta}=\{x\in \Gg | \ad(s)x=\eta(s)x, \forall_{s\in A}\}$. For
each irreducible component of $G$, the set of ${\Bbb R}} \def\fH{{\Bbb H}$-roots is either of type
$\bf C_{\hbox{\scriptsize\bf t}}$ or $\bf BC_{\hbox{\scriptsize\bf t}}$, and of type
$\bf C_{\hbox{\scriptsize\bf t}}$ $\iff$ the corresponding
domain is a tube domain. If $\xi_i$ denote coordinates on $\aa$ dual to
$X_{\mu_i}$, assuming for the moment ${\cal D} $ to be irreducible, the
${\Bbb R}} \def\fH{{\Bbb H}$-roots are explicitly
\begin{equation}\label{e22b.2}
\begin{array}{lcr}\Phi_{{\Bbb R}} \def\fH{{\Bbb H}}: & \pm(\xi_i\pm \xi_j),\ \pm 2 \xi_i \
(1\leq i\leq t,\ i<j) & \mbox{(Type $\bf C_{\hbox{\scriptsize\bf t}}$)} \\
& \pm(\xi_i\pm \xi_j),\ \pm 2 \xi_i, \ \pm\xi_i \
(1\leq i\leq t,\ i<j)
& \mbox{(Type $\bf BC_{\hbox{\scriptsize\bf t}}$)} \\
\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi_{{\Bbb R}} \def\fH{{\Bbb H}}: & \eta_i=\xi_i-\xi_{i+1}, i=1,\ldots,t-1, \mbox{ and }
\eta_t=2\xi_t \mbox{ (Type $\bf C_{\hbox{\scriptsize\bf t}}$) },
\eta_t=\xi_t & \mbox{ (Type $\bf BC_{\hbox{\scriptsize\bf t}}$).}
\end{array}
\end{equation}
Here the simple roots $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi_{{\Bbb R}} \def\fH{{\Bbb H}}$ are with respect to the lexicographical
order on the $\xi_i$. A general ${\Bbb R}} \def\fH{{\Bbb H}$-root system is a disjoint union of
simple root systems. The choice of maximal set of strongly orthogonal roots
determines an order on $\aa$ (the lexicographical order), which
determines, on each simple ${\Bbb R}} \def\fH{{\Bbb H}$-root system, an order as above; this is
called the {\it canonical order}.
\subsection{Real parabolics}
The maximal ${\Bbb R}} \def\fH{{\Bbb H}$-split abelian subalgebra $\aa$,
together with the order on it
(induced by the choice of strongly orthogonal roots), determines a unique
nilpotent Lie algebra of $\Gg$, $\frak n} \def\rr{\frak r=\sum_{\eta\in
\Phi_{{\Bbb R}} \def\fH{{\Bbb H}}^+}\Gg^{\eta}$. Set $A=\exp(\aa),\ N=\exp(\frak n} \def\rr{\frak r)$, and
\begin{equation}\label{e2.1} B:={\cal Z} (A)\rtimes N;
\end{equation}
this is a minimal ${\Bbb R}} \def\fH{{\Bbb H}$-parabolic, the {\it standard} one, uniquely
determined by the choice of strongly orthogonal roots. Every minimal
${\Bbb R}} \def\fH{{\Bbb H}$-parabolic of $G$ is conjugate to $B$. Note that, setting
$M={\cal Z} (A)\cap K$, we have ${\cal Z} (A)=M\times A$, and the group $M$ is the {\it
semisimple anisotropic kernel} of $G$.
Assume again for the moment that ${\cal D} $ is irreducible, and let $\eta_i,\
i=1,\ldots,t$ denote the simple ${\Bbb R}} \def\fH{{\Bbb H}$-roots. Set $\aa_b:=\cap_{j\neq b}
\Ker\eta_j,\ b=1,\ldots, t$, a one-dimensional subspace of $\aa$, and
$A_b:=\exp(\aa_b)$, a one-dimensional ${\Bbb R}} \def\fH{{\Bbb H}$-split subtorus of
$A$. Equivalently, $A_b=\left(\cap_{j\neq b}\Ker\eta_j\right)^0$, where
$\eta_j$ is viewed as a character of $A$. The {\it standard maximal
${\Bbb R}} \def\fH{{\Bbb H}$-parabolic}, $P_b,\ b=1,\ldots, t$, is the group generated by
${\cal Z} (A_b)$ and $N$; equivalently it is the semidirect product (Levi
decomposition)
\begin{equation}\label{e2.2} P_b={\cal Z} (A_b)\rtimes U_b,
\end{equation}
where $U_b$ denotes the unipotent radical. The Lie algebra $\uu_b$ of $U_b$
is the direct sum of the $\Gg^{\eta},\ \eta\in \Phi_{{\Bbb R}} \def\fH{{\Bbb H}}^+,\
\eta_{|\aa_b}\not\equiv 0$. The Lie algebra $\frak z} \def\qq{\frak q(\aa_b)$ of ${\cal Z} (A_b)$ has
a decomposition:
\begin{equation}\label{e2.3} \frak z} \def\qq{\frak q(\aa_b)=\mm_b\oplus \ll_b\oplus
\ll_b'\oplus \aa_b,\quad \ll_b=\sum_{\eta\in
[\eta_{b+1},\ldots,\eta_t]}\Gg^{\eta}+[\Gg^{\eta},\Gg^{-\eta}],\ \
\ll_b'=\sum_{\eta\in
[\eta_{1},\ldots,\eta_{b-1}]}\Gg^{\eta}+[\Gg^{\eta},\Gg^{-\eta}],
\end{equation}
and $\mm_b$ is an ideal of $\mm$, the Lie algebra of the (semisimple)
anisotropic kernel $M$. Both $\ll_b$ and $\ll_b'$ are simple, and the root
system $[\eta_{b+1},\ldots,\eta_t]$ is of type ${\bf C}_{\hbox{\scriptsize\bf t-b}}$
or ${\bf
BC}_{\hbox{\scriptsize\bf t-b}}$, while the root system $[\eta_1,\ldots, \eta_{b-1}]$
is of type
${\bf A}_{\hbox{\scriptsize\bf b-1}}$. Let $L_b,\ L_b'$ denote the analytic groups
with Lie
algebras $\ll_b$ and $\ll_b'$, respectively, and let ${\cal R} _b=L_b'A_b$, a
reductive group (of type ${\bf A}_{\hbox{\scriptsize\bf b-1}}$). We call $L_b$ the
{\it hermitian
factor} of the Levi component and ${\cal R} _b$ the {\it reductive factor}. It
is well known that $L_b$ defines the hermitian symmetric
space which is the $b^{th}$ (standard) boundary component of ${\cal D} $. Indeed,
letting $K_b\subset} \def\nni{\supset} \def\und{\underline L_b$ denote a maximal compact subgroup, ${\cal D} _b=L_b/K_b$ is
hermitian symmetric, and naturally contained in ${\cal D} $ as a subdomain,
${\cal D} _b\subset} \def\nni{\supset} \def\und{\underline {\cal D} $. Let $\gz:{\cal D} \longrightarrow} \def\sura{\twoheadrightarrow \pp^+$ be the Harish-Chandra embedding,
and let ${\bf D}=\gz({\cal D} ),\ {\bf D}_b=\gz({\cal D} _b)$ denote the images; ${\bf
D}_b$ is a bounded symmetric domain contained in a linear subspace (which
can be identified with $\pp_b^+=\ll_{b,\fC}\cap \pp^+$). Let
$o_b=-(E_{\mu_1}+\cdots+E_{\mu_b}),\ 1\leq b\leq t$;
as the elements $o_b$ are in $\pp^+$, one can consider
the orbits $o_b\cdot G$ and $o_b\cdot L_b$. Since for $g\in L_b$ the action
is described by
$o_bg = o_b +\gz(g)$, one has $o_b\cdot L_b = o_b+\gz({\cal D} _b)=o_b+{\bf
D}_b$, and this is the domain ${\bf D}_b$, translated into an affine
subspace ($o_b+\pp_b^+$) of $\pp^+$. One denotes this domain by $F_b:=o_b\cdot
L_b$, and this is the $b^{th}$ {\it standard boundary component} of ${\bf
D}$. $G$ acts by translations on the various $F_b$, and the images are
the {\it boundary components} of ${\bf D}$; one has
$$\overline{\bf D}={\bf D}\cup \{\hbox{ boundary components
}\}={\bf D} \cup \left(\cup_{b=1}^to_b\cdot G\right),$$
and $\overline{\bf D}\subset} \def\nni{\supset} \def\und{\underline \pp^+$ is the compactification of ${\bf D}$ in
the Euclidean topology. For any boundary component $F$ one denotes by
$N(F),\ Z(F)$ and $G(F)$ the normalizer, centralizer and automorphism group
$G(F)=N(F)/Z(F)$, respectively. Then, letting $U(F)$ denote the unipotent
radical of $N(F)$,
\begin{equation}\label{e3.1} N(F_b)=P_b,\quad U(F_b)=U_b,\quad
Z(F_b)=Z_b,\quad G(F_b)=L_b,
\end{equation}
where $Z_b$ is a closed normal subgroup of $P_b$ containing every normal
subgroup of $P_b$ with Lie algebra $\frak z} \def\qq{\frak q_b=\mm_b\oplus\ll_b'\oplus
\aa_b\oplus \uu_b$, which is an ideal in $\pp_b$.
Now consider the general case, ${\cal D} ={\cal D} _1\times \cdots \times {\cal D} _d$,
${\cal D} _i$ irreducible. For each ${\cal D} _i$ we have ${\Bbb R}} \def\fH{{\Bbb H}$-roots $\Phi_{i,{\Bbb R}} \def\fH{{\Bbb H}}$,
of ${\Bbb R}} \def\fH{{\Bbb H}$-ranks $t_i$ and simple ${\Bbb R}} \def\fH{{\Bbb H}$-roots
$\{\eta_{i,1},\ldots,\eta_{i,t_i}\},\ i=1,\ldots, d$. For each factor we
have standard parabolics $P_{i,b_i}$ $(1\leq b_i\leq t_i$) and standard
boundary components $F_{i,b_i}$. The standard parabolics of $G$ and
boundary components of ${\cal D} $ are then
products
\begin{equation}\label{e3.2} P_{\hbox{\scsi \bf b}}=P_{1,b_1}\times \cdots \times
P_{d,b_d},\quad F_{\hbox{\scsi \bf b}}=F_{1,b_1}\times \cdots \times
F_{d,b_d},\quad ({\bf b}=(b_1,\ldots, b_d)),
\end{equation}
and as above $P_{\hbox{\scsi \bf b}}=N(F_{\hbox{\scsi \bf b}})$. Furthermore,
\begin{equation}\label{e3.3} G(F_{\hbox{\scsi \bf b}})=:L_{\hbox{\scsi \bf b}}=L_{1,b_1}\times
\cdots\times L_{d,b_d}.
\end{equation}
As far as the domains are concerned, any of the boundary components
$F_{i,b_i}$ may be the {\it improper} boundary component ${\cal D} _i$,
which is indicated by setting $b_i=0$. Consequently, $P_{i,0}=L_{i,0}=G_i$
and in (\ref{e3.2}) and (\ref{e3.3})
any ${\bf b}=(b_1,\ldots,b_d),\ 0\leq b_i\leq t_i$ are admissible.
\subsection{Fine structure of parabolics}
For real parabolics of hermitian type one has a very useful refinement of
(\ref{e2.2}). This is explained in detail in \cite{SC} and especially in
\cite{S}, \S III.3-4. First we have the decomposition of ${\cal Z} (A_b)$ as
described above,
\begin{equation}\label{e3.4} {\cal Z} (A_b)=M_b\cdot L_b \cdot {\cal R} _b,
\end{equation}
where $M_b$ is compact, $L_b$ is the hermitian Levi factor, ${\cal R} _b$ is
reductive (of type $\bf A_{\hbox{\scriptsize \bf b-1}}$), and the product is almost
direct (i.e., the factors have finite intersection). Secondly, the
unipotent radical decomposes,
\begin{equation}\label{e3.5} U_b={\cal Z} _b\cdot V_b,
\end{equation}
which is a direct product, ${\cal Z} _b$ being the center of $U_b$. The action of
${\cal Z} (A_b)$ on $U_b$ can be explicitly described, and is the basis for the
compactification theory of \cite{SC}. Before we recall this, let us note
the notations used in \cite{SC} and \cite{S} for the decomposition. In
\cite{SC}, we find
\begin{equation}\label{e4.1} P(F)=(M(F)G_h(F)G_{\ell}(F))\rtimes U(F)\cdot
V(F),
\end{equation}
and in \cite{S}, where the author
uses Hermann homomorphisms $\gk:\frak s} \def\cc{\frak c\ll_2({\Bbb R}} \def\fH{{\Bbb H})\longrightarrow} \def\sura{\twoheadrightarrow \Gg$ to
index the boundary component,
\begin{equation}\label{e4.2} B_{\gk}=\left(G_{\gk}^{(1)}\cdot
G_{\gk}^{(2)}\right) \rtimes U_{\gk}V_{\gk}.
\end{equation}
In (\ref{e4.1}), $M=M_b,\ G_h=L_b,\ G_{\ell}={\cal R} _b$, while in (\ref{e4.2}),
$G_{\gk}^{(1)}=M_b\cdot L_b,\ G_{\gk}^{(2)}={\cal R} _b$ in our notations. The
action can be described as follows (\cite{S}, III \S3-4).
\begin{theorem}\label{t4.1} In the decomposition of the standard parabolic
$P_b$ (see (\ref{e3.4}) and (\ref{e3.5}))
$$P_b=(M_b\cdot L_b \cdot {\cal R} _b)\rtimes {\cal Z} _b\cdot V_b,$$
the following statements hold.
\begin{itemize}\item[(i)] The action of $M_b\cdot L_b$ is trivial on
${\cal Z} _b$, while on $V_b$ it is by means of a symplectic representation
$\gr:M_b\cdot L_b \longrightarrow} \def\sura{\twoheadrightarrow Sp(V_b,J_b)$, for a symplectic form $J_b$ on
$V_b$.
\item[(ii)] ${\cal R} _b$ acts transitively on ${\cal Z} _b$ and defines a homogenous
self-dual (with respect to a bilinear form) cone $C_b\subset} \def\nni{\supset} \def\und{\underline {\cal Z} _b$, while
on $V_b$ it acts by means of a representation $\gs:{\cal R} _b\longrightarrow} \def\sura{\twoheadrightarrow GL(V_b,I_b)$
for some complex structure $I_b$ on $V_b$.
\end{itemize}
Furthermore the representations $\gr$ and $\gs$ are compatible in a natural
sense.
\end{theorem}
\section{Holomorphic symmetric embeddings of symmetric domains}
\subsection{Symmetric subdomains}
We continue with the notations of the previous paragraph. Hence $G$ is a
real Lie group of hermitian type (reductive), ${\cal D} $ is the corresponding
domain. We wish to consider reductive subgroups $N\subset} \def\nni{\supset} \def\und{\underline G$, also of
hermitian type, defining domains ${\cal D} _N$, such that the inclusion $N\subset} \def\nni{\supset} \def\und{\underline G$
induces a holomorphic injection of the domains $i:{\cal D} _N\subset} \def\nni{\supset} \def\und{\underline {\cal D} $, and the
$i({\cal D} _N)$ are totally geodesic. First of all we may assume that $K_N$, a
maximal compact subgroup of $N$, is the intersection $K_N=K\cap N$;
equivalently, letting $o\in {\cal D} $ and $o_N\in {\cal D} _N$ denote the base points,
$i(o_N)=o$. Note that conjugating $N$ by an element of $K$ yields an
isomorphic group $N'$ and subdomain $i':{\cal D} _{N'}\subset} \def\nni{\supset} \def\und{\underline {\cal D} $ such that
$i'(o_{N'})=o$, and this defines an equivalence relation on the set of
reductive subgroups $N\subset} \def\nni{\supset} \def\und{\underline G$ as described. For the irreducible hermitian
symmetric domains, the equivalence classes of all such $N$ have been
determined by Satake and Ihara (\cite{S1} for the cases of ${\cal D} $ of type
$\bf I_{p,q},\ II_n, \ III_n$; \cite{I} for the other cases).
Before quoting the results we will need, let us briefly remark on the
mathematical formulation of the conditions. For this, let ${\cal D} ,\ {\cal D} '$ be
hermitian symmetric domains, $G,\ G'$ the automorphism groups, $\Gg,\ \Gg'$
the Lie algebras, $\Gg=\kk\oplus\pp,\ \Gg'=\kk'\oplus\pp'$ the Cartan
decompositions and $\gt,\ \gt'$ the Cartan involutions on $\Gg$ and $\Gg'$,
respectively. To say that for an injection $i_{{\cal D} }:{\cal D} \hookrightarrow} \def\hla{\hookleftarrow {\cal D} '$ of
symmetric spaces,
$i_{{\cal D} }({\cal D} )$ is
totally geodesic in ${\cal D} '$ is to say that
$i_{{\cal D} }$ is induced by an injection $i:\Gg\hookrightarrow} \def\hla{\hookleftarrow \Gg'$ of the Lie
algebras. If this holds, $i_{{\cal D} }$ is said to be {\it strongly
equivariant}. Then, $\gt=\gt'_{|i(\Gg)}$, or $\kk=\Gg\cap \kk',\
\pp=\Gg\cap \pp'$. Since both ${\cal D} $ and ${\cal D} '$ are hermitian
symmetric, there is an element $\xi$ in the center of $\kk$ (resp.
$\xi'$ in the
center of $\kk'$), such that $J=\ad(\xi)$ (resp. $J'=\ad(\xi')$) gives the
complex structure. To say that the injection $i_{{\cal D} }:{\cal D} \hookrightarrow} \def\hla{\hookleftarrow {\cal D} '$ is
holomorphic is the same as saying $i\circ J = J'\circ i$, or equivalently,
\vspace*{.2cm}
$\hbox{(H$_1$)}\hspace*{5.8cm} i\circ \ad(\xi) = \ad(\xi')\circ i.$
\vspace*{.2cm}
\noindent This is the condition utilized by Satake and Ihara in their
classifications. The condition (H$_1$) is clearly implied by
\vspace*{.2cm}(H$_2$)\hspace*{6.5cm}$i(\xi)=\xi',$
\vspace*{.2cm}
\noindent which however, if fulfilled, gives additional information. For
example (\cite{S2}, Proposition 4) if ${\cal D} $ is a tube domain and $i$
satisfies (H$_2$), then ${\cal D} '$ is also a tube domain.
Furthermore, (\cite{S}, Proposition II 8.1), if $i_{{\cal D} }:{\cal D} \longrightarrow} \def\sura{\twoheadrightarrow {\cal D} '$
is a holomorphic map which is strongly equivariant, the corresponding
homomorphism $i$
fulfills (H$_1$), and, moreover, if ${\cal D} $ and ${\cal D} '$ are viewed as bounded
symmetric domains ${\bf D}$, $\bf D'$ via the Harish-Chandra embeddings,
then $i_{\hbf{D}}:{\bf D}\longrightarrow} \def\sura{\twoheadrightarrow {\bf D}'$
is the restriction of a $\fC$-linear map
$i^+:\pp^+\longrightarrow} \def\sura{\twoheadrightarrow (\pp')^+$. If $i_{\fC}:\Gg_{\fC}\longrightarrow} \def\sura{\twoheadrightarrow \Gg_{\fC}'$ denotes the
$\fC$-linear extension of $i$, and $\gs:\Gg_{\fC}\longrightarrow} \def\sura{\twoheadrightarrow \Gg_{\fC},\
\gs':\Gg_{\fC}'\longrightarrow} \def\sura{\twoheadrightarrow \Gg_{\fC}'$ denote the conjugations over $\Gg$ and
$\Gg'$, respectively, then the condition $\gt=\gt'_{|i(\Gg)}$ is equivalent
to the condition $i_{\fC}\circ \gs=\gs'\circ i_{\fC}$. This implies that
$i:(\Gg,\xi)\longrightarrow} \def\sura{\twoheadrightarrow (\Gg',\xi')$ gives rise to a symmetric Lie algebra
homomorphism $(\Gg_{\fC},\gs)\longrightarrow} \def\sura{\twoheadrightarrow (\Gg_{\fC}',\gs')$, and therefore, by
\cite{S}, Proposition I 9.1, to a homomorphism of Jordan triple systems
$i^+:\pp^+\longrightarrow} \def\sura{\twoheadrightarrow (\pp')^+$. It follows (\cite{S}, p.~85) that the following
three categories are equivalent:
\begin{tabular}{cc}$(\hbox{{\script S}} \hbox{{\script D}} )$ & \parbox[t]{15cm}{Category whose objects
are symmetric
domains $({\cal D} ,o)$ with base point $o$, whose morphisms
$\gr_{{\cal D} }:({\cal D} ,o)\longrightarrow} \def\sura{\twoheadrightarrow ({\cal D} ',o')$ are strongly equivariant holomorphic
maps $\gr_{{\cal D} }:{\cal D} \longrightarrow} \def\sura{\twoheadrightarrow {\cal D} '$ with $\gr_{{\cal D} }(o)=o'$.} \\
$(\hbox{{\script H}} \hbox{{\script L}} )$ & \parbox[t]{15cm}{Category whose objects are
semisimple Lie algebras
$(\Gg,\xi)$ of hermitian type (without compact factors), whose morphisms
$\gr:(\Gg,\xi)\longrightarrow} \def\sura{\twoheadrightarrow (\Gg',\xi')$ are homomorphisms satisfying (H$_1$).} \\
$(\hbox{{\script H}} \hbox{{\script J}} )$ & \parbox[t]{15cm}{Category whose objects are
positive definite hermitian
Jordan triple systems $\pp^+$, whose morphisms $\gr_+:\pp^+\longrightarrow} \def\sura{\twoheadrightarrow (\pp')^+$
are $\fC$-linear homomorphisms of Jordan triple systems.}
\end{tabular}
\subsection{Positive-dimensional boundary components}
We now quote some results which we will be using. First, assume we have
fixed $A\subset} \def\nni{\supset} \def\und{\underline G$ as above, and let $F_b\subset} \def\nni{\supset} \def\und{\underline
\overline{\bf D}$ be a standard boundary component of positive dimension,
i.e., if ${\cal D} $ is irreducible, of rank $t$, then $b<t$; if ${\cal D} ={\cal D} _1\times
\cdots\times {\cal D} _d$, then in the notations of (\ref{e3.2}), ${\bf
b}=(b_1,\ldots, b_d)$, we have $b_i<t_i$ for {\it at least} one $i$. If
${\cal D} $ is irreducible, we list in Table \ref{T1} a positive-dimensional
boundary component and a symmetric subdomain ${\cal D} _M\subset} \def\nni{\supset} \def\und{\underline {\cal D} $ with the
property that ${\cal D} _N={\cal D} _F\times {\cal D} '$, where ${\cal D} _F$ is, as a hermitian
symmetric space, isomorphic to the given boundary component. If ${\cal D} $ is
reducible, ${\cal D} ={\cal D} _1\times\cdots\times{\cal D} _d$, and $F_{\hbox{\scsi \bf b}}\subset} \def\nni{\supset} \def\und{\underline
\overline{\bf D}$ is a standard boundary component, we get a subdomain
${\cal D} _N={\cal D} _{N_1}\times \cdots \times {\cal D} _{N_d}$ such that ${\cal D} _{N_i}\subset} \def\nni{\supset} \def\und{\underline
{\cal D} _i$ is of the type just mentioned with respect to the boundary component
$F_{b_i}\subset} \def\nni{\supset} \def\und{\underline \overline{\bf D}_i$.
\begin{table}\caption{\label{T1} Symmetric subdomains incident with
positive-dimensional boundary components}
$$\begin{array}{|c|c|c|c|}\hline {\cal D} & F_b,\ \ (b<t) & {\cal D} _N &
\hbox{(H$_2$)} \\ \hline \hline \bf I_{\hbox{\scriptsize\bf p,q}} & \bf
I_{\hbox{\scriptsize\bf p-b,q-b}} & \bf I_{\hbox{\scriptsize\bf p-b,q-b}}\times
I_{\hbox{\scriptsize\bf b,b}} & p=q \\ \hline \bf II_{\hbox{\scriptsize\bf n}} & \bf
II_{\hbox{\scriptsize\bf n-2b}} & \bf II_{\hbox{\scriptsize\bf n-2b}}\times
II_{\hbox{\scriptsize\bf 2b}} & yes \\ \hline \bf III_{\hbox{\scriptsize\bf n}} & \bf
III_{\hbox{\scriptsize\bf n-b}} & \bf III_{\hbox{\scriptsize\bf n-b}}\times
III_{\hbox{\scriptsize\bf b}} & yes \\ \hline \bf IV_{\hbox{\scriptsize\bf n}} & \bf
IV_{\hbox{\scriptsize\bf 1}} & \bf IV_{\hbox{\scriptsize\bf 1}}\times
IV_{\hbox{\scriptsize\bf 1}} & yes \\ \hline \bf V & \bf I_{\hbox{\scriptsize\bf
5,1}} & \bf I_{\hbox{\scriptsize\bf 5,1}}\times I_{\hbox{\scriptsize\bf 1,1}} &
yes \\ \hline \bf VI & \bf IV_{\hbox{\scriptsize\bf 10}} & \bf
IV_{\hbox{\scriptsize\bf 10}}\times IV_{\hbox{\scriptsize\bf 1}} & yes \\ & \bf
IV_{\hbox{\scriptsize\bf 1}} & \bf IV_{\hbox{\scriptsize\bf 1}}\times
IV_{\hbox{\scriptsize\bf 10}} & yes \\ \hline
\end{array}$$
\end{table}
Next, choose $N\subset} \def\nni{\supset} \def\und{\underline G$ with ${\cal D} _N$
as in Table \ref{T1}, such that $A\subset} \def\nni{\supset} \def\und{\underline N$ is a
maximal ${\Bbb R}} \def\fH{{\Bbb H}$-split torus in $N$, so that we can speak of standard boundary
components of ${\cal D} _N$. Then the subdomains ${\cal D} _N$ listed in Table
\ref{T1} have the following property. For simplicity we will assume from
now on that $G$ is semisimple.
\begin{proposition}\label{p7.1} Given $G$, simple of hermitian type with
maximal ${\Bbb R}} \def\fH{{\Bbb H}$-split torus $A$ and simple ${\Bbb R}} \def\fH{{\Bbb H}$-roots $\eta_i$ ($1\leq
i\leq t=\rank_{{\Bbb R}} \def\fH{{\Bbb H}}G$), let $P_{b}$ and $F_{b}$ denote the
standard maximal ${\Bbb R}} \def\fH{{\Bbb H}$-parabolic and standard boundary component
determined by $\eta_b$ ($b<t$). Let $N\subset} \def\nni{\supset} \def\und{\underline G$ be a symmetric subgroup
with $A\subset} \def\nni{\supset} \def\und{\underline N$, defining a subdomain ${\cal D} _N$ as in Table \ref{T1}, such
that $N=N_{1}\times N_{2}$ and $N_{1}$ is a hermitian Levi factor
of $P_b$.
Let $P_0\times P_{{t_2}}$ be the standard maximal parabolic defined
by the last simple ${\Bbb R}} \def\fH{{\Bbb H}$-root $\eta_{t_2}$ of the second factor in the
decomposition
$N=N_{1}\times N_{2}$. Then if $F:=F_0\times F_{{t_2}}$
($\cong {\cal D} _{N_{1}}\times \{pt.\}$, $t_2=\rank_{{\Bbb R}} \def\fH{{\Bbb H}}N_{2}$) denotes
the corresponding standard boundary component, the equality
$i_N(F)=F_{b}$ holds,
where $i_N:{\cal D} _{N}\longrightarrow} \def\sura{\twoheadrightarrow {\cal D} $ denotes the injection.
\end{proposition}
{\bf Proof:} From construction, $F\cong{\cal D} _{N_{1}}\times \{pt\}
\cong F_{b}$ as a
hermitian symmetric space; to see that they coincide under $i_N$, recall from
(\ref{e2.3}) the root space decomposition of the hermitian Levi component
of $P_{b}$. Since $A\subset} \def\nni{\supset} \def\und{\underline N$ is also a maximal ${\Bbb R}} \def\fH{{\Bbb H}$-split torus of
$N$, in the root system $\Phi(A,N)$ we have the subsystem
$[\eta_{b+1},\ldots, \eta_t]$ giving rise, on the one hand to the hermitian
Levi factor $\ll_b$ in $\pp_{b}$, on the other hand to the Lie algebra
of the first factor $\frak n} \def\rr{\frak r_{1}$ of $N$. From this it follows that
$P_{b}$ stabilizes $i_N(F)$, hence $i_N(F)=F_{b}$. \hfill $\Box$ \vskip0.25cm
Before proceeding to the case of zero-dimensional boundary components, we
briefly explain how the subgroups $N\subset} \def\nni{\supset} \def\und{\underline G$ (which are not unique, of
course) arise in terms of $\pm$symmetric/hermitian forms, at least for the
classical cases. For this, we note that $G$ can be described as follows (we
describe here certain reductive groups; the simple groups are just the
derived groups):
\begin{equation}\label{e7.1}\begin{minipage}{14.5cm}\begin{itemize}\item[I]
$\bf I_{\hbox{\scriptsize\bf p,q}}$:
$G$ is the unitary group of a hermitian form on
$\fC^{p+q}$ of signature $(p,q)$ ($p\geq q$).
\item[II] $\bf II_{\hbox{\scriptsize\bf n}}$:
$G$ is the unitary group of a skew-hermitian form on $\fH^n$.
\item[III] $\bf III_{\hbox{\scsi \bf n}}$: $G$ is the unitary group of a skew-symmetric form
on ${\Bbb R}} \def\fH{{\Bbb H}^{2n}$.
\item[IV] $\bf IV_{\hbox{\scsi \bf n}}$:
$G$ is the unitary group of a symmetric bilinear form on
${\Bbb R}} \def\fH{{\Bbb H}^{n+2}$ of signature $(n,2)$.
\end{itemize}\end{minipage}\end{equation}
Each of the $\pm$symmetric/hermitian forms is isotropic, and if
$t=\rank_{{\Bbb R}} \def\fH{{\Bbb H}}G$, the maximal dimension of a totally isotropic subspace is
$t=q,\ \left[{n\over 2}\right],\ n,\ 2$ in the cases I, II, III, and IV,
respectively. Each maximal real parabolic is the stabilizer of a
totally isotropic
subspace, and using the canonical order on the ${\Bbb R}} \def\fH{{\Bbb H}$-roots as above,
$P_{b}$ stabilizes a totally isotropic subspace of dimension
$b$. Choosing a maximal torus $T$ (resp. a maximal ${\Bbb R}} \def\fH{{\Bbb H}$-split torus
$A\subset} \def\nni{\supset} \def\und{\underline T$) amounts to choosing a basis of $V$ (resp. choosing a subset of
this basis which spans a
maximal totally isotropic subspace), and the standard parabolic is the
stabilizer of a totally isotropic subspace spanned by some part of this
basis. Now let $H\subset} \def\nni{\supset} \def\und{\underline V$ be a totally isotropic subspace with basis
$h_1,\ldots, h_b$. Then there are elements $h_1',\ldots,h_b'$ of $V$ such
that $h(h_i,h_j')=\gd_{ij},\ h(h_i,h_i)=h(h_i',h_i')=0$, and $h_1,\ldots,
h_b,h_1',\ldots, h_b'$ span (over $D$) a vector subspace $W\subset} \def\nni{\supset} \def\und{\underline V$ on which $h$
restricts to a non-degenerate form. Let $W^{\perp}$ denote the orthogonal
complement of $W$ in $V$, $W\oplus W^{\perp}=V$. Then
\begin{equation}\label{e8.1} N:=U(W,W^{\perp};h):=\{g\in U(V,h) |
g(W)\subset} \def\nni{\supset} \def\und{\underline W, g(W^{\perp})\subset} \def\nni{\supset} \def\und{\underline W^{\perp} \}\cong U(W,h_{|W})\times
U(W^{\perp},h_{|{W^{\perp}}}).
\end{equation}
$N$ is a reductive subgroup of $G$, and as one easily sees, its symmetric space
is just the domain denoted ${\cal D} _N$ in Table \ref{T1} above.
The relation
``boundary component $\subset} \def\nni{\supset} \def\und{\underline $ symmetric subdomain'' translates into
``totally isotropic subspace $\subset} \def\nni{\supset} \def\und{\underline $ non-degenerate subspace'', $H\subset} \def\nni{\supset} \def\und{\underline W$,
and {\it because} $h_{|W}$ is non-degenerate, any $g\in U(V,h)$ which
stabilizes $W$ automatically stabilizes its orthogonal complement in $V$ as
in (\ref{e8.1}).
\subsection{Zero-dimensional boundary components}
We now would like to consider the zero-dimensional boundary components,
which correspond in the above picture to maximal totally isotropic
subspaces.
The construction above (\ref{e8.1}) doesn't necessarily work in this case,
as $W^{\perp}$ may be $\{0\}$, and $N=G$.
However, in terms of
domains, given {\it any} subdomain ${\cal D} '\subset} \def\nni{\supset} \def\und{\underline {\cal D} $, it can be translated so
as to contain a given zero-dimensional boundary component. We therefore
place the following three conditions on such a subdomain:
\begin{itemize}\item[1)] The subdomain ${\cal D} '$ has maximal rank
($\rank_{{\Bbb R}} \def\fH{{\Bbb H}}G'=\rank_{{\Bbb R}} \def\fH{{\Bbb H}}G$).
\item[2)] The subdomain ${\cal D} '$ is maximal and $G'$ is a maximal subgroup, or
\item[2')] The subdomain ${\cal D} '$ is maximal of tube type and $G'$ is maximal
with this property.
\item[3')] The subdomain ${\cal D} '$ is maximal irreducible, and $F$ is a
boundary component of ${\cal D} '$.
\end{itemize}
In Table \ref{T2} we list the subdomains (after \cite{I}) ${\cal D} '$ fulfilling
1), 2) and 3') in the column titled ``${\cal D} _N$''.
We have listed also those subgroups fulfilling 1),
2') and 3') in the column titled ``maximal tube''.
\begin{table}\caption{\label{T2} Symmetric subdomains incident with
zero-dimensional boundary components}
$$\begin{array}{|c|c|c|c|} \hline
{\cal D} & {\cal D} _N & \hbox{(H$_2$)} & \hbox{maximal tube} \\ \hline \hline
{\bf I_{\hbox{\scriptsize\bf p,q}}},\ p>q & \bf I_{\hbox{\scriptsize\bf p-1,q}} & no &
\bf I_{\hbox{\scriptsize\bf q,q}}
\\ \hline
\bf I_{\hbox{\scriptsize\bf q,q}} & - & - & - \\ \hline
\bf II_{\hbox{\scriptsize\bf n}},\ n\hbox{ even} & - & - & - \\ \hline
\bf II_{\hbox{\scriptsize\bf n}},\ n\hbox{ odd} & \bf II_{\hbox{\scriptsize\bf n-1}}
& yes & \bf II_{\hbox{\scriptsize\bf n-1}} \\ \hline
\bf III_{\hbox{\scriptsize\bf n}} & - & - & - \\ \hline
\bf IV_{\hbox{\scriptsize\bf n}} & \bf IV_{\hbox{\scriptsize\bf n-1}} & yes & \bf
IV_{\hbox{\scriptsize\bf n-1}} \\ \hline
\bf V & \bf I_{\hbox{\scriptsize\bf 2,4}},\ II_{\hbox{\scriptsize\bf 5}},\
IV_{\hbox{\scriptsize\bf 8}} & yes,\ no,\ no & \bf I_{\hbf{2,2}}, II_{\hbf{4}},
IV_{\hbox{\scriptsize\bf 8}} \\ \hline
\bf VI & \bf I_{\hbox{\scriptsize\bf 3,3}},\ II_{\hbox{\scriptsize\bf 6}} & yes & \bf
I_{\hbox{\scriptsize\bf 3,3}},\ II_{\hbox{\scriptsize\bf 6}} \\ \hline
\end{array}$$
{\small In the column ``${\cal D} _N$'' the subgroups fulfilling 1), 2) and 3')
are listed, and in the column ``maximal tube'' the subgroups fulfilling
1), 2') and 3') (i.e., not necessarily 2)) are listed.}
\end{table}
In Table \ref{T2}, if there is no entry in the column
``${\cal D} _N$'', no such subgroups exist. In these cases it is natural to take the
polydisc ${\cal D} _{N_{\Psi}}$ defined by the maximal set of
strongly orthogonal roots $\Psi=\{\pm\{\mu_1\},\ldots,\pm\{\mu_t\}\}$
as the subdomain
${\cal D} _N$, as there is no irreducible subdomain, and other products already
occur in Table \ref{T1}. Hence for these cases we require the conditions
2'') and 3'') of the introduction. To sum up these facts we make the following
definition.
\begin{definition}\label{d9.1} Let $G$ be a simple real Lie group of
hermitian type, $A$ a fixed maximal
${\Bbb R}} \def\fH{{\Bbb H}$-split torus defined as above by a maximal set of strongly
orthogonal roots, $\eta_i,\ i=1,\ldots, t$ the simple ${\Bbb R}} \def\fH{{\Bbb H}$-roots,
$F_{b}$ a standard boundary component and
$P_{b}$ the corresponding standard maximal ${\Bbb R}} \def\fH{{\Bbb H}$-parabolic. A
reductive subgroup $N\subset} \def\nni{\supset} \def\und{\underline G$ (respectively the subdomain ${\cal D} _{N}\subset} \def\nni{\supset} \def\und{\underline
{\cal D} $) will be called {\it
incident} to $P_{b}$ (respectively to $F_{b}$), if ${\cal D} _N$ is
isomorphic to the corresponding domain of Table \ref{T1} ($b<t$) or Table
\ref{T2} ($b=t$), and if $N$ fulfills:
\begin{itemize}\item $b<t$, then $N$ satisfies 1), 2), 3).
\item $b=t,\ {\cal D} \not\in ({\cal E} {\cal D} )$, then $N$ satisfies 1), 2'), 3').
\item $b=t,\ {\cal D} \in ({\cal E} {\cal D} )$, then $N$ satisfies 1), 2''), 3'').
\end{itemize}
For reducible ${\cal D} ={\cal D} _1\times \cdots \times {\cal D} _d$, we have the product
subgroups $N_{b_1,1}\times \cdots \times N_{b_d,d}$, where
${\cal D} _{N_{b_i,i}}$ is
incident to the standard boundary component $F_{{b_i}}$ of ${\cal D} _i$
(and $N_{0,i}=G_i$).
\end{definition}
Next we briefly discuss uniqueness. We consider first the case of
positive-dimensional boundary components. Let $P_{b}$, $1\leq b< t$
be a standard parabolic and let $L_b$ be the ``standard'' hermitian Levi
factor, i.e., such that $Lie(L_b)=\ll_b$; then
\begin{equation}\label{e10.1} N_b:= L_b\times {\cal Z} _G(L_b)
\end{equation}
is a subgroup having the properties of Proposition \ref{p7.1}, unique since
$L_b$ is unique. We shall refer to this unique subgroup as the {\it
standard} incident subgroup. The different Levi factors
${\cal L} $ in Levi decompositions
$P_b={\cal L} \rtimes {\cal R} _u(P_b)$ are conjugate by elements of ${\cal R} _u(P_b)$, as is
well known. This implies for the hermitian factors
$L={\cal L} ^{herm}\subset} \def\nni{\supset} \def\und{\underline {\cal L} $ (which
are uniquely determined by ${\cal L} $) by Theorem \ref{t4.1} the following.
\begin{lemma}\label{l10A} Two hermitian Levi factors $L,\ L'\subset} \def\nni{\supset} \def\und{\underline P_b$ are
conjugate by an element of $V_b\subset} \def\nni{\supset} \def\und{\underline P_b$.
\end{lemma}
It follows, since $g(L_b\times {\cal Z} _G(L_b))g^{-1}=gL_bg^{-1}\times
{\cal Z} _G(gL_bg^{-1})$, that two subgroups $N,\ N'$, both incident with
$P_b$, are conjugate by an element of $V_b$:
\[\hbox{$N,\ N'$ incident to $P_b$ $\iff$ $N,\ N'$ conjugate (in $G$)
by $g\in V_b$.}\]
\begin{proposition} If $(N,P_b)$ are incident, there is $g\in V_b$ such
that $N$ is conjugate by $g$ to the standard $N_b$ of (\ref{e10.1}).
\end{proposition}
{\bf Proof:} Since $N$ is incident, $N\cong N_1\times N_2$, where $N_1$ is
a hermitian Levi factor of $N$. By Lemma \ref{l10A}, $N_1$ is conjugate by
$g\in V_b$ to $L_b$, the hermitian Levi factor with Lie algebra $\ll_b$
in the notations of the last section. Hence $gNg^{-1}=g(N_1\times
N_2)g^{-1} = gN_1g^{-1}\times gN_2g^{-1}=L_b\times N_{2,b}=N_b$, with
$N_{2,b}={\cal Z} (L_b)$ (this follows from the maximality of $N_b$).
Consequently, $N$ is conjugate by $g\in V_b$ to $N_b$. \hfill $\Box$ \vskip0.25cm
The situation for zero-dimensional boundary components is more complicated,
so we just observe the following. Suppose ${\cal D} \not\in ({\cal E} {\cal D} )$, and that
${\cal D} _N\subset} \def\nni{\supset} \def\und{\underline {\cal D} $ is incident to $F_t$, $F_t$=point. For any $g\in
N(F_t)=P_t,\ g{\cal D} _N={\cal D} '\subset} \def\nni{\supset} \def\und{\underline {\cal D} $ is another subdomain, again incident to
$F_t$. If $g\in P_t\cap N_t$, then $g{\cal D} '={\cal D} _N$. In this sense, letting
$Q_t=P_t\cap N_t$, the coset space $P_t/Q_t$ is a parameter space of
subdomains incident with $F_t$.
Above we have defined the notion of symmetric subgroups incident with a
standard parabolic. Any maximal ${\Bbb R}} \def\fH{{\Bbb H}$-parabolic is conjugate to one and
only one standard maximal parabolic, $P=gP_bg^{-1}$ for some $b$. Let $N_b$
be any symmetric subgroup incident with $P_b$. Then just as one has the
pair $(P_b,N_b)$ one has the pair $(P,N)$,
\begin{equation}\label{e10.3} P=gP_bg^{-1},\quad N=g N_b g^{-1}.
\end{equation}
\begin{definition} \label{d10.1} A pair $(P,N)$ consisting of a maximal
${\Bbb R}} \def\fH{{\Bbb H}$-parabolic $P$ and a symmetric subgroup $N$ is called {\it
incident}, if the groups are conjugate by a common element $g$ as in
(\ref{e10.3}) to a pair
$(P_b,N_b)$ which is incident as in Definition \ref{d9.1}.
\end{definition}
\section{Rational parabolic and rational symmetric subgroups}
\subsection{Notations}
We now fix some notations to be in effect for the rest of the paper. We
will be dealing with algebraic groups defined over $\fQ$, which give rise
to hermitian symmetric spaces, groups of {\it hermitian type}, as we will
say. As we are interested in the automorphism groups of domains, we may,
without restricting generality, assume the group is {\it centerless}, and
{\it simple} over $\fQ$. Henceforth $G$ will denote such an algebraic
group. To avoid complications, we exclude in this paper the following case:
\vspace*{.2cm}{\bf Exclude:}\hspace*{2cm} All non-compact real factors of
$G({\Bbb R}} \def\fH{{\Bbb H})$ are of type $SL_2({\Bbb R}} \def\fH{{\Bbb H})$.
\vspace*{.2cm}
\noindent Finally, we shall
only consider {\it isotropic} groups. This implies the hermitian symmetric
space ${\cal D} $ has no compact factors. By our assumptions, then, we have
\begin{itemize}\item[(i)] $G=Res_{k|\fQ}G'$, $k$ a totally real number
field, $G'$ absolutely simple over $k$.
\item[(ii)] ${\cal D} ={\cal D} _1\times \cdots \times {\cal D} _d$, each ${\cal D} _i$ a non-compact
irreducible hermitian symmetric space, $d=[k:\fQ]$.
\end{itemize}
We now introduce a few notations concerning the root systems involved. Let
$\gS_{\infty}$ denote the set of embeddings $\gs:k\longrightarrow} \def\sura{\twoheadrightarrow {\Bbb R}} \def\fH{{\Bbb H}$; this set is in
bijective correspondence with the set of infinite places of $k$. We
denote the latter by $\nu$, and if necessary we denote the
corresponding embedding by
$\gs_{\nu}$. For each $\gs\in \gS_{\infty}$, the group $^{\gs}G'$ is the
algebraic group defined over $\gs(k)$ by taking the
set of elements $g^{\gs},\ g\in G'$. For each infinite
prime $\nu$ we have
$G_{k_{\nu}}\cong (^{\gs_{\nu}}G')_{{\Bbb R}} \def\fH{{\Bbb H}}$, and the decomposition of ${\cal D} $
above can be written
$${\cal D} =\prod_{\gs\in \gS_{\infty}}{\cal D} _{\gs},\quad
{\cal D} _{\gs}:=(^{\gs}G')_{{\Bbb R}} \def\fH{{\Bbb H}}/K_{(\gs)}=(^{\gs}G')_{{\Bbb R}} \def\fH{{\Bbb H}}^0/K_{(\gs)}^0.$$
Since $G'$ is isotropic, there is a positive-dimensional $k$-split torus
$S'\subset} \def\nni{\supset} \def\und{\underline G'$, which we fix. Then ${^{\gs}S}'$ is a maximal $\gs(k)$-split
torus of $^{\gs}G'$ and there is a canonical isomorphism $S'\ra {^{\gs}S}'$
inducing an isomorphism $\Phi_k=\Phi(S',G')\longrightarrow} \def\sura{\twoheadrightarrow
\Phi_{\gs(k)}({^{\gs}S}',{^{\gs}G}')=:\Phi_{k,\gs}$.
The torus $Res_{k|\fQ}S'$ is
defined over $\fQ$ and contains $S$ as maximal $\fQ$-split torus; in fact
$S\cong S'$, diagonally embedded in $Res_{k|\fQ}S'$. This yields an
isomorphism $\Phi(S,G)\cong \Phi_k$, and the root systems
$\Phi_{\fQ}=\Phi(S,G)$, $\Phi_k$ and $\Phi_{k,\gs}$ (for all $\gs\in
\gS_{\infty}$) are identified
by means of the isomorphisms.
In each group $^{\gs}G'$ one chooses a maximal ${\Bbb R}} \def\fH{{\Bbb H}$-split torus
$A_{\gs}\nni {^{\gs}S}'$, contained in a maximal torus defined over
$\gs(k)$. Fixing an order on $X(S')$ induces one also on $X({^{\gs}S}')$
and $X(S)$. Then, for each $\gs$, one chooses an order on $X(A_{\gs})$
which is compatible with that on $X({^{\gs}S}')$, and $r:X(A_{\gs})\longrightarrow} \def\sura{\twoheadrightarrow
X({^{\gs}S}')\cong X(S)$ denotes the restriction homomorphism. The canonical
numbering on $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi_{{\Bbb R}} \def\fH{{\Bbb H},\gs}$ of simple ${\Bbb R}} \def\fH{{\Bbb H}$-roots of $G$ with respect to
$A_{\gs}$ is compatible by restriction with the canonical numbering of
$\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi_{\fQ}$ (\cite{BB}, 2.8). Recall also that each $k$-root in $\Phi_k$ is
the restriction of at most one simple ${\Bbb R}} \def\fH{{\Bbb H}$-root of $G'({\Bbb R}} \def\fH{{\Bbb H})$ (which is a
simple Lie group). Let $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi_k=\{\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda_1,\ldots,\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda_s\}$; for $1\leq i\leq s$ set
$c(i,\gs)$:= index of the simple ${\Bbb R}} \def\fH{{\Bbb H}$-root of $^{\gs}G'$ restricting on
$\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda_i$. Then $i<j$ implies $c(i,\gs)<c(j,\gs)$ for all $\gs\in \gS$.
Each simple $k$-root defines a unique standard boundary component: for
$b\in \{1,\ldots,s\}$,
\begin{equation}\label{e9.1} F_{\hbox{\scsi \bf b}}:=\prod_{\gs\in
\gS_{\infty}}F_{c(b,\gs)},
\end{equation}
which is the product of standard (with respect to $A_{\gs}$ and
$\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi_{{\Bbb R}} \def\fH{{\Bbb H},\gs}$) boundary components $F_{c(b,\gs)}$ of ${\cal D} _{\gs}$. It
follows that $\overline{F}_{\hbox{\scriptsize\bf j}}\subset} \def\nni{\supset} \def\und{\underline
\overline{F}_{\hbox{\scriptsize\bf i}}$ for $1\leq i\leq j\leq
s$. Furthermore, setting $o_{\hbox{\scsi \bf b}}:=\prod o_{c(b,\gs)}$, then (\cite{BB},
p.~472)
\begin{equation}\label{e9.2} F_{\hbox{\scsi \bf b}}=o_{\hbox{\scsi \bf b}}\cdot L_{\hbox{\scsi \bf b}},
\end{equation}
where $L_{\hbox{\scsi \bf b}}$ denotes the hermitian Levi component (\ref{e3.3})
of the parabolic
$P_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})=N(F_{\hbox{\scsi \bf b}})$. As these are the only boundary components of
interest to us, we will henceforth refer to any conjugates of the
$F_{\hbox{\scsi \bf b}}$ of (\ref{e9.1}) by elements of $G$ as {\it rational boundary
components} (these should more precisely be called rational with respect
to $G$), and to the conjugates of the parabolics $P_{\hbox{\scsi \bf b}}$ as the {\it
rational parabolics}.
\subsection{Rational parabolics}
Let $G'$ be as above, $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi_k=\{\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda_1\ldots,\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda_s\}$ the set of simple
$k$-roots (having fixed a maximal $k$-split torus $S'$ and an order on
$X(S')$). For $b\in \{1,\ldots,s\}$ we have the standard maximal
$k$-parabolic $P_b'$ of $G'$, whose group of ${\Bbb R}} \def\fH{{\Bbb H}$-points is the
normalizer of the standard rational boundary component $F_{c(b)}$ of the domain
${\cal D} '=G_{{\Bbb R}} \def\fH{{\Bbb H}}'/K'$, where $c(b)$ denotes the index of the simple ${\Bbb R}} \def\fH{{\Bbb H}$-root
restricting to $\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda_b$; since $G'$ is absolutely simple, $G_{{\Bbb R}} \def\fH{{\Bbb H}}'$ is simple
and ${\cal D} '$ is irreducible. Hence Theorem
\ref{t4.1} applies to $P_b'({\Bbb R}} \def\fH{{\Bbb H})$. Of the factors given there,
the following are defined over $k$:
the product $M_b'L_b'$ as well as $L_b'$ (but $M_b'$ is {\it not} defined
over $k$, so the $k$-subgroups are (instead of $L_b'$ and $M_b'$) $L_b'$
and ${G_b'}^{(1)}:=M_b'L_b'$), ${\cal R} _b',{\cal Z} _b'$ and $V_b'$. As is well
known, any maximal $k$-parabolic of $G'$ is conjugate to one and only one
of the $P_b'$, and two parabolics are conjugate $\iff$
they are conjugate over $k$. There is a 1-1 correspondence between the set
of $k$-parabolics of $G'$ and the set of $\fQ$-parabolics of $G$, given by
$P'\mapsto Res_{k|\fQ}P'=:P$. The standard maximal $\fQ$-parabolic
$P_{\hbox{\scriptsize\bf b}}$ of $G$ gives a $\fQ$-structure on the real parabolic
$P_{\hbox{\scriptsize\bf b}}({\Bbb R}} \def\fH{{\Bbb H})$, which is the normalizer in ${\cal D} $ of the
standard boundary component $F_{\hbox{\scsi \bf b}}$ as in (\ref{e9.1}) (see also
(\ref{e3.2}) and (\ref{e3.3})), where ${\bf
b}=(c(b,\gs_1),\ldots,c(b,\gs_d))$. In the decomposition of Theorem
\ref{t4.1}, the factors $G_{\hbox{\scriptsize\bf b}}^{(1)}=M_{\hbox{\scriptsize\bf
b}}L_{\hbox{\scriptsize\bf b}},\ {\cal R} _{\hbox{\scriptsize\bf b}}, {\cal Z} _{\hbox{\scriptsize\bf
b}}$ and $V_{\hbox{\scriptsize\bf b}}$ are all defined over $\fQ$. In
particular, for the factor $G_{\hbox{\scriptsize\bf b}}^{(1)}$, which we will
call the $\fQ$-hermitian Levi factor (and similarly, we will call
${G_{b}'}^{(1)}$
the $k$-hermitian Levi factor of $P_{b}'$), we have
\begin{equation}\label{e12.1} G_{\hbox{\scriptsize\bf b}}^{(1)}(\fQ)\cong
\prod_{\gs}({{^{\gs}G}'}_b^{(1)})_{\gs(k)},\quad {\cal Z} _G(G_{\hbox{\scriptsize\bf
b}}^{(1)})(\fQ)\cong \prod_{\gs}({\cal Z} _{(^{\gs}G'_{\gs(k)})}(
({{^{\gs}G}'_b}^{(1)})_{\gs(k)}).
\end{equation}
Furthermore, the hermitian Levi factor $L_{\hbox{\scsi \bf b}}$ is defined over $\fQ$,
and
\[L_{\hbox{\scsi \bf b}}(\fQ)=\prod_{\gs}({^{\gs}L}_b')_{\gs(k)}. \]
We now make a few remarks about the factors of $G({\Bbb R}} \def\fH{{\Bbb H})$ and of
$L_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})$. Since the map $G'\longrightarrow} \def\sura{\twoheadrightarrow {^{\gs}G}'$ is an isomorphism of a
$k$-group onto a $\gs(k)$-group, the algebraic groups (over $\fC$) are
isomorphic, hence the various ${^{\gs}G}'_{{\Bbb R}} \def\fH{{\Bbb H}}$ are all ${\Bbb R}} \def\fH{{\Bbb H}$-forms of
some fixed algebraic group. Similarly, the factors of $L_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})$ are
all ${\Bbb R}} \def\fH{{\Bbb H}$-forms of a single $\fC$-group. However, they need not be
isomorphic, unless the given $\fC$-group has a unique ${\Bbb R}} \def\fH{{\Bbb H}$-form of
hermitian type (like $Sp(2n,\fC)$). Next we note the following.
\begin{lemma}\label{L12a} $L_{\hbox{\scsi \bf b}}$ is anisotropic $\iff$ $b=s$.
\end{lemma}
{\bf Proof:} The group $L_{\hbox{\scsi \bf b}}$ is anisotropic precisely when the
boundary component $F_{\hbox{\scsi \bf b}}$ defined by it contains no other boundary
components $F_{\hbf{c}}^*\subset} \def\nni{\supset} \def\und{\underline F_{\hbox{\scsi \bf b}}^*$, which means $b\geq c$ for all
$c$, or $b=s$. \hfill $\Box$ \vskip0.25cm
In this case the group $L_{\hbox{\scsi \bf b}}$ does not fulfill the assumptions we have
placed on $G$, and our results up to this point are not directly applicable
to $L_{\hbox{\scsi \bf b}}$. Let us see how the phenomenon of compact factors of
$L_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})$ manifests itself in
$F_{\hbox{\scsi \bf b}}=\prod_{\gs}F_{c(b,\gs)}$. Suppose some factor of $L_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})$
is compact, say $L_{1,b}$. Then the symmetric space ${\cal D} _{b,\gs_1}$
of $L_{1,b}$ is compact, so it is not true that ${\cal D} _{b,\gs_1}\cong
F_{c(b,\gs_1)}$, hence it is also not true that ${\cal D} _{\hbox{\scsi \bf b}}\cong F_{\hbox{\scsi \bf b}}$,
where ${\cal D} _{\hbox{\scsi \bf b}}=\prod_{\gs}{\cal D} _{b,\gs}$ is the symmetric space of
$L_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})$. However, letting ${\cal D} _{\hbox{\scsi \bf b}}'$ be the product of all
compact factors, ${\cal D} _{\hbox{\scsi \bf b}}/{\cal D} _{\hbox{\scsi \bf b}}'$ is a symmetric space which
is isomorphic to $F_{\hbox{\scsi \bf b}}$. What happens is that in the product
$F_{\hbox{\scsi \bf b}}=\prod F_{c(b,\gs)}$, all factors $F_{c(b,\gs)}$ are {\it points}
for which ${\cal D} _{b,\gs}$ is {\it compact}. Hence whether this occurs depends
on whether any factors ${\cal D} _{\gs}$ have zero-dimensional (rational)
boundary components or not.
\subsection{Incidence}
We keep the notations used above; $G$ is a simple $\fQ$-group of hermitian
type. Our main definition gives a $\fQ$-form of Definition \ref{d10.1}, and
is the following.
\begin{definition}\label{d12.1} Let $P\subset} \def\nni{\supset} \def\und{\underline G$ be a maximal $\fQ$-parabolic,
$N\subset} \def\nni{\supset} \def\und{\underline G$ a reductive $\fQ$-subgroup. Then we shall say that $(P,N)$ are
{\it incident} (over $\fQ$), if $(P({\Bbb R}} \def\fH{{\Bbb H}),N({\Bbb R}} \def\fH{{\Bbb H}))$ are incident in the
sense of Definition \ref{d10.1}.
\end{definition}
Note that in particular $N$ must itself be of hermitian type, and such that
the Cartan involution of $G({\Bbb R}} \def\fH{{\Bbb H})$ restricts to the Cartan involution of
$N({\Bbb R}} \def\fH{{\Bbb H})$. Furthermore, $N$ must be a $\fQ$-form of a product of groups,
defining domains each of which is as in either Table \ref{T1}
or Table \ref{T2}.
The main result of this paper is the following existence result.
\begin{theorem}\label{t12.1} Let $G$ be $\fQ$-simple of hermitian type
subject to the restrictions above ($G$ is isotropic and $G({\Bbb R}} \def\fH{{\Bbb H})$ is not a
product of $SL_2({\Bbb R}} \def\fH{{\Bbb H})$'s),
$P\subset} \def\nni{\supset} \def\und{\underline G$ a $\fQ$-parabolic. Then there exists a reductive $\fQ$-subgroup
$N\subset} \def\nni{\supset} \def\und{\underline G$ such that $(P,N)$ are incident over $\fQ$, with the exception
of the indices $C^{(2)}_{2n,n}$ for the zero-dimensional boundary
components.
\end{theorem}
We will give the proof in the following sections, where we consider separately
different cases (of the $\fQ$-rank, the dimension of a maximal $\fQ$-split
torus). But before we start, we note here that by definition, if the
theorem holds for {\it standard} parabolics, then it holds for all
parabolics, so it will suffice to consider only standard parabolics. The
case that $G'$ has index $C^{(2)}_{2,1}$ was considered in \cite{hyp}; in
that case there is a unique standard parabolic $P_1$, with zero-dimensional
boundary component; the associated $N_1$ described in \cite{hyp}
has domain ${\cal D} _{N_1}$ which
is not a two-disc, but only a one-dimensional disc.
\section{Split over ${\Bbb R}} \def\fH{{\Bbb H}$ case}
In this paragraph we consider the easiest case. This could loosely be
described as an ${\Bbb R}} \def\fH{{\Bbb H}$-Chevally form.
\begin{definition}\label{d13.1} Let $G'$ be as in the last paragraph,
absolutely simple over $k$, and let $\Phi_k$ be a root system
(irreducible) for $G'$ with respect to a maximal $k$-split torus $S'\subset} \def\nni{\supset} \def\und{\underline
G'$. Let $\Phi_{{\Bbb R}} \def\fH{{\Bbb H}}$ be the root system of $G'({\Bbb R}} \def\fH{{\Bbb H})$ with respect to a
maximal ${\Bbb R}} \def\fH{{\Bbb H}$-split torus $A'$ of the real (simple) group $G'({\Bbb R}} \def\fH{{\Bbb H})$. We
call $G'$ {\it split over ${\Bbb R}} \def\fH{{\Bbb H}$}, if $\Phi_k\cong \Phi_{{\Bbb R}} \def\fH{{\Bbb H}}$ as root
systems, and if the indices of $G'$ and $G'({\Bbb R}} \def\fH{{\Bbb H})$ coincide.
\end{definition}
Note that the indices are independent of the split tori used to form the
root system, so there is no need to assume $S'\subset} \def\nni{\supset} \def\und{\underline A'$ in the above
definition (the notion of isomorphism of indices is obvious). However, one
can always find split tori $S', A'$ such that $S'\subset} \def\nni{\supset} \def\und{\underline A'$. From
$\Phi_k\cong \Phi_{{\Bbb R}} \def\fH{{\Bbb H}}$ it follows then that $S'=A'$, as both tori have
the same dimension.
\begin{lemma}\label{l13.1} Let $G$ be simple over $\fQ$ (=$Res_{k|\fQ}G'$),
${\cal D} =\prod_{\gs\in \gS_{\infty}}{\cal D} _{\gs}$ the domain defined by the real
Lie group $G({\Bbb R}} \def\fH{{\Bbb H})\cong \prod_{\gs\in
\gS_{\infty}}G_{\gs}=:\prod_{\gs}{^{\gs}G'_{{\Bbb R}} \def\fH{{\Bbb H}}}$. If $G'$ is split
over ${\Bbb R}} \def\fH{{\Bbb H}$, then $G_{\gs}=G_{\tau}$ for all $\gs,\tau\in \gS_{\infty}$.
\end{lemma}
{\bf Proof:} For each $\gs$ we have $A_{\gs}\nni {^{\gs}S}'$, so by
assumption $A_{\gs}\cong {^{\gs}S}'$, and for each $\gs$ the map
$\phi:\Phi_k\longrightarrow} \def\sura{\twoheadrightarrow \Phi_{\gs(k)}(^{\gs}G')$ is an isomorphism, and since
$\Phi_k\cong \Phi_{{\Bbb R}} \def\fH{{\Bbb H}}$,
$$\Phi_{\gs(k)}(^{\gs}G')\cong \Phi_{{\Bbb R}} \def\fH{{\Bbb H}}(^{\gs}G_{{\Bbb R}} \def\fH{{\Bbb H}}').$$ It follows
that $\Phi_{{\Bbb R}} \def\fH{{\Bbb H}}\cong \Phi_k\stackrel{\phi}{\cong}
\Phi_{\gs(k)}(^{\gs}G')\cong \Phi_{{\Bbb R}} \def\fH{{\Bbb H}}(^{\gs}G'_{{\Bbb R}} \def\fH{{\Bbb H}})\cong \Phi_{{\Bbb R}} \def\fH{{\Bbb H}}$.
Similarly, since the index of $G'$ is isomorphic to the index of $G'({\Bbb R}} \def\fH{{\Bbb H})$
(which determines the isomorphy class of $G'({\Bbb R}} \def\fH{{\Bbb H})$), the index of
${^{\gs}G}'$ is isomorphic to that of ${^{\gs}G}'({\Bbb R}} \def\fH{{\Bbb H})$. But the index of
$G'({\Bbb R}} \def\fH{{\Bbb H})$ is the same as ${^{\gs}G}'({\Bbb R}} \def\fH{{\Bbb H})$, as an easy case by case check
verifies. For example, for type (I), all factors have the same ${\Bbb R}} \def\fH{{\Bbb H}$-rank
$q$, hence are all isomorphic to $SU(p,q)$. See Examples \ref{examples}
below for the other cases. Hence ${^{\gs}G}'({\Bbb R}} \def\fH{{\Bbb H})\cong {^{\tau}G}'({\Bbb R}} \def\fH{{\Bbb H})$
for all $\gs, \tau$, as claimed. \hfill $\Box$ \vskip0.25cm
{}From this it follows in particular that the (standard) boundary components
are determined by $c(b,\gs)=b,\ \forall_{\gs},\ {\bf b}=(b,\ldots,b),\
1\leq b\leq t=\rank_{\fQ}G=\rank_{k}G'=\rank_{{\Bbb R}} \def\fH{{\Bbb H}}G'({\Bbb R}} \def\fH{{\Bbb H})$. Hence they are
of the form
\begin{equation}\label{e13.1} F_{\hbox{\scsi \bf b}}=\prod_{\gs\in
\gS_{\infty}}F_{b,\gs},
\end{equation}
and $F_{b,\gs}$ is the standard rational boundary component of ${\cal D} _{\gs}$.
\begin{examples}\label{examples}
We now give examples of split over ${\Bbb R}} \def\fH{{\Bbb H}$ groups in each of the cases, and
any such will be of one of the listed types. Let $k$ be a totally real
number field.
\begin{itemize}\item[I.] Let $K|k$ be imaginary quadratic, $V$ a
$K$-vector space of dimension $n=p+q$, and $h$ a hermitian form on $V$
defined over $K$. Then the unitary group $U(V,h)$ is split over ${\Bbb R}} \def\fH{{\Bbb H}$
$\iff$ the hermitian form $h$ has Witt index $q$ and for all infinite
primes, $h_{\nu}$ has signature $(p,q)$.
\item[II.] Let $D|k$ be a totally definite quaternion algebra over $k$
(with the canonical involution), $V$ an $n$-dimensional right vector
space over $D$, $h$ a skew-hermitian form on $V$ defined over $k$. Then
the unitary group $U(V,h)$ is split over ${\Bbb R}} \def\fH{{\Bbb H}$ $\iff$ the skew-hermitian
form has Witt index $[{n\over2}]$ ($n>4$).
\item[III.] Take $G=Sp(2n,k)$.
\item[IV.] Let $V$ be a $(n+2)$-dimensional $k$-vector space, $h$ a
symmetric bilinear form defined over $k$ of Witt index 2. Then if
$U(V,h)$ is of hermitian type, it is split over ${\Bbb R}} \def\fH{{\Bbb H}$.
\item[V.] The Lie algebra in this case is of the form
$\hbox{{\script L}} (\frak C_k,(J_1^b)_k)$, the Tits algebra, where $\frak C_k$ is an
anisotropic octonion algebra and $(J_1^b)_k$ is the Jordan algebra
$\BB^+$ for an associative algebra $\BB$ whose traceless elements with
the Lie product form a Lie algebra of type $\frak s} \def\cc{\frak c\uu(2,1)$; since $G'$ is
split over ${\Bbb R}} \def\fH{{\Bbb H}$, the algebra $\BB^-$ is the Lie algebra of a unitary
group of a $K$-hermitian form ($K|k$ imaginary quadratic as in (I)) of
Witt index 1.
\item[VI.] The Lie algebra is isomorphic to $\hbox{{\script L}} (\AA_k,\JJ_k)$, the Tits
algebra, where $\AA_k$ is a totally indefinite quaternion algebra over
$k$ and $\JJ_k$ is a $k$-form of the exceptional Jordan algebra denoted
$J^b$ by Tits.
\end{itemize}
\end{examples}
\begin{lemma}\label{l14.1} In the notations above, let $N'({\Bbb R}} \def\fH{{\Bbb H})\subset} \def\nni{\supset} \def\und{\underline
G'({\Bbb R}} \def\fH{{\Bbb H})$ be a subgroup such that the Lie algebra $\frak n} \def\rr{\frak r'\subset} \def\nni{\supset} \def\und{\underline \Gg'$ is a
{\it regular} subalgebra, i.e., defined by a closed symmetric set of
roots $\Psi$ of the (absolute) root system $\Phi$ of $G'$. Then $N'$ is
defined over $k$, $N'\subset} \def\nni{\supset} \def\und{\underline G'$.
\end{lemma}
{\bf Proof:} From the isomorphism of the indices of $G'$ and $G'({\Bbb R}} \def\fH{{\Bbb H})$, it
follows that any subalgebra $\Gg'\subset} \def\nni{\supset} \def\und{\underline \Gg$, such that for some subset
$\Psi\subset} \def\nni{\supset} \def\und{\underline \Phi$, the subalgebra $\Gg'$ is given by $\Gg'=\tt+\sum_{\eta\in
\Psi}\Gg^{\eta}$, is defined over ${\Bbb R}} \def\fH{{\Bbb H}$ $\iff$ it is defined over $k$.
The regular subalgebra $\frak n} \def\rr{\frak r'$ is of this type, and it follows that $N'$ is
defined over $k$. \hfill $\Box$ \vskip0.25cm
\begin{corollary}\label{c14.1} Let $N'\subset} \def\nni{\supset} \def\und{\underline G'$ be as in Lemma \ref{l14.1},
$N=Res_{k|\fQ}N'\subset} \def\nni{\supset} \def\und{\underline G$. Then $N$ is defined over $\fQ$.
\end{corollary}
To apply Lemma \ref{l14.1} to ($k$-forms of) subgroups whose domains are
listed in Tables \ref{T1} and \ref{T2}, we need to know which of the
subgroups are defined by regular subalgebras. Ihara in \cite{I} considered
this question, and the result is: all isomorphism classes of groups in
Table \ref{T1} and all isomorphism classes of groups in Table \ref{T2},
with the exception of $SO(n-1,2)\subset} \def\nni{\supset} \def\und{\underline SO(n,2)$ for $n$ even, have
representatives which are defined by (maximal) regular subalgebras.
\begin{corollary}\label{c14.2} Let $G'$ be split over ${\Bbb R}} \def\fH{{\Bbb H}$. Then Theorem
\ref{t12.1} holds for $G=Res_{k|\fQ}G'$.
\end{corollary}
{\bf Proof:} By Lemma \ref{l13.1}, $G({\Bbb R}} \def\fH{{\Bbb H})/K={\cal D} =\prod{\cal D} _{\gs}$, and all
${\cal D} _{\gs}$ are isomorphic to ${\cal D} '=G'({\Bbb R}} \def\fH{{\Bbb H})/K'$; the rational boundary
components are as in (\ref{e13.1}), products of copies of $F_b'$, the
standard boundary component of ${\cal D} '$, and each $\fQ$-parabolic of $G$ is
conjugate to one of $P_b=Res_{k|\fQ}P_b'$, where $P_b'({\Bbb R}} \def\fH{{\Bbb H})=N(F_b')$ is the
standard maximal real parabolic of $G'({\Bbb R}} \def\fH{{\Bbb H})$. Now locate $F_b'$ in Table
\ref{T1} or \ref{T2} as the case may be; the corresponding group $N_b'$ is
isomorphic to one defined by a regular subalgebra of $\Gg_{{\Bbb R}} \def\fH{{\Bbb H}}'$ with the
one exception mentioned above. Then by Lemma \ref{l14.1}, $N_b'$ is defined
over $k$, hence (Corollary) $N_b=Res_{k|\fQ}N_b'$ is defined over $\fQ$,
and is incident with $P_b$. This takes care of all cases except the
exception just mentioned, $\bf IV_{\hbox{\scriptsize\bf n-1}}\subset} \def\nni{\supset} \def\und{\underline
IV_{\hbox{\scriptsize\bf n}}$, $n>3$ even. So let $V$ be a $k$-vector space of
dimension $n+2$, $h$ a symmetric bilinear form on $V$. By assumption, $G'$
is split over ${\Bbb R}} \def\fH{{\Bbb H}$, so the Witt index of $h$ is 2. Let $H\subset} \def\nni{\supset} \def\und{\underline V$ be a
maximal totally isotropic subspace (two-dimensional) defined over $k$, and
$h_1,h_2$ a $k$-basis. Then there are $k$-vectors $h_i'$ such that
$H_1:=<h_1,h_1'>$ and $H_2:=<h_2,h_2'>$ are hyperbolic planes; let
$W=H_1\oplus H_2$ denote their direct sum. From $n>3$, $W$ has codimension
$\geq1$ in $V$. Let $v\in W^{\perp}$ be a $k$-vector, and set:
$$U:=v^{\perp}=\{w\in V|h(v,w)=0\}.$$ Then $W\subset} \def\nni{\supset} \def\und{\underline U$, the dimension of $U$
is $n+2-1=n+1$, and $h_{|U}$ still has Witt index 2. Hence
$$N':=\{g\in U(V,h)|g(U)\subset} \def\nni{\supset} \def\und{\underline U\}$$ is a $k$-subgroup, and $N'({\Bbb R}} \def\fH{{\Bbb H})^0\ifmmode\ \cong\ \else$\isom$\fi} \def\storth{\underline{\perp}
SO(n-1,2)$. This is a group which is incident to a parabolic whose group of
real points is the stabilizer of the zero-dimensional boundary component
$F_{2}'$ of the domain ${\cal D} '$ of type $\bf IV_{\hbox{\scriptsize\bf n}}$. \hfill $\Box$ \vskip0.25cm
This completes the discussion of the split over ${\Bbb R}} \def\fH{{\Bbb H}$ case. We just mention
that, at least in the classical cases, we could have argued case for case
with $\pm$symmetric/hermitian forms as in the proof of the exception above.
Using the root systems simplified the discussion, and, in particular, gives
the desired results for the exceptional groups without knowing their
explicit construction.
\section{Rank $\geq 2$}
In this paragraph we assume $G$ in {\it not} split over ${\Bbb R}} \def\fH{{\Bbb H}$, but that
$\rank_kG'=\rank_{\fQ}G\geq 2$. Under these circumstances, it is known
precisely which $k$-indices are possible for $G'$ of hermitian type.
\begin{proposition}\label{p16.1} Assume $\rank_{\fQ}G\geq 2$ and that $G'$
is not split over ${\Bbb R}} \def\fH{{\Bbb H}$. Then the $k$-index of $G'$ is one of the
following:
\begin{itemize}\item[(I)] ${^2A}^{(d)}_{n,s};\ s\geq2,\ d|n+1,\ 2sd\leq
n+1$; if $d=1$, then $2s<n+1$.
\item[(II)] ${^1D}^{(2)}_{n,s},\ s\geq2,\ s< \ell\ (n=2\ell);\quad
{^2D}^{(2)}_{n,s},\ s\geq 2,\ s<\ell\ (n=2\ell+1)$.
\item[(III)] $C^{(2)}_{n,s},\ s\geq 2,\ s< [{n\over 2}]$.
\item[(IV)] none
\item[(V)] none
\item[(VI)] $E^{31}_{7,2}$.
\end{itemize}
\end{proposition}
{\bf Proof:} All statements are self-evident from the description of the
indices in \cite{T}; in the case (V) there are three possible indices, only
one of which has rank $\geq 2$; this is the split over ${\Bbb R}} \def\fH{{\Bbb H}$ index.
Similarly, in the case (IV), rank $\geq2$ implies split over ${\Bbb R}} \def\fH{{\Bbb H}$. For
type (III), the indices $C^{(1)}$ are also split over ${\Bbb R}} \def\fH{{\Bbb H}$. \hfill $\Box$ \vskip0.25cm There is
only one exceptional index to consider, so we start by dealing with this
case. The index we must discuss is
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There are two simple $k$-roots, $\eta_1$ and $\eta_2$; let $P_{b}'$ be the
corresponding standard maximal $k$-parabolics, $F_{b}'$ the corresponding
standard boundary components of the irreducible domain ${\cal D} '$. Then
$F_{2}'$ is the one-dimensional boundary component, $F_{1}'$ is the
ten-dimensional one. The $k$-root system is of type ${\bf BC}_2$ (since the
highest simple ${\Bbb R}} \def\fH{{\Bbb H}$-root is anisotropic, see \cite{BB}, 2.9). Consider the
decomposition of Theorem \ref{t4.1} for $P_{b}'({\Bbb R}} \def\fH{{\Bbb H})$; in both cases $L'_b$
is non-trivial, and, as mentioned above, $M_b'\cdot L_b'$ is defined over
$k$. Here we have $b=1$ or 2. But for $E_7$, the compact factor $M_b'$ is
in fact {\it absent}\footnote{see \cite{S}, p.~117}, and as $L_b'$ is
defined over $k$, we can set
$$N_b'=L_b'\times {\cal Z} _{G'}(L_b').$$ This is a $k$-subgroup which is a
$k$-form of the corresponding ${\Bbb R}} \def\fH{{\Bbb H}$-subgroup whose domain is listed in
Table \ref{T1}. Now consider $G=Res_{k|\fQ}G'$. It also has two standard
maximal parabolics $P_{\hbox{\scriptsize\bf 1}}$ and $P_{\hbox{\scriptsize\bf 2}}$, and
in each we have a non-trivial hermitian Levi factor\footnote{we note a
change of notation here in that in (\ref{e3.3}), $L_{\hbox{\scsi \bf b}}$ denotes a
real Lie group} $L_{\hbox{\scsi \bf b}}:= Res_{k|\fQ}L_b'$, such that
$$L_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})=\prod_{\gs\in \gS_{\infty}}{^{\gs}(L_b')}_{{\Bbb R}} \def\fH{{\Bbb H}}.$$ Also the
symmetric subgroup $N_{\hbox{\scsi \bf b}}:=Res_{k|\fQ}N_b'$ is defined over $\fQ$ and
satisfies $N_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})=\prod_{\gs\in \gS_{\infty}}{^{\gs}(N_b')}_{{\Bbb R}} \def\fH{{\Bbb H}}$.
It follows that $(P_{\hbox{\scriptsize\bf b}},N_{\hbox{\scsi \bf b}})$ are incident: conditions
1) and 2) follow from the corresponding facts for $(P_b',N_b')$; we should
check 3). But since it is obvious that ${^{\gs}(}L_b')_{{\Bbb R}} \def\fH{{\Bbb H}}\subset} \def\nni{\supset} \def\und{\underline
{^{\gs}(}P_b')_{{\Bbb R}} \def\fH{{\Bbb H}}$ is a hermitian Levi factor, the same holds for
$L_{\hbox{\scsi \bf b}}\subset} \def\nni{\supset} \def\und{\underline P_{\hbox{\scsi \bf b}}$; 3) is satisfied. This completes the proof of
\begin{proposition}\label{p16.2} Theorem \ref{t12.1} is true for the
exceptional groups in the rank$\geq 2$, not split over ${\Bbb R}} \def\fH{{\Bbb H}$ case.
\end{proposition}
We are left with the classical cases. Here we may use the interpretation of
$G({\Bbb R}} \def\fH{{\Bbb H})$ as the unitary group of a $\pm$symmetric/hermitian form as in
(\ref{e7.1}), and $G$ is a $\fQ$-form of this. The precise realisation of
this is the interpretation in terms of central simple algebras with
involution; this is discussed in \cite{W}. More precisely, the algebraic
groups $G'$ which represent the indices of Proposition \ref{p16.1} are
(here we describe reductive groups; the corresponding derived groups are
the simple groups $G'$).
\begin{itemize}\item[(I)] \begin{itemize}\item[$D$:] degree $d$ central
simple division algebra over $K$, $K|k$ an imaginary quadratic
extension, $D$ has a $K|k$-involution (involution of the second kind).
\item[$V$:] right $D$-vector space, of dimension $m$ over $D$, $dm=n+1$.
\item[$h$:] hermitian form $h:V\times V\longrightarrow} \def\sura{\twoheadrightarrow D$ of Witt index $s$, $2s\leq
m$ ($2s<m$ if $d=1$), given by a matrix $H$.
\item[$G'$:] unitary group $U(V,h)=\{g\in GL_D(V)|gHg^*=H\}$.
\end{itemize}
{\bf index:} ${^2A}^{(d)}_{n,s}$.
\item[(II)] \begin{itemize}\item[$D$:] totally definite quaternion division
algebra, central simple over $k$, with canonical involution.
\item[$V$:] right $D$-vector space of dimension $m$ over $D$.
\item[$h$:] skew-hermitian form $h$ of Witt index $s< [{m\over 2}]$,
given by a matrix $H$.
\item[$G'$:] unitary group $U(V,h)=\{g\in GL_D(V)|gHg^*=H\}$.
\end{itemize}
{\bf index:} $D^{(2)}_{m,s}$ ($m$ even), ${^2D}^{(2)}_{m,s}$ ($m$ odd).
\item[(III)] \begin{itemize}\item[$D$:] totally indefinite quaternion
division algebra, central simple over $k$, with the canonical
involution.
\item[$V$:] right $D$-vector space of dimension $m$.
\item[$h$:] hermitian form $h:V\times V\longrightarrow} \def\sura{\twoheadrightarrow D$ of Witt index $s$, $2s\leq
m$, given by a matrix $H$.
\item[$G'$:] unitary group $U(V,h)=\{g\in GL_D(V)|gHg^*=H\}$.
\end{itemize}
{\bf index:} $C^{(2)}_{m,s}$.
\end{itemize}
Finally, we must consider the following ``mixed cases'', which still can
give rise to groups of hermitian type:
\begin{itemize}\item[(II-IV):]\begin{itemize}\item[$D$:] a quaternion
division algebra over $k$, with $D_{\nu}$ definite for
$\nu_1,\ldots,\nu_a$, $D_{\nu}$ indefinite for
$\nu_{a+1},\ldots,\nu_f$, where $f=[k:\fQ]$.
\item[$V$:] same as for (II) above.
\item[$h$:] same as for (II) above, $h$ of Witt index $s$.
\item[$G'$:] same as for (II) above.
\end{itemize}\end{itemize}
$G({\Bbb R}} \def\fH{{\Bbb H})$ is then a product $(SU(n,\fH))^a\times (SO(2n-2,2))^{f-a}$, where
we have taken into account that we are assuming $G$ to be isotropic and of
hermitian type. Note however, that since the factors $SO(2n-2,2)$
corresponding to the primes $\nu_{a+1},\ldots,\nu_f$ have ${\Bbb R}} \def\fH{{\Bbb H}$-split torus
of dimension two, the $k$-rank of $G'$ must be $\leq2$. Hence the only
indices where this can occur are: ${^iD}^{(2)}_{n,1}$ and
${^i}D^{(2)}_{n,2}$, $i=1,2$.
In terms of the spaces $(V,h)$, the standard parabolics are stabilizers of
totally isotropic subspaces $H_b\subset} \def\nni{\supset} \def\und{\underline V$, where $H_1$ is one-dimensional
(over $D$), while $H_s$ is a maximal totally isotropic subspace. The latter
case corresponds to zero-dimensional boundary components. We consider first
the case $H_b,\ b<s$, of which at least $H_1$ exists, because of the
assumption rank $\geq 2$. Fix a basis $h_1,\ldots, h_b$ of $H_b$ of
isotropic vectors $h(h_i,h_i)=0$ for all $i=1,\ldots,b$. Then there exist,
in $V$, elements $h_i',\ i=1,\ldots,b$ with $h(h_i,h_j')=\gd_{ij}$, and
$h_1',\ldots,h_b'$ span a complementary totally isotropic subspace; denote
it by $H_b'$. Then $H:=H_b\oplus H_b'$ is a {\it non-degenerate} space for
$h$, $h_{|H}$ is a non-degenerate form. It follows that $\{g\in GL(V) |
g(H)\subset} \def\nni{\supset} \def\und{\underline H\} = \{g\in GL(V) | g(H^{\perp})\subset} \def\nni{\supset} \def\und{\underline H^{\perp}\}$. In the
following we will work in the (reductive) unitary group $G'=U(V,h)$; for
any subgroup $H\subset} \def\nni{\supset} \def\und{\underline G'$ we can take the intersection $SL(V)\cap H\subset} \def\nni{\supset} \def\und{\underline
SL(V)\cap G'$ to give subgroups of the simple group. Furthermore, up to
Corollary \ref{c19.1} below, we omit the primes in the notations for the
subgroups of $G'$. Set
\begin{equation}\label{e18.1} N=U(H,H^{\perp};h)=\{g\in GL(V)|g(H)\subset} \def\nni{\supset} \def\und{\underline H,\
g(H^{\perp})\subset} \def\nni{\supset} \def\und{\underline H^{\perp}\};
\end{equation}
then $N=U(H,h_{|H})\times U(H^{\perp},h_{|H^{\perp}}),$ and
$U(H,h_{|H})\cong {\cal Z} _G(U(H^{\perp},h_{|H^{\perp}}))$. So setting
$L=U(H^{\perp}, h_{|H^{\perp}})$, we have
\begin{equation}\label{e18.2} N\cong L \times {\cal Z} _G(L).
\end{equation}
Next we note that the basis $h_1,\ldots, h_b$ of $H_b$ determines a unique
${\Bbb R}} \def\fH{{\Bbb H}$-split torus $A_b\subset} \def\nni{\supset} \def\und{\underline A$, where $A$ is the maximal ${\Bbb R}} \def\fH{{\Bbb H}$-split torus
defined by a basis $h_1,\ldots,h_s$ of a maximal totally isotropic subspace
$H_s\nni H_b$, namely the scalars $\ga=\ga\cdot{\bf 1}\in GL(H_b)$,
extended to $GL(V)$ by unity. Taking the centralizer of the torus $A_b$
gives a Levi factor of the parabolic $P_b={\cal N} _G(H_b),\ b<s$ (the normalizer
in $G$ of $H_b$).
\begin{lemma}\label{l18.1} $L=U(H^{\perp},h_{|H^{\perp}})$ is the
$k$-hermitian factor $G_b^{(1)}=M_b\cdot L_b$ of $P_b$ in the
decomposition of $P_b$ as in Theorem \ref{t4.1}.
\end{lemma}
{\bf Proof:} First observe that $L\subset} \def\nni{\supset} \def\und{\underline P_b$, as $H^{\perp}$ is orthogonal
to the totally isotropic subspace, hence $L$ normalizes $H_b$. Since $L$ is
reductive, there is a Levi decomposition of $P_b$ for which $L$ is
contained in the Levi factor. It is clearly of hermitian type, and maximal
with this property. We must explain why the Levi factor is the standard one
${\cal Z} (A_b)$. But this follows from the fact that $H_b$ is constructed by
means of a basis, which in turn was determined by the choice of ${\Bbb R}} \def\fH{{\Bbb H}$-split
torus $A_b$. It therefore suffices to explain the ``compact'' factor $M_b$.
This factor occurs only in the cases $\bf I_{\bf\scriptstyle p,q}$ and $\bf
IV_{\scriptstyle\bf n}$. We don't have to consider the latter case, as this is
split over ${\Bbb R}} \def\fH{{\Bbb H}$ if rank $\geq 2$. So suppose $G\cong U(V,h)$, where
$(V,h)$ is as in (I) above. We first determine the anisotropic kernel. Let
$H_s$ be a maximal totally isotropic subspace, $S:=H_s\oplus H_s'$ as
above. Then $U(S^{\perp}, h_{|S^{\perp}})$ is the anisotropic kernel,
$U(S^{\perp},h_{|S^{\perp}})({\Bbb R}} \def\fH{{\Bbb H})\cong U(md-2sd)$. In particular, for
$m=2s$, there is no anisotropic kernel. Now consider the group
$L=U(H^{\perp},h_{|H^{\perp}})$. Clearly, for $b<s$, we have
\[U(S^{\perp},h_{|S^{\perp}})\subset} \def\nni{\supset} \def\und{\underline U(H^{\perp},h_{|H^{\perp}})=L,\]
so that $L$ contains the anisotropic kernel. Note that
$SU(H^{\perp},h_{|H^{\perp}})({\Bbb R}} \def\fH{{\Bbb H})\cong L_b({\Bbb R}} \def\fH{{\Bbb H})$, while (if $H^{\perp}\neq
\{0\}$)
\[U(H^{\perp},h_{|H^{\perp}})({\Bbb R}} \def\fH{{\Bbb H})/SU(H^{\perp},h_{|H^{\perp}})({\Bbb R}} \def\fH{{\Bbb H})\cong
M_b({\Bbb R}} \def\fH{{\Bbb H})\cong U(1).\] Here we have used that
$U(H^{\perp},h_{|H^{\perp}})\subset} \def\nni{\supset} \def\und{\underline SU(V,h)$, as it is for the group $SU(V,h)$
(and not for $U(V,h)$) that $M_b({\Bbb R}} \def\fH{{\Bbb H})\cong U(1)$ (see \cite{S}, p.~115).
This verifies the Lemma for the groups of type $\bf I$. \hfill $\Box$ \vskip0.25cm
Now note that $L({\Bbb R}} \def\fH{{\Bbb H})\cong(M_b\cdot L_b)({\Bbb R}} \def\fH{{\Bbb H})=M_b({\Bbb R}} \def\fH{{\Bbb H})L_b({\Bbb R}} \def\fH{{\Bbb H})$, so for the
domain defined by $L$ we have
${\cal D} _L=M_b({\Bbb R}} \def\fH{{\Bbb H})L_b({\Bbb R}} \def\fH{{\Bbb H})/M_b({\Bbb R}} \def\fH{{\Bbb H})K_b=L_b({\Bbb R}} \def\fH{{\Bbb H})/K_b\cong F_b$, hence
${\cal D} _N\cong {\cal D} _{N_b}$ as in Table \ref{T1}. Consider also ${\cal Z} _G(L_b)$ and
${\cal Z} _G(M_bL_b)$; both are defined over ${\Bbb R}} \def\fH{{\Bbb H}$, and clearly
${\cal Z} _{G({\Bbb R}} \def\fH{{\Bbb H})}(L_b({\Bbb R}} \def\fH{{\Bbb H}))/M_b({\Bbb R}} \def\fH{{\Bbb H})\cong {\cal Z} _{G({\Bbb R}} \def\fH{{\Bbb H})}(M_b({\Bbb R}} \def\fH{{\Bbb H})L_b({\Bbb R}} \def\fH{{\Bbb H}))$, so
the group $L\times {\cal Z} _G(L)$ (both these factors being defined over $k$)
is, over ${\Bbb R}} \def\fH{{\Bbb H}$,
\begin{eqnarray}\label{e19.1}
L({\Bbb R}} \def\fH{{\Bbb H})\times {\cal Z} _G(L)({\Bbb R}} \def\fH{{\Bbb H}) & \cong & M_b({\Bbb R}} \def\fH{{\Bbb H})\cdot L_b({\Bbb R}} \def\fH{{\Bbb H})\times
{\cal Z} _{G({\Bbb R}} \def\fH{{\Bbb H})}(M_b({\Bbb R}} \def\fH{{\Bbb H})L_b({\Bbb R}} \def\fH{{\Bbb H})) \\ & \cong & M_b({\Bbb R}} \def\fH{{\Bbb H})\cdot L_b({\Bbb R}} \def\fH{{\Bbb H})\times
{\cal Z} _{G({\Bbb R}} \def\fH{{\Bbb H})}(L_b({\Bbb R}} \def\fH{{\Bbb H}))/M_b({\Bbb R}} \def\fH{{\Bbb H}) \nonumber \\ & \cong & L_b({\Bbb R}} \def\fH{{\Bbb H})\times
{\cal Z} _{G({\Bbb R}} \def\fH{{\Bbb H})}(L_b({\Bbb R}} \def\fH{{\Bbb H})).\nonumber
\end{eqnarray}
This completes the proof of
\begin{proposition}\label{p19.1} The subgroup $N$ of (\ref{e18.1})
satisfies $N({\Bbb R}} \def\fH{{\Bbb H})\cong N_b({\Bbb R}} \def\fH{{\Bbb H})$, the latter group being the standard
symmetric subgroup (\ref{e10.1}) standard incident to $P_b({\Bbb R}} \def\fH{{\Bbb H})$.
\end{proposition}
\begin{corollary}\label{c19.1} The parabolic $P_b$ and the symmetric
subgroup $N$ of (\ref{e18.1}) are incident over $k$, i.e.,
$(P_b({\Bbb R}} \def\fH{{\Bbb H}),N({\Bbb R}} \def\fH{{\Bbb H}))$ are incident in the sense of Definition \ref{d9.1}.
\end{corollary}
Up to this point we have been working with the absolutely simple $k$-group;
we now denote this situation by $G'$ as in section 3.1, and consider
$G=Res_{k|\fQ}G'$. Let again primes in the notations denote subgroups of
$G'$, the unprimed notations for subgroups of $G$. As above we set
$P_{\hbox{\scsi \bf b}}:= Res_{k|\fQ}(P_b')$, and we denote the subgroup $N'$ of
(\ref{e18.1}) henceforth by $N_b'$ and set: $N_{\hbox{\scsi \bf b}}:=Res_{k|\fQ}(N_b')$.
Then Corollary \ref{c19.1} tells us that $(P_b'({\Bbb R}} \def\fH{{\Bbb H}),N'({\Bbb R}} \def\fH{{\Bbb H}))$ are
incident. We now claim
\begin{lemma}\label{l19.1} The $\fQ$-groups $(P_{\hbox{\scsi \bf b}},N_{\hbox{\scsi \bf b}})$ are incident.
\end{lemma}
{\bf Proof:} $P_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})$ is a product $P_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})\cong
P_{b,1}({\Bbb R}} \def\fH{{\Bbb H})\times \cdots \times P_{b,d}({\Bbb R}} \def\fH{{\Bbb H})$ corresponding to
(\ref{e9.1}); by assumption $F_{\hbox{\scsi \bf b}}$ is not zero-dimensional. Hence for
at least one factor $P_{b,\gs}({\Bbb R}} \def\fH{{\Bbb H})$ the incident group
${^{\gs}N}'_b({\Bbb R}} \def\fH{{\Bbb H})\cong{^{\gs}L}'_b({\Bbb R}} \def\fH{{\Bbb H})\times
{{\cal Z} }_{{^{\gs}G}'({\Bbb R}} \def\fH{{\Bbb H})}({^{\gs}L}'_b({\Bbb R}} \def\fH{{\Bbb H}))$ is defined. Consequently
$N_{\hbox{\scsi \bf b}}$ is not trivial, and it is clearly a $\fQ$-form of
$N_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})=\prod_{\gs}{^{\gs}N}'_b({\Bbb R}} \def\fH{{\Bbb H})$. \hfill $\Box$ \vskip0.25cm With Corollary
\ref{c19.1} and Lemma \ref{l19.1}, we have just completed the proof of the
following.
\begin{proposition}\label{p19.2} To each standard maximal $\fQ$-parabolic
$P_{\hbox{\scsi \bf b}}$ of $G$ with $b<s$, there is a symmetric $\fQ$-subgroup
$N_{\hbox{\scsi \bf b}}\subset} \def\nni{\supset} \def\und{\underline G$ such that $(P_{\hbox{\scsi \bf b}},N_{\hbox{\scsi \bf b}})$ are incident.
\end{proposition}
Finally, we remark on what happens for the parabolic corresponding to the
zero-dimensional boundary components. We have, in the notations above,
$H=H_s\oplus H_s'$, and $H^{\perp}$ is anisotropic for $h$. It follows
that the group $L$ of Lemma \ref{l18.1} is anisotropic; its semisimple part
is the semisimple anisotropic kernel of $G'$. If $s={1\over 2}\dim V$, then
$H=V$ already, $H^{\perp}=\{0\}$. Hence the group $N$ of (\ref{e18.2}) is
the whole group ($L=1 \Ra {\cal Z} _G(L)=G$). Otherwise it is of the form
$\{\hbox{anisotropic}\}\times \{\hbox{$k$-split}\}$. We list these in Table
\ref{T3}. Note that the domains occuring have ${\Bbb R}} \def\fH{{\Bbb H}$-rank equal to the
$\fQ$-rank of $G$, suggesting this as a possible modification of the
definition of incident:
\begin{itemize}\item[1')] $N$ has ${\Bbb R}} \def\fH{{\Bbb H}$-rank equal to the $\fQ$-rank of
$G$.
\end{itemize}
Viewing things this way, we see that again indices $C^{(1)}$ represent an
exception; for these 1) and 1') are equivalent.
\begin{table}\caption{\label{T3} $k$-subgroups incident with
zero-dimensional boundary components}
$$\begin{array}{|c|c|c|c|c|} \hline \hbox{Index} & L & {\cal Z} _G(L) &
\hbox{subdomains} & {\cal Z} _G(L)({\Bbb R}} \def\fH{{\Bbb H}) \\ \hline \hline {^2A}^{(d)}_{n,s} &
{^2A}^{(d)}_{n-2ds,0} & {^2A}^{(d)}_{2ds-1,s} & \bf
I_{\hbf{p-ds,q-ds}}\times I_{\hbf{ds,ds}} & SU(ds,ds) \\ \hline
{^1D}^{(2)}_{n,s} & {^1D}^{(2)}_{n-2s,0} & {^1D}^{(2)}_{2s,s} & \bf
II_{\hbf{n-s}}\times II_{\hbf{s}} & SU(2s,\fH)\ (\hbox{$n$ even}) \\
\hline {^2D}^{(2)}_{n,s} & {^2D}^{(2)}_{n-2s,0} & {^2D}^{(2)}_{2s,s} &
\bf II_{\hbf{n-s}}\times II_{\hbf{ s}} & SU(2s,\fH)\ (\hbox{$n$ odd})
\\ \hline C^{(1)} & - & G & - & - \\ \hline C^{(2)}_{n,s} &
C^{(2)}_{n-s,0} & C^{(2)}_{s,s} & \bf III_{\hbf{n-s}}\times
III_{\hbox{\scriptsize\bf s}} & Sp(2s,{\Bbb R}} \def\fH{{\Bbb H}) \\ \hline
\end{array}$$
\end{table}
Let us now see which of the subgroups listed in Table \ref{T2} are defined
over $k$. We use the notations $D,\ V, h$ and $G$ as described above in the
cases (I)-(III).
\begin{itemize}\item[(I)] Again $d$ denotes the degree of $D$. In $U(V,h)$
we have the subgroup $U(V',h_{|V'})$ for any codimension one subspace
$V'\subset} \def\nni{\supset} \def\und{\underline V$. Let $W=(V')^{\perp}$ be the one-dimensional (over $D$)
subspace orthogonal to $V'$. Then $U(W,h_{|W})$ is again a unitary group
whose set of ${\Bbb R}} \def\fH{{\Bbb H}$-points is isomorphic to $U(p_W,q_W)$ for some
$p_W,q_W$. Actually each $h_{\nu}$ for each infinite prime $\nu$ gives an
${\Bbb R}} \def\fH{{\Bbb H}$-group $U(p_{\scriptscriptstyle W,\nu},q_{\scriptscriptstyle W,\nu})$. Let
$(p_{\nu},q_{\nu})$ be the signature of $h_{\nu}$ on $V_{\nu}$. Then
$U(V'_{\nu},{h_{\nu}}_{|V'_{\nu}})\cong U(p_{\nu}-p_{\scriptscriptstyle
W,\nu},q_{\nu}-q_{\scriptscriptstyle W,\nu})$. This gives rise to a product
$N=\prod U(p_{\scriptscriptstyle W,\nu},q_{\scriptscriptstyle W,\nu})\times
U(p_{\nu}-p_{\scriptscriptstyle W,\nu},q_{\nu}-q_{\scriptscriptstyle W,\nu})$, and the factors
of the domain ${\cal D} _N$ are of type $\bf I_{\scriptstyle\bf p_{\hbox{$\scriptscriptstyle
W$},\nu},q_ {\hbox{$\scriptscriptstyle W$},\nu}}\times I_{\scriptstyle\bf
p_{\nu}-p_{\hbox{$\scriptscriptstyle W$},\nu},q_{\nu}-q_{\hbox{$\scriptscriptstyle
W$},\nu}}$. In particular, for $p_{\scriptscriptstyle W,\nu}=0$, this is an
irreducible group of type $\bf I_{\scriptstyle\bf
p_{\nu},q_{\nu}-q_{\hbox{$\scriptscriptstyle W$},\nu}}$ and for $q_{\scriptscriptstyle
W,\nu}=0$, of type ${\bf I}_{\scriptstyle\bf p_{\nu}-p_{\hbox{$\scriptscriptstyle
W$},\nu},q_{\nu}}$. Now since $k$ is the degree of $D$, all of
$p_{\nu}, q_{\nu}, p_{\hbox{$\scriptscriptstyle W$},\nu}, q_{\hbox{$\scriptscriptstyle
W$},\nu}$ are divisible by $d$ and the net subdomains these subgroups
(possibly) define are
\begin{equation}\label{eZZ}
\bf I_{\hbf{p-jd,q}},\quad I_{\hbf{p,q-jd}},\quad I_{\hbf{p-id,q-jd}}
\subset} \def\nni{\supset} \def\und{\underline I_{\hbf{p,q}}, \ \ i,j=1,\ldots, s.
\end{equation}
\item[(II)] In $U(V,h)$ we have as above $U(V',h_{|V'})$; now if $h$ is
non-degenerate on $V'$, then $U(V',h_{|V'})\cong U(n-1,D)$, giving
subgroups of the real groups, defined over $k$, of type $U(n-1,\fH)\subset} \def\nni{\supset} \def\und{\underline
U(n,\fH)$, with a corresponding subdomain of type $\bf II_{\scriptstyle\bf
n-1}\subset} \def\nni{\supset} \def\und{\underline II_{\scriptstyle\bf n}$. This occurs at the primes for which $D$ is
definite; at the others $SU(V',h_{|V'})\subset} \def\nni{\supset} \def\und{\underline SU(V,h)$ is of the type
$SO(2n-4,2)\subset} \def\nni{\supset} \def\und{\underline SO(2n-2,2)$ (for $n$=dimension of $V$ over $D$). So we
have maximal $k$-domains
$$\bf II_{\hbf{n-1}}\subset} \def\nni{\supset} \def\und{\underline II_{\hbf{n}},\ (\nu \hbox{ definite}),\quad
\quad IV_{\hbf{2n-4}}\subset} \def\nni{\supset} \def\und{\underline IV_{\hbf{2n-2}},\ (\nu \hbox{ indefinite}).$$
\item[(III)] The index is $C^{(2)}_{n,s}$; this case in considered in more
detail below.
\end{itemize}
{}From this, we deduce
\begin{proposition}\label{p23} Let $G'$ have $\rank_kG'=s\geq2$, not split
over ${\Bbb R}} \def\fH{{\Bbb H}$, and let $P_s'$ be a standard $k$-parabolic defining a
zero-dimensional boundary component, $P_s'({\Bbb R}} \def\fH{{\Bbb H})=N(F)$, and $\dim(F)=0$.
Then there is a $k$-subgroup $N'$ incident with $P_s'$, with the
following exception: Index $C^{(2)}_{2s,s}$.
\end{proposition}
{\bf Proof:} We first deduce for which of the indices listed in Proposition
\ref{p16.1} zero-dimensional boundary components of ${\cal D} '$ are rational
(this is necessary for the zero-dimensional boundary components of ${\cal D} $ to
be rational). We need not consider exceptional cases or type $\bf
IV_{\hbox{\scsi \bf n}}$. We first consider the groups of type ${^2A}$.
\begin{Lemma}\label{L19A} For $G'$ with the index ${^2A}^{(d)}_{n,s}$, let
$G'({\Bbb R}} \def\fH{{\Bbb H})\cong SU(p,q)$. Then the zero-dimensional boundary components are
rational $\iff$ $sd=q$.
\end{Lemma}
{\bf Proof:} Let $H_s$ be an $s$-dimensional (over $D$) totally isotropic
subspace, with basis $h_1,\ldots,h_s$. Let $h_i'\in V$ be vectors such that
$h(h_i,h_j')=\gd_{ij}$, and set $H_s'=<h_1',\ldots,h_s'>$. Then $h$,
restricted to $H:=H_s\oplus H_s'$, is non-degenerate, and
$SU(H^{\perp},h_{|H^{\perp}})$ is the anisotropic kernel. The group
$SU(H,h_{|H})({\Bbb R}} \def\fH{{\Bbb H}) \cong SU(sd,sd)$, while
$SU(H^{\perp},h_{|H^{\perp}})({\Bbb R}} \def\fH{{\Bbb H})\cong SU(p-ds,q-ds)$. This defines the
subdomain of type $\bf I_{\hbf{ds,ds}}\times I_{\hbf{p-ds,q-ds}}$ of Table
\ref{T1}, hence the boundary component, which is the second factor, is
zero-dimensional $\iff$ $q=ds$. \hfill $\Box$ \vskip0.25cm As to indices of type $D$ we observe
the following.
\begin{Lemma}\label{l5.8.1} $\dim(F)=0$ does not occur for the indices
of type (II) in Proposition \ref{p16.1}.
\end{Lemma}
{\bf Proof:} Recall that $D$ is a quaternion division algebra, central
simple over $k$, with the canonical involution, $V$ is an $n$-dimensional
right $D$-vector space, and $h:V\times V\longrightarrow} \def\sura{\twoheadrightarrow D$ is a skew-hermitian form.
Let $\nu_1,\ldots, \nu_a$ denote the infinite primes for which $D_{\nu}$ is
definite, $\nu_{a+1},\ldots, \nu_d$ the primes at which $D_{\nu}$ is split.
Then $G({\Bbb R}} \def\fH{{\Bbb H})$ is a product
$$(SU(n,\fH))^a\times (SO(2n-2,2))^{d-a},$$ where we have taken into
account that $G$ is assumed to be of hermitian type. At each of the first
factors we have the Satake diagram
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\put(770,765){\line( 1, 0){ 60}} \put(850,765){\line( 1, 0){ 60}}
\put(930,765){\line( 1, 0){ 45}} \put(1105,765){\line( 1, 0){ 70}}
\put(1190,775){\line( 4, 3){ 60}}
\end{picture}$$
\hspace*{3cm} for $n$ odd, \hspace*{6.5cm} for $n$ even.
The corresponding ${\Bbb R}} \def\fH{{\Bbb H}$-root systems are then:
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\put(290,735){\circle*{10}} \put( 45,735){\line( 1, 0){ 90}}
\put(165,735){\line( 1, 0){ 50}} \put(310,735){\line( 1, 0){ 45}}
\put(470,770){\line( 5,-6){ 25}} \put(495,730){\line(-5,-6){ 25}}
\put(375,720){\line( 1, 0){110}} \put(375,750){\line( 1, 0){110}} \put(
30,735){\circle{28}} \put(630,735){\circle{28}} \put(1000,750){\line( 1,
0){110}} \put(750,735){\circle{28}} \put(970,735){\circle{28}}
\put(1110,735){\circle{28}} \put(1125,735){$\eta_t$}
\put(835,735){\circle*{10}} \put(865,735){\circle*{10}}
\put(890,735){\circle*{10}} \put(645,735){\line( 1, 0){ 90}}
\put(765,735){\line( 1, 0){ 50}} \put(910,735){\line( 1, 0){ 45}}
\put(1000,720){\line( 1, 0){110}} \put(985,735){\line( 1,-1){ 35}}
\put(985,735){\line( 1, 1){ 35}}
\end{picture}
In particular, the ${\Bbb R}} \def\fH{{\Bbb H}$-root corresponding to the parabolic $P_{t}$ with
$\dim(F_{t})=0$ is the right-most one. On the other hand, the $k$-index is
\begin{equation}\label{eZ1.1}
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25,740){\circle*{22}} \put(185,740){\circle*{22}}
\put(105,740){\circle{22}} \put(285,740){\circle{22}}
\put(540,740){\circle{22}} \put(625,740){\circle*{22}}
\put(740,740){\circle*{10}} \put(760,740){\circle*{10}}
\put(720,740){\circle*{10}} \put(825,740){\circle*{22}}
\put(395,740){\circle*{10}} \put(915,740){\circle*{22}}
\put(921,732){\line( 4,-3){ 60}} \put(990,680){\circle*{20}}
\put(990,800){\circle*{22}} \put( 35,740){\line( 1, 0){ 60}}
\put(115,740){\line( 1, 0){ 60}} \put(195,740){\line( 1, 0){ 80}}
\put(295,740){\line( 1, 0){ 75}} \put(500,740){\line( 1, 0){ 30}}
\put(550,740){\line( 1, 0){ 65}} \put(635,740){\line( 1, 0){ 70}}
\put(705,740){\line(-1, 0){ 5}} \put(785,740){\line( 1, 0){ 40}}
\put(835,740){\line( 1, 0){ 70}} \put(920,750){\line( 4, 3){ 60}}
\end{picture}
\end{equation}
with the $k$-root system
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\put(300,745){\circle*{10}} \put(325,745){\circle*{10}} \put(
80,745){\line( 1, 0){ 90}} \put(200,745){\line( 1, 0){ 50}}
\put(345,745){\line( 1, 0){ 45}} \put( 65,745){\circle{28}} \put(
60,780){$\eta_1$} \put(435,730){\line( 1, 0){110}} \put(415,660){\line(
1, 0){110}} \put(420,745){\line( 1,-1){ 35}} \put(420,745){\line( 1,
1){ 35}} \put(435,760){\line( 1, 0){110}} \put(200,645){(respectively)}
\put(410,645){\circle{28}} \put(550,645){\circle{28}}
\put(510,680){\line( 5,-6){ 25}} \put(535,640){\line(-5,-6){ 25}}
\put(415,630){\line( 1, 0){110}}
\end{picture}$$
from which it is evident that $P_{t}$ is defined over $k$ $\iff$ $s=t$
($=[{n\over 2}]$). But this is the split over ${\Bbb R}} \def\fH{{\Bbb H}$ case. Consequently,
$a=0$ and $D$ is totally indefinite.
So we consider a prime $\nu$ where $D_{\nu}$ is split; the ${\Bbb R}} \def\fH{{\Bbb H}$-index is
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\put(579,701){\circle*{20}} \put( 80,760){\circle{22}}
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\put(240,760){\circle*{22}} \put(160,680){\circle{22}}
\put(120,655){\line( 4, 3){ 30}} \put(80,650){$\eta_2$} \put(
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85,690){\line( 1, 0){ 55}} \put( 85,670){\line( 1, 0){ 55}}
\put(120,705){\line( 4,-3){ 30}} \put(160,650){$\eta_1$}
\end{picture}
$$ the ${\Bbb R}} \def\fH{{\Bbb H}$-root $\eta_2$ corresponding to the two-dimensional totally
isotropic subspace and zero-dimensional boundary component. The $k$-index
is as in (\ref{eZ1.1}), so $\eta_2$ is always anisotropic; the boundary
components are actually one-dimensional. This verifies the statements of
the lemma. \hfill $\Box$ \vskip0.25cm Note that this proves Proposition \ref{p23} for the
indices of type (II).
Now consider index $C^{(2)}_{n,s}$. The $k$-index is
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35,740){\line( 1, 0){ 60}} \put(115,740){\line( 1, 0){ 60}}
\put(195,740){\line( 1, 0){ 80}} \put(295,740){\line( 1, 0){ 75}}
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\put(665,740){\line( 1, 0){ 70}} \put(735,740){\line(-1, 0){ 5}}
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and the $k$-root system is
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\put(300,745){\circle*{10}} \put(325,745){\circle*{10}} \put(
65,745){\circle{28}} \put( 80,745){\line( 1, 0){ 90}}
\put(435,760){\line( 1, 0){110}} \put(200,745){\line( 1, 0){ 50}}
\put(345,745){\line( 1, 0){ 45}} \put(435,730){\line( 1, 0){110}}
\put(420,745){\line( 1,-1){ 35}} \put(420,745){\line( 1, 1){ 35}}
\end{picture}$$
The same reasoning as above shows that $F_{t}$ is rational $\iff$ $2s=t$,
but that is only possible if the index is $C^{(2)}_{2n,n}$. Hence:
\begin{Lemma}\label{l5.8.2}
The only indices of Proposition \ref{p16.1}, case (III), for which
zero-dimensional boundary components occur are $C^{(2)}_{2n,n}$.
\end{Lemma}
This index is that of the unitary group $U(V,h)$, where $V$ is a
$2n$-dimensional vector space over $D$, and $h$ has Witt index $n$. We can
find $n$ hyperbolic planes $V_i$ such that
\[V=V_1\oplus\cdots \oplus V_n.\]
This decomposition is defined over $k$, hence the subgroup
\[N=U(V_1,h_{|V_1})\times \cdots \times U(V_n,h_{|V_n}),\]
which is a product of groups with index $C^{(2)}_{2,1}$, is also defined
over $k$. We have
\begin{equation}\label{E20}N({\Bbb R}} \def\fH{{\Bbb H})\cong \underbrace{Sp(4,{\Bbb R}} \def\fH{{\Bbb H})\times \cdots
\times Sp(4,{\Bbb R}} \def\fH{{\Bbb H})}_{n\ \hbox{\scriptsize times}}
\end{equation}
and the domain ${\cal D} _N$ is of type $\bf (III_{\hbox{\scriptsize\bf 2}})^n$. This
is the exception in the statement of the main theorem.
\vspace*{.3cm}
\noindent{\bf Proof of Proposition \ref{p23}:} We have already completed
the proof for (II) and (III), and as we mentioned above, the exceptional
cases and (IV) need not be considered. It remains to show the existence of
groups of the stated types for indices ${^2A}$. We explained above how one
can find $k$-subgroups $N$ such that ${\cal D} _N$ has irreducible components of
types $\bf I_{\hbox{\scriptsize\bf p-jd,q}}$ (see (\ref{eZZ})). Here we take a
maximal totally isotropic subspace $H_s$, and $H:=H_s\oplus H_s'$ as
described there. Let $H^{\perp}$ denote the orthogonal complement, so that
$SU(H^{\perp},h_{|H^{\perp}})$ is the anisotropic kernel. Then, if
$G'({\Bbb R}} \def\fH{{\Bbb H})=SU(p,q)$, we have
\[SU(H,h_{|H})({\Bbb R}} \def\fH{{\Bbb H})\cong SU(sd,sd),\quad SU(H^{\perp},h_{|H^{\perp}})\cong
SU(p-sd,q-sd).\] Therefore we get a subdomain of type \[\bf
I_{\hbf{sd,sd}}\times I_{\hbf{p-sd,q-sd}},\] which is irreducible $\iff$
$sd=q$; Then $N=\{g\in G\Big| g(H)\subseteq H\}$ is a $k$-subgroup with
$N({\Bbb R}} \def\fH{{\Bbb H})\sim SU(q,q)\times\{\hbox{compact}\}$, and $N$ then fulfills 1), 2')
and 3'). By Lemma \ref{L19A}, this holds precisely when the boundary
component $F_s$ is a point. This completes the proof if $p>q$. It remains
to consider the case where ${\cal D} '$ is of type $\bf I_{\hbf{q,q}}$. In this
case, $q=d\cdot j$ for some $j$, and the hermitian form $h:V\times V\longrightarrow} \def\sura{\twoheadrightarrow D$
has Witt index $j$. The vector space $V$ is then $2j$-dimensional, and it
is the orthogonal direct sum of hyperbolic planes, $V=V_1\oplus \cdots
\oplus V_j$, $\dim_DV_i=2$. Consider the $k$-subgroup
\[ N=\{g\in GL_D(V) \Big| g(V_i)\subset} \def\nni{\supset} \def\und{\underline V_i, i=1,\ldots,j\}.\]
Clearly $N\cong N_1\times \cdots \times N_j$, and each $N_i$ is a subgroup
of rank one with index ${^2A}^{(d)}_{2d-1,1}$. As was shown in \cite{hyp},
in each $N_i$ we have a $k$-subgroup $N_i'\subset} \def\nni{\supset} \def\und{\underline N_i$, with ${\cal D} _{N_i'}$ of
type $({\bf I_{\hbf{1,1}}})^d$. Then
\[ N':=N_1'\times \cdots \times N_j'\]
is a $k$-subgroup with ${\cal D} _{N'}$ of type $(({\bf I_{\hbf{1,1}}})^d)^j=
({\bf I_{\hbf{1,1}}})^{d\cdot j}= ({\bf I_{\hbf{1,1}}})^q$, which is a
maximal polydisc, i.e., satisfies 1), 2'') and 3''). This completes the
proof of Proposition \ref{p23} in this case also. \hfill $\Box$ \vskip0.25cm
\section{Rank one}
We now come to the most interesting and challenging case. In this last
paragraph $G'$ will denote an absolutely simple $k$-group, $G$ the
corresponding $\fQ$-simple group, both assumed to have rank one. There is
only one standard maximal parabolic $P_1'\subset} \def\nni{\supset} \def\und{\underline G'$ in this case, so we may
delete the subscript $_1$ in the notations. Let $P\subset} \def\nni{\supset} \def\und{\underline G$ be the
corresponding $\fQ$-parabolic, so $P({\Bbb R}} \def\fH{{\Bbb H})=P_1({\Bbb R}} \def\fH{{\Bbb H})\times \cdots \times
P_d({\Bbb R}} \def\fH{{\Bbb H})$, where $P_{\nu}({\Bbb R}} \def\fH{{\Bbb H})\subset} \def\nni{\supset} \def\und{\underline {^{\gs_{\nu}}G}'({\Bbb R}} \def\fH{{\Bbb H})$ is a standard
maximal parabolic, say $P_{\nu}({\Bbb R}} \def\fH{{\Bbb H})=N(F_{b_{\nu}}),\ F_{b_{\nu}}\subset} \def\nni{\supset} \def\und{\underline
\overline{{\cal D} }_{\gs_{\nu}}$. As we observed above, the $F_{b_{\nu}}$ are
all hermitian spaces whose automorphism group is an ${\Bbb R}} \def\fH{{\Bbb H}$-form of some
fixed algebraic group. As we are now assuming the rank to be one, it
follows from Lemma \ref{L12a} that $L$ (=$L_{\hbox{\scsi \bf b}}$ in the notations above)
is anisotropic. One way that this may occur was explained there, namely
that if one of the factors $F_{b_{\nu}}$ is a {\it point}, in which case
the symmetric space of $L({\Bbb R}} \def\fH{{\Bbb H})$ has a compact factor. Another possibility
is that all $F_{b_{\nu}}$ are positive-dimensional, in which case $L$ is a
``genuine'' anisotropic group. The type of $F_{b_{\nu}}$ can be determined
from the $k$-index of $G'$ and the ${\Bbb R}} \def\fH{{\Bbb H}$-index of ${^{\gs}_{\nu}G}'$. For
example, for $G'$ of type ${^2A}$, these indices are:
\vspace*{.5cm}
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\put(105,230){$\underbrace{\hspace*{10.5cm}}_{\hbox{$q_{\nu}$
vertices}}$} \put(600,130){The ${\Bbb R}} \def\fH{{\Bbb H}$-index of
${^{\gs_{\nu}}G}'({\Bbb R}} \def\fH{{\Bbb H})$}
\end{picture}
\vspace*{1.5cm} From this we see that the boundary component is of type
$\bf I_{\hbf{p$_{\nu}$-d,q$_{\nu}$-d}}$.
There are basically two quite different cases at hand; the first is that
the boundary components are positive-dimensional, the second occurs when
the boundary components reduce to points. The former can be easily handled
with the same methods as above, by splitting off orthogonal complements.
The real interest is in the latter case, and here a basic role is played by
the {\it hyperbolic planes}, which have been dealt with in detail in
\cite{hyp}. We will essentially reduce the rank one case (at least for the
classical groups) to the case of hyperbolic planes, then we explain how the
results of \cite{hyp} apply to the situation here.
\subsection{Positive-dimensional boundary components}
Let $G', P'$ be as above, and consider the hermitian Levi factor
${G'}^{(1)}=M'L'$, which is defined over $k$. Over ${\Bbb R}} \def\fH{{\Bbb H}$ the factors
$M'({\Bbb R}} \def\fH{{\Bbb H})$ and $L'({\Bbb R}} \def\fH{{\Bbb H})$ are defined. In this section we consider the
situation that the boundary component $F'$ of ${\cal D} '$ defined by $P'$ (i.e.,
$P'({\Bbb R}} \def\fH{{\Bbb H})=N(F')$) is positive-dimensional, or equivalently, that the
hermitian Levi
factor $L'({\Bbb R}} \def\fH{{\Bbb H})$ is non-trivial. As above, we get the following $k$-group
\begin{equation}\label{E23}
N':={G'}^{(1)}\times {\cal Z} _{G'}({G'}^{(1)}).
\end{equation}
The same calculation as in ({\ref{e19.1}) shows that the domain ${\cal D} _{N'}$
defined by $N'$ is the same as that defined by $L'({\Bbb R}} \def\fH{{\Bbb H})\times
{\cal Z} _{G'({\Bbb R}} \def\fH{{\Bbb H})}(L'({\Bbb R}} \def\fH{{\Bbb H}))$. Taking the subgroup $N=Res_{k|\fQ}N'$ defines a
subdomain ${\cal D} _N\subset} \def\nni{\supset} \def\und{\underline {\cal D} $, which is a product ${\cal D} _N={\cal D} _{N,\gs_1}\times \cdots
\times {\cal D} _{N,\gs_f}$. Each factor ${\cal D} _{N,\gs}$ is determined by the
corresponding factor of ${^{\gs}L}'({\Bbb R}} \def\fH{{\Bbb H})$. The ${\Bbb R}} \def\fH{{\Bbb H}$-groups $N'({\Bbb R}} \def\fH{{\Bbb H})$ and
$N({\Bbb R}} \def\fH{{\Bbb H})$ are determined in terms of the data $D,V,h$ as follows.
\begin{itemize}\item[(I)] If $F'\cong {\bf I_{\hbox{\scriptsize\bf p-d,q-d}}}$,
then ${\cal D} _{N'}\cong {\bf I_{\hbox{\scriptsize\bf p-d,q-d}}}\times {\bf
I_{\hbox{\scriptsize\bf d,d}}}$. Note that in terms of the hermitian forms,
this amounts to the following. Since $h$ has Witt index 1, the maximal
totally isotropic subspaces are one-dimensional. Let $H_1=<v>$ be such a
space; we can find a vector $v'\in V$ such that $H=<v,v'>$ is a
hyperbolic plane, that is, $h_{|H}$ has Witt index 1. It follows that
$h_{|H^{\perp}}$ is anisotropic. Consider the subgroup
\begin{equation}\label{e22.0} N_k:=\{g\in U(V,h) | g(H)\subset} \def\nni{\supset} \def\und{\underline H\}.
\end{equation}
It is clear that for $g\in N_k$, it automatically holds that
$g(H^{\perp})\subset} \def\nni{\supset} \def\und{\underline H^{\perp}$, hence
\begin{equation}\label{e22.1} N_k\cong U(H,h_{|H})\times
U(H^{\perp},h_{|H^{\perp}}).
\end{equation}
The first factor has ${\Bbb R}} \def\fH{{\Bbb H}$-points $U(H,h_{|H})({\Bbb R}} \def\fH{{\Bbb H})\cong U(d,d)$, while the
second fulfills $U(H^{\perp},h_{|H^{\perp}})({\Bbb R}} \def\fH{{\Bbb H})\cong U(p-d,q-d)$. Thus
$N_k\cong N'$ as in (\ref{E23}). At any rate, this gives us subdomains of
type
$${\bf I_{\hbox{\scriptsize\bf d,d}}\times I_{\hbox{\scriptsize\bf p-d,q-d}}\subset} \def\nni{\supset} \def\und{\underline }
{\cal D} _{N'},$$
which, in case $d=p=q$ is the whole domain; in all other cases it is a
genuine subdomain as listed in Table \ref{T1}, defined over $k$, and
$(N',P')$ are incident. It follows from this that $(N,P)$ are incident
over $\fQ$. The components $N_{\gs}({\Bbb R}} \def\fH{{\Bbb H})$ of $N_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})$ are determined
as follows. Let $(p_{\nu},q_{\nu})$ be the signature of $h_{\nu}$ (so that
$p_{\nu}+q_{\nu}=dm$ for all $\nu$). This implies
\[ G({\Bbb R}} \def\fH{{\Bbb H})\cong \prod_{\nu}SU(p_{\nu},q_{\nu}).\]
For each factor, we have the boundary component $F_{\gs}\cong SU(p_{\nu}-d,
q_{\nu}-d)/K$, and for each factor for which $q_{\nu}>d$ this is
positive-dimensional. As above, this leads to subdomains, in each factor,
of type $\bf I_{\hbf{d,d}}\times I_{\hbf{p$_{\nu}$-d,q$_{\nu}$-d}}$, so that in
sum
\begin{equation}\label{E22a} {\cal D} _N\cong \prod_{\nu}{\cal D} _{\nu},\quad
\hbox{${\cal D} _{\nu}$ of type $\bf I_{\hbf{d,d}}\times
I_{\hbf{p$_{\nu}$-d,q$_{\nu}$-d}}$}.
\end{equation}
\item[(II)] Here rank 1 means we have the following $k$-index, $D^{(2)}_{n,1}$
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In particular, the boundary component is of type $\bf II_{\hbox{\scriptsize\bf
n-2}}$ if ${\cal D} '$ is of type $\bf II_{\hbox{\scriptsize\bf n}}$. This means
also that the ``mixed cases'' only can occur if ${\cal D} '$ is of type $\bf
II_{\hbox{\scriptsize\bf 4}}$, for then $\bf II_{\hbox{\scriptsize\bf 2}}\cong$
one-dimensional disc. Of course $\bf II_{\hbox{\scriptsize\bf
4}}\cong IV_{\hbox{\scriptsize\bf 6}}$ anyway, so we can
conclude from this that mixed cases do not occur in the hermitian symmetric
setting (for $\fQ$-simple $G$ of rank 1).
The domain ${\cal D} _{N'}$ defined by $N'$ is of
type $\bf II_{\hbox{\scriptsize\bf n-2}}\times II_{\hbox{\scriptsize\bf 2}}$. The
components $N_{\gs}({\Bbb R}} \def\fH{{\Bbb H})$ of $N_{\hbf{1}}({\Bbb R}} \def\fH{{\Bbb H})$ are all of type
$U(n-2,\fH)\times U(2,\fH)\subset} \def\nni{\supset} \def\und{\underline U(n,\fH)$, so the domain ${\cal D} _N$ is ot
type
\begin{equation}\label{E22a.1} {\bf (II_{\hbf{n-2}}\times II_{\hbf{2}})}^f.
\end{equation}
\item[(III)] Here rank 1 implies the index is one of $C^{(1)}_{1,1}$ (which
we have excluded) or $C^{(2)}_{n,1}$. The corresponding
boundary components in these cases are of type $\bf III_{\hbf{n-2}}$.
The case $C^{(2)}_{2,1}$, for which the boundary component is a point,
will be
dealt with later, the others give rise to a subdomain of type $\bf
III_{\hbox{\scriptsize\bf 2}}\times III_{\hbox{\scriptsize\bf n-2}}$. Consequently,
${\cal D} _N$ is of type ${\bf (III_{\hbf{n-2}}\times III_{\hbf{2}})}^f$,
$f=[k:\fQ]$.
\item[(IV)] Here we just have a symmetric bilinear form of Witt index
1. The $k$-index in this case is necessarily of the form
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The corresponding boundary component is a point, a case to be
considered below. Splitting off
an anisotropic vector (defined over $k$)
in this case yields a codimension one
subspace $H^{\perp}$ on which $h$ still has Witt index 1, hence the
stabilizer $N'$ defines a subdomain ${\cal D} _{N'}$ of type $\bf
IV_{\hbox{\scriptsize\bf n-1}}$. ${\cal D} _N$ is then of type ${(\bf
IV_{\hbf{n-1}})}^f$.
\item[(V)] The only index of rank 1 is
$$
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The vertex denoted $\ga_2$ gives rise to the five-dimensional boundary
component. If $\gd$ denotes the lowest root, then, as is well known, $\gd$
is isotropic (does not map to zero in the $k$-root system), so the root
$\gd$ defines a $k$-subalgebra
$\frak n} \def\rr{\frak r_{\gd}:=\Gg^{\gd}+\Gg^{-\gd}+[\Gg^{\gd},\Gg^{-\gd}]\subset} \def\nni{\supset} \def\und{\underline \Gg'$ which is
split over $k$. On the other hand the anisotropic kernel ${\cal K} $ is of type
${^2A}_5$, and ${\cal K} ({\Bbb R}} \def\fH{{\Bbb H})\cong U(5,1)$. Clearly ${\cal K} $ and the $k$-subgroup
$N_{\gd}$ defined by $\frak n} \def\rr{\frak r_{\gd}$ are orthogonal, so we get a $k$-subgroup
\[N'=N_{\gd}\times {\cal K} ,\]
both factors being defined over $k$. The set of ${\Bbb R}} \def\fH{{\Bbb H}$-points is then of type
$N'({\Bbb R}} \def\fH{{\Bbb H})\cong SL_2({\Bbb R}} \def\fH{{\Bbb H})\times SU(5,1)$, and the subdomain ${\cal D} _{N'}$ is
\[{\cal D} _{N'}\cong {\bf I_{\hbox{\scriptsize\bf 1,1}}\times I_{\hbox{\scriptsize\bf
5,1}}}.\]
This is one of the domains listed in Table \ref{T1}, incident to the
five-dimensional boundary component. It follows that ${\cal D} _N$ is a product
of factors of this type.
\item[(VI)] There are no indices of hermitian type with rank one for $E_7$.
\end{itemize}
We sum up these results in the following.
\begin{proposition}\label{p23.1} If the rational boundary components for
$G'$ are positive-dimensional, then Theorem \ref{t12.1} holds for
$G$. The subdomains defined by the symmetric subgroups $N'\subset} \def\nni{\supset} \def\und{\underline G'$ are:
\begin{itemize}\item[(I)] $\bf I_{\hbox{\scriptsize\bf d,d}}\times
I_{\hbox{\scriptsize\bf p-d,q-d}}$.
\item[(II)] $\bf II_{\hbox{\scriptsize\bf n-2}}\times II_{\hbox{\scriptsize\bf 2}}$.
\item[(III)] $\bf III_{\hbox{\scriptsize\bf n-2}}\times III_{\hbox{\scriptsize\bf 2}}$.
\item[(IV)] $\bf IV_{\hbox{\scriptsize\bf n-1}}$ (here there are no
positive-dimensional boundary components).
\item[(V)] $\bf I_{\hbox{\scriptsize\bf 1,1}}\times I_{\hbox{\scriptsize\bf 5,1}}$.
\end{itemize}
Note here $\bf I_{\hbox{\scriptsize\bf 1,1}}\cong II_{\hbox{\scriptsize\bf 2}} \cong
III_{\hbox{\scriptsize\bf 1}}\cong IV_{\hbox{\scriptsize\bf 1}}$.
The corresponding domains ${\cal D} _N$ in ${\cal D} $ defined by the subgroups $N$ are
products of domains of the types listed above.
\end{proposition}
\subsection{Zero-dimensional boundary components}
The restrictions rank equal to one and zero-dimensional boundary componants
are only possible for the domains of type $\bf I_{\hbox{\scriptsize\bf p,q}},\
III_{\hbox{\scriptsize\bf 2}}$ and $\bf IV_{\hbox{\scriptsize\bf n}}$ (see Lemmas
\ref{l5.8.1} and \ref{l5.8.2}). Of these, the
last case requires no further discussion: as above we find a codimension
one $k$-subspace $V'\subset} \def\nni{\supset} \def\und{\underline V$, on which $h$ still is isotropic, and take its
stabilizer as $N'$. This gives a $k$-subgroup $N'\subset} \def\nni{\supset} \def\und{\underline G'$, and defines a
subdomain ${\cal D} _{N'}$ of type $\bf IV_{\hbox{\scriptsize\bf n-1}}$. In the
${^2A}^{(d)}$ case we may assume $d\geq 3$: the $d=1$ case is again easily
dealt with as above. We have a $K$-vector space $V$ ($K|k$ imaginary
quadratic) of dimension $p+q$ and a ($K$-valued) hermitian form $h$ of Witt
index 1 on $V$. By taking a $K$-subspace $V'\subset} \def\nni{\supset} \def\und{\underline V$ of codimension one,
such that $h_{|V'}$ still has Witt index 1, we get the $k$-subgroup $N'$ as
the stabilizer of $V'$. Then the domain ${\cal D} _{N'}$ is either of type $\bf
I_{\hbox{\scriptsize\bf p-1,q}}$ or $\bf I_{\hbox{\scriptsize\bf p,q-1}}$, and by
judicious choice of $V'$ we can assume the first case, which is the domain
listed in Table \ref{T2}. The $d=2$ case is ``lifted'' from the
corresponding $d=2$ case with involution of the first kind: if $D$ is
central simple of degree 2 over $K$ with a $K|k$-involution, then
(\cite{A}, Thm.~10.21) $D=D'\otimes_kK$, where $D'$ is central simple of
degree 2 over $k$ with the canonical involution. Consequently,
$$U(V,h)=U(V'\otimes_kK,h'\otimes_kK)=U(V',h')_K,$$
the group is just the group $U(V',h')$ lifted to $K$. Since $U(V',h')$ has
index $C^{(2)}_{n,1}$, while $U(V',h')_K$ has index $A^{(2)}_{2n-1,1}$, it
follows that the boundary component is a point only if $n\leq 2$. This
implies that if $d=2$, the index is $A^{(2)}_{3,1}$, the domain is $\bf
I_{\hbf{2,2}}\cong IV_{\hbf{4}}$, so $U(V',h')_K$ is isomorphic to an
orthogonal group over $k$ in six variables. As we just saw, in this case
there is a subdomain defined over $k$ of type $\bf IV_{\hbf3}\subset} \def\nni{\supset} \def\und{\underline
IV_{\hbf4}$. So we assume $d\geq 3$. Then, as we have seen, the
boundary component
$F'\cong \bf I_{\hbox{\scriptsize\bf p-d,q-d}}$ will be
zero-dimensional $\iff$ $q=d$ (respectively $F\cong {\bf
I_{\hbf{p$_1$-d,q$_1$-d}}\times \cdots \times I_{\hbf{p$_f$-d,q$_f$-d}}}$
will be
zero-dimensional $\iff$ $q_{\nu}=d,\ \forall_{\nu}$.
Here there are two possibilities:
\begin{itemize}\item[1)] $p=q=d$, the group $N_k$ of (\ref{e22.0}) is
$N_k\cong G'$. This is the case of {\it hyperbolic planes}.
\item[2)] $p>q=d$, the group $N_k$ of (\ref{e22.0}) is over ${\Bbb R}} \def\fH{{\Bbb H}$ just
$N_k({\Bbb R}} \def\fH{{\Bbb H}) = U(d,d)\times U(p-d)\subset} \def\nni{\supset} \def\und{\underline U(p,d)\cong G'({\Bbb R}} \def\fH{{\Bbb H})$.
\end{itemize}
Note that in the second case the domain ${\cal D} _{N_k}$ defined by $N_k$ is of
type $\bf I_{\hbox{\scriptsize\bf d,d}}$, a maximal tube domain in $\bf
I_{\hbox{\scriptsize\bf p,q}}$. So we are also finished in this case. For
completeness, let us quickly go through the details to make sure
nothing unexpected happens.
\begin{proposition}\label{p24.1} Let $G'$ have index ${^2A}^{(d)}_{n,1},\
d=q,\ p>q$, $n+1=p+q$,
and let $P'$ denote the corresponding standard parabolic and
$N'= N_k$, where $N_k\subset} \def\nni{\supset} \def\und{\underline G'$ the symmetric subgroup defined in
(\ref{e22.0}), where $H$
is the hyperbolic plane spanned by the vector which is stabilized by $P'$
and its ortho-complement ($v'$: $h(v,v')=1$). Then $(P',N')$ are
incident, in fact standard incident.
Consequently, $P=Res_{k|\fQ}P'$ and $N=Res_{k|\fQ}N'$ are
incident over $\fQ$.
\end{proposition}
{\bf Proof:} We know that $N'({\Bbb R}} \def\fH{{\Bbb H})\cong U(q,q)\times U(p-q)$
which gives rise to the
maximal tube subdomain $\bf I_{\hbox{\scriptsize\bf q,q}}\subset} \def\nni{\supset} \def\und{\underline I_{\hbox{\scriptsize\bf
p,q}}$ of Table \ref{T2}. We need to check that the standard boundary
component $F'$ stabilized by $P'({\Bbb R}} \def\fH{{\Bbb H})$ is also a standard boundary
component of ${\cal D} _{N'}$; in particular we need the common maximal
${\Bbb R}} \def\fH{{\Bbb H}$-split torus in $P'$ and $N'$. This is seen in (\ref{e22.0}), the
${\Bbb R}} \def\fH{{\Bbb H}$-split torus being contained in the hermitian Levi factor of
$P'({\Bbb R}} \def\fH{{\Bbb H})$, which is contained in $N'({\Bbb R}} \def\fH{{\Bbb H})$.
Consider the group $P'\cap N'$; this
is nothing but the stabilizer of $v$ in $H$, which is a maximal standard
parabolic in $N'$. Since $\nu$ determines the boundary component $F$, both
in $G'({\Bbb R}} \def\fH{{\Bbb H})$ and in $N'({\Bbb R}} \def\fH{{\Bbb H})$, it is clear that $F$ is a boundary component
of ${\cal D} _{N'}$. It follows that $(P',N')$
are incident, and this implies (see the discussion preceeding Proposition
\ref{p19.2}) that $(P,N)\subset} \def\nni{\supset} \def\und{\underline G$ are incident. \hfill $\Box$ \vskip0.25cm
We are left with the following cases: $\bf III_{\hbox{\scriptsize\bf 2}}$ with
index $C^{(2)}_{2,1}$ and $\bf I_{\hbox{\scriptsize\bf q,q}}$ with index
${^2A}^{(d)}_{2d-1,1},\ d\geq3$.
These indices are described in terms of hermitian
forms as follows. Let $D$ be a central simple division algebra
over $K$ ($K=k$ for $d=2$ and $K|k$ is imaginary quadratic if $d\geq3
$) and assume further that $D$ has a $K|k$-involution, $V$ is a
two-dimensional right vector space over $D$ and $h:V\times V\longrightarrow} \def\sura{\twoheadrightarrow D$ is a
hermitian form which is isotropic. Then $d=2$ gives groups with index
$C^{(2)}_{2,1}$, and $d\geq 3$ gives groups with indices
${^2A}^{(d)}_{2d-1,1}$.
\begin{lemma}\label{l25.1} There exists a basis $v_1,v_2$ of $V$ over $D$
such that the form $h$ is given by $h({\bf x},{\bf
y})=x_1\overline{y}_2+x_2\overline{y}_1,\ {\bf x}=(x_1,x_2),\ {\bf
y}=(y_1,y_2)$.
\end{lemma}
{\bf Proof:} Let $v$ be an isotropic vector, defined over $k$. Then there
exists an isotropic
vector $v'$, such that $h(v,v')=1$, hence also
$h(v',v)=1$. Let ${v'}=({v'}_1,{v'}_2)$, and set
$\gd={v'}_1\overline{v'}_2$, so that
$h(v',v')=\gd+\overline{\gd}$. Then the matrix of $h$ with respect to
the basis $v,v'$ is $H'={0\ 1\choose 1\ \ge}$, where
$\ge=\gd+\overline{\gd}$. Now setting
$$w=(w_1,w_2)=(-v_1\overline{\gd}+v_1',-v_2\overline{\gd}+v_2')$$
we can easily verify $h(w,w)=0,\ \
h(v,w)=h(w,v)=1$. Since the change of basis transformation is defined over
$k$, the matrix of the hermitian form with respect
to this $k$-basis $v,w$ is $H={0\ 1\choose 1\ 0}$. \hfill $\Box$ \vskip0.25cm
So as far as the $\fQ$-groups are concerned, we may take the standard
hyperbolic form given by the matrix $H$ as defining the hermitian form on
$V$. We remark that the situation changes when one considers arithmetic
groups, but that need not concern us here. At any rate, a two-dimensional
right $D$-vector space $V$ with a hermitian form as in Lemma \ref{l25.1} is
what we call a {\it hyperbolic plane}, and this case was studied in detail
in \cite{hyp}. There it was determined exactly what kind of
symmetric subgroups
exist. These derive from the existence of splitting subfields $L\subset} \def\nni{\supset} \def\und{\underline D$,
which may be taken to be cyclic of degree $d$ over $K$, if $D$ is central
simple of degree $d$ over $K$. In fact, we have subgroups (\cite{hyp},
Proposition 2.4) $U(L^2,h)\subset} \def\nni{\supset} \def\und{\underline U(D^2,h)$, which give rise to the following
subdomains:
\begin{itemize}
\item[1)] $d=2$; ${\cal D} _L\cong \left(\begin{array}{cc}\tau_1 & 0 \\ 0 &
b^{\zeta_1}\tau_1
\end{array}\right)\times \cdots \times \left(\begin{array}{cc}\tau_1 & 0 \\ 0 &
b^{\zeta_f}\tau_1
\end{array}\right)$, where $\zeta_i:k\longrightarrow} \def\sura{\twoheadrightarrow {\Bbb R}$ denote the distinct real
embeddings of $k$.
\item[2)] $d\geq 3$; ${\cal D} _L\cong \left(\begin{array}{ccc}\tau_1 & & 0 \\ &
\ddots & \\ 0 & & \tau_d\end{array}\right)^f$.
\end{itemize}
In other words, for hyperbolic planes we find subdomains of the following
kinds
\begin{equation}\label{E25}
\bf III_{\hbox{\scriptsize\bf 1}}\subset} \def\nni{\supset} \def\und{\underline III_{\hbox{\scriptsize\bf 2}},\quad
(I_{\hbox{\scriptsize\bf 1,1}})^d\subset} \def\nni{\supset} \def\und{\underline I_{\hbox{\scriptsize\bf d,d}}.
\end{equation}
The latter one is a polydisc, coming from a maximal set of strongly orthogonal
roots, i.e., satisfying 1), 2'') and 3'').
The first case is the only exception to the rule that we have
symmetric subgroups $N'\subset} \def\nni{\supset} \def\und{\underline G'$ with $\rank_{{\Bbb R}} \def\fH{{\Bbb H}}N'=\rank_{{\Bbb R}} \def\fH{{\Bbb H}}G'$.
\vspace*{.2cm}
\noindent{\bf Proof of Theorem \ref{t12.1}:} We have split the set of cases
up into the three considered in \S4, 5 and 6. Corollary \ref{c14.1} proves
\ref{t12.1} for the split over ${\Bbb R}} \def\fH{{\Bbb H}$ case and Proposition \ref{p19.2} for
the rank $\geq 2$ case and positive-dimensional boundary components.
For $\rank \geq 2$ and zero-dimensional boundary components, Proposition
\ref{p23} shows that with the exception given Theorem \ref{t12.1}
holds in this case also. In the case of rank 1, Proposition \ref{p23.1}
verifies \ref{t12.1} for the case that the boundary components are
positive-dimensional, and Proposition \ref{p24.1} took care of the rest of
the cases excepting hyperbolic planes. Then the results of \cite{hyp}
verify \ref{t12.1} for hyperbolic planes, thus completing the proof. \hfill $\Box$ \vskip0.25cm
\vspace*{.2cm}
\noindent{\bf Proof of the Main Theorem:} The first statement is covered by
Theorem \ref{t12.1}. The statements on the domains for the exceptions
follow from (\ref{E20}) and (\ref{E25}). It remains to consider the
condition 4). This is fulfilled for the groups $N$ utilized above
by construction. For the
exceptional cases this is immediate, as we took subgroups defined by
symmetric closed sets of roots. Let us sketch this again for the classical
cases, utilizing the description in terms of $\pm$symmetric/hermitian
forms. The objects $D,\ V,\ h$ and $G'$ will have the meanings as
above. Let $s=\rank_{k}G'$, and let $H_s$ be an $s$-dimensional (maximal)
totally isotropic subspace in $V$, with basis $h_1,\ldots, h_s$. Let
$h_i'\in V$ be vectors of $V$ with $h(h_i,h_j')=\gd_{ij}$,
$H_s'=<h_1',\ldots, h_s'>$ and set $H=H_s\oplus H_s'$. Then $h_{|H}$ is
non-degenerate of index $s$, and $H$ splits into a direct sum of hyperbolic
planes, $H=V_1\oplus \cdots \oplus V_s$. The form $h$ restricted to
$H^{\perp}$ is anisotropic; the semisimple anisotropic kernel is
$SU(H^{\perp},h_{|H^{\perp}})$. Fixing the basis $h_1,\ldots,
h_s,h_1',\ldots, h_s'$ for $H$ amounts to the choice of maximal $k$-split
torus $S'$. For each real prime $\nu$, $(H_{\nu},h_{\nu})$ is a
$2ds$-dimensional ${\Bbb R}} \def\fH{{\Bbb H}$-vecotr space with $\pm$symmetric/hermitian
form. Choosing an ${\Bbb R}} \def\fH{{\Bbb H}$-basis of $H_{\nu}$ amounts to choosing a maximal
${\Bbb R}} \def\fH{{\Bbb H}$-split torus of $SU(H_{\nu},h_{\nu})$, and a choice of basis for a
maximal set of hyperbolic planes (over ${\Bbb R}} \def\fH{{\Bbb H}$) amouts to the choice of
maximal ${\Bbb R}} \def\fH{{\Bbb H}$-split torus. Similarly,
$(H_{\nu}^{\perp},{h_{\nu}}_{|H_{\nu}^{\perp}})$ is an ${\Bbb R}} \def\fH{{\Bbb H}$-vector space,
$h_{|{H^{\perp}_{\nu}}}$ has some index $q_{\nu}$,
and one can find a maximal set of
hyperbolic planes $W_1,\ldots,W_r$, such that
$H_{\nu}^{\perp}=(W_1)_{\nu}\oplus \cdots \oplus (W_r)_{\nu}\oplus W'$,
where ${h_{\nu}}_{|W'}$ is anisotropic over ${\Bbb R}} \def\fH{{\Bbb H}$. A choice of basis of the
$(W_i)_{\nu}$ amounts to the choice of maximal ${\Bbb R}} \def\fH{{\Bbb H}$-split torus, and a
choice of basis, over ${\Bbb R}} \def\fH{{\Bbb H}$, of $V_{\nu}$ amounts to the choice of maximal
torus defined over ${\Bbb R}} \def\fH{{\Bbb H}$. From these descriptions we see that the polydisc
group $N_{\Psi}$ defined by the maximal set of strongly orthogonal roots
$\Psi$ splits into
a component in $SU(H,h_{|H})$ and a component in
$SU(H^{\perp},h_{|H^{\perp}})$, say $N_{\Psi}=N_{\Psi,1}\times
N_{\Psi,2}$. Then $N_{\Psi,2}\subset} \def\nni{\supset} \def\und{\underline SU(H^{\perp},h_{|H^{\perp}})({\Bbb R}} \def\fH{{\Bbb H})$ and
$N_{\Psi,1}\subset} \def\nni{\supset} \def\und{\underline SU(H,h_{|H})$. Since the subgroup
$SU(H^{\perp},h_{|H^{\perp}})$ is contained in all the groups $N$ we have
defined, we need only consider $N_{\Psi,1}$. $H$ is a direct sum of
hyperbolic planes $V_i$, and the question is whether the corresponding
polydisc group is contained in $SU(V_i,h_{|V_i})$. But this is what was
studied in \cite{hyp}; the answer is affirmative. It follows that with the
one exception stated, $C^{(2)}_{2,1}$, $N_{\Psi}\subset} \def\nni{\supset} \def\und{\underline N$. \hfill $\Box$ \vskip0.25cm
|
1995-09-15T05:58:54 | 9505 | alg-geom/9505001 | en | https://arxiv.org/abs/alg-geom/9505001 | [
"alg-geom",
"math.AG",
"math.CO"
] | alg-geom/9505001 | Frank Sottile | Frank Sottile | Pieri's rule for flag manifolds and Schubert polynomials | 21 pages with 1 figure. AMSLaTeX v 1.1 | Ann. de l'Inst. Four., 46 (1996) 89-110 | null | null | null | We establish the formula for multiplication by the class of a special
Schubert variety in the integral cohomology ring of the flag manifold. This
formula also describes the multiplication of a Schubert polynomial by either an
elementary symmetric polynomial or a complete homogeneous symmetric polynomial.
Thus, we generalize the classical Pieri's rule for symmetric
polynomials/Grassmann varieties to Schubert polynomials/flag manifolds. Our
primary technique is an explicit geometric description of certain intersections
of Schubert varieties. This method allows us to compute additional structure
constants for the cohomology ring, which we express in terms of paths in the
Bruhat order on the symmetric group.
| [
{
"version": "v1",
"created": "Tue, 2 May 1995 20:01:16 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Sottile",
"Frank",
""
]
] | alg-geom | \section{Introduction}
Schubert polynomials had their origins in the study
of the cohomology of flag manifolds by
Bernstein-Gelfand-Gelfand~\cite{BGG} and Demazure~\cite{Demazure}.
They were later defined by Lascoux and
Sch\"utzenberger~\cite{Lascoux_Schutzenberger_polynomes_schubert},
who developed a purely combinatorial theory.
For each permutation $w$ in the symmetric group $S_n$ there is a Schubert
polynomial $\frak{S}_w$ in the variables $x_1,\ldots,x_{n-1}$.
When evaluated at certain Chern classes, a Schubert polynomial
gives the cohomology
class of a Schubert subvariety of the manifold of complete flags
in $\Bbb{C}\,^n$.
In this way, the
collection $\{\frak{S}_w\,|\, w\in S_n\}$ of Schubert polynomials
determines an integral basis for the cohomology ring of
the flag manifold.
Thus there exist integer structure constants $c^u_{w\,v}$ such that
$$
\frak{S}_w\cdot\frak{S}_v = \sum_u c^u_{w\,v}\frak{S}_u.
$$
No formula is known, or even conjectured, for these constants.
There are, however, a few special cases in which they are known.
One important case is Monk's rule~\cite{Monk}, which characterizes
the algebra of Schubert polynomials.
While this is usually attributed to Monk,
Chevalley simultaneously established the analogous formula
for generalized flag manifolds in a manuscript that was only
recently published~\cite{Chevalley91}.
Let $t_{k\,k+1}$ be the transposition interchanging $k$ and $k+1$.
Then $\frak{S}_{t_{k\,k+1}} = x_1{+}\cdots{+}x_k=s(x_1,\ldots,x_k)$,
the first elementary symmetric polynomial.
For any permutation $w\in S_n$, Monk's rule states
$$
\frak{S}_w \cdot \frak{S}_{t_{k\,k+1}} \
=\ \frak{S}_w \cdot s_1(x_1,\ldots,x_k)
\ =\ \sum \frak{S}_{w t_{a\,b}},
$$
where $t_{a\,b}$ is the transposition interchanging $a$ and $b$, and
the sum is over all $a\leq k<b$ where $w(a)<w(b)$ and if
$a<c<b$, then $w(c)$ is not between $w(a)$ and $w(b)$.
\smallskip
The classical Pieri's rule computes the product of
a Schur polynomial by either a complete homogeneous
symmetric polynomial or an elementary symmetric polynomial.
Our main result is a formula for Schubert polynomials
and the cohomology of flag manifolds which generalizes both Monk's rule
and the classical Pieri's rule.
Let $s_m(x_1,\ldots,x_k)$ and $s_{1^m}(x_1,\ldots,x_k)$ be
respectively the complete homogeneous
and elementary symmetric polynomials of degree
$m$ in the variables $x_1,\ldots,x_k$ and let $\ell(w)$ be the length of a
permutation $w$.
These polynomials are the cohomology classes of special Schubert varieties.
We will show
\medskip
\noindent{\bf Theorem~\ref{thm:main}.}
{\em Let $k,m,n$ be positive integers, and let $w\in S_n$.
\begin{enumerate}
\item[I.]
$\frak{S}_w\cdot s_m(x_1,\ldots,x_k) =
\sum_{w'} \frak{S}_{w'}$,
the sum over all $w' = w t_{a_1\,b_1}\cdots t_{a_m\,b_m}$, where
$a_i\leq k < b_i$ and
$\ell(w t_{a_1\,b_1}\cdots t_{a_i\,b_i}) = \ell(w) + i$
for $1\leq i\leq m$ with the integers
$b_1,\ldots, b_m$ distinct.
\item[II.]
$\frak{S}_w\cdot s_{1^m}(x_1,\ldots,x_k) =
\sum_{w'} \frak{S}_{w'}$,
the sum over all $w'$ as in {\em I}, except that now
the integers $a_1,\ldots,a_m$ are distinct.
\end{enumerate}
}
\medskip
Both $s_m(x_1,\ldots,x_k)$ and $s_{1^m}(x_1,\ldots,x_k)$
are Schubert polynomials, so Theorem~\ref{thm:main} computes some
of the structure constants in the cohomology ring of the flag manifold.
These formulas were stated in a different form by Lascoux and Sch\"utzenberger
in~\cite{Lascoux_Schutzenberger_polynomes_schubert}, where an
algebraic proof was suggested.
They were later independently conjectured in yet another form by Bergeron and
Billey~\cite{Bergeron_Billey}.
Our formulation facilitates our proofs.
Using geometry, we expose a surprising connection to the classical
Pieri's rule, from which we deduce Theorem~\ref{thm:main}.
These methods enable the determination of additional structure constants.
We further generalize Theorem~\ref{thm:main} to give a formula for the
multiplication of a Schubert polynomial by a hook Schur polynomial,
indicating a relation between multiplication of Schubert
polynomials and paths in the Bruhat order in $S_n$.
This exposition is organized as follows:
Section 2 contains preliminaries about Schubert polynomials
while Section 3 is devoted to the flag manifold.
In Section 4 we deduce our main results from
a geometric lemma proven in Section~5.
Two examples are described in Section~6, illustrating the
geometry underlying the results of Section~5.
We remark that while our results are stated in terms of the
integral cohomology of the complex manifold of complete flags,
our results and proofs are valid for the Chow rings of
flag varieties defined over any field.
We would like to thank Nantel Bergeron and
Sara Billey for suggesting these problems
and Jean-Yves Thibon for showing us the work of Lascoux
and Sch\"utzenberger.
\section{Schubert Polynomials}
In~\cite{BGG,Demazure} cohomology classes of Schubert subvarieties of
the flag manifold were obtained from the class of a point using
repeated correspondences in $\Bbb{P}^1$-bundles, which may be described
algebraically as ``divided differences.''
Subsequently, Lascoux and
Sch\"utzenberger~\cite{Lascoux_Schutzenberger_polynomes_schubert}
found explicit polynomial representatives
for these classes.
We outline Lascoux and Sch\"utzenberger's construction of Schubert
polynomials.
For a more complete account see~\cite{Macdonald_schubert}.
For an integer $n>0$, let $S_n$ be the group of
permutations of $[n] = \{1,2,\ldots,n\}$.
Let $t_{a\, b}$ be the transposition interchanging $a <b$.
Adjacent transpositions $s_i = t_{i\,i{+}1}$
generate $S_n$.
The {\em length} $\ell(w)$ of a
permutation $w$ is the minimal length of a factorization
into adjacent transpositions.
If $w = s_{a_1} s_{a_2}\cdots s_{a_m}$ is such a factorization, then
the sequence $(a_1,\ldots,a_m)$ is a {\em reduced word} for $w$.
The length of $w$ also counts the inversions of $w$,
those pairs $i<j$ where $w(i)>w(j)$.
It follows that $\ell(wt_{a\, b}) = \ell(w){+}1$ if and only if
$w(a)<w(b)$ and whenever $a<c<b$, either $w(c)< w(a)$ or
$w(b)<w(c)$.
For each integer $n>1$, let $R_n = \Bbb{Z}[x_1,\ldots,x_n]$.
The group $S_n$ acts on $R_n$ by permuting the variables.
Let $f\in R_n$ and let $s_i$ be an adjacent transposition.
The polynomial
$f - s_i f$ is antisymmetric in $x_i$ and $x_{i+1}$, and so is
divisible by $x_i - x_{i+1}$.
Thus we may define the linear divided difference operator
$$
\partial_i = (x_i-x_{i+1})^{-1} (1 - s_i).
$$
If $f$ is symmetric in $x_i$ and $x_{i+1}$, then $\partial_i f$ is zero.
Otherwise $\partial_i f$ is symmetric in $x_i$ and $x_{i+1}$.
Divided differences satisfy
\begin{eqnarray*}
\partial_i\circ \partial_i & = & 0 \\
\partial_i \circ\partial_j &=& \partial_j \circ \partial_i \ \
\ \ \ \ \ \ \ \ \ \ \mbox{ if } |i-j|\geq 2\\
\qquad\qquad\partial_{i+1}\circ\partial_i\circ\partial_{i+1} & = &
\partial_i\circ\partial_{i+1}\circ\partial_i
\end{eqnarray*}
It follows that if $(a_1,\ldots,a_p)$ is a reduced word for a permutation
$w$, the composition of divided differences
$\partial_{a_1}\circ\cdots\circ\partial_{a_p}$
depends only upon $w$ and not upon the reduced word chosen.
This defines an operator $\partial_{w}$ for each $w\in S_n$.
Let $w_0$ be the longest permutation in $S_n$, that is
$w_0(j) = n{+}1{-}j$.
For $w \in S_n$, define the {\em Schubert polynomial} $\frak{S}_{w}$
by
$$
\frak{S}_{w} = \partial_{w^{-1}w_0}
\left( x_1^{n-1} x_2^{n-2}\cdots x_{n-1} \right).
$$
The degree of $\partial_i$ is $-1$, so $\frak{S}_{w}$ is homogeneous of
degree $ {n\choose 2} - \ell(w^{-1}w_0) = \ell(w)$.
Let $\cal{S}\subset R_n$ be the ideal generated by the
non-constant symmetric polynomials.
The set $\{\frak{S}_{w}\,|\, w\in S_n\}$ of Schubert polynomials
is a basis for
$\Bbb{Z}\{ x_1^{i_1}\cdots x_{n-1}^{i_{n-1}}\,|\, i_j \leq n{-}j \}$,
a transversal to $\cal{S}$ in $R_n$.
Thus Schubert polynomials are explicit polynomial
representatives of an integral basis for the ring $H_n = R_n/\cal{S}$.
Courting ambiguity, we will use the same notation for
Schubert polynomials in $R_n$ as for their images in the rings
$H_n$.
Recently, other descriptions have been discovered
for Schubert
polynomials~\cite{Bergeron,BJS,Fomin_Kirillov,Fomin_Stanley}.
Combinatorists often define Schubert polynomials $\frak{S}_w$ for all
$w\in S_\infty = \cup_{n=1}^\infty S_n$.
One may show that our results are valid in this wider context.
\smallskip
A {\em partition} $\lambda$ is a decreasing sequence
$\lambda_1 \geq \lambda_2\geq\cdots\geq \lambda_k$
of positive integers, called the {\em parts} of $\lambda$.
Given a partition $\lambda$ with at most $k$ parts,
one may define a Schur polynomial
$s_\lambda = s_\lambda(x_1,\ldots,x_k)$, which is a symmetric
polynomial in the variables $x_1,\ldots,x_k$.
For a more complete treatment of symmetric polynomials and
Schur polynomials, see~\cite{Macdonald_symmetric}.
The collection of Schur polynomials forms an
integral basis for the ring of symmetric polynomials,
$\Bbb{Z}[x_1,\ldots,x_k]^{S_k}$.
The Littlewood-Richardson rule is a formula for the
structure constants $c^\lambda_{\mu\nu}$ of this ring, called
{\em Littlewood-Richardson coefficients} and defined by
$$
s_\mu \cdot s_\nu \ =\ \sum_\lambda \, c^\lambda_{\mu\nu}\, s_\lambda.
$$
If $\lambda$ and $\mu$ are partitions satisfying $\lambda_i \geq \mu_i$
for all $i$, we write $\lambda \supset \mu$.
This defines a partial order on the collection of partitions,
called Young's lattice.
Since $c^\lambda_{\mu\nu} = 0$ unless $\lambda \supset \mu$ and
$\lambda\supset \nu$~(cf. \cite{Macdonald_symmetric}), we see that
$\cal{I}_{n,k}=\{ s_\lambda\,|\, \lambda_1 \geq n-k\}$ is an ideal.
Let $A_{n,k}$ be the quotient ring
$\Bbb{Z}[x_1,\ldots,x_k]^{S_k}/\cal{I}_{n,k}$.
To a partition $\lambda$ we may associate its Young diagram,
also denoted $\lambda$, which is a left-justified
array of boxes in the plane
with $\lambda_i$ boxes in the $i$th row.
If $\lambda \supset \mu$, then the Young diagram of $\mu$ is
a subset of that of $\lambda$, and the skew diagram $\lambda/\mu$
is the set theoretic difference $\lambda-\mu$.
If each column of $\lambda/\mu$ is either empty or a single box, then
$\lambda/\mu$ is a {\em skew row} of {\em length} $m$, where $m$
is the number of boxes in $\lambda/\mu$.
The transpose $\mu^t$ of a partition $\mu$ is the partition whose Young
diagram is the transpose of that of $\mu$.
We call the transpose of a skew row a {\em skew column}.
The map defined by $s_\lambda \mapsto s_{\lambda^t}$ is a ring
isomorphism $A_{n,k} \rightarrow A_{n,n-k}$.
For example, let $\lambda = (5,2,1)$ and $\mu = (3,1)$
then $\lambda/\mu$ is a skew row of length 4 and
$\mu^t = (2,1,1)$.
The following are the Young diagrams of $\lambda$, $\mu$,
$\lambda/\mu$, and $\mu^t$:
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If $w$ has only one {\em descent} ($k$ such that $w(k) >
w(k{+}1)$), then $w$ is said to be {\em Grassmannian}
of descent $k$ and $\frak{S}_{w}$ is the Schur polynomial
$s_{\lambda}(x_1,\ldots,x_k)$.
Here $\lambda$ is the {\em shape} of $w$, the partition
with $k$ parts where $\lambda_{k+1-j} = w(j){-}j$.
For integers $k,m$ define
$r[k,m]$ and $c[k,m]$ to be the Grassmannian permutations of descent $k$
with shapes $(m,0,\ldots,0) = m$ and
$(1^m,0,\ldots,0) = 1^m$, respectively.
These are the $m+1$-cycles
\begin{eqnarray*}
r[k,m] &=&
(k{+}m\,\,\,\,k{+}m{-}1\,\ldots\,k{+}2\,\,\,\,k{+}1\,\,\,\,k)\\
c[k,m] &=&
(k{-}m{+}1\,\,\,\,k{-}m{+}2\,\ldots\,k{-}1\,\,\,\,k\,\,\,\,k{+}1).
\end{eqnarray*}
\section{The Flag Manifold}
Let $V$ be an $n$-dimensional complex vector space.
A {\em flag} ${F\!_{\DOT}\,}$ in $V$ is a sequence
$$
\{0\}\ =\ F_0 \subset F_1 \subset F_2\subset \cdots \subset F_{n-1}
\subset F_n\ =\ V,
$$
of linear subspaces with $\dim_{\Bbb{C}} F_i = i$.
The set of all flags is a $\frac{1}{2}n(n-1)$ dimensional complex
manifold, called the flag manifold
and denoted $\Bbb{F}(V)$.
Over $\Bbb{F}(V)$, there is a tautological flag ${\cal{F}\!_{\DOT}\,}$ of
bundles whose fibre at a point ${F\!_{\DOT}\,}$ is the flag ${F\!_{\DOT}\,}$.
Let $x_i$ be the Chern class of the line bundle
$\cal{F}_i/\cal{F}_{i-1}$.
Then the integral cohomology ring of $\Bbb{F}(V)$ is
$H_n = \Bbb{Z}[x_1,\ldots,x_n]/\cal{S}$, where $\cal{S}$
is the ideal generated by those non-constant polynomials
which are symmetric in
$x_1,\ldots,x_n$.
This description is due to Borel~\cite{Borel}.
Given a subset $S \subset V$, let $\Span{S}$ be its linear span
and for linear subspaces $W\subset U$ let $U-W$ be their
set theoretic difference.
An ordered
basis $f_1,f_2,\ldots,f_n$ for $V$ determines a flag ${E_{\DOT}\,}$; set
$E_i = \Span{f_1,\ldots,f_i}$ for $1\leq i \leq n$.
In this case, write ${E_{\DOT}\,} = \Span{f_1,\ldots,f_n}$
and call $f_1,\ldots,f_n$ a {\em basis} for ${E_{\DOT}\,}$.
A fixed flag ${F\!_{\DOT}\,}$ gives a decomposition due to
Ehresmann~\cite{Ehresmann} of $\Bbb{F}(V)$ into
affine cells indexed by permutations $w$ of $S_n$.
The cell determined by $w$ is
$$
X^{\circ}_w {F\!_{\DOT}\,} \ = \ \{ {E_{\DOT}\,}=\Span{f_1,\ldots,f_n}\,|\,
f_i \in F_{n+1-w(i)}-F_{n-w(i)}, \,1\leq i\leq n\}.
$$
The complex codimension of $X^{\circ}_w {F\!_{\DOT}\,}$ is $\ell(w)$
and its closure is the Schubert subvariety $X_w{F\!_{\DOT}\,}$.
Thus the cohomology ring of $\Bbb{F}(V)$
has an integral basis given by the
cohomology classes\footnote{Strictly speaking, we mean the
classes Poincar\'e dual to the fundamental cycles in homology.}
$[X_w{F\!_{\DOT}\,}]$ of the Schubert subvarieties.
That is, $H^*\Bbb{F}(V) = \bigoplus_{w\in S_n}{\Bbb Z} [X_w{F\!_{\DOT}\,}]$.
Independently, Bernstein-Gelfand-Gelfand~\cite{BGG}
and Demazure~\cite{Demazure}
related this description to Borel's, showing
$[X_w{F\!_{\DOT}\,}] = \partial_{w^{-1}w_0}[\{{F\!_{\DOT}\,}\}]$.
Later, Lascoux and
Sch\"utzenberger~\cite{Lascoux_Schutzenberger_polynomes_schubert}
obtained polynomial
representatives $\frak{S}_w$ for $[X_w{F\!_{\DOT}\,}]$
by choosing
$x_1^{n-1} x_2^{n-2}\cdots x_{n-1}$ for the
representative of the class $[\{{F\!_{\DOT}\,}\}] = \frak{S}_{w_0}$ of a point.
We use the term Schubert polynomial for both the polynomial and
the associated cohomology class.
This Schubert polynomial basis for cohomology diagonalizes
the intersection pairing; If
$\ell(w) + \ell(w') = \dim\Bbb{F}(V) = \frac{1}{2}n(n-1)$, then
$$
\frak{S}_w\cdot \frak{S}_{w'} =
\left\{
\begin{array}{ll} \frak{S}_{w_0} & \mbox{ if } w' = w_0 w\\
0 & \mbox{ otherwise}
\end{array} \right.
$$
\smallskip
For each $k\leq \dim V =n$, the set of all $k$-dimensional subspaces of $V$
is a $k(n{-}k)$ dimensional complex manifold, called the Grassmannian of
$k$-planes in $V$, written $G_kV$.
The cohomology ring of $G_kV$ is a quotient of the ring of symmetric
polynomials in the Chern roots $x_1,\ldots,x_k$ of its tautological
$k$-plane bundle.
This identifies it with the ring $A_{n,k}$ of Section 2.
A fixed flag ${F\!_{\DOT}\,}$ gives a decomposition of $G_kV$ into
cells indexed by partitions $\lambda$ with $k$ parts, none
exceeding $n{-}k$.
The closure of such a cell is the Schubert variety
$$
\Omega_\lambda {F\!_{\DOT}\,} =
\{ H \in G_kV \,|\, \dim H\cap F_{n-k+j-\lambda_j} \geq j
\mbox{ for } 1\leq j\leq k\},
$$
whose codimension is
$\lambda_1{+}{\cdots}{+}\lambda_k = |\lambda|$.
The classes $[\Omega_\lambda{F\!_{\DOT}\,}]$ form a basis for the
cohomology ring of $G_kV$ and $[\Omega_\lambda{F\!_{\DOT}\,}]$
is the Schur polynomial $s_\lambda(x_1,\ldots,x_k)$.
We use the term Schur polynomial for both the polynomial and its
image in the cohomology ring of $G_kV$.
The Schur polynomial $s_{m}$ is the complete homogeneous
symmetric polynomial of degree $m$ in $x_1,\ldots,x_k$.
The Schur polynomial $s_{1^m}$ is the $m$th
elementary symmetric polynomial in $x_1,\ldots,x_k$.
Pieri's rule is a formula for multiplying Schur polynomials by
either $s_m$ or $s_{1^m}$.
For $s_m$, it states
$$
s_\mu \cdot s_m\ = \
\sum s_{\lambda},
$$
the sum over all partitions $\lambda$ with
$n{-}k\geq\lambda_1\geq\mu_1\geq\cdots\geq\lambda_k\geq\mu_k$ and
$|\lambda| = m{+} |\mu|$.
That is, those partitions $\lambda\supset \mu$ with $\lambda/\mu$ a skew row
of length $m$.
To obtain the analogous formula for $s_{1^m}$, use the isomorphism
$A_{n,k} \rightarrow A_{n,n-k}$ given by
$s_\lambda \mapsto s_{\lambda^t}$.
Doing so, we see that
$$
s_\mu \cdot s_{1^m}\ = \ \sum s_\lambda,
$$
the sum over all partitions $\lambda$ with
$\lambda\supset \mu$ with $(\lambda/\mu)^t$ is a skew row of length $m$.
That is, those $\lambda\supset \mu$ with $\lambda/\mu$ a skew column
of length $m$.
\smallskip
If $Y\subset V$ has codimension $d$, then $G_kY \subset G_kV$ is a
Schubert subvariety whose indexing partition is $d^k$, the partition
with $k$ parts each equal to $d$.
It follows that $\Omega_{(n{-}k)^k}{F\!_{\DOT}\,} = \{F_k\}$, so $s_{(n{-}k)^k}$
is the class of a point.
The basis of Schur polynomials diagonalizes the intersection pairing;
For a partition $\lambda$, let $\lambda^c$ be the partition
$(n{-}k{-}\lambda_k,{\ldots},n{-}k{-}\lambda_1)$.
If $|\mu| {+}|\lambda| = k(n{-}k)$, then
$$
s_\lambda \cdot s_\mu =
\left\{
\begin{array}{ll} s_{(n{-}k)^k}& \mbox{ if } \lambda^c = \mu\\
0 & \mbox{ otherwise }
\end{array}
\right ..
$$
We use this to reformulate Pieri's rule.
Suppose $|\mu|+|\lambda|+m = k(n-k)$, then
$$
s_\mu \cdot s_{\lambda^c}\cdot s_m = \left\{
\begin{array}{ll} s_{(n-k)^k} &
\mbox{ if $\lambda/\mu$ is a skew row of length $m$}\\
0 &\mbox{ otherwise}\end{array}\right. .
$$
\smallskip
For $k\leq n$, the association ${E_{\DOT}\,} \mapsto E_k$
defines a map $\pi :\Bbb{F}(V) \rightarrow G_kV$.
The functorial map $\pi^*$ on
cohomology is simply the inclusion into $H_n$
of polynomials symmetric in $x_1,\ldots,x_k$.
That is, $A_{n,k} \hookrightarrow H_n$.
If $\lambda$ is a partition with $k$ parts and $w$
the Grassmannian permutation of descent $k$ and shape $\lambda$,
then $\pi^* s_\lambda = \frak{S}_w$.
Under the Poincar\'e duality isomorphism between homology and
cohomology groups, the functorial map $\pi_*$ on homology induces a
a group homomorphism $\pi_*$ on cohomology.
While $\pi_*$ is not a ring homomorphism, is does satisfy
the projection formula (see Example 8.17 of~\cite{Fulton_intersection}):
$$
\pi_*(\alpha\cdot \pi^* \beta) = (\pi_* \alpha)\cdot \beta,
$$
where $\alpha$ is a cohomology class on $\Bbb{F}(V)$ and $\beta$
is a cohomology class on $G_kV$.
\section{Pieri's Rule for Flag Manifolds}
An open problem is to find the analog
of the Littlewood-Richardson rule for Schubert polynomials.
That is, determine the structure constants $c^u_{w\,v}$ for
the Schubert basis of the cohomology of flag manifolds, which are
defined by
\begin{equation} \label{eq:structure}
\frak{S}_w \cdot \frak{S}_v =
\sum_u c^u_{w\,v} \frak{S}_u.
\end{equation}
These constants are positive integers as they count
the points in a suitable triple intersection
of Schubert subvarieties.
They are are known only in some special cases.
For example, if both $w$ and $v$ are Grassmannian
permutations of descent $k$ so that $\frak{S}_w$ and
$\frak{S}_v$ are
symmetric polynomials in the variables $x_1,\ldots,x_k$, then
(\ref{eq:structure}) is
the classical Littlewood-Richardson rule.
Another case is Monk's rule, which states:
$$
\frak{S}_w\cdot \frak{S}_{t_{k\,k{+}1}}
= \sum \frak{S}_{w t_{a\,b}},
$$
the sum over all $a\leq k <b$ with
$\ell(w t_{a\,b})=\ell(w)+1$.
The Schubert polynomial $\frak{S}_{t_{k\,k{+}1}}$ is
$s_1(x_1,\ldots,x_k)$.
We use geometry to generalize this formula, giving an analog of
the classical Pieri's rule.
\smallskip
Let $w,w' \in S_n$.
Write $w \stackrel{r[k,m]}{\relbar\joinrel\llra} w'$ if
there exist integers $a_1,b_1,\ldots,a_m,b_m$ with
\begin{enumerate}
\item $a_i\leq k <b_i$ for $1\leq i\leq m$ and
$w' = wt_{a_1\,b_1}\cdots t_{a_m\,b_m}$,
\item $\ell(w t_{a_1\,b_1}\cdots t_{a_i\,b_i}) = \ell(w) +i$, and
\item the integers $b_1, b_2,\ldots, b_m$ are distinct.
\end{enumerate}
Similarly, $w\stackrel{c[k,m]}{\relbar\joinrel\llra} w'$
if we have integers $a_1,\ldots,b_m$ as in (1) and (2) where now
\begin{enumerate}
\item[(3)$'$] the integers $a_1,a_2,\ldots, a_m$ are distinct.
\end{enumerate}
Our primary result is the following.
\begin{thm} \label{thm:main}
Let $w \in S_n$.
Then
\begin{enumerate}
\item[I.] For all $k$ and $m$ with $k+m \leq n$, we have \ \
${\displaystyle
\frak{S}_{w}\cdot \frak{S}_{r[k,m]}
= \sum_{w \stackrel{r[k,m]}{\relbar\joinrel\llra} w'}
\frak{S}_{w'}}$.
\smallskip
\item[II.] For all $ m\leq k\leq n$, we have\rule{0pt}{20pt} \ \
${\displaystyle
\frak{S}_{w}\cdot \frak{S}_{c[k,m]}
= \sum_{w \stackrel{c[k,m]}{\relbar\joinrel\llra} w'}
\frak{S}_{w'}}$.
\end{enumerate}
\end{thm}
Theorem~\ref{thm:main} may be alternatively stated in terms of the
structure constants $c^u_{w\,v}$.
\medskip
\noindent{\bf Theorem 1$'\!$.} \ {\em Let $w, w' \in S_n$.
Then
\begin{enumerate}
\item[I.] For all integers $k,m$ with $k+m\leq n$, \ \
${\displaystyle
c^{w'}_{w\, r[k,m]} = \left\{\begin{array}{ll} 1 &\mbox{ if } w\stackrel{r[k,m]}{\relbar\joinrel\llra} w'\\
0 & \mbox{ otherwise}\end{array}\right.
}$.
\item[ II.] For all integers $k,m$ with $m\leq k\leq n$,\rule{0pt}{28pt} \ \
${\displaystyle
c^{w'}_{w\, c[k,m]} = \left\{\begin{array}{ll} 1 &\mbox{ if } w\stackrel{c[k,m]}{\relbar\joinrel\llra} w'\\
0 & \mbox{ otherwise}\end{array}\right.
}$.
\end{enumerate}
}
\bigskip
We first show the equivalence of parts I and II and then establish
part I.
An order $<_k$ on
$S_n$ is introduced, and we show that $c^{w'}_{w\, r[k,m]}$ is 0 unless
$w<_k w'$.
A geometric lemma enables us to compute
$c^{w'}_{w\, r[k,m]}$ when $w<_k w'$.
\begin{lemma}\label{lemma:equivalent}
Let $w_0$ be the longest permutation in $S_n$,
and $k{+}m \leq n$.
Then
\begin{enumerate}
\item $w_0 r[k,m] w_0 = c[n{-}k,m]$.
\item Let $w, w' \in S_n$. Then
$w\stackrel{r[k,m]}{\relbar\joinrel\llra} w'$ if and only if $\,\,\,w_0 w w_0\stackrel{c[n{-}k,m]}{\relbar\joinrel\relbar\joinrel\lllra} w_0w'w_0$.
\item The map induced by $\frak{S}_w \mapsto \frak{S}_{w_0ww_0}$
is an automorphism of $H_n$.
\item Statements {\em I} and {\em II}
of Theorem~\ref{thm:main}$'$ are equivalent.
\end{enumerate}
\end{lemma}
This automorphism $\frak{S}_w \mapsto \frak{S}_{w_0ww_0}$
is the Schubert polynomial analog of the map
$s_\lambda(x_1,\ldots,x_k) \mapsto
s_{\lambda^t}(x_1,\ldots,x_{n-k})$ for Schur polynomials.
\medskip
\noindent{\bf Proof:}
Statements (1) and (2) are easily verified, as $w_0(j) = n+1-j$.
Statement (3) is also immediate, as
$\frak{S}_w \mapsto \frak{S}_{w_0ww_0}$
leaves Monk's rule invariant and Monk's rule characterizes
the algebra of Schubert polynomials.
For (4), suppose $k+m \leq n$ and $w, w' \in S_n$ and
let $\overline{w}$ denote $w_0ww_0$.
The isomorphism $\frak{S}_v \mapsto \frak{S}_{\overline{v}}$
of (3) shows $c^{w'}_{w\, r[k,m]}=
c^{\overline{w'}}_{\overline{w}\,\overline{r[k,m]}}$.
Part (1) shows
$c^{\overline{w'}}_{\overline{w}\,\overline{r[k,m]}} =
c^{\overline{w'}}_{\overline{w}\,c[n{-}k,m]}$.
Then (2) shows the equality of the two statements of
Theorem~\ref{thm:main}$'$.
\QED
Let $<_k$ be the transitive closure of the relation given by
$w <_k w'$, whenever $w' = w t_{a\,b}$ with $a\leq k<b$ and
$\ell(w\, t_{a\,b}) = \ell(w){+}1$.
We call $<_k$ the {\em $k$-Bruhat order},
in~\cite{Lascoux_Schutzenberger_Symmetry}
it is the $k$-colored Ehresmano\"edre.
\begin{lemma}\label{lemma:order}
If $\,c^{w'}_{w\, r[k,m]} \neq 0$, then $w<_k w'$ and
$\ell(w') = \ell(w) + m$.
\end{lemma}
\noindent{\bf Proof:}
By Monk's rule, $w<_kw'$ if and only if $\frak{S}_{w'}$ appears with a
non-zero coefficient when
$\frak{S}_w (\frak{S}_{t_{k\,k{+}1}})^{\ell(w')-\ell(w)}$
is written as a sum of Schubert polynomials.
Since $r[k,m] = t_{k\,k{+}1} \cdot t_{k\,k{+}2}\cdots t_{k\,k{+}m}$,
Monk's rule shows that $\frak{S}_{r[k,m]}$ is a summand of
$(\frak{S}_{t_{k\,k+1}})^m$ with coefficient 1.
Thus the coefficient of $\frak{S}_{w'}$ in the expansion of
$\frak{S}_w \cdot (\frak{S}_{t_{k\,k+1}})^m$ exceeds the
coefficient of $\frak{S}_{w'}$ in $\frak{S}_w \cdot \frak{S}_{r[k,m]}$.
Hence $c_{w\,r[k,m]}^{w'} = 0$ unless
$w<_k w'$ and $\ell(w') = \ell(w) +m$.
\QED
In Section 5 we use geometry to prove the following lemma.
\begin{lemma}\label{lemma:pushforward}
Let $w<_k w'$ be permutations in $S_n$.
Suppose $w' = w t_{a_1\,b_1}\cdots t_{a_m\,b_m}$,
where $a_i\leq k<b_i$, and
$\ell(w t_{a_1\,b_1}\cdots t_{a_i\,b_i}) = \ell(w)+i$.
Let $d = n-k-\#\{b_1,\ldots,b_m\}$.
Then
\begin{enumerate}
\item There is a cohomology class $\delta$ on $G_kV$
such that
$\pi_*(\frak{S}_w\cdot \frak{S}_{w_0w'}) = \delta \cdot s_{d^k}$.
\item If $w\stackrel{r[k,m]}{\relbar\joinrel\llra} w'$, then there are partitions
$\lambda \supset \mu$ where $\lambda/\mu$ is a skew row of length $m$
whose $j$th row has length $\#\{i\,|\, a_i = j\}$
and
$\pi_*(\frak{S}_w\cdot \frak{S}_{w_0w'}) = s_{\mu}\cdot s_{\lambda^c}$.
\end{enumerate}
\end{lemma}
\noindent{\bf Proof of Theorem~\ref{thm:main}$'$:}
By Lemma~\ref{lemma:order}, we need only show that if $w<_k w'$
and $\ell(w')-\ell(w) = m$, then
$$
c^{w'}_{w\,r[k,m]} =
\left\{ \begin{array}{ll} 1 &\mbox{ if } w \stackrel{r[k,m]}{\relbar\joinrel\llra} w'\\
0 & \mbox{ otherwise} \end{array} \right. .
$$
Begin by multiplying the identity $\frak{S}_w\cdot \frak{S}_{r[k,m]} =
\sum_v\, c^v_{w\, r[k,m]}\, \frak{S}_v$
by $\frak{S}_{w_0\, w'}$ and use the intersection
pairing to obtain
$$
\frak{S}_w\cdot\frak{S}_{w_0\, w'}\cdot\frak{S}_{r[k,m]}
\ =\ c^{w'}_{w\,r[k,m]}\, \frak{S}_{w_0}.
$$
Recall that $\frak{S}_{r[k,m]} = \pi^* s_m(x_1,\ldots,x_k)$.
As $\frak{S}_{w_0}$ and $s_{(n-k)^k}$ are the classes of points,
$\pi_*\frak{S}_{w_0} = s_{(n-k)^k}$.
Apply the map $\pi_*$ and then the
projection formula to obtain:
\begin{eqnarray*}
\pi_*(\frak{S}_w\cdot\frak{S}_{w_0\, w'}\cdot \pi^* s_m)
&=& c^{w'}_{w\,r[k,m]}\, \pi_*( \frak{S}_{w_0})\\
\pi_*(\frak{S}_w\cdot\frak{S}_{w_0\, w'}) \cdot s_m
&=& c^{w'}_{w\,r[k,m]} \, s_{(n-k)^k}.
\end{eqnarray*}
By part (1) of Lemma~\ref{lemma:pushforward}, there is a cohomology
class $\delta$ on $G_kV$ with
$$
\pi_*(\frak{S}_w\cdot\frak{S}_{w_0\, w'}) \cdot s_m\
= \ \delta \cdot s_{d^k} \cdot s_m
$$
But $s_{d^k} \cdot s_m = 0$ unless $d+m \leq n-k$.
Since $d = n-k-\#\{b_1,\ldots,b_m\}\geq n-k-m$,
we see that $ c^{w'}_{w\,r[k,m]} =0$ unless
$m = \#\{b_1,\ldots,b_m\}$, which implies
$w \stackrel{r[k,m]}{\relbar\joinrel\llra} w'$.
To complete the proof of Theorem~\ref{thm:main}$'$, suppose that
$w \stackrel{r[k,m]}{\relbar\joinrel\llra} w'$.
By part (2) of Lemma~\ref{lemma:pushforward},
there are partitions
$\lambda \supset \mu$ with $\lambda/\mu$ a skew row of length $m$
where we have
$\pi_*(\frak{S}_w\cdot \frak{S}_{w_0w'}) = s_{\mu}\cdot s_{\lambda^c}$.
Then
$$
\pi_*(\frak{S}_w\cdot\frak{S}_{w_0\, w'}) \cdot s_m\
= \ s_{\mu}\cdot s_{\lambda^c} \cdot s_m \
= \ s_{(n-k)^k},
$$
by the ordinary Pieri's rule for Schur polynomials.
So $c^{w'}_{w\, r[k,m]} = 1$.
\QED
Theorem \ref{thm:main}$'$ determines the structure constants
$c^{w'}_{w\, r[k,m]}$ and $c^{w'}_{w\, c[k,m]}$.
We compute more structure constants.
For $\nu$ a partition with $k$ parts, let $w(\nu)$ be the
Grassmannian permutation of descent $k$ and shape $\nu$.
\begin{thm}
Let $w, w'\in S_n$ and $k\leq n$ be an integer.
Suppose $w\leq_k w'$ and $\ell(w') = \ell(w) +m$.
Let $a_1,b_1,\ldots,a_m,b_m$ be such that
$a_i\leq k <b_i$ where $w' = w t_{a_1\,b_1}\cdots t_{a_m\,b_m}$
and $\ell(w t_{a_1\,b_1}\cdots t_{a_i\,b_i}) = \ell(w) +i$.
Let $\nu$ be a partition with $k$ parts.
\begin{enumerate}
\item If $\,w\stackrel{r[k,m]}{\relbar\joinrel\llra} w'$,
the structure constant $c^{w'}_{w\, w(\nu)}$
equals the Littlewood-Richardson coefficient
$c^\lambda_{\mu\,\nu}$, where $\lambda/\mu$ is a skew row of length
$m$ whose $j$th row has length
$\#\{i \,|\, a_i = j\}$.
\item
If $\,w\stackrel{c[k,m]}{\relbar\joinrel\llra} w'$, the structure constant $c^{w'}_{w\, w(\nu)}$
equals the Littlewood-Richardson coefficient
$c^\lambda_{\mu\,\nu}$, where $\lambda/\mu$ is a skew column of length
$m$ whose $j$th column has length
$\#\{i \,|\, b_i=j\}$.
\end{enumerate}
\end{thm}
\noindent{\bf Proof:} Using the involution
$\frak{S}_{w} \mapsto \frak{S}_{w_0ww_0}$, it suffices to
prove part (1).
We use part (2) of Lemma~\ref{lemma:pushforward} to
evaluate $c^{w'}_{w\,w(\nu)}$.
Recall that $\frak{S}_{w(\nu)} = \pi^*(s_\nu)$. Then
\begin{eqnarray*}
c^{w'}_{w\,w(\nu)}\, s_{(n-k)^k}\ =\
\pi_*(c^{w'}_{w\,w(\nu)}\, \frak{S}_{w_0}) &=&
\pi_*(\frak{S}_w\cdot\frak{S}_{w_0w'}\cdot \frak{S}_{w(\nu)})\\
&=& \pi_*(\frak{S}_w\cdot\frak{S}_{w_0w'}) \cdot s_\nu\\
&=& s_\mu \cdot s_{\lambda^c}\cdot s_\nu\\
&=& c^{\lambda}_{\mu\nu}\, s_{(n-k)^k}. \ \ \ \ \QED
\end{eqnarray*}
The formulas of Theorem~\ref{thm:main} may be formulated as the
sum over certain paths in the $k$-Bruhat order.
We explain this formulation here.
A (directed) path in the $k$-Bruhat order from $w$ to $w'$
is equivalent to a choice of integers $a_1,b_1,\ldots, a_m,b_m$
with $a_i\leq k < b_i$ and if $w^{(0)} = w$ and
$w^{(i)} = w^{(i-1)}\cdot t_{a_i\,b_i}$,
then $\ell(w^{(i)}) = \ell(w) + i$ and $w^{(m)} = w'$.
In this case the path is
$$
w = w^{(0)}<_k w^{(1)} <_k w^{(2)} <_k \cdots <_k w^{(m)} = w'.
$$
\begin{lemma}
Let $w, w' \in S_n$ and $k,m$ be positive integers.
Then
\begin{enumerate}
\item $w\stackrel{r[k,m]}{\relbar\joinrel\llra} w'$ if and only if there is a path in the $k$-Bruhat
order of length $m$ such that
$$
w^{(1)}(a_1) < w^{(2)}(a_2) < \cdots < w^{(m)}(a_m).
$$
\item $w\stackrel{c[k,m]}{\relbar\joinrel\llra} w'$ if and only if there is a path in the $k$-Bruhat
order of length $m$ such that
$$
w^{(1)}(a_1) > w^{(2)}(a_2) > \cdots > w^{(m)}(a_m).
$$
\end{enumerate}
Furthermore, these paths are unique.
\end{lemma}
\noindent{\bf Proof:}
If $w\stackrel{r[k,m]}{\relbar\joinrel\llra} w'$, one may show that
the set of values $\{ w^{(i)}(a_i)\}$ and the set of transpositions
$\{t_{a_i\,b_i}\}$ depend only upon $w$ and $w'$, and not on the
particular path chosen from $w$ to $w'$ in the $k$-Bruhat order.
It is also the case that
rearranging the set $\{ w^{(i)}(a_i)\}$ in order, as in (1),
may be accomplished by interchanging transpositions $t_{a_i\,b_i}$ and
$t_{a_j\,b_j}$ where $a_i\neq a_j$ (necessarily $b_i\neq b_j$).
Both (1) and the uniqueness of this representation follow from these
observations.
Statement (2) follows for similar reasons.
\QED
For a path $\gamma$ in the $k$-Bruhat order, let
$\mbox{end}(\gamma)$ be the endpoint of $\gamma$.
We state a reformulation of Theorem 1.
\begin{cor}[Path formulation of Theorem 1]\label{cor:pieri_paths}
Let $w\in S_n$.
\begin{enumerate}
\item $ \frak{S}_w \cdot \frak{S}_{r[k,m]} \ =\
\sum_\gamma \frak{S}_{\mbox{\scriptsize end}(\gamma)},
$
the sum over all paths $\gamma$ in the $k$-Bruhat order which
start at $w$ such that
$$
w^{(1)}(a_1) < w^{(2)}(a_2) < \cdots < w^{(m)}(a_m),
$$
where $\gamma$ is the path $w <_k w^{(1)} <_k w^{(2)} <_k \cdots
<_k w^{(m)}$.
Equivalently, $c^{w'}_{w\,r[k,m]}$ counts the number of paths
$\gamma$ in the $k$-Bruhat order which start at $w$ such that
$$
w^{(1)}(a_1) < w^{(2)}(a_2) < \cdots < w^{(m)}(a_m).
$$
\item $ \frak{S}_w \cdot \frak{S}_{c[k,m]} \ =\
\sum_\gamma \frak{S}_{\mbox{\scriptsize end}(\gamma)},
$
the sum over all paths $\gamma$ in the $k$-Bruhat order which start at
$w$ such that
$$
w^{(1)}(a_1) > w^{(2)}(a_2) > \cdots > w^{(m)}(a_m),
$$
where $\gamma$ is the path $w<_k w^{(1)} <_k w^{(2)} <_k \cdots
<_k w^{(m)}$.
Equivalently, $c^{w'}_{w\,r[k,m]}$ counts the number of paths
$\gamma$ in the $k$-Bruhat order which start at
$w$ such that
$$
w^{(1)}(a_1) > w^{(2)}(a_2) > \cdots > w^{(m)}(a_m).
$$
\end{enumerate}
\end{cor}
This is the form of the conjectures of
Bergeron and Billey~\cite{Bergeron_Billey}, and it
exposes a link between multiplying
Schubert polynomials and paths in the Bruhat order.
Such a link is not unexpected.
The Littlewood-Richardson rule
for multiplying Schur functions may be expressed
as a sum over certain paths in Young's lattice of
partitions.
A connection between paths in the Bruhat order and the
intersection theory of Schubert varieties is described
in~\cite{Hiller_intersections}.
We believe the eventual description of the structure
constants $c^w_{uv}$ will be in terms
of counting paths of certain types in the Bruhat order
on $S_n$, and will yield new results about the Bruhat order on $S_n$.
Corollary~\ref{cor:hook_enumeration} below is one such result.
\smallskip
Using multiset notation for partitions, $(p,1^{q-1})$
is the hook shape partition whose Young diagram
is the union of a row of length $p$ and
a column of length $q$.
Define $h[k;\,p,q]$ to be the Grassmannian permutation of
descent $k$ and shape $(p,1^{q-1})$.
Then $\frak{S}_{h[k;\,p,q]} = \pi^* s_{(p,1^{q-1})}$.
This permutation, $h[k;\,p,q]$, is the $p+q$-cycle
$$
(k{-}q{+}1\,\,\,k{-}q{+}2\,\ldots\,k{-}1\,\,\,
k\,\,\,k{+}p\,\,\,k{+}p{-}1\,\ldots\,k{+}1).
$$
\begin{thm}\label{thm:hook_formula}
Let $q\leq k$ and $k{+}p \leq n$ be integers.
Set $m = p{+}q{-}1$.
For $w\in S_n$,
$$
\frak{S}_w \cdot \frak{S}_{h[k;\,p,q]} \ =\
\sum \frak{S}_{end(\gamma)},
$$
the sum over all paths $\gamma: w <_k w^{(1)} <_k w^{(2)} <_k \cdots
<_k w^{(m)}$ in the $k$-Bruhat order with
$$
w^{(1)}(a_1) < \cdots < w^{(p)}(a_p)
\ \ \ \mbox{and}\ \ \
w^{(p)}(a_p) > w^{(p{+}1)}(a_{p{+}1})>\cdots > w^{(m)}(a_m).
$$
Alternatively, those paths $\gamma$ with
$$w^{(1)}(a_1) > \cdots > w^{(q)}(a_q)
\ \ \ \mbox{and}\ \ \
w^{(q)}(a_q) <\cdots < w^{(m)}(a_m).
$$
\end{thm}
Setting either $p=1$ or $q=1$, we recover Theorem~\ref{thm:main}.
If we consider the coefficient $c^{w'}_{w\,h[k;p,q]}$
of $\frak{S}_{w'}$ in the
product $\frak{S}_w \cdot \frak{S}_{h[k;p,q]}$, we obtain:
\begin{cor}\label{cor:hook_enumeration}
Let $w, w' \in S_n$, and $p,q$ be positive integers where
$\ell(w')-\ell(w) = p+q-1 = m$.
Then the number of paths
$w <_k w^{(1)} <_k w^{(2)} <_k \cdots
<_k w^{(m)} = w'$
in the $k$-Bruhat order from $w$ to $w'$ with
$$
w^{(1)}(a_1) < \cdots < w^{(p)}(a_p)
\ \ \ \mbox{and}\ \ \
w^{(p)}(a_p) > w^{(p{+}1)}(a_{p{+}1})>\cdots > w^{(m)}(a_m)
$$
equals the number of paths with
$$w^{(1)}(a_1) > \cdots > w^{(q)}(a_q)
\ \ \ \mbox{and}\ \ \
w^{(q)}(a_q) <\cdots < w^{(m)}(a_m).
$$
\end{cor}
\noindent{\bf Proof of Theorem~\ref{thm:hook_formula}:}
By the classical Pieri's rule,
$$
s_{p}\, \cdot \, s_{1^{(q-1)}} \ = \
s_{(p{+}1,1^{q-2})} \, + \, s_{(p,1^{q-1})}.
$$
Expressing these as Schubert polynomials (applying $\pi^*$), we have:
$$
\frak{S}_{r[k,p]}\, \cdot\, \frak{S}_{c[k,q{-}1]}
\ =\ \frak{S}_{h[k;\,p{+}1,q{-}1]} +
\frak{S}_{h[k;\,p,q]}.
$$
Induction on either
$p$ or $q$ (with $m$ fixed) and
Corollary~\ref{cor:pieri_paths} completes the proof.
\QED
\section{Geometry of Intersections}
We deduce Lemma~\ref{lemma:pushforward} by studying certain
intersections of Schubert varieties.
A key fact we use is that if
$X_w{F\!_{\DOT}\,}$ and $X_v{G_{\DOT}}$ intersect generically transversally,
then
$$
[X_w{F\!_{\DOT}\,}\bigcap X_v{G_{\DOT}}] \ = \ [X_w{F\!_{\DOT}\,}]\cdot[X_v{G_{\DOT}}]
\ = \ \frak{S}_w \cdot \frak{S}_v
$$
in the cohomology ring.
Flags ${F\!_{\DOT}\,}$ and ${G_{\DOT}}$ are {\em opposite} if for
$1\leq i \leq n$, $F_i + G_{n-i} = V$.
The set of pairs of opposite flags form the dense orbit of the
general linear group $GL(V)$ acting on the space of all pairs
of flags.
Using this observation and Kleiman's Theorem concerning the
transversality of a general translate~\cite{Kleiman},
we conclude that for any $w,v\in S_n$ and opposite flags ${F\!_{\DOT}\,}$ and
${G_{\DOT}}$,
$X_w{F\!_{\DOT}\,}$ and $X_v{G_{\DOT}}$ intersect generically transversally.
(One may also check this directly by examining the tangent spaces.)
In this case the intersection is either empty or it is
irreducible and contains a
dense subset isomorphic to $(\Bbb{C}^\times\!)^m$,
where $m+ \ell(w) + \ell(v)= \frac{1}{2}n(n-1)$ (cf.~\cite{Deodhar}).
These facts hold for the Schubert subvarieties of $G_kV$ as well.
Namely, if $\lambda$ and $\mu$ are any partitions and
${F\!_{\DOT}\,}$ and ${G_{\DOT}}$ are opposite flags,
then $\Omega_\lambda{F\!_{\DOT}\,} \bigcap \Omega_\mu {G_{\DOT}}$ is either
empty or it is an irreducible,
generically transverse intersection containing a dense subset isomorphic
to $(\Bbb{C}^\times\!)^m$,
where $m+ |\lambda|+|\mu| = k(n-k)$.
Let ${F\!_{\DOT}\,}$ and ${{F\!_{\DOT}}'\,}$ be opposite flags in $V$.
Let $e_1,\ldots,e_n$ be a basis for $V$ such that
$e_i$ generates the one dimensional subspace
$F_{n+1-i}\bigcap F'_i$.
We deduce Lemma~\ref{lemma:pushforward} from the following two
results of this section.
\begin{lemma}\label{lemma:geometry_statementI}
Let $w, w' \in S_n$ with $w <_k w'$ and
$\ell(w')-\ell(w) =m$.
Suppose that
$w' = w t_{a_1\,b_1}\cdots t_{a_m\,b_m}$ with
$a_i\leq k <b_i$ for $1\leq 1\leq m$
and $\ell(w t_{a_1\,b_1}\cdots t_{a_i\,b_i}) = \ell(w)+i$.
Let $\pi : \Bbb{F}(V) \rightarrow G_kV$ be the canonical projection.
Define $Y=\langle e_{w(j)}\,|\,j\leq k\,\mbox{ or }\,w(j)\neq w'(j)\rangle$.
Then $Y$ has codimension $d=n- k - \#\{b_1,\ldots,b_m\}$
and
$$
\pi ( X_w {F\!_{\DOT}\,} \bigcap X_{w_0w'}{{F\!_{\DOT}}'\,} ) \subset G_k Y.
$$
Also, if ${E_{\DOT}\,}=\Span{f_1,\ldots,f_n}
\in X_w {F\!_{\DOT}\,} \bigcap X_{w_0w'}{{F\!_{\DOT}}'\,}$, then we may assume that for
$j>k$ with $w(j) = w'(j)$, we have $f_j = e_{w(j)}$.
\end{lemma}
\begin{lemma}\label{lemma:geometry_statementII}
Let $w, w' \in S_n$
with $w \stackrel{r[k,m]}{\relbar\joinrel\llra} w'$ and let
$a_1,\ldots,b_m$ be as in the statement of
Lemma~\ref{lemma:geometry_statementI}.
Then there exist opposite flags
${G_{\DOT}}$ and ${G_{\DOT}}\!'$ and partitions
$\lambda\supset\mu$, with $\lambda/\mu$ a skew row of length $m$
whose $j$th row has length $\#\{i\,|\, a_i = j\}$
such that
$$
\pi ( X_w {F\!_{\DOT}\,} \bigcap X_{w_0w'}{{F\!_{\DOT}}'\,} )
\quad =\quad \Omega_{\mu}{G_{\DOT}} \bigcap \Omega_{\lambda^c}{G_{\DOT}}\!',
$$
and the map $\pi|_{X_w {F\!_{\DOT}\,} \bigcap X_{w_0w'}{{F\!_{\DOT}}'\,}}
: X_w {F\!_{\DOT}\,} \bigcap X_{w_0w'}{{F\!_{\DOT}}'\,} \rightarrow
\Omega_{\mu}{G_{\DOT}} \bigcap \Omega_{\lambda^c}{G_{\DOT}}\!'$
has degree 1.
\end{lemma}
Lemma~\ref{lemma:geometry_statementII} is the surprising connection
to the classical Pieri's rule that was mentioned in the Introduction.
A typical geometric proof of Pieri's rule for Grassmannians
(see~\cite{Griffiths_Harris,Hodge_Pedoe}) involves showing a
triple intersection of Schubert varieties
\begin{equation}\label{eq:triple_intersection}
\Omega_{\lambda}{G_{\DOT}} \bigcap \Omega_{\mu^c}{G_{\DOT}}\!'
\,\bigcap\, \Omega_m{G_{\DOT}}\!''
\end{equation}
is transverse and consists of a single point,
when ${G_{\DOT}}, {G_{\DOT}}\!'$, and ${G_{\DOT}}\!''$ are in suitably general position.
We would like to construct a proof of Theorem~\ref{thm:main} along those lines,
studying a triple intersection of Schubert subvarieties
\begin{equation}\label{eq:triple_intersection_II}
X_w {G_{\DOT}} \bigcap X_{w_0w'}{G_{\DOT}}\!'
\,\bigcap\, X_{r[k,m]}{G_{\DOT}}\!'',
\end{equation}
where ${G_{\DOT}}, {G_{\DOT}}\!'$, and ${G_{\DOT}}\!''$ are in suitably general position.
Doing so, one observes that the geometry of the intersection
of~(\ref{eq:triple_intersection_II}) is governed entirely by the
geometry of an intersection similar to that in~(\ref{eq:triple_intersection}).
In part, that is because $ X_{r[k,m]}{G_{\DOT}}\!''=\pi^{-1}\Omega_m{G_{\DOT}}\!''$.
This is the spirit of our method, which may be seen most vividly in
Lemmas 14 and 15.
\bigskip
\noindent{\bf Proof of Lemma~\ref{lemma:pushforward}:}
Since ${F\!_{\DOT}\,}$ and ${{F\!_{\DOT}}'\,}$ are opposite flags,
$X_w {F\!_{\DOT}\,} \bigcap X_{w_0w'}{{F\!_{\DOT}}'\,}$
is a generically transverse intersection, so in the cohomology ring
$$
[X_w {F\!_{\DOT}\,} \bigcap X_{w_0w'}{{F\!_{\DOT}}'\,}]\ = \
[X_w {F\!_{\DOT}\,}]\cdot [X_{w_0w'}{{F\!_{\DOT}}'\,}]\ =\ \frak{S}_w\cdot\frak{S}_{w_0w'}.
$$
Let $Y$ be the subspace of Lemma~\ref{lemma:geometry_statementI}.
Since $\pi ( X_w {F\!_{\DOT}\,} \bigcap X_{w_0w'}{{F\!_{\DOT}}'\,} ) \subset G_k Y$,
the class $\pi_*(\frak{S}_w\cdot\frak{S}_{w_0w'})$ is a cohomology class
on $G_kY$.
However, all such classes are of the form $\delta \cdot [ G_k Y]$,
for some cohomology class $\delta$ on $G_kV$.
Since $d$ is the codimension of $Y$, we have $[G_k Y]= s_{d^k}$,
establishing part (1) of Lemma~\ref{lemma:pushforward}.
For part (2), suppose further that $w \stackrel{r[k,m]}{\relbar\joinrel\llra} w'$.
If $\rho$ is the restriction of $\pi$ to
$X_w {F\!_{\DOT}\,} \bigcap X_{w_0w'}{{F\!_{\DOT}}'\,}$, then
$$
\pi_*(\frak{S}_w\cdot\frak{S}_{w_0w'}) \ = \
\pi_*([X_w {F\!_{\DOT}\,} \bigcap X_{w_0w'}{{F\!_{\DOT}}'\,}])
\ = \ \deg \rho\cdot [\pi(X_w {F\!_{\DOT}\,} \bigcap X_{w_0w'}{{F\!_{\DOT}}'\,})].
$$
By Lemma~\ref{lemma:geometry_statementII},
$\deg \rho = 1$ and
$\pi(X_w{F\!_{\DOT}\,} \bigcap x_{w_0w'}{{F\!_{\DOT}}'\,}) =
\Omega_\mu {G_{\DOT}} \bigcap \Omega_{\lambda^c} {G_{\DOT}}\!'$.
Since
${G_{\DOT}}$ and ${G_{\DOT}}\!'$ are opposite flags, we have
$$
\pi_*(\frak{S}_w\cdot\frak{S}_{w_0w'}) \ = \
1\cdot [ \Omega_{\lambda}{G_{\DOT}} \bigcap \Omega_{\mu^c}{G_{\DOT}}\!']
\ = \ [ \Omega_{\lambda}{G_{\DOT}} ] \cdot [\Omega_{\mu^c}{G_{\DOT}}\!']
\ =\ s_\lambda \cdot s_{\mu^c},
$$
completing the proof of Lemma~\ref{lemma:pushforward}.\QED
\bigskip
We deduce Lemma~\ref{lemma:geometry_statementI} from a series of lemmas.
We first make a definition.
Let $W\subsetneq V$ be a codimension 1 subspace and let $e \in V - W$
so that $V = \Span{W,e}$.
For $1\leq p \leq n$, define an expanding map
$\psi_p: \Bbb{F}(W) \rightarrow \Bbb{F}(V)$ as follows
$$
(\psi_p {E_{\DOT}\,})_i = \left\{
\begin{array}{ll} E_i & \mbox{ if } i<p\\
\Span{E_{i-1},e} & \mbox{ if } i \geq p
\end{array}\right. .
$$
Note that if ${E_{\DOT}\,} = \Span{f_1,\ldots,f_{n-1}}$, then
$\psi_p {E_{\DOT}\,}= \Span{f_1,\ldots,f_{p-1},e,f_p,\ldots,f_{n-1}}$.
For $w\in S_n$ and $1\leq p \leq n$, define $w|_p \in S_{n-1}$ by
$$
w|_p (j) = \left\{
\begin{array}{ll}
w(j) & \mbox{ if } j<p \mbox{ and } w(j)<w(p)\\
w(j{+}1) & \mbox{ if } j\geq p \mbox{ and } w(j)<w(p)\\
w(j) - 1 & \mbox{ if } j<p \mbox{ and } w(j)>w(p)\\
w(j{+}1)-1 & \mbox{ if } j\geq p \mbox{ and } w(j)>w(p)
\end{array} \right. .
$$
If we represent permutations as matrices, $w|_p$ is obtained
by crossing out the $p$th row and $w(p)$th column of the matrix
for $w$.
\begin{lemma}\label{lemma:expand}
Let $W\subsetneq V$ and $e\in V - W$ with $V = \Span{W,e}$.
Let ${G_{\DOT}}$ be a complete flag in $W$.
For $1\leq p \leq n$ and
$w\in S_n$,
$$
\psi_p\left( X_{w|_p}{G_{\DOT}} \right) \subset
X_w \left(\psi_{w_0w(p)}({G_{\DOT}})\right).
$$
\end{lemma}
\noindent{\bf Proof:}
Let ${E_{\DOT}\,} \in X_{w|_p}{G_{\DOT}} $. Then $W$ has a basis
$f_1,\ldots,f_{n-1}$ with ${E_{\DOT}\,} = \Span{f_1,\ldots,f_{n-1}}$
and for each $1\leq i \leq n-1$,
$f_i \in G_{n-w|_p(i)}$.
Then we necessarily have
$
\psi_p({E_{\DOT}\,}) = \Span{\phi_1,\ldots,\phi_n}
= \Span{f_1,\ldots,f_{p-1},e,f_p,\ldots,f_{n-1}}$.
Noting
$$
\left(\psi_{w_0w(p)}({G_{\DOT}})\right)_{n+1-j} \ = \ \left\{
\begin{array}{ll} G_{n+1-j} & \mbox{ if } j >w(p) \\
\Span{e,G_{n-j}} & \mbox{ if } j \leq w(p) \end{array}\right. ,
$$
we see that $\phi_i \in
\left(\psi_{w_0w(p)}({G_{\DOT}})\right)_{n+1-w(i)}$.
Thus $\psi_p\left( X_{w|_p}{G_{\DOT}} \right) \subset
X_w \left(\psi_{w_0w(p)}({G_{\DOT}})\right)$.
\QED
\begin{lemma}\label{lemma:expanding}
Let $W\subsetneq V$ and $e\in V - W$ with $V = \Span{W,e}$ and
let ${G_{\DOT}}$ and ${G_{\DOT}}\!'$ be opposite flags in $W$.
Suppose that $w<_kw'$ are permutations in $S_n$ and
$p>k$ an integer such that $w(p) = w'(p)$.
Let $w_0^{(j)}$ is the longest permutation in $S_j$.
Then
\begin{enumerate}
\item $\ell(w'|_p)-\ell(w|_p)=\ell(w')-\ell(w)$ and $w|_p <_k w'|_p$.
\item $\psi_p{\left(X_{w|_p}{G_{\DOT}} \bigcap X_{w_0^{(n-1)}(w'|_p)}{G_{\DOT}}\!'\right)}
\, = \,X_w\left(\psi_{w_0^{(n)}w(p)}({G_{\DOT}}) \right)
\bigcap X_{w_0^{(n)}w'}\left(\psi_{w'(p)}({G_{\DOT}}\!')\right)$.
\item If ${E_{\DOT}\,} \in
X_w\left(\psi_{w_0^{(n)}w(p)}({G_{\DOT}}) \right)
\bigcap X_{w_0^{(n)}w'}\left(\psi_{w'(p)}({G_{\DOT}}\!')\right)$,
then $E_p = \Span{E_{p-1},e}$.
\item If ${F\!_{\DOT}\,}$ and ${{F\!_{\DOT}}'\,}$ are opposite flags
in $V$ and
${E_{\DOT}\,} \in X_w{F\!_{\DOT}\,}\bigcap X_{w_0^{(n)}w'}{{F\!_{\DOT}}'\,}$,
then $E_k \in F_{n-w(p)}+F'_{w(p)-1}$.
\end{enumerate}
\end{lemma}
\noindent{\bf Proof:}
First recall that $\ell(v t_{a\,b}) = \ell(v)+1$ if and only if
$v(a) < v(b)$ and if $a<j<b$, then
$v(j)$ is not between $v(a)$ and $v(b)$.
Thus if $\ell(v t_{a\,b}) = \ell(v)+1$ and $p\not\in \{a,b\}$,
we have $\ell(v t_{a\,b}|_p) = \ell(v|_p)+1$.
Statement (1) follows by induction on
$\ell(w') - \ell(w)$.
For (2), since $(w_0^{(n)}w')|_p = w_0^{(n-1)}(w'|_p)$ and
$w_0^{(n)}w_0^{(n)}w'= w'$, Lemma~\ref{lemma:expand}
shows
$$
\psi_p\left(X_{w|_p}{G_{\DOT}} \bigcap
X_{w_0^{(n-1)}(w'|_p)}{G_{\DOT}}\!'\right)
\subset X_w\left(\psi_{w_0^{(n)}w(p)}({G_{\DOT}})\right)
\bigcap X_{w_0^{(n)}w'}\left(\psi_{w'(p)}({G_{\DOT}}\!')\right).
$$
The flags $\psi_{w_0^{(n)}w(p)}({G_{\DOT}})$ and
$\psi_{w'(p)}({G_{\DOT}}\!')$ are opposite flags in $V$, since
${G_{\DOT}}$ and ${G_{\DOT}}\!'$ are opposite flags in $W$.
Then part (1) shows both sides have the same dimension.
Since $\psi_p$ is injective, they are equal.
To show (3), let ${E_{\DOT}\,} \in
X_w\left(\psi_{w_0^{(n)}w(p)}({G_{\DOT}}) \right)
\bigcap X_{w_0^{(n)}w'}\left(\psi_{w'(p)}({G_{\DOT}}\!')\right)$.
By (2), there is a flag
${E_{\DOT}\!'\,} \in X_{w|_p}{G_{\DOT}} \bigcap
X_{w_0^{(n-1)}(w'|_p)}{G_{\DOT}}\!'$
with $\psi_p({E_{\DOT}\!'\,}) = {E_{\DOT}\,}$,
so $E_p = \Span{E'_{p-1},e} = \Span{E_{p-1},e}$.
For (4), let $W= F_{n-w(p)}+F'_{w'(p)-1}$ and $e$
any nonzero vector in the one dimensional space
$F_{n+1-w(p)}\bigcap F'_{w'(p)}$.
The distinct subspaces in ${F\!_{\DOT}\,}\bigcap W$ define a flag ${G_{\DOT}}$,
and those in ${{F\!_{\DOT}}'\,} \bigcap W$ define a flag ${G_{\DOT}}\!'$.
In fact, $\psi_{w_0^{(n)}w(p)}({G_{\DOT}})={F\!_{\DOT}\,}$
and $\psi_{w(p)}({G_{\DOT}}\!') = {{F\!_{\DOT}}'\,}$, and ${G_{\DOT}}$ and
${G_{\DOT}}\!'$ are opposite flags in $W$.
By (2),
$$
\psi_p\left(X_{w|_p}{G_{\DOT}} \bigcap
X_{w_0^{(n-1)}(w'|_p)}{G_{\DOT}}\!'\right)
= X_w{F\!_{\DOT}\,}
\bigcap X_{w_0^{(n)}w'}{{F\!_{\DOT}}'\,}.
$$
Thus flags in
$X_w{F\!_{\DOT}\,} \bigcap X_{w_0^{(n)}w'}{{F\!_{\DOT}}'\,}$
are in the image of $\psi_p$.
As $k < p$, $\left( \psi_p {E_{\DOT}\,}\right)_k = E_k\subset W$,
establishing part (4).
\QED
\noindent{\bf Proof of Lemma~\ref{lemma:geometry_statementI}:}
Let ${F\!_{\DOT}\,}$ and ${{F\!_{\DOT}}'\,}$ be opposite flags in $V$, let
$w<_k w'$ and let
${E_{\DOT}\,} \in X_w{F\!_{\DOT}\,} \bigcap X_{w_0w'}{{F\!_{\DOT}}'\,}$.
Define a basis $e_1,\ldots,e_n$ for $V$ by
$F_{n+1-j}\bigcap F'_j = \Span{e_j}$ for $1\leq j\leq n$.
Suppose $w' = w t_{a_1\,b_1}\cdots t_{a_m\,b_m}$ with
$a_i\leq k < b_i$.
Let $\{p_1,\ldots,p_d\}$ be the complement of
$\{b_1,\ldots,b_m\}$ in $\{k+1,\ldots,n\}$.
For $1\leq i\leq d$, let
$Y_i = \Span{e_1,\ldots,\widehat{e_{w(p_i)}},\ldots,e_n}
= \Span{e_1,\ldots,e_{w(p_i)-1},e_{w(p_i)+1},\ldots,e_n}$.
Since $w(p_i) = w'(p_i)$ and $k<p_i$, we see that
$Y_i = F_{n-w(p_i)} + F'_{w(p_i)-1}$, so part (4) of
Lemma~\ref{lemma:expanding} shows $E_k \subset Y_i$.
Thus
$$
E_k \in
\bigcap_{i=1}^d Y_i \ = \ \Span{e_{w(j)}\,|\, j<k\mbox{ or }j=b_i}
\ = \ Y.
$$
Since $w(p_i) = w'(p_i)$ for $1\leq i \leq d$,
we have $E_{p_i} = \Span{E_{p_i-1},e_{w(p_i)}}$, by part (3) of
Lemma~\ref{lemma:expanding}.
So if ${E_{\DOT}\,} = \Span{f_1,\ldots,f_n}$, we may assume that
$f_{p_i} = e_{w(p_i)} \in F_{n+1-w(p_i)} \cap F'_{w'(p_i)}$ for
$1\leq i \leq d$, completing the proof.
\QED
To prove Lemma~\ref{lemma:geometry_statementII}, we begin by describing
an intersection in a Grassmannian.
Recall that $\Omega_\lambda{F\!_{\DOT}\,} =
\{H\in G_kV\,|\, \dim H\cap F_{k-j+\lambda_j} \geq j \mbox{ for }
1\leq j\leq k\}$.
\begin{lemma}\label{lemma:grassmannian}
Suppose that $L_1,\ldots,L_k,M \subset V$ with
$V = M \bigoplus L_1\bigoplus\cdots\bigoplus L_k$.
Let $r_j= \dim L_j -1$ and $m = r_1 + \cdots + r_k$.
Then there are opposite flags ${F\!_{\DOT}\,}$ and ${{F\!_{\DOT}}'\,}$ and
partitions $\lambda\supset \mu$ with $\lambda_j - \mu_j = r_j$
and $\lambda/\mu$ a skew row of length $m$ such that in
$G_kV$,
$$
\Omega_\mu{F\!_{\DOT}\,} \bigcap \Omega_{\lambda^c}{{F\!_{\DOT}}'\,} =
\{ H\in G_kV\,|\, \dim H\bigcap L_j = 1 \mbox{ for } 1\leq j \leq k\}.
$$
\end{lemma}
\noindent{\bf Proof:}
Let $\mu_k =0$ and $\mu_j = r_{j+1}+\cdots+r_k$ for
$1\leq j <k$ and $\lambda_j =r_j+\mu_j$ for $1\leq j\leq k$.
Choose a basis $e_1,\ldots,e_n$ for $V$ such that
\begin{eqnarray*}
L_j & =&
\Span{e_{k+1-j+\mu_j},e_{k+2-j+\mu_j},\ldots,e_{k+1+r_j-j+\mu_j}
=e_{k+1-j+\lambda_j}}\\
M &=& \Span{e_{m+k+1},\ldots,e_n}
\end{eqnarray*}
Let ${F\!_{\DOT}\,} = \Span{e_n\ldots,e_1}$ and ${{F\!_{\DOT}}'\,}=\Span{e_1,\ldots,e_n}$.
Then
\begin{center}
$F_{n-k+j-\mu_j}\quad =\quad
M\bigoplus L_1 \bigoplus \cdots \bigoplus L_j \quad\quad$\\
$F'_{n-k+(k+1-j)-\lambda^c_{k+1-j}} \quad = \quad F'_{k+1-j+\lambda_j}
\quad =\quad
L_j\bigoplus \cdots \bigoplus L_k$.
\end{center}
If $H\in \Omega_\mu{F\!_{\DOT}\,} \bigcap \Omega_{\lambda^c}{{F\!_{\DOT}}'\,}$, then
$\dim H\bigcap F_{n-k+j-\mu_j} \geq j$ for $1\leq j\leq k$
and
$$
\dim H\bigcap F'_{n-k+(k+1-j)-\lambda^c_{k+1-j}}\geq k+1-j,
$$
for $1\leq j\leq k$.
Thus for $1\leq j \leq k$,
$$
\dim H\bigcap F_{n-k+j-\mu_j} \bigcap
F'_{n-k+(k+1-j)-\lambda^c_{k+1-j}} \geq 1.
$$
But $F_{n-k+j-\mu_j} \bigcap F'_{n-k+(k+1-j)-\lambda^c_{k+1-j}} = L_j$,
so $\dim H\bigcap L_j\geq 1$ for $1\leq j \leq k$.
Since $L_j\bigcap L_i = \{0\}$ if $j\neq i$, we see that
$\dim H\bigcap L_j =1$.
Thus
$$
\Omega_\mu{F\!_{\DOT}\,} \bigcap \Omega_{\lambda^c}{{F\!_{\DOT}}'\,} \subset
\{ H\in G_kV\,|\, \dim H\bigcap L_j = 1 \mbox{ for } 1\leq j \leq k\}.
$$
We show these varieties have the same dimension,
establishing their equality:
Since $|\lambda| = |\mu| +m$, and ${F\!_{\DOT}\,}$ and ${{F\!_{\DOT}}'\,}$ are opposite
flags, $\Omega_\mu{F\!_{\DOT}\,} \bigcap \Omega_{\lambda^c}{{F\!_{\DOT}}'\,}$
has dimension $m$.
But the map
$H \mapsto (H\bigcap L_1,\ldots,H\bigcap L_k)$
defines an isomorphism between
$\{ H\in G_kV\,|\, \dim H\bigcap L_j = 1 \mbox{ for } 1\leq j \leq k\}$
and $\Bbb{P}L_1 \times \cdots \times \Bbb{P}L_k$,
which has dimension $\sum_j (\dim L_j-1) = m$.
Here, $\Bbb{P}L_j$ is the projective space of one dimensional
subspaces of $L_j$.
\QED
We relate this to intersections of Schubert varieties in the
flag manifold.
\begin{lemma}\label{lemma:intersection_calculation}
Suppose that $w\stackrel{r[k,m]}{\relbar\joinrel\llra} w'$ and
$w' = wt_{a_1\,b_1}\cdots t_{a_m\,b_m}$ with $a_i\leq k<b_i$ and
$\ell(wt_{a_1\,b_1}\cdots t_{a_i\,b_i}) = \ell(w)+i$.
Let ${F\!_{\DOT}\,}$ and ${{F\!_{\DOT}}'\,}$ be opposite flags in $V$ and let
$\Span{e_i} = F_{n+1-i}\bigcap F'_{i}$.
Define
\begin{eqnarray*}
L_j & =& \Span{e_j, e_{w(b_i)}\,|\, a_i = j}\\
M & =&\Span{e_{w(p)}\,|\, k<p \mbox{ and } w(p) = w'(p)}.
\end{eqnarray*}
Then
\begin{enumerate}
\item $\dim L_j = 1 +\#\{i\,|\, a_i = j\}$ and
$ V = M \bigoplus L_1\bigoplus \cdots \bigoplus L_k$.
\item If ${E_{\DOT}\,} \in X_w{F\!_{\DOT}\,} \bigcap X_{w_0w'}{{F\!_{\DOT}}'\,}$, then
$\dim E_k \bigcap L_j = 1$ for $1\leq j\leq k$.
\item
Let $\pi$ be the map induced by ${E_{\DOT}\,} \mapsto E_k$.
Then
$$
\pi : X_w{F\!_{\DOT}\,} \bigcap X_{w_0w'}{{F\!_{\DOT}}'\,}
\rightarrow \{H\in G_kV\,|\, \dim H\bigcap L_j = 1 \mbox{ for }
1\leq j\leq k\}
$$
is surjective and of degree 1.
\end{enumerate}
\end{lemma}
\noindent{\bf Proof:}
Part (1) is immediate.
For (2) and (3), note that both
$\{H\in G_kV\,|\, \dim H\bigcap L_j = 1 \mbox{ for }
1\leq j\leq k\}$ and $X_w{F\!_{\DOT}\,} \bigcap X_{w_0w'}{{F\!_{\DOT}}'\,}$
are irreducible and have dimension $m$.
We exhibit an $m$ dimensional subset of each over which $\pi$ is
an isomorphism.
Let $\alpha = (\alpha_1,\ldots,\alpha_m) \in (\Bbb{C}^\times\!)^m$
be an $m$-tuple of nonzero complex numbers.
We define a basis $f_1,\ldots,f_n$ of $V$ depending upon
$\alpha$ as follows.
$$
f_j = \left\{
\begin{array}{ll}
e_{w(j)} +
{\displaystyle \sum_{i : a_i=j}
\alpha_i e_{w(b_i)}} &
\mbox{ if } j\leq k \\
e_{w(j)} & \mbox{ if } j>k \mbox{ and } j\not\in\{b_1,\ldots,b_m\}\\
\rule{0pt}{22pt}{\displaystyle
\sum_{\shortstack{\scriptsize $i: a_i = a_q$\\ \scriptsize $w(b_i)\geq w(j)$}}
\alpha_i e_{w(b_i)}} & \mbox{ if } j = b_q > k
\end{array} \right. .
$$
Let $i_1<\cdots < i_s$ be those
integers $i_l$ with $a_{i_l} = j$.
Since $t_{a_i\,b_i}$ lengthens the permutation
$wt_{a_1\,b_1}\cdots t_{a_{i-1}\,b_{i-1}}$, we see that
$$
\begin{array}{ccccccc}
w(j) & < & w(b_{i_1}) & < & \cdots & < & w(b_{i_s}) \\
\parallel & & \parallel & & & & \parallel \\
w'(b_{i_1}) & < & w'(b_{i_2}) & < & \cdots & < & w'(j)
\end{array}
$$
Thus the first term in $f_j$ is proportional to $e_{w(j)}$.
Hence $f_j \in F_{n+1-w(j)} - F_{n-w(j)}$, and so
$f_1,\ldots,f_n$ is a basis of $V$ and the flag
${E_{\DOT}\,}(\alpha) = \Span{f_1,\ldots,f_n}$ is in $X_w{F\!_{\DOT}\,}$.
Note that $f'_1,\ldots,f'_n$ is also a basis for ${E_{\DOT}\,}(\alpha)$, where
$f'_j$ is given by
$$
f'_j = \left\{
\begin{array}{ll}
f_j &
\mbox{ if } j\leq k \\
f_j& \mbox{ if } j>k \mbox{ and } j\not\in\{b_1,\ldots,b_m\}\\
f_{a_q} - f_j & \mbox{ if } j = b_q > k
\end{array} \right. .
$$
Here, the last term in each $f'_j$ is proportional to $e_{w'(j)}$,
so $f'_j \in F'_{w'(j)} = F'_{n+1-w_0w'(j)}$,
showing that
${E_{\DOT}\,}(\alpha) \in X_{w_0w'}{{F\!_{\DOT}}'\,}$.
Since $f_j \in L_j$ for $1\leq j\leq k$, we have
$\dim {E_{\DOT}\,}(\alpha) \bigcap L_j = 1$ for $1\leq j\leq k$.
As $\{{E_{\DOT}\,}(\alpha)\,|\, \alpha \in (\Bbb{C}^\times\!)^m\}$
is a subset of $ X_w{F\!_{\DOT}\,}\bigcap X_{w_0w'}{{F\!_{\DOT}}'\,}$ of dimension $m$,
it is dense.
Thus if ${E_{\DOT}\,} \in X_w{F\!_{\DOT}\,}\bigcap X_{w_0w'}{{F\!_{\DOT}}'\,}$, then
$\dim E_k \bigcap L_j = 1$ for $1\leq j\leq k$.
The set $\{({E_{\DOT}\,}(\alpha))_k\,|\, \alpha \in (\Bbb{C}^\times\!)^m\}$
is a dense subset of
$$
\{H\in G_kV\,|\, \dim H\bigcap L_j = 1 \mbox{ for } 1\leq j\leq k\}
\ \simeq \ \Bbb{P}L_1\times\cdots\times\Bbb{P}L_k.
$$
Since $\pi$ is an isomorphism of this set with
$\{{E_{\DOT}\,}(\alpha)\,|\, \alpha \in (\Bbb{C}^\times\!)^m\}$,
the map
$$
\pi: X_w{F\!_{\DOT}\,}\bigcap X_{w_0w'}{{F\!_{\DOT}}'\,} \rightarrow
\{H\in G_kV\,|\, \dim H\bigcap L_j = 1 \mbox{ for } 1\leq j\leq k\}
$$
is surjective of degree 1, proving the lemma.
\QED
We note that Lemma~\ref{lemma:geometry_statementII} is an
immediate consequence of Lemmas~\ref{lemma:grassmannian}
and~\ref{lemma:intersection_calculation}(3).
\section{Examples}
In this section we describe two examples, which should serve to illustrate
the results of Section 5.
This manuscript differs from the version we are submitting
for publication only by the inclusion of this section, and
its mention in the Introduction.
\bigskip
Fix a basis $e_1,\ldots,e_7$ for $\Bbb{C}\,^7$.
This gives coordinates for vectors in $\Bbb{C}\,^7$, where
$(v_1,\ldots,v_7)$ corresponds to
$v_1e_1{+}\cdots{+}v_7e_7$.
Define the opposite flags
${F\!_{\DOT}\,}$ and ${{F\!_{\DOT}}'\,}$ by
$$
{F\!_{\DOT}\,} = \langle e_7,e_6,e_5,e_4,e_3,e_2,e_1\rangle \ \
\mbox{and} \ \
{{F\!_{\DOT}}'\,} = \langle e_1,e_2,e_3,e_4,e_5,e_6,e_7\rangle.
$$
For example, $F_3 = \langle e_7,e_6,e_5 \rangle$ and
$F'_4 = \langle e_1,e_2,e_3,e_4 \rangle$.
Let $w= 5412763$, $w' = 6524713$ and $w'' = 7431652$ be permutations in
$S_7$.
(We denote permutations by the sequence of their values.)
Their lengths are 10, 14, and 14, respectively, and
$w<_4 w'$ and $w<_3 w''$.
We seek to describe the intersections
$$
X_w{F\!_{\DOT}\,} \bigcap X_{w_0w'}{{F\!_{\DOT}}'\,} \ \ \ \
\mbox{and} \ \ \ \ X_w{F\!_{\DOT}\,} \bigcap X_{w_0w''}{{F\!_{\DOT}}'\,}.
$$
Rather than describe each in full, we describe a dense subset of
each which is isomorphic to the torus, $(\Bbb{C}^\times\!)^4$.
This suffices for our purposes.
Recall that the Schubert cell $X^\circ_w{F\!_{\DOT}\,}$ is defined to be
$$
X^{\circ}_w {F\!_{\DOT}\,} = \{ {E_{\DOT}\,}=\Span{f_1,\ldots,f_7}\,|\,
f_i \in F_{8-w(i)}-F_{7-w(i)}, \,1\leq i\leq 7\}.
$$
Using the given coordinates of $\Bbb{C}\,^7$,
we may write a typical element of
$X^\circ_w{F\!_{\DOT}\,}$ in a unique manner.
For each $f_i \in F_{8-w(i)}-F_{7-w(i)}$, the coordinate 7-tuple
for $f_i$ has zeroes in the places $1,\ldots,w(i)-1$
and a nonzero coordinate in its $w(i)$th place, which we
assume to be 1.
We may also assume that the $w(j)$th coordinate of $f_i$ is zero for those
$j<i$ with $w(j)> w(i)$, by subtracting a suitable multiple of $f_j$.
Writing the coordinates of $f_1,\ldots,f_7$ as rows of an array, we
conclude that a typical flag in $X^{\circ}_w {F\!_{\DOT}\,}$ has a unique
representation of the following form:
$$
\begin{array}{ccccccc}
\cdot &\cdot&\cdot&\cdot& 1 & * & * \\
\cdot &\cdot&\cdot& 1 &\cdot& * & * \\
1 & * & * &\cdot&\cdot& * & * \\
\cdot & 1 & * &\cdot&\cdot& * & * \\
\cdot &\cdot&\cdot&\cdot&\cdot&\cdot& 1 \\
\cdot &\cdot&\cdot&\cdot&\cdot& 1 &\cdot\\
\cdot &\cdot& 1 &\cdot&\cdot&\cdot& \cdot
\end{array}
$$
Here, the $i$th column contains the coefficients of $e_i$,
the $\cdot$'s represent 0,
and the $*$'s indicate some complex numbers, uniquely
determined by the flag.
Likewise, flags in $X^{\circ}_{w_0w'} {{F\!_{\DOT}}'\,}$
and $X^{\circ}_{w_0w''} {{F\!_{\DOT}}'\,}$ have unique bases of the forms:
$$
\begin{array}{cccccccc}
& * & * & * & * & * & 1 &\cdot\\
& * & * & * & * & 1 &\cdot&\cdot\\
& * & 1 &\cdot&\cdot&\cdot&\cdot&\cdot\\
& * &\cdot& * & 1 &\cdot&\cdot&\cdot\\
& * &\cdot& * &\cdot&\cdot&\cdot& 1 \\
& 1 &\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
& \cdot&\cdot& 1 &\cdot&\cdot&\cdot& \cdot
\end{array}
\hspace{1in}
\begin{array}{cccccccc}
& * & * & * & * & * & * & 1 \\
& * & * & * & 1 &\cdot&\cdot&\cdot\\
& * & * & 1 &\cdot&\cdot&\cdot&\cdot\\
& 1 &\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
& \cdot& * &\cdot&\cdot& * & 1 &\cdot\\
& \cdot& * &\cdot&\cdot& 1 &\cdot&\cdot\\
& \cdot& 1 &\cdot&\cdot&\cdot&\cdot& \cdot
\end{array}
$$
Let $\alpha,\beta,\gamma$ and $\delta$ be four nonzero complex numbers.
Define bases $f_1,f_2,\ldots,f_7$ and $g_1,g_2,\ldots,g_7$ by the
following arrays of coordinates.
$$
\begin{array}{ccccccccc}
f_1 &= &\cdot& \cdot &\cdot& \cdot & 1 & \alpha &\cdot\\
f_2 &= &\cdot& \cdot &\cdot& 1 & \beta & \cdot &\cdot\\
f_3 &= & 1 & \gamma &\cdot& \cdot & \cdot & \cdot &\cdot\\
f_4 &= &\cdot& 1 &\cdot& \delta & \cdot & \cdot &\cdot\\
f_5 &= &\cdot& \cdot &\cdot& \cdot & \cdot & \cdot & 1 \\
f_6 &= &\cdot& \cdot &\cdot& \cdot & \cdot & \alpha &\cdot\\
f_7 &= &\cdot& \cdot & 1 & \cdot & \cdot & \cdot &\cdot
\end{array}
\hspace{1in}
\begin{array}{ccccccccc}
g_1 & = &\cdot& \cdot & \cdot &\cdot& 1 & \alpha & \beta \\
g_2 & = &\cdot& \cdot & \cdot & 1 &\cdot& \cdot & \cdot \\
g_3 & = & 1 & \gamma & \delta &\cdot&\cdot& \cdot & \cdot \\
g_4 & = &\cdot& \gamma & \delta &\cdot&\cdot& \cdot & \cdot \\
g_5 & = &\cdot& \cdot & \cdot &\cdot&\cdot& \cdot & \beta \\
g_6 & = &\cdot& \cdot & \cdot &\cdot&\cdot& \alpha & \beta \\
g_7 & = &\cdot& \cdot & \delta &\cdot&\cdot& \cdot & \cdot
\end{array}
$$
Let ${E_{\DOT}\,} = \Span{f_1,f_2,\ldots,f_7}$ and
${E_{\DOT}\!'\,} = \Span{g_1,g_2,\ldots,g_7}$.
Considering the left-most nonzero entry in each row, we see that
both ${E_{\DOT}\,}$ and ${E_{\DOT}\!'\,}$ are in $X^\circ_w{F\!_{\DOT}\,}$. To see that
${E_{\DOT}\,} \in X^\circ_{w_0w'}{{F\!_{\DOT}}'\,}$ and
${E_{\DOT}\!'\,} \in X^\circ_{w_0w''}{{F\!_{\DOT}}'\,}$, note that we could
choose
$$
\begin{array}{ccccccccc}
f_6' & = & 1 & \cdot &\cdot&\cdot& \cdot & \cdot&\cdot
\end{array}
\hspace{1in}
\begin{array}{ccccccccc}
g_4' & = & 1 & \cdot &\cdot&\cdot&\cdot& \cdot &\cdot\\
g_5' & = &\cdot& \cdot &\cdot&\cdot& 1 & \alpha &\cdot\\
g_6' & = &\cdot& \cdot &\cdot&\cdot& 1 & \cdot &\cdot\\
g_7' & = & 1 & \gamma &\cdot&\cdot&\cdot& \cdot &\cdot
\end{array}
$$
Replacing the unprimed vectors by the corresponding primed ones
gives alternate bases for ${E_{\DOT}\,}$ and ${E_{\DOT}\!'\,}$.
This shows ${E_{\DOT}\,} \in X^\circ_{w_0w'}{{F\!_{\DOT}}'\,}$ and
${E_{\DOT}\!'\,} \in X^\circ_{w_0w''}{{F\!_{\DOT}}'\,}$.
We use this computation to illustrate Lemmas~\ref{lemma:geometry_statementI}
and~\ref{lemma:geometry_statementII}.
\begin{enumerate}
\item[I.] First note that for
${E_{\DOT}\,} = \Span{f_1,f_2,f_3,f_4,f_5,f_6,f_7}$ as above,
\begin{eqnarray*}
E_3 & \subset & \Span{e_1,e_2,e_5,e_5,e_6}\\
& = & \Span{ e_{w(j)}\,|\, j\leq k\mbox{ or } w(j)\neq w'(j)}\\
& = & Y,
\end{eqnarray*}
the subspace of Lemma~\ref{lemma:geometry_statementI}.
Since this holds for all ${E_{\DOT}\,}$ in a dense subset of
$X_w{F\!_{\DOT}\,} \bigcap X_{w_0w'}{{F\!_{\DOT}}'\,}$, it holds for all ${E_{\DOT}\,}$ in
that intersection.
\item[II.] Recall that $w=5412763$ and note that
$7431652 = w''= w\cdot t_{34}\cdot t_{16}\cdot t_{37}\cdot t_{15}$,
so $w \stackrel{r[3,4]}{\relbar\joinrel\longrightarrow} w''$, and we are in the situation of
Lemma~\ref{lemma:geometry_statementII}.
Let $\mu = (2,2,0)$ and $\lambda = (4,2,2)$ be partitions.
Then $\lambda^c = (2,2,0)$,
and if ${E_{\DOT}\!'\,} = {E_{\DOT}\!'\,}(\alpha,\beta,\gamma,\delta)$ is a
flag in the above form,
then
$$
E_3'(\alpha,\beta,\gamma,\delta) \in
\Omega_{\mu}{F\!_{\DOT}\,} \bigcap \Omega_{\lambda^c} {{F\!_{\DOT}}'\,},
$$
since
$$
\begin{array}{ccccl}
f_1 &\in & F_3 & = & F_{7-3+1-\mu_1} \bigcap F'_{7-3+3-\lambda^c_3}\\
f_2 &\in & \Span{e_4} & =& F_{7-3+1-\mu_2} \bigcap F'_{7-3+3-\lambda^c_2}\\
f_3 &\in & F'_3 & = & F_{7-3+1-\mu_3} \bigcap F'_{7-3+3-\lambda^c_1}.
\end{array}
$$
Furthermore, the map
$\pi: {E_{\DOT}\!'\,} \mapsto E_3'$
is injective for those ${E_{\DOT}\!'\,}(\alpha,\beta,\gamma,\delta)$ given above.
Since that set is dense in
$X_w{F\!_{\DOT}\,} \bigcap X_{w_0w''}{{F\!_{\DOT}}'\,}$,
and the set of $E_3'(\alpha,\beta,\gamma,\delta)$ is dense in
$\Omega_{\mu}{F\!_{\DOT}\,} \bigcap \Omega_{\lambda^c} {{F\!_{\DOT}}'\,}$,
it follows that
$$
\pi : X_w{F\!_{\DOT}\,} \bigcap X_{w_0w''}{{F\!_{\DOT}}'\,} \rightarrow
\Omega_{\mu}{F\!_{\DOT}\,} \bigcap \Omega_{\lambda^c} {{F\!_{\DOT}}'\,}
$$
is surjective and of degree 1.
\end{enumerate}
\smallskip
Note that the description of $X_w{F\!_{\DOT}\,} \bigcap X_{w_0w''}{{F\!_{\DOT}}'\,}$ in II
is consistent with that given for general $w\stackrel{r[k,m]}{\relbar\joinrel\llra} w''$ in the proof of
Lemma~\ref{lemma:intersection_calculation}, part (2).
This explicit description is the key to the understanding we gained
while trying to establish Theorem~\ref{thm:main}
\smallskip
Also note that $w' = w \cdot t_{16}\cdot t_{26}\cdot t_{46}\cdot t_{36}$,
thus $w \stackrel{c[4,4]}{\relbar\joinrel\llra} w'$.
In I above, we give an explicit description of the intersection
$X_w{F\!_{\DOT}\,} \bigcap X_{w_0w'}{{F\!_{\DOT}}'\,}$.
This may be generalized to give a similar description whenever
$w \stackrel{c[k,m]}{\relbar\joinrel\llra} w'$, and may be used to establish Theorem~\ref{thm:main}
in much the same manner as we used the explicit description
of intersections when $w\stackrel{r[k,m]}{\relbar\joinrel\llra} w'$.
|
1998-03-02T21:23:39 | 9505 | alg-geom/9505021 | en | https://arxiv.org/abs/alg-geom/9505021 | [
"alg-geom",
"math.AG"
] | alg-geom/9505021 | null | Fabrizio Catanese and Klaus Hulek | Rational surfaces in P^4 containing a plane curve | 25 pages, LaTeX2e | null | null | null | null | The families of smooth rational surfaces in $\PP^4$ have been classified in
degree $\le 10$. All known rational surfaces in $\PP^4$ can be represented as
blow-ups of the plane $\PP^2$. The fine classification of these surfaces
consists of giving explicit open and closed conditions which determine the
configurations of points corresponding to all surfaces in a given family. Using
a restriction argument originally due independently to Alexander and Bauer we
achieve the fine classification in two cases, namely non-special rational
surfaces of degree 9 and special rational surfaces of degree 8. The first case
completes the fine classification of all non-special rational surfaces. In the
second case we obtain a description of the moduli space as the quotient of a
rational variety by the symmetric group $S_5$. We also discuss in how far this
method can be used to study other rational surfaces in $\PP^4$.
| [
{
"version": "v1",
"created": "Tue, 23 May 1995 10:35:13 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Catanese",
"Fabrizio",
""
],
[
"Hulek",
"Klaus",
""
]
] | alg-geom | \section{Introduction}\label{sectionI}
The families of smooth rational surfaces in ${\Bbb{P}}^4$ have been
classified in degree
$\leq10$ ( \cite{A1}, \cite{I1}, \cite{I2}, \cite{O1}, \cite{O2},
\cite{R1}, \cite{R2}, \cite{PR}). In this thesis Popescu
\cite{P} constructed further examples of rational surfaces in degree $11$. The
existence of these surfaces has been proved in various ways, using linear
systems, vector bundles and sheaves or liaison arguments. All known rational
surfaces can be represented as a blowing-up of ${\Bbb{P}}^2$. Although it would
seem the most natural approach to prove directly that a given linear system is
very ample, this turns out to be a very subtle problem in some cases, in
particular when the surface $S$ in ${\Bbb{P}}^4$ is special
(i.e.~$h^1({\cal O}_S(H))\neq0$). On the other hand, being able to handle the
linear system often means that one knows the geometry of the surface very well.
The starting point of our paper is the observation that every known rational
surface in ${\Bbb{P}}^4$ contains a plane curve $C$. Using the hyperplanes through
$C$ one can construct a residual linear system $|D|$. I.e., we can write
$H\equiv C+D$ with $\dim |D|\geq1$. This situation was studied in particular
by Alexander \cite{A1}, \cite{A2} and Bauer \cite{B}: if $|H|$ restricts to
complete linear systems on $C$ and $D'$ where $D'$ varies in a $1$-dimensional
linear subsystem of $|D|$, then $H$ is very ample on $S$ if and only if it is
very ample on $C$ and the curves $D'$ (cf.~Theorem~(\ref{theo21})). In this
way one can reduce the question of very ampleness of $H$ to the study of
linear systems on curves. In \cite{CFHR} the following curve embedding theorem
was proved which we shall state here only for the (special) case of curves
contained in a smooth surface.
\begin{theorem}\label{theo11}
A divisor $H$ is very ample on $C$ if for every subcurve $Y$ of $C$ of
arithmetic genus $p(Y)$
\noindent $\on{(i)}$ $H.Y\geq 2p(Y)+1$ or
\noindent $\on{(ii)}$ $H.Y\geq 2p(Y)$ and there is no $2$-cycle $\xi$ of $Y$
such that $I_\xi{\cal O}_Y\cong \omega_Y(-H)$.
More generally
\noindent $\on{(iii)}$ If $\xi$ is an $r$-cycle of $C$, then
$H^0(C,{\cal O}_C(H))$ surjects onto $H^0({\cal O}_C(H)\otimes{\cal O}_\xi)$ unless
there is a subcurve $Y$ of $C$ and a morphism $\phi:I_\xi{\cal O}_Y\to
\omega_Y(-H)$ which is "good" (i.e.~$\phi$ is injective with a cokernel of
finite length) and which is not induced by a section of $H^0(Y,\omega_Y(-H))$.
\end{theorem}
The method described above was used in \cite{CF} to characterize exactly all
configurations of points in ${\Bbb{P}}^2$ which define non-special rational surfaces
of degree $\leq8$. In these cases $H.D\geq 2p(D)+1$. This left the case open
of one non-special surface, namely the unique non-special surface of degree
$9$. In this case one has a decomposition $H\equiv C+D$ where $C$ is a plane
cubic, and $|D|$ is a pencil of curves of genus $p(D)=3$ and $H.D=6$.
Section~\ref{sectionII} is devoted to this surface. In Theorem~(\ref{theo22})
we classify all configurations of points in the plane which lead to
non-special surfaces of degree $9$ in ${\Bbb{P}}^4$. This completes the fine
classification of non-special surfaces.
In section~\ref{sectionIII} we show that this method can also be applied to
study special surfaces. We treat the (unique) special surface of degree $8$.
In this case there exists a decomposition $H\equiv C+D$ where $C$ is a conic
and $|D|$ is a pencil of curves of genus $4$ with $H.D=6$. It turns out that
for the general element $D'$ of $|D|$ (but not necessarily for all elements)
$H$ is the canonical divisor on $D'$. In Theorem~(\ref{IIItheo14}) we give a
characterization of these configurations of points which define smooth special
surfaces of degree $8$ in ${\Bbb{P}}^4$. We then use this result to give an
existence proof (in fact we construct the general element in the family) of
these surfaces using only the linear system $|H|$
(Theorem~(\ref{IIItheo17})), and in particular to describe the moduli
space of the above surfaces modulo projective
equivalence (Theorem~(\ref{IIItheo20})).
Finally in section~\ref{sectionIV} we discuss some posibilities how this
method can be used to study other rational surfaces in ${\Bbb{P}}^4$,
suggesting some explicit decompositions $H\equiv C+D$ of the hyperplane
class as the sum of divisors.
\medskip
\noindent {\bf Acknowledgements.} The authors were partially supported by the
DFG-Schwerpunktprogramm "Komplexe Mannigfaltigkeiten" under contract number
Hu337/4-3, the EU HCM project AGE (Algebraic Geometry in Europe) contract
number ERBCHRXCT 940557 and MURST $40\%$. The second author is also
grateful to the Tata-Institute of Fundamental Research for their
hospitality. The final version was written while the first author was
"{\em Professore distaccato}" at the {\em Accademia dei Lincei}.
\section{The non-special rational surface of degree $9$}\label{sectionII}
In this section we want to give an application of Theorem (\ref{theo11})
to non-special rational surfaces. These surfaces have been classified by
Alexander \cite{A1}. Catanese and Franciosi treated all
non-special rational surfaces of degree $\leq8$ by studying suitable
decompositions $H=C+D$ of the embedding linear systems. The crucial
observation here is the following result, originally due to J.~Alexander and
I.~Bauer \cite{B}.
\begin{theorem}[Alexander-Bauer]\label{theo21}
Let $X$ be a smooth projective variety and let $C,D$ be effective divisors with
$\dim|D|\geq1$. Let $H$ be the divisor $H\equiv C+D$. If $\big|H\big||_C$ is
very ample and for all $D'$ in a $1$-dimensional subsystem of $|D|$,
$\big|H\big||_{D'}$ is very ample, then $|H|$ is very ample on $X$.
\end{theorem}
By Alexander's list there is only one non-special rational surface of
degree bigger than $8$. This surface is a ${\Bbb{P}}^2$ blown up in 10 points
$x_1,\ldots,x_{10}$ embedded by the linear system $|H|=|13L-4
\sum_{i=1}^{10}x_i|$. Alexander showed that for general position of the
points $x_i$ the linear system $|H|$ embeds
$S={\tilde{{\Bbb{P}}}}^2(x_1,\ldots,x_{10})$ into ${\Bbb{P}}^4$. Clearly the degree
of $S$ is 9. Here we show that using Theorem (\ref{theo11}) one can also
apply the decomposition method to this surface. In fact we obtain
necessary and sufficient conditions for the position of the points
$x_i$ for $|H|$ to be very ample. Our result is the following
\begin{theorem}\label{theo22} The linear system $|H|= |13L-4\sum x_i|$
embeds the surface
$S ={\tilde {\Bbb{P}}}^2(x_1,\ldots, x_{10})$ into
${\Bbb{P}}^4$ if and only if
\noindent $\on{(0)}$ no $x_i$ is infinitely near,
\noindent $\on{(1)}$ $|L-\sum\limits_{i\in\triangle}x_i|=\emptyset$ for
$|\triangle|\geq4$,
\noindent $\on{(2)}$ $|2L-\sum\limits_{i\in\triangle'}x_i|=\emptyset$ for
$|\triangle'|\geq7$,
\noindent $\on{(3)}$ $|3L-\sum\limits_ix_i|=\emptyset$,
\noindent $\on{(3)}_{ij}'$ $|3L-\sum\limits_{k\neq
i,j}x_k-2x_i|=\emptyset$ for all pairs $(i,j)$,
\noindent $\on{(4)}_{ijk}$ $|4L-2x_i-2x_j-2x_k- \sum\limits_{l\neq
i,j,k}x_l |=\emptyset$ for all triples $(i,j,k)$,
\noindent $\on{(6)}_i$ $|6L-x_i-2\sum\limits_{j\neq i}x_j|=\emptyset$,
\noindent $\on{(10)}_1$ If $D=10L-4x_1-3\sum\limits_{i\geq2}x_i$, then
$\dim|D|=1$.
\end{theorem}
\begin{uremarks} (i) Clearly conditions (0) to (6) are open conditions.
The expected dimension of $|D|$ is 1, hence this condition is also open.
\noindent (ii) The last condition is asymmetrical. If $|H|$ is very ample
condition $(10)_i$ is necessarily fulfilled for all $i$. On the other
hand, our theorem shows that in order to prove very ampleness for $|H|$ it
suffices to check only one of the conditions $(10)_i$.
\end{uremarks}
\begin{Proof} We shall first show that the conditions stated are
necessary. Clearly (0) follows since $H.(x_i-x_j)=0$. Similarly the
ampleness of $H$ immediately implies conditions (1) to (4). Assume the
linear system
$|6L-x_i-2\sum\limits_{j\neq i}x_j|$ contains some element $A$. Then
$H.A=2$, and $p(A)=1$ which contradicts very ampleness of $H$. For (10) we
consider $C\equiv H-D\equiv 3L-\sum\limits_{i\geq2}x_i$. Clearly $|C|$ is
non empty. For $C'\in|C|$ we consider the exact sequence
\setcounter{equation}{10}
\begin{equation}\label{gl11}
0\longrightarrow{\cal O}_S(D)\longrightarrow{\cal O}_S(H)\longrightarrow
{\cal O}_{C'}(H)\longrightarrow 0.
\end{equation} If $h^0({\cal O}_S(D))\geq3$, then either
$h^0({\cal O}_S(H)\geq6$ and $|H|$ does not embed $S$ into ${\Bbb{P}}^4$ or $|H|$
maps $C'$ to a line. But since $p(C)=1$ this means that $|H|$ cannot be
very ample.
Now assume that conditions (0) to $(10)_1$ hold. We shall first show
\begin{align*} h^1({\cal O}_S(D))&=0\tag{I} \\ h^1({\cal O}_S(C))&=0\tag{II}\\
h^0({\cal O}_S(H))&=5 \tag{III}
\end{align*} Ad (I): By condition $(10)_1$ we have $h^0({\cal O}_S(D))=2$.
Clearly
$h^2({\cal O}_S(D))=h^0({\cal O}_S(K-D))=0$. Hence the claim follows from
Riemann-Roch, since $\chi({\cal O}_S(D))=2$.
\noindent Ad (II): We consider $-K\equiv 3L-\sum\limits_ix_i\equiv
C-x_1$. By condition (3) $h^0({\cal O}_S(-K))=0$. Clearly also
$h^2({\cal O}_S(-K))=h^0({\cal O}_S(K))=0$. Hence by Riemann-Roch
$h^1({\cal O}_S(-K))=-\chi ({\cal O}_S(-K))=0$. Now consider the exact sequence
\begin{equation}\label{gl12}
0\longrightarrow{\cal O}_S(-K)\longrightarrow{\cal O}_S(C)\longrightarrow
{\cal O}_S(C)|_{x_1}={\cal O}_{x_1}\longrightarrow 0.
\end{equation} This shows $h^1({\cal O}_S(C))=0$. Note that this also
implies (by Riemann-Roch) that $h^0({\cal O}_S(C))=1$, i.e.~the curve $C'$
is uniquely determined.
\noindent Ad (III): In view of (I) and sequence (\ref{gl11}) it suffices
to show that $h^0({\cal O}_{C'}(H))=3$. By Riemann-Roch on $C'$ this is
equivalent to
$h^1({\cal O}_{C'}(H))=0$. Since $K_{C'}$ is trivial this in term is
equivalent to
$h^0({\cal O}_{C'}(-H))=0$. By condition $(3)'$ the curve $C'$ contains no
exceptional divisor. As a plane curve $C'$ can be irreducible or it can
decompose into a conic and a line or three lines. In view of conditions
(1) and (2), however, $C'$ cannot have multiple components and, moreover
$H$ has positive degree on every component. This proves
$h^0({\cal O}_{C'}(-H))=0$ and hence the claim.
This shows that $|H|$ maps $S$ to ${\Bbb{P}}^4$ and that, moreover, $|H|$
restricts to complete linear systems on $C'$ and all curves $D'\in|D|$.
We shall now show
\begin{align*}
\begin{split} &\ \ \ \,\text{ For every subcurve }A\leq C'\text{ we have }
H.A\geq2p(A)+1
\end{split}\tag{IV}
\end{align*}
\begin{align*}
\begin{split} &\text{For every proper subcurve }B'\subset D'\text{ of an
element}\\ &D'\in |D|\text{ we have }H.B'\geq2p(B')+1
\end{split}\tag{V(i)}
\end{align*}
\begin{align*}
\begin{split} &H\text{ does not restrict to a "$(2+K)$"-divisor on }D',\\
&\text{i.e. }{\cal O}_{D'}(H-K_{D'})\text{ does not have a good section}\\
&\text{defining a degree } 2\text{-cycle}.
\end{split}\tag{V(ii)}
\end{align*}
It then follows from (IV) and \cite[Theorem 3.1]{CF} that
$|H|$ is very ample on $C'$. Because of (V(i)) and (V(ii)) it follows
from Theorem (\ref{theo11}) that $|H|$ is very ample on every element
$D'$ of $|D|$. It then follows from Theorem (\ref{theo21}) that $|H|$ is
very ample.
\noindent Ad (V(ii)): Let $H_{D'}$ be the restriction of $H$ to $D'$, and
denote the canonical bundle of $D'$ by $K_{D'}$. It suffices to show that
$h^0({\cal O}_{D'}(H_{D'}-K_{D'}))=0$. Now
\begin{align*} H_{D'}-K_{D'}&= (H-K-D)|_{D'}\\ &= (C-K)|_{D'}\\ &=
(2C-x_1)|_{D'}.
\end{align*} There is an exact sequence
\begin{equation}\label{gl13}
0\longrightarrow{\cal O}_S(2C-x_1-D)\longrightarrow{\cal O}_S(2C-x_1)
\longrightarrow
{\cal O}_{D'}(H_{D'}-K_{D'})\longrightarrow 0.
\end{equation} Since
$$ 2C-x_1\equiv 6L-x_1-2\sum_{i=2}^{10}x_i
$$ it follows from condition $(6)_1$ that $h^0({\cal O}_S(2C-x_1))=0$.
Clearly
$h^0({\cal O}_S(2C-x_1-D))=0$. Now
$$ 2C-x_1-D\equiv -4L+3x_1+\sum_{i=2}^{10}x_i
$$ resp.
$$ K-(2C-x_1-D)\equiv L-2x_1.
$$ Hence $h^2({\cal O}_S(2C-x_1-D))=h^0({\cal O}_S(K-(2C-x_1-D))=0$. Since
moreover
$\chi({\cal O}_S(2C-x_1-D))=0$ it follows that $h^1({\cal O}_S(2C-x_1-D))=0$.
The assertion follows now from sequence (\ref{gl13}).
\noindent Ad (IV) and (V(i)): We have to show that for all curves $A$
with $A\leq C'$, resp.~$A<D'$, $D'\in |D|$ the following holds
\begin{equation}\label{gl14} H.A\geq2p(A)+1.
\end{equation} We first notice that it is enough to prove (\ref{gl14})
for divisors $A$ with
$p(A)\geq0$. Assume in fact we know this and that $p(A)<0$. Then $A$ is
necessarily reducible. For every irreducible component $A'$ of $A$ we
have $p(A')\geq0$ and hence $H.A'>0$. This shows $H.A>0$ and hence
(\ref{gl14}). Clearly (\ref{gl14}) also holds for the lines $x_i$. Hence
we can assume that $A$ is of the form
\begin{equation}\label{gl15} A\equiv aL-\sum_ib_ix_i\quad\text{with
}1\leq a\leq 10.
\end{equation} Note that
\begin{eqnarray} 2p(A)&=&a(a-3)-\sum_ib_i(b_i-1)+2\label{gl16}\\
H.A&=&13a-4\sum_ib_i.\label{gl17}
\end{eqnarray} We proceed in several steps
\medskip
\noindent {\bf Claim 1 } Let $A$ be as in (\ref{gl15}) with $1\leq a\leq
3$. Assume that $p(A)\geq0$. Then (\ref{gl14}) is fulfilled.
\medskip
\noindent {\em Proof of Claim 1 } After possibly relabelling the $x_i$ we
can assume that $b_1\geq b_2\geq\ldots\geq b_{10}$. If $a=1$ or 2 then
$b_1\leq1$ and $b_2\geq0$. Moreover $p(A)=0$. If $H.A\leq2p(A)$ we get
immediately a contradiction to conditions (1) or (2). If $a=3$ then we
have two cases. Either
$b_1\leq1$, $b_2\geq0$ as above and $p(A)=1$. Then $H.A\leq2p(A)$ violates
condition (3). Or $b_1=2$ or $b_{10}=-1$ and the other $b_i$ are 0 or 1.
Then
$H.A\leq2p(A)$ is only possible for $b_1=2$, but this would violate
condition
$(3)'$.
\medskip
\noindent {\bf Claim 2 } $H$ is ample on $C$ and $D$, i.e.~for every
irreducible component $A$ of $C'$, resp.~$D'$, $D'\in|D|$ we have $H.A>0$.
\medskip
\noindent {\em Proof of Claim 2 } Assume the claim is false. Let $A$ be an
irreducible component with $H.A\leq0$. Since $A$ is irreducible,
$p(A)\geq0$. By (\ref{gl16}), (\ref{gl17}) this leads to the two
inequalities
\begin{eqnarray} 13a&\leq&4\sum b_i\label{gl18}\\
\sum b_i(b_i-1)&\leq&a(a-3)+2.\label{gl19}
\end{eqnarray} Multiplying (\ref{gl19}) by $13^2$ and using (\ref{gl18})
we obtain
\begin{equation}\label{gl20} 169\left(\sum b_i^2-\sum b_i\right)\leq
16\left(\sum b_i\right)^2- 156\sum b_i+338.
\end{equation} Now
\begin{equation}\label{gl21}
\left(\sum b_i\right)^2=10\sum b_i^2-\sum_{i<j}(b_i-b_j)^2
\end{equation} and using this (\ref{gl20}) becomes
\begin{equation}\label{gl22}
\sum_i(9b_i^2-13b_i)+16\sum_{i<j}(b_i-b_j)^2\leq 338.
\end{equation} The function $f(b)=9b^2-13b$ for integers $b$ is non
positive only for $b=0$ or 1. It is minimal for $b=1$. Since $f(1)=-4$ we
derive from (\ref{gl22})
\begin{equation}\label{gl23} 16\sum_{i<j}(b_i-b_j)^2\leq 378
\end{equation} resp.
\begin{equation}\label{gl24}
\sum_{i<j}|b_i-b_j|^2\leq 23.
\end{equation} At this point it is useful to introduce the following
integer valued function
$$
\delta=\delta(A)=\max_{i<j}|b_i-b_j|.
$$ We have to distinguish several cases:
\noindent $\delta\geq3$: Assume there is a pair $(i,j)$ with
$|b_i-b_j|\geq3$. Then for all $k\neq i,j$:
$$ |b_i-b_k|^2+|b_j-b_k|^2\geq5.
$$ Hence
$$
\sum_{i<j}|b_i-b_j|^2\geq9+5\cdot 8=49
$$ contradicting (\ref{gl24}).
\noindent $\delta=2$: After possibly relabelling the $x_i$ we can assume
that
$b_2=b_1+2$ and $b_1\leq b_k\leq b_2$ for $k\geq3$. Then
$$ |b_k-b_1|^2+|b_k-b_2|^2=\left\{\begin{array}{cl} 2&\text{if
}b_k=b_1+1\\ 4&\text{if }b_k=b_1\text{ or }b_k=b_2. \end{array}\right.
$$ Let $t$ be the number of $b_k$ which are either equal to $b_1$ or
$b_2$. Then
\begin{eqnarray*}
\sum_{i<j}|b_i-b_j|^2&\geq&4+4t+2(8-t)+t(8-t)\\ &=&20+t(10-t).
\end{eqnarray*} It follows from (\ref{gl24}) that $t=0$. But then
(\ref{gl22}) gives
$$
\sum(9b_i^2-13b_i)\leq18.
$$ Looking at the values of $f(b)=9b^2-13b$ one sees immediately that
this is only possible for $b_1=-1$ or $b_1=0$. In the first case it
follows from (\ref{gl18}) that $a<0$ which is absurd. In the second case
we obtain $a\leq 3$ and hence we are done by Claim 1.
\noindent $\delta\leq1$: Here we can assume
$$ b_1=\ldots=b_k=m,\quad b_{k+1}=\ldots=b_{10}=m+1.
$$ Since $f(b)\geq42$ for $b\geq3$ it follows immediately from
(\ref{gl22}) that
$m\leq2$. If $m\leq0$ then (\ref{gl18}) gives $a\leq3$ and we are done by
Claim 1. It remains to consider the subcases $m=1$ or 2.
\noindent $m=2$: Since $f(2)=10$ and $f(3)=42$ formula (\ref{gl22})
implies
$$ 10k+42(10-k)+16k(10-k)\leq338.
$$ One checks easily that this is only possible for $k=9$ or 10. In this
case (\ref{gl18}) gives $a\leq6$. If $k=9$ then (\ref{gl18}) gives
$22\leq a(a-3)$, i.e.~$a\geq7$, a contradiction. If $k=10$, then
(\ref{gl18}) implies
$18\leq a(a-3)$. This is only possible for $a=6$. But now the existence
of $A$ would contradict condition (6).
\noindent $m=1$: Since $f(1)=-4$ and $f(2)=10$ formula (\ref{gl22}) reads
$$ -4k+10(10-k)+16k(10-k)\leq338
$$ or equivalently
$$ k(73-8k)\leq119.
$$ It is straightforward to check that this implies $k\leq2$ or $k\geq7$.
If
$k\leq2$ then $\sum b_i(b_i-1)\geq16$ and (\ref{gl19}) shows that
$a\geq6$. On the other hand $\sum b_i\leq19$ and this contradicts
(\ref{gl18}). Now assume
$k\geq7$. Then $\sum b_i\leq13$. It follows from (\ref{gl18}) that either
$a\leq3$ -- and this case is dealt with by Claim 1 -- or $a=4$ and $\sum
b_i=13$. Then $k=7$ and the existence of $A$ contradicts condition (4).
\medskip
\noindent {\em End of proof } It follows immediately from Claim 1 that
(\ref{gl14}) holds for subcurves $A\leq C'$. It remains to consider
subcurves
$A< D'$, $D'\in|D|$. Since $H$ is ample on $D$ we have $H.A>0$, hence it
suffices to consider curves with $p(A)\geq1$. Also by ampleness of $H$ on
$D$ it follows that
\begin{equation}\label{gl25} 1\leq H.A\leq5
\end{equation} since $H.D=6$. Also note that, as an immediate consequence
of (\ref{gl17}):
\begin{equation}\label{gl26} a\equiv H.A\on{mod}4.
\end{equation} Finally we remark the following
\medskip
\noindent {\bf Observation:} If $A<D$ is not one of the exceptional lines
$x_i$, then $H.A\leq4$ implies $b_i\geq0$ for all $i$. Otherwise at most
one
$b_i=-1$ and all other $b_i\geq0$.
This follows from the ampleness of $H$ on $x_i$, since $H.x_i=4$.
{}From now on we set
\begin{equation}\label{gl27} B:=D-A.
\end{equation} By adjunction
\begin{equation}\label{gl28} p(A)+p(B)=p(D)+1-A.B=4-A.B.
\end{equation} We write
$$ B\equiv bL-\sum c_ix_i.
$$ We shall now proceed by discussing the possible values of the
coefficient $a$ of $A$ in decreasing order.
\noindent $a=10$: Then $B=\sum c_ix_i$, $c_i\geq0$ and since $H.B\leq5$
we must have $B=x_i$. Then $A.B=4$ or 5 and $p(A)\leq0$ by (\ref{gl28}).
\noindent $a=9$: By (\ref{gl25}), (\ref{gl26}) we have to consider two
cases
\begin{align*}
H.A=5&, H.B=1 \tag{$\alpha$}\\ H.A=1&, H.B=5.\tag{$\beta$}
\end{align*}
Using our above observation for $B$ in case $(\alpha)$ we find that
$$
B\equiv L-x_i-x_j-x_k.
$$
But now $A.B\geq2$ and hence $p(A)\leq1$. Hence $H.A=5\geq2p(A)+1$.
Using condition (1) we have to consider the following cases for $(\beta)$:
\begin{eqnarray*} B&\equiv&L-x_i-x_j\\ B&\equiv&L-x_i-x_j-x_k+x_l.
\end{eqnarray*} In the first case $A.B\geq4$ and $p(B)=0$, hence
$p(A)\leq0$. In the second case $A.B\geq5$ and $p(B)=-1$, hence again
$p(A)\leq0$.
\noindent $a=8$: Here the only possibility is
$$
H.A=4,\quad H.B=2.
$$
Using our observation for $B$ we find that
$$
B\equiv 2L-x_{i_1}-\ldots-x_{i_6}.
$$
Either the $x_{i_j}$ are all different or we have 1 double point (and
$B$ is a pair of lines) or 3 double points (and $B$ is a double line).
Then $A.B\geq3$ (resp.~4, resp.~8) and $p(B)=0$ (resp.~$-1$, resp.~$-3$).
In either case
$p(A)\leq1$ and hence $H.A\geq2p(A)+1$.
\noindent $a=7$: In this case
$$ H.A=H.B=3.
$$ All coefficients $b_i\geq0$. It is enough to consider divisors $A$ with
$p(A)\geq2$. Together with $H.A=3$ this leads to the following conditions
on the $b_i$:
$$
\sum b_i=22,\quad \sum b_i(b_i-1)\leq26.
$$ Let $\beta_i=\max(0,b_i-1)$. Then these conditions become
$$
\sum\beta_i\geq12,\quad\sum(\beta_i+\beta_i^2)\leq26
$$ and it is easy to check that no solutions exist.
\noindent $a=6$: We now have to consider
$$ H.A=2,\quad H.B=4.
$$ We have to consider divisors $A$ with $p(A)\geq1$. Arguing as in the
case $a=7$ this leads to
$$
\sum b_i=19,\quad \sum b_i(b_i-1)\leq18
$$ resp.
$$
\sum\beta_i\geq9,\quad\sum(\beta_i+\beta_i^2)\leq18.
$$ The only solution is $b_j=1$ for one $b_j$ and $b_i=2$ for $j\neq i$.
But then
$A\in|6L-x_j-2\sum\limits_{i\neq j}x_i|$ contradicting condition (6).
\noindent $a=5$: Then we have two possible cases
\begin{align*} H.A=5&, H.B=1 \tag{$\alpha$}\\ H.A=1&, H.B=5.\tag{$\beta$}
\end{align*} We shall treat $(\alpha)$ first. Then by the ampleness of
$H$ the curve $B$ must be irreducible. Set
$$ B=5L-\sum c_ix_i,\quad c_i\geq0.
$$ Then $H.B=1$ and irreducibility of $B$ gives:
$$
\sum c_i=16,\quad \sum c_i(c_i-1)\leq12.
$$ One easily checks that this is only possible if 6 of the $c_i$ are 2,
and the others are 1. Hence
$$ B\in|5L-2\sum_{i\in\triangle}x_i-\sum_{i\not\in\triangle}x_i|,\quad
|\triangle|=6.
$$ Then $p(B)=0$. Moreover $A.B\geq3$, hence $p(A)\leq1$ and hence
$H.A\geq2p(A)+1$.
In case $(\beta)$ we apply the above argument to $A$ and find $p(A)=0$,
i.e.~again $H.A\geq2p(A)+1$.
\noindent $a=4:$ Then $H.A=4$ and $H.B=2$. We are done if $p(A)\le 1$,
and otherwise $H.A\ge 52-44=8$, a contradiction.
\noindent $1\leq a\leq 3$: This follows immediately from Claim 1.
\noindent $a=0$: The only possibility is $A=x_i$ when nothing is to show.
This finishes the proof of the theorem.
\end{Proof}
\section{The special rational surface of degree $8$ in ${\Bbb{P}}^4$}
\label{sectionIII}
In this section we want to show how the decomposition method can be employed
to obtain very precise geometric information also about special surfaces. We
consider the rational surface in ${\Bbb{P}}^4$ of degree $8$, sectional genus
$\pi=6$ and speciality $h=h^1({\cal O}_S(1))=1$. This surface was first
constructed by Okonek \cite{O2} using reflexive sheaves. In geometric
terms it is ${\Bbb{P}}^2$ blown-up in $16$ points embedded by a linear system of
the form
$$
|H|=|6L-2\sum_{i=1}^4x_i-\sum_{k=5}^{16}y_k|.
$$
Our aim is to study the precise open and closed conditions which the points
$x_i,y_k$ must fulfill for $|H|$ to be very ample. If $|H|$ is very ample, the
exceptional lines $x_i$ are mapped to conics. Their residual intersection with
the hyperplanes gives a {\em pencil} $|D_i|$. Hence we immediately obtain the
(closed) necessary condition
\begin{gather}
|D_i|\equiv |6L-3x_i-2\sum_{j\neq i}x_j-\sum_{k=5}^{16}y_k|\text{ is a pencil}
\tag{$D_i$}
\end{gather}
By Riemann-Roch this is equivalent to $h^1({\cal O}_S(D_i))=1$. We first want to
study the linear system $|H|$ on the elements of the pencil $|D_i|$. Note that
$$
p(D_i)=4,\ H.D_i=6.
$$
If $D=A+B$ is a decomposition of some element $D\in |D_i|$, then
\begin{gather}
p(A)+p(B)+A.B=5\label{IIIgl1}\\
A.H+B.H=6.\label{IIIgl2}
\end{gather}
The first equality can be proved by adjunction, the second is obvious.
\begin{lemma}\label{IIIlemma1}
Assume $|H|$ is very ample. Then for every proper subcurve $Y$ of an element
$D\in |D_i|$, $h^1({\cal O}_Y(H))\leq1$ and $p(Y)\leq3$.
\end{lemma}
\begin{Proof}
Riemann-Roch on $Y$ gives
\begin{gather}
h^0({\cal O}_Y(H))=h^1({\cal O}_Y(H))+H.Y+1-p(Y).\label{IIIgl3}
\end{gather}
Consider the sequence
\begin{gather}
0\longrightarrow{\cal O}_S(H-Y)\longrightarrow{\cal O}_S(H)
\overset{\alpha}{\longrightarrow} {\cal O}_Y(H)
\longrightarrow 0.
\label{IIIgl4}
\end{gather}
Since $h^2({\cal O}_S(H-Y))=h^0({\cal O}_S(K-(H-Y)))=0$ and $h^1({\cal O}_S(H))=1$ we
have $h^1({\cal O}_Y(H))\leq1$. We now consider the rank of the restriction map
$H^0(\alpha)$. Since $Y$ is a curve contained in a hyperplane section
$2\leq\on{rank}(\alpha)\leq4$. If $\on{rank}\alpha=2$, then $Y$ is a
line, hence $p(Y)=0$. Next assume $\on{rank}(\alpha)=3$. In this case $Y$ is a
plane curve of degree $d=Y.H$. Since $Y$ is a proper subcurve of $D$ which is
not a line $2\leq d\leq 5$. Then $h^1({\cal O}_Y(H))=h^0({\cal O}_{{\Bbb{P}}^2}(d-4))$.
Since $h^1({\cal O}_Y(H))\leq1$ this shows in fact $d\leq4$. But then
$p(Y)\leq3$. Finally assume that $\on{rank}(\alpha)=4$, i.e.~$Y$ is a space
curve. By (\ref{IIIgl3})
$$
p(Y)=h^1({\cal O}_Y(H))-h^0({\cal O}_Y(H))+H.Y+1\leq3
$$
since $H.Y\leq5$.
\end{Proof}
\begin{remark}\label{IIIrem2}
Note that the above proof also shows the following: If $Y$ is a proper
subcurve of $D$ with $p(Y)=3$, then $Y$ is a plane quartic with $H_Y=K_Y$ or
$Y$ has degree $5$.
\end{remark}
Before proceeding we note the following result from \cite{CF} which we shall
use frequently in the sequel.
\begin{proposition}\label{IIIprop3}
Let $Y$ be a curve contained in a smooth surface with $p(Y)\leq2$. If $H$
is very ample on
$S$, then
$H.Y\geq 2p(Y)+1$.
\end{proposition}
\begin{Proof}
\cite[Prop.~5.2]{CF}
\end{Proof}
\begin{proposition}\label{IIIprop4}
If $|H|$ is very ample, then every element $D\in |D_i|$ is $2$-connected.
Moreover, either
\noindent $\on{(i)}$ $D$ is $3$-connected or
\noindent $\on{(ii)}$ Every decomposition of $D$ which contradicts
$3$-connectedness is either of the form $D=A+B$ with $H.B=4$, $H_B=K_B$ or of
the form $D=A+B$ with $H.B=5$. In the latter case $B=B'+B''$ with $H.B'=4$,
$H_{B'}=K_{B'}$.
\end{proposition}
\begin{Proof}
Let $D=A+B$. We first consider the case $p(A),p(B)>0$. Since $|H|$ is very
ample, it follows that $H.A\geq3$, $H.B\geq3$. But then $H.A=H.B=3$ and hence
$p(A)=p(B)=1$. By (\ref{IIIgl1}) this shows $A.B=3$.
Now assume $p(A)\leq0$. Since $p(B)\leq3$ by Lemma~(\ref{IIIlemma1}) it
follows from (\ref{IIIgl1}) that $A.B\geq2$. The only case where $A.B=2$ is
possible is $p(A)=0$, $p(B)=3$. In this case $H.B\geq4$ since Riemann-Roch for
$B$ gives
$$
h^0({\cal O}_B(H))=h^1({\cal O}_B(H))+H.B-2
$$
and we know that $h^0({\cal O}_B(H))\geq3$. We first treat the case $H.B=4$. Then
$h^1({\cal O}_B(H))=1$ and $h^0({\cal O}_B(H))=3$. In this case $B$ is a plane
quartic and $H_B=K_B$. Now assume $H.B=5$. If $h^1({\cal O}_B(H))=0$ then $B$ is
a plane quintic. But in this case $p(B)=6$, a contradiction. It remains to
consider the case $h^1({\cal O}_B(H))=1$. By duality $h^0({\cal O}_B(K_B-H))=1$. Let
$\sigma$ be a non-zero section of ${\cal O}_B(K_B-H)$. As usual we can write
$B=Y+Z$ where $Z$ is the maximal subcurve where $\sigma$
vanishes. Note that $Z\neq\emptyset$, since $K_B-H$ has negative
degree. Then $Y.(K_Y-H)\geq0$. By the very ampleness of $H$ this implies
$p(Y)\geq3$ and hence $p(Y)=3$. Then we must have $H.Y=4$ and by the previous
analysis $Y$ is a plane quartic with $H_Y=K_Y$.
\end{Proof}
At this point it is useful to introduce the following concept.
\begin{definition}
We say that an element $D\in |D_i|$ fulfills condition (C) if for every
decomposition $D=A+B$:
\noindent $\on{(i)}$ $p(A),p(B)\leq2$
\noindent $\on{(ii)}$ $H.A\geq 2p(A)+1$, $H.B\geq 2p(B)+1$.
\end{definition}
\begin{remark}\label{IIIrem5}
It follows immediately from (\ref{IIIgl1}) that an element $D\in |D_i|$ which
fulfills condition (C) is $3$-connected.
\end{remark}
For future use we also note
\begin{lemma}\label{IIIlemma6}
Let $D$ be a curve of genus $4$, and let $H$ be divisor on $D$ of degree $6$
with $h^0({\cal O}_D(H))\geq4$. Assume that for every proper subcurve $Y$ of $D$
we have $H.Y\geq 2p(Y)-1$. Then $H$ is the canonical divisor on $D$.
\end{lemma}
\begin{Proof}
By Riemann-Roch and duality $h^0({\cal O}_D(K_D-H))\geq1$. Let $\sigma$ be a
non-zero section of ${\cal O}_D(K_D-H)$. As usual this defines a decomposition
$D=Y+Z$ where $Z$ is the maximal subcurve where $\sigma$ vanishes. If
$Z=\emptyset$ the claim is obvious. Otherwise $(K_D-H).Y\geq Z.Y$ and by
adjunction this gives $H.Y\leq 2p(Y)-2$, a contradiction.
\end{Proof}
Our next aim is to analyze the condition $h^0({\cal O}_S(H))=5$. For this
we introduce the divisor
$$
\Delta_i\equiv H-(L-x_i).
$$
\begin{lemma}\label{IIIlemma7}
The following conditions are equivalent:
\noindent $\on{(i)}$ $h^0({\cal O}_S(H))=5$ (resp.~$h^1({\cal O}_S(H))=1$).
\noindent $\on{(ii)}$ $h^0({\cal O}_D(H))=4$ (resp.~$h^1({\cal O}_D(H))=1$) for some
(every) element $D\in |D_i|$.
\noindent $\on{(iii)}$ $h^0({\cal O}_D(K_D-H))=1$ for some (every) element $D\in
|D_i|$.
\noindent Moreover assume that $D\in |D_i|$ fulfills condition $\on{(C)}$. Then
the following conditions are equivalent to $\on{(i)}$-$\on{(iii)}$:
\noindent $\on{(iv)}$ ${\cal O}_D(H)=K_D$.
\noindent $\on{(v)}$ $\Delta_i|_D\equiv (2L-\sum x_i)|_D$.
\end{lemma}
\begin{Proof}
Since $h^0({\cal O}_S(D_i))\geq1$ we have an exact sequence
$$
0\longrightarrow{\cal O}_S(x_i)\longrightarrow{\cal O}_S(H)
\longrightarrow{\cal O}_D(H) \longrightarrow 0.
$$
Since $h^1({\cal O}_S(x_i))=0$ the equivalence of (i) and (ii) follows. The
equivalence of (ii) and (iii) is a consequence of Serre duality. It follows
from Lemma~(\ref{IIIlemma6}) that (iii) implies (iv) if (C) holds. Conversely
if ${\cal O}_D(H)=K_D$ then $h^0({\cal O}_D(K_D-H))=h^0({\cal O}_D)=1$, since $D$ is
$3$-connected. To show the equivalence of (iv) and (v) note that by adjunction
$$
K_D\equiv (K_S+D)|_D\equiv (3L-2x_i-\sum_{j\neq i}x_j)|_D.
$$
Hence $K_D\equiv H|_D\equiv (\Delta_i+(L-x_i))|_D$ if and only if
$\Delta_i|_D\equiv (K_D-(L-x_i))|_D\equiv (2L-\sum x_i)|_D$.
\end{Proof}
We want to discuss necessary open conditions which must be fulfilled if $|H|$
is ample.
\begin{definition}
We say that $|H|$ fulfills condition (P) if for every divisor $Y$ on $S$ with
$Y.L\leq6$, $p(Y)\leq2$, $H.Y\leq2p(Y)$ the linear system $|Y|$ is empty.
\end{definition}
\begin{remark}\label{IIIrem8}
(i) By Proposition (\ref{IIIprop3}) this condition is necessary for $|H|$
to be very ample.
\noindent (ii) Note that in order to check (P) one only need check {\em
finitely many} open conditions.
\noindent (iii) For $Y.L=0$ condition (P) implies that the only points which
can have infinitely near points are the $x_i$. The only possibility is that at
most one of the points $y_k$ is infinitely near to some point $x_i$.
\noindent (iv) If $Y.L=1$ then (P) implies
$$
|L-\sum_{i\in\triangle}x_i-\sum_{k\in\triangle'}y_k|=\emptyset \text{ for
}2|\triangle|+|\triangle'|\geq 6.
$$
In particular no three of the points $x_i$ can lie on a line.
\noindent (v) If $Y.L=6$ then (P) gives
$$
|D_i-x_j|=\emptyset\ (j\neq i),\quad |D_i-y_k-y_l|=\emptyset\ (k\neq l).
$$
\end{remark}
There are, however, two more open conditions which are not as obvious to see.
\begin{proposition}\label{IIIprop9}
If $|H|$ embeds $S$ into ${\Bbb{P}}^4$ then the following open conditions hold:
\begin{enumerate}
\item[$\on{(Q)}$]
$|D_i-2x_i|=\emptyset, \quad
|D_i-x_i-y_k|=\emptyset, \quad
|D_i-2y_k|=\emptyset$
\item[$\on{(R)}$]
For any effective curve $C$ with $C\equiv L-x_i-x_j-y_k, C\equiv L-x_i-x_j$ or
$ C\equiv y_k$ one has $\dim |D_i-C|\le 0$. Moreover $\dim |H-(L-x_i-x_j)|\le
1$.
\end{enumerate}
\end{proposition}
\begin{Proof}
We start with (R). We already know that $\dim |D_i|=1$. Hence we have to
see that such a curve $C$ is not contained in the plane spanned by the
conic
$x_i$. But this would contradict very ampleness since $C.x_i=1$ or $0$. If
$|H|$ is very ample then it embeds $\Lambda_{ij}=L-x_i-x_j$ as a plane conic
(irreducible or reducible but reduced). The claim then follows from the exact
sequence
$$
0\longrightarrow{\cal O}_S(H-(L-x_i-x_j))\longrightarrow {\cal O}_S(H)
\longrightarrow {\cal O}_{\Lambda_{ij}}(H) \longrightarrow 0.
$$
Next we consider the linear system $|D_i-2x_i|$. Assume there is a curve $B\in
|D_i-2x_i|$. Then $p(B)=-3$. Since $H.B=2$ we have the following
possibilities: $B$ is a reduced conic (either smooth or reducible). Then
$p(B)=0$, a contradiction. If $B$ is the union of $2$ skew lines, then
$p(B)=-1$ which is also not possible. Hence $B$ must be a non-reduced line.
But this is not possible, since the class of $B$ on $S$ is not divisible by
$2$.
The crucial step is to prove the
\begin{claim}
Set $D=D_i$. If $|D|$ contains $y_k+B$, then $B$ is of the form
$B=B'+(L-x_i-x_j-y_k)$ with $H_{B'}=K_{B'}$.
\end{claim}
It follows from Lemma~(\ref{IIIlemma7}) that there exists a non-zero section
$0\neq\sigma\in H^0({\cal O}_D(K_D-H))$. As usual this defines a decomposition
$D=Y+Z$. Since $(K_D-H).y_k=-1$ the curve $Z$ must contain the irreducible
curve $y_k$. Moreover since $y_k.B=2$ and $(K_D-H).B=1$ it follows that $Z$
contains some further curve $Z'$ contained in $B$, i.e.~$B=B'+Z'$. Now as in
proof of Lemma~(\ref{IIIlemma6}) $H.B'\leq2p(B')-2$ and very ampleness of
$|H|$ together with (\ref{IIIlemma1}) implies $p(B')=3$. As in the proof of
Proposition~(\ref{IIIprop4}) one concludes that $H.B'=4$, $H_{B'}=K_{B'}$. In
particular $Z'$ is a line. Since $p(D_i-2y_k)=1$ it follows that $Z'\neq y_k$.
First assume that $Z'.y_k=0$. Then $p(Z'+y_k)=-1$ and $B'.y_k=2$. It follows
from (\ref{IIIgl1}) that $B'.Z'=1$. But now the decomposition $Z'+(B'+y_k)$
contradicts $2$-connectedness. Hence $Z'$ and $y_k$ are two lines meeting in a
point. This gives $p(y_k+Z')=0$, $B'.(y_k+Z')=2$. We can write
$$
Z'=aL-\beta_ix_i-\sum_{j\neq i}\beta_jx_j-y_k-\sum_{l\neq k}\alpha_ly_l.
$$
If $a=0$ then $Z'=x_i-y_k$ or $Z'=x_j-y_k$, $j\neq i$. The first is impossible
since $p(D_i-x_i)=1$ the second contradicts $|D_i-x_j|=\emptyset$. Hence
$1\leq a\leq6$. Since $Z'$ is mapped to a line in ${\Bbb{P}}^4$ we find
$Z'.y_l\leq1$, $Z'.x_j\leq2$, i.e.
\begin{gather}
0\leq\alpha_l\leq1,\quad 0\leq\beta_i,\beta_j\leq2.\label{IIIgl5}
\end{gather}
It follows from (\ref{IIIgl5}) and from $p(Z')=0$ that $a\leq4$; moreover
$p(Z')=0$, $p(B')=3$ and $p(B)=3$ imply $Z'.B'=1$. Using $0\leq\alpha_l\leq 1$
this gives
\begin{gather}
a(6-a)-\beta_i(3-\beta_i)-\sum_{j\neq i}\beta_j(2-\beta_j)=2.
\label{IIIgl6}
\end{gather}
In view of (\ref{IIIgl5}) this shows $a(6-a)\leq7$ and since $a\leq4$ it
follows that $a=1$. Then $\beta_i,\beta_j\leq1$. If $\beta_i=0$ then by
(\ref{IIIgl6}) $\beta_j=1$ for $j\neq i$, but no three of the points $x_i$ can
be collinear by (\ref{IIIgl6}). Hence $\beta_i=1$ and exactly one $\beta_j$ is
$1$. Together with $H.Z'=1$ this gives $Z'=L-x_i-x_j-y_k$ as claimed.
We are now in a position to prove that $|D_i-x_i-y_k|=\emptyset$ and
$|D_i-2y_k|=\emptyset$. For this we have to show that $B'$ cannot contain
$x_i$ or $y_k$. In the first case $B'=x_i+B''$. Then $H.x_i=2$ and
$K_{B'}.x_i=1$ contradicting $H_{B'}=K_{B'}$. Similarly in the second case
$B'=y_k+B''$ with $H.y_k=1$ and $K_{B'}.y_k=0$ giving the same contradiction.
\end{Proof}
Observe for future use that in the following proposition the assumption
that $|H|$ is very ample is not made.
\begin{proposition}\label{IIIprop10}
Assume that the open conditions $\on{(P)}$ and $\on{(Q)}$ hold. Then an
effective decomposition $D=A+B$ either fulfills condition $\on{(C)}$ and
hence is not
$3$-disconnecting or (after possibly interchanging $A$ and $B$) $A=y_k$,
$L-x_i-x_j$ or $L-x_i-x_j-y_k$.
\end{proposition}
\begin{Proof}
Let $D=A+B$. Clearly we can assume $A.L\leq3$. We shall first treat the case
$A.L=0$, i.e.~$A$ is exceptional with respect to the blowing down map
$S\to{\Bbb{P}}^2$. Then $p(A)\leq0$ and $A.H>0$ by (P). By conditions (Q) and
(P) (cf. Remark (III.8)(v)) if $A.H=1$, then either $A=x_j-y_k$ or
$A=x_i-y_k$ or $A=y_k$. In the first two cases $A.B\ge 3$ and $p(B)\le
2$, the third is one of the exceptions stated. If $A.H\ge 2$ then
$p(B)\le 2$ and the claim follows from (P).
Hence we can now write
\begin{eqnarray*}
A&\equiv&aL-\sum\alpha_jx_j-\sum a_ky_k\\
B&\equiv&bL-\sum\beta_jx_j-\sum b_ky_k
\end{eqnarray*}
with $a,b>0$. Using the open conditions from Remark~(\ref{IIIrem8})(v) (which
are a consequence of (P)) and (Q) it follows that
\begin{gather*}
\begin{aligned}
a_k,b_k &\geq -1,\\
\alpha_j,\beta_j &\geq 0,\\
\alpha_i,\beta_i &\geq -1,
\end{aligned}
\quad
\begin{aligned}
a_k+b_k &= 1\\
\alpha_j+\beta_j &= 2\\
\alpha_i+\beta_i &= 3
\end{aligned}
\quad
(j\neq i)
\end{gather*}
and moreover that at most one of the integers $a_k,b_k,\alpha_i,\beta_i$ can
be negative. If $\beta_i=-1$ then $\alpha_i=4$. In this case $A$ cannot be
effective since we have assumed $a\leq3$. If $\alpha_i=-1$ then $\beta_i=4$
and hence $b\geq4$. We have to consider the cases $a=1$ or $2$. In either case
$p(A)\leq0$ and $H.A\geq2p(A)+1$ follows from (P). On the other hand
\begin{eqnarray*}
H.B-(2p(B)+1)&=&(9b-b^2+1)+\sum_{j\neq i}\beta_j(\beta_j-3)+\sum_k b_k(b_k-2)\\
&\geq&(9b-b^2+1)-6-12\geq3
\end{eqnarray*}
since $b=4,5$. Hence we can now assume $\alpha_i,\beta_i\geq0$.
\noindent $\mathbf{a=1}$. We first treat the case $a_k\geq0$ for all $k$. Then
$$
A\equiv L-\sum_{j\in\triangle}x_j-\sum_{k\in\triangle'} y_k.
$$
Clearly $p(A)\leq0$. Let $\delta_{i\triangle}=0$ (resp.~$1$) if
$i\not\in\triangle$ (resp.~$i\in\triangle$). Then
$$
p(B)=|\triangle|+\delta_{i\triangle}.
$$
We only have to treat the cases where $p(B)\geq3$. Then either
$\delta_{i\triangle}=0$, $|\triangle|\geq3$ or $\delta_{i\triangle}=1$,
$|\triangle|\geq2$. In the first case
$$
H.A=6-2|\triangle|-|\triangle'|\leq0
$$
contradicting (P) for $A$. In the second case the only possibilty is
$|\triangle|=2$, $|\triangle'|\leq1$. But then $A=L-x_i-x_j$ or
$L-x_i-x_j-y_k$. Now assume that one $a_k$ is negative. We can assume
$a_{16}=-1$. Then
$$
A\equiv L-\sum_{j\in\triangle}x_j-\sum_{k\in\triangle'}y_k+y_{16}.
$$
In this case $p(A)=-1$ and
$$
p(B)=|\triangle|+\delta_{i\triangle}-1.
$$
Using the same arguments as before we find that $p(B)\leq2$ in all cases.
\noindent $\mathbf{a=2}$. Again we first assume that all $a_k\geq0$. Then
$$
A\equiv 2L-\sum_{j\in\triangle}x_j-\sum_{k\in\triangle'}2x_k-
\sum_{l\in\triangle''}y_l-\sum_{m\in\triangle'''}2y_m.
$$
Clearly $p(A)\leq0$. If $i\not\in\triangle\cup\triangle'$ then $p(B)\leq0$. If
$i\in\triangle$ then $p(B)\leq2$. Now assume that $i\in\triangle'$. In this
case $p(B)\leq2$ with one possible exception: $|\triangle|=3$ and
$|\triangle'''|=0$. But then
$$
A\equiv 2L-2x_i-x_j-x_k-x_l-\sum_{l\in\triangle''}y_l.
$$
In this case $A$ splits into two lines meeting $x_i$. But then one of these
lines must contain $3$ of the points $x_j$ contradicting condition (P).
Finally let $a_{16}=-1$. The above arguments show that in this case
$p(B)\leq2$.
\noindent $\mathbf{a=3}$. Since in this case $p(A),p(B)\leq1$ condition
(C) follows.
\end{Proof}
Propositions~(\ref{IIIprop4}) and (\ref{IIIprop10}) have provided us with a
fairly good understanding of the behaviour of $H$ on the pencil $|D_i|$.
\begin{cor}\label{IIIcor11}
Assume $|H|$ embeds $S$ into ${\Bbb{P}}^4$. For every element $D\in |D_i|$ either:
\noindent $\on{(i)}$ $D$ is $3$-connected and $H_D=K_D$ or
\noindent $\on{(ii)}$ $D=B+(L-x_i-x_j)$ with $H_B=K_B$.
\end{cor}
\begin{remark}\label{IIIrem12}
The conic $L-x_i-x_j$ can be irreducible or reducible in which case it splits
as $(L-x_i-x_j-y_k)+y_k$.
\end{remark}
At this point we can also conclude our discussion about the linear system
$|\Delta_i|=|H-(L-x_i)|$ (cf.~(\ref{IIIlemma7})).
\begin{proposition}\label{IIIprop13}
If $|H|$ embeds $S$ into ${\Bbb{P}}^4$, then $\dim|\Delta_i|=0$.
\end{proposition}
\begin{Proof}
We first claim that the general element $D\in |D_i|$ is $3$-connected. Indeed
if $D$ is not $3$-connected, then $D=B+(L-x_i-x_j)$. The conic $L-x_i-x_j$
spans a plane $E'$. If $E$ is the plane spanned by $x_i$ then $E\neq E'$
since $(L-x_i-x_j).x_i=1$. Hence $D$ is cut out by the hyperplane spanned by
$E$ and $E'$. Varying the index $j$ there are at most $3$ such hyperplanes.
Clearly $L-x_i$ is effective. Consider the exact sequence
$$
0\longrightarrow{\cal O}_S(\Delta_i)\longrightarrow{\cal O}_S(H) \longrightarrow
{\cal O}_S(H)|_{L-x_i} \longrightarrow 0.
$$
Since $H.(L-x_i)=4$ and $p(L-x_i)=0$ it follows that $|H|$ cannot map $L-x_i$
to a plane curve. This shows $h^0({\cal O}_S(\Delta_i))\leq1$.
On the other hand choose an element $D\in|D_i|$ which is $3$-connected. We
have an exact sequence
$$
0\longrightarrow{\cal O}_S(2x_i-L)\longrightarrow{\cal O}_S(\Delta_i)
\longrightarrow {\cal O}_D(\Delta_i) \longrightarrow 0.
$$
Now $h^0({\cal O}_S(2x_i-L))=h^2({\cal O}_S(2x_i-L))=0$ and hence
$h^1({\cal O}_S(2x_i-L))=1$ by Riemann-Roch. Since $|H|$ is ample no $3$ of the
points $x_i$ lie on a line. Hence $|2L-\sum x_i|$ is a base point free pencil.
Since $|(2L-\sum x_i)-D|=\emptyset$ this shows that $|2L-\sum x_i|$ cuts out a
base-point free pencil on $D$. Since $D$ is $3$-connected $(2L-\sum
x_i)|_D\equiv\Delta_i|_D$ by Lemma~(\ref{IIIlemma7}) and hence
$h^0({\cal O}_D(\Delta_i))\geq2$. By the above sequence this implies
$h^0({\cal O}_S(\Delta_i))\geq 1$.
\end{Proof}
We are now ready to characterize very ample linear systems which embed $S$
into ${\Bbb{P}}^4$.
\begin{theorem}\label{IIItheo14}
The linear system $|H|$ embeds $S$ into ${\Bbb{P}}^4$ if and only if
\noindent $\on{(i)}$ The open conditions $\on{(P)}$, $\on{(Q)}$ and $\on{(R)}$
hold.
\noindent $\on{(ii)}$ The following closed conditions hold:
\begin{enumerate}
\item[$\on{(}D_i\on{)}$] $\dim |D_i|=1$
\item[$\on{(}\Delta_i\on{)}$] For a $3$-connected element $D\in|D_i|$ (whose
existence follows from the above conditions) $\Delta_i.D\equiv (2L-\sum
x_i).D$.
\end{enumerate}
\end{theorem}
\begin{remark}\label{IIIrem15}
As the proof will show it is enough to check the closed conditions ($D_i$),
($\Delta_i$) for one $i$.
\end{remark}
\begin{Proof}
We have already seen that these conditions are necessary. Next we shall show
that a $3$-connected element $D\in|D_i|$ exists if the open conditions and
($D_i$) are fulfilled. Assume that no element $D\in|D_i|$ is $3$-connected.
Then by Proposition~(\ref{IIIprop10}) every element $D$ is of the form $D=B+C$
with $C=L-x_i-x_j$, $L-x_i-x_j-y_k$ or $y_k$. But by condition (R) there are
only finitely many such elements in $|D_i|$.
We shall now proceed in several steps.
\noindent {\bf Step 1}: $h^0({\cal O}_S(H))=5$.
We have seen in the proof of Lemma~(\ref{IIIlemma7}) that for a $3$-connected
element $D$ the equality $\Delta_i.D\equiv (2L-\sum x_i).D$ implies
$K_D=H_D$ and hence $h^0({\cal O}_D(K_D-H))=1$, resp.~$h^1({\cal O}_D(H))=1$. Now the
claim follows from the equivalence of (i) and (ii) in Lemma~(\ref{IIIlemma7}).
In order to prove very ampleness of $|H|$ we want to apply the Alexander-Bauer
Lemma to the decomposition
$$
H\equiv D_i+x_i.
$$
We first have to show that $|H|$ cuts out complete linear systems on $x_i$ and
$D\in|D_i|$. Recall that $x_i$ is either a ${\Bbb{P}}^1$ or consists of two
${\Bbb{P}}^1$'s meeting transversally (cf.~Remark~(\ref{IIIrem8})(iii)). Moreover
$H.x_i=2$ and if $x_i$ is reducible then $H$ has degree $1$ on every
component. Hence
$h^0({\cal O}_{x_i}(H))=3$. The claim for $x_i$ then follows from the exact
sequence
$$
0\longrightarrow{\cal O}_S(D_i)\longrightarrow{\cal O}_S(H) \longrightarrow
{\cal O}_{x_i}(H)\longrightarrow 0.
$$
and condition ($D_i$), i.e.~$h^0({\cal O}_S(D_i))=2$. The corresponding claim for
$D$ follows from the sequence
$$
0\longrightarrow{\cal O}_S(x_i)\longrightarrow{\cal O}_S(H) \longrightarrow
{\cal O}_S(H)|_D \longrightarrow 0.
$$
Our above discussion also shows that $|H|$ embeds $x_i$ as a conic (which can
be irreducible or consist of two different lines).
\noindent {\bf Step 2}: If $D\in|D_i|$ is $3$-connected then $H_D=K_D$ and
$|H|$ is very ample on $D$.
We have already seen the first claim. We have to see that $K_D$ is very ample.
For this we consider the pencils $|\Sigma_1|=|L-x_i|$,
resp.~$|\Sigma_2|=|2L-\sum x_j|$. Clearly $|\Sigma_1|$ is base point free and
the same is true for $|\Sigma_2|$ as no three of the points $x_i$ lie on a
line (by (P)). Hence
$$
|\Sigma_1+\Sigma_2|=|3L-2x_i-\sum_{j\neq i}x_j|=|D_i+K_S|
$$
is base point free. By adjunction $(D_i+K_S)|_D\equiv K_D$ and the exact
sequence
$$
0\longrightarrow{\cal O}_S(K_S)\longrightarrow{\cal O}_S(K_S+D_i) \longrightarrow
{\cal O}_D(K_D) \longrightarrow 0
$$
shows that restriction defines an isomorphism $|\Sigma_1+\Sigma_2|\cong
|K_D|$. Let $X$ be the blow-up of ${\Bbb{P}}^2$ in the points $x_j$ and $\pi:S\to X$
the map blowing down the exceptional curves $y_k$. The linear system
$|\Sigma_1+\Sigma_2|$ defines a morphism
$$
f=\phi_{|\Sigma_1+\Sigma_2|}:X\longrightarrow{\Bbb{P}}^3.
$$
It is easy to understand the map $f$: Clearly $f$ contracts the three
$(-1)$-curves $\Lambda_{ij}=L-x_i-x_j$, $j\neq i$. Let $\pi':X\to X'$ be the
map which blows down the curves $\Lambda_{ij}$ (this makes also sense if
$\Lambda_{ij}=(L-x_i-x_j-y_k)+y_k$ where we first contract $y_k$ and then
$L-x_i-x_j-y_k$). Then $X'$ is a smooth surface and we have a commutative
diagram
$$
\unitlength1cm
\begin{picture}(4,2)
\put(0,1.5){$X$} \put(0.5,1.6){\vector(1,0){2.4}} \put(3.1,1.5){$f(X)$}
\put(0.5,1.3){\vector(3,-2){1}} \put(1.5,0.1){$X'$}
\put(2,0.62){\vector(3,2){1}}
\put(0.5,0.7){{$\scriptstyle \pi'$}}
\put(2.5,0.7){{$\scriptstyle f'$}}
\put(1.5,1.7){{$\scriptstyle f$}}
\end{picture}
$$
where $f'$ maps $X'$ isomorphically onto a smooth quadric. This shows that
$\phi_{|K_D|}:D\to{\Bbb{P}}^3$ is the composition of the blowing down maps $\pi:S\to
X$ and $\pi':X\to X'={\Bbb{P}}^1\times{\Bbb{P}}^1$ followed by an embedding of $X'$. Now
$D.y_k=1$, hence $\pi|_D$ can only fail to be an isomorphism if $D$ contains
$y_k$. But this is impossible if $D$ is $3$-connected. Similarly
$D.\Lambda_{ij}=1$ and $D$ cannot contain a component of $\Lambda_{ij}$. Hence
we are done in this case.
It remains to treat the case when $D$ is not $3$-connected.
\noindent {\bf Step 3}: If $D$ is not $3$-connected, then $D=B+(L-x_i-x_j)$,
$H_B=K_B$ and $|H|$ restricts onto $|K_B|$.
We have already seen that $h^0({\cal O}_S(H))=5$ and hence
$h^0({\cal O}_D(K_D-H))=1$. As usual a non-zero section $\sigma$ defines a
decomposition $D=Y+Z$. Our first claim is that $Z$ is different from $0$.
In fact if
$Z=0$ then
$K_D-H$ would be trivial on
$D$. On the other hand $D$ is not $3$-connected, thus it splits as $D=A+B$
with
$A$ as in Proposition~(\ref{IIIprop10}), in particular $p(A)=0$, $A.B=2$.
Then $K_D.A=0$ contradicting $H.A>0$ which follows from (P). Thus $Z$ is
different from $0$ and since the section $\sigma$ defines a good section
$\sigma'$ of
$H^0({\cal O}_Y(K_Y-H))$ it follows that $2p(Y)-2\ge H.Y$, and hence
$p(Y)\ge 3, Y.Z\le 2$. Then Proposition (\ref{IIIprop10}) applies and
$Z=y_k$ or
$L-x_i-x_j-y_k$ or $L-x_i-x_j$.
If $Z=y_k$ or
$L-x_i-x_j$ then
$(K_Y-H).Y=-1$, a contradiction. Hence $Z=L-x_i-x_j$ and $H_Y=K_Y$. We next
claim that $B$ is $2$-connected. Assume we have a decomposition $B=B_1+B_2$
with $B_1.B_2\leq1$. Then $(B_1+B_2).(L-x_i-x_j)=2$, hence we can assume that
$B_1.(L-x_i-x_j)\leq1$. But then $B_1.(B_2+L-x_i-x_j)\leq2$ contradicting
Proposition~(\ref{IIIprop10}). This shows that $h^1({\cal O}_B(K_B))=1$ and
$h^0({\cal O}_B(K_B))=3$. The claim then follows from the exact sequence
$$
0\longrightarrow{\cal O}_S(L-x_j)\longrightarrow{\cal O}_S(H) \longrightarrow
{\cal O}_B(H) \longrightarrow 0.
$$
\noindent {\bf Step 4}: $|H|$ embeds $D$.
Our first claim is that $|H|$ embeds $B$ as a plane quartic. Since $B-y_k$
is not effective by condition (P) and $B.y_k=1$ it follows that the curve
$B$ is mapped isomorphically onto its image under the blowing down map
$\pi:S\to X$. On $X$
$$
B\equiv 5L-2x_i-x_j-2x_k-2x_l,\quad K_B\equiv (2L-x_i-x_k-x_l)|_B.
$$
Thus $|K_B|$ is induced by a standard Cremona transformation centered at
$x_j$, $x_k$ and $x_l$. Again by (P) it follows that $B-\Lambda_{ik}$ for
$k\neq i$ and $B-\Lambda_{kl}$ for $k,l\neq i$ are not effective. Since
$B.\Lambda_{ik}=B.\Lambda_{kl}=1$ it follows that $B$ is mapped isomorphically
onto a plane quartic.
It follows from condition (R) that $|H|$ embeds $\Lambda_{ij}$ as a plane conic
$Q$. The planes containing $B$ and $Q$ intersect in a line and span a ${\Bbb{P}}^3$.
The line of intersection cannot be a component of $Q$ since, by taking
residual intersection with hyperplanes containing $B$, this would contradict
$h^0({\cal O}_S(x_i+y_k))=1$, resp.~$h^0({\cal O}_S(L-x_j-y_k))=1$. Hence the
schematic intersection of the embedded quartic $B$ and the conic $Q$ has
length at most $2$. Let $D'$ be the schematic image of $D$. Then ${\cal O}_{D'}$
is contained in the direct image of ${\cal O}_D$. But the former has colength
$\leq2$ in ${\cal O}_Q\oplus{\cal O}_B$, the latter has colength $2$, thus $D=D'$.
\end{Proof}
\begin{remark}\label{IIIrem16}
We have already remarked that conditions (P) and (Q) lead to finitely many
open conditions. Going through the proof of
Proposition~(\ref{IIIprop10}) one sees that it is sufficient to check that no
decomposition $A+B=D\in|D_i|$ exists where $A$ (or $B$) contradicts one of the
following conditions below: Here $\triangle$ and $\triangle'$ are always
disjoint subsets of
$\{1,\ldots,4\}$ whereas $\triangle''$ is a subset of $\{5,\ldots,16\}$. We
set $\delta_{i\triangle}=1$ (resp.~$0$) if $i\in\triangle$
(resp.~$i\not\in\triangle$). Similarly we define $\delta_{i\triangle'}$.
Moreover $\delta_{m}=1$ for at most one $m\in\{5,\ldots,16\}$ and
$\delta_{m}=0$ otherwise. If $\delta_{m}=1$ then ~$m\not\in\triangle''$.
\begin{enumerate}
\item[(0)]
$|x_j-x_k|=\emptyset$ ($j\neq k$), $|y_k-y_l|=\emptyset$
($k\neq l$), $|y_k-x_j|=\emptyset$, $|x_j-y_k-y_l|=\emptyset$.
\item[(1)]
$|L-\sum\limits_{j\in\triangle}x_j-
\sum\limits_{k\in\triangle''}y_k|=\emptyset$ for
$2|\triangle|+|\triangle'|\geq6$
\item[(2)]
$|2L-\sum\limits_{j\in\triangle}x_j-
\sum\limits_{k\in\triangle''}y_k|=\emptyset$ for
$2|\triangle|+|\triangle'|\geq12$.
\item[(3)]
$|3L-2x_j-\sum\limits_{k\in\triangle}x_k-
\sum\limits_{l\in\triangle''}y_l|=\emptyset$ for
$2|\triangle|+|\triangle''|\geq14$\\
$|3L-\sum\limits_{j\in\triangle}x_j-2y_k-
\sum\limits_{l\in\triangle''}y_l|=\emptyset$ for
$2|\triangle|+|\triangle''|\geq16$\\
$|3L-\sum\limits_{j\in\triangle}x_j-
\sum\limits_{k\in\triangle''}y_k|=\emptyset$ for
$2|\triangle|+|\triangle''|\geq16$
\item[(4)]
$|4L-(3-\delta_{i\triangle}-2\delta_{i\triangle'})x_i-
\sum\limits_{j\neq i \atop j\in\triangle}x_j- 2\sum\limits_{k\neq i \atop
k\not\in(\triangle\cup\triangle')}x_k-
\sum\limits_{l\not\in\triangle''}y_l-\delta_my_m|=\emptyset$ for
$|\triangle|+|\triangle'|+\delta_{i\triangle}+
2\delta_{i\triangle'}-\delta_m\leq5$,
$2|\triangle'|+|\triangle''|-2\delta_{i\triangle}-4\delta_{i\triangle'}+
\delta_m\leq0$, $2|\triangle|+4|\triangle'|+|\triangle''|\leq11$
\item[(5)]
$|5L-(3-\delta_{i\triangle})x_i-\sum\limits_{j\neq i \atop
j\in\triangle}x_j- 2\sum\limits_{k\neq i \atop k\not\in\triangle}x_k-
\sum\limits_{l\not\in\triangle''}y_l-\delta_my_m|=\emptyset$ for
$|\triangle|+\delta_{i\triangle}-\delta_m\leq2$,
$|\triangle''|-2\delta_{i\triangle}+\delta_m\leq0$,
$2|\triangle|+|\triangle''|\leq5$.
\item[(6)]
$|D_i-x_j|=\emptyset$ ($i\neq j$), $|D_i-2x_i|=\emptyset$,
$|D_i-x_i-y_k|=\emptyset$, $|D_i-2y_k|=\emptyset$, $|D_i-y_k-y_l|=\emptyset$
($k\neq l$).
\end{enumerate}
\end{remark}
Now we want to show how Theorem (\ref{IIItheo14}) can be used to prove the
existence of the special surfaces of degree 8 by explicitly constructing a
very ample linear system $|H|$. Let $x_1,\ldots , x_4$ be points in general
position in ${\Bbb{P}}^2$, and blow them up. The linear system
$|5L-x_1-2\sum\limits_{j\ge 2} x_j|$ is 10-dimensional, its elements have
arithmetic genus 3. Let $\Delta_1$ be a general (and hence smooth) element of
the 10-dimensional linear system $|5L-x_1-2\sum\limits_{j\ge 2} x_j|$ on
${\hat{\Bbb{P}}}^2={\Bbb{P}}^2(x_1, \ldots , x_4)$. Note that the image of $\Delta_1$ in
${\Bbb{P}}^2$ is the image of the canonical model of $\Delta_1$ under a standard
Cremona transformation. The linear system $|2L-\sum\limits_j x_j|$ cuts out a
$g^1_3$ on $\Delta_1$, since $H^1({\hat {\Bbb{P}}}^2, {\cal O}_{{\hat
{\Bbb{P}}}^2}(-3L+\sum x_j))=0$. The linear system
$$
|L_0|:=|(6L-3x_1-2\sum\limits_{j\ge 2}
x_j)|_{\Delta_1}-g^1_3|=|(4L-2x_1-\sum\limits_{j\ge 2} x_j)|_{\Delta_1}|
$$
on $\Delta_1$ has degree 12 and dimension 9. The linear system
$|4L-2x_1-\sum\limits_{j\ge 2} x_j|$ on ${\hat{\Bbb{P}}}^2$ cuts out a subsystem of
codimension 1 in $|L_0|$. We consider the variety
$$
{\cal M}:=\{(\Delta_1, \sum y_k);\ \Delta_1 \mbox{ smooth }, \sum y_k \in
|L_0|\}.
$$
${\cal M}$ is rational of dimension 19.
\begin{theorem}\label{IIItheo17}
There is a non-empty open set ${\cal U}$ of the rational variety ${\cal M}$ for
which the linear system $|H|$ embeds $S$ into ${\Bbb{P}}^4$.
\end{theorem}
\begin{Proof}
We have to show that for a general choice of $\Delta_1$ and $\sum y_k \in
|L_0|$ the linear system $|H|$ embeds $S$ into ${\Bbb{P}}^4$. We shall first treat
the closed conditions. Since $\Delta_1$ is smooth we can identify it with its
strict transform on $S$. Consider the exact sequence
$$
0\rightarrow {\cal O}_S(L-2x_1)\rightarrow {\cal
O}_S(D_1)\rightarrow {\cal O}_{\Delta_1}(D_1)\rightarrow 0.
$$
Since $\sum y_k\in |L_0|$ we have
$$
(35) \quad 6L-3x_1-2\sum\limits_{j\ge 2} x_j-\sum y_k\equiv g^1_3 \mbox{ on
} \Delta_1
$$
and hence $h^0({\cal O}_S(D_1))=h^0({\cal O}_{\Delta_1} (D_1))=2$. This is
condition $\on {(D_1)}$. Condition $ (\Delta_1)$ holds by construction.\\
In order to treat the open conditions we will first consider special points in
${\cal M}$ which give us all open conditions but two. These we will then treat
afterwards. The linear system $|4L-2x_1-\sum\limits_{j\ge 2} x_j|$ is free on
${\hat {\Bbb{P}}}^2$. Hence a general element $\Gamma$ is smooth and intersects
$\Delta_1$ transversally in 12 points $y_k$ which neither lie on an exceptional
line, nor on a line of the form $\Lambda_{kl}=L-x_k-x_l$. Moreover a general
element
$\Gamma$ is irreducible. This follows since $\Gamma^2=9$ and $|\Gamma|$ is
not composed of a pencil, since the class of $\Gamma$ is not divisible by 3
on ${\hat {\Bbb{P}}}^2$. Let $\Gamma'$ be the smooth transform of $\Gamma$ on $S$.
Since $\Gamma$ is smooth, $\Gamma'$ is isomorphic to $\Gamma$.
\begin{claim}
$|D_1|=\Gamma'+|2L-\sum\limits_j x_j|.$\\
This follows immediately since $D_1\equiv \Gamma'+(2L-\sum\limits_j x_j)$ and
$\dim |D_1|=1=\dim(\Gamma'+|2L-\sum x_j|).
$
\end{claim}
The only curves contained in an element of $|D_1|$ are $\Gamma'$, conics
$C\equiv 2L-\sum x_j$ and lines $\Lambda_{kl}=L-x_k-x_l$. The latter only
happens for finitely many elements of $|D_1|$. This shows immediately that
conditions (Q) and (R) are fulfilled with the possible exception that
$\dim|H-\Lambda_{1j}|\ge 2$. To exclude this we consider w.l.o.g. the
case $j=2$. Note that $H-\Lambda_{12}\equiv \Delta_2+x_1\equiv\Gamma ' +
\Lambda_{34}+x_1$. Since $\Gamma '$ is smooth of genus 2 and $\Gamma
'.(\Delta_2+x_1)=1$ it follows that $h^0({\cal O}_{\Gamma
'}(\Delta_2+x_1))\le 1$. The claim now follows from the exact sequence
$$
0\rightarrow {\cal O}_S(\Lambda_{34}+x_1)\rightarrow {\cal
O}_S(\Delta_2+x_1)\rightarrow {\cal O}_{\Gamma
'}(\Delta_2+x_1)\rightarrow 0
$$
together with the fact that $h^0({\cal
O}_S(\Lambda_{34}+x_1))=1$.
It remains to consider (P). The curve
$\Gamma'$ contradicts condition (P) since
$p(\Gamma')=2, H.\Gamma'=4$. Similarly the decomposition
$(\Gamma'+\Lambda_{ij})+\Lambda_{kl}$ contradicts (P) if $k, l\neq 1$. On
the other hand the above construction shows that for one (and hence the
general) pair $(\Delta_k, \sum y_k)$ all open conditions given by (P) are
fulfilled for a decomposition $D=A+B$ of an element in
$|D_1|$ with the possible exception of $|\Gamma'|\neq\emptyset$ or
$|D_1-\Lambda_{kl}|\neq\emptyset$ for $k,l\neq 1$. The first case is easy,
we can simply take an element $\sum y_k \in |L_0|$ which is not in the
codimension 1 linear subsystem given by
$|4L-2x_1-\sum\limits_{j\ge2} x_j|$ on ${\hat{\Bbb{P}}}^2$. Next we assume that
there is an element $A\in |D_1-\Lambda_{kl}|$ where $k,l\neq 1$. Then
$A.\Delta_1=2$. Since
$\Delta_1$ cannot be a component of $A$ this means that $A$ intersects
$\Delta_1$ in two points $Q_0, Q_1$. If $j$ is the remaining element of the
set $\{1,\ldots,4\}$ then $L-x_1-x_j\equiv Q_0+Q_1$ on $\Delta_1$ The linear
system $|L|$ cuts out a $g^2_5$ on $\Delta_1$ and is hence complete. Hence
$Q_0 + Q_1$ is the intersection of $\Lambda_{1j}$ with $\Delta_1$. In
particular $\Lambda_{1j}$ intersects $A$ in at least 2 points, namely $Q_0$
and $Q_1$. Since $A.\Lambda_{1j}=0$ this implies that $\Lambda_{1j}$ is a
component of $A$ (we can assume that $\Lambda_{1j}$ is irreducible). Hence
$A=A'+\Lambda_{1j}$ with $A'\in |D_1-\Lambda_{kl}-\Lambda_{1j}|=|\Gamma'|$
and we are reduced to the previous case.
\end{Proof}
\begin{remarks}\label{IIIrem18}
(i) Originally Okonek \cite{O2} constructed surfaces of degree $8$ and
sectional genus $6$ with the help of reflexive sheaves.
\noindent (ii) According to \cite{DES} the rational surfaces of degree $8$
with $\pi=6$ arise as the locus where a general morphism
$\phi:\Omega^3(3)\to{\cal O}(1)\oplus 4{\cal O}$ drops rank by $1$. The space of
such maps has dimension $80$. Taking the obvious group actions into account we
find that the moduli space has dimension $43=19+\dim\on{Aut}{\Bbb{P}}^4$.
Moreover this description shows that the moduli space is irreducible and
unirational.
\noindent (iii) These surfaces are in $(3,4)$-liaison with the Veronese
surface \cite{O2}. Counting parameters one finds again that they depend on $19$
parameters (modulo $\on{Aut}({\Bbb{P}}^4)$).
\noindent (iv) It was pointed out to us by K.~Ranestad that Ellingsrud and
Peskine (unpublished) also suggested a construction of these surfaces via
linear systems. They start with a smooth quartic $K_4=\{f_4=0\}$ and a
smooth quintic
$K_5=\{f_5=0\}$ touching in $4$ points $x_1,\ldots,x_4$. Let
$y_5,\ldots,y_{16}$ be the remaining points of intersection. Let
$$
{\cal I}'={\cal O}_{{\Bbb{P}}^2}\left(-\sum x_i\right),\quad
{\cal I}={\cal O}_{{\Bbb{P}}^2}\left(-2\sum x_i-\sum y_k\right).
$$
Then we have an exact sequence
$$
0\longrightarrow {\cal I}'(-4)\longrightarrow {\cal I}\longrightarrow {\cal O}_{K_4}(-5)
\longrightarrow 0.
$$
Twisting this by ${\cal O}(6)$ and taking global section gives
$$
0\longrightarrow \Gamma({\cal I}'(2)) \longrightarrow \Gamma({\cal O}_S(H))
\longrightarrow \Gamma({\cal O}_{K_4}(1)) \longrightarrow 0.
$$
Since $h^0({\cal I}'(2))=2$ and $h^0({\cal O}_{K_4}(1))=3$ this shows
$h^0({\cal O}_S(H))=5$. One can easily see that $|\Delta_i|\neq\emptyset$ and
$\dim|D_i|\geq1$ in this construction: counting parameters one shows that
$\Delta_i=\{lf_4+f_5=0\}$ for some suitable linear form and that there is
at least a $1$-dimensional family of curves in $|D_i|$ which are of the form
$D=\{qf_4+lf_5\}$ where $q$ is of degree $2$ and $l$ is a linear form. This
construction, too, depends on $19$ parameters.
\end{remarks}
Finally we want to discuss the moduli space of smooth special surfaces of
degree 8 in ${\Bbb{P}}^4$ (modulo Aut ${\Bbb{P}}^4$). Recall the set ${\cal M}$ consisting
of pairs $(\Delta_1, \sum y_k)$ where $\Delta_1 \in |H-(L-x_1)|$ is smooth
and $\sum y_k \in |L_0|$. We have proved in Theorem (\ref{IIItheo17}) that for
a general pair $(\Delta_1, \sum y_k)$ the linear system $|H|$ embeds $S$ into
${\Bbb{P}}^4$. Indeed in this way we obtain the general smooth surface of degree
8 in $ {\Bbb{P}}^4$. The surface $X={\hat {\Bbb{P}}}^2$, i.e. ${\Bbb{P}}^2$ blown up in
$x_1,\ldots, x_4$ is the del Pezzo surface of degree 5. It is well known that
Aut$X\cong S_5$ the symmetric group in 5 letters (Aut $X$ acts transitively on
the 5 maximal sets of disjoint rational curves on $X$, see \cite [Chapter
IV]{M}).
\begin{proposition}\label{IIIprop19}
For general $S$ the only lines contained in $S$ are the $y_k$'s.
\end{proposition}
\begin{Proof}
Let $l$ be a line on S. The statement is clear if $l$ is $\pi$-exceptional as
the $x_i$ are mapped to conics and since we can assume that there are no
infinitesimally near points. If $l$ is not skew to the plane spanned by $x_i$
then $l$ is contained in a reducible member of $|D_i|$. But for general
choice there is no decomposition A+B with A (or B) a line. Hence we can assume
that $l.x_i=0$ for $i=1,\ldots, 4$ and $l.y_k\le 1$ for all $k$. Thus $l\equiv
a L-\sum\limits_{k\in \triangle} y_k$ with $a\le 2$. Since $H.l=1$ we have
either
$a=1$ and $|\triangle|=5$ or $a=2$ and $|\triangle|=11$. In the first case 5 of
the
$y_k$ are collinear. But then it follows with the monodromy argument of
\cite [p.111]{ACGH} that all the $y_k$'s are collinear which is absurd. In
the same way the case $a=2$ would imply that all the $y_k$'s are on a conic
which also contradicts very ampleness of $|H|$.
\end{Proof}
\begin{theorem}\label{IIItheo20}
The moduli space of polarized rational surfaces (S,H) where $|H|$ embeds
$S$ into ${\Bbb{P}}^4$ as a surface of degree 8, speciality 1 and sectional
genus 6 is birationally equivalent to
${\cal M}/S_5$.
\end{theorem}
\begin{Proof}
Let ${\cal V}$ be the open set of ${\cal M}$ where $|H|$ embeds $S$ into
${\Bbb{P}}^4$ and where all the $\Delta_i$'s are smooth. Let $(\Delta_1, \sum
y_k)$ and $(\Delta_1 ', \sum y_k ')$ be two elements which give rise to
surfaces $S, S'\subset {\Bbb{P}}^4$ for which a projective transformation
${\bar g}:S\rightarrow S'$ exists. Since obviously ${\bar g}$ carries
lines to lines, it follows from Proposition (\ref{IIIprop19}) that ${\bar
g}$ is induced by an automorphism $g:X\rightarrow X$ carrying the set
$\{y_k\}$ to $\{y_k'\}$. Conversely, the group $S_5=\mbox { Aut } (X)$
acts on ${\cal V}$ as follows. Let $S$ correspond to $(\Delta_1 \sum
y_k)$ and let $g\in \mbox { Aut } (X)$: Then, since $6L-2\sum
x_j=-2K_X$ which is invariant under the action of $S_5$, we set
$\{y_k'\}=g\{y_k\}, H'=-2K_X-\sum y_k'$. Then $H'$ embeds $S'={\tilde
X}(y_1', \ldots, y_{12}')$ and we set $\Delta_1'$ to be the unique
curve in $|H'-L+y_1|$.
\end{Proof}
\section{Further outlook}\label{sectionIV}
In this section we want to discuss how this method can possibly be applied to
other surfaces. For smooth surfaces of degree $\leq8$ it is rather
straightforward to give a decomposition $H\equiv C+D$ which allows to apply the
Alexander-Bauer lemma. This was done in \cite{B}, \cite{CF} and
section~\ref{sectionIII} of this article. In degree $9$ there is one
non-special surface, which was treated in section~\ref{sectionII} of this
article, and a special surface with sectional genus $\pi=7$ which was found by
Alexander \cite{A2}. Here $S$ is ${\Bbb{P}}^2$ blown up in 15 points
$x_1,\ldots,x_{15}$ and $H\equiv 9L-3\sum\limits_{i=1}^6x_i-
2\sum\limits_{j=7}^9x_j-\sum\limits_{k=10}^{15}x_k$. As pointed out by
Alexander one can take the decomposition $H\equiv C+D$ where $C\equiv
3L-\sum\limits_{i=1}^9x_i$ and $D\equiv H-C$. Then $C$ is a plane cubic and
$|D|$ is a pencil of canonical curves of genus 4.
Rational surfaces of degree 10 were treated by Ranestad \cite{R1}, \cite{R2},
Popescu and Ranestad \cite{PR} and Alexander \cite{A2}. There is one surface
with $\pi=8$. In this case $S$ is ${\Bbb{P}}^2$ blown up in 13 points and $H\equiv
14L-6x_1-4\sum\limits_{i=2}^{10}x_i-2x_{11}-x_{12}-x_{13}$. Following
Alexander \cite{A2} the curve $C\equiv
7L-3x_1-2\sum\limits_{i=2}^{10}x_i-\sum\limits_{j=11}^{13}x_j$ is a plane
quartic and the residual pencil $|D|$ has $p(D)=3$ and degree 6. For sectional
genus $\pi=9$ there are two possibilities. The first is ${\Bbb{P}}^2$ blown in 18
points with $H\equiv 8L-2\sum\limits_{i=1}^{12}x_i-
\sum\limits_{j=13}^{18}x_j$. One can take $C\equiv 4L-\sum\limits_{i=1}^{16}
x_i$ which becomes a plane quartic. For the residual intersection $|D|$ one
finds $p(D)=3$, $H.D=6$. (For more details of this geometrically interesting
situation see \cite[Proposition~2.2]{PR}. The second surface with $\pi=9$ is
more difficult. Again we have ${\Bbb{P}}^2$ blown up in 18 points, but this time
$H\equiv 9L-3\sum\limits_{i=1}^4x_i- 2\sum\limits_{j=5}^{11}x_j-
\sum\limits_{k=12}^{18}x_k$. Clearly $S$ contains plane curves, e.g.~the
conics $x_j$. But then for the residual pencil $|D|$ one has $p(D)=7$, $H.D=9$
and this case seems difficult to handle. Numerically it would be possible
to have a decomposition $H\equiv C+D$ with $C\equiv
3L-\sum\limits_{i=1}^3x_i-
\sum\limits_{j=5}^{11}x_j-x_{12}$ which would be a plane cubic. In this case
$p(D)=4$, $H.D=6$. It might be interesting to check whether one can actually
construct surfaces with such a decomposition.
Of course, one can try and attempt to approach the problem of finding suitable
decompositions $H\equiv C+D$ more systematically. Let us assume $S$ is a
rational surface and $H\equiv C+D$ a decomposition to which the
Alexander-Bauer lemma can be applied. Let $h=h^1({\cal O}_S(H))$ be the
speciality of $S$. Since $C$ is mapped to a plane curve the exact sequence
$$
0\longrightarrow{\cal O}_S(D)\longrightarrow{\cal O}_S(H)\longrightarrow
{\cal O}_C(H)\longrightarrow 0
$$
is exact on global sections, and hence
$$
h=h^1(D)+\delta(C)
$$
where $h^1(D)=h^1({\cal O}_S(D))$ and $\delta(C)=h^1({\cal O}_C(H))$. The analogous
sequence for $D$ and the assumption that $|H|$ restricts to a complete system
on the curves $D'\in|D|$ gives
$$
h=h^1(C)+\delta(D)
$$
where $h^1(C)$ and $\delta(D)$ are defined similarly. In general if $C$ is a
curve of genus $(d-1)(d-2)/2$ and ${\cal O}_C(H)$ is a line bundle of degree $d$
it is difficult to show that $(C,{\cal O}_C(H))$ is a plane curve. Hence it is
natural to assume $H.C\leq4$. In order to be able to control the linear system
$|H|$ on the curves $D'\in|D|$ one is normally forced to assume that $H.D\geq
2p(D)-2$ and $H|_D=K_D$ in case of equality. Hence $\delta(D)=0$ if
$H.D>2p(D)-2$ and $\delta(D)=1$ otherwise. Since $|H|$ is complete on $D$ we
have $h^0({\cal O}_D(H))\leq4$. Now using our assumption that $H.D\geq 2p(D)-2$
and Riemann-Roch on $D$ we find
$$
2p(D)-2\leq H.D\leq p(D)+3+\delta(D)
$$
and from this
$$
p(D)\leq 5+\delta(D).
$$
If $\delta(D)=0$ then $p(D)\leq5$. If $\delta(D)=1$ then $H|_D=K_D$ and
$h^0({\cal O}_D(H))=p(D)$, i.e.~$p(D)\leq4$ in this case. But now
$$
d=H.C+H.D\leq p(D)+7+\delta(D).
$$
This shows that one can find such a decomposition only if the degree $d\leq12$.
The case $d=12$ can only occur for $H.C=4$.
Finally we want to discuss the case $d=11$. In his thesis Popescu \cite{P}
gave three examples of rational surfaces of degree 11. In each case it is
${\Bbb{P}}^2$ blown up in 20 points. The linear systems are as follows:
\begin{eqnarray}
H&\equiv&10L-4x_1-3\sum_{i=2}^4x_i-2\sum_{j=5}^{14}x_j-\sum_{k=15}^{20}x_k
\label{IVgl1}\\
H&\equiv&11L-5x_1-3\sum_{i=2}^7x_i-2\sum_{j=8}^{13}x_j-\sum_{k=14}^{20}x_k
\label{IVgl2}\\
H&\equiv&13L-5x_1-4\sum_{i=2}^8x_i-2\sum_{j=9}^{11}x_j-\sum_{k=12}^{20}x_k
\label{IVgl3}
\end{eqnarray}
In each of these cases $S$ contains a plane quintic. The residual intersection
gives a pencil of rational (cases (\ref{IVgl1}) and (\ref{IVgl2})),
resp.~elliptic (case (\ref{IVgl3})) sextics. Since the linear system $|H|$ is
not complete on the curves of this linear system, one cannot immediately apply
the Alexander-Bauer lemma to this decomposition. One can ask whether there are
decompositions fulfilling the conditions given above. A candidate in case
(\ref{IVgl1}) is given by $C\equiv 4L-x_1-\sum\limits_{i=2}^4x_i-
\sum\limits_{j=5}^{14}x_j-\sum\limits_{k=15}^{17}x_k$ and $D\equiv H-C$. We do
not know whether surfaces with such a decomposition actually occur. In the
other cases one can show that no such decompositions exist.
\bibliographystyle{alpha}
|
1997-11-05T21:51:02 | 9505 | alg-geom/9505003 | en | https://arxiv.org/abs/alg-geom/9505003 | [
"alg-geom",
"math.AG"
] | alg-geom/9505003 | Atsushi Moriwaki | Atsushi Moriwaki | Bogomolov conjecture over function fields for stable curves with only
irreducible fibers (Version 2.0) | 21 pages (with 1 figure), AMSLaTeX version 1.2 (In this version 2.0,
the restriction of characteristic is removed.) | null | null | null | null | Let K be a function field and C a non-isotrivial curve of genus g >= 2 over
K. In this paper, we will show that if C has a global stable model with only
geometrically irreducible fibers, then Bogomolov conjecture over function
fields holds.
| [
{
"version": "v1",
"created": "Thu, 4 May 1995 05:27:50 GMT"
},
{
"version": "v2",
"created": "Thu, 22 Jun 1995 18:45:50 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Moriwaki",
"Atsushi",
""
]
] | alg-geom | \section{Introduction}
Let $k$ be a field,
$X$ a smooth projective surface over $k$, $Y$ a smooth projective curve over
$k$,
and $f : X \to Y$ a generically smooth semistable curve of genus $g \geq 2$
over $Y$.
Let $K$ be the function field of $Y$, $\overline{K}$
the algebraic closure of $K$, and $C$ the generic fiber of $f$.
For $D \in \operatorname{Pic}^1(C)(\overline{K})$, let
$j_D : C_{\overline{K}} \to \operatorname{Pic}^0(C)_{\overline{K}}$
be an embedding defined by $j_D(x) = x - D$.
Then, we have the following conjecture due to Bogomolov.
\begin{Conjecture}[Bogomolov conjecture over function fields]
\label{conj:Geometric:Bogomolov:Conjecture}
If $f$ is non-isotrivial, then, for any
embedding $j_D$, the image $j_D(C(\overline{K}))$ is discrete in terms
of the semi-norm $\Vert \ \Vert_{NT}$ given by
the Neron-Tate height pairing on $\operatorname{Pic}^0(C)(\overline{K})$, i.e.,
for any point $P \in \operatorname{Pic}^0(C)(\overline{K})$,
there is a positive number $\epsilon$ such that the set
\[
\left\{ x \in C(\overline{K}) \mid
\Vert j_D(x) - P \Vert_{NT} \leq \epsilon \right\}
\]
is finite.
\end{Conjecture}
In this paper, we will prove the above conjecture
under the assumption that
the stable model of
$f : X \to Y$ has only geometrically irreducible fibers.
\begin{Theorem}
\label{thm:conj:bogomolov}
If the stable model of $f : X \to Y$ has only geometrically irreducible fibers,
then Conjecture~\ref{conj:Geometric:Bogomolov:Conjecture} holds.
More strongly, there is a positive number $A$ with the following properties.
\begin{enumerate}
\renewcommand{\labelenumi}{(\arabic{enumi})}
\item
${\displaystyle
A \geq \sqrt{\frac{g-1}{12g(2g+1)}\delta}}$,
where $\delta$ is the number of singularities in singular fibers
of $f_{\bar{k}} : X_{\bar{k}} \to Y_{\bar{k}}$.
\item
For any small positive number $\epsilon$, the set
\[
\left\{ x \in C(\overline{K}) \mid \Vert j_D(x) - P \Vert_{NT} \leq
(1 - \epsilon)A \right\}
\]
is finite for any embedding $j_D$ and
any point $P \in \operatorname{Pic}^0(C)(\overline{K})$.
\end{enumerate}
\end{Theorem}
Our proof of Theorem~\ref{thm:conj:bogomolov}
is based on the admissible pairing on semistable curves due to S. Zhang (cf.
\S\ref{sec:metrized:graph:green:function:admissible:pairing}),
Cornalba-Harris-Xiao's inequality over an arbitrary field
(cf. Theorem~\ref{thm:Cornalba-Harris-Xiao:inequality}) and
an exact calculation of a Green function on
a certain metrized graph (cf. Lemma~\ref{lem:green:union:circle}).
The estimation of a Green function also gives the following
result, which strengthen S. Zhang's theorem \cite{Zh}.
\begin{Theorem}[cf. Corollary~\ref{cor:lower:bound:w:w:not:smooth}]
Let $K$ be a number field, $O_K$ the ring of integers,
$f : X \to \operatorname{Spec}(O_K)$ a regular semistable arithmetic surface of genus $g \geq
2$
over $O_K$.
If $f$ is not smooth, then
\[
(\omega_{X/O_K}^{Ar} \cdot \omega_{X/O_K}^{Ar}) \geq \frac{\log 2}{6(g-1)}.
\]
\end{Theorem}
\section{Metrized graph, Green function and admissible pairing}
\label{sec:metrized:graph:green:function:admissible:pairing}
In this section, we recall several facts of metrized graphs,
Green functions and the admissible pairing
on semistable curves. Details can be found in Zhang's paper \cite{Zh}.
Let $G$ be a locally metrized and compact topological space.
We say $G$ is a metrized graph if, for any $x \in G$,
there is a positive number $\epsilon$, a positive integer $d = v(x)$
(which is called the valence at $x$), and
an open neighborhood $U$ of $x$ such that $U$ is isometric to
\[
\left\{ t e^{\frac{2 \pi \sqrt{-1} k}{d}} \in {\Bbb{C}} \mid 0 \leq t < \epsilon,
k \in {\Bbb{Z}} \right\}.
\]
Let $\operatorname{Div}(G)$ be a free abelian group generated by points of $G$.
An element of $\operatorname{Div}(G)$ is called a {\em divisor} on $G$.
Let $F(G)$ be the set of all piecewisely smooth real valued
functions on $G$.
For $f \in F(G)$, we can define the Dirac function $\delta(f)$ associated with
$f$
as follows. If $x \in G$ and $v(x) = n$, then $\delta(f)(x)$ is given by
\[
(\delta(f)(x), g) = g(x) \sum_{i=1}^n \lim_{x_i \to 0} f'(x_i),
\]
where $g \in F(G)$ and $x_i$ is the arc-length parameter of one branch
starting from $x$. The Laplacian $\Delta$ for $f \in F(G)$ is defined by
\[
\Delta(f) = -f'' - \delta(f),
\]
where $f''$ is the second derivative of $f$ in the sense of distribution.
Let $Q(G)$ be a subset of $F(G)$ consisting of piecewisely
quadric polynomial functions.
Let $V$ be a set of vertices of $G$ such that
$G \setminus V$ is a disjoint union of open segments.
Let $E$ be the collection of segments in $G \setminus V$.
We denote by $Q(G, V)$ a subspace of $Q(G)$ consisting of
functions whose restriction to each edge in $E$ are
quadric polynomial functions, and by $M(G, V)$
a vector space of measures on $G$ generated by Dirac functions $\delta_v$ at
$v \in V$ and by Lebesgue measures on edges $e \in E$ arising from
the arc-length parameter.
The fundamental theorem is the following existence of
the admissible metric and the Green function.
\begin{Theorem}[{\cite[Theorem 3.2]{Zh}}]
\label{thm:existence:metric:green}
Let $D = \sum_{x \in G} d_x x$ be a divisor on $G$ such that
the support of $D$ is in $V$.
If $G$ is connected and $\deg(D) \not= -2$, then
there are a unique measure $\mu \in M(G, V)$ and
a unique function $g_{\mu}$ on $G \times G$ with the following properties.
\begin{enumerate}
\renewcommand{\labelenumi}{(\arabic{enumi})}
\item
${\displaystyle \int_{G} \mu = 1}$.
\item
$g_{\mu}(x, y)$ is symmetric and continuous on $G \times G$.
\item
For a fixed $x \in G$, $g_{\mu}(x, y) \in Q(G)$.
Moreover, if $x \in V$, then $g_{\mu}(x, y) \in Q(G, V)$.
\item
For a fixed $x \in G$, $\Delta_y(g_{\mu}(x, y)) = \delta_x - \mu$.
\item
For a fixed $x \in G$, ${\displaystyle \int_G g_{\mu} (x, y) \mu(y) = 0}$.
\item
$g_{\mu}(D, y) + g_{\mu}(y, y)$ is a constant for all $y \in G$,
where $g_{\mu}(D, y) = \sum_{x \in G} d_x g_{\mu}(x, y)$.
\end{enumerate}
Further, if $d_x \geq v(x) - 2$ for all $x \in G$,
then $\mu$ is positive.
\end{Theorem}
The measure $\mu$ in Theorem~\ref{thm:existence:metric:green} is called
the {\em admissible metric} with respect to $D$ and $g_{\mu}$ is called
the {\em Green function} with respect to $\mu$.
The constant $g_{\mu}(D, y) + g_{\mu}(y, y)$ is denoted by $c(G, D)$.
\bigskip
Let $k$ be an algebraically closed field,
$X$ a smooth projective surface over $k$, $Y$ a smooth projective curve over
$k$,
and $f : X \to Y$ a generically smooth semistable curve of genus $g \geq 1$
over $Y$.
Let $\operatorname{CV}(f)$ be the set of all critical values of $f$, i.e.,
$y \in \operatorname{CV}(f)$ if and only if $f^{-1}(y)$ is singular.
For $y \in \operatorname{CV}(f)$, let $G_y$ be the metrized graph of $f^{-1}(y)$
defined as follows.
The set of vertices $V_y$ of $G_y$ is indexed by
irreducible components of the fiber $f^{-1}(y)$ and
singularities of $f^{-1}(y)$ correspond to edges of length $1$.
We denote by $C_v$ the corresponding irreducible curve for a vertex $v$
in $V_y$. Let $K_y$ be a divisor on $G_y$ given by
\[
K_y = \sum_{v \in V_y} (\omega_{X/Y} \cdot C_v) v.
\]
Let $\mu_y$ be the admissible metric with respect to $K_y$ and
$g_{\mu_y}$ the Green function of $\mu_y$.
The admissible dualizing sheaf $\omega_{X/Y}^a$ is defined by
\[
\omega_{X/Y}^a = \omega_{X/Y} - \sum_{y \in \operatorname{CV}(f)} c(G_y, K_y) f^{-1}(y).
\]
Here we define a new pairing $(D \cdot E)_a$ for
$D, E \in \operatorname{Div}(X) \otimes {\Bbb{R}}$ by
\[
(D \cdot E)_a = (D \cdot E) + \sum_{y \in \operatorname{CV}(f)} \left\{ \sum_{v, v' \in V_y}
(D \cdot C_v) g_{\mu_y}(v, v') (E \cdot C_{v'}) \right\}.
\]
This pairing is called the {\em admissible pairing}.
It has lots of properties. For our purpose, the following are important.
\medskip
\begin{enumerate}
\renewcommand{\labelenumi}{(\arabic{enumi})}
\item (Adjunction formula)
If $B$ is a section of $f$, then
$(\omega_{X/Y}^a + B \cdot B)_a = 0$.
\item (Intersection with a fiber)
If $D$ is an ${\Bbb{R}}$-divisor with degree $0$ along general fibers,
then $(D \cdot Z)_a = 0$ for all vertical curves $Z$.
(cf. Proposition~\ref{prop:admissible:with:fiber})
\item (Compatibility with base changes)
The admissible pairing is compatible with base changes.
Namely, let $\pi : Y' \to Y$ be a finite morphism of smooth projective curves,
and
$X'$ the minimal resolution of the fiber product of $X \times_Y Y'$.
We set the induced morphisms as follows.
\[
\begin{CD}
X @<{\pi'}<< X' \\
@V{f}VV @VV{f'}V \\
Y @<<{\pi}< Y'
\end{CD}
\]
Then, for $D, E \in \operatorname{Div}(X) \otimes {\Bbb{R}}$,
$({\pi'}^*(D) \cdot {\pi'}^*(E))_a = (\deg \pi)(D \cdot E)_a$.
Moreover, we have ${\pi'}^*(\omega_{X/Y}^a) = \omega_{X'/Y'}^a$.
Thus, $(\omega_{X'/Y'}^a \cdot \omega_{X'/Y'}^a)_a
= (\deg \pi) (\omega_{X/Y}^a \cdot \omega_{X/Y}^a)_a$.
\end{enumerate}
\medskip\noindent
Using the above properties, we can give the Neron-Tate height paring
in terms of the admissible pairing.
Let $C$ be the generic fiber of $f$, $K$ the function field of $Y$, and
$L, M \in \operatorname{Pic}^0(C)(\overline{K})$. Then, there are
a base change $Y' \to Y$, a semistable model $X'$ of $C$ over $Y'$, and
line bundles ${\cal L}$ and ${\cal M}$ on $X'$ such that
${\cal L}_{\overline{K}} = L$ and ${\cal M}_{\overline{K}} = M$.
Moreover, we can find vertical ${\Bbb{Q}}$-divisors $V$ and $V'$ on $X'$
such that $({\cal L} + V \cdot Z) = ({\cal M} + V' \cdot Z) = 0$
for all vertical curves $Z$ on $X'$. Then, it is easy to see that
\[
\frac{-1}{[k(Y') : k(Y)]}({\cal L} + V \cdot {\cal M} + V')
\]
is well-defined. It is denoted by $(L \cdot M)_{NT}$ and is called
the {\em Neron Tate height pairing}. Moreover, it is easy to see
$(L \cdot L)_{NT} \geq 0$. So $\sqrt{(L \cdot L)_{NT}}$ is denoted
by $\Vert L \Vert_{NT}$. On the other hand, by the definition of the
admissible pairing, we have
\[
({\cal L} + V \cdot {\cal M} + V') = ({\cal L} + V \cdot {\cal M} + V')_a.
\]
Thus, using the second property of the above, we can see that
\[
-[k(Y') : k(Y)] (L \cdot M)_{NT} = ({\cal L}\cdot {\cal M})_a,
\]
which means that the admissible pairing does not depend
on the choice of the compactification of $L$ and $M$, and that
of course
\[
(L \cdot M)_{NT} = \frac{-({\cal L}\cdot {\cal M})_a}{[k(Y') : k(Y)]}.
\]
\medskip
Next, let us consider a height function in terms of the
admissible pairing. Let ${\cal L}$ be an ${\Bbb{R}}$-divisor on $X$ and
$x \in C(\overline{K})$. Then, taking a suitable base change $\pi : Y' \to Y$,
there is a semistable model
$f' : X' \to Y'$ of $C$ such that $x$ is realized as a section $B_x$
of $f'$. We set
\[
h^a_{{\cal L}}(x) = \frac{({\pi'}^*({\cal L}) \cdot B_x)_a}{\deg \pi},
\]
where $\pi' : X' \to X$ is the induced morphism.
One can easily see that $h^a_{{\cal L}}(x)$ is well-defined by the third
property
of the above. The following generic lower estimate of the height function
is important for our purpose.
\begin{Theorem}[{\cite[Theorem 5.3]{Zh}}]
\label{thm:lower:estimate:height}
If $\deg({\cal L}_K) > 0$ and ${\cal L}$ is $f$-nef, then,
for any $\epsilon > 0$, there is a finite subset $S$ of $C(\overline{K})$
such that
\[
h^a_{{\cal L}}(x) \geq \frac{({\cal L} \cdot {\cal L})_a}{2 \deg({\cal L}_K)} -
\epsilon
\]
for all $x \in C(\overline{K}) \setminus S$.
\end{Theorem}
As corollary, we have the following.
\begin{Corollary}[{\cite[Theorem 5.6]{Zh}}]
\label{cor:lower:estimate:NT:metric}
Let $D \in \operatorname{Pic}^1(C)(\overline{K})$. Then, for any $\epsilon > 0$,
there is a finite subset $S$ of $C(\overline{K})$
such that
\[
\Vert D - x \Vert_{NT}^2 \geq
\frac{(\omega_{X/Y}^a \cdot \omega_{X/Y}^a)_a}{4(g-1)} +
\frac{\Vert \omega_C - (2g-2) D \Vert_{NT}^2}{4g(g-1)}
- \epsilon
\]
for all $x \in C(\overline{K}) \setminus S$.
\end{Corollary}
{\sl Proof.}\quad
Let $\pi_1 : Y_1 \to Y$ be a base change of $f : X \to Y$ such that
$D$ is defined over the function field $k(Y_1)$ of $Y_1$.
Let $f_1 : X_1 \to Y_1$ be the semistable model of $C$ over $Y_1$,
$F$ a general fiber of $f_1$, and
${\cal D}$ a compactification of $D$ such that
${\cal D}$ is a ${\Bbb{Q}}$-divisor on $X_1$ and ${\cal D}$ is $f_1$-nef.
Using adjuction formula and applying Theorem~\ref{thm:lower:estimate:height} to
\[
{\cal L} = \omega_{X_1/Y_1}^a + 2 {\cal D} - ({\cal D} \cdot {\cal D})_a F,
\]
we have our corollary.
\QED
\section{Green function of a certain metrized graph}
\label{sec:certain:graph}
In this section, we will construct a Green function of
a certain metrized graph. Let us begin with the following lemma.
\begin{Lemma}
\label{lem:laplacian:on:circle}
Let $C$ be a circle with arc-length $l$.
Fixing a point $O$ on $C$, let $t : C \to [0, l)$ be a coordinate of $C$
with $t(O) = 0$ coming from an arc-length parameterization of $C$.
We set
\[
\phi(t) = \frac{1}{2l} t^2 - \frac{1}{2} |t|
\quad\text{and}\quad
f(x, y) = \phi(t(x) - t(y)).
\]
Then, we have the following.
\begin{enumerate}
\renewcommand{\labelenumi}{(\arabic{enumi})}
\item
$f(x, y)$ is symmetric and continuous on $C \times C$.
\item
$f(x, y)$ is smooth on the outside of the diagonal.
\item
For a fixed $x \in C$,
${\displaystyle \Delta_y(f(x, y)) = \delta_x - \frac{dt}{l}}$.
\end{enumerate}
\end{Lemma}
{\sl Proof.}\quad
We can check them by a straightforward calculation.
\QED
Let $C_1, \ldots, C_n$ be circles and $G$ a metrized graph constructed by
joining $C_i$'s at a point $O$. Let $l_i$ be the arc-length of $C_i$ and
$t_i : C_i \to [0, l_i)$ a coordinate of $C_i$ with $t_i(O) = 0$.
\yes
\par\bigskip
\Draw
\MoveTo(0,0) \MarkLoc(O) \Node(Q)(--$\bullet$--)
\MoveTo(30,60) \MarkLoc(A)
\MoveTo(-30,60) \MarkLoc(B)
\MoveTo(-66.96,-4.02) \MarkLoc(C)
\MoveTo(-36.96,-55.98) \MarkLoc(D)
\MoveTo(36.96,-55.98) \MarkLoc(E)
\MoveTo(66.96,-4.02) \MarkLoc(F)
\Curve(O,A,B,O) \Curve(O,C,D,O) \Curve(O,E,F,O)
\MoveTo(10,7) \Node(G)(--$O$--)
\MoveTo(0,55) \Node(H)(--$C_1$--)
\MoveTo(-47.63,-27.5) \Node(H)(--$C_2$--)
\MoveTo(47.63,-27.5) \Node(H)(--$C_3$--)
\EndDraw
\bigskip
\par\noindent
\else
\fi
{}From now on, we will identify a point on $C_i$ with its coordinate.
As in Lemma~\ref{lem:laplacian:on:circle}, for each $i$, we set
\[
\phi_i(t) = \frac{1}{2l_i} t^2 - \frac{1}{2} |t|.
\]
We fix a positive integer $g$.
Here we consider a measure $\mu$ and a divisor $K$ on $G$ defined by
\[
\mu = \frac{g-n}{g} \delta_O + \sum_{i=1}^n \frac{d t_i}{gl_i}
\quad\text{and}\quad
K = (2g-2)O.
\]
Moreover, let us consider
the following function $g_{\mu}$ on $G \times G$.
\[
g_{\mu}(x, y) =
\begin{cases}
{\displaystyle \phi_i(x - y) - \frac{g-1}{g} \left(\phi_i(x) + \phi_i(y)\right)
+
\frac{L}{12g^2}} &
\text{if $x, y \in C_i$} \\
{\displaystyle \frac{1}{g}\left( \phi_i(x) + \phi_j(y) \right) +
\frac{L}{12g^2}} &
\text{if $x \in C_i$, $y \in C_j$ and $i \not= j$}
\end{cases}
\]
where $L = l_1 + \cdots + l_n$.
Then, we can see the following.
\begin{Lemma}
\label{lem:green:union:circle}
\begin{enumerate}
\renewcommand{\labelenumi}{(\arabic{enumi})}
\item ${\displaystyle \int_G \mu = 1}$.
\item
$g_{\mu}(x, y)$ is symmetric and continuous on $G \times G$.
\item
For a fixed $x \in G$, $\Delta_y(g_{\mu}(x, y)) = \delta_x - \mu$.
\item
For a fixed $x \in G$, ${\displaystyle \int_G g_{\mu}(x, y) \mu(y) = 0}$.
\item
${\displaystyle g_{\mu}(K, y) + g_{\mu}(y, y) = \frac{L(2g-1)}{12g^2}}$
for all $y \in G$.
\end{enumerate}
\end{Lemma}
{\sl Proof.}\quad
(1), (2) These are obvious.
(3) We assume $x \in C_i$. By \cite[Lemma a.4, (a)]{Zh},
\[
\Delta_y(g_{\mu}(x, y)) = \sum_{j=1}^n \Delta_y(\rest{g_{\mu}(x, y)}{C_j}).
\]
Therefore, using Lemma~\ref{lem:laplacian:on:circle},
we get
\begin{align*}
\Delta_y(g_{\mu}(x, y)) & =
\Delta_y(\rest{g_{\mu}(x, y)}{C_i}) + \sum_{j \not= i}^n
\Delta_y(\rest{g_{\mu}(x, y)}{C_j}) \\
& = \left( \delta_x - \frac{dt_i}{l_i} -
\frac{g-1}{g}\left(\delta_O - \frac{d t_i}{l_i}\right)\right) +
\sum_{j \not= i}^n \frac{1}{g}\left( \delta_O - \frac{dt_j}{l_j} \right) \\
& = \delta_x - \mu.
\end{align*}
(4) We assume $x \in C_i$.
Then, by a direct calculation, we can see
\[
\int_{C_j} g_{\mu}(x, t_j) \frac{d t_j}{gl_j} =
\begin{cases}
{\displaystyle -\frac{g-1}{g^2}\phi_i(x) - \frac{l_i}{12g^2} + \frac{L}{12g^3}}
&
\text{if $j = i$} \\
{\displaystyle \frac{1}{g^2} \phi_i(x) - \frac{l_j}{12g^2} + \frac{L}{12g^3}} &
\text{if $j \not= i$}
\end{cases}
\]
Therefore,
\[
\sum_{j=1}^n \int_{C_j} g_{\mu}(x, t_j) \frac{d t_j}{gl_j} =
\frac{n-g}{g}\left( \frac{1}{g} \phi_i(x) + \frac{L}{12g^2} \right).
\]
Hence,
\begin{align*}
\int_G g_{\mu}(x, y) \mu(y) & =
\frac{g-n}{g}g_{\mu}(x, 0) + \sum_{j=1}^n \int_{C_j} g_{\mu}(x, t_j) \frac{d
t_j}{gl_j} \\
& = \frac{g-n}{g}\left( \frac{1}{g} \phi_i(x) + \frac{L}{12g^2} \right) +
\frac{n-g}{g}\left( \frac{1}{g} \phi_i(x) + \frac{L}{12g^2} \right) \\
& = 0.
\end{align*}
(5) Since
\[
g_{\mu}(O, x) = \frac{1}{g} \phi_i(x) + \frac{L}{12g^2} \quad\text{and}\quad
g_{\mu}(x, x) = \frac{-2(g-1)}{g} \phi_i(x) + \frac{L}{12g^2},
\]
(5) follows.
\QED
This lemma says us that
$\mu$ is the admissible metric with respect to $K$,
$g_{\mu}$ is the Green function of $\mu$, and
${\displaystyle c(G, K) = \frac{L(2g-1)}{12g^2}}$.
\section{Cornalba-Harris-Xiao's inequality over an arbitrary field}
In this section, we would like to generalize Cornalba-Harris-Xiao's inequality
to fibered algebraic surfaces over an arbitrary field, namely,
\begin{Theorem}
\label{thm:Cornalba-Harris-Xiao:inequality}
Let $k$ be a field,
$X$ a smooth projective surface over $k$, $Y$ a smooth projective curve over
$k$, and
$f : X \to Y$ a generically smooth morphism with $f_*{\cal{O}}_X = {\cal{O}}_C$.
If the genus $g$ of the generic fiber of $f$ is greater than or equal to $2$
and
$\omega_{X/Y}$ is $f$-nef, then
\[
(\omega_{X/Y} \cdot \omega_{X/Y}) \geq \frac{4(g-1)}{g}
\deg(f_*(\omega_{X/Y})).
\]
\end{Theorem}
The above was proved in \cite{CH} and \cite{Xi} under the assumption
$\operatorname{char}(k) = 0$. Here we prove it using the following result
of Bost.
\begin{Theorem}[{\cite[Theorem III]{Bo}}]
\label{thm:bost:ineq}
Let $k$ be a field, $Y$ a smooth projective curve over $k$, and
$E$ a vector bundle on $Y$. Let
\[
\pi : P =
\operatorname{Proj}\left(
\bigoplus_{n=0}^{\infty} \operatorname{Sym}^n(E)
\right)
\longrightarrow Y
\]
be the projective bundle of $E$ and ${\cal{O}}_P(1)$ the tautological line bundle
on $P$.
If an effective cycle $Z$ of dimension $d \geq 1$ on $P$ is
Chow semistable on the generic fiber of $\pi$, then
\[
\frac{\left({\cal{O}}_P(1)^d \cdot Z \right)}
{d \cdot \left( {\cal{O}}_P(1)^{d-1} \cdot Z \cdot F \right)}
\geq \frac{\deg E}{\operatorname{rk} E},
\]
where $F$ is a general fiber of $\pi$.
\end{Theorem}
\medskip
First of all, let us begin with the following lemmas.
\begin{Lemma}
\label{lem:Chow:semistable:canonical}
Let $K$ be a field, $C$ a smooth projective curve over $K$ of genus $g \geq 2$,
and
$\phi : C \to {\Bbb{P}}^{g -1}$ a morphism given by
the complete linear system $|\omega_C|$. Then $\phi_*(C)$
is a Chow semistable cycle on ${\Bbb{P}}^{g-1}$.
\end{Lemma}
\proof
Let $R$ be the image of $C$ by $\phi$ and $n$ an integer given by
\[
n =
\begin{cases}
1, & \text{if $C$ is non-hyperelliptic}, \\
2, & \text{if $C$ is hyperelliptic}.
\end{cases}
\]
Then, $\phi_*(C) = nR$.
Thus a Chow form of $\phi_*(C)$ is the $n$-th power of
a Chow form of $R$. Therefore, $\phi_*(C)$ is Chow semistable if and only if
$R$ is Chow semistable.
Moreover, Theorem~4.12 in \cite{Mu} says that
Chow semistability of $R$ is derived from linear semistability of $R$ .
Let $V$ be a subspace of $H^0(C, \omega_C)$,
$p : {\Bbb{P}}^{g-1} \dashrightarrow {\Bbb{P}}^{\dim V - 1}$
the projection defined by the inclusion $V \hookrightarrow H^0(C, \omega_C)$,
and $\phi' : C \to {\Bbb{P}}^{\dim V - 1}$ a morphism given by $V$.
Then, $p \cdot \phi = \phi'$.
We need to show that
\addtocounter{Claim}{1}
\begin{equation}
\label{eqn:lem:stable:kernel:Chow:stable}
\frac{2}{n} = \frac{\deg(R)}{g - 1} \leq \frac{\deg(p_*(R))}{\dim V - 1}
\end{equation}
to see linear semistability of $R$.
Since $\deg({\phi'}^*({\cal{O}}(1))) = n \deg(p_*(R))$,
(\ref{eqn:lem:stable:kernel:Chow:stable})
is equivalent to say
\[
2 \leq
\frac{\deg({\phi'}^*({\cal{O}}(1)))}{\dim V - 1}.
\]
On the other hand,
if we denote by
$\omega^V_C$ the image of $V \otimes {\cal{O}}_C \to \omega_C$,
then, by Clifford's lemma, we have
\[
\dim V - 1 \leq \dim |\omega^V_C| \leq \frac{\deg(\omega^V_C)}{2}.
\]
Thus, we get (\ref{eqn:lem:stable:kernel:Chow:stable})
because
$\deg({\phi'}^*({\cal{O}}(1))) = \deg(\omega^V_C)$.
\QED
\begin{Remark}
By \cite[Proposition~4.2]{Bo}, $\phi_*(C)$ is
actually Chow stable when $\operatorname{char}(K) = 0$.
We don't know whether $\phi_*(C)$ is Chow stable if $\operatorname{char}(K) >
0$.
Anyway, semistability is enough for our purpose.
\end{Remark}
\medskip
Let us start the proof of Theorem~\ref{thm:Cornalba-Harris-Xiao:inequality}.
Let
\[
\phi : X \dashrightarrow P = \operatorname{Proj}\left(
\bigoplus_{n=0}^{\infty} \operatorname{Sym}^n(f_*(\omega_{X/Y}))
\right)
\]
be a rational map over $Y$ induced by $f^*f_*(\omega_{X/Y}) \to \omega_{X/Y}$.
Here we take a birational morphism $\mu : X' \to X$ of smooth projective
varieties such that $\phi' = \phi \cdot \mu : X' \to P$
is a morphism.
\[
\begin{CD}
X' @>{\mu}>> X \\
@V{\phi'}VV @VV{\phi}V \\
P @= P
\end{CD}
\]
Then, there is an effective vertical divisor $D$ on $X'$ such that
$\mu^*(\omega_{X/Y}) = {\phi'}^*({\cal{O}}_P(1)) + D$.
Let $Z = {\phi'}_*(X')$. Then, by Lemma~\ref{lem:Chow:semistable:canonical},
$Z$ give a Chow semistable cycle on the generic fiber.
Thus, by Theorem~\ref{thm:bost:ineq}, we have
\[
\frac{\left({\phi'}^*({\cal{O}}_P(1)) \cdot {\phi'}^*({\cal{O}}_P(1))\right)}{4(g-1)}
\geq \frac{\deg(f_*(\omega_{X/Y}))}{g}.
\]
On the other hand, since $\omega_{X/Y}$ is $f$-nef and $(D \cdot D) \leq 0$,
\begin{align*}
\left({\phi'}^*({\cal{O}}_P(1)) \cdot {\phi'}^*({\cal{O}}_P(1))\right) & =
\left( \mu^*(\omega_{X/Y}) - D \cdot \mu^*(\omega_{X/Y}) - D \right) \\
& = \left( \omega_{X/Y} \cdot \omega_{X/Y} \right)
-2 \left(\mu^*(\omega_{X/Y}) \cdot D \right) + (D \cdot D) \\
& \leq \left( \omega_{X/Y} \cdot \omega_{X/Y} \right).
\end{align*}
Therefore, we have our desired inequality.
\QED
\begin{Remark}
\label{rem:semistable:kernel:another:proof}
If $\operatorname{char}(k) = 0$,
we can give another proof of Theorem~\ref{thm:Cornalba-Harris-Xiao:inequality}
according to \cite{Mo2}. A rough idea is the following.
Since the kernel $K$ of $f^*f_*(\omega_{X/Y}) \to \omega_{X/Y}$
is semistable on the generic fiber of $f$ by virtue of \cite{PR}, we can apply
Bogomolov-Gieseker's inequality to $K$, which implies
Cornalba-Harris-Xiao's inequality by easy calculations.
\end{Remark}
\section{Proof of Theorem~\ref{thm:conj:bogomolov}}
In this section, we would like to give the proof of
Theorem~\ref{thm:conj:bogomolov}.
First of all, let us fix notations.
Let $k$ be a field,
$X$ a smooth projective surface over $k$, $Y$ a smooth projective curve over
$k$,
and $f : X \to Y$ a generically smooth semistable curve of genus $g \geq 2$
over $Y$.
Let $K$ be the function field of $Y$,
$\overline{K}$ the algebraic closure of $K$, and $C$ the generic fiber of $f$.
We assume that $f$ is non-isotrivial and
the stable model of $f : X \to Y$ has only geometrically irreducible fibers.
Clearly, for the proof of Theorem~\ref{thm:conj:bogomolov},
we may assume that $k$ is algebraically closed.
Then, we have the following lower estimate of
$(\omega_{X/Y}^a \cdot \omega_{X/Y}^a)_a$.
\begin{Theorem}
\label{thm:lower:bound:admissible:intersection}
Under the above assumptions, $(\omega_{X/Y}^a \cdot \omega_{X/Y}^a)_a$
is positive. Moreover,
\[
(\omega_{X/Y}^a \cdot \omega_{X/Y}^a)_a \geq \frac{(g-1)^2}{3g(2g+1)} \delta,
\]
where $\delta$ is the number of singularities in singular fibers of $f$.
\end{Theorem}
{\sl Proof.}\quad
Let $\operatorname{CV}(f)$ be the set of all critical values of $f$.
For $y \in \operatorname{CV}(f)$, the number of singularities of $f^{-1}(y)$
is denoted by $\delta_y$.
Let $G_y$ be the metrized graph of $f^{-1}(y)$ as in
\S\ref{sec:metrized:graph:green:function:admissible:pairing}.
Then, the total arc-length of $G_y$ is $\delta_y$.
Let $K_y$ be the divisor on $G_y$ coming from $\omega_{X/Y}$
as in \S\ref{sec:metrized:graph:green:function:admissible:pairing},
$\mu_y$ the admissible metric of $K_y$, and $g_{\mu_y}$ the Green
function of $\mu_y$. By the definition of $\omega_{X/Y}^a$
(see \S\ref{sec:metrized:graph:green:function:admissible:pairing}), we have
\[
(\omega_{X/Y}^a \cdot \omega_{X/Y}^a)_a =
(\omega_{X/Y} \cdot \omega_{X/Y}) +
\sum_{ y \in \operatorname{CV}(f)} \left\{ g_{\mu_y}(K_y, K_y) - 2(2g-2) c(G_y, K_y)
\right\}.
\]
On the other hand, $G_y$ is isometric to the graph treated in
\S\ref{sec:certain:graph}. Thus, by Lemma~\ref{lem:green:union:circle},
\begin{align*}
g_{\mu_y}(K_y, K_y) - 2(2g-2) c(G_y, K_y) & =
(2g-2)^2 \frac{\delta_y}{12g^2} - 2(2g-2) \frac{(2g-1) \delta_y}{12g^2} \\
& = -\frac{g-1}{3g} \delta_y.
\end{align*}
Thus
\[
(\omega_{X/Y}^a \cdot \omega_{X/Y}^a)_a =
(\omega_{X/Y} \cdot \omega_{X/Y}) - \frac{g-1}{3g} \delta.
\]
By virtue of Theorem~\ref{thm:Cornalba-Harris-Xiao:inequality} and
Noether formula
\[
\deg(f_*(\omega_{X/Y})) =
\frac{(\omega_{X/Y} \cdot \omega_{X/Y}) + \delta}{12},
\]
we have
\[
( \omega_{X/Y} \cdot \omega_{X/Y} ) \geq \frac{g-1}{2g+1} \delta.
\]
Therefore, we get
\[
(\omega_{X/Y}^a \cdot \omega_{X/Y}^a)_a \geq \frac{(g-1)^2}{3g(2g+1)} \delta.
\]
In particular, $(\omega_{X/Y}^a \cdot \omega_{X/Y}^a)_a > 0$
if $f$ is not smooth. Further, if $f$ is smooth, then
\[
(\omega_{X/Y}^a \cdot \omega_{X/Y}^a)_a = (\omega_{X/Y} \cdot \omega_{X/Y}) > 0
\]
because $f$ is non-isotrivial.
\QED
\bigskip
Let us start the proof of Theorem~\ref{thm:conj:bogomolov}.
We set
\[
A = \sqrt{\frac{(\omega_{X/Y}^a \cdot \omega_{X/Y}^a)_a}{4(g-1)}}.
\]
Then, by Theorem~\ref{thm:lower:bound:admissible:intersection},
$A$ is positive and
\[
A \geq \sqrt{\frac{g-1}{12g(2g+1)} \delta}.
\]
By virtue of Corollary~\ref{cor:lower:estimate:NT:metric},
for any $D \in \operatorname{Pic}^1(C)(\overline{K})$ and
any $P \in \operatorname{Pic}^0(C)(\overline{K})$, there is a finite subset $S$ of
$C(\overline{K})$ such that
\[
\Vert x - D - P \Vert_{NT} > (1-\epsilon)A
\]
for all $x \in C(\overline{K}) \setminus S$.
Therefore, we have
\[
\left\{ x \in C(\overline{K}) \mid \Vert j_D(x) - P \Vert_{NT} \leq
(1 - \epsilon)A \right\} \subset S.
\]
Thus, we get the second property of $A$.
\QED
\section{Effective lower bound of $(\omega \cdot \omega)$
for arithmetic surfaces}
Let $K$ be a number field, $O_K$ the ring of integers,
$f : X \to \operatorname{Spec}(O_K)$ a regular semistable arithmetic surface of genus $g \geq
2$
over $O_K$. In \cite{Mo3}, we proved the following.
\begin{Theorem}
If geometric fibers $X_{\overline{P}_1}, \ldots, X_{\overline{P}_n}$ of $X$
at $P_1, \ldots, P_n \in \operatorname{Spec}(O_K)$ are reducible, then
\[
\left(\omega_{X/O_K}^{Ar} \cdot \omega_{X/O_K}^{Ar} \right) \geq
\sum_{i=1}^n \frac{\log \#(O_K/P_i) }{6(g-1)}.
\]
\end{Theorem}
Using Lemma~\ref{lem:green:union:circle},
we have the following exact lower estimate for
stable curves with only irreducible fibers.
\begin{Theorem}
Assume that the stable model of $f : X \to \operatorname{Spec}(O_K)$ has only
geometric irreducible fibers.
If $\{ P_1, \ldots, P_n \}$ is the set of critical values of $f$, then
\[
(\omega_{X/O_K}^{Ar} \cdot \omega_{X/O_K}^{Ar})
\geq \sum_{i=1}^n \frac{g-1}{3g} \delta_i \log\#(O_K/P_i),
\]
where $\delta_i$ is the number of singularities of the geometric fiber
at $P_i$. Moreover, equality holds if and only if
there is a sequence of distinct points $x_1, x_2, \ldots$ of
$X(\overline{{\Bbb{Q}}})$
such that
\[
\lim_{i \to \infty} \Vert (2g-2) x_i - \omega \Vert_{NT} = 0.
\]
\end{Theorem}
{\sl Proof.}\quad
By virtue of Lemma~\ref{lem:green:union:circle},
\[
(\omega_{X/O_K}^a \cdot \omega_{X/O_K}^a)_a =
(\omega_{X/O_K}^{Ar} \cdot \omega_{X/O_K}^{Ar}) -
\sum_{i=1}^n \frac{g-1}{3g} \delta_i \log \#(O_K/P_i).
\]
Therefore, our theorem follows from \cite[Corollary 5.7]{Zh}.
\QED
Combining the above two theorems, we have the following corollary,
which is a stronger version of S. Zhang's result \cite{Zh}.
\begin{Corollary}
\label{cor:lower:bound:w:w:not:smooth}
If $f : X \to \operatorname{Spec}(O_K)$ is not smooth, then
\[
(\omega_{X/O_K}^{Ar} \cdot \omega_{X/O_K}^{Ar}) \geq \frac{\log 2}{6(g-1)}.
\]
\end{Corollary}
\renewcommand{\thesection}{Appendix \Alph{section}}
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\renewcommand{\theClaim}{\Alph{section}.\arabic{Theorem}.\arabic{Claim}}
\renewcommand{\theequation}{\Alph{section}.\arabic{Theorem}.\arabic{Claim}}
\setcounter{section}{0}
\section{Matrix representation of Laplacian}
In this appendix, we will consider a matrix representation of the Laplacian and
its easy application.
Let $G$ be a metrized graph and $V$ a set of vertices of $G$ such that
$G \setminus V$ is a disjoint union of open segments. Let $E$ be a set of edges
of $G$ by $V$. The length of $e$ in $E$ is denoted by $l(e)$.
Recall that $Q(G, V)$ is a set of continuous functions on $G$ whose
restriction to each edge in $E$ are
quadric polynomial functions, and $M(G, V)$ is
a vector space of measures on $G$ generated by Dirac functions $\delta_v$ at
$v \in V$ and by Lebesgue measures on edges $e \in E$ arising from
the arc-length parameter.
First, we define linear maps $p : Q(G, V) \to {\Bbb{R}}^{V}$
and $q : M(G, V) \to {\Bbb{R}}^{V}$ in the following ways.
If $f \in Q(G, V)$, then $p(f)$ is the restriction to $V$.
If $\delta_v$ is a Dirac function at $v \in V$, then
\[
q(\delta_v)(v') = \begin{cases}
1 & \text{if $v' = v$} \\
0 & \text{if $v' \not= v$}
\end{cases}
\]
If $dt$ is a Lebesgue measure on a edge $e$ in $E$,
then
\[
q(dt)(v) = \begin{cases}
l(e)/2 & \text{if $v$ is a vertex of $e$} \\
0 & \text{otherwise}
\end{cases}
\]
Next let us define a linear map $L : {\Bbb{R}}^{V} \to {\Bbb{R}}^{V}$.
For distinct vertices $v, v'$ in $V$,
let $E(v, v')$ be the set of edges in $E$
whose vertices are $v$ and $v'$. Here we set
\[
a(v, v') = \begin{cases}
0 & \text{if $E(v, v') = \emptyset $} \\
{\displaystyle \sum_{e \in E(v, v')} \frac{1}{l(e)}} & \text{otherwise}
\end{cases}
\]
for $v \not= v'$. Moreover, we set
\[
a(v, v) = -\sum\begin{Sb} v' \in V \\ v' \not= v \end{Sb} a(v, v').
\]
Let $L : {\Bbb{R}}^{V} \to {\Bbb{R}}^{V}$ be a linear map
defined by a matrix $(-a(v, v'))_{v, v' \in V}$, i.e.,
if we denote $q(\delta_v)$ by $e_v$, then
$L(e_v) = -\sum_{v' \in V} a(v, v') e_{v'}$.
Thus, we have the following diagram:
\[
\begin{CD}
Q(G, V) @>{\Delta}>> M(G, V) \\
@V{p}VV @VV{q}V \\
{\Bbb{R}}^{V} @>>{L}> {\Bbb{R}}^{V}
\end{CD}
\]
Then, we can see the following proposition as remarked in \cite[(5.3)]{BGS}.
\begin{Proposition}
\label{prop:commutativity:L:p:q:Delta}
The above diagram is commutative, i.e., $L \circ p = q \circ \Delta$.
\end{Proposition}
{\sl Proof.}\quad
Let $f \in Q(G, V)$.
First, let us consider two special cases of $f$.
Case 1 : A case where $f$ is a linear function on each edge in $E$.
By the definition of $\Delta$, we can see that
\[
\Delta(f) = - \sum_{v \in V}\left(
\sum\begin{Sb} v' \in V \setminus \{ v \} \\
E(v, v') \not= \emptyset \end{Sb}
\left( \sum_{e \in E(v, v')} \frac{f(v') - f(v)}{l(e)} \right)
\right) \delta_v.
\]
On the other hand, by the definition of $a(v, v')$,
\[
\sum\begin{Sb} v' \in V \setminus \{ v \} \\
E(v, v') \not= \emptyset \end{Sb}
\left( \sum_{e \in E(v, v')} \frac{f(v') - f(v)}{l(e)} \right)
= \sum_{v' \in V} a(v, v') f(v').
\]
Therefore, we have
\[
\Delta(f) = \sum_{v \in V} \left(
\sum_{v' \in V} - a(v, v') f(v') \right) \delta_v,
\]
which shows us $q(L(f)) = L(p(f))$.
\medskip
Case 2 : A case where there is $e \in E$ such that $f \equiv 0$ on
$G \setminus e$. Let $v, v'$ be vertices of $e$ and
$\phi : [0, l(e)] \to e$ be the arc-length parameterization of $e$
with $\phi(0) = v$ and $\phi(l(e)) = v'$.
Since $f(v) = f(v') = 0$, $f$ can be written in the form
$f(t) = at(t-l(e))$, where $t$ is the arc-length parameter and
$a$ is a constant. Thus,
\[
\Delta(f) = al(e) \delta_v + al(e) \delta_{v'} - 2a dt.
\]
Therefore, $q(\Delta(f)) = 0$, which means that
$q(\Delta(f)) = L(p(f))$.
\medskip
Let us consider a general case. Let $f_0$
be a continuous function on $G$ such that
$f_0$ is a linear function on each $e \in E$ and
$f_0(v) = f(v)$ for all $v \in V$. Then,
$f - f_0$ can be written by a sum
of functions $f_1, \ldots, f_k$ as in the case 2, i.e.,
\[
f = f_0 + f_1 + \cdots + f_k
\]
and $f_i$ ($1 \leq i \leq k$) is zero on the outside of some edge.
By the previous observation, we know
$q(\Delta(f_i)) = L(p(f_i))$ for all $i = 0, 1, \ldots, k$.
Thus, using linearity of each map, we get our lemma.
\QED
As a corollary, we have the following.
\begin{Corollary}
Let $D = \sum_{v \in V} d_v v$ be a divisor on $G$,
$\mu \in M(G, V)$, and $g \in Q(G, V)$ such that
\[
\int_G \mu = 1 \quad\text{and}\quad
\Delta(g) = \delta_D - (\deg D)\mu.
\]
Then, we have
\[
d_v + \sum_{v' \in V} a(v, v')g(v') = (\deg D)q(\mu)(v)
\]
for all $v \in V$.
\end{Corollary}
{\sl Proof.}\quad
Applying $q$ for $\Delta(g) = \delta_D - (\deg D)\mu$ and
using Proposition~\ref{prop:commutativity:L:p:q:Delta},
we have
\[
q(\delta_D) - L(p(g)) = (\deg D)q(\mu).
\]
Thus, by the definition of $L$, we get our corollary.
\QED
\bigskip
Let $k$ be an algebraically closed field,
$X$ a smooth projective surface over $k$, $Y$ a smooth projective curve over
$k$,
and $f : X \to Y$ a generically smooth semi-stable curve of genus $g \geq 1$
over $Y$.
Let $\operatorname{CV}(f)$ be the set of all critical values of $f$ and $y \in \operatorname{CV}(f)$.
Let $G_y$ be the metrized graph of $f^{-1}(y)$ as in
\S\ref{sec:metrized:graph:green:function:admissible:pairing}.
Let $V_y$ be a set of vertices coming from irreducible curves in $f^{-1}(y)$.
For $v \in V_y$, the corresponding irreducible curve is denoted by $C_v$.
Let $K_y$ be the divisor on $G_y$ defined by
$K_y = \sum_{v \in V_y} (\omega_{X/Y} \cdot C_v) v$,
$\mu_y$ the admissible metric of $K_y$, and $g_{\mu_y}$ the Green
function of $\mu_y$.
In this case, the map $L_y : {\Bbb{R}}^{V_y} \to {\Bbb{R}}^{V_y}$ defined in
the above is given by a matrix $\left(-(C_v \cdot C_{v'})\right)_{v, v' \in
V_y}$.
Thus, the above corollary implies the following proposition.
\begin{Proposition}
\label{prop:admissible:with:fiber}
Let $D$ be an ${\Bbb{R}}$-divisor on $X$ and $C_v$ the irreducible curve
in $f^{-1}(y)$ corresponding to $v \in V_y$. Then,
\[
(D \cdot C_v)_a = (D \cdot F) q(\mu_y)(v),
\]
where $F$ is a general fiber of $f$.
In particular, $(D \cdot C_v)_a$ does not depend on the choice
of compactification of $D$.
\end{Proposition}
\bigskip
|
1996-03-08T06:51:44 | 9410 | alg-geom/9410008 | en | https://arxiv.org/abs/alg-geom/9410008 | [
"alg-geom",
"math.AG"
] | alg-geom/9410008 | null | David B. Jaffe | Applications of iterated curve blowup to set-theoretic complete
intersections in P3 | 57 pages, AMS-LaTeX | null | null | null | null | Let S, T be surfaces in P3. Suppose that S intersect T is set-theoretically a
smooth curve C of degree d and genus g. Suppose that S and T have no common
singular points. Then if C is not a complete intersection, then deg(S), deg(T)
< 2d^4. Fixing (d,g), one can form a finite (shorter) list of all possible
pairs (deg(S),deg(T)). For instance, when (d,g) = (4,0), and assuming for
simplicity that deg(S) <= deg(T):
(deg(S), deg(T)) \in {(3,4), (3,8), (4,4), (4,7), (6,26), (9,48), (10,28)
(12,18), (13,16), (17,220), (18,118), (19,84), (20,67), (22,50), (28,33)}.
Assume characteristic 0. [1] Suppose that S and T have non-overlapping rational
singularities. Then d <= g+3. [2] Suppose that S is normal, and that d>deg(S).
Then C is linearly normal (and so d <= g+3). [3] Suppose that S is a quartic
surface having only rational singularities. Then C is linearly normal. Hard
copy is available from the author. E-mail to [email protected].
| [
{
"version": "v1",
"created": "Wed, 12 Oct 1994 19:33:46 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Jaffe",
"David B.",
""
]
] | alg-geom | \section{#1}}
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\par\noindent {\footnotesize Department of Mathematics and Statistics,
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\par\noindent {\footnotesize Lincoln, NE 68588-0323, USA\ \ (jaffe{\kern0.5pt}@{\kern0.5pt}cpthree.unl.edu)}}
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\par\noindent David B. Jaffe\protect\footnote{Partially supported by
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\begin{document}
{\par\noindent\Large\bf Applications of iterated curve blowup to}
\vskip 0.05in
{\par\noindent\Large\bf set theoretic complete intersections in
$\hbox{{\bbtwo P}}^3$}
\vskip 0.15in
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\newpage
\section*{Introduction}
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\indent
We describe some new results on the set-theoretic complete intersection
problem for projective space curves. Fix an algebraically closed ground
field $k$. Let $S, T \subset \P3$ be surfaces.
Suppose that $S \cap T$ is set-theoretically a smooth curve $C$ of degree
$d$ and genus $g$. For purposes of the introduction, we label the main
results as A, B, Q, X, I, II, and III.\footnote{The actual numbering in the
text is A = \ref{thmA}, B = \ref{thmB}, Q = \ref{thmQ}, X = \ref{thmX},
I = \ref{thmI}, II = \ref{thmII}, III = \ref{thmIII}.}
The results I, II, and III are more technical than A, B, Q, and X.
Suppose that $S$ and $T$ have no common singular points. We discover that
this requirement imposes severe limitations. Indeed, theorem (X) asserts that
if $C$ is not a complete intersection, then
$\deg(S), \deg(T) < 2d^4$. Fixing $(d,g)$, one can in fact form a finite
list of all possible pairs $(\deg(S),\deg(T))$, which is much shorter than
the list implied by theorem (X). For instance, when $(d,g) = (4,0)$, and
assuming for simplicity that $\deg(S) \leq \deg(T)$, we find that
$$(\deg(S), \deg(T)) \in \{ (3,4), (3,8), (4,4), (4,7), (6,26), (9,48),
(10,28),$$
$$(12,18), (13,16), (17,220), (18,118), (19,84), (20,67), (22,50), (28,33)
\}.$$
Very little is known about which of these degree pairs actually correspond
to surface pairs $(S,T)$.
Suppose that $S$ and $T$ have only rational singularities, and that the
ground field $k$ has characteristic zero. We continue to assume that
$S$ and $T$ have no common singular points. Under these conditions, we prove
(A) that $d \leq g + 3$. (The actual statement is somewhat stronger.)
Suppose that $S$ is normal, and that $d > \deg(S)$. Make no assumptions about
how the singularities of $S$ and $T$ meet. Assume that $\mathop{\operatoratfont char \kern1pt}\nolimits(k) = 0$.
We show (Q) that $C$ is linearly normal. In particular, it follows by
Riemann-Roch that $d \leq g+3$.
Suppose that $S$ is a quartic surface having only rational singularities.
Allow $T$ to be an arbitrary surface, and make no assumptions about how the
singularities of $S$ and $T$ meet. Assume that $\mathop{\operatoratfont char \kern1pt}\nolimits(k) = 0$. Under these
conditions, we prove (B) that $C$ is linearly normal.
In other papers\Lspace \Lcitemark 17\Rcitemark \Rspace{},\Lspace \Lcitemark
18\Rcitemark \Rspace{}, we
have proved the following complementary results (in characteristic zero):
if $S$ is has only ordinary nodes as singularities, or is a cone, or
has degree $\leq 3$, then $d \leq g + 3$. It is conceivable (in characteristic
zero) that this
inequality is valid without any restrictions whatsoever on $S$ and $T$, or
even that $C$ is always linearly normal.
Examples of smooth set-theoretic complete intersection\ curves in ${\Bbb C}\kern1pt\P3$ have been constructed by
Gallarati\Lspace \Lcitemark 8\Rcitemark \Rspace{}, Catanese\Lspace \Lcitemark
3\Rcitemark \Rspace{},
Rao (\Lcitemark 27\Rcitemark \ prop.\ 14), and the author\Lspace \Lcitemark
19\Rcitemark \Rspace{}.
To explain the results (I), (II), and (III), and to describe the methods by
which we prove (A), (B), and (X), there are two key ideas which must be
discussed%
.\footnote{We also give an alternate proof of the key ingredient of (X),
which is independent of the main machine of this paper.}
Both of these ideas have to do with the iterated blowing up of curves.
The first idea has to do with certain invariants $p_i = p_i(S,C)$
$(i \in \xmode{\Bbb N})$ which we associate to a pair $(S,C)$ consisting of an abstract
surface $S$ and a smooth curve $C$ on $S$ such that $C \not\IN \mathop{\operatoratfont Sing}\nolimits(S)$. Let
\mp[[ \pi || {\tilde{S}} || S ]] be the blowup along $C$. Then $p_1(S,C)$ is the
sum of the multiplicities of the exceptional curves. (See \S\ref{measure} for
details.) Moreover, $\pi$ admits a unique section ${\tilde{C}}$ over $C$, so we
can define $p_2(S,C) = p_1({\tilde{S}},{\tilde{C}})$, $p_3(S,C) = p_2({\tilde{S}},{\tilde{C}})$, and so forth.
We refer to the sequence $(p_1, p_2, \ldots)$ as the {\it type\/} of $(S,C)$.
It is a sum of local contributions, one for each singular point of $S$ along
$C$, and it is a rather mysterious measure of how singular $S$ is along $C$.
The type depends not only on the particular species of singular points of
$S$ which lie on $C$, but also on the way in which $C$ passes through those
points. For example, the local contribution to the type coming from an
$A_3$ singularity is either $(1,1,1,0,\ldots)$ or $(2,0,\ldots)$, depending
on how $C$ passes through the singular point.
The second idea is the following construction. For this we assume
(as in the first paragraph) that $C = S \cap T$ (in $\P3$) and that
$C \not\IN \mathop{\operatoratfont Sing}\nolimits(S)$, $C \not\IN \mathop{\operatoratfont Sing}\nolimits(T)$. Other than this, no restrictions
are necessary on the singularities of $S$ and $T$.
Let $Y_1$ denote the blowup of $\P3$ along $C$. Let $S_1, T_1 \subset Y_1$
denote the strict transforms of $S$ and $T$ respectively. Let $E_1 \subset Y_1$
be the exceptional divisor, which is a ruled surface over ${\Bbb C}\kern1pt$. Then
$S_1 \cap E_1$ is a curve $C_1$ (mapping isomorphically onto $C$), together
with some rulings. The total number of rulings, counted with multiplicities,
is $p_1(S,C)$. Now let $Y_2$ be the blowup of $Y_1$ along $C_1$. Let
$S_2, T_2, E_2 \subset Y_2$ be as above. Then $S_2 \cap E_2$ is a curve $C_2$
plus $p_2(S,C)$ rulings. Iterate this construction $n$ times, where $n$
is the multiplicity of intersection of $S$ and $T$ along $C$. Then
$S_n \cap T_n$ is a union of strict transforms of rulings. This fact leads
us to theorems (I) and (II), which are statements about the numbers $p_i$.
Theorem (III) is also such a statement, but it does not depend on the
construction we have just described.
We describe theorems (I), (II), and (III). These depend on the data
$(s,t,d,g)$, where $s = \deg(S)$, $t = \deg(T)$. To make this description
as simple as possible, we restrict our attention here to the special case
where $(s,t,d,g) = (4,4,4,0)$.
Theorem (I) has the hypothesis that $\mathop{\operatoratfont Sing}\nolimits(S) \cap \mathop{\operatoratfont Sing}\nolimits(T) = \varnothing$.
Its conclusion (applied to our special case) is that:
$$p_1 = p_2 = p_3 = 8.$%
$Theorem (I) is used in the proofs of (A) and (X).
Theorem (II) has no additional hypotheses.
Its conclusion (applied to our special case) is that:
\begin{eqnarray*}
p_1 & \geq & 8; \\
2p_1 + p_2 & \geq & 24; \\
8p_1 + 3p_2 + p_3 & \geq & 96.
\end{eqnarray*}
Theorem (II) is not used in the proofs of (A) or (B).
Theorem (III) has the hypotheses that $S$ has only rational singularities,
and that $\mathop{\operatoratfont char \kern1pt}\nolimits(k) = 0$. Its conclusion (applied to our special case) is that:
$${1\over2} p_1 + {1\over6} p_2 + {1\over12}p_3 \many+
{1 \over k(k+1)}p_k + \cdots \geq 6.$%
$Theorem (III), or actually a minor variant of it, is used in the proof of (B).
We now mention some open problems and possible ways to improve upon the
results in this paper.
\par\noindent{\bf 1.} Let $(S,C)$ be the local scheme $S$ of a normal surface
singularity, together with a smooth curve $C$ on $S$. There are three
fundamental invariants of $(S,C)$ which are utilized in this paper. Firstly,
there is the type of $(S,C)$. Secondly,
there is the order of $(S,C)$, i.e.\ the smallest positive integer $n$ such
that ${\cal O}_S(nC)$ is Cartier. Thirdly, there is $\Delta(S,C)$, which we
describe in \S\ref{def-section}. What relationships exist between these three
invariants? What is their relationship to the Milnor fiber?
\par\noindent{\bf 2.}
We suspect that (A), (B), and (III) are valid over an arbitrary
algebraically closed field. There are significant difficulties in proving
this which we have not explored fully. The proofs of (I) and (II) do not
depend on the characteristic.
\par\noindent{\bf 3.} The proofs of (A) and (B) use a bound
\pref{bound-formula} on the number of exceptional curves in a minimal
resolution
for a surface $S \subset {\Bbb C}\kern1pt\P3$ having only rational singularities. Formulate
and prove a suitable generalization for arbitrary normal surfaces.
\par\noindent{\bf 4.} Construct examples of surfaces $S, T \subset {\Bbb C}\kern1pt\P3$,
having only rational singularities, meeting set-theoretically along a smooth
curve $C$, such that $\mathop{\operatoratfont Sing}\nolimits(S) \cap \mathop{\operatoratfont Sing}\nolimits(T) \not= \varnothing$. The
only example we know of is where $\deg(S) = \deg(T) = 2$, and $C$ is a line.
\par\noindent{\bf 5.} The generic hypothesis that $C \not\IN \mathop{\operatoratfont Sing}\nolimits(T)$ can
probably be eliminated.
\vspace*{0.1in}
\par\noindent{\footnotesize{\it Acknowledgements.} I thank Dave Morrison
for helpful comments, and Juan Migliore for raising the issue of linear
normality of set-theoretic complete intersections.}
\section*{Conventions}
\addcontentsline{toc}{special}{Conventions}
\begin{arabiclist}
\item We fix an algebraically closed field $k$.
\item A {\it curve\/} [resp.\ {\it surface}] [resp.\ {\it three-fold}] is an
excellent $k$-scheme
such that every maximal chain of irreducible proper closed subsets has length
one [resp.\ two] [resp.\ three]. We make the following additional assumptions:
\begin{itemize}
\item all curves are reduced and irreducible;
\item in part III and the introduction, all surfaces are reduced and
irreducible.
\end{itemize}
\item A surface {\it embeds in codimension one\/} if it can be exhibited
as an effective Cartier divisor on a regular three-fold.
\item A {\it variety} is an integral separated scheme of finite-type over $k$.
\item If $X$ and $Y$ are schemes, then the notation $X \subset Y$ carries the
implicit assumption that $X$ is a {\it closed subscheme\/} of $Y$.
\item In several situations, we use {\it bracketed exponents\/} to denote
{\it repetition\/} in sequences, and we drop trailing zeros, where appropriate.
For example,
$$(2,1^\br{4}) = (2,1,1,1,1,0, \ldots)$%
$and
$$(3^\br{\infty}) = (3,3,\ldots).$$
\item We use the Grothendieck convention regarding projective space
bundles.
\item For any variety $V$, we let $A^k(V)$ denote the group of
codimension $k$ cycles on $V$, modulo {\it algebraic\/} equivalence. When
$d = \dim(V)$ and $V$ is complete, we identify $A^d(V)$ with $\xmode{\Bbb Z}$.
\end{arabiclist}
\vspace{0.25in}
\part{Local geometry of smooth curves on singular surfaces}
\block{Definitions}\label{def-section}
\par\indent\indent We define the category of {\it surface-curve pairs}.
(Sometimes, we use the shorthand term {\it pair\/} for a surface-curve pair.)
An {\it object\/} $(S,C)$
in this category consists of a surface $S$, together with a curve $C \subset S$,
such that $C$ is a regular scheme and $C \not\IN \mathop{\operatoratfont Sing}\nolimits(S)$. A {\it morphism\/}
\mp[[ f || (S',C') || (S,C) ]] is a pair $(S' \mapE{} S, C' \mapE{} C)$ of
morphisms of $k$-schemes, such that the diagram:
\squareSE{C'}{C}{S'}{S%
}commutes, and such that if \mp[[ \phi || C' || C \times_S S' ]] is the
induced map, then $\phi \times_S \mathop{\operatoratfont Spec}\nolimits {\cal O}_{S,C}$ is an isomorphism. Such
a morphism $f$ is {\it cartesian\/} if $\phi$ is an isomorphism. Most
properties
of morphisms of schemes also make sense as properties of morphisms in this
category: the properties are to be interpreted as properties of the morphism
\mapx[[ S' || S ]].
Let $(S,C)$ be a surface-curve pair. We say that:
\begin{itemize}
\item $(S,C)$ is {\it geometric\/} if $S$ is a variety;
\item $(S,C)$ is {\it local\/} if $S$ is a local scheme;
\item $(S,C)$ is {\it local-geometric\/} if $S$ is a local scheme,
essentially of finite type over $k$.
\end{itemize}
To give a local surface-curve pair $(S,C)$ is equivalent to giving the
data $(A,{\xmode{{\fraktur{\lowercase{P}}}}})$, consisting of an excellent local $k$-algebra $A$, of
pure dimension two,
together with a height one prime ${\xmode{{\fraktur{\lowercase{P}}}}} \subset A$ such that $A_{\xmode{{\fraktur{\lowercase{P}}}}}$ and
$A/{\xmode{{\fraktur{\lowercase{P}}}}}$ are regular. We write $(S,C) = \mathop{\operatoratfont Spec}\nolimits(A,{\xmode{{\fraktur{\lowercase{P}}}}})$ to denote this
correspondence.
There are two operations on surface-curve pairs which we will be using.
Firstly, if $(S,C)$ is a local surface-curve pair, then the {\it completion\/}
$({\hat{S}},{\hat{C}})$ makes sense and is also a local surface-curve pair. Indeed,
if $(S,C) = \mathop{\operatoratfont Spec}\nolimits(A,{\xmode{{\fraktur{\lowercase{P}}}}})$, then $A/{\xmode{{\fraktur{\lowercase{P}}}}}$ is regular, and so ${\hat{A}}/{\xmode{{\hat{\fraktur{\lowercase{P}}}}}}$
is regular, since it equals $\widehat{A/{\xmode{{\fraktur{\lowercase{P}}}}}}$, and the completion of a
regular local ring is regular. The reader may also check easily that
${\hat{A}}_{\xmode{{\hat{\fraktur{\lowercase{P}}}}}}$ is regular. Moreover, ${\hat{A}}$ is excellent, since any noetherian
complete local ring is excellent. Note also: there is a canonical morphism
\mapx[[ (S,C) || ({\hat{S}},{\hat{C}}) ]].
Secondly, for any surface-curve pair $(S,C)$, one can define the {\it blowup\/}
$({\tilde{S}},{\tilde{C}})$ of $(S,C)$. This is done by letting \mp[[ \pi || {\tilde{S}} || S ]] be
the blowup of $S$ along $C$, and by letting ${\tilde{C}}$ be the unique section
of $\pi$ over $C$, which exists e.g.\ by (\Lcitemark 11\Rcitemark \ 7.3.5).
There is a
canonical morphism \mapx[[ ({\tilde{S}}, {\tilde{C}}) || (S,C) ]].
Two local surface-curve pairs are {\it analytically isomorphic\/} if their
completions are isomorphic.
If $(S,C)$ is a surface-curve pair, and $p \in C$, we let $(S,C)_p$ denote
the corresponding local surface-curve pair. A {\it configuration\/} is
an element of the free abelian monoid on the set of analytic isomorphism
classes of local-geometric pairs.
Let $(S,C)$ be a geometric surface-curve pair. We may associate the
configuration:
$$\sum_{p \in \mathop{\operatoratfont Sing}\nolimits(S) \cap C} [(S,C)_p]$%
$to $(S,C)$. On occasion, we shall identify $(S,C)$ with the associated
configuration.
We are interested in invariants of a geometric surface-curve pair $(S,C)$
which depend only on the associated configuration. There are four
such invariants which we shall consider:
\begin{arabiclist}
\item The {\it order\/} of $(S,C)$ is the smallest $n \in \xmode{\Bbb N}$ such that
${\cal O}_S(nC)$ is Cartier, or else $\infty$ if ${\cal O}_S(nC)$ is not Cartier for
all $n \in \xmode{\Bbb N}$. If $\mathop{\operatoratfont Sing}\nolimits(S) \cap C = \setof{\vec p1k}$, then
$$\mathop{\operatoratfont order}\nolimits(S,C) = \mathop{\operatoratfont lcm}\nolimits\setof{\mathop{\operatoratfont order}\nolimits(S,C)_{p_1}, \ldots, \mathop{\operatoratfont order}\nolimits(S,C)_{p_k}},$%
$so the computation of the order is a purely local problem. Moreover, at least
if $S$ is normal, the order depends only on the associated configuration.
Indeed, in that case, if $(S,C)$ is a local-geometric pair, then $S$ is
excellent, so ${\hat{S}}$ is normal, and so by (\Lcitemark 6\Rcitemark \ 6.12) one
knows that the
canonical map \mapx[[ \mathop{\operatoratfont Cl}\nolimits(S) || \mathop{\operatoratfont Cl}\nolimits({\hat{S}}) ]] is injective. Hence
$\mathop{\operatoratfont order}\nolimits(S,C) = \mathop{\operatoratfont order}\nolimits({\hat{S}},{\hat{C}})$.
\item The {\it type\/} of $(S,C)$, which is the sequence $(p_i)_{i \in \xmode{\Bbb N}}$
discussed in the introduction, and studied in \S\ref{measure}.
\item Assume that $S$ is normal. We define an invariant $\Delta(S,C) \in {\Bbb Q}\kern1pt$.
Let \mp[[ \pi || {\tilde{S}} || S ]] be a minimal resolution, and let
$\vec E1n \subset {\tilde{S}}$ be the exceptional curves. Let ${\tilde{C}} \subset {\tilde{S}}$ be the strict
transform of $C$. According to (\Lcitemark 26\Rcitemark \ p.\ 241),
there is a unique ${\Bbb Q}\kern1pt$-divisor $E = \sum a_i E_i$ such that
$({\tilde{C}} + E) \cdot E_i = 0$ for all $i$. We define
$\Delta(S,C) = -E^2$. Then $\Delta(S,C)$ is independent of $\pi$. If $S$
is projective, then $\Delta(S,C) = C^2 - {\tilde{C}}^2$, where $C^2$ is
defined in (\Lcitemark 26\Rcitemark \ p.\ 241).
\item Assume that $S$ has only rational double points along $C$. Let
$\Sigma(S,C)$ equal the number of exceptional curves in the minimal
resolution of those singularities of $S$ which lie on $C$.
\end{arabiclist}
\block{The type of a surface-curve pair}\label{measure}
\par\indent\indent We define the {\it type\/} of a surface-curve pair, and
show that it is an analytic invariant, at least when the surface embeds in
codimension one.
\begin{definition}
Let $(S,C)$ be a surface-curve pair. Let $({\tilde{S}},{\tilde{C}})$ be the blowup of
$(S,C)$. Let $\vec E1n \subset {\tilde{S}}$ be the (reduced)
exceptional curves. We define numbers $p_i(S,C)$, for each $i \in \xmode{\Bbb N}$.
Define:
$$p_1(S,C) = \sum_{i=1}^n \mathop{\operatoratfont length}\nolimits {\cal O}_{\pi^{-1}(C), E_i},$%
$where \mp[[ \pi || {\tilde{S}} || S ]] is the blowup map.
For $i \geq 2$, recursively define $p_i(S,C)$ by:
$$p_{i+1}(S,C) = p_i({\tilde{S}}, {\tilde{C}}).$%
$The {\it type\/} of $(S,C)$ is the sequence $(p_1, p_2, \ldots)$.
\end{definition}
It is clear that the computation of the $p_i$ may be reduced to the computation
of the $p_i$ when $(S,C)$ is a local pair.
\begin{remark}\label{goober-peas}
Let $(S,C)$ be a surface-curve pair, and assume that $S$ embeds in codimension
one. Then we have $S \subset T$ for some smooth three-fold $T$. Let
\mp[[ \pi_S || {\tilde{S}} || S ]] and \mp[[ \pi_T || {\tilde{T}} || T ]] be the blowups
of $S$ and $T$ along $C$. Then $\pi_S^{-1}(C) \cong {\tilde{S}} \cap E$, where
$E \subset {\tilde{T}}$ is the exceptional divisor. This fact plays an absolutely
central role in our type computations.
\end{remark}
\begin{remark}
We do not know for which $(S,C)$ we have $p_1(S,C) \geq p_2(S,C)$, and hence
that $p_k(S,C) \geq p_{k+1}(S,C)$ for all $k \geq 1$. Conceivably, these
inequalities may hold whenever $S$ embeds in codimension one, or even
whenever $S$ is Cohen-Macaulay. By explicit calculation,
we shall find in \pref{fantastico} that the inequalities hold if $S$ has only
rational double points along $C$. However, as \pref{type-ex-2} shows,
for some $(S,C)$ one has $p_1(S,C) < p_2(S,C)$.
\end{remark}
\begin{remark}
We consider the following general question. Let $(S,C)$ be a local-geometric
pair. Assume that $S$ is not smooth.
Let $p \in S$ be the closed point. Let $({\tilde{S}},{\tilde{C}})$ be the blowup of
$S$ along $C$. Let \mp[[ \pi || {\tilde{S}} || S ]] be the blowup map.
What is the structure of $X = \RED{\pi^{-1}(p)}$? If
$S$ embeds in codimension one, then $X$ will be a $\P1$. Weird things can
happen if $S$ is not Cohen-Macaulay. For example, in \pref{type-ex-2}, $X$ is isomorphic
to $\mathop{\operatoratfont Proj}\nolimits {\Bbb C}\kern1pt[s,t,u] / (s^3 - t^2u)$, which is a rational curve with a cusp.
In \pref{type-ex-3}, $X$ is the disjoint union of a point and several copies
of $\P1$, which do not meet ${\tilde{C}}$. The isolated point of $X$ is the unique
point of ${\tilde{C}}$ lying over $p$. Assuming only that $S$ is Cohen-Macaulay, we do not know
if $X$ is always isomorphic to $\P1$, or even if it is always connected.
However: if $S$ embeds in codimension two, then $X$ embeds in $\P2$.
\end{remark}
We will prove \pref{analytic-invariant} that the type of a local-geometric
pair $(S,C)$ is an analytic invariant, provided that $S$ embeds in
codimension one. There are some preliminaries.
\begin{lemma}\label{formal-woof-1}
Let \mp[[ f || A || B ]] be a flat, formally smooth homomorphism of Artin
local rings. Assume that $A$ contains a field. Then $\mathop{\operatoratfont length}\nolimits(A) =
\mathop{\operatoratfont length}\nolimits(B)$.
\end{lemma}
\begin{proof}
Let $K$ and $L$ be the residue fields of $A$ and $B$. Let \mp[[ i || K || L ]]
be the induced map. Let ${\xmode{{\fraktur{\lowercase{M}}}}}$ be the maximal ideal of $A$. Then the map
\mapx[[ K = A/{\xmode{{\fraktur{\lowercase{M}}}}} || B/{\xmode{{\fraktur{\lowercase{M}}}}} B ]] is formally smooth, so $B/{\xmode{{\fraktur{\lowercase{M}}}}} B = L$
and so $i$ is formally smooth. A theorem of Cohen
(\Lcitemark 23\Rcitemark \ 28.J) implies that $A$ contains a
coefficient field, which we also denote by $K$. Since $i$ is formally smooth,
$L/K$ is a separable field extension. It follows by the cited theorem that
we may find a coefficient field $L$ for $B$ which contains $f(K)$.
Let ${\overline{A}} = A \o*_K L$. Then $f$ factors as:
\diagramx{A&\mapE{h}&{\overline{A}}&\mapE{g}&B.%
}Since $i$ is formally smooth, so is $h$.
Since both $h$ and $g \circ h$ are formally smooth it follows by
(\Lcitemark 12\Rcitemark \ 17.1.4) that $g$ is formally smooth.
Clearly $B$ is a finite ${\overline{A}}$-module. In particular, $g$ is of finite-type,
so $g$ is smooth. Since $g$ is smooth of relative dimension zero, $g$ is
\'etale. By (\Lcitemark 12\Rcitemark \ 18.1.2), the obvious functor:
\dfunx[[ \'etale ${\overline{A}}$-schemes || \'etale $L$-schemes ]]%
is an equivalence of categories, so $g$ is an isomorphism. Hence
$B \cong A \o*_K L$. Hence $\mathop{\operatoratfont length}\nolimits(A) = \mathop{\operatoratfont length}\nolimits(B)$. {\hfill$\square$}
\end{proof}
\begin{corollary}\label{formal-woof-2}
Let \mp[[ f || X' || X ]] be a flat, formally smooth morphism of irreducible
noetherian schemes. Assume that $X$ is defined over a field. Let $\eta$
and $\eta'$ be the generic points of $X$ and $X'$. Then:
$$\mathop{\operatoratfont length}\nolimits {\cal O}_{X',\eta'} = \mathop{\operatoratfont length}\nolimits {\cal O}_{X,\eta}.$$
\end{corollary}
\begin{remark}
We do not know if \pref{formal-woof-1} and \pref{formal-woof-2} are true
without the hypothesis of being ``defined over a field''.
\end{remark}
\begin{lemma}\label{flat-is-cartesian}
Let \mp[[ f || (S_1,C_1) || (S_2,C_2) ]] be a flat morphism
of surface-curve pairs. Then $f$ is cartesian.
\end{lemma}
\begin{proof}
We must show that the induced map \mp[[ \phi || C_1 || C_2 \times_{S_1} S_2 ]]
is an isomorphism. It suffices to show that $C_1 = C_2 \times_{S_1} S_2$
as closed subschemes of $S_2$. Let \mp[[ \pi || C_2 \times_{S_1} S_2 || C_2 ]]
be the projection map. Because $\pi$ is flat, any irreducible component of
$C_2 \times_{S_1} S_2$ must dominate $C_2$.
(See e.g.{\ }\Lcitemark 14\Rcitemark \ III 9.7.) But
$\phi \times_{S_2} \mathop{\operatoratfont Spec}\nolimits {\cal O}_{S_2,C_2}$ is an isomorphism, so it follows that
$C_2 \times_{S_1} S_2$ is irreducible. Since $C_1 = C_2 \times_{S_1} S_2$
at their generic points, they are equal as closed subschemes of $S_2$. {\hfill$\square$}
\end{proof}
\begin{prop}\label{formal-woof-3}
Let \mp[[ f || (S_1,C_1) || (S_2,C_2) ]] be a formally smooth, flat morphism
of surface-curve pairs. Assume that the induced map
\mapx[[ C_1 || C_2 ]] is bijective. Assume that $S_1$ and $S_2$ embed in
codimension one. Then $(S_1,C_1)$ and $(S_2,C_2)$ have the same type.
\end{prop}
\begin{proof}
The subscript $i$ will always vary through the set $\setof{1,2}$. Because
of our hypothesis on the map \mapx[[ C_1 || C_2 ]],
we may assume that $S_1, S_2$ are local schemes and that
$f$ is a local morphism. Let \mp[[ \pi_i || ({\tilde{S}}_i, {\tilde{C}}_i) || (S_i, C_i) ]]
be the blowup maps. By the universal property of blowing up, and because
$f$ is cartesian by \pref{flat-is-cartesian}, we obtain
a map \mp[[ {\tilde{\lowercase{F}}} || {\tilde{S}}_1 || {\tilde{S}}_2 ]] which makes the diagram:
\diagramx{{\tilde{S}}_1&\mapE{{\tilde{\lowercase{F}}}}&{\tilde{S}}_2\cr
\mapS{\pi_1}&&\mapS{\pi_2}\cr
S_1&\mapE{f}&S_2\cr%
}commute. Furthermore, using the flatness of $f$, we see that this diagram
is cartesian and as a consequence that ${\tilde{\lowercase{F}}}$ is flat and formally smooth.
Since $S_1$ and $S_2$ embed in codimension one, so do ${\tilde{S}}_1$ and ${\tilde{S}}_2$.
Since $p_{k+1}(S_i,C_i) = p_k({\tilde{S}}_i,{\tilde{C}}_i)$ for all $k \geq 1$,
the proof of the proposition will follow
if we can show that $p_1(S_1,C_1) = p_1(S_2,C_2)$.
Let $x_i \in S_i$ be the unique closed points. It is clear that
$\pi_1^{-1}(x_1)$ maps onto $\pi_2^{-1}(x_2)$. Moreover, ${\tilde{\lowercase{F}}}({\tilde{C}}_1) =
{\tilde{C}}_2$.
Let ${\tilde{\lowercase{X}}}_i = \pi_i^{-1}(x_i) \cap {\tilde{C}}_i$. Then ${\tilde{\lowercase{F}}}({\tilde{\lowercase{X}}}_1) = {\tilde{\lowercase{X}}}_2$.
Let $P_i = C_i \times_{S_i} {\tilde{S}}_i$. A little thought shows that there is
a cartesian diagram:
\squareSE{P_1}{P_2}{{\tilde{S}}_1}{{\tilde{S}}_2\makenull{.}%
}Since $S_1$ and $S_2$ embed in codimension one, $P_i = {\tilde{C}}_i \cup E_i$,
where $E_i \cong \P1$ and ${\tilde{C}}_i \cap E_i = {\tilde{\lowercase{X}}}_i$. The equality
$p_1(S_1,C_1) = p_1(S_2,C_2)$ can then be deduced from \pref{formal-woof-2}.
{\hfill$\square$}
\end{proof}
If $(S,C)$ is a local-geometric pair, then $S$ is excellent, so the completion
map \mapx[[ {\hat{S}} || S ]] is formally smooth. Hence we have:
\begin{corollary}\label{analytic-invariant}
If two local-geometric surface-curve pairs embed in codimension one and are
analytically isomorphic, then they have the same type.
\end{corollary}
\begin{remark}
We do not know if \pref{formal-woof-3} and \pref{analytic-invariant} are
true without the hypothesis of ``embedding in codimension one''.
\end{remark}
\block{Examples}
\par\indent\indent
We give three examples which illustrate type computations and pathological
aspects of blowing up.
Cf.\ \pref{fantastico}, where rational double points are dealt with.
The first example illustrates a general conjecture which we cannot yet make
precise: amongst surfaces of given degree in $\P3$, those which occur in
positive characteristic can have ``larger'' type than those which occur in
characteristic zero. Of course, the type also depends on the choice of a curve
on the surface.
More specifically, the example shows that in characteristic two, a quartic
surface (together with a suitably chosen curve) can have $p_1 = p_2 = p_3 = 8$.
We expect that this cannot happen in characteristic zero. If so, it would
follow from (\ref{thmI} = ``I'') that a smooth quartic rational curve
$C \subset {\Bbb C}\kern1pt\xmode{\Bbb P\kern1pt}^3$ cannot be the set-theoretic complete intersection\ of two quartic surfaces, unless $C$ is
contained in the singular locus of one of the surfaces.
On the other hand, Hartshorne
\Lcitemark 15\Rcitemark \Rspace{} and
Samuel\Lspace \Lcitemark 30\Rcitemark \Rspace{} have shown that in positive
characteristic,
the monomial rational quartic curve $C \subset \P3$ is a set-theoretic complete intersection.
See \pref{examplex-char-two} for additional comments.
The second two examples have to do with pairs $(S,C)$ in which $S$ is not
Cohen-Macaulay. These seem to be of some intrinsic interest, but
have no direct relevance to the problem of set-theoretic complete intersections\ in $\P3$. Example two
might be viewed as a statement about
the properties of the singularity at the vertex of the cone over a
space curve. It would be very nice to understand better the connection
between this singularity and the properties of the space curve.
\begin{prop}\label{example-char-two}
Let $k$ be an algebraically closed field of characteristic two.
Let $S \subset \P3$ be the cuspidal cone given by $y^4 - x^3w = 0$. Let
$C \subset S$ be the smooth rational quartic curve given by
$$(s,t) \mapsto (x,y,z,w) = (s^4, s^3t, st^3, t^4).$%
$Then $C$ meets $\mathop{\operatoratfont Sing}\nolimits(S)$ at the unique point $(0,0,0,1)$, and the
type of $(S,C)$ is $(8,8,8)$.
\end{prop}
\begin{proof}
We will calculate
in the category of affine varieties, so we will replace $S$ by an affine
variety, and when we refer to a {\it blowup}, we will actually mean a correctly
chosen affine piece of the blowup. We let $(S_n, C_n)$ denote the \th{n}
iterated blowup of $(S,C)$.
The assertion that $C \cap \mathop{\operatoratfont Sing}\nolimits(S) = \setof{(0,0,0,1)}$ is easily checked.
Taking the affine piece at $w = 1$, we find that $S$ is given by
$y^4 = x^3$ and that $C$ is given by $x = yz$ and $y = z^3$.
Making the change of variable $x \mapsto x + yz$, followed by
$y \mapsto y+z^3$, we obtain the new equation:
$$y^4 + z^{12} = (x+yz+z^4)^3$%
$for $S$ and the equation $x = y = 0$ for $C$.
Blow up $S$ along $C$,
formally substituting $xy$ for $x$. Then $S_1$ is given by:
$$y^3 = x^3y^2 + x^2y^2z + x^2yz^4 + xy^2z^2 + y^2z^3 + yz^6 + xz^8 + z^9.$%
$Intersecting with the exceptional divisor, as in \pref{goober-peas},
corresponds to setting $y = 0$. We obtain $z^8(z + x) = 0$, which tells us
that $p_1(S,C) = 8$, and that $C_1$ is given by $y = 0$, $z+x=0$. Making the
change of variable $z \mapsto z - x$, we obtain the new equation:
$$y^3 = y^2z^3 + yz^6 + x^4yz^2 + x^8z + z^9$%
$for $S_1$, and the equation $y = z = 0$ for $C_1$.
Blow up $S_1$ along
$C_1$, formally substituting $zy$ for $z$. Then $S_2$ is given by:
$$y^2 = y^4z^3 + y^6z^6 + x^4y^2z^2 + y^8z^9 + x^8z.$%
$Setting $y = 0$, we obtain $x^8z = 0$, which tells us that $p_2(S,C) = 8$,
and that $C_2$ is given by $y = z = 0$.
Blow up $S_2$ along $C_2$, formally substituting $zy$ for $z$.
Then the blown up surface $S_3$ is given by:
$$y = y^6z^3 + y^{11}z^6 + x^4y^3z^2 + y^{16}z^9 + x^8z.$%
$Setting $y = 0$, we obtain $x^8z = 0$, which tells us that $p_3(S,C) = 8$,
and that the new curve $C_3$ is given by $y = z = 0$. One can check
that $S_3$ is smooth along $C_3$, so $p_k(S,C) = 0$ for
all $k > 3$. {\hfill$\square$}
\end{proof}
\begin{prop}\label{type-ex-2}
Let $S \subset \xmode{\Bbb A\kern1pt}^4 = \mathop{\operatoratfont Spec}\nolimits {\Bbb C}\kern1pt[x,y,z,w]$ be the cone over the monomial
quartic curve:
$$(s,t) \mapsto (x,y,z,w) = (s^4, s^3t, st^3, t^4)$%
$in $\P3$. Let $C \subset S$ be the ruling given by $y = z = w = 0$.
Then $\mathop{\operatoratfont type}\nolimits(S,C) = (1,2)$. Moreover, if \mp[[ \pi || S_1 || S ]]
denotes the blowup along $C$, and $p \in S$ denotes the unique singular point,
then $\RED{\pi^{-1}(p)} \cong \mathop{\operatoratfont Proj}\nolimits {\Bbb C}\kern1pt[s,t,u](s^3-t^2u)$.
\end{prop}
\begin{proof}
One sees that $S$ is given by the equations $yz = xw$, $x^2z = y^3$,
$z^3 = yw^2$, and $y^2w = xz^2$. The blowup $S_1$ of $S$ along $C$ is
obtained\footnote{In this situation, where $S$ does not embed in codimension
one, it is apparently necessary to look at all of the affine pieces of the
blowup. The details of this are left to the reader. These calculations
are greatly facilitated by the use of a computer program such as Macaulay.}
by formally substituting $z = sy$, $w = ty$. Then $S_1$ is
given by $sy = tx$, $sx^2 = y^2$, $s^3 = t^2$, and $ty = s^2x$. Then:
\disomorx[[ \pi_1^{-1}(C) || \mathop{\operatoratfont Spec}\nolimits {\Bbb C}\kern1pt[x,y,s,t]/(y,tx,sx^2,s^3-t^2,s^2x). ]]%
Set-theoretically,
$$\pi_1^{-1}(C) = V(s,t,y) \cup V(x,y, s^3-t^2).$%
$We have $C_1 = V(s,t,y)$. Thus:
$$p_1(S,C) = \mathop{\operatoratfont length}\nolimits {\Bbb C}\kern1pt[x,s,t]/(tx,sx^2,s^3-t^2,s^2x)_{(x,s^3-t^2)},$%
$which equals one.
The blowup $S_2$ of $S_1$ along $C_1$ is obtained by formally substituting
$s = ay$, $t = by$. Then $S_2$ is given by $ax^2 = y$
and $b = a^2x$. Let \mp[[ \pi_2 || S_2 || S_1 ]] be the
blowup map. Then:
\begin{eqnarray*}
\pi_2^{-1}(C_1) & \cong & \mathop{\operatoratfont Spec}\nolimits {\Bbb C}\kern1pt[a,b,x] / (ax^2,b-a^2x)\\
& \cong & \mathop{\operatoratfont Spec}\nolimits {\Bbb C}\kern1pt[a,x] / (ax^2).
\end{eqnarray*}
This implies that $p_2(S,C) = 2$. Since $S_2$ is smooth, we see that
$\mathop{\operatoratfont type}\nolimits(S,C)$ is as claimed.
\end{proof}
\begin{example}\label{type-ex-3}
Let $S$ be a smooth surface, which is a variety. Fix $n \geq 2$, and let
$\vec p1n \in S$
be distinct (closed) points. Let \mp[[ \pi || S || {\overline{S}} ]] be the morphism
which pinches $\vec p1n$ together, yielding $p \in {\overline{S}}$. Let $C \subset S$
be a smooth curve passing through $p_1$ but not through $\vec p2n$. Then
${\overline{C}} = \pi(C)$ is smooth. Let \mp[[ f || X || {\overline{S}} ]] be the blowup along
${\overline{C}}$. Then $X$ is obtained from $S$ by blowing up $\vec p2n$. Hence
$f^{-1}({\overline{C}})$ is isomorphic to the disjoint union of ${\overline{C}}$ with $n-1$
copies of $\P1$. The type of $(S,C)$ is $(n-1)$.
\end{example}
\block{Classification of rational double point pairs}\label{class}
\par\indent\indent In this section, we assume that $k$ has characteristic
zero. We describe a classification of local-geometric pairs $(S,C)$, up to
analytic isomorphism, where $S$ is the local scheme of a rational double
point singularity.
Let $(S,C)$ be a local-geometric pair corresponding to a rational double point. As is
well-known, such objects $S$ are classified (up to analytic isomorphism)
by A-D-E Dynkin diagrams. Let ${\tilde{S}}$ be the minimal resolution of $S$, and
let ${\tilde{C}} \subset {\tilde{S}}$ be the strict transform of $C$. Let $\vec E1n \subset {\tilde{S}}$
be the exceptional curves, numbered as in
(\Lcitemark 18\Rcitemark \ p.\ 167).
Then ${\tilde{C}}$ meets a unique exceptional curve $E_k$, and we have
${\tilde{C}} \cdot E_k = 1$. Moreover, there are some restrictions on $k$, depending
on $S$. (See\Lspace \Lcitemark 18\Rcitemark \Rspace{}\ 2.2.)
In this way, we are able to define certain local-geometric pairs $A_{n,k}$,
$D_{n,k}$, and $E_{n,k}$. In fact, one can show\Lspace \Lcitemark 20\Rcitemark
\Rspace{}
that these pairs are well-defined, up to analytic isomorphism. We have:
\begin{theorem}\label{the-conjecture}
Let $(S,C)$ be a local-geometric
pair, where $S$ is the local scheme of a rational double point
singularity. Then $(S,C)$ is analytically isomorphic to a unique member of
the following list of local pairs:
\begin{itemize}
\item $A_{n,k}$ (for some positive integers $n,k$ with $k \leq (n+1)/2$);
\item $D_{n,1}$ (for some integer $n \geq 4$);
\item $D_{n,n}$ (for some integer $n \geq 5$);
\item $E_{6,1}$;
\item $E_{7,1}$.
\end{itemize}
\end{theorem}
\begin{remark}
Equations for these pairs may be found in the proof of \pref{fantastico}.
\end{remark}
\block{Invariants of rational double point configurations}\label{invrdp}
\par\indent\indent In this section, we assume that $k$ has characteristic zero.
We will calculate the type of $(S,C)$ in the case where $S$ is the local
scheme of a rational double point singularity.
This depends not only on $S$, but also on $C$. Note that if $S$ is the
local scheme of any rational singularity, and $S$ embeds in a nonsingular
three-fold, then $S$ ``is'' a rational double point.
For each pair of positive integers $(n,k)$ with $k \leq n$, we define a
sequence $\phi(n,k)$ of integers, via the following recursive definition:
$$\phi(n,k) = \cases{ \phi(n,n-k+1),&if $k > {n+1\over2}$;\cr
(k),&if $k = {n+1\over2}$;\cr
(k,\phi(n-k,k)),&if $k < {n+1\over2}$.}$%
$
\begin{examples}
\
\begin{arabiclist}
\item $\phi(n,1) = (1^\br{n})$ for all $n \geq 1$;
\item $\phi(rk,k) = (k^\br{r-1},1^\br{k})$ for all $k \geq 1$, $r \geq 1$
(generalizing 1);
\item $\phi(rk-1,k) = (k^\br{r-1})$ for all $r \geq 2$, $k \geq 1$
(also generalizing 1);
\item $\phi(10,4) = (4,3,1^\br{3}).$
\end{arabiclist}
\end{examples}
Let $a, b \in \xmode{\Bbb N}$. For each integer $n \geq 0$, we define the \th{n}
{\it iterated remainder\/} on division of $a$ by $b$, denoted $\mathop{\operatoratfont rem}\nolimits_n(a,b)$.
Let $\mathop{\operatoratfont rem}\nolimits_0(a,b) = b$, and let $\mathop{\operatoratfont rem}\nolimits_1(a,b)$ be the usual remainder. For
$n \geq 2$, define:
$$\mathop{\operatoratfont rem}\nolimits_n(a,b) = \cases{ \mathop{\operatoratfont rem}\nolimits_1(\mathop{\operatoratfont rem}\nolimits_{n-2}(a,b), \mathop{\operatoratfont rem}\nolimits_{n-1}(a,b)),
&if $\mathop{\operatoratfont rem}\nolimits_{n-1}(a,b) \not= 0$;\cr
0,&if $\mathop{\operatoratfont rem}\nolimits_{n-1}(a,b) = 0$.}$%
$
\par Let $a,b \in \xmode{\Bbb N}$. For each integer $n \geq 1$, we define the \th{n}
{\it iterated quotient\/} of $a$ by $b$, denoted $\mathop{\operatoratfont div}\nolimits_n(a,b)$. Let
$\mathop{\operatoratfont div}\nolimits_1(a,b) = \floor{a/b}$. For $n \geq 2$, define:
$$\mathop{\operatoratfont div}\nolimits_n(a,b) = \cases{ \mathop{\operatoratfont div}\nolimits_1(\mathop{\operatoratfont rem}\nolimits_{n-2}(a,b), \mathop{\operatoratfont rem}\nolimits_{n-1}(a,b)),
&if $\mathop{\operatoratfont rem}\nolimits_{n-1}(a,b) \not= 0$;\cr
0,&if $\mathop{\operatoratfont rem}\nolimits_{n-1}(a,b) = 0$.}$%
$
\begin{prop}\label{key-rem}
Fix $k, n \in \xmode{\Bbb N}$ with $k \leq (n+1)/2$. Let $t$ be the largest integer such
that $\mathop{\operatoratfont rem}\nolimits_t(n-k+1,k) \not= 0$. Let $r_i = \mathop{\operatoratfont rem}\nolimits_i(n-k+1,k)$,
$d_i = \mathop{\operatoratfont div}\nolimits_i(n-k+1,k)$, for various $i$. Then:
$$\phi(n,k) = (r_0^\br{d_1}, r_1^\br{d_2}, \ldots, r_t^\br{d_{t+1}}).$%
$\end{prop}
\begin{sketch}
Define $r_{-1} = n-k+1$. One shows that for all $p \geq 0$,
$$\phi(r_{p-1}+r_p-1, r_p) =
\cases{(r_p^\br{d_{p+1}}, \phi(r_p + r_{p+1}-1, r_{p+1})),
&if $r_{p+1} \not= 0$;\cr
(r_p^\br{d_{p+1}}),&if $r_{p+1} = 0$.\cr}$%
$The result then follows by induction. {\hfill$\square$}
\end{sketch}
\begin{prop}\label{fantastico}
The type of $A_{n,k}$ is $\phi(n,k)$. The type of $D_{n,1}$ is $(2)$. We
have:
$${\begindiagram
\mathop{\operatoratfont type}\nolimits(D_{n,n}) = \cases{({n\over2}),&if $n$ is even\kern1.5pt$;$\cr
({n-1\over2}, 1^\br{n-1}),&
if $n$ is odd.\cr}}$%
$The type of $E_{6,1}$ is $(2,2)$. The type of $E_{7,1}$ is $(3)$.
\end{prop}
\begin{proof}
We let $(S,C)$ correspond to the given pair.
The comments in the first paragraph of the proof of \pref{example-char-two}
apply equally well here.
We make use of the explicit resolutions of rational double points given
in the appendix to\Lspace \Lcitemark 25\Rcitemark \Rspace{}.
First we consider the $A_{n,k}$ case. (We allow $1 \leq k \leq n$.) Then
$S$ is given by $xy - z^{n+1} = 0$, and $C$ is given parametrically by
$x = u^k$, $y = u^{n-k+1}$, $z = u$. [In terms of the notation used in
\Lcitemark 25\Rcitemark \Rspace{}, this may be seen as the image of
$V(u_k=1) \subset W_k$.] After making the change of variable
$x \mapsto x + z^k$ and $y \mapsto y + z^{n-k+1}$, we find that $S$ is
given by:
$$xy + yz^k + xz^{n-k+1} = 0,\eqno(*)$%
$and that $C$ is given by $x = y = 0$. From now on, we assume that
$k \leq {n+1\over2}$. Blow-up along $C$,
formally substituting $yx$ for $y$. Then $S_1$ is given by:
$$xy + yz^k + z^{n-k+1} = 0.$%
$Intersecting with the exceptional divisor, as in \pref{goober-peas},
corresponds to setting $x = 0$. We obtain $z^k(y + z^{n-2k+1}) = 0$. This
tells us that $p_1(S,C) = k$ and that $C_1$ is given by $x = 0$ and
$y + z^{n-2k+1} = 0$. After making the change of variable
$y \mapsto y - z^{n-2k+1}$, and thence $y \mapsto -y$, we obtain the
equation:
$$xy + yz^k + xz^{n-2k+1} = 0\eqno(**)$%
$for $S_1$, and the equation $x = y = 0$ for $C_1$. If $n-2k+1=0$, then
$S_1$ is smooth along $C_1$, and we are done. Otherwise, compare
$(*)$ with $(**)$, to complete the $A_{n,k}$ case.
Now we deal with the case $D_{n,1}$. In terms of the notation used
in\Lspace \Lcitemark 25\Rcitemark \Rspace{}, $C$ is the image of
$V(v_0 = 0) \subset W_0$.
Following\Lspace \Lcitemark 25\Rcitemark \Rspace{}, we would have two cases
($n$ even, $n$ odd), but in fact these two cases are identical in this
situation, after interchanging variables $(x \leftrightarrow y)$.
We find that $S$ is given by:
$$x^2z + y^2 - z^{n-1} = 0,$%
$and that $C$ is given by $y = z = 0$. Blow up along $C$,
formally substituting $zy$ for $z$. Then $S_1$ is given by:
$$x^2z + y - y^{n-2}z^{n-1} = 0.$%
$Setting $y = 0$, we obtain $x^2z = 0$. This tells us that $p_1(S,C) = 2$
and that $C_1$ is given by $y = z = 0$. An easy calculation shows that
$S_1$ is smooth. The result for $D_{n,1}$ follows.
Now we deal with the case $D_{n,n}$. In terms of the notation used in
\Lcitemark 25\Rcitemark \Rspace{}, $C$ is the image of
$V(u_n = 0) \subset W_n$. We may take the same equation for $S$ as we did
in the case $D_{n,1}$. There are two cases:
Case I: $n$ is even. Then $C$ is given parametrically by $x = u^{(n-2)/2}$,
$y = 0$, $z = u$. After making the change of variable
$x \mapsto x + z^{(n-2)/2}$, we find that $S$ is given by:
$$x^2z + y^2 + 2xz^{n/2} = 0,$%
$and that $C$ is given by $x = y = 0$. Blow up along $C$,
formally substituting $xy$ for $x$. Then $S_1$ is given by:
$$x^2yz + y + 2xz^{n/2} = 0.$%
$Setting $y = 0$, we obtain $xz^{n/2} = 0$. This tells us that
$p_1(S,C) = n/2$ and that $C_1$ is given by $x = y = 0$. On checks that
$S_1$ is smooth.
Case II: $n$ is odd. Then $C$ is given parametrically by $x = 0$,
$y = u^{(n-1)/2}$, $z = u$. (In this case, $x$ and $y$ are interchanged from
the notation in\Lspace \Lcitemark 25\Rcitemark \Rspace{}.) After making the
change of variable $y \mapsto y + z^{(n-1)/2}$, we find that $S$ is given by:
$$x^2z + y^2 + 2yz^{(n-1)/2} = 0,$%
$and that $C$ is given by $x = y = 0$. Blow up along $C$,
formally substituting $yx$ for $y$. Then $S_1$ is given by:
$$xz + xy^2 + 2yz^{(n-1)/2} = 0.\eqno(*{*}*)$%
$Setting $x = 0$, we obtain $yz^{(n-1)/2} = 0$. This tells us that:
$$p_1(S,C) = (n-1)/2$%
$and that $C_1$ is given by $x = y = 0$. Now blow-up
along $C_1$, formally substituting $xy$ for $x$. Then the blow-up
$S_2$ is given by:
$$xz + xy^2 + 2z^{(n-1)/2} = 0.$%
$Setting $y = 0$, we obtain $z(x + 2z^{(n-1)/2 - 1}) = 0$. This tells us
that $p_2(S,C) = 1$ and that $C_2$ is given by
($y = 0$ and $x + 2z^{(n-1)/2 - 1} = 0$). After making the change of
variable $x \mapsto x - 2z^{(n-1)/2 - 1}$, we find that $S_2$ is given by:
$$xz + xy^2 - 2y^2z^{(n-1)/2 - 1} = 0,$%
$and that $C_2$ is given by $x = y = 0$.
Now blow up along $C_2$, formally substituting $xy$ for $x$. Then
the blown up surface $S_3$ is given by:
$$xz + xy^2 - 2yz^{(n-1)/2-1} = 0,$%
$$C_3$ is given by $x = y = 0$, and $p_3(S,C) = 1$. Replacing $y$ by
$-y$, we may assume that $S_3$ is given by:
$$xz + xy^2 + 2yz^{(n-1)/2-1} = 0.$%
$This looks like $(*{*}*)$, except that $n$ is now replaced by $n-2$.
Note that if $n = 1$, then $(*{*}*)$ is smooth. A little
thought shows that the asserted type of $D_{n,n}$ is correct. A posteriori,
we see that $(S_1,C_1) = A_{n-1,1}$. A direct proof of this assertion would
of course simplify the proof.
For both $E_{6,1}$ and $E_{7,1}$, we may choose any smooth curve for $C$.
For $E_{6,1}$, $S$ is given by $x^2 - y^3 - z^4 = 0$, and $C$ is given by
$y = 0$, $x + z^2 = 0$. After making the change of variable
$x \mapsto x - z^2$, we obtain the new equation $x^2 - 2xz^2 - y^3 = 0$
for $S$. Then $C$ is given by $x = y = 0$. Blow up along $C$, substituting
$xy$ for $x$. The equation for $S_1$ is:
$$x^2y - 2xz^2 - y^2 = 0.$%
$Setting $y = 0$, we obtain $xz^2 = 0$. Hence $p_1(S,C) = 2$, and $C_1$ is
given by $x = y = 0$. Blow up $S_1$ along $C_1$, substituting $xy$ for $x$.
The equation for $S_2$ is $x^2y^2 - 2xz^2 - y = 0$. Substituting
$y = 0$, we obtain $xz^2 = 0$. Hence $p_2(S,C) = 2$. One checks that
$S_2$ is smooth, so $p_k(S,C) = 0$ for all $k > 2$.
For $E_{7,1}$, $S$ is given by $x^2 + y^3 - yz^3 = 0$, and $C$ is given by
$x = y = 0$. Blow up along $C$, substituting $yx$ for $y$. The equation
for $S_1$ is $x + x^2y^3 - yz^3 = 0$. Setting $x = 0$, we obtain $yz^3 = 0$.
Hence $p_1(S,C) = 3$. As $S_1$ is smooth, we see that the type of $E_{7,1}$
is as claimed. {\hfill$\square$}
\end{proof}
\begin{warning}
Amongst the rational double point\ local-geometric pairs, those of the kind $(S,C) = D_{n,n}$
with $n$ odd ($n \geq 5$) are highly atypical. The following phenomena happen
only for these special pairs:
\begin{romanlist}
\item $\Sigma(S,C) < \sum_{i=1}^\infty p_i(S,C)$;
\item $p_r(S,C) \not= 0$ for some $r > \mathop{\operatoratfont order}\nolimits(S,C)$: see \pref{interesting}.
\end{romanlist}
\end{warning}
The calculation in the proposition allows one to compute not just the
type of a rational double point, but also the precise sequence of
(analytic equivalence classes of) local surface-curve pairs which
arise under successive blowups:
\begin{itemize}\label{blowupAnk}
\item $\mathop{\operatoratfont blowup}\nolimits(A_{n,k})
\cases{\hbox{is smooth},&if $k = {n+1\over2}$;\cr
= A_{n-k,n-2k+1},&if ${n-k+1\over2} < k < {n+1\over2}$;\cr
= A_{n-k,k},&if $k \leq {n-k+1\over2}$;}$
\item $\mathop{\operatoratfont blowup}\nolimits(D_{n,1})$ is smooth;
\item $\mathop{\operatoratfont blowup}\nolimits(D_{n,n})$ is smooth, if $n$ is even;
\item $\mathop{\operatoratfont blowup}\nolimits(D_{n,n}) = A_{n-1,1},$ if $n$ is odd;
\item $\mathop{\operatoratfont blowup}\nolimits(E_{6,1}) = A_{3,2}$;
\item $\mathop{\operatoratfont blowup}\nolimits(E_{7,1})$ is smooth.
\end{itemize}
We now calculate $\mathop{\operatoratfont order}\nolimits(S,C)$, where $S$ is the local scheme of a
rational double point.
\begin{prop}\label{order-calc}
The order of $A_{n,k}$ is the order of ${\overline{\lowercase{K}}}$ in $\xmode{\Bbb Z}/(n+1)\xmode{\Bbb Z}$.
The order of $D_{n,1}$ is $2$. The order of $D_{n,n}$ is $2$ if $n$
is even, and it is $4$ if $n$ is odd. The order of $E_{6,1}$ is $3$.
The order of $E_{7,1}$ is $2$.
\end{prop}
\begin{proof}
Let $(S,C)$ correspond to the given pair.
Some of the orders ($E_{6,1}$, $E_{7,1}$, $D_{n,1}$ ($n$ even) and
$D_{n,n}$ ($n$ even)) can be computed immediately if one knows the
abstract group $\mathop{\operatoratfont Cl}\nolimits(S)$. A list of these groups may be found in
(\Lcitemark 22\Rcitemark \ p.\ 258). We do not use this approach.
Let ${\tilde{S}}$ be the minimal
resolution of $S$. Let $\vec E1n \subset {\tilde{S}}$ be the exceptional curves.
Let ${\tilde{C}} \subset {\tilde{S}}$ denote the strict transform of $C$. There is a unique
${\Bbb Q}\kern1pt$-divisor $E = a_1E_1 \many+ a_nE_n$ such that
${\tilde{C}} \cdot E_i = -E \cdot E_i$ for all $i$.
(See\Lspace \Lcitemark 26\Rcitemark \Rspace{}.) The total transform ${\overline{C}} \subset
{\tilde{S}}$
of $C$ is ${\tilde{C}} + E$; it is a ${\Bbb Q}\kern1pt$-divisor. According to
(\Lcitemark 26\Rcitemark \ p.\ 242), ${\overline{C}}$ is integral
(i.e.\ $E$ is integral, i.e.\ $\vec a1n \in \xmode{\Bbb Z}$) if and only if\ ``$C$ is locally
analytically equivalent to zero''. Since $S$ is a rational double point,
this is equivalent to $[C] = 0$ in $\mathop{\operatoratfont Cl}\nolimits(S)$. As this discussion applies
not just to $C$, but also to positive integer multiples of $C$, we see
that the order of $(S,C)$
is the least positive integer $N$ such that $N(\vec a1n) \in \xmode{\Bbb Z}^n$.
Let $M$ be the inverse of the self-intersection matrix of the $E_i$.
Then $(\vec a1n)$ is the \th{k} column of $M$. This may be computed from
an explicit formula for $M$, which one may find in
(\Lcitemark 18\Rcitemark \ p.\ 169).
In case $(S,C) = A_{n,k}$, one finds that:
$$a_i = \cases{-k(n-i+1)/(n+1),&if $i \geq k$;\cr
ki/(n+1) - i,&if $i \leq k$.\cr}$%
$From this we calculate that $a_1 = k/(n+1) - 1$. The proof for $A_{n,k}$
follows.
In case $(S,C) = D_{n,1}$, one finds that:
$$a_i = \cases{-1,&if $i \leq n-2$;\cr
-1/2,&if $n-1 \leq i \leq n$.\cr}$%
$Hence $\mathop{\operatoratfont order}\nolimits(D_{n,1}) = 2$.
In case $(S,C) = D_{n,n}$, one finds that:
$$a_i = \cases{-i/2,&if $i \leq n-2$;\cr
-(n-2)/4,&if $i = n-1$;\cr
-n/4,&if $i = n$.\cr}$%
$Hence $\mathop{\operatoratfont order}\nolimits(D_{n,n})$ is as claimed.
We now deal with the two exceptional cases. The inverses of the
self-intersection matrices do not appear in
\Lcitemark 18\Rcitemark \Rspace{}, and we omit them here for lack of space.
In case $(S,C) = E_{6,1}$, one finds that:
$$(\vec a1n) = (-{4\over3}, -{5\over3}, -2, -1, -{4\over3}, -{2\over3}),$%
$and in case $(S,C) = E_{7,1}$, one finds that:
\formulaqed{(\vec a1n) = (-{3\over2}, -2, -{5\over2}, -3, -{3\over2}, -2, -1).}
\end{proof}
It is interesting to note that for $D_{n,n}$ ($n$ odd), one has
$p_r(S,C) \not= 0$ for some $r > \mathop{\operatoratfont order}\nolimits(S,C)$. This does not occur for
the other rational double point\ pairs, as we shall see in \pref{interesting}.
\begin{lemma}\label{gcdgcd}
Let $k$ and $N$ be positive integers, with $k < N$. Assume that
$k \nmid N$. Then:
$$\floor{N/k} \leq (N-k) / \gcd(k,N).$$
\end{lemma}
\begin{proof}
First suppose that $k > N/2$. Then $\floor{N/k} = 1$, so we must show that
$\gcd(k,N) \leq N-k$. Indeed, if $x | k$ and $x | N$, then $x | (N-k)$,
so this is clear.
Hence we may assume\ that $k \leq N/2$. Since $k \nmid N$, $\gcd(k,N) \leq k/2$.
Therefore it suffices to show that $(k/2)(N/k) \leq N-k$. This follows
from $k \leq N/2$. {\hfill$\square$}
\end{proof}
\begin{corollary}\label{orderbound}
Let $k$ and $N$ be positive integers, with $k \leq N$. Let $t$ be the
smallest positive integer such that $\mathop{\operatoratfont rem}\nolimits_t(N,k) = 0$. Let
$d_i = \mathop{\operatoratfont div}\nolimits_i(N,k)$, for $i = 1, \ldots, t$. Then:
$$\svec d1t \leq N/\gcd(k,N).$$
\end{corollary}
\begin{proof}
The case $k = N$ is clear, so we may assume\ that $k < N$. If $t = 1$, the result
is clear. Let $r_1 = \mathop{\operatoratfont rem}\nolimits_1(N,k)$. We may assume that $r_1 \not= 0$.
By induction on $t$, we may assume\ that:
$$\svec d2t \leq k/\gcd(r_1,k).$%
$Therefore it suffices to show that:
$$d_1 + {k \over \gcd(r_1,k)} \leq {N \over \gcd(k,N)}.$%
$One sees that $\gcd(r_1,k) = \gcd(k,N)$. Therefore it suffices to show
that $d_1 \leq (N-k) / \gcd(k,N)$. This follows from \pref{gcdgcd}. {\hfill$\square$}
\end{proof}
\begin{prop}\label{interesting}
Let $(S,C)$ be a local-geometric
pair corresponding to a rational double point. Assume that $(S,C) \not= D_{n,n}$ for any
odd integer $n \geq 5$. Then $p_r(S,C) = 0$ for all $r \geq \mathop{\operatoratfont order}\nolimits(S,C)$.
\end{prop}
\begin{proof}
We utilize \pref{fantastico} and \pref{order-calc}. The only nontrivial
case is $(S,C) = A_{n,k}$. We may assume that $k \leq (n+1)/2$.
For any $a,b \in \xmode{\Bbb N}$, let $o(a,b)$ denote the
order of ${\overline{\lowercase{A}}}$ in $\xmode{\Bbb Z}/b\xmode{\Bbb Z}$. In the notation of \pref{key-rem}, we must show
that:
$$\svec d1{t+1} < o(k,n+1).$%
$Translating to the notation of \pref{orderbound}
($N = n+1$), both $d_1$ and $t$ change by $1$. The statement we need is:
$$(d_1 - 1) + \svec d2t < o(k,N).$%
$Since $o(k,N) = N/\gcd(k,N)$, this does follow from \pref{orderbound}. {\hfill$\square$}
\end{proof}
The content of the following proposition may be found in
(\Lcitemark 18\Rcitemark \ proof of 2.3, pp.\ 169-170).
\begin{prop}\label{delta-formulas}
We have:
\begin{eqnarray*}
\Delta(A_{n,k}) & = & k(n-k+1)/(n+1)\\
\Delta(D_{n,1}) & = & 1\\
\Delta(D_{n,n}) & = & n/4\\
\Delta(E_{6,1}) & = & 4/3\\
\Delta(E_{7,1}) & = & 3/2.
\end{eqnarray*}
\end{prop}
\block{Technical lemmas on rational double points}
\par\indent\indent In this section we assume that $k$ has characteristic
zero. We prove various technical relationships between the
invariants of rational double point\ local-geometric pairs. We use these results in part III.
The result \pref{rdp-one} appears to be of intrinsic interest.
\begin{prop}\label{potato-1}
Let $(S,C)$ be a local-geometric pair corresponding to a rational
double point. Write:
$$\mathop{\operatoratfont type}\nolimits(S,C) = (n_1^\br{k_1}, \ldots, n_r^\br{k_r})$%
$with $n_1 > \cdots > n_r \geq 1$ and $k_i \geq 1$ for each $i$. Assume that
$r > 1$. Then $k_r > 1$ and $n_r | n_{r-1}$.
\end{prop}
\begin{proof}
We use \pref{fantastico}. The proposition is clear if $(S,C)$ is of
species $D$ or $E$. Therefore we may assume that $(S,C)$ is of species
$A$. In the notation of \pref{key-rem}, we may write:
$$\mathop{\operatoratfont type}\nolimits(S,C) = (r_0^\br{d_1}, \ldots, r_t^\br{d_{t+1}}).$%
$Since $r_{t+1} = 0$, we have $r_{t-1} = r_t d_{t+1}$. Hence $r_t | r_{t-1}$.
Hence $n_r | n_{r-1}$. Since $r_{t-1} > r_t$, $d_{t+1} > 1$. Hence $k_r > 1$.
{\hfill$\square$}
\end{proof}
\begin{prop}\label{potato-2}
If $\mathop{\operatoratfont type}\nolimits(A_{n,k}) = \mathop{\operatoratfont type}\nolimits(A_{n',k'})$, where $k \leq (n+1)/2$ and
$k' \leq (n'+1)/2$, then $n = n'$ and $k = k'$.
\end{prop}
\begin{proof}
We use \pref{fantastico}.
Write $\mathop{\operatoratfont type}\nolimits(A_{n,k}) = (r_0^\br{d_1}, \ldots, r_t^\br{d_{t+1}})$, as in
\pref{key-rem}. Then $n = n' = r_0(d_1+1) + r_1 - 1$, and $k = k' = r_0$.
{\hfill$\square$}
\end{proof}
\begin{lemma}\label{yechh}
Fix positive integers $k$ and $N$ with $k \leq N/2$. Let $t$ be the smallest
positive integer such that $\mathop{\operatoratfont rem}\nolimits_t(N,k) = 0$. Let $r_i = \mathop{\operatoratfont rem}\nolimits_i(N,k)$,
$d_i = \mathop{\operatoratfont div}\nolimits_i(N,k)$, for various $i$. Then:
\begin{romanlist}
\item If $t = 1$, then $r_0d_1^{-1} = k^2/N$.
\item If $t = 2$, then $(r_0-r_1)d_1^{-1} + r_1(d_1+d_2)^{-1} \leq k^2/N$.
\item If $t \geq 3$, then:
$$(r_0-r_1)d_1^{-1} + (r_1-r_2)(d_1+d_2)^{-1} +
r_2(d_1+d_2+1)^{-1} \leq k^2/N.$%
$\end{romanlist}
\end{lemma}
\begin{proof}
First suppose that $t = 1$. Then $N = d_1 k$. Hence
$k^2/N = k/d_1 = r_0d_1^{-1}$. This proves (i).
Now suppose that $t = 2$. Then $N = r_0d_1 + r_1$ and $r_0 = r_1d_2$.
We must show that:
$${r_0 - r_1 \over d_1} + {r_1 \over d_1 + d_2} \leq {r_0^2\over r_0d_1+r_1}.$%
$Substitute $r_0 = r_1d_2$, and cancel out $r_1$. We must show:
$${d_2 - 1 \over d_1} + {1 \over d_1+d_2} \leq {d_2^2 \over d_1d_2 + 1}.$%
$Eliminating denominators, we find that we must show:
$$(d_1 - 1)(d_2 - 1) \geq 0,$%
$which is certainly true.
Finally, suppose that $t \geq 3$. Then $N = r_0d_1 + r_1$ and
$r_0 = r_1d_2 + r_2$. We must show that:
$${r_0-r_1 \over d_1} + {r_1-r_2 \over d_1+d_2} + {r_2 \over d_1+d_2+1}
\leq {r_0^2 \over r_0d_1 + r_1}.$%
$Substitute $r_0 = r_1d_2 + r_2$. We must show that:
$${r_1d_2 + r_2 - r_1 \over d_1} + {r_1 - r_2 \over d_1 + d_2} +
{r_2 \over d_1 + d_2 + 1} \leq
{(r_1d_2 + r_2)^2 \over r_1d_1d_2 + r_2d_1 + r_1}.$%
$Now cancel denominators. (This is best done with the aid of a computer.)
We must show:
\splitdiagram{d_2r_1^2 - d_1^2d_2r_1^2 - d_1d_2^2r_1^2
+ d_1^2d_2^2r_1^2 - d_2^3r_1^2 + d_1d_2^3r_1^2 - d_1^2r_1r_2 - d_2r_1r_2%
}{ - d_1d_2r_1r_2
+ 2d_1^2d_2r_1r_2 - d_2^2r_1r_2 + d_1d_2^2r_1r_2 + d_1^2r_2^2 \geq 0.}
Equivalently, we must show that:
\splitdiagram{r_1^2d_2[ d_2^2(d_1-1) + d_1(d_1d_2 - d_1 - d_2)+1]%
}{ + r_1r_2[ d_1^2(d_2-1) + d_2^2(d_1-1)+ d_2(d_1^2-d_1-1)] + d_1^2r_2^2 \geq
0.}
If $d_1 \geq 2$ and $d_2 \geq 2$, this is clear. Since $k \leq N/2$,
we have $d_1 \geq 2$. Suppose that $d_2 = 1$. Then the needed inequality
simplifies to:
$$r_1r_2(d_1^2 - 2) + d_1^2r_2^2 \geq 0,$%
$which is true. {\hfill$\square$}
\end{proof}
{}From \pref{yechh}, we obtain the following weaker statement, which we use in
\pref{rdp-one}:
\begin{corollary}\label{weirdness}
Fix positive integers $k$ and $N$, with $k \leq N/2$. Let $t$ be the
smallest positive integer such that $rem_t(N,k) = 0$. Let $r_i = \mathop{\operatoratfont rem}\nolimits_i(N,k)$,
$d_i = \mathop{\operatoratfont div}\nolimits_i(N,k)$, for various $i$. Then:
$$(r_0-r_1)d_1^{-1} \many+ (r_{t-1}-r_t)(\svec d1t)^{-1} \leq k^2/N.$$
\end{corollary}
\begin{warning}
When we use \pref{weirdness}, the symbol $d_1$ will appear to have two
different values, differing by $1$: in the application \pref{rdp-one},
$d_1$ will be smaller by $1$.
\end{warning}
\begin{prop}\label{rdp-one}
Let $(S,C)$ correspond to a rational double point singularity.
Let $p_k = p_k(S,C)$, for each $k \in \xmode{\Bbb N}$.
Then:
$$\sum_{k=1}^\infty{1 \over k(k+1)}p_k \geq \Delta(S,C).$$
\end{prop}
\begin{proof}
We use \pref{delta-formulas} and \pref{fantastico}.
If $(S,C)$ is not of species $A$, then the proposition is proved by the
following table:
{\renewcommand{\arraystretch}{1.2}
\vspace*{0.1in}
\centerline{
\begin{tabular}{||c|c|c||} \hline
singularity & $\sum_{k=1}^\infty {1\over k(k+1)} p_k$
& $\Delta(S,C)$\\ \hline
$D_{n,1}$ & $1$ & $1$\\ \hline
$D_{n,n}$ ($n$ even) & $n/4$ & $n/4$\\ \hline
$D_{n,n}$ ($n$ odd)
& ${n-1\over4}$ + $\sum_{k=2}^n {1\over k(k+1)}$
& $n/4$\\ \hline
$E_{6,1}$ & $4/3$ & $4/3$\\ \hline
$E_{7,1}$ & $3/2$ & $3/2$\\ \hline
\end{tabular} }
\vspace*{0.1in}}
Suppose that $(S,C) = A_{n,k}$ for some $n, k$. We may assume that
$k \leq (n+1)/2$. We must show that:
$$\sum_{j=1}^\infty {1 \over j(j+1)}\phi(n,k)_j \geq k(n-k+1)/(n+1).\eqno(*)$%
$Let $t$ be the largest integer such that $\mathop{\operatoratfont rem}\nolimits_t(n-k+1,k) \not= 0$. For
$i = 0, \ldots, t$, let $r_i = \mathop{\operatoratfont rem}\nolimits_i(n-k+1,k)$,
$d_{i+1} = \mathop{\operatoratfont div}\nolimits_{i+1}(n-k+1,k)$. By \pref{key-rem}, we see that $(*)$ is
equivalent to:
\splitdisplay{r_0 \sum_{j=1}^{d_1} {1 \over j(j+1)} +
r_1 \sum_{j=d_1+1}^{d_1+d_2} {1 \over j(j+1)} \many+
r_t \sum_{j=\svec d1t + 1}^{\svec d1{t+1}} {1 \over j(j+1)}%
}{\geq {k(n-k+1) \over n+1}.%
}Note that for any $a,b \in \xmode{\Bbb N}$ with $a \leq b$,
$$\sum_{j=a+1}^b {1 \over j(j+1)} = {b \over b+1} - {a \over a+1}.$%
$Hence $(*)$ is equivalent to:
\splitdisplay{\left(r_0-r_1\right)\left({d_1 \over d_1+1}\right) \many+
\left(r_{t-1}-r_t\right)\left({\svec d1t \over
\svec d1t + 1}\right)%
}{+ r_t\left({\svec d1{t+1} \over \svec d1{t+1} + 1}\right)
\geq k(n-k+1)/(n+1).%
}This is equivalent to:
\splitdiagram{r_0 - (r_0 - r_1)(d_1 + 1)^{-1} - \cdots - (r_{t-1}-r_t)
(\svec d1t + 1)^{-1}%
}{- r_t(\svec d1{t+1} + 1)^{-1} \geq k(n-k+1)/(n+1).%
}Since $r_0 = k$, this is equivalent to:
\splitdiagram{(r_0 - r_1)(d_1 + 1)^{-1} \many+ (r_{t-1}-r_t)
(\svec d1t + 1)^{-1}}
{ + r_t(\svec d1{t+1} + 1)^{-1} \leq k^2/(n+1).%
}Let $N = n+1$. Then this follows from \pref{weirdness}, and thence completes
the proof. {\hfill$\square$}
\end{proof}
\begin{definition}
Let $(S,C)$ be a local-geometric pair corresponding to a rational double point. Then the
{\it deficiency\/} of $(S,C)$ is:
$$\mathop{\operatoratfont def \kern1pt}\nolimits(S,C) = \Sigma(S,C) - \sum_{i=1}^\infty p_i(S,C).$$
\end{definition}
One always has $\mathop{\operatoratfont def \kern1pt}\nolimits(S,C) \geq 0$, except for $D_{n,n}$, with $n$ odd,
$n \geq 5$.
\part{Iterated curve blowups}
\block{Intersection ring of a blow up}\label{oneblow}
\par\indent\indent
In this section we describe (without proof) the intersection ring of the
blow-up of a nonsingular variety along a nonsingular subvariety, following the
statements given in (\Lcitemark 7\Rcitemark \ 6.7, 8.3.9). There are
two differences between the assertions we make
and the assertions made in\Lspace \Lcitemark 7\Rcitemark \Rspace{}.
Firstly, we work with cycles modulo algebraic equivalence,
rather than modulo rational equivalence. Secondly, we have adjusted the signs
to reflect our convention regarding projective space bundles.
Let $X$ be a nonsingular closed subvariety of a nonsingular variety $Y$.
Let $d = \mathop{\operatoratfont codim}\nolimits(X,Y)$, and assume that $d \geq 2$. Let $N$ be the
normal bundle of $X$ in $Y$.
Let ${\tilde{Y}}$ be the blow-up of $Y$ along $X$. The exceptional divisor
is isomorphic to $\xmode{\Bbb P\kern1pt} N^*$. We use the following diagram to fix notation:
\diagramx{\xmode{\Bbb P\kern1pt} N^*&\mapE{j}&{\tilde{Y}}\cr
\mapS{g}&&\mapS{f}\cr
X&\mapE{i}&Y.\cr%
}Let $F = \mathop{\operatoratfont Ker}\nolimits[ g^*(N^*)\ \mapE{\rm{can}}\ {\cal O}_{\xmode{\Bbb P\kern1pt} N^*}(1)]$. For each $k$,
there is a canonically split exact sequence:
\sesmaps{A^{k-d}(X)}{\delta}{A^{k-1}(\xmode{\Bbb P\kern1pt} N^*) \o+ A^k(Y)}{\beta}{A^k({\tilde{Y}})%
}of cycle groups modulo algebraic equivalence. The maps are given by:
$$\delta(x) = (c_{d-1}(F) \cdot g^*(x), i_*(x))$%
$and
$$\beta({\tilde{\lowercase{X}}}, y) = j_*({\tilde{\lowercase{X}}}) + f^*(y).$%
$
This describes $A^*({\tilde{Y}})$ as an abelian group. The ring structure is
described by the following rules:
\begin{eqnarray*}
(f^*y) \cdot (f^*y') & = & f^*(y \cdot y') \\
(j_*{\tilde{\lowercase{X}}}) \cdot (j_*{\tilde{\lowercase{X}}}')&=&-j_*((c_1 {\cal O}_{\xmode{\Bbb P\kern1pt} N^*}(1))\cdot{\tilde{\lowercase{X}}} \cdot {\tilde{\lowercase{X}}}')
\\
(f^*y) \cdot (j_*{\tilde{\lowercase{X}}}) & = & j_*((g^*i^*y) \cdot {\tilde{\lowercase{X}}}).
\end{eqnarray*}
\block{The intersection ring of an iterated curve blow-up}\label{iterate}
\par\indent\indent The result of this section is:
\begin{theorem}\label{iteration}
Let $Y_0 = \P3$. Let $C_0 \subset Y_0$ be a nonsingular curve of degree $d$
and genus $g$. Let $Y_1$ be the blow-up of $Y_0$ along $C_0$. Choose a
smooth curve $C_1$ which lies on the exceptional divisor $E_1 \subset Y_1$
and which meets each ruling on $E_1$ exactly once. Let $Y_2$ be the
blow-up of $Y_1$ along $C_1$. Iterate this process: $Y_{k+1}$ is obtained
by blowing up a smooth curve $C_k \subset E_k \subset Y_k$. We assume that $C_k$
meets each ruling on $E_k$ exactly once and that for all $k \geq 2$,
$C_k \not= E_k \cap E_{k-1,k}$, where $E_{k-1,k} \subset Y_k$ denotes the
strict transform of $E_{k-1}$. Let $H \subset Y_0$ be a plane.
Let ${\mathbf{\lowercase{H}}} = [H] \in A^1(Y_0)$. Let ${\mathbf{\lowercase{E}}}_k = [E_k] \in A^1(Y_k)$.
Let ${\mathbf{\lowercase{R}}}_k \in A^1(E_k)$ denote the class of a ruling,
which we identify with its image in $A^2(Y_k)$.
Identify ${\mathbf{\lowercase{H}}}$, ${\mathbf{\lowercase{E}}}_k$ and ${\mathbf{\lowercase{R}}}_k$ with their images
in the intersection ring $A^*(Y_n)$ of the $\th{n}$ iterated blow-up $Y_n$.
Then $A^k(Y_n)$ has as a basis:
$$[Y_n]\ \ (k=0);\ {\mathbf{\lowercase{H}}}, \vec\lbE1n\ \ (k=1);\ {\mathbf{\lowercase{H}}}^2, \vec\lbR1n\ \ (k=2);
\ 1\ \ (k=3).$%
$This information, together with the following multiplication rules, completely
describe $A^*(Y_n)$ as a graded ring: ${\mathbf{\lowercase{H}}}^3 = 1$, ${\mathbf{\lowercase{H}}} \cdot {\mathbf{\lowercase{R}}}_k = 0$,
${\mathbf{\lowercase{H}}}^2 \cdot {\mathbf{\lowercase{E}}}_k = 0$, ${\mathbf{\lowercase{E}}}_i \cdot {\mathbf{\lowercase{R}}}_j = -\delta_{i,j}$,
${\mathbf{\lowercase{H}}} \cdot {\mathbf{\lowercase{E}}}_k = d{\mathbf{\lowercase{R}}}_k$,
${\mathbf{\lowercase{E}}}_i \cdot {\mathbf{\lowercase{E}}}_j = -\beta_i{\mathbf{\lowercase{R}}}_j\ \ (\hbox{if } i < j)$,
$${\mathbf{\lowercase{E}}}_k^2 = -d{\mathbf{\lowercase{H}}}^2 - \alpha_{k-1}{\mathbf{\lowercase{R}}}_k - \sum_{i=1}^{k-1} \beta_i{\mathbf{\lowercase{R}}}_i,$%
$where $\alpha_k$ is determined by $[C_k] = c_1{\cal O}_{E_k}(1) - \alpha_k{\mathbf{\lowercase{R}}}_k$
in $A^1(E_k)$, for $k \geq 1$, $\alpha_0 = 2-2g-4d$, and
$\beta_k = \alpha_{k-1} - \alpha_k$, for each $k \geq 1$.
\end{theorem}
We note the following generalization and conceptual reformulation of
\pref{iteration}, whose proof is omitted. It will not be used again.
\begin{theorem}
Let $Y_0$ be a nonsingular complete three-fold. Let $C_0 \subset Y_0$ be a
nonsingular curve. Let $Y_1$ be the blow-up of $Y_0$ along $C_0$. Choose a
smooth curve $C_1$ which lies on the exceptional divisor $E_1 \subset Y_1$
and which meets each ruling on $E_1$ exactly once. Let $Y_2$ be the
blow-up of $Y_1$ along $C_1$. Iterate this process: $Y_{k+1}$ is obtained
by blowing up a smooth curve $C_k \subset E_k \subset Y_k$. We assume that $C_k$
meets each ruling on $E_k$ exactly once and that for all $k \geq 2$,
$C_k \not= E_k \cap E_{k-1,k}$, where $E_{k-1,k} \subset Y_k$ denotes the
strict transform of $E_{k-1}$. Let ${\mathbf{\lowercase{E}}}_k = [E_k] \in A^1(Y_k)$.
Let ${\mathbf{\lowercase{R}}}_k \in A^1(E_k)$ denote the class of a ruling,
which we identify with its image in $A^2(Y_k)$.
Identify ${\mathbf{\lowercase{E}}}_k$ and ${\mathbf{\lowercase{R}}}_k$ with their images
in the intersection ring $A^*(Y_n)$ of the $\th{n}$ iterated blow-up $Y_n$.
Then $A^*(Y_n)$ is the graded $A^*(Y_0)$-algebra generated by
$\vec\lbE1n$ (degree $1$) and $\vec\lbR1n$ (degree $2$), modulo the relations:
$A^1(Y_0) \cdot {\mathbf{\lowercase{R}}}_k = 0$, $A^2(Y_0) \cdot {\mathbf{\lowercase{E}}}_k = 0$,
${\mathbf{\lowercase{E}}}_i \cdot {\mathbf{\lowercase{R}}}_j = -\delta_{i,j}$,
${\mathbf{\lowercase{H}}} \cdot {\mathbf{\lowercase{E}}}_k = ({\mathbf{\lowercase{H}}} \cdot C_0){\mathbf{\lowercase{R}}}_k$\ (for all ${\mathbf{\lowercase{H}}} \in A^1(Y_0)$),
${\mathbf{\lowercase{E}}}_i \cdot {\mathbf{\lowercase{E}}}_j = -\beta_i{\mathbf{\lowercase{R}}}_j\ \ (\hbox{if } i < j)$,
$${\mathbf{\lowercase{E}}}_k^2 = - [C_0] - \alpha_{k-1}{\mathbf{\lowercase{R}}}_k - \sum_{i=1}^{k-1} \beta_i{\mathbf{\lowercase{R}}}_i,$%
$where $\alpha_k$ is determined by $[C_k] = c_1{\cal O}_{E_k}(1) - \alpha_k{\mathbf{\lowercase{R}}}_k$
in $A^1(E_k)$, for $k \geq 1$, $\alpha_0 = \deg(N_{C_0}^*)$, and
$\beta_k = \alpha_{k-1} - \alpha_k$, for each $k \geq 1$.
\end{theorem}
The remainder of this section breaks up into two parts. First we
introduce various notations and conventions which we will use in the proof and
in subsequent sections. Then we prove \pref{iteration}.
There are group homomorphisms \mapx[[ A^i(E_k) || A^{i+1}(Y_k) ]] and
injective ring homomorphisms:
\diagramx{A^*(Y_0)&\mapE{}&A^*(Y_1)&\mapE{}&\cdots&\mapE{}&A^*(Y_n).%
}We systematically identify various elements with their images, via these maps.
Since the latter maps are ring homomorphisms, it is not necessary to
distinguish
between multiplication in $A^*(Y_i)$ and $A^*(Y_j)$, for any $i,j$. On the
other hand, since
the maps \mapx[[ A^i(E_k) || A^{i+1}(Y_k) ]] are not ring homomorphisms, it
is necessary to distinguish between multiplication in $A^*(Y_k)$ and
$A^*(E_k)$.
We do this by using a dot ($\cdot$) to denote multiplication in $A^*(Y_k)$
and brackets ($\inn,$) to denote multiplication in $A^*(E_k)$. No problems
are introduced by the fact that $k$ does not occur explicitly in the bracket
notation.
Let ${\mathbf{\lowercase{C}}}_k = [C_k]$. This is an element of $A^2(Y_k)$, and it is an
element of $A^1(E_k)$ if $k \geq 1$.
Let ${\mathbf{\lowercase{D}}}_k = c_1{\cal O}_{E_k}(1)$. It is an element of $A^1(E_k)$.
For $k\leq n$,
let $E_{k,n} \subset Y_n$ denote the strict transform of $E_k$. In $A^1(Y_n)$
we have ${\mathbf{\lowercase{E}}}_k = [E_{k,n}] \many+ [E_{n,n}]$. (This depends on our
assumption that $C_k \not= E_k \cap E_{k-1,k}$.) {\it In particular, the
reader should observe the following insidious source of error:
${\mathbf{\lowercase{E}}}_k \not= [E_{k,n}]$.} This same sort of error applies to other cycles
which we shall discuss.
In this section we do not fix a particular ruling $R_k \subset E_k$. We do
so in the next section. Having made such a choice, one can then discuss the
strict transform $R_{k,n} \subset Y_n$ of $R_k$.
Let $N_k$ be the normal bundle of $C_k$ in $Y_k$. Then
$E_k \cong \xmode{\Bbb P\kern1pt}(N_{k-1}^*)$.
For each $k = 0, \ldots, n$, we let $\alpha'_k = \deg(N_k^*)$.
For each $k = 1, \ldots, n$, we let $\beta'_k = \alpha'_{k-1} - \alpha_k$.
(We will show that $\alpha_k = \alpha'_k$ and hence that $\beta_k = \beta'_k$.)
\begin{proofnodot}
(of \ref{iteration}).
We make repeated use of the results of \S\ref{oneblow}, without explicitly
referring to them. The abelian group structure of $A^*(Y_n)$ and
the assertions that ${\mathbf{\lowercase{H}}}^3 = 1$, ${\mathbf{\lowercase{H}}} \cdot {\mathbf{\lowercase{R}}}_k = 0$,
${\mathbf{\lowercase{H}}}^2 \cdot {\mathbf{\lowercase{E}}}_k = 0$, and ${\mathbf{\lowercase{E}}}_i \cdot {\mathbf{\lowercase{R}}}_j = -\delta_{i,j}$ are
left to the reader.
We compute ${\mathbf{\lowercase{H}}} \cdot {\mathbf{\lowercase{E}}}_i$. Let $\mu_i = {\mathbf{\lowercase{H}}} \cdot {\mathbf{\lowercase{C}}}_i$. Note that:
$${\mathbf{\lowercase{H}}} \cdot {\mathbf{\lowercase{E}}}_i\ =\ \mu_{i-1} {\mathbf{\lowercase{R}}}_i.$%
$We show that $\mu_k$ is independent of $k$, and in fact equals $d$. First one
checks that ${\mathbf{\lowercase{H}}} \cdot {\mathbf{\lowercase{C}}}_0 = d$. Now we have:
\begin{eqnarray*}
\mu_i & = & {\mathbf{\lowercase{H}}} \cdot {\mathbf{\lowercase{C}}}_i \\
& = & \inn{ ({\mathbf{\lowercase{H}}} \cdot {\mathbf{\lowercase{E}}}_i), {\mathbf{\lowercase{C}}}_i} \\
& = & \inn{\mu_{i-1} {\mathbf{\lowercase{R}}}_i, {\mathbf{\lowercase{C}}}_i} \\
& = & \inn{ \mu_{i-1} {\mathbf{\lowercase{R}}}_i, {\mathbf{\lowercase{D}}}_i - \alpha_i {\mathbf{\lowercase{R}}}_i } \\
& = & \mu_{i-1}.
\end{eqnarray*}
Hence $\mu_k = d$ for all $k$. Hence ${\mathbf{\lowercase{H}}} \cdot {\mathbf{\lowercase{C}}}_i\ =\ d$ and
${\mathbf{\lowercase{H}}} \cdot {\mathbf{\lowercase{E}}}_i\ =\ d{\mathbf{\lowercase{R}}}_i$.
We now work on showing that $\alpha_k = \alpha'_k$. In the process we
calculate
${\mathbf{\lowercase{E}}}_i \cdot {\mathbf{\lowercase{C}}}_j$ for all $i \leq j$, a result we shall need later.
We have:
$$\inn{{\mathbf{\lowercase{D}}}_i,{\mathbf{\lowercase{D}}}_i} = c_1(N_{i-1}^*) = \alpha'_{i-1}.$%
$Further:
\begin{eqnarray*}
{\mathbf{\lowercase{E}}}_i \cdot {\mathbf{\lowercase{C}}}_i & = & -\inn{{\mathbf{\lowercase{D}}}_i, {\mathbf{\lowercase{C}}}_i} \\
& = & -\inn{{\mathbf{\lowercase{D}}}_i, {\mathbf{\lowercase{D}}}_i - \alpha_i{\mathbf{\lowercase{R}}}_i} \\
& = & -(\alpha'_{i-1}-\alpha_i)\ =\ -\beta'_i.
\end{eqnarray*}
Using this we find:
\begin{eqnarray*}
{\mathbf{\lowercase{E}}}_i \cdot {\mathbf{\lowercase{C}}}_{i+1} & = & \inn{({\mathbf{\lowercase{E}}}_i \cdot
{\mathbf{\lowercase{C}}}_i){\mathbf{\lowercase{R}}}_{i+1},{\mathbf{\lowercase{C}}}_{i+1}}\\
& = & {\mathbf{\lowercase{E}}}_i \cdot {\mathbf{\lowercase{C}}}_i.
\end{eqnarray*}
Continuing in this manner, the reader may verify that for $i \leq j$,
${\mathbf{\lowercase{E}}}_i \cdot {\mathbf{\lowercase{C}}}_j = -\beta'_i$.
The class of the canonical divisor on $Y_i$ is given by:
$$[K_{Y_i}] = -4{\mathbf{\lowercase{H}}} + {\mathbf{\lowercase{E}}}_1 \many+ {\mathbf{\lowercase{E}}}_i.$%
$(This may be computed from the formula for the canonical divisor of a
blowup -- see\Lspace \Lcitemark 9\Rcitemark \Rspace{}\ p.\ 608.)
For all $i \geq 0$, we have:
\begin{eqnarray*}\label{quirkalpha}
\alpha'_i & = & c_1 N_i^* \\
& = & -c_1 \det(N_i)\ =\ -[[K_{C_i}] - [K_{Y_i}]|_{C_i}] \\
& = & -[[K_{C_i}] - [K_{Y_i}] \ \cdot {\mathbf{\lowercase{C}}}_i] \\
& = & -[2g-2 - (-4{\mathbf{\lowercase{H}}} + {\mathbf{\lowercase{E}}}_1 \many+ {\mathbf{\lowercase{E}}}_i) \cdot {\mathbf{\lowercase{C}}}_i] \\
& = & -[2g-2 + 4d + \beta'_1 \many+ \beta'_i].
\end{eqnarray*}
{}From this, and from the definition of the $\beta$'s and the $\alpha$'s,
we conclude:
$$\alpha'_k = \alpha_k\ \ \hbox{for all } k \geq 0.$%
$
\par\indent For $i < j$,
$${\mathbf{\lowercase{E}}}_i \cdot {\mathbf{\lowercase{E}}}_j = ({\mathbf{\lowercase{E}}}_i \cdot {\mathbf{\lowercase{C}}}_{j-1}){\mathbf{\lowercase{R}}}_j,$%
$so we obtain the formula ${\mathbf{\lowercase{E}}}_i \cdot {\mathbf{\lowercase{E}}}_j = -\beta_i{\mathbf{\lowercase{R}}}_j$.
We proceed to calculate ${\mathbf{\lowercase{E}}}_k^2$. By the definition of the map $\delta$
given in \S\ref{oneblow}, we have:
\begin{eqnarray*}
{\mathbf{\lowercase{D}}}_i & = & \alpha_{i-1}{\mathbf{\lowercase{R}}}_i + {\mathbf{\lowercase{C}}}_{i-1}\ \ \ (i \geq 1).
\end{eqnarray*}
Continuing to calculate, we find:
\begin{eqnarray*}\label{quirkC}
{\mathbf{\lowercase{C}}}_i & = & {\mathbf{\lowercase{D}}}_i - \alpha_i {\mathbf{\lowercase{R}}}_i\ \ \ (i \geq 1) \\
{\mathbf{\lowercase{D}}}_i & = & \alpha_{i-1}{\mathbf{\lowercase{R}}}_i + {\mathbf{\lowercase{D}}}_{i-1} - \alpha_{i-1}{\mathbf{\lowercase{R}}}_{i-1}
\ \ \ (i \geq 2) \\
{\mathbf{\lowercase{D}}}_i - {\mathbf{\lowercase{D}}}_{i-1} & = & \alpha_{i-1}{\mathbf{\lowercase{R}}}_i - \alpha_{i-1}{\mathbf{\lowercase{R}}}_{i-1}
\ \ \ (i \geq 2) \\
{\mathbf{\lowercase{D}}}_1 & = & \alpha_0 {\mathbf{\lowercase{R}}}_1 + d{\mathbf{\lowercase{H}}}^2 \\
{\mathbf{\lowercase{D}}}_k & = & d{\mathbf{\lowercase{H}}}^2 + \left( \sum_{i=1}^{k-1} (\alpha_{i-1} - \alpha_i)
{\mathbf{\lowercase{R}}}_i \right) + \alpha_{k-1}{\mathbf{\lowercase{R}}}_k\ \ \ (k \geq 1) \\
{\mathbf{\lowercase{E}}}_k^2 & = & -{\mathbf{\lowercase{D}}}_k\ \ \ (k \geq 1) \\
& = & - \left[ d{\mathbf{\lowercase{H}}}^2 + \left( \sum_{i=1}^{k-1} \beta_i{\mathbf{\lowercase{R}}}_i \right)
+ \alpha_{k-1}{\mathbf{\lowercase{R}}}_k\right].
\end{eqnarray*}
\vspace*{-0.25in}
{\hfill$\square$}
\end{proofnodot}
\block{The strict transform of a ruling}
\par\indent\indent The results of this section will be used in the proof of
theorem II \pref{thmII}.
The notations introduced in \S\ref{iterate} remain in effect in this section.
Fix a particular ruling $R_k \subset E_k$, where $1 \leq k \leq n$. We compute
the class of $R_{k,n}$ in $A^2(Y_n)$. A priori, this is a $\xmode{\Bbb Z}$-linear
combination of ${\mathbf{\lowercase{H}}}^2, \vec\lbR1n$, which depends on the particular
choice of $R_k$.
We use the term {\it graph\/} to mean an undirected graph, which we shall
formally view as a reflexive, symmetric relation.
By an {\it augmented graph}, we shall mean a graph, together with a mapping
from the set of vertices of that graph to $\xmode{\Bbb Z}$. If the augmentation map is
injective, we shall refer to the graph as a {\it labeled graph}, with the
obvious connotations.
Let $\Gamma$ be a labeled graph, which we suppose has a maximum vertex $m$.
We define various labeled graphs, coming from $\Gamma$, with maximum vertex
$m+1$.
First we define a labeled graph $\Gamma^+$ by
$\mathop{\operatoratfont vertices}\nolimits(\Gamma^+) = \mathop{\operatoratfont vertices}\nolimits(\Gamma) \cup \setof{m+1}$ and
$\mathop{\operatoratfont edges}\nolimits(\Gamma^+) = \mathop{\operatoratfont edges}\nolimits(\Gamma) \cup \setof{\mathop{\operatoratfont edge}\nolimits(m,m+1)}$.
Now suppose that $\mathop{\operatoratfont edge}\nolimits(l,m) \in \Gamma$. We define a graph
$\Gamma^l$ by $\mathop{\operatoratfont vertices}\nolimits(\Gamma^l) = \mathop{\operatoratfont vertices}\nolimits(\Gamma) \cup \setof{m+1}$ and
$$\mathop{\operatoratfont edges}\nolimits(\Gamma^l) = \mathop{\operatoratfont edges}\nolimits(\Gamma) \cup \setof{\mathop{\operatoratfont edge}\nolimits(l, m+1), \mathop{\operatoratfont edge}\nolimits(m, m+1)}
- \setof{\mathop{\operatoratfont edge}\nolimits(l,m)}.$%
$Intuitively, this construction may be thought of as adding a vertex $(m+1)$
``in the middle'' of the edge from $l$ to $m$.
\begin{definition}
A {\it standard operation\/} is an operation on a labeled graph of the
form $\Gamma \mapsto \Gamma^+$ or $\Gamma \mapsto \Gamma^l$ for some $l$.
A {\it standard labeled graph\/} is a labeled graph obtained from
a one-vertex labeled graph by a finite sequence of standard operations.
\end{definition}
It is not hard to see that given a standard labeled graph, one may
compute the last standard operation which was performed, and thence undo
that operation. It follows that:
\begin{prop}\label{unique-operations}
Let $G$ be a standard labeled graph. Then there is a unique sequence of
standard operations which gives rise to $G$.
\end{prop}
Fix integers $k$ and $m$ with $1 \leq k \leq m \leq n$.
Let $R_k \subset E_k$ be a ruling.
We will show how to associate a certain standard labeled graph
$\Gamma_m(R_k)$ to $R_k$, in such a way that
$[R_{k,m}] \in A^2(Y_m)$ depends only on $\Gamma_m(R_k)$.
To do this, consider the set of all curves $H \subset Y_m$ which are the
strict transforms of some ruling $R_l$ on $E_l$, for some $l$ with
$k \leq l \leq m$. To each such $H$, we may associate an integer, namely $l$.
It may be that $H \subset E_{l',m}$, for some $l'$ with $l' \not= l$ and
$k \leq l' \leq m$, but this does not matter to us. The set of all such
curves $H$ may be viewed as the vertices of a graph $\Gamma_m(k)$: two distinct
vertices are connected by an edge if and only if\ the corresponding two curves on $Y_m$
meet. There is an augmentation on $\Gamma_m(k)$ given by $H \mapsto l$ as
above.
Define $\Gamma_m(R_k)$ to be the maximal connected subgraph of $\Gamma_m(k)$
which contains $R_{k,m}$. The augmentation on $\Gamma_m(k)$ induces an
augmentation on $\Gamma_m(R_k)$. We shall prove shortly \pref{snooker} that
$\Gamma_m(R_k)$ is a labeled graph, and that in fact it is a standard
labeled graph.
\begin{lemma}\label{lemma1}
If two distinct curves $H_1, H_2 \in \Gamma_m(k)$ meet, then they meet at a
unique point, and they meet transversally.
\end{lemma}
\begin{proof}
What we need to show is that if $p \leq q$ are integers
($k \leq p,q \leq m$), and if
$R_p \subset E_p$ and $R_q \subset E_q$ are rulings, and if $R_{p,m}$ meets
$R_{q,m}$ (but $R_{p,m} \not= R_{q,m}$), then in fact $R_{p,m}$ meets
$R_{q,m}$ at a unique point and they
do so transversally. It suffices to show that $R_{p,q}$ meets
$R_q$ in this way. We may assume that $p < q$.
Indeed if $R_{p,q}$ met $R_q$ at more than one point, or if they did not
meet transversally, then the image of $R_{p,q}$ under the map
\mapx[[ Y_q || Y_p ]] would be singular, because this map contracts $R_q$.
{\hfill$\square$}
\end{proof}
\begin{lemma}\label{morsel}
No three distinct curves $H_1, H_2, H_3 \in \Gamma_m(k)$ meet at a common
point.
\end{lemma}
\begin{proof}
We may reduce to showing the following: if $p < q < r$
($k \leq p,q,r \leq m$) and
$R_p \subset E_p$, $R_q \subset E_q$, and $R_r \subset E_r$ are rulings, then
$R_{p,r} \cap R_{q,r} \cap R_r = \varnothing$. We proceed by contradiction:
let $x \in R_{p,r} \cap R_{q,r} \cap R_r$.
We may assume that $r$ is minimal with respect to this assertion.
Let $y$ be the image of $x$ under the map \mapx[[ Y_r || Y_{r-1} ]]. Then
$y \in R_{p,r-1} \cap R_{q,r-1} \cap C_{r-1}$. If $q < r-1$, then
for some ruling $R_{r-1} \subset E_{r-1}$, we have
$y \in R_{p,r-1} \cap R_{q,r-1} \cap R_{r-1}$, thereby contradicting the
minimality of $r$. Hence we may assume that $q = r-1$.
To prove the lemma, it suffices to show that
$\T_y(R_{p,r-1}) + \T_y(R_{q,r-1}) + \T_y(C_{r-1}) = \T_y(Y_{r-1})$.
Since by \pref{lemma1} $R_{p,r-1}$ meets $R_{q,r-1}$ at a unique point,
this will imply that $R_{p,r} \cap R_{q,r} = \varnothing$, thereby yielding
a contradiction. Substituting $q = r-1$, we must show:
$$\T_y(R_{p,r-1}) + \T_y(R_{r-1}) + \T_y(C_{r-1}) = \T_y(Y_{r-1}).\eqno(*)$%
$The curves $R_{r-1}$ and $C_{r-1}$ meet transversally at $y$, tangentially
spanning $\T_y(E_{r-1})$. Therefore, to prove $(*)$, and hence the lemma,
it suffices to show that $R_{p,r-1}$ meets $E_{r-1}$ transversally. This
may be deduced by repeated application of the following two facts, applied
to integers $t$ with $p \leq t \leq r-2$:
\begin{itemize}
\item if $R_{p,t}$ meets $C_t$ transversally (on $Y_t$), then
$R_{p,t+1}$ meets $E_{t+1}$ transversally (on $Y_{t+1}$);
\item if $R_{p,t}$ meets $E_t$ transversally (on $Y_t$), then
$R_{p,t}$ meets any smooth curve on $E_t$ transversally (if at all).
{\hfill$\square$}
\end{itemize}
\end{proof}
\begin{prop}\label{snooker}
Let $k, m \in \xmode{\Bbb Z}$, with $1 \leq k \leq m \leq n$.
Let $R_k \subset E_k$ be a ruling. Let
$\Gamma = \Gamma_m(R_k)$. Then $\Gamma$ is a standard labeled graph with
vertices $[k,m] \cap \xmode{\Bbb Z}$, and provided that $m < n$,
$\Gamma_{m+1}(R_k)$ is obtained from $\Gamma$ by a single standard operation.
\end{prop}
\begin{proof}
By induction, we may assume that $\Gamma$ is a standard labeled graph with
vertices $[k,m] \cap \xmode{\Bbb Z}$. For each $q$ between $k$ and $m$, let
let $R_q \subset E_q$ be the ruling corresponding to the vertex $q \in \Gamma$.
First we show $(*)$ that if $l$ is such that $k \leq l < m$ and $R_{l,m}$ meets
$C_m$, then in fact $R_{l,m}$, $R_m$, and $C_m$ meet at a common point.
Suppose otherwise: $R_{l,m} \cap R_m \cap C_m = \varnothing$. We will obtain
a contradiction. We may choose $m$ to be as small as possible. There are
two cases.
Case (a). We have $l = m-1$. Since $R_{m-1}$ meets $C_{m-1}$ transversally
at a single point, $R_{m-1,m}$ meets $E_m$ at a single point. Since $\Gamma$
is a standard labeled graph,
it is clear that $R_{m-1,m}$ meets $R_m$. Since $R_{m-1,m}$ meets $C_m$,
we see that $R_{m-1,m}$ meets $E_m$ at two distinct points: contradiction.
This proves case (a).
Case (b). We have $l < m-1$. Since $R_{l,m}$ meets $C_m$
(and a fortiori $R_{l,m}$ meets $E_m$), it follows that $R_{l,m-1}$ meets
$C_{m-1}$. By the minimality of $m$,
$R_{l,m-1} \cap R_{m-1} \cap C_{m-1} \not= \varnothing$.
It follows that $R_{l,m}$ and $R_{m-1,m}$ meet a common ruling on $E_m$.
Since $R_{m-1,m}$ meets $R_m$, it is clear that this ruling must be $R_m$.
Hence $R_{l,m}$ meets $R_m$.
Thus $R_{l,m}$ meets both $R_m$ and $C_m$, but the three curves do not
meet at a common point. Hence $R_{l,m}$ meets two distinct rulings on
$E_m$. Hence
$R_{l,m-1}$ meets $C_{m-1}$ at $\geq 2$ distinct points, so $R_l$ meets
$C_l$ at $\geq 2$ distinct points: contradiction. This proves case (b),
and hence $(*)$.
We now proceed with the proof of the proposition. There are two cases.
Case I. For no $l$ (with $k \leq l < m$) is it true that
$R_{l,m}$, $R_m$ and $C_m$ have a point in common. We claim
that $\Gamma_{m+1}(R_k) = \Gamma^+$. It suffices to show that $R_m$ is the
unique curve in $\Gamma$ which meets $C_m$. This follows from $(*)$.
Case II. For some $l$ (with $k \leq l < m$), $R_{l,m}$, $R_m$ and $C_m$ have
a point (say $x$) in
common. We claim that $\Gamma_{m+1}(R_k) = \Gamma^l$. To prove this, we need
to prove two things:
\begin{romanlist}
\item for any $q$ such that $k \leq q < m$ and $q \not= l$,
$R_{q,m}$ does not meet $C_m$;
\item $\T_x(R_{l,m}) + \T_x(R_m) + \T_x(C_m) = \T_x(Y_m)$.
\end{romanlist}
The first assertion follows immediately from $(*)$ and from \pref{morsel}.
The second assertion follows from the proof of \pref{morsel}. {\hfill$\square$}
\end{proof}
\begin{lemma}\label{pine-cone}
Let $G$ be a standard labeled graph, constructed from the single vertex
graph \setof{k} by a sequence of standard operations $+, k^\br{p},\vec o1r$,
for some $p, r \geq 0$, such that $o_1 \not= k$.
Then $G - \setof{k}$ is a standard labeled graph,
which can be constructed from the single vertex graph \setof{k+1} by the
sequence of standard operations:
$$\cases{+^\br{p},\vec o1r,&if $p \geq 1;$\cr
+, \vec o2r,&if $p=0$ and $r \geq 1;$\cr
\varnothing,&if $p = r = 0$.}$$
\end{lemma}
The proof of this lemma is left to the reader.
Let $G$ be a standard labeled graph, with smallest vertex $k$, having at
least two vertices. It is clear that there is a unique $r > k$ such that
$\mathop{\operatoratfont edge}\nolimits(k,r)$ is in $G$. We define the {\it order\/} of $G$ to be $r-k$.
Moreover, if $G$ has order $p$, then $G$ is constructed from the single
vertex graph $\setof{k}$ by a sequence of operations which begins with
$+, k^\br{p-1}$, and whose next operation (if any) is not $k$.
Let $G$ be any standard labeled graph, with vertices $k, \ldots, m$.
We associate a function \mp[[ \mu_G || G || \xmode{\Bbb N} ]], defined by inducting
on $G$: if $G$ is a single vertex graph, then $\mu_G(k) = 1$. If $G$
is any standard labeled graph, then $\mu_{G^+}(j) = \mu_G(j)$ and
$\mu_{G^l}(j) = \mu_G(j)$ for all $j$ with $k \leq j \leq m$, and
$\mu_{G^+}(m+1) = \mu_G(m)$, $\mu_{G^l}(m+1) = \mu_G(m) + \mu_G(l)$.
The fact that $\mu_G$ is well-defined follows from \pref{unique-operations}.
\begin{prop}\label{spitup}
Let $G$ be a standard labeled graph, with smallest vertex $k$, having
at least two vertices. Then:
$$\mu_G = \mu_{\setof{k}} + \sum_{i=1}^{\mathop{\operatoratfont ord}\nolimits(G)}
\mu_{G - \setof{k, \ldots, k+i-1}},$%
$where the functions on the right hand side\ are viewed as functions on $G$, via
extension by zero.
\end{prop}
\begin{sketch}
Use \pref{pine-cone}. If $p = \mathop{\operatoratfont ord}\nolimits(G)$, then
$$G\ \longleftrightarrow\ +, k^\br{p-1}, *$%
$where $*$ is a sequence of standard operations (possibly empty), not beginning
with $k$. The case $p = 1$ is left to the reader. For $p \geq 2$:
$$G - \setof{k}\ \longleftrightarrow\ +^\br{p-1}, *$$
$$G - \setof{k,k+1}\ \longleftrightarrow\ +^\br{p-2}, *$%
$and so forth:
$$G - \setof{k,\ldots,k+p-2}\ \longleftrightarrow\ +, *$%
$$$G - \setof{k,\ldots,k+p-1}\ \longleftrightarrow\ *'$%
$where $*'$ can be determined from \pref{pine-cone}.
We compute $\mu$ in a special case, namely when $*$ is empty. Then:
$$G\ \longleftrightarrow\ (1,1,2,3,\ldots,p)$$
$$G - \setof{k}\ \longleftrightarrow\ (0^\br{1}, 1^\br{p})$$
$$\cdots$$
$$G - \setof{k,\ldots,k+p-1}\ \longleftrightarrow\ (0^\br{p}, 1^\br{1}),$%
$where the sequences on the right are $(\mu(k), \ldots, \mu(k+p))$. In this
case the proposition is clear. The general case is left to the reader. {\hfill$\square$}
\end{sketch}
\begin{corollary}
Fix an integer $k$ with $1 \leq k \leq n$. Let $R_k \subset E_k$ be a ruling.
If $k = n$ then $[R_{k,n}] = {\mathbf{\lowercase{R}}}_n$, and if
$k < n$, then there exists an integer $l$, with $k < l \leq n$, such that
$$[R_{k,n}] = {\mathbf{\lowercase{R}}}_k - \sum_{i=k+1}^l {\mathbf{\lowercase{R}}}_i.$$
\end{corollary}
\begin{sketch}
For each integer $m$ with $k \leq m \leq n$, let $R_m \subset E_m$ be the
ruling which enters into $\Gamma_n(R_k)$. For each integer $l$ with
$k \leq l \leq n$, write:
$$\mu_{\Gamma_n(R_l)} = (\vec bln).$%
$By considering the scheme-theoretic inverse image of $R_l$ under the
map \mapx[[ Y_n || Y_l ]], one can show that:
$${\mathbf{\lowercase{R}}}_l = b_l [R_{l,n}] \many+ b_n [R_{n,n}].$%
$The result then follows from \pref{spitup}. {\hfill$\square$}
\end{sketch}
\begin{corollary}
Let $H \subset Y_n$ be a cycle which is a sum of strict transforms of rulings.
Then $[H]$ is a positive $\xmode{\Bbb Z}$-linear combination of the classes:
$${\mathbf{\lowercase{R}}}_n, ({\mathbf{\lowercase{R}}}_{n-1}-{\mathbf{\lowercase{R}}}_n), ({\mathbf{\lowercase{R}}}_{n-2}-{\mathbf{\lowercase{R}}}_{n-1}-{\mathbf{\lowercase{R}}}_n), \ldots,
({\mathbf{\lowercase{R}}}_1 - {\mathbf{\lowercase{R}}}_2 - \cdots - {\mathbf{\lowercase{R}}}_n).$$
\end{corollary}
\begin{corollary}\label{snort-snort-snort}
Let $H \subset Y_n$ be a cycle which is a sum of strict transforms of rulings.
Then there exists integers $\vec a1n$ such that
$[H] = a_1 {\mathbf{\lowercase{R}}}_1 \many+ a_n {\mathbf{\lowercase{R}}}_n$ and for each integer $k$ with
$1 \leq k \leq n$, we have:
$$\left(\sum_{i=1}^{k-1} 2^{k-i-1} a_i\right) + a_k \geq 0.$$
\end{corollary}
\part{Application to set-theoretic complete intersections}
\block{Theorems I and II}\label{section9}
\par\indent\indent
Let $S, T \subset \xmode{\Bbb P\kern1pt}^3$ be surfaces of degrees $s$ and $t$, respectively.
Write $S_0 = S$, $T_0 = T$.
Assume that $S \cap T$ is set-theoretically a smooth curve $C = C_0$.
Let $d = \deg(C)$. Then $d | st$. Let $n = st/d$.
Assume that $C \not\IN \mathop{\operatoratfont Sing}\nolimits(S)$ and that $C \not\IN \mathop{\operatoratfont Sing}\nolimits(T)$.
Let $Y_0 = \xmode{\Bbb P\kern1pt}^3$. Let $Y_1$ be the blow-up of $Y_0$ along $C_0$. Let
$S_1 \subset Y_1$ be the strict transform of $S$. There is a unique curve
$C_1 \subset S_1$ which maps isomorphically onto $C_0$. Let $Y_2$ be the blow-up
of $Y_1$ along $C_1$. Iterate this process. This puts us in the situation
of \pref{iteration}. Let $p_i = p_i(S,C)$, for $i = 1, \ldots, n-1$.
For each $k = 1, \ldots, n$, let $S_k$ and $T_k$ denote the strict transforms
of $S$ and $T$ on $Y_k$. Since $C \not\IN \mathop{\operatoratfont Sing}\nolimits(S)$ and $C \not\IN \mathop{\operatoratfont Sing}\nolimits(T)$,
it follows that for each $k$ with $0 \leq k \leq n$, $S_k$ meets $T_k$ along
$C_k$ with multiplicity $n - k$. As consequences of this, we see that
$S_n \cap T_n$ is a union of strict transforms of rulings, and that
$[S_k] = s{\mathbf{\lowercase{H}}} - \sum_{i=1}^k {\mathbf{\lowercase{E}}}_i$, $[T_k] = t{\mathbf{\lowercase{H}}} - \sum_{i=1}^k {\mathbf{\lowercase{E}}}_i$.
\label{turnipgreen}
First we derive the formula:
$$\beta_k = ds + (2-4d-2g) - p_k.\eqno(*)$%
$We have $[S_k] \cdot {\mathbf{\lowercase{E}}}_k = {\mathbf{\lowercase{C}}}_k + p_k{\mathbf{\lowercase{R}}}_k$. (See \ref{goober-peas}.)
Combining this with
$[S_k] = s{\mathbf{\lowercase{H}}} - \sum_{i=1}^k {\mathbf{\lowercase{E}}}_i$, ${\mathbf{\lowercase{H}}} \cdot {\mathbf{\lowercase{E}}}_k = d{\mathbf{\lowercase{R}}}_k$
(from \ref{iteration}), and ${\mathbf{\lowercase{C}}}_k = {\mathbf{\lowercase{D}}}_k - \alpha_k{\mathbf{\lowercase{R}}}_k$
(from p.\ \pageref{quirkC}), we obtain:
$$ds{\mathbf{\lowercase{R}}}_k - \sum_{i=1}^k ({\mathbf{\lowercase{E}}}_i \cdot {\mathbf{\lowercase{E}}}_k) = {\mathbf{\lowercase{D}}}_k
- \alpha_k{\mathbf{\lowercase{R}}}_k + p_k{\mathbf{\lowercase{R}}}_k.$%
$Combine this with ${\mathbf{\lowercase{E}}}_i \cdot {\mathbf{\lowercase{E}}}_k = -\beta_i{\mathbf{\lowercase{R}}}_k$ (if $i < k$)
(from \ref{iteration}) and ${\mathbf{\lowercase{E}}}_k^2 = -{\mathbf{\lowercase{D}}}_k$ (from p.\ \pageref{quirkC})
to obtain:
$$ds{\mathbf{\lowercase{R}}}_k + (\svec\beta1{k-1}){\mathbf{\lowercase{R}}}_k = -\alpha_k{\mathbf{\lowercase{R}}}_k + p_k{\mathbf{\lowercase{R}}}_k.$%
$Combine this with the formula:
$$\alpha_k = (2-4d-2g) - (\svec\beta1k)\eqno(**)$%
$from p.\ \pageref{quirkalpha}, to obtain $(*)$.
Since $S_n \cap T_n$ is a union of strict transforms of rulings, it follows
from \pref{snort-snort-snort} that:
$$(s{\mathbf{\lowercase{H}}} - \sum_{i=1}^n {\mathbf{\lowercase{E}}}_i)(t{\mathbf{\lowercase{H}}} - \sum_{i=1}^n {\mathbf{\lowercase{E}}}_i)
= \sum_{l=1}^n a_l {\mathbf{\lowercase{R}}}_l, \eqno(*{*}*)$%
$for some integers $a_l$ such that for each $k$ with $1 \leq k \leq n$, we
have:
$$\left(\sum_{m=1}^{k-1} 2^{k-m-1} a_m\right) + a_k \geq 0.$%
$
We proceed to analyze the consequences of this. The left hand side of
$(*{*}*)$ equals:
$$ st{\mathbf{\lowercase{H}}}^2 - d(s+t)(\sum_{i=1}^n {\mathbf{\lowercase{R}}}_i)
- \sum_{1 \leq i < j \leq n} \beta_i {\mathbf{\lowercase{R}}}_j
- \sum_{1 \leq j < i \leq n} \beta_j {\mathbf{\lowercase{R}}}_i$$
$$ - \sum_{k=1}^n \left[ d{\mathbf{\lowercase{H}}}^2
+ \left( \sum_{i=1}^{k-1} \beta_i {\mathbf{\lowercase{R}}}_i \right)
+ \alpha_{k-1} {\mathbf{\lowercase{R}}}_k \right].$%
$
Then for each $m$ with $1 \leq m \leq n$:
$$-a_m = d(s+t) + 2\sum_{i=1}^{m-1} \beta_i + (n-m) \beta_m + \alpha_{m-1}.$%
$Substituting
$\alpha_k = (2-4d-2g) - (\beta_1 \many+ \beta_k)$, we obtain:
$$-a_m = d(s+t) + \sum_{i=1}^{m-1} \beta_i + (n-m)\beta_m + (2-4d-2g).$%
$
\par\indent In the special case where
$\mathop{\operatoratfont Sing}\nolimits(S) \cap \mathop{\operatoratfont Sing}\nolimits(T) = \varnothing$, we have
$a_m = 0$ for all $m$, with $1 \leq m \leq n$.
Now substitute $\beta_i = ds + (2-4d-2g) - p_i$. We obtain:
$$\left(\sum_{i=1}^{m-1} p_i\right) + (n-m)p_m = d[n(s-4) + t] + (2-2g)n.$%
$This implies:
\begin{theorem}\label{thmI}
{\bf (``I'')}
Let $C \subset \P3$ be a smooth curve of degree $d$ and genus $g$. Suppose that
$C = S \cap T$, where $S$ and $T$ are surfaces of degree $s$ and $t$
respectively. Assume that $\mathop{\operatoratfont Sing}\nolimits(S) \cap \mathop{\operatoratfont Sing}\nolimits(T) = \varnothing$.
Let $n = st/d$.
Let $p_i = p_i(S,C)$, for each $i = 1, \ldots, n-1$, as defined in
\S\ref{measure}. Then:
$$p_1 = \cdots = p_{n-1} = {1 \over n-1}
\left\{d[n(s-4)+t] + (2-2g)n\right\}.$$
\end{theorem}
\begin{example}\label{examplex-char-two}
If $s = t = 4$, $d = 4$, $g = 0$, we obtain $p_1 = p_2 = p_3 = 8$. This
can occur in characteristic two, at least. Indeed, let $(S,C)$ be as in
\pref{example-char-two}, and let $T$ be given by $z^4-xw^3=0$.
\end{example}
We now return to the general case. For each $k$ with $1 \leq k \leq n$,
we have:
\splitdisplay{\left(\sum_{m=1}^{k-1} 2^{k-m-1}
[d(s+t) + \sum_{i=1}^{m-1} \beta_i + (n-m) \beta_m + (2-4d-2g)] \right)%
}{+ [d(s+t) + \sum_{i=1}^{k-1} \beta_i + (n-k) \beta_k +(2-4d-2g)]\leq 0.%
}A simplification yields:
$$2^{k-1}[d(s+t-4)+2-2g]
+ \left( \sum_{i=1}^{k-1} 2^{k-i-1}(n-i+1) \beta_i \right)
+ (n-k) \beta_k \leq 0.$%
$Note that:
$$\sum_{i=1}^{k-1} 2^{k-i-1}(n-i+1) \ =\ (n-1)2^{k-1} + k - n.$%
$Substitute $\beta_i = ds + (2-4d-2g) - p_i$. We obtain:
\begin{theorem}\label{thmII}
{\bf (``II'')}
Let $C \subset \P3$ be a smooth curve of degree $d$ and genus $g$. Suppose that
$C = S \cap T$, where $S$ and $T$ are surfaces of degree $s$ and $t$
respectively. Assume that $C \not\IN \mathop{\operatoratfont Sing}\nolimits(S)$ and $C \not\IN \mathop{\operatoratfont Sing}\nolimits(T)$.
Let $n = st/d$.
Let $p_i = p_i(S,C)$, for each $i = 1, \ldots, n-1$, as defined in
\S\ref{measure}. Then for each $k = 1, \ldots, n-1$, we have:
$$\sum_{i=1}^{k-1} 2^{k-i-1}(n-i+1) p_i + (n-k)p_k
\geq 2^{k-1}\left\{dt + n[d(s-4)+2-2g]\right\}.$$
\end{theorem}
\begin{examples}
\
\begin{itemize}
\item $s = 2$, $t = 3$, $d = 3$, $g = 0$: then the theorem yields the
single inequality $p_1 \geq 1$;
\item $s = 2$, $t = 2$, $d = 1$, $g = 0$: as above the theorem yields
$p_1 \geq 1$;
\item $s = 4$, $t = 4$, $d = 4$, $g = 0$: the theorem yields three
inequalities:
\begin{romanlist}
\item $p_1 \geq 8$;
\item $2p_1 + p_2 \geq 24$;
\item $8p_1 + 3p_2 + p_3 \geq 96$.
\end{romanlist}
\end{itemize}
\end{examples}
\block{Theorems III, Q, and B}
\begin{theorem}\label{thmIII}
{\bf (``III'')}
Let $C \subset {\Bbb C}\kern1pt\P3$ be a smooth curve of degree $d$ and genus $g$ which is
the set-theoretic complete intersection\ of two surfaces $S$, $T$ of degrees $s$, $t$, respectively.
Let $n = st/d$. Let $p_k = p_k(S,C)$, for each $k \in \xmode{\Bbb N}$. Assume that
$S$ has only rational singularities. Assume that $C \not\IN \mathop{\operatoratfont Sing}\nolimits(T)$. Then:
$$\sum_{k=1}^\infty{1 \over k(k+1)}p_k \geq {d^2 \over s} + d(s-4) + 2-2g.$$
\end{theorem}
\begin{proof}
By (\Lcitemark 18\Rcitemark \ 1.1), we know that
$\Delta(S,C) = d^2/s + d(s-4) + 2-2g$. Apply \pref{rdp-one}. {\hfill$\square$}
\end{proof}
\begin{remark}
In the statement of \pref{thmIII}, we do not know if $\sum_{k=1}^\infty$
can be replaced by $\sum_{k=1}^{n-1}$. From (\ref{interesting}),
we see that this can at least be done if $(S,C)$ does not contain any
singularities of type $D_{n,n}$ (with $n$ odd).
\end{remark}
\begin{lemma}\label{bound-formula}
Let $S \subset {\Bbb C}\kern1pt\P3$ be a surface of degree $s$ having only rational
singularities.
Let \mp[[ \pi || {\tilde{S}} || S ]] be a minimal resolution. Let $N$ be the number
of exceptional curves on ${\tilde{S}}$. Then:
$$N \leq {s \over 3}(2s^2 - 6s + 7) - 1.$$
\end{lemma}
\begin{proof}
Clearly $N \leq \mathop{\operatoratfont rank}\nolimits \mathop{\operatoratfont NS}\nolimits({\tilde{S}}) - 1$. Also $\mathop{\operatoratfont rank}\nolimits \mathop{\operatoratfont NS}\nolimits({\tilde{S}}) \leq h^{1,1}({\tilde{S}})$,
so it suffices to show that $h^{1,1}({\tilde{S}}) = {s \over 3}(2s^2-6s+7)$. By
simultaneous resolution of rational double points
\Lcitemark 2\Rcitemark \Rspace{}, and deformation invariance of
Hodge numbers, we may reduce to showing that
$h^{1,1}(S) = {s \over 3}(2s^2-6s+7)$ if $S$ is itself nonsingular. We have:
\begin{eqnarray*}
h^{1,1}(S) & = & h^2(S,{\Bbb Q}\kern1pt) - 2h^2(S,{\cal O}_S) \\
& = & [\chi_{\operatoratfont top}(S) + 4h^1(S,{\cal O}_S) - 2] - 2h^2(S,{\cal O}_S).
\end{eqnarray*}
Using the fact that the top Chern class of the tangent bundle equals the Euler
characteristic (see e.g.{\ }\Lcitemark 1\Rcitemark \ 11.24, 20.10.6), and using
Riemann-Roch, we find:
$$\chi_{\operatoratfont top}(S) = c_2(S) = 12\chi(S) - c_1^2(S)
= 12(1-h^1(S,{\cal O}_S)+h^2(S,{\cal O}_S)) - (4-s)^2s.$%
$The formula for $h^{1,1}$ follows from $h^1(S,{\cal O}_S) = 0$ and
\formulaqed{h^2(S,{\cal O}_S) = {s-1 \choose 3}.}
\end{proof}
\begin{remark}
We do not know if the lemma remains valid if ${\Bbb C}\kern1pt$ is replaced by an
algebraically closed field of positive characteristic.
\end{remark}
\begin{lemma}\label{bungobungo}
Let $(p_k)_{k \in \xmode{\Bbb N}}$ be a sequence of nonnegative integers. Let $n$
be a nonnegative integer. Assume that:
\begin{romanlist}
\item $p_1 \leq 9 - {2\over5}n$
\item $p_1 \geq p_2 \geq p_3 \geq \cdots$
\item $\sum_{k=1}^\infty p_k \leq 19 - n$
\item $n/4 + \sum_{k=1}^\infty {1 \over k(k+1)} p_k \geq 6$.
\end{romanlist}
\par\noindent Then $n=0$ and
$(p_k) \in \setof{ (9,8,2), (9,9), (9,9,1) }.$
\end{lemma}
\begin{proof}
Constraints (i), (iii), and (iv) imply that
$$n/4 + \hbox{$1 \over 2$}\floor{9-\hbox{$2 \over 5$}n}
+ \hbox{$1 \over 6$}(19-n-\floor{9-\hbox{$2 \over 5$}n} )
\geq 6.$%
$It follows that $n \in \setof{0,2}$.
Suppose that $n=2$. Then the same constraints imply that
$$1/2 + \hbox{$1 \over 2$}p_1 + \hbox{$1 \over 6$}(17-p_1)
\geq 6,$%
$so $p_1 = 8$. Now we see that the left hand side\ of (iv) is maximized when
$(p_k) = (8,8,1)$. In that case, the left hand side\ of (iv)
is $5{11\over12}$: contradiction.
Hence $n=0$. Then
$$\hbox{$1 \over 2$}p_1 + \hbox{$1 \over 6$}(19-p_1) \geq 6,$%
$so $p_1 = 9$. If $p_2 \leq 7$,
then the sum in (iv) is bounded by the sum obtained when
$(p_k) = (9,7,3)$. This sum is $< 6$, so $p_2 \not\leq 7$. Hence
$p_2 \in \setof{8,9}$. Etc. {\hfill$\square$}
\end{proof}
\begin{prop}\label{kformula}
Let $C \subset {\Bbb C}\kern1pt\P3$ be a smooth curve of degree $d$ and genus $g$, which
lies on a surface $S \subset {\Bbb C}\kern1pt\P3$ of degree $s$. Assume that $C \not\IN
\mathop{\operatoratfont Sing}\nolimits(S)$.
Let $p_1 = p_1(S,C)$. Let $N$ be the normal bundle of $C$ in ${\Bbb C}\kern1pt{\Bbb P}^3$, and
let $l$ be the maximum degree of a sub-line-bundle
of $N$. Let $k = 3d + (2g-2) - l$. Then $p_1 \leq d(s-1) - k$.
\end{prop}
\begin{proof}
We use the notation of \pref{iteration}. We also use various facts from
\S\ref{section9}, which although apparently dependent on another surface $T$,
actually make sense in this context.
We have $\inn{{\mathbf{\lowercase{C}}}_1, {\mathbf{\lowercase{C}}}_1} \geq \deg(N) - 2l$.
Since $\deg(N) = 4d+2g-2$ and $k = 3d + (2g-2) - l$, we have:
$$\inn{{\mathbf{\lowercase{C}}}_1,{\mathbf{\lowercase{C}}}_1} \geq -2d+2-2g+2k.\eqno(\dag)$%
$Now ${\mathbf{\lowercase{C}}}_1 = {\mathbf{\lowercase{D}}}_1 - \alpha_1{\mathbf{\lowercase{R}}}_1$ in $A^1(E_1)$, and
$\inn{{\mathbf{\lowercase{D}}}_1,{\mathbf{\lowercase{D}}}_1} = 2-2g-4d$, so:
$$\inn{{\mathbf{\lowercase{C}}}_1,{\mathbf{\lowercase{C}}}_1}\ =\ \inn{{\mathbf{\lowercase{D}}}_1,{\mathbf{\lowercase{D}}}_1} - 2\alpha_1\ =
\ 2 - 2g - 4d - 2\alpha_1.$%
$Combining this with $(\dag)$, we obtain $d + \alpha_1 \leq -k$. The
formulas $(*)$ and $(**)$ from \S\ref{section9} imply that
$\alpha_1 = p_1 - ds$. Hence $p_1 \leq d(s-1)-k$. {\hfill$\square$}
\end{proof}
\begin{remark}
This result \pref{kformula} is a strengthening of the very elementary fact
that:
$$\abs{\mathop{\operatoratfont Sing}\nolimits(S) \cap C} \leq d(s-1).$$
\end{remark}
\begin{theorem}\label{thmQ}
{\bf (``Q'')}
Let $C \subset {\Bbb C}\kern1pt{\Bbb P}^3$ be a curve. Assume that $C = S \cap T$
set-theoretically for some surfaces $S$ and $T$. Assume that $S$ is normal.
Assume that $\deg(C) > \deg(S)$. Then $C$ is linearly normal.
\end{theorem}
\begin{proof}
To any Weil divisor $E$ on a normal surface $S$, one can associate a reflexive
${\cal O}_S$-module ${\cal O}_S(E)$. We recall the following result of Sakai from
\Lcitemark 29\Rcitemark \Rspace{}, which is a slightly less general version of
theorem
5.1 of that paper:
\begin{quote}
{\it Let $S$ be a normal projective surface. Let $D$ be a nef Weil divisor on
$S$ with $D^2 > 0$. Then $H^1(S, {\cal O}_S(-D)) = 0$.}
\end{quote}
Since the canonical map \mapx[[ H^0(\P3, {\cal O}_{\P3}(1)) || H^0(S, {\cal O}_S(1)) ]] is
surjective, it suffices to show that the canonical map
\mapx[[ H^0(S, {\cal O}_S(1)) || H^0(S, {\cal O}_C(1)) ]] is surjective. Let $H$ be
a hyperplane section of $S$. From the long exact sequence coming from
\sescomma{{\cal O}_S(H-C)}{{\cal O}_S(H)}{{\cal O}_C(H)%
}we see that it is sufficient to show that $H^1(S, {\cal O}_S(H-C)) = 0$.
Let $d = \deg(C)$. Let $s = \deg(S)$, $t = \deg(T)$, and let $n$ be the
multiplicity of intersection of $S$ with $T$ along $C$, $n = st/d$. Since
$d > s$, we have $t > n$. Hence $(t-n)H$ is a very ample Cartier
divisor. Since $n(C-H) \sim (t-n)H$, the theorem follows from Sakai's
result. {\hfill$\square$}
\end{proof}
\begin{corollary}
Let $C \subset {\Bbb C}\kern1pt{\Bbb P}^3$ be a smooth curve. Assume that $C$ is the set theoretic
complete intersection of two normal surfaces $S$ and $T$, with multiplicity
$\leq 3$. Then $C$ is linearly normal.
\end{corollary}
\begin{proof}
Using the notation of the proof of \pref{thmQ}, we are done
if either $s$ or $t$ is bigger than $n$. Otherwise, $d \leq 3$, and so
$C$ is linearly normal anyway. {\hfill$\square$}
\end{proof}
\begin{remark}
For the case of multiplicity $4$, we must have $C$ linearly normal, except
possibly for the case where $C$ is a rational quartic, which is the set-theoretic complete intersection\ of
two normal quartic surfaces. It is not known if this is possible.
\end{remark}
\begin{theorem}\label{thmB}
{\bf (``B'')}
Let $S, T \subset {\Bbb C}\kern1pt\P3$ be surfaces. Assume that $S \cap T$ is set-theoretically
a smooth curve. Assume that $\deg(S) = 4$ and that $S$ has only rational
singularities. Then $C$ is linearly normal.
\end{theorem}
\begin{proof}
Let $C$ have degree $d$ and genus $g$. By \pref{thmQ}, we may assume that\ $d=4$ and $g=0$.
By\Lspace \Lcitemark 18\Rcitemark \Rspace{}, we may assume that\ $\deg(T) \geq 4$.
Since $\deg(S) \leq \deg(T)$, we may assume that $C \not\IN \mathop{\operatoratfont Sing}\nolimits(T)$, as
follows. Suppose that $C \subset \mathop{\operatoratfont Sing}\nolimits(T)$.
Write $S = V(f)$, $T = V(g)$. Choose $h$ so that
$\deg(fh) = \deg(g)$, and so that $C \not\IN V(h)$. Then
$C \not\IN \mathop{\operatoratfont Sing}\nolimits(V(fh + g))$. Hence we may replace $T$ by $V(fh + g)$.
Write $(S,C) = (S',C') + (S'',C'')$, where
$$(S'',C'') = D_{n_1,n_1} \many+ D_{n_r,n_r},$%
$$\vec n1r$ are odd integers $\geq 5$, and $(S',C')$ is a configuration
which does not involve any such singularities. Let $p_i = p_i(S',C')$.
Let $n = \svec n1r$. We show that the hypotheses
of \pref{bungobungo} are satisfied.
We apply \pref{kformula}, using that fact\Lspace \Lcitemark 5\Rcitemark
\Rspace{} that
$l=7$, concluding that $p_1(S,C) \leq 9$. We have
$p_1(\sum D_{n_i,n_i}) = \sum (n_i-1) / 2)$ by \pref{fantastico}, and
$n_i - 1 \geq {4\over5} n_i$, so $p_1(\sum D_{n_i,n_i}) \geq {2\over5}n$.
Thus hypothesis (i) is satisfied. Hypothesis (ii) holds. Hypothesis
(iii) follows from \pref{bound-formula} and from the fact that $(S',C')$
contains no $D_{m,m}$ pairs with $m$ odd, $m \geq 5$, so that
$\mathop{\operatoratfont def \kern1pt}\nolimits(S',C') \geq 0$. To prove hypothesis (iv), we would like to use
(\ref{thmIII} = ``III''), but that is not quite good enough. By
\pref{rdp-one},
$$\sum_{k=1}^\infty {1\over k(k+1)}p_k \geq \Delta(S',C').$%
$Let $s = \deg(S) = 4$. By (\Lcitemark 18\Rcitemark \ 1.1), we know
that:
$$\Delta(S,C) = d^2/s + d(s-4) + 2 - 2g = 6.$%
$Then $\Delta(S',C') = \Delta(S,C) - \Delta(S'',C'')$. By
\pref{delta-formulas}, $\Delta(S'',C'') = n/4$. Hypothesis (iv) follows.
By \pref{bungobungo}, we conclude that $n = 0$ and that:
$$\mathop{\operatoratfont type}\nolimits(S,C) \in \setof{ (9,8,2), (9,9), (9,9,1) }.$%
$We will use \pref{fantastico}, \pref{potato-1}, and \pref{potato-2}.
Suppose that $\mathop{\operatoratfont type}\nolimits(S,C) = (9,9,1)$. Then for some $p \in C$,
$\mathop{\operatoratfont type}\nolimits(S,C)_p = (r,1,1)$ for some $r \geq 1$. The case $r > 1 $ is
impossible because $\mathop{\operatoratfont type}\nolimits(S,C) - \mathop{\operatoratfont type}\nolimits(S,C)_p = (9-r,9-1)$ and
$9-r \geq 9-1$. Hence $\mathop{\operatoratfont type}\nolimits(S,C)_p = (1,1,1)$. Hence $(S,C)_p = A_{3,1}$.
Since $\Sigma(S,C) \leq 19$ by \pref{bound-formula}, and since
$9+9+1 = 19$, we have $\mathop{\operatoratfont def \kern1pt}\nolimits(S,C) = 0$. Therefore,
since the other singularities of $S$ along $C$ have type
$(k,k)$ for some $k \leq 8$, we see that the other singularities must be
$A_{3k-1,k}$ for some $k \in \setof{1, \ldots, 8}$, depending on the
singular point. Amongst these, only $A_{2,1}$ has deficiency zero.
Hence $(S,C) = 8A_{2,1} + A_{3,1}$. Hence
$\Delta(S,C) = 8({2\over3}) + {3\over4} \not= 6$: contradiction.
Now suppose that $\mathop{\operatoratfont type}\nolimits(S,C) = (9,8,2)$. Then for some $p \in C$,
$$\mathop{\operatoratfont type}\nolimits(S,C)_p \in \setof{ (1,1,1), (2,1,1), (2,2,2) }.$%
$These types are realized by the singularities $A_{3,1}$, $A_{4,2}$,
and $A_{7,2}$, respectively, and by no others. Since $A_{7,2}$ has nonzero
deficiency, it can be excluded. Hence $(S,C)_p \in \setof{A_{3,1},A_{4,2}}$.
If $(S,C)_p = A_{4,2}$, we find (by analogy with the $(9,9,1)$ case) that
$(S,C) = 6A_{2,1} + A_{3,1} + A_{4,2}$. Hence $\Delta(S,C) =
6({2\over3}) + {3\over4} + {6\over5} \not= 6$: contradiction. If
$(S,C)_p = A_{3,1}$, then we may assume that\ $(S,C) = 2A_{3,1} + \hbox{other}$, where
the ``other'' part must have
type $(7,6)$. The only zero-deficiency rational double point\ configuration which realizes
this type is $A_{1,1} + 6A_{2,1}$. Hence
$(S,C) = A_{1,1} + 6A_{2,1} + 2A_{3,1}$. By\Lspace \Lcitemark 24\Rcitemark
\Rspace{},
the sum of the contributions of the singularities must not exceed
$(2/3)\deg(S)(\deg(S)-1)^2 = 24$, where each singularity $p$ contributes
$e(E) - 1/\abs{G}$, $e(E)$ is the topological Euler characteristic of the
exceptional fiber in the minimal resolution of $p$, and $G$ is the order of the
group which defines $p$ as a quotient singularity. In particular, an
$A_n$ singularity contributes $(n+1) - (n+1)^{-1}$. Then the sum of the
contributions is $25$: contradiction.
Suppose that $\mathop{\operatoratfont type}\nolimits(S,C) = (9,9)$. Then each singularity of $S$ along $C$
must have type $(k,k)$ for some $k$, depending on the singular point. Hence
$(S,C)$ must be built up from $E_{6,1}$ and $A_{3k-1,k}$ for various $k$.
Since $\mathop{\operatoratfont def \kern1pt}\nolimits(E_{6,1}) = 2$, we may rule out that case. In fact, there are
only two configurations with deficiency $\leq 1$: either
$(S,C) = 9A_{2,1}$ or else $(S,C) = 7A_{2,1} + A_{5,2}$. In both cases,
$\mathop{\operatoratfont order}\nolimits(S,C) = 3$. Hence we may assume that $\deg(T) = 3$. By
\Lcitemark 18\Rcitemark \Rspace{}, we know that this is impossible. {\hfill$\square$}
\end{proof}
\block{Theorem A}
\begin{lemma}\label{murky-algebra}
Let $s,t,d,g \in \xmode{\Bbb Z}$. Assume that $t \geq s \geq 4$, $d \geq 1$,
and $g \geq 0$. Assume that $d \vert st$. Let $n = st/d$. Assume
that $n \geq 2$. Let $r = d[n(s-4)+t] + (2-2g)n$. Assume that
$r \leq {s \over 3}(2s^2-6s+7)-1$. Then $d \leq g + 3$.
\end{lemma}
\begin{proof}
We assume that $d \geq g + 4$, working toward a contradiction. We have
$2 - 2g \geq 10 - 2d$, so:
$$d[n(s-4) + t] + (10-2d)n \leq {s\over3}(2s^2-6s+7) - 1.\eqno(*)$%
$
\par First suppose that $s = 4$. Then $t \geq 4$, $t \geq d/2$, and
$dt + (10-2d)n \leq 19$. Substituting $n = st/d = 4t/d$ and simplifying,
we obtain:
$$(d^2 - 8d + 40)t \leq 19d.\eqno(\dag)$%
$Since $t \geq d/2$, we have $(d^2 - 8d + 40)(1/2) \leq 19$. It follows
that $d \leq 7$. Hence $d \in \setof{4,5,6,7}$. In each case, $(\dag)$
gives us an upper bound $t_{\rm max}$ for $t$:
{\renewcommand{\arraystretch}{1.0}
\vspace*{0.1in}
\centerline{
\begin{tabular}{||c|c||} \hline
$d$ & $t_{\rm max}$\\ \hline
$4$ & $3$\\ \hline
$5$ & $3$\\ \hline
$6$ & $4$\\ \hline
$7$ & $4$\\ \hline
\end{tabular} }
\vspace*{0.1in}}
The cases $d \in \setof{4,5}$ contradict $t \geq 4$. In case
$d \in \setof{6,7}$, we have $t = 4$, which contradicts our assumption that
$d | st$. Hence $s > 4$.
Now suppose that $s = 5$. Then $t \geq 5$, $t \geq 2d/5$, and
$d(t+n) + (10-2d)n \leq 44$. Substituting $n = st/d = 5t/d$ and simplifying,
we obtain:
$$t \leq 44d / (d^2 - 5d + 50).$%
$This implies that $t < 5$: contradiction. Hence $s \geq 6$.
Since $n = st/d \geq s^2/d$, it follows from $(*)$ that:
$$d[{s^2 \over d}(s-6) + s] + 10{s^2 \over d} \leq {s \over 3}(2s^2-6s+7).$%
$This implies that:
$$s(s-6) + d + 10{s \over d} \leq {1 \over 3}(2s^2 - 6s + 7)\eqno(**)$%
$and in particular that:
$$s(s-6) \leq {1 \over 3}(2s^2 - 6s + 7).$%
$It follows that $s \leq 12$. Hence $6 \leq s \leq 12$. If $s = 12$,
$(**)$ implies that $d + 120/d \leq 2{1\over3}$. This is absurd. In a similar
manner, one may eliminate the cases where $6 \leq s \leq 11$. {\hfill$\square$}
\end{proof}
\begin{theorem}\label{thmA}
{\bf (``A'')}
Let $S, T \subset {\Bbb C}\kern1pt\xmode{\Bbb P\kern1pt}^3$ be surfaces.
Assume that $S$ has only rational singularities.
Assume that $\deg(S) \leq \deg(T)$.
Assume that $S \cap T$ is set-theoretically a smooth curve $C$ of
degree $d$ and genus $g$.
Assume that $\mathop{\operatoratfont Sing}\nolimits(S) \cap \mathop{\operatoratfont Sing}\nolimits(T) = \varnothing$. Then $d \leq g + 3$.
\end{theorem}
\begin{proof}
Let $s = \deg(S)$, $t = \deg(T)$, $n = st/d$.
Let $p_i = p_i(S,C)$. By (\ref{thmI} = ``I''), we have:
$$p_1 = \cdots = p_{n-1} = {1 \over n-1}
\left\{d[n(s-4)+t] + (2-2g)n\right\}.$$
We show that $S$ has no singularities of type $D_{t,t}$
(with $t$ odd, $t \geq 5$), lying on $C$. There are two cases. If $n = 2$,
then $\mathop{\operatoratfont order}\nolimits(S,C) | 2$. But by \pref{order-calc}, the order of $D_{t,t}$
(as above) is $4$. Hence $n > 2$. Hence $p_1 = p_2$. But
$p_1(D_{t,t}) > p_2(D_{t,t})$ (by \ref{fantastico}), so
``$D_{t,t} \notin (S,C)$'' for $t$ odd, $t \geq 5$.
{}From this, it follows that $\svec p1{n-1} \leq \Sigma(S,C)$. Let
$r = d[n(s-4)+t] + (2-2g)n$. By \pref{bound-formula}, we conclude that
$r \leq {s\over3}(2s^2-6s+7)-1$.
The case $n = 1$ corresponds to a complete intersection, and the theorem is
easily verified in this case. Therefore we may assume that\ $n \geq 2$.
By\Lspace \Lcitemark 18\Rcitemark \Rspace{}, it follows that if $s \leq 3$,
then
$d \leq g+3$. Hence we may assume that $s \geq 4$. Therefore
\pref{murky-algebra} applies, and we conclude that $d \leq g + 3$. {\hfill$\square$}
\end{proof}
\begin{corollary}
Let $S, T \subset {\Bbb C}\kern1pt\xmode{\Bbb P\kern1pt}^3$ be surfaces.
Assume that $S$ and $T$ have only rational singularities.
Assume that $S \cap T$ is set-theoretically a smooth curve $C$ of
degree $d$ and genus $g$.
Assume that $\mathop{\operatoratfont Sing}\nolimits(S) \cap \mathop{\operatoratfont Sing}\nolimits(T) = \varnothing$. Then $d \leq g + 3$.
\end{corollary}
\block{Theorem X}
\par\indent\indent
As a corollary of theorem (I), we show:
\begin{theorem}\label{thmX}
{\bf (``X'')}
Let $C \subset \P3$ be a smooth curve. Assume that $C$ is not a complete
intersection. Suppose that $C = S \cap T$ as sets, where $S, T \subset \P3$ are
surfaces. Assume that $\mathop{\operatoratfont Sing}\nolimits(S) \cap \mathop{\operatoratfont Sing}\nolimits(T) = \varnothing$. Then:
$$\deg(S), \deg(T) < 2 \cdot \deg(C)^4.$$
\end{theorem}
\par\noindent First we make a few remarks.
\begin{arabiclist}
\item The proof of theorem (X) depends primarily on the fact that the numbers
$p_k$ in theorem (I) must be integers.
\item The importance of theorem (X) is that an upper bound
is given for the degrees of $S$ and $T$, that this bound is computable,
and that this bound depends only on
the degree of $C$. In the proof, we give the better bounds
$\deg(S) < 2 \cdot \deg(C)^2$, $\deg(T) < 2 \cdot \deg(C)^4$, provided
that $\deg(S) \leq \deg(T)$.
\item Via the bounds in theorem (X), it becomes a computer triviality to
find all possible degrees for $S$ and $T$ which are consistent with the
integrality of the numbers $p_k$ in theorem (I).
\item Doing this when $\deg(C) = 4$, $\mathop{\operatoratfont genus}\nolimits(C) = 0$, and assuming for
efficiency that $\deg(S) \leq \deg(T)$, we find:
$$(\deg(S), \deg(T)) \in \{ (3,4), (3,8), (4,4), (4,7), (6,26), (9,48),
(10,28),$$
$$(12,18), (13,16), (17,220), (18,118), (19,84), (20,67), (22,50), (28,33)
\}.$$
[We have excluded the cases where $\deg(S)$ is $1$ or $2$, which cannot occur.]
\item We do not know which of these pairs of integers can be realized
by pairs of surfaces, as in theorem (X). All that we know is that
$(3,4)$ and $(3,8)$ cannot be realized in characteristic zero, and that
$(4,4)$ can be realized in characteristic two.
\item Theorem (X) is false without the hypothesis that $C$ is
a complete intersection. Counterexample: for any $s \in \xmode{\Bbb N}$,
one can find a smooth curve $D \subset \P2$ of degree $s$ and a line $L \subset \P2$
such that $D \cap L$ is a single point, set-theoretically. Let $S$ and $T$ be
cones over $D$ and $L$, with the same vertex. Then $S \cap T$ is a line,
set-theoretically. Theorem (X) is also false without the hypothesis that
$\mathop{\operatoratfont Sing}\nolimits(S) \cap \mathop{\operatoratfont Sing}\nolimits(T) = \varnothing$.
\end{arabiclist}
Before proceeding with the proof of theorem (X), we need the following
lemma, which was known in characteristic zero, and for
the smooth case, was known in all characteristics. (See proof for
references.)
\begin{lemma}\label{torsion-free}
Let $S \subset \P3$ be a normal surface. Then $\mathop{\operatoratfont Pic}\nolimits(S)/\mathop{\operatoratfont Pic}\nolimits(\P3)$ is
torsion-free.
\end{lemma}
Before proceeding with the proof, we recall some standard material on
differentials for which we do not have a good reference. First of all,
for any scheme $X$, there is a map of sheaves of abelian groups:
\dmap[[ \mathop{\operatoratfont dlog}\nolimits || {\cal O}_X^* || \Omega_X ]]%
given by $f \mapsto df/f$. (All sheaves we shall discuss are sheaves
on the Zariski site.)
Now suppose that $X$ is a normal proper variety, defined
over an algebraically closed field $k$ of positive characteristic $p$. Then we
have an exact sequence:
\diagramx{0&\mapE{}&{\cal O}_X^*&\mapE{F}&{\cal O}_X^*&\mapE{\mathop{\operatoratfont dlog}\nolimits}&\Omega_X\cr%
}of sheaves of abelian groups on $X$, where $F$ denotes the Frobenius map.
The exactness in the middle depends on normality, and may be deduced e.g.\ from
(\Lcitemark 16\Rcitemark \ I\ 4.2). Let ${\cal{D}}$ be the image of
$\mathop{\operatoratfont dlog}\nolimits$. Since $X$ is
proper, $H^0(X, {\cal O}_X^*) = k^*$, so $H^0(X, F)$ is an isomorphism, and we obtain
an isomorphism $H^0(X, {\cal{D}}) \cong \ker H^1(X, F)$. We have
$\ker H^1(X, F) \cong {}_p\mathop{\operatoratfont Pic}\nolimits(X)$. Composing with the canonical injection
\mapx[[ H^0(X, {\cal{D}}) || H^0(X, \Omega_X) ]], we obtain an injective group
homomorphism:
\dmap[[ \psi_X || {}_p\mathop{\operatoratfont Pic}\nolimits(X) || H^0(X, \Omega_X). ]]%
\begin{proofnodot}
(of \ref{torsion-free}).
First we show $(*)$ that $\mathop{\operatoratfont Pic}\nolimits(S)/\mathop{\operatoratfont Pic}\nolimits(\P3)$ has no torsion, except possibly
for $p$-torsion, when the ground field has positive characteristic $p$.
These arguments are very similar to those given by
Lang\Lspace \Lcitemark 21\Rcitemark \Rspace{}. The methods were invented by
Grothendieck
(\Lcitemark 10\Rcitemark \ Expos\'e XI), and further studied
by Hartshorne (\Lcitemark 13\Rcitemark \ \S4.3). We refer the
reader to\Lspace \Lcitemark 21\Rcitemark \Rspace{} or\Lspace \Lcitemark
13\Rcitemark \Rspace{}
for details.
Let $S_n$ be the \th{n} infinitesimal neighborhood of $S$ in $\P3$. Then:
$$\mathop{\operatoratfont Pic}\nolimits(\P3) \cong \displaystyle{\lim_{\overleftarrow{\hphantom{\lim}}}} \mathop{\operatoratfont Pic}\nolimits(S_n).$%
$Moreover, for each $n$ there is an exact sequence of abelian groups:
\les{\mathop{\operatoratfont Pic}\nolimits(S_{n+1})}{\mathop{\operatoratfont Pic}\nolimits(S_n)}{H^2(S, {\cal{J}}^n/{\cal{J}}^{n+1}),%
}where ${\cal{J}}$ is the ideal sheaf of $S$ in $\P3$. Since the $H^2$ term
is a vector space, $(*)$ follows.
{}From now on we may assume that the ground field has positive characteristic
$p$. A standard calculation shows that $H^0(S, \Omega_S) = 0$.
Up to now, we have not used the hypothesis that $S$ is normal. We
now use this hypothesis. Via the map $\psi_S$, defined immediately above
this proof, we see that ${}_p\mathop{\operatoratfont Pic}\nolimits(S) = 0$. (This argument is essentially
that used in\Lspace \Lcitemark 21\Rcitemark \Rspace{}.)
Finally, to complete the proof, we must show that $[O_S(1)]$ does not
have a \th{p} root in $\mathop{\operatoratfont Pic}\nolimits(S)$. The argument given here is essentially
the argument given in (\Lcitemark 4\Rcitemark \ 1.8). For any variety $X$,
there is a natural group homomorphism
\mp[[ H^1(\mathop{\operatoratfont dlog}\nolimits) || \mathop{\operatoratfont Pic}\nolimits(X) || H^1(X, \Omega_X) ]].
Consider this map when $X = S$ and when $X = \P3$. A standard calculation
shows that the map \mapx[[ H^1(\P3, \Omega_{\P3}) || H^1(S, \Omega_S) ]] is
injective. Moreover, one knows that the image of $[{\cal O}_{\P3}(1)]$ in
$H^1(\P3, \Omega_{\P3})$ is not zero.
(See e.g.{\ }\Lcitemark 14\Rcitemark \ Chapter 3, exercise 7.4.)
Hence the image of $[{\cal O}_S(1)]$ in
$H^1(S, \Omega_S)$ is not zero. Hence $[O_S(1)]$ does not have a \th{p}
root in $\mathop{\operatoratfont Pic}\nolimits(S)$. {\hfill$\square$}
\end{proofnodot}
\begin{remark}
Over an algebraically closed field of positive characteristic, let
$S \subset \P3$ be a surface, not necessarily normal. We do not know if
$\mathop{\operatoratfont Pic}\nolimits(S)/\mathop{\operatoratfont Pic}\nolimits(\P3)$ is torsion-free, or even if $\mathop{\operatoratfont Pic}\nolimits(S)$ is torsion-free.
Answers to these questions might be obtained from a general structure
theorem for \hskip 0pt plus 4cm\penalty1000\hskip 0pt plus -4cm\hbox{$\mathop{\operatoratfont Ker}\nolimits[\mathop{\operatoratfont Pic}\nolimits(S)\ \mapE{}\ \mathop{\operatoratfont Pic}\nolimits(\nor{S})]$}, where $S$ is
an arbitrary projective variety.
\end{remark}
\begin{corollary}\label{smooth-stci}
In $\P3$, suppose that $C = S \cap T$ as sets, where $C$ is a curve,
and $S, T$ are surfaces. Assume that $C$ does not meet $\mathop{\operatoratfont Sing}\nolimits(S)$. Then
there exists a surface $T' \subset \P3$ such that $C = S \cap T'$,
scheme-theoretically.
\end{corollary}
\begin{proof}
Since $T \cap \mathop{\operatoratfont Sing}\nolimits(S) = \varnothing$, $S$ is normal. By \pref{torsion-free},
$\mathop{\operatoratfont Pic}\nolimits(S)/\mathop{\operatoratfont Pic}\nolimits(\P3)$ is torsion-free. Hence $[{\cal O}_S(C)] = 0$ in
$\mathop{\operatoratfont Pic}\nolimits(S)/\mathop{\operatoratfont Pic}\nolimits(\P3)$. Hence ${\cal O}_S(C) \cong {\cal O}_S(t)$ for some $t \in \xmode{\Bbb N}$.
Since the canonical map \mapx[[ H^0(\P3, {\cal O}_{\P3}(t)) || H^0(S, {\cal O}_S(t)) ]]
is surjective, it follows that there exists a surface $T'$ of degree $t$ as
claimed. {\hfill$\square$}
\end{proof}
\begin{remark}
Robbiano\Lspace \Lcitemark 28\Rcitemark \Rspace{} proved this in the case
where $S$ is smooth and the ground field has characteristic zero.
\end{remark}
\begin{proofnodot}
(of theorem X).
Let $d = \deg(C)$, $g = \mathop{\operatoratfont genus}\nolimits(C)$, $s = \deg(S)$, $t = \deg(T)$. We may
assume that $s \leq t$. We show that $s < 2d^2$ and $t < 2d^4$. Let
$n = st/d$. By theorem (I), we know that:
$$(n-1) \kern3pt | \kern3pt \setof{ d [ n (s-4) + t ] + (2-2g)n }.$%
$A proof of this fact, independent of (I), is given at the end of this paper.
By \pref{smooth-stci}, we know that $C$ meets $\mathop{\operatoratfont Sing}\nolimits(S)$. Hence
$p_1(S,C) > 0$. Hence the right hand side\ is positive.
Write $d = d_s d_t$, where $d_s, d_t \in \xmode{\Bbb N}$, $d_s|s$, and $d_t|t$. Let
$s_1 = s/d_s$, $t_1 = t/d_t$. Then $n = s_1t_1$, so:
$$(s_1t_1 - 1) | \setof{ d [ s_1t_1 (d_s s_1 - 4) + d_t t_1 ]
+ (2-2g) s_1 t_1 }.$%
$The right hand side\ is divisible by $t_1$, and $\gcd(s_1t_1 - 1, t_1) = 1$, so:
$$(s_1t_1 - 1) | \setof{ d [ s_1 (d_s s_1 - 4) + d_t ] + (2-2g) s_1}.\eqno(*)$%
$Thus for some $k \in \xmode{\Bbb N}$, we have:
$$(s_1t_1 - 1)k = d [ s_1 (d_s s_1 - 4) + d_t ] + (2-2g) s_1.$%
$Reorganizing, we find:
$$(s_1t_1 - 1)k = (d d_s)s_1^2 + (2-2g-4d)s_1 + d d_t. \eqno(**)$%
$Now we have $t \geq s$, so $t_1 \geq (d_s/d_t)s_1$. Hence:
$$\left[s_1^2\left({d_s \over d_t}\right) - 1\right]k
\leq (d d_s)s_1^2 + (2-2g-4d)s_1 + d d_t.$%
$It is conceivable that the left hand side\ of this inequality is negative. This will
not effect the following argument.
Suppose that $k \geq d d_t$. After a short calculation, one finds that
$s_1 \leq d d_t / (2d+g-1)$, and hence that $s \leq d^2/(2d+g-1)$. This
implies that $s < 2d^2$. Hence, in order to prove our assertion that
$s < 2d^2$, we may assume that $k < d d_t$.
\par\noindent From $(**)$ we obtain:
$$(d d_s)s_1^2 + (2 - 2g - 4d - t_1 k)s_1 + (k + d d_t) = 0.$%
$Hence $s_1 | (k + d d_t)$. Hence $s_1 \leq k + d d_t$. Hence
$s_1 < 2d d_t$. Hence $s < 2d^2$.
To complete the proof, we must show that $t < 2d^4$.
The right hand side\ of $(*)$ is nonzero, so:
$$s_1 t_1 - 1 \leq d[s_1(d_s s_1 - 4) + d_t] + (2 - 2g)s_1.$%
$Dividing by $s_1$ and isolating $t_1$, we find:
$$t_1 \leq d [ d_s s_1 - 4 + d_t s_1^{-1} ] + 2 - 2g + s_1^{-1}.$%
$Taking account of $t_1 = t d_t^{-1}$ and $s_1 = s d_s^{-1}$, we obtain:
$$t \leq d_t \setof{ d [ s - 4 + ds^{-1} ] + 2 - 2g } + ds^{-1}.$%
$Since $s < 2d^2$, it follows (with a little work) that $t < 2d^4$. {\hfill$\square$}
\end{proofnodot}
\begin{remark}
We give here an alternate proof of the main ingredient of the proof of (X),
namely that
$$(n-1) \kern3pt | \kern3pt \setof{ d [ n (s-4) + t ] + (2-2g)n }.
\eqno(\dag)$%
$Let ${\tilde{C}}$ be the scheme-theoretic complete intersection of $S$ and $T$.
Let ${\cal{J}}$ be the ideal sheaf of $C$ in ${\tilde{C}}$. Let $p$ be a closed point
of $C$. If ${\cal O}_{S,p}$ is regular, then near $p$, ${\tilde{C}}$ and $C$ are Cartier
divisors on $S$, with ${\tilde{C}} = nC$. Choose an isomorphism
${\cal O}_{S,p} \cong k[[x,y]]$, such that $C$ corresponds to $V(x)$. Then ${\tilde{C}}$
corresponds to $V(x^n)$. Therefore the algebra of conormal invariants
$${\cal{A}}\ =\ {\cal O}_{{\tilde{C}}}/{\cal{J}} \o+ {\cal{J}}/{\cal{J}}^2 \o+ {\cal{J}}^2/{\cal{J}}^3 \o+ \cdots$%
$is a locally free ${\cal O}_C$-module near $p$. But we similarly get the same
conclusion if ${\cal O}_{T,p}$ is regular, so in fact ${\cal{A}}$ is locally free since
$\mathop{\operatoratfont Sing}\nolimits(S) \cap \mathop{\operatoratfont Sing}\nolimits(T) = \varnothing$.
Moreover, there is a line bundle ${\cal{L}}$ on $C$ such that
${\cal{A}} \cong {\cal O}_C \o+ {\cal{L}} \o+ {\cal{L}}^2 \manyo+ {\cal{L}}^{n-1}$. Hence
$\chi({\cal{A}}) = n(1-g) + {n \choose 2}\deg({\cal{L}})$. On the other hand,
$\chi({\cal{A}}) = \chi({\cal O}_{{\tilde{C}}})$, which (via ${\tilde{C}} = S \cap T$) is easily
computed to be $st(4-s-t)/2$. Hence
$$n(1-g) + {n \choose 2}\deg({\cal{L}}) = {st(4-s-t) \over 2}.$%
$Hence
$${n \choose 2} \kern3pt \left| \kern3pt {st(4-s-t) \over 2} - n(1-g).\right.$%
$It is not difficult to verify that this is equivalent to $(\dag)$.
\end{remark}
\vspace{0.25in}
\section*{References}
\addtocontents{toc}{\protect\vspace*{2.25em}}
\addcontentsline{toc}{special}{References}
\ \par\noindent\vspace*{-0.25in}
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\def\Ptest{ }\def\Pcnt{ }\def\Pstr{255--270}%
\def\Qtest{ }\def\Qstr{access via "brieskorn simultaneous resolution annalen"}%
\def\Xtest{ }\def\Xstr{Not on file. The author studies simultaneous resolution
of rational singularities.}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{3}%
\def\Atest{ }\def\Astr{Catanese\Revcomma F\Initper }%
\def\Ttest{ }\def\Tstr{Babbage's conjecture, contact of surfaces, symmetric
determinantal varieties and applications}%
\def\Jtest{ }\def\Jstr{Invent. Math.}%
\def\Vtest{ }\def\Vstr{63}%
\def\Dtest{ }\def\Dstr{1981}%
\def\Ptest{ }\def\Pcnt{ }\def\Pstr{433--465}%
\def\Qtest{ }\def\Qstr{access via "catanese contact"}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{4}%
\def\Atest{ }\def\Astr{Deligne\Revcomma P\Initper }%
\def\Ttest{ }\def\Tstr{{\rm\tolerance=1000 Cohomologie des intersections
compl\`etes, expos\'e\ XI in {\itS\'em\-in\-aire de G\'eom\'e\-trie Al\-g\'e\-bri\-que} (SGA 7)}}%
\def\Stest{ }\def\Sstr{Lecture Notes in Mathematics}%
\def\Itest{ }\def\Istr{Spring\-er-Ver\-lag}%
\def\Ctest{ }\def\Cstr{New York}%
\def\Vtest{ }\def\Vstr{340}%
\def\Dtest{ }\def\Dstr{1973}%
\def\Ptest{ }\def\Pcnt{ }\def\Pstr{39--61}%
\def\Qtest{ }\def\Qstr{access via "deligne intersections"}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{2}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{5}%
\def\Ven{Van de Ven}{}%
\def\Atest{ }\def\Astr{Eisenbud\Revcomma D\Initper %
\Aand A\Initper \Ven}%
\def\Ttest{ }\def\Tstr{On the normal bundles of smooth rational space curves}%
\def\Jtest{ }\def\Jstr{Math. Ann.}%
\def\Vtest{ }\def\Vstr{256}%
\def\Dtest{ }\def\Dstr{1981}%
\def\Ptest{ }\def\Pcnt{ }\def\Pstr{453--463}%
\def\Qtest{ }\def\Qstr{access via "eisenbud normal bundles"}%
\def\Xtest{ }\def\Xstr{Not on file.}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{6}%
\def\Atest{ }\def\Astr{Fossum\Revcomma R\Initper \Initgap M\Initper }%
\def\Ttest{ }\def\Tstr{The Divisor Class Group of a Krull Domain}%
\def\Itest{ }\def\Istr{Spring\-er-Ver\-lag}%
\def\Ctest{ }\def\Cstr{New York}%
\def\Dtest{ }\def\Dstr{1973}%
\def\Qtest{ }\def\Qstr{access via "fossum"}%
\def\Xtest{ }\def\Xstr{I don't have this.}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{7}%
\def\Atest{ }\def\Astr{Fulton\Revcomma W\Initper }%
\def\Ttest{ }\def\Tstr{Intersection Theory}%
\def\Itest{ }\def\Istr{Spring\-er-Ver\-lag}%
\def\Ctest{ }\def\Cstr{New York}%
\def\Dtest{ }\def\Dstr{1984}%
\def\Qtest{ }\def\Qstr{access via "fulton intersection theory"}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{8}%
\def\Atest{ }\def\Astr{Gallarati\Revcomma D\Initper }%
\def\Ttest{ }\def\Tstr{Ricerche sul contatto di superfiche algebriche lungo
curve}%
\def\Jtest{ }\def\Jstr{Acad\'emie royale de Belgique, Classe des Sciences,
M\'emoires, Collection in-$8^0$}%
\def\Vtest{ }\def\Vstr{32}%
\def\Dtest{ }\def\Dstr{1960}%
\def\Ptest{ }\def\Pcnt{ }\def\Pstr{1--78}%
\def\Qtest{ }\def\Qstr{access via "gallarati belgique"}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{2}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{9}%
\def\Atest{ }\def\Astr{Griffiths\Revcomma P\Initper %
\Aand J\Initper Harris}%
\def\Ttest{ }\def\Tstr{Principles of Algebraic Geometry}%
\def\Itest{ }\def\Istr{John Wiley \& Sons}%
\def\Ctest{ }\def\Cstr{New York}%
\def\Dtest{ }\def\Dstr{1978}%
\def\Qtest{ }\def\Qstr{access via "griffiths harris principles" (was "griffiths
harris")}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{10}%
\def\Atest{ }\def\Astr{Grothendieck\Revcomma A\Initper }%
\def\Ttest{ }\def\Tstr{{\rm Cohomologie Locale des Faisceaux Coherents et
Theorems de Lefschetz Locaux et Globaux, in {\it S\'em\-in\-aire de G\'eom\'e\-trie Al\-g\'e\-bri\-que} (SGA 2)}}%
\def\Itest{ }\def\Istr{North-Holland}%
\def\Dtest{ }\def\Dstr{1968}%
\def\Qtest{ }\def\Qstr{access via "grothendieck lefschetz"}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{2}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{11}%
\def\Atest{ }\def\Astr{Grothendieck\Revcomma A\Initper %
\Aand J\Initper \Initgap A\Initper Dieudonn\'e}%
\def\Ttest{ }\def\Tstr{El\'ements de g\'eom\'etrie\ alg\'e\-brique II}%
\def\Jtest{ }\def\Jstr{Inst. Hautes \'Etudes Sci. Publ. Math.}%
\def\Vtest{ }\def\Vstr{8}%
\def\Dtest{ }\def\Dstr{1961}%
\def\Qtest{ }\def\Qstr{access via "EGA2"}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{2}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{12}%
\def\Atest{ }\def\Astr{Grothendieck\Revcomma A\Initper %
\Aand J\Initper \Initgap A\Initper Dieudonn\'e}%
\def\Ttest{ }\def\Tstr{El\'ements de g\'eom\'etrie\ alg\'e\-brique IV (part four)}%
\def\Jtest{ }\def\Jstr{Inst. Hautes \'Etudes Sci. Publ. Math.}%
\def\Vtest{ }\def\Vstr{32}%
\def\Dtest{ }\def\Dstr{1967}%
\def\Qtest{ }\def\Qstr{access via "EGA4-4"}%
\def\Astr{\Underlinemark}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{13}%
\def\Atest{ }\def\Astr{Hartshorne\Revcomma R\Initper }%
\def\Ttest{ }\def\Tstr{Ample Subvarieties of Algebraic Varieties}%
\def\Stest{ }\def\Sstr{Lecture \hskip 0pt plus 4cm\penalty1000\hskip 0pt plus -4cm Notes in Mathematics}%
\def\Itest{ }\def\Istr{Spring\-er-Ver\-lag}%
\def\Ctest{ }\def\Cstr{New York}%
\def\Vtest{ }\def\Vstr{156}%
\def\Dtest{ }\def\Dstr{1970}%
\def\Qtest{ }\def\Qstr{access via "hartshorne ample subvarieties"}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{14}%
\def\Atest{ }\def\Astr{Hartshorne\Revcomma R\Initper }%
\def\Ttest{ }\def\Tstr{Algebraic Geometry}%
\def\Itest{ }\def\Istr{Spring\-er-Ver\-lag}%
\def\Ctest{ }\def\Cstr{New York}%
\def\Dtest{ }\def\Dstr{1977}%
\def\Qtest{ }\def\Qstr{access via "hartshorne algebraic geometry"}%
\def\Astr{\Underlinemark}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{15}%
\def\Atest{ }\def\Astr{Hartshorne\Revcomma R\Initper }%
\def\Ttest{ }\def\Tstr{Complete intersections in characteristic $p > 0$}%
\def\Jtest{ }\def\Jstr{Amer. J. Math.}%
\def\Vtest{ }\def\Vstr{101}%
\def\Dtest{ }\def\Dstr{1979}%
\def\Ptest{ }\def\Pcnt{ }\def\Pstr{380--383}%
\def\Qtest{ }\def\Qstr{access via "hartshorne complete intersections
characteristic"}%
\def\Xtest{ }\def\Xstr{had: (Based on a talk given at the 1964 Woods Hole
conference)}%
\def\Astr{\Underlinemark}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{16}%
\def\Atest{ }\def\Astr{Iversen\Revcomma B\Initper }%
\def\Ttest{ }\def\Tstr{Generic Local Structure in Commutative Algebra}%
\def\Stest{ }\def\Sstr{Lecture Notes in Mathematics}%
\def\Itest{ }\def\Istr{Spring\-er-Ver\-lag}%
\def\Ctest{ }\def\Cstr{New York}%
\def\Vtest{ }\def\Vstr{310}%
\def\Dtest{ }\def\Dstr{1973}%
\def\Qtest{ }\def\Qstr{access via "iversen generic local structure"}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{17}%
\def\Atest{ }\def\Astr{Jaffe\Revcomma D\Initper \Initgap B\Initper }%
\def\Ttest{ }\def\Tstr{Space curves which are the intersection of a cone with
another surface}%
\def\Jtest{ }\def\Jstr{Duke Math. J.}%
\def\Vtest{ }\def\Vstr{57}%
\def\Dtest{ }\def\Dstr{1988}%
\def\Ptest{ }\def\Pcnt{ }\def\Pstr{859--876}%
\def\Qtest{ }\def\Qstr{access via "jaffe another"}%
\def\Htest{ }\def\Hstr{1}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{18}%
\def\Atest{ }\def\Astr{Jaffe\Revcomma D\Initper \Initgap B\Initper }%
\def\Ttest{ }\def\Tstr{On set theoretic complete intersections in $\P3$}%
\def\Jtest{ }\def\Jstr{Math. Ann.}%
\def\Vtest{ }\def\Vstr{285}%
\def\Dtest{ }\def\Dstr{1989}%
\def\Ptest{ }\def\Pcnt{ }\def\Pstr{165--173, 175, 174, 176}%
\def\Qtest{ }\def\Qstr{access via "jaffe on set theoretic annalen"}%
\def\Htest{ }\def\Hstr{2}%
\def\Astr{\Underlinemark}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{19}%
\def\Atest{ }\def\Astr{Jaffe\Revcomma D\Initper \Initgap B\Initper }%
\def\Ttest{ }\def\Tstr{Smooth curves on a cone which pass through its vertex}%
\def\Jtest{ }\def\Jstr{Manu\-scripta Math.}%
\def\Vtest{ }\def\Vstr{73}%
\def\Dtest{ }\def\Dstr{1991}%
\def\Ptest{ }\def\Pcnt{ }\def\Pstr{187--205}%
\def\Qtest{ }\def\Qstr{access via "jaffe vertex"}%
\def\Htest{ }\def\Hstr{5}%
\def\Astr{\Underlinemark}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{20}%
\def\Atest{ }\def\Astr{Jaffe\Revcomma D\Initper \Initgap B\Initper }%
\def\Ttest{ }\def\Tstr{Local geometry of smooth curves passing through rational
double points}%
\def\Jtest{ }\def\Jstr{Math. Ann.}%
\def\Vtest{ }\def\Vstr{294}%
\def\Dtest{ }\def\Dstr{1992}%
\def\Ptest{ }\def\Pcnt{ }\def\Pstr{645--660}%
\def\Qtest{ }\def\Qstr{access via "jaffe local geometry"}%
\def\Htest{ }\def\Hstr{6}%
\def\Astr{\Underlinemark}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{21}%
\def\Atest{ }\def\Astr{Lang\Revcomma W\Initper \Initgap E\Initper }%
\def\Ttest{ }\def\Tstr{Remarks on $p$-torsion of algebraic surfaces}%
\def\Jtest{ }\def\Jstr{Compositio Math.}%
\def\Vtest{ }\def\Vstr{52}%
\def\Dtest{ }\def\Dstr{1984}%
\def\Ptest{ }\def\Pcnt{ }\def\Pstr{197--202}%
\def\Qtest{ }\def\Qstr{access via "william lang torsion"}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{22}%
\def\Atest{ }\def\Astr{Lipman\Revcomma J\Initper }%
\def\Ttest{ }\def\Tstr{Rational singularities, with applications to algebraic
surfaces and unique factorization}%
\def\Jtest{ }\def\Jstr{Inst. Hautes \'Etudes Sci. Publ. Math.}%
\def\Vtest{ }\def\Vstr{36}%
\def\Dtest{ }\def\Dstr{1969}%
\def\Ptest{ }\def\Pcnt{ }\def\Pstr{195--279}%
\def\Qtest{ }\def\Qstr{access via "lipman rational"}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{23}%
\def\Atest{ }\def\Astr{Matsumura\Revcomma H\Initper }%
\def\Ttest{ }\def\Tstr{Commutative Algebra, second edition}%
\def\Itest{ }\def\Istr{Benjamin/Cum\-mings}%
\def\Ctest{ }\def\Cstr{Reading, Massachusetts}%
\def\Dtest{ }\def\Dstr{1980}%
\def\Qtest{ }\def\Qstr{access via "matsumura commutative algebra"}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{24}%
\def\Atest{ }\def\Astr{Miyaoka\Revcomma Y\Initper }%
\def\Ttest{ }\def\Tstr{The maximal number of quotient singularities on surfaces
with given numerical invariants}%
\def\Jtest{ }\def\Jstr{Math. Ann.}%
\def\Vtest{ }\def\Vstr{268}%
\def\Dtest{ }\def\Dstr{1984}%
\def\Ptest{ }\def\Pcnt{ }\def\Pstr{159--171}%
\def\Qtest{ }\def\Qstr{access via "miyaoka quotient"}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{25}%
\def\Atest{ }\def\Astr{Morrison\Revcomma D\Initper \Initgap R\Initper }%
\def\Ttest{ }\def\Tstr{The birational geometry of surfaces with rational double
points}%
\def\Jtest{ }\def\Jstr{Math. Ann.}%
\def\Vtest{ }\def\Vstr{271}%
\def\Dtest{ }\def\Dstr{1985}%
\def\Ptest{ }\def\Pcnt{ }\def\Pstr{415--438}%
\def\Qtest{ }\def\Qstr{access via "morrison rational double points"}%
\def\Xtest{ }\def\Xstr{MR 87e:14011. \par The author considers the problem of
when a curve on a surface can be contracted to a surface having only rational
double points. Has appendix on rational double points.}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{26}%
\def\Atest{ }\def\Astr{Mumford\Revcomma D\Initper }%
\def\Ttest{ }\def\Tstr{The topology of normal singularities of an algebraic
surface and a criterion for simplicity}%
\def\Jtest{ }\def\Jstr{Inst. Hautes \'Etudes Sci. Publ. Math.}%
\def\Vtest{ }\def\Vstr{9}%
\def\Dtest{ }\def\Dstr{1961}%
\def\Ptest{ }\def\Pcnt{ }\def\Pstr{229--246}%
\def\Qtest{ }\def\Qstr{access via "mumford topology of normal"}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{27}%
\def\Atest{ }\def\Astr{Rao\Revcomma A\Initper \Initgap P\Initper }%
\def\Ttest{ }\def\Tstr{On self-linked curves}%
\def\Jtest{ }\def\Jstr{Duke Math. J.}%
\def\Vtest{ }\def\Vstr{49}%
\def\Dtest{ }\def\Dstr{1982}%
\def\Ptest{ }\def\Pcnt{ }\def\Pstr{251--273}%
\def\Qtest{ }\def\Qstr{access via "rao self linked"}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{28}%
\def\Atest{ }\def\Astr{Robbiano\Revcomma L\Initper }%
\def\Ttest{ }\def\Tstr{A problem of complete intersections}%
\def\Jtest{ }\def\Jstr{Nagoya Math. J.}%
\def\Vtest{ }\def\Vstr{52}%
\def\Dtest{ }\def\Dstr{1973}%
\def\Ptest{ }\def\Pcnt{ }\def\Pstr{129--132}%
\def\Qtest{ }\def\Qstr{access via "robbiano complete intersections nagoya"}%
\def\Xtest{ }\def\Xstr{Not on file.}%
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|
1996-01-04T17:05:09 | 9410 | alg-geom/9410002 | en | https://arxiv.org/abs/alg-geom/9410002 | [
"alg-geom",
"math.AG"
] | alg-geom/9410002 | David Reed | David Reed | Topology of Conjugate Varieties | LATEX, 27 pages | null | null | null | null | Serre and Abelson have produced examples of non-homeomorphic conjugate
varieties. We show that if the field of definition of a polarized projective
variety coincides with its field of moduli then all of its conjugates have the
same topological type. This extends the class of varieties known to posses
conjugacy invariant to canonically embedded varieties. We also show that normal
complete interswections in homogeneous varieties have this property.
| [
{
"version": "v1",
"created": "Wed, 5 Oct 1994 10:25:57 GMT"
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""
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\newpage
\title {The Topology of Conjugate Varieties}
\author {David Reed\\
Mathematical Institute\\
24 - 29 St Giles'\\
Oxford OX1 3LB\\
UK\\}
\date{October 1994}
\maketitle
\begin{abstract} Serre \cite{Se:64} and Abelson \cite{Ab:74} have produced
examples of conjugate algebraic
varieties which are not homeomorphic. We show that if the field of definition
of a polarized projective variety coincides with its field of moduli then all
of its conjugates have the same topological type. This immediately extends the
class of varietie
s known to possess invariant topological type to all canonically embedded
varieties. We also show that (normal) complete intersections in projective
space and, more generally in homogeneous varieties, satisfy the condition.
\end{abstract}
\bit{Introduction}
If $V$ is an algebraic variety defined over $k$, a finitely generated extension
of $\Q$, for
each embedding $\sigma :k \rightarrow \C$ we can extend scalars to form the
complex
algebraic variety $V_{\sigma}$ defined by the following cartesian
square (base change or extension of scalars)
\[
\begin{array}{ccc}
V_{\sigma} := V \times _{\Q} \C & \rightarrow & V\\
\downarrow & & \downarrow \\
\Spec \C & \stackrel{\sigma}{\rightarrow} & \Spec k
\end{array}
\]
The complex points of this variety $V_{\sigma}(\C)$ form a topological space
and the
topological type of two such spaces, $V_{\sigma}(\C)$ and $V_{\tau}(\C)$ for
two
different embeddings $k \rightarrow \C$ can be compared. We will refer to
varieties
$V_{\sigma}$ and $V_{\tau}$ obtained in this manner as {\em conjugate
varieties}.
Serre \cite{Se:64} and Abelson \cite{Ab:74} have produced examples of conjugate
varieties whose complex points constitute non-homeomorphic topological spaces.
Since the publication of these papers there has been little published work in
this area. The principal result of the research reported upon here is a
sufficient condition for
the topological spaces of complex points of conjugate varieties to be
homeomorphic. We begin by recalling definitions due to
Matsusaka-Shimura-Koizumi \cite{Ko:72}.
\begin{r}{Definition}
For a divisor $X$ on a projective variety $V$ define the class ${\cal P}(X)$ to
be the class of all divisors $X'$ on $V$ such that there are integers $m, n$
with $mX \equiv nX'$ (algebraic equivalence). If ${\cal P}(X)$ contains an
ample divisor then it
is called a {\em polarization} (this term is also applied to the ample divisor
in the class). An isomorphism of projective varieties $f:V \rightarrow W$ is
said to be a isomorphism of polarized projective varieties if there are
polarizations ${\cal P}$,
${\cal P}'$ on $V$, and $W$ respectively such that the map on divisors induced
by $f$ takes ${\cal P}$ to ${\cal P}'$
\end{r}
\begin{r}{Definition}
The field of moduli for polarized projective variety $(V,{\cal P})$ is the
field $k$ such that for $\sigma \in \Aut(\C)$, $\sigma$ is in fact in $\Aut(\C
/k)$ if and only if $V^{\sigma} \simeq V$ as polarized varieties.\end{r}
Compare this field to the {\em field of definition} of the variety which is the
field $K$ such that $\sigma \in \Aut(\C)$ is in fact in $\Aut(\C /k)$ iff and
only if $V^{\sigma}=V$.
\bl
A discussion of when fields of moduli exist in general for varieties can be
found in \cite{Ko:72}. An example of a variety (a hyperelliptic curve) whose
field of moduli differs from its field of definition can be found in
\cite{Sh:72}.
\begin{s}{Theorem}
\label{big'}
If $V$ is a polarized projective variety defined over $k$, a finitely generated
extension of $\Q$, and if the field of moduli for $V$ coincides with $k$, then
the topological type of $V_{\sigma}(\C)$ is independent of $\sigma$.\end{s}
The proof of this Theorem relies on a strengthened form of Thom's
stratified Isotopy Theorem which is given in \S 2 below. The Theorem itself
is proved in \S3.
It was previously known that certain types of varieties whose topology is
rather easily described, such as non-singular curves, abelian varieties, $K-3$
surfaces and simply connected non-singular surfaces of general type, have
topological types which do n
ot vary under conjugation of their fields of definition. The above Theorem
extends this list to include all canonically embedded varieties.
In \S 4, the criterion given above will be used to show that (normal) complete
intersections in projective space, and, more generally, in homogeneous
varieties, also belong on this list by showing that they satisfy the condition
of the Theorem as well. In
\S 5 we point out a few of the many questions that remain open in this area.
\bit{Stratified Isotopy Theorems and the Topology of Conjugate Varieties}
\bitt{Preliminaries from Algebraic Geometry}
Our intention is to review the proof of the ``stratified isotopy Theorem'' as
it applies to algebraic varieties with a view to establishing that the
stratification described by the Theorem can be defined without extension of the
base fields of the varieti
es involved. This material is essentially contained in \cite{Ve:76}.
In the course of the analysis we will frequently rely on two sets of
well-known and indeed basic facts from algebraic geometry which are stated here
with an emphasis on the relevant fields of definition.
The first set of facts deals with the singular locus of a variety defined over
a field $k$ of characteristic 0 (here we do not need any restriction on the
nature of the extension $k/\Q$).
Let $X$ be a variety of dimension $n$ over a characteristic zero field $k$,
then for any point $x \in X$ the following are equivalent:
\begin{enumerate}
\item $\Omega^1_{X,x}$ (the module of differentials at $x$) is a free module
of rank $n$ over the local ring ${\cal O}_{X,x}$ of $X$ at $x$;
\item ${\cal O}_{X,x}$ is a regular local ring.
\end{enumerate}
and, if either of these conditions obtains at $x$ we say $X$ is {\em smooth} at
$x$.
\begin{s}{Fact}
\label{sm}
There is an everywhere dense Zariski open set $U \subset X$ which is smooth
and the Zariski closed set $X-U$ is defined over $k$.
\end{s}
{\bf Remark}: For a discussion of fields of definition for arbitrary subsets
of schemes see [{\bf EGA IV}, \S 4.8].
The second set of facts is just Hironaka's well-known resolution of
singularities. Once again our only restriction is that we work over
characteristic 0 fields.
Let $X$ be variety defined over $k$ and ${\cal J}$ a coherent sheaf of ideals
defining a closed sub-scheme $D$ then we make the usual:
\begin{r}{Definition}
A blow-up of $X$ at $D$, otherwise known as a monoidal transformation of $X$
with center $D$, is a pair $(P, f)$ consisting of a variety $P$ and a morphism
$f:P \rightarrow X$ such that $f^{-1}({\cal J})$ is an invertible sheaf on $P$
and, for any other p
air $(P',f')$ with $f':P' \rightarrow X$ and $f'^{-1}({\cal J})$ an invertible
sheaf on $P'$, there is a unique morphism $g:P' \rightarrow P$ such that
\commtriang{$P'$}{$P$}{$X$}{$g$}{$f$}{$f'$}
commutes.\end{r}
A general procedure for constructing $P$ is to define
\[
P:=\Proj(\oplus_{d=0}^{\infty} {\cal J}^d)
\]
where we set ${\cal J}^0={\cal O}_X$. There is a natural map $P \rightarrow X$
(given by ${\cal O}_X \rightarrow \oplus {\cal J}^d$) and the universal
property is proved in \cite{H:77} Ch II, \S 7. In particular the field of
definition for $P$ is just t
he field of definition of $D$ or, equivalently, of ${\cal J}$.
\bl
Hironaka has shown,
\begin{s}{Theorem}
\label{m1}
Let $X$ be a variety defined over $k$, characteristic 0, then there is a closed
subscheme $D$ of $X$ such that:
\begin{enumerate}
\item the set of closed points of $D$ is the singular locus of $X$; and
\item if $f:{\tilde X} \rightarrow X$ is the monoidal transform of $X$ at $D$
then $\tilde X$ is smooth.
\end{enumerate}
\end{s}
\pf. \cite{Hi:64} (Main Theorem 1) \fp
For our purposes we note in particular that since $D$ is defined over $k$ by
Fact \re{sm} above we have that $\tilde X$ and $f$ are defined over $k$ as
well.
\begin{r}{Definition}
A divisor with normal crossings $D$ in a smooth variety $X$ is a divisor such
that for any $x \in D \subset X$ with local ring ${\cal O}_{X,x}$ and maximal
ideal ${\bf m}_{X,x}=(z_1, \dots , z_n)$, each component of $D$ passing
through $x$ is described b
y precisely one ideal $(z_i)$.
\end{r}
\begin{s}{Theorem}
\label{ms}
Let $X$ be a smooth variety defined over $k$, $W$ a nowhere dense sub-scheme of
$X$, then there exists a finite set of monoidal transforms
\[
f_i:X_{i+1} \rightarrow X_i
\]
with smooth centers $D_i$, for $0 \leq i < r$ and $X_0=X$ such that
\begin{enumerate}
\item $X_r$ is smooth;
\item if $\bar{f_i}$ is the composition of the $f_j$ for $0 \leq j < i$, then
$D_i \subset {\bar {f_i}}(W)$ for all $i$; and
\item ${\bar{f_r}}^{-1}(W)$ is an invertible sheaf whose support is a divisor
with normal crossings.
\end{enumerate}
\end{s}
\pf. \cite{Hi:64} (Cor 3, to Main Theorem II). \fp
Since we know that the singular locus of a variety $X$ over $k$ is nowhere
dense and hence its inverse image under the monoidal transform $f$ from Theorem
\re{m1} is nowhere dense, we can summarize the above by saying that Hironaka's
resolution of singula
rities starts with an arbitrary variety $X$, defined over $k$ characteristic 0,
and produces a smooth variety $X'$ and a morphism $f:X' \rightarrow X$ such
that the inverse image of the singular locus of $X$ becomes a divisor with
normal crossings in $X'$
and $X'$, $D$ and $f:X \rightarrow X$ are defined over $k$ as well.
\bitt{Stratifications and stratified isotopy in the Real Analytic Category}
The most natural setting for the study of stratifications of singular
spaces is the category of real analytic subspaces of smooth (real analytic)
manifolds and proper maps between them. A brief sketch of the aspects of the
theory used below is give
n here. The application to complex algebraic varieties follows. The best
current reference is \cite{G-M:88}.
Let $M$ be a real analytic manifold, $Z \subset M$ a closed subset and
\[
Z = \bigcup_{i\in S}S_i
\]
($S$ a partially ordered set) a decomposition of $Z$ as a union of a locally
finite collection of disjoint locally closed ``pieces'' or ``strata''
satisfying the {\em boundary} consition
\[
S_i \cap {\bar{S_j}} \neq \emptyset} \def\dasharrow{\to \Leftrightarrow S_i \subset {\bar
{S_j}}\Leftrightarrow i=j\;\;or\;\;i<j
\]
(in the last case we also write $S_i < S_j$). Such a decomposition is called a
{\em Whitney Stratification} if and only if it also satisfies:
\begin{enumerate}
\item each $S_i$ is smooth (not necessarily connected), and
\item each pair $(S_i,S_j)$ satisfies the {\em a} and {\em b} conditions,
namely, if we have a collection of points $\{x_i\} \subset S_i$ such that
$\{x_i\} \rightarrow y \in S_j$ and another set of points $\{y_i\} \subset S_j$
with $\{y_i\} \rightarrow y
$ such that the secant lines ${\overline{x_iy_i}}\rightarrow l$ and the tangent
planes $T_{x_i}S_i \rightarrow \tau$ then we have
\begin{itemize}
\item {\em a}: $T_yS_j \subset \tau$; and
\item {\em b}: $l \subset \tau$
\end{itemize}
\end{enumerate}
These conditions ensure that the pieces $S_i$ ``fit together'' well at an
infinitesimal level (see \cite{B-C-R:87} for examples). The conditions are
local and can be tested by taking local coordinates in $M$ about $y$. The
validity of the conditions is
independent of the choice of coordinate system. It is a theorem
(Hironaka-Hardt) that any subanalytic manifold admits such a stratification.
If a map behaves well with respect to stratifications we say it is a {\em
stratified} map. Specifically, let $Y_1 \subset M_1, \;\; Y_2 \subset M_2$ be
Whitney stratified subsets of manifolds $M_1, \; M_2$, and let $f:M_1
\rightarrow M_2$ be a real analy
tic map such that $f\!\mid Y_1$ is proper and $f(Y_1) \subset Y_2$, then $f$ is
{\em stratified} if for each stratum $A\subset Y_2$ we have $f^{-1}(A)$ a union
of connected components of strata of $Y_1$, say $f^{-1}(A) = \cup S_i$ and $f$
takes each $S_i$
submersively to $A$ (surjection on tangent spaces). There are two key results
on stratified maps which are often referred to as the $1^{st}$ and $2^{nd}$
(stratified) Isotopy Theorems.
\begin{s}{Theorem} For $Z \subset M$ a Whitney stratified subset of a real
analytic manifold, $f:Z \rightarrow \R^n$ proper and such that the restriction
to each stratum
\newline $f\!\mid A : A \rightarrow \R^n$ is a submersion, then there is a
stratum preserving homeomorphism $h:Z \rightarrow \R^n \times (f^{-1}(0) \cap
Z)$ such that
\commrect{$Z$}{$\R^n\times(f^{-1}(0)\cap
Z)$}{$\R^n$}{$\R^n$}{$f$}{$pr_1$}{$h$}{$id$}
\noindent commutes. In particular, the fibers of $f\!\mid Z$ are homeomorphic
by a stratum preserving homeomorphism.
\end{s}
\begin{s}{Theorem}
$A \subset M$, $B \subset N$ subanalytic subsets of real analytic manifolds,
$F: A \rightarrow B$ a proper subanalytic map. Then there exist
stratifications $S$, $T$ of $A$, $B$ into smooth subanalytic manifolds such
that $f$ is stratified with respect t
o $S$ and $T$. Furthermore, given any locally finite collection of subanalytic
subsets $\cal C$ of $A$ (resp. ${\cal D}$ of $B$ we can choose $S$ (resp. $T$)
such that each elements of ${\cal C}$ (resp. ${\cal D}$) is a union of strata
of $S$ (resp. $T$)
{}.
\end{s}
By the $1^{st}$ isotopy Theorem one obtains local topological triviality of the
$f$ along connected components of strata of $B$.
For further discussion and guidance to the literature see \cite{G-M:88} Part I,
Chapter 1, pp. 36 - 44.
We wish to employ this theory in the context of complex algebraic varieties
and to take our stratifications to be constructible sets whose fields of
definition we can control. Following a suggestion of Bernstein, Beilinson,
Deligne \cite{BBD:81} Chapter
6, we find that such a version of stratified isotopy theory has been given by
Verdier.
\bitt{Whitney Stratifications \`{a} la Verdier}
We now define a notion of Whitney stratification which is adapted to algebraic
varieties. The properties {\em a} and {\em b} above will be replaced by a
single property {\em w} which is also local in nature. Hence we will always
assume that {\em smooth}
complex algebraic varieties have been equipped with coordinate charts given by
their underlying real analytic manifold structure. Nothing will depend on the
choice of coordinates (see comments below).
We use the following notion of distnace between sub-vector spaces in a finite
dimensional Euclidean space $E$, ${\delta}(F,G)$
defined by
\[{\delta}(F,G) := \sup_{\small{\begin{array}{ccc} x\! & \in & F \\ \parallel
x \parallel & = & 1 \end{array}}}\!dist (x,G)\]
In particular, $\delta(F,G)=0 \Rightarrow F \subset G$.
\begin{r}{Definition}
A Verdier-Whitney stratification of (the complex points of) a complex algebraic
variety $V$
where $V$ is a $k$-variety
of finite type, $k$ a finitely generated extension of $\Q$, is a finite
disjoint partition
of $V$ by smooth constructible sets $A_i$
\[V=\bigcup_{{i}=1}^n A_{i}\]
such that
\begin{enumerate}
\item the ``boundary property'' holds, namely ${\overline{A_{i}}} \cap A_{j}
\neq \emptyset} \def\dasharrow{\to$ implies ${\overline {A_{i}}} \supset A_{j}$ and
\item if ${\overline {A_{i}}} \supset A_{\j}$ with $i \neq j$ then the pair
$(A_{i},A_{j})$ satisfies
the following property ``{\em w}" at every point $y \in A_{i}$:
Consider $A_{\alpha}$ and $A_{\beta}$ as real analytic manifolds and take
coordinate patches around $y$ to
some
Euclidean space $E$, then there exists a neighborhood $U \subset E$ of (the
image of)
$y$
and a positive real number $C$ such that $\forall x \in U \cap A_{i}$ and $y'
\in U \cap A_{j}$
(here $A_i$ and $A_j$ are taken to mean the images of some small open subsets
around $y$ in $E$) we have
\[{\delta}(T_{y'}{A_{j}},T_x{A_{i}}) \leq C \parallel x - y' \parallel\]
where $T_x{A_{\alpha}}$ is the tangent plane to $A_{\alpha}$ at $x$ and
$\delta$ is as defined above
\end{enumerate} \end{r}
A number of remarks are called for here.
As we are restricting ourselves to algebraic varieties, it is sufficient to
consider
stratifications with finite collections of subsets. The analytic cases
require infinite
collections of subsets. This permits a certain amount of simplification in the
definition and the subsequent arguments.
It also permits us to speak of the (common) field of definition of the
stratification as being the smallest field containing the fields of definition
of the $A_{\alpha}$.
On the other hand the condition {\em w} (so-called by Verdier) replaces the
more familiar
conditions `{\em a} and `{\em b} above. Condition {\em w} implies condition
{\em a} simply because it is a uniform version
of it but {\em w} does not imply {\em b} in general (consider the logarithmic
spiral at 0).
The key fact however is that this implication does hold when $A_{\alpha}$ is a
smooth
subanalytic subspace of a real analytic space and $A_{\beta}$ is a smooth
analytic
subspace of
$\overline {A_{\alpha}}$ (Kuo).
Verification of property {\em w} does not depend on the choice of coordinates
(for this it is important that the strata $A_i$ are required to be smooth).
The next key fact is that using resolution of singularities, we can stratify
arbitrary complex algeb
raic varieties in much the same way as real analytic manifolds.
The first Theorem we will require is,
\begin{s}{Theorem}
\label {3b}
If $V$ is a complex algebraic variety as above and $V_{\beta}$ a finite family
of
constructible subsets of $V$ then there is a Verdier-Whitney stratification of
$V$ such
that each $V_{\beta}$ is obtained as the union of strata. The stratification
is defined over the (common) field of definition of $V$ and the
$V_{\beta}$.\end{s}
The proof of this is based on
\begin{s}{Theorem}
\label {3a}
$V$ as above, $M$, $M'$ smooth, connected, locally closed subsets
such that $M \cap M' = \emptyset} \def\dasharrow{\to$, $M' \subset {\overline M}$ and ${\overline
M} - M'$ is
closed
(all for the Zariski topology), then there is a Zariski open $Y \subset M'$
containing all of the points $y \in M'$ such that $(M, M')$ has the property w
at $y$ and
$M' - V$ is Zariski closed. $Y$ is defined over the (common) field of
definition of $V$, $M$ and $M'$. \end{s}
\pf\quad of \re{3a}. When $M$, $M'$ are locally closed smooth {\em
subanalytic subspaces of a second countable real analytic space} $X$ a
corresponding Theorem is proved by
\begin{enumerate}
\item defining a subset $V \subset M'$ by removing ``bad points'' to arrive at
an open subanalytic subset of $X$ which is dense in $M'$, and then
\item showing that $(M, M')$ has property {\em w} at all points of $V$ by
taking coordinate charts in which $M'$ is an open subset of an affine space
$F$. There is an affine space $G$ such that locally $F\oplus G=X$, there is a
``blow-up'' $W$ of $X$ in w
hich $M'$ is described as a divisor with normal crossings $\prod_1^qz^{n_i}_i$
and there is a map $\pi:W \rightarrow X=F\oplus G$. Additional work involving
an analysis of the matrix representation of $d\pi$ then gives the desired
result.
In what follows we show how $V$ is defined in the complex algebraic case, show
that $V$ is a Zariski open and describe its field of definition. We do not
review the proof that $(M, M')$ has the property {\em w} at all points of $V$
since we are principal
ly interested in showing that the stratifications can be taken to be algebraic
and have the right fields of definition. The reference for all omitted parts
of proofs is \cite{Ve:76}.
We use
resolution of singularities to find a smooth complex algebraic variety $W$ and
a proper
morphism $\pi :W \rightarrow V$ with $\pi(W)={\overline M}$ and
$\pi^{-1}(M')=D \subset W$ is a divisor whose singularities are at worst normal
crossings. By \re{m1} and \re{ms} resolution of singularities takes place
without extension of the field of
definition so that both $W$ and $D$ are defined over the same field as $V$.
The next step is to produce the desired open set by removing ``bad points'', in
this case, the normal crossings singularities.
Consider the subset $D_q \subset D$ of points of $D$ where at least $q$
irreducible
local components of $D$ meet (this set is empty for $q \gg 0$) let ${\tilde
{D_q}}$ be the normalization of $D_q$ (separating the points lying on the
various components)
and let $i:{\tilde{D_q}}\rightarrow D_q$ be the normalization map.
${\tilde{D_q}}$ is
a smooth algebraic variety defined over the same field as $D$ so we can
consider
\[d({\pi}\circ i_q):\; \Omega^1_{D_q} \rightarrow (\pi \circ
i_q)^*\Omega^1_{\overline M}\]
where $\Omega^1$ is the sheaf of differentials. We now have one more
correction to make,
namely we must consider the points where this map is not
surjective so that we do not have a submersion. These form a Zariski closed
subset $R_q \subset D_q$ [EGA IV, \S. 17.15.13] defined by a Jacobian condition
and hence this singular locus is defined over the same field as $D_q$.
$i_q(R_q)$ is closed since $i_q$ is finite and hence a closed map (the
``going-down" Theorem) and the collection of
$i_q(R_q)$ is finite so that $S:=\pi(\bigcup_q i_q(R_q)) \subset {\overline M}$
is Zariski
closed. We thus have that \[Y:=M' \cap (M - S) \subset {\overline M}\] is a
Zariski open
subset of $M$ defined without extension of the base field such that $Y$ is
dense in $M'$
and it is smooth by construction. The proof of \re{3a} now proceeds by
considering the local analysis of
the smooth real analytic varieties undrlying $M$, $M'$ and $Y$ as very briefly
described above.\fp
\pf\quad of Theorem \re{3b}. We need two Lemmas.
\begin{s}{Lemma}
\label{311}
Let $X$ be an algebraic variety, $Y_{\beta}$ a finite family of constructible
subsets of $X$, then there is another finite family of subsets of $X$,
$B_{\alpha}$
satisfying: \begin{enumerate}
\item for all $\alpha$, ${\overline B}_{\alpha}$ and ${\overline
B}_{\alpha}-B_{\alpha}$ are
Zariski closed and the $B_{\alpha}$ are connected and smooth;
\item the $B_{\alpha}$ partition $X$ and each $Y_{\beta}$ is the union of a
collection
of $B_{\alpha}$;
\item ${\overline B}_{\alpha} \cap B_{\beta} \neq \emptyset} \def\dasharrow{\to \Rightarrow
B_{\beta} \subset {\overline B}_{\alpha}$
\end{enumerate} and the $B_{\alpha}$ can be defined without extending the
(common)
field of definition of the $Y_{\beta}$. \end{s}
\pf. Since the $Y_{\beta}$ are locally closed we have that the sets
${\overline Y}_{\beta}$ and ${\overline Y}_{\beta} - Y_{\beta}$ are Zariski
closed and we replace the family $Y_{\beta}$ with the family of Zariski closed
sets $\{X,{\overline Y}_{\beta
},{\overline Y}_{\beta}-Y_{\beta}\}$ which we continue to refer to as
$Y_{\beta}$.
Let $\cal F$ be the largest collection of Zariski closed subsets of $S$ which
is such that: \begin{itemize}
\item for all $\beta$, $Y_{\beta} \in {\cal F}$;
\item ${\cal F}$ is closed under intersection;
\item for all $Z \in {\cal F}$ the irreducible components of $Z$ are in ${\cal
F}$; and
\item $Z \in {\cal F}$ implies that the set of singular points of $Z$, $sing\;
Z \in {\cal F}$.
\end{itemize}
We can get at least one such collection with these properties by taking the
family of sets
consisting of the $Y_{\beta}$ and their singular loci $(Y_{\beta})_{sing}$ and
then
closing this collection under taking of irreducible components (defined over
the field of
definition of the $Y_{\beta}$) and intersections. In particular $\cal F$ can
be
constructed without extending the field of definition of the $Y_{\beta}$.
Then we define
\[B_{\alpha}=Z_{\alpha} - \bigcup_{\stackrel{Z_{\beta}\subset
Z_{\alpha}}{Z_{\beta} \neq Z_{\alpha}}}Z_{\beta}\]
where the $Z_{\gamma}$ are the irreducible sets in $\cal F$. It is clear that
the
$B_{\alpha}$ have the properties set out in the Lemma.\fp
\begin{s}{Lemma}
\label{3l2}
Let $V \subset X$ be a constructible set which is connected and smooth and let
$Z
\subset {\overline V} - V$ be Zariski closed, then there is a (finite)
partition
$B_{\alpha}$ of $Z$ such that such that for all $\alpha$, ${\overline
B}_{\alpha}$,
${\overline B}_{\alpha} - B_{\alpha}$ are Zariski closed in $X$, the
$B_{\alpha}$ are
smooth and connected and the pairs $(V, B_{\alpha})$ have property {\em w}.
The
$B_{\alpha}$ are defined over the (common) field of definition of $X$, $V$ and
$Z$
\end{s}
\pf. By induction. The statement is true for $Z= \emptyset} \def\dasharrow{\to$. Assume $Z
\neq \emptyset} \def\dasharrow{\to$ so we apply Lemma \re{311} to get a finite collection of
smooth connected
subsets $U_{\alpha} \subset Z$, $U_{\alpha} \cap U_{\beta} = \emptyset} \def\dasharrow{\to$
defined
over the field of definition of $Z$such that the ${\overline U}_{\alpha}$,
${\overline U}_{\alpha} -U_{\alpha}$, $Z-U_{\alpha}$ are all Zariski closed and
$Z-\bigcup_{\alpha}U_{\alpha}$ is a Zariski closed set of lower dimension. By
Theorem \re{3a} there is an open dense $W_{\alpha} \subset U_{\alpha}$ such
that the
${\overline U}_{\alpha} - W_{\alpha}$ are Zariski closed and the pairs
$(V, W_{\alpha})$ have the property {\em w}.
Now $Z_1:=Z - \cup W_{\alpha}$ is Zariski closed with dimension lower than $Z$
so
we can apply the induction hypothesis to it. Combine the $B_{\alpha}$ thus
obtained
with the $W_{\alpha}$ to get a new collection of $B_{\alpha}$. Since the
$W_{\alpha}$ provided by Theorem \re{3a} are constructed without extending
fields of
definition we are done.\fp
Returning to the Proof of Theorem \re{3b} we start by restricting to the case
of $V$
irreducible and proceed once again by induction.
For $V= \emptyset} \def\dasharrow{\to$ the Theorem is trivially true so suppose $V \neq
\emptyset} \def\dasharrow{\to$ and
replace the $V_{\beta}$ in the statement of the Theorem with a the family $\{V,
{\overline V}_{\beta}, V - V_{\beta}, {\overline V}_{\beta} - V_{\beta}\}$ as
in the
proof of Lemma \re{311} so that this new family (which we still call
$Y_{\beta}$) is
made up of Zariski closed sets.
Since $V$ is irreducible there is a ${\beta}_0$ such that $Y_{{\beta}_0}$ is
open and
dense in $V$. Hence $V_1:=V-Y_{{\beta}_0}$ is closed and is of lower dimension
than
$V$. We also clearly have that $\beta \neq {\beta}_0 \Rightarrow Y_{\beta}
\subset
X_1$.
Let $B_{\alpha}$ be a partition of $V_1$ coming from Lemma \re{3l2} and apply
Lemma \re{311} to the $Y_{\beta}$ and $B_{\alpha}$ together. This produces a
common refinement $\{C_{\gamma}\}$ which still has property {\em w} because, in
general, if $M$, $M'$ are locally closed subsets of an algebraic variety $X$
which are
smooth with $M' \subset \overline M$ and $M' \cap M = \emptyset} \def\dasharrow{\to$, then if
there is a
locally closed and smooth $M'' \subset M'$, the pair $(M, M'')$ will have the
property
{\em w} if the pair $(M,M')$ does.
Now assume for the moment the following \newline
{\bf Claim}: If $V = V_{\alpha}$ (finite union) and the $V_{\alpha}$ are
Zariski closed,
then if Theorem \re{3b} is true for the $V_{\alpha}$ it is true for $V$.
$V_1$ is a finite union of irreducibles of dimension lower than $V$ so apply
the
induction hypothesis to $V_1$ and the $C_{\gamma}$ to get a Whitney
stratification of
$V_1$ and add $Y_{{\beta_0}}$
If we can now prove the {\bf Claim} we have just made we will both complete the
proof
for the irreducible case and for the general case as well. So let $V=Y \cup Z$
be a union
of irreducibles. Apply the result in the irreducible case to $Z$, $Y_{\beta}
\cap Z$, $Y
\cap Z$ to get a Whitney stratification of $Z$ such that $Y_{\beta} \cap Z$ and
$Y \cap Z$ are unions of strata.
Now apply the result again to $Y$, $Y_{\beta} \cap Y$ and the $A_{\alpha}$ such
that
$A_{\alpha} \subset Y \cap Z$ to get a Whitney stratification of $Y$ such that
the
$A_{\alpha}$ and $Y_{\beta} \cap Y$ are unions of strata. Take $B_{\beta}$ and
those $A_{\alpha}$ such that $A_{\alpha} \subset Z - (Y \cap Z)$. Note that
property
{\em w} still holds by the remark made above. Continue by induction. Nothing in
any of
these procedures requires an extension of fields of definition.\fp
\bitt{Stratified Morphisms}
The next ingredient is the demonstration that quite general algebraic-geometric
morphisms behave well with respect to
stratifications.
Recall that a morphism of stratified spaces $f:X \rightarrow Y$ is called a
{\em stratified} morphism if it is proper and if the inverse image of a stratum
of $Y$ under $f$ is a union of strata of $X$ and each component of these strata
is mapped sunmersiv
ely (as a real analytic manifold) to $Y$.
We now have a version of the $2^{nd}$ isotopy theorem for complex algebraic
varieties.
\begin{s}{Theorem}(Verdier)
\label{3c}
If $f:X \rightarrow Y$ is a morphism of complex algebraic varieties and $f$ is
proper
then there are Verdier-Whitney stratifications $S$ and $T$ of $X$ and $Y$,
defined over the (common) field of
definition of $X$,$Y$ and $f$, such that $f$ is transverse to $S$ and $T$
\end{s}
\pf.
{\em Step 1} We first show that if $X \rightarrow Y$ is proper and $S$ a
stratification of $X$ such as given in Theorem \re{3b}, then there is a Zariski
open $U \subset f(X) \subset Y$ which is smooth in $Y$ and (Zariski) dense in
$f(x)$ such that $f\!\m
id f^{-1}(U) \rightarrow U$ takes the connected components of$S \cap f^{-1}(U)$
submersively to $U$.
Set
\[X_q=\bigcup_{dim\:S_{\alpha} \leq q}S_{\alpha}\]
this is closed, smooth, constructible subset of $X$ and we write $f_q$ for
$f\mid X_q:X_q \rightarrow Y$.
Once again, let $R_q$ be the set of points in $X_q$ where
\[df_q: \Omega^1_{X_q} \rightarrow f^*_q(\Omega^1_Y)\]
is not onto and set
\[U_q=Y-Y_{reg}\cap (\bigcup_{q'\leq q}f_{q'}(R_{q'}))\]
where $Y_{reg}$ is the set of regular points of $Y$ (as noted above this set is
defined
without extension of the field of definition in characteristic 0). The proof
now
proceeds by induction on $q$ to show that
\[f_q\mid U_q:f^{-1}_q(U_q) \rightarrow U_q\]
maps the connected components of to $S\cap f^{-1}_q(U_q)$ to $f_q(U_q)$ where
we always have ${U_q}$ Zariski dense in $Y$. Note that $Y-U_q$
is automatically a Zariski closed subset since the $f_{q'}(R_{q'})$ are.
So let $S_{\alpha} \subset X_{q}$ be a stratum. If dim $S_{\alpha} < q$ then
there is a $q'<q$ such that $f_{q'}$ maps the connected components of $S \cap
f^{-1}_{q'}(U_{q'})$ to $U_{q'}$. Thus we can assume dim$S_{\alpha}=q$ so
$S_{\alpha} \subset X_{
q}$ is
Zariski open and smooth and is a connected component of $W=X_q - X_{q-1}$
which
is
also open and smooth.
Now $R_q \cap f^{-1}(U_{q-1}) \subset W$ since
$\forall q' < q$
\[coker\:(df_{q'}) \rightarrow coker\:(df_q)\]
is surjective over points of $X_{q'}$ as $f_q$ agrees with $f_{q'}$ on
$X_{q'}$.
$f_q(R_q) \cap U_{q-1}$ is a Zariski closed subset and $U_{q-1} - (f_q(R_q)
\cap U_{q-1})$ is dense in $U_{q-1}$ and
$f_q \mid U_q$ takes $S \cap f^{-1}_q(U_q)$ submersively to $f_q(U_q)$.
But $f:X \rightarrow Y$ is
proper so $f_q(R_q)$ is empty for $ q \gg 0$ and we can form $U:= \bigcap_q
U_q$. This is dense in $Y$ with
$Y - U$ an algebraic Zariski closed set and by construction $f$ takes the
connected components of $S\cap f^{-1}(U)$ submersively to $U$.
{\em Step 2} Now take $f:X \rightarrow Y$ with stratifications $S$ and $T$
respectively as guaranteed by Theorem \re{3b}. By step 1 we find a Zariski
open $U \subset f(X)$ smooth in $Y$ and dense in $f(X)$ with $f\!\mid
f^{-1}(U)$ submersive on connecte
d components. We now re-stratify $Y$ per Theorem \re{3b} with $U$ a union of
strata. Take $f^{-1}$ of these strata and restratify $X$. $f$ now behaves as
we want on $f^{-1}(U)$. Consider $f\!\mid X-f^{-1}(U)$, find a $U' \subset
f(X-f^{-1}(U))=f(X)-f(
U) \subset Y$ which is Zariski open, smooth in $Y$ and dense in $f(X)-f(U)$ and
repeat the above restratification process. Since $dim\: U' < dim \:U$ the
procedure terminates after a finite number of steps. Using Theorem \re{3b} we
will always have that
the $U$, $U'$, $\dots$ will be unions of strata and similarly for the
$f^{-1}(U)$, $f^{-1}(U')$, $\dots$ and $f$ will clearly take each connected
component of the stratifications of the $f^{-1}(U)$, $f^{-1}(U')$, $\dots$ to
$f(U)$, $f(U')$, $\dots$
\fp
\bitt{Stratified Isotopy}
Lastly we state a version of the $1^{st}$ isotopy Theorem:
\begin{s}{Theorem}(Thom)
\label{Th}
$X$, $Y$ are real analytic spaces, $S$ and $T$ stratifications,
$f:X \rightarrow Y$ proper and submersive on the connected components of the
strata of $X$.
Set $y_0 \in Y$. Write $X_0=f^{-1}(y_0)$, $S_0=X_0 \cap S$.
Then there is an open neighborhood (in the complex topology) $y_0 \in V
\subset Y$ and a homeomorphism
$\phi:(f^{-1}(V),\;S \cap f^{-1}(V)) \rightarrow (X_0 \times V,\;S_0 \times V)$
preserving the stratifications and compatible with projections to $V$.\end{s}
\pf. Classically this is proved using techniques from differential topology
and is quite difficult. Using the condition {\em w} in place of the more
standard {\em a} and {\em b} conditions Verdier is able to give a fairly
self-contained proof in a few p
ages \cite{Ve:76}. Nonetheless we will pass this over in silence since our
objective is not to see how the result is proved but rather to show how it can
be applied to give useful transversality properties for the algebraically
defined stratifications de
scribed above. See \cite{Ve:76} for the missing details. \fp
When combined with Verdier's results this gives:
\begin{s}{Corollary}
\label{313}
Let $X \stackrel{f}{\rightarrow} Y$ be a proper morphism of algebraic
varieties, then the topological type of the fibers of $f$ over a connected
component of a stratum of $T$ is constant. \end{s}
\pf.
By Theorem \re{3c} there are stratifications $S$ and $T$ of $X$ and $Y$
respectively, defined over the common fields of definition of $X$, $Y$ and $f$,
such that the inverse image of any stratum of $T$ is a union of strata of $S$
$f^{-1}(T_{\alpha}) =
\cup S_{\beta}$ and each connected component of a stratum of $S$ is mapped
submersively onto a stratum of $T$. Thus only the last statement requires
discussion. Consider a connected component of a stratum of $T$, and call it
$W$. Partition $W$ into subs
ets such that the topological type of the fibers of $f$ are constant on each
member of the partition. The sets partitioning $W$ are then open by Theorem
\re{Th} and they are disjoint by construction. Since $W$ is connected only one
of the sets in the pa
rtition is non-empty. \fp
\bit{Principal Results}
\bitt {A Sufficient Condition for Topological Stability Under Conjugation}
The results described above can be applied to give a sufficient criterion for
an algebraic
variety and its conjugates to have the same topological type.
\begin{s}{Theorem}
\label{main}
Let $V$ be a $k$-variety, $k$ a finitely generated extension of $\Q$ and
suppose there exists a family $f:Y \rightarrow B$, that is, a proper morphism
of complex algebraic varieties such that:
\begin{enumerate}
\item all of the conjugate complex algebraic varieties $V_{\sigma}$ are
isomorphic as $k$-varieties to fibers of $f$
(in other words for each $\sigma : k \rightarrow \C$ there is a point
$b_{\sigma} \in B$ and $k$-isomorphism $V_{\sigma} \simeq f^{-1}(b_{\sigma})$);
and
\item $f: Y \rightarrow B$ arises by base extension from $\Q$, that is,
there are $\Q$ varieties $Y_{/{\Q}}$ and $B_{/{\Q}}$ and a morphism
$f_{/{\Q}}: Y_{/{\Q}} \rightarrow B_{/{\Q}}$ such that
\commrect{$Y \simeq Y_{/{\Q}} \times \C$}{$Y_{/{\Q}}$}{$B \simeq B_{/{\Q}}
\times \C$}{$B_{/{\Q}}$}{$f$}{$f_{/{\Q}}$}{${\beta}_Y$}{${\beta}_B$}
commutes,
\end{enumerate}
then the topological type of $V_{\sigma}(\C)$ is independent of $\sigma$.
\end{s}
\pf. By Corollary \re{313} we can stratify $Y$ and $B$ with stratifications
$S$ and $T$
defined over $\Q$ so that the map $f$ is topologically locally trivial over
each connected component of the strata. We need only show therefore that the
points $b_{\sigma}$,$b_{\tau}$ corresponding
to conjugate varieties $V_{\sigma}$, $V_{\tau}$ must lie in a single connected
component of a stratum of $S$.
Since $v_{\sigma}:=f^{-1}(b_{\sigma})$, $v_{\tau}:=f^{-1}(b_{\tau})$ are
$k$-isomorphic to varieties which differ only by conjugation of their field of
definition these fibers of $f$ are mapped to the same subscheme $v_{/{\Q}}$ of
$Y_{/{\Q}}$ by ${\beta}_Y$. This is in turn
mapped to a subscheme of $B_{/{\Q}}$; call it $b_{/{\Q}}$. Now we claim
$v_{/{\Q}}$ and hence $b_{/{\Q}}$ are irreducible. If not, the $v_{\sigma}$
divide up into subsets which are interchanged by some $\phi \in \Aut(\C)$ (each
$v_{\sigma}$ is irreducibl
e since it is isomorphic to a variety $V_{\sigma}$. Thus the largest
irreducible closed subschemes of $Y_{\Q}$ containing the images of these
subsets are not equal to $Y_{\Q}$. But $Y_{\Q}$ is irreducible since $Y$ is,
so we have a contradiction. Thus $
b_{/{\Q}}$ is
defined by a sheaf of prime ideals ${\cal P}$ with local ring ${\cal O}_{\cal
P}$ and
residue field ${\bf k}_{\cal P}$.
Take the inverse image of $b_{/{\Q}}$ under ${\beta}_B$ in $B \simeq B_{/{\Q}}
\times \C$. Call this $b_{/{\C}}$. The points
$b_{\sigma}$ and $b_{\tau}$ lie in $b_{/{\C}}$ by
commutativity of the diagram in Theorem \re{main} and call the residue fields
of these points ${\bf k}_{\sigma}$ and ${\bf k}_{\tau}$. Now ${\beta}_B$ is an
open map so that we have ${\bf k}_{\sigma} \simeq {\bf k}_{\cal P} \otimes \C$
and
similarly for ${\bf k}_{\tau}$.
The stratification $T$ of $B$ is defined over $\Q$ and we claim that points of
a subscheme of $B$, defined over $\Q$ and with residue fields isomorphic to
$k_{\cal P}$ over $\C$ must lie in a single irreducible component of a stratum
of $T$.
Assume not. By Verdier's results $b_{/{\C}}(\C)$, the set of complex points of
$b_{/{\C}}$ is a union of strata $ \cup T_i$ and each $T_i$ is defined over
$\Q$. Assume that we have $b_{\sigma} \in T_1$ and $b_{\tau} \in T_2$. We may
assume $T_1 \cup T_2$ is all of $b_{/{\C}}(\C)$. By the definition of a
stratification, if ${\overline {T_1}} \cap T_2 \neq \emptyset} \def\dasharrow{\to$ then
${\overline {T_1}} \supset T_2$. This implies that one of the points
$b_{\sigma}$ or $b_{\tau}$ lies in a zariski closed subset of $b_{\C}$ and
hence does not have residue field isomorphic to $k_{\cal P}$ over $\C$. So we
must have ${\overline {T_1}} \cap
T_2 = \emptyset} \def\dasharrow{\to$ and similarly ${\overline {T_2}} \cap T_1 = \emptyset} \def\dasharrow{\to$.
But then
$b_{/{\Q}}$ is not irreducible. Thus the points $b_{\sigma}$ and $b_{\tau}$ lie
in a single irreducible component of the stratification of $B$. Finally, over
$\C$, irreducibility in the zariski topology implies connectedness in the
complex topology so $b
_{\sigma}$ and $b_{\tau}$ lie in a single connected component of the
stratification and hence the topological types of $f^{-1}(b_{\sigma})$ and
$f^{-1}(b_{\tau})$ are the same.\fp
{\em Remark}: Shimura has used the existence of non-homeomorphic conjugate
varieties to show that the irreducible components of Chow varieties are not
necessarily defined over $\Q$ \cite{Sh:68}. I am grateful to Professor J-P.
Serre for providing this r
eference.
\bitt{Corollaries}
Following, for example \cite{BBD:81}, it is easy to see that any complex
projective variety $V$ (say defined over a field $k$) can be embedded in a
family
defined over $\Q$. One merely considers the coefficients $c_{\alpha \beta}$
of the
homogeneous ideal defining $V$ as indeterminates. This produces a family
$f: Y \rightarrow S$ over an affine base $S$ with the original variety $V$
isomorphic to
the fiber of $f$ over the point of $S$ corresponding to the $c_{\alpha \beta}$.
This family clearly contains all of the conjugates of $V$, since these are
obtained by conjugating the coefficients in its homogeneous ideal.
Furthermore, since $S$ is affi
ne
this family is actually ``arises via base extension from $\Q$" in the sense of
Theorem \re{main}.
Fortunately we cannot use this technique to show that all varieties have
invariant topological type under conjugation due to the fact that the families
we obtain in this way may not be irreducible. Consider as an example the
hyperelliptic curve ${\cal C
}$ constructed by Shimura \cite{Sh:72}
\[
y^2=a_0x^m + \sum^m_{r=1}(a_rx^{m+r} + (-1)^r{\bar {a_r}}x^{m-r})
\]
If we treat the $a_r$ and $\bar{a_r}$ as independent indeterminates $a_r$,
$b_r$ we get a family of curves $F$ and $\cal C$ lies over a point $p_0$ in the
locus where $a_r=\bar{b_r}$. The family has two components which are
interchanged by complex conjug
ation. We cannot use it in Theorem \re{main} therefore to show that the
conjugates of $\cal C$ are homeomorphic (although, of course, this can be shown
in other ways).
If the field of moduli of $V$ coincides with its field of definition however,
we can use the BBD type family in Theorem \re{main} by virtue of,
\begin{s}{Proposition}
Let $V$ be a projective variety whose field of definition $k$, a finitely
generated extension of $\Q$, coincides with its field of definition. Define
the family $f: Y \rightarrow S$ as above, by letting the coefficients
$\{c_{\alpha \beta}\}$ of the homo
geneous ideal of $V$ vary. Then all of the conjugates of $V$ are isomorphic to
fibers $f^{-1}(b_i)$ where the $b_i$ lie in a single connected component of a
Verdier-Whitney stratification of $S$ and hence have the same topological type
.\end{s}
\pf. Examining the proof of Theorem \re{main} we see that we did not need there
the full assumption that $Y$ is irreducible, but merely that the conjugates of
$V$ do not lie in separate irreducible components of $Y$.
Consider the action of $\sigma \in \Aut(\C)$ on $f:Y \rightarrow S$. Since the
field of moduli of $V$ coincides with its field of definition $k$ we have that
$Y \rightarrow S$ is stable under $\sigma$ if and only if $\sigma \in \Aut (\C
/k)$ (that is, th
e field of definition of $f:Y \rightarrow S$ is $k$). Let $Y=\cup Y_i$ be a
decomposition of $Y$ into irreducible components with fields of definition
$k_i$. If $Y_i \neq Y$ then $k \subset k_i$ and $k \neq k_i$. Let the fibers
of $f$ corresponding to t
he conjugates $V_{\sigma}$ be $v_{\sigma}$ and suppose, for example, that
$v_{\sigma}$ and $v_{\tau}$ were in $Y_i$ and $Y_j$ respectively. Then there is
some $\phi \in \Aut(\C/ k)$ taking $Y_i$ to $Y_j$. but $\phi$ fixes
$v_{\sigma}$ and $v_{\tau}$ so b
oth must be in $Y_i \cap Y_j$. This argument applies to all of the
$v_{\sigma}$ and all components $Y_i$ hence all of the conjugates of $V$ can be
identified with fibers of $f$ lying in a single component $Y_i$ (we may pick
any component). We now replac
e $Y$ with $Y_i$, which is irreducible, and apply the theorem.
\fp
\begin{s}{Corollary}
The topological type of a canonically embedded variety is invariant under
conjugation.\end{s}
\pf. The canonical embedding is given over the field of definition and hence
the field of definition coincides with the field of moduli. \fp
This enlarges the class of varieties previously known to have conjugate
invariant topological type. Further examples are given in \S 4 where we show
that complete intersections in homogeneous varieties have fields of moduli
which coincide with the fields
of definition.
For a proof that Serre's original examples of non-homeomorphic conjugate
varieties do not have fields of definition which coincide with their fields of
moduli see \cite{Re:94}
\bit{Complete Intersection Type Varieties}
\bitt{Generalities}
We can use Theorem \re{main} to exhibit further classes of
varieties which have topological type invariant under conjugation. These
classes of varieties, which include (normal) complete intersection varieties in
projective
space, may be parametrized by vector spaces of sections of vector
bundles and this linear structure provides a natural method of
descending from $\C$ to $\Q$.
The Deformation Theory of these varieties has been studied by a
number of authors including \cite{K-S:58}, \cite{S:75}, \cite{B:83} and
\cite{W:84}
in a series of papers with results extending from smooth hypersurfaces through
to the more general cases. The most general result along these lines is:
\begin{s}{Theorem} (Wehler)
\label{weh}
Let $Z=G/H$ be a non-singular homogeneous complex variety, quotient of a
simple,
simply connected Lie group $G$ by a parabolic subgroup $H$,
$E=\oplus_{j=1}^r{\cal O}_Z(d_j)$ a vector bundle, $s \in H^0(Z,E)$ a
section and $X$ a complex variety described by the zero locus of $s$
such that codim$X=r$ and $X$ is not a $K$-3 surface. Then the vector
space $H^0(Z,E)$ parametrizes a complete set of small deformations of
$X$ and these deformations are given by the family
\[ Y :=\{(z,s) \mid z\in Z, s\in H^0(Z,E), s(z)=0\} \rightarrow
H^0(Z,E)
\]
\end{s}
If we are to apply the Theorem to this case we must show that the family $Y
\rightarrow H^0(Z, E)$ satisfies the conditions of Theorem \re{main}. To do
this we reprove the theorem using algebraic geometric techniques to obtain a
family over $\Q$ which
will have the required properties.
The proofs given here therefore are
modelled on Wehler's but adapted to algebraic rather
than analytic geometry. The key in both the analytic and algebraic
approach is to establish a close connection between the deformation
theory of the objects and their Hilbert schemes as will be explained shortly.
For another approach to establishing the algebraic deformation theory of
complete intersections see \cite{Ma:68}.
\bitt{Comparison of Hilbert Scheme and Deformation Functors}
Let $Z$ be an arbitrary non-singular projective variety over a field
$k$ and $E \rightarrow Z$ an algebraic $k$ vector bundle over $Z$. Consider the
scheme $X
\subset Z $ defined by a global section $s_0 \in H^0(Z,E)$. We construct this
scheme as
follows: the section $s$
defines a map of
sheaves ${\cal O}_Z \rightarrow E$ sending the section ${\bf 1}$ of
${\cal O}_Z$ to the
section $s$ (we are abusing notation here by not distinguishing between the
vector bundle and its associated locally free sheaf). There is a dual
map
$\check s:\check E \rightarrow {\cal O}_Z$ and $X$ is said to be
defined by $s$ if ${\cal O}_X$ fits into an
exact sequence of sheaves
\[\check E \stackrel{\check s}{\rightarrow} {\cal O}_Z \rightarrow
{\cal O}_X \rightarrow 0 \]
$X$ is sometimes referred to as the zero scheme of $s$.
\begin{r}{Definition} The Hilbert functor of $X$ (in $Z$), is
the functor $\Hilb$ from $\cal C$, the category of local artin rings with
residue field $k$ to the category ${\em Sets}$ which assigns to each
object $A$ in $\cal C$ the set of schemes $Y$ which fit into the
following diagram
\commrect{$Z \supset X\simeq Y \times_k A$}{$Y \subset Z \times \Spec
A$}{$\Spec k$}{$\Spec
A$}{}{}{}{}
with $Y$ flat over $\Spec A$.\end{r}
\begin{r}{Definition} The (affine) projective cone (hereinafter simply
``the cone'') $C_X$ of (or on) a projective
variety $X \subset \Pj^n$ is given by
\[
C_X:= \Spec ({\cal O}_X \oplus
{\cal O}_X(1) \oplus {\cal O}_X(2) \oplus \dots )
\]
The vertex $p$ of
$C_X$ is the (closed) subscheme defined by the augmentation ideal,
$ker\:{\epsilon}$, where
\[
\epsilon: ({\cal O}_X \oplus
{\cal O}_X(1) \oplus {\cal O}_X(2) \oplus \dots) \rightarrow {\cal
O}_X
\]
$\Spec ({\epsilon})$ thus defines a map $X \rightarrow C_X$ whose image is
$p$.\end{r}
For future use we recall that the cone with vertex removed
(\'{e}point\'{e})
\[C_X - p \simeq \V({\cal O}_X(-1))\simeq \Spec (\oplus {\cal O}_X(n))\]
where $\V$ denotes
the operation of taking the vector bundle associated to a locally free
sheaf. There is an action of $\G_m$ on $C_X$ (and on $C_X-p$) with integral
weights coming from the action on each tensor power ${\cal O}(n)$.
For details see [{\bf {EGA II}} \S 8.4 - 8.6].
Next we define a deformation functor $\Def_{C_X}$
as the functor which assigns to each object $A$ in $\cal C$ the set of
deformations ${\cal O}_{C_X,p}(A)$ of the $k$-algebra ${\cal
O}_{C_X,p}$ which is the local ring of $C_X$ at $p$.
There is a natural morphism $h$ from the Hilbert functor to this deformation
functor obtained by assigning to a scheme $Y$ as above the local ring
of the vertex of the projective cone on
$Y$. This is naturally a deformation of the local ring of the vertex
of the projective cone on $X$. The morphism thus consists of ``forgetting the
embedding in $\Pj^n$
or, in terms of the underlying rings, forgetting the gradings.
Comparison Theorems between Hilbert functors and Deformation functors
go back (at least) to Schlessinger \cite{Sch:71} and can be found in
\cite{P:74}, \cite{K:79} and \cite{Wa:92}. We will use a version due
to Kleppe which employs Andr\'{e}-Quillen cohomology to avoid unnecessary
smoothness conditions.
Since $X$ is given as the zero-scheme of a section of a vector bundle we want
to describe its Hilbert scheme in the same way. So define functors
$F_{s_0}$ from the category ${\cal C}$ to {\em Sets} which, for any given
section
$s_0 \in H^0(Z,E)$, assign to an object $A$ of ${\cal C}$ the set of zero
schemes in $Z \times \Spec A$ of sections $s_A \in H^0(Z \times\Spec A,
E \times \Spec A)$ which reduce to $s_0$ over the closed point $k$ under
the map $\Spec k \rightarrow \Spec A$.
If we further assume that codim$X$=rank$E$ we have
that $X$ is (at least) a local complete intersection. Moreover, such zero
schemes $s_A$
are flat over
$\Spec A$. By [{\bf EGA IV}
\S 19.3.8] the zero scheme defined by any of
the $s_A$ is a
local complete intersection as well. Thus we obtain elements of ${\Hilb}_X(A)$
as sections of
vector bundles $E \times \Spec A$ and a morphism of functors $F_{s_0}
\stackrel{f}{\rightarrow} \Hilb_X$.
We now study this morphism of functors.
\begin{s}{Proposition}
\label{surj}
Let $Z$, $E$ and $A$ be as above and let $X$ be
the zero set of a section $s_0$ with codim $X$=rank$E$, write ${\cal
I}_X \subset {\cal O}_Z$ for the ideal sheaf of $X$ and suppose that
\[H^1(Z,E \otimes{\cal I}_X)=0\]
then the above morphism of functors
$f: F \rightarrow \Hilb$ is surjective on tangent spaces.\end{s}
\pf. The tangent space to $H$ at $h_0$, is
$\Hilb_{s_0}(k[{\epsilon}])$ and it
is standard that
\[T(H,h_0)=Hom_{{\cal O}_Z}({\cal I}_X, {\cal O}_X)=Hom_{{\cal
O}_X}({\cal I}_X/{\cal I}^2_X, {\cal O}_X)\]
(for the first isomorphism see \cite {G:61}, for the second see
\cite{H:77} Ch. II, \S 8). We also have
\[Hom_{{\cal
O}_X}({\cal I}_X/{\cal I}^2_X, {\cal O}_X)\simeq H^0(X,N_{X/Z}) \simeq
H^0(X,E\!\mid_X)\]
where $N$ is isomorphic to the tangent space
to $\Hilb$. But the tangent space to
$F:=F(k[{\epsilon}])$ is the vector space of sections of
$E \times \Spec(k[{\epsilon}]) \rightarrow Z \times
\Spec(k[{\epsilon}]) $ which reduce to $s_0$
over $k$ and this is just $H^0(Z,E)$.
Consider the standard short exact sequence
\[
0 \rightarrow {\cal I}_X \rightarrow {\cal O}_Z \rightarrow {\cal O}_X
\rightarrow 0
\]
tensor it with $E$ and take cohomology to get
\[0 \rightarrow H^0(Z,E\otimes {\cal I}_X) \rightarrow H^0(Z,E)
\rightarrow H^0(X,E\!\mid_X)\]
\[
\rightarrow H^1(Z,E\otimes {\cal I}_X) \rightarrow \dots \hfill
\]
Hence if $H^1(Z,E\otimes{\cal I}_X)=0$ we have the desired surjectivity.
\fp
So far we have developed portions of a triangle of functors
\commtriang{$F_{s_0}$}{$\Hilb_{s_0}$}{$\Def_{C_X}$}{$f$}{?}{h}
To fill in the morphism marked with ``?'' note that the local ring of the
vertex of the cone of the zero scheme of $s_A$ is naturally a
deformation of the local ring of the vertex of the cone of the zero
scheme of $s_0$. It is clear that the triangle commutes because $g$
is just the composition of $f$ and $h$. We now
invoke the comparison Theorem relating the Hilbert functor and the deformation
functor
\begin{s}{Theorem} Suppose that $V$ is projectively normal and that
$T^1(V)$,the tangent space to $\Def_{C_V}$, is
negatively graded (in a sense to be made precise below), then the
natural morphism of functors
\[h: \Hilb_{V} \rightarrow \Def_{C_V}\]
is smooth.
\end{s}
\pf. See \cite{K:79} \fp
As we will see in the case of the zero schemes of sections of
certain vector bundles, this result together with Proposition \re{surj} will
enable us to establish the
existence of families of the desired type. The codim$X$=rank$E$
condition that we have been placing on our zero schemes of sections
of $E$ ensures projective normality (assuming that the varieties are normal in
the first place)
by a straightforward adaptation of the proof for complete
intersections in projective space. It is the grading condition in the
comparison theorem which
is the more difficult of the two to verify and we will reduce it to a
condition on vanishing of cohomology.
Recall that $\Def_{C_X}$ is the deformation functor of the vertex on the
projective cone $C_X$ over $X$ and that there is an action by $\G_m (k)$ on
$C_X$
with weights ranging over the integers.
$T^1(X):={\Def}_{C_X}(k[{\epsilon}])$ is a vector space and the action
of $\G_m$ on on $C_X$ becomes an action on $T^1(X)$, so that we get a
decomposition $T^1(X)=\oplus_{\nu = -\infty}^{\infty}T^1(\nu )$ and
$T^1(X)$ becomes a graded vector space (as $p$ is an isolated
singularity in $C_X$, in fact $T^1(\nu)=0$ for $\nu \gg 0$ and for $\nu \ll 0$
but we will not need this). We say that $T^1(X)$ is
{\em negatively graded} if $T^1(X)(\nu )=0$ for $\nu > 0$. To demonstrate that
this condition obtains
we show that $\Hilb (k[{\epsilon}])$ too has a grading, which is negative in
this
sense and that the map $Th$ is surjective and respects both gradings.
\begin{s}{Proposition} Let $X$ be the zero scheme of the section $s_0$ of $E
\rightarrow Z$ and assume that $X$ projectvely
normal (e.g. codim $X$ = rank $E$).
Suppose the conclusion of Proposition \re{surj} holds, that is, the morphism of
functors $F_{s_0} \rightarrow
\Hilb_X$ is surjective on tangent spaces. Let $g: F_{s_0}
\rightarrow Def_{C_X}$ be the natural morphism taking an element of
$F_{s_0}(A)$ to an element of $Def_{C_x}(A)$. Denote the map on
tangent spaces by $H^0(Z,E) \stackrel{Tg}{\rightarrow} T^1(X)$. If
this map is surjective then $T^1(X)$ is negatively graded and the morphism of
functors $h$ is smooth.
\end{s}
\pf. We have a triangle of tangent maps
\commtriang{$H^0(Z,E)$}{$\Hilb(k[{\epsilon}])$}{$\Def{C_X}(k[{\epsilon}])$}{$Tf$}{$Tg$}{$Th$}
which commutes because the triangle of underlying maps does and we are
assuming that $Tf$ is onto. If we now further assume that
$H^0(Z,E) \rightarrow T^1(X)$ is onto then $\Hilb(k[{\epsilon}])
\rightarrow T^1(X)$ must also be onto as well.
$\Hilb(k[{\epsilon}])\simeq H^0(X,N_{X/Z})$ as noted above and
$H^0(X,N_{X/Z}) \simeq H^0(C_X,N_{C_X})$ where $C_X$ is the projective cone
over $X$. By projective normality the singularity at the vertex $p$ of
$C_X$ has depth=2 so global sections of $C_X-p$ extend to global sections
of $C_X$. Since $X$ is a local complete intersection, this is true of
all coherent sheaves on $C_X$ by local duality, and so, in particular
\[H^0(C_X,N_{C_X}) \simeq H^0({C_X}-p,N_{C_X}) \simeq H^0(C_X-p,N_{C_X-p})\]
But
\[C_X-p \simeq \V({\cal O}_X(-1)) - {\em zero section}\]
so there is a natural affine
map $\pi:C_X-p \rightarrow X$ and $N_{C_C-p} \simeq \pi^*N_X$ and we can
compute
\[H^0(C_X-p,N_{C_X-p}) \simeq H^0(C_X-p,\pi^*N_X) \simeq
H^0(X,\pi_*\pi^*N_X) \simeq \oplus_{\nu = -\infty}^0 H^0(X,N_X(\nu))\]
where $N_X(\nu):= N_X \otimes {\cal O}(\nu)$ as usual.
The second isomorphism comes from the fact that $\pi$ is affine and
the conclusion that $H^0(X,N_X(\nu))=0$ for $\nu > 0$ comes from the fact that
the sections must extend over all of $C_X$ and hence cannot have any
poles at $p$.
The grading thus produced on $\Hilb (k[{\epsilon}])$ arises from the action of
$\G_m$ on $C_X$ and the morphism $\Hilb \rightarrow \Def$ is contructed
precisely by passing to
the projective cone to go from the varieties in projective space to
abstract varieties. Hence the action of $\G_m$ on $T^1(X)$ and
$\Hilb(k[{\epsilon}])$ is compatible with this morphism. The result is that
$T^1(X)$ must also be negatively graded and hence we can apply the
comparison Theorem to get that this morphism is smooth. \fp
Finally it is not difficult to reduce the requirement ``$H^0(Z,E)
\rightarrow T^1(X)$ surjective'' to a statement of vanishing of
cohomology.
The tangent sheaf to a variety $V$ is defined by $\Theta_V :=
Hom_{{\cal O}_V}(\Omega_V, {\cal O}_V)$, where $\Omega_V$ is the sheaf
of differentials of $V$. Since $C_X \subset \Aff^{n+1}$ and $C_X -p$
is smooth we have that
\[
0 \rightarrow \Theta_{C_X}\!\mid_{C_X-p}\rightarrow \Theta_{\Aff
^{n+1}}\!\mid_{C_X-p} \rightarrow {\cal N}_{C_X}\!\mid_{C_X-p}
\rightarrow 0
\]
is exact. Note that $C-p$ is not affine so there is a longer exact
cohomology sequence
\[
0 \rightarrow H^0(C_X-p, \Theta_{C_X}\!\mid_{C_X-p})
\rightarrow H^0(C_X-p,\Theta_{\Aff^{n+1}\!\mid_{C_X-p}})
\rightarrow\]
\[H^0(C_X-p,{\cal N}_{C_X}\!\mid_{C_X-P}) \rightarrow H^1(C_X-p,
\Theta_{C_X}\!\mid_{C_X-p}) \rightarrow H^1(C_X-p,
\Theta_{\Aff^{n+1}\!\mid_{C_X-p}}) \rightarrow
\]
The theory of the ``cotangent complex'' \cite{Li-Sch 67} gives us the following
exact sequence
\[
0 \rightarrow H^0(C,\Theta_{C_X}) \rightarrow
H^0(C,\Theta_{\Aff^{n+1}}\!\mid_{C_X}) \rightarrow
H^0(C_X,{\cal
N}_{C_X})\]
\[
\rightarrow T^1(X) \rightarrow H^1(X,\Theta_Z\!\mid_X) \rightarrow
\dots \hfill
\]
\noindent Once again since $X$ is projectively normal, sections over $C_X-p$
extend to $C_X$ so that
\[H^0(C_X,\Theta_{C_X}) \simeq
H^0(C_X-p,\Theta_{C_X}\!\mid_{C_X-p})
\]
\[
H^0(C_X,\Theta_{\Aff^{n+1}}\!\mid_{C_X}) \simeq
H^0(C_X-p,\Theta_{\Aff^{n+1}}\!\mid_{{C_X}-p})\]
and
\[H^0(C_X,{\cal N}_{C_X}) \simeq H^0(C_x-p,{\cal N}_{C_X-p})\]
Putting all of this together we see that if
\[
H^1(C_x-p,\Theta_{\Aff^{n+1}}\!\mid_{C_X-p})\simeq
H^1(X,\Theta_Z\!\mid_X)=\emptyset} \def\dasharrow{\to
\]
then
\[
T^1(X) \hookrightarrow H^1(C_X-p,\Theta_{C_X}\!\mid_{C_X-p})=0
\]
To summarize,
\begin{s}{Proposition} For $X$ the zero scheme of a section $S$ bundle $E$ over
a non-singular
projective variety $Z$ with codim $X$ = rank $E$ where
\[H^1(Z, E
\otimes {\cal I}_X)=H^1(X,\Theta_Z\!\mid_X)=0\]
and $X$ is normal we have the commutative triangle
\commtriang{$F_{s_0}$}{$\Hilb_X$}{$\Def_{C_X}$}{}{}{}
where all of the arrows are smooth. \end{s}
\pf. We only need to show that $F_{s_0} \stackrel{f}{\rightarrow} \Hilb_X$ is
smooth since then the third side of the triangle will be smooth by
\cite{Sch:67}. We know that this arrow is surjective on tangent spaces
hence we only need to show that $F_{s_0}$ is less obstructed than
$\Hilb_X$. In other words, if $B \stackrel{\phi}{\rightarrow} A$ is a
surjection in
$\cal C$ with $ker(\phi)^2=0$ and $({\bf m}_B)ker(\phi)=0$ (so $ker(\phi)$ is a
$k$-vector space) and if for $\xi_0 \in \Hilb_X(A)$ which is
in the image of $f$ so, $\xi_0=f(\zeta_0)$, $\zeta_0 \in F_{s_0}(A)$,
there is a $\xi \in \Hilb_X(B)$ such that $\Hilb_X(\phi)(\xi)=\xi_0$
then there is a $\zeta \in F_{s_0}(B)$ such that
$F_{s_0}(\phi)(\zeta)=\zeta_0$. The obstruction to the existence of
such $\zeta$ lies in $Ext^2(L_{X/Z},{\cal O}_X)$ where $L_{X/Z}$ is
the cotangent complex of $X$. This is a two term complex
\[
0 \rightarrow {\cal I}_X/({\cal I}_X)^2 \rightarrow i^*(\Omega^1_Z)
\rightarrow 0
\]
since $X$ is a local complete intersection \cite{I:71} Ch. III, \S 3.2 and the
$Ext^2$ terms vanish
since $H^1(X,\Theta_Z\!\mid_X)=0$. \fp \bl
Since we know that the formal scheme prorepresenting $\Hilb_X$ is
algebraizable \cite{G:61}, we now have the same conclusion for $\Def_{C_X}$.
All
infinitesimal deformations of $X$ come from small deformations and all
small deformations lie in $Z$. Indeed, by versality
$H^0(Z,E)$ is a complete deformation space and the family
\[
Y:=\{(z,s)\mid z \in Z, s \in H^0(Z,E), s(z)=0\} \subset Z \times
H^0(Z,E) \rightarrow H^0(Z,E)
\]
is a universal family both in the sense of the deformation functor and
the Hilbert functor. These results are valid for $X$ defined over any field.
\bitt{Application of the theory}
We now apply this theory to our problem. If a normal $k$-variety $V$ ($k$ a
finitely generated extension of $\Q$) is
given as the zero scheme of a section $s\in H^0(Z,E)$ (everything
defined over $k$) and if we have
\begin{itemize}
\item codim$V$=rank$E$
\item $H^1(Z,E\otimes {\cal I}_V)=H^1(V,\Theta_V)=0$
\end{itemize}
then we know that the $k$-vector space $H^0(Z,E)$ parametrizes a
complete family of deformations of $V$. There is a map
\[
H^0(Z,E) \rightarrow H^0(Z,E)_{/{\Q}} \otimes k
\]
giving a $\Q$ structure to $H^0(Z,E)$. This induces a morphism of
functors
$F^k_{s} \rightarrow F^{\Q}_{s}$.
We observe that this morphism is
smooth since first, if $A' \rightarrow A$ is a surjection in $\cal C$ then
\[
F^k(A') \rightarrow F^k(A) \times_{F^{\Q}(A)} F^{\Q}(A')
\]
must also be onto. To see this note that the arrow $\beta$ in
\[
\begin{array}{ccc}
F^k(A) \times_{F^{\Q}(A)} F^{\Q}(A') & \stackrel{{\beta}'}{\rightarrow} &
F^{\Q}(A')\\
\downarrow & & \downarrow\\
F^k(A) & \stackrel{\beta}{\rightarrow} & F^{\Q}(A)
\end{array}
\]
is onto and hence ${\beta}'$ is also and that $F^k(A') \rightarrow
F^{\Q}(A')$ is onto as well. Secondly the morphism also induces a
bijection on tangent spaces since $H^0(Z,E)_k$ and $H^0(Z,E)_{/{\Q}}$
are vector spaces of the same dimension over fields of the same cardinality.
Thus the projection $H^0(Z,E) \rightarrow H^0(Z,E)_{/{\Q}}$ induces a map of
deformation spaces between the functors $\Def^k$ and ${\Def}^{\Q}$ and
$\Hilb_X^k$ and $\Hilb_X^{\Q}$.
We have a universal family $Y_{/{\Q}} \rightarrow H^0(Z,E)_{/{\Q}}$
and the above ensures that we recover the corresponding universal
family over $k$ by extension of scalars
\[
\begin{array}{ccc}
Y_k & \rightarrow & Y_{/{\Q}}\\
\downarrow & & \downarrow \\
H^0(Z,E)_k & \rightarrow & H^0(Z,E)_{/{\Q}}
\end{array}
\]
But there is a unique extension of scalars from $\Q$ to $\C$ and this
gives a diagram of the
sort described at the beginning of this \S
\commrect{$Y_{/{\C}}\simeq Y_{/{\Q}} \times
\C$}{$Y_{/{\Q}}$}{$H^0(Z,E)_{/{\C}}\simeq
H^0(Z,E)_{/{\Q}}\times
\C$}{$H^0(Z,E)_{/{\Q}}$}{$f$}{$f_{/{\Q}}$}{$\beta_Y$}{$\beta$}
Since we can now consider $V$, $E$, $Z$, and $s$ as objects defined
over the complex numbers it is clear that all of the conjugate
varieties $V_{\sigma}$ can be obtained by conjugating the section $s$, that is,
$V_{\sigma}$ is the zero scheme of $s_{\sigma}:=\sigma (s)$. Hence all
of the conjugates $V$ are in the family $Y$ if $V$ is. Thus the
conditions of Theorem \re{main} are satisfied and the independence of
the topological type from variation with $\sigma$ is ensured.
It only remains to spell out some types of varieties which are defined as the
zero sections of vector bundles satisfying our conditions.
If we assume that $Z$ is homogeneous, that is
\[Z \simeq G/H\]
where $G$ is a simple, simply connected, split algebraic group over
$\Q$, and $H$ a parabolic subgroup, then by algebraic versions of Bott's
vanishing Theorems \cite{Dm:76} we get that $H^1(Z, \Theta_Z)=0$ for all $i >
0$.
If we further suppose that $E\simeq \oplus_{j=1}^r{\cal O}_Z(d_j)$
with $d_j > 0$ and $X$ defined by a section $s$ such that
\begin{itemize}
\item codim $X$=$r$
\item $X$ is not a $K$-3 surface
\item dim $Z \geq 3$
\end{itemize}
then a reworking of calculations of Borcea using an algebraic version of
the Kodaira-Nakano-Akizuki vanishing theorem shows that
$H^1(Z,E\otimes{\cal I}_X)=0$. For details see \cite{Re:94}
If we now assume that $X$ is normal and defined over $k$ that is, the section
$s$ satisfies
\[
s \in H^0(Z,E)_{/{\Q}} \otimes k
\]
\noindent then we have a class of $k$ varieties whose topology
is independent of the embedding of $k$. This class includes complete
intersections in projective space. A somewhat different proof is available for
the special case of complete intersections in projective space which does not
require normality \cite{Re:94}.
We note finally that while the case of
$K$-3 surfaces must be treated separately, the result is the same
since all $K$-3's are homeomorphic (indeed diffeomorphic) and
conjugates of $K$-3 varieties are $K$-3.
\bit{Comments and Open Questions}
The above discussion can be used to shed a bit of light on the nature of the
Serre-Abelson examples. Both authors construct their varieties as
quotients using finite group actions. In both cases the action is
varied under conjugation. The difference lies in the part of the
homotopy type which is affected by conjugation.
In Serre, the variety acted upon is a product of a diagonal
hypersurface by an abelian variety and the group action on the abelian
variety makes the $\pi_1$ of the variety into a module in
demonstrably different ways under conjugation. In Abelson's example
the group acts on complete intersection which is constructed via a
representation of the group and all of this varies under conjugation.
Special choices of the group allow one to demonstrate variation in the
Postnikov tower.
It is not difficult to see, in both cases that the field of moduli of the
varieties thus constructed is smaller than their field of definition. The
group action creates some ``symmetries'' which produce this result.
This suggests the following vague question:
A) Is it possible to produce examples of non-homeomorphic conjugate
varieties without using (finite) group actions?
This question can be made more specific in a number of ways.
Because of the use of group actions the examples of Serre and Abelson
are rather rigid. A small deformation of one of their examples no
longer maintains the structure required to compare it with its
conjugates. One approach might therefore be to ask,
A1) Is it possible to construct examples of non-homemorphic conjugate
varieties which are stable under small deformations? This seems unlikely.
Another way to cut out finite group actions is to ask for
simply-connected examples,
A2) Are there examples of simply connected non-homemorphic conjugate
varieties?
A useful source of new examples may be provided by Shimura varieties.
Note: It may be useful to employ Kollar's
notions of ''essentially large'' fundamental groups here instead.
Finally, along these lines one has the fundamental question,
A4) Are the simply connected covering spaces of conjugate algebraic varieties
analytically isomorphic?
The criterion developed in this paper does not provide an indication
of the minimum '``necessary'' conditions under which the topology of varieties
remains stable under conjugation. Neither does it give any indication of the
``part'' (if any) of the topological type which are conjugations invariant
(over and above the \'{e}tale homotopy type which is clearly invariant). A
Theorem of Deligne \cite{D
e:87} shows that the nilpotent completion of the fundamental group of an
algebraic variety is algebraically determined. One is led to ask,
B) Is the entire rational homotopy type a conjugation invariant?
One might also pose the following question which seems to lie somewhere between
A) and B),
C) Is simple connectivity a conjugation invariant?
\newpage
|
1995-02-24T21:42:49 | 9410 | alg-geom/9410005 | en | https://arxiv.org/abs/alg-geom/9410005 | [
"alg-geom",
"math.AG"
] | alg-geom/9410005 | Lothar Goettsche | Geir Ellingsrud and Lothar G\"ottsche | Variation of moduli spaces and Donaldson invariants under change of
polarization | 44 pages, amslatex, no figures | null | null | null | null | The paper determines the change of moduli spaces of rank $2$ sheaves on
surfaces with $p_g=0$ under change of polarization and the corresponding change
of the Donaldson invariants. In this revised version we have made some minor
stylistic changes in the previous text. In addition we have added a final
chapter of about 20 pages (announced in the previous version), in which the six
lowest order terms (three of them non-zero) of the change are computed
explicitely using computations in the cohomology of Hilbert schemes of points.
| [
{
"version": "v1",
"created": "Fri, 7 Oct 1994 10:45:57 GMT"
},
{
"version": "v2",
"created": "Mon, 19 Dec 1994 13:21:28 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Ellingsrud",
"Geir",
""
],
[
"Göttsche",
"Lothar",
""
]
] | alg-geom | P(n){P(n)}
\def{\text{\rom{\bf b}}}{{\text{\rom{\bf b}}}}
\defS^{(\bb)}{S^{({\text{\rom{\bf b}}})}}
\def{\text{\rom{\bf E}}}{{\text{\rom{\bf E}}}}
\def\Inc#1{Z_{#1}}
\def\inc#1{\zeta_{#1}}
\def\boh#1{pt_{#1}
\def\Boh#1{Pt_{#1}}
\def\mah#1{al_{#1}
\def\Mah#1{Al_{#1}}
\def\pr#1{W_{#1}}
\def\alpha{\alpha}
\def\Gamma{\Gamma}
\def\bar \Gamma{\bar \Gamma}
\def\gamma{\gamma}
\def\bar \eta{\bar \eta}
\def{\cal Z}{{\cal Z}}
\def{pt}{{pt}}
\def\alpha{\alpha}
\def{S^{(n)}}{{S^{(n)}}}
\def{S^{(m)}}{{S^{(m)}}}
\def\<{\langle}
\def\>{\rangle}
\def{\text{\rom{Hilb}}}{{\text{\rom{Hilb}}}}
\def{\Cal W}{{\cal W}}
\def{\hbox{\rom{Tor}}}{{\hbox{\rom{Tor}}}}
\def{\hbox{\rom{Ext}}}{{\hbox{\rom{Ext}}}}
\def\stil#1{\tilde S^#1}
\def\pi{\pi}
\defg{g}
\def\phi{\varphi}
\def\tilde{\tilde}
\def{\bar\al}{{\bar\alpha}}
\begin{document}
\title[Variation of Donaldson invariants]{Variation of moduli spaces
and Donaldson invariants under change of
polarization}
\author{Geir Ellingsrud}
\address{Mathematical Institute\\University of Oslo\{\Bbb P}.~O.~Box~1053\\
N--0316 Oslo, Norway}
\email{ellingsr@@math.uio.no}
\keywords{Moduli spaces, Donalson invariants, Hilbert scheme of points}
\author{Lothar G\"ottsche}
\address{Max--Planck--Institut f\"ur Mathematik\\Gottfried--Claren--Stra\ss e
26\\
D-53225 Bonn, Germany}
\email{lothar@@mpim-bonn.mpg.de}
\maketitle\
\section{Introduction}
Let $S$ be a smooth projective surface over the complex numbers and let
$c_1\in H^2(S,{\Bbb Z})$ and
$c_2\in H^4(S,{\Bbb Z})$ be two classes.
For an ample divisor $H$ on $S$, one can study the
moduli space $M_H(c_1,c_2)$ of $H$-semistable
torsion-free sheaves $E$ on $S$ of rank $2$
with $c_1(E)=c_1$
and $c_2(E)=c_2$. We want to study the change of $M_H(c_1,c_2)$
under variation of $H$. It is known that the ample cone of $S$ has a
chamber structure, and that $M_H(c_1,c_2)$ depends only on the
chamber containing $H$.
In this article we will try to understand how $M_H(c_1,c_2)$ changes,
when $H$ passes
through a wall separating two chambers. The set-theoretic changes of
the subspace
consisting of locally free sheaves and of
$M_H(c_1,c_2)$ have been treated in \cite{Q1} and \cite{Go1} respectively.
We show that the change of the moduli space when $H$ passes through a wall,
can be expressed as a sequence of operations similar to a flip.
In fact the moduli spaces at each step can be identified as moduli spaces
of torsion-free sheaves with a
suitable parabolic structure of length $1$.
We assume that either the geometric genus $p_g(S)$ is $0$ or that $K_S$ is
trivial. We
shall also make suitable hypotheses on the wall, and walls fulfilling this
condition
we call good. This assumption is reasonably weak if the Kodaira
dimension of $S$ is at most $0$, but gets stronger if
e.g., $S$ is of general type.
When the polarization passes through a good wall, each of the steps above
is realized by a smooth blow-up
along a projective bundle
over a product of Hilbert schemes of points on $S$, followed by a smooth
blow-down of the exceptional divisor in another direction.
The change of moduli spaces can be
viewed as a change of GIT quotients, treated in \cite{Th2} and \cite{D-H}.
These results could in principle be applicable, although it
would still take quite some work to do so.
We have however chosen a more direct approach via elementary
transforms of universal families, which is more in the spirit of \cite{Th1},
and which also immediately gives the change of the universal sheaves
needed for the computation of Donaldson invariants.
In the case that $K_S$ is trivial, i.e., $S$ is an abelian or a $K3$ surface,
we see that the change of $M_H(c_1,c_2)$, when $H$ passes through a wall,
is given by elementary transformations of symplectic varieties.
In the case that $p_g(S)=q(S)=0$ we use these results in
order to compute the change of the Donaldson
polynomials under change of polarisation.
The Donaldson polynomials of a $C^\infty$-manifold $M$ of dimension $4$
are defined using a Riemannian metric on $S$,
but in case $b^+(M)>1$ they are known to be independent of the metric, as
long as it is generic.
In case $b_+(M)=1$, (which for an algebraic surface $S$ corresponds to
$p_g(S)=0$), the invariants have been introduced and studied by
Kotschick in \cite{Ko}.
In \cite{K-M} Kotschick and Morgan show that the invariants only
depend on the chamber
of the period point of the metric in the positive cone of $H^2(M,{\Bbb R})$.
They also compute the lowest order term of the change and conjecture
the shape of a formula for the change.
The case we are studying corresponds to $M$ being an
algebraic surface $S$ with $p_g(S)=q(S)=0$ and a wall
lying in the ample cone, in addition we assume
that the wall is good.
In a first step we compute the change of the Donaldson invariants
in terms of
natural cohomology classes on Hilbert schemes of points on $S$ and then we
use some computations
in the cohomology rings of these Hilbert schemes to determine
the six lowest order terms of the change of the Donaldson invariants
explicitly. The results are compatible with the conjecture of \cite{K-M}
(which in particular predicts that three of the terms above are zero.
Parallelly and independently similar results to ours have been
obtained by other authors.
Matsuki and Wentworth show in \cite{M-W} that the change of moduli spaces
of torsion-free sheaves of arbitrary rank on a projective variety
under change of polarisation can be described as a sequence of flips.
In \cite{F-Q} Friedman and Qin obtain very similar results to ours.
\section{Background material}
In this paper let $S$ be a projective surface over ${\Bbb C}$. By the Neron-Severi
group $NS(S)$ of $S$ we mean the group of divisors modulo homological
equivalence, i.e., the image of $Div(S)$ in $H^2(S,{\Bbb Z})$ under the map sending
the class of a divisor $D$ to its fundamental cycle $[D]$. Let
$Div^0(S)$ be its kernel. Let
$c_1\in H^2(S,{\Bbb Z})$ and $c_2\in H^4(S,{\Bbb Z})={\Bbb Z}$ be elements which will be fixed
throughout the paper.
Let $H$ be a polarization of $S$.
As we mostly shall consider stability and semistability in the sense of
Gieseker
and Maruyama we shall write
$H$-stable (resp. $H$-semistable) instead of Gieseker stable (resp.
semistable)
with respect to $H$ and $H$-slope stable (resp. $H$-slope semistable)
instead of
stable (resp. semistable) with respect to $H$ in the sense of Mumford-Takemoto.
Denote by $M_H(c_1,c_2)$
the moduli space of H-semistable torsion-free sheaves $E$ on $S$ of
rank $2$ with $c_1(E)=c_1$ and $c_2(E)=c_2$ and
$M_H^s(c_1,c_2)$ the open subscheme of $M_H^s(c_1,c_2)$ of stable sheaves.
Let $Spl(c_1,c_2)$ be the moduli space of simple
torsion-free sheaves with $c_1(E)=c_1$ and $c_2(E)=c_2$ (see \cite{A-K}).
\begin{nota}
For a sheaf ${\cal F}$ on a scheme $X$ and a divisor $D$ let ${\cal F}(D):={\cal F}\otimes
{\cal O}_X(D)$ .
Many of our arguments will take place over products $S\times X$, where $X$ is a
scheme. We shall denote by $p:S\times X\longrightarrow S$ and $q_X:S\times X\longrightarrow X$ the
two projections and if there is no danger of confusion, we will drop the index
$X$. For a divisor
$D$ on $X$ we denote $D_S:=q_X^*(D)$. For a sheaf ${\cal F}$ on $S\times X$
and a divisor or divisor
class, $D$ on $S$ we denote by ${\cal F}(D)$ the sheaf
${\cal F}(p^*(D))$.
If $X$ is a smooth variety of dimension $n$, we denote the cup product of two
elements $\alpha$ and $\beta$ in $H^*(X,{\Bbb Z})$ by $\alpha\cdot\beta$ and the
degree of a class
$\alpha\in H^{2n}(X,{\Bbb Z})$ by $\int_X\alpha$. For $\alpha,\beta\in H^2(S,{\Bbb Z})$ let
$\<\alpha\cdot\beta\>:=\int_S\alpha\cdot\beta$. We write
$\alpha^2$ for $\<\alpha\cdot\alpha\>$ and, for $\gamma\in H^2(S,{\Bbb Z})$, we put
$\<\alpha,\gamma\>:=\<\alpha\cdot \check\gamma\>$, where
$\check\gamma$ is the Poincar\'e dual of $\gamma$.
\end{nota}
\begin{conve} \label{convent}
If $Y,X$ are schemes and there is a "canonical" map $f:X\longrightarrow Y$,
then for a cohomology class $\alpha\in H^*(Y,{\Bbb Z})$ (resp. for a vector bundle
$E$ on $Y$) we will very often also denote the pull-back via $f$
by $\alpha$ (resp. $E$).
\end{conve}
\begin{defn} \cite{OG2}\label{defpseudofam} Let $B$ be a scheme.
A family of sheaves, ${\cal F}$, on $S$ parametrized by $B$ is a $B$-flat sheaf on
$S\times B$. Two families of sheaves ${\cal F}$ and ${\cal G}$ on $S$ parametrized by
$B$
are called equivalent if there exists an isomorphism
${\cal F}\simeq {\cal G}\otimes q_B^*M$, for some line bundle $M$ on $B$.
Let
$(B_j)_{j\in J}$ be an \'etale cover of $B$ by schemes.
Assume that for each $j\in J$ there is a family ${\cal F}_j$ of sheaves on
$X$ parametrized by $B_j$, and that for each pair $k,l\in J$
the pullbacks of ${\cal F}_k $ and ${\cal F}_l$ to $B_k\times_B B_l$ are equivalent.
Then we will say that the above data defines a pseudo-family of sheaves on $S$
parametrized by $B$. We will denote it by ${\cal F}$.
It is clear what is meant by a map of pseudo-families and by
two pseudo-families being equivalent.
\end{defn}
The main reason to introduce pseudo-families is that the moduli space
$M^s_H(c_1,c_2)$ does not always carry a universal
family of sheaves, but there will always be a universal pseudo-family.
By the universal property of $M^s_H(c_1,c_2)$ a pseudo family of $H$-stable
torsion-free sheaves $E$ on $S$ with $c_1(E)=c_1$,
$c_2(E)=c_2$ parametrized by $B$ gives rise to a morphism $B\longrightarrow
M^s_H(c_1,c_2)$.
\bigskip
{\bf Walls and chambers for torsion-free sheaves}
We now recall some results about walls and chambers from \cite{Q1},
\cite{Q2} and \cite{Go1}.
\begin{defn}\label{defwall}(for the first part see \cite{Q1} Def I.2.1.5)
Let $C_S$ be the ample cone in $NS(S)\otimes {\Bbb R}$.
For $\xi\in NS(S)$ let
$$W^\xi:=C_S\cap\big \{ x\in NS(S)\otimes{\Bbb R} \bigm| \<x\cdot\xi\>=0\big\}.$$
We shall call $W^\xi$ a wall of type $(c_1,c_2)$, and say that it is
defined by $\xi$ if the following conditions are satisfied:
\begin{enumerate}
\item $\xi+c_1$ is divisible by $2$ in
$NS(S)$,
\item $c_1^2-4c_2\le \xi^2<0$,
\item there is a polarisation $H$ with $\< H\cdot \xi\>=0$.
\end{enumerate}
In particular $d_\xi:= (4c_2-c_1^2+\xi^2)/4$ is a nonnegative integer.
An ample divisor $H$ is said to lie in the wall $W$ if $[H]\in W$.
If $D$ is a divisor with $[D]=\xi$, we will also say that
$D$ defines the wall $W$.
A {\it chamber} of type $(c_1,c_2)$ or simply a chamber, is a connected
component of the complement of the union of all the walls of
type
$(c_1,c_2)$. Two different chambers will be said to be
{\it neighbouring chambers} if the intersection of
their closures contains a nonempty open subset
of a wall.
We will call a wall $W$ {\it good}, if $D+K_S$ is not effective
for any divisor $D$ defining the wall $W$.
\end{defn}
If $D$ defines a wall, then neither $D$ nor $-D$ can be effective because
$D$ is orthogonal to an ample divisor.
In particular every wall will be good if $-K_S$ is
effective or
if $[K_S]$ is a torsion class. More generally, a wall $W$ will be good
if there exists an
ample divisor
$H$ in $W$ with $\<K_S \cdot H\>\le 0$.
\begin{defn}\label{defenm}
Let ${\text{\rom{Hilb}}}^l(S)$ be the Hilbert scheme of subschemes of length $l$ on $S$.
For $\alpha\in NS(S)$ and $l\in {\Bbb Z}$, let
$M(1,\alpha,l)$ be the moduli space of rank $1$ torsion-free sheaves
${\cal I}_{Z}(F)$ on $S$
with $c_1({\cal I}_{Z}(F))=[F]=\alpha$, $c_2(F)=length(Z)=l$.
Let
$$T^{n,m}_\xi:=\coprod_{2\alpha=c_1+\xi} M(1,\alpha,n)\times
M(1,c_1-\alpha,m).$$
Let $N_2 \subset NS(S)$ be the subgroup of $2$-torsion elements.
There is a (noncanonical) isomorphism $$T^{n,m}_\xi\simeq N_2\times{\text{\rom{Hilb}}}^n
S\times Div^0(S)\times {\text{\rom{Hilb}}}^mS\times Div^0(S),$$
which depends on the choice of an $\alpha\in NS(S)$ with $2\alpha=c_1+\xi$
and on a representative $F$ in $Div(S)$ for $\alpha$.
For any extension
\begin{equation*}\tag{$\epsilon$}0\longrightarrow A_1\longrightarrow E\longrightarrow A_2\longrightarrow
0\end{equation*} where $A_1$ and $A_2$ are torsion-free rank
one sheaves, we define $\Delta(\epsilon):=\chi(A_1)-\chi(A_2)$. Then if
$\alpha=c_1(A_1)-c_1(A_2)$, the Riemann-Roch theorem gives
$\Delta(\epsilon)=1/2\<(c_1(E)-K_S)\cdot\alpha\>+c_2(A_2)-c_2(A_1)$.
Furthermore for any divisor $D$ we have
$\Delta(\epsilon(D))=\Delta(\epsilon)+\<\alpha\cdot
D\>$, where $ \epsilon(D)$ denotes the extension
$\epsilon$ twisted by the line bundle ${\cal O}(D)$. This follows immediately
from the fact that $c_1(E(D))=c_1(E)+2[D]$.
Assume that $\xi$ defines a wall of type $(c_1,c_2)$,
and that $n$ and $m$ are nonnegative integers with
$n+m=d_\xi=c_2-(c_1^2-\xi^2)/4$.
Let ${\text{\rom{\bf E}}}^{n,m}_\xi$ be the set of sheaves lying in nontrivial extensions
\begin{eqnarray}\label{splitting}
&&0\longrightarrow {\cal I}_{Z_1}(F_1)\longrightarrow E\longrightarrow {\cal I}_{Z_2}(F_2)\longrightarrow 0\end{eqnarray}
where $({\cal I}_{Z_1}(F_1),{\cal I}_{Z_2}(F_2))$ runs through $T^{n,m}_\xi$.
It is easy to see that every sheaf $E\in {{\text{\rom{\bf E}}}}^{n,m}_\xi$ is simple
(\cite{Go1}, lemma 2.3).
Let $$V^{n,m}_\xi:={\text{\rom{\bf E}}}^{n,m}_\xi\setminus\Big(\bigcup_{l,s}
{\text{\rom{\bf E}}}_{-\xi}^{l,s}\Big).$$
\end{defn}
\begin{nota}\label{defaplus}
Assume that $H_+$ and $H_-$ are ample divisors lying in neighbouring
chambers separated
by the wall $W$. Then we define
$$A^+(W):=\Big\{\xi\in NS(S)\Bigm| \xi \text{ defines } W \text{ and } \
\<\xi\cdot H_+\> >0\Big\}$$
and $A^-(W):=-A^+(W)$.\end{nota}
The following proposition mostly comprizes some of the results of
\cite{Go1}, that are generalizations of the corresponding results of \cite{Q1},
\cite{Q2} and
will be important for the rest of the paper.
Note that unlike \cite{Go1} we assume walls to be defined by
classes in $NS(S)$ and not by numerical equivalence classes, and that we
look at moduli spaces with fixed first Chern class and not with fixed
determinant. The proofs in \cite{Go1} stay however valid with very few
changes.
\begin{prop} \label{wall}
\begin{enumerate}
\item For $H$ not lying on a wall, $M_H(c_1,c_2)\setminus
M_H^s(c_1,c_2)$ is independent of $H$ and
$M_H(c_1,c_2)$ depends only on the chamber of $H$.
\noindent For the rest of the proposition we assume that we are in the
situation of \ref{defaplus} and that $\xi\in A^+(W)$.
\item Every $E\in {\text{\rom{\bf E}}}^{n,m}_\xi$ is $H_+$ slope-unstable and the sequence
(\ref{splitting})
is its Harder-Narasimhan filtration with respect to $H_+$.
\item
${\hbox{\rom{Hom}}}({\cal I}_{Z_1}(F),E)={\Bbb C}$. Thus,
for $E\in {\text{\rom{\bf E}}}^{n,m}_\xi$, the sequence \ref{splitting} is the
unique extension
$$0\longrightarrow {\cal I}_{W_1}(F_1)\longrightarrow E\longrightarrow {\cal I}_{W_2}(G_2) \longrightarrow 0$$
with $\<(2F_1-c_1)\cdot H_+\> >0$.
\item In particular we see that, for $\xi,\eta\in A^+(W)$,
the subsets ${\text{\rom{\bf E}}}_\xi^{n,m}$, ${\text{\rom{\bf E}}}_{\eta}^{k,l}$ of
$Spl(c_1,c_2)$ do not intersect, unless
$\xi=\eta$ and $(n,m)=(k,l)$.
\item If $E\in {\text{\rom{\bf E}}}_\xi^{n,m}$ then $E$ is $H_-$-slope stable
if and only if $E\in V_\xi^{n,m}$ and $H_-$-slope unstable otherwise.
\item
On the other hand let $E$ be a torsion-free sheaf with $c_1(E)=c_1$
and $c_2(E)=c_2$, which is $H_-$-semistable and $H_+$-unstable.
Then $E$ is $H_-$-slope stable and $E\in {{\text{\rom{\bf E}}}}^{n,m}_\xi$ for suitable
numbers $n$ and $m$ and
$\xi\in A^+(W)$.
\end{enumerate}
\end{prop}
\begin{pf}
(1) is (\cite{Go1}, theorem 2.9(1)). (2) is easy. (3) follows from (2)
and (\cite{Go1},
lemma 2.3). (4) follows from (2). (5) is (\cite{Go1},
prop 2.5). (6) is (\cite{Go1},
lemma 2.2).
\end{pf}
\section{Parabolic structures and the passage through a wall}
As mentioned in the previous section, $M_H(c_1,c_2)$ depends only on the
chamber to which $H$ belongs. If $H'$ lies in a
neighbouring chamber to $H$ the moduli space $M_{H'}(c_1,c_2)$ will in
most cases be birational to $M_H(c_1,c_2)$, although new components do
occur in some cases. If the wall
separating the two chambers is good, we will describe
the birational transformation in detail by giving an explicit sequence
of blow-ups and blow-downs with smooth
centers which are known.
If the wall is good, but the transformation is not birational, our
arguments give a description of the components which are
added to or deleted from the moduli space.
For the rest of the paper we will assume
that $H_+$ and $H_-$ are ample divisors lying in neighbouring chambers
separated
by the wall $W$, and that $H$ is an ample divisor in the wall
$W$ which lies in the closure of the
chambers containing $H_-$ and $H_+$ respectively
and which does not lie in any other wall. Furthermore
we shall assume that $M=H_+-H_-$ is effective.
By
replacing $H_+$ by a high multiple if necessary,
we can always achieve this.
Our aim is to divide the passage through a wall into a number of smaller
steps. To this purpose we will introduce a finer notion of stability.
The starting point is the observation that unlike slope stability,
Gieseker stability
is not invariant under tensorization by a line bundle.
\begin{lem}\label{tensor}
There is a positive integer $n_0$ such that for all $l\ge n_0$ and all
torsion-free rank $2$ sheaves $E$
on $S$ with $c_1(E)=c_1$, $c_2(E)=c_2$
\begin{enumerate}
\item $E$ is $H_-$-stable (resp. semistable) if and only if $E(-lM)$ is
$H$-stable (resp. semistable).
\item $E$ is $H_+$-stable (resp. semistable) if and only if $E(lM)$ is
$H$-stable (resp. semistable).
\end{enumerate}\end{lem}
\begin{pf}
It will be enough to show (1).
As $H_-$ does not lie on a wall, it is easy to see that $E$ is
$H_-$-(semi)stable if and only if $E(M)$ is.
Also there are only finitely
many $\xi\in NS(S)$ defining the wall $W$.
Therefore lemma \ref{tensor} follows immediately from
lemma \ref{tensor1} and lemma \ref{tensor2} below.\end{pf}
\begin{lem}\label{tensor1}
\begin{enumerate}
\item Assume $E$ is $H_-$-semistable but $H$-unstable.
Then $E\in {{\text{\rom{\bf E}}}}^{n,m}_\xi$ for suitable
$n,m$ and $\xi\in A^+(W)$.
\item Assume $E$ is $H_-$-unstable but $H$-semistable.
Then $E\in {{\text{\rom{\bf E}}}}^{n,m}_{-\xi}$ for suitable
$n,m$ and $\xi\in A^+(W)$.
\end{enumerate}\end{lem}
\begin{pf}
We just prove (1), the proof of (2) being analoguous. By assumption there
is an extension
\begin{equation*}\tag{$\epsilon$}0\longrightarrow {\cal I}_{Z_1}(F_1)\longrightarrow E\longrightarrow
{\cal I}_{Z_2}(F_2)\longrightarrow 0\end{equation*}
with
$\Delta(\epsilon(lH_-))\le 0$ and
$\Delta(\epsilon(lH))>0$ for $l>>0$.
In particular we have
$\<\eta \cdot H_-\>\le 0\le\< \eta \cdot H\>$ where $\eta:=2[F_1]-c_1$.
If $0<\< \eta\cdot H\>$, there would be a wall separating $H_-$ and $H$.
So $\< \eta\cdot H\>=0$, and unless $\eta$ is a torsion
class, it defines a wall in which $H$ lies. As $H$ lies in a unique wall it
must be $W$. Hence $\eta\in A^+(W)$, and $E\in
{{\text{\rom{\bf E}}}}^{n,m}_{\eta}$.
Assume that $\eta$ is a torsion class. Then $F_1$ and $F_2$ are numerically
equivalent, and it is easily verified that
\begin{equation*}
\Delta(\epsilon(lH_-))=\Delta(\epsilon(lH))
\end{equation*}
which is a contradicition.\end{pf}
\begin{lem}\label{tensor2}
Given $n,m,\xi$. Then there exists an integer $k_0$ such that for all
$k>k_0$ and all $E\in {{\text{\rom{\bf E}}}}^{n,m}_\xi$, the sheaf $E(-k M)$ is $H$-stable
if and only $E$ is $H_-$-slope stable.
Otherwise $E$ is both $H_-$-slope unstable and $H$-unstable.
\end{lem}
\begin{pf}
Let $E\in E^{n,m}_\xi$. Then there is an extension
\begin{equation*}\label{split2}\tag{$\epsilon_1$}0\longrightarrow {\cal I}_{Z_1}(F_1)\longrightarrow
E\longrightarrow {\cal I}_{Z_2}(F_2)\longrightarrow 0\end{equation*}
with
$\xi=2[F_1]-c_1$. Assume first that $E$ is $H_-$-slope-stable
and let
\begin{equation*}\tag{$\epsilon_2$}0\longrightarrow {\cal I}_{Y_1}(G_1)\longrightarrow E\longrightarrow
{\cal I}_{Y_2}(G_2)\longrightarrow 0\end{equation*}
be another extension. Put $\eta:=2[G_1]-c_1$. As $E$ is
$H_-$-slope stable
we have $\<\eta\cdot H_-\>< 0$, and because there is no wall
between $H_-$
and $H$, we know that $\<\eta\cdot H\>\le 0$.
For any integers $k$ and $l$
$$\Delta(\epsilon_2(-kM+lH))=\Delta(\epsilon_2)-k\<\eta\cdot
M\>+l\<\eta\cdot H\>.$$
Hence if
$\<\eta\cdot H\><0$ for all extensions $\epsilon_2$ above, then $E(-k M)$
will be $H$-stable for any $k$.
Assume that $\<\eta\cdot H\>=0$. By assumption $H$ is contained in a
single wall $W$, so necessarily $\eta\in A^+(W)$.
Hence by proposition \ref{wall}(3), we get
${\cal I}_{Z_1}(F_1)={\cal I}_{Y_1}(G_1)$. Therefore it suffices
to see that for $k>>0$
and any $l$ we have the inequality
$$\Delta(\epsilon_1(-kM+lH))<0.$$
Now
$$\Delta(\epsilon_1(-kM+lH))=\Delta(\epsilon_1)-k\<\eta\cdot M\>,$$
which is negative for $k>>0$ as $\<\eta\cdot M\>>0$.
To prove the converse, assume that $E$ is not $H_-$-slope stable.
Then by proposition \ref{wall}(5) there is an extension
\begin{equation*}\tag{$\epsilon_3$}0\longrightarrow {\cal I}_{Y_1}(F_2)\longrightarrow E\longrightarrow
{\cal I}_{Y_2}(F_1)\longrightarrow 0.\end{equation*}
Because
$2[F_2]-c_1=-\xi$ we have
$$\Delta(\epsilon_3(-kM+lH))=\Delta(\epsilon_3)+\<-\xi\cdot (-kM+lH)\>=
\Delta(\epsilon_3)+k\<\xi\cdot M\>>0$$
for $k>>0$\end{pf}
{}From now on until the end of this section we fix $n_0$ as in lemma
\ref{tensor},
and we put $C:= (n_0+1)M$.
\begin{defn}\label{defalstable}
Let $a$ be a real number between $0$ and $1$. For any torsion-free sheaf
$E$ we define
$$P_a(E)=((1-a)\chi(E(-C))+a\chi(E(C)))/rk(E).$$
A torsion-free sheaf $E$ on $S$
is called $a$-semistable if and only if every subsheaf
$E'\subset E$
satisfies
$P_a(E'(lH))\le P_a(E(lH))$ for all $l>>0,$
and it is called $a$-stable if strict inequality holds.
\end{defn}
In particular, by lemma \ref{tensor}, $E$ is $0$-semistable if
and only if it is
$H_-$-semistable, and it is $1$-semistable if and only if it is
$H_+$-semistable.
For any extension
\begin{equation*}\tag{$\epsilon$}0\longrightarrow A_1\longrightarrow E\longrightarrow A_2\longrightarrow
0\end{equation*}
we define $\Delta_a(\epsilon):=P_a(A_1)-P_a(A_2)$. Then
$\Delta_a(\epsilon)=\Delta(\epsilon)+(2a-1)\<C\cdot\alpha\>$ where
$\alpha=c_1(A_1)-c_1(A_2)$.
Clearly $\Delta_a(\epsilon(D))=\Delta_a(\epsilon)+\<D\cdot\alpha\>$
for any divisor $D$.
A sheaf $E$ is $a$-stable (resp. $a$-semistable) if
$\Delta_a(\epsilon(lH))<0$ (resp. $\le0$) for all $l>>0$
and for any extension $\epsilon$ whose middle term is $E$.
\begin{rem} It is easy to see that $P_a(E(lH))$ is the parabolic Hilbert
polynomial of the parabolic bundle $(E(C),E(-C),a)$,
(i.e. with a filtration of length $1$). Therefore $E$ is $a$-semistable
if and only if $(E(C),E(-C),a)$ is semistable. In
\cite{Ma-Yo} a coarse quasiprojective moduli space of stable
parabolic sheaves
with fixed Hilbert polynomial is constructed, and by \cite{Yo} there
exists a projective coarse moduli space for
$S$-equivalence classes of semistable parabolic sheaves. In particular
there exists a coarse moduli space $M_{a}(c_1,c_2)$ for
$a$-semistable sheaves $E$ on $S$ with $c_1(E)=c_1$ and
$c_2E=c_2$. We denote by $M_{a}^s(c_1,c_2)$ its open subscheme of stable
sheaves.
\end{rem}
\begin{rem} We see that $M_{H_-}(c_1,c_2)$ and
$M_0(c_1,c_2)$ respectively $M_{H_+}(c_1,c_2)$ and $M_1(c_1,c_2)$
are coarse moduli schemes for the same functor and therefore they are
isomorphic.
\end{rem}
\begin{rem}
The same proof as in the case of $H$-stable sheaves shows that
$M_{a}^s(c_1,c_2)$
carries a universal pseudofamily.
One checks easily that every $E\in M^s_{a}(c_1,c_2)$
is simple. As $M^s_{a}(c_1,c_2)$ and $Spl(c_1,c_2)$
both carry universal
pseudofamilies, ${\cal V}$ and ${\Cal W}$ respectively, there exists a morphism
$f:M^s_{a}(c_1,c_2)\to Spl(c_1,c_2)$ such that $({\hbox{\rom{id}}}_S\times f)^*({\Cal W})={\cal V}$.
Let $M$ be its image. By the same argument there exists a map
$g:M\to M^s_{a}(c_1,c_2)$, with
$({\hbox{\rom{id}}}_S\times f)^*({\hbox{\rom{id}}}_S\times g)^*({\cal V})={\cal V}$. Hence $f$ is an open
embedding.
In particular and what is the most important thing for us,
the tangent space to $M^s_{a}(c_1,c_2)$
at a point $E$ is ${\hbox{\rom{Ext}}}^1(E,E)$.
\end{rem}
\begin{defn}\label{defminiwall} For all $a\in
[0,1]$ let $A^+(a)$ be the set of $(\xi,n,m)\in A^+(W)\times {\Bbb Z}_{\ge
0}^2$ satisfying
\begin{eqnarray}\label{minicond} n+m&=& c_2-(c_1^2-\xi^2)/4,\\ n-m&=&
\<\xi\cdot (c_1-K_S)\>/2
+(2a-1)\<\xi\cdot[C]\>.\end{eqnarray} A number $a$ is called a {\it
miniwall} if $A^+(a)\ne \emptyset$. A {\it minichamber} is a
connected component of the complement of the set of all miniwalls in
$[0,1]$. It is clear that there are finitely many
minichambers. Two minichambers are called neighbouring minichambers if
their closures intersect.
\end{defn}
\begin{rem}Note that $A^+(a)$ is the set of all
$\xi,n,m$ with $\xi\in A^+(W)$ for which there exists a (possibly split)
extension
\begin{equation*}\tag{$\epsilon$}0\longrightarrow A_1\longrightarrow E\longrightarrow A_2\longrightarrow
0\end{equation*}
with
$\xi=c_1(A_1)-c_1(A_2)$, $n=c_2(A_1)$, $m=c_2(A_2)$ and
$\Delta_a(\epsilon)=0$.
\end{rem}
\begin{lem}
Let $0\le a_-< a_+\le 1$ and assume that neither $a_-$ nor $a_+$ is a
miniwall. Let $E$ be $a_-$-semistable and
$a_+$-unstable.
Then there exists a miniwall $a$ between $a_-$ and
$a_+$ and an element $(\xi,n,m)\in A^+(a)$, such that
$E\in {{\text{\rom{\bf E}}}}_{\xi}^{n,m}$.
\end{lem}
\begin{pf} By assumption $E$ is $a_+$-unstable. Hence there is an extension
\begin{equation*}\tag{$\epsilon$}0\longrightarrow A_1\longrightarrow E\longrightarrow A_2\longrightarrow
0\end{equation*}
such that for
all $l>>0$ we
have $\Delta_{a_+}(\epsilon(lH))>0$. Putting $\xi:=c_1(A_1)-c_1(A_2)$ and
using that $E$ is $a_-$-semistable,
we obtain the following inequalities valid for all $l>>0$
$$\Delta_{a_-}(\epsilon(lH))=\Delta_{a_-}(\epsilon)+l\<H\cdot\xi\>\le0<
\Delta_{a_+}(\epsilon(lH))=\Delta_{a_+}(\epsilon)+l\<H\cdot\xi\>.$$
In particular $\<H\cdot\xi\>=0$ and
$\Delta_{a_-}(\epsilon)<0<\Delta_{a_+}(\epsilon)$.
Furthermore $\xi$ is not a torsion class
and $\xi$ defines a wall on which $H$ is lying,
which therefore must be $W$.
There clearly is an $a$ such that $\Delta_a(\epsilon)=0$.
\end{pf}
\begin{lem} \label{stab} Let $a_-<a_+$ be in neighbouring minichambers
separated by the miniwall $a$.
Let $(\xi,n,m)\in A^+(a)$.
\begin{enumerate}
\item Any $E\in {{\text{\rom{\bf E}}}}_\xi^{n,m}$ is $a_-$-stable, strictly
$a$-semistable
and $b$-unstable for all $b>a$.
\item Any $E\in {{\text{\rom{\bf E}}}}_{-\xi}^{m,n}$ is $a_+$-stable, strictly $a$-semistable
and $b$-unstable for all $b<a$.\end{enumerate}
\end{lem}
\begin{pf}
By symmetry it is enough to show (1). Let $E\in {{\text{\rom{\bf E}}}}_\xi^{n,m}$. Then
$E$ is given by an extension
\begin{equation*}\tag{$\epsilon$}0\longrightarrow {\cal I}_{Z_1}(F_1)\longrightarrow E\longrightarrow
{\cal I}_{Z_2}(F_2)\longrightarrow 0\end{equation*}
with $\xi=2[F_1]-c_1$ and $length(Z_1)=n$, $length(Z_2)=m$.
Now if $b>a$ we have
$\Delta_b(\epsilon(lH))=\Delta_a(\epsilon(lH))+2(b-a)\<C\cdot\xi\>=2(b-a)\<C
\cdot\xi\>>0$
since $\Delta_a(\epsilon)=0$ and $\<C\cdot\xi\>>0$. Thus $E$ is $b$-unstable.
Assume that $E$ is not $a_-$-stable. Then it lies in an extension
\begin{equation*}\tag{$\epsilon_1$}0\longrightarrow {\cal I}_{Y_1}(G_1)\longrightarrow E\longrightarrow
{\cal I}_{Y_2}(G_2)\longrightarrow 0,\end{equation*}
for which
$\Delta_{a_-}(\epsilon_1(lH))\ge 0>\Delta_{a}(\epsilon_1(lH))$ for $l>>0$.
Hence we obtain $\<(2G_1-c_1)\cdot H\>\ge \<(2F_1-c_1)\cdot H\>$ and
$P_{a_-}({\cal I}_{Z_1}(F_1+lH))<P_{a_-}({\cal I}_{Y_1}(G_1+lH))$ and thus
$\chi({\cal I}_{Z_1}(F_1+lH-C))<\chi({\cal I}_{Y_1}(G_1+lH-C))$ or
$\chi({\cal I}_{Z_1}(F_1+lH+C))<\chi({\cal I}_{Y_1}(G_1+lH+C))$.
Consequently
${\hbox{\rom{Hom}}}({\cal I}_{Y_1}(G_1),{\cal I}_{Z_1}(F_1))=0$ and the obvious map
${\cal I}_{Y_1}(G_1)\longrightarrow {\cal I}_{Z_2}(F_2)$ is an injection. Hence
$F_2-G_1$ is effective.
If $F_2\ne G_1$, we would have $\<(G_1-F_2)\cdot H\><0$, and, by
$\<\xi \cdot H\>=0$, we would get the contradiction $\<(2G_1-c_1)\cdot H\><0$.
So $G_1=F_2$. By the injectivity of ${\cal I}_{Y_1}(G_1)\longrightarrow {\cal I}_{Z_2}(F_2)$
and the fact that \ref{splitting} is not split, we get
$length(Z_2)<length(Y_1)$ which shoes that $E$ is $a_-$-stable.
A similar argument shows that $E$ is strictly $a$-semistable.
\end{pf}
\begin{rem}\label{stab1} We can also easily see from the above arguments that
in the situation of \ref{stab} any sheaf $E\in M_{a_-}(c_1,c_2)$,
which does not lie in any ${\text{\rom{\bf E}}}_{\xi}^{n,m}$ for $(\xi,n,m)\in A^+(a)$
is $a_-$-stable (resp. semistable) if and only if it is $a$-stable (resp.
semistable).
\end{rem}
\begin{rem}\label{fine} \begin{enumerate}
\item Looking at the proof of \cite{Ma2}
for the sufficient
criterion for the existence of a universal family on $M_H(c_1,c_2)$,
we see that the same proof also works for $M_{a}(c_1,c_2)$ and we
get the same criterion, i.e. if $c_1$ is not divisible by $2$ in
$NS(S)$ or otherwise $4c_2-c_2^2$ is not divisible by $8$ and
$M_a(c_1,c_2)=
M_a^s(c_1,c_2)$, then $M^s_a(c_1,c_2)$ carries a universal family.
\item From the results obtained so far it follows easily that,
under the above conditions
for the Chern classes, $M_a(c_1,c_2)=M^s_a(c_1,c_2)$ if and only if
$a$ is not a miniwall.
\end{enumerate}
\end{rem}
\begin{prop}\label{flip}
\begin{enumerate}
\item $M_0(c_1,c_2)=M_{H_-}(c_1,c_2)$ and $M_1(c_1,c_2)=M_{H_+}(c_1,c_2)$.
\item If $b\in [0,1]$ is not on a miniwall, the moduli space
$M_b(c_1,c_2)$ depends only on the minichamber in which $b$ is
lying, and
$M_b(c_1,c_2)\setminus M^s_b(c_1,c_2)$ is independent of $b$.
\item Let $a_-<a_+$ be in neighbouring minichambers separated
by the miniwall
$a$. Then we have a set-theoretical decomposition
$$M_{a_+}(c_1,c_2)=\left(M_{a_-}(c_1,c_2)\setminus
\coprod_{(\xi,n,m)\in A^+(a)}
{\text{\rom{\bf E}}}^{n,m}_{\xi} \right){\sqcup}
\left(\coprod_{(\xi,n,m)\in A^+(a)}
{\text{\rom{\bf E}}}^{m,n}_{-\xi}\right),$$
and there are morphisms
$$\matrix M_{a_-}(c_1,c_2)&&&&M_{a_+}(c_1,c_2)\cr
&\mapse{\psi_-}&&\mapsw{\psi_+}\cr
&&M_{a}(c_1,c_2)\cr\endmatrix$$
which are open embeddings over
$$M_{a_-}(c_1,c_2)\setminus
\coprod_{(\xi,n,m)\in A^+(a)}
{\text{\rom{\bf E}}}^{n,m}_{\xi} \text{ and }\ \ M_{a_+}(c_1,c_2)\setminus
\coprod_{(\xi,n,m)\in A^+(a)}
{\text{\rom{\bf E}}}^{m,n}_{-\xi}.$$
\end{enumerate}\end{prop}
\begin{pf} (1), (2), (3) follow by putting together the results of this
section. By lemma \ref{stab} all the points of
$M_{a_-}(c_1,c_2)$ and $M_{a_+}(c_1,c_2)$ are
$a$-semistable and hence we get the morphisms
$\psi_-$ and $\psi_+$. The statement that they be open embeddings over the
indicated open subsets, follows from remark
\ref{stab1}.\end{pf}
\section{The normal bundles of the exceptional sets}
Our aim in this and the next chapter is to describe the
passage through a miniwall
which corresponds to a good wall.
We keep the assumptions from the beginning of the previous section.
In addition to those we assume that either $p_g(S)=0$ or
$K_S$ is trivial, and that the wall
$W$ is good.
Let $a$ define a miniwall and let $(\xi,n,m)\in A^+(a)$.
Let $a_-<a_+$ lie in neighbouring minichambers separated
by $a$.
For simplicity of notation we shall assume that $A^+(a)=\{(\xi,n,m)\}$.
Because,
for $(\xi,n_1,m_1)$, $ (\xi_2,n_2,m_2)$ distinct elements of $A^+(a)$,
the sets ${\text{\rom{\bf E}}}_{\xi_1}^{n_1,m_1}$ and ${\text{\rom{\bf E}}}_{\xi_2}^{n_2,m_2}$ are disjoint
by proposition \ref{wall} and our arguments are local in a
neighbourhood of
each ${\text{\rom{\bf E}}}_{\eta}^{l,s}$, this assumption can be made without
loss of generality.
Furthermore we assume for simplicity of notation that $NS(S)$ has no
$2$-torsion. Then the classes $(c_1+\xi)/2$,
$(c_1-\xi)/2\in NS(S)$ are well-defined and
$T_\xi^{n,m}=M(1,(c_1+\xi)/2,n)\times M(1,(c_1-\xi)/2,m)$.
Again this assumption is not important, as otherwise
the components of $E_{\xi}^{n,m}$ and $E_{-\xi}^{m,n}$ are disjoint.
\begin{nota} \label{notsec4} We shall write
$M_-:=M_{a_-}(c_1,c_2)$, $M_+:=M_{a_+}(c_1,c_2)$,
$M^s_-:=M^s_{a_-}(c_1,c_2)$, $M^s_+:=M^s_{a_+}(c_1,c_2)$ and
put ${\text{\rom{\bf E}}}_-:={\text{\rom{\bf E}}}_\xi^{n,m}$ and
${\text{\rom{\bf E}}}_+:={\text{\rom{\bf E}}}^{n,m}_{-\xi}$.
\end{nota}
\begin{defn}
Let $ {\cal F}_1'$ (resp. $ {\cal F}_2'$) be the pull-back of a universal sheaf over
$S\times M(1,(c_1+\xi)/2,n)$ (resp. $S\times M(1,(c_1-\xi)/2,m)$) to
$S\times T$, where
$T:= M(1,(c_1+\xi)/2,n)\times M(1,(c_1-\xi)/2,m)$. Let
$q=q_T:S\times T\to T$ be the projection.
Let ${\cal A}_{-}':={\hbox{\rom{Ext}}}_{q}^1({\cal F}_2',{\cal F}_1')$ and
${\cal A}_{+}':={\hbox{\rom{Ext}}}_{q}^1({\cal F}_1',{\cal F}_2')$
and ${\Bbb P}_{-}:={\Bbb P}({\cal A}_{-}'),$ ${\Bbb P}_{+}:={\Bbb P}({\cal A}_{+}').$ Let
$\pi_-$ (resp. $\pi_+$) be the projections of ${\Bbb P}_-$
(resp. ${\Bbb P}_+$) to $T$ and $\tau_-$ (resp. $\tau_+$) the
tautological sublinebundles of ${\cal A}_-:=\pi_-^*({\cal A}_-')$
(resp. ${\cal A}_+:=\pi_+^*({\cal A}_+')$).
Let ${\cal F}_1:=({\hbox{\rom{id}}}_S\times\pi_-)^*{\cal F}_1'$ and
${\cal F}_2:=({\hbox{\rom{id}}}_S\times \pi_-)^*{\cal F}_2'$.
\end{defn}
\begin{lem}\label{enm}
\begin{enumerate}
\item $ {\cal A}_-'$ is locally free of rank $-\xi(\xi-K_S)/2 +n+m-\chi({\cal O}_S)$
and its formation commutes with arbitrary base change.
\item There is an isomorphism ${\hbox{\rom{Ext}}}^1({\cal F}_2',{\cal F}_1')\longrightarrow H^0(T,{\cal A}_-')$,
hence over $S\times {\Bbb P}_-$ there is a tautological extension
\begin{eqnarray}\label{globext}&&
0\longrightarrow {\cal F}_1\longrightarrow {{\cal E}}\to {\cal F}_2(\tau_-)\longrightarrow 0.\end{eqnarray}
There is a morphism $i_-:{\Bbb P}_-\longrightarrow M_-$ with image ${\text{\rom{\bf E}}}_-$.
\end{enumerate}
\end{lem}
\begin{pf}As $\xi$ defines a wall, ${\hbox{\rom{Hom}}}_{q}({\cal F}_2,{\cal F}_1)$ is fibrewise $0$,
and, as the wall is good, $F_1-F_2+K_S$ is not effective for
$(F_1,F_2)\in T$,
therefore
by Serre duality for the extension groups \cite{Mu2} also
${\hbox{\rom{Ext}}}^2_{q}({\cal F}_2,{\cal F}_1)=0$.
So (1) follows by Riemann-Roch for the extension groups \cite{Mu2}.
Now we apply \cite{La}.
\end{pf}
\begin{prop}\label{normal}
\begin{enumerate}
\item If $p_g(S)=0$ or if $K_S$ is trivial, then
$i_-:{\Bbb P}_-\longrightarrow M_-$ is a closed embedding and
$M_-$ is smooth along ${\text{\rom{\bf E}}}_-$. The irreducible component of
$M_-$ containing
${\text{\rom{\bf E}}}_-$ has the expected dimension.
\item If $p_g(S)=0$, then the normal bundle $N_{{\text{\rom{\bf E}}}_-/M_-}$ of ${\text{\rom{\bf E}}}_-$ in
$M_-$ is equal to ${\cal A}_+(\tau_-)$.
\item If $K_S$ is trivial, then $N_{{\text{\rom{\bf E}}}_-/M_-}=Q^\vee(\tau_-)$,
where $Q$ is the universal quotient bundle on ${\Bbb P}_-={\Bbb P}({\cal A}_-)$.
\end{enumerate}
\end{prop}
\begin{pf} By proposition \ref{wall}(3) and lemma \ref{enm} the map
${\Bbb P}_-\longrightarrow M_-$ is injective with image ${\text{\rom{\bf E}}}_-$. We also
see by proposition \ref{wall} that ${\text{\rom{\bf E}}}_-\subset M_-^s$.
In case $K_S$ is trivial, $Spl(c_1,c_2)$ and thus also
the open subscheme
$M_-^s$ are smooth by \cite{Mu1}.
In order to see that $M_-$ is smooth along ${\text{\rom{\bf E}}}_-$ in the case $p_g(S)=0$,
we have to show that
${\hbox{\rom{Ext}}}^2(E,E)=0$ for any $E\in {{\text{\rom{\bf E}}}}_-$.
So let $E\in {{\text{\rom{\bf E}}}}_-$ be given by a nontrivial extension (\ref{splitting})
\begin{equation*} \tag{$\epsilon$}
0\longrightarrow {\cal I}_{Z_1}(F_1)\longrightarrow E\longrightarrow {\cal I}_{Z_2}(F_2)\longrightarrow 0.\end{equation*}
As the wall $W$ is good, we obtain
by Serre duality and the fact that $p_g(S)=0$ that
${\hbox{\rom{Ext}}}^2({\cal I}_{Z_i}(F_i),{\cal I}_{Z_j}(F_j))=0$ for $i=1,2$ and $j=1,2$.
Hence applying ${\hbox{\rom{Ext}}}^2({\cal I}_{Z_i}(F_i),\cdot)$ to $(\epsilon)$
we get ${\hbox{\rom{Ext}}}^2({\cal I}_{Z_i}(F_i),E)=0$ for $i=1,2$
and this in turn shows that ${\hbox{\rom{Ext}}}^2(E,E)=0$.
We now want to compute the normal bundle to ${\text{\rom{\bf E}}}_-$.
{\it First Case: }$p_g(S)=0$.
Applying ${\hbox{\rom{Hom}}}_{q}(\cdot,\cdot)$ on both sides of
the sequence (\ref{globext})
and denoting by $\pi_i$ the composition of $\pi_-$
with the projection to the $i^{th}$ factor
we get the following exact diagram of locally free sheaves on ${\Bbb P}_-$
\begin{eqnarray}\label{globdiag}&&\matrix
&&0&&0&&0\cr
&&\mapd{}&&\mapd{}&&\mapd{}\cr
0&\mapr{}&T_{{\Bbb P}_-/T}&\mapr{}&{\hbox{\rom{Ext}}}^1_{q}({\cal F}_2(\tau_-),{\cal E})&\mapr{}&
\pi_2^*T_{M(1,(c_1-\xi)/2,m)}&\longrightarrow& 0\cr
&&\mapd{}&&\mapd{\psi}&&\mapd{}\cr
0&\mapr{}&{\hbox{\rom{Ext}}}^1_{q}({\cal E},{\cal F}_1)&\mapr{\varphi}&i_-^*(T_{M_-})&\mapr{}&
{\hbox{\rom{Ext}}}^1_{q}({\cal E},{\cal F}_2)(\tau_-)&\longrightarrow& 0\cr
&&\mapd{}&&\mapd{}&&\mapd{}\cr
0&\mapr{}&\pi_1^*T_{M(1,(c_1+\xi)/2,n)}&\mapr{}
&{\hbox{\rom{Ext}}}^1_{q}({\cal F}_1,{\cal E})&\mapr{}&
{\cal A}_+(\tau_-)&\longrightarrow& 0\cr
&&\mapd{}&&\mapd{}&&\mapd{}\cr
&&0&&0&&0\cr
\endmatrix\end{eqnarray}
To identify the entries in this diagram we have used the following facts.
\begin{enumerate}
\item ${\hbox{\rom{Hom}}}_q({\cal F}_1,{\cal F}_1)={\hbox{\rom{Hom}}}_q({\cal F}_2,{\cal F}_2)={\cal O}_{{\Bbb P}_-}$.
\item If $Q$ is the universal quotient on
${\Bbb P}({\cal A}_-)$, then the relative tangent bundle is
$T_{{\Bbb P}_-/T}=Q(-\tau_-)$, i.e. the cokernel of the natural map
${\cal O}_{{\Bbb P}_-}={\hbox{\rom{Hom}}}_q({\cal F}_1,{\cal F}_1)\longrightarrow {\hbox{\rom{Ext}}}^1_q({\cal F}_2(\tau_-),{\cal F}_1)$.
\item $\pi_2^\ast(T_{M(1,(c_1-\xi)/2,m)})={\hbox{\rom{Ext}}}^1_{q}({\cal F}_2,{\cal F}_2)$ and
$\pi_2^\ast T_{M(1,(c_1+\xi)/2,n)}={\hbox{\rom{Ext}}}^1_{q}({\cal F}_1,{\cal F}_1)$.
\item By Mukai's sheafified Kodaira-Spencer map \cite{Mu1} we have
$i_-^*T_{M_-}={\hbox{\rom{Ext}}}^1_{q}({\cal E},{\cal E}).$ Mukai shows the result only if
$S$ is an abelian or K3-surface, but in his proof he only uses that
$Spl(c_1,c_2)$ is smooth in a neighbourhood of ${\text{\rom{\bf E}}}_-$,
(which we have just seen)
and ${\hbox{\rom{Ext}}}^1_{q}({\cal E},{\cal E})$ is locally free and
compatible with base change.
\end{enumerate}
To show that the sequences in the diagram are exact
we just use standard techniques.
It is enough to check the exactness fibrewise.
One has repeatedly to make use of the fact that $\xi$
defines a good wall,
i.e. if $E\in {{\text{\rom{\bf E}}}}_-$ is given by (\ref{splitting}),
then $F_1-F_2$, $F_2-F_1$, $F_1-F_2+K_S$, $F_2-F_1+K_S$ are not effective,
which implies that
${\hbox{\rom{Hom}}}_q({\cal F}_1,{\cal F}_2(\tau_-))= {\hbox{\rom{Hom}}}_q({\cal F}_2(\tau_-),{\cal F}_1)=
{\hbox{\rom{Ext}}}^2_q({\cal F}_1,{\cal F}_2(\tau_-))= {\hbox{\rom{Ext}}}^2_q({\cal F}_2(\tau_-),{\cal F}_1)=0$.
In addition we use that all $E\in {{\text{\rom{\bf E}}}}_-$ are simple and that
${\hbox{\rom{Ext}}}^2_{q}({\cal F}_2,{\cal F}_2)={\hbox{\rom{Ext}}}^2_{q}({\cal F}_1,{\cal F}_1)=0$.
We also use the vanishings from the proof of the smoothness of
$M_-$ along ${\text{\rom{\bf E}}}_-$.
\noindent{\it Second Case: } $K_S$ is trivial.
We apply essentially the same arguments as in the first case.
Now however we have
${\hbox{\rom{Ext}}}^2_{q}({\cal E},{\cal F}_1)={\hbox{\rom{Ext}}}^2_{q}({\cal F}_2(\tau_-),{\cal E})={\hbox{\rom{Ext}}}^2_{q}({\cal E},{\cal E})=
{\hbox{\rom{Ext}}}^2_{q}({\cal F}_1,{\cal F}_1)={\hbox{\rom{Ext}}}^2_{q}({\cal F}_2,{\cal F}_2)={\cal O}_{{\Bbb P}_-}$,
which follows easily from Mukai's results \cite{Mu1}.
We also notice that by Serre-duality ${\cal A}_+$ is canonically dual to
${\cal A}_-$.
Using all this we again get the diagram (\ref{globdiag}) with the entry
${\cal A}_+(\tau_-)$ in the lower right corner replaced by the
kernel of the natural map
${\cal A}_+(\tau_-)\to {\cal O}_{{\Bbb P}_-}$, i.e. $Q^\vee(\tau_-)$.
\noindent{\it Claim: } The image of the Kodaira-Spencer map
$\kappa:T_{{\Bbb P}_-}\to {\hbox{\rom{Ext}}}^1_{q}({\cal E},{\cal E})$
is $Im(\varphi)$+$Im(\psi)$ (see (\ref{globdiag})).
Note that, by what we have shown so far, the claim implies the theorem.
\noindent{\it Proof of the Claim.}
For dimension reasons it is enough to show that $Im(\varphi)$ and $Im(\psi)$
both are contained in the image of $\kappa$.
We show it for $Im(\varphi)$. It is enough to show this fibrewise.
Let $F_1\in M(1,(c_1+\xi)/2,n)$ and let $({\Bbb P}_-)_{F_1}$ be the
fibre of the projection
$\pi_1:{\Bbb P}_- \longrightarrow M(1,(c_1+\xi)/2,n)$ over $F_1$.
Then
$({\Bbb P}_-)_{F_1}$ is
the space of extensions
$$0\to F_1\to E\to G\to 0$$ with
$G$ running through $M(1,(c_1-\xi)/2,m)$.
Let $x\in ({{\Bbb P}}_-)_{F_1}$ be given by an
extension
\begin{equation}\tag{$\lambda_x$} 0\to F_1\to E\to G_1\to 0.\end{equation}
We will want to show that
$\kappa(T_{({\Bbb P}_-)_{F_1}}(x))=\varphi({\hbox{\rom{Ext}}}^1(G_1,E))$.
The tangent space to $({\Bbb P}_-)_{F_1}$ at $x$ is the
space of first order deformations of $E$ together with an injection
$F_1\to E$. For $t\in T_{({\Bbb P}_-)_{F_1}}(x)$ we get therefore the diagram
$$\matrix
&&0&&0&&0\cr
&&\mapd{}&&\mapd{}&&\mapd{}\cr
0&\mapr{}&F_1&\mapr{}&E&\mapr{}&G_1&\mapr{}&0\cr
&&\mapd{}&&\mapd{}&&\mapd{}\cr
0&\mapr{}&F_1\oplus F_1&\mapr{}&\widetilde
E&\mapr{}&\widetilde G&\mapr{}&0\cr
&&\mapd{}&\mapne{\gamma}&\mapd{}&&\mapd{}\cr
0&\mapr{}&F_1&\mapr{}&E&\mapr{}&G_1&\mapr{}&0\cr
&&\mapd{}&&\mapd{}&&\mapd{}\cr
&&0&&0&&0\cr
\endmatrix
\eqno (*)$$
and we see that $T_{({\Bbb P}_-)_{F_1}}(x)$ can be identified
with the space of diagrams $(*)$.
Furthermore $\kappa(t)$ is the extension class of the middle column
of $(*)$.
{}From $(*)$ we also get a sequence
$0\longrightarrow E\longrightarrow \widetilde E/\gamma(F_1)\longrightarrow G_1\to 0$
such that $\widetilde E$ is defined by pull-back
$$\matrix
\widetilde E/\gamma(F_1)&\mapr{}&G_1\cr
\mapu{}&&\mapu{}\cr
\widetilde E.&\mapr{}&E\cr
\endmatrix$$
This gives a map $\theta:T_{({\Bbb P}_-)_{F_1}}(x)\longrightarrow Ext^1(G_1,E)$, such that the
restriction of
$\kappa$ to $T_{({\Bbb P}_-)_{F_1}}(x)$ is $\varphi\circ\theta$.
To finish the proof we have to see that
$\theta$ is an isomorphism. We give an inverse.
Let
$$0\longrightarrow E\longrightarrow W\longrightarrow G_1\longrightarrow 0$$ be an extension.
We define $\widetilde E$ as the fibre product
$$\matrix
W&\mapr{}&G_1\cr
\mapu{}&&\mapu{}\cr
\widetilde E,&\mapr{}&E\cr
\endmatrix$$
and we see that it lies in a diagram $(*)$.
\end{pf}
\begin{rem}\label{newcomp} Assume $p_g(S)=0$. From
lemma \ref{normal} it follows that the dimension of ${\text{\rom{\bf E}}}_-$ is at
most the expected dimension
$N=(4c_2-c_1^2)-3\chi({\cal O}_S)+q(S)$. We have to distinguish two cases.
\begin{enumerate}
\item $dim({\text{\rom{\bf E}}}_-)<N$ and $dim({\text{\rom{\bf E}}}_+)<N$. Then the change
from $M_-$ to $M_+$ is a birational transformation.
\item $dim({\text{\rom{\bf E}}}_-)=N$ or $dim({\text{\rom{\bf E}}}_+)=N$. We can assume that $dim({\text{\rom{\bf E}}}_-)=N$.
Then by lemma \ref{normal}
${\text{\rom{\bf E}}}_-$ is a smooth connected component of $M_-$, which is isomorphic to
${\Bbb P}_-$. And, ${\cal A}_+(\tau_-)$ being the normal bundle to ${\text{\rom{\bf E}}}_-$,
we have ${\cal A}_+=0$ and therefore ${\text{\rom{\bf E}}}_+=\emptyset$.
This happens if and only if $\<\xi\cdot(\xi-K_S)\>/2+d_\xi=\chi({\cal O}_S)$.
If we allow $NS(S)$ to contain $2$-torsion, we see that all
the connected components of ${\text{\rom{\bf E}}}_-$ are connected components of $M_-$.
\end{enumerate}
\end{rem}
Assume for the following definition and corollary that we are in case (1) of
\ref{newcomp}, i.e. that the change from $M_-$ to $M_+$ is
birational.
\begin{defn} Let $\widetilde M_-$ be the blow-up of
$M_-$ along ${\text{\rom{\bf E}}}_-$ and $D$ the exceptional divisor.
Similarly let $\widetilde M_+$ be the blow up of $M_+$ along ${\text{\rom{\bf E}}}_+$.
Let $\pi_D$, $\pi_{D-}$, $\pi_{D+}$ be the projections
from $D$ to $T$, ${\Bbb P}_-$, ${\Bbb P}_+$ respectively.
\end{defn}
\begin{cor}\begin{enumerate}
\item
If $p_g(S)=0$ then $D$ is isomorphic to
${\Bbb P}_-\times_T {\Bbb P}_+$ and with this identification
${\cal O}(D)|_D={\cal O}(\tau_-+\tau_+)$.
\item If $K_S$ is trivial, then ${\cal A}_-$ and ${\cal A}_+$ are canonically dual and
$D$ is the incidence correspondence
$\{(l,H)\in {\Bbb P}({\cal A}_-)\times_T{\Bbb P}^{\vee}({\cal A}_-) \,|\, l\subset H\}$ and
${\cal O}(D)|_D$ is the restriction of ${\cal O}(\tau_-+\tau_+)$.
\end{enumerate}
\end{cor}
\section{Blow-up construction}
We keep the assumptions and notations of the last section.
In addition we assume that we in case (1) of
\ref{newcomp}, i.e. the map $\widetilde M_-\longrightarrow M_-$ is birational.
In this section we want to show that $\widetilde M_-$ and $\widetilde M_+$
are isomorphic.
We shall construct a morphism $\varphi_+:\widetilde M_-
\to M_+$, which we shall show is the blow-up of $M_+$
along ${\text{\rom{\bf E}}}_+$.
Let $\varphi_-:\widetilde M_-\to M_-$ be the blow-up map and
$j:D\to \widetilde M_-$ be the embedding.
We denote $\widetilde M_-^s:=\varphi_-^{-1}M_-^s$.
Let ${\cal U}_-$ be a universal pseudo-family on $S\times M^s_-$ and
${\cal V}_-:=({\hbox{\rom{id}}}_S\times \varphi_-)^*{\cal U}_-$.
We want to make an elementary transform of ${\cal V}_-$ along $D_S:=S\times D$
to obtain a pseudo-family ${\cal V}_+$ of $a_+$-stable sheaves on
$\widetilde M_-^s$ and
thus the
desired map $\varphi_+$. If ${\cal U}_-$ is a universal family, then also
${\cal V}_+$ will be one.
\begin{nota}
For a sheaf ${\cal H}$ on $S\times{\Bbb P}_-$ (resp. $S\times{\Bbb P}_+$) we will write
${\cal H}_D$ for $({\hbox{\rom{id}}}_S\times \pi_{D-})^*{\cal H}$
(resp. $({\hbox{\rom{id}}}_S\times \pi_{D+})^*{\cal H}$). We also write
${\cal F}_{1D}$ and ${\cal F}_{2D}$ instead of $({\cal F}_{1})_{D}$ and $({\cal F}_{2})_{D}$.
\end{nota}
\begin{defn}
By the universal property of $M_-$ and lemma \ref{enm} there is
a line bundle
$\lambda$ on $D$ such that there is an exact sequence
\begin{eqnarray}
\label{restrsec}&&0\to {\cal F}_{1D}(\lambda)\longrightarrow {\cal V}_-|_{D_S}\longrightarrow
{\cal F}_{2D}(\tau_-+\lambda)\longrightarrow 0,
\end{eqnarray}
indeed there is already a sequence like this on ${\text{\rom{\bf E}}}_-$.
Let $\gamma$ be the composition ${\cal V}_-\longrightarrow {\cal V}_-|_{D_S}\longrightarrow
{\cal F}_{2D}(\tau_-+\lambda).$ Then we put ${\cal V}_+:=ker\gamma$.
Because ${\cal V}_-$ is flat on $S\times \widetilde M^s_-$, and
${\cal F}_{2D}(\lambda+\tau_-)$
is flat on the Cartier divisor $S\times D$, ${\cal V}_+$ is flat over
$S\times \widetilde M^s_-$.
The restrictions of ${\cal V}_+$ and ${\cal V}_-$ to
$S\times \widetilde M^s_-\setminus D$ are naturally isomorphic.
There are diagrams
of sheaves on $S\times \widetilde M_-^s$
\begin{eqnarray}\label{dia1}
&&{\matrix
&&&&0&&0\cr
&&&&\mapd{}&&\mapd{}\cr
0&\mapr{}&{\cal V}_-(-D_S)&\mapr{}&{\cal V}_+&\mapr{}&{\cal F}_{1D}(\lambda)&\mapr{}&0\cr
&&\Big|\Big|&&\mapd{}&&\mapd{}\cr
0&\mapr{}&{\cal V}_-(-D_S)&\mapr{}&{\cal V}_-&\mapr{}&{\cal V}_-|_{D_S}&\mapr{}&0\cr
&&&&\mapd{}&&\mapd{}\cr
&&&&{\cal F}_{2D}(\tau_-+\lambda)&\relgl\joinrel\relgl\joinrel\relgl&{\cal F}_{2D}(\tau_-+\lambda)
&\mapr{}&0\cr
&&&&\mapd{}&&\mapd{}\cr
&&&&0&&0\cr\endmatrix}\end{eqnarray}
\begin{eqnarray}\label{dia2}&&{\matrix
&&&&0&&0\cr
&&&&\mapd{}&&\mapd{}\cr
0&\mapr{}&{\cal V}_+(-D_S)&\mapr{}&{\cal V}_-(-D_S)&\mapr{}
&{\cal F}_{2D}(\lambda-\tau_+)&\mapr{}&0\cr
&&\Big|\Big|&&\mapd{}&&\mapd{}\cr
0&\mapr{}&{\cal V}_+(-D_S)&\mapr{}&{\cal V}_+&\mapr{}&{\cal V}_+|_{D_S}&\mapr{}&0\cr
&&&&\mapd{}&&\mapd{}\cr
&&&&{\cal F}_{1D}(\lambda)&\relgl\joinrel\relgl\joinrel\relgl&{\cal F}_{1D}(\lambda)&\mapr{}&0\cr
&&&&\mapd{}&&\mapd{}\cr
&&&&0&&0\cr\endmatrix}
\end{eqnarray}
By the rightmost column of (\ref{dia2}), $({\cal V}_+)_x\in {\text{\rom{\bf E}}}_+$
for all $x\in D$.
Therefore by proposition \ref{flip}
${\cal V}_+$ is a pseudo-family of $a_+$-stable sheaves
over $\widetilde M^s_-$ and defines
a morphism $\varphi_+:\widetilde M^s_- \to M^s_+$.
We see from the definitions that the restriction of $\varphi_+$ to
$\widetilde M^s_-\setminus D$ is an isomorphism to $M^s_+\setminus {\text{\rom{\bf E}}}_+$,
which coincides with the natural identification
$\widetilde M^s_-\setminus D\simeq M^s_-\setminus
{\text{\rom{\bf E}}}_-\simeq M^s_+\setminus {\text{\rom{\bf E}}}_+$.
As ${\text{\rom{\bf E}}}_-\subset M_-^s$ and ${\text{\rom{\bf E}}}_+\subset M_+^s$,
we see that $\varphi_+$ extends to a morphism
$\widetilde M_-\longrightarrow M_+$, which we still denote by
$\varphi_+$.
\end{defn}
\begin{thm}
$\varphi_+:\widetilde M_-\longrightarrow M_+$ is the blow up of $M_+$ along
${\text{\rom{\bf E}}}_+$.
\end{thm}
\begin{pf} By the above $\varphi_+(D)\subset {\text{\rom{\bf E}}}_+$.
We want to show that $\varphi_+|_D$ is the projection $\pi_{D+}:D\longrightarrow {\text{\rom{\bf E}}}_+$.
For this we have to show that the extension
$$0\longrightarrow{\cal F}_{2D}(\lambda-\tau_+)\longrightarrow {\cal V}_+|_{S\times D}
\longrightarrow {\cal F}_{1D}(\lambda)\mapr{} 0$$ from the rightmost column of (\ref{dia2})
is the pull-back via $\pi_{D+}$ of the tautological extension on ${\Bbb P}_+$
(defined analogously to \ref{globext})
tensorized with ${\cal O}_D(\lambda-\tau_+)$.
It is enough to show this fibrewise.
Let $x=(x_-,x_+)\in D\subset {\Bbb P}_-\times_T {\Bbb P}_+$ and
let $V_-:=({\cal V}_-)_x$ and $V_+:=({\cal V}_+)_x$ be given by extensions
\begin{eqnarray}\label{seq1}
&&0\longrightarrow F_1\longrightarrow V_-\longrightarrow F_2\longrightarrow 0,\\
\label{seq2}
&&0\longrightarrow F_2\longrightarrow V_+\longrightarrow F_1\longrightarrow 0.
\end{eqnarray}
Then $\pi_D(x)$ is the point $(F_1,F_2)\in T$ and $x_-\in ({\Bbb P}_-)_{(F_1,F_2)}=
{\Bbb P}({\hbox{\rom{Ext}}}^1(F_2,F_1))$
is the extension class of (\ref{seq1}).
Then we have to show that $x_+\in ({\Bbb P}_+)_{(F_1,F_2)}={\Bbb P}({\hbox{\rom{Ext}}}^1(F_1,F_2))$
is the extension class of (\ref{seq2}).
Let $R:={\hbox{\rom{Spec}\,}}{\Bbb C}[\epsilon]/(\epsilon^2)$ and
let $t:R\to \widetilde M_-$ be a tangent vector to $\widetilde M_-$ at $x$,
which is not tangent to $D$. Then $t$ factors through $2D$ (i.e. the subscheme
defined by ${\cal I}_D^2$). If we restrict the diagrams (\ref{dia1}), (\ref{dia2})
to $2D_S$, we see that the image of the map
${\cal V}_-(-D_S)|_{2D_S}\longrightarrow {\cal V}_+|_{2D_S}$ is ${\cal I}_{D_S}{\cal V}_-/{\cal I}^2_{D_S}{\cal V}_-$
and the image of the composition
${\cal V}_+(-D_S)|_{2D_S}\longrightarrow{\cal V}_-(-D_S)|_{2D_S}\longrightarrow V_+|_{2D_S}$ is
${\cal I}_{D_S}{\cal F}_{1D}(\lambda)\cdot/{\cal I}_{D_S}^2{\cal F}_{1D}(\lambda)$. Therefore,
by pulling back the diagrams
(\ref{dia1}), (\ref{dia2}) to $S\times R$ via $({\hbox{\rom{id}}}_S\times t)$
and pushing down with the projection
$p:S\times R\to S$, we get the diagrams
\begin{eqnarray}\label{dia3}\matrix
&&&&0&&0\cr
&&&&\mapd{}&&\mapd{}\cr
0&\mapr{}&V_-&\mapr{}&\widetilde V_+&\mapr{}&F_1&\mapr{}&0\cr
&&\Big|\Big|&&\mapd{}&&\mapd{}\cr
0&\mapr{}&V_-&\mapr{}&\overline V_-&\mapr{}&V_-&\mapr{}&0\cr
&&&&\mapd{}&&\mapd{}\cr
&&&&F_2&\relgl\joinrel\relgl\joinrel\relgl&F_2&\mapr{}&0\cr
&&&&\mapd{}&&\mapd{}\cr
&&&&0&&0\cr\endmatrix\end{eqnarray}
\begin{eqnarray}
\label{dia4}
\matrix
&&&&0&&0\cr
&&&&\mapd{}&&\mapd{}\cr
0&\mapr{}&F_1&\mapr{}& V_-&\mapr{}&F_2&\mapr{}&0\cr
&&\Big|\Big|&&\mapd{}&&\mapd{}\cr
0&\mapr{}&F_1&\mapr{}&\widetilde V_+&\mapr{}&V_+&\mapr{}&0\cr
&&&&\mapd{}&&\mapd{}\cr
&&&&F_1&\relgl\joinrel\relgl\joinrel\relgl&F_1&\mapr{}&0\cr
&&&&\mapd{}&&\mapd{}\cr
&&&&0&&0\cr\endmatrix
\end{eqnarray}
The extension class $\delta\in {\Bbb P}({\hbox{\rom{Ext}}}^1(V_-,V_-))$
of the middle row of (\ref{dia3}) is the class of the image of $t$ under
$d\varphi_-:T_{\widetilde M_-}(x)\longrightarrow T_{ M_-}(\varphi_-(x))={\hbox{\rom{Ext}}}^1(V_-,V_-)$.
The image of the composition
$$T_{ M_-}(x)\longrightarrow
\varphi_-^*(T_{M_-}(x))\,\mapr{\rho} \,\varphi_-^*(N_{{\text{\rom{\bf E}}}_-/M_-}(x))$$ is
the tautological subline-bundle of
$\varphi_-^*(N_{{\text{\rom{\bf E}}}_-/M_-}(x))={\hbox{\rom{Ext}}}^1(F_1,F_2)$ and the kernel is $T_{D}(x)$.
Therefore the image of $\rho(\delta)$ in
$({\Bbb P}_+)_{(F_1,F_2)}={\Bbb P}({\hbox{\rom{Ext}}}^1(F_1,F_2))$ is $x_+$.
By (\ref{globdiag}) the map $\rho$ is the composition
$$ {\hbox{\rom{Ext}}}^1(V_-,V_-)\,\mapr{\rho_1}\,{\hbox{\rom{Ext}}}^1(F_1,V_-)\,\mapr{\rho_2}\,
{\hbox{\rom{Ext}}}^1(F_1,F_2)$$ given by applying ${\hbox{\rom{Hom}}}(\cdot,\cdot)$ on both
sides to the
sequence
$0\to F_1\to V_-\to F_2\to 0$. By
(\ref{dia3}) $\rho_1(\delta)$ is the extension class of the
first row of (\ref{dia3}) giving $\widetilde V_+$, and then, by
(\ref{dia4}), $\rho(\delta)$ is the extension class of (\ref{seq2}).
So we see that
$\varphi_+|_D$ is the projection to ${\text{\rom{\bf E}}}_+$.
If for the moment we call $\overline \varphi_+:\widetilde M_+\to M_+$
the blow-up of $M_+$ along ${\text{\rom{\bf E}}}_+$ and $\overline D$ the
exceptional divisor,
we get analogously that $\overline D\simeq {\Bbb P}_-\times_T {\Bbb P}_+$,
(or the incidence correspondence in ${\Bbb P}_-\times_T {\Bbb P}_+$
in case $K_S$ is trivial).
In the same way as above we can construct a morphism
$\overline \varphi_-:\widetilde M_+\longrightarrow M_-$ such that
$\overline \varphi_-|_{\overline D}$
is the projection to ${\text{\rom{\bf E}}}_-$ and
$\overline \varphi_-|_{\widetilde M_+\setminus \overline D}$
is just the natural identification
$\widetilde M_+\setminus \overline D\simeq M_+\setminus
{\text{\rom{\bf E}}}_+\simeq M_-\setminus {\text{\rom{\bf E}}}_-$.
Therefore we have morphisms
$\varphi_-\times \varphi_+:\widetilde M_-\to M_-\times M_+$,
$\overline \varphi_-\times \overline \varphi_+:\widetilde M_+\to M_-\times M_+$,
which by the above are injective and easily seen to be injective on tangent
vectors. Furthermore
$(\varphi_-\times \varphi_+)(\widetilde M_-\setminus D)=
(\overline \varphi_-\times \overline \varphi_+)
(\widetilde M_+\setminus \overline D)$.
Therefore $\widetilde M_-$ and $\widetilde M_+$ are isomorphic
and in fact both isomorphic to the closure of the graph of the obvious
rational map $M_-\to M_+$.
\end{pf}
In the following theorem we put together the main results we
have obtained so far.
\begin{thm}\label{zblowup}
Let $S$ be a surface with either $p_g(S)=0$ or $K_S$ trivial.
Let $c_1\in NS(S)$, $c_2\in {\Bbb Z}$ and put
$N:=4c_2-c_1^2-3\chi({\cal O}_S)+q(S)$.
Let $W$ be a good wall of type $(c_1,c_2)$ and let $H_-$, $H_+$ be
ample divisors on $S$ in neighbouring chambers separated by
$W$. Then for all $a\in [0,1]$ there exist spaces $M_a(c_1,c_2)$
and a finite set of miniwalls dividing $[0,1]$ into
finitely many minichambers
such that the following holds:
\begin{enumerate}
\item $M_0(c_1,c_2)=M_{H_-}(c_1,c_2)$, $M_1(c_1,c_2)=M_{H_+}(c_1,c_2)$.
\item If $a_1$, $a_2$ are in the same minichamber
then $M_{a_1}(c_1,c_2)=M_{a_2}(c_1,c_2)$.
\item If $a_-<a<a_+$ and $a$ is the unique miniwall
between $a_-$ and $a_+$ then
$M_{a_+}(c_1,c_2)$ is obtained from $M_{a_-}(c_1,c_2)$
as follows:
We blow up $M_{a_-}(c_1,c_2)$ along the disjoint smooth subvarieties
${\text{\rom{\bf E}}}_{\xi}^{n,m}$, with $(\xi,n,m)\in A^+(a)$ (see \ref{defminiwall})
which fulfill $0\le dim({\text{\rom{\bf E}}}_{\xi}^{n,m})< N$
and blow-down the exceptional divisors to ${\text{\rom{\bf E}}}_{-\xi}^{m,n}$
respectively.
Then we remove the ${\text{\rom{\bf E}}}_{\xi}^{n,m}$ with $(\xi,n,m)\in{\cal A}^+(a)$ and
$dim({\text{\rom{\bf E}}}_{\xi}^{n,m})= N$
(which are unions of connected components of $M_{a_-}(c_1,c_2)$)
and take the disjoint union with all ${\text{\rom{\bf E}}}_{-\xi}^{m,n}$ with
$(\xi,n,m)\in{\cal A}^+(a)$ and
${\text{\rom{\bf E}}}_{\xi}^{n,m}= \emptyset$.
\item If $H$ is an ample divisor on $W$ which lies in the
closure of both
of the chambers containing $H_-$ and $H_+$, then, for all
$b\in [0,1]$, the space $M_b(c_1,c_2)$
is a moduli space of H-semistable sheaves on $S$ with a
suitable parabolic structure.
\end{enumerate}
\end{thm}
In \cite{Mu1} Mukai defines elementary transforms of a
symplectic variety $X$
as follows. Assume $X$ contains a subvariety $P$, which has codimension
$n$ and is a ${\Bbb P}_n$-bundle over a variety $Y$. Let $\widetilde X$
be the blow-up of $X$ along $P$. Then the exceptional divisor $E$
is isomorphic to the incidence correspondence in $P\times_Y P'$,
where $P'$ is the dual projective bundle to $P$.
One can then blow down $E$ to $P'$ to obtain a smooth
symplectic variety $X'$.
We will for the moment call $Y$ the center of such an
elementary transformation.
So by the above we obtain the following:
\begin{cor}
Let $S$ be a K3-surface or an abelian surface.
Let $H_-,H_+$ be polarisations which both do not lie on a wall.
Then $M_{H_+}(c_1,c_2)$ is obtained from $M_{H_-}(c_1,c_2)$
by a series of elementary transforms, whose centers are of the form
$M(1,(c_1+\xi)/2,n)\times M(1,(c_1-\xi)/2,m)$ for $\xi$
defining a wall between
$H_-$ and $H_+$ and $(n,m)$ running through the nonnegative integers with
$n+m=(4c_2-c_1^2+\xi^2)/4$.
\end{cor}
\begin{rem} If $q(S)\ne 0$ we can also, for $A\in Pic(S)$, $c_2\in {\Bbb Z}$
and an ample divisor $H$,
study the moduli space $\widetilde M_H(A,c_2)$ of rank $2$
torsion-free sheaves
$E$ on $S$ with $det(E)=A$ and $c_2(E)=c_2$. Then there is a morphism
$M_H(c_1,c_2)\longrightarrow Pic^0(S)$, whose fibres are the various $\widetilde
M_H(A,c_2)$ for $A$ with $c_1(A)=c_1$. Then, by restricting our arguments
to the fibres, we get that theorem \ref{zblowup}
also holds with the obvious changes for $\widetilde M_H(A,c_2)$.
\end{rem}
\section{The change of the Donaldson invariants in terms of Hilbert schemes}
In this section we assume that $q(S)=0$.
Let $\gamma_{c_1,c_2,g}$ be the Donaldson polynomial
with respect to a Riemannian metric $g$
associated to the principal $SO(3)$-bundle $P$ on $S$ whose
second Stiefel-Whitney class
$w_2(P)$ is the reduction of $c_1$ mod $2$ and whose first
Pontrjagin class is
$p_1(P)=(c_1^2-4c_2)$. Then $\gamma_{c_1,c_2,g}$
is a homogeneous polynomial on $H_*(S,{\Bbb Q})$ of weight
$2N=2(4c_2-c_1^2-3\chi({\cal O}_S))$,
where the elements of $H_i(S,{\Bbb Q})$ have weight
$4-i$. In case $p_g(S)>0$ it is known that $\gamma_{c_1,c_2,g}$
does not depend on the metric (as long as it is generic).
In [Ko] the invariant has been introduced for $4$-manifolds $M$ with
$b_+(M)=1$.
In [K-M] it has been shown that in case $b_+(M)=1$, $b_1(M)=0$
it depends only on the chamber of the
period point of the metric in the positive cone of $H^2(M,{\Bbb R})$.
The algebro-geometric analogues of the Donaldson polynomials are
defined as follows:
\begin{defn} \label{algana} (\cite{OG1}, \cite{OG2})
Assume that $M_H(c_1,c_2)$ is a fine moduli space, i.e.
$M_H(c_1,c_2)=M_H^s(c_1,c_2)$, and there is a universal sheaf ${\cal U}$ on
$S\times M_H(c_1,c_2)$. We define a linear map
$$\nu_{c_1,c_2,H}:H_i(S,{\Bbb Q})\to H^{4-i}(M_H(c_1,c_2),{\Bbb Q});
\ \ \nu_{c_1,c_2,H}(\alpha):=(c_2({\cal U})-{1\over 4}c_1^2({\cal U}))/\alpha,$$
where $/$ denotes the slant product.
We assume furthermore that $M_H(c_1,c_2)$ is of the expected dimension
$N:=4c_2-c_1^2-3\chi({\cal O}_S)$.
Given classes $\alpha_{s}\in H_{2j_s}(S,{\Bbb Q})$, for $s=1,\ldots,k$
with $2k-\sum_s j_s=N$,
we set
$$\Phi_{c_1,c_2,H}(\alpha_1,\ldots,\alpha_k):=\int\limits_{M_H(c_1,c_2)}
\nu_{c_1,c_2,H}(\alpha_1)\cdot \ldots \cdot\nu_{c_1,c_2,H}(\alpha_k).$$
As $c_1,c_2$ are fixed in our paper, we will write
$\nu_H=\nu_{c_1,c_2,H}$ and $\Phi_{H}=\Phi_{c_1,c_2,H}$.
Let ${pt}\in H_0(S,{\Bbb Z})$ be the class of a point in $S$. Knowing
$\Phi_{H}$ is equivalent to knowing the numbers
$$\Phi_{H,l,r}(\alpha)
:=\int\limits_{M_H(c_1,c_2)}\nu_H(\alpha)^l\cdot\nu_H({pt})^r.$$
for all $l,r$ with $l+2r=N$ and all $\alpha\in
H_2(S,{\Bbb Q})$.
\end{defn}
\begin{defn}
Following \cite{OG2}, we call $M_H(c_1,c_2)$
admissible if the following holds:
\begin{enumerate}
\item $H$ does not lie on a wall of type $(c_1,c_2)$;
\item $dim(M_H(c_1,c_2))=N$,
\item if $c_1$ is divisible by $2$ in $NS(S)$, then
$N> (4c_2-c_1^2)/2$;
\item $dim(M_H(c_1,k))+2(c_2-k)<N$ for all $k<c_2$.
\end{enumerate}
\end{defn}
For admissible $M_H(c_1,c_2)$ the results of
\cite{Mo} and \cite{Li} give
$$\Phi_{H}|_{H^2(S,{\Bbb Q})}=(-1)^{(c_1^2+\<c_1\cdot K_S\>)}
\gamma_{c_1,c_2,g(H)}|_{H^2(S,{\Bbb Q})},$$ where
$g(H)$ is the Fubini-Study metric associated to $H$.
Furthermore if $c_2>>0$, then
$\Phi_{H}=(-1)^{(c_1^2+\<c_1\cdot K_S \>)}\gamma_{c_1,c_2,g(H)}$.
We now want to determine how $\Phi_{H}$ changes, when $H$
passes through a wall.
We assume that
if $c_1$ is divisible by $2$ in $NS(S)$ then
$(4c_2-c_1^2)$ is not divisible by $8$. Then, by the criterion of
\cite{Ma2}, $M_H(c_1,c_2)$ is a fine moduli space, unless
$H$ lies on a wall.
Now we assume that we are in the situation of section 3, i.e.
$H_-$ and $H_+$ are ample divisors lying in neighbouring chambers separated by
$W$, and $H$ a polarization on the wall $W$ not lying on any other
wall and lying in the closure of both the chambers containing $H_-$ and $H_+$.
We assume furthermore that $W$ is a good wall.
For $b\in [0,1]$ we have $M_b(c_1,c_2)$ as in section 3.
\begin{defn} By remark \ref{fine} we see that, for $b$ not on a miniwall,
$M_b(c_1,c_2)=M^s_b(c_1,c_2)$ and there is a universal sheaf on
$M_b(c_1,c_2)$.
Assume that $b\in [0,1]$ does not lie on a miniwall. Then analoguosly to
the definition of
$\Phi_{H}$ and $\Phi_{H,l,r}$ in \ref{algana},
we may define $\Phi_{b}$ and $\Phi_{b,l,r}$ by always replacing
$M_H(c_1,c_2)$ by $M_b(c_1,c_2)$.
\end{defn}
We notice that $\Phi_{H_-}=\Phi_{0}$ and $\Phi_{H_+}=\Phi_{1}$
and it is obvious that $\Phi_{b}$ only depends on the
minichamber containing $b$.
We therefore have to determine the change of $\Phi_b$ when $b$
passes through a miniwall.
We will make the same assumptions as in section 4, i.e. let
$a$ be a miniwall and let $(\xi,n,m)\in A^+(a)$. Let $a_-<a_+$
lie in neighbouring minichambers separated by $a$.
To simplify the notation we will for the moment assume that
$A^+(a)=\{ (\xi,n,m)\}$ and that $H^2(S,{\Bbb Z})$ contains no $2$-torsion.
We also assume that either $p_g(S)=0$ or $K_S$ is trivial.
\begin{nota}\label{notado}
We use the notations and definitions of sections 4 and 5.
If the change is birational, i.e. we are not in case (1) of \ref{newcomp},
we shall write $\widetilde M$ instead of $\widetilde M_-$.
Let $d:=d_\xi=n+m$, $e_-=rk ({\cal A}_-)$, $e_+=rk ({\cal A}_+)$,
then $N=2d+e_-+e_+-1$ if $p_g(S)=0$ and $N=2d+e_-+e_+-2$
if $K_S$ is trivial.
We put $\nu_+:=\nu_{a_+}$, $\nu_-:=\nu_{a_-}$, $\Phi_+:=\Phi_{a_+}$,
$\Phi_-:=\Phi_{a_-}$, $\Phi_{+,l,r}:=\Phi_{a_+,l,r}$ and
$\Phi_{-,l,r}:=\Phi_{a_-,l,r}$.
Note that the condition $q(S)=0$ implies $Pic(S)\simeq NS(S)$.
For $\beta\in NS(S)$ we may therefore denote by ${\cal O}_S(\beta)$
the corresponding
line bundle. Let $q_1,q_2$ be the two projections of $T={\text{\rom{Hilb}}}^n(S)\times
{\text{\rom{Hilb}}}^m(S)$.
\end{nota}
\begin{rem}\begin{enumerate}
\item If the change is birational, then
by the projection formula
$\Phi_+,$ $\Phi_{+,l,r}$ (resp. $\Phi_-$, $\Phi_{-,l,r}$)
coincide with the numbers which are defined analogously by replacing
$M_{a}(c_1,c_2)$ by $\widetilde M$ and the universal sheaf by
${\cal V}_+$ (resp. ${\cal V}_-$).
\item Assume $p_g(S)=0$ and say ${\text{\rom{\bf E}}}_+=\emptyset$.
Let ${\cal E}$ be the universal
sheaf on ${\text{\rom{\bf E}}}_-$ from (\ref{globext}), then we can define
$\sigma_-:H_i(S,{\Bbb Q})\longrightarrow H^{4-i}({\text{\rom{\bf E}}}_-,{\Bbb Q})$ and $\delta_-$
and $\delta_{-,l,r}$
in the same way as $\nu_-$ and $\Phi_-$ and $\Phi_{-,l,r}$ by
replacing $M_-$ by ${\text{\rom{\bf E}}}_-$ and
the universal sheaf on $M_-$ by ${\cal E}$.
Then $\Phi_+-\Phi_-=-\delta_-$.
\end{enumerate}
\end{rem}
\begin{defn}\label{hilbdef}
Let $Z_n(S)\subset S\times {\text{\rom{Hilb}}}^n(S)$ be the universal subscheme.
In $S\times {\text{\rom{Hilb}}}^n(S)\times {\text{\rom{Hilb}}}^m(S)$,
we put ${\cal Z}_1:=({\hbox{\rom{id}}}_S\times q_1)^{-1}(Z_n(S))$,
${\cal Z}_2:=({\hbox{\rom{id}}}_S\times q_2)^{-1}(Z_m(S))$ and denote by
${\cal I}_{{\cal Z}_1}$, ${\cal I}_{{\cal Z}_2}$ the corresponding idealsheaves.
Let $F_1:={\cal O}_S((c_1+\xi)/2)$, $F_2:={\cal O}_S((c_1-\xi)/2)$.
By our assumptions
$T={\text{\rom{Hilb}}}^n(S)\times {\text{\rom{Hilb}}}^m(S)$ and ${\cal F}_1'={\cal I}_{{\cal Z}_1}(F_1)$,
$ {\cal F}_2'={\cal I}_{{\cal Z}_2}(F_2)$.
Let $h_n:{\text{\rom{Hilb}}}^n(S)\longrightarrow S^{(n)}$ be the Hilbert-Chow morphism
\cite{Fo}, where $S^{(n)}$ is the $n$-fold symmetric power of $S$
with the quotient map $\varphi_n:S^n\longrightarrow S^{(n)}$.
For $i=1,\ldots,n$ we denote by $p_i:S^n\to S$
the projection to the $i^{th}$ factor.
We denote
$\Delta_{i}:=\big\{(x,x_1,\ldots,x_n)\in S\times S^n\bigm| x=x_i\big\}$
and $Y_n:=({\hbox{\rom{id}}}_S\times \varphi_n)(\Delta_{1}).$
We have linear maps
\begin{eqnarray*}
\iota_n:H_i(S,{\Bbb Q})\longrightarrow H^{4-i}({\text{\rom{Hilb}}}^n(S),{\Bbb Q});&& \iota_{n}(\alpha)=
[Z_n(S)]/\alpha\ \ \text{ and }\\
\bar\iota_n:H_i(S,{\Bbb Q})\longrightarrow H^{4-i}S^{(n)},{\Bbb Q});&& \bar\iota_{n}(\alpha)=
[Y_n]/\alpha.\end{eqnarray*}
For $\alpha\in H^i(S,{\Bbb Q})$ put $\alpha_{n,m}:=
[{\cal Z}_1]/\alpha+[{\cal Z}_2]/\alpha=q_1^*(\iota_n(\alpha))+q_2^*(\iota_n(\alpha))
\in H^{4-i}(T,{\Bbb Q})$.
\end{defn}
The map $\iota_{n}$ is in fact easy to describe:
\begin{lem} \label{hilbkuenn}
\begin{enumerate}
\item $[Z_n(S)]=({\hbox{\rom{id}}}_S\times h_n)^*([Y_n])$.
\item $({\hbox{\rom{id}}}_S\times\varphi_n)^*([Y_n])=\sum_i[\Delta_i]$
\item For $\alpha\in H^i(S,{\Bbb Q})$ we have
$\iota_n(\alpha)=h_n^*(\bar\iota_n(\alpha))$
and $\varphi_n^*(\bar\iota_n(\alpha))=\sum_{i=1}^n p_i^*(\check
\alpha)$, where $\check
\alpha$ is the Poincar\'e dual of $\alpha$.
\end{enumerate} \end{lem}
\begin{pf}
(1).
Out of codimension $3$ on $S\times {\text{\rom{Hilb}}}^n(S)$ we have ${\cal O}_{Z_n(S)}=
({\hbox{\rom{id}}}_S\times h_n)^*({\cal O}_{Y_n})$. So we get
$[Z_n(S)]=({\hbox{\rom{id}}}_S\times h_n)^*([Y_n]).$
Out of codimension $3$ we also have $({\hbox{\rom{id}}}_S\times \varphi_n)^*({\cal O}_{Y_n})=
\bigoplus_i{\cal O}_{\Delta_{i}}.$
Therefore (2) follows in the same way as (1).
(3) follows immediately from (1) and (2).\end{pf}
\begin{rem}\label{restch} For the total Chern classes we have
$c(({\hbox{\rom{id}}}_S\times j)^*{\cal V}_-)=({\hbox{\rom{id}}}_S\times j)^*c({\cal V}_-)$ and
$c(({\hbox{\rom{id}}}_S\times j)^*{\cal V}_+)=({\hbox{\rom{id}}}_S\times j)^*c({\cal V}_+)$, where,
as above, $j:D\longrightarrow \widetilde M$ is the embedding of the exceptional divisor.
\end{rem}
\begin{pf}
We have to see that ${\hbox{\rom{Tor}}}_k({\cal V}_-,{\cal O}_{S\times D})=0$
for all $k>0$
(and similarly for ${\cal V}_+$). This follows however easily from the flatness
of ${\cal V}_-$ over $\widetilde M_-$.
\end{pf}
\begin{lem} \label{nu}\begin{enumerate}
\item Assume that we are in case (1) of \ref{newcomp}, i.e.
the change of moduli is birational.
Then, for $\alpha\in H_2(S,{\Bbb Q})$, we have
\begin{eqnarray*}\nu_+(\alpha)-\nu_-(\alpha)
&=&-{1\over 2}\<\xi,\alpha\>[D],\\
\nu_{+}(pt)-\nu_{-}({pt})&=&{1\over 4}j_*([\tau_-]-[\tau_+]).
\end{eqnarray*}
\item If ${\text{\rom{\bf E}}}_+=\emptyset$ then
\begin{eqnarray*} \sigma_-(\alpha)&=& {1\over 2}\<\xi,\alpha\>[\tau_-],\\
\sigma_{-}({pt})&=&-{1\over 4}[\tau_-]^2.
\end{eqnarray*}
\end{enumerate}\end{lem}
\begin{pf}By (\ref{dia1}) we have the sequence
$$0\longrightarrow {\cal V}_-\longrightarrow {\cal V}_+(D_S)\longrightarrow {\cal F}_{1D}(\lambda+\tau_-+\tau_+)\longrightarrow 0.$$
Using Riemann-Roch without denominators \cite{Jo} we get
\begin{eqnarray*} c_1({\cal F}_{1D}(\lambda+\tau_-+\tau_+))&=&[D_S]\\
c_2({\cal F}_{1D}(\lambda+\tau_-+\tau_+))&=&
-c_1({\cal F}_{1D}(\lambda)),
\end{eqnarray*}
and thus
\begin{eqnarray*}
c_1({\cal V}_+(D_S))&=&c_1({\cal V}_-)+[D_S],\\
c_2({\cal V}_+(D_S))&=&c_2(V_-)+[D_S]\cdot
c_1(V_-)-({\hbox{\rom{id}}}_S\times j)_*(c_1({\cal F}_{1D}(\lambda)))\\
4c_2({\cal V}_+)-c_1({\cal V}_+)^2&=&4c_2({\cal V}_+(D_S))-c_1({\cal V}_+(D_S))^2\\
&=&
4c_2({\cal V}_-)-c_1({\cal V}_-)^2+2[D_S]\cdot c_1({\cal V}_-)
-[D_S]^2-4({\hbox{\rom{id}}}_S\times j)_*(c_1({\cal F}_{1D}(\lambda))).
\end{eqnarray*}
Let $\alpha\in H_2(S,{\Bbb Q})$. As $[D_S]$ is the pull-back of $[D]$ from
$\widetilde M$, we have
$$([D_S]\cdot c_1({\cal V}_-))/\alpha=[D](c_1({\cal V}_-)/\alpha)=\<c_1,\alpha\>[D].$$
Furthermore
$({\hbox{\rom{id}}}_S\times j)_*c_1({\cal F}_{1D}(\lambda))/\alpha=
j_*(c_1({\cal F}_1(\lambda))/\alpha)$,
where the second slant product is taken on $S\times D$ and
$c_1({\cal F}_1)=\pi_D^*(c_1({\cal F}_1'))=p^*([F_1])$.
So we get $({\hbox{\rom{id}}}_S\times j)_\ast (c_1({\cal F}_1)/\alpha)=\<F_1\cdot \alpha\>[D]$.
As $\lambda$ is the pull-back of a divisor on $D$, we have
$({\hbox{\rom{id}}}_S\times j)_*c_1(\lambda)/\alpha= 0$ and
similarly $[D_S]^2/\alpha=0$.
So we get $\nu_+(\alpha)-\nu_-(\alpha)=-{1\over 2}\<\xi,\alpha\>[D]$
By $c_1({\cal F}_1)=p^*([F_1])$, $c_1({\cal F}_2)=p^*([F_2])$, we get
$c_1({\cal F}_1')/{pt}=c_1({\cal F}_2')/{pt}=0.$
Then the sequence
$$0\longrightarrow {\cal F}_{1D}(\lambda)\longrightarrow {\cal V}_-|_D\longrightarrow {\cal F}_{2D}(\tau_-+\lambda)\longrightarrow 0,$$
and remark \ref{restch} give
\begin{eqnarray*}
(c_1({\cal V}_-)\cdot [D_S])/{pt} &=&
({\hbox{\rom{id}}}_S\times j)_*(c_1({\cal V}_-|_D))/{pt}=j_*([\tau_-]+2[\lambda]),\\
(c_1({\cal F}_{1D}(\lambda)))/{pt} &=& j_*(c_1({\cal F}_1(\lambda))/{pt})=
j_*([\lambda]),\\{}
[D_S]^2/{pt} &=& [D]^2.
\end{eqnarray*}
So we get
\begin{eqnarray*}\nu_+({pt})-\nu_-({pt})&=&{1\over 4}([D]^2
+j_*([2\tau_-+4\lambda])-4j_*([\lambda])) \\
&=&
{1\over 4}j_*([\tau_-]-[\tau_+]).\end{eqnarray*}
(2) can be shown using essentially the same arguments.
\end{pf}
\begin{lem} \label{change1} Let $l+2r=N$.
\begin{enumerate}
\item If we are in case (1)
of \ref{newcomp}, then
\begin{eqnarray*}
&&\Phi_{+,l,r}(\alpha)-\Phi_{-,l,r}(\alpha)\\&&
\quad=\sum_{b=0}^{l}\sum_{c=0}^r (-1)^{r-c+1}2^{b+2c-N}{l\choose b}
{r\choose c}\<\xi,\alpha\>^{l-b}\int\limits_D
\left(\alpha_{n,m}^{b}{pt}_{n,m}^{c}\sum_{s+t=N-b-2c-1}
(-\tau_+)^s\tau_-^t\right)\end{eqnarray*}
\item ${\text{\rom{\bf E}}}_+=\emptyset$, then
\begin{eqnarray*}
&&\Phi_{+,l,r}(\alpha)-\Phi_{-,l,r}(\alpha)\\&&
\quad=\sum_{b=0}^{l}\sum_{c=0}^r (-1)^{r-c+1}2^{b+2c-N}{l\choose b}
{r\choose c}\<\xi,\alpha\>^{l-b}\int\limits_{{\text{\rom{\bf E}}}_-}
\left(\alpha_{n,m}^{b}{pt}_{n,m}^{c}\tau_-^{N-b-2c}\right)\end{eqnarray*}
\end{enumerate}\end{lem}
\begin{pf} (1) By remark \ref{restch} we get for $\alpha\in H_{i}(S,{\Bbb Q})$
that
$[D]\cdot \nu_+(\alpha)= j_*((4c_2({\cal V}_+|_D)-c_1({\cal V}_+|_D)^2)/4\alpha$
(and similar for $\nu_-$).
By the sequences
\begin{eqnarray*}&&0\longrightarrow {\cal F}_{2D}(-\tau_++\lambda)\longrightarrow {\cal V}_+|_{D_S}
\longrightarrow {\cal F}_{1D}(\lambda)\longrightarrow 0\\
&&0\longrightarrow {\cal F}_{1D}(\lambda)\longrightarrow {\cal V}_-|_{D_S}\longrightarrow {\cal F}_{2D}(\tau_-+\lambda)\longrightarrow
0\end{eqnarray*}
we get
\begin{eqnarray*}&&4c_2({\cal V}_+|_D)-c_1({\cal V}_+|_D)^2
=4(c_2({\cal F}_{1D})+c_2({\cal F}_{2D}))-(c_1({\cal F}_{2D})-c_1({\cal F}_{1D})-[\tau_+])^2\\
&&4c_2({\cal V}_-|D)-c_1({\cal V}_-|D)^2
=4(c_2({\cal F}_{1D})+c_2({\cal F}_{2D}))-(c_1({\cal F}_{1D})-c_1({\cal F}_{2D})-[\tau_-])^2
\end{eqnarray*}
By the above we have
$c_1({\cal F}_{1D})=p^*([F_1])$, $c_1({\cal F}_{2D})=p^*([F_2])$,
$c_2({\cal F}_{1D})=({\hbox{\rom{id}}}_S\times \pi_D)^*(c_2({\cal I}_{{\cal Z}_1}))=({\hbox{\rom{id}}}_S\times
\pi_D)^*([{\cal Z}_1])$ and
$c_2({\cal F}_{2D})=({\hbox{\rom{id}}}_S\times \pi_D)^*([{\cal Z}_2])$, where, as above, $\pi_D:D\longrightarrow
T$ is
the projection.
So we have
\begin{eqnarray*}4c_2({\cal V}_+|_D)-c_1({\cal V}_+|_D)^2
&=&4({\hbox{\rom{id}}}_S\times \pi_D)^*([{\cal Z}_1]+[{\cal Z}_2])-(p^*(\xi)+[\tau_+])^2,\\
4c_2({\cal V}_-|_D)-c_1({\cal V}_-|_D)^2
&=&4({\hbox{\rom{id}}}_S\times \pi_D)^*([{\cal Z}_1]+[{\cal Z}_2])-(p^*(\xi)-[\tau_-])^2,
\end{eqnarray*}
and thus for $\alpha\in H_2(S,{\Bbb Q})$:
\begin{eqnarray*}
j^*(\nu_+(\alpha))&=&\alpha_{n,m}+{1\over 2}\<\xi,\alpha\>[-\tau_+]\\
j^*(\nu_-(\alpha))&=&\alpha_{n,m}+{1\over 2}\<\xi,\alpha\>[\tau_-]\\
j^*(\nu_+({pt}))&=&{pt}_{n,m}-{1\over 4}[\tau_+]^2\\
j^*(\nu_-({pt}))&=&{pt}_{n,m}-{1\over 4}[\tau_-]^2
\end{eqnarray*}
We write
\begin{eqnarray*}
&&\Phi_{+,l,r}(\alpha)-\Phi_{-,l,r}(\alpha)=\\
&&\qquad=\int\limits_{\widetilde M}
\left(\nu_+(\alpha)^{l}(\nu_+({pt})^r-\nu_-({pt})^r)+
\nu_+({pt})^r(\nu_+(\alpha)^{l}-\nu_-(\alpha)^{l})\right)\\
&&\qquad=\int\limits_D\left({1\over 4}([-\tau_+]+[\tau_-])
j^*\left(\sum_{s+t=r-1}\nu_+({pt})^s\nu_-({pt})^t
\nu_+(\alpha)^l\right)
\right.\\
&&\qquad\qquad-\left.
{1\over 2}\<\xi,\alpha\>\
j^*\left(\sum_{s+t=l-1}\nu_+(\alpha)^s \nu_-(\alpha)^t
\nu_-({pt})^r\right)
\right).\end{eqnarray*}
Now the claim follows after a straightforward computation.
(2) follows easily from lemma \ref{nu}(2).\end{pf}
\begin{prop}\label{donmin}
\begin{enumerate}
\item If $S$ is a $K3$ surface and $N>0$, then
$\Phi_+=\Phi_-$.
\item If $p_g(S)=0$, then for $\alpha\in H_2(S,{\Bbb Q})$ and $l,r$
with $l+2r=N$ we have
\begin{eqnarray*}
&&\Phi_{+,l,r}(\alpha)-\Phi_{-,l,r}(\alpha)\\
&&\quad=\sum_{b=0}^{l}\sum_{c=0}^r (-1)^{r-c+e_-}2^{b+2c-N}{l\choose b}
{r\choose c}\<\xi,\alpha\>^{l-b}\int\limits_{T}
\left(\alpha_{n,m}^{b} {pt}_{n,m}^{c}
s_{2d-b-2c}({\cal A}_+'\oplus{{\cal A}_-'}^{\vee})\right)
\end{eqnarray*}
\end{enumerate}
\end{prop}
\begin{pf}
(1) It easy to show using Riemann-Roch, that the condition
$N>0$ implies $e_->1$ and $e_+>1$. Therefore, as
$\alpha_{n,m}$ and ${pt}_{n,m}$ are pull-backs from $T$,
it is enough to show that for $k\le e_-+e_+-2$
we have
$$(\pi_D)_*\left(\sum_{s+t=k}(-\tau_+)^s \tau_-^t\right)=0.$$
Now $D$ is the projectivisation ${\Bbb P}(Q)$ where
$Q={\cal A}_-/\tau_-$ over ${\Bbb P}_{-}={\Bbb P}({\cal A}_-')$. Therefore
\begin{eqnarray*}
(\pi_D)_*\left(\sum_{s+t=k}(-\tau_+)^s\tau_-^t\right)&=&
(\pi_{-})_*\left(\sum_{s+t=k}s_{s-e_++2}(Q)\tau_-^t\right)\\
&=&(\pi_{-})_*(s_{k-e_++2}({\cal A}_-))\\
&=&(\pi_{-})_*\pi_-^*(s_{k-e_++2}({\cal A}_-'))=0.\end{eqnarray*}
Here $\pi_{-}:{\Bbb P}_-\longrightarrow T$ is the projection.
(2) We just note that
$\pi_{+})_*((-\tau_+)^k)=s_{k-e_++1}({\cal A}_+')$ and
$(\pi_{-})_*(\tau_-^k)=(-1)^{e_-+1}s_{k-e_-+1}({{\cal A}_-'}^{\vee})$.
Then the result follows immediately from the definitions and
lemma \ref{change1}.
\end{pf}
For the rest of the chapter we assume that $p_g(S)=q(S)=0$.
On the other hand we allow $NS(S)=H^2(S,{\Bbb Z})$ to contain torsion.
\begin{defn}\label{defchange}
Let $\xi\in H^2(S,{\Bbb Z})$ be a class defining a good wall of type
$(c_1,c_2)$.
Let
$d_\xi:=(4c_1-c_1^2+\xi^2)/4$, $e_{\xi}:=-\<\xi\cdot(\xi-K_S)\>/2+d_\xi+1$
and
$$T_\xi:={\text{\rom{Hilb}}}^{d_\xi}(S\sqcup S)=
\coprod_{n+m=d_\xi}{\text{\rom{Hilb}}}^n(S)\times {\text{\rom{Hilb}}}^m(S).$$
Let $q_\xi:S\times T_\xi\longrightarrow T_\xi$ be the projection.
Let $V_\xi$ be the sheaf $p^*({\cal O}_S(-\xi)\oplus{\cal O}_S(-\xi+K_S))$
on $S\times T_\xi$.
Let ${\cal Z}^\xi_{1}$ (resp.${\cal Z}^\xi_{2}$) be the subscheme of
$S\times T_\xi$ which restricted to each component
$S\times{\text{\rom{Hilb}}}^n(S)\times {\text{\rom{Hilb}}}^m(S)$ is the subscheme ${\cal Z}_1$
(resp. ${\cal Z}_2$)
from \ref{hilbdef}. Let ${\cal I}_{{\cal Z}^\xi_{1}}$, ${\cal I}_{{\cal Z}^\xi_{2}}$
be the corresponding ideal sheaves. For $\alpha\in H_i(S,{\Bbb Q})$ let
$\widetilde\alpha\in H^{4-i}(T_\xi,{\Bbb Q})$ be the class whose
restriction to each component
${\text{\rom{Hilb}}}^n(S)\times {\text{\rom{Hilb}}}^m(S)$ of $T_\xi$ is $\alpha_{n,m}$.
Then for all $l,r$ with $l+2r=N$ we define a map
$H_2(S,{\Bbb Q})\longrightarrow {\Bbb Q}$ by
\begin{eqnarray*}
\delta_{\xi,l,r}(\alpha)&:=&
\sum_{b=0}^{l}\sum_{c=0}^r (-1)^{r-c+e_\xi}2^{b+2c-N}{l\choose b}
{r\choose c}\<\xi,\alpha\>^{l-b}\\
&&\int\limits_{T_\xi}
\left(\widetilde\alpha^{b} \widetilde{pt}^{c}s_{2d_\xi-2c-b}
({\hbox{\rom{Ext}}}^1_{q}({\cal I}_{{\cal Z}^\xi_{1}},
{\cal I}_{{\cal Z}^\xi_{2}}\otimes V_\xi)\right)\end{eqnarray*}
\end{defn}
\begin{thm}\label{donch1}
Let $S$ be a surface with $p_g(S)=q(S)=0$.
Let $c_1\in H^2(S,{\Bbb Z})$ and $c_2\in {\Bbb Z}$. Assume that, if
$c_1$ is divisible by $2$ in $H^2(S,{\Bbb Z})$ then
$(4c_2-c_1^2)$ is not divisible by $8$.
Let $W$ be a good wall of type $(c_1,c_2)$ and let $H_-$ and $H_+$ be
ample divisors on $S$
lying in neighbouring chambers separated by
$W$. Let $n_2$ be the number of $2$-torsion points in $H^2(S,{\Bbb Z})$.
Then for all $l,r$ with $l+2r=N=(4c_2-c_1^2)-3$ we have
$$\Phi_{H_+,l,r}-\Phi_{H_-,l,r}=n_2\sum_{\xi\in A^+(W)}
\delta_{\xi,l,r}.$$
Here, as above,
$$A^+(W)=\big\{ \xi\in H^2(S,{\Bbb Z})\bigm | Z\hbox{ defines the wall W and }
\<\xi\cdot H_+\> >0 \}.$$
Therefore we get for a class $\alpha\in H_2(S,{\Bbb Q})$
\begin{eqnarray*}
(\gamma_{c_1,c_2,g(H_+)}- \gamma_{c_1,c_2,g(H_-)})
(\underbrace{{pt},\ldots,{pt}}_r,
\underbrace{\alpha,\ldots,\alpha}_l)
=(-1)^{(c_1^2+\<c_1\cdot K_S \>)}
n_2\sum_{\xi\in A^+(W)}\delta_{\xi,l,r}(\alpha).
\end{eqnarray*}
\end{thm}
\begin{pf}
If $H^2(S,{\Bbb Z})$ contains no $2$-torsion,
and $a_-<a_+$ are in neighbouring minichambers separated by a miniwall
$a$ with $A^+(a)=\{(\xi,n,m)\}$, then proposition \ref{donmin}
computes $\Phi_{a_+,l,r}-\Phi_{a_-,l,r}$.
By Serre duality and the definitions we see that
in the notations of proposition \ref{donmin}
${\cal A}_+'\oplus{{\cal A}_-'}^\vee={\hbox{\rom{Ext}}}^1_{q}({\cal I}_{{\cal Z}_1},{\cal I}_{{\cal Z}_2}\otimes V_\xi)$.
Thus, if for all miniwalls $a$ the set $A^+(a)$ consists of
only one element, the theorem follows.
If $N_2\subset H^2(S,{\Bbb Z})$ is the subgroup of $2$-torsion,
then $T_\xi^{n,m}\simeq N_2\times{\text{\rom{Hilb}}}^n(S)\times {\text{\rom{Hilb}}}^m(S)$. So
the exceptional divisor in $\widetilde M$ has $n_2$
isomorphic components
(or we add $n_2$ isomorphic connected components to $\widetilde M$
or subtract them), and each component gives the same contribution
to $\Phi_{a_+,l,r}-\Phi_{a_-,l,r}$.
Assume that
$A^+(a)=\{(\xi_1,n_1,m_1),\ldots,(\xi_2,n_2,m_2)\}.$
Then, as we have seen above, the ${\text{\rom{\bf E}}}_{\xi_i}^{n_i,m_i}$
are disjoint, and, as the change
$\Phi_{a_+,l,r}-\Phi_{a_-,l,r}$ can be computed on
the exceptional divisor (or the added components),
it is just the sum of the contributions for all $(\xi_i,n_i,m_i)$.
The result now follows by adding up the contributions of
all the miniwalls.
\end{pf}
By the results we have obtained so far, in order
to compute explicitly
the change of the Donaldson invariants, when the polarisation
passes through a good wall $W=W^\xi$, we have
first to determine the Chern classes of the
bundles
${\hbox{\rom{Ext}}}^1_{q_\xi}({\cal I}_{{\cal Z}^\xi_{1}},{\cal I}_{{\cal Z}^\xi_{2}}\otimes V_\xi)$
on $T_\xi$, and then make explicit computations in the
cohomology ring of ${\text{\rom{Hilb}}}^d(S\sqcup S)$.
In the rest of this section we will again use the
assumptions and notations from \ref{notado}, and will
adress the first question, i.e. we express the
Chern classes of the vector bundles
${\hbox{\rom{Ext}}}^1_{q}({\cal I}_{{\cal Z}_{1}},{\cal I}_{{\cal Z}_{2}}\otimes V)$ on
$T={\text{\rom{Hilb}}}^n(S)\times {\text{\rom{Hilb}}}^m(S)$, (where we have written $V:=V_\xi$)
in terms of those of ``standard bundles''.
\begin{defn} Using the projections $p:S\times T \longrightarrow S$ and
$q:S\times T\longrightarrow T$ we associate to
a vector bundle $U$ of rank $r$ on $S$ the vector bundles
$[U]_1:=q_*({\cal O}_{{\cal Z}_2}\otimes p^*(U))$ and
$[U]_2:=
q_*({\cal O}_{{\cal Z}_1}\otimes p^*(U))$ of ranks $rn$
(resp. $rm$) on $T$.
\end{defn}
For a Cohen-Macaulay scheme $Z$, we denote by $\omega_Z$
its dualizing sheaf.
\begin{lem}\label{extlem}
$${\hbox{\rom{Ext}}}^2_q({\cal O}_{{\cal Z}_1},{\cal O}_{{\cal Z}_2}\otimes p^*V)=
q_*(\omega_{{\cal Z}_1}\otimes
\omega_T^{-1}\otimes{\cal O}_{{\cal Z}_2}\otimes p^*V)$$
and ${\hbox{\rom{Ext}}}^i_q({\cal O}_{{\cal Z}_1},{\cal O}_{{\cal Z}_2}\otimes p^*V)=0$ for $i\ne 2$.
\end{lem}
\begin{pf}
Let
\begin{eqnarray}\label{resol}&&0\longrightarrow B_2\longrightarrow B_1
\longrightarrow {\cal O}_{S\times T}\longrightarrow {\cal O}_{{\cal Z}_1}\longrightarrow 0\end{eqnarray}
be a locally free resolution on $S\times T$. We apply
${\cal Hom}(\cdot,{\cal O}_{{\cal Z}_2}\otimes p^*V)$ to
obtain the complex
$$0\longrightarrow {\cal O}_{{\cal Z}_2}\otimes p^*V\longrightarrow B_1^*\otimes
{\cal O}_{{\cal Z}_2}\otimes p^*V\longrightarrow
B_2^*\otimes {\cal O}_{{\cal Z}_2}\otimes p^*V\longrightarrow
0,$$
whose cohomologies are the ${\cal Ext}^i({\cal O}_{{\cal Z}_1},{\cal O}_{{\cal Z}_2}\otimes p^*V)$.
We can arrive at this complex differently,
namely by first dualizing and then tensorizing by ${\cal O}_{{\cal Z}_2}\otimes p^*V$.
By dualizing and using that ${\cal Z}_1$ is Cohen-Macauley we obtain
$$0\longrightarrow {\cal O}_{S\times T}\longrightarrow B_1^*\longrightarrow B_2^*\longrightarrow
\omega_{{\cal Z}_1}\otimes \omega_T^{-1}\longrightarrow 0.$$
Tensorizing by ${\cal O}_{{\cal Z}_2}\otimes p^*V$ gives the sequence
$$0\longrightarrow {\cal O}_{{\cal Z}_2}\otimes p^*V\longrightarrow B_1^*\otimes {\cal O}_{{\cal Z}_2}\otimes p^*V\longrightarrow
B_2^*\otimes {\cal O}_{{\cal Z}_2}\otimes p^*V\longrightarrow
\omega_{{\cal Z}_1}\otimes
\omega_T^{-1}\otimes{\cal O}_{{\cal Z}_2}\otimes p^*V
\longrightarrow 0,$$
which is exact by
the corollaire on p. V.20 in \cite{Se} because
${\cal Z}_1$ and ${\cal Z}_2$ are Cohen-Macaulay and intersect properly.
Hence ${\cal Ext}^2({\cal O}_{{\cal Z}_1},{\cal O}_{{\cal Z}_2}\otimes p^*V)=
\omega_{{\cal Z}_1}\otimes
\omega_T^{-1}\otimes{\cal O}_{{\cal Z}_2}\otimes p^*V$ and
${\cal Ext}^i({\cal O}_{{\cal Z}_1},{\cal O}_{{\cal Z}_2}\otimes p^*V)=0$ for $i<2$.
As ${\cal Z}_2$ and ${\cal Z}_1$ are flat of dimension $0$ over $T$,
the result follows
by applying $q_*$.
\end{pf}
\begin{prop}\label{grot}
In the Grothendieck ring of sheaves on $T$ we have the equality
\begin{eqnarray*}{\hbox{\rom{Ext}}}^1_q({\cal I}_{{\cal Z}_1},{\cal I}_{{\cal Z}_2}\otimes p^*V)&=&
[V]_2+([V^{\vee}(K_S)]_1)^{\vee}
+(H^1(S,{\cal O}_S(-\xi))\oplus H^1(S,{\cal O}_S(-\xi+K_S)))\otimes{\cal O}_T\\
&&\qquad -
q_*(\omega_{{\cal Z}_1}\otimes\omega_T^{-1}\otimes{\cal O}_{{\cal Z}_2}\otimes p^*V).
\end{eqnarray*}
\end{prop}
\begin{pf}{\it Case n=0:}
We will use repeatedly that $\xi$ defines a good wall, so in particular
$q_*(p^*V)=R^2q_*(p^*V)=0$.
We apply
${\hbox{\rom{Hom}}}_q({\cal O}_{S\times T},\cdot)$ to the sequence
$$0\longrightarrow {\cal I}_{{\cal Z}_2}\otimes p^*V\longrightarrow p^*V\longrightarrow {\cal O}_{{\cal Z}_2}\otimes p^*V\longrightarrow 0$$
to obtain
\begin{eqnarray}\label{ex1}
0\longrightarrow [V]_2\longrightarrow {\hbox{\rom{Ext}}}^1_q({\cal O}_{S\times T},{\cal I}_{{\cal Z}_2}\otimes p^*V) \longrightarrow
R^1q_*(p^*V)\longrightarrow 0.
\end{eqnarray}
The surjectivity follows as ${\cal Z}_2$ is flat of dimension $0$ over $T$
and the injectivity by $q_*p^*V=0$.
\noindent {\it General case:}
We apply ${\hbox{\rom{Hom}}}_q(\cdot,{\cal I}_{{\cal Z}_2}\otimes p^*V)$ to the sequence
$0\longrightarrow {\cal I}_{{\cal Z}_1}\longrightarrow{\cal O}_{S\times T}\longrightarrow {\cal O}_{{\cal Z}_1}\longrightarrow 0$
to get
\begin{eqnarray}\label{ex2}
\qquad\qquad 0\to {\hbox{\rom{Ext}}}^1_q({\cal O}_{S\times T},{\cal I}_{{\cal Z}_2}\otimes p^*V)\to
{\hbox{\rom{Ext}}}^1_q({\cal I}_{{\cal Z}_1},{\cal I}_{{\cal Z}_2}\otimes p^*V)\to
{\hbox{\rom{Ext}}}^2_q({\cal O}_{{\cal Z}_1},{\cal I}_{{\cal Z}_2}\otimes p^*V)\to 0.
\end{eqnarray}
The exactness on the left follows from the fact that
$q_*({\cal I}_{{\cal Z}_2}\otimes p^*V)=0$ and so
${\hbox{\rom{Ext}}}^1_q({\cal O}_{{\cal Z}_1},{\cal I}_{{\cal Z}_2}\otimes p^*V)$ is
torsion-free being a subsheaf of the locally free sheaf
$R^1q_*({\cal I}_{{\cal Z}_2}\otimes p^*V)$. Its
support is contained in $q({\cal Z}_1\cap{\cal Z}_2)$ and thus it is the zero sheaf.
We apply ${\hbox{\rom{Hom}}}_q({\cal O}_{{\cal Z}_1},\cdot)$ to
$0\longrightarrow {\cal I}_{{\cal Z}_2}\otimes p^*V\longrightarrow p^*V\longrightarrow {\cal O}_{{\cal Z}_2}\otimes p^*V\longrightarrow 0$
and use
lemma \ref{extlem} to obtain
\begin{eqnarray}\label{ex3}
&&\qquad 0\longrightarrow {\hbox{\rom{Ext}}}^2_q({\cal O}_{{\cal Z}_1},{\cal I}_{{\cal Z}_2}\otimes p^*V)\longrightarrow
{\hbox{\rom{Ext}}}^2_q({\cal O}_{{\cal Z}_1},p^*V)
\longrightarrow
q_*(\omega_{{\cal Z}_1}\otimes\omega_T^{-1}\otimes{\cal O}_{{\cal Z}_2}\otimes p^*V)
\longrightarrow 0.
\end{eqnarray}
By duality
${\hbox{\rom{Ext}}}^2_q({\cal O}_{{\cal Z}_1},p^*V)=q_*({\cal O}_{{\cal Z}_1}\otimes
p^*(V^\vee(K_S))^\vee=[V^\vee(K_S)]_1^\vee$.
Thus the result follows by putting \ref{ex1} to \ref{ex3} together.
\end{pf}
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\section{Explicit computations on Hilbert schemes of points}
The aim of this section is to make theorem \ref{donch1} more explicit.
We want to compute the contributions $\delta_{\xi}$ to the change
of the Donaldson invariants for
a class $\xi$ defining a good wall, in terms of cohomology classes
and intersection
numbers on $S$.
We do not succeed in determining $\delta_{\xi}$ completely. It turns
however out that $\delta_{\xi}$ can be developed in terms of powers of $\xi$
and we will compute the six lowest order terms (as predicted by the
conjecture of Kotschick and Morgan half of them are zero).
\begin{nota} \label{xino}
In this section we fix a class $\xi\in H^2(S,{\Bbb Z})$ which defines a good wall
of type $(c_1,c_2)$ and will therefore drop $\xi$ in our notation.
In particular we write
$d:=d_\xi$, $e:=e_\xi$ and $T:={\text{\rom{Hilb}}}^{d}(S\sqcup S)$. As usual
let $p$ and $q$
be the projections of $S\times T$ to $S$ and $T$ respectively. We write
$V:={\cal O}_S(-\xi)\oplus{\cal O}_S(-\xi+K_S)$, ${\cal Z}_1:={\cal Z}_1^\xi$,
${\cal Z}_2:={\cal Z}_2^\xi$ and $\delta_{l,r}:=\delta_{\xi,l,r}$.
We put $\Gamma:=q_*(\omega_{{\cal Z}_1}\otimes\omega_{S\times T}^{-1}\otimes
{\cal O}_{{\cal Z}_2}\otimes p^*V)$.
\end{nota}
We see by theorem \ref{donch1} that, in order to compute the change
$\delta_{l,r}$, it is enough to compute $\int_T
s({\hbox{\rom{Ext}}}^1_q({\cal I}_{{\cal Z}_1},{\cal I}_{{\cal Z}_2}\otimes p^*V))\cdot \gamma$
for all classes $\gamma\in H^*(T,{\Bbb Q})$
which are pull-backs from $S^{(d)}$ via the natural map
${\text{\rom{Hilb}}}^d(S\sqcup S)\longrightarrow (S\sqcup S)^{(d)}\longrightarrow S^{(d)}$.
By proposition \ref{grot}
we have
\begin{eqnarray*}
\int_T s({\hbox{\rom{Ext}}}^1_q({\cal I}_{{\cal Z}_1},{\cal I}_{{\cal Z}_2}\otimes p^*V))\cdot\gamma&=&
\int_T s([V^\vee(K_S)]_1^\vee \oplus [V]_2)\cdot
\gamma\\
&&\quad +\int_T (c(\Gamma)-1)s([V^\vee(K_S)]_1^\vee
\oplus [V]_2)\cdot\gamma.\\
\end{eqnarray*}
In the first part of this section we compute the first of these two
integrals.
As said in the beginning of this section, we only want to compute
the terms of lowest order of the change of the Donaldson invariants.
This corresponds to restricting our attention to a big open
subset of the Hilbert scheme of points.
\begin{nota} A point $\sigma\in {S^{(n)}}$ is a formal linear
combination $\sum_i m_i x_i$
of points on $S$ with positive integer coefficients and $\sum_i m_i=n$.
The support $supp(\sigma)$ is the set of points $x_i$. For all $i\le n$
let
$$S^{(n)}_i:=\big\{\sigma\in S^{(n)}\bigm|
\# supp(\sigma)\ge n-i+1\big\}.$$
Furthermore, for any variety $X$ with a canonical morphism
$f:X\longrightarrow S^{(n)}$, we denote $f^{-1}S^{(n)}_i$ by $X_i$.
For the universal family $Z_n(S)\subset S\times {\text{\rom{Hilb}}}^n(S)$ we denote by
$Z_n(S)_i$ the preimage of ${\text{\rom{Hilb}}}^n(S)_i$.
\end{nota}
In order to compute the first integral we will use an inductive approach,
which is based on results of \cite{E1},\cite{F-G} and which is
similar to computations in
\cite{Go2} on the Hilbert scheme of $3$ points.
\begin{defn}(\cite{E1},\cite{F-G})
Let $S^{[n-1,n]}\longrightarrow S\times {\text{\rom{Hilb}}}^{n-1}(S)$ be the blow-up
along the universal
family $Z_{n-1}(S)$, and let $F_n$ the exceptional divisor.
Contrary to our conventions in the previous section for any
vector bundle $E$ on $S$ we will denote
by $E[n]$ the vector bundle $q_*({\cal O}_{Z_n(S)}\otimes p^* E)$ on
${\text{\rom{Hilb}}}^n(S)$.
\end{defn}
\begin{thm} \label{snn}(\cite{E1})
$S^{[n-1,n]}$ is smooth. There is a natural morphism
$S^{[n-1,n]}\longrightarrow {\text{\rom{Hilb}}}^n(S)$,
and on $S^{[n-1,n]}$ we have an exact sequence
\begin{eqnarray}\label{hilbseq}
0\longrightarrow V(-F_n)\longrightarrow V[n]\longrightarrow V[{n-1}]\longrightarrow 0,
\end{eqnarray}
where we have used convention \ref{convent}.
\end{thm}
It is easy to see that the induced map $S^{[n-1,n]}\longrightarrow S\times {\text{\rom{Hilb}}}^n(S)$
factors through $Z_n(S)\subset S\times {\text{\rom{Hilb}}}^n(S)$, and that
the map $S^{[n-1,n]}\longrightarrow Z_n(S)$ is an isomorphism over
the open set
$Z_n(S)_1$. We denote by $S^{[n-1,n]}_i$ the preimage of $Z_n(S)_i$.
\begin{lem}\label{hilbnorm}
Let $N^\vee_n$ be the conormal sheaf of $Z_n(S)$ in $S\times {\text{\rom{Hilb}}}^n(S)$.
Then we have an exact sequence on $S^{[n-1,n]}_2$
\begin{eqnarray*}\label{hilbnseq}
0 \longrightarrow N^\vee_n\longrightarrow T_S^\vee\longrightarrow {\cal O}_{F_n}(-F_n)\longrightarrow 0.
\end{eqnarray*}
Here we have used the convention \ref{convent}.
In particular on $S^{[n-1,n]}_2$ we get
$$s(N^\vee_n)=s(T^\vee_S){1-F_n\over 1-2F_n}.$$
\end{lem}
\begin{pf}
It is easy to see that
$S^{[n-1,n]}_2\longrightarrow {\text{\rom{Hilb}}}^n(S)_2$ is a branched $n$-fold cover,
\'etale out of $F_n$ and with ramification of order $1$ along
$F_n$.
So the result follows in the same way as in the proof of (\cite{F-G},
lemma 2.10).
\end{pf}
\begin{lem} \label{discard} Let $i$ be a positive integer
and assume that
$\alpha_1,\alpha_2\in A^{*}({\text{\rom{Hilb}}}^n(S))$
have the same pull-back to $ {\text{\rom{Hilb}}}^n(S)_i$. Then
$$\int\limits_{{\text{\rom{Hilb}}}^n(S)}\alpha_1\cdot \beta=
\int\limits_{{\text{\rom{Hilb}}}^n(S)}\alpha_2\cdot \beta$$
for all $\beta\in H^{4n-4i-2}(S^{(n)},{\Bbb Q})$.
The same result holds if we replace
${\text{\rom{Hilb}}}^n(S)_i$ by $S^{[n-1,n]}_i$.
\end{lem}
\begin{pf}
Let $j:{\text{\rom{Hilb}}}^n(S)\setminus {\text{\rom{Hilb}}}^n(S)_i\longrightarrow Hilb^n(S)$ be the inclusion.
We get $\alpha_1=\alpha_2+j_{*}(\mu)$ for a class
$\mu\in A^*({\text{\rom{Hilb}}}^n(S)\setminus {\text{\rom{Hilb}}}^n(S)_i).$ As the codimension of the
complement of $S^{(n)}_i$ in
$S^{(n)}$ is $2i$, the result follows by the projection formula.
\end{pf}
\begin{nota} \label{tplusminus}
For all $l\ge 1$ we denote by $\Delta_l$ the "small" diagonal
$\{(x,\ldots,x)\ | \ x\in S\}$
and by $[\Delta_l]$ its cohomology class.
We define classes $t_{1-},t_{2-},t_{3-}\in H^*(S,{\Bbb Q})$ by
\begin{eqnarray*}
t_{1-}&:=&1+(2\xi-K_S)+(3\xi^2-3\xi K_S+K_S^2),\\
t_{2-}&:=&3+(18\xi-13K_S)+(63\xi^2-91\xi K_S+33K_S^2+5s_2(S)),\\
t_{3-}&:=&27+(270\xi-237K_S).
\end{eqnarray*}
Here $s_i(S):=s_i(T_S)$ is the $i^{th}$ Segre class of $S$.
We define $t_{1+}$, $t_{2+}$, $t_{3+}$ by replacing $K_S$ by $(-K_S)$
in the definition
of $t_{1-}$, $t_{2-}$, $t_{3-}$ respectively
and put $t_i:=t_{i-}+t_{i+},$ i.e.
\begin{eqnarray*}
t_1&=&2+4\xi+6\xi^2+2K_S^2,\\
t_2&=&6+36\xi+126\xi^2+66K_S^2+10s_2(S),\\
t_3&=&54+540\xi.
\end{eqnarray*}
\end{nota}
\begin{lem}\label{hilbind}
Let $\gamma\in H^{4n-2k}(S^{(n)},{\Bbb Q})$ with $k\le 5$.
Then
\begin{eqnarray*}
n\int\limits_{{\text{\rom{Hilb}}}^n(S)}s(V[n])\cdot\gamma
=\sum_{l=1}^3\int\limits_{S^l\times
{\text{\rom{Hilb}}}^{n-l}(S)} (-1)^{l-1} [\Delta_l]p_1^*t_{l-}\cdot s(V[n-l])\cdot
\gamma,
\end{eqnarray*}
where $p_1:S^l\longrightarrow S$ is the projection to the first factor.
\end{lem}
\begin{pf}
By theorem \ref{snn} we have the identity
$s(V[n])=s(V(-F_n))s(V[n-1])$ on $S^{[n-1,n]}$ and furthermore
$$s(V(-F_n))=\sum_{i,j\ge 0}{i+j+1\choose i+1} s_i(V) F_n^j.$$
So we get
\begin{eqnarray}
n\int_{{\text{\rom{Hilb}}}^n(S)}s(V[n])\cdot\gamma
&=&\int\limits_{S\times{\text{\rom{Hilb}}}^{n-1}(S)}s(V)s(V[{n-1}])\cdot\gamma\label{ha}\\
\label{secint} +\sum_{i,j\ge 0}&&\int\limits_{{S^{[n-1,n]}}}F_n {i+j+2\choose i+1}
s_i(V)F_n^j s(V[{n-1}])\cdot\gamma.\label{hb}
\end{eqnarray}
By using $V={\cal O}_S(-\xi)\oplus{\cal O}_S(-\xi+K_S)$, we
see immediately that $s(V)=t_{1-}$.
We denote for all $i$ by $f_i$ the composition
$$S^{[n-i,n-i+1]}\to S\times {\text{\rom{Hilb}}}^{n-i}(S)\to S\times S^{(n-i)}
\to S^i\times S^{(n-i)} \to S^{(n)},$$ where the second map is induced
by the diagonal map $S\longrightarrow S^i$ and put we $\gamma_i:=f_i^*(\gamma)$.
The integral (\ref{secint})
can be expressed as an integral over $F_n$.
We push it forward to $Z_{n-1}(S)\subset S\times {\text{\rom{Hilb}}}^{n-1}(S)$
and pull
back to $S^{[n-2,n-1]}$. Note that $f_2$ maps $S^{[n-2,n-1]}_i$ to
$S^{(n)}_{i+1}$. So we get, in view of lemma \ref{hilbnorm}
and lemma \ref{discard},
\begin{eqnarray}(\ref{hb})=-\int\limits_{S^{[n-2,n-1]}}\sum_{i,j>0}
{i+j+3\choose i+1} s_i(V) s_j(N^\vee_{n-1})s(V[{n-1}])\cdot \gamma_2.
\label{hc}
\end{eqnarray}
We now again use the identity $s(V[{n-1}])=s(V(-F_{n-1}))s(V[{n-2}])$
on $S^{[n-2,n-1]}$ and obtain
\begin{eqnarray}
(\ref{hc})=\!\!\!\!\!
\int\limits_{S\times {\text{\rom{Hilb}}}^{n-2}(S)} \sum_{i+j+l\le 2}{i+l+3\choose i+1}
s_i(V) s_j(V)s_l(T^\vee_S) s(V[{n-2}])\cdot \gamma_2\label{hd}\\
\ \ \ \ \ \ \ \
+\!\!\!\!\!\!\!\int\limits_{S^{[n-2,n-1]}}\!\!\sum_{i,j,l}\!\!\!\!{i+l+3\choose i+1}
s_i(V) s(V[{n-2}])
\!\left(s_j(V(F_{n-1}))s_l(N^\vee_{n-1})-s_j(V)s_l(T^\vee_S)
\right)\!\cdot\! \gamma_2\label{he}
\end{eqnarray}
By explicit calculation and the definition of $V$, we get
for the first integral
\begin{eqnarray*}&&\sum_{i+j+l\le 2}{i+l+3\choose i+1} s_i(V)
s_j(V)s_l(T^\vee_S)\\
&&\quad=
3+9s_1(V)-4K_S+13s_2(V)+6s_1(V)^2-14s_1(V)K_S+5s_2(S)\\
&&\quad=t_{2-}.
\end{eqnarray*}
Now we compute the integral (\ref{he}).
We use the formula
$$s(N^\vee_{n-1})=s(T^\vee_S){1-F_{n-1}\over 1-2F_{n-1}}$$
and the notation
$$2^{[l]}=\begin{cases} 1&l<0;\\
2^l&l\ge 0.\end{cases}$$
to obtain
\begin{eqnarray*}\qquad
(\ref{he})&=&- \int\limits_{S^{[n-2,n-1]}}\sum_{i,j_1,j_2,k_1,k_2}
F_{n-1}{i+k_1+k_2+3\choose i+1}{j_1+j_2+1\choose j_1+1}
2^{[k_2-1]} \cdot\\
&&\qquad\qquad\qquad\qquad\qquad\qquad
\cdot s_i(V)s_{j_1}(V)s_{k_1}(T^\vee_S)F_{n-1}^{j_2+k_2-1}
s(V[{n-2}])\cdot\gamma_2.
\end{eqnarray*}
This can again be expressed as an integral over $F_{n-1}$.
We push forward to $Z_{n-2}(S)\subset S\times{\text{\rom{Hilb}}}^{n-2}(S)$
and then pull back to
$S^{[n-3,n-2]}$. Note that $f_3$ maps $S^{[n-3,n-2]}_i$ to
$S^{(n)}_{i+2}$. Therefore using lemma \ref{hilbnorm} and lemma
\ref{discard}
to see that we can replace the push-forward of $F_{n-1}^l$ by the pull-back
of $s_{l-2}(T^\vee_S)$ via the projection $S^{[n-3,n-2]}\longrightarrow
S\times{\text{\rom{Hilb}}}^{n-3}(S)$. We then push forward to
$S\times
{\text{\rom{Hilb}}}^{n-3}(S)$ and notice that by theorem \ref{snn} and lemma
\ref{discard} we can replace the push-forward of $s(V[{n-2}])$ by
$s(V)s(V[{n-3}])$.
Putting all this together we obtain
\begin{eqnarray*}
(\ref{he})&=&
\int\limits_{S\times Hilb^{n-3}(S)}\sum_{i+j_1+j_2+k_1+k_2+l\le 3}
{i+k_1+k_2+3\choose i+1}{j_1+j_2+1\choose j_1+1}
2^{[k_2-1]} \cdot\\
&&\qquad\qquad\cdot
s_i(V)s_{j_1}(V)s_l(V)s_{k_1}(T^\vee_S)s_{j_2+k_2-2}(T^\vee_S)
s(V[{n-3}])\cdot\gamma_3.
\end{eqnarray*}
We obtain, again by direct calculation,
\begin{eqnarray*}
\sum_{i+j_1+j_2+k_1+k_2+l\le 3}
&&{i+k_1+k_2+3\choose i+1}{j_1+j_2+1\choose j_1+1}
2^{[k_2-1]} s_i(V)s_{j_1}(V)s_l(V)s_{k_1}(T^\vee_S)s_{j_2+k_2-2}(T^\vee_S)\\&&
=
27+135s_1(V)-102K_S=27+270\xi-237 K_S.
\end{eqnarray*}
This completes the proof.
\end{pf}
\begin{rem}\label{hilbind1}
Let $\gamma\in H^{4n-2k}(S^{(n)},{\Bbb Q})$ with $k\le 5$.
Then the same proof shows
\begin{eqnarray*}
n\int\limits_{{\text{\rom{Hilb}}}^n(S)}s((V^\vee(K_S)[n])^\vee)\cdot\gamma
=\sum_{l=1}^3\int\limits_{S^l\times
{\text{\rom{Hilb}}}^{n-l}(S)} (-1)^{l-1} [\Delta_l]p_1^*t_{l+}\cdot
s((V^\vee(K_S)[n-l])^\vee)\cdot
\gamma.
\end{eqnarray*}
\end{rem}
We will now introduce a compact notation for some symmetric cohomology classes
on $S^n$
that will also help us in organizing our
combinatorical calculations.
\begin{defn}\label{polnota}
We denote by $\frak S_{n}$ the symmetric
group on $n$ letters,
which acts on $S^n$ by permuting the factors.
For $\alpha\in H^{2*}(S^k,{\Bbb Q})$ and $\beta\in H^{2*}(S^l,{\Bbb Q})$
we define $\alpha{{\hbox{$*$}}} \beta\in
H^{2*}(S^{k+l},{\Bbb Q})^{\frak S_{k+l}}$ by putting
$$\alpha{{\hbox{$*$}}} \beta:={1\over (k+l)!} \sum_{\sigma\in \frak S_{k+l}}
(p_{\sigma(1)}\times\ldots\times p_{\sigma(k)})^{{\hbox{$*$}}} \alpha
\cdot
(p_{\sigma(k+1)}\times\ldots\times p_{\sigma(k+l)})^*\beta.$$
It is easy to see that $*$ is a commutative and associative operation.
We will denote
$$\alpha^{{{\hbox{$*$}}} k}:=\underbrace{\alpha{{\hbox{$*$}}} \alpha{{\hbox{$*$}}} \ldots{{\hbox{$*$}}} \alpha}_k.$$
\end{defn}
\begin{rem} \label{starrem}
The following elementary properties of $*$ will be very
important for our further computations:
\begin{enumerate}
\item
For $\alpha\in H^{2*}(S^k,{\Bbb Q})$, $\beta\in H^{2*}(S^l,{\Bbb Q})$
and $w\in H^*(S^{k+l},{\Bbb Q})^{\frak S_{k+l}}$ it follows immediately from the
symmetry of $w$ that
\begin{eqnarray*}\int_{S^{k+l}}(\alpha{{\hbox{$*$}}} \beta)\cdot w
&=&\int_{S^{k+l}}(p_{1}\times\ldots \times p_{k})^*\alpha
\cdot
(p_{k+1}\times\ldots \times p_{k+l})^*\beta\cdot w\\
&=&\sum_{(w_1,w_2)}\int_{S^{k}}\alpha w_1
\cdot \int_{S^{l}}\beta w_2.
\end{eqnarray*}
Here
$w=\sum_{(w_1,w_2)} w_1\cdot w_2$ is the K\"unneth decomposition.
Analogous results hold if more then two factors are multiplied via $*$.
\item Let $1$ denote the neutral element of the ring $H^*(S,{\Bbb Q})$.
Then $1^{{{\hbox{$*$}}} k}$ is the neutral element of $H^*(S^k,{\Bbb Q})$.
\item
It is also easy to see from the definitions
that $*$ fulfills the distributive law
$\alpha {{\hbox{$*$}}} (\beta_1+\beta_2)=\alpha {{\hbox{$*$}}} \beta_1+\alpha {{\hbox{$*$}}} \beta_1$.
In fact $+$ and ${{\hbox{$*$}}} $ make
$\bigoplus_{n\ge 0}H^{2*}(S^n,{\Bbb Q})^{\frak S_n}$ a commutative ring.
\item
In particular the binomial formula holds:
$$\sum_{k+l=n} {n\choose k}\alpha^{{{\hbox{$*$}}} k}{{\hbox{$*$}}}\beta^{{{\hbox{$*$}}} l}
=(\alpha+\beta)^{{{\hbox{$*$}}} n}.$$
\end{enumerate}
\end{rem}
\begin{nota}\label{starnota}
For a class $\alpha\in H^*(S,{\Bbb Q})$ and a postive integer $i$
we denote by $(\alpha)_i:=[\Delta_i]p_1^*\alpha\in H^*(S^i,{\Bbb Q})^{\frak S_i}$,
where $[\Delta_i]$ is the (small) diagonal $\{(x,\ldots,x) \ | \ x\in S\}$
in $S^i$. In particular $(\alpha)_1=\alpha$. We will in the future
write $(\alpha)_i(\beta)_j$ instead of $(\alpha)_i{{\hbox{$*$}}} (\beta)_j$ and
$(\alpha)_i^{m}$ instead of $(\alpha)_i^{{{\hbox{$*$}}} m}$. Furthermore we write
$\alpha^{{{\hbox{$*$}}} m}\beta^{{{\hbox{$*$}}} l}$ and $\alpha^{{{\hbox{$*$}}} m}(\beta)_i$ instead of
$\alpha^{{{\hbox{$*$}}} m}{{\hbox{$*$}}} \beta^{{{\hbox{$*$}}} l}$ and $\alpha^{{{\hbox{$*$}}} m}{{\hbox{$*$}}} (\beta)_i$.
\end{nota}
\begin{prop}\label{erstint}
Let $\gamma\in H^{4d-2k}S^{(d)},{\Bbb Q})$ with $k\le 5$
and $w\in H^{4d-2k}(S^d,{\Bbb Q})$ its pull-back to $S^d$.
Then
\begin{eqnarray*}
&&d!\int\limits_{{\text{\rom{Hilb}}}^d(S\sqcup S)}
s([V^\vee(K_S)]^\vee_1\oplus [V]_2)\cdot \gamma\\
&&\qquad\qquad
= \int\limits_{S^d}\left(t_1^{{{\hbox{$*$}}} d}-{d\choose 2}(t_2)_2 t_1^{{{\hbox{$*$}}} (d-2)}
+2{d\choose 3}(t_3)_3t_1^{{{\hbox{$*$}}} (d-3)}
+3{d\choose 4}(t_2)_2^{2}t_1^{{{\hbox{$*$}}} (d-4)}
\right)\cdot w.
\end{eqnarray*}
\end{prop}
\begin{pf}
Let $n,m$ be nonnegative integers with $n+m=d$.
Let
$\gamma_1\cdot\gamma_2\in H^*(S^{(n)})\times S^{(m)},{\Bbb Q})\setminus 0$ be a
K\"unneth component of the pull-back of $\gamma$ via
$S^{(n)}\times S^{(m)} \longrightarrow S^{(d)}$. Let $w_1\cdot w_2\in
H^{4n-2l}(S^n,{\Bbb Q})^{\frak S_n}\times H^{4n-2r}(S^m,{\Bbb Q})^{\frak S_m}$
be the pull-back of
$\gamma_1\cdot\gamma_2$. Then $0\le l,r\le 5$. By an easy induction
using
lemma \ref{hilbind}, remark \ref{hilbind1}
and remark \ref{starrem} and ignoring all terms of codimension
$\ge 6$ we get
\begin{eqnarray*}
n!\int\limits_{{\text{\rom{Hilb}}}^n(S)}
s((V^\vee(K_S)[n])^\vee)\cdot \gamma_1&=&
\int_{S^n} P_{n}\cdot w_1 \\ {}
m!\int\limits_{{\text{\rom{Hilb}}}^m(S)}
s(V[m])\cdot \gamma_2&=&
\int_{S^m}Q_m\cdot {w_2},
\end{eqnarray*}
where
\begin{eqnarray*}
P_{n}&=&
t_{1+}^{{{\hbox{$*$}}} n}-\sum_{i=2}^n(i-1)(t_{2+})_2t_{1+}^{{{\hbox{$*$}}} (n-2)}+
\sum_{i=2}^{n-2}\sum_{j=i+2}^n(i-1)(j-1)(t_{2+})_2^{2}t_{1+}^{{{\hbox{$*$}}} (n-4)}\\
&&\qquad +\sum_{i=3}^n(i-1)(i-2)(t_{3+})_3 t_{1+}^{{{\hbox{$*$}}} (n-3)}.
\end{eqnarray*} and $Q_m$ is defined analogously to $P_n$ replacing
$n,$ $ t_{1+},$ $t_{2+}$ and $t_{3+}$ by
$m,$ $ t_{1-},$ $t_{2-}$ and $t_{3-}$ respectively.
Applying again remark \ref{starrem} we obtain
$$n!m!\int\limits_{{\text{\rom{Hilb}}}^n(S)\times {\text{\rom{Hilb}}}^m(S)}
s((V^\vee(K_S)[n])^\vee)s(V[m])\cdot \gamma=
\int\limits_{S^d}(P_n{{\hbox{$*$}}} Q_m)\cdot w,
$$
and thus
$$
d!\int\limits_{{\text{\rom{Hilb}}}^d(S\sqcup S)}
s([V^\vee(K_S)]^\vee_1)s([V]_2)\cdot \gamma\\
=
\sum_{n+m=d} {d\choose n} \int\limits_{S^d} (P_n{{\hbox{$*$}}}
Q_m)\cdot w$$
Finally we have
\begin{eqnarray*}
&&\sum_{n+m=d} {d\choose n} P_n{{\hbox{$*$}}} Q_m\\
&&\quad =\sum_{n+m=d} {d\choose n}
\Bigg(t_{1+}^{{{\hbox{$*$}}} n} t_{1-}^{{{\hbox{$*$}}} m}-{n\choose 2}
(t_{2+})_2t_{1+}^{{{\hbox{$*$}}}(n-2)}{{\hbox{$*$}}} t_{1-}^{{{\hbox{$*$}}} m}
-{m\choose 2}(t_{2-})_2t_{1+}^{*n} t_{1-}^{{{\hbox{$*$}}} (m-2)}\\
&&\quad\quad + 3{n\choose 4}(t_{2+})_2^{ 2}
t_{1+}^{{{\hbox{$*$}}}(n-4)}t_{1-}^{{{\hbox{$*$}}} m}+
{n\choose 2}{m\choose 2}(t_{2+})_2 (t_{2-})_2
t_{1+}^{{{\hbox{$*$}}}(n-2)}t_{1-}^{{{\hbox{$*$}}}(m-2)}
+ 3{m\choose 4}(t_{2-})_2^{ 2} t_{1+}^{{{\hbox{$*$}}} n} t_{1-}^{{{\hbox{$*$}}}(m-4)}\\
&&\quad\quad +2{n\choose 3}(t_{3+})_3 t_{1+}^{{{\hbox{$*$}}} (n-3)}
t_{1-}^{{{\hbox{$*$}}} m}
+2{m\choose 3}(t_{3-})_3 t_{1+}^{{{\hbox{$*$}}} n} t_{1-}^{{{\hbox{$*$}}}(m-3)}\Bigg)\\
&&=\quad t_1^{{{\hbox{$*$}}} d}-{d\choose 2}(t_2)_2 t_1^{{{\hbox{$*$}}} (d-2)}
+3{d\choose 4}(t_2)_2^{ 2} t_1^{{{\hbox{$*$}}} (d-4)}
+2{d\choose 3}(t_3)_3 t_1^{{{\hbox{$*$}}} (d-3)}.
\end{eqnarray*}
\end{pf}
\def{pt}{{pt}}
\def\alpha{\alpha}
\def{S^{(n)}}{{S^{(n)}}}
\def{S^{(m)}}{{S^{(m)}}}
\def{S^{(n)}}{{S^{(n)}}}
\def{S^{(m)}}{{S^{(m)}}}
\def\<{\langle}
\def\>{\rangle}
\def{\text{\rom{Hilb}}}{{\text{\rom{Hilb}}}}
\def{\Cal W}{{\Cal W}}
\def{\hbox{\rom{Tor}}}{{\hbox{\rom{Tor}}}}
\def\Hilb^{n} (S){{\text{\rom{Hilb}}}^{n} (S)}
\def\Hilb^{m} (S){{\text{\rom{Hilb}}}^{m} (S)}
\def\tilde S^{n}{\tilde S^{n}}
\def\Hilb^{m} (S){{\text{\rom{Hilb}}}^{m} (S)}
\def\tilde S^{m}{\tilde S^{m}}
\def\pi{\pi}
\defg{g}
\def\phi{\varphi}
\def\tilde{\tilde}
\def{\bar\al}{{\bar\alpha}}
\def\Star#1{{(#1)}_*}
\def{\ti{\cal Z_2}}{{\tilde{\cal Z_2}}}
\def\Delta^0{\Delta^0}
\def{\cal E}#1#2{E_{#1#2}}
\def\Ez#1#2{E^0_{#1#2}}
\def\bE#1#2{\bar E_{#1#2}}
\def\bEz#1#2{\bar E^0_{#1#2}}
\def\bar F{\bar F}
Now we want to compute the second integral
$$\int_T(c(\Gamma)-1)s([V^\vee(K_S)]_1^\vee)s([V]_2)\cdot\gamma$$
for $\gamma\in H^{4d-2k}(S^{(d)},{\Bbb Q})$ with $k\le 5$,
The conventions of \ref{convent} stay in effect.
\begin{defn}\label{diazwei}
Let $n,m$ be nonnegative integers with $n+m=d$.
We consider the following diagram
$$\CD
S\times \Hilb^{n} (S) \times\Hilb^{m} (S) & @>q
>> &\Hilb^{n} (S) \times \Hilb^{m} (S) &@>g>> &{S^{(n)}}\times {S^{(m)}}\cr
@AA{\tilde\phi}A&&@A{\phi}AA&&@A\phi'AA\cr
S\times \tilde S^{n} \times\tilde S^{m} & @>\tilde q>> &\tilde S^{n} \times \tilde S^{m}&@>\tig>>
&S^{n}\times S^{m}\cr
\endCD$$
Here, as above, $q$ and $\tilde q$ are the projections, $g:\Hilb^{n} (S) \times \Hilb^{m} (S)
\longrightarrow
{S^{(n)}}\times {S^{(m)}}$ is the product of the Hilbert-Chow morphisms and
$\varphi':S^{n}\times S^{m}
\longrightarrow {S^{(n)}}\times {S^{(m)}}$ is the product of the quotient maps,
and all the other varieties and maps are defined via pull-back.
For $i=1,2$ we put $\widetilde Z_i:=\tilde\varphi^{-1}({\cal Z}_i)$ and $\widetilde
Z_{1,2}:=\widetilde Z_1\cap \widetilde Z_2$,
(i.e. the scheme-theoretic intersection).
\end{defn}
\begin{nota} We denote by
\begin{eqnarray*}&&({S^{(n)}}\times {S^{(m)}})_{*}
:=\Big\{ (\sigma_+,\sigma_-)\in
{S^{(n)}}\times{S^{(m)}}
\Bigm|\\
&&\qquad\qquad\qquad\qquad\qquad\# supp(\sigma_+)\ge n-1,\,
\# supp(\sigma_-)\ge m-1,\,
\# supp(\sigma_++\sigma_-)\ge
d-2\Big\}.
\end{eqnarray*}
Furthermore for all $X$ with a natural morphism
$f:X\longrightarrow {S^{(n)}}\times {S^{(m)}}$ we denote
$X_{*}:=f^{-1}({S^{(n)}}\times {S^{(m)}})_{*}$.
We put $$\Gamma_{n,m}:=\varphi^*(\Gamma|_{(\Hilb^{n} (S)\times \Hilb^{m} (S))_{*}})$$
(see 7.1 for the definition of $\Gamma$).
For $1\le i\le n$ (resp. $1\le j\le m$) we denote by $p_{i+}$ (resp.
$p_{j-}$) the projection from
$S^{n}\times S^{m}$ onto the $i^{th}$ factor of $S^{n}$
(resp. the $j^{th}$ factor of $S^{m}$).
For $\epsilon =+,-$, $\eta =+,-$
we put
\begin{eqnarray*}\Delta_{0,i}^{\epsilon }&:=&\Bigm\{(x,x_1^+,\ldots x_{n}^+,
x_1^-,\ldots x_{m}^-)\in S\times S^{n}\times S^{m}\Bigm|
x=x_i^\epsilon \Big\}\\
\Delta_{i,j}^{\epsilon \eta }&:=&\Big\{(x_1^+,\ldots x_{n}^+,
x_1^-,\ldots x_{m}^-)\in S^{n}\times S^{m}\Bigm| x_i^\epsilon
=x_j^\eta \Big\}\\
\Delta_{0,i,j}^{\epsilon \eta }&:=&\Big\{(x,x_1^+,\ldots x_{n}^+,
x_1^-,\ldots x_{m}^-)\in S\times S^{n}\times S^{m}\Bigm|
x=x_i^\epsilon =x_j^\eta \Big\}\\
\end{eqnarray*}
We will also denote by $\Delta_{i,j}^{+-}$,
$\Delta_{0,i,j}^{+-}$,
$\Delta^{\epsilon }_{0,i}$
the pull-backs $\tilde g^{-1}(\Delta_{i,j}^{+- })$,
$({\hbox{\rom{id}}}_S\times \tilde g)^{-1}(\Delta_{0,i,j}^{+-})$,
$({\hbox{\rom{id}}}_S\times \tilde g)^{-1}(\Delta^{\epsilon }_{0,i})$.
We denote $D_{i,j}:=
\tilde g^{-1}(\Delta^{++}_{i,j})$ and $E_{i,j}:=
\tilde g^{-1}(\Delta^{--}_{i,j})$.
$D_{i,j}$ and $E_{i,j}$ are divisors (see below), we denote
$F_{i}:=\sum_{j<i}D_{i,j}$ and $G_{i}:=\sum_{j<i}E_{i,j}$.
\end{nota}
\begin{rem}\label{hilbfacts}
The following easy facts will be used throughout the computation.
\begin{enumerate}
\item It is well known that $({\text{\rom{Hilb}}}^{n}(S)\times {\text{\rom{Hilb}}}^{m}(S))_{*}$ is
obtained from
$(S^{n}\times S^{m})_{*}$ by blowing up all the $\Delta^{++}_{i,j}$
and $\Delta^{--}_{i,j}$ and taking the quotient by the action of the
product of the symmetric groups
$\frak S_{n}\times \frak S_{m}$. It follows that in fact
$(\tilde S^{n}\times \tilde S^{m})_{*}$
is just the blow up of $(S^{n}\times S^{m})_{*}$ along the
(disjoint) smooth subvarieties $(\Delta^{++}_{i,j})_*$
and $(\Delta^{--}_{i,j})_*$ and the $(D_{i,j})_*$ and $(E_{i,j})_*$ are the
exceptional
divisors.
\item It is also easy to see that
$(\widetilde Z_1)_{*}=\bigcup_{i=1}^{n} (\Delta_{0,i}^{+})_*$,
$(\widetilde Z_2)_{*}=\bigcup_{j=1}^{m} (\Delta_{0,j}^{-})_*$
and therefore
$$(\widetilde Z_{1,2})_{*}=\bigcup_{i=1}^{n}\bigcup_{j=1}^{m}
(\Delta_{0,i,j}^{+-})_{*}.$$
(We mean here the scheme theoretic union, i.e. the scheme defined by
the intersection of the ideals).
\item
For $i\ne j$ we have (scheme-theoretically)
\begin{eqnarray*}
(\Delta_{0,i}^{+})_{*}\cap \Delta_{0,j}^{+}&=&
(\Delta_{0,i}^{+})_{*}\cap D_{i,j},\\
(\Delta_{0,i}^{-})_{*}\cap \Delta_{0,j}^{-}&=&
(\Delta_{0,i}^{-})_{*}\cap E_{i,j}.
\end{eqnarray*}
\end{enumerate}
\end{rem}
\begin{lem}\label{comalg}
\begin{enumerate}
\item Let $X$ be a smooth variety, and let $Y$ and $Z$ be
Cohen-Macauley subschemes
of $X$ such that the ideal ${\cal I}_{Z/(Y\cup Z)}$ of $Z$ in
$Y\cup Z$ is ${\cal O}_Y(-D)$ for a divisor $D$ on $Y$.
Then in the Grothendieck ring of $X$ we have
\begin{eqnarray*}{\cal O}_{Y\cup Z}&=&{\cal O}_Y(-D)+{\cal O}_Z\ \hbox{ and }\\
\omega_X^{-1}\otimes \omega_{Y\cup Z}&=&\omega_X^{-1}\otimes\omega_Y(D)
+\omega_X^{-1} \otimes\omega_Z.\end{eqnarray*}
\item Let $f:X\longrightarrow Y$ be a morphism between smooth varieties. Let
$Z\subset Y$ be a Cohen-Macauley subscheme of codimension $2$ and
assume $W:=f^{-1}(Z)$ has pure codimension $2$ in $X$. Then
$$f^{*}(\omega_Y^{-1}\otimes \omega_Z)=\omega_X^{-1}\otimes \omega_W.$$
\item Let $X$ be a smooth variety and $Y$ and $Z$ Cohen-Macauley
subschemes of codimension
$2$ intersecting properly. Then in the Grothendieck ring of $X$
we have
$${\cal O}_{Y}\otimes{\cal O}_{Z}={\cal O}_{Y\cap Z}.$$
\end{enumerate}
\end{lem}
\begin{pf}
(1) The first identity follows from the standard exact sequence
$$0\longrightarrow {\cal O}_Y(-D)\longrightarrow {\cal O}_{Y\cup Z}\longrightarrow {\cal O}_Z\longrightarrow 0.\eqno (*)$$
Now we dualize $(*)$ and use that for a two codimensional
Cohen-Macauley subscheme
$W\subset X$ we have
$${\cal Ext}^i({\cal O}_W,{\cal O}_X)=\begin{cases} 0& i<2,\\
\omega_X^{-1}\otimes \omega_X& i=2\end{cases}$$
to obtain the sequence
$$0\longrightarrow \omega_X^{-1}\otimes
\omega_Z\longrightarrow \omega_X^{-1}\otimes \omega_{Y\cup Z}\longrightarrow
\omega_X^{-1}\otimes\omega_Y(F)\longrightarrow 0$$ and thus the second identity.
(2) We take a locally free resolution
$$0\longrightarrow B\longrightarrow A\longrightarrow {\cal O}_Y\longrightarrow {\cal O}_Z\longrightarrow 0.$$
Pulling it back we obtain the sequence
$$0\longrightarrow f^*B\longrightarrow f^*A\longrightarrow {\cal O}_X\longrightarrow {\cal O}_W\longrightarrow 0,$$
which stays exact by the Hilbert-Birch theorem (see e.g. \cite{P-S}
lemma 3.1).
Dualizing we obtain the exact sequence
$$0\longrightarrow {\cal O}_X\longrightarrow f^*A\longrightarrow f^*B\longrightarrow
\omega_X^{-1} \otimes\omega_Z\longrightarrow 0.$$
We can also arrive at this sequence differently, by first dualizing and
then pulling back. This way we obtain the sequence
$$0\longrightarrow {\cal O}_X\longrightarrow f^*A\longrightarrow f^*B\longrightarrow
f^*(\omega_X^{-1} \otimes\omega_Z)\longrightarrow 0,$$
and (2) follows.
(3) By the corollaire on p. 20 in \cite{Se} we have
${\hbox{\rom{Tor}}}_i({\cal O}_Y,{\cal O}_Z)=0$ for $i>0$, and (3) follows.
\end{pf}
\begin{lem}\label{grot2}
In the Grothendieck ring of
$(\widetilde S^{n}\times \widetilde S^{m})_{*}$
we have the equality
$$\varphi^*(\Gamma_{n,m})=\sum_{i=1}^{n}\sum_{j=1}^{m}\Big(
{\cal O}_{\Delta_{i,j}^{+-}}(F_i-G_j-p_{i+}^*\xi)+
{\cal O}_{\Delta_{i,j}^{+-}}(F_i-G_j-p_{i+}^*(\xi+K_S))\Big).$$
\end{lem}
\begin{pf}
Using remark \ref{hilbfacts}(2) and remark \ref{hilbfacts}(3) and applying
lemma \ref{comalg}(1) inductively we obtain in the Grothendieck ring of
$S\times(\widetilde S^n\times \widetilde S^m)_*$ the equalities
\begin{eqnarray}
\label{gr1}{\cal O}_{\widetilde Z_1}&=&\sum_{i=1}^n
{\cal O}_{\Delta_{0,i}^+}(-F_i)\\
\label{gr2}{\cal O}_{\widetilde Z_2}&=&\sum_{j=1}^m
{\cal O}_{\Delta_{0,j}^-}(-G_j)\ \hbox{ and}
\end{eqnarray}
\begin{eqnarray*}
\widetilde\varphi^*(\omega_{S\times{\text{\rom{Hilb}}}^n(S)\times{\text{\rom{Hilb}}}^m(S)}^{-1}\otimes
\omega_{{\cal Z}_1})&=&
\omega_{S\times\widetilde S^n\times \widetilde S^m}^{-1}
\otimes\omega_{\widetilde Z_1}\\
&=&\sum_{i=1}^n
\omega_{S\times\widetilde S^n\times \widetilde S^m}^{-1}
\omega_{\Delta_{0,i}^+}(F_i)\\
&=&\sum_{i=1}^n
{\cal O}_{\Delta_{0,i}^+}(-p_{i+}^*K_S+F_i),
\end{eqnarray*}
where in the third and the last line we have used
lemma \ref{comalg}(2).
Now using lemma \ref{comalg}(3) and tenzorizing by
$p^*V$ we obtain in the Grothendieck ring of
$S\times(\widetilde S^n\times \widetilde S^m)_*$ the equality
$$({\hbox{\rom{id}}}_S\times \varphi)^*(\omega_T^{-1}\otimes \omega_{{\cal Z}_1}\otimes
{\cal O}_{{\cal Z}_2}\otimes p^*V)
=\sum_{i=1}^{n}\sum_{j=1}^{m}\Big({\cal O}_{\Delta_{0,i,j}^{+-}}(F_i-G_j
-p_{i+}^*\xi)+ {\cal O}_{\Delta_{0,i,j}^{+-}}(F_i-G_j
-p_{i+}^*(\xi+K_S)\Big)
.$$
The morphism $\varphi:(\widetilde S^{n}\times \widetilde S^{m})_{*}
\longrightarrow {\text{\rom{Hilb}}}^{n}(S)\times {\text{\rom{Hilb}}}^{m}(S)$
is flat.
Therefore we get by (\cite{Ha} prop.III.9.3)
\begin{eqnarray*}
&&\varphi^*q_*(\omega_T^{-1}\otimes \omega_{Z_1}\otimes {\cal O}_{Z_2}\otimes p^*V)\\
&&\qquad =
\tilde q_*({\hbox{\rom{id}}}_S\times \varphi)^*(\omega_T^{-1}\otimes
\omega_{Z_1}\otimes {\cal O}_{Z_2}\otimes p^*V)\\
&&\qquad=\sum_{i=1}^{n}\sum_{j=1}^{m}\tilde q_*(
({\cal O}_{\Delta_{0,i,j}^{+-}}(F_i-G_j
-p_{i+}^*\xi)+ {\cal O}_{\Delta_{0,i,j}^{+-}}(F_i-G_j
-p_{i+}^*(\xi+K_S))\\
&&\qquad
=\sum_{i=1}^{n}\sum_{j=1}^{m}({\cal O}_{\Delta_{i,j}^{+-}}(F_i-G_j
-p_{i+}^*\xi)+ {\cal O}_{\Delta_{i,j}^{+-}}(F_i-G_j -p_{i+}^*(\xi+K_S)),
\end{eqnarray*}
in the Grothendieck ring of
$(\widetilde S^{n}\times \widetilde S^{m})_{*}$. The last identity
follows from the fact that the projection
$\tilde q|_{\Delta_{0,i,j}^{+-}}:{\Delta_{0,i,j}^{+-}}\longrightarrow \Delta_{i,j}^{+-} $
is an isomorphism.
\end{pf}
\begin{lem}\label{rrsd} Let $X$ be a smooth variety and let
$i:Y\longrightarrow X$ be the closed
embedding of a smooth subvariety of codimension $2$ with conormal bundle
$N^\vee$. Let
$D$ be a divisor on $Y$. Then $$c(i_*({\cal O}_Y(-D)))=1-i_*\Big(\sum_{k,l\ge 0}
{k+l+1\choose l} D^l s_k(N^\vee)\Big).$$
\end{lem}
\begin{pf}
This is a straightforward application of Riemann-Roch without denominators
\cite{Jo}.
\end{pf}
\begin{lem}\label{cgamma}
Let $1\le k\le 5$. Then
$c_k(\Gamma_{n,m}|_{(\widetilde S^n\times \widetilde S^m)_*})$
is the part of degree $k$ of
$$\matrix
\displaystyle -\sum_{(i,j)}
[\Delta_{i,j}^{+-}]\Big(2+4p_{i+}^*\xi+4(G_j-F_i)
+p_{i+}^*(6\xi^2+3s_2(S)-K_S^2)
+12p_{i+}^*\xi(G_j-F_i)\cr
\displaystyle+6(F_i^2+G_j^2)+p_{i+}^*(24\xi^2+12s_2(S)-4K_S^2)
(G_j-F_i)
+24p_{i+}^*\xi(G_j^2+F_i^2)+8(G_j^3-F_i^3)\Big)\cr
+\displaystyle \sum_{(i,j)\ne (i_1,j_1)}
[\Delta_{i,j}^{+-}][\Delta_{i_1,j_1}^{+-}](4+8p_{i+}^*\xi+8p_{i_1+}^*\xi)
\endmatrix\eqno (7.20.1)$$
Here $(i,j)$ and $(i_1,j_1)$ run through
$\{1,\ldots, n\}\times \{1,\ldots, m\}$.
\end{lem}
\begin{pf}
We compute on $(\widetilde S^n\times \widetilde S^m)_*$.
We notice that $[\Delta_{i,j}^{+-}]$ is just the pull-back of
the corresponding class in $S^{n}\times S^{m}$ via $\tilde g$
and the conormal bundle of $\Delta_{i,j}^{+-}$ is just the pull-back
of the conormal bundle, i.e. $p_{i+}^*(T^\vee_S)$.
Furthermore we note that on
$(\widetilde S^{n}\times \widetilde S^{m})_{*}$ we have
$[\Delta_{i,j}^{+-}]\cdot F_i\cdot G_j=0$.
Therefore we obtain by lemma \ref{rrsd}
after some calculation
that for $1\le k\le 5$ the Chern class
$c_k({\cal O}_{\Delta_{i,j}^{+-}}(-p_{i+}^*\xi +F_i-G_j))$
is the part of degree $k$ of
\begin{eqnarray*}&&-[\Delta_{i,j}^{+-}]\Big(1+p_{i+}^*(2\xi-K_S)+2(G_j-F_i)
+p_{i+}^*(3\xi^2-3\xi K_S+s_2(S))
\\
&&\qquad+p_{i+}^*(6\xi-3K_S)(G_j-F_i)+3(G_j^2+F_i^2)\\
&&\qquad
+p_{i+}^*(12\xi^2-12K_S\xi+4s_2(S))(G_j-F_i)
+p_{i+}^*(12\xi-6K_S)(G_j^2+F_i^2)+4(G_j^3-F_i^3)\Big).\\
\end{eqnarray*}
Analogously we obtain that $c_k({\cal O}_{\Delta_{i,j}^{+-}}
(-p_{i+}^*(\xi+K_S) +F_i-G_j))$ is the part of degree $k$ of
\begin{eqnarray*}
&&1-[\Delta_{i,j}^{+-}]\Big(1+p_{i+}^*(2\xi+K_S)+2(G_j-F_i)
+p_{i+}^*(3\xi^2+3\xi K_S+s_2(S))\\
&&\qquad
+p_{i+}^*(6\xi+3K_S)(G_j-F_i)+3(G_j^2+F_i^2)\\
&&\qquad
+p_{i+}^*(12\xi^2+12K_S\xi+4s_2(S))(G_j-F_i)
+p_{i+}^*(12\xi+6K_S)(G_j^2+F_i^2)+4(G_j^3-F_i^3)\Big).\\
\end{eqnarray*}
We notice that
$[\widetilde\Delta_{i,j}^{+-}]^2=[\Delta_{i,j}^{+-}]p_{i+}^*(c_2(S))$.
Thus, by
multiplying out, we get that $c_k({\cal O}_{\Delta_{i,j}^{+-}}(-p_{i+}^*\xi
-F_i+G_j)\oplus {\cal O}_{\Delta_{i,j}^{+-}} (-p_{i+}^*(\xi+K_S) -F_i+G_j))$
is the
part of degree $k$ of
\begin{eqnarray*}
&&1-[\Delta_{i,j}^{+-}]\Big(2+4p_{i+}^*\xi+4(G_j-F_i)\\
&&\qquad +p_{i+}^*(6\xi^2+3s_2(S)-K_S^2)+
12p_{i+}^*\xi(G_j-F_i)+6(G_j^2+F_i^2)\\
&&\qquad +p_{i+}^*(24\xi^2+12s_2(S)-4K_S^2)(G_j-F_i)
+24p_{i+}^*\xi(G_j^2+F_i^2)+8(G_j^3-F_i^3)\Big).\\
\end{eqnarray*}
Now we take the product over all $i,j$. We use that on
$(\tilde S^{n}\times\tilde S^{m})_{*}$ we have
$[\Delta_{i_1,j_1}^{+-}]\cdot [\Delta_{i_2,j_2}^{+-}]\cdot
F_i=[\Delta_{i_1,j_1}^{+-}]\cdot [\Delta_{i_2,j_2}^{+-}]\cdot G_j=0$ unless
$\{i_1,j_1\}=\{i_2,j_2\}$, and obtain the result.
\end{pf}
\begin{rem}
\label{cstand}
\begin{enumerate}
\item In the Grothendieck ring of
$(\widetilde S^{n}\times \widetilde S^{m})_{*}$ we
have
\begin{eqnarray*}
\varphi^*([V]_2|_{{\text{\rom{Hilb}}}^n(S)\times {\text{\rom{Hilb}}}^m(S)})
&=& \sum_{j=1}^{m} p_{j-}^*V(-G_j),\\
\varphi^*([V^\vee(K_S)]^\vee|_{{\text{\rom{Hilb}}}^n(S)\times {\text{\rom{Hilb}}}^m(S)})
&=& \sum_{i=1}^{n} p_{i+}^*(V(-K_S))(F_i).\\
\end{eqnarray*}
\item Therefore, for $l\le 3$,
$s_l(\varphi^*([V]_2|_{{\text{\rom{Hilb}}}^n(S)\times {\text{\rom{Hilb}}}^m(S)}))$
is the part of degree $l$ of
$$\matrix\displaystyle\prod_{j=1}^{m}p_{j-}^*t_{1-}
+\sum_{1\le j\le j_1\le m}\Big(
+2E_{j,j_1}
+p_{j-}^*(10\xi-5K_S)E_{j,j_1}+3E_{j,j_1}^2)\cr
+p_{j-}^*(30\xi^2-30\xi K_S+9K_S^2)E_{j,j_1}
+p_{j-}^*(18\xi-9K_S)E_{j,j_1}^2+4E_{j,j_1}^3 \Big)
\displaystyle\prod_{j_2\not\in \{j,j_1\}} p_{j_2-}^*t_{1-},
\endmatrix\eqno (7.21.1)$$
and $s_l(\varphi^*([V^\vee(K_S)])^\vee|_{{\text{\rom{Hilb}}}^n(S)\times {\text{\rom{Hilb}}}^m(S)}
)$ is the part of degree $l$ of
$$\matrix
\displaystyle\prod_{i=1}^{n}p_{i+}^*t_{1+} +
\sum_{1\le i\le i_1\le n}
\Big( -2D_{i,i_1}
-p_{i+}^*(10\xi+5K_S)D_{i,i_1}+3D_{i,i_1}^2 \cr
-p_{i+}^*(30\xi^2+30\xi K_S+9K_S^2)D_{i,i_1}
+p_{i+}^*(18\xi+9K_S)D_{i,i_1}^2-4D_{i,i_1}^3\Big)
\displaystyle\prod_{i_2\not\in \{i,i_1\}}
p_{i_2+}^*t_{1+}.
\endmatrix
\eqno (7.21.2)$$
\end{enumerate}
\end{rem}
\begin{pf} (1) follows from the formulas $\ref{gr1}$, $\ref{gr2}$
by tensorizing with $p^*V$ (resp. $p^*(V^\vee(K_S))$) and pushing
down via
$\tilde q_*$.
(2) is just a straightforward computation
using that $E_{i,j}\cdot E_{k,l}=D_{i,j}\cdot D_{k,l}=0$ for $\{i,j\}\ne
\{k,l\}$.
\end{pf}
\begin{rem}\label{discard2}
Let $k\le 5$ and $\gamma\in H^{4d-2k}(\tilde S^{n}\times \tilde
S^{m},{\Bbb Q})$ and assume that $\alpha_1,\alpha_2\in A^*(\tilde S^{n}\times \tilde
S^{m})$ have the same pull-back to $(\tilde S^{n}\times \tilde
S^{m})_{*}$. Then, for all $i\le n$, $j\le m$, we get
analogously to lemma \ref{discard}
$$\int\limits_{\tilde S^{n}\times \tilde
S^{m}}\Delta_{i,j}^{+-}\cdot(\alpha_1-\alpha_2)\cdot\gamma=0.$$
\end{rem}
\begin{prop}\label{intzwei}
Let $\gamma\in H^{4d-2k}(S^{(d)},{\Bbb Q})$ with $k\le 5$, and
let $w\in H^{4d-2k}(S^d,{\Bbb Q})$ be the pull-back of $\gamma$ to $S^d$.
Then
\begin{eqnarray*}
&&d!\int\limits_{{\text{\rom{Hilb}}}^d(S\sqcup S)}
(c(\Gamma)-1)s([V^\vee(K_S)]_1^\vee)s([V]_2)\cdot\gamma\\
&&\qquad =
\int_{S^d}\Big( -d(d-1)(2+12\xi+42\xi^2+3s_2(S)+K_S^2)_2
t_1^{{{\hbox{$*$}}} (d-2)}\\
&&\qquad\quad\qquad\quad +d(d-1)(d-2)(30+260\xi)_3 t_1^{{{\hbox{$*$}}} (d-3)}\\
&&\qquad\quad\qquad\quad
+2d(d-1)(d-2)(d-3)(2+12\xi)_2^{ 2} t_1^{{{\hbox{$*$}}} (d-4)}\Big)\cdot w,
\end{eqnarray*}
and, with
\begin{eqnarray*}
R_d&:=&t_1^{{{\hbox{$*$}}} d}-d(d-1)(5+30\xi+105\xi^2+8s_2(S)+34K_S^2)_2
t_1^{{{\hbox{$*$}}} (d-2)}\\
&&\quad+d(d-1)(d-2)(48+440\xi)_3 t_1^{{{\hbox{$*$}}} (d-3)}\\
&&\quad +{d(d-1)(d-2)(d-3)\over 2}
(5+30\xi)_2^{ 2} t_1^{{{\hbox{$*$}}} (d-4)},\end{eqnarray*}
we get
\begin{eqnarray*}
d!\int\limits_{{\text{\rom{Hilb}}}^d(S\sqcup S)}
c(\Gamma)s([V^\vee(K_S)]_1^\vee)s([V]_2)\cdot\gamma
=\int\limits_{S^d}R_d\cdot w.
\end{eqnarray*}
\end{prop}
\begin{pf}
We fix $n$ and $m$ with $n+m=d$ and start by computing on
$(\widetilde S^{n}\times \widetilde S^{m})$. Using remark
\ref{discard2} we can restrict our
attention to $(\widetilde S^{n}\times \widetilde S^{m})_{*}$.
We multiply out the formulas (7.21.1),(7.21.2) and (7.20.1)
and push down to $S^{n}\times S^{m}$.
We shall use the following facts:
On $(\tilde S^{n}\times \tilde S^{m})_{*}$ any of
$D_i$ and $E_j$,
gives zero when multiplied by
$[\Delta_{i,j}^{+-}][\Delta_{i_1,j_1}^{+-}]$.
Furthermore
$\tilde g_*(D_{i,j})=\tilde g_*E_{i,j}=0$,
$\tilde g_*(D_{i,i_1}^2)=-[\Delta_{i,i_1}^{++}]$,
$\tilde g_*(E_{j,j_1})^2=-[\Delta_{j,j_1}^{--}]$,
$\tilde g_*(D_{i,i_1}^3)=[\Delta_{i,i_1}^{++}]p_{i+}^*(K_S)$,
$\tilde g_*(E_{j,j_1}^3)=[\Delta_{j,j_1}^{--}]]p_{j-}^*(K_S)$.
Below we collect the result of the push-down in ten terms according to
the factors that they contain {\it before} the push-down.
All the summands contain at least one diagonal factor
$[\Delta_{i,j}^{+-}]$ and at most two diagonal factors
$[\Delta_{i,j}^{+-}]$,
$[\Delta_{i_1,j_1}^{+-}]$. The first seven terms come from summands
containing precisely one factor $[\Delta_{i,j}^{+-}]$. So to define these
summandss we can fix $i$ and $j$. The first
term corresponds to summands not containing any exceptional divisor $D_{i,i_1}$
or $E_{j,j_1}$.
The second to seventh summands correspond in that order
to the push-downs of the terms containing only powers of $D_{i,i_1}$ with
$i_1<i$, $E_{j,j_1}$ with $j_1<j$, $D_{i,i_1}$ with
$i_1>i$, $E_{j,j_1}$ with $j_1>j$, $D_{i_1,i_2}$ with
$i\not\in \{i_1,i_2\}$ and $E_{j_1,j_2}$ with $j\not\in \{j_1,j_2\}$.
Notice that on $(\widetilde S^{n}\times \widetilde S^{m})_{*}$
the class
$[\Delta_{i,j}^{+-}]D_{i_1,i_2}E_{j_1,j_2}$ is zero for all
$i_1,i_2,j_1,j_2$ and
$[\Delta_{i,j}^{+-}]D_{i_1,i_2}D_{i_3,i_4}
=[\Delta_{i,j}^{+-}]E_{j_1,j_2}E_{j_3,j_4}=0$
unless $\{i_1,i_2\}=\{i_3,i_4\}$ (resp. $\{j_1,j_2\}=\{j_3,j_4\}$).
The last three summands correspond to terms containing
two diagonal factors $[\Delta_{i,j}^{+-}][\Delta_{i_1,j_1}^{+-}]$.
In that order they correspond to the possibilties that
$j=j_1$, that $i=i_1$ and finally that $i\ne i_1$ and $j\ne j_1$.
After a long but elementary computation we get that, if
$k\le 5$,
$\tilde g_*(\varphi^*(c_k(\Gamma_{n,m})-1)s([V^\vee(K_S)[n]^\vee)s(V[m]))$
is the part of degree $k$ of
\begin{eqnarray*}
&&\sum_{(i,j)}\Bigg(
- [\Delta_{i,j}^{+-}] p_{i+}^*(2+12\xi+42\xi^2+3s_2(S)+K_S^2)
\prod_{i_1\ne i}p_{i_1+}^*(t_{1+})
\prod_{j_1\ne j} p_{j_1-}^*(t_{1-})\\
&&\qquad\quad
+\sum_{i_1<i}
[\Delta_{i,j}^{+-}][\Delta_{i,i_1}^{++}]
p_{i+}^*(20+160\xi+90K_S)
\prod_{i_2\not \in \{i,i_1\}}p_{i_2+}^*t_{1+}
\prod_{j_1\ne j} p_{j_1-}^*t_{1-}\\
&&\qquad\quad
+\sum_{j_1<j}[\Delta_{i,j}^{+-}][\Delta_{j,j_1}^{--}]
p_{i+}^*(20+160 \xi-90K_S)
\prod_{i_1\ne i }p_{i_1+}^*t_{1+}
\prod_{j_2\not \in \{j,j_1\}}
p_{j_2-}^*t_{1-}
\\
&&\qquad\quad
+\sum_{i_1>i}
[\Delta_{i,j}^{+-}][\Delta_{i,i_1}^{++}]
p_{i+}^*(6+60\xi+20K_S)
\prod_{i_2\not \in \{i,i_1\}}p_{i_2+}^*t_{1+}
\prod_{j_1\ne j} p_{j_1-}^*t_{1-}\\
&&\qquad\quad
+\sum_{j_1>j}[\Delta_{i,j}^{+-}][\Delta_{j,j_1}^{--}]
p_{j-}^*(6+60 \xi-20K_S)
\prod_{i_1\ne i }p_{i_1+}^*t_{1+}
\prod_{j_2\not \in \{j,j_1\}}
p_{j_2-}^*t_{1-}\\
&&\qquad\quad
+\sum_{i_1\ne i}\sum_{i_2\ne i,i_2< i_1}\Bigg(
[\Delta_{i,j}^{+-}][\Delta_{i_1,i_2}^{++}]
p_{i+}^*(2+12 \xi)
p_{i_1+}^*(3+18\xi+13K_S)\\
&&\qquad\quad\cdot
\prod_{i_3\not\in\{ i,i_1,i_2\} }p_{i_3+}^*t_{1+}
\prod_{j_1\ne j}
p_{j_1-}^*t_{1-}\Bigg)
+\sum_{j_1\ne j}\sum_{j_2\ne j,j_2< j_1}\Bigg(
[\Delta_{i,j}^{+-}][\Delta_{j_1,j_2}^{--}]
p_{j-}^*(2+12 \xi)\\
&&\qquad\quad\cdot
p_{j_1-}^*(3+18\xi-13K_S)
\prod_{i_1\ne j}p_{i_1+}^*t_{1+}
\prod_{j_3\not\in\{ j,j_1,j_2\} }p_{j_3-}^*t_{1-}\Bigg)
\\
&&\qquad\quad
+\sum_{i_1<i}
[\Delta_{i,j}^{+-}][\Delta_{i_1,j}^{+-}]
p_{i+}^*(4+40\xi+4K_S)
\prod_{i_2\not \in \{i,i_1\}}p_{i_2+}^*t_{1+}
\prod_{j_1\ne j} p_{j_1-}^*t_{1-}\\
&&\qquad\quad
+\sum_{j_1<j}[\Delta_{i,j}^{+-}][\Delta_{i,j_1}^{+-}]
p_{j-}^*(4+40 \xi-4K_S)
\prod_{i_1\ne i }p_{i_1+}^*t_{1+}
\prod_{j_2\not \in \{j,j_1\}}
p_{j_2-}^*t_{1-}\\
&&\qquad\quad
+\sum_{i_1<i}\sum_{j_1\ne j}
[\Delta_{i,j}^{+-}][\Delta_{i_1,j_1}^{+-}]
p_{i+}^*(2+12 \xi)
p_{i_1+}^*(2+12\xi)
\prod_{i_2\not\in\{ i,i_1\} }p_{i_2+}^*t_{1+}
\prod_{j_2\not \in \{j,j_1\}}
p_{j_2-}^*t_{1-}\Bigg).
\end{eqnarray*}
Now we want to translate this result into the notation \ref{starnota}.
Using remark \ref{starrem} and notation \ref{starnota} we see that for
$w\in H^{4d-2k}(S^d,{\Bbb Q})^{\frak S_{d}}$ and $a\in H^*(S,{\Bbb Q})$
we have
\begin{eqnarray*}
\int_{S^d}[\Delta^{+-}_{i,j}] p_{i+}^*a\prod_{i_1\ne i}p_{i_1+}^*t_{1+}
\prod_{j_1\ne i}p_{j_1-}^*t_{1-}\cdot w
&=&\int_{S^d}(a)_2 t_{1+}^{{{\hbox{$*$}}} (n-1)} t_{1-}^{{{\hbox{$*$}}} (m-1)} \cdot w.
\end{eqnarray*}
Now assume $j\ne j_1$. Then
\begin{eqnarray*}
\int_{S^d}[\Delta^{+-}_{i,j}][\Delta^{+-}_{i,j_1}]p_{i+}^*a
\prod_{i_1\ne i}
p_{i_1+}^*t_{1+}
\prod_{j_2\not\in \{j,j_1\}}p_{j_2-}^*t_{1-}\cdot w &=&
\int_{S^d}(a)_3 t_{1+}^{{{\hbox{$*$}}} (n-1)} t_{1-}^{{{\hbox{$*$}}} (m-2)}\cdot w.
\end{eqnarray*}
We also see that
$[\Delta^{+-}_{i,j}][\Delta^{+-}_{i,j_1}]
=[\Delta^{+-}_{i,j}][\Delta^{--}_{j,j_1}]$ and
$[\Delta^{+-}_{i,j}][\Delta^{+-}_{i_1,j}]
=[\Delta^{+-}_{i,j}][\Delta^{++}_{i,i_1}]$.
If $i\ne i_1$ and $j\ne j_1$ we get similarly
\begin{eqnarray*}
\int_{S^d}[\Delta^{+-}_{i,j}][\Delta^{+-}_{i_1,j_1}]
p_{i+}^*a_1 p_{i_1+}^*a_2\!
\!\!\!\!\!\prod_{i_2\not\in \{i,i_1\}}\!\!\!\!\!
p_{i_2+}^*t_{1+}
\!\!\!\!\!\prod_{j_2\not\in \{j,j_1\}}\!\!\!\!\!p_{j_2-}^*t_{1-}\cdot w
\!\!&=&\! \int_{S^d}\!\!(a_1)_2(a_2)_2t_{1+}^{{{\hbox{$*$}}} (n-2)}
t_{1-}^{{{\hbox{$*$}}} (m-2)}\cdot w.
\end{eqnarray*}
We can translate our result into this notation and simplify it by
collecting the terms number $2,4,8$ and the terms
$3,5,9$ respectively. So we get for
$w\in H^{4d-2k}(S^d,{\Bbb Q})^{\frak S_{d}}$ with $k\le 5$:
\begin{eqnarray*}
&&\int_{S^d}\tilde g_*\big(\varphi^*\big((c(\Gamma)-1)
s([V^\vee(K_S)[n]^\vee)s(V[m])\big)\big)\cdot w\\
&&\quad =\int_{S^d}\Bigg(-nm
(2+12\xi+42\xi^2+3s_2(S)+K_S^2)_2
t_{1+}^{{{\hbox{$*$}}} (n-1)}
t_{1-}^{{{\hbox{$*$}}} (m-1)}\\
&&\qquad\quad
+{n\choose 2}m
(30+260\xi+114K_S)_3
t_{1+}^{{{\hbox{$*$}}} (n-2)}
t_{1-}^{{{\hbox{$*$}}} (m-1)}\\
&&\qquad\quad
+{m\choose 2}n(30+260\xi-114K_S)_3
t_{1+}^{{{\hbox{$*$}}} (n-1)}
t_{1-}^{{{\hbox{$*$}}} (m-2)}\\
&&\qquad\quad
+2{n\choose 2}{m\choose 2}
(2+12 \xi)_2^{ 2}
t_{1+}^{{{\hbox{$*$}}} (n-2)} t_{1-}^{{{\hbox{$*$}}} (m-2)}\\
&&\qquad\quad
+mn{n-1\choose 2}
(2+12 \xi)_2
(3+18\xi+13K_S)_2 t_{1+}^{{{\hbox{$*$}}} (n-3)}
t_{1-}^{{{\hbox{$*$}}} (m-1)}\\
&&\qquad\quad
+nm{m-1\choose 2}
(2+12 \xi)_2
(3+18\xi-13K_S)_2 t_{1-}^{{{\hbox{$*$}}} (m-3)}
t_{1+}^{{{\hbox{$*$}}} (n-1)}\Bigg)\cdot w
\end{eqnarray*}
Now we sum over all $m,n$
and keep in mind that the map
$\varphi:(\widetilde S^{n}\times \widetilde S^{m})_{*}\longrightarrow
{\text{\rom{Hilb}}}^{n}(S)\times {\text{\rom{Hilb}}}^{m}(S)$ has degree $m! n!$. So we obtain
\begin{eqnarray*}
&&d!\int_{{\text{\rom{Hilb}}}^d(S\sqcup S)}
(c(\Gamma)-1)s([V^\vee(K_S)]_1^\vee)s([V]_2)\cdot\gamma\\
&&\qquad
=\sum_{n+m=d} {d\choose n}
\int_{S^d}\tilde
g_*\big(\varphi^*\big((c(\Gamma)-1)s([V^\vee(K_S)[n]^\vee)s(V[m])\big)\big)
\cdot w\\
&&\qquad
=\sum_{n+m=d} \int_{S^d}\Bigg(- d(d-1) {d-2\choose n-1}
(2+12\xi+42\xi^2+3s_2(S)+K_S^2)_2
t_{1+}^{{{\hbox{$*$}}} (n-1)}
t_{1-}^{{{\hbox{$*$}}} (m-1)}\\
&&\qquad\quad +d(d-1)(d-2)
{d-3\choose n-2}(30+260\xi)_3
t_{1+}^{{{\hbox{$*$}}} (n-2)}
t_{1-}^{{{\hbox{$*$}}} (m-1)}\\
&&\qquad\quad
+2{d(d-1)(d-2)(d-3)}{d-4\choose n-2}
(2+12 \xi)_2^{ 2}
t_{1+}^{{{\hbox{$*$}}} (n-2)}
t_{1-}^{{{\hbox{$*$}}} (m-2)}\Bigg)\cdot w\\
&&\qquad
=\int_{S^d}\Big(-d(d-1)(2+12\xi+42\xi^2+3s_2(S)+K_S^2)_2 t_1^{{{\hbox{$*$}}} (d-2)}\\
&&\qquad\quad +d(d-1)(d-2)(30+260\xi)_3 t_1^{{{\hbox{$*$}}} (d-3)}\\
&&\qquad\quad +2{d(d-1)(d-2)(d-3)}(2+12 \xi)_2^{ 2} t_1^{{{\hbox{$*$}}} (d-4)}\Big)\cdot
w.
\end{eqnarray*}
This shows the first formula. The second follows by combining this formula with
proposition \ref{erstint}.
\end{pf}
Now we have described the intersection numbers
$\int_T s({\hbox{\rom{Ext}}}^1_q({\cal I}_{Z_1},{\cal I}_{Z_2}\otimes p^*V)\cdot\gamma$,
and are in a position to
finish our computation of the leading terms of the change of the Donaldson
invariants $\delta_{l,r}(\alpha)$.
We first want to compute a formula for the change of $\delta_{N,0}(\alpha)$ and
then compute how one has to modify this formula to get $\delta_{l,r}(\alpha)$.
The reason that the computation of $\delta_{N,0}(\alpha)$ is easier,
is the following fact:
\begin{rem} \label{pteinszwei}
Let $l,j,k$ be positive integers, $\alpha\in H^2(S,{\Bbb Q})$,
$\beta\in H^{2i}(S,{\Bbb Q})$ and
$\gamma\in H^*(S^k,{\Bbb Q})^{\frak S_{k}}$. Then we get
\begin{eqnarray}\label{pteinszw1}
\qquad\qquad\int_{S^{k+j}} (\beta)_j{{\hbox{$*$}}} \gamma\cdot
(p_1^*\alpha+\ldots +p_{k+j}^*\alpha)^l= j^{2-i}\int_{S^{k+j}}
\beta{{\hbox{$*$}}} pt^{{{\hbox{$*$}}} (j-1)}{{\hbox{$*$}}} \gamma\cdot
(p_1^*\alpha+\ldots +p_{k+j}^*\alpha)^l
\end{eqnarray}
\end{rem}
\begin{pf} For the diagonal $\Delta_{j}\subset S^j$
and a class $\alpha\in H^2(S,{\Bbb Q})$, we have
$(p_1^*\alpha+\ldots +p_j^*\alpha)\cdot
[\Delta_{j}]=jp_1^*(\alpha)[\Delta_{j}]$.
By remark \ref{starrem} the left hand side of (\ref{pteinszw1}) is equal to
\begin{eqnarray*}
\left(\int_{S^{k}} \Delta_{j}p_1^*\beta\cdot
(p_1^*\alpha+\ldots +p_{k}^*\alpha)^{2-i}\right)\left(\int_{S^{j}}\gamma\cdot
(p_1^*\alpha+\ldots +p_{j}^*\alpha)^{l+i-2}\right).
\end{eqnarray*}
So the result follows.
\end{pf}
\begin{nota}
We denote by $q_S$ the quadratic form on $H_2(S,{\Bbb Z})$
and, for $\gamma\in H^2(S,{\Bbb Q})$, we let $L_{\gamma}$ be the linear form on
$H_2(S,{\Bbb Q})$ given by
$\alpha\mapsto \<\gamma,\alpha\>$.
For a class $\beta\in H_i(S,{\Bbb Q})$ we denote
$\bar \beta:=p_1^*\check \beta+ \ldots +p_d^*\check \beta\in
H^{4-i}(S^d,{\Bbb Q})$, where
as above, $\check\beta$ is the
Poincar\'e dual of $\beta$.
Note that by lemma \ref{hilbkuenn} and definition \ref{defchange}
$\varphi^*(\widetilde\beta|_{{\text{\rom{Hilb}}}^{n}(S)\times {\text{\rom{Hilb}}}^{m}(S)})$ is the pullback of
$\bar \beta$.
Let $N=4c_2-c_1^2-3$ again be the expected dimension of $M_H(c_1,c_2)$.
\end{nota}
\begin{lem}\label{formel1}
For all $x,y\ge 0$ and all $\alpha\in H_2(S,{\Bbb Q})$ we have
\begin{eqnarray*}\int_{S^d}\xi^{{{\hbox{$*$}}} x} {pt}^{{{\hbox{$*$}}} y} 1^{{{\hbox{$*$}}} (d-x-y)}
\cdot \bar\alpha^{2d-x-2y}={(2d-x-2y)!\over 2^{d-x-y}}
q_S(\alpha)^{d-x-y}\<\xi,\alpha\>^x.\end{eqnarray*}
\end{lem}
\begin{pf}
By remark \ref{starrem} we have
\begin{eqnarray*}
\int_{S^d}\!\!\!\! \xi^{{{\hbox{$*$}}} x} {pt}^{{{\hbox{$*$}}} y}1^{{{\hbox{$*$}}}(d-x-y)}
\cdot{\bar\alpha^{N-m}}=
\int_{S^d} \!\!\! p_1^{*}\xi\cdot \ldots \cdot p_x^{*}\xi
\cdot p_{x+1}^{*}{pt}\cdot
\ldots\cdot p_{x+y}^{*}{pt}
\cdot (p_1^*\check\alpha +\ldots +p_d^*\check\alpha)^{N-m},
\end{eqnarray*}
and it is easy to see that this is just
${(2d-x-2y)!\over 2^{d-x-y}}
q_S(\alpha)^{d-x-y}\<\xi,\alpha\>^x.$
\end{pf}
\begin{thm} \label{chthm1} In the polynomial ring on
$H^*(S,{\Bbb Q})$ we have
$$\delta_{\xi,N,0}\equiv(-1)^{e_\xi}\sum_{k=0}^2
{N!\over (N-2d+2k)!(d-k)!}Q_{k}(N,d,K_S^2) L_{\xi/2}^{N-2d+2k}q_S^{d-k}\
\hbox{ modulo } L_\xi^{N-2d+6},$$ where, by convention
${1\over m!}=0$ for $m<0$ and
\begin{eqnarray*}
Q_{0}(N,d,K_S^2)&=& 1\\
Q_{1}(N,d,K_S^2)&=& 2N+2K_S^2-2d+8\\
Q_{2}(N,d,K_S^2)&=& 2N^2-4dN+4NK_S^2+21N+2d^2
-4dK_S^2 -18d+2(K_S^2)^2+18K_S^2+49.
\end{eqnarray*}
\end{thm}
\begin{pf}
Let $R_d\in H^*(S^d,{\Bbb Q})^{\frak S_{d}}$ be the class from
proposition \ref{intzwei}
with
$$d!\int\limits_{{\text{\rom{Hilb}}}^d(S\sqcup S)}
c(\Gamma)s([V^\vee(K_S)]_1^\vee)s([V]_2)\cdot \gamma=\int\limits_{S^d}
R_d\cdot {w}.$$
By remark \ref{pteinszwei}
there is a class $U'_d$ which is a linear combination of classes of the form
$\xi^{{{\hbox{$*$}}} x}{pt}^{{{\hbox{$*$}}} y} 1^{{{\hbox{$*$}}} (d-x-y)}$ with
$\int_{S^d}
R_d\cdot\bar \alpha^b=\int_{S^d}
U'_d\cdot {\bar \alpha^b}$
for all $\alpha\in H_2(S,{\Bbb Q})$.
We write $U'_d:=\sum_{x,y\ge 0}u_{x,y}\xi^{{{\hbox{$*$}}} x}{pt}^{{{\hbox{$*$}}} y} 1^{{{\hbox{$*$}}} (d-x-y)}$
and $U_d:=\sum_{x+y\le 2}u_{x,y}\xi^{{{\hbox{$*$}}} x}{pt}^{{{\hbox{$*$}}} y} 1^{{{\hbox{$*$}}} (d-x-y)}$
By definition \ref{defchange} and theorem \ref{donch1}
we see that
\begin{eqnarray*}
\delta_{\xi,N,0}(\alpha)=\sum_{i=0}^{2d}A_i\cdot\<\xi,\alpha\>\int_{S^d}
\{U'_d\}_i\bar\alpha^{2d-i},
\end{eqnarray*} where $\{\ \}_i$ denotes the part of degree $i$, and the
$A_i$ are suitable rational numbers.
Thus
$\delta_{\xi,N,0}$ modulo $L_\xi^{n-2d+6}$ is already determined by $U_d$.
As $S$ is a surface
with $p_g(S)=q(S)=0$, we have $12=12\chi({\cal O}_S)=K_S^2+c_2(S)$
and thus we can replace
$s_2(S)$ by $2K_S^2-12$.
So, using
proposition \ref{intzwei}, we obtain after a short calculation
that
\begin{eqnarray*}
U_d&=&
2^{d}1^{{{\hbox{$*$}}} d}+2^{d+1}d \cdot1^{{{\hbox{$*$}}} (d-1)}{{\hbox{$*$}}}\xi
+2^{d+1}d(d-1)1^{{{\hbox{$*$}}} (d-2)} \xi^{{{\hbox{$*$}}} 2}\\
&&\quad+2^d d(3\xi^2+K_S^2-5d+5)1^{{{\hbox{$*$}}} (d-1)} {{\hbox{$*$}}} {pt}\\
&&\quad+2^d d(d-1)(6\xi^2+2K_S^2-10d+5)1^{{{\hbox{$*$}}} (d-2)}{{\hbox{$*$}}} \xi{{\hbox{$*$}}} {pt}\\
&&\quad+2^{d-2} d(d-1)(18(\xi^2)^{2}+12\xi^2K_S^2+2(K_S^2)^{ 2}
-60d\xi^2-20dK_S^2+50d^2\\
&&\quad +15\xi^2-10K_S^2-34d-36){pt}^{{{\hbox{$*$}}} 2} 1^{{{\hbox{$*$}}} (d-2)},
\end{eqnarray*}
where we view $\xi^2$ and $K_S^2$ as integers and not as cohomology classes.
Now we apply definition \ref{defchange} and lemma \ref{formel1}.
Then, after some computation, we get the result with
$Q_0(N,d,K_S^2)$, $Q_1(N,d,K_S^2)$, $Q_2(N,d,K_S^2)$
replaced by
\begin{eqnarray*}
P_0(N,d,K_S^2,\xi^2)&=&1\\
P_1(N,d,K_S^2,\xi^2)&=&8N-26d+6\xi^2+2K_S^2+26\\
P_2(N,d,K_S^2,\xi^2)&=&18(\xi^2)^2+12(\xi^2)(K_S^2)+2(K_S^2)^2+48N\xi^2
-156d\xi^2\\
&&\quad-52dK_S^2+338d^2 +16K_S^2N+32N^2-208d
N+207\xi^2\\
&&\quad+54K_S^2+264N
-882d+508.
\end{eqnarray*}
We notice that by definition $d=(4c_2-c_1^2+\xi^2)/4$
and $N=4c_2-c_1^2-3$ and thus
$\xi^2=4d-N-3$.
Substituting this into the $P_i(N,d,K_S^2,\xi^2)$ we obtain the result.
\end{pf}
We see that the result is compatible with the conjecture
of Kotschick and Morgan. In fact it suggests a slightly sharper statement.
\begin{conj} In the polynomial ring on $H^2(S,{\Bbb Q})$ we have
$$\delta_{\xi,N,0}=(-1)^{e_\xi}\sum_{k=0}^d
{N!\over (N-2d+2k)!(d-k)!}Q_{k}(N,d,K_S^2) L_{\xi/2}^{N-2d+2k}q_S^{d-k},$$
where $Q_{k}(N,d,K_S^2)$ is a polynomial of degree
$k$ in $N,d,K_S^2$, which is independent of $S$ and $\xi$.
\end{conj}
Now we want to compute $\delta_{l,r}$ in general.
We shall see that there is reasonably simple relationship between the
formula for $\delta_{N,0}$ and that for $\delta_{l,r}$
(with $l+2r=N$),
which is however obscured by the existence of
a correction term coming from the failure of
remark \ref{pteinszwei} for classes of the form
$\bar\alpha^{k-2}\bar {pt}$ (instead of $\bar\alpha^{k}$).
\begin{lem}\label{formel2}
\begin{enumerate}
\item
For all $x,y\ge 0$, all $c\le r$ and all $\alpha\in H_2(S,{\Bbb Q})$ we
have with $m:=2d-2c-x-2y$:
\begin{eqnarray*}
\int_{S^d}
\xi^{{{\hbox{$*$}}} x} {pt}^{{{\hbox{$*$}}} y}1^{{{\hbox{$*$}}} (d-x-y)}\cdot
{\bar{pt}^c\bar\alpha^{m}}={(d-x-y)!\over
(d-x-y-c)!}{m!\over 2^{d-x-y-c}}q_S(\alpha)^{d-x-y-c}
\<\xi,\alpha\>^{x}
\end{eqnarray*}
\item
\begin{eqnarray*}
\int_{S^d}(1)_2 1^{{{\hbox{$*$}}} (d-2)}\cdot {\bar{pt}\, \bar\alpha^{2d-4}}
={4d-6\over d-1}\int_{S^d}{pt}{{\hbox{$*$}}} 1^{{{\hbox{$*$}}} (d-1)}\cdot
{\bar{pt}\, \bar\alpha^{2d-4}}
\end{eqnarray*}
\end{enumerate}
\end{lem}
\begin{pf}
(1) By remark \ref{starrem} we have
\begin{eqnarray*}
&&\int_{S^d}\xi^{{{\hbox{$*$}}} x} {pt}^{{{\hbox{$*$}}} y}\cdot{\bar{pt}^c\bar\alpha^{m}}
=
\int_{S^d} p_1^{*}\xi\ldots p_x^{*}\xi \cdot p_{x+1}^{*}{pt}\ldots
p_{x+y}^{*}{pt}
\cdot (p_1^*\check\alpha +\ldots +p_d^*\check\alpha)^{m}
\cdot (p_1^*\check{pt} +\ldots +p_d^*\check{pt})^{c}
\end{eqnarray*}
and it is elementary to show that this is just
$$
{(d-x-y)!\over
(d-x-y-c)!}{m!\over 2^{d-x-y-c}}q_S(\alpha)^{d-x-y-c}
\<\xi,\alpha\>^{x}.
$$
(2) By remark \ref{starrem} and remark \ref{pteinszwei} we have
\begin{eqnarray*}&&\int_{S^d} (1)_2 1^{{{\hbox{$*$}}} (d-2)}\cdot{\bar{pt}\,
\bar\alpha^{2d-4}}\\
&&\qquad
=\int_{S^d} [\Delta_{1,2}]
\cdot (p_1^* \check\alpha +\ldots +p_d^*\check\alpha)^{2d-4}
\cdot (p_1^*\check{pt} +\ldots +p_d^*\check{pt})\\
&&\qquad=2\int_{S^{d-2}} (p_1^*\check\alpha +\ldots
+p_{d-2}^*\check\alpha)^{2d-4}+(d-2)\int_{S^{d-1}} [\Delta_{1,2}]
\cdot (p_1^*\check\alpha +\ldots +p_d^*\check\alpha)^{2d-4}\\
&&\qquad=(4d-6)\int_{S^{d-2}}\bar\alpha^{2d-4}\\
&&\qquad=(4d-6)\int_{S^{d-1}}{pt}*1^{{{\hbox{$*$}}} (d-2)}\bar\alpha^{2d-4}\\
&&\qquad={4d-6\over
d-1}\int_{S^d}{pt}{{\hbox{$*$}}} 1^{ {{\hbox{$*$}}} (d-1)}\cdot{\bar{pt}\,\bar\alpha^{2d-4}}
\end{eqnarray*}
\end{pf}
\begin{thm} \label{chthm2}
Let $l,r$ be nonnegative integers with $l+2r=N$.
Then in the polynomial ring on $H^*(S,{\Bbb Q})$ we get
$$
\delta_{\xi,l,r}\equiv\sum_{c=0}^2{(-1)^{r-c+{e_\xi}}\over 2^{-3c+2r}}
{r\choose c}\sum_{k=c}^{2}
{l!\over (l-2d+2k)!(d-k)!} Q_{k-c,c}(l,d,K_S^2,\xi^2)
L_{\xi/2}^{l-2d+2k}q_S^{d-k}$$ modulo $ \xi^{N-2d+6},$
where
\begin{eqnarray*}
Q_{m,c}(l,d,K_S^2,\xi^2)&=& P_{m}(l,d,K_S^2,\xi^2)+21mc\ \hbox{ for }
m+c\le 2
\end{eqnarray*}
Here the $P_{i}(N,d,K_S^2,\xi^2)$ are the polynomials from the
proof of theorem \ref{chthm1}.
\end{thm}
\begin{pf}
For $i\le r$ and a class $\gamma\in H^*(S^d,{\Bbb Q})^{\frak S_{d}}$
we denote by $W_{l,r,c}(\gamma)$ the map that associates to
$\alpha\in H_2(S,{\Bbb Q})$ the number
\begin{eqnarray*}
\sum_{b=0}^{l}(-1)^{r-c+e_\xi}2^{b+2c-N}{l\choose
b}{r\choose c}\<\xi,\alpha\>^{l-b}
\int_{S^d}\gamma\cdot{\bar\alpha^b\bar{pt}^c}.
\end{eqnarray*}
Let $R_d$, $U_d\in H^*(S^d,{\Bbb Q})^{\frak S_d}$ be the classes from the proof of
\ref{chthm1}.
By thm \ref{donch1} and proposition \ref{intzwei} we get
$$\delta_{l,r}\equiv\sum_{c=0}^r
W_{l,r,c}(R_d).$$
By lemma \ref{pteinszwei} we see that $W_{l,r,0}(R_d)\equiv W_{l,r,0}(U_d)$
modulo $L_\xi^{N-2d+6}$.
Furthermore we get modulo $L_\xi^{N-2d+6}$
\begin{eqnarray*}
W_{l,r,k}(R_d)&\equiv& 0 \ \hbox{ for } k>2\\
W_{l,r,2}(R_d)&\equiv & W_{l,r,2}(2^d 1^{{{\hbox{$*$}}} d})\\
W_{l,r,1}(R_d)&\equiv& W_{l,r,1}(\bar R_d),
\end{eqnarray*}
where $\bar R_d=t_1^{{{\hbox{$*$}}} d}-5\cdot 2^{d-2}d(d-1)(1)_21^{{{\hbox{$*$}}} (d-2)}$.
By lemma \ref{formel2}(2)
$W_{l,r,1}(\bar R_d)\equiv W_{l,r,1}(\bar U_d)$ where
$$\bar U_d=
t_1^{{{\hbox{$*$}}} d}-5\cdot 2^{d-2}d(4d-6){pt}{{\hbox{$*$}}} 1^{{{\hbox{$*$}}}(d-1)}.$$
Now the result follows by applying lemma \ref{formel2}(1)
and some computation.
\end{pf}
\begin{rem}
Using $\xi^2=4d-2r-l-3$ we get equivalently
\begin{eqnarray*}
Q_{0,0}(l,d,K_S^2,\xi^2)&=&Q_{0,1}(l,d,K_S^2,\xi^2)=
Q_{0,2}(l,d,K_S^2,\xi^2)=1\\
Q_{1,0}(l,d,K_S^2,\xi^2)&=&2l-2d-12r+2K_S^2+8\\
Q_{2,0}(l,d,K_S^2,\xi^2)&=&72r^2-24rl+24dr-24K_S^2r+2l^2-4dl+4K_S^2l\\
&&\quad +2d^2-4dK_S^2+2(K_S^2)^2-198r+21l-18d+18K_S^2+49\\
Q_{1,1}(l,d,K_S^2,\xi^2)&=&2 l - 2 d - 12 r +2K_S^2+ 29.\\
\end{eqnarray*}
\end{rem}
\begin{rem} We see that our results contain as a special case
the formulas for the change for $d\le 2$.
In the case that $d=3$ we notice that the spaces $X_*$ and
$Y_{*}$ (for $X$ and $Y$ schemes with a natural morphism
to $S^{(d)}$ and $S^{(n)}\times
S^{(m)}$ respectively)
just coincide with $X$ respectively $Y$. Therefore our computations are valid
on the whole of ${\text{\rom{Hilb}}}^d(S\sqcup S)$ and our methods will also give complete
formulas for the change of the Donaldson invariants in case $d=3$. We
however do not carry out the elementary but long computations here.
\end{rem}
|
1994-11-23T06:20:13 | 9410 | alg-geom/9410001 | en | https://arxiv.org/abs/alg-geom/9410001 | [
"alg-geom",
"hep-th",
"math.AG"
] | alg-geom/9410001 | Victor Batyrev | Victor V. Batyrev and Dimitrios I. Dais | Strong McKay Correspondence, String-theoretic Hodge Numbers and Mirror
Symmetry | 42 pages, Latex | null | null | null | null | In the revised version of the paper, we correct misprints and add some new
statements.
| [
{
"version": "v1",
"created": "Tue, 4 Oct 1994 20:42:03 GMT"
},
{
"version": "v2",
"created": "Tue, 22 Nov 1994 22:27:36 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Batyrev",
"Victor V.",
""
],
[
"Dais",
"Dimitrios I.",
""
]
] | alg-geom | \section{Introduction}
Throughout this paper by an {\em algebraic variety} (or simply {\em
variety}) we mean an integral, separated algebraic scheme over ${\bf C}$.
By a {\em compact algebraic variety} we mean the representative of a
complete variety within the analytic category. The
{\em singular} {\em locus} of an algebraic variety
$X$ is denoted by ${\rm Sing}\,X$. The
words {\em smooth variety} and {\em manifold} are used interchangeably. By
the word {\em singularity} we sometimes intimate a singular point and
sometimes
the underlying space of a neighbourhood or the germ of a singular point,
but its meaning will be always clear from the context. Following Danilov
\cite{danilov}, \S 13.3, we shall say that an $x \in X$ is a
{\em toroidal singularity} of $X$, if there is an analytic isomorphism
between the germ $(X, x)$ and the germ corresponding to the toric singularity
$({\bf A}_{\sigma}, p_{\sigma})$ (see also \S 4).
Our main tool will be certain algebraic varieties
with special Gorenstein singularities, primarily having in mind
the Calabi-Yau varieties. A {\em Calabi-Yau
variety} is defined to be a normal projective algebraic variety $X$
with trivial canonical sheaf ${\omega}_X$ and $H^i(X, {\cal O}_X) = 0$,
$0 < i < {\rm dim}_{\bf C}\,X$, which, in addition, can have at most
{\em canonical Gorenstein singularities}. (For the notion of
{\em canonical singularity} we refer to \cite{reid1}.) If ${\rm Sing}\,X =
\emptyset$, then $X$ is called, as usual, {\em Calabi-Yau manifold}.
In this paper we shall attempt to realize some Hodge-theoretical
invariants used by physicists for
singular varieties being related to the mirror symmetry phenomenon.
The necessity of working with singular varieties becomes unavoidable
from the fact that, in many examples of pairs $X$, $X^*$ of mirror symmetric
Calabi-Yau manifolds, at least one of the two manifolds $X$ or $X^*$ is
obtained as {\em a crepant desingularization} of a singular
Calabi-Yau variety \cite{batyrev1,morrison}.
Here, by a crepant desingularization of a
Gorenstein variety $Z$, we mean a birational morphism $\pi\,: \,
Z' \rightarrow Z$, such that $\pi^*(\omega_Z) \cong \omega_{Z'}$,
where $\omega_Z$ and $\omega_{Z'}$ denote the canonical sheaves
on $Z$ and $Z'$ respectively. 3-dimensional Gorenstein
quotient singularities and their crepant desingularizations
have been studied in
\cite{bertin,ito1,ito2,markushevich,markushevich2,roan0,roan1,roan2,roan3,roan-y
au,reid2,yau}.
The most known physical cohomological invariant of
singular varieties obtained as
quotient-spaces of certain compact manifolds by actions of finite groups
is the so called
{\em physicists Euler number} \cite{dixon}. It has been investigated
by several mathematicians in \cite{atiyah,got2,hirzebruch,roan,roan1,reid2}.
Let $X$ be a smooth simplectic manifold over ${\bf C}$ having
an action of a finite group $G$ such that the simplectic
volume form $\omega$ is $G$-invariant. For any $g \in G$, we set $X^g
: = \{ x \in X \mid g(x) = x \}$.
Physicists have proposed the following formula for computing
the {\em orbifold Euler number} \cite{dixon}:
\begin{equation}
e(X,G) = \frac{1}{\mid G \mid} \sum_{gh = hg} e(X^g \cap X^h).
\label{euler.phys}
\end{equation}
It is expected that $e(X,G)$ coincides with the usual Euler number
$e(\widehat{X/G})$ of a crepant desingularization $\widehat{X/G}$ of
the quotient space $X/G$ provided such a desingularization exists.
For a volume-invariant linear action on ${\bf C}^n$ of a finite group $G$,
the corresponding conjectural local properties
of crepant desingularizations were formulated by M. Reid \cite{reid2}:
\begin{conj}
{\rm (generalized McKay correspondence)} Let $X = {\bf C}^n$,
$G$ an arbitrary finite subgroup in $SL(n, {\bf C})$. Assume that
$Y = X/G$ admits a crepant desingularization $\pi \,: \, \hat{Y}
\rightarrow Y$. Then $H^*(\pi^{-1}(0), {\bf C})$ has a basis consisting of
classes of algebraic cycles $Z_c \subset \pi^{-1}(0)$ which are
in $1$-to-$1$ correspondence with conjugacy classes $c$ of $G$.
In particular, we obtain for the Euler number
\[ e(\hat{Y}) = e(\pi^{-1}(0))
= \# \{ \mbox{\rm conjugacy classes in $G$} \}. \]
\label{general}
\end{conj}
\begin{rem}
{\rm For $n =2$ an one-to-one correspondence between the nontrivial
irreducible representations of a subgroup $G \subset SL(2, {\bf C})$ and
the irreducible components of $\pi^{-1}(0)$ was discovered by McKay
\cite{mckay} and investigated in \cite{gonzalez,knorrer,sandro-infirri}. }
\end{rem}
Our first purpose is to use some stronger version of Conjecture \ref{general}
in order to give an analogous
interpretation for the {\em physicists Hodge numbers} $h^{p,q}(X,G)$
of orbifolds considered by C. Vafa \cite{vafa} and E. Zaslow \cite{zaslow}.
Let $X$ be a smooth compact
K\"ahler manifold of dimension $n$ over ${\bf C}$ being equipped with
an action of a finite group $G$, such that $X$ has a $G$-invariant
volume form. Let $C(g) : = \{ h \in G \mid hg = gh\}$. Then the action
of $C(g)$ on $X$ can be restricted on $X^g$. For any point $x \in X^g$, the
eigenvalues of $g$ in the holomorphic tangent space $T_x$ are roots of unity:
\[ e^{2\pi \sqrt{-1} \alpha_1}, \ldots, e^{2\pi \sqrt{-1} \alpha_d} \]
where $0 \leq \alpha_j <1 $ $(j =1, \ldots, d)$ are locally constant
functions on $X^g$ with values in ${\bf Q}$.
We write $X^g = X_1(g) \cup \cdots \cup
X_{r_g}(g)$, where $X_1(g), \ldots, X_{r_g}(g)$ are the smooth connected
components of $X^g$. For each $ i \in \{1, \ldots, r_{g} \}$,
the {\em fermion shift number} $F_i(g)$ is defined
to be equal to the value of $\sum_{1 \leq j \leq n}
{\alpha_j}$ on the connected component $X_i(g)$.
We denote by $h^{p,q}_{C(g)}(X_i(g))$ the dimension of the subspace of
$C(g)$-invariant elements in $H^{p,q}(X_i(g))$. We set
\[ h^{p,q}_g(X,G) : = \sum_{i =1}^{r_g} h^{p - F_i(g),q -
F_i(g)}_{C(g)}(X_i(g)).\]
The {\em orbifold Hodge numbers} of $X/G$ are defined by the
formula (3.21) in \cite{zaslow}:
\begin{equation}
h^{p,q}(X,G): = \sum_{\{ g \}} h^{p,q}_g(X,G)
\label{phys.form}
\end{equation}
where $\{g\}$ runs over the conjugacy classes of $G$, so that $g$ represents
$\{ g \}$. As we shall see in Corollary \ref{c.des},
these numbers coincide with
the {\em usual} Hodge numbers of a crepant desingularization of $X/G$.
One of our next intentions is to convince the reader of the existence
of some {\em new cohomology theory} $H^*_{\rm st}(X)$ of more general
algebraic varieties $X$ with mild Gorenstein singularities.
Since this cohomology is inspired from the string
theory, we call $H^*_{\rm st}(X)$ the {\em string cohomology of} $X$.
For compact varieties $X$, we expect
that the string cohomology groups $H^*_{\rm st}(X)$ will satisfy the
Poincar\'{e} duality and will be endowed with a pure Hodge structure.
The role of crepant resolutions for the string-cohomology
$H^*_{\rm st}(X)$ is analogous to that one of small resolutions
for the intersection cohomology $IH^*(X)$ with middle perversity.
Physicists compute orbifold Hodge numbers without using
crepant desingularizations. From mathematical point of view, however,
crepant desingularizations seem to be rather helpful, although they
have some disadvantages. Firstly, they might not exist
(at least in dimension $\geq 4$) and ,secondly, even if they
exist, they might be not unique.
The consistency of the physical approach naturally suggests
the formulation of the following
conjecture (which can be verified for the toric case by
Theorem \ref{invariants}):
\begin{conj}
Hodge numbers of smooth crepant resolutions do not depend on the choice of
such a resolution.
\end{conj}
Let us briefly review the rest of the paper. In Section 2, we consider an
example showing the importance of
the ``physical Hodge numbers'' in connection with
the mirror duality.
In Section 3, we remind basic properties of
$E$-polynomials. In Section 4, we study the Hodge structure of
the exceptional loci of local crepant toric resolutions. In Section 5,
we formulate
the conjecture concerning the strong McKay correspondence and we prove that
it is true for $2$- and $3$-dimensional Gorenstein quotient singularities,
as well as for abelian Gorenstein quotient singularities of arbitrary
dimension.
This correspondence will be used in Section
6 in order to give the formal definition of
the string-theoretic Hodge numbers and to study their main properties.
In Section
7, we give some applications relating to the mirror symmetry and
formulate the string-theoretic Hodge diamond-mirror conjecture
for Calabi-Yau complete intersections in $d$-dimensional
toric Fano varieties. This conjecture will be proved
in Section 8 for the case of $\Delta$-regular hypersurfaces in
toric Fano varieties ${\bf P}_{\Delta}$ which are defined by
$d$-dimensional reflexive simplices $\Delta$ (for arbitrary $d$); it gives
the mirror duality {\em for all string-theoretic Hodge numbers}
$h^{p,q}_{\rm st}$ of abelian quotients of
Calabi-Yau Fermat-type hypersurfaces which are embedded in $d$-dimensional
weighted projective spaces. This duality agrees with the mirror
construction proposed by Greene and Plesser \cite{greene0,greene1,greene}
and the polar duality of reflexive polyhedra proposed in \cite{batyrev1}.
\bigskip
{\bf Acknowledgements.} We would like to express our thanks to D. Cox, A.
Dimca,
H. Esnault, L. G\"ottsche, Yu. Ito, D. Kazhdan, M. Kontsevich,
D. Markushevich,
Yu. Manin, K. Oguiso,
M. Reid, A. V. Sardo-Infirri, D. van Straten and E. Viehweg
for fruitful discussions, suggestions and remarks.
\section{Hodge numbers
and mirror symmetry}
At the beginning we shall state some introductory questions
which could be considered also
as another motivation for the paper. These questions are related
to singular varieties of dimension $\geq 4$ which arose
as examples of the mirror duality
\cite{batyrev1,candelas,greene,schimmrigk1,schimmrigk2,schimmrigk3}.
If two $d$-dimensional Calabi-Yau manifolds $X$ and $Y$
form a mirror pair, then for all $0 \leq p, q \leq d$
their Hodge numbers must satisfy
the relation
\begin{equation}
h^{p,q}(X) = h^{d-p,q}(Y).
\label{duality}
\end{equation}
However, it might happen that a mirror pair consists of two
$d$-dimensional Calabi-Yau varieties $X$ and $Y$ having singularities. In
this case, the duality (\ref{duality}) is expected to take place
not for $X$ and $Y$
themselves, but for their crepant desingularizations $\hat{X}$ and
$\hat{Y}$, if such desingularizations exist. Using the existence of smooth
crepant desigularizations of Gorenstein toroidal singularities
in dimension $\leq 3$, one can check the relations (\ref{duality})
for many examples of $3$-dimensional mirror pairs
\cite{batyrev1,roan0}. But there are difficulties to prove
(\ref{duality}) {\em for all} $p,q$ and $d \geq 4$, even if
one heuristically
knows a mirror pair of singular Calabi-Yau varieties, for instance, as
an orbifold. The main problem in dimension
$d \geq 4$ is due to the existence of many {\em terminal} Gorenstein
quotient singularities, i.e., to singularities
which obviously do not admit any crepant resolution.
In \cite{batyrev1}, the first author constructed the so called {\em maximal
projective crepant partial desingularizations} (MPCP-desingularizations)
of singular Calabi-Yau hypersurfaces in toric varieties.
Using MPCP-desingularizations, the relation
(\ref{duality}) was proved for $h^{1,1}$ and $h^{d-1,1}$ in \cite{batyrev1}.
We shall show later that MPCP-desingularizations are sufficient to
establish (\ref{duality}) for $q=1$ and arbitrary $p$ in the case of
$d$-dimensional Calabi-Yau hypersurfaces in toric varieties (see
\ref{p1}, \ref{p1cor}).
Although MPCP-desingularizations always exist, it is important
to stress that they are not sufficient
to prove (\ref{duality}) for all $p,q$, and $d \geq 4$, because of the
following two properties which can be easily illustrated
by means of various examples:
\begin{itemize}
\item In general, a MPCP-desingularization of a
Gorenstein toroidal singularity is
not a manifold, but a variety with Gorenstein terminal
abelian quotient singularities.
\item Cohomology and Hodge numbers of different MPCP-desingularizations
might be different.
\end{itemize}
\bigskip
It turns out that, even for $3$-dimensional Calabi-Yau manifolds,
the mirror construction inspired from the superconformal field theory demands
consideration of higher dimensional manifolds with singularities
\cite{batyrev-borisov,candelas,schimmrigk1,schimmrigk2,schimmrigk3}.
In this case, we again meet difficulties if we wish to obtain analogues
of the duality in (\ref{duality}). Let us explain them for the example
which was discussed in \cite{candelas}.
Let $E_0$ be the unique elliptic curve having an authomorphism of order
$3$ with $3$ fixed points $p_0, p_1, p_2 \in E_0$. We consider
the natural
diagonal action of $G \cong {\bf Z}/3{\bf Z}$ on $Z = E_0\times E_0 \times
E_0$. The quotient $X = Z/G$ is a singular Calabi-Yau variety whose
smooth crepant resolution $\hat{X}$ has Hodge numbers
\[ h^{1,1}(\hat{X}) = 36,\;\; h^{2,1} (\hat{X}) = 0. \]
As the mirror partner of $X$, it has been proposed the $7$-dimensional
orbifold $Y$ obtained from the quotient of the Fermat-cubic
$(W\, :\, z_0^3 + \cdots z_8^3 = 0)$ in
${\bf P}^8$ by the order 3 cyclic group
action defined by the matrix
\[ g = {\rm diag}( 1,1,1,e^{2\pi \sqrt{-1}/3},e^{2\pi \sqrt{-1}/3} ,e^{2\pi
\sqrt{-1}/3},
e^{-2\pi \sqrt{-1}/3},e^{-2\pi \sqrt{-1}/3},e^{-2\pi \sqrt{-1}/3}). \]
By standard methods, counting $G$-invariant monomials in the
Jacobian ring, one immediately verifies that $h^{4,3}(Y) = 30$.
One could expect that a crepant resolution of singularities of $Y$
along the $3$ elliptic curves
\[ C_0 = \{ z_3 = \cdots = z_8 = 0 \} \cap Y, \]
\[ C_1 = \{ z_0 = z_1= z_2 = z_6 = z_7 = z_8 = 0 \} \cap Y, \]
\[ C_2 = \{ z_0 = \cdots = z_5 = 0 \} \cap Y \]
would give the missing $6$ dimensions to $h^{4,3}(Y)$ in order to obtain
$36$ (this would be the analogue of (\ref{duality})).
But also this hope must be given up because of a very simple reason: all
singularities along $C_0, C_1, C_2$ are terminal, i.e.,
they do not
admit any smooth crepant resolution.
\begin{ques}
What could be that suitable mathematical reasoning which would give
back the missing $6$
in the above example?
\end{ques}
{}From the viewpoint of physicists, one should consider $Y$ as
an orbifold quotient of $W$ by $G = \{ e, g, g^{-1} \}$.
By physicists' formula (\ref{phys.form}),
\[ h^{4,3}(W,G) = h^{4,3}_e(W,G) + h^{4,3}_g(W,G) +
h^{4,3}_{g^{-1}}(W,G). \]
It is clear that $h^{4,3}_g(W,G) = h^{4,3}_{g^{-1}}(W,G)$ and
$h^{4,3}_e(W,G) = h^{4,3}(Y) = 30$. So, it remains to compute
$h^{4,3}_g(W,G)$. Notice that $W^g = C_0 \cup C_1 \cup C_2$; i.e.,
$W_i(g) = C_i$ $( i =0,1,2)$. Moreover, $g$ acts on the tangent
space $T_w$ of a point $w \in W^g$ by the matrix
\[ {\rm diag}( 1,e^{2\pi \sqrt{-1}/3},e^{2\pi \sqrt{-1}/3} ,e^{2\pi
\sqrt{-1}/3},
e^{-2\pi \sqrt{-1}/3},e^{-2\pi \sqrt{-1}/3},e^{-2\pi \sqrt{-1}/3}). \]
Therefore, $F_i(g) = 3$ $({\rm for}\; i=0,1,2)$.
So $h^{4,3}_g(W,G) = \sum_{i =0}^2 h^{4-F_i(g),3-F_i(g)}(C_i) = 3$
and the required $6$ is indeed present!
\begin{ques}
Is there a local version of the formula {\rm (\ref{phys.form})}
for the underlying space of
a quotient singularity extending that of \ref{general}?
\end{ques}
We shall answer both questions in Sections 5 and 6.
\section{$E$-polynomials of algebraic varieties}
In this section we recall some basic properties of the {\em
$E$-polynomials} of (not necessarily smooth or compact) {\em algebraic
varieties}. $E$-polynomials are defined by means of the mixed Hodge
structure (MHS) of rational
cohomology groups with compact supports \cite{dan.hov}.
As we shall see below, these polynomials obey to similar additive and
multiplicative laws as those of the {\em usual} Euler characteristic, which
enables us to compute all the Hodge numbers coming into question in a
very convenient way.
As Deligne shows in \cite{deligne}, the cohomology
groups $H^k(X, {\bf Q})$ of a (not necessarily smooth or compact) algebraic
variety $X$ carry a natural MHS. By similar methods, one can
determine a canonical MHS by considering $H^k_c(X, {\bf Q})$, i.e., the
cohomology groups {\em with compact supports}. Compared with
$H^k(X, {\bf Q})$, the MHS on $H^k_c(X, {\bf Q})$ presents some additional
technical advantages. One of them is the existence of the
following exact sequence:
\begin{prop}
Let $X$ be an algebraic variety and $Y \subset X$ a closed subvariety. Then
there is an exact sequence
\[ \ldots \rightarrow H^k_c(X \setminus Y, {\bf Q}) \rightarrow
H^k_c(X, {\bf Q}) \rightarrow H^k_c(Y, {\bf Q}) \rightarrow \cdots \]
consisting of $MHS$-morphisms.
\label{exact-s}
\end{prop}
\begin{dfn} {\rm Let $X$ be an algebraic variety over ${\bf C}$
which is not necessarily compact or smooth. Denote by
$h^{p,q}(H^k_c(X, {\bf C}))$ the dimension of the $(p,q)$-Hodge component of
the $k$-th cohomology with compact supports.
We define:
\[e^{p,q}(X) := \sum_{k \geq 0} (-1)^k h^{p,q}(H_c^k(X, {\bf C})). \]
The polynomial
\[ E(X; u,v) : = \sum_{p,q} e^{p,q}(X) u^p v^q \]
is called the {\em E-polynomial} of $X$}.
\label{e-poly}
\end{dfn}
\begin{rem}
{\rm If the Hodge structure of $X$ in \ref{e-poly} is
{\em pure}, then the coefficients $e^{p,q}(X)$ of the E-polynomial
of $X$ are related to the usual Hodge numbers
by $e^{p,q}(X) = (-1)^{p+q}h^{p,q}(X)$. In fact, the E-polynomial
(in the general case) can be regarded as a notional refinement of the
{\em virtual Poincar\'{e} polynomial} $E(X; -u,-u)$ and, of course,
of the {\em Euler characteristic with compact supports} $e_c(X): =
E(X, -1,-1)$. It should be also mentioned, that $e_c(X) = e(X)$, i.e.,
that $e_c(X)$ is equal to the {\em usual} Euler characteristic of $X$
(cf. \cite{fulton}, pp. 141-142). }
\end{rem}
Using Proposition \ref{exact-s}, one obtains:
\begin{prop}
Let $X$ be a disjoint union of locally closed subvarieties $X_i$ $(i \in I)$.
Then
\[E(X;u,v) = \sum_{i \in I} E(X_i;u,v). \]
\label{proper1}
\end{prop}
\begin{dfn}
{\rm Let $X$ be a disjoint union of locally closed
subvarieties $X_i$ $(i \in I)$. We shall write $X_{i'} < X_i$, if
$X_{i'} \neq X_i$ and $X_{i'}$ is contained in the Zariski closure
$\overline{X}_i$ of $X_i$.}
\end{dfn}
\begin{prop}
For any $i_0 \in I$, one has
\[ E(X_{i_0}; u,v) = \sum_{k \geq 0} (-1)^k \sum_{X_{i_k} < \cdots <
X_{i_1} < X_{i_0} } E(\overline{X}_{i_k}; u,v). \]
\label{stra1}
\end{prop}
\noindent
{\em Proof.} By \ref{proper1}, we get
\[ E(X_{i_0}; u,v) = E(\overline{X}_{i_0}; u,v) -
E(\overline{X}_{i_0} \setminus X_{i_0}; u,v). \]
Moreover,
\[ E(\overline{X}_{i_0} \setminus X_{i_0}; u,v) =
\sum_{X_{i_1} < X_{i_0}} E(X_{i_1}; u,v). \]
Repeating the same procedure for $i_1 \in I$, we obtain:
\[ E(X_{i_1}; u,v) = E(\overline{X}_{i_1}; u,v) -
E(\overline{X}_{i_1} \setminus X_{i_1}; u,v), \]
\[ E(\overline{X}_{i_1} \setminus X_{i_1}; u,v) =
\sum_{X_{i_2} < X_{i_1}} E(X_{i_2}; u,v), \;\; \; \mbox{\rm etc. } \dots \]
This leads to the claimed formula. \hfill $\Box$
\bigskip
Applying the K\"unneth formula, we get:
\begin{prop}
Let $\pi\,: \, X \rightarrow Y$ be a locally trivial fibering in Zariski
topology. Denote by $F$ the fiber over a closed point in $Y$. Then
\[ E(X;u,v) = E(Y;u,v) \cdot E(F;u,v). \]
\label{proper2}
\end{prop}
We shall use \ref{proper1} and \ref{proper2} in the following situation.
Let $\pi\,:\, {X}' \rightarrow X$ be a proper birational morphism of
algebraic varieties ${X}'$ and $X$. Let us further assume that ${X}'$ is
smooth and $X$ has a stratification by locally closed subvarieties
$X_i$ $(i \in I)$, such that each $X_i$ is smooth and
the restriction of $\pi$ on $\pi^{-1}(X_i)$ is a locally trivial
fibering over $X_i$ in Zariski topology. Using
\ref{proper1} and \ref{proper2}, we can compute all Hodge
numbers of ${X}'$ as follows:
\begin{prop}
Let $F_i$ $(i \in I)$ denote the fiber over a closed point of $X_i$. Then
\[ E({X}'; u,v) = \sum_{i \in I} E(X_i; u,v) \cdot E(F_i;u,v). \]
\label{formula}
\end{prop}
We shall next deal with the case in which
$\pi\,:\, \tilde{X} \rightarrow X$ represents
a crepant resolution of
singularities of an algebraic variety $X$ having only
Gorenstein singularities. The problem of main interest
is to characterize the $E$-polynomials $E(F_i; u,v)$ in terms of
singularities of $X$ along the $X_i$'s. This problem will be
solved in the case when $X$ has Gorenstein toroidal or quotient
singularities.
\section{Local crepant toric resolutions}
We shall compute here the $E$-polynomials of the fibers of
crepant toric resolution mappings of Gorenstein toric
singularities by using
their combinatorial description in terms of convex cones.
It is assumed that the reader is familiar with the theory of toric
varieties as it is presented, for instance, in the expository article
of Danilov \cite{danilov}, or in the books
of Oda \cite{oda} and Fulton \cite{fulton}.
Let $M$, $N$ be two free abelian groups of rank $d$,
which are dual to each other, and let
$M_{\bf R}$ and $N_{\bf R}$ be their real scalar extensions.
The type of every $d$-dimensional
Gorenstein toroidal singularity can be described combinatorially
by a $d$-dimensional cone $\sigma = \sigma_{\Delta}
\subset N_{\bf R}$ which supports
a $(d-1)$-dimensional lattice polyhedron $\Delta \subset N_{\bf R}$
\cite{reid1}. This
lattice polyhedron $\Delta$ can be defined as $\{ x \in \sigma
\mid \langle x, m_{\sigma} \rangle = 1 \}$ for some uniquely
determined element
$m_{\sigma} \in M$. Let
$\check{\sigma} \subset M_{\bf R}$ be dual to $\sigma$ and
set ${\bf A}_{\sigma} := {\rm Spec}\, {\bf C}
\lbrack \check{\sigma} \cap M \rbrack$. Then ${\bf A}_{\sigma}$ is a
$d$-dimensional affine toric variety with only Gorenstein singularities.
We denote by $p= p_{\sigma}$ the unique torus invariant closed point in
${\bf A}_{\sigma}$.
\begin{dfn}
{\rm A finite collection ${\cal T} = \{ \theta \}$ of simplices with
vertices in $\Delta \cap N$ is called a {\em triangulation}
of $\Delta$ if the
following properties are satisfied:
(i) if $\theta'$ is a face of $\theta \in {\cal T}$, then $\theta' \in
{\cal T}$;
(ii) the intersection of any two simplices $\theta_1', \theta_2' \in
{\cal T}$ is either empty, or a common face of both of them;
(iii) $\Delta = \bigcup_{\theta \in {\cal T}} \theta$.
}
\end{dfn}
Every triangulation ${\cal T}$ of $\Delta$ gives rise to a
partial crepant toric
desingularization
$\pi_{\cal T}\, : \, X_{\cal T} \rightarrow {\bf A}_{\sigma}$ of
${\bf A}_{\sigma}$, so that
$X_{\cal T}$ has at most abelian quotient Gorenstein singularities.
\begin{dfn}
{\rm A simplex $\theta \subset \Delta \subset
\{ x \in N_{\bf R} \mid \langle x, m_{\sigma} \rangle =1 \}$
is called {\em regular} if its vertices form a part of a
${\bf Z}$-basis of $N$. }
\end{dfn}
\noindent
It is known (see, for instance, \cite{oda}, Thm. 1.10, p.15) that $X_{\cal T}$
is smooth
if and only if all simplices in ${\cal T}$
are regular.
\begin{theo}
Assume that $\Delta$ admits a triangulation ${\cal T}$ into regular
simplices; i.e., that the corresponding toric variety $X_{\cal T}$
in the crepant resolution
\[ \pi_{\cal T}\, : \, X_{\cal T} \rightarrow {\bf A}_{\sigma} \]
is smooth. Then $F = \pi_{\cal T}^{-1}(p)$ can be stratified
by affine spaces.
\label{stratification}
\end{theo}
\noindent
{\em Proof.} Let $\theta_0$ be an arbitrary $(d-1)$-dimensional simplex
in ${\cal T}$ with vertices $ e_1, \ldots, e_d$.
Choose an 1-parameter multiplicative
subgroup $G_{\omega} \subset ({\bf C}^*)^d$
whose action on ${\bf A}_{\sigma}$
is defined by a weight-vector $\omega \in \sigma \cap N$, so that
$\omega = \omega_1 e_1 + \cdots + \omega_d e_d$, where
$\omega_1, \ldots, \omega_d$ are positive integers. The action of
$G_{\omega}$ on ${\bf A}_{\sigma}$ extends naturally to an action
on $X_{\cal T}$.
If $\{ \theta_0, \theta_1, \ldots, \theta_s \}$ denotes the set of all
$(d-1)$-dimensional simplices in ${\cal T}$, then
$\sigma = \bigcup_{i =0}^s \sigma_{\theta_i}$, and
$X_{\cal T}$ is canonically covered by the corresponding
$G_{\omega}$-invariant open subsets $U_0, \ldots, U_s$, so that
$U_i \cong {\bf C}^d$. Denote by $p_i$ $(i = 0,1, \ldots, s)$ the
unique torus invariant point in $U_i$. We assume that $\omega$ has been
already chosen in such a way, that $p_i$ is the unique $G_{\omega}$-invariant
point in $U_i$. We consider a multiplicative parameter $t$ on $G_{\omega}$
for which the action of $G_{\omega}$ on $U_0$ is defined as follows:
\[ t \cdot (x_1, \dots, x_d): = (t^{\omega_1}x_1, \ldots, t^{\omega_d}x_d). \]
Furthermore, we set:
\[ X_i : = \{ x = (x_1, \ldots, x_d) \in X_{\cal T} \mid \lim_{t \rightarrow
\infty}
t(x) = p_i \}. \]
Since $\pi_{\cal T}(p_i) = p$, we have $X_i \subset F$. By compactness of $F$,
for every point $x \in F$, there exists $\lim_{t \rightarrow \infty}
t(x)$ which is a $G_{\omega}$-invariant point; i.e.,
$\lim_{t \rightarrow \infty}
t(x) = p_i$ for some $i$ $(0 \leq i \leq s)$. So
$\bigcup_{i =0}^s X_i = F$. Obviously, $X_i \subset U_i$.
Moreover, $X_i \cap X_j = \emptyset$ for $i \neq j$.
If we now choose appropriate torus coordinates
$y_1, \ldots, y_d$ on $U_i$, so that $G_{\omega}$ acts by
\[ t\cdot (y_1, \ldots, y_k, y_{k+1}, \ldots, y_d) =
(t^{\lambda_1} y_1, \ldots, t^{\lambda_k} y_k, t^{\lambda_{k+1}}y_{k+1},
\ldots, t^{\lambda_d}y_d ) \]
with $\lambda_1, \ldots, \lambda_k$ positive and
$\lambda_{k+1}, \ldots, \lambda_d$ negative, $X_i$ is defined by
the equations $y_1 = \ldots = y_k = 0$. Therefore, $X_i$ is isomorphic to
an affine space.
\hfill $\Box$
\medskip
Let $l(k\Delta)$ denote the number of lattice points of $k\Delta$.
Then the {\em Ehrhart power series}
\[ P_{\Delta}(t): = \sum_{k \geq 0} l(k\Delta) t^k \]
can be considered as a numerical characteristic
of the toric singularity at $p_{\sigma}$.
It is well-known (see, for instance,
\cite{batyrev1}, Thm 2.11, p.356) that $P_{\Delta}(t)$ can be
always written in the form:
\[ P_{\Delta}(t) = \frac{\psi_0(\Delta) + \psi_1(\Delta)t + \cdots
+ \psi_{d-1}(\Delta)t^{d-1}}{(1-t)^d}, \]
where $\psi_0(\Delta) = 1$
and $\psi_1(\Delta), \ldots, \psi_{d-1}(\Delta)$ are certain
nonnegative integers.
\begin{theo}
Let $\Delta$ be as in {\rm \ref{stratification}}, and
$F = \pi_{\cal T}^{-1}(p)$.
Then the cohomology groups
$ H^{2i}(F, {\bf C}), \;\; i = 0, \ldots, d-1$
are generated by the $(i,i)$-classes of algebraic cycles, and
$H^{j}_c(F, {\bf C}) = 0$ for odd values of $j$ . Moreover,
$h^{i,i}(F) = \psi_i(\Delta) \;\; i = 0, \ldots, d-1$.
In particular, the dimensions $h^{i,i}(F) = {\rm dim}\, H^{2i}(F,
{\bf C})$ $( 0 \leq i \leq d-1)$
do not depend on the choice of the triangulation ${\cal T}$.
\label{invariants}
\end{theo}
\noindent
{\em Proof.} The first statement follows immediately from
Theorem \ref{stratification}.
Since $F$ is compact, we have $H^i(F,{\bf C}) = H^i_c(F, {\bf C})$.
Therefore, it is sufficient to compute the $E$-polynomial
\[ E(F; u,v) = \sum_{p,q} e^{p,q}(F) u^p v^q. \]
Since $X_{\cal T}$ is a toric variety, it admits a natural stratification
by strata which are isomorphic to algebraic tori $T_{\theta}$ corresponding
to regular subsimplices $\theta \in {\cal T}$, such that
\[ \mbox{\rm dim}\, T_{\theta} + \mbox{\rm dim}\, \theta = d-1. \]
The natural stratification of $X_{\cal T}$ induces a stratification of $F$.
Notice that $\pi_{\cal T} (T_{\theta}) = p_{\sigma}$ (i.e.,
$T_{\theta} \in F$) if and only if $\theta$ does not belong to the
boundary of $\Delta$.
If $a_i$ denotes the number of $i$-dimensional regular simplices
of ${\cal T}$ which do not belong to the boundary of $\Delta$, then
$a_i$ can be identified with the number of
$(d-1-i)$-dimensional tori in the natural stratification of
$\pi_{\cal T}^{-1}(p)$. By \ref{proper1}, we get:
\[ E(F; u,v) = \sum_{\pi_{\cal T} (T_{\theta}) = p} E(T_{\theta}; u, v). \]
Since $E(({\bf C}^*)^k; u,v) = (uv -1 )^k$, we
obtain
\[ E(F; u,v) = a_0(u v -1 )^{d-1} +
a_1(u v -1 )^{d-2} + \cdots + a_{d-1}. \]
Now we compute $P_{\Delta}(t)$ by using the numbers $a_i$.
If $\theta \in {\cal T}$ is a $i$-dimensional regular simplex, then
\[ l(k\theta) = {k+i \choose k}; \;\;\;\;\;\; {\rm i.e.,}\; \;\;\;\;\;
P_{\theta}(t) = \frac{1}{(1-t)^{i+1}}. \]
Applying the usual inclusion-exclusion principle
for the counting of lattice points of $k\Delta$, we obtain:
\[ l(k\Delta) = \sum_{i =0}^{d-1}
\sum_{{\rm dim}\, \theta=d-1-i} (-1)^i l(k\theta), \]
where $\theta$ runs over all regular simplices in ${\cal T}$ which do not
belong to the boundary of $\Delta$. Thus,
\[ P_{\Delta}(t) =
\frac{a_{d-1}}{(1-t)^d} - \frac{a_{d-2}}{(1 - t)^{d-1}} + \cdots +
(-1)^{d-1} \frac{a_0}{(1 - t)} \]
and the polynomial
\[ \psi_0(\Delta) + \psi_1(\Delta)t + \cdots \psi_{d-1}(\Delta)t^{d-1}
= P_{\Delta}(t) (1-t)^d \]
is equal to
\[ a_{d-1} + a_{d-2}(t -1) + \cdots + a_0(t-1)^{d-1}. \]
The latter coincides with the $E$-polynomial $E(F; u,v)$ after
making the substitution $t = uv$. Hence,
$\psi_i(\Delta) = e^{i,i}(F)$ ($0 \leq i \leq d-1$).
\hfill $\Box$
\begin{dfn}
{\rm Let $\Delta$ be a $(d-1)$-dimensional lattice polyhedron
defining a $d$-dimensional Gorenstein toric
singularity $p \in {\bf A}_{\sigma}$. Then
\[ S(\Delta;uv): = \psi_0(\Delta) + \psi_1(\Delta)uv +
\cdots + \psi_{d-1}(\Delta)(uv)^{d-1} \]
will be called the {\em $S$-polynomial} of the Gorenstein
toric singularity at $p$. }
\end{dfn}
\begin{coro}
The Euler number $e(F)$ equals $S(\Delta, 1) = (d-1)! {\rm vol}(\Delta)$.
\label{eu.number}
\end{coro}
\noindent
{\em Proof.} By definition of $P_{\Delta}(t)$,
\[ \psi_0(\Delta) + \psi_1(\Delta) + \cdots + \psi_{d-1}(\Delta)
= (d-1)! {\rm vol}(\Delta). \]
Obviously, the left hand side equals $e(F)$.
\hfill $\Box$
\begin{rem}
{\rm It is known that the coefficient
$\psi_{d-1}(\Delta)$ equals $l^*(\Delta)$, i.e., the number
of rational points in the interior of
$\Delta$ (see \cite{dan.hov}, pp. 292-293). }
\label{lead}
\end{rem}
\section{Gorenstein
quotient singularities}
Let $G$ be a finite subgroup of $SL(d, {\bf C})$. We shall use the fact
that any element $ g \in G$ is obviously conjugate to a diagonal matrix.
\begin{dfn}
{\rm If an element $g \in G$ is conjugate to
\[ {\rm diag}( e^{2\pi \sqrt{-1}\alpha_1}, \ldots, e^{2\pi \sqrt{-1}\alpha_d}
)
\]
with $\alpha_i \in {\bf Q} \cap [0,1)$, then the sum
\[ wt(g): = \alpha_1 + \cdots + \alpha_d \]
will be called the {\em weight} of the element $g \in G$.
The number $ht(g): = {\rm rk} (g - e)$ will be called the {\em height} of
$g$. }
\end{dfn}
\begin{prop}
For any $g \in G$, one has
\[ wt(g) + wt(g^{-1}) = ht(g) = ht(g^{-1}). \]
\label{dualit}
\end{prop}
\noindent
{\em Proof.} Let
$g = {\rm diag}( e^{2\pi \sqrt{-1}\alpha_1}, \ldots, e^{2\pi \sqrt{-1}\alpha_d}
)$,
$g^{-1} = {\rm diag}( e^{2\pi \sqrt{-1}\beta_1}, \ldots, e^{2\pi
\sqrt{-1}\beta_d} )$.
Then $ht(g)$ equals the number of nonzero elements in
$\{ \alpha_1, \ldots, \alpha_d \}$. On the other hand,
$\alpha_i + \beta_i = 1$ if $\alpha_i \neq 0$, and
$\alpha_i + \beta_i = 0$ otherwise. Hence
$\sum_{i =1}^d (\alpha_i + \beta_i) = ht(g)$.
\hfill $\Box$
\begin{conj}
{\rm (strong McKay correspondence)}
Let $G \subset SL(d, {\bf C})$ be a finite group. Assume
that $X = {\bf C}^d/G$ admits a smooth crepant desingularization
$\pi \,: \, \hat{X}
\rightarrow X$ and $F:= \pi^{-1}(0)$. Then $H^*(F, {\bf C})$ has a
basis consisting of classes of algebraic cycles $Z_{\{g\}} \subset F$
which are
in $1$-to-$1$ correspondence with the conjugacy classes $\{g\}$ of $G$,
so that
\[ {\rm dim}\, H^{2i}(F, {\bf C}) =
\# \{ \mbox{\rm conjugacy classes $\{g\} \subset G$,
such that $wt(g) = i$} \}. \]
\label{strong1}
\end{conj}
Now we give several evidences in support of Conjecture \ref{strong1}.
\begin{theo}
Let $G \subset SL(d, {\bf C})$ be a finite abelian group. Suppose
that $X = {\bf C}^d/G$ admits a smooth crepant toric desingularization
$\pi \,: \, \hat{X}
\rightarrow X$ and $F: = \pi^{-1}(0)$. Then $H^*(F, {\bf C})$ has a
basis consisting of classes of algebraic cycles $Z_g \subset F$
which are
in $1$-to-$1$ correspondence with the elements $g$ of $G$, so that
\[ {\rm dim}\, H^{2i}(F, {\bf C}) =
\# \{ \mbox{\rm elements $g \in G$,
such that $wt(g) = i$} \}. \]
In particular, the Euler number of $F$ equals $\mid G \mid$.
\label{strong}
\end{theo}
\noindent
{\em Proof.} Let $N \subset {\bf R}^d$ be the free abelian group
generated by ${\bf Z}^d \subset {\bf R}^d$ and all
vectors $(\alpha_1, \ldots, \alpha_d)$ where
$g = {\rm diag}( e^{2\pi \sqrt{-1}\alpha_1}, \ldots, e^{2\pi \sqrt{-1}\alpha_d}
)$ runs
over all the elements of $G$. Then $N$ is a full sublattice
of ${\bf R}^d = {\bf N}_{\bf R}$, ${\bf Z}^d$ is a subgroup
of finite index in $N$, and $N/{\bf Z}^d$ is canonically isomorphic to $G$.
Let $M = {\rm Hom}(N, {\bf Z})$. We identify
${\bf Z}^d$ with ${\rm Hom}({\bf Z}^d, {\bf Z})$ by using the dual
basis. $M$ is a canonical sublattice of ${\bf Z}^d$ and
can be identified with the set of all Laurent monomials in
variables $t_1, \ldots, t_d$ which are $G$-invariant.
Therefore, the cone $\sigma$ defining the affine toric variety
$X = {\bf A}_{\sigma}$ is the positive $d$-dimensional octant
${\bf R}_{\geq 0}^d \subset {\bf R}^d = N_{\bf R}$.
Furthermore, the element $m_{\sigma} \in M$,
which was mentioned at the beginning of the previous section,
equals $(1,\ldots, 1) \in {\bf Z}^d$.
Now if $S: = {\bf C} \lbrack \sigma \cap N \rbrack $ and if for any
$x \in \sigma \cap N$, we define a {\em degree}
${\rm deg}\, x : = \langle x, m_{\sigma} \rangle$,
$S$ becomes a graded ring, so that
\[ n_1 := (1, 0, \ldots ,0), \ldots, n_d : = (0,\ldots, 0,1) \]
form a regular sequence of elements of degree $1$ in $S$.
This means that $S/(n_1, \ldots, n_d)$ has a monomial basis
corresponding to those elements of $N$ which are not in ${\bf Z}^d$.
The element $(\alpha_1, \ldots, \alpha_d) \in N$ corresponds
precisely to
the element $g = {\rm diag}( e^{2\pi \sqrt{-1}\alpha_1}, \ldots, e^{2\pi
\sqrt{-1}\alpha_d}
) \in G$. Moreover,
\[ \langle (\alpha_1, \ldots, \alpha_d), m_{\sigma} \rangle = w(g). \]
Thus, the Poincar\'{e} series of the quotient ring $S/(n_1, \ldots, n_d)$
equals
\[ \psi_0(\Delta) + \psi_1(\Delta)t + \cdots + \psi_{d-1}(\Delta)t^{d-1} \]
with coefficients
\[ \psi_i(\Delta) = \# \{ \mbox{\rm elements $g \in G$
such that $wt(g) = i$} \} \]
and $\sigma = \sigma_{\Delta}$ as in \S 4.
The proof is completed after making use of Theorem \ref{invariants}
and Corollary \ref{eu.number}.
\hfill $\Box$
\begin{exam}
{\rm For an abelian finite group $G \subset SL(3, {\bf C})$, the quotient
$X = {\bf C}^3 /G$ admits always smooth crepant toric desingularizations
coming from the full triangulations of the corresponding triangle $\Delta$
which is determined by $n_1, n_2, n_3$. All these triangulations contain only
regular simplices and each of them differs from another one by finitely
many elementary transformations (cf. \cite{oda}, Prop. 1.30 (ii)). In
particular, if $G$ is a cyclic group generated by
\[ {\rm diag}( e^{\frac{2\pi \sqrt{-1}\lambda_1}{|G|} },
e^{\frac{2\pi \sqrt{-1} \lambda_2}{|G|}},
e^{\frac{2\pi \sqrt{-1}\lambda_3}{|G|} }) \]
with
\[ 0 < \lambda_1, \lambda_2, \lambda_3 < | G |, \;
\lambda_1 + \lambda_2 + \lambda_3 = | G |,\;
{\rm gcd}(\lambda_1, \lambda_2, \lambda_3) =1, \]
then:
\[ {\rm dim}\, H^0(F, {\bf C}) = 1, \;{\rm dim}\, H^1(F, {\bf C}) =
{\rm dim}\, H^3(F, {\bf C}) = {\rm dim}\, H^5(F, {\bf C}) = 0, \]
\[ {\rm dim}\, H^2(F, {\bf C}) = \frac{1}{2} \left( |G| + \sum_{i =1}^3
{\rm gcd}(\lambda_i, |G|) \right) - 2 \]
and
\[ {\rm dim}\, H^4(F, {\bf C}) = \frac{1}{2} \left( |G| - \sum_{i =1}^3
{\rm gcd}(\lambda_i, |G|) \right) +1. \]}
\end{exam}
\begin{prop}
The Conjecture \ref{strong1} is true for $d \leq 3$.
\end{prop}
\noindent
{\em Proof.} If $d = 2$, then $wt(g) = 1$ unless $g = e$. The
number of the conjugacy classes with weight $1$ is equal to
the number of nontrivial irreducible representations of $G$. Since
the exceptional locus $F$ of a crepant resolution is a tree of
rational curves, ${\rm dim}\, H^0(F, {\bf C})$= 1, and
${\rm dim}\, H^2(F, {\bf C})$ is the number of
irreducible components of $F$. By the classical McKay correspondence
\cite{gonzalez,knorrer,mckay},
we obtain the statement \ref{strong1}.
If $d=3$, we use the result of Roan \cite{roan3} about the existence of
crepant resolutions and the Euler number of the exceptional locus. Let
$F$ be the exceptional locus over $0$ of a crepant resolution
$\pi \, : \, \hat{X} \rightarrow X$. Then $F$ is a strong
deformation retract of
$\hat{X}$; i.e., $H^i(F, {\bf C}) = H^i(\hat{X}, {\bf C})$. On the other
hand, $H^4(\hat{X}, {\bf C})$ is Poincar\'{e} dual to $H_c^2(\hat{X}, {\bf
C})$.
Note that ${\rm dim}\, H^4(F, {\bf C})$ is nothing but
the number of irreducible
$2$-dimensional components of $F$. Since $H^2(\hat{X}, {\bf Z})$ is
isomorphic to the Picard group of $\hat{X}$, ${\rm dim}\,
H^2(\hat{X}, {\bf C})$ is equal to the number of $\pi$-exceptional divisors.
Moreover, the subspace $H^2_c(\hat{X}, {\bf C}) \subset
H^2(\hat{X}, {\bf C})$ is spanned exactly by the classes of those exceptional
divisors whose image under $\pi$ is $0$. Therefore,
\[ {\rm dim}\, H^2(\hat{X}, {\bf C}) -
{\rm dim}\, H^4(\hat{X}, {\bf C}) = \]
\[ = \# \{ \mbox{\rm exceptional divisors $E \subset \hat{X}$,
such that $ \pi (E) $ is a curve on $X$} \}. \]
By the classical McKay correspondence in dimension $2$,
\[ {\rm dim}\, H^2(\hat{X}, {\bf C}) -
{\rm dim}\, H^4(\hat{X}, {\bf C}) = \]
\[ = \# \{ \mbox{\rm conjugacy classes $\{g\} \subset G$,
such that $wt(g) = 1$ and $ht(g) = 2$} \}. \]
By \cite{roan3},
\begin{equation}
1 + {\rm dim}\, H^2(\hat{X}, {\bf C}) + {\rm dim}\, H^4(\hat{X}, {\bf C})
= \# \{ \mbox{\rm all conjugacy classes $\{g\} \subset G$} \}.
\label{euler3}
\end{equation}
By \ref{dualit},
\[ \# \{ \mbox{\rm conjugacy classes $\{g\} \subset G$,
with $wt(g) = 1$ and $ht(g) = 3$} \} = \]
\[ = \# \{ \mbox{\rm conjugacy classes $\{g\} \subset G$,
with $wt(g) = 2$ and $ht(g) = 3$} \}. \]
Hence,
\[ \# \{ \mbox{\rm conjugacy classes $\{g\} \subset G$,
with $wt(g) = 2$ and $ht(g) = 3$ } \} =
{\rm dim}\, H^4(\hat{X}, {\bf C}). \]
Notice that if $wt(g)= 2$, then the height of $g$ must be equal to $3$.
Thus,
\[ {\rm dim}\, H^4(F, {\bf C}) =
\# \{ \mbox{\rm conjugacy classes $\{g\} \subset G$,
such that $wt(g) = 2$ } \} . \]
Finally,
\[ {\rm dim}\, H^2(F, {\bf C}) =
\# \{ \mbox{\rm conjugacy classes $\{g\} \subset G$,
such that $wt(g) = 1$ } \} \]
follows immediately from (\ref{euler3}).
\hfill $\Box$
\bigskip
\begin{dfn}
{\rm Let $G$ be a finite subgroup of $SL(d, {\bf C})$
and $0 \in {\bf C}^d/G$ the corresponding $d$-dimensional Gorenstein toric
singularity. If we denote by $\psi_i(G)$ the
number of the conjugacy classes of $G$ having the weight $i$,
then
\[ S(G;uv): = \psi_0(G) + \psi_1(G)uv +
\cdots + \psi_{d-1}(G)(uv)^{d-1} \]
will be called the {\em $S$-polynomial} of the regarded Gorenstein
quotient singularity at $0$. }
\end{dfn}
\begin{dfn}
{\rm Let $G$ be a finite subgroup of $SL(d, {\bf C})$
and $0 \in {\bf C}^d/G$ the corresponding $d$-dimensional Gorenstein toric
singularity. If we denote by $\tilde{\psi}_i(G)$ the
number of the conjugacy classes of $G$ having the weight $i$ and
the height $d$, then
\[ \tilde{S}(G;uv): = \tilde{\psi}_0(G) + \tilde{\psi}_1(G)uv +
\cdots + \tilde{\psi}_{d-1}(G)(uv)^{d-1} \]
will be called the {\em $\tilde{S}$-polynomial} of the Gorenstein
quotient singularity at $0$. }
\end{dfn}
By \ref{dualit}, we easily obtain:
\begin{prop}
The $\tilde{S}$-polynomial satisfies the following reciprocity
relation:
\[ \tilde{S}(G; uv) = (uv)^d \tilde{S}(G; (uv)^{-1}). \]
\label{dualit1}
\end{prop}
\section{String-theoretic Hodge numbers}
Let $X$ be a compact $d$-dimensional Gorenstein variety
with ${\rm Sing}\,X$ consisting of at most toroidal or quotient singularities.
\begin{dfn}
{\rm Let $x \in {\rm Sing}\,X$. We say that the $d$-dimensional
singularity at $x$ has the {\em splitting codimension $k$}, if $k$
is the maximal number for which
the analytic germ at $x$ is locally isomorphic to the
product of ${\bf C}^{d-k}$ and a $k$-dimensional
toric singularity defined by
a $(k-1)$-dimensional lattice polyhedron $\Delta'$ or, correspondingly,
to the product of ${\bf C}^{d-k}$ and the underlying space
${\bf C}^k/G'$ of a
$k$-dimensional quotient singularity defined by a finite subgroup
$G' \subset SL(k, {\bf C})$. For simplicity, we also
say that the singularity at $x$ is defined by $\Delta'$, or by $G'$.}
\end{dfn}
Using standard arguments, we can easily show that $X$ is
always stratified by locally closed
subvarieties $X_i$ $(i \in I)$, such that the germs of the singularities
of $X$ along $X_i$ are analytically isomorphic to that of a
Gorenstein toric singularity defined by means of
a $(k-1)$-dimensional lattice
polytope $\Delta_i$ or to that of a quotient
singularity defined by means of a finite subgroup $G_i$ of $SL(k, {\bf C})$,
respectively, where $k$ denotes the splitting codimension of singularities
on $X_i$.
\begin{dfn}
We denote by $S(X_i; uv)$ the $S$-polynomial $S(\Delta_i; uv)$
or $S(G_i; uv)$. Analogously, $\tilde{S}(X_i; uv)$ will denote
the $\tilde{S}$-polynomial $\tilde{S}(G_i; uv)$ if $X_i$ has only
Gorenstein quotient singularities.
\end{dfn}
\begin{dfn}
{\rm Suppose that $X$ has at most quotient Gorenstein singularities.
A stratification $X = \bigcup_{i \in I} X_i$, as above, is called
{\em canonical}, if for every $i \in I$ and every $x \in X_i$, there
exists an open subset $U \cong {\bf C}^d/G_i$ in $X$ and
an element $g \in G_i$, such that
$\overline{X_i} \cap U = ({\bf C}^d)^g/C(g)$, where $({\bf C}^d)^g$ is the
set of $g$-invariant points of ${\bf C}^d$.}
\label{stratif}
\end{dfn}
\begin{rem}
{\rm An algebraic variety is called {\em V-variety} if it has at most
quotient singularities. A {\em Gorenstein $V$-variety} (abbreviated
{\em $GV$-variety}) is a $V$-variety having at most Gorenstein
quotient singularities. The notion
of $V$-variety (or $V$-manifold) was first
introduced by Satake \cite{satake}. The existence and the uniqueness
of the canonical stratification for a $V$-variety was proved
by Kawasaki in \cite{kawasaki}. ( Note that our {\em canonical} stratification
in \ref{stratif} is not the first, but the second stratification of $X$
defined by Kawasaki in \cite{kawasaki}, p. 77.) }
\end{rem}
\begin{prop}
Suppose that $X$ is a $GV$-variety and
$X = \bigcup_{i \in I} X_i$ is its canonical stratification. Then
for any $i_0 \in I$, one has:
\[ S(X_{i_0}; uv) = \tilde{S}(X_{i_0}; uv) +
\sum_{X_{i_0} < X_{i_1}} \tilde{S}(X_{i_1}; uv). \]
\label{can.strat}
\end{prop}
\noindent
{\em Proof}. It is sufficient to prove the
corresponding local statement; i.e., we can assume, without loss of
generality, that
$X_{i_0} = {\bf C}^k/G_{i_0}$. For simplicity, we set
$Y = {\bf C}^k$, $Z = X_{i_0}$. Denote by $\pi$ the
natural mapping $Y \rightarrow Z$. For $g \in G_{i_0}$,
the image $Z(g) : = \pi(Y^g) \subset Z$
depends only on the conjugacy class of $g$.
Since $ht(g)$ equals the codimension of $Z(g)$ in $Z$, we
obtain
\[ S(X_{i_0}; uv) = \tilde{S}(X_{i_0}; uv) +
\sum_{X_{i_0} < X_{i_1}} \tilde{S}(X_{i_1}; uv). \]
\hfill $\Box$
\begin{coro}
Suppose that $X$ is a $GV$-variety and
$X = \bigcup_{i \in I} X_i$ is its canonical stratification. Then
for any $i_0 \in I$ one has:
\[ \tilde{S}(X_{i_0}; uv) =
\sum_{k \geq 0} (-1)^k \sum_{X_{i_0} < \cdots < X_{i_k}} {S}(X_{i_k}; uv). \]
\label{stra2}
\end{coro}
\noindent
{\em Proof.} By \ref{can.strat}, we have
\[ \tilde{S}(X_{i_0}; uv) = S(X_{i_0}; uv) - \sum_{X_{i_0} <
X_{i_1}} \tilde{S}(X_{i_1}; uv). \]
After that we apply \ref{can.strat} to $X_{i_1}$:
\[ \tilde{S}(X_{i_1}; uv) = S(X_{i_1}; uv) - \sum_{X_{i_1} <
X_{i_2}} \tilde{S}(X_{i_2}; uv), \;\;\; {\rm etc} \ldots \]
The repetition of this procedure completes
the proof of the assertion. \hfill $\Box$
\begin{dfn}
{\rm Let $X$ be a stratified variety with at most Gorenstein
toroidal or quotient singularities. We shall call the polynomial
\[ E_{\rm st}(X; u,v):= \sum_{i \in I}
E(X_i; u,v) \cdot S(X_i; uv) \]
the {\em string-theoretic $E$-polynomial of $X$}.
Let us write $E_{\rm st}(X;u,v)$ in the following expanded form:
\[ E_{\rm st}(X; u,v) = \sum_{p,q} a_{p,q} u^p v^q. \]
The numbers $h^{p,q}_{\rm st}(X): = (-1)^{p+q} a_{p,q}$
will be called the {\em string-theoretic Hodge numbers of $X$}.
Correspondingly,
\[ e_{\rm st}(X): = E_{\rm st}(X; -1,-1) = \sum_{p,q} (-1)^{p+q}
h^{p,q}_{\rm st}(X) \]
will be called the {\em string-theoretic Euler number of $X$}. }
\end{dfn}
\begin{rem}
{\rm If $X$ admits
a smooth crepant toroidal desingularization $\pi\, : \,
\hat{X} \rightarrow X$, then, by \ref{formula} and \ref{invariants},
the $E$-polynomial of $\hat{X}$ equals
\[ E(\hat{X}; u, v) = \sum_{i \in I}
E(X_i; u, v)\cdot E(F_i; u, v) \]
where $F_i$ denotes a the special fiber $\pi^{-1}(x)$ over
a point $x \in X_i$.}
\label{crep1}
\end{rem}
By \ref{crep1}, we obtain:
\begin{theo}
If $X$ admits a smooth crepant toroidal desingularization
$\hat{X}$, then the string-theoretic Hodge numbers
$h^{p,q}_{\rm st}(X)$ coincide with the ordinary
Hodge numbers $h^{p,q}(\hat{X})$. In particular, the numbers
$h^{p,q}_{\rm st}(X)$ are nonnegative and satisfy the Poincar\'{e} duality
$h^{p,q}_{\rm st}(X) = h^{d-p,d-q}_{\rm st}(X)$.
\end{theo}
The next theorem will play an important role in the forthcoming statements:
\begin{theo}
Suppose that $X$ is a $GV$-variety and
$X = \bigcup_{i \in I} X_i$ denotes its canonical stratification. Then
\[ E_{\rm st}(X; u,v) = \sum_{i \in I}
E(\overline{X}_i; u,v) \cdot \tilde{S}(X_i; uv). \]
\label{second.f}
\end{theo}
\noindent
{\em Proof.} By \ref{stra1}, we get
\[ E(X_{i_0}; u,v) = \sum_{k \geq 0} (-1)^k \sum_{X_{i_k} < \cdots <
X_{i_1} < X_{i_0} } E(\overline{X}_{i_k}; u,v). \]
Therefore,
\[E_{\rm st}(X; u,v) = \sum_{i_0 \in I}
\left( \sum_{k \geq 0} (-1)^k \sum_{X_{i_k} < \cdots <
X_{i_1} < X_{i_0} } E(\overline{X}_{i_k}; u,v) \right)
\cdot S(X_{i_0}; uv) = \]
\[ = \sum_{i_k \in I} E(\overline{X}_{i_k}; u,v) \cdot
\left( \sum_{k \geq 0} (-1)^k \sum_{X_{i_k} < \cdots <
X_{i_1} < X_{i_0} } S(X_{i_0}; uv) \right). \]
By \ref{stra2}, we have
\[ \tilde{S}(X_{i_k}; uv) = \sum_{k \geq 0} (-1)^k \sum_{X_{i_k} < \cdots <
X_{i_1} < X_{i_0} } S(X_{i_0}; uv). \]
This implies the required formula.
\hfill $\Box$
\begin{coro}
Suppose that $X$ is a $GV$-variety.
Then the numbers
$h^{p,q}_{\rm st}(X)$ are nonnegative and satisfy the Poincar\'{e} duality
$h^{p,q}_{\rm st}(X) = h^{d-p,d-q}_{\rm st}(X)$.
\label{main.prop}
\end{coro}
\noindent
{\em Proof.} Since $\overline{X}_i$ itself is a $V$-variety, one has
$h^{p,q}(\overline{X}_i) \geq 0$, as well as the Poincar\'{e} duality
\[ E(\overline{X}_i; u,v) = (uv)^{{\rm dim}\,\overline{X}_i}
E(\overline{X}_i; u^{-1},v^{-1}). \]
On the other hand, by \ref{dualit1}, we obtain
\[ \tilde{S}(X_i; uv) = (uv)^{{\rm dim}\,\overline{X}_i}
\tilde{S}(X_i; (uv)^{-1}). \]
This implies
\[ E_{\rm st}(X; u,v) = (uv)^{{\rm dim}\,X}
E_{\rm st}(X; u^{-1},v^{-1}). \]
Since $\tilde{S}(X_i; uv)$ is a polynomial of $uv$ with
nonnegative coefficients, we conclude that $h^{p,q}_{\rm st}(X) \geq 0$.
\hfill $\Box$
\begin{theo}
Suppose that $X$ has at most toroidal Gorenstein singularities. Let
$\pi \, : \, \hat{X} \rightarrow X$ be a $MPCP$-desingularization of
$X$. Then
\[ E_{\rm st}(X;, u,v) = E_{\rm st}(\hat{X}; u,v). \]
Moreover,
\[ h^{p,1}_{\rm st}(X) = h^{p,1}(\hat{X}), \;\;\;\; \mbox{\rm for all $p$}. \]
\label{MPCP-desing}
\end{theo}
\noindent
{\em Proof. } Let $X = \bigcup_{i \in I} X_i$ be a stratification of
$X$, such that
\[ E_{\rm st}(X; u,v) = \sum_{i \in I}
E(X_i; u,v) \cdot S(\Delta_i; uv) \]
and $\pi\, : \, \hat{X} \rightarrow X$ be a MPCP-desingularization of
$X$. We set $\hat{X}_i : = \pi^{-1}(X_i)$. Then $\hat{X}_i$ has the
natural stratification by products
$X_i \times ({\bf C}^*)^{{\rm codim}\, \theta}$ induced by the
triangulation
\[ \Delta_i = \bigcup_{\theta \in {\cal T}_i} \theta. \]
Thus,
\[ E_{\rm st}(\hat{X}; u,v) = \sum_{i \in I}
\left(\sum_{\theta \in {\cal T}_i} (uv-1)^{{\rm codim}\, \theta}
E(X_i; u,v) \cdot S(\theta; uv) \right). \]
By counting lattice points in $k\Delta_i$, we obtain
\[ S(\Delta_i; uv) = \sum_{\theta \in {\cal T}_i}
(uv-1)^{{\rm codim}\, \theta}
S(\theta; uv). \]
Hence,
\[ E_{\rm st}(X;, u,v) = E_{\rm st}(\hat{X}; u,v). \]
Since $\hat{X}$ has only terminal ${\bf Q}$-factorial singularities,
for any $\theta \in {\cal T}_i$ we obtain
\[ \psi_1(\theta) = 0;\;\; \mbox{\rm i.e.,} \;\;
S(\theta; uv) = 1 + \psi_2(\theta) (uv)^2 + \cdots . \]
Therefore, the coefficient of $u^p v$ in $E_{\rm st}(\hat{X}; u,v)$
coincides with the coefficient of $u^p v$ in the usual
$E$-polynomial $E(\hat{X}; u,v)$. As $\hat{X}$ is a $V$-variety,
the Hodge structure in $H^*(\hat{X}, {\bf C})$ is pure, and
\[ h^{p,1}_{\rm st}(X) = h^{p,1}(\hat{X}),\; \mbox{\rm for all $p$}. \]
\hfill $\Box$
\begin{coro}
Suppose that $X$ has at most toroidal Gorenstein singularities.
Then the numbers
$h^{p,q}_{\rm st}(X)$ are nonnegative and satisfy the Poincar\'{e} duality
$h^{p,q}_{\rm st}(X) = h^{d-p,d-q}_{\rm st}(X)$.
\end{coro}
\noindent
{\em Proof.} By \ref{MPCP-desing}, it is sufficient to prove the statement
for a $MPCP$-desingularization $\hat{X}$ of $X$. The latter follows from
\ref{main.prop}. \hfill $\Box$
\begin{theo}
Let $X$ be a smooth compact
K\"ahler manifold of dimension $n$ over ${\bf C}$ being equipped with
an action of a finite group $G$, such that $X$ has a $G$-invariant
volume form. Then the orbifold Hodge numbers
$h^{p,q}(X,G)$ which were defined in the introduction
coincide with the string-theoretic Hodge numbers
$h^{p,q}_{\rm st}(X/G)$.
Moreover,
\[ e(X,G) = e_{\rm st}(X/G). \]
\label{str-eul}
\end{theo}
\noindent
{\em Proof.} We use the canonical stratification of $Y= X/G$:
\[ Y = \bigcup_{i \in I} Y_i. \]
For every stratum $Y_i$, there exists an element $g_i \in G$,
such that $\overline{Y}_i = X^{g_i}/C(g_i)$. We note that
\[ E(\overline{Y}_i; u,v) = \sum_{p,q} (-1)^{p+q}
{\rm dim} H^{p,q}(X^{g_i})^{C(g_i)} u^pv^q. \]
Now the equality
\[ h^{p,q}_{\rm st}(X/G) = h^{p,q}(X,G) \]
follows from Theorem \ref{second.f}.
\newline
In order to get $e(X,G) = e_{\rm st}(X/G)$, it remains to prove
the equality
\[ e(X,G) = \sum_{p,q} (-1)^{p+q} h^{p,q}(X,G). \]
We shall make use of the notation which was introduced in \S 1.
Since $\{ g \}$ expresses a system of representatives for
$G/C(g)$ and the number of conjugacy classes of $G$ equals
\[ \frac{1}{\mid G \mid} \sum_{g \in G} \mid C(g) \mid, \]
one can rewrite the physicists Euler number (\ref{euler.phys}) as
\[ e(X, G) = \frac{1}{\mid G \mid} \sum_{g} | C(g) | \cdot
e(X^g/C(g)) = \sum_{\{ g \}} e(X^g/C(g)), \]
where $\{g\}$ runs over all conjugacy classes of $G$ with $g$ representing
$\{ g \}$.
We show that
\[ \sum_{p,q} (-1)^{p+q} h^{p,q}_g(X,G) =
e(X^g/C(g)). \]
This follows from the equalities
\[ \sum_{p,q} (-1)^{p+q} h^{p,q}_g(X,G) = \sum_{i =1}^{r_g}
\sum_{p,q} (-1)^{p+q - 2 F_i(g)} h^{p - F_i(g),q - F_i(g)}_{C(g)}(X_i(g)) = \]
\[ = \sum_{p,q} (-1)^{p+q} h^{p,q}_{C(g)}(X^g) = e(X^g/C(g)). \]
\hfill $\Box$
\bigskip
\begin{coro}
Suppose that $X/G$ has a crepant desingularization $\widehat{X/G}$
and that the strong
McKay correspondence $($Conjecture \ref{strong1}$)$ holds true
for the singularities occuring along every stratum of $X/G$. Then
\[ h^{p,q}(\widehat{X/G}) = h^{p,q}_{\rm st}(X/G). \]
\label{c.des}
\end{coro}
\begin{exam}
{\rm Let us first give a $3$-dimensional example of
an orbit space (with a simple acting group) containing both abelian and
non-abelian quotient singularities, and which was proposed by F. Hirzebruch.
We consider the Fermat quintic $X = \{ [z_1, \ldots, z_5] \in {\bf P}^4
\mid \sum_{i =1}^5 z_i^5 = 0 \}$ and let the alternating group ${\cal A}_5$
act on it coordinatewise. The group ${\cal A}_5$ has five conjugacy
classes: the trivial, one consisting of all $20$ $3$-cycles, one consisting
of the $15$ products of disjoint transpositions, and two more conjugacy
classes of $5$-cycles, each of which has $12$ elements. Note that the action
of the elements of these last two conjugacy classes is fixed point free.
Each of the $20$ $3$-cycles fixes a plane quintic and two additional points.
Correspondingly, each of the $15$ products of disjoint traspositions fixes a
plane quintic and a projective line (without common points). As
$X/{\cal A}_5$ is a Calabi-Yau variety, the generic points of the
$1$-dimensional components of ${\rm Sing}\, X/{\cal A}_5$ are compound
Du Val points \cite{reid1}. Up to the above mentioned
$40$ additional points coming from the $3$-cycles and having isotropy
groups $\cong {\bf Z}/3{\bf Z}$, there exist $175$ more fixed points
on $X$ creating (after appropriate group identifications)
{\em dissident} points on
$X/{\cal A}_5$ (we follow here the terminology of M. Reid).
Namely, the $25$ points of the intersection locus of the $20$ plane quintics
(with isotropy groups $\cong {\cal A}_4$), further $125$ points
lying in the intersection locus of the $15$ plane quintics
(with isotropy groups $\cong {\cal S}_3$), as well as $15 + 10 = 25$ points
coming from the intersection of the projective lines (with isotropy groups
isomorphic to the Kleinian four-group and to
${\cal S}_3$ respectively). Using Ito's results \cite{ito1,ito2}, we can
construct global crepant desingularizations $\pi\, : \,
\widehat{X/{\cal A}_5} \rightarrow X/{\cal A}_5$. By \ref{c.des},
$h^{p,q}(\widehat{X/{\cal A}_5}) = h^{p,q}_{\rm st}({X/{\cal A}_5})$.
Thus, for the
computation of $h^{p,q}(\widehat{X/{\cal A}_5})$, we just need to choose two
representatives, say $(123)$ and $(12)(34)$, of the two non-freely acting
conjugacy classes. We have:
\begin{itemize}
\item
$h^{p,q}({X/{\cal A}_5}) =
h^{p,q}_{\{1\}}({X,{\cal A}_5})$ equals $\delta_{p,q}$
( $=$ Kronecker symbol) for
$p + q \neq 3$, $h^{p,q}_{\{1\}}({X,{\cal A}_5}) =
1$ for $(p,q) \in \{ (3,0), (0,3) \}$ and $h^{p,q}_{\{1\}}({X,{\cal A}_5}) =
5$ for
$(p,q) \in \{ (2,1), (1,2) \}$;
\item
$h^{p,q}_{\{(123)\}} (X, {\cal A}_5)$
equals $2$ for $(p,q) \in \{ (1,1), (2,2) \}$, $h^{p,q}_{\{(123)\}}
(X, {\cal A}_5) =6$ for
$(p,q) \in \{ (2,1), (1,2) \}$, $h^{p,q}_{\{(123)\}} (X, {\cal A}_5) =
0$ otherwise;
\item
$h^{p,q}_{\{(12)(34)\}} (X, {\cal A}_5)$ equals $2$ for $1 \leq p,q \leq 2$
and $0$ otherwise.
\end{itemize}
Thus, we get:
\[ h^{2,1}_{\rm st}(X/{\cal A}_5) = h^{1,2}_{\rm st}(X/{\cal A}_5) = 13, \]
\[ h^{1,1}_{\rm st}(X/{\cal A}_5) = h^{2,2}_{\rm st}(X/{\cal A}_5) = 5. \]
In particular, $e(\widehat{X/{\cal A}_5}) = e_{\rm st}(X/{\cal A}_5) =
-16$, in agreement with the calculations of physicists (cf. \cite{KS}, p. 57).
}
\end{exam}
\begin{exam}
{\rm Let $X^{(n)} : = X^n/{\cal S}_n$ be the $n$-th symmetric power
of a smooth
projective surface $X$. As it is known (see, for instance,
\cite{got2}, p.54 or \cite{hirzebruch}, p.258), $X^{(n)}$ is
endowed with a canonical crepant desingularization $X^{[n]}: =
{\rm Hilb}^n(X) \rightarrow X^{(n)}$ given by the Hilbert scheme
of finite subschemes of length $n$. In \cite{got1,got2}, G\"ottsche
computed the Poincar\'{e} polynomial of $X^{[n]}$. In particular, his
formula for the Euler number gives:
\[ \sum_{n =0}^{\infty} e(X^{[n]}) t^n =
\prod_{k =1}^{\infty}(1-t^k)^{-e(X)}. \]
Using power series comparison and the above formula, Hirzebruch and
H\"ofer gave in \cite{hirzebruch} a formal proof of the equality
$e(X^{[n]}) = e(X^{(n)}, {\cal S}_n)$. In fact, for the proof of the
validity of {\em orbifold Euler formulae} of this kind, it is enough to
check locally that the Conjecture \ref{general} of M. Reid is true (cf.
\cite{roan3},
Lemma 1). Our results \ref{str-eul} and \ref{c.des} say more: in order to
obtain the equality $ h^{p,q}(X^{[n]}) = h^{p,q}(X^{(n)},G)$
it is sufficient to verify locally our ``strong'' McKay correspondence.
The latter has been checked by G\"ottsche in \cite{got3}. The numbers
$h^{p,q}(X^{[n]})$ can be computed by means of the Hodge
polynomial $h(X^{[n]}; u,v) := E(X^{[n]}; -u,-v)$.
If $\Pi(n)$ denotes the set of all finite series
$(\alpha) = ( \alpha_1, \alpha_2, \ldots )$ of
nonnegative integers with $\sum_i i \alpha_i = n$, then the conjugacy
class of a permutation $\sigma \in {\cal S}_n$ is determined by
its type $(\alpha) = ( \alpha_1, \alpha_2, \ldots ) \in \Pi(n)$, where
$\alpha_i$ expresses the number of cycles of length $i$ in $\sigma$.
G\"ottsche and Soergel
\cite{GS,got2} proved that
\[ h(X^{[n]}; u,v) = \sum_{(\alpha) \in \Pi(n)} (uv)^{n- \mid \alpha \mid}
\prod_{k =1}^{\infty} h(X^{(\alpha_k)}; u,v), \]
where $\mid \alpha \mid : = \alpha_1 + \alpha_2 + \cdots $ denotes the
sum of the members of $(\alpha) \in \Pi(n)$.
(Similar formulae can be obtained for the even-dimensional Kummer varieties
of higher order, cf. \cite{got2,got3,GS}.) }
\end{exam}
\section{Applications to quantum cohomology $\;\;\;\;\;\;\;\;\;\;\;
\;\;\; $ and mirror symmetry}
{}From now on, and throughout this section, we use the notion of
{\em reflexive polyhedron} being
introduced in \cite{batyrev1}.
\begin{prop}
Let $\Delta$ be a reflexive polyhedron of dimension $d$. Then
\[ S(\Delta,t) = (t -1)^d +
\sum_{\begin{array}{c} {\scriptstyle 0 \leq {\rm dim}\,\theta \leq d-1}
\\ {\scriptstyle \theta \subset \Delta} \end{array}} S(\theta,t)
\cdot (t -1)^{{\rm dim}\, \theta^*}. \]
\label{relation}
\end{prop}
\noindent
{\em Proof. }
Denote by $\partial \Delta$ the $(d-1)$-dimensional boundary of $\Delta$ which
is homeomorphic to $(d-1)$-dimensional sphere. Let
$l(k \cdot \partial \Delta)$ be the number of lattice points belonging
to the boundary of $k\Delta$. The reflexivity of $\Delta$ implies:
\[ l(k \cdot \partial \Delta) = \sum_{0 \leq {\rm dim}\, \theta \leq d-1}
(-1)^{d-1 - {\rm dim}\, \theta} l(k\theta), \; \; \; \mbox{\rm for
$k > 0$}. \]
Since the Euler number of a $(d-1)$-dimensional sphere is $1 + (-1)^{d-1}$,
we obtain
\[ (-1)^{d-1} + (1-t)P_{\Delta}(t)\;\; = \;\;
(-1)^{d-1} \sum_{0 \leq {\rm dim}\, \theta \leq d-1}
(-1)^{{\rm dim}\, \theta} P_{\theta}(t), \]
i.e.,
\[ (-1)^{d-1} + \frac{S(\Delta;t)}{(1-t)^d}\;\; = \;\;
(-1)^{d-1} \sum_{0 \leq {\rm dim}\, \theta \leq d-1}
(-1)^{{\rm dim}\, \theta}
\frac{S(\theta;t)}{(1-t)^{{\rm dim}\, \theta +1}}. \]
This implies the required equality.
\hfill $\Box$
\medskip
We prove the following relation between
the polar duality of lattice polyhedra
and string-theoretic cohomology:
\begin{theo}
Let ${\bf P}_{\Delta}$ be a $d$-dimensional Gorenstein toric Fano variety
corresponding to a $d$-dimensional reflexive polyhedron $\Delta$. Then
\[ E_{\rm st}({\bf P}_{\Delta}; u, v) =
(1- uv)^{d+1} P_{\Delta^*} (uv) \]
where $\Delta^*$ is the dual reflexive polyhedron.
\end{theo}
\noindent
{\em Proof.} ${\bf P}_{\Delta}$ has
a natural stratification being defined by the strata $T_{\theta}$,
where $\theta$ runs over all the faces of $\Delta$.
On the other hand, the Gorenstein
singularities along $T_{\theta}$ are determined
by the dual face $\theta^*$
of the dual polyhedron $\Delta^*$ (cf. \cite{batyrev1}, 4.2.4). We set
$S(\theta^*, uv) =1$ if $\theta = \Delta$. Then
\[ E_{\rm st}({\bf P}_{\Delta}; u, v) = \sum_{\theta \subset \Delta}
E(T_{\theta}; u,v) \cdot S(\theta^*; uv). \]
Note that
\[ E(T_{\theta}; u,v) = (uv -1)^{{\rm dim}\, \theta}, \]
and that, for ${\rm dim}\, \theta < d$, one has by definition:
\[ S(\theta^*; uv) = (1 - uv)^{{\rm dim}\, \theta^* +1} P_{\theta^*}(uv). \]
If we apply Proposition \ref{relation} to the
dual reflexive polyhedron $\Delta^*$, then,
using ${\rm dim}\, \theta + {\rm dim}\, \theta^* = d-1$, we obtain
the desired formula for $E_{\rm st}({\bf P}_{\Delta}; u, v)$.
\hfill $\Box$
\bigskip
\begin{coro}
The string-theoretic Euler number of ${\bf P}_{\Delta}$ is equal to
$d!({\rm vol}\, \Delta^*)$.
\end{coro}
\begin{rem}
{\rm The quantum cohomology ring of a smooth toric variety was described
in \cite{batyrev00}. It was proved that the usual cohomology
of a smooth toric manifold can be obtained as a limit of the quantum
cohomology ring. On the other hand, one can immediately extend
the description of the quantum cohomology ring to arbitrary
(possibly singular) toric variety (cf. \cite{batyrev00}, 5.1).
In particular, one can easily show that
${\rm dim}\, QH^*_{\varphi}({\bf P}_{\Delta}, {\bf C})
= d! ({\rm vol}\, \Delta^*)$, for
any $d$-dimensional reflexive polyhedron. Comparing dimensions, we
see that, for singular toric Fano varieties ${\bf P}_{\Delta}$,
the limit of the quantum cohomology ring is not the usual
cohomology ring, but rather the cohomology of a smooth crepant
desingularization, if such a desingularization exists
(cf. \cite{batyrev00}, 6.5). By our general philosophy,
we should consider the string-theoretic Hodge numbers
$h^{p,p}_{\rm st}({\bf P}_{\Delta})$ as the Betti numbers
of a limit of the quantum cohomology ring
$QH^*_{\varphi}({\bf P}_{\Delta}, {\bf C})$. }
\end{rem}
\bigskip
Let $\overline{Z}_f: = \overline{Z}_{f_1} \cap \cdots \cap \overline{Z}_{f_r}$
be a generic $(d-r)$-dimensional Calabi-Yau
complete intersection variety, which is embedded
in a Gorenstein toric Fano variety
${\bf P}_{\Delta}$ corresponding to a
$d$-dimensional reflexive polyhedron $\Delta = \Delta_1 + \cdots \Delta_r$,
where $\Delta_i$ is the Newton polyhedron of $f_i$ $(i = 1, \ldots, r)$.
Assume that the lattice polyhedra $\Delta_1, \ldots, \Delta_r$ are
defined by a {\em nef-partition} of vertices of the dual reflexive polyhedron
$\Delta^* = {\rm Conv}\{ \nabla_1, \ldots , \nabla_r\}$. (For definitions
and notations the reader is referred to \cite{batyrev-borisov,borisov}.)
Denote by $\overline{Z}_g : = \overline{Z}_{g_1}
\cap \cdots \cap \overline{Z}_{g_r}$
a generic Calabi-Yau complete intersection variety in
the Gorenstein toric Fano variety ${\bf P}_{\nabla^*}$, which is
defined by the reflexive polyhedron $\nabla^* = {\rm Conv}\{
\Delta_1, \ldots ,
\Delta_r \}$, where $\nabla_i$ is the Newton polyhedron of
$g_i$ $(i = 1, \ldots, r)$.
\begin{conj} {\rm (Mirror duality of string-theoretic Hodge numbers)}
The string-theoretic $E$-polynomials of $\overline{Z}_f$ and
$\overline{Z}_g$ obey to the following reciprocity law:
\[ E_{\rm st}(\overline{Z}_f; u,v) =
(-u)^{d-r}E_{\rm st}(\overline{Z}_g;u^{-1},v). \]
Equivalently, the string-theoretic Hodge numbers of $\overline{Z}_f$ and
$\overline{Z}_g$ are related to each other by:
\[ h^{p,q}_{\rm st}(\overline{Z}_f) =
h^{d-r-p,q}_{\rm st}(\overline{Z}_g),
\;\; \mbox{ {\rm for all $p,q$}}. \]
\label{symmetry}
\end{conj}
\noindent
We want to show some evidences in support of Conjecture \ref{symmetry} for
Calabi-Yau hypersurfaces ($r =1$).
\begin{theo}
Let $\overline{Z}_f$ be a $\Delta$-regular Calabi-Yau hypersurface in
${\bf P}_{\Delta}$. Then
\[ E_{\rm st}(\overline{Z}_f; 1,v) =
\frac{S(\Delta^*;v)}{v} + (-1)^{d-1} \frac{S(\Delta;v)}{v} + \]
\[ + \sum_{\begin{array}{c} {\scriptstyle 1 \leq {\rm dim}\,\theta \leq d-2}
\\ {\scriptstyle \theta \subset \Delta} \end{array}}
\frac{(-1)^{{\rm dim}\, \theta-1}}{v} \left(
S(\theta; v) \cdot
S(\theta^*; v) \right) - \]
\[ - \sum_{\begin{array}{c} {\scriptstyle {\rm dim}\,\theta = d-1}
\\ {\scriptstyle \theta \subset \Delta} \end{array}}
(-1)^{d-1} \frac{S(\theta,v)}{v} -
\sum_{\begin{array}{c} {\scriptstyle {\rm dim}\,\theta^* = d-1}
\\ {\scriptstyle \theta^* \subset \Delta^*} \end{array}}
\frac{S(\theta^*,v)}{v}. \]
\label{formul0}
\end{theo}
\begin{coro}
\[ E_{\rm st}(\overline{Z}_f; 1,v) =(-1)^{d-1}
E_{\rm st}(\overline{Z}_g; 1,v). \]
\end{coro}
At first we need the following formula:
\begin{prop} Let $\theta$ be a face of $\Delta$ and
${\rm dim}\, \theta \geq 1$. Then
\[ E(Z_{f,\theta}; 1,v) = \frac{ (v-1)^{{\rm dim}\, \theta}}{v}
+ (-1)^{{\rm dim}\, \theta-1}
\frac{S(\theta,v)}{v} . \]
\label{e-f}
\end{prop}
\noindent
{\em Proof. } It follows from the formula
of Danilov and Khovanski\^i (\cite{dan.hov}, Remark 4.6):
\[ (-1)^{{\rm dim}\, \theta -1} \sum_p e^{p,q}(Z_{f,\theta}) =
(-1)^q { n \choose q + 1 } + \psi_{q+1}(\theta). \]
\hfill $\Box$
\noindent
{\bf Proof of Theorem \ref{formul0}}. By definition,
\[ E_{\rm st}(\overline{Z}_f; 1,v) \; = \; E(Z_{f, \Delta}; 1,v) \; + \;
\sum_{\begin{array}{c}
{\scriptstyle {\rm dim}\,\theta = d-1} \\
{\scriptstyle \theta \subset \Delta} \end{array}}
E(Z_{f, \theta}; 1,v) \;\; + \]
\[ + \sum_{\begin{array}{c}
{\scriptstyle 1 \leq {\rm dim}\,\theta \leq d-2 } \\
{\scriptstyle \theta \subset \Delta} \end{array}}
E(Z_{f, \theta}; 1,v) \cdot S(\theta^*; v). \]
Substituting the expressions which follow from \ref{e-f}, we get:
\[ E(\overline{Z}_{f, \Delta}; 1,v) = \frac{(v-1)^{d}}{v}
+ (-1)^{d-1} \frac{S(\Delta,v)}{v} + \]
\[ + \sum_{\begin{array}{c}
{\scriptstyle {\rm dim}\,\theta = d-1} \\ {\scriptstyle
\theta \subset \Delta} \end{array} }
\left( (-1)^{d-2} \frac{S(\theta,v)}{v} + \frac{(v-1)^{d-1}}{v} \right) + \]
\[ + \sum_{\begin{array}{c}
{\scriptstyle 1 \leq {\rm dim}\,\theta \leq d-2} \\
{\scriptstyle \theta \subset \Delta} \end{array}}
\left( (-1)^{{\rm dim}\, \theta - 1} \frac{S(\theta,v)}{v} +
\frac{(v-1)^{{\rm dim}\, \theta}}{v} \right) \cdot S(\theta^*; uv). \]
It remains to use \ref{e-f} and \ref{relation}.
\hfill $\Box$
\bigskip
\begin{dfn}
{\rm For a face $\theta$ of $\Delta$, we denote by
${\bf v}(\theta)$ the normalized volume of $\theta$:
$({\rm dim}\,\theta)! {\rm vol}(\theta)$. }
\end{dfn}
\begin{coro}
Let $\Delta$ be a $d$-dimensional reflexive polyhedron. Then
\[ e_{\rm st} (\overline{Z}_f) = \sum_{i =1}^{d-2}
\sum_{{\rm dim}\, \theta = i}
(-1)^i {\bf v}(\theta)\cdot {\bf v}(\theta^*). \]
In particular,
\[ e_{\rm st} (\overline{Z}_f) = (-1)^{d-1} e_{\rm st} (\overline{Z}_g). \]
\end{coro}
We remark that \ref{symmetry} is evident if $q =0$, because
$h^{p,0}_{\rm st} (\overline{Z}_f) = 1$, for $q = 0, d-1$ and
$h^{p,0} _{\rm st} (\overline{Z}_f) = 0$ otherwise.
For $q = 1$ $(r=1)$, and $p \in \{ 1,d-2\}$,
Conjecture \ref{symmetry} is proved by Theorem \ref{MPCP-desing}
combined with Thm. 4.4.3 from \cite{batyrev1}. We generalize this
for arbitrary values of $p$.
\begin{theo}
For a face $\theta$ of $\Delta$, we denote by
$l^*(\theta)$ the number of lattice points in the relative interior of
$\theta$. Assume that $d \geq 5$. Then for $ 2 \leq p \leq d-3$ one has
\[ h^{p,1}_{\rm st}(\overline{Z}_f) =
\sum_{{\rm codim}\, \theta = p } l^*(\theta)\cdot
l^*(\theta^*). \]
By the duality among faces, one has
\[ h^{p,1}_{\rm st}(\overline{Z}_f) =
h^{d-1-p,1}_{\rm st}(\overline{Z}_g). \]
\label{p1}
\end{theo}
\noindent
{\em Proof.} By the Poincar\'{e} duality, it is enough to compute
$h^{d-1-p,d-2}_{\rm st}(\overline{Z}_f) =
h^{p,1}_{\rm st}(\overline{Z}_f)$. We use
\[ E_{\rm st} (\overline{Z}_f;u,v) =
\sum_{\theta \subset \Delta} E(Z_{f,\theta};u,v)\cdot
S(\theta^*; uv). \]
By \ref{lead},
\[ S(\theta^*; uv) = l^*(\theta^*)(uv)^{{\rm dim}\, \theta^*} +
\mbox{\rm $\{$lower order terms in $uv$$\}$}. \]
On the other hand, by \cite{dan.hov}, Prop. 3.9,
\[e^{p,q}(Z_{f,\theta}) =0\;\; \mbox{if
$p + q > {\rm dim}\, \theta - 1 =
{\rm dim}\, Z_{f,\theta}$ and $p \neq q$}. \]
Hence, the only possible case in which we can meet the monomial of
type $u^{d-1-p}v^{d-2}$ within the product
$ E(Z_{f,\theta};u,v)\cdot
S(\theta^*; u,v)$ is that occuring by consideration of
the product of the term $l^*(\theta^*)(uv)^{{\rm dim}\, \theta^*}$ from
$S(\theta^*; uv)$ and the term
\[ e^{0, {\rm dim}\, \theta -1}(Z_{f,\theta}) v^{{\rm dim}\, \theta -1}, \]
where ${\rm dim}\, \theta^* = d - 1 - p$.
As it is known (cf. \cite{dan.hov}, Prop. 5.8.):
\[ e^{0, {\rm dim}\, \theta -1}(Z_{f,\theta})
= (-1)^{{\rm dim}\, \theta -1}l^*(\theta). \]
Therefore,
\[ h^{d-1-p,d-2}_{\rm st}(\overline{Z}_f) = l^*(\theta) \cdot
l^*(\theta^*). \]
\hfill $\Box$
\begin{coro}
Let $\hat{Z}_f$ be a MPCP-desingularization of $\overline{Z}_f$.
Assume that $d \geq 5$. Then, for $ 2 \leq p \leq d-3$, one has
\[ h^{p,1}(\hat{Z}_f) =
\sum_{{\rm codim}\, \theta = p } l^*(\theta)\cdot
l^*(\theta^*). \]
\label{p1cor}
\end{coro}
\noindent
{\em Proof.} It follows from Theorem \ref{p1} and
Theorem \ref{MPCP-desing}. \hfill $\Box$
\section{Duality of string-theoretic Hodge numbers
for the Greene-Plesser construction}
In \cite{greene0,greene1} B. Greene and R. Plesser proposed
an explicit construction of mirror pairs of Calabi-Yau orbifolds
which are obtained as abelian quotients of Fermat hypersurfaces
in weighted projective spaces. As it was shown in \cite{batyrev1},
5.5, the Greene-Plesser construction can be interpreted
in terms of the polar duality of {\em reflexive simplices}.
The main purpose of this section is to
verify the mirror duality of all string-theoretic Hodge
numbers for this construction.
{}From now on, we assume that $\Delta$ and $\Delta^*$ are
$d$-dimensional reflexive simplices.
We shall prove Conjecture \ref{symmetry} for
$\Delta$-regular Calabi-Yau hypersurfaces in
${\bf P}_{\Delta}$ and ${\bf P}_{\Delta^*}$.
(We remind that, for this kind of hypersurfaces and for $d = 4$,
Conjecture \ref{symmetry} was proved in
\cite{roan0,batyrev1}.)
\begin{dfn}
{\rm Let $\Theta$ be a $k$-dimensional lattice simplex. We denote
by $\tilde{S}(\Theta; uv)$ the $\tilde{S}$-polynomial of the
$(k+1)$-dimensional abelian quotient singularity defined by
$\Theta$. We denote the corresponding finite abelian subgroup
of $SL(k+1,{\bf C})$ by $G_{\Theta}$ (in the sence of \S 4,5).}
\end{dfn}
Our main statement is an immediate consequence of
the following:
\begin{theo}
Let $\overline{Z}_f$ be a $\Delta$-regular Calabi-Yau hypersurface in
${\bf P}_{\Delta}$. Then
\[ E_{\rm st}(\overline{Z}_f; u,v) =
\frac{1}{uv}\tilde{S}(\Delta^*;uv) + (-1)^{d-1} \frac{u^{d}}{v}
\tilde{S}(\Delta; u^{-1}v) + \]
\[ + \sum_{\begin{array}{c} {\scriptstyle 1 \leq {\rm dim}\,\theta \leq d-2}
\\ {\scriptstyle \theta \subset \Delta} \end{array}}
(-1)^{{\rm dim}\, \theta-1} \left(
\frac{u^{{\rm dim}\, \theta}}{v} \tilde{S}(\theta; u^{-1}v) \cdot
\tilde{S}( \theta^*; uv) \right). \]
\label{formul}
\end{theo}
\noindent
Indeed, if we apply Theorem \ref{formul} to the dual polyhedron
$\Delta^*$, then we get
\[ E_{\rm st}(\overline{Z}_g; u,v) =
\frac{1}{uv}\tilde{S}(\Delta;uv) + (-1)^{d-1} \frac{u^d}{v}
\tilde{S}(\Delta^*; u^{-1}v) + \]
\[ + \sum_{\begin{array}{c} {\scriptstyle 1 \leq {\rm dim}\,\theta^* \leq d-2}
\\ {\scriptstyle \theta^* \subset \Delta^*} \end{array}}
(-1)^{{\rm dim}\, \theta^* -1} \left(
\frac{u^{{\rm dim}\, \theta^*}}{v} \tilde{S}(\theta^*; u^{-1}v) \cdot
\tilde{S}( \theta; uv) \right). \]
Now the required equality
\[ E_{\rm st}(\overline{Z}_f; u,v) =(-u)^{d-1}
E_{\rm st}(\overline{Z}_g; u^{-1},v) \]
follows evidently from the $1$-to-$1$ correspondence
$\theta \leftrightarrow \theta^*$ $( 1 \leq {\rm dim}\, \theta,\,
{\rm dim}\, \theta^* \leq d-1)$ and
from the property: ${\rm dim}\, \theta + {\rm dim}\, \theta^* = d-1$.
\bigskip
For the proof of Theorem \ref{formul}, we need some preliminary
facts.
\begin{prop} Let $\theta$ be a face of $\Delta$ and
${\rm dim}\, \theta \geq 1$. Then
\[ E(Z_{f,\theta}; u,v) = \frac{ (uv-1)^{{\rm dim}\, \theta} -
(-1)^{{\rm dim}\, \theta}}{uv} + (-1)^{{\rm dim}\, \theta-1}
\left( \sum_{\begin{array}{c} {\scriptstyle {\rm dim}\, \tau \geq 1} \\
{\scriptstyle \tau \subset \theta} \end{array}}
\frac{u^{{\rm dim}\, \tau}}{v} \tilde{S}(\tau; u^{-1}v) \right) . \]
\label{e-ff}
\end{prop}
\noindent
{\em Proof. } By \cite{dan.hov}, Prop. 3.9, the natural mapping
\[ H^i_c(Z_{f, \theta}) \rightarrow H^{i+1}_c(T_{\theta}) \]
is an isomorphism if $i > {\rm dim}\, \theta -1$ and
surjective if $i = {\rm dim}\, \theta -1$. Moreover,
$H^i_c(Z_{f, \theta}) = 0$ if $i < {\rm dim}\, \theta -1$.
In order to compute the mixed Hodge structure in
$H^{{\rm dim}\, \theta -1}_c(Z_{f, \theta})$, we use
the explicit description of the weight filtration in
$H^{{\rm dim}\, \theta -1}_c(Z_{f, \theta})$ (see \cite{batyrev0}).
Note that if we choose a $\theta$-regular Laurent polynomial $f$
containing only ${\rm dim}\, \theta + 1$ monomials
associated with vertices of
$\theta$ (such a polynomial $f$ defines a Fermat-type
hypersurface $\overline{Z}_f$ in ${\bf P}_{\theta}$), then
the corresponding Jacobian ring $R_f$ has a monomial
basis. Thus, the weight filtration on $R_f$ can be described
in terms of the partition of
monomials in $R_f$ which is defined by the faces $\tau \subset \theta$.
To get the claimed formula, it suffices to
identify the partition of monomials in $R_f$ with
the height-partition of elements of the
finite abelian group $G_{\theta} \subset SL({\rm dim}\, \theta +1,
{\bf C})$ and its subroups $G_{\tau} \subset G_{\theta}$.
Another way to obtain the same result is to use the formulae
of Danilov and Khovanski\^i (cf. \cite{dan.hov}, \S 5.6,5.7)
which are valid for
an arbitrary simple polyhedron $\Delta$.
\hfill $\Box$
\begin{prop}
Let $\theta$ be a face of $\Delta$ and
${\rm dim}\, \theta \geq 1$. Then
\[ S(\theta; t) = 1 + \sum_{\begin{array}{c}
{\scriptstyle {\rm dim}\, \eta \geq 1} \\ {\scriptstyle
\eta \subset \theta} \end{array} }
\tilde{S}( \eta; t). \]
\label{can}
\end{prop}
\noindent
{\em Proof. } It is similar to that of \ref{can.strat}. \hfill
$\Box$
\begin{prop}
We fix a face $\tau \subset \Delta$ and a face $\eta \subset \Delta^*$,
such that: $\tau$ is a face of $\eta^*$. Then
\[ \sum_{\theta,\; \tau \subset \theta \subset \eta^*}
(-1)^{{\rm dim}\, \theta} = (-1)^{{\rm dim}\, \tau}\;\;
\mbox{if $ \tau = \eta^*$ } \]
and
\[ \sum_{\theta,\; \tau \subset \theta \subset \eta^*}
(-1)^{{\rm dim}\, \theta} = 0\;\;
\mbox{if $ \tau \neq \eta^*$. } \]
\label{sum1}
\end{prop}
\noindent
{\em Proof.}
If $\eta^* = \tau$, this is obvious.
For ${\rm dim}\, \eta^* > {\rm dim}\, \tau $,
the number of faces $\theta \subset \Delta$, for which
$\tau \subset \theta \subset \eta^*$, is equal to
${ {\rm dim}\, \eta^* - {\rm dim}\, \tau
\choose {\rm dim}\, \theta - {\rm dim}\, \tau }$.
It remains to use the equality
\[ \sum_{\theta, \tau \subset \theta \eta^*}
(-1)^{ {\rm dim}\, \theta } = (-1)^{{\rm dim}\, \tau }
\left( \sum_{ i =0}^{{\rm dim}\, \eta^* - {\rm dim}\, \tau}
(-1)^i { {\rm dim}\, \eta^* - {\rm dim}\, \tau \choose i} \right) = 0. \]
\hfill $\Box$
\begin{prop}
\[ \frac{1}{uv} \tilde{S}(\Delta; uv) =
\frac{(uv)^d -1}{uv -1}\;\;\; +
\sum_{\begin{array}{c} {\scriptstyle 1 \leq {\rm dim}\,\tau \leq d-2} \\
{\scriptstyle \tau \subset \Delta} \end{array}}
\left( \frac{(uv)^{{\rm dim}\, \tau^*} -1}{uv -1} \right) \cdot
\tilde{S}( \eta; uv). \]
\label{tilde-s}
\end{prop}
\noindent
{\em Proof.}
By Proposition \ref{relation}, we have
\[ (-1)^{d-1} + \frac{S(\Delta;t)}{(1-t)^d}\;\; = \;\;
(-1)^{d-1} \sum_{0 \leq {\rm dim}\, \theta \leq d-1}
(-1)^{{\rm dim}\, \theta}
\frac{S(\theta;t)}{(1-t)^{{\rm dim}\, \theta +1}}. \]
Applying Proposition \ref{can} to both sides of this equality, we get
\[ (-1)^{d-1}\;\; + \;\; \frac{1}{(1-t)^d}\;\; +
\sum_{\begin{array}{c} {\scriptstyle {\rm dim}\, \tau \geq 1}
\\ {\scriptstyle \tau \subset \Delta} \end{array}}
\frac{\tilde{S}(\tau; t)}{(1-t)^d} \;\; = \]
\[ = \;\; (-1)^{d-1} \sum_{0 \leq {\rm dim}\, \theta \leq d-1}
\frac{(-1)^{{\rm dim}\, \theta}}{( 1- t)^{{\rm dim}\, \theta + 1}} \;\; + \]
\[ + \;\; (-1)^{d-1} \sum_{0 \leq {\rm dim}\, \theta \leq d-1}
(-1)^{{\rm dim}\, \theta}
\sum_{\begin{array}{c} {\scriptstyle {\rm dim}\, \tau \geq 1}
\\ {\scriptstyle \tau \subset \theta} \end{array}}
\frac{\tilde{S}(\tau; t)}{(1-t)^{{\rm dim}\, \theta +1}}. \]
As the number of $k$-dimensional faces of $\Delta$ equals ${ d+1 \choose
k+1 }$, we have
\[ - \;\; (-1)^{d-1} \; - \; \frac{1}{(1-t)^d} +
(-1)^{d-1} \sum_{0 \leq \theta \leq d-1}
\frac{(-1)^{{\rm dim}\, \theta}}{( 1- t)^{{\rm dim}\, \theta + 1}} \; = \]
\[ -\;\; (-1)^{d-1} \; - \; \frac{1}{(1-t)^d} \;\; + \;\; \sum_{k =0}^{d-1}
\frac{(-1)^k}{(1-t)^{k+1}} { d+1 \choose k+1 } \; = \;
(-1)^d \frac{t^{d+1} - t}{(t-1)^{d+1}} \]
and we can deduce that:
\[ \frac{\tilde{S}(\Delta, t)}{(1-t)^d} \; +
\sum_{{\rm dim}\, \tau = d-1} \frac{\tilde{S}(\tau, t)}{(1-t)^d} \; +
\sum_{1 \leq {\rm dim}\, \tau \leq d-2}
\frac{\tilde{S}(\tau, t)}{(1-t)^d} \; = \]
\[ = (-1)^d \frac{t^{d+1} - t}{(t-1)^{d+1}}\;\; +
\sum_{{\rm dim}\, \tau = d-1} \frac{\tilde{S}(\tau, t)}{(1-t)^d} \;\; + \]
\[ + \sum_{{\rm dim}\, \theta = d-1} \sum_{\begin{array}{c}
{\scriptstyle 1 \leq {\rm dim}\,\tau \leq d-2} \\
{\scriptstyle \tau \subset \theta} \end{array}}
\frac{\tilde{S}(\tau, t)}{(1-t)^d} \;\; + \]
\[ + \;\; (-1)^{d-1}
\sum_{ 1 \leq {\rm dim}\, \theta \leq d-2} (-1)^{{\rm dim}\, \theta}
\sum_{\begin{array}{c} {\scriptstyle {\rm dim}\, \tau \geq 1}
\\ {\scriptstyle\tau \subset \theta} \end{array} }
\frac{\tilde{S}(\tau, t)}{(1- t)^{{\rm dim}\, \theta + 1}}. \]
The terms containing $\tilde{S}(\tau, t)$, with
${\rm dim}\, \tau = d-1$, have the same contribution
to the right and left hand sides. The coefficient of
$\tilde{S}(\tau, t)$
$( 1 \leq {\rm dim}\, \tau \leq d-2 )$ in the right hand side of the
last equality is
\[ (-1)^{d-1} \sum_{\begin{array}{c}
{\scriptstyle {\rm dim}\, \theta \leq d-2}
\\ {\scriptstyle \tau \subset \theta} \end{array}}
(-1)^{{\rm dim}\, \theta} \frac{1}{(1-t)^{{\rm dim}\, \theta + 1}} \;\; = \]
\[ = \;\; \frac{(-1)^d}{(t-1)^{d+1}} \left( t^{d- {\rm dim}\, \tau} - 1 -
(d - {\rm dim}\,\tau ) (t-1) \right). \]
Correspondingly, the coefficient of $\tilde{S}(\tau, t)$
$(1 \leq {\rm dim}\, \tau \leq d-2)$ in the left hand side equals
\[ \frac{d - 1 - {\rm dim}\, \tau}{(1-t)^d}. \]
Finally, using ${\rm dim}\, \tau + {\rm dim}\, \tau^* = d -1$, we
obtain:
\[ \frac{\tilde{S}(\Delta, t)}{(1-t)^d} \; = \;
(-1)^d \frac{t^{d+1} - t}{(t-1)^{d+1}} \; + \;
(-1)^d \sum_{1 \leq \tau \leq d-2}
\tilde{S}(\tau, t)
\frac{(t^{{\rm dim}\, \tau^* + 1} - t)}{ (t - 1)^{d + 1} }. \]
\hfill $\Box$
\noindent
{\bf Proof of Theorem \ref{formul}}. By definition,
\[ E_{\rm st}(\overline{Z}_f; u,v) \; = \; E(Z_{f, \Delta}; u,v) \; + \;
\sum_{\begin{array}{c}
{\scriptstyle {\rm dim}\,\theta = d-1} \\
{\scriptstyle \theta \subset \Delta} \end{array}}
E(Z_{f, \theta}; u,v) \;\; + \]
\[ + \sum_{\begin{array}{c}
{\scriptstyle 1 \leq {\rm dim}\,\theta \leq d-2 } \\
{\scriptstyle \theta \subset \Delta} \end{array}}
E(Z_{f, \theta}; u,v) \cdot S(\theta^*; uv). \]
Substituting the expressions which were found out in \ref{e-ff} for
the $E$-polynomials of the above three summands, we get:
\[ E(Z_{f, \Delta}; u,v) = \frac{ (uv-1)^{d} - (-1)^{d}}{uv}
+ (-1)^{d-1}
\left( \sum_{\begin{array}{c} {\scriptstyle {\rm dim}\, \tau \geq 1} \\
{\scriptstyle \tau \subset \Delta} \end{array}}
\frac{u^{{\rm dim}\, \tau}}{v} \tilde{S}(\tau; u^{-1}v) \right) , \]
\newline
\[ \sum_{\begin{array}{c}
{\scriptstyle {\rm dim}\,\theta = d-1} \\ {\scriptstyle
\theta \subset \Delta} \end{array} }
E(Z_{f, \theta}; u,v) \;\; = \;\; \sum_{\begin{array}{c}
{\scriptstyle {\rm dim}\,\theta = d-1} \\
{\scriptstyle \theta \subset \Delta} \end{array}}
\frac{ (uv-1)^{{\rm dim}\, \theta} -
(-1)^{{\rm dim}\, \theta}}{uv} \;\; + \]
\[ + \sum_{\begin{array}{c}
{\scriptstyle {\rm dim}\,\theta = d-1} \\
{\scriptstyle \theta \subset \Delta} \end{array}} (-1)^{{\rm dim}\, \theta-1}
\left( \sum_{\begin{array}{c}
{\scriptstyle {\rm dim}\, \tau \geq 1} \\
{\scriptstyle \tau \subset \theta} \end{array}}
\frac{u^{{\rm dim}\, \tau}}{v} \tilde{S}(\tau; u^{-1}v) \right), \]
and
\[ \sum_{\begin{array}{c}
{\scriptstyle 1 \leq {\rm dim}\,\theta \leq d-2} \\
{\scriptstyle \theta \subset \Delta} \end{array}}
E(Z_{f, \theta}; u,v) \cdot S(\theta^*; uv) \;\; =
\;\; \sum_{\begin{array}{c}
{\scriptstyle 1 \leq {\rm dim}\,\theta \leq d-2} \\
{\scriptstyle \theta \subset \Delta} \end{array}}
\frac{ (uv-1)^{{\rm dim}\, \theta} -
(-1)^{{\rm dim}\, \theta}}{uv} \;\; + \]
\[ + \sum_{\begin{array}{c}
{\scriptstyle 1 \leq {\rm dim}\,\theta \leq d-2} \\
{\scriptstyle \theta \subset \Delta} \end{array}} (-1)^{{\rm dim}\, \theta-1}
\left( \sum_{\begin{array}{c}
{\scriptstyle {\rm dim}\, \tau \geq 1} \\
{\scriptstyle \tau \subset \theta}
\end{array}}
\frac{u^{{\rm dim}\, \tau}}{v} \tilde{S}(\tau; u^{-1}v)
\right) \cdot
\left( 1 + \sum_{
\begin{array}{c} {\scriptstyle {\rm dim}\, \eta \geq 1} \\
{\scriptstyle \eta \subset \theta^*} \end{array}}
\tilde{S}( \eta; uv) \right). \]
Hence, $ E_{\rm st}(\overline{Z}_f; u,v)$
can be written as the sum of the following $4$ terms $E_i$ $( i =1,2,3,4)$:
\newline
\[ E_1 = \sum_{\begin{array}{c}
{\scriptstyle 1 \leq {\rm dim}\, \theta} \\
{\scriptstyle \theta \subset \Delta} \end{array}}
\frac{ (uv-1)^{{\rm dim}\, \theta} -
(-1)^{{\rm dim}\, \theta}}{uv}, \]
\newline
\[ E_2 = \sum_{\begin{array}{c}
{\scriptstyle 1 \leq {\rm dim}\, \theta} \\
{\scriptstyle \theta \subset \Delta}
\end{array}}
(-1)^{{\rm dim}\, \theta-1}
\left( \sum_{\begin{array}{c}
{\scriptstyle {\rm dim}\, \tau \geq 1} \\
{\scriptstyle \tau \subset \theta}
\end{array}}
\frac{u^{{\rm dim}\, \tau}}{v} \tilde{S}(\tau; u^{-1}v) \right) ,\]
\[ E_3 = \sum_{\begin{array}{c}
{\scriptstyle 1 \leq {\rm dim}\,\theta \leq d-2} \\
{\scriptstyle \theta \subset \Delta} \end{array}}
\left( \frac{ (uv-1)^{{\rm dim}\, \theta} -
(-1)^{{\rm dim}\, \theta}}{uv} \right) \cdot
\left( \sum_{\begin{array}{c}
{\scriptstyle {\rm dim}\, \eta \geq 1} \\
{\scriptstyle \eta \subset \theta^*} \end{array}}
\tilde{S}( \eta; uv) \right), \]
and
\[ E_4 = \sum_{\begin{array}{c}
{\scriptstyle 1 \leq {\rm dim}\,\theta \leq d-2} \\
{\scriptstyle \theta \subset \Delta}
\end{array}}
(-1)^{{\rm dim}\, \theta-1}
\left( \sum_{\begin{array}{c}
{\scriptstyle {\rm dim}\, \tau \geq 1} \\
{\scriptstyle \tau \subset \theta} \end{array}}
\frac{u^{{\rm dim}\, \tau}}{v} \tilde{S}(\tau; u^{-1}v) \right)
\cdot
\left( \sum_{\begin{array}{c}
{\scriptstyle {\rm dim}\, \eta \geq 1} \\
{\scriptstyle \eta \subset \theta^*}\end{array}}
\tilde{S}( \eta; uv) \right). \]
By \ref{sum1}, we can simplify the multiple summation into a single
sum:
\[ E_4 = \sum_{\begin{array}{c}
{\scriptstyle 1 \leq {\rm dim}\,\theta \leq d-2} \\
{\scriptstyle \theta \subset \Delta} \end{array}}
(-1)^{{\rm dim}\, \theta-1} \left(
\frac{u^{{\rm dim}\, \theta}}{v} \tilde{S}(\theta; u^{-1}v) \cdot
\tilde{S}( \theta^*; uv) \right). \]
If we make use of the combinatorial identity
\[ \sum_{\begin{array}{c}
{\scriptstyle 1 \leq {\rm dim}\, \theta} \\
{\scriptstyle \theta \subset \Delta} \end{array}}
a^{{\rm dim}\, \theta} = \sum_{k =2}^{d+1} { d+1 \choose k } a^{k-1} =
a^{-1} \left( (a+1)^{d+1} - 1 - (d+1) a \right), \]
we obtain:
\[ E_1 = \sum_{\begin{array}{c}
{\scriptstyle 1 \leq {\rm dim}\, \theta} \\
{\scriptstyle \theta \subset \Delta}\end{array}}
\frac{ (uv-1)^{{\rm dim}\, \theta} -
(-1)^{{\rm dim}\, \theta}}{uv} = \]
\[ = [uv(uv - 1)]^{-1} \left( (uv)^{d+1} - 1 - (d+1)(uv -1) \right) +
d (uv)^{-1} = \frac{(uv)^d -1}{uv -1}. \]
By \ref{sum1}, we get
\[ E_2 = \sum_{\begin{array}{c}
{\scriptstyle 1 \leq {\rm dim}\, \theta} \\
{\scriptstyle \theta \subset \Delta} \end{array}}
(-1)^{{\rm dim}\, \theta-1}
\sum_{\begin{array}{c}
{\scriptstyle {\rm dim}\, \tau \geq 1} \\
{\scriptstyle \tau \subset \theta} \end{array}}
\frac{u^{{\rm dim}\, \tau}}{v} \tilde{S}(\tau; u^{-1}v) =
(-1)^{d-1} \frac{u^d}{v} \tilde{S}(\Delta; u^{-1}v). \]
\noindent
It remains to compute $E_3$. As above for $E_1$, we have
\[ \sum_{\begin{array}{c}
{\scriptstyle 1 \leq {\rm dim}\, \theta} \\
{\scriptstyle \theta \subset \eta^*} \end{array}}
\frac{ (uv-1)^{{\rm dim}\, \theta} -
(-1)^{{\rm dim}\, \theta}}{uv} =
\frac{(uv)^{{\rm dim}\, \eta^*} -1}{uv -1}. \]
Hence, by \ref{tilde-s},
\[ E_3 = \sum_{\begin{array}{c}
{\scriptstyle 1 \leq {\rm dim}\,\eta \leq d-2} \\
{\scriptstyle \eta \subset \Delta^*}
\end{array}}
\left( \frac{(uv)^{{\rm dim}\, \eta^*} -1}{uv -1} \right) \cdot
\tilde{S}( \eta; uv) = \frac{1}{uv}\tilde{S}(\Delta^*;uv) -
\frac{(uv)^d -1}{uv -1}. \]
Finally, we get altogether
\[ E_{\rm st}(\overline{Z}_f; u,v) = \frac{1}{uv}\tilde{S}(\Delta^*;uv)
+ (-1)^{d-1} \frac{u^d}{v}
\tilde{S}(\Delta; u^{-1}v) + \]
\[ + \sum_{\begin{array}{c}
{\scriptstyle 1 \leq {\rm dim}\,\theta \leq d-2} \\
{\scriptstyle \theta \subset \Delta}
\end{array}}
(-1)^{{\rm dim}\, \theta-1} \left(
\frac{u^{{\rm dim}\, \theta}}{v} \tilde{S}(\theta; u^{-1}v) \cdot
\tilde{S}( \theta^*; uv) \right). \]
\hfill $\Box$
\begin{exam}
{\rm The polar duality between reflexive simplices shows
(cf. \cite{batyrev1}, Thm. 5.1.1.) that the family of all smooth
Calabi-Yau hypersurfaces $X_{d+1}$ of degree $d+1$ in ${\bf P}^d$
has as its mirror partner the one-parameter family
$\{ Q_{d+1}(\lambda)/G_{d+1} \}$, where
\[ Q_{d+1}(\lambda) := \{ [z_0, \ldots, z_d ] \in {\bf P}^d \mid
\sum_{i=0}^d z_i^{d+1} - (d+1)\lambda\prod_{ i=0}^d z_i = 0 \} \]
denotes the so called {\em Dwork pencil} and
$G_{d+1}$ the acting finite abelian group
\[ G_{d+1} := \{ (\alpha_0, \ldots, \alpha_d) \in
({\bf Z}/(d+1){\bf Z})^{d+1} \mid \prod_{i =0}^d \alpha_i = 1 \} /
\{\rm scalars \}, \]
which is abstractly isomorphic to $({\bf Z}/(d+1){\bf Z})^{d-1}$.
The moduli space ${\bf P}^1 \setminus \{ 0,1, \infty \}$ of
$\{ Q_{d+1}(\lambda)/G_{d+1} \}_{\lambda}$ can be described by
means of the parameter $\lambda^{d+1}$ (cf. \cite{greene},
\S 3.1, \cite{morrison}, \S 5, and \cite{morrison1} \S 11).
Since Conjecture \ref{symmetry} is true for the case being
under consideration, the quotient $Q_{d+1}(\lambda)/G_{d+1}$ has
the following string-theoretic Hodge numbers:
\[ h^{p,q}_{\rm st} (Q_{d+1}(\lambda)/G_{d+1}) =
h^{p,q}(Q_{d+1}(\lambda),G_{d+1}) = h^{d-1-p,q}(X_{d+1}) = \delta_{d-1-p,q},
\;\;\; \mbox{\rm for $p \neq q$}; \]
\[ h^{p,p}_{\rm st} (Q_{d+1}(\lambda)/G_{d+1}) =
h^{p,p}(Q_{d+1}(\lambda),G_{d+1}) = h^{d-1-p,p}(X_{d+1}) \;= \]
\[ = \; \sum_{i =0}^p (-1)^i { d+1 \choose i }
{ (p+1 -i)d + p \choose d } + \delta_{2p,d-1} . \]
In particular, the string-theoretic Euler number is given by:
\[ e_{\rm st} (Q_{d+1}(\lambda)/G_{d+1}) =
e(Q_{d+1}(\lambda),G_{d+1}) = - e(X_{d+1})\; = \]
\[ = \; \frac{1}{d+1} \left(
(-1)^{d+2} \cdot d^{d+1} + 1 \right) - d - 1. \]
The first two equalities follow from Lefschetz hyperplane section
theorem and from the ``four-term formula'' (cf. \cite{hirzebruch1},
\S 2.2 ). The third one can be obtained directly by computing the
$(d-1)$-th Chern class of $X_{d+1}$. }
\end{exam}
|
1994-10-12T05:20:12 | 9410 | alg-geom/9410006 | en | https://arxiv.org/abs/alg-geom/9410006 | [
"alg-geom",
"math.AG"
] | alg-geom/9410006 | null | Barbara Fantechi, Rita Pardini | Automorphisms and moduli spaces of varieties with ample canonical class
via deformations of abelian covers | 30 pages, LaTeX | null | null | null | null | By a recent result of Viehweg, projective manifolds with ample canonical
class have a coarse moduli space, which is a union of quasiprojective
varieties. In this paper, we prove that there are manifolds with ample
canonical class that lie on arbitrarily many irreducible components of the
moduli; moreover, for any finite abelian group $G$ there exist infinitely many
components $M$ of the moduli of varieties with ample canonical class such that
the generic automorphism group $G_M$ is equal to $G$. In order to construct the
examples, we use abelian covers, i.e. Galois cover whose Galois group is finite
and abelian. We prove two results about abelian covers: first, that if the
building data are sufficiently ample, then the natural deformations surject on
the Kuranishi family of $X$; second, that if the building data are sufficiently
ample and generic, then $Aut(X)=G$.
| [
{
"version": "v1",
"created": "Tue, 11 Oct 1994 15:51:18 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Fantechi",
"Barbara",
""
],
[
"Pardini",
"Rita",
""
]
] | alg-geom | \section{Introduction} Coverings of algebraic varieties are a classical theme
in algebraic geometry, since Riemann's description of curves as branched
covers of the projective line. Double covers were used by the Italian school
to construct examples that shed light on the theory of surfaces and to
describe special classes of surfaces, as in the case of Enriques surfaces.
More recently, cyclic coverings have been extensively applied by several
authors to the study of surfaces of general type; it will be enough to recall
the work of Horikawa, Persson and Xiao Gang. Abelian covers have been used
by Hirzebruch to give examples of surfaces of general type on and near the
line $c_1^2=3c_2$; Catanese and Manetti have used bidouble and iterated
double covers, respectively, of $\P^1\times \P^1$ to construct explicitly
connected components of the moduli space of surfaces of general type.
In \cite{Pa1}, the second author has given a complete description of abelian
covers of algebraic varieties in terms of the so-called building data, namely
of certain line bundles and divisors on the base of the covering, satisfying
suitable compatibility relations. Natural deformations of an abelian cover
$f:X\to Y$ are also introduced there and it is shown that they are complete,
if $Y$ is rigid, regular and of dimension $\ge 2$, and if the building data
are sufficiently ample. (Natural deformations are obtained by modifying the
equations defining $X$ inside the total space of the bundle $f_*\O_X$).
In this paper we study natural deformations of an abelian cover $f:X\to Y$
and prove that they are complete for varieties of dimension at least two if
the branch divisors are sufficiently ample. The result requires no
assumption on
$Y$, and in particular also holds when the cover has obstructed
deformations; this is a key technical step towards the moduli space
constructions described below.
We then turn to the study of the automorphism group of the cover. Since the
automorphism group of a variety of general type is finite, one would expect
that in the case of a Galois cover it coincides with the Galois group, at
least if the cover is generic. Our main theorem \ref{mainthm} shows that this
is indeed the case for an abelian cover, if the branch divisors are generic
and sufficiently ample.
We construct explicitly coarse moduli spaces of abelian covers and
complete families of natural deformations for a fixed base of the cover $Y$;
this is useful if one wants to investigate the birational structure of the
components of the moduli obtained by the methods of this paper.
The main application of the results described so far is the study of moduli
of varieties with ample canonical class. Recently Viehweg proved the
existence of a coarse moduli space for varieties with ample canonical class
of arbitrary dimension, generalizing Gieseker's result for surfaces. Given
an irreducible component
$M$ of the moduli space of varieties with ample canonical class, the
automorphism group $G_M$ of a generic variety in $M$ is well-defined. In
contrast with the case of curves (where this group is trivial for $g\ge 3$),
it was already known in the case of surfaces that there exist infinitely
many components
$M$ of the moduli with nontrivial automorphism group $G_M$; it is easy to
construct examples such that $G_M$ contains an involution, and Catanese
gave examples where $G_M$ contains a subgroup isomorphic to ${\bf Z}_2\times
{\bf Z}_2$. There are also, of course, easy examples of components $M$ where
$G_M$ is trivial (for instance the hypersurfaces of degree $d\ge 5$ in $\P^3$).
As a first application of theorem \ref{mainthm} we prove that for any finite
abelian group $G$ there are infinitely many irreducible components $M$ of the
moduli of varieties with ample canonical class such that $G_M=G$; notice that
we precisely determine $G_M$ instead of just bounding it from below.
We also prove that there are varieties with ample canonical class lying on
arbitrarily many irreducible components of the moduli. We distinguish these
components by means of their generic automorphism group; there are examples
both in the equidimensional and in the non-equidimensional case. In the
surface case, this answers a question raised by Catanese in \cite{Ca2}.
Let $S$ be a surface of general type; Xiao has given explicit upper bounds
both for the cardinality of $Aut(S)$ and of an abelian subgroup of $Aut(S)$,
in terms of the invariants of $S$ (\cite{Xi1}, \cite{Xi2}). Some upper
bounds are also known for a higher-dimensional variety $X$ with
ample canonical class, although sharp bounds are still
lacking. It seems interesting to ask whether these bounds can be improved by
considering instead of $Aut(X)$ the group
$Aut_{\rm gen}(X)$, namely the intersection in
$Aut(X)$ of the images of the generic automorphism groups $G_M$ of all
irreducible components $M$ of the moduli space containg $X$ (in particular,
if $X$ lies in a unique component $M$, then $Aut_{\rm gen}(X)=G_M$).
As a first step towards the computation of a sharp bound for
$\#Aut_{\hbox{\rm gen}}(S)$, we show that such a bound cannot be ``too
small''; in fact we give a sequence of surfaces $S_n$ of general type, whose
Chern numbers tend to infinity with $n$, and such that
$\#Aut_{\hbox{\rm gen}}(S_n)\ge 2^{-4}K_{S_n}^2$.
The paper goes as follows: in section 2 we collect some results from the
literature and set up the notation. In section 3 we prove that, if the
branch divisors are
sufficiently ample, then infinitesimal natural
deformations are complete. In section 4 we prove (theorem \ref{mainthm}) that
the
automorphism group of an abelian cover coincides with the Galois group if the
building data are sufficiently ample and generic. To do this, we prove some
results on extensions of automorphisms, which we believe should be of
independent interest. The proof of \ref{mainthm} is based on a degeneration
argument and requires an explicit partial desingularization, contained in
section 7. Section 5 contains the construction of a
coarse moduli space for abelian covers of a given variety $Y$ and of a
complete family of natural deformations. Finally, in section 6 we apply
the results of sections 3 and 4 to
the study of moduli spaces of varieties with ample canonical class, as stated
above.
\smallskip
\noindent{\em Acknowledgements}. This work was supported by the italian MURST
60\% funds. The first author would also like to thank the Max-Planck-Institut
f\"ur Mathematik (Bonn) for hospitality and the italian CNR for support.
\section{Notation and conventions} All varieties will be complex, and smooth
and projective unless the contrary is explicitly stated.
For a projective morphism of schemes $Y\to S$, $Hilb_S(Y)$ will be the
relative Hilbert scheme (see \cite{FGA}, expos\'e 221). When $Y$ is smooth
over $S$, $Hilb^{\rm div}_S(Y)$ will be the (open and closed) subscheme of
$Hilb_S(Y)$ parametrizing divisors (see \cite{Fo} for a proof of this).
When $S$ is a point, it will be omitted
from the notation.
For $Y$ a smooth
projective variety, let
$c_1:Pic(Y)\to H^2(Y,{\bf Z})$ be the map associating to a line bundle its first
Chern class; let
$NS(Y)$ be the image in
$H^2(Y,{\bf Z})$ of
$Pic(Y)$, and
$Pic^\xi(Y)$ the inverse image of $\xi\in NS(Y)$. Let $q(Y)=\dim
H^1(Y,\O_Y)$ be the dimension of $Pic^0(Y)$.
Let ${\cal X}\to B$ be any flat family, with integral fibres. Then there are open
subschemes $Aut_{{\cal X}/B}$ and $Bir_{{\cal X}/B}$ of the relative Hil\-bert sche\-me
\hbox{$Hilb_B({\cal X}\times_B{\cal X})$} parametrizing fibrewise the (graphs of)
automorphisms and birational automorphisms of the fibre (\cite{FGA},
\cite{Ha}).
We denote the cardinality of a (finite) set $S$ by $\#S$; for each integer
$m\ge 2$, let $\zeta_m=e^{2\pi i/m}$.
\smallskip
\noindent{\em Notation for abelian covers}. The following notation will be
used freely throughout the paper: we collect it here for the reader's
convenience.
$G$ will be a finite abelian group, $G^*$ its dual; the
order of an element $g$ will be denoted by $\ord{g}$.
Let $I_G$ be the set of all pairs $(H,\psi)$ where $H$ is a cyclic subgroup
of $G$ with at least two elements and $\psi$ is a generator of $H^*$. There
is a bijection between $I_G$ and $G\setminus 0$ given by $(H,\psi)\mapsto g$
where $g\in H$ is such that $\psi(g)=\zeta_{\#H}$. For $\chi\in G^*$,
$i=(H_i,\psi_i)\in I_G$, let $\re^i_\chi$ be the unique integer such that
$0\le \re^i_\chi<m_i$ (where $m_i=\#H_i$) and
$\chi_{|H_i}=\psi_i^{\re^i_\chi}$ (cfr.\ \cite{Pa1}, remark 1.1 on p.~195,
where $\re^i_\chi$ is denoted by $f_{H,\psi}(\chi)$). Let
$\eps^i_{\chi,\chi'}=[(\re^i_\chi+\re^i_{\chi'})/m_i]$, where $[r]$ is the integral part of
a real number $r$; note that $\eps^i_{\chi,\chi'}$ is either $0$ or $1$.
A basis of $G$ will be a sequence of elements of $G$, $(e_1,\dots,e_s)$,
such that $G$ is the direct sum of
the (cyclic) subgroups generated by the $e_j$'s, and such that $\ord{e_j}$
divides $\ord{e_{j+1}}$ for each $j=1,\ldots,s-1$.
Given a basis
$(e_1,\dots,e_s)$ of $G$, we will call dual basis of
$G^*$ the $s$-tuple
$(\chi_1,\ldots,\chi_s)$, where $\chi_j(e_i)=1$ if $i\ne j$ and
$\chi_i(e_i)=\zeta_{\ord{e_i}}$.
We will write $\re^i_j$ instead of $\re^i_{\chi_j}$, for all
$j=1,\ldots,s$; for $\chi=\chi_1^{\alpha_1}\cdots\chi_s^{\alpha_s}$, let
$$q^i_\chi=\left[\sum_{j=1}^s\frac{\alpha_i\re^i_j}{m_i}\right].$$
Note that, unlike $\re^i_\chi$, $q^i_\chi$ depends on the choice of the basis
and not only on $\chi$ and $i$.
\begin{lem} Let $G$ be as above, and let $I\subset I_G$ be a subset with $k$
elements {\rm(}which we denote by $1,\ldots,k${\rm)} such that the natural
map $H_1\oplus
\ldots \oplus H_k\to G$ is surjective. Then the $k\times s$ matrix
$(\re^i_j)$ has rank $s$ over ${\bf Q}$.
\end{lem}
\noindent{\sc Proof.~}
Let $g_i$ be the element corresponding to $(H_i,\psi_i)$ via the bijection
$I_G\leftrightarrow G\setminus 0$ described above. Then, for any
$i=1,\ldots,k$ and for any $j=1,\ldots,s$, one has $\re^i_j/m_i=\lambda_{ij}/n_j$,
where $n_j=\ord{e_j}$ and $g_i=\sum \lambda_{ij}e_j$, with $0\le \lambda_{ij}<n_j$ and
$\lambda_{ij}\in{\bf Z}$ . So the matrix $(\re^i_j)$ has the same rank over ${\bf Q}$ as the
matrix $\lambda_{ij}$. On the other hand $\lambda_{ij}$ is the matrix associated to
the natural map $H_1\oplus
\ldots \oplus H_k\to G$, which is surjective. Let $p$ be a prime factor of
$n_1$, hence of all of the $n_j$'s. Then the map ${\bf Z}_p^k\to {\bf Z}_p^s$
represented by the matrix $(\lambda_{ij})\ \hbox{\rm mod}\; p$ is also surjective, hence the
matrix $(\lambda_{ij})$ has an $s\times s$ minor whose determinant is nonzero
modulo $p$. This implies that the determinant is nonzero, hence the result.
\ $\Box$\par\smallskip
\smallskip
Let $X$ be any projective variety. A {\sl deformation} of $X$ over a pointed
analytic space
$(T,o)$ will be a flat, proper map ${\cal X}\to T$, together with an isomorphism of
the special fibre ${\cal X}_o$ with $X$.
Deformations modulo isomorphism are a
contravariant functor $Def_X$ from the category $\hbox{\sl Ansp}_0$ of pointed analytic
spaces to the category $\hbox{\sl Sets}$, where the functoriality is given by pullback.
More generally, given a contravariant functor $F:\hbox{\sl Ansp}_0\to \hbox{\sl Sets}$, we will
use the same letter $F$ to denote the induced functor on the categories $\hbox{\sl Germs}$
of germs of analytic spaces and $\hbox{\sl Art}^*$ of finite length spaces supported in a
point (i.e. $Spec$'s of local Artinian ${\bf C}$-algebras). For the properties of
functors on $\hbox{\sl Art}^*$, we refer the reader to \cite{schl}.
\smallskip
Let $M$ be an irreducible component of the moduli space of
(projective) manifolds with ample canonical class. As the automorphism group
is semicontinuous (see corollary 4.5), it makes sense to speak of the
automorphism group of a generic manifold in $M$; we will denote it by $G_M$.
Note that for any $X$ such that $[X]\in M$, there is a natural identification
of $G_M$ with a subgroup of $Aut(X)$. If $X$ is a minimal surface of general
type, we denote the intersection in
$Aut(X)$ of $G_M$ for all components $M$ containing $[X]$ by $Aut_{\hbox{\rm
gen}}(X)$; it is the largest subgroup $H$ of $Aut(X)$ such that the action of
$H$ extends to any small deformation of $X$.
\section{Deformations of abelian covers}
In this section we
introduce natural deformations of a smooth abelian cover and prove
that infinitesimal natural deformations
are complete, if the branch
divisors are sufficiently ample and the dimension is at least two.
We start by recalling from \cite{Pa1} some fundamental results on abelian
covers; the reader will find there a more detailed exposition and proofs of
the following statements.
Let $G$ be a finite abelian group and
let
$I$ be a subset of
$I_G$: we will use freely throughout the paper the notation introduced in
section 2. Let
$Y$ be a smooth projective variety: a $(G,I)$-cover of
$Y$ is a normal variety $X$ and a Galois cover $f:X\to Y$ with
Galois group $G$ and branch divisors $D_i$ (for $i\in I)$ having
$(H_i,\psi_i)$ as inertia group and induced character (see \cite{Pa1} for
details). $X$ is smooth if and only if the $D_i$'s are smooth, their union is
a normal crossing divisor, and, whenever
$D_{i_1},\ldots,D_{i_k}$ have a common point, the natural map
$H_{i_1}\oplus\ldots\oplus H_{i_k}\to G$ is injective. The cover is said to be
{\em totally ramified} if the natural map $\bigoplus_{i\in I}H_i\to G$ is
surjective. Note that each abelian cover can be factored as the composition of
a totally ramified with an unramified cover.
Let $M_i=\O_Y(D_i)$. The vector bundle $f_*\O_X$ on $Y$ splits naturally as sum
of eigensheaves
$L_\chi^{-1}$ for $\chi\in G^*$, and multiplication in the $\O_Y$-algebra
$f_*\O_X$ induces isomorphisms \begin{equation}
\label{bdata}L_\chi\otimes L_{\chi'}=L_{\chi\chi'}\otimes\Bigotimes_{i\in
I} M_i^{\otimes \eps^i_{\chi,\chi'}}\qquad\quad \hbox{for all
$\chi,\chi'\in G^*\setminus 1$}.
\end{equation}
Denote $L_{\chi_j}$ by $L_j$, and let $n_j=\ord{\chi_j}$. The isomorphisms
above induce isomorphisms
\begin{equation} \label{rbdata} L_j^{\otimes n_j} =\Bigotimes_{i\in
I}M_i^{\otimes\reb^i_j}\qquad\quad \hbox{for all $j=1,\ldots,s$}.
\end{equation}
The $(D_i,L_\chi)$ are the {\em building data} of the cover; the $(D_i,L_j)$
are the {\em reduced building data}. The sheaves $L_\chi$ can be recovered from
the reduced building data by setting, for
$\chi=\chi_1^{\alpha_1}\cdots\chi_s^{\alpha_s}$,
\begin{equation} \label{chidarbd}
L_\chi=\Bigotimes_{j=1}^s L_j^{\alpha_j}\otimes\Bigotimes_{i\in
I}M_i^{-q^i_\chi}.
\end{equation}
Conversely, for each choice of $(D_i,L_\chi)$ (resp.\ $(D_i,L_j)$) satisfying
equation (\ref{bdata}) (resp.\ (\ref{rbdata})), there exists a unique cover
having these as (reduced) building data. Note that equations (\ref{rbdata})
have
a solution in
$Pic(Y)$ (viewing the line bundles $M_i$'s as parameters and the
$L_j$'s as variables) if and only if their images via $c_1$ have a solution
in $\hbox{\sl NS}(Y)$.
\begin{assu}\label{totram} In this paper all $(G,I)$-covers will be totally
ramified.
Unless otherwise stated, $f:X\to Y$ will be a
$(G,I)$-cover, with reduced building data $(D_i,L_j)$. We will also assume
that
$X$ and
$Y$ are smooth, of dimension $\ge 2$, and that $X$ has ample canonical class.
\end{assu}
We say that a property holds whenever a line bundle $L$ (or a divisor $D$) is
sufficiently ample if it holds whenever $c_1(L)$ (or $c_1(D)$) belongs to a
(given) suitable translate of the ample cone. It is easy to see that
assumption \ref{totram} implies the following: if all of the $D_i$'s are
sufficiently ample then so is $L_\chi$ for any $\chi\ne 1$. Moreover, if $V$
is a vector bundle, $V\otimes L$ is ample for any sufficiently ample $L$.
Let $S=\{({i,\chi})\in I\times G^*|\chi_{|H_i}\ne\psi_i^{-1}\}$. Given a
$(G,I)$-cover
$X\to Y$ as above, together with sections
$s_{i,\chi}$ of
$H^0(M_i\otimes L_\chi^{-1})$ for all $({i,\chi})\in S$, a natural
deformation of
$X$ was defined in
\cite{Pa1}, \S 5.
We now give a functorial (and more general) version of that definition in
order to be able to apply standard techniques from deformation theory.
\begin{defn}\label{natdef}{\rm A {\em natural deformation of the reduced
building data} of
$f:X\to Y$ over $(T,o)\in\hbox{\sl Ansp}_0$ is $({\cal Y},{\cal M}_i,\L_j,s_{{i,\chi}},\phi_j)$
where:\begin{enumerate}
\item $i\in I$, $j=1,\ldots,r$, and $({i,\chi})\in S$;
\item ${\cal Y}\to T$ is a deformation of $Y$ over $T$;
\item $\L_j$ and ${\cal M}_i$ are line bundles on ${\cal Y}$
such that $\L_j$ restricts to $L_j$ and ${\cal M}_i$ to $M_i$ over $o$;
\item $
\phi_j:\L_j^{\otimes n_j}\to \bigotimes {\cal M}_i^{\otimes \reb^i_j}$ is an
isomorphism whose restriction to ${\cal Y}_o$ coincides with the isomorphism
$L_j^{\otimes n_j}\to \bigotimes M_i^{\otimes \reb^i_j}$ given by
multiplication;
\item $s_{{i,\chi}}$ is a section of $\L_\chi^{-1}\otimes{\cal M}_i$, where $
\L_\chi=\Bigotimes_{j=1}^s \L_j^{\alpha_j}\otimes\Bigotimes_{i\in
I}{\cal M}_i^{-q^i_\chi}$;
\item $s_{{i,\chi}}$ restricts over ${\cal Y}_o$ to $s_{i,\chi}^0$, where
$s_{i,\chi}^0=0$ if
$\chi\ne 1$, and $s^0_{i,1}$ is a section of $M_i$ defining
$D_i$.
\end{enumerate}
We will say that a deformation is {\em Galois} if $s_{i,\chi}=0$ for $\chi\ne
1$.}
\end{defn}
Natural deformations modulo isomorphism define a contravariant functor
$\hbox{\rm Dnat}_X:\hbox{\sl Ansp}_0\to
\hbox{\sl Sets}$, and Galois deformations are a subfunctor $\hbox{\rm Dgal}_X$. Note that the
inclusion
$\hbox{\rm Dgal}_X\hookrightarrow \hbox{\rm Dnat}_X$ is naturally split.
We now extend formulas in \S 5 of \cite{Pa1} to define a natural
transformation of functors
$\hbox{\rm Dnat}_X\to \defor X$.
\begin{defn}\label{trasnat}{\rm Let $T$ be a germ of an analytic space, and let
$$({\cal Y},\L_j,{\cal M}_i,\phi_j,s_{i,\chi})\in\hbox{\rm Dnat}_X(T).$$ Let $V$ be the
total space of the vector bundles $\bigoplus_{\chi\in G^*}\L_\chi$, and let
$\pi:V\to {\cal Y}$ be the natural projection. For a line bundle $\L$ on ${\cal Y}$,
denote its pullback to $V$ by $\bar \L$, and analogously for sections and
isomorphisms. Each of the line bundles $\bar\L_\chi$ has a tautological section
$\sigma_\chi$.
For each pair $(\chi,\chi')\in G^*\times G^*$, the isomorphisms $\phi_j$
induce isomorphisms $$\phi_{\chi,\chi'}:\L_\chi\otimes\L_{\chi'}
\to \L_{\chi\chi'}\otimes\Bigotimes {\cal M}_i^{\eps^i_{\chi,\chi'}}.$$
Let $\tau_i\in H^0(V,\bar{\cal M}_i)$ be defined by $$
\tau_i=\sum_{\{\chi|({i,\chi})\in S\}}\bar s_{i,\chi}\sigma_\chi.$$
Define a section $\rho_{\chi,\chi'}$ of $\bar\L_\chi\otimes\bar\L_{\chi'}$ by
$$\rho_{\chi,\chi'}=\sigma_\chi\sigma_{\chi'}-
\bar\phi_{\chi,\chi'}^*(\sigma_{\chi\chi'}{\textstyle\prod}\tau_i^{\eps^i_{\chi,\chi'}}).$$
Then the zero locus of all the $\rho_{\chi,\chi'}$ is naturally a deformation
${\cal X}\to T$ of
$X$ over $T$ (in particular $X$ can be naturally identified with the fibre of
${\cal X}\to T$ over the closed point). This is proven in
\cite{Pa1} in the case where the deformation of
$Y$, $L_j$ and $M_i$ is the trivial one, but it is easy to see that the same
proof works in our generalized setting. The deformation ${\cal X}\to T$ so obtained
is
called the {\em natural deformation of $X$} associated to the given natural
deformation
of the reduced building data.}
\end{defn}
It is now clear why Galois deformations were called that way:
\begin{rem} Let ${\cal X}\to T$ be a deformation of $X$ induced by a
Galois deformation $({\cal Y},\ldots)$ of the reduced building data; ${\cal X}$ has a
canonical structure of $(G,I)$-cover of ${\cal Y}$, induced by the action of $G$ on
the total space of the line bundle $\L_\chi$ given by the character $\chi$.
\end{rem}
The restrictions to the category $\hbox{\sl Art}^*$ of the functors $\hbox{\rm Dnat}_X $
and $\hbox{\rm Dgal}_X$ satisfy Schlessinger's conditions for the existence of a
projective hull (see \cite{schl}); in fact, they can be described (as usual in
deformation
theory) in terms of tangent and obstruction spaces. If $F:\hbox{\sl Art}^*\to \hbox{\sl Sets}$ is a
contravariant functor, then we denote its tangent (resp.~obstruction) space by
$T^1(F)$ (resp.~$T^2(F)$), when this makes sense.
\setcounter{equation}{0}
\begin{lem}\label{tdefnat} There is a natural action of $G$ on $\hbox{\rm Dnat}_X$,
whose invariant locus is $\hbox{\rm Dgal}_X $; the decomposition of $T^l(\hbox{\rm Dnat}_X )$
according to characters, for $l=1,2$, is the following:
\begin{eqnarray} &&T^l(\hbox{\rm Dgal}_X )=T^l(\hbox{\rm Dnat}_X )^{\rm
inv}=H^l(Y,T_Y(-\log{\textstyle\sum} D_i));\label{tgal}\\ &&T^l(\hbox{\rm Dnat}_X
)^{\chi}=\Bigoplus_{i\in S_\chi}
H^{l-1}(Y,\O_Y(D_i)\otimes L_\chi^{-1})\qquad\hbox{for $\chi\ne1$;}
\end{eqnarray} where $S_\chi=\{i\in I|({i,\chi})\in S\}$.
\end{lem}
\noindent{\sc Proof.~} An element $g\in G$ acts by $$({\cal Y},{\cal M}_i,\L_j,s_{i,\chi},\phi_j)
\mapsto
({\cal Y},{\cal M}_i,\L_j,\chi(g)s_{i,\chi},\phi_j).$$ It is clear that $\hbox{\rm Dgal}_X$ is
contained in the invariant locus. It is not difficult to show the other
inclusion using the fact that the cover is totally ramified.
We now study separately tangent and obstructions spaces corresponding to
the different characters. For the trivial character, i.e. $\hbox{\rm Dgal}_X$, the
functor is isomorphic to the deformation functor of the data $(Y,M_i,s_i)$;
(\ref{tgal}) is then well known (see \cite{We}).
Fix a nontrivial character $\chi$. Then the problem reduces to studying the
deformations of the zero section of a line bundle, given a
deformation of the base and of the bundle. The statement can then be
proven by applying the following lemma.
\ $\Box$\par\smallskip
\begin{lem} Let $o\in B'\subset B\in\hbox{\sl Art}^*$ be schemes of length $1,n,n+1$
respectively for some $n$; for schemes, etc.~over $B$ denote the restriction to
$B'$ by a prime and the restriction to $o$ by ${}_o$. Let
${\cal Y}\to B$ be a smooth projective morphism, $\L$ a line bundle on ${\cal Y}$; let
$s'$ be a section of $\L'$, such that $s'_o=0$. Then the obstruction to lifting
$s'$ to a section $s$ of $\L$ lies in $H^1({\cal Y}_o,\L_o)$, and two liftings differ
by an element of $H^0({\cal Y}_o,\L_o)$.
\end{lem}
\noindent{\sc Proof.~} Let $\{U_\alpha\}$ be an affine open cover of $Y={\cal Y}_o$ such that $L$ is
trivial on each $U_\alpha$. Let $U_{\al\be}$ be $U_\alpha\cap U_\beta\subset U_\alpha$.
As $Y$ is smooth, we have that ${\cal Y}$ is covered by open subsets $V_\alpha$
isomorphic to
$U_\alpha\times B$, glued via $B$-isomorphisms $\phi_{{\al\be}}:U_{\al\be}\times B\to
U_{\be\al}\times B$ satisfying the cocycle condition and restricting to the
identity over $o$. Let $g_{{\al\be}}$ be transition functions for $\L$ with
respect to the open cover $V_\alpha$.
The section $s'$ can be described by functions $s'_\alpha$ on $U_\alpha\times B'$
such that, on $U_{\al\be}\times B'$, $$
s'_\alpha=g'_{\al\be}(s'_\beta\circ\phi_{\al\be}).$$
Extend $s'_\alpha$ arbitrarily to a function $s_\alpha$ on $U_\alpha\times B$;
any other extension is of the form $s_\alpha+\varepsilon\sigma_\alpha$, where $\varepsilon=0$ is an
equation of $B'$ in $B$ and $\sigma_\alpha$ is a function on $U_\alpha$ (as $\varepsilon f=0$
for any function $f$ in the ideal of $o$ in $B$).
If an extension $s$ of $s'$ exists, then there must be functions $\sigma_\alpha$ on
$U_\alpha$ such that, on $U_{\al\be}\times B$,
$$
s_\alpha+\varepsilon\sigma_\alpha=g_{\al\be}((s_\beta+\varepsilon\sigma_\beta)\circ\phi_{\al\be}).$$
Let $u_{\al\be}=s_\alpha-g_{\al\be}(s_\beta\circ\phi_{\al\be})$. The restriction of
$u_{\al\be}$ to $U_{\al\be}\times B'$ is zero, hence $u_{\al\be}$ is divisible by
$\varepsilon$: let $u_{\al\be}=\varepsilon v_{\al\be}$.
One can verify, using the fact that $s_o=0$, that $v_{\al\be}$ is a cocycle
in $H^1(Y,\L_o)$: it is enough to check that $$
u_{\al\be}+g_{\al\be}(u_{\be\gamma}\circ\phi_{\al\be})=u_{\al\gamma} $$
on $U_{{\al\be}\gamma}$, for all triples $\alpha,\beta,\gamma$ of indices of the cover.
It is then immediate to verify that $v_{\al\be}$ is the obstruction to lifting
$s'$ to ${\cal Y}$, and the statement about the difference of two liftings can be
proven in a similar way.
\ $\Box$\par\smallskip
We now recall some properties of $\defor X$. Let $\defg X:\hbox{\sl Ansp}_0\to \hbox{\sl Sets}$
be
the functor of deformations of $X$ together with the $G$ action.
\begin{lem} There is a natural action of $G$ on $\defor X$, whose
invariant locus is $\defg X$.
\end{lem}
\noindent{\sc Proof.~} Let ${\cal X}\to T$ be a deformation of $X$ over $(T,o)$; there is a given
isomorphism $i:X\to {\cal X}_o$. The action of an element $g\in G$ is given by
replacing $i$ with $i\circ \phi(g)$, where $\phi:G\to Aut(X)$ is the natural
action.
It is clear that if $G$ acts on a deformation ${\cal X}\to T$, then this belongs to
$\defg X$. The other implication follows from \cite{Ca2}, \S 7 or directly
from the fact that the automorphisms of $X$ and of its deformations are rigid.
\ $\Box$\par\smallskip
Note that, as $X$ is of general type,
the $G$-action on $\defor X$ induces an action on the Kuranishi family
${\cal X}\to B$ of $X$; the restriction of the Kuranishi family to the fixed
locus $B^G$ is universal for the functor $\defg X$ (compare (\cite{Pi},
(2.8) p.~19, \cite{Ca2}, \S 7).
Recall the following result from \cite{Pa1}.
\setcounter{equation}{0}
\begin{lem} Let $X$ be a smooth $(G,I)$-cover of $Y$ with building data
$(D_i,L_\chi)$. Then the decomposition according to characters of
$H^l(X,T_X)$ is as follows:
\begin{eqnarray} &&H^l(X,T_X)^{\rm inv}=H^l(T_Y(-\log \sum_{i\in I} D_i))\\
&&H^l(X,T_X)^\chi=H^l(T_Y(-\log \!\!\!\sum_{i\in S_\chi}\!\!\! D_i)\otimes
L_\chi^{-1})\qquad \hbox{\ if $\chi\ne1$}
\end{eqnarray}
where $S_\chi$ is the same as in lemma \rm{\ref{tdefnat}}.
\end{lem}
\noindent{\sc Proof.~} This follows immediately from proposition 4.1. in \cite{Pa1}.
\ $\Box$\par\smallskip
\setcounter{equation}{0}
\begin{cor}\label{tdef} Assume that, for all $\chi\in G^*\setminus 1$, the
bundles $L_\chi$ and
$\Omega_Y^1\otimes L_\chi$ are ample. Then there are natural exact
sequences, for all $\chi\in G^*\setminus 1$:
\begin{eqnarray} &&0\to \textstyle\bigoplus\limits_{i\in S_\chi}
H^0(Y,\O(D_i)\otimes L_\chi^{-1})\to H^1(X,T_X)^\chi\to 0.\\ &&0\to
\textstyle\bigoplus\limits_{i\in S_\chi} H^1(Y,\O(D_i)\otimes L_\chi^{-1})\to
H^2(X,T_X)^\chi.
\end{eqnarray}
\end{cor}
\noindent{\sc Proof.~} Fix $\chi\ne 1$, let $D=\sum_{i\in S_\chi}D_i$, and consider the following
diagram of sheaves with exact rows and columns: $$
\begin{array}{ccccccccc}
&&&& 0 && 0 \\
&&&& \downarrow && \downarrow \\
&&&& \Bigoplus_{i\in S_\chi}^{\phantom{S_\chi}}\O_Y & = & \Bigoplus_{i\in
S_\chi}\O_Y\\
&&&& \downarrow && \downarrow \\
0 & \longrightarrow & T_Y(-\log D) & \longrightarrow &{\cal P}^*&\longrightarrow&
\Bigoplus_{i\in S_\chi}^{\phantom{S_\chi}}\O_Y(D_i)&\longrightarrow&0\\
&&\Vert&& \downarrow && \downarrow \\
0 & \longrightarrow & T_Y(-\log D) & \longrightarrow &T_Y&\longrightarrow&
\Bigoplus_{i\in S_\chi}^{\phantom{S_\chi}}\O_{D_i}(D_i)&\longrightarrow&0\\
&&&& \downarrow && \downarrow \\
&&&& 0 && 0
\end{array}$$
where ${\cal P}$ is the prolongation bundle associated to the normal crossing
divisor $D$. By the previous lemma, it is enough to prove that the first two
cohomology groups of ${\cal P}^*\otimes L_\chi^{-1}$ vanish; this follows from the
corresponding vanishing for $L_\chi^{-1}$ and $T_Y\otimes L_\chi^{-1}$, and
the latter is just Kodaira vanishing (it is here that one needs the assumption
$\dim Y\ge 2$).
\ $\Box$\par\smallskip
The natural transformation of functors $\hbox{\rm Dnat}_X\to \defor
X$ defined in \ref{trasnat} is equivariant with respect to the natural actions
of $G$ on these functors. Therefore, there is a commutative diagram
$$\begin{array}{ccc}
\hbox{\rm Dgal}_X & \longrightarrow & \defg X \\
\downarrow & &\downarrow \\
\hbox{\rm Dnat}_X & \longrightarrow & \defor X
\end{array}
$$ where the vertical arrows are injections. The following theorem shows
that the horizontal arrows are smooth morphisms of functors when the branch
divisors are sufficiently ample.
This was proven in \cite{Pa1} under the hypothesis that $Y$ be rigid and
regular; in this case natural deformations are unobstructed, and it is
enough to check the surjectivity of the Kodaira-Spencer map. In the general
case one has to take into account the
obstructions as well.
\begin{thm}\label{complete} Let $f:X\to Y$ be a totally ramified $(G,I)$-cover
with building
data $D_i$, $L_\chi$, such that $X$ and $Y$ are smooth of dimension $\ge 2$
and that $X$ is of general type. Assume that for all $\chi\in G^*\setminus 1$
the
bundles $L_\chi$ and $\Omega_Y^1\otimes L_\chi$ are ample. Then the natural
map of
functors (from $\hbox{\sl Art}^*$ to $\hbox{\sl Sets}$) $\hbox{\rm Dnat}_X \to \defor X$ is smooth, and so is
the induced map
$\hbox{\rm Dgal}_X $ and
$\defg X$.
\end{thm}
\noindent{\sc Proof.~} By a well-known criterion, smoothness of a natural transformation of
functors is implied by surjectivity of the induced map on tangent spaces,
and injectivity on obstruction spaces.
This is immediate by lemma \ref{tdefnat} and corollary \ref{tdef}, and by
the fact that the map between tangent (obstruction) spaces induced by the
map of functors is the natural one.
\ $\Box$\par\smallskip
\section{Main theorem} In this section we will prove that the automorphism
group of an abelian cover is precisely the Galois group, provided that the
branch divisors are sufficiently ample and generic. The proof depends on the
construction of an explicit partial resolution of some singular covers, which
will be given in section 7.
Although the result is in some sense expected, the proof is rather involved
and the techniques applied are, we believe, of independent interest.
The following lemma is inspired by a similar result of McKernan (\cite{McK}).
\begin{lem} \label{mac} Let $\Delta$ be the unit disc in ${\bf C}$,
$\Delta^*=\Delta\setminus\{0\}$. Let $p:{\cal X}\to \Delta$ be a flat map, smooth over
$\Delta^*$, whose fibres are integral
projective varieties of non negative
Kodaira dimension. Assume we are given a section $\sigma$ of
$Aut_{{\cal X}/\Delta^*}$. If there exists a resolution of singularities $\varepsilon:\tilde
{\cal X}\to{\cal X}$ such that each divisorial component of the exceptional locus has
Kodaira dimension $-\infty$, then $\sigma$ can be (uniquely) extended to a
section of $Bir_{{\cal X}/\Delta}$.
\end{lem}
\noindent{\sc Proof.~} The section $\sigma$ induces a birational map $\phi:{\cal X}\hbox{\rm-}\,\hbox{\rm-}\,\hbox{\rm-}\!\!\!\!\!> {\cal X}$ over
$\Delta$; the uniqueness of the extension follows from this. Let
$\tilde\phi:\tilde {\cal X}\hbox{\rm-}\,\hbox{\rm-}\,\hbox{\rm-}\!\!\!\!\!>\tilde {\cal X}$ be the induced birational map, and let
$\Gamma$ be a resolution of the closure of the graph of $\tilde\phi$; let
$p_1$, $p_2$ be the natural projections of $\Gamma$ on $\tilde{\cal X}$ (such that
$p_2=\tilde\phi\circ p_1$), and let $q_i=\varepsilon\circ p_i$.
The strict transform ${\cal X}_0'$ of ${\cal X}_0$ in $\Gamma$ via $q_1^{-1}$ has
positive Kodaira dimension, hence it cannot be contracted by $p_2$, which is
a birational morphism with smooth image. Therefore the restriction of $p_2$
to ${\cal X}_0'$ is birational (because ${\cal X}_0'$ is not contained in the exceptional
locus of $p_2$) onto some irreducible divisor ${\cal X}_0''$ in ${\cal X}$.
As ${\cal X}''_0$ is birational to ${\cal X}_0$ it cannot be of Kodaira dimension
$-\infty$; hence it is not contained in the exceptional locus of $\varepsilon$.
Therefore $\varepsilon({\cal X}_0'')$ is a divisor contained in ${\cal X}_0'$, hence it is
${\cal X}_0'$ by irreducibility, and the map $\varepsilon:{\cal X}_0''\to{\cal X}_0$ is birational.
So the birational map $\phi$ can be extended to ${\cal X}_0$ by the birational map
$q_2\circ \left(q_{1|{\cal X}_0'}\right)^{-1}$.
\ $\Box$\par\smallskip
\begin{lem} \label{trick}In the same hypotheses of lemma {\rm \ref{mac}},
assume moreover that there is a line bundle $L$ on ${\cal X}$, flat over $\Delta$,
whose restriction to ${\cal X}_t$ is very ample for all $t$, and such that
$h^0({\cal X}_t,L_{|{\cal X}_t})$ is constant in $t$. If the action of $\sigma$ can be
lifted to an action on $L$, then $\sigma$ can be uniquely extended to a
section of $Aut_{{\cal X}/\Delta}$.
\end{lem}
\noindent{\sc Proof.~} Let $N$ be the rank of the vector bundle $p_*L$ on $\Delta$; choosing a
trivializing basis yields an embedding ${\cal X}\hookrightarrow\P^{N-1}\times \Delta$. The
automorphisms $\phi_t$ of ${\cal X}_t$ are restrictions to ${\cal X}_t$ of nondegenerate
projectivities of $\P^{N-1}$; their limit, as $t\to 0$, is a well-defined,
possibly degenerate projectivity $\phi_0$. This gives an extension of $\phi$
to an open set of ${\cal X}_0$; this must now be birational by the previous lemma,
which in turn implies that $\phi_0$ is nondegenerate (as ${\cal X}_0$ is not
contained in a hyperplane), and therefore that $\phi_0$ is a morphism.
Applying the same argument to $\phi^{-1}$ concludes the proof.
\ $\Box$\par\smallskip
\begin{rem} The hypothesis that $\sigma$ acts on $L$ is obviously verified
if $L_{|{\cal X}_t}$ is a pluricanonical bundle for all $t\ne 0$.
\end{rem}
\begin{prop}\label{prop.aut
Let $p:{\cal X}\to \Delta$
be a flat family of integral projective varieties of general type, smooth
over $\Delta^*$. Assume that there is a line bundle $L$ on ${\cal X}$, flat over
$\Delta$, with $L_t:=L_{|{\cal X}_t}$ ample on ${\cal X}_t$, and $Aut({\cal X}_t)$ acts on $L_t$
for $t\ne 0$. Assume
moreover that for any $m$-th root base change $\rho_m:\Delta\to \Delta$ the
pullback $\rho_m^*{\cal X}$ admits a resolution having only divisors of negative
Kodaira dimension in the exceptional locus. Then $Aut_{X/\Delta}$ is proper over
$\Delta$, and the cardinality of the fibre is an upper semi-continuous function.
\end{prop}
\noindent{\sc Proof.~} After replacing $L$ with a suitable multiple and maybe shrinking $\Delta$,
we can assume that $L_t$ is very ample on ${\cal X}_t$, and that $h^0({\cal X}_t,L_t)$
is constant in $t$. The map $Aut_{{\cal X}/\Delta}\to \Delta$ is obviously quasi-finite
(because the
fibres are of general type) and the fibres are reduced (because automorphism
groups are always reduced in char.~$0$). It is enough to prove that given a
map of a pointed curve $(C,P)$ to $\Delta$ and a lifting of the map to
$Aut_{{\cal X}/B}$ out of $P$, the lifting can be extended to $P$.
Via restriction to an open set we can assume that $C$ is the unit disc $\Delta$,
$P$ is the origin and $\Delta\to \Delta$ is the map $z\to z^m$; we can then apply
lemma \ref{trick} to conclude the proof.
\ $\Box$\par\smallskip
\begin{cor}\label{gen.aut} Let ${\cal X}\to B$ be a smooth family of varieties
having ample canonical bundle. Then the scheme $Aut_{X/B}$ is proper over
$B$, and the cardinality of the fibre is an upper semi-continuous function.
\end{cor}
\noindent{\sc Proof.~} We can apply the previous proposition with $L=K_{{\cal X}/\Delta}$.
\ $\Box$\par\smallskip
\begin{thm} \label{mainthm} Let $Y$ be a smooth projective variety, and $X$ a
smooth $(G,I)$-cover with ample canonical bundle, with covering data
$L_\chi$, $D_i$. Let $H=\O_Y(1)$ for some embedding of $Y$ in $\P^{N-2}$;
assume that the linear system $$|D_1-m_1NH|$$ is base-point-free. Assume also
that the ${\bf Q}$-divisor $$ M=K_Y-(m_1-1)NH+\sum_{i\in I}\frac{(m_i-1)
}{m_i}D_i$$ is ample on $Y$. Then, for a generic choice of $D_1$ in its linear
system, $X$
has automorphism group isomorphic to $G$.
\end{thm}
\newcommand{1}{1}
\noindent{\sc Proof.~} Let $d$ be the number of automorphisms of a generic cover with the given
covering data (cfr.~corollary \ref{gen.aut}). It is enough to show that $d\le
\#G$, the other inequality being obvious.
Let $H$ be as in the statement of the theorem, and let ${\cal H}\subset |H|$ be
the (not necessarily complete) linear system giving the embedding; let
$H_1,\ldots,H_N$ be $N$ projectively independent divisors in ${\cal H}$. Assume
that the $H_i$'s are generic, in particular that they are smooth and that
their union with all of the $D_i$'s has normal crossings. Let
$m=m_1$, $D=D_1$.
The strategy of the proof is the following: start from a generic cover $X$ of
$Y$, and construct a sequence of manifolds $X_1,\ldots,X_N$ and of subgroups
$G_k$ of $Aut(X_k)$ such that $$
\#Aut(X)\le \#G_1\le\ldots\le \#G_N\qquad{\rm and}\qquad G_N=G.$$ In fact,
$X_k$ will be a $(G,I)$-cover of $Y$ with covering data $D^{(k)},D_2,\ldots$,
$L_\chi^{(k)}$, where $L_\chi^{(k)}=L_\chi-k\re^1_\chi H$ and $D^{(k)}$ is a
generic divisor in $|D-kmH|$ (recall that $\re^i_\chi$ was defined as the
unique integer $a$ satisfying $0\le a\le m_i-1$ and $\chi_{|H_i}=\psi_i^a$).
We let $G_k$ be the group of automorphisms of $X_k$ preserving the inverse
images of the curves $H_1,\ldots,H_k$ in $Y$.
We therefore want to prove the following:\begin{enumerate}
\item $\#Aut(X)\le \#G_1$;
\item $\#G_k\le \#G_{k+1}$;
\item $G_N=G$.
\end{enumerate}
\smallskip
\noindent {\sc First step:} $\#Aut(X)\le \#G_1$. Let $D^{(1)}$ be a generic
divisor in $|D-m H|$, and choose equations $f_1$, $g$ and $h_1$ for $H_1$,
$D$ and $D^{(1)}$ respectively. Define divisors ${\cal D}_i$ on $Y\times {\bf C}$ by
${\cal D}_i=D_i\times {\bf C}$ for $i\ne 1$, ${\cal D}_1=\{(1-t)f_1^mh_1+tg=0\}$; let
${\cal X}^1$ be the corresponding abelian cover. ${\cal X}^1_0$ is a singular variety
(singular along the inverse image of the curve $H_1$ in $Y$), with
smooth normalization $X_1$ (see \cite{Pa1}, step 1 of normalization algorithm
of p.~203). Note that $X_1$ is of general type by the ampleness assumption on
$M$.
By proposition \ref{reslemma}, the family ${\cal X}^1$ and each $n$-th root base
change of ${\cal X}^1$ admit a resolution with only divisors of Kodaira dimension
$-\infty$ in the exceptional locus. Moreover, the pull-back of $(\#G)M$
restricts to the $\#G$-canonical bundle on the smooth fibres of ${\cal X}^1$ (cfr the
proof of prop.~4.2 in \cite{Pa1}, p.~208). Applying proposition \ref{prop.aut}
gives that $Aut_{{\cal X}^1/{\bf C}}$ is proper over ${\bf C}$, and hence that $\#Aut(X)\le
Aut({\cal X}^1_0)$ (as we assumed $X$ to be generic). On the other hand it is clear
that each automorphism of ${\cal X}^1_0$ lifts to the normalization $X_1$, yielding
an automorphism which maps to itself the inverse image of the singular locus,
i.e., the inverse image of the curve $H_1$.
\smallskip
\noindent {\sc Second step:} $\#G_{k-1}\le \#G_{k}$. We use a similar
construction; let $X_{k-1}$ be as above, let $h_{k-1}$ be an equation of
$D^{(k-1)}$, $f_k$ an equation of $H_k$, and $h_k$ an equation of $D^{(k)}$.
Define a $(G,I)$-cover ${\cal X}^k$ of $Y\times
{\bf C}$ branched over $D_i\times {\bf C}$ for $i\ne 1$, and over
${\cal D}_1^{(k)}=\{(1-t)f_k^mh_k+th_{k-1}=0\}$; ${\cal X}^k_0$ is singular along the
inverse image $C_k$ of $H_k$, and its normalization is $X_k$; again $X_k$ is
of general type.
Again by proposition \ref{reslemma} the family ${\cal X}^{(k)}$ and all its $n$-th
root base changes have a resolution with only uniruled components in the
exceptional locus; the same argument as before proves the result.
\smallskip\noindent {\sc Final step:} $G_N=G$. Let $\pi:X_N\to Y$ be the
covering map: $G_N$ is the group of automorphisms of $Y$ fixing the inverse
images of the curves $H_1,\ldots,H_N$. Every element of $G_N$ preserves
$\pi^*\left({\cal H}\right)$, hence induces an automorphism of $Y$; this
automorphism must be the identity as it induces the identity on ${\cal H}$.
Therefore $G_N$ must coincide with $G$.
\ $\Box$\par\smallskip
\begin{rem} In theorem {\rm \ref{mainthm}} we can replace the assumption that
the linear system $|D_1-m_1NH|$ be base point free by asking that for each
$i\in I$ $$ |D_i-m_iN_iH|$$ be base point free, with $N_i$ nonnegative
integers with sum $N${\rm ;} we then get that, for a generic choice of the
$D_i$'s such that $N_i\ne 0$, $Aut(X)=G$.
\end{rem}
\begin{ex}{\rm One might wonder whether it is always true that a generic
abe\-lian cover of general type has no ``extra automorphisms". Here is an easy
example where this is not the case. Consider a ${\bf Z}_3$-cover of
$\P^1$, branched over two pairs of distinct points, with opposite characters.
A generic such cover is a smooth genus $2$ curve, hence its automorphism
group cannot be ${\bf Z}_3$. }
\end{ex}
\begin{ex}{\rm Here is a slightly more complicated example of
extra automorphisms, which works in any
dimension. Let $Y$ be a principally polarized abelian variety, and let $L$
be a principal polarization; assume that $L$ is symmetric, i.e. invariant
under the natural involution $\sigma(y)=-y$ on $Y$. The sections of
$L^{\otimes 2}$ are all symmetric, and the associated linear
system has no base points. Let $G={\bf Z}_2^s$, with the canonical basis
$e_1,\ldots,e_s$. Choose $I=\{1,\ldots,s\}$, and let $H_i$ be the subgroup
generated by $e_i$, for $i=1,\ldots,s$.
\par
The equations for the reduced building data become $L_j^{\otimes
2}=\O_Y(D_j)$; we choose the solution $L_j=L$, $M_i=L^{\otimes 2}$ for all
$i,j$. We are in fact constructing a fibred product of double covers. Choose
the $D_i$'s to be generic divisors in the linear system $|L^{\otimes 2}|$.
Each of them must be symmetric; this implies that the involution $\sigma$ can
be lifted to an involution of $X$, which is an automorphism not contained in
the Galois group of the cover.
\par
Note
that in this case the total branch divisor can become arbitrarily large, still
all
$(G,I)$-covers have an automorphism group bigger than $G$. }
\end{ex}
\section{Moduli spaces of abelian covers and global constructions}
In this section we will explicitly
construct a coarse moduli space for abelian covers of a smooth variety $Y$
and a complete space of natural deformations. Although some of the material in
this section is implicit in
\cite{Pa1}, we find it important to state it in a precise and explicit way.
In particular we will apply theorem \ref{complete} to construct (under
suitable ampleness assumptions) a family of natural deformations which maps
dominantly to the moduli (theorem 5.12).
Let $Y$ be a smooth, projective variety, $G$ an abelian group, $I$ a subset
of $I_G$. A {\em family
of smooth $(G,I)$-covers} of $Y$ over a base scheme $T$ is a smooth, proper
map
${\cal X}\to T$ and an action of $G$ on ${\cal X}$ compatible with the projection on $T$,
together with a $T$-isomorphism of the quotient ${\cal X}/G$ with $Y\times T$,
such that for each $t\in T$ the induced cover ${\cal X}_t\to Y$ is a
$(G,I)$-cover. Two families over
$T$ are {\em {\rm (}strictly\/{\rm )} isomorphic} if there is a
$G$-equivariant isomorphism inducing on the quotient $Y\times T$ the identity
map.
A (coarse) moduli space $\ZZ$ for smooth $(G,I)$-covers of $Y$ is a
scheme structure on the set of smooth $(G,I)$-covers modulo isomorphisms,
such that for any family of $(G,I)$-covers of $Y$ with base $T$ the induced
map $T\to
\ZZ$ is a morphism.
\begin{thm} There is a coarse moduli space of $(G,I)$-covers of $Y$, which is
a Zariski open set $\ZZ=\ZZ(Y,G,I)$ in the closed subvariety of
$$\prod_{\chi\in G^*\setminus 1}Pic(Y)\times \prod_{i\in I} Hilb^{\rm div}(Y)$$ of
all the $(L_\chi,D_i)$ satifying the relations {\rm (\ref{bdata})}. The open
set $\ZZ$ is the set of $(L_\chi,D_i)$'s which satisfy the additional
conditions:
\begin{enumerate}
\item each $D_i$ is smooth and the union of the $D_i$'s is a divisor with
normal crossings;
\item whenever $D_{i_1}, \ldots, D_{i_k}$ meet, the natural map
$H_{i_1}\oplus\cdots\oplus H_{i_k}\to G$ is injective.
\end{enumerate}
\end{thm}
\noindent{\sc Proof.~} The set $\ZZ$ parametrizes the smooth abelian covers of $Y$ by
\cite{Pa1}, theorem 2.1. The fact that the induced maps from a family of
abelian covers to $\ZZ$ are morphisms follows from the corresponding property
of the Hilbert schemes and Picard groups.
\ $\Box$\par\smallskip
Proposition 2.1 of \cite{Pa1} implies:
\begin{rem} For any basis $\chi_1,\ldots,\chi_s$ of $G^*$, the natural map
$$\ZZ\to\prod_{j=1}^s Pic(Y)\times\prod_{i\in I} Hilb^{\rm div}(Y)$$ induced by
projection is an isomorphism with its image.
\end{rem}
$\ZZ$ decomposes as the disjoint union of infinitely many quasiprojective
varieties $Z(\xi_i,\eta_\chi)=Z(\xi_i,\eta_\chi)(Y,G,I)$, where $\eta_\chi$,
$\xi_i$ are the Chern classes of $L_\chi$ and $\O(D_i)$, respectively. We now
give an explicit description of $Z(\xi_i,\eta_\chi)$ under the assumption
that the
$\xi_i$'s are sufficiently ample.
\setcounter{equation}{0}
\begin{prop}\label{moduli} Let $\xi_i$, $\eta_\chi$ be cohomology classes
satisfying the following relations {\rm(}compare {\rm(\ref{bdata})):}
\begin{equation}
\label{Chbdata}\eta_\chi+\eta_{\chi'}=\eta_{\chi\chi'}+\sum_{i\in I}
\eps^i_{\chi,\chi'}\xi_i\qquad\quad \hbox{for
all $\chi,\chi'\in G^*\setminus 1$}.
\end{equation} Assume moreover that $\xi_i-c_1(K_Y)$ is the class of an ample
line bundle for all $i\in I$. Then $Z(\xi_i,\eta_\chi)$ is an open set in a
smooth fibration (with fibre a product of projective spaces) over an abelian
variety $A(\xi_i,\eta_\chi)$ isogenous to $Pic^0(Y)^{\#I}$. $Z(\xi_i,
\eta_\chi)$ is nonempty iff
there are smooth effective divisors $D_i$, with $c_1(D_i)=\xi_i$, such that
their union has normal crossings.
\end{prop}
\noindent{\sc Proof.~} Let $A=A(\xi_i,\eta_\chi)\subset\prod_{i\in
I}Pic^{\xi_i}(Y)\times \prod_{\chi\in G^*\setminus 1}Pic^\chi(Y)$ be the image
of $Z(\xi_i,\eta_\chi)$; by
equations (\ref{rbdata}) the natural map $A\to \prod_{i\in I}Pic^{\xi_i}(Y)$
is a finite \'etale cover of degree $(2q)^{\#G}$, where $q$ is the
irregularity of $Y$. So each connected component of $A$ is an abelian
variety, isogenous to $Pic^0(Y)^{\#H}$. The fact that $A$ is connected is a
consequence of the covering being totally ramified. In fact, choose a basis
$\chi_1,\ldots,\chi_s$ of $G^*$, and consider the diagram
$$
\begin{array}{ccc} A&\longrightarrow&\prod_{i\in I}Pic^{\xi_i}(Y)\\
\downarrow& &\downarrow\\
\prod_{j=1}^sPic^{\eta_j}(Y)&\longrightarrow&\prod_{j=1}^sPic^{\ord{\chi_j}
\eta_j}\\
\end{array}$$ with maps given by $$
\begin{array}{ccc} (M_i,L_j)&\mapsto&(M_i)\\
\downarrow& & \downarrow\\ (L_j)&\mapsto&(L_j^{\otimes n_j}=\otimes M_i^
{\reb^i_{j}}).\\
\end{array}$$ The diagram is a fibre product of (connected) abelian varieties;
to prove that $A$ is connected is equivalent to proving that
$\pi_1(\prod_{i\in I}Pic^{\xi_i}(Y))$ surjects on
$$\pi_1(\prod_{j=1}^sPic^{\ord{\chi_j}\eta_j})/
\pi_1(\prod_{j=1}^sPic^{\eta_j}(Y));$$ this is in turn equivalent to proving
that $G^*$ injects in $\oplus_{i\in I}H_i^*$, which follows by dualizing
from assumption \ref{totram}.
Let ${\cal P}_i$ on $A\times Y$ be the pullback of the Poincar\'e line bundles
from $Pic^{\xi_i}(Y)\times Y$; the pushforward of ${\cal P}_i$ to $A$ is a vector
bundle $E_i$ because of the ampleness condition (the rank of $E_i$ can be
computed by Riemann-Roch). The moduli space $Z(\xi_i,\eta_\chi)$ is an open
set of the fibred product of the $\P(E_i)$.
\ $\Box$\par\smallskip
\begin{rem} {\rm If $q(Y)$ is not zero, then the components
$Z(\xi_i,\eta_\chi)$ are uniruled, but not unirational.}
\end{rem}
\begin{rem}{\rm In general $Z(\xi_i,\eta_\chi)$ is a coarse
but not a fine moduli space, i.e., it does not carry a universal family.
Keeping the notation of proposition \ref{moduli}, let ${\cal V}$ be the total
space of the fibred product of the $E_i$'s, and let ${\cal V}^o$ the inverse image
of
$Z(\xi_i,\eta_\chi)$; we have a natural abelian cover of $Y\times {\cal V}^o$,
which is a complete family of smooth covers of $Y$ with the given data.}
\end{rem}
There is a natural action of $Aut(Y)$ on the moduli space of $(G,I)$-covers
$\ZZ$, given by
$$\phi(D_i,L_\chi)=(\phi(D_i),(\phi^{-1})^*L_\chi)\qquad\qquad\hbox{for
$\phi\in Aut(Y)$.}$$ The automorphism group of $G$ acts naturally on $G^*$
(by $\Phi(\chi)=\chi\circ \Phi^{-1}$) and on $I_G$ (by $\Phi
(H,\psi)=(\Phi(H),\psi\circ \Phi^{-1})$); given a subset $I$ of $I_G$, let
$Aut_I(G)$ be the set of automorphisms of $G$ preserving $I$. There is a
natural action of $Aut_I(G)$ on $\ZZ$, induced by the natural action of this
group on the indexing sets $G^*\setminus 1$ and $I$.
\begin{prop} If the classes $\xi_i$'s are ample enough {\rm (}so that theorem
{\rm \ref{mainthm}} applies to some cover in $Z(\xi_i,\eta_j)${\rm),} then
the quotient
of
$Z(\xi_i,\eta_j)$ by the natural action of
$Aut(Y)\times Aut_I(G)$ maps birationally to its image in the moduli of
manifolds with ample canonical class.
\end{prop}
\noindent{\sc Proof.~} That the natural map to the moduli factors via this action is clear.
Viceversa, given a generic cover $X$ in $Z(\xi_i,\eta_j)$, by theorem
\ref{mainthm} its automorphism group is isomorphic to $G$; so it can be
identified uniquely as a $(G,I)$-cover up to isomorphisms of $G$ and of $Y$.
\ $\Box$\par\smallskip
\begin{defn}{\rm Let
${\cal Y}\to T$ be a deformation of $Y$ over a simply connected pointed analytic
space $(T,o)$. As $T$ is simply connected, the cohomology of every fibre
${\cal Y}_t$ is canonically isomorphic with that of $Y$. Then the varieties
$Z(\xi_i,\eta_\chi)({\cal Y}_t,G,I)$ (resp.\ $A(\xi_i,\eta_\chi)({\cal Y}_t,G,I)$) for
$t\in T$ glue to a global variety
$\ZZ_T(\xi_i,\eta_\chi)=\ZZ_T(\xi_i,\eta_\chi)({\cal Y},G,I)$ (resp.\
${\cal A}_T(\xi_i,\eta_\chi)$), surjecting on the
locus on $T$ where the classes
$\xi_i$ {\rm(}and hence also the $\eta_\chi${\rm)} stay of type $(1,1)$. The
global varieties are constructed by
replacing the Hilbert and Picard schemes in the construction of
$Z(\xi_i,\eta_\chi)$ and $A(\xi_i,\eta_\chi)$ with their relative versions.
The previous results can all be extended to this relative setting.
}
\end{defn}
For each smooth $(G,I)$-cover $f:X\to Y$, the natural deformations of
the reduced building data such that the induced deformations of
$(Y,L_j,M_i)$ is trivial are parametrized naturally by $\prod_{({i,\chi})\in
S}H^0(Y,M_i\otimes L_\chi^{-1})$, as in \S 5 of \cite{Pa1}.
\begin{thm} Let ${\cal Y}\to T$ be a deformation of $Y$ over a germ $(T,o)$, and
assume that the $\xi_i$'s stay of type $(1,1)$ on $T$. Then there is a
quasiprojective morphism ${\cal W}_T(\xi_i,\eta_\chi)\to {\cal A}_T(\xi_i,\eta_\chi)$
whose fibre over a point parametrizing line bundles $(L_j,M_i)$ on ${\cal Y}_t$
is canonically isomorphic to $\prod_{({i,\chi})\in
S}H^0(Y,M_i\otimes L_\chi^{-1})$.
\end{thm}
\noindent{\sc Proof.~} The theorem follows, by taking suitable fibre products, from the
following two lemmas.
\ $\Box$\par\smallskip
\begin{lem} Let $Y$ be a smooth projective variety, and $\xi\in NS(Y)$. Then
there exists a morphism of schemes $\pi:W^\xi(Y)\to Pic^\xi(Y)$ such that the
fibre over a point $[L]$ is naturally isomorphic to the vector space
$H^0(Y,L)$. For any choice of the Poincar\'e line bundle ${\cal P}$ on $Y\times
Pic^\xi(Y)$, there exists such a $W^\xi(Y)$ with the property that the line
bundle $\pi^*{\cal P}$ on
$Y\times W^\xi(Y)$ has a tautological section.
\end{lem}
Let ${\cal P}$ be the Poincar\'e line bundle on $Y\times Pic^\xi(Y)$, and let
$p:Y\times Pic^\xi(Y)\to Pic^\xi(Y)$ and $q:Y\times Pic^\xi(Y)\to Y$ be the
projections; if
$p_*({\cal P})$ is a vector bundle, it is enough to take $W$ to be the
total space of this vector bundle.
It is also clear that if $\xi-c_1(K_Y)$ is an ample class, then $p_*({\cal P})$
is indeed a vector bundle. For the general case, let $A$ be a line bundle on
$Y$ such that $c_1(A)+\xi-c_1(K_Y)$ is ample, and such that there exists an
$s\in H^0(Y,A)$ defining an effective, smooth divisor $D$. Let $\pi:V\to
Pic^\xi(Y)$ be
the total space of the vector bundle $p_*({\cal P}\otimes q^*A)$, and let
$\sigma:\O_{Y\times V}\to
\pi^*({\cal P}\otimes q^*A)$ be the tautological section.
For every $y\in D$, let $\sigma_y$ be the induced section of $\pi^*({\cal P}\otimes
q^*A)|_{\{y\}\times V}$; let $W_y\subset V$ be the divisor defined by
$\sigma_y$. Let $W=W^\xi(Y)$ be the intersection of all $W_y$'s for $y\in D$:
then
$\sigma/s$ is regular on $W$, and defines the required tautological
section.
\ $\Box$\par\smallskip
\begin{lem}{Let ${\cal Y}\to T$ be a deformation of $Y$ over
a germ of analytic space $T$, and assume that $\xi$ stays of type $(1,1)$
over $T$. Then, after maybe replacing $T$ with a Zariski-open subset, the
spaces $W^\xi({\cal Y}_t)$ glue together to a quasiprojective morphism
$W^\xi_T({\cal Y})\to Pic^\xi_T({\cal Y})$.}
\end{lem}
\noindent{\sc Proof.~} After possibly restricting $T$, we can extend $A$ to a line bundle ${\cal A}$
over
${\cal Y}$, and $s$ to a section of
${\cal A}$. The rest of the proof remains the same, using the fact that the
relative Picard scheme exists and carries a Poincar\'e line bundle.
\ $\Box$\par\smallskip
We now want to describe explicitly
$W^\xi(Y)$ in the case $\xi=0$, which we will use repeatedly later.
\begin{rem}{\rm For any deformation ${\cal Y}\to T$ over a germ of analytic space,
$W^0_T({\cal Y})$ is naturally isomorphic to the union in $Pic^0_T({\cal Y})\times {\bf C}$
of
$j(T)\times{\bf C}$ and
$Pic^0_T({\cal Y})\times
\{0\}$, where $j:T\to Pic^0_T({\cal Y})$ is the zero section.}
\end{rem}
In particular $W^0(Y)$ is reducible when $q(Y)\ne 0$; this reflects the fact
that the deformations, as pair (line bundle, section), of
$(\O_Y,0)$ are obstructed; one can either deform the line bundle or the
section, but not both at the same time. This remark will be used to construct
examples of manifolds lying in several components of the moduli in section 6.
\begin{thm} {\rm (i)} Let $Y$ be a smooth projective variety, and let $X\to Y$
be a smooth $(G,I)$-cover such that theorem {\rm \ref{complete}} holds. Then
there exists a pointed analytic space $({\cal W},w)$ and a natural deformation of
the reduced building data of $X$ over ${\cal W}$ such that the induced map of germs
from $({\cal W},w)$ to the Kuranishi family of $X$ {\rm(}defined as in {\rm
3.3}{\rm)} is surjective.
\par\noindent
{\rm (ii)} One can choose ${\cal W}$ to be a quasi-projective scheme, and then the
induced rational map from ${\cal W}$ to the moduli of manifolds with ample canonical
class is dominant onto each component of the moduli containing $[X]$.
\end{thm}
\noindent{\sc Proof.~}
(i) Let ${\cal Y}\to T$ be the
restriction of the Kuranishi family of $Y$ to the locus where all the
$\xi_i$'s stay of type $(1,1)$. Let ${\cal W}={\cal W}_T(\xi_i,\eta_\chi)$,
${\cal Y}_{\cal W}={\cal Y}\times_T{\cal W}$. Over ${\cal Y}_{\cal W}$ there are tautological line bundles
$\L_j$, ${\cal M}_i$ and tautological sections $s_{{i,\chi}}$ of
${\cal M}_i\otimes\L_\chi^{-1}$ (where $\L_\chi$ is defined as in \ref{natdef});
moreover
$\L_j^{\otimes n_j}$ is isomorphic to $\Bigotimes {\cal M}_i^{\reb^i_j}$. ${\cal W}$
parametrizes data $({\cal Y}_t,L_j,M_i,s_{{i,\chi}})$ such that $t\in
T$, $L_j$ and $M_i$ are line bundles on ${\cal Y}_t$ satisfying (\ref{rbdata}) and
having Chern classes $\eta_j,\xi_i$, and $s_{i,\chi}$ are sections of
$L_\chi\otimes M_i^{-1}$.
Let $w\in {\cal W}$ be a point corresponding to the reduced building data of $X$:
that is, assume that $w$ corresponds to the data $({\cal Y}_o,L_j,M_i,s_{{i,\chi}})$,
where
$s_{i,\chi}=0$ for all
$\chi\ne 1$, $o$ is the chosen point in $T$, and the sections $s_{i,0}$
define divisors $D_i$ such that $(L_j,D_i)$ are the reduced building data of
$X$.
Choose arbitrarily isomorphisms $\Phi_j:\L_j^{\otimes n_j}\to \Bigotimes
{\cal M}_i^{\otimes\reb^i_j}$, extending the isomorphism over $w$ induced by
multiplication in $\O_X$.
By theorem \ref{complete}, together with Artin's results on approximation of
analytic mappings (see \cite{Ar}), it is enough to show that every natural
deformation of the reduced building data of $X$ over a germ of analytic
space can be obtained as pullback from $({\cal W},w)$.
It is clear that all small deformations of the data
$(Y,L_j,M_i,s_{i,\chi})$ can be obtained as pullback from $W$. So it is enough
to prove that, up to isomorphism of natural deformations, we can choose the
$\phi_j$'s arbitrarily. This is proven in lemma \ref{lautnonconta}.
\noindent (ii) Start by noting that one can construct a deformation ${\cal Y}\to B$
of $Y$ over a pointed quasi-projective variety $(B,o)$, such that the germ of
$B$ at $o$ maps surjectively to the locus in the Kuranishi family of $Y$ where
the classes $\xi_i$'s stay of type $(1,1)$. In fact, choose any $\chi\in
G^*\setminus 1$, and let $L$ be a sufficiently big multiple of $L_\chi$;
assume in particular that $L$ is very ample and that all its higher cohomology
groups vanish. Let $N=\dim H^0(Y,L)-1$; choosing a basis of $ H^0(Y,L)$ gives
an embedding of $Y$ in $\P^N$. Take the union of the irreducible components of
the Hilbert scheme of $\P^N$ containing $b=[Y]$, and consider inside it the
open locus $B'$ of points corresponding to smooth subvarieties. Then the
natural map from the germ of $B'$ at $b$ to the Kuranishi family of $X$
surjects on the locus where $\eta_\chi$ stays of type $(1,1)$. Let $B$ be the
closed subscheme of $B'$ where also the classes $\xi_i$ stay of type $(1,1)$.
Let ${\cal Y}\to B$ be the universal family; by replacing $B$ with an \'etale open
subset we can assume that ${\cal Y}\to B$ has a section. Then (compare for instance
\cite{Mu}, p.~20) there exists a global projective morphism ${\cal A}\to B$ and line
bundles ${\cal M}_i$, $\L_j$ on ${\cal Y}\times_B{\cal A}$, such that ${\cal A}_b$ parametrizes line
bundles $(M_i,L_j)$ on ${\cal Y}_b$ such that firstly, they satisfy the usual
compatibility conditions, and secondly, the Chern classes of $(M_i,L_j)$ lie
in the orbit of $(\xi_i,\eta_j)$ via the monodromy action of $\pi_1(B,b)$.
Mimicking the proof in the germ case, and replacing $B$ by an \'etale open
subset if necessary, one can find a quasi-projective morphism ${\cal W}\to {\cal A}$ whose
fibre over a point corresponding to line bundles $(M_i,L_j)$ on ${\cal Y}_b$ is
isomorphic to $\prod H^0({\cal Y}_b,M_i\otimes L_\chi^{-1})$ for $({i,\chi})\in S$,
together with tautological sections $\sigma_{{i,\chi}}$ of the pullbacks to
${\cal Y}\times_B{\cal W}$ of
${\cal M}_i\otimes\L_\chi^{-1}$.
Let $w\in {\cal W}$ be a point corresponding to the building data of $X$ as before.
Again (possibly passing to an \'etale open subset) one can extend the
multiplications isomorphisms
$\phi_j$ to isomorphisms
$\Phi_j:\L_j^{\otimes n_j}\to \Bigotimes {\cal M}_i^{\otimes\reb^i_j}$.
Putting everything together, we have a natural deformation of the building
data of $X$ over $({\cal W},w)$; this induces by (3.3) a rational map to the moduli
of manifolds with ample canonical class, which is a morphism on the open
subset of ${\cal W}$ where the natural deformation of $X$ is smooth. Applying the
same methods as in (i) implies that the map from ${\cal W}$ to the moduli is
dominant on each irreducible component containing $[X]$.
\ $\Box$\par\smallskip
\begin{lem}\label{lautnonconta}
Let $T$ be a germ of analytic space. For any
$({\cal Y},{\cal M}_i,\L_j,s_{{i,\chi}},\phi_j)\in\hbox{\rm Dnat}_X(T)$, and for any other admissible
choice of isomorphisms $\phi_j':\L_j^{\otimes n_j}\to\bigotimes M_i^{\otimes
\re^i_j}$, there exist sections $s'_{i,\chi}$ such that $
({\cal Y},{\cal M}_i,\L_j,s_{{i,\chi}},\phi_j)$ is isomorphic to
$({\cal Y},{\cal M}_i,\L_j,s'_{{i,\chi}},\phi'_j)$.
\end{lem}
\noindent{\sc Proof.~}
It is enough to show that there are automorphisms $\psi_i$ of ${\cal M}_i$ such
that the composition $(\bigotimes \psi_i^{\otimes \reb^i_j})\circ \phi_j$
equals $\phi_j'$; in fact in this case one can choose
$s'_{i,\chi}=\psi_i^*(s_{i,\chi})$, for all $({i,\chi})\in S$.
As both $\phi_j$ and $\phi_j'$ are isomorphisms, $\phi_j=f_j\phi_j'$, where
$f_j$ is an invertible function on ${\cal Y}$ restricting to $1$ on the central
fibre.
Finding the $\psi_i$'s is equivalent to finding functions $g_i$'s on ${\cal Y}$ such
that $g_i$ restricts to $1$ on the central fibre and $f_j=\prod
g_i^{\reb^i_j}$,
for all $j=1,\ldots,s$. The existence of such $g_i$'s follows from the fact
that the matrix
$\re^i_j$ has rank equal to $s$, which in turn is implied by the cover
being totally ramified (see lemma 2.1).
\ $\Box$\par\smallskip
\setcounter{equation}{0}
\begin{rem}{\rm There is a natural action of $(C^*)^{\#I}$ on the functor of
natural deformations, which is the identity on $({\cal Y},\L_j,{\cal M}_i,\phi_j)$ and
acts on
$\sigma_{i,\chi}$
by
\begin{equation}\label{C*action} (\lambda_i)_{i\in
I}(\sigma_{j,\chi})=\prod_{i\in I}\lambda_i^{\delta_{ij}m_i-\re^i_\chi}\cdot
\sigma_{j,\chi};\end{equation} This action has the property that the induced
flat maps ${\cal X}\to T$ are invariant under it; in particular the
natural map from ${\cal W}(\xi_i,\eta_j)$ to the moduli factors
through the corresponding action.
}
\end{rem}
\section{Applications to moduli}
In this chapter we want to apply the results on deformation theory together
with theorem \ref{mainthm} to study the generic automorphism group of some
components of the moduli spaces of manifolds with ample canonical class,
components containing suitable abelian covers with sufficiently ample branch
divisors.
To begin with, we study the case of simple cyclic covers (i.e., those for
which the Galois group
$G$ is cyclic and there is only one irreducible branch divisor).
\begin{prop}\label{cyclic} Let $f:X\to Y$ be a smooth simple cyclic cover,
with Galois group ${\bf Z}_m$, and reduced building data $D$ and $L$ (where $D$ is
a smooth divisor and $L$ is a line bundle satisfying $mL\equiv D$). Assume
that $D$ is sufficiently ample. Let $M$ be an irreducible component of the
moduli space of surfaces of general type containing $X$. Then $G_M$ is
trivial if $m\ge 3$, and $G_M=G$ if $m=2$.
\end{prop}
\noindent{\sc Proof.~} In case $m=2$, it is easy to check that $H^i(X,T_X)$ is $G$-invariant
for $i=1,2$; hence the natural map $\hbox{\rm Dgal}_X
\to \hbox{\rm Dnat}_X$ is surjective,
and all deformations are Galois. By theorem \ref{mainthm}, $Aut(X)=G$ for a
generic choice of $D$ in its linear system.
If $m\ge 3$, assume without loss of generality that $D$ is generic in its
linear system. Let $(G,\chi)$ be the element of $I_G$ corresponding to the
only nonempty branch divisor. Then the natural deformations of $X$ such that
$Y$ and $\O(D)$ are fixed are parametrized by $$
\Bigoplus_{i=0}^{m-2} H^0(Y,L^{-i}(D))=\Bigoplus_{i=0}^{m-2} H^0(Y,L^{m-i});$$
in particular they are unobstructed. Moreover, given any nontrivial element
$g$ of the Galois group $G$, it acts on the (necessarily nonzero) summand
$H^0(Y,L^{m-1})$ as multiplication by $\chi(g)$, hence nontrivially;
therefore $g$ does not extend to the generic deformation. By genericity
however $Aut(X)=G$, hence by semicontinuity of the automorphism group the
proof is complete.
\ $\Box$\par\smallskip
Hence, to get nontrivial examples, and to prove the results on the
moduli claimed in the introduction, it is necessary to study more general
abelian covers.
\begin{constr}\label{construction} {\rm Let $s$ be an integer $\ge2$. Let
$d_1,\ldots,d_s$ be integers $\ge 2$, such that $d_i|d_{i+1}$ for $i\le s-1$;
let
$d_0=d_s$, and define integers $b_i$ by requiring that $b_id_i=d_0$, for all
$i=1,\ldots,s$. Let $G={\bf Z}_{d_1}\times\cdots\times {\bf Z}_{d_s}$, and let
$e_1,\ldots,e_s$ be the canonical basis of $G$; let $\chi_1,\ldots,\chi_s$ be
the dual basis of $G^*$.
\par Let $e_0:=-(e_1+\ldots+e_s)$, and let $H_i$ be the subgroup generated by
$e_i$; for $i=0,\ldots,s$, let $\psi_i\in H_i^*$ be the unique character such
that $\psi_i(e_i)=\zeta_{d_i}$; note that, for each
$j=1,\ldots,s$ and $i\ne 0$ we have
$\re^i_j=\delta_{ij}$, while $\re^0_j=b_j(d_j-1)$. Moreover $\ord{e_i}=d_i$,
for
$i=0,\ldots,s$. Let $I=\{0,\ldots,s\}$; identify $I$ with a subset of $I_G$
via
$i\mapsto (H_i,\psi_i)$.
\par Fix a smooth projective variety $Y$ of dimension $d$, and assume that
$s\ge d\ge 2$. Let $f:X\to Y$ be a $(G,I)$-cover of $Y$, with branch divisors
$D_i$. Equations (\ref{rbdata}) become $$
L_j^{\otimes d_j}=M_j\otimes M_0^{\otimes (d_j-1)}$$
for all $j=1,\ldots,s$, hence they can be solved by letting
$L_j=M_0\otimes F_j$, $M_j=M_0\otimes F_j^{\otimes d_j}$, for all $j=1,\ldots
,s$.
\par We compute explicitly $L_\chi$ for $\chi\in G^*$, using equation
(\ref{chidarbd}). Let $\chi\in G^*$, and write
$\chi=\chi_1^{\alpha_1}\cdots\chi_s^{\alpha_s}$, with $0\le
\alpha_j< d_j$. One gets $$
L_{\chi}=\Bigotimes_{j=1}^s F_j^{\otimes \alpha_j}\otimes M_0^{\otimes
N_\chi},$$ where
$N_\chi=-[(-\alpha_1b_1-\ldots-\alpha_sb_s)/d_0]$. In particular $N_\chi$ is an
integer $\ge 0$; $N_\chi=0$ if and only if $\chi=0$, $N_\chi=1$ if and only
if $\sum(\alpha_ib_i)\le d_0$.
In the following we will always assume that $c_1(F_j)=0$, for $j=1,\ldots,s$;
let $\xi=c_1(M_0)$.
Assume also that $X$ is a smooth $(G,I)$-cover, that is that the divisors
$D_i$ are smooth and their union has normal crossings.\par
\setcounter{equation}{0} In the surface case, one can compute the Chern
invariants of the cover $X$:
\begin{eqnarray*}
K_X^2/\#G&=&\left(K_Y+(s-(d_0^{-1}+\ldots+d_s^{-1}))\xi\right)^2\\
c_2(X)/\#G&=&c_2(Y)-((s+1)-(d_0^{-1}+\ldots+d_s^{-1}))\xi K_Y+\\
&&\left({s+2\choose 2}+
\sum_{i=0}^sd_i^{-1}+\sum_{0\le i<j\le s}d_i^{-1}d_j^{-1}\right)\xi^2.
\end{eqnarray*} The first equality follows from \cite{Pa1}, proposition 4.2;
the second from the additivity of the Euler characteristic, by decomposing
$Y$ in locally closed subsets according to whether a point lies in $2$, $1$
or no branch divisor. Note that no other possibilities can occur, as we
assume that the union of the branch divisors has normal crossings. The second
equality could also be derived by Noether's formula and proposition 4.2 in
\cite{Pa1}. }
\end{constr}
\begin{lem}\label{keylemma} Let $f:X\to Y$ be a $(G,I)$-cover as in
construction \ref{construction}. Assume that $q(Y)$ is
nonzero, that
$\xi\in NS(Y)$ is sufficiently ample, that $F_j=\O_Y$ (for $j=1,\ldots,s$),
and that $D_i\in |M_i|=|M_0|$ is generic {\rm(}for
$i=0,\ldots,s${\rm)}. Then, for each
$k=0,\ldots, s$, there exists a component $M_k$ of the moduli of
manifolds with ample canonical class, containing $X$, such that the generic
automorphism group $G_{M_k}\subset G$ is equal to
$G_k={\bf Z}_{d_{1}}\times\ldots\times{\bf Z}_{d_k}$.
\end{lem}
\noindent{\sc Proof.~} By assumption $X$ has ample canonical class,
$Aut(X)=G$ and the natural deformations of $X$ are complete. Assume first that
$Y$ is rigid. Let $\chi\in G^*$ be such that
$N_\chi=1$, and let $({i,\chi})\in S$; these are the only values of
${i,\chi}$ (with $\chi$ nontrivial) for which $M_i\otimes L_\chi^{-1}$ can have
sections, i.e.~can contribute to non-Galois deformations. In fact
$c_1(M_i\otimes L_\chi^{-1})=0$, hence it has sections if and only if it is
trivial (compare remark 5.11). The condition that the line bundle $M_i\otimes
L_\chi^{-1}$ be trivial can be expressed, in terms of the $F_j$'s, as
\begin{equation}\label{trivial}
\sum_j \alpha_j F_j= d_i F_i.\end{equation}
Let $T_k\subset Pic^0(Y)^s$ be the locus where $F_i=0$ for all $i>k$. Note
that $F_i=0$ for all $i>k$ implies that $M_i=M_0$ for all $i>k$, and that
$M_i\otimes L_\chi^{-1}$ is trivial for any $({i,\chi})$ such that $N_\chi=1$,
$i>k$ and $\chi$ restricted to $G_k$ is trivial.
For a
generic choice of
$(F_j)\in T_k$, the line bundles $M_i\otimes L_\chi^{-1}$ are nontrivial for
each $\chi$ such that $\chi_{|G_k}\ne 1$; in fact, for any such $\chi$ there
exists $j_0\le k$ such that $\alpha_{j_0}> 0$, hence the coefficient of $F_{j_0}$
in (\ref{trivial}) is nonzero (being either $\alpha_{j_0}>0$ or
$\alpha_{j_0}-d_{j_0}<0$).
On the other hand, for each $j>k$, one has
$(0,\chi_j)\in S$ and $M_0\otimes L_j^{-1}$ is trivial (in fact one has to
exclude here the case where $d_0$ is equal to $2$, and hence all $d_i$'s are;
this case needs a slightly different analysis, see below). Hence for every
$g\in G\setminus G_k$, and for any $(G,I)$-cover with building data in $T_k$,
there are natural deformations of the cover to which the action of $g$ does
not extend.
Therefore the $(G,I)$-covers whose building data are in $T_k$, together with
their
natural deformations such that $s_{{i,\chi}}=0$ for all $\chi$ acting
nontrivially on $G_k$, form an irreducible component of the Kuranishi family
of $X$; in fact they are parametrized by an irreducible variety, and at some
point they are complete (at least at all points corresponding to
$(G,I)$-covers with a generic choice of the $F_j$'s for $j\le k$). The generic
element of this component has therefore automorphism group $G_k$.
In the case where $d_0=2$, $(0,\chi_j)\notin S$; however, if $k\ne s-1$, we
can consider $M_{j'}\otimes L_j^{-1}$ instead of $M_0\otimes L_j^{-1}$, where
$j'$ is any index $>k$ and different from $j$. If $k=s-1$, let
$\chi=\chi_1+\chi_s$; then $N_\chi=1$ (as $s\ge 2$), and $(0,\chi)\in S$. As
$\chi(e_s)\ne 0$, there are natural deformations to which the action of $G$
does not extend.
The same argument applies if $Y$ is non-rigid, by replacing $Pic^0(Y)^s$ with
$Pic^0_T({\cal Y})^s$, where ${\cal Y}\to T$ is the restriction of the Kuranishi family of
$Y$ to the locus where $\xi$ stays of type $(1,1)$.
\ $\Box$\par\smallskip
\begin{rem} {\rm We can find a $Y$ of arbitrary dimension and an ample class
$\xi$ such that deformations of $Y$ for which $\xi$ stays of type $(1,1)$ are
unobstructed; for instance, by taking $Y$ a product of curves of genus at
least two and $\xi$ the canonical class.}
\end{rem}
\begin{thm} Let $d\ge 2$ be an integer. Given any integer $N$, there exists a
point in the moduli space of manifolds of dimension $d$ with ample canonical
class which is contained in at least
$N$ distinct irreducible components.
\end{thm}
\setcounter{equation}{0}
\noindent{\sc Proof.~} Without loss of generality, assume that $N\ge d$. Choose arbitrarily
integers $d_1,\ldots,d_N$, each of them $\ge2$ and such that $d_i|d_{i+1}$.
Let
$(Y,L)$ be as in lemma \ref{keylemma}; then for each $k=1,\ldots,N$ there
exists a component of the moduli containing $X$ and having generic
automorphism group isomorphic to ${\bf Z}_{d_1}\times\ldots\times{\bf Z}_{d_k}$. Hence
$X$ lies in at least $N$ different irreducible components of the moduli.
\ $\Box$\par\smallskip
In the case of surfaces, this result gives a strong negative answer to the
open problem (ii) on page 485 of \cite{Ca1}.
\begin{thm} Let $G$ be a finite abelian group, and $d\ge 2$ an integer. Then
there exist infinitely many components $M$ of the moduli space of manifolds
of dimension $d$ with ample canonical class such that
$G_M=G$.
\end{thm}
\noindent{\sc Proof.~} Write $G$ as ${\bf Z}_{d_1}\times\ldots\times{\bf Z}_{d_k}$, with $d_i|d_{i+1}$. If
$k\ge d$, let
$s=k$; if
$k<d$, let $s=d$ and let $d_{k+1}=\ldots=d_s=d_k$.
Choose $(Y,\xi)$ as in
lemma \ref{keylemma}. Applying the lemma to $(Y,i\xi)$ for $i\ge 1$
gives the claimed result.
\ $\Box$\par\smallskip
In the case of surfaces, another natural question concerns the cardinality of
the automorphism group. Xiao proved in \cite{Xi1} that if $X$ is a minimal
surface of general type,
$\#G\le 52K_X^2+32$ for all abelian subgroups $G$ of $Aut(X)$; it is not
known whether this bound is sharp, but he gives examples to the effect that
any better bound must still be linear in $K_X^2$. It seems natural to ask if
there is a smaller bound if one replaces $Aut(X)$ by $Aut_{\hbox{\rm
gen}}(X)$, the intersection in $Aut(X)$ of $G_M$, for each irreducible
component $M$ containg $X$. Notice that in Xiao's examples the generic
automorphism group is obviously smaller, so a better bound should be possible.
We prove here that such a bound cannot be less than linear in $K_X^2$.
\begin{prop} There exists a sequence $S_n$ of minimal surfaces of general
type such that
\begin{enumerate}
\item $k_n=K_{S_n}^2$ tends to infinity with $n$;
\item $S_n$ lies on a unique irreducible component, $M_n$;
\item $\#G_{M_n}> 2^{-4}k_n$.
\end{enumerate}
\end{prop}
\noindent{\sc Proof.~} Let $n\ge 2$ be an integer. Apply contruction \ref{construction} with
$s=2$, $d_1=d_2=n$, $Y$ a principally polarized abelian surface with
$NS(Y)=
{\bf Z}$ and
$\xi$ equal to the double of the class of the principal polarization. Choose
$S_n$ to be a cover branched over divisors $D_i$ whose linear equivalence
classes are generic; then all infinitesimal deformations must be Galois, and
the Kuranishi family of $S_n$ is smooth. So $G_{M_n}$ must contain ${\bf Z}_n^2$,
hence $\#G_{M_n}\ge n^2$. On the other hand,
$k_n=16(n-1)^2$. Note
that as we only want to bound $G_M$ from below, we don't need to apply
theorem \ref{mainthm}, which would have forced us to choose as class $\xi$ a
higher multiple of the principal polarization.\ $\Box$\par\smallskip
\begin{rem} {\rm Using the computation of Chern numbers for construction
\ref{construction}, one can determine where the examples constructed so far
lie in the geography of surfaces of general type. For instance by setting all
$d_i$'s equal to $m$ and letting $s$ and $m$ go to infinity, one gets a
sequence of examples where $K^2/c_2$ tends to $2$ from below.}
\end{rem}
\section{Resolution of singularities}
\begin{rem} \label{resnc} Let $\pi:X\to Y$ be a $(G,I)$-cover with $Y$ smooth
and branch locus with normal crossings. Let $Z\to X$ be a resolution of
singularities; then the exceptional locus of $Z$ has uniruled divisorial
components.
\end{rem}
\noindent{\sc Proof.~} The question is local on $Y$, so we can assume that $Y$ is affine and
that the line bundles $L_\chi$ and $\O(D_i)$ are trivial. Let $G'$ be the
abelian group with $\#I$ generators $e_1,\ldots,e_s$, and relations
$m_ie_i=0$ (where $m_i=\#H_i$). There exists a smooth $G'$-cover $X'$ of $Y$
branched over the
$D_i$ such that the inertia subgroup of $D_i$ is generated by $e_i$, and such
that the map $V\to Y$ factors via $X$. Let $Z'$ be a resolution of
singularities of the fibre product $Z\times_X X'$; we have a commutative
diagram $$
\begin{array}{ccc} Z' & \to & X'\\
\downarrow & & \downarrow\\ Z&\to &X\\
\end{array}
$$ Let $E$ be an irreducible divisorial component of the exceptional locus of
$Z\to X$; its strict transform $E'$ in $Z'$ must be contracted in $X'$ as
$X'\to X$ is finite. As $X'$ is smooth and $Z'\to X'$ is birational, $E'$
must be ruled by \cite{Ab}, therefore $E$ must be uniruled.
\ $\Box$\par\smallskip
\begin{lem} Let ${\cal Y}\to \Delta$ be a family of smooth manifolds, ${\cal X}\to {\cal Y}$ an
abelian cover branched on divisors which are all smooth except $D$, of
branching order $n$, which has local equation $f^nh+tg=0$ with $f$, $t$, $h$,
$g$ local coordinates on $Y$ (and $t$ coordinate on $\Delta$). Then there exists
a morphism $\tilde{\cal Y}\to {\cal Y}$ such that:
\begin{enumerate}
\item $\tilde{\cal Y}\to {\cal Y}$ is a composition of blowups with smooth center;
\item the normalization $\tilde {\cal X}$ of the induced cover of $\tilde {\cal Y}$ is an
abelian cover of $\tilde {\cal Y}$ branched over a normal crossing divisor;
\item the exceptional divisors of $\tilde {\cal X}\to {\cal X}$ have Kodaira dimension
$-\infty$.
\end{enumerate}
\end{lem}
\noindent{\sc Proof.~} We will construct $\tilde{\cal Y}$ by successive blowups; a local coordinate
and its strict transform after the blowup will be denoted by the same letter.
At each blowing-up step one checks that the normalization of the last
introduced exceptional divisor has Kodaira dimension $-\infty$ (further
blowups change the situation only up to birational maps).
The strategy of the proof is as follows; each blowup introduces a divisor
which is a $\P^r$ bundle (for $r=1,2$), and we prove that the induced cover
of the generic $\P^r$ has Kodaira dimension $-\infty$. We can assume that the
Galois group coincides with the inertia subgroup $H$ of $D$; if this is not
the case, consider the factorization ${\cal X}\to {\cal X}/H\to Y$, and note that the map
${\cal X}/H\to Y$ is unramified near generic points of $D$, hence after blowing up
the inverse image of the generic $\P^r$ is an unramified cover, which is
therefore a disjoint union of copies of $\P^r$.
We first prove the result on the locus where $h\ne 0$ (this is all one needs
if ${\cal Y}$ is a threefold). By changing local coordinates one can assume $h=1$.
Let $n$ be the order of $H$. We distinguish two cases: $n$ even and $n$ odd.
Let $E_1,E_2,\ldots$ be the subsequent exceptional divisors.
\smallskip
\noindent{\sc Case of $n$ even.} Blow up at each step the singular locus
$t=f=g=0$ and look at the $f$ chart. At the first step one obtains $$
z^n=f^2(f^{n-2}+tg)$$ and the total transform of the branch locus $D$ is
$D+2E_1$. The covering restricted to $E_1$ is the composition of a totally
ramified cover of degree $n/2$ and of a double cover ramified over $D\cap
E_1$ which is (on each $\P^2$ in $E_1$) a (possibly reducible) conic. Hence
the cover of $E_1$ is fibered in two-dimensional quadrics (maybe singular).
At the $k$-th step ($1<k\le n/2$) we have $$ z^n=f^{2k}(f^{n-2k}+tg)$$ and
the total transform of $D$ is $$ D+2E_1+\ldots+2kE_k.$$ Again $D$ cuts out a
(possibly reducible) conic on the $\P^2$ fibration of $E_k$; moreover,
$E_k\cap E_i=\emptyset$ if $i<k-1$, and $E_k\cap E_{k-1}$ is (fibrewise) a
line which is not contained in $D$.
If $\xi$ is a generator of the group $H$, the induced cover of $E_k$ is the
composite of a totally ramified cover and of a
cyclic cover of degree $r$, where $r$ is the cardinality of $H/\< \xi^{2k}\>$;
the
cover is ramified on each $\P^2$ on a conic and on a line. The pairs
(inertia group, character) for the branch divisors correspond, via the
bijection defined in \S 2, to $\xi$ for the conic and to $\xi^{-2}$ for the
line.
The canonical bundle of the cover is (fibrewise) the pullback of a multiple
of a line in $\P^2$, the multiple being $$ -3+2\left(\frac{r-1}{r}\right)
+\left(\frac{r/2-1}{r/2}\right)<0
$$ if $r$ is even and $$
-3+2\left(\frac{r-1}{r}\right)+\left(\frac{r-1}{r}\right)<0$$ if $r$ is odd;
in both cases the anticanonical bundle of the cover is ample and the surface
must be of Kodaira dimension $-\infty$.
\smallskip
\noindent{\sc Case of $n$ odd.} Start by blowing up the singular locus
$t=f=g=0$. At the first step the total transform of $D$ is $D+2E_1$ and the
cover of $E_1$ is totally ramified (as $2$ is prime with $n$), hence the
cover is again $E_1$. If $2k<n$ the same formulas as before hold; we can
repeat the previous argument where $r$ is necessarily odd.
Look now at the $k=(n-1)/2$ case. The total transform of $D$ is $$
D+2E_1+\ldots+(n-1)E_{(n-1)/2}.$$ The strict transform of $D$ is now smooth;
$E_{(n-1)/2}\cap D$ is fibered in singular conics, and we blow up the
singular locus. The center of this blowup does not meet $E_k$ for
$k<(n-1)/2$, and $D$ and $E_{(n-1)/2}$ have the same tangent space there.
Therefore after blowing one gets an exceptional divisor $E_{(n+1)/2}$
intersecting both $E_{(n-1)/2}$ and $D$ in the same line. The equation (in
the $g$ chart) becomes $$ z^n=f^{n-1}g^n(f+tg).$$
The cover of $E_{(n+1)/2}$ is a
$\P^1$-bundle ramified on a generic $\P^1$ with opposite characters on the
same divisor, hence when normalizing it splits completely. The components of
the total transform of $D$ are smooth, but they meet non-transversally along
the $\P^1$-bundle $f=g=0$.
We now blow up the locus $f=g=0$ and call the exceptional divisor
$F$; the total transform of $D$ is $$
D+2E_1+\ldots+(n-1)E_{(n-1)/2}+nE_{(n+1)/2}+2nF,$$ and $F$ is a $\P^1$-bundle
over a $\P^1$-bundle. The covering of the generic $\P^1$-fibre of $F$ is
ramified of degree $n$ over two points (corresponding to $F\cap D$ and $F\cap
E_{(n+1)/2}$) with opposite characters, hence is again isomorphic to $\P^1$.
\smallskip In both cases the fact that the divisors are smooth and
transversal can be checked at each step out of the center of the next blowup.
We now work in the neighborhood of a point where $h=0$. If $n$ is even, one can
perform the same blowups as in the previous case and check that the same
arguments work. If $n$ is odd, one can perform the first $(n-1)/2$ blowups
as before. After them, the total transform of $D$ has equation
$f^{n-1}(fh+tg)$. In particular (the strict transform of) $D$ is not smooth
any more; we blow up its singular locus, and get a smooth exceptional divisor
$\bar E$. The total transform of $D$ is $$
D+2E_1+\ldots+(n+1)\bar E$$
and is given (in local equations in the $h$ chart) by $$
f^{n-1}h^{n+1}(f+tg).$$
Let $\xi$ be a generator of $H$; the induced cover of $\bar E$ is
cyclic with group $H/\<\xi^{n+1}\>$, hence it is totally ramified and
therefore of Kodaira dimension $-\infty$, being a $\P^2$-bundle.
We are not done because the divisors $D$ and $E_{(n-1)/2}$
are not transversal along $f=g=t=0$; but now we can apply the previous
blowup procedure again.
\ $\Box$\par\smallskip
\begin{prop}\label{reslemma} Let ${\cal X}\to {\cal Y}\to \Delta$ be an abelian cover,
branched over all smooth divisors except one, which has local equation
$f^mh+tg$, where $f,t,g$ are coordinates and $m$ is the order of branching
(where $t$ is the coordinate on $\Delta$). Then ${\cal X}$ and all its transforms via
an $n$-th root base change admit a resolution of singularities such that the
divisorial components of the exceptional divisor all have Kodaira dimension
$-\infty$.
\end{prop}
\noindent{\sc Proof.~} The statement without the base change has already been proved; let
$\tilde {\cal X}$ be such a resolution. By Hironaka's resolution of singularities
(\cite{Hi}, p.~113, lines 8--4 from the bottom) we can assume that
$\tilde{\cal X}_0$ is a normal crossing divisor. Let now $\rho_n:\Delta\to \Delta$ be the
map $t\mapsto t^n$. There is a natural birational mapping $\rho_n^*\tilde
{\cal X}\to \rho_n^*{\cal X}$; moreover $\rho_n^*\tilde {\cal X}$ is a cyclic cover of the
manifold $\tilde {\cal X}$ ramified over $\tilde{\cal X}_0$, which has normal
crossings, hence by remark \ref{resnc} $\rho_n^*\tilde {\cal X}$ has a resolution
such that the
divisorial components of the exceptional divisor all have Kodaira dimension
$-\infty$.
\ $\Box$\par\smallskip
|
1996-03-08T06:53:35 | 9410 | alg-geom/9410010 | en | https://arxiv.org/abs/alg-geom/9410010 | [
"alg-geom",
"math.AG"
] | alg-geom/9410010 | null | David B. Jaffe | Functorial structure of units in a tensor product | 23 pages, AMS-LaTeX. Hard copy is available from the author. E-mail
to [email protected] | null | null | null | null | We study the units in a tensor product of rings. For example, let k be an
algebraically closed field. Let A and B be reduced rings containing k, having
connected spectra. Let u \in A tensor_k B be a unit. Then u = a tensor_k b for
some units a \in A and b \in B. Here is a deeper result, stated for simplicity
in the affine case only. Let k be a field, and let f: R --> S be a homomorphism
of f.g. k-algebras such that Spec(f) is dominant. Assume that every irreducible
component of Spec(R_red) or Spec(S_red) is geometrically integral and has a
rational point. Let B --> C be a faithfully flat homomorphism of reduced
k-algebras. For A a k-algebra, define Q(A) to be (S tensor_k A)^*/(R tensor_k
A)^*. Then Q satisfies the following sheaf property: the sequence
0 --> Q(B) --> Q(C) --> Q(C tensor_B C)
is exact. This and another result are used in the proof of the following
statement from "The kernel of the map on Picard groups induced by a faithfully
flat homomorphism" by R. Guralnick, D. Jaffe, W. Raskind, R. Wiegand:
Let K/k be an algebraic field extension and let A be a f.g. k-algebra. Assume
resolution of singularities. Then there is a finite extension E/k contained in
K/k such that Pic(A tensor_k E) --> Pic(A tensor_k K) is injective.
| [
{
"version": "v1",
"created": "Wed, 12 Oct 1994 21:45:50 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Jaffe",
"David B.",
""
]
] | alg-geom | \section{#1}}
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\vskip 0.15in
\par\noindent {\footnotesize Department of Mathematics and Statistics,
University of Nebraska}
\par\noindent {\footnotesize Lincoln, NE 68588-0323, USA\ \ (jaffe{\kern0.5pt}@{\kern0.5pt}cpthree.unl.edu)}}
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\par\noindent David B. Jaffe\protect\footnote{Partially supported by
the National Science Foundation.}
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\begin{document}
\vskip 0.15in
\def\cat{(abelian group)-valued $k$-functors}{\cat{(abelian group)-valued $k$-functors}}
\par\noindent{\Large\bf Functorial structure of units in a tensor product}
\vspace{0.15in}
\def\arabic{footnote}}\setcounter{footnote}{0}{\fnsymbol{footnote}
\vspace{0.1in}
\block{Introduction}
Let $k$ be a field, and let $A$ be a finitely generated\ $k$-algebra.\footnote{All rings
in this paper are commutative.} We explore the
structure of the functor from \cat{$k$-algebras} to \cat{abelian groups} given
by $B \mapsto (A \o*_k B)^*$. More generally, if $S$ is a $k$-scheme of finite
type, not necessarily affine, we study the functor $\mu(S)$ given by
$B \mapsto (\Gamma(S,{\cal O}_S) \o*_k B)^*$. This was done in
(\Lcitemark 8\Rcitemark \ 4.5) for the case where $k$ is algebraically
closed and $S$ is a variety.
We make the assumption that every irreducible component of $\RED{S}$ is
geometrically integral and has a rational point. We summarize these
properties by saying that $S$ is {\it geometrically stable}. If $S$ is
any $k$-scheme of finite type, we can always find a finite extension $k'$ of
$k$ such that $S \times_k \mathop{\operatoratfont Spec}\nolimits(k')$ is geometrically stable as a $k'$-scheme.
With the assumption that $S$ is geometrically stable,
we find that $\mu(S)$ fits into an exact sequence
\ses{{\Bbb G}_m^r \times U \times \xmode{\Bbb Z}^n}{\mu(S)}{I%
}in which $I$ is a sheaf (for the fpqc topology), $I(B) = 0$ for every reduced
$k$-algebra $B$, and $U$ admits a finite filtration with successive quotients
isomorphic to ${\Bbb G}_a^{\kern1pt\beta}$, for various $\beta \in \xmode{\Bbb N} \cup
\setof{\infty}$.
We summarize these properties by saying that $I$ is {\it nilpotent\/} and
$U$ is {\it additive}. In the sequence, $\xmode{\Bbb Z}^n$ denotes the constant sheaf
associated to the abelian group $\xmode{\Bbb Z}^n$, or equivalently, the functor which
represents the constant group scheme associated to the abelian group $\xmode{\Bbb Z}^n$.
Moreover, suppose we have a dominant morphism \hbox{\mp[[ f || S || T ]],} in
which both $S$ and $T$ are geometrically stable. There is an
induced morphism of functors \mp[[ \mu(f) || \mu(T) || \mu(S) ]]. Let
$Q = \mathop{\operatoratfont Coker}\nolimits[\mu(f)]$. We find that $Q$ also fits into an exact sequence as
shown above, except that ${\Bbb G}_m^r \times U \times \xmode{\Bbb Z}^n$ is replaced by an
extension of a finitely generated\ abelian group (i.e.\ the associated constant sheaf) by
${\Bbb G}_m^r \times U$, $U$ is pseudoadditive (see p.\ \pageref{pseudoadditive-def}),
and we do not know if $I$ is a sheaf. Correspondingly,
we do not know if $Q$ is a sheaf, but we do know at least that
$Q|_{\smallcat{reduced $k$-algebras}}$ is a sheaf and moreover that the
canonical map \mapx[[ Q || Q^+ ]] is a monomorphism.
Specializing to the affine case, we see for example that if $A$ is a subalgebra
of a $k$-algebra $C$ (and $\mathop{\operatoratfont Spec}\nolimits(A)$, $\mathop{\operatoratfont Spec}\nolimits(C)$ are geometrically stable),
then the functor given by $B \mapsto (C \o* B)^*/(A \o* B)^*$ fits into such an
exact sequence.
We have thus far described the content of the first theorem
\pref{tori-result-generalized} of this paper. Now we describe the second
theorem \pref{kernel-pic-nilimmersion}, which is an application of the first.
Let $X$ be a geometrically stable $k$-scheme. Let
\hbox{\mp[[ i || X_0 || X ]]} be a nilimmersion, such that the ideal sheaf
${\cal{N}}$ of $X_0$ in $X$ has square zero. Let $P$ be the functor from
\cat{$k$-algebras} to \cat{abelian groups} given by
$$P(B) = \mathop{\operatoratfont Ker}\nolimits[ \mathop{\operatoratfont Pic}\nolimits(X \times_k \mathop{\operatoratfont Spec}\nolimits(B))\ \rightarrow
\ \mathop{\operatoratfont Pic}\nolimits(X_0 \times_k \mathop{\operatoratfont Spec}\nolimits(B))].$%
$Of course, if $X$ is affine, $P = 0$, but in general $P$ is not zero. We
find that $P$ fits into an exact sequence
\sescomma{D \o+ I}{U}{P%
}in which $I$ is nilpotent (except possibly not a sheaf), $U$ is
pseudoadditive,
and $D$ is the constant sheaf associated to a finitely generated\ abelian group.
Although this theorem does not imply that $P$ is a sheaf, it does imply that
if \mp[[ f || B || C ]] is a faithfully flat homomorphism of reduced
$k$-algebras, then $P(f)$ is injective \pref{sheaf-kernel-pic-nilimmersion}.
In fact, this holds even if ${\cal{N}}^2 \not= 0$.
We indicate the idea of the proof of the second theorem. We have an exact
sequence
\splitdiagram{H^0(X,{\cal O}_X^*)&\mapE{}&H^0(X_0,{\cal O}_{X_0}^*)%
}{\mapE{}&H^1(X,{\cal{N}})&\mapE{}&
\mathop{\operatoratfont Ker}\nolimits[\mathop{\operatoratfont Pic}\nolimits(X)\ \rightarrow\ \mathop{\operatoratfont Pic}\nolimits(X_0)]&\mapE{}&0.%
}Functorializing this yields an exact sequence:
\sescomma{\mathop{\operatoratfont Coker}\nolimits[\mu(i)]}{{\Bbb G}_a^{\kern1pt\beta}}{P%
}in which $\beta = h^1(X,{\cal{N}})$. The first theorem tells us what
$\mathop{\operatoratfont Coker}\nolimits[\mu(i)]$ is like. The second theorem is deduced from this.
Finally, we describe a theorem about the Picard group, whose proof in
\Lcitemark 7\Rcitemark \Rspace{} uses both theorems of this paper. Let $k$
be a field, and let
$X$ be a separated $k$-scheme of finite type. Then there exists a finite
field extension $k^+$ of $k$ such that for every algebraic extension $L$ of
$k^+$, the canonical map \mapx[[ \mathop{\operatoratfont Pic}\nolimits(X_L) || \mathop{\operatoratfont Pic}\nolimits(X_{L^a}) ]] is injective.
\vspace{0.1in}
\par\noindent{{\bf Acknowledgements.}\ Bob Guralnick supplied the
neat proof of \pref{unit-lemma-generalized}. Faltings kindly provided example
\pref{Faltings}, thereby correcting an error.
\vspace{0.1in}
\par\noindent{\bf Conventions.}
\begin{alphalist}
\item A {\it $k$-functor\/} is a functor from \cat{$k$-algebras} to \cat{sets}.
(The usage of the term {\it $k$-functor\/} here is slightly different from the
usage in\Lspace \Lcitemark 8\Rcitemark \Rspace{}.) If $V$ is a $k$-scheme, then
we also
let $V$ denote the representable $k$-functor given by
$V(B) = \Morkschemes(\mathop{\operatoratfont Spec}\nolimits(B),V)$.
\item A $k$-functor $F$ is a {\it sheaf\/} (by which we mean
{\it sheaf for the fpqc topology}) if for every faithfully flat homomorphism
\mp[[ p || B || C ]], the canonical map\label{Psi-place}
\dmap[[ \Psi_{F,p} || F(B) || \setof{x \in F(C): F(i_1)(x) = F(i_2)(x)} ]]%
is bijective, where \mp[[ i_2, i_2 || C || C \o*_B C ]] are given by
$c \mapsto c \o* 1$ and $c \mapsto 1 \o* c$, respectively.
\item The superscript $+$ is used to denote {\it associated sheaf}.
\item If $k$ is a field, $X$ is a $k$-scheme, and $L$ is a field extension of
$k$, we let $X_L$ denote $X \times_k \mathop{\operatoratfont Spec}\nolimits(L)$. We let $k^a$ denote an
algebraic closure of $k$.
\item If $X$ is a scheme, we let $\Gamma(X)$ denote $\Gamma(X,{\cal O}_X)$, and we
let $\Gamma^*(X)$ denote $\Gamma(X,{\cal O}_X)^*$.
\item If $B$ is a ring, $\mathop{\operatoratfont Nil}\nolimits(B)$ denotes its nilradical.
\item $k$-functors are said to be
{\it (abelian group)-valued\/} if they take values in
\cat{abelian groups} rather than \cat{sets}.
\end{alphalist}
We give some definitions which are adapted from
\Lcitemark 8\Rcitemark \Rspace{}\ pp.\ 173, 180. If $k$ is a field and $X$,
$Y$
are $k$-schemes, then $\mathop{\mathbf{Hom}}\nolimits(X,Y)$ denotes the $k$-functor given by
$$B \mapsto \Morkschemes(X \times_k \mathop{\operatoratfont Spec}\nolimits(B), Y).$%
$
An (abelian group)-valued $k$-functor $F$ is {\it nilpotent\/} if it is a sheaf
and $F(B) = 0$ for every reduced $k$-algebra $B$. We say that $F$ is
{\it subnilpotent\/} if it can be embedded as a subsheaf of a nilpotent
$k$-functor.
An (abelian group)-valued $k$-functor is {\it discrete\/} if it is a constant
sheaf. We also say that such a functor is (for example)
{\it discrete and finitely generated}, if it is the constant sheaf associated to a
finitely generated\ abelian group.
\block{Additive $k$-functors}
Let $k$ be a field. We need to consider a countably-infinite-dimensional
analog of unipotent group schemes over $k$. Actually, what we will be
considering is more restrictive, as we shall only be considering the analog of
unipotent group schemes over $k$ which are smooth, connected, and moreover
which are $k$-solvable. (See\Lspace \Lcitemark 10\Rcitemark \Rspace{}\ \S5.1.)
The simplest infinite dimensional example is the (abelian group)-valued
$k$-functor ${\Bbb G}_a^\infty$ given by $B \mapsto \o+_{i=1}^\infty B$. However,
this is not good enough for our purposes, since in positive characteristic
one can have nontrivial extensions of ${\Bbb G}_a$ by ${\Bbb G}_a$. (See
e.g.{\ }\Lcitemark 13\Rcitemark \ p.\ 67, exercise 8 or
\Lcitemark 12\Rcitemark \Rspace{}\ VII\ \S2.)
We want to define a class of objects which is closed under extension.
\begin{definition}
Let $k$ be a field, and let $F$ be an (abelian group)-valued $k$-functor. Then
$F$ is {\it strictly additive\/} if it is isomorphic to ${\Bbb G}_a^\alpha$ for some
$\alpha \in \setof{0,1,\ldots,\infty}$, and $F$ is {\it additive\/} if it
admits a filtration:
$$0\ =\ F_0\ \subset F_1\ \subset \cdots \subset F_n\ =\ F,$%
$whose successive quotients are strictly additive.
\end{definition}
This terminology is not perfect, but it is at least consistent with the
usage of the word {\it additive\/} in\Lspace \Lcitemark 8\Rcitemark \Rspace{}:
by (\ref{ext-results}\ref{additive-additive-char0}), it will follow that if
$k$ has characteristic zero, then additive $\Longrightarrow$ strictly additive.
We define the {\it dimension\/} of an additive $k$-functor $F$ to be the sum
of the dimensions of the successive quotients in a filtration of $F$, as
in the definition of additive. Thus $\dim(F) \in \setof{0,1,\ldots,\infty}$.
If $F$ is additive, we define its {\it period\/} to be the smallest $n$ for
which there exists a filtration as in the definition of additive.
A direct sum of countably many additive $k$-functors need not be additive,
even if the summands are finite-dimensional. Also, we shall not concern
ourselves with uncountable direct sums (e.g.\ of ${\Bbb G}_a$), as they seem not
to arise in practice. The following two statements are easily checked:
\begin{prop}\label{extension-of-additive}
Let $k$ be a field. Let
\ses{F'}{F}{F''%
}be an exact sequence of (abelian group)-valued $k$-functors, in which
$F'$ and $F''$ are additive. Then $F$ is additive.
\end{prop}
\begin{prop}\label{quotient-of-additive}
Let $k$ be a field of characteristic zero. Let
\ses{F'}{F}{F''%
}be an exact sequence of (abelian group)-valued $k$-functors, in which
$F'$, $F$ are additive. Then $F''$ is additive.
\end{prop}
\begin{prop}\label{lemma-two}
Let $k$ be a field. Let
\ses{F'}{F}{F''%
}be an exact sequence of (abelian group)-valued $k$-functors, in which
$F'$, $F$ are additive and finite-dimensional. Then $F''$ is additive.
\end{prop}
\begin{proof}
By (\Lcitemark 3\Rcitemark \ 11.17), $(F'')^+$ is representable. Let
\mp[[ p || F || (F'')^+ ]] be the canonical map, which is fpqc-surjective.
By (\Lcitemark 11\Rcitemark \ Theorem 10), there exists a morphism
\mp[[ \sigma || (F'')^+ || F ]] of $k$-functors such that
$p \circ \sigma = 1_{(F'')^+}$. Hence $F'' = (F'')^+$. Let $X'$, $X$, and
$X''$ be the group schemes which represent $F'$, $F$, and $F''$, respectively.
Then we have an exact sequence
\ses{X'}{X}{X''%
}in \cat{commutative $k$-group schemes}. Since $F$ is additive and
finite-dimensional, $X$ admits a series whose factors are copies of the
group scheme ${\Bbb G}_a$. Hence $X''$ admits a series whose factors are group scheme
quotients of ${\Bbb G}_a$. But any quotient of ${\Bbb G}_a$ is $0$ or ${\Bbb G}_a$
(\Lcitemark 10\Rcitemark \ 2.3), so $X''$ admits a series whose
factors are the group scheme ${\Bbb G}_a$. From the argument at the beginning of
the proof (showing that under certain circumstances
fpqc-surjective $\Longrightarrow$ surjective), we see that $F''$ admits a
series whose factors are the (abelian group)-valued $k$-functor ${\Bbb G}_a$.
Hence $F''$ is additive. {\hfill$\square$}
\end{proof}
Unfortunately, \pref{quotient-of-additive} fails in positive characteristic.
We will give an example of this, but there are a couple of preliminaries:
\begin{lemma}\label{strictly-additive-splitting}
If $F$ and $G$ are strictly additive and \mp[[ \pi || F || G ]] is an
epimorphism in \cat{(abelian group)-valued $k$-functors}, then $\pi$ splits.
\end{lemma}
\begin{sketch}
The lemma is clear if $\mathop{\operatoratfont char \kern1pt}\nolimits(k) = 0$, so we may assume that\ $\mathop{\operatoratfont char \kern1pt}\nolimits(k) = p > 0$.
We do the case where $F = G = {\Bbb G}_a^\infty$; the proof in the other cases
is the same. Let $e_i$ denote the element of ${\Bbb G}_a^\infty(k)$
which has $1$ in the \th{i} spot and $0$'s elsewhere. Let $A = k[t]$.
For each $i$, choose $f_i \in {\Bbb G}_a^\infty(A)$ such that $\pi(f_i) = te_i$.
Let $g_i$ be the part of $f_i$ involving only the monomials
$t,t^p,t^{p^2},\ldots$. Since the monomials which appear in an expression
for $\pi$ also have this form (with various $t$), it follows that
$\pi(g_i) = te_i$.
Regard $g_i$ as a function of $t$. Define \mp[[ \sigma || G || F ]] by
$\sigma(be_i) = g_i(b)$, where $B$ is a $k$-algebra and $b \in B$. Then
$\sigma$ splits $\pi$. {\hfill$\square$}
\end{sketch}
\begin{corollary}\label{strictly-additive-quotient}
If $F$ is strictly additive and $G$ is additive, and
\hbox{\mp[[ \pi || F || G ]]}
is an epimorphism in \cat{(abelian group)-valued $k$-functors}, then $G$ is strictly additive.
\end{corollary}
\begin{proof}
Induct on the period of $G$. If $\mathop{\operatoratfont period}\nolimits(G) \leq 1$ we are done. Otherwise,
we can find $G' \subset G$ and an exact sequence
\sesmaps{G'}{}{G}{p}{H%
}in which $H$ is strictly additive, $G'$ is additive, and
$\mathop{\operatoratfont period}\nolimits(G') < \mathop{\operatoratfont period}\nolimits(G)$. By \pref{strictly-additive-splitting},
$p \circ \pi$ splits. Hence $p$ splits. Hence $\mathop{\operatoratfont period}\nolimits(G) = \mathop{\operatoratfont period}\nolimits(G')$:
contradiction. {\hfill$\square$}
\end{proof}
\begin{example}\label{Faltings}
(provided by G.\ Faltings)
\par\noindent Let \mp[[ f || {\Bbb G}_a^\infty || {\Bbb G}_a^\infty ]] be given by
$(x_1,x_2,x_3,\ldots) \mapsto (x_1, x_2 - x_1^p, x_3 - x_2^p, \ldots)$.
Then $f$ is a monomorphism. Let $F'' = \mathop{\operatoratfont Coker}\nolimits(f)$. If $F''$ were additive,
then by \pref{strictly-additive-quotient} $F''$ would be strictly additive,
and so by \pref{strictly-additive-splitting} $f$ would split. However, this
is clearly not the case. Hence $F''$ is not additive.
\end{example}
The following generalization of {\it additive\/} allows us to work around the
behavior illustrated by the example:
\begin{definition}\label{pseudoadditive-def}
An (abelian group)-valued $k$-functor $P$ is {\it pseudoadditive\/} if
for some $n \in \xmode{\Bbb N}$ there exists an exact sequence
\Rowseven{0}{U_1}{U_2}{\cdots}{U_n}{P}{0%
}in \cat{(abelian group)-valued $k$-functors}\ in which $\vec U1n$ are additive.
\end{definition}
By example \pref{Faltings}, one cannot always take $n=1$, i.e.\ additive
$\not=$ pseudoadditive. We do not know if one can always take $n=2$.
\begin{prop}\label{pseudoadditive-is-sheaf}
If $P$ is pseudoadditive, then $P$ is a sheaf.
\end{prop}
\begin{proof}
Let ${\cal{C}}$ be the class of (abelian group)-valued $k$-functors $F$ with the
property that for any faithfully flat homomorphism \mapx[[ B || C ]] of
$k$-algebras, the usual \v Cech complex
\sRowsix{0}{F(B)}{F(C)}{F(C \o*_B C)}{F(C \o*_B C \o*_B C)}{\cdots%
}is exact. Then ${\Bbb G}_a^\alpha \in {\cal{C}}$ for all $\alpha$. If
\ses{F'}{F}{F''%
}is an exact sequence and any two of $F', F, F''$ are in ${\cal{C}}$, then so is
the third. Hence $P$ is in ${\cal{C}}$, so $P$ is a sheaf. {\hfill$\square$}
\end{proof}
\block{Extensions in \cat{(abelian group)-valued $k$-functors}}
For any objects $F_1, F_2$ in an abelian category, one can define an abelian
group $\mathop{\operatoratfont Ext}\nolimits^1(F_1, F_2)$, whose elements are isomorphism classes of extensions
\sesdot{F_2}{F}{F_1%
}(The general theory is described in
\Lcitemark 9\Rcitemark \Rspace{}\ Ch.\ VII, among other places.) Also, we will
refer
to such an exact sequence as defining an {\it extension of $F_1$ by $F_2$}.
In particular, the theory applies to \cat{(abelian group)-valued $k$-functors}.
We shall say that an exact sequence
\sesmaps{F_2}{}{F}{\pi}{F_1%
}in this category is {\it set-theoretically split\/} if there exists a morphism
of $k$-functors \mp[[ \sigma || F_1 || F ]] such that
$\pi \circ \sigma = 1_{F_1}$. We also refer to
{\it set-theoretically split extensions}. Let $\mathop{\operatoratfont Ext}\nolimits^1_s(F_1,F_2)$ denote
the subgroup of elements of $\mathop{\operatoratfont Ext}\nolimits^1(F_1,F_2)$ which correspond to
set-theoretically split extensions.
To compute $\mathop{\operatoratfont Ext}\nolimits^1_s(F_1,F_2)$, we copy (with appropriate but minor changes)
some definitions
which may be found in (\Lcitemark 12\Rcitemark \ VII\ \S4).
For this discussion, fix $F_1$ and $F_2$. A {\it symmetric factor system\/} is
a morphism \mp[[ f || F_1 \times F_1 || F_2 ]] of $k$-functors such that
\begin{eqnarray*}
0 & = & f(y,z) - f(x+y,z) + f(x,y+z) - f(x,y)\\
f(x,y) & = & f(y,x)
\end{eqnarray*}
for all $k$-algebras $B$ and all $x,y,z \in F_1(B)$. If
\mp[[ g || F_1 || F_2 ]] is a morphism of $k$-functors, then there is a
symmetric factor system $\delta g$ defined by
$$\delta g(x,y) = g(x+y) - g(x) - g(y);$%
$such a system is called {\it trivial}. The group structure on $F_2$ makes
the set of symmetric factor systems into a group. Then by standard arguments,
$\mathop{\operatoratfont Ext}\nolimits^1_s(F_1,F_2)$ is isomorphic to the group of symmetric factor systems,
modulo the subgroup of trivial factor systems.
If $F_1$ and $F_2$ are sheaves, then one can also compute the group
$\mathop{\operatoratfont Ext}\nolimits^1_{\operatoratfont fpqc}(F_1,F_2)$, i.e.\ the group of isomorphism classes of
extensions of $F_1$ by $F_2$ in
\cat{(abelian group)-valued $k$-functors which are sheaves}. If moreover
$F_1$ and $F_2$ are represented by commutative group schemes $X_1$ and $X_2$
of finite type over $k$, then [see\Lspace \Lcitemark 4\Rcitemark \Rspace{}\ 5.4
and
\Lcitemark 2\Rcitemark \Rspace{}\ 3.5, 7.3(ii)]
$\mathop{\operatoratfont Ext}\nolimits^1_{\operatoratfont fpqc}(F_1,F_2) = \mathop{\operatoratfont Ext}\nolimits^1(X_1,X_2)$, where
the latter $\mathop{\operatoratfont Ext}\nolimits$ group is computed relative to the abelian category
\cat{commutative group schemes of finite type over $k$}. For arbitrary
sheaves $F_1$, $F_2$, we have $\mathop{\operatoratfont Ext}\nolimits^1(F_1,F_2) \subset \mathop{\operatoratfont Ext}\nolimits^1_{\operatoratfont fpqc}(F_1,F_2)$,
but not equality in general, as may be seen e.g.\ from the exact sequence
\sesmaps{\xmode{\Bbb Z}/p\xmode{\Bbb Z}}{}{{\Bbb G}_a}{t\ \mapsto\ t^p - t}{{\Bbb G}_a%
}in the group scheme category, where $\mathop{\operatoratfont char \kern1pt}\nolimits(k) = p > 0$.
\begin{prop}\label{ext-results}
Let $k$ be a field. We consider objects and morphisms in
\cat{(abelian group)-valued $k$-functors}. Then:
\begin{alphalist}
\item If \mapx[[ W || G ]] is an epimorphism, and $V$ is additive, then the
induced map \mapx[[ \mathop{\operatoratfont Mor}\nolimits_{\smallcat{$k$-functors}}(V,W) ||
\mathop{\operatoratfont Mor}\nolimits_{\smallcat{$k$-functors}}(V,G) ]] is surjective.
\item\label{additive-any}
If $V$ is additive, then $\mathop{\operatoratfont Ext}\nolimits^1_s(V,F) = \mathop{\operatoratfont Ext}\nolimits^1(V,F)$ for all $F$.
\item\label{Z-sheaf} If $F$ is a sheaf then $\mathop{\operatoratfont Ext}\nolimits^1(\xmode{\Bbb Z},F) = 0$.
\item\label{Gm-Ga} If we have an exact sequence
\ses{U_1}{U_2}{P%
}in which $U_1$ and $U_2$ are additive, then $\mathop{\operatoratfont Ext}\nolimits^1({\Bbb G}_m, P) = 0$.
\item\label{nilpotent-discrete} $\mathop{\operatoratfont Ext}\nolimits^1(I,D) = 0$ if $I$ is subnilpotent and
$D$ is discrete.
\item\label{additive-additive-char0} $\mathop{\operatoratfont Ext}\nolimits^1(U,V) = 0$ if $U$ and $V$ are
additive and $k$ has characteristic zero.
\end{alphalist}
\end{prop}
\begin{proof}
{\bf (a):\ }
First we prove this when $V = {\Bbb G}_a^\alpha$ for some $\alpha$. If
$\alpha < \infty$, the claim is immediate. Otherwise, the essential point is
that in a commutative diagram
we can (exercise) fill in a dotted arrow as shown.
Now suppose that $V$ is arbitrary. From what we have just shown, it follows
that $\mathop{\operatoratfont Ext}\nolimits^1_s({\Bbb G}_a^\alpha,F) = \mathop{\operatoratfont Ext}\nolimits^1({\Bbb G}_a^\alpha,F)$ for all $F$. In turn,
this implies that $V \cong {\Bbb G}_a^{\kern1pt\beta}$ in \cat{$k$-functors}, for some
$\beta$.
Hence (a) holds when $V$ is arbitrary.
\vspace{0.1in}
\par\noindent{\bf (b):\ } follows immediately from (a).
\vspace{0.1in}
\par\noindent{\bf (c):\ }
We have to show that if \mp[[ \pi || H || \xmode{\Bbb Z} ]] is an epimorphism in
\cat{(abelian group)-valued $k$-functors}, and $H$ is a sheaf, then
$\pi$ splits. For each $n \in \xmode{\Bbb Z}$, let $y_n \in \xmode{\Bbb Z}(k)$ correspond to the
constant map \mapx[[ \mathop{\operatoratfont Spec}\nolimits(k) || \xmode{\Bbb Z} ]] of topological spaces with value $n$,
and choose $x_1 \in H(k)$ such that $\pi(x_1) = y_1$. For each $n \in \xmode{\Bbb Z}$,
define $x_n \in H(k)$ to be $n x_1$. Define \mp[[ \sigma || \xmode{\Bbb Z} || H ]] as
follows. For any ring $B$, an element $\lambda \in \xmode{\Bbb Z}(B)$ corresponds to a
locally constant map \mapx[[ \mathop{\operatoratfont Spec}\nolimits(B) || \xmode{\Bbb Z} ]] of topological spaces, and
therefore we may write
$B = B_1 \times \cdots \times B_n$ in such a way that $\lambda$ is induced by
$(y_{r_1},\ldots,y_{r_n})$ for suitable $\vec r1n \in \xmode{\Bbb Z}$. Since $H$ is a
sheaf for the Zariski topology, there is a unique element
$x_{r_1,\ldots,r_n} \in H(k^n)$ whose image in $H(k)$ under the \th{i}
projection map is $x_{r_i}$. Now set $\sigma(\lambda)$ equal to the image
of $x_{r_1,\ldots,r_n}$ under the canonical map
\mapx[[ H(k^n) || H(B_1 \times \cdots \times B_n) ]]. This defines $\sigma$, and thus
proves that $\pi$ splits.
\vspace{0.1in}
\par\noindent{\bf (d):\ }
First suppose that $\mathop{\operatoratfont char \kern1pt}\nolimits(k) = 0$. Then $P \cong {\Bbb G}_a^\alpha$ for some $\alpha$.
If $\alpha < \infty$, the statement follows from
[\Lcitemark 10\Rcitemark \ 5.1.1(i)].
We have $\mathop{\operatoratfont Ext}\nolimits^1({\Bbb G}_m,{\Bbb G}_a^\infty) = \mathop{\operatoratfont Ext}\nolimits^1_s({\Bbb G}_m,{\Bbb G}_a^\infty)$.
Therefore an extension of ${\Bbb G}_m$ by ${\Bbb G}_a^\infty$ corresponds to a symmetric
factor system \mp[[ f || {\Bbb G}_m \times {\Bbb G}_m || {\Bbb G}_a^\infty ]]. For $n \gg 0$, we
can find a morphism \mp[[ f_n || {\Bbb G}_m \times {\Bbb G}_m || {\Bbb G}_a^n ]] through which $f$
factors. But then $f_n$ is a symmetric factor system, and so $f_n$ is
trivial, since we already know that $\mathop{\operatoratfont Ext}\nolimits^1({\Bbb G}_m,{\Bbb G}_a^n) = 0$. Hence $f$ is
trivial. Hence $\mathop{\operatoratfont Ext}\nolimits^1({\Bbb G}_m,{\Bbb G}_a^\infty) = 0$.
Now suppose that $k$ has characteristic $p > 0$.
Since ${\Bbb G}_m$ and $P$ are sheaves, it suffices to show that
$\Ext_{\op fpqc}^1({\Bbb G}_m,P) = 0$. For $n$ sufficiently large, multiplication by
$p^n$ gives a zero map from $P$ to $P$. It follows from the fpqc-exact
sequence
\sesmapsone{\mu_{p^n}}{}{{\Bbb G}_m}{{p^n}}{{\Bbb G}_m%
}that it is enough to show $\mathop{\operatoratfont Hom}\nolimits(\mu_{p^n}, P) = 0$. Let
\mp[[ f || \mu_{p^n} || P ]] be a morphism. Let $H$ be the fiber product of
$\mu_{p^n}$ and $U_2$ over $P$. Then we have an exact sequence:
\diagramno{(*)}{\rowfive{0}{U_1}{H}{\mu_{p^n}}{1.}%
}We will show that this sequence splits. We can do this by showing that
$\mathop{\operatoratfont Ext}\nolimits^1(\mu_{p^n},U_1) = 0$, but by the definition of additive, it is clearly
enough to show that $\mathop{\operatoratfont Ext}\nolimits^1(\mu_{p^n}, {\Bbb G}_a^\alpha) = 0$ for all $\alpha$.
Arguing as in the characteristic zero case, one sees further that it is further
enough to show that $\mathop{\operatoratfont Ext}\nolimits^1(\mu_{p^n}, {\Bbb G}_a) = 0$. This is a special case of
\Lcitemark 10\Rcitemark \Rspace{}\ 5.1.1(d). Hence $(*)$ splits.
Hence there exists a morphism \mp[[ \sigma || \mu_{p^n} || U_2 ]] such that
$\pi \circ \sigma = f$, where \mp[[ \pi || U_2 || P ]] is the given map.
Now I claim that $\sigma = 0$. For this (arguing as above), it is enough
to show that $\mathop{\operatoratfont Hom}\nolimits(\mu_{p^n}, {\Bbb G}_a) = 0$. It is enough to do this when
$k = k^a$, and then the statement is well-known. Hence $f = 0$. Hence
$\mathop{\operatoratfont Hom}\nolimits(\mu_{p^n}, P) = 0$, which completes the proof.
\vspace{0.1in}
\par\noindent{\bf (e):\ }
Let
\ses{D}{L}{I%
}be an exact sequence of (abelian group)-valued $k$-functors.
Define a $k$-functor $I'$ by
$I'(B) = \mathop{\operatoratfont Ker}\nolimits[L(B)\ \rightarrow\ L(\RED{B})]$. Then $I'$ defines a
splitting of the sequence.
\vspace{0.1in}
\par\noindent{\bf (f):\ } It suffices to show that
$\mathop{\operatoratfont Ext}\nolimits^1({\Bbb G}_a^\alpha, {\Bbb G}_a^{\kern1pt\beta}) = 0$ for all $\alpha, \beta$.
Moreover, since
$\mathop{\operatoratfont Ext}\nolimits^1$ converts a coproduct in the first variable into a product,
we may assume that\ $\alpha = 1$. By (b), it suffices to show that
$\mathop{\operatoratfont Ext}\nolimits^1_s({\Bbb G}_a, {\Bbb G}_a^{\kern1pt\beta}) = 0$. Arguing as in (d), it suffices to
show
that $\mathop{\operatoratfont Ext}\nolimits^1_s({\Bbb G}_a,{\Bbb G}_a^n) = 0$, and moreover we may as well take $n=1$.
Suppose we have an exact sequence
\sesdot{{\Bbb G}_a}{X}{{\Bbb G}_a%
}By (\Lcitemark 10\Rcitemark \ 3.9 ter.), $X \cong {\Bbb G}_a^2$. But
(in characteristic zero) morphisms from ${\Bbb G}_a^n$ to ${\Bbb G}_a^m$ are in bijective
correspondence with vector space homomorphisms from $k^n$ to $k^m$, so the
sequence splits. {\hfill$\square$}
\end{proof}
\block{Functorial structure of units in a tensor product}
The main purpose of this section is to prove \pref{tori-result-generalized},
which generalizes (\Lcitemark 8\Rcitemark \ 4.5). The
preparatory lemmas are similar to those in
(\Lcitemark 8\Rcitemark \ \S4),
and we shall omit their proofs if the proofs of the corresponding statements
in\Lspace \Lcitemark 8\Rcitemark \Rspace{} carry over with minor changes.
\begin{lemma}\label{unit-lemma-generalized}
Let $k$ be an algebraically closed field. Let $A$ and $B$ be reduced rings
containing $k$, having connected spectra. Let $u \in A \o*_k B$ be a unit.
Then $u = a \o* b$ for some units $a \in A$ and $b \in B$.
\end{lemma}
\begin{proof}
The statement generalizes (\Lcitemark 8\Rcitemark \ 4.2), but we
give a new and simpler proof, due to Guralnick.
Let $X$ be the set of maximal ideals of $A$, and let $Y$ be the set of maximal
ideals of $B$. Let $x_0 \in X$, $y_0 \in Y$. We will prove the lemma by
showing that for all $x \in X$, $y \in Y$, we have:
$$u(x,y)\ =\ {u(x,y_0) u(x_0,y) \over u(x_0, y_0)}.\eqno(*)$%
$For this we may suppose that $k$ is uncountable.
By a variant of a result of Roquette (see
\Lcitemark 7\Rcitemark \Rspace{}\ 1.5) the group $B^*/k^*$ is finitely generated, so
$F = \setof{f \in B^*: f(y_0) = 1}$ is countable. For each $f \in F$, let
$$Q(f)\ =\ \setof{x \in X: u(x,y) = u(x,y_0)f(y)\hbox{\ for all\ } y \in Y}.$%
$Then $Q(f)$ is a closed subset of $X$. For any given $x \in X$, the
function on $Y$ given by $y \mapsto u(x,y)/u(x,y_0)$ sends $y_0$ to $1$
and so lies in $F$. Hence $X = \cup_{f \in F}Q(f)$.
Since $k$ is uncountable, it follows that if $I$ is an irreducible component
of $X$, then $I \subset Q(f)$ for some $f \in F$. Hence for any fixed $y \in Y$,
the function \mp[[ g_y || X || k ]] given by $x \mapsto u(x,y)/u(x,y_0)$ is
constant on each irreducible component of $X$. Since $X$ is connected, $g_y$
is constant. Then $u(x,y) = u(x,y_0)g_y(x_0)$, which proves $(*)$. {\hfill$\square$}
\end{proof}
\begin{corollary}\label{second-generalized}
Let $k$ be an algebraically closed field. Let $A$ and $B$ be rings containing
$k$, having connected spectra. Assume that $A$ is reduced. Let ${\xmode{{\fraktur{\lowercase{M}}}}} \subset A$
be a maximal ideal such that $A/{\xmode{{\fraktur{\lowercase{M}}}}} = k$. Then $(A \o*_k B)^*$ is the direct
sum of the two subgroups $A^*B^*$ and $1 + ({\xmode{{\fraktur{\lowercase{M}}}}} \o* \mathop{\operatoratfont Nil}\nolimits(B))$%
.\footnote{Statements 4.3 and 4.4 from\Lspace \Lcitemark 8\Rcitemark \Rspace{}
should
also have the hypothesis that $A/{\xmode{{\fraktur{\lowercase{M}}}}} = k$.}
\end{corollary}
\begin{corollary}\label{third-generalized}
Let $k$ be an algebraically closed field. Let $A$ and $B$ be rings
containing $k$. Assume that $A$ is reduced and has a connected spectrum.
Let ${\xmode{{\fraktur{\lowercase{M}}}}} \subset A$ be a maximal ideal such that $A/{\xmode{{\fraktur{\lowercase{M}}}}} = k$.
\begin{itemize}
\item For any decomposition
$B = B_1 \times \cdots \times B_n$, there is a subgroup:
$$\mu(\vec B1n) = \oplus_{i=1}^n [A^*B_i^* \o+ (1 + ({\xmode{{\fraktur{\lowercase{M}}}}} \o* \mathop{\operatoratfont Nil}\nolimits(B_i))]$%
$of $(A \o*_k B)^*$.
\item For any $x \in (A \o*_k B)^*$, there exists a decomposition
$B = B_1 \times \cdots \times B_n$ such that $x \in \mu(\vec B1n)$.
\item If $B$ has only finitely many idempotent elements (e.g.\ if $B$ is
noetherian), we can write $B = B_1 \times \cdots \times B_n$ for
rings $B_i$ having connected spectra. Then $\mu(\vec B1n) = (A \o*_k B)^*$.
\end{itemize}
\end{corollary}
Let $k$ be a field, and let $S$ be a $k$-scheme of finite type. Let
$F = \mathop{\mathbf{Hom}}\nolimits(S,{\Bbb G}_m)$. Then
$$F(B)\ =\ \Gamma(S \times \mathop{\operatoratfont Spec}\nolimits(B), {\cal O}^*_{S \times \mathop{\operatoratfont Spec}\nolimits(B)})
\ =\ [\Gamma(S,{\cal O}_S) \o*_k B]^*,$%
$by (\Lcitemark 6\Rcitemark \ 9.3.13 (i)).
In particular, if $S = \mathop{\operatoratfont Spec}\nolimits(A)$, then $F(B) = (A \o*_k B)^*$. The next
theorem gives an abstract description of $F$, and thus (in effect) a
description of how units in a tensor product $A \o*_k B$ vary as $B$ varies.
First, for convenience, we encapsulate the following definition:
\begin{definition}
Let $k$ be a field. A $k$-scheme $S$ is {\it geometrically stable\/} if $(1)$
it is of finite type, and $(2)$ every irreducible component of $\RED{S}$ is
geometrically integral and has a rational point.
\end{definition}
\begin{theorem}\label{tori-result-generalized}
Let $k$ be a field. Define an (abelian group)-valued $k$-functor $F$ to be of
type $(*)$ if there exist exact sequences
\seslabcomma{R}{F}{I}{\dag%
}\seslab{{\Bbb G}_m^r \times U}{R}{L}{\dag\dag%
}in \cat{(abelian group)-valued $k$-functors}, in which
$r \geq 0$, $U$ is pseudoadditive, $I$ is subnilpotent, and $L$ is discrete
and finitely generated.
\par\noindent{\bf\rm (a):\ }
Let $S$ be a geometrically stable $k$-scheme. Then $\mathop{\mathbf{Hom}}\nolimits(S,{\Bbb G}_m)$ is of type
$(*)$ and we have $r =$ the number of connected components of $S$. Also $U$
is additive of dimension $\dim_k \mathop{\operatoratfont Nil}\nolimits[\Gamma(S,{\cal O}_S)]$. Also,
$I$ is nilpotent, $L$ is free, and $(\dag\dag)$ splits.
\par\noindent{\bf\rm (b):\ }
Let $S$ and $T$ be geometrically stable $k$-schemes, and let
\mp[[ f || S || T ]] be a dominant morphism of $k$-schemes.
Then the cokernel of $\mathop{\mathbf{Hom}}\nolimits(f,{\Bbb G}_m)$ is of type $(*)$. Moreover, $r$ equals
the number of connected components of $S$ minus the number of connected
components of $T$.
\end{theorem}
\begin{corollary}\label{tori-quotient-sheaf}
Let $S$ and $T$ be geometrically stable $k$-schemes, and let
\mp[[ f || S || T ]] be a dominant morphism of $k$-schemes. Let $Q$ be the
cokernel of $\mathop{\mathbf{Hom}}\nolimits(f,{\Bbb G}_m)$. Then the canonical map \mapx[[ Q || Q^+ ]] is a
monomorphism, and $Q|_{\smallcat{reduced $k$-algebras}}$ is a sheaf, in the
sense that if \mp[[ p || B || C ]] is a faithfully flat homomorphism of
reduced $k$-algebras, then $\psi_{Q,p}$ (see p.\ \pageref{Psi-place}) is
bijective.
\end{corollary}
\begin{remarks}
\
\begin{romanlist}
\item In part (a) of the theorem, one can choose $(\dag)$ so that it splits
if $S$ is reduced, but probably not in general.
\item In part (b), the sequence $(\dag\dag)$ does not always split. For
an example, take $k$ to be an imperfect field of characteristic $p$,
let $u \in k - k^p$, and let $f$ be $\mathop{\operatoratfont Spec}\nolimits$ of the ring map
\mapx[[ k[t,t^{-1}] || k[x,x^{-1}] \times k ]], given by
$t \mapsto (x^p,u)$.
\item The hypothesis that the schemes in the theorem be geometrically stable
can be weakened slightly, as is indicated in the proof. They presumably
can be weakened further, but we do not know what is possible in this
direction.
\item\label{tori-result-remark-three}
We suspect that $I$ in part (b) of the theorem is a sheaf (and thus
satisfies the definition of {\it nilpotent}). If true, this would imply
(in the corollary) that the cokernel of $\mathop{\mathbf{Hom}}\nolimits(f,{\Bbb G}_m)$ is a sheaf. To
prove that $I$ is a sheaf, it would be sufficient (at least in the
case where $S$ and $T$ are connected) to show that if $C$ is
a subalgebra of a reduced $k$-algebra $A$, then the
(abelian group)-valued $k$-functor given by
$$B\ \mapsto\ {1 + A \o* \mathop{\operatoratfont Nil}\nolimits(B) \over 1 + C \o* \mathop{\operatoratfont Nil}\nolimits(B)}$%
$ is a sheaf.
\item In part (b), we have $U$ pseudoadditive with $n=2$, as in the definition
of pseudoadditive. However, it is conceivable that $U$ is always
additive.
\end{romanlist}
\end{remarks}
\begin{proofnodot}
(of \ref{tori-result-generalized}.)
The hypothesis that $S$ be geometrically stable is chosen for simplicity
and we note here some consequences which are in fact sufficient to prove the
theorem:
\vspace{0.05in}
\par\circno{A}:\ every connected component of $S$ has a rational
point;
\vspace{0.05in}
\par\circno{B}:\ $\RED{S}$ is geometrically reduced
[by\Lspace \Lcitemark 5\Rcitemark \Rspace{}\
4.6.1(e)].
\vspace{0.05in}
\par\noindent Moreover, \circno{A}\ and \circno{B}\ also imply:
\vspace{0.05in}
\par\circno{C}:\ every connected component of $S$ is geometrically
connected [by\Lspace \Lcitemark 5\Rcitemark
\Rspace{}\ 4.5.14];
\vspace{0.05in}
\par\circno{D}:\ if $Q$ is a connected component of $S$, then $k$ is
integrally closed in $\Gamma(\RED{Q})$.
\par\noindent All of these comments apply equally to $T$.
Now we want to reduce to the affine case. This is not literally possible,
because $\Gamma(S)$ need not be finitely generated\ as a $k$-algebra. What we can do is
reformulate the theorem in terms of a certain class of $k$-algebras. This
class is chosen simply to serve the needs of the proof: a $k$-algebra
is {\it good\/} if it is of the form $\Gamma(S)/N$, where $S$ is a
geometrically stable $k$-scheme and $N \subset \Gamma(S)$ is a nilpotent ideal.
Here is a reformulation of the theorem in terms of good $k$-algebras:
\begin{quote}
\par\noindent{\bf\rm (a):\ }
Let $A$ be a good $k$-algebra. Then $B \mapsto (A \o* B)^*$ is of type
$(*)$ and we have $r =$ the number of connected components of $\mathop{\operatoratfont Spec}\nolimits(A)$,
$\dim(U) = \dim_k \mathop{\operatoratfont Nil}\nolimits(A)$. Also, $L$ is free and $I$ is nilpotent.
\par\noindent{\bf\rm (b):\ }
Let \mp[[ \phi || C || A ]] be a homomorphism of good $k$-algebras.
Assume that $\mathop{\operatoratfont Ker}\nolimits(\phi)$ is nilpotent. Then the cokernel of the morphism from
$B \mapsto (C \o* B)^*$ to $B \mapsto (A \o* B)^*$ is of type $(*)$.
Moreover, $r$ equals the number of connected components of $\mathop{\operatoratfont Spec}\nolimits(A)$ minus
the number of connected components of $\mathop{\operatoratfont Spec}\nolimits(C)$.
\end{quote}
Since the map \mapx[[ (C \o* B)^* || (C/\mathop{\operatoratfont Ker}\nolimits(\phi) \o* B)^* ]] is surjective,
we may reduce to the case where $\phi$ is {\it injective}. It was the need for
this reduction which lead to the introduction of good $k$-algebras in the
proof.
We proceed to build a diagram involving (abelian group)-valued $k$-functors,
which we associate to $A$, and which is functorial in $A$.
Let $G_A$ be given by $G_A(B) = (\RED{A} \o* B)^*$. Write
$\RED{A} = A_1 \times \cdots \times A_r$, where $\vec A1r$ have connected spectra.
We can identify $A \o* B$ with $(A_1 \o* B) \times \cdots \times (A_r \o* B)$. Let
$F_A$ be the sheaf associated to the subfunctor of $G_A$ given by
$B \mapsto
\setof{(a_1 \o* b_1, \ldots, a_r \o* b_r): a_i \in A_i^*, b_i \in B^*}$.
Let $E_A$ be the subfunctor of $F_A$ given by
$E_A(B) = \setof{(\vec b1r): \vec b1r \in B^*}$. Let
$D_A = F_A/E_A$. Let $I_A = G_A/F_A$.
Define $H_A$ by $H_A(B) = (A \o* B)^*$.
Let \mp[[ p || H_A || G_A ]] be the canonical map, and let $U_A$ be its kernel.
We have $U_A(B) = 1 + \mathop{\operatoratfont Nil}\nolimits(A) \o* B$. For each $n \in \xmode{\Bbb N}$, let
$U_A^n$ be given by $U_A^n(B) = 1 + \mathop{\operatoratfont Nil}\nolimits(A)^n \o* B$.
Here is the diagram of (abelian group)-valued $k$-functors which we have
built:
\diagramx{&&&&0\cr
&&&&\mapS{}\cr
&& 0 &&U_A & = \kern10pt
\hbox to 0pt{$U_A^1 \kern10pt \supset \kern10pt U_A^2
\kern10pt \supset \kern10pt \cdots$}\cr
&& \mapS{} && \mapS{}\cr
&& E_A && H_A \cr
&& \mapS{} && \mapS{}\cr
\rowfive{0}{F_A}{G_A}{I_A}{0}\cr
&& \mapS{} && \mapS{} && \vbox to 0pt{\box5}\cr
&& D_A && 0\cr
&& \mapS{}\cr
&& 0}
We proceed to analyze the various components of this diagram. In particular,
we will show that \circno1\ $G_A \cong F_A \times I_A$ and $I_A$ is nilpotent,
\circno2\ $E_A \cong {\Bbb G}_m^r$,
where $r$ is the number of connected components of $\mathop{\operatoratfont Spec}\nolimits(A)$, \circno3\ $D_A$
is represented by a constant group scheme (corresponding to a finitely generated\ free abelian
group), and that \circno4\ $U_A$ is additive. For these purposes,
we may assume that\ $\mathop{\operatoratfont Spec}\nolimits(A)$ is connected. Then $E_A(B) = B^*$.
\vspace{0.05in}
\par\noindent\circno1\ Let
${\xmode{{\fraktur{\lowercase{M}}}}}_A \subset \RED{A}$ be a maximal ideal such that $A/{\xmode{{\fraktur{\lowercase{M}}}}}_A = k$. (This is
possible by \circno{A}.) Define a
subfunctor $I'_A$ of $G_A$ by $I'_A(B) = 1 + {\xmode{{\fraktur{\lowercase{M}}}}}_A \o* \mathop{\operatoratfont Nil}\nolimits(B)$.
Let \mp[[ \psi || F_A \o+ I'_A || G_A ]] be the canonical map. We will show
that $\psi$ is an isomorphism. If $k$ is algebraically closed, this follows
from \pref{third-generalized}. But both the source and the target of $\psi$
are sheaves, so it follows (using \circno{B}\ and \circno{C}) that $\psi$ is an
isomorphism for any $k$. Thus $I'_A \cong I_A$, so $I_A$ is nilpotent.
\vspace{0.05in}
\par\noindent\circno2\ We have $E_A \cong {\Bbb G}_m$.
\vspace{0.05in}
\par\noindent\circno3\ Write $A = \Gamma(S)/N$, as in the definition of
good. Let ${\cal{N}}$ be the nilradical of $S$. Then we have an exact sequence:
\les{H^0(S,{\cal{N}})}{H^0(S,{\cal O}_S)}{H^0(\RED{S},{\cal O}_{\RED{S}})%
}and so $\RED{A}$ is a subring of $H^0(\RED{S},{\cal O}_{\RED{S}})$. Since $k$ is
integrally closed in $H^0(\RED{S},{\cal O}_{\RED{S}})$ by \circno{D}, it follows by
(\Lcitemark 7\Rcitemark \ 1.5),that $D_A(k)$ is free abelian of
finite rank. In fact, $D_A$ is represented by the corresponding constant group
scheme.
\vspace{0.05in}
\par\noindent\circno4\ For each $n$ we have
$${U_A^n \over U_A^{n+1}}(B)
\ =\ {1 + \mathop{\operatoratfont Nil}\nolimits(A)^n \o* B \over 1 + \mathop{\operatoratfont Nil}\nolimits(A)^{n+1} \o* B}
\ \cong\ {\mathop{\operatoratfont Nil}\nolimits(A)^n \over \mathop{\operatoratfont Nil}\nolimits(A)^{n+1}} \o* B$%
$as (abelian group)-valued $k$-functors.
Hence $U_A$ is additive.
Now we describe $H_A$, making a number of non-canonical choices.
We have $F_A \cong E_A \times D_A$ non-canonically, e.g.\ by
(\ref{ext-results}\ref{Z-sheaf}), but it is easily proved directly. Hence
$F_A \cong {\Bbb G}_m^r \times \xmode{\Bbb Z}^n$ for some $n$. Also, we have shown that
$G_A \cong F_A \times I_A$. Therefore we have an exact sequence
\sesmapsdot{U_A}{}{H_A}{q}{{\Bbb G}_m^r \times \xmode{\Bbb Z}^n \times I_A%
}Let $M = q^{-1}({\Bbb G}_m^r \times \xmode{\Bbb Z}^n)$. Then we have an exact sequence
\sesdot{U_A}{M}{{\Bbb G}_m^r \times \xmode{\Bbb Z}^n%
}By (\ref{ext-results}\ref{Z-sheaf}\ref{Gm-Ga}) this sequence splits.
This proves (a).
Now, to prepare for proving (b), we analyze the functorial behavior of each
basic component of the big diagram shown above. Let \mp[[ \phi || C || A ]] be
an injective homomorphism of good $k$-algebras.
First we analyze $E_\phi$. It is a monomorphism,
corresponding to a map \mapx[[ {\Bbb G}_m^{r_1} || {\Bbb G}_m^{r_2} ]], for some $r_1$ and
$r_2$, which is given by an $r_2 \times r_1$ matrix of $0$'s and $1$'s.
The cokernel of $E_\phi$ is isomorphic to ${\Bbb G}_m^{r_2 - r_1}$.
Now we analyze $D_\phi$. Let us show that $D_\phi$ is a monomorphism.
Since its source and target are constant sheaves, it suffices to show that
$D_\phi(k^a)$ is injective. The assertion then boils down to showing that
if one has a dominant morphism \mp[[ \psi || V || W ]] of reduced schemes of
finite type over an algebraically closed field $k$, and $W$ is connected, and
\mp[[ g || W(k) || k ]] is a non-constant regular function, then
$g \circ \psi(k)$ is not constant on each connected component of $V$. This is
clear, so $D_\phi$ is a monomorphism. The cokernel of $D_\phi$ is the
constant sheaf associated to a finitely generated\ abelian group.
We show that $I_\phi$ is a monomorphism. In the process of doing so, we
justify remark \pref{tori-result-remark-three} from
p.\ \pageref{tori-result-remark-three}. Also, once we know that $I_\phi$ is a
monomorphism, it will follow immediately that $\mathop{\operatoratfont Coker}\nolimits(I_\phi)$ is subnilpotent.
We may assume that $\mathop{\operatoratfont Spec}\nolimits(C)$ is connected.
We have a canonical map \mapx[[ 1 + \RED{C} \o* \mathop{\operatoratfont Nil}\nolimits(B) || I_C(B) ]], and
likewise for $A$. From our discussion of $I'$, it is clear that these maps
are surjective. Letting $X_C(B)$ and $X_A(B)$ denote their kernels, we have a
commutative diagram with exact rows:
\diagramx{\sesonerow{X_C(B)}{1 + \RED{C} \o* \mathop{\operatoratfont Nil}\nolimits(B)}{I_C(B)}\cr
&& \mapS{} && \mapS{} && \mapS{}\cr
\sesonerowdot{X_A(B)}{1 + \RED{A} \o* \mathop{\operatoratfont Nil}\nolimits(B)}{I_A(B)}}
We describe $X_C(B)$.
Let $x \in X_C(B)$. Locally (for the fpqc topology) on $B$, we may write
$x = c \o* b = 1 + \sum_{i=1}^s c_i \o* b_i$, where $c \in \RED{C}^*$,
$b \in B^*$, $c_i \in \RED{C}$, and $b_i \in \mathop{\operatoratfont Nil}\nolimits(B)$. It follows that $b$
must lie in the $k$-linear span of $1$ and the $b_i$, and in particular that
we may write $b = 1 + n$, where $n \in \mathop{\operatoratfont Nil}\nolimits(B)$. Passing to
$\RED{C} \o* \RED{B}$, we see then that $c = 1$. Hence $x = 1 + n$. From
this it follows that $X_C(B) = 1 + \mathop{\operatoratfont Nil}\nolimits(B)$.
Similarly, we have
$X_A(B) = (1+\mathop{\operatoratfont Nil}\nolimits(B)) \times \cdots \times (1+\mathop{\operatoratfont Nil}\nolimits(B))$, with one copy for each
connected component of $\mathop{\operatoratfont Spec}\nolimits(A)$. It follows (details omitted) that the
canonical map
\dmapx[[ {X_A(B) \over X_C(B)} ||
{1 + \RED{A} \o* \mathop{\operatoratfont Nil}\nolimits(B) \over 1 + \RED{C} \o* \mathop{\operatoratfont Nil}\nolimits(B)} ]]%
is injective, and hence that $I_\phi$ is a monomorphism.
Assume now that $\mathop{\operatoratfont Spec}\nolimits(A)$ and $\mathop{\operatoratfont Spec}\nolimits(C)$ are connected.
Let ${\xmode{{\fraktur{\lowercase{M}}}}}_C$ be the preimage of ${\xmode{{\fraktur{\lowercase{M}}}}}_A$ under the map
\mapx[[ \RED{C} || \RED{A} ]] induced by $\phi$. From our discussion of $I'$,
it is clear that $\mathop{\operatoratfont Coker}\nolimits(I_\phi)$ is isomorphic to the cokernel of the morphism
given at $B$ by \mapx[[ 1 + {\xmode{{\fraktur{\lowercase{M}}}}}_C \o* \mathop{\operatoratfont Nil}\nolimits(B) || 1 + {\xmode{{\fraktur{\lowercase{M}}}}}_A \o* \mathop{\operatoratfont Nil}\nolimits(B) ]].
In turn this implies remark (iii).
We have exact sequences
\sescomma{\mathop{\operatoratfont Coker}\nolimits(U_\phi)}{\mathop{\operatoratfont Coker}\nolimits(H_\phi)}{\mathop{\operatoratfont Coker}\nolimits(G_\phi)%
}\sescomma{\mathop{\operatoratfont Coker}\nolimits(F_\phi)}{\mathop{\operatoratfont Coker}\nolimits(G_\phi)}{\mathop{\operatoratfont Coker}\nolimits(I_\phi)%
}\sesdot{\mathop{\operatoratfont Coker}\nolimits(E_\phi)}{\mathop{\operatoratfont Coker}\nolimits(F_\phi)}{\mathop{\operatoratfont Coker}\nolimits(D_\phi)%
}Since $\mathop{\operatoratfont Coker}\nolimits(U_\phi)$ is clearly pseudoadditive, part (b) of the theorem
follows from these sequences and (\ref{ext-results}\ref{Gm-Ga}). {\hfill$\square$}
\end{proofnodot}
\block{Line bundles becoming trivial on pullback by a nilimmersion}
If $X$ is a $k$-scheme, we let $\mathop{\mathbf{Pic}}\nolimits(X)$ denote the $k$-functor given by
$B \mapsto \mathop{\operatoratfont Pic}\nolimits(X \times_k \mathop{\operatoratfont Spec}\nolimits(B))/\mathop{\operatoratfont Pic}\nolimits(B)$. Then $\mathop{\mathbf{Pic}}\nolimits$ itself defines a
functor whose source is \opcat{$k$-schemes}. If \mp[[ f || X || Y ]] is a
morphism of $k$-schemes of finite type, such that $X$ and $Y$ each have a
rational point, then $\mathop{\operatoratfont Ker}\nolimits[\mathop{\mathbf{Pic}}\nolimits(f)]$ is isomorphic to the $k$-functor given by
$$B \mapsto \mathop{\operatoratfont Ker}\nolimits[\mathop{\operatoratfont Pic}\nolimits(Y \times_k \mathop{\operatoratfont Spec}\nolimits(B))
\ \rightarrow\ \mathop{\operatoratfont Pic}\nolimits(X \times_k \mathop{\operatoratfont Spec}\nolimits(B))].$$
\begin{theorem}\label{kernel-pic-nilimmersion}
Let $k$ be a field, and let $X$ be a geometrically stable $k$-scheme.
Let \mp[[ i || X_0 || X ]] be a nilimmersion, such that the ideal sheaf ${\cal{N}}$
of $X_0$ in $X$ has square zero. Then there is an exact sequence
of (abelian group)-valued $k$-functors
\sescomma{D \o+ I}{P}{\mathop{\operatoratfont Ker}\nolimits[\mathop{\mathbf{Pic}}\nolimits(i)]%
}in which $D$ is discrete and finitely generated, $I$ is subnilpotent, and $P$ is
pseudoadditive.
\end{theorem}
\begin{remarks}
\
\begin{alphalist}
\item If $X$ is affine, $\mathop{\operatoratfont Ker}\nolimits[\mathop{\mathbf{Pic}}\nolimits(i)] = 0$.
\item If $X$ is proper over $k$, $D = 0$ and $I = 0$, so
$\mathop{\operatoratfont Ker}\nolimits[\mathop{\mathbf{Pic}}\nolimits(i)] \cong P$. Also, $P$ is additive and finite-dimensional.
One way to get examples is to take $k$
to be algebraically closed, $Y$ to be a projective variety over $k$, and
${\cal{M}}$ to be a coherent ${\cal O}_Y$-module with $H^1(Y,{\cal{M}}) \not= 0$. Make
${\cal O}_Y \o+ {\cal{M}}$ into a coherent ${\cal O}_Y$-algebra via the rule ${\cal{M}}^2 = 0$.
Let $X = \mathop{\mathbf{Spec}}\nolimits({\cal O}_Y \o+ {\cal{M}})$, and let \mp[[ i || \RED{X} || X ]] be the
inclusion. Then $\mathop{\operatoratfont Ker}\nolimits[\mathop{\mathbf{Pic}}\nolimits(i)] \cong {\Bbb G}_a^\alpha$, where
$\alpha = h^1(Y,{\cal{M}})$.
\item In the non-affine, non-proper case, we have not determined exactly what
can happen. In particular, we do not know if $D$ can be nonzero.
If $k$ has characteristic zero, then $D \cong \xmode{\Bbb Z}^n$ for some $n$. If $k$
has characteristic $p > 0$, then $D \cong \xmode{\Bbb Z}^n \o+ (\xmode{\Bbb Z}/p\xmode{\Bbb Z})^m$ for some
$n$ and some $m$.
\item Conceivably the theorem holds without the assumption that ${\cal{N}}^2 = 0$.
To prove this, one would at least have to understand
$\mathop{\operatoratfont Coker}\nolimits[\mathop{\mathbf{Pic}}\nolimits(i)]$ in the case where ${\cal{N}}^2 = 0$, which we do not.
\item We will find an exact sequence
\Rowsix{0}{U_1}{U_2}{{\Bbb G}_a^{\kern1pt\beta}}{P}{0%
}in which $U_1$ and $U_2$ are additive, for some $\beta$. This is
stronger than saying that $P$ is pseudoadditive. Perhaps $P$ is always
additive.
\end{alphalist}
\end{remarks}
\begin{proof}
We have an exact sequence of sheaves of abelian groups on $X$
\Rowfive{0}{{\cal{N}}}{{\cal O}_X^*}{{\cal O}_{X_0}^*}{1,%
}and thus an exact sequence of abelian groups
\splitdiagram{H^0(X,{\cal O}_X^*)&\mapE{}&H^0(X_0,{\cal O}_{X_0}^*)%
}{\mapE{}&H^1(X,{\cal{N}})&\mapE{}&
\mathop{\operatoratfont Ker}\nolimits[\mathop{\operatoratfont Pic}\nolimits(X)\ \rightarrow\ \mathop{\operatoratfont Pic}\nolimits(X_0)]&\mapE{}&0.%
}In fact, everything is functorial in $k$, and we thus obtain an exact
sequence of (abelian group)-valued $k$-functors
\splitdiagram{\mathop{\mathbf{Hom}}\nolimits(X,{\Bbb G}_m)&\mapE{}&\mathop{\mathbf{Hom}}\nolimits(X_0,{\Bbb G}_m)&\mapE{}&
{\Bbb G}_a^{\kern1pt\beta}}{\mapE{}&
\mathop{\operatoratfont Ker}\nolimits[\mathop{\mathbf{Pic}}\nolimits(X)\ \rightarrow\ \mathop{\mathbf{Pic}}\nolimits(X_0)]&\mapE{}&0,%
}where $\beta = \dim_k[H^1(X,{\cal{N}})]$.
Let $K = \mathop{\operatoratfont Ker}\nolimits[\mathop{\mathbf{Pic}}\nolimits(X)\ \rightarrow\ \mathop{\mathbf{Pic}}\nolimits(X_0)]$,
$L = \mathop{\operatoratfont Coker}\nolimits[\mathop{\mathbf{Hom}}\nolimits(X,{\Bbb G}_m)\ \rightarrow\ \mathop{\mathbf{Hom}}\nolimits(X_0,{\Bbb G}_m)]$.
We have an exact sequence
\sesdot{L}{{\Bbb G}_a^{\kern1pt\beta}}{K%
}According to (\ref{tori-result-generalized}b), there are exact sequences
\sescomma{P'}{R}{Q%
}\sescomma{R}{L}{I%
}in which $Q$ is discrete and finitely generated, $P'$ is additive, and $I$ is subnilpotent.
By definition,
$\mathop{\operatoratfont Coker}\nolimits[ P'\ \rightarrow\ {\Bbb G}_a^{\kern1pt\beta}]$ is pseudoadditive. Hence by
(\ref{ext-results}\ref{nilpotent-discrete}), we have an exact sequence
\sescomma{Q \times I}{P}{K%
}in which $P$ is pseudoadditive. {\hfill$\square$}
\end{proof}
\begin{corollary}\label{sheaf-kernel-pic-nilimmersion}
Let $k$ be a field, and let $X$ be a geometrically stable $k$-scheme.
Let \mp[[ i || X_0 || X ]] be a nilimmersion. Let
$F = \mathop{\operatoratfont Ker}\nolimits[\mathop{\mathbf{Pic}}\nolimits(i)]$. Let \mp[[ p || B || C ]] be a faithfully flat ring
homomorphism.
\begin{alphalist}
\item If $B$ and $C$ are reduced, then the canonical map
\mapx[[ F(B) || F(C) ]] is injective.
\item Assume that $k$ has characteristic zero and that the ideal sheaf of $X_0$
in $X$ has square zero. Assume that $B$ is normal and
that $C$ is \'etale over $B$. Then $\Psi_{F,p}$ (see p.\ \pageref{Psi-place})
is bijective.
\end{alphalist}
\end{corollary}
\begin{proof}
Let ${\cal{N}}$ be the ideal sheaf of $X_0$ in $X$. First suppose that
${\cal{N}}^2 = 0$. Then (a) follows from \pref{kernel-pic-nilimmersion}. For (b),
let $S = \mathop{\operatoratfont Spec}\nolimits(B)$. Since $\mathop{\operatoratfont char \kern1pt}\nolimits(k) = 0$, $D$ is torsion-free. Therefore
it suffices to show that in the category of
(abelian group)-valued $k$-functors
which are sheaves for the \'etale topology, the sequence
\ses{\xmode{\Bbb Z}^n}{{\Bbb G}_a^\alpha}{({\Bbb G}_a^\alpha/\xmode{\Bbb Z}^n)^+%
}is exact when evaluated at $B$. (Here we let $({\Bbb G}_a^\alpha/\xmode{\Bbb Z}^n)^+$ denote the
quotient in this category.) Since $H^1(\et{S}, \xmode{\Bbb Z}) = 0$ by
(\Lcitemark 1\Rcitemark \ 3.6(ii)), we are done.
Now we prove the general case of (a). For each $m$, let $X_m$ be the closed
subscheme of $X$ defined by ${\cal{N}}^m$. Choose $n \in \xmode{\Bbb N}$ so that ${\cal{N}}^n = 0$.
Let $K_m = \mathop{\operatoratfont Ker}\nolimits[\mathop{\mathbf{Pic}}\nolimits(X_{m+1})\ \rightarrow\ \mathop{\mathbf{Pic}}\nolimits(X_m)]$.
Let $F_m = \mathop{\operatoratfont Ker}\nolimits[\mathop{\mathbf{Pic}}\nolimits(X_m)\ \mapE{}\ \mathop{\mathbf{Pic}}\nolimits(X_0)]$. We have an exact sequence
\Rowfour{0}{K_m}{F_{m+1}}{F_m.%
}Then \mapx[[ K_m(B) || K_m(C) ]] is injective by
\pref{kernel-pic-nilimmersion}.
By induction on $m$, it follows that \mapx[[ F_m(B) || F_m(C) ]] is injective
for all $m$. Taking $m = n$, we get (a). {\hfill$\square$}
\end{proof}
\begin{problem}
Is the functor $F$ a sheaf?
\end{problem}
\section*{References}
\addcontentsline{toc}{section}{References}
\ \par\noindent\vspace*{-0.25in}
\hfuzz 5pt
\bgroup\Resetstrings%
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|
1996-02-27T06:25:21 | 9410 | alg-geom/9410007 | en | https://arxiv.org/abs/alg-geom/9410007 | [
"alg-geom",
"math.AG",
"math.GT"
] | alg-geom/9410007 | Zhenbo Qin | Robert Friedman and Zhenbo Qin | Flips of moduli spaces and transition formulas for Donaldson polynomial
invariants of rational surfaces | 56 pages, amstex | null | null | OSU Math 1995-11 | null | We study the change of moduli spaces of Gieseker-semistable torsion free
rank-$2$ sheaves on algebraic surfaces as we vary the polarizations. When the
surfaces are rational with an effective anti-canonical divisor, the moduli
spaces are linked by a series of flips (blowups and blowdowns). Using these
results, we compute the transition formulas for Donaldson polynomial invariants
of rational surfaces. Part of the work is also obtained independently by
Matsuki-Wentworth and Ellingsrud-G{\" o}ttsche.
| [
{
"version": "v1",
"created": "Wed, 12 Oct 1994 19:02:58 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Friedman",
"Robert",
""
],
[
"Qin",
"Zhenbo",
""
]
] | alg-geom | \section{1. Introduction.}
In \cite{7}, Donaldson has defined polynomial invariants for
smooth simply connected 4-manifolds with $b_2^+\geq 3$.
These invariants have also been defined for 4-manifolds with $b_2^+=1$
in \cite{24, 17, 18}, along lines suggested by the work of
Donaldson in \cite{5}. In this case, however,
they depend on an additional piece of information, namely a chamber
defined on the positive cone of $H^2(X; \Ar)$ by
a certain locally finite set of walls. Explicitly,
let $X$ be a simply connected, oriented, and closed smooth
$4$-manifold with $b_2^+ = 1$ where $b_2^+$ is the number
of positive eigenvalues of the quadratic form $q_X$
when diagonalized over $\Ar$. Let
$$\Omega_X = \{\, x \in H^2(X, \Ar)\mid x^2 > 0 \,\}$$
be the positive cone. Fix a class $\Delta$ in $H^2(X, \Zee)$ and
an integer $c$ such that $d = 4c - \Delta ^2 - 3$ is nonnegative.
A {\sl wall of type $(\Delta, c)$\/} is a nonempty
hyperplane:
$$W^\zeta = \{\, x \in \Omega_X\mid x \cdot \zeta = 0 \,\}$$
in $\Omega_X$ for some class $\zeta \in H^2(X, \Zee)$
with $\zeta \equiv \Delta \pmod 2$
and $\Delta ^2 - 4c \le \zeta^2 < 0$.
The connected components of the complement in $\Omega_X$ of the walls of type
$(\Delta, c)$ are the {\sl chambers of type $(\Delta, c)$}.
Then the Donaldson polynomial invariants of $X$ associated to
$\Delta$ and $c$ are defined with respect to chambers of type $(\Delta, c)$.
The invariants only depend on the class $w=\Delta \bmod 2\in H^2(X;
\Zee/2\Zee)$
and the integer $p = \Delta ^2 - 4c$, and we shall often refer to walls and
chambers of type $(w,p)$ as well. We shall write $D^X _{w,p}(\Cal C)$
for the Donaldson polynomial corresponding to the $SO(3)$ bundle $P$ with
invariants $w_2(P) = w$ and $p_1(P) = p$, depending on the chamber $\Cal C$.
A basic question is then the following: Suppose that $\Cal C_+$ and $\Cal C_-$
are separated by a single wall $W^\zeta$. Here there may be more than one class
$\zeta$ of type $(\Delta, c)$ defining $W^\zeta$. Then find a formula for the
difference
$$\delta ^X_{w,p}(\Cal C_+, \Cal C_-) = D^X _{w,p}(\Cal C_+) - D^X _{w,p}(\Cal
C_-).$$
We shall refer to such a difference as a {\sl transition formula}.
There has been considerable interest in the above problem. The first
result in this direction is due to Donaldson in \cite{5},
who gave a formula in case $\Delta = 0$ and $c = 1$.
Kotschick \cite{17} showed that, on the part of the symmetric algebra generated
by
$2$-dimensional classes,
$\delta ^X_{w,p}(\Cal C_+, \Cal C_-) = \pm \zeta^d$ for
$\zeta^2 = -(4c - \Delta^2)=p$, and that $\delta ^X_{w,p}(\Cal C_+, \Cal C_-)$
is in fact always divisible by $\zeta$, except when $p=-5$ and $\zeta ^2 = -1$
(cf\. also Mong \cite{24} for some partial results along these lines).
For a rational ruled surface
$X$, all the transition formulas for $\Delta = 0$ and $2 \le c \le 4$
have been determined in \cite{24, 33, 22}. Using a gauge-theoretic approach,
Yang \cite{35} settled the problem for $\Delta = 0$ and $c = 2$, and computed
the degree $5$ Donaldson polynomials for rational surfaces. The known
examples and the work of Kotschick and Morgan \cite{18} raise
the following rather natural conjecture:
\medskip\noindent
{\bf Conjecture.} {\it The transition formula $\delta ^X_{w,p}(\Cal C_+, \Cal
C_-)$ is a homotopy invariant of the pair $(X, \zeta)$; more precisely,
if $\phi$ is an oriented homotopy equivalence from $X'$ to $X$, then
$$\delta^{X'}_{\phi ^*w,p}(\phi ^*(\Cal C_+), \phi ^*(\Cal
C_-))=\phi ^*\delta ^{X}_{w,p}(\Cal C_+, \Cal C_-).$$}
\medskip
We remark that this conjecture is essentially equivalent to the following
statement: the transition formula $\delta ^X_{w,p}(\Cal C_+, \Cal C_-)$ is a
polynomial in $\zeta$ and the quadratic form $q_X$ with coefficients
involving only $\zeta^2$, homotopy invariants of $X$ (i.e\. $b^-_2(X)$), and
universal constants.
Our goal in this paper is to study the corresponding problem in
algebraic geometry. More precisely, let $X$ be an algebraic surface (not
necessarily with $b_2^+(X)= 1$) and let $L$ be an ample line bundle on $X$. We
can then identify the moduli space of $L$-stable rank two bundles $V$ on $X$
with
$c_1(V)=\Delta$ and $c_2(V)=c$ with the moduli space of equivalence classes of
ASD connections on $X$ with respect to a Hodge metric on $X$ corresponding to
$L$. Let $\frak M_L(\Delta, c)$ be the Gieseker compactification of this moduli
space. It is known that $\frak M_L(\Delta, c)$ changes as we change $L$, and
that $\frak M_L(\Delta, c)$ is constant on a set of chambers for the ample cone
of $X$ which are defined in a way analogous to the definition of chambers for
$\Omega _X$ given above. Using the recent result of Morgan \cite{25} and
Li \cite{21} that the Donaldson polynomial of an algebraic surface
can be evaluated using the Gieseker compactification ${\frak M}_L(\Delta, c)$
of the moduli space of stable bundles, we shall work on
${\frak M}_L(\Delta, c)$ for suitable choices of $L$ and in particular
analyze the change in ${\frak M}_L(\Delta, c)$ for $L\in \Cal C_+$ or
$L\in \Cal C_-$, where $\Cal C_\pm$ are two adjacent chambers.
It turns out that we can obtain $\frak M _{L_+}(\Delta, c)$ from
$\frak M _{L_-}(\Delta, c)$ by a series of blowups and blowdowns
(flips). Our results are thus very similar to those of
Thaddeus in \cite{31}. Thaddeus \cite{32} and also Dolgachev-Hu \cite{3}
have developed a general picture for the variation of GIT quotients
after a change of polarization, and although our methods are somewhat different
it seems quite possible that they fit into
their general framework. We have also found it convenient to borrow
some of Thaddeus' notation.
Next we shall apply our results on the change in the moduli spaces to determine
the transition formula for Donaldson polynomials in case
$X$ is a rational surface with
$-K_X$ effective. We shall give explicit formulas for
$\delta ^X_{w,p}(\Cal C_+, \Cal C_-)$ in case the nonnegative integer
$\ell _\zeta = (\zeta ^2 - p)/4 \leq 2$. These formulas are in agreement
with the above conjecture, in the sense that the transition formula is
indeed a polynomial in $\zeta$ and $q_X$ with coefficients involving only
$\zeta^2$, $K_X^2$, and universal constants. We shall also give a
formula in principle for $\delta ^X_{w,p}(\Cal C_+, \Cal C_-)$ in general
(see Theorem 5.4),
but to make this formula explicit involves more knowledge of the enumerative
geometry of $\Hilb^nX$ than seems to be available at present.
In case $-K_X$ is effective, the moduli spaces are (essentially) smooth and
the centers of the blowup are smooth as well; in fact they are
$\Pee ^k$-bundles over $\Hilb ^{n_1}X\times \Hilb ^{n_2}X$ for appropriate $k$,
$n_1$ and $n_2$. In this way, we obtain general formulas which can be made
explicit for low values of $n$. For instance, we show the following
(see Theorem 6.4 for details):
\theorem{} Assume that the wall $W^\zeta$ is defined only by
$\pm \zeta$ with $\ell_\zeta = 1$ and that $\Cal C_\pm$ lies
on the $\pm$-side of $W^\zeta$. Then, on the subspace of the symmetric algebra
generated by $H_2(X)$,
$\delta^X_{w, p}(\Cal C_-, \Cal C_+)$ is equal to
$$(-1)^{{(\Delta \cdot K_X + \Delta^2) + (\zeta \cdot K_X - \zeta^2)} \over 2}
\cdot \left \{ d(d - 1) \cdot \left(\zeta \over 2 \right)^{d - 2} \cdot q_X
+ (2K_X^2 + 2d + 6) \cdot \left ({\zeta \over 2} \right )^d \right \}.$$
\endstatement
Along the direction of the work of Kronheimer and Mrowka \cite{19, 20},
we also consider the difference of Donaldson polynomial invariants
involving the natural generator $x \in H_0(X; \Zee)$. More precisely,
let $\nu$ be the corresponding $4$-dimensional class in
the instanton moduli space. For $\alpha \in H_2(X; \Zee)$,
we give a formula for the difference
$\delta^X_{w, p}(\Cal C_-, \Cal C_+)(\alpha^{d - 2}, \nu)$
in Theorem 5.5. It is worth to point out that the similarity between
Theorem 5.4 and Theorem 5.5 may indicate that there exists
a deep relation between $\delta^X_{w, p}(\Cal C_-, \Cal C_+)(\alpha^d)$ and
$\delta^X_{w, p}(\Cal C_-, \Cal C_+)(\alpha^{d - 2}, \nu)$,
and suggest a way to generalize the notion of {\sl simple type} in
\cite{19, 20} from the case of $b_2^+ > 1$ to the case of $b_2^+ = 1$.
For instance, modulo some lower degree terms,
$\delta^X_{w, p}(\Cal C_-, \Cal C_+)(\alpha^{d - 2}, \nu)$
can be obtained from $(-1/4) \cdot \delta^X_{w, p}(\Cal C_-, \Cal
C_+)(\alpha^d)$ by replacing $d$ by $(d - 2)$
(see Theorem 5.13 and Theorem 5.14). In fact, based on some heuristic
arguments, it seems reasonable to conjecture that
$\delta^X_{w, p}(\Cal C_-, \Cal C_+)(\alpha^{d - 2}, \nu)$
is a combination of $\delta^X_{w, p'}(\Cal C_-, \Cal C_+)(\alpha^{d - 4k})$
for various nonnegative integers $k$ if the degrees are properly arranged.
We hope to return to this issue in future.
Our paper is organized as follows. In section 2, we study rank two
torsion free sheaves which are semistable with respect to ample
divisors in $\Cal C_-$ but not semistable with respect to ample
divisors in $\Cal C_+$. When the surface $X$
is rational with $-K_X$ effective, these sheaves are parametrized by an open
subset of a union of projective bundles over the product of two Hilbert
schemes
of points in
$X$. More precisely, if $\zeta$ defines the wall separating $\Cal C_-$ from
$\Cal
C_+$, define
$E_\zeta^{n_1, n_2}$ be the set of all isomorphism classes of nonsplit
extensions of the form
$$0 \to \scrO _X(F )\otimes I_{Z_1} \to V \to \scrO _X(\Delta -F ) \otimes
I_{Z_2} \to 0,$$
where $F$ is a divisor class such that $2F-\Delta \equiv \zeta$ and $Z_1$ and
$Z_2$ are two zero-dimensional subschemes of $X$ with $\ell (Z_i) = n_i$
such that $n_1 + n_2 = \ell _\zeta$. In case $X$ is rational,
$E_\zeta^{n_1, n_2}$ is a $\Pee ^N$ bundle over
$\Hilb ^{n_1}X\times \Hilb ^{n_2}X$, and the set of points of $E_\zeta^{n_1,
n_2}$ lying in $\frak M_{L_-}(\Delta, c)$ but not in $\frak M_{L_+}(\Delta,
c)$
is a Zariski open subset of $E_\zeta^{n_1, n_2}$. The main technical difficulty
is that it is hard to control the rational map from $E_\zeta^{n_1, n_2}$ to
$\frak M_{L_-}(\Delta, c)$, and in particular this map is not a morphism. The
general picture that we establish is the following: first, the map $E_\zeta^{0,
\ell_\zeta}\dasharrow\frak M_{L_-}(\Delta, c)$ is a morphism, and it is
possible
to make an elementary transformation, or {\sl flip\/}, along its image. The
result is a new space for which the rational map $E_\zeta^{1,
\ell_\zeta - 1}\dasharrow\frak M_{L_-}(\Delta, c)$ becomes a morphism, and it
is
possible to make a flip along {\sl its\/} image. We continue in this way until
we reach $\frak M_{L_+}(\Delta, c)$.
It seems rather difficult to see that the above picture holds directly. Instead
we shall proceed as follows. We define abstractly a sequence of moduli spaces,
indexed by an integer $k$ with $0\leq k \leq \ell _\zeta+1$, such that the
moduli space for $k=0$ is $\frak M_{L_-}(\Delta, c)$, the moduli space for
$k=\ell _\zeta+1$ is $\frak M_{L_+}(\Delta, c)$, and moreover the
$k^{\text{th}}$ moduli space contains an embedded copy of $E_\zeta^{k, \ell
_\zeta -k}$ such that the flip along this copy yields the $(k+1)^{\text{st}}$
moduli space. Thus the picture is very similar to that developed independently
by Thaddeus in \cite{31}. To define our sequence of moduli spaces,
we define $(L_0, \boldsymbol \zeta, \bold k)$-semistability in section 3 for
rank two torsion free sheaves, where $L_0$ is any ample divisor contained in
the common face of
$\Cal C_+$ and $\Cal C_-$,
$\boldsymbol
\zeta$ is the set of classes of type $(\Delta, c)$ defining the common wall
of $\Cal C_+$ and $\Cal C_-$, and $\bold k$ is a set of integers.
We show that $\frak M _{L_-}(\Delta, c)$ and $\frak M _{L_+}(\Delta, c)$
are linked by the moduli spaces $\frak M_0 ^{(\boldsymbol \zeta, \bold k)}$
where the data $\bold k$ is allowed to vary. When the surface $X$ is
rational with $-K_X$ effective, we can obtain $\frak M _{L_+}(\Delta, c)$
from $\frak M _{L_-}(\Delta, c)$ by a series of flips.
The fact that all $(L_0, \boldsymbol \zeta, \bold k)$-semistable
rank two torsion free sheaves do form a moduli space
$\frak M_0 ^{(\boldsymbol \zeta, \bold k)}$ in the usual sense
is proved in section 4 where we introduce an equivalent notion of stability
called {\sl mixed stability}. Our method follows Gieseker's GIT argument in
\cite{13}. Roughly speaking, the goal of mixed stability is to define stability
for a sheaf of the form $V\otimes \Xi$, where $V$ is a torsion free sheaf but
$\Xi$ is just a $\Bbb Q$-divisor. To make this idea precise, given
actual divisors
$H_1$ and $H_2$ and positive weights $a_1$ and $a_2$, we shall define a notion
of stability which ``mixes" stability for $V\otimes H_1$ with stability for
$V\otimes H_2$, together with weightings of the stability condition for
$V\otimes H_i$. The effect of this definition will be formally the same as if
we had defined stability of $V\otimes \Xi$, where $\Xi$ is the $\Bbb
Q$-divisor
$$\frac{a_1}{a_1+a_2}H_1 + \frac{a_2}{a_1+a_2}H_2.$$
In section 5, using our results on flips of moduli spaces,
we give a formula for the transition formula of Donaldson polynomials
when $X$ is rational with $-K_X$ effective,
and compute the leading term in the transition formula.
In section 6, we obtain explicit transition formulas when $\ell_\zeta \le 2$.
Some of the material in our section 2 has been worked out independently
by Hu and Li \cite{16} and G\"ottsche \cite{14}.
Moreover Ellingsrud and G\"ottsche \cite{8} have recently studied the change
in the moduli space by similar methods and have obtained results
very similar to ours. Using very different methods, the results in
Section 4 have also been obtained by Matsuki and Wentworth \cite{23},
who also consider the case of higher rank. They use branched covers of
the surface $X$ to study the change in the moduli space.
We expect that a minor modification of the arguments in Section 4 of
this paper will also handle the case of higher rank.
\section{Conventions and notations}
We fix some conventions and notations for the rest of this paper.
Let $X$ be a smooth algebraic surface. We shall be primarily interested
in the case where $X$ is simply connected and $-K_X$ is effective and
nonzero. Thus necessarily $X$ is a rational surface. However much of the
discussion
in sections 1--4 will also apply to the general case.
Stability and semistability with respect to an ample line bundle $L$
will always be understood to mean Gieseker
stability or semistability unless otherwise noted. We shall not mention the
choice of $L$ explicitly if it is clear from the context.
Recall that a torsion free sheaf $V$ of rank two is Gieseker $L$-stable
if and only if, for every rank one subsheaf $W$ of $V$,
either $\mu _L(W) < \mu _L(V)$ or $\mu _L(W) = \mu _L(V)$ and
$2\chi(W) < \chi(V)$, where $\mu _L$ is the normalized degree
with respect to $L$. Semistability is similarly defined, where the second
inequality is also allowed to be an equality.
For a torsion free sheaf $V$, we use $V\ddual$ to stand for its double dual.
For two divisors $D_1$ and $D_2$ on $X$, the notation $D_1 \equiv D_2$
means that $D_1$ and $D_2$ are numerically equivalent, that is,
$D_1 \cdot D = D_2 \cdot D$ for any divisor $D$.
For a locally free sheaf (or equivalently a vector bundle) $\Cal E$
over a smooth variety $Y$, we use $\Pee(\Cal E)$ to denote
the associated projective space bundle,
that is, $\Pee(\Cal E)$ is the {\bf Proj} of $\oplus_{d \ge 0} S^d(\Cal E)$.
Fix a divisor $\Delta$ and an integer $c$. Let $\Cal C_-$ and $\Cal C_+$
be two adjacent chambers of type $(\Delta, c)$ separated by the wall
$W^\zeta$. We assume that $\zeta \cdot \Cal C_- < 0<\zeta \cdot \Cal C_+$.
Let $L_\pm\in \Cal C_\pm$ be an ample line bundle,
so that $L_- \cdot \zeta < 0 < L_+\cdot \zeta$, and denote by
$\frak M_\pm$ the moduli space $\frak M _{L_\pm}(\Delta, c)$
of rank two Gieseker semistable torsion free sheaves $V$ with
$c_1(V) = \Delta$ and $c_2(V) = c$. Let $L_0$ be any ample divisor
contained in the interior of the intersection of $W^\zeta$ and
the closures of $\Cal C_\pm$. Let $\zeta = \zeta _1,
\dots, \zeta _n$ be all the positive rational multiples of $\zeta$
such that $\zeta_i$ is an integral class of type $(w,p)$ which also
defines the wall $W^\zeta$. In sections 5--6, we will assume that
$n = 1$ for notational simplicity.
Finally, we point out that our $\mu$-map is half of the $\mu$-map
used in \cite{17, 18} (see (viii) and (ix) in Notation 5.1).
Thus our transition formula differs from the one defined in \cite{18}
by a universal constant.
\medskip\noindent
{\it Acknowledgements.} We would like to thank Hong-Jie Yang for invaluable
access to his calculations, which helped to keep us on the right track.
The second author would like to thank Wei-ping Li and Yun-Gang Ye for
helpful discussions, and the Institute for Advanced Study at
Princeton for its hospitality and financial support through NSF grant
DMS-9100383 during the academic year 1992--1993 when part of this work was
done.
\section{2. Preliminaries on the moduli space.}
In this section, we study rank two torsion free sheaves which are
related to walls. These sheaves arise naturally from the comparison of
$L_-$-semistability and $L_+$-semistability. We will show that
when the surface $X$ is rational with $-K_X$ effective, the moduli spaces
$\frak M_\pm$ are smooth at the points corresponding to these sheaves.
We start with the following lemma, which for simplicity is just stated for
$L_-$-stability.
\lemma{2.1} Let $V$ be a rank two torsion free sheaf on $X$ with
$c_1(V) = \Delta$ and $c_2(V) = c$. If $V$ is $L_-$-semistable,
then exactly one of the following holds:
\roster
\item"{(i)}" Both $V$ and $V\ddual$ are $L_-$-stable and Mumford
$L_-$-stable.
\item"{(ii)}" $V$ sits in an exact sequence
$$0 \to \scrO_X(F_1)\otimes I_{Z_1} \to V \to \scrO_X(F_2)
\otimes I_{Z_2} \to 0$$
where $2F_1 \equiv \Delta \equiv 2F_2$, and $Z_1$ and $Z_2$ are
zero-dimensional subschemes of $X$ such that
$\ell (Z_1) \geq \ell(Z_2)$. Moreover in this case $V$ is $L$-semistable for
every choice of an ample line bundle $L$ and $V$ is strictly
$L_\pm$-semistable if and only if $\ell(Z_1) = \ell (Z_2)$.
\endroster
\endstatement
\proof Suppose that $V$ is (Gieseker) $L_-$-semistable. The vector bundle
$V\ddual$ satisfies $c_1(V\ddual ) = \Delta$ and $c_2(V\ddual) \leq c$.
Standard arguments \cite{10} show that $V\ddual$ is Mumford
$L_-$-semistable. If $V\ddual$ is strictly Mumford
$L_-$-semistable, then by \cite{10, 30},
either $L_-$ must lie on a wall of type $(\Delta, c)$ or
if $\scrO_X(F_1)$ is a destabilizing sub-line bundle then $\Delta \equiv 2F_1$.
Since by assumption $L_-$ does not lie on a wall of type $(\Delta, c)$,
either $V\ddual$ is Mumford $L_-$-stable or there is an exact sequence
$$0 \to \scrO_X(F_1) \to V\ddual \to \scrO_X(F_2) \otimes I_Z \to 0,$$
where $F_2 = \Delta - F_1 \equiv F_1$ and $Z$ is a zero-dimensional subscheme
of $X$. If $V\ddual$ is Mumford $L_-$-stable, then $V$ is Mumford
$L_-$-stable and therefore $L_-$-stable. Thus case (i) holds. Otherwise
$\scrO_X(F_1) \cap V$ is of the form $\scrO_X(F_1)\otimes I_{Z_1}$ for
some $Z_1$ and $V/\scrO_X(F_1)\otimes I_{Z_1}$ is a subsheaf of
$\scrO_X(F_2) \otimes I_Z$ and thus of the form
$\scrO_X(F_2) \otimes I_{Z_2}$ for some $Z_2$. Thus we are in case (ii) of the
lemma. Since $\mu(\scrO_X(F_1)\otimes I_{Z_1}) = \mu (V)$
and $V$ is semistable, we have
$$2\chi(\scrO_X(F_1)\otimes I_{Z_1}) \leq \chi (V)=
\chi(\scrO_X(F_1) \otimes I_{Z_1})+ \chi(\scrO_X(F_2)\otimes I_{Z_2}).$$
Hence $\chi(\scrO_X(F_2)\otimes I_{Z_2}) -
\chi(\scrO_X(F_1)\otimes I_{Z_1}) \geq 0$.
As $F_1 \equiv F_2$ and $\chi(\scrO_X(F_i)\otimes I_{Z_i})
= \chi(\scrO_X(F_i)) - \ell(Z_i)$, we must then have
$\ell(Z_1) - \ell (Z_2) \geq 0$. The last sentence of (ii) is a
straightforward argument left to the reader.
\endproof
If $V$ satisfies the conclusions of (2.1)(ii), we shall call $V$ {\sl
universally semistable}.
Next we shall compare stability for $L_-$ and $L_+$.
\lemma{2.2} Let $V$ be a torsion free rank two sheaf with $c_1(V) = \Delta$ and
$c_2(V) = c$.
\roster
\item"{(i)}" If $V$ is $L_-$-stable but $L_+$-unstable,
then there exist a divisor class $F$ and two zero-dimensional subschemes
$Z_-$ and $Z_+$ of $X$ and an exact sequence
$$0 \to \scrO _X(F )\otimes I_{Z_-} \to V \to \scrO _X(\Delta -F ) \otimes
I_{Z_+} \to 0,$$ where $L_-\cdot (2F-\Delta) < 0 < L_+\cdot (2F-\Delta) $.
Moreover the divisor $F$, the schemes $Z_-$ and $Z_+$, and the map $F \otimes
I_{Z_-} \to V$ are unique mod scalars, and $\zeta = 2F - \Delta$ defines
a wall of type $(\Delta, c)$.
\item"{(ii)}" Conversely, suppose that there is a nonsplit exact sequence as
above. Then $V$ is simple. Moreover, $V$ is not $L_-$-stable if and
only if it is $L_-$-unstable if and only if there exist subschemes
$Z'$ and $Z''$ and an exact sequence
$$0 \to \scrO _X(\Delta - F )\otimes I_{Z'} \to V \to \scrO _X(F )
\otimes I_{Z''} \to 0,$$
if and only if $V\ddual$ is a direct sum $\scrO _X(F) \oplus
\scrO _X(\Delta -F )$. In this case the scheme $Z'$ strictly contains
the scheme $Z_+$, $\ell (Z') > \ell (Z_+)$
and $\ell (Z')+\ell (Z'') = \ell (Z_-)+\ell (Z_+)$.
Finally, if $Z_-=\emptyset$ then $V$ is always $L_-$-stable.
\endroster
\endstatement
\proof We first show (i). Suppose that $V$ is $L_-$-stable but $L_+$-unstable.
Then by (2.1) $V\ddual$ is also $L_-$-stable and $L_+$-unstable.
By \cite{30}, there is a uniquely determined line bundle $\scrO_X(F)$ and
a map $\scrO _X(F) \to V\ddual$ with torsion free quotient such that
$L_-\cdot (2F-\Delta) < 0 < L_+\cdot (2F-\Delta) $.
Moreover $\zeta = 2F - \Delta$ defines a wall of type $(\Delta, c)$.
The subsheaf $\scrO_X(F) \cap V$ of $V\ddual$ is
a subsheaf of $\scrO_X(F)$ and agrees with it away from finitely many points.
Thus $\scrO_X(F) \cap V = \scrO _X(F )\otimes I_{Z_-}$
for some well-defined subscheme $Z_-$. Moreover the quotient
$V\Big/[\scrO _X(F )\otimes I_{Z_-}]$ is a subsheaf
of $\scrO _X(\Delta -F ) \otimes I_Z$ for some zero-dimensional subscheme
$Z$, and agrees with $\scrO _X(\Delta -F ) $ away from finitely many points.
Thus the quotient is of the form $\scrO _X(\Delta -F ) \otimes I_{Z_+}$
for some zero-dimensional subscheme $Z_+$. The uniqueness is clear.
To see (ii), suppose that $V$ is given as a nonsplit exact sequence
$$0 \to \scrO _X(F )\otimes I_{Z_-} \to V \to \scrO _X(\Delta -F ) \otimes
I_{Z_+} \to 0$$ as above, where $L_-\cdot (2F-\Delta) < 0 < L_+\cdot
(2F-\Delta)
$. Again by (2.1), $V$ is $L_-$-semistable if and only if it is $L_-$-stable if
and only if
$V\ddual$ is
$L_-$-stable. Now taking double duals of the above exact sequence, there is an
exact sequence
$$0 \to \scrO _X(F ) \to V\ddual \to \scrO _X(\Delta -F ) \otimes I_Z \to 0$$
for some zero-dimensional scheme $Z$. Moreover, by \cite{30}, $V\ddual$ is
$L_-$-unstable if and only if the above exact sequence splits, and in
particular
if and only if $Z=\emptyset$ and $V\ddual = \scrO _X(F)\oplus \scrO _X(\Delta
-F
)$. In this case, the map $\scrO _X(\Delta -F )
\to V\ddual$ induces a map $\scrO _X(\Delta -F )\otimes I_{Z'} \to V$ for some
ideal sheaf $I_{Z'}$. We may clearly assume that the quotient is torsion free,
in
which case it is necessarily of the form $\scrO _X(F )
\otimes I_{Z''}$ with $\ell (Z')+\ell (Z'') = \ell (Z_-)+\ell (Z_+)$. Using the
nonzero map
$\scrO _X(\Delta -F )\otimes I_{Z'} \to
\scrO _X(\Delta -F )\otimes I_{Z_+}$, we see that there is an inclusion $I_{Z'}
\subseteq I_{Z_+}$; moreover this inclusion must be strict since the defining
exact sequence for $V$ is nonsplit. Thus $Z'$ strictly contains $Z_+$ and in
particular $\ell (Z') > \ell (Z_+)$. Conversely, if there exists a nonzero map
$\scrO _X(\Delta - F )\otimes I_{Z'} \to V$, then there is a nonzero map $\scrO
_X(\Delta - F ) \to V\ddual$ and thus $V\ddual $ is the split extension.
We next show that $V$ is simple. If $V$ is stable then it is simple. If $V$ is
not stable, then $V\ddual = \scrO _X(F)\oplus \scrO _X(\Delta -F )$. There is
an
inclusion $\Hom (V, V) \subseteq
\Hom (V\ddual, V\ddual)$. If $V\ddual$ is split, then $\Hom (V\ddual, V\ddual)
=
\Cee \oplus \Cee$. In this case, using a nonscalar endomorphism of $V$, it is
easy to see that we can split the exact sequence defining $V$.
Finally suppose that $Z_- = \emptyset$ in the notation of (2.2). If $V$ is
$L_-$-unstable, then we can find $Z'$ with $\ell (Z') > \ell (Z_+)$ and a
subscheme $Z''$ such that $\ell (Z') + \ell (Z'') = \ell (Z_+)$. Thus $\ell
(Z')
\leq \ell (Z_+)$, a contradiction. It follows that $V$ is $L_-$-stable.
\endproof
For the rest of this section, we shall assume that $-K_X$ is effective and
nonzero and that $q(X)=0$. Thus $X$ is a rational surface.
\lemma{2.3} Suppose that $\frak M_\pm$ is nonempty. Suppose that $(w,p)\neq
(0,0)$, or equivalently that $\frak M_\pm$ does not consist of a single point
corresponding to a twist of the trivial vector bundle. Then the open subset of
$\frak M_\pm$ corresponding to Mumford stable rank two vector bundles is
nonempty
and dense. Every component of $\frak
M_\pm$ has dimension
$4c-\Delta ^2 -3=-p-3$. The points of $\frak M_\pm$ corresponding to
$L_\pm$-stable sheaves $V$ are smooth points.
\endstatement
\proof Suppose that $\frak M_\pm$ is nonempty, and let
$V$ correspond to a point of $\frak M_\pm$. Then by general theory (e.g\.
Chapter 7 of \cite{10}), $\frak M_\pm$ is smooth of dimension $4c-\Delta ^2
-3=-p-3$
at $V$ if $V$ is stable and $\Ext^2(V, V) = 0$, since $h^2(X; \scrO
_X) = 0$. Moreover, setting $W = V\ddual$, there is a surjection from $H^2(X;
Hom(W, W))$ to $\Ext^2(V, V)$. Thus to show that $\Ext^2(V, V) =0$ it suffices
to show that $H^2(X; Hom(W, W))=0$. Now $H^2(X; Hom(W,
W))$ is dual to $H^0(X; Hom (W,W) \otimes K_X)$. Since $-K_X$ is
effective, there is an inclusion of $H^0(X; Hom (W,W) \otimes K_X)$ in $H^0(X;
Hom (W,W))$. If $W$ is stable, then $H^0(X; Hom (W,W)) \cong \Cee$ and $H^0(X;
Hom (W,W) \otimes K_X) = 0$. Thus
$\frak M_\pm$ is smooth at $V$. Standard theory \cite{1, 10} also shows that
every
torsion free sheaf $V$ for which $V\ddual$ is stable is smoothable. Thus the
set
of locally free sheaves is nonempty and dense in the component containing $V$
in
this case.
Now consider a $V$ such that $W=V\ddual$ is not stable.
Using the exact sequence
$$0 \to \scrO_X(F) \to W \to \scrO_X(F) \otimes I_Z \to 0$$
for $W$ which was given in the course of the proof of (2.1),
it is easy to check that there is an exact sequence
$$0 \to \Hom (I_Z, W\otimes \scrO_X(-F)\otimes K_X) \to \Hom (W,W \otimes
K_X) \to H^0(W\otimes \scrO_X(-F)\otimes K_X).$$
Since $-K_X$ is effective and nonzero, $H^0(W\otimes \scrO_X(-F)\otimes
K_X)=\Hom (I_Z, W\otimes \scrO_X(-F)\otimes K_X) =0$. Thus $\Hom (W,W
\otimes K_X)=0$ as well. Once again $V$ is smoothable.
Now we claim that a general smoothing $V'$ of $V$ is Mumford stable.
For otherwise by the proof of (2.1) there is an exact sequence
$$0 \to \scrO_X(F) \to V' \to \scrO_X(F) \otimes I_Z \to 0$$
as above, with $\ell (Z) \leq \ell (\emptyset) =0$. In this case $V'$
is an extension of $\scrO_X(F)$ by $\scrO_X(F)$, forcing $w=p=0$ and
(since $h^1(\scrO_X)= 0$) $V' = \scrO_X(F) \oplus \scrO_X(F) $.
\endproof
It is natural
to make the following conjecture, which is true for geometrically ruled $X$ by
\cite{29} and is verified in certain other cases by \cite{34}.
\proclaim{ Conjecture 2.4} If $X$ is a rational surface with
$-K_X$ effective, then for every choice of
$L$, $\Delta$ and $c$, ${\frak M}_L(\Delta, c)$ is
either empty or irreducible.
\endstatement
Let us fix some notations for the rest of this paper.
\definition{Definition 2.5} Let $X$ be an algebraic surface (not necessarily
rational), and let $\zeta$ be a fixed numerical equivalence class defining a
wall of type $(\Delta, c)$. Set
$\ell _\zeta = (4c-\Delta ^2 +\zeta ^2)/4 = (\zeta ^2-p)/4$.
Choose two nonnegative integers $n_-$ and $n_+$ with
$n_-+ n_+ = \ell _\zeta$, and let $E_\zeta^{n_-, n_+}$ be the set of
all isomorphism classes of nonsplit extensions of the form
$$0 \to \scrO _X(F )\otimes I_{Z_-} \to V \to \scrO _X(\Delta -F ) \otimes
I_{Z_+} \to 0$$
with $\zeta \equiv 2F -\Delta$ and $\ell (Z_\pm) = n_\pm$.
\enddefinition
\medskip
We remark that since $\zeta \equiv \Delta \pmod 2$
and $\Delta ^2 - 4c \le \zeta^2 < 0$, $\ell _\zeta$ is a nonnegative integer.
If $V$ corresponds to a point of $E_\zeta^{n_-, n_+}$, then $V$ is
$L_+$-unstable since $L_+\cdot \zeta > 0$. By (2.2)(ii), $V$ is simple, and
if it is $L_-$-semistable then it is actually stable. By (2.3), if
$X$ is a rational surface with
$-K_X$ effective, then $\frak M_-$ is smooth in a neighborhood of a point
corresponding to a sheaf $V$ lying in $E_\zeta^{n_-, n_+}$ for some $\zeta,
n_-,
n_+$. We shall now study $E_\zeta^{n_-, n_+}$
in more detail for rational surfaces.
\lemma{2.6} Suppose that $-K_X$ is effective and that $q(X) =0$. For $Z_-$ and
$Z_+$ two fixed zero-dimensional subschemes of
$X$ of lengths $n_-$ and $n_+$ respectively,
$$\dim \Ext^1(\scrO _X(\Delta -F ) \otimes
I_{Z_+} , \scrO _X(F )\otimes I_{Z_-}) =n_-+ n_+ + h(\zeta)= \ell _\zeta +
h(\zeta),$$ where
$$h(\zeta ) = h^1(X; \scrO _X(2F-\Delta)) = \frac{(\zeta \cdot K_X)}{2} -
\frac{\zeta ^2}{2} -1.$$
\endstatement
\proof Note that $\Hom(\scrO _X(\Delta -F ) \otimes
I_{Z_+} , \scrO _X(F )\otimes I_{Z_-})\subseteq H^0(\scrO_X(2F-\Delta)) = 0$,
since $L_- \cdot (2F-\Delta ) < 0$. Likewise $\Ext^2(\scrO _X(\Delta -F )
\otimes
I_{Z_+} , \scrO _X(F )\otimes I_{Z_-})$ is Serre dual to $\Hom (\scrO _X(F
)\otimes I_{Z_-} , \scrO _X(\Delta -F ) \otimes I_{Z_+}\otimes K_X) \subseteq
H^0(\scrO_X(\Delta -2F) \otimes K_X) \subseteq H^0(\scrO_X(\Delta -2F))$,
since $-K_X$ is effective. Thus as $L_+ \cdot (\Delta - 2F) < 0$, $\Ext^2(\scrO
_X(\Delta -F ) \otimes I_{Z_+} , \scrO _X(F )\otimes I_{Z_-})=0$ as well. If we
set $\chi (\scrO _X(\Delta -F ) \otimes
I_{Z_+} , \scrO _X(F )\otimes I_{Z_-}) = \sum _i (-1)^i\dim \Ext^i(\scrO
_X(\Delta -F ) \otimes I_{Z_+} , \scrO _X(F )\otimes I_{Z_-})$, then $\chi
(\scrO _X(\Delta -F ) \otimes I_{Z_+} , \scrO _X(F )\otimes I_{Z_-}) =
-\dim \Ext^1(\scrO _X(\Delta -F ) \otimes I_{Z_+} , \scrO _X(F )\otimes
I_{Z_-})$. Now a standard argument \cite{27} shows that
$$\gather
\chi (\scrO_X(\Delta -F ) \otimes I_{Z_+} , \scrO _X(F )\otimes I_{Z_-})\\
= \int_X\ch(\scrO _X(\Delta -F ) \otimes I_{Z_+})\spcheck\cdot
\ch(\scrO _X(F)\otimes I_{Z_-})\cdot \Todd_X.
\endgather$$
Here given a class $a = \sum a_i \in \bigoplus _iA^i(X)$, we denote by
$a\spcheck$ the class $\sum _i (-1)^ia_i$. An easy computation gives
$$\gather
\int _X\ch(\scrO_X(\Delta -F ) \otimes I_{Z_+})\spcheck\cdot
\ch(\scrO _X(F )\otimes I_{Z_-})\cdot \Todd_X\\
=\int _X\ch(\scrO_X(\Delta -F )\spcheck\cdot \ch(\scrO
_X(F )\cdot\Todd _X - \ell (Z_-) - \ell(Z_+).
\endgather$$
Reversing the above argument, we see that
$$\align
\int _X\ch(\scrO_X(\Delta -F )\spcheck\cdot \ch(\scrO
_X(F )\cdot\Todd _X &= \chi (\scrO_X(2F-\Delta))\\
=- h^1(X; \scrO _X(2F-\Delta)) &=
\frac{\zeta ^2}{2}-\frac{(\zeta \cdot K_X)}{2} +1 = -h(\zeta ).
\endalign$$
Putting these together we see that $\dim \Ext^1(\scrO _X(\Delta -F ) \otimes
I_{Z_+} , \scrO _X(F )\otimes I_{Z_-})$ is equal to $n_-+ n_+ + h(\zeta)$.
\endproof
Let us describe the scheme structure on $E_\zeta^{n_-, n_+}$ more carefully.
For $Z_-$ and $Z_+$ fixed, the set of extensions in $E_\zeta^{n_-, n_+}$
corresponding to $Z_-$, $Z_+$, is equal to $\Pee \Ext^1(\scrO _X(\Delta -F )
\otimes I_{Z_+} , \scrO _X(F )\otimes I_{Z_-})$. To make a universal
construction, let $H_{n_\pm} = \Hilb ^{n_\pm}X$. Let $\Cal Z_{n_\pm}$ be the
universal codimension two subscheme of $X\times H_{n_\pm}$. Let $\pi _1, \pi
_2$
be the projections of $X\times H_{n_-}\times H_{n_+}$ to $X$, $H_{n_-}\times
H_{n_+}$ respectively, and let $\pi _{1,2}$, $\pi _{1,3}$ be the projections of
$X\times H_{n_-}\times H_{n_+}$ to $X\times H_{n_-}$, $X\times H_{n_+}$
respectively. Define
$$\Cal E_\zeta ^{n_-, n_+} = Ext ^1_{\pi _2}(\pi _1^*\scrO _X(\Delta -F )
\otimes \pi _{1,3}^*I_{\Cal Z_{n_+}}, \pi _1^*\scrO _X(F )\otimes \pi
_{1,2}^*I_{\Cal Z_{n_-}}).$$
The previous lemma and standard base change results show that $\Cal E_\zeta
^{n_-, n_+}$ is locally free of rank $h(\zeta)+\ell _\zeta$ over $H_{n_-}\times
H_{n_+}$. We set $E_\zeta^{n_-, n_+} = \Pee ((\Cal E_\zeta ^{n_-,
n_+})\spcheck)$, if $h(\zeta)+\ell _\zeta > 0$. Moreover by standard facts
about relative Ext sheaves there is an exact sequence
$$\gather
0 \to R^1\pi _2{}_*Hom\Big(\pi _1^*\scrO _X(\Delta -F )
\otimes \pi _{1,3}^*I_{\Cal Z_{n_+}}, \pi _1^*\scrO _X(F )\otimes \pi
_{1,2}^*I_{\Cal Z_{n_-}}\Big) \to \Cal E_\zeta
^{n_-, n_+} \to \\
\to \pi _2{}_*Ext ^1\Big(\pi _1^*\scrO _X(\Delta -F )
\otimes \pi _{1,3}^*I_{\Cal Z_{n_+}}, \pi _1^*\scrO _X(F )\otimes \pi
_{1,2}^*I_{\Cal Z_{n_-}}\Big) \to 0.
\endgather$$
\corollary{2.7} With $X$ as in \rom{(2.6)}, if $h(\zeta) +\ell _\zeta= h^1(X;
\scrO _X(2F-\Delta)+\ell_\zeta
\neq 0$,
$E_\zeta^{n_-, n_+}$ is a $\Pee ^{N_\zeta}$-bundle over
$H_{n_-}\times H_{n_+}$, where $N_\zeta = \dim \Ext^1 - 1
= h(\zeta)+\ell _\zeta -1$. Thus if $h(\zeta)+\ell _\zeta\neq 0$, then $\dim
E_\zeta^{n_-, n_+} = 3\ell _\zeta + h(\zeta)-1$. Moreover in this case
$E_{-\zeta}^{n_+, n_-}$ is a $\Pee ^{N_{-\zeta}}$-bundle over
$H_{n_+}\times H_{n_-}$, and $N_\zeta + N_{-\zeta} + 2\ell _\zeta = -p-4$. If
$h(\zeta) +\ell _\zeta =0$, then $E_\zeta^{0, 0} = \emptyset$ and
$E_{-\zeta}^{0, 0} =
\Pee ^{-p-3}$ is a component of $\frak M_+$. Finally this last case arises if
and only if
$\zeta ^2 = p$ and $\zeta \cdot K_X = \zeta ^2+2 = p+2$.
\endstatement
\proof Note that $N_\zeta \geq 0$ unless $h(\zeta) +\ell _\zeta =0$. Under this
assumption, we have
$$N_\zeta + N_{-\zeta} + 2\ell _\zeta = 4\ell _\zeta - \zeta ^2 -4= -p-4.$$
The case where $h(\zeta) +\ell _\zeta =0$ is similar.
Moreover if $h(\zeta) +\ell_\zeta =0$, then it follows from (2.2)(ii) that
all of the sheaves $V$ corresponding to points of $E_{-\zeta}^{0, 0}$
are $L_+$-stable. By (2.2)(i) the map $E_{-\zeta}^{0, 0}\to \frak M_+$
is one-to-one. Since $\frak M_+$ is of dimension $-p-3$ and smooth at
points corresponding to the sheaves in $E_{-\zeta}^{0, 0}\to \frak M_+$,
the map $E_{-\zeta}^{0, 0}\to \frak M_+$ must be
an embedding onto a component of $\frak M_+$.
The final statement follows from the formulas
$\zeta ^2 = 4\ell_\zeta +p$ and $\dsize h(\zeta)=
\frac{(\zeta \cdot K_X)}{2} -\frac{\zeta ^2}{2} -1$.
\endproof
If $h(\zeta)+\ell_\zeta\neq 0$, then by Lemma 2.2 there is a rational map from
$E_\zeta^{n_-, n_+}$ to the moduli space $\frak M_-$ which is birational onto
its image. However this map will not in general be a morphism if $n_->0$
(see \cite{16}). We shall study this more carefully in the next sections.
Let us also remark that standard theory gives a universal sheaf $\Cal V$ over
$E_\zeta^{n_-, n_+}$:
\proposition{2.8} Let $\rho \: X\times E_\zeta^{n_-, n_+} \to
X\times H_{n_-}\times H_{n_+}$ be the natural projection, and let $\pi_2\:
X\times E_\zeta^{n_-, n_+} \to E_\zeta^{n_-, n_+}$ be the projection. Then
there
is a coherent sheaf $\Cal V$ over
$X\times E_\zeta^{n_-, n_+}$ and an exact sequence
$$\gather
0 \to
\rho ^*\left(\pi _1^*\scrO _X(F )\otimes \pi
_{1,2}^*I_{\Cal Z_{n_-}}\right)\otimes \pi_2^*\scrO_{E_\zeta^{n_-, n_+}}(1)\\
\to \Cal V \to \rho ^*\left(\pi _1^*\scrO _X(\Delta -F )
\otimes \pi _{1,3}^*I_{\Cal Z_{n_+}}\right) \to 0.\qed
\endgather$$
\endstatement
\noindent {\bf Remark 2.9.} Very similar results hold in the case where $-K_X$
is effective and nonzero (corresponding to certain elliptic ruled surfaces) or
$K_X=0$ (corresponding to $K3$ or abelian surfaces). For example, in the case
of
a $K3$ surface $X$, the moduli space is smooth of dimension $-p-6$ away from
the
sheaves which are strictly semistable for every ample divisor (although there
exist components consisting entirely of non-locally free sheaves for small
values of $-p$). In this case however $h(\zeta ) = -\zeta ^2/2 -2$ and $N_\zeta
+ N_{-\zeta} + 2\ell _\zeta = -p-6$, which is equal to the dimension $d$ of the
moduli space instead of to $d-1$. For example, if $\ell _\zeta = 0$, then
$N_\zeta = N_{-\zeta} = d/2$. In this case $E_\zeta ^{0,0} \cong \Pee ^{d/2}$
is
a maximal isotropic submanifold of the symplectic manifold $\frak M_-$. In
other words, the natural holomorphic $2$-form $\omega$ on $\frak M_-$ vanishes
on $E_\zeta ^{0,0}$ and identifies the normal bundle of $E_\zeta ^{0,0}$ in
$\frak M_-$ with the cotangent bundle of $E_\zeta ^{0,0}$.
\section{3. Flips of moduli spaces.}
In this section, we begin by assuming again that $X$ is an arbitrary algebraic
surface. Let $\zeta = \zeta _1, \dots, \zeta _n$ be the positive
rational multiples of $\zeta$ such that $\zeta_i$ is an integral class
also defining the wall $W^\zeta$. Our goal in this section is to deal
with the problem that
there is only a rational map in general from $E_{\zeta_i}^{n_-, n_+}$ to
$\frak M_-$. We shall do so by finding a sequence of spaces between
$\frak M_-$ and $\frak M_+$, each one given by blowing up and down
the previous one, such that for an appropriate member of the sequence
the rational map $E_{\zeta_i}^{n_-, n_+}\dasharrow \frak M_-$
becomes a morphism (and a smooth embedding in the case of rational surfaces).
Throughout the rest of this paper, $L_0$ shall denote any ample divisor
contained in the interior of the intersection of $W^\zeta$ and
the closures of $\Cal C_\pm$. Recall that we have defined
universal semistability after the proof of (2.1).
\definition{Definition 3.1} Let $k$ be an integer. A rank two torsion free
sheaf $V$ with $c_1(V) = \Delta$ and $\Delta ^2 - 4c_2(V) = p$ is
{\sl $(L_0, \zeta, k)$-semistable\/} if $V$ is Mumford
$L_0$-semistable and if it is strictly Mumford semistable, then either it is
universally semistable or, for all divisors
$F$ such that $2F-\Delta \equiv \zeta$, we have the following:
\roster
\item"{(i)}" If there exists an exact sequence
$$0 \to \scrO _X(F )\otimes I_{Z_1} \to V \to \scrO _X(\Delta -F ) \otimes
I_{Z_2} \to 0,$$
then $\ell (Z_2) \leq k$ and thus $\ell (Z_1) \geq \ell _\zeta - k$.
\item"{(ii)}" If there exists an exact sequence
$$0 \to \scrO _X(\Delta - F )\otimes I_{Z_1} \to V \to \scrO _X(F )
\otimes I_{Z_2} \to 0,$$
then $\ell (Z_1) \geq k+1$ and thus $\ell (Z_2) \leq \ell _\zeta - k-1$.
\endroster
Likewise, setting $\boldsymbol \zeta = (\zeta _1, \dots,\zeta _n)$ and $\bold k
= (k_1, \dots, k_n)$, we say that $V$ is {\sl $(L_0, \boldsymbol \zeta,
\bold k)$-semistable\/} if
$V$ is $(L_0, \zeta _i, k_i)$-semistable for every $i$. Let $\frak
M_0^{(\boldsymbol \zeta, \bold k)}$ denote the set of isomorphism classes of
$(L_0, \boldsymbol \zeta, \bold k)$-semistable rank two sheaves $V$ with
$c_1(V)
= \Delta$ and $\Delta ^2 - 4c_2(V) = p$.
\enddefinition
Next we give some easy properties of $(L_0, \boldsymbol \zeta, \bold
k)$-semistability.
\lemma{3.2}
\roster
\item"{(i)}" If $k_i \geq \ell _{\zeta_i}$ for all $i$, and $V$ is not
universally semistable, then $V$ is
$(L_0, \boldsymbol \zeta, \bold k)$-semistable if and only if it is
$L_-$-stable. Likewise if $k_i \leq -1$ for all $i$ and $V$ is not
universally semistable, then $V$ is
$(L_0, \boldsymbol \zeta, \bold k)$-semistable
if and only if it is$L_+$-stable.
\item"{(ii)}" If $k_i \geq \ell _{\zeta_i}$ for all $i$,
then $\frak M_0^{(\boldsymbol \zeta, \bold k)} = \frak M_-$.
Likewise if $k_i \leq -1$ for all $i$,
then $\frak M_0^{(\boldsymbol \zeta, \bold k)} = \frak M_+$.
\item"{(iii)}" For $n_2 > k_i$, $\frak M_0 ^{(\boldsymbol \zeta, \bold k)}
\cap E_{\zeta_i}^{n_1, n_2}= \emptyset$.
\item"{(iv)}" There is an injection $E_{\zeta_i}^{\ell_{\zeta_i}-k_i,
k_i} \to \frak M_0 ^{(\boldsymbol \zeta, \bold k)}$.
Likewise there is an injection $E_{-\zeta_i}^{k_i+1, \ell _{\zeta_i} - k_i-1}
\to \frak M_0 ^{(\boldsymbol \zeta, \bold k)}$. Finally,
the images of $E_{\zeta_i}^{\ell _{\zeta_i} -k_i, k_i}$ and
$E_{\zeta_j}^{\ell _{\zeta_j}-k_j, k_j}$ are disjoint if $i\neq j$.
\endroster
\endstatement
\proof If $k_i\geq \ell _{\zeta_i}$ for all $i$, then the condition that
$\ell(Z_2) \leq \ell _{\zeta_i}$ and $\ell (Z_1) \geq 0$ are trivially always
satisfied and the conditions
$\ell(Z_2) \leq -1$ and $\ell (Z_1) \geq \ell _{\zeta_i} +1$ are vacuous. A
similar argument handles the case $k_i \leq -1$ for all $i$. It is easy to see
that this implies (i). Statement (ii) follows from (i), and
(iii) follows from the definitions. As for (iv), let
$V\in E_{\zeta_i}^{\ell _{\zeta_i} -k_i, k_i}$. To decide if $V$ is in $\frak
M_0 ^{(\boldsymbol \zeta, \bold k)}$, we look for potentially destabilizing
subsheaves with torsion free quotient. Similar arguments as in \cite{30}
show that the only potentially destabilizing subsheaves with
torsion free quotient must be either $\scrO _X(F )\otimes I_{Z_1}$
or $\scrO _X(\Delta - F )\otimes I_{Z}$. By hypothesis, there is a
unique subsheaf of $V$ of the form
$\scrO _X(F )\otimes I_{Z_1}$, and it is not destabilizing.
If there is a subsheaf of the form $\scrO _X(\Delta - F )\otimes I_{Z}$
with torsion free quotient, then by Lemma 2.2 we have
$\ell (Z) > \ell (Z_2) = k_i$ and so $\ell (Z) \geq k_i + 1$.
Hence such a subsheaf is also not destabilizing.
Thus by Definition 3.1 $V$ is $(L_0, \boldsymbol \zeta, \bold k)$-semistable.
The fact that the map $E_{\zeta_i}^{\ell _{\zeta_i} -k_i, k_i} \to
\frak M_0 ^{(\boldsymbol \zeta, \bold k)}$ is one-to-one and that
$E_{\zeta_i}^{\ell _{\zeta_i} -k_i, k_i}$ and
$E_{\zeta_j}^{\ell _{\zeta_j} -k_j, k_j}$ are disjoint if $i\neq j$ also
follow from similar arguments in \cite{30}. The statement about
$E_{-\zeta_i}^{k_i+1, \ell _{\zeta_i} - k_i-1}$ is similar.
\endproof
Next suppose that we are given two integral vectors $\bold k$ and $\bold k'$
and
a subset $I$ of $\{1 \dots, n\}$ such that $k_i' = k_i$ if $i\notin I$ and
$k_i'
= k_i-1$ if $i\in I$. We investigate the change as we pass from
$\frak M_0 ^{(\boldsymbol
\zeta,
\bold k)}$ to $\frak M_0 ^{(\boldsymbol \zeta, \bold k')}$.
\lemma{3.3} The set of sheaves $V$ in $\frak M_0 ^{(\boldsymbol \zeta, \bold
k)}$
which are not $(L_0, \boldsymbol \zeta, \bold k')$-semistable is exactly the
image
of
$\bigcup _{i\in I}E_{\zeta_i}^{\ell _{\zeta_i}-k_i, k_i}$. Likewise the set of
$V\in
\frak M_0 ^{(\boldsymbol \zeta, \bold k')}$ which are not
$(L_0, \boldsymbol \zeta, \bold k)$-semistable is exactly the image of
$\bigcup _{i\in I}E_{-\zeta_i}^{k_i,\ell _{\zeta_i} -k_i}$.
\endstatement
\proof If $V$ is $(L_0, \boldsymbol \zeta, \bold k)$-semistable but not $(L_0,
\boldsymbol \zeta, \bold k')$-semistable, then $V$ must be Mumford strictly
$L_0$-semistable. Suppose that the
$(L_0, \boldsymbol \zeta, \bold k')$-destabilizing subsheaf is of the form
$\scrO
_X(F )\otimes I_{Z_1}$, where $F$ corresponds to $\zeta _i$ for some $i\in I$.
Then
$\ell (Z_2)
\leq k_i$ (since $V\in
\frak M_0 ^{(\boldsymbol \zeta, \bold k)}$) but
$\ell (Z_2) \geq k_i$ (since the subsheaf is $(L_0, \boldsymbol \zeta, \bold
k')$-destabilizing, for $k_i' = k_i-1$) so that $\ell (Z_2) = k_i$. Thus
$V\in E_\zeta^{\ell _{\zeta_i} -k_i, k_i}$. The other possibility is that the
destabilizing subsheaf is of the form $\scrO _X(\Delta - F )\otimes I_{Z_1}$.
Here we need $\ell (Z_1) \geq k_i+1$ but $\ell (Z_1) < k_i$ and there are no
such
sheaves. The statement about $\frak M_0 ^{(\boldsymbol \zeta, \bold k')}$
follows
by symmetry.
\endproof
We shall now describe a sequence of actual moduli spaces $\frak M_0
^{(\boldsymbol \zeta, \bold k)}$ for which the integral vector $\bold k$
change in the way described before the statement of (3.3).
\definition{Definition 3.4} Suppose that $\zeta _i = r_i\zeta _1$, where $r_i$
is a positive rational number. Given $t\in \Bbb Q$, let $t_i = r_it$, so that
$t_1=t$. Suppose that $\dsize
\frac{\ell _{\zeta _i} + t_i}2$ is not an integer for any
$i$. In this case, define
$$k_i(t) = \fracwithdelims[]{\ell _{\zeta _i} + t_i}{2},$$
where $[x]$ is the greatest integer function, and define $\bold k(t)$ to be
the vector formed by the
$k_i(t)$. A rational number $t$ is {\sl $\zeta _i$-critical\/} if
$\dsize \frac{\ell_{\zeta _i} + t_i}2 \in \Zee$ and $-1 \leq \dsize \frac{\ell
_{\zeta _i} + t_i}2 \leq \ell _{\zeta _i}$. We shall also say that $t_i$ is
{\sl $\zeta _i$-critical\/}. Finally $t$ is {\sl
$\boldsymbol
\zeta$-critical\/} if it is $\zeta _i$-critical for some $i$. Note that there
are
only finitely many such $t$.
\enddefinition
\medskip
Given $t \in \Bbb Q$, let $I(t) = \{\, i: t {\text{ is $\zeta
_i$-critical}}\,\}$. Suppose that $\varepsilon$ is chosen so that, for every
$i$, either there is no $\zeta _i$-critical rational number in $[t_i -
r_i\varepsilon, t_i +
r_i\varepsilon]$ or
$t_i$ is the unique
$\zeta _i$-critical rational number in $[t_i - r_i\varepsilon, t_i +
r_i\varepsilon]$. Equivalently either there is no $\boldsymbol
\zeta$-critical number in $[t-\varepsilon, t+\varepsilon]$ or $t$ is the unique
$\boldsymbol
\zeta$-critical number in $[t-\varepsilon, t+\varepsilon]$. Then we clearly
have:
$$k_i(t - \varepsilon) = \cases k_i(t+
\varepsilon) , &\text{if
$i\notin I(t)$}\\
k_i(t + \varepsilon) -1, &\text{if $i\in I(t)$.}
\endcases$$
In particular if there is no
$\boldsymbol
\zeta$-critical number in $[t-\varepsilon, t+\varepsilon]$, so that $I(t)=
\emptyset$, then
$k_i(t -
\varepsilon) = k_i(t + \varepsilon)$ for every $i$. Further note that if $t\gg
0$, then $k_i(t) > \ell _{\zeta _i}$ for every $i$, and if $t\ll 0$, then
$k_i(t) <-1$ for every
$i$.
We then have the following theorem, whose proof will be given in the next
section:
\theorem{3.5} For all $t\in \Bbb Q$ which are not $\boldsymbol
\zeta$-critical, there exists a natural structure of a projective scheme on
$\frak
M_0 ^{(\boldsymbol
\zeta, \bold k(t))}$ for which it is a coarse moduli space.
\endstatement
\medskip
The proof of (3.5) will also show that $\frak M_0 ^{(\boldsymbol
\zeta, \bold k(t))}$ has the usual properties of a coarse moduli space: all
sheaves corresponding to points of $\frak M_0 ^{(\boldsymbol
\zeta, \bold k(t))}$ will turn out to be simple (as they will turn out to be
stable for an appropriate notion of stability), a classical or formal
neighborhood of a point of
$\frak M_0 ^{(\boldsymbol
\zeta, \bold k(t))}$ may be identified with the universal deformation space of
the corresponding sheaf, and there exists a universal sheaf locally in the
classical or
\'etale topology around every point of $\frak M_0 ^{(\boldsymbol
\zeta, \bold k(t))}$.
For the rest of this section, we shall again restrict to the case where $X$ is
a
rational surface with $-K_X$ effective, unless otherwise noted. Let $\zeta =
\zeta _i$ for some $i$ and let $\bold k=\bold k(t)$ for some $t$ which
is not
$\boldsymbol \zeta$-critical. The first step is to make some infinitesimal
calculations concerning the differential of the map
$E_\zeta^{\ell _\zeta -k, k}\to
\frak M_0 ^{(\boldsymbol \zeta, \bold k)}$ and the normal bundle to its image.
\proposition{3.6} The map $E_\zeta^{\ell _\zeta -k, k}\to
\frak M_0 ^{(\boldsymbol \zeta, \bold k)}$ is an immersion. The normal bundle
$\Cal N_\zeta ^{\ell _\zeta -k, k}$ to $E_\zeta^{\ell _\zeta -k, k}$ in $\frak
M_0 ^{(\boldsymbol \zeta, \bold k)}$ is exactly $\rho ^*\Cal E _{-\zeta}^{k,
\ell _\zeta -k}\otimes
\scrO_{E_\zeta ^{\ell _\zeta -k, k}}(-1)$, in the notation of the previous
section.
\endstatement
\proof Since every sheaf in $\frak M_0 ^{(\boldsymbol \zeta, \bold k)}$ is
actually stable and therefore simple (which was also proved in (2.2)) we may
identify an analytic neighborhood of $V\in \frak M_0 ^{(\boldsymbol \zeta,
\bold
k)}$ with the germ of the universal deformation space for $V$, i\.e\. with
$\Ext
^1(V,V)$. Let us now calculate the tangent space to
$E_\zeta ^{\ell _\zeta -k, k}$ at $V$: suppose that $\xi \in
\Ext^1(\scrO _X(\Delta -F ) \otimes I_{Z_2} , \scrO _X(F)\otimes I_{Z_1})
=\Ext^1$ is a nonzero extension class corresponding to $V$, where $\ell (Z_1) =
\ell _\zeta -k$ and $\ell (Z_2) = k$. Let
$H_{\ell _\zeta -k} = \Hilb ^{\ell _\zeta -k}X$ and $H_k =
\Hilb ^kX$. Then there is the following exact sequence for the tangent
space to $E_\zeta ^{\ell _\zeta -k, k}$ at $\xi$:
$$0\to \Ext ^1/\Cee \cdot \xi \to T_\xi E_\zeta ^{\ell _\zeta -k, k}\to
T_{Z_1}H_{\ell _\zeta -k}
\oplus T_{Z_2}H_k \to 0.$$
Note further that the tangent space to $\Hilb ^nX$ at $Z$ is
equal to $\Hom(I_Z,\scrO_Z)$, which we may further canonically identify with
$\Ext^1(I_Z, I_Z)$ since $X$ is rational and by a local calculation. We then
have the following:
\proposition{3.7} For all nonzero $\xi \in \Ext^1$, the natural map from a
neighborhood of
$\xi$ in $E_\zeta ^{\ell _\zeta -k, k}$ to $\frak M_0 ^{(\boldsymbol \zeta,
\bold
k)}$ is an immersion at
$\xi$. The image of $T_\xi E_\zeta ^{\ell _\zeta -k, k}$ in $\Ext ^1(V,V)$ is
exactly the kernel of the natural map $\Ext ^1(V,V)\to \Ext ^1(\scrO
_X(F)\otimes I_{Z_1},
\scrO _X(\Delta -F ) \otimes I_{Z_2})$, and the normal space to $E_\zeta
^{\ell _\zeta -k, k}$ at $\xi$ in $\frak M_0 ^{(\boldsymbol \zeta, \bold k)}$
may
be canonically identified with
$\Ext ^1 (\scrO _X(F)\otimes I_{Z_1}, \scrO _X(\Delta -F ) \otimes I_{Z_2})$.
\endstatement
\noindent
{\it Proof.} Consider the natural map from
$\Ext ^1(V,V)$ to
$\Ext ^1 (\scrO _X(F)\otimes I_{Z_1}, \scrO _X(\Delta -F ) \otimes I_{Z_2})$.
We claim that this map is surjective and will describe its kernel in more
detail. The map factors into two maps:
$$\gather
\Ext ^1(V,V) \to \Ext^1(V, \scrO _X(\Delta -F ) \otimes I_{Z_2}) \\
\Ext^1(V, \scrO _X(\Delta -F ) \otimes I_{Z_2}) \to \Ext ^1 (\scrO
_X(F)\otimes I_{Z_1}, \scrO _X(\Delta -F ) \otimes I_{Z_2}).
\endgather$$
The cokernel of the first map is contained in $\Ext^2(V, \scrO _X(F)\otimes
I_{Z_1})$. To see that this group is zero, apply Serre duality: it suffices to
show that $\Hom(\scrO _X(F)\otimes I_{Z_1}, V\otimes K_X) =0$. From the
defining exact sequence for $V$, we have an exact sequence
$$\gather
0 \to \Hom(\scrO _X(F)\otimes I_{Z_1},\scrO _X(F)\otimes I_{Z_1}\otimes
K_X)
\to \Hom(\scrO _X(F)\otimes I_{Z_1}, V\otimes K_X)\\
\to \Hom(\scrO _X(F)\otimes
I_{Z_1},\scrO _X(\Delta -F ) \otimes I_{Z_2}).
\endgather$$
The first term is just $H^0(K_X) = 0$ and the third is contained in
$H^0(\scrO_X(\Delta -2F) \otimes K_X) =0$. Thus $\Hom(\scrO _X(F)\otimes
I_{Z_1}, V\otimes K_X) =0$. The vanishing of the
cokernel of the second map, namely $\Ext ^2(\scrO _X(\Delta -F ) \otimes
I_{Z_2}, \scrO _X(\Delta -F )
\otimes I_{Z_2})$, is similar. Thus $\Ext ^1(V,V)\to \Ext ^1 (\scrO
_X(F)\otimes
I_{Z_1}, \scrO _X(\Delta -F ) \otimes I_{Z_2})$ is onto. If $K$ is the kernel,
then arguments as above show that there is an exact sequence
$$0 \to \Ext^1(V, \scrO _X(F)\otimes I_{Z_1}) \to K \to \Ext^1(\scrO _X(\Delta
-F ) \otimes I_{Z_2}, \scrO _X(\Delta -F ) \otimes I_{Z_2})\to 0.$$
Here $\Ext^1(\scrO _X(\Delta
-F ) \otimes I_{Z_2}, \scrO _X(\Delta -F ) \otimes I_{Z_2}) = \Ext^1(I_{Z_2},
I_{Z_2})$ is the tangent space to $H_k$. Moreover, there is an exact
sequence
$$\gather
\Hom(\scrO _X(F)\otimes I_{Z_1}, \scrO _X(F)\otimes I_{Z_1}) \to
\Ext^1(\scrO _X(\Delta -F ) \otimes I_{Z_2},\scrO _X(F)\otimes I_{Z_1}) \to \\
\to \Ext^1(V, \scrO _X(F)\otimes I_{Z_1}) \to \Ext^1(\scrO _X(F)\otimes
I_{Z_1},
\scrO _X(F)\otimes I_{Z_1})\to 0.
\endgather$$
The last term is $\Ext^1(I_{Z_1}, I_{Z_1})$ which is the tangent space to
$H_{\ell _\zeta -k}$ at $Z_1$, and the first two terms combine to give
$\Ext^1/\Cee \cdot \xi$. Thus the kernel $K$ looks very much like
the tangent space to $E_\zeta ^{\ell _\zeta -k, k}$ at $\xi$ and both spaces
have the same dimension.
Let us describe
the tangent space to $E_\zeta ^{\ell _\zeta -k, k}$ at $\xi$ and the
differential of the map $E_\zeta ^{\ell _\zeta -k, k}$ to $\frak M_0
^{(\boldsymbol \zeta, \bold k)}$ in more intrinsic terms. It is easy to see
that
a $\Spec
\Cee[\epsilon]$-valued point of $E_\zeta ^{\ell _\zeta -k, k}$ which
restricts to $\xi$ defines two codimension two subschemes $\Cal
Z_1\subseteq X\times
\Spec\Cee[\epsilon]$,
$\Cal Z_2\subseteq X\times \Spec\Cee[\epsilon]$, flat over
$\Spec\Cee[\epsilon]$, restricting to $Z_i$ over $X$, and an extension
$\Cal V$ over $X\times \Spec\Cee[\epsilon]$ of the form
$$0 \to \pi _1^* \scrO _X(F)\otimes I_{\Cal Z_1} \otimes \to \Cal V \to
\pi _1^*\scrO _X(\Delta -F ) \otimes I_{\Cal Z_2}\to 0.$$
Conversely such a choice of $\Cal Z_1$, $\Cal Z_2$ and $\Cal V$ define
a $\Spec \Cee[\epsilon]$-valued point of $E_\zeta ^{\ell _\zeta -k, k}$.
Thus there is a commutative diagram with exact rows and columns:
$$\CD
@. 0 @. 0 @. 0 @.\\
@. @VVV @VVV @VVV @. \\
0 @>>> \scrO _X(F)\otimes I_{Z_1} @>>> V @>>> \scrO _X(\Delta -F ) \otimes
I_{Z_2} @>>> 0\\
@. @VVV @VVV @VVV @. \\
0 @>>> \pi _1^* \scrO _X(F)\otimes I_{\Cal Z_1} @>>> \Cal V @>>> \pi _1^*\scrO
_X(\Delta -F ) \otimes I_{\Cal Z_2}@>>> 0\\
@. @VVV @VVV @VVV @. \\
0 @>>> \scrO _X(F)\otimes I_{Z_1} @>>> V @>>> \scrO _X(\Delta -F ) \otimes
I_{Z_2} @>>> 0\\
@. @VVV @VVV @VVV @. \\
@. 0 @. 0 @. 0 @.
\endCD$$
Here the extension $\Cal V$ of $V$ by $V$, viewed as a point of $\Ext
^1(V,V)$, corresponds to the Kodaira-Spencer map of the deformation $\Cal V$
of $V$. Likewise the left and right hand columns give classes in $\Ext
^1(I_{Z_1}, I_{Z_1})$ and $\Ext ^1(I_{Z_2}, I_{Z_2})$ corresponding to $\Cal
Z_1$ and $\Cal Z_2$. A straightforward diagram chase shows that if $\Cal V$
fits into this commutative diagram then the image of the extension class $\xi
\in \Ext ^1(V,V)$ corresponding to $\Cal V$ in $\Ext ^1 (\scrO
_X(F)\otimes I_{Z_1}, \scrO _X(\Delta -F ) \otimes I_{Z_2})$ is zero. To see
the converse, that every element in the kernel $K$ of the map $\Ext ^1(V,V) \to
\Ext ^1 (\scrO _X(F)\otimes I_{Z_1}, \scrO _X(\Delta -F ) \otimes I_{Z_2})$ is
the image of a tangent vector to $E_\zeta ^{\ell _\zeta -k, k}$ at $\xi$, use
the arguments above which show that there is a surjection from $K$ to
$$\Ext^1(\scrO _X(\Delta -F ) \otimes I_{Z_2}, \scrO _X(\Delta -F )
\otimes I_{Z_2}) = \Ext ^1(I_{Z_2}, I_{Z_2}).$$
Thus there is an induced extension of $\scrO _X(\Delta -F ) \otimes I_{Z_2}$
by $\scrO _X(\Delta -F ) \otimes I_{Z_2}$, necessarily of
the form $\scrO _X(\Delta -F ) \otimes I_{\Cal Z_2}$,
and a map from $\Cal V$ to
$\scrO _X(\Delta -F ) \otimes I_{\Cal Z_2}$, necessarily a surjection. The
kernel of this surjection then defines an extension $\scrO _X(F)\otimes
I_{\Cal Z_1}$ of $\scrO _X(F)\otimes I_{Z_1}$ by $\scrO _X(F)\otimes I_{Z_1}$.
It follows that $K$ is in the image of the tangent space to
$E_\zeta ^{\ell _\zeta -k, k}$ at $\xi$. By counting dimensions the map
on tangent spaces from
$T_\xi E_\zeta ^{\ell _\zeta -k, k}$ to $\Ext ^1(V,V)$ is injective, showing
that the map from $E_\zeta ^{\ell _\zeta -k, k}$ to $\frak M_0^{(\boldsymbol
\zeta, \bold k)}$ is an immersion and identifying the normal space at $\xi$.
\qed
Let us continue the proof of Proposition 3.6.
To give a global description of the normal bundle to
$E_\zeta ^{\ell _\zeta -k, k}$ in $\frak M_0^{(\boldsymbol \zeta, \bold
k)}$, recall by standard deformation
theory \cite{10} that the pullback of the tangent bundle of $\frak
M_0^{(\boldsymbol \zeta, \bold k)}$ to $E_\zeta ^{\ell _\zeta -k, k}$ is
just $Ext ^1_{\pi_2}(\Cal V, \Cal V)$,
where $\Cal V$ is the universal sheaf over
$X\times E_\zeta ^{\ell _\zeta -k, k}$ described in (2.8) and $\pi_2\: X \times
E_\zeta ^{\ell _\zeta -k, k} \to E_\zeta ^{\ell _\zeta -k, k}$ is
the second projection. Moreover the
calculations above globalize to show that the normal bundle is exactly
$$Ext ^1_{\pi_2}(\rho ^*\left(\pi _1^*\scrO _X(F )\otimes \pi
_{1,2}^*I_{\Cal Z_1}\right)\otimes \pi_2^*\scrO_{E_\zeta^{\ell _\zeta -k,
k}}(1),
\rho ^*\left(\pi _1^*\scrO _X(\Delta -F )
\otimes \pi _{1,3}^*I_{\Cal Z_2}\right)),$$
where $\rho \: X\times E_\zeta^{\ell _\zeta -k, k} \to
X \times H_{\ell _\zeta -k}\times H_k$ is the
natural projection. Using standard base change results and the projection
formula, we see that this sheaf is equal to
$$\rho ^*Ext ^1_{\pi _2}(\pi _1^*\scrO _X(F )\otimes \pi
_{1,2}^*I_{\Cal Z_1}, \pi _1^*\scrO _X(\Delta -F )
\otimes \pi _{1,3}^*I_{\Cal Z_2})\otimes \scrO_{E_\zeta^{\ell _\zeta -k,
k}}(-1),$$ which is the same as $\rho ^*\Cal E _{-\zeta}^{k,\ell _\zeta
-k}\otimes
\scrO_{E_\zeta ^{\ell _\zeta -k, k}}(-1)$.
\endproof
Finally, to compare the moduli space $\frak M_0 ^{(\boldsymbol \zeta, \bold
k(t+\varepsilon))}$ with $\frak M_0 ^{(\boldsymbol \zeta, \bold
k(t-\varepsilon))}$, where $t$ is the unique
$\boldsymbol \zeta$-critical point in $[t-\varepsilon, t +\varepsilon]$, we
shall
need the following result which is a straightforward generalization of (A.2) of
\cite{11}.
\proposition{3.8} Let
$X$ be a smooth projective scheme or compact complex manifold, and let $T$ be
smooth. Suppose that $\Cal
V$ is a rank two reflexive sheaf over $X\times T$, flat over $T$. Let $D$ be
a reduced divisor on $T$, not necessarily smooth and let $i\: D \to T$ be the
inclusion. Suppose that $L$ is a line bundle on $X$ and that $\Cal Z$ is a
codimension two subscheme of $X\times D$, flat over $D$. Suppose further that
$\Cal V \to i_*\pi _1^*L\otimes I_{\Cal Z}$ is a surjection, and let $\Cal V'$
be its kernel: $$0 \to \Cal V' \to \Cal V \to i_*\pi _1^*L\otimes I_{\Cal Z}
\to 0.$$ Then there is a line bundle $M$ on $X$ and a subscheme
$\Cal Z'$ of $X\times D$ codimension at least two, flat over $D$, with the
following properties:
\roster
\item"{(i)}" $\Cal V'$ is reflexive and flat over $T$.
\item"{(ii)}" There are exact sequences
$$\align
0 \to \pi _1^*M \otimes I_{\Cal Z'} \to &\Cal V|D \to \pi _1^*L\otimes
I_{\Cal Z} \to 0 ;\\
0\to \pi _1^*L\otimes I_{\Cal Z}\otimes \scrO_D(-D) \to &\Cal V'|D \to \pi
_1^*M
\otimes I_{\Cal Z'}
\to 0,
\endalign$$
which restrict for each $t\in D$ to give
exact sequences
$$\align 0 \to M\otimes I_{Z'} \to &V_t \to L\otimes I_Z \to 0;\\ 0\to
L\otimes I_Z \to &(V_t)' \to M\otimes I_{Z'} \to 0.
\endalign$$ Here $Z$ is the subscheme of $X$ defined by $\Cal Z$ for the
slice $X\times
\{t\}$ and $Z'_t$ is likewise defined by $\Cal Z'$.
\item"{(iii)}" If $D$ is smooth, then the extension class corresponding to
$(V_t)'$ in $\Ext ^1(M\otimes I_W, L\otimes I_Z)$ is defined by the image of
the normal vector to $D$ at $t$ under the composition of the Kodaira-Spencer
map from the tangent space of $T$ at $t$ to
$\Ext ^1(V_t, V_t)$, followed by the natural map $\Ext ^1(V_t, V_t) \to \Ext
^1(M\otimes I_{Z'}, L\otimes I_Z)$. \endroster
\endstatement
\medskip
Here $\Cal V'$ is called the {\sl elementary modification\/} of $\Cal V$ along
$D$. This construction has the following symmetry: if we make the elementary
modification of $\Cal V'$ along $D$ corresponding to the surjection $\Cal V'
\to i_*\bigl(\pi _1^*M \otimes I_{\Cal Z'}\bigr)$, then the result is $\Cal V
\otimes \scrO_{X\times T}(-(X\times D))$.
Here is the typical way that we will apply the above: given $X$, let $M$ be a
smooth manifold and $Y$ a submanifold of $M$. Let $T$ be the
blowup of $M$ along $Y$ and let $D$ be the exceptional divisor.
Let $\pi \: T \to M$ be the natural map. Then, given $\xi \in D$,
the image in the normal space to $\pi(\xi)$ of the normal direction at $\xi$ to
$D$ under $\pi _*$ may be identified with the line in the normal space
corresponding to $\xi$.
We can now state the main result as follows:
\theorem{3.9} Suppose that $t$ is the unique
$\boldsymbol \zeta$-critical point in $[t-\varepsilon, t +\varepsilon]$. If
$h(\pm\zeta_i) + \ell _{\pm\zeta_i} \neq 0$ for every $i$, then
the rational map
$\frak M_0 ^{(\boldsymbol \zeta, \bold k(t+\varepsilon))}
\dasharrow
\frak M_0 ^{(\boldsymbol \zeta, \bold k(t-\varepsilon))}$ is
obtained as follows. For every $i$, fixing $\zeta _i =\zeta$ and
$k_i(t+\varepsilon) = k$, blow up
$E_\zeta^{\ell _\zeta -k, k}$ in
$\frak M_0 ^{(\boldsymbol \zeta, \bold k(t+\varepsilon))}$. Then
the exceptional divisor $D$ is a $\Pee ^{N_\zeta}\times \Pee
^{N_{-\zeta}}$-bundle over $\Hilb ^{\ell _\zeta -k}X\times
\Hilb ^kX$. Moreover this divisor can be contracted in two different ways.
Contracting the $\Pee ^{N_{-\zeta}}$ fibers for all possible $\zeta$ gives
$\frak
M_0 ^{(\boldsymbol
\zeta, \bold k(t+\varepsilon))}$. Contracting the $\Pee
^{N_{\zeta}}$ fibers for all possible $\zeta$ gives $\frak M_0 ^{(\boldsymbol
\zeta, \bold k(t-\varepsilon))}$. Moreover the morphism from the
blowup to
$\frak M_0 ^{(\boldsymbol \zeta, \bold k(t-\varepsilon))}$ is
induced by an elementary modification as in \rom{(3.8)}, and the image of the
the component of the exceptional divisor which is the blowup of $E_\zeta^{\ell
_\zeta -k, k}$ is
$E_{-\zeta}^{k, \ell _\zeta -k}$. Finally the construction is symmetric.
Similar statements hold if $h(\pm\zeta_i) + \ell _{\pm\zeta_i}=0$ for some $i$,
where we must also add in or delete an extra component coming from $\pm\zeta
_i$.
\endstatement
\proof Begin by blowing up $E_\zeta^{\ell _\zeta -k, k}$ in $\frak M_0
^{(\boldsymbol \zeta, \bold k(t+\varepsilon))}$ for all possible
$\zeta$. For simplicity we shall just write down the argument in case there is
only one
$\zeta$; the general case is just additional notation. Let
$\widetilde{\frak M}_0^{(\boldsymbol \zeta, \bold k(\bold
t+\varepsilon))}$ denote the blowup and $D$ the exceptional divisor.
Note that the normal bundle
$\Cal N_\zeta ^{\ell _\zeta -k, k}$ to $E_\zeta^{\ell _\zeta -k, k}$ in $\frak
M_0 ^{(\boldsymbol \zeta, \bold k(\bold
t+\varepsilon))}$ is $\rho ^*\Cal E _{-\zeta}^{k, \ell _\zeta -k}\otimes
\scrO_{E_\zeta ^{\ell _\zeta -k, k}}(-1)$, where $\rho\: E_\zeta ^{\ell _\zeta
-k, k} \to \Hilb ^{\ell _\zeta -k}X\times \Hilb ^kX$ is the projection. In
particular $\Cal N_\zeta ^{\ell _\zeta
-k, k}$ restricts to each fiber $\Pee ^{N_\zeta}$ to a bundle of the form
$\left[\scrO_{\Pee ^{N_\zeta}}^N\right]\otimes \scrO_{\Pee ^{N_\zeta}}(-1)$,
and an easy calculation using (2.7) shows that $N= N_{-\zeta}+1$. It follows
that
the fibers of the induced map from
$D$ to $\Hilb ^{\ell _\zeta -k}X\times \Hilb ^kX$ are naturally $\Pee
^{N_\zeta}\times \Pee ^{N_{-\zeta}}$. Moreover it is easy to see that
$\scrO(D)|\Pee ^{N_\zeta}= \scrO_{\Pee ^{N_\zeta}}(-1)$, using for example the
fact that $\scrO(D)|\Pee ^{N_\zeta}\times \Pee ^{N_{-\zeta}}= \scrO(a,-1)$ for
some $a$ and the fact that
$$\Cal N_\zeta ^{\ell _\zeta
-k, k}|\Pee ^{N_\zeta} = R^0\pi _1{}_*[\scrO(-D)|\Pee ^{N_\zeta}\times \Pee
^{N_{-\zeta}}]= \left[\scrO_{\Pee ^{N_\zeta}}^{N_{-\zeta}+1}\right]\otimes
\scrO_{\Pee ^{N_\zeta}}(-a).$$
For the rest of the argument, we assume that there exists a universal family
on $X\times \frak M_0^{(\boldsymbol \zeta,
\bold k(\bold t+\varepsilon))}$. Of course, such a family will only exist
locally in the classical or \'etale topology, but this will suffice for the
argument. Let $\Cal U$ be the pullback of the universal family to
$X\times \widetilde{\frak M}_0^{(\boldsymbol \zeta,
\bold k(\bold t+\varepsilon))}$. Locally again we may assume that the
restriction of $\Cal U$ to $X\times D$ is the pullback of the universal
extension $\Cal V$ of (2.8):
$$\gather
0 \to
\rho ^*\left(\pi _1^*\scrO _X(F )\otimes \pi
_{1,2}^*I_{\Cal Z_{n_-}}\right)\otimes \pi_2^*\scrO_{E_\zeta^{n_-, n_+}}(1)\\
\to \Cal V \to \rho ^*\left(\pi _1^*\scrO _X(\Delta -F )
\otimes \pi _{1,3}^*I_{\Cal Z_{n_+}}\right) \to 0.
\endgather$$
Now consider the effect of making an elementary
transformation of $\Cal U$ on
$X\times \widetilde{\frak M}_0^{(\boldsymbol \zeta,
\bold k(\bold t+\varepsilon))}$ along the divisor $D$, using the morphism from
$\Cal U$ to the pullback of $\rho ^*\left(\pi _1^*\scrO _X(\Delta -F ) \otimes
\pi _{1,3}^*I_{\Cal Z_k}\right)$ given by considering the pullback of the
universal extension. Applying
(3.8) to the elementary transformation $\Cal U'$, we see that the fiber of
$\Cal U'$ at a point of the fiber $\Pee ^{N_\zeta}\times \Pee
^{N_{-\zeta}}$ lying over a point $(Z_1, Z_2)\in \Hilb ^{\ell _\zeta -k}X\times
\Hilb ^kX$ is given by a nonsplit extension of the form
$$0 \to \scrO_X(\Delta - F)\otimes I_{Z_2} \to U \to \scrO_X(F)\otimes I_{Z_1}
\to 0.$$
Moreover the extension class corresponding to $U$ is given by the projectivized
normal vector in $\Pee ^{N_{-\zeta}}$. Thus it is independent of the first
factor
$\Pee ^{N_\zeta}$ and the set of all possible such classes is parametrized by
the
second factor $\Pee
^{N_{-\zeta}}$. There is then an induced morphism from $\widetilde{\frak
M}_0^{(\boldsymbol \zeta, \bold k(t+\varepsilon))}$ to $\frak
M_0^{(\boldsymbol \zeta, \bold k(t-\varepsilon))}$ and clearly it
has the effect of contracting
$D$ along its first ruling and has the property that the image of $D$ is
exactly $E_{-\zeta}^{k, \ell _\zeta -k}$. We leave the symmetry of the
construction to the reader. This concludes the proof of (3.9).
\endproof
\noindent {\bf Remark 3.10.} In the $K3$ or abelian case, the arguments of this
section show that the rational map $\frak M_0 ^{(\boldsymbol \zeta, \bold
k(\bold
t+\varepsilon))} \dasharrow \frak M_0 ^{(\boldsymbol \zeta, \bold
k(t-\varepsilon))}$ is a Mukai elementary transformation \cite{26, 28}.
\medskip
We can also use (3.8) to analyze the rational map from $E_\zeta ^{n_-, n_+}$
to $\frak M_-$, in the case where it is not a morphism. For simplicity we shall
only consider the case of
$E_\zeta ^{1,0}$, i\.e\.
$\ell _\zeta = 1$. In this case $Z_- = p\in X$ and
$I_{Z_-} = \frak m_p$ is the maximal ideal sheaf of $p$. Moreover $\Ext
^1(\scrO
_X(\Delta -F ), \scrO _X(F)\otimes
\frak m_p) = H^1(\scrO _X(2F-\Delta)\otimes \frak m_p)$. There is an exact
sequence
$$0 \to H^0(\Cee _p) \to H^1(\scrO _X(2F-\Delta)\otimes \frak m_p) \to
H^1(\scrO _X(2F-\Delta)) \to 0.$$
Moreover, for $p$ fixed, the extensions $V$ corresponding to a split extension
for $V\ddual$ are exactly the kernel of the map from $H^1(\scrO
_X(2F-\Delta)\otimes \frak m_p)$ to $H^1(\scrO _X(2F-\Delta))$, i\.e\. the
image of $H^0(\Cee _p)$. The normal space is thus identified with $H^1(\scrO
_X(2F-\Delta))$. Now if the extension for $V\ddual$ is split, then there is a
map $\scrO _X(\Delta -F )\otimes \frak m_p \to V$ with quotient $\scrO _X(F)$.
This way of realizing $V$ as an extension gives a surjection $\Ext ^1(V,V) \to
\Ext ^1(\scrO _X(\Delta -F )\otimes \frak m_p, \scrO _X(F))$, and we must look
at the image of the normal space $H^1(\scrO _X(2F-\Delta))$ in this extension
group. On the other hand, we have an exact sequence
$$0\to H^1(\scrO _X(2F-\Delta)) \to \Ext ^1(\scrO _X(\Delta -F )\otimes \frak
m_p, \scrO _X(F)) \to H^0(\Cee _p) \to 0$$
coming from the long exact Ext sequence, and it is an easy diagram chase to see
that the induced map $\Ext ^1(\scrO _X(\Delta -F ), \scrO _X(F)\otimes
\frak m_p) \to \Ext ^1(\scrO _X(\Delta -F )\otimes \frak m_p, \scrO _X(F))$
factors through the map $\Ext ^1(\scrO _X(\Delta -F ), \scrO _X(F)\otimes
\frak m_p) \to H^1(\scrO _X(2F-\Delta))$ and that the image is exactly the
natural subgroup $H^1(\scrO _X(2F-\Delta))$ of $\Ext ^1(\scrO _X(\Delta -F
)\otimes \frak m_p, \scrO _X(F))$.
The above has the following geometric interpretation: the locus $U$ in $E_\zeta
^{1,0}$ of $L_-$-unstable sheaves is in fact a section of $E_\zeta ^{1,0}$. If
we blow up this section and then make the elementary transformation, the result
is exactly the set of elements of $E_\zeta ^{0,1}$ corresponding to nonlocally
free sheaves. This set is already a divisor in $E_\zeta ^{0,1}$. There is thus
a morphism from the blowup of $E_\zeta ^{1,0}$ along $U$ to $\frak M_-$ which
is an embedding into $\frak M_-$. Its image $(E_\zeta ^{1,0})'$ in $\frak M_-$
meets $E_\zeta ^{0,1}$ exactly along the divisor in $E_\zeta ^{0,1}$ of
nonlocally free sheaves.
We can now give a picture of the birational map from $\frak M_-$ to $\frak M_+$
in this case. Begin with the subvariety $E_\zeta ^{0,1}$ in $\frak M_-$ and
blow it up. Let $D^{0,1}$ be the exceptional divisor, ruled in two different
ways. As $E_\zeta ^{0,1}$ meets
$(E_\zeta ^{1,0})'$ along a divisor, the proper transform of $(E_\zeta
^{1,0})'$
in the blowup is again
$(E_\zeta ^{1,0})'$. Making the elementary modification along $D^{0,1}$, we
then
blow down $D^{0,1}$ to get a new moduli space. This moduli space then contains
$E_\zeta ^{1,0}$. At this point we can then blow up $E_\zeta ^{1,0}$ and
contract the new exceptional divisor $D^{1,0}$ to obtain $\frak M_+$ (a few
extra details need to be checked here concerning the Kodaira-Spencer class).
Note again the symmetry of the situation. In principle we could hope to carry
through this analysis to the case where $\ell _\zeta >1$ as well, but we run
into trouble with the birational geometry of $\Hilb ^nX$. Somehow the
construction of our auxiliary sequence of moduli spaces has eliminated the
necessity for understanding this birational geometry in detail.
\section{4. Mixed stability and mixed moduli spaces.}
Our goal in this section is to give a proof of Theorem 3.5 (for an arbitrary
algebraic surface $X$). By way of motivation for our construction, let us
analyze Gieseker semistability more closely. In the notation of the last
section, we suppose that $L_0$ is an ample line bundle lying on a unique wall
$W$ of type $(w,p)$, and let $\zeta _1, \dots, \zeta _n$ be the integral
classes
of type $(w,p)$ defining $W$. Let $V$ be an
$L_0$-semistable rank two sheaf. Thus either $V$ is Mumford $L_0$-stable or it
is Mumford strictly semistable. In the second case, let $\scrO_X(F)\otimes
I_{Z_1}$ be a destabilizing subsheaf and suppose that there is an exact
sequence
$$0 \to \scrO_X(F)\otimes I_{Z_1} \to V \to \scrO_X(\Delta -F)\otimes
I_{Z_2}\to
0.$$
Let $\zeta = 2F-\Delta$. We shall assume that $\zeta =\zeta _i$ for some $i$,
or equivalently that $\zeta$ is not numerically equivalent to zero (i.e\., $V$
is not universally semistable). By assumption
$\mu _{L_0}(V)
\geq
\mu _{L_0}(\scrO_X(F)\otimes I_{Z_1})$, and so $\chi(V) \geq 2\chi
(\scrO_X(F)\otimes I_{Z_1})$. Since $\chi (V) = \chi (\scrO_X(F)\otimes
I_{Z_1})
+ \chi (\scrO_X(\Delta -F)\otimes I_{Z_2})$, we may rewrite this last condition
as
$$\chi(\scrO_X(\Delta -F)\otimes I_{Z_2}) - \chi (\scrO_X(F)\otimes I_{Z_1})
\geq 0.$$
Now from the exact sequence
$$0 \to \scrO_X(F)\otimes I_{Z_1} \to \scrO_X(F) \to \scrO_{Z_1} \to 0,$$
we see that $\chi(\scrO_X(F)\otimes I_{Z_1}) = \chi(\scrO_X(F)) - \ell (Z_1)$,
and similarly $\chi(\scrO_X(\Delta -F)\otimes I_{Z_2}) = \chi(\scrO_X(\Delta
-F)) - \ell(Z_2)$. By Riemann-Roch,
$$\align
\chi(\scrO_X(\Delta -F)) - \chi(\scrO_X(F)) &= \frac12((\Delta -F)^2 -
(\Delta -F)\cdot K_X - F^2 + F\cdot K_X)\\
&= \frac12(\Delta ^2 - 2\Delta \cdot F + \zeta \cdot K_X) \\
&= \frac12\zeta\cdot(K_X-\Delta)=t.
\endalign$$
Thus we have the following conditions on $Z_1$ and $Z_2$:
$$\align
\ell (Z_2) - \ell (Z_1) &\leq t;\\
\ell (Z_2) + \ell (Z_1) &= \ell _\zeta,
\endalign$$
and so $2\ell (Z_2) \leq \ell _\zeta +t$.
Setting $k= \dsize\fracwithdelims[]{\ell _\zeta +t}2$, we have $\ell (Z_2) \leq
k$. Applying a similar analysis to a subsheaf of the form $\scrO_X(\Delta -F)
\otimes I_{Z_1}$ shows that, if there is such a subsheaf, with a torsion free
quotient $\scrO_X(F)\otimes I_{Z_2}$, then
$$\ell (Z_2)\leq \frac{\ell _\zeta - t}2 = \ell _\zeta - \frac{\ell _\zeta
+t}2.$$ In particular, if $\dsize\frac{\ell _\zeta +t}2$ is not an integer,
then
this condition becomes $\ell (Z_2) \leq \ell _\zeta - k-1$. Thus, provided
$\dsize\frac{\ell _\zeta +t}2$ is not an integer for every $\zeta$ defining the
wall $W$ (i.e\. $t$ is not $\zeta$-critical for every $\zeta$), $V$ is $(L_0,
\zeta, k)$-semistable for
$k =\dsize\fracwithdelims[]{\ell _\zeta +t}2$ and indeed $V$ is $(L_0,
\boldsymbol\zeta, \bold k)$-semistable, where $\bold k$ is defined in the
obvious way. Conversely, assuming that $t$ is not $\zeta$-critical for every
$\zeta$,
$V$ is Gieseker $L_0$-semistable, indeed Gieseker $L_0$-stable, if it is $(L_0,
\boldsymbol\zeta, \bold k)$-semistable for $\bold k$ as above.
We would like to produce a similar condition where $t$ is allowed to be any
rational number which is not $\zeta$-critical. One way to think of
this problem is to consider the analogous problem where we replace $\Delta$ by
$\Delta + 2 \Xi$ and make the corresponding change in $c$, so that $\Delta$ and
$p$ remain the same. This corresponds to twisting $V$ by $\scrO_X(\Xi)$, and
$t$
is replaced by
$t - \zeta \cdot \Xi$. In particular, we see that the notion of Gieseker
stability is rather sensitive to twisting by a line bundle. Moreover if $W$ is
defined by exactly one
$\zeta$ such that there exists a divisor $\Xi$ with $\zeta \cdot \Xi=1$, for
example if $\zeta$ is primitive and $p_g(X)=0$, it is easy to see that we can
construct the appropriate moduli spaces as Gieseker moduli spaces
corresponding to twists of $V$ by various multiples of
$\Xi$. In general however we will need to consider a problem which is roughly
analogous to allowing twists of $V$ by a $\Bbb Q$-divisor $\Xi$. This is the
goal of the following definition of mixed stability:
\definition{Definition 4.1} Let $X$ be an algebraic surface and let $L_0$ be an
ample line bundle on $X$. Fix line bundles $H_1$ and $H_2$ on $X$ and positive
integers $a_1$ and $a_2$. For every torsion free sheaf $V$ on $X$ of rank $r$,
define
$$p_{V; H_1, H_2, a_1, a_2}(n)= \frac{a_1}{r}\chi(V\otimes H_1 \otimes L_0^n) +
\frac{a_2}{r}\chi(V\otimes H_2 \otimes L_0^n).$$
A torsion free sheaf $V$ is {\sl$(H_1, H_2, a_1, a_2)$ $L_0$-stable\/} if,
for all subsheaves $W$ of $V$ with $0< \rank W < \rank V$ and for all $n \gg
0$,
$$p_{V; H_1, H_2, a_1, a_2}(n) > p_{W; H_1, H_2, a_1, a_2}(n).$$
$(H_1, H_2, a_1, a_2)$ $L_0$-semistable and unstable are defined similarly.
\enddefinition
The usual arguments show the following:
\lemma{4.2} If $V$ is $(H_1, H_2, a_1, a_2)$ $L_0$-stable, then it is simple.
\qed
\endstatement
In the case of rank two on a surface $X$ (which is the only case
which shall concern us), $V$ is $(H_1, H_2, a_1, a_2)$ $L_0$-stable if and only
if, for all rank one subsheaves $W$, and for all $n\gg 0$, we have
$$a_1(\chi(V\otimes H_1 \otimes L_0^n) - 2\chi(W\otimes H_1 \otimes L_0^n))+
a_2(\chi(V\otimes H_2 \otimes L_0^n) - 2\chi(W\otimes H_2 \otimes L_0^n)) >0.$$
In particular, if $V$ is $(H_1, H_2, a_1, a_2)$ $L_0$-stable then either
$V\otimes
H_1$ or $V\otimes H_2$ is stable, and a similar statement holds for
semistability. A short calculation shows that the coefficient of
$n$ in the above expression (which is a degree two polynomial in $n$) is
$(a_1+a_2)(L_0\cdot (c_1(V) - 2F))$ and that the constant term is
$$(a_1+a_2)(\chi(V) - 2\chi(W)) + a_1H_1\cdot (c_1(V) - 2F) + a_2H_2\cdot
(c_1(V)
- 2F).$$ Thus $V$ is $(H_1, H_2, a_1, a_2)$ $L_0$-stable (resp\. semistable)
if
and only if it is either Mumford $L_0$-stable or Mumford strictly semistable
and
the above constant term is positive (resp\. nonnegative). It is easy to see,
comparing this with the discussion at the beginning of this section, that
formally
this is the same as requiring that $V\otimes \Xi$ is (Gieseker)
$L_0$-stable or semistable, where $\Xi$ is the $\Bbb Q$-divisor
$$\frac{a_1}{a_1+a_2}H_1 + \frac{a_2}{a_1+a_2}H_2.$$ Thus for example taking
$H_2=0$ and replacing $H_1$ by a positive integer multiple we see that we can
take for $\Xi$ an arbitrary $\Bbb Q$-divisor.
Let us explicitly relate mixed stability to our previous notion of $(L_0,
\boldsymbol \zeta, \bold k)$-semistability:
\lemma{4.3} Given $\Delta$ and $c$ and the corresponding $w$ and $p$, let $L_0$
be
an ample divisor lying on a unique wall of type $(w,p)$ and let $V$ be a rank
two torsion free sheaf with $c_1(V) = \Delta$ and $c_2(V)=c$. Let $\Xi$ be the
$\Bbb Q$-divisor $\dsize \frac{a_1}{a_1+a_2}H_1 + \frac{a_2}{a_1+a_2}H_2$ and
suppose that the rational number $t_i = \frac12\zeta_i\cdot(K_X-\Delta) -
\zeta_i
\cdot \Xi$ is not $\zeta_i$-critical for every $\zeta_i$ of type $(w,p)$
defining
$W$. Then, with $t=t_1$,
$V$ is $(L_0, \boldsymbol \zeta, \bold k(t))$-semistable
if and only if it is $(H_1, H_2, a_1, a_2)$
$L_0$-semistable if and only if it is $(H_1, H_2, a_1, a_2)$
$L_0$-stable.
\endstatement
\proof Using the additivity of the polynomials $p_{V; H_1, H_2, a_1, a_2}$ over
exact sequences, it is easy to check that $V$ is $(H_1, H_2, a_1, a_2)$
$L_0$-semistable if and only if it is Mumford $L_0$-semistable, and for every
Mumford destabilizing subsheaf of the form $\scrO_X(F)\otimes I_{Z_1}$, either
$V$ is universally semistable or we have
$$\chi (V) - 2\chi (\scrO_X(F)\otimes I_{Z_1}) - \zeta _i\cdot \Xi >0,$$
where $\zeta_i = 2F-\Delta$. Using our calculations above, this works out to
$$\ell (Z_2) - \ell (Z_1) \leq \frac12\zeta_i\cdot(K_X-\Delta) - \zeta_i
\cdot \Xi =t_i.$$
Equivalently since $\ell (Z_1) + \ell (Z_2) = \ell _{\zeta _i}$, this becomes
$\ell (Z_2) \leq \dsize \fracwithdelims[]{\ell _{\zeta _i}+t_i}2$. Thus $V$ is
$(H_1, H_2, a_1, a_2)$
$L_0$-semistable if and only if it is $(L_0,
\boldsymbol \zeta,
\bold k(t))$-semistable. Moreover, since $t$ is not $\zeta_i$-critical, the
inequalities are automatically strict, so that $V$ is also $(H_1, H_2, a_1,
a_2)$
$L_0$-stable.
\endproof
Now choosing a $\Xi_0$ such that $\zeta _1\cdot \Xi _0 \neq 0$, every
rational number $t$ is of the form $\frac12\zeta_1\cdot(K_X-\Delta) - \zeta_1
\cdot r\Xi_0$ for some rational number $r$. Thus Theorem 3.5 will follow
from Lemma 4.3 and from the more general result below:
\theorem{4.4} Let $X$ be an algebraic surface $X$ and let $L_0$ be an ample
line
bundle on $X$. Given a divisor $\Delta$ and an integer $c$, line bundles $H_1$
and $H_2$ on
$X$ and positive integers $a_1$ and $a_2$, suppose that every rank
two torsion free sheaf
$V$ with $c_1(V) = \Delta$, $c_2(V) = c$ which is
$(H_1, H_2, a_1, a_2)$ $L_0$-semistable is actually $(H_1, H_2, a_1, a_2)$
$L_0$-stable. Then there exists a projective coarse moduli space
$\frak M _{L_0}(\Delta, c;H_1, H_2, a_1, a_2)$ of isomorphism classes of rank
two torsion free sheaves
$V$ with $c_1(V) = \Delta$, $c_2(V) = c$, which are
$(H_1, H_2, a_1, a_2)$ $L_0$-semistable.
\endstatement
\proof The argument will follow the arguments in \cite{13} as closely as
possible,
and we shall assume a familiarity with that paper.
Suppose that $V$ is $(H_1, H_2, a_1, a_2)$ $L_0$-semistable. Then either
$V\otimes H_1$ or $V\otimes H_2$ is $L_0$-semistable, and thus by \cite{13},
Lemma 1.3 the set of all such $V$ is bounded. We may thus choose an $n$ such
that, for all
$V$ which are $(H_1, H_2, a_1, a_2)$ $L_0$-semistable, $V\otimes H_i \otimes
L_0^n$ is generated by its global sections and has no higher cohomology, for
$i=1,2$. Fix such an $n$ for the moment, and let $d_i = h^0(V\otimes H_i
\otimes
L_0^n)$. Then
$d_i$ is independent of $V$ and $V$ is a quotient of $(H_i^{-1}\otimes
L_0^{-n})^{\oplus d_i}$. Let $Q_i$ be the open subset of the corresponding Quot
scheme associated to $(H_i^{-1}\otimes L_0^{-n})^{\oplus d_i}$ consisting of
quotients which are rank two torsion free sheaves $V_i$ with $c_1(V_i) =
\Delta$
and
$c_2(V_i) = c$, and such that $V_i\otimes H_i \otimes L_0^n$ is generated by
its
global sections and has no higher cohomology. We will write a point of $Q_i$ as
$V_i$, suppressing the surjection
$(H_i^{-1}\otimes L_0^{-n})^{\oplus d_i} \to V_i$. Inside
$Q_1\times Q_2$, we have the closed subscheme
$I_0$ consisting of quotients $V_1$ and $V_2$ such that $\dim \Hom (V_1, V_2)
\geq 1$. There is also the open subvariety $I_0'$ of $I_0$ consisting of $(V_1,
V_2)$ with $\dim \Hom (V_1, V_2) =1$. Using the universal sheaves $\Cal U_i$
over $X\times Q_i$, we can construct a $\Cee ^*$ bundle $I$ over $I_0'$ whose
points are $(V_1, V_2, \varphi)$, where $\varphi\: V_1 \to V_2$ is a nonzero
homomorphism, unique up to scalars.
For $i=1,2$, let
$E_i$ be a fixed vector space of dimension equal to $d_i = h^0(V\otimes H_i
\otimes L_0^n)$. Fix once and for all an isomorphism $(H_i^{-1}\otimes
L_0^{-n})^{\oplus d_i} \cong (H_i^{-1}\otimes L_0^{-n})\otimes E_i$.
A surjection $(H_i^{-1}\otimes L_0^{-n})^{\oplus d_i}\to V_i$ then gives a map
$E_i \to H^0(V_i\otimes H_i \otimes L_0^n)$ and via such a surjection a basis
$v_1,\dots, v_{d_1}$ of $E_1$ gives $d_1$ sections of $V_1
\otimes H_1 \otimes L_0^n$ and similarly for a basis $w_1, \dots, w_{d_2}$ of
$E_2$. Moreover
$GL(d_i)$ acts on $(H_i^{-1}\otimes L_0^{-n})^{\oplus d_i}$ and on $Q_i$. By
the universal property of the Quot scheme, this action extends to a
$GL(d_i)$-linearization of the universal sheaf $\Cal U_i$ over $X\times Q_i$.
Thus there is a right action of $GL(d_1) \times GL(d_2)$ on $I$, and it is easy
to see that the elements $(\lambda\Id, \lambda\Id)$ act trivially. Let
$F_i$ be the fixed vector space
$H^0(\Delta \otimes H_i^2 \otimes L_0^{2n})$, and $F$ the fixed vector space
$H^0(\Delta \otimes H_1 \otimes H_2 \otimes L_0^{2n})$. Let
$$U= \Hom(\bigwedge ^2E_1, F_1) \oplus \Hom(\bigwedge ^2E_2, F_2)\oplus \Hom
(E_1\otimes E_2, F).$$
(The factor $\Hom (E_1\otimes E_2, F)$ is there to make sure that the
destabilizing subsheaves for $V\otimes H_1$ and $V\otimes H_2$ are
in fact the same.) Note that $GL(d_1) \times GL(d_2)$ operates on
the right on $U$ and $\Pee U$. For example, the pair $(\lambda\Id, \mu\Id)$
acts on the triple $(T_1, T_2, T) \in U$ via
$(T_1, T_2, T) \mapsto (\lambda ^2T_1, \mu ^2T_2, \lambda\mu T)$.
Thus $(A_1, A_2)$ acts trivially on $\Pee U$ if and only if
$(A_1, A_2) = (\lambda\Id, \lambda \Id)$. Given a quintuple
$\underline{V} = (V_1, V_2,\psi _1, \psi_2, \varphi)$, where
$V_i \in Q_i$, $\psi _i\: E_i \to H^0(V_i \otimes H_i \otimes L_0^n)$
is an isomorphism, and $\varphi\: V_1 \to V_2$ is a nonzero map,
we will define a point
$(T_1(\underline{V}), T_2(\underline{V}), T(\underline{V})) \in \Pee U$.
To do so, fix an isomorphism $\alpha _2 \: \det V_2 \to \scrO_X(\Delta)$,
and set $\alpha _1 = \alpha _2 \circ \det\varphi$. (Thus $\alpha_1 = 0$
if $\varphi$ is not an isomorphism.) Given $v,v' \in E_1$ and
$w, w' \in E_2$, identify $v,v'$ with their images in
$H^0(V_i \otimes H_1 \otimes L_0^n)$ and similarly for $w,w'$, and let
$$\align
T_1(\underline{V})(v\wedge v') &= \alpha _1(v\wedge v') =\alpha
_2\circ
\det\varphi (v\wedge v') \in H^0(\Delta \otimes H_1^2\otimes L_0^{2n});\\
T_2(\underline{V})(w\wedge w') &= \alpha _2(w\wedge w') \in H^0(\Delta \otimes
H_2^2\otimes L_0^{2n});\\
T(\underline{V})(v\otimes w) &= \alpha _2(\varphi(v)\wedge w) \in H^0(\Delta
\otimes H_1 \otimes H_2 \otimes L_0^{2n}).
\endalign$$
Changing $\alpha _2$ by a nonzero scalar $\lambda$ multiplies
$(T_1(\underline{V}), T_2(\underline{V}), T(\underline{V}))$ by $\lambda$, so
that
the induced element of $\Pee U$ is well defined. Similarly, if we replace
$\varphi$ by
$\lambda\varphi$, then $(T_1(\underline{V}), T_2(\underline{V}),
T(\underline{V}))$ is replaced by $(\lambda ^2T_1(\underline{V}),
T_2(\underline{V}),
\lambda T(\underline{V}))$. It is easy to check that the map $\underline{V}
\mapsto T(\underline{V})$ induces a morphism from
$I$ to
$\Pee U$ which is $GL(d_1) \times GL(d_2)$-equivariant. Further note that we
can
define
$(T_1(\underline{V}), T_2(\underline{V}), T(\underline{V}))$ more generally if
we are given the data $\underline{V}$ of two rank two torsion free sheaves
$V_1$
and $V_2$ with $\det V_i = \Delta$, a morphism $\varphi\: V_1 \to V_2$, and
linear maps
$\psi _i\: E_i \to H^0(V_i \otimes H_i \otimes L_0^n)$, not necessarily
isomorphisms, although it is possible for $(T_1(\underline{V}),
T_2(\underline{V}), T(\underline{V}))$ to be zero in this case.
We have not yet introduced the extra parameters $a_1$ and $a_2$. To do so,
define $G(a_1, a_2) \subset GL(d_1) \times GL(d_2)$ as follows:
$$G(a_1, a_2) = \{\, (A_1,A_2)\mid \det A_1^{a_1}\det A_2^{a_2} = \Id\,\}.$$
Thus unlike Thaddeus we don't change the polarization or the linearization but
the actual group which we use to determine stability; still our construction
could
probably be interpreted in his general framework. Fixing $a_1$ and $a_2$ for
the
rest of the discussion, we shall denote $G(a_1, a_2)$ by $G$. Since $a_1$ and
$a_2$ are positive, the matrix $(\lambda\Id, \lambda\Id)$ lies in $G$ if and
only
if
$\lambda$ is an $m^{\text{th}}$ root of unity, where $m = a_1d_1+a_2d_2$. Thus
a
quotient of
$G$ by a finite group acts faithfully on $\Pee U$. Moreover, the problem
of finding a good quotient of
$\Pee U$ (for an appropriate open subset of $\Pee U$) for $G$ is the same as
that of finding a good quotient of $\Pee U$ for $GL(d_1) \times GL(d_2)$, since
$$G\cdot \Cee ^*(\Id, \Id) = GL(d_1) \times GL(d_2).$$
This last statement follows since $G$ clearly contains $SL(d_1) \times SL(d_2)$
and since $\Cee ^*\times \Cee ^*$ is generated by its diagonal subgroup and by
the
subgroup
$$\{\,(\lambda, \mu):
\lambda ^{a_1d_1}\mu ^{a_2d_2} = 1\,\}.$$
We may thus apply the general machinery of GIT to the group $G$ acting on $\Pee
U$. A one parameter subgroup of $G$ is given by a basis $\{v_i\}$ of $E_1$, a
basis $\{w_k\}$ of $E_2$ and weights $n_i$, $m_k\in \Zee$, such that
$v_i^\lambda
= \lambda ^{n_i}v_i$, $w_k ^\lambda = \lambda ^{m_k}w_k$, and
$$a_1\sum _in_i + a_2 \sum _km_k = 0.$$
We shall always arrange our choice of basis so that $n_1 \leq n_2 \leq \dots
\leq n_{d_1}$ and $m_1 \leq m_2 \leq \dots \leq m_{d_2}$. Given
$(T_1, T_2, T)
\in U$ and a one parameter subgroup of $G$ as above, we see that $\lim
_{\lambda
\to 0}(T_1, T_2, T)^\lambda =0$ if and only if $T_1(v_i\wedge v_j) = 0$ for
every
pair of indices
$i,j$ such that $n_i+n_j \leq 0$, $T_2(w_k\wedge w_\ell) = 0$ for every pair of
indices $k, \ell$ such that $m_k+m_\ell \leq 0$, and $T(v_i\otimes w_j) = 0$
for
every pair $i,k$ such that $n_i+m_k\leq 0$. Likewise the condition that $\lim
_{\lambda
\to 0}(T_1, T_2, T)^\lambda$ exists is similar, replacing the $\leq$ by strict
inequality. Finally note that if $n_i+n_j \leq 0$, then $n_1+n_j \leq 0$, if
$m_k+m_\ell \leq 0$ then $m_1+m_\ell \leq 0$, and if $n_i+m_k\leq 0$ then
$n_1+m_k\leq 0$ and $n_i+m_1\leq 0$.
We then have the following:
\lemma{4.5}
\roster
\item"{(i)}" Suppose that we are given the data $\underline{V}$ of two rank two
torsion free sheaves $V_1$ and $V_2$ with $\det V_i = \Delta$, a morphism
$\varphi\: V_1 \to V_2$, and a linear map
$E_i \to H^0(V_i \otimes H_i \otimes L_0^n)$, not necessarily an isomorphism.
If $E_i \to H^0(V_i \otimes H_i \otimes L_0^n)$ is not injective for some
$i$ or if $\varphi$ is not an isomorphism, then $(T_1(\underline{V}),
T_2(\underline{V}), T(\underline{V}))$ is either zero or $G$-unstable.
\item"{(ii)}" For $n$ sufficiently large depending only on $\Delta$ and $c$ and
for
$V$ a rank two torsion free sheaf with $\det V = \Delta$ and $c_2(V) = c$,
$V$ is $(H_1, H_2, a_1, a_2)$
$L_0$-unstable if and only if $(T_1(\underline{V}), T_2(\underline{V}),
T(\underline{V}))$ is $G$-unstable for all choices of data $\underline{V}$ such
that $E_i \to H^0(V_i \otimes H_i \otimes L_0^n)$ is injective and $\varphi\:
V_1 \to V_2\cong V$ is an isomorphism, and
$V$ is $(H_1, H_2, a_1, a_2)$
$L_0$-strictly semistable if and only if $(T_1(\underline{V}),
T_2(\underline{V}), T(\underline{V}))$ is $G$-strictly semistable for all such
$\underline{V}$. Thus
$V$ is $(H_1, H_2, a_1, a_2)$
$L_0$-stable if and only if $(T_1(\underline{V}), T_2(\underline{V}),
T(\underline{V}))$ is $G$-stable for all such $\underline{V}$.
\endroster
\endstatement
\proof First let us prove (i). We may assume that $(T_1(\underline{V}),
T_2(\underline{V}), T(\underline{V}))\neq 0$. Suppose for example that $v_1
\in E_1
\mapsto 0
\in H^0(V_1\otimes H_1\otimes L_0^n)$. Complete $v_1$ to a basis of $E_1$ and
choose a basis $\{w_k\}$ for $E_2$. Then
$T_1(\underline{V})(v_1\wedge v_i) = 0$ for all $i$ and
$T(\underline{V})(v_1\otimes w_k) =0$ for all $k$. Define a one parameter
subgroup of $G$ as follows: let $v_1^\lambda = \lambda ^{-N}v_1$, $v_i^\lambda
=
\lambda ^av_i$ for $i>1$, and $w_k^\lambda = \lambda ^bw_k$ for all $k$.
Clearly
$\lim _{\lambda \to 0}(T_1(\underline{V}), T_2(\underline{V}),
T(\underline{V}))^\lambda =0$ provided that $a$ and $b$ are positive, so that
$(T_1(\underline{V}), T_2(\underline{V}), T(\underline{V}))$ is $G$-unstable
provided that the one parameter subgroup so constructed lies in $G$, or on
other
words provided that
$$a_1(-N + a(d_1-1))+ a_2bd_2 =0.$$
It thus suffices to take $a$ an arbitrary positive integer, $b=a_1$, and $N=
a(d_1-1) + a_2d_2$.
The argument in case $\varphi$ has a kernel is similar: in this case let $v_1
\in \Ker \varphi$. Then $T_1(\underline{V})=0$ and
$T(\underline{V})(v_1\otimes
w_k) =0$ for all $k$, so that the previous argument handles this case also.
Next we show (ii). Let $p_{V\otimes H_i}$ be the usual normalized Hilbert
polynomial of $V\otimes H_i$, and similarly for $p_{W\otimes H_i}$, where $W$
is a rank one subsheaf of $V$. Thus $p_{V\otimes H_i}$ and $p_{W\otimes
H_i}$ have the same leading term. Given a polynomial
$p$, let $\Delta p$ denote the difference polynomial. In our case, all of the
polynomials $p$ that occur are quadratic polynomials with the same fixed degree
two term. Thus if $p_1$ and
$p_2$ are two such polynomials, then
$p_1(n) > p_2(n)$ for all $n\gg 0$ if and only if the linear term of $p_1$ is
greater than or equal to the linear term of $p_2$, and if the linear terms are
equal then the constant term of $p_1$ is greater than the constant term of
$p_2$.
In this last case, where the linear terms are also equal, we see that $p_1(n) >
p_2(n)$ for all
$n\gg 0$ if and only if $p_1(n) > p_2(n)$ for some $n$. Finally the linear
term of $p_1$ is greater than or equal to the linear term of $p_2$ if and only
if
$\Delta p_1(n) \geq \Delta p_2(n)$ for all $n$, which we shall write as $\Delta
p_1\geq \Delta p_2$. Thus if $\Delta p_1 \geq \Delta p_2$ and $p_1(n) > p_2(n)$
for some $n$, then $p_1(n) > p_2(n)$ for all $n\gg 0$. If $\Delta p_1 =
\Delta p_2$, then $p_1(n) > p_2(n)$
for some $n$ if and only if $p_1(n) > p_2(n)$ for all $n$.
We shall show that, for sufficiently large $n$, if $V$ is $(H_1, H_2, a_1,
a_2)$
$L_0$-semistable and $\underline{V}$ corresponds to data where $E_i \to
H^0(V\otimes H_i\otimes L_0^n)$ is injective and $\varphi$ is an isomorphism,
then
$(T_1(\underline{V}), T_2(\underline{V}), T(\underline{V}))$ is
$G$-semistable.
Note that $V$ is Mumford semistable. First we may choose
$n$ so that
$V\otimes H_i$ is generated by its global sections and has no higher
cohomology,
and so $\chi (V\otimes H_i \otimes L_0^n) = h^0(V\otimes H_i \otimes L_0^n) =
d_i$. Hence, since $E_i \to H^0(V\otimes H_i\otimes L_0^n)$ is injective, it
is an isomorphism. Let
$W$ be a rank one subsheaf of $V$. Since $V\otimes H_i$ is Mumford semistable,
$\Delta p_{W\otimes H_i} \leq \Delta p_{V\otimes H_i}$. Now the proof of (3) of
Lemma 1.2 in \cite{13} shows that
there exists an $N$ so that, for all $n\geq N$, with $d_i$ as above, if
$W$ is a rank one subsheaf of
$V$ and such that $h^0(W\otimes H_i \otimes L_0^n )
\geq d_i/2$ for at least one
$i$ ($i=1,2$), then in fact $\Delta p_{V\otimes H_i} = \Delta
p_{W\otimes H_i}$ for all such $W$, and thus $\mu _{L_0}(V) = \mu _{L_0}(W)$.
It is then easy to see that there is a twist $W\otimes H_i\otimes L_0^{-k}$,
depending only on $L_0$ and $\Delta$, such that $h^0((V/W)\otimes H_i\otimes
L_0^{-k}) = 0$. The proof of Proposition 3.1 in \cite{13} shows that in this
case
$h^1(W\otimes H_i \otimes L_0^{-k})$ is bounded by $Q$, where $Q$ is some
universal bound for the numbers
$h^1(V\otimes H_i\otimes L_0^{-k})$ as $V\otimes H_i$ ranges over the
appropriate set of
$L_0$-semistable sheaves Thus by (4) of Lemma 1.2 in \cite{13}, the $W$
satisfying the condition that $h^0(W\otimes H_i \otimes L_0^n )
\geq d_i/2$ for at least one $i$ form a bounded family, and we may
choose $n$ so large, depending only on $L_0$, $\Delta$, $c$, such that
$h^j(W\otimes H_i \otimes L_0^n)=0$ for $j\geq 1$ and $i=1,2$.
Now suppose that
$(T_1(\underline{V}), T_2(\underline{V}), T(\underline{V}))$ is $G$-unstable.
Then there exists a one parameter subgroup of $G$ as above such that
$\lim_{\lambda \to 0} (T_1(\underline{V}), T_2(\underline{V}),
T(\underline{V}))^\lambda =0$. Let
$$\align
s_1&= \#\{\, j: T_1(\underline{V})(v_1\wedge v_j) =0\,\} \geq
\max \{\, j: n_1 + n_j \leq 0\,\};\\
s_2 &= \#\{\, j: T_2(\underline{V})(w_1\wedge
w_\ell) =0\,\} \geq
\max \{\, \ell: m_1 + m_\ell \leq 0\,\}.
\endalign$$
Since $a_1\sum _in_i + a_2 \sum _km_k =0$, at least one of $n_1, m_1$ is
negative. By symmetry we may assume that $n_1$ is negative, and that $n_1 \leq
m_1$. Since for
$j\leq s_1$, $v_1\wedge v_j$ is zero as a section of
$\det (V\otimes H_1\otimes L_0^n)$, the sections corresponding to
$v_j$, $1\leq j \leq s_1$, are all sections of a rank one subsheaf $W_1$ of
$V$. Likewise the sections $w_\ell$, $1\leq \ell \leq s_2$, if there are any
such, are all sections of a rank one subsheaf $W_2$ of $V$.
The condition that $T(\underline{V})(v_1\otimes w_1) =0$
insures that $W_1$ and $W_2$ are contained in a saturated
rank one subsheaf $W$, if $s_2\neq 0$, otherwise we shall just take for $W$ the
saturated rank one subsheaf containing $W_1$. Moreover
$h^0(W\otimes H_1\otimes L_0^n)
\geq s_1$ and $h^0(W\otimes H_2\otimes L_0^n) \geq s_2$. Suppose that we show
that
$$a_1(d_1 - 2s_1) + a_2(d_2 - 2s_2) <0.$$
Thus in particular $s_i\geq d_i/2$ for at least one $i$. By our choice of $n$
and the previous paragraph, if $s_i\geq d_i/2$ for at least one $i$,
then $h^0(W\otimes H_i\otimes L_0^n) = \chi (W\otimes H_i\otimes L_0^n)$
and furthermore $\mu _{L_0}(V) = \mu _{L_0}(W)$.
Thus
$$h^0(W\otimes H_i \otimes L_0^n)
= \chi (W\otimes H_i \otimes L_0^n) \geq s_i$$
for $i=1,2$ and so
$p_{V; H_1, H_2, a_1, a_2}(n) < p_{W; H_1, H_2, a_1, a_2}(n)$.
On the other hand, $p_{V; H_1, H_2, a_1, a_2}$ and $p_{W; H_1, H_2, a_1, a_2}$
are two quadratic polynomials with the same linear and quadratic terms
(since $\mu _{L_0}(V) = \mu _{L_0}(W)$),
and $p_{V; H_1, H_2, a_1, a_2}(n) < p_{W; H_1, H_2, a_1, a_2}(n)$
for one value of $n$. Thus the constant term of $p_{W; H_1, H_2, a_1, a_2}$
must be larger than that of $p_{V; H_1, H_2, a_1, a_2}$.
This contradicts the $(H_1, H_2, a_1, a_2)$ $L_0$-semistability of $V$.
To see that $a_1(d_1 - 2s_1) + a_2(d_2 - 2s_2) <0$, let
$$t_1 = \#\{\, j: n_j +m_1 \leq 0\,\} \leq s_1.$$
Here $t_1 \leq s_1$ since $T(\underline{V})(v_j \otimes w_1) = 0$
implies that $v_j$ and $w_1$ are contained in a rank one subsheaf of $V$,
necessarily $W$, and thus that $v_1\wedge v_j = 0$. Let
$$t_2 = \#\{\, \ell: n_1 +m_\ell \leq 0\,\} \leq s_2.$$ We
have assumed that $n_1 \leq m_1$. Then consider the expression
$$a_1\sum _j (n_1+ n_j) + a_2\sum _\ell (n_1 + m_\ell).$$
On the one hand from the definition of the one parameter subgroup we have
$$a_1\sum _j (n_1+ n_j) + a_2\sum _\ell (n_1 + m_\ell)
= a_1d_1n_1 + a_2d_2n_1.$$
On the other hand, to estimate $\sum _j (n_1+ n_j)$, we can ignore the positive
terms where $n_1+n_j \geq 0$ and each of the $s_1$ negative terms are at least
$n_1 + n_1 \geq 2n_1$. Thus $\sum _j (n_1+ n_j) \geq 2s_1n_1$.
Since $n_1 <0$, this term is $\geq 2s_1n_1$.
Also this inequality is strict or $n_1 + n_i \leq 0$ for
every $i$, which would say that every section of $V\otimes H_1 \otimes L_0^n$
is
really a section of $W\otimes H_1 \otimes L_0^n$ contradicting the fact that
$V\otimes H_1 \otimes L_0^n$ is generated by global sections. So $\sum _j (n_1+
n_j) < 2s_1n_1$. Likewise we claim that $\sum _\ell (n_1 + m_\ell) \geq
2s_2n_1$. Here, to estimate $\sum _\ell (n_1 + m_\ell)$, we may ignore the
terms with $n_1 + m_\ell$ positive, leaving $t_2$ terms $n_1+m_\ell$ which are
$\leq 0$, and moreover each such term is at least $n_1+m_1 \geq 2n_1$. Thus
$\sum _\ell (n_1 + m_\ell) \geq 2t_2n_1$, and since $t_2 \leq s_2$ and $n_1
<0$, we have $2t_2n_1 \geq 2s_2n_1$.
Putting this together we have
$$\align
a_1d_1n_1 + a_2d_2n_1 &= a_1\sum _j (n_1+ n_j) + a_2\sum _\ell
(n_1 + m_\ell) \\ &> a_1(2s_1n_1)+ a_2(2s_2n_1),
\endalign$$
so that
$$a_1(d_1 - 2s_1)n_1 + a_2(d_2 - 2s_2)n_1 >0.$$ As $n_1 <0$, we must have
$a_1(d_1 - 2s_1) + a_2(d_2 - 2s_2) <0$, as desired.
We have thus shown that, if $(T_1(\underline{V}), T_2(\underline{V}),
T(\underline{V}))$ is $G$-unstable, then $V$ is $(H_1, H_2, a_1, a_2)$
$L_0$-unstable. A very similar argument
handles the $G$-strictly semi\-stable case.
Now we turn to the converse statement, that if $V$ is $(H_1, H_2, a_1, a_2)$
$L_0$-unstable then $(T_1(\underline{V}), T_2(\underline{V}),
T(\underline{V}))$ is $G$-unstable. Suppose instead that
$$(T_1(\underline{V}), T_2(\underline{V}), T(\underline{V}))$$
is $G$-semistable. Let $W$ be a rank one subsheaf of $V$ such that
$p_{W; H_1, H_2, a_1, a_2}(m) > p_{V; H_1, H_2, a_1, a_2}(m)$ for all $m\gg 0$.
We may assume that the quotient $W' = V/W$ is torsion free.
Thus $p_{W'; H_1, H_2, a_1, a_2}(m) < p_{V; H_1, H_2, a_1, a_2}(m)$
for all $m\gg 0$, and so
$\Delta p_{W'\otimes H_i} \leq \Delta p_{V\otimes H_i}$.
Now we have the map $E_i \to H^0(V\otimes H_i \otimes L_0^n)$.
Consider $E_i \cap H^0(W\otimes H_i \otimes L_0^n)\subseteq E_i$.
Let $\dim E_i \cap H^0(W\otimes H_i \otimes L_0^n) = s_i$.
Suppose first that $a_1(d_1 - 2s_1) + a_2(d_2 - 2s_2) <0$.
We claim that in this case
$(T_1(\underline{V}), T_2(\underline{V}), T(\underline{V}))$ is $G$-unstable,
a contradiction. To see this, choose a basis $v_1, \dots, v_{d_1}$ for
$E_1$ such that
$$v_i \in E_1\cap H^0(W\otimes H_1 \otimes L_0^n)$$
for $i\leq s_1$, and similarly choose a basis $w_1, \dots, w_{d_2}$ for
$E_2$ such that $w_k \in E_2\cap H^0(W\otimes H_2 \otimes
L_0^n)$ for $i\leq s_2$. Thus, if $i,j\leq s_1$ then
$T_1(\underline{V})(v_i\wedge v_j) =0$; if $k, \ell \leq s_2$ then $T_2(
\underline{V})(w_k\wedge w_\ell) = 0$; if $i\leq s_1$ and $k\leq s_2$ then
$T(\underline{V})(v_i\otimes w_k) =0$.
We will try to find a one parameter subgroup of $G$ of the form
$$v_i^\lambda = \cases \lambda ^{-m}v_i, &\text{for $i \leq s_1$;} \\
\lambda ^nv_i , &\text{for $i > s_1$,}
\endcases$$ and similarly
$$w_k^\lambda = \cases \lambda ^{-m}w_k, &\text{for $i \leq s_2$;} \\
\lambda ^nw_k , &\text{for $i > s_2$.}
\endcases$$
It is easy to check that $\lim _{\lambda \to 0}(T_1(\underline{V}),
T_2(\underline{V}), T(\underline{V}))^\lambda=0$ if and only if $n>m$.
What we must arrange is the condition
$$a_1(-ms_1 + n(d_1-s_1)) + a_2(-ms_2 +n(d_2-s_2)) = 0.$$
Now consider the linear function with rational coefficients
$$f(t) = a_1(-s_1 + t(d_1-s_1)) + a_2(-s_2 +t(d_2-s_2)).$$
Since the coefficient of $t$ is strictly positive $f(t)$ is increasing, and
$$\align f(1) &= a_1(-s_1 + (d_1-s_1)) + a_2(-s_2 +(d_2-s_2))\\
&= a_1(d_1 - 2s_1) + a_2(d_2 - 2s_2) <0.
\endalign$$
Thus there is a rational number $t= n/m>1$ such that $f(t) = 0$,
and this gives the desired choice of $n$ and $m$.
Thus if $a_1(d_1 - 2s_1) + a_2(d_2 - 2s_2) <0$, then
$(T_1(\underline{V}), T_2(\underline{V}), T(\underline{V}))$ is $G$-unstable,
contradicting our hypothesis.
The other possibility is that $a_1(d_1 - 2s_1) +
a_2(d_2 - 2s_2) \geq 0$. In this case $d_i \geq 2 s_i$ for at least one $i$.
Recalling that we have the quotient $W'$ of $V$ by $W$, it then follows that
for
such an $i$ the image of $E_i$ in $H^0(W' \otimes H_i \otimes L_0^n)$ must have
dimension at least $d_i/2$. Arguing as in Proposition 3.2 of \cite{13},
it then follows from Lemma 1.2 of \cite{13} that
$\Delta p_{W'\otimes H_i} = \Delta p_{V\otimes H_i}$ and so that $V$ is
Mumford $L_0$-semistable and $\mu _{L_0}(V) = \mu _{L_0}(W)$.
Moreover, after enlarging $n$ if necessary (independently of $V$)
we may assume that $h^j(V\otimes H_i \otimes L_0^n) = 0$ for $j>0$.
In particular, $d_i = \dim H^0(V\otimes H_i \otimes L_0^n)$ for $i=1,2$,
and $E_i \to H^0(V \otimes H_i \otimes L_0^n)$ is an isomorphism;
so $s_i = h^0(W \otimes H_i \otimes L_0^n)$.
As $\mu _{L_0}(V) = \mu _{L_0}(W)$, the polynomials
$p_{W; H_1, H_2, a_1, a_2}$ and $p_{V; H_1, H_2, a_1, a_2}$ have
the same terms in degree one and two, and thus since
$p_{W; H_1, H_2, a_1, a_2}(m) > p_{V; H_1, H_2, a_1, a_2}(m)$ for some $m$
the same is true for all $m$, in particular for $m=n$. Moreover,
for a general choice of a smooth curve $C$ in the linear system corresponding
to $L_0$, there is a fixed bound on the line bundle $W\otimes H_i|C$.
A standard argument as in the proof of (2) of Lemma 1.2 of \cite{13}
shows that, for $n$ sufficiently large but independent of $V$,
we have $H^2(W\otimes H_i\otimes L_0^n) =0$. Thus
$s_i = h^0(W \otimes H_i \otimes L_0^n) \geq p_{W\otimes H_i}(n)$.
It follows that
$$\align
a_1(d_1 - 2s_1) + a_2(d_2 - 2s_2)&\leq a_1(d_1 - 2p_{W\otimes H_1}(n)) +
a_2(d_2 - 2p_{W\otimes H_2}(n))\\
&= 2(p_{V; H_1, H_2, a_1, a_2}(n) - p_{W; H_1, H_2, a_1, a_2}(n)) <0.
\endalign$$
This contradicts the assumption that $a_1(d_1 - 2s_1) +
a_2(d_2 - 2s_2) \geq 0$. It then follows that $(T_1(\underline{V}),
T_2(\underline{V}), T(\underline{V}))$ is $G$-unstable.
The strictly semistable case is similar.
\endproof
We may now finish the proof of Theorem 4.4. Let $\Pee U_{\text{ss}}$ be the set
of $G$-semistable points of $\Pee U$. Let $I_{\text{ss}}$ be the inverse image
of $\Pee U_{\text{ss}}$ under the morphism $I \to \Pee U$. Since every
semistable sheaf is stable, $I_{\text{ss}}$ is a $\Cee ^*$-bundle over
its image in $Q_1\times Q_2$. Moreover the representable functor
corresponding to $I_{\text{ss}}$ is easily seen to be formally smooth
over the moduli functor. Arguments very similar to those for Lemma 4.3 and 4.5
of
\cite{13} show that the morphism
$I_{\text{ss}} \to \Pee U_{\text{ss}}$ is one-to-one and proper, and thus in
particular finite. Thus we may construct a quotient $\frak M _{L_0}(\Delta,
c;H_1, H_2, a_1, a_2)$ of $I_{\text{ss}}$ by $G$. This quotient maps in a
one-to-one and proper way to the GIT quotient of $\Pee U_{\text{ss}}$ and is
therefore projective. By the discussion at the beginning of the proof of
Theorem
4.4 the points of $\frak M _{L_0}(\Delta, c;H_1, H_2, a_1, a_2)$ may be
identified with isomorphism classes of $(H_1, H_2, a_1, a_2)$
$L_0$-semistable rank two sheaves. Standard arguments then show that $\frak M
_{L_0}(\Delta, c;H_1, H_2, a_1, a_2)$ has the usual properties of a coarse
moduli space.
\endproof
\section{5. The transition formula for Donaldson polynomial invariants.}
From now on, we will assume that the surface $X$ is rational with
$-K_X$ effective, and will study the transition formula of
Donaldson polynomial invariants:
$$\delta ^X_{w,p}(\Cal C_+, \Cal C_-)
= D^X _{w,p}(\Cal C_+) - D^X _{w,p}(\Cal C_-)$$
where $\Cal C_-$ and $\Cal C_+$ are two adjacent chambers separated by
a single wall $W^{\zeta}$ of type $(w, p)$
or equivalently of type $(\Delta, c)$. For simplicity, we assume that
the wall $W^{\zeta}$ is only represented by $\pm \zeta$ since
the general case just involves additional notation. We use $\frak
M_0^{(k)}$ to stand for the moduli space $\frak M_0^{(\boldsymbol \zeta, k)}$.
When $\ell_\zeta = 0$, we also assume that
$$h(\zeta) = h^1(X; \scrO _X(2F-\Delta) \neq 0$$
(see Corollary 2.7). The special case when $\ell_\zeta = h(\zeta) = 0$
will be treated in Theorem 6.1. By Theorem 3.9 and Lemma 3.2 (ii),
we have the following diagram:
$$\matrix
& &{\widetilde{\frak M}}_0^{(\ell_\zeta)}&&&&\ldots&&&&{\widetilde{\frak
M}}_0^{(0)}&&\\ &\swarrow&&\searrow&&\swarrow&&\searrow&&\swarrow&&\searrow\\
\frak M_0^{(\ell_\zeta)} & & && \frak M_0^{(\ell_\zeta - 1)} &
&&& \frak M_0^{(0)} & &&& \frak M_0^{(-1)}\\
\Vert \quad&&&&&&&&&&&& \Vert \quad\\
\frak M_- &&&&&&&&&&&&\frak M_+\\
\endmatrix
$$
where the morphism ${\widetilde{\frak M}}_0^{(k)} \to \frak M_0^{(k)}$
is the blowup of $\frak M_0^{(k)}$ at $E_\zeta^{\ell_\zeta -k, k}$,
and the morphism ${\widetilde{\frak M}}_0^{(k)} \to \frak M_0^{(k - 1)}$
is the blowup of $\frak M_0^{(k - 1)}$ at $E_{-\zeta}^{k, \ell_\zeta -k}$.
Next, we collect and establish some notations. Recall that in section 2
we have constructed the bundle $\Cal E_\zeta^{\ell_\zeta -k, k}$ over
$ H_{\ell_\zeta - k} \times H_{k}$, where $H_k = \Hilb^k X$.
\medskip
\noindent
{\bf Notation 5.1}. Let $\zeta$ define a wall of type $(w, p)$.
\roster
\item"{(i)}" $\lambda_k$ is the tautological line bundle over
$E_\zeta^{\ell_\zeta - k, k} = \Pee((\Cal E_\zeta^{\ell_\zeta -k,
k})\spcheck)$; for simplicity, we also use $\lambda_k$ to denote its first
Chern class;
\item"{(ii)}" $\rho_k: X \times E_\zeta^{\ell_\zeta - k, k} \to
X \times H_{\ell_\zeta - k} \times H_{k}$ is the natural projection;
\item"{(iii)}" $p_k: \widetilde{\frak M}_0^{(k)} \to \frak M_0^{(k)}$ is the
blowup of $\frak M_0^{(k)}$ at $E_\zeta^{\ell_\zeta -k, k}$;
\item"{(iv)}" $q_{k - 1}: \widetilde{\frak M}_0^{(k)} \to \frak M_0^{(k-1)}$ is
the contraction of $\widetilde{\frak M}_0^{(k)}$ to $\frak M_0^{(k-1)}$;
\item"{(v)}" $\Cal N_k$ is the normal bundle of $E_\zeta^{\ell_\zeta -k, k}$
in $\frak M_0^{(k)}$; by Proposition 3.7, we have
$$\Cal N_k = \rho_k^*\Cal E_{-\zeta}^{k, \ell_\zeta -k}
\otimes \lambda_k^{-1};$$
\item"{(vi)}" $D_k = \Pee(\Cal N_k\spcheck)$ is the exceptional divisor in
$\widetilde{\frak M}_0^{(k)}$;
\item"{(vii)}" $\xi_k = \Cal O_{\widetilde{\frak M}_0^{(k)}}(-D_k)|D_k$ is
the tautological line bundle on $D_k$; again, for simplicity, we also use
$\xi_k$ to denote its first Chern class;
\item"{(viii)}" $\mu^{(k)}(\alpha) = -{1 \over 4} p_1(\Cal U^{(k)})/\alpha$
where
$\alpha \in H_2(X; \Zee)$ and $\Cal U^{(k)}$ is a universal sheaf
over $X \times \frak M_0^{(k)}$.
Let $\mu^{(\ell_\zeta)}(\alpha) = \mu_-(\alpha)$ and
that $\mu^{(-1)}(\alpha) = \mu_+(\alpha)$.
\item"{(ix)}" $\nu^{(k)} = -{1 \over 4} p_1(\Cal U^{(k)})/x$ where
$x \in H_0(X; \Zee)$ is the natural generator. Let
$\nu^{(\ell_\zeta)}= \nu_-$ and that $\nu^{(-1)} = \nu_+$.
\endroster
Note that, in (viii) and (ix) above, the sheaf $\Cal U^{(k)}$ is only defined
locally in the classical topology. However, since it is defined on the level of
the Quot scheme a straightforward argument shows that $p_1(\Cal U^{(k)})$ is a
well-defined element in the rational cohomology of $X \times \frak M_0^{(k)}$,
at least in the complement of the universally semistable sheaves. In case there
are universally semistable sheaves, then the work of Li \cite{21} extends the
$\mu$-map to $\frak M_0^{(k)}$, at least for the two-dimensional algebraic
classes. We can then extend the $\mu$-map to the 4-dimensional class via a
blowup
formula due to O'Grady (unpublished). Moreover, there
is a universal sheaf
$\Cal V_k$ over
$X \times E_\zeta^{\ell_\zeta - k, k}$. In what follows, we shall work as if
there were a universal sheaf $\Cal U^{(k)}$, and leave it to the reader to
check
that our final Chern class calculations can be verified directly even when no
universal sheaf exists.
In the following lemma, we study the restrictions of
$p_k^*\mu^{(k)}(\alpha)$ and $p_k^*\nu^{(k)}$ to $D_k$.
\lemma{5.2} Let $\alpha \in H_2(X; \Zee)$ and $a = (\zeta \cdot \alpha)/2$.
Let $\tau_1$ and $\tau_2$ be the projections of $E_\zeta^{\ell_\zeta - k, k}$
to $H_{\ell_\zeta - k}$ and $H_k$ respectively. Then,
$$\align
&(\operatorname{Id} \times p_{k})^*c_1(\Cal U^{(k)})|(X \times D_k)
= \pi _1^*\Delta + (p_k|D_k)^*\lambda_k\\
&p_k^*\mu^{(k)}(\alpha)|D_k =
(p_k|D_k)^*\left[\tau_1^*([{\Cal Z_{\ell_\zeta - k}}]/\alpha)
+ \tau_2^*([{\Cal Z_{k}}]/\alpha) - a \lambda_k\right]\\
&p_k^*\nu^{(k)}|D_k = {1 \over 4} (p_k|D_k)^* \left [
4 \tau_1^*([{\Cal Z_{\ell_\zeta - k}}]/x) + 4 \tau_2^*([{\Cal Z_{k}}]/x)
- \lambda_k^2 \right ]. \\
\endalign$$
\endstatement
\proof Note that $\Cal U^{(k)}|X \times E_\zeta^{\ell_\zeta - k, k}
= \Cal V_k$, where the sheaf $\Cal V_k$ is constructed by Proposition 2.8
and sits in the exact sequence:
$$0 \to \pi _1^*\scrO _X(F )\otimes \rho_k ^*\pi
_{1,2}^*I_{\Cal Z_{\ell_\zeta - k}} \otimes \pi_2^*\lambda_k
\to \Cal V_k \to \pi _1^*\scrO _X(\Delta -F )
\otimes \rho_k ^*\pi _{1,3}^*I_{\Cal Z_{k}} \to 0.$$
Thus, $c_1(\Cal V_k) = \pi _1^*\Delta + \pi_2^*\lambda_k$ and
$(\operatorname{Id} \times p_{k})^*c_1(\Cal U^{(k)})|(X \times D_k)
= \pi _1^*\Delta + (p_k|D_k)^*\lambda_k$. Moreover,
$c_2(\Cal V_k) = \rho_k ^*\pi_{1,2}^*[{\Cal Z_{\ell_\zeta - k}}]
+ \rho_k ^*\pi_{1,3}^*[{\Cal Z_{k}}] + (\pi _1^*F + \pi_2^*\lambda_k)
\cdot \pi _1^*(\Delta - F)$.
Since $p_k^*\mu^{(k)}(\alpha)|D_k =
(p_k|D_k)^*[\mu^{(k)}(\alpha)|E_\zeta^{\ell_\zeta - k, k}]
= (p_k|D_k)^*[-{1 \over 4} p_1(\Cal V_k)/\alpha]$, we have
$$p_k^*\mu^{(k)}(\alpha)|D_k =
(p_k|D_k)^*\left[\tau_1^*([{\Cal Z_{\ell_\zeta - k}}]/\alpha)
+ \tau_2^*([{\Cal Z_{k}}]/\alpha) - a \lambda_k\right].$$
Similarly, $p_k^*\nu^{(k)}|D_k = {1 \over 4} (p_k|D_k)^* \left [
4 \tau_1^*([{\Cal Z_{\ell_\zeta - k}}]/x) + 4 \tau_2^*([{\Cal Z_{k}}]/x)
- \lambda_k^2 \right ]$.
\endproof
It follows from the work of Morgan \cite{25} and Li \cite{21},
together with unpublished work of Morgan, that
$D^X _{w,p}(\Cal C_\pm)(\alpha^d) = \delta(\Delta) \cdot \mu_\pm(\alpha)^d$ and
$$D^X _{w,p}(\Cal C_\pm)(\alpha^{d - 2}, x)
= \delta(\Delta) \cdot \mu_\pm(\alpha)^{d - 2} \cdot \nu_\pm$$
where $d = -p - 3$,
$\delta(\Delta) = (-1)^{{{(\Delta^2 + \Delta \cdot K_X)}/2}}$
is the difference between the complex orientation and the standard orientation
on the instanton moduli space (see \cite{6}), and $x \in H_0(X; \Zee)$
is the natural generator. Strictly speaking, their methods only handle the case
of $D^X _{w,p}(\Cal C_\pm)(\alpha^d)$. To handle the case of $D^X _{w,p}(\Cal
C_\pm)(\alpha^{d - 2}, x) $, one needs a blowup formula in algebraic geometry,
which has been established by O'Grady (unpublished). To compute the
differences
$$\mu_+(\alpha)^d - \mu_-(\alpha)^d \quad \text{and} \quad
\mu_+(\alpha)^{d - 2} \cdot \nu_+ - \mu_-(\alpha)^{d - 2} \cdot \nu_-,$$
we need to know how $\mu^{(k)}(\alpha)$ and $\mu^{(k - 1)}(\alpha)$
are related, and also how $\nu^{(k)}$ and $\nu^{(k - 1)}$ are related.
The following lemma handles this problem.
\lemma{5.3} For $\alpha \in H_2(X; \Zee)$ and the natural generator
$x \in H_0(X; \Zee)$, we have
$$\align
&q_{k - 1}^*\mu^{(k-1)}(\alpha) = p_k^*\mu^{(k)}(\alpha) - aD_k\\
&q_{k - 1}^*\nu^{(k - 1)} = p_k^*\nu^{(k)} - {1 \over 4}
[D_k^2 + 2(p_k|D_k)^*\lambda_k].\\
\endalign$$
\endstatement
\proof From the construction,
the sheaf $(\operatorname{Id} \times q_{k - 1})^*\Cal U^{(k - 1)}$
on $X \times \widetilde{\frak M}_0^{(k)}$ is
the elementary modification of $(\operatorname{Id} \times p_{k})^*\Cal U^{(k)}$
along the divisor $X \times D_k$, using the surjection from
$(\operatorname{Id} \times p_{k})^*\Cal U^{(k)}$ to the pullback of
$\rho_k^*(\pi_1^*\Cal O_X(\Delta - F) \otimes
\pi_{1, 3}^*I_{\Cal Z_k})$:
$$0 \to (\operatorname{Id} \times q_{k - 1})^*\Cal U^{(k - 1)}
\to (\operatorname{Id} \times p_{k})^*\Cal U^{(k)}$$
$$\to (\operatorname{Id} \times p_{k}|D_k)^* \rho_k^*(\pi_1^*\Cal O_X(\Delta
- F) \otimes \pi_{1, 3}^*I_{\Cal Z_k}) \to 0$$
where $(2F - \Delta) = \zeta$ and $\pi_1$ is the natural projection
$X \times H_{\ell_\zeta - k} \times H_{k} \to X$. Note that
$(\operatorname{Id} \times p_{k}|D_k)^* \rho_k^*(\pi_1^*\Cal O_X(\Delta - F)
\otimes \pi_{1, 3}^*I_{\Cal Z_k})$ is a sheaf supported on $X \times D_k$,
and that its first and second Chern classes are equal to $(X \times D_k)$
and $(X \times D_k^2) - \pi_1^*(\Delta - F) \cdot (X \times D_k)$ respectively.
It follows that
$$\align
&(\operatorname{Id} \times q_{k - 1})^* c_1(\Cal U^{(k - 1)}) =
(\operatorname{Id} \times p_{k})^* c_1(\Cal U^{(k)}) - (X \times D_k)\\
&(\operatorname{Id} \times q_{k - 1})^* c_2(\Cal U^{(k - 1)}) =
(\operatorname{Id} \times p_{k})^* c_2(\Cal U^{(k)}) -
(\operatorname{Id} \times p_{k})^* c_1(\Cal U^{(k)}) \cdot (X \times D_k)\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad
+ \pi_1^*(\Delta - F) \cdot (X \times D_k).\\
\endalign$$
By Lemma 5.2, $(\operatorname{Id} \times p_{k})^*c_1(\Cal U^{(k)})
\cdot (X \times D_k) = (\Delta \times D_k) + (X \times (p_k|D_k)^*\lambda_k)$.
Thus,
$$\align
(\operatorname{Id} \times q_{k - 1})^* p_1(\Cal U^{(k - 1)})
&= (\operatorname{Id} \times p_{k})^* p_1(\Cal U^{(k)}) + (X \times D_k^2)
- 4 (\Delta - F) \times D_k\\
&\qquad\qquad
+ 2 (\operatorname{Id} \times p_{k})^*c_1(\Cal U^{(k)}) \cdot (X \times D_k)\\
&= (\operatorname{Id} \times p_{k})^* p_1(\Cal U^{(k)})
+ 2(2F - \Delta) \times D_k\\
&\qquad\qquad + X \times [D_k^2 + 2(p_k|D_k)^*\lambda_k].\\
\endalign$$
Now the conclusions follow from some straightforward calculations.
\endproof
In the next two theorems, we will give formulas for the differences
$[\mu_+(\alpha)]^d - [\mu_-(\alpha)]^d$ and
$[\mu_+(\alpha)]^{d - 2} \cdot \nu_+ - [\mu_-(\alpha)]^{d - 2} \cdot \nu_-$
in terms of the intersections in $H_{\ell_\zeta - k} \times H_k$
and the Segre classes of the vector bundles
$\Cal E_{\zeta}^{\ell_\zeta - k, k} \oplus
(\Cal E_{-\zeta}^{k, \ell_\zeta -k})\spcheck$ on $H_{\ell_\zeta - k} \times
H_k$,
where $k = 0, 1, \ldots, \ell_\zeta$.
The arguments are a little complicated, but the idea is that we are trying
to get rid of the exceptional divisors $D_k$ as well as
the Chern classes of the tautological line bundles $\xi_k$ and $\lambda_k$.
\theorem{5.4} Let $\zeta$ define a wall of type $(w, p)$,
and $d = (-p - 3)$. For $\alpha \in H_2(X; \Zee)$,
put $a = (\zeta \cdot \alpha)/2$. Then,
$[\mu_+(\alpha)]^d - [\mu_-(\alpha)]^d$ is equal to
$$\sum_{j = 0}^{2\ell_\zeta}~ {d \choose j} \cdot
(-1)^{h(\zeta) + \ell_\zeta + j} \cdot a^{d - j}
\cdot \sum_{k = 0}^{\ell_\zeta}~
([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j
\cdot s_{2\ell_\zeta - j}(\Cal E_{\zeta}^{\ell_\zeta - k, k} \oplus
(\Cal E_{-\zeta}^{k, \ell_\zeta -k})\spcheck).$$
\endstatement
\proof By Lemma 5.3, we have
$q_{k - 1}^*\mu^{(k-1)}(\alpha) = p_k^*\mu^{(k)}(\alpha) - aD_k$.
Since $p_k$ and $q_{k - 1}$ are birational morphisms,
$[p_k^*\mu^{(k)}(\alpha)]^d = [\mu^{(k)}(\alpha)]^d$ and
$[q_{k - 1}^*\mu^{(k-1)}(\alpha)]^d = [\mu^{(k-1)}(\alpha)]^d$. Thus,
$[\mu^{(k - 1)}(\alpha)]^d - [\mu^{(k)}(\alpha)]^d$ is equal to
$$\align
&\quad \sum_{i = 1}^d~ {d \choose i} \cdot
[p_k^*\mu^{(k)}(\alpha)|D_k]^{d - i} \cdot (-D_k|D_k)^{i - 1} \cdot (-a^i) \\
&= \sum_{i = 1}^d~ {d \choose i} \cdot
[p_k^*\mu^{(k)}(\alpha)|D_k]^{d - i} \cdot \xi_k^{i - 1} \cdot (-a^i). \\
\endalign$$
By Lemma 5.2, $p_k^*\mu^{(k)}(\alpha)|D_k =
(p_k|D_k)^*([{\Cal Z_{\ell_\zeta - k}}]/\alpha
+ [{\Cal Z_{k}}]/\alpha - a \lambda_k)$. So we have
$$\align
&\quad [\mu^{(k - 1)}(\alpha)]^d - [\mu^{(k)}(\alpha)]^d \\
&= \sum_{i = 1}^d~ {d \choose i} \cdot \sum_{j = 0}^{2\ell_\zeta}~
{{d - i} \choose j} \cdot
([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot
(-a\lambda_k)^{d - i - j} \cdot \xi_k^{i - 1} \cdot (-a^i)\\
&= \sum_{j = 0}^{2\ell_\zeta}~ \sum_{i = 1}^{d - j}~ {d \choose j}
\cdot {{d - j} \choose i} \cdot (-a^{d - j}) \cdot
([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot
\xi_k^{i - 1} \cdot (-\lambda_k)^{d - i - j}\\
&= \sum_{j = 0}^{2\ell_\zeta}~ {d \choose j} \cdot (-a^{d - j}) \cdot
([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j
\cdot \sum_{i = 1}^{d - j}~ {{d - j} \choose i} \cdot
\xi_k^{i - 1} \cdot (-\lambda_k)^{d - i - j}\\
&= \sum_{j = 0}^{2\ell_\zeta}~ {d \choose j} \cdot (-a^{d - j}) \cdot
([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j
\cdot \sum_{i = 0}^{d - 1 - j}~ {{d - j} \choose {i + 1}} \cdot
\xi_k^{i} \cdot (-\lambda_k)^{d - 1 - j - i}\\
\endalign$$
Now, our formula follows from the following claim by summing
$k$ from $0$ to $\ell_\zeta$.
\claim{}
$$([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j
\cdot \sum_{i = 0}^{d - 1 - j}~ {{d - j} \choose {i + 1}} \cdot
\xi_k^{i} \cdot (-\lambda_k)^{d - 1 - j - i}$$
$$= ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j
\cdot (-1)^{h(\zeta) + \ell_\zeta + j - 1} \cdot
s_{2\ell_\zeta - j}(\Cal E_{\zeta}^{\ell_\zeta - k, k}
\oplus (\Cal E_{-\zeta}^{k, \ell_\zeta -k})\spcheck).$$
\endstatement
\par\noindent
{\it Proof.} For simplicity, on the exceptional divisor $D_k$, we put
$$\sigma_s = ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j
\cdot \sum_{i = 0}^s {{s + 1} \choose {i + 1}} \cdot \xi_k^i \cdot
(-\lambda_k)^{s - i}.$$
So we must compute $\sigma_{d - 1 - j}$. Notice the relation
$$\sigma_s + \lambda_k \cdot \sigma_{s - 1}
= ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j
\cdot (\xi_k - \lambda_k)^s.$$
Thus for $0 \le t \le s$, we have
$$\sigma_s = (-\lambda_k)^t \cdot \sigma_{s - t} +
([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j
\cdot \sum_{i = 0}^{t - 1} (\xi_k - \lambda_k)^{s - i} \cdot (-\lambda_k)^i.$$
Put $s = d - 1 - j$ and $t = s - {N_{-\zeta}} = d - 1 - j - {N_{-\zeta}}$,
where ${N_{-\zeta}} = \ell_{-\zeta} + h(-\zeta) - 1
= \ell_{\zeta} + h(-\zeta) - 1$ as defined in Corollary 2.7. Then,
$\sigma_{d - 1 - j}$ is equal to
$$(-\lambda_k)^{d - 1 - j - {N_{-\zeta}}} \cdot \sigma_{N_{-\zeta}} +
([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j
\cdot \sum_{i = 0}^{d - 2 - j - {N_{-\zeta}}}
(\xi_k - \lambda_k)^{(d - 1 - j) - i} \cdot (-\lambda_k)^i.$$
Since $\dim E_\zeta^{\ell_\zeta - k, k} = d - 1 - {N_{-\zeta}}$,
we see that $(-\lambda_k)^{d - 1 - j - {N_{-\zeta}}} \cdot
\sigma_{N_{-\zeta}}$ is equal to
$$\align
&\quad (-\lambda_k)^{d - 1 - j - {N_{-\zeta}}} \cdot
([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j
\cdot \sum_{i = 0}^{N_{-\zeta}} {{{N_{-\zeta}} + 1} \choose {i + 1}}
\cdot \xi_k^i \cdot (-\lambda_k)^{{N_{-\zeta}} - i}\\
&= (-\lambda_k)^{d - 1 - j - {N_{-\zeta}}} \cdot
([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j
\cdot \xi_k^{N_{-\zeta}}\\
&= ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot
(-\lambda_k)^{d - 1 - j - {N_{-\zeta}}} \cdot (\xi_k -
\lambda_k)^{N_{-\zeta}}\\
\endalign$$
since the restriction of $\xi_k$ to a fiber of
$D_k \to E_\zeta^{\ell_\zeta - k, k}$ is a hyperplane. Therefore,
$$\sigma_{d - 1 - j} =
([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot
\sum_{i = 0}^{d - 1 - j - {N_{-\zeta}}} (\xi_k - \lambda_k)^{(d - 1 - j) - i}
\cdot (-\lambda_k)^i.$$
Now, we shall simplify $(\xi_k - \lambda_k)^{(d - 1 - j) - i}$.
Since $\xi_k$ is the tautological line bundle on
$D_k = \Pee(\Cal N_k\spcheck)$, the line bundle $(\xi_k \otimes
\lambda_k^{-1})$
is the tautological line bundle on
$$\Pee(\Cal N_k\spcheck \otimes \lambda_k^{-1}) =
\Pee[((\rho_k|E_\zeta^{k, \ell_\zeta - k})\spcheck\Cal E_{-\zeta}^{k,
\ell_\zeta -k})\spcheck].$$
Since ${N_{-\zeta}} + 1$ is the rank of $\Cal E_{-\zeta}^{k, \ell_\zeta -k}$,
it follows that
$$(\xi_k - \lambda_k)^{1 + {N_{-\zeta}}} = -\sum_{j = 1}^{1 + {N_{-\zeta}}}
c_j(\Cal E_{-\zeta}^{k, \ell_\zeta -k})
\cdot (\xi_k - \lambda_k)^{1 + {N_{-\zeta}} - j}.$$
One verifies that in general, for $u' \ge {N_{-\zeta}}$, one has
$$(\xi_k - \lambda_k)^{u'} =
s_{u' - {N_{-\zeta}}}(\Cal E_{-\zeta}^{k, \ell_\zeta -k}) \cdot
(\xi_k - \lambda_k)^{{N_{-\zeta}}} +
O\left((\xi_k - \lambda_k)^{{N_{-\zeta}} - 1}\right)$$
where $s_i(\Cal E_{-\zeta}^{k, \ell_\zeta -k})$ is the $i^{\text{th}}$ Segre
class of $\Cal E_{-\zeta}^{k, \ell_\zeta -k}$. Therefore,
since $(d - 1 - j) - i \ge {N_{-\zeta}}$, we see that
$(\xi_k - \lambda_k)^{(d - 1 - j) - i}$ is equal to
$$s_{d - 1 - j - i - {N_{-\zeta}}}(\Cal E_{-\zeta}^{k, \ell_\zeta -k}) \cdot
(\xi_k - \lambda_k)^{{N_{-\zeta}}}
+ O\left((\xi_k - \lambda_k)^{{N_{-\zeta}} - 1}\right)$$
and that $([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot
(\xi_k - \lambda_k)^{(d - 1 - j) - i} \cdot (-\lambda_k)^i$ is equal to
$$([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot
\left[s_{d - 1 - j - i - {N_{-\zeta}}}(\Cal E_{-\zeta}^{k, \ell_\zeta -k})
\cdot (\xi_k - \lambda_k)^{{N_{-\zeta}}}\right] \cdot (-\lambda_k)^i$$
$$= ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot
s_{d - 1 - j - i - {N_{-\zeta}}}(\Cal E_{-\zeta}^{k, \ell_\zeta -k})
\cdot (-\lambda_k)^i.$$
Next, we note that $([{\Cal Z_{\ell_\zeta - k}}]/\alpha +
[{\Cal Z_{k}}]/\alpha)^j \cdot
s_{d - 1 - j - i - {N_{-\zeta}}}(\Cal E_{-\zeta}^{k, \ell_\zeta -k})$
is a cycle on $E_{\zeta}^{\ell_\zeta -k, k}$ pulled-back from
$H_{\ell_\zeta -k} \times H_k$. So this term is zero unless
$d - 1 - i - {N_{-\zeta}} \le 2\ell_\zeta$, that is,
$i \ge d - 1 - {N_{-\zeta}} - 2\ell_\zeta$.
Note that by Corollary 2.7, $d - 1 - {N_{-\zeta}} - 2\ell_\zeta = {N_\zeta}$
and ${N_\zeta} + 1 = h(\zeta) + \ell_\zeta$ is the rank of
$\Cal E_{\zeta}^{\ell_\zeta -k, k}$.
Since $\lambda_k$ is the tautological line bundle on
$E_{\zeta}^{\ell_\zeta -k, k} = \Pee((\Cal E_{\zeta}^{\ell_\zeta -k,
k})\spcheck)$, we see as before that
$$\lambda_k^i = s_{i - {N_\zeta}}(\Cal E_{\zeta}^{\ell_\zeta -k, k}) \cdot
\lambda_k^{N_\zeta} + O\left(\lambda_k^{{N_\zeta} - 1}\right).$$
Putting all these together, we conclude that $\sigma_{d - 1 - j}$ is equal to
$$\align
&\quad ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot
\sum_{i = {N_\zeta}}^{d - 1 - j - {N_{-\zeta}}}~
s_{d - 1 - j - i - {N_{-\zeta}}}(\Cal E_{-\zeta}^{k, \ell_\zeta -k})
\cdot (-1)^i \cdot s_{i - {N_\zeta}}(\Cal E_{\zeta}^{\ell_\zeta -k, k})\\
&= ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j
\cdot \sum_{i = 0}^{2\ell_\zeta - j}~ (-1)^{i + {N_\zeta}} \cdot
s_{(2\ell_\zeta - j) - i}(\Cal E_{-\zeta}^{k, \ell_\zeta -k}) \cdot
s_{i}(\Cal E_{\zeta}^{\ell_\zeta -k, k})\\
&= ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j
\cdot (-1)^{j + {N_\zeta}} \cdot\sum_{i = 0}^{2\ell_\zeta - j}~
s_{(2\ell_\zeta - j) - i}((\Cal E_{-\zeta}^{k, \ell_\zeta -k})\spcheck) \cdot
s_{i}(\Cal E_{\zeta}^{\ell_\zeta -k, k})\\
&= ([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j
\cdot (-1)^{j + {N_\zeta}} \cdot
s_{2\ell_\zeta - j}(\Cal E_{\zeta}^{\ell_\zeta -k, k} \oplus
(\Cal E_{-\zeta}^{k, \ell_\zeta -k})\spcheck) \qed\\
\endalign$$
This completes the proof of the Theorem.
\endproof
For the difference $[\mu_+(\alpha)]^{d - 2} \cdot \nu_+
- [\mu_-(\alpha)]^{d - 2} \cdot \nu_-$, we have the following.
\theorem{5.5} Let $\zeta$ define a wall of type $(w, p)$,
and $d = -p - 3$. For $\alpha \in H_2(X; \Zee)$,
put $a = (\zeta \cdot \alpha)/2$. Then, $[\mu_+(\alpha)]^{d - 2} \cdot \nu_+
- [\mu_-(\alpha)]^{d - 2} \cdot \nu_-$ is equal to
$${1 \over 4} \cdot \sum_{j = 0}^{2 \ell_\zeta} {{d - 2} \choose j} \cdot
(-1)^{h(\zeta) + \ell_\zeta - 1 + j} \cdot a^{d - 2 - j} \cdot$$
$$\sum_{k = 0}^{\ell_\zeta}
([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot
\left [ s_{2 \ell_\zeta - j} -
4 ([{\Cal Z_{\ell_\zeta - k}}] + [{\Cal Z_{k}}])/x \cdot
s_{2 \ell_\zeta - 2 - j} \right ]$$
where $s_i$ stands for the $i^{\text{th}}$ Segre class of
$\Cal E_{\zeta}^{\ell_\zeta - k, k}
\oplus (\Cal E_{-\zeta}^{k, \ell_\zeta -k})\spcheck$.
\endstatement
\proof By Lemma 5.3, we have $q_{k - 1}^*\mu^{(k-1)}(\alpha)
= p_k^*\mu^{(k)}(\alpha) - aD_k$ and
$$q_{k - 1}^*\nu^{(k - 1)}
= p_k^*\nu^{(k)} - {1 \over 4} [D_k^2 + 2 (p_k|D_k)^*\lambda_k].$$
It follows that $[\mu^{(k-1)}(\alpha)]^{d - 2} \cdot \nu^{(k-1)} -
[\mu^{(k)}(\alpha)]^{d - 2} \cdot \nu^{(k)} = I_1 + I_2$ where
$$\align
&I_1 = [\mu^{(k)}(\alpha) - aD_k]^{d - 2} \cdot
{1 \over 4} [-D_k^2 - 2 (p_k|D_k)^*\lambda_k] \\
&\quad = [\mu^{(k)}(\alpha)|D_k + a\xi_k]^{d - 2} \cdot
{1 \over 4} (\xi_k - 2 \lambda_k) \\
&I_2 = \sum_{i = 1}^{d - 2} {{d - 2} \choose i}
\cdot \mu^{(k)}(\alpha)^{d - 2 - i} \cdot (-aD_k)^i \cdot \nu^{(k)} \\
&\quad = \sum_{i = 1}^{d - 2}
{{d - 2} \choose i} \cdot [\mu^{(k)}(\alpha)|D_k]^{d - 2 - i}
\cdot \xi_k^{i - 1} \cdot (-a^i) \cdot (\nu^{(k)}|D_k).\\
\endalign$$
First of all, since $\mu^{(k)}(\alpha)|D_k =
([{\Cal Z_{\ell_\zeta - k}}]/\alpha
+ [{\Cal Z_{k}}]/\alpha - a \lambda_k)$, we see that
$$\align
I_1 &= \left [([{\Cal Z_{\ell_\zeta - k}}]/\alpha
+ [{\Cal Z_{k}}]/\alpha) + a (\xi_k - \lambda_k) \right]^{d - 2}
\cdot {1 \over 4} (\xi_k - 2 \lambda_k)\\
&= {1 \over 4} \sum_{j = 0}^{2 \ell_\zeta} {{d - 2} \choose j} \cdot
a^{d - 2 - j} \cdot
([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j
\cdot (\xi_k - \lambda_k)^{d - 2 - j} \cdot (\xi_k - 2 \lambda_k)\\
&= {1 \over 4} \sum_{j = 0}^{2 \ell_\zeta} {{d - 2} \choose j} \cdot
a^{d - 2 - j} \cdot
([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot \\
&\quad\quad\quad\quad \cdot \left [ (\xi_k - \lambda_k)^{d - 1 - j} -
\lambda_k \cdot (\xi_k - \lambda_k)^{d - 2 - j} \right ]\\
&= {1 \over 4} \sum_{j = 0}^{2 \ell_\zeta} {{d - 2} \choose j} \cdot
a^{d - 2 - j} \cdot
([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot \\
&\quad\quad\quad\quad \cdot \left [
s_{d - 1 - j - {N_{-\zeta}}}(\Cal E_{-\zeta}^{k, \ell_\zeta -k}) -
\lambda_k \cdot
s_{d - 2 - j - {N_{-\zeta}}}(\Cal E_{-\zeta}^{k, \ell_\zeta -k}) \right ].\\
\endalign$$
Next, by Lemma 5.2, we have $\nu^{(k)}|D_k = {1 \over 4}
\left [ 4 [{\Cal Z_{\ell_\zeta - k}}]/x +
4 [{\Cal Z_{k}}]/x - \lambda_k^2 \right ]$. Thus,
as in the proof of Theorem 5.4, we can verify that $I_2$ is equal to
$$\align
&\quad {1 \over 4} \left [ 4 [{\Cal Z_{\ell_\zeta - k}}]/x +
4 [{\Cal Z_{k}}]/x - \lambda_k^2 \right ] \cdot
\sum_{i = 1}^{d - 2} {{d - 2} \choose i} \cdot
[\mu^{(k)}(\alpha)|D_k]^{d - 2 - i} \cdot \xi_k^{i - 1} \cdot (-a^i) \\
&= {1 \over 4} \left [ 4 [{\Cal Z_{\ell_\zeta - k}}]/x +
4 [{\Cal Z_{k}}]/x - \lambda_k^2 \right ] \cdot
\sum_{j = 0}^{2 \ell_\zeta} {{d - 2} \choose j} \cdot (-a^{d - 2 - j}) \cdot \\
&\quad\quad\quad \cdot
([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot
\sum_{i = 0}^{2 \ell_\zeta + {N_\zeta} - 2 - j}
s_{d - 3 - j - i - {N_{-\zeta}}}(\Cal E_{-\zeta}^{k, \ell_\zeta -k}) \cdot
(- \lambda_k)^i\\
&= {1 \over 4} \sum_{j = 0}^{2 \ell_\zeta} {{d - 2} \choose j} \cdot
(-a^{d - 2 - j}) \cdot
([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot \\
&\cdot \left [ 4 ([{\Cal Z_{\ell_\zeta - k}}] + [{\Cal Z_{k}}])/x \cdot
(-1)^{j + {N_\zeta}} \cdot s'
- \sum_{i = 0}^{2 \ell_\zeta + {N_\zeta} - 2 - j}
s_{d - 3 - j - i - {N_{-\zeta}}}(\Cal E_{-\zeta}^{k, \ell_\zeta -k}) \cdot
(- \lambda_k)^{i + 2} \right ] \\
&= {1 \over 4} \sum_{j = 0}^{2 \ell_\zeta} {{d - 2} \choose j} \cdot
a^{d - 2 - j} \cdot
([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot \\
&\cdot \left [ \sum_{i = 0}^{2 \ell_\zeta + {N_\zeta} - 2 - j}
s_{d - 3 - j - i - {N_{-\zeta}}}(\Cal E_{-\zeta}^{k, \ell_\zeta -k}) \cdot
(- \lambda_k)^{i + 2} - 4 ([{\Cal Z_{\ell_\zeta - k}}] +
[{\Cal Z_{k}}])/x \cdot (-1)^{j + {N_\zeta}} \cdot s' \right ] \\
\endalign$$
where $s'$ stands for
$s_{2 \ell_\zeta - 2 - j}(\Cal E_{\zeta}^{\ell_\zeta - k, k}
\oplus (\Cal E_{-\zeta}^{k, \ell_\zeta -k})\spcheck)$. Thus,
$I_1 + I_2$ is equal to
$$\align
&\quad {1 \over 4} \sum_{j = 0}^{2 \ell_\zeta} {{d - 2} \choose j} \cdot
a^{d - 2 - j} \cdot
([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot \\
&\cdot \left [ \sum_{i = -2}^{2 \ell_\zeta + {N_\zeta} - 2 - j}
s_{d - 3 - j - i - {N_{-\zeta}}}(\Cal E_{-\zeta}^{k, \ell_\zeta -k}) \cdot
(- \lambda_k)^{i + 2} - 4 ([{\Cal Z_{\ell_\zeta - k}}] +
[{\Cal Z_{k}}])/x \cdot (-1)^{j + {N_\zeta}} \cdot s' \right ] \\
&= {1 \over 4} \sum_{j = 0}^{2 \ell_\zeta} {{d - 2} \choose j} \cdot
a^{d - 2 - j} \cdot
([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot \\
&\quad \cdot \left [ (-1)^{j + {N_\zeta}} \cdot s'' -
4 ([{\Cal Z_{\ell_\zeta - k}}] + [{\Cal Z_{k}}])/x \cdot
(-1)^{j + {N_\zeta}} \cdot s' \right ] \\
&= {1 \over 4} \sum_{j = 0}^{2 \ell_\zeta} {{d - 2} \choose j} \cdot
(-1)^{h(\zeta) + \ell_\zeta - 1 + j} \cdot a^{d - 2 - j} \cdot
([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot \\
&\quad \cdot
\left [ s'' - 4 ([{\Cal Z_{\ell_\zeta - k}}] + [{\Cal Z_{k}}])/x \cdot
s' \right ] \\
\endalign$$
since ${N_\zeta} = h(\zeta) + \ell_\zeta - 1$, where $s''$ stands for
$s_{2 \ell_\zeta - j}(\Cal E_{\zeta}^{\ell_\zeta - k, k}
\oplus (\Cal E_{-\zeta}^{k, \ell_\zeta -k})\spcheck)$. Letting $k$ run from
$0$ to $\ell_\zeta$, we obtain the desired formula.
\endproof
\par\noindent
{\bf Remark 5.6.} For the sake of convenience, we record here the following
relation among the Chern classes and the Segre classes of a vector bundle:
$$s_n = -c_1 \cdot s_{n - 1} - c_2 \cdot s_{n - 2} - \ldots - c_n$$
with the convention that $s_0 = 1$. We refer to \cite{12} for details.
\medskip
In the next section, using Theorem 5.4 and Theorem 5.5,
we shall compute $[\mu_+(\alpha)]^d - [\mu_-(\alpha)]^d$
and $[\mu_+(\alpha)]^{d - 2} \cdot \nu_+
- [\mu_-(\alpha)]^{d - 2} \cdot \nu_-$
explicitly when $0 \le \ell_\zeta \le 2$.
In principle, Theorem 5.4 and Theorem 5.5 give formulas for
these differences in terms of certain intersections
in $H_{\ell_\zeta -k} \times H_k$. However, it is difficult
to evaluate these intersection numbers in general. In the following,
we shall compute the term
$$S_j = \sum_{k = 0}^{\ell_\zeta}~
([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j
\cdot s_{2\ell_\zeta - j}(\Cal E_{\zeta}^{\ell_\zeta - k, k} \oplus
(\Cal E_{-\zeta}^{k, \ell_\zeta -k})\spcheck) \eqno (5.7)$$
in the special cases when
$j = 2\ell_\zeta$ and $2\ell_\zeta - 1$. We start with a simple lemma.
\lemma{5.8} Let $\alpha, \beta \in H_2(X; \Zee)$. Then
$$\align
&([\Cal Z_k]/\alpha)^{2k} =
{{(2k)!} \over {2^k \cdot k!}} \cdot (\alpha^2)^k\\
&([\Cal Z_k]/\alpha)^{2k - 1} \cdot ([\Cal Z_k]/\beta) =
{{(2k)!} \over {2^k \cdot k!}} \cdot (\alpha^2)^{k - 1}
\cdot (\alpha \cdot \beta)\\
&([\Cal Z_k]/\alpha)^{2k - 2} \cdot ([\Cal Z_k]/\beta)^2 = \\
&{{(2k - 2)!} \over {2^{k - 1} \cdot (k - 1)!}} \cdot (\alpha^2)^{k - 1}
\cdot \beta^2
+ {{(2k - 2)!} \over {2^{k - 2} \cdot (k - 2)!}} \cdot (\alpha^2)^{k - 2}
\cdot (\alpha \cdot \beta)^2.\\
\endalign$$
\endstatement
\proof The first equality is well-known (see \cite{28} for instance).
The other statements follow from the first one by considering
$$ ([\Cal Z_k]/\alpha + [\Cal Z_k]/\beta)^{2k}= {{(2k)!} \over {2^k \cdot k!}}
\cdot ((\alpha + \beta)^2)^k,$$
and formally equating the terms involving $(2k-1)$ copies of $\alpha$
and one $\beta$ or $(2k-2)$ copies of $\alpha$ and two copies of $\beta$.
\endproof
The next result computes the term (5.7) when $j = 2\ell_\zeta$.
\proposition{5.9} Let $\zeta$ define a wall of type $(w, p)$,
and $\alpha \in H_2(X; \Zee)$. Then,
$$S_{2\ell_\zeta} = \sum_{k = 0}^{\ell_\zeta}~
([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^{2\ell_\zeta} =
{{(2\ell_\zeta)!} \over {\ell_\zeta !}} \cdot (\alpha^2)^{\ell_\zeta}.$$
\endstatement
\par\noindent
{\it Proof.} This follows in a straightforward way from Lemma 5.8 (i):
$$\align
&\quad \sum_{k = 0}^{\ell_\zeta}~
([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^{2\ell_\zeta}\\
&= \sum_{k = 0}^{\ell_\zeta}~{{2\ell_\zeta} \choose {2k}} \cdot
([{\Cal Z_{\ell_\zeta - k}}]/\alpha)^{2(\ell_\zeta - k)} \cdot
([{\Cal Z_{k}}]/\alpha)^{2k} \\
&= \sum_{k = 0}^{\ell_\zeta}~{{2\ell_\zeta} \choose {2k}} \cdot
\left [{{(2\ell_\zeta - 2k)!} \over {2^{\ell_\zeta - k} \cdot
(\ell_\zeta - k)!}} \cdot (\alpha^2)^{\ell_\zeta - k} \right ] \cdot
\left [{{(2k)!} \over {2^k \cdot k!}} \cdot (\alpha^2)^k \right ]\\
&= \sum_{k = 0}^{\ell_\zeta}~ {\ell_\zeta \choose k} \cdot
{{(2\ell_\zeta)!} \over {2^{\ell_\zeta} \cdot \ell_\zeta !}}
\cdot (\alpha^2)^{\ell_\zeta} \\
&= {{(2\ell_\zeta)!} \over {\ell_\zeta !}} \cdot (\alpha^2)^{\ell_\zeta}\qed\\
\endalign$$
To compute the term (5.7) when $j = (2\ell_\zeta - 1)$,
we study $\Cal E_{-\zeta}^{k, \ell_\zeta -k}$ and
$\Cal E_{\zeta}^{\ell_\zeta -k, k}$,
and evaluate their first Chern classes. We begin with a general lemma.
\lemma{5.10} Let $Z, W$ be codimension $2$ cycles in
a smooth variety $Y$.
\roster
\item"{(i)}" If $Z \subseteq W$, then $Hom(I_W, I_Z) = \Cal O_Y$;
\item"{(ii)}" If $(Z - Z \cap W)$ is open and dense in $Z$,
then $Hom(I_W, I_Z) = I_Z$;
\item"{(iii)}" If $Z$ and $W$ are local complete intersections meeting
properly,
then there is an exact sequence:
$$0 \to Ext ^1(I_W, I_Z) \to \Cal O _W \otimes \det
N_W \to \Cal O _{W \cap Z} \otimes \det N_W \to 0$$
where $N_W$ is the normal bundle of $W$ in $Y$;
\item"{(iv)}" Assume that $Z \cap W$ is nowhere dense in $W$ and that
$W$ is smooth at a generic point. Then, as a sheaf on $W$,
$Ext^1(I_W, I_Z)$ is of rank $1$; thus,
$$c_0(Ext^1(I_W, I_Z)) = c_1(Ext^1(I_W, I_Z)) = 0,
\quad c_2(Ext^1(I_W, I_Z)) = -[W].$$
\endroster
\endstatement
\proof (i) Applying the functor $Hom(I_W, \cdot)$ to
the exact sequence
$$0 \to I_Z \to \Cal O_Y \to \Cal O_Z \to 0,$$
we obtain $0 \to Hom(I_W, I_Z) \to Hom(I_W, \Cal O_Y) = \Cal O_Y$.
Thus, $Hom(I_W, I_Z) = I_U$ for some closed subscheme $U$ of $Y$.
On the other hand, since $Z \subseteq W$,
$$H^0(Y; Hom(I_W, I_Z)) = \Hom(I_W, I_Z) \ne 0.$$
Thus, $U$ must be empty, and $Hom(I_W, I_Z) = \Cal O_Y$.
(ii) As in the proof of (i), $Hom(I_W, I_Z) = I_U$ for
some closed subscheme $U$ of $Y$. Applying the functor
$Hom(\cdot, I_Z)$ to the exact sequence
$$0 \to I_W \to \Cal O_Y \to \Cal O_W \to 0,$$
we get $0 \to I_Z \to Hom(I_W, I_Z) = I_U \to Ext^1(\Cal O_W, I_Z)$.
Thus, $U \subseteq Z$; moreover, since $Ext^1(\Cal O_W, I_Z) = 0$
on $(X - W)$, we have $(Z - Z \cap W) = (U - U \cap W)$. So
$$(Z - Z \cap W) \subseteq U \subseteq Z.$$
Since $(Z - Z \cap W)$ is open and dense in $Z$, it follows that $U = Z$.
(iii) We begin with the local identification:
let $R$ be a regular local ring, and let $Z$ and $W$ be two codimension $2$
local complete intersection subschemes of $R$ meeting properly.
Applying the functor $Hom_R(\cdot, I_Z)$ to the Koszul resolution of $W$
$$0 \to R \to R \oplus R \to I_W \to 0$$
gives $I_Z \oplus I_Z \to I_Z \to Ext_R^1(I_W, I_Z) \to 0$.
It follows that $Ext_R^1(I_W, I_Z) = I_Z/(I_Z \cdot I_W)$.
Since $Z$ and $W$ are codimension $2$ local complete intersections
meeting properly, we have $I_Z \cdot I_W = I_Z \cap I_W$.
Thus, $Ext^1_R(I_W, I_Z) \cong I_Z/(I_Z \cap I_W)$,
and we can fit it into an exact sequence
$$0 \to Ext^1_R(I_W, I_Z) \to R/I_W \to R/(I_W + I_Z) \to 0.$$
Here $(I_W + I_Z)$ corresponds to the intersection $W \cap Z$.
The identification of $Ext_R^1(I_W, I_Z)$ and $I_Z/(I_Z \cap I_W)$
is not canonical. Globally we must correct by $\det N_W$.
Thus globally we have an exact sequence:
$$0 \to Ext ^1(I_W, I_Z) \to \Cal O _W \otimes \det
N_W \to \Cal O _{W \cap Z} \otimes \det N_W \to 0.$$
(iv) It is clear that $Ext^1(I_W, I_Z)$ is a sheaf supported on $W$.
To show that it has rank $1$ as a sheaf on $W$, it suffices to verify
that it has rank $1$ at a generic point $w$ of $W$.
Since $Z \cap W$ is nowhere dense in $W$ and
$W$ is smooth at a generic point, we may assume that
$w \not \in Z$ and that $w$ is a smooth point of $W$.
Then it follows from (iii) that $Ext^1(I_W, I_Z)$ is of rank $1$ at $w$.
\endproof
\lemma{5.11} Let $Hom = Hom(I_{\Cal Z_k},
I_{\Cal Z_{\ell_\zeta - k}})$,
$Ext^1 = Ext^1(I_{\Cal Z_k}, I_{\Cal Z_{\ell_\zeta - k}})$,
$\pi_1$ and $\pi_2$ be the projections from
$X \times (H_{\ell_\zeta - k} \times H_k)$ to $X$ and
$(H_{\ell_\zeta - k} \times H_k)$ respectively.
\roster
\item"{(i)}" There exist a row exact sequence and a column exact sequence:
$$\matrix
&0&\\
&\downarrow&\\
&\pi_{2*}(\pi_1^*\Cal O_X(\zeta) \otimes
\Cal O_{\Cal Z_{\ell_\zeta - k}}) &\\
&\downarrow&\\
0 \to & R^1\pi_{2*} \left (\pi_1^*\Cal O_X(\zeta) \otimes Hom \right )
& \to \Cal E_{\zeta}^{\ell_\zeta -k, k} \to
\pi_{2*} \left (\pi_1^*\Cal O_X(\zeta) \otimes Ext^1 \right ) \to 0; \\
&\downarrow&\\
&[\Cal O_{H_{\ell_\zeta - k} \times H_k}]^{\oplus~ h(\zeta)} &\\
&\downarrow&\\
&0&\\
\endmatrix$$
\item"{(ii)}" $c_1 \left(R^1\pi_{2*} (\pi_1^*\Cal O_X(\zeta)
\otimes Hom)\right) = [\Cal Z_{\ell_\zeta - k}]/(\zeta - K_X/2)
+ \pi_{2*}[c_3(\Cal O_{\Cal Z_{\ell_\zeta - k}})]/2$;
\item"{(iii)}" $c_1 \left(\pi_{2*} (\pi_1^*\Cal O_X(\zeta) \otimes
Ext^1 )\right ) = [\Cal Z_k]/(\zeta - K_X/2)
+ \pi_{2*}[c_3(Ext^1)]/2$.
\endroster
\endstatement
\proof (i) Note that the bundle $\Cal E_\zeta ^{\ell_\zeta -k, k}$
is defined as
$$Ext^1_{\pi _2}(\pi_1^*\scrO _X(\Delta - F) \otimes I_{\Cal Z_k},
\pi _1^*\scrO _X(F)\otimes I_{\Cal Z_{\ell_\zeta -k}})
= Ext^1_{\pi _2}(I_{\Cal Z_k},
\pi _1^*\scrO _X(\zeta) \otimes I_{\Cal Z_{\ell_\zeta -k}}).$$
Since $R^2\pi_{2*}(\pi_1^*\Cal O_X(\zeta) \otimes Hom) = 0$,
the row exact sequence follows from standard facts about relative Ext sheaves.
To see the column exact sequence, we use Lemma 5.10 (ii)
and apply the functor $\pi_{2*}$ to the exact sequence
$$0 \to \pi _1^*\scrO _X(\zeta) \otimes I_{\Cal Z_{\ell_\zeta -k}} \to
\pi _1^*\scrO _X(\zeta) \to \pi _1^*\scrO _X(\zeta) \otimes
\Cal O_{\Cal Z_{\ell_\zeta -k}} \to 0.$$
(ii) Note that $Hom = I_{\Cal Z_{\ell_\zeta -k}}$ and that
$R^i\pi_{2*}(\pi_1^*\Cal O_X(\zeta) \otimes Hom) = 0$
for $i = 0, 2$. By the Grothendieck-Riemann-Roch Theorem, we have
$$\align
&\quad -\ch \left (R^1\pi_{2*}(\pi_1^*\Cal O_X(\zeta)
\otimes Hom)\right )\\
&= \pi_{2*}\left (\ch (\pi_1^*\Cal O_X(\zeta)
\otimes I_{\Cal Z_{\ell_\zeta -k}}) \cdot \pi_1^*\Todd (T_X) \right )\\
&= \pi_{2*}\left (\pi_1^*\ch (\Cal O_X(\zeta)) \cdot
\ch (I_{\Cal Z_{\ell_\zeta -k}})
\cdot \pi_1^*\Todd (T_X) \right ).\\
\endalign$$
Now, the conclusion follows by comparing the degree $1$ terms and
by the fact that
$$\ch (I_{\Cal Z_{\ell_\zeta -k}}) =
1 - \ch (\Cal O_{\Cal Z_{\ell_\zeta - k}})
= 1 - [\Cal Z_{\ell_\zeta - k}] -
{c_3(\Cal O_{\Cal Z_{\ell_\zeta - k}}) \over 2}
+ (\text{terms with degree} \ge 4).$$
(iii) We have $R^i\pi_{2*}(\pi_1^*\Cal O_X(\zeta) \otimes Ext^1) = 0$
for $i = 1, 2$. By Lemma 5.10 (iv),
$$\ch (Ext^1) = [\Cal Z_k] +
{c_3(Ext^1) \over 2} + (\text{terms with degree} \ge 4).$$
Again, using the Grothendieck-Riemann-Roch Theorem, we obtain
$$\align
&\quad \ch \left (\pi_{2*}(\pi_1^*\Cal O_X(\zeta)
\otimes Ext^1)\right )\\
&= \pi_{2*}\left (\ch (\pi_1^*\Cal O_X(\zeta)
\otimes Ext^1) \cdot \pi_1^*\Todd (T_X) \right )\\
&= \pi_{2*}\left (\pi_1^*\ch (\Cal O_X(\zeta)) \cdot
\ch (Ext^1) \cdot \pi_1^*\Todd (T_X) \right ).\\
\endalign$$
Then, our conclusion follows by comparing the degree $1$ terms.
\endproof
Now, we can compute the term (5.7) for $j = 2\ell_\zeta - 1$.
\proposition{5.12} Let $\alpha \in H_2(X; \Zee)$ and
$a = (\zeta \cdot \alpha)/2$. Then,
$$\align
S_{2\ell_\zeta - 1} &= \sum_{k = 0}^{\ell_\zeta}~
([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^{2\ell_\zeta - 1}
\cdot s_1(\Cal E_{\zeta}^{\ell_\zeta - k, k} \oplus
(\Cal E_{-\zeta}^{k, \ell_\zeta -k})\spcheck) \\
&= (-4) \cdot {{(2\ell_\zeta)!} \over {\ell_\zeta !}} \cdot
(\alpha^2)^{\ell_\zeta - 1} \cdot a.\\
\endalign$$
\endstatement
\par\noindent
{\it Proof.} By the symmetry between $k$ and $(\ell_\zeta - k)$,
we see that $S_{2\ell_\zeta - 1}$ is equal to
$$\sum_{k = 0}^{\ell_\zeta}~
([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^{2\ell_\zeta - 1}
\cdot {{s_1(\Cal E_{\zeta}^{\ell_\zeta - k, k} \oplus
(\Cal E_{-\zeta}^{k, \ell_\zeta -k})\spcheck) +
s_1(\Cal E_{\zeta}^{k, \ell_\zeta -k} \oplus
(\Cal E_{-\zeta}^{\ell_\zeta - k, k})\spcheck)} \over 2}.$$
From Lemma 5.11, we conclude that $c_1(\Cal E_{\zeta}^{\ell_\zeta - k, k})$
is equal to
$$([\Cal Z_{\ell_\zeta - k}] + [\Cal Z_k])/(\zeta - K_X/2) +
{{\pi_{2*}[c_3(\Cal O_{\Cal Z_{\ell_\zeta - k}}) +
c_3(Ext^1(I_{\Cal Z_k}, I_{\Cal Z_{\ell_\zeta - k}}))]} \over 2}.$$
Since $s_1(\Cal E_{\zeta}^{\ell_\zeta - k, k} \oplus
(\Cal E_{-\zeta}^{k, \ell_\zeta -k})\spcheck) =
c_1(\Cal E_{-\zeta}^{k, \ell_\zeta -k}) -
c_1(\Cal E_{\zeta}^{\ell_\zeta - k, k})$, we see that
$${{s_1(\Cal E_{\zeta}^{\ell_\zeta - k, k} \oplus
(\Cal E_{-\zeta}^{k, \ell_\zeta -k})\spcheck) +
s_1(\Cal E_{\zeta}^{k, \ell_\zeta -k} \oplus
(\Cal E_{-\zeta}^{\ell_\zeta - k, k})\spcheck)} \over 2} =
(-2) \cdot ([\Cal Z_{\ell_\zeta - k}] + [\Cal Z_k])/\zeta$$
where the $c_3$'s are cancelled out. Therefore, by Lemma 5.8,
$$\align
S_{2\ell_\zeta - 1}&= \sum_{k = 0}^{\ell_\zeta}~
([{\Cal Z_{\ell_\zeta - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^{2\ell_\zeta - 1}
\cdot (-2) \cdot ([\Cal Z_{\ell_\zeta - k}]/\zeta + [\Cal Z_k]/\zeta)\\
&= (-2) \cdot \sum_{k = 0}^{\ell_\zeta}~ [
{{2\ell_\zeta - 1} \choose {2k}} \cdot
([{\Cal Z_{\ell_\zeta - k}}]/\alpha)^{2\ell_\zeta - 2k - 1}
\cdot ([{\Cal Z_{k}}]/\alpha)^{2k} \cdot [\Cal Z_{\ell_\zeta - k}]/\zeta\\
&\quad\quad\quad\quad + {{2\ell_\zeta - 1} \choose {2k - 1}} \cdot
([{\Cal Z_{\ell_\zeta - k}}]/\alpha)^{2\ell_\zeta - 2k}
\cdot ([{\Cal Z_{k}}]/\alpha)^{2k - 1} \cdot [\Cal Z_k]/\zeta]\\
&= (-4) \cdot \sum_{k = 1}^{\ell_\zeta}~
{{2\ell_\zeta - 1} \choose {2k - 1}} \cdot
([{\Cal Z_{\ell_\zeta - k}}]/\alpha)^{2\ell_\zeta - 2k}
\cdot ([{\Cal Z_{k}}]/\alpha)^{2k - 1} \cdot [\Cal Z_k]/\zeta\\
&= (-4) \cdot {{(2\ell_\zeta)!} \over {\ell_\zeta !}} \cdot
(\alpha^2)^{\ell_\zeta - 1} \cdot a \qed\\
\endalign$$
It is possible, but far more complicated, to compute (5.7) for
$j = 2\ell_\zeta - 2$.
Next, we shall draw some consequences from our previous computations.
Recall that $q_X$ denotes the intersection form of $X$,
and that
$$\delta(\Delta) = (-1)^{{{\Delta^2 + \Delta \cdot K_X} \over 2}}$$
is the difference between the complex orientation and the standard orientation
on the instanton moduli space (see \cite{6}). Theorem 5.13 below has
already been obtained by Kotschick and Morgan \cite{18} for
any smooth $4$-manifold with $b_2^+ = 1$.
\theorem{5.13} Let $\zeta$ define a wall of type $(w, p)$,
and $d = -p - 3$. Then,
$$[\mu_+(\alpha)]^d - [\mu_-(\alpha)]^d \equiv
(-1)^{h(\zeta) + \ell_\zeta} \cdot
{{d!} \over {\ell_\zeta! \cdot (d - 2\ell_\zeta)!}}
\cdot a^{d - 2\ell_\zeta} \cdot (\alpha^2)^{\ell_\zeta}
\pmod {a^{d - 2\ell_\zeta + 2}}$$
for $\alpha \in H_2(X; \Zee)$, where $a = (\zeta \cdot \alpha)/2$.
In other words,
$$\delta^X_{w, p}(\Cal C_-, \Cal C_+) \equiv
\delta(\Delta) \cdot (-1)^{h(\zeta) + \ell_\zeta} \cdot
{{d!} \over {\ell_\zeta! \cdot (d - 2\ell_\zeta)!}}
\cdot \left ( {\zeta\over 2} \right )^{d - 2\ell_\zeta} \cdot q_X^{\ell_\zeta}
\pmod {\zeta^{d - 2\ell_\zeta + 2}}.$$
\endstatement
\par\noindent
{\it Proof.} By Theorem 5.4 and our notation (5.7), we have
$$[\mu_+(\alpha)]^d - [\mu_-(\alpha)]^d \equiv
\sum_{j = 2\ell_\zeta - 1}^{2\ell_\zeta}~ {d \choose j} \cdot
(-1)^{h(\zeta) + \ell_\zeta + j} \cdot a^{d - j} \cdot S_j
\pmod {a^{d - 2\ell_\zeta + 2}}.$$
By Proposition 5.12, $S_{2\ell_\zeta - 1}$ is divisible by $a$. Therefore,
$$[\mu_+(\alpha)]^d - [\mu_-(\alpha)]^d \equiv
{d \choose {2\ell_\zeta}} \cdot
(-1)^{h(\zeta) + \ell_\zeta} \cdot a^{d - 2\ell_\zeta} \cdot S_{2\ell_\zeta}
\pmod {a^{d - 2\ell_\zeta + 2}}.$$
Now, our conclusion follows from Proposition 5.9 and the fact that
$$\gamma_{\pm}(\alpha^d) = \delta(\Delta) \cdot \mu_{\pm}(\alpha)^d.\qed$$
The following is proved by using a similar method.
\theorem{5.14} Let $\zeta$ define a wall of type $(w, p)$.
For $\alpha \in H_2(X; \Zee)$, let $a = (\zeta \cdot \alpha)/2$.
Then, modulo $a^{d - 2\ell_\zeta}$, $[\mu_+(\alpha)]^{d - 2} \cdot \nu_+ -
[\mu_-(\alpha)]^{d - 2} \cdot \nu_-$ is equal to
$${1 \over 4} \cdot (-1)^{h(\zeta) + \ell_\zeta - 1} \cdot
{{(d - 2)!} \over {\ell_\zeta! \cdot (d - 2 - 2\ell_\zeta)!}}
\cdot a^{d - 2 - 2\ell_\zeta} \cdot (\alpha^2)^{\ell_\zeta}.$$
\endstatement
\par\noindent
{\it Proof.} By Theorem 5.5, $[\mu_+(\alpha)]^{d - 2} \cdot \nu_+ -
[\mu_-(\alpha)]^{d - 2} \cdot \nu_-$ is equal to
$${1 \over 4} \cdot \sum_{j = 2 \ell_\zeta - 1}^{2 \ell_\zeta}
{{d - 2} \choose j} \cdot (-1)^{h(\zeta) + \ell_\zeta - 1 + j}
\cdot a^{d - 2 - j} \cdot S_j$$
modulo $a^{d - 2\ell_\zeta}$, where $S_j$ is the notation introduced in (5.7).
By Proposition 5.12, $S_{2 \ell_\zeta - 1}$ is divisible by $a$;
by Proposition 5.9, we have
$$S_{2\ell_\zeta} = {{(2\ell_\zeta)!} \over {\ell_\zeta !}}
\cdot (\alpha^2)^{\ell_\zeta}.$$
Therefore, modulo $a^{d - 2\ell_\zeta}$,
$[\mu_+(\alpha)]^{d - 2} \cdot \nu_+ -
[\mu_-(\alpha)]^{d - 2} \cdot \nu_-$ is equal to
$${1 \over 4} \cdot (-1)^{h(\zeta) + \ell_\zeta - 1} \cdot
{{(d - 2)!} \over {\ell_\zeta! \cdot (d - 2 - 2\ell_\zeta)!}}
\cdot a^{d - 2 - 2\ell_\zeta} \cdot (\alpha^2)^{\ell_\zeta}. \qed$$
\section{6. The formulas when $\ell_\zeta = 0, 1, 2$.}
In this section, we shall compute
$[\mu_+(\alpha)]^d - [\mu_-(\alpha)]^d$ and
$$[\mu_+(\alpha)]^{d - 2} \cdot \nu_+
- [\mu_-(\alpha)]^{d - 2} \cdot \nu_-$$
by assuming that $\ell_\zeta = 0, 1, 2$.
Our first result, Theorem 6.1 below, was first obtained by Mong and
Kotschick \cite{17}.
\theorem{6.1} Let $\zeta$ define a wall of type $(w, p)$ with
$\ell_\zeta = 0$. Then,
$$[\mu_+(\alpha)]^d - [\mu_-(\alpha)]^d = (-1)^{h(\zeta)}
\cdot \left ({{\zeta \cdot \alpha} \over 2} \right )^d$$
for $\alpha \in H_2(X; \Zee)$. In other words,
$\delta^X_{w, p}(\Cal C_-, \Cal C_+) =
\delta(\Delta) \cdot (-1)^{h(\zeta)} \cdot ({\zeta}/2)^d$.
\endstatement
\par\noindent
{\it Proof.} There are two cases: $h(\zeta)> 0$ and $h(\zeta)= 0$.
In the first case when $h(\zeta)> 0$,
the formula follows immediately from Theorem 5.4. In the second case
when $h(\zeta)= 0$, we must have $\zeta^2 = p$
and $\zeta \cdot K_X = \zeta^2 + 2 = p + 2$ by Corollary 2.7.
Then $\frak M_+$ consists of $\frak M_-$ and an additional connected component
$E_{-\zeta}^{0, 0} \cong \Pee^{-p - 3}$. We have constructed
a universal sheaf $\Cal U$ over $X \times E_{-\zeta}^{0, 0}$:
$$0 \to \pi_1^*\Cal O_X(\Delta - F) \otimes \pi_2^*\lambda \to
\Cal U \to \pi_1^*\Cal O_X(F) \to 0$$
where $F$ is the unique divisor satisfying $(2F - \Delta) = \zeta$,
$\lambda$ is the line bundle corresponding to a hyperplane in
$E_{-\zeta}^{0, 0} \cong \Pee^{-p - 3}$,
and $\pi_1$ and $\pi_2$ are the natural projections of
$X \times E_{-\zeta}^{0, 0}$. Thus for $\alpha \in H_2(X; \Zee)$, we have
$$\mu_+(\alpha) = \mu_-(\alpha) - {1 \over 4} \cdot p_1(\Cal U)/\alpha
= \mu_-(\alpha) + a \lambda$$
where $a = ({\zeta \cdot \alpha})/2$. Since $h(\zeta) = 0$, we conclude that
$$\mu_+(\alpha)^d = \mu_-(\alpha)^d +
\left ({{\zeta \cdot \alpha} \over 2} \right )^d
= \mu_-(\alpha)^d + (-1)^{h(\zeta)} \cdot
\left ({{\zeta \cdot \alpha} \over 2} \right )^d. \qed$$
The proof of the next result is similar to the proof of Theorem 6.1.
\theorem{6.2} Let $\zeta$ define a wall of type $(w, p)$ with
$\ell_\zeta = 0$, let $d = -p - 3$. Then, for $\alpha \in H_2(X; \Zee)$,
we have
$$[\mu_+(\alpha)]^{d - 2} \cdot \nu_+ - [\mu_-(\alpha)]^{d - 2} \cdot \nu_-
= {1 \over 4} \cdot (-1)^{h(\zeta) - 1} \cdot
\left ({{\zeta \cdot \alpha} \over 2} \right)^{d - 2}. \qed$$
\endstatement
Next, we shall study the difference $\delta^X_{w, p}(\Cal C_-, \Cal C_+)$
when $\ell_\zeta = 1$. In this case, we have to know (5.7)
for $j = 2, 1, 0$. In view of Propositions 5.9 and 5.12,
it suffices to calculate (5.7) for $j = 0$. The following lemma
deals with this.
\lemma{6.3} Let $\zeta$ define a wall of type $(w, p)$ with
$\ell_\zeta = 1$. Then
$$S_0 = \sum_{k = 0}^1~ s_2(\Cal E_{\zeta}^{1 - k, k} \oplus
(\Cal E_{-\zeta}^{k, 1 - k})\spcheck) = (6 \zeta^2 + 2K_X^2).$$
\endstatement
\par\noindent
{\it Proof.} First, we compute the Chern classes of $\Cal E_{\zeta}^{1, 0}$.
Let notations be as in Lemma 5.11,
and set $\ell_\zeta = 1$ and $k = 0$ in Lemma 5.11.
Then $Ext^1 = 0$. Since $(H_{\ell_\zeta - k} \times H_k) = X$,
the codimension $2$ cycle $\Cal Z_1$ is exactly the diagonal in
$X \times (H_{\ell_\zeta - k} \times H_k) = X \times X$. Thus,
$\pi_{2*}(\pi_1^*\Cal O_X(\zeta) \otimes \Cal O_{\Cal Z_{\ell_\zeta - k}})
= \Cal O_X(\zeta)$. By Lemma 5.11 (i), the bundle $\Cal E_{\zeta}^{1, 0}$
sits in an exact sequence:
$$0 \to \Cal O_X(\zeta) \to \Cal E_{\zeta}^{1, 0} \cong
R^1\pi_{2*} \left (\pi_1^*\Cal O_X(\zeta) \otimes Hom \right )
\to \Cal O_X^{\oplus~ h(\zeta)} \to 0.$$
Thus, $c_1(\Cal E_{\zeta}^{1, 0}) = \zeta$ and
$c_2(\Cal E_{\zeta}^{1, 0}) = 0$.
Next, we compute the Chern classes of $\Cal E_{\zeta}^{0, 1}$.
Let $\ell_\zeta = 1$ and $k = 1$ in Lemma 5.11. Then,
$Ext^1 = \det (N)$ where $N$ is the normal bundle
of $\Cal Z_1$ in $X \times X$. Thus,
$$\pi_{2*} \left (\pi_1^*\Cal O_X(\zeta) \otimes Ext^1 \right )
= \Cal O_X(\zeta - K_X).$$
By Lemma 5.11 (i), the bundle $\Cal E_{\zeta}^{0, 1}$
sits in an exact sequence:
$$0 \to \Cal O_X^{\oplus~ h(\zeta)} \to \Cal E_{\zeta}^{0, 1}
\to \Cal O_X(\zeta - K_X) \to 0.$$
Thus, $c_1(\Cal E_{\zeta}^{0, 1}) = \zeta - K_X$ and
$c_2(\Cal E_{\zeta}^{0, 1}) = 0$. Replacing $\zeta$ by $-\zeta$ gives
$c_1(\Cal E_{-\zeta}^{0, 1}) = -\zeta - K_X$
and $c_2(\Cal E_{-\zeta}^{0, 1}) = 0$.
It follows that $c_1(\Cal E_{\zeta}^{1, 0} \oplus
(\Cal E_{-\zeta}^{0, 1})\spcheck) = 2\zeta + K_X$ and that
$$c_2(\Cal E_{\zeta}^{1, 0} \oplus (\Cal E_{-\zeta}^{0, 1})\spcheck)
= \zeta \cdot (\zeta + K_X) = \zeta^2 + \zeta \cdot K_X.$$
So we conclude that the Segre class
$s_2(\Cal E_{\zeta}^{1, 0} \oplus (\Cal E_{-\zeta}^{0, 1})\spcheck)$ is equal
to
$$c_1(\Cal E_{\zeta}^{1, 0} \oplus (\Cal E_{-\zeta}^{0, 1})\spcheck)^2
- c_2(\Cal E_{\zeta}^{1, 0} \oplus (\Cal E_{-\zeta}^{0, 1})\spcheck)
= 3\zeta^2 + 3 \zeta \cdot K_X + K_X^2.$$
Replacing $\zeta$ by $-\zeta$ gives
$s_2(\Cal E_{-\zeta}^{1, 0} \oplus (\Cal E_{\zeta}^{0, 1})\spcheck) =
3\zeta^2 - 3 \zeta \cdot K_X + K_X^2$. Therefore,
$$\align
S_0 &= \sum_{k = 0}^1~ s_2(\Cal E_{\zeta}^{1 - k, k} \oplus
(\Cal E_{-\zeta}^{k, 1 - k})\spcheck)\\
&= s_2(\Cal E_{\zeta}^{1, 0} \oplus (\Cal E_{-\zeta}^{0, 1})\spcheck)
+ s_2(\Cal E_{\zeta}^{0, 1} \oplus (\Cal E_{-\zeta}^{1, 0})\spcheck)\\
&= s_2(\Cal E_{\zeta}^{1, 0} \oplus (\Cal E_{-\zeta}^{0, 1})\spcheck)
+ s_2((\Cal E_{\zeta}^{0, 1})\spcheck \oplus \Cal E_{-\zeta}^{1, 0})\\
&= (3\zeta^2 + 3 \zeta \cdot K_X + K_X^2) +
(3\zeta^2 - 3 \zeta \cdot K_X + K_X^2)\\
&= 6 \zeta^2 + 2K_X^2.\qed\\
\endalign$$
Now we can compute the difference $\delta^X_{w, p}(\Cal C_-, \Cal C_+)$
when $\ell_\zeta = 1$.
\theorem{6.4} Let $\zeta$ define a wall of type $(w, p)$ with
$\ell_\zeta = 1$. Then,
$$[\mu_+(\alpha)]^d - [\mu_-(\alpha)]^d = (-1)^{h(\zeta) + 1} \cdot
\left \{ d(d - 1) \cdot a^{d - 2} \cdot \alpha^2 +
(2K_X^2 + 2d + 6) \cdot a^d \right \}$$
for $\alpha \in H_2(X; \Zee)$, where $a = (\zeta \cdot \alpha)/2$.
In other words, $\delta^X_{w, p}(\Cal C_-, \Cal C_+)$ is equal to
$$\delta(\Delta) \cdot (-1)^{h(\zeta) + 1} \cdot
\left \{ d(d - 1) \cdot \left(\zeta \over 2 \right)^{d - 2} \cdot q_X +
(2K_X^2 + 2d + 6) \cdot \left ({\zeta \over 2} \right )^d \right \}.$$
\endstatement
\par\noindent
{\it Proof.} From 5.4, 5.9, 5.12, and 6.3, we conclude that
$$\align
&\quad [\mu_+(\alpha)]^d - [\mu_-(\alpha)]^d\\
&= (-1)^{h(\zeta) + 1} \cdot d(d - 1) \cdot a^{d - 2} \cdot \alpha^2
+ (-1)^{h(\zeta) + 1} \cdot 8d \cdot a^d\\
&\quad\quad\quad\quad\quad\quad\quad\quad
+ (-1)^{h(\zeta) + 1} \cdot a^d \cdot (6 \zeta^2 + 2K_X^2)\\
&= (-1)^{h(\zeta) + 1} \cdot
\left \{ d(d - 1) \cdot a^{d - 2} \cdot \alpha^2 +
(2K_X^2 + 2d + 6) \cdot a^d \right \}. \qed\\
\endalign$$
For $[\mu_+(\alpha)]^{d - 2} \cdot \nu_+
- [\mu_-(\alpha)]^{d - 2} \cdot \nu_-$, we have the following.
\theorem{6.5} Let $\zeta$ define a wall of type $(w, p)$ with
$\ell_\zeta = 1$, let $d = -p - 3$. For $\alpha \in H_2(X; \Zee)$,
let $a = (\zeta \cdot \alpha)/2$. Then,
$[\mu_+(\alpha)]^{d - 2} \cdot \nu_+
- [\mu_-(\alpha)]^{d - 2} \cdot \nu_-$ is equal to
$${1 \over 4} \cdot (-1)^{h(\zeta)} \cdot \left [
(d - 2)(d - 3) \cdot a^{d - 4} \cdot \alpha^2
+ (2K_X^2 + 2d - 18) \cdot a^{d - 2} \right ].$$
\endstatement
\par\noindent
{\it Proof.} By Theorem 5.5,
$[\mu_+(\alpha)]^{d - 2} \cdot \nu_+
- [\mu_-(\alpha)]^{d - 2} \cdot \nu_-$ is equal to
$$\align
&\quad {1 \over 4} \cdot \sum_{j = 0}^{2} {{d - 2} \choose j} \cdot
(-1)^{h(\zeta) + j} \cdot a^{d - 2 - j} \cdot S_j
- (-1)^{h(\zeta)} \cdot a^{d - 2} \cdot \sum_{k = 0}^1
([{\Cal Z_{1 - k}}] + [{\Cal Z_{k}}])/x \\
&= {1 \over 4} \cdot \sum_{j = 0}^{2} {{d - 2} \choose j} \cdot
(-1)^{h(\zeta) + j} \cdot a^{d - 2 - j} \cdot S_j
- (-1)^{h(\zeta)} \cdot 2a^{d - 2}.\\
\endalign$$
By Proposition 5.9, Proposition 5.12, and Lemma 6.3, we have
$$S_2 = 2 \alpha^2, S_1 = -8a, S_0 = 6 \zeta^2 + 2K_X^2.$$
Therefore, we conclude that $[\mu_+(\alpha)]^{d - 2} \cdot \nu_+
- [\mu_-(\alpha)]^{d - 2} \cdot \nu_-$ is equal to
$${1 \over 4} \cdot (-1)^{h(\zeta)} \cdot \left [
(d - 2)(d - 3) \cdot a^{d - 4} \cdot \alpha^2
+ (2K_X^2 + 2d - 18) \cdot a^{d - 2} \right ]. \qed$$
In the rest of this section, we assume that $\ell_\zeta = 2$. The following
standard facts about double coverings can be found in \cite{2, 10}.
\lemma{6.6} Let $\phi: Y_1 \to Y_2$ be a double covering between
two smooth projective varieties with
$\phi_*\Cal O_{Y_1} = \Cal O_{Y_2} \oplus L^{-1}$
where $L$ is a line bundle on $Y_2$.
\roster
\item"{(i)}" $K_{Y_1} = \phi^*(K_{Y_2} \otimes L)$ and
$L^{\otimes 2} = \Cal O_{Y_2}(B)$
where $B$ is the branch locus in $Y_2$ and is the image of
the fixed set of the involution $\iota$ on $Y_1$;
\item"{(ii)}" If $D$ is a divisor on $Y_1$, then $\phi_*(\Cal O_{Y_1}(D))$
is a rank $2$ bundle on $Y_2$ with
$c_1(\phi_*(\Cal O_{Y_1}(D))) = \phi_*D - L$ and
$$c_2(\phi_*(\Cal O_{Y_1}(D))) = {1 \over 2} \cdot \left [(\phi_*D)^2
- \phi_*(D^2) - \phi_*D \cdot L \right].$$
\endroster
\endstatement
Next, we recall some standard facts about the Hilbert scheme
$H_2 = \Hilb ^2(X)$. Let $\Delta_0 \subset X \times X$ be the diagonal,
and let $\iota$ be the obvious involution on
$\tilde H_2 = \operatorname{Bl}_{\Delta_0} (X\times X)$,
the blowup of $X\times X$ along $\Delta_0$.
Let $E$ be the exceptional divisor of the blowup in $\tilde H_2$.
Then, $H_2 = \tilde H_2/\iota$ and the branch locus lies under $E$.
Let $\tilde \Cal Z_2 \subset X \times \tilde H_2$ be the pullback of
the codimension $2$ cycle $\Cal Z_2 \subset X \times H_2$.
Then, $\tilde \Cal Z_2$ splits into a union of two cycles $\tilde H_{12}$
and $\tilde H_{13}$ in $X \times \tilde H_2$, which are the proper transforms
in $X \times \tilde H_2$ of the two morphisms of $X \times X$ into
$X \times (X \times X)$: the first maps the first factor
in $X \times X$ diagonally into $X \times X$ which is the product of
the first and second factors in $X \times (X \times X)$,
while the second maps the first factor
in $X \times X$ diagonally into $X \times X$ which is the product of
the first and third factors in $X \times (X \times X)$.
Thus each $\tilde H_{1j}$ is isomorphic to
$\operatorname{Bl}_{\Delta_0} (X\times X)$, and the projection of each
to $\tilde H_2$ is an isomorphism. If $\alpha \in H_2(X; \Zee)$, then
$$[\tilde \Cal Z_2]/\alpha = \alpha \otimes 1 + 1 \otimes \alpha
= \alpha \otimes 1 + \iota^*(\alpha \otimes 1) \eqno(6.7)$$
where $\alpha \otimes 1$ and $1 \otimes \alpha$ are the pull-backs
of $\alpha$ by the two projections of $\tilde H_2$ to $X$.
Fix $x \in X$. Let $\tilde X_x$ be the pull-back of
$X \times x \subset X \times X$ to $\tilde H_2$.
Then, $\tilde X_x$ is isomorphic to the blow-up of $X$ at $p$ with
the exceptional divisor $(\tilde X_x \cap E)$; moreover,
$$[\tilde \Cal Z_2]/x = \tilde X_x + \iota^* \tilde X_x. \eqno(6.8)$$
It is known (see p. 685 in \cite{9}) that $\Cal Z_2$ is smooth.
Let $B$ be the branch locus of the natural double covering from
$\Cal Z_2$ to $H_2$. Then, $B \sim 2L$ for some divisor $L$ on $H_2$,
and the pull-back of $B \subset H_2$ to $\tilde H_2$ is $2E$.
Let $i: \Cal Z_2 \to X \times H_2$ be the embedding,
and $\pi_1$ and $\pi_2$ be the natural projections of $X \times H_2$
to $X$ and $H_2$ respectively.
In the following, we compute the Chern and Segre classes of
$\Cal E_{\zeta}^{2 - k, k}$ for $k = 0, 1, 2$.
The method is to use Lemma 5.11 together with Lemma 6.6.
We start with $\Cal E_{\zeta}^{2, 0}$.
\lemma{6.9} $c_3(\Cal E_{\zeta}^{2, 0}) = c_4(\Cal E_{\zeta}^{2, 0}) = 0$,
$c_1(\Cal E_{\zeta}^{2, 0}) = [\Cal Z_2]/\zeta - L$,
and
$$c_2(\Cal E_{\zeta}^{2, 0}) = {1 \over 2} \left [([\Cal Z_2]/\zeta)^2
- \zeta^2 \cdot X_x - [\Cal Z_2]/\zeta \cdot L \right]$$
where $x$ is any point on $X$, and $X_x$ stands for $[\Cal Z_2]/x$.
\endstatement
\proof Let notations be as in Lemma 5.11, and let $\ell_{\zeta} = 2$ and
$k = 0$. Then, $ Ext^1 = 0$. By Lemma 5.11 (i),
$\Cal E_{\zeta}^{2, 0}$ sits in an exact sequence
$$0 \to (\pi_2 \cdot i)_*(\pi_1 \cdot i)^*\Cal O_X(\zeta) \to
\Cal E_{\zeta}^{2, 0} \to [\Cal O_{H_2}]^{\oplus~ h(\zeta)} \to 0.$$
Since $(\pi_2 \cdot i)_*(\pi_1 \cdot i)^*\Cal O_X(\zeta)$ has rank $2$,
$c_3(\Cal E_{\zeta}^{2, 0}) = c_4(\Cal E_{\zeta}^{2, 0}) = 0$.
By Lemma 6.6 (ii),
$$c_1(\Cal E_{\zeta}^{2, 0}) = (\pi_2 \cdot i)_*(\pi_1 \cdot i)^*\zeta - L
= [\Cal Z_2]/\zeta - L$$
since $(\pi_2 \cdot i)_*(\pi_1 \cdot i)^*\zeta = [\Cal Z_2]/\zeta$;
moreover, we have
$$\align
c_2(\Cal E_{\zeta}^{2, 0}) &=
{1 \over 2} \left [((\pi_2 \cdot i)_*(\pi_1 \cdot i)^*\zeta)^2
- (\pi_2 \cdot i)_*((\pi_1 \cdot i)^*\zeta)^2 -
(\pi_2 \cdot i)_*(\pi_1 \cdot i)^*\zeta \cdot L \right]\\
&= {1 \over 2} \left [([\Cal Z_2]/\zeta)^2 - \zeta^2 \cdot X_x
- [\Cal Z_2]/\zeta \cdot L \right]\\
\endalign$$
since $(\pi_2 \cdot i)_*((\pi_1 \cdot i)^*\zeta)^2 = \zeta^2 \cdot
(\pi_2 \cdot i)_*(\pi_1 \cdot i)^*x = \zeta^2 \cdot [\Cal Z_2]/x
= \zeta^2 \cdot X_x$.
\endproof
The following follows from Lemma 6.9 and Remark 5.6.
\corollary{6.10} The Segre classes of the bundle $\Cal E_{\zeta}^{2, 0}$
are given by
$$\align
&s_1(\Cal E_{\zeta}^{2, 0}) = L - [\Cal Z_2]/\zeta\\
&s_2(\Cal E_{\zeta}^{2, 0}) = {1 \over 2} \left[[[\Cal Z_2]/\zeta]^2
- 3 [\Cal Z_2]/\zeta \cdot L + 2L^2 + \zeta^2 \cdot X_x \right]\\
&s_3(\Cal E_{\zeta}^{2, 0}) = [\Cal Z_2]/\zeta]^2 \cdot L -
2 [\Cal Z_2]/\zeta \cdot L^2 + L^3 - \zeta^2 \cdot X_x \cdot [\Cal Z_2]/\zeta
+ \zeta^2 \cdot X_x \cdot L\\
&s_4(\Cal E_{\zeta}^{2, 0}) = {(\zeta^2)^2 \over 2} - 5\zeta^2
- {5 \over 2} \zeta \cdot K_X + (6\chi(\Cal O_X) - K_X^2).\\
\endalign$$
Here we have identified degree $4$ classes with the corresponding integers.
\endstatement
\par\noindent
{\it Proof.} Since the computation is straightforward, we only calculate
$s_4(\Cal E_{\zeta}^{2, 0})$. For simplicity,
let $c_i$ denote the $i^{\text{th}}$ Chern class of $\Cal E_{\zeta}^{2, 0}$.
Note that $c_3 = c_4 = 0$ by Lemma 6.9. Thus,
$s_4(\Cal E_{\zeta}^{2, 0}) = c_1^4 - 3 c_1^2 c_2 + c_2^2$ by Remark 5.6.
Therefore,
$$\align
s_4(\Cal E_{\zeta}^{2, 0}) &= ([\Cal Z_2]/\zeta - L)^4 -
3 ([\Cal Z_2]/\zeta - L)^2 \cdot {1 \over 2} \left [([\Cal Z_2]/\zeta)^2
- \zeta^2 \cdot X_x - [\Cal Z_2]/\zeta \cdot L \right]\\
&\quad\quad\quad\quad\quad\quad
+ {1 \over 4} \left [([\Cal Z_2]/\zeta)^2
- \zeta^2 \cdot X_x - [\Cal Z_2]/\zeta \cdot L \right]^2\\
&= L^4 - {5 \over 2} \cdot [\Cal Z_2]/\zeta \cdot L^3 +
{7 \over 4} \cdot ([\Cal Z_2]/\zeta)^2 \cdot L^2 +
{3 \over 2}\zeta^2 \cdot X_x \cdot L^2\\
&\quad\quad\quad\quad\quad\quad
- {1 \over 4} ([\Cal Z_2]/\zeta)^4 + {1 \over 4}(\zeta^2)^2 \cdot X_x^2
+ \zeta^2 \cdot ([\Cal Z_2]/\zeta)^2 \cdot X_x\\
\endalign$$
since $([\Cal Z_2]/\zeta)^3 \cdot L = 0 =
[\Cal Z_2]/\zeta \cdot L \cdot X_x$. Now, we need a claim.
\claim{} Let $\alpha, \beta \in H_2(X; \Zee)$. Then, we have the following:
\roster
\item"{(i)}" $[\Cal Z_2]/\alpha \cdot [\Cal Z_2]/\beta \cdot X_x
= \alpha \cdot \beta$;
\item"{(ii)}" $X_x^2 = 1$;
\item"{(iii)}" $X_x \cdot L^2 = -1$;
\item"{(iv)}" $L^4 = 6\chi(\Cal O_X) - K_X^2$;
\item"{(v)}" $[\Cal Z_2]/\alpha \cdot L^3 = \alpha \cdot K_X$;
\item"{(vi)}" $[\Cal Z_2]/\alpha \cdot [\Cal Z_2]/\beta \cdot L^2
= - 2(\alpha \cdot \beta)$.
\endroster
\endstatement
\par\noindent
{\it Proof.} Let $\pi: \tilde H_2 \to H_2 = \tilde H_2/\iota$
be the quotient map. By (6.8), we have
$$\pi^*X_x = \pi^*([\Cal Z_2]/x) = [\tilde \Cal Z_2]/x =
(\tilde X_x + \iota^* \tilde X_x).$$
(i) Recall from (6.7) that
$\pi^*([\Cal Z_2]/\alpha) = [\tilde \Cal Z_2]/\alpha =
\alpha \otimes 1 + 1 \otimes \alpha$. Thus,
$$\align
[\Cal Z_2]/\alpha \cdot [\Cal Z_2]/\beta \cdot X_x
&= {1 \over 2} \cdot \pi^*([\Cal Z_2]/\alpha) \cdot \pi^*([\Cal Z_2]/\beta)
\cdot \pi^*X_x \\
&= {1 \over 2} \cdot (\alpha \otimes 1 + 1 \otimes \alpha) \cdot
(\beta \otimes 1 + 1 \otimes \beta) \cdot (\tilde X_x + \iota^* \tilde X_x)\\
&= \alpha \cdot \beta.\\
\endalign$$
(ii) Let $x_1 \in X$ be a point different from $x$. Then,
$$\align
X_x^2 &= X_x \cdot X_{x_1} =
{1 \over 2} \cdot \pi^*(X_x) \cdot \pi^*(X_{x_1}) \\
&= {1 \over 2} \cdot (\tilde X_x + \iota^* \tilde X_x) \cdot
(\tilde X_{x_1} + \iota^* \tilde X_{x_1}) \\
&= 1.\\
\endalign$$
(iii) Since $B \sim 2L$ and $\pi^*(B) = 2E$, $\pi^*(L) \sim E$. Thus,
$$X_x \cdot L^2 = {1 \over 2} \cdot (\tilde X_x + \iota^* \tilde X_x)
\cdot E^2 = \tilde X_x \cdot E^2 = (\tilde X_x \cdot E)^2 = -1.$$
(iv) Since $E = \Pee(N\spcheck)$ where $N$ is the normal bundle of $\Delta_0$
in $X \times X$, $-E|E = \xi$ is the tautological line bundle on $E$.
Since $N = T_{\Delta_0}$,
$$\xi^2 = -(\pi|E)^*c_1(N) \cdot \xi - c_2(N) =
(\pi|E)^*K_{\Delta_0} \cdot \xi + (K_X^2 - 12 \chi(\Cal O_X)).$$
It follows that $\xi^3 = (2K_X^2 - 12 \chi(\Cal O_X)) \cdot \xi$. Therefore,
$$L^4 = {1 \over 2} \cdot E^4 = -{1 \over 2} \cdot \xi^3
= 6\chi(\Cal O_X) - K_X^2.$$
(v) Note that $(\alpha \otimes 1)|E = (\pi|E)^*\alpha$
since ${\Delta_0} \cong X$. Thus,
$$[\Cal Z_2]/\alpha \cdot L^3 = {1 \over 2} \cdot
(\alpha \otimes 1 + 1 \otimes \alpha) \cdot E^3
= (\alpha \otimes 1) \cdot E^3
= (\pi|E)^*\alpha \cdot \xi^2 = \alpha \cdot K_X.$$
(vi) Again since $(\alpha \otimes 1)|E = (\pi|E)^*\alpha
= (1 \otimes \alpha)|E$, we have
$$\align
[\Cal Z_2]/\alpha \cdot [\Cal Z_2]/\beta \cdot L^2
&= {1 \over 2} \cdot (\alpha \otimes 1 + 1 \otimes \alpha) \cdot
(\beta \otimes 1 + 1 \otimes \beta) \cdot E^2\\
&= -2 \cdot (\pi|E)^*\alpha \cdot (\pi|E)^*\beta \cdot \xi \\
&= -2 (\alpha \cdot \beta).\qed \\
\endalign$$
We continue the calculation of $s_4(\Cal E_{\zeta}^{2, 0})$.
By Lemma 5.8 (i), $([\Cal Z_2]/\zeta)^4 = 3 (\zeta^2)^2$.
It follows from the above Claim with a straightforward computation that
$$s_4(\Cal E_{\zeta}^{2, 0}) = {(\zeta^2)^2 \over 2} - 5\zeta^2
- {5 \over 2} \zeta \cdot K_X + (6\chi(\Cal O_X) - K_X^2). \qed$$
Next, we compute the Chern and Segre classes of $\Cal E_{\zeta}^{0, 2}$
on $H_2$.
\lemma{6.11} $c_3(\Cal E_{\zeta}^{0, 2}) = c_4(\Cal E_{\zeta}^{0, 2}) = 0$,
$c_1(\Cal E_{\zeta}^{0, 2}) = [\Cal Z_2]/(\zeta - K_X) + L$,
and
$$c_2(\Cal E_{\zeta}^{0, 2}) = {1 \over 2} \left [
L \cdot [\Cal Z_2]/(\zeta - K_X) + [[\Cal Z_2]/(\zeta - K_X)]^2
- (\zeta - K_X)^2 \cdot X_x \right].$$
\endstatement
\proof Let $\ell_{\zeta} = 2$ and $k = 2$ in Lemma 5.11. By Lemma 6.6 (i),
$$(\det T_{\Cal Z_2})^{-1} = \Cal O_{\Cal Z_2}(K_{\Cal Z_2})
= (\pi_2 \cdot i)^*\Cal O_{H_2}(K_{H_2} + L).$$
Let $N_{\Cal Z_2}$ be the normal bundle of $\Cal Z_2$ in $X \times H_2$.
Since $\Cal Z_2$ is smooth and has codimension $2$ in $X \times H_2$,
$Ext^1 = Ext^1(I_{\Cal Z_2}, \Cal O_{X \times H_2})$
is isomorphic to
$$\det N_{\Cal Z_2} = i^*\det T_{X \times H_2} \otimes
(\det T_{\Cal Z_2})^{-1} =
\Cal O_{\Cal Z_2}((\pi_2 \cdot i)^*L - (\pi_1 \cdot i)^*K_X).$$
By Lemma 5.11 (i), $\Cal E_{\zeta}^{0, 2}$ sits in an exact sequence
$$0 \to [\Cal O_{H_2}]^{\oplus~ h(\zeta)} \to \Cal E_{\zeta}^{0, 2} \to
(\pi_2 \cdot i)_*\Cal O_{\Cal Z_2}((\pi_2 \cdot i)^*L +
(\pi_1 \cdot i)^*(\zeta - K_X)) \to 0.$$
Note that $(\pi_2 \cdot i)_*(\pi_2 \cdot i)^*L = 2L$. Thus, by Lemma 6.6 (ii),
$$c_1(\Cal E_{\zeta}^{0, 2}) = (\pi_2 \cdot i)_*\left[(\pi_2 \cdot i)^*L +
(\pi_1 \cdot i)^*(\zeta - K_X)\right ] - L = [\Cal Z_2]/(\zeta - K_X) + L.$$
Also, Lemma 6.6 (ii) together with a straightforward calculation gives
$$c_2(\Cal E_{\zeta}^{0, 2}) = {1 \over 2} \left [
L \cdot [\Cal Z_2]/(\zeta - K_X) + [[\Cal Z_2]/(\zeta - K_X)]^2
- (\zeta - K_X)^2 \cdot X_x \right ]$$
where we have used the projection formula
$$(\pi_2 \cdot i)_*[(\pi_2 \cdot i)^*L \cdot (\pi_1 \cdot i)^*(\zeta - K_X)]
= L \cdot (\pi_2 \cdot i)_*(\pi_1 \cdot i)^*(\zeta - K_X)$$
and the fact that $(\pi_2 \cdot i)_*(\pi_2 \cdot i)^*L^2 = 2L^2$.
\endproof
The following follows from Lemma 6.11 and Remark 5.6.
\corollary{6.12} The Segre classes of $\Cal E_{\zeta}^{0, 2}$ are given by
$$\align
&s_1(\Cal E_{\zeta}^{0, 2}) = [\Cal Z_2]/(K_X - \zeta) - L\\
&s_2(\Cal E_{\zeta}^{0, 2}) = {1 \over 2} \left[[[\Cal Z_2]/(\zeta - K_X)]^2
+ 3 [\Cal Z_2]/(\zeta - K_X) \cdot L + 2L^2
+ (\zeta - K_X)^2 \cdot X_x \right]\\
&s_3(\Cal E_{\zeta}^{0, 2}) = -[[\Cal Z_2]/(\zeta - K_X)]^2 \cdot L -
2 [\Cal Z_2]/(\zeta - K_X) \cdot L^2 - L^3\\
&\quad\quad\quad\quad\quad\quad
- (\zeta - K_X)^2 \cdot X_x \cdot [\Cal Z_2]/(\zeta - K_X)
- (\zeta - K_X)^2 \cdot X_x \cdot L\\
&s_4(\Cal E_{\zeta}^{0, 2}) =
{((K_X - \zeta)^2)^2 \over 2} - 5(K_X - \zeta)^2
- {5 \over 2} (K_X - \zeta) \cdot K_X + (6\chi(\Cal O_X) - K_X^2).\\
\endalign$$
\endstatement
\proof The calculation of $s_4(\Cal E_{\zeta}^{0, 2})$ is similar
to that of $s_4(\Cal E_{\zeta}^{2, 0})$ in Corollary 6.10.
\endproof
Note that $s_4(\Cal E_{\zeta}^{0, 2})$ may be
obtained from $s_4(\Cal E_{\zeta}^{2, 0})$ by replacing $\zeta$
by $K_X - \zeta$, and indeed this holds more generally for $s_i$ when we add
the sign $(-1)^i$.
Now we compute the Chern and Segre classes of
$\Cal E_{\zeta}^{1, 1}$ on $X \times X$.
\lemma{6.13} Let $\tau_1$ and $\tau_2$ be the two natural projections
of $X \times X$ to $X$, let ${\Delta_0}$ be the diagonal in $X \times X$, and
let $j\: \Delta _0 \to X\times X$ be the inclusion. Then
$$\align
&c_1(\Cal E_{\zeta}^{1, 1}) = \tau_1^*\zeta + \tau_2^*(\zeta - K_X) \\
&c_2(\Cal E_{\zeta}^{1, 1}) = \tau_1^*\zeta \cdot \tau_2^*(\zeta - K_X)
+ {\Delta_0} \\
&c_3(\Cal E_{\zeta}^{1, 1}) = \tau_1^*\zeta \cdot {\Delta_0} -
\tau_2^*(\zeta - K_X) \cdot {\Delta_0} - j_*K_{\Delta_0} \\
&c_4(\Cal E_{\zeta}^{1, 1}) = -{{K_{\Delta_0}^2} \over 2}.\\
\endalign$$
\endstatement
\proof Let $\ell_{\zeta} = 2$ and $k = 1$ in Lemma 5.11. Recall that
$\pi_1$ and $\pi_2$ are the natural projections of $X \times (X \times X)$
to $X$ and $(X \times X)$ respectively.
\claim{1} $\pi_{2*}\left (\pi_1^*\Cal O_X(\zeta)
\otimes Ext^1 \right) \cong \tau_2^*\Cal O_X(\zeta - K_X)
\otimes I_{\Delta_0}$.
\endstatement
\par\noindent
{\it Proof.} Let $\Delta_{12}$ be the diagonal in $X \times X$ which is formed
by the first and second factors in $X \times (X \times X)$, and let
$\Delta_{13}$ be the diagonal in $X \times X$ which is formed
by the first and third factors in $X \times (X \times X)$. Then,
$\Delta_{12} \times X$ and $\Delta_{13} \times X$ are smooth codimension $2$
subvarieties in $X \times (X \times X)$. Here it is understood that
the factor $X$ in $\Delta_{13} \times X$ is embedded as the second
factor in $X \times (X \times X)$. Moreover,
$\Delta_{12} \times X$ and $\Delta_{13} \times X$ intersect properly
along the diagonal $\Delta_{123}$ in $X \times X \times X$.
Thus, from Lemma 5.10 (iii), we conclude that
$$Ext^1 = Ext^1(I_{\Delta_{13} \times X}, I_{\Delta_{12} \times X})
\cong I \otimes \det N$$
where $N$ is the normal bundle $\Delta_{13} \times X$ in
$X \times (X \times X)$, and $I$ is the ideal sheaf of $\Delta_{123}$
in $\Delta_{13} \times X$. Now, the restriction of $\pi_2$ to
$\Delta_{13} \times X$ gives an isomorphism from $\Delta_{13} \times X$
to $X \times X$. Via this isomorphism,
$\Delta_{123}$ in $\Delta_{13} \times X$ is identified with
the diagonal ${\Delta_0}$ in $X \times X$, $\det N$ is identified with
$\tau_2^*(-K_X)$, and the restriction
$\pi_1^*\Cal O_X(\zeta)|(\Delta_{13} \times X)$
is identified with $\tau_2^*(\zeta)$. Therefore,
$$\pi_{2*}\left (\pi_1^*\Cal O_X(\zeta) \otimes Ext^1 \right) \cong
\pi_{2*}\left (\pi_1^*\Cal O_X(\zeta) \otimes I \otimes \det N \right)
= \tau_2^*\Cal O_X(\zeta - K_X) \otimes I_{\Delta_0}. \qed$$
Note that $\pi_{2*}(\pi_1^*\Cal O_X(\zeta) \otimes
\Cal O_{\Delta_{12} \times X}) = \tau_1^*(\zeta)$.
Thus by Lemma 5.11 (i) and Claim 1, we have a row exact sequence and
a column exact sequence
$$\matrix
&0&\\
&\downarrow&\\
&\tau_1^*(\zeta) &\\
&\downarrow&\\
0 \to & R^1\pi_{2*} \left (\pi_1^*\Cal O_X(\zeta) \otimes Hom \right )
& \to \Cal E_{\zeta}^{1, 1} \to
\tau_2^*\Cal O_X(\zeta - K_X) \otimes I_{\Delta_0} \to 0. \\
&\downarrow&\\
&[\Cal O_{X \times X}]^{\oplus~ h(\zeta)} &\\
&\downarrow&\\
&0&\\
\endmatrix \eqno (6.14)$$
In the next claim, we compute the Chern classes of $I_{\Delta_0}$.
Clearly, $c_0(I_{\Delta_0}) = 1$.
\claim{2} $c_1(I_{\Delta_0}) = 0$, $c_2(I_{\Delta_0}) = {\Delta_0}$,
$c_3(I_{\Delta_0}) = -j_*K_{\Delta_0}$,
$c_4(I_{\Delta_0}) = {K_{\Delta_0}^2}/2$.
\endstatement
\proof Note that $\Todd (N_{\Delta_0})^{-1} = 1 + K_{\Delta_0}/2 +
(K_{\Delta_0}^2/4 - \chi (\Cal O_{\Delta_0}))$.
By a formula on p.288 of \cite{12}
(a special case of the Grothendieck-Riemann-Roch Theorem),
$$\align
\ch (j!\Cal O_{\Delta_0}) &= j_*(\Todd (N_{\Delta_0})^{-1} \cdot
\ch (\Cal O_{\Delta_0})) = j_*(\Todd (N_{\Delta_0})^{-1})\\
&= {\Delta_0} + {{j_*K_{\Delta_0}} \over 2} +
j_*\left({{K_{\Delta_0}^2} \over 4} - \chi (\Cal O_{\Delta_0}) \right).\\
\endalign$$
Since $\ch (j!\Cal O_{\Delta_0})$ is just equal to
$\ch (j_*\Cal O_{\Delta_0})$, we obtain
$$\ch (I_{\Delta_0}) = \ch (\Cal O_{X \times X}) -
\ch (j_*\Cal O_{\Delta_0})
= 1 - {\Delta_0} - {{j_*K_{\Delta_0}} \over 2} -
j_*\left({{K_{\Delta_0}^2} \over 4} - \chi (\Cal O_{\Delta_0}) \right).$$
From this, the Chern classes of $I_{\Delta_0}$ follows immediately.
In particular,
$$c_4(I_{\Delta_0}) = {{{\Delta_0}^2} \over 2} +
j_*\left({{3K_{\Delta_0}^2} \over 2} - 6\chi (\Cal O_{\Delta_0}) \right)
= {K_{\Delta_0}^2 \over 2}$$
since ${\Delta_0}^2 = c_2(T_X) = 12 \chi (\Cal O_X) - K_X^2$
(see the Example 8.1.12 in \cite{12}).
\endproof
Now the calculation of the Chern classes of $\Cal E_{\zeta}^{1, 1}$ follows
from
(6.14) and Claim 2. In particular,
$$\align
c_4(\Cal E_{\zeta}^{1, 1}) &= -\tau_1^*\zeta \cdot \tau_2^*(\zeta -
K_X) \cdot {\Delta_0} - \tau_1^*\zeta \cdot j_*K_{\Delta_0}
+ \tau_2^*(\zeta - K_X)^2 \cdot {\Delta_0}\\
&\quad\quad\quad\quad + 2 \tau_2^*(\zeta - K_X) \cdot j_*K_{\Delta_0}
+ {{j_*K_{\Delta_0}^2} \over 2}= -{{K_{\Delta_0}^2} \over 2}\\
\endalign$$
since $\tau_1^*\zeta \cdot \tau_2^*(\zeta - K_X)
\cdot {\Delta_0} = \zeta \cdot (\zeta - K_X)$ and
$\tau_1^*\zeta \cdot j_*K_{\Delta_0} = \zeta \cdot K_X$.
\endproof
The next result follows immediately from Lemma 6.13 and Remark 5.6.
\corollary{6.15} Let notations be the same as in Lemma 6.13. Then
$$\align
&s_1(\Cal E_{\zeta}^{1, 1}) = -\tau_1^*\zeta - \tau_2^*(\zeta - K_X) \\
&s_2(\Cal E_{\zeta}^{1, 1}) = \tau_1^*\zeta^2 +
\tau_1^*\zeta \cdot \tau_2^*(\zeta - K_X) + \tau_2^*(\zeta - K_X)^2
- {\Delta_0} \\
&s_3(\Cal E_{\zeta}^{1, 1}) = -\tau_1^*\zeta^2 \cdot \tau_2^*(\zeta - K_X)
- \tau_1^*\zeta \cdot \tau_2^*(\zeta - K_X)^2\\
&\quad\quad\quad\quad\quad + \tau_1^*\zeta \cdot {\Delta_0}
+ 3 \tau_2^*(\zeta - K_X) \cdot {\Delta_0} + j_*K_{\Delta_0}\\
&s_4(\Cal E_{\zeta}^{1, 1}) = (12 \zeta \cdot K_X - 12 \zeta^2 - 3K_X^2).\qed
\endalign$$
\endstatement
We can now work out (5.7) explicitly for $\ell_\zeta = 2$ and $j = 2, 1, 0$.
For simplicity, let
$$S_j = \sum_{k = 0}^2~ S_{j, k} = \sum_{k = 0}^2~
([{\Cal Z_{2 - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j
\cdot s_{4 - j}(\Cal E_{\zeta}^{2 - k, k} \oplus
(\Cal E_{-\zeta}^{k, 2 - k})\spcheck). \eqno (6.16)$$
\lemma{6.17} $S_2 = 64a^2 + (12 \zeta^2 + 4K_X^2 - 20)\alpha^2$
where $a = (\zeta \cdot \alpha)/2$.
\endstatement
\proof Note that $s_i(\Cal E_{-\zeta}^{k, 2 - k})$ (respectively, $S_{j, k}$)
can be obtained from $s_i(\Cal E_{\zeta}^{k, 2 - k})$
(respectively, $(-1)^j \cdot S_{j, 2 - k}$) by replacing $\zeta$ by $-\zeta$.
Also, $S_{2, 2}$ is equal to
$$([{\Cal Z_2}]/\alpha)^2 \cdot s_2(\Cal E_{\zeta}^{0, 2} \oplus
(\Cal E_{-\zeta}^{2, 0})\spcheck)
= ([{\Cal Z_2}]/\alpha)^2 \cdot \left [
s_2(\Cal E_{\zeta}^{0, 2}) - s_1(\Cal E_{\zeta}^{0, 2}) \cdot
s_1(\Cal E_{-\zeta}^{2, 0}) + s_2(\Cal E_{-\zeta}^{2, 0}) \right].$$
Therefore, by Corollary 6.10 and Corollary 6.12, we obtain
$$S_{2, 2} + S_{2, 0} = 32a^2 + (6 \zeta^2 + 2K_X^2 - 12)\alpha^2
+ 2(\alpha \cdot K_X)^2.$$
Let $\tau_1$ and $\tau_2$ be the projections of $X \times X$ to $X$.
Then, by Corollary 6.15,
$$\align
S_{2, 1} &= (\tau_1^*\alpha + \tau_2^*\alpha)^2
\cdot s_2(\Cal E_{\zeta}^{1, 1} \oplus (\Cal E_{-\zeta}^{1, 1})\spcheck)\\
&= (\tau_1^*\alpha + \tau_2^*\alpha)^2 \cdot \left[
s_2(\Cal E_{\zeta}^{1, 1}) - s_1(\Cal E_{\zeta}^{1, 1}) \cdot
s_1(\Cal E_{-\zeta}^{1, 1}) + s_2(\Cal E_{-\zeta}^{1, 1}) \right]\\
&= 32a^2 + (6 \zeta^2 + 2K_X^2 - 8)\alpha^2 - 2(\alpha \cdot K_X)^2.\\
\endalign$$
It follows that $S_2 = (S_{2, 2} + S_{2, 0}) + S_{2, 1} =
64a^2 + (12 \zeta^2 + 4K_X^2 - 20)\alpha^2$.
\endproof
Next, adopting the same method as in the proof of Lemma 6.17,
we compute the values of $S_1$ and $S_0$ in the next two lemmas respectively.
\lemma{6.18} $S_1 = -(48 \zeta^2 + 16K_X^2 - 120) a$
where $a = (\zeta \cdot \alpha)/2$.
\endstatement
\proof In view of (6.16), we have to compute $S_{1, 2}, S_{1, 1}$,
and $S_{1, 0}$. Note that $S_{1, 0}$ can be obtained from $-S_{1, 2}$
by replacing $\zeta$ by $-\zeta$. Using Corollary 6.10 and Corollary 6.12,
we see that $(S_{1, 2} + S_{1, 0}) = -(24 \zeta^2 + 8K_X^2 - 72) a
- 6(\zeta \cdot K_X) (\alpha \cdot K_X)$.
Let $\tau_1$ and $\tau_2$ be the projections of $X \times X$ to $X$.
Then, by Corollary 6.15,
$$S_{1, 1} = (\tau_1^*\alpha + \tau_2^*\alpha)
\cdot s_3(\Cal E_{\zeta}^{1, 1} \oplus (\Cal E_{-\zeta}^{1, 1})\spcheck)
= -(24 \zeta^2 + 8K_X^2 - 48) a + 6(\zeta \cdot K_X) (\alpha \cdot K_X).$$
It follows that $S_1 = (S_{1, 2} + S_{1, 0}) + S_{1, 1} =
- (48 \zeta^2 + 16K_X^2 - 120) a$.
\endproof
\lemma{6.19} $S_0 = 18 (\zeta^2)^2 + (14K_X^2 - 105) \zeta^2
+ [2(K_X^2)^2 - 50K_X^2 + 96]$.
\endstatement
\par\noindent
{\it Proof.} We need to compute $S_{0, 2}, S_{0, 1}$, and $S_{0, 0}$.
Again, $S_{0, 0}$ can be obtained from $S_{0, 2}$
by replacing $\zeta$ by $-\zeta$. Using Corollary 6.10 and Corollary 6.12,
we see that
$$(S_{0, 2} + S_{0, 0}) = 9 (\zeta^2)^2 + (8K_X^2 - 63) \zeta^2
+ [(K_X^2)^2 - 43K_X^2 + 60].$$
By Corollary 6.15, $S_{0, 1} = 9 (\zeta^2)^2 + (6K_X^2 - 42) \zeta^2
+ [(K_X^2)^2 - 7K_X^2 + 36]$. Therefore,
$$S_0 = (S_{0, 2} + S_{0, 0}) + S_{0, 1} = 18 (\zeta^2)^2 +
(14K_X^2 - 105) \zeta^2 + [2(K_X^2)^2 - 50K_X^2 + 96]. \qed$$
Now we can calculate the difference $\delta^X_{w, p}(\Cal C_-, \Cal C_+)$
when $\ell_\zeta = 2$.
\theorem{6.20} Let $\zeta$ define a wall of type $(w, p)$ with
$\ell_\zeta = 2$. Then
$$[\mu_+(\alpha)]^d - [\mu_-(\alpha)]^d = (-1)^{h(\zeta)} \cdot \left \{
g_0 \cdot a^d
+ g_1 \cdot a^{d - 2} \cdot \alpha^2
+ g_2 \cdot a^{d - 4} \cdot (\alpha^2)^2 \right \}$$
for $\alpha \in H_2(X; \Zee)$, where $a$ stands for $(\zeta \cdot \alpha)/2$
and
$$\align
&g_2 = {{d!} \over {2! \cdot (d - 4)!}}\\
&g_1 = {d \choose 2} \cdot (4K_X^2 + 4d + 8)\\
&g_0 = 2d^2 + 2d \cdot K_X^2 + 2 (K_X^2)^2 + 13d + 20K_X^2 + 21.\\
\endalign$$
In other words, the difference $\delta^X_{w, p}(\Cal C_-, \Cal C_+)$
is equal to
$$\delta(\Delta) \cdot (-1)^{h(\zeta)} \cdot \left \{
g_0 \cdot \left({\zeta \over 2} \right)^d
+ g_1 \cdot \left({\zeta \over 2} \right)^{d - 2} \cdot q_X
+ g_2 \cdot \left({\zeta \over 2} \right)^{d - 4} \cdot q_X^2 \right \}.$$
\endstatement
\par\noindent
{\it Proof.} In view of Theorem 5.4 and the notation (6.16), we have
$$[\mu_+(\alpha)]^d - [\mu_-(\alpha)]^d = \sum_{j = 0}^{4}~ {d \choose j}
\cdot (-1)^{h(\zeta) + j} \cdot a^{d - j} \cdot S_j.$$
Now, $S_4$ and $S_3$ are given by Proposition 5.9 and Proposition 5.12
respectively; $S_2, S_1$, and $S_0$ are computed in the previous three lemmas.
So it follows that the coefficient
of $(-1)^{h(\zeta)} \cdot a^{d - 4} \cdot (\alpha^2)^2$ is equal to
$$g_2 = {{d!} \over {2! \cdot (d - 4)!}}.$$
Similarly, also keeping in mind that $\zeta^2 = (p + 8)
= (5 - d)$, we have
$$\align
&g_1 = {d \choose 2} \cdot (12 \zeta^2 + 4K_X^2 + 16d - 52)
= {d \choose 2} \cdot (4K_X^2 + 4d + 8)\\
&g_0 = 64 \cdot {d \choose 2}
+ (48 \zeta^2 + 16K_X^2 - 120) \cdot d~ + \\
&\quad\quad\quad + \left [18(\zeta^2)^2 + 14 \cdot \zeta^2 \cdot K_X^2
+ 2 (K_X^2)^2 - 105 \zeta^2 - 50 K_X^2 + 96 \right ]\\
&\quad= 2d^2 + 2d \cdot K_X^2 + 2 (K_X^2)^2 + 13d + 20K_X^2 + 21. \qed \\
\endalign$$
\corollary{6.21} Let $\zeta$ define a wall of type $(w, p)$ with
$\ell_\zeta \le 2$. Then, the difference $\delta^X_{w, p}(\Cal C_-, \Cal C_+)$
of Donaldson polynomial invariants is
a polynomial in $\zeta$ and $q_X$ with coefficients involving only
$\zeta^2$, homotopy invariants of $X$, and universal constants.
\endstatement
\proof Follows from Theorems 6.1, 6.4, and 6.20.
\endproof
Finally, we compute the difference $[\mu_+(\alpha)]^{d - 2} \cdot \nu_+
- [\mu_-(\alpha)]^{d - 2} \cdot \nu_-$ for $\ell_\zeta = 2$.
\theorem{6.22} Let $\zeta$ define a wall of type $(w, p)$ with
$\ell_\zeta = 2$, and let $d = -p - 3$. Then,
$[\mu_+(\alpha)]^{d - 2} \cdot \nu_+
- [\mu_-(\alpha)]^{d - 2} \cdot \nu_-$ is equal to
$${1 \over 4} \cdot (-1)^{h(\zeta) + 1} \cdot \left \{
{\tilde g}_0 \cdot a^{d - 2}
+ {\tilde g}_1 \cdot a^{d - 4} \cdot \alpha^2
+ {\tilde g}_2 \cdot a^{d - 6} \cdot (\alpha^2)^2 \right \}$$
for $\alpha \in H_2(X; \Zee)$, where $a$ stands for $(\zeta \cdot \alpha)/2$
and
$$\align
&{\tilde g}_2 = {{(d - 2)!} \over {2! \cdot (d - 6)!}}\\
&{\tilde g}_1 = {{d - 2} \choose 2} \cdot (4K_X^2 + 4d - 40)\\
&{\tilde g}_0 = 2d^2 + 2d \cdot K_X^2 + 2 (K_X^2)^2 - 35d - 28K_X^2 - 99.\\
\endalign$$
\endstatement
\proof By Theorem 5.5, $[\mu_+(\alpha)]^{d - 2} \cdot \nu_+
- [\mu_-(\alpha)]^{d - 2} \cdot \nu_-$ is equal to
$${1 \over 4} \cdot \sum_{j = 0}^4 {{d - 2} \choose j} \cdot
(-1)^{h(\zeta) + 1 + j} \cdot a^{d - 2 - j} \cdot S_j
- \sum_{j = 0}^2 {{d - 2} \choose j} \cdot
(-1)^{h(\zeta) + 1 + j} \cdot a^{d - 2 - j} \cdot T_j$$
where for simplicity we have defined $T_j$ as
$$T_j = \sum_{k = 0}^2 T_{j, k} = \sum_{k = 0}^2
([{\Cal Z_{2 - k}}]/\alpha + [{\Cal Z_{k}}]/\alpha)^j \cdot
([{\Cal Z_{2 - k}}] + [{\Cal Z_{k}}])/x \cdot
s_{2 - j}(\Cal E_{\zeta}^{2 - k, k}
\oplus (\Cal E_{-\zeta}^{k, 2 -k})\spcheck).$$
Next, we compute $T_0$. Using Corollary 6.10 and Corollary 6.12, we obtain
$$\align
T_{0, 0} &= X_x \cdot s_2(\Cal E_{\zeta}^{2, 0}
\oplus (\Cal E_{-\zeta}^{0, 2})\spcheck) \\
&= X_x \cdot \left [s_2(\Cal E_{\zeta}^{2, 0}) - s_1(\Cal E_{\zeta}^{2, 0})
\cdot s_1(\Cal E_{-\zeta}^{0, 2}) + s_2(\Cal E_{-\zeta}^{0, 2}) \right ]\\
&= (3\zeta^2 + 3 \zeta \cdot K_X + K_X^2 - 3).\\
\endalign$$
Note that $T_{0, 2}$ can be obtained from $T_{0, 0}$
by replacing $\zeta$ by $-\zeta$. Thus,
$$T_{0, 2} = (3\zeta^2 - 3 \zeta \cdot K_X + K_X^2 - 3).$$
Similarly, using Corollary 6.15, we get $T_{0, 1} = (6\zeta^2 + 2K_X^2 - 4)$.
Therefore,
$$T_0 = \sum_{k = 0}^2 T_{0, k} = (12\zeta^2 + 4K_X^2 - 10).$$
By similar but much simpler arguments, we conclude that
$T_1 = -16a$ and $T_2 = 4 \alpha^2$.
From (5.9), (5.12), (6.17), (6.18), and (6.19), we have
$$\align
&S_4 = 12 (\alpha^2)^2, \\
&S_3 = -48 a \cdot \alpha^2,\\
&S_2 = 64a^2 + (12 \zeta^2 + 4K_X^2 - 20)\alpha^2, \\
&S_1 = -(48 \zeta^2 + 16K_X^2 - 120) a, \\
&S_0 = 18 (\zeta^2)^2 + (14K_X^2 - 105) \zeta^2
+ [2(K_X^2)^2 - 50K_X^2 + 96].\\
\endalign$$
Putting all these together, we see that
$[\mu_+(\alpha)]^{d - 2} \cdot \nu_+
- [\mu_-(\alpha)]^{d - 2} \cdot \nu_-$ is equal to
$${1 \over 4} \cdot (-1)^{h(\zeta) + 1} \cdot \left \{
{\tilde g}_0 \cdot a^{d - 2}
+ {\tilde g}_1 \cdot a^{d - 4} \cdot \alpha^2
+ {\tilde g}_2 \cdot a^{d - 6} \cdot (\alpha^2)^2 \right \}$$
where ${\tilde g}_0, {\tilde g}_1$, and ${\tilde g}_2$ are as defined in the
statement of Theorem 6.22 above.
\endproof
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\enddocument
|
1994-10-28T05:20:14 | 9410 | alg-geom/9410028 | en | https://arxiv.org/abs/alg-geom/9410028 | [
"alg-geom",
"math.AG"
] | alg-geom/9410028 | Rick Miranda | Bruce Crauder and Rick Miranda | Quantum Cohomology of Rational Surfaces | 31 pages, AMS-LaTeX Version 1.1 | null | null | null | null | In this article formulas for the quantum product of a rational surface are
given, and used to give an algebro-geometric proof of the associativity of the
quantum product for strict Del Pezzo surfaces, those for which $-K$ is very
ample. An argument for the associativity in general is proposed, which also
avoids resorting to the symplectic category. The enumerative predictions of
Kontsevich and Manin concerning the degree of the rational curve locus in a
linear system are recovered. The associativity of the quantum product for the
cubic surface is shown to be essentially equivalent to the classical
enumerative facts concerning lines: there are $27$ of them, each meeting $10$
others.
| [
{
"version": "v1",
"created": "Thu, 27 Oct 1994 15:01:42 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Crauder",
"Bruce",
""
],
[
"Miranda",
"Rick",
""
]
] | alg-geom | \section*{Introduction}
The purpose of this article is to give an algebro-geometric
description
of the quantum cohomology ring for a general rational surface
$X$.
By a ``general'' rational surface we mean one in which all linear
systems
have the expected dimension, and in which the locus of rational
curves
in each linear system also has the expected dimension.
We understand that it is not known whether the general blowup of
the plane in ten or more general points
is general in this sense;
however we proceed anyway, developing the quantum cohomology ring
based on the expected linear systems.
In this sense the theory is entirely a numerical one.
We view the genericity assumption on the surface $X$
as capable of replacing the genericity assumptions
for the symplectic geometry on which the quantum theory is usually
based
(see for example \cite{ruan-tian,siebert-tian} and a forthcoming
paper
of Grassi \cite{grassi}
which also addresses aspects of the quantum cohomology of
rational surfaces).
It seems to us desirable to have a description of the quantum
product
which is based on algebraic geometry rather than symplectic
geometry,
and this is what we have tried to offer in the case of rational
surfaces.
In Section \ref{sectionX3class} we explain how classes in the
triple product $X^3$ are used to define classes on $X$
itself,
via the K\"unneth formula; this is formal.
In Section \ref{sectionTPclass} we introduce the ``triple-point''
classes,
or Gromov-Witten classes,
from which the quantum product is defined.
Gromov-Witten classes were first rigorously defined by Ruan
\cite{ruan} using symplectic deformations.
We have found it convenient to define these classes
(which measure rational curves on $X$ with $3$ marked points
satisfying certain geometric conditions)
in terms of the locus of such curves inside linear systems on
$X$,
rather than using the space of maps from ${\Bbb P}^1$ to $X$.
Other definitions (see
\cite{kontsevich-manin,ruan,siebert-tian,witten1,witten2})
use the space of maps instead; for rational surfaces this seems
unnecessary, at least for the definitions,
and slightly less transparent for our construction.
In Section \ref{sectiondefqntm} we give the definition
of the quantum cohomology ring.
As noted above, the quantum product is defined in terms of classes
of loci of rational curves on $X$; not all such loci
or cohomology classes appear in the definition,
and we call those that do ``relevant'' classes.
In Section \ref{strictrelevantsection} we enumerate the relevant
classes
for the general strict Del Pezzo surfaces,
that is, for the general blow-up of the plane at $6$ or fewer
points.
(These are the surfaces which are Fano surfaces in the strictest sense,
namely that $-K$ is very ample.)
In Section \ref{sectionformula} we give explicit formulas
for the quantum product in terms of the relevant classes on
$X$,
and using these formulas in Section \ref{sectionordinary} we show
that there is a natural map from the quantum cohomology ring
to the ordinary cohomology ring.
In Section \ref{sectionexamples} we explicitly compute
the quantum cohomology rings using the formulas
for the minimal rational strict Del Pezzo surfaces,
namely ${\Bbb P}^2$, ${\Bbb F}_0 = {\Bbb P}^1\times{\Bbb P}^1$,
and ${\Bbb F}_1$.
Quantum cohomology seems to be an admixture of homology and
cohomology;
as such, a functoriality property seems elusive
(e.g., should it be contravariant or covariant?).
We show in Section \ref{sectionfunctor}
that there is at least some degree of functoriality
for a blowup. We use this to give a new proof of the associativity
of the quantum product for the general strict Del Pezzo surface
in Section \ref{sectionSDPassoc}; the functoriality property is
strong
enough so that it suffices to check associativity on the general
$6$-fold
blowup of the plane.
Moreover the results hold without the genericity assumptions on the
$6$-fold blowup mentioned above, although it is known in this case
that the general $n$-fold blowup of the plane is general in the
above sense
for $n \leq 9$.
We view this section as the primary technical contribution
of the paper, namely the verification of the associativity of the
quantum
product using algebro-geometric techniques rather than methods from
symplectic geometry.
Associativity in general is a tricky business;
the original heuristic arguments from physics
have been made precise only using symplectic geometry
(\cite{mcduff-salamon,ruan-tian}). We have given in Section
\ref{sectionassoc}
an alternate approach using algebraic geometry.
The sketch which we offer here is related to the argument
given in \cite{witten2}.
Finally in Section \ref{sectionenum} we draw some of the
enumerative
consequences of the associativity of the quantum product.
These lead in particular to the formulas appearing in
\cite{kontsevich-manin}, interpreted properly.
As a small application we show that associativity for the general
cubic
surface is essentially equivalent to the standard enumerative facts
concerning lines: there are $27$ of them and each meets $10$
others.
The authors would like to thank Igor Dolgachev,
who inspired the second author to think about these questions
with an excellent lecture on the subject. We also profited
greatly from conversations with Antonella Grassi, Sheldon Katz,
and David Morrison.
\section{Classes on the triple product}
\label{sectionX3class}
Let $X$ be a general rational surface;
by this we mean firstly that
if $X$ is obtained by blowing up $n$ points $p_i$,
creating exceptional curves $E_i$, then for every $d$ and $m_i$,
the linear system $|dH - \sum_{i=1}^n m_i E_i|$ has the expected
dimension
(where $H$ is the pullback from the plane of the line class);
this expected dimension is ${1\over 2}[d^2+3d - \sum_i m_i(m_i+1)]$
unless this is negative.
Secondly we also assume that the locus of irreducible rational
curves in this
system has codimension equal to the arithmetic genus of the general
curve
in the system, which is the expected codimension.
The ordinary integral cohomology $H^*(X) = H^*(X,{\Bbb Z})$
is
\begin{eqnarray*}
H^0(X) &=& {\Bbb Z}[X], \\
H^1(X) &=& \{0\}, \\
H^2(X) &\cong& {\Bbb Z}^\rho, \\
H^3(X) &=& \{0\}, \\
H^4(X) &=& {\Bbb Z}[p], \text{ and } \\
H^i(X) &=& \{0\} \text{ for } i \geq 5,
\end{eqnarray*}
where here $[X]$ is the fundamental class of the surface
$X$ itself,
and $[p]$ is the class of a single point.
The Picard number $\rho$ is the rank of the $H^2$ group.
The triple product $X^3 = X \times X \times X$
is then an algebraic six-fold.
Its cohomology is (by the K\"unneth Theorem)
the triple tensor product of the cohomology of $X$,
and is therefore generated over ${\Bbb Z}$ by tensors of the form
$\alpha \otimes \beta \otimes \gamma$,
where $\alpha$, $\beta$, and $\gamma$ are either $[X]$, $[p]$,
or generators for $H^2(X,{\Bbb Z})$.
In particular all cohomology is even-dimensional.
Suppose that $[A]$ is a cohomology class in the triple product,
so that $[A] \in H^{2d}(X^3)$,
where $d$ is the complex codimension of the class.
Suppose that we choose $\alpha$ and $\beta$ to be homogeneous
elements
of $H^*(X)$, of degrees $2a$ and $2b$,
such that
\begin{equation}
\label{dimcondition}
4 \leq a + b + d \leq 6.
\end{equation}
In this case if we let $c = 6 - a - b - d$,
then for any class $\gamma \in H^{2c}(X)$,
the class $(\alpha\otimes\beta\otimes\gamma)$
will have the complementary dimension to the class $[A]$,
and therefore the intersection product
\[
[A] \cdot (\alpha \otimes \beta \otimes \gamma) \in {\Bbb Z}
\]
will be defined.
We therefore obtain a linear functional
\[
\Phi_{[A]}(\alpha,\beta): H^{2c}(X) \to {\Bbb Z}
\]
which by duality must be represented
by intersection with a cohomology class in $H^{4-2c}(X)$.
Call this cohomology class $\phi_{[A]}(\alpha,\beta)$;
in this case we have by definition
that for all $\gamma \in H^{2c}(X)$,
\[
[A] \cdot (\alpha \otimes \beta \otimes \gamma)
= \phi_{[A]}(\alpha,\beta) \cdot \gamma,
\]
and indeed by duality this characterizes the class
$\phi_{[A]}(\alpha,\beta)$.
Note that the intersection on the left side of this formula
is intersection in the cohomology of the triple product $X^3$,
while the intersection on the right side is intersection
in the cohomology of $X$.
The element $\phi_{[A]}(\alpha,\beta)$, by definition,
is linear in both $\alpha$ and $\beta$.
For notational sanity we declare $\phi_{[A]}(\alpha,\beta) = 0$
unless we have the dimension condition that $4 \leq a + b + d \leq
6$.
\begin{example}
\label{phiDelta}
Let $[\Delta]$ be the class of the diagonal
$\Delta \subset X \times X \times X$.
Since $\Delta$ has complex codimension $d = 4$,
$[\Delta] \in H^8(X^3)$.
Let $a$, $b$, and $c$ be non-negative integers such that $a+b \leq
2$
(this is (\ref{dimcondition}))
and $c = 2 - a - b$.
In this case if $\alpha$, $\beta$, and $\gamma$ are classes in
$H^*(X)$
of degrees $a$, $b$, and $c$ respectively,
the intersection product $\alpha \cdot \beta \cdot \gamma$
is defined in ${\Bbb Z}$;
this is just cup product to $H^4(X)$, then taking the degree.
In particular it is equal to $(\alpha \cup \beta) \cdot \gamma$.
Moreover $[\Delta] \cdot \alpha\otimes\beta\otimes\gamma$
is equal to this triple intersection.
Hence we see that $\phi_{[\Delta]}(\alpha,\beta) =
\alpha\cup\beta$.
\end{example}
\begin{example}
Let $[A]$ be the fundamental class of $X^3$;
this has codimension $d = 0$,
and lies in $H^0(X^3)$.
Indeed, it is equal to $[X]\otimes[X]\otimes[X]$.
Suppose that $a$, $b$, and $c=6-a-b$ are possible
complex codimensions in $X$,
satisfying (\ref{dimcondition}),
which is that $4 \leq a+b \leq 6$;
since $X$ is a surface, we must have $a = b = c = 2$.
If $\alpha$, $\beta$, and $\gamma$ are classes in $H^4(X)$,
by linearity for the computation we may take all three classes
equal to the
class $[p]$ of a point.
In this case we obviously have $[A] \cdot [p]\otimes[p]\otimes[p]
= 1$.
Therefore $\phi_{[A]}([p],[p]) = [X]$,
and unless both $\alpha$ and $\beta$ lie in $H^4(X)$,
$\phi_{[A]}(\alpha,\beta) = 0$.
\end{example}
\begin{example}
\label{exCCC}
Let $C$ be an divisor on $X$,
and let $[A] = [C] \otimes [C] \otimes [C] \in H^6(X^3)$.
Suppose that $a$, $b$, and $c=6-a-b$ are possible
complex codimensions in $X$,
satisfying (\ref{dimcondition}),
which is that $1 \leq a+b \leq 3$,
and $\alpha$, $\beta$, and $\gamma$ are classes in $H^*(X)$
of degrees $a$, $b$, and $c$ respectively.
Then $[A] \cdot \alpha\otimes\beta\otimes\gamma =
([C]\cdot \alpha)([C]\cdot \beta)([C]\cdot \gamma)$,
which is zero unless $a=b=c=1$,
and $\alpha$, $\beta$, $\gamma$ are divisor classes.
Therefore in this case
$\phi_{[A]}(\alpha,\beta) = ([C]\cdot \alpha)([C]\cdot \beta)[C]$
for divisor classes $\alpha$ and $\beta$,
and is zero otherwise.
\end{example}
\begin{example}
Let $C$ be an divisor on $X$,
and let $[A] = [C] \otimes [C] \otimes [X] \in H^4(X^3)$.
Suppose that $\alpha$ and $\beta$ are divisor classes in
$H^2(X)$.
Then $[A] \cdot \alpha\otimes\beta\otimes[p] =
([C]\cdot \alpha)([C]\cdot \beta)$.
Therefore in this case
$\phi_{[A]}(\alpha,\beta) = ([C]\cdot \alpha)([C]\cdot
\beta)[X]$
for divisor classes $\alpha$ and $\beta$,
and is zero otherwise.
\end{example}
\begin{example}
Let $C$ be an divisor on $X$,
and instead let $[A] = [C] \otimes [X] \otimes [C] \in
H^4(X^3)$.
Suppose that $\alpha$ and $\gamma$ are divisor classes in
$H^2(X)$.
Then $[A] \cdot \alpha\otimes[p]\otimes\gamma =
([C]\cdot \alpha)([C]\cdot \gamma)$.
Therefore in this case
$\phi_{[A]}(\alpha,[p]) = ([C]\cdot \alpha)[C]$
for a divisor class $\alpha$
(and is zero otherwise).
\end{example}
\begin{example}
Let $C$ be an divisor on $X$,
and let $[A] = [C] \otimes [X] \otimes [X] \in
H^2(X^3)$.
Suppose that $\alpha$ is a divisor classes in $H^2(X)$.
Then $[A] \cdot \alpha\otimes[p]\otimes[p] =
([C]\cdot \alpha)$.
Therefore in this case
$\phi_{[A]}(\alpha,[p]) = ([C]\cdot \alpha)[X]$
for a divisor class $\alpha$
(and is zero otherwise).
\end{example}
\begin{example}
Let $C$ be an divisor on $X$,
and instead let $[A] = [X] \otimes [X] \otimes [C] \in
H^2(X^3)$.
Then for any divisor class $\gamma$ in $H^2(X)$,
we have $[A] \cdot [p]\otimes[p]\otimes\gamma =
([C]\cdot \gamma)$.
Therefore in this case
$\phi_{[A]}([p],[p]) = [C]$.
\end{example}
\section{Three-point classes on the triple product}
\label{sectionTPclass}
Fix a divisor class $[C] \in H^2(X)$,
such that $|C|$ has no fixed components and is non-empty.
Define the locus ${\cal R}_{[C]} \subset |C|$
representing irreducible rational curves with only nodes as
singularities.
Inside the product ${\cal R}_{[C]} \times X^3$
consider the incidence correspondence
\[
{\cal S}_{[C]} = \{(C,p_1,p_2,p_3)\;|\; \text{ the points }p_i
\text{ are distinct smooth points on the curve }C\}.
\]
Let $\overline{{\cal S}}_{[C]}$ be the closure of this subvariety
inside $|C|\times X^3$.
The second projection gives a regular map
$\pi:\overline{{\cal S}}_{[C]} \to X^3$.
We define the {\em three-point class} $[A_{[C]}]$
to be $\pi_*[\overline{{\cal S}}_{[C]}]$,
the image of the fundamental class.
(This is a priori in homology,
but we consider it in cohomology
via duality.)
Note that $\dim {\cal S}_{[C]} = 3 + \dim {\cal R}_{[C]}$.
A degenerate version of this locus is obtained
when we allow the cohomology class $[C]\in H^2(X)$ to be
trivial;
we declare in this case that the three-point class $[A_{[0]}]$ is
the class
of the diagonal $\Delta$.
It is an elementary matter to compute the dimensions
of these three-point classes,
in terms of the numerical characters of the class $[C]$.
Since $X$ is a general rational surface,
the general member of the linear system $|C|$
is smooth, and the system has the expected dimension,
which is $(C\cdot C) +1-p_a(C)$;
here $p_a(C)$ is the arithmetic genus, and equals
$p_a(C) = 1 + (C \cdot C + K_{X})/2$ by Riemann-Roch.
Imposing a node on a member of $|C|$ is one condition;
hence the locus of nodal rational curves ${\cal R}_{[C]}$
has dimension $\dim |C| - p_a(C) = (C \cdot C) + 1 - 2p_a(C)$
by our general assumption on $X$.
Therefore the dimension of the incidence locus ${\cal S}_{[C]}$
is $(C \cdot C) + 4 - 2p_a(C)$,
which we may re-write as
\[
\dim {\cal S}_{[C]} = 2 - (C \cdot K_{X}).
\]
To be more explicit,
suppose that $[C] = dH - \sum_{i=1}^n m_i E_i$,
where $H$ is the pullback of the line class from ${\Bbb P}^2$
and $E_i$ is the class of the exceptional curve over the blown up
point $p_i$.
Then $K_{X} = -3H + \sum_i E_i$,
so that $(C \cdot K_{X}) = -3d + \sum_i m_i$.
Hence we have
\[
\dim {\cal S}_{[C]} = 3d +2 - \sum_i m_i.
\]
The only classes of curves on $X$ which are not of this form
are the classes of the exceptional curves $E_i$ themselves.
Here $|E_i|$ is a single point (the only member is $E_i$ itself)
and $E_i$ is a smooth rational curve;
so $\dim {\cal R}_{[E_i]} = 0$ and $\dim {\cal S}_{[E_i]} = 3$.
(Actually the formula holds in this case also,
with $d=0$, $m_i = -1$, and $m_j = 0$ for $j \neq i$.)
\begin{definition}
A class $[C] \in H^2(X)$ is {\em relevant} (for quantum
cohomology)
if either $[C] = 0$ or ${\cal R}_{[C]} \neq \emptyset$
and $\dim {\cal S}_{[C]} \leq 6$.
\end{definition}
If $\dim {\cal S}_{[C]} > 6$, then the image of the fundamental
class
of its closure in the six-dimensional variety $X^3$
is trivial. Therefore all non-relevant classes induce a trivial
three-point class $[A_{[C]}]$.
Given the class $[A_{[C]}]$, they induce as noted in the previous
section
classes $\phi_{[A_{[C]}]}(\alpha,\beta)$ in the cohomology of
$X$.
We will abbreviate the notation for these classes and write simply
$\phi_{[C]}(\alpha,\beta)$.
We note that there is an obvious $S_3$-symmetry to the three-point
classes,
in the sense that
\[
[A_{[C]}] \cdot \alpha_1 \otimes \alpha_2 \otimes \alpha_3
=
[A_{[C]}] \cdot
\alpha_{\sigma(1)} \otimes \alpha_{\sigma(1)} \otimes
\alpha_{\sigma(1)}
\]
for any permutation $\sigma \in S_3$.
This is simply because the locus ${\cal S}_{[C]}$ is
$S_3$-invariant.
As a consequence of this, we see that the $\phi$-classes are
symmetric:
\[
\phi_{[C]}(\alpha,\beta) = \phi_{[C]}(\beta,\alpha)
\]
and of course they are bilinear in $\alpha$ and $\beta$.
\begin{example}
\label{phi0}
If we start with the trivial class $[C] = 0$,
then the three-point class $[A_{[0]}]$ is the class of the
diagonal.
Hence as we have noted above,
for any classes $\alpha$ and $\beta$ in $H^*(X)$,
\[
\phi_0(\alpha,\beta) = \alpha \cup \beta.
\]
\end{example}
\begin{example}
Suppose that $E$ is a $(-1)$-curve on $X$,
that is, a smooth rational curve with $(E \cdot E) = -1$.
The only member of the linear system $|E|$ is the curve $E$ itself;
the locus ${\cal R}_{[E]}$ of nodal rational curves in $|E|$
is the whole system $|E| = \{E\}$.
The incidence locus ${\cal S}_{[E]} = \{E\} \times E \times E
\times E$
with the large diagonal removed;
its closure $\overline{{\cal S}}_{[E]} = \{E\} \times E \times E
\times E$.
Hence the image under the projection to $X^3$
is $E\times E\times E$, and the class $[A_{[E]}] =
[E]\otimes[E]\otimes[E]$.
Hence by the computation in Example \ref{exCCC},
we have
\[
\phi_{[E]}(\alpha,\beta) = ([E]\cdot \alpha)([E]\cdot \beta)[E]
\]
for divisor classes $\alpha$ and $\beta$,
and is zero unless $\alpha$ and $\beta$ are both in $H^2(M)$.
\end{example}
\begin{example}
More generally suppose that $E$ is an irreducible curve on $X$
whose class is relevant, and $(E\cdot E) - 2p_a(E) = -1$,
so that $\dim {\cal R}_{[E]} = 0$ and is therefore a finite set;
say it has $d$ members $E_1,\dots,E_d$.
The incidence locus ${\cal S}_{[E]} =
\bigcup_{i=1}^d \{E_i\} \times E_i \times E_i \times E_i$
with the large diagonal removed;
its closure $\overline{{\cal S}}_{[E]} =
\bigcup_{i=1}^d \{E_i\} \times E_i \times E_i \times E_i$.
Hence the image under the projection to $X^3$
is $\bigcup_{i=1}^d E_i\times E_i\times E_i$,
and the class $[A_{[E]}] = d [E]\otimes[E]\otimes[E]$.
Hence by the computation in Example \ref{exCCC},
we have
\[
\phi_{[E]}(\alpha,\beta) = d ([E]\cdot \alpha)([E]\cdot \beta)[E]
\]
for divisor classes $\alpha$ and $\beta$,
and is zero unless $\alpha$ and $\beta$ are both in $H^2(M)$.
This generalizes the previous example, where $d=1$.
\end{example}
\begin{example}
Suppose that $F$ is a fiber in a ruling on $X$,
that is, a smooth rational curve with $(F \cdot F) = 0$.
The linear system $|F|$ is a pencil;
the locus ${\cal R}_{[F]}$ of irreducible nodal rational curves in
$|F|$
is an open dense subset of the whole system $|F|$
(it is the set of smooth members of $|F|$).
The incidence locus ${\cal S}_{[F]}$ has complex dimension $4$;
an element is obtained by choosing a member of $|F|$,
then three points on this member.
The complementary classes in $H^*(X^3)$ have complex
codimension $4$,
that is, they are the classes in $H^8(X^3)$.
This group is generated by the classes
$[p]\otimes[p]\otimes[X]$,
$\alpha\otimes\beta\otimes[p]$ for divisor classes $\alpha$ and
$\beta$,
and the associated classes obtained by symmetry.
The intersection product
$[A_{[F]}] \cdot [p]\otimes[p]\otimes[X] = 0$,
since there is a no curve in the system through two general points
$p$.
The intersection product
$[A_{[F]}] \cdot \alpha\otimes\beta\otimes[p] =
(F\cdot \alpha)(F \cdot \beta)$;
forcing the curve to pass through the general point $p$ gives a
unique
curve in the system, and the choice of the other two points,
which must lie in the divisor $\alpha$ and $\beta$ respectively,
gives the result above.
Therefore we have
\[
\phi_{[F]}(\alpha,\beta) = ([F]\cdot \alpha)([F]\cdot \beta)[X]
\]
for divisor classes $\alpha$ and $\beta$. Moreover by the symmetry
we also have
$[A_{[F]}] \cdot \alpha\otimes[p]\otimes\gamma =
(F\cdot \alpha)(F \cdot \gamma)$, so that (after taking symmetry
into account)
\[
\phi_{[F]}([p],\alpha) = \phi_{[F]}(\alpha,[p]) = ([F]\cdot
\alpha)[F]
\]
for a divisor class $\alpha$. All other $\phi$-classes are zero.
\end{example}
\begin{example}
Suppose that $F$ gives a relevant class on $X$
with $(F\cdot F) - 2p_a(F) = 0$;
then the locus ${\cal R}_{[F]}$ of nodal rational curves in $|F|$
forms a curve.
Denote by $d$ the degree of this curve in the projective space
$|F|$.
The incidence locus ${\cal S}_{[F]}$ has complex dimension $4$;
an element is obtained by choosing a member of ${\cal R}_{[F]}$,
then three points on this member.
The complementary classes are the classes in $H^8(X^3)$;
This group as above is generated by the classes
$[p]\otimes[p]\otimes[X]$,
$\alpha\otimes\beta\otimes[p]$ for divisor classes $\alpha$ and
$\beta$,
and the associated classes obtained by symmetry.
The intersection product
$[A_{[F]}] \cdot [p]\otimes[p]\otimes[X] = 0$,
since there is a no curve in the system through two general points
$p$.
The intersection product
$[A_{[F]}] \cdot \alpha\otimes\beta\otimes[p] =
d (F\cdot \alpha)(F \cdot \beta)$;
forcing the curve to pass through the general point $p$
gives $d$ curves in the system,
and the choice of the other two points,
which must lie in the divisor $\alpha$ and $\beta$ respectively,
contributes $(F\cdot \alpha)$ and $(F\cdot \beta)$ respectively
to the number of choices.
Therefore we have
\[
\phi_{[F]}(\alpha,\beta) = d ([F]\cdot \alpha)([F]\cdot
\beta)[X]
\]
for divisor classes $\alpha$ and $\beta$. Moreover by the symmetry
we also have
$[A_{[F]}] \cdot \alpha\otimes[p]\otimes\gamma =
d (F\cdot \alpha)(F \cdot \gamma)$,
so that (after taking symmetry into account)
\[
\phi_{[F]}([p],\alpha) = \phi_{[F]}(\alpha,[p]) = d([F]\cdot
\alpha)[F]
\]
for a divisor class $\alpha$. All other $\phi$-classes are zero.
This generalizes the previous example, where $d=1$.
\end{example}
\begin{example}
Suppose that $L$ is a smooth rational curve on $X$
with $(L \cdot L) = 1$.
The linear system $|L|$ is a net;
the locus ${\cal R}_{[L]}$ of nodal (i.e. smooth) rational curves
in $|L|$
is an open dense subset of $|L|$.
The incidence locus ${\cal S}_{[L]}$ has complex dimension $5$;
an element is obtained by choosing a smooth member of $|L|$,
then three points on this member.
The complementary classes in $H^*(X^3)$ have complex
codimension $5$,
that is, they are the classes in $H^{10}(X^3)$.
This group is generated by the classes
$[p]\otimes[p]\otimes\alpha$,
for a divisor classes $\alpha$,
and the associated classes obtained by symmetry.
The intersection product
$[A_{[L]}] \cdot [p]\otimes[p]\otimes\alpha = (L \cdot \alpha)$,
since through two general points there is a unique member $L_0$ of
$|L|$,
whose third point can be any of the points
where $L_0$ meets the divisor $\alpha$.
Therefore we have
\[
\phi_{[L]}(\alpha,[p]) = (L\cdot \alpha)[X]
\]
for a divisor classes $\alpha$, and
\[
\phi_{[L]}([p],[p]) = [L].
\]
All other $\phi$-classes are zero.
\end{example}
\begin{example}
Again we may generalize the above in case
$L$ induces any relevant class with $(L \cdot L) -2 p_a(L) = 1$.
The locus ${\cal R}_{[L]}$ of nodal (i.e. smooth) rational curves
in $|L|$
is a surface inside the projective space $|L|$;
let $d$ be the degree of this surface.
The incidence locus ${\cal S}_{[L]}$ has complex dimension $5$.
The complementary classes in $H^*(X^3)$
are the classes in $H^{10}(X^3)$,
which is generated by the classes
$[p]\otimes[p]\otimes\alpha$,
for a divisor classes $\alpha$,
and the associated classes obtained by symmetry.
The intersection product
$[A_{[L]}] \cdot [p]\otimes[p]\otimes\alpha = d (L \cdot \alpha)$,
since through two general points there are $d$ members of ${\cal
R}_{[L]}$,
whose third point can be any of the points
where the member meets the divisor $\alpha$.
Therefore we have
\[
\phi_{[L]}(\alpha,[p]) = d (L\cdot \alpha)[X]
\]
for a divisor classes $\alpha$, and
\[
\phi_{[L]}([p],[p]) = d [L].
\]
All other $\phi$-classes are zero.
This generalizes the previous example, where $d=1$.
\end{example}
\begin{example}
Suppose that $C$ is a smooth rational curve on $X$
with $(C \cdot C) = 2$.
The linear system $|C|$ is a web (that is, it is $3$-dimensional);
the locus ${\cal R}_{[C]}$ of nodal (i.e. smooth) rational curves
in $|C|$
is again an open dense subset of $|C|$.
The incidence locus ${\cal S}_{[C]}$ has complex dimension $6$;
an element is obtained by choosing a smooth member of $|C|$,
then three points on this member.
The complementary classes in $H^*(X^3)$ have complex
codimension $6$,
that is, they are the classes in $H^{12}(X^3)$.
This group has rank one, and is generated by the class
$[p]\otimes[p]\otimes[p]$.
The intersection product
$[A_{[C]}] \cdot [p]\otimes[p]\otimes[p] = 1$,
since through three general points there is a unique member of
$|C|$.
Indeed, we have that $[A_{[C]}] =
[X]\otimes[X]\otimes[X]$.
Therefore we have
\[
\phi_{[C]}([p],[p]) = [X].
\]
All other $\phi$-classes are zero.
\end{example}
\begin{example}
\label{phiC2=2}
Again if $C$ is a relevant class with $(C\cdot C) - 2p_a(C) = 2$,
then the locus ${\cal R}_{[C]}$ of nodal rational curves in $|C|$
is a threefold of $|C|$;
let $d$ be the degree of this threefold.
The incidence locus ${\cal S}_{[C]}$ has complex dimension $6$;
and the complementary classes in $H^*(X^3)$ have complex
codimension $6$,
that is, they are the classes in $H^{12}(X^3)$,
which is generated by the class
$[p]\otimes[p]\otimes[p]$.
The intersection product
$[A_{[C]}] \cdot [p]\otimes[p]\otimes[p] = d$,
since through three general points
there is are $d$ members of ${\cal R}_{[C]}$.
Indeed, we have that $[A_{[C]}] = d
[X]\otimes[X]\otimes[X]$.
Therefore we have
\[
\phi_{[C]}([p],[p]) = d [X].
\]
All other $\phi$-classes are zero.
This generalizes the previous example, where $d=1$.
\end{example}
We offer the following example which shows that the above
phenomenon
occurs, namely that there are relevant classes
which come from singular rational curves.
\begin{example}
Let $X$ be the blow-up of the plane ${\Bbb P}^2$
at $5$ general points $p_1,\dots p_5$.
Let $H$ denote the line class on $X$
and $E_i$ denote the exceptional curve lying over $p_i$.
Consider the anti-canonical class $C = -K_{X} = 3H -
\sum_{i=1}^5 E_i$.
This is the linear system of cubics passing through the $5$ points
$p_i$.
Note that $(C\cdot C) = 4$ and $p_a(C) = 1$
so that $(C\cdot C) - 2p_a(C) = 2$.
We have $\dim {\cal S}_{[C]} = 2 - (C \cdot K_{X}) = 6$,
so $[C]$ is a relevant class.
The map
\[
\pi_2:\overline{{\cal S}_{[C]}} \to X^3
\]
has as its general fiber over a triple $(q_1,q_2,q_3)$
those nodal rational curves in the linear system $|C|$
through the three points $q_1$, $q_2$, and $q_3$.
This is exactly the set of nodal rational cubics in the plane
passing through the eight points $p_1,\dots p_5$, $q_1,\dots q_3$.
The system of cubics through these general eight points
forms a pencil of genus one curves,
which has exactly $12$ singular members.
Hence the map $\pi_2:\overline{{\cal S}_{[C]}} \to X^3$
is generically finite of degree $12$,
and so pushing down the fundamental class we see that
\[
[A_{[C]}] = 12[X]\otimes[X]\otimes[X].
\]
Hence
\[
[A_{[C]}] \cdot [p]\otimes[p]\otimes[p] = 12 \text{ and }
\phi_{[C]}([p],[p]) = 12[X].
\]
All other $\phi$-classes are zero.
\end{example}
\section{Definition of quantum cohomology}
\label{sectiondefqntm}
We denote by $\operatorname{Eff}(X)$
the cone of effective divisor classes in $H^2(X)$.
It forms a semigroup under addition.
Let $Q = {\Bbb Z}[[\operatorname{Eff}(X)]]$
be the completion of the integral group ring over $\operatorname{Eff}(X)$;
$Q$ is a ${\Bbb Z}$-algebra.
It is customary to formally introduce a multi-variable $q$
and to write the module generators of $Q$ as elements $q^{[D]}$,
where $[D]$ is an effective cohomology class in $\operatorname{Eff}(X)$.
With this notation, every element of $Q$ can be written
as a formal series
\[
\sum_{[D]\in\operatorname{Eff}(X)} n_{[D]} q^{[D]}
\]
with integral coefficients $n_{[D]}$,
and divisor class exponents $[D] \in \operatorname{Eff}(X)$.
In this way multiplication in the ring $Q$
is induced by the relations that
\[
q^{[D_1]}q^{[D_2]} = q^{[D_1+D_2]}
\]
for divisors $D_1$ and $D_2$ on $X$,
and the ordinary distributive and associative laws.
Define the {\em quantum cohomology ring} of $X$ to be
\[
H^*_Q(X) = H^*(X) \otimes_{\Bbb Z} Q
\]
as a free abelian group.
Moreover it is also a $Q$-module, with the obvious structure.
The multiplication $\operatorname{\ast_Q}$ on $H^*_Q(X)$,
called the {\em quantum product},
is determined by knowing the products of (homogeneous) elements
from
$H^*(X)$,
since the rest comes from linearity and the $Q$-module structure.
For two homogeneous elements $\alpha$ and $\beta$ in $H^*(X)$
define
\[
\alpha \operatorname{\ast_Q} \beta = \sum_{[C]} \phi_{[C]}(\alpha,\beta) q^{[C]},
\]
the sum begin taken over
the relevant cohomology classes in $H^2(X)$.
We remark that if there are only
finitely many relevant classes in $H^2(X)$,
then the quantum cohomology ring may be formulated as a polynomial
ring
instead of a power series ring;
in other words, one may take $Q = {\Bbb Z}[\operatorname{Eff}(X)]$
to be simply the integral semigroup ring instead of its completion.
This is the case for a Del Pezzo surface $X$.
We have an immediate identification for the identity of the quantum
product:
\begin{lemma}
The fundamental class $[X]$ is the identity for the quantum
product.
In other words, for every class $\alpha \in H^*(X)$,
\[
\phi_0([X],\alpha) = \alpha
\]
and if $[C] \neq 0$, then
\[
\phi_{[C]}([X],\alpha) = 0.
\]
\end{lemma}
\begin{pf}
The $[C]= 0$ statement follows from the computation in Example
\ref{phiDelta};
we have
\[
\phi_0([X],\alpha) = \phi_{[\Delta]}([X],\alpha)
= [X]\cup\alpha = \alpha
\]
since $[X]$ is the identity for the ordinary cup product.
If $[C] \neq 0$, then
$\dim {\cal S}_{[C]} = 3 + \dim {\cal R}_{[C]} \geq 3$ if $[C]$ is
relevant.
Therefore the three-point class $[A_{[C]}]$
lies in $H^{2k}(X^3)$ for $k \leq 3$.
Any complementary class of the form
$[X]\otimes\beta\otimes\gamma$
must have $\deg(\beta)+\deg(\gamma) = 12-2k \leq 6$.
Suppose first that $\dim {\cal S}_{[C]} = 3$,
so that ${\cal R}_{[C]}$ is a finite set and
the three-point class $[A_{[C]}]$
lies in $H^{6}(X^3)$.
The only complementary classes involving the fundamental class
$[X]$
have the form $[X]\otimes\beta\otimes[p]$ for some divisor
class $\beta$.
But $[A_{[C]}] \cdot [X]\otimes\beta\otimes[p]$ counts the
number of curves
in ${\cal S}_{[C]}$ whose second point lies in the divisor $\beta$
and whose third point equals $p$;
since $p$ is a general point and ${\cal R}_{[C]}$ is a finite set,
there are no curves in ${\cal R}_{[C]}$ through $p$
and this intersection number is zero.
Hence $\phi_{[C]}([X],\alpha) = 0$ for any $\alpha$.
Suppose next that $\dim {\cal S}_{[C]} = 4$,
so that the locus ${\cal R}_{[C]}$ is one-dimensional and
the three-point class $[A_{[C]}]$
lies in $H^{4}(X^3)$.
The only complementary classes involving the fundamental class
$[X]$
have the form $[X]\otimes[p]\otimes[p]$.
The intersection product
$[A_{[C]}] \cdot [X]\otimes[p]\otimes[p]$ counts the number of
curves
in ${\cal S}_{[C]}$ whose second and third point are specified
general points;
since ${\cal R}_{[C]}$ is one-dimensional,
there are no curves in ${\cal R}_{[C]}$ through two specified
general points,
and this intersection number is zero.
Hence again $\phi_{[C]}([X],\alpha) = 0$ for any $\alpha$.
Finally if $\dim {\cal S}_{[C]} \geq 5$,
there are no complementary classes in the cohomology of $X^3$
of the form $[X]\otimes\beta\otimes\gamma$ at all.
\end{pf}
\section{Relevant classes on strict Del Pezzo surfaces}
\label{strictrelevantsection}
Let $X$ be a general strict Del Pezzo surface,
that is, $X = {\Bbb F}_0 \cong {\Bbb P}^1\times {\Bbb P}^1$
or a general blowup of the plane such that $-K_{X}$ is very
ample.
This amounts to having $X \cong {\Bbb F}_0$
or $X$ being a blowup of the plane at $n \leq 6$ general points.
It is an elementary matter to compute the relevant classes
on such a strict Del Pezzo surface $X$.
The results are shown in the tables below.
In the first few columns are the numerical characters of the class:
the bidegree in the case of ${\Bbb F}_0$
and the integers $d$ and $m_i$ for a class of the form $dH - \sum_i
m_i E_i$
on a blowup of the plane; here $d$ is the degree and $m_i$
is the multiplicity of the curves at the associated blown up point.
The final three columns contain the quantity $C^2-2p_a(C)$
(on which relevance is based), the arithmetic genus $p_a(C)$,
and the number of such classes up to permutations of the $E_i$'s.
\begin{center}
\begin{tabular}{c|c|c|c}
\multicolumn{4}{c}{Relevant nonzero classes on ${\Bbb P}^2$} \\
\hline
degree & $C^2-2p_a(C)$ & $p_a(C)$ & $\#$ \\
1 & 1 & 0 & 1 \\
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{c|c|c|c}
\multicolumn{4}{c}{Relevant classes on ${\Bbb F}_0$} \\ \hline
bidegree & $C^2-2p_a(C)$ & $p_a(C)$ & $\#$ \\
(0,1) & 0 & 0 & 1 \\
(1,0) & 0 & 0 & 1 \\
(1,1) & 2 & 0 & 1 \\
\end{tabular}
\end{center}
Denote by $X_n$ a general blowup of ${\Bbb P}^2$ at $n$ points.
\begin{center}
\begin{tabular}{c|c|c|c|c}
\multicolumn{5}{c}{Relevant classes on $X_1$} \\ \hline
degree & $m_1$ & $C^2-2p_a(C)$ & $p_a(C)$ & $\#$ \\
0 & -1 & -1 & 0 & 1 \\
1 & 1 & 0 & 0 & 1 \\
1 & 0 & 1 & 0 & 1 \\
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{c|cc|c|c|c}
\multicolumn{6}{c}{Relevant nonzero classes on $X_2$} \\ \hline
degree & $m_1$ & $m_2$ & $C^2-2p_a(C)$ & $p_a(C)$ & $\#$ \\
0 & -1 & 0 & -1 & 0 & 2 \\
1 & 1 & 1 & -1 & 0 & 1 \\
1 & 1 & 0 & 0 & 0 & 2 \\
1 & 0 & 0 & 1 & 0 & 1 \\
2 & 1 & 1 & 2 & 0 & 1 \\
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{c|ccc|c|c|c}
\multicolumn{7}{c}{Relevant classes on $X_3$} \\ \hline
degree & $m_1$ & $m_2$ & $m_3$ & $C^2-2p_a(C)$ & $p_a(C)$ & $\#$ \\
0 & -1 & 0 & 0 & -1 & 0 & 3 \\
1 & 1 & 1 & 0 & -1 & 0 & 3 \\
1 & 1 & 0 & 0 & 0 & 0 & 3 \\
1 & 0 & 0 & 0 & 1 & 0 & 1 \\
2 & 1 & 1 & 1 & 1 & 0 & 1 \\
2 & 1 & 1 & 0 & 2 & 0 & 3 \\
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{c|cccc|c|c|c}
\multicolumn{8}{c}{Relevant nonzero classes on $X_4$} \\ \hline
degree & $m_1$ & $m_2$ & $m_3$ & $m_4$
& $C^2-2p_a(C)$ & $p_a(C)$ & $\#$ \\
0 & -1 & 0 & 0 & 0 & -1 & 0 & 4 \\
1 & 1 & 1 & 0 & 0 & -1 & 0 & 6 \\
1 & 1 & 0 & 0 & 0 & 0 & 0 & 4 \\
2 & 1 & 1 & 1 & 1 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 \\
2 & 1 & 1 & 1 & 0 & 1 & 0 & 4 \\
2 & 1 & 1 & 0 & 0 & 2 & 0 & 6 \\
3 & 2 & 1 & 1 & 1 & 2 & 0 & 4 \\
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{c|ccccc|c|c|c}
\multicolumn{9}{c}{Relevant classes on $X_5$} \\ \hline
degree & $m_1$ & $m_2$ & $m_3$ & $m_4$ & $m_5$
& $C^2-2p_a(C)$ & $p_a(C)$ & $\#$ \\
0 & -1 & 0 & 0 & 0 & 0 & -1 & 0 & 5 \\
1 & 1 & 1 & 0 & 0 & 0 & -1 & 0 & 10 \\
2 & 1 & 1 & 1 & 1 & 1 & -1 & 0 & 1 \\
1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 5 \\
2 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 5 \\
1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 \\
2 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 10 \\
3 & 2 & 1 & 1 & 1 & 1 & 1 & 0 & 5 \\
2 & 1 & 1 & 0 & 0 & 0 & 2 & 0 & 10 \\
3 & 1 & 1 & 1 & 1 & 1 & 2 & 1 & 1 \\
3 & 2 & 1 & 1 & 1 & 0 & 2 & 0 & 20 \\
4 & 2 & 2 & 2 & 1 & 1 & 2 & 0 & 10 \\
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{c|cccccc|c|c|c}
\multicolumn{10}{c}{Relevant classes on $X_6$} \\ \hline
degree & $m_1$ & $m_2$ & $m_3$ & $m_4$ & $m_5$ & $m_6$
& $C^2-2p_a(C)$ & $p_a(C)$ & $\#$ \\
0 & -1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 6 \\
1 & 1 & 1 & 0 & 0 & 0 & 0 & -1 & 0 & 15 \\
2 & 1 & 1 & 1 & 1 & 1 & 0 & -1 & 0 & 6 \\
1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 \\
2 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 15 \\
3 & 2 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 6 \\
1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 \\
2 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 20 \\
3 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
3 & 2 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 30 \\
4 & 2 & 2 & 2 & 1 & 1 & 1 & 1 & 0 & 20 \\
5 & 2 & 2 & 2 & 2 & 2 & 2 & 1 & 0 & 1 \\
2 & 1 & 1 & 0 & 0 & 0 & 0 & 2 & 0 & 15 \\
3 & 1 & 1 & 1 & 1 & 1 & 0 & 2 & 1 & 6 \\
3 & 2 & 1 & 1 & 1 & 0 & 0 & 2 & 0 & 60 \\
4 & 2 & 2 & 1 & 1 & 1 & 1 & 2 & 1 & 15 \\
4 & 2 & 2 & 2 & 1 & 1 & 0 & 2 & 0 & 60 \\
4 & 3 & 1 & 1 & 1 & 1 & 1 & 2 & 0 & 6 \\
5 & 2 & 2 & 2 & 2 & 2 & 1 & 2 & 1 & 6 \\
5 & 3 & 2 & 2 & 2 & 1 & 1 & 2 & 0 & 60 \\
6 & 3 & 3 & 2 & 2 & 2 & 2 & 2 & 0 & 15 \\
\end{tabular}
\end{center}
The table of relevant classes on $X_6$ has some features
which will be useful below. Let us collect them in the following
lemma.
Note that the anti-canonical class $-K$ on the surface
has $d = 3$ and $m_i = 1$ for each $i = 1,\ldots,6$.
\begin{lemma}
\label{X6relevantlemma}
Let $X_6$ denote a general $6$-fold blowup of the plane
(that is, a general cubic surface in ${\Bbb P}^3$).
\begin{enumerate}
\item All relevant classes $[C]$ on $X_6$
have arithmetic genus $p_a(C) \leq 1$.
\item The anti-canonical class $-K$ is the unique relevant class
$[C]$ on $X_6$
with $C^2 - 2p_a(C) = 1$ and $p_a(C) = 1$.
\item There are exactly $27$ relevant classes $[E]$ on $X_6$
with $E^2 - 2p_a(E) = -1$; all have $p_a(E) = 0$.
These are the classes of the $27$ lines on the cubic surface.
\item There are exactly $27$ relevant classes $[F]$ on $X_6$
with $F^2 - 2p_a(F) = 0$; all have $p_a(F) = 0$.
Each such class $F$ is obtained from a unique relevant class $E$
with $E^2 - 2p_a(E) = -1$ by subtracting $E$ from the anticanonical
class:
$F = -K - E$.
Any two such classes $F$, $G$ satisfy $0 \leq (F \cdot G) \leq 2$;
$(F\cdot G) = 0$ if and only if $F = G$.
If $(F \cdot G) = 1$ then $C = F+G$ is a relevant class
with $C^2 - 2p_a(C) = 2$ and $p_a(C) = 0$.
If $(F \cdot G) = 2$ then $C = F+G$ is a relevant class
with $C^2 - 2p_a(C) = 2$ and $p_a(C) = 1$.
\item There are exactly $72$ relevant classes $[L]$ on $X_6$
with $L^2 - 2p_a(L) = 1$ and $p_a(L) = 0$.
For each such class $[L]$ and each relevant class $E$ with $E^2 -
2p_a(E) = -1$
we have $0 \leq (L \cdot E) \leq 2$.
If $(L\cdot E) = 1$ then $C = L+E$ is a relevant class
with $C^2 - 2p_a(C) = 2$ and $p_a(C) = 0$.
If $(L \cdot E) = 2$ then $C = L+E$ is a relevant class
with $C^2 - 2p_a(C) = 2$ and $p_a(C) = 1$.
\item There are exactly $216$ relevant classes $[C]$ on $X_6$
with $C^2 - 2p_a(C) = 2$ and $p_a(C) = 0$.
Each such class $C$ can be written uniquely (up to order) as $C =
F + G$,
where $F$ and $G$ are classes with $F^2 - 2p_a(F) = G^2 - 2p_a(G)
= 0$.
\item There are exactly $27$ relevant classes $[C]$ on $X_6$
with $C^2 - 2p_a(C) = 2$ and $p_a(C) = 1$.
Each such class $C$ is obtained from a unique relevant class $E$
with $E^2 - 2p_a(E) = -1$ by adding $E$ to the anticanonical class:
$C = -K + E$.
\end{enumerate}
\end{lemma}
The proof of the above lemma is left to the reader.
All of the statements can be easily shown by careful examination
of the table of relevant classes on $X_6$;
many of the statements are also elementary consequences of
intersection theory on rational surfaces.
\section{A formula for the quantum product}
\label{sectionformula}
It is obvious that the computation of the quantum product
depends on knowing the intersection numbers
\[
[A] \cdot (\alpha \otimes \beta \otimes \gamma) \in {\Bbb Z}
\]
for the relevant three-point loci $A$,
and for generators $\alpha$, $\beta$, and $\gamma$ of $H^*(X)$.
{}From the computations made in Examples \ref{phi0}-\ref{phiC2=2},
the degree of the closure of the locus ${\cal R}_{[C]}$ is an
important
number for the quantum product. We will denote this degree by
$d_{[C]}$:
\[
d_{[C]} = \text{degree of }\overline{{\cal R}_{[C]}}
\text{ in the projective space }|C|.
\]
The following lemma is then immediate from the computations
made in Examples \ref{phi0}-\ref{phiC2=2}.
\begin{lemma}
Let $X$ be a Del Pezzo surface,
and suppose that $C$ gives a relevant class on $X$
(or $C = 0$).
We denote by $(C \cdot C)$ the self-intersection of $C$
and by $p_a(C)$ its arithmetic genus.
Let $d_{[C]}$ denote the degree of the closure of the locus ${\cal
R}_{[C]}$
in the projective space $|C|$.
Then:
\begin{itemize}
\item in case $C=0$, the three-point class $[A_0]$
is the class of the small diagonal
and has real codimension $8$ in $X^3$.
The classes
$[X]\otimes[X]\otimes[p]$ and
$[D_1]\otimes[D_2]\otimes[X]$
(for divisors $D_i$) generate the complementary space
$H^4(X^3)$
(together with the classes obtained from these by permutations).
We have:
\begin{itemize}
\item[] $[A_0] \cdot ([X]\otimes[X]\otimes[p]) = 1$ and
\item[] $[A_0] \cdot ([D_1]\otimes[D_2]\otimes[X]) = (D_1\cdot
D_2)$.
\end{itemize}
\item in case $(C\cdot C) -2p_a(C)= -1$,
the image of ${\cal S}_{[C]}$ has real codimension $6$ in
$X^3$.
The classes
$[X]\otimes[D]\otimes[p]$ and $[D_1]\otimes[D_2]\otimes[D_3]$
(for divisors $D$, $D_i$) generate the complementary space
$H^6(X^3)$
(together with the classes obtained from these by permutations).
We have:
\begin{itemize}
\item[] $[A_{[C]}] \cdot ([X]\otimes[D]\otimes[p]) = 0$ and
\item[] $[A_{[C]}] \cdot ([D_1]\otimes[D_2]\otimes[D_3]) =
d_{[C]} (C \cdot D_1)(C \cdot D_2)(C \cdot D_3)$.
\end{itemize}
In this case $[A_{[C]}] = d_{[C]} [C]\otimes[C]\otimes[C] \in
H^6(X^3)$.
\item in case $(C\cdot C) -2p_a(C) = 0$,
the image of ${\cal S}_{[C]}$ has real codimension $4$ in
$X^3$.
The classes
$[X]\otimes[p]\otimes[p]$ and $[D_1]\otimes[D_2]\otimes[p]$
(for divisors $D_i$) generate the complementary space
$H^8(X^3)$
(together with the classes obtained from these by permutations).
We have:
\begin{itemize}
\item[] $[A_{[C]}] \cdot ([X]\otimes[p]\otimes[p]) = 0$ and
\item[] $[A_{[C]}] \cdot ([D_1]\otimes[D_2]\otimes[p]) =
d_{[C]} (C \cdot D_1)(C \cdot D_2)$.
\end{itemize}
In this case $[A_{[C]}] =
d_{[C]} [C]\otimes[C]\otimes[X] +
d_{[C]} [C]\otimes[X]\otimes[C] +
d_{[C]} [X]\otimes[C]\otimes[C]
\in H^4(X^3)$.
\item in case $(C\cdot C) -2p_a(C) = 1$,
the image of ${\cal S}_{[C]}$ has real codimension $2$ in
$X^3$.
The classes
$[D]\otimes[p]\otimes[p]$
(for divisors $D$) generate the complementary space
$H^{10}(X^3)$
(together with the classes obtained from these by permutations).
We have:
\begin{itemize}
\item[] $[A_{[C]}] \cdot ([D]\otimes[p]\otimes[p]) = d_{[C]} (C
\cdot D)$.
\end{itemize}
In this case $[A_{[C]}] =
d_{[C]} [C]\otimes[X]\otimes[X] +
d_{[C]} [X]\otimes[C]\otimes[X] +
d_{[C]} [X]\otimes[X]\otimes[C]
\in H^2(X^3)$.
\item in case $(C\cdot C) = 2$,
the image of ${\cal S}_{[C]}$ is all of $X^3$
(and therefore has codimension zero).
The class
$[p]\otimes[p]\otimes[p]$
generates the complementary space $H^{12}(X^3)$.
We have:
\begin{itemize}
\item[] $[A_{[C]}] \cdot ([p]\otimes[p]\otimes[p]) = d_{[C]}$
\end{itemize}
and $[A_{[C]}] = d_{[C]} [X]\otimes[X]\otimes[X] \in
H^0(X^3)$.
\end{itemize}
\end{lemma}
This gives the following descriptions
of the classes $\phi_{[C]}(\alpha,\beta)$:
\begin{corollary}
Let $X$ be a Del Pezzo surface,
and suppose that $C$ gives a relevant class on $X$
(or $C = 0$).
We denote by $(C \cdot C)$ the self-intersection of $C$
and by $p_a(C)$ the arithmetic genus.
Let $d_{[C]}$ denote the degree of the closure of the locus ${\cal
R}_{[C]}$
in the projective space $|C|$.
Then:
\begin{itemize}
\item in case $C=0$,
\begin{itemize}
\item[] $\phi_0([X],[X]) = [X]$,
\item[] $\phi_0([X],[p]) = [p]$,
\item[] $\phi_0([X],[D]) = [D]$ for a divisor $D$, and
\item[] $\phi_0([D_1],[D_2]) = (D_1\cdot D_2)[p]$ for divisors
$D_i$.
\item[] The classes $\phi_{[0]}([D],[p]) = \phi_{[0]}([p],[p]) =
0$.
\end{itemize}
\item in case $(C\cdot C) -2p_a(C) = -1$,
\begin{itemize}
\item[] $\phi_{[C]}([D_1],[D_2]) = d_{[C]} (C \cdot D_1)(C \cdot
D_2) [C]$.
\item[] If $\alpha$ and $\beta$ are homogeneous elements of
$H^*(X)$,
$\phi_{[C]}(\alpha,\beta) = 0$
unless both $\alpha$ and $\beta$ lie in $H^2(X)$.
\end{itemize}
\item in case $(C\cdot C) -2p_a(C) = 0$,
\begin{itemize}
\item[] $\phi_{[C]}([D_1],[D_2]) = d_{[C]} (C \cdot D_1)(C \cdot
D_2)[X]$,
and
\item[] $\phi_{[C]}([D],[p]) = d_{[C]} (C \cdot D) [C]$.
\item[] $\phi_{[C]}([p],[p]) = 0$ and
$\phi_{[C]}([X],\beta) = 0$ for all $\beta$.
\end{itemize}
\item in case $(C\cdot C) -2p_a(C) = 1$,
\begin{itemize}
\item[] $\phi_{[C]}([p],[p]) = d_{[C]} [C]$ and
\item[] $\phi_{[C]}([D],[p]) = d_{[C]} (C \cdot D) [X]$.
\item[] $\phi_{[C]}([D_1],[D_2]) = 0$ and
$\phi_{[C]}([X],\beta) = 0$ for all $\beta$.
\end{itemize}
\item in case $(C\cdot C) -2p_a(C) = 2$,
\begin{itemize}
\item[] $\phi_{[C]}([p],[p]) = d_{[C]} [X]$.
\item[] If $\alpha$ and $\beta$ are homogeneous elements of
$H^*(X)$,
$\phi_{[C]}(\alpha,\beta) = 0$
unless both $\alpha$ and $\beta$ lie in $H^4(X)$.
\end{itemize}
\end{itemize}
\end{corollary}
Finally we deduce the formulas for the quantum product.
\begin{proposition}
\label{quantumproductformulas}
Let $X$ be a Del Pezzo surface.
\begin{enumerate}
\item The class $[X]$ is an identity for the quantum product.
\item For two divisors $D_1$ and $D_2$,
\[
[D_1] \operatorname{\ast_Q} [D_2] = (D_1 \cdot D_2)[p] q^{[0]}
+ \sum\begin{Sb} E \text{ relevant}\\{E^2-2p_a(E) = -1}\end{Sb}
d_{[E]}(E \cdot D_1)(E \cdot D_2) [E] q^{[E]}
\]\[
+ \sum\begin{Sb} F \text{ relevant}\\{F^2-2p_a(F) = 0}\end{Sb}
d_{[F]} (F \cdot D_1)(F \cdot D_2) [X] q^{[F]}
\]
where the sum is taken over the linear systems (not over the curves
actually).
\item For a divisor $D$,
\[
[D] \operatorname{\ast_Q} [p] =
\sum\begin{Sb} F \text{ relevant}\\{F^2-2p_a(F) = 0}\end{Sb}
d_{[F]} (F \cdot D) [F] q^{[F]}
+ \sum\begin{Sb} L \text{ relevant}\\{L^2-2p_a(L) = 1}\end{Sb}
d_{[L]} (L \cdot D) [X] q^{[L]}
\]
where the sum is again taken over the linear systems.
\item
\[
[p] \operatorname{\ast_Q} [p] = \sum\begin{Sb} L \text{ relevant}\\{L^2-2p_a(L) =
1}\end{Sb}
d_{[L]} [L] q^{[L]}
+ \sum\begin{Sb} C \text{ relevant}\\{C^2-2p_a(C) = 2}\end{Sb}
d_{[C]} [X] q^{[C]}
\]
where the sum is again taken over the linear systems.
\end{enumerate}
\end{proposition}
\section{The relationship with ordinary cohomology}
\label{sectionordinary}
The effective cone $\operatorname{Eff}(X)$ in $H^2(X)$ is a proper cone,
in the sense that it contains no subgroups of $H^2(X)$.
Hence there is an ``augmentation'' ring homomorphism
\[
G: H^*_Q(X) \to H^*(X)
\]
defined by sending a quantum cohomology class $\sum_{[D]} \alpha_D
q^{[D]}$
to the coefficient $\alpha_0$ of the $q^{[0]}$ term.
\begin{proposition}
Let $X$ be a Del Pezzo surface.
Then the map $G$ is a ring homomorphism
from the quantum cohomology ring $H^*_Q(X)$ (with the quantum
product)
to the integral cohomology ring $H^*(X)$ (with the cup
product).
\end{proposition}
This is clear from the formulas for the quantum product given
in Proposition \ref{quantumproductformulas}.
\section{Examples}
\label{sectionexamples}
\begin{example}
Let $X = {\Bbb P}^2$, the complex projective plane.
\end{example}
Then $H^2(X) = {\Bbb Z}[L]$, where $[L]$ is the class of a
line.
The quantum products determining the multiplication are
\[
[L] \operatorname{\ast_Q} [L] = [p]q^{[0]},
\]
\[
[L] \operatorname{\ast_Q} [p] = [X]q^{[L]}, \text{ and }
\]
\[
[p] \operatorname{\ast_Q} [p] = [L]q^{[L]}.
\]
We may identify $q^{[0]}$ and $[X]$ with $1$ and $q^{[L]}$ with
$q$;
if we write $\ell$ for the class $[L]$, the above relations are
that
$\ell^2 = [p]$, $\ell^3 = q$, and $\ell^4 = \ell q$.
Hence the quantum cohomology ring is isomorphic to
\[
H^*_Q({\Bbb P}^2) = {\Bbb Z}[\ell,q]/(\ell^3 - q).
\]
\begin{example}
Let $X = {\Bbb P}^1 \times {\Bbb P}^1$, the smooth quadric
surface.
\end{example}
Then $H^2(X) = {\Bbb Z}[F_1] \oplus {\Bbb Z}[F_2]$,
where the classes $[F_i]$ are those of the two rulings on $X$.
The quantum products determining the multiplication are
\[
[F_1] \operatorname{\ast_Q} [F_1] = [X]q^{[F_2]},
\]
\[
[F_1] \operatorname{\ast_Q} [F_2] = [p]q^{[0]},
\]
\[
[F_2] \operatorname{\ast_Q} [F_2] = [X]q^{[F_1]},
\]
\[
[F_1] \operatorname{\ast_Q} [p] = [F_2]q^{[F_2]},
\]
\[
[F_2] \operatorname{\ast_Q} [p] = [F_1]q^{[F_1]}, \text{ and }
\]
\[
[p] \operatorname{\ast_Q} [p] = [X]q^{[F_1+F_2]}.
\]
Denote $[F_i]$ by $f_i$, and $q^{[F_i]}$ by $q_i$.
These relations then become
$f_1^2 = q_2$, $f_1f_2 = [p]$, $f_2^2 = q_1$,
$f_1[p] = f_2q^2$, $f_2[p] = f_1q_1$, and $[p]^2 = q_1q_2$.
Hence the quantum cohomology ring is isomorphic to
\[
H^*_Q({\Bbb P}^1 \times {\Bbb P}^1) =
{\Bbb Z}[f_1,f_2,q_1,q_2]/(f_1^2-q_2,f_2^2-q_1).
\]
\begin{example}
Let $X = {\Bbb F}_1$, the blowup of the plane at one point.
\end{example}
Then $H^2(X) = {\Bbb Z}[E] \oplus {\Bbb Z}[F]$,
where $[E]$ is the class of the exceptional curve
and $[F]$ is the class of the fiber.
The only other class with a smooth rational curve
of self-intersection at most $2$ is the class $[L] = [E]+[F]$;
it has self-intersection $1$.
The quantum products determining the multiplication are
\[
[E] \operatorname{\ast_Q} [E] = -[p]q^{[0]} + [E]q^{[E]} + [X]q^{[F]},
\]
\[
[E] \operatorname{\ast_Q} [F] = [p]q^{[0]} - [E]q^{[E]},
\]
\[
[F] \operatorname{\ast_Q} [F] = [E]q^{[E]},
\]
\[
[E] \operatorname{\ast_Q} [p] = [F]q^{[F]},
\]
\[
[F] \operatorname{\ast_Q} [p] = [X]q^{[E]+[F]}, \text{ and }
\]
\[
[p] \operatorname{\ast_Q} [p] = [L]q^{[E]+[F]}.
\]
Denote $[E]$ by $e$, $q^{[E]}$ by $q$,
$[F]$ by $f$, $q^{[F]}$ by $r$, and $[p]$ by $p$.
These relations then become
$e^2 = -p+eq+r$, $ef = p-eq$, $f^2 = eq$,
$ep = fr$, $fp = qr$, and $p^2 = (e+f)qr$.
We may eliminate $p$ from the generators since $p = ef+eq$;
after doing so, the first and third relations become
$e^2 = r - ef$ and $f^2 = eq$,
and the other relations formally follow from these two.
Hence the quantum cohomology ring is isomorphic to
\[
H^*_Q({\Bbb F}_1) = {\Bbb Z}[e,f,q,r]/
(e^2+ef-r,f^2-eq).
\]
\section{A functoriality property}
\label{sectionfunctor}
Let $\pi:X \to Y$ be a general blowup
of a Del Pezzo surface $Y$ at a
single point $p$,
with an exceptional curve $E$.
Then we have $\pi\times\pi\times\pi: X^3 \to Y^3$.
Suppose that $C$ is an irreducible curve in $Y$,
such that its cohomology class $[C]$
is relevant for the quantum cohomology of $Y$.
We may assume that $p$ is not on $C$.
Then $[\pi^{-1}(C)] = \pi^*[C]$ as a class on $X$,
and has the same self-intersection and arithmetic genus as does
$[C]$;
therefore it is a relevant class for the quantum cohomology for
$X$.
In other words, the three-point classes $[A_{[C]}]$ on $Y^3$
and $[A_{\pi^*[C]}]$ on $X^3$ are both defined.
\begin{lemma}
With the above notation and conditions,
if $[C] \neq 0$, then
\[
[A_{\pi^*[C]}] = {(\pi\times\pi\times\pi)}^*([A_{[C]}]).
\]
\end{lemma}
\begin{pf}
What is clear is that the nodal rational curve loci
${\cal R}_{[C]}$ and ${\cal R}_{\pi^*[C]}$ are birational,
so that $\pi\times\pi\times\pi$ induces a birational map
\[
{\cal S}_{\pi^*[C]} \to {\cal S}_{[C]}.
\]
since we are blowing up a point which is not on the general member
of ${\cal R}_{[C]}$.
This implies immediately that
\[
\pi\times\pi\times\pi([A_{\pi^*[C]}]) = [A_{[C]}]
\]
as classes on $Y^3$.
We want to investigate the pull-back, not the image;
however this at least says that
$[A_{\pi^*[C]}]$ and ${(\pi\times\pi\times\pi)}^*([A_{[C]}])$
will differ only on the exceptional part of the map
$\pi\times\pi\times\pi$,
i.e., only over the fundamental locus
${\cal E} = (\{p\}\times Y \times Y) \cup
(Y\times \{p\} \times Y) \cup
(Y\times Y \times \{p\})$.
We can take the different cases up one by one.
If $(C \cdot C) - 2p_a(C) = -1$,
then the rational curve locus ${\cal R}_{[C]}$ is a finite set,
and by assumption the point $p$ is not on any member;
hence the loci in question are disjoint from the fundamental loci
for $\pi\times\pi\times\pi$, and there is nothing to prove.
Similarly if $(C\cdot C) -2p_a(C)= 2$, then $[A_{[C]}] =
d_{[C]}[Y^3]$
and $[A_{\pi^*[C]}] = d_{\pi^*[C]}[X^3]$;
since $d_{[C]} = d_{\pi^*[C]}$, the result follows in this case.
Suppose that $(C \cdot C) -2p_a(C)= 0$, so that ${\cal R}_{[C]}$ is
a curve.
Let $C_1, \dots C_d$ be the members of ${\cal R}_{[C]}$ through $p$
(here $d = d_{[C]}$).
Then $[A_{[C]}]$ intersects the fundamental locus ${\cal E}$
exactly in the union of the loci
$(\{p\} \times C_j \times C_j) \cup
(C_j\times \{p\} \times C_j) \cup
(C_j\times C_j \times \{p\})$;
over this locus in $X^3$ is the union of the loci
$(E \times (E + \overline{C_j}) \times (E + \overline{C_j})) \cup
((E + \overline{C_j})\times \{p\} \times (E + \overline{C_j})) \cup
((E + \overline{C_j})\times (E + \overline{C_j}) \times \{p\})$
(where $(E + \overline{C_j})$ means the union of the exceptional
curve $E$
with the proper transform of $C_j$).
Note that this is a union of three-folds in $X^3$;
since in this case $[A_{\pi^*[C]}]$ is the class of a four-fold,
we have no extra contribution to the pull-back class.
The final case of $(C \cdot C) -2p_a(C) = 1$ is similar;
here the locus $[A_{[C]}]$ intersects the fundamental locus ${\cal
E}$
in a three-fold, over which lies a four-fold in $X^3$;
since $[A_{\pi^*[C]}]$ is the class of a five-fold,
again we have no extra contribution to the pull-back class.
\end{pf}
This implies the following.
\begin{corollary}
\label{pi*phi}
With the above notations, if $[C] \neq 0$, then
for any homogeneous classes $\alpha$ and $\beta$ in $H^*(Y)$,
we have
\[
\pi^*(\phi_{[C]}(\alpha,\beta)) =
\phi_{[\pi^*C]}(\pi^*\alpha,\pi^*\beta).
\]
\end{corollary}
\begin{pf}
Choose a class $\gamma$ of the correct dimension on $Y$,
and compute
\begin{eqnarray*}
\phi_{[\pi^*C]}(\pi^*\alpha,\pi^*\beta) \cdot \pi^*\gamma
&=& [A_{\pi^*[C]}] \cdot \pi^*\alpha\otimes \pi^*\beta \otimes
\pi^*\gamma \\
&=& {(\pi\times\pi\times\pi)}^*[A_{[C]}] \cdot
{(\pi\times\pi\times\pi)}^*(\alpha\otimes\beta\otimes\gamma \\
&=& [A_{[C]}] \cdot (\alpha\otimes\beta\otimes\gamma \\
&=& \phi_{[C]}(\alpha,\beta) \cdot \gamma.
\end{eqnarray*}
Moreover if $\phi_{[C]}(\alpha,\beta)$ is a class in $H^2(Y)$,
and $E$ is the exceptional curve for the map $\pi$, then
\begin{eqnarray*}
\phi_{[\pi^*C]}(\pi^*\alpha,\pi^*\beta) \cdot E
&=& [A_{\pi^*[C]}] \cdot \pi^*\alpha\otimes \pi^*\beta \otimes E \\
&=& {(\pi\times\pi\times\pi)}^*[A_{[C]}] \cdot
\pi^*\alpha\otimes \pi^*\beta \otimes E \\
&=& 0.
\end{eqnarray*}
Hence as far as intersections go,
the class $\phi_{[\pi^*C]}(\pi^*\alpha,\pi^*\beta)$
is behaving exactly like the class
$\pi^*(\phi_{[C]}(\alpha,\beta))$.
However this class is defined in terms of its intersection
behaviour,
and so the equality as claimed holds.
\end{pf}
Note that we in any case have the formula
\[
\pi^*(\phi_{0}(\alpha,\beta)) = \phi_{0}(\pi^*\alpha,\pi^*\beta)
\]
since $\phi_{0}(\pi^*\alpha,\pi^*\beta) = \pi^*\alpha \cup
\pi^*\beta$
(this is cup product on $X$)
which is in turn equal to $\pi^*(\alpha \cup \beta)$
since $\pi^*$ is a ring homomorphism on ordinary cohomology.
Since $\pi^*(0) = 0$, we view this as the ``$[C] = 0$'' case
of Corollary \ref{pi*phi}.
Putting this together we derive the following version of
functoriality
for the quantum ring:
\begin{corollary}
\label{functoriality}
Let $\alpha$ and $\beta$ be ordinary cohomology classes in
$H^*(Y)$.
Then for any relevant class $[C]$ in $H^2(Y)$,
the $q^{\pi^*[C]}$-term of $\pi^*\alpha \operatorname{\ast_Q} \pi^*\beta$
is equal to $\pi^*$ of the $q^{[C]}$-term of $\alpha\operatorname{\ast_Q}\beta$.
\end{corollary}
Another way of saying this is to define the quantum pullback
\[
\pi_Q^*:H^*_Q(Y) \to H^*_Q(X)
\]
by setting
\[
\pi^*_Q(\sum_{[D]} c_{[D]} q^{[D]} ) = \sum_{[D]} \pi^*(c_{[D]})
q^{\pi^*[D]}.
\]
This is NOT in general a ring homomorphism.
But the above corollary says that
for classes $\alpha$ and $\beta$ in $H^*(Y)$,
the two quantum cohomology classes
\[
\pi^*(\alpha) \operatorname{\ast_Q} \pi^*(\beta) \text{ and }
\pi^*_Q(\alpha \operatorname{\ast_Q} \beta)
\]
differ only in the $q^{[D]}$ terms for those classes $[D]$ on
$X$
which are NOT pullbacks from $Y$.
That is, they agree on all the $q^{[\pi^*C]}$ terms,
for any effective classes $C$ on $Y$.
\section{Associativity of the quantum product for strict Del Pezzo
surfaces}
\label{sectionSDPassoc}
In this section we will use
the formulas of Proposition \ref{quantumproductformulas}
to check the associative law for the quantum product
for general strict Del Pezzo surfaces.
These are the surfaces ${\Bbb P}^2$,
${\Bbb F}_0 = {\Bbb P}^1 \times {\Bbb P}^1$,
and $X_n$ (the $n$-fold general blowup of ${\Bbb P}^2$)
for $n \leq 6$.
The first reduction is to note that it suffices to prove the
associative law
for the general surface $X_6$, the six-fold blowup of the plane.
(This is the general cubic surface in ${\Bbb P}^3$.)
This is due to the functoriality property
stated in Corollary \ref{functoriality};
if associativity holds on a blowup $X$ of a surface $Y$,
then in fact it must hold on $Y$.
By the tables of relevant classes on these surfaces
given in Section \ref{strictrelevantsection},
we note that all relevant classes have arithmetic genus at most
one.
(This is no longer true if one blows up $7$ general points in the
plane;
the class of quartics double at one point and passing through $6$
others
is relevant on $X_7$, and has arithmetic genus $2$.)
Hence for the strict Del Pezzo surface case,
the following lemma suffices to give us all the relevant $d_{[C]}$
numbers.
\begin{lemma}
Suppose that $X$ is a general rational surface,
and $[C]$ is a relevant class on $X$.
Then
\[
d_{[C]} = \left\{\begin{array}{cl}
1 & \text{ if }\;\; p_a(C) = 0, \text{ and } \\
12 & \text{ if }\;\; p_a(C) = 1.
\end{array}\right.
\]
\end{lemma}
\begin{pf}
If $p_a(C) = 0$, then the locus ${\cal R}_{[C]}$
of irreducible rational curves in the linear system $|C|$
is an open subset of $|C|$
(it is the subset parametrizing the smooth curves of $|C|$).
Hence its closure is the entire linear system $|C|$,
and therefore has degree one.
If $p_a(C) = 1$, then the locus ${\cal R}_{[C]}$
of irreducible rational curves in the linear system $|C|$
is an open subset of the discriminant locus of $|C|$.
Its degree is the number of irreducible nodal rational curves
in a general pencil of curves in $|C|$.
Such a general pencil, after blowing up the base points,
will give a fibration of elliptic curves on a rational surface;
the degree of ${\cal R}_{[C]}$ is the number of singular fibers
of this fibration.
This is $12$, by standard Euler number considerations.
\end{pf}
Let us begin with checking associativity for a triple product of
the form
$[p] \operatorname{\ast_Q} [p] \operatorname{\ast_Q} [D]$ for a divisor $D$.
For notational convenience let us define
\[
s(C) = (C \cdot C) - 2 p_a(C)
\]
for an irreducible curve $C$ in a relevant class $[C]$.
We have
\begin{eqnarray*}
([p] \operatorname{\ast_Q} [p]) \operatorname{\ast_Q} [D] &=&
(\sum_{s(L)= 1} d_{[L]}[L] q^{[L]} +
\sum_{s(C)= 2} d_{[C]}[X] q^{[C]}) \operatorname{\ast_Q} [D] \\
&=& \sum_{s(L) = 1} d_{[L]}([L]\operatorname{\ast_Q}[D]) q^{[L]} +
\sum_{s(C)= 2} d_{[C]}[D] q^{[C]} \\
&=& \sum_{s(C)= 2} d_{[C]}[D] q^{[C]} +
\sum_{s(L)= 1} d_{[L]}(L \cdot D)[p] q^{[L]} + \\
&& + \sum_{s(L)= 1}\sum_{s(E)= -1}
d_{[L]} d_{[E]} (E \cdot L)(E \cdot D) [E] q^{[E+L]}\\
&& +
\sum_{s(L) = 1}\sum_{s(F)= 0}
d_{[L]} d_{[F]} (F \cdot L)(F \cdot D) [X] q^{[F+L]})
\end{eqnarray*}
while
\begin{eqnarray*}
[p] \operatorname{\ast_Q} ([p] \operatorname{\ast_Q} [D]) &=&
[p]\operatorname{\ast_Q} (\sum_{s(F)= 0} d_{[F]}(F \cdot D) [F] q^{[F]}
+ \sum_{s(L)= 1} d_{[L]} (L \cdot D) [X] q^{[L]}) \\
&=& \sum_{s(F)= 0} d_{[F]}(F \cdot D) ([p]\operatorname{\ast_Q} [F]) q^{[F]}
+ \sum_{s(L)= 1} d_{[L]} (L \cdot D) ([p]\operatorname{\ast_Q} [X]) q^{[L]} \\
&=& \sum_{s(L)= 1} d_{[L]} (L \cdot D) [p] q^{[L]} +\\
&&+\sum_{s(F)= 0}\sum_{s(G)= 0}
d_{[F]}d_{[G]}(F \cdot D)(G \cdot F)[G] q^{[F+G]} + \\
&& + \sum_{s(F)= 0}\sum_{s(L)= 1}
d_{[F]}d_{[L]}(F \cdot D)(L \cdot F)[X] q^{[F+L]}.
\end{eqnarray*}
Comparing terms in the above two expressions we see that
this particular triple product is associative if and only if
\begin{eqnarray}
\label{ppD}
\sum_{s(C)= 2} d_{[C]} [D] q^{[C]} &+ &
\sum_{s(L)= 1}\sum_{s(E)= -1}
d_{[L]} d_{[E]} (E \cdot L)(E \cdot D) [E] q^{[E+L]} \\
&=& \sum_{s(F)= 0}\sum_{s(G)= 0}
d_{[F]}d_{[G]}(F \cdot D)(G \cdot F)[G] q^{[F+G]}. \nonumber
\end{eqnarray}
Of course only relevant classes are included in the above sums.
By Lemma \ref{X6relevantlemma},
if $L$ and $E$ are relevant classes with $s(L) = 1$ and $s(E) =
-1$,
then $0 \leq (L\cdot E) \leq 2$; if $(L \cdot E) = 0$
then there is no contribution in (\ref{ppD}).
If $(L\cdot E) \neq 0$ then $L+E$ is a relevant class with $s(L+E)
= 2$;
if $L \neq -K$ then $p_a(L+E) = (L\cdot E) - 1$,
and if $L = -K$ then $(L \cdot E) = 1$ and $p_a(L+E) = 1$.
Similarly if $F$ and $G$ are relevant classes with $s(F) = s(G) =
0$,
then $0 \leq (F\cdot G) \leq 2$; if $(F \cdot G) = 0$
then there is no contribution in (\ref{ppD}).
If $(F\cdot G) \neq 0$ then $F+G$ is a relevant class with $s(F+G)
= 2$
and $p_a(F+G) = (F \cdot G) - 1$.
Therefore the only terms which can appear in the associativity
formula
(\ref{ppD}) are those $q^{[C]}$ terms for relevant classes $[C]$
with $s(C) = 2$ (and $p_a(C) \leq 1$).
In fact we have the following.
\begin{lemma}
The associativity of the triple product $p \operatorname{\ast_Q} p \operatorname{\ast_Q} D$
on $X_6$ is equivalent to the following two formulas:
\begin{itemize}
\item[(a)] For every relevant class $C$ with $s(C) = 2$ and $p_a(C)
= 0$,
and every divisor $D$,
\[
[D] +
\sum\begin{Sb} E \\ s(E) = -1 \\ (C \cdot E) = 0 \end{Sb}
(D \cdot E) [E] =
\sum\begin{Sb} (F,G) \\ s(F) = s(G) = 0 \\ F+G = C \end{Sb}
(D \cdot F) [G].
\]
\item[(b)] For every relevant class $C$ with $s(C) = 2$ and $p_a(C)
= 1$,
\[
6[D] + 6(K+C \cdot D) [K+C] +
\sum\begin{Sb} E \\ s(E) = -1 \\(C \cdot E) = 1 \end{Sb}
(D \cdot E) [E] =
\sum\begin{Sb} (F,G) \\ s(F) = s(G) = 0 \\ F+G = C\end{Sb}
(D \cdot F) [G].
\]
\end{itemize}
\end{lemma}
\begin{pf}
As noted above, we may decompose (\ref{ppD}) into the $q^{[C]}$
terms
fixing a relevant class $C$ with $s(C) = 2$. The two cases of the
lemma
correspond to the two possibilities for $p_a(C)$.
If $p_a(C) = 0$, then the only pairs $(L,E)$ with $s(L) = 1$ and
$s(E) = -1$
having $L + E = C$ must have $L \neq -K$ (and therefore $d_{[L]} =
1$).
Moreover $(L \cdot E) = 1$ (else $p_a(C) = 1$ by Lemma
\ref{X6relevantlemma}).
Hence $(C \cdot E) = (L + E \cdot E) = 1 - 1 = 0$.
Conversely for any class $E$ with $(C \cdot E) = 0$,
the class $L = C-E$ has $s(L) = 1$ and occurs in the sum.
Therefore this $(L,E)$ sum with $L+E = C$
is a sum over those $E$'s with $(C \cdot E) = 0$.
In this case $d_{[E]} = (L \cdot E) = 1$ also, so these
contributions
can be ignored.
Similarly, if $p_a(C) = 0$, then if $C = F+G$ with $s(F) = s(G) =
0$,
then $d_{[F]} = d_{[G]} = (F\cdot G) = 1$.
This then produces the equation of part (a).
Suppose then that $p_a(C) = 1$.
Then $d_{[C]} = 12$, and in the $(L,E)$ sum, $L = -K$ is a
possibility.
The $E$ that pairs with $L = -K$ is of course $E = K+C$,
and has $(E \cdot L) = (E \cdot -K) = 1$.
This gives a term $12(K+C\cdot D)[K+C]$ to the $(L,E)$ sum.
If $L \neq -K$ and $L+E = C$,
then by Lemma \ref{X6relevantlemma} we have $(L\cdot E) = 2$,
or, equivalently, $(C \cdot E) = 1$.
Conversely, any class $E$ with $s(E) = -1$ and $(C \cdot E) = 1$
occurs, and is paired with the class $L = C-E$.
Therefore again this sum can be written as a sum over such $E$'s,
each $E$ giving the term $2(E\cdot D)[E]$
(since $d_{[L]} = d_{[E]} = 1$ and $(L\cdot E) = 2$).
Finally, in the $(F,G)$ sum,
for two such classes to sum to a $C$ with $p_a(C) = 1$,
we must have $(F \cdot G) = 2$ by Lemma \ref{X6relevantlemma};
since $d_{[F]}=d_{[G]}=1$, each such pair $(F,G)$ contributes a
term
of the form $2 (F\cdot D)[G]$.
Dividing all terms by two produces the equation of part (b).
\end{pf}
It remains to prove these two formulas.
We begin with (a).
\begin{lemma}
Let $C$ be a relevant class on $X_6$ with $s(C) = 2$ and $p_a(C) =
0$.
Then for any divisor $D$ on $X_6$,
\[
[D] +
\sum\begin{Sb} E \\ s(E) = -1 \\ (C \cdot E) = 0 \end{Sb}
(D \cdot E) [E] =
\sum\begin{Sb} (F,G) \\ s(F) = s(G) = 0 \\ F+G = C \end{Sb}
(D \cdot F) [G].
\]
\end{lemma}
\begin{pf}
By Lemma \ref{X6relevantlemma},
the class $C$ can be written uniquely (up to order) as $C = F+G$,
with $s(G) = s(G) = 0$; if we do so, we see that
the right-hand side of the above equation consists of only the two
terms
$(D \cdot F) [G] + (D \cdot G) [F]$.
Therefore we must actually show that
\[
[D] = (D \cdot F) [G] + (D \cdot G) [F] -
\sum\begin{Sb} E \\ s(E) = -1 \\ (C \cdot E) = 0 \end{Sb}
(D \cdot E) [E].
\]
The two pencils $|F|$ and $|G|$ on $X_6$ give a birational map
$\pi:X_6 \to {\Bbb F}_0$, realizing $X_6$ as a general five-fold
blowup
of ${\Bbb F}_0$. The only curves $E$ on $X_6$ with $s(E) = -1$
which do not meet $C = F+G$ are the five exceptional curves for
this blowup;
call these five curves $E_1,\ldots,E_5$.
Note that the seven classes $[F],[G],[E_1],\ldots,[E_5]$
generate the Picard group over ${\Bbb Z}$;
the intersection matrix is unimodular.
Now the above formula is exactly the writing of the class $[D]$
in terms of these generators.
\end{pf}
Finally we address the equation (b).
\begin{lemma}
Let $C$ be a relevant class on $X_6$ with $s(C) = 2$ and $p_a(C) =
1$.
Then for any divisor $D$ on $X_6$,
\[
6[D] + 6(K+C \cdot D) [K+C] +
\sum\begin{Sb} E \\ s(E) = -1 \\(C \cdot E) = 1 \end{Sb}
(D \cdot E) [E] =
\sum\begin{Sb} (F,G) \\ s(F) = s(G) = 0 \\ F+G = C\end{Sb}
(D \cdot F) [G].
\]
\end{lemma}
\begin{pf}
Let $\hat{E}$ denote the class $K+C$;
we have $s(\hat{E}) = -1$ as noted above.
Any class $E$ with $s(E) = -1$ and $(C \cdot E) = 1$
must therefore have $(E \cdot \hat{E}) = 0$ and conversely;
therefore the $E$ sum above is a sum over those $E$'s
with $(E \cdot \hat{E}) = 0$.
On the other side, suppose that $F+G=C$ with $s(F)=s(G)=0$.
By Lemma \ref{X6relevantlemma}, we must have $(F \cdot G) = 2$,
and so $(F\cdot C) = (G \cdot C) = 2$.
Then $(F \cdot \hat{E}) = (F \cdot K+C) = (F \cdot K) + 2 = 0$
since $(F \cdot K) = -2$ by the genus formula.
Similarly $(G \cdot \hat{E}) = 0$.
Therefore in the two pencils $|F|$ and $|G|$,
the curve $\hat{E}$ occurs in a singular fiber of each.
Hence there are unique curves $E_F$ and $E_G$
with $s(E_F) = s(E_G) = -1$
such that $F = \hat{E} + E_F$ and $G = \hat{E} + E_G$.
Moreover $(E_F \cdot E_G) = 1$ since $(F \cdot G) = 2$.
Note that the three curves $\hat{E}$, $E_F$, and $E_G$ form a
triangle
on $X_6$ (considered as a cubic surface).
Conversely, given a triangle of curves $\hat{E}$, $E_F$, and $E_G$
with $s(E_F) = s(E_G)= -1$,
we obtain a unique pair $(F,G)$ with $s(F) = s(G) = 0$ and $F+G =
C$
by setting $G = \hat{E}+E_G$ and $F = \hat{E} + E_G$.
Therefore the $(F,G)$ sum above can be made into a sum over such
triangles;
for a fixed $\hat{E}$ there are five such \cite{beauville,reid}.
Therefore the equation in question may be written as
\[
6[D] + 6(\hat{E} \cdot D) [\hat{E}] +
\sum\begin{Sb} E \\ s(E) = -1 \\(\hat{E} \cdot E) = 0 \end{Sb}
(D \cdot E) [E] =
\sum\begin{Sb} \text{five triangles}\\ \hat{E}+E_F+E_G\end{Sb}
(D \cdot \hat{E}+E_F) [\hat{E}+E_G] + (D \cdot \hat{E}+E_G)
[\hat{E}+E_F].
\]
We first claim that the above equation holds when $D = \hat{E}$.
In this case the first two terms on the left side cancel,
while each term in the two sums are clearly zero.
Next we claim that the equation holds when $D = \hat{E} + E'$,
for any curve $E'$ with $s(E') = -1$ and $(\hat{E}\cdot E') = 1$.
In this case $(\hat{E} \cdot D) = 0$
so the second term of the equation drops out.
Of those $E$'s which satisfy $(\hat{E} \cdot E) = 0$,
there are exactly $8$ which meet $E'$
and contribute to the sum on the left-hand side;
these are exactly the other curves in the four triangles containing
$E'$
which do not involve $\hat{E}$.
These come in pairs, and if $E_1$ and $E_2$ form a pair,
then they contribute
$(D\cdot E_1)[E_1] + (D\cdot E_2)[E_2] = [E_1+E_2]$.
However each triangle is equivalent to $-K$,
so this pair's contribution may be written as $[-K - E']$;
hence this sum reduces to $-4[K+E']$.
Hence the entire left-hand side is equal to
$6[\hat{E} + E'] -4[K+E'] = 6[\hat{E}] - 4[K] + 2[E']$.
On the right-hand side,
if we have a triangle $E+E_F+E_G$,
$(D \cdot \hat{E}+E_F) = (E' \cdot \hat{E}+E_F) = 1 + (E'\cdot
E_F)$
and similarly for the $E_G$ term.
Now $E'$ is part of a triangle containing $\hat{E}$, say $\hat{E}
+ E' + E''$;
the other four triangles are disjoint from $E'$
(except for the curve $\hat{E}$).
For one of these four triangles, we obtain a contribution of
$[\hat{E}+E_G] + [\hat{E}+E_F] = [\hat{E} - K]$.
For the triangle with $E'$ and $E''$,
we have a contribution of
$2[\hat{E}+E']$.
Thus the right-hand sum reduces to $6[\hat{E}] - 4[K] + 2[E']$.
As noted above, this is equal to the left-hand side;
therefore the equation holds for this $D$.
The proof now finishes by remarking that the Picard group of $X_6$
is generated rationally by $\hat{E}$ and the classes $\hat{E}+E'$
considered above.
Since the equation is linear in $D$, and is true for these
generators,
it is true for all divisors $D$.
\end{pf}
Let us now address the associativity for a triple product of the
form
$[p]\operatorname{\ast_Q}[D_1]\operatorname{\ast_Q}[D_2]$ for divisors $D_i$. We have
\[
[p]\operatorname{\ast_Q}([D_1]\operatorname{\ast_Q}[D_2]) =
\]
\[
{\renewcommand{\arraystretch}{1.5}
\begin{array}{ll}
= & [p]\operatorname{\ast_Q}((D_1 \cdot D_2)[p] q^{[0]}
+ \sum_{s(E) = -1} d_{[E]}(E \cdot D_1)(E \cdot D_2) [E] q^{[E]}
+ \sum_{s(F) = 0} d_{[F]}(F \cdot D_1)(F \cdot D_2) [X]
q^{[F]}) \\
= & (D_1 \cdot D_2)[p]\operatorname{\ast_Q}[p] q^{[0]}
+ \sum_{s(E) = -1} d_{[E]}(E \cdot D_1)(E \cdot D_2) [p]\operatorname{\ast_Q}[E]
q^{[E]} \\
& + \sum_{s(F) = 0} d_{[F]}(F \cdot D_1)(F \cdot D_2)
[p]\operatorname{\ast_Q}[X] q^{[F]}
\\
= & (D_1 \cdot D_2)(\sum_{s(L) = 1}d_{[L]}[L] q^{[L]} + \sum_{s(C)
= 2}
d_{[C]}[X] q^{[C]}) \\
& + \sum_{s(E) = -1} d_{[E]}(E \cdot D_1)(E \cdot D_2)
(\sum_{s(F) = 0} d_{[F]}(F \cdot E) [F] q^{[F]}
+\sum_{s(L) = 1} d_{[L]}(L \cdot E) [X] q^{[L]}) q^{[E]} \\
& + \sum_{s(F) = 0} d_{[F]}(F \cdot D_1)(F \cdot D_2) [p]q^{[F]}
\\
= & \sum_{s(L) = 1} d_{[L]}(D_1 \cdot D_2) [L] q^{[L]}
+ \sum_{s(C) = 2} d_{[C]}(D_1 \cdot D_2) [X] q^{[C]} \\
& + \sum_{s(E) = -1}\sum_{F^2 = 0}
d_{[E]}d_{[F]} (E \cdot D_1)(E \cdot D_2)(F \cdot E) [F]
q^{[E]+[F]} \\
& + \sum_{s(E) = -1}\sum_{L^2 = 1}
d_{[E]}d_{[L]}(E \cdot D_1)(E \cdot D_2)(L \cdot E) [X]
q^{[E]+[L]} \\
& + \sum_{s(F) = 0} d_{[F]}(F \cdot D_1)(F \cdot D_2) [p]q^{[F]}
\end{array}
}
\]
while
\[
([p]\operatorname{\ast_Q}[D_1])\operatorname{\ast_Q}[D_2]) =
\]
\[
{\renewcommand{\arraystretch}{1.5}
\begin{array}{ll}
= & (\sum_{s(F) = 0} d_{[F]}(F \cdot D_1) [F] q^{[F]}
+ \sum_{s(L) = 1} d_{[L]}(L \cdot D_1) [X] q^{[L]}) \operatorname{\ast_Q}
[D_2] \\
= & \sum_{s(F) = 0} d_{[F]}(F \cdot D_1) ([F]\operatorname{\ast_Q}[D_2]) q^{[F]}
+ \sum_{s(L) = 1} d_{[L]}(L \cdot D_1) [D_2] q^{[L]} \\
= & \sum_{s(F) = 0} d_{[F]}(F \cdot D_1)
( (F \cdot D_2)[p] q^{[0]}
+ \sum_{s(E) = -1} d_{[E]}(E \cdot F)(E \cdot D_2) [E] q^{[E]} \\
& + \sum_{s(G) = 0} d_{[G]}(G \cdot F)(G \cdot D_2) [X] q^{[F]}
) q^{[F]} \\
& + \sum_{s(L) = 1} d_{[L]}(L \cdot D_1) [D_2] q^{[L]} \\
= & \sum_{s(F) = 0} d_{[F]}(F \cdot D_1) (F \cdot D_2)[p] q^{[F]}
\\
& + \sum_{s(F) = 0} \sum_{s(E) = -1}
d_{[F]}d_{[E]}(F \cdot D_1)(E \cdot F)(E \cdot D_2) [E] q^{[E+F]}
\\
& + \sum_{s(F) = 0} \sum_{s(G) = 0}
d_{[F]}d_{[G]}(F \cdot D_1)(G \cdot F)(G \cdot D_2) [X]
q^{[F+G]} \\
& + \sum_{s(L) = 1} d_{[L]}(L \cdot D_1) [D_2] q^{[L]} . \\
\end{array}
}
\]
Therefore associativity of this triple product is equivalent to the
identity
\[
{\renewcommand{\arraystretch}{1.5}
\begin{array}{ll}
& \sum_{s(L) = 1} d_{[L]}(D_1 \cdot D_2) [L] q^{[L]}
+ \sum_{s(C) = 2} d_{[C]}(D_1 \cdot D_2) [X] q^{[C]} \\
& + \sum_{s(E) = -1}\sum_{s(F) = 0}
d_{[F]}d_{[E]}(E \cdot D_1)(E \cdot D_2)(F \cdot E) [F] q^{[E+F]}
\\
& + \sum_{s(E) = -1}\sum_{s(L) = 1}
d_{[L]}d_{[E]}(E \cdot D_1)(E \cdot D_2)(L \cdot E) [X]
q^{[E+L]} \\
=&\\
& \sum_{s(F) = 0} \sum_{s(E) = -1}
d_{[F]}d_{[E]}(F \cdot D_1)(E \cdot F)(E \cdot D_2) [E] q^{[E+F]}
\\
& + \sum_{s(F) = 0} \sum_{s(G) = 0}
d_{[F]}d_{[G]}(F \cdot D_1)(G \cdot F)(G \cdot D_2) [X]
q^{[F+G]} \\
& + \sum_{s(L) = 1} d_{[L]}(L \cdot D_1) [D_2] q^{[L]} .
\end{array}
}
\]
Comparing those terms with coefficients in $H^0(X)$ and those
in
$H^2(X)$, we see that
$ p \operatorname{\ast_Q} (D_1 \operatorname{\ast_Q} D_2) = (p \operatorname{\ast_Q} D_1) \operatorname{\ast_Q} D_2$ if and
only if the following two equations hold:
\begin{equation}
\label{pDD1}
{\renewcommand{\arraystretch}{1.5}
\begin{array}{ll}
& \sum_{s(L) = 1} d_{[L]}(D_1 \cdot D_2) [L] q^{[L]} \\
& + \sum_{s(E) = -1}\sum_{s(F) = 0}
d_{[F]}d_{[E]}(E \cdot D_1)(E \cdot D_2)(F \cdot E) [F] q^{[E+F]}
\\
=&\\
& \sum_{s(L) = 1} d_{[L]}(L \cdot D_1) [D_2] q^{[L]} \\
& + \sum_{s(F) = 0} \sum_{s(E) = -1}
d_{[F]}d_{[E]}(F \cdot D_1)(E \cdot F)(E \cdot D_2) [E] q^{[E+F]}
\end{array}
}
\end{equation}
and
\begin{equation}
\label{pDD2}
{\renewcommand{\arraystretch}{1.5}
\begin{array}{ll}
& \sum_{s(C) = 2} d_{[C]}(D_1 \cdot D_2) q^{[C]} \\
& + \sum_{s(E) = -1}\sum_{s(L) = 1}
d_{[L]}d_{[E]}(E \cdot D_1)(E \cdot D_2)(L \cdot E) q^{[E+L]} \\
=&\\
& \sum_{s(F) = 0} \sum_{s(G) = 0}
d_{[F]}d_{[G]}(F \cdot D_1)(G \cdot F)(G \cdot D_2) q^{[F+G]} .\\
\end{array}
}
\end{equation}
\begin{lemma}
Equation (\ref{pDD2}) follows from (\ref{ppD}).
\end{lemma}
\begin{pf}
In fact, it is obtained from (\ref{ppD})
by setting $D = D_1$ and
dotting with $D_2$.
\end{pf}
The proof of associativity in the $p \operatorname{\ast_Q} D_1 \operatorname{\ast_Q} D_2$ case
now follows from the lemma below.
\begin{lemma}
Equation (\ref{pDD1}) holds for $X_6$.
\end{lemma}
\begin{pf}
We will show that (\ref{pDD1})
follows from two types of relations,
one of which holds generally for generic rational surfaces
and the other of which is special to the cubic surface.
Note that every relevant class $F$ with $s(F) = 0$
can be written uniquely as $F = -K - E_F$,
where $E_F$ is a relevant class with $s(E_F) = -1$
(see Lemma \ref{X6relevantlemma}).
Therefore for any relevant class $E$ with $s(E) = -1$,
we must have $0 \leq (F \cdot E) \leq 2$;
moreover $(F \cdot E) = 2$ if and only if $E = E_F$,
and $(F \cdot E) = 1$ if and only if $(E \cdot E_F) = 1$.
In this latter case $F+E$ is a relevant class with $s(F+E) = 1$
and $p_a(F+E) = 0$.
Therefore (\ref{pDD1}) can be analyzed by considering only these
types of
$q$-terms. We begin by considering a term of the form $q^{[L]}$,
where $L$ is a relevant class with $s(L) = 1$ and $p_a(L) = 0$.
Note that in this case if $L=E+F$,
then $(E \cdot F) = 1$, $(E \cdot L) = 0$ and $(F\cdot L) = 1$.
Moreover all classes contributing to this term have $d=1$.
Thus considering the coefficent of $q^{[L]}$ in (\ref{pDD1})
gives the equation
\[
(D_1 \cdot D_2)[L] +
\sum\begin{Sb} s(E) = -1 \\ s(F) = 0 \\ E+F=L \end{Sb}
(E \cdot D_1)(E \cdot D_2)[F]
= (L \cdot D_1)[D_2] +
\sum\begin{Sb} s(E) = -1 \\ s(F) = 0 \\ E+F=L \end{Sb}
(E \cdot D_2)(F \cdot D_1)[E]
\]
Conversely we note that if $(L \cdot E) = 0$ for $E$ an exceptional
curve,
then by Riemann-Roch, $L\equiv E+F$ for some $F$.
Also if $E+F\equiv E'+F'$, then
$E\equiv E'$ and $F\equiv F'$ or $E$ and $E'$ are disjoint;
this follows from the fact that
$0 = (E' \cdot L) = (E' \cdot E) + (E' \cdot F) \geq (E' \cdot E)$
since $F$ moves in a pencil.
For each such class $L$,
it is easy to see that there are exactly $6$ classes $E$
with $s(E) = -1$ and $(E \cdot L) = 0$.
Therefore there is a disjoint basis of $\operatorname{Pic}(X)$,
$[L], [E_1], \ldots, [E_6]$ where $E_1, \dots , E_6$ are
exceptional curves,
and since $(L \cdot E_i) = 0$,
$L$ can be written as $F_i + E_i$ for each $i$.
In this basis, the equation above is equivalent to:
\[
(D_1 \cdot D_2)[L] - (L \cdot D_1)[D_2] =
\sum_i ((E_i \cdot D_2)(F_i \cdot D_1)[E_i] -
(E_i \cdot D_1)(E_i\cdot D_2)[F_i])
\]
and the right-hand side of this equation is equal to
\begin{eqnarray*}
&=& \sum_i ((E_i \cdot D_2)(F_i \cdot D_1)[E_i] -
(E_i \cdot D_1)(E_i\cdot D_2)[F_i]) \\
&=& \sum_i (E_i \cdot D_2)
((L \cdot D_1)-(E_i \cdot D_1)) [E_i] - (E_i \cdot D_1) [L-E_i]) \\
&=& \sum_i (E_i \cdot D_2)
(- (E_i \cdot D_1)[L] +
((L \cdot D_1)-(E_i \cdot D_1) + (E_i \cdot D_1)) [E_i]) \\
&=& (L \cdot D_1) \sum_i (E_i \cdot D_2)[E_i]
- \sum_i (E_i \cdot D_2)(E_i \cdot D_1)[L]
\end{eqnarray*}
and so we must show that
\[
(D_1 \cdot D_2)[L] - (L \cdot D_1)[D_2] =
(L \cdot D_1) \sum_i (E_i \cdot D_2)[E_i]
- \sum_i (E_i \cdot D_2)(E_i \cdot D_1)[L].
\]
On the other hand
$[D_2] = (L \cdot D_2)[L] - \sum_i(E_i \cdot D_2)[E_i]$.
Plugging this expression into the above equation,
we obtain an expression all in terms of the basis
$[L],[E_1],\ldots,[E_6]$.
The coefficients of $[E_i]$ on the two sides are obviously equal,
to $(L \cdot D_1)(E_i \cdot D_2)$.
Hence we must only check the coefficient of $[L]$, and so
the above equation follows from the identity
\[
(D_1 \cdot D_2) =
(L \cdot D_1)(L \cdot D_2) - \sum_i(E_i \cdot D_1)(E_i \cdot D_2)
\]
which is immediate from writing $[D_1]$ and $[D_2]$
in terms of the basis $[L],[E_1],\ldots,[E_6]$.
To complete the proof of the lemma,
we need to consider the coefficient of $q^{[-K]}$
in (\ref{pDD1});
$L = -K$ is the unique relevant class with $s=1$ and $p_a = 1$.
In the $E,F$ sums, we may sum over the $E$'s only, setting $F =
-K-E$;
noting that in this case $(E \cdot F) = 2$,
equating the coefficients of $q^{[-K]}$ in (\ref{pDD1})
and dividing by two gives
\begin{eqnarray}
6 (K \cdot D_1)[D_2] - 6(D_1 \cdot D_2)[K] &= &
\sum_{s(E)= -1}
((-K-E)\cdot D_1)(E \cdot D_2)[E] - (E \cdot D_2)(E \cdot
D_1)[-K-E]\nonumber
\\
&=& \sum_{s(E)= -1}
(-K \cdot D_1)(E \cdot D_2)[E] - (E \cdot D_2)(E \cdot D_1)[-K].
\label{pDD-K}
\end{eqnarray}
As we saw in the proof above for the associativity of the triple
product
$p \operatorname{\ast_Q} p \operatorname{\ast_Q} D$, $-K$ is
linearly equivalent to any triangle of exceptional curves;
moreover precisely ten exceptional curves meet any given
exceptional curve.
Thus for all exceptional curves $E'$,
\begin{eqnarray*}
5[-K] &=& 5[E'] + \sum_{(E \cdot E') = 1} [E] \\
&=& 6[E'] + \sum_E (E \cdot E')[E].
\end{eqnarray*}
Thus for all exceptional curves $E'$ and $E''$,
\[
6[E'] + \sum_E (E \cdot E')[E] = 6[E''] + \sum_E (E \cdot E'')[E]
\]
and so
\[
6[E'] = 6[E''] + \sum_E (E \cdot E'')[E] - \sum_E (E \cdot E')[E].
\]
Intersecting with $D_2$ and noting that $(-K \cdot E') = 1$ for all
$E'$,
we have
\[
6(E' \cdot D_2) = ( 6(E'' \cdot D_2) + \sum_E (E \cdot D_2)(E \cdot
E''))
(-K \cdot E') - \sum_E (E \cdot D_2)(E \cdot E').
\]
Here this equation holds for all exceptional curves $E'$,
which generate $\operatorname{Pic}(X_6)$, so
\[
6[D_2] = (6(E'' \cdot D_2) + \sum_E(E \cdot D_2)(E \cdot E''))[-K]
- \sum_E (E \cdot D_2)[E].
\]
Intersecting now with $D_1$ and again noting that $(-K \cdot
E'')=1$, we have
\[
(6(D_1 \cdot D_2)+ \sum_E (E \cdot D_2)(E \cdot D_1))(-K \cdot E'')
=
(-K \cdot D_1)(6(E'' \cdot D_2) + \sum_E(E \cdot D_2)(E \cdot
E'')).
\]
As this is true for all $E''$, which generate $\operatorname{Pic}(X_6)$, we see
that
\[
(6(D_1 \cdot D_2)+ \sum_E (E \cdot D_2)(E \cdot D_1))[-K]
= (-K \cdot D_1)(6[D_2] + \sum_E(E \cdot D_2)[E]).
\]
This can be re-written as
\begin{eqnarray*}
6(D_1 \cdot D_2)[-K] - 6(-K \cdot D_1)[D_2]
&=& \sum_E (-K \cdot D_1)(E \cdot D_2)[E] - (E \cdot D_1)(E \cdot
D_2)[-K]\\
&=& \sum_E (E \cdot D_2)((-K \cdot D_1)[E] - (E \cdot D_1)[-K]).
\end{eqnarray*}
This is exactly the desired equation (\ref{pDD-K}).
\end{pf}
To conclude our proof of associativity for the quantum product on
$X_6$,
we must deal with triple quantum products
of the form $D_1 \operatorname{\ast_Q} D_2 \operatorname{\ast_Q} D_3$ for divisors $D_i$.
Note that $d_{[C]} = 1$ whenever $[C]$ is a relevant class on $X_6$
with $s(c) \leq 0$;
we then compute:
\begin{eqnarray*}
D_1 \operatorname{\ast_Q} (D_2 \operatorname{\ast_Q} D_3) & =& D_1 \operatorname{\ast_Q} ( (D_2 \cdot D_3)[p]q^0
+ \!\!\sum_{s(E)= -1} (E \cdot D_2)(E \cdot D_3)[E]q^E
+ \!\!\sum_{s(F)= 0} (F \cdot D_2)(F \cdot D_3)[X]q^F) \\
& = & (D_2 \cdot D_3)(D_1 \operatorname{\ast_Q} [p])q^0
+ \sum_{s(E)= -1} (E \cdot D_2)(E \cdot D_3)(D_1 \operatorname{\ast_Q} [E])q^E \\
&&+ \sum_{s(F)= 0} (F \cdot D_2)(F \cdot D_3)[D_1]q^F \\
& = & \sum_{s(F)= 0} (D_2 \cdot D_3)(F \cdot D_1)[F]q^F
+ \sum_{s(L)= 1} d_{[L]}(D_2 \cdot D_3)(L \cdot D_1)[X]q^L \\
&&+ \sum_{s(F)= 0} (F \cdot D_2)(F \cdot D_3)[D_1]q^F \\
&&+ \sum_{s(E)= -1} (E \cdot D_2)(E \cdot D_3)
( (E \cdot D_1)[p]q^E +
\sum_{s(E')= -1} (E' \cdot D_1)(E \cdot E')[E']q^{E+E'} \\
&&+ \sum_{s(F)= 0} (F \cdot D_1)(F \cdot E)[X]q^{E+F} ) \\
& = & \text{(dimension zero terms:)}
\sum_{s(E)= -1} (E \cdot D_1)(E \cdot D_2)(E \cdot D_3)[p]q^E \\
& + & \text{(dimension two terms:)}
\sum_{s(F)= 0} ( (D_2 \cdot D_3)(F \cdot D_1)[F] +
(F \cdot D_2)(F \cdot D_3)[D_1] ) q^F \\
&& + \sum_{s(E)=s(E')= -1}
(E \cdot D_2)(E \cdot D_3)(E' \cdot D_1)(E \cdot
E')[E']q^{E+E'} \\
& + & \text{(dimension four terms:)}
\sum_{s(L)= 1} d_{[L]}(D_2 \cdot D_3)(L \cdot D_1)[X]q^L \\
&& + \sum\begin{Sb} s(E)= -1 \\ s(F)= 0 \end{Sb}
(E \cdot D_2)(E \cdot D_3)(F \cdot D_1)(F \cdot E)[X]q^{E+F}
\\
\end{eqnarray*}
On the other hand
\begin{eqnarray*}
(D_1 \operatorname{\ast_Q} D_2) \operatorname{\ast_Q} D_3 & =& D_3 \operatorname{\ast_Q} (D_1 \operatorname{\ast_Q} D_2) \\
&=& \text{(dimension zero terms:)}
\sum_{s(E)= -1} (E \cdot D_3)(E \cdot D_1)(E \cdot D_2)[p]q^E \\
& + & \text{(dimension two terms:)}
\sum_{s(F)= 0} ( (D_1 \cdot D_2)(F \cdot D_3)[F] +
(F \cdot D_1)(F \cdot D_2)[D_3] ) q^F \\
&& + \sum_{s(E)=s(E')= -1}
(E \cdot D_1)(E \cdot D_2)(E' \cdot D_3)(E \cdot
E')[E']q^{E+E'} \\
& + & \text{(dimension four terms:)}
\sum_{s(L)= 1} d_{[L]}(D_1 \cdot D_2)(L \cdot D_3)[X]q^L \\
&& + \sum\begin{Sb} s(E)= -1 \\ s(F)= 0 \end{Sb}
(E \cdot D_1)(E \cdot D_2)(F \cdot D_3)(F \cdot E)[X]q^{E+F}
\\
\end{eqnarray*}
(This is obtained from the previous by permuting indices.)
Comparing terms, we see that the dimension zero terms are identical
and that the dimension four terms follow from Equation (\ref{pDD1})
(obtained by considering the $p \operatorname{\ast_Q} D_1 \operatorname{\ast_Q} D_2$ product)
intersected with $D_3$.
Comparing terms in dimension two, we need to show that
\begin{equation}
\label{DDD}
{\renewcommand{\arraystretch}{1.5}
\begin{array}{ll}
& \sum_{s(F)= 0}
( (D_1 \cdot D_2)(F \cdot D_3)[F] + (F \cdot D_1)(F \cdot
D_2)[D_3] )q^F \\
& + \sum_{s(E)=s(E')= -1}
(E \cdot D_1)(E \cdot D_2)(E' \cdot D_3)(E' \cdot E) [E']
q^{E+E'} \\
=&\\
& \sum_{s(F)= 0}
( (D_2 \cdot D_3)(F \cdot D_1)[F] + (F \cdot D_2)(F \cdot
D_3)[D_1] ) q^F \\
& + \sum_{s(E)=s(E')= -1}
(E' \cdot D_1)(E \cdot D_2)(E \cdot D_3)(E \cdot E')[E']q^{E+E'}.
\end{array}
}
\end{equation}
Now if $E$ and $E'$ are disjoint, then there is no contribution to
either side. If $E = E'$, then the coefficients of $q^{E+E'}$ are
seen to be equal. If $E$ meets $E'$, then $(E \cdot E')=1$ and
$E+E' \equiv F$ for some $F$.
Hence the equality above follows from the equality of the
coefficients of $q^F$
for a particular class $[F]$ with $s(F) = 0$.
Hence associativity is implied by the following lemma.
\begin{lemma}
For all relevant classes $[F]$ with $s(F)=0$,
\begin{eqnarray*}
&&(D_1 \cdot D_2)(F \cdot D_3)[F] + (F \cdot D_1)(F \cdot D_2)[D_3]
-(D_2 \cdot D_3)(F \cdot D_1)[F] - (F \cdot D_2)(F \cdot D_3)[D_1]
\\
&=&
\sum_{E+E' \equiv F}((E' \cdot D_1)(E \cdot D_2)(E \cdot D_3) - (E
\cdot D_1)(E \cdot D_2)(E' \cdot D_3))[E'].
\end{eqnarray*}
\end{lemma}
\begin{pf}
Our fixed curve $F$ gives $X$ a structure of ruled surface
with $F$ as general fiber.
Each $E+E'$ summing to $F$ is a reducible fiber
with respect to that ruling (there are five such reducible fibers).
Note that there is a basis, $[F],[G],[E_1],\dots,[E_5]$ of
$\operatorname{Pic}(X_6)$
such that $G$ has self-intersection zero,
the $E_i$'s have self-intersection -1, $(F \cdot G) =1$,
and all other intersections are zero
(this is equivalent to $X_6$ having
${\Bbb F}_0 = {\Bbb P}^1\times {\Bbb P}^1$ as a minimal ruled
model).
We now write the right hand side of the above equation
in terms of the basis.
Note that $E'_i \equiv F - E_i$
and that the role of $E'$ is played by both $E'$ and $E$.
We have
\begin{eqnarray*}
&& \sum_{E+E' \equiv F}((E' \cdot D_1)(E \cdot D_2)(E \cdot D_3) -
(E
\cdot D_1)(E \cdot D_2)(E' \cdot D_3))[E'] \\
&=&
\sum_i ((F-E_i) \cdot D_1)(E_i \cdot D_2)(E_i \cdot D_3) - (E_i
\cdot D_1)(E_i \cdot D_2)((F-E_i) \cdot D_3))[F-E_i] \\
&& + \sum_i (E_i \cdot D_1)((F-E_i) \cdot D_2)((F-E_i) \cdot D_3)
-
((F-E_i) \cdot D_1)((F-E_i) \cdot D_2)(E_i \cdot D_3))[E_i].
\end{eqnarray*}
The coefficient of each $[E_i]$ in this expression
contains a sum of twelve products,
all of which cancel except for
$(E_i \cdot D_1)(F \cdot D_2)(F \cdot D_3) -
(F \cdot D_1)(F \cdot D_2)(E_i \cdot D_3)$.
The coefficient of the $[F]$ term is seen to be
$\sum_i ( (F \cdot D_1)(E_i \cdot D_2)(E_i \cdot D_3)-
(E_i \cdot D_1)(E_i \cdot D_2)(F \cdot D_3) )$.
Thus we may re-write the equation of the lemma as
\begin{eqnarray*}
&&(D_1 \cdot D_2)(F \cdot D_3)[F] + (F \cdot D_1)(F \cdot D_2)[D_3]
-(D_2 \cdot D_3)(F \cdot D_1)[F] - (F \cdot D_2)(F \cdot D_3)[D_1]
\\
&=& \\
&&
[F]\sum_i ( (F \cdot D_1)(E_i \cdot D_2)(E_i \cdot D_3)-
(E_i \cdot D_1)(E_i \cdot D_2)(F \cdot D_3) ) \\
&& + \sum_i ((E_i \cdot D_1)(F \cdot D_2)(F \cdot D_3) -
(F \cdot D_1)(F \cdot D_2)(E_i \cdot D_3)) [E_i].
\end{eqnarray*}
To show that this linear equivalence is true,
it suffices to show equality when dotted with a basis of $\operatorname{Pic}(X)$.
It is easy to see
that equality holds when the expression above is dotted with $F$ or
any $E_i$.
Dotting with $G$, and recalling that $(F\cdot G) = 1$
and $(G \cdot E_i)= 0$ yields the following expression:
\begin{eqnarray*}
\label{mess}
&&(D_1 \cdot D_2)(F \cdot D_3) + (F \cdot D_1)(F \cdot D_2)(G\cdot
D_3)
-(D_2 \cdot D_3)(F \cdot D_1) - (F \cdot D_2)(F \cdot D_3)(G \cdot
D_1) \\
&=& \\
&&
\sum_i ( (F \cdot D_1)(E_i \cdot D_2)(E_i \cdot D_3)-
(E_i \cdot D_1)(E_i \cdot D_2)(F \cdot D_3) ).
\end{eqnarray*}
On the other hand, the divisors $D_j$, $j=1,2,3$
are written in terms of the basis as
\[
[D_j] \equiv (D_j \cdot G)[F] + (D_j \cdot F)[G] - \sum_i (D_j
\cdot E_i)[E_i].
\]
Substituting these expressions into $(D_1 \cdot D_2)$ and $(D_2
\cdot D_3)$
of (\ref{mess}) and collecting terms proves the result.
\end{pf}
This completes our analysis of the associativity of the quantum
product
for $X_6$, and therefore for all general strict Del Pezzo surfaces.
We have proved the following.
\begin{theorem}
The quantum product $\operatorname{\ast_Q}$ is associative for
${\Bbb P}^2$, ${\Bbb F}_0$, and $X_1,\ldots,X_6$.
\end{theorem}
\section{Associativity in general}
\label{sectionassoc}
In this section we offer an algebro-geometric approach
to proving the associativity of the quantum product for a general
rational surface.
This approach avoids the reliance on perturbing
to a non-integrable almost complex structure
(see \cite{mcduff-salamon,ruan-tian});
we work with the existing complex/algebraic structure.
The associativity of the quantum product
is implied by checking associativity
for triple products of homogeneous generators for $H^*(X)$.
In other words, we must check that if $\alpha$, $\beta$, and
$\gamma$
are homogeneous classes in $H^*(X)$, then
\begin{equation}
\label{assoc1}
\alpha \operatorname{\ast_Q} (\beta \operatorname{\ast_Q} \gamma) =
(\alpha \operatorname{\ast_Q} \beta) \operatorname{\ast_Q} \gamma.
\end{equation}
We need not check the formula
when one of the constituents is the identity $[X]$,
or when they are all equal.
\begin{lemma}
\label{assoclemma}
The associativity of the quantum product
is equivalent to the following identity:
\[
\sum_{([C_1],[C_2]):[C_1+C_2]=[D]}
\phi_{[C_2]}(\alpha,\delta) \cdot \phi_{[C_1]}(\beta,\gamma)
=
\sum_{([C_1],[C_2]):[C_1+C_2]=[D]}
\phi_{[C_2]}(\gamma,\delta) \cdot \phi_{[C_1]}(\alpha,\beta).
\]
This identity must hold for all divisor classes $[D]$
and all homogeneous classes $\alpha$, $\beta$, $\gamma$, and
$\delta$
in $H^*(X)$.
\end{lemma}
\begin{pf}
Expanding the two sides of (\ref{assoc1}), we have
\begin{eqnarray*}
\alpha \operatorname{\ast_Q} (\beta \operatorname{\ast_Q} \gamma) &=&
\alpha \operatorname{\ast_Q} ( \sum_{[C_1]} \phi_{[C_1]}(\beta,\gamma) q^{[C_1]})
\\
&=& \sum_{[C_1]} \sum_{[C_2]}
\phi_{[C_2]}( \alpha,\phi_{[C_1]}(\beta,\gamma) ) q^{[C_1+C_2]}
\end{eqnarray*}
while
\begin{eqnarray*}
(\alpha \operatorname{\ast_Q} \beta) \operatorname{\ast_Q} \gamma &=&
( \sum_{[C_1]} \phi_{[C_1]}(\alpha,\beta) q^{[C_1]} ) \operatorname{\ast_Q}
\gamma \\
&=& \sum_{[C_1]} \sum_{[C_2]}
\phi_{[C_2]}(\phi_{[C_1]}(\alpha,\beta),\gamma ) q^{[C_1+C_2]}.
\end{eqnarray*}
For these to be equal,
they must have equal coefficients for all terms $q^{[D]}$.
Therefore associativity of the quantum product is equivalent to
having
\[
\sum_{([C_1],[C_2]):[C_1+C_2]=[D]}
\phi_{[C_2]}( \alpha,\phi_{[C_1]}(\beta,\gamma) )
=
\sum_{([C_1],[C_2]):[C_1+C_2]=[D]}
\phi_{[C_2]}(\phi_{[C_1]}(\alpha,\beta),\gamma )
\]
for all homogeneous classes $\alpha$, $\beta$, and $\gamma$ in
$H^*(X)$
and all divisor classes $[D]$.
The equality is equivalent to knowing that
for all homogeneous $\delta \in H^*(X)$,
the intersection products with $\delta$ are equal, i.e.,
\[
\sum_{([C_1],[C_2]):[C_1+C_2]=[D]}
\phi_{[C_2]}( \alpha,\phi_{[C_1]}(\beta,\gamma) ) \cdot \delta
=
\sum_{([C_1],[C_2]):[C_1+C_2]=[D]}
\phi_{[C_2]}(\phi_{[C_1]}(\alpha,\beta),\gamma ) \cdot \delta.
\]
(Here a dot product is taken to be zero
unless the codimensions of the classes are complementary.)
These intersection products can be computed on $X^3$,
and we then are requiring that
\[
\sum_{([C_1],[C_2]):[C_1+C_2]=[D]}
[A_{[C_2]}] \cdot
(\alpha\otimes\phi_{[C_1]}(\beta,\gamma)\otimes\delta)
=
\sum_{([C_1],[C_2]):[C_1+C_2]=[D]}
[A_{[C_2]}] \cdot (\phi_{[C_1]}(\alpha,\beta) \otimes\gamma \otimes
\delta).
\]
By the symmetry of the $[A]$-classes we may rewrite this as
\[
\sum_{([C_1],[C_2]):[C_1+C_2]=[D]}
[A_{[C_2]}] \cdot
(\alpha\otimes\delta\otimes\phi_{[C_1]}(\beta,\gamma))
=
\sum_{([C_1],[C_2]):[C_1+C_2]=[D]}
[A_{[C_2]}] \cdot ( \gamma \otimes
\delta\otimes\phi_{[C_1]}(\alpha,\beta)),
\]
which we may then reformulate as
\[
\sum_{([C_1],[C_2]):[C_1+C_2]=[D]}
\phi_{[C_2]}(\alpha,\delta) \cdot \phi_{[C_1]}(\beta,\gamma)
=
\sum_{([C_1],[C_2]):[C_1+C_2]=[D]}
\phi_{[C_2]}(\gamma,\delta) \cdot \phi_{[C_1]}(\alpha,\beta).
\]
This is the desired identity.
\end{pf}
Let $[C]$ be a relevant class on $X$,
and let ${\cal R}_{[C]}$ denote the locus of irreducible nodal
rational curves
in the linear system $|C|$.
Recall that $d_{[C]}$ is the degree of the closure $\overline{{\cal
R}_{[C]}}$.
Suppose that $[C]$ decomposes as $[C] = [C_1] +[C_2]$ where
$[C_1]$ and $[C_2]$ are relevant classes.
We may form the following locus
\begin{eqnarray*}
{\cal S}_{[C_1],[C_2]} &= \{&
(C_1,C_2,x_1,y_1,x_2,y_2,z)
\in {\cal R}_{[C_1]} \times {\cal R}_{[C_2]} \times X^5 \;|\;
\\
&& C_1 \text{ and } C_2 \text{ meet transversally at } z, \\
&& x_1 \text{ and } y_1 \text{ are smooth points of }C_1, \text{
and } \\
&& x_2 \text{ and } y_2 \text{ are smooth points of }C_2
\}.
\end{eqnarray*}
There is a natural map
\[
{\cal S}_{[C_1],[C_2]} \to X^6
\]
sending $(C_1,C_2,x_1,y_1,x_2,y_2,z)$ to $(x_1,y_1,z,x_2,y_2,z)$.
Call the image of the fundamental class $[A^6_{[C_1],[C_2]}]$.
Related to this is the map
\[
{\cal S}_{[C_1]} \times {\cal S}_{[C_2]} \to X^6
\]
sending a pair
$((C_1,x_1,y_1,z_1),(C_2,x_2,y_2,z_2))$ to
$(x_1,y_1,z_1,x_2,y_2,z_2)$.
The image of the fundamental class of this map is clearly
$[A_{[C_1]}] \otimes [A_{[C_2]}]$.
If we denote by $\pi_{36}:X^6 \to X^2$
the projection onto the third and sixth coordinates,
we see that
\[
[A^6_{[C_1],[C_2]}] = ([A_{[C_1]}] \otimes [A_{[C_2]}])
\cup \pi_{36}^*([\Delta])
\]
where $\Delta \subset X^2$ is the diagonal.
Finally consider the natural map
\[
{\cal S}_{[C_1],[C_2]} \to X^4
\]
sending $(C_1,C_2,x_1,y_1,x_2,y_2,z)$ to $(x_1,y_1,x_2,y_2)$.
Call the image of the fundamental class $[A^4_{[C_1],[C_2]}]$.
\begin{lemma}
With the above notation,
\[
\phi_{[C_1]}(\alpha,\beta) \cdot \phi_{[C_2]}(\gamma,\delta) =
[A^4_{[C_1],[C_2]}] \cdot
\alpha\otimes\beta\otimes\gamma\otimes\delta.
\]
\end{lemma}
\begin{pf}
Write $[\Delta] = \sum_i u_i \otimes v_i$ in $H^4(X^2)$,
where the $u_i$ and $v_i$ are classes in $H^*(X)$.
Denote by $\pi_{1245}:X^6 \to X^4$ the projection onto the
first, second, fourth, and fifth factors.
Then
\begin{eqnarray*}
\phi_{[C_1]}(\alpha,\beta) \cdot \phi_{[C_2]}(\gamma,\delta) &=&
[\phi_{[C_1]}(\alpha,\beta) \otimes \phi_{[C_2]}(\gamma,\delta)]
\cdot [\Delta] \\
&=& \sum_i [\phi_{[C_1]}(\alpha,\beta) \otimes
\phi_{[C_2]}(\gamma,\delta)]
\cdot[u_i\otimes v_i] \\
&=& \sum_i (\phi_{[C_1]}(\alpha,\beta) \cdot u_i)
(\phi_{[C_2]}(\gamma,\delta) \cdot v_i) \\
&=& \sum_i([A_{[C_1]}] \cdot \alpha\otimes\beta\otimes u_i)
([A_{[C_2]}] \cdot \gamma\otimes\delta\otimes v_i) \\
&=& \sum_i ([A_{[C_1]}] \otimes [A_{[C_2]}])\cdot
(\alpha\otimes\beta\otimes u_i \otimes \gamma\otimes\delta\otimes
v_i) \\
&=& ([A_{[C_1]}] \otimes [A_{[C_2]}])\cup
(\alpha\otimes\beta\otimes X \otimes \gamma\otimes\delta\otimes
X)
\cup \pi_{36}^*([\Delta]) \\
&=& ([A_{[C_1]}] \otimes [A_{[C_2]}])\cup
\pi_{1245}^*(\alpha\otimes\beta\otimes \gamma\otimes\delta)
\cup \pi_{36}^*([\Delta]) \\
&=& [A^6_{[C_1],[C_2]}] \cdot
\pi_{1245}^*(\alpha\otimes\beta\otimes \gamma\otimes\delta) \\
&=& [A^4_{[C_1],[C_2]}] \cdot
(\alpha\otimes\beta\otimes \gamma\otimes\delta)
\end{eqnarray*}
as claimed.
\end{pf}
Next we introduce the space
\begin{eqnarray*}
{\cal S}^4_{[C]} &= \{&
(C,x_1,x_2,x_3,x_4)
\in {\cal R}_{[C]} \times X^4 \;|\; \\
&& x_i \text{ are smooth points of }C
\}.
\end{eqnarray*}
There is a natural projection to $X^4$;
we denote the image of the fundamental class by $[A^4_{[C]}]$.
We note that we can consider the spaces ${\cal S}_{[C_1],[C_2]}$
as lying in the closure of the space ${\cal S}^4_{[C_1+C_2]}$,
via the natural map
\[
\psi:{\cal S}_{[C_1],[C_2]} \to \overline{ {\cal S}^4_{[C_1+C_2]}
}
\]
defined by sending $(C_1,C_2,x_1,y_1,x_2,y_2,z)$ to
$(C_1+C_2,x_1,y_1,x_2,y_2)$.
Actually, this map $\psi$ may be finite-to-one,
if $(C_1\cdot C_2) \geq 2$.
The image points represent the addition of an extra node to a curve
which already has arithmetic genus zero;
this then breaks the curve into two components.
One expects that the boundary of ${\cal S}^4_{[C]}$ in its closure
will have exactly the images of these loci
$\psi({\cal S}_{[C_1],[C_2]})$
with $[C_1]+[C_2] = [C]$
as components.
Now the space ${\cal S}^4_{[C]}$ has a cross-ratio function on it,
\[
\operatorname{CR}:{\cal S}^4_{[C]} \to {\Bbb P}^1,
\]
defined by sending $(C,x_1,x_2,x_3,x_4)$
to the cross-ratio
$(x_1-x_3)(x_2-x_4)/(x_1-x_4)(x_2-x_3)$.
(Here a coordinate is chosen on the normalization of $C$.)
Again one expects that this cross-ratio function will extend to the
closure
$\overline{ {\cal S}^4_{[C_1+C_2]} }$,
or at least to a model of the closure which is birational
on the boundary divisors $\psi({\cal S}_{[C_1],[C_2]})$
with $[C_1]+[C_2] = [C]$.
For distinct points, the cross-ratio takes values in
${\Bbb P}^1 - \{0,1,\infty\}$.
For the general coalescence of two of the four points,
the cross-ratio takes value $0$ when $x_1 = x_3$,
value $1$ when $x_1 = x_2$, and value $\infty$ when $x_1 = x_4$.
Therefore $\operatorname{CR}^{-1}(1)$ on the closure
should be the space where the first two points $x_1$ and $x_2$
come together; when this happens,
the curve will split, with $x_1$ and $x_2$ moving to points
on one curve and $x_3$ and $x_4$ lying on the other.
Therefore if we denote by $[A^4_{[C]}(\lambda)]$
the class of the image of $\operatorname{CR}^{-1}(\lambda)$ in $X^4$,
we have that
\[
[A^4_{[C]}(1)] = \sum\begin{Sb} ([C_1],[C_2]) \\ [C_1+C_2]=[C]
\end{Sb}
[A^4_{[C_1],[C_2]}]
\]
Therefore
\begin{eqnarray*}
\sum\begin{Sb} ([C_1],[C_2]) \\ [C_1+C_2]=[C] \end{Sb}
\phi_{[C_2]}(\alpha,\delta) \cdot \phi_{[C_1]}(\beta,\gamma)
&=& \sum\begin{Sb} ([C_1],[C_2]) \\ [C_1+C_2]=[C] \end{Sb}
[A^4_{[C_1],[C_2]}]
\cdot \beta\otimes\gamma\otimes\alpha\otimes\delta \\
&=& [A^4_{[C]}(1)] \cdot
\beta\otimes\gamma\otimes\alpha\otimes\delta.
\end{eqnarray*}
Now as $\lambda$ varies, the classes $[A^4_{[C]}(\lambda)]$
are rationally equivalent; hence the intersection product is the
same.
Therefore
\[
[A^4_{[C]}(1)] \cdot \beta\otimes\gamma\otimes\alpha\otimes\delta
=
[A^4_{[C]}(0)] \cdot \beta\otimes\gamma\otimes\alpha\otimes\delta.
\]
This class $[A^4_{[C]}(0)]$ can be analyzed
by studying the maps
\[
\tilde{\psi}:{\cal S}_{[C_1],[C_2]} \to \overline{ {\cal
S}^4_{[C_1+C_2]} }
\]
defined by sending $(C_1,C_2,x_1,y_1,x_2,y_2,z)$ to
$(C_1+C_2,x_1,x_2,y_1,y_2)$.
This is just the map $\psi$ above, followed
by a permutation of the four points;
but we see that if we denote the image of these classes by
$[\tilde{A}^4_{[C_1],[C_2]}]$, then
\[
[A^4_{[C]}(0)] = \sum\begin{Sb} ([C_1],[C_2]) \\ [C_1+C_2]=[C]
\end{Sb}
[\tilde{A}^4_{[C_1],[C_2]}]
\]
since the cross-ratio being zero represents when the first and
third
points come together, and this should be modelled by having them
split off
to one of the component curves (in this case $C_1$).
This class $[\tilde{A}^4_{[C_1],[C_2]}]$ is related to the original
version
$[A^4_{[C_1],[C_2]}]$ by the relation that
\[
[A^4_{[C_1],[C_2]}] \cdot
\beta\otimes\alpha\otimes\gamma\otimes\delta
=
[\tilde{A}^4_{[C_1],[C_2]}] \cdot
\beta\otimes\gamma\otimes\alpha\otimes\delta
\]
because of the permutation
which relates the maps $\psi$ and $\tilde{\psi}$.
Hence
\begin{eqnarray*}
[A^4_{[C]}(0)] \cdot \beta\otimes\gamma\otimes\alpha\otimes\delta
&=&
\sum\begin{Sb} ([C_1],[C_2]) \\ [C_1+C_2]=[C] \end{Sb}
[\tilde{A}^4_{[C_1],[C_2]}] \cdot
\beta\otimes\gamma\otimes\alpha\otimes\delta \\
&=& \sum\begin{Sb} ([C_1],[C_2]) \\ [C_1+C_2]=[C] \end{Sb}
[A^4_{[C_1],[C_2]}] \cdot
\beta\otimes\alpha\otimes\gamma\otimes\delta \\
&=& \sum\begin{Sb} ([C_1],[C_2]) \\ [C_1+C_2]=[C] \end{Sb}
\phi_{[C_1]}(\beta,\alpha) \cdot \phi_{[C_2]}(\gamma,\delta)
\end{eqnarray*}
We conclude that
\[
\sum\begin{Sb} ([C_1],[C_2]) \\ [C_1+C_2]=[C] \end{Sb}
\phi_{[C_2]}(\alpha,\delta) \cdot \phi_{[C_1]}(\beta,\gamma)
=
\sum\begin{Sb} ([C_1],[C_2]) \\ [C_1+C_2]=[C] \end{Sb}
\phi_{[C_1]}(\beta,\alpha) \cdot \phi_{[C_2]}(\gamma,\delta)
\]
which is equivalent to the associative law for the quantum product
by Lemma \ref{assoclemma}, using the symmetry of the
$\phi$-classes.
The reader will note that the approach given above
to the proof of associativity
relies on the existence of a model of the closure of
the spaces ${\cal S}^4_{[C]}$,
which has rather nice properties:
the boundary is well-understood in terms of splittings of $C$ as
$C = C_1 + C_2$,
and the cross-ratio function extends nicely to it.
The existence of such a model we only conjecture,
and have not attempted to construct it in this paper.
\section{Enumerative consequences of associativity}
\label{sectionenum}
We will now extract several enumerative consequences
from the associative law for the quantum product
which have been noted by Kontsevich and Manin in
\cite{kontsevich-manin}.
Let us change notation somewhat and introduce the integer
\[
k(C) = k([C]) =\begin{cases}
(-K\cdot C) & \text{ if the class $[C]$ is relevant on $X$}\\
0 & \text{if $[C]$ is not relevant.}
\end{cases}
\]
We note that by the adjunction formula, if $[C]$ is a relevant
class,
then $k(C) = s(C) + 2$
(recall that $s(C) = (C\cdot C) - 2p_a(C)$).
Moreover $k(-)$ is linear in relevant
classes:
\[
k(C_1+C_2) = k(C_1)+k(C_2).
\]
Also, the subscript notation for the degree $d_{[C]}$ of the
rational curve
locus ${\cal R}_{[C]}$ in the linear system $|C|$
is too cumbersome; we will switch notation and call this degree
$N(C)$
(following the notation in \cite{kontsevich-manin}).
The equations which were seen in Section \ref{sectionSDPassoc}
to be equivalent to the associative law
for the quantum product for strict Del Pezzos
are actually equivalent to associativity for any rational surface
$X$.
These were (\ref{ppD}), (\ref{pDD1}), and (\ref{DDD}).
With the above notation, they can be written as
\begin{eqnarray}
\label{assocc1}
\sum_{k(C)= 4} N(C) [D] q^{[C]} &+ &
\sum_{k(L)= 3}\sum_{k(E)= 1}
N(L) N(E) (E \cdot L)(E \cdot D) [E] q^{[E+L]} \\
&=& \sum_{k(F)= 2}\sum_{k(G)= 2}
N(F)N(G)(F \cdot D)(G \cdot F)[G] q^{[F+G]} \nonumber
\end{eqnarray}
for any divisor class $[D]$,
\begin{equation}
\label{assoc2}
{\renewcommand{\arraystretch}{1.5}
\begin{array}{ll}
& \sum_{k(L) = 3} N(L)(D_1 \cdot D_2) [L] q^{[L]} \\
& + \sum_{k(E) = 1}\sum_{k(F) = 2}
N(F)N(E)(E \cdot D_1)(E \cdot D_2)(F \cdot E) [F] q^{[E+F]} \\
=&\\
& \sum_{k(L) = 3} N(L)(L \cdot D_1) [D_2] q^{[L]} \\
& + \sum_{k(F) = 2} \sum_{k(E) = 1}
N(F)N(E)(F \cdot D_1)(E \cdot F)(E \cdot D_2) [E] q^{[E+F]}
\end{array}
}
\end{equation}
for any divisor classes $[D_1]$ and $[D_2]$, and
\begin{equation}
\label{assoc3}
{\renewcommand{\arraystretch}{1.5}
\begin{array}{ll}
& \sum_{k(F)= 2} N(F)
( (D_1 \cdot D_2)(F \cdot D_3)[F] + (F \cdot D_1)(F \cdot
D_2)[D_3] )q^F \\
& + \sum_{k(E)=k(E')= 1} N(E)N(E')
(E \cdot D_1)(E \cdot D_2)(E' \cdot D_3)(E' \cdot E) [E']
q^{E+E'} \\
=&\\
& \sum_{k(F)= 2}N(F)
( (D_2 \cdot D_3)(F \cdot D_1)[F] + (F \cdot D_2)(F \cdot
D_3)[D_1] ) q^F \\
& + \sum_{k(E)=k(E')= 1} N(E)N(E')
(E' \cdot D_1)(E \cdot D_2)(E \cdot D_3)(E \cdot E')[E']q^{E+E'}.
\end{array}
}
\end{equation}
for any divisor classes $[D_1]$, $[D_2]$, and $[D_3]$.
(Note that (\ref{DDD}) had no $N(-)$ numbers; all were equal to one
in the case of $X_6$, but in general they must be included of
course.)
Suppose that $X = X_n$, the general $n$-fold blowup of the
plane.
We take as a basis for $\operatorname{Pic}(X)$ the classes
$[H]$, $[E_1], \dots, [E_n]$,
where $[H]$ is the class of the pullback of a line from ${\Bbb
P}^2$,
and $E_i$ is the exceptional curve over the $i$-th point $p_i$
which is blown up.
In this case every divisor class can be written as
\[
[D] = d[H] - \sum_{i=1}^n m_i [E_i]
\]
which we will abbreviate to $[D] = (d;m_1,\dots,m_n)$.
Note then that the anticanonical class $[-K] = (3;1^n)$
where we use the exponential notation for repeated $m_i$'s,
as is rather standard.
Hence if $[D] = (d;m_1,\dots,m_n)$ then $(-K\cdot D) = 3d-\sum_i
m_i$.
With this notation we see that quantum cohomology
is a completely numerically based theory.
A class $[D] = (d;m_1,\dots,m_n)$ is relevant if and only if
$d^2 + 3d \geq \sum_i m_i^2 + \sum_i m_i$
and $k(D) \leq 4$.
(The first condition is that the expected dimension of $|D|$ is
non-negative,
so that there will be curves in $|D|$;
the second is the relevance condition, that the dimension of the
locus
of rational curves in $|D|$ is not more than $3$.)
Suppose that we take the associativity condition (\ref{assocc1}),
set $D = H$, and dot with $H$: we obtain
\begin{eqnarray}
\label{assoc1D=HdotH}
\sum_{k(C)= 4} N(C) q^{[C]} &= &
\sum_{k(F)= 2}\sum_{k(G)= 2}
N(F)N(G)(F \cdot H)(G \cdot F)(G \cdot H) q^{[F+G]} \\
&& - \sum_{k(L)= 3}\sum_{k(E)= 1}
N(L) N(E) (E \cdot L){(E \cdot H)}^2q^{[E+L]} \nonumber
\end{eqnarray}
We want to use this to develop a recursive formula for the degrees
$N(C)$
if possible.
The $q^{[C]}$ terms of the (\ref{assoc1D=HdotH}) are
\begin{eqnarray}
\label{assoc1qC}
N(C) &= &
\sum\begin{Sb} (F,G) \\ k(F)=k(G)=2 \\ F+G \equiv C \end{Sb}
N(F)N(G)(F \cdot H)(G \cdot F)(G \cdot H) \\
&& - \sum\begin{Sb} (E,L) \\ k(E)=1, k(L) = 3 \\ E+L \equiv C
\end{Sb}
N(L) N(E) (E \cdot L){(E \cdot H)}^2 \nonumber
\end{eqnarray}
To relate this recursive formula to those of Kontsevich and Manin
(Claims 5.2.1 and 5.2.3b of \cite{kontsevich-manin}), we write this
as
\begin{eqnarray}
\label{assoc2qC}
& & \\
N(C) &= &
\sum\begin{Sb} (C_1,C_2) \\ C_1+C_2 \equiv C \end{Sb}
N(C_1)N(C_2)(C_1 \cdot C_2)(H \cdot C_1)
( (H \cdot C_2)\delta_{k(C_1)-2} - (H \cdot C_1)\delta_{k(C_1)-1}
) \nonumber
\end{eqnarray}
where $\delta_n = \begin{cases} 1 & \text{ if $n=0$ } \\
0 & \text{ if $n\ne 0$ } \end{cases}$. This expression is
equivalent to Claim 5.2.3b of \cite{kontsevich-manin} in the case
for which $k(C) = 4$, assuming that their convention for
$\left( \begin{array}{c} 0 \\ n \end{array} \right)$ is that
$\left( \begin{array}{c} 0 \\ n \end{array} \right) = \delta_n$.
Note that \ref{assoc2qC} is not valid when $k(C) \ne 4$, as may be
seen when $C \equiv -K_X$ on $X=X_6$. It is unclear what 5.2.3b of
\cite{kontsevich-manin} means in this case, since $(-K \cdot C) =
3$ and so 5.2.3b involves terms of the form $\left(
\begin{array}{c} -1 \\ n \end{array} \right)$ where $n \le 0$. Note
also
that the
same $C$ is indecomposible in the semi-group of numerically
effective curves and yet has $N(C)=12$, contrary to the expectation
expressed in 5.2.3b of \cite{kontsevich-manin}.
Our next goal is to compute $N(d)$, the degree of the locus of
rational curves of degree $d$ in the plane.
This is not a relevant class on the plane, unless $d = 1$.
Since forcing a curve to pass through a generically chosen point
is a linear condition on the linear system, we have
\[
N(d;m_1,\dots,m_n,1) = N(d;m_1,\dots,m_n).
\]
Hence by induction we have that
\[
N(d) = N(d;1^{3d-4})
\]
in particular.
Now on the surface $X_{3d-4}$,
the class $C = (d;1^{3d-4})$ is a relevant class;
in fact $k(d;1^{3d-4}) = 4$ and so \ref{assoc1qC} may be used to
compute $N(d)$. We now need to understand those $k(E) = 1$ and
$k(L) = 3$ classes which sum to $C$,
and those $k(F) = k(G) = 2$ classes which sum to $C$.
First consider the class $E_i$ itself, which has $k(E) = 1$.
However $(E_i \cdot H) = 0$,
so these $k(E)=1$ classes do not contribute to the recursive
formula of
(\ref{assoc1qC}).
Hence we may assume $E$ is a relevant class with all $m_i$'s
non-negative.
In this case since $C = (d;1^{3d-4})$, all $m_i$'s for $E$
(and for the complementary $k=3$ class $L$) must be $0$ or $1$.
Therefore $E = (e;1^{3e-1})$ for some $e$ with $1 \leq e \leq d-1$,
where this notation means that $3e-1$ of the $m_i$'s are $1$,
and all the others are zero.
(There are
{\small $\left(\begin{array}{c} 3d-4 \\ 3e-1 \end{array}\right)$}
such classes.)
The complementary class $L$ is of the form $L = (d-e;1^{3d-3e-3})$
where the $1$'s occur in the complementary positions.
Note that with this notation
$(E\cdot L) = e(d-e)$ and $(E \cdot H) = e$,
with $N(E) = N(e)$ and $N(L) = N(d-e)$;
so the second sum above reduces to
\[
\sum_{E+L\equiv C}
N(L) N(E) (E \cdot L){(E \cdot H)}^2
=
\sum_{e=1}^{d-1}
\left(\begin{array}{c} 3d-4 \\ 3e-1 \end{array}\right)
N(e) N(d-e) (d-e) e^3.
\]
Now suppose that $F$ is a $k=2$ class;
again all its multiplicity numbers $m_i$ must be zero or one,
and so $F$ must have the form $F = (e;1^{3e-2})$
for some $e$ with $1 \leq e \leq d-1$;
there are
{\small $\left(\begin{array}{c} 3d-4 \\ 3e-2 \end{array}\right)$}
such classes.
The complementary class $G$ is $G = (d-e;1^{3d-3e-2})$,
where again the $1$'s occur in the complementary positions.
Note that with this notation
$(F\cdot G) = e(d-e)$, $(F \cdot H) = e$, and $(G \cdot H) = d-e$;
also $N(F) = N(e)$ and $N(G) = N(d-e)$.
Hence the first sum above reduces to
\[
\sum_{F+G\equiv C}
N(F)N(G)(F \cdot H)(G \cdot F)(G \cdot H)
=
\sum_{e=1}^{d-1}
\left(\begin{array}{c} 3d-4 \\ 3e-2 \end{array}\right)
N(e) N(d-e) e^2{(d-e)}^2.
\]
Collecting terms gives the following recursion relation for the
degrees $N(d)$:
\begin{equation}
\label{km5.2.1}
N(d) = \sum_{e=1}^{d-1} e^2(d-e)
\left[ (d-e) \left(\begin{array}{c} 3d-4 \\ 3e-2 \end{array}\right)
- e \left(\begin{array}{c} 3d-4 \\ 3e-1 \end{array}\right) \right]
N(e) N(d-e).
\end{equation}
This is exactly the enumerative prediction made by Kontsevich and
Manin
(Claim 5.2.1 of \cite{kontsevich-manin}).
We note here that from our point of view
this prediction follows from the associativity of the quantum
product
for arbitrarily large blowups of the plane; it is not enough to
know
it just for the plane.
To further illustrate the geometric and enumerative significance of
associativity of quantum cohomology, we return to $X_6$.
\begin{proposition}
Associativity of quantum cohomology on strict Del Pezzo surfaces is
equivalent to the fact that there are 27 exceptional curves on
$X_6$, each of which meets precisely 10 others.
\end{proposition}
\begin{pf}
In Section \ref{sectionSDPassoc}, associativity for $X_6$ (and so
all strict Del Pezzos) was shown using the fact that the number and
mutual disposition of exceptional curves are as above. To complete
our proof, it suffices to show that if $\hat E$ is an exceptional
curve on $X_6$, $m$ is the number of other exceptional curves
meeting $\hat E$ and $e$ is the total number of exceptional curves,
then $m=10$ and $e=27$. Consider the associativity relation
\ref{assoc2} with $ L \equiv D_1 \equiv D_2 \equiv -K $. Allowing
ourselves the knowledge that $N(F)=N(E)=1$ for all cases for which
$F+E \equiv -K$ and collecting terms, \ref{assoc2} implies that
\begin{eqnarray}
0 & = & \sum\begin{Sb} (E,F) \\ k(F)=2, k(E)=1 \\ E+F \equiv -K
\end{Sb} (E \cdot -K)( (F \cdot -K)[E] - (E \cdot -K)[F] )
\nonumber \\
& = & \sum_{k(E)=1} 2[E] - [-K-E] \nonumber \\
& = & \sum_{k(E)=1} ( 3[E] + [K] ) \nonumber
\end{eqnarray}
Intersecting with our fixed exceptional curve $\hat E$, we have $
0 = \sum_E ( 3 (E \cdot \hat E) - 1)$ and so $0 = -4 + 2m - (e-
(m+1))$ and thus $e = 3m - 3$.
On the other hand, applying the associativity relation
\ref{assocc1}
with $C \equiv -K + \hat E$, we get for all divisors $D$
\[
12[D] = \sum_{F+G=C} 1 \cdot 1 \cdot (F \cdot D) \cdot 2 [G]
- \sum\begin{Sb} L+E \equiv -K+\hat E \\ p_a(L) = 0
\end{Sb}
1 \cdot 1 \cdot 2 \cdot (E \cdot D)[E]
- 12 (\hat E \cdot D)[\hat E]
\]
which is equivalent, after noting that $F+G \equiv C$ if and only
if $F \equiv \hat E + E_F$ for some exceptional curve $E_F$, to
\[
6( [D] + (\hat E \cdot D)[\hat E] ) =
\sum\begin{Sb} \hat E + E_F \\ \hat E + E_G \end{Sb}
( (\hat E + E_F) \cdot D)[\hat E + E_G]
- \sum_{(E \cdot \hat E) = 0} (E \cdot D)[E].
\]
Letting $D \equiv -K$ and intersecting with $-K$ yields $24 = 4m -
(e-(m+1))$ and so $e = 5m -23$, which combined with our previous
equation yields $e=27$ and $m=10$.
\end{pf}
Note finally that
$N(d;m_1, \dots , m_n)$ is invariant under permutations and Cremona
transformations. The former is obvious and the latter follows from
the fact that $k$ and the decomposition of a curve into sums of
curves are invariant under Cremona transformations.
Also of course the arithmetic genus of a class is
invariant under symmetries and Cremona transformations.
It is tempting to conjecture that the number $N(d;m_1, \dots ,
m_n)$
depends only on the genus.
However a recent computation of A. Grassi \cite{grassi}
shows that this is not the case in general for classes
with arithmetic genus at least $2$.
|
1994-10-27T05:20:16 | 9410 | alg-geom/9410027 | en | https://arxiv.org/abs/alg-geom/9410027 | [
"alg-geom",
"math.AG"
] | alg-geom/9410027 | Heath Martin | Heath Martin and Juan Migliore | Submodules of the deficiency modules and an extension of Dubreil's
Theorem | 18 pages, LaTeX, version 2.09 | null | null | null | null | In its most basic form, Dubreil's Theorem states that for an ideal $I$
defining a codimension $2$, arithmetically Cohen--Macaulay subscheme of
projective $n$-space, the number of generators of $I$ is bounded above by the
minimal degree of a minimal generator plus $1$. By introducing a new ideal $J$
which is the complete intersection of $n-1$ general linear forms, we are able
to extend Dubreil's Theorem to an ideal $I$ defining a locally Cohen--Macaulay
subscheme $V$ of any codimension. Our new bound involves the lengths of the
Koszul homologies of the cohomology modules of $V$, with respect to the ideal
$J$, and depends on a careful identification of the module $(I \cap J)/IJ$ in
terms of the maps in the free resolution of $J$. As a corollary to this
identification, we also give a new proof of a theorem of Serre which gives a
necessary and sufficient condition to have the equality $I \cap J = IJ$ in the
case where $I$ and $J$ define disjoint schemes in projective space.
| [
{
"version": "v1",
"created": "Wed, 26 Oct 1994 15:04:24 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Martin",
"Heath",
""
],
[
"Migliore",
"Juan",
""
]
] | alg-geom | \section{When does $I \cap J = IJ$?}
Let $S= k[x_0, \dots, x_n]$ be a polynomial ring over the
algebraically closed field $k$. Let $I$ and $J$ be ideals
defining subschemes $V$ and $Y$, respectively, of the
projective space $\Bbb P^n_k = {\Bbb P^n}$ over $k$. In particular,
both $I$ and $J$ are homogeneous, saturated ideals.
In this section, we will derive a relationship between
the quotient module $(I \cap J)/IJ$ and the cohomology
of $V$, when $V$ and $Y$ meet in the expected dimension.
In general, if $V$ is a subscheme of ${\Bbb P^n}$, with saturated
homogeneous defining ideal $I = I_V$, the cohomology
modules of $V$ (or, less precisely, of $I$) are defined,
for $i=0, \dots, n-1$, by
$$
H^i_*({\cal I}_V) = H^i_*(V) = \bigoplus_j H^i({\Bbb P^n}, {\cal I}_V(j)),
$$
where ${\cal I}_V = \widetilde{I_V}$ is the ideal sheaf of $V$.
These are all graded $S$-modules. Moreover,
$H^0_*(V) = I_V$ and $H^i_*(V) = 0$ for $i > \dim V + 1$.
Usually, when $i = 1, \dots, \dim V$,
we will call $H^i_*(V)$ a deficiency module. This name
comes from the fact that the $H^i_*(V)$, $i = 1, \dots, \dim V$,
measure
the failure of $V$ to be an arithmetically Cohen--Macaulay
subscheme, since they vanish whenever $V$ is aCM.
We will also have need to use the cohomology of modules.
If $M$ is a (finitely generated) $S$-module,
let $\widetilde{M}$ be its sheafification. Then, exactly as in
the case of ideal sheaves, we define the cohomology module of $\widetilde{M}$
to be
$$
H^i_*(\widetilde{M}) = \bigoplus_j H^i({\Bbb P^n}, \widetilde{M}(j)).
$$
These are again graded $S$-modules.
We note here for future reference
that the cohomology modules of $M$ are related to the local
cohomology modules $H^i_{\goth m}(M)$ of $M$ with respect to the homogeneous
maximal ideal ${\goth m}$ as follows:
\begin{equation}\label{local-coh}
0 \rightarrow H^0_{\goth m}(M) \rightarrow M \rightarrow H^0_*(\widetilde{M})
\rightarrow H^1_{\goth m}(M) \rightarrow 0
\end{equation}
$$
H^i_{\goth m}(M) \cong H^i_*(\widetilde{M})\mbox{\quad\quad for $i>1$.}
$$
See \cite[Chapter 0]{SV:buchsbaum} for a good discussion of
graded and local cohomology.
In this paper, we will sometimes require subschemes of ${\Bbb P^n}$ to
be locally Cohen--Macaulay and equidimensional. This
is equivalent to saying that all the cohomology modules
have finite length, except of course for the top cohomology
$H^{d+1}_*(V)$, $d = \dim V$.
By Serre's vanishing theorem, this is again equivalent to
having $[H^i_*(V)]_j = 0$ for $j \ll 0$, and $1 \le i \le d$, since
in any case the cohomology modules vanish in high degrees.
Now, let $I$ and $J$ be as above, let $s = \mathop{\rm pd\,} J$ be the
projective dimension of $J$, and write a minimal graded free resolution
of $J$ as follows:
\begin{equation}
\begin{array}{ccccccccccccccccc}
0 & \rightarrow & F_s &
\buildrel {\phi_s} \over \longrightarrow
& \dots
& {\buildrel {\phi_2} \over \rightarrow}
& F_1&
\buildrel {\phi_1} \over \longrightarrow & F_0 &
\rightarrow & J & \rightarrow & 0.
\end{array}
\end{equation}
where $F_j = \bigoplus_i S(-a_{ji})$ are free modules.
For each $j = 1, \dots, s$, let $K_j$ be the $j$-th syzygy module, so that
there are short exact sequences
$$
0 \rightarrow K_{j+1} \buildrel {\psi_{j+1}} \over \longrightarrow
F_j \buildrel {\eta_j} \over \longrightarrow K_j \rightarrow 0,
$$
where the maps $\psi_j$ and $\eta_j$ are the canonical inclusions and
projections, respectively. Note that for $j = s$, we have $K_s = F_s$,
$\psi_s = \phi_s$ and $\eta_s = id$.
For $S$-modules $M$ and $N$, and a map $f : M \to N$, we denote
by $f^i : H^i_*(\widetilde{M \otimes I}) \to H^i_*(\widetilde{N \otimes I})$
the map
induced on cohomology by $f \otimes id : M \otimes I \to N \otimes I$.
Our main technical result for this paper is the following Theorem.
\begin{thm}\label{main:technical}
Suppose the ideals $I$ and $J$ as above define
disjoint subschemes $V$ and $Y$, respectively.
Then for each $i \ge 1$, there are isomorphisms
$$
\ker \psi_i^1 \cong \mathop{\rm Tor}\nolimits_i^S(S/I, S/J)
$$
and, for each $i,j \ge 1$, a long exact sequence
\begin{equation}\label{main:sequence}
0 \rightarrow \mathop{\rm im\,} \psi_{i+1}^j \rightarrow \ker \phi_i^j \rightarrow
\ker \psi_i^j \rightarrow \ker \psi_{i+1}^{j+1} \rightarrow
{\mathop{\rm coker\,}} \phi_i^j \rightarrow {\mathop{\rm coker\,}} \psi_i^j \rightarrow 0.
\nonumber
\end{equation}
\end{thm}
\begin{proof} We remark that for $i > s$, both statements
are trivial, since then $\phi_i = \psi_i$ is the zero map. Also,
if $j > \dim S/I$, then because $H^j_*(V) = 0$
again the second statement is trivial.
Now, let $\mu : I \otimes J \to IJ$ be
the natural surjection, and note that $\ker \mu \cong \mathop{\rm Tor}\nolimits_2^S(S/I, S/J)$.
This follows, for instance, by tensoring
$$
0 \rightarrow I \rightarrow S \rightarrow S/I \rightarrow 0
$$
with $J$, comparing the resulting sequence with
$$
0 \rightarrow IJ \rightarrow J \rightarrow J/IJ \rightarrow 0
$$
via the multiplication map, and using that
$\mathop{\rm Tor}\nolimits_1^S(S/I, J) \cong \mathop{\rm Tor}\nolimits_2^S(S/I, S/J)$. Note especially that
$\ker \mu$ has finite length since it is annihilated by $I+J$.
In particular, by sheafifying and taking cohomology of the
short exact sequence
$$
0 \rightarrow \ker \mu \rightarrow I \otimes J \rightarrow IJ
\rightarrow 0,
$$
we see that $H^i_*(\widetilde{I \otimes J}) \cong H^i_*(\widetilde{IJ})$
for all $i \ge 0$.
Next, using the functorial map $M \to H^0_*(\widetilde{M})$ for any
$S$-module $M$, we get a commutative diagram
\begin{equation}\label{eq:1}
\begin{array}{ccccccccc}
&&&& 0 && 0 \\
&&&& \downarrow && \downarrow \\
&&&& H^0_{\goth m}(I \otimes J) & \rightarrow & H^0_{\goth m}(IJ) \\
&&&& \downarrow && \downarrow \\
0 &\rightarrow & \mathop{\rm Tor}\nolimits_2^S(S/I, S/J) & \rightarrow & I \otimes J &
{\buildrel {\mu} \over {\longrightarrow}} & IJ & \rightarrow & 0 \\
&&&& \downarrow && \downarrow \\
&& 0 & \rightarrow & H^0_*(\widetilde{I \otimes J}) &\rightarrow &
H^0_*({\widetilde{IJ}}) & \rightarrow & 0 \\
&&&& \downarrow && \downarrow \\
&&&& H^1_{\goth m}(I \otimes J) & \rightarrow &H^1_{\goth m}(IJ) \\
&&&& \downarrow && \downarrow \\
&&&& 0 && 0
\end{array}
\end{equation}
But $IJ$ is an ideal, so $H^0_{\goth m}(IJ) = 0$. Hence the kernel
of the map $I \otimes J \to H^0_*(\widetilde{I \otimes J})$
is $\mathop{\rm Tor}\nolimits_2^S(S/I, S/J)$.
Now, with these preliminaries out of the way, we prove the
isomorphisms by induction on $i$. For $i = 1$, tensor the exact
sequence
$$
0 \rightarrow K_1 {\buildrel {\psi_1} \over \longrightarrow }
F_0 \rightarrow J \rightarrow 0
$$
by $I$. This yields an exact sequence
$$
0 \rightarrow \mathop{\rm Tor}\nolimits_1^S(I, J) \rightarrow K_1 \otimes I
{\buildrel {\psi_1 \otimes 1} \over {\hbox{$\hbox to .35in{\rightarrowfill}$} }}
F_0 \otimes I \rightarrow J \otimes I \rightarrow 0.
$$
Since $\mathop{\rm Tor}\nolimits_1^S(I, J) \cong \mathop{\rm Tor}\nolimits_3^S(S/I, S/J)$ is annihilated
by $I+J$, in particular it has finite length. Thus, taking cohomology
and comparing with the original sequence gives a diagram
\begin{equation}\nonumber
\setlength{\arraycolsep}{1pt}
\begin{array}{ccccccccccccc}
0 & \rightarrow & \mathop{\rm Tor}\nolimits_3^S(S/I, S/J) & \rightarrow & K_1 \otimes I &
\rightarrow & F_0 \otimes I & \rightarrow & J \otimes I &
\rightarrow & 0 \\
&&&& \downarrow &&\downarrow&&\downarrow \\
&& 0 & \rightarrow & H^0_*(\widetilde{K_1 \otimes I}) & \rightarrow &
H^0_*({\widetilde{F_0 \otimes I}}) & \rightarrow &
H^0_*(\widetilde{J \otimes I}) & \rightarrow & \ker \psi_1^1 &
\rightarrow & 0.
\end{array}
\end{equation}
Here, the middle vertical map is an isomorphism, since $I$ is
saturated. Thus the snake lemma shows that there are exact sequences
$$
0 \rightarrow \mathop{\rm Tor}\nolimits_3^S(S/I, S/J) \rightarrow K_1 \otimes I \rightarrow
H^0_*(\widetilde{K_1 \otimes I}) \rightarrow \mathop{\rm Tor}\nolimits_2^S(S/I, S/J)
\rightarrow 0,
$$
and
$$
0 \rightarrow \mathop{\rm Tor}\nolimits_2^S(S/I, S/J) \rightarrow J \otimes I \rightarrow
H^0_*(\widetilde{J \otimes I}) \rightarrow \ker \psi_1^1
\rightarrow 0.
$$
But from the above discussion, the last sequence implies the short exact
sequence
$$
0 \rightarrow IJ \rightarrow H^0_*(\widetilde{IJ}) \rightarrow \ker \psi_1^1
\rightarrow 0.
$$
Since $I$ and $J$ define disjoint varieties, we have
$H^0_*(\widetilde{IJ}) = I \cap J$. Thus, the above sequence
shows that
$$
\ker \psi_1^1 \cong {{I \cap J} \over {IJ}} \cong \mathop{\rm Tor}\nolimits_1^S(S/I, S/J).
$$
By induction, we may assume that $\ker \psi_i^1 \cong \mathop{\rm Tor}\nolimits_i^S(S/I, S/J)$,
and that there is an exact sequence
$$
0 \rightarrow \mathop{\rm Tor}\nolimits_{i+2}^S(S/I, S/J) \rightarrow K_i \otimes I
\rightarrow H^0_*(\widetilde{K_i \otimes I}) \rightarrow
\mathop{\rm Tor}\nolimits_{i+1}^S(S/I, S/J) \rightarrow 0.
$$
Tensoring the exact sequence
$$
0 \rightarrow K_{i+1} {\buildrel {\psi_{i+1}} \over \hbox{$\hbox to .35in{\rightarrowfill}$} }
F_i \rightarrow K_i \rightarrow 0
$$
with $I$ yields
$$
0 \rightarrow \mathop{\rm Tor}\nolimits_1^S(K_i, I) \rightarrow K_{i+1} \otimes I
\rightarrow F_i \otimes I \rightarrow K_i \otimes I \rightarrow 0.
$$
Here, $\mathop{\rm Tor}\nolimits_1^S(K_i, I) \cong \mathop{\rm Tor}\nolimits_{i+1}^S(I, J) \cong \mathop{\rm Tor}\nolimits_{i+3}^S(S/I, S/J)$.
In particular, it is finite length. Hence, taking cohomology and comparing
yields a diagram
\begin{equation}\nonumber
\setlength{\arraycolsep}{1pt}
\begin{array}{ccccccccccccc}
0 & \rightarrow & \mathop{\rm Tor}\nolimits_{i+3}^S(S/I, S/J) & \rightarrow & K_{i+1} \otimes I &
\rightarrow & F_i \otimes I & \rightarrow & K_i \otimes I &
\rightarrow & 0 \\
&&&& \downarrow &&\downarrow&&\downarrow \\
&& 0 & \rightarrow & H^0_*(\widetilde{K_{i+1} \otimes I}) & \rightarrow &
H^0_*({\widetilde{F_i \otimes I}}) & \rightarrow &
H^0_*(\widetilde{K_i \otimes I}) & \rightarrow & \ker \psi_{i+1}^1 &
\rightarrow & 0.
\end{array}
\end{equation}
But by the inductive hypothesis, we know the kernel and cokernel
of the right-hand vertical map. Thus the snake lemma implies that
$$
\ker \psi_{i+1}^1 \cong \mathop{\rm Tor}\nolimits_{i+1}^S(S/I, S/J),
$$
and that there is a long exact sequence
$$
0 \rightarrow \mathop{\rm Tor}\nolimits_{i+3}^S(S/I, S/J) \rightarrow K_{i+1} \otimes I
\rightarrow H^0_*(\widetilde{K_{i+1} \otimes I}) \rightarrow
\mathop{\rm Tor}\nolimits_{i+2}^S(S/I, S/J) \rightarrow 0,
$$
which finishes the proof of the isomorphisms.
Next, we show that the long exact sequence exists.
Fix an $i \ge 1$. Thus there is an exact sequence
$$
0 \rightarrow K_{i+1}
{\buildrel {\psi_{i+1}} \over \hbox{$\hbox to .35in{\rightarrowfill}$} } F_i
{\buildrel {\eta_{i}} \over \hbox{$\hbox to .35in{\rightarrowfill}$} } K_i \rightarrow 0.
$$
Tensor this sequence with $I$, to obtain
$$
0 \rightarrow \mathop{\rm Tor}\nolimits_1^S(K_i, I) \rightarrow K_{i+1} \otimes I
{\buildrel {\psi_{i+1} \otimes 1} \over \hbox{$\hbox to .35in{\rightarrowfill}$} } F_i \otimes I
{\buildrel {\eta_{i} \otimes 1} \over \hbox{$\hbox to .35in{\rightarrowfill}$} } K_i \otimes I
\rightarrow 0,
$$
and note that
$\mathop{\rm Tor}\nolimits_1^S(K_i , I) = \mathop{\rm Tor}\nolimits_{i+3}^S(S/I, S/J)$, has finite length.
Thus, after sheafifying and taking cohomology, at the $j$-th stage
this yields isomorphisms
\begin{eqnarray*}
\ker \eta_i^j &\cong& \mathop{\rm im\,} \psi_{i+1}^j \\
{\mathop{\rm coker\,}} \eta_i^j &\cong& \ker \psi_{i+1}^{j+1}.
\end{eqnarray*}
Now, using the functoriality of tensor products and of cohomology,
we obtain a commutative square
\begin{equation}
\begin{array}{ccc}
H^j_*(\widetilde{F_i \otimes I}) & \widetilde{\hbox{$\hbox to .35in{\rightarrowfill}$} } &
H^j_*(\widetilde{F_i \otimes I}) \\
\mapdown{\eta_i^j} && \mapdown{\phi_i^j} \\
H^j_*(\widetilde{K_i \otimes I}) &
{\buildrel {\psi_i^j} \over {\longrightarrow}} &
H^j_*(\widetilde{F_{i-1} \otimes I}).
\end{array}
\end{equation}
Applying the snake lemma to the columns, and using the two isomorphisms
above shows that there is a sequence
$$
0 \rightarrow \mathop{\rm im\,} \psi_{i+1}^j \rightarrow \ker \phi_i^j \rightarrow
\ker \psi_i^j \rightarrow \ker \psi_{i+1}^{j+1} \rightarrow
{\mathop{\rm coker\,}} \phi_i^j \rightarrow {\mathop{\rm coker\,}} \psi_i^j \rightarrow 0,
$$
which is what we claimed.
\end{proof}
We note that this greatly extends the arguments in
\cite[Section 1]{Mig:submodules}. The situation there was much
simpler in that it only considered the case that $J$ was
codimension $2$ and arithmetically Cohen--Macaulay (so most of
the terms in the sequence~(\ref{main:sequence}) vanish), and
only the case $i=j=1$ was studied, so it focused on
$\ker \psi_1^1 = (I \cap J)/IJ$. Our
extension makes no assumptions on the Cohen--Macaulayness of $J$,
nor on its codimension. Of course, our conclusion is much
more complicated, reflecting the fact that so much information
is encoded in the free resolution of $J$.
As an application of this technical result, in the next theorem
we give a proof of a statement due to Serre on when there
is an equality $I \cap J = IJ$.
\begin{thm}{\rm \cite[Corollaire, p. 143]{Serre}}
Suppose the ideals $I$ and $J$
define disjoint subschemes of ${\Bbb P}^n$. Then $IJ = I \cap J$ if
and only if $\dim S/I + \dim S/J = \dim S$ and
both $S/I$ and $S/J$ are Cohen--Macaulay.
\end{thm}
\begin{proof} Suppose first that $S/I$ and $S/J$ are
Cohen--Macaulay with $\dim S/I + \dim S/J = \dim S$.
Then by the Auslander--Buchsbaum formula,
$s = \mathop{\rm pd\,} J = \dim S/I - 1$, and moreover $H^i_*(\widetilde{I}) = 0$
for $i = 1, \dots, s$. In particular, $\ker \phi_i^i = 0 = {\mathop{\rm coker\,}} \phi_i^i$
for $i = 1, \dots, s$. Since $\psi_s = \phi_s$, by reverse induction
the sequence (\ref{main:sequence}) with $j=i$ shows
that $\ker \psi_i^i = 0$ for $i = 1, \dots, s$.
Thus $(I \cap J)/IJ = \ker \psi_1^1 = 0$.
Conversely, since the subschemes defined by $I$ and $J$ are disjoint,
we have $\dim S/I + \dim S/J \le \dim S$ Hence
\begin{equation}\label{ineqs}
\dim S/I \le \dim S - \dim S/J \leq \dim S - \mathop{\rm depth\,} S/J = s+1
\end{equation}
where the last equality is by the Auslander--Buchsbaum formula. Now,
if $IJ = I \cap J$, then $\mathop{\rm Tor}\nolimits_1^S(S/I, S/J) = 0$, and
so by rigidity, $\mathop{\rm Tor}\nolimits_i^S(S/I, S/J) = 0$ for $i \ge 1$.
Hence the isomorphisms of Theorem~\ref{main:technical}
show that $\ker \phi_s^1 = 0$. But this implies that
$H^1_*(I) = 0$. Thus also $\ker \phi_i^1 = 0 = {\mathop{\rm coker\,}} \phi_i^1$
for all $i = 1, \dots, s$, and since $\ker \psi_1^1 = (I \cap J)/IJ = 0$,
the exact sequence (\ref{main:sequence}) with $j = 1$, $i = 1, \dots, s$
implies that $\ker \psi^2_i = 0$, for $i = 2, \dots, s$.
Since $\psi_s = \phi_s$, this shows $\ker \psi_s^2 = 0$ and
hence also $H^2_*(\widetilde{I}) = 0$. Continuing inductively,
we see that $\ker \psi_i^j = 0$ for all $i$ and $j$ with $i \ge j$.
In particular, since $\psi_s = \phi_s$ we get that
$H^j_*(\widetilde{I}) = 0$ for $j = 1, \dots, s$.
Let $d = \dim S/I$. We have seen that $d-1 \le s$. If this
inequality were strict, then in particular $H^d_*(\widetilde{I}) = 0$,
which is impossible. Hence we have $d-1 = s$ and $H^j_*(\widetilde{I}) = 0$
for $j = 1, \dots, d-1$; that is $S/I$ is Cohen--Macaulay.
But furthermore, each of the inequalities in (\ref{ineqs}) is
actually in equality. This shows both that $\dim S/J = \mathop{\rm depth\,} S/J$,
i.e., $S/J$ is Cohen--Macaulay, and that $\dim S/I + \dim S/J = \dim S$,
which finishes the proof.
\end{proof}
\section{An Extension of Dubreil's Theorem}
In this section, we wish to use the results of Section 1 to
extend a theorem of Dubreil on the number of generators of
certain ideals. Let $\nu(I)$ denote the minimal number of
generators of $I$, and let $\alpha(I)$ denote the
least degree of a minimal generator. In its most
basic form, Dubreil's Theorem states:
\begin{thm} Let $I$ be a homogeneous ideal of $k[x,y]$. Then
$\nu(I) \le \alpha(I) + 1$.
\end{thm}
See \cite{DGM:Dubreil} for a proof of this; note however that it
is essentially a consequence of the Hilbert--Burch theorem.
Dubreil's theorem is easily extended to the case that $I$
is a codimension $2$ arithmetically Cohen--Macaulay ideal in
any polynomial ring $k[x_0, \dots, x_n]$; again, see \cite{DGM:Dubreil}
for the details. On the other hand, when $I$ is not arithmetically
Cohen--Macaulay, or when $I$ is not codimension $2$, not much is
known in this direction. However, in the case of an ideal defining a
subscheme of $\Bbb P^3$, the following theorem of Migliore
shows that the general case will involve the cohomology of the
subscheme.
\begin{thm}{\rm \cite[Corollary 3.3]{Mig:submodules}}
Suppose $I$ defines a subscheme $V$ of $\Bbb P^3$,
of codimension at least $2$. Let $A = (L_1, L_2)$ be the
complete intersection of two general linear forms, and
let $K_A$ denote the submodule of $H^1_*(V)$
annihilated by $A$. Then
$$
\nu(I) \le \alpha(I) + 1 + \nu(K_A).
$$
\end{thm}
We note in particular that this formula is valid both for
the case that $V$ is codimension $2$, not necessarily
arithmetically Cohen--Macaulay, and the case that $V$ is
codimension $3$. In the latter case, even though
$H^1_*(V)$ is not finitely generated, we still have
that at least $K_A$ is finitely generated
(see \cite[Theorem 2.1]{Mig:submodules} or our Lemma~\ref{finite-length}),
so the theorem still has useful content. Furthermore, in
case $I$ defines an arithmetically Buchsbaum curve, so
that $H^1_*(V)$ is a $k$-vector space, $K_A = H^1_*(V)$
and $\nu(K_A) = \dim_k H^1_*(V)$. Thus the Buchsbaum
case is particularly easy to calculate in examples. In this
special case, the bound can be obtained from \cite{Amasaki:structure}.
In this section, we will give a generalization of
Dubreil's Theorem to ideals defining subschemes of ${\Bbb P^n}$ of
arbitrary codimension. As an easy consequence, we recover by
our methods the above two theorems, and also part of a result of
Chang, \cite{Chang:charac}, on
the number of generators of an ideal defining a Buchsbaum
codimension $2$ subscheme of ${\Bbb P^n}$, which again seems
to be the best understood case. Our generalization is a
corollary to the technical statement Theorem~\ref{main:technical}
in Section~1, underscoring the usefulness of identifying
the difference between intersections and products.
Our generalization will be based on the Koszul homologies
of the cohomology modules of an ideal $I$ defining a subscheme
of projective space. As such, we will make some general
remarks concerning Koszul homology. These comments are
basic, and can be found, for instance, in \cite{Mat}.
We first set the notation. If $R$ is a ring, and
$y_1, \dots, y_s$ elements of $R$, we let ${\Bbb K}((y_1, \dots, y_s);R)$
denote the Koszul complex with respect to $y_1, \dots, y_s$.
If $M$ is an $R$-module, put
${\Bbb K}((y_1, \dots, y_s); M) = {\Bbb K}((y_1, \dots, y_s);R) \otimes M$,
the Koszul complex on $M$ with respect to $y_1, \dots, y_s$.
Set ${\Bbb H}_i((y_1, \dots, y_s); M)$ to be the $i$-th homology
module of ${\Bbb K}((y_1, \dots, y_s); M)$; this is the
Koszul homology on $M$ with respect to $y_1, \dots, y_s$.
We will need the following facts:
\begin{remark}\label{Koszul}
\begin{enumerate}
\item If $y_1, \dots, y_n$ forms a regular sequence on $M$,
then ${\Bbb K}((y_1, \dots, y_n); M)$ is acyclic.
\item Let $J = (y_1, \dots, y_s)$. Then for each $i = 0, \dots, s$,
$J \subseteq \ann {\Bbb H}_i((y_1, \dots, y_s);M)$.
\item Suppose there is a short exact sequence of $R$-modules
$$
0 \rightarrow M_1 \rightarrow M_2 \rightarrow M_3 \rightarrow 0.
$$
Then there is a long exact sequence on Koszul homology
$$
\begin{array}{c}
\cdots \rightarrow
{\Bbb H}_{i+1}((y_1, \dots, y_n); M_3) \rightarrow
{\Bbb H}_i((y_1, \dots, y_n); M_1)
\hspace{1.5in} \\
\hspace{1.5in} \rightarrow
{\Bbb H}_i((y_1, \dots, y_n); M_2) \rightarrow
{\Bbb H}_i((y_1, \dots, y_n); M_3) \rightarrow \cdots.
\end{array}
$$
\item For each $i = 0, \dots, s$, there is an isomorphism
$$
{\Bbb H}_i((0, y_2, \dots, y_s); M) \cong {\Bbb H}_i((y_2, \dots, y_s); M)
\oplus {\Bbb H}_{i-1}((y_2, \dots, y_s); M).
$$
\end{enumerate}
\end{remark}
Throughout this section, let $I$ be the saturated defining
ideal of a locally Cohen--Macaulay, equidimensional subscheme
$V$ of ${\Bbb P}^n$; put $d=\dim V$. Let $J = (L_1, \dots, L_{n-1})$ be
the complete intersection of $n-1$ general linear forms. In
particular, ${\Bbb K}((L_1, \dots, L_{n-1}); R)$ is a
free resolution of $S/J$.
Recall that the highest non-zero cohomology module
$H^{d+1}_*(\widetilde{I})$ is never finitely generated,
when $d \ge 0$. In the next result, we show that nonetheless,
most of its Koszul homologies are finitely generated.
As general notation, for a module $M$ and an ideal $A$,
let $M_A$ denote the submodule of $M$ which is annihilated
by $A$; that is, $M_A = (0 :_M A)$.
\begin{prop}\label{finite-length} Let $I$ and $J = (L_1, \dots, L_{n-1})$
be as above. Then
the Koszul homology
${\Bbb H}_i((L_1, \dots, L_{n-1});H^{d+1}_*(\widetilde{I}))$
is finitely generated for each $i \ge d+2$. In particular,
if $d \ge 0$, the Koszul homology has finite length.
\end{prop}
\begin{proof} By changing coordinates if necessary, we may,
without loss of generality,
assume that $L_i = x_{i-1}.$ We will prove the proposition
by using induction on $d$. If $d = -1$, i.e., $I$
defines the empty subscheme, then
$H^0_*(\widetilde{I}) = I$ is already finitely generated,
so each of its Koszul homologies is also finitely generated.
Suppose $d \ge 0$. The exact sequence
$$
0 \rightarrow I {\buildrel {x_0} \over \longrightarrow} I
\rightarrow I/x_0 I \rightarrow 0
$$
induces the long exact sequence on cohomology
\begin{equation}\label{endcohomology}
\begin{array}{rcl}
0 \rightarrow A \rightarrow H^{d}_*(\widetilde{I/x_0 I})
& \hbox{$\hbox to .45in{\rightarrowfill}$} &
H^{d+1}_*(\tilde{I}) {\buildrel {x_0} \over \longrightarrow}
H^{d+1}_*(\tilde{I}) \rightarrow 0 \\
& \searrow \hfill \nearrow \\
& H^{d+1}_*(\tilde{I})_{(x_0)} \\
& \nearrow \hfill \searrow \\
0 && 0
\end{array}
\end{equation}
where $A = H^{d}_*(\widetilde{I}) / x_0 H^{d}_*(\widetilde{I})$.
Note in particular that $A$ is finitely generated, since it
is the quotient of two finitely generated modules.
{}From the right-hand part of this sequence, we obtain a long exact
sequence of Koszul homology (see Remark~\ref{Koszul})
$$
\begin{array}{c}
\cdots {\buildrel {x_0} \over \longrightarrow}
{\Bbb H}_{i+1}((x_0, \dots, x_{n-2}); H^{d+1}_*(\widetilde{I}))
\rightarrow
{\Bbb H}_i((x_0, \dots, x_{n-2}); H^{d+1}_*(\widetilde{I})_{(x_0)})
\hspace*{1in}\\
\hspace*{1in}\rightarrow
{\Bbb H}_i((x_0, \dots, x_{n-2}); H^{d+1}_*(\widetilde{I}))
{\buildrel {x_0} \over \longrightarrow}
{\Bbb H}_i((x_0, \dots, x_{n-2}); H^{d+1}_*(\widetilde{I}))
\rightarrow \cdots.
\end{array}
$$
But for any module $M$, ${\Bbb H}_i((x_0, \dots, x_{n-2}); M)$
is annihilated by $x_0$, so the long exact sequence breaks into
short exact sequences
\begin{center}
\makebox[\textwidth][l]{$\qquad 0 \longrightarrow
{\Bbb H}_{i+1}((x_0, \dots, x_{n-2}); H^{d+1}_*(\widetilde{I}))
\longrightarrow
{\Bbb H}_i((x_0, \dots, x_{n-2}); H^{d+1}_*(\widetilde{I})_{(x_0)})
\longrightarrow$}
\makebox[\textwidth][r]{${\Bbb H}_i((x_0, \dots, x_{n-2});
H^{d+1}_*(\widetilde{I}))
\longrightarrow 0.\qquad$}
\end{center}
Thus to show that ${\Bbb H}_i((x_0, \dots, x_{n-2}); H^{d+1}_*(\widetilde{I}))$
is finitely generated for $i \ge d+2$, it will suffice to show
that ${\Bbb H}_i((x_0, \dots, x_{n-2}); H^{d+1}_*(\widetilde{I})_{(x_0)})$
is finitely generated for $i \ge d+2$. However, $x_0$ kills
$H^{d+1}_*(\widetilde{I})_{(x_0)}$, and so
{\setlength{\arraycolsep}{0pt}
\begin{eqnarray*}
{\Bbb H}_i((x_0, \dots, x_{n-2}); H^{d+1}_*(\widetilde{I})_{(x_0)})
=&& {\Bbb H}_i((0, x_1, \dots, x_{n-2}); H^{d+1}_*(\widetilde{I})_{(x_0)}) \\
= {\Bbb H}_i((x_1, \dots, &&x_{n-2}); H^{d+1}_*(\widetilde{I})_{(x_0)})
\oplus {\Bbb H}_{i-1}((x_1, \dots, x_{n-2});
H^{d+1}_*(\widetilde{I})_{(x_0)}),
\end{eqnarray*}}
and we can calculate this over $R = S/(x_0) = k[x_1, \dots, x_n]$.
Now, the left-hand part of (\ref{endcohomology}) yields a long
exact sequence of Koszul homology
\begin{equation}\label{sequence1}
\begin{array}{c}
\cdots \rightarrow
{\Bbb H}_{j+1}((x_1, \dots, x_{n-2}); H^{d+1}_*(\widetilde{I})_{(x_0)})
\rightarrow {\Bbb H}_j((x_1, \dots, x_{n-2}); A)
\rightarrow
\hspace*{1in} \\
\hspace*{1in}
{\Bbb H}_j((x_1, \dots, x_{n-2}); H^d_*(\widetilde{I/x_0 I}))
\rightarrow
{\Bbb H}_j((x_1, \dots, x_{n-2}); H^{d+1}_*(\widetilde{I})_{(x_0)})
\rightarrow\cdots.
\end{array}
\end{equation}
Here, the saturation of $I/x_0 I \subseteq R$ defines a subscheme
$\overline{V}$ of ${\Bbb P}^{n-1}$, with $\dim \overline{V} = d-1$.
Thus, by the induction hypothesis,
${\Bbb H}_j(H^d_*(\widetilde{I/ x_0 I}))$ is finitely
generated for each $j \ge d+1$. In particular, since $i \ge d+2$,
this is true for $j = i, i-1$. Also, since $A$ is finitely
generated, all of its Koszul homologies are also finitely generated.
But then (\ref{sequence1})
shows that both
${\Bbb H}_i((x_1, \dots, x_{n-2}); H^{d+1}_*(\widetilde{I})_{(x_0)})$
and ${\Bbb H}_{i-1}((x_1, \dots, x_{n-2}); H^{d+1}_*(\widetilde{I})_{(x_0)})$
are finitely generated. This implies that
${\Bbb H}_i((x_0, \dots, x_{n-2}); H^{d+1}_*(\widetilde{I}))$
is finitely generated, which finishes the proof of the first
statement.
For the second statement, recall that the Serre vanishing theorem
says that $H^{d+1}(\widetilde{I}(t))$ vanishes for large $t$, and
hence the Koszul homology ${\Bbb H}_i((L_1, \dots, L_{n-1});
H^{d+1}_*(\widetilde{I}))$
also vanishes in high degrees. But since it is finitely generated,
it must also vanish in low degrees, and we can conclude that
it must have finite length.
\end{proof}
\begin{thm}\label{extended-Dubreil}
Suppose $I$ defines a locally Cohen--Macaulay, equidimensional
subscheme $V$ of dimension $d$ of ${\Bbb P^n}$.
Let $J = (L_1, \dots, L_{n-1})$ be generated by $n-1$ general linear
forms, and let
$\Bbb H_i((L_1, \dots, L_{n-1}); H^j_*(\widetilde{I}))$
be the Koszul homologies of
$H^j_*(\widetilde{I})$ with respect to $J$. Then
\begin{equation}\label{dubreil:formula}
\nu(I) \le \alpha(I) + 1 + \sum_{i=1}^{n-2}
\dim \Bbb H_{i+1}((L_1, \dots, L_{n-1});H^i_*(V)).
\end{equation}
\end{thm}
\begin{proof} Note that we have an exact sequence
$$
0 \rightarrow {{I \cap J} \over {IJ}} \rightarrow {I \over {IJ}}
\rightarrow {{I+J} \over J} \rightarrow 0.
$$
Hence,
$\nu(I) = \nu(I/IJ) \le \nu({{I+J}\over J}) + \nu({{I \cap J} \over IJ})$.
Now, since $(I+J)/J$ is an ideal in $S/J$, which is a polynomial
ring in two variables, Dubreil's Theorem applies, and says
that $\nu((I+J)/J) \le \alpha((I+J)/J) + 1$. Since $J$ is
generated by general linear forms, $\alpha((I+J)/J) = \alpha(I)$,
and so $\nu((I+J)/J) \le \alpha(I) + 1$.
Thus it only remains to estimate $\nu((I \cap J)/IJ)$.
Note that $I$ and $J$ define disjoint schemes, since
$J$ is generated by general linear forms, and that
the Koszul complex ${\Bbb K}((L_1, \dots, L_{n-1});S)$ is
a free resolution of $S/J$. In particular,
the isomorphisms and exact sequences of Theorem~\ref{main:technical}
hold.
For each $i = 1, \dots, n-1$,
let $P_i = \ker \phi_i^i/\mathop{\rm im\,} \psi_{i+1}^i$. Note then, that
$\Bbb H_{i+1}(H^i_*(V))$ naturally maps surjectively onto $P_i$,
since $\mathop{\rm im\,} \psi_{i+1}^i$ contains $\mathop{\rm im\,} \phi_{i+1}^i$. Thus,
in particular, $\dim P_i \le \dim \Bbb H_{i+1}(H^i_*(V))$. Hence
it follows from Theorem~\ref{main:technical} that
\begin{eqnarray}
\nu((I \cap J) / IJ) \le \dim ((I \cap J)/IJ) &\le& \dim \Bbb H_2(H^1_*(V))
+ \dim \ker\psi_2^2
\nonumber \\
&\le& \dim \Bbb H_2(H^1_*(V)) + \dim \Bbb H_3(H^2_*(V)) +
\dim \ker \psi_3^3 \nonumber \\
&\vdots & \nonumber\\
&\le& \sum_{i=1}^{n-2} \dim \Bbb H_{i+1}(H^i_*(V)). \nonumber
\end{eqnarray}
\end{proof}
\begin{remark} Since $H^i_*(V) = 0$ whenever $i > \dim V + 1$,
many of the terms in the formula~(\ref{dubreil:formula}) vanish.
For instance, if $V$ is a curve in ${\Bbb P}^5$, there are only
two terms coming from the cohomology of $V$.
\end{remark}
\begin{remark} In general, for a finite length graded module,
$\nu(M)$ is much less than $\dim M$, and so we would like to be able
to replace ``$\dim$'' by ``$\nu$'' throughout in the above formula.
However, counting minimal generators is much more difficult in
general than counting vector space dimensions.
\end{remark}
One important case in which we can replace ``$\dim$'' by ``$\nu$'' is
when all but the top cohomology of $V$ is annihilated by the
maximal ideal. Recall the definition:
\begin{definition} A subscheme $V$ of ${\Bbb P^n}$ of dimension $d$
is said to be {\em arithmetically Buchsbaum} if $H^i_*(V)$
is annihilated by the maximal ideal for each $i=1, \dots, d$,
and if for each general linear subspace $Y$ of ${\Bbb P^n}$,
the cohomology $H^i_*(V \cap Y)$ is annihilated by the maximal ideal
for $i = 1, \dots, \dim V \cap Y$.
\end{definition}
Note that the condition on linear subspaces of $V$ is required,
for there are examples of subschemes $V$ whose cohomology
is annihilated by the maximal ideal which have
hypersurface sections whose cohomology is not annihilated
by the maximal ideal; see for instance, \cite{Miy:graded}.
In general, we will not require the
full strength of this definition, only that
the cohomologies are annihilated by the maximal ideal.
Such subschemes are called {\em quasi--Buchsbaum}.
The next corollary was obtained for the case $n=3$ in \cite{Mig:submodules},
and a somewhat better bound is stated in \cite{Chang:charac} for
arithmetically Buchsbaum subschemes of ${\Bbb P}^n$.
\begin{cor} Let $I$ define a codimension $2$ subscheme
of ${\Bbb P^n}$ which is quasi-Buchsbaum. Then
$$
\nu(I) \le \alpha(I) + 1 +
\sum_{i=1}^{n-2} {{n-1} \choose {i+1}} \dim_k H^i_*(V).
$$
\end{cor}
\begin{proof} We only have to note that since $H^i_*(V)$ is
annihilated by the maximal ideal for $i = 1, \dots, n-1$, and since
every non-zero entry in a matrix representation for $\phi_i$
is a linear form, then for each $i = 1, \dots, n-1$,
$\Bbb H_{i+1}(H^i_*(V)) = \ker \phi_i^i$ is a
direct sum of ${{n-1} \choose {i+1}}$
copies of (twists of) $H^i_*(V)$.
\end{proof}
More generally, we can apply the same kind of analysis to quasi-Buchsbaum
subschemes of arbitrary codimension, except that we now have to
consider the top cohomology as well.
\begin{cor}\label{number}
Let $V$ be a $d$-dimensional quasi-Buchsbaum subscheme of ${\Bbb P^n}$,
defined by the saturated ideal $I$. Let $J = (L_1, \dots, L_{n-1})$
be generated by $n-1$ general linear forms, and let
$\Bbb H_i(H^j_*(V))$ be the Koszul homologies of $H^j_*(V)$
with respect to $J$. Then
$$
\nu(I) \le \alpha(I) + 1 +
\sum_{i=1}^{d} {{n-1} \choose {i+1}} \dim_k H^i_*(V)
+ \dim \Bbb H_{d+2}(H^{d+1}_*(V)).
$$
\end{cor}
\begin{proof} Again, since $H^i_*(V)$ is annihilated by every
linear form for $i = 1, \dots, d$, then $\Bbb H_{i+1}(H^i_*(V))$
is just a direct sum of ${{n-1} \choose {i+1}}$ copies of
$H^i_*(V)$.
\end{proof}
\begin{remark} By Lemma~\ref{finite-length}, even though
$H^{d+1}_*(V)$ does not have finite length, the Koszul
homology $\Bbb H_{d+2}(H^{d+1}_*(V))$ does have
finite length, and so this corollary really does
give a finite bound on the number of generators of $I$.
\end{remark}
\begin{example} Unfortunately, the bound in
Theorem~\ref{extended-Dubreil} does not seem to be very sharp.
For example, in ${\Bbb P}^4$, let $V$ be the union of a conic and a line not
meeting the plane of the conic.
Then $V$ is arithmetically Buchsbaum,
and $\dim_k H^1_*(V) = 1$. Also, $\alpha(I_V) = 2$,
and $\nu(I_V) = 7$. However, if $J$ is generated by three
general linear forms, then $H^2_*(V)_J$ is at least $2$,
as can be seen, for instance, by a calculation using
the inductive procedure in Proposition~\ref{finite-length}.
So $\alpha(I_V) + 1 + 3\dim_k H^1_*(V) + \dim H^2_*(V)_J \geq 8$,
but $\nu(I_V) = 7$.
\end{example}
\begin{remark} During the final preparation of this paper,
the authors received the preprint \cite{Hoa} of Hoa,
which contains bounds on the number of generators of
an ideal based in part on the cohomology of the ideal, but
involving different invariants of the ideal than
our bounds. Neither Hoa's bounds nor our bounds seem to be
particularly sharp in general.
\end{remark}
\section{On the Least Degree of Surfaces Containing Certain Buchsbaum
Subschemes}
In this section, we want to use the bound given in Section~2 to
extend a result of Amasaki on
the minimal degree of the minimal generators of an ideal $I$
defining a codimension $2$ Buchsbaum subscheme of ${\Bbb P^n}$.
In \cite{Amasaki:structure}, Amasaki showed that if $C$ is
a Buchsbaum curve in $\Bbb P^3$, and if $N = \dim_k H^1_*(C)$
is the Buchsbaum invariant of $C$, then $\alpha(I) \ge 2N$.
A different proof was subsequently given in \cite{GM:generators}
based on combining the upper bound estimate for $\nu(I)$ of
Corollary~\ref{number} in the case of curves in $\Bbb P^3$
together with a lower bound estimate coming from a determination
of the free resolution of the ideal $I$ from a free resolution
of $H^1_*(C)$. Also, Chang extended Amasaki's bound to
a codimension $2$ Buchsbaum subscheme of any ${\Bbb P^n}$ in \cite{Chang:charac},
based on a structure theorem for the locally free resolution of
the ideal sheaf associated to the subscheme.
Here, we would like to use our methods to give a different
proof of Amasaki's bound for certain codimension $2$ subschemes
of ${\Bbb P^n}$. Specifically, we will give a lower bound for $\alpha(I)$
in terms of $H^1_*(V)$ for a codimension $2$ subscheme of $V$
for which $H^1_*(V)$ is annihilated by the maximal ideal,
and $H^i_*(V) = 0$ for $i = 2, \dots, \dim V$.
Note that these quasi-Buchsbaum schemes are in fact Buchsbaum, since
if $H$ is a general hyperplane defined by a linear form $L$,
the from the standard exact sequence
$$
0 \rightarrow I {\buildrel {\times L} \over {\longrightarrow}} I
\rightarrow I/LI \rightarrow 0,
$$
it is easy to see that $H^i_*(V \cap H) \cong H^i_*(V)$
for $1 \le i \le \dim V - 1$.
Our method of proof will be to follow the lines of
\cite{GM:generators}. That is, we will combine our
upper bound estimate of $\nu(I)$ with a lower bound
estimate for $\nu(I)$ based on the free resolution of $I$.
We begin with an extension of a result in \cite{Rao}.
\begin{prop} Suppose $V$ is a codimension $2$ subscheme
of ${\Bbb P^n}$, such that $H^i_*(V) = 0$ for all $i = 2, \dots, \dim V$.
Let
$$
0 \rightarrow L_{n+1} {\buildrel {\sigma_{n+1}} \over \longrightarrow} L_n
{\buildrel {\sigma_n} \over \longrightarrow} L_{n-1} \rightarrow \dots
{\buildrel {\sigma_1} \over \longrightarrow} L_0
\rightarrow H^1_*(V) \rightarrow 0
$$
be the minimal free resolution of the finite length module $H^1_*(V)$.
Then the saturated defining ideal $I = I_V$ of $V$ has a minimal
free resolution
$$
0 \rightarrow L_{n+1} {\buildrel {\sigma_{n+1}} \over \longrightarrow}
L_n {\buildrel {\sigma_{n}} \over \longrightarrow}
L_{n-1} {\buildrel_{\sigma_{n-1}} \over \longrightarrow}
\dots {\buildrel {\sigma_4} \over \longrightarrow}
L_4 {\buildrel {{[\sigma_3\; 0]}} \over \hbox{$\hbox to .35in{\rightarrowfill}$} }
L_3 \oplus \bigoplus_{i=1}^r S(-b_i)
\rightarrow \oplus_{j=1}^p S(-a_j)
\rightarrow I \rightarrow 0,
$$
for some $r \ge 0$, and $p = \nu(I)$.
\end{prop}
\begin{proof} Write a minimal free resolution of $S/I$ as
follows:
$$
0 \rightarrow F_n {\buildrel {\phi_n} \over \longrightarrow} F_{n-1}
{\buildrel {\phi_{n-1}} \over \longrightarrow} \dots
{\buildrel {\phi_{2}} \over \longrightarrow} F_1
\rightarrow S \rightarrow S/I \rightarrow 0,
$$
and let $E_i$ be the $i$-th syzygy module. Sheafifying and dualizing
the short exact sequence
$$
0 \rightarrow F_n {\buildrel {\phi_{n-1}} \over \longrightarrow}
F_{n-1} \rightarrow E_{n-1} \rightarrow 0
$$
yields the exact sequence
$$
0 \rightarrow {\cal E}_{n-1}^\vee \rightarrow {\cal F}_{n-1}^\vee
{\buildrel {\phi_{n-1}^\vee} \over \hbox{$\hbox to .35in{\rightarrowfill}$} }
{\cal F}_n^\vee \rightarrow 0.
$$
Taking cohomology then gives a sequence
$$
0 \rightarrow H^0_*({\cal E}_{n-1}^\vee) \rightarrow
F_{n-1}^\vee {\buildrel {\phi_{n-1}^\vee} \over \longrightarrow}
F_n^\vee \rightarrow
H^1_*({\cal E}_{n-1}^\vee) \rightarrow 0.
$$
Note, though, that
$H^1_*({\cal E}_{n-1}^\vee) \cong \mathop{\rm Ext}\nolimits_S^{n+1}(H^1_*(V), S))$.
Next, for each $i = 2, \dots, n-2$, consider the sequence
$$
0 \rightarrow E_{i+1} \rightarrow F_{i} \rightarrow E_{i} \rightarrow 0.
$$
Sheafifying, dualizing and taking cohomology gives an exact sequence
$$
0 \rightarrow H^0_*({\cal E}_i^\vee) \rightarrow F_{i}^\vee
\rightarrow H^0({\cal E}_{i+1}^\vee) \rightarrow
H^1_*({\cal E}_i^\vee) \rightarrow 0.
$$
But $H^1_*({\cal E}_i^\vee) = \mathop{\rm Ext}\nolimits_S^{n+1}(H^{n-i}_*(V), S) = 0$,
by assumption. Hence, we can paste together all these exact sequences
to get a long exact sequence
$$
F_2^\vee {\buildrel {\phi_{3}^\vee} \over \hbox{$\hbox to .35in{\rightarrowfill}$} }
F_3^\vee {\buildrel {\phi_{4}^\vee} \over \hbox{$\hbox to .35in{\rightarrowfill}$} }
\dots {\buildrel {\phi_{n}^\vee} \over \hbox{$\hbox to .35in{\rightarrowfill}$} }
\mathop{\rm Ext}\nolimits_S^{n+1}(H^1_*(V), S) \rightarrow 0.
$$
However, a minimal free resolution of $\mathop{\rm Ext}\nolimits_S^{n+1}(H^1_*(V), S)$
is given by just dualizing the resolution of $H^1_*(V)$, and
so we see that $F_i = L_{i+1}$ and
$\phi_i = \sigma_{i+1}$ for $i = 3, \dots, n$, and
$F_2 = L_3 \oplus \bigoplus_{i=1}^r S$ and $\phi_2 = [\sigma_3\; 0]$,
for some $r \ge 0$. This finishes the proof.
\end{proof}
\begin{cor} Let $V$ be as in the previous proposition, and
let $I$ be its saturated defining ideal. Then
$\nu(I) \ge 1 + \sum_{i=3}^{n+1} (-1)^i \mathop{\rm rank\,} L_i$.
\end{cor}
\begin{proof} With the notation as in the statement of the Proposition,
we have
$$
\nu(I) = p = 1 + \mathop{\rm rank\,} L_3 + r -\mathop{\rm rank\,} L_4 + \mathop{\rm rank\,} L_5 + \dots
\ge 1 + \sum_{i=3}^{n+1} (-1)^i \mathop{\rm rank\,} L_i.
$$
\end{proof}
\begin{cor} In addition to the assumptions of the Proposition,
suppose that $H^1_*(V)$ is annihilated by the maximal ideal.
Let $N = \dim_k H^1_*(V)$. Then $\alpha(I) \ge (n-2)N$.
\end{cor}
\begin{proof} A minimal free resolution of $H^1_*(V)$ is
just a direct sum of $N$ copies of the Koszul complex
resolving $k = S/{\goth m}$. Thus, $\mathop{\rm rank\,} L_i = N{{n+1} \choose i}$.
By the previous Corollary and Corollary~\ref{number},
we have
$$
1 + \sum_{i=3}^{n+1} (-1)^i N{{n+1} \choose i} \le 1 + \alpha(I)
+ (n+1)N.
$$
A simple arithmetic calculation then reduces this
to $(n-2)N \le \alpha(I)$, as claimed.
\end{proof}
|
1996-03-08T06:54:15 | 9410 | alg-geom/9410031 | en | https://arxiv.org/abs/alg-geom/9410031 | [
"alg-geom",
"math.AG"
] | alg-geom/9410031 | null | Robert Guralnick, David Jaffe, Wayne Raskind, and Roger Wiegand | On the Picard group: torsion and the kernel | 27 pages, AMS-LaTeX | null | null | null | null | For a homomorphism f: A --> B of commutative rings, let D(A,B) denote
Ker[Pic(A) --> Pic(B)]. Let k be a field and assume that A is a f.g. k-algebra.
We prove a number of finiteness results for D(A,B). Here are four of them.
1: Suppose B is a f.g. and faithfully flat A-algebra which is geometrically
integral over k. If k is perfect, we find that D(A,B) is f.g. (In positive
characteristic, we need resolution of singularities to prove this.) For an
arbitrary field k of positive characteristic p, we find that modulo p-power
torsion, D(A,B) is f.g.
2: Suppose B = A tensor k^sep. We find that D(A,B) is finite.
3: Suppose B = A tensor L, where L is a finite, purely inseparable extension.
We give examples to show that D(A,B) may be infinite.
4: Assuming resolution of singularities, we show that if K/k is any algebraic
extension, there is a finite extension E/k contained in K/k such that D(A
tensor E,A tensor K) is trivial.
The remaining results are absolute finiteness results for Pic(A).
5: For every n prime to char(k), Pic(A) has only finitely many elements of
order n.
6: Structure theorems are given for Pic(A), in the case where k is absolutely
f.g.
All of these results are proved in a more general form, valid for schemes.
Hard copy is available from the authors.
| [
{
"version": "v1",
"created": "Mon, 31 Oct 1994 17:15:08 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Guralnick",
"Robert",
""
],
[
"Jaffe",
"David",
""
],
[
"Raskind",
"Wayne",
""
],
[
"Wiegand",
"Roger",
""
]
] | alg-geom | \section{The kernel under a separable extension}
We start by recalling some material on Galois actions, leading up to an
application of the Hochschild-Serre spectral sequence to the computation of
$\operatorname{Pic}$. This material is more or less standard, but it is not available in the
literature in quite the form we need. In particular, we want the statement of
\pref{the-kernel} to be free of noetherian hypotheses. Later (see \ref{belch})
this will be important, because a noetherian scheme $X$ may have $\Gamma(X)$
non-noetherian, and our proofs about $\operatorname{Pic}(X)$ depend on understanding
$\operatorname{Pic} \Gamma(X)$.
If $Y$ is a scheme, and $G$ is a group (or just a set), we let $Y \times G$
denote the scheme which is a disjoint union of copies of $Y$, one for each
$g \in G$.
\begin{definition}\label{Galois-defn}
Let $X$ be a scheme, and let $Y$ be a finite \'etale $X$-scheme. Suppose a
finite group $G$ acts on the right of $Y$ as an $X$-scheme.\footnote{Hereafter
we say simply that {\it $G$ acts on $Y/X$.}}
Then this action is {\it Galois\/} if the map
$Y \times G \to Y \times_X Y$ given by $(y,g) \mapsto (y, yg)$ is an
isomorphism of schemes.%
\footnote{If $y \in Y$, then $yg \in Y$, and $g$ induces an isomorphism of
$k(y)$ with $k(yg)$. Therefore we get maps
\mp[[ \sigma_1, \sigma_2 || \operatorname{Spec}(k(y)) || Y ]], such that
$\pi \circ \sigma_1 = \pi \circ \sigma_2$, where \mp[[ \pi || Y || X ]] is the
structure map.
By the universal property of the fiber product, we obtain a morphism
\mapx[[ \operatorname{Spec}(k(y)) || Y \times_X Y ]], whose image is by definition the point
$(y,yg)$.}
\end{definition}
It is important to note that $G$ cannot be recovered from $Y \rightarrow X$ in
general: for instance this is the case if $Y$ consists of a disjoint union of
copies of $X$.
If we have a Galois action of $G$ on $Y/X$, and $X' \to X$ is any morphism, we
get a Galois action of $G$ on $Y \times_X X'$ as an $X'$-scheme.
Let $A \subset B$ be rings, and let a finite group $G$ act on
$B/A$, meaning that $G$ acts on (the left of) $B$ as an $A$-algebra. Then we
shall call this action {\it Galois\/} if the action of $G$ on
$\operatorname{Spec}(B)/\operatorname{Spec}(A)$ is Galois with respect to definition \pref{Galois-defn}.
For some
definitions, stated directly for rings, see \cite{KO2, Chapter II, \S5}.
In the case where $A$ and $B$
are fields, the action of $G$ on $B/A$ is Galois if and only if $B/A$ is a
Galois extension (in the usual sense), and $G = \operatorname{Aut}_A(B)$.
We now recall the {\it Hochschild-Serre spectral sequence}. This may be
found in Milne \cite{Mi1, p.\ 105}. Although Milne refers only to locally
noetherian schemes, the argument he presents is valid for any scheme. For
clarity, however, we note that $X_{\hbox{\footnotesize\'et}}$ as used here means the (small) \'etale
site (on an arbitrary scheme $X$), as defined in \cite{G2}.
Let $X$ be a scheme, let $Y$ be a finite
\'etale $X$-scheme, and let a Galois action of a finite group $G$ on $Y/X$
be given. Let ${\cal{F}}$ be a sheaf (of abelian groups) for the \'etale topology
on $X$. Then the Hochschild-Serre spectral sequence is:
$$E^{p,q}_2 = H^p(G, H^q(Y_{\hbox{\footnotesize\'et}}, \pi^*{\cal{F}}))
\ \Longrightarrow \ H^{p+q}(X_{\hbox{\footnotesize\'et}}, {\cal{F}}),$$%
where $\pi: Y \to X$ is the structure morphism.
Apply this with ${\cal{F}} = {\Bbb G}_{\op m}$. One has an exact sequence:
$$0 \to E^{1,0}_2 \to H^1(X_{\hbox{\footnotesize\'et}},{\Bbb G}_{\op m}) \to E^{0,1}_2.$$%
The first term is $H^1(G,\Gamma(Y)^*)$. The middle term is $\operatorname{Pic}(X)$.
The last term is $H^0(G,\operatorname{Pic}(Y))$, which embeds in $\operatorname{Pic}(Y)$. Hence:
\begin{proposition}\label{the-kernel}
Let \mp[[ f || Y || X ]] be a finite \'etale morphism of schemes, and suppose
we have a Galois action of a finite group $G$ on $Y/X$. Then
$$\operatorname{Ker}[\operatorname{Pic}(f)]\ \cong\ H^1(G,\Gamma(Y)^*).$$
\end{proposition}
In particular \cite{Sw2, (4.2)}, if $A \subset B$ are rings, and we are given a
Galois action of a finite group $G$ on $B/A$, then $\operatorname{D}(A,B) \cong \operatorname{H}^1(G,B^*)$.
The following example shows that one has to be a bit careful about
generalizing the proposition to the non-Galois case.
\begin{example}
Let $A$ be a complete discrete valuation ring with fraction field $K$, and let
$L$ be a finite Galois extension of $K$ with Galois group $G$. Denote the
integral closure of $A$ in $L$ by $B$. We would like there to be an exact
sequence:
$$0\to H^1(G,B^*)\to \operatorname{Pic}(A)\to \operatorname{Pic}(B),$$
and hence conclude that $H^1(G,B^*) = 0$. But this is not the case in general:
$H^1(G,B^*)$ has order equal to the ramification index of $B/A$.
\end{example}
We recall some easy facts about cohomology of groups:
\begin{proposition}\label{group-cohomology-basics}
Let $G$ be a finite group and $M$ a left $\Bbb Z G$-module.
\begin{enumerate}
\item If $M$ is finitely generated, then $\operatorname{H}^n(G,M)$ is finite for every $n > 0$.
\item If $M$ has trivial $G$-action then
$H^1(G,M) \cong \operatorname{Hom}_{\smallcat{groups}}(G,M)$.
\end{enumerate}
\end{proposition}
\begin{proof}
(1) Since $\operatorname{H}^n(G,M) = \operatorname{Ext}^n_{\Bbb Z G}(\Bbb Z,M)$ and $\Bbb Z G$ is left Noetherian it is
clear that all the cohomology groups are finitely
generated modules (over $\Bbb Z G$ or, equivalently, over $\Bbb Z$).
Moreover, they are annihilated by $|G|$ by \cite{Br, Chap.\ III, (10.2)}.
(2) is clear from the representation of $\operatorname{H}^1(G,M)$ in terms of crossed
homomorphisms (or, see \cite{Bab, \S23}). \qed
\end{proof}
\begin{proposition}\label{90}
Let a finite group $G$ act on a field $K$. Then $H^1(G,K^*)$ is finite,
and (Hilbert's Theorem 90) it is $0$ if the group action is faithful.
\end{proposition}
\begin{proof}
Let $H$ be the subgroup of $G$ which acts trivially on $K$, and let
$\overline{G} = G/H$. Let $k = K^{\overline{G}}$. Then the extension $K/k$ is Galois, with
Galois group $\overline{G}$. By \pref{the-kernel},
$H^1(\overline{G},K^*) \cong \operatorname{Ker}[\operatorname{Pic}(k) \to \operatorname{Pic}(K)]$, which is $0$. We are done
if $G$ acts faithfully on $K$.
We have the inflation-restriction exact sequence
$$0 \to \operatorname{H}^1(\overline{G}, K^*) \to \operatorname{H}^1(G,K^*) \to \operatorname{H}^1(H,K^*),$$
so it is enough to show that $\operatorname{H}^1(H,K^*)$ is finite, which is clear from
\pref{group-cohomology-basics}(2). \qed
\end{proof}
The next result is a variant of a well-known result due to Roquette \cite{Ro}:
\begin{proposition}\label{roquette}
Let $K$ be a field, $X$ a $K$-scheme of finite type, and $\Lambda$ the
integral closure of $K$ in $A:=\Gamma(X)_{\op{red}}$. Then $\Lambda$ is
finite-dimensional as a $K$-vector space,
and $A^*/\Lambda^*$ is a finitely generated\ free abelian group.
\end{proposition}
\begin{proof}
Let $\{U_1,\dots,U_m\}$ be an affine open cover of $X$, and set
$R_i = \Gamma(U_i)_{\op{red}}$. Each $R_i$ is a a reduced $K$-algebra of
finite type. We have an embedding $A \to B := R_1\times\dots \times R_m$.
Let $P_1,\dots,P_n$ be the minimal prime ideals of $B$,
and let $C_j$ be the normalization of the domain $B/P_j$.
Then each $C_j$ is a normal domain of finite type over $K$, and
$A \subset C_1\times\dots\times C_n$.
Let $\Delta_j$ be the integral closure of $K$ in $C_j$. By the
usual formulation of Roquette's theorem (see \cite{L, Chapter 2, (7.3)} or
\cite{Kr, (1.4)}) $C_j^*/\Delta_j^*$ is finitely generated\ for each $j$. We have
$\Lambda = A \cap (\prod_j\Delta_j)$.
Therefore $A^*/\Lambda^*$ embeds in the finitely generated
group $\prod_j(C_j^*/\Delta_j^*)$. Obviously $A^*/\Lambda^*$ is torsion-free,
and since it is finitely generated, it is free.
The fraction field $K_j$ of $C_j$ is a finitely generated\ field extension of $K$.
Each $\Delta_j$, being $0$-dimensional, reduced and
connected, is a field algebraic over $K$. Since $K_j/K$
is finitely generated, so is $\Delta_j/K$. Therefore
$\prod_j\Delta_j$ is a finite-dimensional $K$-algebra,
and hence so is its subalgebra $\Lambda$. \qed
\end{proof}
\begin{theorem}\label{kernel2b}
Let $X$ be a scheme of finite type over a field $k$. Let $f: Y \to X$ be a
finite, \'etale, surjective morphism of schemes. If $k$ has positive
characteristic, assume that $X$ is reduced. Then $\operatorname{Ker}[\operatorname{Pic}(f)]$ is finite.
\end{theorem}
\begin{remark}
If $X$ is affine, $\operatorname{Pic}(X) = \operatorname{Pic}(X_{\op{red}})$, and so the assumption about
$X$ being reduced (in positive characteristic) is not needed. In general
it is: see \pref{finite-etale-infinite}.
\end{remark}
\begin{proofnodot}
(of \ref{kernel2b}.)
We may assume that $X$ is connected. Let $Y_0$ be a connected component
of $Y$. Then $f|_{Y_0}$ is finite and \'etale, and since $X$ is
connected, it is surjective. Therefore we may reduce to the case where $Y$
is connected. By \cite{Mur, (4.4.1.8)}, there
exists a scheme $Y'$ over $Y$ such that $Y' \to X$ is finite \'etale\
surjective and the
action of the finite group $\operatorname{Aut}(Y'/X)$ on $Y'/X$ is Galois. Therefore
we may assume that\ in fact there is a Galois action of a finite group $G$ on $Y/X$.
By \pref{the-kernel}, $\operatorname{Ker}[\operatorname{Pic}(f)] \cong H^1(G,\Gamma(Y)^*)$. We will
complete the proof by showing that $H^1(G,\Gamma(Y)^*)$ is finite. This
will depend only on the fact that we have a finite group $G$ acting on
an algebraic scheme $Y$, which is reduced if the characteristic is positive.
Let $B = \Gamma(Y)$. Let $\Lambda$ be the integral closure of $k$ in $B_{\op{red}}$.
By \pref{roquette}, $\Lambda$ is a finite-dimensional $k$-algebra and
$B_{\op{red}}^*/\Lambda^*$ is finitely generated. By
\pref{group-cohomology-basics}(1) we know that
$H^1(G,B_{\op{red}}^*/\Lambda^*)$ is finite.
We show that $H^1(G,\Lambda^*)$ is finite.
Write $\Lambda = \prod_{j\in J}F_j$, where each $F_j$ is a finite-dimensional
field extension of $k$ (and $J$ is a finite index set). Since the action
of $G$ preserves idempotents, we can define an action of $G$ on $J$
by the rule $F_{gj} = gF_j$. Let $I$ be any orbit of $G$ on $J$, and
look at $\Upsilon := \prod_{i\in I}F_i$. Since $\Lambda^*$ is the
direct sum of the groups $\Upsilon^*$ (over the various orbits of
$G$ on $J$), it is enough to show that $\operatorname{H}^1(G,\Upsilon^*)$ is finite.
Fix $i \in I$, let $H$ be the isotropy subgroup of $i$, and put $F =
F_i$. It follows from \cite{Br, Chap. III, (5.3), (5.9), (6.2)} that
$\operatorname{H}^1(G,\Upsilon^*) \cong \operatorname{H}^1(H,F^*)$, which is finite by \pref{90}.
Hence $H^1(G,\Lambda^*)$ is finite.
Running the long exact sequence of
cohomology coming from the exact sequence
$$1 \to \Lambda^* \to B_{\op{red}}^* \to B_{\op{red}}^*/\Lambda^* \to 1,$$
we conclude that $H^1(G,B_{\op{red}}^*)$ is finite. Thus we are done if $X$ is
reduced.
We may assume that $\mathop{\operator@font char \kern1pt}\nolimits(k) = 0$. Let $N$ be a $G$-stable nilpotent ideal
of $B$. (For instance, we might take $N$ to be the nilradical of $B$.) It is
enough to show that the canonical map \mapx[[ H^1(G,B^*) || H^1(G,(B/N)^*) ]]
is injective. If $N^r = 0$, note that we can factor the map
\mapx[[ B || B/N ]] as
\diagramx{B & \mapE{} & B/N^{r-1} & \mapE{} & B/N^{r-2} & \mapE{} &
\cdots & \mapE{} & B/N,}%
and $G$ acts on everything in the sequence, so we can reduce to the case where
$N$ has square zero. We have an exact sequence
\diagramx{0&\mapE{} & N & \mapE{} & B^* & \mapE{} & (B/N)^* & \mapE{} & 1,}%
and it is enough to show that $H^1(G,N) = 0$. On the one hand, $H^1(G,N)$ is
annihilated by $|G|$, and so is torsion. On the other hand,
$H^1(G,N)$ is an $B[G]$-module, thus a $\Bbb Q$-vector space, and so is
torsion-free. Hence $H^1(G,N) = 0$. \qed
\end{proofnodot}
\begin{remark}\label{belch}
In the theorem, if $Y = X_L$ for some finite separable field
extension $L/k$, we will show that
there is no need to assume $X$ is reduced, even in positive
characteristic. Indeed in that case, $\Gamma(X_L) = \Gamma(X)_L$, so we have
a Galois action of $G$ on $\operatorname{Spec} \Gamma(Y) / \operatorname{Spec} \Gamma(X)$. Applying
\pref{the-kernel} twice, we conclude that
$\operatorname{Ker}[\operatorname{Pic}(f)] = \operatorname{Ker}[\operatorname{Pic}(\Gamma(f))]$. But $\operatorname{Pic}$ of a ring is the same
as $\operatorname{Pic}$ of its reduction, so if $C = \Gamma(X)$, we conclude
that $\operatorname{Ker}[\operatorname{Pic}(f)] = \operatorname{Ker}[\operatorname{Pic}(C_{\op{red}}) \to \operatorname{Pic}((C_{\op{red}})_L)]$. Applying
\pref{the-kernel} once again, we see that
$\operatorname{Ker}[\operatorname{Pic}(f)] = H^1(G, (C_{\op{red}})_L) = H^1(G,B_{\op{red}})$.
Now the remainder of the proof of the theorem goes through.
\end{remark}
\begin{problem}\label{fidofido}
The proof of the theorem shows that if $k$ is a field, $A$ is a
finitely generated\ $k$-algebra (reduced if $\mathop{\operator@font char \kern1pt}\nolimits(k)\not=0$), and a finite group $G$ acts on
$A$, then $H^1(G,A^*)$ is finite. Under the same hypotheses, for which
$n \in \cal N$ is the set $H^1(G,\operatorname{GL}_n(A))$ finite?
\end{problem}
\begin{theorem}\label{makes-injective}
Let $k$ be a field, and let $K/k$ be a Galois extension of fields, not
necessarily finite. Let $S$ be a $k$-scheme of finite type, and assume that
each connected component of $S$ is geometrically connected.
Let $\Lambda$ be the integral closure of $K$ in $\Gamma(S_K)_{\op{red}}$.
Assume that $\Gamma(S)_{\op{red}}^* \Lambda^* = \Gamma(S_K)_{\op{red}}^*$. Then the
canonical map $\operatorname{Pic}(S) \to \operatorname{Pic}(S_K)$ is injective.
\end{theorem}
\begin{proof}
It is enough to show that for each finite Galois extension
$L/k$ with $k \subset L \subset K$, the map $\operatorname{Pic}(S) \to \operatorname{Pic}(S_L)$ is
injective. Let $G = \operatorname{Gal}(L/k)$. Let $B_0 = \Gamma(S_L)_{\op{red}}$.
Let $\Lambda_0$ be the integral closure of $L$ in $B_0$.
The condition of the theorem implies that
$\Gamma(S)_{\op{red}}^* \Lambda_0^* = B_0^*$. Therefore the action of $G$ on
$B_0^*/\Lambda_0^*$ is trivial. Hence
$H^1(G,B_0^*/\Lambda_0^*) \cong \operatorname{Hom}_{\smallcat{groups}}(G, B_0^*/\Lambda_0^*)$
by \pref{group-cohomology-basics}(2). Since $G$ is finite and
$B_0^*/\Lambda_0^*$ is torsion-free [by \pref{roquette}],
$H^1(G,B_0^*/\Lambda_0^*) = 0$.
Since $S$ is geometrically connected, $S_L$ is connected, and so $\Lambda_0$ is
a field. Since $G$ acts faithfully on $\Lambda_0$,
$H^1(G,\Lambda_0^*) = 0$ by \pref{90}.
Hence $H^1(G,B_0^*) = 0$. Arguing as in \pref{belch}, one obtains the
theorem. \qed
\end{proof}
\begin{theorem}\label{separable-extension}
Let $k$ be a field and $S$ a scheme of finite type
over $k$. Let $K/k$ be a separable algebraic field extension.
Then the kernel of the map $\operatorname{Pic}(S) \to \operatorname{Pic}(S_K)$ (induced by the projection
$\pi: S_K \to S$) is a finite group.
\end{theorem}
\vspace{0.08in}
\par\noindent{\bf Affine version of Theorem \ref{separable-extension}.}
\ {\it Let $k$ be a field, and let $A$ be a finitely generated\ $k$-algebra. Let $K/k$ be a
separable algebraic field extension. Then the kernel of the induced map
$\operatorname{Pic}(A) \to \operatorname{Pic}(A_K)$ is finite.}
\vspace{0.1in}
\begin{proofnodot}
(of \ref{separable-extension}). By passing to the normal closure, we may
assume that $K/k$ is Galois (possibly infinite). Let $B = \Gamma(S_K)_{\op{red}}$,
and let $\Lambda$ be the integral closure of $K$ in $B$. By \pref{roquette},
$B^*/\Lambda^*$ is finitely generated. Therefore we can find a finite Galois
extension $L$ of $k$ (with $L \subset K$) such that the canonical map
$(\Gamma(S_L)_{\op{red}})^* \to B^*/\Lambda^*$ is surjective. We can also choose
$L$ so that each connected component of $S_L$ is geometrically connected. By
\pref{kernel2b} and \pref{belch}, the kernel of $\operatorname{Pic}(S) \to \operatorname{Pic}(S_L)$ is
finite, and by \pref{makes-injective}, the map
$\operatorname{Pic}(S_L) \to \operatorname{Pic}(S_K)$ is injective, so we are done. \qed
\end{proofnodot}
We will see in \S\ref{examples-section} that \pref{separable-extension} can
fail if $K/k$ is not
assumed to be separable. On the other hand, assuming resolution of
singularities, we will show in \S4 that for any separated scheme $S$ of finite
type over $k$, and any algebraic field extension $K/k$,
there is an intermediate field $E/k$ of finite
degree over $k$ such that $\operatorname{Pic}(S_E) \to \operatorname{Pic}(S_K)$ is one-to-one.
We conclude this section by showing that the kernel is not always
trivial. In fact, any finite abelian group can occur. If, in the
following construction, one takes $K/k = \Bbb C/\Bbb R$, one
obtains the familiar example $A = \Bbb R[X,Y]/(X^2+Y^2-1)$. The
Picard group of $A$ has order two (generated by the M\"obius
band), whereas $A\otimes_{\Bbb R}\Bbb C \cong \Bbb C[U,U^{-1}]$, which
has trivial Picard group.
\begin{example}
Let $K/k$ be a finite Galois extension with Galois
group $G$. There is a domain $A$ of finite type over $k$ such that
$\operatorname{Pic}(A) \cong G/[G,G]$ but $\operatorname{Pic}(A\otimes_kK)$ is trivial.
\end{example}
\begin{proof}
Make $\Bbb Z G$ into a $G$-module via the left regular action.
We have an exact sequence of $G$-modules
$$0 \to \Bbb Z \mapE{\sigma} \Bbb Z G \to L \to 0,$$%
where $G$ acts trivially on $\Bbb Z$, $\sigma$ takes $1$ to $\sum_{g\in G}g$,
and $L$ is defined by the sequence. Let $F = \operatorname{Im}(\sigma)$, which is the
set of fixed points of $\Bbb Z G$ under the action of $G$. Clearly $L$ is a
free $\Bbb Z$-module of rank $|G| - 1$.
Form the group ring $B = K[L]$. Since $L$ is a free abelian
group, $B$ is isomorphic to the Laurent polynomial ring in
$|G|-1$ variables and hence has trivial Picard group. Note that
$G$ acts on $B$ by acting as the Galois group on $K$ and by
the $G$-module structure on $L$. Let $A = B^G$. Then
$A\otimes_kK \cong B$ by \cite{Sw1, (2.5)}, and the action of $G$ on
$B/A$ is Galois. Using (\ref{the-kernel}) we get
$\operatorname{Pic}(A) = \operatorname{D}(A,B) = \operatorname{H}^1(G,B^*)$. But $B^* \cong
K^* \oplus L$, so $\operatorname{Pic}(A) \cong \operatorname{H}^1(G,K^*) \oplus
\operatorname{H}^1(G,L) \cong \operatorname{H}^1(G,L)$, since $G$ acts faithfully on $K^*$ and so
$H^1(G,K^*) = 0$ by \pref{90}. We will show that $\operatorname{H}^1(G,L) \cong G/[G,G]$.
We know \cite{Br, Chap. III. (6.6)} that $\operatorname{H}^i(G,\Bbb Z G) = 0$ for $i>0$.
Therefore, by the long exact sequence of cohomology we have
$\operatorname{H}^1(G,L) \cong \operatorname{H}^2(G,\Bbb Z)$. But
$\operatorname{H}^2(G,\Bbb Z) \cong \operatorname{H}^1(G,\Bbb Q/\Bbb Z) \cong
\operatorname{Hom}_{\smallcat{groups}}(G, \Bbb Q/\Bbb Z) \cong G/[G,G]$.
(See \cite{Bab, \S23}.) \qed
\end{proof}
\section{Torsion in Picard groups}\label{torsion-section}
We note that $_n\operatorname{Pic}(R)$ can be infinite even for $R$ an domain finitely generated\ over
a field. For example, take $R = k[T^2,T^3]$, where $k$ is an
infinite field of characteristic $p > 0$. Then $\operatorname{Pic}(R)$ is
isomorphic to to the additive group of $k$ and is therefore an
infinite group of exponent $p$. As long as we avoid the
characteristic, however, this cannot happen. First we need the following
lemma (cf.\ \cite{Bas1, IX, (4.7)}):
\begin{lemma}\label{kernel-is-torsion}
Let \mp[[ f || X || S ]] be a finite flat morphism of schemes, of constant
degree $d > 0$. Then $\operatorname{Ker}[\operatorname{Pic}(f)]$ is $d$-torsion.
\end{lemma}
\begin{proof}
Let $\cal M \in \operatorname{Pic}(S)$. Then $f_*f^*\cal M \cong \cal M \o* f_* \cal O_X$ as
$\cal O_S$-modules. If moreover $\cal M \in \operatorname{Ker}[\operatorname{Pic}(f)]$, then
$f_* f^* \cal M \cong f_* \cal O_X$. Hence $\cal M \o* f_* \cal O_X \cong f_* \cal O_X$. Apply
$\wedge^d$, yielding
$\cal M^{\o* d} \o* \wedge^d(f_*\cal O_X) \cong \wedge^d(f_*\cal O_X)$, and hence
$\cal M^{\o* d} \cong \cal O_S$. \qed
\end{proof}
Now let us generalize to the proper case. First we need:
\begin{lemma}\label{is-locally-free}
Let \mp[[ f || X || S ]] be a proper flat morphism of noetherian
schemes. Then $f_*\cal O_X$ is a locally free $\cal O_S$-module.
\end{lemma}
\begin{proof}
We may assume that $S$ is affine. Let $W = \mathop{\mathbf{Spec}}\nolimits(f_*\cal O_X)$
(cf.\ \cite{Ha, II, exercise 5.17}), and label morphisms
\diagramx{X & \mapE{\varphi} & W & \mapE{h} & S.}%
Since $f$ is proper, $f_*\cal O_X$ is coherent \cite{EGA$3_1$, 3.2.1}, and so it is
enough to show that
$h$ is flat. Therefore it is enough to show that for any injection $i$ of
coherent $\cal O_S$-modules, $h^*(i)$ is also injective. By
construction, $\varphi_* \cal O_X = \cal O_W$, and so
$\varphi_* \varphi^* h^*(i) = h^*(i)$ by the projection formula
\cite{Ha, II, exercise 5.1d}. Since $f$ is
flat, $f^*(i)$ [which equals $\varphi^* h^*(i)$] is injective. Since
$\varphi_*$ is left exact, $\varphi_* \varphi^* h^*(i)$ is also injective.
\qed
\end{proof}
\begin{corollary}\label{proper-bounded}
Let \mp[[ f || X || S ]] be a surjective proper flat morphism of noetherian
schemes. Then $\operatorname{Ker}[\operatorname{Pic}(f)]$ is a bounded torsion group.
\end{corollary}
\begin{proof}
We may assume that $S$ is connected.
By \pref{is-locally-free}, $f_*\cal O_X$ is a locally free $\cal O_S$-module.
Factor $f$ as in the proof of \pref{is-locally-free}. By the projection
formula, $\varphi_* \varphi^*$ is the identity, so $\operatorname{Pic}(\varphi)$ is
injective.
Apply \pref{kernel-is-torsion}. \qed
\end{proof}
\begin{theorem}\label{finitely-many}
Let $S$ be a scheme of finite type over a field $k$ and let $n \in \Bbb N$ be
prime to the characteristic of $k$. Then $_n\operatorname{Pic}(S)$ is finite.
\end{theorem}
\vspace{0.03in}
\par\noindent{\bf Affine version of Theorem \ref{finitely-many}.}
\ {\it Let $k$ be a field, and let $A$ be a finitely generated\ $k$-algebra. Let $n \in \Bbb N$
be prime to the characteristic of $k$. Then $_n\operatorname{Pic}(A)$ is finite.}
\vspace{0.1in}
\begin{proofnodot}
(of \ref{finitely-many}). Suppose first that $k$ is separably closed.
The argument in this case seems to be fairly well known and was pointed
out to us several years ago by David Saltman and Tim Ford.
We consider the Kummer sequence \cite{SGA4$1\over2$, p.\ 21, (2.5)}:
\diagramx{1 & \mapE{} & \mu_n & \mapE{} & {\Bbb G}_{\op m} & \mapE{n} & {\Bbb G}_{\op m} & \mapE{} & 1}%
This is an exact sequence of sheaves for the \'etale topology
on $S$. Taking \'etale cohomology, we get an exact sequence
\diagramno{(*)}{\operatorname{H}^1(S_{\hbox{\footnotesize\'et}},\mu_n) & \mapE{} & \operatorname{H}^1(S_{\hbox{\footnotesize\'et}},{\Bbb G}_{\op m}) & \mapE{n} &
\operatorname{H}^1(S_{\hbox{\footnotesize\'et}},{\Bbb G}_{\op m}).}%
Now $\operatorname{H}^1(S_{\hbox{\footnotesize\'et}},{\Bbb G}_{\op m})\cong\operatorname{Pic}(S)$ by \cite{SGA4$1\over2$, p.\ 20, (2.3)}.
Since $k$ is separably closed, $\mu_n$ is isomorphic to the
constant sheaf $\Bbb Z/n\Bbb Z$. (See \cite{SGA4$1\over2$, p.\ 21, (2.4)}.)
Therefore $\mu_n$ is constructible \cite{SGA4$1\over2$, p.\ 43, (3.2)}. By
\cite{SGA4$1\over2$, p.\ 236, (1.10)} $\operatorname{H}^1(S_{\hbox{\footnotesize\'et}},\mu_n)$ is finitely generated.
Hence the kernel of the map $n$ in $(*)$ is finite, so the theorem holds when
$k$ is separably closed. Apply \pref{separable-extension}. \qed
\end{proofnodot}
As a consequence of \pref{proper-bounded} and \pref{finitely-many}, we have:
\begin{corollary}\label{proper-Pic-finite}
Let $S$ be a scheme of finite type over a field $k$. Let \mp[[ f || X || S ]]
be a proper flat surjective morphism of schemes. Assume that over each
connected component of $S$, the rank of the locally free sheaf $f_*\cal O_X$
is invertible in $k$. Then $\operatorname{Ker}[\operatorname{Pic}(f)]$ is finite.
\end{corollary}
We can use (\ref{finitely-many}) to answer a question of S.\ Montgomery about
outer automorphism groups of Azumaya algebras:
\begin{corollary}
Let $k$ be a field. Let $R$ be a finitely generated\ $k$-algebra. Let $A$ be a (not
necessarily commutative) Azumaya algebra over $R$ of degree $d$. Then
$\operatorname{Out}_R(A)$ is finite whenever $d$ is not a multiple of
the characteristic of $k$.
\end{corollary}
\begin{proof}
There is an embedding (see e.g.\ \cite{DI}) of $\operatorname{Out}_R(A)$ in $\operatorname{Pic}(R)$,
and in fact the image of $\operatorname{Out}_R(A)$ is contained in $_d \operatorname{Pic}(R)$.
(See \cite{KO1}.) \qed
\end{proof}
In fact, one can show that the finiteness of $_{d^e}\operatorname{Pic}(R)$ for all $e$
is equivalent to the finiteness of $\operatorname{Out}_R(M_{d^e}(R))$ for
all $e$ (see \cite{BG}).
Finally, we consider $n$-torsion in a normal algebraic scheme. It turns
out that this is finite, even if $n$ is not prime to the characteristic,
at least assuming that resolution of singularities holds. To prove this, we
need to know what happens to $\operatorname{Pic}$ of a normal
scheme when its singular locus is deleted. The kernel is described by the
following ``folklore'' lemma, which we state in greater generality for later
application \pref{seminormal-S2}:
\begin{lemma}\label{S2}
Let $X$ be a noetherian $S_2$ scheme%
\footnote{Recall that a noetherian scheme $X$ is by definition $S_2$ if for
every $x \in X$, $\operatorname{depth} \cal O_{X,x} \geq \min\setof{\operatorname{dim} \cal O_{X,x},2}$.},
and let $C \subset X$ be a closed subset
of codimension $\geq 2$. Let $U = X - C$. Then the canonical map
$\operatorname{Pic}(X) \to \operatorname{Pic}(U)$ is injective.
\end{lemma}
\begin{proof}
Let $\cal L$ be a line bundle on $X$ which becomes trivial on $U$. For any
line bundle $\cal M$ on $X$, the long exact sequence of local cohomology gives us
$$\operatorname{H}^0_C(X,\cal M) \to \operatorname{H}^0(X,\cal M) \mapE{\rho_{\cal M}} \operatorname{H}^0(U,\cal M) \to \operatorname{H}^1_C(X,\cal M).$$%
Since $X$ is $S_2$, the end terms vanish \cite{G1, (1.4), (3.7), (3.8)}.
Let $\phi: \cal O_U \to \cal L|_U$ be an isomorphism. Since $\rho_{\cal L}$
is an isomorphism, we can lift $\phi$ to a morphism
$\psi: \cal O_X \to \cal L$. Since $\rho_{\cal L^{-1}}$ is an isomorphism,
we can lift $\phi^{-1}$ to a morphism $\psi': \cal L \to \cal O_X$. Since
$\rho_{\cal O_X}$ and $\rho_{\cal L}$ are isomorphisms, $\psi' \circ \psi$
and
$\psi \circ \psi'$ are the identity maps. \qed
\end{proof}
\begin{lemma}\label{alg-closed-regular}
Let $k$ be an algebraically closed field. Assume that resolutions of
singularities exist for varieties over $k$. Let $X$ be a normal $k$-scheme
of finite type. Let $n \in {\Bbb N}$. Then $_n \operatorname{Pic}(X)$ is finite.
\end{lemma}
\begin{sketch}
We may assume that $X$ is connected. By \pref{S2}, we may replace $X$ by
$X_{\operatoratfont reg}$ and so assume that $X$ is regular. If we further replace
$X$ by a nonempty open subscheme, we kill a finitely generated\ subgroup of $\operatorname{Pic}(X)$. In
this way we may reduce to the case where $X$ is affine. Since we have
resolution of singularities, we can embed $X$ as an open subscheme of a
regular $k$-scheme ${\overline{X}}$. Then $\operatorname{Pic}(X)$ is the quotient of $\operatorname{Pic}({\overline{X}})$
by a finitely generated\ subgroup. Now $\operatorname{Pic}^0({\overline{X}})$ is the group $A(k)$ of $k$-valued points
of an abelian variety $A$ over $k$, and $\operatorname{Pic}({\overline{X}})/\operatorname{Pic}^0({\overline{X}})$ is
finitely generated\ (see e.g.\ \cite{K, (5.1)}). Therefore it suffices to show that $_n A$ is
finite. This is well-known \cite{Mum, p.\ 39}. \qed
\end{sketch}
\section{Faithfully flat extensions}
The main results of this section are \pref{faithfully-flat-fg},
\pref{faithfully-flat-fg-imperfect}, and \pref{open-cover}, which give
information
about the kernel of the map on Picard groups induced by a faithfully flat
morphism of algebraic schemes. First we consider the case where the target of
the morphism is normal, in which case we can weaken the hypothesis of faithful
flatness.
\begin{theorem}\label{normal-faithfully-flat}
Let $k$ be a field. Let $X$ be a normal $k$-scheme of finite type. Let
\mp[[ f || Y || X ]] be a dominant morphism of finite type.
Then $\operatorname{Ker}[\operatorname{Pic}(f)]$ is finitely generated\ (if $\mathop{\operator@font char \kern1pt}\nolimits(k) = 0$) and is the
direct sum of a finitely generated\ group and a bounded $p$-group (if $\mathop{\operator@font char \kern1pt}\nolimits(k) = p > 0$).
If $k$ is algebraically closed, and resolution of singularities holds, then
$\operatorname{Ker}[\operatorname{Pic}(f)]$ is finitely generated.
\end{theorem}
\begin{proof}
We may assume that $X$ is connected. By \pref{S2}, we may assume that\ $X$ is regular.
Then we may replace $X$ by any nonempty open subscheme. In particular,
we may assume that\ $X$ is affine.
Moreover, by replacing $Y$ by a suitable open subscheme, we may assume that\ $Y$
is affine too. We may assume that $Y$ is a regular integral scheme. We
may embed $Y$ as an open subscheme of an $X$-scheme ${\overline{Y}}$ which is projective
over $X$ and is an integral scheme as well. Replace ${\overline{Y}}$ be its
normalization.
Now by again replacing $X$ by a nonempty open subscheme, we may
assume (by generic flatness) that the morphism \mapx[[ {\overline{Y}} || X ]] is flat;
certainly we may assume that\ it is surjective. Call this morphism $\varphi$. By
\pref{proper-bounded},
$\operatorname{Ker}[\operatorname{Pic}(\varphi)]$ is a bounded torsion group. Hence by
\pref{finitely-many}, $\operatorname{Ker}[\operatorname{Pic}(\varphi)]$ is finite (if $\mathop{\operator@font char \kern1pt}\nolimits(k) = 0$) and
is the direct sum of a finite group and a bounded $p$-group
(if $\mathop{\operator@font char \kern1pt}\nolimits(k) = p > 0$). [By \pref{alg-closed-regular}, if $k$ is
algebraically closed, then $\operatorname{Ker}[\operatorname{Pic}(\varphi)]$ is always finite.]
By \pref{S2} and \cite{Ha, II, (6.5c), (6.16)},
$\operatorname{Ker}[\operatorname{Pic}({\overline{Y}}) \rightarrow \operatorname{Pic}(Y)]$ is finitely generated. The theorem follows. \qed
\end{proof}
Now we want to see what happens when we consider a faithfully flat morphism
\mapx[[ Y || X ]], where $X$ is not necessarily normal. The issue is
complicated by nilpotents, even in the affine case. The problem [see examples
\pref{reduction-not-faithfully-flat}, \pref{rnff2} below] is that one can have
a domain $A$, and a finitely generated\ faithfully flat $A$ algebra $B$, such that $B_{\op{red}}$ is
not flat over $A$.
\begin{lemma}\label{B-cap-K-if-flat}
Let $A$ be a reduced ring, with total ring of fractions $K$. Let $B$ be a
faithfully flat $A$-algebra. Then (inside $B \o*_A K$) $B \cap K = A$.
\end{lemma}
\begin{proof}
Let $b \in B \cap K$, so we have an equation of the form $bu = v$, for some
$u, v \in A$ with $u$ a non-zero-divisor. By the equational criterion for
faithful flatness \cite{Bour, Ch.\ I, \S3, $\operatoratfont{n}^\circ$ 7, Prop.\ 13},
$b \in A$. \qed
\end{proof}
\begin{example}\label{reduction-not-faithfully-flat}
Let $k$ be a field of characteristic $2$. Let $A = k[s,t]/(s^2-t^3)$.
Let $B = A[x,y]/(x^2-s,y^2-t)$. Then $B$ is a faithfully flat $A$-algebra.
Now $B \cong k[x,y]/(x^4-y^6)$, so the nilradical of $B$ is generated by
$(x^2-y^3)$. Hence $B_{\op{red}} = A[x,y]/(x^2-s,y^2-t,x^2-y^3)$. Since
$s=ty$ in $B_{\op{red}}$, we have $y \in B_{\op{red}} \cap A_{\op{nor}}$ and $y \notin A$, so it
follows from \pref{B-cap-K-if-flat} that $B_{\op{red}}$ is not flat over $A$.
One can also prove this directly by taking $M = A/(t)$ and checking that
the map \mapx[[ M || M \o*_A B_{\op{red}} ]] is not injective.
\end{example}
\begin{example}[shown to us by Bill Heinzer and Sam Huckaba]\label{rnff2}
It is known that there is a smooth curve $C \subset {\Bbb C}\kern1pt{\Bbb P}^3$ of degree $8$ and
genus $5$ which is set-theoretically the intersection of two surfaces $S$, $T$,
but which is not arithmetically Cohen-Macaulay (see \cite{Bar}, \cite{Hu}).
Let $\tilde C = S \cap T$,
scheme-theoretically. Choose lines $L, L'$ which are noncoplanar and do not
meet $C$. Then projection from $L$ onto $L'$ defines a
nonconstant morphism \mapx[[ \tilde C || \Bbb P^1 ]].
Let $A$ be the homogeneous coordinate ring of $\Bbb P^1$, and let
$B$ be the homogeneous coordinate ring of $\tilde C$. Then $A = \Bbb C[x,y]$, $B$
is Cohen-Macaulay, $A \subset B$, and $B$ is module-finite over $A$, so $B$ is
faithfully flat over $A$. (See \cite{Ma, p.\ 140}.) On the other hand,
$B_{\op{red}}$ is not Cohen-Macaulay, so $B_{\op{red}}$ is {\it not\/} flat over $A$.
\end{example}
It is not clear if the behavior illustrated by the characteristic $p$
example can be mimicked in characteristic zero:
\begin{problem}\label{furry-friend}
Let $k$ be an algebraically closed field of characteristic zero. Let
$A$ be a finitely generated, reduced $k$-algebra. Let $B$ be a faithfully flat and
finitely generated\ $A$-algebra. Do we have $B_{\op{red}} \cap A_{\op{nor}} = A$?
\end{problem}
\begin{theorem}\label{faithfully-flat-fg}
Let $k$ be a perfect field. Assume that resolutions of singularities exist
for varieties over $k^a$. Let $X$ and $Y$ be geometrically integral
$k$-schemes of finite type. Let \mp[[ f || Y || X ]] be a faithfully flat
morphism of $k$-schemes. Then $\operatorname{Ker}[\operatorname{Pic}(f)]$ is finitely generated.
\end{theorem}
We are not sure to what extent the hypothesis ``geometrically integral''
can be relaxed. Certainly if $X$ is not affine, nonreduced, and $Y$ is
disconnected, one can have trouble \pref{nonreduced-cover-example}.
In positive characteristic, the
assumption that $Y$ is reduced is needed \pref{random-rabbits}.
In characteristic zero, we do not know if it is necessary to assume that $Y$ is
reduced. [If the answer to \pref{furry-friend} is yes, then we do not
need to assume $Y$ reduced.] We do not know if it is necessary to
assume that $X$ and $Y$ are geometrically irreducible. Cf.\ \pref{open-cover}.
\vspace{0.1in}
\par\noindent{\bf Affine version of Theorem \ref{faithfully-flat-fg}.}
\ {\it Let $k$ be a perfect field. Assume that resolutions of singularities
exist for varieties over $k^a$. Let $A$ be a finitely generated\ $k$-algebra.
Let $B$ be a finitely generated\ and faithfully flat $A$-algebra, which is geometrically
integral over $k$. Then $\operatorname{Ker}[\operatorname{Pic}(A) \rightarrow \operatorname{Pic}(B)]$ is finitely generated.}
\vspace{0.1in}
\begin{proofnodot}
(of \ref{faithfully-flat-fg}).
By \pref{separable-extension}, $\operatorname{Ker}[\operatorname{Pic}(X) \rightarrow \operatorname{Pic}(X^{\operatoratfont a})]$ is
finite, so we may assume that\ $k$ is algebraically closed.
By \pref{normal-faithfully-flat}, the theorem holds when $X$ is normal. To
complete the proof, we need to show that
$\operatorname{Ker}[\operatorname{Pic}(f)] \cap \operatorname{Ker}[\operatorname{Pic}(X) \to \operatorname{Pic}(X_{\op{nor}})]$ is finitely generated. If we relax the
assumptions on $Y$, by assuming only that each connected component of $Y$ is
geometrically integral, it is enough to do the two cases:
\begin{romanlist}
\item $Y$ is an ``open cover'' of $X$;
\item $X$, $Y$ are both affine.
\end{romanlist}
Let $P$ be the fiber product of $X_{\op{nor}}$ and $Y$ over $X$. Then $P$ is reduced.
Let \mp[[ \pi || X_{\op{nor}} || X ]] and \mp[[ \tau || P || Y ]] be the canonical
maps. Let ${\cal{C}}$ be the quotient sheaf $(\pi_*\cal O_{X_{\op{nor}}}^*)/\cal O_X^*$, and
similarly let ${\cal{D}} = (\tau_*\cal O_P^*)/\cal O_{Y}^*$. (Note that the canonical map
\mapx[[ \cal O_{Y} || \tau_*\cal O_P ]] is injective.)
We have a commutative diagram with exact rows (and some maps labelled):
\diagramx{1 & \mapE{} & {\Gamma(X_{\op{nor}})^*/\Gamma(X)^*} & \mapE{i}
& \Gamma({\cal{C}}) & \mapE{}
& \operatorname{Ker}[\operatorname{Pic}(X) \to \operatorname{Pic}(X_{\op{nor}})] & \mapE{} & 1\cr
&& \mapS{\lambda} && \mapS{\delta} && \mapS{}\cr
1 & \mapE{} & {\Gamma(P)^*/\Gamma(Y)^*} & \mapE{} & \Gamma({\cal{D}})
& \mapE{}
& \operatorname{Ker}[\operatorname{Pic}(Y) \to \operatorname{Pic}(P)] & \mapE{} & 1\makenull{.}}%
We will be done with the proof if we can show that $\delta$ is injective
and that $\operatorname{Coker}(\lambda)$ is finitely generated.
We show that $\delta$ is injective. In case (i) this is clear, since
${\cal{C}}$ is a sheaf. So we may assume that\ $X$, $Y$ are both affine.
Let $A = \Gamma(X)$, $B = \Gamma(Y)$. Let $\alpha \in \Gamma({\cal{C}})$.
Then there exist elements $\vec f1n \in A$ with $(\vec f1n) = (1)$ and
elements $\alpha_i \in (A_{\op{nor}})_{f_i}^*$ $(i=1,\ldots,n)$,
$\beta_{ij} \in A_{f_i f_j}^*$ such that
$\alpha_i = \beta_{ij}\alpha_j$ in $(A_{\op{nor}})_{f_i f_j}^*$ and $\alpha$
induces $\vec \alpha1n$. Suppose $\alpha$ maps to $1 \in \Gamma({\cal{D}})$.
Then for each $i$ we have elements $b_i \in B_{f_i}^*$ such that
$b_i = \alpha_i$ in $B_{f_i} \o*_A A_{\op{nor}}$ for each $i$. By
\pref{B-cap-K-if-flat}, applied with $A_{f_i}$ substituted for $A$, we see that
$\alpha_i \in A_{f_i}$. Hence $\alpha = 1$. Hence $\delta$ is injective.
To complete the proof, we need to show that
$$\operatorname{Coker}(\lambda) = {\Gamma(P)^* \over \Gamma(Y)^* \Gamma(X_{\op{nor}})^*}$$%
is finitely generated. For this, we may assume that $Y$ is connected. By \pref{roquette}, it
is enough to show that the map \mapx[[ P || X_{\op{nor}} ]] is bijective on connected
components. This is true since $X$ and $Y$ are integral schemes. \qed
\end{proofnodot}
We note that the proof breaks down if we do not assume that $Y$
is geometrically irreducible. Indeed, without this hypothesis, the
map \mapx[[ P || X_{\op{nor}} ]] may not be bijective on connected components:
\begin{example}
Let $k$ be a field of characteristic $\not= 2$. Let $A$ be the subring
$k[t^2, t+t^{-1}]$ of
$k[t,t^{-1}]$. Let $B = A[x]/(x^2-t^2)$. Then $B$ is a faithfully flat
$A$-algebra. We will see that (i) $\operatorname{Spec}(B)$ is connected, but (ii)
$\operatorname{Spec}(B \o*_A A_{\op{nor}})$ is not connected. First note that
$t = (t^2+1)/(t+t^{-1})$, from which it follows that $A_{\op{nor}} = k[t,t^{-1}]$.
Then $B \o*_A A_{\op{nor}} \cong A_{\op{nor}} \times A_{\op{nor}}$, so (ii) holds.
Let $z = t^2 + 1$, $y = t+t^{-1}$. Then $A \cong k[y,z]/(y^2+z^2-y^2z)$, so
$$B \cong k[x,y,z]/(x^2-z+1, y^2+z^2-y^2z) \cong k[x,y]/((x^2+1)^2-x^2y^2).$$%
{}From this one sees that $\operatorname{Spec}(B)$ has two smooth components, meeting at
the two points $(\pm i,0)$. Hence $\operatorname{Spec}(B)$ is connected.
\end{example}
\begin{theorem}\label{faithfully-flat-fg-imperfect}
Let $k$ be a field of characteristic $p > 0$. Let $X$ and $Y$ be
geometrically irreducible $k$-schemes of finite type. Let
\mp[[ f || Y || X ]] be a faithfully flat morphism of $k$-schemes.
Then $\operatorname{Ker}[\operatorname{Pic}(f)]$ is the direct sum of a finitely generated\ group and a $p$-group.
\end{theorem}
\vspace{0.05in}
\par\noindent{\bf Affine version of Theorem
\ref{faithfully-flat-fg-imperfect}.}
\ {\it Let $k$ be a field of characteristic $p > 0$. Let $A$ be a
finitely generated\ $k$-algebra. Let $B$ be a finitely generated\ and faithfully flat $A$-algebra such
that $\operatorname{Spec}(A \o*_k k^a)$ is irreducible. Then the group
$\operatorname{Ker}[\operatorname{Pic}(A) \rightarrow \operatorname{Pic}(B)]$
is the direct sum of a finitely generated\ group and a $p$-group.}
\vspace{0.1in}
\begin{proofnodot}
(of \ref{faithfully-flat-fg-imperfect}).
It follows from \pref{kernel-is-torsion} and \pref{finitely-many} that
$\operatorname{Ker}[\operatorname{Pic}(X) \to \operatorname{Pic}(X^a)]$ is the direct sum of a finitely generated\ group and a
$p$-group, so we may assume that\ $k$ is algebraically closed. Since
$\operatorname{Ker}[\operatorname{Pic}(X) \to \operatorname{Pic}(X_{\op{red}})]$ is a $p$-group, we may assume that\ $X$ is reduced.
If $Y$ is reduced, we are done by \pref{faithfully-flat-fg}. In the
general case, we modify the argument of the proof of
\pref{faithfully-flat-fg}. Since the scheme $P$ in that argument may be
nonreduced, we do not get that $\operatorname{Coker}(\lambda)$ is finitely generated, but rather
$(*)$ it is finitely generated\ mod $p$-power torsion. We still get that
$$M := {\Gamma(P)_{\op{red}}^* \over \Gamma(Y)_{\op{red}}^* \Gamma(X_{\op{nor}})^*}$$%
is finitely generated. Let $N$ be the nilradical of $\Gamma(P)$. Then the kernel of the
canonical map \mapx[[ \operatorname{Coker}(\lambda) || M ]] is a quotient of $1+N$, and
hence is a bounded $p$-group, so $(*)$ follows. \qed
\end{proofnodot}
\begin{remark}
By \pref{bounded}, it will follow that the $p$-group of the
theorem is actually a bounded $p$-group, provided that resolution of
singularities is valid.
\end{remark}
\begin{theorem}\label{open-cover}
Let $X$ be a reduced scheme of finite type over a perfect field. Let
$\vec U1n$ be an open cover of $X$. Then the canonical map
\dmapx[[ \operatorname{Pic}(X) || \operatorname{Pic}(U_1) \times \cdots \times \operatorname{Pic}(U_n) ]]
has finitely generated\ kernel.
\end{theorem}
For the case where $X$ is normal (in fact, any noetherian normal scheme), this
is easy, following roughly from \pref{S2}.
For the general case, one can follow the proof of \pref{faithfully-flat-fg},
taking $Y$ to be the disjoint union of the $U_i$'s;
one need only adjust the last sentence.
\section{Examples where the kernel is not finitely generated}%
\label{examples-section}
Let \mp[[ f || Y || X ]] be a faithfully flat morphism of noetherian schemes.
The results \ref{kernel2b}, \ref{separable-extension}, \ref{proper-Pic-finite},
\ref{normal-faithfully-flat}, \ref{faithfully-flat-fg}, and \ref{open-cover}
all give conditions under which $\operatorname{Ker}[\operatorname{Pic}(f)]$ is finitely generated. While it is
reasonable to think that there are unifying and more general results with
the same conclusion, we do not know what form such results should take.
With this in mind, we give in this section a varied collection of examples
in which $\operatorname{Ker}[\operatorname{Pic}(f)]$ is {\it not\/} finitely generated.
We mention an obviously related question, about which we know very little.
For which faithfully flat proper morphisms \mp[[ f || Y || X ]] is it the
case that the map
\dmapx[[ \setofh{iso.\ classes of vector bundles on $X$} ||
\setofh{iso.\ classes of vector bundles on $Y$} ]]%
is finite-to-one? Cf.\ \pref{fidofido}, \pref{woofwoof}.
Returning to $\operatorname{Pic}$,
first we see that there are examples with $X$ algebraic of positive
characteristic, and $f$ a finite \'etale\ morphism. For these, by
\pref{kernel2b}, $X$ must be nonreduced.
\begin{example}\label{finite-etale-infinite}
Let $k$ be an algebraically closed field of characteristic $p > 0$. Let
\mp[[ \lambda || F || E ]] be a finite \'etale\ morphism of varieties over $k$,
of degree $p$, and suppose we have a Galois action of a finite group $G$
on $F/E$. Let $R = k[\epsilon]/(\epsilon^p)$.
Let $X = E \times_k R$, $Y = F \times_k R$, and let \mp[[ f || Y || X ]] be
the induced morphism. Then $\operatorname{Ker}[\operatorname{Pic}(f)] \cong H^1(G, \Gamma(Y)^*)$ by
\pref{the-kernel}. Since $\Gamma(Y)$ is just $R$, and $G$ acts trivially on
it, we have by \pref{group-cohomology-basics}(2) that
$\operatorname{Ker}[\operatorname{Pic}(f)] \cong \operatorname{Hom}(\Bbb Z/p\Bbb Z, R^*)$, which is not finitely generated, since
$(1+c\epsilon)^p = 1$ in $R$ for each $c \in k$.
To get examples of such morphisms $\lambda$,
let $E$ be an elliptic curve over $k$ which is not supersingular. Then
the Tate module $T_p(E)$ is isomorphic
to the $p$-adic integers $\Bbb Z_p$. According to a theorem of Serre-Lang
\cite{SGA1, XI, (2.1)}, the $p$-primary part of the algebraic fundamental
group $\pi_1(E)$ is $T_p(E)$. Therefore there exists a surjective
homomorphism \mapx[[ \pi_1(E) || \Bbb Z/p\Bbb Z ]], and so there exists a variety $F$
and a morphism $\lambda$ as indicated. More concretely, this may be seen
as follows. The multiplication by $p$ map $E\ \mapE{p}\ E$ factors through
$E^{(p)}$, the scheme defined in essence by raising the coefficients in
the equation defining $E$ to the \th{p} power. The induced map
\mapx[[ E^{(p)} || E ]], called the {\it Verschiebung}, is exactly $\lambda$.
\end{example}
Now we see that there are examples with $X$ reduced and algebraic, even over an
algebraically closed field (of positive characteristic),
and $f$ a finite flat morphism:
\begin{example}\label{random-rabbits}
Take example \pref{reduction-not-faithfully-flat}, and use $X = \operatorname{Spec}(A)$,
$Y = \operatorname{Spec}(B)$. Since the map \mapx[[ A || B_{\op{red}} ]] factors through $A_{\op{nor}}$,
it follows that $\operatorname{Ker}[\operatorname{Pic}(f)] = k$, which is not finitely generated\ if $k$ is infinite.
\end{example}
Now we see that there are examples with $X$ algebraic and $Y$ \'etale, even
in characteristic zero. Indeed, one may take $Y$ to be an ``affine open
cover''
of $X$:
\begin{example}\label{nonreduced-cover-example}
Let $T$ be a projective variety over an algebraically closed field $k$,
and let ${\cal{F}}$ be a coherent sheaf on $T$ with $H^1(T,{\cal{F}}) \not= 0$.
Make ${\cal{A}} := \cal O_T \oplus {\cal{F}}$ into an $\cal O_T$-algebra by forcing
${\cal{F}} \cdot {\cal{F}} = 0$. Let $X = \mathop{\mathbf{Spec}}\nolimits({\cal{A}})$. Let $Y$ be an affine open
cover of $X$, i.e.\ the disjoint union of the schemes in such a cover.
Then $\operatorname{Ker}[\operatorname{Pic}(X) \to \operatorname{Pic}(X_{\op{red}})] \subset \operatorname{Ker}[\operatorname{Pic}(X) \to \operatorname{Pic}(Y)]$, and
$\operatorname{Ker}[\operatorname{Pic}(X) \to \operatorname{Pic}(X_{\op{red}})]$ is a nonzero vector space (over $k$), so
$\operatorname{Ker}[\operatorname{Pic}(X) \to \operatorname{Pic}(Y)]$ is not finitely generated.
\end{example}
Now we give families of examples in which $X$ and $Y$ can be chosen to be
reduced noetherian schemes of characteristic zero, but the morphism is not of
finite type:
\begin{example}
Given any ring $A$ there exists a faithfully flat extension $B$ with
$\operatorname{Pic}(B) =1$: take $B=A[x]$ localized at the set of primitive polynomials
(i.e.\ polynomials such that the coefficients generate the unit ideal).
For the fact that $\operatorname{Pic}(B)=1$, see \cite{EG, (5.4), (3.5), (2.6 with $R=S$)}.
\end{example}
\begin{example}
Let $k$ be a field and let $X = \operatorname{Spec}(k[t^2,t^3])$. Let
$$Y = \operatorname{Spec}(k[t,t^{-1}]) \times k[t^2,t^3]_{(t^2,t^3)}),$$%
which is the disjoint union of $X_{\op{reg}}$ and $\operatorname{Spec} \cal O_{X,x}$, where $x$ is
the singular point of $X$. Then $\operatorname{Ker}[\operatorname{Pic}(f)] = k$.
\end{example}
Now we will give examples based on purely inseparable base extension. For
purely inseparable extensions we cannot use Galois cohomology to control the
Picard group. Instead, we use differentials, following the ideas in Samuel's
notes on unique factorization domains \cite{Sam}. We build a purely inseparable
form of the affine line whose Picard group is infinite.
We need the following version of a result of Samuel \cite{Sam, 2.1, p.\ 62}:
\begin{lemma}\label{son-of-samuel}
Let $B$ be a domain of characteristic $p > 0$ with fraction field $Q(B)$. Let
$\mathop{\delta}$ be a $\Bbb Z$-linear
derivation of $B$ with $A$ the subring of invariants of $\mathop{\delta}$ (i.e.\ the
elements with $\mathop{\delta}(a)=0$). Let $\mathop{\Delta}$ be the logarithmic derivative
of $\mathop{\delta}$ defined on $Q(B)^*$ (i.e. $\mathop{\Delta}(b)=\mathop{\delta}(b)/b)$. If $M_1, M_2$ are
invertible ideals of $A$ with $M_iB = b_iB$ for each $i$ ($b_1,b_2 \in B$),
then $M_1 \cong M_2$ if and only if $\Delta(b_1)-\Delta(b_2) \in \Delta(B^*)$.
\end{lemma}
\begin{proof}
If $M_1 \cong M_2$, then $b_1=aub_2$ for some $a$ in the quotient
field of $A$ and $u \in B^*$. Thus, $\Delta(b_1)=\Delta(b_2) + \Delta(u)$.
Conversely, if $\Delta(b_1)-\Delta(b_2) = \Delta(u)$ for some
$u \in B^*$, then replacing $b_1$ by $b_1u$ allows us to assume
that $\Delta(b_1)=\Delta(b_2)$ whence $\delta(b_1^{-1}b_2 )=0$.
Set $a=b_1^{-1}b_2$. Thus, $a$ is in the quotient field of $A$.
Replacing $M_1$ by $aM_1$ allows us to assume that $M_1B=M_2B$.
We claim that this implies $M_1=M_2$. It suffices to check this
locally and so we may assume that each $M_i$ is principal. Let
$a_i$ be a generator for $M_i$. It follows that $a_1/a_2$ is a unit
in $B$. We have $\delta(a_1/a_2) = 0$ and $a_1/a_2 \in B$, so $a_1/a_2 \in A$.
Similarly, $a_2/a_1 \in A$, so $a_1/a_2 \in A^*$, and hence $M_1 = M_2$. \qed
\end{proof}
\begin{example}\label{deranged-derivations}
Let $k$ be a separably closed imperfect field of
characteristic $p$. Let $\alpha \in k - k^p$. Let
$q$ be a power of $p$ with $q > 2$, and let $K = k(\alpha^{1/q})$.
Let $A = k[X,Y]/(X^q-X-\alpha Y^q)$. Set $r=q/p$. Then $A$ is a
Dedekind domain and $P=\operatorname{Pic}(A)$ is a group of exponent $q$ with $P/{_r}P$
infinite. Also, $A_K \cong K[Z]$. Hence the kernel of the map
$\operatorname{Pic}(A) \rightarrow \operatorname{Pic}(A_K)$ is infinite of exponent $q$.
\end{example}
\begin{proof}
Let $\beta = \alpha^{1/q}$. In the polynomial ring $B = K[Z]$, put $x = Z^q$
and $y = \beta^{-1}(Z^q - Z)$. Note that $Z = x-\beta y$, so that
$K[Z] = K[x,y]$. Moreover, we can identify $A$ with the subring $k[x,y]$ of
$B$. Then $A_K = B$. Thus $A$ is a Dedekind domain and $\operatorname{Pic}(A_K)$ is
trivial. It follows that $\operatorname{Pic}(A)$ has exponent dividing $q$.
Let $V=\{a \in k : a^q -a \in \alpha k^q\}$. Since $k$ is separably closed,
for every $b \in k$ there is an element $a\in V$ such that
$$a^q - a -\alpha b^q = 0.$$
Thus $V$ is infinite. Given $a \in V$, define $b \in k$
by the above equation. Let $M(a)$ be the maximal ideal $(x-a,y-b)$ of $A$.
Set $c= a- \beta b=a^{1/q}$. Note that $M(a)B=(Z-c)B$.
Let $a_1$ and $a_2$ be distinct elements of $V$. Let $b_i$ and $c_i$ be the
corresponding elements defined above. It suffices to show that
$M(a_1)^r$ and $M({a_2})^r$ are nonisomorphic.
Let $\gamma = \alpha^{1/p}$. Define a derivation $\mathop{\delta}$ on $k[\gamma]$ with
$\mathop{\delta}(k)=0$ and $\mathop{\delta}(\gamma)= \gamma$. Extend this derivation to $A_0=A[\gamma]$
(and to its quotient field) by taking $\mathop{\delta}$ trivial on $A$. Set $W=Z^r$. Since
$Z=x - \beta y$, $W=x^r -\gamma y^r$. Thus, $\mathop{\delta}(W)= -\gamma y^r = W-W^q$. Let
$\mathop{\Delta}$ denote the logarithmic derivative of $\mathop{\delta}$.
The following observation will be useful: $\mathop{\delta}(c_i^r)=-\gamma b_i^r=
c_i^r -c_i^{rq}$. Thus
$$\mathop{\Delta}(W-c_i^r) ={{W-W^q-c_i^r+c_i^{rq}}\over{W-c_i^r}}=
1-(W-c_i^r)^{q-1}.$$ Thus, $$\Delta(W-c_1^r) - \Delta(W-c_2^r) =
(W - c_2^r)^{q-1} - (W - c_1^r)^{q-1}$$
is a polynomial in $Z$ of degree $r(q-2) > 0$ as long as $ q > 2$.
Let $J=(W-c_i^r) A_0$ and $I=M(a_i)^r A_0$. Then $I^pA_0=J^pA_0$, since after
extension to B, they become equal. Now $A_0$ is a normal domain, so its group
of invertible fractional ideals is torsion-free. Hence $I A_0 = J A_0$,
i.e.\ $M(a_i)^rA_0=(W-c_i^r)A_0$.
Therefore, it suffices to show (by \ref{son-of-samuel})
that $\Delta(W-c_1^r) - \Delta(W-c_2^r) \ne \Delta(u)$
for any $u \in A_0^*$. Since $\Delta(u)$ is a constant
in $Z$, the result follows by the previous paragraph. \qed
\end{proof}
Now we give an alternate (more geometric) explanation of the preceeding result.
It will show that the kernel of \mapx[[ \operatorname{Pic}(A) || \operatorname{Pic}(A_K) ]] is infinite
and $q$-torsion, but not that the kernel has exponent $q$.
Let $U = \operatorname{Spec}(A)$, and let $V = \operatorname{Proj}(k[X,Y,T]/(X^q-X T^{q-1}-\alpha Y^q))$.
Then $U$ is an open subscheme of $V$. We have a commutative diagram
\squareSE{\operatorname{Pic}(V)}{\operatorname{Pic}(U)}{\operatorname{Pic}(V_K)}{\operatorname{Pic}(U_K)\makenull{.}}%
Of course $\operatorname{Pic}(U_K) = 0$ since $A_K$ is a polynomial ring. Since $U$ is
obtained from $V$ by deleting the single regular point $[X,Y,T] = [1,0,0]$,
\mapx[[ \operatorname{Pic}(V) || \operatorname{Pic}(U) ]] is surjective with cyclic kernel, and in fact one
sees that the kernel is infinite cyclic. On the other hand, the map
\mapx[[ \operatorname{Pic}(V) || \operatorname{Pic}(V_K) ]] is injective, as is well-known.%
\footnote{More generally, it is even true that if $K/k$ is any field extension,
$V$ is a projective $k$-scheme, and one has two coherent $\cal O_V$-modules which
become isomorphic over $V_K$, then they were already isomorphic over $V$
-- see \cite{Wi, (2.3)}.} Hence the kernel of \mapx[[ \operatorname{Pic}(U) || \operatorname{Pic}(U_K) ]]
is exactly $\operatorname{Pic}^0(V)$.
By results of Grothendieck \cite{FGA, (2.1), (3.1)},
there exists a commutative group scheme $P$
of finite type over $k$ such that $\operatorname{Pic}^0(V) = P(k)$. Since $\operatorname{dim}(V) = 1$,
$H^2(V,\cal O_V) = 0$, so $P$ is smooth and $\operatorname{dim}(P) = h^1(V,\cal O_V)$ by
\cite{FGA, \#236, 2.10(ii, iii)}. Now $h^1(V,\cal O_V)$ is just the arithmetic
genus of a plane curve of degree $q$, which is $(q-1)(q-2)/2$. In particular,
since $q \geq 3$, we have $h^1(V,\cal O_V) > 0$. Hence $P$ is
positive-dimensional. Now since $P$ is a geometrically integral scheme of
finite type over a separably closed field $k$, $P(k)$ is Zariski dense in
$P(k^a)$ -- see \cite{Se, discussion on p.\ 107}. In particular, since $P$ is
positive-dimensional, it follows that $P(k)$ is infinite. Hence
$\operatorname{Pic}^0(V)$ is infinite. Hence the kernel of the map
\mapx[[ \operatorname{Pic}(U) || \operatorname{Pic}(U_K) ]] is infinite; it is $q$-torsion by
\pref{kernel-is-torsion}. This completes the alternate proof of
\pref{deranged-derivations}, except that we have not shown that the kernel
has exponent $q$. \qed
\vspace{0.03in}
We will see in the next section (at least assuming resolution of singularities)
that for a separated scheme $X$ of finite type over a field $k$, the kernel of
the map $\operatorname{Pic}(X) \to \operatorname{Pic}(X_{k^{\text a}})$ is always bounded (i.e., torsion with
finite exponent). The Picard group itself, however, can have infinite
exponent. For example, if $X$ is any smooth affine or projective curve
of positive genus over an algebraically closed field, then the torsion
subgroup of $\operatorname{Pic}(X)$ is unbounded, because then $\operatorname{Pic}(X)$ is the quotient of
an abelian variety by a finitely generated\ subgroup.
\section{Eventual vanishing of the kernel of Pic}\label{eventual-section}
Let $k$ be a field. The main theorem of this section
\pref{eventually-injective} asserts that if $X$ is a separated $k$-scheme of
finite type, then there exists
a finite field extension $k^+$ of $k$, such that for every algebraic field
extension $L$ of $k^+$, the canonical map $\operatorname{Pic}(X_L) \to \operatorname{Pic}(X_{L^{\text a}})$
is injective.
This statement has a sheaf-theoretic formulation, which we consider, in part
because it figures in the proof. Let $F$ be an
(abelian group)-valued $k$-functor, meaning a functor from
$\cat{$k$-algebras}$ to $\cat{abelian groups}$.
Let $p: B \to C$ be a faithfully flat homomorphism of $k$-algebras.
There are maps $i_1,i_2: C \to C\otimes_BC$, given by
$c \mapsto c \o* 1$ and $c \mapsto 1 \o* c$, respectively. If the sequence
\diagramno{(*)}{0 & \mapE{} & F(B) & \mapE{F(p)} & F(C) & \mapE{F(i_1)-F(i_2)}
& F(C\o*_BC)}%
is exact for all $p$, then one says that $F$ is a {\it sheaf\/}
(for the fpqc [faithfully flat quasi-compact] topology).
This is often too much to ask, and so one may look only at certain maps $p$, or
ask only that $F(p)$ be injective. To relate this to the theorem
\pref{eventually-injective}, let
$F$ be given by $B \mapsto \operatorname{Pic}(X_B)/\operatorname{Pic}(B)$. Of course if $B$ is a field,
we have $F(B) = \operatorname{Pic}(X_B)$. What the theorem says
is that if we enlarge $k$ sufficiently (replacing it by a finite extension),
then $F(L \to L^{\text a})$ is injective, for all algebraic extensions $L$ of $k$.
In this sense, $F$ becomes close to being a sheaf, if we allow for the
enlargement of $k$. However, in general, one cannot get $(*)$ exact
in an analogous manner. More precisely, for suitable $k$ and $X$, one cannot
find a finite extension $k^+$ of $k$ such that for every algebraic extension
$L$ of $k^+$, if $p:L \to L^{\text a}$ is the canonical map, then $(*)$
is exact. As an example, let $k$ be a separable closure of $\Bbb F_q(t)$,
for some prime $q$, and let $X = \operatorname{Spec}(k[x,y]/(y^2-x^3))$. Then
$\operatorname{Pic}(X_L) = L$ for every extension field $L$ of $k$. For any finite extension
$k^+$ of $k$, there is some $a \in k^+ - (k^+)^q$. If $L = k^+$, one finds
that $a^{1/q}$ lies in the kernel of $F(i_1)-F(i_2)$, but not in the image of
$F(p)$.
\begin{definition}
Let $k$ be a field. A $k$-scheme $S$ is {\it geometrically stable\/}
if (1) it is of finite type, and (2) every irreducible component of $S_{\op{red}}$ is
geometrically integral and has a rational point.
\end{definition}
\begin{theorem}\label{eventually-injective}
Let $k$ be a field; assume that resolutions of singularities exist for
varieties over $k^a$. Let $X$ be a separated $k$-scheme of finite type.
Then there exists a finite field extension $k^+$ of
$k$, such that for every algebraic field extension $L$ of $k^+$, the
canonical map $\operatorname{Pic}(X_L) \to \operatorname{Pic}(X_{L^{\text a}})$ is injective.
\end{theorem}
\vspace{0.1in}
\par\noindent{\bf Proof of (\ref{eventually-injective})}
In the course of the proof, we will refer to {\it enlarging\/} $k$, by
which we mean that $k$ is to be replaced by a suitably large finite field
extension, contained in $k^{\text a}$. This is done only finitely many times. Then,
at the end
of the proof, the $k$ we have is really the $k^+$ of which the theorem speaks.
We may treat $L$ as an extension of $k$ which is contained in $k^{\text a}$.
By induction on $\operatorname{dim}(X)$, we may assume that the theorem holds
when $X$ is replaced by a scheme of strictly smaller dimension.
(The case of dimension zero is trivial.)
\vspace{0.1in}
\par\noindent{\bf Step 1. The case of a geometrically normal scheme}
\vspace{0.1in}
If $T$ is a $k$-scheme, we have let $T^{\text a}$ denote $T_{k^{\text a}}$.
However, in some places in the next paragraph we shall define a $k^{\text a}$-scheme
$T^{\text a}$, even though $T$ has not yet been defined; we will then
proceed to construct a $k$-scheme $T$ such that $T^{\text a} = T_{k^{\text a}}$.
Suppose $X$ is geometrically normal. By \cite{N}, there is a proper
$k^{\text a} $-scheme ${\overline{X}}^{\text a}$ which contains $X^{\text a} $ as a dense open subscheme. (Note
that ${\overline{X}}$ does not yet been defined.) After normalizing ${\overline{X}}^{\text a}$
we may assume that\ ${\overline{X}}^{\text a}$ is normal. Let $\pi^{\text a} : {\overline{Y}}^{\text a} \to {\overline{X}}^{\text a}$ be a
{\it resolution of singularities}, by which we mean that
${\overline{Y}}^{\text a}$ is regular, $\pi^{\text a}$ is a proper morphism, and $\pi^{\text a}$ is an
isomorphism over ${\overline{X}}^{\text a} - \operatorname{Sing}({\overline{X}}^{\text a})$. (Note that ${\overline{Y}}$ and $\pi$ have not
yet been defined.)
By looking at the equations defining ${\overline{X}}^{\text a}$, ${\overline{Y}}^{\text a}$, and
$\pi^{\text a}$, we can (after enlarging $k$ if necessary) find a $k$-scheme
${\overline{X}}$
(containing $X$ as a dense open subscheme), a $k$-scheme ${\overline{Y}}$,
and a morphism
$\pi: {\overline{Y}} \to {\overline{X}}$ such that $\pi \times_k k^{\text a} = \pi^{\text a}$.
Then ${\overline{X}}$ is geometrically normal, ${\overline{Y}}$ is
geometrically regular, and (by faithfully flat descent
\cite{EGA4, (2.7.1)(vii)}) $\pi$ is proper. Let $Y = \pi^{-1}(X)$.
By enlarging $k$ if necessary, we may assume that if $C_1,\dots,C_n$ are the
irreducible codimension one components of ${\overline{Y}}-Y$, and if $p: {\overline{Y}}^{\text a}
\to {\overline{Y}}$ is the natural map, then $p^{-1}(C_1), \dots, p^{-1}(C_n)$ are the
irreducible codimension one components of ${\overline{Y}}^{\text a} - Y^{\text a} $.
Let $d : \operatorname{Pic}({\overline{Y}}^{\text a} ) \to \operatorname{Pic}(Y^{\text a} )$,
$e : \operatorname{Pic}({\overline{Y}}) \to \operatorname{Pic}({\overline{Y}}^{\text a} )$, $f : \operatorname{Pic}({\overline{Y}}) \to \operatorname{Pic}(Y)$
and $h : \operatorname{Pic}(Y) \to \operatorname{Pic}(Y^{\text a} ) $ be the canonical maps.
For any normal proper $k$-scheme $V$ of finite type, the canonical
map $\operatorname{Pic}(V) \to \operatorname{Pic}(V^{\text a} ) $ is injective. (See
\cite{Mi3, (6.2)}.) In particular, $e$ is injective. Since
$$[p^{-1}(C_1)],\ldots,[p^{-1}(C_n)]$$%
generate $\operatorname{Ker}(d)$, it follows
that $[C_1], \ldots, [C_n]$ generate $\operatorname{Ker}(de)$. Since $f$ is surjective,
it follows that $h$ is injective.
A slight modification of the argument above shows that the canonical
map $h_L : \operatorname{Pic}(Y_L) \to \operatorname{Pic}(Y^{\text a} ) $ is injective for every algebraic
extension $L/k$. Let $\cal L$ be a line
bundle on $X_L$ that becomes trivial on $X^{\text a} $. Since $h_L$ is
injective,
$\cal L$ becomes trivial on pullback to $Y_L$. Let $r : X^{\text a} \to X_L $
be the canonical map. Then the restriction of
$\cal L$ to $X_L - r(\operatorname{Sing}(X^{\text a} ))$ is trivial. By (\ref{S2})
$\cal L$ is trivial. This completes the proof when $X$ is geometrically
normal.
\vspace{0.1in}
\par\noindent{\bf Step 2. The case of a geometrically reduced scheme}
\vspace{0.1in}
If $X$ is geometrically reduced we may assume, by enlarging $k$ if
need be, that the normalization
$X_{\op{nor}}$ is geometrically normal. Let $\pi : X_{\op{nor}} \to X $ be
the canonical map. Let $\cal I = [\cal O_X : \pi_*\cal O_{X_{\op{nor}}}]$ be the
conductor of $X_{\op{nor}} $ into $X$. This
is a coherent sheaf of ideals in
$\cal O_X$. Let $X/\cal I$ denote $\hbox{\bf Spec}(\cal O_X/\cal I)$, and let
$X_{\op{nor}}/\cal I$ denote $(X/\cal I) \times_X X_{\op{nor}} $. By further enlarging
$k$, we may assume that the pullback of $\cal I$ to $X^{\text a} $ is the
conductor of $(X_{\op{nor}} )^{\text a} $
into $X^{\text a} $. Enlarging $k$ still further, we may assume that
$X_{\op{nor}} $, $X/\cal I$, and $X_{\op{nor}} /\cal I$ are all geometrically stable.
Let $F_1$ and $F_2$ be the (abelian group)-valued $k$-functors defined by
$$
F_1(B) = \frac{\Gamma((X_{\op{nor}} )_B)^*}{\Gamma(X_B)^*} \text{\ \
and\ \ }
F_2(B) = \frac{\Gamma((X_{\op{nor}} /\cal I)_B)^*}{ \Gamma((X/\cal I)_B)^*}.
$$
Let $G = F_2/F_1$, the quotient in $\cat{(abelian group)-valued $k$-functors}$.
We have a commutative diagram
$$
\begin{CD}
{} @. 0 @. 0 @. {}
@. {}\\
@. @VVV @VVV @.
@. \\
0 @>>> F_1(L) @>>> F_2(L) @>>> G(L) @>>> 0 \\
@. @VVV @VVV @VVV
@. \\
0 @>>> F_1(L^{\text a}) @>>> F_2(L^{\text a}) @>>> G(L^{\text a}) @>>> 0 \\
@. @VVV @VVV @VVV
@. \\
0 @>>> F_1(L^{\text a}\o*_LL^{\text a}) @>>> F_2(L^{\text a}\o*_LL^{\text a}) @>>>
G(L^{\text a}\o*_LL^{\text a}) @>>> 0
\end{CD}
$$
with exact rows. Now we use \cite{J2, (4.5)}:
\vspace{0.1in}
\par\noindent{\bf Theorem}\ \ {\it
Let $S$ and $T$ be geometrically stable $k$-schemes, and let $f: S \to T$ be a
dominant morphism of $k$-schemes. Let $Q$ be the (abelian group)-valued
$k$-functor given by $Q(A) = \Gamma(S_A)^* / \Gamma(T_A)^*$. Let $p: B \to C$
be a faithfully flat homomorphism of reduced $k$-algebras. Then the sequence
$$0 \to Q(B) \to Q(C) \to Q(C \otimes_B C)$$%
is exact.}
\vspace{0.1in}
This implies that the first two columns are exact. It follows that the
canonical map $G(L) \to G(L^{\text a})$ is injective.
We need a scheme-theoretic version of Milnor's Mayer-Vietoris sequence
\cite{Bas1, Chap.\ IX, (5.3)}, which may be found in
\cite{We, (7.8)(i)}, and which implies that there is an exact sequence
$$0 \to F_1(L) \to F_2(L) \to \operatorname{Pic}(X_L) \to
\operatorname{Pic}((X_{\op{nor}})_L) \times \operatorname{Pic}((X/\cal I)_L) \eqno(\diamondsuit)$$\label{diamond}%
for each algebraic extension $L/k$. By the induction
hypothesis announced near the beginning of the proof,
we may assume that\ $\operatorname{Pic}((X/\cal I)_L) \to \operatorname{Pic}((X/\cal I)^{\text a} )$ is injective. Also, by
Step 1, $\operatorname{Pic}((X_{\op{nor}})_L) \to \operatorname{Pic}((X_{\op{nor}})^{\text a})$ is injective.
Since $G(L) \to G(L^{\text a})$ is injective, it follows that
$\operatorname{Pic}(X_L) \to \operatorname{Pic}(X^{\text a})$ is injective. Thus the theorem
is true for geometrically reduced schemes.
\vspace{0.1in}
\par\noindent{\bf Step 3. Deal with the nonreduced case}
\vspace{0.1in}
This step may of course be ignored if $X$ is affine. Otherwise, we
use the following result \cite{J2,(5.2)(a)}:
\vspace{0.1in}
\par\noindent{\bf Theorem}\ \ {\it Let $k$ be a field and $X$ a geometrically
stable $k$-scheme. Let $i: X_0 \to X$ be a nilimmersion. For any
$k$-algebra $A$, let $\kappa(A)$ be the kernel of the natural map
$\operatorname{Pic}(X\times_kA) \to \operatorname{Pic}(X_0\times_kA)$. Let $A \to B$
be a faithfully flat homomorphism of reduced $k$-algebras. Then the
induced map $\kappa(A) \to \kappa(B)$ is injective.}
\vspace{0.1in}
To complete the proof of (\ref{eventually-injective}), we may assume, by
enlarging $k$ if need be, that $X$ is geometrically stable and that $k$
is big enough so that the conclusion is valid for the (geometrically
reduced) scheme $X_{\op{red}}$. We have the following commutative
diagram, for any algebraic extension $L/k$:
$$
\begin{CD}
\operatorname{Pic}(X_L) @>\alpha>> \operatorname{Pic}((X_{\op{red}})_L)\\
@V\gamma VV @V\delta VV\\
\operatorname{Pic}(X^{\text a}) @>\beta>> \operatorname{Pic}(X_{\op{red}}^{\text a})
\end{CD}
$$
Taking $(A \to B) = (L \to k^{\text a})$ in \cite{J2,(5.2)(a)}, cited above, we see
that $\gamma$ is one-to-one on $\operatorname{Ker}(\alpha)$. Since
$\delta$ is injective, so is $\gamma$. \qed
\begin{remark}\label{woofwoof}
Let $F$ be a finite field of characteristic different from $5$ and containing
a primitive fifth root of unity $\zeta$. Let $Y\subset {\Bbb P}^3_F$
be the Fermat quintic given by the equation:
$$x^5+y^5+z^5+t^5=0.$$%
Then the group of fifth roots of unity acts on $Y$ by sending $(x,y,z,t)$
to $(x,\zeta y,\zeta ^2z,\zeta^3t)$. This action has no fixed points, and
the quotient $X$ is a smooth projective surface which is called the
{\it Godeaux surface}. For any finite extension $E/F$,
$\operatorname{CH}^2(X_E)_{\operatoratfont tors}=\Bbb Z/5\Bbb Z$ \cite{KS1, Proposition 9}. On the other hand,
over an algebraic closure ${\overline F}$, we have
$\operatorname{CH}^2(X_{\overline F})_{\operatoratfont tors})=0$ \cite{Mi2}. Thus
for each $E$ there exists a finite extension $H$ over which the $\Bbb Z/5\Bbb Z$
dies, but a new one comes to take its place. Now for any smooth surface over a
field, the natural map $K_0(X)\to \operatorname{CH}^*(X)$
is an isomorphism between the Grothendieck ring and the Chow ring. Hence
we produce an element of $K_0(X)$ with similar properties. Thus the theorem
does not hold with $K_0$ in place of $\operatorname{Pic}$.
\end{remark}
\begin{corollary}\label{bounded}
Let $k$ be a field; assume that resolutions of singularities exist for
varieties over $k^a$. Let $X$ be a $k$-scheme of finite type.
Then the kernel of the map $\operatorname{Pic}(X) \to \operatorname{Pic}(X^a)$ is a bounded torsion group.
\end{corollary}
\begin{proof}
One can (details omitted) use \pref{open-cover} to reduce to the case where
$X$ is separated. Let $K = k^{\text a}$.
Let $k^+$ be as in \pref{eventually-injective}. Then $\operatorname{Pic}(X) \to
\operatorname{Pic}(X_K)$ and $\operatorname{Pic}(X) \to \operatorname{Pic}(X_{k^+})$ have the same kernel. By
(\ref{kernel-is-torsion}) the kernel has exponent dividing $[k^+ : k]$. \qed
\end{proof}
\section{Finitely Generated Fields}\label{abs-section}
We say a field $k$ is {\it absolutely finitely generated\/}
if it is finitely generated over its prime subfield. In this section we will
study the structure of $\operatorname{Pic}(X)$, where $X$ is a scheme of finite type over an
absolutely finitely generated field. We begin with a result
that is presumably well known, but for which we have found
no reference.
\begin{proposition}\label{normal-over}
Let $X$ be a normal scheme of finite
type over $\Bbb Z$ or over an absolutely finitely generated
field $k$. Then $\operatorname{Pic}(X)$ is finitely generated.
\end{proposition}
\begin{proof}
A theorem due to Roquette
\cite{Ro}, \cite{L, Chap. 2, (7.6)} handles the case of
schemes of finite type over $\Bbb Z$. Suppose now that
$X$ is of finite type over the absolutely finitely generated
field $k$.
By \pref{S2} the map $\operatorname{Pic}(X) \to \operatorname{Pic}(X - \operatorname{Sing}(X))$ is
injective. Therefore we may assume that\ $X$ is regular. There exists
a finitely generated $\Bbb Z$-algebra $A \subset k$ and an
$A$-scheme $X_0$ of finite type such that $X\cong X_0\times_A k$.
We have $X \subset X_0$. Since $X$ is regular, $\cal O_{X_0,x}$ is
regular for every $x\in X$. Since $X_0$ is excellent, its
regular locus is open, so there exists an open subscheme of $X_0$
which is regular and contains $X$. By replacing $X_0$ by this
subscheme, we may assume that $X_0$ is regular. The map on
divisor class groups $\operatorname{Cl}(X_0) \to \operatorname{Cl}(X)$ is certainly surjective,
and since both $X_0$ and $X$ are regular, the map
$\operatorname{Pic}(X_0) \to \operatorname{Pic}(X)$ is surjective.
Since $\operatorname{Pic}(X_0)$ is finitely generated (by Roquette's theorem),
so is $\operatorname{Pic}(X)$. \qed
\end{proof}
The following examples show, in contrast, that the torsion
subgroup of $\operatorname{Pic}(X)$ need not be finite if $X$ is not normal.
\begin{example}
Let $B = \Bbb Q[x,x^{-1}]$, and put $A = \Bbb Q+(x-1)^2B$. Then $A$ is a
one-dimensional domain, finitely generated as a $\Bbb Q$-algebra, and
$\operatorname{Pic}(A) \cong \Bbb Q/\Bbb Z$. In particular, $\operatorname{Pic}(A)$ is an unbounded torsion group.
\end{example}
\begin{proof}
We note that $I:= (x-1)^2B$ is the conductor of $B$ into $A$. Therefore by
Milnor's Mayer-Vietoris exact sequence \cite{Bas1, Chap. IX, (5.3)}
[or see ($\diamondsuit$, p.\ \pageref{diamond}) for the scheme-theoretic
version], we have $\operatorname{Pic}(A) \cong (B/I)^*/((A/I)^*U)$,
where $U$ is the image of $B^*$ in $(B/I)^*$. But $(A/I)^* =
\Bbb Q^*$, so $\operatorname{Pic}(A) \cong (B/I)^*/U$. Now
$$B^* = \{sx^j:s\in\Bbb Q^*,j\in \Bbb Z\}\cong \Bbb Q^*\oplus \Bbb Z,$$%
and $(B/I)^* \cong \Bbb Q^*\oplus W$, where $W =\{1+s(x-1):s\in\Bbb Q\}
\cong \Bbb Q$. By keeping track of these identifications, one
easily gets $\operatorname{Pic}(A) \cong \Bbb Q/\Bbb Z$. \qed
\end{proof}
By a slight modification we get an example of finite type over $\Bbb Z$:
\begin{example}
Fix a positive integer $m$, put
$B = \Bbb Z[x,x^{-1},\frac{1}{m}]$, and let $A = \Bbb Z[\frac{1}{m}]+(x-1)^2B$.
Then $A$ is a two-dimensional domain finitely generated as a $\Bbb Z$-algebra, and
$\operatorname{Pic}(A) \cong \bigoplus_{p|m}\Bbb Z_{p^{\infty}}$.
\end{example}
The pathology in the examples above stems from the fact that $B/I$ is not
reduced.
Before stating our main finiteness theorems
[\pref{seminormal-S2} and \pref{absolutely-p}] we record the following result
from \cite{CGW, (7.4)}:
\begin{theorem}\label{CGW}
Let $k$ be an absolutely finitely generated\ field and let $\Lambda$ be a finite-dimensional
reduced $k$-algebra. Let $E_1$ and $E_2$ be intermediate
subalgebras of $\Lambda/k$.
\begin{enumerate}
\item If $k$ has positive characteristic $p$, then $\Lambda^*/E_1^*E_2^*$
is a direct sum of a countably generated free abelian group, a
finite group, and a bounded $p$-group.
\item If $\Lambda/k$ is separable, then $\Lambda^*/E_1^*E_2^*$ is
a direct sum of a countably generated free abelian group and a finite group.
\end{enumerate}
\end{theorem}
\begin{theorem}\label{seminormal-S2}
Let $k$ be a field finitely generated over $\Bbb Q$ and let $X$ be a reduced
$k$-scheme of finite type
which is seminormal and $S_2$. Then $\operatorname{Pic}(X)$ is isomorphic to the direct sum
of a free abelian group and a finite abelian group.
\end{theorem}
\begin{proof}
Let $\cal I$ be the conductor of $X_{\op{nor}}$ into $X$
(see \S\ref{eventual-section}, Step 2).
Let $X/\cal I := \hbox{\bf Spec}(\cal O_X/\cal I)$ and
$X_{\op{nor}}/\cal I := (X/\cal I) \times_X X_{\op{nor}}$ denote the
corresponding closed subschemes. Since $X$ is seminormal, it follows
\cite{T, (1.3)} that $X/\cal I$ is reduced.
Let $Q$ be the non-normal locus of $X/\cal I$, which has codimension $\geq 2$
in $X$. By \pref{S2}, the canonical map $\operatorname{Pic}(X) \to \operatorname{Pic}(X-Q)$ is injective.
Therefore we may replace $X$ by $X-Q$ and start the proof over, with the
added assumption that $X/\cal I$ is normal.
Let $D$ be the kernel of the map
$\phi: \operatorname{Pic}(X) \to \operatorname{Pic}(X/\cal I) \times \operatorname{Pic}(X_{\op{nor}})$.
By (\ref{normal-over}), the target of this morphism is finitely generated.
Let $\Lambda$ be the integral closure of $k$ in $\Gamma(X_{\op{nor}}/\cal I)$,
$E_1$ the integral closure of $k$ in
$\Gamma(X/\cal I)$, and $E_2$ the image in $\Lambda$ of the
integral closure of $k$ in $\Gamma(X_{\op{nor}})$. Using \pref{roquette}
and the exact sequence ($\diamondsuit$, p.\ \pageref{diamond}) with $L = k$,
we see that there is an exact sequence
$$\hbox{finitely generated} \to \Lambda^*/E_1^*E_2^* \to D
\to \hbox{finitely generated}.$$%
By (\ref{CGW})(2), $\Lambda^*/E_1^*E_2^*$ is free $\oplus$ finite. It follows
that
$D$ and thence $\operatorname{Pic}(X)$ is free $\oplus$ finite. \qed
\end{proof}
\begin{theorem}\label{absolutely-p}
Let $k$ be an absolutely finitely generated
field of positive characteristic $p$, and let $X$ be a $k$-scheme
of finite type. Then $\operatorname{Pic}(X)$ has the form
$$\hbox{(countably generated free abelian group)} \oplus
\hbox{(bounded $p$-group)} \oplus \hbox{(finite group)}.$$
\end{theorem}
\begin{proof}
Let $\cal C$ be the class of abelian groups having the form ascribed to
$\operatorname{Pic}(X)$ in the theorem. We leave to the reader to verify that $\cal C$
is closed under formation of subgroups and extensions.
Induct on $\operatorname{dim}(X)$; the case where $\operatorname{dim}(X) = 0$ is trivial. We will reduce
to the case where $X$ is reduced. For this, it is enough to show $(*)$ that
if $\cal J \subset \cal O_X$ is a square-zero ideal,
$X_0 = \hbox{\bf Spec}(\cal O_X/\cal J)$, and $\operatorname{Pic}(X_0) \in \cal C$, then
$\operatorname{Pic}(X) \in \cal C$. The standard exact sequence of sheaves on $X$
$$\begin{CD}
0 @>>> \cal J @>{a\kern2pt \mapsto 1+a}>> \cal O_X^* @>>> (\cal O_X)/\cal J @>>> 1
\end{CD}$$
yields on taking cohomology an exact sequence
$$H^1(X,\cal J) \to H^1(X,\cal O_X^*) \to H^1(X,(\cal O_X/\cal J)^*),$$%
from which $(*)$ follows, since $H^1(X,\cal J)$ is an $\Bbb F_p$-vector space.
Therefore we may assume that $X$ is reduced.
Let $\cal I, D, \phi$, etc.\ be as in the proof of \pref{seminormal-S2}.
(Here we do not know that $X/\cal I$ is normal.) By induction and
\pref{normal-over}, the target of $\phi$ is in $\cal C$, and therefore
$\operatorname{Im}(\phi)$ is also. Since $\cal C$ is closed under extensions, it suffices
to show that $D \in \cal C$. Arguing as in the proof of
\pref{seminormal-S2}, with \pref{CGW}(2) replaced by \pref{CGW}(1), we see that
this is the case. \qed
\end{proof}
\begin{remark}
For $k = \Bbb F_p$, it was shown in \cite{J1, (10.11)} that $\operatorname{Pic}(X)$ has the
form
$$\left( \oplus_{n=1}^\infty F \right) \oplus \hbox{(finitely generated\ abelian group)},$$
where $F$ is a finite $p$-group.
\end{remark}
\section{Complements on $K_0(X)$}
Let $k$ be a field, and let $X$ be a $k$-scheme of finite type.
Let $K_0(X)$ denote the Grothendieck group of vector bundles on $X$.
In this section, which is purely expository, we consider the analog for
$K_0(X)$ of the absolute finiteness results for $\operatorname{Pic}(X)$ proved in sections
\ref{torsion-section} and \ref{abs-section}. Consider the following table:
\vspace{0.15in}
\vspace{0.05in}
\begin{verbatim}
k alg. k abs. f.g. k finite k abs f.g.
closed of char. p>0 of char. 0
X arbitrary [1] finite [3] finite [4] f.g.
n-torsion for torsion mod mod p^n-torsion
all n p^n-torsion, for some n
invertible in k for some n
X regular [2] finite
n-torsion [5] f i n i t e l y g e n e r a t e d
for all n
\end{verbatim}
If we view this as a collection of statements about $\operatorname{Pic}(X)$, then all
five statements are true\footnote{For statement \circno2, we have used
resolution of singularities.}, as we have seen in the preceeding sections.
{\bf From now on, regard the table as a table of five conjectures about
$K_0(X)$.} The numbering of these conjectures is not related to the numbering
of results in the introduction.
It would be surprising if all five of these conjectures held. There is no
field over which any of them are known to hold, even if one restricts
attention to smooth projective or smooth affine schemes $X$. Conjecture
\circno5\ would follow from a conjecture of Bass \cite{Bas2, \S9.1}
to the effect that $K_i(X)$ is finitely generated\ for
all $i \geq 0$ and all regular schemes $X$ which are of finite type over $\Bbb Z$.
Over an arbitrary field $k$, it is not clear what sort of finiteness statement
might hold for $K_0(X)$: there are examples of infinite $n$-torsion in the
Chow groups of smooth projective varieties over a field of characteristic
zero \cite{KM}.
If $X$ is smooth of dimension $n$, then the operation of taking Chern classes
defines a homomorphism of graded rings \mapx[[ \operatorname{Gr}[K_0(X)] || \operatorname{CH}^*(X) ]],
which becomes an isomorphism after tensoring by $\Bbb Z[1 / (n-1)!]$.
(See \cite{F, (15.3.6)}.) It follows that for any almost
any question about $K_0(X)$, there is a parallel question about the groups
$\operatorname{CH}^q(X)$, $q = 1, \ldots, n$. Note that when $q = 1$, $\operatorname{CH}^1(X) = \operatorname{Pic}(X)$.
For the remainder of this section, suppose that $X$ is smooth, projective,
and of dimension $n$; we give a partial discussion of results and conjectures
about $\operatorname{CH}^q(X)$, for $q \geq 2$.
First suppose that $k$ is algebraically closed. Some things are known when
$q \in \setof{2,n}$: (i) The group ${}_m \operatorname{CH}^2(X)$ is finite if $m$ is
invertible in $k$ \cite{Ra, (3.1)}; (ii) the group
${}_m \operatorname{CH}^n(X)$ is finite for every $m$. This follows from Roitman's
theorem (See e.g.{\ } \cite{Ra, (3.2)}.) Hence
\circno2\ holds for smooth projective surfaces.
Now suppose that $k$ is a number field. Bloch has conjectured that
$\operatorname{CH}^q(X)$ is finitely generated\ for
all $q$. All results in this direction assume at least that $H^2(X,\cal O_X) = 0$.
With this hypothesis, it has been shown that the torsion subgroup of $\operatorname{CH}^2(X)$
is finite \cite{CR}. Moreover, if $X$ is a surface which is not
of general type, it is known \cite{Sal}, \cite{CR} that $\operatorname{CH}^2(X)$ is finitely generated.
Hence conjecture \circno5\ holds for a smooth projective surface over a
number field which is not of general type.
Finally, suppose that $k$ is a finite field. Again, it is conjectured that
$\operatorname{CH}^q(X)$ is finitely generated\ for all $q$. What is known is that $\operatorname{CH}^n(X)$ is
finitely generated\ (in fact $\operatorname{CH}_0(X)$ is finitely generated\ for any scheme $X$ of finite type over $\Bbb Z$
\cite{KS2}), and that the torsion subgroup of $\operatorname{CH}^2(X)$ is finite \cite{CSS},
\ cf.{\ }\cite{CR, (3.7)}. The first assertion implies that conjecture
\circno5\ holds for any smooth projective surface over a finite field.
|
1994-10-28T05:20:20 | 9410 | alg-geom/9410029 | en | https://arxiv.org/abs/alg-geom/9410029 | [
"alg-geom",
"math.AG"
] | alg-geom/9410029 | Marco Andreatta | M. Andreatta, M. Mella | Contractions on a manifold polarized by an ample vector bundle | 18 pages, LateX | null | null | null | null | A complex manifold $X$ of dimension $n$ together with an ample vector bundle
$E$ on it will be called a {\sf generalized polarized variety}. The adjoint
bundle of the pair $(X,E)$ is the line bundle $K_X + det(E)$. We study the
positivity (the nefness or ampleness) of the adjoint bundle in the case $r :=
rank (E) = (n-2)$. If $r\geq (n-1)$ this was previously done in a series of
paper by Ye-Zhang, Fujita, Andreatta-Ballico-Wisniewski.
If $K_X+detE$ is nef, then by the Kawamata-Shokurov base point free theorem,
it supports a contraction; i.e. a map $\pi :X \longrightarrow W$ from $X$ onto
a normal projective variety $W$ with connected fiber and such that $K_X +
det(E) = \pi^*H$, for some ample line bundle $H$ on $W$. We describe those
contractions for which $dimF \leq (r-1)$. We extend this result to the case in
which $X$ has log terminal singualarities. In particular this gives the Mukai's
conjecture1 for singular varieties. We consider also the case in which $dimF =
r$ for every fibers and $\pi$ is birational. Hard copies of the paper are
available.
| [
{
"version": "v1",
"created": "Thu, 27 Oct 1994 15:54:28 GMT"
}
] | 2015-06-30T00:00:00 | [
[
"Andreatta",
"M.",
""
],
[
"Mella",
"M.",
""
]
] | alg-geom | \section*{Introduction}
An algebraic variety $X$ of dimension $n$
(over the complex field) together with an ample vector bundle $E$
on it will be called a {\sf generalized polarized variety}.
The adjoint bundle of the pair $(X,E)$
is the line bundle $K_X + det(E)$.
Problems concerning adjoint bundles have drawn a lot of attention
to algebraic geometer: the classical case is when $E$
is a (direct sum of) line bundle (polarized variety), while the generalized
case was
motivated by the solution of Hartshorne-Frankel conjecture by Mori (
\cite{Mo})
and by consequent conjectures of Mukai (\cite{Mu}).
\par
A first point of view is to study the positivity (the nefness or
ampleness) of the adjoint line bundle in the case $r = rank (E)$
is about $n = dim X$.
This was done in a sequel of papers for $r\geq (n-1)$
and for smooth manifold $X$ ([Ye-Zhang], [Fujita],
[Andreatta-Ballico-Wisniewski]).
In this paper we want to discuss the next case, namely when $rank (E) = (n-2)$,
with $X$ smooth; we obtain a complete answer which is described in the
theorem (4.1). This is divided in three cases, namely when
$K_X + det(E)$ is not nef, when it is nef and not big and
finally when it is nef and big but not ample.
If $n=3$ a complete picture is already contained in the famous paper
of Mori (\cite{Mo1}), while the particular case in which
$E = \oplus^{(n-2)} (L)$
with $L$ a line bundle was also studied (\cite{Fu1}, \cite{So};
in the singular case see \cite{An}). The part 1 of the theorem was proved
(in a slightly weaker form) by Zhang (\cite{Zh}) and, in the case $E$ is
spanned
by global sections, by Wisniewski (\cite{Wi2}).
\par
Another point of view can be the following: let $(X,E)$ be
a generalized polarized variety with $X$ smooth and $rankE=r$.
If $K_X + det(E)$ is nef, then by the Kawamata-Shokurov
base point free theorem it supports a contraction (see (1.2));
i.e. there exists a map $\pi :X \rightarrow W$ from $X$ onto a
normal projective variety $W$ with connected fiber and such that
$K_X + det(E) = \pi^*H$ for some ample line bundle $H$ on $W$.
It is not difficult to see that, for every fiber $F$ of
$\pi$, we have $dimF \geq (r-1)$, equality holds only if
$dimX > dimW$. In the paper we study the "border" cases:
we assume that $dimF = (r-1)$ for every fibers
and we prove that $X$ has a ${\bf P}^r$-bundle
structure given by $\pi$ (theorem (3.2)). We consider also the case
in which $dimF = r$ for every fibers and $\pi$ is birational,
proving that $W$ is smooth and that $\pi$ is a blow-up of a
smooth subvariety (theorem (3.1)).
This point of view was discussed in the case $E = \oplus^r L$
in the paper [A-W].
\par
Finally in the section (4) we extend the theorem (3.2) to the
singular case, namely for projective variety $X$
with log-terminal singularities. In particular this gives
the Mukai's conjecture1 for singular varieties.
\bigskip
\section{Notations and generalities}
\addtocounter{subsection}{1
We use the standard notations from algebraic geometry.
Our language is compatible with that of [K-M-M] to which we refer constantly.
We just explain some special definitions
and propositions used frequently.
In particular in this paper $X$ will always stand
for a smooth complex projective variety
of dimension $n$. Let
$Div(X)$ the group of Cartier divisors on $X$;
denote by $K_X$ the {\sf canonical divisor} of $X$, an element of
$Div(X)$ such that ${\cal O}_{X}(K_X) = \Omega^n_{X}$.
Let $N_1(X)=\frac{\{1-cycles\}}{\equiv}\otimes {\bf R}$,
$N^1(X)= \frac{\{divisors\}}{\equiv}\otimes {\bf R}$ and
$\overline {<NE(X)>}=\overline{\{\mbox{effective 1-cycles}\}}$;
the last is a closed
cone in $N_1(X)$. Let also
$\rho(X)=dim_{{\bf R}}N^1(X)<\infty$.
\medskip
Suppose that $K_X$ is not nef, that is there exists an effective curve $C$
such that $K_X\cdot C<0$.
\begin{Theorem}\cite{KMM}
Let $X$ as above and $H$ a nef Cartier divisor such that
$F:= H^{\bot} \cap \overline {<NE(X)>} \setminus \{0\}$
is entirely contained in the set
$\{Z\in N_1(X) :K_X\cdot Z<0\}$,
where $H^{\bot} = \{Z:H\cdot Z=0\}$.
Then there exists a projective morphism $\varphi:X\rightarrow W$ from $X$ onto a normal
variety $W$ with the following properties:
\begin{itemize}
\item[{i})] For an irreducible curve $C$ in $X$, $\varphi(C)$
is a point if and only if $H.C = 0$, if and only if
$cl(C) \in F$.
\item[{ii})] $\varphi$ has only connected fibers
\item[{iii})] $H = \varphi^*(A)$ for some ample divisor $A$
on $W$.
\item[{iv})] The image $\varphi^* :Pic(W) \rightarrow Pic(X)$ coincides with
$\{D \in Pic(X): D.C = 0 \mbox{ \rm for all } C \in F\}.$
\end{itemize}
\label{contractionth}
\end{Theorem}
\begin{Definition} The following terminology is mostly used (\cite{KMM},
definition 3-2-3).
Referring to the above theorem,
the map $\varphi$ is called a {\sf contraction}
(or an
{\sf extremal contraction}); the set $F$ is an {\sf extremal face},
while the Cartier
divisor $H$ is a {\sf supporting divisor} for the map $\varphi$ (or the face $F$).
If $dim_{{\bf R}}F = 1$ the face $F$ is called an {\sf extremal ray}, while $\varphi$
is called an {\sf elementary contraction}.
\end{Definition}
\begin{remark} We have also (\cite{Mo1}) that if $X$ has an extremal ray $R$
then there exists a rational curve $C$ on $X$ such that
$0< -K_X \cdot C\leq n+1$ and
$R=R[C]:=\{D\in <NE(X)>: D\equiv \lambda C, \lambda\in {\bf R}^+\}$.
Such a curve is called an {\sf extremal curve}.
\end{remark}
\begin{remark}\label{biraz} Let $\pi:X\rightarrow V$ denote a contraction of an
extremal
face $F$, supported by $H=\pi^*A$([iii]\ref{contractionth}) . Let
$R$ be an extremal ray in $F$ and $\rho:X\rightarrow W$ the contraction of $R$.
Since $\pi^*A\cdot R=0$, $\pi^*A$ comes from $Pic (W)$
([iv]\ref{contractionth}). Thus $\pi$ factors trough $\rho$.
\end{remark}
\begin{Definition} To an extremal ray $R$ we can associate:
\begin{itemize}
\item[{i})] its {\sf length} $l(R):=min\{ -K_X\cdot C;$
for $C$ rational curve and $C\in R\}$
\item[{ii})] the {\sf locus} $E(R):=\{$the locus
of the curves whose numerical classes are in $R\}\subset X$.
\end{itemize}
\end{Definition}
\begin{Definition} It is usual to divide the elementary contractions
associated to an
extremal ray $R$ in three types according to the dimension of $E(R)$:
more precisely we say that $\varphi$ is of {\sf fiber type}, respectively
{\sf divisorial type}, resp. {\sf flipping type}, if
$dim E(R) = n$, resp. $n-1$, resp. $< n-1$.
Moreover an extremal ray is said not nef if
there exists an effective $D\in Div(X)$ such that $D\cdot C<0$.
\end{Definition}
The following very useful inequality was proved in \cite{Io} and \cite{Wi3}.
\begin{Proposition} Let $\varphi$ the contraction of an extremal ray
$R$, $E^{\prime}(R)$ be any irreducible component of the exceptional
locus and $d$ the dimension of a fiber of the contraction
restricted to $E^{\prime}(R)$. Then
$$ dim E^{\prime}(R)+d\geq n+l(R)-1.$$
\label{diswis}
\end{Proposition}
\addtocounter{subsection}{1 Actually it is very useful to understand when a contraction is
elementary
or in other words when the locus of two distinct extremal rays are
disjoint. For this we will use in this paper the following results.
\begin{Proposition}\cite[Corollary 0.6.1]{BS} Let $R_1$ and $R_2$ two distinct
not nef
extremal rays such that $l(R_1)+l(R_2)>n$.
Then $E(R_1)$ and $E(R_2)$ are disjoint.
\label{birelementare}
\end{Proposition}
Something can be said also if $l(R_1)+l(R_2)=n$:
\begin{Proposition}\cite[Theorem 2.4]{Fu3}
Let $\pi:X\rightarrow V$ as above and suppose $n\geq 4$ and $l(R_i)\geq n-2$.
Then the exceptional loci corresponding to different extremal rays,
are disjoint with each other.
\label{n=4}
\end{Proposition}
\begin{Proposition}\cite{ABW1} Let $\pi:X\rightarrow W$ be a contraction of
a face such that $dimX > dim W$. Suppose that for every rational curve $C$
in a general fiber of $\pi$ we have
$-K_X\cdot C\geq (n+1)/2$
Then $\pi$ is an elementary contraction except if
\begin{itemize}
\item[a)] $-K_X\cdot C=(n+2)/2$ for some rational curve $C$ on $X$,
$W$ is a point, $X$ is a Fano manifold
of pseudoindex $(n+2)/2$ and $\rho(X)=2$
\item[b)] $-K_X\cdot C=(n+1)/2$
for some rational curve $C$, and $dim$W$\leq 1$
\end{itemize}
\label{fibelementare}
\end{Proposition}
The following definition is used in the theorem:
\begin{Definition} Let $L$ be an an
ample line bundle on $X$. The pair $(X,L)$ is called a scroll (respectively
a quadric fibration, respectively a del Pezzo fibration)
over a normal variety
$Y$ of dimension $m$ if there exists a surjective morphism
with connected fibers
$\phi: X \rightarrow Y$ such that
$$K_X+(n-m+1)L \approx p^*{\cal L}$$
(respectively $K_X+(n-m)L \approx p^*{\cal L}$;
respectively $K_X+(n-m-1)L \approx p^*{\cal L}$)
for some ample line bundle ${\cal L}$ on $Y$.
$X$ is called a classical scroll (respectively
quadric bundle) over a projective
variety $Y$ of dimension $r$ if there exists a surjective morphism
$\phi : X\rightarrow Y$ such that every fiber is isomorphic to ${\bf P}^{n-r}$
(respectively to a quadric in ${\bf P}^{(n-r+1)}$)
and if there exists a vector bundle $E$ of rank $(n-r+1)$ (respectively
of rank $n-r+2$) on $Y$ such that $X\simeq {\bf P}(E)$
(respectively exists an embedding of $X$ as a subvariety of ${\bf P}(E)$).
\end{Definition}
\section{A technical construction}
\label{tech}
Let $E$ be a vector bundle of rank $r$ on $X$ and assume
that $E$ is ample, in the sense of Hartshorne.
\begin{remark} Let $f:{\bf P}^1\rightarrow X$
be a non constant map, and $C=f({\bf P}^1)$,
\label{mha}
then $detE\cdot C\geq r$.
\par
In particular if there exists a curve $C$ such that
$(K_X+detE).C \leq 0$ (for instance if $(K_X+detE)$
is not nef) then there exists
an extremal ray $R$ such that $l(R) \geq r$.
\end{remark}
\addtocounter{subsection}{1
\label{sopra}
Let $Y={\bf P}(E)$ be the associated projective space bundle,
$p:Y \rightarrow X$ the natural map onto $X$ and
${\xi}_E$ the tautological bundle of $Y$.
Then we have the formula for the canonical bundle
$K_Y=p^*(K_X+detE)-r{\xi}_E$.
Note that $p$ is an elementary contraction; let $R$ be the associated extremal
ray.
Assume that $K_X+detE$ is nef but not ample and that it is
the supporting divisor of an elementary contraction $\pi:X\rightarrow W$.
Then $\rho(Y/W) = 2$ and $-K_Y$ is $\pi\circ p$-ample.
By the relative Mori theory
over $W$ we have that there exists a ray on $NE(Y/W)$, say
$R_1$, of length $\geq r$, not contracted by $p$, and a relative
elementary contraction
$\varphi:Y\rightarrow V$. We have thus the following commutative diagram.
\begin{equation}
\label{dia1}
\matrix{{\bf P}(E)=Y&\mapright\varphi&V\cr
\mapdown{p}&&\mapdown\psi\cr X&\mapright\pi&W}
\end{equation}
where $\varphi$ and $\psi$ are elementary contractions.
Let $w\in W$ and let $F(\pi)_w$ be an irreducible component of $\pi^{-1}(w)$;
choose
also $v$ in $\psi^{-1}(w)$ and let $F(\varphi)_v$ be an irreducible
component of $\varphi ^{-1}(v)$
such that $p(F(\varphi)_v) \cap F(\pi)_w \not= \emptyset$; then
$p(F(\varphi)_v) \subset F(\pi)_w$. This is true by the commutativity of the
diagram.
Since $p$ and $\varphi$ are elementary contractions of different extremal
rays we have that $dim(F(\varphi)\cap F(p))=0$,
that is curve contracted by $\varphi$ cannot be contracted by $p$.
In particular this implies that $dim p(F(\varphi)_v) = dim F(\varphi)_v$;
therefore
$$dimF(\varphi)_v\leq dimF(\pi)_w.$$
\begin{remark} If $dimF(\varphi)_v=dimF(\pi)_w$,
then $dim F(\psi)_w:=dim(\psi^{-1}(w)) = r-1$;
if this holds for every $w \in W$ then $\psi$ is equidimensional.
\end{remark}
\noindent{\bf Proof. } Let $Y_w$ be an irreducible component of
$p^{-1}\pi^{-1}(w)$ such that $\varphi (Y_w) = F(\psi)_w$.
Then $dim F(\psi)_w = dim Y_w - dimF(\varphi)_v = dim Y_w -
dimF(\pi)_w = dimF(p) = (r-1)$.
\par\hfill $\Box$\par
\addtocounter{subsection}{1 {\bf Slicing techniques}
\label{adj}
Let $H = \varphi ^*(A)$ be a supporting divisor for $\varphi$ such that
the linear system $|H|$ is base point free.
We assume as in (\ref{sopra}) that $( K_X + detE)$ is nef
and we refer to the diagram (\ref{dia1}). The divisor
$K_Y+r{\xi}_E =p^*( K_X + detE)$ is nef on $Y$ and therefore
$m(K_Y+r{\xi}_E+aH)$, for $m\gg 0$, $a\in{\bf N}$,
is also a good supporting divisor for $\varphi$.
Let $Z$ be a smooth n-fold obtained by intersecting $r-1$ general divisor
from the linear system H, i.e. $Z = H_1\cap \dots \cap H_{r-1}$ (this is what
we call a {\sf slicing});
let $H_i = \varphi^{-1} A_i$.
Note that the map ${\varphi}^{\prime}=\varphi_{|Z}$
is supported by $m|(K_Y+r{\xi}_E+a\varphi^*A)_{|Z}|$,
hence, by adjunction, it is supported by $K_Z+rL$, where $L={{\xi}_E}_{|Z}$.
Let $p^{\prime}=p_{|Z}$; by construction $p^{\prime}$ is finite.
If $T$ is (the normalization of) $\varphi (Z)$ and
$\psi^{\prime} :T \rightarrow W$ is the map obtained restricting $\psi$
then we have from (\ref{dia1}) the following diagram
\begin{equation}
\label{dia2}
\matrix{Z&\mapright{\varphi\prime}&T\cr
\mapdown{p\prime}&&\mapdown{\psi\prime}\cr X&\mapright\pi&W}
\end{equation}
In general one has a good comprehension of the map $\varphi^{\prime}$
(for instance in the case $r = (n-2)$ see the results in \cite{Fu1} or in
\cite{An}). The goal is to "transfer" the information that we have on
$\varphi^{\prime}$ to the map $\pi$. The following proposition
is the major step in this program.
\begin{Proposition} Assume that
$\psi$ is equidimensional (in particular this is the case if
for every non trivial fiber we have
$dimF(\varphi)=dimF(\pi)$). Then $W$ has the same singularities of $T$.
\label{fujita}
\end{Proposition}
\begin{proof} By hypothesis any irreducible reduced component $F_i$ of a non
trivial
fiber $F(\psi)$
is of dimension $r-1$; this implies also that
$F_i=\varphi(F(p))$ for some fiber of $p$.
Now, let us follow an argument as in
\cite[Lemma 2.12]{Fu1}.
We can assume that the divisor $A$ is very ample; we will choose
$r-1$ divisors $A_i \in |A|$ as above such that, if
$T = {\bigcap_{i}} A_i$, then $T \cap \psi^{-1}(w)_{red}= N$
is a reduced 0-cycle and $Z = H_1\cap \dots \cap H_{r-1}$ is a
smooth n-fold, where $H_i = \varphi^{-1} A_i$. This can be done by
Bertini theorem. Moreover the number of points in $N$ is given by
$A^{r-1}\cdot \psi^{-1}(w)_{red}=\sum_i A^{r-1}\cdot F_i=\sum_i d_i$.
Note that, by projection formula, we have
$A^{r-1}\cdot F_i= \varphi^*A^{r-1}\cdot F(p)$;
moreover, since $p$ is a projective bundle, the last number is constant
i.e. $\varphi^*A^{r-1}\cdot F(p) = d$ for all fiber $F(p)$,
that is the $d_i$'s are constant.
Now take a small enough neighborhood $U$ of $w$, in the metric topology,
such that any
connected component $U_{\lambda}$ of $\psi^{-1}(U)\cap T$
meets $\psi^{-1}(w)$ in a single point. This is
possible because $\psi^{\prime}:=\psi_{|T}: T \rightarrow W$
is proper and finite over $w$.
Let $\psi_{\lambda}$ the restriction of $\psi$ at
$U_{\lambda}$ and $m_{\lambda}$ its degree.
Then $deg{\psi}^{\prime}=\sum m_{\lambda}\geq \sum_i d_i = \sum_i d$
and equality holds if and only if $\psi$ is not ramified at $w$
(remember that $\sum_i d_i$ is the number of $U_{\lambda}$).
The generic $F(\psi)_w$ is irreducible and generically
reduced.
Note that we can choose
$\tilde{w}\in W$ such that $\psi^{-1}(\tilde{w})=
\varphi(F(p))$ and
$deg{\psi}^{\prime}=A^{r-1}\cdot\psi^{-1}(\tilde{w})$, the latter is
possible by the choice of generic sections of $|A|$.
Hence, by projection formula
$deg\psi^{\prime}= A^{r-1}\cdot \psi^{-1}(\tilde{w})=
\varphi^*A^{r-1}\cdot F(p)=d$,
that is $m_{\lambda}=1$ and the fibers are irreducible.
Since $W$ is normal we can conclude, by Zarisky's Main theorem, that $W$ has
the
same singularity as $T$.
\par\hfill $\Box$\par
\section{Some general applications}
As an application of the above construction we will prove the following
proposition; the case $r = (n-1)$ was proved in
\cite {ABW2}.
\begin{Proposition} Let $X$ be a smooth projective complex variety and $E$ be
an ample vector bundle of rank $r$ on $X$. Assume that $K_X+detE$ is nef and
big but
not ample and let $\pi:X\rightarrow W$ be the
contraction supported by $K_X+detE$.
Assume also that $\pi$ is a divisorial elementary contraction,
with exceptional divisor $D$,
and that $dim F\leq r$ for all fibers $F$.
Then $W$ is smooth, $\pi$ is the blow up of a
smooth subvariety $B: = \pi (D)$ and
$E =\pi^*E^{\prime}\otimes[-D]$, for
some ample $E^{\prime}$ on $W$. \label{bd}
\end{Proposition}
\noindent{\bf Proof. } Let $R$ be the extremal ray contracted by $\pi$ and $F:=F(\pi)$ a fiber.
Then $l(R)\geq r$ and thus $dimF\geq r$ by proposition (\ref{diswis}).
Hence all the fibers of $\pi$ have dimension $r$.
Consider the commutative diagram (\ref{dia1}); let $R_1$ be
the ray contracted by $\varphi$. Since $l(R_1)\geq r$, again
by proposition (\ref{diswis}), we have that $dimF(\varphi) \geq r$ (note that
$R_1$ is not nef). Therefore,
since $dimF(\varphi) \leq dimF$, we have that
$dimF(\varphi) = dimF = r$, $l(R)=l(R_1)=r$ and ${\xi}_E\cdot C_1=1$, where $C_1$ is
a (minimal) curve in the ray $R_1$.
Via slicing we obtain the map $\varphi^{\prime}:Z\rightarrow T$ which is supported by
$K_Z+r{\xi}_E{}_{|Z}$. This last map is very well understood:
namely by \cite[Th 4.1 (iii)]{AW} it follows that $T$ is smooth and
$\varphi^{\prime}$ is a blow up along a smooth subvariety.
By proposition (\ref{fujita}) also $W$ is smooth.
Therefore $\pi$ is a birational morphism between smooth varieties
with exceptional locus a prime divisor and with equidimensional
non trivial fibers; by \cite[Corollary 4.11]{AW} this implies that
$\pi$ is a blow up of a smooth subvariety in $W$.
We want to show that $E =\pi^*E^{\prime}\otimes[-D]$.
Let $D_1$ be the exceptional divisor of $\varphi$; first we claim that
${\xi}_E+D_1$ is a good supporting divisor for $\varphi$.
To see this observe that $({\xi}_E+D_1)\cdot C_1=0$, while $({\xi}_E+D_1)\cdot C>0$
for any curve $C$ with $\varphi(C)\not= pt$ (in fact ${\xi}_E$ is ample and $D_1\cdot
C\geq 0$ for such a curve).
Thus ${\xi}_E+D_1=\varphi^*A$ for some ample $A\in Pic(V)$; moreover by
projection formula $A\cdot l=1$, for any line $l$ in the fiber of $\psi$.
Hence by Grauert theorem $V={\bf P}(E^{\prime})$ for some ample vector bundle
$E^{\prime}$ on $W$. This yields, by the commutativity of diagram (1), to
$E\otimes D=p_*({\xi}_E+D_1)=p_*\varphi^*A=\pi^*\psi_*A=\pi^*E^{\prime}$.
\par\hfill $\Box$\par
We now want to give a similar proposition for the fiber type case.
\begin{Theorem} Let $X$ be a smooth projective complex variety and $E$
be an ample vector bundle of rank $r$ on $X$. Assume that $K_X+detE$
is nef and let $\pi:X\rightarrow W$ be the
contraction supported by $K_X+detE$.
Assume that $r\geq (n+1)/2$ and
$dim F\leq r-1$ for any fiber $F$ of $\pi$. Then $W$ is smooth,
for any fiber $F\simeq {\bf P}^{r-1}$ and $E_{|F}=\oplus^r{\cal O}(1)$.
\label{relscroll}
\end{Theorem}
\noindent{\bf Proof. } Note that by proposition (\ref{diswis}) $\pi$ is a contraction of
fiber type and all the fibers have dimension $r-1$.
Moreover the contraction is elementar, as it follows from
proposition (\ref{fibelementare}).
We want to use an inductive argument to prove the thesis.
If $dim W=0$ then this is Mukai's conjecture1; it was proved
by Peternell, Koll\'ar, Ye-Zhang (see for instance \cite{YZ}).
Let the claim be true for dimension $m-1$.
Note that the locus over which the fiber is not ${\bf P}^{r-1}$
is discrete and $W$ has isolated singularities.
In fact take a general hyperplane section $A$ of $W$,
and $X^{\prime}=\pi^{-1}(A)$ then $\pi_{|X^{\prime}}:X^{\prime}\rightarrow A$
is again a contraction supported by $K_{X^{\prime}}+det E_{|X^{\prime}}$,
such that $r\geq ((n-1)+1)/2$. Thus by induction $A$ is smooth, hence $W$
has isolated singularities.
\par
Let $U$ be an open disk in the complex topology, such that $U\cap SingW=\{0\}$.
Then by lemma below \ref{scroll} we have locally, in the complex topology, a
$\pi$-ample line bundle $L$ such that restricted to the general fiber is
${\cal O}(1)$. As in \cite[Prop. 2.12]{Fu1} we can prove
that $U$ is smooth and all the fibers are ${\bf P}^{r-1}$.
\par\hfill $\Box$\par
\begin{Lemma}
\label{scroll}
Let $X$ be a complex manifold and $(W,0)$ an analityc germ such that
$W\setminus \{0\}\simeq \Delta^m\setminus \{0\}$.
Assume we have an holomorphic map $\pi:X\rightarrow W$ with $-K_X$ $\pi$-ample;
assume also that $F\simeq {\bf P}^r$ for all fibers of $\pi$, $F\not=
F_0=\pi^{-1}(0)$,
and that $codim F_0\geq 2$.
Then there exists a line bundle $L$ on $X$ such that $L$ is $\pi$-ample and
$L_{|F}={\cal O}(1)$.
\end{Lemma}
\noindent{\bf Proof. } (see also \cite[pag 338, 339]{ABW2})
Let $W^*=W\setminus \{0\}$ and $X^*=X\setminus F_0$.
By abuse of notation call $\pi=\pi_{|X^*}:X^*\rightarrow W^*$;
it follows immediately that $R^1\pi_*{\bf Z}_{X^*}=0$ and $R^2\pi_*{\bf Z}_{X^*}={\bf Z}$.
If we look at Leray spectral sequence, we have that:
$$
E^{0,2}_2= {\bf Z}\mbox{ and } E^{p,1}_2= 0 \mbox{ for any p.}$$
Therefore $d_2:E^{0,2}_2\rightarrow E^{2,1}_2$ is the zero map
and moreover we have the following exact sequence
$$0\rightarrow E^{0,2}_{\infty}\rightarrow E^{0,2}_2\stackrel{d_3}{\rightarrow} E^{3,0}_2,$$
since the only non zero map from $ E^{0,2}_2$ is $d_3$ and hence
$E^{0,2}_{\infty}=kerd_3$.
On the other hand we have also, in a natural way, a surjective map
$H^2(X^*,{\bf Z})\rightarrow E^{0,2}_{\infty}\rightarrow 0$.
Thus we get the following exact sequence
$$ H^2(X^*,{\bf Z})\stackrel{\alpha}{\rightarrow} E^{0,2}_2\rightarrow
E^{3,0}_2=H^3(W^*,{\bf Z}).$$ We want to show that $\alpha$ is surjective.
If $dimW := w\geq 3$ then $H^3(W^*,{\bf Z})=0$ and we have done. Suppose $w=2$ then
$H^3(W^*,{\bf Z})={\bf Z}$; note
that the restriction of $-K_X$ gives a non zero class (in fact it is
$r+1$ times the generator) in $E^{0,2}_2$ and is mapped to zero in $E^{0,3}_2$
thus the mapping
$E^{0,2}_2\rightarrow E^{3,0}_2$ is the zero map and $\alpha$ is surjective.
Since $F_0$ is of
codimension at least 2 in $X$ the restriction map $H^2(X,{\bf Z})\rightarrow
H^2(X^*,{\bf Z})$ is a bijection. By the vanishing of $R_i\pi_*{\cal O}_X$ we get
$H^2(X,{\cal O}_X)=H^2(W,{\cal O}_W)=0$ hence also $Pic(X)\rightarrow H^2(X,{\bf Z})$ is surjective.
Let $L\in Pic(X)$ be a preimage of a generator of $E^{0,2}_2$.
By construction $L_t$ is ${\cal O}(1)$, for $t\in W^*$. Moreover
$(r+1)L=-K_X$ on $X^*$ thus, again by the codimension of $X^*$, this is true
on $X$ and $L$ is $\pi$-ample.
\par\hfill $\Box$\par
\section{An approach to the singular case}
The following theorem arose during a discussion between us
and J.A. Wisniewski; we would like to thank him.
The idea to investigate this argument came from a preprint of Zhang
[Zh2] where he proves the following result under
the assumption that $E$ is spanned by global sections.
For the definition of log-terminal
singularity we refer to \cite{KMM}.
\begin{Theorem} Let $X$ be an n-dimensional log-terminal projective
variety and $E$ an ample vector bundle of rank $n+1$, such that
$c_1(E)=c_1(X)$.
Then $(X,E)=({\bf P}^n,\oplus^{n+1}{\cal O}_{{\bf P}^n}(1))$.
\end{Theorem}
\noindent{\bf Proof. } We will prove that $X$ is smooth, then we can apply proposition
(\ref{relscroll}).
We consider also in this case the associated projective space bundle $Y$
and the commutative diagram
\begin{equation}
\matrix{{\bf P}(E)=Y&\mapright\varphi&V\cr
\mapdown{p}&&\mapdown\psi\cr X&\mapright\pi&pt}
\end{equation}
as in (\ref{dia1}); it is immediate that $Y$ is a weak Fano variety
(i.e. $Y$ is Gorenstein, log-terminal and $-K_Y$ is ample; in particular
it has Cohen-Macaulay singularities); moreover, as in (3.1),
$dimF(\varphi) \leq dimF(\pi) = n$ and the map $\varphi$ is supported by $K_Y+(n+1)H$,
where
$H ={\xi}_E + A$, with ${\xi}_E$ the tautological line bundle and $A$ a
pull back of a ample line bundle from $V$.
It is known that a contraction supported
by $K_Y+rH$ on a log terminal variety has to have fibers
of dimension $\geq (r-1)$ and of dimension
$\geq r$ in the birational case (\cite [remark 3.1.2]{AW}).
Therefore in our case
$\varphi$ can not be birational and all fibers have dimension $n$;
moreover, by the Kobayashi-Ochiai criterion the general fiber is
$F\simeq {\bf P}^n$.
We want to adapt the proof of \cite[Prop 1.4]{BS};
to this end we have only to show
that there are no fibers of $\varphi$ entirely contained in $Sing(Y)$. Note
that, by construction, $Sing(Y)\subset p^{-1}(Sing X)$
hence no fibers $F$ of $\varphi$
can be contained in $Sing(Y)$.
Hence the same proof of \cite[Prop 1.4]{BS} applies and we can prove that
$V$ is nonsingular and $\varphi:Y\rightarrow V$ is a classical scroll.
In particular $Y$ is nonsingular and therefore also $X$ is nonsingular.
\par\hfill $\Box$\par
More generally we can prove the following.
\begin{Theorem} Let $X$ be an n-dimensional log-terminal projective
variety and $E$ be an ample vector bundle of rank $r$. Assume that $K_X+det E$
is nef and let $\pi:X\rightarrow W$ be the contraction supported by $K_X+det E$.
Assume also that for any fiber $F$ of $\pi$ $dimF\leq r-1$, that $r\geq
(n+1)/2$
and $codim Sing(X)>dim W$.
Then $X$ is smooth and for any fiber $F\simeq {\bf P}^{r-1}$.
\end{Theorem}
\noindent{\bf Proof. } The proof that $X$ is smooth is as in the theorem above and then
we use proposition (\ref{relscroll})
\par\hfill $\Box$\par
\section{Main theorem}
This section is devoted to the proof of the following theorem.
\begin{Theorem} Let $X$ be a smooth projective variety over the complex
field of dimension $n \geq 3$ and $E$ an ample vector bundle on $X$
of rank $r= (n-2)$. Then we have
\begin{itemize}
\item[1)] $K_X + det(E)$ is nef unless $(X,E)$ is one of the following:
\begin{itemize}
\item[{i})] there exist a smooth $n$-fold, $W$, and
a morphism $\phi : X \rightarrow W$ expressing $X$ as a blow up of a
finite set $B$ of points and an ample vector bundle $E'$ on
$W$ such that $E = \phi^*E'\otimes[-\phi^{-1}(B)]$.
\par\noindent
Assume from now on that $(X,E)$ is not as in (i) above (that is eventually
consider the new pair $(W,E')$ coming from (i)).
\item[{ii})] $X = {\bf P}^n$ and $E =\oplus^{(n-2)}{\cal O}(1)$ or
$\oplus^{2}{\cal O}(2)\oplus^{(n-4)}{\cal O}(1)$ or
${\cal O}(2)\oplus^{(n-3)}{\cal O}(1)$ or
${\cal O}(3)\oplus^{(n-3)}{\cal O}(1)$.
\item[{iii})] $X = {\bf Q}^n$ and $E =\oplus^{(n-2)}{\cal O}(1)$ or
${\cal O}(2)\oplus^{(n-3)}{\cal O}(1)$ or ${\bf E}(2)$ with ${\bf E}$
a spinor bundle on ${\bf Q}^n$.
\item[{iv})] $X = {\bf P}^2 \times {\bf P}^2$ and $E = \oplus^2{\cal O}(1,1)$
\item[{v})] $X$ is a del Pezzo manifold with $b_2 = 1$, i.e.
$Pic(X)$ is generated by an ample line bundle ${\cal O}(1)$ such that
${\cal O}(n-1) = {\cal O}(-K_X)$ and $E = \oplus^{(n-1)}{\cal O}(1)$.
\item[{vi})] $X$ is a classical scroll or a quadric bundle over
a smooth curve $Y$.
\par
\item[{vii})] $X$ is a fibration over a smooth surface
$Y$ with all fibers isomorphic to ${\bf P}^{(n-2)}$.
\end{itemize}
\item[2)] If $K_X + det(E)$ is nef then it is big unless
there exists a morphism $\phi : X \rightarrow W$ onto a
normal variety $W$ supported by (a large multiple of) $K_X + det(E)$
and $dim(W) \leq 3$; let $F$ be a general fiber of $\phi$ and
$E^{\prime}=E_{|F}$.
We have the following according to
$s = dim W$:
\begin{itemize}
\item[{i})] If $s = 0$ then $X$ is a Fano manifold and
$K_X + det(E) = 0$. If $n\geq 6$ then $b_2(X) = 1$ except if
$X={\bf P}^3\times{\bf P}^3$ and $E=\oplus^4{\cal O}(1,1)$.
\item[{ii})] If $s = 1$ then $W$ is a smooth curve and $\phi$ is a
flat (equidimensional) map.
Then $(F,E')$ is one of the pair described in \cite{PSW},
in particular $F$ is either ${\bf P}^n$ or a quadric or a del Pezzo variety.
If $n \geq 6$ then $\pi$ is an elementary contraction.
If the general fiber is $P^{n-1}$ then $X$ is a classical scroll while
if the general fiber is ${\bf Q}^{n-1}$ then $X$ is a quadric bundle.
\item[{iii})] If $s = 2$ and $n \geq 5$ then $W$ is a smooth surface,
$\phi$ is a flat map and $(F,E^{\prime})$ is one of the pair described
in the Main Theorem of \cite{Fu2}.
If the general fiber is ${\bf P}^{n-2}$ all the fibers are ${\bf P}^{n-2}$.
\item[{iv})] If $s = 3$ and $n \geq 5$ then $W$ is a smooth
3-fold and all fibers are isomorphic to ${\bf P}^{n-3}$.
\end{itemize}
\end{itemize}
\item[3)] Assume finally that $K_X + det(E)$ is nef and big but not ample.
Then a high multiple of $K_X + det(E)$ defines a birational map,
$\varphi :X \rightarrow X'$, which contracts an "extremal face" (see section 2).
Let $R_i$, for $i$ in a finite set of index, the extremal rays
spanning this face; call
$\rho_i: X \rightarrow W$ the contraction associated to one of the $R_i$.
Then we have that each $\rho_i$ is birational and divisorial; if $D$ is
one of the exceptional divisors
(we drop the index) and $Z = \rho (D)$
we have that $dim(Z) \leq 1$
and the following possibilities occur:
\begin{itemize}
\item[{i})] $dimZ = 0$, $D = {\bf P}^{(n-1)}$ and $D_{|D} = {\cal O}(-2)$
or ${\cal O}(-1)$;
moreover, respectively, $E_{|D} =\oplus^{n-2}{\cal O}(1)$ or
$E_{|D} =\oplus^{n-1}{\cal O}(1)\oplus {\cal O}(2)$.
\item[{ii})] $dimZ = 0$, $D$ is a (possible singular) quadric,
${\bf Q}^{(n-1)}$, and $D_{|D} = {\cal O}(-1)$;
moreover $E_{|D} =\oplus^{n-2}{\cal O}(1)$.
\item[{iii})] $dimZ = 1$, $W$ and $Z$ are smooth projective varieties
and $\rho$ is the blow-up of $W$ along $Z$.
Moreover $E_{|F} =\oplus^{n-2}{\cal O}(1)$.
\end{itemize}
If $n > 3$ then $\varphi$ is a composition of "disjoint" extremal
contractions as in i), ii) or iii).
\label{main}
\end{Theorem}
\noindent{\bf Proof. } Proof of part 1) of the theorem
Let $(X,E)$ be a generalized polarized variety
and assume that $K_X + det(E)$ is not nef.
Then there exist on $X$ a finite number of extremal rays, $R_1, \dots , R_s$,
such that $(K_X + det(E))^.R_i < 0$ and therefore, by the remark in section
(2),
$l(R_i) \geq (n-1)$.
Consider one of this extremal rays, $R = R_i$, and let $\rho : X \rightarrow Y$
be its associated elementary contraction. Then $L := -(K_X+det(E))$ is
$\rho$-ample and also the vector bundle $E_1 := E \oplus L$ is
$\rho$-ample; moreover $K_X + det(E_1) = {\cal O}_X$ relative to $\rho$.
We can apply the theorem in \cite{ABW2} which study the positivity of
the adjoint bundle in the case of $rank E_1 = (n-1)$. More precisely
we need a relative version of this theorem, i.e. we do not assume
that $E_1$ is ample but that it is $\rho$-ample
(or equivalently a local statement in a neighborhood
of the exceptional locus of the extremal ray $R$).
We just notice that the theorem in \cite{ABW2} is true also in the relative
case
and can be proved exactly with the
same proof using the relative minimal model theory (see [K-M-M]; see also the
section 2
of the paper \cite{AW} for a discussion of the local set up).
Assume first that $\rho$ is birational, then $K_X + det(E_1)$ is $\rho$-nef and
$\rho$-big; note also that, since $l(R_i) \geq (n-1)$, $\rho$ is divisorial.
Therefore we are in the (relative) case C of the theorem
in \cite{ABW2} (see also the proposition \ref{bd}
with $r = (n-1)$); this implies that $Y$ is smooth and $\rho$ is the blow up
of a point in $Y$.
Since $l(R_i) \geq (n-1)$, the exceptional loci
of the birational rays are pairwise disjoint by proposition
(\ref{birelementare}).
This part give the point {\sf (i)} of the
theorem \ref{main}; i.e. the birational extremal rays have
disjoint exceptional loci which are divisors isomorphic
to ${\bf P}^{(n-1)}$ and which contract
simultaneously to smooth distinct points on a $n$-fold $W$. The
description of $E$ follows trivially (see also \cite{ABW2}).
If $\rho$ is not birational then we are in the case
B of the theorem in \cite{ABW2}; from this we
obtain similarly as above the other cases of the theorem \ref{main},
with some trivial
computations needed to recover $E$ from $E_1$.
\par\hfill $\Box$\par
\bigskip
Proof of the part 2) of the theorem
Let $K_X+detE$ be nef but not big; then it is the supporting
divisor of a face $F = (K_X+detE)^{\bot}$. In particular we can
apply the theorems of section (\ref{tech}): therefore there
exist a map $\pi:X\rightarrow W$ which is given by a high multiple
of $K_X+detE$ and which contracts the curves in the face. Since
$K_X+detE$ is not big we have that $dimW<dimX$.
Moreover for every rational curve $C$
in a general fiber of $\pi$ we have $-K_X\cdot C \geq (n-2)$ by the remark
in section (\ref{tech}).
We apply proposition
(\ref{fibelementare}), which, together with the above inequality on
$-K_X\cdot C$, says that $\pi$ is an elementary contraction
if $n\geq 5$ unless
either $n=6$, $W$ is a point and $X$ is a Fano manifold of pseudoindex
$4$ and $\rho(X) = 2$ or $n = 5$ and $dimW \leq 1$.
By proposition (\ref{diswis}) we have the inequality
$$n+dimF\geq n+n-2-1;$$
in particular it follows that $dim W\leq 3$.
\addtocounter{subsection}{1 Let $dimW=0$, that is $K_X+detE=0$ and therefore $X$ is a Fano manifold.
By what just said above we have that $b_2(X)=1$ if $n \geq 6$ with an exception
which will be treated in the following lemma.
\begin{Lemma} Let $X$ be a $6$ dimensional projective manifold,
$E$ is an ample vector bundle on $X$ of rank $4$ such that $K_X+detE=0$.
Assume moreover that $b_2 \geq 2$.
Then $X={\bf P}^3\times{\bf P}^3$ and $E=\oplus^4{\cal O}(1,1)$.
\label{slice}
\end{Lemma}
\noindent{\bf Proof. }
The lemma is a slight generalizzation of \cite[Prop B]{Wi1} for
dimension $6$; the poof is similar
and we refer to this paper. In particular as in \cite {Wi1}
we can see that $X$ has two extremal rays whose contractions, $\pi_i$,$i
=1,2$,
are of fiber type with equidimensional fibers onto
3-folds $W_i$ and with general fiber $F_i\simeq {\bf P}^3$.
We claim that the $W_i$ are smooth and thus $W_i\simeq {\bf P}^3$.
First of all note that $W_i$ can have only isolated singularity and only
isolated points over which the fiber is not ${\bf P}^{n-3}$; in fact
let $S$ be a general hyperplane section of $W_i$ and $T_i=\pi_i^*(S)$, then
$(\pi_i)_{|T_i}$ is an extremal contraction, by proposition
\ref{fibelementare};
hence by \cite[Prop 1.4.1]{ABW2} $S$ is smooth; moreover the contraction is
supported by $K_{T_i}+det E_{T_i}$ hence all fibers are
${\bf P}^3$ by the main theorem of \cite{ABW2}.
Now we are (locally) in the hypothesis of lemma \ref{scroll} so we get,
locally in the complex topology, a tautological bundle and
we can conclude, by \cite[Prop 2.12]{Fu1}, that $W_i$ is smooth.
Let $T = H_1 \cap H_2$, where $H_i$ are two general
elements of $\pi_1^*({\cal O}(1)$. $T$ is smooth, we claim that
$T\simeq {\bf P}^1\times {\bf P}^3$. In fact $\pi_{1 _{|T}}$
makes $T$ a projective bundle over a line (since $H^2({\bf P}^1,{\cal O}^*)=0$),
that is $T={\bf P}({\cal F})$. Moreover $\pi_{2_{|T}}$ is onto ${\bf P}^3$,
therefore the claim follows. Therefore we conclude
that $\pi_i^*{\cal O}_{{\bf P}^3}(1)_{|F_i}\simeq {\cal O}_{{\bf P}^3}(1)$ for $i=1,2$.
This implies by Grauert Theorem that the two fibrations are classical scroll,
that is
$X={\bf P}({\cal F}_i)$, for $i=1,2$; moreover
computing the canonical class of $X$ the ${\cal F}_i$ are ample
and the lemma easily follows.
\end{proof}
\addtocounter{subsection}{1 Let $dimW=1$. Then $W$ is a smooth curve and $\pi$ is a flat map.
Let $F$ be a general
fiber, then $F$ is a smooth Fano manifold and $E_{|F}$ is
an ample vector bundle on $F$ of rank $(n-2) = dimF - 1$
such that $-K_F = det(E_{|F})$. These pairs $(F, E_{|F})$
are classified in the Main Theorem of \cite{PSW}; in particular if $dimF \geq
5$
$F$ is either ${\bf P}^{(n-1)}$ or ${\bf Q}^{(n-1)}$ or a
del Pezzo manifold with $b_2(F) = 1$.
Moreover if $n \geq 6$ then $\pi$
is an elementary contraction by proposition (\ref{fibelementare}).
\noindent{\bf Claim } Let $n\geq 6$ and assume that the general fiber is ${\bf P}^{n-1}$, then
$X$ is a classical scroll and $E_{|F}$ is the same for all $F$.
(See also \cite {Fu2}) Let $S= W\setminus U$ be the locus
of points over which the fiber is not ${\bf P}^{n-1}$. Over $U$ we have a
projective
fiber bundle. Since $H^2(U,{\cal O}^*)=0$ we
can associate this ${\bf P}$-bundle to a vector bundle ${\cal F}$ over $U$.
Let $Y={\bf P}({\cal F})$ and
$H$ the tautological bundle;
by abuse of language let $H$ the extension of $H$ to $X$. Since
$\pi$ is elementary $H$ is an ample line bundle on $X$.
Therefore by semicontinuity $\Delta(F,H_F)\geq \Delta(G,H_G)$, for
any fiber $G$, where $\Delta(X,L)$ is Fujita delta-genus. In our case
this yields $0=\Delta(F,H_F)\geq \Delta(G,H_G)\geq 0$. Moreover by
flatness $(H_G)^{n-1}=(H_F)^{n-1}=1$ and Fujita classification allows to
conclude.
The possible vector bundle restricted to the fibers are all decomposables,
hence they are rigid, that is $H^1(End(E))=\oplus_i H^1(End({\cal O}(a_i))=
\oplus_i H^1({\cal O}(-a_i))=0$. Hence the decomposition is the same
along all fibers of $\pi$.
\noindent{\bf Claim } Let $n\geq 6$ and assume that the general fiber is ${\bf Q}^{n-1}$.
Then $X$ is a quadric bundle.
Let as above $S=W\setminus U$ be the locus of points over which the fiber is
not
a smooth quadric.
Let $X^*=\pi^{-1}(U)$ then we can embed $X^*$ in a fiber bundle of projective
spaces over $U$,
since it is locally trivial. Associate this $P$-bundle over $U$
to a projective bundle and argue as before.
\par\hfill $\Box$\par
\addtocounter{subsection}{1 Let now $dimW=2$ and assume that $n\geq 5$;
then $\pi$ is an elementary contraction.
This implies first, by
\cite[Prop. 1.4.1]{ABW2}, that $W$
is smooth; secondly that $\pi$ is equidimensional, hence flat and
the general fiber
is ${\bf P}^{n-2}$ or ${\bf Q}^{n-2}$, see \cite{Fu2}.
\noindent{\bf Claim } Let $n\geq 5$ and the general fiber is ${\bf P}^{n-2}$ then
for any fiber $F\simeq {\bf P}^{n-2}$ and
$E_{|F}$ is the same for all $F$.
Let $S\subset W$
be the locus of singular fibers, then $dimS\leq 0$ since
$W$ is normal. Let $U\subset W$
be an open set, in the complex topology, with $U\cap S=\{0\}$ and let
$V\subset X$ such that $V=\pi^{-1}(U)$. We are in the hypothesis
of lemma \ref{scroll}
thus we get a "tautological" line bundle $H$ on
$V$ and we conclude by \cite[Prop. 2.12]{Fu1}.
There are two possible restriction of $E$ to the fiber,
namely $E_{|F}\simeq {\cal O}(2)\oplus(\oplus^{n-1}{\cal O}(1))$ or
$E_{|F}$ is the tangent bundle. As observed by Fujita in \cite{Fu2} this two
restrictions have a different behavior in the diagram (\ref{dia1}),
in the former $\varphi$ is birational while in the latter it is of fiber
type. Hence the restriction has to be constant along all the fibers.
\par\hfill $\Box$\par
\addtocounter{subsection}{1 Let finally $dimW=3$; the general fiber is ${\bf P}^{n-3}$
(see for instance \cite{Fu2}).
Assume that $n\geq 5$, therefore
$\pi$ is elementary; we claim that all fibers are ${\bf P}^{n-3}$.
Since $\pi$ is elementary any fiber $G$ has $cod G\geq 2$.
Let $S\subset W$
be the locus of point over which the fiber is not ${\bf P}^{n-3}$;
$dimS\leq 0$ since a generic linear
space section can not
intersect $S$, by the above.
Let $U\subset W$
be an open set, in the complex topology, with $U\cap S=\{0\}$ and let
$V\subset X$ such that $\pi(V)=U$.
Then by lemma \ref{scroll} we get a "tautological" line bundle $H$ on
$V$; $\pi: V\rightarrow U$ is supported by $K_V+(n-2)H$.
Thus by \cite[Th 4.1]{AW} $U$ is smooth and all the fibers are ${\bf P}^{n-3}$
( we use that $n\geq 5$).
\par\hfill $\Box$\par
Proof of the part 3) of the theorem
In the last part of the theorem we assume that $K_X+detE$ is nef and big but
not ample.
Then $K_X+detE$ is a supporting divisor of an extremal face, $F$; let $R_i$ the
extremal rays spanning this face. Fix one of this ray, say $R = R_i$ and
let $\pi:X\rightarrow W$ be the elementary contraction associated to $R$.
We have $l(R)\geq n-2$; this implies first that the
exceptional loci are
disjoint if $n > 3$, proposition
(\ref{n=4}). Secondly, by the inequality (\ref{diswis}),
we have
$$dimE(R)+dimF(R)\geq 2n - 3.$$
Therefore $dimE(R)=n-1$ and either $dimF(R) = n-1$ or $dimF(R) = n-2$;
if $Z := \rho (E)$ and $D=E(R)$ this implies that either $dimZ = 0$ or $1$.
If $dimZ = 1$ then $dim F(\pi) = n-2$ for all fibers (note that since the
contraction $\pi$ is elementary there cannot be fiber of dimension
$(n-1)$); thus we can apply
proposition
(\ref{bd}) with $r = (n-2)$. This will give the case 3-(iii) of the theorem.
Consider again the construction in section (\ref{tech}),
in particular we refer to
the diagram (\ref{dia1}). Let $S$ be the extremal ray contracted
by $\varphi$; note that $l(S)\geq n-2$
and that the inequality (\ref{diswis}) gives
$$dimE(S)+dimF(S)\geq 3n - 6;$$
in particular, since $dim F(S) \leq dim F(R) $,
we have two cases, namely $dimE(S) = 2n-5$ and $dimF(S) = (n-1)$ or
$dimE(S) = 2n-4$ and $dimF(S) = (n-1)$ or $(n-2)$.
The case in which $dimE(S) = 2n-5$ will not occur. In fact, after "slicing",
(see \ref{adj}),
we would obtain a map $\varphi^{\prime}=\varphi_{|Z}$ which would be a small
contraction supported by a divisor of the type $K_Z+(n-2)L$
but this is impossible by the classification of \cite[Th 4]{Fu1}
(see also \cite{An}).
\medskip
Hence $dimE(S)=2n-4$, that is also $\varphi$ is divisorial.
Suppose that the general fiber of $\varphi$, $F(S)$, has
dimension $(n-2)$. After slicing we obtain a map
${\varphi}^{\prime}=\varphi_{|Z}: Z \rightarrow T$
supported by $K_Z+(n-2)L$, where $L={{\xi}_E}_{|Z}$.
This map contracts divisors $D$ in $Z$ to curves; by
(\cite[Th 4]{Fu1})
we know that every fiber $F$ of this map is ${\bf P}^{(n-2)}$ and that
$D_{|F} = {\cal O}(-1)$ (actually this map is a blow up of a smooth curve in a
smooth variety).
In particular there are curves in $Y$, call them
$C$, such that $-E(S).C = 1$. We will discuss this case in
a while.
Suppose then the general fiber of $\varphi$, $F(S)$, has
dimension $(n-1)$; therefore all fibers have dimension $(n-1)$.
Slicing we obtain a map
${\varphi}^{\prime}=\varphi_{|Z}: Z \rightarrow T$
supported by $K_Z+(n-2)L$, where $L={{\xi}_E}_{|Z}$.
This map contracts divisors $D$ in $Z$ to points; by (\cite{Fu1})
we know that these divisors are either ${\bf P}^{(n-1)}$
with normal bundle ${\cal O}(-2)$ or ${\bf Q}^{(n-1)}\subset {\bf P}^n$ with normal bundle
${\cal O}(-1)$.
In the latter case we have as above that there are curves $C$ in $Y$,
such that $-E(S).C = 1$.
In these cases observe that $E(S)\cdot \tilde{C}=0$, where $\tilde{C}$
is a curve in the fiber of $p$. Hence $E(S)=p^*(-M)$ for some $M\in Div(X)$.
Let $l$ be an extremal curve of $E(S)$. Then, by projection formula, we have
$-1=E(S)\cdot l=-M\cdot mC$ and thus $M$ generates $Im[Pic(X)\rightarrow Pic(D)]$,
hence
$M$ is $\pi$-ample;
note that in general it does not generate $Pic(D)$.
We study now the Hilbert polynomial of $M_{|D}$ to show
that $\Delta(D,M_{|D})=0$, where $\Delta(X,L)$ is
Fujita delta genus.
Let ${\cal O}_D(-K_X) \simeq{\cal O}_D(pM)$,
where $p=l(R)\geq n-2$, and ${\cal O}_D(-D)\simeq{\cal O}_D(qM)$
for some $p,q\in {\bf N}$. By adjunction
formula $\omega_D\simeq {\cal O}_D(-(p+q)M)$. By \cite[Lemma 2.2]{Ando} or
\cite[pag 179]{BS}, Serre duality and relative vanishing
we obtain that $q\leq 2$, the Hilbert polynomial is
$$P(D,M_{|D})= \frac{a}{(n-1)!}(t+1)\cdots(t+(n-2))(t+c)$$
and the only possibilities are $a=1, c=n-1, q=1 or 2$
and $a=2, c=(n-1)/2, q=1$. In particular $\Delta(D,M_{|D})=0$ and,
by Fujita classification, $D$ is equal to ${\bf P}^{(n-1)}$
or to ${\bf Q}^{(n-1)}\subset {\bf P}^n$. Now the rest of the claim in
3) i) and ii) follows easily.
It remains the case in which
${\varphi}^{\prime}=\varphi_{|Z}: Z \rightarrow T$
contracts divisors $D= {\bf P}^{(n-1)}$
with normal bundle ${\cal O}(-2)$ to points.
We can apply the above proposition (\ref{fujita})
and show that the singularities of $W$ are the same as those of
$T$. Then, as in (\cite{Mo1}), this means that we can
factorize $\pi$ with the blow up of the singular point.
Let $X^{\prime}=Bl_{w}(W)$, then we have a birational map $g:X\rightarrow
X^{\prime}$. Note that $X^{\prime}$ is smooth and that $g$ is finite.
Actually
it is an isomorphism outside $D$ and cannot contract any curve of
$D$. Assume to the contrary that $g$ contracts a curve $B\subset D$;
let $N\in Pic(X^{\prime})$ be an ample divisor then we have
$g^*N\cdot B=0$ while $g^*N\cdot C\not=0$ contradiction.
Thus by Zarisky's main theorem $g$ is an
isomorphism. This gives a case in 3)i).
\small
|
2006-02-10T17:52:27 | 9410 | alg-geom/9410030 | en | https://arxiv.org/abs/alg-geom/9410030 | [
"alg-geom",
"math.AG"
] | alg-geom/9410030 | Oleg Viro | Oleg Viro | Self-linking number of a real algebraic link | 7 pages, AMS-LaTeX with 6 pictures in the format of *.EPS files | null | null | null | null | For a nonsingular real algebraic curve in the 3-dimensional projective space
and sphere a new numeric characteristic is introduced. It takes integer values,
is invariant under rigid isotopy, multiplied by -1 under mirror reflection. In
a sense it is a Vassiliev invariant of degree 1 and a counter-part of a link
diagram writhe.
| [
{
"version": "v1",
"created": "Sun, 30 Oct 1994 20:55:15 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Viro",
"Oleg",
""
]
] | alg-geom | \section{Introduction}\label{sI} In the classical knot theory by a link one
means a smooth closed 1-dimensional submanifold of the 3-dimensional
sphere $S^3$, i.~e. several disjoint circles smoothly embedded into
$S^3$. A classical link may emerge as
the set of real points of a real algebraic curve. First, it gives rise
to questions about relations between invariants of the same curve
which are provided by link theory and algebraic geometry.
Second, it suggests to develop a theory
parallel to the classical link theory, but taking into account the
algebraic nature of objects. From this viewpoint it is more
natural to consider real algebraic links up to isotopy consisting of
real algebraic links, which belong to the same continuous family of
algebraic curves, rather than up to smooth isotopy in the class of
classical links. I call an isotopy of the former kind a {\it rigid
isotopy\/} following a terminology established by Rokhlin \cite{R} in
a similar study of real algebraic plane projective curves and extended
later to various other situations (see, e.g., \cite{Viro New pr.}).
Of course, there is a forgetting functor: any real
algebraic link can be considered as a classical link and a rigid
isotopy as a smooth isotopy. It is interesting, how much is lost under
this transition.
In this note I point out a characteristic of a real
algebraic link which is lost. It is unexpectedly simple. In an obvious
sense it is a nontrivial Vassiliev invariant of degree 1 on the class
of real algebraic knots.\footnote{Recall that a knot is a link
consisting of one component.} In the classical knot theory the lowest
degree of a nontrivial Vassiliev knot invariant is 2. Thus there is an
essential difference between classical knot theory and the theory of
real algebraic knots.
The characteristic of real algebraic links which is defined below is
very similar to self-linking number of framed knots. I
call it also {\it self-linking number.\/} Its definition looks
like a refinement of an elementary definition of the writhe of a knot
diagram, but taking into consideration the imaginary part of the knot.
\section{Self-linking of a nonalgebraic knot}\label{slknonalg}In the
classical theory, a self-linking number of a knot is defined only if the
knot is equipped with an additional structure like framing or just a
vector field nowhere tangent to the knot.\footnote{A framing is a pair
of orthogonal to each other normal vector fields on a knot. There is an
obvious construction which makes a framing from a nontangent vector
field and establishes one to one correspondence between homotopy
classes of framings and nontangent vector fields. The vector fields are
more flexible and relevant to the case.} The self-linking number is the
linking number of the knot oriented somehow and its copy obtained by a
small shift in the direction specified by the vector field. It does not
depend on the choice of orientation, since reversing the orientation of
the knot is compensated by reversing the induced orientation of
its shifted copy. Of course, it depends on the homotopy class of the
vector field.
A knot has no natural preferable homotopy class of framings
which would allow to speak about a self-linking number of the
knot without a special care on choice of a framing.\footnote{Moreover,
the self-linking number is used to define a natural class of framings:
namely, framings with the self-linking number zero.} Some framings
appear naturally in geometric situations. For example, if one fixes a
generic projection of a knot to a plane, the vector field of directions
of the projection appears. The corresponding self-linking number is
called the {\it writhe\/} of the knot. However, it depends on the
choice of projection and changes under isotopy.
The linking number is a Vassiliev invariant of order 1 of
two-component oriented links. That means that it changes by a constant
(in fact, by 2) when the link experiences a homotopy with a generic
appearance of an intersection point of the components. Whether the
linking number increases or decreases depends only on the local picture
of orientations near the double point: when it passes from
$\vcenter{\epsffile{f01.eps}}$ through
$\vcenter{\epsffile{f02.eps}}$ to
$\vcenter{\epsffile{f03.eps}}$, the linking number
increases by 2. Generalities on Vassiliev invariants see, e.~g., in
\cite{V}
In a sense the linking number is
the only Vassiliev invariant of degree 1 of two-component oriented
links: any Vassiliev invariant of degree 1 of two-component oriented
links is a linear function of the linking number. Similarly, the
self-linking number is a Vassiliev invariant of degree 1 of framed
knots (it changes by 2 when the knot experiences a homotopy with a
generic appearance of a self-intersection point) and it is the only
Vassiliev of degree 1 of framed knots in the same sense. Necessity of
framing for definition of self-linking number can be formulated now
more rigorously: only constants are Vassiliev invariants of degree 1 of
(nonframed) knots.
The definition of the writhe, which is mimicked below, runs as follows:
for each crossing point of the knot projection one defines a {\it local
writhe\/} equal to $+1$ if near the point the knot diagram looks like
$\vcenter{\epsffile{f03.eps}}$ and $-1$ if it looks like
$\vcenter{\epsffile{f01.eps}}$.
Then one sums up the local writhes over all double points of the
projection. The sum is the writhe.
A continuous change of the projection may cause vanishing of a crossing
point. It happens under the first Reidemeister move shown in
the left hand half of Figure \ref{f1}. This move changes the writhe by
$\pm 1$.
\section{How algebraicity enhances self-linking number}\label{Genrem}If
a link is algebraic then its projection to a plane is algebraic, too. A
generic projection has only ordinary double points. The total number
of complex double points is constant. The number of real double points
can change, but only by an even number. A real double point cannot turn
alone into imaginary one, as it seems happen under the first
Reidemeister move. Under the algebraic version of the first
Reidemeister move the double point stays in the real domain, but
becomes solitary, like the only real point of the curve $x^2+y^2=0$.
The algebraic version of the first Reidemeister move is shown in the
right hand half of Figure \ref{f1}. It is not difficult to prove that
the corresponding family of plane curves can be transformed by a local
diffeomorphism to the family of real rational cubic curves
$y^2=x^2(t-x)$ with $t\in \Bbb R$.
\begin{figure}[h]
\centerline{\epsffile{f1.eps}}
\caption{Topological and real algebraic versions of the first
Reidemeister move}
\label{f1}
\end{figure}
A solitary double point of the projection is not image of any real
point of the link. It is the image of two imaginary complex conjugate
points of the complexification of the link. The preimage of the point
in the 3-space under the projection is a real line. It is disjoint from
the real part of the link, but intersects its complexification in a
couple of complex conjugate imaginary points.
In the next section with any solitary double point of the projection,
a local writhe equal to $\pm1$ is associated. It is done in such a way
that the local writhe of the crossing point vanishing in the first
Reidemeister move is equal to the local writhe of the borning solitary
double point. In the case of an algebraic knot the sum of local
writhes of all double points, both solitary and crossings, does not
depend on the choice of projection and is invariant under rigid
isotopy. This sum is the self-linking number.
There are two types of generic deformations of an algebraic link
changing the rigid isotopy type. One of them is exactly as in the
category of classical links: two pieces of the set of real points come
to each other and pass through each other. A generic projection of
the link experiences an isotopy. No events happen besides that one
crossing point becomes for a moment the image of a double point of the
link and then turns back into a crossing point, but with the opposite
writhe. Another type has no counter-part in the topological context.
Two complex conjugate imaginary branches pass through each other. At
the moment of passing they intersect in a real isolated double point.
A generic projection of the link experiences an isotopy. No events
happen besides that one solitary double point becomes for a
moment the image of an isolated double point of the link and then turns
back into a usual solitary double point, but with the opposite writhe.
It is clear that the self-linking number of an algebraic knot changes
under both modifications by $\pm2$ with the sign depending only on the
local structure of the modification near the double point. It means
that the self-linking number is the Vassiliev invariant of degree 1.
A construction similar to the construction of the self-linking number
of an algebraic knot can be applied to algebraic {\it links.\/} However
in this case it is necessary either to orient the link or to exclude
from the sum the crossings where the branches belong to distinct
components of the set of real points. In fact, the local writhe depends
on the orientations of the branches, but if the branches belong to the
same component orientations of the branches can be induced from the
same orientation of the component. It is easy to see that the result
does not depend on the choice of orientation of the component.
In the case of knots, self-linking number defines a natural class of
framings, since for knots homotopy classes of framings are enumerated
by their self-linking numbers and we can choose the framing having the
self-linking number equal to the algebraic self-linking number
constructed here. I do not know any direct construction of this
framing. Moreover, there seems to be a reason for absence of such a
construction. In the case of links the construction above gives a
single number, while framings are enumerated by sequences of numbers
with entries corresponding to components.
The construction of this paper can be applied to algebraic links in the
sphere $S^3$. Although from the
viewpoint of knot theory this is the most classical case, from the
viewpoint of algebraic geometry the case of curves in the projective
space is simpler, and I will start from it. The case of spherical
links is postponed to the Section \ref{srS}.
\section{Real algebraic projective links}\label{s0}Let $A$ be a
nonsingular real algebraic curve in the 3-dimensional projective space.
Then the set $\Bbb R A$ of its real points is a smooth closed 1-dimensional
submanifold of $\Bbb R P^3$, i.~e. a smooth projective link. The set $\Bbb C A$
of its complex points is a smooth complex 1-dimensional submanifold of
$\Bbb C P^3$.
Let $c$ be a point of $\Bbb R P^3$. Consider the projection
$p_c:\Bbb C P^3\smallsetminus c\to \Bbb C P^2$ from $c$. Assume that $c$ is such that
the restriction to $\Bbb C A$ of $p_c$ is generic. This means
that it is an immersion without triple points and at each double point
the images of the branches have distinct tangent lines. As it
follows from well-known theorems, those $c$'s for which this is the
case form an open dense subset of $\Bbb R P^3$ (in fact, it is the
complement of a 2-dimensional subvariety).
The real part $p_c(\Bbb C A)\cap\Bbb R P^2$ of the image consists of the image
$p_c(\Bbb R A)$ of the real part and, maybe, several solitary points, which
are double points of $p_c(\Bbb C A)$.
There is a purely topological construction which assigns a local writhe
equal to $\pm1$ to a crossing belonging to the image of only one
component of $\Bbb R A$. This construction is well-known in the case of
classical knots. Here is its projective version. I borrow it from
Drobotukhina's paper \cite{Dr} on generalization of Kauffman brackets
to links in the projective space.
Let $K$ be a smooth connected one-dimensional submanifold of $\Bbb R P^3$,
and $c$ be a point of $\Bbb R P^3\smallsetminus K$. Let $x$ be a generic double
point of the projection $p_c(K)\subset \Bbb R P^2$ and $L\subset \Bbb R P^3$ be
the line which is the preimage of $x$ under the projection. Denote by
$a$ and $b$ the points of $L\cap \Bbb R P^3$.
\begin{figure}[t]
\centerline{\epsffile{f2.eps}}
\caption{Construction of the frame $v$, $l$, $w'$.}
\label{f2}
\end{figure}
The points $a$ and $b$ divides the line $L$ into two segments.
Choose one of them and denote it by $S$. Choose an orientation of $K$.
Let $v$ and $w$ be tangent vectors of $K$ at $a$ and $b$ respectively
directed along the selected orientation of $K$.
Let $l$ be a vector tangent to $L$ at $a$ and directed inside $S$.
Let $w'$ be a vector at $a$ such that it is tangent to the plane
containing $L$ and $w$ and is directed to the same side of $S$ as $w$
(in an affine part of the plane containing $S$ and $w$). See Figure
\ref{f2}.
The triple $v$, $l$, $w'$ is a base of the tangent space $T_a\Bbb R P^3$.
The value taken by the orientation of $\Bbb R P^3$ on this frame is the
local writhe of $x$. Its definition involves several choices. However
it is easy to prove that the result does not depend on them.
Let $A$, $c$ and $p_c$ be as in the beginning of this Section
and let $s\in\Bbb R P^2$ be a solitary double point of $p_c$. Here is a
construction assigning $\pm1$ to $s$. I will call the
result also a {\it local writhe\/} at $s$.
Denote the preimage of $s$ under $p_c$ by $L$. This is a real line in
$\Bbb R P^3$ connecting $c$ and $s$. It intersects $\Bbb C A$ in two imaginary
complex conjugate points, say, $a$ and $b$. Since $a$ and $b$ are
conjugate they belong to different components of $\Bbb C L\smallsetminus\Bbb R L$.
Choose one of the common points of $\Bbb C A$ and $\Bbb C L$, say, $a$. The
natural orientation of the component of $\Bbb C L\smallsetminus\Bbb R L$ defined
by the complex structure of $\Bbb C L$ induces orientation on $\Bbb R L$ as on
the boundary of its closure. The image under $p_c$ of the local branch
of $\Bbb C A$ passing through $a$ intersects the plane of the projection
$\Bbb R P^2$ transversally at $s$. Take the local orientation of the plane
of projection such that the local intersection number of the plane and
the image of the branch of $\Bbb C A$ is $+1$.
Thus the choice of one of two points of $\Bbb C A\cap\Bbb C L$ defines an
orientation of $\Bbb R L$ and a local orientation of the plane of
projection $\Bbb R P^2$ (we can speak only on a local orientation of
$\Bbb R P^2$, since the whole $\Bbb R P^2$ is not orientable). The plane of
projection intersects\footnote{We may think on the plane of projection
as embedded into $\Bbb R P^3$. If you would like to think on it as on the
set of lines of $\Bbb R P^3$ passing through $c$, please, identify it in a
natural way with any real projective plane contained in $\Bbb R P^3$ and
disjoint from $c$. All such embeddings $\Bbb R P^2\to\Bbb R P^3$ are isotopic.}
transversally $\Bbb R L$ in $s$. The local orientation of the plane,
orientation of $\Bbb R L$ and the orientation of the ambient $\Bbb R P^3$
determine the intersection number. This is the local writhe.
It does not depend on the choice of $a$. Indeed, if one chose
$b$ instead, then both the orientation of $\Bbb R L$ and the local
orientation of $\Bbb R P^2$ would be reversed. The orientation of $\Bbb R
L$ would be reversed, because $\Bbb R L$ receives opposite
orientations from different halves of $\Bbb C L\smallsetminus\Bbb R L$. The local
orientation of $\Bbb R P^2$ would be reversed, because the complex
conjugation involution $%\operatorname{conj: \Bbb C P^2\to\Bbb C P^2$ preserves the complex
orientation of $\Bbb C P^2$, preserves $\Bbb R P^2$ (point-wise) and maps one
of the branches of $p_c(\Bbb C A)$ at $s$ to the other reversing its
complex orientation.
Now for any real algebraic projective link $A$ choose a point
$c\in\Bbb R P^3$ such that the projection of $A$ from $c$ is
generic and sum up writhes at all crossing points of the projection
belonging to image of only one component of $\Bbb R A$ and writhes of all
solitary double points. The sum is called the {\it self-linking number
of $A$.\/}
It does not depend on the choice of projection. Moreover it is
invariant under {\it rigid isotopy\/} of $A$. By rigid isotopy we
mean an isotopy made of nonsingular real algebraic curves. The effect
of a movement of $c$ on the projection can be achieved by a
rigid isotopy defined by a path in the group of projective
transformations of $\Bbb R P^3$. Therefore the following theorem implies
both independence of the self-linking number on the choice of
projection and its invariance under rigid isotopy.
\begin{thm}\label{mainth} For any two rigidly isotopic real algebraic
projective links $A_1$ and $A_2$ such that their projections from the
same point $c\in\Bbb R P^3$ are generic, the self-linking numbers of $A_1$
and $A_2$ defined via $c$ are equal.\end{thm}
To prove this statement, first replace any rigid isotopy by a generic
one. As in purely topological situation of classical links, any generic
rigid isotopy may be decomposed to a composition of rigid isotopies,
each of which makes a local standard move of the projection. There
are 5 local standard moves. They are similar to the
Reidemeister moves. The first of these 5 moves is shown in the right
hand half of Figure \ref{f1}. The next two coincide with the second and
third Reidemeister moves. The fourth move is similar to the second
Reidemeister move: also two double points of projection come to each
other and disappear. However the double points are solitary. The fifth
move is similar to the third Reidemeister move: also
a triple point appears for a moment. But at this triple point only one
branch is real, the other two are imaginary conjugate to each other.
In this move a solitary double point traverses a real branch.
Only in the first, fourth and fifth moves solitary double points are
involved. The invariance under the second and the third move follows
from well-known fact of knot theory that the topological writhe is
invariant under the second and third Reidemeister moves. Thus we have
to prove that:\begin{enumerate}
\item in the first move the writhe of
vanishing crossing point is equal to the writhe of the borning
solitary point,
\item in the fourth move the writhes of the vanishing
solitary points are opposite and
\item in the fifth move the writhe of the
solitary point does not change.
\end{enumerate}
The proof is not complicated, but would take room inappropriate in this
short note.
The same construction may be applied to real algebraic curves in
$\Bbb R P^3$ having singular imaginary points, but no real singularities.
In the construction we can avoid usage of projections from the points
such that some singular point is projected from it to a real point.
Indeed, for any imaginary point there exists only one real line passing
through it (the line connecting the point with its complex
conjugate), thus we have to exclude a finite number of real lines.
\section{Real algebraic links in sphere}\label{srS} The
three-dimensional sphere $S^3$ is a real algebraic variety. It is a
quadric in the four-dimensional real affine space. A stereographic
projection is a birational isomorphism of $S^3$ onto $\Bbb R P^3$. It
defines a diffeomorphism between the complement of the center of
projection in $S^3$ and a real affine space.
Given a real algebraic link in $S^3$, one may choose a real point of
$S^3$ from the complement of the link and project the link from this
point to an affine space. Then include the affine space into the
projective space and apply the construction above. The image has no
real singular points, therefore we can use the remark from the end of
the previous section.
\section{Other generalizations}\label{sGeneralizations} It is difficult
to survey all possible generalizations. Here I indicate only two
directions.
First, consider the most straightforward generalization. Let $L$ be a
nonsingular real algebraic $(2k-1)$-dimensional subvariety in the
projective space of dimension $4k-1$. Its generic projection to $\Bbb R
P^{4k-2}$ has only ordinary double points. At each double point either
both branches of image are real or they are imaginary complex
conjugate. If set of real points is orientable then one can repeat
everything from Section \ref{s0} with obvious changes and obtain a
definition of a numeric invariant generalizing the self-linking number
defined in Section \ref{s0}.
Let $M$ be a nonsingular three-dimensional real algebraic variety with
oriented set of real points equipped with a real algebraic fibration
over a real algebraic surface $F$ with fiber a projective line. There
is a construction which assigns to a real algebraic link (i.~e., a
nonsingular real algebraic curve in $M$) with a generic projection to
$F$ an integer, which is invariant under rigid isotopy, multiplied by
$-1$ under reversing of the orientation of $M$ and is a Vassiliev
invariant of degree 1. This construction is similar to that of Section
\ref{s0}, but uses, instead of projection to $\Bbb R P^2$, an algebraic
version of Turaev's shadow descriptions of links \cite{T}.
|
1994-11-14T06:20:12 | 9410 | alg-geom/9410019 | en | https://arxiv.org/abs/alg-geom/9410019 | [
"alg-geom",
"math.AG"
] | alg-geom/9410019 | Bernd Siebert | B.Siebert and G.Tian | Recursive relations for the cohomology ring of moduli spaces of stable
bundles | 15 pages, Latex | null | null | null | null | The authors learnt that similar results have been independently found by
D.Zagier, V.Baranovsky and V.Balaji/A.King/P.Newstead. The corresponding
references have been added (and some typos corrected).
| [
{
"version": "v1",
"created": "Thu, 20 Oct 1994 02:15:58 GMT"
},
{
"version": "v2",
"created": "Fri, 11 Nov 1994 16:16:22 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Siebert",
"B.",
""
],
[
"Tian",
"G.",
""
]
] | alg-geom | \section{Method and notation}
Let us now fix a Riemann surface $\Sigma$ of genus $g$ and a line
bundle $L$ on $\Sigma$ of odd degree. We write
${\calm_g}={{\cal M}}_g(\Sigma,L)$. Generators for $H^*({\calm_g},{\Bbb Q})$ occur as
coefficients in the K\"unneth decomposition of a characteristic class
associated to the universal bundle ${\cal U}$ over ${\calm_g}\times\Sigma$
\[
c_2({\mbox{\rm End}\skp\skp}{\cal U})\ =\ -\beta+4\psi+2\alpha\otimes\omega\,.
\]
Here $\omega\in H^2(\Sigma,{\Bbb Z})$ is the normalized volume form and $\psi$
is the part of type $(3,1)$. Choosing a standard basis $e_i$ for
$H^1(\Sigma,{\Bbb Z})$, $i=1,\ldots,2g$ (s.th.\ $e_i e_j=0$ unless $i\equiv
j(g)$ and $e_i e_{i+g}[\Sigma]=1$), $\psi$ decomposes further
$\psi=\sum_{i=1}^{2g} \psi_i\otimes e_i$. The classes $\alpha\in
H^2({\calm_g},{\Bbb Q})$, $\psi_i \in H^3({\calm_g},{\Bbb Q})$ and $\beta\in
H^4({\calm_g},{\Bbb Q})$ are actually integral and generate the cohomology ring
\cite{newstead1}, \cite{atiyah-bott}.
There is an interesting subring of $H^*({{\cal N}}_g,{\Bbb Q})$ to which the
intersection pairing may easily be reduced by geometric arguments as
noted by Thaddeus \cite{thaddeus}. Namely, any orientation preserving
diffeomorphism of $\Sigma$ induces a diffeomorphism of ${{\cal N}}_g$ by
acting on $\pi_1$. The corresponding action on $H^*({\calm_g},{\Bbb Q})$ leaves
$c_2({\mbox{\rm End}\skp\skp}{\cal U})$ fixed. Thus $\alpha$ and $\beta$ are invariant and the
$\psi_i$ transform dually to the $e_i$. It is not hard to show that the
ring of such transformations is precisely the subring generated by
$\alpha$, $\beta$ and a newly defined class
$\gamma:=2\sum_{i=1}^{2g}\psi_i\psi_{i+g}$, or more intrinsically
$\psi^2=\gamma\omega$ (in view of the functional equation for the
generating function $\Phi$ it might be more natural to take twice this
class, but to be consistent with the work of Newstead and Thaddeus we
keep this definition).
On the other hand, there is a method introduced by Mumford to construct
relations among the generators, cf.\ \cite{atiyah-bott}: Letting $L$
vary among the line bundles of fixed (odd) degree $D$ (the space of
which we denote by ${\mbox{\rm Pic}\skp}^D(\Sigma)$), one gets a moduli space
$\tilde{{\cal M}}(\Sigma)$ and a fibration
$\tilde{{\cal M}}(\Sigma)\rightarrow{\mbox{\rm Pic}\skp}^D(\Sigma)$ with fibre ${\cal N}_g$.
In rational cohomology this fibration is trivial, i.e.\
$H^*(\tilde{{\cal M}}(\Sigma),{\Bbb Q})\simeq H^*({\cal N}_g,{\Bbb Q})\otimes_{\Bbb Q}
H^*({\mbox{\rm Pic}\skp}^D(\Sigma),{\Bbb Q})$. But ${\mbox{\rm Pic}\skp}^D(\Sigma)\simeq{\mbox{\rm Pic}\skp}^0(\Sigma)$ and
$H^*({\mbox{\rm Pic}\skp}^0(\Sigma),{\Bbb Q})$ is a free alternating algebra in $2g$
generators ${\hat e}_i$ of degree 1 (the push-forwards of $e_i$ under
the Jacobi map $\Sigma\rightarrow{\mbox{\rm Pic}\skp}^0(\Sigma)$). Now let
$\tilde{\cal U}$ be the universal bundle over
${\tilde{\cal M}}_g\times\Sigma$ and let $\tilde\pi:{\tilde{\cal M}}_g
\times\Sigma\rightarrow{\tilde{\cal M}}_g$ be the projection. If $D=4g-3$,
$R^1\tilde\pi_*\tilde{\cal U}=0$ and $\tilde\pi_*\tilde{\cal U}$ is locally
free of rank $2g-1$. By Grothendieck-Riemann-Roch then the Chern
classes of $\tilde\pi_*\tilde{\cal U}=\tilde\pi_!\,\tilde{\cal U}$ are
expressed as polynomials in $\alpha$, $\beta$, $\psi_i$ and $\hat
e_i$. Equating to zero the coefficients of $\hat e_{i_1}\ldots\hat
e_{i_\nu}$ in $c_r(\tilde\pi_*\tilde{\cal U})$, $r\ge 2g$ (which vanish
for rank reasons) gives a number of relations among the generators
$\alpha$, $\beta$ and $\psi_i$. The point of letting $L$ vary is of
course to lower the degrees of the relations by up to $2g$. The
smallest degree of a relation we thus obtain is $4g-2g=2g$.
The conjecture of Mumford which Kirwan recently succeeded to prove as
remarked in the introduction, is that this set of relations is
complete. But in view of the explicit formula for the intersection
pairing of Thaddeus the authors could not believe in this end of the
story. In fact, computer evidence (up to $g=18$ using ``Macaulay''
\cite{mcly}) showed that for the subring generated by $\alpha$,
$\beta$, $\gamma$ there should be only three independent relations of
degrees $g$, $g+1$, $g+2$ respectively, coming from the lowest degree
equation $c_{2g}=0$. Unfortunately, explicit computations are rather
arduous, e.g.\ due to the presence of the odd degree classes $\hat
e_i$.
To find relations of low degrees without bringing ${\mbox{\rm Pic}\skp}(\Sigma)$ into
the game we first remark that because the ${{\cal M}}(\Sigma,L)$ are all
diffeomorphic as long as we fix the genus of $\Sigma$ and the degree of
$L$ modulo 2, we may restrict ourselves to a hyperelliptic curve
$\Sigma$. In this case, there is a closed embedding
$\varphi:{\calm_g}\hookrightarrow G(g+3,2g+2)$ into a Grassmannian as follows
\cite{ramanan}: Let $p:\Sigma\rightarrow{\rm I \! P}^1$ represent $\Sigma$ as
two-fold covering of ${\rm I \! P}^1$, $\iota:\Sigma\rightarrow\Sigma$ the
corresponding hyperelliptic involution (s.th.\ $p\circ\iota=p$) and
$B\subset{\rm I \! P}^1$ the branch locus of $p$ ($\sharp B=2g+2$). Now let $L$
be a line bundle of degree $2g+1$ (as opposed to $d=4g-3$ in Mumford's
method) and let $E$ be a stable 2-bundle over $\Sigma$ with determinant
$L$. Applying $\iota^*$ to $E\otimes\iota^* E$ and switching factors
induces an involution $J: E\otimes\iota^*E\rightarrow E\otimes\iota^*
E$. Denote by $p_*(E\otimes \iota^*E)^\natural$ the $J$-anti-invariant
subsheaf of $p_*(E\otimes\iota^* E)$. One shows
$h^0({\rm I \! P}^1,p_*(E\otimes\iota^* E)^\natural)=g+3$ (loc.\ cit.,
Prop.~2.2). In a branch point $t\in B$ we have the identification
\[
p_*(E\otimes\iota^* E)^\natural_t\ \simeq\ p_*(E\wedge E)_t
\ =\ (p_*L)_t\,.
\]
The map $\varphi:{\calm_g}\rightarrow G(g+3,2g+2)$ is then defined as
\[
E\longmapsto\left(H^0\Big({\rm I \! P}^1,p_*(E\otimes\iota^* E)^\natural
\Big)\Big|_B\subset H^0({\rm I \! P}^1,p_* L|_B)\simeq\cz^{2g+2}\right)\,.
\]
Now let $S$ and $Q$ be the universal bundle and the universal quotient
bundle on $G(g+3,2g+2)$ respectively. The key observation is that $Q$
has rank $g-1$! Similarly to Mumford's method we ``just'' have to
express the Chern classes of $\varphi^* Q$ in terms of $\alpha$, $\beta$,
$\gamma$ (essentially by Grothendieck-Riemann-Roch of course) to get
relations $c_r(\varphi^* Q)=0$, $r=g,g+1,g+2$.
The Chern class computations are based on the following exact
sequence.
\begin{lemma}\label{exact}
On ${\calm_g}\times\Sigma$, there is an exact sequence
\[
0\longrightarrow\hat p^*\hat p_*({\cal U}\otimes\iota^*{\cal U})^\natural
\longrightarrow{\cal U}\otimes\iota^*{\cal U}\longrightarrow
S^2{\cal U}|_{{\calm_g}\times p^{-1}(B)}\longrightarrow 0\,,
\]
where $\hat p={\mbox{\rm Id}\skp}\times p:{\calm_g}\times\Sigma\rightarrow{\calm_g}\times{\rm I \! P}^1$.
\end{lemma}
{\em Proof. }
The restriction map $\hat p^*\hat p_* E\rightarrow E$ is injective for
any locally free sheaf $E$ by flatness of $p$. Composing this map with
the inclusion $\hat p^*\hat
p_*({\cal U}\otimes\iota^*{\cal U})^\natural\hookrightarrow \hat p^*\hat
p_*({\cal U}\otimes\iota^*{\cal U})$ yields exactness at the first place.
Outside of the branch points this map is obviously an isomorphism (by
anti-invariance elements of $\hat p^*\hat p_*({\cal U}\otimes
\iota^*{\cal U})^\natural$ are uniquely determined by their behaviour on
one branch). It is then a matter of linear algebra to check that at a
branch point $y\in\Sigma$ the cokernel is given by mapping a germ of
sections of ${\cal U} \otimes\iota^*{\cal U}$ to
$(s+J(s))(y)\in{\cal U}\otimes\iota^*{\cal U} |_{{\calm_g}\times p^{-1}(B)}\simeq
S^2{\cal U}|_{{\calm_g}\times p^{-1}(B)}$.
{\hfill$\Diamond$}\vspace{1.5ex}
\section{Computations of Chern classes}
This section contains the computational heart of the paper, summarized
in the following proposition. We adopt the convention that in writing
Chern classes or Chern characters as analytic functions in certain
cohomology classes we understand to evaluate on these classes the
corresponding power series expansion.
\begin{prop}\label{chernquot}
Denote by $c(\varphi^*Q)=\sum_{i\ge0}c_i(\varphi^*Q)$ the total Chern class
of $\varphi^*Q$. Then
\[
c(\varphi^*Q)=(1-\beta)^{-1/2}\exp\bigg[\alpha+\Big(\alpha+
\frac{2\gamma}{\beta}\Big)\sum_{m\ge1}\frac{\beta^m}{2m+1}\bigg]\,.
\]
\end{prop}
Note that the $c_i(\varphi^*(Q))$ are really {\em polynomials} in $\alpha$,
$\beta$, $\gamma$, the denominator $\beta$ cancels. Also, if one
prefers, one could write $({\mbox{\rm arctanh}}(\sqrt{\beta})/\sqrt{\beta})-1$
(with $\sqrt{\beta}$ formally adjoint to $H^*({\calm_g})$) instead of the
infinite sum.
Before turning to the proof we will need some preparations. Letting
$\pi:{\calm_g}\times{\rm I \! P}^1\rightarrow{\calm_g}$ and $\tilde\pi=\pi\circ\hat
p:{\calm_g}\times \Sigma\rightarrow{\calm_g}$ be the projections, we know
$\varphi^*S={\tilde\pi}_*( {\cal U}\otimes\iota^*{\cal U})^\natural\otimes
H^{-1}$ with $H\in{\mbox{\rm Pic}\skp}({\calm_g}) \simeq{\Bbb Z}$ the ample generator and
${\tilde\pi}_*( {\cal U}\otimes\iota^*{\cal U})^\natural:=\pi_*(\hat p_*
({\cal U}\otimes\iota^*{\cal U})^\natural)$ \cite{ramanan}. To apply
Grothendieck-Riemann-Roch to this sheaf we first need a closed formula
for the Chern character of ${\cal U}\otimes\iota^*{\cal U}$. We will use a
compuational trick which the authors learned from \cite[...]{kirwan} to
represent most of the terms in exponential form. This will be
convenient later when we transform back to Chern classes. For that fix
a large number $N$ such that $\beta^N=0$ (e.g.\ for dimension reasons)
and let $\mu_1,\ldots,\mu_N\in\cz$ be such that
\[
N_k(\mu_1,\ldots,\mu_N)\ :=\ \sum_{\nu=1}^N\mu_\nu^k\ =\ \frac{1}{k+1}
\ \ \ \mbox{for } 1\le k\le N.
\]
The existence of $\mu_1,\ldots,\mu_N$ is clear either by an elementary
argument or from the finiteness of the map
$(N_1,\ldots,N_N):\cz^N\rightarrow\cz^N$.
\begin{lemma}
Formally adjoining $\sqrt{\beta}$ and $\alpha':=\alpha+2\gamma/\beta$
(both of real degree 2) to $H^{2*}({\calm_g},{\Bbb Q})$ the following holds:
\begin{eqnarray*}
{\mbox{\rm ch}\skp}({\cal U}\otimes\iota^*{\cal U})&=&e^\alpha\bigg[\Big(
2+e^{\sqrt{\beta}}+e^{-\sqrt{\beta}}\Big)(1+D\omega)-2\alpha\omega\\
&&-\,\alpha'\omega\sum_\nu
\Big(e^{\mu_\nu\sqrt{\beta}}+e^{-\mu_\nu\sqrt{\beta}}-2\Big)\bigg]\,.
\end{eqnarray*}
\end{lemma}
{\em Proof. }
We have
$c({\cal U}^*\otimes{\cal U})=1+c_2({\mbox{\rm End}\skp\skp}{\cal U})=1-\beta+4\psi+2\alpha\omega$
with $\psi=\sum_i\psi_i\otimes e_i$. Note that
$\iota^*:H^1(\Sigma)\rightarrow H^1(\Sigma)$ is just multiplication by
$-1$. In fact, if $\delta$ is a closed $1$-form on $\Sigma$, then
$\delta+\iota^*\delta$ is closed and $\iota^*$-invariant, hence
$\delta+\iota^*\delta=p^*df=dp^*f$ for $f\in{\cal C}^\infty({\rm I \! P}^1)$ since
$H^1({\rm I \! P}^1)=0$. But any orientation preserving diffeomorphism leaves
$c({\mbox{\rm End}\skp\skp}{\cal U})$ unchanged, hence $\iota^*\psi_i=-\psi_i$. Now it is
almost clear and can be easily verified by a standard Chern class
computation that
$c_2({\cal U}^*\otimes\iota^*{\cal U})=-\beta+2\alpha\omega$, i.e.\ that the
factor $4\psi$ drops out. Instead, there is a non-trivial $c_4$, namely
$c_4({\cal U}^*\otimes\iota^*{\cal U})=4\psi^2=4\gamma\omega$. Summarizing,
we have
\[
c({\cal U}^*\otimes\iota^*{\cal U})\ =\ 1+(-\beta+2\alpha\omega)
+4\gamma\omega\,.
\]
Next, for any bundle $E$ with only $c_2$ and $c_4$ non-vanishing
\[
{\mbox{\rm ch}\skp}_{2k}(E)\ =\ 2(-1)^k c_2(E)^{k-2}[c_2(E)^2-kc_4(E)],\ \ \
{\mbox{\rm ch}\skp}_{2k+1}(E)=0
\]
(induction on $k$). Thus for $k\ge2$ ($k=1$: ${\mbox{\rm ch}\skp}_2({\cal U}^*\otimes
\iota^*{\cal U} )=-2c_2=2\beta-4\alpha\omega$)
\begin{eqnarray*}
{\mbox{\rm ch}\skp}_{2k}({\cal U}^*\otimes\iota^*{\cal U})&=&2(-1)^k\left((-\beta)^{k-2}
+(k-2)(-\beta)^{k-3}2\alpha\omega\right)(\beta^2-4\alpha\omega-4k
\gamma\omega)\\
&=&2\beta^k-4k(\alpha\beta+2\gamma)\beta^{k-2}\omega\,.
\end{eqnarray*}
Formally adjoining $\sqrt{\beta}$ and $\alpha'=\alpha+2\gamma/\beta$
($\beta$ in the denominator always cancels in the following) we get
\begin{eqnarray*}
{\mbox{\rm ch}\skp}({\cal U}^*\otimes\iota^*{\cal U})&=&2+2\sum_{k\ge0}\frac{\beta^k}{(2k)!}
-2(\alpha\beta+2\gamma)\omega\sum_{k\ge2}2k\frac{\beta^{k-2}}{(2k)!}
-\frac{4\alpha\omega}{2}\\
&=&2+e^{\sqrt{\beta}}+e^{-\sqrt{\beta}}-2\alpha\omega-2\alpha'\omega
\sum_{k\ge1}\frac{1}{2k+1}\frac{\beta^k}{(2k)!}\,.
\end{eqnarray*}
The computational trick consists in writing
\[
2\sum_{k\ge1}\frac{1}{2k+1}\frac{\beta^k}{(2k)!}\ =\
\sum_{\nu=1}^N\Big(e^{\mu_\nu\sqrt{\beta}}
+e^{-\mu_\nu\sqrt{\beta}}-2\Big)\,.
\]
Finally using $c_1({\cal U})=\alpha+D\omega$ ($D=2g+1$) together with the
isomorphim ${\cal U}\simeq{\cal U}^*\otimes\det{\cal U}$ and the
multiplicativity of the Chern character we deduce
${\mbox{\rm ch}\skp}({\cal U}\otimes\iota^*{\cal U})=e^\alpha
{\mbox{\rm ch}\skp}({\cal U}^*\otimes\iota^*{\cal U})$ which upon inserting the previous
computations gives the stated formula.
{\hfill$\Diamond$}\vspace{1.5ex}
\vspace{10pt}
Pushing-forward the exact sequence from Lemma~\ref{exact} we get
\[
0\longrightarrow \hat p_*({\cal U}\otimes\iota^*{\cal U})^\natural
\otimes\hat p_*{\cal O}\longrightarrow \hat p_*({\cal U}\otimes\iota^*{\cal U})
\longrightarrow\hat p_*\left(S^2{\cal U}|_{{\calm_g}\times p^{-1}(B)}\right)
\longrightarrow0
\]
(for the first term apply the projection formula).
\begin{lemma}
Denoting $\bar\omega$ the normalized volume form on ${\rm I \! P}^1$ the
following holds
\begin{eqnarray*}
{\mbox{\rm ch}\skp}\left(\hat p_*({\cal U}\otimes\iota^*{\cal U})^\natural\right)
&=&e^\alpha\bigg[(2+e^{\sqrt{\beta}}+e^{-\sqrt{\beta}})
(1-\frac{\bar\omega}{2})\\
&&+\,\bar\omega\Big(-\alpha-\frac{\alpha'}{2}\sum_\nu\big(
e^{\mu_\nu\sqrt{\beta}}+e^{-\mu_\nu\sqrt{\beta}}-2\big)
+(g+1)\Big)\bigg]\,.
\end{eqnarray*}
\end{lemma}
{\em Proof. }
{}From the above exact sequence, we see
\[
{\mbox{\rm ch}\skp}\left(\hat p_*({\cal U}\otimes\iota^*{\cal U})^\natural\right)
\ =\ \left({\mbox{\rm ch}\skp} \hat p_*({\cal U}\otimes\iota^*{\cal U})-{\mbox{\rm ch}\skp}
\hat p_*(S^2{\cal U}|_{{\calm_g}\times p^{-1}(B)})\right)/{\mbox{\rm ch}\skp}(\hat p_*{\cal O})
\]
which by Grothendieck-Riemann-Roch applied to $\hat p$ and writing
$\bar g=g-1$ and ${\mbox{\rm ch}\skp}({\cal U}\otimes\iota^*{\cal U})-{\mbox{\rm ch}\skp}(S^2
{\cal U}|_{{\calm_g}\times p^{-1}(B)})=A+B\omega$, equals
\begin{eqnarray*}
\hat p_*\left[(A+B\omega)(1-\bar g\omega)\right]/
\hat p_*(1-\bar g\omega)
\ =\ \hat p_*\Big(A+(B-\bar g A)\omega\Big)/(2-\bar g\bar\omega)\\
=\ \Big(2A+(B-\bar g A)\bar\omega\Big)\frac{1}{2}(1+
\frac{\bar g}{2}\bar\omega)\ =\ A+B\frac{\bar\omega}{2}\,.
\end{eqnarray*}
To compute the Chern character of $S^2{\cal U}|_{{\calm_g}\times p^{-1}(B)}$
we restrict the exact sequence of Lemma~\ref{exact} to ${\calm_g}\times
p^{-1}(B)$. Then since $\hat p^*\hat
p_*({\cal U}\otimes\iota^*{\cal U})^\natural|_{{\calm_g}\times p^{-1} (B)}$
$\simeq\ \det{\cal U}|_{{\calm_g}\times p^{-1}(B)}$ we get
\[
{\mbox{\rm ch}\skp}(S^2{\cal U}|_{{\calm_g}\times p^{-1}(B)})\ =\ \left[{\mbox{\rm ch}\skp}({\cal U}\otimes\iota^*
{\cal U})-{\mbox{\rm ch}\skp}(\det{\cal U})\right](2g+2)\omega\,.
\]
Thus
\begin{eqnarray*}
A+B\omega&=&{\mbox{\rm ch}\skp}({\cal U}\otimes\iota^*{\cal U})\Big(1-(2g+2)\omega\Big)
+{\mbox{\rm ch}\skp}(\det{\cal U})(2g+2)\omega\\
&=&e^\alpha\bigg[(2+e^{\sqrt{\beta}}+e^{-\sqrt{\beta}})
(1-\omega)\\
&&+\,\omega\Big(-\alpha-\frac{\alpha'}{2}\sum_\nu\big(e^{\mu_\nu
\sqrt{\beta}}+e^{-\mu_\nu\sqrt{\beta}}-2\big)+(g+1)\Big)\bigg]\,,
\end{eqnarray*}
hence the claim.
{\hfill$\Diamond$}\vspace{1.5ex}
\vspace{10pt}
\noindent
{\em Proof of Proposition~\ref{chernquot}}: It follows from
Proposition~2.2 of \cite{ramanan} applied to ${\cal U}\otimes\iota^*{\cal U}$
that $R^1\pi_*(\hat p_*({\cal U}\otimes\iota^*{\cal U})^\natural)=0$. The
Grothendieck-Riemann-Roch theorem for pushing-forward
$\hat p_*({\cal U}\otimes\iota^*{\cal U})^\natural$ by
$\pi:{\calm_g}\times{\rm I \! P}^1\rightarrow{\calm_g}$ thus reads
\begin{eqnarray*}
\lefteqn{{\mbox{\rm ch}\skp}(\tilde\pi_*({\cal U}\otimes\iota^*{\cal U})^\natural)\ =\
\pi_*\left({\mbox{\rm ch}\skp}(\hat p_*({\cal U}\otimes\iota^*{\cal U})^\natural)\cdot
(1+\bar\omega)\right)}\hspace{1cm}\\
&=&e^\alpha\bigg[\frac{1}{2}(2+e^{\sqrt{\beta}}+e^{-\sqrt{\beta}})-\alpha
-\frac{\alpha'}{2}\sum_\nu\Big(e^{\mu_\nu\sqrt{\beta}}+
e^{-\mu_\nu\sqrt{\beta}}-2\Big)+(g+1)\bigg]\,.
\end{eqnarray*}
Plugging in the relation expressing the pull-back of S, i.e.\
$\varphi^*S=\tilde\pi_*({\cal U}\otimes\iota^*{\cal U})^\natural\otimes H^{-1}$,
we get
\begin{eqnarray*}
{\mbox{\rm ch}\skp}\varphi^*Q&=&(2g+2)-{\mbox{\rm ch}\skp}\varphi^*S\ =\
(2g+2)-{\mbox{\rm ch}\skp}\tilde\pi_*({\cal U}\otimes\iota^*{\cal U})^\natural/{\mbox{\rm ch}\skp}(H)\\
&=&(g-1)-\frac{1}{2}\Big(e^{\sqrt{\beta}}+e^{-\sqrt{\beta}}\Big)+\alpha
+\frac{\alpha'}{2}\sum_\nu\Big(e^{\mu_\nu\sqrt{\beta}}
+e^{-\mu_\nu\sqrt{\beta}}-2\Big)\,.
\end{eqnarray*}
Next we need to make use of the computational trick of Kirwan again: Assume
$M\gg 0$ s.th.\ $\alpha^M=\alpha'^M=0$ and find $\lambda_\kappa\in\cz$ with
$N_1(\lambda_1,\ldots,\lambda_M)=1$, $N_k(\lambda_1,\ldots,\lambda_M)=0$
for $2\le k\le M$. Then $\alpha=\sum_\kappa(e^{\lambda_\kappa\alpha}-1)$,
$\alpha'=\sum_\kappa(e^{\lambda_\kappa\alpha'}-1)$. Inserting we get
\begin{eqnarray*}
{\mbox{\rm ch}\skp}\varphi^*Q&=&(g-1)-\frac{1}{2}\Big(e^{\sqrt{\beta}}+e^{-\sqrt{\beta}}\Big)
+\sum_\kappa(e^{\lambda_\kappa\alpha}-1)\\
&&+\frac{1}{2}\sum_{\kappa,\nu}\left(e^{\lambda_\kappa\alpha'+\mu_\nu
\sqrt{\beta}}+e^{\lambda_\kappa\alpha'-\mu_\nu\sqrt{\beta}}
-e^{\mu_\nu\sqrt{\beta}}-e^{-\mu_\nu\sqrt{\beta}}-2(e^{\lambda_\kappa
\alpha'}-1)\right)\,.
\end{eqnarray*}
This is a sum of exponentials and as such easily transformed into the
corresponding total Chern class:
\begin{eqnarray*}
c(\varphi^*Q)&=&\Big[\Big(1+\sqrt{\beta}\Big)\Big(1-\sqrt{\beta}\Big)
\Big]^{-1/2}
\prod_\kappa(1+\lambda_\kappa\alpha)\\
&&\cdot\,\bigg[\prod_{\kappa,\nu}
\frac{1+\mu_\nu\sqrt{\beta}+\lambda_\kappa\alpha'}{1+\mu_\nu\sqrt{\beta}}
\cdot
\frac{1-\mu_\nu\sqrt{\beta}+\lambda_\kappa\alpha'}{1-\mu_\nu\sqrt{\beta}}
\cdot\frac{1}{(1+\lambda_\kappa\alpha')^2}\bigg]^{1/2}\!\!.
\end{eqnarray*}
To get rid of the $\lambda_\kappa$ we observe that $\sigma_k(\lambda_1,
\ldots,\lambda_M)=1/k!$ \cite[p.862]{kirwan}. The product over $\kappa$ can
thus be carried out, e.g.
\begin{eqnarray*}
\lefteqn{\prod_{\kappa,\nu}\frac{1+\mu_\nu\sqrt{\beta}+\lambda_\kappa
\alpha'}{1+\mu_\nu\sqrt{\beta}}\ =\
\prod_{\kappa,\nu}\bigg(1+\lambda_\kappa\frac{\alpha'}{1+\mu_\nu\sqrt{\beta}}
\bigg)\ =\ \prod_\nu\exp\frac{\alpha'}{1+\mu_\nu\sqrt{\beta}}}\hspace{4cm}\\
&=&\exp\alpha'\sum_\nu\sum_{l\ge0}\Big(-\mu_\nu\sqrt{\beta}\Big)^l\ =\
\exp\alpha'\sum_{l\ge0}\frac{(-\sqrt{\beta})^l}{l+1}\,.
\end{eqnarray*}
Inserting into our last formula we thus find (the term $(1+\lambda_\kappa
\alpha')^{-2}$ cancels the summand for $l=0$)
\begin{eqnarray*}
c(\varphi^*Q)&=&(1-\beta)^{-1/2}\exp\bigg(\alpha+\frac{\alpha'}{2}\sum_{l\ge1}
\frac{(-\sqrt{\beta})^l+(\sqrt{\beta})^l}{l+1}\bigg)\\
&=&(1-\beta)^{-1/2}\exp\bigg(\alpha+\alpha'\sum_{m\ge1}\frac{1}{2m+1}
\beta^m\bigg)
\end{eqnarray*}
as claimed.
{\hfill$\Diamond$}\vspace{1.5ex}
\vspace{10pt}
It is convenient to gather the relations in a generating function.
\begin{defi}\label{genfct}
We define $\Phi\in{\Bbb Q}[\alpha,\beta,\gamma][\![t]\!]$ by
\[
\Phi(t)\ :=\ (1-\beta t^2)^{-1/2}
\exp\bigg[\alpha t+\Big(\alpha+\frac{2\gamma}{\beta}
\Big)t\sum_{m\ge 1}\frac{\beta^m t^{2m}}{2m+1}\bigg]\,.
\]
\end{defi}
For later use let us also state a functional equation that $\Phi$ obeys.
This equation is actually equivalent to the recursion formula
to be proved below (Proposition~\ref{recursion}).
\begin{prop}\label{fctleqn}
$\Phi$ obeys the following differential equation:
\[
\Phi'(t)\ =\ \frac{\alpha+\beta t+2\gamma t^2}{1-\beta t^2}
\cdot\Phi(t)\,.
\]
\end{prop}
{\em Proof. }
Direct computation.
{\hfill$\Diamond$}\vspace{1.5ex}
\vspace{10pt}
Let us add that with the same methods, it is not hard to deduce also
a closed formula for the Chern classes of ${\cal N}_g$. The result is:
\[
c({\cal N}_g)\ =\ (1-\beta)^g \exp\Big(\frac{-8\gamma}{1-\beta}\Big)\cdot
c(\varphi^*Q)^2\,.
\]
Note the simple dependence on the genus!
\section{A minimal set of relations}
We have remarked in the introduction that the three generating
relations $f_1^g$, $f_2^g$, $f_3^g$ of degrees $g$, $g+1$ and $g+2$ are
uniquely determined by their initial terms $\alpha^g$,
$\alpha^{g-1}\beta$ and $\alpha^{g-1}\gamma$ respectively (w.r.t.\ the
reverse lexicographic order; we will prove this as an easy consequence
of the recursion relations, see Proposition~\ref{inf}). It is then an
exercise in calculus to find the following
\begin{defi}
Writing $\Phi^{(r)}=\displaystyle\frac{d^r\Phi}{dt^r}(0)$
we define for $g\ge1$
\begin{eqnarray*}
f_1^g&:=&\Phi^{(g)}\\
f_2^g&:=&\frac{1}{g^2}\left(\Phi^{(g+1)}-\alpha\Phi^{(g)}\right)\\
f_3^g&:=&\frac{1}{2g(g+1)}\left(\Phi^{(g+2)}-\alpha\Phi^{(g+1)}
-(g+1)^2\beta\Phi^{(g)}\right)\,.
\end{eqnarray*}
\end{defi}
\vspace{10pt}
We are now in a position to prove the recursion relations.
\begin{prop}\label{recursion}
$(f_1^1,f_2^1,f_3^1)=(\alpha,\beta,\gamma)$ and inductively for $g\ge1$
\begin{eqnarray*}
f_1^{g+1}&=&\alpha f_1^g+g^2 f_2^g\\
f_2^{g+1}&=&\beta f_1^g+\frac{2g}{g+1}f_3^g\\
f_3^{g+1}&=&\gamma f_1^g\,.
\end{eqnarray*}
\end{prop}
{\em Proof. }
The first claim is by direct check. Next, the recursion relations for
$f_1^{g+1}$ and $f_2^{g+1}$ are immediate consequences of their
definition. All the work is thus shifted to the innocuous looking
formula for $f_3^{g+1}$. What we have to show is the vanishing of
\begin{eqnarray*}
\lefteqn{2(g+1)(g+2)\left(f_3^{g+1}-\gamma
f_1^g\right)}\hspace{1cm}\\
&=&\Phi^{(g+3)}-\alpha\Phi^{(g+2)}-(g+2)^2\beta\Phi^{(g+1)}-2(g+1)(g+2)
\gamma\Phi^{(g)}\\
&=&(g+2)!\left[(g+3)\varphi_{g+3}-\alpha\varphi_{g+2}-(g+2)\beta\varphi_{g+1}
-2\gamma\varphi_g\right]
\end{eqnarray*}
with $\varphi_k$ the $k$-th Taylor coefficient of $\Phi$ at $t=0$.
Multiplying with $t^{g+3}$ and taking the sum this will follow from
\[
\sum_{g\ge1}(g+3)\varphi_{g+3}t^{g+3}\ =\ \alpha
t\sum_{g\ge1}\varphi_{g+2}t^{g+2} +\beta
t^2\sum_{g\ge1}(g+2)\varphi_{g+1}t^{g+1}+2\gamma t^3\sum_{g\ge1}\varphi_g t^g
\]
which is the part of order larger 3 of
\[
t\cdot\Phi'\ =\ \alpha t\Phi+\beta t^2(\Phi\cdot t)'+2\gamma t^3\Phi
\ =\ (\alpha t+\beta t^2+2\gamma t^3)\Phi+\beta t^3\Phi'\,,
\]
the functional equation for $\Phi$ (Proposition~\ref{fctleqn}).
{\hfill$\Diamond$}\vspace{1.5ex}
\section{The Leitideal}
The decisive step in the proof of completeness of our relations is that
the Leitideal (initial ideal) can be computed completely and has a
particularly simple form. In all that follows we use the (graded)
reverse lexicographic order in ${\Bbb Q}[\alpha,\beta,\gamma]$ and write
${\mbox{\rm In}\skp}(f)$ (${\mbox{\rm In}\skp}({\cal I})$) for the initial term of
$f\in{\Bbb Q}[\alpha,\beta,\gamma]$ (resp.\ the initial ideal of an ideal
${\cal I}\subset{\Bbb Q}[\alpha,\beta,\gamma]$). As warm-up let us check that
the initial terms of the $f_i^g$ are as promised in the last chapter:
\begin{prop}\label{inf}
In the reverse lexicographic order ${\mbox{\rm In}\skp}(f_1^g)=\alpha^g$,
${\mbox{\rm In}\skp}(f_2^g)=\alpha^{g-1}\beta$, ${\mbox{\rm In}\skp}(f_3^g)=\alpha^{g-1}\gamma$.
\end{prop}
{\em Proof. }
By induction on $g$. $g=1$ is clear by the first line of
Proposition~\ref{recursion}. Applying our recursion relations and the
induction hypothesis, we get
$f_1^{g+1}=\alpha^{g+1}+\alpha^g\beta+\ldots\,$,
$f_2^{g+1}=\alpha^g\beta+\frac{2g}{g+1}\alpha^{g-1}\gamma+\ldots\,$,
$f_3^{g+1}=\alpha^g\gamma+\ldots\,$, where $\ldots$ mean terms of lower
order.
{\hfill$\Diamond$}\vspace{1.5ex}
Now setting ${\cal I}_g:=(f_1^g,f_2^g,f_3^g)\subset
{\Bbb Q}[\alpha,\beta,\gamma]$, the ideal spanned by $f_i^g$, $i=1,2,3$, then
\begin{prop}\label{inideal}
${\mbox{\rm In}\skp}({\cal I}_g)=(\alpha^a\beta^b\gamma^c,a+b+c\ge g)$.
\end{prop}
{\em Proof. }
By induction on $g$, $g=1$ being trivially true. From
\begin{eqnarray*}
\gamma f_1^g&=&f_3^{g+1}\\
g^2\gamma f_2^g&=&\gamma f_1^{g+1}-\alpha\gamma f_1^g
\ =\ \gamma f_1^{g+1}-\alpha f_3^{g+1}\\
\frac{4g}{g+1}\gamma f_3^g&=&\gamma f_2^{g+1}-\beta\gamma f_1^g
\ =\ \gamma f_2^{g+1}-\beta f_3^{g+1}
\end{eqnarray*}
we see that $\gamma{\cal I}_g\subset{\cal I}_{g+1}$ (this is also clear from
the observation that $\gamma\in H^*({\cal M}_{g+1})$ is Poincar\'e dual to
$2g$ copies of ${\calm_g}$, cf.\ below). By induction hypothesis the claim
is thus true for $c>0$. But in any homogeneous expression (with
$\alpha$, $\beta$, $\gamma$ having weights $1$, $2$, $3$ respectively)
the monomials containing $\gamma$ have lower order than those without.
Therefore, we can reduce modulo $\gamma$ (indicated by a bar) and have
only to show ${\mbox{\rm In}\skp}(\bar{{\cal I}_g})=(\bar\alpha^a\bar\beta^b,a+b\ge g)$.
Modulo $\gamma$ the recursion relations read
\[
\bar{f}_1^{g+1}=\bar\alpha\bar{f}_1^g+g^2\bar{f}_2^g,\ \ \
\bar{f}_2^{g+1}=\bar\beta\bar{f}_1^g\,.
\]
Now we are able to repeat the argument from above with $\bar\beta$
instead of $\gamma$, because
\begin{eqnarray*}
\bar\beta\bar{f}_1^g&=&\bar{f}_2^{g+1}\\
g^2\bar\beta\bar{f}_2^g&=&\bar\beta\bar{f}_1^{g+1}-\bar\alpha\bar\beta
\bar{f}_1^g\ =\ \bar\beta\bar{f}_1^{g+1}-\bar\alpha\bar{f}_2^{g+1}\,.
\end{eqnarray*}
This leaves us with the case $b=0$, $c=0$, which is clearly true since
$\alpha^g={\mbox{\rm In}\skp}(f_1^g)$ is the smallest power of $\alpha$ contained in
${\cal I}_g$ (for $\deg f_i^g\le g$, $i=1,2,3$).
{\hfill$\Diamond$}\vspace{1.5ex}
\section{Completeness}
The strategy of showing that ${\cal I}_g=(f_1^g,f_2^g,f_3^g)\subset
\cz[\alpha,\beta,\gamma]$ is really the ideal of relations between
$\alpha$, $\beta$, $\gamma$ is a simple dimension count. But although
the subring $\langle\alpha,\beta,\gamma\rangle\subset H^*({\calm_g},{\Bbb Q})$
generated by $\alpha$, $\beta$, $\gamma$ is the invariant ring under
the action of the orientation preserving diffeomorphisms, the authors
do not know a direct way to compute
$\dim_{\Bbb Q}\langle\alpha,\beta,\gamma\rangle$. Instead we are viewing the
even cohomology $H^{2*}({\calm_g},{\Bbb Q})$ as module over
$\cz[\alpha,\beta,\gamma]/ {\cal I}_g$ and check injectivity of the
structure map $\cz[\alpha,\beta,\gamma]/ {\cal I}_g\rightarrow
H^{2*}({\calm_g},{\Bbb Q})$ by refining the basis $\{\alpha^a
\beta^b\gamma^c\mid a+b+c<g\}$ of $\cz[\alpha,\beta,\gamma]/{\cal I}_g$ to
a basis of $H^{2*}({\calm_g},{\Bbb Q})$. As a by-product we will actually find
an explicit basis of the latter, which in a sense explains the
inductive formulas for the even Betti numbers found by Newstead
\cite{newstead2}.
\begin{prop}\label{generators}
$H^{2*}({\calm_g},{\Bbb Q})$ is generated by elements of the form\\[5pt]
\hspace*{2cm} $\alpha^a\beta^b\psi_{i_1}\ldots\psi_{i_{2l}}$
with $a+b+2l<g-1$, \\
\hspace*{1cm}and $\alpha^a\beta^b\gamma^k\psi_{i_1}\ldots\psi_{i_{2l}}$
with $a+b+k+2l=g-1$, $k\ge0$,\\[5pt]
where $1\le i_1<\ldots<i_{2l}\le2g$.
\end{prop}
As an intermediate notion between the $\psi_i$ and $\gamma$ let us
introduce the classes $\gamma_j:=\psi_j\psi_{j+g}$, $j=1,\ldots,g$
(then $\gamma=2\sum_j\gamma_j$). Each of the $\gamma_j$ is Poincar\'e
dual to a diffeomorphic image $N_j$ of ${\cal M}_{g-1}$ (by ``contracting
a handle'', cf.\ no.26 in \cite{thaddeus}). Moreover, ${\cal U}|N_j$ is
topologically a universal bundle on ${\cal M}_{g-1}$, so $\alpha$,
$\beta$, $\gamma$ restrict to generators $\hat\alpha$, $\hat\beta$,
$\hat\psi_i$ ($i\neq j,j+g$) of $H^*({\cal M}_{g-1},{\Bbb Q})$
($\psi_j|N_j=0=\psi_{j+g}|N_j$ since $\psi_j\gamma_j=0
=\psi_{j+g}\gamma_j$ trivially). We will also use the fact that
intersection proucts $\alpha^a\beta^b\psi_{i_1}\ldots\psi_{2l}[{\calm_g}]$
($a+2b+3l=3g-3$) are zero unless
$\{i_1,\ldots,i_{2l}\}=\{j_1,j_1+g,\ldots,j_l,j_l+g\}$ in which case
\[
\alpha^a\beta^b\psi_{i_1}\ldots\psi_{i_{2l}}[{\calm_g}]
=\frac{1}{g!}\alpha^a\beta^b\gamma^l[{\calm_g}]\,,
\]
i.e.\ only depending on the length of the sequence $(j_1,\ldots,j_l)$,
cf.\ \cite{thaddeus}.
\vspace{15pt}
\noindent
{\em Proof of proposition.}
We want to refine the result of Proposition~\ref{inideal} that a
monomial $\alpha^a\beta^b\gamma^c$ is equivalent (= may be reduced
modulo ${\cal I}_g$) to a polynomial of lower order. For this we will use
the reverse lexicographic order
$\alpha>\beta>\psi_1>\ldots>\psi_{2g}>\gamma_1>\ldots>\gamma_{2g}>
\gamma$.
Let $1\le i_1<\ldots<i_k\le2g$ and $1\le j_1<\ldots<j_k\le g$ with
$\{i_1,\ldots,i_k\}\cap\{j_1,j_1+g,\ldots,j_l,j_l+g\}=\emptyset$.
\vspace{5pt}
\noindent
{\em Claim:} If $a+b+k+l\ge g$ then
$\alpha^a\beta^b\psi_{i_1}\ldots\psi_{i_k}
\gamma_{j_1}\ldots\gamma_{j_l}$ is equivalent to a polynomial of lower
order modulo ${\cal I}_g$, which can be taken of the form
$F(\alpha,\beta)\,
\psi_{i_1}\ldots\psi_{i_k}\gamma_{j_1}\ldots\gamma_{j_l}$.
\vspace{-7pt}
\noindent
The claim certainly holds if $k+l=0$ by Proposition~\ref{inideal}. For
$l>0$ let $\iota:{\cal M}_{g-1}\hookrightarrow{\calm_g}$ have image $N_{j_k}$
(Poincar\'e dual to $\gamma_{j_k}$) and use a hat to denote pull-back
by $\iota$. By descending induction on $g$ then ($l<g$ because
$\gamma_1\ldots\gamma_g=0$ for dimension reasons),
\begin{eqnarray*}
\lefteqn{\iota^*\left(\alpha^a\beta^b\psi_{i_1}\ldots\psi_{i_k}
\gamma_{j_1}\ldots\gamma_{j_{l-1}}\right)
\ =\ {\hat\alpha}^a{\hat\beta}^b{\hat\psi_{i_1}}\ldots{\hat\psi_{i_k}}
{\hat\gamma_{j_1}}\ldots{\hat\gamma_{j_{l-1}}} }\\
&=&F(\hat\alpha,\hat\beta)\,{\hat\psi_{i_1}}\ldots{\hat\psi_{i_k}}
{\hat\gamma_{j_1}}\ldots{\hat\gamma_{j_{l-1}}}
\ =\ \iota^*\left(F(\alpha,\beta)\,\psi_{i_1}\ldots\psi_{i_k}
\gamma_{j_1}\ldots\gamma_{j_{l-1}}\right)
\end{eqnarray*}
with $\mbox{order}(F)<a+b$. This means $\alpha^a\beta^b\psi_{i_1}\ldots
\psi_{i_k}\gamma_{j_1}\ldots\gamma_{j_l}=F(\alpha,\beta)\,\psi_{i_1}
\ldots\psi_{i_k}\gamma_{j_1}$ $\ldots\gamma_{j_l}$ as wanted. Finally
the case $l=0$, $k>0$: Set $\bar\imath_k=i_k+g$ if $i_k\le g$ and
$\bar\imath_k=i_k-g$ if $i_k>g$. By the previous case, we get
$\alpha^a\beta^b\psi_{i_1}\ldots
\psi_{i_k}\psi_{\bar\imath_k}=F(\alpha,\beta)\psi_{i_1}\ldots
\psi_{i_k}\psi_{\bar\imath_k}$. Then the above remarks on the
intersection product show
\[
\left(\alpha^a\beta^b\psi_{i_1}\ldots\psi_{i_k}
-F(\alpha,\beta)\,\psi_{i_1}\ldots\psi_{i_k}\right)\cdot
A[{\calm_g}]\ =\ 0
\]
for all $A\in H^*({\calm_g})$,
i.e.\ $\alpha^a\beta^b\psi_{i_1}\ldots\psi_{i_k}
=F(\alpha,\beta)\psi_{i_1}\ldots\psi_{i_k}$. This proves the claim.\\
The proposition is now clearly reduced to a second
\vspace{5pt}
\noindent
{\em Claim:} Let $M=\alpha^a\beta^b\psi_{i_1}\ldots\psi_{i_k}
\gamma_{j_1}\ldots\gamma_{j_l}$ with $a+b+k+l=g-2$. Then for all $1\le
i,j \le g$, $M\gamma_i-M\gamma_j$ may be reduced to lower order modulo
${\cal I}_g$.
\vspace{5pt}
\noindent
In fact, from the above we know already
$M\gamma_i\gamma_j=F\gamma_i\gamma_j$ in $H^*({\calm_g})$ with
$\mbox{order}(F)<g-2$. $F\gamma_i-F\gamma_j$ is our candidate for the
lower order term. If $\{i,i+g,j,j+g\}\cap\{i_1,\ldots,i_k\} =\emptyset$
and $A=\alpha^{a'}\beta^{b'}\psi_{i_1}\ldots\psi_{i_k}$ then
\[
\left(M\gamma_i-M\gamma_j\right)\cdot A[{\calm_g}]\ =\ 0\
=\ \left(F\gamma_i-F\gamma_j\right)\cdot A[{\calm_g}]
\]
again by the symmetry in the $\gamma_i$ of the intersection pairing. On
the other hand
\begin{eqnarray*}
(M\gamma_i-M\gamma_j)\gamma_j[{\calm_g}]
&=&M\gamma_i\gamma_j A[{\calm_g}]
\ =\ F\gamma_i\gamma_j A[{\calm_g}]\\
&=& (F\gamma_i-\gamma_j)\gamma_j[{\calm_g}]
\end{eqnarray*}
and analogously with $\gamma_i$. Thus $M\gamma_i=M\gamma_j+
F(\gamma_i-\gamma_j)$ modulo ${\cal I}_g$ as claimed.\\
(Note: This argument fails for the $\psi_i$ because of the presence of
$\psi_{i+g}$ respectively $\psi_{i-g}$.)
{\hfill$\Diamond$}\vspace{1.5ex}
The only thing we finally need to do is to count the number of generators
and compare with the inductive formula for the Betti numbers found by
Newstead.
\begin{prop}\label{dimcount}
The generators for $H^{2*}({\calm_g},{\Bbb Q})$ in Proposition~\ref{generators}
are linearly independent up to the middle dimension.
\end{prop}
{\em Proof. }
Let $G_r$ be the set of generators of (real) degree $r$ from
Proposition~\ref{generators}, $g_r:=\sharp G_r$. We will show that for
$s\displaystyle\le\left[\frac{3g-1}{2}\right]$
\[
g_{2s+4}\ =\ g_{2s}+\sum_{l=s-g+1}^{[s/3]}{2g\choose 2l}
\]
which together with $g_0=1$ and $g_2=1$ is exactly Newstead's formula
for the even Betti numbers \cite{newstead1}.
Define a map $\varphi:G_{2s}\rightarrow G_{2s+4}$ by
\[
\alpha^a\beta^b\psi_{i_1}\ldots\psi_{i_{2l}}\longmapsto
\alpha^a\beta^{b+1}\psi_{i_1}\ldots\psi_{i_{2l}}\ \ \
\mbox{for }a+b+2l<g-1
\]
and
\begin{eqnarray*}
\alpha^a\beta^b\gamma^k\psi_{i_1}\ldots\psi_{i_{2l}}\longmapsto
\alpha^{a-1}\beta^b\gamma^{k+1}\psi_{i_1}\ldots\psi_{i_{2l}}
&&\mbox{ for }a+b+k+2l=g-1,
\end{eqnarray*}
with $k\ge0$ and $a>0$.
Note that the case $a=k=0$ does not occur (then $b+2l=g-1$ and
$2b+3l=s$ imply $s-2g+2=-l\le0$ which contradicts
$s\le(3g-1)/2$). Now $G_{2s+4}\setminus{\mbox{\rm im}\skp\skp}\varphi=\{\alpha^a\psi_{i_1}\ldots
\psi_{i_{2l}}\mid a+2l\le g-1, a+3l=s\}$ s.th.\ $l$ runs from
$s-g+1$ to $[s/3]$ ($a$ is determined through $a+3l=s$). The contribution
for $l$ fixed is then precisely $2g\choose2l$.
{\hfill$\Diamond$}\vspace{1.5ex}
|
1994-10-21T05:20:16 | 9410 | alg-geom/9410020 | en | https://arxiv.org/abs/alg-geom/9410020 | [
"alg-geom",
"math.AG"
] | alg-geom/9410020 | Bas Edixhoven | Bas Edixhoven | On the prime-to-$p$ part of the groups of connected components of
N\'eron models | 21 pages, LaTeX, 94-23 | null | null | null | null | This article improves certain results of Dino Lorenzini concerning the groups
of connected components of special fibres of N\'eron models of abelian
varieties. Lorenzini has shown the existence of a four step filtration on the
prime-to-$p$ part ($p$ is the residue characteristic), and proved certain
bounds for the successive quotients. We improve these bounds and show that our
bounds are sharp. As an application, we give a complete classification of the
groups that can arise as the prime-to-$p$ part of the group of connected
components of the Neron model of an abelian variety with given dimensions for
the abelian, toric and unipotent part. Hard copies of this preprint are
available.
| [
{
"version": "v1",
"created": "Thu, 20 Oct 1994 09:55:48 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Edixhoven",
"Bas",
""
]
] | alg-geom | \section{\@startsection {section}{1}{\z@}{-3.5ex plus -1ex minus
-.2ex}{2.3ex plus .2ex}{\large\bf}}
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\title{On the prime-to-$p$ part of the groups of connected components
of N\'eron models.}
\author{Bas Edixhoven}
\newtheorem{theorem}[subsection]{Theorem.}
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\begin{document}
\maketitle
\tableofcontents
\section{Introduction.}
The aim of this article was originally to improve certain results of Dino
Lorenzini concerning the groups of connected components of special fibres of
N\'eron models of abelian varieties. Let $D$ be a strictly henselian
discrete valuation ring, $K$ its field of fractions, $k$ its residue field and
$A_K$ an abelian variety over $K$ with N\'eron model $A$ over $D$. Let $p\geq0$
be the characteristic of $k$ and let $\Phi_{(p)}$ denote the prime-to-$p$ part
of the group of connected components $\Phi$ of $A_k$.
In \cite{Dino1} Lorenzini shows the existence of a functorial four step
filtration on $\Phi_{(p)}$ and he proves certain properties satisfied by
this filtration. In particular, he gives bounds for the successive quotients.
These bounds are of the following type. For a prime $l$ and a finite abelian
group $G$ of $l$-power order, say $G\cong\oplus_{i\geq1}{\msy Z}/l^{a_i}{\msy Z}$
with $a_1\geq a_2\geq\cdots$, he defines
$\delta_l'(G):=l^{a_1}-1+(l-1)\sum_{i\geq2}a_i$.
Then he gives bounds for the
$\delta_l'$ of certain successive quotients in terms of the dimensions of the
toric and abelian variety parts of the special fibres of N\'eron models of
$A_K$ over various extensions of~$D$. In \cite[Remark~2.16]{Dino1} he remarks
that the bounds might possibly be improved by replacing $\delta_l'$ by
an other invariant $\delta_l$ defined as follows: for $G$ as above one
has $\delta_l(G)=\sum_{i\geq1}(l^{a_i}-1)$. This improvement is exactly
what we do in this article. The results can be found in \S\ref{section3}.
Needless to say, we follow very much the approach of \cite{Dino1} in order
to prove these sharper bounds. In fact, only Lemma~2.13 of \cite{Dino1} has
to be changed, so the proof we give is rather short. We have taken this
opportunity to weaken slightly the hypotheses of Lorenzini's results
(he supposes $D$ to be complete and $k$ to be algebraically closed).
In \S\ref{section2} we recall Lorenzini's filtration. In \S\ref{section3} we
state and prove the bounds on the $\delta_l$ of the $l$-parts of certain
successive quotients and in \S\ref{section5} we show by some examples that
the bounds of \S\ref{section3} are sharp; \S\ref{section4} is used
to show some results on finite abelian groups that are needed in
the other sections.
After all this work it turned out that a complete classification of the
possible $\Phi_{(p)}$ for abelian varieties whose reduction has toric part,
abelian variety part and unipotent part of fixed dimensions was in reach.
The result, which is surprisingly simple to state, can be found in
Thm.~\ref{thm61}.
In this article we will frequently speak of the abelian variety part,
the toric part and the unipotent part of the fibre over $k$ of a N\'eron
model over $D$. Since we are only interested in the characteristic polynomials
of certain endomorphisms on the toric and abelian variety part, it suffices
to define these parts after base change to an algebraic closure of $k$, and
up to isogeny. Over the algebraic closure of $k$ we can apply Chevalley's
theorem; in \cite[Thm.~9.2.1, Thm.~9.2.2]{BLR} one finds what is needed, and
even more.
I would like to thank Xavier Xarles for indicating a mistake in an earlier
version of this article, and Rutger Noot for his help concerning the proofs
of Lemma's~\ref{lemma410} and~\ref{lemma411}.
\section{Lorenzini's filtration.} \label{section2}
Let $D$ be a discrete valuation ring, let $K$ be its field of fractions and
$k$ its residue field. Let $A_K$ be an abelian variety over $K$, $A$ its
N\'eron model over $D$ and $\Phi:=A_k/A_k^0$ the finite \'etale group scheme
over $k$ of connected components of the special fibre $A_k$. Let $p\geq0$ be
the characteristic of $k$ and let $\Phi_{(p)}$ be the prime-to-$p$ part of
$\Phi$; if $p=0$ we define $\Phi_{(p)}$ to be equal to $\Phi$.
In this section we will briefly recall the construction in \cite{Dino1} of a
descending filtration
\begin{eqn} \label{eqn21}
\Phi_{(p)} = \Phi_{(p)}^0 \supset \Phi_{(p)}^1 \supset \Phi_{(p)}^2 \supset
\Phi_{(p)}^3 \supset \Phi_{(p)}^4 = 0
\end{eqn}
which is functorial in $A_K$ and invariant under base change by automorphisms
of $D$.
Since $\Phi_{(p)}$ is the direct sum of its $l$-parts $\Phi_l$, with $l$
ranging through the primes different from $p$, it suffices to describe the
filtration on each $\Phi_l$. We replace $D$ by its strict henselization and
view the group scheme $\Phi$ over the separably closed field $k$ as just a
group.
Let $l\neq p$ be a prime number. Let $K\to{K^{\rm s}}$ be a separable closure,
let ${D^{\rm s}}$ be the integral closure of $D$ in ${K^{\rm s}}$ and let ${\overline{k}}$ be
the residue field of ${D^{\rm s}}$; note that ${\overline{k}}$ is an algebraic closure of
$k$ and that $k\to{\overline{k}}$ is purely inseparable. The first step in the
construction of the filtration is the description of $\Phi_l$ in
terms of the Tate module $U_l:={\rm T}_l(A({K^{\rm s}}))$ with its action by
$I:={\rm Gal}({K^{\rm s}}/K)$ given in Prop.~11.2 of \cite{Grothendieck1}:
\begin{eqn} \label{eqn22}
\Phi_l = \left(U_l\otimes{\msy Q}/{\msy Z}\right)^I/
\left(U_l^I\otimes{\msy Q}/{\msy Z}\right)
\end{eqn}
The long exact cohomology sequence of the short exact sequence
$$
0\to U_l\to U_l\otimes{\msy Q}\to U_l\otimes{\msy Q}/{\msy Z}\to 0
$$
of continuous $I$-modules gives a canonical isomorphism
(see \cite[(11.3.8)]{Grothendieck1})
\begin{eqn} \label{eqn23}
\Phi_l = {\rm tors}({\rm H}^1(I,U_l))
\end{eqn}
where for $M$ any abelian group, ${\rm tors}(M)$ denotes the subgroup of
torsion elements. Let ${I_{\rm t}}$ be the quotient of $I$ corresponding to
the maximal tamely ramified extension of $D$, and let $P$ be the kernel
of $I\to{I_{\rm t}}$. Then ${I_{\rm t}}$ is canonically isomorphic to
$\prod_{q\neq p}{\rm T}_q({{\rm G}_m}(k))=\prod_{q\neq p}{\msy Z}_q(1)$ and $P$ is a pro-$p$
group. The Hochschild-Serre spectral sequence shows that
\begin{eqn}\label{eqn24}
\Phi_l = {\rm tors}({\rm H}^1(I,U_l)) = {\rm tors}({\rm H}^1({I_{\rm t}},U_l^P)) =
{\rm tors}((U_l^P)_{I_{\rm t}})(-1) = {\rm tors}((U_l)_I)(-1)
\end{eqn}
with the lower indices ${I_{\rm t}}$ and $I$ denoting coinvariants and ``$(-1)$''
a Tate twist. Let $N_l$ be the submodule of $U_l$ which is generated
by the elements $\sigma(x)-x$ with $\sigma$ in $I$ and $x$ in $U_l$. Then
by definition we have $(U_l)_I=U_l/N_l$. As in \cite[\S2.5]{Grothendieck1}, we
define $V_l:=U_l^I$. Then $V_l$, which is called the fixed part of $U_l$, is
canonically isomorphic to ${\rm T}_l(A_k(k))$.
Let $A_K'$ be the dual of $A_K$, i.e., $A_K'={\rm Pic}^0_{A_K/K}$. We will denote
by $A'$ the N\'eron model over $D$ of $A_K'$, by $\Phi'$ its group of
connected components, etc.
Let $\langle{\cdot},{\cdot}\rangle\colon U_l\times U_l'\to{\msy Z}_l(1)$ be the Weil
pairing.
For any $y$ in $V'_l$, $\sigma$ in $I$ and $x$ in $U_l$ we have
$\langle\sigma(x)-x,y\rangle = \langle\sigma(x),y\rangle-\langle x,y\rangle =
\sigma(\langle x,\sigma^{-1}(y)\rangle) - \langle x,y\rangle = 0$. It follows that
$N_l$ is contained in the orthogonal ${V_l'}^\perp$ of $V_l'$ in $U_l$.
Since $U_l/{V_l'}^\perp$ is torsion free, we conclude that
\begin{eqn} \label{eqn25}
\Phi_l = {\rm tors}\left({V_l'}^\perp/N_l\right)(-1)
\end{eqn}
\begin{rmk} \label{rmk26}
In the proof of Thm.~\ref{thm33} we will see that ${V_l'}^\perp/N_l$ is in
fact a finite group, hence we have $\Phi_l = ({V_l'}^\perp/N_l)(-1)$.
\end{rmk}
Now it is clear that any filtration on ${V_l'}^\perp$ induces a filtration
on $\Phi_l$. As in \cite[\S2.5]{Grothendieck1}, we define $W_l\subset V_l$
to be the submodule corresponding to the maximal torus in $A_k$.
Let ${\widetilde{W}}_l\subset{\widetilde{V}}_l\subset V_l$ be the submodules called the
essentially toric part and the essentially fixed part in
\cite[\S4.1]{Grothendieck1}; if $G/k'$ is the connected component of the
special fibre of a semi-stable N\'eron model of $A_K$ over a suitable
sub-extension of $K\to{K^{\rm s}}$ then ${\widetilde{V}}_l$ corresponds to ${\rm T}_l(G({\overline{k}}))$
and ${\widetilde{W}}_l$ to the Tate module of the maximal torus in $G$.
We denote by $t$, $a$ and $u$ the dimensions of the toric part, the
abelian variety part and the unipotent part of $A^0_{\overline{k}}$; we denote by
${\tilde{t}}$ and ${\tilde{a}}$ the analogous dimensions of any semi-stable reduction
of $A_K$. Note that $t+a+u={\tilde{t}}+{\tilde{a}}=\dim(A_K)$. An easy application
of the Igusa-Grothendieck orthogonality theorem (which states that
$W_l=V_l\cap{V_l'}^\perp$, see \cite[Thm.~2.4]{Grothendieck1}, or
\cite[Thm.~3.1]{Oort1}), gives us the following filtration of ${V_l'}^\perp$,
in which the successive quotients are torsion free and of the indicated rank:
\begin{eqn} \label{eqn27}
{V_l'}^\perp \;\;\stackrel{{\tilde{t}}-t}{\supset}\;\; {\widetilde{V}}_l\cap{V_l'}^\perp
\;\;\stackrel{2({\tilde{a}}-a)}{\supset}\;\; {\widetilde{W}}_l
\;\;\stackrel{{\tilde{t}}-t}{\supset}\;\; W_l
\;\;\stackrel{t}{\supset}\;\; 0
\end{eqn}
Lorenzini's filtration (\ref{eqn21}) on $\Phi_l$ is the filtration induced by
(\ref{eqn25}) and (\ref{eqn27}). Note that in fact any finite sub-extension
of $K\to{K^{\rm s}}$ induces a filtration on $\Phi_l$ as above; see
\cite[Thm.~3.1]{Dino1} for results concerning those filtrations. The reason
we only consider the filtration coming from extensions over which $A_K$ has
semi-stable reduction is that only that filtration matters for the bounds
on $\Phi_{(p)}$ of the next section.
\section{Bounds on $\Phi_{(p)}$.} \label{section3}
We keep the notation of the previous section. Recall that $K$ is strictly
henselian. First we define some invariants of finite abelian groups and fix
some notation needed to state our results.
\begin{definition} \label{def31}
For $l$ a prime number and $a=(a_1,a_2,\ldots)$ a sequence of integers
$a_i\geq0$ with $a_i=0$ for $i$ big enough, let
$\delta_l(a):=\sum_i(l^{a_i}-1)$. For $l$ a prime number and
$G\cong\oplus_i{\msy Z}/l^{a_i}{\msy Z}$ a finite abelian group of $l$-power order let
$\delta_l(G):=\delta_l(a)$, where $a:=(a_1,a_2,\ldots)$.
For $G$ a finite abelian group let $\delta(G):=\sum_l\delta_l(G_l)$,
where $G=\oplus_l G_l$ is the decomposition of $G$ into groups of prime
power order.
\end{definition}
\begin{notation} \label{notation32}
Let ${\widetilde{K}}$ be the minimal sub-extension of ${K^{\rm s}}$ over which $A_K$ has
semi-stable reduction; it corresponds to the kernel of $I$ acting on
${\widetilde{V}}$, see \cite[\S4.1]{Grothendieck1}.
We define ${K^{\rm t}}$ to be the maximal tame extension in ${\widetilde{K}}$, and
for all $l\neq p$ we let $K_l$ denote the maximal sub-extension of ${\widetilde{K}}$
whose degree over $K$ is a power of $l$. We denote by ${\tilde{t}}$, ${\tilde{a}}$,
${t_{\rm t}}$, ${a_{\rm t}}$, ${u_{\rm t}}$, $t_l$, $a_l$ and $u_l$ the dimensions of the
toric parts, the abelian variety parts and the unipotent parts
of the corresponding N\'eron models of $A_K$. For each prime $l\neq p$ we
let $I_{(l)}$ be the subgroup of $I$ such that $I/I_{(l)}$ is the quotient
${\msy Z}_l(1)$ of ${I_{\rm t}}$.
Let $A^{\rm t}$ be the N\'eron model of $A_K$ over the ring of integers ${D^{\rm t}}$
of ${K^{\rm t}}$. Then ${\rm Gal}({K^{\rm t}}/K)$ acts (from the right) on $A^{\rm t}$,
compatibly with its right-action on ${\rm Spec}({D^{\rm t}})$. This action induces an
action of ${\rm Gal}({K^{\rm t}}/K)$ on the special fibre $A^{\rm t}_k$. Let $\sigma$ be
a generator of the cyclic group ${\rm Gal}({K^{\rm t}}/K)$. Let $l\neq p$ be a
prime number and $i\geq1$ an integer. Let $f_{l,i}$ denote the cyclotomic
polynomial whose roots are the roots of unity of order $l^i$.
We define $m_{{\rm a},l,i}$ and $m_{{\rm t},l,i}$ to be the multiplicities of
$f_{l,i}$ in the characteristic polynomials of $\sigma$ on the abelian variety
part and on the toric part, respectively, of $A^{\rm t}_k$ (say one lets $\sigma$
act on ${\rm T}_l(A^{\rm t}_k({\overline{k}}))\otimes{\msy Q}$).
Let $m_{l,i}:=m_{{\rm a},l,i}+m_{{\rm t},l,i}$. Finally, for $j\geq1$ we define
$p_{{\rm a},l,j}:=|\{i\geq1\;|\;m_{{\rm a},l,i}\geq j\}|$,
$p_{{\rm t},l,j}:=|\{i\geq1\;|\;m_{{\rm t},l,i}\geq j\}|$ and
$p_{l,j}:=|\{i\geq1\;|\;m_{l,i}\geq j\}|$. For an interpretation of
$p_{{\rm a},l}=(p_{{\rm a},l,1},p_{{\rm a},l,2},\ldots)$ in terms of
$m_{{\rm a},l}=(m_{{\rm a},l,1},m_{{\rm a},l,2},\ldots)$ etc. using partitions, see the
beginning of the proof of Lemma~\ref{lemma45}.
\end{notation}
\begin{theorem} \label{thm33}
Let $l\neq p$ be a prime number and consider the filtration (\ref{eqn21})
on $\Phi_l$. With the notations above, we have:
\begin{enumerate}
\item The group $\Phi_l^3$ can be generated by $t$ elements.
\item $\delta_l(\Phi_l^2/\Phi_l^3) \leq \delta_l(p_{{\rm t},l}) \leq t_l-t$.
\item $\delta_l(\Phi_l^1/\Phi_l^2) \leq \delta_l(p_{{\rm a},l}) \leq 2(a_l-a)$.
\item $\delta_l(\Phi_l/\Phi_l^1) \leq \delta_l(p_{{\rm t},l}) \leq t_l-t$.
\item $\delta_l(\Phi_l/\Phi_l^2) \leq \delta_l(p_l) \leq (t_l-t)+2(a_l-a)$.
\item $\delta_l(\Phi_l^1/\Phi_l^3) \leq \delta_l(p_l) \leq (t_l-t)+2(a_l-a)$.
\end{enumerate}
\end{theorem}
\begin{corollary} \label{cor34}
\begin{enumerate}
\item The group $\Phi_{(p)}^3$ can be generated by $t$ elements.
\item $\delta(\Phi_{(p)}^2/\Phi_{(p)}^3) \leq {t_{\rm t}}-t.$
\item $\delta(\Phi_{(p)}^1/\Phi_{(p)}^2) \leq 2({a_{\rm t}}-a).$
\item $\delta(\Phi_{(p)}/\Phi_{(p)}^1) \leq {t_{\rm t}}-t.$
\item $\delta(\Phi_{(p)}/\Phi_{(p)}^2) \leq ({t_{\rm t}}-t)+2({a_{\rm t}}-a).$
\item $\delta(\Phi_{(p)}^1/\Phi_{(p)}^3) \leq ({t_{\rm t}}-t)+2({a_{\rm t}}-a).$
\end{enumerate}
\end{corollary}
\begin{prf}{{\bf Proof} {\rm (of Thm.~\ref{thm33})}}
We begin with some generalities. We always have $M_I=(M_{I_{(l)}})_{{\msy Z}_l(1)}$.
The functors $M\mapsto M_{I_{(l)}}$ and $M\mapsto M^{I_{(l)}}$ are exact on
the category of finitely generated ${\msy Z}_l$-modules with continuous
$I_{(l)}$-action, and, for such modules, the canonical map
$M^{I_{(l)}}\to M_{I_{(l)}}$ is an isomorphism, hence $M_{I_{(l)}}$ is torsion
free if $M$ is torsion free. For $M$ a finitely generated ${\msy Z}_l$-module with
continuous ${\msy Z}_l(1)$-action we have $M_{{\msy Z}_l(1)}=M/(\sigma-1)M$ and
$M^{{\msy Z}_l(1)}=M[\sigma-1]$, where $\sigma$ is any topological generator
of ${\msy Z}_l(1)$.
Next we recall some general facts on the action of $I$ on $U_l$.
Let ${\widetilde{I}}$ denote the subgroup ${\rm Gal}({K^{\rm s}}/{\widetilde{K}})$ of $I$.
Then ${\widetilde{I}}$ acts trivially on ${\widetilde{V}}_l=U_l^{\widetilde{I}}$ and on
$U_l/{\widetilde{V}}_l$; the action of ${\widetilde{I}}$ on $U_l$ factors through the biggest
pro-$l$ quotient ${\msy Z}_l(1)$ of ${\widetilde{I}}$ and is given by an isogeny
$U_l/{\widetilde{V}}_l\to{\widetilde{W}}_l(-1)$ (see \cite[\S9.2, Thm.~10.4]{Grothendieck1}).
It follows that $N_l\cap{\widetilde{W}}_l$ is open in ${\widetilde{W}}_l$. The group $I$
acts on ${\widetilde{V}}_l$ via its finite quotient ${\rm Gal}({\widetilde{K}}/K)=I/{\widetilde{I}}$; this
action can be described in terms of an action of $I/{\widetilde{I}}$ on the special
fibre of the N\'eron model of $A_{\widetilde{K}}$ (see \cite[\S4.2]{Grothendieck1}).
Dually, $I$ acts with finite image on $U_l/{\widetilde{W}}_l$.
As promised in Remark.~\ref{rmk26} we will show that $\Phi_l={V_l'}^\perp/N_l$.
It suffices to show that ${V_l'}^\perp$ and $N_l$ have the same rank.
We have ${\rm rank}(U_l/{V_l'}^\perp)=t+2a$. From the generalities at the beginning
of the proof it follows that
$U_l/N_l=(U_l)_I=((U_l)_{I_{(l)}})_{{\msy Z}_l(1)}=(U_l^{I_{(l)}})_{{\msy Z}_l(1)}$.
Let $\sigma$ be a topological generator of ${\msy Z}_l(1)$. The exact sequence
\begin{eqn} \label{eqn35}
0 \longrightarrow U_l^I \longrightarrow U_l^{I_{(l)}}
\;\;\stackrel{\sigma-1}{\longrightarrow} \;\; U_l^{I_{(l)}} \longrightarrow
\left(U_l^{I_{(l)}}\right)_{{\msy Z}_l(1)} \longrightarrow 0
\end{eqn}
shows that ${\rm rank}((U_l)_I)={\rm rank}(U_l^I)={\rm rank}(V_l)=t+2a$. In order to prove
Thm.~\ref{thm33} we may neglect the Tate twist in~(\ref{eqn25}).
By definition, we have $\Phi_l^3=W_l/N_l\cap W_l$. Since $W_l$
is a free ${\msy Z}_l$ module of rank~$t$, $\Phi_l^3$ can be generated by $t$
elements.
Let us now consider $\Phi_l^2/\Phi_l^3$.
Since $\Phi_l^2={\widetilde{W}}_l/{\widetilde{W}}_l\cap N_l$, the group $\Phi_l^2/\Phi_l^3$ is
a quotient of $({\widetilde{W}}_l/W_l)_I=(({\widetilde{W}}_l/W_l)_{I_{(l)}})_{{\msy Z}_l(1)}$.
Lemma~\ref{lemma44} implies that
$\delta_l(\Phi_l^2/\Phi_l^3)\leq
\delta_l((({\widetilde{W}}_l/W_l)_{I_{(l)}})_{{\msy Z}_l(1)})$.
By the generalities above, $({\widetilde{W}}_l/W_l)_{I_{(l)}}$ is isomorphic as
${\msy Z}_l(1)$-module to ${\widetilde{W}}_l^{{\rm Gal}({\widetilde{K}}/K_l)}/W_l$. Note that
${\widetilde{W}}_l^{{\rm Gal}({\widetilde{K}}/K_l)}$ is the Tate module of the toric part of the
special fibre of the N\'eron model of $A_K$ over the ring of integers of $K_l$,
and that $W_l={\widetilde{W}}_l^{{\rm Gal}({\widetilde{K}}/K)}$. It follows that for all $i\geq1$
the multiplicity of $f_{l,i}$ in the characteristic polynomial of a generator
$\sigma$ of ${\rm Gal}({K^{\rm t}}/K)$ on $({\widetilde{W}}_l/W_l)_{I_{(l)}}$ is $m_{{\rm t},l,i}$
and that $1$ is not a root of this characteristic polynomial.
Applying Lemma~\ref{lemma45} and Cor.~\ref{cor46} gives the second part of the
theorem.
The proof of parts 3, 4, 5 and 6 of the theorem follows the same lines.
For example, $\Phi_l^1/\Phi_l^2$ is a quotient of
$(({\widetilde{V}}_l\cap{V_l'}^\perp)/{\widetilde{W}}_l)_I$. The group $I$ acts with finite
image on $({\widetilde{V}}_l\cap{V_l'}^\perp)/{\widetilde{W}}_l$. The Grothendieck-Igusa
orthogonality theorem \cite[Thm.~2.4]{Grothendieck1} shows that
$({\widetilde{V}}_l\cap{V'_l}^\perp/{\widetilde{W}}_l)_{I_{(l)}}$ has rank $2(a_l-a)$.
We have
\begin{eqn} \label{eqn36}
\left(\frac{{\widetilde{V}}_l\cap{V_l'}^\perp}{{\widetilde{W}}_l}\right)_I\otimes{\msy Q} =
\frac{({\widetilde{V}}_l\cap{V_l'}^\perp)^I}{{\widetilde{W}}_l^I}\otimes{\msy Q} =
\frac{V_l\cap{V_l'}^\perp}{W_l}\otimes{\msy Q} =
\frac{W_l}{W_l}\otimes{\msy Q} = 0
\end{eqn}
which shows that the hypotheses of Lemma~\ref{lemma45} are satisfied.
Since ${\widetilde{V}}_l\cap{V_l'}^\perp/{\widetilde{W}}_l$ is isogenous to ${\widetilde{V}}_l/V_l$,
the multiplicities of the $f_{l,i}$ in the characteristic polynomial of a
generator $\sigma$ of ${\rm Gal}({K^{\rm t}}/K)$ on
$({\widetilde{V}}_l\cap{V_l'}^\perp/{\widetilde{W}}_l)_{I_{(l})}$ are precisely the
$m_{{\rm a},l,i}$.
The proof of part 6 is entirely similar to the proofs of parts~2 and 3.
For parts~4 and 5 one notes that ${V'_l}^\perp/{\widetilde{V}}_l\cap {V'_l}^\perp$
is dual to ${\widetilde{W}}_l'/W_l'$, that ${V'_l}^\perp/{\widetilde{W}}_l$ is dual to
${\widetilde{V}}_l'/V_l'$ and one uses that $A_K$ and $A_K'$ are isogenous.
\end{prf}
\begin{prf}{{\bf Proof} {\rm (of Cor.~\ref{cor34})}}
One just considers the factorization into irreducible factors of the
characteristic polynomial of a generator $\sigma$ of ${\rm Gal}({K^{\rm t}}/K)$ acting
on the semi-abelian variety part of~$A^{\rm t}_k$.
\end{prf}
\section{Some abelian group theory.} \label{section4}
In this section we prove some results needed in the proof of Thm.~\ref{thm33}.
We fix a prime number $l$ and consider finite ${\msy Z}_l$-modules, i.e., finite
abelian groups of $l$-power order. Recall that there is a bijection between
the set of isomorphism classes of finite ${\msy Z}_l$-modules and the set of
partitions (i.e., sequences $m=(m_1,m_2,\ldots)$ of non-negative integers
such that $m_1\geq m_2\geq\cdots$ and $m_i=0$ for $i$ big enough):
a finite ${\msy Z}_l$-module $M$ corresponds to the partition $m=(m_1,m_2,\ldots)$
which satisfies $M\cong\oplus_{i\geq1}{\msy Z}/l^{m_i}{\msy Z}$.
To any partition $m$ we attach the number
$\delta_l(m):=\sum_{i\geq1}(l^{m_i}-1)$. Note that with these definitions,
we have $\delta_l(M)=\delta_l(m)$, with $\delta_l(M)$ as in Def.~\ref{def31}.
\begin{lemma} \label{lemma41}
Let $0\to B\to E\to A\to 0$ be an extension of finite ${\msy Z}_l$-modules. Let
$b=(b_1,b_2,\ldots)$, $e$ and $a$ denote their invariants.
Define $n_i:=a_i+b_i$ and $n=(n_1,n_2,\ldots)$. Let $m=(m_1,m_2,\ldots)$ be
the invariant of $A\oplus B$, i.e., $m$ is the sequence obtained by
reordering $(a_1,b_1,a_2,b_2,\ldots)$. Then we have $m\leq e\leq n$,
with ``$\leq$'' the lexicographical ordering.
\end{lemma}
\begin{proof}
Let us first prove that $e\leq n$. We use induction on $|E|$. We have
$e_1\leq a_1+b_1=n_1$ since $l^{a_1+b_1}$ kills $E$. If $e_1<n_1$ there is
nothing to prove, so we suppose that $e_1=n_1$. Choose any element $x$ in
$E$ of order $l^{e_1}$ and consider the subgroup it generates. We get a
diagram
\begin{subeqn} \label{eqn411}
\begin{array}{ccccccccc}
& & 0 & & 0 & & 0 & & \\
& & \downarrow & & \downarrow & & \downarrow & & \\
0 & \to & {\msy Z}/l^{b_1}{\msy Z} & \to & {\msy Z}/l^{e_1}{\msy Z} & \to & {\msy Z}/l^{a_1}{\msy Z} & \to & 0 \\
& & \downarrow & & \downarrow & & \downarrow & & \\
0 & \to & B & \to & E & \to & A & \to & 0 \\
& & \downarrow & & \downarrow & & \downarrow & & \\
0 & \to & B' & \to & E' & \to & A' & \to & 0 \\
& & \downarrow & & \downarrow & & \downarrow & & \\
& & 0 & & 0 & & 0 & &
\end{array}
\end{subeqn}
in which the rows and columns are exact. Now the columns are split, since
$l^{b_1}$ is the exponent of $B$, etc. Hence $b':=(b_2,b_3,\ldots)$,
$e':=(e_2,e_3,\ldots)$ and $a':=(a_2,a_3,\ldots)$ are the invariants of $B'$,
$E'$ and $A'$, respectively. The proof is finished by induction.
Let us now prove that $e\geq m$. By passing to Pontrjagin duals, if necessary,
we may assume that $b_1\geq a_1$. Then $m_1=b_1$. If $e_1>m_1$ there is
nothing to prove, hence we suppose that $e_1=m_1=b_1$. We choose any
element $x$ in $B$ of order $l^{b_1}$. Just as above we find a diagram
\begin{subeqn} \label{eqn412}
\begin{array}{ccccccccc}
& & 0 & & 0 & & & & \\
& & \downarrow & & \downarrow & & & & \\
& & {\msy Z}/l^{b_1}{\msy Z} & \stackrel{\rm id}{\to} & {\msy Z}/l^{b_1}{\msy Z} & & & & \\
& & \downarrow & & \downarrow & & & & \\
0 & \to & B & \to & E & \to & A & \to & 0 \\
& & \downarrow & & \downarrow & & \downarrow & & \\
0 & \to & B' & \to & E' & \to & A & \to & 0 \\
& & \downarrow & & \downarrow & & & & \\
& & 0 & & 0 & & & &
\end{array}
\end{subeqn}
in which the columns are split. Induction finishes the proof.
\end{proof}
\begin{rmk} \label{rmk42}
It would be nice to have a complete description of the possible invariants
of extensions $E$ of finite ${\msy Z}_l$-modules $A$ by $B$ in terms of the
invariants of $A$ and $B$. In particular, are there more restrictions than
the following: those in Lemma~\ref{lemma41}, $e_i\geq a_i$ and $e_i\geq b_i$
for all~$i$, and the minimal number of generators for $E$ does not exceed the
sum of those numbers for $A$ and $B$? As Hendrik Lenstra pointed out to me,
the problem can be phrased in terms of Hall polynomials, see for example
\cite{MacDonald1}.
\end{rmk}
\begin{lemma} \label{lemma43}
Suppose that $a=(a_1,a_2,\ldots)$ and $b=(b_1,b_2,\ldots)$ are partitions
of $N$ (i.e., $\sum_{i\geq1}a_i=N=\sum_{i\geq1}b_i$) and that $a\geq b$
in the lexicographical ordering. Then $\delta_l(a)\geq\delta_l(b)$, with
equality if and only if $a=b$.
\end{lemma}
\begin{proof}
Consider the set $X$ of all partitions of $N$ with its lexicographical
ordering. From the inequality
$$
(l^{n+1}-1) + (l^{m-1}-1) > (l^n-1) + (l^m-1)
$$
satisfied for any integers $n\geq m$ it follows that $\delta_l\colon X\to{\msy Z}$
is strictly increasing.
\end{proof}
\begin{lemma} \label{lemma44}
\begin{enumerate}
\item For $M$ a finite ${\msy Z}_l$-module we have $\delta_l(M)\geq0$, with
equality if and only if $M=0$.
\item Let $0\to M'\to M\to M''\to 0$ be a short exact sequence of finite
${\msy Z}_l$-modules. Then $\delta_l(M)\geq\delta_l(M')+\delta_l(M'')$, with
equality if and only if the sequence is split.
\item Let $0\to M'\to M\to M''\to 0$ be a short exact sequence of finite
${\msy Z}_l$-modules. Suppose that $M$ is killed by $l^a$ and that $|M'|=l^{b}$.
Then $\delta_l(M)\leq \delta_l(M'')+b(l^a-l^{a-1})$.
\end{enumerate}
\end{lemma}
\begin{proof}
This follows directly from Lemmas~\ref{lemma41} and \ref{lemma43}.
\end{proof}
\begin{lemma} \label{lemma45}
Let $M$ be a finitely generated free ${\msy Z}_l$-module with an automorphism
$\sigma$ of finite order. Suppose that $M/(\sigma-1)M$ is finite, or,
equivalently, that the automorphism $\sigma\otimes1$ of the ${\msy Q}_l$-vector
space $M\otimes{\msy Q}$ does not have $1$ as eigenvalue. For $i\geq1$ let
$m_i$ be the multiplicity, in the characteristic polynomial of $\sigma$,
of the cyclotomic polynomial $f_i$ whose roots are the roots of unity of
order $l^i$. For each $j\geq1$, let $p_j:=|\{i\geq1\;|\; m_i\geq j\}|$.
Then $\delta_l(M/(\sigma-1)M)\leq \sum_{i\geq1}(l^{p_i}-1)$.
\end{lemma}
\begin{proof}
Let $q=(q_1,q_2,\ldots)$ be the partition obtained by reordering
$(m_1,m_2,\ldots)$. Then $p:=(p_1,p_2,\ldots)$ is what is usually called
the conjugate of $q$: when viewing partitions as Young diagrams, $p$ and
$q$ are obtained from each other by interchanging rows and columns.
In particular, we have $\sum_{i\geq1}p_i=\sum_{i\geq1}m_i$.
Let $n$ be the order of $\sigma$. Then $M$ becomes a module over the ring
${\msy Z}_l[x]/(x^n-1)$. Let us write $n=l^rn'$ with $n'$ not divisible by $l$.
Then ${\msy Z}_l[x]/(x^n-1)$ is the product of the ring ${\msy Z}_l[x]/(x^{l^r}-1)$
by another ring $R$ and $x-1$ is invertible in $R$. This implies that
$M$ is the direct sum of two modules, one over ${\msy Z}_l[x]/(x^{l^r}-1)$ and
the other over $R$, and that the module over $R$ does not contribute
to $M/(\sigma-1)M$. Hence we have reduced the problem to the case where the
order of $\sigma$ is $l^r$.
Let $i_1<i_2<\cdots<i_{p_1}$ denote the integers $i\geq1$ such that
$m_i>0$. For $1\leq j\leq p_1$, let $F_j:=f_{i_j}$ be the corresponding
cyclotomic polynomials, and let $F:=F_1{\cdot}F_2\cdots F_{p_1}$.
Since $1$ is not an eigenvalue of $\sigma$ on $M\otimes{\msy Q}$, and $M$ is
torsion free as ${\msy Z}_l$-module, $M$ is a module over the ring
$A:={\msy Z}_l[x]/(F)$. For any $A$-module $N$, we define $\overline{N}:=N/(x-1)N$.
Let us first note that for all $j$ we have $F_j(1)=l$. It follows that
$\overline{A}={\msy Z}/l^{p_1}{\msy Z}$. For $N$ an $A$-module, $\overline{N}$ is
an $\overline{A}$-module, hence $l^{p_1}$ annihilates $\overline{N}$.
We claim that $|\overline{M}|=l^{\sum_{i\geq1}p_i}$. To prove this, note
that $|\overline{M}|=|\det(\sigma-1)|_l^{-1}$, with $|\cdot|_l$ the $l$-adic
absolute value on ${\msy Q}_l$, normalized by $|l|_l=1/l$. So in order to
compute $|\overline{M}|$ we may replace $M$ by any $\sigma$-stable lattice
$M'$ in $M\otimes{\msy Q}\cong\oplus_{i\geq1}({\msy Q}_l[x]/(f_i))^{m_i}$. Taking
$M':=\oplus_{i\geq1}({\msy Z}_l[x]/(f_i))^{m_i}$ and noting that $f_i(1)=1$ gives
the result.
Let $a=(a_1,a_2,\ldots)$ be the invariant of $\overline{M}$, i.e.,
$\overline{M}\cong\oplus_{i\geq1}{\msy Z}/l^{a_i}{\msy Z}$ and $a_1\geq a_2\geq\cdots$.
Note that $a$ and $p$ are partitions of the same number, hence in view of
Lemma~\ref{lemma43}, it suffices to show that $a\leq p$ in the lexicographical
ordering. Since $l^{p_1}$ annihilates $\overline{M}$, we have $a_1\leq p_1$.
If $a_1<p_1$ there is nothing to prove, so we assume that $a_1=p_1$. Let
$y$ be in $M$ such that its image $\overline{y}$ in $\overline{M}$ corresponds
to $(1,0,0,\ldots)$. Let $A'$ denote the submodule $Ay$ of $M$. Since
$M$ is free as a ${\msy Z}_l$-module, $A'$ is free as a ${\msy Z}_l$-module, and we
have $A'={\msy Z}_l[x]/(G)$, with $G$ dividing $F$. Let $0\leq s\leq p_1$ be the
number of irreducible factors of $G$. Then we have $\overline{A'}={\msy Z}/l^s{\msy Z}$.
We have a short exact sequence
\begin{subeqn} \label{eqn451}
0 \to A' \to M \to M' \to 0
\end{subeqn}
of $A$-modules, with $M'$ not necessarily free as ${\msy Z}_l$-module.
Multiplication by $x-1$ on this sequence induces an exact sequence
\begin{subeqn} \label{eqn452}
0 \to M'[x-1] \to \overline{A'} \to \overline{M} \to \overline{M'} \to 0
\end{subeqn}
The element $\overline{1}$ of $\overline{A'}$, which is annihilated by $l^s$,
is mapped to $\overline{y}$ which has annihilator $l^{p_1}$. It follows that
$s=p_1$, that $G=F$ and that $M'[x-1]=0$. Let us now consider the finite
$A$-module ${\rm tors}(M')$. Multiplication by $x-1$ acts injectively, hence
bijectively. Since $x-1$ is in the maximal ideal of $A$, it follows that
${\rm tors}(M')=0$, hence that $M'$ is free as ${\msy Z}_l$-module. The proof is now
finished by induction on ${\rm rank}(M)$, since
$\overline{M'}\cong\oplus_{i\geq2}{\msy Z}/l^{a_i}{\msy Z}$ and the partition $p'$
obtained from $M'$ is $(p_2,p_3,\ldots)$.
\end{proof}
\begin{rmk}\label{rmk46}
Lemma~\ref{lemma45} can be seen as a bound on the cohomology group
${\rm H}^1({\msy Z}/n{\msy Z},M)$, where $1$ in ${\msy Z}/n{\msy Z}$ acts on $M$ via $\sigma$. It is
an interesting question, raised by Xavier Xarles, to obtain similar
bounds for non-cyclic groups.
\end{rmk}
\begin{corollary} \label{cor46}
Let $M$ be a finitely generated free ${\msy Z}_l$-module with an automorphism
$\sigma$ of finite order. Suppose that $M/(\sigma-1)M$ is finite.
Then $\delta_l(M/(\sigma-1)M)\leq{\rm rank}(M)$.
\end{corollary}
\begin{proof}
We use the notation of the beginning of the proof of Lemma~\ref{lemma45}.
Then one has:
\begin{subeqn} \label{eqn461}
{\rm rank}(M)\geq \sum_{i\geq1}m_i\phi(l^i) \geq \sum_{i\geq1}q_i\phi(l^i)
= \sum_{i\geq1}\sum_{j=1}^{p_i}\phi(l^j) = \sum_{i\geq1}(l^{p_i}-1)
\end{subeqn}
The proof is finished by applying Lemma~\ref{lemma45}.
\end{proof}
\begin{lemma} \label{lemma47}
Let $M$ be a finitely generated free ${\msy Z}_l$-module with an automorphism
$\sigma$ of finite order. Suppose that $M/(\sigma-1)M$ is finite and that
$\delta_l(M/(\sigma-1)M)={\rm rank}(M)$. Then $M$ is a direct sum of
${\msy Z}_l$-modules of the type ${\msy Z}_l[x]/(f_1{\cdot}f_2\cdots f_r)$ with
$\sigma$ acting as multiplication by $x$ and where $f_i$ denotes the
cyclotomic polynomial whose roots are the roots of unity of order $l^i$.
\end{lemma}
\begin{proof}
The proof is by induction on ${\rm rank}(M)$ and consists of an inspection of the
proofs of Lemma~\ref{lemma45} and Cor.~\ref{cor46}. First of all we must have
that $n'=1$. Secondly, we note that $m_i=q_i$ for all $i\geq1$ since the
inequalities in (\ref{eqn461}) are equalities (here we use that
$\sum_{i\geq1}m_i=\sum_{i\geq1}q_i$ and that $\phi(l^i)<\phi(l^j)$ if
$1\leq i<j$). So $m$ is the conjugate partition of $p$, hence
$A={\msy Z}_l[x]/(F)$ with $F=f_1{\cdot}f_2\cdots f_{p_1}$. The formula for the
number of elements of $|\overline{M}|$ in the proof of Lemma~\ref{lemma45}
shows that $a$ and $p$ are partitions of the same number. By the hypotheses
of the lemma we are proving, we have $\delta_l(a)=\delta_l(p)$.
Lemma~\ref{lemma43} implies that $a=p$. The end of the proof of
Lemma~\ref{lemma45} shows that $A'=A$ and that $M'$ is free as ${\msy Z}_l$-module.
By induction on ${\rm rank}(M)$, we know that $M'$ is of the indicated type.
It remains to show that the short exact sequence
(\ref{eqn451}) splits. To do that, it is sufficient to show that
${\rm Ext}^1_A(A_i,A)=0$, where $A_i={\msy Z}_l[x]/(f_1\cdots f_i)$ with $i\leq p_1$.
This ${\rm Ext}^1$ is easily computed using the projective
resolution
\begin{subeqn}\label{eqn471}
\cdots \longrightarrow A \;\;\stackrel{f}{\longrightarrow}\;\;A
\;\;\stackrel{g}{\longrightarrow}\;\;A
\;\;\stackrel{f}{\longrightarrow}\;\;A \longrightarrow A_i \longrightarrow 0
\end{subeqn}
with $f=f_1\cdots f_i$ and $g=f_{i+1}\cdots f_{p_1}$.
\end{proof}
The following lemmas will be used in \S\ref{section5} and \S\ref{section6}.
\begin{lemma} \label{lemma48}
Let $M$ be a finite ${\msy Z}_l$-module and let
$$
M=M^0\supset M^1\supset M^2\supset\cdots\supset M^r=0
$$
be a strictly descending filtration. Suppose that for all $i$ with
$0\leq i\leq r-2$ the group $M^i/M^{i+2}$ is cyclic. Then $M$ is cyclic.
\end{lemma}
\begin{proof}
For $r\leq2$ there is nothing to prove. If we know the result for $r=3$, the
general case follows by induction since then $M^0/M^3$ is cyclic and
the filtration $M^0\supset M^2\supset M^3\supset\cdots\supset M^r$ has length
$r-1$. So assume now that $r=3$. Let $x$ be an element of $M^0$ such that its
image in $M^0/M^2$ is a generator. Then a certain multiple $ax$ of $x$ gives a
generator of $M^1/M^2$. Since $M^1$ is cyclic, and $M^1/M^2$ a non-trivial
quotient, $ax$ is a generator of $M^1$. The subgroup of $M^0$ generated by $x$
contains $M^1$ and its quotient by $M^1$ is $M^0/M^1$. We conclude that $x$
generates~$M^0$.
\end{proof}
\begin{rmk}\label{remark49}
The proof of Lemma~\ref{lemma48} generalizes immediately to a proof of the
following assertion: let $A$ be a local ring and $M$ an $A$-module with
a finite strictly descending filtration $M^i$ such that the $M^i/M^{i+2}$ are
cyclic, then $M$ is cyclic.
\end{rmk}
\begin{lemma}\label{lemma410}
Let $0\to B\to E\to A\to 0$ be a short exact sequence of finite ${\msy Z}_l$-modules
with invariants $b$, $e$ and $a$. Let $t\geq0$ be an integer and suppose that
$B$ is generated by $t$ elements. Then for all $i\geq1$ we have
$a_i\geq e_{i+t}$.
\end{lemma}
\begin{proof}
For a partition $p$, let $p'$ denote its conjugate. Then for all $i\geq1$
we have $l^{a_i'}=|A[l^i]A[l^{i-1}]|$. Let $d$ be the endomorphism of the
set of partitions defined by: $d(p)_i=p_{i+1}$ for all $i\geq1$. Let $d'$
be the conjugate of $d$: $d'(p)=d(p')'$. Then $d'(p)_i=\max(0,p_i-1)$.
When viewing a partition $p$ as a Young diagram in which the $p_i$ are the
lengths of the columns, $d$ and $d'$ remove the longest column and row,
respectively. Note that for a finite ${\msy Z}_l$-module $M$ with invariant $m$,
the submodule $lM$ has invariant $d'(m)$. Note that $d$ and $d'$ commute.
In the rest of this proof we will consider the partial
ordering on the set of partitions in which $p\leq q$ if and only if for all
$i\geq1$: $p_i\leq q_i$. Note that $p\leq q$ is equivalent to $p'\leq q'$.
Below we will use that $p\geq q$ if and only if: $p_1'\geq q_1'$ and
$d'(p)\geq d'(q)$. We will also use that if $N$ and $M$ are finite
${\msy Z}_l$-modules with invariants $n$ and $m$ such that $N$ is a subquotient of
$M$, then $n\leq m$.
The proof of the lemma is by induction on $|E|$. What we have to prove is that
$a\geq d^t(e)$. The exact sequence $0\to B[l]\to E[l]\to A[l]$ shows that
$a_1'\geq e_1'-t$. Note that $d^t(e)'_1=\max(0,e_1'-t)$, hence we have
$a_1'\geq d^t(e)'_1$. The exact sequence
$0\mapsto B\cap lE\to lE\to lA$ shows (induction hypothesis) that
$d'(a)\geq d^t(d'(e))=d'(d^t(e))$. The two inequalities just proved imply
that $a\geq d^t(e)$.
\end{proof}
\begin{lemma}\label{lemma411}
Let $l$ be a prime. Let $0\to B\to E\to A$ be a short exact sequence of
finite ${\msy Z}_l$-modules with invariants $b$, $e$ and $a$, respectively.
Then
\begin{subeqn}\label{eqn4111}
\delta_l(b) + \delta_l(a) \geq
\sum_{i\geq1}\left(
\frac{l^{\lfloor e_i/2\rfloor}+l^{\lceil e_i/2\rceil}}{2}-1\right)
\end{subeqn}
where for any real number $x$, $\lfloor x\rfloor$ and $\lceil x\rceil$ denote
the largest (resp. smallest) integer $\leq x$ (resp. $\geq x$).
\end{lemma}
\begin{proof}
We have $\sum_i(a_i+b_i)=\sum_i e_i$. Lemma~\ref{lemma41} asserts that
$a+b\geq e$ in the lexicographical ordering. Consider the set $S$ of all
pairs $(r,s)$ of partitions, such that $r+s\geq e$ and
$\sum_i(r_i+s_i)=\sum_i e_i$. Let $f\colon S\to {\msy Z}$ be the map which sends
$(r,s)$ to $\delta_l(r)+\delta_l(s)$. We will show that $f$ achieves its
minimum at all $(r,s)$ in $S$ with the property that, for all $i\geq1$,
one has $\{r_i,s_i\}=\{\lfloor e_i/2\rfloor,\lceil e_i/2\rceil\}$.
Suppose now that $(r,s)$ is an element of $S$ where $f$ has a minimum.
We have to show that $|r_i-s_i|\leq1$ for all $i\geq1$. Suppose that this is
not the case. Let $j\geq1$ be minimal for the property that $|r_j-s_j|>1$
and $|r_i-s_i|\leq1$ for all $i<j$. We may and do suppose that $r_j-s_j>1$.
Note that if $j>1$ we have $s_{j-1}>s_j$. We define $r'$ and $s'$ as
follows: $(r'_i,s'_i)=(r_i-1,s_i+1)$ if $i\geq j$ and $r_i=r_j$;
in all other cases $(r'_i,s'_i)=(r_i,s_i)$. Note that $r'$ and $s'$ are
partitions, that $\sum_i(r'_i+s'_i)$ is equal to $\sum_i e_i$ and that
$f(r',s')$ is strictly smaller than $f(r,s)$.
\end{proof}
\section{Examples.} \label{section5}
The aim of this section is to give examples that show that the bounds in
Thm.~\ref{thm33} and Cor.~\ref{cor34} are sharp, in a sense that will become
clear in the examples. The examples we construct here will play an important
role in \S\ref{section6}.
We give our examples over the field $K:={\msy C}((q))$ of formal Laurent series
over the complex numbers with its usual valuation, but it is easy to get
similar examples in mixed characteristic, or equal characteristic $p>0$.
The building stones of our examples are the following. For each integer
$n\geq1$ we let $E_n$ be the so-called Tate elliptic curve ``${{\rm G}_m}/q^{n{\msy Z}}$''
over $K$ as described in \cite[\S VII]{DeligneRapoport} or in
\cite[\S6]{Mumford} ($E_n$ is obtained from the analytic family of elliptic
curves over the punctured unit disc with coordinate $q$ whose fibres are the
${\msy C}^*/q^{n{\msy Z}}$, by base change from the field of finite tailed convergent
Laurent series to $K$). It is well known that the special fibre of the N\'eron
model of $E_n$ over $D:={\msy C}[[q]]$ is an extension of ${\msy Z}/n{\msy Z}$ by the
multiplicative group.
For each prime $l$ and integer $r\geq0$ we define the ring
$\Lambda_{l,r}:={\msy Z}[x]/(f_{l,1}\cdots f_{l,r})$, where as before $f_{l,i}$ is
the polynomial whose roots are the roots of unity of order $l^i$.
When $l>2$, we let $A_{l,r}$ be an abelian variety over ${\msy C}$ obtained as
follows: we choose an isomorphism of ${\msy R}$-algebras between
$\Lambda_{l,r}\otimes{\msy R}$ and a product of a number of copies of ${\msy C}$ and
define $A_{l,r}:=(\Lambda_{l,r}\otimes{\msy R})/\Lambda_{l,r}$ (it is well known
that the trace form on $\Lambda_{l,r}$ implies the existence of a
polarization). The first three examples will be isogenous to twists of
products of copies of $E_{n}$ and of $A_{l,r,K}$. Of course
Lemma~\ref{lemma47} tells us how to cook up the required examples.
\begin{example}\label{example51}
Let $d\geq0$ and let $G$ be any finite abelian group that can be generated by
$d$ elements. Then $G\cong\oplus_{i=1}^d{\msy Z}/n_i{\msy Z}$, say.
For $A_K:=\prod_{i=1}^d E_{n_i}$ one has $\Phi=\Phi^3=\oplus_{i=1}^t{\msy Z}/n_i{\msy Z}$
and we have $d=t=\dim(A_K)$.
\end{example}
\begin{example}\label{example52}
Now consider parts~2 and 4 of Thm.~\ref{thm33}. Let $l$ be a prime.
For $i\geq1$ we let $B_{l,i}$ be the abelian variety $E_1\otimes\Lambda_{l,i}$
over ${\msy C}$, i.e., $B_{l,i}$ is a direct sum of copies of $E_1$, indexed by
some ${\msy Z}$-basis of $\Lambda_{l,i}$, and $\Lambda_{l,i}$ acts on $B_{l,i}$
according to its action on itself. In particular, multiplication by $x$ in
$\Lambda_{l,i}$ induces an automorphism $\sigma$ of $B_{l,i}$. Note that
$\sigma$ has order $l^i$. Let $C_{l,i}$ be the twist of $B_{l,i,K}$ over
$K(q^{1/l^i})$ by $\sigma$, i.e., $C_{l,i}$ is the quotient of the $K$-scheme
$B_{l,i,K}\times_{{\rm Spec}(K)}{\rm Spec}(K(q^{1/l^i}))$ by the group
${\rm Gal}(K(q^{1/l^i})/K)={\msy Z}/l^i{\msy Z}$ (here we choose a root of unity of order $l^i$)
which acts by $a\mapsto\sigma^a$ on the first
factor and via its natural action on the second factor.
We will now compute the group of connected
components $\Psi$ of the N\'eron model of $C_{l,i}$ over $D$, using
(\ref{eqn24}).
First of all we have ${\rm T}_l(E_1({K^{\rm s}}))={\msy Z}_l(1)\oplus{\msy Z}_l$, with $I$
acting via its quotient ${\msy Z}_l(1)$ in the following way: an element of $I$ with
image $a$ in ${\msy Z}_l(1)$ acts as multiplication by the matrix
$({1\atop0}{a\atop1})$. By construction,
${\rm T}_l(C_{l,i}({K^{\rm s}}))={\rm T}_l(E_1({K^{\rm s}}))\otimes\Lambda_{l,i}$
and an element in $I$ with image $a$ in ${\msy Z}_l(1)$ acts as
$({1\atop0}{a\atop1})\otimes x^a$. Since $C_{l,i}$ has ${\tilde{a}}=t=0$, we have
$\Psi_l^1=\Psi_l^2$ and $\Psi_l^3=0$. The filtration
${\msy Z}_l(1)\subset{\rm T}_l(E_1({K^{\rm s}})$ induces the filtration
${\widetilde{W}}_l\subset{\rm T}_l(C_{l,i}({K^{\rm s}}))$. It follows that $\Psi_l$ is the
cokernel of $({x-1\atop0}{x\atop x-1})$ and that $\Psi_l/\Psi_l^1$ and
$\Psi_l^2/\Psi_l^3$ are both isomorphic to $\Lambda_{l,i}/(x-1)={\msy Z}/l^i{\msy Z}$.
An analogous computation shows that $\Psi=\Psi_l$. One can show that
$\Psi$ is isomorphic to ${\msy Z}/l^i{\msy Z}\oplus{\msy Z}/l^i{\msy Z}$ if $l>2$ and to
${\msy Z}/2^{i+1}{\msy Z}\oplus{\msy Z}/2^{i-1}{\msy Z}$ if $l=2$.
Let $G$ be a finite abelian group of $l$-power order, say with invariant
$a=(a_1,a_2,\ldots)$. Then for $A_K:=\prod_{i\geq1}C_{l,a_i}$ we have
$\Phi_l/\Phi_l^1\cong\Phi_l^2/\Phi_l^3\cong G$ and
$\delta_l(G)=t_l=\dim(A_K)$. We remark that abelian varieties over $K$ that
are isogeneous to $A_K$ provide examples with $\Phi_l/\Phi_l^1$ not isomorphic
to $\Phi_l^2/\Phi_l^3$.
\end{example}
\begin{example}\label{example53}
For $l>2$ prime and $i\geq0$ we let $D_{l,i}$ be the abelian
variety over $K$ obtained by twisting $A_{l,i,K}$ over $K(q^{1/l^i})$ by the
automorphism $\sigma$ of $A_{l,i}$ which is induced from the multiplication
by $x$ in $\Lambda_{l,i}$. Then we have
${\rm T}_l(D_{l,i}({K^{\rm s}}))={\rm T}_l(A_{l,i}({\msy C}))=\Lambda_{l,i}\otimes{\msy Z}_l$, and an
element in $I$ with image $a$ in ${\msy Z}_l(1)$ acts as $x^a$. Let $\Psi_l$
denote the group of connected components of attached to $D_{l,i}$. In this
case we have ${\tilde{t}}=a=0$, hence $\Psi_l=\Psi_l^1$ and $\Psi_l^2=0$.
By (\ref{eqn24}) we have $\Psi_l=\Lambda_{l,i}/(x-1)={\msy Z}/l^i{\msy Z}$.
Suppose now that $l\neq2$. Let $G$ be a finite abelian group of $l$-power
order, say with invariant $a=(a_1,a_2,\ldots)$. Then for
$A_K:=\prod_{i\geq1}D_{l,a_i}$ we have
$\Phi_l=\Phi_l^1$, $\Phi_l^2=0$, $\Phi_l^1/\Phi_l^2\cong G$ and
$\delta_l(G)=2a_l=2\dim(A_K)$.
The case $l=2$ is a little bit different because $m_{{\rm a},2,1}$ is always even.
\end{example}
\begin{example} \label{example54}
Let $l$ be prime and let $r>0$ and $s>0$ be positive integers.
We will construct an abelian variety $A_K$ with $t=a=0$,
${\tilde{t}}=l^r-1$, ${\tilde{a}}=(l^{r+s}-l^r)/2$ and $\Phi=\Phi_l\cong{\msy Z}/l^{2r+s}{\msy Z}$.
It follows from Thm.~\ref{thm33} that in such an example $\Phi_l/\Phi_l^1$ and
$\Phi_l^2$ are cyclic of order $l^r$, that $\Phi_l^1/\Phi_l^2$ is
cyclic of order $l^s$ and that $\Phi_l/\Phi_l^2$ and $\Phi_l^1$ are cyclic of
order $l^{r+s}$. Hence this example shows that, as far as the exponent is
concerned, the two-fold extension $\Phi_l/\Phi_l^3$ can be arbitrary.
As in the previous examples, $f_{l,i}$ will denote the polynomial whose
roots are the roots of unity of order $l^i$, and $\Lambda_{l,r}$ is the
ring ${\msy Z}[x]/(f_{l,1}\cdots f_{l,r})$.
Let $\Lambda_{l,r,s}:={\msy Z}[x]/(f_{l,r+1}\cdots f_{l,r+s})$. Let $D_{l,r,s}$ be
an abelian variety over $K$ obtained by replacing $\Lambda_{l,i}$ by
$\Lambda_{l,r,s}$ and $q^{1/l^i}$ by $q^{1/l^{r+s}}$ in the construction of
$D_{l,i}$ in Example~\ref{example53}.
Let $C_{l,r}$ be as in Example~\ref{example52}. Our example $A_K$ will be
isogeneous to $C_{l,r}\times D_{l,r,s}$.
Let $V:={\rm T}_l((C_{l,r}\times D_{l,r,s})({K^{\rm s}}))\otimes{\msy Q}$. Then $V$ is a
${\msy Q}_l$-vector space with an action of $I={\rm Gal}({K^{\rm s}}/K)$. We have an
isomorphism of ${\msy Q}_l$-vector spaces with $I$-action
\begin{subeqn}\label{eqn541}
\Lambda_{l,r}\otimes{\msy Q}_l \;\oplus\; \Lambda_{l,r,s}\otimes{\msy Q}_l \;\oplus\;
\Lambda_{l,r}\otimes{\msy Q}_l \;\;\tilde{\longrightarrow}\;\; V
\end{subeqn}
such that an element of $I$ with image $a$ in ${\msy Z}_l(1)$ acts via
\begin{subeqn}\label{eqn542}
\left(\begin{array}{ccc}x^a&0&ax^a\\0&x^a&0\\0&0&x^a\end{array}\right)
\end{subeqn}
Let
\begin{subeqn}\label{eqn543}
V=V^0\supset V^1\supset V^2\supset V^3=0
\end{subeqn}
be the filtration (\ref{eqn27}) on $V$. Then $V^2$ is simply the first
term in (\ref{eqn541}) and $V^1$ is the sum of the first two terms.
For any ${\msy Z}_l$-lattice $M$ in $V$ let $M^i:=M\cap V^i$.
To get our example $A_K$, it suffices to find an $I$-invariant ${\msy Z}_l$-lattice
$M$ in $V$ such that $M^1$ and $M/M^2$ are isomorphic, as ${\msy Z}_l[I]$-modules,
to $\Lambda_{l,r+s}\otimes{\msy Z}_l$, where an element of $I$ with image $a$ in
${\msy Z}_l(1)$ acts on $\Lambda_{l,r+s}\otimes{\msy Z}_l$ as $x^a$. Namely, since $M$ is
$I$-invariant, $M$ is the $l$-adic Tate module of an abelian variety $A_K$
which is isogeneous to $C_{l,r}\times D_{l,r,s}$; for $A_K$ one has
$\Phi_l/\Phi_l^2$ and $\Phi_l^1$ cyclic of order $l^{r+s}$, hence $\Phi_l$
cyclic of order $l^{2r+s}$ by Lemma~\ref{lemma48}.
Let us now try to find such a $M$. Note that we have canonical projections
$\Lambda_{l,r+s}\to\Lambda_{l,r}$ and $\Lambda_{l,r+s}\to\Lambda_{l,r,s}$
which induce an embedding $\Lambda_{l,r+s}\otimes{\msy Z}_l\subset V^1$.
We will first show that we only have to
look among the sublattices $M$ with $M^1=\Lambda_{l,r+s}\otimes{\msy Z}_l$.
Namely, if $M$ is an $I$-invariant ${\msy Z}_l$-lattice in $V$ of the type we are
looking for, then for a suitable element of the form $v=(a,b,a)$ of $V^*$
(here we consider $V$ as a ${\msy Q}_l$-algebra and $V^*$ denotes the group of units
of $V$) $vM$ is isomorphic to $M$ as ${\msy Z}_l[I]$-module and has
$(vM)^1=\Lambda_{l,r+s}\otimes{\msy Z}_l$. Since the ${\msy Z}_l[I]$-module structure
determines the filtration, we also have $(vM)/(vM)^2\cong M/M^2$.
\par From now on we only consider $M$ with $M^1=\Lambda_{l,r+s}\otimes{\msy Z}_l$.
Such $M$ are determined by their image in $V/M^1$. So we look for an
$I$-invariant torsion free ${\msy Z}_l$-submodule $N$ of
$V/M^1=V^1/M^1\oplus\Lambda_{l,r}\otimes{\msy Q}_l$ whose image in
$\Lambda_{l,r}\otimes{\msy Q}_l$ is a lattice and for whose associated $M$ we have
$M/M^2\cong\Lambda_{l,r+s}\otimes{\msy Z}_l$. It follows that such a $N$ is
isomorphic, via the canonical projection, to its image in
$\Lambda_{l,r}\otimes{\msy Q}_l$. Lemma~\ref{lemma47} implies that this image is
isomorphic to $\Lambda_{l,r}\otimes{\msy Z}_l$, hence of the form
$z{\cdot}\Lambda_{l,r}\otimes{\msy Z}_l$ for some $z$ in
$(\Lambda_{l,r}\otimes{\msy Q}_l)^*$. We conclude that $N$ is of the form
${\rm im}(\alpha)$, where
\begin{subeqn}\label{eqn544}
\alpha\colon \Lambda_{l,r}\otimes{\msy Z}_l \longrightarrow
V^1/M^1\;\oplus\;\Lambda_{l,r}\otimes{\msy Q}_l, \quad
a\mapsto(\phi(a),za)
\end{subeqn}
with $\phi\colon\Lambda_{l,r}\otimes{\msy Z}_l\to V^1/M^1$ a morphism of
${\msy Z}_l$-modules, and $z\in(\Lambda_{l,r}\otimes{\msy Q}_l)^*$. For a given pair
$(\phi,z)$, let $N_{\phi,z}$ denote the image of the corresponding $\alpha$.
Let us first study what it means for $(\phi,z)$ that $N_{\phi,z}$ is
$I$-invariant. Using that $N_{\phi,z}$ is $I$-invariant if and only if
it is invariant under the matrix in (\ref{eqn542}) with $a$ replaced by $1$,
one easily sees that $N_{\phi,z}$ is $I$-invariant if and only if
\begin{subeqn}\label{eqn545}
\forall a\in\Lambda_{l,r}\otimes{\msy Z}_l\colon \quad
\phi(xa)=x\phi(a)+\overline{(xza,0)}
\end{subeqn}
To find out which $(\phi,z)$ satisfy (\ref{eqn545}), we write out everything
in terms of the ${\msy Z}_l$-basis $(1,x,\ldots,x^{l^r-2})$ of
$\Lambda_{l,r}\otimes{\msy Z}_l$. Let $y=(y_1,y_2)$ be in $V^1$ such that
$\phi(1)=\overline{y}$. One then checks that
\begin{subeqn}\label{eqn546}
\phi(x^i) = x^i\overline{y} + ix^i\,\overline{(z,0)}, \quad
\mbox{for $0\leq i\leq l^r-2.$}
\end{subeqn}
Applying (\ref{eqn545}) with $a=x^{l^r-2}$, and using that
$\sum_{i=0}^{l^r-1}x^i=0$ in $\Lambda_{l,r}$, gives
\begin{subeqn}\label{eqn547}
(xg_r'(x)z,g_r(x)y_2) \in \Lambda_{l,r+s}\otimes{\msy Z}_l
\end{subeqn}
where $g_r=f_{l,1}\cdots f_{l,r}$ and $g_r'$ is the derivative of $g_r$. The
conclusion is that $N_{\phi,z}$ is $I$-invariant if and only if $\phi$ is
given by (\ref{eqn546}) and $(y,z)$ satisfies (\ref{eqn547}). For a given
such pair $(y,z)$, let $M_{y,z}$ denote the lattice $M$ in $V$ corresponding
to $N_{\phi,z}$.
It remains now to be seen that there exist $(y,z)$ satisfying (\ref{eqn547})
such that $M_{y,z}/M_{y,z}^2$ is isomorphic to $\Lambda_{l,r+s}\otimes{\msy Z}_l$,
or, equivalently, such that $(M_{y,z}/M_{y,z}^2)_I$ is cyclic.
In order to have a useful description of $M_{y,z}$, we lift
$\phi$ to $V^1$ as follows: let
${\widetilde{\phi}}\colon\Lambda_{l,r}\otimes{\msy Z}_l\to V^1$ be the morphism of
${\msy Z}_l$-modules such that
\begin{subeqn}\label{eqn548}
{\widetilde{\phi}}\colon x^i\mapsto x^iy + ix^i(z,0),
\quad\mbox{for $0\leq i\leq l^r-2$}
\end{subeqn}
Then we have an isomorphism of ${\msy Z}_l$-modules:
\begin{subeqn}\label{eqn549}
\beta\colon \Lambda_{l,r+s}\otimes{\msy Z}_l\;\oplus\;\Lambda_{l,r}\otimes{\msy Z}_l
\;\;\tilde{\longrightarrow}\;\; M\subset V,\quad
(a,b)\mapsto (a,0)+({\widetilde{\phi}}(b),zb)
\end{subeqn}
Let $\tau$ be an element of $I$ with image $1$ in ${\msy Z}_l(1)$. Then $\tau$ acts
on $V$ by the matrix in (\ref{eqn542}) with $a=1$. One computes that in order
to make $\beta$ invariant under $I$, one must let $\tau$ act on the source of
$\beta$ in (\ref{eqn549}) by
\begin{subeqn}\label{eqn5410}
\tau\colon (a,b)\mapsto \left(xa+(xzb,0)+x{\widetilde{\phi}}(b)-{\widetilde{\phi}}(xb),xb\right)
\end{subeqn}
Using this formula, we can study $M_{y,z}/M_{y,z}^2$. Recall that
$\Lambda_{l,r+s}\otimes{\msy Z}_l$ is the image in $V^1$ of the sum of the two
canonical projections from $\Lambda_{l,r+s}\otimes{\msy Z}_l$ to
$\Lambda_{l,r}\otimes{\msy Z}_l$ and $\Lambda_{l,r,s}\otimes{\msy Z}_l$. It follows that
\begin{subeqn}\label{eqn5411}
M_{y,z}/M_{y,z}^2 \cong N:=
\Lambda_{l,r,s}\otimes{\msy Z}_l \;\oplus\; \Lambda_{l,r}\otimes{\msy Z}_l
\end{subeqn}
with $\tau$ acting on $N$ by
\begin{subeqn}\label{eqn5412}
\tau\colon (a,b)\mapsto (xa+x{\overline{\phi}}(b)-{\overline{\phi}}(xb),xb)
\end{subeqn}
where
${\overline{\phi}}\colon\Lambda_{l,r}\otimes{\msy Z}_l\to V^1/V^2=\Lambda_{l,r,s}\otimes{\msy Q}_l$
is ${\widetilde{\phi}}$ composed with the projection $V^1\to V^1/V^2$;
we have ${\overline{\phi}}(x^i)=x^iy_2$ for $0\leq i\leq l^r-2$.
Note that $\Phi_l/\Phi_l^2\cong N/(\tau-1)N$. Hence $\Phi_l/\Phi_l^2$ is cyclic
if and only if the endomorphism $\tau-1$ of the ${\msy F}_l$-vector space
$N\otimes{\msy F}_l$ has corank $1$. Now $N\otimes{\msy F}_l$ is the direct sum of
${\msy F}_l[\varepsilon]/(\varepsilon^{l^{r+s}-l^r})$ and ${\msy F}_l[\varepsilon]/(\varepsilon^{l^r-1})$,
with $\varepsilon=x-1$. The matrix of $\tau-1$ with respect to the direct sum of
the bases $(1,\varepsilon,\ldots,\varepsilon^{l^{r+s}-l^r-1})$ and
$(1,\varepsilon,\ldots,\varepsilon^{l^r-2})$ is of the form
\begin{subeqn}\label{eqn5413}
\renewcommand{\baselinestretch}{1}
\left(
\begin{array}{cccc|cccc}
0& & & & & & & \\
1&0& & & &A& & \\
&\ddots&\ddots& & & & & \\
& &1&0& & & & \\
\hline
& & & &0& & & \\
& & & &1&0& & \\
& & & & &\ddots&\ddots& \\
& & & & & &1&0
\end{array}
\right)
\end{subeqn}
It follows that $\tau-1$ has corank $1$ if and only if the upper right
coefficient of $A$ is not zero, or, equivalently, if and only if there
exists $b$ in $\Lambda_{l,r}\otimes{\msy Z}_l$ such that $x{\overline{\phi}}(b)-{\overline{\phi}}(xb)$
is a unit in $\Lambda_{l,r,s}\otimes{\msy Z}_l$. A computation shows that
$x{\overline{\phi}}(x^{l^r-2})-{\overline{\phi}}(x^{l^r-1})=g_r(x)y_2$. Now recall that we are
free to choose $y=(y_1,y_2)$ in $V^1$ and $z$ in
$(\Lambda_{l,r}\otimes{\msy Q}_l)^*$ as long as $(y,z)$ satisfies (\ref{eqn547}).
Note that $g_r(x)$ and $g_r'(x)$ are units in $\Lambda_{l,r,s}\otimes{\msy Q}_l$
and $\Lambda_{l,r}\otimes{\msy Q}_l$, respectively. Hence we can choose
$y_2=g_r(x)^{-1}$ and $z=x^{-1}g_r'(x)^{-1}$.
\end{example}
\begin{example}\label{example55}
Our final example is the analog of Example~\ref{example54} in the case
${\tilde{a}}=0$. More precisely, let $l$ be a prime and $r\geq0$ an integer.
Then there exists an abelian variety $A_K$ with $t={\tilde{a}}=0$,
${\tilde{t}}=l^r-1$ and $\Phi=\Phi_l\cong{\msy Z}/l^{2r}{\msy Z}$.
Let $C_{l,r}$ be as in Example~\ref{example52}. The abelian variety $A_K$
can be found in the isogeny class of $C_{l,r}$ in the same way as used in
Example~\ref{example54}. In this case the construction is somewhat easier,
since the filtration on $V$ has only two steps ($V^1=V^2$), so we leave the
details to the reader. Let us just mention that all formulas up to
(\ref{eqn5410}) remain valid (in adapted form), and after (\ref{eqn5410})
one shows that $M/(\tau-1)M$ can be cyclic with the same method as used to
show that $N/(\tau-1)N$ can be cyclic.
\end{example}
\section{Classification of the $\Phi_{(p)}$.} \label{section6}
The aim of this section is to prove the following theorem.
\begin{theorem}\label{thm61}
Let $D$ be a strictly henselian discrete valuation ring of residue
characteristic $p\geq0$. Let $G$ be a finite commutative group of order
not divisible by $p$. For each prime $l\neq p$, let
$m_l:=(m_{l,1},m_{l,2},\ldots)$ be the partition corresponding to the
$l$-part $G_l$ of $G$ (i.e., $G_l\cong\oplus_{i\geq1}{\msy Z}/l^{m_{l,i}}{\msy Z}$ and
$m_{l,1}\geq m_{l,2}\cdots$). Let $d$, $t$, $a$ and $u$ be non-negative
integers such that $d=t+a+u$. Then there exists an abelian variety over
the field of fractions of $D$, of dimension $d$, toric rank $t$, abelian
rank $a$ and unipotent rank $u$ which has $\Phi_{(p)}\cong G$, if and
only if
\begin{subeqn}\label{eqn611}
u \geq \sum_{l\neq p}\sum_{i\geq t+1}\left(
\frac{l^{\lfloor m_{l,i}/2\rfloor}+l^{\lceil m_{l,i}/2\rceil}}{2}-1\right)
\end{subeqn}
where for any real number $x$, $\lfloor x\rfloor$ and $\lceil x\rceil$ denote
the largest (resp. smallest) integer $\leq x$ (resp. $\geq x$).
\end{theorem}
\begin{proof}
We will start by showing that if $A_K$ is as indicated in the theorem, then
(\ref{eqn611}) holds. Let $l\neq p$ be a prime. Let $f_l$ be the map from
the set of partitions to ${\msy R}$ defined by
\begin{subeqn}\label{eqn612}
f_l(m) = \sum_{i\geq1} \left(\frac{l^{\lfloor m_i/2\rfloor}+
l^{\lceil m_i/2\rceil}}{2}-1\right)
\end{subeqn}
Then $f_l$ is strictly increasing for the partial ordering in which $a\geq b$
if and only if $a_i\geq b_i$ for all~$i$. One easily sees that $f_l$ is
increasing for the lexicographical ordering on the set of partitions of a
fixed number, but we won't use that.
Consider the filtration
\begin{subeqn}\label{eqn613}
\Phi_l \supset \Phi_l^1 \supset \Phi_l^3 \supset 0
\end{subeqn}
induced by (\ref{eqn21}). Theorem~\ref{thm33} shows that
\begin{subeqn}\label{eqn614}
2(t_l-t + a_l-a) \geq \delta_l(\Phi_l/\Phi_l^1) + \delta_l(\Phi_l^1/\Phi_l^3)
\end{subeqn}
Let $n_l$ be the invariant of $\Phi_l/\Phi_l^3$. Lemma~\ref{lemma410}
shows that for all $i\geq1$ we have $n_{l,i}\geq m_{l,i+t}$, or, in the
terminology of the proof of that lemma, that $n_l\geq d^t(m_l)$ in the partial
ordering. Lemma~\ref{lemma411} says that
\begin{subeqn}\label{eqn615}
\delta_l(\Phi_l/\Phi_l^1)+\delta_l(\Phi_l^1/\Phi_l^3)\geq 2f_l(n_l)
\end{subeqn}
It follows that
\begin{subeqn}\label{eqn616}
2(t_l-t+a_l-a) \geq 2f_l(n_l)
\end{subeqn}
Summing over all $l\neq p$ and dividing by $2$ gives (\ref{eqn611}).
It remains to show that all groups $G$ satisfying (\ref{eqn611}) can
occur as the $\Phi_{(p)}$ of an abelian variety $A_K$ over the field of
fractions $K$ of $D$ of dimension $d$, toric rank $t$, abelian rank $a$ and
unipotent rank $u$. It is sufficient to show that all groups $G$ satisfying
\begin{subeqn}\label{eqn617}
u = \lceil\left(\sum_{l\neq p}\sum_{i\geq t+1}\left(
\frac{l^{\lfloor m_{l,i}/2\rfloor}+l^{\lceil m_{l,i}/2\rceil}}{2}-1\right)
\right)\rceil
\end{subeqn}
occur in such a way, since one can replace $A_K$ by the product of $A_K$ with
an abelian variety $B_K$ which has unipotent reduction and trivial group
of connected components.
Let us first suppose that $K={\msy C}((q))$. Let $d$, $t$, $a$, $u$ and $G$ be as
in the theorem, and suppose that they satisfy (\ref{eqn617}). We have
$G\cong \oplus_{i\geq1}{\msy Z}/n_i{\msy Z}$ with $n_i\geq1$ and $n_{i+1}|n_i$ for all
$i$. Let $B_K$ be of the type described in Example~\ref{example51}:
it has dimension $t$, completely toric reduction and group of connected
components ${\msy Z}/n_1{\msy Z}\oplus\cdots\oplus{\msy Z}/n_t{\msy Z}$. The abelian variety $A_K$
we are constructing will be of the form
\begin{subeqn}\label{eqn618}
A_K=B_K\times\prod_{l\neq p}C_{K,l}, \qquad{\rm with}\quad
\dim(C_{K,l})=\lceil\left(\sum_{i\geq t+1}\left(
\frac{l^{\lfloor m_{l,i}/2\rfloor}+l^{\lceil m_{l,i}/2\rceil}}{2}-1\right)
\right)\rceil
\end{subeqn}
and such that all $C_{K,l}$ have unipotent reduction. Note that in fact
such an $A_K$ has unipotent rank $u$, since for $l\neq2$ the function $f_l$
defined above has integer values. For $l\neq2$ we define
\begin{subeqn}\label{eqn619}
C_{K,l} = \prod_{i>t} C_{K,l,i}
\end{subeqn}
where $C_{K,l,i}$ is the abelian variety constructed in Example~\ref{example54}
with $r=(m_{l,i}-1)/2$ and $s=1$ if $m_{l,i}\neq1$ is odd, where $C_{K,l,i}$
is the abelian variety constructed in Example~\ref{example53} with $i=1$ if
$m_{l,i}=1$, and $C_{K,l,i}$ is the abelian variety constructed in
Example~\ref{example55} with $r=m_{l,i}/2$ if $m_{l,i}$ is even.
Note that the group of connected components of
the reduction of $C_{K,l,i}$ is cyclic of order~$l^{m_{l,i}}$. For $l=2$ the
construction of $C_{K,l}$ is a bit different. Let $r\geq0$ be maximal such
that $m_{2,i}>1$ for all $i\leq r$. Then $C_{K,2}$ will be of the form
\begin{subeqn}\label{eqn6110}
C_{K,2} = D_{K,2} \times \prod_{t<i\leq r}C_{K,2,i}
\end{subeqn}
where $C_{K,2,i}$ is defined as $C_{K,l,i}$ but with $l$ replaced by $2$, and
where $D_{K,2}$ is as follows. Let $v$ be the number of $i>t$ such that
$m_{2,i}=1$. If $v$ is even we let $D_{K,2}$ be the product of $v/2$
elliptic curves which have unipotent reduction and group of connected
components isomorphic to ${\msy Z}/2{\msy Z}\times{\msy Z}/2{\msy Z}$. If $v$ is odd we let
$D_{K,2}$ be the product of $(v-1)/2$ elliptic curves with unipotent reduction
and group of connected components isomorphic to ${\msy Z}/2{\msy Z}\times{\msy Z}/2{\msy Z}$ and one
elliptic curve with unipotent reduction and group of connected components
cyclic of order~$2$. One verifies easily that $A_K$ has all the desired
properties.
To finish the proof of the theorem, we have to show that similar examples
exist over any strictly henselian discrete valuation ring $D$ with residue
characteristic $p$. Since our examples are products of the examples of
\S\ref{section5}, it suffices to show that the examples in \S\ref{section5}
exist over $D$. Since we do not suppose $D$ complete, we cannot use a
Tate curve ``${{\rm G}_m}/q^{\msy Z}$'' with $q$ a uniformizer of~$D$. Instead we can
use any elliptic curve $E$ over $K$ which has toric reduction and trivial
group of connected components. Then $I$ acts on the Tate module
${\rm T}_l(E({K^{\rm s}}))$ through its quotient ${\msy Z}_l(1)$ and for a suitable choice
of a ${\msy Z}_l$-basis of ${\rm T}_l(E({K^{\rm s}}))$, an element of $I$ with image $a$
in ${\msy Z}_l(1)$ acts as $({1\atop0}{a\atop1})$. It follows that
Examples~\ref{example51} and \ref{example52} with $E_1$ replaced by $E$ still
work. To make Example~\ref{example53} work over $D$, it is enough to show
that for all $l\neq p$ and $r>0$ such that $l^r>2$, there exists an abelian
scheme over $D$ of relative dimension $l^{r-1}(l-1)/2$ and with an action by
${\msy Z}[x]/(f_{l,r})$. Once one has these abelian schemes, the constructions
of \S\ref{section5} can be carried out over~$D$. The fact that such abelian
schemes exist is a consequene of the theory of abelian varieties of
``CM-type''. Fix an $l$ and $r$ as above. The moduli scheme over ${\msy Z}[1/l]$ of
abelian schemes of the desired type, with a suitable polarization and
$l$-power level structure, is finite etale and not empty. Another way to
prove the desired existence is to consider isogeny factors over
${\msy Q}(\zeta_{l^r})$ of the jacobian of the Fermat curve of degree $l^r$.
\end{proof}
\section{Further remarks and questions.} \label{section7}
Although Theorem~\ref{thm61} gives a complete classification of the
prime-to-$p$ parts of the groups of connected components of special fibres of
N\'eron models with some fixed invariants, there are still questions left.
For example, it is clear
that the groups of connected components $\Phi$ have functorial additional
structure coming from the fact that the category of abelian varieties has
an involution: every abelian variety has its dual. More precisely, suppose
that $\Phi$ comes from the abelian variety $A_K$. Let $A'_K$ be the
dual of $A_K$ and denote its group of connected components by $\Phi'$.
Then there are several pairings with values in ${\msy Q}/{\msy Z}$, conjecturally perfect
and the same, between $\Phi$ and $\Phi'$; see
\cite[\S\S1.2, 1.3, 11.2]{Grothendieck1}, \cite{Milne1}, \cite[\S3]{Dino1},
\cite{Moret-Bailly1} and
\cite[Prop.~3.3]{Oort1}. Let us note by the way that the last reference is
clearly wrong since it says that the pairing has values in $({\msy Q}/{\msy Z})(1)$;
the mistake in the proof is that the direct sum decomposition in the unique
displayed formula in it is not unique. Anyway, for each of the remaining
pairings we get a filtration
\begin{eqn}\label{eqn71}
\Phi_{(p)} = {\Phi'}_{(p)}^{4,\perp} \supset {\Phi'}_{(p)}^{3,\perp} \supset
{\Phi'}_{(p)}^{2,\perp} \supset {\Phi'}_{(p)}^{1,\perp} \supset
{\Phi'}_{(p)}^{0,\perp} = 0
\end{eqn}
It would be interesting to know the common refinement of this filtration
with (\ref{eqn21}). Also, it would be of interest to prove that the various
pairings are the same up to a determined sign. Some relations between
the two filtrations (\ref{eqn21}) and (\ref{eqn71}) on the $l$-part for
$l\neq p$ can be found in \cite[Thm.~3.21]{Dino1}, under the hypothesis that
$A_K$ has a polarization of degree prime to~$l$.
Let us consider the functor from the category of abelian varieties over $K$
to the category of finite abelian groups which associates to each abelian
variety the group of connected components of the special fibre of its
N\'eron model. A rather vague question one can ask is through what categories
of abelian groups endowed with some extra structure this functor factors.
We have seen for example that there is a filtration of four steps on the
prime-to-$p$ part, but as we have just remarked that is certainly not all
there is.
Lorenzini has shown \cite[Thm.~3.22]{Dino1}, under the hypothesis that there
is a polarization of degree prime to $l$, that ${\Phi'}_l^{2,\perp}$ is the
prime-to-$p$ part of the kernel of the map from $\Phi$ to the group of
connected components of $A_L$, where $K\to L$ is any extension over which
$A_K$ has semi-stable reduction. It would be interesting to generalize this.
Even in the case in which $A_K$ acquires semi-stable reduction after a
tamely ramified extension $K\to L$, when the theory of \cite{Edixhoven1}
applies, I have not been able to give a description of the filtration
(\ref{eqn21}) in terms of the special fibre of the N\'eron model of $A_L$
with its action of ${\rm Gal}(K/L)$.
The $p$-part of $\Phi$ remains difficult. For example, one expects a bound
for its order in terms of the dimension of $A_K$ if the toric part of the
reduction is zero, but even in the case of potentially good reduction I don't
know of any such bound (of course, if $A_K$ is the jacobian of a curve with a
rational point, the usual bound, i.e., the bound we have when $k$ is of
characteristic zero, holds, since one can apply Winters's theorem
\cite{Winters1}).
In a forthcoming article \cite{EdiLiuLor} one can
find a generalization of a result of McCallum (unpublished) which says that
in the case of potentially good reduction the $p$-part is annihilated by the
degree of any extension after which one obtains semi-stable reduction, but
not in general by the exponent of the Galois group of such an extension.
Work in progress by Bosch and Xarles, using a rigid analytic uniformization of
N\'eron models, seems to imply that there is a four-step functorial filtration
on the whole of~$\Phi$, for which three of the four successive quotients can
be described in terms of the ineria group acting on the character group of the
toric part of the semi-stable reduction. The remaining successive quotient
comes from an abelian variety, obtained by Raynaud's extension, which has
potentially good reduction. This part is still a mystery.
|
1994-10-17T05:20:11 | 9410 | alg-geom/9410015 | en | https://arxiv.org/abs/alg-geom/9410015 | [
"alg-geom",
"math.AG"
] | alg-geom/9410015 | Bert van Geemen | Bert van Geemen and Emma Previato | On the Hitchin System | 23 pages, LaTeX Version 2.09 <7 Dec 1989> | null | null | null | null | The Hitchin system is a completely integrable hamiltonian system (CIHS) on
the cotangent space to the moduli space of semi-stable vector bundles over a
curve. We consider the case of rank-two vector bundles with trivial
determinant. Such a bundle $E$ defines a divisor $D_E$ in the Jacobian of the
curve and for any smooth point of $D_E$ we define a cotangent vector (a Higgs
field). The Hitchin map on these Higgs fields is then determined in terms of
the Gauss map on the divisor $D_E$. We apply the results to the $g=2$ case and
show how Hitchin's system is related to classical line geometry in $\PP^3$.
| [
{
"version": "v1",
"created": "Fri, 14 Oct 1994 19:08:38 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"van Geemen",
"Bert",
""
],
[
"Previato",
"Emma",
""
]
] | alg-geom | \section{\@startsection {section}{1}{\z@}{-3.5ex plus -1ex minus
-.2ex}{1.5ex plus .2ex}{\large\bf}
\def\secdef\empsubsection{\emppsubsection*}{\@startsection{subsection}{2}{\z@}{-3.25ex plus -1ex
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\newcommand{\vspace{\baselineskip}\noindent{\bf Proof.}$\;\;$}{\vspace{\baselineskip}\noindent{\bf Proof.}$\;\;$}
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\title{On the Hitchin System}
\author{{\sc Bert van Geemen}\\
University of Utrecht, The Netherlands\\
{\sc Emma Previato}\\
Boston University, USA
\thanks {Research partially supported by NSF Grant
DMS-9105221 at Boston University and DMS-9022140 at MSRI.}
}
\date{}
\begin{document}
\maketitle
\section{Introduction}
\secdef\empsubsection{\emppsubsection*}{}
What is known as the Hitchin system is a completely integrable hamiltonian
system (CIHS)
involving vector bundles over algebraic curves, identified by Hitchin in
(\cite{H1},
\cite{H2}). It was recently generalized by Faltings \cite{F}.
In this paper we only consider the case of rank-two vector bundles with
trivial determinant. In that case the Hitchin system corresponding to a curve
$C$ of genus $g$ is obtained as follows. Let
$$
{\cal M}:=\{ E\rightarrow C:\;E\;\mbox{a semi-stable rank two bundle},\;
\wedge^2E\cong {\cal O}\;\}/\sim_{\rm S}
$$
be the moduli space of (S-equivalence classes of) semi-stable rank-two vector
bundles on $C$. Then ${\cal M}$ is a projective variety (singular if $g>2$) of
dimension $3g-3$.
The locus of stable bundles ${\cal M}^s$ is the set of smooth
points of ${\cal M}$ for $g>2$. The cotangent space of ${\cal M}$ at a stable
bundle $E$
is :
$$
T^*_E{\cal M}=Hom_0(E,E\otimes K),\mbox{with}\quad
Hom_0(E,E\otimes K):=H^0(C,{\cal E}nd_0(E)\otimes K)
$$
where ${\cal E}nd_0(E)$ is the sheaf of endomorphisms of $E$ with trace zero
and $K$ is the canonical bundle on $C$.
A $\Phi\in Hom_0(E,E\otimes K)$ is called a Higgs field. The determinant of a
Higgs field $det(\Phi)\in Hom(\wedge^2E,\wedge^2(E\otimes K))=H^0(C,2K)$ gives
a map
$$
det:T^*_E{\cal M}=Hom_0(E,E\otimes K)\longrightarrow
H^0(C,2K),
$$
which globalizes to a map on $T^*{\cal M}^s$. Hitchin considered the map:
$$
H:T^*{\cal M}^s\longrightarrow H^0(C,2K),\quad
\Phi\mapsto det(\Phi)
$$
and showed that it is a CIHS
in the following sense: the functions
on $T^*{\cal M}^s$
that one obtains by choosing any basis
in $H^0(C,2K)$ are a complete set of hamiltonians in involution
(with respect to the natural symplectic
structure on a cotangent bundle). Since $det$ is homogeneous of degree two in
the fibre variables ($det(t\Phi)=t^2det(\Phi))$, one can define a
(rational) projective Hitchin map
$$
\overline{H}:{\Bbb P} T^*{\cal M}^s\longrightarrow{\Bbb P} H^0(C,2K)=|2K|
$$
and it is in fact this map that we consider.
\secdef\empsubsection{\emppsubsection*}{}
In the first two sections of this paper we define and study a set of Higgs
fields
associated to any semi-stable bundle $E$. These results are then applied to the
$g=2$ case;
they
may also be of independent interest for studying
moduli spaces of Higgs bundles, which are pairs $(E,\Phi)$
as above, with certain restrictions.
To study ${\cal M}$ as a projective variety
(see \cite{NR}, \cite{B1}, \cite{B2})
one associates to any $E\in {\cal M}$
a divisor $D_E$ in the
Jacobian of $C$
using
which, questions
on
rank two bundles are rephrased in terms of line bundles (and extensions).
We exhibit a natural map
$$
\phi_E:D^{sm}_E\longrightarrow {\Bbb P} Hom_0(E,E\otimes K),\qquad
\xi\mapsto \Phi_\xi
$$
(with $D_E^{sm}$ the smooth points of $D_E$)
and we are able to compute $det(\Phi_\xi)$. The result is best stated in a
diagram (the stable case of
Proposition \ref{detPhi}): For any stable bundle $E$ the following diagram
commutes:
\begin{equation}\label{diagram}
\begin{array}{rcccl}
&&{\Bbb P} T_E^*{\cal M}&&\\
&\phi_E\nearrow&&\searrow\bar{H}&\\
D_E^{sm} &&&&{\Bbb P} H^0(C,2K)\\
&\psi_E\searrow&&\nearrow Sq&\\
&&{\Bbb P} H^0(K)&&
\end{array}
\end{equation}
here $\psi_E$ is just the Gauss map of the divisor $D_E$ inside the Jacobian
and
$Sq(\omega)=\omega\otimes \omega$.
Thus the divisor $D_E$ (rather, its image in ${\Bbb P} T_E^*{\cal M}$) plays an
important role in the study of the fibers of $\bar{H}$ over the quadratic
differentials which are squares of one forms. However, our
fiberwise approach (for the map $T^*{\cal M}\rightarrow {\cal M}$) is, in a
sense, perpendicular to Hitchin's approach which studies the fibers of
$H:T^*{\cal M}\rightarrow H^0(C,2K)$. That approach
establishes that such a fiber, over a (general) quadratic differential $\eta$
is the Prym
variety associated with a `spectral' double cover $C_\eta\rightarrow C$
defined by $\eta$ (\cite{H2},\,\cite{BNR}). It would be interesting to relate
our results to a study of the fibers of $H$.
\secdef\empsubsection{\emppsubsection*}{}
In the remaining sections we apply these results to investigate the
case $g=2$. Then the space ${\cal M}$ is isomorphic to ${\Bbb P}^3$ (\cite{NR}),
so that we look for a CIHS on $T^*{\Bbb P}^3$ (and also on the open subset
$T^*{\Bbb C}^3$).
Using information on $\phi_E$ from $\S$\ref{phiE}
we work out the maps of the diagram (1.2.1)
in the genus two case in $\S$\ref{g=2}.
Here we encounter some classical algebraic geometry of curves of genus two
and three. It turns out that finding $H$ explicitly involves
a problem in line geometry in ${\Bbb P}^3$
(a sketch of the solution in fact appears in
in J.H. Grace's article ``Line Geometry'' in
the Encyclopaedia Britannica, 1911).
We can thus make an educated guess as to what the explicit
hamiltonians
should be. A computer calculation (using the {\it Mathematica}
system) showed that our candidates actually define a CIHS. We
are not able to show that our hamiltonians define the Hitchin map, but we
can prove that Hitchin's hamiltonians and ours differ by multiplication by
functions from the base (an open set in ${\Bbb P}^3$). For a more precise result we
would have to extend the results of section \ref{g=2} to enlarge the open set
in the base were those results hold, or we would need further information on
Hitchin's
system.
\secdef\empsubsection{\emppsubsection*}{Acknowledgements.} The first named author
wishes to thank: the University of Pavia for a six month stay where much of
the work on this paper was done, the NSF for supporting visits to Boston
University under Grant DMS-9105221, and J. de Jong for helpful discussions.
The second named author wishes to acknowledge:
Carolyn Gordon's invitation to MSRI for two
weeks of the special year in Differential Geometry 1993/94
(research at MSRI supported in part by NSF grant \# DMS 9022140), and
participation in the LMS/Europroj Workshop ``Vector bundles in
algebraic geometry'' (Durham, 1993; organizers N. Hitchin, P. Newstead
and W.M. Oxbury), on which occasion N. Hitchin provided generous
insight.
\section{Higgs fields}
\secdef\empsubsection{\emppsubsection*}{}
We fix some notations and recall some basic facts.
In this paper $C$ will be a smooth, irreducible projective curve
of genus $g>1$ over ${\Bbb C}$
and $E$ will be a rank two semi-stable bundle on $C$ with trivial
determinant.
Since $\wedge^2E\cong {\cal O}$, we have:
$$
E\wedge E={\cal O}_C,\qquad {\rm so}\quad
E\cong E^*:={\cal H}om(E,{\cal O}_C),\quad e\mapsto [f\mapsto e\wedge f]
$$
thus $E$ is self-dual.
This gives isomorphisms:
$$
E\otimes E\cong E^*\otimes E={\cal E}nd(E),\qquad
S^2E\cong {\cal E}nd_0(E)
$$
with ${\cal E}nd_0(E)\subset {\cal E}nd(E)$ the sheaf of endomorphisms of
trace zero.
We recall that
$End_0(E):=H^0(C,{\cal E}nd_0(E))=0$ for a stable bundle $E$, the only
endomorphisms of $E$ being
scalar multiples of the identity. Thus:
$ H^0(C,S^2E)=0$.
For a vector space $V$, we let ${\Bbb P} V$ be the space of one dimensional
linear subspaces of $V$.
\secdef\empsubsection{\emppsubsection*}{}\label{inv}
We will construct Higgs fields by relating $E$ to line bundles. Such a
connection is provided by the following results.
Let $E$ be a semi-stable rank two bundle on $C$ with $det(E)={\cal O}$.
Associated to $E$ is a divisor (\cite{B1}, 2.2):
$$
D_E:=\{\xi\in Pic^{g-1}(C):\;\dim H^0(\xi\otimes E)>0\;\}.
$$
With its natural scheme structure, $D_E$ is linearly equivalent to $2\Theta$.
Here $\Theta$ is the natural theta divisor:
$$
\Theta:=\{\xi\in Pic^{g-1}(C):\;\dim H^0(C,\xi)>0\;\},\qquad
D_E\in |2\Theta|.
$$
On $Pic^{g-1}(C)$ there is a natural involution:
$$
\iota:Pic^{g-1}(C)\longrightarrow Pic^{g-1}(C),\qquad
\xi\mapsto K\otimes
\xi^{-1}.
$$
All divisors in $|2\Theta|$ are invariant under the involution $\iota$. For a $D_E$
that is easy to check since by Riemann-Roch and Serre duality:
$$
\dim H^0(\xi\otimes E)=\dim H^1(\xi\otimes E)=
\dim H^0(\xi^{-1}\otimes K\otimes E).
$$
Note that $H^0(\xi^{-1}\otimes K\otimes E)=Hom(\xi,E\otimes K)$ and that, since
$E$ is self-dual,
$H^0(\xi\otimes E)=Hom(E,\xi)$.
Thus we have:
$$
\xi\in D_E\Longleftrightarrow Hom(E,\xi)\neq 0 \Longleftrightarrow
Hom(\xi,E\otimes K)\neq 0.
$$
\secdef\empsubsection{\emppsubsection*}{}
For any semi-stable $E$ (cf.\ \cite{L}, Cor. V.6):
$$
\xi\in D^{sm}_E\;\Longrightarrow\; \dim H^0(\xi\otimes E)=1.
$$
Thus for $\xi\in D^{sm}_E$ there
are unique (up to scalar multiple) maps:
$$
\pi:E\longrightarrow \xi,\qquad
\tau: \xi\longrightarrow E\otimes K.
$$
The composition
$$
\tau\circ \pi:\;E\longrightarrow E\otimes K
$$
is an element of $Hom(E,E\otimes K)$, defined (up to scalar multiple) by
$\xi$.
\secdef\empsubsection{\emppsubsection*}{Definitions.} \label{defpp}
Let $E$ be a semi-stable rank-two bundle on $C$ with $det(E)={\cal O}$.
We define rational maps:
$$
\phi_E:\; D_E^{sm}\longrightarrow {\Bbb P} Hom_0(E,E\otimes K),
$$
$$
\xi\mapsto \Phi_\xi:=
\tau\circ \pi-(1/2)(id_E\otimes tr(\tau\circ \pi)):
E\otimes{\cal O}\longrightarrow E\otimes K
$$
and
$$
\psi_E:\; D_E^{sm}\longrightarrow {\Bbb P} H^0(C,K)\;={\Bbb P} Hom(\xi,\xi\otimes K),
$$
$$
\psi_E(\xi)=(\pi\otimes id_K)\circ\tau:\;
\xi\stackrel{\tau}{\longrightarrow} E\otimes K
\stackrel{\pi\otimes 1}{\longrightarrow}\xi\otimes K .
$$
\secdef\empsubsection{\emppsubsection*}{} Now we have a large supply of Higgs fields, the $\Phi_\xi$'s.
It is surprisingly easy to determine $\psi_E$. In Proposition \ref{detPhi}
we will see how that already determines the Hitchin map to a large extent.
Recall that the cotangent bundle to $Pic^{g-1}(C)$ is trivial:
$$
T^*Pic^{g-1}(C)\cong Pic^{g-1}(C)\times H^1(Pic^0(C),{\cal O})^*.
$$
For a smooth point $\xi$ in a divisor $D\subset Pic^{g-1}(C)$ the tangent
space
to $D$ at $\xi$ is then defined by an element of $H^1(Pic^0(C),{\cal O})^*$, unique
up to scalar multiple. The corresponding morphism $D^{sm}\rightarrow {\Bbb P}
H^1(Pic^0(C),{\cal O})^*$ is called the Gauss map.
\secdef\empsubsection{\emppsubsection*}{Proposition.}\label{psig}
The map $\psi_E$ is the Gauss map on $D_E\subset Pic^{g-1}(C)$.
$$
\psi_E:D_E^{sm}\longrightarrow {\Bbb P} H^1(Pic^0(C),{\cal O})^*={\Bbb P} H^0(C,K).
$$
In particular, $\psi_E$ is a morphism.
\vspace{\baselineskip}\noindent{\bf Proof.}$\;\;$
By \cite{L}, Prop.\ V.2, we know that for $\xi\in D_E^{sm}$ the space
$$
T_\xi D_E\subset T_\xi Pic^{g-1}(C)= H^1(Pic^0(C),{\cal O})=H^1(C,{\cal O})
$$
is defined by the image of the cup-product map
$$
H^0(C,E\otimes\xi)\otimes H^0(C,E\otimes \iota(\xi))\longrightarrow
H^0(C,K)\cong H^1(C,{\cal O})^*.
$$
This map coincides with the composition:
$$
Hom(E\otimes K,\xi\otimes K)\otimes Hom(\xi,E\otimes K)
\longrightarrow Hom(\xi,\xi\otimes K)\cong H^0(K),
$$
and in our case we recover the definition of $\psi_E$:
$$
(\pi\otimes 1)\otimes \tau\mapsto (\pi\otimes 1)\circ\tau =\psi_E(\xi).
$$
(One may in fact also consider $Hom(E,\xi_\eta)$ where $\xi_\eta$ is a
deformation of $\xi$ given by $\eta\in H^1(C,{\cal O})$. Then $\eta\in T_\xi D_E$
iff
$\pi\in Hom(E,\xi)$ lifts to $Hom(E,\xi_\eta)$ iff $\pi\cup \eta=0\in
Ext^1(E,\xi)=H^1(E\otimes\xi)$, which gives the statement above. The
justification for this argument is given in \cite{L}, II.)
{\unskip\nobreak\hfill\hbox{ $\Box$}\par}
\secdef\empsubsection{\emppsubsection*}{} We are interested in computing the determinant of the Higgs
field $\Phi_\xi$. Since the maps in \ref{defpp} are only defined up to scalar
multiple,
we consider
$$
det: Hom_0(E,E\otimes K) \longrightarrow {\Bbb P} H^0(C, 2K),
\qquad \Phi\mapsto \langle det(\Phi)\rangle.
$$
Let
$$
Sq: H^0(C,K)\longrightarrow H^0(C, 2K),\qquad
\omega\mapsto\omega^{\otimes 2}.
$$
\secdef\empsubsection{\emppsubsection*}{Proposition.}\label{detPhi}
For a semi-stable $E$ and $\xi\in D_E^{sm}$ we have:
$$
det(\Phi_\xi)=\psi_E(\xi)^{\otimes 2}\qquad
(\in {\Bbb P} H^0(C, 2K)).
$$
Thus, the compositions $det\circ\phi_E$ and $Sq\circ\psi_E$ coincide:
$$
\begin{array}{cccccc}
det\circ \phi_E: & D_E^{sm}&\stackrel{\phi_E}{\longrightarrow}&
{\Bbb P} Hom_0(E,E\otimes K)&
\stackrel{det}{\longrightarrow}& {\Bbb P} H^0(C, 2K),\\
Sq\circ\psi_E:& D_E^{sm}&\stackrel{\psi_E}{\longrightarrow} &{\Bbb P} H^0(C,K)&
\stackrel{Sq}{\longrightarrow} &{\Bbb P} H^0(C, 2K).
\end{array}
$$
\vspace{\baselineskip}\noindent{\bf Proof.}$\;\;$
Since $\psi_E:D_E^{sm}\rightarrow {\Bbb P} H^0(C,K)$ is a morphism,
$\psi_E(\xi)$ is (represented by) a non-zero differential form
for each $\xi\in D_E^{sm}$.
Define a canonical divisor on $C$ by:
$$
K_\xi:=(\psi_E(\xi)),\qquad {\rm let}\quad U:=C-Support(K_\xi),
$$
On the open set $U$, the map (defined by)
$\psi_E(\xi):\xi\rightarrow \xi\otimes K$ is an isomorphism. Its
inverse, composed with $\tau:\xi\rightarrow E\otimes K$, gives a
map $\xi\otimes K\rightarrow E\otimes K$ which splits the map
$\pi\otimes1:E\otimes K\rightarrow \xi\otimes K$.
$$
\begin{array}{ccc}
\xi & {}_{\psi_E(\xi)}& \\
\tau\Big\downarrow\phantom{\tau}&\searrow
&\\
E\otimes K &\stackrel{\pi\otimes 1}{\longrightarrow}& \xi\otimes K
\end{array}
$$
Thus over $U$, the bundle $E$ splits:
$$
{E}_{|U}\cong L\oplus\xi_{|U},\qquad{\rm with}\quad
L:=\ker(\pi_U:E_{|U}\longrightarrow \xi_{|U})
$$
and $L$ is a line bundle on $U$.
Since $\Phi_\xi:=\tau\pi-(1/2)tr(\tau\pi)$, we get
$\Phi_{\xi|U}(L)=0$ so that:
$$
\Phi_{\xi|U}=\left(\begin{array}{cc}
-(1/2)\psi_E(\xi)&\ast\\0&(1/2)\psi_E(\xi)
\end{array}\right).
$$
Then $det(\Phi_{\xi{|U}})=-(1/4)\psi_E(\xi)^{\otimes 2}$, which
does not
vanish at any point of $U$.
If $D_E$ is irreducible,
the image of the Gauss map on $D^{sm}_E$ contains an open subset of
${\Bbb P} H^0(C,K)$.
Thus, for general $\xi$ on such a $D_E$,
$2K_\xi$ is the only divisor in ${\Bbb P} H^0(C,2K)$ with support in $C-U$.
Since $det(\Phi_\xi)\in H^0(C,2K)$ and since its divisor must have support in
$C-U$, we conclude that
$$
(det(\Phi_\xi))=2K_\xi.
$$
For general $C$, the rank of the N\' eron-Severi
group of $Pic^{g-1}(C)$ is one,
and then the reducible divisors in $|2\Theta|$ are a subvariety of dimension $g$
(they are unions of two translates of $\Theta$).
Since $\Delta({\cal M})$ (see \ref{delta}) has dimension $3g-3$ (\cite{B1}),
the divisor $D_E$ is irreducible for general $C$.
Thus if we work in a family
of $D_E$'s over a general family of curves containing the given $D_E$,
the maps $\xi\mapsto det(\Phi_\xi)$ and $\xi\mapsto \psi_E(\xi)^{\otimes
2}$ agree on a non-empty open subset, hence must agree everywhere.
{\unskip\nobreak\hfill\hbox{ $\Box$}\par}
\section{The map $\phi_E$} \label{phiE}
\secdef\empsubsection{\emppsubsection*}{}\label{delta}
In this section we consider only stable bundles $E$ and we
study the map $\phi_E:D_E^{sm}\rightarrow {\Bbb P} T^*_E{\cal M}$.
We use the codifferential of the map:
$$
\Delta: {\cal M}\longrightarrow |2\Theta|,\qquad E\mapsto D_E
$$
to relate the cotangent bundles of ${\cal M}$ and of the projective space
$|2\Theta|$.
First of all we recall some facts on the cotangent bundle to a projective
space
and on the dual of $|2\Theta|$ (following \cite{NR2}, $\S$ 3).
\secdef\empsubsection{\emppsubsection*}{} \label{dual}
Let $V$ be a vector space.
The dual of the Euler sequence on ${\Bbb P} V$ gives:
$$
0\longrightarrow T^*{\Bbb P} V\longrightarrow V^*\otimes{\cal O}(-1)\longrightarrow
{\cal O}
\longrightarrow 0,
$$
the last non-trivial map is given by
$(\ldots,s_i,\ldots)\mapsto\ldots+x_is_i+\ldots$ over
$(\ldots :x_i:\ldots)\in{\Bbb P} V$.
Taking the associated projective bundles
we have an isomorphism:
$$
{\Bbb P} T^*{\Bbb P} V\cong I:=\{\;(x,h)\in {\Bbb P} V\times {\Bbb P} V^*:\;
x\in h\;\},
$$
the variety $I$ is called the incidence bundle.
\secdef\empsubsection{\emppsubsection*}{}\label{delta2}
To identify the dual of $|2\Theta|$ we use the map:
$$
\delta:Pic^{g-1}(C)\longrightarrow |2\Theta_0|,\qquad
\xi\mapsto D_\xi:=L^*_\xi\Theta +L^*_{\iota(\xi)}\Theta,
$$
here $\Theta_0$ is (any) symmetric theta divisor in $Pic^0(C)$ (the linear
equivalence class of $2\Theta_0$ is independent of the choice) and
$$
L_\alpha:Pic(C)\longrightarrow Pic(C),\qquad
\beta\mapsto \alpha\otimes\beta
$$
is translation by $\alpha$ in $Pic(C)$.
Pulling back the linear forms on $|2\Theta_0|$ gives an isomorphism
$$
\delta^*:H^0(|2\Theta_0|,{\cal O}(1))=H^0(Pic^0(C),2\Theta_0)^*
\stackrel{\cong}{\longrightarrow} H^0(Pic^{g-1}(C),2\Theta).
$$
Projectivizing gives
$\delta^*:|2\Theta_0|^*\stackrel{\cong}{\longrightarrow} |2\Theta|$.
In fact, there is a commutative diagram:
$$
\begin{array}{ccc}
& & |2\Theta|^*\\
&{}^\nu\nearrow\phantom{{}^\nu}& \\
Pic^{g-1}(C)& &\;\;\downarrow (\delta^*)^*\\
& {}_\delta\searrow\phantom{{}_\delta}&\\
& &|2\Theta_0|
\end{array}
$$
where $\nu$ is the natural map (see \cite{NR2} and \cite{B2} \S 2 for a
variant). From now on, $I$ will be the incidence bundle:
$$
I:={\Bbb P} T^*|2\Theta|\subset |2\Theta|\times |2\Theta_0|.
$$
We will denote by
$(d\Delta)^*$ the projectivized codifferential of $\Delta$
(see \ref{delta}):
$$
(d\Delta)^*:{\Bbb P} T^*|2\Theta|= I\longrightarrow {\Bbb P} T^*{\cal M}.
$$
\secdef\empsubsection{\emppsubsection*}{Lemma.}\label{dd}
Let $D\in |2\Theta|$ and let $\xi\in Pic^{g-1}(C)$. Then
$$
(D,D_\xi)\in I\;\Longleftrightarrow\; \xi\in D.
$$
\vspace{\baselineskip}
\noindent
{\bf Proof.}$\;\;$
For $\xi\in Pic^{g-1}(C)$, $\nu(\xi)\in |2\Theta|^*$
is the hyperplane in
$|2\Theta|$ consisting of the
divisors passing through $\xi$.
Thus $D\in |2\Theta|$ and $\xi\in |2\Theta|^*$ are incident iff $\xi\in D$.
The dual of the isomorphism $\delta^*$ maps $\nu(\xi)$ to $D_\xi$ so the result
follows.
{\unskip\nobreak\hfill\hbox{ $\Box$}\par}
\secdef\empsubsection{\emppsubsection*}{} We recall that for non-hyperelliptic curves,
the map $\Delta$ has degree one over its image \cite{B1}
(so it is locally an isomorphism with its image for generic $E$) and is an
embedding for the general curve as recently announced by Y. Laszlo
and also by
S. Brivio and A. Verra jointly.
In case $g=2$ the map is an isomorphism \cite{NR} but for hyperelliptic curves
of genus greater than two
the map is 2:1 and `ramifies' along a subvariety of dimension $2g-1$
\cite{B1}.
\secdef\empsubsection{\emppsubsection*}{Proposition.}\label{glue}
Let $E$ be a stable bundle such that the map $\Delta$ is locally at $E$ an
isomorphism with its image. Then
the rational map
$$
\phi_E:D_E\longrightarrow {\Bbb P} T^*_E{\cal M},\qquad \xi\mapsto \Phi_{\xi},
$$
is the left-hand column in the diagram:
$$
\begin{array}{ccc}
D_E&\hookrightarrow&Pic^{g-1}(C)\\
\Bigg\downarrow&&\Bigg\downarrow\delta\\
{\Bbb P} T^*_{D_E}|2\Theta| &\hookrightarrow& |2\Theta_0|\\
(d\Delta_E)^*\Bigg\downarrow\phantom{(d\Delta_E)^*} & &\\
{\Bbb P} T^*_E {\cal M}&&\\
\end{array}
$$
where the last vertical arrow is a linear projection given by the dual of the
differential of $\Delta$ at $E\in{\cal M}$.
\vspace{\baselineskip}
\noindent
{\bf Proof.}$\;\;$
Let $\xi\in D_E$. Then $\delta(\xi)=D_\xi\in|2\Theta_0|=|2\Theta|^*$ corresponds to a
hyperplane $H_\xi\subset |2\Theta|$. By Lemma \ref{dd}, $\xi\in D_E$ implies
$D_E\in H_\xi\subset |2\Theta|$.
This says that $H_\xi$ passes through
$D_E=\Delta(E)\in\Delta({\cal M})\subset |2\Theta|$ and
(by the assumption on local isomorphism)
defines a codimension
$\leq 1$ subspace in $T_E{\cal M}$. We must show that
$\Phi_\xi\;(\in T^*_E{\cal M})$ is the defining equation for this subspace.
We first determine $H_\xi\cap \Delta({\cal M})$;
the pull-back $\Delta^*H_\xi$ will be the divisor
$\tilde{D}_\xi$ defined below; in particular, the subspace of $T_E{\cal M}$
defined by
$H_\xi$ is $T_E\tilde{D}_\xi$.
We recall from \cite{B1} that
$$
Pic({\cal M})\cong{\Bbb Z},\qquad{\rm and}\quad {\cal L}:=\Delta^*({\cal O}(1))
$$
is the ample generator of this group. Moreover, the natural map
$$
{\cal M}\longrightarrow {\Bbb P} H^0({\cal M},{\cal L})^*
$$
actually coincides with $\Delta$.
Define for $\xi\in Pic^{g-1}(C)$:
$$
\tilde{D}_\xi:=\{E\in{\cal M}:\; H^0(C,E\otimes\xi)\neq 0\;\}.
$$
This divisor, with its natural scheme structure, is defined by a section of
$H^0({\cal M},{\cal L})$.
Restriction to the Kummer variety of $Pic^0(C)$ (= locus of non-stable bundles)
in ${\cal M}$ induces the isomorphism (\cite{B1}):
$$
{\Bbb P} H^0({\cal M},{\cal L})\stackrel{\cong}{\longrightarrow} |2\Theta_0|,
\qquad {\rm and}\quad \tilde{D}_\xi\mapsto D_\xi
$$
(indeed, for $L\in Pic^0(C)$ one has $H^0((L\oplus L^{-1})\otimes\xi)>0$ iff
$L\in L_\xi^*\Theta$ or $L\in L_{\iota(\xi)}^*\Theta$ iff $L\in D_\xi$).
Now, by definition, the hyperplane $H_\xi$ intersects $Pic^0(C)$ in $D_\xi$ so
that
$H_\xi$ intersects ${\cal M}$ in $\tilde{D}_\xi$, as desired.
We must now show that $\Phi_\xi$ defines the subspace $T_E \tilde{D}_\xi$ of
$T_E{\cal M}$. The divisor $\tilde{D}_\xi$ is (the closure of) the image of the
(rational) map:
$$
\rho:{\Bbb P} H^1(\xi^{-2})\longrightarrow {\cal M},\qquad
\epsilon\mapsto [E_\epsilon]
$$
where $E_\epsilon$ is the extension defined by $\epsilon\in
Ext^1(\xi,\xi^{-1})=H^1(\xi^{-2})$,
\begin{equation}
\label{ext}
0\longrightarrow \xi^{-1}\longrightarrow E_\epsilon\stackrel{\pi}{\longrightarrow}
\xi
\longrightarrow 0.
\end{equation}
These maps were studied in detail by Bertram in \cite{B}.
We will now assume $\pi:E\rightarrow \xi$ to be surjective, so $\xi^{-1}$ is a
subbundle of $E$. By specialization the result
follows for all $\xi\in D_E$, all $E$.
We tensor the sequence \ref{ext} by $E$,
obtaining the following sequence for $S^2E$:
$$
0\longrightarrow \xi^{-2}\longrightarrow
S^2E \longrightarrow \xi\otimes E
\longrightarrow 0.
$$
Since ${\cal E}nd_0(E)\cong S^2E$,
the differential of $\rho$ is the natural map:
$$
d\rho:H^1(\xi^{-2})/H^0(\xi\otimes E)\longrightarrow H^1(S^2E)=T_E{\cal M},
\qquad
{\rm so}\quad
T_E \tilde{D}_\xi=Im(d\rho).
$$
Dualizing the sequence above and tensoring it by $K$ we get:
$$
0\longrightarrow \xi^{-1}\otimes E\otimes K\longrightarrow
S^2E\otimes K \longrightarrow \xi^{ 2}\otimes K
\longrightarrow 0.
$$
Now we have:
$$
\begin{array}{rcl}
\langle\Phi_\xi\rangle
&=& Im(H^0(\xi^{-1}\otimes E\otimes K)=\langle\tau\rangle
\longrightarrow H^0(S^2E\otimes K))\\
&=& \ker(H^1(E\otimes\xi)\longleftarrow H^1(S^2E))^*\\
&=& Im(H^1(S^2E)\longleftarrow H^1(\xi^{-2}))^*.
\end{array}
$$
Therefore we have indeed, as claimed:
$$
T_E \tilde{D}_\xi=\ker(\Phi_\xi).
$$
{\unskip\nobreak\hfill\hbox{ $\Box$}\par}
\secdef\empsubsection{\emppsubsection*}{}\label{covD}
The previous proposition shows that the line bundle
${\cal O}_{Pic^{g-1}(C)}(2\Theta)_{|D_E}\cong {\cal O}_{D_E}(D_E)$ plays an essential role as
regards
the map $\phi_E$. Recall that the divisor $D_E$ is invariant under
the involution $\iota$ (see \ref{inv}).
The following lemma shows how $\iota$ acts on the global sections of this line
bundle.
\secdef\empsubsection{\emppsubsection*}{Lemma.}\label{split}
The involution $\iota$ gives a splitting in an invariant and an
anti-invariant part:
$$
\begin{array}{rcccc}
H^0(D_E,{\cal O}_{D_E}(D_E))&=& H^0(D_E,{\cal O}_{D_E}(D_E))_+&\oplus&
H^0(D_E,{\cal O}_{D_E}(D_E))_-\\
&\cong&H^0(Pic^{g-1}(C),{\cal O}(D_E))/\langle s_E\rangle & \oplus&
H^1(Pic^{g-1}(C),{\cal O}).
\end{array}
$$
Here $s_E\in H^0(Pic^{g-1}(C),{\cal O}(D_E))$ is a section with divisor
$(s_E)=D_E$.
The map $\phi_E$ factors over the natural map
$$
D_E\longrightarrow {\Bbb P} H^0(D_E,{\cal O}_{D_E}(D_E))_+^*\;(\cong {\Bbb P}
T^*_{D_E}|2\Theta|).
$$
The map $\psi_E$ (the Gauss map)
is the natural rational map:
$$
D_E\longrightarrow {\Bbb P} H^0(D_E,{\cal O}_{D_E}(D_E))_-^*\cong
{\Bbb P} H^1(Pic^{g-1}(C),{\cal O})^*.
$$
Therefore both $\phi_E$ and $\psi_E$ factor over $\bar{D}_E$.
\vspace{\baselineskip}\noindent{\bf Proof.}$\;\;$
The exact sequence of sheaves on $Pic^{g-1}(C)$:
$$
0\longrightarrow {\cal O}\stackrel{\cdot s_E}{\longrightarrow}
{\cal O}(D_E)\longrightarrow {\cal O}_{D_E}(D_E)\longrightarrow 0
$$
gives the cohomology sequence:
$$
0\longrightarrow H^0(Pic^{g-1}(C),{\cal O}(D_E))/\langle s_E \rangle
\longrightarrow H^0(D_E,{\cal O}_{D_E}(D_E))
\longrightarrow H^1(Pic^{g-1}(C),{\cal O})\longrightarrow 0.
$$
It is well known that $\iota^*$ acts as the identity on
$H^0(Pic^{g-1}(C),{\cal O}(D_E))$ and as minus the identity on
$H^1(Pic^{g-1}(C),{\cal O})$. The remaining assertions are standard.
{\unskip\nobreak\hfill\hbox{ $\Box$}\par}
\section{The genus two case}\label{g=2}
\secdef\empsubsection{\emppsubsection*}{} To determine the Hitchin map in the genus two case, we study
first
the divisors $D_E$ for general stable $E$ and we
study three of the four maps from the diagram \ref{diagram} (see
\ref{lower2}, \ref{upper}).
This leads to quite classical geometry involving for instance \' etale double
covers and tangent
conics. Then we can easily determine the fourth map, which is Hitchin's map
(projectivized and restricted to ${\Bbb P} T^*_E{\cal M}$).
We then `rigidify' our construction using the classical Proposition
\ref{prym}.
From now on, $C$ will be a genus two curve.
\secdef\empsubsection{\emppsubsection*}{}
The main result of \cite{NR} is that the map
$$
\Delta:{\cal M}\stackrel{\cong}{\longrightarrow} |2\Theta|\cong{\Bbb P}^3,\qquad
E\mapsto D_E
$$
is an isomorphism, so ${\cal M}\cong {\Bbb P}^3$.
In particular, any element in $|2\Theta|$ is a $D_E$ for some $E$.
Since this linear system is base-point free, the divisors,
now in fact curves,
$D_E$ are smooth and have genus 5
for general stable $E$
(a description of the singular curves in $|2\Theta|$ can be found in
\cite{Verra}).
\secdef\empsubsection{\emppsubsection*}{}\label{cov}
Each divisor in $|2\Theta|$ is fixed by $\iota$ and
for general $E$,
the involution $\iota$ restricted to $D_E$ is a fixed-point free involution on
a smooth curve. The induced covering
$$
\pi_E:D_E\longrightarrow \bar{D}_E:=D_E/\iota
$$
is an \' etale 2:1 covering (the associated Prym variety is $Jac(C)$,
see for example \cite{Verra}, p.\ 438).
In particular, for general $E$, $\bar{D}_E$ is a smooth genus three curve.
From now on we will consider only such $E$.
\secdef\empsubsection{\emppsubsection*}{}\label{bunde}
The kernel of the map $\pi_E^*:Pic(\bar{D}_E)\rightarrow Pic(D_E)$ is generated
by a point $\alpha$ of order two.
One has:
$$
\pi_E^* K_{\bar{D}_E}\cong K_{D_E},\quad
\pi_{E*}{\cal O}_{D_E}\cong{\cal O}_{\bar{D}_E}\oplus\alpha.
$$
The adjunction formula on $Pic^{g-1}(C)$ shows ${\cal O}_{D_E}(D_E)\cong
K_{D_E}$.
The involution $\iota$ gives a splitting in an invariant and an
anti-invariant part:
$$
H^0(D_E,K_{D_E})\cong H^0(\bar{D}_E,\pi_* K_{D_E})\cong\;
H^0(\bar{D}_E,K_{\bar{D}_E})\oplus
H^0(\bar{D}_E,K_{\bar{D}_E}\otimes\alpha),
$$
(projection formula)
which coincides with the splitting given in Lemma \ref{split}. In particular,
since $H^1(Pic^{g-1}(C),{\cal O})\cong H^1(C,{\cal O})\cong H^0(C,K)^*$, we have a
natural identification
$$
H^0(C,K)\cong H^0(\bar{D}_E,K_{\bar{D}_E}\otimes\alpha)^*.
$$
\secdef\empsubsection{\emppsubsection*}{} \label{lower2}
We will now write $C_3$ for $\bar{D}_E$, $K_3$ for $K_{\bar{D}_E}$.
The Gauss map on $D_E$ then factors over $C_3$, and on $C_3$ coincides with the
natural map
$$
C_3\longrightarrow {\Bbb P} H^0(C_3,K_3\otimes\alpha)^*\cong
{\Bbb P} H^1(Pic^{g-1}(C),{\cal O})^* \cong {\Bbb P}^1
$$
which is therefore also essentially $\psi_E$.
The map $Sq:|K|\rightarrow |2K|$ from diagram \ref{diagram}
corresponds to the second Veronese map which embeds ${\Bbb P}^1$ as a conic in
${\Bbb P}^2$. The (three dimensional) space $S^2H^0(C_3,K_3\otimes\alpha)^*$ may be
identified with a quotient of the (six dimensional) $H^0(C_3,2K_3)^*$
(note $2(K_3\otimes\alpha)\equiv 2K_3$), thus we have a diagram (where the last
map is a linear projection):
$$
\begin{array}{ccccccccc}
D_E& &\stackrel{\psi_E}{\longrightarrow}&
&|K|&\stackrel{Sq}{\longrightarrow}&|2K|& & \\
\Big|\,\!\Big|&& && \;\;\Big\downarrow \cong && \;\Big\downarrow\cong& & \\
D_E&\stackrel{\pi_E}{\longrightarrow}\!&C_3&\!\rightarrow&\!{\Bbb P}
H^0(C_3,K_3\otimes\alpha)^*&\longrightarrow&\!{\Bbb P} S^2H^0(C_3,K_3\otimes\alpha)^*&
\leftarrow &\!{\Bbb P} H^0(C_3,2K_3)^*.
\end{array}
$$
\secdef\empsubsection{\emppsubsection*}{}\label{upper}
As $\Delta:
{\cal M}{\rightarrow} |2\Theta|={\Bbb P}^3$,
is an isomorphism, the cotangent bundle to ${\cal M}$ is the incidence
bundle. The map $(d\Delta)^*$ induces an isomorphism.
$$
(d\Delta)^*:I=\{(x,h)\in {\Bbb P}^3\times({\Bbb P}^3)^*:\;x\in h\;\}
\stackrel{\cong}{\longrightarrow} {\Bbb P} T^*{\cal M} \quad
{\rm and}\quad {\Bbb P} T_{D_E}^*|2\Theta|\cong {\Bbb P} T^*_E{\cal M}.
$$
From Lemma \ref{split} and \ref{bunde} we get:
$$
\phi_E:D_E\stackrel{\pi_E}{\longrightarrow} C_3
\stackrel{\kappa}{\longrightarrow} {\Bbb P} H^0(C_3, K_3)^*\cong {\Bbb P} T_E^*{\cal M}
$$
where $\kappa$ is just the canonical map.
\secdef\empsubsection{\emppsubsection*}{}\label{kum}
We show how the various $C_3$'s ($=\bar{D}_E$'s) fit together as $E$ moves
over ${\Bbb P}^3\;(={\cal M})$.
The image $S$ of the map
$$
\delta:Pic^{g-1}(C)\longrightarrow S\subset |2\Theta_0|
= {\Bbb P} H^0(Pic^{g-1}(C),2\Theta)^*=({\Bbb P}^3)^*.
$$
(see \ref{delta2})
is the Kummer surface of $Pic^{g-1}(C)$:
$$
S\cong Pic^{g-1}(C)/\iota,\qquad \iota:L\mapsto K\otimes L^{-1}.
$$
The surface $S$ is a quartic surface and its singular locus consists of the 16
fixed points of $\iota$ (which are the theta characteristics on $C$).
Moreover, we can view ${\Bbb P} T_E^*{\cal M}$ as a plane in $({\Bbb P}^3)^*$:
$$
{\Bbb P} T_E^*{\cal M}=\{h\in({\Bbb P}^3)^*:\; E\in h\;\}.
$$
Proposition \ref{glue} shows:
$$
{\Bbb P} T_E^*{\cal M}\cap S\,=\; \phi_E(D_E),\qquad
({\rm with}\quad \phi_E:D_E\longrightarrow {\Bbb P} T_E^*{\cal M}).
$$
Thus $\phi_E(D_E)$ is a hyperplane section
of the Kummer surface $S$, hence a quartic plane curve.
For general $E$, this curve will be smooth (i.e.\ the curve $\bar{D}_E$ is
non-hyperelliptic).
We consider only these $E$.
\secdef\empsubsection{\emppsubsection*}{}\label{hf3}
The curve $C_3$ is now non-hyperelliptic by assumption, so
the canonical map $\kappa$
is an embedding, and the image of $C_3$ is a smooth quartic in ${\Bbb P}^2$.
Pull-back along $\kappa$ gives an isomorphism
$H^0(C_3,2K_3)\cong H^0({\Bbb P}^2,{\cal O}(2))$.
Let $s,\,t$ be a basis of $H^0(C_3,K_3\otimes\alpha)$. We define
conics $Q_i$ in ${\Bbb P}^2$
by:
$$
s\otimes s=Q_1,\quad s\otimes t=Q_2,\quad t\otimes t=Q_3
$$
and the essential part (that is, on $D_E/\iota=C_3$) of $Sq\circ\psi_E$ is
now:
$$
C_3\longrightarrow {\Bbb P} S^2H^0(C_3,K_3\otimes\alpha)^*\cong {\Bbb P}^2,
$$
$$
x\mapsto (s^2(x):st(x):t^2(x))=(Q_1(\kappa(x)):
Q_2(\kappa(x)):Q_3(\kappa(x))).
$$
Since $Sq\circ \psi_E=\bar{H}\circ\phi_E$,
we conclude that the Hitchin map is given by:
$$
\bar{H}:{\Bbb P} T_E^*{\cal M}
\longrightarrow {\Bbb P} H^0(2K)=|2K|,\qquad
p\mapsto (Q_1(p):Q_2(p):Q_3(p))
$$
(since $\kappa(C_3)=\phi_E(D_E)$ has degree 4, spans ${\Bbb P} T_E^*{\cal M}$
and $\bar{H}$ has quadratic coordinate functions, $\bar{H}$
is determined by its restriction to $\phi_E(D_E)$).
Note that the inverse image in ${\Bbb P} T_E^*{\cal M}$
of the conic $Sq|K|\subset |2K|$ under $\bar{H}$ is a quartic curve containing
$\phi_E(D_E)$ and thus is equal to $\phi_E(D_E)$.
\secdef\empsubsection{\emppsubsection*}{}\label{bit}
We study the construction above a little more closely and exhibit natural (up
to
scalar multiple) subsets of $H^0(C_3,K_3\otimes\alpha)^*$ and $H^0(K)$ which
correspond under the isomorphism $H^0(C_3,K_3\otimes\alpha)^*\cong H^ 0(K)$
that we found in \ref{bunde}.
For any $a,\,b\in{\Bbb C}$ we have a section $as+bt\in H^0(C_3,K_3\otimes\alpha)$,
let:
$$
Q_{(a:b)}:=S^2(as+bt)=a^2Q_1+2abQ_2+b^2Q_3.
$$
The $Q_{(a:b)}$ are a quadratic system of conics, each of which is
tangent to $C_3$ (that is, has even intersection multiplicity at each
intersection point) because $Q_{(a:b)}$ cuts out twice the divisor of the
section $as+bt\in H^0(K_3\otimes \alpha)$.
There are 6 conics $H_i$, $i\in\{1,\ldots ,6\}$, in the quadratic system of
tangent conics which
split as pairs of bitangents. They correspond to the six points
$$
\langle s_i\rangle=\langle a_is+b_it\rangle\in{\Bbb P}^1=
{\Bbb P} H^0(K_3\otimes\alpha),\qquad{\rm with}\quad
det(a_i^2Q_1+2a_ib_iQ_2+b_i^2Q_3)=0
$$
(where we now view the $Q_i$ as $3\times 3$ matrices).
In this way obtain 12 bitangents of $C_3$. The other 16 bitangents are best
seen
by identifying $C_3$ with a hyperplane section of the Kummer surface $S$
(\ref{kum}).
In fact the divisor $\Theta$ and its translates by points of order two map
to conics in $S$, the plane through such a conic intersects $S$ in a double
conic, and thus intersects the plane in
which $\bar{D}_E$ lies in a bitangent of $\bar{D}_E$.
The following classical result relates these six points
$\langle s_i\rangle$ to the Weierstrass points of the curve $C$.
\secdef\empsubsection{\emppsubsection*}{Proposition.}\label{prym}
Under the natural isomorphism from \ref{bunde}:
$$\
{\Bbb P} H^0(C_3,K_3\otimes\alpha)\stackrel{\cong}{\longrightarrow} {\Bbb P} H^0(K)^*,
$$
the six points which correspond to pairs of bitangents are mapped to the six
linear maps corresponding to the Weierstrass points $p_i$
of $C$:
$$
\langle s_i\rangle\mapsto \langle [\omega\mapsto \omega(p_i)]\rangle
\qquad\qquad (\omega\in H^0(K)).
$$
\vspace{\baselineskip}\noindent{\bf Proof.}$\;\;$
We recall the way $D_E$ and $C$ can be recovered from
$H^0(C_3,K_3\otimes\alpha)$.
It will be shown that the double cover of ${\Bbb P} H^0(C_3,K_3\otimes\alpha)$
branched over the 6 points corresponding to pairs of bitangents is
isomorphic to $C$, which proves the Proposition.
Since we have $(st)^{ 2}=(s^ {2})( t^{2})$ on
$C_3$, the quartic equation of $C_3$ must be:
$$
C_3:\quad Q_1Q_3-Q_2^2=0.
$$
The curve $C_5$
defined by:
$$
s^2=Q_1,\quad st=Q_2,\quad t^2=Q_3 \qquad(\subset{\Bbb P}^4)
$$
is a canonically embedded genus 5
curve in ${\Bbb P}^4$ (with coordinates $x,y,z,s,t$ and where $x,y,z$
are a basis of $H^0(C_3,K_3)$).
Projection onto ${\Bbb P}^2$ defines a 2:1 unramified covering $\pi:C_5\rightarrow
C_3$ and clearly $\pi^*( K_{3}\otimes\alpha)\cong K_{C_5}$, so $\pi$ is
defined by $\alpha$ and $C_5\cong D_E$.
From the theory of Prym varieties we have (cf.\ \cite{Mumford}, $\S$ 6):
$$
Nm^{-1}(K_3)=P^+\cup P^-
\qquad{\rm with}\quad Nm:\; Pic^4(C_5)\longrightarrow Pic^4(C_3)
$$
and $P^+,\,P^-$ are both isomorphic to $J(C)$, the Prym variety of the cover
$C_5\rightarrow C_3$ (\cite{Verra}, p.\ 438).
Here we have:
$$
P^+:=\{ L\in Pic^4(C_5):\; Nm(L)=K_3,\;\;
\dim H^0(C_5,L)\equiv 0\;{\rm mod}\;2\},\quad{\rm and}\quad
\tilde{\Theta}\cap P^+=\Xi,
$$
where $\tilde{\Theta}$ is the theta divisor in $Pic^4(C_5)$ and where $\Xi$ is the
theta divisor of the Prym variety (actually the intersection has multiplicity
2), so in our case $\Xi=C$.
A point of $C$ thus corresponds to a $g^1_4$ on $C_5$ with norm $K_3$. The
$g^1_4$'s on $C_5$ are cut out by rulings of quadrics in the ideal of $C_5$ of
rank $\leq 4$. The hyperelliptic involution on $C$ corresponds to the
permutation
of the rulings in the rank 4 quadrics, so the Weierstrass points correspond to
the $g^1_4$'s from rank 3 quadrics.
To a section $as+bt\in H^0(K_3\otimes\alpha)$ corresponds
a quadric of rank $\leq 4$ in the ideal of $C_5$ given by:
$$
(as+bt)^2=a^2Q^2+2abQ_2+b^2Q_3,\quad{\rm so}\quad
(as+bt)^2=Q_{(a:b)}.
$$
Such a quadric has rank 3 iff $det(Q_{(a:b)})=0$. Thus these rank 4 quadrics
are parametrized by $|K_3\otimes\alpha|$ and there are 6 rank 3 quadrics that
correspond to the pairs of bitangents.
Each quadric is a cone over a 2:1 cover of the plane $s=t=0$ branched along the
conic $Q_{(a:b)}$. The rulings of a rank four quadric
are the two irreducible components in the inverse image of lines tangent to the
conic $Q_{(a:b)}$; they are interchanged by the covering involution
($s,\,t\mapsto -s,\,-t$).
Any ${\Bbb P}^2$ in such a rank 4 quadric thus projects to a line tangent to the
conic $Q_{(a:b)}$ and the divisor cut out by the ${\Bbb P}^2$ on $C_5$ maps onto
the divisor cut out on $C_3$ by that tangent line.
Hence the norm of the $g^1_4$'s obtained from these quadrics is $K_3$.
This shows that $C$ is indeed the double cover of $|K_3\otimes\alpha|$ branched
over the
six points corresponding to pairs of bitangents.
{\unskip\nobreak\hfill\hbox{ $\Box$}\par}
\secdef\empsubsection{\emppsubsection*}{}
The proposition allows us to make `consistent' choices for the coordinate
functions of the Hitchin map as $E$ varies.
Let $s_i\in H^0(C_3,K_3\otimes\alpha)$ be the six sections which correspond to
bitangents. Then $H_i$ restricts to $s_i^2$ on $C_3$,
so if we put:
$$
\bar{H}:{\Bbb P} T^*_E{\cal M} \longrightarrow {\Bbb P}^2,\qquad
p\mapsto (H_1(p):H_2(p):H_3(p))
$$
then we have a choice of coordinate functions for $\bar{H}$ which makes sense
for any (general) $E$.
The only remaining problem is that we can can still multiply each $H_i$ by a
function on ${\cal M}\cong{\Bbb P}^3$ which has poles and zeros in the locus where
the map $D_E\rightarrow \bar{D}_E$ is not an
\' etale 2:1 map of smooth curves.
\section{Computing the Hitchin map}\label{compute}
\secdef\empsubsection{\emppsubsection*}{}
In the previous section we saw that the
polynomials $H_i$ on ${\Bbb P}^3\times ({\Bbb P}^3)^*$ defining the Hitchin map
have the property:
for any general $q\in{\Bbb P}^3$,
$$
(H_i=0)\cap {\Bbb P} T^*_q{\Bbb P}^3=l_i\cup l_i'\qquad
({\Bbb P} T^*_q{\Bbb P}^3\subset{\Bbb P} T^*{\Bbb P}^3=I=\{(x,h)\in{\Bbb P}^3\times({\Bbb P}^3)^*:\;x\in
h\;\}),
$$
where $l_i$ and $l_i'$ form the pair of bitangents to the smooth curve
$$
C_3:=S\cap {\Bbb P} T^*_q{\Bbb P}^3
$$
(see \ref{kum}) corresponding to $s_i\in H^0(C_3,K_3\otimes\alpha)$
(i.e.\ $(s_i^2)=C_3\cap (l_i\cup l_i'$)). Here $S\subset ({\Bbb P}^3)^*$
is the Kummer surface of $Pic^{g-1}(C)$ and $\alpha$ is the bundle of order two
defined by the \' etale double cover of $C_3$ obtained by pull-back from the
map
$Pic^{g-1}(C)\rightarrow S$.
We now consider the problem of finding such polynomials.
\secdef\empsubsection{\emppsubsection*}{}
This problem was actually solved a century ago using the relation between
Kummer surfaces and Quadratic line complexes. The classical solution
is as follows.
A line in $({\Bbb P}^3)^*$ with Klein coordinates (see \ref{linec})
$(x_1:\ldots:x_6)\in{\Bbb P}^5$ is a bitangent to the Kummer surface $S$
occurring in one of the six pairs
iff there is an $i\in\{1,\ldots ,6\}$
such that the following two equations are satisfied (see Proposition
\ref{bip}):
$$
x_i=0,\qquad
\sum_{j\neq i} \frac{x_j^2}{\lambda_i-\lambda_j}=0
\qquad\qquad(j\in\{1,\ldots ,6\}),
$$
where the $\lambda_i$ correspond to the Weierstrass points of the curve $C$:
$$
C:\quad y^2=(x-\lambda_1)\ldots (x-\lambda_6).
$$
We will show in \ref{biteq} how to derive the $H_i$ by `restricting'
these two equations to the incidence bundle $I$.
\secdef\empsubsection{\emppsubsection*}{}\label{linec} We start with some definitions from line geometry.
The Pl\"ucker coordinates of the line $l=\langle
(Z_0:\ldots: Z_3),\;(W_0:\ldots :W_3)\rangle\subset({\Bbb P}^3)^*$ are:
$$
p_{ij}:=Z_iW_j-W_iZ_j\quad{\rm and}\quad
G:\;\;p_{01}p_{23}-p_{02}p_{13}+p_{03}p_{12}=0
$$
is the equation of the Grassmannian of lines, embedded in ${\Bbb P}^5$.
The Klein coordinates of a line are:
$$
\begin{array}{lll}
X_1=p_{01}+p_{23},\quad& X_3=i(p_{02}+p_{13}),\quad&X_5=p_{03}+p_{12}\\
X_2=i(p_{01}-p_{23}),&X_4=p_{02}-p_{13},&X_6=i(p_{03}-p_{12}).
\end{array}
$$
Note that each $X_i$ corresponds to a non-degenerate alternating bilinear form
in the $Z_i,\,W_i$. These six bilinear forms give sections $\Phi_i$ of the
bundle projection ${\Bbb P} T^*{\Bbb P}^3\longrightarrow {\Bbb P}^3$:
$$
\Phi_i:{\Bbb P}^3\longrightarrow {\Bbb P} T^*{\Bbb P}^3=I\subset{\Bbb P}^3\times({\Bbb P}^3)^*,
\quad q\mapsto (q,\epsilon_i(q)):=(q,X_i(q,-)).
$$
That $\Phi_i(q)\in I$ follows from the fact that $X_i$ is alternating:
$X_i(q,q)=0$. Explicitly, if $q=(x:y:z:t)\in{\Bbb P}^3$, then the
$\epsilon_i=\epsilon_i(q)\in{\Bbb P}^{3*}$ have the dual coordinates:
$$
\begin{array}{lll}
\epsilon_1=(y:-x:t:-z),\quad& \epsilon_3=(z:t:-x:-y),\quad&\epsilon_5=(t:z:-y:-x)\\
\epsilon_2=(y:-x:-t:z),&\epsilon_4=(z:-t:-x:y),&\epsilon_6=(t:-z:y:-x).
\end{array}
$$
\secdef\empsubsection{\emppsubsection*}{}\label{biteq}
We show how to take care of the first equation.
Let $q=(x:y:z:t)\in{\Bbb P}^3$.
As the incidence bundle is the cotangent bundle we have:
$$
T^*_q{\Bbb P}^3=\{(u:v:w:s)\in ({\Bbb P}^3)^*:\; xu+yv+zw+ts=0\;\}.
$$
Note that we can rewrite the equation to obtain:
$$
T^*_q{\Bbb P}^3=\{p\in({\Bbb P}^3)^*:\;X_i(\epsilon_i(q),p)=0\;\}.
$$
This implies that the lines in $T^*_q{\Bbb P}^3$ with $X_i=0$ (which form a linear
line complex) are exactly the lines passing through the point $\epsilon_i(q)$
(cf.\ \cite{GH}, p. 759-760).
In particular, if $T^*_q{\Bbb P}^3\cap S$ is a smooth quartic curve,
and $l_i,\;l'_i$ are a pair of bitangents as before,
then both lines have $X_i=0$ and thus they must intersect in $\epsilon_i(q)$.
Let now $p\in T^*_q{\Bbb P}^3,\; p\neq \epsilon_i(q)$.
The condition that $p\in l_i\cup l'_i$ is equivalent to demanding that the
line $\langle\epsilon_i(q),p\rangle$
is one of these two bitangents.
The $i$-th Klein coordinate of this line is zero because it passes through
$\epsilon_i(q)$.
Thus for these lines the first equation is verified. We conclude:
$$
p\in l_i\cup l_i'\subset {\Bbb P} T^*_q{\Bbb P}^3\;
\Longleftrightarrow\;
H_i(p,q):=\sum_{j\neq i} \frac{x_j^2}{\lambda_i-\lambda_j}=0,\quad
{\rm with}\quad x_j:=X_j(\langle\epsilon_i(q),p\rangle),
$$
the Klein coordinates of the line $\langle\epsilon_i(q),p\rangle\subset ({\Bbb P}^3)^*$.
The coordinates of
$\epsilon_i(q)$ are linear in those of $q$ and so the Pl\"ucker coordinates of
$\langle\epsilon_i(q),p\rangle$
are homogeneous of bidegree (1,1) in those of $q$ and $p$. Thus $H_i$ is given
by a homogeneous polynomial of bidegree (2,2) on ${\Bbb P}^3\times ({\Bbb P}^3)^*$.
\secdef\empsubsection{\emppsubsection*}{}
On the open subset $T^*{\Bbb C}^3={\Bbb C}^3\times ({\Bbb C}^3)^*$ of $T^*{\Bbb P}^3$
one can obtain a CIHS from the polynomials $H_i$ as follows.
Let $(x,y,z)$ be coordinates on ${\Bbb C}^3$ and let $(u,v,w)$ be the dual
coordinates on ${\Bbb C}^{3*}$. Then, the inclusion of cotangent bundles
followed by the (rational) projectivization map is given by
$$
T^*{\Bbb C}^3 \longrightarrow T^*{\Bbb P}^3\longrightarrow {\Bbb P} T^*{\Bbb P}^3
$$
$$
(q,p):=((x,y,z),(u,v,w))\mapsto
(\tilde{q},\tilde{p}):=((x:y:z:1),(u:v:w:-(xu+yv+zw))).
$$
(Note that the last coordinate is obtained from the incidence condition.)
Now we define:
$$
H^a_i(p,q)=\sum_{j\neq i}
\frac{X_j(\langle
\epsilon_i(\tilde{q})\,,\,(u:v:w:-(xu+yv+zw))\rangle)^2}{\lambda_i-\lambda_j}.
$$
The $H_i$ are homogeneous of degree (2,2), and the last coordinate has degree
one in $x,\,y,\,z$ so the $H^a_i$ will have degree $\leq 4$ in the $x,\,y,\,z$
(and need not be homogeneous in these variables), but they
are still homogeneous of degree $2$ in the $u,\,v,\,w$.
With the help of a computer,
one can explicitly write down the polynomials $H^a_i$ (the expressions are
rather long though).
To verify that these polynomials actually Poisson commute (with respect to the
standard two form
$dx\wedge du+dy\wedge dv+dz\wedge dw$) we again used the computer (after
normalizing
three of the $\lambda_i$'s by a linear fractional transformation). This then
allows us to
conclude that the map $H^a:T^*{\Bbb C}^3\rightarrow {\Bbb C}^3$ (whose coordinate
functions are any three of the six $H_i^a$'s) is a CIHS.
It seems reasonable to expect that the CIHS defined by these $H^a_i$
is actually Hitchin's system, but we could not establish that.
\section{Quadratic Line Complexes}\label{Qcomp}
\secdef\empsubsection{\emppsubsection*}{}
In this section we recall how the equations for the bitangents are determined.
We summarize the results we need from \cite{GH}, Chapter 6 and follow
\cite{Hu}.
Let $G\subset {\Bbb P}^5$ be the Grassmannian of lines in ${\Bbb P}^3$, so $G$ is
viewed as a quadric in ${\Bbb P}^5$. For $x\in G$
we denote by $l_x$ the corresponding line in ${\Bbb P}^3$. For $p\in{\Bbb P}^3$ and
$h\subset {\Bbb P}^3$ a plane we
define
$$
\sigma(p):=\{x\in G:\; p\in l_x\},\qquad \sigma(h):=\{x\in G:\; l_x\subset h\}.
$$
Both $\sigma(p)$ and $\sigma(h)$ are isomorphic to ${\Bbb P}^2$, in fact any (linear)
${\Bbb P}^2$ in $G$ is either a $\sigma(p)$ or a $\sigma(h)$. Let $L$ be a line
in $G$, then $L$ is the intersection of a (unique) $\sigma(p)$ with
a (unique) $\sigma(h)$:
$$
L=\sigma(p)\cap\sigma(h)=\{x\in G:\; p\in l_x\subset h\}.
$$
We will sometimes write $h=h_L$, $p=p_L$ and $L=L_{p,h}$.
Thus the points on the line $L$ (in $G$) correspond to the lines (in ${\Bbb P}^3$)
in a pencil in
$h$ with `focus' $p$.
\secdef\empsubsection{\emppsubsection*}{} A quadratic line complex $X$ is the intersection of $G$ with
another quadric $F$; we assume $X$ to be smooth.
$$
X:=G\cap F.
$$
For any $p\in{\Bbb P}^3$,
the intersection of $\sigma(p)={\Bbb P}^2\subset G$ with the quadric $F$ is a conic
in
$\sigma(p)$. Let
$$
S:=\{p\in{\Bbb P}^3:\; \sigma(p)\cap F\;{\rm is}\;{\rm singular}\},
$$
then $S$ is a Kummer surface.
\secdef\empsubsection{\emppsubsection*}{}
If $\sigma(p)\cap F$ is singular, it is the union of two
lines $L,\,L'$ or it is a double line. The double lines correspond to the 16
singular
points of $S$. The points $x\in L$ correspond to the lines in a plane
$h_L$ passing through $p$, similarly the points in $L'$ correspond to lines
in a plane $h_{L'}$ passing through $p$. These pencils are called
confocal pencils (having the same focus $p$). Note that the line $l=h_L\cap
h_{L'}$ lies in both these pencils; it is called a singular line of the
complex $X$. This line $l$ corresponds to the intersection of $L$ and
$L'$ in $G$: $[l]=L\cap L'$.
The singular lines of $X$ form a smooth surface $\Sigma$ in $G$.
$$
\Sigma:=\{x\in X:\;l_x\;\mbox{is a singular line in}\;X\}.
$$
The set $\Sigma$ is determined in \cite{GH}, p.\ 767-769:
$$
x\in\Sigma\; \Longleftrightarrow\; T_xF=T_{x'}G\quad \mbox{for some}\;x'\in G.
$$
\secdef\empsubsection{\emppsubsection*}{}\label{kc}
In Klein coordinates $X_i$ the relation between the points $x$ and $x'$ above
assumes a very simple form. Any quadratic line complex $X$ can be given by
(\cite{GH},p.\ 789):
$$
G:\;\; X_1^2+X_2^2+\dots +X_6^2=0,\quad
F:\;\;\lambda_1X_1^2+\lambda_2X_2^2+\ldots +\lambda_6X_6^2=0,
\quad X=G\cap F.
$$
Then $S$ is the Kummer variety associated with the genus two curve
$$
C:\quad y^2=(x-\lambda_1)\ldots (x-\lambda_6).
$$
The equations defining the surface $\Sigma$ are then
(\cite{GH}, p.\ 769)
$$
\Sigma=G\cap F\cap F_2,\qquad
F_2:\quad \lambda_1^2X_1^2+\dots +\lambda_6^2X_6^2=0.
$$
Let now
$$
x=(x_1:\ldots:x_6)\in \Sigma\;\Longrightarrow\;\;
T_xF:\quad\lambda_1x_1X_1+\ldots+\lambda_6x_6X_6=0.
$$
Defining
$$
x':=(\lambda_1x_1:\ldots:\lambda_6x_6),\;\Longrightarrow\;\;
\quad
x'\in G,\quad
T_{x'}G:\quad\lambda_1x_1X_1+\ldots+\lambda_6x_6X_6=0,
$$
so $T_xF=T_{x'}G$ and $x'$ satisfies the required condition.
\secdef\empsubsection{\emppsubsection*}{Lemma.}\label{lembit}
Let $x\in \Sigma\subset G$ and let $x'\in G$ as above. Define a line:
$$
L:=\langle x,\,x'\rangle\;\subset {\Bbb P}^5.
$$
Then we have $L\subset G$ and
$$
L=L_{p,h}\qquad {\rm with}\quad p\in S,\quad h=T_pS,
$$
so that the points $y\in L$ correspond to the lines $l_y\subset {\Bbb P}^3$ with
$p\in
l_y\subset T_pS$. For $i\in\{1,\ldots ,6\}$ let
$$
\{[l_i]\}:=\;(X_i=0) \cap L\qquad(\in G\subset {\Bbb P}^5).
$$
Then $l_i$ is a bitangent line to $S$. Moreover,
if $p$ is a general point of $S$ then any bitangent to $S$ passing
through $p$ is one of the six $l_i$'s.
\vspace{\baselineskip}\noindent{\bf Proof.}$\;\;$
Since $x\in T_{x'}G$ we have $L\subset G$ (this is also easily verified using
the three equations defining $\Sigma$). Then $L=L_{p,h}$ with $p=l_x\cap
l_{x'}$.
We claim that $p\in S$. Since $\sigma(p)$ is a linear subspace in $G$ passing
through $x'$ we have $\sigma(p)\subset T_{x'}G$, and thus also
$\sigma(p)\subset T_xF(=T_{x'}G)$. Thus $\sigma(p)$ is tangent to $F$ at $x$, so
$\sigma(p)\cap F$ is singular in $x$. Therefore $p\in S$ (and $l_x$ is the singular
line of $X$ passing through $p$).
Any point on $L$, distinct from $x$, can be written as:
$$
x_\lambda:=\lambda x+x'=(\ldots:(\lambda +\lambda_i)x_i:\ldots)\qquad (\lambda\in {{\Bbb C}}).
$$
It is easy to check by substitution that $x\in \Sigma\Rightarrow
x_\lambda\in\Sigma_\lambda$ with:
$$
\Sigma_\lambda:= G\cap F^{(\lambda)}\cap F^{(\lambda)}_2,\quad(\lambda\neq -\lambda_i)
$$
and where we define:
$$
F^{(\lambda)}:\;(\lambda+\lambda_1)^{-1}X_1^2+\ldots+(\lambda+\lambda_6)^{-1}X_6^2=0,\quad
F^{(\lambda)}_2:\;(\lambda+\lambda_1)^{-2}X_1^2+\ldots+(\lambda+\lambda_6)^{-2}X_6^2=0.
$$
Thus $x_\lambda$ corresponds to a singular line for the quadratic complex
$X_\lambda:=G\cap F^{(\lambda)}$.
As above, there exists thus a point $x'_\lambda\in G$ with:
$$
T_{x_\lambda}F^{(\lambda)}=T_{x'_\lambda}G,\qquad
x'_\lambda:=(\ldots:x_i:\ldots)=x\in G.
$$
Therefore $x_\lambda$ and $x'_\lambda$ lie on the line $L\subset G$ and thus the point
$p$ is a point of $S_\lambda$, the Kummer surface associated to the quadratic line
complex $X_\lambda$. This holds for all singular lines $x$ of $X$ (and thus for all
points $p\in S$), therefore we conclude:
$$
S=S_\lambda.
$$
In particular, there is a one-dimensional family of quadratic line complexes
$X_\lambda$ which give rise to the same Kummer surface, the so called Klein variety
(see \cite{NR} for a modern treatment).
Now we can determine $h$. Each $x_\lambda\in L$ is a singular line for a
quadratic line complex defining $S$. Then the line in ${\Bbb P}^3$ corresponding to
it is tangent to $S$ at the (unique; cf. the verification on p.\ 767 of
\cite{GH})
point $p_\lambda\in S$
with $x_\lambda=Sing(\sigma(p_\lambda)\cap F)$ (cf.\ \cite{GH},
p.\ 764-765, p. 791).
In our case, $p_\lambda=p$ for all $\lambda$, so we conclude that $L$ is the pencil of
lines in ${\Bbb P}^3$ that are tangent to $S$ at $p$, which implies $h=T_pS$.
The lines from $L$ that are bitangent to $S$ are thus the bitangents of the
curve $T_pS\cap S$.
This is, in general, a plane quartic curve with a node at $p$, so there are six
bitangents to $S$ in the pencil $L$. These must then correspond to the values
$\lambda=-\lambda_i$, since for other values the lines $x_\lambda$ are singular lines of a
smooth quadratic line complex and cannot be tangent to $S$ at other points.
Thus the bitangent lines in $T_pS$ passing through $p$
correspond to the points on $L$ with exactly one Klein coordinate
equal to zero.
{\unskip\nobreak\hfill\hbox{ $\Box$}\par}
\secdef\empsubsection{\emppsubsection*}{Remarks.} Viewing $S$ as $Pic^{g-1}(C)/\iota$, the divisors
$T_pS\cap S$ (for $p$ smooth) correspond to the divisors
$$
D_\beta:=L_\beta^*\Theta+L_{-\beta}^*\Theta\in |2\Theta|,\qquad(
\beta \in Pic^0(C))
$$
with $2\beta\neq{\cal O}$.
These are the union of two copies of $C\,(=\Theta)$ meeting in two points. These
two points, and the two copies, are interchanged by $\iota$, the quotient is a
nodal curve isomorphic to $C$ with the two points identified.
The normalization of $T_pS\cap S$ is thus isomorphic with $C$. The six
bitangents to $C$ correspond to the lines spanned by $p$ and (the image in
$T_pS\cap S$) of a Weierstrass point of $C$.
This, once again, establishes the connection between bitangents to $S$
(and its plane sections) and Weierstrass points on $C$.
The 16 non-reduced divisors $L_\beta^*(2\Theta)\in |2\Theta|$ (so $2\beta={\cal O}$) map to
double conics.
A point on a double conic is `exceptional' for the Lemma since
any line tangent to $S$ at a point of the conic lies in the plane of the conic
and is thus a bitangent to $S$.
These (double) conics are called the tropes
of the Kummer surface.
The `self'-duality of the Kummer surface $S=\delta(Pic^{g-1}(C))$
fits in nicely with the map
$\delta:Pic^{g-1}(C)\rightarrow |2\Theta|_0$ from \ref{delta}, the duality
between $|2\Theta_0|$ and $|2\Theta|$, and the map $\delta':Pic^0(C)\rightarrow
|2\Theta|,\;
\beta\mapsto D_\beta$. In fact, it identifies the tangent planes to points of
$S$ (which cut out $D_\beta$) with the points $\pm \beta \in Pic^0(C)/\pm 1
\cong \delta'(Pic^0(C))$ which is the Kummer surface of $Pic^0(C)$.
This surface
is isomorphic to $S=Pic^{g-1}(C)/\iota$, but the `self'-duality
is however not an
isomorphism; it is a birational isomorphism which blows up
double points and blows down tropes.
The special case that the line $L$ in the Lemma
actually lies in $X$ is studied in \cite{GH}, p.\ 791-796 (note that they fix
the
quadratic line complex whereas in the proof of the Lemma we consider a family
of complexes).
\secdef\empsubsection{\emppsubsection*}{Proposition.}\label{bip}
For $x\in G$ the line $l_x$ is a bitangent to a general point of $S$
iff for some $i\in\{1,\ldots ,6\}$ one has:
$$
x_i=0,\qquad
\sum_{j\neq i} \frac{x_j^2}{\lambda_i-\lambda_j} =0.
$$
\vspace{\baselineskip}
\noindent{\bf Proof.}$\;\;$
In Lemma \ref{lembit} we saw that any such
bitangent $l_z$ of $S$ has one Klein coordinate equal to zero,
we will assume it is the first one. Then
$$
z=(0:(-\lambda_1+\lambda_2)x_2:\ldots :(-\lambda_1+\lambda_6)x_6)\quad
{\rm with}\quad
x=(x_1:x_2:\ldots:x_6)\in\Sigma,
$$
in particular $x\in X=G\cap F$.
Substituting the coordinates of $z$ in the second equation we get:
$$
(\lambda_1-\lambda_2)x^2_2+\ldots +(\lambda_1-\lambda_6)x_6^2=\lambda_1(x_1^2+\ldots +x_6^2)-
(\lambda_1x_1^2+\ldots+\lambda_6x_6^2)=0.
$$
Conversely, let $z=(0:z_2:\ldots :z_6)\in G$ satisfy also the second equation
above, so:
$$
z_2^2+\ldots +z_6^2=0,\quad
(\lambda_1-\lambda_2)^{-1}z^2_2+\ldots +(\lambda_1-\lambda_6)^{-1}z_6^2=0.
$$
Then we define:
$$
x_i:=(\lambda_1-\lambda_2)^{-1}z_i,\quad(2\leq i\leq 6),\qquad
x_1:=\sqrt{-(x_2^2+\ldots +x_6^2)},
$$
here the choice of the square root does not matter. Define
$$
x:=(x_1:x_2:\ldots :x_6),\quad{\rm so}\quad x_1^2+\ldots +x_6^2=0
$$
and we have $x\in G$ (the quadric defined by $X_1^2+\ldots +X_2^2$.)
We claim that $l_x$ is a singular line of $S$. For this we verify that $x$
satisfies the other two quadratic equations defining $\Sigma$. First of all:
$$
\begin{array}{rcl}
\lambda_1x_1^2+\lambda_2x_2^2+\ldots+\lambda_6x_6^2&=&
-\lambda_1(x_2^2+\ldots +x_6^2)+\lambda_2x_2^2+\ldots+\lambda_6x_6^2\\
&=&(\lambda_1-\lambda_2)x_2^2+\ldots+(\lambda_1-\lambda_6)x_6^2\\
&=&(\lambda_1-\lambda_2)^{-1}z^2_2+\ldots
+(\lambda_1-\lambda_6)^{-1}z_6^2\\
&=&0,
\end{array}
$$
so $x\in F$. We use these two relations on the $x_i$'s to obtain the third:
$$
\begin{array}{rcl}
\lambda_1^2x_1^2+\ldots+\lambda_6^2x_6^2&=&
(\lambda_1^2x_1^2+\ldots+\lambda_6^2x_6^2)-2\lambda_1(\lambda_1x_1^2+\ldots+\lambda_6x_6^2)
+\lambda_1^2(x_1^2+\ldots+x_6^2)\\
&=&
(\lambda_1-\lambda_2)^2x_2^2+\ldots +(\lambda_1-\lambda_6)^2x_6^2\\
&=& z_2^2+\ldots +z_6^2\\
&=&0,
\end{array}
$$
since $z=(0:z_2:\ldots:z_6)\in G$. Thus $x\in F_2$ and we conclude
$x\in\Sigma$,
so we verified that $l_x$ is a singular line.
Note that (with notation from \ref{kc}):
$$
x'=(\lambda_1x_1:\ldots:\lambda_6x_6),\quad {\rm so}\;\; z=-\lambda_1x+x'
$$
and thus $l_z$ is indeed a bitangent to $S$ (see the proof of the Lemma).
{\unskip\nobreak\hfill\hbox{ $\Box$}\par}
|
1994-10-06T05:20:22 | 9410 | alg-geom/9410003 | en | https://arxiv.org/abs/alg-geom/9410003 | [
"alg-geom",
"math.AG"
] | alg-geom/9410003 | Valery Alexeev | Valery Alexeev | Moduli spaces $M_{g,n}(W)$ for surfaces | 21 pages, written in AMS-LaTeX | null | null | null | null | We construct and prove the projectiveness of the moduli spaces which are
natural generalizations to the case of surfaces of the following:
1) $M_{g,n}$, the moduli space of $n$-marked stable curves,
2) $M_{g,n}(W)$, the moduli space of $n$-marked stable maps to a variety $W$.
| [
{
"version": "v1",
"created": "Wed, 5 Oct 1994 11:27:39 GMT"
}
] | 2015-06-30T00:00:00 | [
[
"Alexeev",
"Valery",
""
]
] | alg-geom | \section{Introduction}
\label{sec:introduction}
\begin{say}
\label{say:moduli spaces for curves}
We recall the following coarse moduli spaces in the case of curves:
\begin{enumerate}
\item $M_{g}$, parameterizing nonsingular curves of genus $g\ge2$
and its compactification $\overline{M_{g}}$, parameterizing
Mumford-Deligne moduli-stable curves, see Mumford \cite{Mumford77},
\item spaces $M_{g,n}$, $2g-2+n>0$, for stable
$n$-pointed curves, see Knudsen \cite{Knudsen83},
\item a moduli space $M_{g,n}(W)$ of stable maps from reduced curves
to a variety $W$, see Kontsevich \cite{Kontsevich94}.
\end{enumerate}
It is well known that $\overline{M_{g}}$ and $M_{g,n}$ are
projective, $M_{g}$ is quasi-projective.
\end{say}
\begin{say}
For surfaces, Gieseker \cite{Gieseker77} established the existence
of a quasi-projective scheme parameterizing surfaces with at worst
Du Val singularities, ample canonical class $K$ and fixed $K^{2}$,
this is a straightforward analog of $M_{g}$ and we will denote it
by $M_{K^{2}}$. A geometrically meaningful compactification of this
space, $\overline{M^{sm}_{K^{2}}}$, was constructed by Koll\'ar and
Shepherd-Barron in \cite{KollarShepherdBarron88} as a separated
algebraic space. It is a moduli space of smoothable stable (not in
the G.I.T. sense) surfaces of general type. In \cite{Kollar90}
Koll\'ar has shown that if the class of smoothable stable surfaces
with a fixed $K^2$ is bounded then $\overline{M^{sm}_{K^{2}}}$ is in
fact a projective scheme (Corollary 5.6). Finally, the boundedness
was proved by the author in \cite{Alexeev94b}.
\end{say}
\begin{say}
The main purpose of this paper is to construct analogs of $M_{g,n}$
and $M_{g,n}(W)$ in the case of surfaces, and to prove their
projectiveness. After this is done, we touch on a connection between
our moduli spaces and the standard moduli spaces of K3 and Abelian
surfaces.
\end{say}
\begin{say}
An idea of ``$M_{g,n}$ for surfaces'' occured to me when I mentioned
that my boundedness theorem 9.2 \cite{Alexeev94b} is strictly
stronger than what was used for $\overline{M^{sm}_{K^{2}}}$. Then,
looking at the definition of $M_{g,n}(W)$ in \cite{Kontsevich94} I
realized that this is simply a relative version of the same scheme,
and can be done for surfaces too.
\end{say}
\begin{say}
The basic construction of a moduli space as an algebraic space used
here is the same as in \cite{Mumford82}, \cite{Kollar85},
\cite{Viehweg94} and elsewhere. For the hardest question involved,
proof of local closedness, we refer to a result of Koll\'ar
\cite{Kollar94}.
For proving that our moduli spaces are projective schemes, rather
than mere algebraic spaces, we use Koll\'ar's Ampleness Lemma 3.9
\cite{Kollar90}, which can be applied in a straightforward way to a
variety of complete moduli problems.
\end{say}
\begin{say}
Kontsevich and Manin \cite{KontsevichManin94} use the moduli spaces
$M_{g,n}(W)$ to define Gromov-Witten classes of varieties in the
``quantum cohomology'' theory. Hence one distant application of
``$M_{g,n}(W)$ for surfaces'' might be ``higher'' GW-classes of
schemes.
\end{say}
\begin{notation}
All schemes are assumed to be at least Noetherian and defined over a
fixed algebraically closed field $k$ of characteristic zero.
Obstacles to extending the results to positive characteristic are
discussed in the last section. In most cases, we prefer the
additive notation to the multiplicative one, for divisors and line
bundles. All moduli spaces in this paper are coarse.
\end{notation}
\begin{ack}
It is a pleasure to acknowledge useful discussions that I had with
F.Campana, Y.Kawamata, J.Koll\'ar, Yu.Manin, E.Sernesi and
V.V.Shokurov while working on this paper.
\end{ack}
\section{Overview}
\label{sec:overview}
\begin{say}
One possible approach to solving an algebro-geometric moduli
problem goes through the following steps:
\begin{enumerate}
\item defining the objects that we are trying to parameterize,
\item giving the right definition for a moduli functor,
\item establishing properties of this functor,
\item constructing a moduli space in some fashion,
\item proving finer facts about this space.
\end{enumerate}
In our treatment, we will follow two guiding principles well
understood in the field:
\begin{principle}
Most moduli spaces exist in the category of algebraic spaces.
\end{principle}
\begin{principle}
Most complete and separated moduli spaces are projective.
\end{principle}
\end{say}
\begin{say}
Moduli spaces of nonsingular curves $M_g$ and of Deligne-Mumford
stable curves $\overline{M_{g}}$ of genus $g$ provide a textbook
illustration of how this works in practice. Since nonsingular curves
can degenerate into singular ones in an uncorrectable way, $M_g$ is
not complete. There are many different ways to compactify it but the
one we are interested in here is adding more curves and trying to
solve an enlarged moduli problem. It turns out that the curves one
has to add are Deligne-Mumford moduli-stable curves which are
defined as connected and complete reduced curves with ordinary nodes
only such that every smooth rational irreducible component
intersects others in at least 3 points and every irreducible
component of arithmetical genus one intersects the rest in at least
1 point. The latter condition has two equivalent meanings:
\begin{enumerate}
\label{enu:properties of deligne-mumford curves}
\item the automorphism group $\operatorname{Aut} (X)$ is finite (and this property
is a must for the Geometric Invariant Theory),
\item the dualizing sheaf $\omega_X$ is ample.
\end{enumerate}
To arrive at this answer, one can look at the way the good
limits are obtained. One considers a family ${\cal X}$ of curves over
a marked curve, or the specter of a DVR ring, $({\cal S},0)$ with a
nonsingular general fibre and a degenerate special fibre. Then by
the Semistable Reduction Theorem, after making a finite base change
${\cal S}'\to{\cal S}$ and resolving the singularities of
${\cal X}'$, the central fibre will be a reduced curve with
simple nodes. Following (1) above one should contract all the
rational curves $E$ in the central fiber that intersect the rest
only at 1 or 2 points. These have self-intersection numbers $E^2=-1$
and $E^2=-2$ respectfully. Contracting $(-1)$-curves leaves the
ambient space, which is a surface, nonsingular. Contracting
$(-2)$-curves introduces very simple surface singularities, called
Du Val or rational double. The central fiber has nodes only.
\end{say}
\begin{say}
One can recognize that the above is a field of the Minimal Model
Program. In fact, we have just constructed the canonical model, in
dimension 2, of ${\cal X}'$ over ${\cal S}'$. So, to generalize
$\overline{M_{g}}$ to the surfaces of general type we have to
apply the Minimal Model Program in dimension 3. This was done by
Koll\'ar and Shepherd-Barron in \cite{KollarShepherdBarron88}. By
that time, the end of 1980-s, all the necessary for this
construction tools from MMP in dimension 3 were already available.
The new objects that one has to add are defined as connected reduced
surfaces with semi-log canonical singularities and ample tensor
power of the dualizing sheaf $(\omega_X^N)^{**}$, where $^{**}$
means taking the self-dual. Using the additive notation, we say that
a $\Bbb Q$-Cartier divisor $K_X$ is ample.
\end{say}
\begin{say}
At the present time, the {\em log} Minimal Model Program in
dimension 3 is in a pretty good shape. Let us see what kind of
statements we can get using its principles. Keeping in line with
what we did before, we now consider pairs $(X,B)$ of surfaces $X$
and divisors $B=\sum_{j=1}^{n} B_j$ with ample $K_X+B$. A
construction very similar to the one above, with Semistable
Reduction Theorem and, this time {\em log} canonical model, shows
that we again get a complete moduli functor. What about
singularities of the pair $(X,B)$? Why, they ought to be semi-log
canonical, of course!
What is the analog of this in dimension one? That is easy to answer
and we get something very familiar. The divisor
$B=\sum_{j=1}^{n} B_j$ becomes a set of marked points. Semi-log
canonical means that the curve has nodes only, and that marked
points are distinct and lie in the nonsingular part. These are
exactly the $n$-marked semistable curves of Knudsen
\cite{Knudsen83}.
\end{say}
\begin{say}
Another possible generalization would be looking not at absolute
curves (or surfaces) $X$ (or pairs $(X,B)$) but doing it in the
relative setting. In other words, let us consider maps $X\to W$ to a
fixed projective scheme $W$ with $K_X$ (resp. $K_X+B$) {\em
relatively\/} ample. The only modification in the above
construction will be that we have to apply the relative version of
the (log) Minimal Model Program over ${\cal S}'\times W$ instead of over
${\cal S}'$. What we get for curves is the moduli space $M_{g,n}(W)$
introduced by Kontsevich in \cite{Kontsevich94}.
\end{say}
\begin{say}
Now that we have outlined the objects we will be dealing with, let
us return back to the basic example of $\overline{M_g}$. We recall
two different approaches to constructing it.
\end{say}
\begin{approach}[G.I.T.]
One first proves that moduli-stable curves are asymptotically
Hilbert-stable, \cite{Mumford77}. Then the standard G.I.T. machinery
produces a quasi-projective moduli space. Since it is complete, it is
actually projective.
\end{approach}
\begin{approach}
Using a fairly general argument (\cite{Mumford82}, p.172) one proves
the existence of a moduli space in the category of algebraic spaces.
To a family of curves $\pi:{\cal X}\to{\cal S}$ one can in a natural way
associate line bundles on ${\cal S}$ which are defined as
$\det(\pi_*\omega^k)$. They descend to ($\Bbb Q$-)line bundles
$\lambda_k$ on $\overline{M_g}$ and one can further show that
$\lambda_k$ are ample for $k\ge1$.
\end{approach}
\begin{say}
As mentioned in \cite{ShepherdBarron83} and \cite{Kollar90}, for
surfaces of general type the first approach fails. By
\cite{Mumford77} 3.19 in order to be asymptotically Chow- or
Hilbert-stable a surface has to have singularities of multiplicities
at most 7. On the other hand, the semi-stable limits described above
have semi-log canonical singularities and it looks like they must be
included in any reasonable complete moduli problem. These semi-log
canonical singularities include all quotient singularities, for
example, and can have arbitrarily high multiplicities.
\end{say}
\begin{say}
The second approach is what we will be using here. After
establishing the existence as an algebraic space, we will use
Koll\'ar's Ampleness Lemma \cite{Kollar90} to prove that it is
projective. The Ampleness Lemma is a general scheme that shows
projectiveness once some good properties of the moduli functor are
established: local closedness, completeness, separateness,
semipositiveness, finite reduced automorphism groups, and,
crucially, boundedness. The projectiveness will be the only
``finer'' property of the obtained moduli spaces that we will
consider.
\end{say}
\section{The objects}
\label{sec:the objects}
\begin{say}
The main objects into the consideration will be
{\em stable maps of pairs\/} $g:(X,B)\to W$, where
\begin{enumerate}
\item $W\subset\Bbb P$ is a fixed projective scheme,
\item $X$ is a connected projective surface,
\item $B=\sum_{j=1}^n B_j$ is a divisor on $X$, $B_j$ are reduced
but not necessarily irreducible,
\item the pair $(X,B)$ has semi-log canonical singularities,
\item the divisor $K_X+B$ is relatively $g$-ample.
\end{enumerate}
The precise definitions follow.
\end{say}
\begin{say}
For a normal variety $X$, $K_{X}$ or simply $K$ will always denote
the class of linear equivalence of the canonical Weil divisor. The
corresponding reflexive sheaf ${\cal O}_X(K_X)$ is defined as
$i_*(\Omega_U^{\dim X})$, where $i:U\to X$ is the embedding of the
nonsingular part of $X$.
\end{say}
\begin{defn}
An {\em $\Bbb R$-divisor\/} $D=\sum d_{j}D_{j}$ is a linear
combination of prime Weil divisors with real coefficients, i.e.\ an
element of $N^{1}\otimes \Bbb R$. An $\Bbb R$-divisor is said to be
$\Bbb R$-Cartier if it is a combination of Cartier divisors with
real coefficients, i.e.\ if it belongs to the image of
$Div(X)\otimes\Bbb R \to N^{1}(X)\otimes\Bbb R$ (this map is of
course injective for normal varieties). The $\Bbb Q$-divisors and
$\Bbb Q$-Cartier divisors are defined in a similar fashion.
\end{defn}
\begin{defn}
Consider an $\Bbb R$-divisor
$K+B=K_{X}+\sum b_{j}B_{j}$ and assume that
\begin{enumerate}
\item $K+B$ is $\Bbb R$-Cartier,
\item $0\le b_{j}\le1$.
\end{enumerate}
For any resolution $f:Y\to X$ look at the natural formula
\begin{eqnarray}
\label{defn:codiscrepancies}
K_{Y}+B^{Y}= f^{*}(K_{X}+\sum b_{j}B_{j})=
K_{Y}+\sum b_{j}f^{-1}B_{j} + \sum b_{i}F_{i}
\end{eqnarray}
or, equivalently,
\begin{eqnarray}
\label{defn:log discrepancies}
K_{Y}+\sum b_{j}f^{-1}B_{j} + \sum F_{i}=
f^{*}(K_{X}+\sum b_{j}B_{j}) + \sum a_{i}F_{i}
\end{eqnarray}
Here $f^{-1}B_{j}$ are the proper preimages of $B_{j}$ and $F_{i}$
are the exceptional divisors of $f:Y\to X$.
The coefficients $b_{i},b_{j}$ are called codiscrepancies, the
coefficients $a_{i}=1-b_{i},a_{j}=1-b_{j}$ -- log discrepancies.
\end{defn}
\begin{rem}
In fact, $K+B$ is not a usual $\Bbb R$-divisor but rather a special
gadget consisting of a linear class of a Weil divisor $K$ (or a
corresponding reflexive sheaf) and an honest $\Bbb R$-divisor $B$.
This, however, does not cause any confusion.
\end{rem}
\begin{defn}
A pair $(X,B)$ (or a divisor $K+B$) is said to be
\begin{enumerate}
\item log canonical, if the log discrepancies $f_{k}\ge0$
\item Kawamata log terminal, if $f_{k}>0$
\item canonical, if $f_{k}\ge1$
\item terminal, if $f_{k}>1$
\end{enumerate}
for every resolution $f:Y\to X$, $\{k\}=\{i\} \cup \{j\}$.
\end{defn}
\begin{say}
The notion of {\em semi-log canonical\/} is a generalization of
{\em log canonical\/} to the case of varieties that are singular in
codimension 1. The basic observation here is that for a curve with a
simple node the definition of the log discrepancies still makes
sense and gives $a_1=a_2=0$, so it can also be considered to
be (semi-)log canonical. No new Kawamata semi-log terminal
singularities appear, however.
Recall that according to Serre's criterion normal is equivalent to
Serre's condition $S_2$ and regularity in codimension 1. So, if
we do allow singularities in codimension 1, $S_2$ will be exactly
what we will need to keep.
\end{say}
\begin{defn}
\label{defn:semi-log canonical}
Let $X$ be a reduced (but not necessarily irreducible)
equidimensional scheme which satisfies Serre's condition $S_{2}$ and
is Gorenstein in codimension 1. Let $B=\sum b_{j}B_{j}$, $0\le
b_{j}\le1$ be a linear combination with real coefficients of
codimension 1 subvarieties none of irreducible components of which
is contained in the singular locus of $X$. Denote by ${\cal O}(K_X)$ the
reflexive sheaf $i_*(\omega_U)$, where $i:U\to X$ is the open subset
of Gorenstein points of $X$ and $\omega_U$ is the dualizing sheaf of
$U$. We can again consider a formal combination of $K_X$ and an
$\Bbb R$-divisor $B$, and there is a good definition for $K_X+\sum
b_jB_j$ to be $\Bbb R$-Cartier. It means that in a neighborhood of
any point $P\in X$ we can choose a section $s$ of ${\cal O}(K_X)$ such
that the divisor $(s)+\sum b_jB_j$ is a formal combination with real
coefficients of Cartier divisors with no components entirely in the
singular set.
A pair $(X,B)$ (or a divisor $K_{X}+B$) is said to be semi-log
canonical if, similar to the above,
\begin{enumerate}
\item $K_{X}+B$ is $\Bbb R$-Cartier,
\item for any morphism $f:Y\to X$ which is birational on every
irreducible component, and with a nonsingular $Y$, in the natural
formula
\begin{displaymath}
f^*(K_{X}+B)=K_{Y}+f^{-1}B+ \sum b_{i}F_{i}
\end{displaymath}
with $F_{i}$ being irreducible components of the exceptional set,
all $b_{i}\le 1$ (resp.~ $a_{i}=1-b_{i}\ge0$).
\end{enumerate}
As before, the coefficients $b_i,b_j$ are called codiscrepancies,
the coefficients $a_i,a_j$ -- the log discrepancies.
\end{defn}
\begin{rem}
In the case when $(X,B)$ has a good semi-resolution (for example,
for surfaces) this definition is equivalent to that of
\cite{KollarShepherdBarron88}, \cite{FAAT} chapter 12. In our
opinion, it is more natural to give a definition which is
independent of the existence of a semi-resolution.
\end{rem}
\begin{rem}
For surfaces the condition $S_2$ is of course equivalent to
Cohen-Macaulay.
\end{rem}
\begin{defn}
By the Kleiman's criterion, the ampleness for proper schemes is a
numerical condition, hence it extends to $\Bbb R$-Cartier divisors.
If coefficients of $B$ are rational, $K_{X}+B$ is $g$-ample iff for
some positive integer $n$ the divisor $n(K_X+B)$ is Cartier and
$g$-ample in the usual sense.
\end{defn}
\begin{rem}
Below we will only consider the case when $B$ is reduced, i.e.\ all
the coefficients $b_j=1$. See the last section for the discussion on
non-integral coefficients.
\end{rem}
\begin{exmp}
If $X$ is a curve then $(X,B)$ is semi-log canonical iff the only
singularities of $X$ are simple nodes and $B$ consists of distinct
points lying in the nonsingular part of $X$. $K_X+B$ is relatively
ample iff every smooth rational component of $X$ mapping to a point
on $W$ has at least 3 points of intersection with the rest of $X$, or
points in $B_j$, and every component of arithmetical genus 1 has at
least 1 such point. In the absolute case, i.e. when $W$ is a point,
this is the usual definition of a stable curve with marked points.
Every $B_j$ can also be considered as a group of unordered points.
\end{exmp}
\begin{exmp}
The only codimension 1 semi-log canonical singularities are normal
crossings.
\end{exmp}
\begin{exmp}
If $X$ is a nonsingular surface then $(X,B)$ is semi-log canonical
iff $B$ has only normal intersections.
\end{exmp}
\begin{exmp}
For the case when $X$ is a surface and $B$ is empty the semi-log
canonical singularities over $\Bbb C$ were classified in
\cite{KollarShepherdBarron88}. They are (modulo analytic
isomorphism): nonsingular points, Du Val singularities, cones over
nonsingular elliptic curves, cusps or degenerate cusps (which are
similar to cones over singular curves of arithmetical genus 1),
double normal crossing points $xy=0$, pinch points $x^{2}=y^{2}z$,
and all cyclic quotients of the above. If $B$ is nonempty then the
singularities of $X$ are from the same list and, in addition, there
are different ways $B$ can pass through them. For normal $X$ the
list could be found in \cite{Alexeev92} for example.
\end{exmp}
\begin{say}
The following describes an easy reduction of semi-log canonical
singularities to log canonical, cf. \cite{FAAT} 12.2.4.
\end{say}
\begin{lem}
\label{lem:reduction of semi-log canonical to log canonical}
Let $(X,B)$ be as in the definition \ref{defn:semi-log canonical}
and denote by $\nu:X^{\nu}\to X$ its normalization. Assume that
$K_X+B$ is $\Bbb R$-Cartier. Then $(X,B)$ is semi-log canonical iff
$(X^{\nu},\nu^{-1}B+cond(\nu))$ is log canonical, and they have the
same log discrepancies.
\end{lem}
\begin{pf}
Clear from the definition.
\end{pf}
\begin{say}
The next theorem explains how semi-log canonical surfaces appear in
families (cf. \cite{KollarShepherdBarron88} 5.1). But first we will
need an auxiliary definition.
\end{say}
\begin{defn}
Let $f:({\cal X},{\cal B})\to {\cal S}$ be a 3-dimensional one-parameter family.
Let ${\cal B}=\sum b_{j}{\cal B}_{j}$ with $0\le b_{j} \le1$ be an
$\Bbb R$-divisor and assume that ${\cal X}$ and all ${\cal B}_j$ are flat over
${\cal S}$ and that $K_{{\cal X}}+{\cal B}$ is $\Bbb R$-Cartier. We will say that
the pair $({\cal X},{\cal B})$ (or the divisor $K_{{\cal X}}+{\cal B}$) is {\em
$f$-canonical} if in the definition of log discrepancies for all
exceptional divisors with $f(F_i)$ a closed point on ${\cal S}$ one has
for the corresponding log discrepancy $a(F_i)\ge1$ (resp.
$b(F_i)\le0$). This condition does not say anything about log
discrepancies of divisors that map surjectively onto ${\cal S}$.
\end{defn}
\begin{thm}
\label{thm:family is good iff central fiber is good}
Let $f:({\cal X},{\cal B})\to {\cal S}$ be a 3-dimensional one-parameter family
over a pointed curve or a specter of a DVR (a discrete valuation
ring). Let ${\cal B}=\sum b_{j}{\cal B}_{j}$ with $0\le b_{j} \le1$ be an
$\Bbb R$-divisor and assume that ${\cal X}$ is irreducible, ${\cal X}$ and all
${\cal B}_j$ are flat over ${\cal S}$ and that the fibers satisfy Serre's
condition $S_2$ and are Gorenstein in codimension 1 (note that this
implies that ${\cal X}$ itself is Cohen-Macaulay and is Gorenstein in
codimension 1). Further assume that $K_{{\cal X}}+{\cal B}$ is $\Bbb
R$-Cartier. Then the following is true:
\begin{enumerate}
\item If $K_{{\cal X}_{0}}+{\cal B}_{0}$ is semi-log canonical then
$K_{{\cal X}}+{\cal B}$ is log canonical and $f$-canonical.
\item Under assumptions of (1), the general fiber is also semi-log
canonical.
\item Suppose that there exists a birational morphism
$\mu: {\cal Y} \to {\cal X}$ with a nonsingular ${\cal Y}$ such that all
exceptional divisors of $\mu$ and strict preimages of ${\cal B}_{i}$
have normal crossings and such that the central fiber is reduced.
Then the opposite to (1) is true.
\end{enumerate}
\end{thm}
\begin{pf}
The proof is an application of the adjunction formula.
(1) The log adjunction theorem \cite{FAAT} 17.12 and
\ref{lem:reduction of semi-log canonical to log canonical} imply
that $K_{{\cal X}_0}+{\cal B}_{0}$ is semi-log canonical iff
$K_{{\cal X}}+{\cal B}+{\cal X}_0$ is. Now, the connection between the log
discrepancies of the divisors $K_{{\cal X}}+{\cal B}$ and $K_{{\cal X}}+{\cal B}+{\cal X}_0$
is clear. For a divisor $E$ mapping onto ${\cal S}$ the log discrepancies
are the same. For $E$ mapping to a central point of ${\cal S}$ the
difference is the coefficient of $E$ in the central fiber of
${\cal Y}\to{\cal S}$, and so is at least 1.
(3) Here the differences between the log discrepancies over the
central fiber are all equal to 1.
(2) is by adjunction.
\end{pf}
\begin{say}
Finally, we show how to pass from a relatively ample $K+B$ to an
ample divisor.
\end{say}
\begin{lem}
\label{lem:absolute ampleness}
Let $g:(X,B)\to W\subset \Bbb P$ be a map, where $X$ is a
projective surface and $B=\sum b_jB_j$ is an $\Bbb R$-divisor on
$X$. Assume that $K_X+B$ is semi-log canonical and is relatively
$g$-ample. Then $K_X+B+4H$ is ample, where $H=g^*{\cal O}(1)$.
\end{lem}
\begin{pf}
It is enough to prove that the restriction on the normalization of
$X$ is ample, therefore by
\ref{lem:reduction of semi-log canonical to log canonical} we can
assume that $X$ is normal and that $(X,B)$ is log canonical.
We show that $K_X+B+3H$ is nef (numerically effective) and this
implies the statement. Indeed, $K_X+B+MH$ is ample for $M\gg0$ and
$K_X+B+4H$ is a weighted average of the above two divisors.
Assume that $K_X+B+3H$ is not nef. Then the Cone Theorem, which
holds for arbitrary normal surfaces, tells us that there exists an
irreducible curve $C$ generating an extremal ray and such that
\begin{displaymath}
(K_X+B)C<0
\end{displaymath}
This is possible only if $C$ does not map to a point. But then
$C\cdot3H\ge3$ and $(K_X+B)C\ge-3$ by a theorem on the length
of extremal curves, see \cite{MiyaokaMori86}.
In dimension 2 the latter statement is very elementary. Let
$f:Y\to X$ be a minimal resolution of singularities of $X$. Then,
if $X\ne\Bbb P^2$, one necessarily has $(f^{-1}C)^2\le0$ and
\begin{eqnarray*}
(K_X+B)C \ge (K_Y+f^{-1}B)f^{-1}C \ge \\
(K_Y+f^{-1}C)f^{-1}C = 2p_a(f^{-1}C)-2 \ge -2.
\end{eqnarray*}
And the case of $X=\Bbb P^2$ is clear.
\end{pf}
\section{Definition and properties of a moduli functor}
\label{sec:definition and properties of a moduli functor}
\begin{say}
Below we give a few general definitions for moduli functors. They
are fairly standard (see e.g. \cite{Viehweg94}, \cite{Kollar90}) but
we need to make slight modifications to adapt them to our situation.
\end{say}
\begin{say}
The moduli functor for a moduli problem of polarized schemes is
normally constructed in the following way. One fixes a class ${\cal C}$
of schemes $X/k$ with a polarization, i.e. an ample line bundle, $L$
and some extra structure and subject to some conditions. Then for an
arbitrary scheme ${\cal S}/k$ one defines ${{\cM\cC}} ({\cal S})$ as the set of all
(relatively) polarized flat families over ${\cal S}$ with all geometric
fibers from ${\cal C}$ and, possibly, subject to more conditions. The
families are considered modulo an equivalence relation. Usually it
is an isomorphism between ${\cal X}_1/{\cal S}$ and ${\cal X}_2/{\cal S}$ with whatever
extra structure they have and a fiber-wise linear equivalence
between ${\cal L}_1$ and ${\cal L}_2$. In other cases it is an algebraic or a
numerical, or a numerical up to a scalar equivalence, instead of
linear.
Sometimes, it is also useful considering a $\Bbb Q$-polarization
$L$ on $X$, i.e a reflexive sheaf such that $(L^{\otimes N})^{**}$
is an ample line bundle.
\end{say}
\begin{say}
The above definition is intentionally vague since extra structures
and conditions vary greatly from one moduli problem to another.
Instead of trying to cover all future generalizations, we will
formulate general principles and, when nontrivial, say exactly how
they specialize to our situation.
\end{say}
\begin{defn}
The class ${\cal S}$ is said to be {\em bounded\/} if there exists a
scheme $({\cal X},{\cal L})$ with an extra structure, and a morphism
$F:{\cal X}\to{\cal S}$ to a scheme ${\cal S}$ of finite type such that all
elements of ${\cal C}$ appear as geometric fibers of $F$, not necessarily
in a one-to-one way.
There are two important variations of this definition. There is the
{\em polarized\/} boundedness, when one requires $F$ to be
projective and ${\cal L}$ to restrict to the given polarization $L$ on a
fiber, versus {\em non-polarized\/}. One can also consider
boundedness {\em in the narrow sense\/}, requiring that all fibers
of $F$ belong to ${\cal C}$, or {\em in the wide sense\/}, asking only
for some of the fibers to be from ${\cal C}$.
Here we make the choice of the polarized boundedness in the wide
sense.
\end{defn}
\begin{defn}
A moduli functor ${{\cM\cC}} $ is said to be {\em separated\/} if every
one-parameter family in ${{\cal M}{\cal C}({\cal S}_{gen})}$, where ${\cal S}_{gen}$ is
a generic point of a DVR, has at most one extension to ${\cal S}$.
\end{defn}
\begin{defn}
A moduli functor ${{\cM\cC}} $ is said to be {\em complete\/} if every
one-parameter family in ${{\cal M}{\cal C}({\cal S}_{gen})}$, where ${\cal S}_{gen}$ is
a generic point of a DVR, has at least one extension after a finite
cover ${{\cal S}'\to{\cal S}}$.
\end{defn}
\begin{defn}
\label{defn:local closedness}
A class ${\cal C}$ is said to be {\em locally closed\/} if for
every flat family $F:({\cal X},{\cal L})\to{\cal S}$ with an extra structure
there exist locally closed subschemes ${\cal S}_l\subset{\cal S}$ with the
following universal property:
\begin{itemize}
\item A morphism of schemes ${\cal T}\to{\cal S}$ factors through
$\coprod S_l$ iff $({\cal X},{\cal L})\underset{{\cal S}}{\times}{\cal T} \to{\cal T}$
belongs to ${{\cM\cC}} ({\cal T})$.
\end{itemize}
\end{defn}
\begin{defn}
The class ${\cal C}$ is said to {\em have finite reduced automorphisms\/}
if every object in ${\cal C}$ has a finite and reduced (the latter is
automatic in characteristic 0) group of automorphisms.
\end{defn}
\begin{defn}
A moduli functor ${{\cM\cC}} $ is said to be {\em functorially polarizable\/}
if for every family $({\cal X},{\cal L})$ in {$\cM\cC(\cS)$} there exists an equivalent
family $({\cal X},{\cal L}^c)$ such that
\begin{enumerate}
\item if $({\cal X}_1,{\cal L}_1)$ and $({\cal X}_2,{\cal L}_2)$ are equivalent, then
$({\cal X}_1,{\cal L}^c_1)$ and $({\cal X}_2,{\cal L}^c_2)$ are isomorphic,
\item for any base chance $h:{{\cal S}'\to{\cal S}}$, $({\cal X}',{{\cal L}'}^c)$ and
$({\cal X}',h^*({\cal L}^c))$ are isomorphic.
\end{enumerate}
The main example of a functorial polarization is delivered by the
polarization $\omega_{{\cal X}/{\cal S}}$ for canonically polarized manifolds.
\end{defn}
\begin{defn}
A functorial polarization ${\cal L}^c$ is said to be {\em semipositive\/}
if there exists a fixed $k_{0}$ such that whenever ${\cal S}$ is a
complete smooth curve and $f:({\cal X},{\cal L})\to{\cal S}$ an element in {$\cM\cC(\cS)$} ,
then for all $k\ge k_{0}$ the vector bundles $f_*(k{\cal L}^c)$ are
semipositive, i.e.\ all their quotients have nonnegative degrees.
This definition will be slightly modified for our purposes, we will
also require semipositiveness of restrictions of ${\cal L}^c$ to certain
divisors ${\cal B}_j$ on ${\cal X}$.
\end{defn}
\begin{say}
The following is the class that we will be considering from now on.
\end{say}
\begin{defn}
\label{defn:the class}
The elements of the class ${\cal C}^N={\cal C}^N_{(K+B)^2,(K+B)H,H^2}$ are
{\em stable maps of pairs\/} $g:(X,B,L_N)\to W$, where
\begin{enumerate}
\item $W\subset\Bbb P$ is a fixed projective scheme,
\item $X$ is a connected projective surface,
\item $B=\sum_{j=1}^n B_j$ is a divisor on $X$, $B_j$ are reduced
but not necessarily irreducible,
\item the pair $(X,B)$ has semi-log canonical singularities,
\item the divisor $K_X+B$ is relatively $g$-ample,
\item $(K_X+B)^2=C_1, (K_X+B)H=C_2, H^2=C_3$ are fixed,.
\item $L_N={\cal O}(N(K_X+B+5H))$, where $H=g^*{\cal O}_W(1)$. Here $N$ is a
positive integer such that for every map as above $L_N$ is a line
bundle. For example, we can choose $N$ to be the minimal positive
integer satisfying this condition. The existence of such an $N$
will be proved in \ref{thm:boundedness of maps}, and it is ample
by \ref{lem:absolute ampleness}.
\end{enumerate}
\end{defn}
\begin{say}
The classes ${\cal C}^N$ and ${\cal C}^M$ for different $N,M$ are in a
one-to-one correspondence between each other, and the only
difference is the polarizations.
As a consequence, the polarization in our functor plays a secondary
role. We will switch from a polarization $L_N$ to its multiple $L_M$
when it will be convenient.
\end{say}
\begin{thm}
\label{thm:boundedness of maps}
For some $M>0$ the class ${\cal C}^M$ is bounded.
\end{thm}
\begin{pf}
We start with the boundedness theorem which gives what we want in
the absolute case.
\begin{thm}[\cite{Alexeev94b}, 9.2]
\label{thm:absolute boundedness}
Fix a constant $C$ and a set ${\cal A}$ satisfying the descending
chain condition. Consider all surfaces $X$ with an $\Bbb R$-divisor
$B=\sum b_jB_j$ such that the pair $(X,B)$ is semi-log canonical,
$K_X+B$ is ample, $b_j\in {\cal A}$ and $(K_X+B)^2=C$. Then the class
$\{(X,\sum b_jB_j)\}$ is bounded.
\end{thm}
Apply this theorem with the set ${\cal A}=\{1\}$ to $K_X+B+D$, where $D$
is a general member of the linear system $|4H|$. Since this linear
system is base point free, the pair $(X,B+D)$ also has semi-log
canonical singularities. Therefore, all pairs $(X,B)$ satisfying
the conditions of the theorem can be embedded by a linear system
$|M(K_X+B+4L)|$ for a fixed large divisible $M$ in a fixed
projective space $\Bbb P^{d_1}$. Every map $g:X\to W$ is defined by
its graph $\Gamma_g$. Consider a Veronese embedding of $W$ by
$|{\cal O}_W(M)|$ in some $\Bbb P^{d_2}$ and then look at the graphs
$\Gamma_g$ in a Segre embedding
$\Bbb P^{d_1}\times\Bbb P^{d_2} \subset \Bbb P^{d_3}$. Note that
${\cal O}_{\Bbb P^{d_3}}(1)$ restricted on $X\simeq\Gamma_g$ is
$L_M=M(K_X+B+5H)$.
$L^2$ is fixed, hence by the boundedness theorem
\ref{thm:boundedness of maps} above there are
only finitely many possibilities for Hilbert polynomials
$\chi({\cal O}_{\Gamma_g}(t))$. By the same theorem, there are also only
finitely many possibilities for Hilbert polynomials
$\chi({\cal O}_{B_j}(t))$. Therefore, all elements of our class
$g:(X,B)\to W$ are parameterized by finitely many products of
Hilbert schemes. In each product, we have to extract a subscheme
parameterizing subschemes of $\Bbb P^{d_1}\times W$ and with fixed
${\cal O}_{\Bbb P^{d_1}}(1)^2$,
${\cal O}_{\Bbb P^{d_1}}(1)\cdot{\cal O}_{\Bbb P^{d_2}}(1)$ and
${\cal O}_{\Bbb P^{d_2}}(1)^2$, and these are obviously closed algebraic
conditions. We also need to extract the graphs, i.e subschemes
mapping isomorphically to $\Bbb P^{d_1}$, and this is an open
condition.
The resulting scheme will parameterize the maps, including all maps
from the class ${\cal C}^M$. This proves the theorem.
\end{pf}
\begin{say}
We won't need the boundedness of the class ${\cal C}^N$ itself, although
it will follow from the proof of the local closedness
\ref{thm:local closedness}.
\end{say}
\begin{defn}
\label{defn:moduli functor 1}
There are several ways to define the moduli functor for our class.
The one we use here is the most straightforward one (cf.
\cite{KollarShepherdBarron88}, \cite{Viehweg94} in the absolute case
with $B=\emptyset$). For any scheme ${\cal S}/k$,
${{\cM\cC}} ^N={{\cM\cC}} ^N_{(K+B)^2,(K+B)H,H^2}$ is given by
\begin{displaymath}
{{{\cM\cC}} ^N({\cal S})}=
\left\{
\begin{aligned}
& \text{all families }
f:({\cal X},{\cal L})\to{\cal S}
\text{ with a divisor } {\cal B}=\sum_{j=1}^N {\cal B}_j
\text{ on } X, \\
& \text{a map }
g:{\cal X}\to W
\text{ and a line bundle } {\cal L}
\text{ such that every }\\
& \text{geometric fiber belongs to } {\cal C},
X \text{ and all } {\cal B}_j
\text{ are flat over } {\cal S} \\
\end{aligned}
\right\}
\end{displaymath}
Two families over ${\cal S}$ are equivalent if they are isomorphic
fiber-wise.
In this functor we consider a sub-functor ${{\cM\cC}} {'}^N$, requiring
in addition that for each $s$ there exists a
1-dimensional family from ${{\cM\cC}} ^N$ with the central fiber
${\cal X}_s$ and an irreducible general fiber ${\cal X}_g$ such that:
\begin{enumerate}
\item ${\cal X}_g$ is irreducible,
\item the pair $({\cal X}_g,0)$ is (Kawamata) log terminal.
\end{enumerate}
This is similar to the smoothability condition for
$\overline{M_{K^2}^{sm}}\subset\overline{M_{K^2}}$ (see
\cite{Kollar90}) and is necessary due to the technical reasons.
Consider a one parameter family of maps. Then we would like the
ambient 3-fold to be irreducible since MMP is not developed for
non-irreducible varieties yet. We would also want the 3-fold to have
log terminal singularities because they are Cohen-Macaulay in
characteristic 0.
\end{defn}
\begin{say}
A little disadvantage of the above definition is that even though
${{\cM\cC}} ^{N,irr}$ and, say, ${{\cM\cC}} ^{2N,irr}$ are the same on the closed
points, the corresponding moduli spaces can potentially have
different scheme structures, the second one could be strictly
larger. So, in fact, we have not one but infinitely many moduli
spaces. It would be better if we had a formula for the minimal $N$
in terms of $(K+B)^2,(K+B)H,H^2$. We know, however, only that such
an $N$ exists.
\end{say}
\begin{say}
A different solution was suggested (again, in the absolute case with
$B=\emptyset$) by Koll\'ar in \cite{Kollar90},\cite{Kollar94}. In a
sense, it produces a moduli space with the ``minimal'' scheme
structure.
We introduce some necessary notation first.
\end{say}
\begin{defn}
Let $F:{\cal X}\to{\cal S}$ be a projective family of graphs of maps
$(X,B)\to W$. Assume that every fiber is Gorenstein in codimension 1
and satisfies Serre's condition $S_2$.
Denote by $i:{\cal U}\hookrightarrow{\cal X}$ the open subset where $f$ is
Gorenstein and the divisors ${\cal B}_j$ are Cartier. Note that on every
fiber one has $\operatorname{codim} _{{\cal X}_s}({\cal X}_s-{\cal U}_s)\ge2$. Define the sheaves
${\cal L}_{{\cal U},k}$ and ${\cal L}_k$ by
\begin{displaymath}
{\cal L}_{{\cal U},k}={\cal O}_{{\cal U}}(k(K_{{\cal U}/{\cal S}}+{\cal B}+g^*{\cal O}_W(5))
\end{displaymath}
\begin{displaymath}
{\cal L}_k=i_*{\cal L}_{{\cal U},k}
\end{displaymath}
It follows that the sheaves ${\cal L}_k$ on ${\cal X}$ are coherent.
\end{defn}
\begin{notationnum}
Let $f:{\cal X}\to{\cal S}$ be a morphism of schemes,
$i:{\cal U}\hookrightarrow{\cal X}$ be the immersion of an open set and ${\cal F}$ be
a coherent sheaf on ${\cal U}$ which is flat over ${\cal S}$. For a base
change $h:{\cal S}'\to{\cal S}$ we obtain
${\cal X}^h:={\cal X}\underset{{\cal S}}{\times}{\cal S}'$,
${\cal U}^h:={\cal U}\underset{{\cal S}}{\times}{\cal S}'$ etc. Denote the induced
morphism ${\cal U}^h\to {\cal U}$ by $h_{{\cal U}}$ and set ${\cal F}^h:=h_{{\cal U}}^*{\cal F}$.
The induced morphism ${\cal X}^h\to{\cal X}$ is denoted by $h_X$
One says that
{\em the push forward of ${\cal F}$ commutes with a base change\/}
$h:{\cal S}'\to{\cal S}$ if the natural map $h_X^*(i_*{\cal F})\to i^h_*{\cal F}^h$ is an
isomorphism.
\end{notationnum}
\begin{defn}
Define ${{\cM\cC}} ^{all}={{\cM\cC}} ^{all}_{(K+B)^2,(K+B)H,H^2}$ by
\begin{displaymath}
{{{\cM\cC}} ^{all}({\cal S})}=
\left\{
\begin{aligned}
& \text{all families }
f:{\cal X}\to{\cal S}
\text{ with a divisor } {\cal B}=\sum_{j=1}^N {\cal B}_j
\text{ on } {\cal X}
\text{ and} \\
& \text{a map }
g:{\cal X}\to W
\text{ such that every geometric fiber belongs to }
{\cal C}, \\
& {\cal X} \text{ and all } {\cal B}_j
\text{ are flat over } {\cal S},
\text{ and for each } k \\
& i_*{\cal L}_{{\cal U},k}
\text{ commutes with arbitrary base changes}
\end{aligned}
\right\}
\end{displaymath}
As above, one can consider a sub-functor
${{\cM\cC}} {'}^{all}\subset{{\cM\cC}} ^{all}$.
We will not go into detailed discussion of this functor.
\end{defn}
\begin{say}
One can see that if we require that $i_*{\cal L}_{{\cal U},k}$ commutes with
arbitrary base changes only for $k=N$ instead of all positive $k$,
then we get the previous definition of the moduli functor. Indeed,
if a line bundle ${\cal L}$ exists, then ${\cal L}_N={\cal L}+f^*{\cal E}$ for some
invertible sheaf ${\cal E}$ on ${\cal S}$. Then for every $h:{\cal S}'\to{\cal S}$ the
two sheaves $i^h_*{\cal L}_{{\cal U},N}^h$ and
$h_X^*(i_*{\cal L}_{{\cal U},N})=h_X^*({\cal L}_N)$ on ${\cal X}'$ are both reflexive
and coincide on $h_X^{-1}({\cal U})$, hence everywhere.
Vice versa, if $i_*{\cal L}_{{\cal U},N}$ commutes with base changes, then
${\cal L}_N$ is flat and for every closed point $s\in{\cal S}$
\begin{displaymath}
{\cal L}_N\big|_{{\cal X}_s}={\cal O}_{{\cal X}_s}(N(K+B+g^*{\cal O}_W(5)))
\end{displaymath}
Since the latter restriction is locally free for every $s$ and the
sheaves ${\cal O}_{{\cal X}}$, ${\cal L}_N$ are coherent and flat over ${\cal S}$, it
follows by \cite{Matsumura86} 22.5, 22.3 that ${\cal L}_N$ is locally
free.
\end{say}
\begin{say}
Now let us show that our moduli functor ${{\cM\cC}} {'}^N$ has all the good
properties listed above. We start with the local closedness. The
main technical result we will be using is the following theorem.
\end{say}
\begin{thm}[Koll\'ar \cite{Kollar94}]
\label{kollar's flattening decomposition}
With the above notations, assume that $f:{\cal X}\to{\cal S}$ is projective,
$i_*{\cal F}$ is coherent and that for every point $s\in{\cal S}$ the sheaf
${\cal F}_s$ on the fiber ${\cal X}_s$ satisfies Serre's condition $S_2$.
Then there exist locally closed subschemes ${\cal S}_l\subset{\cal S}$ such
that for any morphism $h:{\cal T}\to{\cal S}$ the following are equivalent:
\begin{enumerate}
\item $h$ factors through ${\cal T}\to\coprod{\cal S}_l\to{\cal S}$,
\item $i^h_*{\cal F}^h$ commutes with all future base changes.
\end{enumerate}
\end{thm}
\begin{thm}
\label{thm:local closedness}
The functors ${{\cM\cC}} ^N$ and ${{\cM\cC}} {'}^N$ are locally closed.
\end{thm}
\begin{pf}
Let $F:{\cal X}\to{\cal S}$ be an arbitrary projective family of graphs of
maps $(X,B)\to W$. First, after the flattening decomposition (see
\cite{Mumford66} lecture 8) of ${\cal S}$ into locally closed subschemes,
we can assume that ${\cal X}$ and ${\cal B}_j$ are flat over ${\cal S}$ if they are
not already.
Consider a one-parameter sub-family ${\cal X}_{{\cal R}}\to{\cal R}$ and a point $P$
on the central fiber ${\cal X}_0$. Then ${\cal X}_0$ is Cohen-Macaulay at $P$
iff the 3-fold ${\cal X}_{{\cal R}}$ is. The property of a
local ring to be Cohen-Macaulay
is open (\cite{Matsumura86} 24.5) and the morphism $F$ is
projective. Therefore, if ${\cal X}_0$ is Cohen-Macaulay then there exists
an open neighborhood of ${\cal R}$, and also of ${\cal S}$, that contains
exactly the points over which the fibers are Cohen-Macaulay.
The property of a local ring to be Gorenstein is also open
(\cite{Matsumura86} 24.6) and by the same argument there exists a
closed subset $Z$ of non-Gorenstein points in ${\cal X}$. Give it the
structure of a reduced scheme. Then we have to throw away all fibers
on which the Hilbert polynomial of
${\cal O}_Z\underset{{\cal O}_{{\cal S}}}{\otimes}k(s)$ has degree $\ge1$. There
are only finitely many possible Hilbert polynomials and the
condition on the degree is obviously closed.
At this point we use the previous theorem
\ref{kollar's flattening decomposition} to the sheaf ${\cal L}_{{\cal U},N}$
to conclude that there exist locally closed subschemes
${\cal S}_l\subset{\cal S}$ such that every map $h:{\cal T}\to{\cal S}$ with
${\cal X}\underset{{\cal S}}{\times}{{\cal T}}\in{{\cM\cC}} ({\cal T})$ factors through
$\coprod{\cal S}_l$. ${\cal S}_l$ are disjoint, so
we can concentrate on one of them. If $P$ is a point of ${\cal S}$
and some $h$ as in the definition does not factor through ${\cal S}-P$,
then the fiber of $F$ over $P$ has to be a pair $(X,B)$ from our
class. The sheaf ${\cal L}_N$ on $X\underset{{\cal S}}{\times}S_l$ is flat
over $S_l$ and its restriction to the fiber over $P$ is locally
free. Hence, it has to be locally free in a neighborhood of the
fiber. Therefore, for each $S_l$ if we denote by $U_l\subset S_l$
the open set over which ${\cal L}_N$ is locally free, then $h:{\cal T}\to{\cal S}$
has to factor through $\coprod{\cal U}_l$. Now we can apply
\ref{thm:family is good iff central fiber is good}(2) to conclude that
there exist open subsets ${\cal V}_l\subset{\cal U}_l$ containing all
the points over which the fibers have semi-log canonical
singularities. Also, ${{\cM\cC}} {'}^N\subset{{\cM\cC}} ^N$ is evidently closed
and we end up with a disjoint union of locally closed subschemes.
There is one more thing one has to take care of: the polarization
${\cal O}_{\Bbb P^{d_3}}(1)$ on the fibers has to coincide with ${\cal L}_N$
or its fixed multiple ${\cal L}_M$. Standard semi-continuity theorems for
$h^0$ in flat families show that there exists a closed subset where
the two sheaves are the same. One can also define the scheme
structure on it, see lemma 1.26 \cite{Viehweg94}.
\end{pf}
\begin{lem}
For the functors ${{\cM\cC}} ^N$ and ${{\cM\cC}} {'}^N$ the polarization ${\cal L}_N$ is
functorial.
\end{lem}
\begin{pf}
$K_{{\cal U}/{\cal S}}$ of a flat family commutes with base changes, and so
do ${\cal O}({\cal B}_j)$ and $g^*{\cal O}_W(1)$. Therefore, ${\cal L}_{{\cal U},k}$ are
functorial. By the definitions of the functor ${{\cM\cC}} $ the same is
true for ${\cal L}_k$ (resp. ${\cal L}_N$).
\end{pf}
\begin{thm}
${{\cM\cC}} {'}^N$ is
\begin{enumerate}
\item separated,
\item complete,
\item have finite and reduced automorphisms.
\end{enumerate}
\end{thm}
\begin{pf}
The first two properties have code names in the Minimal Model
Program: ``uniqueness and existence of the log canonical model''.
It is enough to check them in the case when the general fiber is
irreducible and has log terminal singularities.
(1) Let ${\cal S}$ be a specter of a DVR or a pointed curve. Two families
in ${{\cM\cC}} ({\cal S})$ that coincide outside of $0$ are birationally
isomorphic. \ref{thm:family is good iff central fiber is good}(1)
implies that they are both log canonical and both are relative log
canonical models over ${\cal S}\times W$ for the same divisor, hence
isomorphic. If ${\cal Y}\to{\cal S}$ is a common resolution then the divisor
is
\begin{displaymath}
K_{{\cal Y}}+f^{-1}{\cal B} + \sum{\cal E}_i
\end{displaymath}
where ${\cal E}_i$ are exceptional divisors that do not map to a central
point $0\in{\cal S}$.
(2) If there is a family over ${\cal S}-0$, we can complete it over $0$
somehow. Then by a variant of the Semistable Reduction Theorem,
after a finite base change, there is a resolution ${\cal Y}$ of
singularities such that the central fiber is reduced and all
exceptional divisors and ${\cal B}_j$ have normal crossings. Consider the
log canonical model for the same divisor as above, relative over
${\cal S}\times W$. It exists by \cite{KeelMcKernanMatsuki93} for
example. This log canonical model has the same fibers as $(X,B)$
outside $0$. It has log terminal
singularities only, which are Cohen-Macaulay in dimension 3 and
characteristic 0. Therefore, the central fiber is also Cohen-Macaulay
and it is from our class ${\cal C}$ by
\ref{thm:family is good iff central fiber is good}(3).
We also have to show that the sheaf ${\cal L}_N$ for this family is
locally free. It amounts to proving that the Hilbert polynomials
$h_1(t)$ of the sheaf $L_{N,0}$ on the special fiber, and $h_2(t)$
of the sheaf $L_{N,g}$ of the general fiber coincide. Both sheaves
are locally free. But the log canonical model is constructed by
applying the Base Point Freeness theorem, and by the very
construction we have that some ${\cal L}_M$ for a large divisible $M$
is locally free on ${\cal X}$. Therefore the polynomials $h_1(M/Nt)$ and
$h_2(M/Nt)$ are the same,
and that means that $h_1(t)$ and $h_2(t)$ are also the same.
(3) In the absolute case, the fact that $K+B$ is ample and log
canonical implies that the automorphism group is finite by
\cite{Iitaka82}. In the relative case we apply the same theorem to
$K_X+B+D$, $D\in|4H|$ general, which is ample by lemma
\ref{lem:absolute ampleness}. We are working in characteristic 0
and so the group scheme $\operatorname{Aut} X$ is reduced.
\end{pf}
\begin{thm}
\label{thm:semipositiveness}
The functors ${{\cM\cC}} ^N$ and ${{\cM\cC}} {'}^N$ are semipositive.
\end{thm}
\begin{pf}
One has the following
\begin{thm}[Koll\'ar \cite{Kollar90} 4.12]
\label{thm:kollar's semipositiveness}
Let $Z$ be a complete variety over a field of characteristic zero.
Assume that $Z$ satisfies Serre's condition $S_2$ and that it is
Gorenstein in codimension one. Let $Z\to C$ be a map onto a smooth
curve. Assume that the general fiber of $f$ has only semi-log
canonical singularities, and further that $K$ of the general fiber
is ample. Then $f_*{\cal O}(kK_{Z/C})$ is semipositive for $k\ge1$.
\end{thm}
For the sheaves ${\cal L}_N= O_{{\cal X}}(N(K_{{\cal X}/{\cal S}}+{{\cal B}}))$ with empty
${\cal B}$ in the absolute case this is exactly what we need. Analyzing
the proof of \ref{thm:kollar's semipositiveness} shows that it works
with very minor changes in the case of a non-empty reduced ${\cal B}$. In
the relative case instead of $K_{{\cal X}/{\cal S}}+{{\cal B}}$ we consider
$K_{{\cal X}/{\cal S}}+{{\cal B}}+5H$, $H=g^*{\cal O}_W(1)$. We can think of $5H$ simply
as of an additional component of the boundary ${\cal B}$. If a member of
the linear system $|5H|$ is chosen generically, on the general fiber
of $f$ the pair $(X,B+5H)$ will still be semi-log canonical.
For the positiveness of the sheaves ${\cal L}_N\Big|_{{\cal B}_j}$ we use the
log adjunction formula, see \cite{Shokurov91} or \cite{FAAT} chapter
16. We get the following semi-log canonical divisors on ${\cal B}_j$:
\begin{displaymath}
K_{{\cal X}}+{\cal B}\Big|_{{\cal B}_j}=K_{{\cal B}_j}+\sum(1-1/m_{k}){\cal M}_{k}
\end{displaymath}
for some Weil divisors ${\cal M}_{k}$ on ${\cal B}_j$ and $m_k\in\Bbb
N\cup\{\infty\}$. So, here we need a more general semipositiveness
theorem, with nonempty ${\cal B}$ that has fractional coefficients. The
situation is saved by the fact that the relative dimension of
${\cal B}_j$ over ${\cal S}$ equals 1, and the semipositiveness for this case
is proved in \cite{Kollar90} 4.7.
\end{pf}
\section{Existence and projectivity of a moduli space}
\label{sec:existence and projectivity of a moduli space}
\begin{thm}
\label{thm:existence as an algebraic space}
The functor ${{\cM\cC}} ={{\cM\cC}} {'}^N$ is coarsely represented by a proper
separated algebraic space of finite type ${{\bold M \bold C}} ={{\bold M \bold C}} {'}^N$.
\end{thm}
\begin{pf}
The proof is essentially the same as in \cite{Mumford82}, p.172. We
remind that we are working in characteristic zero, and over $\Bbb C$
the argument is easier.
The class ${\cal C}^M$ is bounded, and we can embed all graphs $\Gamma_g$
of the maps $g$ by a linear system $|M(K_X+B+5H)|$ in $\Bbb
P^{d_1}\times\Bbb P^{d_2} \subset \Bbb P^{d_3}$ as in
\ref{thm:boundedness of maps} for a large divisible $M$. By taking
$M$ even larger we can assume that all $X=\Gamma_g$ and all
$B_j\subset X$ are projectively normal, $h^0(M(K_X+B+5H))$ is
locally constant and there are no higher cohomologies.
$(\Gamma_g,B)$ are parameterized, not in a one-to-one way, by some
scheme that we will denote by ${\cal H}$. For any family in
${{\cM\cC}} ^N({\cal T})$, the embedding by a relatively very ample linear
system $|M(K_X+B+4H)|$ defines a non-unique map ${\cal T}\to{\cal H}$. By
\ref{thm:local closedness} there exists a disjoint union of locally
closed subschemes ${\cal S}=\coprod{\cal S}_l\hookrightarrow{\cal H}$ with a
universal property, and ${\cal T}\to{\cal H}$ factors through ${\cal S}$. We
conclude that the coarse moduli space ${{\bold M \bold C}} $ is a categorial
quotient of ${\cal S}$ by an equivalence relation $R$, described as
follows.
$R$ is a set of pairs $(h,G)$, where $h\in{\cal S}$ and $G$ corresponds
to a different embedding of $X$ in $\Bbb P^{d_1}$, i.e. $G$ varies
in a group $PGL(d_1+1)$. There is a natural map
$F:R\to{\cal S}\times{\cal S}$.
Every fiber of $\pi_1\circ F$ is isomorphic to $PGL(d_1+1)$ and this
map is obviously smooth. The map $F$ is quasi-finite and unramified
because its fibers are automorphism groups of objects in ${\cal C}$,
and these are finite reduced. The fact that ${{\cM\cC}} $ is also proper
implies that $F$ is finite.
The rest of the proof is the same as in \cite{Mumford82}, p.172
verbatim. By taking the transversal sections locally the question is
reduced to the case of a finite equivalence relation dominated by a
map $F':R\to{\cal H}'\times{\cal H}'$ with $\pi_{1}\circ F'$ \'etale, and then the
quotient is easily constructed as an algebraic space.
Finally, since ${{\cM\cC}} $ is proper, so is ${{\bold M \bold C}} $.
\end{pf}
\begin{thm}
\label{thm:projectiveness}
The moduli space ${{\bold M \bold C}} ={{\bold M \bold C}} {'}^N$ is projective.
\end{thm}
\begin{pf}
The proof follows the general scheme of \cite{Kollar90}. By the
very construction of ${{\bold M \bold C}} $, there exists a subscheme
${\cal S}\subset{\cal H}$ of a product of Hilbert schemes, with the
corresponding universal family $V_{{\cal S}}\to{\cal S}$, that maps to ${{\bold M \bold C}} $.
One starts by constructing a {\em finite\/} morphism from a scheme
$Y\to{{\bold M \bold C}} $ with a universal family $f:V_Y\to Y$. This is done
locally by cutting ${\cal S}\to{{\bold M \bold C}} $ transversally, then adding more
copies of these sections, so that the automorphisms do not obstruct
gluing the local pieces together, see \cite{Kollar90} 2.7. The only
properties of the class ${\cal C}$ used in this construction are
boundedness and finiteness of automorphisms, which we have.
Next step is to consider the line bundles
\begin{displaymath}
\lambda_M=\det(f_*{\cal L}_M\oplus f_*{\cal L}_M\big|_{{\cal B}_j})
\end{displaymath}
on $Y$ for $M$ large divisible, where
\begin{displaymath}
{\cal L}_M={\cal O}_V(M(K_{V/Y}+{\cal B}+g^*{\cal O}_W(5))).
\end{displaymath}
These line bundles do not descend to ${{\bold M \bold C}} $ because of
automorphisms, but since the objects of ${\cal C}$ have finite groups of
automorphisms and ${\cal C}$ is bounded, for every $M$ there is a finite
power of $\lambda_M$ that does come from a line bundle on ${{\bold M \bold C}} $. To
prove that ${{\bold M \bold C}} $ is projective it is enough to show that one of
$\lambda_M$ is ample, which is achieved by the following theorem.
For simplicity we formulate it only in characteristic 0.
\begin{notationnum}
Let $Y$ be a scheme and let $W$ be a vector bundle of rank $w$
with structure group $\rho:G\to GL_w$. Let $q:W\to Q$ be a
quotient vector bundle of rank $k$. Let $Gr(w,k)/G$ denote the set
of $G$-orbits on the $k$-dimensional quotients of a
$w$-dimensional vector space. The natural map of sets
\begin{displaymath}
u_{Gr}:\{\text{closed points of }X\}\to
Gr(w,k)/G
\end{displaymath}
is called the {\em classifying map}.
One says that the classifying map is {\em finite\/} if
\begin{enumerate}
\item every fiber of $u_{Gr}$ is finite, and
\item for every $y\in Y$ only finitely many elements of $G$ leave
$\ker q_y$ invariant.
\end{enumerate}
\end{notationnum}
\begin{thm}[Koll\'ar's Ampleness Lemma, \cite{Kollar90} 3.9]
Let $Y$ be a proper algebraic space and let $W$ be a semipositive
vector bundle with structure group $G$. Let $Q$ be a quotient
vector bundle of $W$. Assume that
\begin{enumerate}
\item $G$ is reductive,
\item the classifying map is finite.
\end{enumerate}
Then $\det Q$ is ample. In particular, $Y$ is projective.
\end{thm}
This is what it translates to in our situation. The sheaves are
\begin{displaymath}
W=\operatorname{Sym} ^j(f_*{\cal L}_M)\oplus\operatorname{Sym} ^j(f_*{\cal L}_M\big|_{{\cal B}_j})
\end{displaymath}
and
\begin{displaymath}
Q=f_*L_{jM}\oplus f_*L_{jM}\big|_{{\cal B}_j},
\end{displaymath}
$q$ is the multiplication map.
By \ref{thm:semipositiveness} we already know that $Q$ is
semipositive, and so is $W$ since symmetric powers of a semipositive
sheaf are semipositive.
Recall that the universal family $U_Y$ over $Y$ is embedded into a
product of $Y$ and
\begin{displaymath}
\Bbb P^{d_1}\times W \subset
\Bbb P^{d_1}\times\Bbb P^{d_2} \subset \Bbb P^{d_3}
\end{displaymath}
and that the sheaf $L_{M}$ is the restriction of
${\cal O}_{P^{d_3}}(1)$ in this embedding.
The group $G$ acting on $W$ is $GL_{d_1+1}\times GL_1$. If every
fiber ${\cal X}={\cal\Gamma}_g$ together with all $B_j$ can be uniquely
reconstructed from the map $W_s\to Q_s$, then the fibers of $u_{Gr}$
will be exactly the same as fibers of $Y\to{{\bold M \bold C}} $, hence finite.
For this to be true we need the following:
\begin{enumerate}
\item every fiber in $\Bbb P^{d_3}$ is set-theoretically defined by
degree $\le j$ equations,
\item the multiplication maps $\operatorname{Sym} ^j(f_*{\cal L}_M)\to f_*L_{jM}$
and $\operatorname{Sym} ^j(f_*{\cal L}_M\big|_{{\cal B}_J})\to f_*L_{jM}\big|_{{\cal B}_J}$
are surjective.
\end{enumerate}
(1) holds if $j$ is large enough. (2) is satisfied because we have
chosen $M$ so large that all $X$ and $B_j$ are projectively
normal in $\Bbb P^{d_3}$.
Finally, the second condition in the definition of finiteness of the
classifying map is satisfied because all graphs $(\Gamma_g,B)=(X,B)$
in $\Bbb P^{d_3}$ have finite groups of automorphisms.
\end{pf}
\section{Related questions}
\label{sec:related questions}
\begin{say}
Let us see how our moduli spaces are related to some others. For
example, consider the moduli space ${\cal M}_{L^2}$ of K3 surfaces $X$
with a polarization $L$ with a fixed square.
Compare it with ${\cal M}_{(K+B)^2}$, where $W=pt$, $B=B_1$ is one
reduced divisor and $(K+B)^2=L^2$ is the same number.
${\cal M}_{(K+B)^2}$ contains an open subset $U$ parameterizing K3
surfaces with reduced divisors having normal intersections only, and
we have a map $F:U\to{\cal M}_{H^2}$. A well-known result (Saint-Donat
\cite{SaintDonat74}) says that every ample linear system $|L|$ on a
K3 surface contains at least one reduced divisor with normal
intersections, therefore $F$ is surjective. In fact, ${\cal M}_{H^2}$ is
a quotient of $U$ modulo an obvious equivalence relation $R$:
$(X_1,B_1)\underset{R}{\sim}(X_2,B_2)$ iff $X_1$, $X_2$ are
isomorphic and $B_1$, $B_2$ are linearly equivalent.
There is a natural map $G:R\to U\times U$. $\pi_1\circ G$ is smooth
and its fibers are open subsets in $\Bbb P^{h^0(H)-1}$. The
situation is very similar to what we had in theorem
\ref{thm:existence as an algebraic space}, except this time the
quotient $U/R$ is not proper. The obvious way to try to obtain a
compactification of ${\cal M}_{H^2}$ is to consider the closure
$\overline{U}$ of $U$ in ${\cal M}_{(K+B)^2}$, then somehow define
the closure $\overline{R}$ of $R$, and ask if it has good enough
properties enabling one to construct $\overline{U}/\overline{R}$ and
to prove that it is projective. Alternatively, one can ask if the
closure of $\overline{G}(R)$ in $\overline{U}\times\overline{U}$ has
good properties.
The situation resembles what happens for elliptic curves. The
natural compactification of the moduli space ${\cal M}_1=\Bbb A^1_k$ is
$\Bbb P_k^1$, and the infinite point corresponds not to one but to
many degenerations: wheels of rational curves of lengths $1\dots n$
if we consider ${\cal M}_1$ as a factor of ${\cal M}_{1,n}$. Similarly, the
boundary points of ${\cal M}_{H^2}$ should correspond to many different
degenerations of smooth K3 surfaces with geometric divisors,
properly identified.
The first thing to ask on this way is:
\begin{question}
Is it possible to define an equivalence relation
$\overline{G}:\overline{R}\to\overline{U}\times\overline{U}$,
so that the morphism $\pi_1\circ\overline{G}$ is smooth or at
least flat?
\end{question}
Even if this is done, there are problems with taking the quotient.
There does not seem to exist in the literature a ready-to-use method
that would cover our situation. There is, on one hand, a theorem of
M.Artin (see \cite{Artin69} 7.1, \cite{Artin74b} 6.3) that shows that
if $\overline{G}:\overline{R}\to\overline{U}\times\overline{U}$ were
a monomorphism (which it is not) with flat projections, then the
quotient would be defined as an algebraic space. In this case it
would also easily follow that the quotient is actually projective.
On the other hand, there is the method of \cite{Mumford82}, p.172
that we used in the previous section, in which the equivalence
relation is smooth, and the map $\overline{G}$ is finite. Natural
degenerations of K3 surfaces can have infinite groups of
automorphisms, however.
I think that the question deserves a more detailed consideration.
\end{say}
\begin{say}
Similarly to K3 surfaces, for any principally polarized Abelian
variety $A$ with a theta divisor $\Theta$ the pair $(A,\Theta)$ has
log canonical singularities, see \cite{Kollar93}. So. the previous
discussion applies to principally polarized Abelian surfaces too.
One can also ask what happens if the polarization is not principal.
\end{say}
\begin{say}
It goes without saying that the projectivity theorem
\ref{thm:projectiveness} applies in the case of curves, with
significant simplifications. Therefore, the moduli spaces $M_{g,n}(W)$
of \cite{Kontsevich94} are also projective.
\end{say}
\begin{say}
Most ${{\bold M \bold C}} _{(K+B)^2,(K+B)H,H^2}$ are definitely not irreducible and
not even connected. They are subdivided according to various
invariants, such as the numerical or homological type of $g(X)$ and
$g(B_j)$, intersection numbers $(K+B)B_j$ etc.
One can also get by fixing only one number, $(K+B+4H)^2$. Then there
are only finitely many possibilities for other invariants.
\end{say}
\begin{say}
The boundedness theorem \ref{thm:absolute boundedness} is in fact
even stronger than what we used here: it applies to the case when
the coefficients $b_j$ belong to an arbitrary set $\cal A$ that
satisfies the descending chain condition. One, perhaps, would want
to define even more general moduli spaces. There are two obstacles,
however. First, the semipositiveness theorem \ref{thm:kollar's
semipositiveness} for the case of fractional coefficients seems to
be quite hard to prove, but probably still possible. The second
obstacle is a fundamental one: for proving the semipositiveness
theorems for ${\cal L}_k|_{{\cal B}_j}$ we used the log adjunction formula. It
basically just says $K+B|_B=K_B$, and here the coefficient 1 of $B$
is important.
\end{say}
\begin{say}
The places where assumption about the characteristic 0 was
used:
\begin{enumerate}
\item MMP in dimension 3. This is not serious since we worked in the
situation of the relative dimension 2. For surfaces log MMP is
characteristic free, and perhaps it is true for families of
surfaces in generality needed. For the case $B=\emptyset$ see
\cite{Kawamata91}
\item The semipositiveness theorem
\ref{thm:kollar's semipositiveness} requires characteristic 0.
Since we are dealing with a case of relative dimension 2 only,
this also probably can be dealt with.
\item A group scheme in characteristic 0 is reduced, hence
smooth. This was used in the proof of
\ref{thm:existence as an algebraic space}. Perhaps, the argument
could be strengthened.
\item The argument of \cite{Mumford82} p.172 is a whole lot more
complicated in characteristic $p>0$.
\end{enumerate}
\end{say}
\begin{say}
It should be possible to prove the semipositiveness theorems and the
Ampleness Lemma, as well as the \cite{Mumford82} p.172 argument,
entirely in the relative situation $/W$, without appealing to
absolutely ample divisors. The moduli spaces obtained should be then
projective over $W$.
\end{say}
\begin{say}
One can see that most of the theorems that we proved for the functor
${{\cM\cC}} {'}^N$ apply to the functor ${{\cM\cC}} {'}^{all}$ as well.
\end{say}
\makeatletter \renewcommand{\@biblabel}[1]{\hfill#1.}\makeatother
|
1994-10-27T05:20:12 | 9410 | alg-geom/9410026 | en | https://arxiv.org/abs/alg-geom/9410026 | [
"alg-geom",
"math.AG"
] | alg-geom/9410026 | null | Severinas Zube | Exceptional vector bundle on Enriques surfaces | 12 pages, LATEX | null | null | null | null | The main purpose in this paper is to study exceptional vector bundles on
Enriques surfaces.
| [
{
"version": "v1",
"created": "Wed, 26 Oct 1994 10:51:35 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Zube",
"Severinas",
""
]
] | alg-geom | \subsection{ Introduction}
The purpose of this note is to study exceptional\ vector\ bundles on Enriques surfaces.
Exceptional bundle E on a surface with irregularity $q=h^1(O_S)$ and geometric
genus
$p_g=h^2(O_S)$ is the bundle with the following properties:\\
$Ext^0(E,E)=\bbbc
,Ext^1(E,E)=q, Ext^2(E,E)=p_g$. On an Enriques surface Kim and Naie in
[Ki1],[Ki2],[N] have been studied extremal bundles which are very similar
to exceptional\ \ bundles. Extremal bundle on Enriques surface by definition is a
simple $Ext^0(E,E)=\bbbc$, rigid $Ext^1(E,E)=\bbbc$ with the following
condition:$Ext^2(E,E)=\bbbc$. From the Riemann-Roch theorem easily follows
that any exceptional\ bundle has odd rank and any extremal even rank. In [Ki2],
Kim characterized extremal bundles on Enriques surfaces. They exist only
on nodal surfaces and satisfy $c_1^2=4n-2,c_2=n$ for $n>4$. They have also
geometrical meaning. It turns out that the existence of extremal bundles
is closely related to the embedding a general Enriques surface in the
Grassmannian G(2,n+1).\\
The main result in this paper is to give the necessary and sufficient
conditions for the existence exceptional\ bundles on Enriques surfaces. The
statement is similar to the theorem 4 in [Ki2]. The proof fill a
gap in the proof of this theorem 4. Also I give constructions of them
by using some
constructions of Enriques surfaces and by using modular operations
(reflections) which is
described in the last section. I think the reflection is very useful to
construct and study moduli of sheaves on Enriques surfaces.
\subsection{Enriques surfaces}
A smooth irreducible surface $S$, such that $h^1 (O_S )=h^2 (O_S )=0 $ and
$2K_S \sim O_S $, is called a Enriques surface.
Recall that a divisor $D$ on a smooth surface $X$ is said to be $nef$
if $DC \geq 0$ for every curve $C$ on $X$ . The following useful properties
will be used throughout, sometimes without explicit mention:
(A)([C,D] Corollary 3.1.3) If $D$ is a $nef$ divisor and $D^2 > 0 $ , then
$H^1 (O_S (-D)) = 0$ and $\chi(O(D)) -1 =dim|D| = \frac{D^2 }{2}$. \label{222}
(B)([C,D] Proposition 3.1.4) If $ \mid D \mid $ has no fixed components, then
one of the following holds: \nonumber \\
(i) $D^2 > 0$ and there exist an irreducible curve $C$ in $ \mid D \mid $.
\nonumber \\
(ii) $D^2 =0$ and there exist a genus 1 pencil $ \mid P \mid $ such that $D
\sim kP$ for some $k \geq 1 $. \label{333}
(C)([C,D]Chapter 4, appendix , corollary 1. and corollary 2.) If $D^2 \geq 6$
and $D$ is $nef$ then $D$ is ample , $2D$ is generated by its global sections,
$3D$ is very ample.
The Enriques surface $S$ is called nodal (resp. unnodal) if there are
(resp. not) a smooth (-2)-curve
contained in $S$. A general Enriques surface is an unnodal.
Every Enriques surface admits an elliptic fibration over ${\rm I\!P} ^1$
with exactly two multiple fibers F, F' and
an elliptic pencil $\mid 2F\mid =\mid 2F'\mid$ with $K_S=F-F'$.
On a general Enriques surface
there are ten different elliptic pencils $\mid 2F_1\mid ,\mid 2F_2\mid,...,
\mid 2F_{10}\mid $ (see [CD]).
\addtocounter{subsection}{1}
\subsection{Mukai lattice}
It is convenient to describe discrete invariants of sheaves and
bundles on a K3 or an Enriques surface $X$ in the form of vectors in the
algebraic Mukai lattice
$$M(X) = H^0 (X,{\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}) \oplus Pic X \oplus H^4 (X,{\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}) = {\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$} \oplus Pic
X \oplus {\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$} \ni v=(r,D,s)$$
with inner product $< , >$
$$<(r,D,s), (r',D',s')> = rs'+s'r-D.D'$$
To each sheaf $E$ on $X$ with $c_1(E) = D, c_2(E) \in H^4 (X, {\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}) =
{\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}$ we associate the vector
$$v(E) = \left( rk(E),D, \frac{1}{2}D^2 - c_2 + rk(E)
\frac{\chi(O_X )}{2}\right) , $$
where $\chi(O_X )=1-q+p_g $ is equal to 2 for a K3 surface and 1 for
an Enriques surface.
This formula,the Riemann-Roch theorem yield the equalities:
\begin{eqnarray}
<v(F),v(E)>=<v(E),v(F)>&=&\chi(E,F)\nonumber\\
= dim Ext^0 (E,F) - dim Ext^1 (E,F)&+& dim Ext^2 (E,F)
\end{eqnarray}
Because $K_X $ is numerically equal to zero we have
that $\chi(E,F) = \chi (F,E)$ on a K3 or an Enriques surface.
For the short exact sequence
$$0 \to F \to E \to G \to 0$$
we have the following equalities
$$v(E) = v(F) + v(G) = (r(F)+r(G), c_1 (F) +c_1 (G),s(F) + s(G)).$$
\addtocounter{section}{1}
\subsection{Exceptional bundles}
{\bf Definition:} E is an exceptional sheaf on surface S if:\\
${~~~~~~~~~~~~~~}dim Ext^0 (E,E) = 1, dim Ext^1 (E,E) = q , dim Ext^2 (E,E) =
p_g$.\newline
From the Riemann-Roch theorem we have:
$$\chi(E,E) = r^2 \chi(O_S )+(r-1)c_1^2 -2rc_2.$$
Since $H^2 (S,{\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$} )$ is even lattice and $\chi(O_S)=1$ for an Enriques
surface S we see that E has odd rank if E is
an exceptional bundle.
For the description of exceptional vector bundles we need the following
result of Kuleshov.
\newtheorem{ttt}{Theorem [Ku]}[subsection]
\begin{ttt}
Let X be a smooth a K3 surface and let $H$ be an arbitrary ample divisor
on X, and $v=(r,D,s), r > 0$ is an exceptional vector (i.e $v^2 =2$)
belonging to the Mukai lattice on X. Then there exist a simple ,
$\mu_H $-semi-stable bundle E which realize the vector v (i.e.v=v(E)).
\end{ttt}
I wish to start from some useful facts. The first one is about torsion
free sheaves with some homological condition.
\newtheorem{Mukai}{Proposition}[subsection]
\begin{Mukai}
Let {\it E} be a torsion free sheaf on a smooth surface S and $dim Ext^1
({\it E,E}) =1$ or 0. Then {\it E} is locally free.
\end{Mukai}
{\bf Proof:} \ We have the following exact sequence:
$$0 \to {\it E} \to {\it E}^{**} \to M \to 0, $$
where ${\it E}^{**}$ is double dual of {\it E} and cokernel M is of finite
length. Now I use Mukai [M] result
( see Corollary 2.11 and 2.12) and obtain the following inequality:
$$ dim Ext^1 ({\it E}^{**},{\it E}^{**}) + dim Ext^1
(M,M) \leq dim Ext^1 ({\it E},{\it E}) $$
Because $v^2 (M)=0$ we have that $ dim Ext^1 (M,M)$ is equal to
$2dim End_{O_S} (M)$.\\
Since $Ext^1 ({\it E},{\it E})=0$ we
obtain that M=0 and ${\it E} = {\it E}^{**}$. And hence ${\it E}$ is
locally free the statement follows. $\odot$
Now I wish formulate the main result.
\newtheorem{exc}[ttt]{Theorem
\begin{exc}
Let S be a smooth Enriques surface S, $v=(r,D,s) \in M(S)\\ ( r > 0 )$ and
$v^2=1$ then: \\
(i)\ There is an ample divisor H such that $D\cdot H$ and r have not common
divisor greater than 1 (i.e. $(D\cdot H,r) = 1$). \\
(ii)\ For any ample divisor H with condition $(D\cdot H,r)=1$ exist an
exceptional
vector bundle E and only one such that v(E)=v and E is H-stable.
\end{exc}
{\bf Proof:}\ \ (i) Because $v^2=1$ we have $2rs-D^2=1$. This means that $D \not\subset
r'\cdot H^2 (S,{\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$} ) $ for any $r'$ such that $( r',r ) > 1 $ (here ( , )
means the greatest common divisor). Since our
lattice $H^2 (S,{\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$} ) $ is unimodular there is $X \in H^2 (S,{\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$} ) $
such that $(X\cdot D,r)=1$. Hence $H_k =X + krH$ will be very ample for any
ample divisor H and
$k\gg 0$ . And $H_k$ satisfy our condition $(H_k \cdot D,r)=1$. This prove
the first statement.\\
(ii) Let a universal covering space of S be X which is a K3 surface and
let $\pi$ be the quotient map. Consider the vector $\hat v =(r,D',s')=\pi
^*(v)$ on K3
surface X. It turns out that $\hat v ^2=2$ so by theorem of Kuleshov there is
an exceptional vector bundle F on X such that $\hat v (F)=\hat v $ and F is
$\hat H = \pi
^*(H)$-semi-stable. In fact F is $\hat H $-stable. Indeed, we have
$(\hat H \cdot D',r) =(2H\cdot D,r) = 1$ (recall that r is even number
as we notice above) and for
any subsheaf W with $0 < rank(W) < rank(F) $ the following:
$$\frac{c_1 (W)\cdot
\hat H }{rank(W)}<\frac{D'\cdot \hat H }{rank(F)},$$ therefore F is $\hat H$
-stable .
So F and $\sigma ^* (F)$ both are $\hat H$ -stable, where $\sigma$ is the free
involution on X such that $X/\sigma = S.$ It is easy to see that
$\chi(F, \sigma ^* (F)) = 2$. Hence there is non zero homomorphism from F to
$\sigma ^*
(F)$ which should be isomorphism because both vector bundles have the same
determinant and are $\hat H$ -stable. This isomorphism means that there is a
vector
E on the surface S such that $\pi ^*(E)=F$. Of course, the vector bundle E is
H-stable and v(E)=v.
Assume that there is another H-stable vector bundle $\hat E$ and
$v(E)=v(\hat E )$. Then $\chi(E,\hat E )=1$, therefore there is non trivial map
from $\phi:E\to \hat E $ or by Serre duality $\rho:\hat E \to E\otimes K$. In
both
cases it should be isomorphisms by stability assumption. But then $E=\hat
E$ or
$det\rho$ gives non zero element of canonical class. This contradiction
prove that H-stable bundle is unique.$\odot$ \\
{\bf Notice} that from condition
$Ext^1(E,E)=0$ we get only that moduli space of E contain only discrete set of
bundles. But by (ii) such E is only one so moduli space consist of only one
point.
\paragraph{Remark}: It is not clear (to my ) whether an exceptional
H-stable vector bundle E is G-stable for any another ample divisor G. By
theorem it can happen only if $(G\cdot c_1 (E),rk(E)) > 1$.
\addtocounter{section}{1}
\subsection{Examples}
I wish to give some explicit examples of exceptional\ \ vector\ \ bundles. The one way
to construct them is as in the theorem above to find a stable, invariant,
exceptional\ vector\ bundle on a K3 surface.
Consider Horikawa's representation of Enriques surface (see for
details [BPV]).
We introduce coordinates $(z_0:z_1:z_2:z_3)$ on ${{\rm I\!P} ^3}$ such that a
quadric $Q={\rm I\!P} ^1 \times {\rm I\!P} ^1$ is embedded by
$$z_0=x_0y_0,\ \ z_1=x_1y_1,\ \ z_2=x_0y_1,\ \ z_3=x_1y_0.$$
If we define the involution $\tau$ on ${{\rm I\!P} ^3}$ by $\tau
(z_0:z_1:z_2:z_3)=(z_0:z_1:-z_2:-z_3)$, then Q is $\tau$ invariant with
$\tau$ acting on Q by
$$\tau ((x_0:x_1)(y_0:y_1)=(x_0:-x_1)(y_0:-y_1).$$
On Q the involution $\tau$ has the four fixed points
$$(x_0:x_1)(y_0:y_1)=(1:0)(1:0) , (1:0)(0:1),(0:1)(0:1) , (0:1)(1:0).$$
Take a polynomial of bidegree (4,4) which define a $\tau$ invariant curve
B. Assume that B have not any of fixed points. Consider a surface X which
is a double cover of Q
ramified over B. It turns out that X is a K3 surface and the involution
$\tau$ induce the involution $\sigma$ on X which is without fixed points.
Hence $X/\sigma =S$ is an Enriques surface. Notice that general Enriques
surface can be obtain by this construction. Let $\pi :X \to S$ be
factorization map and $\phi :X \to Q$ the double cover ramified over B.
If E is exceptional\ vector\ bundle on Q then $\phi ^* (E)=\hat E $ is an exceptional\ . Indeed , by
projection formula we have $H^i (End_{O_X}(\hat E ) = H^i (End_{O_Q}(E) \oplus
H^i (End_{O_Q}(E) \otimes (-K_Q))$ (recall that $B=-2K_Q$). Since, by the
Serre duality, we have $H^i (End_{O_Q}(E) \otimes
(-K_Q))=H^{2-i}(End_{O_Q}(E))$ therefore $\hat E$ is an exceptional\ \ vector\ \ bundle on
the K3 surface X. This
bundle $\hat E$ is $\sigma$ invariant because E is $\tau$ invariant
on quadric Q.
(Indeed, E is rigid, therefore E is $PGL(2)\times PGL(2)$-homogeneous.)
Because $\hat E$ is $\sigma$ invariant, there is a vector\ \ bundle F on S such
that
$\hat E =\pi ^* (F)$. It is easy to see that $v^2 (F)=1$ and F is rigid,
therefore
F is exceptional\ \ vector\ \ bundle on Enriques surface S. I am not able to say anything
about the stability of $\hat E$ and F. This produce a lot of exceptional\ \ vector\
bundles because we know how to construct all exceptional\ \ vector\ \ bundles on smooth
quadric Q.
In the similar way we can consider quartic X in ${\rm I\!P} ^3$ which is
defined by the equation $z_0^4+z_1^4-z_2^4-z_3^4$. This is a smooth
K3 surface (see for details in [GH]). Let T be an automorphism on ${\rm I\!P} ^3$
defined as follows:
$$T: (z_0,z_1,z_2,z_3) \to (z_0,\sqrt{-1}z_1,-z_2,-\sqrt{-1}z_3).$$
This automorphism T has 4 fixed points on ${\rm I\!P} ^3$ no one of which lay on
the surface X , $T^2$ has 2 fixed lines:
$$l_1=(z_0=z_2=0), \ l_2=(z_1=z_3=0). $$
These lines intersect the surface X in 8 points $p_1,....,p_8.$ Consider
blow-up of X in these 8 points ${\bar X} \to X$. Let ${\bar T}$ denote induced
automorphism on ${\bar X}$. It turns out that
$X'={\bar X}/\{ {\bar T}^{2n} \} $ is a K3 surface and $\bar T$ acts on
$X'$
as an involution
without fixed points. So $X'/\bar T$ is an Enriques surface S.
Now we can get an exceptional\ \
vector\ \ bundle on S from any exceptional\ \ vector\ bundle on ${\rm I\!P} ^3$ because each exceptional\
\ vector\ \ bundle on ${\rm I\!P} ^3$ is a homogeneous, therefore it is T invariant.
Of course, we get the vector\ bundle on $X'$ which is an exceptional\ and the bundle
descend to S also as an exceptional\ . This procedure gives us a lot exceptional\ \ vector\ \
bundles on S and we can describe it because we know constructions of exceptional\
bundles on ${\rm I\!P} ^3$.
Now I wish discuss about the ability to construct exceptional
collections on Enriques surface. Recall that by definition
${E_1,E_2,...,E_n}$ is an exceptional\ \
collection if $Ext^i (E_k,E_j)=0$ for any i and $k>j.$ In particular, we
have that $\chi (E_k,E_j)=0$. But on an Enriques surface we have $\chi
(E,F)=\chi
(F,E)$ for any sheaves E and F. Hence for an exceptional\ \ collection on Enriques
surface should be true the following:
$$ Ext^i (E_k,E_j)=0 , \forall \ i \ and\ k>j;\ \ \ \ \chi (E_a,E_b)=0\ if\
a\not=
b.$$
On a general Enriques surface exist exceptional\ \ colection with ten bundles. Indeed,
it is well known that
on general Enriques surface there are ten different elliptic pencils say
$\mid 2F_1 \mid ,...,\mid 2F_{10} \mid $ (see [CD]). It is easy to see that
$Ext^i (F_k,F_j)=0 , \forall \ i \ and\ k\not= j,$
therefore ${F_1,F_2,...,F_{10}}$ is an exceptional\ \ collection. It will be very
interesting to describe the orthogonal category in the derived category
D(S) (which is finite )
of all sheaves on Enriques surface S. This orthogonal category should have
only two independent elements.
\addtocounter{section}{1}
\subsection{ Modular operations }
There are some natural modular operations from one moduli space to
another which gives an isomorphism
of tangent bundles of moduli spaces.For example $E\longleftrightarrow
E^*, E \longleftrightarrow E\otimes D$, where D is a line bundle.
On an Enriques surface we have the very interesting modular
operation which I call a reflection. This operation is similar to the
reflection on a K3 surface (see [T] 4.10,4.11). I wish to describe it.
First of all this reflection acts on Mukai lattice in the following way:
$$v=(r,D,s) \longleftrightarrow R(v)=\hat v =\left( 2s,D+\left(
s+\frac{r}{2}\right)
K_S,\frac{r}{2}\right).$$
Notice that $v^2 = 2rs-D^2 = \hat v ^2 $.\\
Now I describe it on the level of sheaves.
Assume that a torsion free sheaf E is generated by its section,v(E)=v=(r,D,s)
and
$\chi(E\otimes K)=h^0 (E\otimes K),\\
h^1 (E\otimes K)= h^1 (E) = 0$. Notice
that from this we have $h^2 (E\otimes K)=\\
h^2 (E) = 0, h^0 (E)=\chi
(E)=\chi (E\otimes K)$. For example, if we twist any stable bundle by
sufficiently large very ample divisor then our condition will be
satisfied.
Consider the following exact sequence :
\begin{equation}
0 \to \bar E^* \to H^0 (E)\otimes O_S\stackrel{ev}{ \to } E \to 0 ,
\label{ebundle}
\end{equation}
where $ev: H^0 (E)\otimes O_S \to E \to 0$ is the canonical evaluation map
(surjective by assumption).
For the convenience denote $H=H^0 (E),\ h=dim H$ and consider the dual
sequence:
\begin{equation}
0 \to E^* \to H^* \otimes O_S \to \bar E \to 0. \label{edual}
\end{equation}
By our conditions and Serre duality we have that
$$h^1 (\bar E )=h^2 (E^* )=h^0 (E \otimes K) =\chi (E\otimes K)=h.$$
Consider the following sequence:
\begin{equation}
0\to H^1(\bar E ) \otimes K_S \to \hat E \to \bar E \to 0, \label{extention}
\end{equation}
where $\hat E$ is given by universal extension element $id \in Ext^1 (\bar E,
H^1(\bar E )) = End(H^1(\bar E )).$ Denote $R(E)=\hat E $.
As an easy consequence of two sequences (\ref{ebundle}) and
(\ref{extention}) our
assumptions and Serre duality is the
following
\newtheorem{PP}{Proposition}[subsection]
\begin{PP}
Assume E is the sheaf as above then sheaves $\bar E$ and $\hat E $
from sequences (\ref{ebundle}) and (\ref{extention}) satisfies the following
properties: \\
1. ${\bar E}$ is globally generated by sections.\\
2. $\chi(\bar E)=0,\ h^0 (\bar E)=h=h^1 (\bar E), \ h^2 (\bar E)=0.$\\
3. $\chi(\bar E\otimes K)=0,\ h^i (\bar E\otimes K_S)=0, \ for \ \forall
i>0$.\\
4. $v(\hat E )=\hat v =R(v)=
\left( 2s,D+\left( s+\frac{r}{2}\right) K_S,\frac{r}{2}\right) \ and\\
h^0 ( \hat E )= h^0 ( \hat E \otimes K)=h,\ h^i ( \hat E )=
h^i ( \hat E \otimes K)=0 \ for \ \forall i>0$.\\
5.$Hom(E,E)=Hom(\bar E,\bar E)=Hom(\hat E ,\hat E ),Ext^2(\bar E,\bar E)=0.$\\
6. $rank(E)=rank(\hat E)\pmod{2}$. Moreover, if $rank(E)=2k+1$ and
both E and $\hat E$ are H-stable then
$Ext^2(\hat E ,\hat E )=Ext^2(E,E)=0,Ext^1(\hat E ,\hat E )=Ext^1(E,E)$.
\label{PP}
\end{PP}
{\bf Proof:} \ \
1. The sheaf $H^* \otimes O_S$ in the middle of the sequence
(\ref{edual})
is globally generated by sections so $\bar E$ is too.\\
\noindent 2.The corresponding long in cohomology to the sequence
(\ref{edual})
gives us $h^0 (\bar E)=h$ because by Serre duality $h^0
(E^*)=h^2 (E\otimes K)=0$ and $h^1 (E^*)=h^1 (E\otimes K)=0$. Also
$h^2(\bar E)=0$ and $h^1 (\bar E)=h$ as we already noticed. Hence
$\chi(\bar E)=0.$\\
\noindent 3.Consider the sequence (\ref{edual}) twisted by K:
\begin{equation}
0 \to E^*\otimes K \to H^* \otimes K_S \to \bar E\otimes K \to 0.
\label{etwistK}
\end{equation}
In the same way the corresponding long exact sequence in cohomology gives:
$h^0(\bar E \otimes K)=h^1( E^* \otimes K)=h^1( E) $ and $\ h^1(\bar E \otimes
K)=0$ because $h^2(E^* \otimes K)=h^0(E)=h=h^2(H^*\otimes K);\ h^2(\bar
E\otimes K)
=h^0(\bar E^*)$ (by the sequence (\ref{ebundle})).\\
\noindent 4. An easily calculation shows that
$v(\hat E )=\hat v =\left( 2s,c_1(E)+\left( s+\frac{r}{2}\right)
K_S,\frac{r}{2}\right).$
Since the sequence (\ref{extention}) is the
universal extension we have $h^1( \hat E )=h^2( \hat E )=0$, therefore, by the
Riemann-Roch theorem, we obtain that $h^0( \hat E )=h$. If we twist
(\ref{extention}) by K and
then use properties of $\bar E\otimes K$ we easily get that\\ $h^0(
\hat E \otimes K)=h,\ h^1(\hat E \otimes K)=h^2(\hat E \otimes K)=0$. \\
\noindent 5. Applying $Hom(\bar E,*)$ to (\ref{edual}) we get the long
exact sequence:
\begin{eqnarray*}
0\to Ext^0(\bar E,E^*)\to & Ext^0(\bar E,H^*\otimes O_S) & \to Ext^0(\bar
E,\bar E) \to \\
\to Ext^1(\bar E,E^*)\to & Ext^1(\bar E,H^*\otimes O_S) & \to Ext^1(\bar
E,\bar E) \to \\
\to Ext^2(\bar E,E^*)\to & Ext^2(\bar E,H^*\otimes O_S) & \to Ext^2(\bar
E,\bar E) \to0
\end{eqnarray*}
By Serre duality and the statement 3 Ext groups in the middle are\\
$H^i(\bar E\otimes K)\otimes
H^* =0$. Hence we have $ Ext^1(\bar E,E^*)= Ext^0(\bar E,\bar E) ,\\
Ext^2(\bar E,\bar E) =0.$
Also applying $Hom(*,E^*) $ to (\ref{edual}) we get:
\begin{eqnarray*}
0\to Ext^0(\bar E,E^*)\to & Ext^0(H^*\otimes O_S,E^*) & \to Ext^0(E^*,E^*) \to
\\
\to Ext^1(\bar E,E^*)\to & Ext^1(H^*\otimes O_S,E^*) & \to Ext^1(E^*,E^*)
\to
\end{eqnarray*}
By our assumption and Serre duality the middle Ext groups are \\
$Ext^0(H^*\otimes O_S,E^*)=H^2(E\otimes K) = 0 = Ext^1(H^*\otimes
O_S,E^*)=H^1(E\otimes K)$.\\
Hence $Hom(E^*,E^*)=Hom(E,E)=Hom(\bar E,\bar E).$
Now applying $Hom(\hat E ,*)$ to (\ref{extention}) we get:
\begin{eqnarray*}
0\to Ext^0(\hat E ,H\otimes K)\to & Ext^0(\hat E ,\hat E ) & \to Ext^0(\hat E
,\bar E) \to \\
0\to Ext^1(\hat E ,H\otimes K)\to & Ext^1(\hat E ,\hat E ) & \to Ext^1(\hat E
,\bar E) \to
\end{eqnarray*}
Since $Ext^0(\hat E ,H\otimes K)=H^2(\hat E )\otimes H=0$ we obtain that
$Ext^0(\hat E ,\hat E )= Ext^0(\hat E ,\bar E)$. And, in the similar way, after
applying
$Hom(*,\bar E)$, we get that $Ext^0(\hat E ,\hat E )= Ext^0(\bar E,\bar
E)$.
6.If rank(E) is odd then $s(E)\in \frac{1}{2}{\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}$ but
$s(E)\not\subset {\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$} $, therefore $rank(\hat E)\\
= 2s(E)$
is odd number too. If rank(E) is even then $s(E)\in {\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}$, therefore
$rank(\hat E) = 2s(E)$
is even.Consider any non zero element $\phi$ in $Ext^2(E,E)=\\Ext^0(E,E\otimes
K)^*$
by the stability assumption this should be an isomorphism but then $det\phi$
is non zero element of canonical class. This contradiction shows that
$Ext^2(E,E)=0$ and $Ext^2(\hat E ,\hat E )=0$. Because $v^2=\hat v ^2$ we have
$Ext^1(E,E)=Ext^1(\hat E ,\hat E )$.$\odot$\\
I am able to reverse this operation in the following situation.
Consider a sheaf F and the following exact sequence:
\begin{equation}
0\to H^0( F\otimes K_S ) \otimes K_S \stackrel{ev}{\to} F \to \bar F \to 0,
\label{fx}
\end{equation}
where $ ev:H^0( F\otimes K_S ) \otimes K_S \to F$ is the canonical evaluation
map.
Assume that $\bar F$ is globally generated a torsion free sheaf then we have
the
following exact sequence:
\begin{equation}
0 \to \hat F ^* \to H^0 (\bar F) \otimes O_S \stackrel{ev}{\to} \bar
F \to 0. \label{ff}
\end{equation}
Denote $R(F)=\hat F$.
Under these assumptions we can prove the similar result:
\newtheorem{P'}[PP]{Proposition
\begin{P'}
Assume F is the sheaf as above then sheaves $\bar F$ and $\hat F$ from
sequences
(\ref{ff}) and (\ref{fx}) satisfies the following properties: \\
1. $\chi(\bar F)=0,\ h^0 (\bar F)=h=h^1 (\bar F), \ h^2 (\bar F)=0.$\\
2. $\chi(\bar F\otimes K)=0,\ h^i (\bar F\otimes K_S)=0, \ for \ \forall
i>0$.\\
3. $v(\hat F )=\hat v =\left( 2s,D+\left( s+\frac{r}{2}\right)
K_S,\frac{r}{2}\right) \ and\ h^0 ( \hat F )= h^0 ( \hat F \otimes K)=h,\\
\ \ \ h^i ( \hat F )=
{~~}h^i ( \hat F \otimes K)=0 \ for \ \forall i>0$.\\
4.$Hom(F,F)=Hom(\bar F,\bar F)=Hom(\hat F ,\hat F ),Ext^2(\bar F,\bar F)=0.$\\
5. $rank(F)=rank(\hat F)\pmod{2}$. Moreover, if $rank(F)=2k+1$ and
both F and $\hat F$ are H-stable then
$Ext^2(\hat F ,\hat F )=Ext^2(F,F)=0,Ext^1(\hat F ,\hat F )=Ext^1(F,F)$.
\label{P'}
\end{P'}
Notice that from both propositions follows that R(R(E))=E.
\paragraph{Remark:} The reflection always exist in the derived category of
sheaves
on surface S. It does not matter whether the evaluation map
$ H^0( F ) \otimes O_S \stackrel{ev}{\to} F$ is a surjective or an
injective map.\\
Now I wish to give a few examples of reflections.\\
1. Assume we have a smooth curve C on S, A a globally generated
divisor on the curve C with the properties $h^1 (O_C(A))=h^1(O_C(A\otimes
K_S))=0$.
We consider the following exact sequence:
\begin{equation}
0 \to E(C,A)^* \to H^{0}(O_C(A)) \otimes O_S\stackrel{ev}{\to} O_{C}(A)
\to 0 \label{E*}
\end{equation}
The dual sequence to (\ref{E*}) is :
\begin{equation}
0 \to H^{0}(A)^{*} \otimes O_S \to E(C,A) \to O_{C}(C) \otimes A^* \to 0
\end{equation}
By our assumption, this sequence and Serre duality on the curve C,
we obtain that
$h^1(E(C,A))=h^1(O_C(C-A))=h^0(O_C(A)$ and get
the following sequence:
\begin{equation}
0 \to H^1(E(C,A))\otimes K_S \to \hat E \to E(C,A) \to 0
\end{equation}
So we have $R(O_C(A))=\hat E $. \\
If C is (-2)-curve and $A=O_C$,
we see that $R(O_C)=F$ is an extremal rank 2 vector bundle (i.e.
$Ext^0(F,F)=Ext^2(F,F)=\bbbc, Ext^1(F,F)=0.)$
2.If $\mid 2F\mid,\ \mid 2G\mid$ are two elliptic pencils on S then a pencil
$\mid F+G\mid$ has two different base points x and y. From the standard
sequence:
\begin{equation}
0 \to J_{x+y} (F+G) \to O(F+G)\to O_{x+y}(F+G)\to 0, \label{jj}
\end{equation}
we see that $h^1(J_{x+y} (F+G))=2$. This gives us the following bundle E,
defined by the universal extension element $id\in End(H^1(J_{x+y} (F+G))$:
\begin{equation}
0 \to H^1(J_{x+y} (F+G)) \otimes K_S \to E\to J_{x+y} (F+G) \to 0, \label{ej}
\end{equation}
By lemmas 1.1,1.2 in [T], E is a simple bundle. An easily calculation
shows that E is an exceptional\ \ bundle. Also by \ref{P'}, we see that $R(E)=O(F+G)$.
Notice that $O(F+G)$ is not globally generated by section, therefore we cannot
use \ref{PP} to this bundle to produce $R(O(F+G))$, but we can do this in
the derived category. Since $R(E)=O(F+G)$, we also obtain that
$R(O(F+G))=E$.
3.If we consider a divisor $O(aF+bG)$ for $a\geq b\geq 2$ then this divisor
will be an ample on a general Enriques surface. Hence $R(O(aF+bG))$ is an exceptional\ \
bundle of rank $2ab+1$.
\paragraph{References} : \newline
[BPV] W.Bart, C.Peter, A.Van de Ven. Compact complex surfaces.
Berlin, Heidelberg,New York :Springer 1984. \newline
[CD] F. Cossec ; I. Dolgachev "Enriques Surfaces 1", Birkh\"{a}user 1989.
\newline
[GH] Griffiths,Ph., Harris J.,Principles of algebraic geometry, New
York\newline
(1978) \newline
[Ki1] Kim,Hoil.:Exceptional bundles on nodal Enriques surfaces,\newline
Bayreuth
preprint(1991). \newline
[Ki2] Kim,Hoil.:Exceptional bundles and moduli spaces of stable vector
\newline
bundles on Enriques surfaces,Bayreuth
preprint(1991). \newline
[Ku] Kuleshov,S.A:An existence theorem for exceptional\ \ bundles on K3 \newline
surfaces,Math.USSR Izvestia,vol.34,373-388.(1990). \newline
[M] Mukai,S.On the moduli space of bundles on K3 surfaces I, in Vector
\newline
Bundles, ed. Atiay et all, Oxford University Press, Bombay, 341-413(1986).
\newline
[N] Daniel Naie. Special rank two vector bundles over Enriques surfaces,
\newline
preprint. \newline
[T] A.N. Tyurin "Cycles, curves and vector bundles on algebraic surfaces."
Duke Math.J. 54, 1-26,(1987). \newline
Department of geometry and topology,
Faculty of mathematics,
Vilnius university,
Naugarduko g.24,
2009 Vilnius, Lithuania.
e-mail:[email protected]
\end{document}
|
1994-10-14T05:20:36 | 9410 | alg-geom/9410014 | en | https://arxiv.org/abs/alg-geom/9410014 | [
"alg-geom",
"math.AG"
] | alg-geom/9410014 | null | Dmitri Zaitsev | On the linearization of the automorphism groups of algebraic domains | 10 pages, LaTeX | null | null | null | null | Let $D$ be a domain in $C^n$ and $G$ a topological group which acts
effectively on $D$ by holomorphic automorphisms. In this paper we are
interested in projective linearizations of the action of $G$, i.e. a linear
representation of $G$ in some $C^{N+1}$ and an equivariant imbedding of $D$
into $\P^N$ with respect to this representation. The domains we discuss here
are open connected sets defined by finitely many real polynomial inequalities
or connected finite unions of such sets. Assume that the group $G$ acts by
birational automorphisms.
Our main result is the equivalence of the following conditions:
1) there exists a projective linearization, i.e. a linear representation of
$G$ in some $\C^{N+1}$ and a biregular imbedding $i\colon \P^n \hookrightarrow
\P^N$ such that the restriction $i|_D$ is $G$-equivariant.
2) $G$ is a subgroup of a Lie group $\hat G$ of birational automorphisms of
$D$ which extends the action of $G$ and has finitely many connected components;
3) $G$ is a subgroup of a Nash group $\hat G$ of birational automorphisms of
$D$ which extends the action of $G$ to a Nash action $\hat G\times D\to D$;
4) $G$ is a subgroup of a Nash group $\hat G$ such that the action $G\times
D\to D$ extends to a Nash action $\hat G\times D\to D$;
5) the degree of the automorphism $\phi_g\colon D\to D$
| [
{
"version": "v1",
"created": "Thu, 13 Oct 1994 15:26:54 GMT"
}
] | 2015-06-30T00:00:00 | [
[
"Zaitsev",
"Dmitri",
""
]
] | alg-geom | \section{Linearization Theorem and applications}
Let $D$ be a domain in $\C^n$ and
$G$ a topological group which acts effectively
on $D$ by holomorphic automorphisms. In this paper we are interested in
projective linearizations of the action of $G$, i.e. a linear representation
of $G$
in some $\C^{N+1}$ and an equivariant imbedding of $D$ into $\P^N$ with
respect to this representation. Since $G$ acts effectively, the representation
in $\C^{N+1}$ must be faithful.
In our previous paper~\cite{Z}, however, we considered
an example of a bounded domain $D\subset\C^2$ with an effective action of a
finite covering $G$ of the group $SL_2(\R)$. In this case the group $G$ doesn't
admit a faithful representation. The example shows that a linearization
in the above sense doesn't exist in general.
In the present paper we give a criterion for the existence of the
projective linearization for birational automorphisms.
The domains we discuss here are open connected sets defined by finitely
many real polynomial inequalities or connected finite unions of such sets.
These domains are called {\it algebraic}.
For instance, in the above example the domain $D$ is algebraic.
\begin{Def}\label{Nash}
\begin{enumerate}
\item A {\bf Nash map} is a real analytic map
$$f=(f_1,\ldots,f_m)\colon U\to \R^m$$ (where $U\subset\R^n$ is open)
such that for each of the components $f_k$ there is a
non-trivial polynomial $P_k$ with $$P_k(x_1,\ldots,x_n,f_k(x_1,\ldots,x_n))=0$$
for all $(x_1,\ldots,x_n)\in U$.
\item A {\bf Nash manifold} $M$ is a real analytic manifold with finitely many
coordinate charts $\phi_i\colon U_i\to V_i$ such that $V_i\subset\R^n$ is
Nash for all $i$ and the transition functions are Nash
(a Nash atlas).
\item A {\bf Nash group} is a Nash manifold with a group operation
$(x,y)\mapsto xy^{-1}$ which is Nash with respect to all Nash coordinate
charts.
\end{enumerate}
\end{Def}
In the above example the group $SL_2(\R)$ and its finite covering $G$
are Nash groups (The universal covering of $SL_2(\R)$ is a so-called
locally Nash group).
Moreover, the action $G\times D\to D$ is also Nash.
Since the linearization doesn't exist here,
we need a stronger condition on the action of $G$.
A topological group $G$ is said to be
a {\it group of birational automorphisms}
of a domain $D\subset\C^n$ if we are given an effective (continuous)
action $G\times D\to D$ such that every element
$g\in G$ defines an automorphism of $D$ which extends to a birational
automorphism of $\C^n$. By the {\it degree} of a Nash map $f$ we mean the
minimal natural number $d$ such that all polynomials $P_k$
in Definition~\ref{Nash} can be chosen such that their degrees don't
exceed $d$.
Finally, under a {\it biregular} map between two algebraic
varieties we understand an isomorphism in sense of algebraic geometry.
Our main result is the following linearization criterion.
It will be proved in section~\ref{proof}.
\begin{Th}\label{main}
Let $D\subset\C^n$ be a algebraic domain and $G$ a group
of birational automorphisms of $D$. The following properties are equivalent:
\begin{enumerate}
\item $G$ is a subgroup of a Lie group $\hat G$ of birational automorphisms
of $D$ which extends the action of $G$ and has finitely many connected
components;
\item $G$ is a subgroup of a Nash group $\hat G$ of birational
automorphisms of $D$ which extends the action of $G$ to a Nash action
$\hat G\times D\to D$;
\item $G$ is a subgroup of a Nash group $\hat G$ such that the action
$G\times D\to D$ extends to a Nash action $\hat G\times D\to D$;
\item the degree of the automorphism $\phi_g\colon D\to D$
defined by $g\in G$ is bounded;
\item there exists a projective linearization,
i.e. a linear representation of $G$ in some $\C^{N+1}$ and
a biregular imbedding $i\colon \P^n \hookrightarrow \P^N$ such that
the restriction $i|_D$ is $G$-equivariant.
\end{enumerate}
\end{Th}
We finish this section with applications of Theorem~\ref{main}.
In the previous paper~(\cite{Z}) we gave sufficient conditions on $D$ and $G$
such that $G$ is a Nash group and the action $G\times D\to D$ is Nash.
The condition on $D$ is to be bounded and
to have a non-degenerate boundary in the following sense.
\begin{Def}\label{deg}
A boundary of a domain $D\subset\C^n$ is called {\bf non-degenerate}
if it contains a smooth point where the Levi-form is non-degenerate.
\end{Def}
The group $G$ is taken to be the group $Aut_a(D)$ of all
holomorphic Nash (algebraic) automorphisms of $D$. We proved in~\cite{Z} that,
if $D$ is a algebraic bounded domain with non-degenerate boundary,
the group $Aut_a(D)$ is closed in $Aut(D)$ and carries
a unique structure of a Nash group such that
the action $Aut_a(D)\times D\to D$ is Nash with respect to this structure.
Now let $G=Aut_b(D)\subset Aut_a(D)$ be the group of all birational
automorphisms of $D$. Then $G$ satisfies the property~3 in Theorem~\ref{main}
with $\hat G = Aut_a(D)$. By the property~2, $G$ is a subgroup of a Nash
group of birational automorphisms of $D$.
Since $G$ contains all the birational automorphisms of $D$, $G$ is itself
a Nash group with the Nash action on $D$. We obtain the following corollary.
\begin{Cor}\label{aut-b}
Let $D\subset\subset\C^n$ be a bounded algebraic domain with non-degenerate
boundary.
Then the group $Aut_b(D)$ of all birational automorphisms of
$D$ is Nash with the Nash action on $D$ which
admits a projective linearization, i.e. there exist a representation
of $Aut_b(D)$ in some $\C^{N+1}$ and
a biregular imbedding $i\colon \P^n \hookrightarrow \P^N$ such that
the restriction $i|_D$ is $Aut_b(D)$-equivariant.
\end{Cor}
Furthermore, S.~Webster (see \cite{W}) established the following
sufficient conditions on $D$ which make its automorphisms birational.
Let $D$ be a algebraic domain. The theory of semialgebraic sets
(see Benedetti-Risler~\cite{BR}) implies that the boundary
$\partial D$ is contained in finitely many irreducible real hypersurfaces.
Several of them, let say $M_1,\ldots,M_k$, have generically
non-degenerate Levi forms. If $\partial D$ is non-degenerate in sense of
Definition~\ref{deg},
such hypersurfaces exist. The complexifications ${\cal M}_i$'s
of $M_i$'s are defined
to be their complex Zariski closures in $\C^n\times\overline{\C^n}$ where
$M_i$'s are totally real imbedded via the diagonal map $z\mapsto (z,\bar z)$.
It follows that ${\cal M}_i$'s are the irreducible complex hypersurfaces.
Furthermore, the so-called Segre varieties $Q_{iw}$'s,
$w\in \C^n$ are defined by
$$ Q_{iw} := \{z\in \C^n \mid (z, \bar w) \in {\cal M}_i \}.$$
The complexifications and Segre varieties are the important biholomorphic
invariants of a domain $D$ and play a decisive role in the reflection
principle.
Now we are ready to formulate the conditions of S.~Webster.
\begin{Def}
A algebraic domain is said to satisfy the condition $(W)$
if for all $i$ the Segre varieties
$Q_{iw}$ uniquely determine $z\in\C^n$ and $Q_{iw}$ is an irreducible
hypersurface in $\C^n$ for all $z$ off a proper subvariety $V_i\subset\C^n$.
\end{Def}
The Theorem of S.~Webster (see \cite{W}, Theorem~3.5) can be formulated
in the following form:
\begin{Th}\label{bir}
Let $D\subset\C^n$ be a algebraic domain with non-degenerate boundary
which satisfies the condition $(W)$. Further,
let $f\in Aut(D)$ be an automorphism which is holomorphically
extendible to a smooth boundary point
with non-degenerate Levi-form. Then $f$ is birationally extendible to the
whole $\C^n$.
\end{Th}
Since every Nash automorphism $f\in Aut_a(D)$ extends holomorphically
to generic boundary points, we obtain the following Corollary.
\begin{Cor}
Let $D\subset\C^n$ be a bounded algebraic domain which satisfies
the condition $(W)$. Then the whole
group $Aut_a(D)$ is projective linearizable,
i.e. there exist a representation
of $Aut_a(D)$ in some $\C^{N+1}$ and
a biregular imbedding $i\colon \P^n \hookrightarrow \P^N$ such that
the restriction $i|_D$ is $Aut_a(D)$-equivariant.
\end{Cor}
To obtain the extendibility of the whole group $Aut(D)$ of holomorphic
automorphisms, we consider
the {\it algebraic} domains in sense of Diederich-Forn\ae ss (see \cite{DF}).
\begin{Def}
A domain $D\subset\subset C^n$ is called {\bf algebraic} if there exists
a real polynomial $r(z,\bar z)$ such that $D$ is a connected component of the
set $$\{z\in\C^n \mid r(z,\bar z)<0 \}$$
and $dr(z)\ne 0$ for $z\in\partial D$.
\end{Def}
The following fundamental result for such domains is due to
K.~Diederich and J.~E.~Forn\ae ss (see \cite{DF}).
\begin{Th}\label{hol}
Let $D\subset\subset\C^n$ be an algebraic domain. Then $Aut_a(D)=Aut(D)$.
\end{Th}
Thus we obtain the linearization of the whole automorphism group $Aut(D)$.
\begin{Th}\label{alg}
Let $D\subset\subset \C^n$ be an algebraic domain which satisfies
the condition $(W)$. Then the group $Aut(D)$ is projective linearizable,
i.e. there exist a representation
of $Aut(D)$ in some $\C^{N+1}$ and
a biregular imbedding $i\colon \P^n \hookrightarrow \P^N$ such that
the restriction $i|_D$ is $Aut(D)$-equivariant.
\end{Th}
Further corollaries are devoted to the constructions of complexifications.
\section{Complexifications}
Using the linearization criterion we establish here existences of
complexifications.
To every real Lie group $G$ one can associate its complexification
(see Hochschild~\cite{Ho}) defined as follows.
\begin{Def}\label{cpx}
Let $G$ be a real Lie group. A complex Lie group $G^{\C}$ together with
a Lie homomorphism $\imath\colon G \to G^{\C}$ is called a
{\bf complexification} of $G$ if for a given Lie homomorphism
$\phi$ from $G$ into a complex Lie group $H$, there exists exactly one
holomorphic Lie homomorphism $\phi^{\C}\colon G^{\C} \to H$ such that
$\phi=\phi^{\C}\circ\imath$.
A real Lie group $G$ is called {\bf holomorphically extendible} if the map
$\imath\colon G\to G^{\C}$ is injective.
\end{Def}
A complexification always exists and is unique up to biholomorphisms
(see Hochschild~\cite{Ho} and Heinzner~\cite{He} and \cite{He1}).
Further, one defines the complexification of an action
(see Heinzner~\cite{He}).
\begin{Def}\label{G-cpx}
Let a real Lie group $G$ act on a complex space $X$ by holomorphic
automorphisms. A complex space $X^{\C}$ together with a holomorphic action
of $G^{\C}$ and a $G$-equivariant map $\imath\colon X\to X^{\C}$ is called
a $G$-{\bf complexification} of $X$ if to every holomorphic $G$-equivariant
map $\phi\colon X\to Y$ into another complex space $Y$ with a holomorphic
action of $G^{\C}$ there exists exactly one holomorphic $G^{\C}$-equivariant
map $\phi^{\C}$ such that $\phi=\phi^{\C}\circ\imath$.
\end{Def}
A $G$-complexification is unique up to biholomorphic $G^{\C}$-equivariant
maps provided it exists. P.~Heinzner proved in \cite{He} the existence of
a $G$-complexification of $X$ with properties that $\imath\colon X\to X^{\C}$
is an open imbedding
and $X^{\C}$ is Stein in case $G$ is compact and $X$ is a Stein space.
Now the projective linearization in Theorem~\ref{main} implies
the existence of complexifications in our situation.
\begin{Cor}\label{main1}
Let $D\subset\C^n$ be a algebraic domain and $G$ a Lie group
of birational automorphisms of $D$ which satisfies one of the equivalent
properties in Theorem~\ref{main}. Then the group $G$ is
holomorphically extendible and there exists a smooth $G$-complexification
$D^{\C}$ of $D$ such the map $\imath\colon D\to D^{\C}$ is an open imbedding.
\end{Cor}
{\bf Proof.} By property~5 in Theorem~\ref{main}, $G$ is a subgroup of the
complex Lie group $GL_N(\C)$. By Definition~\ref{cpx}, $G$ is
holomorphically extendible. Let $i$ be the embedding of
$\C^n\supset D$, given by Theorem~\ref{main}. Since the stability group
$H\subset GL_N(\C)$ of the complex projective variety
$X:=i(\P^n)$ is a complex Lie group and $G\subset H$,
the Definition~\ref{cpx} yields a holomorphic
action of $G^{\C}$ on $X$. We claim that
$D^{\C}:=G^{\C}\cdot D\subset X$ is the required $G$-complexification
of $D$. Indeed, let $\phi\colon D\to Y$ be a $G$-equivariant holomorphic
map into another complex space $Y$ with a holomorphic
action of $G^{\C}$. To define the required in Definition~\ref{G-cpx}
map $\phi^{\C}$ we take a point $z\in D^{\C}$ which is always of the form
$z=Ax$ with $A\in G^{\C}$ and $x\in D$. Then we set
$\phi^{\C}(z):=A\phi(x)$. Why is $\phi^{\C}(z)$ independent of the
representation $z=Ax$? Because the holomorphic map $A\mapsto A\phi(x)$ is
determined by values on the maximal totally real subgroup $G$:
for $A\in Aut_b(D)$ one has $A\phi(x)=\phi(Ax)$. We obtain a well-defined
$G^{\C}$-equivariant map $\phi^{\C}(z)\colon D\to D^{\C}$ with
the property $\phi=\phi^{\C}\circ\imath$ (because for $z\in D$ one can
choose $A=1$). The holomorphicity of $\phi^{\C}$ is obtained by fixing $A$
in the formula $\phi^{\C}(z):=A\phi(x)$. \hfill $\square$
For the algebraic domains we obtain the following Corollaries.
\begin{Cor}
Let $D\subset\subset\C^n$ be a algebraic domain with non-degenerate boundary.
Then the group $Aut_b(D)$ is
holomorphically extendible and there exists an $Aut_b(D)$-complexification
of $D$.
\end{Cor}
\begin{Cor}
Let $D\subset\subset\C^n$ be a algebraic domain with non-degenerate boundary
which satisfies the condition $(W)$.
Then the group $Aut_a(D)$ is
holomorphically extendible and there exists an $Aut_a(D)$-complexification
of $D$.
\end{Cor}
\begin{Cor}
Let $D\subset\subset \C^n$ be an algebraic domain which satisfies
the condition $(W)$. Then the group $Aut(D)$ is
holomorphically extendible and there exists an $Aut(D)$-complexification
of $D$.
\end{Cor}
\section{Proof of the main Theorem}\label{proof}
Let $D$ be a algebraic domain and $G$ a group of birational automorphisms
of $D$. We prove the equivalence of the properties in Theorem~\ref{main}
in the direction of the following two chains:
$2\Longrightarrow 3\Longrightarrow 4\Longrightarrow 5\Longrightarrow 2$
and $2\Longrightarrow 1\Longrightarrow 4$.
\fbox{$2\Longrightarrow 3$.} The proof is trivial. \hfill $\square$
\bigskip
\fbox{$3\Longrightarrow 4$.} Let $G$ be a subgroup of
a Nash group $\hat G$ such that the action
$G\times D\to D$ extends to a Nash action $\hat G\times D\to D$.
We prove the statement for arbitrary Nash manifold $\hat G$ and
Nash map $\hat G\times D\to D$
by induction on $\dim G$. It is obvious for $\dim G=0$.
Let $U\subset \hat G$ be a Nash coordinate chart and
$\phi_i(g)\colon D\to \R$ be the $i$th coordinate of $\phi_g\colon D\to D$
for $g\in U$. Since the map $\phi_i\colon U\times D\to \R$ is Nash, it
satisfies a polynomial equation
$P(g,x,\phi_i(g,x))\equiv 0$. This yields polynomial equations of the same
degree for all $g\in U$ outside a proper Nash submanifold. This submanifold
has lower dimension and the statement is true for it by induction.
In summary, we obtain the boundness of the degree for the whole
neighborhood $U$ and, since the Nash atlas is finite, for $G$. \hfill$\square$
\bigskip
\fbox{$4\Longrightarrow 5$.} Here is a sketch of the proof.
The idea is to imbed the group
$G$ into a complex algebraic variety so that the action on $D$
is given by a rational mapping. Using this mapping we construct
a collection of homogeneous polynomials on $\C^{n+1}$ which generate
a finite-dimensional linear subspace, invariant with respect to
the action of $G$. These polynomials
yield the required projective linearization.
The imbedding of $G$ is obtained by associating to every element
$g\in G$ the Chow coordinates of the complex Zariski closure of the graph of
the automorphism defined by $g$ (see Shafarevich, \cite{S} ,page 65).
The main problem here is that the Chow scheme $C$ has infinitely many disjoint
components parameterized by dimensions and degrees of subvarieties.
In order to concern finitely many components of $C$ we have required
the degree of automorphisms $\phi_g\colon D\to D$ to be bounded.
To every $g\in G$ one associates the $n$-dimensional
(complex) Zariski closure $\tilde\Gamma_g\subset \P^n\times\P^n$ of the
graph $\Gamma_g\subset D\times D$ of the automorphism defined by
$g$. To regard $\tilde\Gamma_g$ as a
subvariety of some $P^N$ let us consider the Segre imbedding:
\begin{equation}\label{segre}
v([z_0,...,z_n],[w_0,...,w_n])=[{z_iw_j}]_
{0 \le i \le n, 0 \le j \le n},
\end{equation}
$$v\colon \P^n\times \P^n \to \P^{n^2+2n}.$$
We set $N=n^2+2n$ and obtain a family of
subvarieties $\rho(g):=v(\tilde\Gamma_g)\subset \P^N$ parameterized by $g\in
G$.
The family $V$ of all algebraic subvarieties of $\P^n$ is parameterized
by the Chow scheme $C$ (see Shafarevich, \cite{S} ,page 65).
Different automorphisms $g\in G$ define different subvarieties
$v(\tilde\Gamma_g)\subset \P^N$ and one obtains an imbedding $\rho$of
$G$ in the Chow scheme $C$.
The (complex) dimension of the subvarieties $v(\tilde\Gamma_g)$ is $n$.
The degree of $v(\tilde\Gamma_g)\subset\P^N$ is the
intersection number with $N-n$ generic linear hyperplanes
$\{L_1=0\},\ldots,\{L_{N-n}=0\}$. It is equal to the intersection number of
$\tilde\Gamma_g\subset\P^N\times\P^N$ with divisors $v^*L_1,\ldots,v^*L_{N-n}$.
By the Bezout theorem this intersection number is bounded.
Thus, $G$ lies in fact in finitely
many components of the Chow scheme $C$. Let $C_0$ denote the union of
these components and $V_0$ the corresponding family of subvarieties of $P^N$.
We obtain an imbedding $\rho$ of $G$ in a complex projective variety $C_0$.
\begin{Lemma}
The imbedding $\rho\colon G\to C_0$ is continuous.
\end{Lemma}
{\bf Proof.} Assume the contrary. Then there exists a sequence
$g_n\to g$ in $G$ such that no subsequence $\rho(g_{n(k)})$ converges
to $\rho(g)$. On the other hand, since the degree of $\rho(g_n)$ is bounded,
there exists a subsequence $\rho(g_{n(k)})$ which converges in $C_0$.
This follows from the Theorem of Bishop (see e.g. F.~Campana, \cite{Cam}).
Let $A\in C_0$ be the limes cycle of this subsequence. Our cycles lie
in $v(\P^n\times\P^n)$ and we identify them with the preimages in
$\P^n\times\P^n$. Since the action $G\times D\to D$ is continuous,
the cycle $A$ contains the graph of $\phi_g$ and therefore its Zariski
closure $\rho(g)$.
We claim that $A=\rho(g)$. This yields a contradiction with the choice
of $\rho(g_{n(k)})$. Indeed, otherwise there exists
a horizontal of vertical $n$-dimensional projective subspace
$H\subset\P^n\times\P^n$ ($H=\{z\}\times\P^n$ or $H=\P^n\times\{w\}$)
such that the intersection number of $A$ and $H$ is more than one.
Then the intersection number of $\rho(g_{n(k)})$ and $H$ is also more than
one which contradicts to the birationality of $\phi_{g_{n(k)}}$. \hfill
$\square$
Further let $\tilde G$ be the complex Zariski closure of $G$ in $C_0$.
\begin{Lemma}
The action $\phi\colon G\times D\to D$ extends to a rational map
$\tilde\phi \colon \tilde G\times \P^n \to \P^n$.
\end{Lemma}
{\bf Proof.} We begin with the construction of the graph
$\Gamma_{\tilde\phi} \subset \tilde G\times \P^n \to \P^n$ of $\tilde\phi$.
For this we regard $\P^n \times \P^n$ as a subset of $P^N$
(via the Segre imbedding $v$ in (\ref{segre})). We then define
$\Phi = \Gamma_{\tilde\phi}$ to be the intersection of the Chow family $V_0$
with $\tilde G\times \P^n \to \P^n$. This is a complex algebraic variety.
Moreover, for $g\in G$ and $x\in D$ the fibre $\Phi_{(g,x)}\subset\P^n$
consists of the single point $g(x)$. Since the set $G\times D$ is
Zariski dense in $\tilde G\times \P^n$, this is true for every generic fibre of
$\Phi$. This means that $\Phi$ is the graph of a rational map
$\tilde\phi \colon \tilde G\times \P^n \to \P^n$. \hfill $\square$
The projective variety $\tilde G$ is imbedded in a projective space $\P^m$.
The map $\tilde\phi$ can be extended to a rational map from
$\P^m \times \P^n$ into $\P^n$. Such map is given by $n+1$ polynomials
$P_1(x,y),\dots,P_{n+1}(x,y)$, homogeneous separately in
$x\in\C^{m+1}$ and $y\in\C^{n+1}$.
Let $h$ be a fixed homogeneous polynomial on $\C^{n+1}$. Then the function
$$(x,y)\mapsto h(P_1(x,y),\ldots,P_{n+1}(x,y))$$
is a separately homogeneous polynomial on $\C^{m+1} \times \C^{n+1}$.
The algebra $\C_h[x,y]$ of such polynomials is equal to the tensor
product $\C_h[x]\otimes\C_h[y]$. Therefore there exist polynomials
$\varphi_i\in \C_h[x]$, $\psi_i\in \C_h[y]$, $i=1,\ldots,l$ such that
$$h(P_1(x,y),\ldots,P_{n+1}(x,y)) = \sum_{i=1}^l \varphi_i(x) \psi_i(y).$$
For $x=g\in G$ fixed we obtain
$$\alpha_*(f^{-1}) h = \sum_{i=1}^l c_i \psi_i(y),$$
where $\alpha_*$ denotes the associated action of $G$ on homogeneous
polynomials. In other words, the orbit of $h$ via the action of $G$
is contained in the finite-dimensional subspace
$<\psi_1,\ldots,\psi_l> \subset \C_h[y]$. The linear hull of this orbit
is a finite-dimensional $G$-invariant subspace containing $h$.
We choose mow sufficiently many polynomials $h_j$, $j=1,\ldots,s$ which
separate the points of $\C^{n+1}$ and such that neither $h_j$ nor
the differentials
$dh_j$ nowhere vanish simultaneously. They lie in a finite-dimensional
$G$-invariant subspace $L\subset \C_h[y]$. Let $(p_1,\ldots,p_{N+1})$
be a collection of homogeneous polynomials which yields a basis of $L$.
The required representation of $G$ is the action on $L$ and the polynomial
map $(p_1,\ldots,p_{u+1})\colon \C^{n+1}\to \P^N$
defines the required projective linearization. \hfill $\square$
\bigskip
\fbox{$5\Longrightarrow 2$.} Assume we are given a projective linearization
of the action of $G$
on $D$. It follows that the given representation of $G$ is faithful and
we identify $G$ with its image in $GL_{N+1}(\C)$.
We define now the group $\hat G\supset G$ to be the subgroup of all
$g\in GL_{N+1}$ such that $g(i(D))=i(D)$. It follows that $G\subset\hat G$.
We wish to prove that $\hat G$ is a Nash subgroup of
$GL_{N+1}(\C)$. For the proof we use the technique of
semialgebraic sets and maps which are closely related to the Nash
manifolds and maps.
The semialgebraic subsets of $\R^n$ are the sets of the form
$\{P_1=\cdots=P_k, Q_1<0,\ldots,Q<s\}$ and finite unions of them where
$P_1,\ldots,P_k$ and $Q_1,\ldots,Q_s$ are real polynomials on $\R^n$.
More generally, the semialgebraic subsets of a Nash manifold $M$ are the
subsets which have semialgebraic intersections with every Nash coordinate
chart.
The semialgebraic maps between semialgebraic sets are any maps with
semialgebraic graphs. The Nash submanifolds of $\R^n$ are exactly
semialgebraic real analytic submanifolds and the Nash maps are
semialgebraic real analytic maps.
Now the graph $\Gamma\subset Gl_{N+1}\times i(D) \to \P^N$ of the restriction
to $i(D)$ of the linear action of $GL_{N+1}$ on $\P^N$ is a semialgebraic
subset. The condition $g(i(D))=i(D)$ on $g$ defines a semialgebraic subset
$\hat G\subset Gl_{N+1}$. We proved this in the previous paper
(see \cite{Z}, Lemma~6.2). Since $\hat G$ is a subgroup, it is Nash.
Thus, $G$ in a subgroup of the Nash group $\hat G$ of birational automorphisms
of $i(D)\cong D$ with required properties.\hfill $\square$
\bigskip
\fbox{$2\Longrightarrow 1$.} A Nash open subset of $\R^n$ is semialgebraic
and has therefore finitely many connected components
(see Benedetti-Risler, \cite{BR}, Theorem~2.2.1). The Nash group $\hat G$
admits a finite Nash atlas and has also finitely many components.\hfill
$\square$
\bigskip
\fbox{$1\Longrightarrow 4$.} Assume $G$ is a subgroup of a Lie group $\hat G$
of birational automorphisms of $D$ with finitely many connected components.
Consider the complex coordinates $\phi_i\colon \hat G\times D\to \C$ of the
action of $\hat G$. For fixed $g\in \hat G$ the map
$\phi_i(g)\colon D\to \C$ extends to a rational map
$\tilde\phi_i(g)\colon \C^n\to \C$. These extensions define a map
$\tilde\phi_i\colon \hat G\times\C^n\to \C$.
A priori we don't
know whether this new map is real analytic or even continuous.
To prove this we use the following result of Kazaryan (\cite{Ka}):
\begin{Prop}\label{Kaz}
Let $D'$ be a domain in $\C^n$ and let $E\subset D'$ be a
nonpluripolar\footnote{a subset $E\subset D'$ is called {\it nonpluripolar}
if
there are no plurisubharmonic functions $f\colon D'\to \R\cup\{-\infty\}$
such that $f|_E\equiv -\infty$}
subset. Let $D''$ be an open set in a
complex manifold $X$. If $f$ is a meromorphic function on $D'\times D''$
such that $f(g,\cdot)$ extends to a meromorphic function on $X$
for all $g\in E$, then $f$ extends to a meromorphic function in
a neighborhood of $E\times X\subset D'\times X$.
\end{Prop}
\begin{Lemma}
The map $\tilde\phi_i\colon \hat G\times\C^n\to \C$ is real analytic.
\end{Lemma}
{\bf Proof.} The question of real analyticity of
$\tilde\phi_i$ is local with respect to $\hat G$ so we can take a
real analytic coordinate neighborhood $E$ in $\hat G$, regarded as
an open subset of $\R$ . The map $\phi_i$ is real analytic
in $E\times D$ and extends therefore to a holomorphic function
in a neighborhood $D'\times D''$ of $E\times D''$ in the complex
manifold $\C\times X$. Here we must replace $D$ by a bit smaller
neighborhood $D''\subset D$.
The set $E$, being an open subset of $\R$ , is nonpluripolar.
By Proposition~\ref{Kaz}, $\phi_i$ extends to a
meromorphic functions in a neighborhood of $E\times X$.
The restriction $\tilde\phi_i$ is therefore real analytic. \hfill $\square$
According to the
construction of Chow scheme (see Shafarevich, \cite{S},p.65)
every graph $\Gamma_{\tilde\phi_i}\subset\C^{2n}\subset\P^{2n}$ has
its Chow coordinate in the Chow scheme C. The Chow
coordinates yield a continuous mapping $f\colon \hat G \to C$. This is
the universal property of the Chow scheme. It follows from
the Theorems of D. Barlet on universality of the Barlet
space and on the equivalence of the latter to the Chow
scheme in case of projective space (see Barlet, \cite{B}).
Since $\hat G$ has finitely many components, the image in $C$ is
has also this property. But the degree of variety is constant on
the components of $C$. This implies that the degree of the variety in
$\P^{2n}$ associated to $g\in \hat G$ is bounded. This implies that
the degrees of defining polynomials are bounded and the statement is proven.
\hfill $\square$
|
1996-03-08T06:52:37 | 9410 | alg-geom/9410009 | en | https://arxiv.org/abs/alg-geom/9410009 | [
"alg-geom",
"math.AG"
] | alg-geom/9410009 | null | David B. Jaffe | Coherent functors, with application to torsion in the Picard group | 46 pages, AMS-LaTeX | null | null | null | null | Let A be a commutative noetherian ring. Call a functor <<commutative
A-algebras>> --> <<sets>> coherent if it can be built up (via iterated finite
limits) from functors of the form B \mapsto M tensor_A B, where M is a f.g.
A-module. When such a functor F in fact takes its values in <<abelian groups>>,
we show that there are only finitely many prime numbers p such that _p F(A) is
infinite, and that none of these primes are invertible in A. This (and related
statements) yield information about torsion in Pic(A). For example, if A is of
finite type over Z, we prove that the torsion in Pic(A) is supported at a
finite set of primes, and if _p Pic(A) is infinite, then the prime p is not
invertible in A. These results use the (already known) fact that if such an A
is normal, then Pic(A) is finitely generated. We obtain a parallel result for a
reduced scheme X of finite type over Z. We show that the groups which can occur
as the Picard group of a scheme of finite type over a finite field all have the
form (finitely generated) + sum_{n=1}^infty F, where F is a finite p-group.
Hard copy is available from the author. E-mail to [email protected].
| [
{
"version": "v1",
"created": "Wed, 12 Oct 1994 20:53:28 GMT"
}
] | 2015-06-30T00:00:00 | [
[
"Jaffe",
"David B.",
""
]
] | alg-geom | \section{#1}}
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\par\noindent {\footnotesize Department of Mathematics and Statistics,
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\par\noindent {\footnotesize Lincoln, NE 68588-0323, USA\ \ (jaffe{\kern0.5pt}@{\kern0.5pt}cpthree.unl.edu)}}
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\footnote{{\it 1991 Mathematics Subject Classification.}
Primary: 14C22, 18A25, 14K30, 18A40.}
\footnote{{\it Key words and phrases.\/} Coherent functor, representable
functor, Picard group.}
\footnote{Partially supported by the National Science Foundation.}
\def\arabic{footnote}}\setcounter{footnote}{0{\fnsymbol{footnote}}\setcounter{footnote}{0}
\par\noindent{\bf\LARGE Coherent functors, with application
to torsion in the Picard group}
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\vspace*{15pt}
\par\noindent David B.\ Jaffe \makeaddress
\vspace*{0.2in}
\medskip
\par\noindent{\bf\large Abstract}
\medskip
Let $A$ be a commutative noetherian ring. We investigate a class of functors
from \cat{commutative $A$-algebras} to \cat{sets}, which we call
{\it coherent}. When such a functor $F$ in fact takes its values in
\cat{abelian groups}, we show that there are only finitely many prime numbers
$p$ such that ${}_p F(A)$ is infinite, and that none of these primes are
invertible in $A$. This (and related statements) yield information about
torsion in $\mathop{\operatoratfont Pic}\nolimits(A)$. For example, if $A$ is of finite type over $\xmode{\Bbb Z}$, we
prove that the torsion in $\mathop{\operatoratfont Pic}\nolimits(A)$ is supported at a finite set of primes,
and if ${}_p \mathop{\operatoratfont Pic}\nolimits(A)$ is infinite, then the prime $p$ is not invertible in $A$.
These results use the (already known) fact that if such an $A$ is normal, then
$\mathop{\operatoratfont Pic}\nolimits(A)$ is finitely generated. We obtain a parallel result for a reduced scheme $X$ of
finite type over $\xmode{\Bbb Z}$. We classify the groups which can occur as the Picard
group of a scheme of finite type over a finite field.
\medskip
\par\noindent{\bf\large Coherent functors (introductory remarks)}
\medskip
Let us say that an {\it $A$-functor\/} is a functor from the category of
commutative $A$-algebras to \cat{sets}. Some such $A$-functors have
additional structure: they are actually functors from
\cat{commutative $A$-algebras} to \cat{groups}. We refer to such functors as
{\it group-valued\/} $A$-functors. We will also consider $A$-functors $F$
such that $F(B)$ is a $B$-module for every $B$; these {\it module-valued\/}
$A$-functors are discussed later in the introduction. For now, all
$A$-functors which we consider will be treated as set-valued functors.
An $A$-functor is {\it coherent\/} if it may be built up as an iterated finite
limit of functors of the form ${\underline{M}}$, given by ${\underline{M}}(B) = M \o*_A B$, where $M$
is a finitely generated\ $A$-module. We do not know if every coherent functor may be
expressed as a finite limit of such functors ${\underline{M}}$. However, the analogous
question regarding module-valued\ functors is answered affirmatively below.
The idea of {\it coherent functor\/} was originally devised by Auslander
\Lcitemark 5\Rcitemark \Rspace{}, in a somewhat different setting; his
notion of coherence applied to functors from an abelian category to
\cat{abelian groups}. Later Artin\Lspace \Lcitemark 2\Rcitemark \Rspace{}
transposed Auslander's notion to a setting closer to that given here. Artin
also raised a question about coherence of higher direct images as functors.
This question is considered in \S\ref{higher-section}.
If an $A$-functor is representable by a commutative $A$-algebra of finite
type, then it is coherent. There are many examples of non-representable
$A$-functors which are coherent. For example, if $M$ is a finitely generated\ $A$-module,
then $B \mapsto \mathop{\operatoratfont Aut}\nolimits_{B-{\operatoratfont mod}}(M \o*_A B)$ defines a coherent $A$-functor.
More examples may be found in \S\ref{examples-section}.
A {\it module-valued\ $A$-functor\/} is an (abelian group)-valued $A$-functor $F$,
together with the following additional structure: for each commutative
$A$-algebra $B$, $F(B)$ has the structure of a $B$-module, such that for any
homomorphism \mapx[[ B_1 || B_2 ]] of commutative $A$-algebras, the induced
map \mapx[[ F(B_1) || F(B_2) ]] is a homomorphism of $B_1$-modules.
The module-valued\ $A$-functors form an abelian category.
A module-valued\ $A$-functor $F$ is {\it module-coherent\/} if there exists a homomorphism
\mp[[ f || M || N ]] of finitely generated\ $A$-modules such that $F$ is isomorphic to the
module-valued\ $A$-functor given by $B \mapsto \ker(f \o*_A B)$. The module-coherent\ $A$-functors
form a full subcategory of the category of module-valued\ $A$-functors.
Most examples of module-valued\ $A$-functors are induced naturally by functors from
\cat{$A$-modules} to \cat{$A$-modules}. Certainly for many purposes it makes
more sense to study the latter sort of functor. On the other hand, (as pointed
out by Artin\Lspace \Lcitemark 2\Rcitemark \Rspace{}) one can set up a
correspondence
between module-coherent\ $A$-functors and functors from \cat{$A$-modules} to
\cat{$A$-modules} which satisfy an analogous coherence axiom. This creates a
bridge to the ideas of Auslander\Lspace \Lcitemark 5\Rcitemark \Rspace{} and
Grothendieck (\Lcitemark 18\Rcitemark \ \S7). We have not exploited this point
of view.
A key result is that if \mp[[ \sigma || F || G ]] is a
morphism of module-coherent\ $A$-functors, then $\ker(\sigma)$ and $\mathop{\operatoratfont coker}\nolimits(\sigma)$ are
module-coherent. (These are to be computed in \cat{module-valued\ $A$-functors}.) One deduces
easily from this that any finite limit or finite colimit of module-coherent\ $A$-functors
is module-coherent. In particular, the iterated finite limit construction which we used
in the definition of {\it coherent\/} $A$-functor is not necessary here,
although in the body of the paper we find it convenient to begin with a
definition of module-coherent\ which uses iterated finite limits.
Let $X$ be a noetherian scheme. An {\it $X$-functor\/} is a functor from
\hskip 0pt plus 4cm\penalty1000\hskip 0pt plus -4cm\hbox{\opcat{$X$-schemes}} to \cat{sets}. We define the notion of
coherent $X$-functor by analogy with the definition for $A$-functors. When
$X = \mathop{\operatoratfont Spec}\nolimits(A)$, the theory of coherent $X$-functors is identical to the theory
of coherent $A$-functors. Similarly, we define module-coherent\ $X$-functors. We show
that the property of being a module-coherent\ $X$-functor is local on $X$, assuming that
$X$ is separated. We conjecture that the property of being a coherent
$X$-functor is local on $X$.
\medskip
\par\noindent{\bf\large Finiteness theorems (introductory remarks)}
\medskip
If $F$ is an (abelian group)-valued coherent $X$-functor, we prove that there
are only finitely many primes $p$ such that ${}_p F(X)$ is infinite, and that
none of these primes are invertible in $\Gamma(X,{\cal O}_X)$. A stronger form of
this statement holds if $X$ is essentially of finite type over $\xmode{\Bbb Z}$ or over
$\xmode{\Bbb Z}_p$ for some prime $p$. For example, if $X$ is of finite type over ${\Bbb Q}\kern1pt$ or
over ${\Bbb Q}\kern1pt_p$, then the torsion subgroup of $F(X)$ is finite.
Assuming that $X$ is reduced and that the canonical map \mapx[[ \nor{X} || X ]]
is finite, consider the quotient sheaf $F = {\cal O}_{\nor{X}}^*/{\cal O}_X^*$ on $X$. We
extend $F$ to an (abelian group)-valued $X$-functor, also denoted here by $F$.
We do not know if $F$ is coherent, but we are able (more or less) to find an
(abelian group)-valued coherent $X$-functor $G$, and a morphism
\mp[[ \psi || F || G ]] such that $\psi(X)$ is injective.
We say ``more or less'' because the actual proof works via a
sequence of partial normalizations. The end result however is the same:
there are only finitely many primes $p$ such that ${}_p F(X)$ is infinite,
and none of these primes are invertible in $\Gamma(X,{\cal O}_X)$. There is also a
stronger form for certain $X$ as discussed in the previous paragraph.
It follows that if the group $Q = \Gamma({\cal O}_{\nor{X}}^*)/\Gamma({\cal O}_X^*)$ is
finitely generated, and if $K = \mathop{\operatoratfont Ker}\nolimits[\mathop{\operatoratfont Pic}\nolimits(X)\ \mapE{}\ \mathop{\operatoratfont Pic}\nolimits(\nor{X})]$, then there are only
finitely many primes $p$ such that ${}_p K$ is infinite, and none of these
primes are invertible in $\Gamma(X,{\cal O}_X)$. This holds for instance if $X$ is
of finite type over $\xmode{\Bbb Z}$, thereby yielding one of the results stated in the
summary.
When $X$ is of finite type over a field $k$, it has been shown
\Lcitemark 21\Rcitemark \Rspace{} that ${}_n \mathop{\operatoratfont Pic}\nolimits(X)$ is finite for every $n$
which is invertible in $k$. For
a finite field $k$ of characteristic $p$, we prove a strengthened form of this
statement: modulo $p$-power torsion, $\mathop{\operatoratfont Pic}\nolimits(X)$ is finitely generated. We completely describe
the structure of $\mathop{\operatoratfont Pic}\nolimits(X)$ as an abstract abelian group.
The following related result is relevant. Claborn\Lspace \Lcitemark
12\Rcitemark \Rspace{} has shown that
every abelian group occurs as the Picard group of some Dedekind domain over
${\Bbb Q}\kern1pt$. (See also\Lspace \Lcitemark 15\Rcitemark \Rspace{}\ \S14.) In
particular, for suitable $X$,
$\mathop{\operatoratfont Pic}\nolimits(X)$ itself has infinite $n$-torsion, for every $n$.
It would be interesting to know to what extent the results of this paper
on $\mathop{\operatoratfont Pic}\nolimits(X)$ can be obtained via \'etale cohomology.
\vspace{0.1in}
\par\noindent{\footnotesize{\it Acknowledgements.} I thank Deligne
and Ogus for much help on \S\ref{higher-section}.}
\vspace*{0.1in}
\par\noindent{\bf Conventions}
\
\begin{itemize}
\item $A$ denotes an arbitrary commutative noetherian ring (unless specified
otherwise); $B$ usually denotes an arbitrary commutative $A$-algebra;
\item $X$ denotes an arbitrary noetherian scheme;
\item By a {\it Zariski sheaf}, we mean a sheaf for the Zariski topology.
\item If $S$, $T$ are sets, and $M$ is an abelian group, by a
{\it left exact sequence}
\diagramx{S&\rightarrowtail&T&\mapE{f}&M\cr%
}we mean that the map from $S$ to $T$ is injective, and that $S$ is the
kernel of the map from $T$ to $M$, meaning that
$S = \setof{t \in T: f(t) = 0}$. Similar language applies when $S$, $T$,
and $M$ are functors.
\end{itemize}
\block{Coherent functors}
In this section we develop the basic theory of coherent functors.
We have already defined the notion of {\it $A$-functor\/} in the introduction.
These form a category \cat{$A$-functors} whose morphisms
are natural transformations.
If $M$ is an $A$-module, then there is an $A$-functor ${\underline{M}}$ given by
${\underline{M}}(B) = M \o*_A B$. If a given $A$-functor $F$ is isomorphic to ${\underline{M}}$
for some finitely generated\ $A$-module $M$, we shall say that $F$ is {\it strictly coherent}.
\begin{definition}
Let ${\cal{C}}$ be a category. Let $S$ be a collection of objects in ${\cal{C}}$.
Let $S_0 = S$, and for each $n \geq 0$, let $S_{n+1}$ be the collection of
all objects of ${\cal{C}}$, which may be obtained as limits (in ${\cal{C}}$) of diagrams
involving finitely many objects in $S_n$ and finitely many morphisms. Let
$S_\infty = \cup_{n=0}^\infty S_n$. Then we say that the objects in $S_\infty$
are {\it iterated finite limits\/} of objects in $S$.
\end{definition}
\begin{definition}
An $A$-functor is {\it coherent\/} if it may be obtained as an iterated finite
limit of strictly coherent $A$-functors, where the limits are taken
in \cat{$A$-functors}.
\end{definition}
We may define $\cat{coherent $A$-functors}$: it is a full subcategory of
\cat{$A$-functors}, which may be thought of as the
{\it finite completion\/} of the subcategory
\cat{strictly coherent $A$-functors} of \cat{$A$-functors}.
Let ${\cal{C}}_A^0$ denote the collection of strictly coherent
$A$-functors. For each $n \geq 0$, let ${\cal{C}}_A^{n+1}$ denote the
collection of $A$-functors which may be obtained as finite limits (inside
\cat{$A$-functors}) of objects in ${\cal{C}}_A^n$. We have:
$${\cal{C}}_A^0 \subset {\cal{C}}_A^1 \subset {\cal{C}}_A^2 \subset \cdots.$%
$Let ${\cal{C}}_A = \cup_{n=0}^\infty {\cal{C}}_A^n$. Then the objects in
${\cal{C}}_A$ are exactly the coherent $A$-functors.
\begin{definition}
Let $F$ be a coherent $A$-functor. Then the {\it level\/} of $F$ is the
smallest integer $n$ such that $F \in {\cal{C}}_A^n$.
\end{definition}
In some proofs, we will need to induct on the level of a given coherent
$A$-functor. This process will be facilitated by the following lemma, whose
proof is left to the reader:
\begin{lemma}\label{lesx-exists}
Let $F$ be a coherent $A$-functor of level $n \geq 1$. Then there exists
a left exact sequence:
\lesx{F}{G}{{\underline{M}}%
}in which $G$ is a coherent $A$-functor of level $n-1$ and $M$ is a
finitely generated\ $A$-module. Moreover, $F$ may be embedded as a
subfunctor of a strictly coherent $A$-functor.
\end{lemma}
If $k$ is a field, then every coherent $k$-functor is representable, and from
this one deduces easily that every coherent $k$-functor has level $\leq 1$.
Later \pref{main-theorem} we shall prove that a large class of coherent
$k$-functors have level $\leq 1$. However, we do not know the answer to the
following basic question:
\begin{problemx}
Does every coherent $A$-functor have level $\leq 1$?
\end{problemx}
Note that for a given $A$-functor $F$, this is the case
if and only if\ there exist finitely generated\ $A$-modules $M$ and $N$,
together with a morphism \mp[[ \phi || {\underline{M}} || {\underline{N}} ]] of $A$-functors such that
$F$ is the ``kernel'' of $\phi$, meaning that
$$F(B) = \setof{x \in M \o*_A B: \phi(x) = 0}.$%
$The maps $\phi(B)$ need not be homomorphisms of $B$-modules.
In some situations, for a given coherent $A$-functor $F$, it will be necessary
to consider the set ${\cal{S}} = \setof{\vec M1n}$ of all $A$-modules which enter
into its construction. This set is not uniquely determined by $F$. Also it
does not carry information about multiplicity: ${\cal{S}}$ might consist of a single
module $M$, but many copies of $M$ might enter into the construction of $F$.
If $F$ has level $0$, we can choose ${\cal{S}}$ to have one element. If $F$ has
level $1$, then we may view $F$ as a limit of strictly coherent $A$-functors,
and thus we may choose ${\cal{S}}$ to consist of the corresponding modules. If $F$
has level $2$, then $F$ is a limit of level $1$ coherent $A$-functors
$\vec F1k$, and we may choose ${\cal{S}}$ to be the union of the sets corresponding
(as just considered) to $\vec F1k$. In any case, we shall say that $F$ is
{\it built up from\/} $\vec M1n$. Conversely, given an arbitrary class ${\cal{S}}$
of finitely generated\ $A$-modules, we may speak of coherent $A$-functors which are
{\it built up from\/} ${\cal{S}}$, meaning that such $A$-functors are built up from
finite subsets of ${\cal{S}}$.
\block{Module-coherent functors}
In this section we develop the basic theory of module-coherent functors.
The definition given initially will not be the same as that given in
the introduction. It is only after considerable work that we will find
\pref{main-theorem} that the two definitions agree.
We have already defined the notion of {\it module-valued\/} $A$-functor
in the introduction. If $F$ and $G$ are module-valued\ $A$-functors, then
a {\it morphism\/} \mp[[ \sigma || F || G ]] is a natural transformation of
functors, in the following sense. It is a system of homomorphisms
\mp[[ \sigma(B) || F(B) || G(B) ]] of $B$-modules, for each commutative
$A$-algebra $B$, such that for any $A$-algebra homomorphism
\mp[[ f || B_1 || B_2 ]], the diagram:
\diagramx{F(B_1)&\mapE{\sigma(B_1)}&G(B_1)\cr
\mapS{F(f)}&&\mapS{G(f)}\cr
F(B_2)&\mapE{\sigma(B_2)}&G(B_2)\cr%
}commutes.
With this definition of morphism, the module-valued\ $A$-functors form a
category, which is abelian. Kernels and cokernels are computed in the
obvious way; if \mp[[ \sigma || F || G ]] is a morphism, we have:
$$[\ker(\sigma)](B)\ =\ \ker(\sigma(B))
\ =\ \setof{x \in F(B): \sigma(x) = 0},$%
$$$[\mathop{\operatoratfont coker}\nolimits(\sigma)](B)\ =\ \mathop{\operatoratfont coker}\nolimits(\sigma(B))\ =\ G(B)/\sigma(F(B)).$%
$One sees that $\sigma$ is a {\it monomorphism\/} if and only if\ $\sigma(B)$ is
injective for every $B$, and $\sigma$ is an {\it epimorphism\/}
if and only if\ $\sigma(B)$ is surjective for every $B$.
Evidently, any module-valued\ $A$-functor may be viewed also as an $A$-functor. In some
situations we shall want to consider \mp[[ \phi || F || G ]] in which
$F$ and $G$ are module-valued\ $A$-functors but $\phi$ is a morphism of $A$-functors,
not necessarily preserving the module structure. For clarity, we may say that
$\phi$ is {\it linear}, if we wish to assume that it is a morphism of
module-valued\ $A$-functors. In this section, all morphisms are linear.
If $M$ is an $A$-module, then ${\underline{M}}$ is a module-valued\ $A$-functor. If a given
module-valued\ $A$-functor $F$ is isomorphic to ${\underline{M}}$ for some finitely generated\ $A$-module $M$, we
shall say that $F$ is {\it strictly module-coherent}.
\begin{definition}
A module-valued\ $A$-functor is {\it module-coherent\/} if it may be obtained as an iterated finite
limit of strictly module-coherent\ $A$-functors.
These limits are all taken in \cat{module-valued\ $A$-functors}.
\end{definition}
We will show \pref{main-theorem}, that in fact this definition is equivalent
to the (much simpler) definition of module-coherent\ given in the introduction.
\begin{problemx}
If a module-valued\ $A$-functor $F$ is coherent (when thought of simply as an
$A$-functor), does it follow that $F$ is module-coherent?
\end{problemx}
We may define $\cat{module-coherent\ $A$-functors}$: it is a full subcategory of
\cat{module-valued\ $A$-functors}, which may be thought of as the
{\it finite completion\/} of the subcategory \cat{strictly module-coherent\ $A$-functors}
of \cat{module-valued\ $A$-functors}.
Let ${\cal{M}}{\cal{C}}_A^0$ denote the collection of strictly
module-coherent\ $A$-functors. For each $n \geq 0$, let ${\cal{M}}{\cal{C}}_A^{n+1}$ denote the
collection of module-valued\ $A$-functors which may be obtained as finite limits (inside
\cat{module-valued\ $A$-functors}) of objects in ${\cal{M}}{\cal{C}}_A^n$. We have:
$${\cal{M}}{\cal{C}}_A^0 \subset {\cal{M}}{\cal{C}}_A^1 \subset {\cal{M}}{\cal{C}}_A^2 \subset \cdots.$%
$Let ${\cal{M}}{\cal{C}}_A = \cup_{n=0}^\infty {\cal{M}}{\cal{C}}_A^n$. Then the objects in
${\cal{M}}{\cal{C}}_A$ are
exactly the module-coherent\ $A$-functors.
\begin{definition}
Let $F$ be a module-coherent\ $A$-functor. Then the {\it level\/} of $F$ is the
smallest integer $n$ such that $F \in {\cal{M}}{\cal{C}}_A^n$.
\end{definition}
To show that our definition of module-coherent\ is equivalent to the definition given
in the introduction, we will show \pref{main-theorem} that the level of
$F$ is always $\leq 1$. In the meantime, however, we will employ induction
on the level of a given module-coherent\ $A$-functor. For this we use the
following analog of \pref{lesx-exists}, whose proof is left to the reader:
\begin{lemma}\label{les-exists}
Let $F$ be a module-coherent\ $A$-functor of level $n \geq 1$. Then there exists
a left exact sequence:
\les{F}{G}{{\underline{M}}%
}in which $G$ is a module-coherent\ $A$-functor of level $n-1$ and $M$ is a finitely generated\
$A$-module.
Moreover, $F$ may be embedded as a
sub-module-valued functor of a strictly module-coherent\ $A$-functor.
\end{lemma}
There is a functor $i_A$ from \cat{finitely generated\ $A$-modules} to
\hskip 0pt plus 4cm\penalty1000\hskip 0pt plus -4cm\hbox{\cat{module-coherent\ $A$-functors}}, given by $M \mapsto {\underline{M}}$.
It is easily seen that $i_A$ is fully
faithful, so we may view module-coherent $A$-functors as a sort of
generalization of finitely generated\ $A$-modules. The functor $i_A$ is cocontinuous:
it preserves colimits. However, $i_A$ does not carry
monomorphisms to monomorphisms and is not continuous: it does not preserve
limits. For example, if $J \subset A$ is an ideal, then $J$ is the kernel of
the canonical map \mapx[[ A || A/J ]] of modules, but ${\underline{J}}$ is not the kernel
of the induced map \mp[[ \phi || {\underline{A}} || \underline{A/J} ]]. The kernel of $\phi$ is
instead given by $B \mapsto JB$.
Let $H$ be a module-coherent\ $A$-functor. It would be very convenient if
there existed an epimorphism \mapx[[ \underline{A^n} || H ]] for some $n$.
Unfortunately, this is not always the case. For example, it is not the case
if $H(B) = \mathop{\operatoratfont Ann}\nolimits_B(x)$, where $A = {\Bbb C}\kern1pt[x]$. (The module-coherence of this
functor follows from example \pref{annihilator} of \S\ref{examples-section}.)
As a compromise, we are lead to the following notion:
\begin{definition}
A module-valued\ $A$-functor $F$ is {\it linearly representable\/} if there
exists a left exact sequence:
\les{F}{\underline{A^n}}{\underline{A^k}%
}in \cat{module-valued\ $A$-functors}, for some $n, k \geq 0$.
\end{definition}
If $F$ is linearly representable, it is module-coherent, and it is
representable by an $A$-algebra of the form
$$A[\vec x1n]/(\vec f1k),$%
$where $\vec f1k$ are linear and homogeneous in $\vec x1n$.
The following proposition is a basic tool, because it exhibits any
module-coherent\ $A$-functor as a quotient of ``something simple''.
\begin{prop}\label{dog-eats-dog}
Let $F$ be a module-coherent\ $A$-functor. Then there exists a linearly
representable $A$-functor $R$ and an epimorphism \mapx[[ R || F ]].
\end{prop}
There are some preliminaries.
\begin{lemma}\label{gerbil}
Let $F$ be a linearly representable $A$-functor, represented by
$C = A[\vec x1n]/(\vec f1k)$, where $\vec f1k$ are linear and homogeneous.
We assume that the module structure on $F$ is the canonical one, induced
from the embedding in $\underline{A^n}$ defined by $\vec x1n$. Let $C_1$ denote the
degree $1$ part of $C$. Let $N$ be an $A$-module. Then morphisms from
$F$ to ${\underline{N}}$ are in bijective correspondence with elements of $N \o*_A C_1$.
\end{lemma}
\begin{proof}
We can think of $F$ and ${\underline{N}}$ as functors from the category of commutative
$A$-algebras
to \cat{sets}. If we take this point of view, then some morphisms
(i.e.\ natural transformations) from $F$ to ${\underline{N}}$ define morphisms in
\cat{module-valued $A$-functors}, and some do not. Those which do will be
called {\it linear}, for purposes of this proof. The linear morphisms are
those which preserve the module structure.
If we take this point of view, then morphisms from $F$ to ${\underline{N}}$ are in
bijective correspondence with elements of ${\underline{N}}(C) = N \o*_A C$, and it is clear
that the elements of $N \o*_A C_1$ define linear morphisms. To complete the
proof, we must show that if an element $\eta \in N \o*_A C$ corresponds to a
linear morphism, then $\eta \in N \o*_A C_1$.
The grading of $C$ induces a grading of $N \o*_A C$.
Let $\eta_1 \in N \o*_A C_1$ denote the degree $1$ part of $\eta$. Let
$\eta_0 = \eta - \eta_1$. Then the degree $1$ part of $\eta_0$ is $0$ and
$\eta_0$ defines a linear morphism \mp[[ \psi || F || {\underline{N}} ]]. We must show
that $\psi = 0$.
Let $B$ be a commutative $A$-algebra.
Let $D = B[t]$. Then $\psi(dx) = d\psi(x)$ for all $d \in D$ and all
$x \in F(D)$. In particular, $\psi(tx) = t\psi(x)$ for all $x \in F(B)$. We
have $\psi(x) \in N \o*_A B$, $\psi(tx) \in N \o*_A D = (N \o*_A B)[t]$. Since
$\psi(tx) = t\psi(x)$, $\psi(tx)$ is a homogeneous linear polynomial in $t$.
The element $tx \in F(D)$ defines a ring homomorphism \mp[[ \rho || C || D ]],
which maps each generator $x_i$ of $C$ to a homogeneous linear polynomial
in $t$. The map $\rho$ induces a map \mapx[[ N \o*_A C || N \o*_A D ]],
which sends $\eta_0$ to $\psi(tx)$. But $\eta_0$ has no linear part, so it
follows that $\psi(tx)$ has no linear part. Hence $\psi(tx) = 0$. Hence
$t\psi(x) = 0$, so $\psi(x) = 0$. Hence $\psi = 0$. {\hfill$\square$}
\end{proof}
\begin{prop}\label{make-arrow}
Let $P$ be an $A$-module. Suppose given a diagram:
\diagramx{\rowfour{0}{F}{\underline{A^n}}{\underline{A^k}}\cr
&&\mapS{\phi}\cr
&&{\underline{P}}\cr%
}in \cat{module-valued\ $A$-functors}, with the row exact. Then there exists a
morphism \mp[[ h || \underline{A^n} || {\underline{P}} ]] which makes the diagram commute.
\end{prop}
\begin{proof}
Let $C = A[\vec x1n]$,
$${\overline{C}} = A[\vec x1n]/(\vec f1k),$%
$where $\vec f1k$ are homogeneous linear elements determined by the given map
from $\underline{A^n}$ to $\underline{A^k}$. Then $F$ represents ${\overline{C}}$. According to
\pref{gerbil}, $\phi$ corresponds to an element of $P \o*_A {\overline{C}}_1$. The
canonical map \mapx[[ P \o*_A C_1 || P \o*_A {\overline{C}}_1 ]] is surjective, so $h$
exists. {\hfill$\square$}
\end{proof}
\begin{corollary}\label{limit-of-linearly-rep}
Let ${\cal{D}}$ be a finite diagram in \cat{module-valued $A$-functors}, in
which the objects are linearly representable. Then the limit of ${\cal{D}}$
is linearly representable.
\end{corollary}
\begin{proof}
It is clear that a product of finitely many linearly representable
$A$-functors is linearly representable. Since any linearly representable
$A$-functor embeds in $\underline{A^r}$ for some $r$, we may reduce to showing that
if $F$ is linearly representable and \mp[[ \varphi || F || \underline{A^r} ]] is
a morphism, then $\ker(\varphi)$ is linearly representable. Let
\lesmaps{F}{}{\underline{A^n}}{h}{\underline{A^k}%
}be as in the definition of {\it linearly representable}. By
\pref{make-arrow},
$\varphi$ extends to a morphism \mp[[ \psi || \underline{A^n} || \underline{A^r} ]].
Hence $\ker(\varphi) = \ker(\psi) \cap \ker(h)$,
so $\ker(\varphi)$ is linearly representable. {\hfill$\square$}
\end{proof}
\begin{proofnodot}
(of \ref{dog-eats-dog}.)
For purposes of the proof, let us say that a module-valued\ $A$-functor $G$
{\it dominates\/} a module-valued\ $A$-functor $F$ if there exists an epimorphism
\mapx[[ G || F ]], and that a module-valued\ $A$-functor $F$ is
{\it linearly-affine-dominated\/}
if it is dominated by a linearly representable $A$-functor.
Let $n$ be the level of $F$. The case $n = 0$ is clear -- in that case $F$
is dominated by $\underline{A^r}$ for some $r$.
Suppose that $n \geq 1$. By \pref{les-exists}, we may find a left exact
sequence:
\les{F}{G}{{\underline{M}}%
}in which $G$ is a module-coherent\ $A$-functor of level $n-1$ and $M$ is a
finitely generated\ $A$-module. By induction on $n$, we may assume that\ there exists a linearly
representable $A$-functor $H$ and an epimorphism \mapx[[ H || G ]]. Let $P$ be
the fiber product of $H$ with $F$ over $G$. Then $P$ dominates $F$, so it
suffices to show that $P$ is linearly-affine-dominated.
We have a left exact sequence:
\lesdot{P}{H}{{\underline{M}}%
}Choose an epimorphism \mp[[ h || A^r || M ]]. Let $L$ be the fiber product
of $H$ with $\underline{A^r}$ over ${\underline{M}}$. Let $Q$ be the fiber product of $L$ with
$P$ over $H$. We have a diagram with cartesian squares, in which some maps
are labelled:
\diagramx{Q&\rightarrowtail&L&\mapE{\sigma}&\underline{A^r}\cr
\mapS{}&&\mapS{}&&\mapS{\pi}\cr
P&\rightarrowtail&H&\mapE{f}&{\underline{M}}\makenull{.}\cr%
}The bottom row (but not the top) is exact. The vertical arrows are all
epimorphisms. Since $Q$ dominates $P$, it suffices
to show that $Q$ is linearly-affine-dominated.
Let $K = \ker(\pi)$. Then $Q = \sigma^{-1}(K)$, so we have a cartesian diagram
\squareSE{Q}{K}{L}{\underline{A^r}\makenull{.}%
}We will show that $K$ and $L$ are linearly-affine-dominated. It will follow
(by taking suitable fiber products) that $Q$ is dominated by a fiber product
of linearly-representable $A$-functors. By \pref{limit-of-linearly-rep}, it
will follow that $Q$ is linearly-affine-dominated.
There is a canonical epimorphism \mapx[[ \underline{\ker(h)} || K ]].
Choose an epimorphism \mapx[[ A^s || \ker(h) ]]. Then $K$ is dominated by
$\underline{A^s}$. To complete the proof, we will show that $L$ is
linearly-affine-dominated.
By \pref{gerbil}, it follows that $f$ factors through $\pi$; let
\mp[[ g || H || \underline{A^r} ]] be such that $f = \pi \circ g$. Then
\begin{eqnarray*}
L(B) & = & \setof{(x,y) \in H(B) \times \underline{A^r}(B): f(x) = \pi(y)}\\
& = & \setof{(x,y) \in H(B) \times \underline{A^r}(B): \pi(g(x) - y) = 0}.
\end{eqnarray*}
Hence the morphism of module-valued\ $A$-functors \mapx[[ H \times K || L ]] given by
$$(x,y) \mapsto (x,g(x)+y)$%
$is an isomorphism. Since $K$ is linearly-affine-dominated, and $H$ is
linearly representable, it follows that $L$ is linearly-affine-dominated. {\hfill$\square$}
\end{proofnodot}
\begin{theorem}\label{coherence-of-cokernel}
Let \mp[[ \varphi || F || G ]] be a morphism of module-coherent\ $A$-funct\-ors. Then
$\mathop{\operatoratfont Coker}\nolimits(\varphi)$ is module-coherent.
\end{theorem}
\begin{proof}
Let $n$ be the level of $G$.
First suppose that $n = 0$, so we may assume that\ $G = {\underline{M}}$ for some finitely generated\ $A$-module $M$.
Choose an epimorphism \hbox{\mapx[[ A^m || M ]]} of $A$-modules and thus an
epimorphism \mapx[[ \underline{A^m} || {\underline{M}} ]].
Let $F'$ be the fiber product of $F$ with $\underline{A^m}$ over ${\underline{M}}$. Let
$C = \mathop{\operatoratfont Coker}\nolimits(\varphi)$. Then we have a right exact sequence:
\resmapsdot{F'}{f}{\underline{A^m}}{}{C%
}By \pref{dog-eats-dog}, we may assume that\ $F'$ is linearly representable. Choose a
left exact sequence:
\lesdot{F'}{\underline{A^n}}{\underline{A^k}%
}By \pref{make-arrow}, there exists a morphism \mapx[[ \underline{A^n} || \underline{A^m} ]]
which makes the following diagram commute:
$$\diagram F' \rto^f \dto & \underline{A^m} \rto & C \rto & 0 \\
\underline{A^n} \dto \urto \\
\underline{A^k}\makenull{.} \enddiagram$%
$Let $D$ be the co-fiber product of $A^m$ and $A^k$ over $A^n$, computed in
the category of $A$-modules. Then in fact
${\underline{D}}$ is the co-fiber product of $\underline{A^m}$ and $\underline{A^k}$ over $\underline{A^n}$.
Let \mp[[ g || \underline{A^m} || {\underline{D}} ]] be the canonical map. Then
$\ker(g) = \mathop{\operatoratfont Im}\nolimits(f)$, so $C$ is isomorphic to $\mathop{\operatoratfont Im}\nolimits(g)$, which is module-coherent
by example \pref{image} from \S\ref{examples-section}. This completes the
case $n = 0$.
Now suppose that $n \geq 1$. By \pref{les-exists}, we may choose a left exact
sequence:
\les{G}{H}{{\underline{M}}%
}in which $H$ is a module-coherent\ $A$-functor of level $n-1$ and $M$ is a finitely generated\
$A$-module.
Abusing notation slightly, we have a left exact sequence:
\lesdot{G/F}{H/F}{{\underline{M}}%
}By induction on $n$, we may assume that\ $H/F$ is module-coherent. But then $\mathop{\operatoratfont Coker}\nolimits(\varphi) = G/F$ is
exhibited as the kernel of a morphism of module-coherent\ $A$-functors, so it too is module-coherent.
{\hfill$\square$}
\end{proof}
\begin{corollary}\label{result-A}
If \mp[[ \phi || F || G ]] is a morphism of module-coherent\ $A$-functors,
then $\mathop{\operatoratfont Im}\nolimits(\phi)$ is module-coherent.
\end{corollary}
\begin{proof}
Let $K = \mathop{\operatoratfont Ker}\nolimits(\phi)$. Then $K$ is module-coherent. Hence \hskip 0pt plus 4cm\penalty1000\hskip 0pt plus -4cm $\mathop{\operatoratfont Coker}\nolimits[K \mapE{} F]$ is
module-coherent\ by \pref{coherence-of-cokernel}, but this equals $\mathop{\operatoratfont Im}\nolimits(\phi)$. {\hfill$\square$}
\end{proof}
The next result says that the definition of module-coherent\ given in this section
coincides with the simpler definition given in the introduction.
\begin{corollary}\label{main-theorem}
Let $F$ be a module-coherent\ $A$-functor. Then $F$ has level $\leq 1$.
\end{corollary}
\begin{proof}
By \pref{les-exists}, we may embed $F$ as a subfunctor of ${\underline{M}}$, for some
finitely generated\ $A$-module $M$. Let $Q = {\underline{M}}/F$. By \pref{coherence-of-cokernel},
$Q$ is module-coherent. By \pref{les-exists}, we may embed $Q$ as a subfunctor of ${\underline{N}}$,
for some finitely generated\ $A$-module $N$. Hence $F$ is the kernel of a morphism from
${\underline{M}}$ to ${\underline{N}}$. {\hfill$\square$}
\end{proof}
\block{Quasi-coherent functors}
In this section we sketch a theory (parallel to the last two sections) of
quasi-coherent\ and module-quasi-coherent\ $A$-functors. The results of this section will be used
in \S\ref{global-section}. In particular, it is the case that
quasi-coherent\ $A$-functors (which are not coherent) are useful in the study of coherent
$A$-functors. However, the reader interested only in the Picard
group results may ignore this section and everything from \pref{locally-mc}
to the end of \S\ref{global-section}. The reason for this is explained in
the paragraph preceding \pref{locally-mc}.
If a given $A$-functor is isomorphic to ${\underline{M}}$ for some $A$-module $M$, we
shall say that $F$ is {\it strictly quasi-coherent}. Similarly,
if a given module-valued\ $A$-functor $F$ is isomorphic to ${\underline{M}}$
for some $A$-module $M$, we shall say that $F$ is {\it strictly module-quasi-coherent}.
An $A$-functor is {\it quasi-coherent\/} if it may be obtained as an iterated finite limit
of strictly coherent $A$-functors. These limits are all taken in
\cat{$A$-functors}. We shall not have much more to say about quasi-coherent\ $A$-functors
per se in this paper.
A module-valued\ $A$-functor is {\it module-quasi-coherent\/} if it may be obtained as an iterated finite
limit of strictly module-quasi-coherent\ $A$-functors. These limits are all taken in
\cat{module-valued\ $A$-functors}.
The rest of this section is about module-quasi-coherent\/ $A$-functors. All morphisms will be
linear.
We may define the {\it level\/} of a module-quasi-coherent\ $A$-functor, as we have done for
module-coherent\ $A$-functors. As we shall see \pref{main-theorem-mqc}, any module-quasi-coherent\
$A$-functor has level $\leq 1$.
The analog of \pref{les-exists} for module-quasi-coherent\ $A$-functors is:
\begin{lemma}\label{les-exists-qc}
Let $F$ be a module-quasi-coherent\ $A$-functor of level $n \geq 1$. Then there exists
a left exact sequence:
\les{F}{G}{{\underline{M}}%
}in which $G$ is a module-quasi-coherent\ $A$-functor of level $n-1$ and $M$ is an $A$-module.
Moreover, $F$ may be embedded as a
sub-module-valued functor of a strictly module-quasi-coherent\ $A$-functor.
\end{lemma}
We make the following convention: if $N$ is a set, then $A^N$ denotes a
{\it direct sum\/} of copies of $A$, one for each element of $N$. If
$S$ is a subset of $N$, then we may view $A^S$ as a submodule of $A^N$, and
thence we may view $\underline{A^S}$ as a subfunctor of $\underline{A^N}$.
\begin{prop}\label{make-arrow-generalized}
Let $P$ be an $A$-module. Let $N$ and $K$ be sets. Suppose given a diagram:
\diagramx{0&\mapE{}&F&\mapE{}&\underline{A^N}&\mapE{g}&\underline{A^K}\cr
&&\mapS{\phi}\cr
&&{\underline{P}}\cr%
}in \cat{module-valued $A$-functors}, with the row exact. Then there exists a
morphism \mp[[ h || \underline{A^N} || {\underline{P}} ]] which makes the diagram commute.
\end{prop}
\begin{proof}
Let $\mathop{\operatoratfont fin}\nolimits(N)$ denote the collection of finite subsets of $N$.
Then
$$\setof{\underline{A^S}}_{S \in \mathop{\operatoratfont fin}\nolimits(N)}$%
$forms a directed system of subfunctors of $\underline{A^N}$,
whose union is $\underline{A^N}$. Let $F_S = F \cap \underline{A^S}$ for each $S$.
Then $\setof{F_S}_{S \in \mathop{\operatoratfont fin}\nolimits(N)}$ forms a directed system of subfunctors
of $F$, whose union is $F$.
It is clear that $g|_{\underline{A^S}}$ factors through the subfunctor $\underline{A^{S^*}}$
of $\underline{A^K}$, for some finite subset $S^*$ of $K$. It follows that
$F_S$ is linearly representable by a ring $\overline{C_S} = A[\sets xsS]/I_S$, where
$I_S$ is generated by linear homogeneous elements. Let $C_S$ be the
polynomial ring $A[\sets xsS]$.
By \pref{gerbil}, $\phi|_{F_S}$ corresponds to an element of
$P \o*_A (\overline{C_S})_1$. By lifting this element to an element of
$P \o*_A (C_S)_1$, we see that $\phi|_{F_S}$ can be extended to a
morphism \mp[[ h_S || \underline{A^S} || {\underline{P}} ]]. As $S$ varies, we have to choose
these extensions $h_S$ so that they are compatible with each other. To do
this is equivalent to showing that the canonical map:
\dmapx[[ \invlim{S \in \mathop{\operatoratfont fin}\nolimits(N)} P \o*_A (C_S)_1 ||
\invlim{S \in \mathop{\operatoratfont fin}\nolimits(N)} P \o*_A (\overline{C_S})_1 ]]%
is surjective. In general, it is not true that an inverse limit of surjective
module maps is surjective, but (\Lcitemark 4\Rcitemark \ 10.2) it is the case
if
the transition maps in the system of kernels are surjective. To show this, it
suffices to show that if $S, S' \in \mathop{\operatoratfont fin}\nolimits(N)$, with $S \subset S'$, then the
canonical map \mapx[[ P \o*_A (I_{S'})_1 || P \o*_A (I_S)_1 ]] is surjective.
This follows from the fact that the canonical map \mapx[[ I_{S'} || I_S ]] is
surjective. {\hfill$\square$}
\end{proof}
\begin{definition}
A module-valued $A$-functor $F$ is {\it linearly quasi-representable\/} if
there exist sets $N$ and $K$ and a left exact sequence:
\les{F}{\underline{A^N}}{\underline{A^K}%
}in \cat{module-valued\ $A$-functors}.
\end{definition}
Using \pref{make-arrow-generalized} one may prove the following analog
of \pref{limit-of-linearly-rep}:
\begin{corollary}\label{limit-of-linearly-rep-gen}
Let ${\cal{D}}$ be a finite diagram in \cat{module-valued\ $A$-functors}, in
which the objects are linearly quasi-representable. Then the limit of ${\cal{D}}$
is linearly quasi-representable.
\end{corollary}
Evidently, any linear quasi-representable $A$-functor is module-quasi-coherent.
\begin{prop}\label{rat-eats-rat}
Let $F$ be a module-quasi-coherent\ $A$-functor. Then there exists a linearly
quasi-representable $A$-functor $R$ and an epimorphism \mapx[[ R || F ]].
\end{prop}
\begin{sketch}
Take the proof of \pref{dog-eats-dog}, and modify it in the following ways.
In the various places where $A^r$ is written, one has to allow $r$ to be an
arbitrary set. Do the same with $A^s$. Change each reference to {\it module-coherent\/}
to {\it module-quasi-coherent}. Drop the assumption that $M$ is finitely generated. The construction of $g$
requires the use of \pref{make-arrow-generalized}. Use
\pref{limit-of-linearly-rep-gen} instead of \pref{limit-of-linearly-rep}. Use
\pref{les-exists-qc} instead of \pref{les-exists}. {\hfill$\square$}
\end{sketch}
\begin{prop}\label{quasi-coherence-of-cokernel}
Let \mp[[ \varphi || F || G ]] be a morphism of module-quasi-coherent\ $A$-functors. Then
$\mathop{\operatoratfont Coker}\nolimits(\varphi)$ is module-quasi-coherent.
\end{prop}
\begin{sketch}
Take the proof of \pref{coherence-of-cokernel}, and modify it in the following
ways. Change each reference to {\it module-coherent\/} to {\it module-quasi-coherent}.
Drop the assumption that $M$ is finitely generated. In the notations $A^m$, $A^n$, and
$A^k$, one must allow $m$, $n$, and $k$ to be arbitrary sets. Use
\pref{make-arrow-generalized} instead of \pref{make-arrow}. Use
\pref{rat-eats-rat} instead of \pref{dog-eats-dog}. Use
\pref{les-exists-qc} instead of \pref{les-exists}. Modify reference to example
\pref{image} from \S\ref{examples-section} appropriately. {\hfill$\square$}
\end{sketch}
\begin{corollary}\label{result-A-MQC}
Let \mp[[ \phi || F || G ]] be a morphism of module-quasi-coherent\ $A$-functors. Then
$\mathop{\operatoratfont Im}\nolimits(\phi)$ is module-quasi-coherent.
\end{corollary}
\begin{corollary}\label{main-theorem-mqc}
Let $F$ be a module-quasi-coherent\ $A$-functor. Then $F$ has level $\leq 1$.
\end{corollary}
\begin{lemma}\label{noetherian}
Let $F$ be a module-coherent\ $A$-functor. Let $\sets F\lambda\Lambda$ be a system of
subfunctors of $F$, whose union is $F$. Then $F = F_{\lambda_0}$ for some
$\lambda_0 \in \Lambda$.
\end{lemma}
\begin{proof}
By \pref{dog-eats-dog}, we may find a linearly-representable $A$-functor $H$,
and an epimorphism \mp[[ \pi || H || F ]]. Let
$H_\lambda = \pi^{-1}(F_\lambda)$,
for each ${\lambda \in \Lambda}$. (That is, $H_\lambda$ is the fiber product of $F_\lambda$
with $H$ over $F$.) Then $H$ is the union of the $H_\lambda$.
Identify $H$ with the functor representing some $A$-algebra $C$:
$H(B) = \mathop{\operatoratfont Hom}\nolimits_{A-{\operatoratfont alg}}(C,B)$. Choose $\lambda_0 \in \Lambda$ such that
$1_C \in H_{\lambda_0}(C)$. It follows that there exists a natural
transformation of functors \mp[[ s || H || H_{\lambda_0} ]] (possibly not
preserving module structures) such that $i \circ s = 1_H$, where
\mp[[ i || H_{\lambda_0} || H ]] is the inclusion. Hence $i$ is an
isomorphism. Hence $H_{\lambda_0} = H$. Since $\pi$ is an epimorphism, it
follows that $F_{\lambda_0} = F$. {\hfill$\square$}
\end{proof}
\begin{remark}\label{not-coh-example}
If $F$ is module-coherent\ and $G$ is module-quasi-coherent, and \mp[[ \varphi || F || G ]] is a
morphism, then $\ker(\varphi)$ need not be module-coherent. For an example,
let $A = \xmode{\Bbb Z}$, $F = \underline{\xmode{\Bbb Z}}$, $G = \underline{{\Bbb Q}\kern1pt}$, and let
$\varphi$ be the map given by $n \mapsto n$. Indeed,
$$\ker(\varphi)(B)\ =\ \setofh{$b \in B: nb = 0$ for some $n \in \xmode{\Bbb N}$}.$%
$Then $\ker(\varphi)$ is the direct limit of its subfunctors
$F_n = \setof{b \in B: nb = 0}$, for $n \geq 1$,
and since $\ker(\varphi) \not= F_n$ for all $n$, it follows from
\pref{noetherian} that $\ker(\varphi)$ is not module-coherent.
\end{remark}
\block{Examples}\label{examples-section}
First we give some examples, all of which are easily seen to satisfy the
simple definition of module-coherent\ which we gave in the introduction.
\begin{arabiclist}
\item $B \mapsto M \o*_A B$, where $M$ is a finitely generated $A$-module;
\item\label{kernel}
$B \mapsto \mathop{\operatoratfont Ker}\nolimits(f \o*_A B)$, where \mp[[ f || M || N ]] is a homomorphism
of finitely generated\ $A$-modules
\par[We shall denote this $A$-functor by $\mathop{\underline{\operatoratfont Ker}}\nolimits(f)$.]
\item $B \mapsto IB$, where $I$ is an ideal of $A$
\par[Consider the kernel of the map \mapx[[ {\underline{A}} || \underline{A/I} ]].]
\item\label{image}
$B \mapsto \mathop{\operatoratfont Im}\nolimits(f \o*_A B)$, where \mp[[ f || M || N ]] is a homomorphism
of finitely generated $A$-modules.
\par[We shall denote this functor by $\mathop{\underline{\operatoratfont Im}}\nolimits(f)$. Let
\mp[[ g || N || N/\mathop{\operatoratfont Im}\nolimits(f) ]] be the canonical map. Then
$\mathop{\underline{\operatoratfont Im}}\nolimits(f) = \mathop{\underline{\operatoratfont Ker}}\nolimits(g)$, so $\mathop{\underline{\operatoratfont Im}}\nolimits(f)$ is module-coherent.]
\item\label{hom-example}
$B \mapsto \mathop{\operatoratfont Hom}\nolimits_{B-{\operatoratfont mod}}(M \o*_A B, N \o*_A B)$, where
$M$ and $N$ are finitely generated $A$-modules
\par[We shall denote this functor by $\mathop{\underline{\operatoratfont Hom}}\nolimits(M, N)$. To see why it is
module-coherent, choose a presentation $A^k \mapE{} A^n \mapE{} M \mapE{} 0$ for
$M$. Consider the induced map \mp[[ f || \mathop{\operatoratfont Hom}\nolimits(A^n, N) || \mathop{\operatoratfont Hom}\nolimits(A^k, N) ]].
Then $\mathop{\underline{\operatoratfont Hom}}\nolimits(M, N)$ is isomorphic to $\mathop{\underline{\operatoratfont Ker}}\nolimits(f)$.]
\item\label{end-example}
$B \mapsto \mathop{\operatoratfont End}\nolimits_{B-{\operatoratfont mod}}(M \o*_A B)$, where $M$ is a
finitely generated\ $A$-module \par[We shall denote this functor by $\mathop{\underline{\operatoratfont End}}\nolimits(M)$.]
\item\label{annihilator}
$B \mapsto \mathop{\operatoratfont Ann}\nolimits_B(I)$, where $I \subset A$ is a fixed ideal
\par[We shall denote this functor by $\mathop{\underline{\operatoratfont Ann}}\nolimits(I)$. The point is that
$\mathop{\operatoratfont Ann}\nolimits_B(I) = \mathop{\operatoratfont Hom}\nolimits_{B-{\operatoratfont mod}}(B/I,B)$;
use example \pref{hom-example}.]
\end{arabiclist}
\par\noindent Now we give some simple examples of coherent $A$-functors.
\begin{arabiclist}
\setcounter{arabicctr}{7}
\item\label{representable-example}
$B \mapsto \mathop{\operatoratfont Hom}\nolimits_{A-{\operatoratfont alg}}(C,B)$, where $C$ is a
commutative $A$-algebra of finite type
\par[Choose a presentation $C = A[\vec x1n]/(\vec f1k)$. Then
$\vec f1k$ define a morphism of $A$-functors
\mapx[[ \underline{A^n} || \underline{A^k} ]], whose kernel is the given functor.]
\item $B \mapsto \setof{x \in B: x^2 \in IB}$
\par[Consider the kernel of the map \mapx[[ {\underline{A}} || \underline{A/I} ]]
given by $x \mapsto x^2$.];
\item $B \mapsto \setof{a \in A^n: f(a) = 0}$, where $f \in M[\vec x1n]$
\par[Consider kernels of maps \mapx[[ \underline{A^n} || {\underline{M}} ]]; this
generalizes the preceding example.]
\end{arabiclist}
The coherence of the remaining examples of this section follows without great
difficulty from the tools developed so far. However, the remaining examples
seem to be deeper, in the sense that their coherence cannot be deduced directly
from the definitions.
Presumably, all of the usual linear algebra operations ($\mathop{\operatoratfont Hom}\nolimits$, $\o*$,
$\Lambda^n$, $\ldots$) have analogs for module-valued\ $A$-functors. A thorough study
(which we do not give) would include definitions of these operations and an
analysis of which preserve module-coherence. We restrict our attention to
some special cases.
Let $F$ and $G$ be module-valued\ $A$-functors. Then there is a module-valued\ $A$-functor
$F \o* G$, given by $B \mapsto F(B) \o*_B G(B)$. If $F$ and $G$ are
module-coherent, one can ask if $F \o* G$ is module-coherent. It turns out \pref{tensor-mc} that
this is not the case. However, there is the following special case:
\begin{prop}\label{tensor-basic}
Let $F$ be a module-coherent\ $A$-functor, and let $M$ be a finitely generated\ $A$-module. Then
$F \o* {\underline{M}}$ is module-coherent.
\end{prop}
\begin{proof}
Choose a right exact sequence:
\res{A^k}{A^n}{M%
}of $A$-modules. We obtain a right exact sequence:
\res{F \o* \underline{A^k}}{F \o* \underline{A^n}}{F \o* {\underline{M}}%
}of module-valued\ $A$-functors, and thence a right exact sequence:
\resdot{F^k}{F^n}{F \o* {\underline{M}}%
}Since $F^k$ and $F^n$ are module-coherent, it follows by \pref{coherence-of-cokernel}
that $F \o* {\underline{M}}$ is module-coherent. {\hfill$\square$}
\end{proof}
To show that tensor products do not (in general) preserve module-coherence, we
need the following lemma, which will also be used in a counterexample presented
in \S\ref{higher-section}.
\begin{lemma}\label{ding-dong}
Let $A$ be a noetherian local ring of dimension $d$, having maximal ideal
${\xmode{{\fraktur{\lowercase{M}}}}}$. Let $F$ be a module-coherent\ $A$-functor. Then there exists a constant $c$, such
that for each $n \in \xmode{\Bbb N}$, and every ideal $I$ of $A$ with ${\xmode{{\fraktur{\lowercase{M}}}}}^n \subset I$, we
have $\mu[F(A/I)] \leq c n^d$, where $\mu$ gives the minimal number of
generators of an $A$-module.
\end{lemma}
\begin{remark}
Perhaps the bound $c n^d$ can be replaced by $c n^{d-1}$.
\end{remark}
\begin{proofnodot}
(of \ref{ding-dong}.)
We may assume that $F = \mathop{\underline{\operatoratfont Ker}}\nolimits(f)$, for some homomorphism \mp[[ f || M || N ]]
of finitely generated\ $A$-modules. Let $\lambda$ denote {\it length}. Choose a surjection
\mapx[[ A^s || M ]]. Since we have
$${\xmode{{\fraktur{\lowercase{M}}}}}^{n+1}M\ \subset\ {\xmode{{\fraktur{\lowercase{M}}}}} f^{-1}(IN)\ \subset\ f^{-1}(IN) \subset\ M,$%
$it follows that:
\begin{eqnarray*}
\mu[\mathop{\operatoratfont Ker}\nolimits(f \o*_A A/I)] & = & \mu[f^{-1}(IN)/IM]\ \leq\ \mu[f^{-1}(IN)]\\
& & =\ \mu[f^{-1}(IN)/{\xmode{{\fraktur{\lowercase{M}}}}} f^{-1}(IN)]\\
& & =\ \lambda[f^{-1}(IN)/{\xmode{{\fraktur{\lowercase{M}}}}} f^{-1}(IN)]\\
& & \leq\ \lambda(M/{\xmode{{\fraktur{\lowercase{M}}}}}^{n+1}M)
\ \leq\ \lambda(A^s/{\xmode{{\fraktur{\lowercase{M}}}}}^{n+1} A^s)\\
& & =\ s\lambda(A/{\xmode{{\fraktur{\lowercase{M}}}}}^{n+1}).
\end{eqnarray*}
The lemma follows from the theory of the Hilbert-Samuel polynomial. {\hfill$\square$}
\end{proofnodot}
Now we show that the tensor product of two module-coherent\ $A$-functors need not be module-coherent:
\begin{prop}\label{tensor-mc}
Let $A = {\Bbb C}\kern1pt[[s,t,u]]$, and let $I$ be the ideal $(s)$ of $A$. Then the
module-valued\ $A$-functor $\mathop{\underline{\operatoratfont Ann}}\nolimits(I) \o* \mathop{\underline{\operatoratfont Ann}}\nolimits(I)$ is not module-coherent.
\end{prop}
\begin{proof}
Let $F$ denote the given functor. Fix $n \in \xmode{\Bbb N}$, and let $B = A/(s,t,u)^n$.
Then a minimal generating set for $\mathop{\operatoratfont Ann}\nolimits_B(s)$ is
$$\setof{s^i t^j u^k}_{0 \leq i,j,k \leq n-1,\ i+j+k = n-1},$%
$which has cardinality $n(n+1)/2$.
Then $\mu[F(B)] = \mu[\mathop{\operatoratfont Ann}\nolimits_B(s)]^2 = [n(n+1)/2]^2$. By \pref{ding-dong},
$F$ is not module-coherent. {\hfill$\square$}
\end{proof}
Let $M$ and $N$ be finitely generated\ $A$-modules. For $n \geq 0$, one can ask if
the functor $\mathop{\underline{\operatoratfont Tor}}\nolimits_{\kern1pt n}(M,N)$ given by
$B \mapsto \mathop{\operatoratfont Tor}\nolimits_n^B(M \o*_A B, N \o*_A B)$ is module-coherent. This seems unlikely for
$n \geq 2$ (but we do not have a counterexample). For $n = 0$,
\pref{tensor-basic} applies. For $n = 1$, we have:
\begin{prop}
Let $M$ and $N$ be finitely generated\ $A$-modules. Then $\mathop{\underline{\operatoratfont Tor}}\nolimits_1(M,N)$ is module-coherent.
\end{prop}
\begin{proof}
Choose an epimorphism \mapx[[ A^n || M ]] and thence a short exact sequence:
\ses{K}{\underline{A^n}}{{\underline{M}}%
}in which $K$ is module-coherent. One obtains a left exact sequence:
\lesdot{\mathop{\underline{\operatoratfont Tor}}\nolimits_1(M,N)}{K \o* {\underline{N}}}{\underline{A^n} \o* {\underline{N}}%
}The corollary follows now from \pref{tensor-basic}. {\hfill$\square$}
\end{proof}
If $F$ and $G$ are module-valued\ $A$-functors, we let $\mathop{\mathbf{Hom}}\nolimits(F,G)$ denote the
module-valued\ $A$-functor given by
$$B \mapsto \mathop{\operatoratfont Hom}\nolimits_{\kern2pt{\operatoratfont module-valued}\ B-{\operatoratfont functors}} (F, G),$%
$where $F$ and $G$ may be viewed as module-valued\ $B$-functors because any
$B$-algebra is an $A$-algebra. In a natural way, $\mathop{\mathbf{Hom}}\nolimits(F,G)$ is itself a
module-valued\ $A$-functor. We let $\mathop{\mathbf{End}}\nolimits(F)$ denote $\mathop{\mathbf{Hom}}\nolimits(F,F)$.
\begin{example}\label{uHomHOM}
Let $M$ and $N$ be finitely generated\ $A$-modules. Then there is a
canonical isomorphism of module-valued\ $A$-functors
\dmapx[[ \mathop{\underline{\operatoratfont Hom}}\nolimits(M,N) || \mathop{\mathbf{Hom}}\nolimits({\underline{M}},{\underline{N}}). ]]
\end{example}
\begin{prop}\label{HOM-coherent}
Let $F$ and $G$ be module-coherent\ $A$-functors. Then \hskip 0pt plus 4cm\penalty1000\hskip 0pt plus -4cm $\mathop{\mathbf{Hom}}\nolimits(F,G)$ is module-coherent.
\end{prop}
\begin{proof}
First observe that the bifunctor
\dmap[[ \mathop{\mathbf{Hom}}\nolimits || \opcat{module-valued $A$-functors} \times
\cat{module-valued $A$-functors} || \cat{module-valued $A$-functors} ]]%
is left exact in both variables.
By \pref{main-theorem}, there is a left exact sequence:
\les{G}{\underline{M_1}}{\underline{M_2}%
}for some finitely generated\ $A$-modules $M_1$ and $M_2$. This yields a left exact sequence:
\lesdot{\mathop{\mathbf{Hom}}\nolimits(F,G)}{\mathop{\mathbf{Hom}}\nolimits(F,\underline{M_1})}{\mathop{\mathbf{Hom}}\nolimits(F,\underline{M_2})%
}Thus it suffices to show that $\mathop{\mathbf{Hom}}\nolimits(F,\underline{M_i})$ is module-coherent\ for each $i$.
Let $M = M_i$.
It follows from \pref{dog-eats-dog} that there exists a right exact sequence
\res{L_2}{L_1}{F%
}in which $L_1$ and $L_2$ are linearly representable. We obtain a left exact
sequence:
\lesdot{\mathop{\mathbf{Hom}}\nolimits(F,{\underline{M}})}{\mathop{\mathbf{Hom}}\nolimits(L_1,{\underline{M}})}{\mathop{\mathbf{Hom}}\nolimits(L_2,{\underline{M}})%
}Therefore we may reduce to proving that $\mathop{\mathbf{Hom}}\nolimits(L,{\underline{M}})$ is module-coherent\ for any linearly
representable $A$-functor $L$. The $A$-functor $L$ is representable by a ring
$C = A[\vec x1n]/(\vec f1k)$, where $\vec f1k$ are homogeneous linear
polynomials in $\vec x1n$. From \pref{gerbil}, it follows that
$\mathop{\mathbf{Hom}}\nolimits(L,{\underline{M}}) \cong \underline{M \o*_A C_1}$, so $\mathop{\mathbf{Hom}}\nolimits(L,{\underline{M}})$ is module-coherent. {\hfill$\square$}
\end{proof}
\begin{corollary}
For any module-coherent\ $A$-functor $F$, $\mathop{\mathbf{End}}\nolimits(F)$ is module-coherent.
\end{corollary}
If $F$ is a module-valued\ $A$-functor, we let $\mathop{\mathbf{Aut}}\nolimits(F)$ denote the $A$-functor given by
$$B \mapsto \mathop{\operatoratfont Aut}\nolimits_{\kern2pt{\operatoratfont module-valued} \ B-{\operatoratfont functors}} (F),$%
$where $F$ is viewed as a $B$-functor. Then $\mathop{\mathbf{Aut}}\nolimits(F)$ is an $A$-functor.
\begin{corollary}
For any module-coherent\ $A$-functor $F$, $\mathop{\mathbf{Aut}}\nolimits(F)$ is coherent.
\end{corollary}
\begin{proof}
Take the kernel of the map
\dmapx[[ \mathop{\mathbf{End}}\nolimits(F) \times \mathop{\mathbf{End}}\nolimits(F) || \mathop{\mathbf{End}}\nolimits(F) \times \mathop{\mathbf{End}}\nolimits(F) ]]%
given by
$(\alpha,\beta) \mapsto (\alpha \circ \beta - \mathop{\operatoratfont id}\nolimits, \beta \circ \alpha - \mathop{\operatoratfont id}\nolimits)$.
{\hfill$\square$}
\end{proof}
By an {\it algebra-valued\/} $A$-functor $F$, we shall mean a module-valued\ $A$-functor
$F$ which has the additional structure of a $B$-algebra on $F(B)$, for each
$B$. We do not assume that these algebras $F(B)$ are commutative.
If $F$, $G$, and $H$ are module-valued\ $A$-functors, then $\mathop{\mathbf{Bil}}\nolimits(F \times G, H)$ will
denote the functor of bilinear maps from $F \times G$ to $H$, which sends $B$
to the $B$-module consisting of all morphisms of $B$-functors from
$F \times G$ to $H$ which are bilinear. Then $\mathop{\mathbf{Bil}}\nolimits(F \times G, H)$ is a
module-valued\ $A$-functor.
\begin{corollary}
Let $F$, $G$, and $H$ be module-coherent\ $A$-functors. Then
\hskip 0pt plus 4cm\penalty1000\hskip 0pt plus -4cm\hbox{$\mathop{\mathbf{Bil}}\nolimits(F \times G, H)$} is module-coherent.
\end{corollary}
\begin{proof}
We may identify $\mathop{\mathbf{Bil}}\nolimits(F \times G, H)$ with $\mathop{\mathbf{Hom}}\nolimits(F,\mathop{\mathbf{Hom}}\nolimits(G,H))$. {\hfill$\square$}
\end{proof}
If $F$ and $G$ are algebra-valued $A$-functors, we let $\mathop{{\mathbf{Hom}}_{\kern1pt\operatoratfont alg}(F,G)$ denote
the $A$-functor given by
$$B \mapsto \mathop{\operatoratfont Hom}\nolimits_{\kern2pt{\operatoratfont algebra-valued} \ B-{\operatoratfont functors}} (F, G),$%
$where $F$ and $G$ are viewed as algebra-valued $B$-functors. Similarly, we
may define $\mathop{{\mathbf{Aut}}_{\kern1pt\operatoratfont alg}(F)$.
\begin{corollary}
Let $R$ and $S$ be module-coherent, algebra-valued $A$-functors. Then
$\mathop{{\mathbf{Hom}}_{\kern1pt\operatoratfont alg}(R,S)$ is coherent.
\end{corollary}
\begin{proof}
Let \mp[[ \mu_R || R \times R || R ]] and \mp[[ \mu_S || S \times S || S ]]
denote the multiplication maps. Then $\mathop{{\mathbf{Hom}}_{\kern1pt\operatoratfont alg}(R,S)$ is the kernel of the map:
\dmapx[[ \mathop{\mathbf{Hom}}\nolimits(R,S) || \mathop{\mathbf{Bil}}\nolimits(R \times R, S) ]]%
given by $f \mapsto [f \circ \mu_R] - [\mu_S \circ (f \times f)]$. {\hfill$\square$}
\end{proof}
\begin{example}
Let $R$ and $S$ be module-finite $A$-algebras (not necessarily commutative).
Then the $A$-functor defined by
$$B \mapsto \mathop{\operatoratfont Hom}\nolimits_{B-{\operatoratfont alg}}(R \o*_A B, S \o*_A B)$%
$is coherent.
\end{example}
\begin{corollary}\label{autalg-is-coherent}
For any module-coherent, algebra-valued $A$-functor $R$, \hskip 0pt plus 4cm\penalty1000\hskip 0pt plus -4cm $\mathop{{\mathbf{Aut}}_{\kern1pt\operatoratfont alg}(R)$ is coherent.
\end{corollary}
\begin{example}
Let $R$ be a module-finite $A$-algebra (not necessarily commutative). Then the
$A$-functor given by $B \mapsto \mathop{\operatoratfont Aut}\nolimits_{B-{\operatoratfont alg}}(R \o*_A B)$
is coherent.
\end{example}
\block{The global case}\label{global-section}
We have defined the notion of $X$-functor in the introduction; these form a
category \cat{$X$-functors}. If $X = \mathop{\operatoratfont Spec}\nolimits(A)$, then there is a canonical
equivalence of categories between \cat{$A$-functors which are Zariski sheaves}
and \cat{$X$-functors which are Zariski sheaves}; we can pass back and forth
freely between these two categories. Similarly, for an arbitrary noetherian
scheme $X$, we may (instead of looking at $X$-functors which are Zariski
sheaves) look at functors which are Zariski sheaves and whose source is
$$\opcat{$X$-schemes which are quasi-compact}.$$
Let \mp[[ \phi || X || Y ]] be a morphism of noetherian schemes. We
consider pull-back and push-forward of functors:
\begin{itemize}
\item Let $F$ be a $Y$-functor. Then there is an $X$-functor $\phi^*F$, given
by $(\phi^*F)(T) = F(T)$ for all $X$-schemes $T$. Sometimes we will
write $\phi|_X$ instead of $\phi^*F$, and refer to the
{\it restriction\/} of $F$ to $X$.
\item Let $G$ be an $X$-functor. Then there is a $Y$-functor $\phi_*G$, given
by $(\phi_*G)(S) = F(X \times_Y S)$, for all $Y$-schemes $S$.
\item Let $F$ be a $Y$-functor. Then the $Y$-functor $\phi_*\phi^*(F)$ is
given by $S \mapsto F(X \times_Y S)$, for all $Y$-schemes $S$. Instead
of writing $\phi_*\phi^*(F)$, we may refer to ``$F|_X$, viewed as a
$Y$-functor.''
\end{itemize}
\par\noindent Note that both $\phi^*$ and $\phi_*$ are {\it exact\/} functors.
If $Y$ is an $X$-scheme, \mp[[ \pi || Y || X ]] is a morphism of schemes, and
${\cal{M}}$ is a quasi-coherent\ ${\cal O}_X$-module, we let ${\cal{M}}_Y$ denote $\pi^*{\cal{M}}$.
For any such ${\cal{M}}$, there is an $X$-functor ${\underline{\cal M}}$ given by
${\underline{\cal M}}(Y) = \Gamma(Y,{\cal{M}}_Y)$.
An $X$-functor is {\it strictly coherent\/} if it is isomorphic to ${\underline{\cal M}}$
for some coherent ${\cal O}_X$-module ${\cal{M}}$. An $X$-functor is {\it coherent\/}
if it is an iterated finite limit of strictly coherent $X$-functors, where the
limits are taken in \cat{$X$-functors}. Similarly, one may define
{\it quasi-coherent\/} $X$-functors, by allowing ${\cal{M}}$ to be quasi-coherent.
We may define the {\it level\/} of a coherent $X$-functor, exactly as we have
done for coherent $A$-functors. We may also define the {\it level\/} of a
quasi-coherent\ $X$-functor, and it is distantly conceivable that there exists a coherent
$X$-functor whose level is lower when viewed as a quasi-coherent\ $X$-functor.
It is very important to note that if $X = \mathop{\operatoratfont Spec}\nolimits(A)$, then coherent $X$-functors
are essentially the same as coherent $A$-functors. This follows from the
fact that coherent $X$-functors are sheaves for the Zariski topology. Indeed
we have:
\begin{prop}\label{sheaf}
Let $F$ be a quasi-coherent\ $X$-functor. Then $F$ is a sheaf for the fpqc topology.
\end{prop}
\begin{proof}
If $F = {\underline{\cal M}}$, for some quasi-coherent\ ${\cal O}_X$-module ${\cal{M}}$, then the statement is true.
The proposition follows because any limit of sheaves is a sheaf. {\hfill$\square$}
\end{proof}
\begin{definition}
A {\it module-valued\ $X$-functor\/} is an (abelian group)-valued $X$-functor $F$,
together with the structure of a $\Gamma(Y,{\cal O}_Y)$-module on each set $F(Y)$,
with the property that for each map of $X$-schemes \mapx[[ Y_1 || Y_2 ]], the
induced map \mapx[[ F(Y_2) || F(Y_1) ]] is a homomorphism of
$\Gamma(Y_2, {\cal O}_{Y_2})$-modules.
\end{definition}
The module-valued\ $X$-functors form an abelian category. When we have $X = \mathop{\operatoratfont Spec}\nolimits(A)$,
there is a canonical equivalence of categories between:
$$\cat{module-valued\ $A$-functors which are Zariski sheaves}$%
$and
$$\cat{module-valued\ $X$-functors which are Zariski sheaves}.$%
$
If ${\cal{M}}$ is a quasi-coherent\ ${\cal O}_X$-module, we let ${\underline{\cal M}}$ denote the module-valued\ $X$-functor
given by $Y \mapsto \Gamma(Y,{\cal{M}}_Y)$. A module-valued\ $X$-functor $F$ is
{\it strictly module-coherent\/} if there exists a coherent ${\cal O}_X$-module ${\cal{M}}$ such that
$F \cong {\underline{\cal M}}$.
\begin{definition}
A module-valued\ $X$-functor is {\it module-coherent\/} if it may be obtained as an iterated finite
limit of strictly module-coherent\ $X$-functors. These limits are all taken in
\cat{module-valued\ $X$-functors}.
\end{definition}
In a similar way, one may define module-quasi-coherent\ $X$-functors.
If $X = \mathop{\operatoratfont Spec}\nolimits(A)$, then module-coherent\ $X$-functors are essentially the same as
module-coherent\ $A$-functors. For arbitrary $X$, the theory of module-coherent\ $X$-functors runs
parallel to the theory of module-coherent\ $A$-functors, but there is one difference. When
one takes the cokernel of a morphism of module-coherent\ $X$-functors, it is necessary to
take the associated sheaf (with respect to\ the Zariski topology), in order to obtain a
module-coherent\ $X$-functor.
The {\it level\/} of a module-coherent\ (or module-quasi-coherent) $X$-functor is defined analogously to
the definition of level for a module-coherent\ $A$-functor. As in that case, we will find
ultimately that the level is always $\leq 1$.
An $X$-functor $F$ is {\it locally coherent\/} if $F$ is a Zariski sheaf and if
there exists an open cover $\vec U1n$ of $X$ such that $F|_{U_i}$ is a coherent
$U_i$-functor for each $i$. Similarly, a module-valued\ $X$-functor $F$ is
{\it locally module-coherent\/} if $F$ is a Zariski sheaf and if there exists an
open cover $\vec U1n$ of $X$ such that $F|_{U_i}$ is a module-coherent\ $U_i$-functor for
each $i$. We will show shortly that any locally module-coherent\ $X$-functor is module-coherent,
assuming that $X$ is separated.
(One can also define {\it locally quasi-coherent\/} and {\it locally module-quasi-coherent\/}
$X$-functors.) We do not know the answer to the analogous question for locally
coherent $X$-functors:
\begin{conjecture}\label{locally-coherent-conjecture}
Every locally coherent $X$-functor is coherent.
\end{conjecture}
If the conjecture were true, it would follow immediately that if an $X$-functor
$F$ represents an affine $X$-scheme of finite type, then $F$ is coherent.
(This is true if $X$ is affine: see example \pref{representable-example} from
\S\ref{examples-section}.)
More generally, one could ask:
\begin{problemx}
Let \mp[[ \phi || Y || X ]] be a faithfully flat morphism of noetherian
schemes. Let $F$ be an $X$-functor, which is a sheaf for the fpqc
topology. Assume that $\phi^*F$ is a coherent $Y$-functor. Does it follow
that $F$ is a coherent $X$-functor?
\end{problemx}
We now consider push-forward and pull-back of coherent and quasi-coherent
$X$-functors. These operations also make sense for module-valued\ $X$-functors.
\begin{prop}\label{pull}
Let \mp[[ \phi || X || Y ]] be a morphism of noetherian schemes. Let
$F$ be a coherent $Y$-functor. Then $\phi^*F$ is coherent. Similarly,
if $F$ is a module-coherent\ $Y$-functor, then $\phi^*F$ is module-coherent.
\end{prop}
\begin{proof}
Suppose that $F$ is a coherent $Y$-functor. (The parallel case for
module-coherent\ $Y$-functors is left to the reader.) Let $n$ be the level of $F$.
First suppose that $n = 0$, so $F \cong {\underline{\cal M}}$ for some coherent
${\cal O}_Y$-module ${\cal{M}}$. But then $\phi^*F \cong \underline{\phi^*{\cal{M}}}$, so $\phi^*F$
is coherent. Now suppose that $n \geq 1$. By an unstated analog of
\pref{lesx-exists}, there is a left exact sequence:
\lesx{F}{G}{{\underline{\cal M}}%
}of $X$-functors in which $G$ is coherent of level $n-1$ and ${\cal{M}}$ is a
coherent ${\cal O}_X$-module. By induction on $n$, we may assume that\ $\phi^*G$ is coherent.
Since $\phi^*$ is an exact functor, we have a left exact sequence:
\lesxdot{\phi^*F}{\phi^*G}{\phi^*{\underline{\cal M}}%
}Hence $\phi^*F$ is coherent. {\hfill$\square$}
\end{proof}
\begin{prop}\label{push}
Let \mp[[ \phi || X || Y ]] be a morphism of noetherian schemes.
Let $F$ be an $X$-functor.
\begin{alphalist}
\item If $F$ is quasi-coherent\ and $\phi$ is affine, then $\phi_*F$ is quasi-coherent.
\item If $F$ is coherent and $\phi$ is finite, then $\phi_*F$ is coherent.
\item Parallel statements apply if $F$ is a module-valued\ $X$-functor.
\end{alphalist}
\end{prop}
\begin{proof}
(a): Let $n$ be the level of $F$. If $n = 0$, $F = {\underline{\cal M}}$ for some
quasi-coherent\ ${\cal O}_X$-module ${\cal{M}}$, so
$(\phi_*F)(T) = \Gamma(X \times_Y T, {\cal{M}}_{X \times_Y T})$ for any $Y$-scheme
$T$. By (\Lcitemark 20\Rcitemark \ I:9.1.1), it follows that
$(\phi_*F)(T) \cong \Gamma(T, (\phi_*{\cal{M}})_T)$. Hence $\phi_*F$ is coherent.
Now suppose that $n \geq 1$. By the (unstated) analog of
\pref{lesx-exists} for quasi-coherent\ $X$-functors, there is a left exact sequence:
\lesx{F}{G}{{\underline{\cal M}}%
}in which $G$ is a quasi-coherent\ $X$-functor of level $n-1$ and ${\cal{M}}$ is a
quasi-coherent\ ${\cal O}_X$-module.
Then we have a left exact sequence:
\lesx{\phi_*F}{\phi_*G}{\phi_*{\underline{\cal M}}%
}By induction on $n$, we may assume that\ $\phi_*G$ is quasi-coherent. By the $n=0$ case, we may
identify $\phi_*{\underline{\cal M}}$ with $\underline{\phi_*{\cal{M}}}$. Hence $\phi_*F$ is quasi-coherent.
Parts (b) and (c) are left to the reader. {\hfill$\square$}
\end{proof}
The next result is key, since it permits us to reduce to the affine case, and
thereby obtain the analogs of the results for module-valued\ $A$-functors. In
particular, it will follow that many examples of $X$-functors are coherent.
However, the reader interested only in the Picard group results may ignore the
next result and its corollaries, since for purposes of the finiteness result
\pref{coherent-implies-linear}, it is sufficient to know that a given
$X$-functor is {\it locally coherent}.
\begin{theorem}\label{locally-mc}
Assume that $X$ is separated. Let $F$ be a locally
module-quasi-coherent\ [resp.\ locally module-coherent\/] $X$-functor. Then $F$ is module-quasi-coherent\ [resp.\ module-coherent].
\end{theorem}
\begin{proof}
In the course of the proof we refer to sheaves, which shall always mean
sheaves for the {\it Zariski topology}. We work not with $X$-functors, but
with functors whose source is \opcat{$X$-schemes which are quasi-compact}, as
discussed briefly at the beginning of this section.
Let $\vec U1n$ be as in the definition of {\it locally module-quasi-coherent\/}
(or {\it locally module-coherent}).
By \pref{pull}, we may assume that\ each $U_i$ is affine. Since $X$ is separated, it
follows that the open subschemes $U_i \cap U_j$ are affine and that the
inclusions of $U_i$ in $X$ and of $U_i \cap U_j$ in $X$ are affine morphisms.
First we prove the module-quasi-coherent\ case. (This will be needed for the module-coherent\ case.)
Regard $F|_{U_i}$ and $F|_{U_i \cap U_j}$ as module-valued\ $X$-functors. It follows
from \pref{push} and \pref{pull} that these are module-quasi-coherent.
Because $F$ is a sheaf, we have a left exact sequence:
\les{F}{\prod_{i=1}^n F|_{U_i}}{\prod_{1 \leq i < j \leq n} F|_{U_i \cap U_j}%
}of module-valued\ $X$-functors. Hence $F$ is module-quasi-coherent.
Now we show that if $F$ is a module-quasi-coherent\ $X$-functor, then there exists a morphism
\mp[[ \phi || {\cal{M}} || {\cal{N}} ]] of quasi-coherent\ ${\cal O}_X$-modules such that
$F \cong \mathop{\underline{\operatoratfont Ker}}\nolimits(\phi)$. By an unstated analog of \pref{les-exists},
we may embed $F$ as a sub-module-valued-functor of
${\underline{\cal M}}$ for some quasi-coherent\ ${\cal O}_X$-module ${\cal{M}}$. Let $G = ({\underline{\cal M}}/F)^*$, where
the superscript $*$ denotes sheafification.
By \pref{quasi-coherence-of-cokernel}, it follows that
$G$ is locally module-quasi-coherent, so (by the first part of the proof) $G$ is module-quasi-coherent. Embed
$G$ as a sub-module-valued-functor of ${\underline{\cal N}}$ for some quasi-coherent\ ${\cal O}_X$-module ${\cal{N}}$.
Let \mp[[ \phi || {\cal{M}} || {\cal{N}} ]] be the induced map. Then $F = \mathop{\underline{\operatoratfont Ker}}\nolimits(\phi)$,
as required.
Now we begin the proof of the module-coherent\ case. By what we have already shown,
we may assume that\ there exists a morphism \mp[[ \phi || {\cal{M}} || {\cal{N}} ]] of
quasi-coherent\ ${\cal O}_X$-modules such that
$F = \mathop{\underline{\operatoratfont Ker}}\nolimits(\phi)$. We will show that there exist coherent sub-${\cal O}_X$-modules
${\cal{M}}_0 \subset {\cal{M}}_1 \subset {\cal{M}}$ and ${\cal{N}}_0 \subset {\cal{N}}$ such that
$\phi({\cal{M}}_0) \subset {\cal{N}}_0$ and such that in the induced diagram:
\diagramx{\underline{{\cal{M}}_1}\cr
\mapN{g}\cr
\underline{{\cal{M}}_0}&\mapE{f}&\underline{{\cal{N}}_0}\cr%
}we have $F \cong g[\ker(f)]^*$.
Let us verify that the construction of this data will complete the proof. We
must show that $g[\ker(f)]^*$ is module-coherent. Let ${\cal{P}}$ denote the co-fiber product
of ${\cal{M}}_1$ with ${\cal{N}}_0$ over ${\cal{M}}_0$, taken in the category of
quasi-coherent\ ${\cal O}_X$-modules. Then in fact the induced diagram
\diagramx{\underline{{\cal{M}}_1}&\mapE{h}&{\underline{\cal P}}\cr
\mapN{g}&&\mapN{}\cr
\underline{{\cal{M}}_0}&\mapE{f}&\underline{{\cal{N}}_0}\cr%
}is cocartesian, if it is viewed as a diagram in
$$\cat{module-valued\ $X$-functors which are sheaves}.$%
$Since we have $\ker(h) \cong g[\ker(f)]^*$, the theorem will follow.
It remains to construct the data. Certainly, for any choice of data,
there is a canonical morphism
\dmap[[ \psi || g[\ker(f)]^* || F ]]%
of module-valued\ $X$-functors.
We work on choosing ${\cal{M}}_0$. Let $\sets {\cal{M}}\lambda\Lambda$ be the coherent
sub-${\cal O}_X$-modules of ${\cal{M}}$. Let $H_\lambda$ be the sheafified image of the
map \mapx[[ \underline{{\cal{M}}_\lambda} || {\underline{\cal M}} ]]. Then the $H_\lambda$ form a
directed system of module-valued\ subfunctors of ${\underline{\cal M}}$ (which are sheaves), whose union
is ${\underline{\cal M}}$. (The validity of the last assertion depends on the simplifying
assumption made in the first paragraph of this proof, to the effect that we
work only with quasi-compact $X$-schemes.)
Let $F_\lambda = F \cap H_\lambda$. Then the $F_\lambda$
form a directed system of module-valued\ subfunctors of $F$ (which are sheaves),
whose union is $F$. Then it follows from \pref{noetherian} that
$F = F_\lambda$ for some ${\lambda \in \Lambda}$. Let ${\cal{M}}_0 = {\cal{M}}_\lambda$. Then
$F$ is contained in the sheafified image of the map
\mapx[[ \underline{{\cal{M}}_0} || {\underline{\cal M}} ]].
Now we work on choosing ${\cal{N}}_0$. Let $\sets {\cal{N}}\lambda\Lambda$ be the
coherent sub-${\cal O}_X$-modules of ${\cal{N}}$ which contain $\phi({\cal{M}}_0)$. Let
$I_\lambda$ be the sheafified image of the map
\mapx[[ \ker[\underline{{\cal{M}}_0}\ \mapE{}\ \underline{{\cal{N}}_\lambda}] || {\underline{\cal M}} ]]. Then the
$I_\lambda$ form a directed system of module-valued\ subfunctors of $F$ (which are
sheaves). We will show that the union of the $I_\lambda$ is $F$. Let
$J_\lambda = \ker[\underline{{\cal{M}}_0}\ \mapE{}\ \underline{{\cal{N}}_\lambda}]$. It suffices to
show that $\ker[\underline{{\cal{M}}_0}\ \mapE{}\ {\underline{\cal N}}]$ is the union of the $J_\lambda$.
Let $Y$ be a quasi-compact $X$-scheme. Let $\alpha \in \Gamma[({\cal{M}}_0)_Y]$,
and assume that $\alpha \mapsto 0$ in $\Gamma({\cal{N}}_Y)$.
We must show that there exists some ${\cal{N}}_\lambda$ such that
$\alpha \mapsto 0$ in $\Gamma[({\cal{N}}_\lambda)_Y]$. By
(\Lcitemark 20\Rcitemark \ I.6.9.9), we know that ${\cal{N}}$ is the direct limit of
the
${\cal{N}}_\lambda$. It follows that ${\cal{N}}_Y$ is the direct limit of the
$({\cal{N}}_\lambda)_Y$. This implies the statement about $\alpha \mapsto 0$, and
hence that the union of the $I_\lambda$ is $F$. Arguing as in the preceding
paragraph, we see that for some ${\lambda \in \Lambda}$, we have $I_\lambda = F$. Let
${\cal{N}}_0 = {\cal{N}}_\lambda$. It follows now that now matter how we choose
${\cal{M}}_1$, the map $\psi$ will be an epimorphism, when viewed as a morphism
in the category of module-valued\ $X$-functors which are sheaves.
Now we work on choosing ${\cal{M}}_1$. Let $G = \ker(f)$. Then $G$ is module-coherent. Let
$K = \ker[G\ \mapE{}\ {\underline{\cal M}} ]$. Then $K = \ker[G\ \mapE{}\ F ]$, so $K$ is
locally module-coherent. Let $\sets {\cal{M}}\lambda\Lambda$ be
the coherent sub-${\cal O}_X$-modules of ${\cal{M}}$ which contain ${\cal{M}}_0$. (These
${\cal{M}}_\lambda$ are not the same as those defined earlier.) Let
$K_\lambda = \ker[ G\ \mapE{}\ \underline{{\cal{M}}_\lambda} ]$. Then the
$K_\lambda$ form a directed system of module-valued\ subfunctors of $K$ (which are
sheaves). Arguing as in the preceding paragraph, we see that the union of
the $K_\lambda$ is $K$. But $K$ is locally module-coherent,
so it follows from \pref{noetherian} that $K_\lambda = K$ for some ${\lambda \in \Lambda}$.
Let ${\cal{M}}_1 = {\cal{M}}_\lambda$.
Then $\psi$ is a monomorphism. Hence $\psi$ is an isomorphism. {\hfill$\square$}
\end{proof}
\begin{corollary}\label{mump}
Assume that $X$ is separated.
Let \mp[[ \phi || F || G ]] be a morphism of module-coherent\ $X$-funct\-ors. Then
the Zariski sheaf associated to $\mathop{\operatoratfont Coker}\nolimits(\phi)$ is module-coherent.
\end{corollary}
\begin{corollary}
Assume that $X$ is separated. Then the category of module-coherent\ $X$-functors is
abelian.
\end{corollary}
\begin{corollary}
Assume that $X$ is separated.
Let $F$ be a module-coherent\ $X$-functor. Then $F$ has level $\leq 1$. That is,
there exists a left exact sequence:
\les{F}{{\underline{\cal M}}}{{\underline{\cal N}}%
}in which ${\cal{M}}$ and ${\cal{N}}$ are coherent ${\cal O}_X$-modules.
\end{corollary}
The constructions $\mathop{\mathbf{Hom}}\nolimits$, $\mathop{\mathbf{Aut}}\nolimits$ and so forth which we defined in
\S\ref{examples-section} make sense for $X$-functors. For example,
if $F$ and $G$ are module-valued $X$-functors, we let
$\mathop{\mathbf{Hom}}\nolimits(F,G)$ denote the $X$-functor given by
$$Y \mapsto \mathop{\operatoratfont Hom}\nolimits_{\kern2pt{\operatoratfont module-valued}\ Y-{\operatoratfont functors}}
(F|_Y, G|_Y).$%
$
\begin{prop}
Assume that $X$ is separated.
Let $F$ and $G$ be module-coherent\ $X$-functors. Then \hskip 0pt plus 4cm\penalty1000\hskip 0pt plus -4cm $\mathop{\mathbf{Hom}}\nolimits(F,G)$ is module-coherent.
\end{prop}
\begin{proof}
By \pref{HOM-coherent} and \pref{locally-mc}, it suffices to show that
$\mathop{\mathbf{Hom}}\nolimits(F,G)$ is a Zariski sheaf. This can be directly checked from the
definition. {\hfill$\square$}
\end{proof}
Similarly, we have:
\begin{corollary}\label{lump}
Assume that $X$ is separated. Then
for any module-coherent\ $X$-functor $F$, $\mathop{\mathbf{End}}\nolimits(F)$ is module-coherent, and $\mathop{\mathbf{Aut}}\nolimits(F)$ is coherent.
Let $R$ and $S$ be module-coherent, algebra-valued $X$-functors. Then
$\mathop{{\mathbf{Hom}}_{\kern1pt\operatoratfont alg}(R,S)$ and $\mathop{{\mathbf{Aut}}_{\kern1pt\operatoratfont alg}(R)$ are coherent.
\end{corollary}
Now we consider the extent to which a module-quasi-coherent\ $A$-functor can be viewed as a
direct limit of module-coherent\ $A$-functors. These considerations will enter into an
analysis of extensions of module-coherent\ $X$-functors, which will be the last topic
discussed in this section.
Unfortunately, it is not the case that every module-quasi-coherent\ $A$-functor $H$ is the
direct limit of its module-coherent\ subfunctors. For an example, let $H = \mathop{\operatoratfont Im}\nolimits(\phi)$,
where $\phi$ is as in remark \pref{not-coh-example}. If there existed a
directed system $\sets H\lambda\Lambda$ of module-coherent\ subfunctors of $H$, with union
$H$, it would follow by \pref{noetherian}, applied with $F = \underline{\xmode{\Bbb Z}}$,
$F_\lambda = \phi^{-1}(H_\lambda)$, that $\phi$ factors through a
module-coherent\ subfunctor of $\underline{{\Bbb Q}\kern1pt}$, and hence that $\ker(\phi)$ is module-coherent, which is not
the case.
\begin{definition}
A module-valued\ $A$-functor $C$ is {\it bar-module-coherent\/} if it is module-quasi-coherent\ and if
there exists a module-coherent\ $A$-functor $F$, together with an epimorphism
\mapx[[ F || C ]].
\end{definition}
A bar-module-coherent\ $A$-functor need not be module-coherent. For an example, let $H = \mathop{\operatoratfont Im}\nolimits(\phi)$,
where $\phi$ is as in remark \pref{not-coh-example}. Then $H$ is bar-module-coherent, but
not module-coherent, since otherwise $\ker(\phi)$ would be module-coherent.
\begin{lemma}\label{directed-system-of-images}
Let $F$ be a module-quasi-coherent\ $A$-functor. Then there exists a directed
system $\sets F\lambda\Lambda$ of bar-module-coherent\ subfunctors of $F$, with union $F$.
\end{lemma}
\begin{proof}
Choose $A$-modules $M$, $N$ and a left exact sequence:
\lesmaps{F}{}{{\underline{M}}}{h}{{\underline{N}}%
}of module-valued $A$-functors, in which $h$ is induced by a homomorphism
\mp[[ \phi || M || N ]] of $A$-modules. Let ${\cal{S}}$ denote the
collection $\setof{(M_\lambda,N_\lambda)}_{{\lambda \in \Lambda}}$ consisting of all pairs
\hskip 0pt plus 4cm\penalty1000\hskip 0pt plus -4cm\hbox{$(M_\lambda,N_\lambda)$} in which $M_\lambda$ is a finitely generated\ submodule of
$M$, $N_\lambda$ is a finitely generated\ submodule of $N$, and
$\phi(M_\lambda) \subset N_\lambda$. For each ${\lambda \in \Lambda}$,
let \mp[[ h_\lambda || \underline{M_\lambda} || \underline{N_{\lambda}} ]] be the
induced morphism of $A$-functors. Let $K_\lambda = \ker(h_\lambda)$. Then
$K_\lambda$ is module-coherent. There is a canonical map
\mp[[ f_\lambda || K_\lambda || F ]]. Let $F_\lambda = \mathop{\operatoratfont Im}\nolimits(f_\lambda)$. Then
$F_\lambda$ is bar-module-coherent, and the $F_\lambda$ form a directed system of
subfunctors of $F$.
Let $B$ be a commutative $A$-algebra, and let $c \in F(B)$. Then $c \in M_B$.
Choose a finitely generated\ submodule $M_\lambda \subset M$ and an element
$c_\lambda \in (M_\lambda)_B$ such that $c_\lambda \mapsto c$. There exists a
finitely generated\ submodule $N_\lambda$ of $N$ such that $\phi(M_\lambda) \subset N_\lambda$
and such that $c_\lambda \mapsto 0$ in $(N_\lambda)_B$. It follows that
$\cup_{{\lambda \in \Lambda}}F_\lambda = F$. {\hfill$\square$}
\end{proof}
We close this section with some questions and a result about extensions:
\begin{problemx}
Let
\Rowfive{1}{F'}{F}{F''}{1%
}be a short exact sequence of group-valued $X$-functors. Assume that $F'$ and
$F''$ are coherent. Is $F$ coherent?
\end{problemx}
\begin{problemx}
Let
\ses{F'}{F}{F''%
}be a short exact sequence of module-valued\ $X$-functors. Assume that $F'$ and $F''$
are
module-coherent. Is $F$ module-coherent?
\end{problemx}
We can prove this if we assume that $F$ is module-quasi-coherent\ and that $X$ is separated:
\begin{prop}\label{extension-is-mc}
Assume that $X$ is separated. Let
\ses{F'}{F}{F''%
}be a short exact sequence of module-valued\ $X$-functors. Assume that $F'$ and $F''$
are module-coherent. Assume that $F$ is module-quasi-coherent. Then $F$ is module-coherent.
\end{prop}
\begin{proof%
}By \pref{locally-mc}, we may reduce to working with $A$-functors. By
\pref{directed-system-of-images}, $F$ is the direct limit of its
bar-module-coherent\ subfunctors. By \pref{noetherian}, it follows that there exists a
bar-module-coherent\ subfunctor $B$ of $F$ such that $B$ maps onto $F''$. Since $F'$ is
module-coherent, we see that $F$ is itself bar-module-coherent. Choose a module-coherent\ $A$-functor $C$ and
an epimorphism \mp[[ \beta || C || F ]].
Let $P$ be the fiber product of $F'$ and $C$ over $F$. Then we have a
commutative diagram with exact rows:
\diagramx{\rowfive{0}{P}{C}{F''}{0}\cr
&&\mapS{\alpha}&&\mapS{\beta}&&\mapS{=}\cr
\rowfive{0}{F'}{F}{F''}{0}\makenull{.}%
}Hence $\ker(\alpha) \cong \ker(\beta)$. Since $C$ and $F''$ are module-coherent, so is
$P$. Since $F'$ and $P$ are module-coherent, so is $\ker(\alpha)$. Hence $\ker(\beta)$
is module-coherent. Since $\ker(\beta)$ and $C$ are module-coherent, it follows by
\pref{coherence-of-cokernel} that $F$ is also module-coherent. {\hfill$\square$}
\end{proof}
This fact will be used in the proof of \pref{mulch-material}.
\block{Continuity}\label{continuity-section}
We consider the extent to which quasi-coherent\ $A$-functors preserve limits, and briefly,
the extent to which they preserve colimits. We prove that a module-quasi-coherent\ $A$-functor
which preserves products is module-coherent. Although these topics do not play
much of a role in the subsequent parts of this paper, they are very natural.
Some important examples of limit and colimit preservation which have
arisen previously are Grothendieck's theorem on formal functions
(see e.g.{\ }\Lcitemark 22\Rcitemark \ III:11), and Grothendieck's
notion of functors which are locally of finite presentation
(\Lcitemark 1\Rcitemark \ 1.5), which enters into Artin's
criterion for representability (\Lcitemark 3\Rcitemark \ 3.4).
We begin by recalling some definitions about continuity of functors.
Let ${\cal{C}}$ and ${\cal{A}}$ be complete (meaning small-complete) categories, and let
\mp[[ F || {\cal{C}} || {\cal{A}} ]] be any functor. Then $F$ is {\it continuous\/} if
it {\it preserves limits}, i.e.\ if for every small category ${\cal{D}}$,
and every functor \mp[[ H || {\cal{D}} || {\cal{C}} ]], the canonical map
\dmapx[[ F \left( \displaystyle{\lim_{\overleftarrow{\hphantom{\lim}}}}\ H \right) || \displaystyle{\lim_{\overleftarrow{\hphantom{\lim}}}}(F \circ H) ]]%
is an isomorphism. (See e.g.{\ }\Lcitemark 27\Rcitemark \ V.4.) It is also
of interest to know if $F$ preserves more restricted sorts of limits,
e.g.\ does it preserve arbitrary products, or does it preserve equalizers.
These conditions correspond to placing appropriate restrictions on ${\cal{D}}$.
For particular sorts of limits, one can usually rephrase the continuity
condition in a simpler way. For example, $F$ preserves products if and only if\ for
every set \sets XiI of objects in ${\cal{C}}$, the canonical map
\dmapx[[ F \left( \prod_{i \in I} X_i \right) || \prod_{i \in I} F(X_i) ]]%
is an isomorphism.
Just as one can check a category for completeness by checking if it has
products and equalizers, so one can check a functor for continuity by
checking if it preserves products and equalizers.
We will study the continuity properties of $A$-functors. There are two general
observations to be made. The first observation is that an
(abelian group)-valued $A$-functor or a module-valued\ $A$-functor is continuous (or
preserves a particular kind of limit) if and only if\ the same statement holds for the
underlying functor from \cat{commutative $A$-algebras} to \cat{sets}. The
second observation is the following lemma, whose proof is left to the reader:
\begin{lemma}\label{kernel-preserves}
Let $M$ be an $A$-module and let
\lesx{F}{G}{{\underline{M}}%
}be a left exact sequence of $A$-functors. If ${\underline{M}}$ and $G$ preserve a
particular type of limit, then so does $F$.
\end{lemma}
The phrase ``particular type of limit'' is to be construed as referring to
a class of limits which is constrained by some restriction on the categories
${\cal{D}}$ and/or the functors \mp[[ H || {\cal{D}} || \cat{commutative $A$-algebras} ]]
which are allowed.
We proceed to investigate the extent to which quasi-coherent\ $A$-functors preserve
various types of limits. First we consider finite limits, that is limits in
which the category ${\cal{D}}$ has only finitely many objects and morphisms.
The most important examples are finite products (including terminal objects),
and equalizers. Also, if a functor preserves finite products and equalizers,
then it preserves all finite limits. As for finite products, one sees
easily (using \ref{kernel-preserves}) that:
\begin{prop}\label{number-1}
Let $F$ be a quasi-coherent\ $A$-functor. Then $F$ preserves finite products.
\end{prop}
Now we consider equalizers. An $A$-module $M$ is flat if and only if\ the functor
\fun[[ \o*_A M || $A$-modules || $A$-modules ]] preserves equalizers,
so the analogous fact for $A$-functors is hardly surprising:
\begin{prop}\label{commutes-eq-iff-flat}
Let $M$ be an $A$-module. Then the $A$-functor ${\underline{M}}$ preserves
equalizers if and only if\ $M$ is flat.
\end{prop}
\begin{proof}
The case where $M$ is flat is left to the reader. So suppose that $M$ is not
flat. Then (see e.g.{\ }\Lcitemark 28\Rcitemark \ 3.53) there exists an
ideal $I \subset A$ such that the induced map \mp[[ \phi || M \o*_A I || M ]] is
not injective. Let $y \in \ker(\phi) - \setof{0}$. Let $\vec a1n$ be
generators for $I$. Choose $\vec m1n \in M$ such that
$y = \sum_{i=1}^n m_i \o* a_i$. Let
$B = A[\vec x1n]/(\setof{x_i x_j}_{1 \leq i,j \leq n})$. Let
\mp[[ f,g || B || A ]] be the $A$-algebra maps given by $f(x_i) = a_i$
and $g(x_i) = 0$ for each $i$. Let $p \in M \o*_A B$ be
$\sum_{i=1}^n m_i \o* x_i$. Then $(f \o*_A M)(p) = (g \o*_A M)(p)$. Let
$\mathop{\operatoratfont Eq}\nolimits(f,g)$ be the equalizer of $f$ and $g$. Since $y \not= 0$, it follows
(after a little work) that $p$ does not lie in the image of the canonical map
\mp[[ \lambda || \mathop{\operatoratfont Eq}\nolimits(f,g) \o*_A M || \mathop{\operatoratfont Eq}\nolimits(f \o*_A M, g \o*_A M) ]], and hence
that $\lambda$ is not an isomorphism. Hence ${\underline{M}}$ does not preserve the
equalizer of $f$ and $g$. {\hfill$\square$}
\end{proof}
It follows that ${\underline{M}}$ preserves finite limits if and only if\ $M$ is flat. In
addition, \pref{kernel-preserves} yields the following corollary:
\begin{corollary}\label{number-2}
Any quasi-coherent\ $A$-functor which is built up from flat modules will preserve
finite limits.
\end{corollary}
It is worth noting that if $F$ is an $A$-functor which preserves finite limits,
then $F$ is a sheaf with respect to\ the fppf topology; indeed the sheaf axioms may be
viewed as a statement about continuity. While quasi-coherent\ $A$-functors do not always
preserve finite limits, we do know that they are sheaves with respect to\ the
fppf topology.
\begin{remark}\label{number-4}
It is not difficult to see that any coherent $A$-functor preserves
products. Any coherent $A$-functor which is built up from finitely generated\ projective
$A$-modules will also preserve equalizers, and hence all limits. Of
course, the $A$-functors which are built up in this way are exactly the
$A$-functors which are representable by an $A$-algebra of finite type.
\end{remark}
In general, coherent $A$-functors do not preserve inverse limits. For example,
if $A = \xmode{\Bbb Z}$, then the $A$-functor $\underline{\xmode{\Bbb Z}/2\xmode{\Bbb Z}}$ does not preserve the limit of
\diagramx{\xmode{\Bbb Z}[x]/(x^2) & \mapW{x\kern2pt\mapsto\kern2pt 3x} & \xmode{\Bbb Z}[x]/(x^2)
& \mapW{x\kern2pt\mapsto\kern2pt 3x} & \cdots.%
}The transition maps here are not surjective. One might hope that the limit
would be preserved if the transition maps were surjective, or at least if the
system satisfied the Mittag-Leffler condition (\Lcitemark 17\Rcitemark \
0:13.1).
Unfortunately this is not the case:
\begin{example}
Let $A = \xmode{\Bbb Z}$. Let
$$B_n = \xmode{\Bbb Z}[x_1, x_2,\ldots, y, z_1,\ldots,z_{n-1}]
/ (2x_1,2x_2,\ldots,Q),$%
$where $Q$ denotes the set of homogeneous quadratic polynomials in all of the
given variables. Form an inverse system of commutative $A$-algebras
\diagramx{B_1 & \mapW{} & B_2 & \mapW{} & B_3 & \mapW{} & \cdots%
}in which (for each $n > 1$) the transition map \mapx[[ B_n || B_{n-1} ]] is
given by $x_k \mapsto x_{k+1}$, $y \mapsto x_1 + y$, $z_1 \mapsto x_1$,
$z_k \mapsto z_{k-1}$ for $k \geq 2$. The transition maps of this system are
surjective.
The elements $2y, 2y, \ldots$ form a coherent sequence. Each element in
this sequence is divisible by $2$, but there is no coherent sequence which when
multiplied by $2$ yields $2y, 2y, \ldots$. Hence the $A$-functor $\underline{\xmode{\Bbb Z}/2\xmode{\Bbb Z}}$
does not preserve the limit of this system.
\end{example}
{}From (\Lcitemark 4\Rcitemark \ 10.13) and \pref{kernel-preserves} it follows
that:
\begin{prop}\label{number-5}
Let $F$ be a coherent $A$-functor. Let $B$ be a commutative noetherian
$A$-algebra. Let $I \subset B$ be an ideal. Then $F$ preserves the limit of
\diagramx{B/I & \mapW{} & B/I^2 & \mapW{} & B/I^3 & \cdots.}
\end{prop}
There may well be interesting situations in which coherent functors preserve
inverse limits, other than those given in \pref{number-5} and \pref{number-4}.
Our next objective is to show that a module-quasi-coherent\ $A$-functor which preserves
products is module-coherent. This as well as \pref{commutes-eq-iff-flat} allow one to use
knowledge about continuity to deduce some information about how a
quasi-coherent\ $A$-functor is built up.
\begin{lemma}\label{fg-sufficient}
Let $L_1$ and $L_2$ be module-quasi-coherent\ subfunctors of a module-quasi-coherent\ $A$-functor $H$. If
$L_1(B) \subset L_2(B)$ for every finitely generated\ commutative $A$-algebra $B$, then
$L_1 \subset L_2$.
\end{lemma}
\begin{proof}
Let $G = L_1 / (L_1 \cap L_2)$. Since $G$ is module-quasi-coherent, we have $G \cong \mathop{\underline{\operatoratfont Ker}}\nolimits(f)$
for some homomorphism \mp[[ f || M || N ]] of $A$-modules. Then
$\mathop{\operatoratfont Ker}\nolimits(f \o*_A B) = 0$ for every finitely generated\ commutative $A$-algebra $B$, from which it
follows that $\mathop{\operatoratfont Ker}\nolimits(f \o*_A B) = 0$ for {\it every\/} commutative $A$-algebra
$B$. Hence $G = 0$. {\hfill$\square$}
\end{proof}
\begin{lemma}\label{factor-fg}
Let \mp[[ \phi || H || N ]] be a homomorphism of $A$-modules. Assume that $H$
is finitely generated. Let $B$ be a commutative $A$-algebra. Let $h \in \mathop{\operatoratfont Ker}\nolimits(\phi \o*_A B)$.
Then there exists a finitely generated\ submodule $N_0$ of $N$ such that $\phi$ factors
through $N_0$ and such that $h \mapsto 0$ in $N_0 \o*_A B$.
\end{lemma}
\begin{proof}
Certainly $N$ is the direct limit of its finitely generated\ submodules which contain
$\phi(H)$. The statement follows from the fact that tensor products commute
with direct limits. {\hfill$\square$}
\end{proof}
\begin{prop}\label{product-preserving}
Let $F$ be a module-quasi-coherent\ $A$-functor which preserves products. Then $F$ is module-coherent.
\end{prop}
\begin{proof}
All tensor products in this proof are over $A$.
We may assume that $F = \mathop{\underline{\operatoratfont Ker}}\nolimits(f)$ for some homomorphism \mp[[ f || M || N ]] of
$A$-modules. We will show that there exists a finitely generated\ submodule $L \subset M$ such
that if \mp[[ i || L || M ]] is the inclusion, then $\mathop{\underline{\operatoratfont Ker}}\nolimits(f) \subset \mathop{\underline{\operatoratfont Im}}\nolimits(i)$.
Choose a complete set of isomorphism class representatives
$\sets B\lambda\Lambda$ for the finitely generated\ commutative $A$-algebras. Form the
disjoint union $T = \coprod_{{\lambda \in \Lambda}} \mathop{\operatoratfont Ker}\nolimits(f \o* B_\lambda)$. For any
$t \in T$, let $\lambda(t)$ denote the corresponding element of $\Lambda$. Let
$B = \prod_{t \in T} B_{\lambda(t)}$. The elements $t \in T$ define an element
$x \in \prod_{t \in T} \mathop{\operatoratfont Ker}\nolimits(f \o* B_{\lambda(t)})$. Since $F$ preserves
products, $x \in \mathop{\operatoratfont Ker}\nolimits(f \o* B)$. Write $x = \sum_{j=1}^r m_j \o* b_j$, where
$\vec m1r \in M$, $\vec b1r \in B$. Let $L$ be the submodule of $M$ generated
by $\vec m1r$. The expansion of $x$ defines an element ${\tilde{\lowercase{X}}} \in L \o* B$
with the property that ${\tilde{\lowercase{X}}} \mapsto x$. From this it follows that
$\mathop{\operatoratfont Ker}\nolimits(f \o* B_\lambda) \subset \mathop{\operatoratfont Im}\nolimits(i \o* B_\lambda)$ for every ${\lambda \in \Lambda}$. By
\pref{fg-sufficient}, we have $\mathop{\underline{\operatoratfont Ker}}\nolimits(f) \subset \mathop{\underline{\operatoratfont Im}}\nolimits(i)$.
Since ${\tilde{\lowercase{X}}} \mapsto 0$ in $N \o* B$, it follows from \pref{factor-fg} that
there exists a finitely generated\ submodule $N_0$ of $N$ such that \mapx[[ L || N ]] factors
through $N_0$ and such that ${\tilde{\lowercase{X}}} \mapsto 0$ in $N_0 \o* B$.
For any $t \in T$, let $\lambda = \lambda(t)$, and let
${\tilde{\lowercase{T}}} \in L \o* B_\lambda$ be the image of ${\tilde{\lowercase{X}}}$ under the canonical map
\mp[[ \pi_t || L \o* B || L \o* B_\lambda ]] which projects onto the \th{t}
factor. Then $(i \o* B_\lambda)({\tilde{\lowercase{T}}}) = t$, and ${\tilde{\lowercase{T}}} \mapsto 0$ in
$N_0 \o* B_\lambda$. Let ${\underline{K}} = \mathop{\underline{\operatoratfont Ker}}\nolimits(L\ \mapE{}\ N_0)$. Then
${\underline{K}}(B_\lambda)$ maps {\it onto\/} $F(B_\lambda)$. By \pref{fg-sufficient},
the map \mp[[ \psi || {\underline{K}} || F ]] is an epimorphism. In particular, $F$ is the
image of a module-coherent\ $A$-functor.
Let $Q = \mathop{\operatoratfont Ker}\nolimits(\psi)$. Since ${\underline{K}}$ is module-coherent, it preserves products. Since $F$
also preserves products, it follows by \pref{kernel-preserves} that $Q$
preserves products. Replaying the first part of the proof, with $F$ replaced
by $Q$, we see that $Q$ is also the image of a module-coherent\ $A$-functor. Hence $F$ is
the cokernel of a map of module-coherent\ $A$-functors, so $F$ is module-coherent. {\hfill$\square$}
\end{proof}
The last objective of this section is to consider (briefly) the extent to which
coherent functors preserve colimits.
Whether or not coherent functors preserve finite colimits and coproducts does
not seem to be an interesting question. One reason for this is that the
forgetful functor from \cat{groups} to \cat{sets} does not preserve coproducts
or finite colimits. (One can substitute various other categories for
\cat{groups} with the same outcome.) Therefore a functor from
\cat{commutative $A$-algebras} to \cat{groups} might preserve such colimits,
but the induced functor from \cat{commutative $A$-algebras} to \cat{sets}
might not. With either interpretation, preservation of coproducts or finite
colimits seems like a bizarre requirement.
On the other hand, the forgetful functor \funx[[ groups || sets ]] does
preserve direct limits\footnote{The reader is reminded that direct limits are
a type of {\it colimit}.}, and the same statement is valid with various other
categories substituted for \cat{groups}. Therefore, the situation for
direct limits is just like the situation which holds for all limits: an
(abelian group)-valued $A$-functor or a module-valued\ $A$-functor preserves direct
limits if and only if\ the same statement holds for the underlying functor from
\cat{commutative $A$-algebras} to \cat{sets}.
The analog of \pref{kernel-preserves} for direct limits is valid, and since
tensor products commute with direct limits, it follows that any
quasi-coherent\ $A$-functor preserves direct limits. Artin remarks that nearly all
$A$-functors which occur in practice do this; in Artin's terminology an
$A$-functor which preserves direct limits is said to be
{\it locally of finite presentation} (\Lcitemark 1\Rcitemark \ 1.5).
This condition enters into his criterion for representability
(\Lcitemark 3\Rcitemark \ 3.4).
\block{Coherence of higher direct images as functors}\label{higher-section}
In this section we consider a question which was posed (in an equivalent
form) by Artin\Lspace \Lcitemark 2\Rcitemark \Rspace{}:
\begin{problem}
Let $X$ be a proper $A$-scheme, let ${\cal{F}}$ be a coherent sheaf on $X$, and
fix $n \geq 0$. Is the $A$-functor $H = H^n_{\cal{F}}$ given by
$B \mapsto H^n(X_B, {\cal{F}}_B)$ module-coherent?
\end{problem}
Taking the \u Cech resolution of ${\cal{F}}$ relative to some affine open cover of
$X$ yields a complex $K$ of $A$-modules, and by (\Lcitemark 17\Rcitemark \
1.4.1) we have
$H^n(X_B, {\cal{F}}_B) \cong H^n(K \o* B)$ for all commutative $A$-algebras $B$. It
follows that at least $H$ is {\it module-quasi-coherent}. The issue of whether $H$ is module-coherent\ is
quite subtle. We will show that if ${\cal{F}}$ is $A$-flat, or $A$ is a Dedekind
domain, then $H$ is module-coherent. By \pref{product-preserving}, we know also that $H$
is module-coherent\ if and only if\ $H$ preserves products, but it is not clear how to use this
statement. We will give an example which shows that in general $H$ is not module-coherent.
Let us say that a module-valued\ $A$-functor $F$ is {\it upper semicontinuous\/} if
for every commutative $A$-algebra $B$, and every ${\xmode{{\fraktur{\lowercase{P}}}}} \in \mathop{\operatoratfont Spec}\nolimits(B)$, there
is a neighborhood $U$ of ${\xmode{{\fraktur{\lowercase{P}}}}}$ such that
$\dim_{k({\xmode{{\fraktur{\lowercase{Q}}}}})} F(k({\xmode{{\fraktur{\lowercase{Q}}}}})) \leq \dim_{k({\xmode{{\fraktur{\lowercase{P}}}}})} F(k({\xmode{{\fraktur{\lowercase{P}}}}}))$ for all ${\xmode{{\fraktur{\lowercase{Q}}}}} \in
U$.
Similarly, one defines {\it lower semicontinuous\/} by reversing the
inequality. If $M$ is a finitely generated\ $A$-module, then it follows by Nakayama's lemma
that ${\underline{M}}$ is upper semicontinuous. If \mp[[ f || P_1 || P_2 ]] is a map of
finitely generated\ projective $A$-modules, then $\mathop{\underline{\operatoratfont Ker}}\nolimits(f)$ is upper semicontinuous, whereas
$\mathop{\underline{\operatoratfont Im}}\nolimits(f)$ is lower semicontinuous. For $I \subset A$ an ideal,
$\mathop{\underline{\operatoratfont Ker}}\nolimits(A\ \mapE{}\ A/I)$ is in general not upper semicontinuous. (But it is
lower semicontinuous.) If the numerator of
a quotient is upper semicontinuous and the denominator is lower semicontinuous,
then the quotient is itself upper semicontinuous. It follows that if $K_0$ is
a complex of finitely generated\ free $A$-modules, and $n \in \xmode{\Bbb Z}$, then the module-valued\ $A$-functor
given by $B \mapsto H^n(K_0 \o* B)$ is upper semicontinuous. It is also module-coherent.
\begin{prop}\label{flat-implies-cohomologically-coherent}
If $X$ is proper over $A$ and ${\cal{F}}$ is a coherent sheaf on $X$ which is
$A$-flat, then the module-valued\ $A$-functor $H^n_{\cal{F}}$ given by
$B \mapsto H^n(X_B, {\cal{F}}_B)$ is module-coherent\ and upper semicontinuous.
\end{prop}
\begin{proofnodot}
(pointed out to me by Deligne and Ogus; cf.{\ }\Lcitemark 23\Rcitemark .)
Let $K$ be the \u Cech complex discussed above. It is bounded and flat.
Since $X$ is proper over $A$, the complex $K$ has finitely generated\ cohomology modules.
It follows that there exists a complex $K_0$ of finitely generated\ free $A$-modules which is
bounded above and a quasi-isomorphism \mp[[ \phi || K_0 || K ]]. Since
$K$ is flat and bounded above, there is a spectral sequence
$$E_{p,q}^2\ =\ \mathop{\operatoratfont Tor}\nolimits_p(H^{-q}(K), B)\ \Longrightarrow\ H^{-p-q}(K \o* B).$%
$(See e.g.{\ }\Lcitemark 28\Rcitemark \ 11.34.)
Similarly, one has such a spectral sequence for $K_0$. Moreover, $\phi$
induces a morphism from the spectral sequence for $K_0$ to the spectral
sequence for $K$. Since $\phi$ is a quasi-isomorphism, the induced maps
\mapx[[ H^{-q}(K_0) || H^{-q}(K) ]] are isomorphisms, and so the induced maps
\mapx[[ \mathop{\operatoratfont Tor}\nolimits_p(H^{-q}(K_0), B) || \mathop{\operatoratfont Tor}\nolimits_p(H^{-q}(K), B) ]] are isomorphisms.
Hence $H^n(\phi)$ is an isomorphism. Hence the module-valued\ $A$-functor
$B \mapsto H^n(X_B, {\cal{F}}_B)$ is isomorphic to the module-valued\ $A$-functor
$B \mapsto H^n(K_0 \o* B)$. {\hfill$\square$}
\end{proofnodot}
The {\it upper semicontinuity\/} part of the proposition is of course the
usual theorem on upper semicontinuity of cohomology
(see e.g.{\ }\Lcitemark 22\Rcitemark \ III\ 12.8).
\begin{theorem}\label{mulch-material}
If $X$ is proper over a Dedekind domain $A$ and ${\cal{F}}$ is a coherent sheaf on
$X$, then the module-valued\ $A$-functor $H^n_{\cal{F}}$ given by $B \mapsto H^n(X_B, {\cal{F}}_B)$
is module-coherent.
\end{theorem}
Before proving this, there are some preliminaries. By a {\it truncated discrete valuation ring},
we shall mean a ring $A$ of the form $R/I$ where $R$ is a discrete valuation ring\ and $I$ is a
proper nonzero ideal. By a {\it uniformizing parameter\/} for $A$, we shall
mean the image in $A$ of a uniformizing parameter for $R$.
\begin{lemma}\label{mulch-material-lemma}
Let $A$ be a truncated discrete valuation ring\ with uniformizing parameter $t$. Let $C$ be a
bounded complex of $A$-modules. Assume that $H^n(C \o* A/(t^l))$ is finitely generated\ for
all $n$ and all $l$. Then for each $n$, the $A$-functor $F$ given by
$B \mapsto H^n(C \o* B)$ is module-coherent.
\end{lemma}
\begin{proof}
Any module $M$ over $A$ is a direct sum of cyclic modules. (See
e.g.{\ }\Lcitemark 11\Rcitemark \ Ch.\ VII\ \S2\ exercise 12(b).) It
follows that if $M_0$ is a finitely generated\ submodule of $M$, then there exists a
finitely generated\ direct summand $M_1$ of $M$ with $M_0 \subset M_1$.
For each $l$, let $A_l$ denote $A/(t^l)$.
By working from low indices to high indices, one can construct a subcomplex
$C_0$ of $C$ with the properties that for each $n$:
\begin{alphalist}
\item for each $l$,
$\mathop{\operatoratfont Ker}\nolimits[ d^n \o* A_l ] \subset C_0^n \o* A_l + \mathop{\operatoratfont Im}\nolimits[ d^{n-1} \o* A_l ]$ and
\item $C_0^n$ is finitely generated\ and is a direct summand of $C^n$.
\end{alphalist}
Property (b) comes from the first paragraph.
Let ${\overline{C}} = C/C_0$. It follows from (b) that the sequence
\ses{C_0}{C}{{\overline{C}}%
}is universally exact. Since any $A$-module is a direct sum of cyclic modules,
it follows from (a) that the induced map
\mapx[[ H^n(C_0 \o* B) || H^n(C \o* B) ]] is surjective for every $n$ and
every $B$, and hence that we have short exact sequences:
\sesdot{H^{n-1}({\overline{C}} \o* B)}{H^n(C_0 \o* B)}{H^n(C \o* B)%
}Now construct a subcomplex ${\overline{C}}_0 \subset {\overline{C}}$ in the same way that
we constructed $C_0 \subset C$. It follows that $F$ is expressible as the
cokernel of a map of module-coherent\ functors, and hence that $F$ is itself module-coherent. {\hfill$\square$}
\end{proof}
Now we prove \pref{mulch-material}, using arguments provided by Deligne.
\begin{proof}
There is an exact sequence of coherent sheaves on $X$:
\ses{{\cal{F}}'}{{\cal{F}}}{{\cal{F}}''%
}in which ${\cal{F}}'$ is $A$-torsion and ${\cal{F}}''$ is $A$-torsion-free. Since $A$
is a Dedekind domain, ${\cal{F}}''$ is $A$-flat. Hence this
sequence remains exact after tensoring over $A$ by anything, so by
the long exact sequence of cohomology and \pref{extension-is-mc}, it suffices
to prove the theorem when ${\cal{F}}$ is either $A$-torsion or $A$-flat. The
second case is taken care of by \pref{flat-implies-cohomologically-coherent}.
Therefore we may assume that\ ${\cal{F}}$ is $A$-torsion. Then in fact $a{\cal{F}} = 0$ for some
nonzero $a \in A$. We may then reduce to the case where $A$ is a truncated
discrete valuation ring. Apply \pref{mulch-material-lemma} to the \u Cech complex of ${\cal{F}}$.
{\hfill$\square$}
\end{proof}
Finally we give a counterexample which shows that in general, $H^n_{\cal{F}}$ is not
module-coherent. The counterexample to be given here is based on
examples constructed by H.\ Cohen\Lspace \Lcitemark 13\Rcitemark \Rspace{}.
The same
examples appear in his thesis\Lspace \Lcitemark 14\Rcitemark \Rspace{}, where
Cohen remarks briefly
that the techniques therein yield a counterexample to Artin's problem.
However, he says nothing more about the matter. It seems likely that Cohen
and/or Verdier did construct a counterexample, but it has now (apparently) been
lost.
Let $k$ be a field, let $A = k[[s,t]]$,
and let $X = \P3_A$, with coordinates $x,y,z,w$. Let ${\cal{F}}$ be the cokernel of
the map \mapx[[ {\cal O}_X || {\cal O}_X(1) ]] given by multiplication by $sx-ty$%
.\footnote{In terms of Cohen's construction, we have $r = 3$, $c_0 = s$,
$c_1 = -t$, and $m = 0$.}
Let $n = 1$. We will show that $H$ is not module-coherent, using \pref{ding-dong}.
In his proposition 2, Cohen shows (in effect)
that if $B$ is a commutative $A$-algebra, and
$R = B[x^{\pm 1}, y^{\pm 1}, z^{\pm 1}, w^{\pm 1}]$,
then $H(B)$ is isomorphic to the degree $0$ part of the quotient of
$$\left\{ {f \over x^a y^b z^c w^d} \in R: f \in B[x,y,z,w],\ a,b,c,d \in \xmode{\Bbb N},
\hbox{\ and\ } (sx-ty)f = 0 \right\}$%
$by the sub-$B[x,y,z,w]$-module generated by
$$\left\{ {f \over yzw}, {f \over xzw}, {f \over xyw}, {f \over xyz} \in R:
f \in B[x,y,z,w] \hbox{\ and\ } (sx-ty)f = 0 \right\}.$%
$
Let $B = A/(s^k, t^k)$, for some $k \in \xmode{\Bbb N}$. We proceed to compute $H(B)$.
\begin{lemma}
The $B[x,y,z,w]$-module $\mathop{\operatoratfont Ann}\nolimits_{B[x,y,z,w]}(sx-ty)$ is generated by
$$(st)^{k-1},\ (st)^{k-2} \sum_{i=0}^1 (sx)^i (ty)^{1-i},\ \ldots,
\ (st)^0 \sum_{i=0}^{k-1} (sx)^i (ty)^{k-1-i}.$$
\end{lemma}
\begin{sketch}
Let $C = k[s,t,x,y]/(s^k,t^k)$. Since $B[x,y,z,w]$ is a flat $C$-algebra, it
suffices to show that the $C$-module $\mathop{\operatoratfont Ann}\nolimits_C(sx-ty)$ admits the given
generators. Let $f \in \mathop{\operatoratfont Ann}\nolimits_C(sx-ty)$. Write
$$f = \sum_{0 \leq i,j \leq k-1} f_{ij} s^i t^j,$%
$where $f_{ij} \in k[x,y]$. The following assertions are easily checked:
\begin{itemize}
\item $f_{i,0} = 0$ if $0 \leq i \leq k-2$;
\item $f_{0,j} = 0$ if $0 \leq j \leq k-2$;
\item $f_{i,j-1} = (x/y) f_{i-1,j}$ if $1 \leq i,j \leq k-1$.
\end{itemize}
Hence $f$ is completely determined by $f_{0,k-1}, \ldots, f_{k-1,k-1}$.
The lemma follows. {\hfill$\square$}
\end{sketch}
{}From this lemma it follows that
$H(B)$ is isomorphic to the sub-$B$-module of $R$ generated by
$$\bigcup_{j=5}^k \left\{ {(st)^{k-j} \sum_{i=0}^{j-1} (sx)^i (ty)^{j-1-i}
\over x^a y^b z^c w^d}: a,b,c,d \in \xmode{\Bbb N} \hbox{\ and\ }
a+b+c+d = j-1 \right\}.$%
$Since this is a minimal generating set, we have:
$$\mu[H(B)]\ =\ \sum_{j=5}^k {j-2 \choose 3}\ \geq\ O(k^3).$%
$Since $(s,t)^{2k-1} \subset (s^k, t^k)$, it follows from \pref{ding-dong} that $H$
is not module-coherent.
\block{Global sections%
}\label{torsion-section}
Let us say that a group is {\it linear\/} if it may be embedded as a
subgroup of $\mathop{\operatoratfont GL}\nolimits_n(k_1) \times \cdots \times \mathop{\operatoratfont GL}\nolimits_n(k_r)$ for some $n$ and some fields
$\vec k1r$. We shall want to have some control over the fields: a group is
{\it $X$-linear\/} if there exist points $\vec x1r \in X$ (not necessarily
closed) and finitely generated\ field extensions $k_i$ of $k(x_i)$ (for each $i$) such that
the group may be embedded as a subgroup of
$\mathop{\operatoratfont GL}\nolimits_n(k_1) \times \cdots \times \mathop{\operatoratfont GL}\nolimits_n(k_r)$ for some $n$.
The purpose of this section is two-fold. The first purpose is to develop a
tool (the next theorem) for proving that groups are $X$-linear. The second
purpose is to study the torsion in $X$-linear groups. Our result is
\pref{linear-implies-finite-torsion}, or in a slightly different form
\pref{linear-bound}.
\begin{theorem}\label{coherent-implies-linear}
Let $G$ be a group-valued locally coherent $X$-functor. Then $G(X)$ is
$X$-linear.
\end{theorem}
Before proving this theorem, we will study the torsion in $X$-linear groups.
For any abelian group $H$, one can try to determine for which $n \in \xmode{\Bbb N}$
one has $\abs{{}_nH} < \infty$. If $n = p_1^{k_1} \cdots p_r^{k_r}$, where
$\vec p1r$ are prime numbers, then $\abs{{}_nH} < \infty$ if and only if\
$\abs{{}_{p_i} H} < \infty$ for each $i$. Therefore we may as well restrict to
the problem of determining when $\abs{{}_pH} < \infty$, where $p$ is prime.
If $C$ is a commutative ring, let us say that a morphism
\mp[[ \pi || X || \mathop{\operatoratfont Spec}\nolimits(C) ]] is {\it essentially of finite type\/} if there
exists a commutative ring $D$, a homomorphism \mp[[ \phi || C || D ]]
which is essentially of finite type, and a morphism of finite type
\mp[[ \pi_0 || X || \mathop{\operatoratfont Spec}\nolimits(D) ]], such that $\pi = \mathop{\operatoratfont Spec}\nolimits(\phi) \circ \pi_0$.
Note that if $X$ is essentially of finite type over $\xmode{\Bbb Z}$, and $x \in X$, then
$k(x)$ is a finitely generated\ field extension of its prime subfield.
\begin{prop}\label{linear-implies-finite-torsion}
Let $H$ be an $X$-linear abelian group.
\begin{alphalist}
\item There are only finitely many prime numbers $p$ such that $H$ has infinite
$p$-torsion. Moreover, such a $p$ cannot be invertible in
$\Gamma(X,{\cal O}_X)$.
\item If $X$ is essentially of finite type over $\xmode{\Bbb Z}$ or over $\xmode{\Bbb Z}_p$ (for some
prime number $p$), then there exist prime numbers $\vec p1n$, none of
which are invertible in $\Gamma(X,{\cal O}_X)$, such that the subgroup of $H$
consisting of torsion prime to $p_1 \cdot \ldots \cdot p_n$ is finite.
\end{alphalist}
\end{prop}
Clearly, if in the above proposition, $X$ is essentially of finite type over
$\xmode{\Bbb Z}_p$, then the list of primes $\vec p1n$ may be taken to be the single prime
$p$. Also we have:
\begin{corollary}
Let $H$ be an $X$-linear abelian group. If $X$ is essentially of finite
type over ${\Bbb Q}\kern1pt$ or over ${\Bbb Q}\kern1pt_p$ (for some prime number $p$), then the torsion
subgroup of $H$ is finite.
\end{corollary}
We now state a generalization of the proposition to the non-abelian case:
\begin{prop}\label{linear-bound}
Let $H$ be an $X$-linear group.
\begin{alphalist}
\item Let $n \in \xmode{\Bbb N}$ be invertible in $\Gamma(X,{\cal O}_X)$. Then there exists some
$N \in \xmode{\Bbb N}$, such that whenever $K$ is an $n$-torsion abelian subgroup of
$H$, we have $\abs{K} \leq N$.
\item If $X$ is essentially of finite type over $\xmode{\Bbb Z}$ or over $\xmode{\Bbb Z}_p$ (for some
prime number $p$), then there exist prime numbers $\vec p1n$, none of
which are invertible in $\Gamma(X,{\cal O}_X)$, and some $N \in \xmode{\Bbb N}$, such that
if $K$ is an abelian subgroup of $H$, and every element of $K$ is torsion
prime to $p_1 \cdot \ldots \cdot p_n$, then $\abs{K} \leq N$.
\end{alphalist}
\end{prop}
\begin{lemma}\label{injective-on-roots}
Let $(A,{\xmode{{\fraktur{\lowercase{M}}}}},k)$ be a local ring. Let $n \in \xmode{\Bbb N}$, and assume that $n$ is
invertible in $A$. Then the canonical map:
\dmapx[[ \setofh{\th{n} roots of unity in $A$}
|| \setofh{\th{n} roots of unity in $k$} ]]%
is injective.
\end{lemma}
\begin{proof}
Let $n \in \xmode{\Bbb N}$, $x \in A$, $a \in {\xmode{{\fraktur{\lowercase{M}}}}}$, and suppose that $x^n = (x+a)^n = 1$.
Then $0 = [(x+a)^n - x^n] = a[nx^{n-1} + c]$, for some $c \in {\xmode{{\fraktur{\lowercase{M}}}}}$. But
$nx^{n-1}$ is a unit, so $nx^{n-1}+c$ is a unit, so $a = 0$. {\hfill$\square$}
\end{proof}
It is known (\Lcitemark 11\Rcitemark \ \S14, \#7, Cor.\ 2 to Prop.\ 17)
that a field finitely generated\ over its prime subfield (as a field extension) contains
only finitely many roots of unity. This also holds for a field finitely generated\ over
${\Bbb Q}\kern1pt_p$. We will need a modest generalization of these statements:
\begin{lemma}\label{root-bound}
Let $K$ be a finitely generated\ field extension of $F$, where $F$ is ${\Bbb Q}\kern1pt$, or ${\Bbb F}\kern1pt_p$,
or ${\Bbb Q}\kern1pt_p$, for some prime number $p$. Then there exists a constant $c$ such
that for every $m \in \xmode{\Bbb N}$, and every finite field extension $L$ of $K$ with
$[L:K] \leq m$, the number of roots of unity in $L$ is $\leq c m$
(if $F = {\Bbb Q}\kern1pt$), and is $\leq c^m$ if $F$ is ${\Bbb F}\kern1pt_p$ or ${\Bbb Q}\kern1pt_p$.
\end{lemma}
\begin{proof}
Let $\vec x1r$ be a transcendence basis for $K$ over $F$. Let
$s = [K:F(\vec x1r)]$. If $F = {\Bbb Q}\kern1pt$, let $c = 2s$. If $F = {\Bbb F}\kern1pt_p$, let
$c = p^s$. If $F = {\Bbb Q}\kern1pt_p$, let $c = 2sp^s$.
Let $L_0$ be the subfield of $L$ consisting of
elements algebraic over $F$. Then $[L:F(\vec x1r)] = s[L:K] \leq sm$, so
$[L_0:F] \leq sm$. If $F = {\Bbb Q}\kern1pt$ or $F = {\Bbb F}\kern1pt_p$, it is clear that
the given $c$ works.
Suppose that $F = {\Bbb Q}\kern1pt_p$.
Extend the standard absolute value on ${\Bbb Q}\kern1pt_p$ to $L_0$. Let $(A,{\xmode{{\fraktur{\lowercase{M}}}}},k)$ be the
valuation ring of $L_0$. If $x \in L_0$ is a root of unity, $\abs{x} = 1$, so
$x \in A$. Since $k$ is an extension of ${\Bbb F}\kern1pt_p$ of degree $\leq sm$, the
number of elements in $k$ is bounded by $p^{sm}$. By
\pref{injective-on-roots}, it follows that for any $r \in \xmode{\Bbb N}$ which is prime to
$p$ (and hence invertible in $A$), the number of \th{r} roots of unity in $A$
is $\leq p^{sm}$.
For $p \not= 2$, ${\Bbb Q}\kern1pt_p$ has no \th{p} roots of unity other than $1$ (see
\Lcitemark 24\Rcitemark \Rspace{}\ p.\ 20\ exercise 14). For $p = 2$, ${\Bbb Q}\kern1pt_p$
contains no square root of $-1$. For any field $M$, let $M'$ denote its
subfield generated by
\setof{x \in M: x^{(p^n)} = 1 \hbox{\ for some\ } n \in \xmode{\Bbb N}}. Then
${\Bbb Q}\kern1pt_p' = {\Bbb Q}\kern1pt$. Hence $[L_0':{\Bbb Q}\kern1pt] \leq sm$. Hence
$$\abs{\setof{x \in L_0: x^{(p^n)} = 1 \hbox{\ for some\ } n \in \xmode{\Bbb N}}}
\ \leq\ 2sm.$%
$Hence the number of roots of unity in $L_0$ is $\leq (2sm)p^{sm} \leq c^m$.
{\hfill$\square$}
\end{proof}
\begin{proofnodot}
(of \ref{linear-bound})
For part (a), we may assume that\ $H \subset \mathop{\operatoratfont GL}\nolimits_r(k)$ for some field $k$, where $n$ is
invertible in $k$. Let $\vec g1l \in \mathop{\operatoratfont GL}\nolimits_r(k)$ be distinct commuting elements
with $g_i^n = 1$ for each $i$. We need to prove that there is some $N \in \xmode{\Bbb N}$
(independent of $\vec g1l$) such that $l \leq N$. Let $C$ be the subalgebra of
$\mathop{\operatoratfont Mat}\nolimits_{r \times r}(k)$ generated by $\vec g1l$. Then $C$ is a commutative,
artinian $k$-algebra, and $\mathop{\operatoratfont Spec}\nolimits(C)$ has at most $r$ components, since
$\mathop{\operatoratfont Mat}\nolimits_{r \times r}(k)$ has at most $r$ distinct nonzero orthogonal idempotents.
It follows from \pref{injective-on-roots} that the equation $x^n = 1$ has at
most $n^r$ solutions in $C$. Let $N = n^r$.
For part (b), the field $k$ will be a finitely generated\ field extension of ${\Bbb Q}\kern1pt$, ${\Bbb F}\kern1pt_p$,
or ${\Bbb Q}\kern1pt_p$, for some prime number $p$.
Construct $C$ as in the preceding paragraph. We have to bound
$l$ in terms of $r$ alone, and not in terms of $n$. We may assume that $C$ is
local. The residue field $L$ of $C$ is a finite extension of $k$, and
$[L:k] \leq r^2$. Apply \pref{root-bound} and \pref{injective-on-roots}. {\hfill$\square$}
\end{proofnodot}
We now work towards a proof of \pref{coherent-implies-linear}.
\begin{lemma}\label{artinian-exists}
Let $M$ be a finitely generated\ $A$-module. Then there exists a commutative artinian
$A$-algebra $B$, such that $B$ is essentially of finite type over $A$, and such
that the canonical map \mapx[[ M || M \o*_A B ]] is injective.
\end{lemma}
\begin{proof}
Let us say that an ideal $I \subset A$ is {\it good\/} if
there exists a commutative artinian $(A/I)$-algebra $B_{[I]}$ which is
essentially of finite type over $A/I$ such that the canonical map
\dmapx[[ M \o*_A (A/I) || M \o*_A B_{[I]} ]]%
is injective.
Let $I \subset A$ be an ideal, and suppose that every ideal properly containing
$I$ is good. To prove the lemma, it suffices (by a sort of noetherian
induction) to show that $I$ is good. Replacing $A$ by $A/I$, we may assume that\ $I = 0$.
Choose a primary decomposition $0 = Q_1 \cap \cdots \cap Q_r$ of $0$ in
$M$. Then the map \mapx[[ M || M/Q_1 \times \cdots \times M/Q_r ]] is injective and each
module $M/Q_i$ has a unique associated prime ${\xmode{{\fraktur{\lowercase{P}}}}}_i$.
Since each module $M/Q_i$ admits a filtration with quotients isomorphic to
$A/{\xmode{{\fraktur{\lowercase{P}}}}}_i$, it follows that there exists an integer $N$ such that the maps
\mapx[[ M/Q_i || (M/Q_i) \o*_A A/{\xmode{{\fraktur{\lowercase{P}}}}}_i^N ]] are injective. Note also that
the maps \mapx[[ M/Q_i || (M/Q_i) \o*_A A_{{\xmode{{\fraktur{\lowercase{P}}}}}_i} ]] are injective.
We may assume that $\vec \lfP1r$ are arranged so that $\vec \lfP1k$ are
minimal primes of $A$ and $\vec {\xmode{{\fraktur{\lowercase{P}}}}}{k+1}r$ are not. Then
${\xmode{{\fraktur{\lowercase{P}}}}}_{k+1}^N,\ldots,{\xmode{{\fraktur{\lowercase{P}}}}}_r^N$ are nonzero. Let
\formulaqed{B = A_{{\xmode{{\fraktur{\lowercase{P}}}}}_1} \times A_{{\xmode{{\fraktur{\lowercase{P}}}}}_k} \times B_{[{\xmode{{\fraktur{\lowercase{P}}}}}_{k+1}^N]}
\times B_{[{\xmode{{\fraktur{\lowercase{P}}}}}_r^N]}.}
\end{proof}
We use Witt rings in this and the next section. The reader may find treatments
of the subject in
(\Lcitemark 26\Rcitemark \ VIII\ exercises\ 42--44),
(\Lcitemark 29\Rcitemark \ II\ \S5,\ \S6), and\Lspace \Lcitemark 9\Rcitemark
\Rspace{}. Some
remarks about notation are in order. There is a version of the Witt ring which
does {\it not\/} depend on the choice of a prime number $p$, as discussed for
example in (\Lcitemark 26\Rcitemark \ VIII\ exercise\ 42). It
seems reasonable to denote this version of the Witt ring by $W(A)$. There is a
second version of the Witt ring which does depend on the choice of a prime
number $p$, as discussed for example in
(\Lcitemark 26\Rcitemark \ VIII\ exercise\ 43). To avoid
confusion, we will denote this version of the Witt ring by $W^p(A)$, as is done
in (\Lcitemark 9\Rcitemark \ p.\ 179). However, the ring we denote by $W^p(A)$
is the same as the ring denoted $W(A)$ in\Lspace \Lcitemark 29\Rcitemark
\Rspace{}. We will be
using the ring $W^p(A)$, as well as the truncated version $W^p_n(A)$.
\begin{lemma}\label{witt-coherent}
Let $k$ be a perfect field of positive characteristic $p$. Fix $n \in \xmode{\Bbb N}$, and
let $A = W^p_n(k)$. Let $F$ be a coherent $A$-functor. Define a $k$-functor
${\tilde{F}}$ by ${\tilde{F}}(B) = F(W^p_n(B))$, for all (commutative) $k$-algebras $B$. Then
${\tilde{F}}$ is coherent.
\end{lemma}
\begin{proof}
We let\ $\tilde{}$\ denote the operation which is in effect defined in the
statement. Let $r$ be the level of $F$.
Suppose that $r = 0$, so we may assume that\ $F = {\underline{M}}$ for some finitely generated\ $A$-module $M$. Since
$A = W^p(k)/(p^n)$, and $W^p(k)$ is a complete discrete valuation ring\ with uniformizing
parameter $p$, it follows that $M$ may be expressed as a direct sum of modules
of the form $A/(p^i)$, where $i \in \setof{1,\ldots,n}$. We may assume that in
fact $M = A/(p^i)$. Then:
$${\tilde{F}}(B)\ =\ A/(p^i) \o*_A W_n^p(B)\ =\ W_i^p(B),$%
$which may be identified (as a set) with $B^i$. Hence ${\tilde{F}}$ is coherent.
Now suppose that $r \geq 1$. By \pref{lesx-exists}, we may choose a
left exact sequence:
\lesx{F}{G}{{\underline{N}}%
}of $A$-functors where $G$ is a coherent $A$-functor of level $r-1$
and $N$ is a finitely generated\ $A$-module. We obtain a left exact sequence:
\lesx{{\tilde{F}}}{{\tilde{G}}}{\tilde{{\underline{N}}}%
}of $k$-functors. By induction on $r$, we may assume that\ ${\tilde{G}}$ is coherent,
and by the case $r = 0$, $\tilde{{\underline{N}}}$ is coherent. Hence ${\tilde{F}}$ is coherent.
{\hfill$\square$}
\end{proof}
\begin{proofnodot}
(of \ref{coherent-implies-linear})
Since $G$ is locally coherent, it is a sheaf with respect to\ the Zariski topology, so we
may immediately reduce to the case where $X$ is affine, say $X = \mathop{\operatoratfont Spec}\nolimits(A)$. We
work with $A$-functors.
By \pref{lesx-exists}, there exists a finitely generated\ $A$-module $M$ and an
embedding $G\ \includeE{}\ {\underline{M}}$ of $A$-functors. (Note that this map need
not preserve the group structure; otherwise the proof would be much easier!)
By \pref{artinian-exists}, we can choose an artinian $A$-algebra $B$ which
is essentially of finite type over $A$ such that
the map \mapx[[ {\underline{M}}(A) || {\underline{M}}(B) ]] is injective. Let $G|_B$ denote the
$B$-functor given by $G|_B(C) = G(C)$. Then $G|_B$ is coherent by \pref{pull}.
Therefore it suffices to show that $G|_B$ has the desired property. Replacing
$A$ by $B$, we may reduce to the case where $A$ is artinian. Since $A$ is a
product of Artin local rings, we may in fact reduce to the case where $A$ is an
Artin local ring.
First suppose that $A$ is a field. Then $A$ is a coherent
$A$-functor, so $G$ is representable by an affine group scheme of finite type
over $A$. Hence (\Lcitemark 10\Rcitemark \ 11.11) $G(A)$ embeds in $\mathop{\operatoratfont GL}\nolimits_n(A)$
for
some $n$.
Now suppose that $A$ contains a field. Then from the Cohen structure theorem
for complete local rings, we know that $A$ contains a coefficient field $k$.
Since $A$ is artinian, it follows that $A$ is module-finite over $k$. By
\pref{push}, we may reduce to the case $A = k$.
Finally, suppose that $A$ is an Artin local ring which does not contain a
field. Then $A$ has mixed characteristic. Let ${\xmode{{\fraktur{\lowercase{M}}}}}$ be its maximal ideal,
and let $k$ be its residue field. By (\Lcitemark 20\Rcitemark \ 0.6.8.3),
there exists a
(commutative) faithfully flat noetherian local $A$-algebra $({\tilde{A}},{\xmode{{\tilde{\fraktur{\lowercase{M}}}}}}, {\tilde{\lowercase{K}}})$
with ${\tilde{\lowercase{K}}}$ being an algebraic closure of $k$, such that ${\xmode{{\fraktur{\lowercase{M}}}}}{\tilde{A}} = {\xmode{{\tilde{\fraktur{\lowercase{M}}}}}}$.
{}From the latter fact, it follows that ${\tilde{A}}$ is artinian. Since $G$ is
coherent, by \pref{sheaf} it is a sheaf for the fpqc topology, so
\mapx[[ F(A) || F({\tilde{A}}) ]] is injective. Hence we may assume that\ $A = {\tilde{A}}$
and so $k$ is algebraically closed. By the Cohen structure theorem,
$A \cong W^p_n(k)[[\vec x1r]]/I$ for some $r,n \in \xmode{\Bbb N}$ and some ideal $I$.
Since $W^p_n(k)$ maps onto the residue field of $A$, and since
$A$ is artinian, it follows that $A$ is module-finite over $W^p_n(k)$.
By \pref{push}, we may reduce to the case $A = W^p_n(k)$. Apply
\pref{witt-coherent} to reduce to the case $A = k$. {\hfill$\square$}
\end{proofnodot}
\block{Global sections -- arithmetic case}
In this section we refine the results of the last section, in the special case
where $X$ is of finite type over $\xmode{\Bbb Z}$. In particular,
(\ref{linear-implies-finite-torsion}b) is supplanted by
\pref{arith-linear-structure}.
Let us say that a group is
{\it arithmetically linear\/} if it may be embedded as a subgroup of
$\mathop{\operatoratfont GL}\nolimits_n(C)$, for some $n$ and some finitely generated\ commutative $\xmode{\Bbb Z}$-algebra $C$.
We will prove:
\begin{theorem}\label{coherent-implies-arithmetically-linear}
Assume that $X$ is of finite type over $\xmode{\Bbb Z}$. Let $G$ be a group-valued locally
coherent $X$-functor. Then $G(X)$ is arithmetically linear. Moreover, if
$n$ is invertible in $\Gamma(X,{\cal O}_X)$, then the $n$-torsion in $G(X)$ is
finite.
\end{theorem}
First we analyze the structure of arithmetically linear abelian groups.
Recall that an abelian group $H$ is {\it bounded\/} if $nH = 0$ for some
$n \in \xmode{\Bbb N}$. It is known [see\Lspace \Lcitemark 16\Rcitemark \Rspace{}\ 11.2 or
\Lcitemark 11\Rcitemark \Rspace{}\ Ch.\ VII\ \S2\ exercise 12(b)] that any
bounded
abelian group is a direct sum of cyclic groups. Thus one may characterize the
bounded abelian groups as those which can be expressed as direct sums of cyclic
groups, in which the orders of the summands are bounded.
For purposes of this paper, let us say that an abelian group $H$ is
{\it cobounded\/} if it may be embedded as a subgroup of a direct sum of
(possibly infinitely many) copies of $\xmode{\Bbb Z}[1/n]$ for some $n \in \xmode{\Bbb N}$. This is
equivalent to saying that $H$ is torsion-free and that $H \o*_\xmode{\Bbb Z} \xmode{\Bbb Z}[1/n]$ is a
free $\xmode{\Bbb Z}[1/n]$-module for some $n$.
\begin{remark}
We do not know of a structure theorem for abelian groups which are countable
and cobounded. Certainly such groups can be rather complicated. For
example, not every subgroup $H$ of $\o+_{k=1}^\infty \xmode{\Bbb Z}[1/2]$ can be expressed
as a direct sum of copies of $\xmode{\Bbb Z}$ and copies of $\xmode{\Bbb Z}[1/2]$; consider:
$$H\ = \ \Bigl\{ a \in \o+_{k=1}^\infty \xmode{\Bbb Z}[1/2]:
\sum_{k=1}^\infty {a_k \over k} \in \xmode{\Bbb Z} \Bigr\}.$%
$Let $L$ be the maximal $2$-divisible subgroup of $H$. Then $H/L \cong {\Bbb Q}\kern1pt$.
\end{remark}
Let us say that an abelian group $H$ is {\itbounded\/\kern1.5pt$\times$\discretionary{}{}{\kern1.5pt}cobounded\/} if $H \cong B \times C$ for
some bounded group $B$ and some cobounded group $C$. We shall see
\pref{arith-linear-structure} that arithmetically linear abelian groups are
the same as countable (bounded\/\kern1.5pt$\times$\discretionary{}{}{\kern1.5pt}cobounded) abelian groups.
\begin{lemma}\label{bxc}
\
\begin{alphalist}
\item Let $M$ be a bounded\/\kern1.5pt$\times$\discretionary{}{}{\kern1.5pt}cobounded\ abelian group, and let $H$ be a subgroup of $M$. Then
$H$ is bounded\/\kern1.5pt$\times$\discretionary{}{}{\kern1.5pt}cobounded.
\item Let
\ses{M'}{M}{M''%
}be a short exact sequence of abelian groups, in which $M'$ and $M''$ are bounded\/\kern1.5pt$\times$\discretionary{}{}{\kern1.5pt}cobounded.
Then $M$ is bounded\/\kern1.5pt$\times$\discretionary{}{}{\kern1.5pt}cobounded.
\end{alphalist}
\end{lemma}
\begin{proof}
Part (a) follows from (\Lcitemark 16\Rcitemark \ 50.3): if the torsion
subgroup of an abelian group is bounded, then it is a direct summand.
For part (b),
write $M' = B' \times C'$ and $M'' = B'' \times C''$ where $B'$, $B''$ are
bounded and $C'$, $C''$ are cobounded. Let $M_{\operatoratfont tor}$ be the torsion
subgroup of $M$. We have a left exact sequence:
\les{B'}{M_{\operatoratfont tor}}{B''%
}from which it follows that $M_{\operatoratfont tor}$ is bounded, and hence that
$M_{\operatoratfont tor}$ is a direct summand of $M$. Therefore it suffices to show
that $M/M_{\operatoratfont tor}$ is cobounded. We have an exact sequence:
\ses{C'}{M/M_{\operatoratfont tor}}{\overline{B''} \times C''%
}in which $\overline{B''}$ is a quotient of $B''$ and hence is bounded. Choose
$n \in \xmode{\Bbb N}$ such that $n\overline{B''} = 0$, and such that $n$ satisfies the property
of $n$ in the definition of cobounded, for both $C'$ and $C''$. Tensoring by
$\xmode{\Bbb Z}[1/n]$ yields an exact sequence:
\sesdot{C'[1/n]}{(M/M_{\operatoratfont tor})[1/n]}{C''[1/n]%
}By construction, $C'[1/n]$ and $C''[1/n]$ are submodules of free
$\xmode{\Bbb Z}[1/n]$-modules and hence are themselves free. Hence $(M/M_{\operatoratfont tor})[1/n]$
is free so $M/M_{\operatoratfont tor}$ is cobounded. {\hfill$\square$}
\end{proof}
\begin{lemma}\label{pork}
Let $A$ be a commutative $\xmode{\Bbb Z}$-algebra of finite type. Let $M$ be a
finitely generated\ $A$-module. Then the abelian group $M$ is bounded\/\kern1.5pt$\times$\discretionary{}{}{\kern1.5pt}cobounded.
\end{lemma}
\begin{proof}
As an abelian group, we may embed $M$ as a subgroup of the additive group of
the symmetric algebra of $M$, which is a finitely generated\ $A$-algebra. Therefore we may
reduce to the case $M = A$.
Take a primary decomposition $0 = {\xmode{{\fraktur{\lowercase{Q}}}}}_1 \cap \cdots \cap {\xmode{{\fraktur{\lowercase{Q}}}}}_m$ of $0$ in $A$.
Then the canonical map \mapx[[ A || \prod_{i=1}^n A/{\xmode{{\fraktur{\lowercase{Q}}}}}_i ]] is injective, so
we may reduce to the case where $A$ has a unique associated prime.
Let $q$ be the characteristic of $A$. First suppose that $q > 0$. Then
$qA = 0$ so $A$ is bounded.
Now suppose that $q = 0$. By the Noether normalization lemma, there exist
elements $\vec x1r \in A \o*_\xmode{\Bbb Z} {\Bbb Q}\kern1pt$ which are algebraically independent over
${\Bbb Q}\kern1pt$ and such that $A \o*_\xmode{\Bbb Z} {\Bbb Q}\kern1pt$ is module-finite over ${\Bbb Q}\kern1pt[\vec x1r]$. We may
assume that $\vec x1r \in A$. Let $S = \mathop{\operatoratfont Spec}\nolimits(\xmode{\Bbb Z})$, $X = \mathop{\operatoratfont Spec}\nolimits(A)$,
$Y = \mathop{\operatoratfont Spec}\nolimits(\xmode{\Bbb Z}[\vec x1r])$, so we have a morphism \mp[[ \phi || X || Y ]] of
$S$-schemes. If $\eta \in S$ is the generic point, then $\phi_\eta$ is finite,
so it follows from (\Lcitemark 19\Rcitemark \ 8.1.2(a), 8.10.5(xii), 8.11.1,
9.6.1(vii)) that
for some $n \in \xmode{\Bbb N}$, $\phi \o*_S \mathop{\operatoratfont Spec}\nolimits(\xmode{\Bbb Z}[1/n])$ is finite, i.e.\ that
$A_n = A \o*_\xmode{\Bbb Z} \xmode{\Bbb Z}[1/n]$ is a module-finite $\xmode{\Bbb Z}[1/n, \vec x1r]$-algebra. Since
$A$ has characteristic zero and has a unique associated prime, it follows that
$A_n$ is a torsion-free $\xmode{\Bbb Z}[1/n, \vec x1r]$-module and that the canonical map
\mapx[[ A || A_n ]] is injective. Hence $A$ embeds (as an abelian group) in
$(\xmode{\Bbb Z}[1/n, \vec x1r])^k$ for some $k$, so $A$ is cobounded. {\hfill$\square$}
\end{proof}
\begin{prop}\label{arith-linear-structure}
Let $H$ be an abelian group. Then $H$ is arithmetically linear if and only if\ it is
countable and bounded\/\kern1.5pt$\times$\discretionary{}{}{\kern1.5pt}cobounded.
\end{prop}
\begin{proof}
First suppose that $H$ is countable and bounded\/\kern1.5pt$\times$\discretionary{}{}{\kern1.5pt}cobounded. For some $n,k \in \xmode{\Bbb N}$ and some
prime numbers $\vec p1k$, we may embed $H$ as a subgroup of a countable direct
sum $K$ of copies of
$$\xmode{\Bbb Z}[1/n] \o+ \left( \o+_{i=1}^k \xmode{\Bbb Z}/p_i^n\xmode{\Bbb Z} \right).$%
$Then $K$ is the additive group of the ring
$$A = \xmode{\Bbb Z}[1/n,t] \times (\xmode{\Bbb Z}/p_1^n\xmode{\Bbb Z})[t] \times \cdots \times (\xmode{\Bbb Z}/p_k^n\xmode{\Bbb Z})[t],$%
$so $K$ may be embedded as a subgroup of $\mathop{\operatoratfont GL}\nolimits_2(A)$. Hence $H$ is
arithmetically linear.
Now suppose that $H$ is arithmetically linear. The countability of $H$ is
clear. Embed $H$ as a subgroup of $\mathop{\operatoratfont GL}\nolimits_r(A)$, for some finitely generated\ commutative
$\xmode{\Bbb Z}$-algebra $A$. Let $R$ be the sub-$A$-algebra of $\mathop{\operatoratfont Mat}\nolimits_{r \times r}(A)$
generated by $H$. Then $R$ is a finite $A$-algebra, and $H$ is a subgroup of
$R^*$. By (\ref{bxc}a), it suffices to show that $R^*$ is bounded\/\kern1.5pt$\times$\discretionary{}{}{\kern1.5pt}cobounded. We may as
well view $R$ as an arbitrary finitely generated\ commutative $\xmode{\Bbb Z}$-algebra.
Let $J$ be the nilradical of $R$. For each $n \in \xmode{\Bbb N}$, there is a short
exact sequence of abelian groups:
\diagramx{0&\mapE{}&J^n/J^{n+1}&\mapE{}&(R/J^{n+1})^*&\mapE{}&(R/J^n)^*&
\mapE{}&1.%
}The group $(R/J)^*$ is finitely generated\ (see \ref{units-over-Z}), and hence is bounded\/\kern1.5pt$\times$\discretionary{}{}{\kern1.5pt}cobounded.
Hence
by (\ref{bxc}b), it suffices to show that $J^n/J^{n+1}$ is bounded\/\kern1.5pt$\times$\discretionary{}{}{\kern1.5pt}cobounded. Apply
\pref{pork}. {\hfill$\square$}
\end{proof}
We now work towards a proof of
\pref{coherent-implies-arithmetically-linear}.
\def\overline{S}^{\kern2pt\lower3pt\hbox{$\scriptstyle -1$}}{\overline{S}^{\kern2pt\lower3pt\hbox{$\scriptstyle -1$}}}
\begin{lemma}\label{localize-now}
Fix $n \in \xmode{\Bbb N}$ and a prime number $p$. Let $S \subset W_n^p(A)$ be a
multiplicatively closed set. Let \mp[[ \mu || W_n^p(A) || A ]] be the
canonical map. Let ${\overline{S}} = \mu(S)$. Then the canonical map
\mp[[ i || W_n^p(A) || W_n^p(\overline{S}^{\kern2pt\lower3pt\hbox{$\scriptstyle -1$}} A) ]] factors through $S^{-1}W_n^p(A)$.
\end{lemma}
\begin{proof}
Let \mp[[ \nu || W_n^p(\overline{S}^{\kern2pt\lower3pt\hbox{$\scriptstyle -1$}} A) || \overline{S}^{\kern2pt\lower3pt\hbox{$\scriptstyle -1$}} A ]] be the canonical map. Let
$f \in S$. Then the image of $\mu(f)$ in $\overline{S}^{\kern2pt\lower3pt\hbox{$\scriptstyle -1$}} A$ is invertible, so
$\nu(i(f))$ is invertible. But $\nu$ is surjective and has nilpotent kernel,
so $i(f)$ is invertible. {\hfill$\square$}
\end{proof}
Note that if $A$ is any ${\Bbb F}\kern1pt_p$-algebra, there is a canonical map
\mapx[[ W_n^p({\Bbb F}\kern1pt_p) || W_n^p(A) ]], and since $W_n^p({\Bbb F}\kern1pt_p) = \xmode{\Bbb Z}/p^n\xmode{\Bbb Z}$, we see
that $p^n = 0$ in $W_n^p(A)$.
\begin{prop}\label{witt-ring-of-poly-ring}
For any $n, k \in \xmode{\Bbb N}$, and any prime number $p$, the ring
$$W_n^p({\Bbb F}\kern1pt_p[\vec t1k])$%
$is isomorphic to the subring of
$(\xmode{\Bbb Z}/p^n\xmode{\Bbb Z})[t_1^{1/p^{n-1}},\ldots,t_k^{1/p^{n-1}}]$ generated by the elements
$p^r t_i^{j/p^r}$ for $0 \leq r \leq n-1$, $1 \leq i \leq k$, and
$1 \leq j \leq p-1$.
\end{prop}
\begin{proof}
Let $R = {\Bbb F}\kern1pt_p[t_1^{p^{-\infty}},\ldots,t_k^{p^{-\infty}}]$, and let
$V = W_n^p(R)$. Let
$$W = (\xmode{\Bbb Z}/p^n\xmode{\Bbb Z})[t_1^{p^{-\infty}},\ldots,t_k^{p^{-\infty}}].$%
$It is easily seen that there exists a unique ring homomorphism
\mp[[ \eta || W || V ]] with the property that
$\eta(t_i^{p^{-m}}) = (t_i^{p^{-m}},0,\ldots,0)$ for each $i$ and each
$m \geq 0$.
We show that $\eta$ is surjective. Let \mp[[ \mu || V || R ]] be the
canonical map. Since $R$ is perfect, $\mathop{\operatoratfont Ker}\nolimits(\mu) = (p)$. Because of this,
because $p^n = 0$ in $V$, and because $t_i^{1/p^r}$ (for various $i$, $r$)
generate $R$ as a $\xmode{\Bbb Z}$-algebra, it follows that the $\eta(t_i^{1/p^r})$
generate $V$ as a $\xmode{\Bbb Z}$-algebra. Hence $\eta$ is surjective.
We show that $\eta$ is injective. Suppose otherwise. Let $x \in \mathop{\operatoratfont Ker}\nolimits(\eta)$,
$x \not= 0$. We may assume that $px = 0$. Then $x = p^{n-1}y$ for some
$y \in W$; we may assume that\ the coefficients which appear in $y$ lie in the set
\setof{1,\ldots,p-1}. It follows that the $\th{0}$ component of $\eta(y)$ is
nonzero. Hence $p^{n-1} \eta(y) \not= 0$, so $\eta(x) \not= 0$: contradiction.
Hence $\eta$ is injective.
We return to the proof of the proposition. Let
$A_n = W_n^p({\Bbb F}\kern1pt_p[\vec t1k])$. From what we have just done, it follows that
$A_n$
may be identified with a subring of $W$. Since
$\eta(p^r t_i^{j/p^r}) = (0,\ldots,0,t_i^j,0,\ldots,0)$, where $t_i^j$ appears
in the \th{r} spot, it follows that $p^r t_i^{j/p^r} \in A_n$, for each $i$,
$r$, and $j$. To complete the proof, we must show that these elements generate
$A_n$. Let $A_n'$ be the subring of $A_n$ generated by the elements
$p^r t_i^{j/p^r}$. Consider the canonical map \mp[[ \tau || A_n || A_{n-1} ]].
By induction on $n$ we may assume that\ the elements $p^r t_i^{j/p^r}$ generate $A_{n-1}$,
and so that $\tau(A_n') = A_{n-1}$. Let $f \in {\Bbb F}\kern1pt_p[\vec t1k]$. Write
$f = \sum_I a_I t^I$, where $I$ is a multi-index. Since
$a_I \in \setof{0,\ldots,p-1}$, we may view $a_I$ as an element of
$\xmode{\Bbb Z}/p^n\xmode{\Bbb Z}$. Let ${\tilde{\lowercase{F}}} = p^{n-1} \sum_I a_I (t^I)^{1/p^{n-1}} \in W$. Then
${\tilde{\lowercase{F}}}$ corresponds to the element $(0,\ldots,0,f)$ of $A_n$. Hence
$\mathop{\operatoratfont Ker}\nolimits(\tau) \subset A_n'$. Since $\tau(A_n') = A_{n-1}$, it follows that
$A_n' = A_n$. {\hfill$\square$}
\end{proof}
\begin{corollary}\label{witt-finite}
Fix $n \in \xmode{\Bbb N}$ and a prime number $p$. Let $A$ be an ${\Bbb F}\kern1pt_p$-algebra of finite
type. Then $W_n^p(A)$ is a $\xmode{\Bbb Z}$-algebra of finite type.
\end{corollary}
\begin{proof}
Choose a surjection \mp[[ \pi || {\Bbb F}\kern1pt_p[\vec x1k] || A ]]. Then $W_n^p(\pi)$ is
a surjection, and since \pref{witt-ring-of-poly-ring} $W_n^p({\Bbb F}\kern1pt_p[\vec x1k])$
is of finite type over $\xmode{\Bbb Z}$, so is $W_n^p(A)$. {\hfill$\square$}
\end{proof}
The following result is a variant of \pref{witt-coherent}.
\begin{lemma}\label{sneezewort}
Fix $n \in \xmode{\Bbb N}$ and a prime number $p$. Assume that $A$ is an ${\Bbb F}\kern1pt_p$-algebra of
finite type. Let $C = W_n^p(A)$. (By \pref{witt-finite} $C$ is noetherian.)
Let $F$ be a coherent $C$-functor, built up from
$\setof{C/(p^k)}_{1 \leq k \leq n}$. Let $G$ be the $A$-functor given by
$G(B) = F(W_n^p(B))$. Let ${\tilde{G}}$ be the sheaf associated to $G$ for the ffqc
topology. Then ${\tilde{G}}$ is representable.
\end{lemma}
\begin{remarks}
The functor $G$ is not in general coherent, as it is not in general an ffqc
sheaf. For an arbitrary coherent $C$-functor $F$, with no restrictions on how
it is built up, it may be that the corresponding functor ${\tilde{G}}$ is always
representable.
\end{remarks}
\begin{proofnodot}
(of \ref{sneezewort}.)
Since any limit of finitely many representable functors is representable,
we may assume that\ $F = \underline{C/(p^k)}$ for some $k$. Then we may describe $G$ by:
$$G(B)\ =\ {\setof{(\vec b0{n-1}): b_i \in B \hbox{\ for all\ } i} \over
\setof{(0,\ldots,0,d_k^{p^k},\ldots,d_{n-1}^{p^k}):
d_i \in B \hbox{\ for all\ } i}},$%
$where it is to be understood that this quotient of abelian groups takes place
with respect to\ the abelian group structure on $W_n^p(B)$. For for any
$b \in B$, there exists a faithfully flat ring extension
\mp[[ \phi || B || B' ]] such that $\phi(b)$ is a \th{(p^k)} power. Therefore
to complete the proof, it suffices to show that the $A$-functor $H$ given by
$$H(B)\ =\ {\setof{(\vec b0{n-1}): b_i \in B \hbox{\ for all\ } i} \over
\setof{(0,\ldots,0,e_k,\ldots,e_{n-1}):
e_i \in B \hbox{\ for all\ } i}}$%
$is representable. (Then we will have $H = {\tilde{G}}$.) But
\begin{eqnarray*}
H(B) & \cong & W_n^p(B) / \mathop{\operatoratfont Ker}\nolimits[W_n^p(B)\ \mapE{}\ W_k^p(B)]\\
& \cong & W_k^p(B)\ \cong\ B^k,
\end{eqnarray*}
so $H$ is representable. {\hfill$\square$}
\end{proofnodot}
Let $A = {\Bbb F}\kern1pt_p[\vec t1k]$. Let $C = (\xmode{\Bbb Z}/p^n\xmode{\Bbb Z})[\vec t1k]$. There is a
canonical map \mp[[ \phi || C || W_n^p(A) ]] given by setting
$\phi(t_i) = (t_i,0,\ldots,0)$ for each $i$.
\begin{prop}\label{bindweed}
Let $A = {\Bbb F}\kern1pt_p[\vec t1k]$. Let $C = (\xmode{\Bbb Z}/p^n\xmode{\Bbb Z})[\vec t1k]$.
Let $S$ be a multiplicatively closed subset of $C$. Let \mp[[ \pi || C || A ]]
be the canonical map. Let $F$ be a coherent $S^{-1}C$-functor. Assume that
$F$ is built up from $\setof{S^{-1}C/(p^r)}_{1 \leq r \leq n}$. Let
${\overline{S}} = \pi(S)$. Let $G$ be the ffqc sheaf associated to the
${\overline{S}}^{-1}A$-functor given by $B\mapsto F(W_n^p(B))$. (This makes sense by
\ref{localize-now}.) Then:
\begin{alphalist}
\item $G$ is representable;
\item the canonical map \mp[[ i || F(S^{-1}C) || G({\overline{S}}^{-1}A) ]] is injective.
\end{alphalist}
\end{prop}
\begin{proof}
{\bf (a):} Let $D = W_n^p({\overline{S}}^{-1}A)$.
Let \mp[[ \phi || \mathop{\operatoratfont Spec}\nolimits(D) || \mathop{\operatoratfont Spec}\nolimits(S^{-1}C) ]] be
the canonical map. Then $\phi^*F$ is a $D$-functor, which is coherent (see
\ref{pull}), and is in fact built up from $\setof{D/(p^r)}_{1 \leq r \leq n}$.
Apply \pref{sneezewort}.
{\bf (b):} The construction is functorial in $F$, so we may reduce
to the case where $F = \underline{S^{-1}C/(p^r)}$. In that case, one sees that $i$ is
isomorphic to the canonical map
\mapx[[ S^{-1}(\xmode{\Bbb Z}/p^r\xmode{\Bbb Z})[\vec t1k] || W_r^p({\overline{S}}^{-1}{\Bbb F}\kern1pt_p[\vec t1k]) ]]. We may
reduce to showing that the canonical map
\mp[[ j || (\xmode{\Bbb Z}/p^r\xmode{\Bbb Z})[\vec t1k] || W_r^p({\Bbb F}\kern1pt_p[\vec t1k]) ]] is injective.
This follows from the proof of \pref{witt-ring-of-poly-ring}. {\hfill$\square$}
\end{proof}
\begin{lemma}\label{generic-representability}
Assume that $A$ is a domain. Let $F$ be a coherent $A$-functor. Then
for some $f \in A - \setof{0}$, the pullback [along
\mapx[[ \mathop{\operatoratfont Spec}\nolimits(A_f) || \mathop{\operatoratfont Spec}\nolimits(A) ]]{}] of $F$ to $A_f$ is representable
by an $A$-algebra of finite type.
\end{lemma}
\begin{sketch}
The functor $F$ is built up from finitely many $A$-modules, each finitely generated. Pick
some $f \in A - \setof{0}$ such that the localization of each such module at
$f$ is free. {\hfill$\square$}
\end{sketch}
\begin{lemma}\label{generic-linearity}
Assume that $X$ is integral. Let $G$ be an affine group scheme of finite type
over $X$. Then there exists some $n \in \xmode{\Bbb N}$, a nonempty open subscheme
$U \subset X$, and a closed immersion \mapx[[ G_U || \mathop{\operatoratfont GL}\nolimits_n(U) ]] of $U$-schemes
which is also a homomorphism.
\end{lemma}
\begin{sketch}
Let $\eta$ be the generic point of $X$. For some $n \in \xmode{\Bbb N}$, we have a closed
immersion (and a homomorphism) \mp[[ h || G_\eta || \mathop{\operatoratfont GL}\nolimits_n(X)_\eta ]] of
$\mathop{\operatoratfont Spec}\nolimits k(\eta)$-schemes. There is a nonempty open subscheme $V \subset X$ and a
closed immersion \mapx[[ G_V || \mathop{\operatoratfont GL}\nolimits_n(V) ]] of $V$-schemes which induces $h$.
Replacing $V$ by a sufficiently small nonempty open subscheme $U$, we obtain
(by restriction) a homomorphism (and a closed immersion)
\mapx[[ G_U || \mathop{\operatoratfont GL}\nolimits_n(U) ]] of $U$-schemes. {\hfill$\square$}
\end{sketch}
\begin{lemma}\label{soup-bone}
Let $A = (\xmode{\Bbb Z}/p^n\xmode{\Bbb Z})[\vec t1k]$. Let $M$ be a finitely generated\ $A$-module. Then there
exists a non-zero-divisor $f \in A$ and positive integers $\vec l1r$ such
that $M_f \cong A_f/(p^{l_1}) \manyo+ A_f/(p^{l_r})$ as $A_f$-modules.
\end{lemma}
\begin{sketch}
Let $S$ be the set of non-zero-divisors of $A$. In $S^{-1}A$, the only
elements (up to associates) are $1,p,\ldots,p^{n-1},0$. Hence every ideal of
$S^{-1}A$ is principal, so every finitely generated\ $S^{-1}A$-module is a direct sum of
cyclic modules, necessarily of the form $S^{-1}A/(p^j)$ for various $j$. The
lemma follows. {\hfill$\square$}
\end{sketch}
\begin{proofnodot}
(of \ref{coherent-implies-arithmetically-linear}.)
The comment about what happens when $n$ is invertible in $\Gamma(X,{\cal O}_X)$
follows from (\ref{linear-implies-finite-torsion}b), so a direct proof is
omitted.
Some of the steps here follow the proof of \pref{coherent-implies-linear}.
We may assume that $X$ is affine, $X = \mathop{\operatoratfont Spec}\nolimits(A)$. Choose $M$ and $B$ as in the
proof of \pref{coherent-implies-linear}. Write $B = S^{-1}C$ for some
finitely generated\ $A$-algebra $C$ and some multiplicatively closed subset $S \subset C$.
Certainly we may replace $A$ by $C$, so $B = S^{-1}A$. Replacing $A$ by $A_g$
for some suitably chosen $g \in S$, we may assume that\ the connected components of
$\mathop{\operatoratfont Spec}\nolimits(A)$ are irreducible and that they correspond bijectively with the points
of $\mathop{\operatoratfont Spec}\nolimits(S^{-1}A)$. Write $A = A_1 \times \cdots \times A_m$, where $\mathop{\operatoratfont Spec}\nolimits(A_i)$ is
irreducible for each $i$. Localizing further if necessary, we may assume that\ $A_i$ has a
unique associated prime for each $i$.
We may reduce to the following situation: $A$ has a unique associated prime,
$S^{-1}A$ is an Artin local ring. It suffices to show that for some $f \in S$,
$G(A_f)$ is arithmetically linear. Let $r$ be the characteristic of $A$.
Since $A$ has a unique associated prime and $S^{-1}A$ is an Artin local ring,
every non-nilpotent element of $A$ lies in $S$.
First suppose that $r = 0$. By Noether normalization, we may find
algebraically
independent elements $\vec x1s \in A$ such that $A \o*_\xmode{\Bbb Z} {\Bbb Q}\kern1pt$ is module-finite
over ${\Bbb Q}\kern1pt[\vec x1s]$. As in the proof of \pref{pork}, there is some $n \in \xmode{\Bbb N}$
such that $A \o*_\xmode{\Bbb Z} \xmode{\Bbb Z}[1/n]$ is a module-finite $\xmode{\Bbb Z}[1/n, \vec x1s]$-algebra.
Every nonzero element of $\xmode{\Bbb Z}[\vec x1s]$ lies in $S$. We may replace $A$ by
$A \o*_\xmode{\Bbb Z} \xmode{\Bbb Z}[1/n]$. By (\ref{push}b), the pushforward of $G$ to
$\xmode{\Bbb Z}[1/n, \vec x1s]$ is coherent. By \pref{generic-representability}, there
is some $f \in \xmode{\Bbb Z}[1/n,\vec x1s] - \setof{0}$ such that the pullback of
$G$ to $\xmode{\Bbb Z}[1/n,\vec x1s,f^{-1}]$ is representable by an affine
$\xmode{\Bbb Z}[1/n,\vec x1s,f^{-1}]$-algebra of finite type. Apply
\pref{generic-linearity}.
Now suppose that $r > 0$. Then $r = p^m$ for some prime $p$ and some $m$. By
the Noether normalization theorem, we may find algebraically independent
elements $\vec x1s \in A$ such that $A \o*_\xmode{\Bbb Z} {\Bbb F}\kern1pt_p$ is module-finite over
${\Bbb F}\kern1pt_p[\vec x1s]$. It follows that $A$ is module-finite over
$E = (\xmode{\Bbb Z}/p^m\xmode{\Bbb Z})[\vec x1s]$. By (\ref{push}b), the pushforward $H$ of $G$ to
$E$ is coherent, so we may reduce to the case where $A = (\xmode{\Bbb Z}/p^m\xmode{\Bbb Z})[\vec x1s]$.
Let $\vec M1d$ be $A$-modules from which $G$ can be built up. By
\pref{soup-bone}, there is some $f \in S$ such that each $(M_i)_f$ is a direct
sum of modules of the form $A_f/(p^r)$, for various $r$. Let $G_f$ be the
pullback of $G$ to $A_f$. Then $G_f$ is built up from
$\setof{A_f/(p^r)}_{1 \leq r \leq n}$. The theorem follows now by applying
\pref{bindweed} and \pref{generic-linearity}. {\hfill$\square$}
\end{proofnodot}
\begin{problemx}
Which arithmetically linear groups arise as $G(X)$, for some scheme $X$ of
finite type over $\xmode{\Bbb Z}$ and some group-valued coherent $X$-functor $G$?
\end{problemx}
Certain groups can be shown to be quotients of arithmetically linear abelian
groups by finitely generated\ subgroups. For example, in the next section we shall see that
this is the case for $\mathop{\operatoratfont Pic}\nolimits(X)$, where $X$ is a reduced scheme of finite type
over $\xmode{\Bbb Z}$. Although it may in fact be the case that $\mathop{\operatoratfont Pic}\nolimits(X)$ is itself
arithmetically linear, we have not been able to show this, so we are lead to
the following lemma:
\begin{lemma}\label{arith-linear-over-finite}
Let $G$ be an arithmetically linear abelian group. Let $H$ be a
finitely generated\ subgroup of $G$. Then the torsion subgroup of $G/H$ is supported at a
finite set of primes.
\end{lemma}
\begin{proof}
Write $G = B \times C$, where $B$ is bounded and $C$ is cobounded. For any
abelian group $M$, let $M[1/n]$ denote $M \o*_\xmode{\Bbb Z} \xmode{\Bbb Z}[1/n]$. Choose $n \in \xmode{\Bbb N}$
such that $nB = 0$ and such that $C[1/n]$ is a free $\xmode{\Bbb Z}[1/n]$-module. Let
$Q = G/H$. It suffices to show that the torsion subgroup of $Q[1/n]$ is
supported at a finite set of primes. We have an exact sequence:
\ses{H[1/n]}{C[1/n]}{Q[1/n]%
}of $\xmode{\Bbb Z}[1/n]$-modules. Since $H[1/n]$ is contained in a finitely generated\ direct summand of
$C[1/n]$, it suffices to show that for any finitely generated\ $\xmode{\Bbb Z}[1/n]$-module $M$, the
torsion subgroup of $M$ is supported at a finite set of primes. This is easily
checked. {\hfill$\square$}
\end{proof}
\block{Application to the Picard group}
Let $M$ be a finitely generated\ $A$-module. Let $\mathop{\underline{\operatoratfont PGL}}\nolimits(M)$ denote the Zariski sheaf
associated to the $A$-functor given by
$$B \mapsto {\mathop{\operatoratfont Aut}\nolimits_B(M \o*_A B) \over B^*}.$%
$More generally, let ${\cal{M}}$ be a coherent ${\cal O}_X$-module. Let $\mathop{\underline{\operatoratfont PGL}}\nolimits({\cal{M}})$
denote the Zariski sheaf associated to the $X$-functor given by
$$Y\ \mapsto\ {\mathop{\operatoratfont Aut}\nolimits_Y({\cal{M}}_Y) \over \Gamma({\cal O}_Y^*)}.$%
$Since $\mathop{\underline{\operatoratfont Aut}}\nolimits({\cal{M}})$ acts by conjugation on $\mathop{\underline{\operatoratfont End}}\nolimits({\cal{M}})$, we obtain
a canonical morphism of group-valued $X$-functors:
\dmap[[ \psi || \mathop{\underline{\operatoratfont PGL}}\nolimits({\cal{M}}) || \mathop{{\mathbf{Aut}}_{\kern1pt\operatoratfont alg}(\mathop{\underline{\operatoratfont End}}\nolimits({\cal{M}})).\footnote{For the
actual arguments which we use, one could substitute the simpler functor
$\mathop{\mathbf{Aut}}\nolimits(\mathop{\underline{\operatoratfont End}}\nolimits({\cal{M}}))$, but we use $\mathop{{\mathbf{Aut}}_{\kern1pt\operatoratfont alg}(\mathop{\underline{\operatoratfont End}}\nolimits({\cal{M}}))$ instead for asthetic
reasons because it makes $\psi$ closer to an isomorphism.} ]]%
The $X$-functor $\mathop{{\mathbf{Aut}}_{\kern1pt\operatoratfont alg}(\mathop{\underline{\operatoratfont End}}\nolimits({\cal{M}}))$ is locally coherent by
\pref{autalg-is-coherent} and example \pref{end-example} from
\S\ref{examples-section}; it is coherent, at least assuming that $X$ is
separated, by \pref{lump}.
We would like to show that $\mathop{\underline{\operatoratfont PGL}}\nolimits({\cal{M}})$ is coherent. We could do this by
showing that $\psi$ is an isomorphism. However, other than the case where
${\cal{M}}$ is locally free, we do not know if $\psi$ is an isomorphism. Therefore,
we settle for showing that (under certain special circumstances) $\psi(X)$ is
injective. This is a weak substitute for showing that $\mathop{\underline{\operatoratfont PGL}}\nolimits({\cal{M}})$ is
coherent.\footnote{In trying to show that $\psi$ is a monomorphism, one comes
to the following question: Let $M$ be a finitely generated\ $A$-module. Let
$\sigma$ be an automorphism of $M$ as an $A$-module. Assume that for every
commutative $A$-algebra $B$, $\sigma \o*_A B$ lies in the center of
$\mathop{\operatoratfont End}\nolimits_B(M \o*_A B)$. Does it follow that $\sigma$ is a homothety, i.e.\ that
$\sigma$ is given by multiplication by an element of $A$?}
First we prove a lemma, then a corollary which
says something directly about $\psi$.
\begin{lemma}\label{homothety}
Assume that $A$ is reduced. Let $A'$ be a ring, with
$A \subset A' \subset \nor{A}$. Assume that the ideal $[A:A']$ of $A$ is prime. Let
$\sigma$ be an endomorphism of $A'$ as an $A$-module. Assume that for every
${\xmode{{\fraktur{\lowercase{P}}}}} \in \mathop{\operatoratfont Spec}\nolimits(A)$, $\sigma \o*_A k({\xmode{{\fraktur{\lowercase{P}}}}})$ is a homothety of
$A' \o*_A k({\xmode{{\fraktur{\lowercase{P}}}}})$ as a $k({\xmode{{\fraktur{\lowercase{P}}}}})$-module. Then $\sigma$ is a homothety of $A'$
as an $A$-module.
\end{lemma}
\begin{proof}
First we show that $\sigma$ is given by multiplication by $\sigma(1)$.
By subtracting the endomorphism of $A'$ given by multiplication by $\sigma(1)$,
we may reduce to showing that if $\sigma(1) = 0$, then $\sigma = 0$. Let
$\vec\lfP1r$ be the minimal primes of $A$. Using the fact that
$\sigma \o*_A k({\xmode{{\fraktur{\lowercase{P}}}}}_i)$ is a homothety, we conclude that
$\sigma \o*_A k({\xmode{{\fraktur{\lowercase{P}}}}}_i) = 0$. Let $x \in A'$. Then $\sigma(x) \mapsto 0$
in $A' \o*_A k({\xmode{{\fraktur{\lowercase{P}}}}}_i)$. Since the map
\dmapx[[ A' || \oplus_{i=1}^r A' \o*_A k({\xmode{{\fraktur{\lowercase{P}}}}}_i) ]]%
is injective, it follows that $\sigma(x) = 0$, and hence that $\sigma = 0$.
Hence (reverting to the original problem), we see that $\sigma$ is given
by multiplication by $\sigma(1)$.
Let $Q = A'/A$, which is an $A$-module. Let ${\xmode{{\fraktur{\lowercase{P}}}}} = [A:A']$.
Let $\overline{\sigma(1)}$ denote the image of $\sigma(1)$ in $Q$. It follows
that $\overline{\sigma(1)} \mapsto 0$ in $Q \o*_A k({\xmode{{\fraktur{\lowercase{P}}}}})$. But by the construction
of ${\xmode{{\fraktur{\lowercase{P}}}}}$, the canonical map \mapx[[ Q || Q \o*_A k({\xmode{{\fraktur{\lowercase{P}}}}}) ]] is injective.
Hence $\overline{\sigma(1)} = 0$. Hence $\sigma(1) \in A$. {\hfill$\square$}
\end{proof}
\begin{corollary}\label{key-embedding}
Assume that $A$ is reduced. Let $A'$ be a ring, with
$A \subset A' \subset \nor{A}$. Assume that the ideal $[A:A']$ of $A$ is prime.
Assume that $A'$ is a finitely generated\ $A$-module. Let
\dmap[[ \psi || \mathop{\underline{\operatoratfont PGL}}\nolimits(A') || \mathop{{\mathbf{Aut}}_{\kern1pt\operatoratfont alg}(\mathop{\underline{\operatoratfont End}}\nolimits(A')) ]]%
be the canonical (conjugation) morphism of group-valued $A$-functors. Then
$\psi(A)$ is injective.
\end{corollary}
\begin{proof}
Consider the morphism \mp[[ \psi_0 || \mathop{\underline{\operatoratfont Aut}}\nolimits(A') || \mathop{{\mathbf{Aut}}_{\kern1pt\operatoratfont alg}(\mathop{\underline{\operatoratfont End}}\nolimits(A')) ]]
which induces $\psi$. Then $\ker(\psi_0(A))$ consists of those automorphisms
$\sigma$ of $A'$ as an $A$-module with the property that
$\sigma \o*_A B \in Z[\mathop{\operatoratfont End}\nolimits_B(A' \o*_A B)]$ for every commutative $A$-algebra
$B$. If $B$ is a field, it follows that for such a $\sigma$,
$\sigma \o*_A B$ is given by multiplication by an element of $B$.
By \pref{homothety}, such a $\sigma$ is itself a homothety. Hence
$\ker(\psi_0(A)) = A^*$.
Let $f \in A - \setof{0}$. Then the ideal $[A_f:A'_f]$ of $A_f$ is prime
and it follows that $\ker(\psi_0(A_f)) = A_f^*$. Hence the map
\dmapx[[ {\mathop{\operatoratfont Aut}\nolimits_{A_f}(A' \o*_A A_f) \over A_f^*} || \mathop{{\mathbf{Aut}}_{\kern1pt\operatoratfont alg}(\mathop{\underline{\operatoratfont End}}\nolimits(A'))(A_f) ]]%
is injective. Considering the sheafification which occurs in the definition
of $\mathop{\underline{\operatoratfont PGL}}\nolimits(A')$, we see that $\psi(A)$ is injective. {\hfill$\square$}
\end{proof}
\begin{lemma}\label{filtration-of-normalization}
Let $C$ be an overring of $A$, with $C$ finitely generated\ as an $A$-module.
Then there exists a chain:
$$A\ =\ A_0\ \subset\ A_1\ \subset\ \cdots\ \subset\ A_n\ =\ C$%
$of rings such that for each $k = 1, \ldots, n$, the ideal $[A_{k-1}:A_k]$
of $A_{k-1}$ is prime.
\end{lemma}
\begin{proof}
We may assume that $C \not= A$. For each $y \in C - A$, let
$I_y = [A:A[y]]$. (All conductors are to be computed as ideals in $A$.)
Choose $y \in C - A$ so that $I_y$ is maximal amongst all such ideals.
We will show that $I_y$ is prime. This will complete the proof.
Pick $a \in A$ such that $[I_y:a]$ is prime. Choose $n$ so that $y$ satisfies
a monic polynomial of degree $n$ with coefficients in $A$. Then
\hskip 0pt plus 4cm\penalty1000\hskip 0pt plus -4cm\hbox{$I_y = [A: y,y^2,\ldots,y^{n-1}]$}, and
$[I_y:a] = [A: ay,ay^2,\ldots,ay^{n-1}]$.
First we show that $ay \notin A$. Suppose otherwise. Since
$[I_y:a] \not= A$, $a \notin I_y$. Hence $ay^k \notin A$, for some $k$
with $2 \leq k \leq n-1$. Consider all pairs $(r,s) \in \xmode{\Bbb N}^2$,
$s < n$, such that $a^r y^s \notin A$. Choose such a pair $(r_0,s_0)$ such
that the ratio $s_0/r_0$ is as small as possible. (This is possible because
$ay \in A$, and hence $a^r y^s \in A$ for all $r \geq s$.)
Let $(r,s) \in \xmode{\Bbb N}^2$ be such that $a^r y^s \notin A$. We will show that
$s_0/r_0 \leq s/r$. Suppose otherwise: $s_0/r_0 > s/r$. We may assume that
$s \geq n$. We have:
$$a^r y^s = \sum_{i = 0}^{n-1} c_i a^r y^i$%
$for suitable $c_i \in A$. For $i$ in the given range, $s/r > i/r$, so
$s_0/r_0 > i/r$. Hence $a^r y^i \in A$. Hence $a^r y^s \in A$: contradiction.
Hence $s_0/r_0 \leq s/r$.
Let $z = a^{r_0} y^{s_0}$.
We have $I_y \subset I_z$, so by the maximality of $I_y$, we have $I_y = I_z$.
For any $k \in \xmode{\Bbb N}$, $az^k = a^{r_0 k+1} y^{s_0 k}$, and
$${s_0 k \over r_0 k + 1} < {s_0 \over r_0}.$%
$Hence $az^k \in A$. Hence $a \in I_z$. Hence $a \in I_y$: contradiction.
Hence $ay \notin A$.
Note that $ay$ satisfies a monic
polynomial of degree $n$ with coefficients in $A$. We have:
$$I_y\ \subset\ [I_y:a]\ \subset\ [A:ay,(ay)^2,\ldots,(ay)^{n-1}]\ =\ I_{ay},$%
$so by the maximality of $I_y$ we must have $I_y = I_{ay}$ and hence $I_y$ is
prime. {\hfill$\square$}
\end{proof}
Let us say that an abelian group $G$ is {\it pseudo-$X$-linear\/} if there
exists a filtration
$$0 = G_0 \subset G_1 \subset \cdots \subset G_m = G,$%
$with the property that $G_i/G_{i-1}$ is $X$-linear for each $i$. It is
conceivable that every such group $G$ is $X$-linear.
\begin{theorem}\label{filtration}
Assume that $X$ is reduced, and that the canonical map
\mp[[ \pi || \nor{X} || X ]] is finite. Let $C$ be the quotient sheaf
$\pi_*{\cal O}_{\nor{X}}^*/{\cal O}_X^*$. Then the group $C(X)$ is pseudo-$X$-linear. If
$X$ is of finite type over $\xmode{\Bbb Z}$, then $C(X)$ is arithmetically linear.
\end{theorem}
\begin{proof}
The comment about what happens when $X$ is of finite type over $\xmode{\Bbb Z}$ is left
to the reader; the proof given below works with appropriate changes, provided
that one uses in addition (\ref{bxc}b) and \pref{arith-linear-structure}. One
uses \pref{coherent-implies-arithmetically-linear} instead of
\pref{coherent-implies-linear}.
The following two facts are easily verified: any subgroup of a
pseudo-$X$-linear abelian group is pseudo-$X$-linear, and any product of
finitely many pseudo-$X$-linear abelian groups is pseudo-$X$-linear. It
follows that we may reduce to the case where
$X$ is affine. Then by \pref{filtration-of-normalization}, we may reduce to
the following situation: $A$ and $A'$ are reduced noetherian rings, with
$A \subset A' \subset \nor{A}$ (and $\nor{A}$ is module-finite over $A$), and the
ideal $[A:A']$ of $A$ is prime. We must show that if $F$ is the Zariski sheaf
associated to the $A$-functor given by $B \mapsto (A' \o*_A B)^*/B^*$, then
$F(A)$ is $X$-linear. But $F(A) \cong [\mathop{\underline{\operatoratfont PGL}}\nolimits(A')](A)$, so by
\pref{key-embedding}, it suffices to show that if $G = \mathop{{\mathbf{Aut}}_{\kern1pt\operatoratfont alg}(\mathop{\underline{\operatoratfont End}}\nolimits(A'))$,
then $G(A)$ is $X$-linear. But $G$ is coherent by \pref{autalg-is-coherent},
so the theorem follows from \pref{coherent-implies-linear}. {\hfill$\square$}
\end{proof}
One can check that \pref{linear-implies-finite-torsion} applies to a
pseudo-$X$-linear abelian group, so one obtains the following corollary:
\begin{corollary}\label{finiteness-of-C}
Assume that $X$ is reduced, and that the canonical map
\mp[[ \pi || \nor{X} || X ]] is finite. Let $C$ be the quotient sheaf
$\pi_*{\cal O}_{\nor{X}}^*/{\cal O}_X^*$. Then:
\begin{alphalist}
\item There are only finitely many prime numbers $p$ such that ${}_p C(X)$ is
infinite. Moreover, such a $p$ cannot be invertible in $\Gamma(X,{\cal O}_X)$.
\item If $X$ is essentially of finite type over $\xmode{\Bbb Z}$ or $\xmode{\Bbb Z}_p$ (for some prime
number $p$), then there exist prime numbers $\vec p1n$, none of which are
invertible in $\Gamma(X,{\cal O}_X)$, such that the subgroup of $C(X)$
consisting of torsion prime to $p_1 \cdot \ldots \cdot p_n$ is finite.
\end{alphalist}
\end{corollary}
\begin{corollary}\label{pic-main-theorem}
Assume that $X$ is reduced, and that the canonical map
\mp[[ \pi || \nor{X} || X ]] is finite. Let
$Q = \Gamma({\cal O}_{\nor{X}}^*)/\Gamma({\cal O}_X^*)$. Let $K$ be the kernel of the
canonical map \mapx[[ \mathop{\operatoratfont Pic}\nolimits(X) || \mathop{\operatoratfont Pic}\nolimits(\nor{X}) ]].
\begin{alphalist}
\item Fix $n \in \xmode{\Bbb N}$, and assume that $n$ is invertible in $\Gamma(X,{\cal O}_X)$.
If $Q$ is finitely generated, or more generally if it admits an $n$-divisible subgroup
with finitely generated\ cokernel, then ${}_n K$ is finite.
\item If $Q$ admits a divisible subgroup with finitely generated\ cokernel, then there are
only finitely many prime numbers $p$ such that ${}_p K$ is infinite.
\end{alphalist}
\end{corollary}
\begin{proof}
Any finitely generated\ projective module of constant rank over a semilocal ring is free, so
any line bundle on $\nor{X}$ may be trivialized by an open cover pulled back
from $X$. (This argument was shown to me by R.\ Wiegand.) Hence
$R^1\pi_*({\cal O}_{\nor{X}}^*) = 0$, and so by the Leray spectral sequence we see
that $H^1(X,\pi_* {\cal O}_{\nor{X}}^*) \cong \mathop{\operatoratfont Pic}\nolimits(\nor{X})$.
Run the long exact sequence of cohomology on
\sesdot{{\cal O}_X^*}{\pi_*{\cal O}_{\nor{X}}^*}{C%
}We obtain a short exact sequence:
\sesdot{Q}{C(X)}{K%
}By \pref{finiteness-of-C}, it suffices to show that if
\ses{M'}{M}{M''%
}is a short exact sequence of abelian groups, $\abs{{}_n M} < \infty$, and
there exists an $n$-divisible subgroup $H \subset M'$ such that $M'/H$ is finitely generated,
then $\abs{{}_n M''} < \infty$. This is easily proved -- see
\Lcitemark 21\Rcitemark \Rspace{}. {\hfill$\square$}
\end{proof}
\begin{example}
Let $X = \mathop{\operatoratfont Spec}\nolimits {\Bbb Q}\kern1pt[x,y]/(x^2 - 2y^2)$. Then
$\Gamma({\cal O}_{\nor{X}}^*) / \Gamma({\cal O}_X^*) \cong {\Bbb Q}\kern1pt[\sqrt{2}]^*/{\Bbb Q}\kern1pt^*$, which is
not finitely generated.
\end{example}
We consider what happens when $X$ is of finite type over $\xmode{\Bbb Z}$. We need the
following well-known result, which is apparently due to Roquette. A proof of
the key case ($X$ integral, affine) may be found in
(\Lcitemark 8\Rcitemark \ p.\ 39).
\begin{prop}\label{units-over-Z}
Let $X$ be a reduced scheme of finite type over $\xmode{\Bbb Z}$. Then the group
$\Gamma(X,{\cal O}_X)^*$ is finitely generated.
\end{prop}
\begin{theorem}\label{rabbit-food}
Let $X$ be a reduced scheme of finite type over $\xmode{\Bbb Z}$. Then the torsion
subgroup of $\mathop{\operatoratfont Pic}\nolimits(X)$ is supported at a finite set of primes, and if
${}_p \mathop{\operatoratfont Pic}\nolimits(X)$ is infinite, then the prime $p$ is not invertible in
$\Gamma(X,{\cal O}_X)$.
\end{theorem}
\begin{proof}
By (\Lcitemark 25\Rcitemark \ 2.7.6) we know that $\mathop{\operatoratfont Pic}\nolimits(\nor{X})$ is finitely generated.
Therefore it suffices to show that if
$K = \mathop{\operatoratfont Ker}\nolimits[\mathop{\operatoratfont Pic}\nolimits(X)\ \mapE{}\ \mathop{\operatoratfont Pic}\nolimits(\nor{X})]$, then $K$ is supported at a finite
set of primes, and if ${}_p K$ is infinite, then the prime $p$ is not
invertible in $\Gamma(X,{\cal O}_X)$.
By \pref{filtration}, \pref{units-over-Z} and the argument of
\pref{pic-main-theorem}, one sees that $K$ is isomorphic to the quotient of
an arithmetically linear group by a finitely generated\ subgroup. Hence
\pref{arith-linear-over-finite} tells us that the torsion subgroup of $\mathop{\operatoratfont Pic}\nolimits(X)$
is supported at a finite set of primes. The last assertion of the theorem
follows from (\ref{pic-main-theorem}a). {\hfill$\square$}
\end{proof}
For $X$ a non-reduced scheme of finite type over $\xmode{\Bbb Z}$, we do not know if the
torsion subgroup of $\mathop{\operatoratfont Pic}\nolimits(X)$ is supported at a finite set of primes. However,
for any commutative noetherian ring $A$, the canonical map
\mapx[[ \mathop{\operatoratfont Pic}\nolimits(A) || \mathop{\operatoratfont Pic}\nolimits(\RED{A}) ]] is an isomorphism, so we have:
\begin{corollary}\label{yyyyy}
Let $A$ be a finitely generated\ commutative $\xmode{\Bbb Z}$-algebra. Then the torsion subgroup of
$\mathop{\operatoratfont Pic}\nolimits(A)$ is supported at a finite set of primes, and if ${}_p \mathop{\operatoratfont Pic}\nolimits(A)$ is
infinite, then the prime $p$ is not invertible in $A$.
\end{corollary}
Let $A$ be a finitely generated\ commutative $\xmode{\Bbb Z}$-algebra. If is natural to ask if there
exist prime numbers
$\vec p1n$, none of which are invertible in $A$, such that the subgroup of
$\mathop{\operatoratfont Pic}\nolimits(A)$ consisting of torsion prime to $\vec p1n$ is finite.
Unfortunately, the answer is no. For a counterexample, see
(\Lcitemark 21\Rcitemark \ 6.3).
Now we consider what happens when $X$ is of finite type over ${\Bbb F}\kern1pt_p$.
\begin{theorem}\label{F-sub-p}
Let $X$ be a scheme of finite type over ${\Bbb F}\kern1pt_p$. Then there exists a
finitely generated\ abelian group $H$ and a finite $p$-group $F$ such that
$$\mathop{\operatoratfont Pic}\nolimits(X)\ \cong\ H\ \o+ \ [\o+_{n=1}^\infty F].$$
\end{theorem}
\begin{sketch}
Consider the class ${\cal{C}}$ of groups of the form ascribed to $\mathop{\operatoratfont Pic}\nolimits(X)$ in the
theorem. The groups in ${\cal{C}}$ are all bounded\/\kern1.5pt$\times$\discretionary{}{}{\kern1.5pt}cobounded. One can check without great
difficulty that ${\cal{C}}$ is closed under formation of subgroups, quotient groups,
and extensions.
First suppose that $X$ is reduced. Since $\mathop{\operatoratfont Pic}\nolimits(\nor{X})$ is finitely generated, it suffices
to show that $K = \mathop{\operatoratfont Ker}\nolimits[\mathop{\operatoratfont Pic}\nolimits(X)\ \mapE{}\ \mathop{\operatoratfont Pic}\nolimits(\nor{X})]$ is in ${\cal{C}}$. We may
assume that $X$ is affine. In fact, it suffices to show that if
$X' = \mathop{\operatoratfont Spec}\nolimits(A')$ is a partial normalization of $X = \mathop{\operatoratfont Spec}\nolimits(A)$, and if
$[A':A]$ is prime, then $K' = \mathop{\operatoratfont Ker}\nolimits[\mathop{\operatoratfont Pic}\nolimits(X)\ \mapE{}\ \mathop{\operatoratfont Pic}\nolimits(X')]$ is in ${\cal{C}}$.
The proof of \pref{rabbit-food} shows that $K'$ is a quotient of an
arithmetically linear group. In fact, the arguments used to arrive at this
result show that there exists an ${\Bbb F}\kern1pt_p$-algebra $A$ of finite type such that
$K'$ is a quotient of an abelian subgroup $H \subset \mathop{\operatoratfont GL}\nolimits_n(A)$. Therefore there
exists an ${\Bbb F}\kern1pt_p$-algebra $C$ of finite type such that $K'$ is a quotient of a
subgroup of $C^*$. The usual methods show that $C^*$ may be built up via
extensions from a finitely generated\ abelian group and some ${\Bbb F}\kern1pt_p$-vector spaces. It follows
that $C^* \in {\cal{C}}$, and hence that any quotient of $C^*$ is in ${\cal{C}}$.
Now suppose that $X$ is arbitrary, not necessarily reduced. Let ${\cal{J}}$ be the
nilradical of $X$. There is an exact sequence:
\Rowfive{0}{{\cal{J}}^n/{\cal{J}}^{n+1}}{({\cal O}_X/{\cal{J}}^{n+1})^*}{({\cal O}_X/{\cal{J}}^n)^*}{1%
}of sheaves of abelian groups on $X$. Let $X_n$ be the closed subscheme of $X$
corresponding to the ideal ${\cal{J}}^n$. By taking cohomology, one sees that
$$\mathop{\operatoratfont Ker}\nolimits[\mathop{\operatoratfont Pic}\nolimits(X_{n+1}) \ \mapE{}\ \mathop{\operatoratfont Pic}\nolimits(X_n)]$%
$is $p$-torsion. The theorem follows. {\hfill$\square$}
\end{sketch}
Finally, we compute the Picard group of a few simple examples. The
calculations are an easy consequence of an exact sequence of Milnor
(\Lcitemark 6\Rcitemark \ IX\ 5.3) applied to the cartesian square
\squareSE{A}{\nor{A}}{A/{\xmode{{\fraktur{\lowercase{C}}}}}}{\nor{A}/{\xmode{{\fraktur{\lowercase{C}}}}}\makenull{,}%
}where ${\xmode{{\fraktur{\lowercase{C}}}}}$ is the conductor of $\nor{A}$ into $A$. The exact sequence is:
\splitdiagram{A^*&\mapE{}&\nor{A}^* \times (A/{\xmode{{\fraktur{\lowercase{C}}}}})^*&\mapE{}%
&(\nor{A}/{\xmode{{\fraktur{\lowercase{C}}}}})^*}{\mapE{}&\mathop{\operatoratfont Pic}\nolimits(A)&\mapE{}%
&\mathop{\operatoratfont Pic}\nolimits(\nor{A}) \times \mathop{\operatoratfont Pic}\nolimits(A/{\xmode{{\fraktur{\lowercase{C}}}}})&\mapE{}&\mathop{\operatoratfont Pic}\nolimits(\nor{A}/{\xmode{{\fraktur{\lowercase{C}}}}}).%
}Let $p$ be a prime number. The first four rings given below may be
viewed as subrings of $\xmode{\Bbb Z}[t,x]$.
\vspace*{0.1in}
\begin{center}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{||c|c||} \hline
ring $A$ & structure of $\mathop{\operatoratfont Pic}\nolimits(A)$ \\ \hline\hline
$\xmode{\Bbb Z}[t^2,t^3]$ & $\xmode{\Bbb Z}$ \\ \hline
$\xmode{\Bbb Z}[t^2, t^3, x]$ & free abelian of countably infinite rank \\ \hline
$\xmode{\Bbb Z}[pt, t^2, t^3]$ & ${\Bbb F}\kern1pt_p$ \\ \hline
$\xmode{\Bbb Z}[pt, t^2, t^3, x]$ & ${\Bbb F}\kern1pt_p$-vector space of countably infinite rank
\\ \hline
$\xmode{\Bbb Z}[1/p, t^2, t^3]$ & $\xmode{\Bbb Z}[1/p]$ \\ \hline
${\Bbb F}\kern1pt_p[t^2, t^3, x]$ & ${\Bbb F}\kern1pt_p$-vector space of countably infinite rank
\\ \hline
\end{tabular}
\end{center}
\section*{References}
\addcontentsline{toc}{section}{References}
\ \par\noindent\vspace*{-0.25in}
\hfuzz 5pt
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\def\Ftest{ }\def\Fstr{1}%
\def\Atest{ }\def\Astr{Artin\Revcomma M\Initper }%
\def\Ttest{ }\def\Tstr{Algebraic approximation of structures over complete
local rings}%
\def\Jtest{ }\def\Jstr{Inst. Hautes \'Etudes Sci. Publ. Math.}%
\def\Vtest{ }\def\Vstr{36}%
\def\Dtest{ }\def\Dstr{1969}%
\def\Ptest{ }\def\Pcnt{ }\def\Pstr{23--58}%
\def\Qtest{ }\def\Qstr{access via "artin algebraic approximation"}%
\def\Xtest{ }\def\Xstr{In bound volume.}%
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\def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{2}%
\def\Atest{ }\def\Astr{Artin\Revcomma M\Initper }%
\def\Ttest{ }\def\Tstr{Letter to Grothendieck}%
\def\Dtest{ }\def\Dstr{Nov.\ 5, 1968}%
\def\Qtest{ }\def\Qstr{access via "artin letter grothendieck"}%
\def\Astr{\Underlinemark}%
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\def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{3}%
\def\Atest{ }\def\Astr{Artin\Revcomma M\Initper }%
\def\Ttest{ }\def\Tstr{Algebraization of formal moduli: I}%
\def\Btest{ }\def\Bstr{Global Analysis: Papers in Honor of K.\ Kodaira}%
\def\Itest{ }\def\Istr{Princeton Univ. Press}%
\def\Dtest{ }\def\Dstr{1969}%
\def\Ptest{ }\def\Pcnt{ }\def\Pstr{21--71}%
\def\Qtest{ }\def\Qstr{access via "artin formal moduli one"}%
\Refformat\egroup%
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}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{2}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{4}%
\def\Atest{ }\def\Astr{Atiyah\Revcomma M\Initper \Initgap F\Initper %
\Aand I\Initper \Initgap G\Initper Macdonald}%
\def\Ttest{ }\def\Tstr{Introduction to Commutative Algebra}%
\def\Itest{ }\def\Istr{Addison-Wesley}%
\def\Ctest{ }\def\Cstr{Reading, Mass.}%
\def\Dtest{ }\def\Dstr{1969}%
\def\Qtest{ }\def\Qstr{access via "atiyah macdonald"}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{4}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{5}%
\def\Atest{ }\def\Astr{Auslander\Revcomma M\Initper }%
\def\Ttest{ }\def\Tstr{Coherent functors}%
\def\Btest{ }\def\Bstr{Proceedings of the Conference on Categorical Algebra (La
Jolla, 1965)}%
\def\Etest{ }\def\Estr{S\Initper Eilenberg%
\Ecomma D\Initper \Initgap K\Initper Harrison%
\Ecomma S\Initper MacLane%
\Eandd H\Initper R\"ohrl}%
\def\Ptest{ }\def\Pcnt{ }\def\Pstr{189--231}%
\def\Itest{ }\def\Istr{Spring\-er-Ver\-lag}%
\def\Ctest{ }\def\Cstr{New York}%
\def\Qtest{ }\def\Qstr{access via "auslander coherent functors"}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{6}%
\def\Atest{ }\def\Astr{Bass\Revcomma H\Initper }%
\def\Ttest{ }\def\Tstr{Algebraic K-Theory}%
\def\Itest{ }\def\Istr{W.\ A.\ Benjamin}%
\def\Ctest{ }\def\Cstr{New York}%
\def\Dtest{ }\def\Dstr{1968}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{7}%
\def\Atest{ }\def\Astr{Bass\Revcomma H\Initper }%
\def\Ttest{ }\def\Tstr{Some problems in ``classical'' algebraic K-theory}%
\def\Stest{ }\def\Sstr{Lecture Notes in Mathematics}%
\def\Itest{ }\def\Istr{Spring\-er-Ver\-lag}%
\def\Ctest{ }\def\Cstr{New York}%
\def\Vtest{ }\def\Vstr{342}%
\def\Dtest{ }\def\Dstr{1973}%
\def\Ptest{ }\def\Pcnt{ }\def\Pstr{3--73}%
\def\Qtest{ }\def\Qstr{access via "bass classical problems"}%
\def\Xtest{ }\def\Xstr{I don't have this.}%
\def\Astr{\Underlinemark}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{8}%
\def\Atest{ }\def\Astr{Bass\Revcomma H\Initper }%
\def\Ttest{ }\def\Tstr{Introduction to some methods of algebraic K-theory}%
\def\Jtest{ }\def\Jstr{Conference Board of the Mathematical Sciences Regional
Conference Series in Mathematics}%
\def\Vtest{ }\def\Vstr{20}%
\def\Dtest{ }\def\Dstr{1974}%
\def\Qtest{ }\def\Qstr{access via "bass introduction methods"}%
\def\Xtest{ }\def\Xstr{MR 50:441. (pamphlet on shelf) \par Let $R$ be either
$\xmode{\Bbb Z}$ or $F_q[t]$. Let $A = R[\vec t2d]$, where $d \geq 1$. Then $\mathop{\operatoratfont GL}\nolimits_n(A)$ is
a finitely generated group for $n \geq d+2$. Also, let $A$ be any semilocal
ring or $\xmode{\Bbb Z}$ or $k[t]$, where $k$ is any field. Then $\SL_n(A)$ is generated
by elementary matrices for any $n \geq 1$. (See 2.6, 2.8.)}%
\def\Astr{\Underlinemark}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{9}%
\def\Atest{ }\def\Astr{Bergman\Revcomma G\Initper \Initgap M\Initper }%
\def\Ttest{ }\def\Tstr{Ring schemes; the Witt scheme}%
\def\Btest{ }\def\Bstr{Lectures on Curves on an Algebraic Surface {\rm by David
Mumford}}%
\def\Ptest{ }\def\Pcnt{ }\def\Pstr{171---187}%
\def\Itest{ }\def\Istr{Princeton Univ.\ Press}%
\def\Dtest{ }\def\Dstr{1966}%
\def\Qtest{ }\def\Qstr{access via "bergman witt scheme"}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{10}%
\def\Atest{ }\def\Astr{Bertin\Revcomma J\Initper \Initgap E\Initper }%
\def\Ttest{ }\def\Tstr{{\rm\tolerance=1000 Generalites sur les preschemas en
groupes, expos\'e\ ${\rm VI}_{\rm B}$ in {\itS\'em\-in\-aire de G\'eom\'e\-trie Al\-g\'e\-bri\-que} (SGA 3)}}%
\def\Stest{ }\def\Sstr{Lecture Notes in Mathematics}%
\def\Itest{ }\def\Istr{Spring\-er-Ver\-lag}%
\def\Ctest{ }\def\Cstr{New York}%
\def\Vtest{ }\def\Vstr{151}%
\def\Dtest{ }\def\Dstr{1970}%
\def\Ptest{ }\def\Pcnt{ }\def\Pstr{318--410}%
\def\Qtest{ }\def\Qstr{access via "bertin preschemas"}%
\def\Xtest{ }\def\Xstr{This includes the following result, announced by
Raynaud, but apparently not proved in this volume: Theorem (11.11.1) Let $S$
be a regular noetherian scheme of dimension $/leq 1$. Let \map(\pi,G,S) be an
$S$-group scheme of finite-type. Assume that $\pi$ is a flat, affine morphism.
Then $G$ is isomorphic to a closed subgroup of $\mathop{\mathbf{Aut}}\nolimits(V)$, where $V$ is a
vector bundle on $S$.}%
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\def\Ftest{ }\def\Fstr{11}%
\def\Atest{ }\def\Astr{Bourbaki\Revcomma N\Initper }%
\def\Ttest{ }\def\Tstr{Elements of Mathematics (Algebra II, Chapters 4--7)}%
\def\Itest{ }\def\Istr{Spring\-er-Ver\-lag}%
\def\Ctest{ }\def\Cstr{New York}%
\def\Dtest{ }\def\Dstr{1990}%
\def\Qtest{ }\def\Qstr{access via "bourbaki algebra part two"}%
\def\Xtest{ }\def\Xstr{I don't have this.}%
\Refformat\egroup%
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\def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{12}%
\def\Atest{ }\def\Astr{Claborn\Revcomma L\Initper }%
\def\Ttest{ }\def\Tstr{Every abelian group is a class group}%
\def\Jtest{ }\def\Jstr{Pacific J. Math.}%
\def\Vtest{ }\def\Vstr{18}%
\def\Dtest{ }\def\Dstr{1966}%
\def\Ptest{ }\def\Pcnt{ }\def\Pstr{219--222}%
\def\Qtest{ }\def\Qstr{access via "claborn"}%
\def\Xtest{ }\def\Xstr{I don't have this.}%
\Refformat\egroup%
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\def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{13}%
\def\Atest{ }\def\Astr{Cohen\Revcomma H\Initper }%
\def\Ttest{ }\def\Tstr{Un faisceau qui ne peut pas \^etre d\'etordu
universellement}%
\def\Jtest{ }\def\Jstr{C. R. Acad. Sci. Paris S\'er. I Math.}%
\def\Vtest{ }\def\Vstr{272}%
\def\Dtest{ }\def\Dstr{1971}%
\def\Ptest{ }\def\Pcnt{ }\def\Pstr{799--802}%
\def\Qtest{ }\def\Qstr{access via "cohen faisceau article"}%
\Refformat\egroup%
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\def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{14}%
\def\Atest{ }\def\Astr{Cohen\Revcomma H\Initper }%
\def\Ttest{ }\def\Tstr{Detorsion universelle de faisceaux coherents}%
\def\otest{ }\def\ostr{thesis (Docteur $3^\circ$ Cycle)}%
\def\Dtest{ }\def\Dstr{1972}%
\def\Itest{ }\def\Istr{Universit\'e de Paris}%
\def\Ctest{ }\def\Cstr{Orsay}%
\def\Qtest{ }\def\Qstr{access via "cohen thesis"}%
\def\Astr{\Underlinemark}%
\Refformat\egroup%
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}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{15}%
\def\Atest{ }\def\Astr{Fossum\Revcomma R\Initper \Initgap M\Initper }%
\def\Ttest{ }\def\Tstr{The Divisor Class Group of a Krull Domain}%
\def\Itest{ }\def\Istr{Spring\-er-Ver\-lag}%
\def\Ctest{ }\def\Cstr{New York}%
\def\Dtest{ }\def\Dstr{1973}%
\def\Qtest{ }\def\Qstr{access via "fossum"}%
\def\Xtest{ }\def\Xstr{I don't have this.}%
\Refformat\egroup%
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\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
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\def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{16}%
\def\Atest{ }\def\Astr{Fuchs\Revcomma L\Initper }%
\def\Ttest{ }\def\Tstr{Abelian Groups}%
\def\Itest{ }\def\Istr{Hungarian Academy of Sciences}%
\def\Ctest{ }\def\Cstr{Budapest}%
\def\Dtest{ }\def\Dstr{1958}%
\def\Qtest{ }\def\Qstr{access via "fuchs abelian groups 1958"}%
\def\Xtest{ }\def\Xstr{I don't have this.}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{2}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{17}%
\def\Atest{ }\def\Astr{Grothendieck\Revcomma A\Initper %
\Aand J\Initper \Initgap A\Initper Dieudonn\'e}%
\def\Ttest{ }\def\Tstr{El\'ements de g\'eom\'etrie\ alg\'e\-brique III (part one)}%
\def\Jtest{ }\def\Jstr{Inst. Hautes \'Etudes Sci. Publ. Math.}%
\def\Vtest{ }\def\Vstr{11}%
\def\Dtest{ }\def\Dstr{1961}%
\def\Qtest{ }\def\Qstr{access via "EGA3-1"}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{2}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{18}%
\def\Atest{ }\def\Astr{Grothendieck\Revcomma A\Initper %
\Aand J\Initper \Initgap A\Initper Dieudonn\'e}%
\def\Ttest{ }\def\Tstr{El\'ements de g\'eom\'etrie\ alg\'e\-brique III (part two)}%
\def\Jtest{ }\def\Jstr{Inst. Hautes \'Etudes Sci. Publ. Math.}%
\def\Vtest{ }\def\Vstr{17}%
\def\Dtest{ }\def\Dstr{1963}%
\def\Qtest{ }\def\Qstr{access via "EGA3-2"}%
\def\Astr{\Underlinemark}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{2}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{19}%
\def\Atest{ }\def\Astr{Grothendieck\Revcomma A\Initper %
\Aand J\Initper \Initgap A\Initper Dieudonn\'e}%
\def\Ttest{ }\def\Tstr{El\'ements de g\'eom\'etrie\ alg\'e\-brique IV (part three)}%
\def\Jtest{ }\def\Jstr{Inst. Hautes \'Etudes Sci. Publ. Math.}%
\def\Vtest{ }\def\Vstr{28}%
\def\Dtest{ }\def\Dstr{1966}%
\def\Qtest{ }\def\Qstr{access via "EGA4-3"}%
\def\Xtest{ }\def\Xstr{bertini: 9.7.7 A proper monomorphism of finite
presentation is a closed immersion (8.11.5).}%
\def\Astr{\Underlinemark}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{2}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{20}%
\def\Atest{ }\def\Astr{Grothendieck\Revcomma A\Initper %
\Aand J\Initper \Initgap A\Initper Dieudonn\'e}%
\def\Ttest{ }\def\Tstr{El\'ements de g\'eom\'etrie\ alg\'e\-brique I}%
\def\Itest{ }\def\Istr{Springer-Verlag}%
\def\Ctest{ }\def\Cstr{New \hskip 0pt plus 4cm\penalty1000\hskip 0pt plus -4cm York}%
\def\Dtest{ }\def\Dstr{1971}%
\def\Qtest{ }\def\Qstr{access via "EGA1"}%
\def\Astr{\Underlinemark}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{4}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{21}%
\def\Atest{ }\def\Astr{Guralnick\Revcomma R\Initper %
\Acomma D\Initper \Initgap B\Initper Jaffe%
\Acomma W\Initper Raskind%
\Aandd R\Initper Wiegand}%
\def\Ttest{ }\def\Tstr{The kernel of the map on Picard groups induced by a
faithfully flat homomorphism}%
\def\Rtest{ }\def\Rstr{preprint}%
\def\Qtest{ }\def\Qstr{access via "gang of four"}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{22}%
\def\Atest{ }\def\Astr{Hartshorne\Revcomma R\Initper }%
\def\Ttest{ }\def\Tstr{Algebraic Geometry}%
\def\Itest{ }\def\Istr{Spring\-er-Ver\-lag}%
\def\Ctest{ }\def\Cstr{New York}%
\def\Dtest{ }\def\Dstr{1977}%
\def\Qtest{ }\def\Qstr{access via "hartshorne algebraic geometry"}%
\Refformat\egroup%
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{23}%
\def\Atest{ }\def\Astr{Illusie\Revcomma L\Initper }%
\def\Ttest{ }\def\Tstr{G\'en\'eralit\'es sur les conditions de finitude dans
les cat\'egories d\'eriv\'ees, expos\'e\ I in {\itS\'em\-in\-aire de G\'eom\'e\-trie Al\-g\'e\-bri\-que} (SGA) 6}%
\def\Stest{ }\def\Sstr{Lecture Notes in Mathematics}%
\def\Itest{ }\def\Istr{Spring\-er-Ver\-lag}%
\def\Ctest{ }\def\Cstr{New York}%
\def\Vtest{ }\def\Vstr{225}%
\def\Dtest{ }\def\Dstr{1971}%
\def\Ptest{ }\def\Pcnt{ }\def\Pstr{78--159}%
\def\Qtest{ }\def\Qstr{access via "illusie finiteness first"}%
\Refformat\egroup%
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|
1994-10-26T05:20:13 | 9410 | alg-geom/9410025 | en | https://arxiv.org/abs/alg-geom/9410025 | [
"alg-geom",
"math.AG",
"math.CV"
] | alg-geom/9410025 | null | Dmitri Zaitsev | On the automorphism groups of algebraic bounded domains | 29 pages, LaTeX, Mathematischen Annalen, to appear | null | null | null | null | Let $D$ be a bounded domain in $C^n$. By the theorem of H.~Cartan, the group
$Aut(D)$ of all biholomorphic automorphisms of $D$ has a unique structure of a
real Lie group such that the action $Aut(D)\times D\to D$ is real analytic.
This structure is defined by the embedding $C_v\colon Aut(D)\hookrightarrow
D\times Gl_n(C)$, $f\mapsto (f(v), f_{*v})$, where $v\in D$ is arbitrary. Here
we restrict our attention to the class of domains $D$ defined by finitely many
polynomial inequalities. The appropriate category for studying automorphism of
such domains is the Nash category. Therefore we consider the subgroup
$Aut_a(D)\subset Aut(D)$ of all algebraic biholomorphic automorphisms which in
many cases coincides with $Aut(D)$. Assume that $n>1$ and $D$ has a boundary
point where the Levi form is non-degenerate. Our main result is theat the group
$Aut_a(D)$ carries a unique structure of an affine Nash group such that the
action $Aut_a(D)\times D\to D$ is Nash. This structure is defined by the
embedding $C_v\colon Aut_a(D)\hookrightarrow D\times Gl_n(C)$ and is
independent of the choice of $v\in D$.
| [
{
"version": "v1",
"created": "Tue, 25 Oct 1994 14:13:08 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Zaitsev",
"Dmitri",
""
]
] | alg-geom | \section{Introduction}\label{in}
Let $D$ be a domain in $\C^n$ and $Aut(D)$ be the group of
all biholomorphic automorphisms of $D$. Let $v\in D$ be fixed and define
the map $C_v\colon Aut(D) \to D\times Gl(n)$ by $f\mapsto (f(v),f_{*v})$.
The theorem of H.~Cartan (see Narasimhan, \cite{N}, p.~169) can be stated
as follows.
\begin{samepage}
\begin{Th}
Let $D$ be bounded. Then:
\begin{enumerate}
\item The group $Aut(D)$ possesses a natural Lie group structure
compatible with the compact-open topology such that the action
$Aut(D)\times D\to D$ is real analytic.
\item For all $v\in D$ the map $C_v$ is a real-analytic homeomorphism
onto its image.
\end{enumerate}
\end{Th}
\end{samepage}
The domains we discuss here are open connected sets defined by finitely
many real polynomial inequalities or connected finite unions of such sets.
These are the domains in the so-called ``semi-algebraic category''
defined below (Definition~\ref{def-s-a}).
For this reason we call them ``semi-algebraic domains''.
In this paper we are interested in the algebraic nature of the image of
$Aut(D)$ and its subgroups in $D\times Gl(n)$ under the map $C_v$.
{\bf Example.1.} The simplest example of a semi-algebraic domain is the
unit disk $D=\{|z|<1\}$. For this domain
$$Aut(D)=PGL_2(\R)_+:= \{A\in PGL_2(\R) \mid \det A > 0 \}.$$
We see that $Aut(D)$, as a subgroup of $PGL_2(\R)$, is defined by an
inequality and therefore is not an algebraic subgroup. In fact, the
group $Aut(D)$ here does not admit algebraic structure as a Lie group.
To show this, assume that there is a Lie isomorphism
$\varphi\colon PGL_2(\R)_+\to G$, where $G$ is a real algebraic group.
It continues to an isomorphism of complexifications
$\varphi^{\C}\colon PGL_2(\C) \to G^{\C}$. The latter, being a Lie isomorphism
between semi-simple complex algebraic groups, is algebraic. Since
$G\subset G^{\C}$ is real algebraic, so is its preimage
$(\varphi^{\C})^{-1}(G)=PGL_2(\R)_+$. On the other hand,
$PGL_2(\R)_+\subset PGL_2(\C)$ is not real Zariski closed. This is a
contradiction.
\par\hfill {\bf Q. E. D.}
{\bf Example.2.} More generally let $D$ be a bounded homogeneous
domain in $\C^n$. By the classification theorem of Vinberg, Gindikin
and Pyatetskii-Shapiro (see \cite{VGP}, Theorem~6, p.~434), $D$ is
biholomorphic to a homogeneous Siegel domain of the 1st or the 2nd kind.
Such a domain is defined algebraically in terms
of a homogeneous convex cone (\cite{VGP}) .
Rothaus (\cite{R}) gave a procedure for constructing
all homogeneous convex cones. The construction implies that
all homogeneous convex cones, and therefore all homogeneous Siegel domains
of the 1st and 2nd kind, are defined by finitely many polynomial inequalities.
The Siegel domains are unbounded but they are birationally equivalent to
bounded domains which are also defined by finitely many polynomial inequalities
and therefore are semi-algebraic. Thus, $D$ is biholomorphic to a bounded
semi-algebaic domain.
The automorphism group $Aut(D)$ of a bounded homogeneous domain was
discussed by Kaneyuki (see \cite{K}) where he proved in particular
that the identity component $Aut(D)^0$ is isomorphic to an identity
component of a real algebraic group (\cite{K}, Theorem.3.2., p.106).
Let $x_0\in D$ be a fixed point. Since $D=Aut(D)/Iso(x_0)$
and the isotropy group $Iso(x_0)$
is compact, the automorphism group $Aut(D)$ has finitely many components.
Together with the result of Kaneyuki this implies that $Aut(D)$ is
isomorphic to an open subgroup of a real algebraic group.
In the above examples the automorphism group $Aut(D)$ is isomorphic to
an open subgroup of a real algebraic group. Therefore, it admits a faithful
representation. For general domains however the automorphism group does not
admit a faithfull representation.
{\bf Example.3.} Let
$$D:=\{(z,w)\in\C^2 \mid |z|^2+|w|^2<1, w\ne 0 \} $$
be the unit ball in $\C^2$ with a unit disk removed. We consider $D$ as
a subset of $\P^2$ with homogeneous coordinates $\xi_0,\xi_1,\xi_2$,
$z=\xi_1/\xi_0$, $w=\xi_2/\xi_0$. The automorphism group of $D$
is $Aut(D)=SU(1,1)\times S^1$ with the action on $D$ given by
$$ (A,\tau)(\xi_0,\xi_1,\xi_2) = \pmatrix{
A & 0 \atop 0 \cr
0 \, 0 & \tau \cr} \pmatrix{
\xi_0 \cr
\xi_1 \cr
\xi_2 \cr },\quad A\in SU(1,1),\quad \tau \in S^1. $$
The isomorphism $K\colon C\mapsto JCJ^{-1}$ between $SL_2(\R)$ and
$SU(1,1)$ ($J=\pmatrix{-i&1\cr i&1\cr}$) yelds an effective action of
$SL_2(\R)$ on $D$. Furthermore, each map $j_x\colon SL_2(\R)\to D$,
$j_x(C):=Cx$ induces an isomorphism between fundamental groups
$\pi_1(SL_2(\R))=\pi_1(D)=\Bbb Z$.
This implies that the induced action of a finite covering of $SL_2(\R)$
on $D$ lifts to an effective action on the finite covering $D'$ of $D$ of the
same degree. But no covering of $SL_2(\R)$ admits a faithfull
representation (because every such representation factorizes through a
representation of $SL_2(\C)$). On the other hand, the map
$(z,w)\mapsto (z,\root d \of w)$ defines
an isomorphism between a finite covering of $D$ of degree $d$ and a bounded
semi-algebraic domain $\tilde D\subset\C^2$.
In this example the group $Aut(D)$ is not isomorphic to an open subset
of an algebraic group. Moreover, even in case it is,
the action $Aut(D)\times D\to D$ can be ``far from algebraic''.
This phenomena is shown in the following example.
{\bf Example.4.} Let $F=\C/\Lambda$ be a complex elliptic curve and
${\cal P}\colon F\to \P^1$ the Weierstra\ss\ ${\cal P}$-function which
defines a $2-1$ ramified covering over $P^1$. The strip
$\{z\in\C \mid c-\epsilon<{\rm Im}z<c+\epsilon\}$ covers a ``circle strip''
$\tilde D\subset \C/\Lambda$. Let $D$ be the projection of $\tilde D$
on $P^1$. If the constant $c$ is generic and $\epsilon$ is small enough,
the projection of $\tilde D$ is biholomorphic. The real algebraic group
$S^1\subset \C/\Lambda$ acts by translations on $\tilde D$ which yields
an effective action on $D$. Since the domain $D$ is bounded by real elliptic
curves, it is therefore semi-algebraic. However the action of $S^1$ is
expressed in terms of the Weierstra\ss\ ${\cal P}$-function and is not
algebraic. In fact there is no homomorphism of $S^1$ in a real algebraic
group $G$ such that the action $S^1\times D\to D$ is given by restrictions
of polynomials on $G$. This follows from the classification of 1-dimensional
Nash (semi-algebraic) groups given by Madden and Stanton (see \cite{MS}).
We see therefore that even in simple cases the class of
real algebraic groups and their subgroups is not large enough
to describe the group $Aut(D)$ and its action on $D$.
Consequently, we consider a larger class of groups where {\em defining
inequalities} are allowed.
\begin{Def}
\begin{enumerate}
\item A {\bf Nash function} is a real analytic function
$f=(f_1,\ldots,f_m)\colon U\to \R^m$ (where $U$ is an open semi-algebraic
subset of $\R^n$) such that for each of the components $f_k$ there is a
nontrivial polynomial $P$ with $P(x_1,\ldots,x_n,f_k(x_1,\ldots,x_n))=0$
for all $(x_1,\ldots,x_n)\in U$.
\item A {\bf Nash manifold} $M$ is a real analytic manifold with finitely many
coordinate charts $\phi_i\colon U_i\to V_i$ such that $V_i\subset\R^n$ is
semi-algebraic for all $i$ and the transition functions are Nash
(a Nash atlas).
\item A Nash manifold is called {\bf affine} if it can be Nash
(locally closed) imbedded in $\R^N$ for some $N$.
\item A {\bf Nash group} is a Nash manifold with a group operation
$(x,y)\to xy^{-1}$ which is Nash with respect to every Nash coordinate chart.
\end{enumerate}
\end{Def}
\begin{Rem}
The simplest example of a Nash manifold which is not affine is
the quotient $\R/\Bbb Z$ with the Nash structure inherited from the standard
Nash structure on $\R$.
For the classification of such groups in the one-dimensional case see
J.~J.~Madden and C.~M.~Stanton in \cite{MS}.
\end{Rem}
Roughly speaking, the goal of this paper is to prove that the automorphism
group $Aut(D)$ of a semi-algebraic domain $D$ has a natural Nash group
structure such that the action $Aut(D)\times D\to D$ is also Nash. For this
we need a certain non-degeneracy condition on the boundary of $D$.
To give the reader a flavour of the main result, we first mention
an application for the {\it algebraic} domains introduced by Diederich and
Forn\ae ss (\cite{DF}), for which this condition is automatically satisfied.
\begin{Def}(see Diederich-Forn\ae ss, \cite{DF})
A domain $D\subset\subset C^n$ is called {\bf algebraic} if there exists
a real polynomial $r(z,\bar z)$ such that $D$ is a connected component of the
set
$$\{z\in\C^n \mid r(z,\bar z)<0 \}$$
and $dr(z)\ne 0$ for $z\in\partial D$.
\end{Def}
\begin{Th}\label{cor}
Let $D\subset\subset\C^n$, $n>1$, be an algebraic domain.
The group $Aut(D)$ possesses a unique structure of an affine Nash group
so that the action $Aut(D)\times D\to D$ is Nash.
For all $v\in D$, $C_v\colon Aut(D)\to D\times Gl(n)$ is a Nash
isomorphism onto its image.
\end{Th}
\begin{Cor}
Let $D$ be as in Theorem~\ref{cor}. Then the group $Aut(D)$
has finitely many connected components.
\end{Cor}
\begin{Rem}
In general the number of components of $Aut(D)$ can be infinite.
For example, let $H:=\{z\in\C \mid {\rm Im}z=0\}$ be the upper half-plane and
$$\hat H:=H\cup \bigcup_{n\in\Bbb Z} B_{\epsilon}(n),$$
where $B_{\epsilon}(n)$ is the ball with centre $n$ and radius
$\epsilon<1/2$. Let $D\subset\C^2$ be the union of $H\times H$ and
$(H+i)\times \hat H$. Then $D$ is biholomorphic to a simply connected
bounded domain. The flat pieces of the boundary of $D$ admit canonical
foliations $z=\rm const$ and $w=\rm const$. The latters induce foliations
of $D$ of the same form which are preserved by the automorphisms
(see Remmert and Stein \cite{RS}).
By this argument one shows that $Aut(D)=\R\oplus\Bbb Z$.
\end{Rem}
For the formulation of our main result we need the following condition on
$D$, which is automatically satisfied for all bounded domains with smooth
boundary, in particular, for all algebraic domains.
\begin{Def}\label{L}
\begin{enumerate}
\item
A domain $D\subset\C^n$ or its boundary
is called {\bf Levi-non-degenerate},
if there exists a point $x_0\in\partial D$ and a neighborhood $U\subset\C^n$
of $x_0$ such that
$$D\cap U=\{z\in U \mid \varphi(z)<0 \}$$
for a $C^2$-function
$\varphi$ with $d\varphi\ne 0$ and such that the Levi form in $x_0$
$$L_r(x_0):=\sum_{k,l=1}^n
{\partial^2 r \over \partial z_i \, \partial \bar z_j} dz_k \otimes d\bar z_l$$
restricted to the holomorphic tangent space of $\partial D$ in $x_0$ is
non-degenerate;
\item
A domain $D\subset\C^n$ or its boundary
is called {\bf completely Levi-non-degenerate},
if every boundary point outside a real analytic subset of dimension $2n-2$
is non-degenerate in the above sense.
\end{enumerate}
\end{Def}
Now let $D$ be semi-algebraic and
consider the subset $Aut_a(D)\subset Aut(D)$ of all
(biholomorphic) automorphisms which are Nash.
In the following sense
Nash automorphisms are branches of algebraic maps
(see Proposition~\ref{bra}).
\begin{Def}\label{br}
Let $D$ be a domain in $\C^n$. A holomorphic map $f\in Aut(D)$
is a {\bf branch of an algebraic map} if
there exists a complex $n$-dimensional algebraic subvariety
$G\subset\C^n\times\C^n$ which contains the graph of $f$.
\end{Def}
The following is the main result of this paper.
\begin{Th}\label{main}
Let $D\subset\subset\C^n$, $n>1$, be a semi-algebraic
Levi-non-degenerate domain. Then
\begin{enumerate}
\item $Aut_a(D)$ is a (closed) Lie subgroup of $Aut(D)$,
\item $Aut_a(D)$ possesses a unique structure of an affine Nash group
so that the action $Aut_a(D)\times D\to D$ is Nash.
\item For all $v\in D$, $C_v\colon Aut_a(D)\to D\times Gl(n)$
is a Nash isomorphism onto its image.
\end{enumerate}
\end{Th}
Theorem~\ref{cor} is now a corollary of Theorem~\ref{main}. This is
a consequence of the following result of K.~Diederich and
J.~E.~Forn\ae ss (\cite{DF}).
{\bf Theorem~1.4}
{\it Let $D\subset\subset\C^n$ be an algebraic domain. Then $Aut_a(D)=Aut(D)$.}
{\bf Remark.} The proof of Theorem~1.4 makes use of the reflection
principle (see S.~Pin\v cuk, \cite{P}) and basic
methods of S.~Webster (\cite{W}) which are also fundamental in the
present paper. Due to the results of K.~Diederich and S.~Webster (\cite{DW})
and of S.~Webster (\cite{W}) on the continuation of automorphisms,
Theorem~\ref{main} can be applied to every situation where the
automorphisms can be $C^\infty$ extended to the boundary.
K.~Diederich has informed the author that he and S.~Pin\v cuk recently
proved that, under natural non-degeneracy conditions on the boundary,
automorphisms of domains are always almost everywhere continuously
extendable. Applying this along with the reflection method, one would
expect $Aut(D)=Aut_a(D)$ for $D$ a semi-algebraic domain with $\partial D$
completely Levi-non-degenerate. Thus Theorem~\ref{main} applied to this
situation would show that $Aut(D)$ is an affine Nash group acting
semi-algebraically on $D$.
{\bf Acknowledgement.} On this occasion I would like to thank my teacher
A.~T.~Huckleberry for formulating the problem, for calling my attention
to the relevant literature and for numerous very useful discussions.
\section{Real semi-algebraic sets}\label{sa}
Here we present some basic properties of semi-algebraic sets which
will be used in the proof of Theorem~\ref{main}.
For the proofs we refer to Benedetti-Risler \cite{BR}.
\begin{samepage}
\begin{Def}\label{def-s-a}
A subset $V$ of \/ $\R^n$ is called {\bf semi-algebraic}
if it admits some representation of the form
$$V = \bigcup_{i=1}^s \bigcap_{j=1}^{r_i} V_{ij}$$
where, for each $i=1,\ldots,s$ and $j=1,\ldots,r_i$,
$V_{ij}$ is either $\{ x\in\R^n \mid P_{ij}(x)<0 \}$ or
$\{ x\in\R^n \mid P_{ij}(x)=0 \}$ for a real polynomial $P_{ij}$.
\end{Def}
\end{samepage}
As a consequence of the definition it follows that finite unions and
intersections of semi-algebraic sets are always semi-algebraic. Moreover,
closures, boundaries, interiors (see Proposition~\ref{cl})
and connected components of semi-algebraic sets are semi-algebraic.
Further, the number of connected components is finite
(see Corollary~\ref{con}). Finally, any semi-algebraic set admits
a finite semi-algebraic stratification (see Definition~\ref{str} and
Proposition~\ref{strat}).
The natural morphisms in the category of semi-algebraic set are {\bf
semi-algebraic maps}:
\begin{Def}\label{map}
Let $X\subset\R^n$ and $Y\subset\R^n$ be semi-algebraic sets.
A map $f\colon X\to Y$ is called {\bf semi-algebraic} if the graph
of $f$ is a semi-algebraic set in $\R^{m+n}$.
\end{Def}
\begin{samepage}
\begin{Def}\label{str}
A {\bf stratification} of a subset $E$ of $\R^n$ is a
partition $\{A_i\}_{i\in I}$ of $E$ such that
\begin{enumerate}
\item each $A_i$ (called a {\bf stratum}) is a real analytic
locally closed submanifold of $\R^n$;
\item if $\overline{A_i} \cap A_j \ne \emptyset$,
then $\overline{A_i} \supset A_j$ and $\dim A_j < \dim A_i$ (frontier
condition).
\end{enumerate}
A stratification is said to be {\bf finite} if there is a finite
number of strata and to be {\bf semi-algebraic} if furthermore
each stratum is also a semi-algebraic set.
\end{Def}
\end{samepage}
\begin{Prop}\label{strat}
Every semi-algebraic set $E\subset\R^n$ admits a semi-algebraic
stratification.
\end{Prop}
\begin{Cor}\label{con}
Every semi-algebraic set has a finite number of connected components
and each such component is semi-algebraic.
\end{Cor}
\begin{Prop}\label{cl}
Let $X$ be a semi-algebraic set in $\R^m$. Then the closure $\bar X$,
its interior $\mathop{X}\limits^0$, and its boundary $\partial X$
are semi-algebraic sets.
\end{Prop}
Using Proposition~\ref{strat}, dimension of a semi-algebraic
set is defined to be the maximal dimension of its stratum.
This is independent of
the choice of a finite stratification.
\begin{Prop}\label{as}
Let $Y\subset\R^m$ be a semi-algebraic set of $\dim Y\le k$.
Then it is contained in some real algebraic set $Z$ with ${\rm dim}Z\le k$.
\end{Prop}
The following results on images and local triviality of semi-algebraic maps
will play an important role in the present paper.
\begin{Th}[Tarski-Seidenberg]\label{TS}
Let $f\colon X \to Y$ be a semi-algebraic map. Then the image
$f(X)\subset Y$ is semi-algebraic set.
\end{Th}
Further we need the Theorem on local triviality
(see Benedetti-Risler \cite{BR}, Theorem~2.7.1, p.~98).
\begin{samepage}
\begin{Th}\label{tr}
Let $X$ and $Y$ be semi-algebraic sets and let $f\colon X\to Y$
be a continuous semi-algebraic map. Fix a finite semi-algebraic partition
of $X$, $\{X_1,\ldots,X_h\}$. Then there exists
\begin{enumerate}
\item a finite semi-algebraic stratification $\{Y_1,\ldots,Y_k\}$ of $Y$;
\item a collection of semi-algebraic sets $\{F_1,\ldots,F_k\}$ and,
for every $i=1,\ldots,k$, a finite semi-algebraic partition
$\{ F_{i1},\ldots,F_{ir} \}$ of $F_i$ (typical fibres);
\item a collection of semi-algebraic homeomorphisms
$$ g_i\colon f^{-1}(Y_i) \to Y_i\times F_i, \quad i=1,\ldots,k$$
such that
\begin{enumerate}
\item the diagram
$$\def\baselineskip20pt\lineskip3pt\lineskiplimit3pt{\baselineskip20pt\lineskip3pt\lineskiplimit3pt}
\def\mapright#1{{\buildrel #1 \over \longrightarrow}}
\def\mapdown#1{\Big\downarrow \rlap{$\scriptstyle#1$}} \matrix{
f^{-1}(Y_i) & \mapright{g_i} & Y_i\times F_i \cr
\mapdown{f} & &\mapdown{p} \cr
Y_i & \mapright{id} & Y_i \cr
}$$
is commutative ($p\colon Y_i\times F_i \to Y_i$ is the natural projection);
\item for every $i=1,\ldots,k$ and every $h=1,\ldots,r$,
$$ g_i(f^{-1}(Y_i)\cap X_h) = Y_i \times F_{ih}. $$
\end{enumerate}
\end{enumerate}
\end{Th}
\end{samepage}
\begin{Rem}
In the above setting one says that
$f$ is a trivial semi-algebraic map over $Y_i$ with
typical fibre $F_i$ and structure homeomorphisms $g_i$.
\end{Rem}
Further, we shall use the following proposition from real algebraic
geometry (see \cite{BR}, Proposition~3.2.4).
\begin{Prop}\label{ri}
Let $V\subset\R^n$ be a real algebraic variety. The set of singular
points is an algebraic set properly contained in $V$.
\end{Prop}
The following Proposition provides a motivation for Definition~\ref{br}.
\begin{Prop}\label{bra}
The Nash automorphisms $f\in Aut_a(D)$ are branches of algebraic maps.
\end{Prop}
{\bf Proof.}
Let $f\in Aut_a(D)$ be a Nash automorphism. Then $f$ is holomorphic and
semi-algebraic. Every coordinate $f_j\colon D\to \C$ is also
holomorphic and, by Theorem~\ref{TS}, semi-algebraic.
By Proposition~\ref{as}, there
exist real algebraic sets $Z_j$ of (real) codimension $2$ with
$\Gamma_{f_j}\subset Z_j$. By Proposition~\ref{ri}, there exists a regular
point $(w_0,f(w_0))\in \Gamma_{f_j}$ where
$\Gamma_{f_j}$ is locally given
by two real polynomials $P_1(z,\bar z)$, $P_2(z,\bar z)$ or by a complex one
$P(z,\bar z):=P_1(z,\bar z)+iP_2(z,\bar z)$, such that $dP\ne 0$.
The latter property implies that
either $\partial P\ne 0$ or $\bar\partial P\ne 0$.
We can assume that $\partial P\ne 0$, otherwise $P$ can be replaced with $\bar
P$.
Let $P(z,\bar z)=P'(z)+P''(z,\bar z)$ be a decomposition of $P$ in
the holomorphic part $P'$ and the remainder $P''$ which
consists only of terms with non-trivial
powers of $\bar z$. The part $P'$ is not zero because
$dP'=\partial P\ne 0$.
We wish to prove that $\Gamma_{f_j}$ is locally defined by
the holomorphic polynomial $P'(z)$. The identity
$$P(w,f_j(w),\bar w, \overline{f_j(w)}) \equiv 0$$
for all $w$ near $w_0$ implies in particular the vanishing of
the Taylor coefficients of $w^k$ for all multi-indices $k$.
But these coefficients are just
$${1\over |k|!}{\partial^{|k|} P'(w,f_j(w)) \over \partial w^k}$$
and their vanishing means the vanishing of $P'(z,f_j(z))$.
Thus the graph $\Gamma_{f_j}$ is locally defined by
a holomorphic polynomial $P_j$
and the polynomials $P_1,\ldots,P_n$ define the $n$-dimensional
algebraic variety required in the definition of a branch of an algebraic map.
\nopagebreak\par\hfill {\bf Q. E. D.}
We shall also make frequent use of Chevalley's theorem on
constructible sets (see Mumford \cite{M}, p.~72).
\begin{Def}\label{cons}
A subset $A\in\C^n$ is called {\bf constructible} if it is a finite
union of locally closed complex algebraic subvarieties.
\end{Def}
\begin{Rem}
Constructible sets are semi-algebraic.
\end{Rem}
\begin{Th}[Chevalley]\label{Chev}
Let $X$ and $Y$ be affine varieties and $f\colon X\to Y$ any morphism.
Then $f$ maps constructible sets in $X$ to constructible sets in $Y$.
\end{Th}
\section{A scheme of the proof}
The proof of Theorem~\ref{main} can be divided in two steps.
The essential ingredient for the first step is the method of S.~Webster
(see \cite{W}) based on the reflection principle
(see S.~Pin\v cuk, \cite{P}). We use it to construct
an appropriate {\it family of graphs} of automorphisms from $Aut_a(D)$.
In fact we construct a constructible family $F$ which fibres contain
automorphisms from $Aut_a(D)$.
This is carried out in sections~\ref{wm} and \ref{we}.
In the second step we show using the family constructed in
the first step that the set $C_v(Aut_a(D))$ is Nash.
Taking a neighborhood where $C_v(Aut_a(D))$ is not empty and closed and
taking its pullback in $Aut(D)$ we obtain a neighborhood where $Aut_a(D)$
is closed, which implies statement~1. in Theorem~\ref{main}.
In fact we prove that the {\it exact}
family of graphs
$$\Gamma:=\{(C_v(f),w,f(w) \mid f\in Aut_a(D), \, w\in\C^n \}$$
is Nash. If we identify $Aut_a(D)$ with $C_v(Aut_a(D))$,
$\Gamma$ is the graph of the action $Aut_a(D)\times D\to D$. This
proves statement~3. To obtain statement~2., we observe that group operation
can be defined {\em semi-algebraically} in terms of $\Gamma$.
Here we use the theory of semi-algebraic sets and their morphisms.
\section{Reflection principle}\label{wm}
Let $D\subset\subset\C^n$ be a semi-algebraic Levi-non-degenerate domain.
By Proposition~\ref{cl}, the boundary $\partial D$ is
semi-algebraic. By Proposition~\ref{as}, there exists a real algebraic
set $H$ of dimension $2n-1$ which contains $\partial D$. Let
$H_i$ be irreducible components of $H$ of dimension $2n-1$.
By Definition~\ref{L}, the Levi form of some component of $H$, let say
of $H_1$, is not everywhere degenerate.
To every irreducible hypersurface $H_i$ we associate a real Zariski open
set $U_i\subset\C^n$ and a real polynomial $r_i(z,\bar z)$ with
$H_i\cap U_i= \{r_i=0\} \cap U_i$ and $dr\ne 0$ on $U_i$.
By Proposition~\ref{ri}, such $U_i$'s and $r_i$'s exist.
Let $f\in Aut_a(D)$ be any fixed map which is, by Proposition~\ref{bra}
a branch of an
algebraic map and $V\subset\C^n\times\C^n$ be the corresponding
$n$-dimensional algebraic subvariety which contains the graph of $f$.
We wish to extend $f$ in a neighborhood of a boundary point $x\in H_1$.
Outside a proper complex algebraic subvariety $V'\subset V$ the variety
$V$ defines $f$ and $f^{-1}$ as possibly multiple-valued algebraic maps.
Since $\dim_{\C} V=n$, $\dim_{\C} V'\le n-1$. Since $\dim_{\R} H_1=2n-1$,
there exists a point $x\in H_1\cap U_1$ and
a neighborhood $U\subset U_1$ of $x$ such
that $V$ is a trivial covering over $U$
and all sections of this covering define biholomorphic maps onto their images.
One of these maps coincides with $f$ over $D\cap U$. This map yields the
desired extension of $f$. Since $f$ is an automorphism of $D$,
it maps $H_1\cap U$ into $\partial D$.
{\bf Notation.} Let $i=i(f)$ be such that $f(H_1\cap U)\subset H_i$.
Let $w_0\in H_1\cap U$ be
an arbitrary point such that for $w'_0=f(w_0)$ one has $dr_i(w'_0)\ne 0$.
We use the notation
$$ r:=r_1, r':=r_i, H:=H_1 \hbox{, and } H':=H_i.$$
Then $f(H\cap U)\subset H'$ and we have a relation
\begin{equation}\label{pz}
r'(f(z),\bar f(\bar z)) = g(z,\bar z) r(z,\bar z),
\end{equation}
where $g(z,\bar z)$ is real analytic.
Let $z=x+iy$, where $x$ and $y$ are real coordinate vectors. Since
the functions in (\ref{pz}) are given by power series in $(x,y)$,
they are still defined for complex vectors $x$ and $y$ near $w_0$.
This is equivalent to varying $z$ and $\bar z$
independently. The relation (\ref{pz}) persists:
\begin{equation}\label{pzw}
r'(f(z),\bar f(\bar w)) = g(z,\bar w) r(z,\bar w).
\end{equation}
Now we consider the spaces $Z:=\C^n$, $Z':=\C^n$, $W:=\C^n$
and $W':=\C^n$ and define
the complexifications ${\cal H}\subset Z\times\bar W$ and
${\cal H'}\subset Z'\times\bar W'$ by
\begin{equation}\label{wz}
r(z, \bar w) = 0, \quad r'(z', \bar w') = 0.
\end{equation}
The so-called {\em Segre complex varieties} associated to
the points $w\in W$ are defined by
$$ Q_w = \{ z\in Z \mid r(z,\bar w)=0 \}. $$
Since $g(z,\bar w)$ is holomorphic for $z$ and $w$ near $w_0$,
we see from (\ref{pzw}) that the map $f^{\C}:=f\times\bar f$ takes
$\{z\} \times Q_z$ into $\{z'\} \times Q'_{z'}$,
where $z'=f(z)$. Hence, the family of complex hypersurfaces
$\{z\} \times Q_z$ is invariantly related to $H$.
Since $r(z,\bar z)$ is real, we have
$$ r(z,\bar w) = \bar r(\bar w, z) = \overline{r(w,\bar z)}, $$
so that $z\in Q_w \iff w\in Q_z$. Also $z\in Q_z \iff z\in H$.
Since $r$ is real and
\newpage
\noindent $dr=\partial r + \bar\partial r$
does not vanish at $w_0$ (resp. $r'$ is real and
$dr'=\partial r' + \bar\partial r'$ does not vanish at $w'_0$), we have
\begin{equation}\label{dr}
\partial r(z,\bar w)\ne 0, \quad \partial r'(z',\bar w')\ne 0
\end{equation}
for $z$ and $w$ near $w_0$ and for $z'$ and $w'$ near $w'_0$.
The relation (\ref{dr}) implies that $Q_w$ is non-singular
in $z$ if $(z,\bar w)$ is near $(w_0,\bar w_0)$.
Let $\pi_z(\bar w)$ denote the complex tangent
space $T_z Q_w$ as an element in the grassmanian $G_{n,n-1}$.
It follows that $\pi_z$ is an
antiholomorphic map from $\{z\}\times Q_z$ to $G_{n,n-1}$.
S.~Webster proves the following fact (\cite{W}, p.~55, Lemma~1.1):
\begin{Lemma}\label{inv}
The antiholomorphic map $\pi_z(\bar w)$ is locally
invertible near the points of $H$ where the Levi form is non-degenerate.
\end{Lemma}
Since the Levi form of $U\cap H$ is non-degenerate, and
a biholomorphic map preserves this property, the Levi form of $H'$ is
non-degenerate around $w'_0=f(w_0)\in H'$.
By Lemma~\ref{inv}, $\pi_z$ and $\pi'_{z'}$ are
locally invertible for $(z,\bar w)$ near $(w_0,\bar w_0)$ and
$(z',\bar w')$ near $(w'_0,\bar w'_0)$.
The first step of Webster's method is to describe
the map $f$ between $Q_z$ and $Q'_{z'}$. We have seen
that $f$ takes $Q_w$ into $Q'_{w'}$, $w'=f(w)$. We take $w\in Q_z$.
Then all $Q_w$'s pass through the point $z$ and all $Q'_{w'}$'s through $z'$.
Therefore the differential $f_{*z}\in Gl(n)$ takes
$T_zQ_w$ into $T_{z'}Q'_{w'}$,
i.e. $\pi_z(\bar w)$ into $\pi_{z'}(\bar w')$. These considerations mean
that the restriction $f|_{Q_z}$ can be decomposed as follows (\cite{W}, p.~56):
\begin{equation}\label{dec}
w \stackrel{\vphantom{\pi_{z'}^{-1}}\pi_z}{\longmapsto} T_zQ_w
\stackrel{\vphantom{\pi_{z'}^{-1}}J_{n-1}(f_{*z})}{\longmapsto}
T_{z'}Q'_{w'} \stackrel{\pi_{z'}^{-1}}{\longmapsto} w',
\end{equation}
where the second map $J_{n-1}(f_{*z})\colon G_{n,n-1}\to G'_{n,n-1}$
is the natural map between grassmanians which is induced by the differential
$f_{*z}\colon T_zZ \to T_{z'}Z'$.
The decomposition (\ref{dec}) implies that for $w\in Q_z$ and $w'=f(w)$,
$z'=f(z)$ and $q=f_{*z}\in Gl(n)$ the following relation is satisfied:
\begin{equation}\label{wqw}
\pi'_{z'}(\bar w') = J_{n-1}(q)(\pi_z(\bar w)).
\end{equation}
This relation expresses the restriction $\phi:=f|_{Q_z}$
in terms of parameters $(z,z',q)\in Z\times Z'\times Gl(n)$.
This is of great importance for our parametrization of the graphs of
elements of $Aut_a(D)$. Hence we underline this fact by introducing
the notation
\begin{equation}\label{phi}
\phi(\bar z,\bar z',\bar q)\colon Q_z \to Q'_{z'}
\end{equation}
for $\phi:=f|_{Q_z}$. We write the conjugate variables for arguments
of $\phi$ in order to emphasize that $\phi$ depends holomorphically on them.
The idea of the second step is to express the map $f$ in terms of
restrictions $f|_{Q_z}$. This is done separately for $n=2$ and
$n\ge 3$.
\subsection{The case $n\ge 3$}
Let $z_0\in U\cap Q_{w_0}$ be any point.
Lemma~\ref{inv} gives points $v_1,\ldots,v_n\in Q_{z_0}$ near
$w_0$ such that all $Q_{v_j}$ are non-singular and transverse in $z_0$ and
the algebraic curve $\gamma$, defined by
\begin{equation}\label{zv0}
r(z,\bar v_1) = \cdots = r(z,\bar v_{n-1}) = 0,
\end{equation}
is transverse to $Q_{w_0}$.
Every $w$ near $w_0$ lies in some $Q_z,z\in\gamma$, namely for
$z\in \gamma\cap Q_w$. Therefore, to describe $f(w)$ we need only to
consider restrictions $f|_{Q_z},z\in \gamma$.
Given the values $z'=f(z)$ and differentials $q=f_{*z}\in Gl(n)$,
the map $f|_{Q_z}$ is determined by (\ref{wqw}).
Since $\gamma\subset Q_{v_1}$, the values $z'=f(z)$ along
$\gamma$ are determined, in turn, due to (\ref{wqw}) by parameters
$(v_1,v'_1,l_1)=(v_1,f(v_1),f_{*v_1})$.
Namely, we use the map $\phi$ in (\ref{phi}) and set
\begin{equation}\label{zz'}
z'=\phi(\bar v_1,\bar v'_1,\bar l_1)(z).
\end{equation}
Further, the differentials $f_{*z}$ along $\gamma$ can be expressed
in terms of parameters $(v_1,v'_1,l_1)=(v_1,f(v_1),f_{*v_1})$.
Consider the differential 1-forms
$$\theta_\alpha=\partial r(z,\bar v_\alpha).$$
They define a frame in the cotangent spaces.
Let
$$\{Y_j=Y_j(z,\bar v_1,\ldots,\bar v_{n-1},\bar w),j=1,\ldots,n\}$$
be the dual vector field frame.
This frame has rational coefficients in the
variables $(z,\bar v_1,\ldots,\bar v_{n-1},\bar w)$ and satisfies
the conditions
\begin{equation}\label{Y}
\left.\matrix{
\hbox{$Y_1$ is transverse to $Q_{v_1}$ and tangent to $Q_{v_2}$,} \cr
\hbox{$Y_2$ is transverse to $Q_{v_2}$ and tangent to $Q_{v_1}$,} \cr
\hbox{$Y_3,\ldots,Y_n$ are tangent to $Q_{v_1} \cap Q_{v_2}$.}\cr
}\right\rbrace
\end{equation}
Similar differential $1$-forms $\theta'_\alpha$
and frame vector fields $Y'_j$ are constructed for $H'$.
Relative to these two frame fields
$$ f_{*z}Y_l = \sum q_{lj}Y'_j, $$
where
\begin{equation}\label{mat}
[q_{lj}]= \left[ \matrix{ q_{11} & 0 & 0 \cr
0 & q_{22} & 0 \cr
q_{1\beta} & q_{2\beta} & q_{\alpha\beta} \cr } \right],
\end{equation}
$\alpha,\beta=3,\ldots,n$.
The functions $q_{11}$, $q_{1\beta}$, $q_{\alpha\beta}$ are determined by
values of $f$ along $Q_{v_1}$, i.e. by $\phi(v_1,v'_1,l_1)$
(where $v'_1=f(v_1)$ and $l_1=f_{*v_1}$).
Similarly, $q_{22}$ and $q_{2\beta}$ are determined by
$\phi(v_2,v'_2,l_2)$. These dependencies can be expressed by
relations
\begin{equation}\label{q}
\left.\matrix{
q_{11} & = &\theta'_1(\phi_{*z}(v_1,v'_1,l_1)Y_1), \cr
q_{1\beta} & = &\theta'_1(\phi_{*z}(v_1,v'_1,l_1)Y_\beta), \cr
q_{\alpha\beta}& = &\theta'_\alpha(\phi_{*z}(v_1,v'_1,l_1)Y_\beta), \cr
q_{22} & = &\theta'_2(\phi_{*z}(v_2,v'_2,l_2)Y_2), \cr
q_{2\beta} & = &\theta'_2(\phi_{*z}(v_2,v'_2,l_2)Y_\beta), \cr
q_{12} & = & q_{21} = q_{\alpha 1} = q_{\alpha 2} = 0. \cr
}\right\rbrace.
\end{equation}
Thus, the map $f$ is completely determined by parameters
$v_j\in V_j:=\C^n$, $v'_j\in V'_j:=\C^n$ and $l_i\in L_j:=Gl(n)$,
$j=1,\ldots,n$.
\subsection{The case $n=2$}
In case $n=2$ there are no frames with
properties~(\ref{Y}) and another construction (\cite{W},~p.~58) is needed.
By Lemma~\ref{inv}, two points $\zeta_1,\zeta_2\in Q_{w_0}$ can be chosen
such that $Q_{\zeta_1}$ and $Q_{\zeta_2}$ are non-singular and transverse in
$w_0$. Then choose $v_1\in Q_{\zeta_1}$ and $v_2\in Q_{\zeta_2}$ such that
each $Q_{v_j}$ is non-singular in $\zeta_j$ and transverse to
$Q_{w_0}$ there. Now fix $v_1$ and $v_2$ and let $z_1$ and $z_2$ move
along $Q_{v_1}$ and $Q_{v_2}$ respectively. For $z_1$ and $z_2$ near
$\zeta_1$ and $\zeta_2$, it follows that $Q_{z_1}$ and $Q_{z_2}$ are still
transverse near $w_0$ and intersect each other in a single point $w$
there. Conversely, for given $w$ near $w_0$, $Q_w$ intersects
each $Q_{v_j}$ transversely in a point $z_j$ near $\zeta_j$.
In this way a local biholomorphic correspondence between $w\in W(:=\C^n)$ and
$(z_1,z_2)\in Q_{v_1}\times Q_{v_2}$ is obtained. It is defined by relations
\begin{equation}\label{etaw}
r(w,\bar{z}_j) = 0 \quad (\iff z_j\in Q_w), \quad j=1,2;
\end{equation}
\begin{equation}\label{etav}
r(v_j,\bar{z}_j) = 0 \quad (\iff z_j\in Q_{v_j}).
\end{equation}
Further, set $w'_0:=f(w_0)$, $\zeta'_j:=f(\zeta_j)$, $v'_j:=f(v_j)$.
All transverse properties are preserved by the biholomorphic map $f$. Again,
one obtains a local biholomorphic correspondence between $w'\in W'$ and
$(z'_1,z'_2)\in Q_{v'_1}\times Q_{v'_2}$, which is defined by
\begin{equation}\label{etaw'}
r'(w',\bar{z}'_j) = 0 \quad (\iff w'\in Q'_{z'_j});
\end{equation}
\begin{equation}\label{etav'}
r'(v'_j,\bar{z}'_j) = 0 \quad (\iff z'_j\in Q'_{v'_j}).
\end{equation}
Since the Segre varieties $Q_z$ are invariant with respect to $f$,
if $z'_j:=f(z_j)$, one obtains the corresponding point $w'=f(w)$.
Thus, $f$ can be decomposed in the following way:
\begin{equation}\label{dec1}
W \longrightarrow Q_{v_1}\times Q_{v_2} \stackrel{f\times f}
{\longrightarrow} Q'_{v'_1}\times Q'_{v'_2} \longrightarrow W'.
\end{equation}
The middle map here is in fact $f|_{Q_{v_1}} \times f|_{Q_{v_2}}$ which
is equal to
$\phi(\bar v_1,\bar v'_1,\bar l_1) \times \phi(\bar v_2,\bar v'_2,\bar l_2)$,
where $\phi$ is the map (\ref{phi}) and $l_j:=f_{*v_j}$. In other words
we have a relation between $z_j$ and $z'_j$:
\begin{equation}\label{zz'1}
z'_j = \phi(\bar v_j,\bar v'_j,\bar l_j)(z_j)
\end{equation}
Thus, $f$ is completely determined by parameters
$v_j\in V_j:=\C^n$, $v'_j\in V'_j:=\C^n$ and $l_i\in L_j:=Gl(n)$, $j=1,2$.
\subsection{Reflection principle with parameters}\label{we}
The local construction recalled in previous paragraph
is in fact global because of its algebraic nature. The map $f$
was locally expressed in terms of parameters
$(v,v',l)=(v,f(v),f_{*v})\in P$, where $v:=(v_1,\ldots,v_n)$ and
$$P:=\prod_{j=1}^n (V_j\times V'_j\times L_j).$$
Using the same algebraic relations
globally, we shall obtain a constructible family
$F\subset P\times W\times W'$
such that the graph $\Gamma_f$ is an open subset of the closure of the fibre
$\overline{F_p}$ for generic $v$ and $p=(v,f(v),f_{*v})$
(this will be made precise below).
We start with construction of a constructible family for the map
$$\phi(z,z',q)\colon Q_z \to Q'_{z'}$$
in (\ref{phi}). For this we consider the
constructible subset
$$\Phi \subset \overline{Gl(n)} \times \bar Z \times W \times \bar Z' \times
W'$$
defined by relations (\ref{wz}), (\ref{dr}), and (\ref{wqw}).
The relations (\ref{dr}) provide the existence of
$\pi_z(\bar w)$ and $\pi'_{z'}(\bar w')$ respectively. Furthermore, we
have seen that
$$(\overline{f_{*z}},\bar z,w,\overline{f(z)},f(w))\in \Phi$$ for
$(z,\bar w)\in \cal H$ near $(w_0,\bar w_0)$, and $\pi_z$ and $\pi_{z'}$ are
local invertible there (Lemma~\ref{inv}). To provide this local
invertibility ``globally'',
we assume, changing if necessary to a smaller constructible subset of $\Phi$,
that
\begin{equation}\label{dpi}
\det {\partial \pi_z(\bar w) \over \partial \bar w} \ne 0,
\quad \det {\partial \pi'_{z'}(\bar w') \over \partial \bar w'} \ne 0.
\end{equation}
The set $\Phi$ defines now a family of possibly multiple-valued maps
$$\phi(\bar z,\bar z',\bar q)\colon Q_z \to Q'_{z'}.$$
Consider the complexifications ${\cal H}\subset Z\times \bar W$ and
${\cal H'}\subset Z'\times \bar W'$. Then $\Phi$ is a subset in
$\overline{Gl(n)} \times \bar{\cal H} \times \bar{\cal H}'$.
\begin{Lemma}\label{f1}
The projection
$\delta\colon \Phi \to \overline{Gl(n)} \times \bar Z' \times \bar {\cal H}$
has finite fibres and is locally biholomorphic.
\end{Lemma}
{\bf Proof.} We need to prove that
$w'\in\delta^{-1}(\bar q,\bar z',\bar z,w)$
depends locally holomorphically on $(\bar q,\bar z',\bar z,w)$.
For this it is enough to observe, that, by (\ref{dpi}),
$\pi_{z'}(\bar w)$ is locally invertible and, by (\ref{wqw}),
$\bar w'=(\pi'_{z'})^{-1}(J_{n-1}(q)(\pi_z(\bar w)))$. Since the above
fibres are constructible, they are finite.
\par\hfill {\bf Q. E. D.}
In the following let $\phi$ denote the multiple-valued map defined
by $\Phi$. Since every value of $\phi$ is, by Lemma~\ref{f1},
locally holomorphic in $w$,
we can discuss its differential $\phi_*=\phi(\bar q,\bar z',\bar z,w)_*$
which is also possibly multiple-valued.
For the construction of the required family we need to consider
auxiliary parameter spaces $A:=Z\times Z'\times Gl(n)$
for $n>2$ and $A:=(Z\times Z')^2$ for $n=2$.
Let $F\subset \bar A \times P \times W\times W'$ be
the constructible subset defined by relations (\ref{wz}),
(\ref{wqw}), (\ref{zv0}), (\ref{zz'}) and (\ref{q})
in case $n>2$ and by (\ref{etaw}), (\ref{etav}), (\ref{etaw'}), (\ref{etav'})
and (\ref{zz'1}) in case $n=2$.
Passing if necessary to a constructible subset, we can require that
in case $n>3$ all $Q_{v_j}$'s and $Q_w$ are transverse in $z$ and
all $Q'_{v'_j}$'s and $Q'_{w'}$ are transverse in $z'$.
In case $n=2$ we require that each $Q_{v_j}$ is transverse to $Q_w$ in $z_j$
and $Q_{z_j}$'s are transverse in $w$ and, similarly,
each $Q'_{v'_j}$ is transverse to $Q'_{w'}$ in $z'_j$
and $Q'_{z'_j}$'s are transverse in $w'$.
Further, by Theorem~\ref{Chev} of Chevalley,
the projection $\pi(F)$ of $F$ on $P\times W\times W'$ is also constructible.
We don't have in general a local biholomorphic
property as in Lemma~\ref{f1} for $\pi(F)$,
but we still can prove the finiteness:
\begin{Lemma}\label{f2}
The projection $\sigma\colon \pi(F) \to P\times W$ has finite fibres.
\end{Lemma}
{\bf Proof.} Let fix $(p,w)\in P\times W$.
Let $(p,w,w')\in F$ be any point. By the construction of $\pi(F)$,
there exist points $a\in A$ such that $(\bar a,p,w,w')\in F$.
{\em Case $n>2$} Let $a=(z,z',q)$. We constructed $F$ such that
$Q_{v_1},\ldots,Q_{v_{n-1}}$ and $Q_w$ are transversal in $z$. Then, by
(\ref{wz}) and (\ref{zv0}), the set of possible $z\in Z$ is discrete and
therefore finite. Further, by (\ref{zz'}) and (\ref{q}), only finitely
many $z'$'s and $q$'s are possible. Here we use Lemma~\ref{f1}.
Now $w'\in W'$ is determined by (\ref{wqw}), which implies finiteness of
the set of $w'$'s.
{\em Case $n=2$} Let $a=(z_1,z'_1,z_2,z'_2)$. By definition of $F_2$,
$Q_w$ and $Q_{v_j}$ are transverse in $z_j$. Then there are only finitely
many possible intersections $z_j$. By (\ref{zz'1}),
the number of possible $z'_j$'s is also finite. Finally, since $Q'_{z'_j}$ are
transverse in $w'$, the number of possible $w'$'s is also finite.
\par\hfill {\bf Q. E. D.}
Now let $f\in Aut_a(D)$ be fixed and $H=H_1$, $H'=H_i$ for $i=i(f)$
(we defined $i(f)$ by the condition $f(H_1\cap U)\subset H_i$ for some open
$U\subset\C^n$ with $H_1\cap U\ne\emptyset$).
By our construction,
$(v,f(v),f_{*v},w,f(w))\in \pi(F)$ for all $(v,w)$ in some open subset
$U_1\subset V\times W$. Here we wish to point out that the family
$\pi(F)$ depends on the
index $i=i(f)$. To include all automorphisms $f\in Aut_a(D)$, we just
consider the finite union of $\pi(F)_i\subset P\times W\times W'$ for
all possible $i=i(f)$, $f\in Aut_a(D)$ and denote it again by $F$.
The set $F$ is constructible, i.e. a finite union
of locally closed algebraic subvarieties. It follows that the set
$$E(f):= \{(v,w)\in D^{n+1} \mid (v,f(v),f_{*v},w,f(w))\notin F \} $$
is analytically constructible, i.e. a finite union of locally closed
analytic subvarieties. If $(v,w)$ is outside the closure of $E(f)$, we have
$(p,w,f(w))\in F$ for $p=(v,f(v),f_{*v})\in P$. This means that
the graph $\Gamma_f:=\{(w,f(w)) \mid w\in D \}$ lies in the closure of
the fibre $F_p$.
Now we wish to prove main result of this section.
\begin{samepage}
\begin{Prop}\label{par}
Let $P$, $W$ and $W'$ be as above. There exist constructible subsets
$F\subset P\times W\times W'$ and $E\subset P\times V\times W$ such that
\begin{enumerate}
\item The projection $\sigma\colon F\to P\times W$ has finite fibres;
\item For every fixed $f\in Aut_a(D)$ there exists a proper subset
$E(f)\subset D^{n+1}$, such that for all
$(v,w)\in (D^{n+1})\backslash E(f)$ and $p=(v,f(v),f_{*v})\in P$
one has $(p,w,f(w))\in F$, the graph $\Gamma_f$ is a subset
of the closure of the fibre $F_p$ and $E(f)\subset E_p$;
\item $E_p$ is of complex codimension at least $1$ in $V\times W$.
\end{enumerate}
\end{Prop}
\end{samepage}
We need the following lemma.
\begin{Lemma}\label{clos}
Let $A$, $B$ and $C\subset A\times B$ be constructible subsets of
arbitrary algebraic varieties. Then the
{\bf fibrewise} closure of $C$ in $A\times B$,
i.e. the union of closures of the fibres
$C_a := (\{a\} \times B) \cap C$, $a\in A$ is constructible.
\end{Lemma}
The proof is based on the following fact
(see Mumford, \cite{M}, Corollary~1, p.71). Recall that a morphism is
{\it dominating} if its image is dense.
\begin{Prop}\label{mor}
Let $X$ and $Y$ be two complex algebraic varieties,
$f\colon X\to Y$ be a dominating morphism and $r=\dim X-\dim Y$.
Then there is a nonempty open set $U\subset Y$ such that, for all $y\in U$,
$f^{-1}(y)$ is a nonempty ``pure'' $r$-dimensional set, i.e. all its
components have dimension $r$.
\end{Prop}
{\bf Proof of Lemma~\ref{clos}.}
We first observe that given two constructible subsets
$C_1,C_2\subset A\times B$ which have constructible fibrewise closures,
the union $C=C_1\cup C_2$ has also this property. Changing to locally
closed irreducible components, we can assume that $A$, $B$ and $C$ are
irreducible algebraic varieties.
Now we prove the statement by induction on dimension of $A$.
In case $\dim A=0$ the fibrewise closure of $C$ is just the closure of $C$
which is constructible.
Let $\pi\colon \bar C\to A$ denote the projection of the closure
$\bar C$ on $A$. We can assume $\pi$
to be dominant, otherwise $A$ is replaced by the closure of $\pi(\bar C)$
which has a smaller dimension. Now we apply Proposition~\ref{mor} to
the projection $\pi$ and obtain an open subset $U\subset A$, such that
the fibre's over $U$ have pure dimension $\dim C-\dim A$. We have a
partition $A=U\cup (A\backslash U)$ of $A$ and the corresponding
partition $C=(C_1\cup C_2)$
($C_1:=C\cap (U\times B)$, $C_2:=C\cap ((A\backslash U)\times B)$).
By the above observation, it is enough to
prove the statement for $C_1$ and $C_2$ separately.
The statement for $C_2$ follows by induction, because
$\dim (A\backslash U)< \dim A$. Therefore we can assume $A=U$.
Now we consider the irreducible components $C_i$ of
$\bar C \backslash C$, $\dim C_i<\dim C$.
If $S_i:=\overline{\pi(C_i)}\ne A$ for some $i$, then we replace $A$ by
$A\backslash S_i$ and correspondingly $C$ by $C\cap\pi^{-1}(A\backslash S_i)$.
Thus we may assume
that $\pi\colon C_i\to A$ is dominaiting for all $i$.
Then we can apply Proposition~\ref{mor} to
every $C_i$ and obtain a number of open sets $U_i\subset A$.
Let $U$ be the intersection of all $U_i$'s. Since $A$ is irreducible,
$U$ is not empty. Again, proceeding by induction, we can reduce the
statement to the case $A=U$. But in this case the fibres of $\bar C$ are
of pure dimension $\dim C-\dim A$ and the fibres of
$\bar C \backslash C$ have smaller dimension. This implies that
the fibrewise closure of $C$ coincides with the usual closure $\bar C$
which is constructible.
\par\hfill {\bf Q. E. D.}
{\bf Proof of Proposition~\ref{par}.} Statement~1 follows from Lemma~\ref{f2}.
It follows from the local Webster's construction (section~\ref{wm}) that
$(p,w,f(w))\in F$ ($p=(v,f(v),f_{*v})$) for all $(v,w)$ in an open subset
$U\subset D^{n+1}$. This means that $U$ lies in the complement of the
`` exceptional set'' $E(f)$. Let
$\Omega(f):=D^{n+1}\backslash E(f)$, i.e.
$$\Omega(f)= \{(v,w)\in D^{n+1} \mid (v,f(v),f_{*v},w,f(w))\in F \}. $$
The set $E$ must be globally defined independently of any automorphism
$f\in Aut_a(D)$. For this it is necessary to define $\Omega(f)$ in another
way. Changing if necessary to a constructible subset of $F$,
we may assume that the projection
$\sigma_p \colon F_p\to W$ is locally biholomorphic and $\Omega(f)$ still
contains an open subset $U\subset D^{n+1}$. Then the
differentials $\partial w' \over \partial w$ are certainly defined.
We now define the family $F'\subset F\times Gl(n)$
of differentials by adding values of
$\partial w' \over \partial w$:
$$F':= \{ (p,w,w',q)\in F\times Gl(n) \mid q={
\partial w' \over \partial w} \}$$
This is a constructible set and we have
$(v,f(v),f_{*v},w,f(w),f_{*w})\in F'$ for $(v,w)$
in some open set $U\subset D^{n+1}$.
Now we write the definition of $\Omega(f)$ in the form:
$$\Omega(f):= \{ (v,w) \in V\times W
\mid \hbox{ for } v'=f(v),l=f_{*v},w'=f(w): $$
$$ (v,v',l,w,w')\in F \}. $$
Now we define a set $\Omega$ which contains $\Omega(f)$ for all $f\in
Aut_a(D)$:
\begin{samepage}
$$\Omega:= \{ (p,v,w,v',w',l) \in P\times V\times W\times V'\times W'\times
Gl(n)
\mid $$
$$\forall j: (p,v_j,v'_j,l_j)\in F' \land (p,w,w')\in F \land
(v,v',l,w,w')\in F \}. $$
\end{samepage}
For $f\in Aut_a(D)$, $(v,w)\in U$, and $p=(v,f(v),f_*v)$,
we have
$$(v,w,f(v),f(w),f_{*v})\in \Omega_p.$$
Let $\Omega'$ be the fibrewise closure of $\Omega$, i.e. the union of all
closures
of $\Omega_p$, $p\in P$. By Lemma~\ref{clos}, $\Omega'$ is constructible.
Finally, we define
$E\subset P\times V\times W$ to be the projection of $\Omega'\backslash \Omega$
on $P\times V\times W$. By Theorem~\ref{Chev} of Chevalley, $E$ is
constructible.
We now wish to prove that every fibre $E_p$ is of codimension at
least $1$. For this we note that for $p\in P$ fixed the projection of
$\Omega_p$ on $V\times W$ has finite fibres,
i.e. $\dim \Omega_p\le \dim (V\times W)$.
This implies $\dim (\Omega'\backslash \Omega)_p < \dim (V\times W)$
and $E_p$ is of codimension at least $1$ as required in statement~3.
We take now any $(v,w)\in D^{n+1}$ outside $E(f)$ and set $p=(v,f(v),f_{*v})$.
For the proof of statement~2, consider $f\in Aut_a(D)$, take
$(v,w)\in D^{n+1}\backslash E(f)$, and set $p=(v,f(v),f_*v)$.
We shall prove that $E(f)\subset E_p$.
Let $(v_0,w_0)\in E(f)$ be any point. If $(v,w)\in U$, we have
$$(v,f(v),f_{*v},w,f(w),f_{*w})\in F',$$
which implies
$$(p, v, w, f(v), f(w), f_{*v}) \in \Omega.$$
Here $(v,w)\in U$ is arbitrary. Since $f$ is holomorphic,
we have this property globally for all $(v,w)\in D^{n+1}$ if we replace
$\Omega$ with its fibrewise closure $\Omega'$. In particular, we have
$$(v_0,w_0,f(v_0),f(w_0),f_{*v_0}) \in \Omega'_p.$$
Since $(v_0,w_0)\in E(f)$, the point
$$(v_0,f(v_0),f_{*v_0},w_0,f(w_0))$$
does not lie in $F$. This implies that
$$(v_0,w_0,f(v_0),f(w_0),f_{*v_0})$$
does not lie in $\Omega_p$ and then
it is in $\overline{\Omega_p}\backslash \Omega_p$. This means $(v_0,w_0)\in
E_p$,
which is required. The proof of Proposition~\ref{par} is finished.
\par\hfill {\bf Q. E. D.}
\section{The choice of parameters}
In the previous section we proved the existence of a constructible
algebraic family $F\subset P\times W\times W'$ with the
property that
for all $f\in Aut_a(D)$ there exists a point $p\in P$ such that
\begin{equation}\label{subs}
\Gamma_f\subset \overline{F_p}.
\end{equation}
The goal of this section is to choose for every $f$ appropriate parameter $p$
with this property and obtain a map $\imath$ from $Aut_a(D)$ in the
corresponding
parameter space $P$. The first idea is to take some generic
$v\in V(:=V_1\times\cdots\times V_n=\C^{n^2})$ and to define
$p=(v,f(v),f_{*v})$. If $(v,w)\notin E(f)$, Proposition~\ref{par} yields the
required property (\ref{subs}). However, if we wish to define a global map
$Aut_a(D)\to P$, the condition $(v,w)\notin E(f)$ must be satisfied for
all $f\in Aut_a(D)$. Unfortunately, this is not true in general. It is
therefore necessary to take sufficiently many points
$(v_\mu,w_\mu) \in V\times W$ instead of
one $(v,w)$, such that $(v_\mu,w_\mu) \notin E(f)$ is always true at least
for one $\mu$. In fact, we prove the following Proposition.
\begin{samepage}
\begin{Prop}\label{par1}
There exists a natural number $N$, a constructible subset
$F\subset P^N\times W\times W'$ and
a collection of points $v_1,\ldots,v_m\in D$, $m=nN$, such that
\begin{enumerate}
\item the projection $\sigma \colon F\to P^N\times W$ has finite fibres,
\item for all $f\in Aut_a(D)$ the graph $\Gamma_f$ is a subset
of the closure $\overline{F_{\imath(f)}}$, where the map
$\imath\colon Aut(D)\to P^N$ is given by
$\imath(f) = (v_1,f(v_1),f_{*v_1},\ldots,v_m,f(v_m),f_{*v_m})$.
\end{enumerate}
\end{Prop}
\end{samepage}
\begin{Rem}\label{any-v}
Once the set $v_1,\ldots,v_m$ is chosen, we can add to it finitely
many other $v$'s and not change the statement of Proposition~\ref{par1}.
\end{Rem}
Before we start with the proof we need a technical lemma.
\begin{Lemma}\label{pts}
Let $A$, $B$, $C\subset A\times B$ be constructible sets and
every fibre $C_a:=\{b\in B\mid (a,b)\in C \}$ be of codimension at least one.
Then there
exists a finite number of points $b_\mu\in B$, $\mu=1,\ldots,s$
such that for every
$a\in A$ there is a point $b_\mu\notin C_a$.
\end{Lemma}
{\bf Proof.} We first prove the statement for $A$ a locally closed
irreducible subvariety by induction on dimension of $A$.
If $\dim A=0$, the statement is obvious.
Assume it to be proven for $\dim A < d$. By definition of constructible sets,
$C$ is a finite union of locally closed subvarieties
$C_\alpha=U_\alpha \cap F_\alpha$ where $U_\alpha$ are Zariski open and
$F_\alpha$ are closed subvarieties. The subvarieties $F_\alpha$ are not
open, otherwise a fibre $C_a$ would contain an open subset.
So we can choose a point
$$(a_0,b_0)\in (\cap_\alpha U_\alpha) \backslash (\cup_\alpha F_\alpha)$$
The set $A'$ of points $a\in A$, such that $b_0\in C_a$,
is the projection on $A$ of the
intersection $(A\times \{b_0\})\cap C$, which is constructible.
There is an entire neighborhood of $a_0$ in the complement and,
hence, $A'$ has lower dimension than $A$.
Now we use induction for all irreducible components of the
closure $\overline{A'}$. This yields a number of points $b_\mu$. These
points together with $b_0$ satisfy the required condition.
To prove the statement in case $A$ is constructible we note, that
$A$ is by Definition~\ref{cons} a finite union of locally closed $A_\alpha$'s.
For every $A_\alpha$ with $C_\alpha:=(A_\alpha \times B) \cap C$
the statement of Lemma gives a finite set of points $b_\mu$. The union of
these finite sets for all $\alpha$ satisfies the required property.
\par\hfill {\bf Q. E. D.}
{\bf Proof of Proposition~\ref{par1}.}
Now we apply Lemma~\ref{pts} to our situation. Let $P'$ be the
constructible subset of all parameters $p\in P$ such that
$E_p\subset V\times W$ is of codimension at least $1$.
Then we set in Lemma~\ref{pts} $A:=P'$, $B:=V\times W$ and
$C:=E \cap (P'\times V\times W)$.
The statement of Lemma yields a number of points
$$(v^{(\mu)},w_\mu) \in D^{n+1},\quad \mu=1,\ldots,N.$$
For every $f\in Aut_a(D)$ and $(v,w)\notin E(f)$ we have by condition~3
in Proposition~\ref{par}, $E(f)\subset E_p$ for $p=(v,f(v),f_{*v})$.
Then for some $\mu=1,\ldots,N$ we have $(v^{(\mu)},w_\mu)\notin E_p$, i.e.
$(v^{(\mu)},w_\mu)\notin E(f)$ and, by condition~2 in
Proposition~\ref{par}, $(p,w,f(w))\in F$. We obtain
$m=Nn$ points $v_1,\ldots,v_m$.
Now we construct the required family $F$ to be the union of
the sets $F_\mu$ defined by
\begin{equation}\label{fnu}
F_\mu := \{ (p_1,\ldots,p_N,w,w')\in P^N\times W\times W'
\mid (p_\mu,w,w')\in F \},
\end{equation}
Statement~1 in Proposition~\ref{par1} follows from condition~1
in Proposition~\ref{par}. Let $\imath\colon Aut(D) \to P^N$
be the map defined by
\begin{equation}\label{imath}
\imath(f) := (v_1,f(v_1),f_{*v_1},\ldots,v_m,f(v_m),f_{*v_m}).
\end{equation}
It is in fact a product of Cartan maps $C_v\colon f\mapsto (f(v),f_{*v})$
and is therefore a homeomorphism onto its image.
Statement~2 in Proposition~\ref{par1} follows now from the
above choice of $v_j$'s.
\par\hfill {\bf Q. E. D.}
\section{Defining conditions for $Aut_a(D)$}
In Proposition~\ref{par1} we constructed a map
$\imath\colon Aut(D) \to P^N$.
Our goal here is to give semi-algebraic description of the
image $\imath(Aut_a(D))$ and to prove the following Proposition.
\begin{Prop}\label{Ga}
The image $\imath(Aut_a(D))$ and the set of all graphs
$$ \Gamma := \{(\imath(f),w,f(w)) \mid f\in Aut_a(D) \land w\in D \} $$
are semi-algebraic.
\end{Prop}
\subsection{Reduction to a fixed pattern.}
In Proposition~\ref{par1} we obtained a constructible family
$F\subset P^N\times W\times W'$. Our goal now is to find
a stratification of $P^N\times W$ such that $F$ has a simplier
form over each stratum. This is done by applying Theorem~\ref{tr} on
local triviality of semi-algebraic morphisms.
To simplify the notation we shall write $P$ for $P^N$.
We first consider the projection $\sigma\colon F\to P\times W$.
Since we are interested only in points over $P\times D\subset P\times W$,
we write $F\subset P\times D\times W'$ for the intersection with
$P\times D\times W'$. Since $D$ is semi-algebraic, $F$ is semi-algebraic.
The projection $\sigma\colon F\to P\times D$
is a continuous semi-algebraic map (see Definition~\ref{map}) and
we can apply Theorem~\ref{tr} on local triviality.
Theorem~\ref{tr} yields a
finite semi-algebraic stratification $\{Y_1,\ldots,Y_h\}$ of $P\times D$
(see Definition~\ref{str}),
a collection of semi-algebraic typical fibres $\{E_1,\ldots,E_h\}$
and a collection of semi-algebraic structural homeomorphisms
\begin{equation}\label{E}
\tilde e_i\colon Y_i\times E_i \to \sigma^{-1}(Y_i), \quad i=1,\ldots,h
\end{equation}
(the $\tilde e_i$'s here are the inverses of the $g_i$'s in Theorem~\ref{tr}).
By statement~1 in Proposition~\ref{par1}, every typical fibre $E_i$ is finite.
The semi-algebraic stratification $\{Y_i\}$ of the product $P\times D$
defines a stratification of every fibre $\{p\}\times D$. This stratification
depends on $p\in P$ and the qualitative picture (e.g. the number of open
strata) can also depend on $p$. However, by changing to a partition of $P$
we reduce this general case to the case of fixed stratification of
$\{p\}\times D$, a fixed {\em pattern}.
For this we apply Theorem~\ref{tr} again to the projection
$\rho\colon P\times D \to P$ and partition $\{Y_1,\ldots,Y_h\}$ of
$P\times D$. We obtain a finite semi-algebraic stratification
$\{P_1,\ldots,P_r\}$ of $P$, a collection of semi-algebraic typical fibres
$\{G_1,\ldots,G_r\}$, for every $l=1,\ldots,r$ a finite semi-algebraic
partition $\{G_{l1},\ldots,G_{lh} \}$ of $G_l$ and a collection of
semi-algebraic structural homeomorphisms
\begin{equation}\label{G}
g_l\colon P_l\times G_l \to \rho^{-1}(P_l), \quad l=1,\ldots,r,
\end{equation}
such that
\begin{equation}\label{Gi}
g_l(P_l\times G_{li}) = (\rho^{-1}(P_l)) \cap Y_i, \quad l=1,\ldots,r,
\quad i=1,\ldots,h.
\end{equation}
\subsection{The set of all automorphisms.}
Here we discuss the set of all automorphisms of $D$ the graphs of which
are contained in the closures of fibres of our family $F$.
By condition~1 in
Proposition~\ref{par1}, only finitely many automorphisms can be contained
in the closure of a fixed fibre.
On the other hand, a fixed automorphism can be contained
in closures of a multitude of fibres.
Without loss of generality we assume, that $\{G_{l1},\ldots,G_{lh} \}$
is a finite semi-algebraic stratification of $G_l$ and $G_{li}$ are connected
(see Proposition~\ref{strat} and Corollary~\ref{con}).
Now for fixed $p\in P_l$ we wish to determine if the fibre
$F_p\subset D\times W'$ is related to some $f\in Aut(D)$.
Our procedure for doing this is semi-algebraic:
over the fixed decomposition $D=\sqcup_i G_{il}$
(in fact only over open $G_{il}$'S) we consider the pieces of $F_p$,
determined by the trivialization of it with typical fibres $E_i$.
The condition that certain of these pieces fit together to form a graph
of an automorphism proves to be semi-algebraic.
Among the strata $G_{li}$, $i=1,\ldots,h$ we
choose the open one's, which are assumed to be $G_{li}$, $i=1,\ldots,t$,
$t\le h$. By Proposition~\ref{par1},
the projection $\sigma \colon F\to P\times D$ has finite fibres
so the typical fibres $E_i$ are all finite.
Let us fix a $t$-tuple
$e=(e_1,\ldots,e_t)\in E:=E_1\times\cdots\times E_t$.
The number
of possible $t$-tuples is finite. Further, we define the maps $\xi_{e,p}$ over
each open $(Y_i)_p := \{w\in D \mid (p,w)\in Y_i\}\subset D$ by
\begin{equation}\label{xi}
(p,w,\xi_{e,p}(w)) = \tilde e_i(p,w,e_i), \quad i=1,\ldots,t,
\end{equation}
where $\tilde e_i\colon Y_i\times E_i \to \sigma^{-1}(Y_i)$ are the
trivialization morphisms in~(\ref{E}).
\begin{Prop}\label{pl}
Let $P_l$ and $e\in E$ be fixed.
The set $P_{e,l}$ of all parameters $p\in P_l$ such that
the map $\xi_{e,p}$ extends to a biholomorphic automorphism from $Aut_a(D)$
is semi-algebraic.
\end{Prop}
We begin with three lemmas. The first one is a semi-algebraic version of
Lemma~\ref{clos} on constructible sets.
\begin{Lemma}\label{clos-s-a}
Let $A$, $B$ and $C\subset A\times B$ be semi-algebraic sets. Then the
``fibrewise'' closure of $C$, i.e. the union of closures of the fibres
$C_a :=(\{a\} \times B) \cap C$, $a\in A$ is semi-algebraic.
\end{Lemma}
{\bf Proof.} Let $X:=A\times B$ and consider the partition
$X_1:=C$, $X_2:=(A\times B)\backslash C$ of $X$. Apply Theorem~\ref{tr}
to the projection of $X$ on $A$. To obtain the
closures of fibres we take the closures of typical fibres $F_{i1}$ in
$F_i$ and their images in $X$ under the structural trivializing
homeomorphisms. The images are semi-algebraic by Theorem~\ref{TS}.
The union of these images for all $i$ is semi-algebraic
and is exactly the ``fibrewise'' closure of $C$.
\nopagebreak\par\hfill {\bf Q. E. D.}
\begin{Lemma}\label{sub}
Let $A$, $B$ and $C,D\subset A\times B$ be semi-algebraic sets. Then
the set of $a\in A$ such that $C_a \subset D_a$ is semi-algebraic.
\end{Lemma}
{\bf Proof.} The complement of the required set in $A$ coincides
with the projection on $A$ of the difference $C\backslash D$. The difference
of semi-algebraic set is semi-algebraic, the projection is semi-algebraic
by the Tarski-Seidenberg theorem (Theorem~\ref{TS}).
\par\hfill {\bf Q. E. D.}
\begin{Lemma}\label{1}
Let $A$, $B$, $C$, $E\subset A\times B$ and $G\subset E\times C$ be
semi-algebraic sets. Then the
set of all $a\in A$, such that for all $b\in E_a$ the fibre $G_{(a,b)}$
consists of exactly one point, is also semi-algebraic.
\end{Lemma}
{\bf Proof.} We apply the Theorem~\ref{tr} on local trivialization
to the projection of $G$ on $E$. This yields a partition $\{Y_i\}$ of $E$.
The set $E'\subset E$ of one-point fibres $G_{(a,b)}$
is then the finite union of $Y_i$'s
such that the corresponding typical fibres $F_i$ consist of one point.
It follows that $E'$ is semi-algebraic. The required set in $A$ coincides with
the
set of $a\in A$ such that $C_a\subset E'_a$.
The latter set is semi-algebraic by Lemma~\ref{sub}.
\par\hfill {\bf Q. E. D.}
{\bf Proof of Proposition~\ref{pl}.} We first consider the condition that
$\xi_{e,p}$ extends to a well-defined continuous map on $D$.
This means that for any point
$w\in \overline{(Y_i)_p} \cap \overline{(Y_j)_p}$, $i,j=1,\ldots,t$,
the limits of graphs of $\xi_{e,p}$ over $(Y_i)_p$ and $(Y_j)_p$
coincide over $w$ and consist of one point.
For every stratum
$G_{ls}$, $G_{li}$, $G_{lj}$, $s=1,\ldots,h$, $i,j=1,\ldots,t$, with
$$G_{ls}\subset \overline{(Y_i)_p} \cap \overline{(Y_j)_p}$$
we write these conditions in a form
\begin{equation}\label{cont}
\left. \matrix{
i) & \overline{\Gamma_i(p)} \cap (B(p)\times W') =
\overline{\Gamma_j(p)} \cap (B(p)\times W'), \cr\cr
ii) & \forall w\in B(p) :
\#(\overline{\Gamma_i(p)} \cap (\{w\}\times W')) = 1
} \right\rbrace,
\end{equation}
where
$$B(p) := g_l(\{p\} \times G_{ls}) \subset \{p\} \times D$$
and
$\Gamma_i(p) := \tilde e_i((Y_i)_p\times \{e_i\})$ is the graph of $\xi_{e,p}$
over $(Y_i)_p$.
Now, by Lemmas~\ref{clos-s-a} and \ref{sub},
the set $\{p\in P_l\mid i)$ in (\ref{cont}) is satisfied $\}$ is a
semi-algebraic subset of $P_l$. For the condition ii) in (\ref{cont})
we set in Lemma~\ref{1}
$A:=P'$, $B:=D$, $C:= W'$,
$$E:=\{(p,w)\in P'\times D \mid w\in B(p) \}$$
and
$$G:=\{(p,w,w')\in E\times W' \mid
w\in \overline{\Gamma_i(p)} \cap (B(p)\times W')\}.$$
Then, by Lemma~\ref{1}, the set $\{p\in P_l\mid ii)$ in (\ref{cont})
is satisfied $\}$ is a semi-algebraic subset of $P_l$.
Without loss of generality, $i)$ and $ii)$ are satisfied for $p\in P_l$.
Thus, the closures of graphs of $\xi_{e,p}$ over $\{p\} \times D$ yield
well-defined maps $\xi_p\colon D\to W'$ (We do not know yet, whether or not
these maps are continuous).
The next condition on $\xi_{e,p}$ is
\begin{equation}\label{D}
\xi_{e,p}(D) = D,
\end{equation}
which is, by Lemma~\ref{sub}, a semi-algebraic condition.
Now, if conditions (\ref{cont}) and (\ref{D}) are satisfied,
we can prove that $\xi_{e,p}$ is continuous.
For this let $U_p\subset D$ denote the union of all
$(Y_i)_p$'s, $i=1,\ldots,t$.
This is an open dense subset of $D$ where $\xi_{e,p}$ is continuous.
Fix a point $w_0\in D$. By (\ref{cont}), $\xi_{e,p}(w_0)$ is the only limit
value of $\xi_{e,p}(w)$ for $w\in U_p$. Since $\xi_{e,p}$ is bounded, we have
\begin{equation}\label{lim}
\xi_{e,p}(w_0) = \lim_{{w\to w_0 \atop w\in U}} \xi_{e,p}(w),
\end{equation}
which means $\xi_{e,p}$ is continuous.
Thus, we obtained a family of continuous maps $\xi_{e,p}$ from $D$ onto
$D$, which are holomorphic outside some real analytic locally closed
subvariety of codimension~$1$. By the theorem on removable singularities,
$\xi_{e,p}$ is holomorphic on $D$.
Further, by the theorem of Osgood (see \cite{N}, Theorem~5, Chapter~5)
$\xi_{e,p}$ is biholomorphic if and only if it is injective. This is the
condition on fibres:
\begin{equation}\label{inj}
\#(\xi_{e,p}^{-1}(y))=1,\, y\in D.
\end{equation}
The set $\{p\in P_l\mid$ (\ref{inj}) is satisfied $\}$ is,
by Lemma~\ref{1}, semi-algebraic
(we set $A:=P'$, $B:= W'$, $C:=D$, $E:=P'\times D\subset A\times B$ and
$G$ is the family of graphs of $\xi_{e,p}$). This finishes the proof of
Proposition~\ref{pl}.
\par\hfill {\bf Q. E. D.}
\subsection{The image $\imath(Aut_a(D))$ and the set of associated graphs.}
Here we wish to prove Proposition~\ref{Ga}. Let $P_{e,l}$
be the semi-algebraic subsets from Proposition~\ref{pl}.
We obtain a diagram:
\begin{equation}\label{di}
\def\baselineskip20pt\lineskip3pt\lineskiplimit3pt{\baselineskip20pt\lineskip3pt\lineskiplimit3pt}
\def\mapup#1{\Big\uparrow \rlap{$\scriptstyle#1$}} \matrix{
& & P \cr
& \nearrow & \mapup{\imath} \cr
P_{e,l} & \to & Aut_a(D) \cr
p & \mapsto & \xi_{e,p} \cr },
\end{equation}
where the map from $P_{e,l}$ into $P$ is the usual inclusion.
We define $P'_{e,l}\subset P_{e,l}$
to be the subset of all points $p\in P_{e,l}$,
for which the diagram is commutative. This condition means
$p=(v,v',l)=(v,\xi_{e,p}(v),(\xi_{e,p})_{*v})$ and is therefore semi-algebraic.
Therefore, $P'_{e,l}$ is semi-algebraic. The semi-algebraic property of
$\imath(Aut_a(D))$ is a consequence of the following observation.
\begin{Lemma}
$$\imath(Aut_a(D))=\bigcup_{{e\in E \atop l=1,\ldots,r}} P'_{e,l}.$$
\end{Lemma}
{\bf Proof.} Let $p\in \imath(Aut_a(D))$, i.e.
$p=\imath(f)$ for some $f\in Aut_a(D)$. Then, by Proposition~\ref{par1},
$\Gamma_f\subset \overline{F_p}$.
We have $p\in P_l$ for some $l=1,\ldots,r$.
The graph $\Gamma_f$ defines sections in $F$
over every connected open stratum $(Y_i)_p$, $i=1,\ldots,t$. This means
that for some choice $e\in E$ we have $f=\xi_{e,p}$.
Then $\xi_{e,p}\in Aut_a(D)$, which implies $p\in P_{e,l}$. Further, the
equality $f=\xi_{e,p}$ means that diagram~(\ref{di}) is commutative for $p$.
Then $p\in P'_{e,l}$ and the inclusion in one direction is proven.
Conversely, let $e\in E$ be fixed and $p\in P'_{e,l}$.
Since $p\in P_{e,l}$,
$f:=\xi_{e,p}$ is an automorphism in $Aut_a(D)$.
The commutativity of diagram~(\ref{di}) means
$p=\imath(f)$. This implies $p\in \imath(Aut_a(D))$, which proves the
inclusion in other direction.
\nopagebreak\par\hfill {\bf Q. E. D.}
{\bf Proof of Proposition~\ref{Ga}.}
The family
$$\Gamma:=\{(\imath(f),w,f(w)) \mid f\in Aut_a(D) \land w\in D \}$$
over $P'_{e,l}$ coincides now with the family of graphs of $\xi_{e,p}$.
The latter is,
by construction, semi-algebraic and Proposition~\ref{Ga} is proven.
\nopagebreak\par\hfill {\bf Q. E. D.}
\section{Semi-algebraic structures on $Aut_a(D)$}
Here we finish the proof of Theorem~\ref{main}.
In previous section we considered imbeddings $\imath\colon Aut(D)\to P^N$.
Here we wish to change to Cartan imbeddings $C_v(f):=(f(v),f_{*v})$.
By Proposition~\ref{par1}, $\imath$ is given by
$\imath(f)=(v,f(v),f_{*v})$, where $v=(v_1,\ldots,v_m)\in D^m$.
The image $C_{v_j}(Aut_a(D))$ is equal to the projection of
$\imath(Aut_a(D))$ on $V'_j\times L_j$ (we use our notations
$v'_j=f(v_j)\in V'_j$, $l_j=f_{*v_j}\in L_j$). By Theorem~\ref{TS} of
Tarski-Seidenberg, $C_{v_j}(Aut_a(D))$ is semi-algebraic. Further, it is
semi-algebraically isomorphic to $\imath(Aut_a(D))$.
By Remark~\ref{any-v}, any $v$ and $v'$ can be
among $v_j$'s. Thus we obtain the following result.
\begin{Prop}\label{stru}
Let $v\in D$ be any point. The image $C_v(Aut_a(D))$ is semi-algebraic
and this semi-algebraic structure is independent of $v\in D$.
\end{Prop}
Now we fix some $v\in D$ and denote by $K$ the image
$C_v(Aut_a(D))\subset D\times Gl(n)$. The family
$\Gamma'\subset C_v(Aut_a(D))\times D^2$ of graphs over $C_v(Aut_a(D))$
is a projection of $\Gamma$ and is therefore semi-algebraic.
To simplify our notation we set $P:=D\times Gl(n)$, and $\Gamma:=\Gamma'$.
Statements~2 and 3 in Theorem~\ref{main} can
now be formulated as follows:
\begin{Lemma}\label{s-a}
With respect to the group operation of $Aut_a(D)$,
$K$ is a Nash group and the action on $D$ is Nash.
\end{Lemma}
{\bf Proof.} We consider the graph of the operation
$(x,y)\mapsto xy^{-1}$ in $K^3$. For this, start with the family
$\Gamma\subset K\times D\times D$ and define a new family
$\Gamma_1\subset K^3\times D^3$ by
\begin{equation}\label{G1}
\Gamma_1 := \{ (x,y,z,w,w',w'')\in K^3\times D^3 \mid
(y,w',w)\in \Gamma \land (x,w',w'')\in \Gamma \}.
\end{equation}
The conditions in (\ref{G1}) express the fact that $y^{-1}\in K$
transforms $w$ in $w'$ and $x\in K$ transforms $w'$ in $w''$. The projection
$\Gamma_2$ of $\Gamma_1$ on $K^3\times D^2$ (with coordinates $(x,y,z,w,w'')$)
is, by the Theorem of Tarski-Seidenberg, semi-algebraic. Now the
condition $z=xy^{-1}$ means that the graphs of $z$ and $xy^{-1}$ coincide,
i.e. the fibres $(\Gamma_2)_{(x,y,z)}$ and $(\Gamma_3)_{(x,y,z)}$ coincide,
where $\Gamma_3 := \{(x,y,z,w,w'') \mid (z,w,w'')\in \Gamma \}$ is an
extension of $\Gamma$.
By Lemma~\ref{sub}, the coincidence of fibres is
a semi-algebraic condition on $(x,y,z)\in K^3$. This proves that the
graph of the correspondence $(x,y)\mapsto xy^{-1}$ is semi-algebraic, which
means that the group operation is semi-algebraic. Since the latter is also
real analytic by Theorem of Cartan, $K$ is an affine Nash group.
Furthermore, the graph $\Gamma$ of the action of $K$ on $D$ is semi-algebraic
and real analytic and therefore Nash.
\par\hfill {\bf Q. E. D.}
It remains to prove statement~1 in Theorem~\ref{main}
which asserts that $Aut_a(D)$ is a Lie subgroup of $Aut(D)$.
{\bf Proof of statement~1. } We begin with the semi-algebraic set $K$.
By Proposition~\ref{strat}, it admits a finite semi-algebraic stratification.
Let $x\in K$ be a point in a stratum of maximal dimension. Then there is
a neighborhood $U_x\subset P$ of $x$, such that $K\cap U_x$ is a closed real
analytic submanifold of $U_x$. The preimage
$K':=C_v^{-1}(K\cap U_x)$ is a closed real analytic submanifold in the
neighborhood $U_f:=C_v^{-1}(U_x)$ of $f:=C_v^{-1}(x)$. Since
$Aut_a(D)$ is a subgroup of $Aut(D)$, we see that
$Aut_a(D)\cap (f^{-1} \cdot U_f) = f^{-1} \cdot K'$ is a closed real
analytic submanifold in the neighborhood $f^{-1} \cdot U_f$ of the unit
$id\in Aut(D)$. This implies that $Aut_a(D)$ is a real analytic
subgroup of $Aut(D)$. \hfill {\bf Q. E. D.}
|
1998-03-16T20:39:38 | 9710 | alg-geom/9710020 | en | https://arxiv.org/abs/alg-geom/9710020 | [
"alg-geom",
"math.AG"
] | alg-geom/9710020 | null | Victor V. Batyrev | Birational Calabi--Yau n-folds have equal Betti numbers | AMS-LaTeX, 11 pages, to appear in in Proc. European Algebraic
Geometry Conference (Warwick, 1996) | null | null | null | null | Let X and Y be two smooth projective n-dimensional algebraic varieties X and
Y over C with trivial canonical line bundles. We use methods of p-adic analysis
on algebraic varieties over local number fields to prove that if X and Y are
birational, they have the same Betti numbers.
| [
{
"version": "v1",
"created": "Thu, 16 Oct 1997 09:24:33 GMT"
},
{
"version": "v2",
"created": "Mon, 16 Mar 1998 19:39:38 GMT"
}
] | 2007-05-23T00:00:00 | [
[
"Batyrev",
"Victor V.",
""
]
] | alg-geom | \section{Introduction}
The purpose of this note is to show how to use the elementary
theory of $p$-adic integrals on algebraic varieties to prove
cohomological properties of birational algebraic varieties over
$\C$. We prove the following theorem, which was used by
Beauville in his recent explanation of a Yau--Zaslow formula for
the number of rational curves on a K3 surface \cite{Beauville}
(see also \cite{FGS,YZ}):
\begin{thm}
\label{main-th}
Let $X$ and $Y$ be smooth $n$-dimensional irreducible
projective algebraic varieties over $\C$. Assume that the
canonical line bundles $\Om^n_X$ and $\Om^n_Y$ are trivial and
that $X$ and $Y$ are birational. Then $X$ and $Y$ have the same
Betti numbers, that is,
\[
H^i(X,\C) \cong H^i(Y,\C) \quad \text{for all $i\ge0$.}
\]
\end{thm}
Note that Theorem~\ref{main-th} is obvious for $n =1$, and for $n=2$,
it follows from the uniqueness of minimal models of surfaces with
$\kappa\ge0$, that is, from the property that any birational map between
two such minimal models extends to an isomorphism \cite{KMM}. Although
$n$-folds with $\kappa\ge0$ no longer have a unique minimal models for
$n\ge3$, Theorem~\ref{main-th} can be proved for $n=3$ using a result
of Kawamata (\cite{kawamata}, \S6): he showed that any two birational
minimal models of $3$-folds can be connected by a sequence of flops
(see also \cite{kollar}), and simple topological arguments show that if
two projective $3$-folds with at worst $\Q$-factorial terminal
singularities are birational via a flop, then their singular Betti
numbers are equal. Since one still knows very little about flops in
dimension $n\ge4$, it seems unlikely that a consideration of flops
could help to prove Theorem~\ref{main-th} in dimension $n\ge4$.
Moreover, Theorem~\ref{main-th} is false in general for projective
algebraic varieties with at worst $\Q$-factorial Gorenstein terminal
singularities of dimension $n\ge4$. For this reason, the condition in
Theorem~\ref{main-th} that $X$ and $Y$ are smooth becomes very
important in the case $n\ge4$. We remark that in the case of
holomorphic symplectic manifolds some stronger result is obtained in
\cite{H}.
\bigskip
\noindent
{\bf Acknowledgements:}
The author would like to thank Professors A. Beauville, B. Fantechi,
L. G\"ottsche, K. Hulek, Y. Kawamata, M. Kontsevich, S. Mori, M. Reid
and D. van Straten for fruitful discussions and stimulating e-mails.
\section{Gauge forms and $p$-adic measures}
Let $F$ be a local number field, that is, a finite extension of the
$p$-adic field $\Q_p$ for some prime $p\in\Z$. Let $R\subset F$ be the
maximal compact subring, $\mathfrak q\subset R$ the maximal ideal,
$F_{\mathfrak q}=R/\mathfrak q$ the residue field with $|F_{\mathfrak q}|=q=p^r$.
We write
\[
N_{F/{\Q}_p}\colon F\to\Q_p
\]
for the standard norm, and $\| \cdot \|\colon F \to \R_{\ge 0}$
for the multiplicative $p$-adic norm
\[
a \mapsto \|a\|=p^{-\operatorname{Ord}(N_{F/\Q_p}(a))}.
\]
Here $\operatorname{Ord}$ is the $p$-adic valuation.
\begin{dfn} Let $\mathcal X$ be an arbitrary flat reduced algebraic
$S$-scheme, where $S=\operatorname{Spec} R$. We denote by $\mathcal X(R)$ the set of
$S$-morphisms $S\to\mathcal X$ (or sections of $\mathcal X\to S$). We call $\mathcal X(R)$
the set of $R$-{\em integral points} in $\mathcal X$. The set of sections of
the morphism $\mathcal X\times_S\operatorname{Spec} F\to\operatorname{Spec} F$ is denoted by $\mathcal X(F)$ and
called the set of $F$-{\em rational points} in $\mathcal X$.
\end{dfn}
\begin{rem}\label{point}
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item If $\mathcal X$ is an affine $S$-scheme, then one can
identify $\mathcal X(R)$ with the subset
\[
\bigl\{x\in\mathcal X(F) \bigm|
\text{$f(x)\in R$ for all $f\in\Ga(\mathcal X,\mathcal O_{\mathcal X})$}\bigr\}
\subset \mathcal X(F).
\]
\item If $\mathcal X$ is a projective (or proper) $S$-scheme, then
$\mathcal X(R)=\mathcal X(F)$.
\end{enumerate}
\end{rem}
Now let $X$ be a smooth $n$-dimensional algebraic variety over
$F$. We assume that $X$ admits an extension $\mathcal X$ to a regular
$S$-scheme. Denote by $\Om^n_X$ the canonical line bundle of
$X$ and by $\Om^n_{\mathcal X/S}$ the relative dualizing sheaf on
$\mathcal X$.
Recall the following definition introduced by Weil \cite{weil}:
\begin{dfn} A global section $\om\in\Ga(\mathcal X,\Om^n_{\mathcal X/S})$ is
called a {\em gauge form} if it has no zeros in $\mathcal X$. By
definition, a gauge form $\om$ defines an isomorphism
$\mathcal O_\mathcal X\cong\Om^n_{\mathcal X/S}$, sending $1$ to $\om$. Clearly, a
gauge form exists if and only if the line bundle $\Om^n_{\mathcal X/S}$
is trivial.
\end{dfn}
Weil observed that a gauge form $\om$ determines a canonical
$p$-adic measure $\dd\mu_\om$ on the locally compact $p$-adic
topological space $\mathcal X(F)$ of $F$-rational points in $\mathcal X$.
The $p$-adic measure $\dd\mu_\om$ is defined as follows:
Let $x\in\mathcal X(F)$ be an $F$-point, $t_1, \dots, t_n$ local $p$-adic
analytic parameters at $x$. Then $t_1, \dots , t_n$ define a $p$-adic
homeomorphism $\theta \colon U \to\A^n(F)$ of an open subset
$\mathcal U\subset\mathcal X(F)$ containing $x$ with an open subset
$\theta(\mathcal U)\subset\A^n(F)$. We stress that the subsets
$\mathcal U\subset\mathcal X(F)$ and $\theta(\mathcal U)\subset\A^n(F)$ are considered to be
open in the $p$-adic topology, not in the Zariski topology. We write
\[
\om=\theta^*\left(g \dd t_1 \wedge \cdots
\wedge \dd t_n\right),
\]
where $g=g(t)$ is a $p$-adic analytic function on $\theta(\mathcal U)$
having no zeros. Then the $p$-adic measure $\dd\mu_\om$ on $\mathcal U$
is defined to be the pullback with respect to $\theta$ of the
$p$-adic measure $\|g(t)\|\bdt$ on $\theta(\mathcal U)$, where $\bdt$ is the
standard $p$-adic Haar measure on $\A^n(F)$ normalized by the condition
\[\int_{\A^n(R)} \bdt =1.
\]
It is a standard exercise using the Jacobian to check that two
$p$-adic measures $\dd\mu_\om',\dd\mu_\om''$ constructed as above on
any two open subsets $\mathcal U',\mathcal U''\subset\mathcal X(F)$ coincide on the
intersection $\mathcal U'\cap\,\mathcal U''$.
\begin{dfn} The measure $\dd\mu_\om$ on $\mathcal X(F)$ constructed above is
called the {\em Weil $p$-adic measure} associated with the gauge form
$\om$.
\end{dfn}
\begin{thm}[\cite{weil}, Theorem~2.2.5] Let $\mathcal X$ be a regular
$S$-scheme, $\om$ a gauge form on $\mathcal X$, and $\dd\mu_\om$ the
corresponding Weil $p$-adic measure on $\mathcal X(F)$. Then
\[
\int_{\mathcal X(R)} \dd\mu_\om=\frac{|\mathcal X(F_\mathfrak q)|}{q^n},
\]
where $\mathcal X({F_\mathfrak q})$ is the set of closed points of $\mathcal X$ over
the finite residue field $F_\mathfrak q$.
\label{weil1}
\end{thm}
\begin{pf} Let
\[
\fie\colon \mathcal X(R)\to\mathcal X({F_\mathfrak q})
\quad\text{given by}\quad
x \mapsto \xbar\in\mathcal X({F_\mathfrak q})
\]
be the natural surjective mapping. The proof is based on the idea that
if $\xbar\in\mathcal X({F_\mathfrak q})$ is a closed ${F_\mathfrak q}$-point of $\mathcal X$ and
$g_1,\dots,g_n$ are generators of the maximal ideal of $\xbar$ in
$\mathcal O_{\mathcal X,\xbar}$ modulo the ideal $\mathfrak q$, then the elements
$g_1,\dots,g_n$ define a $p$-adic analytic homeomorphism
\[
\ga\colon\fie^{-1}(\xbar)\to\A^n(\mathfrak q),
\]
where $\fie^{-1}(\xbar)$ is the fiber of $\fie$ over $\xbar$ and
$\A^n(\mathfrak q)$ is the set of all $R$-integral points of $\A^n$ whose
coordinates belong to the ideal $\mathfrak q \subset R$. Moreover, the
$p$-adic norm of the Jacobian of $\ga$ is identically equal to
$1$ on the whole fiber $\fie^{-1}(\xbar)$. In order to see the latter
we remark that the elements define an \'etale morphism $g\colon V \to
\A^n$ of some Zariski open neighbourhood $V$ of $\xbar\in\mathcal X$. Since
$\fie^{-1}(\xbar) \subset V(R)$ and $g^*(\dd t_1,\wedge \cdots \wedge
\dd t_n)=h \om$, where $h$ is invertible in $V$, we obtain that $h$ has
$p$-adic norm $1$ on $\fie^{-1}(\xbar)$. So, using the $p$-adic analytic
homeomorphism $\ga$, we obtain
\[
\int_{\fie^{-1}(\xbar)} \dd\mu_\om=\int_{\A^n(\mathfrak q)}\bdt
=\frac{1}{q^n}
\]
for each $\xbar\in\mathcal X({F_\mathfrak q})$.
\ifhmode\unskip\nobreak\fi\quad $\square$\end{pf}
Now we consider a slightly more general situation. We assume only that
$\mathcal X$ is a regular scheme over ${S}$, but do not assume the existence
of a gauge form on $\mathcal X$ (that is, of an isomorphism
$\mathcal O_\mathcal X\cong\Om^n_{\mathcal X/S}$). Nevertheless under these weaker
assumptions we can define a unique natural $p$-adic measure $\dd\mu$ at
least on the compact $\mathcal X(R)\subset\mathcal X(F)$ -- although possibly not on
the whole $p$-adic topological space $\mathcal X(F)$!
Let $\mathcal U_1,\dots,\mathcal U_k$ be a finite covering of $\mathcal X$ by Zariski open
$S$-subschemes such that the restriction of $\Om^n_{\mathcal X/S}$ on each
$\mathcal U_i$ is isomorphic to $\mathcal O_{\mathcal U_i}$. Then each $\mathcal U_i$ admits a gauge
form $\om_i$ and we define a $p$-adic measure $\dd\mu_i$ on each compact
$\mathcal U_i(R)$ as the restriction of the Weil $p$-adic measure
$\dd\mu_{\om_i}$ associated with $\om_i$ on $\mathcal U_i(F)$. We note that
the gauge forms $\om_i$ are defined uniquely up to elements
$s_i\in\Ga(\mathcal U_i,\mathcal O^*_\mathcal X)$. On the other hand, the $p$-adic norm
$\|s_i(x)\|$ equals $1$ for any element $s_i\in\Ga( \mathcal U_i, \mathcal O^*_\mathcal X)$
and any $R$-rational point $x\in\mathcal U_i(R)$. Therefore, the $p$-adic
measure on $\mathcal U_i(R)$ that we constructed does not depend on the choice
of a gauge form $\om_i$. Moreover, the $p$-adic measures $\dd\mu_i$ on
$\mathcal U_i(R)$ glue together to a $p$-adic measure $\dd\mu$ on the whole
compact $\mathcal X(R)$, since one has
\[
\mathcal U_i(R)\cap\,\mathcal U_j(R)=(\mathcal U_i\cap\,\mathcal U_j)(R)
\quad\text{for $i,j=1,\dots,k$}
\]
and
\[
\mathcal U_1(R)\cup\cdots\cup\,\mathcal U_k(R)=
(\mathcal U_1\cup\cdots\cup\,\mathcal U_k)(R)=\mathcal X(R).
\]
\begin{dfn}
\label{can-m}
The $p$-adic measure constructed above defined on the set $\mathcal X(R)$ of
$R$-integral points of a $S$-scheme $\mathcal X$ is called the {\em canonical
$p$-adic measure}.
\end{dfn}
For the canonical $p$-adic measure $\dd\mu$, we obtain the same
property as for the Weil $p$-adic measure $\dd\mu_\om$:
\begin{thm}
\label{integ2}
\[
\int_{\mathcal X(R)} \dd\mu=\frac{|\mathcal X({F_\mathfrak q})|}{q^n}.
\]
\end{thm}
\begin{pf} Using a covering of $\mathcal X$ by some Zariski open subsets
$\mathcal U_1,\dots, \mathcal U_k$, we obtain
\[
\int\limits_{\mathcal X(R)}
\dd\mu=\sum_{i_1}\int\limits_{\mathcal U_{i_1}(R)} \kern-2mm\dd\mu
-\sum_{i_1<i_2}\kern2mm\int\limits_{(\mathcal U_{i_1}\cap\,\mathcal U_{i_2})(R)}
\kern-4mm\dd\mu
\kern2mm+\cdots+\kern2mm
(-1)^k \kern-4mm\int\limits_{(\mathcal U_1\cap\cdots\cap\,\mathcal U_{k})(R)}
\kern-4mm\dd\mu
\]
and
\begin{multline*}
\bigl|\mathcal X({F_\mathfrak q})\bigr|=
\sum_{i_1}\bigl|\mathcal U_{i_1}({F_\mathfrak q})\bigr|-\sum_{i_1<i_2}\bigl|
(\mathcal U_{i_1}\cap\,\mathcal U_{i_2})({F_\mathfrak q})\bigr| \\
+\cdots+(-1)^k\bigl|(\mathcal U_1\cap\cdots\cap\,\mathcal U_{k})({F_\mathfrak q})\bigr|.
\end{multline*}
It remains to apply Theorem~\ref{weil1} to every intersection
$\mathcal U_{i_1}\cap\cdots\cap\,\mathcal U_{i_s}$.
\ifhmode\unskip\nobreak\fi\quad $\square$\end{pf}
\begin{thm}
\label{m-zero}
Let $\mathcal X$ be a regular integral $S$-scheme and $\mathcal Z\subset \mathcal X$ a closed
reduced subscheme of codimension $\ge1$. Then the subset $\mathcal Z(R) \subset
\mathcal X(R)$ has zero measure with respect to the canonical $p$-adic measure
$\dd\mu$ on $\mathcal X(R)$.
\end{thm}
\begin{pf} Using a covering of $\mathcal X$ by Zariski open affine subsets
$\mathcal U_1, \dots, \mathcal U_k$, we can always reduce to the case when $\mathcal X$ is
an affine regular integral $S$-scheme and $\mathcal Z\subset\mathcal X$ an
irreducible principal divisor defined by an equation $f=0$, where $f$
is a prime element of $A=\Ga(\mathcal X, \mathcal O_\mathcal X)$.
Consider the special case $\mathcal X=\A^n_S=\operatorname{Spec} R[X_1,\dots,X_n]$ and
$\mathcal Z=\A^{n-1}_{S} =\operatorname{Spec} R[X_2,\dots,X_n]$, that is, $f=X_1$. For every
positive integer $m$, we denote by $\mathcal Z_m(R)$ the subset in
$\A^n(R)$ consisting of all points $x=(x_1,\dots,x_d)\in R^n $ such
that $x_1\in\mathfrak q^m$. One computes the $p$-adic integral in the
straightforward way:
\[
\int\limits_{\mathcal Z_m(R)} \bdx=\int\limits_{\A^1(\mathfrak q^m)}
\kern-2mm {\dd x_1}\kern2mm
\prod_{i=2}^n \left(\int_{\A^1(R)}{\dd x_i}\right)
=\frac{1}{q^m}.
\]
On the other hand, we have
\[
\mathcal Z(R)=\bigcap_{m =1}^{\infty} \mathcal Z_m(R).
\]
Hence
\[
\int_{\mathcal Z(R)} \bdx=\lim_{m \to\infty }\int_{\mathcal Z_m(R)} \bdx=0,
\]
and in this case the statement is proved. Using the Noether
normalization theorem reduces the more general case to the above
special one.
\ifhmode\unskip\nobreak\fi\quad $\square$\end{pf}
\section{The Betti numbers}
\begin{prop}
\label{main-th2}
Let $X$ and $Y$ be birational smooth projective
$n$-dimensional algebraic varieties over $\C$ having trivial canonical
line bundles. Then there exist Zariski open dense subsets $U\subset
X$ and $V \subset Y$ such that $U$ is isomorphic to $V$ and
$\operatorname{codim}_X(X\setminus U),\operatorname{codim}_Y(Y\setminus V)\ge2$.
\end{prop}
\begin{pf}
Consider a birational rational map $\fie\colon X\dasharrow Y$. Since
$X$ is smooth and $Y$ is projective, $\fie$ is regular at the general
point of any prime divisor of $X$, so that there exists a maximal
Zariski open dense subset $U\subset X$ with $\operatorname{codim}_X(X\setminus
U)\ge2$ such that $\fie$ extends to a regular morphism $\fie_0\colon
U\to Y$. Since
$\fie^*\om_Y$ is proportional to $\om_X$, the morphism $\fie_0$ is
\'etale, that is, $\fie_0$ is an open embedding of $U$ into the maximal
open subset $V \subset Y$ where $\fie^{-1}$ is defined. Similarly
$\fie^{-1}$ induces an open embedding of $V$ into $U$, so we conclude
that $\fie_0$ is an isomorphism of $U$ onto $V$.
\ifhmode\unskip\nobreak\fi\quad $\square$\end{pf}
\bigskip
\begin{pfof}{Theorem~\ref{main-th}} Let $X$ and $Y$ be smooth
projective birational varieties of dimension $n$ over $\C$ with trivial
canonical bundles. By Proposition~\ref{main-th2}, there exist Zariski
open dense subsets $U\subset X$ and $V\subset Y$ with
$\operatorname{codim}_X(X\setminus U)\ge2$ and $\operatorname{codim}_Y(Y\setminus V)\ge2$ and an
isomorphism $\fie\colon U \to V$.
By standard arguments, one can choose a finitely generated
$\Z$-subalgebra $\mathcal R\subset\C$ such that the projective varieties $X$
and $Y$ and the Zariski open subsets $U\subset X$ and $V\subset Y$ are
obtained by base change $*\times_\mathcal S\operatorname{Spec}\C$ from regular projective
schemes $\mathcal X$ and $\mathcal Y$ over $\mathcal S:=\operatorname{Spec}\mathcal R$ together with Zariski open
subschemes $\mathcal U\subset\mathcal X$ and $\mathcal V\subset\mathcal Y$ over $\mathcal S$. Moreover,
one can choose $\mathcal R$ in such a way that both relative canonical line
bundles $\Om^n_{\mathcal X/\mathcal S}$ and $\Om^n_{\mathcal Y/\mathcal S}$ are trivial, both
codimensions $\operatorname{codim}_\mathcal X(\mathcal X\setminus\mathcal U)$ and
$\operatorname{codim}_\mathcal Y(\mathcal Y\setminus\mathcal V)$ are $\ge2$, and the isomorphism
$\fie\colon U\to V$ is obtained by base change from an isomorphism
$\Phi\colon\mathcal U\to\mathcal V$ over $\mathcal S$.
For almost all prime numbers $p\in\N$, there exist a regular
$R$-integral point $\pi\in\mathcal S \times_{\operatorname{Spec}\Z}{\operatorname{Spec}\Z_p}$, where $R$
is the maximal compact subring in a local $p$-adic field $F$; let $\mathfrak q$
be the maximal ideal of $R$. By an appropriate choice of
$\pi\in\mathcal S\times_{\operatorname{Spec}\Z}{\operatorname{Spec}\Z_p}$, we can ensure that both $\mathcal X$
and $\mathcal Y$ have good reduction modulo $\mathfrak q$. Moreover, we can assume that
the maximal ideal $I(\overline{\pi})$ of the unique closed point in
\[
S: =\operatorname{Spec} R \stackrel{\pi}{\hookrightarrow}
\mathcal S\times_{\operatorname{Spec}\Z}{\operatorname{Spec}\Z_p}
\]
is obtained by base change from some maximal ideal
$J(\overline{\pi})\subset\mathcal R$ lying over the prime ideal $(p)\subset\Z$.
Let $\om_\mathcal X$ and $\om_\mathcal Y$ be gauge forms on $\mathcal X$ and $\mathcal Y$
respectively and $\om_\mathcal U$ and $\om_\mathcal V$ their restriction to $\mathcal U$
(respectively $\mathcal V$). Since $\Phi^*$ is an isomorphism over $\mathcal S$,
$\Phi^*\om_\mathcal Y$ is another gauge form on $\mathcal U$. Hence there exists a
nowhere vanishing regular function $h\in\Ga(\mathcal U,\mathcal O^*_\mathcal X)$ such that
\[
\Phi^* \om_\mathcal V=h \om_\mathcal U.
\]
The property $\operatorname{codim}_\mathcal X(\mathcal X\setminus \mathcal U)\ge2$ implies that $h$ is
an element of $\Ga(\mathcal X, \mathcal O^*_\mathcal X)=\mathcal R^*$. Hence, one has $\| h(x) \|
=1$ for all $x\in\mathcal X(F)$, that is, the Weil $p$-adic measures on
$\mathcal U(F)$ associated with $\Phi^* \om_\mathcal V$ and $\om_\mathcal U$ are the same.
The latter implies the following equality of the $p$-adic integrals
\[
\int_{\mathcal U(F)} \dd\mu_\mathcal X=\int_{\mathcal V(F)} \dd\mu_\mathcal Y.
\]
By Theorem~\ref{m-zero} and Remark~\ref{point}, (ii), we obtain
\[
\int_{\mathcal U(F)} \dd\mu_\mathcal X=\int_{\mathcal X(F)} \dd\mu_\mathcal X =\int_{\mathcal X(R)}
\dd\mu_\mathcal X
\]
and
\[\int_{\mathcal V(\mathcal F)} \dd\mu_\mathcal Y=\int_{\mathcal Y(\mathcal F)} \dd\mu_\mathcal Y=\int_{\mathcal Y(R)} \dd\mu_\mathcal Y.
\]
Now, applying the formula in Theorem~\ref{integ2}, we come to the
equality
\[
\frac{|\mathcal X({F_\mathfrak q})|}{q^n}=\frac{|\mathcal Y({F_\mathfrak q})|}
{q^n}.
\]
This shows that the numbers of $F_\mathfrak q$-rational points in $\mathcal X$ and
$\mathcal Y$ modulo the ideal $J(\overline{\pi})\subset\mathcal R$ are the same. We
now repeat the same argument, replacing $R$ by its cyclotomic extension
$\mathcal R^{(r)}\subset\C$ obtained by adjoining all complex $(q^r-1)$th
roots of unity; we deduce that the projective schemes $\mathcal X$ and $\mathcal Y$
have the same number of rational points over $F_\mathfrak q^{(r)}$, where
$F_\mathfrak q^{(r)}$ is the extension of the finite field $F_\mathfrak q$ of degree
$r$. We deduce in particular that the Weil zeta functions
\[
Z(\mathcal X,p,t)=\exp \left(\sum_{r =1}^{\infty}
|\mathcal X({F_\mathfrak q^{(r)}})| \frac{t^r}{r} \right)
\]
and
\[
Z(\mathcal Y, p, t) =\exp \left(\sum_{r =1}^{\infty}
|\mathcal Y({F_\mathfrak q^{(r)}})| \frac{t^r}{r} \right)
\]
are the same. Using the Weil conjectures proved by Deligne
\cite{Deligne} and the comparison theorem between the \'etale and
singular cohomology, we obtain
\begin{equation}
\label{eq_zeta}
Z(\mathcal X,p,t)=\frac
{P_1(t)P_3(t)\cdots P_{2n-1}(t)}
{P_0(t)P_2(t)\cdots P_{2n}(t)}
\end{equation}
and
\[
Z(\mathcal Y,p, t)=\frac{ Q_1(t) Q_3(t) \cdots Q_{2n-1}(t)}{
Q_0(t) Q_2(t) \cdots Q_{2n}(t) },
\]
where $P_i(t)$ and $Q_i(t)$ are polynomials with integer coefficients
having the properties
\begin{equation}
\label{eq_betti}
\deg P_i(t)=\dim H^i(X, \C), \quad \deg Q_i(t)=\dim H^i(Y, \C)
\quad \text{for all $i\ge0$.}
\end{equation}
Since the standard archimedean absolute value of each root of
polynomials $P_i(t)$ and $Q_i(t)$ must be $q^{-i/2}$ and
$P_i(0)=Q_i(0)=1$ for all $i\ge0$, the equality $Z(\mathcal X,p,t)=Z(\mathcal Y,p,t)$
implies $P_i(t)=Q_i(t)$ for all $i\ge0$. Therefore, we have $\dim
H^i(X,\C)=\dim H^i(Y,\C)$ for all $i\ge0$.
\ifhmode\unskip\nobreak\fi\quad $\square$\end{pfof}
\section{Further results}
\begin{dfn} Let $\fie\colon X\dasharrow Y$ be a birational map
between smooth algebraic varieties $X$ and $Y$. We say that $\fie$ {\em
does not change the canonical class}, if for some Hironaka resolution
$\al\colon Z\to X$ of the indeterminacies of $\fie$ the composite
$\al\circ\fie$ extends to a morphism $\be\colon Z\to Y$ such that
$\be^*\Om^n_Y\cong\al^*\Om^n_X$.
\end{dfn}
The statement of Theorem~\ref{main-th} can be generalized to the case of
birational smooth projective algebraic varieties which do not necessary
have trivial canonical classes as follows:
\begin{thm}
\label{main-th3}
Let $X$ and $Y$ be irreducible birational smooth $n$-dimensional
projective algebraic varieties over $\C$. Assume that the exists a
birational rational map $\fie\colon X\dasharrow Y$ which does not
change the canonical class. Then $X$ and $Y$ have the same Betti
numbers.
\end{thm}
\begin{pf} We repeat the same arguments as in the proof of
Theorem~\ref{main-th} with the only difference that instead of the Weil
$p$-adic measures associated with gauge forms we consider the canonical
$p$-adic measures (see Definition~\ref{can-m}). Using the birational
morphisms $\al\colon\mathcal Z\to\mathcal X$ and $\be\colon\mathcal Z\to\mathcal Y$ having the
property
\[
\be^*\Om^n_{\mathcal Y/S} \cong \al^*\Om^n_{\mathcal X/S},
\]
we conclude that for some prime $p\in\N$, the integrals of the canonical
$p$-adic measures $\mu_\mathcal X$ and $\mu_\mathcal Y$ over $\mathcal X(R)$ and $\mathcal Y(R)$ are
equal, since there exists a dense Zariski open subset $\mathcal U\subset\mathcal Z$ on
which we have $\al^*\mu_\mathcal X=\be^*\mu_\mathcal Y$. By Theorem~\ref{integ2}, the
zeta functions of $\mathcal X$ and $\mathcal Y$ must be the same.
\ifhmode\unskip\nobreak\fi\quad $\square$\end{pf}
Another immediate application of our method is related to the
McKay correspondence \cite{R}.
\begin{thm}
Let $G \subset\operatorname{SL}(n,\C)$ be a finite subgroup. Assume that there
exist two different resolutions of singularities on $W:=\C^n/G$:
\[
f\colon X \to W, \quad g\colon Y \to W
\]
such that both canonical line bundles $\Om^n_X$ and $\Om^n_Y$
are trivial. Then the Euler numbers of $X$ and $Y$ are the same.
\end{thm}
\begin{pf} We extend the varieties $X$ and $Y$ to regular schemes over
a scheme $\mathcal S$ of finite type over $\operatorname{Spec}\Z$. Moreover, one can choose
$\mathcal S$ in such a way that the birational morphisms $f$ and $g$ extend to
birational $\mathcal S$-morphisms
\[
F\colon \mathcal X \to\mathcal W, \quad G\colon \mathcal Y \to\mathcal W,
\]
where $\mathcal W$ is a scheme over $\mathcal S$ extending $W$. Using the same
arguments as in the proof of Theorem~\ref{main-th}, one obtains that
there exists a prime $p\in\N$ such that $Z(\mathcal X,p,t)=Z(\mathcal Y,p,t)$. On the
other hand, in view of (\ref{eq_betti}), the Euler number is determined
by the Weil zeta function (\ref{eq_zeta}) as the degree of the numerator
minus the degree of the denominator. Hence $e(X)=e(Y)$.
\ifhmode\unskip\nobreak\fi\quad $\square$\end{pf}
With a little bit more work one can prove even more precise
statement:
\begin{thm} Let $G\subset\operatorname{SL}(n,\C)$ be a finite subgroup and
$W:=\C^n/G$. Assume that there exists a resolution
\[
f\colon X \to W
\]
with trivial canonical line bundle $\Om^n_X$. Then the Euler
number of $X$ equals the number of conjugacy classes in $G$.
\end{thm}
\begin{rem} As we saw in the proof of Theorem~\ref{main-th2}, the Weil
zeta functions of $Z(\mathcal X,p,t)$ and $Z(\mathcal Y,p,t)$ are equal for almost
all primes $p\in\operatorname{Spec}\Z$. This fact being expressed in terms of the
associated $L$-functions indicates that the isomorphism $H^i(X,\C)\cong
H^i(Y,\C)$ for all $i\ge0$ we have established must have some more deep
motivic nature. Recently Kontsevich suggested an idea of a motivic
integration \cite{K}, developed by Denef and Loeser \cite{DL}. In
particular, this technique allows to prove that not only the Betti
numbers, but also the Hodge numbers of $X$ and $Y$ in \ref{main-th}
must be the same.
\end{rem}
|
1997-10-01T13:48:56 | 9710 | alg-geom/9710001 | en | https://arxiv.org/abs/alg-geom/9710001 | [
"alg-geom",
"math.AG"
] | alg-geom/9710001 | Jonathan Fine | Jonathan Fine | Convex polytopes and linear algebra | LaTeX2e. 14 pages | null | null | null | null | This paper defines, for each convex polytope $\Delta$, a family $H_w\Delta$
of vector spaces. The definition uses a combination of linear algebra and
combinatorics. When what is called exact calculation holds, the dimension
$h_w\Delta$ of $H_w\Delta$ is a linear function of the flag vector $f\Delta$.
It is expected that the $H_w\Delta$ are examples, for toric varieties, of the
new topological invariants introduced by the author in "Local-global
intersection homolog" (preprint alg-geom/9709011).
| [
{
"version": "v1",
"created": "Wed, 1 Oct 1997 11:48:56 GMT"
}
] | 2007-05-23T00:00:00 | [
[
"Fine",
"Jonathan",
""
]
] | alg-geom | \section{Introduction}
The goal, towards which this paper is directed, is as follows. Suppose
$\Delta$ is a convex polytope. One wishes to construct from $\Delta$
vector spaces whose dimension is a combinatorial invariant of $\Delta$.
The smaller the dimension of these spaces, the better. The convex
polytope $\Delta$ has both a linear structure, due to the ambient affine
linear space, and a combinatorial structure, due to the incidence
relations among the faces. The construction in the paper uses both
structures to produce many `vector-weighted inclusion-exclusion formulae',
to each one of which corresponds a complex of vector spaces. When such a
complex exactly computes its homology (a concept to be explained later)
the result is a vector space of the type that is sought. The proof of
exact calculation, which is not attempted in this paper, is expected to be
difficult.
In a special case, part of this problem has already been solved. If
$\Delta$ has rational vertices then from $\Delta$ a projective algebraic
variety $\PDelta$ can be constructed, and the middle perversity
intersection homology (mpih) Betti numbers $h\Delta$ are combinatorial
invariants of $\Delta$, by virtue of the Bernstein-Khovanskii-MacPherson
formula \cite{bib.JD-FL.IHNP,bib.KF.IHTV,bib.RS.GHV}, that express the
$h_i$ as linear functions of the flag vector $f\Delta$ of $\Delta$. In
this case Braden and MacPherson (personal communication) have proved an
exact calculation result. Their proof relies on deep results in algebraic
geometry, and in particular on Deligne's proof \cite{bib.PD.WC1,bib.PD.WC2}
of the Weil conjectures. Elsewhere \cite{bib.JF.LGIH}, the author has defined
local-global intersection homology groups. The construction in this paper
corresponds to the extension and unwinding (\cite{bib.JF.LGIH}, formulae
(8)--(10)) of the extended $h$-vector defined in that paper. This
correspondence, which is an exercise in combinatorics, is left to the
reader. It might also be presented elsewhere. This paper has been written
to be independent of \cite{bib.JF.LGIH}.
Central to this paper is the study of flags. They too have linear and
combinatorial strucutre. A \emph{flag} $\delta$ on a convex polytope
$\Delta$ is a sequence
\[
\delta = ( \delta_1 \subset \delta_2 \subset \dots
\subset \delta_r \subset \Delta )
\]
of faces $\delta_i$ of $\Delta$, each strictly contained in the next. The
\emph{dimension} $d = \dim \delta$ is the sequence
\[
d = ( d_1 < d_2 < \dots < d_r < n = \dim \Delta )
\]
of the dimensions $d_i$ of the \emph{terms} $\delta_i$ of $\delta$.
Altogether there are $2^n$ possible flag dimensions. The number $r$ is
the \emph{order} $\ord \delta$ of the flag $\delta$ (and of the dimension
vector $d$). If $\delta$ is a face of $\Delta$, the order one flag whose
only term is $\delta$, namely $(\delta\subset\Delta)$, will also be
denoted by $\delta$. The empty flag will be denoted by $\Delta$.
Similarly, if $V$ is a vector space then a flag $U$ on $V$ is a sequence
\[
U = ( U_1 \subset U_2 \subset \dots \subset U_r \subset V )
\]
of subspaces $U_i$ of $V$, each stricly contained in the next. The
\emph{dimension} $d=\dim V$ is the sequence
\[
d = ( d_1 < d_2 < \dots < d_r < n = \dim V )
\]
where now the $d_i$ are the dimensions of the $U_i$. Throughout $V$ will
be the vector space $\langle\Delta\rangle$ spanned by the vectors that lie
on $\Delta$. Each face $\delta_i$ similarly determines a subspace
$U_i=\langle\delta_i\rangle$ of $V$. Thus, each flag $\delta$ of faces on
$\Delta$ determines a flag $\langle\delta\rangle$ of subspaces of
$V=\langle\Delta\rangle$.
The construction of this paper is, in general terms, as follows. Suppose
$U$ is a flag on $V$. From $U$ many vector spaces can be constructed.
This paper constructs for each $U$, and for each $w$ lying in an as yet
unspecified index set, a vector space $U_w$. Now let $U'$ be obtained
from $U$ by deleting from $U$ one of its terms $U_i$. It so happens that
this deletion operator induces a natural map
\[
U_w \to U'_w
\]
between the associated $w$-spaces. (In the simplest case, which
corresponds to mpih, all the $U_w$ are subspaces of a single space $V_w$,
and the maps are inclusions. In general there is no such a global space
$V_w$.) Now suppose $U'$ is obtained from $U$ by deleting two or more
terms. By choosing an order for the deletion of these terms, a map
$U_w \to U'_w$ can be obtained. The single-deletion map is natural (or
geometric) in that the induced multiple deletion maps are independent of
the choice of deletion order.
Suppose now that such spaces $U_w$ have been defined for every flag on
$V$, and that $\Delta$ is a convex polytope whose vector space
$\langle\Delta\rangle$ is $V$. Each flag $\delta$ on $\Delta$ thus
determines a flag $\langle\delta\rangle$ on $V$, and thus a vector space
$\langle\delta\rangle_w$, or $\delta_w$ for short. These vector spaces
will be assembled into a complex, according to the order $r$ of $\delta$.
Define the space $\Delta(w,r)$ of \emph{$w$-weighted $r$-flags} to be the
direct sum of the $\delta_w$, where $\delta$ has order $r$. Each vector
in $\Delta(w,r)$ can be thought of as a formal sum $\sum v_\delta
[\delta]$, where $[\delta]$ is a formal object representing an $r$-term
flag $\delta$, and where the coefficient $v_\delta$ is drawn from the
vector space $\delta_w$.) By the assumptions of the previous paragraph,
the deletion operator on flags determines a differential
\[
\bound: \Delta(w,r) \to \Delta(w,r-1)
\]
and so induces a complex
\[
0 \to \Delta(w,r) \to \Delta(w,r-1) \to \dots
\Delta(w,1) \to \Delta(w,0) \to 0
\]
of vector spaces. (As is usual, $\bound=\sum (-1)^{i+1}\partial_i$, where
$\partial_i$ is the operator induced by deletion of the $i$-th term.
Because the maps $U_w\to U'_w$ are natural, $\bound^2$ is zero, and thus
one indeed has a complex.)
\emph{Exact calculation} is when this complex is exact except at one
point, say $\Delta(w,j)$. In that case the homology $H_w\Delta$ at that
point is a suitable alternating sum of the dimensions of the
$\Delta(w,i)$, and thus of the $\delta_w$. By construction the dimension
of $\delta_w$ will depend only on the dimension vector $d$ of $\delta$,
and thus (provided exact calculation holds) one has that the dimension
$h_w\Delta$ of $H_w\Delta$ is a linear function of the flag vector
$f\Delta$ of $\Delta$, and so is a combinatorial invariant. (This is
because the $f_d\Delta$ component of $f\Delta$ counts how many $d$-flags
there are on $\Delta$. Each $f_d\Delta$ contributes, up to an alternating
sign, the quantity $\lambda_d=\dim\delta_w$ to $h_\Delta$, where
$\delta_w$ is a coefficient space due to any flag $\delta$ of dimension
$d$.)
Convex polytopes are not the only combinatorial objects for which the
concept of a flag can be defined. In \cite{bib.JF.QTHGFV,bib.JF.SFV}, the
author defines flag vectors for $i$-graphs, or more generally any object
that is a union of cells (or edges), and which can be shelled. There
seems to be no reason why the general form of the construction described
here cannot also be applied in this new context. This is not to say the
the proper choice of the vector spaces $\delta_w$ to be associated to the
flags $\delta$ is not expected to be a deep question. Both for convex
polytopes and for $i$-graphs there are subtle and presently unknown
combinatorial inequalities on the flag vectors. An exact homology theory,
using vector spaces such as the $\delta_w$, is the only method the author
can envision, that will lead to the proof such inequalities.
The exposition is organised as follows. The next section (\S2) give the
definition of exact homology. The deletion operator applied to flags
yields the flag complex~(\S3). Next comes a complex~(\S4) that
corresponds to middle perversity intersection homology. The local-global
variant is more complicated, and is the substance of the paper. First the
coefficient spaces to be used are defined~(\S5), then the maps between
them~(\S6), and finally the local-global homology~(\S7).
The purpose of \S\S3--7 is to present the definition as the result of a
study of the geometric resources and constraints. Conversely, in \S8
decorated bar diagrams are used to reformulate the definition, and allow
the properties to be demonstrated, in a concise manner. Finally, \S9
provides a wider discussion of what has been done, and what remains to be
done.
The reader may at first find \S\S5--7 somewhat abstruse. They contain a
study of the linear algebra of a segmented flag of vector spaces. Once
the problem is understood, the key definitions come out in a fairly
natural way. In \S8, the same definitions are presented, but this time
via coordinates. Here, the definitions are clear, but may appear somewhat
arbitrary. Each point of view informs the other. This paper attempts to
show how the definitions arise naturally out of the logic of the
situation, and thus places \S\S5--7 before \S8. The reader may wish to
reverse this order.
The basic devices are the combinatorics of flags, and linear algebra. The
exterior algebra on a vector space is widely used. The fibre of the
moment map from a projective toric variety to its defining polytope is
always a product of circles, and so the homology of the fibre is
isomorphic to an exterior algebra. This fact is the beginning of the
connection between the linear algebra of flags and the existence of cycles
on the toric variety. The reader does not need to know this.
Throughout this paper $\Delta$ will be a convex polytope of some fixed
dimension~$n$, and $V$ will be the vector space spanned by the vectors
lying on $\Delta$. It will do no harm to think of $\Delta$ as lying in
$V$.
\section{Exact Homology}
This section explains the concept of exact homology. Suppose that a
sequence
\[
A =
( 0 \to A_n \to A_{n-1} \to \dots \to A_1 \to A_0 \to 0 )
\]
of vector spaces is given, together with a map
\[
\bound:A_i \to A_{i-1}
\]
between successive terms. If, for each $i$, the composite map
\[
\bound \circ \bound = \bound^2 : A_i \to A_{i-2}
\]
is the zero map, then $A$ is called a \emph{complex}, and $d$ its
\emph{boundary map}. If $A$ is a complex then the statement
\[
\im (\bound:A_{i+1} \to A_i )
\> \subseteq \>
\ker (\bound:A_i \to A_{i-1})
\]
restates the condition $\bound^2=0$, and the quotient of the above kernel
by the image is called the $i$-th homology $H_iA$ of the complex $A$. This
formalism originated in the definition, via chains and cycles, of the
homology groups $H_iX$ of a topological space $X$. There, most of the
homology groups were expected to be non-zero. The present use will be
different.
Suppose $A$ is a complex. If all the homology groups $H_iA$ are zero
(i.e.~at each $A_i$ the kernel and image are equal) then $A$ is called a
(long) \emph{exact sequence}, or \emph{exact} for short. If $A$ is exact
then the alternating sum
\begin{equation}
\label{eqn:chi}
\sum \nolimits _{i=0}^{n}\> (-1)^i \dim A_i
\end{equation}
of the dimensions of the $A_i$ (assumed finite) will be zero. To prove
this, introduce in each $A_i$ a subspace $B_i$ that is a complement to
$\bound A_{i+1}$ in $A_i$. The dimension of $A_i$ is the sum of the
dimensions of $\bound A_{i+1}$ and of $B_i$. The exactness assumption
implies that the restricted form
\[
\bound:B_i \to \bound A_i \subseteq A_{i-1}
\]
of the boundary map is an isomorphism. Thus, the contributions of $B_i$
and $\bound A_i$ to the alternating sum are equal but opposite, and so the
result follows.
Suppose $A$ is a complex. If $H_iA$ is zero then $A$ is said to be
\emph{exact at $A_i$}. Now suppose that $A$ is known to be exact at all
its $A_i$ except perhaps one, say $A_r$. In this case the alternating sum
(\ref{eqn:chi}) of the dimensions gives not zero, but the dimension of the
only non-zero homology $H_rA$ of the complex $A$, multiplied by $(-1)^r$.
If a complex is exact at all but one location, we shall say that it
\emph{exactly computes} the homology at that location.
Now suppose that the $A_i$ are constructed from the convex polytope
$\Delta$, and that their dimensions depend only on the combinatorial
structure of $\Delta$. The same will then be true for $H_rA$, provided
that for each $\Delta$ the complex exactly computes its homology at
$A_r$. If this holds we shall say that $H_rA$ is an \emph{exact homology}
group of $\Delta$. It is of course one thing to define a complex $A$, as
is done in this paper, and quite another to prove its exactness. This is
expected to require new concepts and methods.
\section{The flag complex}
The convex polytope $\Delta$ has both a combinatorial structure (incidence
relations among faces) and a linear structure (the vectors lying on a face
$\delta$ span a subspace $\langle\delta\rangle$ of $V$). In general both
structures will be used to define the complex $A$. This section defines a
complex that uses the combinatorial structure alone. The general case
will arise by allowing vectors to be used instead of numbers in the
construction that follows.
Recall the definition, in \S1, of a flag $\delta$ on $\Delta$, its
dimension vector $d$, and its order $r$. Now suppose $\delta$ is a flag on
$\Delta$, of order $r$. By removing one or more of the terms $\delta_i$
from $\delta$, new flags can be obtained, of lower order. Let
$\partial_j$ be the \emph{deletion operator} that removes from a flag
$\delta$ the $j$-th term.
The operators $\partial_j$ do not commute. Removing say the $2$nd term
from a sequence, and then say the $4$th, gives the same final result as
does first removing the $5$th and then the $2$nd. This is because
removing the $2$nd term will cause the subsequent items to move down one
place in the sequence. The equation
\[
\partial_j \partial_k = \partial_{k+1} \partial_j
\qquad \mbox{for $j<k$}
\]
is an example of the \emph{commutation law} for these deletion operators.
Now let $A_i$ consist of all formal weighted sums $x$ of the form
\[
x = \sum \nolimits _ { \ord \delta = i } x_\delta [\delta]
\]
where the coefficients $x_\delta$ are numbers and, as indicated, the sum
is over all flags $\delta$ on $\Delta$, which have $i$ terms. Here
$[\delta]$ denotes $\delta$ considered as a formal object. Where
confusion will not then result, $\delta$ will be written in its place.
Thus, a vector $x$ in $A_i$ is a formal sum of $i$-term flags, with numeric
coefficients $x_\delta$.
The deletion operation $\partial_j$ induces a map $A_i \to A_{i-1}$. (It
is zero if $j$ is larger than $i$.) Now use the formula
\[
\bound = \partial_1 - \partial_2 + \partial_3 + \dots
+ (-1)^{i-1}\partial_i + \ldots
\]
to define a map $\bound:A_i \to A_{i-1}$. It follows immediately from the
commutation laws, that $\bound^2=0$, and so $A$ is a complex. It will be
called the \emph{flag complex} of $\Delta$. It depends only on the
combinatorial structure of $\Delta$.
\section{Global coefficient spaces}
Instead of using numeric coefficients for the formal sums that constitute
the space $A_i$, one could instead write
\begin{equation}
\label{eqn.sum.vdelta.delta}
v = \sum \nolimits _ { \ord \delta = i } v_\delta \delta
\end{equation}
where now $v_\delta$ is to lie in a vector space $\Lambda(\delta)$ that is
in some way associated to $\delta$. Thus, $A_i$ is again to be formal
sums of order $i$ flags, where now the coefficients are to be vectors
rather than numbers. This section will describe the simplest way of
constructing such \emph{coefficient spaces} $\Lambda(\delta)$.
Suppose $\delta'$ is obtained from $\delta$ by the deletion of one or more
terms. For such a definition to produce a complex, there must also be
a natural map
\[
\Lambda(\delta) \to \Lambda(\delta')
\]
between the corresponding coefficient spaces. One way to do this, which
will be used in this section, is to have this map be an inclusion. Thus,
all the coefficients will lie in the global coefficient space
$\Lambda(\Delta)$ associated to the empty flag. Each flag $\delta$ will
then define a subspace $\Lambda(\delta)$ of $\Lambda(\Delta)$.
For this to work, one must have that $\Lambda(\delta)$ is a subspace of
$\Lambda(\delta')$. One way to do this is to have each individual face
$\delta_j$ in $\delta$ define a condition (or set of conditions) on
$\Lambda(\Delta)$. Now define $\Lambda(\delta)$ to be those vectors in
$\Lambda(\Delta)$ that satisfy the condition(s) due to the faces
$\delta_j$ in $\delta$. Provided the conditions due to $\delta_j$ depend
only on the face $\delta_j$ (and not on its location in $\delta$, or
whatever), it will follow automatically that $\delta'$ will provide fewer
conditions, and so it will be certain that $\Lambda(\delta')$ will
contain $\Lambda(\delta)$ as a subspace.
To summarise this section so far, suppose that a vector space
$\Lambda(\Delta)$ is given, and for each face $\delta$ of $\Delta$ a
subspace $\Lambda(\delta)$ is given (here $\delta$ stands for the flag
which has $\delta$ as its only term). From this a complex $A$ can be
constructed. First define $\Lambda(\delta)$ to be the intersection of the
spaces $\Lambda(\delta_j)$ associated to the terms $\delta_j$ of
$\delta$. Next define $A_i$ to be all formal sums
(\ref{eqn.sum.vdelta.delta}, where the coefficients $v_\delta$ are to lie
in $\Lambda(\delta)$. Finally, the boundary map
\[
\bound (v_\delta \delta ) = v_\delta [\partial_1\delta]
- v_\delta [\partial_2\delta]
+ \cdots
\]
is defined just as before.
To complete such a definition, one must provide a vector space
$\Lambda(\Delta)$, and derive from each face $\delta\subset\Delta$ a
subspace $\Lambda(\delta)$ of $\Lambda(\Delta)$.
Recall that $V$ stands for the span of the vectors lying on $\Delta$. Let
$V^*$ be the dual space of linear functions. That such a linear function
$\alpha$ is constant on $\delta$ (i.e.~zero on the vectors lying on
$\delta$) describes a subspace $\delta^\perp$ of $V^*$.
Exterior algebra will now be used. Fix a degree $r$, and let
$\Lambda(\Delta)$ be the $r$-fold exterior product $\Lambda^r=
\bigwedge^rV^*$ of arbitrary linear functions on $V$. The face $\delta$
defines a filtration of $\Lambda(\Delta)$ in the following way. For
each decomposition $r=s+t$ one can take the span of expressions of the
form
\[
\alpha_1 \wedge \alpha_2 \wedge \dots \wedge \alpha_{s}
\quad \wedge \quad
\beta_1 \wedge \beta_2 \wedge \dots \wedge \beta_{t}
\]
where the $\alpha_i$ are to vanish on $\delta$. No conditions are placed
on the $\beta_i$. The result is of course a subspace of
$\Lambda(\Delta)=\bigwedge^rV^*$.
To conclude this definition it is enough, for each face $\delta$ of
$\Delta$, to choose one of these subspaces of $\Lambda(\Delta)$. One
would like the resulting complex to produce an exact homology group, a
matter which is presently not well understood, and which involves concepts
that lie outside the scope of this paper.
The condition
\[
s > \codim \delta - s
\]
is satisfied by some smallest value of $s$. (The \emph{codimension}
$\codim\delta$ is defined as usual to be $\dim\Delta-\dim\delta$.) Use
this value to define for each face $\delta$ the subspace
$\Lambda(\delta)$. If the resulting $s$ is greater than $r$, then
$\Lambda(\delta)$ is taken to be zero.
This choice of spaces corresponds to middle perversity intersection
homology (for the associated toric variety $\PDelta$, if it exists). There
is little doubt that this gives the correct choice of subspaces, for
reasons that will be discussed in the final section.
\section{Local coefficient spaces}
To produce a complex $A$ one requires a coefficient space
$\Lambda(\delta)$ for each flag $\delta$ on $\Delta$, and natural maps
$\Lambda(\delta) \to \Lambda(\delta')$ whenever $\delta'$ is obtained
from $\delta$ by the deletion of one or more terms. The previous section
assumed the maps were inclusions (and so all the $\Lambda(\delta)$ were
subspaces of $\Lambda(\Delta)$). This section will relax this assumption,
to obtain the coefficient spaces that will later be used to define further
complexes. In the next section, the boundary map will be defined.
Suppose $V_1$ is a subspace of $V$. A basic construction of the previous
section was to use $V_1$ to define subspaces (in fact a filtration) of the
$r$-fold exterior produce $\bigwedge^rV^*$. Such a construction is in
fact forced upon us, provided we assume that the deletion operator on
flags induces inclusion, and also that $\Lambda(\Delta)$ is
$\bigwedge^rV^*$.
Relaxing this assumption allows the following. Given $V_1\subset V$ one
can form the vector spaces $V_1$ and $V/V_1$ and then form the tensor
product
\[
\bigwedge \nolimits ^{r_1} V_1^*
\otimes
\bigwedge \nolimits ^{r_2} V_1^\perp
\]
of the corresponding exterior products. Here, $V_1^\perp$ consists of the
$\alpha$ in $V^*$ that vanish on $V_1$. It is of course naturally
isomorphic to $(V/V_1)^*$.
In this way $V$ can be broken into two or more segments, within each of
which the construction of the previous section can be applied. For
example, each subspace $U$ of $V_1$ will determine a filtration of the
first factor $\bigwedge ^{r_1} V_1^*$ above. Similarly, if $U$ lies beween
$V_1$ and $V$, a filtration of the second factor $\bigwedge ^{r_2}
V_1^\perp$ will arise.
All the coefficient spaces $\Lambda(\delta)$ will be obtained in this way.
Given a flag $\delta$ use some (perhaps all or none) of its terms
$\delta_i$ to segment $V$. This gives a tensor product of exterior
algebras. For each factor choose a component, i.e.~a degree. The
resulting space is used in the same way as $\Lambda(\Delta)$ was, in the
previous section. Note that this space depends on the flag, or more
exactly the subflag used to segment $V$. There is no longer one global
space, in which all the coefficient vectors lie.
Now choose a term $\delta_j$ of $\delta$. This can be used to filter
$\bigwedge^{r_{k+1}} (V_{k+1}/V_k)^*$, where $\delta_k$ is the largest
segmenting face contained in $\delta_j$. (If there is none such, set
$k=0$ and use $\bigwedge^{r_1}V_1^*$ instead.) Now, as before, use the
condition
\[
s_j > \codim \delta_j - s_j
\]
to select a term in the filtration. Here, however,
\[
\codim \delta_j = \dim \delta_{k+1} - \dim \delta_j
\]
is to be the codimension of $\delta_j$ not within $\Delta$ but within its
segment.
The above filtration is to be applied for all the terms in the flag,
including those used to segment. Such terms can produce of course only
a trivial filtration. However, when
\[
r_k > ( \dim \delta_{k} - \dim \delta_{k-1} ) - r_k
\]
holds, the whole of $\bigwedge^{r_{k}} (V_{k}/V_{k-1})$ is to be used, as
the coefficient space for $[\delta]$. If the condition fails, zero is the
only coefficient to be used with $[\delta]$.
This concludes the definition of the coefficient spaces $\Lambda(\delta)$.
Note that to specify such a space the following is required, in addition
to $\delta$. One must select some (or all or none) of the faces of
$\delta$, to be used for segmentation. One must also specify a degree
$r_k$ for each segment. A more explicit notation might be
\[
\Lambda ( \delta , -r , s )
\]
where $r$ is the sequence $(r_1, r_2, \ldots )$ of degrees, and $s$ gives
the subflag $\delta_s$ of $\delta$ that is used to segment $\delta$. The
minus sign in $-r$ is to distinguish this notation from
$\Lambda(\delta,r,s)$, which will be introduced later. As already noted,
each degree must be greater than half the length of the corresponding
segment, for the coefficient space to be non-zero.
\section{The boundary map}
Suppose that $\delta$ is a flag of $\Delta$, and that some segmentation
$s$ of $\delta$ is chosen, which breaks $\delta$ (and $V$) into $l$
segments. Suppose also that a multi-degree $r=(r_1,\dots, r_l)$ is given.
The construction of the previous section yields a coefficient space
$\Lambda(\delta,-r,s)$. Now suppose that $\delta'$ is obtained from
$\delta$ be removing one of the terms $\delta_j$ from $\delta$. Provided
$\delta_j$ was not used to segment $\delta$, the space
$\Lambda(\delta',-r,s)$ will contain $\Lambda(\delta,-r,s)$.
One could obtain a complex by not allowing deletion to occur only at the
faces $\delta_j$ used to segment $\delta$, or rather setting the result of
deleting such a face to be zero. The resulting complex and its homology
will not however properly speaking be an invariant of the polytope
$\Delta$. Rather, for each choice of a segmenting flag $\delta_s$ one
will have a complex, and any invariant of $\Delta$ so defined will simply
be the direct sum of these flag contributions.
The purpose of this section is to define a map from $\Lambda(\delta)$ to
$\Lambda(\delta')$, where $\delta'$ is obtained from $\delta$ by the
removal of a segmenting face $\delta_i$. This will have the effect of
producing a single global complex, that will in general be indecomposable.
It can be thought of as a gluing together of the local complexes of the
previous paragraph.
Here is an example. Suppose one has subspaces $V_1 \subset V_2 \subset
V$, where $V_i$ is the span $\langle\delta_i\rangle$ of vectors lying on
the face $\delta_i$. The segmentation of $V$ due to $V_1$ will be
compared to that due to $V_2$. In both cases, the raw materials are
firstly linear functions $\alpha$ defined on $V_i$, and secondly linear
functions $\beta$ vanishing on $V_i$ (and defined on the whole of $V$).
Now suppose that $\alpha_2$ is defined on $V_2$. Because $V_1\subset
V_2$, the linear function $\alpha_2$ can be restricted to give $\alpha_1$
defined on $V_1$. This is straightforward. Now suppose that $\beta_2$
vanishes on $V_2$ (and is defined on $V$). Again because $V_1 \subset
V_2$, the linear function $\beta_2$ also vanishes on $V_1$. Thus there is
a linear map (restriction of range $\otimes$ relaxation of condition)
\begin{equation}
\label{eqn:rrmap}
\bigwedge \nolimits ^r V_2^*
\otimes \bigwedge \nolimits ^s V_2^\perp
\to
\bigwedge \nolimits ^r V_1^*
\otimes \bigwedge \nolimits ^s V_1^\perp
\end{equation}
between the basic spaces associated to the two segmentations.
This map has an interesting relation to the conditions used to define the
coefficient spaces $\Lambda(\delta)$. When the condition
\[
s > \codim V_i - s
\]
holds, the coefficient space due to the flag $(\delta_i)$ will be one of
the above tensor products. When the condition fails, the coefficient
space is zero.
Because $V_1$ is a subspace of $V_2$, the condition for $(\delta_1)$ is
more onerous than that for $(\delta_2)$, and so the map (\ref{eqn:rrmap})
is going in the wrong direction, to map the one coefficient space to the
other.
Regarding conditions however, the situation is different. The trick is to
think of the conditions that define the coefficient spaces
$\Lambda(\delta_i)$ to be themselves subspaces of an exterior algebra
(or more exactly a tensor product of such). If $U$ is a vector space of
dimension $l+m$, then each subspace of $\bigwedge^l U$ determines a
subspace of $\bigwedge^m U$, and vice versa. This is via the
nondegenerate pairing
\[
\bigwedge \nolimits ^l U \otimes \bigwedge \nolimits ^m U
\to \bigwedge \nolimits ^{l+m} U
\]
provided by the exterior algebra. By a slight abuse of language, this
will be called duality. (The value space $\bigwedge^{l+m}U$ is has
dimension one, but is not naturally isomorphic to $\bfR$.)
It is now necessary to formulate the conditions using this new point of
view. Suppose $U\subset V$ is a subspace, and $W\subseteq \bigwedge^r
V^*$ is spanned by
\[
\alpha_1 \wedge \dots \wedge \alpha_ l
\quad \wedge \quad
\beta_1 \wedge \dots \wedge \beta_ m
\]
where the $\alpha_i$ are to vanish on $U$. Consider the space
$W'\subseteq\bigwedge^{r'}V^*$ spanned by
\[
\alpha_1 \wedge \dots \wedge \alpha_{l'}
\quad \wedge \quad
\beta_1 \wedge \dots \wedge \beta_{m'}
\]
where $r+r'=\dim V$, $l+l'= \codim U + 1 $, and as before the $\alpha_i$
vanish on $U$; this is the subspace of the complementary component of the
exterior algebra, determined mutually by $W$.
The $\beta_i$ are arbitrary, and so by a further application of duality,
they can be dropped from the definition of the condition space. Thus,
take the span of
\[
\alpha_1 \wedge \dots \wedge \alpha_{l'}
\]
for $l+l'=\codim U + 1$, $\alpha_i$ vanishing on $U$, as the condition
space for $W$.
We now return to the change of segmentation map (\ref{eqn:rrmap}). As
already noted, the conditions for $V_2$ are less onerous that those for
$V_1$. The condition space for $V_2$ is either zero (no conditions) or
$\bigwedge^{\codim V_2} V_2^\perp$ (zero is the only solution to the
conditions), and similarly for $V_1$. The map (\ref{eqn:rrmap}) will in
either case respect these conditions. (To make sense of this statement,
one should think of the conditions as being an ideal in the exterior
algebra, and then the image under (\ref{eqn:rrmap}) of the one ideal is
contained in the other.)
The goal of this section is now in sight. It is the conditions that are
respected by the natural map that is due to change of segmentation, not
the coefficient spaces defined by the conditions. Interpret each
coefficient vector $v_\delta$ as a linear function, taking values in a
tensor product of top-degree exterior products, that vanishes on the
condition space.
Recall that the example $V_1\subset V_2\subset V$ gives rise to a natural
map
\[
\bigwedge \nolimits ^r V_2^*
\otimes \bigwedge \nolimits ^s V_2^\perp
\to
\bigwedge \nolimits ^r V_1^*
\otimes \bigwedge \nolimits ^s V_1^\perp
\]
which respects conditions. Now let $v_\delta$ be a coefficient vector on
$V_1$, interpreted as a linear function on the range of the above map.
Using the map, this linear function on the range can be pulled back to
give a linear function on the domain. Because $v_\delta$ vanishes on the
$V_1$ condition space, the pull-back vanishes on the $V_2$ condition
space. By duality, the pull-back linear function is associated to a
unique coefficient vector $v'_\delta$ for the $V_2$ segmentation. This is
an example of a \emph{change of segmentation component} of the boundary
map. Note that it takes a $V_1$-segmentation coefficient to a $V_2$ such.
In other words, under boundary the segmenting term(s) may move rightwards,
or in other words, increase in dimension.
To finish, there are some details to be taken care of. First, although
the one-dimensional value spaces for $V_1$ and $V_2$ are not the same,
they are naturally isomorphic. This is good enough. Secondly, because
the map (\ref{eqn:rrmap}) is natural, the argument that shows $d^2=0$
works just as before. Thirdly, the above argument has been applied only
to the conditions due to the segmenting faces. The reader may wish to
show that it works also for the conditions, as imposed in the previous
section. (An alternative way of defining the map and checking the
statements made will be outlined, as part of the discussion of bar
diagrams.)
The final matter concerns the degree of the coefficient $v_\delta$. The
map (\ref{eqn:rrmap}) preserves the degree of (tensor products of)
exterior powers. It induces the map on coefficients via duality and
pull-back, and so as (tensor products of) exterior powers the coefficients
$v_\delta$ on $V_1$ and $v_\delta'$ on $V_2$ will have different degrees.
However, they will by definition have the same \emph{co-degree}, by which
is meant the amount by which they fall short of being of top degree. For
this reason, in the rest of the paper the coefficient spaces will be
indexed by co-degree. For this the notation $\Lambda(\delta,r,s)$ will be
used. Here, $\delta$ stands for a flag that has been broken into $l$
segments by segmentating data $s$, and $r=(r_1,r_1,\ldots,r_l)$ provides a
\emph{co-degree} for each segment. The same flag can perhaps be segmented
in many ways, evn if $l$ is fixed. The boundary map preserves co-degree.
\section{Local-global homology}
The main definition of this paper can now be given. Recall that if
$\delta$ is a flag on the convex polytope $\Delta$, and $s$ breaks
$\delta$ (and $V$) into $l$ segments, and if a multi-component co-degree
$r=(r_1,\dots,r_l)$ is given, then from all this a coefficient space
$\Lambda(\delta,r,s)$ has been defined. Recall also that if $\delta'$ is
obtained from $\delta$ by removal of the $i$-th term from $\delta$, then
there is a natural map
\[
\partial_i : \Lambda(\delta,r,s) \to \Lambda(\delta',r,s')
\]
such that the usual definition of $\bound$ will produce a boundary map on
\[
A_r = (0 \to A_{r,n} \to A_{r,n-1} \to
\dots \to A_{r,1} \to A_{r,0} \to 0 )
\]
where $A_{r,i}$ is the direct sum of the $\Lambda(\delta_r)$ for
$\ord \delta=i$.
(The following detail is important. When $\delta_i$ is removed from
$\delta$ to obtain $\delta'$, one might as a result have to change the
segmentation $s$. This happens when $\delta_i$ is used to segment
$\delta$. In this case the next term $\delta_{i+1}$ is used to segment
$\delta'$. If $\delta_{i+1}$ is already used by $s$ to segment $\delta$,
or does not exist because $\delta_i$ is the last term in $\delta$, then
this component of the boundary map is treated as zero. In this way, one
obtains either a satisfactory $s'$, or the zero map.)
Thus, for each co-degree $r=(r_1,\ldots,r_l)$, a complex $A_r$ has been
defined. The boundary map $\bound$ of $A_r$ may cause the terms of the
segmenting flag $\delta_s$ to move rightwards. This observation leads to
the following. One can define subcomplexes of $A_r$ by placing conditions
of the segmentation subflags that are to be used. These conditions must
of course allow the movement to the right of the segmentation terms.
One way to formulate this is to introduce a multi-dimension $s=(s_1 <
\dots < s_{l-1})$ and allow only those segmentations $\delta$ to be used,
for which the dimension $d=(d_1<\dots<d_l)$ is term by term at least as
large as $s$. In this way one obtains for each $r$ and $s$ a complex
$A_{r,s}$. Of course, if $s$ is too great compared to $r$, then the
complex will be zero. The complex $A_{r,s}$ is the $A_w$ mentioned in
\S1.)
We now define the $(r,s)$ \emph{homology space} $H_{r,s}\Delta$ of
$\Delta$ to be the homology of $A_{r,s}$ at the level where $\ord\delta=i$
is equal to the total degree of $\Lambda(\delta,r,s)$. In other words, it
is the homology at $A_{r,s;i}$, where $ i + r_1 + \dots + r_l = n$, for
$r$ gives the co-degree. If $A_{r,s}$ exactly computes $H_{r,s}\Delta$
then its dimension $h_{r,s}\Delta$ is of course a linear function of the
flag vector. The next section clarifies this.
As mentioned in the introduction, these spaces correspond to the
local-global intersection homology spaces of $\PDelta$, introduced in
\cite{bib.JF.LGIH}.
\section{Decorated bar diagrams}
The previous discussion made no use of coordinates, and does not give the
dimension of the various $\Lambda(\delta, r, s)$ spaces involved in the
construction of local-global homology. This section provides another
approach. The contribution made by a flag $\delta$ to the homology
$H_{r,s}\Delta$, and the maps between these contributions, can be
described using coordinates, via the use of decorated bar diagrams. (Part
of the theory of bar diagrams was first published in the survey
paper~\cite{bib.MB.TASI}.)
Suppose, for example, that the dimension $n$ of $\Delta$ is eleven, and
that $\delta$ is a flag of dimension $d=(3 < 5 < 9 < 11)$. The
(undecorated) \emph{bar diagram}
\[
\bardiagram{...|..|....|..}
\]
expresses this situation. There are eleven dots, and a bar `\bardiagram{|}'
is placed after the $d_i$-th dots. Each bar represents a term $\delta_i$ of
$\delta$. Now choose a basis $e_1, \dots , e_n$ of $V$ such that for each
$i$, the initial sequence $e_1, \dots , e_j$ (with $j=d_i$) is a basis for
the span $\langle\delta_i\rangle$ of the vectors lying on the $i$-th face.
Finally, let each dot represent not $e_i$ but the corresponding linear
function $\alpha_i$, that vanishes on each $e_j$ except $e_i$. Call this
a \emph{system of coordinates} for $V$, that is \emph{subordinate} to
$\delta$.
By construction, each dot represent a linear function that vanishes on the
faces $\delta_i$ of $\delta$ represented by the bars that lie to its left.
So that we can speak more concisely, we shall think of the dots and bars
as actually being the linear functions and terms of the flag respectively.
Each subset of the dots represents an element of the exterior algebra
generated by the linear functions on $V$. To represent the selection of a
subset, promote the chosen dots `\bardiagram{.}' into circles `\bardiagram{o}'.
Thus
\begin{equation}
\label{eqn:dbd}
\bardiagram{..o|o.|o.o.|oo}
\end{equation}
represents (or more concisely is) a degree six element of the exterior
algebra. It is an example of a \emph{decorated bar diagram}.
(Each circle represents an element in the homology of the torus, that is
the generic or central fibre of the moment map. As the fibre is moved
towards a face that lies to the left of the circle, so the circle shrinks
to a point. Thus, circles represent $1$-cycles with specified vanishing
properties. Similarly, the promotion of say $6$ dots to circles produces
a diagram that represent a $6$-cycle in the generic fibre, with specific
shrinking properties, as the fibre is moved to the faces of the flag. This
will be important, in the topological interpretation of exact homology.)
The exterior form (\ref{eqn:dbd}) has certain vanishing properties, with
respect to the faces of $\delta$. The condition of \S4 is equivalent to
the following: That between each bar and the right hand end of the
diagram, there should be strictly more `\bardiagram{o}'s than `\bardiagram{.}'s.
Our example satisfies this condition, and so is an \emph{admissable}
decorated bar diagram.
Now fix an unsegmented degree $r=(r_1)$ and define the coefficient space
$\Lambda(\delta,r)$ to be the span of the admissable $r$-circle bar
diagrams (considered as exterior forms). (Because there is no
segmentation, $s$ is trivial, and will be omitted.) For the
$\Lambda(\delta,r)$ to come together to produce a complex, the following
must hold. First, $\Lambda(\delta,r)$ as a subspace of the degree~$r$
forms should not depend on the choice of a basis subordinate to $\delta$.
Second, the boundary map should not depend on the basis (or in other words
should be covariant for such change). Third, when a bar is removed from an
admissable diagram, the result should also be admissable. The last two
conditions are immediately seen to be true.
There are two ways to see that the first requirement (that
$\Lambda(\delta,r)$ not move when the subordinate basis is changed) is
true. One method is to show that the span $\Lambda(\delta,s)$ of the
admissable diagrams is the solution set to a problem that can be
formulated without recourse to use of a basis. This is the approach taken
in \S\S3--7. The other method is to show directly that
$\Lambda(\delta,r)$ does not move, under change of subordinate basis.
Any change of subordinate basis can be obtained as a result of applying
the following moves. First, one can multiply basis elements by non-zero
scalars. Second, one can permute the basis elements (dots and circles),
provided so doing does not cause a dot or circle to pass over a bar.
Thirdly, one can increase a basis element by some multiple of another
basis element, that lies to its right.
It is clear that applying either of the first two moves to the basis will
not change $\Lambda(\delta,r)$. Regarding the third move, if a diagram
such as (\ref{eqn:dbd}) is admissable, then the result of moving one or
more `\bardiagram{o}'s to the right, perhaps over bars, will also be
admissable. This is obvious, from the nature of the conditions defining
admissability. The third type of move makes such changes. Thus, the span
$\Lambda(\delta,r)$ of the admissable decorated diagrams does not move.
It has now been shown that the admissable bar diagrams, such as
(\ref{eqn:dbd}), define coefficient spaces $\Lambda(\delta,r)$ that can
be assembled to produce a complex. It is left to the reader, to check that
it is exactly the same complex, as was defined in \S4.
The remainder of this section is devoted to the description of the
segmented form of the above construction. As before, let
\[
\bardiagram{...|..|....|..}
\]
denote a flag of dimension $d=(3 < 5 < 9 < 11)$, and now choose some of
the bars, say just the second, to segment the diagram. The result
is
\[
\bardiagram{...|..} \qquad \bardiagram{|....|..}
\]
or more concisely
\[
\bardiagram{...|..!....|..}
\]
where the promotion of a `\bardiagram{|}' to a `\bardiagram{!}' indicates that is
is being used for segmentation. Note that each segmenting face is also
used as the first face in the following segment.
As before, one can promote some of the `\bardiagram{.}'s to `\bardiagram{o}'s to
obtain a \emph{decorated bar diagram}, which will be \emph{admissable} if
between any bar (either `\bardiagram{|}' or `\bardiagram{!}') and the end of the
segment, there are more `\bardiagram{o}'s than `\bardiagram{.}'s.
To be able to assemble the span of the admissable diagrams into a complex,
certain requirements must be met. They have already been formulated. The
first is that the span should not depend on the choice of a subordinate
basis. Scalar and permutation moves on the basis clearly leave the span
unchanged, as in the single segment case. Adding to one basis vector
another, lying to its right, has no effect on the span, provided the
second lies in the same segment as the first.
Now consider the situation, where one of the basis linear forms is
changed, by adding to it another basis linear form, as before lying to the
right, but this time in a different segment. The way to have this have no
effect on the span is to have the dots and circles in a segmented diagram
represent not linear functions on $V$, but rather linear functions on the
(span of vectors lying on) the face at the right end of the segment (which
is $V$ for the last segment).
Thus, by having each decorated diagram (possibly segmented) represent an
element in a tensor product of exterior algebras, the span of all the
admissable diagrams becomes a space that does not move, when the
subordinate basis is changed. This agrees with \S5.
For these spaces to be assembled into a complex, the boundary map must be
well defined. As in the single segment situation, all is well when a
non-segmenting term (a `\bardiagram{|}' rather than a `\bardiagram{!}') is removed
from a decorated bar diagram.
The removal of a `\bardiagram{!}' terms has a more subtle effect, for the
segmentation will have to change. Whatever rule is used, it must respect
admissability of decorated diagrams, and it must respect change of
subordinate basis.
The rule, which we state without prior justification, is this: whenever
something such as
\[
\bardiagram{!ooooo|}
\]
occurs in a decorated diagram, one can obtain a component of the boundary
by replacing it with
\[
\bardiagram{ooooo!}
\]
while the result of removing any other `\bardiagram{!}' terms is zero.
(The number of `\bardiagram{o}'s is not relevant, that one has
`\bardiagram{!}' followed by some circles, and then a `\bardiagram{|}'
is.)
It is clear that this rule will, from admissable diagrams, generate only
admissable diagrams. This is because it can but only cause some extra
`\bardiagram{o}'s to appear, whenever the `\bardiagram{.}' and `\bardiagram{o}' counts
are to be compared. The same is of course not true, for the replacement
of `\bardiagram{!.o.|}' by `\bardiagram{.o.!}' and similar situations.
The next task is to show that this part of the boundary respects change of
subordinate basis. Given a fragment such as `\bardiagram{!.o.|}' that
contributes zero to this part of the boundary, whatever change of basis is
made, the contribution will still be zero. Consider now a fragment such
as `\bardiagram{!ooo|}'. Scalar and permutation moves will have no effect
on the boundary. This is obvious. The only way that adding something
that lies to the right can have any effect is if that something lies to
the right of the whole fragment `\bardiagram{!ooo|}'. This is because the
exterior algebra is antisymmetric. Consider now the boundary contribution
`\bardiagram{ooo!}'. As already described, the `\bardiagram{!}' induces a
restriction of linear functions, and so the just considered change of
basis will after all have no effect on the boundary map.
This concludes the presentation of the complex $A$ via decorated bar
diagrams. The details of co-degree (number of `\bardiagram{.}'s in each
segment) and so forth are as in the earlier exposition~(\S5--7). It is left
to the reader to verify that the two approaches lead to exactly the same
family $A_{r,s}$ of complexes.
\section{Summary and conclusions}
The flag vector of a convex polytope satisfies subtle linear inequalities,
and also non-linear inequalities, that are at present not known. Exact
homology seems to be the only general method available to us, to prove
such results. The difficulty with purely combinatorial means is firstly
that there are no natural maps, other than the deletion operators, between
the flags on a polytope, and secondly that the convexity of the polytope
has to be allowed to enter into the discussion in a significant way. This
said, very few results relating to exact homology are known at present.
For the usual middle perversity intersection homology theory, the three
approachs mentioned in the introduction are known to have significant
areas of agreement. The recursive formula in \cite{bib.RS.GHV} for $h\PDelta$
can be unwound to express each Betti number as a sum of contributions due
to flags. These contributions are exactly the same as those implicit in
the definitions of \S4. The method of decorated bar diagrams provides a
reformulation of the recursive formula for $h\PDelta$, where now one
simply counts the number of valid diagrams. As was indicated in \S8, once
the valid diagrams are known, it is not then hard to determine what the
corresponding space of coefficients should be. Thus, the derivation from
\cite{bib.RS.GHV} of \S4 is without difficulty. Finally, via the
identification of the exterior algebra $\bigwedge^\bullet V^*$ with the
homology of the generic fibre of the moment map, one can interpret the
complex in \S4 in terms of the construction of intersection homology
cycles on $\PDelta$, and relations between them. This last will be
presented elsewhere.
The important property of intersection homology (for middle perversity
only) is that it seems to produce exact homology groups. That there are
formula for such Betti numbers, and moreover as somewhat geometric
alternating sums, supports this view. The local-global construction
defined in \S\S5--7 can, via the moment map, be translated into a class of
cycles on $\PDelta$, and relations between them. These cycles and
relations have special properties with respect to the strata of $\PDelta$.
In this way one can translate the definition of local-global homology in
this paper into a topological one. Indeed, if intersection homology were
not already known, it could have been discovered via its similarity to the
$H\Delta$ theory presented here.
There is a subtlety connected to the concept of a linear function of the
flag vector. The flag vectors of convex polytopes span a proper subspace
of the space of all possible flag vectors. Now, a linear function on a
subspace is not the same as a linear function on the whole space. The
latter contains more information. The constructions of this paper provide
linear functions of arbitrary (not necessarily polytope) flag vectors.
As noted, they agree with the formulae (8)--(10) of~\cite{bib.JF.LGIH}.
The construction presented here of $H\Delta$ is probably not the only one.
In particular, it ought to be possible to define a theory, with the same
expected Betti numbers, but where $H\Delta$ is built out of the $H\delta$,
for all proper faces $\delta\subset\Delta$, together with perhaps a little
gluing information. This can certainly be done in the simple case, where
it corresponds to the natural formula in that context for $h\Delta$ in
terms of the face vector. For general polytopes, such a theory would
correspond to a linear function on arbitrary flag vectors, which agrees on
polytope flag vectors with the $h$-vector presented in this paper. Such a
theory may be part of a geometric proof of exactness of homology.
One cannot, of course, prove that which is not true; and exact homology
cannot hold for a complex if the expected Betti number turns out to be
negative for some special polytope. Bayer (personal communication) has an
example of a 5~dimensional polytope (the bipyramid on the cylinder on a
3-simplex) where this in fact happens. This is discussed further in
\cite{bib.JF.LGIH}. All this indicates that there are as yet unrealised
subtleties in the concept and proof of exact calculation.
The central definitions of this paper, namely of the coefficient spaces
$\Lambda(\delta,r,s)$ and the maps between them, use only linear algebra
and the deletion operator on flags. Put another way, the largest part of
this paper has been the study of the flag
\[
V_1 \subset V_2 \subset \dots \subset V_m \subset V
\]
of vector spaces associated to a flag $\delta$ of faces on $\Delta$, and
the vector spaces that can be defined from it. Given the requirements of
exact homology, there is little else available that one could study.
However, these definitions are very closely linked to the usual
intersection homology theory and also its local-global variant. These
connections indicate that there may be unremarked subtleties in the linear
algebra of a flag, and undiscovered simplicity in intersection homology. A
better understanding of the intersection homology of Schubert varieties
would be very useful.
Finally, as mentioned in the Introduction, one could wish for a similar
theory that applies to $i$-graphs, and similar combinatorial objects.
This will probably require a different sort of linear algebra.
|
1997-10-10T23:53:30 | 9710 | alg-geom/9710014 | en | https://arxiv.org/abs/alg-geom/9710014 | [
"alg-geom",
"math.AG"
] | alg-geom/9710014 | Kai Behrend | Kai Behrend | The product formula for Gromov-Witten invariants | LaTeX | null | null | null | null | We prove that the system of Gromov-Witten invariants of the product of two
varieties is equal to the tensor product of the systems of Gromov-Witten
invariants of the two factors.
| [
{
"version": "v1",
"created": "Fri, 10 Oct 1997 21:53:29 GMT"
}
] | 2007-05-23T00:00:00 | [
[
"Behrend",
"Kai",
""
]
] | alg-geom | \subsection{Introduction}
\newcommand{\mbox{$\tilde{\GG}_s(V)$}}{\mbox{$\tilde{\GG}_s(V)$}}
\newcommand{\mbox{$\tilde{\GG}_s(W)$}}{\mbox{$\tilde{\GG}_s(W)$}}
\newcommand{\mbox{$\tilde{\GG}_s(0)$}}{\mbox{$\tilde{\GG}_s(0)$}}
\newcommand{\mbox{$\tilde{\GG}_s(V\times W)$}}{\mbox{$\tilde{\GG}_s(V\times W)$}}
Let $V$ and $W$ be smooth and projective varieties over the field
$k$. In this article we treat the question how to express the
Gromov-Witten invariants of $V\times W$ in terms of the Gromov-Witten
invariants of $V$ and $W$.
On an intuitive level, the answer is quite obvious. For example,
assume $V=W={\Bbb P}^1$ and let us ask the question how many curves in
${\Bbb P}^1\times{\Bbb P}^1$ of genus $g$ and bidegree $(d_1,d_2)$ pass through
$n=2(d_1+d_2)+g-1$ given points $P_1,\ldots,P_n$ of ${\Bbb P}^1\times
{\Bbb P}^1$ in general position. The answer is given by the Gromov-Witten
invariant $$I^{{\Bbb P}^1\times{\Bbb P}^1}_{g,n}(d_1,d_2)(\gamma^{\otimes n}),$$
where $\gamma\in H^4({\Bbb P}^1\times{\Bbb P}^1,{\Bbb Q})$ is the cohomology class
Poincar\'e dual to a point.
We rephrase the question by asking how many triples
$(C,x_1,\ldots,x_n,f)$, where $C$ is a curve of genus $g$,
$x_1,\ldots,x_n$ are marked points on $C$ and $f:C\to{\Bbb P}^1\times{\Bbb P}^1$
is a morphism of bidegree $(d_1,d_2)$ exist (up to isomorphism) which
satisfy $f(x_i)=P_i$, for all $i=1,\ldots,n$. Now a morphism
$f:C\to{\Bbb P}^1\times{\Bbb P}^1$ of bidegree $(d_1,d_2)$ is given by two
morphisms $f_1:C\to{\Bbb P}^1$ and $f_2:C\to{\Bbb P}^1$ of degrees $d_1$ and
$d_2$, respectively. The requirement that $f(x_i)=P_i$ translates into
$f_1(x_i)=Q_i$ and $f_2(x_i)=R_i$, if we write the components of $P_i$
as $P_i=(Q_i,R_i)$. The family of all marked curves
$(C,x_1,\ldots,x_n)$ admitting such an $f_1$ is some cycle, say
$\Gamma_1$, in $\overline{M}_{g,n}$. Of course, the family of all curves
$(C,x_1,\ldots,x_2)$ admitting an $f_2$ as above is another cycle
$\Gamma_2$ in $\overline{M}_{g,n}$ and the family of all
$(C,x_1,\ldots,x_n)$ admitting an $f_1$ and an $f_2$ is the
intersection $\Gamma_1\cdot\Gamma_2$. So the Gromov-Witten number we
are interested in is
\[I^{{\Bbb P}^1\times{\Bbb P}^1}_{g,n}(d_1,d_2)(\gamma^{\otimes n}) =
\Gamma_1\cdot\Gamma_2.\]
In fact, the dual cohomology classes of $\Gamma_1$ and $\Gamma_2$ are
Gromov-Witten invariants themselves, namely
$I^{{\Bbb P}^1}_{g,n}(d_1)(\tilde\gamma^{\otimes n})$ and
$I^{{\Bbb P}^1}_{g,n}(d_2)(\tilde\gamma^{\otimes n})$, where
$\tilde\gamma\in H^2({\Bbb P}^1,{\Bbb Q})$ is the cohomology class dual to a
point. Thus we have
\[I^{{\Bbb P}^1\times{\Bbb P}^1}_{g,n}(d_1,d_2)(\gamma^{\otimes n}) =
I^{{\Bbb P}^1}_{g,n}(d_1)(\tilde\gamma^{\otimes n}) \cup
I^{{\Bbb P}^1}_{g,n}(d_2)(\tilde\gamma^{\otimes n}) \] in
$H^{\ast}(\overline{M}_{g,n}),{\Bbb Q})$. This is the simplest instance of the
product formula, which we shall prove in this article. (Note that we
have identified, as usual, top degree cohomology classes on $\overline
M_{g,n}$ with their integrals over the fundamental cycle $[\overline
M_{g,n}]$.)
We get a more general statement by letting $V$ and $W$ be arbitrary
smooth projective varieties over $k$. We fix cohomology classes
$\gamma_1,\ldots,\gamma_n\in H^{\ast}(V)$ and
$\epsilon_1,\ldots,\epsilon_n\in H^{\ast}(W)$, which we assume to be
homogeneous, for simplicity. Then the product formula says that
\begin{eqnarray}\label{bpf}
\lefteqn{
I^{V\times W}_{g,n}(\beta)
(\gamma_1\otimes\epsilon_1\otimes\ldots
\otimes\gamma_n\otimes\epsilon_n)}\nonumber\\
& = & (-1)^s
I^{V}_{g,n}(\beta_V)(\gamma_1\otimes\ldots \otimes\gamma_n) \cup
I^{W}_{g,n}(\beta_W)(\epsilon_1\otimes\ldots \otimes\epsilon_n)
\end{eqnarray}
in $H^{\ast}(\overline{M}_{g,n},{\Bbb Q})$. Here $\beta\in H_2(V\times W)^+$ and
$\beta_V={p_V}_{\ast}\beta$, $\beta_W={p_W}_{\ast}\beta$, where $p_V$ and
$p_W$ are the projections onto the factors of $V\times W$. The sign
is given by
\[s=\sum_{i>j}\deg\gamma_i\deg\epsilon_j.\]
This formula is already stated in \cite{KM} as a property expected of
Gromov-Witten invariants. In the case of $g=0$ and $V$ and $W$ (and
hence $V\times W$) convex, it is not difficult to prove, once the
properties of stacks of stable maps are established, as they are, for
example, in \cite{BM}. Essentially, the above intuitive argument can
then be translated into a rigorous proof. In the general case, the
enumerative meaning of Gromov-Witten invariants is much less clear,
since one has to use `virtual' fundamental classes to define
them. (This is done in \cite{BF} and \cite{gwi} or \cite{litian}.) So
the theorem follows from properties of virtual fundamental
classes. This is what we prove in the present paper.
Formula~(\ref{bpf}) has been used by various authors to understand the
quantum cohomology of a product. (See \cite{KMK}, \cite{KMZ} and
\cite{Kauf}.) By Formula~(\ref{bpf}), the codimension zero
Gromov-Witten invariants (ie.\ those that are numbers, like
$I^{{\Bbb P}^1\times{\Bbb P}^1}_{g,n}(d_1,d_2)(\gamma^{\otimes n})$, above) of a
product are determined by the Gromov-Witten invariants of higher
codimension of the factors and by the intersection theory of
$\overline{M}_{g,n}$.
To explain the treatment in this article, let us
reformulate~(\ref{bpf}) by saying that
\[\begin{array}{ccc}
h(V\times W)^{\otimes n} & \stackrel{I^{V\times W}_{g,n}(\beta)}
{\longrightarrow} & h(\overline M_{g,n}) \\
\parallel & & \rdiagup{\Delta^{\ast}} \\
h(V)^{\otimes n}\otimes h(W)^{\otimes n} &
\stackrel{I^{V}_{g,n}(\beta_V)\otimes
I^{W}_{g,n}(\beta_W)}{\longrightarrow} &
h(\overline M_{g,n})\otimes h(\overline M_{g,n}),
\end{array}\]
where $\Delta:\overline{M}_{g,n}\to \overline{M}_{g,n}\times\overline{M}_{g,n}$ is the
diagonal, commutes. Here we have passed to the motivic Gromov-Witten
invariants. These are homomorphisms between DMC-motives. (These are
like Chow motives, except that they are made from smooth and proper
Deligne-Mumford stacks, instead of varieties. For details see
\cite{BM}, Section~8.)
To summarize all of their functorial properties, Gromov-Witten
invariants where defined in \cite{BM} as natural transformations
between the functors $h(V)^{\otimes S}$ and $h(\overline M)$, which are
functor from a certain graph category $\mbox{$\tilde{\GG}_s(V)$}_{\mbox{\tiny cart}}$ to the category of
graded DMC-motives. To explain, let us start by reviewing some graph
theory. The category $\tilde{\GG}_s=\mbox{$\tilde{\GG}_s(0)$}$ is the category of stable
modular graphs (graphs whose vertices are labeled with genuses; see
\cite{BM}, Definition~1.5) with so called extended isogenies as
morphisms. An {\em extended isogeny }is either a morphism gluing two
tails to an edge, or it is a proper isogeny (or a composition of the
two). An {\em isogeny }is a morphism which contracts various edges or
tails or both. (The name isogeny comes from the fact that such
morphisms do not affect the genus of the components of the graphs
involved.) For the definition of composition of extended isogenies,
see \cite{BM}, Page~36.
The category $\mbox{$\tilde{\GG}_s(V)$}_{\mbox{\tiny cart}}$ is called the {\em cartesian extended
isogeny category }over $V$. The most important objects of $\mbox{$\tilde{\GG}_s(V)$}_{\mbox{\tiny cart}}$
are pairs $(\tau,(\beta_i)_{i\in I})$, where $\tau$ is a stable
modular graph and $(\beta_i)_{i\in I}$ is a family of $H_2(V)^+$
markings on $\tau$. This means that each $\beta_i$ is a function
$\beta_i:V\t\to H_2(V)^+$, where $V\t$ is the set of vertices of
$\tau$. (The indexing set $I$ is finite.) The fundamental property of
$\mbox{$\tilde{\GG}_s(V)$}_{\mbox{\tiny cart}}$ is that it is {\em fibered } over $\tilde{\GG}_s$. This
means that there is a functor $\mbox{$\tilde{\GG}_s(V)$}_{\mbox{\tiny cart}}\to\tilde{\GG}_s$ (projection
onto the first component) and that given an object
$(\tau,(\beta_i)_{i\in I})$ of $\mbox{$\tilde{\GG}_s(V)$}_{\mbox{\tiny cart}}$ and a morphism
$\phi:\sigma\to\tau$ there exists, up to isomorphism, a unique object
$(\sigma,(\gamma_j)_{j\in J})$ of $\mbox{$\tilde{\GG}_s(V)$}_{\mbox{\tiny cart}}$ together with a
morphism $\Phi:(\sigma,(\gamma_j)_{j\in J}):\to(\tau,(\beta_i)_{i\in
I})$ covering $\phi:\sigma\to\tau$. When constructing $\Phi$, the
basic non-obvious case is that where $\phi$ contracts a non-looping
edge of $\sigma$ and $I$ has only one element. Then we have the graph
$\tau$ with an $H_2(V)^+$-marking $\beta$ and there are two vertices
$v_1$, $v_2$ of $\sigma$ corresponding to one vertex $w$ of
$\tau$. Then $(\sigma,(\gamma_j)_{j\in J})$ is defined such that $J$
counts the ways to write $\beta(w)=\beta_1+\beta_2$ in $H_2(V)^+$ and
$\gamma_j$ assigns $\beta_1$ to $v_1$ and $\beta_2$ to $v_2$, and
otherwise does not differ from $\beta$.
Things get more complicated, if one also considers the less important
objects of $\mbox{$\tilde{\GG}_s(V)$}_{\mbox{\tiny cart}}$. These are of the form $(\tau,(\tau_i)_{i\in
I})$, where, as above, $\tau$ is a stable modular graph, but now each
$\tau_i$ is a stable $H_2(V)^+$-marked graph (as opposed to an
$H_2(V)^+$-marked stable graph), together with a {\em stabilizing
morphism }$\tau_i\to\tau$. For the complete picture, see \cite{BM},
Definition~5.9.
On objects, the morphisms $h(V)^{\otimes S}$ and $h(\overline M)$ from
$\mbox{$\tilde{\GG}_s(V)$}_{\mbox{\tiny cart}}$ to $(\mbox{graded DMC-motives})$ are defined as follows:
For an object $(\tau,(\beta_i)_{i\in I})$ of $\mbox{$\tilde{\GG}_s(V)$}_{\mbox{\tiny cart}}$ we have
\[h(V)^{\otimes S}(\tau,(\beta_i)_{i\in I})=h(V)^{\otimes S\t},\]
where $S\t$ is the set of tails of $\tau$ and
\[h(\overline M)(\tau,(\beta_i)_{i\in I})=h(\overline M(\tau)),\]
where $$\overline M(\tau)=\prod_{v\in V\t}\overline{M}_{g(v),F\t(v)}$$ and
$F\t(v)$ is the set of flags meeting the vertex $v$ of $\tau$. So both
of these functors only depend on the first component $\tau$ of
$(\tau,(\beta_i)_{i\in I})$. For the definition of these functors on
morphisms, see \cite{BM}, Section~9. Note that $h(V)^{\otimes S}$
actually comes with a twist (ie.\ a degree shift) $\chi\dim V$. This
we shall ignore here, to shorten notation and since nothing
interesting happens to it, anyway.
The {\em Gromov-Witten transformation } of $V$ is now defined as a
natural transformation
\[I^V:h(V)^{\otimes S}\longrightarrow h(\overline M)\]
of functors from $\mbox{$\tilde{\GG}_s(V)$}_{\mbox{\tiny cart}}$ to $(\mbox{graded DMC-motives})$. In
this paper (Theorem~\ref{st}), we shall prove that
\[I^{V\times W}=I^V\cup I^W,\]
where $I^V\cup I^W$ is defined as $\Delta^{\ast}(I^V\otimes I^W)$.
Since Gromov-Witten invariants are defined in terms of virtual
fundamental classes on moduli stacks of stable maps, this theorem
follows from a certain compatibility between virtual fundamental
classes. This is our main result (Theorem~\ref{pt}) and takes up most
of this paper.
\subsection{Virtual Fundamental Classes}
Fix a ground field $k$. For a smooth projective $k$-variety $V$ let
$\mbox{$\tilde{\GG}_s(V)$}$ be the category of extended isogenies of stable
$H_2(V)^+$-graphs bounded by the characteristic of $k$ (see \cite{BM},
Definition~5.6 and Example~II following Definition~5.11). Let
$J(V,\tau)\in A_{\dim(V,\tau)}(\overline{M}(V,\tau))$, for $\tau\in\mathop{\rm ob}\mbox{$\tilde{\GG}_s(V)$}$,
be the `virtual fundamental class', or orientation (\cite{BM},
Definition~7.1) of $\overline{M}$ over $\mbox{$\tilde{\GG}_s(V)$}$ constructed in \cite{gwi},
Theorem~6, using the techniques from \cite{BF}.
Now let us consider two smooth projective $k$-varieties $V$ and $W$;
denote the two projections by $p_V:V\times W\to V$ and $p_W:V\times
W\to W$. If $\tau$ is a stable $H_2(V\times W)^+$-graph, we denote by
${p_V}_{\ast}(\tau)$ and ${p_W}_{\ast}(\tau)$ the stabilizations of $\tau$
with respect to ${p_V}_{\ast}:H_2(V\times W)^+\to H_2(V)^+$ and
${p_W}_{\ast}:H_2(V\times W)^+\to H_2(W)^+$ (see \cite{BM}, Remark~1.15),
by $\tau^s$ the absolute stabilization of $\tau$.
Applying the functor $\overline{M}$ to the commutative diagram
\[\comdia{(V\times
W,\tau)}{}{(W,{p_W}_{\ast}(\tau))}{}{}{}{(V,{p_V}_{\ast}(\tau))}{}{(\mathop{\rm Spec}\nolimits
k,\tau^s)} \]
in ${\frak V}\GG_s$ (see \cite{BM}, Remark~3.1 and the remark following
Theorem~3.14) we get a commutative diagram of proper Deligne-Mumford
stacks
\[\comdia{\overline{M}(V\times W,\tau)}{}{\overline{M}(W,{p_W}_{\ast}(\tau))} {}{}{}
{\overline{M}(V,{p_V}_{\ast}(\tau))}{}{\overline{M}(\tau^s).}\]
In general, this diagram is not cartesian; let $P$ be the cartesian
product
\[\comdia{P}{}{\overline{M}(W,{p_W}_{\ast}(\tau))} {}{}{}
{\overline{M}(V,{p_V}_{\ast}(\tau))}{}{\overline{M}(\tau^s).}\]
Rewrite these diagrams as follows:
\[\begin{array}{ccccc}
\overline{M}(V\times W,\tau) & \stackrel{h}{\longrightarrow} & P &
\longrightarrow & \overline{M}(V,{p_V}_{\ast}\tau)\times\overline{M}(W,{p_W}_{\ast}\tau)
\\ & \searrow & \ldiag{} & & \rdiag{} \\
& & \overline{M}(\tau^s) & \stackrel{\Delta}{\longrightarrow} &
\overline{M}(\tau^s)\times\overline{M}(\tau^s).\end{array}\]
To shorten notation, write $J(V\times W)=J(V\times W,\tau)$,
$J(V)=J(V,{p_V}_{\ast}\tau)$ and $J(W)=J(W,{p_W}_{\ast}\tau)$.
\begin{them} \label{pt}
We have
\[\Delta^{!}(J(V)\times J(W))=h_{\ast}(J(V\times W)).\]
\end{them}
For a stable $A$-graph $\tau$ ($A=H_2(V\times W)^+$, $H_2(V)^+$ etc.)
we denote by ${\frak M}(\tau)$ the algebraic $k$-stack of $\tau$-marked
prestable curves, forgetting the $A$-structure, and thinking of $\tau$
simply as a (possibly not stable) modular graph. We consider the
diagram
\begin{equation} \label{bigd} \begin{array}{cccccc}
& \overline{M}(V\times W,\tau) & \stackrel{h}{\longrightarrow} & P &
\longrightarrow & \overline{M}(V,{p_V}_{\ast}\tau)\times\overline{M}(W,{p_W}_{\ast}\tau)
\\ \phantom{M}/ & \rdiag{c} & & \rdiag{} & & \rdiag{a} \\ b\mid &
{\frak D}(\tau) & \stackrel{l}{\longrightarrow} & \PP &
\stackrel{\phi}{\longrightarrow} &
{\frak M}({p_V}_{\ast}\tau)\times{\frak M}({p_W}_{\ast}\tau) \\ \phantom{nM}\searrow&
\rdiag{e} & \searrow & \rdiag{} & & \rdiag{s\times s} \\ & {\frak M}(\tau) &
& \overline{M}(\tau^s) & \stackrel{\Delta}{\longrightarrow} &
\overline{M}(\tau^s)\times\overline{M}(\tau^s). \end{array}\end{equation} Here
$s\times s$ is given by stabilizations and $\PP$ is defined as the
fibered product of $\Delta$ and $s\times s$. The morphisms $a$ and $b$
are given by forgetting maps, retaining only marked curves.
The algebraic stack ${\frak D}(\tau)$ is defined as follows.
For a $k$-scheme $T$ the groupoid ${\frak D}(\tau)(T)$ has as objects
diagrams
\begin{equation}\label{ddd}
\begin{array}{ccc}
(C,x) & \longrightarrow & (C'',x'') \\
\ldiag{} & & \\
(C',x') & & \end{array}\end{equation}
where $(C,x)$ is a $\tau$-marked prestable curve over $T$, $(C',x')$ a
${p_V}_{\ast}(\tau)$-marked prestable curve over $T$ and $(C'',x'')$ a
${p_W}_{\ast}(\tau)$-marked prestable curve over $T$. The arrow
$(C,x)\to(C',x')$ is a morphism of marked prestable curves covering
the morphism $\tau\to{p_V}_{\ast}(\tau)$ of modular graphs. Similarly,
$(C,x)\to(C'',x'')$ is a morphism of marked prestable curves covering
$\tau\to{p_W}_{\ast}(\tau)$. This concept has not been defined in
\cite{BM}; the definition (in this special case) is as follows. Let us
explain it for the case of $W$ instead of $V$, since this will lead to
less confusion of notation with the set of vertices of a graph. The
morphism $\tau\to{p_W}_{\ast}(\tau)$ is given by a combinatorial morphism
of $0$-marked graphs $a:{p_W}_{\ast}(\tau)\to\tau$ (see \cite{BM},
Definition~1.7). So there are maps $a:V_{{p_W}_{\ast}(\tau)}\to V_{\tau}$
and $a:F_{{p_W}_{\ast}(\tau)}\to F_{\tau}$. The morphism
$(C,x)\to(C'',x'')$ is given by a family
$p=(p_v)_{v\in V_{{p_W}_{\ast}(\tau)}}$ of morphisms of prestable curves
(\cite{BM}, Definition~2.1) $p_v:C_{a(v)}\to C''_v$ such that for
every $i\in F_{{p_W}_{\ast}(\tau)}$ we have
$p_{\partial(i)}(x_{a(i)})=x''_i$.
\newcommand{{p_V}}{{p_V}}
\newcommand{{p_W}}{{p_W}}
There are morphisms of stacks $e:{\frak D}(\tau)\to{\frak M}(\tau)$,
${\frak D}(\tau)\to{\frak M}({p_V}_{\ast}\tau)$ and ${\frak D}(\tau)\to{\frak M}({p_W}_{\ast}\tau)$,
given, respectively, by mapping Diagram~(\ref{ddd}) to $(C,x)$,
$(C',x')$ and $(C'',x'')$. Let us denote the product of the latter two
by
\[\tilde{\Delta}:{\frak D}(\tau)\longrightarrow
{\frak M}({p_V}_{\ast}\tau)\times{\frak M}({p_W}_{\ast}\tau). \]
\begin{lem} \label{lem2}
In Diagram~(\ref{ddd}) both morphisms induce isomorphisms on
stabilizations.
\end{lem}
\begin{pf}
This follows from the fact that any morphism of stable marked curves
(with identical dual graphs) is an isomorphism. This fact is proved
in \cite{BM}, at the very end of the proof of Theorem~3.6, which
immediately precedes Definition~3.13.
\end{pf}
By this lemma there is a commutative diagram
\[\begin{array}{ccc}
{\frak D}(\tau) & \stackrel{\tilde{\Delta}}{\longrightarrow} &
{\frak M}({p_V}_{\ast}\tau)\times {\frak M}({p_W}_{\ast}\tau) \\
\ldiag{e} & & \\
{\frak M}(\tau) & & \rdiag{s\times s} \\
\ldiag{s} & & \\
\overline{M}(\tau^s) & \stackrel{\Delta}{\longrightarrow} &
\overline{M}(\tau^s)\times\overline{M}(\tau^s), \end{array}\]
which gives rise to the morphism $l:{\frak D}(\tau)\to\PP$ of
Diagram~(\ref{bigd}).
\begin{prop} \label{podd}
The morphisms $\Delta$ and $\tilde{\Delta}$ are proper regular local
immersions. Their natural orientations satisfy
$$l_{\ast}[\tilde{\Delta}]=(s\times s)^{\ast}[\Delta].$$
\end{prop}
\begin{pf}
Let $S_1$ and $S_2$ be finite sets, set $S=S_1\amalg S_2$. Let the
modular graph ${p_V}_{\ast}(\tau)'$ be obtained from ${p_V}_{\ast}(\tau)$ by
adding (in any fashion) $S_1$ to the set of tails of
${p_V}_{\ast}(\tau)$. Similarly, let ${p_W}_{\ast}(\tau)'$ be obtained form
${p_W}_{\ast}(\tau)$ by adding $S_2$ to the set of tails, arbitrarily. Now
let $\tau'$ be obtained from $\tau$ by adding the set $S$ to the tails
of $\tau$ in the unique way such that $\tau\to{p_V}_{\ast}(\tau)$ induces a
morphism $\tau'\to{p_V}_{\ast}(\tau)'$, which gives the inclusion
$S_1\subset S$ on tails, and $\tau\to{p_W}_{\ast}(\tau)$ induces a morphism
$\tau'\to{p_W}_{\ast}(\tau)'$, which gives the inclusion $S_2\subset S$ in
tails.
With these choices we have a cartesian diagram of $k$-stacks
\begin{equation}\label{poddi}
\comdia{\overline{M}(\tau')}{\delta} {\overline{M}({p_V}_{\ast}(\tau)') \times
\overline{M}({p_W}_{\ast}(\tau)')} {}{}{\chi} {{\frak D}(\tau)} {\tilde{\Delta}}
{{\frak M}({p_V}_{\ast}(\tau)) \times {\frak M}({p_W}_{\ast}(\tau)).}
\end{equation}
The proof that this is the case is similar to the proof of
Proposition~\ref{prop5}, below. The morphism $\chi$ in
Diagram~(\ref{poddi}) is a local presentation of ${\frak M}({p_V}_{\ast}(\tau))
\times {\frak M}({p_W}_{\ast}(\tau))$, (see \cite{gwi}, remarks following
Lemma~1). Moreover, by choosing $S$ and the primed graphs correctly,
any point of ${\frak M}({p_V}_{\ast}(\tau)) \times {\frak M}({p_W}_{\ast}(\tau))$ can be
assumed to be in the image of $\chi$. So to prove that
$\tilde{\Delta}$ is a proper regular local immersion, it suffices to
prove that $\delta$ is a proper regular local immersion. Properness is
clear; the stacks $\overline{M}(\tau')$, $\overline{M}({p_V}_{\ast}(\tau)')$ and
$\overline{M}({p_W}_{\ast}(\tau)')$ are proper. The regular local immersion
property follows from injectivity on tangent spaces which can be
proved by a deformation theory argument.
The proof for $\Delta$ is comparatively trivial.
To prove the fact about the orientations, first note that $s\times s$
is flat (see \cite{gwi}, Proposition~3) and so $\phi$ is a regular local
immersion and $(s\times s)^{\ast}[\Delta]=[\phi]$. To prove that
$l_{\ast}[\tilde{\Delta}]=[\phi]$, it suffices to identify dense open
substacks ${\frak D}(\tau)'\subset{\frak D}(\tau)$ and $\PP'\subset\PP$ such that
$l$ induces an isomorphism ${\frak D}(\tau)'\to\PP'$. We define ${\frak D}(\tau)'$
to be the open substack of ${\frak D}(\tau)$ over which $C_v\to C_v'$ is an
isomorphism for all $v\in V_{{p_V}_{\ast}(\tau)}$ and $C_v\to C_v''$ is an
isomorphism for all $v\in V_{{p_W}_{\ast}(\tau)}$. We define $\PP'$ to be
the pullback via $\Delta$ of ${\frak M}({p_V}_{\ast}(\tau))' \times
{\frak M}({p_W}_{\ast}(\tau))'$, where ${\frak M}({p_V}_{\ast}(\tau))'$ is the open substack
over which $(C_v,(x_i)_{i\in F(v)})$ is stable, for all $v\in
V_{\tau^s}$, similarly for ${\frak M}({p_W}_{\ast}(\tau))'$. Note the slight
abuse of notation; we have denoted vertices of different graphs by the
same letter.
\end{pf}
\begin{lem}
The morphism $e:{\frak D}(\tau)\to{\frak M}(\tau)$ is \'etale.
\end{lem}
\begin{pf}
Similar to the proof of \cite{gwi}, Lemma~7.
\end{pf}
To complete Diagram~(\ref{bigd}), define a morphism $\overline{M}(V\times
W,\tau)\to{\frak D}(\tau)$ by mapping a stable $(V\times W,\tau)$-map
$(C,x,f)$ first to the diagram
\[\begin{array}{ccc}
(C,x,f) & \longrightarrow & (C,x,{p_W}\mathbin{{\scriptstyle\circ}} f)^{\mbox{\tiny stab}} \\
\ldiag{} & & \\
(C,x,{p_V}\mathbin{{\scriptstyle\circ}} f)^{\mbox{\tiny stab}} & & \end{array}\]
and then passing to the underlying prestable curves.
\begin{prop}\label{prop5}
The diagram
\[\comdia{\overline{M}(V\times W,\tau)} {}
{\overline{M}(V,{p_V}_{\ast}\tau)\times\overline{M}(W,{p_W}_{\ast}\tau)} {c}{}{a}{{\frak D}(\tau)}
{\tilde{\Delta}} {{\frak M}({p_V}_{\ast}\tau)\times{\frak M}({p_W}_{\ast}\tau)}\]
is cartesian.
\end{prop}
\begin{pf}
We have to construct a morphism from the fibered product of
$\tilde{\Delta}$ and $a$ to $\overline{M}(V\times W,\tau)$. So let there be
given a Diagram~(\ref{ddd}), representing an object of ${\frak D}(\tau)(T)$,
for a $k$-scheme $T$. Moreover, let there be given families of maps
$(f')_{v\in V_{{p_V}_{\ast}\tau}}$, $f_v':C_v'\to V$ and $(f'')_{v\in
V_{{p_W}_{\ast}\tau}}$, $f_v'':C_v''\to W$, making $(C',x',f')$ and
$(C'',x'',f'')$ stable maps. We need to construct a stable map from
$(C,x)$ to $V\times W$. So let $v\in V_{\tau}$ be a vertex of
$\tau$.
Let us construct a map $h_v:C_v\to W$. In case $v$ is in the
image of $V_{{p_W}_{\ast}\tau}\to V_{\tau}$, and $w\mapsto v$ under this
map, we take $h_v$ to be the composition
$$C_v\stackrel{{p_W}}{\longrightarrow}
C_w''\stackrel{f_w''}{\longrightarrow} W.$$ In case $v$ is not in the
image of $V_{{p_W}_{\ast}\tau}\to V_{\tau}$, then $v$ partakes in a long
edge or a long tail associated to and edge $\{i,\overline{i}\}$ or a tail
$i$ of ${p_W}_{\ast}\tau$ (see the discussion of {\em stabilizing
morphisms}, Definition~5.7, in \cite{BM} for this terminology). Then
we define $f_v:C_v\to W$ to be the composition
$$C_v\longrightarrow T\stackrel{x_i''}{\longrightarrow}
C_{\partial(i)}''\stackrel{f_{\partial(i)}''}{\longrightarrow} W.$$
In the same manner, construct a map $g_v:C_v\to V$. Finally, let
$f_v:C_v\to V\times W$ be the product $g_v\times h_v$. Then the family
$(f_v)_{v\in V_{\tau}}$ makes $(C,x,f)$ a stable map over $T$ to
$V\times W$. One checks that $(C,x,{p_V}\mathbin{{\scriptstyle\circ}} f)^{\mbox{\tiny stab}}=(C',x',f')$ and
$(C,x,{p_W}\mathbin{{\scriptstyle\circ}} f)^{\mbox{\tiny stab}}=(C'',x'',f'')$, using the universal mapping
property of stabilization and the fact already alluded to in the proof
of Lemma~\ref{lem2}.
\end{pf}
Let $E^{\scriptscriptstyle\bullet}(V)=E^{\scriptscriptstyle\bullet}(V,{p_V}_{\ast}\tau)$ and
$E^{\scriptscriptstyle\bullet}(W)=E^{\scriptscriptstyle\bullet}(W,{p_W}_{\ast}\tau)$ denote the relative obstruction
theories for $\overline{M}(V,{p_V}_{\ast}\tau)\to{\frak M}({p_V}_{\ast}\tau)$ and
$\overline{M}(W,{p_W}_{\ast}\tau)\to{\frak M}({p_W}_{\ast}\tau)$, respectively, which were
defined in \cite{gwi}. As in \cite{BF} Proposition~7.4 there is an
induced obstruction theory $E^{\scriptscriptstyle\bullet}(V)\boxplus E^{\scriptscriptstyle\bullet}(W)$ for the
morphism $a$. Pulling back via $\tilde{\Delta}$ (as in \cite{BF}
Proposition~7.1) we get an induced obstruction theory
$\tilde{\Delta}^{\ast}(E^{\scriptscriptstyle\bullet}(V)\boxplus E^{\scriptscriptstyle\bullet}(W))$ for the morphism
$c$.
On the other hand, we have the relative obstruction theory
$E^{\scriptscriptstyle\bullet}(V\times W)=E^{\scriptscriptstyle\bullet}(V\times W,\tau)$ for the morphism $b$. Since
$e:{\frak D}(\tau)\to{\frak M}(\tau)$ is \'etale, we may think of $E^{\scriptscriptstyle\bullet}(V\times
W)$ as a relative obstruction theory for $c$.
\begin{prop}
The two relative obstruction theories
$\tilde{\Delta}^{\ast}(E^{\scriptscriptstyle\bullet}(V)\boxplus E^{\scriptscriptstyle\bullet}(W))$ and $E^{\scriptscriptstyle\bullet}(V\times
W)$ for the morphism $c$ are naturally isomorphic.
\end{prop}
\begin{pf}
Let ${\cal C}(V\times W,\tau)\to\overline{M}(V\times W,\tau)$,
${\cal C}(V,{p_V}_{\ast}\tau)\to\overline{M}(V,{p_V}_{\ast}\tau)$ and
${\cal C}(W,{p_W}_{\ast}\tau)\to\overline{M}(W,{p_W}_{\ast}\tau)$ be the universal
curves. Recall from \cite{gwi} that they are constructed by gluing the
curves associated to the vertices of a graph according the the edges
of that graph. Let us denote the pullbacks of the latter two universal
curves to $\overline{M}(V\times W,\tau)$ by ${\cal C}(V)$ and ${\cal C}(W)$,
respectively. We have maps $f_{V\times W}$, $f_V$ and $f_W$,
constructed from the universal stable maps, which fit into the
following commutative diagram:
\[\begin{array}{ccccc}
V & \stackrel{{p_V}}{\longleftarrow} & V\times W &
\stackrel{{p_W}}{\longrightarrow} & W \\
\ldiagup{f_V} & & \rdiagup{f_{V\times W}} & & \rdiagup{f_W} \\
{\cal C}(V) & \stackrel{q_{V}}{\longleftarrow} & {\cal C}(V\times W,\tau) &
\stackrel{q_W}{\longrightarrow} & {\cal C}(W) \\
& \sediag{\pi_V} & \rdiag{\pi_{V\times W}} & \swdiag{\pi_W} & \\
&& \overline{M}(V\times W,\tau) & & \end{array}\]
By base change it is clear that
$$\tilde{\Delta}^{\ast}(\dual{E^{\scriptscriptstyle\bullet}(V)}\boxplus\dual{E^{\scriptscriptstyle\bullet}(W)})
= R{\pi_V}_{\ast} f_V^{\ast} T_V\oplus R{\pi_W}_{\ast} f_W^{\ast} T_W.$$
For any vector bundle $F$ on ${\cal C}(W)$ the canonical homomorphism $F\to
R{q_W}_{\ast} q_W^{\ast} F$ is an isomorphism. Of course, the same property
is enjoyed by $q_V$. Hence we have a canonical isomorphism
\begin{eqnarray*}
\lefteqn{R{\pi_V}_{\ast} f_V^{\ast} T_V\oplus R{\pi_W}_{\ast} f_W^{\ast} T_W
\stackrel{\textstyle\sim}{\longrightarrow} }\\
&&R{\pi_V}_{\ast} R{q_V}_{\ast} q_V^{\ast} f_V^{\ast} T_V\oplus R{\pi_W}_{\ast}
R{q_W}_{\ast} q_W^{\ast} f_W^{\ast} T_W \\
& = & R{\pi_{V\times W}}_{\ast} f_{V\times W}^{\ast}(T_V\boxplus T_W) \\
& = & \dual{E^{\scriptscriptstyle\bullet}(V\times W)}.
\end{eqnarray*}
To conclude, we have a canonical isomorphism
$$\tilde{\Delta}^{\ast}(\dual{E^{\scriptscriptstyle\bullet}(V)}\boxplus\dual{E^{\scriptscriptstyle\bullet}(W)}) \stackrel{\textstyle\sim}{\longrightarrow}
\dual{E^{\scriptscriptstyle\bullet}(V\times W)}$$
and by dualizing
$$E^{\scriptscriptstyle\bullet}(V\times W) \stackrel{\textstyle\sim}{\longrightarrow}
\tilde{\Delta}^{\ast}({E^{\scriptscriptstyle\bullet}(V)}\boxplus{E^{\scriptscriptstyle\bullet}(W)}).$$
\end{pf}
By this proposition, we have
\begin{eqnarray*}
J(V\times W) & = & [\overline{M}(V\times W,\tau), E^{\scriptscriptstyle\bullet}(V\times W)] \\
& = & [\overline{M}(V\times W,\tau),
\tilde{\Delta}^{\ast}(E^{\scriptscriptstyle\bullet}(V)\boxplus E^{\scriptscriptstyle\bullet}(W))] \\
& = & \tilde{\Delta}^{!} [ \overline{M}(V,{p_V}_{\ast}\tau) \times
\overline{M}(W,{p_W}_{\ast}\tau) , E^{\scriptscriptstyle\bullet}(V)\boxplus E^{\scriptscriptstyle\bullet}(W)] \\
& & \quad\quad\mbox{(by \cite{BF} Proposition~7.2)} \\
& = & \tilde{\Delta}^{!} (
[\overline{M}(V,{p_V}_{\ast}\tau),E^{\scriptscriptstyle\bullet}(V)] \times
[\overline{M}(W,{p_W}_{\ast}\tau),E^{\scriptscriptstyle\bullet}(W)]) \\
& & \quad\quad\mbox{(by \cite{BF} Proposition~7.4)} \\
& = & \tilde{\Delta}^{!} (J(V)\times J(W)).
\end{eqnarray*}
So we may now calculate as follows:
\begin{eqnarray*}
\tilde{\Delta}^{!} (J(V)\times J(W)) & = & a^{\ast}(s\times s)^{\ast}
[\Delta] (J(V)\times J(W)) \\
& = & a^{\ast} l_{\ast}[\tilde{\Delta}] (J(V)\times J(W)) \\
& & \quad\quad\mbox{(by Proposition~\ref{podd})} \\
& = & h_{\ast} \tilde{\Delta}^{!}(J(V)\times J(W)) \\
& = & h_{\ast} J(V\times W),
\end{eqnarray*}
which is the product property. This finishes the proof of
Theorem~\ref{pt}.
\subsection{Gromov-Witten Transformations}
Theorem~\ref{pt} easily implies that the system of Gromov-Witten
invariants for $V\times W$ is equal to the tensor product (see
\cite{KM}, 2.5) of the systems of Gromov-Witten invariants for $V$ and
$W$, respectively. To get the full `operadic' picture, we need a few
graph theoretic preparations.
\begin{prop} \label{psi}
There is a natural functor of categories fibered over $\mbox{$\tilde{\GG}_s(0)$}$
\[\Psi:\mbox{$\tilde{\GG}_s(V\times W)$}_{{\mbox{\tiny cart}}} \longrightarrow
\mbox{$\tilde{\GG}_s(V)$}_{{\mbox{\tiny cart}}}\times_{\mbox{$\tilde{\GG}_s(0)$}}\mbox{$\tilde{\GG}_s(W)$}_{{\mbox{\tiny cart}}}.\]
This functor is cartesian.
\end{prop}
\begin{pf}
Let $p:V\to W$ be a morphism of smooth projective varieties over
$k$. We shall construct a natural functor of fibered categories over
$\mbox{$\tilde{\GG}_s(0)$}$
$$\Psi_p:\mbox{$\tilde{\GG}_s(V)$}_{{\mbox{\tiny cart}}}\longrightarrow\mbox{$\tilde{\GG}_s(W)$}_{{\mbox{\tiny cart}}}.$$
This functor will be cartesian.
For an object $(\tau,(\overline{a}_i,\tau_i)_{i\in I})$ of $\mbox{$\tilde{\GG}_s(V)$}_{{\mbox{\tiny cart}}}$
let the image under $\Psi_p$ be $(\tau,(\overline{b}_i,p_{\ast}(\tau_i))_{i\in
I})$. Here $p_{\ast}(\tau_i)$ is the stabilization of $\tau_i$ covering
$p_{\ast}:H_2(V)^+\to H_2(W)^+$. It comes with a natural morphism
$\overline{b}_i:p_{\ast}(\tau_i)\to \tau$. To make $\overline{b}_i$ a stabilizing
morphism, we have to endow it with an orbit map (see \cite{BM},
Definition~5.7). Let $\overline{a}_i^m:E\t\cup S\t\to E_{\tau_i}\cup
S_{\tau_i}$ be the orbit map of $\overline{a}_i:\tau_i\to \tau$. Let
$\{f,\overline{f}\}$ be an edge of $\tau$. Then there exists a unique factor
$\{f',\overline{f}'\}$ of the long edge of $p_{\ast}(\tau_i)$ associated to
$\{f,\overline{f}\}$, such that $\overline{a}_i^m(\{f,\overline{f}\})$ is a factor of
the long edge of $\tau_i$ associated to $\{f',\overline{f}'\}$. We set
$\overline{b}_i^m(\{f,\overline{f}\})=\{f',\overline{f}'\}$. This defines the orbit map
$\overline{b}_i^m$ of $\overline{b}_i$ on edges. On tails it is defined entirely
analogously.
This achieves the definition of $\Psi_p$ on objects. We leave it to
the reader to explicate the action of $\Psi_p$ on morphisms; it boils
down to checking that the {\em pullback }of \cite{BM}, Definition~5.8
is compatible with applying $p_{\ast}$.
Now we define $\Psi$ by
\begin{eqnarray*}
\Psi:\mbox{$\tilde{\GG}_s(V\times W)$}_{{\mbox{\tiny cart}}}& \longrightarrow &
\mbox{$\tilde{\GG}_s(V)$}_{{\mbox{\tiny cart}}}\times_{\mbox{$\tilde{\GG}_s(0)$}}\mbox{$\tilde{\GG}_s(W)$}_{{\mbox{\tiny cart}}} \\
(\tau,(\tau_i)) & \longmapsto &
(\Psi_{{p_V}}(\tau,(\tau_i)),\Psi_{{p_W}}(\tau,(\tau_i))).
\end{eqnarray*}
\end{pf}
Let us denote, for any $V$, the Gromov-Witten transformation for $V$
(see \cite{BM}, Theorem~9.2), by
\[I^V:h(V)^{\otimes S}(\chi\dim V)\longrightarrow h(\overline{M}).\]
Recall that $I^V$ is a natural transformation between functors
\[\mbox{$\tilde{\GG}_s(V)$}_{{\mbox{\tiny cart}}}\longrightarrow(\mbox{graded DMC-motives}),\]
where the two functors $h(V)^{\otimes S}(\chi\dim V)$ and $h(\overline{M})$
are induced from functors (with the same names)
$\mbox{$\tilde{\GG}_s(0)$}\to(\mbox{DMC-motives})$, constructed in \cite{BM}, Section~9.
\begin{numrmk} \label{lem8}
By our various definitions we have
\[h(V)^{\otimes S}(\chi\dim V)\otimes h(W)^{\otimes S}(\chi\dim W) =
h(V\times W)^{\otimes S}(\chi\dim V\times W)\]
as functors $\mbox{$\tilde{\GG}_s(0)$}\to(\mbox{\rm DMC-motives})$.
\end{numrmk}
The transformations $I^V$ and $I^W$ induce a transformation
\[I^V\otimes I^W: h(V)^{\otimes S}(\chi\dim V) \otimes h(W)^{\otimes
S} (\chi\dim W) \longrightarrow h(\overline{M})\otimes h(\overline{M})\]
between functors
\[\mbox{$\tilde{\GG}_s(V)$}_{{\mbox{\tiny cart}}}\times_{\mbox{$\tilde{\GG}_s(0)$}}\mbox{$\tilde{\GG}_s(W)$}_{{\mbox{\tiny cart}}}\longrightarrow (\mbox{graded
DMC-motives}).\]
It is defined as follows. Let $((\tau,(\tau_i)_{i\in
I}),(\tau,(\sigma_j)_{j\in J}))$ be an object of
$\mbox{$\tilde{\GG}_s(V)$}_{{\mbox{\tiny cart}}}\times_{\mbox{$\tilde{\GG}_s(0)$}}\mbox{$\tilde{\GG}_s(W)$}_{{\mbox{\tiny cart}}}$. The value of $I^V\otimes I^W$
on this object is the morphism
\[I^V(\tau,(\tau_i))\otimes I^W(\tau,(\sigma_j)): \]\[
h(V)^{\otimes
S\t}(\chi(\tau)\dim V)\otimes h(W)^{\otimes S\t}(\chi(\tau)\dim W)
\longrightarrow h(\overline{M}(\tau))\otimes h(\overline{M}(\tau)).\]
Composing with $\Delta^{\ast}:h(\overline{M})\otimes h(\overline{M})\to h(\overline{M})$
we get the transformation
$$\Delta^{\ast}(I^V\otimes I^W):h(V)^{\otimes S}(\chi\dim V) \otimes
h(W)^{\otimes S} (\chi\dim W) \longrightarrow h(\overline{M}),$$
which we shall also denote by $I^V\cup I^W=\Delta^{\ast}(I^V\otimes
I^W)$.
Pulling back via the functor $\Psi$ of Proposition~\ref{psi} and using
Remark~\ref{lem8}, we may think of $I^V\cup I^W$ as a natural
transformation
\[I^V\cup I^W:h(V\times W)^{\otimes S}(\chi\dim V\times W)
\longrightarrow h(\overline{M})\]
between functors $\mbox{$\tilde{\GG}_s(V\times W)$}_{{\mbox{\tiny cart}}}\to(\mbox{graded DMC-motives})$.
\begin{them} \label{st}
We have
\[I^V\cup I^W=I^{V\times W}.\]
\end{them}
\begin{pf}
This follows from Theorem~\ref{pt} and the identity principle for
DMC-motives, Proposition~8.2 of \cite{BM}.
\end{pf}
|
2000-03-23T17:27:49 | 9710 | alg-geom/9710029 | en | https://arxiv.org/abs/alg-geom/9710029 | [
"alg-geom",
"math.AG"
] | alg-geom/9710029 | Tomas L. Gomez | Tomas L. Gomez | Irreducibility of the moduli space of vector bundles on surfaces and
Brill-Noether theory on singular curves | Revised PhD thesis (Princeton, 1997), 64 pages, 1 figure, LaTeX2e,
Xy-pic | null | null | null | null | We prove the irreducibility of the moduli space of rank 2 semistable torsion
free sheaves (with a generic polarization and any value of c_2) on a K3 or a
del Pezzo surface. In the case of a K3 surface, we need to prove a result on
the connectivity of the Brill-Noether locus for singular curves on the surface.
In the case of a del Pezzo surface, we reduce the problem to the case of P^2 by
first relating the moduli spaces of the plane and the blown-up plane, and then
studying how the moduli space changes when we change the polarization.
| [
{
"version": "v1",
"created": "Mon, 27 Oct 1997 10:11:47 GMT"
},
{
"version": "v2",
"created": "Thu, 23 Mar 2000 16:27:48 GMT"
}
] | 2007-05-23T00:00:00 | [
[
"Gomez",
"Tomas L.",
""
]
] | alg-geom | \chapter*{Acknowledgments}
I would like to thank my advisor, Robert Friedman, for suggesting me
this problem, for many discussions and for his encouragement. I am
very grateful to him for introducing me to the study of moduli spaces
of vector bundles. I consider myself very fortunate for having found
his generous help when I was looking for a thesis problem.
I would also like to thank Ignacio Sols Lucia, who introduced me to
algebraic geometry. With the time that he
generously dedicated to teach me, he made possible
the transition from my physics undergraduate background to algebraic geometry.
Thanks for his patience and his friendship.
I have also benefited from discussions with R. Lazarsfeld, R.
MacPherson and many other people as well as from many seminars and
courses at Princeton.
I want to thank the ``Banco de Espa{\~n}a'' for the fellowship ``Beca
para la ampliaci{\'o}n de estudios en el extranjero'' that supported
my graduate studies. Without their generous support, this thesis
couldn't have been made.
\chapter{Introduction}
In this chapter we will explain the main results of the thesis using
as little mathematical background as possible. We will always work
over the complex numbers, i.e all manifolds will be complex manifolds.
Also we assume that manifolds are projective, i.e. there is an
embedding in $\mathbb{P}^n_\mathbb{C}$ (in particular they are K{\"a}hler).
$X$ will be a variety (or a manifold with singularities).
We will consider holomorphic vector bundles over
$X$ (the transition functions are assumed to be holomorphic).
To distinguish non-isomorphic vector bundles of fixed rank
we can define certain
invariants called Chern classes. These are cohomology classes
$c_i(V)\in H^{2i}(X,\mathbb{Z})$, $1\leq i\leq \dim_\mathbb{C} X$. We have
$c_i(V)=0$ if $i>\operatorname{rank}(V)$. Even after fixing these discrete
invariants, we can have continuous families of non-isomorphic vector
bundles. I.e., to specify the isomorphism class of a vector bundle it is
not enough to fix some discrete invariants, but we have to fix also
some continuous parameters.
For fixed rank and Chern classes we would like to define a variety
called the moduli space of vector bundles. Each point of this variety
will correspond to a different vector bundle. In this thesis we will
study the irreducibility of this space for rank 2 vector bundles over
certain surfaces.
Unfortunately, in order to construct this moduli space
we have to restrict our attention to \textit{stable}
vector bundles (this is not a very strong restriction, since it can be
proved that in some
sense all vector bundles can be constructed starting from stable
ones). There are different notions of stability (see chapter
\ref{Preliminaries}). Here we will only discuss \textit{Mumford
stability} (also called slope stability). Let $X$ be a projective
variety and $H$ an ample divisor. For a vector
bundle $V$ we define the slope of $V$ with respect to $H$ as:
$$
\mu_H (V) = \frac { c_1(V) H^{n-1}}{\operatorname{rank}(V)}
$$
(the product is the intersection product or cup product in cohomology)
$V$ is called $H$-stable if for every subbundle $W$ of $V$ we
have
$$
\mu_H (W) < \mu_H (V).
$$
It can be proved that there is a space $\mathfrak{M}^0_H(r,c_i)$, called the
moduli space of $H$-stable vector bundles of rank $r$ and Chern
classes $c_i$ (if the rank is clear from the context we will drop it
from the notation). In many situations it will be a variety (maybe with
singularities) but in general it can have a very singular behavior.
In general $\mathfrak{M}^0_H(r,c_i)$ is not compact. To get a compact moduli
space we need to consider a larger class of objects. Instead of vector
bundles we consider torsion free sheaves (they can be thought as
``singular'' vector bundles that fail to be locally a product on a
subvariety of X). Also we have to relax the stability condition, and
we will consider \textit{Gieseker semistable} sheaves (see chapter
\ref{Preliminaries} for the definition). The moduli space of Gieseker
semistable torsion free sheaves is compact, and we denote it by
$\mathfrak{M}_H(r,c_i)$.
There is an important relationship between the theory of holomorphic
vector bundles and gauge theory: there is a bijection between the set of
$H$-stable vector bundles and differentiable bundles with a
Hermite-Einstein connection. If $c_1=0$ and $\dim_\mathbb{C}=2$ then a
Hermite-Einstein connection is the same thing as an anti-selfdual (ASD)
connection. Note that the metric of the manifold appears in the ASD
equation. This is reflected in the fact that the stability condition
depends on the polarization. This relationship has been used, for
instance, to calculate Donaldson polynomials for the study of the
topology of four-manifolds.
Now we are going to consider some particular cases.
Let $C$ be a
curve (i.e., a Riemann surface) of genus $g$. Then the moduli space of
rank $r$ vector
bundles on $C$ is a smooth variety of dimension $(g-1)r^2+1$.
For line bundles (rank=1) over a variety $X$ ($\dim_\mathbb{C} X=n$) the
moduli space is called the Jacobian and we have
a very explicit description of it. First we note that
all line bundles are stable (because they don't have subbundle).
Recall that $q=h^1({\mathcal{O}}_X)=b_1/2$, where $b_1$ is the first Betti number
(if $X$ is simply connected then $b_1=0$. If $X$ is a curve
then $b_1=2g$). The Jacobian $J$ is the moduli space of line bundles
with $c_1=0$ and it is of the form $\mathbb{C}^q/\mathbb{Z}^{2q}$, where
$\mathbb{Z}^{2q}$ is a lattice in $\mathbb{C}^q$. If $c_1 \neq 0$ then the moduli
space $J^{c_1}$ is isomorphic to the Jacobian, but the isomorphism is
not canonical.
If $X$ is a curve $c_1$ is called the degree $d$. We define some
subsets of $J^d$ as follows
$$
W^a_d=\{L \in J^d: \dim(H^0(L))=a+1\}
$$
The study of the properties of these subsets is called Brill-Noether theory.
If the curve is generic, then $W^a_d$ is a subvariety of the
expected dimension $\rho(a,d)=g-(a+1)(g-d+a)$. If $\rho(a,d)>0$ then
(for any curve) $W^a_d$ is connected. If the curve is
singular, one has to consider also torsion free sheaves in order to get
a compact moduli space (in this case the moduli space will be
singular). This moduli space has been constructed, but little is known
about its Brill-Noether theory.
In this thesis, to
prove the irreducibility of the moduli space of rank 2 vector bundles
on a $K3$ surface, we will need the connectivity of $W^a_d$ for certain
singular curves that lie
in the surface. In chapter \ref{bn},
theorem I,
we will prove that if $\rho(a,d)>0$, $W^a_d$ is still connected for
singular curves satisfying certain conditions.
If $X$ is a complex surface ($\dim_\mathbb{C} X=2$), then in general the
moduli space of vector bundles of rank $r$ is very singular, but in
many situations it will be a variety (maybe with singularities) of
the expected dimension
$$
\dim
\mathfrak{M}^0_H(r,c_1,c_2)=2rc_2-(r-1)c_1^2-(r^2-1)\chi({\mathcal{O}}_X)+h^1({\mathcal{O}}_X).
$$
By a slight abuse of language we have denoted by $c_2$ the integral
of the second Chern class
on the variety $\int_X c_2(V)$, and by $c_1^2$ the integral $\int_X
c_1(V) \wedge c_1(V)$. Recall that $\chi({\mathcal{O}}_X)=1-h^1({\mathcal{O}}_X)+h^2({\mathcal{O}}_X)$.
In this thesis we will consider the case rank=2. For rank 2,
$\mathfrak{M}^0_H(c_1,c_2)$
is known to be irreducible and of the expected dimension if $c_2>N$,
where $N$ is a constant that
depends on $X$, $c_1$ and $H$. In chapter \ref{k3} we present a
new proof of the theorem of O'Grady
\smallskip\noindent
\textbf{Theorem II.}
\textit{Let $X$ be a projective $K3$
surface (a $K3$ surface is a simply connected complex surface with
$c_1(T_X)=0$, where $T_X$ is the tangent bundle). Let $H$ be a
generic polarization (see chapter \ref{Preliminaries} for a
definition). Assume that $c_1$ is a nonzero primitive element of $H^2(X,\mathbb{Z})$
(under this condition $\mathfrak{M}^0_H(c_1,c_2)$ is smooth of the
expected dimension).}
\textit{Then $\mathfrak{M}^0_H(c_1,c_2)$ is irreducible.}
\smallskip
We should note that in chapter \ref{k3} we work with the moduli space
of Gieseker semistable torsion free sheaves $\mathfrak{M}_H(c_1,c_2)$.
But it can be shown that under the conditions of the theorem the
points of $\mathfrak{M}_H(c_1,c_2)$ that are not in
$\mathfrak{M}^0_H(c_1,c_2)$ are in a subvariety of positive codimension,
then the irreducibility of one of them is equivalent to the
irreducibility of the other.
Using the relationship of holomorphic vector bundles with gauge theory
this theorem can be stated as follows:
\begin{theorem}
Let $X$ be a projective $K3$ surface with a generic K{\"a}hler
metric $g$. Then the moduli space of
$SO(3)$ anti-selfdual connections (with fixed instanton number $k$ and
second Stiefel-Whitney class $w_2$) is smooth
and connected.
\end{theorem}
In chapter \ref{dp} we study moduli spaces of vector bundles on
\textit{del Pezzo} surfaces. A del Pezzo surface is a surface whose
anticanonical bundle is ample. It can be shown that these are all the
del Pezzo surfaces: $\mathbb{P}^2$, $\mathbb{P}^1\times\mathbb{P}^1$, or the projective
plane $\mathbb{P}^2$ blown up at most at 8 generic points. For rank 2 and
fixed Chern classes, the moduli space of stable vector bundles is
known to be empty or irreducible for $X=\mathbb{P}^2$ or $\mathbb{P}^1\times\mathbb{P}^1$.
In chapter \ref{dp} we prove this result for a projective plane
$\mathbb{P}^2$ blown up at most at 8 generic points, and then we get the
general result:
\smallskip\noindent
\textbf{Theorem III.}
\textit{Let $X$ be a del Pezzo surface with a generic polarization
$H$. Fix some Chern classes $c_1$ and $c_2$. Then
$\mathfrak{M}^0_H(c_1,c_2)$ is irreducible (or empty).}
\smallskip
As in the $K3$ case, in this case irreducibility is equivalent to
connectedness, and we have the same result for $\mathfrak{M}_H(c_1,c_2)$.
We can also translate this theorem into gauge theory language:
\begin{theorem}
Let $X$ be a del Pezzo surface with a generic K{\"a}hler metric $g$.
Then the moduli space of $SO(3)$ anti-selfdual connections (with fixed
Stiefel-Whitney class $w_2$ and instanton number $k$) is
connected (or empty). The same is true for $SU(2)$ anti-selfdual
connections with fixed instanton number $k$.
\end{theorem}
\chapter{Preliminaries}
\label{Preliminaries}
\section{Brill-Noether theory}
Let $C$ be a smooth curve of genus $g$ (we will always assume that the
base field is $\mathbb{C}$), $J(C)$ its Jacobian, and $W^r_d(C)$ the
Brill-Noether locus corresponding to line bundles $L$ of degree $d$ and
$h^0(L) \geq r+1$ (see \cite{ACGH}). The expected dimension of this
subvariety is $\rho(r,d)= g -
(r+1)(g-d+r)$. Fulton and Lazarsfeld \cite{F-L} proved that $W^r_d(C)$
is connected
when $\rho > 0$. We are going to generalize this result for certain singular
curves, but before stating our result
(theorem I), we
need to recall some concepts.
Let $C$ be an integral curve (not necessarily smooth). We still have a
generalized Jacobian $J(C)$, defined as
the variety parametrizing line bundles, but it
will not be complete in general. Define the degree of a rank
one torsion-free sheaf on $C$ to be
$$
\deg(A)=\chi(A)+p_a-1,
$$
where $p_a$ is the arithmetic genus of $C$.
One can define a scheme $\overline J^d(C)$
parametrizing rank one torsion-free sheaves on $C$ of degree $d$
(see \cite{AIK}, \cite {D},
\cite {R}). If $C$ lies on a surface,
then $\overline
J^d(C)$ is integral, and furthermore the generalized
Jacobian $J(C)$ is an open set in $\overline J^d(C)$,
and then $\overline J^d(C)$
is a natural compactification of $J(C)$.
We will need to consider
families of sheaves parametrized by a scheme $T$, and furthermore the
curve will vary as we vary the parameter $t\in T$.
All this can be done
using a relative version of $\overline J^d(C)$, but we will proceed in a
different way. We will use the fact that all these curves are going to
lie on a fixed surface $S$. Then we will think of the coherent sheaves on $C$
as torsion sheaves on $S$ (all sheaves in this paper will be coherent).
To define precisely which sheaves we will
consider we need some notation. For any sheaf $F$ on $S$, let $d(F)$
be the dimension of its support. We say that $F$ has pure dimension
$n$ if for any subsheaf $E$ of $F$ we have $d(E)=d(F)=n$. Note that if
the support is irreducible, then having pure dimension $n$ is
equivalent to being torsion-free when considered as a sheaf on its
support. The following theorem follows from \cite[theorem 1.21]{S}.
\smallskip
\begin{theorem}[Simpson]
Let $C$ be an integral curve on a surface $S$.
Let $\overline {\mathcal{J}}^d_{|C|}$ be the functor which associates to any scheme
$T$ the set of equivalence classes of sheaves ${\mathcal{A}}$ on $S\times T$ with
(a) ${\mathcal{A}}$ is flat over $T$.
(b) The induced sheaf $A_t$ on each fiber $S\times \{t\}$ has pure
dimension 1, and its support is an integral curve in the linear system
$|C|$.
(c) If we consider $A_t$ as a sheaf on its support, it is
torsion-free and has rank one and degree $d$.
Sheaves ${\mathcal{A}}$ and ${\mathcal{B}}$ are equivalent if there exists a line
bundle $L$ on $T$ such that ${\mathcal{A}} \cong {\mathcal{B}} \otimes p_T^*L$, where
$p_T:S \times T \to T$ is the projection on the second factor.
Then there is a coarse moduli space that we also denote by $\overline
{\mathcal{J}}^d_{|C|}$. I.e., the points of $\overline {\mathcal{J}}^d_{|C|}$ correspond to
isomorphism classes of sheaves, and for any family ${\mathcal{A}}$ of such
sheaves parametrized by $T$, there is a morphism
$$
\phi :T \to \overline {\mathcal{J}}^d_{|C|}
$$
such that $\phi(t)$ corresponds to the isomorphism class of $A_t$.
\end{theorem}
Note that $\overline{\mathcal{J}}^d_{|C|}$ parametrizes pairs $(C',A)$ with $C'$
an integral curve linearly equivalent to $C$ and $A$ a torsion-free
rank one sheaf on $C$.
We denote by $\pi :\overline{\mathcal{J}}^d_{|C|} \to U \subset |C|$ the
obvious projection giving
the support of each sheaf, where $U$ is the open subset of $|C|$
corresponding to integral curves.
A family of curves on a surface $S$ parametrized by a curve $T$ is a
subvariety ${\mathcal{C}} \subset S \times T$, flat over $T$, such that the
fiber ${\mathcal{C}}|_t=C_t$ over each $t\in T$ is a curve on $S$.
Analogously, a family of sheaves on a surface $S$ parametrized by a
curve $T$ is a sheaf ${\mathcal{A}}$ on $S\times T$, flat over $T$. For each
$t\in T$ we will denote the corresponding member of the family by
$A_t={\mathcal{A}}|_t.$
Altman, Iarrobino and Kleiman
\cite{AIK} proved the following theorem
\begin{theorem}[Altman--Iarrobino--Kleiman]
With the same notation as before, $\overline {\mathcal{J}}^d_{|C|}$
is flat over $U$ and its
geometric fibers are integral. The subset of $\overline {\mathcal{J}}^d_{|C|}$
corresponding to line bundles (i.e., the relative generalized Jacobian) is
open and dense in $\overline {\mathcal{J}}^d_{|C|}$.
\end{theorem}
We also
consider the family of generalized Brill-Nother loci $\overline
{\mathcal{W}}^r_{d,|C|} \subset \overline {\mathcal{J}}^d_{|C|}$, and the projection
$q:\overline{\mathcal{W}}^r_{d,|C|} \to U$.
We can define the generalized Brill-Noether locus $\overline W^r_d(C)$
as the set of points in $\overline J^d(C)$ corresponding to sheaves $A$
with $h^0(A) \geq r+1$ (note that it is complete because of the
upper semicontinuity of $h^0(\cdot)$). There is also a determinantal
description that
gives a scheme structure. This description is a straightforward
generalization of the description for smooth curves (see \cite
{ACGH}), but we are only interested in the connectivity of $\overline
W^r_d(C)$, so we can give it the reduced scheme structure.
We will consider curves that lie on a surface $S$ with the following
property:
\bigskip
$h^1(\SO_S)=0$ \textit{, and }$-K_S$ \textit{ is generated by global
sections.}\hfill (*)
\bigskip
We will need this condition to prove proposition \ref{bn1.5}.
For instance, $S$ can be a K3 surface or a del Pezzo surface with
$K_S^2\neq 1$.
Now we can state the theorem that we are going to prove in chapter
\ref{bn}.
\smallskip
\noindent\textbf{Theorem I.}
\textit{ Let $C$ be a reduced irreducible curve of arithmetic genus
$p_a$ that lies in a surface $S$
satisfying \textup{(*)}.
Let $\overline J^d(C)$, $d>0$, be the compactification of the generalized
Jacobian. Then for any $r \geq 0$ such that $\rho(r,d)= p_a -
(r+1)(p_a-d+r)>0$, the generalized Brill-Noether subvariety $\overline
W^r_d(C)$ is nonempty and connected.}
\smallskip
\section{Moduli space of torsion free sheaves}
Let $X$ be a smooth projective variety over $\mathbb{C}$ of dimension $n$. We are
interested in
constructing a moduli space that will parametrize torsion free sheaves
on $X$, with some fixed rank $r$ and Chern classes $c_i$. To do so, we
have to introduce some notion of stability.
\begin{definition}\textup{\textbf{(Mumford-Takemoto stability)}}
Fix a polarization $H$. For any nonzero torsion free sheaf $F$ we define
the slope of $F$ with respect to $H$ as
$$
\mu_H(F)={\frac {c_1(F)\cdot H^{n-1}} {\operatorname{rk} (F)}}
$$
We will say that a torsion free sheaf $V$ on $X$ is Mumford H-stable
(resp. semistable) if for every nonzero subsheaf $W$ of $V$ we have
$$
\mu_H(W) < \mu_H(V) \ \ \ (\text{resp. }\leq ).
$$
\end{definition}
Using this definition one can construct the coarse moduli space of
Mumford stable vector bundles. This notion of stability turns out to
have an analog in differential geometry: Mumford stable holomorphic
vector bundles are in one to one correspondence with differentiable
vector bundles with a Hermite-Einstein connection (the choice of a
polarization is replaced by the choice of a Riemannian metric). This
correspondence
was proved by Narasimhan and Seshadri \cite{N-S} if $X$ is a curve,
by Donaldson \cite{Do1,D-K} for a surface, and it was then generalized for
any dimension by \cite{U-Y}. This correspondence has been useful to
calculate differentiable invariants of 4-manifolds (the so called
Donaldson invariants).
The moduli space of Mumford stable sheaves is in general not compact. To
define a compactification we have to introduce a refined notion of
stability.
\begin{definition}\textup{\textbf{(Gieseker stability).}}
Fix a polarization $H$. For any nonzero torsion free sheaf $F$ define
the Hilbert polynomial of $F$ with respect to $H$ as
$$
p_H(F)(n) =\frac {\chi(F \otimes {\mathcal{O}}_X(nH))}{rank(F)}.
$$
Given two polynomials $f$ and $g$, we will write $f\prec g$ (resp.
$\preceq$) if $f(n)<g(n)\ ($resp. $\leq)$ for $n\gg0$.
We will say that $V$ is Gieseker stable (resp. semistable) if for
every nonzero torsion free subsheaf $W$ we have
$$
p_H(W) \preceq p_H(V) \ \ \ (\text{resp. }\preceq ).
$$
\end{definition}
Using Hirzebruch-Riemann-Roch theorem we can see that Gieseker
semistability implies Mumford semistability, and Mumford stability
implies Gieseker stability.
In order to have a separated moduli space it is not enough to consider
isomorphism classes of sheaves. We will introduce the notion of
S-equivalence. For any Gieseker semistable sheaf $V$ there is a
filtration (\cite{G,Ma})
$$
0 =V_0 \subseteq V_1\subseteq \cdots \subseteq V_t=V
$$
such that $V_i/V_{i-1}$ is stable and $p_H(V_i)=p_H(V)$. We define
$\operatorname{gr} (V)=\oplus (V_i/V_{i-1})$. It can be proved that $\operatorname{gr}(V)$ doesn't
depend on
the filtration chosen. We will say that $V$ is S-equivalent to $V'$ if
$\operatorname{gr}(V)=\operatorname{gr}(V')$.
There is another characterization of S-equivalence that is more
illuminating from the point of view of moduli problems. Assume that we
have a family of Gieseker semistable sheaves parametrized by a curve
$T$. I.e., we have a sheaf ${\mathcal{V}}$ on $X \times T$, flat over $T$
inducing torsion free
Gieseker stable sheaves ${\mathcal{V}}|_t$ on the slices $X \times \{t\}$. Assume
that for one point $0\in T$ we have ${\mathcal{V}}|_0\cong V$ and for the rest of
the points ${\mathcal{V}}|_t$ is isomorphic to some other fixed $V'$. We will
say that $V$ and $V'$ are equivalent. The equivalence relation
generated by this definition is S-equivalence.
It can be proved that there is a coarse moduli space for
S-equivalence classes of Gieseker semistable torsion free sheaves
(with fixed rank and Chern classes). This moduli space is projective.
In general the moduli space can be very singular, but if $X$ is a surface
and for fixed rank $r$, $c_1$ and polarization, it is
known that for $c_2$ large enough the singular locus is a proper
subset of positive codimension of the moduli space
\cite{Do2,F,Z,G-L2}. The moduli space has the expected dimension
$$
2rc_2-(r-1)c_1^2-(r^2-1)\chi({\mathcal{O}}_X),
$$
and is irreducible
\cite{G-L1,G-L2,O1,O2}. In the rank two case it is also known, again
for $c_2$ large enough, that the moduli space is normal and has local
complete intersection singularities at points corresponding to stable
sheaves, and if the surface $X$ is of general type (with some
technical conditions), then also the moduli space is of general type
\cite{L2}.
It is natural to ask what is the effect of changing the choice of
polarization. From now on we will assume that $X$ is a surface $S$ and
that the rank is 2 unless otherwise stated. We will denote the moduli
space of rank 2 torsion free sheaves that are Gieseker semistable with
respect to the polarization $H$ by $\mathfrak{M}_H(c_1,c_2)$.
\begin{definition}
Fix $S$, the first and second Chern classes $c_1$, $c_2$.
Let $\zeta$ be some class in $H^2(S,\mathbb{Z})$ with
$$
\zeta\equiv c_1\ (\text{mod}\ 2),\ \ c_1^2-4c_2\leq \zeta^2 <0.
$$
The wall of type $(c_1,c_2)$ associated to $\zeta$ is a hyperplane of
$H^2(S,\mathbb{Q})$ with
nonempty intersection with the ample cone $\Omega_S$
$$
W^\zeta=\{x\in \Omega_S |\ x \cdot \zeta =0\}.
$$
The connected components of the complement of the walls in the ample
cone are called chambers.
\smallskip
A polarization $H$ is called $(c_1,c_2)$-generic if it doesn't lie on
a wall (i.e., it lies in a chamber).
\end{definition}
The walls of type $(c_1,c_2)$ are known to be locally finite on the
ample cone \cite{F-M}.
If a polarization is $(c_1,c_2)$-generic it is easy to see that
Mumford and Gieseker stability coincide, and furthermore there are no
strictly semistable sheaves. Stable sheaves are \textit{simple}
($\operatorname{Hom}(V,V)=\mathbb{C}$). If $-K_S$ is effective then this fact and the
Kuranishi local model for the moduli space proves that the moduli space is
smooth of the expected dimension (if not empty) \cite{F}.
If $H_1$ and $H_2$ are two generic polarizations in the same chamber,
then every sheaf that is $H_1$-stable is also $H_2$-stable, and we can
identify the corresponding moduli spaces \cite{F,Q1,Q2}.
If we restrict our attention to some particular class of algebraic
surfaces we can obtain more properties of the moduli space (without
the condition on $c_2$).
The moduli space of sheaves on a $K3$ surface with a generic polarization
has been studied by Mukai
\cite{M} when the expected dimension is 0 or 2. In particular he
proved that the moduli space is irreducible. O'Grady \cite{O3} has proved
irreducibility for any $c_2$ (and any rank), as well as having
obtained results about the Hodge structure.
In chapter \ref{k3} (theorem II), we give a new
proof of the irreducibility for any $c_2$ (and rank 2) based on our
results about Brill-Noether theory on singular curves.
Now let's consider the case $S=\mathbb{P}^2$ and rank 2.
Tensoring with a line bundle we can assume that $c_1$ is either $0$ or
$1$. If $c_1=0$, then the moduli space is empty for $c_2<2$ and is
irreducible of the expected dimension for $c_2\geq2$. If $c_1=1$ then
the moduli space is empty for $c_2<1$ and irreducible of the expected
dimension for $c_2\geq1$.
In the case $S=\mathbb{P}^1\times\mathbb{P}^1$ (and also rank 2), if we take a
$(c_1,c_2)$-generic polarization, the moduli space is also known to be
either empty or irreducible. We will generalize this result for any
del Pezzo surface (i.e., a surface with $-K_S$ ample) in chapter
\ref{dp} (theorem III).
\chapter{Connectivity of Brill-Noether loci for singular curves}
\label{bn}
\setcounter{proposition}{0}
Recall (see chapter \ref{Preliminaries}) that we are going to study the
Brill-Noether locus of singular irreducible curves that lie on a
smooth surface $S$ satisfying
\bigskip
$h^1(\SO_S)=0$ \textit{, and }$-K_S$ \textit{ is generated by global
sections.}\hfill (*)
\bigskip
Now we state the theorem that we are going to prove:
\smallskip
\noindent\textbf{Theorem I.}
\textit{ Let $C$ be a reduced irreducible curve of arithmetic genus
$p_a$ that lies in a surface $S$
satisfying \textup{(*)}.
Let $\overline J^d(C)$, $d>0$, be the compactification of the generalized
Jacobian. Then for any $r \geq 0$ such that $\rho(r,d)= p_a -
(r+1)(p_a-d+r)>0$, the generalized Brill-Noether subvariety $\overline
W^r_d(C)$ is nonempty and connected.}
\smallskip
\begin{remark}
\label{remark}
\textup{If $r \leq d-p_a$, by Riemann-Roch inequality we have
$\overline W^r_d(C)
= \overline
J^d(C)$, and this is connected. Then, in order to prove theorem I we
can assume
$r>d-p_a$. Note that if $A$ corresponds to a point in $\overline
W^r_d(C)$ with $r>d-p_a$, then by Riemann-Roch theorem $h^1(A) > 0$.}
\end{remark}
\bigskip
\textbf{Outline of the proof of theorem I}
\bigskip
Note that $\overline W^r_d(C)$ is the fiber of $q$ over the point $u_0
\in |C|$ corresponding to the curve $C$.
Let $U$ be the open subset of $|C|$ corresponding to integral curves,
and $V$ the subset of smooth curves. Define $(\overline
{\mathcal{W}}^r_d)_V$ to be the Brill-Noether locus of sheaves with smooth
support, i.e. $(\overline{\mathcal{W}}^r_d)_V=q^{-1}(V)$.
By \cite{F-L}, the restriction
$q^{}_V: (\overline {\mathcal{W}}^r_d)_V \to V$ has connected fibers. We want to
use this fact to show that $\overline W^r_d(C)$ is connected. Let $A$
be a rank one
torsion-free sheaf on $C$ corresponding to a point in $\overline
W^r_d(C)$,
and assume that it is generated by global sections. We think of $A$
as a torsion sheaf on $S$. Then we have a
short exact sequence on $S$
$$
0 \to E \stackrel{f_0} \to H^0(A) \otimes \SO_S \to A \to 0,
$$
where the map on the right is evaluation. This sequence has already
appeared in the literature (see \cite{La}, \cite{Ye}).
Our idea is to deform $f_0$ to a family $f_t$. The cokernel of $f_t$
will define
a family of sheaves $A_t$ with $h^0(A_t) \geq h^0(A)$ (because
$h^0(E)=0$), and then for each $t$ the point in $\overline
{\mathcal{J}}^d_{|C|}$ corresponding to $A_t$ lies in $\overline {\mathcal{W}}^r_{d,|C|}$.
Assume that there are 'enough' homomorphisms from $E$ to $H^0 \otimes
\SO_S$ and the family $f_t$ can be chosen
general enough, so that for a general $t$, the support of $A_t$ is
smooth (the details of this construction are in section
\ref{bnParticular case}). The family $A_t$ shows that the point
in $\overline
W^r_d(C)$ corresponding to $A$ is in the closure of $(\overline
{\mathcal{W}}^r_d)_V$ in $\overline {\mathcal{J}}^d_{|C|}$. It can be shown that this
closure has connected
fibers. Let $X$ be the fiber over $u_0$ of this closure. Then all sheaves
for which this construction works are in the
connected component $X$ of $\overline
W^r_d(C)$. If this could be
done for all sheaves in $\overline
W^r_d(C)$ this would finish the proof, but there are sheaves for which
this construction doesn't work. For these sheaves we show in
section \ref{bnGeneral case} that they can be deformed (keeping the
support $C$ unchanged) to a sheaf for which a refinement of this
construction works.
This shows that all points in $\overline W^r_d(C)$ are in the connected
component $X$.
\section{The main lemma}
\label{bnMain lemma}
The precise statement that we will use to prove theorem I is
the following lemma.
\begin{lemma}
\label{bn0.2}
Let $C$ be an integral complete curve in a surface $S$. Assume that for each
rank one
torsion-free sheaf $A$ on $C$ with $h^0(A)=r+1>0$ and $\deg(A)=d>0$
such that $\rho(r,d)>0$ we
have the following data:
\smallskip
(a) A family of curves ${\mathcal{C}}$ in $S$ parametrized by an irreducible
curve $T$ (not necessarily complete).
(b) A connected curve $T'$ (not necessarily irreducible nor complete)
with a map $\psi:T' \to T$.
(c) A rank one torsion-free sheaf ${\mathcal{A}}$ on ${\mathcal{C}}' = {\mathcal{C}} \times _T T'$,
flat over $T'$,
inducing rank one torsion-free sheaves on the fibers of ${\mathcal{C}}' \to T'$.
\smallskip
Assume that the following is satisfied:
\smallskip
(i) ${\mathcal{C}}|_{t^{}_0} \cong C$ for some $t^{}_0 \in T$, ${\mathcal{C}}|_t$ is linearly
equivalent to $C$ for all $t\in T$, and ${\mathcal{C}}|_t$
is smooth for $t \neq t^{}_0$.
(ii) One irreducible component of $T'$ is a finite cover of $T$,
and the rest of the components of $T'$ are mapped to $t^{}_0 \in T$.
(iii) ${\mathcal{A}}|_{t'_0} \cong A$ for some $t'_0 \in T'$ mapping to
$t^{}_0 \in T$.
(iv) $h^0({\mathcal{A}}|_{t'}) \geq r+1$ for all $t' \in T'$.
\smallskip
Then the generalized
Brill-Noether subvariety $\overline W^r_d(C)$ of the compactified
generalized Jacobian $\overline J^d(C)$ is connected.
\end{lemma}
\begin{proof}
We will use the notation introduced in the previous section. The map
$q:\overline {\mathcal{W}}^r_{d,|C|} \to U$ is a projective morphism. Recall
that $\overline W^r_d(C)$ is the fiber of $q$ over $u_0$, where $u_0$
is the point corresponding to $C$. By
\cite{F-L} the morphism $q$ has connected
fibers over $V$, thus a general fiber of $q$ is connected, and we
want to prove that the fiber over $u_0 \in U$ is also
connected.
$$
\begin{array}{ccc}
\overline W^r_d(C) & \hookrightarrow & \overline {\mathcal{W}}^r_{d,|C|} \\
\fcndown{} & {} & \fcndown{q} \\
u_0 & \hookrightarrow & U
\end{array}
$$
Let $\overline {\mathcal{W}}^r_{d,|C|} \stackrel{q'} \to U' \stackrel{g} \to U$ be
the Stein factorization of $q$ (see \cite[III Corollary 11.5]{H}),
i.e. $q'$ has connected fibers and
$g$ is a finite morphism. A general fiber of $q$ is connected, and
then $U'$ has one irreducible component $Z$ that maps to $U$
birationally.
The subset $U$ is open in $|C|$ and hence normal, the restriction
$g|_Z:Z \to U$
is finite and birational, $Z$ and
$U$ are integral, thus by
Zariski's main theorem (see \cite[III Corollary 11.4]{H}) each fiber
of $g|_Z$ consists of just one point. Let
$z_0$ be the point of $Z$ in the fiber $g^{-1}(u_0)$.
\textbf{\textit{Claim.}} Let $y_0$ be a point in the fiber
$q^{-1}(u_0)=
\overline W^r_d(C)$. Then $y_0$ is
mapped by $q'$ to $z_0$.
This claim implies that that $\overline
W^r_d(C)$ is connected. Now we will prove the claim.
Let $A$ be the sheaf on $S$ corresponding to the point $y_0$. Let $T'$, $T$,
$t'_0\in T'$, $t^{}_0\in T$, $\psi:T' \to T$ be the
curves points and morphism given by the hypothesis of the lemma.
Let $\phi:T' \to
\overline{\mathcal{J}}^d_{|C|}$ be the morphism given by the universal property
of the moduli space $\overline{\mathcal{J}}^d_{|C|}$. Item (iv) imply that the
image of $\phi$ is in $\overline {\mathcal{W}}^r_{d,|C|}$.
\centerline{
\xymatrix{
{}\save[]+<0.7cm,0cm>*{t'_0\in} \restore
& T' \ar[d]_\psi \ar[r]^\phi & \overline{\mathcal{W}}^r_{d,|C|}
\ar[d]^{q'} \ar@(dl,ul)[dd]_q
& {}\save[]+<-0.7cm,0cm>*{\ni y_0} \restore\\
{}\save[]+<0.7cm,0cm>*{t^{}_0\in} \restore& T & U' \ar[d]^{g}&
Z \ar@{_{(}->}[l] \ar[dl]^{g|_Z} & {}\save[]-<0.6cm,0cm>*{\ni z^{}_0} \restore\\
& & U & {}\save[]+<-0.8cm,0cm>*{\ni u^{}_0} \restore}
}
\begin{figure}[ht]
\centerline{\epsfig{file=figure.eps,height=3in,width=4.5in,
bbllx=0in,bblly=9.187in,bburx=4.5in,bbury=12.187in,clip=}}
\end{figure}
The restriction of $q' \circ \phi$ to $T' \setminus \psi^{-1}(t^{}_0)$
maps to $Z$, because for $t' \in T' \setminus
\psi^{-1}(t^{}_0)$ the sheaf ${\mathcal{A}}|_{t'}$ has smooth support by item (i).
Items (c) and (i) imply that $g\circ q' \circ \phi
(\psi^{-1}(t^{}_0))=u_0$ . Thus $q' \circ \phi (\psi^{-1}(t^{}_0))$ is a
finite number of points (because it is in the fiber of $g$ over $u_0$).
The facts that $q' \circ \phi(T' \setminus \psi^{-1}(t^{}_0))$ is in
$Z$ and that $q' \circ \phi (\psi^{-1}(t^{}_0))$ is a
finite number of points imply that $q' \circ \phi (\psi^{-1}(t^{}_0))$ is
also in $Z$ (because by item (b) the curve $T'$ is connected and thus also its
image under $q'\circ \phi$), and in fact $q' \circ \phi
(\psi^{-1}(t^{}_0))=z_0$ because $q' \circ \phi (\psi^{-1}(t^{}_0))$ is in
the fiber of $g$ over $u_0$.
By item (ii), $t'_0 \in \psi^{-1}(t^{}_0)$. Then $q' \circ \phi
(t'_0)=z_0$, and by item (iii) we have $y_0=\phi(t'_0)$, then
$q'(y_0)= q'(\phi(t'_0))=z_0$ and the claim is proved.
\end{proof}
In section \ref{bnParticular case} we will construct this family under
some assumptions on $A$ (proposition \ref{bn1.5}),
and in section \ref{bnGeneral case} we will show how to use that to
construct a family for any $A$. Note that because of remark \ref{remark} we can
assume $h^1(A)>0$.
\section{A particular case}
\label{bnParticular case}
Given a rank one torsion-free sheaf $A$ on an integral curve lying on
a surface $S$, we
define another sheaf $A^*$ that is going to be some sort of dual. Let
$j$ be the inclusion of the curve $C$ in the surface $S$. We
define $A^*$ as follows:
$$
A^*= Ext^1(j_*A,\omega_S).
$$
The operation $A \to A^*$ is a contravariant functor.
Note that the support of $A^*$ is
$C$. It will be clear from the
context when we are referring to $A^*$ as a torsion sheaf on $S$ or as a sheaf
on $C$. In the case in which $A$ is a line bundle, then $A^*=A^\vee
\otimes \omega_C$. Now we prove some properties of this ``dual''.
\begin{lemma}
\label{bn1.1}
Let $A$ be a rank one torsion-free sheaf on an integral curve lying on
a surface. Then $A^{**}=A$
\end{lemma}
\begin{proof}
First observe that if $L$ is a line bundle on $C$, then $(A \otimes
L)^* \cong A^* \otimes L^\vee$.
To see this, take an injective resolution of $\omega_S$
$$
0 \to \omega_S \to {\mathcal{I}}_0 \to {\mathcal{I}}_1 \to \cdots
$$
Now we use this resolution to calculate the $Ext$ sheaf.
$$
Ext^1(A\otimes L,\omega_S) = h^1(Hom(A \otimes L,{\mathcal{I}}_\bullet))=
h^1(L^\vee\otimes Hom(A,{\mathcal{I}}_\bullet)) =
$$
$$
=L^\vee\otimes h^1
(Hom(A,{\mathcal{I}}_\bullet))=L^\vee\otimes Ext^1(A,\omega_S)
$$
The third equality follows from the fact that $Hom(A,{\mathcal{I}}_\bullet)$
is supported on the curve and $L^\vee$ is locally free
It follows that $(L\otimes A)^{**}\cong L\otimes A^{**}$, and then
proving the lemma for $A$ is equivalent to proving it for $L\otimes A$.
Multiplying with an appropriate very ample line bundle, we can assume
that $A$ is generated by global sections. Then we have an exact
sequence
\begin{equation}
0 \to E \to V \otimes \SO_S \to A \to 0,
\label{eqbn1.1}
\end{equation}
where $V=H^0(A)$. The following lemma proves that $E$ is locally free.
\begin{lemma}
\label{bn1.2}
Let $M$ be a torsion-free sheaf on an integral curve $C$ that lies on a
smooth surface $S$. Let $j:C \to S$ be the inclusion. Let $F$ be a
locally free sheaf on the surface. Let
$f:F \to j_* M$ be a surjection. Then
the elementary transformation $F'$ of $F$, defined as the kernel of
$f$
\begin{equation}
\label{eqbn1.1bis}
0 \to F' \to F \stackrel{f}{\to} j_* M \to 0,
\end{equation}
is a locally free sheaf.
\end{lemma}
\begin{proof}
$M$ is torsion-free sheaf on $C$, and then $j_* M$ has depth at least one,
and because $S$ is smooth of dimension 2, this implies that the
projective dimension of $j_* M$ is at most one
($Ext^i(j_* M,{\mathcal{O}}_S)=0$
for $i \geq 2$). Now $Ext^i(F,{\mathcal{O}}_S)=0$ for $i \geq 1$ because $F$ is
locally free, and then from the
exact sequence \ref{eqbn1.1bis}, we get
$$
0 \to Ext^i(F',\SO_S) \to Ext^{i+1}(j_* M,\SO_S) \to 0,
\quad i\geq 1,
$$
and then $Ext^i(F',{\mathcal{O}}_S)=0$ for $i \geq 1$, and
this implies that $F'$ is locally free.
\end{proof}
In particular, $E^{\vee\vee}=E$. Applying the
functor $Hom(\cdot,\omega_S)$ twice to the sequence \ref{eqbn1.1}, we get
$$
0 \to E \to V \otimes \SO_S \to A^{**} \to 0.
$$
Comparing with \ref{eqbn1.1} we get the result (because the map on the
left is the same for both sequences).
\end{proof}
\begin{lemma}
\label{bn1.3}
$\operatorname{Ext}^1(A,\omega_S) \cong H^0(A^*)$, and this is dual to $H^1(A)$.
\end{lemma}
\begin{proof}
The local to global spectral sequence for Ext gives the following
exact sequence
$$
0 \to H^1(Hom(A,\omega_S)) \to \operatorname{Ext}^1(A,\omega_S) \to H^0(A^*)
\to H^2(Hom(A,\omega_S))
$$
But $Hom(A,\omega_S)=0$ because $A$ is supported in $C$ and then the first
and last terms in the sequence are zero and we have the desired
isomorphism.
\end{proof}
Now we will prove a lemma that we will need. The proof can also be
found in \cite{O}, but for convenience we reproduce it here.
\begin{lemma}
\label{bn1.4}
Let $E$ and $F$ be two vector bundles of rank $e$ and $f$ over a
smooth variety $X$. Assume that
$E^\vee \otimes F$ is generated by global sections.
If $\phi: E \to F$ is a sheaf morphism, we define $D_k(\phi)$ to be
the subset of X where $\operatorname{rk}(\phi_x) \leq k$ (there is an obvious
determinantal description of $D_k(\phi)$ that gives a scheme
structure). Let $d_k$ be the expected dimension of $D_k(\phi)$
$$
d_k=\dim (X) - (e-k)(f-k).
$$
Then there is a Zariski dense set $U$ of $\operatorname{Hom}(E,F)$ such that if
$\phi\in U$, then
we have that $D_k(\phi) \setminus
D_{k-1}(\phi)$ is smooth of the expected dimension (if $d_k < 0$ then it
will be empty).
\end{lemma}
\begin{proof}
Let $M_k$ be the set of matrices of dimension $e \times f$ and of rank at
most $k$
(there is an obvious determinantal description that gives a
scheme structure to this subvariety). It is well known that the
codimension of $M_k$ in the space of all matrices is
$(e-k)(f-k)$, and that the singular locus of $M_k$ is $M_{k-1}$.
Now, because $E^\vee \otimes F$ is generated by global sections,
we have a surjective morphism
$$
H^0(E^\vee \otimes F) \otimes {\mathcal{O}}_X \to E^\vee \otimes F
$$
that gives a morphism of maximal rank between the varieties defined as
the total space of the previous vector bundles
$$
p: X \times H^0(E^\vee \otimes F) \to
\mathbb{V} (E^\vee \otimes F).
$$
Define $\Sigma_k \subset \operatorname{V}(E^\vee \otimes F)$ as
the set such that $\operatorname{rk}(\phi_x) \leq k$. The fiber of $\Sigma_k$ over
any point in $X$ is obviously $M_k$. Define $Z_k$ to be $p^{-1}(\Sigma_k)$. The
fact that $p$ has maximal rank implies that $Z_k$ has codimension
$(e-k)(f-k)$ in $X \times H^0(E^\vee \otimes F)$ and that the
singular locus of $Z_k$ is $Z_{k-1}$.
Now observe that the restriction of the projection
$$
q|_{Z_k \setminus Z_{k-1}} :Z_k \setminus Z_{k-1} \to H^0(E^\vee
\otimes F)
$$
has fiber ${q|_{Z_k \setminus Z_{k-1}}}^{-1}(\phi) \cong D_k(\phi) \setminus
D_{k-1}(\phi)$. Finally, by generic smoothness, for a general $\phi \in
H^0(E^\vee \otimes F)$ this is smooth of the expected dimension
(or empty).
\end{proof}
Now we will construct the deformation of $A$ that we described in
the section \ref{bnMain lemma}
in the particular case in which both $A$ and $A^*$ are generated by
global sections.
\begin{proposition}
\label{bn1.5}
Let $A$ be a rank one torsion-free sheaf on an integral curve $C$
lying on a surface $S$ with $h^1(\SO_S)=0$ and $-K_S$ generated by
global sections. Denote $j:C \hookrightarrow S$. If $A$ and $A^*$ are both generated by
global sections, then there exists a
(not necessarily complete) smooth irreducible curve $T$ and a sheaf
${\mathcal{A}}$ on $S
\times T$ flat over $T$, such that
\smallskip
(a) the sheaf
induced on the fiber of $S \times T \to T$ over some $t^{}_0 \in T$ is $j_* A$
(b) the sheaf $A_t$ induced on the fiber over any $t\in T$ with $t \neq
t^{}_0$ is supported on a
smooth curve $C_t$ and it is a rank one torsion-free sheaf when
considered as a sheaf on $C_t$
(c) $h^0(A_t) \geq h^0(A)$ for every $t\in T$.
\end{proposition}
Note that these are the hypothesis of lemma \ref{bn0.2} for the particular
case in which both $A$ and $A^*$ are generated by global sections. We
will lift this condition in the next section.
\begin{proof}
The fact that $A$ is generated by global sections implies that there is
an exact sequence
$$
0 \to E \stackrel{f_0}{\to} V \otimes \SO_S \to A \to 0 \; \; \; \; \;
V=H^0(A),
$$
with $E$ locally free (by proposition \ref{bn1.2}). Taking global sections in
this sequence we see that $H^0(E)=0$, because
$$
0 \to H^0(E) \to V \stackrel{\cong}{\to} H^0(A).
$$
Consider a curve $T$ mapping to $\operatorname{Hom}(E,V \otimes \SO_S)$ with $t^{}_0 \in
T$ mapping to $f_0$ (so that item (a) is satisfied). Denote by $f_t$
the morphism given for $t \in T$
by this map. After shrinking $T$ we can assume that $f_t$ is still
injective. Let $\pi_1$ be the projection of $S \times T$ onto the
first factor and let ${\mathcal{E}}=\pi_1^*E$. Using the universal sheaf and
morphism on $\operatorname{Hom}(E,V \otimes
\SO_S)$ we can construct (by pulling back to $S \times T$) an exact
sequence on $S \times T$
$$
0 \to {\mathcal{E}} \stackrel{f}{\to} V \otimes {\mathcal{O}}_{S \times T} \to {\mathcal{A}} \to 0
$$
that restricts for each $t$ to an exact sequence
\begin{equation}
\label{eqbn1.2}
0 \to E \stackrel{f_t}{\to} V \otimes \SO_S \to A_t \to 0,
\end{equation}
where $A_t$ is a sheaf supported in the degeneracy locus of $f_t$. It
is clear that $\deg(A)=\deg(A_t)$.
Now we are going to prove that if the curve $T$ and the mapping to
$\operatorname{Hom}(E, V \otimes \SO_S)$ are chosen generically, the quotient of the map
gives the desired deformation.
The flatness of ${\mathcal{A}}$ over $T$ follows from the fact that it has a
short resolution and from the local criterion of flatness (we
can apply \cite[III Lemma 10.3.A]{H}).
The condition on $h^0(A_t)$ follows because $H^0(E)=0$ and we have a
sequence
$$
0 \to H^0(E)=0 \to V \to H^0(A_t),
$$
and then $h^0(A) \leq h^0(A_t)$. This proves item (c).
Using the long exact sequence obtained
by applying $Hom(\cdot,\SO_S)$ to \ref{eqbn1.2}, and the fact that $E$ is
locally free, we obtain that
$Ext^i(A_t,\SO_S)$ vanishes for $i \geq 2$, and so the projective dimension of
$A_t$ is 1, and this implies that $A_t$, when considered as a sheaf on
its support $C_t$, is torsion-free.
We have to prove that we can choose the curve $T$ and the map
to $\operatorname{Hom}(E,V \otimes \SO_S)$ such that $C_t$ is smooth for $t \neq t^{}_0$
(here we will use that $A^*$ is
generated by global sections).
First note that $Ext^1(A,\SO_S)$ is generated by global sections,
because $Ext^1(A,\SO_S)=A^*\otimes \omega_S^{-1}$, and both $A^*$ and
$\omega_S^{-1}$ are generated by global sections.
Now we see that $E^\vee$ is generated by global sections, because
we have
$$
0 \to V^\vee \otimes \SO_S \to E^\vee \to Ext^1(A,\SO_S) \to 0,
$$
$Ext^1(A,\SO_S)$ is generated by global sections and $H^1(V^\vee \otimes
\SO_S)=0$. Then $E^\vee \otimes (V \otimes \SO_S)$ is generated by
global sections.
Now apply lemma \ref{bn1.4} with $F=V \otimes \SO_S$. Then $n=m=r+1$,
$k=r$ and the
expected dimension is 1. And the lemma gives that for $\phi$ in a
Zariski open subset of
$\operatorname{Hom}(E, V\otimes \SO_S)$, the degeneracy locus $D_r(\phi)$ of $\phi$ is
smooth away from the locus $D_{r-1}(\phi)$ where $\phi$ has rank
$r-1$, but again by
lemma \ref{bn1.4} the locus $D_{r-1}(\phi)$ is empty. This proves item
(b).
\end{proof}
\section{General case}
\label{bnGeneral case}
Now we don't assume that $A$ satisfies the properties of the
particular case (i.e., $A$ and $A^*$ now might not be generated by
global sections). We will find a new sheaf that satisfies those
conditions. We know how to deform this new sheaf, and we will show how
we can use this deformation to construct a deformation of the original $A$.
We start with a rank one torsion-free sheaf $A$ with $h^0(A)$,
$h^1(A)>0$ on an integral curve
$C$ lying on a surface.
First we define $A'$ as the base point free part of $A$, i.e. $A'$ is
the image of the evaluation map
$$
H^0(A)\otimes {\mathcal{O}}_C \to A.
$$
We have assumed that $h^0(A)>0$, and then $A'$ is a (nonzero) rank one
torsion-free sheaf.
Obviously, $H^0(A)=H^0(A')$. We have a short exact sequence
$$
0 \to A' \to A \to Q \to 0,
$$
where $Q$ has support of dimension 0. Now consider ${A'}^*$,
and define $B$ to be
its base point free part. We have $h^0({A'}^*)=h^1(A')=h^1(A)+h^0(Q)
\geq h^1(A)>0$. The first equality by lemma \ref{bn1.3}, and the last
inequality by assumption. Then $B$ is a (nonzero) rank one
torsion-free sheaf. Finally define $A''$ to be equal to $B^*$.
\begin{lemma}
\label{bn2.1}
Both $A''$ and ${A''}^*$ are generated by global sections.
\end{lemma}
\begin{proof}
Since $B$ is the base point free part of ${A'}^*$, we have a sequence
$$
0 \to B \to {A'}^* \to R \to 0
$$
where $R$ has support of dimension zero. Applying $Hom(\cdot,\omega_S)$
we get
$$
0 \to A' \to B^*=A'' \to \widetilde R \to 0 \; \; \; \; \; \widetilde
R=Ext^2(R,\omega_S),
$$
whose associated cohomology long exact sequence gives
$$
0 \to H^0(A') \to H^0(B^*) \to H^0(\widetilde R) \to H^1(A') \to
H^1(B^*) \to 0.
$$
To see that $A''$ is generated by global sections, it is enough to
prove that the last map is an isomorphism, because
then the first three terms make a short exact sequence, and the fact
that $A'$ and $\widetilde R$ are generated by global sections (the first
by definition, the second because its support has dimension zero) will
imply that $B^*$ (that is equal to $A''$ by definition) is generated
by global sections.
To prove that the last map is an isomorphism, we only need to show
that $h^1(A')=h^1(B^*)$, and this is true because
$$
h^1(A')=h^0({A'}^*)=h^0(B)=h^0(B^{**})=h^1(B^*).
$$
The first equality is by lemma
\ref{bn1.3}, the second because $B$ is the base point free part of
${A'}^*$, the
third by lemma \ref{bn1.1}, and
the last again by lemma \ref{bn1.3}.
To see that ${A''}^*$ is generated by global sections, note that by
definition ${A''}^*=B^{**}=B$, and this is generated by global sections.
\end{proof}
We started with a rank one torsion-free sheaf $A$ with $h^0(A)$ and
$h^1(A)>0$, and we
have constructed new sheaves $A'$ and $A''$ with (nontrivial) maps $A'
\to A$ and $A' \to A''$. They give rise to exact sequences
\begin{eqnarray}
0 \to A' \to A \to Q \to 0 \nonumber \\
\label{eqbn2.1}
0 \to A' \to A'' \to \widetilde Q \to 0
\end{eqnarray}
\begin{lemma}
\label{equalit}
With the previous definitions we have $h^0(A')=h^0(A)$
and $h^1(A'')=h^1(A')$.
\end{lemma}
\begin{proof}
By construction $h^0(A')=h^0(A)$ and $h^0({A''}^*)=h^0({A'}^*)$. By
lemma \ref{bn1.3} this last equality is equivalent to
$h^1(A'')=h^1(A')$.
\end{proof}
As $A''$ and ${A''}^*$ are generated by global sections, then by
proposition \ref{bn1.5} the sheaf $A''$ can be
deformed in a family $A''_t$ in such a way that the support of a
general member of the
deformation is smooth. The idea now is to find (flat) deformations of
$A'$ and $A$, so that for every $t$ we still have maps like \ref{eqbn2.1}.
{}From the existence of these maps we will be able to obtain the
condition that $h^0(A_t) \geq h^0(A)$,
then we will be able to apply lemma \ref{bn0.2} and then
theorem I will be proved. The details are in
section \ref{bnProof of the main theorem}. We will start by showing
how the condition on
$h^0(A_t)$ is obtained, and then how we can find the deformations of
$A'$ and $A$.
\begin{proposition}
\label{bn2.2}
Let $A$, $A'$, $A''$ be rank one torsion-free sheaves on an integral
curve $C$. Assume that they fit into exact sequences like \ref{eqbn2.1}
and that $h^0(A')=h^0(A)$ and $h^1(A')=h^1(A'')$.
Let $P$ be a curve (not necessarily complete), and
let ${\mathcal{A}}$, ${\mathcal{A}}'$, and ${\mathcal{A}}''$ be sheaves on $S \times P$, flat over
$P$, inducing for each
$p \in P$ rank one torsion-free sheaves $A^{}_p$, $A'_p$, $A''_p$,
supported on a curve $C_p$ of $S$, where $A^{}_{p^{}_0}=A$, $A'_{p^{}_0}=A'$,
and
$A''_{p^{}_0}=A''$ for some $p^{}_0 \in P$. Assume that $h^0(A''_p) \geq
h^0(A''_{p^{}_0})$ for all $p\in P$ and that we have
short exact sequences
$$
0 \to {\mathcal{A}}' \to {\mathcal{A}} \to {\mathcal{Q}} \to 0
$$
$$
0 \to {\mathcal{A}}' \to {\mathcal{A}}'' \to \widetilde {\mathcal{Q}} \to 0
$$
with ${\mathcal{Q}}$ and $\widetilde {\mathcal{Q}}$ flat over $P$ (i.e., the induced
sheaves $Q_p$, $\widetilde Q_p$ have constant length, equal to $l(Q)$
and $l(\widetilde Q)$ respectively).
Then we have $h^0(A^{}_p) \geq h^0(A^{}_{p^{}_0})$ for all $p\in P$.
\end{proposition}
\begin{proof}
For each $p \in P$ we have sequences
$$
0 \to A'_p \to A^{}_p \to Q^{}_p \to 0
$$
$$
0 \to A'_p \to A''_p \to \widetilde Q^{}_p \to 0.
$$
The maps on the left are injective because they are nonzero and
the sheaves have rank one and are torsion-free.
Using the associated long exact sequences and the hypothesis we
have
$$
h^0(A^{}_p) \geq h^0(A'_p) \geq h^0(A''_p)-l(\widetilde Q^{}_p) \geq h^0(A'')-
l(\widetilde Q)=h^0(A')=h^0(A).
$$
\end{proof}
It only remains to prove that those sheaves can be ``deformed along'',
and that those deformations are flat, i.e. that given $A$, $A'$ and
$A''$ we can construct ${\mathcal{A}}'$ and ${\mathcal{A}}''$. This is proved in the following
propositions.
\begin{proposition}
\label{bn2.3}
Let $L$ and $M$ be rank one torsion-free sheaves on an integral curve
$C$ that lies on a surface $S$. Assume we have
a short exact sequence
\begin{equation}
\label{eqbn2.2}
0 \to L \to M \to Q \to 0.
\end{equation}
Assume furthermore that we are given a sheaf ${\mathcal{M}}$ on $S \times P$
(where $P$
is a connected but not necessarily irreducible curve) that is a
deformation of $M$, flat over $P$. I.e., ${\mathcal{M}}|_{p^{}_0}
\cong M$ for some $p^{}_0 \in P$, and for all $p\in P$ we have that
$M_p={\mathcal{M}}|_p$ are torsion-free
sheaves on $C_p$,
where $C_p$ is a curve on $S$.
Then, there is a connected curve $P'$
with a map $f:P' \to P$ and a
sheaf ${\mathcal{L}}'$ over $S \times P'$ with the following properties:
One irreducible component of $P'$ is a finite cover of $P$ and the
rest of the components map to $p^{}_0 \in
P$. The sheaf ${\mathcal{L}}'$ is a deformation of $L$, in the sense that
${\mathcal{L}}'|_{p'_0} \cong L$
for some $p'_0 \in P'$ mapping to $p^{}_0 \in P$, the sheaf ${\mathcal{L}}'$ is flat over
$P'$ and induces rank one torsion-free sheaves on the fibers over
$P'$. And if we define ${\mathcal{M}}'$ to be the pullback of ${\mathcal{M}}$ to $S
\times P'$, there exists an exact sequence
$$
0 \to {\mathcal{L}}' \to {\mathcal{M}}' \to {\mathcal{Q}}' \to 0,
$$
inducing short exact sequences
$$
0 \to L'_{p'} \to M'_{p'} \to Q'_{p'} \to 0
$$
for every $p'\in P'$.
\end{proposition}
\begin{proof}
If the support of $Q$ were in the
smooth part of the curve, we would have $M \cong L \otimes {\mathcal{O}}_C(D)$, with
$D$ an effective divisor of degree $l(Q)$. Then, if we are given a
deformation $M_p$ of $M$, we only need to find a deformation $D_p$ of the
effective divisor $D$, with the only condition that $D_p$ is an
effective divisor on $C_p$, with degree $l(Q)$. This can easily be
done if we are in the analytic category. In general we might need to
do a base change of the parametrizing curve $P$ and we will obtain a
finite cover $P'$ of $P$ (What we are doing is
moving a dimension zero and length $l(Q)$ subscheme of $S$, with the
only restriction that for each $p$ the corresponding scheme is in $C_p$).
Then we only need to define $L_{p'}=M_{p'} \otimes
{\mathcal{O}}_{C_{p'}}(-D_{p'})$ and the
proposition would be proved (with $P'$ a finite cover of $P$).
To be able to apply this, we will have to make first a deformation of
$L$, keeping $M$ fixed, until we get $Q$ to be supported in the smooth
part of $C$ (the curve $C$ also remains fixed in this deformation).
This is the reason for the need of the curve $P'$ with some
irreducible components mapping to $p^{}_0$.
We will prove this by induction on the length of the intersection of
the support of $Q$ and the singular part of $C$.
\begin{lemma}
\label{bn2.4}
Let $L$ and $M$ be rank one torsion-free sheaves on an integral curve
$C$ that lies on a surface $S$. Assume we have
a short exact sequence
$$
0 \to L \to M \to Q \to 0.
$$
Assume that $Q=R \oplus Q'$ where $Q'$ has length $l(Q)-1$ and it is
supported in the smooth part of $C$, and $R$ has length one as it is
supported in a singular point of $C$ (``the length of the
intersection of the
support of $Q$ and the singular part of $C$ is one'').
Then there is a flat deformation $L_y$ of $L$ parametrized by a
connected curve $Y$ (it might not be
irreducible) such that $L_{y_0}=L$ for
some $y_0\in Y$ and for every $y\in Y$ there is an exact sequence
$$
0 \to L_y \to M \to Q_y \to 0
$$
and there is some $y_1\in Y$ such that the support of $Q_{y_1}$ is in
the smooth part of $C$.
\end{lemma}
\begin{proof}
In this situation, the exact sequence \ref{eqbn2.2}
gives rise to another exact sequence
$$
0 \to L \otimes I_Z^\vee \to M \to R \to 0
$$
where the map on the right is the composition of $M \to Q$ and the
projection $Q \to R$, and we denote by $I_Z$ the ideal sheaf of the
support $Z$ of $Q'$.
Because $Z$ is in the smooth part of $C$, $I_Z$ is an invertible sheaf.
Note that $Q'$ is the quotient of ${\mathcal{O}}_C$ by this ideal sheaf.
Define $\widehat L$ to be $L \otimes I_Z^\vee$. If we know how to
make a flat deformation $\widehat L_y$ of $\widehat L$ so that the quotient
$R_y$ is
supported in the smooth part of $C$ for some $y_1\in Y$,
then we can construct a deformation $L_y$ of $L$ defined as
$$
L_y = \widehat L_y \otimes I_Z.
$$
Note that this deformation is also flat. The cokernel $Q_y$ of $L_y
\to M$ is supported in the smooth part of $C$ for the
points $y\in Y$ for which $R_y$ is supported in the smooth part of $C$.
This shows that to prove the lemma we can assume that $Q$ has length
one and its support is a singular point of $C$, i.e. $Q={\mathcal{O}}_x$, where
$x$ is a singular point of $C$.
Consider the scheme $\operatorname{Quot}^1(M)$ representing the functor of quotients
of $M$ of length 1. If the support $x$ of the quotient $Q$ is in the
smooth part of $C$, then there is only one surjective map (up to
scalar) because $\dim \operatorname{Hom} (M,Q)=1$, whose kernel is $M\otimes
{\mathcal{O}}_C(-x)$.
If $x$ is in the singular part, then in general $\dim\operatorname{Hom}(M,Q)>1$, and
the quotients are parametrized by $\mathbb{P}\operatorname{Hom}(M,Q)$ (the universal bundle
is flat
over $\mathbb{P}\operatorname{Hom}(M,Q)$). We
want to show that $\operatorname{Quot}^1(M)$ is connected by constructing a flat
family of quotients $M \to \widetilde Q_{\tilde c}$ (the family $\widetilde
Q_{\tilde c}$ will
be parametrized by an open set of the normalization $\widetilde C$ of $C$)
such that for a general ${\tilde c}$ the support
of $\widetilde Q_{\tilde c}$ is in the smooth part of $C$, and for some point
$\tilde c_0$ the support of
$\widetilde Q_{\tilde c_0}$ is a singular point of $C$.
Consider the normalization $\widetilde C$ of $C$, and let $F$ be an
open set of $\widetilde C$
\begin{equation}
\label{eqbn2.3}
\CD
F @>j>> C \times F \\
@. @V{\pi_1}VV \\
@. C \\
\endCD
\end{equation}
Where $\pi_1$ is the projection to the first factor and $j=(\nu,i)$,
the morphism
$\nu:F\hookrightarrow \widetilde C\to C $ being the restriction to $F$ of the
normalization map and $i$ the
identity map. Note that $j$
is a closed immersion, and its image is just $C \times_C F \cong F$.
Let $\tilde c_0$ be a point of $\widetilde C$ in $\nu^{-1}(x)$ (the
family is
going to be parametrized by an open neighborhood $F$ of $\tilde c_0$).
We have to
construct a surjection of $\widetilde {\mathcal{M}}=\pi_1^* M$ onto $\widetilde {\mathcal{Q}} = j_*
{\mathcal{O}}_{F}$. Note that $\widetilde{\mathcal{Q}}|_{C \times \tilde c }=\widetilde
Q_{\tilde c} \cong {\mathcal{O}}_{\nu
(\tilde c)}$ and that
$\widetilde{\mathcal{Q}}$ is flat over $F$.
Now, to define that quotient, it is enough to define it in the
restriction to the image of $j$ (because this is exactly
the support of $\widetilde {\mathcal{Q}}$). So the map we have to define is
$$
j^*\widetilde{\mathcal{M}} \to {\mathcal{O}}_{F} \; .
$$
But $j^*\widetilde{\mathcal{M}}=\nu^*M$ is a rank one sheaf on the smooth curve
$F$, so it is the direct sum of a line bundle and a torsion
part $T$. Shrinking $F$ if necessary, the line bundle part is
isomorphic to ${\mathcal{O}}_F$, and we have
$$
j^*\widetilde{\mathcal{M}} \cong T \oplus {\mathcal{O}}_F \; ,
$$
and then to define the quotient we just take an isomorphism in the
torsion-free part. This finishes the proof of the lemma.
\end{proof}
Now we go to the general case: the intersection of the support of $Q$
with the singular part of $C$ has length $n$. We are going to see how
this can be reduced to the case $n=1$.
Take a surjection from $Q$ to a sheaf $Q'$ of length $n-1$, such that
$Q$ is isomorphic to $Q'$ at the smooth points. The kernel $R$ of this
surjection will have length 1, and will be supported in a singular
point of $C$. It is isomorphic to ${\mathcal{O}}_x$, for some singular point $x$.
We have a diagram
$$
\CD
@. @. 0 @. 0 @. \\
@. @. @VVV @VVV @. \\
0 @>>> L @>>> L'@>>> R @>>> 0 \\
@. @| @VVV @VVV @. \\
0 @>>> L @>>> M @>>> Q @>>> 0 \\
@. @. @VVV @VVV @. \\
@. @. Q' @= Q' @. \\
@. @. @VVV @VVV @. \\
@. @. 0 @. 0 @. \\
\endCD
$$
Observe that $L$, $L'$ and $R$ satisfy the hypothesis of lemma \ref{bn2.4},
so we
can find deformations $L_y$, $R_y$ (parametrized by some curve $Y$ and
with $L_{y_0}=L$ and $R_{y_0}=R$ for some $y_0 \in Y$) such
that for some $y_1 \in Y$ we have that the support of the
corresponding sheaf $R_{y_1}$ is
a smooth point of $C$. All the maps of the previous diagram can be
deformed along. To do this, we change $L$ by $L_y$, $R$ will be
deformed to $R_y$ and $L'$ is kept constant. Then $Q$ is deformed to a
family $Q_y$ defined as $M/L_y$. The cokernel of $R_y \to Q_y$ will be
$Q_y/R_y=M/L'=Q'$, and hence we keep it constant.
Then for each $y$ we still have a commutative
diagram, and furthermore it is easy to see that all deformations
are flat (note that $R_y$ is a flat deformation and $Q'$ is kept
constant, and then $Q_y$ is a flat deformation). An important point is
that $M$ remains fixed, and the
injection $L \to M$ is deformed to $L_y \to M$.
$$
\CD
@. @. 0 @. 0 @. \\
@. @. @VVV @VVV @. \\
0 @>>> L_y @>>> L'@>>> R_y @>>> 0 \\
@. @| @VVV @VVV @. \\
0 @>>> L_y @>>> M @>>> Q_y @>>> 0 \\
@. @. @VVV @VVV @. \\
@. @. Q' @= Q' @. \\
@. @. @VVV @VVV @. \\
@. @. 0 @. 0 @. \\
\endCD
$$
For $y_1$ we have that the length of the intersection
of the support
of $Q_{y_1}$ with the singular part of $C$ is $n-1$. We repeat the process
(starting now with $L_{y_1}$, $M$ and $Q_{y_1}$), until all the points of the
support of $Q$ are moved to the smooth part of $C$. This finishes
the proof of the proposition.
\end{proof}
The following proposition is similar to proposition \ref{bn2.3},
but now the roles of
$L$ and $M$ are changed: we are given a deformation of $L$ and we have
to deform $M$ along.
\begin{proposition}
\label{bn2.5}
Let $L$ and $M$ be rank one torsion-free sheaves on an integral curve
$C$ that lies on a surface $S$. Assume we have
a short exact sequence
\begin{equation}
\label{eqbn2.4}
0 \to L \to M \to Q \to 0.
\end{equation}
Assume furthermore that we are given a sheaf ${\mathcal{L}}$ on $S \times P$
(where $P$ is a connected but not necessarily irreducible curve) that
is a deformation of $L$, flat over $P$ , i.e., ${\mathcal{L}}|_{p^{}_0}
\cong L$ for some $p^{}_0\in P$, and for all $p\in P$, we have that
$L_p={\mathcal{L}}|_p$ are torsion-free sheaves on $C_p$,
where $C_p$ is a curve on $S$.
Then, there is a connected curve $P'$
with a map $f:P' \to P$ and a
sheaf ${\mathcal{M}}'$ over $S \times P'$ with the following properties:
One irreducible component of $P'$ is a finite cover of $P$ and
the rest of the components map to $p^{}_0 \in
P$. The sheaf ${\mathcal{M}}'$ is a deformation of $M$, in the sense
that ${\mathcal{M}}'|_{p'_0} \cong M$
for some $p'_0 \in P'$ mapping to $p^{}_0 \in P$, the sheaf ${\mathcal{M}}'$ is
flat over
$P'$ and induces rank one torsion-free sheaves on the fibers over
$P'$. And if we define ${\mathcal{L}}'$ to be the pullback of ${\mathcal{L}}$ to $S
\times P'$, there exists an exact sequence
$$
0 \to {\mathcal{L}}' \to {\mathcal{M}}' \to {\mathcal{Q}}' \to 0,
$$
inducing short exact sequences
$$
0 \to L'_{p'} \to M'_{p'} \to Q'_{p'} \to 0
$$
for every $p'\in P'$.
\end{proposition}
\begin{proof}
The proof is very similar to the
proof of proposition \ref{bn2.3}. Again we start by observing that if the
support of $Q$
were in the smooth part of the curve, we would have $M \cong L \otimes
{\mathcal{O}}_C (D)$, with $D$ an effective divisor. Then if we are given a
flat deformation $L_p$ of $L$, we find a deformation $D_p$ of $D$ as
in the first part, and the proposition would be proved. So again we
need a lemma that deforms $Q$ so that its support is in the smooth
part of $C$.
\begin{lemma}
\label{bn2.6}
Let $L$ and $M$ be rank one torsion-free sheaves on an integral curve
$C$ that lies on a surface $S$. Assume we have
a short exact sequence
$$
0 \to L \to M \to Q \to 0.
$$
Assume that the part of $Q$ with support in the smooth part of $C$ has
length $l(Q)-1$, i.e. $Q=R \oplus Q'$, where $R$ has length one and is
supported in a singular point of $C$ and $Q'$ has length $l(Q)-1$ and
is supported in the smooth part of $C$. Then there is a flat
deformation $M_y$ of $M$ parametrized by a curve $Y$, such that for
every $y\in Y$ there is an exact sequence
$$
0 \to L \to M_y \to Q_y \to 0
$$
with $M_y$ a torsion-free sheaf, and there is some $y_1 \in Y$ such
that the support of $Q_{y_1}$ is in
the smooth part of $C$.
\end{lemma}
\begin{proof}
Arguing as in the proof of lemma \ref{bn2.4}, we see that it is enough to
prove the case $l(Q)=1$, and $Q={\mathcal{O}}_x$ for $x$ a singular point of
$C$, then we can assume that the extension of the hypothesis of the
lemma is
\begin{eqnarray}
0 \to L \to M \to {\mathcal{O}}_x \to 0.
\label{assumeext}
\end{eqnarray}
Now we will consider all extensions of ${\mathcal{O}}_x$ (for $x$ any point in $C$)
by $L$. If $x$ is a smooth point, then there is only one extension
that is not trivial (up to equivalence)
$$
0 \to L \to M \cong L\otimes {\mathcal{O}}_C(x) \to {\mathcal{O}}_x \to 0.
$$
All these extensions are then parametrized by the smooth part of $C$.
But if $x$ is a singular point, we could have more extensions, because
in general
$s=\dim \operatorname{Ext}^1({\mathcal{O}}_x,L)>1$. They will be parametrized by a projective
space $\mathbb{P} ^{s-1}$. We call this space $E_x$.
Note that there is a universal extension on $C \times E_x$ that is
flat over $E_x$.
We denote by $e_1$ the
point in $E_x$ corresponding to the extension \ref{assumeext}.
Assume that $\widetilde Q_{\tilde c}$ is a
family of torsion sheaves on $C$ with length 1,
parametrized by a curve $F$ such that for a general point
$\tilde c\in F$ of the parametrizing curve the support of $\widetilde Q_{\tilde
c}$ in $C$ is a smooth point, and
for a special point $\tilde c_0\in F$ the support of
$\widetilde Q_{\tilde c_0}$ is a
singular point.
Now assume that we can construct a flat family (parametrized by $F$) of
nontrivial
extensions of $\widetilde Q_{\tilde c}$ by $L$. The extension corresponding
to $\tilde c_0$ gives a point $e_2$ in $E_x$. The space $E_x$ is a
projective space, thus connected, and then there is a curve containing
$e_1$ and $e_2$. Using this curve (together with the universal
extension for $E_x$) and the curve $F$ (together with the family of
extensions that it parametrizes) we construct the
curve $Y$ that proves the lemma.
Now we need to construct $F$. As in the proof of lemma \ref{bn2.4},
the parametrizing curve $F$ will be an affine neighborhood of $\tilde
c_0$ in the normalization $\widetilde C$ of $C$, where $\tilde c_0$ is a
point that maps to the singular point $x$ of $C$. Consider again the
diagram \ref{eqbn2.3} of the proof of lemma \ref{bn2.4}. The
family will be given by an extension of $\widetilde{\mathcal{Q}} =
j_*{\mathcal{O}}_F$ by $\widetilde{\mathcal{L}}=\pi^*_1 L$ on $C \times F$. These extensions are
parametrized by the group $\operatorname{Ext}^1(\widetilde{\mathcal{Q}},\widetilde{\mathcal{L}})$. The following lemma gives
information about this group and relates this extension with the
extensions that we get after restriction for each slice $C \times
\tilde c$.
We will call $\widetilde Q_{\tilde c}$ and $\widetilde L_{\tilde c}$ the restrictions of
$\widetilde{\mathcal{Q}}$ and $\widetilde{\mathcal{L}}$ to
the slice $C \times {\tilde c}$. Note that the restriction $\widetilde
L_{\tilde c}$ is isomorphic to $L$.
\begin{lemma}
\label{bn2.7}
With the previous notation, we have
1) $\operatorname{Ext}^1(\widetilde{{\mathcal{Q}}},\widetilde{\mathcal{L}}) \cong H^0(Ext^1(\widetilde{\mathcal{Q}},\widetilde{\mathcal{L}}))$
2) $Ext^1(\widetilde{\mathcal{Q}},\widetilde{\mathcal{L}})$ has rank zero outside of the support of
$\widetilde{\mathcal{Q}}$, and
rank 1 on the smooth points of the support of $\widetilde{\mathcal{Q}}$
3) Let $I$ be the ideal sheaf corresponding to a slice $C \times
{\tilde c}$. Then the natural map
$$
Ext^1_{{\mathcal{O}}_{C \times F}} (\widetilde{\mathcal{Q}},\widetilde{\mathcal{L}}) \otimes {\mathcal{O}}_{C \times F}/I \to
Ext^1_{{\mathcal{O}}_{C \times {\tilde c}}}(\widetilde Q_{\tilde c},\widetilde L_{\tilde c})
$$
is injective.
\end{lemma}
\begin{proof}
Item 1 follows from the fact that
$Hom(\widetilde{\mathcal{Q}},\widetilde{\mathcal{L}})=0$ and the exact sequence
$$
0 \to H^1(Hom(\widetilde{\mathcal{Q}},\widetilde{\mathcal{L}})) \to \operatorname{Ext}^1(\widetilde{\mathcal{Q}},\widetilde{\mathcal{L}}) \to
H^0(Ext^1(\widetilde{\mathcal{Q}},\widetilde{\mathcal{L}})) \to H^2(Hom(\widetilde{\mathcal{Q}},\widetilde{\mathcal{L}})).
$$
To prove item 2 note
that the stalk of $Ext^1(\widetilde{\mathcal{Q}},\widetilde{\mathcal{L}})$ at a point $p$ is isomorphic to
$\operatorname{Ext}^1(R/I,R)$, where $R$ is the local ring
at the point $p$, and $I$ is the ideal defining the support or $\widetilde{\mathcal{Q}}$.
The ideal $I$ is principal if the point $p$ is smooth, then $R/I$ has
a free resolution
$$
0 \to I \to R \to R/I \to 0
$$
and it follows that $\operatorname{Ext}^1(R/I,R)\cong R/I$.
For item 3, consider the exact sequence
$$
0 \to \widetilde{\mathcal{Q}} \stackrel{\cdot f}{\to} \widetilde{\mathcal{Q}} \to \widetilde Q_{\tilde c} \to 0
$$
where the first map is multiplication by the local equation $f$ of the
slice $C \times {\tilde c}$. Applying $Hom(\cdot,\widetilde L_{\tilde c})$ we get
$$
Hom(\widetilde{\mathcal{Q}},\widetilde L_{\tilde c})=0 \to Ext^1(\widetilde Q_{\tilde c},\widetilde
L_{\tilde c}) \to
Ext^1(\widetilde{\mathcal{Q}},\widetilde L_{\tilde c}) \to
Ext^1(\widetilde{\mathcal{Q}},\widetilde L_{\tilde c}),
$$
but the last map is zero. To see this, take a locally free resolution
of $\widetilde{\mathcal{Q}}$. The map induced on the resolution by the multiplication with
the equation $f$ is just multiplication by the same $f$ on each term
$$
\CD
{\mathcal{F}}^\bullet @>>> \widetilde{\mathcal{Q}} @>>> 0 \\
@V{\cdot f}VV @V{\cdot f}VV @. \\
{\mathcal{F}}^\bullet @>>> \widetilde{\mathcal{Q}} @>>> 0 \\
\endCD
$$
A local section of the sheaf $Ext^i(\widetilde{\mathcal{Q}},\widetilde L_{\tilde c})$ is
represented by some
local section $\varphi(\cdot)$ of $Hom({\mathcal{F}}^i,\widetilde L_{\tilde c})$,
and the endomorphism
induced by multiplication by $f$ on $Ext^i(\widetilde {\mathcal{Q}},\widetilde L_{\tilde
c})$ is given by
precomposition with multiplication $\varphi(f\cdot)$, but $\varphi$ is
a morphism of sheaves of modules and then this is equal to $f
\varphi(\cdot)$, and this is equal to zero because $f \widetilde L_{\tilde
c}=0$.
Then we have that
\begin{equation}
\label{iso}
Ext^1(\widetilde Q_{\tilde c},\widetilde L_{\tilde c}) \cong
Ext^1(\widetilde{\mathcal{Q}},\widetilde L_{\tilde c}).
\end{equation}
Taking the exact sequence
$$
0 \to \widetilde{\mathcal{L}} \stackrel{\cdot f}{\to} \widetilde{\mathcal{L}} \to \widetilde L_{\tilde c} \to 0
$$
and applying $Hom(\widetilde{\mathcal{Q}},\cdot)$ we get
$$
Ext^1(\widetilde{\mathcal{Q}},\widetilde{\mathcal{L}}) \stackrel{\cdot f}{\to} Ext^1(\widetilde{\mathcal{Q}},\widetilde{\mathcal{L}}) \to
Ext^1(\widetilde{\mathcal{Q}},\widetilde L_{\tilde c})
$$
and using this and the isomorphism \ref{iso} we have an injection
$$
Ext^1(\widetilde{\mathcal{Q}},\widetilde{\mathcal{L}}) \otimes {\mathcal{O}}_{C \times F}/I \cong
Ext^1(\widetilde{\mathcal{Q}},\widetilde{\mathcal{L}})/(f\cdot Ext^1(\widetilde{\mathcal{Q}},\widetilde{\mathcal{L}})) \hookrightarrow
Ext^1(\widetilde Q_{\tilde c},\widetilde L_{\tilde c}).
$$
\end{proof}
Now we are going to construct the family of extensions. By item 2 of
the lemma
the sheaf ${\mathcal{E}}=Ext^1(\widetilde {\mathcal{Q}},\widetilde {\mathcal{L}})$ is isomorphic to
${\mathcal{O}}_X \oplus T({\mathcal{E}})$ (shrinking $F$ if necessary) where $X$ is the
support of $\widetilde{\mathcal{Q}}$ and $T({\mathcal{E}})$
is the torsion part. Take a nonvanishing section of the torsion-free
part, and by item 1 this gives a nonzero element $\psi$ of
$\operatorname{Ext}^1(\widetilde{\mathcal{Q}},\widetilde{\mathcal{L}})$. This element gives a nontrivial extension
$$
0 \to \widetilde{\mathcal{L}} \to \widetilde{\mathcal{M}} \to \widetilde{\mathcal{Q}} \to 0.
$$
Observe that $\widetilde{\mathcal{M}}$ is flat over the base, because both $\widetilde{\mathcal{L}}$ and
$\widetilde{\mathcal{Q}}$ are
flat.
By items 3 and 1 we have that the image of $\psi$ under the
restriction map
$$
\operatorname{Ext}^1(\widetilde{\mathcal{Q}},\widetilde{\mathcal{L}}) \to \operatorname{Ext}^1(\widetilde Q_{\tilde c},L)
$$
is nonzero for any ${\tilde c}$ (recall that $\widetilde L_{\tilde c}=L$ for all
${\tilde c}$), and this means that the
extensions that we
obtain after restriction to the corresponding slices
\begin{equation}
\label{eqbn2.5}
0 \to L \to \widetilde M_{\tilde c} \to \widetilde Q_{\tilde c} \to 0
\end{equation}
are non trivial. Furthermore $\widetilde M_{\tilde c}$ is torsion-free. To prove this
claim, let $T(\widetilde M_{\tilde c})$
be the torsion part of $\widetilde M_{\tilde c}$. The map $L \to T(\widetilde
M_{\tilde c})$ coming
from \ref{eqbn2.5} is zero, because $L$ is torsion-free, i.e.
$T(\widetilde M_{\tilde c})$ injects in $\widetilde Q_{\tilde c}$. Then we have
$$
\widetilde Q_{\tilde c} \cong {\frac {\widetilde M_{\tilde c}} L} \cong
{\frac{\widetilde M_{\tilde c}/T(\widetilde M_{\tilde c}) \oplus T(\widetilde M_{\tilde c})} L}
\cong {\frac{\widetilde M_{\tilde c}/T(\widetilde M_{\tilde c})} L} \oplus T(\widetilde
M_{\tilde c}).
$$
$\widetilde Q_{\tilde c}$ doesn't decompose as the direct sum of two sheaves,
and then one of these summands must be zero.
The first summand cannot be zero, because this would
imply that $L \cong \widetilde M_{\tilde c}/T(\widetilde M_{\tilde c})$ and then
$\widetilde M_{\tilde c} \cong L \oplus
\widetilde Q_{\tilde c}$, contradicting the hypothesis that the extension is
not trivial.
Then we must have $T(\widetilde M_{\tilde c})=0$, and the claim is proved.
\end{proof}
Now we are going to consider the general case, in which the part of
$Q$ supported in singular points has length $n$. We are going to see
that this can be reduced to the case $n=1$, in a similar way to
proposition \ref{bn2.3}.
Let $R={\mathcal{O}}_x$, where $x$ is a singular point in the support of $Q$,
and take a surjection from $Q$ to $R$. We have a diagram
$$
\CD
@. @. @. 0 @. \\
@. @. @. @VVV @. \\
@. @. @. L'/L @. \\
@. @. @. @VVV @. \\
0 @>>> L @>>> M @>>> Q @>>> 0 \\
@. @VVV @| @VVV @. \\
0 @>>> L' @>>> M @>>> R @>>> 0 \\
@. @. @. @VVV @. \\
@. @. @. 0 @. \\
\endCD
$$
Note that $L'$, $M$ and $R$ satisfy the hypothesis of lemma \ref{bn2.6}, then
we can find (flat) deformations $M_y$ and $R_y$ parametrized by a
curve $Y$ such that for some $y_1\in Y$ we have that the support of the
corresponding sheaf $R_{y_1}$ is a smooth point of $C$. All sheaves and
maps can be deformed along.
To do this we define $Q_y=M_y/L$ (we have $L \hookrightarrow L' \hookrightarrow M_y$, thus
this quotient is well defined). The kernel of $Q_y \to R_y$ is $L'/L$. Then
$Q_y$ is a flat deformation (being the extension of a flat deformation
$R_y$ by a constant and hence flat deformation $L'/L$).
Then for each $y$ we have a commutative
diagram
$$
\CD
@. @. @. 0 @. \\
@. @. @. @VVV @. \\
@. @. @. L'/L @. \\
@. @. @. @VVV @. \\
0 @>>> L @>>> M_y @>>> Q_y @>>> 0 \\
@. @VVV @| @VVV @. \\
0 @>>> L' @>>> M_y @>>> R_y @>>> 0 \\
@. @. @. @VVV @. \\
@. @. @. 0 @. \\
\endCD
$$
Observe that the length of the part of $Q_{y_1}$ supported in singular
points is $n-1$, so repeating this process we can deform $Q$ until its
support lies in the smooth part of $C$. This finishes the proof of the
proposition.
\end{proof}
\section{Proof of theorem I}
\label{bnProof of the main theorem}
In this section we will prove theorem I:
\begin{proof}
Nonemptyness follows from the fact that the Brill-Nother loci for
smooth curves is nonempty, and by upper semicontinuity of
$h^0(\cdot)$. By remark \ref{remark} we can assume $r>d-p_a$.
We will prove theorem I by applying lemma \ref{bn0.2}.
We start
with a rank one torsion-free sheaf $A$ corresponding to a point in
$\overline W^r_d$, $d>0$, $r\geq 0$, with $\rho(r,d)>0$ (recall that
we are assuming $r>d-p_a$). We have $h^0(A)$, $h^1(A)>0$. As we
explained at the
beginning of the section \ref{bnGeneral case}, we call $A'$ its base point
free part. Then we take $B$ to be the base point free part of ${A'}^*$,
and finally define $A''$ to be $B^*$.
By lemma \ref{bn2.1}, $A''$ and ${A''}^*$ are rank one locally free
sheaves on $C$ generated by global
sections. Then by proposition \ref{bn1.5} we find a deformation
${\mathcal{A}}''$ of $A''$ parametrized by a some smooth irreducible curve $T$.
The support of ${\mathcal{A}}''$ defines a family of curves ${\mathcal{C}}$ parametrized
by the irreducible curve $T$. note that ${\mathcal{C}}|_t$ is smooth for
$t\neq0$.
By the definition of $A'$ and $A''$ we have exact sequences
\begin{equation}
0 \to A' \to A \to Q \to 0
\label{short1}
\end{equation}
\begin{equation}
0 \to A' \to A'' \to \widetilde Q \to 0
\label{short2}
\end{equation}
with $h^0(A')=h^0(A)$ and $h^1(A'')=h^1(A')$ (lemma \ref{equalit}).
If we look at \ref{short2} we see that we are in the situation of
proposition \ref{bn2.3}, with $L=A'$, $M=A''$, ${\mathcal{M}}={\mathcal{A}}''$, $P=T$.
Then we get a family ${\mathcal{A}}'$ (parametrized by some connected but in
general not irreducible curve).
Now we use this family ${\mathcal{A}}'$ and the sequence \ref{short1} to apply
\ref{bn2.5} with $L=A'$, $M=A$ and ${\mathcal{L}}={\mathcal{A}}'$. We get a new family
${\mathcal{A}}$. We denote by $T'$ the curve parametrizing the family ${\mathcal{A}}$.
This family satisfies all the hypothesis of lemma \ref{bn0.2}
(item (iv) is given by proposition \ref{bn2.2}), and
then theorem I is proved.
\end{proof}
\chapter{Irreducibility of the moduli space for $K3$ surfaces}
\label{k3}
In this chapter we will prove the following theorem:
\smallskip
\noindent\textbf{Theorem II.}
\textit{
With the notation of chapter \ref{Preliminaries}, if $L$ is a primitive
nonzero element of
$\operatorname{Pic}(S)$, and $H$ is an $(L,c_2)$-generic polarization, then
$\mathfrak{M}_H(L,c_2)$ is irreducible.}
\smallskip
Due to the fact that the moduli space is smooth, irreducibility is
equivalent to connectedness.
\bigskip
\textbf{Outline of the proof of theorem II}
\bigskip
First we will prove the theorem for the case in which Pic$(S)=\mathbb{Z}$
For $H$ to be $(L,c_2)$-generic we need $L$ to be an odd multiple of
a generator of Pic$(S)$, and tensoring the vector bundles with a line
bundle we can assume that $H=L$ is a generator of Pic$(S)$. After proving the
theorem for this case, in section \ref{General K3 surface} we show,
by considering families of
surfaces, that if the result is true for Pic$(S)=\mathbb{Z}$, then it is also
true under the conditions of the theorem (this part is very similar to
an argument in \cite{G-H}). From now on we will assume
that Pic$(S)=\mathbb{Z}$ and that $H=L$ is the ample generator.
The proof is divided into two parts. In section \ref{Small second Chern
class} we handle the case in which
$c_2 \leq \frac{1}{2} L^2 + 3$. First we see (proposition \ref{2.1}) that
the sheaves
satisfying this inequality are exactly those which are nonsplit
extensions of the form
$$ 0 \to \SO_S \to V \to L \otimes I_Z \to 0,$$
with $l(Z)=c_2$. Then we study the set $X \subset \Hilb ^{c_2} (S)$ for which there
exist nonsplit extensions like these above, and we see,
using theorem I,
that it is connected (proposition \ref{2.2}).
Finally we use this to
prove (proposition \ref{2.4}) the connectedness of $\FM (L,c_2)$ for
$\dim \FM (L,c_2)>0$ (if the dimension is zero the result is known \cite{M}).
Note that for $c_2=\frac{1}{2} L^2 + 3$ we have $\dim \FM (L,c_2) =
L^2+6>0$, and then we can continue the proof by induction on $c_2$.
Let $C(n)$ be the set of irreducible components of $\mathfrak{M}(L,n)$.
We construct a map
$$\Phi _n :C(n) \to C(n+1).$$
To define this map, take a sheaf $E$ in a component $A$ of $\mathfrak{M}(L,n)$.
Take a point $p \in S$ and a surjection $E \to {\mathcal{O}}_p$. Let $F$ be the
kernel
$$0 \to F \to E \to {\mathcal{O}}_p \to 0.$$
$F$ is clearly stable, and $c_2(F)=c_2(E)+1$. Now we define $\Phi _n
(A)$ to be the component in which $F$ lies. It is easy to see that
this is independent of all the choices made, so that $\Phi _n$ is well
defined.
Now we assume that $\FM (L,c_2)$ is irreducible for $c_2 < n$. We are
going to see that if every connected component of $\mathfrak{M}(L,n)$ has a
non-locally free sheaf $F$, $\Phi _{n-1}$ is surjective, and then by
induction $\mathfrak{M}(L,n)$ will be irreducible.
Let $B$ be a component of $\mathfrak{M}(L,n)$ with non-locally free sheaves. By
lemma \ref{3.3}, it has a non-locally free sheaf $F$ such that
$F^{\vee\vee} \in \mathfrak{M}(L,n-1)$. By smoothness of the moduli
space, $F^{\vee\vee}$ is in only one irreducible component.
Call this component $A$. By construction $\Phi_{n-1}(A)=B$.
In other words, we have seen that if $\mathfrak{M}(L,n-1)$ is irreducible, then
there is only one component $\mathfrak{M}_0$ of $\mathfrak{M}(L,n)$ that has sheaves
that are not locally free, and then to prove that the later has only
one component, it will be enough to check that every component has a
non-locally free sheaf.
We divide the possible values of $c_2$ in regions labeled by $n \geq
1$, with $c_2$ satisfying
$$
((n-1)^2 + (n-1) + \frac{1}{2}) L^2 + 3 < c_2 \leq (n^2 + n + \frac{1}{2})
L^2 + 3.
$$
If $V$ is locally free, we prove that then $V$ fits in a short exact
sequence
$$ 0 \to L^{\otimes -m} \to V \to L^{\otimes m+1} \otimes I_{Z_m} \to 0
$$
with $0 \leq m \leq n$ (proposition \ref{3.1}). We call it an
extension of type $m$. We will also say that $V$ is of type $m$.
Next (proposition \ref{3.3}) we show that the set of sheaves that are not
locally free has positive codimension, and then we prove (proposition
\ref{3.4}) that the generic sheaf is a vector bundle of type $n$.
But this is not enough, and we need more information about the generic
vector bundle. Let $C$ be the set of vector
bundles $V$ such that for any exact sequence
\begin{equation}
0 \to L^{\otimes -n} \to V \to L^{\otimes n+1} \otimes I_{Z_n} \to 0,
\label{eq0.1}
\end{equation}
$L^{\otimes n+1} \otimes I_{Z_n}$ has no sections whose zero locus
is an irreducible reduced curve. In proposition \ref{3.7} we prove that
this set has positive codimension. The reason
to look at
this set is because it is precisely because of these sheaves that we
cannot apply the generalization of Fulton-Lazarsfeld's theorem to
proof that the set of type $n$ vector bundles is connected. But now we
know that we can ignore $C$, because it has positive codimension,
and then conclude that the generic vector bundle $V$ sits in an
extension like \ref{eq0.1} such that $L^{\otimes 2n+1} \otimes
I_{Z_n}$ has a section
whose zero locus is an irreducible reduced curve. In proposition \ref{3.6}
we prove that those vector bundles make a connected set. We will need
the induction hypothesis to prove this proposition.
\section{Preliminaries}
\label{secPreliminaries}
In this section we will prove some propositions that will be useful later.
\begin{lemma}
\label{1.1}
Let $S$ be a smooth surface and $C$ a smooth (not necessarily
complete) curve. Let $p$ be a point
in the curve and $j:S \hookrightarrow S \times C$ the corresponding injection.
Let $L$ be a line bundle on $S$ and $I_W$ an ideal sheaf on $S$
corresponding to a subscheme of dimension zero. Let ${\mathcal{V}}$ be a family
of rank two sheaves on $S$, i.e. a sheaf on $S \times C$ flat over $C$. If we
have the following elementary transformation:
$$ 0 \to {\mathcal{W}} \to {\mathcal{V}} \to j_*(L \otimes I_W) \to 0 $$
then ${\mathcal{W}}$ is a flat family of rank two sheaves on $S$, and furthermore
$$ c_i({\mathcal{W}}_{p'}) = c_i({\mathcal{V}}_{p'})$$
for $i=1,2$ and $p'$ any point of $C$.
\end{lemma}
\begin{proof}
We calculate the Chern classes of $j_*(L \otimes I_W)$ by the
Grothendieck-Riemann-Roch theorem, and then the classes of ${\mathcal{W}}$ by
Whitney's formula. The fact that ${\mathcal{W}}$ is flat is proved in \cite{F}.
\end{proof}
Now we will apply this lemma to take limits of stable extensions. Let
$S$ be a smooth surface with $\operatorname{Pic} (S)=\mathbb{Z}$. Consider a family of
extensions parametrized by a curve $T$
$$ 0 \to L^{\otimes -n} \to V_t \to L^{\otimes n+1} \otimes I_Z \to 0,$$
where L is a generator of $\operatorname{Pic} (S)$, $t \in T$, and $Z$ is a subscheme
of dimension zero.
Assume that $V_t$ is stable for $t \not= 0$, where $0$ is some fixed
point of T, and unstable for $t = 0$. This defines a map $\varphi :
T-\{0\} \to \mathfrak{M}$ to the moduli space
of stable sheaves. By properness of $\mathfrak{M}$, this can be extended to a
map $\varphi : T \to \mathfrak{M}$, i.e. we can take the limit of the family as $t$
goes to $0$ and we obtain a stable sheaf corresponding to $\varphi (0)$.
\begin{proposition}
\label{1.2}
The stable sheaf $V'$ corresponding to $\varphi (0)$ is not locally free or can
be written as an extension
\begin{equation}
0 \to L^{\otimes -m} \to V' \to L^{\otimes m+1} \otimes I_{Z'} \to 0
\label{eq1.1}
\end{equation}
with $m < n$.
\end{proposition}
\begin{proof}
$V_0$ is unstable, so we have
$$ 0 \to L^{\otimes a} \otimes I_W \to V_0 \to L^{\otimes 1-a} \otimes I_{W'}
\to 0$$
with $0 < a \leq n$. Consider the elementary transformation
$$ 0 \to {\mathcal{W}} \to {\mathcal{V}} \to j_*[L^{\otimes 1-a} \otimes I_{W'}] \to 0.$$
By lemma \ref{1.1} we have a new family ${\mathcal{W}}$. By standard arguments the
member $W_0$ of the new family corresponding to $t=0$ can be written
as an extension
\begin{equation}
0 \to L^{\otimes 1-a} \otimes I_{W'} \to W_0 \to L^{\otimes a} \otimes I_W
\to 0.
\label{eq1.2}
\end{equation}
Note that $0 \geq 1-a > -n$. If $W_0$ is not stable, repeat the
process: unstability gives an injective map $L^{\otimes a'}\otimes I_{W''}
\to W_0$, and by \ref{eq1.2} we have $a' < a$. Then in each step $1-a$
grows. We are going to see that eventually
we are going to get a stable sheaf. Assume we reach $1-a=0$ and $W_0$
is still unstable. The destabilizing sheaf has to be $L \otimes
I_{Z_d}$ with $l(Z_d)>l(W)$ and gives a short exact sequence
$$ 0 \to L \otimes I_{Z_d} \to W_0 \to I_{Z'_d} \to 0.$$
Note that $l(Z'_d)<l(W')$. Performing the corresponding
elementary transformation we
get a new family $\overline {\mathcal{W}}$ and the sheaf corresponding to $0$
sits in an exact sequence
\begin{equation}
0 \to I_{Z'_d} \to \overline W_0 \to L\otimes I_{Z_d} \to 0.
\label{eq1.3}
\end{equation}
This sequence is like \ref{eq1.2} but with $0\leq l(Z'_d)<l(W')$. If we still
don't get a stable sheaf, repeat this. In each step $l(Z'_d)$
decreases, but this must stop because if $l(Z'_d)=0$, the sheaf
given by \ref{eq1.3} is stable, as the following lemma shows.
Now, once we have obtained a stable sheaf, if it is not locally free,
we are done. If it is locally free, then necessarily the subscheme
$W'$ is empty, and we get an extension
like \ref{eq1.1} as desired.
\end{proof}
\begin{lemma}
\label{1.3}
Let $V$ be a torsion free sheaf on a surface $S$ with $\operatorname{Pic} (S)=\mathbb{Z}$,
given by an extension
$$0 \to \SO_S \to V \to L \otimes I_Z \to 0,$$
where $L$ is the effective generator of $\operatorname{Pic} (S)$. Then $V$ is
stable.
\end{lemma}
\begin{proof}
A destabilizing
subsheaf should be of the form $L^{\otimes m} \otimes I_W$, with $m>0$. By
standard arguments, it is enough to check stability with subsheaves
whose quotients are torsion free, so we can assume this.
The composition $L^{\otimes m} \otimes I_W \to V \to L \otimes I_Z$ is
nonzero, because otherwise it would factor through $\SO_S$, but this is
impossible because $m>0$. Then $m=1$ and we have $I_W \hookrightarrow I_Z$.
Furthermore, $l(W) > l(Z)$ because if $W=Z$, the sequence would split.
Then we have a sequence
$$ 0 \to L \otimes I_W \to V \to I_{W'} \to 0,$$
but we reach a contradiction because $c_2 = l(W) + l(W') > l(Z) +
l(W') = c_2 + l(W')$. Then there is no destabilizing subsheaf, and $V$
is stable.
\end{proof}
\begin{proposition}
\label{1.4}
Let $S$ be a smooth $K3$ surface with Picard group $\operatorname{Pic} (S)=\mathbb{Z}$.
If $\dim \operatorname{Ext} ^1(L'
\otimes I_Z,L) \geq 2$, then there is a nonsplit extension
\begin{equation}
0 \to L \to V \to L' \otimes I_Z \to 0
\label{eq1.4}
\end{equation}
such that V is not locally free.
\end{proposition}
\begin{proof}
We have an exact sequence
$$ 0 \to H^1(L\otimes(L')^{-1}) \to \operatorname{Ext} ^1(L' \otimes I_Z, L) \to H^0({\mathcal{O}}_Z).$$
If $L=L'$, then $H^1(L\otimes(L')^{-1})=0$ because $S$ is a K3 surface. If $L \neq
L'$, then due to the fact that $\operatorname{Pic}(S)=\mathbb{Z}$, applying Kodaira's
vanishing theorem we also have $H^1(L\otimes(L')^{-1})=0$. We have then an injection
$$ 0 \to \operatorname{Ext} ^1(L' \otimes I_Z,L) \stackrel{f}{\rightarrow} H^0({\mathcal{O}}_Z).$$
An extension corresponding to $\xi$ is locally free iff the section
$f(\xi)$ generates the sheaf ${\mathcal{O}}_Z$, i.e., iff
$$ f(\xi) \notin W=\{ s \in H^0({\mathcal{O}}_Z) : 0=s \otimes k(p) \in
H^0({\mathcal{O}}_p) \text { for some } p \in \operatorname{Supp}(Z) \}.$$
$W$ is a union of codimension 1 linear subspaces, hence if $\dim
\operatorname{Ext} ^1(L' \otimes I_Z, L) \geq 2$, then $\dim \text{im}(f) \cap
W > 0$, and we have a nonzero $\xi$ corresponding to an extension
\ref{eq1.4} with $V$ not locally free.
\end{proof}
Usually we will apply the following corollary
\begin{corollary}
\label{1.5}
Let $S$ be a smooth $K3$ surface with $\operatorname{Pic} (S)=\mathbb{Z}$. If
$$
\dim \operatorname{Ext} ^1 (L^{\otimes n+1}\otimes I_Z,L^{\otimes -n}) \geq 2,$$
and there is a stable extension
$$0 \to L^{\otimes -n} \to V \to L^{\otimes n+1} \otimes I_Z \to 0,$$
then there is a sheaf $V'$, in the same irreducible component of
$\FM (L,c_2)$ as $V$, that is not locally free or sits in an extension
$$0 \to L^{\otimes -m} \to V \to L^{\otimes m+1} \otimes I_Z \to 0$$
for some $m<n$.
\end{corollary}
\begin{proof}
There is an open set
in $\mathbb{P}(\operatorname{Ext} ^1 (L^{\otimes n+1} \otimes I_Z,L^{\otimes -n}))$
whose points correspond to
stable extensions, due to the openness of the stability condition.
All these points get mapped to the same irreducible
component of $\FM (L,c_2)$. By proposition \ref{1.4}, there is an extension $V$
that is not locally free. If it is not stable, we can take a curve as
in proposition \ref{1.2}, and applying the proposition we get a family of
stable sheaves. All get mapped to the same component of $\FM (L,c_2)$,
and the sheaf corresponding to $t=0$ has the required properties.
\end{proof}
\section{Small second Chern class}
\label{Small second Chern class}
In this section we will consider the case in which $c_2 \leq \frac{1}{2}
L^2 +3$. Recall that we are assuming that $S$ is a K3
surface with $\operatorname{Pic} (S)=\mathbb{Z}$. In this case we have the following
characterization of the stable torsion free sheaves.
\begin{proposition}
\label{2.1}
Let $V$ be a torsion free stable rank two sheaf with $c_1=L$, $c_2
\leq \frac{1}{2} L^2 + 3$. Then $V$ fits in an exact sequence
\begin{equation}
0 \to \SO_S \to V \to L \otimes I_Z \to 0.
\label{eq2.1}
\end{equation}
Conversely, every nonsplit extension of $L \otimes I_Z$ by $\SO_S$
is a torsion free stable sheaf.
\end{proposition}
\begin{proof}
Take $V$ stable. Using the Riemann-Roch theorem,
$$ h^0(V) + h^2(V) \geq \frac{L^2}{2} - c_2 + 4 \geq 1.$$
If $h^2(V)$ were different from zero, by Serre duality we would have
$\text {Hom} (V,{\mathcal{O}}) \not= 0$, contradicting stability because this
would give a map $V \to \SO_S$ with image $L^{\otimes -n} \otimes I_Z$ ($n
\geq 0$) and kernel $L^{\otimes n+1} \otimes I_{Z'}$.
Then $h^0(V) \not= 0$. Take a section of $V$. By stability, the
quotient of the section is torsion free, and we have an extension like
\ref{eq2.1}. The extension is not split because $V$ is stable.
The converse is lemma \ref{1.3}.
\end{proof}
Now that we know that all sheaves can be written as extensions of $L
\otimes I_Z$ by $\SO_S$, the obvious strategy is to construct families
of extensions $\mathbb{P}(\Ext ^1 (L \otimes I_Z, \OOS))$ for each $Z$ such that $\dim \Ext ^1 (L \otimes I_Z, \OOS) \geq
1$. Ideally we would like to put all these families together in a
bigger family parametrized by a variety $M$. This $M$ would map to
$\FM (L,c_2)$ surjectively, so it would be enough to prove the
connectedness of $M$, and because $M$ maps to $\Hilb ^{c_2} (S)$ with connected
fibers, it would be enough to prove that the set $X=\{ Z \in \Hilb ^{c_2} (S) :
\dim \Ext ^1 (L \otimes I_Z, \OOS) \geq 1\}$ (i.e., the image of the map $M \to \Hilb ^{c_2} (S)$) is
connected.
Unfortunately we cannot construct $M$ because $\dim \Ext ^1 (L \otimes I_Z, \OOS)$ is not
constant. We will use a somewhat more elaborate argument to bypass
this difficulty, but we will still use the connectivity of $X$, that
we prove in the following proposition.
\begin{proposition}
\label{2.2}
The set $X=\{ Z \in \Hilb ^{c_2} (S) :\dim \Ext ^1 (L \otimes I_Z, \OOS) \geq 1\}$ is connected.
\end{proposition}
\begin{proof}
By Serre duality and looking at the sequence
$$ 0 \to H^0(L \otimes I_Z) \to H^0(L) \to H^0({\mathcal{O}}_Z) \to H^1(L
\otimes I_Z) \to 0,$$
we have $\dim \Ext ^1 (L \otimes I_Z, \OOS) \geq 1\ \iff h^0(L \otimes I_Z)\geq \frac{1}{2}
L^2 +3-c_2$. Now consider the following commutative diagram
$$
\CD
@. @. 0 @. 0 @.\\
@. @. @VVV @VVV @.\\
0 @>>> \SO_S @>>> L \otimes I_Z @>>> j_*(\omega _C \otimes I_Z) @>>> 0\\
@. @| @VVV @VVV @.\\
0 @>>> \SO_S @>>> L @>>> L| _C=j_*\omega _C @>>> 0\\
@. @. @VVV @VVV @.\\
@. @. {\mathcal{O}}_Z @= {\mathcal{O}}_Z @.\\
@. @. @VVV @VVV @.\\
@. @. 0 @. 0 @.\\
\endCD
$$
where $C \in \mathbb{P}(H^0(L\otimes I_Z))$ (maybe $C$ is singular, but we know it is
irreducible and reduced because $\operatorname{Pic} (S)=\mathbb{Z}$ and $L$ is a generator of
the group), $j:C \hookrightarrow S$ is the inclusion, and $\omega_C=L|_C$ is the
dualizing sheaf on $C$.
Using the top row we get $h^0(L \otimes I_Z)\geq \frac{1}{2}
L^2 +3-c_2 \iff h^0(\omega _C \otimes I_Z) \geq \frac{1}{2}
L^2 +2-c_2$. This condition can be restated in terms of Brill-Noether
sets $W^r_d$:
$$
\omega _C \otimes I_Z \in W^r_d
$$
where $r=\frac{1}{2}L^2 +1-c_2$, and $d=L^2 -c_2$.
By a theorem of Fulton and Lazarsfeld \cite {F-L}, the Brill-Noether
set $W^r_d$
of a smooth curve is nonempty and connected if the expected dimension
$\rho (r,d) = g-(r+1)(g-d+r)$ is greater than zero. In the case of
an irreducible reduced curve lying on a $K3$ the generalized Jacobian can be
compactified, and the connectedness result is still true (theorem I).
In our case we have
$$ \rho (r,d)=2c_2-\frac{L^2}{2}-3= \frac{{\dim \FM (L,c_2)}}{2} > 0,$$
(recall that for $\dim \FM (L,c_2)=0$ the irreducibility of the moduli space
is known by the work of Mukai \cite{M}) and we can apply the theorem.
Now consider the variety
$$N = \{ (Z,C): Z \subset C, \dim \Ext ^1 (L \otimes I_Z, \OOS) \geq 1\} \subset \Hilb ^{c_2} (S)
\times \mathbb{P}(H^0(L))$$
and the projections
$$
\CD
N @>p_2>> \mathbb{P}(H^0(L)) \\
@Vp_1VV @. \\
\Hilb ^{c_2} (S) @. \\
\endCD
$$
By theorem I, $p_2$ is surjective with connected
fibers. Then $N$ is connected, and also the image of $p_1$, that is
equal to $X$.
\end{proof}
For the following proposition we will need this lemma:
\begin{lemma}
\label{2.3}
Let $T$ be a smooth curve, $p$ a point in $T$ and $S$ a variety. Consider the
diagram
$$
\CD
S @>v>> S\times T \\
@VgVV @VfVV \\
p @>u>> T \\
\endCD
$$
Then for every coherent torsion free sheaf ${\mathcal{F}}$ on $S\times T$, there
exist a natural map
$$
(f_* {\mathcal{F}} )(p) \to H^0({\mathcal{F}} _{S\times \{p\} }),
$$
where ${\mathcal{F}}(p)=v^* {\mathcal{F}}$. Furthermore, this map is injective.
\end{lemma}
\begin{proof}
The question is local in $T$, so we can assume that $T$ is affine,
$T=\operatorname{Spec} A$, and there is an element $x\in A$ such that $p$ is the
zero locus of $x$. We have
\begin{equation}
0 \to {\mathcal{F}} \stackrel{\cdot x}{\to} {\mathcal{F}} \to {\mathcal{F}} / x\cdot {\mathcal{F}} \to 0,
\label{eq2.2}
\end{equation}
where the map on the left is multiplication by $f^*x$. On the other hand
we have the sequence
$$
0 \to {\mathcal{O}}_{S \times T}(-f^*p) \to {\mathcal{O}}_{S\times T} \to {\mathcal{O}}_{S\times \{p\}
} \to 0. $$
Tensoring with ${\mathcal{F}}$ is right exact, so we get an exact sequence
$$
{\mathcal{F}} \otimes {\mathcal{O}}_{S \times T}(-f^*p) \to {\mathcal{F}} \to {\mathcal{F}}_p \to 0.
$$
Note that the image of the left map is $x\cdot {\mathcal{F}}$, and then we
conclude that ${\mathcal{F}} / x\cdot {\mathcal{F}}$ is isomorphic to ${\mathcal{F}}_p$. Taking
cohomology in the sequence \ref{eq2.2} we get
$$
{H^0({\mathcal{F}})}/(x\cdot H^0({\mathcal{F}})) \hookrightarrow H^0({\mathcal{F}}_p),
$$
but the first group is exactly $(f_* {\mathcal{F}})|_p$.
\end{proof}
Finally we can prove:
\begin{proposition}
\label{2.4}
The moduli space $\FM (L,c_2)$ of torsion free, rank two sheaves with
$c_2 \leq \frac{1}{2}L^2 + 3$ over a
K3 surface with $\operatorname{Pic} (S)=\mathbb{Z}$ is connected (hence irreducible, because
we know it is smooth).
\end{proposition}
\begin{proof}
We have a stratification of $X$
$$
X=\bigcup _{r \geq 1} H_r, H_r=\{ Z \in \Hilb ^{c_2} (S) : \dim \Ext ^1 (L \otimes I_Z, \OOS)
=r\}.
$$
On each stratum $H_r$ we can construct a projective bundle $M_r \to
H_r$ with fiber $\mathbb{P} (\Ext ^1 (L \otimes I_Z, \OOS))$, because the dimension of the group is
constant. Each point of $M_r$ corresponds to an extension (up to weak
isomorphism of extensions). We have then morphisms $M_r \to \mathfrak{M}
(L,c_2)$ with fiber $\mathbb{P}(H^0(V))$ over $V$ (see proposition \ref{3.8}). We have
$h^0(V)=\frac{1}{2}L^2 +3-c_2 + h^1(L \otimes I_Z)$, and a
corresponding stratification of $\FM (L,c_2)$
$$
\FM (L,c_2)=\bigcup _{r \geq 1} \mathfrak{M} _r ,\ \ \
\mathfrak{M} _r = \{V \in \mathfrak{M} (L,c_2) : h^0(V) = \frac{1}{2}L^2 +3-c_2
+ r \}
$$
(the reason for this dependence on $r$ in the definition is that $h^0(V)
= \frac{1}{2}L^2 +3-c_2+h^1(L\otimes I_Z)$. The condition is
equivalent to requiring that if $V \in
\FM (L,c_2)$ and $V$ is an extension of $L\otimes I_Z$ by $\SO_S$, then
$h^1(L\otimes I_Z)=r$. The previous formula proves that $r$ only
depends on $V$).
Note that $M_r$ can be thought also as a projective bundle over $\mathfrak{M}
_r$ with fiber $\mathbb{P}(H^0(V))$.
To prove that $\FM (L,c_2)$ is connected, we will show that for any two
sheaves $V_a \in \mathfrak{M} _a$, $V_b \in \mathfrak{M} _b$, we can construct a family
${\mathcal{V}}$ of stable sheaves on a connected parameter space, with ${\mathcal{V}} |_0
= V_a$, ${\mathcal{V}} |_1 = V_b$
Due to the fact that $X$ is connected, it is enough to prove this for
$V_a$, $V_b$ given by extensions
$$
\CD
0 \to \SO_S @>s_a>> V_a \to L \otimes I_{Z_a} \to 0 \\
0 \to \SO_S @>s_b>> V_b \to L \otimes I_{Z_b} \to 0
\endCD
$$
such that $Z_a \in H_a$, $Z_b \in H_b$ and $Z_a \in \overline H_b$,
the closure of $H_b$.
Take a curve $f:T \to \Hilb ^{c_2} (S)$ with $f(0)=Z_a$, $f(1)=Z_b$ and
$\operatorname{im} (T-\{0\}) \in H_b$. $T$ doesn't need to be complete. By shrinking
$T$ to a smaller open set, we can assume that there is a lift
$f$ to a map $f:T\setminus\{0\} \to M_b$. This gives a family of sheaves $\widetilde
{\mathcal{V}}$ and sections $s_t \in H^0({\mathcal{V}}|_t)$ parametrized by $T\setminus\{0\}$.
We want to extend this to a
family ${\mathcal{V}}$ of sheaves and sections parametrized by $T$, in such a way
that the
cokernel of $s_0:\SO_S \to {\mathcal{V}} |_0$ is $L \otimes I_{Z_a}$. Maybe ${\mathcal{V}} |_0$
won't be isomorphic to $V_0$, but at least both are extensions of the
same sheaf $L \otimes I_{Z_a}$ by $\SO_S$, and then they are in the
same connected component, and this is enough.
This family gives a morphism $T-\{0\}\to \FM (L,c_2)$ that extends to a
unique $T \to \FM (L,c_2)$ by properness (see section \ref{secPreliminaries},
just before
proposition \ref{1.2}). With this we have already
extended the family $\widetilde {\mathcal{V}}$ to a family ${\mathcal{V}}$ parametrized by
$T$, and we only need to extend the sections $s_t$. Shrinking $T$ to a
smaller neighborhood of $\{0\}$ if
necessary, we can assume that $\pi _2{}_* {\mathcal{V}}$ is trivial ($\pi _1$
and $\pi _2$ are the first and second projections of $S \times T$).
Then the sections $s_t$ fit together to give ${\mathcal{O}}_{S \times (T-\{0\})}
\to \widetilde {\mathcal{V}}$, i.e. an element
$\tilde s \in ({\pi_2} _* {\mathcal{V}})(T-\{0\})$. This can be extended to some
$s\in ({\pi_2} _* {\mathcal{V}})(T)$ that is nonzero on the fiber of $t=0$.
Using the previous lemma, we have an injection $({\pi_2} _* {\mathcal{V}})|_{t=0}
\to H^0({\mathcal{V}} |_0)$ that gives a nonzero section $s_0$ of ${\mathcal{V}} |_0$. Now
we only have to check that the cokernel of this section is $L \otimes
I_{Z_a}$. We have a short exact sequence over $S \times T$
$$
0 \to {\mathcal{O}}_{S \times T} \stackrel{s}{\to} {\mathcal{V}} \to {\mathcal{Q}} \to 0.
$$
Then ${\mathcal{Q}}$ is torsion free, flat over $T$, and then it is of the form
$${\mathcal{Q}} = {\pi _1}^* (L) \otimes I_{\mathcal{Z}} \otimes {\pi _2}^* (L')$$
for some line bundle $L'$ over $T$ and some subscheme ${\mathcal{Z}}$ of $S \times T$
flat over $T$. This subscheme gives a morphism $g:T \to \Hilb ^{c_2} (S)$.
By construction
$f(t)=g(t)$ for $t \not= 0$, and by properness the equality also holds
for $t=0$, then $Z_0 = Z$ as desired.
\end{proof}
\section{Large second Chern class}
\label{Large second Chern class}
In this section we will handle the case in which $c_2$ is large.
\begin{proposition}
\label{3.1}
Assume that $c_2$ satisfies
$$
((n-1)^2 + (n-1) + \frac{1}{2}) L^2 + 3 < c_2 \leq (n^2 + n + \frac{1}{2})
L^2 + 3.
$$
If $V$ is locally
free, then $V$ fits in a short exact sequence
$$
0 \to L^{\otimes -m} \to V \to L^{\otimes m+1} \otimes I_{Z_m} \to 0,
l(Z_m) = c_2 + (m+1) m L^2 ,
$$
with $0 \leq m \leq n$. We will call such an exact sequence an
extension of type $m$.
\end{proposition}
\begin{proof}
For any sheaf $V$,
$h^2(V\otimes L^{\otimes n}) = \dim \operatorname{Hom}(V, L^{\otimes -n}) = 0$ by stability,
and then using the Riemann-Roch theorem we have
\begin{equation}
h^0(V\otimes L^{\otimes n}) \geq \frac{L^2}{2} - c_2 + 4 +(n + n^2) L^2 \geq 1,
\label{eq3.1}
\end{equation}
so that we have an inclusion $L^{\otimes -n} \hookrightarrow V$. If $V$ is locally
free, this will give an exact sequence on type $m$, with $m \leq n$.
\end{proof}
\begin{proposition}
\label{3.2}
If a component $\mathfrak{M}'$ has sheaves of type $m$, with $0 \leq m \leq
n-1$, then it has sheaves that are not locally free.
\end{proposition}
\begin{proof}
Choose $m$ such that there is no $V$ of type $m'$ for $m' < m$. Let
$V$ be of type $m$. By Serre duality
$$
\dim \operatorname{Ext} ^1(L^{\otimes m+1} \otimes I_{Z_m}, L^{\otimes -m}) = h^1(L^{\otimes 2m+1}
\otimes I_{Z_m}),
$$
and this is greater than 2. By proposition \ref{1.5}, $\mathfrak{M}'$ has a sheaf
that is not locally free or is of type $m'<m$, but the later cannot
happen because of the choice of m.
\end{proof}
\begin{lemma}
\label{3.3}
Let $X$ be an irreducible component of $\FM (L,c_2)$. Let $X_0$ be the
subset corresponding to non-locally free sheaves. If $X_0$ is not
empty, then it has codimension one. Furthermore, there is a dense
subset $Y$ of $X_0$ such that for any $F \in Y$, we have
$$
c_2(F^{\vee\vee})=c_2(F)+1.$$
\end{lemma}
\begin{proof}
By \cite {O1}, prop. 7.1.3, we know that
\begin{equation}
\operatorname{codim}(X_0,X)\leq 1.
\label{bound}
\end{equation}
On the other hand, let $F\in X_0$. It fits in an exact sequence
$$
0 \to F \to F^{\vee\vee} \to {\mathcal{Q}} \to 0
$$
where ${\mathcal{Q}}$ is an Artinian sheaf
with length $l=H^0({\mathcal{Q}})=c_2(F^{\vee\vee})-c_2(F)$. We use this to
bound the dimension of $X_0$ by a parameter count. First we choose a
locally free sheaf $E\in \mathfrak{M}(L,c_2-l)$. These requires
$4(c_2-l)-L^2-6$ parameters. Now we have to choose a quotient to a
sheaf of length $l$ concentrated on a subset of dimension zero. These
quotients are parametrized by the Grothendieck Quot scheme $\operatorname{Quot}
(E,l)$, whose dimension is $3l$ (this follows from \cite {L1}
Appendix, where it is proved that $\operatorname{Quot} ^0(E,l)$, the Quot scheme
corresponding to quotients supported in $l$ distinct points, is dense
in $\operatorname{Quot}(E,l)$).
Define the following stratification on $X_0$:
$$
X_0=\bigcup_{l\geq 1} X_0^l \text{, }X_0^l=\{F:
c_2(F^{\vee\vee})-c_2(F)=l\}
$$
Then we have
$$
\dim (X_0^l) \leq 4(c_2-l)-L^2-6 + 3l
$$
and together with the previous bound \ref{bound} we obtain that
$Y=X_0^1$ is dense
and $\operatorname{codim}(X_0,X)=1$.
\end{proof}
\begin{proposition}
\label{3.4}
If $c_2$ satisfies the inequalities of the hypothesis of proposition
\ref{3.1}, then there is an open dense set on $\FM (L,c_2)$ that
corresponds to sheaves of type n.
\end{proposition}
\begin{proof}
We will prove this by showing that the codimension of sheaves of type
$m \leq n-1$ is greater than zero.
We will divide the proof into two cases:
\textbf{Case 1.}
\textit{Extensions of type $m$ with $Z_m$ such that $h^0(L^{\otimes 2m+1} \otimes
I_{Z_m}) = 0$.}
We have then $h^1(L^{\otimes 2m+1} \otimes I_{Z_m}) = c_2 -(m^2 +m+\frac{1}{2})
L^2 -2$. The dimension of the family $M$ of extensions of this kind is
bounded (via Serre duality) by
$$
\dim M \leq 2 l(Z_m) + h^1(L^{\otimes 2m+1} \otimes I_{Z_m}) -1 = 3 c_2
+(m^2 +m-\frac{1}{2}) L^2 -3.
$$
There is a map $\pi:M \to \FM (L,c_2)$ with fiber over each $V$ equal to
$\mathbb{P}(H^0(V\otimes L^{\otimes n}))$, and we can give a bound to its dimension (see proof
of proposition \ref{3.1}).
$$
\dim \mathbb{P}(H^0(V\otimes L^{\otimes n})) \geq (n^2 + n + \frac{1}{2}) L^2 - c_2 + 3
$$
Then the
dimension of the image of $\pi$ is bounded by
\begin{eqnarray*}
\dim (\operatorname{im} \pi) & \leq & \dim (M) - \min \dim \mathbb{P}(H^0(V\otimes L^{\otimes n})) \\
& \leq & 4 c_2 - L^2 -6 - 2nL^2
\end{eqnarray*}
And then
$$
\operatorname{codim} (\operatorname{im} \pi) \geq 2nL^2 >0
$$
\textbf{Case 2.}
\textit{Extensions of type $m$ with $Z_m$ such that
$h^0(L^{\otimes 2m+1} \otimes I_{Z_m}) \geq 1$.}
Let $M$ be the family of stable extensions with $Z_m$ satisfying this
inequality. Let $l = l(Z_m) = c_2 + (m+1)mL^2$.
The subscheme $Z_m$ is in a curve $C$ defined as the zeroes of a section
of $L^{\otimes 2m+1}$, i.e. $Z_m
\in \operatorname{Hilb} ^l (C)$. Although this curve will be reducible and
not reduced in general, the fact that $C$ is in a smooth surface
allows us to prove:
\begin{lemma}
\label{3.5}
$\dim \operatorname{Hilb} ^d (C) = d$
\end{lemma}
\begin{proof}
We have a natural stratification of $\operatorname{Hilb} ^d (C)$ given by the number
of points in the support of a subscheme
$$
\operatorname{Hilb} ^d (C) = \bigcup _{1 \leq r \leq d} \operatorname{Hilb} ^d _r (C),
$$
where $\operatorname{Hilb} ^d _r (C) = \{ Z : \# \operatorname{Supp} Z = r\}$. We have natural maps
giving the support of a subscheme:
$$
z^C _r : \operatorname{Hilb} ^d _r(C) \to \operatorname{Sym} ^r(C).
$$
In the same way we define maps $z^S _r$, when we consider subschemes
of the surface $S$. Clearly, if $x \in \operatorname{Sym} ^r (C) \subset \operatorname{Sym} ^r
(S)$, then $(z^C _r)^{-1} (x) \subset (z^S _r)^{-1} (x)$. Taking this
into account, and using a result of Iarrobino about zero dimensional
subschemes of a smooth surface \cite {I}:
\begin{eqnarray*}
\dim \operatorname{Hilb} ^d _r (C) &\leq & \dim \operatorname{Sym} ^r (C) + \operatorname{max.dim} (z^C _r)^{-1}
(x) \\
&\leq & \dim \operatorname{Sym} ^r (C) + \operatorname{max.dim} (z^S _r)^{-1} (x)\\
&= & r + d - r = d
\end{eqnarray*}
This gives $\dim \operatorname{Hilb} ^d (C) \leq d$, and the opposite direction is
trivial. This finishes the proof the lemma.
\end{proof}
Now we are going to bound the dimension of the set
$$
H' = \{ Z \in \operatorname{Hilb}^l (S): h^0(L^{\otimes 2m+1} \otimes I_Z) \geq 1\}.
$$
Consider the diagram
$$
\CD
Y = \{ (Z,C) \in \operatorname{Hilb}^l \times \mathbb{P}(H^0(L^{\otimes 2m+1}) : Z \in C\}
@>p_2>>
\mathbb{P}(H^0(L^{\otimes 2m+1})) \\
@Vp_1VV @. \\
\operatorname{Hilb} ^l (S) @. \\
\endCD
$$
We have $H' = \operatorname{im} (p_1)$ and the fiber of $p_2$ is $\operatorname{Hilb} ^l (C)$. $p_2$
is clearly surjective and the
fiber of $p_1$ is $\mathbb{P} (H^0(L^{\otimes 2m+1} \otimes I_Z))$. Then
$$
\dim H' = \dim \mathbb{P} (H^0(L^{\otimes 2m+1})) + \dim \operatorname{Hilb} ^l (C) - \dim \mathbb{P}
(H^0(L^{\otimes 2m+1} \otimes I_Z)) = 2l-1.$$
Again we have a map $\pi:M \to \FM (L,c_2)$ with fiber
$\mathbb{P} (H^0(V\otimes L^{\otimes m}))$, and then
$$
\operatorname{codim} (\operatorname{im} \pi) \geq (2m+1) L^2 +1 > 0.
$$
\end{proof}
As a corollary to this proposition we learn that to prove
connectedness of $\FM (L,c_2)$ it is enough to prove that all type $n$
sheaves are in one component.
\begin{proposition}
\label{3.6}
All stable extensions of $L^{\otimes n+1} \otimes I_Z$ by $L^{\otimes -n}$ such that
$L^{\otimes 2n+1} \otimes I_Z$ has
a section corresponding to an integral curve, are in one component.
\end{proposition}
\begin{proof}
Define the sets
\begin{eqnarray*}
\widetilde X_r & = & \{Z \in \Hilb ^{c_2 + n(n+1)L^2} (S) : \dim \Ext ^1 (L^{\otimes n+1} \otimes I_Z, L^{\otimes -n}) = r \text { and} \\
& &L^{\otimes 2n+1} \otimes I_Z \text { has a section
corresponding to an integral curve} \} \\
& & \\
X_r &= &\{ Z \in \widetilde X_r : \text { there is a stable extension of
} L^{\otimes n+1} \otimes I_Z \text { by } L^{\otimes -n} \} \\
\widetilde M_r &=& \{ \text {extensions of } L^{\otimes n+1} \otimes I_Z
\text {by } L^{\otimes -n} \text { with } Z \in X_r \}\\
& & \\
M_r &=& \{ m \in M_r : m \text { corresponds to a stable extension} \} \\
& & \\
N_r &=& \{(Z,C) \in \Hilb ^{c_2 + n(n+1)L^2} (S) \times \mathbb{P}(H^0(L^{\otimes 2n+1})) : Z \subset C \\
& & \text { and } \dim \Ext ^1 (L^{\otimes n+1} \otimes I_Z, L^{\otimes -n}) = r \} \\
& & \\
U &=& \{ C \in \mathbb{P}(H^0(L^{\otimes 2m+1})) : C \text { is irreducible and reduced} \}
\end{eqnarray*}
We construct $\widetilde M_r$ as parameter spaces of universal families of
extensions by standard techniques. These techniques require that the
dimension of the $\operatorname{Ext} ^1$ group is constant on the whole family. This
why we have to introduce the subscript $r$ and break
everything into pieces according to the dimension of the group $\operatorname{Ext}
^1$. We also consider the unions
$$
\widetilde M = \bigcup _{r \geq 1} \widetilde M_r \text {, }
M = \bigcup _{r \geq 1} M_r \text {, }
X = \bigcup _{r \geq 1} X_r \text {, } \ldots
$$
Note that $X$, being a subset of $\Hilb ^{c_2 + n(n+1)L^2} (S)$, has a natural scheme
structure. This is also true for $N \subset C
\times \mathbb{P}(H^0(L^{\otimes 2n+1}))\ $. On the other hand, for $M_r$
there is no natural way of
``putting them together'', so we take just the disjoint union.
We have the following maps
$$
\CD
\widetilde M @<<< M @>>> \FM (L,c_2) \\
@VVV @VVV @. \\
\widetilde X @<<< X @. \\
@A{p_1}AA @. @. \\
N @>{p_2}>> \mathbb{P}(H^0(L^{\otimes 2n+1})) @. \\
\endCD
$$
By construction $\widetilde X = p_1 {p_2}^{-1}(U)$.
Now we are going to prove that the fibers of $p_2$ over $U$ are
nonempty and connected. For each point in $N$ we have a commutative
diagram
$$
\CD
@. @. 0 @. 0 @.\\
@. @. @VVV @VVV @.\\
0 @>>> \SO_S @>>> L^{\otimes 2n+1} \otimes I_Z @>>>
j_*(\omega _C \otimes I_Z) @>>> 0\\
@. @| @VVV @VVV @.\\
0 @>>> \SO_S @>>> L^{\otimes 2n+1} @>>>
L^{\otimes 2n+1}|_C=j_*\omega _C @>>> 0\\
@. @. @VVV @VVV @.\\
@. @. {\mathcal{O}}_Z @= {\mathcal{O}}_Z @.\\
@. @. @VVV @VVV @.\\
@. @. 0 @. 0 @.\\
\endCD
$$
We argue in the same way as in proposition \ref{2.2}. Here we have
$$
r=(n^2+n+\frac{1}{2})L^2 +1 -c_2 \text{, }
d=(3n^2+3n+1)L^2-c_2
$$
$$ \rho (r,d)=2c_2-\frac{L^2}{2}-3=\frac{{\dim \FM (L,c_2)}}{2} > 0.$$
But now we don't know if $p_2$ is surjective with connected fibers,
because theorem I only applies
for irreducible reduced curves. This is the reason why we introduce
the open set $U$. For the fibers on $U$ we can apply the theorem, and
we conclude that $p_2$ is surjective over $U$ with connected fibers.
Then $\widetilde X=p_1 {p_2}^{-1}(U)$ is connected.
\textbf{Case 1.}
\textit{$p_1 {p_2}^{-1} (U) = X$.}
If $X_1 = X$, then we can construct a (connected, because $X$ is
connected) family $M_1$ parametrizing all sheaves with the
required properties, and we are done.
Now, if $X_1 \not= X$, then there are extensions with $r \geq 2$. By
corollary \ref{1.5}, $M_r$ with $r \geq 2$ is mapped to $\mathfrak{M} _0$, the
irreducible component that has sheaves that are not locally free. There is
only one irreducible component with this property, because by induction
hypothesis the moduli space when the second Chern class is smaller
than $c_2$ is
irreducible (see the outline of the proof). Now
we have to show that all the connected components of $M_1$ also go to
this component $\mathfrak{M} _0$.
The connectivity of $X=\widetilde X$ and the fact that $\dim \Ext ^1 (L^{\otimes n+1} \otimes I_Z, L^{\otimes -n})$ is
upper semicontinuous allows us to take a curve $f:T \to X$ with
$f(T-\{0\})$ in any given connected component of $X_1$ and $f(0) \in
X_r$, with $r \geq 2$.
Lift $f$ to a map $f:T-\{0\} \to M_1$. Note that $M_1$ won't be in
general a projective bundle because we have removed the points
corresponding to unstable extensions, but these make a closed set, and
(maybe after restricting $T$ to a smaller open set) we can construct
the lift without hitting this set.
$M_1$ maps to $\FM (L,c_2)$, and then we have a map $\phi : T-\{0\} \to
\FM (L,c_2)$. As in the proof of proposition \ref{2.4}, this gives us a
family of stable sheaves and sections parametrized by $T-\{0\}$ that
we can extend to a family parametrized by $T$. Now there are two
possibilities:
If $\phi(0)$ is of type $n$, then we have a family of extensions
$$
0 \to L^{\otimes -n} \to V_t \to \SO_S(L^{\otimes n+1} \otimes I_{Z_t}) \to 0 $$
and a corresponding map $\psi : T \to \Hilb ^{c_2 + n(n+1)L^2} (S)$, $t \mapsto Z_t$. By
construction $\psi(t) = f(t)$ for $t \not= 0$, and by properness also
for $t=0$. The extension corresponding to $t=0$ has to be in $M_r$
with $r \geq 2$, and then $M_1$ is also mapped to $\mathfrak{M} _0$.
On the other hand, if $\phi(0)$ is not a vector bundle of type $n$,
then it is either of type $m$ for $m<n$ or it is not locally free.
In either case, we conclude that $M_1$ is also mapped to $\mathfrak{M} _0$
\textbf{Case 2.}
\textit{$p_1 {p_2}^{-1} (U) \not= X$.}
Again, $M_r$, $r \geq 2$, gets mapped to $\mathfrak{M} _0$.
No connected component of $X_1$ can be closed, because by
connectedness of $\widetilde X$ and upper semicontinuity of $\dim
\Ext ^1 (L^{\otimes n+1} \otimes I_Z, L^{\otimes -n})$, we would have $\widetilde X = X_1$, and then $X = \widetilde
X$, contrary to the hypothesis.
Now we can prove that every connected component of $M_1$ gets mapped
to $\mathfrak{M} _0$. Take the corresponding connected component of $X_1$. Take
a curve $f:T \to \widetilde X$, with $f(T-\{0\})$ in the given
connected component of $X_1$, and $f(0) \notin X_1$. As in the
previous case, lift $f$ to a map $f:T-\{0\} \to M_1$, and now the proof
finishes like the end of case 1.
\end{proof}
\begin{proposition}
\label{3.7}
The set of sheaves $V$ of type $n$ such that for any extension
$$
0 \to L^{\otimes -n} \to V \to L^{\otimes n+1} \otimes I_Z \to 0,$$
$L^{\otimes 2n+1} \otimes I_Z$ has no section whose zero locus is an
integral curve, has positive codimension.
\end{proposition}
\begin{proof}
Define $\widetilde P=\{ Z \in \Hilb ^{c_2 + n(n+1)L^2} (S) : L^{\otimes 2n+1} \otimes I_Z$ has no
sections whose
zero locus is an irreducible reduced curve $\}$.
For each point of $\widetilde P$ we have a family of extension of type $n$
given by the projectivization of the corresponding $\operatorname{Ext} ^1$ group.
Writing $\widetilde P = \cup \widetilde P_r$, with $r$ equal to the
dimension of the group, we can
construct a family of extensions $\widetilde M_r^P$ for each $r$. As is
the previous proposition, let $P_r \subset \widetilde P_r$ be the
subset that has stable extensions.
We have a natural map $\pi_1:M_r^P \to \FM (L,c_2)$, where $M_r^P$ is the
subset of $\widetilde M_r^P$ corresponding to stable sheaves.
\begin{lemma}
\label{3.8}
The fiber of $\pi_1$ over $V \in \FM (L,c_2)$ is
$\mathbb{P} (H^0(V\otimes L^{\otimes n}))$.
\end{lemma}
\begin{proof}
The fiber consists of all extensions giving the same $V$. Now, given a
point in $\mathbb{P} (H^0(V\otimes L^{\otimes n}))$, we have an injection
$f: L^{\otimes -n} \hookrightarrow V$ (up
to scalar). $V$
is locally free and of type $n$, then the quotient is torsion free
and we get an element $Z_f$ of $\Hilb ^{c_2 + n(n+1)L^2} (S)$:
$$0 \to L^{\otimes -n} \to V \to L^{\otimes n+1} \otimes I_{Z_f} \to 0.$$
This defines a map from $\mathbb{P} (H^0(V\otimes L^{\otimes n}))$ to the fiber of $\pi$. It is
clearly surjective. Now we will check that it is also injective.
If $Z_f = Z_{f'}$, then $f$ and $f'$ have to differ at most by scalar
multiplication, because all nonsplit extensions of $L^{\otimes n+1} \otimes
I_{Z_f} = L^{\otimes n+1} \otimes I_{Z_{f'}}$ by $L^{\otimes -n}$ that give the same $V$
are weak isomorphic, so we get a diagram:
$$
\CD
0 @>>> L^{\otimes -n} @>f>> V @>>> L^{\otimes n+1} \otimes I_{Z_f} @>>> 0 \\
@. @V{\alpha}VV @V{\cong}VV @V{\beta}VV @. \\
0 @>>> L^{\otimes -n} @>f'>> V @>>> L^{\otimes n+1} \otimes I_{Z_{f'}} @>>> 0 \\
\endCD
$$
where $\alpha$ is multiplication by scalar.
\end{proof}
In $\mathbb{P} (H^0(L^{\otimes 2n+1}))$ we have a subvariety $Y$ corresponding to
reducible curves. This subvariety is the image of the natural map
$$\bigcup _{0<a<2n+1} \mathbb{P}(H^0(L^{\otimes a})) \times
\mathbb{P}(H^0(L^{\otimes 2n+1-a})) \to
\mathbb{P}(H^0(L^{\otimes 2n+1}))$$
We define the set
\begin{eqnarray*}
\widetilde N_r^P & = &\{ (Z,C) \in X_r \times \mathbb{P}(H^0(L^{\otimes 2n+1})): \\
& &Z \subset C, \dim (p_1)^{-1}(Z)=(n^2+n+
\frac{1}{2})L^2-c_2 +1+r \}
\end{eqnarray*}
and the maps
$$
\CD
\widetilde N_r^P @>p_2>> \mathbb{P}(H^0(L^{\otimes 2n+1})) \\
@Vp_1VV @. \\
\Hilb ^{c_2 + n(n+1)L^2} (S) @. \\
@A{\pi_2}AA @. \\
M_r^P @>{\pi_1}>> \FM (L,c_2) \\
\endCD
$$
By construction we have $P_r \subset \widetilde P_r \subset
p_2 p_1^{-1} (Y)$. Then
$$
\dim P_r \leq \dim p_2((p_1)^{-1} (Y)) = \dim Y + \dim\operatorname{fiber} (p_2) -
\dim\operatorname{fiber} (p_1),
$$
where $\dim Y$ is the maximum of the dimensions of its irreducible
components. Finally
$$\operatorname{codim} (\operatorname{im} \pi_1) = \dim \FM (L,c_2) - \dim P_r -\dim\operatorname{fiber} \pi_2 +
\dim\operatorname{fiber} \pi_1,$$
and putting everything together we have $\operatorname{codim} (\operatorname{im} \pi_1) >
(2n-a)(a-1)L^2$ for every $0<a<n+1$, and then $\operatorname{codim} (\operatorname{im} \pi_1) > 0$.
\end{proof}
\section{General K3 surface (proof of theorem II)}
\label{General K3 surface}
In this section we finally prove theorem II by showing that if the
result is true for a surface $S$ with $\operatorname{Pic}(S)=\mathbb{Z}$, then it also
holds under the hypothesis of theorem II. The idea is to deform the
given surface to a generic surface with $\operatorname{Pic}(S)=\mathbb{Z}$. We also deform
the moduli space, and then the irreducibility of the moduli space for
the deformed surface will imply the irreducibility for the surface we
started with. This is very similar to an argument in
\cite{G-H}.
Because we are going to vary the surface, in this
section we will denote the moduli space of semistable sheaves with
$\mathfrak{M}_H(S,L,c_2)$, where $S$ is the surface on which the sheaves are
defined.
\smallskip
\begin{proof2} \textit{of theorem II}
Recall that we have a surface $S$ with a $(L,c_2)-$generic polarization $H$.
By 2.1.1 in \cite{G-H}, there is a connected family of surfaces ${\mathcal{S}}$
parametrized by a curve $T$ and a line bundle ${\mathcal{L}}$ on ${\mathcal{S}}$ such that
$({\mathcal{S}}_0,{\mathcal{L}}_0) = (S,L)$ and $\operatorname{Pic}({\mathcal{S}}_t)={\mathcal{L}}_t \cdot \mathbb{Z}$
for $t\neq 0$. By proposition 2.3 in \cite{G-H}, there is a connected smooth
proper family ${\mathcal{Z}} \to T$ such that ${\mathcal{Z}}_0 \cong \mathfrak{M}_H(S,{\mathcal{L}}_0,c_2)$
(note that the polarization is $H$ and not ${\mathcal{L}}_0$)
and ${\mathcal{Z}}_t \cong \mathfrak{M}_{{\mathcal{L}}_t}({\mathcal{S}}_t,{\mathcal{L}}_t,c_2)$ for $t \neq 0$.
By propositions \ref{2.4} and \ref{3.4} we know that ${\mathcal{Z}}_t$ is irreducible
for $t \neq 0$, and then by an argument parallel to lemma \ref{bn0.2}, we
obtain that ${\mathcal{Z}}_0$ is connected, but ${\mathcal{Z}}_0$ is smooth
(because $H$ is generic), and then this implies that ${\mathcal{Z}}_0$ is
irreducible.
\end{proof2}
\chapter{Irreducibility of the moduli space for del Pezzo surfaces}
\label{dp}
In this chapter we will consider the case in which $S$ is a del Pezzo
surface. We will prove the following theorem (see chapter
\ref{Preliminaries} for the notation).
\smallskip
\noindent\textbf{Theorem III.}
\textit{
Let $S$ be a del Pezzo surface. Let $\mathfrak{M}_L(S,c_1,c_2)$ be the moduli
space of rank 2, Gieseker L-semistable torsion free sheaves with Chern
classes $c_1$, $c_2$, with $L$ a $(c_1,c_2)$-generic polarization.
Then $\mathfrak{M}_L(S,c_1,c_2)$ is either empty or irreducible.}
\smallskip
As we explained in chapter \ref{Preliminaries}, this is already known
for $\mathbb{P}^2$ and $\mathbb{P}^1 \times \mathbb{P}^1$, so we will assume from now on
that $S$ is a surface isomorphic to $\mathbb{P}^2$ blown up at
most at 8 points in general position. We denote the blow up map
$\pi:S \to \mathbb{P}^2$. We will denote by $H$ the effective generator of
$\operatorname{Pic}(\mathbb{P}^2)$. We also denote by $H$ the pullback $\pi^*(H)$.
$\mathfrak{M}_L(X,c_1,c_2)$ will denote the moduli space of $L$-semistable
torsion free rank two sheaves on $X$ with Chern classes $c_1$, $c_2$.
In the case $X=\mathbb{P}^2$ we will drop $L$ from the notation, because
there is only one possible polarization.
This is a particular case of the conjecture of Friedman and Qin
\cite{F-Q} that states that the moduli
space of stable vector bundles on a rational surface with effective
anticanonical line bundle is either empty or irreducible (for any
choice of polarization and Chern classes).
\begin{proposition}
\label{dp1}
$\mathfrak{M}_{L_0}(S,c_1,c_2)$ is either irreducible or empty, where
$L_0$ is a polarization of $S$ that lies in a chamber whose closure
contains $H$.
\end{proposition}
\begin{proof}
By \cite{B}, $\mathfrak{M}_{L_0}(S,bH+a_1E_1+\cdots+a_nE_n,c_2)$ is
birational to a $(\mathbb{P}^1)^m$ bundle over $\mathfrak{M}(\mathbb{P}^2,bH,c_2)$ (where $m$
is the number of $a_i$'s that are odd), but
it is well known that this moduli space is either irreducible or
empty.
\end{proof}
To prove this statement for any generic polarization, we will
need some lemmas about the following system of equations on integer
numbers:
$$
\left.
\begin{array}{rl}
a_1^2+ \cdots +a_8^2 &= x+b^2\\
-a_1- \cdots -a_8&=x-2+3b
\end{array}
\right \}
\ \ \
\begin{array}{c}
(\dagger)
\end{array}
$$
where $x$ is some given (integer) number, and $b$,$a_1,\ldots,a_8$
are the unknowns.
\begin{lemma}
\label{dp2}
If $x\geq3$ then any integer solution of $(\dagger)$ with
$b>0$ has $b\leq2$.
\end{lemma}
\begin{proof}
This is obtained by an elementary argument.
We can interpret geometrically these equations as the intersection of
a one-sheeted hyperboloid and a plane. We will look first at real
solutions. Rewrite $(\dagger)$ as
$$
\left.
\begin{array}{rl}
b &= \frac {1}{3}(-a_1 - \cdots -a_8 +2-x) \\
a_1^2+ \cdots +a_8^2 &= x+\frac {1}{9}(-a_1 - \cdots -a_8 +2-x)^2
\end{array}
\right \}
$$
The first equation defines a function, and the second equation is a
constrain. Using the method of Lagrange multipliers we obtain that the
maximum and minimum values of $b$ are at points of the form $a_i=t$
for some $t$. Then $(\dagger)$ become
$$
\left .
\begin{array}{rl}
8t^2 & =b^2+x \\
-8t-3b & =-2+x
\end{array}
\right \}
$$
Looking at the real solutions of these equations, we find
\begin{equation}
\label{bpm}
b^\pm = \frac {-6(x-2)\pm \sqrt{36(x-2)^2-4((x-2)^2-8x)}}{2}.
\end{equation}
For $x\geq 3$, $b^-$ is always negative.
If we want to have solutions with $b>0$
we need $b^+>0$. Using \ref{bpm}, this is equivalent to
$(x-2)^2<8x$, and this implies $x<(12+\sqrt{128})/2<12$. This bound,
together with the hypothesis $3\leq x$ and \ref{bpm} implies
$b^+<3$, but if we are only interested in integer solutions this gives
$b\leq 2$.
\end{proof}
\begin{lemma}
\label{dp3}
If $1\leq b \leq 2$, then the integer solutions of $(\dagger)$ (up to
permutation of $a_i$) are:
\begin{eqnarray*}
b=1,\ a_1=\cdots=a_{x+1}=-1,\ a_{x+2}=\cdots=a_8=0 & \\
b=2,\ a_1=\cdots=a_{x+4}=-1,\ a_{x+5}=\cdots=a_8=0 & \! .
\end{eqnarray*}
\end{lemma}
\begin{proof}
In both cases ($b=1$ or $b=2$), the right hand sides of the two
equations are equal, and then, substracting the equations we have
$$
\sum a_i^2+a_i=0.
$$
But for $a_i$ integer we have $a_i^2+a_i\geq0$, and then $a_i$ must be
equal to -1 or 0.
Now, looking at the equations we see that the number of nonzero
$a_i$'s is given by $x+b^2$, and we obtain the result.
\end{proof}
\begin{theorem}
\label{dp4}
$\mathfrak{M}_L(S,c_1,c_2)$, for any generic polarization $L$, is either empty
or irreducible.
\end{theorem}
\begin{proof}
We will denote by $L_0$ a polarization lying in a chamber whose
closure contains $H$.
If $\mathfrak{M}_L(S,c_1,c_2)$ has more than one irreducible component, then
there must be a wall between $L$ and $L_0$ that created the extra
component. Recall that a wall $W^\zeta$ is a hyperplane in the ample
cone perpendicular to a class $\zeta$ with $\zeta \equiv c_1$ (mod 2),
and $c_1^2-4c_2\leq\zeta^2<0$. By \cite{F-Q}, if $L_1\cdot\zeta>0
>L_2\cdot\zeta$, then the sheaves that are $L_1$-unstable and
$L_2$-stable make an irreducible family of dimension $N_\zeta+2l_\zeta$, where
$N_\zeta=h^1(\zeta)+l_\zeta-1$ and $l_\zeta=(4c_2-c_1^2+\zeta^2)/4$.
In the case of a rational surface we have
$$
h^1(\zeta)=\frac{\zeta\cdot K_S}{2}-\frac{\zeta^2}{2}-1.
$$
The wall creates a new component if $N_\zeta+2l_\zeta$ is equal to the
dimension of the moduli space, in our case $4c_2-c_1^2-3$. For a
rational surface we have $N_\zeta+N_{-\zeta}+2l_\zeta=4c_2-c_1^2-4$,
and then this condition is equivalent to $N_{-\zeta}=-1$. Denoting
$\zeta=bH+a_1E_1+\cdots+a_nE_n$ and $x=4c_2-c_1^2$ we get the system
of equations ($\dagger$) ($n\leq8$ so without loss of generality we can study
the equations with $n=8$). Furthermore, this wall will create a new
component if
$L_0\cdot\zeta<0<L\cdot\zeta$. By the definition of chamber, the last
equality is equivalent to $0<H\cdot\zeta$, and this
translates to $b>0$.
We will prove the proposition by showing that for given $(S,c_1,c_2)$
there is at most one such wall in the ample cone and that in this
case $\mathfrak{M}_{L_0}(S,c_1,c_2)$ is empty, so that $\mathfrak{M}_L(S,c_1,c_2)$ is
always either empty or only has one irreducible component.
If the moduli space is not empty, its dimension should be
nonnegative, and this translates to $x\geq3$. Then by lemmas \ref{dp2}
and \ref{dp3} we know all the solutions of ($\dagger$), i.e. all the
walls creating components.
By tensoring with a line bundle (and relabeling the exceptional
curves), we can assume that $c_1$ is either $H+E_1+\cdots+E_m$ or
$E_1+\cdots+E_m$ for some $m\leq8$. Now we will use the condition $c_1
\equiv \zeta$ (mod 2).
In the first case this implies that the only
possible solution for ($\dagger$) is $\zeta=H-E_1-\cdots-E_m$, and $m=x+1$.
Then $4c_2-c_1^2=x$ implies $c_2=0$. By \cite{B},
$\mathfrak{M}_{L_0}(S,H+E_1+\cdots+E_m,0)$ is birational to a $(\mathbb{P}^1)^m$ bundle over
$\mathfrak{M}(\mathbb{P}^2,H,0)$, but it is well known that this moduli space is
empty, then the same is true for $\mathfrak{M}_{L_0}(S,H+E_1+\cdots+E_m,0)$.
In the second case ($c_1=E_1+\cdots+E_m$), the condition $c_1 \equiv
\zeta$ (mod 2) implies that the only possible solution is
$\zeta=2H-E_1-\cdots-E_m$ with $m=x+4$. Then $c_2=-1$ and then
$\mathfrak{M}_{L_0}(S,c_1,c_2)$ is empty (because its expected dimension is
negative).
\end{proof}
This techniques can also be used to study the irreducibility of the
moduli space of stable torsion free sheaves on $\mathbb{P}^2$
with more than 8 blown up points. We obtain equations similar to
$(\dagger)$, but with more variables. Unfortunately now we don't have
a bound on $b$ like the one given by lemma \ref{dp2}, and then it
becomes more difficult to classify the solutions. This is still work
in progress and it will appear elsewhere.
|
1997-12-09T23:22:42 | 9710 | alg-geom/9710004 | en | https://arxiv.org/abs/alg-geom/9710004 | [
"alg-geom",
"math.AG"
] | alg-geom/9710004 | Uli Walther | Uli Walther (University of Minnesota) | Algorithmic Computation of Local Cohomology Modules and the
Cohomological Dimension of Algebraic Varieties | 20 pages, amsart, uses amstex, amssymb, xypic.tex, corrected some
typos | null | null | null | null | In this paper we present algorithms that compute certain local cohomology
modules associated to a ring of polynomials containing the rational numbers. In
particular we are able to compute the local cohomological dimension of
algebraic varieties in characteristic zero. Our approach is based on the theory
of D-modules.
| [
{
"version": "v1",
"created": "Fri, 3 Oct 1997 20:05:38 GMT"
},
{
"version": "v2",
"created": "Tue, 9 Dec 1997 22:22:37 GMT"
}
] | 2007-05-23T00:00:00 | [
[
"Walther",
"Uli",
"",
"University of Minnesota"
]
] | alg-geom | \section{Introduction}
\subsection{} Let $R$ be a commutative Noetherian ring, $I$ an ideal in
$R$ and $M$ an
$R$-module. The $i$-th {\em local
cohomology functor} with respect to $I$ is the $i$-th right
derived functor of the functor $H^0_I(-)$ which sends $M$ to the
$I$-torsion $\bigcup_{k=1}^\infty (0:_MI^k)$ of $M$ and is denoted by
$H^i_I(-)$. Local
cohomology was introduced by Grothendieck as an algebraic analog of
(classical) relative cohomology. A brief introduction to local
cohomology may be found in appendix 4 of \cite{E}.
The {\em
cohomological dimension} of $I$ in $R$, denoted by $\operatorname{cd}(R,I)$, is the
smallest integer $c$ such that the local cohomology modules
$H^q_I(M)=0$ for all $q>c$ and all $R$-modules $M$. If $R$ is the
coordinate ring of an affine variety $X$ and $I\subseteq R$ is the defining
ideal of the Zariski closed subset $V\subseteq X$ then the {\em local
cohomological dimension} of $V$ in $X$ is defined as $\operatorname{cd}(R,I)$.
It is not hard to show that if $X$ is smooth, then the integer
$\dim(X)-\operatorname{cd}(R,I)$ depends only on $V$ but neither on $X$ nor on the
embedding $V\hookrightarrow X$.
\subsection{} Knowledge of local cohomology modules provides interesting
information, illustrated by the following three situations. Let $I\subseteq
R$ and $c=\operatorname{cd}(R,I)$. Then
$I$ cannot be generated by fewer than $c$ elements. In fact, no ideal $J$
with the same radical as $I$ will be generated by fewer than $c$ elements.
Let $H^i_{dR}$ stand for the $i$-th de Rham cohomology group.
A second application is a family of results commonly known as Barth
theorems which are a generalization of the classical
Lefschetz theorem that states that if $Y\subseteq {\Bbb P}^n_{\Bbb C}$ is a hypersurface
then $H^i_{dR}({\Bbb P}^n_{\Bbb C})\to
H^i_{dR}(Y)$ is an isomorphism for $i<\dim (Y)-1$ and injective for
$i=\dim (Y)$. For example, let $Y\subseteq {\Bbb P}_{\Bbb C}^n$ be a closed subset and $I\subseteq
R={\Bbb C}[x_0,\ldots,x_n]$ the
defining ideal of $Y$.
Then $H^i_{dR}({\Bbb P}^n_{\Bbb C})\to H^i_{dR}(Y)$ is an isomorphism for $i\le
\operatorname{depth}_{{\cal O}_{{\Bbb P}^n_{\Bbb C}}}({\cal O}_Y)-\operatorname{cd}(R,I)$ (compare \cite{Og}, 4.7 and
\cite{DRCAV}, the theorem after III.7.6).
Finally, it is also a consequence of the work of Ogus and Hartshorne
(\cite{Og}, 2.2, 2.3 and \cite{DRCAV}, IV.3.1) that if $I\subseteq
R={\Bbb C}[x_0,\ldots,x_n]$ is the defining
ideal of a complex smooth variety $V\subseteq {\Bbb P}^n_{\Bbb C}$ then, for $i<n-\operatorname{codim} (V)$,
$\dim_{\Bbb C}\operatorname{soc}_R
(H^0_{\frak m}(H^{n-i} (R)))$ equals $\dim_{\Bbb C} H^i_x(\tilde V,{\Bbb C})$ where $H^i_x(\tilde
V,{\Bbb C})$ stands for the $i$-th singular cohomology group of the affine
cone $\tilde V$ over $V$ with support in the vertex $x$ of $\tilde V$ and
with coefficients in ${\Bbb C}$ ($\operatorname{soc}_R(M)$ denotes the socle
$(0:_M{\frak m})\subseteq M$ for any
$R$-module $M$).
\subsection{}
The cohomological dimension has been studied by many authors, for
example R.~Hartshorne (\cite{CDAV}), A.~Ogus (\cite{Og}),
R.~Hartshorne and R.~Speiser
(\cite{H-Sp}), C.~Peskine and L.~Szpiro (\cite{P-S}), G.~Faltings
(\cite{F}),
C.~Huneke and G.~Lyubeznik (\cite{Hu-L}).
Yet
despite this extensive effort, the
problem of finding an algorithm for the computation of cohomological
dimension remained open. For the determination of $\operatorname{cd}(R,I)$ it is in
fact enough to find an algorithm to
decide whether or not the local cohomology module $H^i_I(R)=0$ for
given $i, R, I$. This is because $H^q_I(R)=0$ for all $q>c$ implies
$\operatorname{cd}(R,I)\le c$ (see \cite{CDAV}, section 1). In \cite{L-Fmod}
G.~Lyubeznik gave an
algorithm for deciding whether or not
$H^i_I(R)=0$ for all $I\subseteq R=K[x_1,\ldots,x_n]$ where
$K$ is a field of positive characteristic.
One
of the main purposes of
this work is to produce such an algorithm in the case where $K$ is a field
containing the rational numbers and $R=K[x_1,\ldots,x_n]$.
Since in such a situation
the local cohomology
modules $H^i_I(R)$ have a natural
structure of finitely generated left $D(R,K)$-modules (\cite{L-Dmod}), $D(R,K)$
being the ring
of $K$-linear differential operators of $R$, explicit computations may be
performed.
Using
this finiteness we employ the theory of
Gr\"obner bases
to develop
algorithms that give a representation of $H^i_I(R)$ and $H^i_{\frak m} (H^j_I(R))$
for all triples $i,j\in {\Bbb N},I\subseteq R$ in terms of generators and
relations over
$D(R,K)$ (where ${\frak m}=(x_1,\ldots,x_n)$). This also leads to an
algorithm for the computation of the invariants
$\lambda_{i,j}(R/I)=\dim_K\operatorname{soc}_R(H^i_{\frak m}(H^{n-j}_I(R)))$ introduced in
\cite{L-Dmod}.
We remark that if $R$ is an arbitrary finitely generated $K$-algebra
and $I$ is an ideal in $R$
then, if
$R$ is
regular, our algorithms can be used to determine $\operatorname{cd}(R,I)$ for
all ideals $I$ of $R$,
but if $R$ is not regular, then the problem of algorithmic
determination of $\operatorname{cd}(R,I)$ remains open (see also the comments in
subsection \ref{singular_spaces}).
\subsection{}
The outline of the paper is as follows.
The next section is devoted to a short survey of results on
local cohomology and $D$-modules as they apply to our work, as well as
their interrelationship.
In section \ref{sec-gb} we
review the theory of Gr\"obner bases as it applies to $A_n$ and
modules over the Weyl algebra. Most of that section should be
standard and readers interested in proofs and more details are
encouraged to look at the book by D.~Eisenbud (\cite{E}, chapter 15
for the commutative case)
or the fundamental article \cite{KR-W} (for the more general situation.
In section \ref{sec-mal-kash} we generalize some results due to
B.~Malgrange and
M.~Kashiwara on $D$-modules and their localizations. The purpose of
sections \ref{sec-mal-kash} and \ref{sec-oaku} is to find a
representation of $R_f\otimes N$ as a cyclic $A_n$-module if $N$
is a given holonomic $D$-module (for a definition and some properties
of holonomic modules, see subsection \ref{D-modules} below).
Many of the essential ideas in section \ref{sec-oaku} come from
T.~Oaku's work \cite{Oa}.
In
section \ref{sec-lc} we describe our main results, namely
algorithms that for
arbitrary $i,j,k,I$
determine the structure of
$H^k_I(R), H^i_{\frak m} (H^j_I(R))$ and find $\lambda_{i,j}(R/I)$.
Some of these algorithms have been
implemented in the
programming language C and the theory is illustrated with examples.
The final section is devoted to comments on
implementations, effectivity and examples.
It is a pleasure to thank my advisor Gennady Lyubeznik for suggesting the
problem of algorithmic computation of cohomological dimension
to me and pointing out that the theory of $D$-modules might be
useful for its solution.
\section{Preliminaries}
\label{sec-prelim}
\subsection{Notation} Throughout we shall use the following notation: $K$ will
denote a field
of characteristic zero, $R=K[x_1,\ldots,x_n]$ the ring of polynomials
over $K$ in $n$ variables, $A_n=K\langle
x_1,\partial_1,\ldots,x_n,\partial_n\rangle$ the Weyl algebra over $K$ in $n$
variables, or, equivalently, the ring of $K$-linear differential operators on
$R$, ${\frak m}$ will stand for the maximal ideal $(x_1,\ldots,x_n)$ of
$R$, $\Delta$ will denote the maximal left ideal $(\partial_1,\ldots,\partial_n)$
of $A_n$ and $I$ will stand for the ideal $(f_1,\ldots,f_r)$ in $R$.
All tensor products in this work will be over $R$ and all
$A_n$-modules (resp.~ideals) will be left modules (resp.~left ideals).
\subsection{Local cohomology}
It turns out that
$H^k_I(M)$ may be
computed as follows. Let $C^\bullet(f_i)$ be the complex $0\to
R\stackrel{1\to\frac{1}{1}}{\longrightarrow} R_{f_i}\to 0$. Then
$H^k_I(M)$ is the $k$-th
cohomology group of the {\em \v Cech complex} defined by
$C^\bullet(M;f_1,\ldots,f_r)
=\bigotimes_1^r C^\bullet(f_i)\otimes M$.
Unfortunately,
explicit
calculations are complicated by the fact that $H^k_I(M)$ is rarely
finitely generated as $R$-module. This
difficulty disappears for $H^k_I(R)$ if we
enlarge the ring to $A_n$, in essence because $R_f$ is finitely
generated over $A_n$
for all $f\in R$.
\subsection{$D$-modules} \mylabel{D-modules}
A good introduction to $D$-modules is the book by Bj\"ork, \cite{B}.
Let $f\in R$. Then the $R$-module $R_f$ has a
structure as left $A_n$-module: $x_i(\frac{g}{f^k})=\frac{x_ig}{f^k},
\partial_i(\frac{g}{f^k})=\frac{\partial_i(g)f-k\partial_i(f)g}{f^{k+1}}$. This
may be thought of as a special case of localizing an $A_n$-module: if
$M$ is an $A_n$-module and $f\in R$ then $R_f\otimes_R M$ becomes an
$A_n$-module via $\partial_i(\frac{g}{f^k}\otimes
m)=\partial_i(\frac{g}{f^k})\otimes m+\frac{g}{f^k}\otimes
\partial_i m$. Localization of $A_n$-modules lies at the heart of our arguments.
Of particular interest are the {\em
holonomic} modules which are those finitely generated $A_n$-modules $N$
for which $\operatorname{Ext}^j_{A_n}(N,A_n)$ vanishes
unless $j=n$. Holonomic modules are
always cyclic and of finite length over $A_n$. Besides that, if
$N=A_n/L$, $f\in
R$, $s$ is an indeterminate and $g$ is
some fixed generator of $N$, then
there is a nonzero polynomial $b(s)$ in $K[s]$ and an
operator $P(s)\in A_n[s]$ such that $P(s)(f\cdot f^s\otimes g)=b(s)\cdot
f^s\otimes g$. The unique monic polynomial that divides
all other polynomials satisfying an identity of this type is called the
{\em Bernstein polynomial} of $L$ and $f$ and denoted by $b_f^L(s)$.
Any operator $P(s)$ that satisfies $P(s)f^{s+1}\otimes
g=b_f^L(s)f^s\otimes g$ we shall call a {\em Bernstein operator} and refer
to the roots of $b_f^L(s)$ as {\em Bernstein roots} of $f$ on $A_n/L$.
Localizations of
holonomic modules at a single element (and hence at any finite number
of elements) of $R$ are holonomic (see \cite{B}, section 5.9) and
in particular cyclic over $A_n$, generated by $f^{-a}g$ for
sufficiently large $a\in {\Bbb N}$ (see also our proposition
\ref{kashiwara}). So the complex $C^\bullet(N;f_1,\ldots,f_r)$
consists of holonomic $A_n$-modules whenever $N$ is holonomic.
This facilitates the use
of Gr\"obner bases as
computational tool for maps between holonomic modules and their
localizations.
As a
special case we note that
localizations of $R$ are holonomic, and hence finite, over $A_n$ (since
$R=A_n/\Delta$ is holonomic).
\subsection{The \v Cech complex}
In \cite{L-Dmod} it is shown that local cohomology modules are not only
$D$-modules but in fact holonomic: we know already that the modules in
the \v Cech complex are holonomic, it suffices to show that the maps are
$A_n$-linear. All maps in the \v Cech complex are direct sums of
localization maps. Suppose $R_f$ is generated by $f^s$ and
$R_{fg}$ by
$(fg)^t$. We may replace $s,t$ by their minimum $u$ and then we
see that the inclusion $R_f\to R_{fg}$ is nothing but the map
$A_n/\operatorname{ann}(f^u)\to A_n/\operatorname{ann}((fg)^u)$ sending the coset of the operator
$P$ to the coset of the operator $P\cdot g^u$. So
$C^i(N;f_1,\ldots,f_r)\to C^{i+1}(N;f_1,\ldots,f_r)$ is an
$A_n$-linear map between holonomic modules for every holonomic $N$.
One can prove that kernels and cokernels of $A_n$-linear maps between
holonomic modules are holonomic. Holonomicity of $H^k_I(R)$
follows.
In the same way it can be seen that $H^i_{\frak m} (H^j_I(R))$ is holonomic for
$i,j\in{\Bbb N}$ (since $H^j_I(R)$ is holonomic).
\section{Gr\"obner bases of modules over the Weyl algebra}
\mylabel{sec-gb}
In this section we review some of the concepts and results related to
the Buchberger algorithm in modules over Weyl
algebras. It turns out that with a little care many of the important
constructions from the theory of commutative Gr\"obner bases carry
over to our case. For an introduction into non-commutative monomial
orders and related
topics, \cite{KR-W} is a good source.
Let us agree that every time we write an element in $A_n$, we
write it as a sum of terms $c_{\alpha\beta}x^\alpha \partial^\beta$ in multi-index
notation. That is, $\alpha,\beta\in {\Bbb N}^n$, $c_{\alpha\beta}$ are scalars,
$x^\alpha=x_1^{\alpha_1}\cdot \ldots\cdot x_n^{\alpha_n},
\partial^\beta=\partial_1^{\beta_1}\cdot\ldots\cdot \partial_n^{\beta_n}$ and in
every monomial
we write first the powers of $x$
and then the powers of the differentials. Further, if
$m=c_{\alpha\beta}x^\alpha\partial^\beta, c_{\alpha\beta}\in K$, we will
say that $m$ has degree $\deg
m=|\alpha+\beta|$ and an operator $P\in A_n$ has degree equal to the largest
degree of any monomial occuring in $P$.
Recall that a {\em monomial order} $<$ in $A_n$ is a total order on the
monomials of $A_n$, subject to $m<m'\Rightarrow mm''<m'm''$ for all nonzero
monomials $m,m',m''$.
Since the product of two monomials in our notation is not likely to be
a monomial (as $\partial_i x_i=x_i\partial_i+1$) it is not obvious that such
orderings exist at all. However, the commutator of any two monomials
$m_1,m_2$
will be a polynomial of degree at most $\deg m_1+\deg m_2-2$. That means
that the degree of an operator and its component of maximal degree
is independent of the way it is
represented. Thus we may for example introduce a monomial order on
$A_n$ by taking
any monomial order on $\tilde A_n=K[x_1,\ldots,x_n,\partial_1,\ldots,\partial_n]$ (the
polynomial ring in $2n$ variables) that refines the partial order
given by total degree, and saying that $m_1>m_2$ in $A_n$
if and only if $m_1>m_2$ in $\tilde A_n$.
Let $<$ be a monomial order on $A_n$.
Let $G=\bigoplus_1^dA_n\cdot \gamma_i$ be the free
$A_n$-module on the symbols $\gamma_1,\ldots,\gamma_d$. We define a
monomial order on $G$ by $m_i\gamma_i>m_j\gamma_j$ if either $m_i>m_j$
in the order on $A_n$, or $m_i=m_j$ and $i>j$.
The largest monomial $m\gamma$ in an element $g\in G$ will be denoted by
$\operatorname{in}(g)$. Of fundamental importance is
\begin{alg}[Remainder]
\mylabel{remainder}
Let $h$ and $\_ g=\{g_i\}_1^s$ be elements of $G$. Set $h_0=h, \sigma_0=0,
j=0$ and let $\varepsilon_i=\varepsilon(g_i)$ be symbols. Then
\[
\begin{array}{llll}
&{\tt Repeat}&&\\
&&{\tt If} \operatorname{in}(g_i)|\operatorname{in}(h_j) {\tt \,set}&\\
&&&\{h_{j+1}:=h_j-\frac{\operatorname{in}(h_j)}{\operatorname{in}(g_i)}g_i,\\
&&&\sigma_{j+1}:=\sigma_j+\frac{\operatorname{in}(h_j)}{\operatorname{in}(g_i)}\varepsilon_i,\\
&&&j:=j+1\}\\
&{\tt Until}&{\tt No} \operatorname{in}(g_i)|\operatorname{in}(h_j).
\end{array}
\]
The result is $h_a$, called a {\em remainder $\Re(h,\_ g)$ of $h$
under division by $\_ g$}, and an expression $\sigma_a=\sum_{i=1}^s
a_i\varepsilon_i$
with $a_i\in A_n$.
By Dickson's lemma (\cite{KR-W}, 1.1) the
algorithm terminates. It is worth mentioning that $\Re(h,\_ g)$ is not
uniquely determined, it depends on which $g_i$ we pick amongst those
whose initial term divides the initial term of $h_j$.
Note that if $h_a$ is zero, $\sigma_a$ tells us how to write $h$ in
terms of $\_ g$. Such a $\sigma_a$ is called a \em{standard
expression for $h$} with respect to $\{g_1,\ldots,g_s\}$.
\end{alg}
\begin{df}\mylabel{schreyer}
If $\operatorname{in}(g_i)$ and $\operatorname{in}(g_j)$ involve the same basis element of $G$,
then we set $s_{ij}=m_{ji}g_i-m_{ij}g_j$ and
$\sigma_{ij}=m_{ji}\varepsilon_i-m_{ij}\varepsilon_j$ where
$m_{ij}=\frac{\operatorname{lcm}(\operatorname{in}(g_j),\operatorname{in}(g_i))}{\operatorname{in}(g_j)}$. Otherwise,
$\sigma_{ij}$ and $s_{ij}$ are defined to be zero. $s_{ij}$ is the
{\em Schreyer-polynomial} to $g_i$ and $g_j$.
Suppose $\Re(s_{ij},\_ g)$ is zero for all $i,j$. Then we call $\_ g$
a {\em Gr\"obner basis} for the module $A_n\cdot(g_1,\ldots,g_s)$.
\end{df}
The following proposition (\cite{KR-W}, Lemma 3.8) indicates the
usefulness of Gr\"obner
bases.
\begin{prop}
\mylabel{gb-char}
Let $\_ g$ be a finite set of elements of $G$. Then $\_ g$ is a Gr\"obner
basis if and only if $h\in A_n\_ g$ implies $\exists i:
\operatorname{in}(g_i)|\operatorname{in}(h)$.\hfill$\Box$
\end{prop}
Computation of Gr\"obner bases over the Weyl algebra works just as
over polynomial rings:
\begin{alg}[Buchberger]
\mylabel{buchberger}
Input: $\_ g=\{g_1,\ldots,g_s\}\subseteq G$.
Output: a Gr\"obner basis for $A_n\cdot(g_1,\ldots,g_s)$.
Begin.
\[
\begin{array}{llll}
&{\tt Repeat}&&\\
&&{\tt If\,} h=\Re(s_{ij},\underline g)\not =0&\\
&&&{\tt add\,} h {\tt \,to \,} \_ g\\
&{\tt Until}&{\tt \, all\,} \Re(s_{ij},\_ g)=0.\\
&{\tt Return } \,\,\_ g.
\end{array}
\]
\indent End.
\end{alg}
\subsection{}
\mylabel{free-kernel}
Now we shall describe the construction of kernels of $A_n$-linear maps
using Gr\"obner bases. Again, this is similar to the commutative
case and we first consider the case of a map between free $A_n$-modules.
Let $E=\bigoplus_1^sA_n\varepsilon_i, G=\bigoplus_1^rA_n\gamma_j$ and $\phi:E\to G$
be a
$A_n$-linear map. Assume $\phi(\varepsilon_i)=g_i$. Suppose that in order to
make $\_ g$ a Gr\"obner basis we have to add $g'_1,\ldots,g'_{s'}$ to
$\underline
g$ which
satisfy $g'_i=\sum_{k=1}^s a_{ik}g_k$. We get an induced map
$\diagram
\bigoplus_1^{s+s'}A_n\varepsilon_i\dto_\pi\drto_{\tilde\phi}\\
\bigoplus_1^sA_n\varepsilon_i\rto_\phi &\bigoplus_1^rA_n\gamma_j
\enddiagram
$ where $\pi$ is the identity on $\varepsilon_i$ for $i\le s$ and sends
$\varepsilon_{i+s}$ into $\sum_{k=1}^s a_{ik}\varepsilon_k$. Of course,
$\tilde\phi=\phi\pi$.
The kernel of $\phi$ is just the image of the kernel of $\tilde\phi$
under $\pi$. So in order to find kernels of maps between free modules
one may assume that the generators of the source are mapped to a
Gr\"obner basis and replace $\phi$ by $\tilde\phi$. Recall from
definition \ref{schreyer} that $\sigma_{ij}=m_{ji}\varepsilon_i-m_{ij}\varepsilon_j$ or
zero, depending on the leading terms of $g_i$ and $g_j$.
\begin{prop}
\mylabel{syz}
Assume that $\{g_1,\ldots,g_s\}$ is a Gr\"obner
basis. Let $s_{ij}=\sum d_{ijk}g_k$ be standard expressions for the Schreyer
polynomials. Then $\{\sigma_{ij}-\sum_k d_{ijk}\varepsilon_k\}_{1\le i<j\le s}$
generate the kernel
of $\phi:\bigoplus_1^sA_n\varepsilon_i\to \bigoplus_1^rA_n\gamma_j$, sending
$\varepsilon_i$ to $g_i$.
\end{prop}
The proof proceeds exactly as in the commutative case, see for example
\cite{E}, section 15.10.8.
\subsection{}
\mylabel{kernel}
We explain now how to find a set of generators for the kernel of an
arbitrary
$A_n$-linear map. Let $E, G$ be as in subsection \ref{free-kernel} and suppose
$A_n(p_1,\ldots,p_a)=P\subseteq E,
A_n(q_1,\ldots,q_b)=Q\subseteq G$ and $\phi:\bigoplus_1^s
A_n\varepsilon_i/P\to \bigoplus_1^r
A_n\gamma_j/Q$. It will
be sufficient to consider the case $P=0$ since we may lift $\phi$ to the free
module $E$ surjecting onto $E/P$.
Let as before $\phi(\varepsilon_i)=g_i$. A kernel element in $E$ is a sum
$\sum_ia_i\varepsilon_i, a_i\in A_n$, which if $\varepsilon_i$ is replaced by
$g_i$ can be written in terms
of the generators $q_j$ of $Q$. Let $\_
\beta=\{\beta_1,\ldots,\beta_c\}$ be such that $\_ g\cup \_ q\cup
\_\beta$ is a Gr\"obner basis for $A_n(\_ g,\_ q)$. We may assume
that the $\beta_i$ are the results of applying algorithm
\ref{buchberger} to $\_
g\cup\_ q$. Then from algorithm $\ref{remainder}$ we have expressions
\begin{equation}
\label{star}
\beta_i=\sum_j c_{ij}g_j+\sum_k c'_{ik}q_k,
\end{equation}
with $c_{ij}, c'_{ik}\in A_n$.
Furthermore, by proposition \ref{syz}, algorithm \ref{buchberger} returns a
generating set $\_\sigma$
for the
syzygies on $\_g\cup\_ q\cup\_\beta$. Write
\begin{equation}
\sigma_i=\sum_j a_{ij}\varepsilon_{g_j}+\sum_k a'_{ik}\varepsilon_{q_k}+\sum_l
a''_{il}\varepsilon_{\beta_l}
\end{equation}
and eliminate the last sum using the relations (\ref{star}) to obtain
syzygies
\begin{equation}
\tilde \sigma_i=\sum_j a_{ij}\varepsilon_{g_j}+\sum_k a'_{ik}\varepsilon_{q_k}+\sum_l
a''_{il}\left(\sum_v c_{lv}\varepsilon_{g_v}+\sum_w c'_{lw}\varepsilon_{q_w}\right).
\end{equation}
These will then form a generating set for the syzygies on
$\_g\cup \_q$. Cutting off the $q$-part of these syzygies
we get a set of forms
\[
\left\{\sum_j a_{ij}\varepsilon_{g_j}+\sum_l
a''_{il}\left(\sum_v c_{lv}\varepsilon_{g_v}\right)\right\}
\]
which generate the kernel of the map
$E\to G/Q$.
\subsection{}
The comments in this subsection will find their application in algorithm
\ref{lclc-alg} which computes the structure of $H^i_{\frak m}(H^j_I(R))$ as
$A_n$-module.
Let
\mylabel{double-kernel}
\[
\diagram
M_3'\rto^\alpha& M_3\rto^{\alpha '}& M_3''\\
M_2'\rto^\beta\uto_{\phi'}& M_2\rto^{\beta'}\uto_{\psi'}& M_2''\uto_{\rho'}\\
M_1'\rto^\gamma\uto_{\phi}& M_1\rto^{\gamma'}\uto_\psi& M_1''\uto_{\rho}
\enddiagram
\]
be a commutative diagram of $A_n$-modules.
Note that the row cohomology of the column co\-ho\-mo\-lo\-gy at $N$ is
given by
\[
\left[\ker(\psi ')\cap {\beta
'}^{-1}(\operatorname{im}\rho)+\operatorname{im}(\psi)\right]\,\,
/\,\,\left[\beta(\ker(\phi '))+\operatorname{im}(\psi )\right].
\]
In
order to compute this we need to be able to find:
\begin{itemize}
\item preimages of submodules,
\item kernels of maps,
\item intersections of submodules.
\end{itemize}
It is apparent that the second and third calculation is a special case
of the first: kernels are preimages of zero, intersections are
images of preimages (if ${A_n}^r\stackrel{\phi}{\rightarrow} {A_n}^s/M
\stackrel{\psi}{\leftarrow} {A_n}^t$ is given, then $\operatorname{im}(\phi)\cap
\operatorname{im}(\psi)=\psi(\psi^{-1}(\operatorname{im}(\phi)))$ ).
So suppose in the situation $\phi:{A_n}^r/M\to {A_n}^s/N$,
$\psi:{A_n}^t/P\to {A_n}^s/N$ we
want to find the preimage under $\psi$ of the image of $\phi$. We may
reduce to the case where $M$ and $P$ are zero and then lift
$\phi,\psi$ to maps into ${A_n}^s$. The elements $x$ in
$\psi^{-1}(\operatorname{im}
\phi)\subseteq {A_n}^t$ are exactly the elements in
$\ker({A_n}^t\stackrel{\psi}{\to} {A_n}^s/N\to {A_n}^s/(N+\operatorname{im}\phi))$ and this
kernel can be found according to the comments in \ref{kernel}.
\section{$D$-modules after Kashiwara and Malgrange}
\mylabel{sec-mal-kash}
The purpose of this and the following section is as follows. Given $f\in
R$ and an ideal $L\subseteq A_n$ such that $A_n/L$ is holonomic and $L$ is
$f$-saturated (i.e.~$f\cdot P\in L$ only if $P\in L$), we want
to compute the structure of the module
$R_f\otimes A_n/L$. It turns out that it is useful to know the ideal
$J^L(f^s)$ which consists of the operators $P(s)\in A_n[s]$ that
annihilate $f^s\otimes \bar 1\in M:=R_f[s]f^s\otimes A_n/L$ where the bar
denotes cosets in $A_n/L$.
In order to find $J^L(f^s)$, we will consider the module
$M$ over the ring $A_{n+1}=A_n\langle
t,\partial_t\rangle$. It will turn out in \ref{malgrange} that one can
easily compute the ideal $J^L_{n+1}(f^s)\subseteq A_{n+1}$ consisting of all
operators that kill $f^s\otimes\bar 1$. In section \ref{sec-oaku}
we will then show how to compute $J^L(f^s)$ from $J_{n+1}^L(f^s)$.
The second basic fact in this section (proposition \ref{kashiwara})
shows how to
compute the structure of $R_f\otimes A_n/L$ as $A_n$-module once
$J^L(f^s)$ is known.
\subsection{}
Consider $A_{n+1}=A_n\langle t,\partial_t\rangle$, the Weyl algebra in
$x_1,\ldots,x_n$ and the new variable $t$. B.~Malgrange has defined an
action of $t$
and $\partial_t$ on $M=R_f[s]\cdot f^s\otimes_R A_n/L$ by $t(g(x,s)\cdot
f^s\otimes \bar P)=g(x,s+1)f\cdot f^s\otimes \bar P$ and $\partial_t(g(x,s)\cdot
f^s\otimes \bar P)=\frac{-s}{f}g(x,s-1)\cdot f^s\otimes \bar P$ for
$\bar P\in A_n/L$. $A_n$ acts on $M$ as expected, the variables by
multiplication
on the left, the $\partial_i$ by the product rule.
One checks that this actually
defines an structure of $M$ as a left $A_{n+1}$-module and that
$-\partial_tt$ acts as
multiplication by $s$.
We denote by $J^L_{n+1}(f^s)$ the ideal in $A_{n+1}$ that annihilates the
element $f^s\otimes \bar 1$ in $M$. Then we have an induced morphism
of $A_{n+1}$-modules $A/J^L_{n+1}(f^s)\to M$ sending
$P+J^L_{n+1}(f^s)$ to
$P(f^s\otimes\bar 1)$.
The operators $t$ and $\partial_t$ were introduced in \cite{M}. The
following lemma generalizes lemma 4.1 in \cite{M} (as well as
part of the proof given there) where the special case
$L=(\partial_1,\ldots,\partial_n), A_n/L=R$ is considered.
Note that $J^L_{n+1}(f^s)$ makes perfect sense even if
$L$ is not holonomic.
\begin{lem}
\mylabel{malgrange}
Suppose that $L=A_n\cdot (P_1,\ldots,P_r)$ is $f$-saturated (i.e., if
$f\cdot P\in L$, then $P\in L$).
With the above definitions, $J^L_{n+1}(f^s)$ is the
ideal generated by $f-t$ together with the images of the $P_j$ under
the automorphism $\phi$ of $A_{n+1}$ induced by $x\to x, t\to t-f$.
\pf
The automorphism sends $\partial_i$ to $\partial_i+f_i\partial_t$ and $\partial_t$ to
$\partial_t$. So if we write $P_j=P_j(\partial_1,\ldots,\partial_n)$, then $\phi
P_j=P_j(\partial_1+f_1\partial_t,\ldots,\partial_n+f_n\partial_t)$.
One checks that $(\partial_i+f_i\partial_t)(f^s\otimes \bar Q)=f^s\otimes \bar{\partial_i
Q}$ for all differential operators $Q$ so that
$\phi(P_j(\partial_1,\ldots,\partial_n))(f^s\otimes
\bar 1)=f^s\otimes \bar{P_j(\partial_1,\ldots,\partial_n)}=0$. By
definition, $f\cdot
(f^s\otimes \bar 1)=t\cdot (f^s\otimes \bar 1)$. So $t-f\in
J^L_{n+1}(f^s)$ and $\phi(P_j)\in J^L_{n+1}(f^s)$ for $i=1,\ldots,r$.
Conversely let $P(f^s\otimes \bar 1)=0$. We may assume, that $P$ does not contain
any $t$ since we can eliminate $t$ using $f-t$. Now rewrite $P$ in
terms of $\partial_t$ and the $\partial_i+f_i\partial_t$. Say, $P=\sum
c_{\alpha\beta}\partial_t^\alpha
x^\beta Q_{\alpha\beta}(\partial_1+f_1\partial_t,\ldots,\partial_n+f_n\partial_t)$,
where the $Q_{\alpha\beta}$ are polynomials in $n$ variables and
$c_{\alpha\beta}\in K$. Application to
$f^s\otimes \bar 1$ results in $\sum \partial_t^\alpha(f^s\otimes
c_{\alpha\beta}x^\beta \bar{Q_{\alpha\beta}(\partial_1,\ldots,\partial_n)})$.
Let $\bar\alpha$ be the largest $\alpha\in{\Bbb N}$ for which there is a
nonzero $c_{\alpha\beta}$ occuring in $P=\sum
c_{\alpha\beta}\partial_t^\alpha
x^\beta Q_{\alpha\beta}(\partial_1+f_1\partial_t,\ldots,\partial_n+f_n\partial_t)$.
We show that the sum of terms that contain
$\partial_t^{\bar\alpha}$ is in $A_{n+1}\cdot \phi(L)$ as
follows. In
order for $P(f^s\otimes\bar 1)$ to vanish, the sum of terms with the
highest $s$-power, namely $s^{\bar\alpha}$, must vanish,
and so $\sum_\beta c_{\bar\alpha\beta}(-1/f)^{\bar\alpha}f^s\otimes
x^\beta Q_{\bar\alpha\beta}(\partial_1,\ldots,\partial_n)\in
R_ff^s\otimes L$ as $R_f$ is $R$-flat.
It follows, that $\sum_\beta c_{\bar\alpha\beta}x^\beta
Q_{\bar\alpha\beta}(\partial_1,\ldots,\partial_n)\in L$ ($L$ is
$f$-saturated!) and hence $\sum_\beta
\partial_t^{\bar\alpha}c_{\bar\alpha\beta}
x^\beta
Q_{\bar\alpha\beta}(\partial_1+f_1\partial_t,\ldots,\partial_n+f_n\partial_t)\in
A_{n+1}\cdot \phi(L)$.
So by the first part,
$P-\sum_\beta c_{\bar\alpha\beta}\partial_t^{\bar\alpha} x^\beta
Q_{\bar\alpha\beta}(\partial_1+f_1\partial_t,\ldots,\partial_n+f_n\partial_t)$ kills
$f^s\otimes \bar 1$, but is of
smaller degree in $\partial_t$ than $P$ was.
The claim follows.\hfill$\Box$
\end{lem}
\subsection{}
\mylabel{jt-js}
Let $J^L(f^s)$ stand for the ideal in $A_n[s]\cong A_n[-\partial_tt]$
that kills $f^s\otimes\bar 1\in R_f[s]f^s\otimes_R A_n/L$. Note that
$J^L(f^s)=J^L_{n+1}(f^s)\cap A_n[-\partial_tt]$. Again, we may talk about
$J^L(f^s)$ independently of the holonomicity of $L$.
We will in the next section show how the lemma can be used to
determine $J^L(f^s)$. Now we show why $J^L(f^s)$ is useful, generalizing
\cite{K}, proposition 6.2.
Recall that the Bernstein polynomial $b^L_f(s)$ is defined to be the
monic generator of the ideal of polynomials $b(s) \in K[s]$ for which
there exists an operator $P(s)\in A_n[s]$ such that $P(s)(f^{s+1}\otimes \bar
1)=b(s)f^s\otimes \bar 1$ (\cite{B}, chapter 1), and that $b_f^L(s)$
will exist for example if $L$ is holonomic.
\begin{prop}
\mylabel{kashiwara}
If $L$ is holonomic and $a\in \Bbb Z$ is such that no integer root of
$b_f^L(s)$ is smaller than
$a$, then we have isomorphisms
\begin{equation}
R_f\otimes A_n/L\cong A_n[s]/J^L(f^s)|_{s=a}\cong A_n\cdot f^a\otimes \bar
1.
\end{equation}
\pf
We mimick the proof given by Kashiwara, who proved the proposition for
the case $L=(\partial_1,\ldots,\partial_n), A_n/L=R$ (\cite{K}, proposition 6.2).
Let us first prove the second equality. Certainly $J^L(f^s)|_{s=a}$
kills $f^a\otimes\bar 1$. So we have to show that if $P(f^a\otimes\bar
1)=0$ then
$P\in J^L(f^s)+A_n[s]\cdot(s-a)$. To that end note that
$st$ acts as $t(s-1)$ which means that $t\cdot
(A_n[s]/J^L(f^s))$ is a left $A_n[s]$-module. Identify $A_n[s]/J^L(f^s)$
with ${\cal N}^L_f:=A_n[s]\cdot (f^s\otimes\bar 1)$.
By
definition, $b^L_f(s)$ is the minimal polynomial for which there is
$P(s)$ with $b^L_f(s)(f^s\otimes \bar 1)=P(s)f^{s+1}=t\cdot
P(s-1)(f^s\otimes\bar 1)$. So $b^L_f(s)$ multiplies
$A_n[s]\cdot(f^s\otimes\bar 1)$ into $t\cdot A_n[s](f^s\otimes\bar
1)$ and whenever the polynomial $b(s)\in K[s]$ is relatively prime to
$b_f^L(s)$
its action on ${\cal N}_f^L/t\cdot {\cal N}_f^L$ is injective.
Since by hypothesis $s-a+j$ is not a divisor of
$b^L_f(s)$ for $0<j\in \Bbb N$,
\begin{equation}
\label{kash-eqn}
(s-a+j){\cal N}^L_f\cap t\cdot {\cal N}^L_f\subseteq (s-a+j)t\cdot
{\cal N}^L_f.
\end{equation}
So $(s-a+m){\cal N}^L_f\cap
t^m{\cal N}^L_f\subseteq (s-a+m)t{\cal N}^L_f\cap t^m{\cal N}^L_f=t[(s-a+m-1)
{\cal N}^L_f\cap
t^{m-1}{\cal N}^L_f]$ whenever $m\geq 1$.
We show now by induction on $m$ that $(s-a+m){\cal N}^L_f\cap
t^m{\cal N}^L_f\subseteq (s-a+m)t^m{\cal N}^L_f$ for $m\geq 1$. The claim
is clear for $m=1$ from equation (\ref{kash-eqn}). So let $m>1$. The
inductive hypothesis states that $(s-a+m-1){\cal N}^L_f\cap
t^{m-1}{\cal N}^L_f\subseteq (s-a+m-1)t^{m-1}{\cal N}^L_f$. The previous
paragraph shows that $(s-a+m){\cal N}^L_f\cap t^m{\cal N}^L_f\subseteq
t\left[(s-a+m-1){\cal N}^L_f\cap t^{m-1}{\cal N}^L_f\right]$. Combining these two
facts we get
\begin{eqnarray*}
(s-a+m){\cal N}^L_f\cap t^m{\cal N}^L_f&\subseteq & t(s-a+m-1)t^{m-1}{\cal N}^L_f\\
&=&(s-a+m)t^m{\cal N}^L_f.
\end{eqnarray*}
Now if $P(s)\in A_n[s]$ is of degree $m$ in the $\partial_i$ and
$P(a)(f^a\otimes \bar 1)=0$,
then
$P(s+m)\cdot f^m+J^L(f^s)\in (s-a+m)\cdot {\cal N}^L_f$ because we can
interprete
$P(s+m)(f^{s+m}\otimes \bar 1)$ as a polynomial in $s+m$ with root
$a$. But then
$P(s+m) (f^{s+m}\otimes\bar 1)=P(s+m)(f^{m}f^s\otimes\bar 1)$ is in
\[
(s-a+m){\cal N}^L_f\cap t^m{\cal N}^L_f\subseteq
(s-a+m)t^m{\cal N}^L_f,
\]
implying $P(s+m)(f^{s+m}\otimes \bar 1)=(s-a+m)Q(s)(f^{s+m}\otimes\bar 1)$
for some $Q(s)\in A_n[s]$ (note
that $J^L(f^s)$ kills $f^s\otimes\bar 1$). In other words, $P(s)-(s-a)Q(s-m)\in
J^L(f^s)$.
For the first isomorphism we have to show that
$A_n\cdot(f^a\otimes\bar 1)=R_f\otimes A_n/L$. It suffices
to show that every term of the form $f^mf^a\otimes\bar Q$ is in the
module generated by $(f^a\otimes\bar 1)$ for all $m\in \Bbb
Z$. Furthermore, we may assume that $Q$ is a monomial in
$\partial_1,\ldots,\partial_n$.
Existence and definition of $b^L_f(s)$ provides an operator $P(s)$ with
$[b^L_f(s-1)]^{-1}P(s-1)(f^s\otimes \bar 1)=f^{-1}f^s\otimes \bar
1$. As $b_f^L(a-m)\not=0$ for
all $0<m\in \Bbb N$ we have $f^{m}f^a\otimes\bar 1\in A_n\cdot
(f^a\otimes \bar
1)$ for all $m$. Now let $Q$ be a monomial in $\partial_1,\ldots,\partial_n$
of $\partial$-degree $j>0$ and assume that
$f^mf^a\otimes \bar{Q'}\in A_n\cdot (f^a\otimes \bar 1)$ for all
$m$ and all operators $Q'$ of $\partial$-degree lower than $j$.
Then $Q=\partial_i Q'$ for some $1\le i\le
n$. Fix $m\in{\Bbb Z}$. By assumption on $j$, for some $P'$ we have
$P'(f^a\otimes \bar
1)=f^mf^a\otimes\bar{Q'}$. So
\begin{equation}
f^mf^a\otimes\bar Q=
\partial_iP'(f^a\otimes\bar 1)-f_i\cdot(a+m)f^{m-1}f^a\otimes \bar{Q'}
\in A_n\cdot (f^a\otimes \bar 1).
\end{equation}
The claim follows by induction.
This completes the proof of the proposition.\hfill$\Box$
\end{prop}
We remark that if any $a\in{\Bbb Z} $ satisfies the conditions of the
proposition, then so does every integer smaller than $a$.
\section{An algorithm of Oaku}
\mylabel{sec-oaku}
The purpose of this section is to review and generalize an algorithm
due to Oaku.
In \cite{Oa} (algorithm 5.4.), Oaku
showed how to construct a generating set for $J^L(f^s)$ in the case where
$L=(\partial_1,\ldots,\partial_n)$. According to \ref{jt-js}, $J^L(f^s)$ is the
intersection of $J^L_{n+1}(f^s)$ with $A_n[-\partial_tt]$.
We shall explain how one may calculate $J\cap
A_n[-\partial_tt]$ whenever $J\subseteq A_{n+1}$ is any given ideal and as a
corollary develop an algorithm that for $f$-saturated $A_n/L$
computes $J^L(f^s)$. The proof
follows closely Oaku's argument.
On $A_{n+1}[y_1,y_2]$ define weights $w(t)=w(y_1)=1,
w(\partial_t)=w(y_2)=-1, w(x_i)=w(\partial_i)=0$. If $P=\sum_i P_i\in
A_{n+1}[y_1,y_2]$ and all $P_i$ are monomials, then we will write
$(P)^h$ for the operator $\sum_i P_i\cdot y_1^{d_i}$ where
$d_i=\max_j(w(P_j))-w(P_i)$ and call it the {\em $y_1$-homogenization}
of $P$.
Note that the
Buchberger algorithm preserves homogeneity
in the following sense: if a set of generators for an ideal is given
and these generators are homogeneous with respect to the weights above,
then any new generator for the ideal constructed with the classical
Buchberger algorithm will also be homogeneous. (This is a consequence
of the facts that the $y_i$ commute with all other variables and that
$\partial_t t=t\partial_t+1$ is homogeneous of weight zero.)
\begin{prop}
\mylabel{oaku}
Let $J=A_{n+1}\cdot(Q_1,\ldots,Q_r)$ and let
$y_1,y_2$ be two new
variables.
Let $I$ be the left ideal in $A_{n+1}[y_1]$ generated by the
$y_1$-homogenizations $(Q_i)^h$ of the $Q_i$, relative to the weight
$w$ above, and let $\tilde
I=A_{n+1}[y_1,y_2]\cdot (I,1-y_1y_2)$. Let $G$ be a Gr\"obner basis
for $\tilde I$ under a monomial order that eliminates $y_1,y_2$. For
each $P\in G$ set $P'=t^{-w(P)}P$ if $w(P)<0$ and $P'=\partial_t^{w(P)}P$ if
$w(P)>0$ and let $G'=\{ P': P\in G\}$. Then
$G_0=G'\cap A_n[-\partial_tt]$ generates $J\cap A_n[-\partial_tt]$.
\pf
Note first that $G$ consists of $w$-homogeneous operators and so $w(P)$ is
well defined for $P\in G$.
Suppose $P\in G_0$. Hence $P\in\tilde I$. So $P=Q_{-1}\cdot (1-y_1y_2)+\sum
a_i\cdot (Q_i)^h$ where the $a_i$ are in $A_{n+1}[y_1,y_2]$. Since $P\in
A_n[-\partial_tt]$, the
substitution $y_i\to 1$ shows that $P=\sum a_i(1,1)\cdot
(Q_i)^h(1,1)=\sum a_i(1,1)\cdot Q_i\in J$. Therefore $G_0\subseteq J\cap
A_n[-\partial_tt]$.
Now assume that $P\in J\cap A_n[-\partial_tt]$. So $P$ is
$w$-homogeneous of weight 0. Also, $P\in J$ and $J$ is contained in
$I(1)$, the ideal of operators $Q(1)\subseteq A_{n+1}$ for which $Q(y_1)\in
I$. By
lemma \ref{oaku-lemma} below (taken from \cite{Oa}), $y_1^a
P\in I$ for some $a\in \Bbb
N$. Therefore $P=(1-(y_1y_2)^a)P+(y_1y_2)^aP\in \tilde I$.
Let $G=\{P_1,\ldots,P_b,P_{b+1},\ldots,P_c\}$ and assume that $P_i\in
A_{n+1} $ if and only if $i\le b$.
Buchberger algorithm gives a
standard expression $P=\sum a_iP_i$ with all $\operatorname{in}(a_iP_i)\le \operatorname{in}
(P)$. That implies that $a_{b+i}$ is zero for positive $i$ and $a_i$
does not contain $y_1,y_2$ for any $i$.
Since $P, P_i$ are $w$-homogeneous,
the same is true for all $a_i$, from Buchberger algorithm. In fact,
$w(P)=w(a_i)+w(P_i)$ for all $i$. As $w(P)=0$ (and $t, \partial_t$ are
the only variables with nonzero weight that may appear in $a_i$) we
find $a'_i\in A_n$ with $a_i=a_i'\cdot t^{-w(P_i)}$ or $a_i=a_i'\cdot
\partial_t^{w(P_i)}$,
depending on
whether $w(P_i)$ is negative or positive.
It follows that $P=\sum_1^b a_iP_i=\sum_1^b a_i'P_i'\in A_n[-\partial_tt]\cdot
G_0$. \hfill$\Box$
\end{prop}
\begin{lem}
\mylabel{oaku-lemma}
Let $I$ be a $w$-homogeneous ideal in $A_{n+1}[y_1]$ with respect to the
weights introduced before the proposition and $I(1)$ defined as in the
proof of the proposition. Assume $P\in A_{n+1}$ is a $w$-homogeneous
operator. Then
$P\in I(1)$ implies
$y_1^aP\in I$ for some $a$.
\pf
Note first that $y_1\to 1$ will not lead to cancellation of terms in any
homogeneous operator as $w(y_1)\not =0$.
If $P\in I(1)$, $P=\sum Q_i(1)$, with all $Q_i$ $w$-homogeneous in
$I$. Then the
$y_1$-homogenization of $Q_i(1)$ will be a divisor of $Q_i$ and the
quotient will be some power of $y_1$, say
$y_1^{\eta_i}$. Homogenization of the equation
$P=\sum Q_i(1)$ results in $y_1^{\eta}P=\sum Q_i(1)^h$ (since $P$ is
homogeneous) so that
\[\parbox{12.65cm}
{\hfill$y_{1}^{\eta+\max(\eta_i)}P=\sum y_1^{\max(\eta_i)-\eta_i}Q_i\in
I.$\hfill$ \Box$} \]
\end{lem}
So we have
\begin{alg}
\mylabel{ann-fs}
Input: $f\in R, L\subseteq A_n$ such that $L$ is
$f$-satuarated.
Output: Generators for $J^L(f^s)$.
Begin
\begin{enumerate}
\item For each generator $Q_i$ of $L$ compute the image $\phi(Q_i)$
under $x_i\to
x_i, t\to t-f, \partial_i\to \partial_i+f_i\partial_t,\partial_t\to\partial_t$. Add $t-f$
to the list.
\item Homogenize all $\phi(Q_i)$ with respect to the new variable
$y_1$ relative to the weight $w$ introduced before proposition \ref{oaku}.
\item Compute a Gr\"obner basis for the ideal generated by
$(\phi(Q_1))^h$, $\ldots$, $(\phi(Q_r))^h$, $1-y_1y_2$, $t-y_1f$
in $A_{n+1}[y_1,y_2]$
using an order that eliminates $y_1,y_2$.
\item Select the operators $\{ P_j\}_1^b$ in this basis which do not
contain $y_1, y_2$.
\item For each $P_j$, $1\le j\le b$, if $w(P_j)>0$ replace $P_j$ by
$P_j'=\partial_t^{w(P_j)}P_j$. Otherwise replace $P_j$ by
$P_j'=t^{-w(P_j)}P_j$.
\item Return the new operators $\{P_j'\}_1^b$.
\end{enumerate}
End.
\end{alg}
In order to guarantee existence of the Bernstein polynomial $b^L_f(s)$
we assume for our next result that $L$ is holonomic.
\begin{cor}
\mylabel{b-poly}
Suppose $L$ is a holonomic ideal.
If $J^L(f^s)$ is known or it is known that $L$ is $f$-saturated, then the
Bernstein polynomial $b_f^L(s)$ of $R_f\otimes_R
A_n/L$ can be found from
$(b^L_f(s))=A_n[s]\cdot(J^L(f^s),f)\cap K[s]$.
Moreover, if $K\subseteq {\Bbb C}$,
suppose $b^L_f(s)=s^d+b_{d-1}s^{d-1}+\ldots+b_0$ and define
$B=\max_{i}\{|b_i|^{1/(d-i)}\}$.
In order to
find the smallest integer root of $b^L_f(s)$, one
only needs to check the integers between $-2B$ and $2B$.
If in particular $L=(\partial_1,\ldots,\partial_n)$, it suffices to check the
integers between $-b_{d-1}$ and -1.
\pf
If $L$ is $f$-saturated, propositions \ref{malgrange} and \ref{oaku}
enable us to find $J^L(f^s)$. The first part follows then easily from the definition of $b^L_f(s)$:
as $(b_f^L(s)-P_f^L\cdot f)(f^s\otimes\bar 1)=0$ it is clear that
$b_f^L(s)$ is in $K[s]$ and in
$A_n[s](J^L(f^s),f)$. Using an elimination order on $A_n[s]$,
$b^L_f(s)$ will be (up to a scalar
factor) the unique element in the reduced Gr\"obner basis for
$J^L(f^s)+(f)$ that
contains no $x_i$ nor $\partial_i$.
Now suppose $K\subseteq {\Bbb C}$,
$|s|=2B\rho$ where $B$ is as defined above and $\rho>1$.
Assume
also that $s$ is a root of $b_f^L(s)$. We find
\begin{eqnarray}
(2B\rho)^d=|s|^d&=&|-\sum_0^{d-1}b_is^i|
\le\sum_0^{d-1}B^{d-i}|s|^i\\
&=&B^d\sum_0^{d-1}(2\rho)^i
\le B^d((2\rho)^d-1),
\end{eqnarray}
using $\rho\geq 1$.
By contradiction, $s$ is not a root.
The
final claim is a consequence of Kashiwara's work \cite{K} where
it is proved that if $L=(\partial_1,\ldots,\partial_n)$ then all roots of
$b_f^L(s)$ are negative and hence
$-b_{n-1}$ is a
lower bound for each single root.
\hfill$\Box$
\end{cor}
For purposes of reference we also list algorithms that compute the
Bernstein polynomial to a holonomic module and the localization of a
holonomic module.
\begin{alg}
\mylabel{b-poly-L}
Input: $f\in R, L\subseteq A_n$ such that $A_n/L$ is holonomic and
$f$-torsionfree.
Output: The Bernstein polynomial $b^L_f(s)$.
Begin
\begin{enumerate}
\item Determine $J^L(f^s)$ following algorithm \ref{ann-fs}.
\item Find a reduced Gr\"obner basis for the ideal
$J^L(f^s)+A_n[s]\cdot f$
using an elimination order for $x$ and $\partial$.
\item Pick the unique element in that basis contained in $K[s]$ and
return it.
\end{enumerate}
End.
\end{alg}
\begin{alg}
\mylabel{D/L-loc-f}
Input: $f\in R, L\subseteq A_n$ such that $A_n/L$ is holonomic and
$f$-torsionfree.
Output: Generators for an ideal $J$ such that $R_f\otimes A_n/L\cong A_n/J$.
Begin
\begin{enumerate}
\item Determine $J^L(f^s)$ following algorithm \ref{ann-fs}.
\item Find the Bernstein polynomial $b_f^L(s)$ using algorithm
\ref{b-poly-L}.
\item Find the smallest integer root $a$ of $b_f^L(s)$ (using corollary
\ref{b-poly}, if $K\subseteq {\Bbb C}$).
\item Replace $s$ by $a$ in all generators for $J^L(f^s)$ and
return these generators.
\end{enumerate}
End.
\end{alg}
The algorithms \ref{ann-fs} and \ref{b-poly-L} appear already in
\cite{Oa} in the special case $L=(\partial_1,\ldots,\partial_n), A_n/L=R$.
\section{Local cohomology as $A_n$-module}
\mylabel{sec-lc}
In this section we will combine the results from the previous sections to
obtain algorithms that compute for given $i,j,k\in {\Bbb N}, I\subseteq R$ the
local cohomology modules $H^k_I(R), H^i_{\frak m}(H^j_I(R))$ and the
invariants $\lambda_{i,j}(R/I)$ associated to $I$.
\subsection{Computation of $H^k_I(R)$}
\mylabel{subsec-lc}
Here we will describe an algorithm that takes in a finite
set of polynomials $\underline f=\{f_1,\ldots,f_r\}\subset R$ and
returns a
presentation of $H^k_I(R)$ where $I=(f_1,\ldots,f_r)$. In particular,
if $H^k_I(R)$ is zero, then the algorithm will return the zero
presentation.
Consider the \v Cech complex associated to $f_1,\ldots,f_r$ in
$R$,
\begin{equation}
\label{cechcomplex}
0\to R\to \bigoplus_1^r R_{f_i}\to \bigoplus_{1\le i<j\le r}R_{f_if_j}
\to\cdots\to R_{f_1\cdot\ldots\cdot f_r}\to 0.
\end{equation}
Its $k$-th cohomology group is the local cohomology module
$H^k_I(R)$.
The map
\begin{equation}
\label{cechmap}
C^k=\bigoplus\limits_{1\le i_1<\cdots<i_k\le
r}R_{f_{i_1}\cdot\ldots\cdot f_{i_k}}\to \bigoplus\limits_{1\le
j_1<\cdots<j_{k+1}\le
r}R_{f_{j_1}\cdot\ldots\cdot f_{j_{k+1}}}=C^{k+1}
\end{equation}
is the sum of maps
\begin{equation}
\label{cechmap-parts}
R_{f_{i_1}\cdot\ldots\cdot f_{i_k}}\to R_{f_{j_1}\cdot\ldots\cdot
f_{j_{k+1}}}
\end{equation}
which are either zero (if $\{i_1,\ldots,i_k\}\not\subseteq
\{j_1,\ldots,j_{k+1}\}$) or send $\frac{1}{1}$ to
$\frac{1}{1}$, up to sign.
Recall that $A_n/\Delta=
A_n/A_n\cdot(\partial_1,\ldots,\partial_n)\cong R$ and identify
$R_{f_{i_1}\cdot\ldots\cdot f_{i_k}}$ with
$A_n/J^\Delta((f_{i_1}\cdot\ldots\cdot f_{i_k})^s)|_{s=a}$ and
$R_{f_{j_1}\cdot\ldots\cdot f_{j_{k+1}}}$ with
$A_n/J^\Delta((f_{j_1}\cdot\ldots\cdot f_{j_{k+1}})^s)|_{s=b}$ where
$a,b$ are sufficiently small integers. By
proposition \ref{kashiwara} we may assume that $a=b\le 0$. Then the map
(\ref{cechmap}) is in the nonzero case multiplication from the right by
$(f_l)^{-a}$ where $l=\{j_1,\ldots,j_{k+1}\}\setminus \{i_1,\ldots,i_k\}$,
again up to sign.
It follows that the matrix representing the map $C^k\to C^{k+1}$ in
terms of $A_n$-modules is very easy to write down once the annihilator
ideals and Bernstein polynomials for all $k$- and $(k+1)$-fold products
of the $f_i$ are known: the entries are 0 or $\pm f_l^{-a}$ where
$f_l$ is the new factor.
Let $\Theta^r_k$ be the set of $k$-element subsets of $1,\ldots,r$ and
for $\theta\in \Theta^r_k$ write $F_\theta$ for the product $\prod_{i\in
\Theta^r_k}f_{i}$.
We have demonstrated the correctness and finiteness of the following
algorithm.
\begin{alg}
\mylabel{lc-alg}
Input: $f_1,\ldots,f_r\in R; k\in {\Bbb N}$.
Output: $H_I^k(R)$ in terms of generators and relations as finitely
generated $A_n$-module.
Begin
\begin{enumerate}
\item Compute the annihilator ideal $J^\Delta((F_\theta)^s)$
and the Bernstein
polynomial $b^\Delta_{F_\theta}(s)$ for all $(k-1)$-, $k$- and $(k+1)$-fold
products of
${f_1}^s,\ldots,{f_r}^s$ as in \ref{ann-fs} and \ref{b-poly-L} (so
$\theta$ runs through $\Theta^r_{k-1}\cup \Theta^r_k\cup \Theta^r_{k+1}$).
\item Compute the smallest integer root $a_\theta$ for each
$b^\Delta_{F_\theta}(s)$, let $a$
be the minimum and replace $s$ by $a$ in all the annihilator ideals.
\item Compute the two matrices $M_{k-1},M_k$ representing the
$A_n$-linear maps
$C^{k-1}\to C^k$ and $C^k\to C^{k+1}$ as explained in subsection
\ref{subsec-lc}.
\item Compute a Gr\"obner basis $G$ for the kernel of the map
\[
\bigoplus_{\theta\in\Theta_k^r}A_n\to \bigoplus_{\theta\in\Theta^r_k}
A_n/J^\Delta((F_\theta)^s)|_{s=a}\stackrel{M_k}{\longrightarrow}
\bigoplus_{\theta\in
\Theta^r_{k+1}}A_n/J^\Delta((F_\theta)^s)|_{s=a} \] as in \ref{kernel}.
\item Compute a Gr\"obner basis $G_0$ for the module
\[
\operatorname{im}(M_{k-1})+\bigoplus_{\theta\in \Theta^r_k}
J^\Delta((F_\theta)^s)|_{s=a}\subseteq \bigoplus_{\theta\in\Theta^r_k}
A_n/J^\Delta((F_\theta)^s)|_{s=a}.
\]
\item Compute the remainders of all elements of $G$ with
respect to lifts of $G_0$ to $ \bigoplus_{\theta\in\Theta_k^r}A_n$.
\item Return these remainders and $G_0$.
\end{enumerate}
End.
\end{alg}
The nonzero elements of $G$ generate the quotient $G/G_0\cong
H^k_I(R)$ so that
$H^k_I(R)=0$ if and only if all returned remainders are zero.
\subsection{Computation of $H^i_{\frak m}( H^j_I(R))$}
As a second application of Gr\"ob\-ner basis computations over the
Weyl algebra we
show now how to compute $H^i_{\frak m} (H^j_I(R))$.
Note that we cannot apply lemma \ref{malgrange} to $A_n/L=H^j_I(R)$
since $H^j_I(R)$ may well
contain some torsion.
As in the previous sections, $C^j(R;f_1,\ldots,f_r)$ denotes the $j$-th
module in the
\v Cech complex to $R$ and $\{f_1,\ldots,f_r\}$.
Let $C^{\bullet\bullet}$ be the double complex with
$C^{i,j}=C^i(R;x_1,\ldots,x_n)\otimes_R C^j(R;f_1,\ldots,f_r)$, the
vertical maps $\phi^{\bullet\bullet}$ induced by the identity on the
first factor and the
usual \v Cech maps on the second, whereas the horizontal maps
$\xi^{\bullet\bullet} $ are induced
by the \v Cech maps on the first factor and the identity on the
second. Since $C^i(R;x_1,\ldots,x_n)$ is $R$-projective, the column
co\-ho\-mo\-lo\-gy of $C^{\bullet\bullet}$ at $(i,j)$ is
$C^i(R;x_1,\ldots,x_n)\otimes_RH^j_I(R)$ and the induced horizontal maps
in the $j$-th row are
simply the maps in the \v Cech complex $C^\bullet(H^j_I(R);x_1,\ldots,x_n)$.
It follows that
the row cohomology of the column cohomology at $(i_0,j_0)$ is
$H^{i_0}_{\frak m}(H^{j_0}_I(R))$.
Now note that $C^{i,j}$ is a direct sum of modules $R_g$ where
$g=x_{\alpha_1}\cdot\ldots\cdot x_{\alpha_i}\cdot
f_{\beta_1}\cdot\ldots\cdot f_{\beta_j}$. So the whole double complex
can be rewritten in terms of $A_n$-modules and $A_n$-linear maps using
\ref{D/L-loc-f}:
\[
\diagram
{\,C^{i-1,j+1}\,}{\rto^{\,\,\xi^{i-1,j+1}}}&
C^{i,j+1}\rto^{\xi^{i,j+1}}&
C^{i+1,j+1}\\
C^{i-1,j}\rto^{\xi^{i-1,j}}\uto_{\phi^{i-1,j}}&
C^{i,j}\rto^{\xi^{i,j}}\uto_{\phi^{i,j}}&
C^{i+1,j}\uto_{\phi^{i+1,j}}\\
C^{i-1,j-1}\rto^{\xi^{i-1,j-1}}\uto_{\phi^{i-1,j-1}}&
C^{i,j-1}\rto^{\xi^{i,j-1}}\uto_{\phi^{i,j-1}}&
C^{i+1,j-1}\uto_{\phi^{i+1,j-1}}
\enddiagram
\]
Using the comments in subsection \ref{double-kernel}, we may now
compute the modules $H^i_{\frak m} (H^j_I(R))$. More
concisely, we have the following
\begin{alg}
\mylabel{lclc-alg}
Input: $f_1,\ldots,f_r\in R; i_0,j_0\in \Bbb N$.
Output: $H^{i_0}_{\frak m} (H^{j_0}_I(R))$ in terms of generators and relations as
finitely generated $A_n$-module.
Begin.
\begin{enumerate}
\item For $i=i_0-1, i_0, i_0+1$ and $j=j_0-1,j_0,j_0+1$ compute the
annihilators $J^\Delta((F_\theta\cdot X_{\theta'})^s)$ and Bernstein
polynomials $b^\Delta_{F_\theta\cdot X_{\theta'}}(s)$ of $F_\theta\cdot
X_{\theta'}$
where $\theta \in \Theta^r_j, \theta'\in \Theta^n_i$ and $X_{\theta'}$
denotes in analogy to $F_\theta$ the product $\prod_{\alpha\in
\theta'}x_\alpha$.
\item Let $a$ be the minimum integer root of the product of all these
Bernstein polynomials and replace $s$ by $a$ in all the annihilators
computed in the previous step.
\item Compute the matrices to the $A_n$-linear maps
$\phi^{i,j}:C^{i,j}\to
C^{i,j+1}$
and $\xi^{i,j}:C^{i,j}\to C^{i+1,j}$, again for $i=i_0-1,i_0,i_0+1$ and
$j=j_0-1,j_0,j_0+1$.
\item Compute Gr\"obner bases for the modules
\[
G=\ker(\phi^{i_0,j_0})\cap
\left[
(\xi^{i_0,j_0})^{-1}(\operatorname{im}(\phi^{i_0+1,j_0-1}))\right]+\operatorname{im}(\phi^{i_0,j_0-1})
\]
and
$G_0=\xi^{i_0-1,j_0}(\ker(\phi^{i_0-1,j_0}))+\operatorname{im}(\phi^{i_0,j_0-1})$.
\item Compute the remainders of all elements of $G$ with
respect to $G_0$
and return these remainders together with $G_0$.
\end{enumerate}
End.
\end{alg}
The elements of $G$ will be generators for $H^{i_0}_{\frak m}
(H^{j_0}_I(R))$ and
the elements of $G_0$ generate the relations that are not
syzygies.
\subsection{Computation of $\lambda_{i,n-j}(R/I)$}
In \cite{L-Dmod} it has been shown that $H^i_{\frak m} (H^j_I(R))$ is an injective
${\frak m}$-torsion $R$-module of finite socle dimension $\lambda_{i,n-j}$
(which depends only on $i,j$ and $R/I$) and so
isomorphic to $(E_R(K))^{\lambda_{i,n-j}}$ where $E_R(K)$ is the
injective hull of $K$ over $R$. We
are now in a position that allows computation of these invariants of $R/I$.
For, let $H^i_{\frak m} (H^j_I(R))$ be generated by $g_1,\ldots,g_l\in {A_n}^d$
modulo the relations $h_1,\ldots,h_e\in {A_n}^d$. Let $H$ be the module
generated by the $h_i$. We know that
$(A_n\cdot g_1+H)/H$ is ${\frak m}$-torsion and so it is possible (with trial
and error) to find a multiple of $g_1$, say $mg_1$ with $m$ a monomial
in $R$, such that $(A_n\cdot mg_1+H)/H$ is nonzero but $x_img_1\in H$ for
all $1\le i\le n$. Then the element $mg_1+H/H$ has annihilator equal
to ${\frak m}$ and hence generates an $A_n$-module isomorphic to
$A_n/A_n\cdot {\frak m}\cong E_R(K)$. The injection $A_n\cdot mg_1+H/H\hookrightarrow
A_n\cdot(g_1,\ldots,g_l)+H/H$ splits as map of $R$-modules because of
injectivity and so the cokernel
$A_n(g_1,\ldots,g_l)/A_n(mg_1,h_1,\ldots,h_e)$ is isomorphic to
$(E_R(K))^{\lambda_{i,n-j}-1}$.
Reduction of the $g_i$ with respect to a Gr\"obner basis of the new
relation module and repetition of the previous will lead to
the determination of $\lambda_{i,n-j}$.
\subsection{Local cohomology in ambient spaces different from ${\Bbb A}^n_K$}
\mylabel{singular_spaces}
If $A$ equals $K[x_1,\ldots,x_n]$, $I\subseteq A$, $X=\operatorname{Spec} (A)$ and $V=\operatorname{Spec}(A/I)$,
knowledge of $H^i_I(A)$ for all $i\in {\Bbb N}$ answers of course the
question about the local cohomological dimension of $V$ in $X$. It is
worth mentioning, that if $W\subseteq X$ is a smooth variety containing $V$
then our algorithm \ref{lc-alg} for the computation of $H^i_I(A)$ also
leads to a determination of the local cohomological dimension of $V$
in $W$. Namely, if $J$ stands for the
defining ideal of $W$ in $X$ so that $R=A/J$ is the affine
coordinate ring of $W$ and if we set $c=\operatorname{ht}(J)$, then it can be
shown that
$H^{i-c}_{I}(R)=\operatorname{Hom}_A(R,H^i_I(A))$ for all $i\in{\Bbb N}$.
As $H^i_I(A)$ is
$I$-torsion (and hence $J$-torsion), $\operatorname{Hom}_A(R,H^i_I(A))$ is zero if
and only if
$H^i_I(A)=0$. It follows that the local cohomological dimension of $V$
in $W$ equals $\operatorname{cd}(A,I)-c$ and $\{q\in {\Bbb N}:H^q_I(A)\not =0\}=\{q\in
{\Bbb N}:H^{q-c}_I(R)\not =0\}$.
If however $W$ is not smooth, no algorithms for the computation of
either $H^i_I(R)$ or $\operatorname{cd}(R,I)$ are known, irrespective of the
characteristic of the base field.
\section{Implementation and examples}
Some of the algorithms described above have been implemented as C-scripts
and tested on some examples.
\subsection{}
The algorithm \ref{ann-fs} with $L=\Delta$ has been implemented by
Oaku using the package Kan (see \cite{T}) which
is a postscript language for computations in the Weyl algebra and in
polynomial rings. An implementation for general $L$ is written by the current
author and part of a program that deals exclusively with computations
around local cohomology (\cite{W}). \cite{W} is theoretically able to compute
$H^i_I(R)$ for arbitrary $i, R={\Bbb Q}[x_1,\ldots,x_n], I\subseteq R$ in the
above described terms of generators and relations, using algorithm
\ref{lc-alg}. It is expected that in the near future \cite{W} will
work for $R=K[x_1,\ldots,x_n]$ where $K$ is an arbitrary field of
characteristic zero and also algorithms
for computation of $H^i_{\frak m} (H^j_I(R))$ and $\lambda_{i,j}(R)$ will be
implemented, but see the comments in
\ref{efficiency} below.
\begin{ex}
\label{example}
Let $I$ be the ideal in $R=K[x_1,\ldots,x_6]$ that is generated by the
$2\times 2$ minors $f,g,h$ of the matrix
$\left(\begin{array}{ccc}x_1&x_2&x_3\\x_4&x_5&x_6\end{array}\right)$.
Then $H_I^i(R)=0$ for $i<2$ and
$H^2_I(R)\ne 0$ because $I$ is a height 2 prime and $H^i_I(R)=0$ for $i>3$
because $I$ is
3-generated, so the only remaining case is $H^3_I(R)$. This module
in
fact does
not vanish, but until the discovery of our algorithm, its non-vanishing was a
rather non-trivial fact. Our algorithm provides the first known proof of this
fact by direct calculation.
Previously, Hochster pointed out that $H^3_I(R)$ is nonzero,
using the fact that the map $K[f,g,h]\to R$ splits (compare \cite{Hu-L},
Remark 3.13) and Bruns and Schw\"anzl (\cite{Br-S}, the corollary to
Lemma 2) provided a
topological proof of the nonvanishing of $H^3_I(R)$ via \'etale
cohomology.
Both of these proofs are quite
ingenious and work only in very special situations.
Using the
program \cite{W}, one finds that $H^3_I(R)$ is isomorphic to a cyclic
$A_6$-module
generated by $1\in A_6$ subject to relations $x_1=\ldots =x_6=0$.
This is a straightforward
computational proof of the non-vanishing of $H^3_I(R)$. Of course this proof
gives
more than simply the non-vanishing. Since the quotient of $A_6$ by the
left ideal generated by $x_1,\dots,x_6$ is known to be isomorphic as an
$R$-module to
$E_R(R/(x_1,\dots,x_6))$, the injective hull of $R/(x_1,\dots,x_6)=K$ in
the category
of $R$-modules, our proof implies that
$H^3_I(R)\cong E_R(K)$.
\end{ex}
\subsection{}
\mylabel{efficiency}
Computation of Gr\"obner bases in many variables is in general a time-
and space consuming enterprise. Already in (commutative) polynomial
rings the worst case performance for the number of elements in reduced
Gr\"obner bases
is doubly exponential in the number of variables and the degrees of
the generators. In the (relatively small)
example above $R$ is of dimension 6,
so that the intermediate ring $A_{n+1}[y_1,y_2]$ contains 16
variables. In view of these facts the following idea
has proved useful.
The general context in which lemma \ref{malgrange} and proposition
\ref{kashiwara} were stated allows successive localization of $R_{fg}$
in the following way. First one computes $R_f$ according to
algorithm \ref{D/L-loc-f} as quotient of $A_n$ by a certain holonomic
ideal $L=J^\Delta(f^s)|_{s=a}, a\ll 0$.
Then $R_{fg}$ may be
computed using \ref{D/L-loc-f} again since $R_{fg}\cong R_g\otimes
A_n/L$. (Note that all
localizations of $R$ are automatically $f$-torsion free for $f\in R$
as $R$ is a domain.) This process
may be iterated for products with any finite number of factors.
Note
also that the exponents for the various factors might be different.
This requires some care as the following situations illustrate. Assume
first that $-1$ is the smallest integer root of the Bernstein polynomials
of $f$ and $g$ (both in $R$) with respect to the holonomic module
$R$. Assume further that $R_{fg}\cong A_n\cdot f^{-2}g^{-1}\supsetneq
A_n\cdot (fg)^{-1}$. Then $R_f\to R_{fg}$ can be written as
$A_n/\operatorname{ann}(f^{-1})\to A_n/\operatorname{ann}(f^{-2}\cdot g^{-1})$ sending $P\in A_n$ to
$P\cdot f\cdot g$.
Suppose on the other hand that we are interested in $H^2_I(R)$ where
$I=(f,g,h)$ and we know that $R_f=A_n\cdot f^{-2}\supsetneq A_n\cdot
f^{-1}, R_g=A_n\cdot g^{-2}$ and $R_{fg}=A_n\cdot f^{-1}g^{-2}$. (In
fact, the degree 2 part of the \v Cech complex of example
\ref{example} consists of such localizations.) It is tempting to write
the embedding $R_f\to R_{fg}$ with the use of a Bernstein operator (if
$P_f(s) f^{s+1}=b^\Delta_f(s)f^s$ then take $s=-2$) but as $f^{-1}$ is not a
generator for $R_f$, $b^\Delta_f(-2)$ will be zero. In other words, we must
write $R_{fg}$ as $A_n/\operatorname{ann}((fg)^{-2})$ and then send $P\in
\operatorname{ann}(f^{-2})$ to $P\cdot g^2$.
The two examples indicate how to write the \v Cech complex in terms
of generators and relations over $A_n$ while making sure that the maps
$C^k\to C^{k+1}$ can be made explicit using the $f_i$: the exponents
used in $C^i$ have to be at least as big as those in $C^{i-1}$ (for the
same $f_i$).
\begin{rem}
\label{remark}
We suspect that for all holonomic $fg$-torsionfree
modules $M=A_n/L$
we have (with $R_g\otimes M\cong A_n/L'$):
\[\min\{s\in{\Bbb Z}:b_f^L(s)=0\}\le \min\{s\in{\Bbb Z}:b_f^{L'}(s)=0\}.\]
This would have two nice consequences.
First of all, it would
guarantee, that successive localization at the factors of a product
does not lead to matrices in the \v Cech complex with entries of
higher degree than localization at the product at once.
Secondly, if \ref{remark} were known to be true, we could proceed as
follows for the computation of $C^i(R;f_1,\ldots,f_r)$. First compute
$J^\Delta((f_i)^s)$ for all $i$, find all minimal integer
Bernstein roots $\beta_i$
of $f_i$ on $R$ and substitute them into the appropriate annihilator
ideals. If from now on we want to use algorithm \ref{D/L-loc-f} in
order to compute $R_{f_{i_1}\cdot\ldots\cdot f_{i_k}\cdot
f_{i_{k+1}}}$ from $R_{f_{i_1}\cdot\ldots\cdot f_{i_k}}$ then we can
skip steps 2 and 3 of \ref{D/L-loc-f} as the remark gives us a lower bound for
the minimal integer Bernstein root of $f_{i_{k+1}}$ on
$R_{f_{i_1}\cdot\ldots\cdot f_{i_k}}$. (From the comments before
\ref{remark} it is also clear that we cannot hope to use a larger
value.)
\end{rem}
The advantage of localizing $R_{fg}$ as $(R_f)_g$ is twofold. For
one, it allows the exponents of the various factors to be distinct
which is useful for the subsequent cohomology computation: it helps
to keep the degrees of the maps small. (So for example $R_{x\cdot
(x^2+y^2)}$ can be written as $A_n\cdot
x^{-1} (x^2+y^2)^{-2}$ instead
of $A_n\cdot (x^{-2}\cdot (x^2+y^2)^{-2})$.
On the other hand,
since the computation of Gr\"obner bases is doubly exponential it
seems to be advantageous to break a big problem (localization at a
product) into many ``easy'' problems (successive localization).
An extreme case of this behaviour is our example \ref{example}: if we
compute $R_{fgh}$ as $((R_f)_g)_h$, the calculation uses
approximately 2.5 kB and lasts 32 seconds on a Sun
workstation using \cite{W}. If one tries to localize $R$ at the
product of the three generators at once, \cite{W} crashes after about
30 hours having used up the entire available memory (1.2 GB).
|
1997-10-19T12:37:44 | 9710 | alg-geom/9710022 | en | https://arxiv.org/abs/alg-geom/9710022 | [
"alg-geom",
"hep-th",
"math.AG"
] | alg-geom/9710022 | null | Victor V. Batyrev, Ionunt Ciocan-Fontanine, Bumsig Kim, and Duco van
Straten | Conifold Transitions and Mirror Symmetry for Calabi-Yau Complete
Intersections in Grassmannians | 36 pages, LaTeX 2.09 | null | 10.1016/S0550-3213(98)00020-0 | null | null | In this paper we show that conifold transitions between Calabi-Yau 3-folds
can be used for the construction of mirror manifolds and for the computation of
the instanton numbers of rational curves on complete intersection Calabi-Yau
3-folds in Grassmannians. Using a natural degeneration of Grassmannians
$G(k,n)$ to some Gorenstein toric Fano varieties $P(k,n)$ with conifolds
singularities which was recently described by Sturmfels, we suggest an explicit
mirror construction for Calabi-Yau complete intersections $X \subset G(k,n)$ of
arbitrary dimension. Our mirror construction is consistent with the formula for
the Lax operator conjectured by Eguchi, Hori and Xiong for gravitational
quantum cohomology of Grassmannians.
| [
{
"version": "v1",
"created": "Sun, 19 Oct 1997 10:37:44 GMT"
}
] | 2009-10-30T00:00:00 | [
[
"Batyrev",
"Victor V.",
""
],
[
"Ciocan-Fontanine",
"Ionunt",
""
],
[
"Kim",
"Bumsig",
""
],
[
"van Straten",
"Duco",
""
]
] | alg-geom | \section{Introduction}
One of the simplest ways to connect moduli spaces of
two Calabi-Yau $3$-folds $X$ and $Y$ is a so called {\em conifold
transition} that attracted interest of physicists
several years ago in connection
with {\em black hole condensation} \cite{S,GMS,CGGK}.
The idea of the conifold transition goes back to Miles Reid \cite{R},
who proposed to connect the moduli spaces of two Calabi-Yau $3$-folds
$X$ and $Y$ by choosing a
point $x_0$ on the moduli space of complex structures on $X$ corresponding
to a Calabi-Yau $3$-fold $X_0$
whose singularities consist of finitely many nodes. If $Y$ is a small
resolution of singularities on $X_0$
which replaces the nodes by a union of ${\bf P}^1$'s with normal
bundle ${\cal O}(-1) \oplus {\cal O}(-1)$, one obtains another
smooth Calabi-Yau $3$-fold $Y$.
Let $p$ be the number of nodes on $X_0$, and let $\alpha$ be the number of
relations between the homology classes of the $p$ vanishing $3$-cycles
on $X$ shrinking to nodes in $X_0$. Then the Hodge numbers of $X$ and $Y$
are related by the following equations \cite{C}:
\[ h^{1,1}(Y) = h^{1,1}(X) + \alpha, \]
\[ h^{2,1}(Y) = h^{2,1}(X) -p + \alpha. \]
The Hodge numbers of mirrors $X^*$ and $Y^*$ of $X$ and $Y$
must satisfy the equations
\[ h^{1,1}(X) = h^{2,1}(X^*), \; h^{1,1}(X^*) = h^{2,1}(X) \]
and
\[ h^{1,1}(Y) = h^{2,1}(Y^*), \; h^{1,1}(Y^*) = h^{2,1}(Y). \]
It is natural to expect that the moduli spaces of
mirrors $X^*$ and $Y^*$ are again connected in the same
simplest way: i.e., that
$X^*$ can be obtained by a small
resolution of some Calabi-Yau $3$-fold $Y^*_0$ with $p^*$ nodes and
$\alpha^*$ relations, corresponding
to a point $y_0^*$ on the moduli space of complex structures on $Y^*$.
Hence, as suggested in \cite{CGGK,GMS,LS} and \cite{DM1},
the conifold transition can be used to find mirrors of $X$,
provided one knows mirrors $Y^*$ of $Y$.
For this to work, one then needs
$$p^*=\alpha+\alpha^*=p, $$
i.e., $X_0$ and $Y_0^*$ have the same
number of nodes and complementary number of relations between them.
We remark that even for the simplest
family of Calabi-Yau $3$-folds, quintic hypersurfaces in ${\bf P}^4$,
it is an open problem to determine all possible values of $p$
\cite{Straten}.
One of the problems solved in this paper is an explicit geometric
construction of mirrors $X^*$ for
Calabi-Yau complete intersections
$3$-folds $X$ in Grassmannians $G(k,n)$
(this was only known for quartics in
$G(2,4)$, as a particular example of complete intersections
in projective space \cite{LT}). Our method is based on connecting $X$ via a
conifold transition to complete intersections $Y$ in a toric manifold.
This manifold is a small crepant desingularization
$\widehat{P(k,n)}$ of a Gorenstein toric
Fano variety $P(k,n)$, which in turn is a flat degeneration of $G(k,n)$ in its
Pl\"ucker embedding, constructed by Sturmfels (see \cite{St}, Ch. 11).
Since one knows how to construct mirrors for Calabi-Yau complete
intersections in $\widehat{P}(k,n)$ \cite{BS,LB},
it remains to find an appropriate specialization of the toric mirrors $Y^*$
for $Y$ to conifolds $Y_0^*$ whose small resolutions provide mirrors $X^*$ of
$X$. The choice of the $1$-parameter
subfamily of $Y_0^*$ among toric mirrors $Y^*$
is determined by the monomial-divisor mirror correspondence and the
embedding
$${\bf Z} \cong Pic(P(k,n)) \hookrightarrow Pic(\widehat{P}(k,n))
\cong {\bf Z}^{ 1 + (k-1)(n-k-1)}.$$
We expect that this method of mirror constructions can be applied
to all Calabi-Yau $3$-folds whose moduli spaces are connected by
conifolds transitions to the web of Calabi-Yau complete intersections in
Gorenstein toric Fano varieties. This web has been studied in
\cite{ACJM,AKMS,BLS} as a generalization of the earlier results
on Calabi-Yau complete intersection in products of projective
spaces and in weighted projective spaces \cite{CDLS,CGH}.
In order to obtain the instanton numbers
of rational curves on Calabi-Yau complete intersections
in Grassmannians, we compute a
generalized hypergeometric series $\Phi_X(z)$,
describing the monodromy invariant period of $X^*$,
by specializing a $(1 + (k-1)(n-k-1))$-dimensional generalized
(Gelfand-Kapranov-Zelevinski) $GKZ$-hypergeometric series for
the main period of toric mirrors $Y^*$ to
a single monomial parameter $z$.
Since $h^{1,1}(X) =1$, the corresponding Picard-Fuchs differential
system for periods of $X^*$ reduces to an ordinary
differential equation ${\cal D} \Phi =0$
of order $4$ for $\Phi_X(z)$. The Picard-Fuchs
differential operator $P$ can be computed from the
recurrent relation satisfied by the coefficients
of the series $\Phi_X(z)$. Applying the same computational
algorithm as in \cite{BS}, one computes the instanton numbers of
rational curves on all possible
Calabi-Yau complete intersection $3$-folds
$X \subset G(k,n)$. The numbers of lines and conics on
these Calabi-Yau $3$-folds have
been verified by S.-A. Str{\o}mme
using classical methods and the Schubert package for MapleV.
Another new ingredient of the present
paper is the so called {\em Trick with the Factorials}.
This is a naive form of a
{\em Lefschetz hyperplane section theorem} in quantum cohomology,
which goes back to Givental's idea \cite{G2}
about the relation between solutions of quantum
${\cal D}$-module for Fano manifolds $V$ and complete intersections $X \subset V$.
The validity of this procedure
has been established recently for all homogeneous spaces by B. Kim
in \cite{K3}. If the Trick with the Factorials
works for a Fano manifold $V$, one is able to compute
the instanton numbers of rational curves on Calabi-Yau complete
intersections $X \subset V$ without knowing a mirror $X^*$ for $X$, provided
one knows a special regular solution
$A_V$ to the quantum ${\cal D}$-module for $V$.
In the case of Grassmannians we
conjecture in \ref{flagmirror} that this special solution $A_{G(k,n)}(q)$
to the quantum ${\cal D}$-module determined by the small quantum cohomology
of $G(k,n)$)
can be obtained from
a natural specialization of the $GKZ$-hypergeometric series
associated with the Gorenstein toric degeneration $P(k,n)$ of $G(k,n)$.
Conjecture \ref{flagmirror} has been
checked by direct computation for all Grassmanians containing
Calabi-Yau $3$-folds
$X$ as complete intersections. In fact,
there is no essential difficulty in checking the conjecture
in each particular case at hand, because such a check involves only
calculations in the small quantum cohomology ring of
$G(k,n)$, whose structure is well-known \cite{Ber}.
This last result implies that the instanton numbers for
rational curves on $3$-dimensional
Calabi-Yau complete intersections in Grassmannians are correct
in all computed cases.
We remark that our conjecture \ref{flagmirror} on
the coincidence of $A_{G(k,n)}(q)$ with the
specialization of the multidimensional generalized $GKZ$-hypergeometric
series corresponding to the Gorenstein toric Fano variety
$P(k,n)$ strongly supports the idea that
Gromov-Witten invariants of $G(k,n)$ and
complete intersections $X \subset G(k,n)$
behave well under flat deformation and conifold transitions.
Using the degeneration of $G(k,n)$ to $P(k,n)$, we
propose in arbitrary dimension an explicit construction for mirrors of
Calabi-Yau complete intersections $X \subset G(k,n)$
whose monodromy invariant period
coincide with the power series $\Phi_X(z)$
obtained by applying the Trick with the Factorials
to $A_{G(k,n)}(q)$. We observe that our mirror construction
is consistent with the formula
for the Lax operator of Grassmannians conjectured by
Eguchi, Hori and Xiong in \cite{EHX}.
Many results formulated in this paper have been generalized and proved
in \cite{BCKS} for toric degenerations of
partial flag manifolds which have been introduced and investigated
by N. Gonciulea and V. Lakshmibai in
\cite{GL0,GL1,GL2}. These results are most easily interpreted in terms of
certain {\em diagrams} associated to a partial flag manifold, generalizing
the one used in \cite{G2} for the case of the complete flag manifold.
\section{Simplest Examples}
\subsection{Quartics in $G(2,4)$}
First we illustrate our method by analyzing a simple case, for which the mirror
construction is already known: the case of
quartics in $G(2,4)$, the Grassmannian of
$2$-planes in ${\bf C} ^4$ \cite{BS,LT}. The
Pl\"{u}cker embedding realizes
the Grassmannian $G(2,4)$ as a nonsingular quadric in ${\bf P}^5$, defined
by the homogeneous equation:
\[ z_{12} z_{34} - z_{13}z_{24} + z_{14}z_{23} = 0, \]
where $z_{ij}$ $(1\leq i<j \leq 4)$ are homogeneous
coordinates on ${\bf P}^5$. Let $P(2,4) \subset {\bf P}^5$
be the $4$-dimensional Gorenstein toric Fano variety defined by
the quadratic equation
\[ z_{13}z_{24} = z_{14}z_{23}. \]
Denote by $X$ the intersection of $G(2,4)$ with a generic
hypersurface $H$ of degree $4$ in ${\bf P}^5$, so that $X$ is
a nonsingular Calabi-Yau hypersurface in $G(2,4)$.
Its topological invariants are
$h^{1,1}(X) = 1$, $h^{2,1}(X) = 89$, and $\chi(X)= -176$. Denote by
$X_0$ the intersection of $P(2,4)$ with a generic hypersurface $H$ of
degree $4$ in ${\bf P}^5$. Then $X_0$ is a Calabi-Yau $3$-fold with
$4$ nodes which are the intersection points of $H$ with
the line $l \subset P(2,4)$
of conifold singularities. Considering $X_0$ as a deformation
of $X$, it follows from general formulas proved in \ref{formul-2}
that the homology classes of the vanishing $3$-cycles on $X$ shrinking
to $4$ nodes in $X_0$ satisfy a single relation. Denote by
$Y$ a simultaneous small resolution of all $4$ nodes. One obtains this
resolution by restriction of a small toric resolution of
singularities in $P(2,4)$: $\rho\,: \; \widehat{P}(2,4) \rightarrow P(2,4)$.
The smooth toric variety $\widehat{P}(2,4)$ is a toric ${\bf P}^3$-bundle
over ${\bf P}^1$:
\[ \widehat{P}(2,4) = {\bf P}_{{\bf P}^1}({\cal O} \oplus {\cal O}
\oplus {\cal O}(1) \oplus {\cal O}(1)) \]
and the morphism $\rho$ contracts a $1$-parameter family
of sections of this ${\bf P}^3$-bundle with
the normal bundle ${\cal O} \oplus {\cal O}(-1) \oplus {\cal O}(-1)$.
A smooth Calabi-Yau hypersurface $Y \subset \widehat{P}(2,4)$ has a natural
$K3$-fibration over ${\bf P}^1$ and the following
topological invariants: $\chi(Y) = -168$, $h^{1,1}(Y) = 2$, and
$h^{2,1}(Y)=86$.
The Gorenstein toric Fano variety $P(2,4)$ can be
described by a $4$-dimensional fan $\Sigma(2,4) \subset {\bf R}^4$ consisting
of cones over the faces of a $4$-dimensional reflexive polyhedron
$\Delta(2,4)$ with $6$ vertices:
\[ u_{1,0} := f_{1,1}, \; u_{2,0} = f_{2,1} - f_{1,1}, \;
u_{2,1}: = f_{2,2} - f_{1,2}, \]
\[ v_{2,2} := - f_{2,2}, \;
v_{2,1}: = f_{2,2} - f_{2,1}, \;
v_{1,1} = f_{1,2} - f_{1,1}, \]
where $\{ f_{1,1}, f_{1,2}, f_{2,1}, f_{2,2} \}$ is a basis
of the lattice ${\bf Z}^4 \subset {\bf R}^4$.
The regular fan $\widehat{\Sigma}(2,4)$ defining the smooth projective
toric variety $\widehat{P}(2,4)$
is obtained by a subdivision of $\Sigma(2,4)$.
The combinatorial structure of $\widehat{\Sigma}(2,4)$ is defined by the
following primitive collections (see notations in \cite{Ba1}):
\[ {\cal R} = \{ u_{1,0}, v_{1,1}, u_{2,1}, v_{2,2} \} , \;
{\cal C}_{1,1} = \{ v_{2,1}, u_{2,0} \}. \]
The fan $\widehat{\Sigma}(2,4)$ contains $8$ cones of dimension $4$, obtained
by deleting one vector from each primitive collection.
The primitive relations corresponding to ${\cal R}_0$ and ${\cal C}_{1,1}$
are
\[ u_{1,0} + v_{1,1} + u_{2,1} + v_{2,2} = 0 \]
and
\[ v_{2,1} + u_{2,0} = v_{1,1} + u_{2,1}. \]
Let ${\bf P}_{\Delta(2,4)}$ be the Gorenstein toric Fano variety
associated with the reflexive polyhedron $\Delta(2,4)$.
By the toric method of \cite{Ba2}, the mirror $Y^*$ of $Y$ can be obtained as
a crepant desingularization of the closure in ${\bf P}_{\Delta(2,4)}$
of an affine hypersurface
$Z_f$ with the equation
\[ f(X) = -1 + a_1X_1 + a_2X_2 + a_3 X_3 + a_4 X_4 + a_5(X_1X_2X_3)^{-1} +
a_6(X_4^{-1}X_1X_2), \]
where $a_1, \ldots, a_6$ are some complex numbers
(the Newton polyhedron of $f$ is isomorphic
to $\Delta(2,4)$).
We choose a subfamily of Laurent polynomials $f_0$ with coefficients
$\{a_i \}$ satisfying an additional monomial equation
\[ a_1 a_2 = a_4 a_6. \]
The affine Calabi-Yau hypersurfaces $Z_{f_0}$ of this subfamily are
not $\Delta(2,4)$-regular anymore, because the closures
$\overline{Z}_{f_0}$ in ${\bf P}_{\Delta(2,4)}$ have a singular
intersection with the stratum
$T_{\Theta} \subset {\bf P}_{\Delta(2,4)}$ corresponding
to the face
\[ \Theta = {\rm Conv}\left\{ (1,0,0,0),\, (0,1,0,0), \,
(0,0,0,1),\,(1,1,0,-1) \right\}. \]
Without loss of generality, we can assume that $a_1=a_2=a_3=a_4 =1$
(this condition can be satisfied using the action of $({\bf C}^*)^4$ on
$X_1, \ldots, X_4$). Thus we obtain a $2$-parameter family of Laurent
polynomials defining $Z_f$:
\[ f(X) = -1 + X_1 + X_2 + X_3 + X_4 + a_5(X_1X_2X_3)^{-1} +
a_6(X_4^{-1}X_1X_2), \]
and a $1$-parameter subfamily of Laurent polynomials
\[ f_0(X) = -1 + X_1 + X_2 + X_3 + X_4 + a_5(X_1X_2X_3)^{-1} +
(X_4^{-1}X_1X_2) \]
defining $Z_{f_0}$.
The monodromy invariant period $\Phi$
of the toric hypersurface $Z_f$ can be computed by the residue theorem:
\[ \Phi_X(a_5,a_6) = \frac{1}{(2\pi i )^4} \int_{\gamma} \frac{1}{(-f)}
\frac{dX_1}{X_1} \wedge \cdots \wedge \frac{dX_4}{X_4}. \]
By this method, we obtain the generalized
hypergeometric series corresponding to $f(X)$:
\[ \Phi_X(a_5,a_6) = \sum_{k, l \geq 0} \frac{(4k + 4l)!}{(k!)^2 (l!)^2
((k+l)!)^2} a_5^{k+l} a_6^l. \]
By the substitution $a_6 =1$ $(a_1a_2 = a_4a_6)$
and $a_5 = z$, we obtain the series corresponding to
the $1$-parameter family of Laurent polynomials
$f_0$:
\[ \Phi_X(z) = \sum_{m \geq 0} \frac{(4m)!}{(m!)^2}
\left(\sum_{k+l = m} \frac{1}{(k!)^2 (l!)^2} \right)z^m. \]
Using the identity
\[ \sum_{k+l =m} \frac{(m!)^2}{(k!)^2 (l!)^2} = { 2m \choose m}, \]
we transform $\Phi_X(z)$ to the form
\[ \Phi_X(z) = \sum_{m \geq 0} \frac{(4m)!(2m)!}{(m!)^6} z^m. \]
This is a well-known series, satisfying a Picard-Fuchs
differential equation
\[ \left( D^4 - 16z(2 D +1)^2(4 D + 1)(4 D + 3)\right)
\Phi_X(z) = 0,\;\;
D = z\frac{\partial}{\partial z}, \]
predicting the instanton numbers of rational curves on $X$ (cf. \cite{LT}).
The correctness of these numbers
now follows from the work of Givental, \cite{G1}.
\subsection{Complete intersections of type $(1,1,3)$ in $G(2,5)$}
\vskip 10pt
Let $z_{ij}$ $(1 \leq i < j \leq 5$ ) be homogeneous
coordinates on the projective space ${\bf P}^9$.
The Grassmannian $G(2,5)$ of $2$-planes in ${\bf C}^5$
can be identified with the subvariety
in ${\bf P}^9$
defined by the quadratic equations:
\[ z_{23}z_{45} - z_{24}z_{35} + z_{25}z_{34}= 0, \]
\[ z_{13}z_{45} - z_{14}z_{35} + z_{15}z_{34}= 0, \]
\[ z_{12}z_{45} - z_{14}z_{35} + z_{15}z_{34}= 0, \]
\[ z_{12}z_{35} - z_{13}z_{25} + z_{15}z_{23}= 0, \]
\[ z_{12}z_{34} - z_{13}z_{24} + z_{14}z_{23}= 0. \]
We associate with $G(2,5)$ a $6$-dimensional Gorenstein
toric Fano variety $P(2,5) \subset {\bf P}^9$ defined by the
equations
\[ z_{24}z_{35} = z_{25}z_{34}, \; z_{14}z_{35} = z_{15}z_{34},\;
z_{14}z_{35} = z_{15}z_{34}, \]
\[ z_{13}z_{25} = z_{15}z_{23}, \; z_{13}z_{24} = z_{14}z_{23}. \]
The following statement is due to Sturmfels (see \cite{St}, Example 11.9 and
Proposition 11.10):
\begin{prop}
The Gorenstein toric Fano variety
$P(2,5)$ is a degeneration of the Grassmannian $G(2,5)$, i.e.,
$P(2,5)$ is the special fibre of a
flat family whose generic fibre is $G(2,5)$. \hspace*{\fill}\hbox{$\Box$}
\end{prop}
The toric variety $P(2,5)$ can be described by a fan
$\Sigma(2,5) \subset {\bf R}^6$ consisting of cones over the faces of
a $6$-dimensional reflexive polyhedron $\Delta(2,5)$ with $9$ vertices
\[ u_{1,0} := f_{1,1}\; u_{2,i}: = f_{2,i+1} - f_{1,i+1}, \,\; i = 0, 1, 2,\]
\[ v_{2,3} := - f_{2,3}, \; v_{i,j}: = f_{i,j+1} - f_{i,j}, \, \;
i = 1, 2,\; j =1,2, \]
where $\{ f_{1,1}, f_{1,2}, f_{1,3}, f_{2,1}, f_{2,2}, f_{2,3} \}$
is a basis of the lattice ${\bf Z}^6 \subset {\bf R}^6$.
There exists a subdivison
of the fan $\Sigma(2,5)$ into a regular
fan $\widehat{\Sigma}(2,5)$ defined by the primitive collections:
\[ {\cal R} = \{ u_{1,0}, v_{1,1}, v_{1,2}, u_{2,2}, v_{2,3} \}, \]
\[ {\cal C}_{1,1} = \{ u_{2,0}, v_{2,1} \}, \;
{\cal C}_{1,2} = \{ u_{2,1}, v_{2,2} \}, \]
i.e., $\widehat{\Sigma}(2,5)$
contains exactly $20$ cones of dimension $6$ generated
by the $6$-element sets obtained by taking all but one of the
vectors from each
primitive collection.
The primitive relations corresponding to
${\cal R}$, ${\cal C}_{1,1}$ and ${\cal C}_{1,2}$ are
\[ u_{1,0} + v_{1,1} + v_{1,2} + u_{2,2} + v_{2,3} = 0, \]
\[ u_{2,0} + v_{2,1} = v_{1,1} + u_{2,1}, \; \;
u_{2,1} + v_{2,2} = v_{1,2} + u_{2,2}. \]
Denote by $\widehat{P}(2,5)$ the smooth toric variety associated with
the fan $\widehat{\Sigma}(2,5)$. It is easy to check that $P(2,5)$ is a
Gorenstein toric Fano variety and $\widehat{P}(2,5)$ is a small
crepant resolution
of singularities of $P(2,5)$. The toric manifold
$\widehat{P}(2,5)$ has nonnegative
first Chern class and it can be identified with a
toric bundle over ${\bf P}^1$ with the $5$-dimensional fiber
\[ F: = {\bf P}_{{\bf P}^1} ({\cal O} \oplus {\cal O} \oplus {\cal O} \oplus
{\cal O}(1) \oplus {\cal O}(1) ) \]
There is another description of $P(2,5)$. We remark that
variables $z_{12}$ and $z_{45}$ do not appear in the equations for $P(2,5)$.
Thus $P(2,5)$ is a cone over a Gorenstein $4$-dimensional toric Fano
variety
\[ P'(2,5) : = P(2,5) \cap \{ z_{12} = z_{45} = 0\} \subset {\bf P}^7. \]
We can describe $P'(2,5)$ by a $4$-dimensional fan $\Sigma'(2,5)$
consisting of cones over a $4$-dimensional reflexive polyhedron
$\Delta'(2,5)$ with $7$ vertices
\[ e_1 = (1,0, 0,0), \; e_2= (0,1,0,0),\; e_3 = (-1,-1, 0,0), \]
\[ e_4 = (0,0, 1,0), \; e_5= (0,0,0,1),\; e_6 = (0,0,-1,-1), \]
\[ e_7 = (1,1,1,1). \]
The only singularities of $P'(2,5)$ are
nodes along two lines $l_1, l_2 \in P'(2,5) \subset {\bf P}^7$
corresponding to the
$3$-dimensional cones
$$ \sigma_1 = {\bf R}_{\geq 0} < e_1, e_2, e_6, e_7 >\;\;
\mbox{\rm and } \;\;\sigma_2 = {\bf R}_{\geq 0}
< e_4, e_5, e_3, e_7 >$$
in $\Sigma'(2,5)$.
Subdividing each of these cones into the union of $2$ simplicial ones, we
obtain a small crepant resolution $\widehat{P'}(2,5)$ of singularities
of $P'(2,5)$. The smooth toric $4$-fold $\widehat{P'}(2,5)$ can be identified
with the blow up of a point on ${\bf P}^2 \times {\bf P}^2$.
Let $X = X_{1,1,3} \subset G(2,5)$ be a
smooth $3$-dimensional Calabi-Yau complete intersection
of $3$ hypersurfaces of degrees $1$, $1$ and $3$ in ${\bf P}^9$
with $G(2,5)$. One can compute $h^{1,1}(X) = 1$, $h^{2,1}(X) = 76$,
and $\chi(X) = -150$. Now let $X_0$ be the intersection of $P'(2,5)$ with
a generic hypersurface $H \subset {\bf P}^7$ of degree $3$. Then
$X_0$ is a deformation of $X$,
having $6$ nodes obtained from the intersections
$H \cap l_1$ and $H \cap l_2$. The $3$ nodes on each intersection
$H \cap l_i$ $(i =1,2)$ are described by $3$ vanishing $3$-cycles on
$X$, satisfying a single
linear relation. Resolving singularities of $X_0$, we obtain
another smooth Calabi-Yau $3$-fold $Y$ with
\[ h^{1,1}(Y) = h^{1,1}(X) + 2 = 3, \;\;
h^{2,1}(Y) = h^{2,1}(X) + 2 - 6 = 72. \]
The mirror $Y^*$ of the Calabi-Yau $3$-fold $Y$
can be obtained by the toric construction
\cite{Ba2}. The Calabi-Yau $3$-fold $Y^*$ is a toric
desingularization $\widehat{Z}_f$ of a $\Delta'(2,5)$-compactification of a
generic hypersurface $Z_f$ in $({\bf C}^*)^4$
defined by a Laurent polynomial $f(X)$ with the Newton polyhedron
$\Delta'(2,5)$:
\[ f(X) = -1 + a_1 X_1 + a_2 X_2 + a_3 (X_1X_2)^{-1} + a_4 X_3 + a_5X_4+ \]
\[ + a_6(X_3X_4)^{-1} + a_7X_1X_2X_3X_4. \]
As it was shown in \cite{Ba2}, one has
$h^{1,1}(\widehat{Z}_f) = h^{2,1}(Y) = 72$ and
$h^{2,1}(\widehat{Z}_f) = h^{1,1}(Y) = 3$.
We identify the mirror $X^*$ of $X$
with a desingularization $\widehat{Z}_{f_0}$ of
a $\Delta'(2,5)$-compactification $\overline{Z}_{f_0}$ of a
generic hypersurface $Z_{f_0}$ in $({\bf C}^*)^4$
defined by Laurent polynomials $f_0$ whose
coefficients $\{ a_i\} $ satisfy two additional
monomial equations
$$ a_1a_2 = a_6a_7\ \ \mbox{\rm and} \ \ a_4 a_5 = a_3 a_7. $$
Without loss of generality, we can put $a_1 = a_2 = a_4 = a_7$. So one
obtains
\[ f(X) = -1 + X_1 + X_2 + a_3 (X_1X_2)^{-1} + X_3 + a_5X_4 \]
\[ + a_6(X_3X_4)^{-1} + X_1X_2X_3X_4 \]
and
\[ f_0(X) = -1 + X_1 + X_2 + a_3 (X_1X_2)^{-1} + X_3 + a_3X_4 \]
\[ + (X_3X_4)^{-1} + X_1X_2X_3X_4. \]
It is easy to see that the Laurent polynomial $f_0$
is not $\Delta'(2,5)$-regular (this regularity fails
exactly for two $2$-dimensional faces $\Theta_1
:= {\rm Conv}(e_1,e_2,e_6,e_7)$
and $\Theta_2:= {\rm Conv}(e_4,e_5,e_3,e_7)$
of $\Delta'(2,5)$ (see definition of $\Delta$-regularity in
\cite{Ba2}).
The $4$-dimensional
Gorenstein toric Fano variety ${\bf P}_{\Delta'(2,5)}$ associated with
the reflexive polyhedron $\Delta'(2,5)$-closure has singularities
of type $A_2$ along of the $2$-dimensional strata $T_{\Theta_1}$ and
$T_{\Theta_2}$. The projective hypersurfaces $\overline{Z}_{f_0} \subset
{\bf P}_{\Delta'(2,5)}$
defined by the equation $f_0 =0$ have non-transversal intersections with
$T_{\Theta_1}$ and $T_{\Theta_2}$ (each intersection is a union of
two rational curves with a single normal crossing point). After toric
resolution of $A_2$-singularities along $T_{\Theta_i}$
on ${\bf P}_{\Delta'(2,5)}$, we obtain $3$ new $2$-dimensional
strata over each $T_{\Theta_i}$. This shows that
we cannot resolve all singularities of $\overline{Z}_{f_0}$ by a toric
resolution of singularities on the ambient toric variety
${\bf P}_{\Delta'(2,5)}$.
Let $Y_0^* := \widehat{Z}_{f_0}$ be the pullback of $\overline{Z}_{f_0}$ under
a $MPCP$-desingularization
$$\rho\, : \, \widehat{{\bf P}}_{\Delta'(2,5)} \rightarrow
{\bf P}_{\Delta'(2,5)}.$$
Then $Y_0^*$ is a Calabi-Yau 3-fold with
$3 + 3 = 6$ nodes obtained as singular points of intersections
of $Y_0^*$ with
the $6$ strata of dimension $2$ in $\widehat{{\bf P}}_{\Delta'(2,5)}$
over $T_{\Theta_1},\; T_{\Theta_2} \subset \widehat{{\bf P}}_{\Delta'(2,5)}$.
One can show that the
vanishing $3$-cycles associated with the $3$ nodes over each
$T_{\Theta_i}$ $(i=1,2)$ satisfy $2$ linear relations (see \ref{formul-2}).
If $X^*$ denotes a small resolution of
these $6$ nodes on $Y_0^*$, then
\[ h^{1,1}(X^*) = h^{1,1}(\widehat{Z}_{f}) + 4 = 76 \]
and
\[ h^{2,1}(X^*) = h^{2,1}(\widehat{Z}_f) + 4 - 6 = 1. \]
Thus the Hodge numbers of $X^*$ and $X$ satisfy the mirror duality.
Finally, we explain the computation of the instanton numbers
of rational curves
of degree $m$ in the case of
Calabi-Yau complete intersections $X$ of type $(1,1,3)$
in $G(2,5)$. As shown in \cite{BS}, one obtains
the following monodromy invariant period for $Z_f$:
$$\Phi(a_3,a_5,a_6) = \sum_{k,l,n \geq 0}
\frac{(3k+3l+3n)!}{(k!)^2(n!)^2l!(k+l)!(l+n)!}a_3^{k+l}a_5^n a_6^{n+l}.$$
By the substitution $a_3 = a_5 =z$ and $a_6 = 1$, we obtain the monodromy
invariant period for $Z_{f_0}$:
\[ \Phi_X(z) = \left( \sum_{k + l +n = m}
\frac{(3m)!}{(k!)^2(n!)^2l!(k+l)!(l+n)!} \right) z^m. $$
It remains to apply to the series $\Phi_X(z)$ the general algorithm
from \cite{BS} (see 6.2 and 7.1 for details, and the instanton numbers).
\vskip 10pt
\section{Toric Degenerations of Grassmannians}
In this section we review without proof some results, which we prove
for arbitrary partial flag manifolds in \cite{BCKS}.
\subsection{The toric variety $P(k,n)$ and its singular locus}
Let $G(k,n)$ be the Grassmannian of
$k$-dimensional ${\bf C}$-vector subspaces
in a $n$-dimensional complex vector space ($k < n$).
Denote by
$$X_{i,j} \;\; i = 1, \ldots, k, \; j = 1, \ldots, n-k$$
$k(n-k)$
independent variables. We denote by $T(k,n)$ the algebraic torus
$ {\rm Spec}\, {\bf C}[X_{i,j},X_{i,j}^{-1} ]\cong
({\bf C}^*)^{k(n-k)}$ of dimension $k(n-k)$.
We put $N(k,n): = {\bf Z}^{k(n-k)} $ to be a free abelian group of rank
$k(n-k)$ with a fixed ${\bf Z}$-basis $f_{i,j}$
$(i = 1, \ldots, k, \; j = 1, \ldots, n - k)$.
Define the set of $2(k-1)(n-k-1) +
n$ elements
in $N(k,n)$ as follows:
\[ u_{1,0} := f_{1,1},\; u_{i,j}: = f_{i,j+1} - f_{i-1,j+1},
\, \; i = 2, \ldots, k,\;
j =0, \ldots, n-k-1\; \]
\[ v_{k,n-k}: = - f_{k,n-k}, \;
v_{i,j}: = f_{i,j+1} - f_{i,j}, \,\; i = 1, \ldots, k,\;
j =1, \ldots, n-k-1. \]
We set $N(k,n)_{\bf R} = N(k,n) \otimes {\bf R}$.
\begin{definition}
{\rm Define a convex polyhedron
$\Delta(k,n) \subset N(k,n)_{\bf R}$ as the convex hull
of all lattice points $\{ u_{i,j}, v_{i',j'} \}$.
We set $\Sigma(k,n) \subset N(k,n)_{\bf R}$ to be the fan over all
proper faces of the polyhedron $\Delta(k,n)$.}
\label{polyh}
\end{definition}
\begin{definition}
{\rm Define $P(k,n)$ to be the toric variety associated with
the fan $\Sigma(k,n)$. }
\end{definition}
\begin{theorem}
The polyhedron $\Delta(k,n)$ is reflexive. In particular,
$P(k,n)$ is a Gorenstein toric Fano variety.
\end{theorem}
\begin{definition}
{\rm Let $\widehat{\Sigma}(k,n)$ be a complete regular fan
whose $1$-dimensional cones are generated by the lattice vectors
$\{ u_{i,j}, v_{l,m}\} $ and whose combinatorics
is defined by the following $1 + (k-1)(n-k-1)$ primitive collections:
\[ {\cal R}_0: = \{ u_{1,0}, v_{1,1}, v_{1,2}, \ldots, v_{1,{n-k-1}},
u_{2, n-k-1}, u_{3, n-k -1}, \ldots, u_{k,n-k-1}, v_{k,n-k} \}, \]
\[ {\cal C}_{i,j} = \{ u_{k+1-i, j-1}, v_{k+1-i, j},\;
\; i=1, \ldots, k-1,\; j =1, \ldots, n-k-1 \}.\]
In particular, the fan $\widehat{\Sigma}(k,n)$
consists of $n2^{(k-1)(n-k-1)}$
cones of dimension $k(n-k)$.
}
\end{definition}
\begin{remark}
{\rm We notice that the lattice vectors $u_{i,j}$ and $v_{l,m}$ satisfy the
following $1+ (k-1)(n-k-1)$ independent primitive relations:
\[ u_{1,0} + v_{1,1} + \cdots + v_{1,{n-k-1}} +
u_{2, n-k-1} + \cdots + u_{k,n-k-1} + v_{k,n-k} = 0, \]
\[ u_{k+1-i, j-1} + v_{k+1-i, j} = u_{k+1-i, j} + v_{k-i,j}, \]
\[ i=1, \ldots, k-1,\; j =1, \ldots, n-k-1 .\]
According to Theorem 4.3 in \cite{Ba1}, the toric variety
$\widehat{\Sigma}(k,n)$ can be obtained as $ (k-1)(n-k-1)$-times
iterated toric bundle over ${\bf P}^1$'s: we start with ${\bf P}^{n-1}$
and construct on each step a toric bundle over ${\bf P}^1$
whose fiber is the toric variety constructed in the previous step.
At each stage of this process, we obtain a smooth projective toric variety
with the nonnegative first Chern class which is divisible by $n$.
In particular we obtain that the smooth projective toric
variety $\widehat{P}(k,n)$ defined by the fan $\widehat{\Sigma}(k,n)$
has Picard number $1 + (k-1)(n-k-1)$. Moreover, the first Chern class
$\widehat{c}_1(k,n)$ of
$\widehat{P}(k,n)$ is nonnegative and it is divisible by $n$
in ${\rm Pic}(\widehat{P}(k,n))$.
}
\end{remark}
\begin{definition}
We denote by
$\widehat{P}(k,n)$ $( 1 \leq i \leq k-1, \;
1 \leq j \leq n- k-1)$ $(k-1)(n-k-1)$ codimension-$2$ subvarieties
of $\widehat{P}(k,n)$ corresponding to the $2$-dimensional cones $
\sigma_{ij} \in \widehat{\Sigma}(k,n)$:
\[ \sigma_{ij} = {\bf R}_{\geq 0} <u_{k+1-i, j-1},
v_{k+1-i, j} >. \]
\end{definition}
\begin{theorem}
The small contraction $\rho\, : \, \widehat{P}(k,n) \rightarrow {P}(k,n)$
defined by the semi-ample anticanonical divisor
on $\widehat{P}(k,n)$
contracts smooth toric varieties
$\widehat{W}_{i,j}$ to codimension-$3$ toric subvarieties $W_{i,j}
\subset {P}(k,n)$ whose open strata consist of conifold singularities,
i.e.,
singularities whose $3$-dimensional cross-sections are isolated nondegenerate
quadratic singularities (nodes, ordinary double points).
\label{sing-l}
\end{theorem}
The proof of a generalized version of \ref{sing-l} for arbitrary
partial flag manifolds is contained in \cite{BCKS}(Th. 3.1.4).
\subsection{The flat degeneration of
$G(k,n)$ to $P(k,n)$}
\begin{definition}
{\rm Denote by $A(k,n)$ the set of all sequences of integers
\[ a= (a_1, a_2, \ldots, a_k) \in {\bf Z}^k \]
satisfying the condition
\[ 1 \leq a_1 < a_2 < \cdots < a_k \leq n. \]
We consider $A(k,n)$ as a partially ordered set
with the following natural partial order:
\[ a = (a_1, \ldots, a_k) \prec a'= (a_1', \ldots, a_k') \]
if and only if $a_i \leq a_i'$ for all $i =1, \ldots, k$.
We set
\[ \min{(a,a')} : = (\min{(a_1,a_1')}, \ldots, \min{(a_k,a_k')}) \]
and
\[ \max{(a,a')} : = (\max{(a_1,a_1')}, \ldots, \max{(a_k,a_k')}). \] }
\end{definition}
\begin{theorem}
There exists a natural one-to-one correspondence between
faces of codimension $1$ of the polyhedron $\Delta(k,n)$
and
elements of $A(k,n)$.
\end{theorem}
\noindent
{\em Proof.} See \cite{BCKS} (Th. 2.2.3).
\begin{theorem}
The first Chern class
of the Gorenstein
toric Fano variety ${P}(k,n)$ is equal to $n[H]$,
where $[H]$ is the class of the ample generator of
${\rm Pic}({P}(k,n)) \cong {\bf Z}$. Moreover, there
exists a natural one-to-one correspondence
between the elements of the monomials basis of
\[ H^0({P}(k,n), {\cal O}(H)) \]
and elements of
$A(k,n)$.
In particular,
\[ {\rm dim} \, H^0({P}(k,n), {\cal O}(H)) = { n \choose k }. \]
\label{gl-sections}
\end{theorem}
\noindent
{\em Proof.} See \cite{BCKS} (Prop. 3.2.5).
\begin{theorem}
The ample line bundle ${\cal O}(H)$ on $P(k,n)$
defines a projective embedding into the
projective space ${\bf P}^{{ n \choose k}-1}$
whose homogeneous coordinates $z_a$ are naturally indexed
by elements $a \in A(k,n)$. Moreover,
the image of $P(k,n)$ in ${\bf P}^{{ n \choose k}-1}$
is defined by the quadratic homogeneous binomial equations
\[ z_a z_{a'} - z_{min(a,a')}z_{max(a,a')} \]
for all pairs $(a,a')$ of non-comparable elements $a, a' \in
A(k,n)$.
\end{theorem}
\noindent
{\em Proof.} See \cite{BCKS} (Th. 3.2.13).
\begin{example}
{\rm The following ${ n \choose 4 }$ quadratic
equations in homogeneous coordinates
$\{ z_{i,j} \}$ $( 1\leq i < j \leq n)$ are defining equations
for the toric variety $P(2,n)$ in ${\bf P}^{{ n \choose 2}-1}$:
\[ z_{i_1,i_4}z_{i_2,i_3} - z_{i_1,i_3}z_{i_2,i_4} = 0, \;\;
( 1 \leq i_1 < i_2 < i_3 < i_4 \leq n). \]
}
\end{example}
The following theorem is due to B. Sturmfels (\cite{St}, Prop. 11.10.)
\begin{theorem}
There exists a natural flat deformation of the
Pl\"ucker-embedded Grassmannian
$$G(k,n) \subset {\bf P}^{{ n \choose k}-1}$$
whose special fiber is isomorphic to the subvariety defined
quadratic homogeneous binomial equations
\[ z_a z_{a'} - z_{min(a,a')}z_{max(a,a')} \]
for all pairs $(a,a')$ of noncomparable elements $a, a' \in
A(k,n)$.
\end{theorem}
\begin{corollary}
The toric variety $P(k,n) \subset {\bf P}^{{ n \choose k}-1}$
is isomorphic to a flat degeneration of the Pl\"ucker
embedding of the Grassmannian $G(k,n)$.
\label{def-gr}
\end{corollary}
\section{Equations for Mirror Manifolds}
\subsection{The mirror construction}
Recall the definition of nef-partions for Gorenstein toric
Fano varieties and the mirror construction
for Calabi-Yau complete intersections associated with
nef-partitions \cite{LB} (we will follow the notations in \cite{BB2}).
\begin{definition}
{\rm Let $\Delta \subset M_{\bf R}$ be a reflexive polyhedron,
$\Delta^* \subset N_{\bf R}$ its
dual, and $\{ e_1, \ldots ,e_l\}$ the set of vertices
of $\Delta^*$ corresponding to torus invariant divisors
$D_1, \ldots, D_l$ on the Gorenstein toric Fano
variety ${\bf P}_{\Delta}$. We set $I := \{ 1, \ldots, l \}$.
A partition $I= J_1 \cup \cdots \cup J_r$ of $I$ into
a disjoint union of subsets $J_i \subset I$
is called a {\bf nef-partition}, if
\[ \sum_{j \in J_i} D_j \]
is a semi-ample Cartier divisor on ${\bf P}_{\Delta}$
for all $i =1, \ldots, r$.}
\end{definition}
\begin{definition}
{\rm Let $I= J_1 \cup \cdots \cup J_r$ be a nef-partition.
We define the polyhedron $\nabla_i$ $(i =1, \ldots, r)$
as the convex hull of $0 \in \Delta$ and all vertices
$e_j$ with $j \in J_i$. By $\Delta_i \subset M_{\bf R}$
$(i =1, \ldots, r)$ we denote the
supporting polyhedron for global sections of the corresponding
semi-ample invertible sheaf ${\cal O}( \sum_{j \in J_i} D_j)$ on
${\bf P}_{\Delta}$.
For each $i =1, \ldots, r$, we denote by $g_i$ (resp.
by $h_i$) a generic Laurent polynomial with the Newton polyhedron
$\Delta_i$ (resp. $\nabla_i$). }
\end{definition}
The mirror construction in \cite{LB} says that the mirror
of a compactified generic Calabi-Yau complete intersection
$g_1 = \cdots = g_r = 0$ is a compactified generic Calabi-Yau
complete intersection defined by the equations
$h_1 = \cdots = h_r = 0$.
Now we specialize the above mirror construction for the case
$\Delta = \Delta^*(k,n)$, $\Delta^* = \Delta(k,n)$, and ${\bf P}_{\Delta} =
P(k,n)$, where
$\Delta(k,n)$ is a reflexive polyhedron defined in \ref{polyh},
$\Delta^*(k,n)$ its polar-dual reflexive polyhedron and $P(k,n)$ the
Gorenstein toric Fano degeneration of the Grassmannian $G(k,n)$.
\begin{definition}
{\rm Define the following $n$ subsets
$E_1, \ldots, E_n$ of the set of vertices $\{ u_{i,j}, v_{i',j'} \}$
of the polyhedron $\Delta(k,n)$:
\[ E_1 : =\{u_{1,0}\}, \;
E_i = \{ u_{i,0}, u_{i,1}, \ldots, u_{i,n-k-1} \}, \; i =2, \ldots,k, \]
\[ E_{k+j}: = \{ v_{1,j}, v_{2,j}, \ldots,
v_{k,j} \}, \; j =1, \ldots, n-k-1, \;
E_n: = \{ v_{k,n-k}\}. \]}
\end{definition}
\begin{proposition}
Let $D(E_i) \subset P(k,n)$ $(i =1, \ldots, n)$ be the torus
invariant divisor whose irreducible components have multiplicity $1$
and correspond to vertices of $\Delta(k,n)$ from the subset $E_i$.
Then the class of $D(E_i)$ is an ample generator of $Pic(P(k,n))$.
\label{gener}
\end{proposition}
\noindent
{\em Proof.} By a direct computation, one obtains that
for all $i, j \in \{ 1, \ldots, n \}$ the difference
$D(E_i) - D(E_j)$ is a principal divisor, i.e,
all divisors $D(E_1), \ldots, D(E_n)$ are linearly equivalent.
On the other hand,
\[ D(E_1) + \cdots + D(E_n) \]
is the ample anticanonical divisor on $P(k,n)$. By \ref{gl-sections},
the anticanonical divisor on $P(k,n)$ is linearly equivalent
to $nH$, where $H$ is an ample generator of $Pic(P(k,n))$.
Hence, each divisor $D(E_i)$ is linearly equivalent to $H$.
\hspace*{\fill}\hbox{$\Box$}
\begin{definition}
{\rm Let $1 \leq d_1 \leq \cdots \leq d_r$ be positive integers
satisfying the equation
$$d_1 + \cdots + d_r = n$$ and
$I:= \{ 1, \ldots, n \}$. We denote by
$X:= X_{d_1, \ldots, d_r} \subset G(k,n)$ a Calabi-Yau
complete intersection of hypersurfaces of
degrees $d_1, \ldots, d_r$ with $G(k,n) \subset
{\bf P}^{ { n \choose k } -1}$. Consider a partition
$I= J_1 \cup \cdots \cup J_r$ of $I$ into
a disjoint union of subsets $J_i \subset I$ with $|J_i| = d_i$. }
\label{J's}
\end{definition}
\begin{definition}
{\rm Let $\nabla_{J_i}$ $(i =1, \ldots, r)$ be the
convex hull of $0 \in N(k,n)_{\bf R}$ and all vertices
of $\Delta(k,n)$ contained in the union
\[ \bigcup_{j \in J_i } E_j. \]
We denote by $h_{J_i}(X)$ a generic Laurent polynomial
in variables $X_{i',j'} := X^{f_{i',j'}}$ $( 1 \leq i' \leq k, \;
1 \leq j' \leq n-k)$ having $\nabla_{J_i}$ as a Newton polyhedron. }
\label{nablas}
\end{definition}
By \ref{gener}, one immediately obtains the following:
\begin{corollary}
Let $Y:= Y_{d_1, \ldots, d_r} \subset P(k,n)$ a Calabi-Yau
complete intersection of hypersurfaces of
degrees $d_1, \ldots, d_r$ with the Gorenstein toric Fano
variety $P(k,n) \subset {\bf P}^{ { n \choose k } -1}$. Then the mirror
$Y^*$ of $Y$ $($according to \cite{BS} and \cite{LB}$)$
is a compactified generic Calabi-Yau complete intersection
defined by the equations
\[ h_{J_1}(X) = \cdots = h_{J_r}(X) = 0. \]
\end{corollary}
\begin{definition} {\rm Define $n$
Laurent polynomials in $k \times (n-k)$ variables $X_{i,j} := X^{f_{i,j}}$
as follows:
\[ p_1(X) = a_{1,0} X^{u_{1,0}}, \;
p_i(X) = \sum_{j =0}^{n-k-1} a_{i,j}X^{u_{i,j}}, \; i =2, \ldots,k, \]
\[ p_{k+j}(X) = \sum_{i =1}^{k} b_{i,j}X^{v_{i,j}}, \; j =1, \ldots, n-k-1, \;
p_n(X) = b_{k,n-k}X^{v_{k,n-k}}, \]
where $a_{i,j}$ and $b_{l,m}$ are generically choosen complex numbers.
In particular, the Newton polyhedron of $p_i(X)$ is the convex hull of
$E_i$. }
\end{definition}
\begin{conjecture}
Let $I = \{1, \ldots, n\} = J_1 \cup \cdots \cup J_r$ be a partition
of $I$ into
a disjoint union of subsets $J_i \subset I$ with $|J_i| = d_i$ as in
$($\ref{J's}$)$ and $Y^*_0$ be a Calabi-Yau compactification of
a general complete intersection
in $({\bf C}^*)^{k(n-k)}$ defined by the equations
\[ 1 - \sum_{j \in J_i} p_j(X) = 0 \;\; ( i =1, \ldots, n), \]
where the coefficients
$a_{i,j}$ and $b_{l,m}$ satisfy the
following $(k-1)(n-k-1)$ conditions
\[ a_{k+1 -i, j-1} b_{k+1 -i,j} = a_{k+1 -i,j}b_{k-i,j}. \]
Then a minimal desingularization $X^*$ of $Y_0^*$
is a mirror of a generic Calabi-Yau complete
intersection $X:= X_{d_1, \ldots, d_r} \subset G(k,n)$.
\label{mirror-c}
\end{conjecture}
\begin{example}
{\rm If $X: =X_{1,1,3} \subset G(2,5)$, we take $J_1 = \{1\}$,
$J_2 = \{5\}$ and $J_3 = \{2,3,4 \}$. Then the mirror construction for $X$
proposed by \ref{mirror-c} coincides with the one considered
in 2.2. }
\end{example}
\subsection{Lax operators of Grassmannians}
In the paper \cite{EHX} Eguchi, Hori, and Xiong have computed
the Lax operator $L$ for various Fano manifolds V:
projective spaces, Del Pezzo surfaces and Grassmannians. In particular
for $V = {\bf P}^n$ the corresponding Lax operator $L$ is given by the
formula:
\[ L = X_1 + X_2 + \cdots + X_n + qX_1^{-1} X_2^{-1}
\cdots X_n^{-1}, \]
where $\log q $ is an element of $H_2({\bf P}^n)$.
On the other hand, if $Z$ is an affine hypersurface
defined by the equation
$ L(X_1, \ldots, X_n) = 1$
in the algebraic torus
$ T \cong ({\bf C}^*)^n = {\rm Spec}\,
{\bf C} [ X_1^{\pm 1}, \ldots, X_n^{\pm 1}]$,
then, according to \cite{Ba2},
a suitable compactification of $Z$ is a Calabi-Yau variety which
is mirror dual to Calabi-Yau hypersurfaces of degree $n+1$ in ${\bf P}^n$.
\begin{remark}
{\rm It is natural to suggest that the last observation
can be used
as a guiding principle for the construction of mirror manifolds
of Calabi-Yau hypersurfaces $X$ in Fano manifolds $V$.}
\end{remark}
Let $V$ be a Fano manifold of dimension $n$. Denote by $P$ (resp. by
$[V]$) the class of
unity (resp. the class of the normalized by unity volume form on $V$)
in the cohomology ring $H^*(V)$. Let
\[ \omega = \frac{dX_1}{X_1} \wedge \cdots \wedge \frac{dX_n}{X_n} \]
be the invariant differential $n$-form on the $n$-dimensional algebraic
torus $T \cong ({\bf C}^*)^n$. According to
\cite{EHX}, the Lax operator $L(X)$ of the Fano manifold $V$
is a Laurent polynomial in $X_1, \ldots, X_n$ with coefficients
in the group algebra ${\bf Q}[ H_2(V,{\bf Z})]$ satisfying
for all $m \geq 0$ the equation
\[ \langle \sigma_m([V])P) \rangle = \frac{1}{m+1}
\int_{\gamma} L^{m+1}(X) \omega. \]
where $\sigma_m([V])$ is the $m$-gravitational descendent of $[V]$ on
the moduli spaces of stable maps of curves of genus $g =0$ to $V$,
$ \langle \sigma_m([V])P) \rangle$ is the corresponding two point
correlator function, and $\gamma$ is the standard
generator of $H_n(T, {\bf Z})$.
For the case $V= G(r,s)$ $(n = r(s-r))$ the following was conjectured
in \cite{EHX}:
\begin{conjecture}
The Lax operator of the Grassmannian $G(r,s)$ has the following form
\[ L(X) = X_{[1,1]} +
\sum_{\begin{array}{c} {\scriptstyle 1 \leq a \leq s-r
} \\
{\scriptstyle 1 \leq b \leq r } \end{array} }
X_{[a,b]}^{-1}(X_{[a+1,b]} + X_{[a,b+1]})
+ q X_{[s-r,r]}^{-1}, \]
where $\log q \in H_2(G(r,s))$ and $X_{a,b} = 0$ if $a > s-r$ or $b > r$.
\label{lax}
\end{conjecture}
\begin{proposition}
Let $P(r,s)$ be the toric degeneration of the
Grassmannian $G(r,s)$. Then the equation $L(X) = 1$ defines
a $1$-parameter subfamily in the family of
toric mirrors of Calabi-Yau hypersurfaces in $P(r,s)$
$($see {\rm \cite{Ba2}}$)$.
\end{proposition}
{\it Proof:} According to \cite{Ba2}, we have to identify the Newton
polyhedron of the Laurent polynomial $L(X)$ in Conjecture \ref{lax} with
the reflexive polyhedron $\Delta(r,s)$. The latter follows immediately
from the explicit description of $\Delta(r,s)$ in \ref{polyh} and from the
$1$-to-$1$-correspondence
$f_{i,j} \leftrightarrow X_{[j,i]}$.\hspace*{\fill}\hbox{$\Box$}
\vskip 10pt
\begin{proposition}
The equations for the mirrors to
Calabi-Yau hypersurfaces conjectured in \ref{mirror-c} in
$G(r,s)$ coincide with the equations $L(X) =1$ where
$L(X)$ is the Lax operator conjectured for $G(r,s)$
in \cite{EHX}.
\end{proposition}
\noindent
{\it Proof:} It is easy to see that the coefficients of the polynomial
$L(X)$ satisfy all $r(s-r)$ monomial relations which reduce to the
equality $1 \cdot 1 = 1 \cdot 1$. On the other hand, using
the action of the $r(s-r)$-dimensional torus on the coefficients of the
Laurent polynomial
\[ 1 - (p_1(X) + \cdots + p_s(X)) \]
defining the mirror in \ref{mirror-c}, one can reduce to only one independent
parameter, for instance, the unique coefficient $b_{r,s-r}$ of $p_s(X) =
b_{r,s-r}X_{r,s-r}^{-1}$. By setting $q : = b_{r,s-r}$ and
$X_{[i,j]}: = X_{j,i}$, we can identify the variety $Y_0^*$ in
\ref{mirror-c} with a toric compactification of the affine hypersurface
$L(X) =1$.
\hspace*{\fill}\hbox{$\Box$}
\vskip 10pt
Using the explicit description of the multiplicative
structure of the small quantum cohomology of $G(k,n)$,
it is not difficult to check Conjecture \ref{lax} for each
given $r$ and $s$:
\begin{example}
{\rm The Lax operator of the
Grassmannian $G(2,4)$ is
\[ X_{[1,1]} + X_{[1,1]}^{-1}(X_{[2,1]} + X_{[1,2]}) +
X_{[2,1]}^{-1}X_{[2,2]} + X_{[1,2]}^{-1}X_{[2,2]}
+ q X_{[2,2]}^{-1}. \]
Its Newton polyhedron is isomorphic to $\Delta(2,4)$ from 2.1.}
\end{example}
\begin{example}
{\rm The Lax operator of the
Grassmannian $G(2,5)$ is
\[ X_{[1,1]} +
X_{[1,1]}^{-1}(X_{[2,1]} + X_{[1,2]}) +
X_{[2,1]}^{-1}( X_{[3,1]} + X_{[2,2]} ) +
X_{[1,2]}^{-1} X_{[2,2]} +
X_{[2,2]}^{-1} X_{[3,2]} + q X_{[3,2]}^{-1}. \]
Its Newton polyhedron is isomorphic to $\Delta(2,5)$ from 2.2.}
\end{example}
\section{Hypergeometric series}
\subsection{The Trick with the Factorials}\label{trick}
If $X$ is a Calabi-Yau the complete intersection of
hypersurfaces of degree $l_1, l_2, \ldots, l_r$ in ${\bf P}^n$, then the
generalised hypergeometric series
$$ \Phi_X(q)=\sum_{m=0}^{\infty}
\frac{(l_1m)!(l_2m)!\ldots(l_rm)!}{(m!)^{n+1}}q^m$$
is main period of its mirror $X^*$.
As is well-known, one can obtain the instanton numbers
for $X$ by a formal manipulation with this series,
see e.g. \cite{BS} and \ref{instanton}. More precisely, one
transforms the Picard-Fuchs differential operator $P$ annihilating the series
$\Phi_X$ to the form $D^2 \frac{1}{K(q)}D^2$
(where $D=q\partial/\partial q$) and reads off
the the $n_d$ from the power series expansion of the function $K$:\\
$$K(q)=l_1 l_2 \ldots l_r +\sum_{d=1}^{\infty} n_d d^3 \frac{q^d}{1-q^d}.$$
It is important to observe that the power
series $\Phi_X$ can be obtained from a power series
$$A_V(q)=\sum_{m=0}^{\infty} \frac{1}{(m!)^{n+1}}q^m$$
by the multiplication of its $m$-th
coefficient by the product
of factorials $(l_1m)!(l_2m)!\ldots(l_rm)!$.
On the other hand, the power series $A_V$ can be characterized
as the unique series
$A_V=1+\ldots$ solving the differential equation
$((q\frac{\partial}{\partial q})^{n+1}-q)A_V=0$
associated with the small quantum cohomology of ${\bf P}^n$.
This differential equation arizes as the reduction of the first order
differential {\em system}
$$q\frac{\partial}{\partial q} \vec{S} = p \circ \vec{S}$$
for a $H^*({\bf P}^n)$-valued function $\vec{S}=S_0+S_1p+\ldots+S_np^n,$
where $p \in H^2({\bf P}^{n+1})$ is an ample generator,
$\{1, p, p^2, \ldots, p^n\}$
is a basis for $H^*({\bf P}^n)$, and $p \circ$ is the operation of
{\em quantum multiplication} with $p$ in the small quantum cohomology of
${\bf P}^{n}$. Since it is well-known that the small
quantum cohomology ring of ${\bf P}^n$ is
defined by the relation $(p \circ)^{n+1} -q=0$, one finds immediately
comes to the differential equation.
In particular, we see that the function $A_V$ is uniquely determined
by the small quantum cohomology ring of $V={\bf P}^n$.
It is natural to try to use these ideas to obtain $\Phi_X$ from $A_V$
for varieties other than ${\bf P}^n$, for example
for Grassmannians or other Fano varieties.
If it works, this method allows one to
find instanton numbers without knowing an explicit mirror manifold.
We will formulate this trick in
some generality below.
Let $V$ be a smooth projective variety, which for reasons of
simplicity of exposition is assumed to have only even cohomology and
and that $H^2(V,{\bf Z} ) \cong H_2(V,{\bf Z}) \cong {\bf Z}$. Let
$p$ be the ample generator of $H^2(V, {\bf Z})$,
$\gamma$ a positive generator
for $H_2(V, {\bf Z})$. We denote by $1_V \in
H^0(V)$ the fundamental class of $V$ and by $<-,->$
the Poincar\'e pairing. The small quantum cohomology ring
$QH^*(V)$ of $V$ is the free ${\bf Q}[[q]]$-module $H^*(V,{\bf Q}[[q]])$
with a new multiplication $\circ$ determined by
$<A \circ B,C>=<A,B,C>=\sum_{m=0}^{\infty}<A,B,C>_mq^m>$
where $$<A,B,C>_m=I^V_{0,3,m\gamma}=\int_{[\overline{M}_{0,3}]}e^*_1(A)
\cup e_2^*(B) \cup e^*_3(C)$$
are the {\em $3$-point, genus 0, Gromov-Witten
invariants}, see \cite{FP}. The operator of quantum multiplication
with the ample generator $p \in H^2(V, {\bf Z})$ defines the
{\em Quantum Differential System}, see e.g. \cite{G1}:
$$\frac{\partial}{\partial t}\vec{S}=p \circ \vec{S}$$
where $\vec{S}$ is an series in the variable
$t=log $ with coefficients from $H^*(V,{\bf Q})$.
The {\em Quantum Cohomology ${\cal D}$-module}is the ${\cal D}$-module
generated by the top components
$<\vec{S}, 1_V>$ of all solutions $\vec{S}$ to the above differential system.
In the case under consideration, it will be of the form ${\cal D}/{\cal D} P$, for a
certain differential operator $P$.
\begin{definition}{\em
The {\em $A$-series of $V$} is the unique solution of the
Quantum Cohomology ${\cal D}$-module of the form
$A_V=\sum_{m=1}^{\infty} a_m q^m$
with $a_0=1$.
}
\end{definition}
Let $X$ be the intersection of hypersurfaces
of degree $l_1, l_2, \ldots, l_r$ in $V$.
In other words, $X$ is the zero-set
of a generic section of the decomposable bundle
${\cal E}:={\cal O}(l_1p)\oplus {\cal O}(l_2p)\oplus\ldots\oplus{\cal O}(l_rp)$.
\begin{definition}
{\em Let $A_V=\sum_{m=1}^{\infty} a_m q^m$ be the
$A$-series of a Fano manifolds $V$.
Define the {\em ${\cal E}$-modification of $A_V$} as
follows:
$$\Phi_{{\cal E}}(q): =\sum_{m=0}^{\infty} a_m \prod_{i=1}^r (m l_i)! q^m.$$
}
\end{definition}
\begin{definition}
{\em Assume that $X \subset V$ has trivial canonical class, i.e,
$X$ is a Calabi-Yau variety.
We say that the {\em Trick with the Factorials works}, if the
function $\Phi_{{\cal E}}$ is equal to the monodromy invariant period
$\Phi_X$ of the mirror family $X^*$ in some algebraic parametrization.}
\end{definition}
If the Trick with the Factorials works,
the usual formal manipulation (see \cite{BS}, \ref{instanton}) with
the series $\Phi_{{\cal E}}$ produces the instanton numbers for $X$!\\
\begin{remark}
{\rm (i) It is possible to formulate the Trick with the Factorials in much
greater generality \cite{K3}, \cite{BCKS}.
(ii) The $A$-series $A_V$ very well can be identically $1$,
but if $V$ is Fano, it will contain interesting information
and it is in such cases that the Trick with the Factorials
has a chance to work.
(iii) A better formulation uses instead of $A_V$ a certain
cohomology-valued series $S_V$, whose components make up a complete
solution set to the quantum ${\cal D}$-module. Instead of the
factorially modified series $\Phi_{{\cal E}}$
one has a factorially modified cohomological function $F_{{\cal E}}$.
We say that Trick with the Factorials works, if $S_V$ and
$F_{{\cal E}}$ differ by a coordinate change \cite{K3}, \cite{BCKS}.
Such a theorem is a form of the Lefschets hyperplane section theorem
in quantum cohomology.
(iv) Givental's mirror theorem for toric varieties, \cite{G3},
implies that the
Trick with the Factorials works for complete intersections in toric
varieties.
(v) More generally, it follows from a recent theorem of Kim,
\cite{K3}, that the
Trick with the Factorials works for arbitrary homogeneous spaces.
(vi) E. Tj{\o}tta has applied the Trick with the Factorials
succesfully in a
non-homogeneous case, \cite{tjotta}.}
\end{remark}
\subsection{Hypergeometric solutions for Grassmannians}
In this paragraph we apply the above ideas to the case of
Grassmannians.
In \cite{BCKS}, we describe a simple rule for writing down the
$GKZ$-hypergeometric series $A_{P(k,n)}$ assiciated with the
Gorenstein toric Fano variety
$P(k,n)$ in terms of the combinatorics of a certain
graph. Here we give a formula for $A_{P(k,n)}$ without going into
the details:
$$A_{P(k,n)}(q,\tilde{q})=
\sum_{s_{i,j}\ge 0}\frac{1}{(m!)^{n}}
\prod_{i=1}^{k-1}\prod_{j=1}^{n-k-1}{s_{i+1,j}
\choose s_{i,j}}{s_{i,j+1}\choose s_{i,j}}q^m \tilde{q}_{i,j}^{s_{i,j}}$$
where we put $s_{i,j}=m$ if $i > k-1$ or $j > n-k-1$.
\begin{example}
{\rm $G(2,5)$:
$$ A_{P(2,5)}(q, \tilde{q})=
\sum_{m,r,s \ge 0}\frac{1}{(m!)^5}{m \choose r}{s \choose r}
{m \choose s}^2q^m \tilde{q}_1^r\tilde{q}_2^s.$$
}
\end{example}
\vskip 10pt
\begin{example}
{\rm
$G(3,6)$ :
$$ A_{P(3,6)}(q, \tilde{q})=
\sum_{m,r,s,t,u}\frac{1}{(m!)^6}{s \choose r}{t \choose r}
{m \choose s}{u \choose s}{u \choose t}{m \choose t}
{m \choose u}^2 q^m \tilde{q}_1^r \tilde{q}_2^s \tilde{q}_3^u \tilde{q}_4^v.$$
}
\end{example}
\vskip 10pt
We conjecture an explicit general formula
for the series $A_{G(k,n)}(q)$ of an arbitrary Grassmannian:
\begin{conjecture}
Let $A_{P(k,n)}(q, \tilde{q})$ be the
A-hypergeometric series of the toric variety $\widehat{P(k,n)}$ as above. Then
\[A_{G(k,n)}(q) = A_{P(k,n)}(q, {\bf 1}). \]
\label{flagmirror}
\end{conjecture}
Using the explicit formulas for multiplication in the quantum cohomology
of Grassmannians \cite{Ber}, one can write down the
Quantum Differential System for $G(k,n)$ and reduce this
first order system to a higher
order differential equation satisfied by its components. In particular, one
can write down the differential operator $P$ annihilating the
component $<\vec{S},1>$ of any solution $\vec{S}$.
Below we record some of the (computer aided)
calculations of the operator $P$ we did ($D$ denotes the
operator $\partial/\partial t=q \partial/\partial q$):
$$
\begin{array}{|lcl|}
\hline
& & \\
G(2,4)&:&D^5-2q(2D+1)\\
& & \\
G(2,5)&:&D^7(D-1)^3-qD^3(11D^2+11D+3)-q^2 \\
& & \\
G(2,6)&:&D^9(D-1)^5-qD^5(2D+1)(13D^2+13D+4)\\
& & \\
& &-3q^2(3D+4)(3D+2)\\
& & \\
G(3,6)&:&D^{10}(D-1)^4-qD^4(65D^4+130D^3+105D^2+40D+6)\\
& & \\
& &+4q^2(4D+3)(4D+5)\\
& & \\
\hline
\end{array}
$$
The operator for $G(2,7)$ is:
$$
\begin{array}{c}
D^{11}(D-1)^7(D-2)^7(D-3)^7(D-4)^3\\ \\
-\frac{1}{3}qD^7(D-1)^7(D-2)^7(D-3)^3(173D^4+340D^3+272D^2+102D+15)\\ \\
-\frac{2}{9}q^2D^7(D-1)^7(D-2)^3(1129D^4+5032D^3+7597D^2+4773D+1083)\\ \\
+\frac{2}{9}q^3D^7(D-1)^3(843D^4+2628D^3+2353D^2+675D+6)\\ \\
-\frac{1}{9}q^4D^3(295D^4+608D^3+478D^2+174D+26)+\frac{1}{9}q^5,\\
\end{array}
$$
while the one for $G(2,8)$ takes about two pages.
Clearly, since both the structure of the quantum cohomology
ring and the hypergeometric series are very explicit, one
should seek a better way to prove Conjecture \ref{flagmirror}.
Nevertheless, using the above operators one obtains by direct computation
the following:
\begin{theorem} The conjecture \ref{flagmirror} is true for
$G(2,4)$, $G(2,5)$, $G(2,6)$, $G(2,7)$, $G(3,6)$. \hspace*{\fill}\hbox{$\Box$}
\label{grasscheck}
\end{theorem}
\section{Complete Intersection Calabi-Yau $3$-folds}
\subsection{Conifold transitions and mirrors}
Now we turn our attention to the main point of the
paper, namely the construction, via
conifold transitions, of mirrors for Calabi-Yau $3$-folds $X$
which are complete
intersections in Grassmannians $G(k,n)$.
By \ref{sing-l}, the singular locus of a generic $3$-dimensional complete intersection $X_0$
of $P(k,n)$ with
$r$ hypersurfaces $H_1, \ldots, H_r$ of degrees $d_1, \ldots, d_r$
($[H_i] = d_i[H]$, $i =1, \ldots, r$) consists of
\[ p = d_1 d_2 \cdots d_r \left( \sum_{i =1,\, j =1}^{k-1,\,n-k-1}
d(W_{i,j}) \right) \]
nodes, where $d(W_{i,j})$ is the degree of $W_{i,j}$ with respect to
the generator $H$ of the Picard group of $P(k,n)$.
On the other hand, by \ref{def-gr}, $X_0$ is a flat degeneration
of the smooth Calabi-Yau $3$-fold $X \subset G(k,n)$.
The small crepant resolution $\widehat{P}(k,n) \longrightarrow P(k,n)$ of
the ambient toric variety
induces a small crepant resolution $Y\longrightarrow X_0$. Hence $Y$ is a {\em smooth}
Calabi-Yau complete intersection in the toric variety $\widehat{P}(k,n)$,
which is obtained from $X$ by a {\em conifold transition}.
\begin{theorem} Let $p$ be the number of nodes
of $X_0$, and let $\alpha = (k-1)(n-k-1)$.
Then the Hodge numbers of $X$ and $Y$ are related by
$$h^{1,1}(Y)=h^{1,1}(X)+\alpha$$
and $$h^{2,1}(Y)=h^{2,1}(X)+\alpha -p.$$
\label{formul-2}
\end{theorem}
\noindent
{\it Proof:} By construction, $Y$ is a complete intersection of general sections of
big {\it semiample} line bundles on $\widehat{P}(k,n)$
(i.e., line bundles which are generated by global
sections and big). Using the explicit formula for
$h^{1,1}(Y)$ from (\cite{BB},
Corollary 8.3) and the fact that the only boundary lattice points
of $\Delta(k,n)$ are its vertices, we obtain the isomorphism
${\rm Pic}(Y)\cong{\rm Pic}(\widehat{P})$,
which gives the first relation.
On the other hand, the $p$ vanishing
3-cycles on X that shrink to nodes in
the degeneration must satisfy $\alpha$ linearly
independent relations by \cite{C}, and the second relation follows. \hspace*{\fill}\hbox{$\Box$}
\vskip 10pt
The mirror construction for complete intersection
Calabi-Yau manifolds in toric
varieties given in \cite{Ba2, BB} provides us
with the mirror family of Calabi-Yau manifolds $Y^*$.
The generic member of this family is nonsingular
(it is obtained by a MPCP-resolution of the ambient toric variety).
There is a natural isomorphism of the Hodge groups
$H^{1,1}(Y)\longrightarrow H^{2,1}(Y^*)$ (see \cite{Ba2, BB}).
During the conifold transition from $X$ to $Y$,
we have increased the "K\"ahler moduli", that is,
the rank of $H^{1,1}$. This says that we should really
look at the one-parameter subfamily of mirrors given by the subspace
of $H^{2,1}(Y^*)$ corresponding via the isomorphism above
to the divisors on $Y$ which come from $X$.
For this reason, the generalized hypergeometric series
$\Phi_X$ of $X^*$ is a specialization
of the monodromy invariant period integral of the mirror family $Y^*$
to the subfamily $Y_0^*$ defined in \ref{mirror-c}.
Let $\nabla_{J_1}, \ldots, \nabla_{J_r}$ be convex polyhedra as in
\ref{nablas}. Denote by $\nabla(k,n)$ the Minkowski sum of
$\nabla_{J_1}, \ldots, \nabla_{J_r}$. Then $\nabla(k,n)$ is
a reflexive polyhedron and ${\bf P}_{\nabla(k,n)}$ is
a Gorenstein toric Fano variety
defined by a nef-partition corresponding to the equation
\[ d_1 + \cdots + d_r = n. \]
\begin{conjecture}
After a MPCP-desingularization of the ambient toric variety
${\bf P}_{\nabla(k,n)}$,
the general member $Y_0^*$ of the special $1$-parameter subfamily
is a Calabi-Yau variety with the same number $p$
of nodes as $X_0$, satisfying
$\alpha -p$ relations. A small resolution
$X^*$ of $Y^*_0$ is a mirror of $X$.
\label{toricmirror}
\end{conjecture}
\begin{remark}
{\rm The statement \ref{toricmirror} has been easily checked for the two
simplest cases of Section 2, where the toric mirror construction reduces
to a hypersurface case. However, singularities of $Y_0^*$
are more difficult control for $4$ remaining cases which can not be
reduced to Calabi-Yau hypersurfaces in $4$-dimensional Gorenstein
toric Fano varieties.}
\end{remark}
\subsection{The computation of instanton numbers}\label{instanton}
We denote
by $X_{d_1,\ldots, d_r}\subset G(k,n)$ a
Calabi-Yau complete
intersection of $r$ hypersurfaces of degrees $d_1, \ldots, d_r$
with the Grassmannian $G(k,n) \subset {\bf P}^{{n \choose k} -1}$.
We denote by $Y$ the toric Calabi-Yau complete intersection
in $\widehat{P}(k,n)$ obtained by
a conifold transition via resolution of $p$ nodes on the
degeneration $X_0$ of $X$ $(h^{1,1}(X) = 1$), and by
$\alpha$ the number of relations satisfied by
the homology classes of the corresponding $p$ vanishing $3$-cycles on $X$.
Now we list all cases of Calabi-Yau complete intersection
$3$-folds $X$ in Grassmannians and collect the information about
topological invariants of $X$
and their conifold modifications $Y$.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|} \hline
$X$ & $h^{2,1}(X)$ &
$\chi(X)$ & $h^{1,1}(Y)$ & $h^{2,1}(Y)$ & $\chi(Y)$ & $\alpha$ & $p$ \\
\hline
$X_4 \subset G(2,4)$ & $89$ & $- 176$ & $2$ & $86$ & $-168$ & $1$ & $4$ \\
\hline
$X_{1,1,3} \subset G(2,5)$ & $76$ & $-150$ & $3$ & $72$ & $-138$ & $2$
& $6$ \\
\hline
$X_{1,2,2} \subset G(2,5)$ & $61$ & $-120$ & $3$ & $55$ & $ -104$ & $2$
& $8$ \\
\hline
$X_{1,1,1,1,2} \subset G(2,6)$ & $59$ & $-116$ & $4$ & $52$ & $-96$ & $3$
& $10$ \\
\hline
$X_{1, \ldots, 1} \subset G(2,7)$ & $50$ & $ -98$ & $5$ & $40$ & $-70$ & $4$
& $14$ \\
\hline
$X_{1, \ldots, 1} \subset G(3,6)$ & $49$ & $-96$ & $5$ & $37$ & $-64$ & $4$ &
$16$ \\
\hline
\end{tabular}
\end{center}
Recall the (standard) formal procedure used to compute
the instanton numbers. (More details can
be found e.g in \cite{BS}.)
We set
$$\Phi_X(z) := \sum_{m \geq 0} b_mz^m $$
to be the generalized hypergeometric series (with variable $z$)
corresponding to the
monodromy invariant period of the mirror $X^*$. As was explained in
\ref{trick}, one can start with the the $A$-series for the
grassmannian, and apply the Trick with the Factorials to find the
coefficients $b_m$.
Then $\Phi_X(z)$ satisfies a Picard-Fuchs differential equation
\[ P \Phi_X(z) = 0, \]
where
$P$ is a differential operator of order $4$ having a maximal unipotent
monodromy at $z= 0$. We compute $P$ by finding an explicit
recursion relation among coefficients $b_m$ of the
generalized hypergeometric series $\Phi_X(z)$. To bring $P$ into the
form $D^2 \frac{1}{K}D^2$, one has to
change the coordinate $z$ to $q = {\rm exp}(\Phi_1(z)/\Phi_X(z))$,
where $\Phi_1$ is the logarithmic solution of $P$. To obtain $K$, it is
convenient to use the Yukawa coupling. In the coordinate $z$ it has form
\[ K_{zzz} =
\frac{K_z^{(3)}}{\Phi_X^2(z)} \left( \frac{dz}{z} \right)^{\otimes 3}, \]
where $K_z^{(3)}$ is some rational function of $z$ that can be determined directly
from $P$. The Yukawa coupling in coordinate $q$ then is of the form
\[ K_{qqq} = K_q^{(3)}
\left( \frac{dq}{q} \right)^{\otimes 3}, \]
where
\[ K_q^{(3)} = n_0 + \sum_{m =1}^{\infty} n_m \frac{m^3 q^m}{1 -q^m} \]
and $n_m$ are the instanton numbers for rational curves of degree
$m$ on $X$.
From proposition \ref{grasscheck} and
Kim's Quantum Hyperplane Theorem (\cite{K3}), we have the following
\begin{theorem}
The virtual numbers of rational curves on a general complete intersection
Calabi-Yau three-fold in a Grassmannian are the ones listed in the
tables of the next section.
\end{theorem}
\section{Picard-Fuchs Operators and Yukawa Couplings}
{\tiny
\noindent
\subsection{$X_{1,1,3} \subset G(2,5)$}
\begin{center}
\begin{tabular}{|c|c|} \hline
& \\
$ b_m $ & ${\displaystyle
\frac{(m!)(m!)(3m)!}{(m!)^5}\sum_{r,s}{m\choose r}{s \choose r}
{m\choose s}^2} $ \\
& \\
\hline
& \\
${P}$ & $ D^4 - 3z (3 D +2)( 3 D + 1)
(11 D^2 + 11 D + 3) $\\
& $ - 9z^2 (3 D + 5)(3 D + 2)
(3 D + 4)(3 D + 1)$ \\
& \\
\hline
& \\
$ K_z^{(3)}$ & ${\displaystyle \frac{15}{1-11\cdot3^3z - 3^9z^2} }$ \\
& \\
\hline
& \\
$ n_m$ & $n_1 =540 ,\; n_2 =12555 , \; n_3 =621315 ,\; n_4 =44892765 ,\;
n_5 = 3995437590 $ \\
& \\
\hline
\end{tabular}
\end{center}
}
\noindent
{\tiny
\subsection{$X_{1,2,2} \subset G(2,5)$}
\begin{center}
\begin{tabular}{|c|c|} \hline
& \\
$ b_m $ & ${\displaystyle \frac{ (m!)(2 m)!)^2}{(m!)^5} \sum_{r,s }
{m\choose r}{s \choose r}{m\choose s}^2}$ \\
& \\
\hline
& \\
${P}$ & $ D^4 - 4z (11 D^2 + 11 D + 3)
(1 + 2 D)^2$\\
& $ - 16z^2 (2 D + 3)^2 (1 + 2 D)^2$ \\
& \\
\hline
& \\
$ K_z^{(3)}$ & ${\displaystyle \frac{20}{1 - 11 \cdot 2^4z - 2^8z^2} }$ \\
& \\
\hline
& \\
$ n_m$ & $n_1 =400 ,\; n_2 =5540 , \; n_3 = 164400,\; n_4 =7059880 ,\;
n_5 = 373030720 $ \\
& \\
\hline
\end{tabular}
\end{center}
}
{\small
The locus of conifold singularities in the toric variety $P(2,5)$ consists
of $2$ codimesion-$3$ toric strata of degree $1$. This gives $6$ nodes
on the generic complete intersection of type $(1,1,3)$ in
$P(2,5) \subset {\bf P}^{10}$ and $8$ nodes on the generic
complete intersection of type $(1,2,2)$ in
$P(2,5) \subset {\bf P}^{10}$
}
\noindent
\subsection{$X_{1,1,1,1,2} \subset G(2,6)$}
{\tiny
\begin{center}
\begin{tabular}{|c|c|} \hline
& \\
$ b_m $ & ${\displaystyle \frac{ (m!)^4(2 m)!)}{(m!)^6} \sum_{r,s,t}
{m\choose r}{s\choose r}{m\choose s}{t\choose s}{m\choose t}^2}$\\
& \\
\hline
& \\
${P}$ & $ D^4 - 2z (4 + 13 D + 13 D^2)(1 +
2 D)^2 $\\
& $ -12 z^2 (3 D +2)( 2 D + 3)(1 + 2 D)( 3 D + 4)$ \\
& \\
\hline
& \\
$ K_z^{(3)}$ & ${\displaystyle \frac{28}{1-26\cdot2^2z -
27\cdot 2^4z^2} }$ \\
& \\
\hline
& \\
$ n_m$ & $n_1 =280 ,\; n_2 =2674 , \; n_3 =48272 ,\; n_4 = 1279040,\;
n_5 = 41389992 $ \\
& \\
\hline
\end{tabular}
\end{center}
}
{\small
The locus of conifold singularities in the toric variety $P(2,6)$ consists
of $2$ codimesion-$3$ toric strata of degree $2$ and
$1$ codimension-$3$ toric stratum of degree $1$. This gives $10$ nodes
on the generic complete intersection of type $(1,1,1,1,2)$ in
$P(2,6) \subset {\bf P}^{14}$.
}
\noindent
\subsection{$X_{1,1,1,1,1,1,1} \subset G(2,7)$}
{\tiny
\begin{center}
\begin{tabular}{|c|c|} \hline
& \\
$ b_m $ & ${\displaystyle \frac{ (m!)^7}{(m!)^7} \sum_{r,s,t,u }
{m\choose r}{s\choose r}{m\choose s}{t\choose s}
{m\choose t}{u\choose t}{m\choose u}^2}$\\
& \\
\hline
& \\
${P}$ & $ 9 D^4 - 3z (15 + 102 D + 272 D^2 +
340 D^3 + 173 D^4) $\\
& $ - 2z^2 (1083 + 4773 D + 7597 D^2 + 5032 D^3 +
1129 D^4)$ \\
& $ + 2z^3 (6 + 675 D + 2353 D^2 + 2628 D^3 + 843 D^4)$ \\
& $ - z^4(26 + 174 D + 478 D^2 + 608 D^3 + 295 D^4) +
z^5 ( D + 1)^4 $ \\
& \\
\hline
& \\
$ K_z^{(3)}$ & ${\displaystyle \frac{42-14z}{1-57z -289z^2 + z^3} }$ \\
& \\
\hline
& \\
$ n_m$ & $n_1 = 196,\; n_2 =1225 , \; n_3 =12740 ,\; n_4 =198058 ,\;
n_5 = 3716944 $ \\
& \\
\hline
\end{tabular}
\end{center}
}
{\small
The locus of conifold singularities in the toric variety $P(2,7)$ consists
of $2$ codimesion-$3$ toric strata of degree $2$ and
$2$ codimension-$3$ toric stratum of degree $5$. This gives $14$ nodes
on the generic complete intersection of type $(1,1,1,1,1,1)$ in
$P(2,7) \subset {\bf P}^{20}$.
}
\noindent
\subsection{$X_{1,1,1,1,1,1} \subset G(3,6)$}
{\tiny
\begin{center}
\begin{tabular}{|c|c|} \hline
& \\
$ b_m $ & ${\displaystyle \frac{ (m!)^6}{(m!)^6} \sum_{r,s,t,u }
{s \choose r}{t \choose r}
{m \choose s}{u \choose s}{u \choose t}{m \choose t}
{m \choose u}^2}$\\
& \\
\hline
& \\
${P}$ & $ D^4 - z (6 + 40 D + 105 D^2 +
130 D^3 + 65 D^4) $\\
& $ +4z^2 (4 D + 5)(4 D +3)( D + 1)^2$ \\
& \\
\hline
& \\
$ K_z^{(3)}$ & ${\displaystyle \frac{42}{1 - 65 z - 64z^2} }$ \\
& \\
\hline
& \\
$ n_m$ & $n_1 =210 ,\; n_2 =1176 , \; n_3 =13104 ,\; n_4 = 201936 ,\;
n_5 =3824016 $ \\
& \\
\hline
\end{tabular}
\end{center}
}
{\small
The locus of conifold singularities in the toric variety $P(3,6)$ consists
of $2$ codimesion-$3$ toric strata of degree $2$ and
$2$ codimension-$3$ toric strata of degree $6$. This gives $16$ nodes
on the generic complete intersection of type $(1,1,1,1,1,1)$ in
$P(3,6) \subset {\bf P}^{19}$.
}
\vskip .2truein
\noindent
\section{Acknowledgement}
\noindent
We would like to thank
S. Katz, S.-A. Str{\o}mme, E. R{\o}dland and E. Tj{\o}tta
for helpful discussions and
the Mittag-Leffler Institute for hospitality.
The second and third named authors
have been supported by Mittag-Leffler Institute postdoctoral fellowships.
\newpage
\vskip .2truein
|
1998-03-03T08:56:28 | 9710 | alg-geom/9710007 | en | https://arxiv.org/abs/alg-geom/9710007 | [
"alg-geom",
"math.AG"
] | alg-geom/9710007 | null | Shu Kawaguchi and Atsushi Moriwaki | Inequalities for semistable families of arithmetic varieties | Version 3.0 (75 pages), the new version of the paper titled "Relative
Bogomolov's inequality in the arithmetic case" | null | null | null | null | In this paper, we will consider a generalization of Bogomolov's inequality
and Cornalba-Harris-Bost's inequality to semistable families of arithmetic
varieties under the idea that geometric semistability implies a certain kind of
arithmetic positivity. The first one is an arithmetic analogue of the relative
Bogomolov's inequality proved by the second author. We also establish the
arithmetic Riemann-Roch formulae for stable curves over regular arithmetic
varieties and generically finite morphisms of arithmetic varieties.
| [
{
"version": "v1",
"created": "Tue, 7 Oct 1997 03:36:41 GMT"
},
{
"version": "v2",
"created": "Thu, 20 Nov 1997 00:47:59 GMT"
},
{
"version": "v3",
"created": "Tue, 3 Mar 1998 07:56:27 GMT"
}
] | 2007-05-23T00:00:00 | [
[
"Kawaguchi",
"Shu",
""
],
[
"Moriwaki",
"Atsushi",
""
]
] | alg-geom | \section*{Introduction}
\renewcommand{\theTheorem}{\Alph{Theorem}}
In this paper, we will consider a generalization
of Bogomolov's inequality and Cornalba-Harris-Bost's inequality
to the case of semistable families of arithmetic varieties.
An underlying idea of these inequalities as in
\cite{Bo}, \cite{BGS}, \cite{Ga}, \cite{MiBi}, \cite{MoBG},
\cite{MoABG}, \cite{MoBU}, \cite{MorFh}, \cite{SoVan},
and \cite{Zh}
is that geometric semistability implies a certain kind of
arithmetic positivity.
The first one is related to the semistability of vector bundles,
and the second one involves the Chow (or Hilbert) semistability
of cycles.
\medskip
First of all, let us consider Bogomolov's inequality.
Let $X$ and $Y$ be smooth algebraic varieties over
an algebraically closed field of characteristic zero, and
$f : X \to Y$ a semi-stable curve.
Let $E$ be a vector bundle of rank $r$ on $X$,
and $y$ a point of $Y$.
In \cite{MoRB}, the second author proved that if
$f$ is smooth over $y$ and $\rest{E}{X_{\bar{y}}}$
is semistable, then $\operatorname{dis}_{X/Y}(E) = f_* \left(
2r c_2(E) - (r-1)c_1^2(E) \right)$
is weakly positive at $y$.
In the first half of this paper, we would like to consider an
arithmetic analogue of the above result.
Let us fix regular arithmetic varieties $X$ and $Y$,
and a semistable curve $f : X \to Y$.
Since we have a good dictionary for
translation from a geometric case to an arithmetic case,
it looks like routine works.
There are, however, two technical difficulties
to work over the standard dictionary.
The first one is how to define a push-forward
of arithmetic cycles
in our situation. If $f_{{\mathbb{Q}}} : X_{{\mathbb{Q}}} \to Y_{{\mathbb{Q}}}$
is smooth, then,
according to Gillet-Soul\'{e}'s arithmetic intersection theory
\cite{GSArInt}, we can get the push-forward
$f_* : \widehat{\operatorname{CH}}^{p+1}(X) \to \widehat{\operatorname{CH}}^p(Y)$.
We would not like to restrict ourselves to
the case where $f_{{\mathbb{Q}}}$ is smooth because
in the geometric case,
the weak positivity of $\operatorname{dis}_{X/Y}(E)$ gives
wonderful applications to analyses of
the boundary of the moduli space of stable curves.
Thus the usual push-forward
for arithmetic cycles is insufficient
for our purpose.
A difficulty in defining the push-forward arises from a fact:
if $f_{{\mathbb{C}}} : X({\mathbb{C}}) \to Y({\mathbb{C}})$ is not smooth,
then $(f_{{\mathbb{C}}})_*(\eta)$ is not necessarily $C^{\infty}$
even for a $C^{\infty}$ form $\eta$.
This suggests us that we need to extend the usual arithmetic
Chow groups defined by Gillet-Soul\'{e} \cite{GSArInt}.
For this purpose, we will introduce
an arithmetic $L^1$-cycle of
codimension $p$, namely, a pair $(Z, g)$ such that
$Z$ is a cycle of codimension $p$,
$g$ is a current of type $(p-1,p-1)$, and
$g$ and $dd^c(g) + \delta_{Z({\mathbb{C}})}$ are
represented by locally integrable forms.
Thus, dividing by the usual arithmetical rational
equivalence, an arithmetic Chow group,
denoted by $\widehat{\operatorname{CH}}_{L^1}^p$, consisting
of arithmetic $L^1$-cycles of codimension $p$ will be defined
(cf. \S\ref{subsec:var:arith:chow}).
In this way, we have the natural push-forward
\[
f_* : \widehat{\operatorname{CH}}_{L^1}^{p+1}(X) \to \widehat{\operatorname{CH}}_{L^1}^p(Y)
\]
as desired (cf. Proposition~\ref{prop:push:forward:arith:cycle}).
The second difficulty is the existence of
a suitable Riemann-Roch formula in our situation.
As before, if $f_{{\mathbb{Q}}} : X_{{\mathbb{Q}}} \to Y_{{\mathbb{Q}}}$ is smooth,
we have the arithmetic Riemann-Roch theorem due to
Gillet-Soul\'{e} \cite{GSRR}. If we ignore Noether's formula,
then, under the assumption that
$f_{{\mathbb{Q}}} : X_{{\mathbb{Q}}} \to Y_{{\mathbb{Q}}}$ is smooth,
their Riemann-Roch formula can be written in the following form:
\begin{multline*}
\widehat{{c}}_1 \left( \det Rf_*(E), h_Q^{\overline{E}} \right) -
\operatorname{rk} (E) \widehat{{c}}_1 \left( \det Rf_*({\mathcal{O}}_X), h_Q^{\overline{{\mathcal{O}}}_X}
\right) \\ = f_* \left( \frac{1}{2} \left(
\widehat{{c}}_1 (\overline{E})^2 -
\widehat{{c}}_1 (\overline{E}) \cdot \widehat{{c}}_1 (\overline{\omega}_{X/Y}) \right)
- \widehat{{c}}_2 (\overline{E})
\right)
\end{multline*}
where $\overline{E} = (E, h)$ is
a Hermitian vector bundle on $X$ and
$\overline{\omega}_{X/Y}$ is the dualizing sheaf of $f : X \to Y$
with a Hermitian metric.
If we consider a general case
where $f_{{\mathbb{Q}}} : X_{{\mathbb{Q}}} \to Y_{{\mathbb{Q}}}$
is not necessarily smooth,
the right hand side in the
above equation is well defined and
sits in $\widehat{\operatorname{CH}}_{L^1}^1(X)_{{\mathbb{Q}}}$.
On the other hand, the left hand side is rather complicated.
If we admit singular fibers of $f_{{\mathbb{C}}} : X({\mathbb{C}}) \to Y({\mathbb{C}})$,
then the Quillen metric $h_Q^{\overline{E}}$ is no longer
$C^{\infty}$. According to \cite{BBQm},
it extends to a generalized metric. Thus, we may define
$\widehat{{c}}_1 \left( \det Rf_*(E), h_Q^{\overline{E}} \right)$
(cf. \S\ref{subsec:arith:div:gen:metric}).
In general, this cycle is not an $L^1$-cycle.
However, using Bismut-Bost's formula \cite{BBQm},
we can see that
\[
\widehat{{c}}_1 \left( \det Rf_*(E), h_Q^{\overline{E}} \right) -
\operatorname{rk} (E) \widehat{{c}}_1 \left( \det Rf_*({\mathcal{O}}_X), h_Q^{\overline{{\mathcal{O}}}_X}
\right)
\]
is an element of $\widehat{\operatorname{CH}}_{L^1}^1(Y)$. Thus, we have a way to establish
a Riemann-Roch formula in the arithmetic Chow group
$\widehat{\operatorname{CH}}_{L^1}^1(Y)_{{\mathbb{Q}}}$. Actually,
we will prove the above formula in our situation
(cf. Theorem~\ref{thm:arith:Riemann:Roch:stable:curves}).
The idea of comparing two sides in $\widehat{\operatorname{CH}}_{L^1}^1(Y)_{{\mathbb{Q}}}$
is the tricky Lemma~\ref{lem:criterion:linear:equiv:B:cycle}.
Let us go back to our problem.
First of all, we need to define an arithmetic analogue of
weak positivity.
Let $\alpha$ be an element of $\widehat{\operatorname{CH}}_{L^1}^1(Y)_{{\mathbb{Q}}}$, $S$ a subset of $Y({\mathbb{C}})$,
and $y$ a closed point of $Y_{{\mathbb{Q}}}$.
We say $\alpha$ is semi-ample at $y$ with respect to $S$ if
there are an arithmetic $L^1$-cycle $(E, f)$ and a positive integer
$n$ such that
(1) $dd^c(f) + \delta_{E({\mathbb{C}})}$ is $C^{\infty}$ around each $z \in S$,
(2) $E$ is effective,
(3) $y \not\in \operatorname{Supp}(E)$,
(4) $f(z) \geq 0$ for all $z \in S$, and
(5) $n \alpha$ coincides with the class of $(E, f)$ in $\widehat{\operatorname{CH}}_{L^1}^1(Y)_{{\mathbb{Q}}}$.
Moreover, $\alpha$ is said to be weakly positive at $y$ with respect to $S$
if it is the limit of semi-ample cycles at $y$ with respect to $S$
(for details, see \S\ref{subsec:wp:div}).
For example, if $Y = \operatorname{Spec}(O_K)$, $y$ is the generic point, and
$S = Y({\mathbb{C}})$, then,
$\alpha$ is weakly positive at $y$ with respect to $S$
if and only if $\widehat{\operatorname{deg}}(\alpha) \geq 0$,
where $K$ is a number field and $O_K$ is the ring of integers in $K$
(cf. Proposition~\ref{prop:wp:for:curve}).
Let $(E, h)$ be a Hermitian vector bundle of rank $r$ on $X$, and
$\widehat{\operatorname{dis}}_{X/Y}(E, h)$ the arithmetic discriminant divisor of
$(E, h)$ with respect
to $f : X \to Y$, that is, the element of $\widehat{\operatorname{CH}}_{L^1}^1(Y)$ given by
$f_* \left( 2r \widehat{{c}}_2(E, h) - (r-1)\widehat{{c}}_1(E, h)^2 \right)$.
We assume that $f$ is smooth over $y$ and $\rest{E}{X_{\bar{y}}}$ is
poly-stable.
In the case where $\dim X = 2$ and $Y = \operatorname{Spec}(O_K)$,
Miyaoka \cite{MiBi}, Moriwaki \cite{MoBG,MoABG,MoBU}, and Soul\'{e} \cite{SoVan}
proved that $\widehat{\operatorname{deg}} \left( \widehat{\operatorname{dis}}_{X/Y}(E, h) \right) \geq 0$, consequently,
$\widehat{\operatorname{dis}}_{X/Y}(E, h)$ is weakly positive at $y$ with respect to $Y({\mathbb{C}})$.
One of the main theorems of this paper is the following generalization.
\begin{Theorem}[cf. Theorem~\ref{thm:relative:Bogomolov:inequality:arithmetic:case}]
\label{thm:A:relative:Bogomolov:inequality}
Under the above assumptions, $\widehat{\operatorname{dis}}_{X/Y}(E, h)$
is weakly positive at $y$ with respect to any subsets $S$ of $Y({\mathbb{C}})$
with the following
properties: \textup{(1)} $S$ is finite, and
\textup{(2)} $f_{{\mathbb{C}}}^{-1}(z)$ is smooth and $\rest{E_{{\mathbb{C}}}}{f_{{\mathbb{C}}}^{-1}(z)}$ is
poly-stable for all $z \in S$.
In particular, if the residue field of $x$ is $K$,
and the canonical morphism $\operatorname{Spec}(K) \to X$
induced by $x$ extends to $\tilde{x} : \operatorname{Spec}(O_K) \to X$,
then $\widehat{\operatorname{deg}}\left( \tilde{x}^*\left(\widehat{\operatorname{dis}}_{X/Y}(E, h)\right)\right) \geq 0$.
\end{Theorem}
\medskip
Next, let us consider Cornalba-Harris-Bost's inequality.
Motivated by the work of Cornalba and Harris \cite{CoHa}
in the geometric case,
Bost \cite[Theorem~I]{Bo} proved that, roughly speaking,
if $X(\overline{{\mathbb{Q}}}) \subset {\mathbb{P}}^{r-1}(\overline{{\mathbb{Q}}})$ has
the $\operatorname{\mathbf{SL}}_r(\overline{{\mathbb{Q}}})$ semi-stable Chow point,
then the height of $X$ has a certain kind of positivity.
We call this result Cornalba-Harris-Bost's inequality.
Zhang \cite{Zh} then gave precision to it and
also showed the converse of Bost's result.
Further, Gasbarri \cite{Ga} considered a wide range of actions
instead of $\operatorname{\mathbf{SL}}_r(\overline{{\mathbb{Q}}})$-action.
In the second half of this paper,
we would like to consider a relative version of
Cornalba-Harris-Bost's inequality.
First, let us fix a terminology.
Let $V$ be a set, $\phi$ a non-negative function on $V$, and
$S$ a finite subset of $V$.
We define the geometric mean $\operatorname{g.\!m.}(\phi; S)$ of
$\phi$ over $S$ to be
\[
\operatorname{g.\!m.}(\phi; S) = \left( \prod_{s \in S} \phi(s) \right)^{1/\#(S)}.
\]
Then, the following is our solution.
\begin{Theorem}[cf. Theorem~\ref{thm:semistability:imply:average:semi-ampleness}]
\label{thm:intro:B}
Let $Y$ be a regular projective arithmetic variety, and
$\overline{E} = (E,h)$ a Hermitian vector bundle of rank $r$.
Let $\pi : {\mathbb{P}}(E) = \operatorname{Proj} (\bigoplus_{n \geq 0} \operatorname{Sym}^n(E^{\lor})) \to Y$
be the projection and $\overline{{\mathcal{O}}_{E}(1)}$
the tautological line bundle with the quotient metric induced from $f^*(h)$.
Let $X$ be an effective cycle in ${\mathbb{P}} (E)$
such that $X$ is flat over $Y$
with the relative dimension $d$
and degree $\delta$ on the generic fiber.
For each irreducible component $X_i$ of $X_{red}$,
let $\tilde{X}_i \to X_i$ be a proper birational morphism
such that $(\tilde{X}_i)_{{\mathbb{Q}}}$ is smooth over ${\mathbb{Q}}$.
Let $Y_0$ be the maximal open set of $Y$ such that
the induced morphism $\tilde{X}_i \to Y$ is smooth over $Y_0$
for every $i$.
Let $(B, h_B)$ be a line bundle equipped with
a generalized metric on $Y$ given by the equality:
\[
\widehat{{c}}_1(B, h_B) =
r \pi_* \left( \widehat{{c}}_1(\overline{{\mathcal{O}}_{E}(1)})^{d+1} \cdot (X,g_X) \right)
+ \delta (d+1) \widehat{{c}}_1(\overline{E}).
\]
\textup{(}Here we postpone the definition of $g_X$, i.e.,
a suitable compactification of $X$ in the arithmetic sense.\textup{)}
Then, $h_B$ is $C^{\infty}$ over $Y_0$. Moreover,
there are a positive integer
$e=e(r,d,\delta)$, a positive integer $l=l(r,d,\delta)$,
a positive constant $C=C(r,d,\delta)$, and
sections $s_1, \ldots, s_l \in H^0(Y, B^{\otimes e})$
with the following properties.
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item
$e$, $l$, and $C$ depend only on $r$, $d$, and $\delta$.
\item
For a closed point $y$ of $Y_{{\mathbb{Q}}}$, if $X_y$ is Chow semistable,
then $s_i(y) \not= 0$ for some $i$.
\item
For all $i$ and all closed points $y$ of $(Y_0)_{{\mathbb{Q}}}$,
\[
\operatorname{g.\!m.}\left( \left( h_B^{\otimes e} \right)(s_i, s_i);\
O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}(y)\right)
\leq C,
\]
where $O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}(y)$ is the orbit of $y$
by the Galois action in $Y_0(\overline{{\mathbb{Q}}})$.
\end{enumerate}
\end{Theorem}
Compared with the geometric analogue
(cf. Remark~\ref{rem:geom:analog:Cornalba-Harris-Bost}),
the difficult part of this theorem is
the estimate of the geometric mean of the norm over
the Galois orbits of closed points.
We will do this by reducing to the absolute case.
For this purpose, we have to associate $X$ with a `nice' Green current $g_X$.
How do we do? One way is to fix a K\"{a}hler metric
$\mu \in A^{1,1}({\mathbb{P}}(E)_{{\mathbb{R}}})$ and
to attach a $\mu$-normalized Green current for $X$,
namely, a Green current $g$ such that
$dd^c g + \delta_X = H(\delta_Y)$ and $H(g_Y) = 0$,
where $H : D^{p,p}({\mathbb{P}}(E)_{{\mathbb{R}}}) \to H^{p,p}({\mathbb{P}}(E)_{{\mathbb{R}}})$ is
the harmonic projection (cf. \cite[2.3.2]{BGS}).
This construction however is not suitable for our purpose
because it does not behave well
when restricted on fibers.
Thus we are led to define an $\Omega$-normalized Green form
which is given, roughly speaking, by attaching a Green form fiberwisely
(Here $\Omega = {c}_1(\overline{{\mathcal{O}}_{E}(1)})$).
Precisely, an $\Omega$-normalized Green form $g_X$ for $X$ is
characterized by the following three conditions;
(i) $g_X$ is an $L^1$-form on ${\mathbb{P}}(E)$,
(ii) $dd^c([g_X]) + \delta_X
= \sum_{i=0}^{d} \left[ \pi^*(\gamma_i) \wedge \Omega^i \right]$.
where $\gamma_i$ is a $d$-closed $L^1$-form of type
$(d-i,d-i)$
on $Y$ ($i=0, \ldots, d$).
(iii) $\pi_*(g_X \wedge \Omega^{r - d}) = 0$
(cf. Proposition~\ref{prop:normalized:Green:form}).
Then we can show that it has a desired property
when restricted on fibers
(cf. Remark~\ref{rem:norm:Green:general:fiber}).
Suppose now $X$ is regular.
Let $i : X \to {\mathbb{P}}(E)$ be the inclusion map
and $f : X \to Y$ the restriction of $\pi$.
If we set $\overline{L} = i^*(\overline{{\mathcal{O}}_{E}(1)})$,
then
$\pi_* ( \widehat{{c}}_1(\overline{{\mathcal{O}}_{E}(1)})^{d+1} \cdot (X,g_X) )
= f_*(\widehat{{c}}_1(\overline{L})^{d+1})$
(cf. Proposition~\ref{prop:when:Bost:divisor:smooth}).
Since $f_*(\widehat{{c}}_1(\overline{L})^{d+1})$ is in general
only an element of $\widehat{\operatorname{CH}}^1_{L^1}(Y)$,
the above equality explains why we need to consider $(X,g_X)$ in
the enlarged arithmetic Chow group $\widehat{\operatorname{CH}}^{r-d-1}_{L^1}({\mathbb{P}}(E))$.
Moreover, a similar equality when $X$ is not necessarily regular
shows that
$\pi_* ( \widehat{{c}}_1(\overline{{\mathcal{O}}_{E}(1)}) \cdot (X,g_X) )$
is independent of the choice of an $\Omega$-normalized Green
form $g_X$ for $X$
(cf. Proposition~\ref{prop:when:Bost:divisor:smooth}).
Suppose now $Y=\operatorname{Spec}(O_K)$, $y$ is the generic point, and
$X_y$ is Chow semistable,
where $K$ is a number field.
In this case,
there exists a generic resolution of $X$
smooth over $y$. Then Theorem~\ref{thm:intro:B} tells us that
\[
r \widehat{\operatorname{deg}} ( \widehat{{c}}_1(\overline{L})^{d+1} )
+ \delta (d+1) \widehat{\operatorname{deg}} (\overline{E})
+ [K:{\mathbb{Q}}] \alpha(r,d,\delta) \geq 0
\]
for some constant $\alpha(r,d,\delta)$ depending only on
$r$, $d$ and $\delta$,
which is nothing but Theorem~I of Bost \cite{Bo}.
We can also think a wide range of actions like \cite{Ga}.
Namely, let $\rho : \operatorname{GL}_r \to \operatorname{GL}_R$ be a morphism of group schemes
such that there is an integer $k$ with $\rho(t I_r) = t^k I_R$ for any $t$,
and that $\rho$ commutes with the transposed morphism.
For a Hermitian vector bundle $\overline{E}$,
we then get the associated Hermitian vector bundle $\overline{E}^{\rho}$
(cf. \S\ref{subsec:associated:herm:vb}).
If $X$ is a flat cycle on ${\mathbb{P}}(E^{\rho})$
and $y$ is a closed point of $Y_{{\mathbb{Q}}}$,
then $\operatorname{\mathbf{SL}}_r(\overline{{\mathbb{Q}}})$ acts a Chow form ${\Phi_X}_y$.
The stability of ${\Phi_X}_y$ under this action yields a similar inequality
(cf. Theorem~\ref{thm:semistability:imply:average:semi-ampleness}).
Finally, in \S\ref{section:Bogomolov:to:Bost}
we make a comparison between
the relative Bogomolov's inequality
(Theorem~\ref{thm:relative:Bogomolov:inequality:arithmetic:case})
and the relative Cornalba-Harris-Bost's inequality
(Theorem~\ref{thm:semistability:imply:average:semi-ampleness}).
\renewcommand{\theTheorem}{\arabic{section}.\arabic{subsection}.\arabic{Theorem}}
\section{Locally integrable forms and their push-forward}
\subsection{Locally integrable forms}
\setcounter{Theorem}{0}
Let $M$ be an $n$-dimensional orientable differential manifold.
We assume that $M$ has a countable basis of open sets.
Let $\omega$ be a $C^{\infty}$ volume element of $M$, and
$C_c^0(M)$ the set of all complex valued continuous functions on $M$
with compact supports.
Then, there is a unique Radon measure $\mu_{\omega}$
defined on the topological $\sigma$-algebra of $M$ such that
\[
L\!\!\!\int_M f d\mu_{\omega} = \int_M f \omega
\]
for all $f \in C_c^0(M)$, where
${\displaystyle L\!\!\!\int_M f d\mu_{\omega}}$ is the Lebesgue integral arising
from the measure $\mu_{\omega}$.
Let $f$ be a complex valued
function on $M$. We say $f$ is {\em locally integrable},
denoted by $f \in L^1_{\operatorname{loc}}(M)$, if $f$ is measurable
and, for any compact sets $K$,
\[
L\!\!\!\int_K |f| d\mu_{\omega} < \infty.
\]
Let $\omega'$ be another $C^{\infty}$ volume form on $M$.
Then, there is a positive $C^{\infty}$ function $a$ on $M$
with $\omega' = a \omega$.
Thus,
\[
L\!\!\!\int_K |f| d\mu_{\omega'} = L\!\!\!\int_K |f| a d\mu_{\omega},
\]
which shows us that local integrability does not depend on the choice
of the volume form $\omega$. Moreover, it is easy to see that,
for a measurable complex valued function $f$ on $M$,
the following are equivalent.
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item
$f$ is locally integrable.
\item
For each point $x \in M$, there is an open neighborhood
$U$ of $x$ such that the closure of $U$ is compact and
${\displaystyle L\!\!\!\int_U |f| d\mu_{\omega} < \infty}$.
\end{enumerate}
Let $\Omega_M^p$ be a $C^{\infty}$ vector bundle consisting
of $C^{\infty}$ complex valued $p$-forms.
Let $\pi_p : \Omega_M^p \to M$ be the canonical map.
We denote $C^{\infty}(M, \Omega_M^p)$
(resp. $C^{\infty}_c(M, \Omega_{M}^p)$) by $A^p(M)$ (resp. $A_c^p(M)$).
Let $\alpha$ be a section of $\pi_p : \Omega_M^p \to M$.
We say $\alpha$ is
{\em locally integrable}, or simply an {\em $L^1$-form}
if, at any point of $M$,
all coefficients of $\alpha$ in terms of
local coordinates are locally integrable functions.
The set of all locally integrable $p$-forms is denoted by
$L^1_{\operatorname{loc}}(M, \Omega^p_M)$.
For an maximal form $\alpha$ on $M$,
there is a unique function $g$ on $M$ with
$\alpha = g \omega$.
We denote this function $g$ by $c_{\omega}(\alpha)$.
Let us define the Lebesgue integral of locally integrable
$n$-forms with compact support.
Let $\alpha$ be an element of $L^1_{\operatorname{loc}}(M, \Omega^n_M)$ such that
the support of $\alpha$ is compact.
Then $c_{\omega}(\alpha) \in L^1_{\operatorname{loc}}(M)$ and
$\operatorname{supp}(c_{\omega}(\alpha))$ is compact. Thus,
${\displaystyle L\!\!\!\int_{M} c_{\omega}(\alpha) d\mu_{\omega}}$ exists.
Let $\omega'$ be another $C^{\infty}$ volume element of $M$.
Then, there is a positive $C^{\infty}$ function $a$ on $M$
with $\omega' = a \omega$. Here $a c_{\omega'}(\alpha) = c_{\omega}(\alpha)$.
Thus,
\[
L\!\!\!\int_{M} c_{\omega'}(\alpha) d\mu_{\omega'} =
L\!\!\!\int_{M} c_{\omega'}(\alpha) a d\mu_{\omega} =
L\!\!\!\int_{M} c_{\omega}(\alpha) a d\mu_{\omega}.
\]
Hence, ${\displaystyle L\!\!\!\int_{M} c_{\omega}(\alpha) d\mu_{\omega}}$
does not depend on the choice of the volume form $\omega$.
Thus, the Lebesgue integral of $\alpha$ is defined by
\[
L\!\!\!\int_M \alpha = L\!\!\!\int_{M} c_{\omega}(\alpha) d\mu_{\omega}.
\]
Moreover, we denote by $D^p(M)$ the space of
currents of type $p$ on $M$.
Then, there is the natural homomorphism
\[
[\ ] : L^1_{\operatorname{loc}}(M, \Omega^p_M) \to D^p(M)
\]
given by ${\displaystyle [\alpha](\phi) = L\!\!\!\int_M \alpha \wedge \phi}$
for $\phi \in A_c^{n-p}(M)$.
It is well known that the kernel of $[ \ ]$ is
$\{ \alpha \in L^1_{\operatorname{loc}}(M, \Omega^p_M) \mid
\alpha = 0 \ (\operatorname{a.e.}) \}$.
A topology on $D^p(M)$ is defined in the following way.
For an sequence $\{ T_n \}_{n=1}^{\infty}$ in $D^p(M)$,
$T_n \to T$ as $n \to \infty$ if and only if
$T_n(\phi) \to T(\phi)$ as $n \to \infty$
for each $\phi \in A_c^{n-p}(M)$.
For an element $T \in D^n(M)$,
by abuse of notation,
we denote by $c_{\omega}(T)$
a unique distribution $g$ on $M$ given by
$T = g \omega$.
\begin{Proposition}
\label{prop:criterion:loc:int}
Let $T$ be a current of type $p$ on $M$.
Then, the following are equivalent.
\begin{enumerate}
\renewcommand{\labelenumi}{(\arabic{enumi})}
\item
$T$ is represented by a $L^1$-form.
\item
For any $\phi \in A^{n-p}(M)$,
$c_{\omega}(T \wedge \phi)$ is represented by
a locally integrable function.
\end{enumerate}
\end{Proposition}
{\sl Proof.}\quad
(1) $\Longrightarrow$ (2):
Let $\phi \in A^{n-p}(M)$.
Then, by our assumption, for any point $x \in M$,
there are an open neighborhood $U$ of $x$,
$C^{\infty}$ functions $a_1, \ldots, a_r$ on $U$, and
locally integrable functions $b_1, \ldots, b_r$ on $U$ such that
\[
\rest{c_{\omega}(T \wedge \phi)}{U} = \sum_{i=1}^r [a_i b_i].
\]
Thus, if $K$ is a compact set in $U$, then
\[
L\!\!\!\int_K \left|\sum_{i=1}^r a_i b_i \right| d\mu_{\omega} \leq
L\!\!\!\int_K \sum_{i=1}^r |a_i| |b_i| d\mu_{\omega} \leq
\max_i \sup_{x \in K} \{ |a_i(x)| \} \sum_{i=1}^r L\!\!\!\int_K |b_i|
d\mu_{\omega} < \infty.
\]
Thus, we get (2).
\medskip
(2) $\Longrightarrow$ (1):
Before starting the proof, we would like to claim
the following fact.
Let $\{ U_{\alpha} \}_{\alpha \in A}$ be an open covering of $M$
such that $A$ is at most a countable set.
Let $\lambda_{\alpha}$ be a locally integrable form $U_{\alpha}$
with $\lambda_{\alpha} = \lambda_{\beta} \ (\operatorname{a.e.})$ on
$U_{\alpha} \cap U_{\beta}$ for
all $\alpha, \beta \in A$. Then, there is a locally integrable
form $\lambda$ on $M$ such that $\lambda = \lambda_{\alpha} \ (\operatorname{a.e.})$
on $U_{\alpha}$ for all $\alpha \in A$.
Indeed, let us fix a map $a : M \to A$ with $x \in U_{a(x)}$
and define a form $\lambda$ by $\lambda(x) = \lambda_{a(x)}(x)$.
Then, $\lambda$ is our desired form because
for each $\alpha \in A$,
\[
\{ x \in U_{\alpha} \mid \lambda(x) \not= \lambda_{\alpha}(x) \}
\subseteq
\bigcup_{\beta \in A \setminus \{ \alpha \}}
\{ x \in U_{\alpha} \cap U_{\beta} \mid
\lambda_{\beta}(x) \not= \lambda_{\alpha}(x) \}
\]
and the right hand side has measure zero.
\medskip
Let $U$ be an open neighborhood of a point $x \in M$ and
$(x_1, \ldots, x_n)$ a local coordinate of $U$ such that
$dx_1 \wedge \cdots \wedge dx_n$ coincides with the orientation
by $\omega$.
Then, there is a positive $C^{\infty}$ function $a$ on $U$
with $\omega = a dx_1 \wedge \cdots \wedge dx_n$ over $U$.
We set
\[
T = \sum_{i_1 < \cdots < i_p }
T_{i_1 \cdots i_p} dx_{i_1} \wedge \cdots \wedge dx_{i_p}
\]
for some distributions $T_{i_1 \cdots i_p}$.
We need to show that $T_{i_1 \cdots i_p}$
is represented by a locally integrable function.
Since $M$ has a countable basis of open sets, by the above claim,
it is sufficient to check that
$T_{i_1 \cdots i_p}$ is represented by
an integral function on every compact set $K$ in $U$.
Let $f$ be a non-negative $C^{\infty}$ function on $M$ such that
$f = 1$ on $K$ and $\operatorname{supp}(f) \subset U$.
Choose $i_{p+1}, \ldots, i_{n}$ such that
$\{ i_1, \ldots, i_{n} \} = \{ 1, \ldots, n \}$.
Here we set $\phi = f a dx_{i_{p+1}} \wedge \cdots \wedge dx_{i_n}$.
Then, $\phi \in A^{n-p}(M)$ and
\[
T \wedge \phi =
\epsilon T_{i_1 \cdots i_p} f a dx_{1} \wedge \cdots \wedge dx_{n}
= \epsilon T_{i_1 \cdots i_p} f \omega,
\]
where $\epsilon = 1$ or $-1$ depending on
the orientation of $\{ x_{i_1}, \ldots, x_{i_n} \}$.
By our assumption, there is a locally integrable function
$h$ on $M$ with $c_{\omega}(T \wedge \phi) = [h]$.
Thus, $[\epsilon h] = T_{i_1 \cdots i_p} f$.
Therefore, $T_{i_1 \cdots i_p}$ is represented by
$\epsilon h$ on $K$ because $f = 1$ on $K$.
Thus, we get (2).
\QED
\subsection{Push-forward of $L^1$-forms as current}
\label{subsec:push:forward:L1:current}
\setcounter{Theorem}{0}
First of all, we recall the push-forward of currents.
Let $f : M \to N$ be a proper morphism
of orientable manifolds with the relative dimension $d = \dim M - \dim N$.
Then,
\[
f_* : D^p(M) \to D^{p-d}(N)
\]
is defined by $(f_*(T))(\phi) = T(f^*(\phi))$ for
$\phi \in A_c^{\dim N - p + d}(N)$.
It is easy to see that $f_*$ is a continuous homomorphism.
Let us begin with the following lemma.
\begin{Lemma}
\label{lem:push:forward:product}
Let $F$ be an orientable compact differential manifold and
$Y$ an orientable differential manifold. Let $\omega_F$
\textup{(}resp. $\omega_Y$\textup{)} be
a $C^{\infty}$ volume element of $F$
\textup{(}resp. $Y$\textup{)}.
Let $p : F \times Y \to Y$
be the projection to the second factor.
Then, we have the following.
\begin{enumerate}
\renewcommand{\labelenumi}{(\arabic{enumi})}
\item
If $g$ is a continuous function on $F \times Y$,
then $\int_F g \omega_F$ is a continuous function on $Y$.
\item
If $\alpha$ is a continuous maximal form on $F \times Y$,
then $p_*([\alpha])$ is represented by
a unique continuous from.
This continuous form is denoted by $\int_p \alpha$.
\item
For a continuous function $g$ on $F \times Y$,
\[
\left| c_{\omega_Y}\left( \int_{p} g \omega_F \wedge \omega_Y \right)
\right|
\leq c_{\omega_Y} \left( \int_{p} |g| \omega_F \wedge \omega_Y \right).
\]
\end{enumerate}
\end{Lemma}
{\sl Proof.}\quad
(1) This is standard.
\medskip
(2) Since $\omega_F \wedge \omega_Y$ is a volume form on
$F \times Y$, there is a continuous function $g$ on
$F \times Y$ with $\alpha = g \omega_F \wedge \omega_Y$.
Thus, it is sufficient to show that
\[
p_*([\alpha]) = \left[ \left(\int_F g \omega_F \right) \omega_Y \right].
\]
Indeed, by Fubini's theorem,
for $\phi \in A_c^0(Y)$,
\[
p_*([\alpha])(\phi) = \int_{F \times Y} \phi \alpha
= \int_Y \left( \int_F g \omega_F \right) \phi \omega_Y
= \left[ \left(\int_F g \omega_F \right) \omega_Y \right](\phi).
\]
\medskip
(3) This is obvious because
\[
\left| \int_F g \omega_F \right| \leq
\int_F |g| \omega_F.
\]
\QED
\begin{Corollary}
\label{cor:cont:ineq:integral:fiber}
Let $f : X \to Y$ be a proper, surjective and smooth morphism
of connected complex manifolds. Let $\omega_X$ and
$\omega_Y$ be volume elements of $X$ and $Y$ respectively. Then,
\begin{enumerate}
\renewcommand{\labelenumi}{(\arabic{enumi})}
\item
For a continuous maximal form $\alpha$ on $X$,
$f_*([\alpha])$ is represented by a unique continuous form.
We denote this continuous form by $\int_{f} \alpha$.
\item
For any continuous functions $g$ on $X$,
\[
\left| c_{\omega_Y}\left( \int_{f} g \omega_X \right) \right|
\leq c_{\omega_Y} \left( \int_{f} |g| \omega_X \right).
\]
\end{enumerate}
\end{Corollary}
{\sl Proof.}\quad
(1)
This is a local question on $Y$.
Thus, we may assume that there are a compact complex manifold
$F$ and a differomorphism $h : X \to F \times Y$
such that the following diagram is commutative:
\[
\begin{CD}
X @>{\sim}>{h}> F \times Y \\
@V{f}VV @VV{p}V \\
Y @= Y,
\end{CD}
\]
where $p : F \times Y \to Y$ is the natural projection.
Hence, (1) is a consequence of (2) of Lemma~\ref{lem:push:forward:product}.
\medskip
(2)
First, we claim that if the above inequality holds for some special
volume elements $\omega_X$ and $\omega_Y$, then
the same inequality holds for any volume elements.
Let $\omega'_{X}$ and $\omega'_{Y}$ be another volume elements
of $X$ and $Y$ respectively. We set $\omega'_X = a \omega_X$ and
$\omega'_Y = b \omega_Y$. Then, $a$ and $b$ are positive $C^{\infty}$
functions.
Let $g$ be any continuous function on $X$. Then, by our assumption,
\[
\left| c_{\omega_Y}\left( \int_{f} g \omega'_X \right) \right| =
\left| c_{\omega_Y}\left( \int_{f} g a \omega_X \right) \right| \leq
c_{\omega_Y} \left( \int_{f} |g| a \omega_X \right) =
c_{\omega_Y} \left( \int_{f} |g| \omega'_X \right).
\]
On the other hand, for any maximal forms $\alpha$ on $Y$,
\[
c_{\omega_Y}\left( \alpha \right) =
b c_{\omega'_Y}\left( \alpha \right).
\]
Thus, we get our claim.
Hence, as in the proof of (1),
using the differomorphism $h$ and
(3) of Lemma~\ref{lem:push:forward:product},
we can see (2).
\QED
\begin{Remark}
In the situation of Corollary~\ref{cor:cont:ineq:integral:fiber},
if $\alpha$ is a $C^{\infty}$-form on $X$,
then $f_*([\alpha])$ is represented by a unique $C^{\infty}$-form.
\end{Remark}
\begin{Proposition}
\label{prop:loc:int:integral:fiber}
Let $f : X \to Y$ be a proper and surjective morphism
of connected complex manifolds.
Let $U$ be a non-empty Zariski open set of $Y$ such that
$f$ is smooth over $U$.
Let $\alpha$ be a compactly supported continuous maximal form on $X$.
If we set
\[
\lambda =
\begin{cases}
{\displaystyle \int_{f^{-1}(U) \to U} \alpha}
& \text{on $U$}, \\
{} & {} \\
0 & \text{on $Y \setminus U$,}
\end{cases}
\]
then $\lambda$ is integrable.
Moreover, $f_*([\alpha]) = [\lambda]$.
\end{Proposition}
{\sl Proof.}\quad
Let $\omega_X$ and $\omega_Y$ be volume forms of $X$ and $Y$
respectively. Let $h$ be a function on $Y$ with
$\lambda = h \omega_Y$. Then, $h$ is continuous on $U$ by
Corollary~\ref{cor:cont:ineq:integral:fiber}.
Moreover, let $g$ be a continuous function on $X$ with
$\alpha = g \omega_X$.
We need to show that
$h$ is an integrable function.
First note that $\int_X |g| \omega_X < \infty$
because $g$ is a compactly supported continuous function.
Let $\{ U_n \}_{n=1}^{\infty}$ be a sequence of
open sets such that $\overline{U_n} \subset U$,
$\overline{U_n}$ is compact,
$U_1 \subseteq U_2 \subseteq \cdots \subseteq U_n \subseteq \cdots$, and
$\bigcup_{n=1}^{\infty} U_n = U$.
Here we set
\[
h_n(y) = \begin{cases}
|h(y)| & \text{if $y \in U_n$} \\
0 & \text{otherwise}.
\end{cases}
\]
Then, $0 \leq h_1 \leq h_2 \leq \cdots \leq h_n \leq \cdots$ and
${\displaystyle \lim_{n \to \infty} h_n(y) = |h(y)|}$.
By Corollary~\ref{cor:cont:ineq:integral:fiber},
\[
\left| \rest{h}{U} \right| \leq
c_{\omega_{Y}}\left(
\int_{f^{-1}(U) \to U} |g| \omega_X \right).
\]
Thus,
\begin{align*}
\int_{U_n} |h| \omega_Y
& \leq \int_{U_n}
c_{\omega_{Y}}\left(
\int_{f^{-1}(U_n) \to U_n}
|g| \omega_X \right) \omega_Y
= \int_{U_n} \int_{f^{-1}(U_n) \to U_n}
|g| \omega_X \\
& = \int_{f^{-1}(U_n)}
|g| \omega_X \leq \int_X |g| \omega_X.
\end{align*}
Therefore,
\[
L\!\!\!\int_{Y} h_n d\mu_{\omega_Y} = \int_{U_n} |h| \omega_Y \leq
\int_{X}
|g| \omega_X < \infty.
\]
Thus, by Fatou's theorem,
\[
L\!\!\!\int_{Y} |h| d\mu_{\omega_Y} =
\lim_{n \to \infty} L\!\!\!\int_{Y} h_n d\mu_{\omega_Y}
\leq
L\!\!\!\int_{X}
|g| \omega_X < \infty.
\]
Hence, $h$ is integral.
Let $\phi$ be any element of $A^{0}_c(Y)$.
Then, since
${\displaystyle \lim_{n \to \infty} \mu_{\omega_Y}(Y \setminus U_n) = 0}$
and $h \phi$ is integrable,
by the absolute continuity of Lebesgue integral,
\[
\lim_{n \to \infty} L\!\!\!\int_{Y \setminus U_n} h \phi
d\mu_{\omega_Y} = 0.
\]
Thus,
\begin{align*}
L\!\!\!\int_Y \lambda \phi &=
\lim_{n \to \infty} \left(
L\!\!\!\int_{U_n} h \phi d\mu_{\omega_{Y}} +
L\!\!\!\int_{Y \setminus U_n} h \phi d\mu_{\omega_{Y}}
\right) \\
& = \lim_{n \to \infty}
L\!\!\!\int_{U_n} h \phi d\mu_{\omega_{Y}}
= \lim_{n \to \infty}
\int_{U_n} h \phi \omega_Y
= \lim_{n \to \infty} \int_{U_n} \lambda \phi.
\end{align*}
In the same way,
\[
\int_X \alpha f^*(\phi) =
\lim_{n \to \infty}
\int_{f^{-1}(U_n)} \alpha f^*(\phi).
\]
On the other hand, we have
\[
\int_{U_n} \lambda \phi =
\int_{f^{-1}(U_n)} \alpha f^*(\phi).
\]
Hence
\begin{align*}
f_*([\alpha])(\phi) & = [\alpha](f^*(\phi))
= \int_X \alpha \wedge f^*(\phi)
= \lim_{n \to \infty} \int_{f^{-1}(U_n)} \alpha f^*(\phi) \\
& = \lim_{n \to \infty} \int_{U_n} \lambda \phi
= L\!\!\!\int_{Y} \lambda \phi
= [\lambda](\phi)
\end{align*}
Therefore, $f_*([\alpha]) = [\lambda]$.
\QED
Let $X$ be an equi-dimensional complex manifold, i.e.,
every connected component has the same dimension.
We denote by $A^{p,q}(X)$ the space of $C^{\infty}$ complex valued
$(p,q)$-forms on $X$. Let $A^{p,q}_c(X)$ be the subspace
of compactly supported forms.
Let $D^{p,q}(X)$ be the space of currents on $X$ of type $(p,q)$.
As before, there is a natural homomorphism
\[
[\ ] : L^1_{\operatorname{loc}}(\Omega_{X}^{p,q}) \to D^{p,q}(X).
\]
Then, as a corollary of Proposition~\ref{prop:loc:int:integral:fiber},
we have the following main result of this section.
\begin{Proposition}
\label{prop:push:forward:B:pq}
Let $f : X \to Y$ be a proper morphism of equi-dimensional complex manifolds.
We assume that every connected component of $X$ maps surjectively to
a connected component of $Y$.
Let $\alpha$ be an $L^1$-form of type $(p+d,q+d)$ on $X$, where
$d = \dim X - \dim Y$.
Then there is a $\lambda \in L^1_{\operatorname{loc}}(\Omega_Y^{p,q})$
with $f_*([\alpha]) = [\lambda]$.
\end{Proposition}
{\sl Proof.}\quad
Clearly we may assume that $Y$ is connected.
Since $f$ is proper, there are finitely many connected components
of $X$, say, $X_1, \ldots, X_e$.
If we set $\alpha_i = \rest{\alpha}{X_i}$ and $f_i = \rest{f}{X_i}$ for
each $i$, then
$f_*([\alpha]) = (f_1)_*([\alpha_1]) + \cdots + (f_e)_*([\alpha_e])$.
Thus, we may assume that $X$ is connected.
Further, since $f_*([ \alpha \wedge f^*(\phi) ]) = f_*([\alpha]) \wedge \phi$
for all $\phi \in A^{\dim Y - p, \dim Y - q}(Y)$,
we may assume that $\alpha$ is a maximal form
by Proposition~\ref{prop:criterion:loc:int}.
Let $g$ be a locally integrable function on $X$ with
$\alpha = g \omega_X$. Since the question is local with respect to $Y$,
we may assume that $g$ is integrable. Thus,
since $C_c^0(Y)$ is dense on $L^1(Y)$ (cf. \cite[Theorem~3.14]{Ru}),
there is a sequence
$\{ g_n \}_{n=1}^{\infty}$ of compactly supported continuous functions on $X$
such that
\[
\lim_{n \to \infty} L\!\!\!\int_X |g_n - g| d\mu_{\omega_X} = 0.
\]
By Proposition~\ref{prop:loc:int:integral:fiber}, for each $n$,
there is an integrable function $h_n$ on $Y$
such that $f_*([g_n \omega_X]) = [h_n \omega_Y]$.
Moreover, by (2) of Corollary~\ref{cor:cont:ineq:integral:fiber},
\[
|h_n - h_m| \leq c_{\omega_Y} \left(
\int_{f^{-1}(U) \to U} |g_n - g_m | \omega_X \right)
\]
over $U$. Thus, we can see
\[
L\!\!\!\int_Y |h_n - h_m| d\mu_{\omega_Y} \leq
L\!\!\!\int_X |g_n - g_m| d\mu_{\omega_X}
\]
for all $n, m$.
Hence, $\{ h_n \}_{n=1}^{\infty}$ is a Cauchy sequence
in $L^1(Y)$. Therefore,
by the completeness of $L^1(Y)$,
there is an integrable function $h$ on $Y$
with $h = \lim_{n \to \infty} h_n$ in $L^1(Y)$.
Then, for any $\phi \in A_c^{0,0}(Y)$,
\[
\lim_{n \to \infty} L\!\!\!\int_Y h_n \phi \omega_Y =
L\!\!\!\int_Y h \phi \omega_Y
\quad\text{and}\quad
\lim_{n \to \infty} L\!\!\!\int_X g_n f^*(\phi) \omega_X =
L\!\!\!\int_X g f^*(\phi) \omega_X.
\]
Thus,
\begin{align*}
f_*([\alpha])(\phi) & =
L\!\!\!\int_X g f^*(\phi)\omega_X =
\lim_{n \to \infty} L\!\!\!\int_X g_n f^*(\phi) \omega_X \\
& = \lim_{n \to \infty} L\!\!\!\int_Y h_n \phi \omega_Y =
L\!\!\!\int_Y h \phi \omega_Y = [h\omega_Y](\phi).
\end{align*}
Therefore, $f_*([\alpha]) = [h\omega_Y]$.
\QED
\section{Variants of arithmetic Chow groups}
\subsection{Notation for arithmetic varieties}
\label{subsec:notation:arith:variety}
\setcounter{Theorem}{0}
An {\em arithmetic variety $X$} is
an integral scheme which is flat and quasi-projective
over $\operatorname{Spec}({\mathbb{Z}})$, and
has the smooth generic fiber $X_{{\mathbb{Q}}}$.
Let us consider the ${\mathbb{C}}$-scheme $X \otimes_{{\mathbb{Z}}} {\mathbb{C}}$.
We denote the underlying analytic space of $X \otimes_{{\mathbb{Z}}} {\mathbb{C}}$
by $X({\mathbb{C}})$. We may view $X({\mathbb{C}})$ as the set of all
${\mathbb{C}}$-valued points of $X$.
Let $F_{\infty} : X({\mathbb{C}}) \to X({\mathbb{C}})$ be the anti-holomorphic
involution given by the complex conjugation.
For an arithmetic variety $X$,
every $(p,p)$-form $\alpha$ on $X({\mathbb{C}})$ is
always assumed to be compatible with $F_{\infty}$, i.e.,
$F_{\infty}^*(\alpha) = (-1)^p \alpha$.
Let $E$ be a locally free sheaf on $X$ of finite rank, and
$\pi : \pmb{E} \to X$ the vector bundle associated with $E$, i.e.,
$\pmb{E} = \operatorname{Spec}\left( \bigoplus_{n=0}^{\infty} \operatorname{Sym}^n(E) \right)$.
As before, we have the analytic space $\pmb{E}({\mathbb{C}})$ and
the anti-holomorphic involution $F_{\infty} : \pmb{E}({\mathbb{C}}) \to \pmb{E}({\mathbb{C}})$.
Then, $\pi_{{\mathbb{C}}} : \pmb{E}({\mathbb{C}}) \to X({\mathbb{C}})$ is a holomorphic
vector bundle on $X({\mathbb{C}})$, and the following diagram is commutative:
\[
\begin{CD}
\pmb{E}({\mathbb{C}}) @>{F_{\infty}}>> \pmb{E}({\mathbb{C}}) \\
@V{\pi_{{\mathbb{C}}}}VV @VV{\pi_{{\mathbb{C}}}}V \\
X({\mathbb{C}}) @>>{F_{\infty}}> X({\mathbb{C}})
\end{CD}
\]
Here note that
$F_{\infty} : \pmb{E}({\mathbb{C}}) \to \pmb{E}({\mathbb{C}})$
is anti-complex linear at each fiber.
Let $h$ be a $C^{\infty}$ Hermitian metric of $\pmb{E}({\mathbb{C}})$.
We can think $h$ as a $C^{\infty}$ function on
$\pmb{E}({\mathbb{C}}) \times_{X({\mathbb{C}})} \pmb{E}({\mathbb{C}})$.
For simplicity, we denote by $F_{\infty}^*(h)$
the $C^{\infty}$ function
$\left( F_{\infty} \times_{X({\mathbb{C}})} F_{\infty} \right)^*(h)$
on $\pmb{E}({\mathbb{C}}) \times_{X({\mathbb{C}})} \pmb{E}({\mathbb{C}})$.
Then, $\overline{F_{\infty}^*(h)}$ is
a $C^{\infty}$ Hermitian metric of $\pmb{E}({\mathbb{C}})$.
We say {\em $h$ is invariant under $F_{\infty}$}
if $F_{\infty}^*(h) = \overline{h}$.
Moreover, the pair $(E, h)$ is called {\em a Hermitian vector bundle on $X$}
if $h$ is invariant under $F_{\infty}$.
Note that even if $h$ is not invariant under $F_{\infty}$,
$h + \overline{F_{\infty}^*(h)}$ is an invariant metric.
\subsection{Variants of arithmetic cycles}
\setcounter{Theorem}{0}
\label{subsec:var:arith:chow}
Let $X$ be an arithmetic variety.
We would like to define three types of arithmetic cycles,
namely, arithmetic $A$-cycles, arithmetic $L^1$-cycles, and
arithmetic $D$-cycles.
In the following definition,
$g$ is compatible with $F_{\infty}$ as mentioned in
\S\ref{subsec:notation:arith:variety}.
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item
(arithmetic $A$-cycle on $X$ of codimension $p$) :
a pair $(Z, g)$ such that
$Z$ is a cycle on $X$ of codimension $p$ and
$g$ is represented by a Green form $\phi$ of $Z({\mathbb{C}})$, namely,
$\phi$ is a $C^{\infty}$ form on $X({\mathbb{C}}) \setminus
\operatorname{Supp}(Z({\mathbb{C}}))$ of logarithmic type along $\operatorname{Supp}(Z({\mathbb{C}}))$
with $dd^c([\phi]) + \delta_{Z({\mathbb{C}})} \in A^{p,p}(X({\mathbb{C}}))$.
\item
(arithmetic $L^1$-cycle on $X$ of codimension $p$) :
a pair $(Z, g)$ such that
$Z$ is a cycle on $X$ of codimension $p$ and,
there are $\phi \in L^1_{\operatorname{loc}}(\Omega_{X({\mathbb{C}})}^{p-1,p-1})$ and
$\omega \in L^1_{\operatorname{loc}}(\Omega_{X({\mathbb{C}})}^{p,p})$ with
$g = [\phi]$ and $dd^c(g) + \delta_{Z({\mathbb{C}})} = [\omega]$.
\item
(arithmetic $D$-cycle on $X$ of codimension $p$) :
a pair $(Z, g)$ such that
$Z$ is a cycle on $X$ of codimension $p$ and
$g \in D^{p-1,p-1}(X({\mathbb{C}}))$.
\end{enumerate}
The set of all arithmetic $A$-cycles (resp.
$L^1$-cycles, $D$-cycles) of codimension $p$
is denoted by $\widehat{Z}_A^p(X)$ (resp.
$\widehat{Z}_{L^1}^p(X)$, $\widehat{Z}_D^p(X)$).
Let $\widehat{R}^p(X)$ be the subgroup of $\widehat{Z}^p(X)$ generated
by the following elements:
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item
$((f), - [\log |f|^2])$,
where $f$ is a rational function on some
subvariety $Y$ of codimension $p-1$ and $[\log |f|^2]$
is the current defined by
\[
[\log |f|^2](\gamma) =
L\!\!\!\int_{Y({\mathbb{C}})} (\log |f|^2)\gamma.
\]
\item
$(0, \partial(\alpha) + \bar{\partial}(\beta))$,
where $\alpha \in D^{p-2, p-1}(X({\mathbb{C}}))$,
$\beta \in D^{p-1, p-2}(X({\mathbb{C}}))$.
\end{enumerate}
Here we define
\[
\begin{cases}
\widehat{\operatorname{CH}}_A^p(X) = \widehat{Z}_A^{p}(X)/\widehat{R}^p(X) \cap \widehat{Z}_A^{p}(X), \\
\widehat{\operatorname{CH}}_{L^1}^p(X) = \widehat{Z}_{L^1}^{p}(X)/\widehat{R}^p(X) \cap \widehat{Z}_{L^1}^{p}(X), \\
\widehat{\operatorname{CH}}_D^p(X) = \widehat{Z}_D^{p}(X)/\widehat{R}^p(X).
\end{cases}
\]
\begin{Proposition}
\label{prop:AChow:equal:Chow}
The natural homomorphism $\widehat{\operatorname{CH}}_A^p(X) \to \widehat{\operatorname{CH}}^p(X)$
is an isomorphism.
\end{Proposition}
{\sl Proof.}\quad
Let $(Z, g) \in \widehat{Z}^p(X)$.
By \cite[Theorem~1.3.5]{GSArInt}, there is
a Green form $g_Z$ of $Z({\mathbb{C}})$.
Then, $dd^c(g - [g_Z]) \in A^{p,p}(X({\mathbb{C}}))$.
Hence, by \cite[Theorem~1.2.2]{GSArInt},
there are $a \in A^{d,d}(X({\mathbb{C}}))$ and
$v \in \operatorname{Image}(\partial) + \operatorname{Image}(\bar{\partial})$
with $g - [g_Z] = [a] + v$.
Since $g - [g_Z]$ is compatible with $F_{\infty}$,
replacing $a$ and $v$ by $(1/2)(a + (-1)^pF_{\infty}^*(a))$ and
$(1/2)(v + (-1)^p F_{\infty}^*(v))$ respectively,
we may assume that $a$ and $v$ are compatible with $F_{\infty}$.
Here, $g_Z + a$ is a Green form of $Z$. Thus,
$(Z, [g_Z + a]) \in \widehat{Z}_A^p(X)$.
Moreover, since
$(Z, g) - (Z, [g_Z + a]) \in \widehat{R}^p(X)$,
our proposition follows.
\QED
Let $f : X \to Y$ be a proper morphism of arithmetic varieties
with $d = \dim X - \dim Y$. Then, we have a homomorphism
\[
f_* : \widehat{Z}_D^{p+d}(X) \to \widehat{Z}_D^{p}(Y)
\]
defined by $f_*(Z, g) = (f_*(Z), f_*(g))$.
In the same way as in the proof of \cite[Theorem~3.6.1]{GSArInt},
we can see $f_*(\widehat{R}^{p+d}(X)) \subseteq \widehat{R}^p(Y)$.
Thus, the above homomorphism induces
\[
f_* : \widehat{\operatorname{CH}}_D^{p+d}(X) \to \widehat{\operatorname{CH}}_D^{p}(Y).
\]
Then we have the following.
\begin{Proposition}
\label{prop:push:forward:arith:cycle}
If $f$ is surjective, then
$f_* : \widehat{\operatorname{CH}}_D^{p+d}(X) \to \widehat{\operatorname{CH}}_D^{p}(Y)$ gives rise to
\[
f_* : \widehat{\operatorname{CH}}_{L^1}^{p+d}(X) \to \widehat{\operatorname{CH}}_{L^1}^{p}(Y).
\]
In particular, we have the homomorphism
$f_* : \widehat{\operatorname{CH}}^{p+d}(X) \to \widehat{\operatorname{CH}}_{L^1}^{p}(Y)$.
\end{Proposition}
{\sl Proof.}\quad
Clearly we may assume that $p \geq 1$.
It is sufficient to show that if $(Z, g) \in \widehat{Z}_{L^1}^{p+d}(X)$,
then $(f_*(Z), f_*(g)) \in \widehat{Z}_{L^1}^{p}(Y)$.
By the definition of $L^1$-arithmetic cycles,
$g$ and $dd^c(g) + \delta_{Z({\mathbb{C}})}$ are
represented by $L^1$-forms.
Thus, by Proposition~\ref{prop:push:forward:B:pq},
there is an $\omega \in L^1_{\operatorname{loc}}(\Omega_{Y({\mathbb{C}})}^{p,p})$ with
\[
f_*\left( dd^c(g) + \delta_{Z({\mathbb{C}})} \right) = [\omega].
\]
On the other hand,
\[
f_*\left( dd^c(g) + \delta_{Z({\mathbb{C}})} \right) =
dd^c(f_*(g)) + \delta_{f_*(Z({\mathbb{C}}))}.
\]
Moreover, by Proposition~\ref{prop:push:forward:B:pq},
$f_*(g)$ is represented by an $L^1$-form on $Y({\mathbb{C}})$.
Thus, $(f_*(Z), f_*(g))$ is an element of $\widehat{Z}_{L^1}^p(Y)$.
\QED
\subsection{Scalar product for arithmetic $L^1$-cycles and arithmetic $D$-cycles}
\renewcommand{\theequation}{\arabic{section}.\arabic{subsection}.\arabic{Theorem}}
\setcounter{Theorem}{0}
Let $X$ be a regular arithmetic variety.
The purpose of this subsection is to give a scalar product on
$\widehat{\operatorname{CH}}_D^{*}(X)_{{\mathbb{Q}}} = \bigoplus_{p \geq 0} \widehat{\operatorname{CH}}_D^p(X)_{{\mathbb{Q}}}$ by
the arithmetic Chow ring
$\widehat{\operatorname{CH}}^{*}(X)_{{\mathbb{Q}}} = \bigoplus_{p \geq 0} \widehat{\operatorname{CH}}^p(X)_{{\mathbb{Q}}}$.
Roughly speaking, the scalar product is defined by
\[
(Y, f) \cdot (Z, g) =
( Y \cap Z, f \wedge \delta_Z + \omega((Y,f)) \wedge g)
\]
for $(Y, f) \in \widehat{Z}^p(X)$ and $(Z, g) \in \widehat{Z}_D^q(X)$.
This definition, however, works only under the assumption
that $Y$ and $Z$ intersect properly.
Usually, by using Chow's moving lemma,
we can avoid the above assumption.
This is rather complicated, so that in this paper we try to
use the standard arithmetic intersection theory to define
the scalar product.
Let $x \in \widehat{\operatorname{CH}}^p(X)$, $(Z, g) \in \widehat{Z}_D^q(X)$, and
$g_Z$ a Green current for $Z$.
First we shall check that
\[
x \cdot [(Z, g_Z)] + [(0, \omega(x) \wedge (g-g_Z))]
\]
in $\widehat{\operatorname{CH}}_D^{p+q}(X)_{{\mathbb{Q}}}$ does not depend on the choice of $g_Z$.
For, let $g'_Z$ be another Green current for $Z$.
Then, there are $\eta \in A^{p-1,p-1}(X({\mathbb{C}}))$, and
$v \in \operatorname{Image}(\partial) + \operatorname{Image}(\bar{\partial})$ with
$g'_Z = g_Z + [\eta] + v$.
Then, since $[(0, [\eta] + v)] \in \widehat{\operatorname{CH}}^p(X)$,
\begin{align*}
x \cdot [(Z, g'_Z)] + [(0, \omega(x) \wedge (g-g'_Z))] & =
x \cdot [(Z, g_Z)] + x \cdot [(0, [\eta] + v)] \\
& \qquad\qquad + [(0, \omega(x) \wedge (g-g_Z-[\eta]-v))] \\
& = x \cdot [(Z, g_Z)] + [(0, \omega(x) \wedge ([\eta] + v))] \\
& \qquad\qquad + [(0, \omega(x) \wedge (g-g_Z-[\eta]-v))] \\
& = x \cdot [(Z, g_Z)] + [(0, \omega(x) \wedge (g-g_Z))].
\end{align*}
Thus, we have the bilinear homomorphism
\[
\widehat{\operatorname{CH}}^p(X) \times \widehat{Z}_D^q(X) \to \widehat{\operatorname{CH}}_D^{p+q}(X)_{{\mathbb{Q}}}
\]
given by $x \cdot (Z, g) = x \cdot [(Z, g_Z)] + [(0, \omega(x) \wedge (g-g_Z))]$.
Moreover, if $(Z, g) \in \widehat{R}^q(X)$,
then, by \cite[Theorem~4.2.3]{GSArInt}, $x \cdot (Z, g) = 0$ in $\widehat{\operatorname{CH}}^{p+q}(X)_{{\mathbb{Q}}}$.
Thus, the above induces
\addtocounter{Theorem}{1}
\begin{equation}
\label{eqn:scalar:product:1}
\widehat{\operatorname{CH}}^p(X) \otimes \widehat{\operatorname{CH}}_D^q(X) \to \widehat{\operatorname{CH}}_D^{p+q}(X)_{{\mathbb{Q}}},
\end{equation}
which may give rises to a natural scalar product of
$\widehat{\operatorname{CH}}_D^{*}(X)_{{\mathbb{Q}}}$ over
the arithmetic Chow ring $\widehat{\operatorname{CH}}^{*}(X)_{{\mathbb{Q}}}$.
To see that this is actually a scalar product, we need to check
that
\[
(x \cdot y) \cdot z = x \cdot (y \cdot z)
\]
for all $x \in \widehat{\operatorname{CH}}^p(X)$, $y \in \widehat{\operatorname{CH}}^q(X)$ and $z \in \widehat{\operatorname{CH}}_D^r(X)$.
If $z \in \widehat{\operatorname{CH}}^r(X)$, then this is nothing more than
the associativity of the product of
the arithmetic Chow ring (cf. \cite[Theorem~4.2.3]{GSArInt}).
Thus, we may assume that $z = [(0, g)]$ for some $g \in D^{r-1,r-1}(X({\mathbb{C}}))$.
Then, since
\[
(x \cdot y) \cdot z = [(0, \omega(x \cdot y) \wedge g)] =
[(0, \omega(x) \wedge \omega(y) \wedge g)]
\]
and
\[
x \cdot (y \cdot z) = x \cdot [(0, \omega(y) \wedge g)] =
[(0, \omega(x) \wedge \omega(y) \wedge g)],
\]
we can see $(x \cdot y) \cdot z = x \cdot (y \cdot z)$.
Therefore, we get the natural scalar product.
Moreover, (\ref{eqn:scalar:product:1}) induces
\addtocounter{Theorem}{1}
\begin{equation}
\label{eqn:scalar:product:2}
\widehat{\operatorname{CH}}^p(X) \otimes \widehat{\operatorname{CH}}_{L^1}^q(X) \to \widehat{\operatorname{CH}}_{L^1}^{p+q}(X)_{{\mathbb{Q}}}.
\end{equation}
Indeed, if $(Z, g) \in \widehat{Z}_{L^1}^{q}(X)$ and $g_Z$ is a Green form of $Z$,
then,
\[
x \cdot [(Z, g)] = x \cdot [(Z, g_Z)] + [(0, \omega(x) \wedge (g - g_Z))].
\]
Thus, in order to see that $x \cdot [(Z, g)] \in \widehat{\operatorname{CH}}_{L^1}^{p+q}(X)_{{\mathbb{Q}}}$,
it is sufficient to check that
\[
\begin{cases}
\omega(x) \wedge (g - g_Z)
\in L^1_{\operatorname{loc}}(\Omega_{X({\mathbb{C}})}^{p+q-1,p+q-1}), \\
dd^c \left( \omega(x) \wedge (g - g_Z) \right)
\in L^1_{\operatorname{loc}}(\Omega_{X({\mathbb{C}})}^{p+q,p+q}).
\end{cases}
\]
The first assertion is obvious because $g$ and $g_Z$ are $L^1$-forms.
Further, we can easily see the second assertion because
\[
dd^c \left( \omega(x) \wedge (g - g_Z) \right) =
\pm \omega(x) \wedge dd^c(g - g_Z) =
\pm \omega(x) \wedge (\omega(g) - \omega(g_Z)).
\]
Gathering all observations,
we can conclude the following proposition,
which is a generalization of \cite[Theorem~4.2.3]{GSArInt}.
\begin{Proposition}
\label{prop:module:structure:arith:D:cycle}
$\widehat{\operatorname{CH}}_{L^1}^{*}(X)_{{\mathbb{Q}}}$ and $\widehat{\operatorname{CH}}_D^{*}(X)_{{\mathbb{Q}}}$ has a natural
module structure over the
arithmetic Chow ring $\widehat{\operatorname{CH}}^{*}(X)_{{\mathbb{Q}}}$.
\end{Proposition}
Moreover, we have the following projection formula.
\begin{Proposition}
\label{prop:projection:formula:regular:smooth}
Let $f : X \to Y$ be a proper morphism of regular arithmetic varieties such that
$f_{{\mathbb{Q}}} : X_{{\mathbb{Q}}} \to Y_{{\mathbb{Q}}}$ is smooth.
Then, for any $\alpha \in \widehat{\operatorname{CH}}^p(Y)$ and $\beta \in \widehat{\operatorname{CH}}_{L^1}^q(X)$,
\[
f_* (f^*(\alpha) \cdot \beta) = \alpha \cdot f_* (\beta)
\]
in $\widehat{\operatorname{CH}}_{L^1}^{p + q -(\dim X - \dim Y)}(Y)_{{\mathbb{Q}}}$.
\end{Proposition}
{\sl Proof.}\quad
If $\alpha \in \widehat{\operatorname{CH}}^p(Y)$ and $\beta \in \widehat{\operatorname{CH}}^q(X)$,
then this is well known (cf. \cite{GSArInt}). Thus,
we may assume that
$\beta = (0,[\phi]) \in \widehat{Z}_{L^1}^q(Y)$.
Then
\begin{align*}
f_* (f^*(\alpha) \cdot \beta)
& = f_* ((0,\omega (f^*(\alpha)) \wedge [\phi]) \\
& = (0, [f_*\left( \omega (f^*(\alpha) \wedge \phi) \right)]).
\end{align*}
On the other hand,
\begin{equation*}
\alpha \cdot f_* (\beta)
= \alpha \cdot (0,[f_*(\phi)])
= (0, \omega(\alpha) \wedge [f_*(\phi)]).
\end{equation*}
Since $f_* (\omega (f^*(\alpha))) = \omega(\alpha)$,
we have proven the projection formula.
\QED
\renewcommand{\theequation}{\arabic{section}.\arabic{subsection}.\arabic{Theorem}.\arabic{Claim}}
\subsection{Scalar product, revisited (singular case)}
\setcounter{Theorem}{0}
Let $X$ be an arithmetic variety.
Here $X$ is not necessarily regular.
Let $\operatorname{Rat}_X$ be the sheaf of rational functions on $X$.
We denote $H^0(X, \operatorname{Rat}_X^{\times}/{\mathcal{O}}_X^{\times})$ by $\operatorname{Div}(X)$.
An element of $\operatorname{Div}(X)$ is called {\em a Cartier divisor on $X$}.
For a Cartier divisor $D$ on $X$, we can assign a divisor $[D] \in Z^1(X)$
in a natural way. This gives rise to the homomorphism
\[
c_X : \operatorname{Div}(X) \to Z^1(X).
\]
Note that $c_X$ is neither injective nor surjective
in general. An exact sequence
\[
1 \to {\mathcal{O}}_X^{\times} \to \operatorname{Rat}_X^{\times} \to \operatorname{Rat}_X^{\times}/{\mathcal{O}}_X^{\times} \to 1
\]
induces to the homomorphism $\operatorname{Div}(X) \to H^1(X, {\mathcal{O}}_X^{\times})$.
For a Cartier divisor $D$ on $X$, the image of $D$ by
the above homomorphism induces the line bundle on $X$.
We denote this line bundle by ${\mathcal{O}}_X(D)$.
Here we set
\[
\widehat{\operatorname{Div}}(X) = \{ (D, g) \mid
\text{$D \in \operatorname{Div}(X)$ and $g$ is a Green function for $D({\mathbb{C}})$ on $X({\mathbb{C}})$} \}.
\]
Similarly, we can define
$\widehat{\operatorname{Div}}_{L^1}(X)$ and $\widehat{\operatorname{Div}}_D(X)$.
The homomorphism $c_X : \operatorname{Div}(X) \to Z^1(X)$ gives rise to
the homomorphism $\hat{c}_X : \widehat{\operatorname{Div}}(X) \to \widehat{Z}^1(X)$.
Then, we define $\widehat{\operatorname{Pic}}(X)$, $\widehat{\operatorname{Pic}}_{L^1}(X)$, and $\widehat{\operatorname{Pic}}_D(X)$ as follows.
\[
\begin{cases}
\widehat{\operatorname{Pic}}(X) = \widehat{\operatorname{Div}}(X)/\hat{c}_X^{-1}(\widehat{R}^1(X)), \\
\widehat{\operatorname{Pic}}_{L^1}(X) = \widehat{\operatorname{Div}}_{L^1}(X)/\hat{c}_X^{-1}(\widehat{R}^1(X)), \\
\widehat{\operatorname{Pic}}_D(X) = \widehat{\operatorname{Div}}_D(X)/\hat{c}_X^{-1}(\widehat{R}^1(X)).
\end{cases}
\]
Note that if $X$ is regular, then
\[
\widehat{\operatorname{Pic}}(X) = \widehat{\operatorname{CH}}^1(X),
\quad
\widehat{\operatorname{Pic}}_{L^1}(X) = \widehat{\operatorname{CH}}_{L^1}^1(X)
\quad\text{and}\quad
\widehat{\operatorname{Pic}}_D(X) = \widehat{\operatorname{CH}}_D^1(X).
\]
Let $(E, h)$ be a Hermitian vector bundle on $X$. Then,
by virtue of \cite[Theorem~4]{GSRR},
we have a cap product of $\widehat{\operatorname{ch}}(E, h)$ on $\widehat{\operatorname{CH}}^{*}(X)_{{\mathbb{Q}}}$,
i.e., a homomorphism
$\widehat{\operatorname{CH}}^{*}(X)_{{\mathbb{Q}}} \to \widehat{\operatorname{CH}}^{*}(X)_{{\mathbb{Q}}}$
given by $x \mapsto \widehat{\operatorname{ch}}(E, h) \cap x$
for $x \in \widehat{\operatorname{CH}}^{*}(X)_{{\mathbb{Q}}}$.
In the same way as before, we can see that
the above homomorphism extends to
\[
\widehat{\operatorname{CH}}_D^{*}(X)_{{\mathbb{Q}}} \to \widehat{\operatorname{CH}}_D^{*}(X)_{{\mathbb{Q}}}
\quad\text{and}\quad
\widehat{\operatorname{CH}}_{L^1}^{*}(X)_{{\mathbb{Q}}} \to \widehat{\operatorname{CH}}_{L^1}^{*}(X)_{{\mathbb{Q}}}
\]
as follows. If $(Z, g) \in \widehat{Z}_D^p(X)$ and $g_Z$ is a Green current of $Z$,
then
\[
\widehat{\operatorname{ch}}(E, h) \cap (Z, g) = \widehat{\operatorname{ch}}(E, h) \cap (Z, g_Z) +
a(\operatorname{ch}(E, h) \wedge (g - g_Z)).
\]
In particular,
we have intersection pairings
\[
\widehat{\operatorname{Pic}}(X)_{{\mathbb{Q}}} \otimes \widehat{\operatorname{CH}}_D^{p}(X)_{{\mathbb{Q}}} \to \widehat{\operatorname{CH}}_D^{p+1}(X)_{{\mathbb{Q}}}
\quad\text{and}\quad
\widehat{\operatorname{Pic}}(X)_{{\mathbb{Q}}} \otimes \widehat{\operatorname{CH}}_{L^1}^{p}(X)_{{\mathbb{Q}}} \to \widehat{\operatorname{CH}}_{L^1}^{p+1}(X)_{{\mathbb{Q}}}.
\]
For simplicity, the cap product ``$\cap$'' is denoted by the dot ``$\cdot$''.
Note that
\[
\widehat{\operatorname{Pic}}(X)_{{\mathbb{Q}}} \otimes \widehat{\operatorname{CH}}_D^{p}(X)_{{\mathbb{Q}}} \to \widehat{\operatorname{CH}}_D^{p+1}(X)_{{\mathbb{Q}}}
\]
is actually defined by
\[
(D, g) \cdot (Z, f) = (D \cdot Z, g \wedge \delta_Z + \omega(g) \wedge f)
\]
if $D$ and $Z$ intersect properly.
Then, we have the following projection formula.
\begin{Proposition}
\label{prop:projection:formula:line:bundle}
Let $f : X \to Y$ be a proper morphism of arithmetic varieties.
Let $(L, h)$ be a Hermitian line bundle on $Y$, and
$z \in \widehat{\operatorname{CH}}_D^p(X)$. Then
\[
f_*(\widehat{{c}}_1 (f^*L, f^*h) \cdot z) = \widehat{{c}}_1(L, h) \cdot f_*(z).
\]
\end{Proposition}
{\sl Proof.}\quad
Let $(Z, g)$ be a representative of $z$.
Clearly, we may assume that $Z$ is reduced and irreducible.
We set $T = f(Z)$ and $\pi = \rest{f}{Z} : Z \to T$.
Let $s$ be a rational section of $\rest{L}{T}$.
Then, $\pi^*(s)$ gives rise to a rational section of $\rest{f^*(L)}{Z} =
\pi^* \left( \rest{L}{T} \right)$.
Thus, $\widehat{{c}}_1 (f^*L, f^*h) \cdot z$ can be represented by
\[
\left( \operatorname{div}(\pi^*(s)),
\left[ -\log \pi^*\left( \rest{h}{T} \right)(\pi^*(s), \pi^*(s)) \right]
+ c_1(f^*L, f^*h) \wedge g \right),
\]
where
$\left[ -\log \pi^*\left( \rest{h}{T} \right)(\pi^*(s), \pi^*(s)) \right]$
is the current given by
\[
\left[ -\log \pi^*\left( \rest{h}{T} \right)(\pi^*(s), \pi^*(s)) \right](\phi)
= \int_{Z({\mathbb{C}})}
\left( - \log \pi^*\left( \rest{h}{T} \right)(\pi^*(s), \pi^*(s)) \right) \phi.
\]
If we set
\[
\deg(\pi) =
\begin{cases}
0 & \text{if $\dim T < \dim Z$} \\
\deg(Z \to T) & \text{if $\dim T = \dim Z$,}
\end{cases}
\]
then
\begin{align*}
\int_{Z({\mathbb{C}})}
\left( - \log \pi^*\left( \rest{h}{T} \right)(\pi^*(s), \pi^*(s)) \right)
f^*(\psi)
& = \int_{Z({\mathbb{C}})} \pi^* \left( \left(
-\log \left( \rest{h}{T} \right) (s, s) \right) \psi \right) \\
& = \deg(\pi) \int_{T({\mathbb{C}})} \left( -\log \left( \rest{h}{T} \right) (s, s) \right) \psi
\end{align*}
for a $C^{\infty}$-form $\psi$ on $Y({\mathbb{C}})$.
Thus, we have
\[
f_* \left[ -\log \pi^*\left( \rest{h}{T} \right)(\pi^*(s), \pi^*(s)) \right]
= \deg(\pi) \left[ -\log \left( \rest{h}{T} \right) (s, s) \right].
\]
Therefore,
\begin{align*}
f_*(\widehat{{c}}_1 (f^*L, f^*h) \cdot z)
& =
\left( \deg(\pi) \operatorname{div}(s), \deg(\pi) \left[ -\log \left( \rest{h}{T} \right) (s, s) \right] +
c_1(L, h) \wedge f_*(g) \right) \\
& = \widehat{{c}}_1 (L, h) \cdot (\deg(\pi) T, f_*(g)) = \widehat{{c}}_1 (L, h) \cdot f_*(z).
\end{align*}
Hence, we get our proposition.
\QED
Let $Z$ be a quasi-projective integral scheme over ${\mathbb{Z}}$.
Then, by virtue of Hironaka's resolution of singularities \cite{Hiro},
there is a proper birational morphism $\mu : Z' \to Z$
of quasi-projective integral schemes over ${\mathbb{Z}}$ such that
$Z'_{{\mathbb{Q}}}$ is non-singular.
The above $\mu : Z' \to Z$ is called a {\em generic resolution of
singularities of $Z$}.
As a corollary of the above projection formula,
we have the following proposition.
\begin{Proposition}
\label{prop:formula:restriction:intersection}
Let $X$ be a arithmetic variety, and
$\overline{L}_1 = (L_1, h_1), \ldots, \overline{L}_{n} = (L_{n}, h_{n})$
be Hermitian line bundles on $X$.
Let $(Z, g)$ be an arithmetic $D$-cycle on $X$, and
$Z = a_1 Z_1 + \cdots + a_r Z_r$ the irreducible decomposition as cycles.
For each $i$, let $\tau_i : Z'_i \to Z_i$ be a proper birational morphism
of quasi-projective integral schemes. We assume that if $Z_i$ is horizontal with respect to
$X \to \operatorname{Spec}({\mathbb{Z}})$, then $\tau_i$ is a generic resolution of
singularities of $Z_i$. Then, we have
\[
\widehat{{c}}_1(\overline{L}_1) \cdots \widehat{{c}}_1(\overline{L}_n) \cdot (Z, g) =
\sum_{i=1}^r a_i {\mu_i}_* \left(
\widehat{{c}}_1(\mu_i^* \overline{L}_1) \cdots \widehat{{c}}_1(\mu_i^* \overline{L}_n)
\right) +
a(c_1(\overline{L}_1) \wedge \cdots \wedge c_1(\overline{L}_n) \wedge g)
\]
in $\widehat{\operatorname{CH}}_D^*(X)_{{\mathbb{Q}}}$,
where $\mu_i$ is the composition of
$Z'_i \overset{\tau_i}{\longrightarrow} Z_i \hookrightarrow X$
for each $i$.
\end{Proposition}
{\sl Proof.}\quad
We prove this proposition by induction on $n$.
First, let us consider the case $n = 1$.
Clearly we may assume that $Z$ is integral, i.e., $Z = Z_1$.
Let $h_1$ be the Hermitian metric of $\overline{L}_1$, and
$s$ a rational section of $\rest{L_1}{Z}$.
Then,
\[
\left(\operatorname{div}(s),
- \log (\rest{h_1}{Z})(s, s) + c_1(\overline{L}_1) \wedge g \right)
=
\left(\operatorname{div}(s),
- \log (\rest{h_1}{Z})(s, s) \right) + a(c_1(\overline{L}_1) \wedge g)
\]
is a representative of $\widehat{{c}}_1(\overline{L}_1) \cdot (Z, g)$.
Moreover,
\[
\left(\operatorname{div}(\tau_1^*(s)),
- \log \tau_1^*(\rest{h_1}{Z})(\tau_1^*(s), \tau_1^*(s)) \right)
\]
is a representative of $\widehat{{c}}_1(\mu_1^* \overline{L}_1)$.
Hence, we have our assertion in the case $n=1$ because
\[
\left({\mu_1}_* (\operatorname{div}(\tau_1^*(s)),
- \log \tau_1^*(\rest{h_1}{Z})(\tau_1^*(s), \tau_1^*(s)) \right)
= (\operatorname{div}(s), - \log (\rest{h_1}{Z})(s, s)).
\]
Thus, we may assume that $n > 1$.
Therefore, using Proposition~\ref{prop:projection:formula:line:bundle} and
hypothesis of induction,
\begin{align*}
\widehat{{c}}_1(\overline{L}_1) \cdots \widehat{{c}}_1(\overline{L}_n) \cdot (Z, g) & =
\widehat{{c}}_1(\overline{L}_1) \cdot
\left( \widehat{{c}}_1(\overline{L}_2) \cdots \widehat{{c}}_1(\overline{L}_n) \cdot (Z, g) \right) \\
& =
\sum_{i=1}^r a_i \widehat{{c}}_1(\overline{L}_1) {\mu_i}_* \left(
\widehat{{c}}_1(\mu_i^* \overline{L}_2) \cdots \widehat{{c}}_1(\mu_i^* \overline{L}_n)
\right) + \\
& \qquad\qquad\qquad\qquad
\widehat{{c}}_1(\overline{L}_1) a(c_1(\overline{L}_2) \wedge \cdots \wedge c_1(\overline{L}_n) \wedge g) \\
& =
\sum_{i=1}^r a_i {\mu_i}_* \left(
\widehat{{c}}_1(\mu_i^* \overline{L}_1) \cdots \widehat{{c}}_1(\mu_i^* \overline{L}_n)
\right) +
a(c_1(\overline{L}_1) \wedge \cdots \wedge c_1(\overline{L}_n) \wedge g).
\end{align*}
\QED
\subsection{Injectivity of $i^*$}
\setcounter{Theorem}{0}
Let $X$ be an arithmetic variety, $U$ a non-empty
Zariski open set of $X$, and
$i : U \to X$ the inclusion map.
Then, there is a natural homomorphism
\[
i^* : \widehat{Z}_{L^1}^1(X) \to \widehat{Z}_{L^1}^1(U)
\]
given by $i^*(D, g) = (\rest{D}{U}, \rest{g}{U({\mathbb{C}})})$.
Since $i^* \left( \widehat{(f)} \right) = \widehat{(\rest{f}{U})}$
for any non-zero rational functions $f$ on $X$, the above
induces the homomorphism
\[
i^* : \widehat{\operatorname{CH}}_{L^1}^1(X) \to \widehat{\operatorname{CH}}_{L^1}^1(U).
\]
Then, we have the following useful lemma.
\begin{Lemma}
\label{lem:criterion:linear:equiv:B:cycle}
If $X \setminus U$ does not contain any irreducible components of
fibers of $X \to \operatorname{Spec}({\mathbb{Z}})$, then
\[
i^* : \widehat{\operatorname{CH}}_{L^1}^1(X) \to \widehat{\operatorname{CH}}_{L^1}^1(U).
\]
is injective. In particular,
$i^* : \widehat{\operatorname{CH}}_{L^1}^1(X)_{{\mathbb{Q}}} \to \widehat{\operatorname{CH}}_{L^1}^1(U)_{{\mathbb{Q}}}$
is injective.
\end{Lemma}
{\sl Proof.}\quad
Suppose that $i^*(\alpha) = 0$ for some $\alpha \in \widehat{\operatorname{CH}}_{L^1}^1(X)$.
Let $(D,g) \in \widehat{Z}_{L^1}^1(X)$ be a representative of $\alpha$.
Since $i^*(\alpha) = 0$, there is a non-zero rational function
$f$ on $X$ with
\[
(\rest{D}{U}, \rest{g}{U({\mathbb{C}})}) = (\rest{(f)}{U}, \rest{-[\log |f|^2]}{U({\mathbb{C}})}).
\]
Pick up $\phi \in L^1_{\operatorname{loc}}(X({\mathbb{C}}))$ with $g = [\phi]$.
Then, the above implies that
$\rest{[\phi]}{U({\mathbb{C}})} = \rest{-[\log |f|^2]}{U({\mathbb{C}})}$.
Thus, $\phi = - \log |f|^2 \ (\operatorname{a.e.})$.
Therefore, we have
\addtocounter{Claim}{1}
\begin{equation}
\label{eqn:1:lem:criterion:linear:equiv:B:cycle}
g = [\phi] = - [\log |f|^2].
\end{equation}
Here, $dd^c(g) + \delta_{D({\mathbb{C}})} = [h]$ for some
$h \in L^1_{\operatorname{loc}}(\Omega_{X({\mathbb{C}})}^{1,1})$ and
$dd^c(-[\log |f|^2]) + \delta_{(f)({\mathbb{C}})} = 0$.
Thus, by (\ref{eqn:1:lem:criterion:linear:equiv:B:cycle}),
$\delta_{D({\mathbb{C}})} - \delta_{(f)({\mathbb{C}})} =[h]$.
This shows us that
$h = 0 \ (\operatorname{a.e.})$ over
$X({\mathbb{C}}) \setminus \left( \operatorname{Supp}(D({\mathbb{C}})) \cup \operatorname{Supp}((f)({\mathbb{C}})) \right)$.
Hence $h = 0 \ (\operatorname{a.e.})$ on $X({\mathbb{C}})$.
Therefore, we have $D({\mathbb{C}}) = (f)({\mathbb{C}})$, which implies
$D = (f)$ on $X_{{\mathbb{Q}}}$.
Thus, $D - (f)$ is a linear combination of
irreducible divisors lying on finite fibers.
On the other hand, $D = (f)$ on $U$ and
$X \setminus U$ does not contain any irreducible components of
fibers.
Therefore, $D = (f)$.
Hence $\alpha = 0$ because $(D, g) = \widehat{(f)}$.
\QED
\section{Weakly positive arithmetic divisors}
\subsection{Generalized metrics}
\setcounter{Theorem}{0}
\label{subsec:gen:metric}
Let $X$ be a smooth algebraic scheme over ${\mathbb{C}}$
and $L$ a line bundle on $X$.
We say $h$ is {\em a generalized metric on $L$} if there is a
$C^{\infty}$ Hermitian metric $h_0$ of $L$ over $X$ and
$\varphi \in L^1_{\operatorname{loc}}(X)$ with $h = e^{\varphi}h_0$.
To see when a metric of a line bundle defined over
a dense Zariski open set
extends to a generalized metric, the following lemma is useful.
\begin{Lemma}
\label{lem:criterion:gen:metric}
Let $X$ be a smooth algebraic variety over ${\mathbb{C}}$ and
$L$ a line bundle on $X$. Let $U$ be a non-empty Zariski open set of $X$ and
$h$ a $C^{\infty}$ Hermitian metric of $L$ over $U$.
We fix a non-zero rational section $s$ of $L$.
Then, $h$ extends to a generalized metric of $L$ on $X$ if and only if
$\log h(s, s) \in L^1_{\operatorname{loc}}(X)$.
\end{Lemma}
{\sl Proof.}\quad
If $h$ extends to a generalized metric of $L$ on $X$, then
$\log h(s, s) \in L^1_{\operatorname{loc}}(X)$ by the definition of generalized metrics.
Conversely, we assume that $\log h(s, s) \in L^1_{\operatorname{loc}}(X)$.
Let $h_0$ be a $C^{\infty}$ Hermitian metric of $L$ over $X$. Here we consider
the function $\phi$ given by
\[
\phi = \frac{h(s,s)}{h_0(s, s)}.
\]
Let $y \in U$ and $\omega$ be a local frame of $L$ around $y$.
If we set $s = f \omega$ for some meromorphic function $f$ around $y$, then
\[
\phi = \frac{h(s,s)}{h_0(s, s)} =
\frac{|f|^2 h(\omega, \omega)}{|f|^2 h_0(\omega, \omega)} =
\frac{h(\omega, \omega)}{h_0(\omega, \omega)}.
\]
This shows us that $\phi$ is a positive $C^{\infty}$ function on $U$ and
$h = \phi h_0$ over $U$.
On the other hand,
\[
\log \phi = \log h(s, s) - \log h_0(s, s).
\]
Here since $\log h(s, s), \log h_0(s, s) \in L^1_{\operatorname{loc}}(X)$,
we have $\log \phi \in L^1_{\operatorname{loc}}(X)$.
Thus, if we set $\varphi = \log \phi$, then $\varphi \in L^1_{\operatorname{loc}}(X)$ and
$h = e^{\varphi} h_0$.
\QED
\subsection{Arithmetic $D$-divisors and generalized metrics}
\label{subsec:arith:div:gen:metric}
\setcounter{Theorem}{0}
Let $X$ be an arithmetic variety, $L$ a line bundle on $X$,
and $h$ a generalized metric of $L$ on $X({\mathbb{C}})$ with
$F_{\infty}^*(h) = \overline{h} \ (\operatorname{a.e.})$.
We would like to define $\widehat{{c}}_1(L, h)$ as an element
of $\widehat{\operatorname{CH}}_D^1(X)$.
Let $s, s'$ be two non-zero rational sections of $L$, and
$f$ a non-zero rational function on $X$
with $s' = fs$.
Then, it is easy to see that
\[
(\operatorname{div}(s'), [-\log h(s', s')]) =
(\operatorname{div}(s), [-\log h(s,s)]) + \widehat{(f)}
\]
in $\widehat{Z}_D^1(X)$.
Thus, we may define $\widehat{{c}}_1(L, h)$ as the class of
$(\operatorname{div}(s), [-\log h(s,s)])$ in $\widehat{\operatorname{CH}}_D^1(X)$.
Let us consider the homomorphism
\[
\omega : \widehat{Z}_D^p(X) \to D^{p,p}(X({\mathbb{C}}))
\]
given by $\omega(Z, g) = dd^c(g) + \delta_{Z({\mathbb{C}})}$.
Since $\omega\left( \widehat{R}^{p}(X) \right) = 0$,
the above homomorphism induces the homomorphism
$\widehat{\operatorname{CH}}_D^p(X) \to D^{p,p}(X({\mathbb{C}}))$.
Hence, we get the homomorphism
$\widehat{\operatorname{CH}}_D^p(X)_{{\mathbb{Q}}} \to D^{p,p}(X({\mathbb{C}}))$
because $D^{p,p}(X({\mathbb{C}}))$ has no torsion.
By abuse of notation,
we denote this homomorphism by $\omega$.
\begin{Proposition}
\label{prop:B:cycle:produce:hermitian:line:bundle}
Let $X$ be an arithmetic variety,
$(Z, [\phi]) \in \widehat{\operatorname{Div}}_D(X)$ with
$\phi \in L^1_{\operatorname{loc}}(X({\mathbb{C}}))$, and
$1$ a rational section of ${\mathcal{O}}_X(Z)$
with $\operatorname{div}(1) = Z$.
Then, there is a unique generalized
metric $h$ of ${\mathcal{O}}_X(Z)$ such that
$F_{\infty}^*(h) = \overline{h} \ (\operatorname{a.e.})$ and
$[-\log h(1, 1)] = [\phi]$.
\textup{(}Here uniqueness of $h$ means that
if $h'$ is another generalized metric with
the same property, then
$h = h' \ (\operatorname{a.e.})$.\textup{)} Moreover,
$\omega(Z, [\phi])$
is $C^{\infty}$ around $x \in X({\mathbb{C}})$ if and only if
$h$ is $C^{\infty}$ around $x$.
We denote this line bundle
$({\mathcal{O}}_X(Z), h)$ with the generalized metric $h$ by ${\mathcal{O}}_Z((Z, [\phi]))$.
With this notation,
for $(Z_1, [\phi_1]), (Z_2, [\phi_2]) \in \widehat{\operatorname{Div}}_D(X)$
with $\phi_1, \phi_2 \in L^1_{\operatorname{loc}}(X({\mathbb{C}}))$,
if $(Z_1, [\phi_1]) \sim (Z_2, [\phi_2])$, then
${\mathcal{O}}_X((Z_1, [\phi_1]))$ is isometric to
${\mathcal{O}}_X((Z_2, [\phi_2]))$ at every point around which
$\omega(Z_1, [\phi_1]) = \omega(Z_2, [\phi_2])$ is a
$C^{\infty}$ form.
\end{Proposition}
{\sl Proof.}\quad
First, let us see uniqueness.
Let $h$ and $h'$ be generalized metrics
of ${\mathcal{O}}_X(Z)$ with
$[-\log h(1, 1)] = [-\log h'(1, 1)] = [\phi]$.
Take $a \in L^1_{\operatorname{loc}}(X({\mathbb{C}}))$
with $h' = e^a h$. Then,
by our assumption, $a = 0 \ (\operatorname{a.e.})$.
Thus, $h = h' \ (\operatorname{a.e.})$.
Pick up an arbitrary point $x \in X({\mathbb{C}})$.
Let $s$ be a local basis of ${\mathcal{O}}_X(Z)$ around $x$.
Then, there is a non-zero
rational rational function $f$ on $X({\mathbb{C}})$
with $1 = f s$. Let us consider
\[
\exp(-\phi - \log |f|^2)
\]
around $x$.
Let $s'$ be a another local basis of ${\mathcal{O}}_X(Z)$
around $x$. We set $s' = us$ and $1 = f' s'$.
Then,
\[
\exp(-\phi - \log |f'|^2) =
\exp(-\phi - \log |f/u|^2) =
|u|^2 \exp(-\phi - \log |f|^2),
\]
which means that if we define the generalized
metric $h$ by
\[
h(s, s) = \exp(-\phi - \log |f|^2),
\]
then $h$ is patched globally, and $h$ is a generalized metric
by Lemma~\ref{lem:criterion:gen:metric}.
Moreover,
\[
-\log h(1,1) = -\log h(fs, fs)
= -\log \left( |f|^2 h(s,s) \right) = \phi.
\]
Here, since $F_{\infty}^*(\phi) = \phi \ (\operatorname{a.e.})$,
we can see $F_{\infty}^*(h) = \overline{h} \ (\operatorname{a.e.})$.
Thus, we can construct our desired metric.
We set $b = \omega(Z, [\phi]) \in D^{1,1}(X({\mathbb{C}}))$.
Then, since $1 = fs$ around $x$,
we have $Z = (f)$ around $x$. Thus,
since $dd^c([\phi]) + \delta_{Z({\mathbb{C}})} = b$ and
$dd^c(-[\log |f|^2]) + \delta_{(f)} = 0$,
\[
dd^c(-[\phi + \log |f|^2]) =
\delta_{Z({\mathbb{C}})} - b - \delta_{(f)}
= - b
\]
around $x$. Therefore,
\begin{align*}
\text{$h$ is $C^{\infty}$ around $x$}
& \Longleftrightarrow
\text{$-\phi - \log |f|^2$ is $C^{\infty}$ around $x$} \\
& \Longleftrightarrow
\text{$dd^c(-[\phi + \log |f|^2])$ is $C^{\infty}$
around $x$} \qquad
\text{($\because$ \cite[Theorem~1.2.2]{GSArInt})}\\
& \Longleftrightarrow
\text{$b$ is $C^{\infty}$ around $x$}
\end{align*}
Finally, let us consider the last assertion.
By our assumption, there is a rational function $z$ on
$X$ such that
\[
(Z_1, [\phi_1]) = (Z_2, [\phi_2]) + \widehat{(z)}.
\]
Then,
$Z_1 = Z_2 + (z)$ and
$\phi_1 = \phi_2 - \log |z|^2$.
Let us consider the homomorphism
$\alpha : {\mathcal{O}}_X(Z_1) \to {\mathcal{O}}_X(Z_2)$
defined by $\alpha(s) = zs$.
Then, $\alpha$ is an isomorphism.
Let $1$ be the unit in the rational function field of $X$.
Then, $1$ gives rise to canonical rational sections of
${\mathcal{O}}_X(Z_1)$ and ${\mathcal{O}}_X(Z_2)$.
Let $x$ be a point of $X({\mathbb{C}})$ such that
$\omega(Z_1, [\phi_1])$ is $C^{\infty}$
around $x$,
and $s$ a local basis of
${\mathcal{O}}_X(Z_1)$ around $x$.
Then, $\alpha(s) = zs$
is a local basis of ${\mathcal{O}}_X(Z_2)$ around
$x$. Choose a rational function $f$ with
$1 = fs$. Then, $1 = z^{-1}f\alpha(s)$.
Thus, if $h_1$ and $h_2$ are metrics of
${\mathcal{O}}_X((Z_1, [\phi_1]))$ and ${\mathcal{O}}_X((Z_2, [\phi_2]))$
respectively, then
\[
h_1(s, s) = \exp(-\phi_1 -\log |f|^2) =
\exp( -\phi_2 - \log |z^{-1}f|^2) =
h_2(\alpha(s), \alpha(s))
\]
Hence, $\alpha$ is an isometry.
\QED
\subsection{Semi-ampleness and small sections}
\setcounter{Theorem}{0}
Let $X$ be an arithmetic variety and $S$ a subset of $X({\mathbb{C}})$.
We set
\[
\widehat{\operatorname{CH}}_{L^1}^1(X;S)_{{\mathbb{Q}}} = \{ \alpha \in \widehat{\operatorname{CH}}_{L^1}^1(X)_{{\mathbb{Q}}} \mid
\text{$\omega(\alpha)$ is $C^{\infty}$ around $z$ for
all $z \in S$} \}.
\]
In the same way, we can define
$\widehat{\operatorname{CH}}_{L^1}^1(X;S)$, $\widehat{Z}_{L^1}^1(X;S)$,
$\widehat{Z}_{L^1}^1(X;S)_{{\mathbb{Q}}}$, $\widehat{\operatorname{Div}}_{L^1}(X;S)$,
$\widehat{\operatorname{Div}}_{L^1}(X;S)_{{\mathbb{Q}}}$, $\widehat{\operatorname{Pic}}_{L^1}(X;S)$ and
$\widehat{\operatorname{Pic}}_{L^1}(X;S)_{{\mathbb{Q}}}$.
Let $x$ be a closed point of $X_{{\mathbb{Q}}}$.
An element $\alpha$ of $\widehat{\operatorname{CH}}_{L^1}^1(X;S)_{{\mathbb{Q}}}$
is said to be {\em semi-ample at $x$ with respect to $S$} if
there are a positive integer $n$ and $(E, g) \in \widehat{Z}_{L^1}^1(X;S)$
with the following properties:
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item
$E$ is effective and $x \not\in \operatorname{Supp}(E)$.
\item
$g(z) \geq 0$ for each $z \in S$.
(Note that $g(z)$ might be $\infty$.)
\item
$n \alpha$ coincides with $(E,g)$ in $\widehat{\operatorname{CH}}_{L^1}^1(X;S)_{{\mathbb{Q}}}$.
\end{enumerate}
We remark that $\alpha \in \widehat{\operatorname{CH}}_{L^1}^1(X;S)_{{\mathbb{Q}}}$ by the condition (c).
Moreover, it is easy to see that
if $\alpha_1$ and $\alpha_2$ are semi-ample at $x$ with respect to $S$,
so is $\lambda \alpha_1 + \mu \alpha_2$ for all
non-negative rational numbers
$\lambda$ and $\mu$.
In terms of the natural action of $\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})$
on $X(\overline{{\mathbb{Q}}})$, we can consider the orbit
$O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}(x)$ of $x$.
If we fix an embedding $\overline{{\mathbb{Q}}} \to {\mathbb{C}}$, we have
an injection $X(\overline{{\mathbb{Q}}}) \to X({\mathbb{C}})$.
It is easy to see that the image of $O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}(x)$
does not depend on the choice of the embedding $\overline{{\mathbb{Q}}} \to {\mathbb{C}}$.
By abuse of notation, we denote this image by
$O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}(x)$.
Then, $O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}(x)$ is one of the examples
of $S$.
\medskip
Let $U$ be a Zariski open set of $X$,
and $F$ a coherent ${\mathcal{O}}_X$-module such that $F$ is locally free over $U$.
Let $h_F$ be a $C^{\infty}$ Hermitian metric of $F$ over $U({\mathbb{C}})$.
We assume that $S \subseteq U({\mathbb{C}})$.
For a closed point $x$ of $U_{{\mathbb{Q}}}$, we say
$(F, h_F)$ is
{\em generated by small sections at $x$ with respect to $S$}
if there are sections $s_1, \ldots, s_n \in H^0(X, F)$ such that
$F_x$ is generated by $s_1, \ldots, s_n$, and that
$h_F(s_i, s_i)(z) \leq 1$ for all $1 \leq i \leq n$ and
$z \in S$.
\begin{Proposition}
\label{prop:comparion:semiample:gen:small:sec}
We assume that $S \subseteq U({\mathbb{C}})$.
For an element $(Z, g)$ of $\widehat{\operatorname{Div}}_{L^1}(X;S)$,
$(Z,g)$ is semi-ample at $x$ with respect to $S$ if and only if
there is a positive integer $n$ such that
${\mathcal{O}}_X(n(Z,g))$ is generated by small sections at $x$
with respect to $S$.
\end{Proposition}
{\sl Proof.}\quad
First, we assume that $(Z,g)$ is semi-ample at $x$ with respect to $S$.
Then, there is $(E, f) \in \widehat{Z}_{L^1}^1(X;S)$ and a positive integer $n$
such that $n(Z, g) \sim (E, f)$,
$E$ is effective, $x \not\in \operatorname{Supp}(E)$, and
$f(z) \geq 0$ for each $z \in S$.
Note that $E$ is a Cartier divisor.
Then, by Proposition~\ref{prop:B:cycle:produce:hermitian:line:bundle},
${\mathcal{O}}_X(n(Z,g)) \simeq {\mathcal{O}}_X((E, f))$.
Moreover, if $h$ is the metric of ${\mathcal{O}}_X((E, f))$ and
$1$ is the canonical section of ${\mathcal{O}}_X(E)$ with $\operatorname{div}(1) = E$,
then $-\log(h(1,1)) = f$.
Here $f(z) \geq 0$ for each $z \in S$.
Thus, $h(1,1)(z) \leq 1$ for each $z \in S$.
Therefore, ${\mathcal{O}}_X((E, f))$ is generated by small sections at $x$
with respect to $S$.
Next we assume that ${\mathcal{O}}_X(n(Z,g))$ is generated by small sections
at $x$ with respect to $S$ for some positive integer $n$.
Then, there is a section $s$ of ${\mathcal{O}}_X(nZ)$ such that
$h(s,s)(z) \leq 1$ for each $z \in S$.
Thus, if we set $E = \operatorname{div}(s)$ and $f = -\log h(s,s)$,
then we can see $(Z,g)$ is semi-ample at $x$ with respect to $S$.
\QED
\begin{Proposition}
\label{prop:finite:sup:imply:globalsup}
Let $U$ be a Zariski open set of $X$,
and $L$ a line bundle on $X$.
Let $h$ be a $C^{\infty}$ Hermitian metric of $L$ over $U({\mathbb{C}})$.
Fix a closed point $x$ of $U_{{\mathbb{Q}}}$.
If $X$ is projective over ${\mathbb{Z}}$, then the followings are equivalent.
\begin{enumerate}
\renewcommand{\labelenumi}{(\arabic{enumi})}
\item
$(L, h)$ is generated by small sections at $x$ with respect to $U({\mathbb{C}})$.
\item
$(L, h)$ is generated by small sections at $x$ with respect to any
finite subsets $S$ of $U({\mathbb{C}})$.
\end{enumerate}
\end{Proposition}
{\sl Proof.}\quad
Clearly, (1) implies (2).
So we assume (2).
First of all, we can easily find $z_1, \ldots, z_n \in U({\mathbb{C}})$ such that,
for any $s \in H^0(X({\mathbb{C}}), L_{{\mathbb{C}}})$, if $s(z_1) = \cdots = s(z_n) = 0$,
then $s = 0$.
Thus, if we set
\[
\Vert s \Vert = \sqrt{h(s, s)(z_1)} + \cdots + \sqrt{h(s, s)(z_n)}
\]
for each $s \in H^0(X({\mathbb{C}}), L_{{\mathbb{C}}})$, then $\Vert \ \Vert$ gives rise to
a norm on $H^0(X({\mathbb{C}}), L_{{\mathbb{C}}})$.
Here we set
\[
B_z = \{ s \in H^0(X,L) \mid \text{$h(s,s)(z) \leq 1$} \}
\]
for each $z \in U({\mathbb{C}})$.
Then, since $H^0(X, L)$ is a discrete subgroup of $H^0(X({\mathbb{C}}), L_{{\mathbb{C}}})$,
$\bigcap_{i=1}^{n} B_{z_i}$ is a finite set.
Thus, adding finite points $z_{n+1}, \ldots, z_{N} \in U({\mathbb{C}})$
to $z_1, \ldots, z_n$ if necessary, we have
\[
\bigcap_{z \in U({\mathbb{C}})} B_z = \bigcap_{i=1}^N B_{z_i}.
\]
By our assumption, there is a section $s \in H^0(X, L)$ such that
$s(x) \not= 0$ and $h(s, s)(z_i) \leq 1$ for all $i=1, \ldots, N$.
Then, $s \in \bigcap_{i=1}^N B_{z_i} = \bigcap_{z \in U({\mathbb{C}})} B_z$.
Thus, we get (2).
\QED
\begin{comment}
\begin{Example}
We note that even if a Hermitian line bundle
$\overline{L}=(L,h_L)$ on an arithmetic variety $X$
is generated by locally small sections at every $x \in X({\mathbb{C}})$,
it does not necessarily have a non-zero global section
whose sup-norm is less than or equal to $1$. For example,
let $X = {\mathbb{P}}^1_{{\mathbb{Z}}}$ and $L = {\mathcal{O}}(1)$.
We shall give a metric on $L$ as follows.
First let $U$ be a neighborhood of a point $(4:3) \in {\mathbb{P}}^1({\mathbb{C}})$
such that
\[
0 < \frac{\vert x_0-x_1 \vert}{\sqrt{\vert x_0 \vert^2 +\vert x_1 \vert^2 }}
\leq \frac{1}{4}
\]
for every $(x_0:x_1) \in U$. Let $f$ be a smooth positive
function on ${\mathbb{P}}^1({\mathbb{C}})$
such that $1 \leq f \leq 4$, $f((4:3)) = 4$
and $f = 1$ on the compliment of $U$.
We define a metric $\Vert\cdot\Vert$ on $L$ by
\[
\Vert s \Vert (x) = f(x)
\frac{\vert ax_0 +bx_1 \vert}{\sqrt{\vert x_0 \vert^2 +\vert x_1 \vert^2 }}
\]
for $s = aX +bY \; (a,b \in {\mathbb{Z}})$ and $x=(x_0:x_1) \in {\mathbb{P}}^1({\mathbb{C}})$.
Now we shall show that, for this metric,
$L$ is generated by locally small sections
for every $x \in {\mathbb{P}}^1({\mathbb{C}})$.
If $x \in U$, then we take the global section
$x_0-x_1$. In this case, by the definition of $U$ and $f$,
$0 < \Vert x_0-x_1 \Vert (x) \leq 1$.
If $x$ is in the compliment of $U$,
then we take the global section $x_0$ or $x_1$.
One of them does not vanish at $x$,
and it is easy to see that the value is less than or equal to $1$
since $f(x) = 1$.
Next, let us see that
\[
\sup_{x \in {\mathbb{P}}^1({\mathbb{C}})} \Vert s \Vert(x) >1
\]
for any non-zero global sections $s = aX_0 +bX_1 \; (a,b \in {\mathbb{Z}})$.
If $(a,b) = (\pm 1 , 0)$, then
\[
\sup_{x \in {\mathbb{P}}^1({\mathbb{C}})} \Vert s \Vert(x) \geq
\Vert s \Vert ((4:3)) = 4 \cdot \frac {4}{5} > 1.
\]
If $(a,b) = (0, \pm 1)$, then
\[
\sup_{x \in {\mathbb{P}}^1({\mathbb{C}})} \Vert s \Vert(x) \geq
\Vert s \Vert ((4:3)) = 4 \cdot \frac {3}{5} > 1.
\]
If $(a,b) \not= (\pm 1 , 0)$ or $(0, \pm 1)$, then
\[
\sup_{x \in {\mathbb{P}}^1({\mathbb{C}})} \Vert s \Vert(x) \geq
\Vert s \Vert ((a : b)) \geq \sqrt{a^2 + b^2} >1.
\]
\end{Example}
\end{comment}
\subsection{Restriction to arithmetic curves}
\setcounter{Theorem}{0}
Let $X$ be an arithmetic variety,
$S$ a subset of $X({\mathbb{C}})$,
$x$ a closed point of $X_{{\mathbb{Q}}}$,
$K$ the residue field of $x$,
and $O_K$ the ring of integers in $K$.
We assume that
the orbit of $x$ by $\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})$ is contained in $S$,
namely, $O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}(x) \subseteq S$, and
that the canonical morphism $\operatorname{Spec}(K) \to X$
induced by $x$ extends to
$\tilde{x} : \operatorname{Spec}(O_K) \to X$.
\begin{Proposition}
\label{prop:homo:ChowB:to:Chow:on:OK}
There is a natural homomorphism
\[
\tilde{x}^* : \widehat{\operatorname{Pic}}_{L^1}(X;S)_{{\mathbb{Q}}} \to \widehat{\operatorname{CH}}^1(\operatorname{Spec}(O_K))_{{\mathbb{Q}}}
\]
such that the restriction of $\tilde{x}^*$ to
$\widehat{\operatorname{Pic}}(X)_{{\mathbb{Q}}}$ coincides with the usual pull-back
homomorphism.
\end{Proposition}
{\sl Proof.}\quad
Let $\alpha \in \widehat{\operatorname{Pic}}_{L^1}(X;S)_{{\mathbb{Q}}}$.
Choose $(Z, g) \in \widehat{\operatorname{Div}}_{L^1}(X;S)$ and a positive integer $e$ such that
the class of $(1/e)(Z, g)$ in $\widehat{\operatorname{Pic}}_{L^1}(X;S)_{{\mathbb{Q}}}$
coincides with $\alpha$.
We would like to define $\tilde{x}^*(\alpha)$ by
\[
(1/e) \widehat{{c}}_1\left( \tilde{x}^*({\mathcal{O}}_X((Z, g))) \right).
\]
For this purpose, we need to check that the above does not depend on
the choice $(Z, g)$ and $e$.
Let $(Z', g')$ and $e'$ be another $L^1$-cycle of codimension $1$ and
positive integer such that
the class of $(1/e')(Z', g')$ is $\alpha$.
Then, there is a positive integer $d$ such that
$de'(Z, g) \sim de(Z', g')$.
Thus, by Proposition~\ref{prop:B:cycle:produce:hermitian:line:bundle},
${\mathcal{O}}_Z(de'(Z, g))$ is isometric to ${\mathcal{O}}_Z(de(Z', g'))$.
Hence,
\begin{align*}
de' \widehat{{c}}_1\left( \tilde{x}^*({\mathcal{O}}_X((Z, g))) \right) & =
\widehat{{c}}_1\left( \tilde{x}^*({\mathcal{O}}_X(de'(Z, g))) \right) \\
& = \widehat{{c}}_1\left( \tilde{x}^*({\mathcal{O}}_X(de(Z', g'))) \right) \\
& = de \widehat{{c}}_1\left( \tilde{x}^*({\mathcal{O}}_X((Z', g'))) \right).
\end{align*}
Therefore,
\[
(1/e) \widehat{{c}}_1\left( \tilde{x}^*({\mathcal{O}}_X((Z, g))) \right)
= (1/e') \widehat{{c}}_1\left( \tilde{x}^*({\mathcal{O}}_X((Z', g'))) \right).
\]
Thus, we can define $\tilde{x}^*$.
\QED
\subsection{Weak positivity of arithmetic $L^1$-divisors}
\label{subsec:wp:div}
\setcounter{Theorem}{0}
Let $X$ be an arithmetic variety,
$S$ a subset of $X({\mathbb{C}})$, and $x$ a closed point of $X_{{\mathbb{Q}}}$.
Let $\alpha \in \widehat{\operatorname{CH}}_{L^1}^1(X;S)_{{\mathbb{Q}}}$ and
$\{ \alpha_n \}_{n=1}^{\infty}$ a sequence of
elements of $\widehat{\operatorname{CH}}_{L^1}^1(X;S)_{{\mathbb{Q}}}$.
We say $\alpha$ is the limit of $\{ \alpha_n \}_{n=1}^{\infty}$
as $n \to \infty$, denoted by
${\displaystyle \alpha = \lim_{n \to \infty} \alpha_n}$,
if there are
(1) $Z_1, \ldots, Z_{l_1} \in \widehat{\operatorname{CH}}_{L^1}^1(X;S)_{{\mathbb{Q}}}$,
(2) $g_1, \ldots, g_{l_2} \in L^1_{\operatorname{loc}}(X({\mathbb{C}}))$ with
$a(g_j) \in \widehat{\operatorname{CH}}_{L^1}^1(X;S)_{{\mathbb{Q}}}$ for all $j$,
(3) sequences $\{ a_n^{1} \}_{n=1}^{\infty}, \ldots,
\{ a_n^{l_1} \}_{n=1}^{\infty}$ of rational numbers, and
(4) sequences $\{ b_n^{1} \}_{n=1}^{\infty}, \ldots,
\{ b_n^{l_2} \}_{n=1}^{\infty}$ of real numbers
with the following properties:
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item
$l_1$ and $l_2$ does not depend on $n$.
\item
${\displaystyle \lim_{n \to \infty} a_n^{i} =
\lim_{n \to \infty} b_n^{j} = 0}$ for all $1 \leq i \leq l_1$ and
$1 \leq j \leq l_2$.
\item
${\displaystyle
\alpha = \alpha_n +
\sum_{i=1}^{l_1} a_n^{i}Z_i + \sum_{j=1}^{l_2} a(b_n^{j} g_j)}$
in $\widehat{\operatorname{CH}}_{L^1}^1(X;S)_{{\mathbb{Q}}}$ for all $n$.
\end{enumerate}
It is easy to see that if
${\displaystyle \alpha = \lim_{n \to \infty} \alpha_n}$ and
${\displaystyle \beta = \lim_{n \to \infty} \beta_n}$
in $\widehat{\operatorname{CH}}_{L^1}^1(X;S)_{{\mathbb{Q}}}$, then
${\displaystyle \alpha + \beta =
\lim_{n \to \infty} (\alpha_n + \beta_n)}$.
An element $\alpha$ of $\widehat{\operatorname{CH}}_{L^1}^1(X;S)_{{\mathbb{Q}}}$
is said to be {\em weakly positive at $x$ with respect to $S$} if
it is the limit of semi-ample ${\mathbb{Q}}$-cycles at $x$ with respect to $S$,
i.e., there is a sequence $\{ \alpha_n \}_{n=1}^{\infty}$ of
elements of $\widehat{\operatorname{CH}}_{L^1}^1(X;S)_{{\mathbb{Q}}}$ such that
$\alpha_n$'s are semi-ample at $x$ with respect to $S$ and
${\displaystyle \alpha = \lim_{n \to \infty} \alpha_n}$.
Let $K$ be the residue field of $x$ and
$O_K$ the ring of integers in $K$.
We assume that $O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}(x) \subseteq S$, and
the canonical morphism $\operatorname{Spec}(K) \to X$ induced by $x$ extends to
$\tilde{x} : \operatorname{Spec}(O_K) \to X$.
Then, we have the following proposition.
\begin{Proposition}
\label{prop:non:negative:wp:div:via:beta}
If $X$ is regular and an element $\alpha$ of $\widehat{\operatorname{CH}}_{L^1}^1(X;S)_{{\mathbb{Q}}}$ is
weakly positive at $x$ with respect to $S$,
then $\widehat{\operatorname{deg}}(\tilde{x}^*(\alpha)) \geq 0$.
\end{Proposition}
{\sl Proof.}\quad
Take $Z_1, \ldots, Z_{l_1}$,
$g_1, \ldots, g_{l_2}$,
$\{ a_n^{1} \}_{n=1}^{\infty}, \ldots,
\{ a_n^{l_1} \}_{n=1}^{\infty}$,
$\{ b_n^{1} \}_{n=1}^{\infty}, \ldots,
\{ b_n^{l_2} \}_{n=1}^{\infty}$, and
$\{ \alpha_n \}_{n=1}^{\infty}$ as in the previous
definition of weak positive arithmetic divisors.
Then,
\[
\widehat{\operatorname{deg}} ( \tilde{x}^*(\alpha) ) = \widehat{\operatorname{deg}} ( \tilde{x}^*(\alpha_n) ) +
\sum_{i=1}^{l_1} a_n^{i} \widehat{\operatorname{deg}} (\tilde{x}^*(Z_i)) +
\sum_{j=1}^{l_2} b_n^{j} \widehat{\operatorname{deg}}(\tilde{x}^*a(g_j)).
\]
Thus, since ${\displaystyle \lim_{n \to \infty} a_n^{i} =
\lim_{n \to \infty} b_n^{j} = 0}$ for all $1 \leq i \leq l_1$ and
$1 \leq j \leq l_2$ and $\widehat{\operatorname{deg}} ( \tilde{x}^*(\alpha_n) ) \geq 0$
for all $n$, we have $\widehat{\operatorname{deg}} ( \tilde{x}^*(\alpha) ) \geq 0$.
\QED
\subsection{Characterization of weak positivity}
\setcounter{Theorem}{0}
Let $X$ be a regular arithmetic variety, $S$ a subset of $X({\mathbb{C}})$, and
$x$ a closed point of $X_{{\mathbb{Q}}}$.
For an element $\alpha \in \widehat{\operatorname{CH}}_{L^1}^1(X)_{{\mathbb{Q}}}$,
we say $\alpha$ is {\em ample at $x$ with respect to $S$}
if there are $(A, f) \in \widehat{Z}_{L^1}^1(X;S)$ and
a positive integer $n$ such that
$A$ is an effective and ample Cartier divisor on $X$, $x \not\in \operatorname{Supp}(A)$,
$f(z) > 0$ for all $z \in S$, and
$n \alpha$ is equal to $(A, f)$ in $\widehat{\operatorname{CH}}_{L^1}^1(X)_{{\mathbb{Q}}}$.
First, let us consider the case where $X = \operatorname{Spec}(O_K)$.
\begin{Proposition}
\label{prop:wp:for:curve}
We assume that $X = \operatorname{Spec}(O_K)$,
$x$ is the generic of $X$, and
$S = X({\mathbb{C}})$.
For an element $\alpha \in \widehat{\operatorname{CH}}^1(X;S)_{{\mathbb{Q}}}$, we have the following.
\begin{enumerate}
\renewcommand{\labelenumi}{(\arabic{enumi})}
\item
$\alpha$ is ample at $x$ with respect to $S$ if and only if
$\widehat{\operatorname{deg}}(\alpha) > 0$.
\item
$\alpha$ is weakly positive at $x$ with respect to $S$ if and only if
$\widehat{\operatorname{deg}}(\alpha) \geq 0$.
\end{enumerate}
\end{Proposition}
{\sl Proof.}\quad
(1)
Clearly, if $\alpha$ is ample at $x$ with respect to $S$,
then $\widehat{\operatorname{deg}}(\alpha) > 0$. Conversely,
we assume that $\widehat{\operatorname{deg}}(\alpha) > 0$.
We take a positive integer $e$ and a Hermitian line bundle
$(L, h)$ on $X$ such that $\widehat{{c}}_1(L, h) = e \alpha$.
Then, $\widehat{\operatorname{deg}}(L, h) > 0$. Thus, by virtue of Riemann-Roch formula and
Minkowski's theorem, there are a positive integer $n$ and
a non-zero element $s$ of $L^{\otimes n}$
with $(h^{\otimes n})(s, s)(z) < 1$ for all
$z \in S$. Thus, $\alpha$ is ample at $x$ with respect to $S$.
\medskip
(2)
First, we assume that $\alpha$ is weakly positive at $x$
with respect to $S$.
Then, by Proposition~\ref{prop:non:negative:wp:div:via:beta},
$\widehat{\operatorname{deg}}(\alpha) \geq 0$.
Next, we assume that $\widehat{\operatorname{deg}}(\alpha) \geq 0$.
Let $\beta$ be an element of $\widehat{\operatorname{CH}}^1(X;S)_{{\mathbb{Q}}}$ such that
$\beta$ is ample at $x$ with respect to $S$.
Then, for any positive integer $n$,
$\widehat{\operatorname{deg}}(\alpha + (1/n) \beta) > 0$. Thus, $\alpha + (1/n)\beta$
is ample at $x$ with respect to $S$ by (1).
Hence, $\alpha$ is weakly positive at $x$ with respect to $S$.
\QED
Before starting a general case,
let us consider the following lemma.
\begin{Lemma}
\label{lem:make:semiample:by:ample}
We assume that $S$ is compact.
Let $\alpha$ be an element of $\widehat{\operatorname{CH}}_{L^1}^1(X)_{{\mathbb{Q}}}$ such that
$\alpha$ is ample at $x$ with respect to $S$.
Then, we have the following.
\begin{enumerate}
\renewcommand{\labelenumi}{(\arabic{enumi})}
\item
Let $\beta$ be an element of $\widehat{\operatorname{CH}}_{L^1}^1(X;S)_{{\mathbb{Q}}}$.
Then, there is a positive number $\epsilon_0$ such that
$\alpha + \epsilon \beta$ is semi-ample at $x$ with respect to $S$
for all rational numbers $\epsilon$ with $|\epsilon| \leq \epsilon_0$.
\item
Let $g$ be a locally integrable function on $X({\mathbb{C}})$
with $a(g) \in \widehat{\operatorname{CH}}_{L^1}^1(X;S)_{{\mathbb{Q}}}$.
Then, there is a positive number
$\epsilon_0$ such that
$\alpha + a(\epsilon g)$ is semi-ample at $x$ with respect to $S$
for all real numbers $\epsilon$ with $|\epsilon| \leq \epsilon_0$.
\end{enumerate}
\end{Lemma}
{\sl Proof.}\quad
(1)
First, we claim that there is a positive number $t_0$ such that
$t \alpha + \beta$ is semi-ample at $x$ with respect to $S$ for
all rational numbers $t \geq t_0$.
Let us choose $(A, f) \in \widehat{Z}_{L^1}^1(X;S)$ and
a positive integer $n_0$ such that
$A$ is an effective and ample Cartier divisor on $X$, $x \not\in \operatorname{Supp}(A)$,
$f(z) > 0$ for all $z \in S$, and
$n_0 \alpha$ is equal to $(A, f)$ in $\widehat{\operatorname{CH}}_{L^1}^1(X)_{{\mathbb{Q}}}$.
Moreover, we choose $(D, g) \in \widehat{Z}_{L^1}^1(X;S)$ and
a positive integer $e$
such that $e \beta$ is equal to $(D, g)$ in
$\widehat{\operatorname{CH}}_{L^1}^1(X)_{{\mathbb{Q}}}$.
Since $A$ is ample, there is a positive integer $n_1$
such that ${\mathcal{O}}_X(n_1 A) \otimes {\mathcal{O}}_X(D)$ is generated by global
sections at $x$.
Thus, there are $(Z, h) \in \widehat{Z}_{L^1}^1(X;S)_{{\mathbb{Q}}}$ such that
$Z$ is effective,
$x \not\in \operatorname{Supp}(Z)$ and $(Z, h) \sim n_1(A, f) + (D, g)$.
We would like to find a positive integer $n_2$
with $n_2 f(z) + h(z) \geq 0$
for all $z \in S$.
Let $U$ be an open set of $X({\mathbb{C}})$ such that $S \subseteq U$, and
$\omega(A, f)$ and $\omega(Z,h)$ are $C^{\infty}$ over $U$.
We set $\phi = \exp(-f)$ and $\psi = \exp(-h)$.
Then, $\phi$ and $\psi$ are continuous on $U$, and
$0 \leq \phi < 1$ on $S$.
Since $n_2 f + h = -\log(\phi^{n_2} \psi)$, it is sufficient to find
a positive integer $n_2$ with $\phi^{n_2} \psi \leq 1 $ on $S$.
If we set $a = \sup_{z \in S} \phi(z)$ and $b = \sup_{z \in S} \psi(z)$,
then $0 \leq a < 1$ and $0 \leq b$ because $S$ is compact.
Thus, there is a positive integer $n_2$ with $a^{n_2} b \leq 1$.
Therefore, $\phi^{n_1} \psi \leq 1$ on $S$.
Here we set
$t_0 = (n_1+n_2)n_0e^{-1}$.
In order to see that $t\alpha + \beta$ is semi-ample at $x$
with respect to $S$ for $t \geq t_0$,
it is sufficient to show that
$(n_1+n_2)n_0 \alpha + e \beta$
is semi-ample at $x$ with respect to $S$ because $e t \geq (n_1+n_2)n_0$.
Here
\begin{align*}
(n_1 + n_2)n_0 \alpha + e \beta
& \sim n_2(A, f) + \left( n_1(A, f) + (D, g) \right) \\
& \sim n_2(A, f) + (Z, h) \\
& = (n_2 A + Z, n_2 f + h),
\end{align*}
$x \not\in \operatorname{Supp}(n_2 A + Z)$, and
$(n_2 f + h)(z) \geq 0$ for all $z \in S$.
Thus, $(n_1 + n_2)n_0 \alpha + e \beta$
is semi-ample at $x$ with respect to $S$.
Hence, we get our claim.
In the same way, we can find a positive number $t_1$ such that
$t \alpha - \beta$ is semi-ample with respect to $S$
for all $t \geq t_1$.
Thus, if we set $\epsilon_0 = \min \{ 1/t_0, 1/t_1 \}$, then we have (1).
\medskip
(2)
In the same way as in the proof of (1),
we can find a positive number $\epsilon_0$ such that
$(f + \epsilon n_0 g)(z) \geq 0$ for all $z \in S$ and
all real number $\epsilon$ with $|\epsilon| \leq \epsilon_0$.
Thus we have (2)
because $n_0( \alpha + a(\epsilon g)) \sim (A, f + \epsilon n_0 g)$.
\QED
\begin{Proposition}
\label{prop:characterization:wp:div}
We assume that $S$ is compact.
Let $\beta$ be an element of $\widehat{\operatorname{CH}}_{L^1}^1(X;S)_{{\mathbb{Q}}}$.
Then the following are equivalent.
\begin{enumerate}
\renewcommand{\labelenumi}{(\arabic{enumi})}
\item
$\beta$ is weakly positive at $x$ with respect to $S$.
\item
$\beta + \alpha$ is semi-ample at $x$ with respect to $S$
for any ample $\alpha \in \widehat{\operatorname{CH}}_{L^1}^1(X;S)_{{\mathbb{Q}}}$ at $x$
with respect to $S$.
\end{enumerate}
\end{Proposition}
{\sl Proof.}\quad
(1) $\Longrightarrow$ (2):
Since $\beta$ is weakly positive at $x$ with respect to $S$,
there is a sequence of $\{ \beta_n \}$ such that
$\beta_n \in \widehat{\operatorname{CH}}_{L^1}^1(X;S)_{{\mathbb{Q}}}$,
$\beta_n$'s are semi-ample at $x$ with respect to $S$,
and $\lim_{n \to \infty} \beta_n = \beta$.
Take $Z_1, \ldots, Z_{l_1}$,
$g_1, \ldots, g_{l_2}$,
$\{ a_n^{1} \}_{n=1}^{\infty}, \ldots,
\{ a_n^{l_1} \}_{n=1}^{\infty}$, and
$\{ b_n^{1} \}_{n=1}^{\infty}, \ldots,
\{ b_n^{l_2} \}_{n=1}^{\infty}$
as in the definition of the limit in $\widehat{\operatorname{CH}}_{L^1}^1(X;S)_{{\mathbb{Q}}}$.
Then, by Lemma~\ref{lem:make:semiample:by:ample},
there is a positive number $\epsilon_0$ such that
$\alpha + \epsilon Z_i$'s are semi-ample at $x$ with respect to $S$
for all rational numbers $\epsilon$ with $|\epsilon| \leq \epsilon_0$,
and $\alpha + a(\epsilon g_j)$'s are semi-ample at $x$
with respect to $S$
for all real numbers $\epsilon$ with $|\epsilon| \leq \epsilon_0$.
We choose $n$ such that $(l_1 + l_2)|a_n^{i}| \leq \epsilon_0$ and
$(l_1 + l_2)|b_n^{j}| \leq \epsilon_0$ for all $i$ and $j$.
Then,
\[
\beta + \alpha = \beta_n +
\sum_{i=1}^{l_1} \frac{\alpha + (l_1 + l_2)a_n^{i} Z_i}{l_1 + l_2} +
\sum_{j=1}^{l_2} \frac{\alpha + a((l_1+ l_2)b_n^{j} g_j)}{l_1 + l_2}.
\]
Here, $\alpha + (l_1 + l_2)a_n^{i} Z_i$ and
$\alpha + a((l_1+ l_2)b_n^{j} g_j)$ are semi-ample $x$
with respect to $S$.
Thus, we get the direction (1) $\Longrightarrow$ (2).
\medskip
(2) $\Longrightarrow$ (1):
Let $\alpha$ be an element of $\widehat{\operatorname{CH}}_{L^1}^1(X)_{{\mathbb{Q}}}$ such that
$\alpha$ is ample at $x$ with respect to $S$.
We set $\beta_n = \beta + (1/n)\alpha$.
Then, by our assumption, $\beta_n$ is semi-ample at $x$
with respect to $S$.
Further, $\beta = \lim_{n \to \infty} \beta_n$.
\QED
\subsection{Small sections via generically finite morphisms}
\setcounter{Theorem}{0}
Let $g : V \to U$ be a proper and \'{e}tale morphism
of complex manifolds.
Let $(E, h)$ be a Hermitian vector bundle on $V$.
Then, a Hermitian metric $g_*(h)$ of $g_*(E)$
is defined by
\[
g_*(h)(s, t)(y) = \sum_{x \in g^{-1}(y)} h(s, t)(x)
\]
for any $y \in U$ and $s, t \in g_*(E)_y$.
\begin{Proposition}
\label{prop:find:small:section}
Let $X$ be a scheme such that
every connected component of $X$ is a arithmetic variety.
Let $Y$ be a regular arithmetic variety, and
$g : X \to Y$ a proper and generically finite morphism such that
every connected component of $X$ maps surjectively to $Y$.
Let $U$ be a Zariski open set of $Y$ such that
$g$ is \'{e}tale over $U$.
Let $S$ be a subset of $U({\mathbb{C}})$ and
$y$ a closed point of $U_{{\mathbb{Q}}}$.
Then, we have the following.
\begin{enumerate}
\renewcommand{\labelenumi}{(\arabic{enumi})}
\item
Let $\phi : E \to Q$ be a homomorphism of
coherent ${\mathcal{O}}_X$-modules such that
$\phi$ is surjective over $g^{-1}(U)$, and
$E$ and $Q$ are locally free over $g^{-1}(U)$.
Let $h_E$ be a $C^{\infty}$ Hermitian metric
of $E$ over $g^{-1}(U)({\mathbb{C}})$, and $h_Q$ the quotient metric
of $Q$ induced by $h_E$.
If $(g_*(E), g_*(h_E))$ is generated by small sections at $y$
with respect to $S$, then
$(g_*(Q), g_*(h_Q))$ is generated by small sections at $y$
with respect to $S$.
\item
Let $E_1$ and $E_2$ be coherent ${\mathcal{O}}_X$-modules such that
$E_1$ and $E_2$ are locally free over $g^{-1}(U)$.
Let $h_1$ and $h_2$ be $C^{\infty}$ Hermitian metrics
of $E_1$ and $E_2$ over $g^{-1}(U)({\mathbb{C}})$.
If $(g_*(E_1), g_*(h_1))$ and $(g_*(E_2), g_*(h_2))$
are generated by small sections
at $y$ with respect to $S$,
then so is $(g_*(E_1 \otimes E_2), g_*(h_1 \otimes h_2))$.
\item
Let $E$ be a coherent ${\mathcal{O}}_X$-module such that
$E$ is locally free over $g^{-1}(U)$.
Let $h_E$ be a $C^{\infty}$ Hermitian metric of $E$ over
$g^{-1}(U)({\mathbb{C}})$.
If $(g_*(E), g_*(h_E))$
is generated by small sections at $y$ with respect
to $S$, then
$(g_*(\operatorname{Sym}^n(E)), g_*(\operatorname{Sym}^n(h_E)))$
is generated by small sections at $y$
with respect to $S$.
\textup{(}For the definition of $\operatorname{Sym}^n(h_E)$,
see \S\textup{\ref{subsec:formula:chern:sym:power}}.\textup{)}
\item
Let $F$ be a coherent ${\mathcal{O}}_Y$-module such that
$F$ is locally free over $U$.
Let $h_F$ be a $C^{\infty}$ Hermitian metric of $F$ over $U({\mathbb{C}})$.
Since $\rest{\det(F)}{U}$ is canonically isomorphic to
$\det(\rest{F}{U})$, $\det(h_F)$ gives rise to a $C^{\infty}$
Hermitian metric of $\det(F)$ over $U({\mathbb{C}})$.
If $(F, h_F)$ is generated by small sections at $y$ with respect to $S$,
then so is $(\det(F), \det(h_F))$.
\end{enumerate}
\end{Proposition}
{\sl Proof.}\quad
(1)
By our assumption,
$g_*(\phi) : g_*(E) \to g_*(Q)$ is surjective over $U$.
Let $s_1, \ldots, s_l \in H^0(Y, g_*(E)) = H^0(X, E)$
such that $g_*(E)_{y}$ is generated by
$s_1, \ldots, s_l$, and that $g_*(h_E)(s_i, s_i)(z) \leq 1$
for all $i$ and $z \in S$.
Then, $g_*(Q)_y$ is generated by
$g_*(\phi)(s_1), \ldots, g_*(\phi)(s_l)$.
Moreover,
by the definition of the quotient metric $h_Q$,
\[
g_*(h_Q)(g_*(\phi)(s_i), g_*(\phi)(s_i))(z) =
\sum_{x \in g^{-1}(z)}
h_Q(\phi(s_i), \phi(s_i))(x) \leq
\sum_{x \in g^{-1}(z)}
h_E(s_i, s_i)(x) \leq 1
\]
for all $z \in S$.
Hence, $g_*(Q)$ is generated by small sections at $y$
with respect to $S$.
\medskip
(2)
Since $g$ is \'{e}tale over $U$,
$\alpha : g_*(E_1) \otimes g_*(E_2) \to g_*(E_1 \otimes E_2)$
is surjective over $U$.
By our assumption, there are
$s_1, \ldots, s_l \in H^0(Y, g_*(E_1))$ and
$t_1, \ldots, t_m \in H^0(Y, g_*(E_2))$ such that
$g_*(E_1)_y$ (resp. $g_*(E_2)_y$)
is generated by $s_1, \ldots, s_l$ (resp. $t_1, \ldots, t_m$),
and that $g_*(h_1)(s_i, s_i)(z) \leq 1$ and
$g_*(h_2)(t_j, t_j)(z) \leq 1$ for
all $i$, $j$ and $z \in S$.
Then, $g_*(E_1 \otimes E_2)_y$ is generated by
$\{ \alpha(s_i \otimes t_j) \}_{i,j}$.
Moreover,
{\allowdisplaybreaks
\begin{align*}
g_*(h_1 \otimes h_2)(\alpha(s_i \otimes t_j),
\alpha(s_i \otimes t_j))(z) & =
\sum_{x \in g^{-1}(z)} (h_1 \otimes h_2)(s_i \otimes t_j,
s_i \otimes t_j)(x) \\
& = \sum_{x \in g^{-1}(z)}
h_1(s_i, s_i)(x) h_2(t_j,t_j)(x) \\
& \leq
\left( \sum_{x \in g^{-1}(z)} h_1(s_i, s_i)(x) \right)
\left( \sum_{x \in g^{-1}(z)} h_2(t_j, t_j)(x) \right) \\
& \leq 1
\end{align*}}
for all $z \in S$.
Thus, we get (2).
\medskip
(3) This is a consequence of (1) and (2).
\medskip
(4)
Let $r$ be the rank of $F$.
Since $F$ is generated by small sections at $y$
with respect to $S$,
there are $s_1, \ldots, s_r \in H^0(Y, F)$ such that
$F \otimes \kappa(y)$ is generated by $s_1, \ldots, s_r$ and
$h(s_i, s_i)(z) \leq 1$
for all $i$ and $z \in S$.
Let us consider an exact sequence:
\[
0 \to F_{tor} \to F \to F/F_{tor} \to 0.
\]
Then, $\det(F) = \det(F/F_{tor}) \otimes \det (F_{tor})$.
Noting that $F_{tor} = 0$ on $U$,
let $g$ be a Hermitian metric of $\det(F/F_{tor})$ over $U({\mathbb{C}})$
given by $\det(h_F)$.
Then, there is a Hermitian metric $k$ of $\det(F_{tor})$ over $U({\mathbb{C}})$
such that
$(\det(F), \det(h_F)) = (\det(F/F_{tor}), g) \otimes (\det (F_{tor}), k)$
over $U({\mathbb{C}})$. If we identify $\det (F_{tor})$ with ${\mathcal{O}}_{Y}$ over $U$,
$k$ is nothing more than the canonical metric of ${\mathcal{O}}_{Y}$ over $U({\mathbb{C}})$.
Let us fix a locally free sheaf $P$ on $Y$ and
a surjective homomorphism $P \to F_{tor}$.
Let $P'$ be the kernel of $P \to F_{tor}$.
Here $\left( \bigwedge^{\operatorname{rk} P'} P' \right)^{*}$
is an invertible sheaf on $Y$ because $Y$ is regular.
Thus we may identify $\det(F_{tor})$ with
\[
\bigwedge^{\operatorname{rk} P} P \otimes
\left( \bigwedge^{\operatorname{rk} P'} P' \right)^{*}.
\]
Further, a homomorphism
$\bigwedge^{\operatorname{rk} P'} P' \to \bigwedge^{\operatorname{rk} P} P$
induced by $P' \hookrightarrow P$
gives rise to
a non-zero section $t$ of $\det(F_{tor})$ because
\[
\operatorname{\mathcal{H}\textsl{om}}\left(\bigwedge^{\operatorname{rk} P'} P', \bigwedge^{\operatorname{rk} P} P\right) =
\operatorname{\mathcal{H}\textsl{om}}\left(\bigwedge^{\operatorname{rk} P'} P', {\mathcal{O}}_Y\right) \otimes \bigwedge^{\operatorname{rk} P} P.
\]
Here $F_{tor} = 0$ on $U$. Thus,
$\det(F_{tor})$ is canonically isomorphic to ${\mathcal{O}}_{Y}$ over $U$.
Since $P' = P$ over $U$, under the above isomorphism,
$t$ goes to the determinant of
$P' \overset{\operatorname{id}}{\longrightarrow} P$,
namely $1 \in {\mathcal{O}}_{Y}$ over $U$.
Thus, $k(t,t)(z) = 1$ for each $z \in S$.
Let $\overline{s}_i$ be the image of $s_i$ in $F/F_{tor}$.
Then, $\overline{s}_1 \wedge \cdots \wedge \overline{s}_r$
gives rise to a section $s$ of $\det(F/F_{tor})$.
Thus, $s \otimes t$ is a section of $\det(F)$.
By our construction, $(s \otimes t)(y) \not= 0$.
Moreover, using Hadamard's inequality,
\begin{align*}
\det(h_F)(s \otimes t, s \otimes t)(z) & =
g(s, s)(z) \cdot k(t,t)(z) =
\det \left( h( s_i, s_j)(z) \right) \\
& \leq
h(s_1, s_1)(z) \cdots h(s_r, s_r)(z) \leq 1
\end{align*}
for each $z \in S$.
Thus, we get (4).
\QED
\section{Arithmetic Riemann-Roch for generically finite morphisms}
\subsection{Quillen metric for generically finite morphisms}
\setcounter{Theorem}{0}
Before starting
Proposition~\ref{prop:Quillen:metric:generalised:gen:finite:morph},
we recall the tensor product of two
matrices, which we will use in the proof.
For an
$r\times r$ matrix $A=(a_{ij})$
and an
$n\times n$ matrix $B=(b_{kl})$,
consider the following $rn\times rn$ matrix
\begin{equation*}
\begin{pmatrix}
a_{11}B & a_{12}B & \cdots & a_{1r}B \\
a_{21}B & a_{22}B & \cdots & a_{2r}B \\
\vdots & \vdots & \ddots & \vdots \\
a_{r1}B & a_{r2}B & \cdots & a_{rr}B \\
\end{pmatrix}.
\end{equation*}
This matrix, denoted by
$A\otimes B$,
is called {\em the tensor product of $A$ and $B$}.
Then for
$r \times r$ matrices $A,A'$
and
$n \times n$ matrices $B,B'$,
we immediately see
\begin{equation*}
(A \otimes B) (A' \otimes B') = AA' \otimes BB', \cr
\det (A \otimes B) = (\det A)^n (\det B)^r.
\end{equation*}
Let $X$ be a smooth algebraic scheme over ${\mathbb{C}}$,
$Y$ a smooth algebraic variety over ${\mathbb{C}}$, and
$f : X \to Y$ a proper and generically finite morphism.
We assume that every connected component of $X$ maps surjectively to
$Y$. Let $W$ be the maximal open set of $Y$
such that $f$ is \'{e}tale over there.
Let $(E, h)$ be a Hermitian vector
bundle on $X$ such that on every connected component of $X$,
$E$ has the same rank $r$.
\begin{Proposition}
\label{prop:Quillen:metric:generalised:gen:finite:morph}
With notation and assumptions being as above, the Quillen metric
$h_Q^{\overline E}$ on $\det Rf_*(E)$ over $W$
extends to a generalized metric on $\det Rf_*(E)$ over $Y$.
\end{Proposition}
{\sl Proof.}\quad
Let $n$ be the degree of $f$.
Since $f$ is \'{e}tale over $W$,
$f_*(E)$ is a locally free sheaf of rank $rn$ and
$R^if_*(E) = 0$ for $i \ge 1$ over there. Thus
\[
\det Rf_*(E) \vert {}_W = \bigwedge^{rn} f_*(E) \vert {}_W.
\]
If $y \in W$ is a complex point and
$X_y=\{x_1,x_2,\cdots,x_n\}$ the fiber of $f$ over $y$,
then we have
\[
\det Rf_*(E)_y = \det H^0(X_y,E).
\]
The Quillen metric on $\det Rf_*(E)$ over $W$ is defined as follows.
On $H{}^0(X_y,E)$ the $L^2$-metric is defined by the formula:
\[
h_{L^2}(s,t) = \sum_{\alpha=1}^n h(s,t)(x_\alpha),
\]
where $s,t \in H{}^0(X_y,E)$.
This metric naturally induces the $L^2$-metric on
$\det H{}^0(X_y,E)$. Since $X_y$ is zero-dimensional,
there is no need for zeta function regularization
to obtain the Quillen metric.
Thus the Quillen metric $h_Q^{\overline E}$
on $\det Rf_*(E) \vert {}_W$ is defined by
the family of Hermitian line bundles
$\{\det H^0(X_y,E)\}_{y \in W}$
with the induced $L^2$-metrics pointwisely.
To see that the Quillen metric over $W$
extends to a generalized metric over $Y$, let
$s_1,s_2,\cdots,s_r$ be rational sections of $E$ such that
at the generic point of every connected component of $X$,
they form a basis of $E$. Also let
$\omega_1,\omega_2,\cdots,\omega_n$ be rational
sections of $f_*({\mathcal{O}}_X)$ such that at the generic point
they form a basis of
$f_*({\mathcal{O}}_X)$. Since
\addtocounter{Claim}{1}
\begin{equation}
\label{eqn:2:prop:Quillen:metric:generalised:gen:finite:morph}
\det Rf_*(E) = \left(\bigwedge^{rn} (f_*(E))
\right)^{**}.
\end{equation}
over $Y$, we can regard $\bigwedge_{ik}s_i\omega_k=s_1\omega_1\operatornamewithlimits{\wedge}
s_1\omega_2\operatornamewithlimits{\wedge}\cdots\operatornamewithlimits{\wedge}
s_1\omega_n\operatornamewithlimits{\wedge}\cdots\operatornamewithlimits{\wedge} s_r\omega_n$
as a non-zero rational section of $\det Rf_*(E)$.
Shrinking $W$, we can find a non-empty Zariski open set $W_0$ of $W$
such that $s_i$'s and $\omega_j$'s has no poles or zeros over $f^{-1}(W_0)$.
To proceed with our argument, we need the following lemma.
\begin{Lemma}
\label{lem:formula:for:Quillen:metric}
Let $L$ be the total quotient field of $X$, and
$K$ the function field of $Y$.
Then,
\[
\log h_Q^{\overline E} \left( \bigwedge_{ik}s_i\omega_k,
\bigwedge_{ik}s_i\omega_k \right) =
r \log \left| \det (\operatorname{Tr}_{L/K}(\omega_i \cdot \omega_j)) \right|
+ f_* \log \det (h(s_i, s_j))
\]
over $W_0$.
\end{Lemma}
{\sl Proof.}\quad
Let $y \in W_0$ be a complex point, and
$\{x_1,x_2,\ldots,x_n\}$ the fiber of $f^{-1}(y)$ over $y$. Then,
{\allowdisplaybreaks
\newcommand{\smash{\hbox{\Large 0}}}{\smash{\hbox{\Large 0}}}
\newcommand{\smash{\lower1.7ex\hbox{\Large 0}}}{\smash{\lower1.7ex\hbox{\Large 0}}}
\begin{align*}
\log h_Q^{\overline E}
\left( \bigwedge_{ik}s_i\omega_k,\bigwedge_{ik}s_i\omega_k \right)(y)
& = \log \det
\left(
\sum_{\alpha =1}^n
h(s_i\omega_k,s_j\omega_l)(x_{\alpha})
\right)_{ij,kl} \\
& = \log \det
\left(
\sum_{\alpha =1}^n
\omega_k(x_\alpha) h(s_i,s_j)(x_{\alpha})
\overline{\omega_l(x_\alpha)}
\right)_{ij,kl} \\
& = \log \det \left(
(I_r \otimes \Omega)
\begin{pmatrix}
H(x_1) & & \smash{\lower1.7ex\hbox{\Large 0}} \\
& \ddots & \\
\smash{\hbox{\Large 0}} & & H(x_n) \\
\end{pmatrix}
\overline{{}^t(I_r \otimes \Omega)}
\right) \\
& = \log \det \left\{
\vert \det (\Omega) \vert ^{2r}
\prod_{\alpha=1}^n
\det \left(h(s_i,s_j)(x_{\alpha})\right)_{ij}
\right\} \\
& = r \log \det \vert \det (\Omega) \vert ^{2} +
\sum_{\alpha=1}^n
\log \det \left(h(s_i,s_j)\right)(x_{\alpha}),
\end{align*}}
where
$\Omega = (\omega_k(x_\alpha))_{k \alpha}$
and
$H(x_\alpha) = (h(s_i,s_j)(x_{\alpha}))_{ij}$.
On the other hand, we have
\[
\sum_{\alpha=1}^n
\log \det \left(h(s_i,s_j)\right) (x_{\alpha})
= \left(f_* \log \det \left(h(s_i,s_j)\right)\right)(y).
\]
Moreover, using the
following Lemma~\ref{lemma:trace:global:local}, we have
\[
\vert \det (\Omega) \vert ^{2} =
\vert \det (\Omega {}^t \Omega) \vert =
\left\vert \det \left(\sum_{\alpha=1}^n \omega_k(x_\alpha)
\omega_l(x_\alpha) \right)_{kl} \right\vert
= \vert \det \left(\operatorname{Tr}_{L/K}(\omega_k\cdot \omega_l)\right)_{kl} \vert.
\]
Thus we get the lemma.
\QED
\begin{Lemma}
\label{lemma:trace:global:local}
Let $f : \operatorname{Spec}(B) \to \operatorname{Spec}(A)$
be a finite \'{e}tale morphism of regular affine schemes.
Let ${\mathfrak m}$ be the maximal ideal of $A$ and
${\mathfrak n}_1,{\mathfrak n}_2,\cdots,{\mathfrak n}_n$ the prime ideals lying over
${\mathfrak m}$. Assume that $\kappa ({\mathfrak m})$ is algebraically closed
and hence $\kappa ({\mathfrak n}_i)$ is \textup{(}naturally\textup{)}
isomorphic to $\kappa ({\mathfrak m})$ for each $1 \leq i \leq n$.
Let $b$ be an element of $B$ and $b({\mathfrak n}_i)$ the value of
$b$ in $\kappa ({\mathfrak n}_i) \cong \kappa ({\mathfrak m})$.
Then
\[
\operatorname{Tr}_{B/A}(b)({\mathfrak m}) = \sum_{i=1}^n b({\mathfrak n}_i)
\]
in $\kappa ({\mathfrak m})$,
where $\operatorname{Tr}_{B/A}(b)({\mathfrak m})$ is
the value of $\operatorname{Tr}_{B/A}(b)$ in $\kappa ({\mathfrak m})$.
\end{Lemma}
{\sl Proof.}\quad
It is easy to see that every ${\mathfrak n}_i$ is the maximal ideal and that
${\mathfrak m}B = {\mathfrak n}_1{\mathfrak n}_2 \cdots {\mathfrak n}_n$.
Let $\widehat{A}$ be the completion of $A$ with respect to ${\mathfrak m}$,
$\widehat{B}$ the completion of $B$ with respect to ${\mathfrak m}B$, and
$\widehat{B_i}$ the completion of $B$
with respect to ${\mathfrak n}_i$ for each $1 \leq i \leq n$.
Then by Chinese remainder theorem,
$\widehat{B} = \prod_{i=1}^n \widehat{B_i}$ as an $\widehat{A}$-algebra.
Note that $\widehat{A} /{\mathfrak m} \widehat{A} = \kappa ({\mathfrak m})$
and $\widehat{B_i} /{\mathfrak n}_i \widehat{B_i} = \kappa ({\mathfrak n}_i)$.
Since $\widehat{A} \to \widehat{B_i}$ is \'{e}tale and $\kappa ({\mathfrak m})
\cong \kappa ({\mathfrak n}_i)$, we have
$\widehat{A} \cong \widehat{B_i}$.
Let $e_1=(1,0,\cdots,0),e_2=(0,1,\cdots,0),\cdots,e_n=(0,0,\cdots,1)
\in \prod_{i=1}^n \widehat{B_i}
=\widehat{B}$ be a free basis of $\widehat{B}$ over
$\widehat{A}$. We put $b e_i = b_i e_i$ with $b_i \in
\widehat{B_i} \cong \widehat{A}$ for each $1 \leq i \leq n$.
Then $b_i \equiv b({\mathfrak n}_i) \pmod{{\mathfrak n}_i}$.
Now the lemma follows from
\[
\operatorname{Tr}_{B/A}(b) = \operatorname{Tr}_{\widehat{B} / \widehat{A}}(b)
= \sum_{i=1}^n b_i
\]
in $\widehat{A}$.
\QED
Let us go back to the proof of
Proposition~\ref{prop:Quillen:metric:generalised:gen:finite:morph}.
Since $\rest{\det (\operatorname{Tr}_{L/K}(\omega_i \cdot \omega_j))}{W_0}$ extends to a rational
function $\det (\operatorname{Tr}_{L/K}(\omega_i \cdot \omega_j))$ on $Y$,
\[
\log \left |
\det (\operatorname{Tr}_{L/K}(\omega_i \cdot \omega_j)) \right|
\in L^1_{\operatorname{loc}}(Y).
\]
Moreover,
by Proposition~\ref{prop:push:forward:B:pq},
$f_* \log \det (h(s_i, s_j)) \in L^1_{\operatorname{loc}}(Y)$.
Thus, by Lemma~\ref{lem:formula:for:Quillen:metric},
\[
\rest{\log h_Q^{\overline E}
\left( \bigwedge_{ik}s_i\omega_k, \bigwedge_{ik}s_i\omega_k \right)}{W_0}
\]
extends to a locally integrable function on $Y$.
Hence by Lemma~\ref{lem:criterion:gen:metric}
the Quillen metric over $W$
extends to a generalized metric over $Y$.
\QED
\begin{Remark}
\label{rmk:Quillen:metric:continuous:finite:morphism}
In the above situation, Let $W'$ be a open set of $Y$ such that
$f$ is flat and finite over there.
Then the Quillen metric extends to a continuous function over $W'$
by the same formula as in
(\ref{lem:formula:for:Quillen:metric})
\end{Remark}
\subsection{Riemann-Roch for generically finite morphisms}
\setcounter{Theorem}{0}
In this subsection, we formulate
the arithmetic Riemann-Roch theorem
for generically finite morphisms.
\begin{Theorem}
\label{thm:arith:Riemann:Roch:gen:finite:morphism}
Let $X$ be a scheme such that every connected component of $X$ is an arithmetic variety.
Let $Y$ be a regular arithmetic variety, and
$f : X \to Y$ a proper and generically finite morphism such that
every connected component of $X$ maps surjectively to $Y$.
Let $(E, h)$ a Hermitian vector bundle on $X$ such that
on each connected component of $X$, $E$ has the same rank $r$. Then,
\[
\widehat{{c}}_1 \left( \det Rf_*(E), h_Q^{\overline{E}} \right) -
r \widehat{{c}}_1 \left( \det Rf_*({\mathcal{O}}_X), h_Q^{\overline{{\mathcal{O}}}_X}
\right) \in
\widehat{\operatorname{CH}}_{L^1}^1(Y)
\]
and
\[
\widehat{{c}}_1 \left( \det Rf_*(E), h_Q^{\overline{E}} \right) -
r \widehat{{c}}_1 \left( \det Rf_*({\mathcal{O}}_X), h_Q^{\overline{{\mathcal{O}}}_X}
\right) = f_* \left( \widehat{{c}}_1 (E, h) \right)
\]
in $\widehat{\operatorname{CH}}_{L^1}^1(Y)_{{\mathbb{Q}}}$,
where $h_Q^{\overline{E}}$ and $h_Q^{\overline{{\mathcal{O}}}_X}$ are
the Quillen metric of $\det Rf_*(E)$ and $\det Rf_*({\mathcal{O}}_X)$ respectively.
\end{Theorem}
{\sl Proof.}\quad
Let $X = \coprod_{\alpha \in A} X_{\alpha}$ be the decomposition into connected
components of $X$. Since $f$ is proper, $A$ is a finite set.
We set $f_{\alpha} = \rest{f}{X_{\alpha}}$ and $(E_{\alpha}, h_{\alpha}) = \rest{(E, h)}{X_{\alpha}}$.
Then
\[
Rf_*(E) = \bigoplus_{\alpha \in A} R (f_{\alpha})_*(E_{\alpha}),\quad
Rf_*({\mathcal{O}}_X) = \bigoplus_{\alpha \in A} R (f_{\alpha})_*({\mathcal{O}}_{X_{\alpha}}),\quad
\widehat{{c}}_1 (E, h) = \sum_{\alpha \in A} \widehat{{c}}_1 (E_{\alpha}, h_{\alpha}).
\]
Hence we have the following:
\[
\begin{cases}
{\displaystyle \widehat{{c}}_1 \left( \det Rf_*(E), h_Q^{\overline{E}} \right) =
\sum_{\alpha \in A} \widehat{{c}}_1 \left( \det R(f_{\alpha})_*(E_{\alpha}),
h_Q^{\overline{E}_{\alpha}} \right)}, \\
{\displaystyle \widehat{{c}}_1 \left( \det Rf_*({\mathcal{O}}_X), h_Q^{\overline{{\mathcal{O}}}_X} \right) =
\sum_{\alpha \in A} \widehat{{c}}_1 \left( \det R(f_{\alpha})_*({\mathcal{O}}_{X_{\alpha}}),
h_Q^{\overline{{\mathcal{O}}}_{X_{\alpha}}} \right)}, \\
{\displaystyle f_* \left( \widehat{{c}}_1 (E, h) \right) =
\sum_{\alpha \in A} f_* \left( \widehat{{c}}_1 (E_{\alpha}, h_{\alpha}) \right)}.
\end{cases}
\]
Thus, we may assume that $X$ is connected, i.e.,
$X$ is an arithmetic variety.
\medskip
Let $K=K(Y)$ and $L=K(X)$
be the function fields of
$Y$ and $X$ respectively.
Let $n$ be the degree of $f$
and
$\omega_1,\omega_2,\cdots,\omega_n$
rational functions on $X$
such that at the generic point they form a basis of $K$-vector space $L$.
Further, let $s_1,s_2,\ldots,s_r$
be rational sections of $E$
such that at the generic point they form a basis of
$L$-vector space $E_L$.
Then $s_1\omega_1\operatornamewithlimits{\wedge} s_1\omega_2\operatornamewithlimits{\wedge}\cdots\operatornamewithlimits{\wedge}
s_1\omega_n\operatornamewithlimits{\wedge}\cdots\operatornamewithlimits{\wedge} s_r\omega_n$,
$s_1\operatornamewithlimits{\wedge}\cdots\operatornamewithlimits{\wedge} s_r$
and
$\omega_1\operatornamewithlimits{\wedge}\cdots\operatornamewithlimits{\wedge}\omega_n$
are non-zero rational sections of
$\det f_*(E)$,
$\det (E)$
and
$\det f_*({\mathcal{O}}_X)$ respectively.
Here we shall prove the following equality in
$\widehat{Z}_D^1(Y)$:
\addtocounter{Claim}{1}
\begin{multline}
\label{eqn:desired:thm:arith:Riemann:Roch:gen:finite:morphism}
\left(\operatorname{div} \left(\bigwedge_{ik}s_i\omega_k\right),
\left[-\log h_Q^{\overline E}
\left(\bigwedge_{ik}s_i\omega_k,\bigwedge_{ik}s_i\omega_k\right)\right]
\right) \\
- r
\left(\operatorname{div}\left(\bigwedge_k\omega_k\right),
\left[-\log h_Q^{\overline {\mathcal{O}}_X}
\left(\bigwedge_{k}\omega_k,\bigwedge_{k}\omega_k\right)\right]
\right) \\
= f_*\left(\operatorname{div}\left(\bigwedge_i s_i\right),
\left[-\log \det h\left(\bigwedge_i s_i,\bigwedge_i s_i\right)\right]
\right),
\end{multline}
where
$\bigwedge_{ik}s_i\omega_k=s_1\omega_1\operatornamewithlimits{\wedge}
s_1\omega_2\operatornamewithlimits{\wedge}\cdots\operatornamewithlimits{\wedge}
s_1\omega_n\operatornamewithlimits{\wedge}\cdots\operatornamewithlimits{\wedge} s_r\omega_n$,
$\bigwedge_k\omega_k
=\omega_1\operatornamewithlimits{\wedge}\cdots\operatornamewithlimits{\wedge}\omega_n$ and
$\bigwedge_i s_i=s_1\operatornamewithlimits{\wedge}\cdots\operatornamewithlimits{\wedge} s_r$.
First we shall show the equality of divisors.
Let $Y_0$ be the maximal Zariski open set
of $X$ such that $f$ is flat over $Y_0$.
Then, $\operatorname{codim}_Y(Y \setminus Y_0)\ge 2$
by \cite[III,Proposition~9.7]{Hartshorne}.
Since $f$ is generically finite,
$f$ is in fact finite over $Y_0$.
Then $Z^1(Y)=Z^1(Y_0)$ and thus
it suffices to prove the equality of divisors over $Y_0$.
Since it suffices to prove it locally, let
$U=\operatorname{Spec}(A)$
be an affine open set of $Y_0$
and $f^{-1}(U)=\operatorname{Spec}(B)$
the open set of $X_0=f^{-1}(Y_0)$.
Shrinking $U$ if necessary,
we may assume that
$B$ is a free $A$-module of rank $n$
and that
$E$ is a free $B$-module of rank $r$.
Let $d_1,d_2,\cdots,d_n$
be a basis of $B$ over $A$,
and
$e_1,e_2,\cdots,e_r$
be a basis of $E$ over $B$.
Note that $K$ and $L$ are the
quotient fields of $A$ and $B$ respectively.
In the following we freely identify a rational function (or
section) by the corresponding element at the generic point.
In this sense, we set
\begin{align*}
\omega_k = \sum_{l=1}^{n}a^{kl}d_l \quad (k=1,2,\cdots,n) \\
s_i = \sum_{j=1}^{r}\sigma_{ij}e_j \quad (i=1,2,\cdots,r),
\end{align*}
where
$a^{kl} \in K\,(1\le k,l\le n)$
and
$\sigma_{ij}\in L\,(1\le i,j\le r)$.
For each
$\sigma_{ij}(1\le i,j\le r)$,
let
$T_{\sigma_{ij}} : L \to L$
be multiplication by
$\sigma_{ij}$.
With respect to a basis
$\omega_1,\omega_2,\cdots,\omega_n$
of $L$ over $K$,
$T_{\sigma_{ij}}$ gives rise to
the matrix
$(c_{ij}^{kl})_{1\le k,l\le n}\in M_n(K)$
defined by
\[
\sigma_{ij}\omega_k
=\sum_{l=1}^n c_{ij}^{kl}\omega_l
\quad (k=1,2,\cdots,n).
\]
We also denote this matrix by
$T_{\sigma_{ij}}$.
Then,
{\allowdisplaybreaks
\begin{align*}
\bigwedge_{ik}s_i\omega_k
& = \bigwedge_{ik}\left(\sum_{j=1}^{r}
\sigma_{ij}e_j\right)\omega_k
= \bigwedge_{ik}\left(\sum_{j=1}^{r}\sum_{l=1}^{n}
c_{ij}^{kl}\right)e_j\omega_l \\
& = \det (c_{ij}^{kl})_{ik,jl}
\bigwedge_{ik}e_i\omega_k
= \det (c_{ij}^{kl})_{ik,jl}
\bigwedge_{ik}e_i\left(\sum_{l=1}^{n}a^{kl}d_l\right) \\
& = \det (c_{ij}^{kl})_{ik,jl}
\bigwedge_{ik}\left(\sum_{j=1}^{r}
\delta_{ij}a^{kl}\right)e_j \omega_l
= \det (c_{ij}^{kl})_{ik,jl}
\det (\delta_{ij}a^{kl})_{ik,jl}
\bigwedge_{ik}e_i d_l.
\end{align*}}
On the other hand, since the matrices
$T_{\sigma_{ij}}$ and $T_{\sigma_{i'j'}}$
commute with each other, we have
{\allowdisplaybreaks
\begin{align*}
\det (c_{ij}^{kl})_{ik,jl}
& = \det
\begin{pmatrix}
T_{\sigma_{11}} & T_{\sigma_{12}} & \cdots &
T_{\sigma_{1r}} \\
T_{\sigma_{21}} & T_{\sigma_{22}} & \cdots &
T_{\sigma_{2r}} \\
\vdots & \vdots & \ddots & \vdots \\
T_{\sigma_{r1}} & T_{\sigma_{r2}} & \cdots &
T_{\sigma_{rr}} \\
\end{pmatrix}\\
& = \det \left(
\sum_{\tau\in {\mathfrak{S}}_r}
\operatorname{sign}(\tau)
T_{\sigma_{1\tau(1)}}\cdot\cdots\cdot
T_{\sigma_{r\tau(r)}}
\right) \\
& = \det(T_{\det(\sigma_{ij})_{ij}}) \\
& = \operatorname{Norm}_{L/K}(\det(\sigma_{ij})_{ij}).
\end{align*}}
Moreover, we have
\begin{align*}
\det (\delta_{ij}a^{kl})_{ik,jl}
& = \det (I_r \otimes (a^{kl})_{kl}) \\
& = (\det (a^{kl})_{kl})^r.
\end{align*}
>From the above three equalities,
$\operatorname{div}\left(\bigwedge_{ik}s_i\omega_k\right)$
is given by the rational function
\[
\operatorname{Norm}_{L/K}(\det(\sigma_{ij})_{ij})
(\det (a^{kl})_{kl})^r.
\]
Further
\[
\bigwedge_{i}s_i = (\det (\sigma_{ij})_{ij})
\bigwedge_{i} e_k
\quad\text{and}\quad
\bigwedge_{k}\omega_k = (\det (a^{kl})_{kl})
\bigwedge_{k} d_k.
\]
Hence we have
\[
\operatorname{div}\left(\bigwedge_{ik}s_i\omega_k\right)
-r\left(\operatorname{div}\left(\bigwedge_k\omega_k\right)\right)
= f_*\left(\operatorname{div}\left(\bigwedge_i s_i\right)\right).
\]
Next we shall show the equality of currents.
Since all the currents in the equality come from
locally integrable functions by
Proposition~\ref{prop:push:forward:B:pq} and
Proposition~\ref{prop:Quillen:metric:generalised:gen:finite:morph},
it suffices to show the equality over a non-empty Zariski open set of
every connected component of $Y({\mathbb{C}})$.
So let $W_0$
be a non-empty Zariski open set of a connected component of $Y({\mathbb{C}})$
such that $f_{{\mathbb{C}}}$ is \'{e}tale
and that
$s_i\, (1 \le i \le r)$ or $\omega_k \,(1 \le k \le n)$
have no poles or zeroes over there.
Then over $W_0$
all these currents are defined by $C^{\infty}$ functions.
Let
$y \in Y({\mathbb{C}})$
be a complex point
and
$x_1,x_2,\cdots,x_n$
be the fiber $f_{{\mathbb{C}}}^{-1}(y)$ over $y$.
>From the proof of Lemma~\ref{lem:formula:for:Quillen:metric},
as $C^{\infty}$ functions around $y$,
{\allowdisplaybreaks
\begin{align*}
-\log h_Q^{\overline E}
\left(\bigwedge_{ik}s_i\omega_k,\bigwedge_{ik}s_i\omega_k\right)(y)
& = -\log \det \left\{
\vert \det (\Omega) \vert ^{2r}
\prod_{\alpha=1}^n
\det \left(h(s_i,s_j)(x_{\alpha})\right)_{ij}
\right\},
\end{align*}}
where
$\Omega = (\omega_k(x_\alpha))_{k \alpha}$
and
$H(x_\alpha) = (h(s_i,s_j)(x_{\alpha}))_{ij}$. Also,
{\allowdisplaybreaks
\begin{align*}
-\log h_Q^{\overline {\mathcal{O}}_X}
\left(\bigwedge_{k}\omega_k,\bigwedge_{k}\omega_k\right)(y)
& = -\log \det
\vert \det (\Omega) \vert ^{2}.
\end{align*}}
On the other hand, by the definition of the push-forward $f_*$,
\begin{align*}
f_* \left(
-\log \det h\left(\bigwedge_i s_i,\bigwedge_i s_i\right)
\right)(y)
& = \sum_{\alpha =1}^n
-\log \det h\left(\bigwedge_i s_i,\bigwedge_i s_i\right)(x_\alpha) \\
& = \sum_{\alpha =1}^n
-\log \det \left(
h(s_i,s_j)(x_\alpha)
\right)_{ij}.
\end{align*}
Hence we have the desired equality of
currents by the above three equalities.
Thus we have showed the equality
(\ref{eqn:desired:thm:arith:Riemann:Roch:gen:finite:morphism}).
Since the right hand side belongs in fact to
$\widehat{Z}_{L^1}^1(Y)$,
the left hand side must also belong to
$\widehat{Z}_{L^1}^1(Y)$,
and thus we have the equality in
$\widehat{Z}_{L^1}^1(Y)$. \QED
\section{Arithmetic Riemann-Roch for stable curves}
\subsection{Bismut-Bost formula}
\setcounter{Theorem}{0}
Let $X$ be a smooth algebraic variety over ${\mathbb{C}}$,
$L$ a line bundle on $X$,
and $h$ a generalized metric of $L$ over $X$.
Let $s$ be a rational section of $L$. Then, by the definition of
the generalized metric $h$, $-\log h(s, s)$ gives rise to a current
$-[\log h(s,s)]$. Moreover, it is easy to see that a current
\[
dd^c(-[\log h(s,s)]) + \delta_{\operatorname{div}(s)}
\]
does not depend on the choice of $s$. Thus, we define $c_1(L, h)$ to be
\[
c_1(L, h) = dd^c(-[\log h(s,s)]) + \delta_{\operatorname{div}(s)}.
\]
Let $f : X \to Y$ be a proper morphism of smooth algebraic varieties
${\mathbb{C}}$ such that every fiber of $f$ is a reduced and connected curve
with only ordinary double singularities.
We set $\Sigma = \{ x \in X \mid \text{$f$ is not smooth at $x$.} \}$ and
$\Delta = f_*(\Sigma)$.
Let $|\Delta|$ be the support of $\Delta$.
We fix a Hermitian metric of $\omega_{X/Y}$. Then, in \cite{BBQm},
Bismut and Bost proved the following.
\begin{Theorem}
\label{thm:Quillen:metric:stable:curve}
Let $\overline{E} = (E, h)$ be a Hermitian vector bundle on $X$.
Then, the Quillen metric $h_Q^{\overline{E}}$ of $\det Rf_*(E)$ on
$Y \setminus |\Delta|$ gives
rise to a generalized metric of $\det Rf_*(E)$ on $Y$.
Moreover,
\[
c_1 \left( \det Rf_*(E), h_Q^{\overline{E}} \right) = - f_* \left[
\operatorname{td}({\overline{\omega}_{X/Y}}^{-1}) \operatorname{ch}(\overline{E}) \right]^{(2,2)}
- \frac{\operatorname{rk} E}{12} \delta_{\Delta}.
\]
\end{Theorem}
\subsection{Riemann-Roch for stable curves}
\setcounter{Theorem}{0}
In this subsection, we prove the arithmetic Riemann-Roch theorem
for stable curves.
\begin{Theorem}
\label{thm:arith:Riemann:Roch:stable:curves}
Let $f : X \to Y$ be a projective morphism of regular arithmetic varieties
such that every fiber of $f_{{\mathbb{C}}} : X({\mathbb{C}}) \to Y({\mathbb{C}})$ is a reduced
and connected curve with only ordinary double singularities.
We fix a Hermitian metric of the dualizing sheaf $\omega_{X/Y}$.
Let $\overline{E} = (E, h)$ be a Hermitian vector bundle on $X$.
Then,
\[
\widehat{{c}}_1 \left( \det Rf_*(E), h_Q^{\overline{E}} \right) -
\operatorname{rk} (E) \widehat{{c}}_1 \left( \det Rf_*({\mathcal{O}}_X), h_Q^{\overline{{\mathcal{O}}}_X}
\right) \in
\widehat{\operatorname{CH}}_{L^1}^1(Y)
\]
and
\begin{multline*}
\widehat{{c}}_1 \left( \det Rf_*(E), h_Q^{\overline{E}} \right) -
\operatorname{rk} (E) \widehat{{c}}_1 \left( \det Rf_*({\mathcal{O}}_X), h_Q^{\overline{{\mathcal{O}}}_X}
\right) \\ = f_* \left( \frac{1}{2} \left(
\widehat{{c}}_1 (\overline{E})^2 -
\widehat{{c}}_1 (\overline{E}) \cdot \widehat{{c}}_1 (\overline{\omega}_{X/Y}) \right)
- \widehat{{c}}_2 (\overline{E})
\right)
\end{multline*}
in $\widehat{\operatorname{CH}}_{L^1}^1(Y)_{{\mathbb{Q}}}$,
where $h_Q^{\overline{E}}$ and $h_Q^{\overline{{\mathcal{O}}}_X}$ are
the Quillen metric of $\det Rf_*(E)$ and $\det Rf_*({\mathcal{O}}_X)$ respectively.
\end{Theorem}
{\sl Proof.}\quad We prove the theorem in two steps.
{\bf Step 1.}\quad
First, we assume that $f_{{\mathbb{Q}}} : X_{{\mathbb{Q}}} \to Y_{{\mathbb{Q}}}$ is
smooth. In this case, by \cite{GSRR},
\[
\widehat{{c}}_1 \left( \det Rf_*(E), h_Q^{\overline{E}} \right) =
f_* \left( \widehat{\operatorname{ch}}(E, h)\widehat{\operatorname{td}}(Tf, h_f) -
a(\operatorname{ch}(E_{{\mathbb{C}}})\operatorname{td}(Tf_{{\mathbb{C}}})R(Tf_{{\mathbb{C}}}))
\right)^{(1)}.
\]
in $\widehat{\operatorname{CH}}^1(Y)_{{\mathbb{Q}}}$.
Since
\[
\widehat{\operatorname{ch}}(\overline{E}) = \operatorname{rk}(E) + \widehat{{c}}_1(\overline{E}) +
\left( \frac{1}{2} \widehat{{c}}_1(\overline{E})^2 -
\widehat{{c}}_2(\overline{E}) \right) + \text{(higher terms)}
\]
and
\[
\widehat{\operatorname{td}}(Tf, h_f) = 1 - \frac{1}{2} \widehat{{c}}_1(\overline{\omega}_{X/Y}) +
\widehat{\operatorname{td}}_2(Tf, h_f) + \text{(higher terms)},
\]
we have
\[
\left( \widehat{\operatorname{ch}}(E, h)\widehat{\operatorname{td}}(Tf, h_f) \right)^{(2)} =
\frac{1}{2} \left(
\widehat{{c}}_1 (\overline{E})^2 -
\widehat{{c}}_1 (\overline{E}) \cdot \widehat{{c}}_1 (\overline{\omega}_{X/Y}) \right)
- \widehat{{c}}_2 (\overline{E}) + \operatorname{rk}(E) \widehat{\operatorname{td}}_2(Tf, h_f).
\]
On the other hand,
since the power series $R(x)$ has no constant term, the $(1,1)$ part of
\[
\operatorname{ch}(E_{{\mathbb{C}}})\operatorname{td}(Tf_{{\mathbb{C}}})R(Tf_{{\mathbb{C}}})
\]
is $\operatorname{rk}(E) R_1(Tf_{{\mathbb{C}}})$, where $R_1(Tf_{{\mathbb{C}}})$ is the $(1,1)$ part of
$R(Tf_{{\mathbb{C}}})$. Therefore, we obtain
\addtocounter{Claim}{1}
\begin{multline}
\label{eqn:1:thm:arith:Riemann:Roch:stable:curves}
\widehat{{c}}_1 \left( \det Rf_*(E), h_Q^{\overline{E}} \right) =
f_* \left( \frac{1}{2} \left(
\widehat{{c}}_1 (\overline{E})^2 -
\widehat{{c}}_1 (\overline{E}) \cdot \widehat{{c}}_1 (\overline{\omega}_{X/Y}) \right)
- \widehat{{c}}_2 (\overline{E})
\right) \\
+ \operatorname{rk}(E) f_* \left( \widehat{\operatorname{td}}_2(Tf, h_f) - a(R_1(Tf_{{\mathbb{C}}})) \right).
\end{multline}
Applying (\ref{eqn:1:thm:arith:Riemann:Roch:stable:curves})
to the case $(E, h) = ({\mathcal{O}}_X, h_{can})$,
we have
\addtocounter{Claim}{1}
\begin{equation}
\label{eqn:2:thm:arith:Riemann:Roch:stable:curves}
\widehat{{c}}_1 \left( \det Rf_*({\mathcal{O}}_X), h_Q^{\overline{{\mathcal{O}}}_X} \right) =
f_* \left( \widehat{\operatorname{td}}_2(Tf, h_f) - a(R_1(Tf_{{\mathbb{C}}})) \right).
\end{equation}
Thus, combining (\ref{eqn:1:thm:arith:Riemann:Roch:stable:curves})
and (\ref{eqn:2:thm:arith:Riemann:Roch:stable:curves}),
we have our formula in the case where
$f_{{\mathbb{Q}}} : X_{{\mathbb{Q}}} \to Y_{{\mathbb{Q}}}$ is smooth.
{\bf Step 2.}\quad
Next, we consider the general case.
The first assertion is a consequence of Theorem~\ref{thm:Quillen:metric:stable:curve} because
using Theorem~\ref{thm:Quillen:metric:stable:curve},
\begin{multline*}
{c}_1 \left( \det Rf_*(E), h_Q^{\overline{E}} \right) -
\operatorname{rk} (E) {c}_1 \left( \det Rf_*({\mathcal{O}}_X), h_Q^{\overline{{\mathcal{O}}}_X}
\right) \\
=
- f_* \left[
\operatorname{td}({\overline{\omega}_{X/Y}}^{-1}) \operatorname{ch}(\overline{E})
\right]^{(2,2)}
+ \operatorname{rk} (E) f_* \left[
\operatorname{td}({\overline{\omega}_{X/Y}}^{-1}) \operatorname{ch}(\overline{{\mathcal{O}}}_X)
\right]^{(2,2)}
\end{multline*}
belongs to $L^1_{\operatorname{loc}}(\Omega_{Y({\mathbb{C}})}^{1,1})$
by Proposition~\ref{prop:push:forward:B:pq}.
The second assertion is a consequence of the useful Lemma~\ref{lem:criterion:linear:equiv:B:cycle}.
In fact, both sides of the second assertion are arithmetic $L^1$-cycles
on Y by the first assertion and the Proposition~\ref{prop:push:forward:arith:cycle}:
If we take
$\Delta=\{y \in Y_{{\mathbb{Q}}} \mid \text{$f_{{\mathbb{Q}}}$ is not smooth over $y$} \}$
and define $\overline {\Delta}$ to be the closure of $\Delta$ in $Y$, then
the compliment $U=Y \setminus \overline {\Delta}$
contains no irreducible
components of fibers of $Y \to \operatorname{Spec}({\mathbb{Z}})$
and $f_{{\mathbb{C}}}$ is smooth over $U({\mathbb{C}})$:
The arithmetical linear equivalence of both sides restricted to $U$ is
a consequence of Step~1.
Thus by Lemma~\ref{lem:criterion:linear:equiv:B:cycle}, we also have our
formula in the general case.
\QED
\section{Asymptotic behavior of analytic torsion}
\renewcommand{\theTheorem}{\arabic{section}.\arabic{Theorem}}
\renewcommand{\theequation}{\arabic{section}.\arabic{Theorem}}
Let $M$ be a compact K\"{a}hler manifold of dimension $d$,
$\overline{E} = (E, h_E)$ a flat vector bundle of rank $r$ on $M$
with a flat metric $h_E$, and
$\overline{A} = (A, h_A)$ a Hermitian vector bundle on $M$.
For $0 \leq q \leq d$, let $\Delta_{q,n}$ be the Laplacian on
$A^{0,q}\left( \operatorname{Sym}^n(\overline{E}) \otimes \overline{A}
\right)$ and
$\Delta_{q,n}'$
the restriction of $\Delta_{q,n}$
to $\operatorname{Image} \partial \oplus \operatorname{Image} \overline{\partial}$. Let
$\sigma(\Delta_{q,n}') = \{ 0<\lambda_1 \leq \lambda_2 \leq \cdots \}$
be the sequence of eigenvalues of
$\Delta_{q,n}'$.
Here we count each eigenvalue up to its multiplicity.
Then, the associated zeta function $\zeta_{q,n}(s)$ is given by
\[
\zeta_{q,n}(s) = \operatorname{Tr}\left[ (\Delta_{q,n}')^{-s} \right] =
\sum_{i=1}^{\infty} \lambda_i^{-s}.
\]
It is well known that
$\zeta_{q,n}(s)$ converges absolutely for $\Re(s)>d$
and that it has a meromorphic continuation to the whole complex plane
without pole at $s=0$. The analytic torsion
$T \left( \operatorname{Sym}^n(\overline{E}) \otimes \overline{A} \right)$
is defined by
\[
T \left( \operatorname{Sym}^n(\overline{E}) \otimes \overline{A} \right)
= \sum_{q=0}^{d} (-1)^q q \zeta_{q,n}'(0).
\]
In the following we closely follow \cite[\S 2]{Vojta}.
The Theta function associated with $\sigma(\Delta_{q,n}')$ is
defined by
\[
\Theta_{q,n}(t) = \operatorname{Tr} \left[ \exp (-t \Delta_{q,n}') \right]
= \sum_{i=1}^{\infty} e^{- \lambda_i t}.
\]
By taking Mellin transformation, we have, for $\Re(s)>d$,
\[
\zeta_{q,n}(s) =
\frac{1}{\Gamma(s)}
\int_0^{\infty} \Theta_{q,n}(t) t^s \frac{dt}{t}.
\]
We put
\[
\tilde \zeta_{q,n}(s) =
\frac{1}{\operatorname{rk} (\operatorname{Sym}^n(E))} n^{-d} \frac{1}{\Gamma(s)}
\int_0^{\infty} \Theta_{q,n}\left(\frac{t}{n}\right) t^s \frac{dt}{t}.
\]
Then we have
\[
\frac{1}{\operatorname{rk} (\operatorname{Sym}^n(E))} n^{-d} \zeta_{q,n}(s)
= n^{-s} \tilde \zeta_{q,n}(s)
\]
and thus
\addtocounter{Theorem}{1}
\begin{equation}
\label{eqn:1:relation:zeta:zero}
\frac{1}{\operatorname{rk} (\operatorname{Sym}^n(E))} n^{-d} \zeta_{q,n}'(0)
= -(\log n) \tilde \zeta_{q,n}(0) + \zeta_{q,n}'(0)
\end{equation}
Bismut and Vasserot~\cite[(14),(19)]{BVAT} showed that
$\Theta_{q,n}(t)$ has the following properties
(note that these
parts of \cite{BVAT} do not depend on the assumption of
positivity of a line bundle, as indicated in Vojta
\cite[Proposition~2.7.3]{Vojta}):
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item For every $k \in {\mathbb{N}}$, $0 \leq q \leq d$ and $n \in {\mathbb{N}}$,
there are real numbers $a_{q,n}^j\;(-d \leq j \leq k)$
such that
\[
\frac{1}{\operatorname{rk} (\operatorname{Sym}^n(E))} n^{-d} \Theta_{q,n}\left(\frac{t}{n}\right)
= \sum_{j=-d}^{k} a_{q,n}^j t^j + o(t^k)
\]
as $t \downarrow 0$,
with $o(t^k)$ uniform with respect to $n \in {\mathbb{N}}$.
\item For every $0
\leq q \leq d$ and $j \geq -d$, there are real numbers
$a_{q}^j$ such that
\[
a_{q,n}^j = a_{q}^j + O\left(\frac{1}{\sqrt n}\right)
\]
as $n \to \infty$.
\end{enumerate}
Also by (b),
we can replace the $o(t^k)$ in
(a) by $O(t^{k+1})$ and still have the uniformity statement.
Thus we can write, for every $k \in {\mathbb{N}}$,
\[
\frac{1}{\operatorname{rk} (\operatorname{Sym}^n(E))} n^{-d} \Theta_{q,n}\left(\frac{t}{n}\right)
= \sum_{j=-d}^{k} a_{q,n}^k t^j + \rho_{q,n}^k(t)
\]
with $\rho_{q,n}^k(t) = o(t^{k+1})$. Then
{\allowdisplaybreaks
\begin{align*}
\tilde \zeta_{q,n}(s) & =
\frac{1}{\operatorname{rk} (\operatorname{Sym}^n(E))} n^{-d} \frac{1}{\Gamma(s)}
\int_1^{\infty} \Theta_{q,n}\left(\frac{t}{n}\right) t^s \frac{dt}{t} \\
& \qquad + \frac{a_{q,n}^j}{\Gamma(s)}
\int_0^1 t^{j+s-1} dt
+
\sum_{j=-d}^{k} \frac{1}{\Gamma(s)}
\int_0^1 \rho_{q,n}^k(t) dt \\
& =
\frac{1}{\operatorname{rk} (\operatorname{Sym}^n(E))} n^{-d} \frac{1}{\Gamma(s)}
\int_1^{\infty} \Theta_{q,n}\left(\frac{t}{n}\right) t^s \frac{dt}{t} \\
& \qquad + \sum_{j=-d}^{k} \frac{a_{q,n}^j}{\Gamma(s) (j+s)}
+ \frac{1}{\Gamma(s)}
\int_0^1 \rho_{q,n}^k(t)t^s \frac{dt}{t}.
\end{align*}
}
In the last expression, the first integral is holomorphic
for all $s \in {\mathbb{C}}$, while the second integral is holomorphic for
$\Re (s) > -k-1$; the middle term is a meromorphic function in the
whole complex plane.
Putting $k=0$ and $s=0$ in the above equation, we have
\addtocounter{Theorem}{1}
\begin{equation}
\label{eqn:3:zeta:zero}
\tilde \zeta_{q,n}(0) = a_{q,n}^0.
\end{equation}
Moreover, by differentiating the above equation
when $k=0$, we have
\addtocounter{Theorem}{1}
\begin{multline}
\label{eqn:4:diff:zeta:zero}
\tilde \zeta_{q,n}'(0) =
\frac{1}{\operatorname{rk} (\operatorname{Sym}^n(E))} n^{-d}
\int_1^{\infty} \Theta_{q,n}\left(\frac{t}{n}\right) \frac{dt}{t} \\
+ \sum_{j=-d}^{-1} \frac{a_{q,n}^j}{j} - a_{q,n}^0 \Gamma'(1)
+ \frac{1}{\Gamma(s)}
\int_0^1 \rho_{q,n}^0(t) \frac{dt}{t}.
\end{multline}
We have now the following Proposition.
\begin{Proposition}
\label{prop:lower:bound:zeta:function}
There exists a constant c such that for all $n \in {\mathbb{N}}$,
\[
\zeta_{q,n}'(0) \geq -c n^{d+r-1} \log n
\]
\end{Proposition}
{\sl Proof.}\quad
By (\ref{eqn:1:relation:zeta:zero}),
(\ref{eqn:3:zeta:zero}) and (\ref{eqn:4:diff:zeta:zero}), we have
\begin{multline*}
\zeta_{q,n}'(0)
=
- \operatorname{rk} (\operatorname{Sym}^n(E)) n^d (\log n) a_{q,n}^0 \\
+
\operatorname{rk} (\operatorname{Sym}^n(E)) n^d \left(
\frac{1}{\operatorname{rk} (\operatorname{Sym}^n(E))} n^{-d}
\int_1^{\infty} \Theta_{q,n}\left(\frac{t}{n}\right) \frac{dt}{t}
\right. \\
\left.
+ \sum_{j=-d}^{-1} \frac{a_{q,n}^j}{j} - a_{q,n}^0 \Gamma'(1)
+ \frac{1}{\Gamma(s)}
\int_0^1 \rho_{q,n}^0(t) \frac{dt}{t}
\right)
\end{multline*}
In the first term of the right hand side,
$a_{q,n}^0$ is bounded with respect to $n$ by (b).
In the second term of the right hand side, the first integral
is non-negative;
the sum of $a_{q,n}^j$'s
is bounded with respect to $n$ by (b);
the term $- a_{q,n}^0 \Gamma'(1)$
is also bounded with respect to $n$ by (b);
the second integral is also bounded with respect to $n$,
for $\rho_{q,n}^0(t)=O(t)$ uniformly with respect to $n$.
Moreover,
\[
\operatorname{rk} (\operatorname{Sym}^n(E)) = \binom{n+r-1}{r-1}=O(n^{r-1})
\]
as $n \to \infty$. Thus, there is a constant $c$
such that for all $n \in {\mathbb{N}}$,
\[
\zeta_{q,n}'(0) \geq -c n^{d+r-1} \log n.
\]
\QED
\medskip
In the following sections, we only need the case of $d=1$, namely
where $M$ is a compact Riemann surface. In this case,
the above Proposition~\ref{prop:lower:bound:zeta:function}
gives an asymptotic upper bound of analytic torsion.
\begin{Corollary}
\label{cor:asymp:analytic:torsion}
Let $C$ be a compact Riemann surface,
$\overline{E} = (E, h_E)$ a flat vector bundle of rank $r$ on $C$
with a flat metric $h$, and
$\overline{A} = (A, h_A)$ a Hermitian vector bundle on $C$.
Then, there is a constant $c$ such that for all $n \in {\mathbb{N}}$,
\[
T \left( \operatorname{Sym}^n(\overline{E}) \otimes \overline{A} \right)
\leq c n^r \log n.
\]
\end{Corollary}
{\sl Proof.}\quad
Since $\dim C = 1$
\[
T \left( \operatorname{Sym}^n(\overline{E}) \otimes \overline{A} \right)
= -\zeta_{1,n}'(0).
\]
Now the corollary follows from
Proposition~\ref{prop:lower:bound:zeta:function}.
\QED
\renewcommand{\theTheorem}{\arabic{section}.\arabic{subsection}.\arabic{Theorem}}
\renewcommand{\theequation}{\arabic{section}.\arabic{subsection}.\arabic{Theorem}.\arabic{Claim}}
\section{Formulae for arithmetic Chern classes}
\subsection{Arithmetic Chern classes of symmetric powers}
\label{subsec:formula:chern:sym:power}
\setcounter{Theorem}{0}
Let $M$ be a complex manifold and $(E, h)$
a Hermitian vector bundle on $M$.
Since $E^{\otimes n}$ has the natural Hermitian metric
$h^{\otimes n}$,
we can define a Hermitian metric $\operatorname{Sym}^n(h)$ of $\operatorname{Sym}^n(E)$
to be the quotient metric of $E^{\otimes n}$ in terms of
the natural surjective homomorphism $E^{\otimes n} \to \operatorname{Sym}^n(E)$.
We denote $(\operatorname{Sym}^n(E), \operatorname{Sym}^n(h))$ by $\operatorname{Sym}^n(E, h)$.
If $x \in M$ and $\{ e_1, \ldots, e_r \}$ is an
orthonormal basis of $E_x$ with respect to $h_x$,
then it is easy to see that
\[
(\operatorname{Sym}^n(h))_x \left( e_1^{\alpha_1} \cdots e_r^{\alpha_r},
e_1^{\beta_1} \cdots e_r^{\beta_r} \right) =
\begin{cases}
{\displaystyle \frac{\alpha_1 ! \cdots \alpha_r !}{n !}} &
\text{if $(\alpha_1, \ldots, \alpha_r) = (\beta_1, \ldots, \beta_r)$}, \\
0 & \text{otherwise}.
\end{cases}
\]
Then we have the following proposition.
\begin{Proposition}
\label{prop:chern:class:sym:power}
Let $X$ be an arithmetic variety and
$\overline{E} = (E, h)$ a Hermitian vector
bundle of rank $r$ on $X$. Then, we have the following.
\begin{enumerate}
\renewcommand{\labelenumi}{(\arabic{enumi})}
\item
${\displaystyle \widehat{{c}}_1 \left( \operatorname{Sym}^n(\overline{E}) \right) =
\frac{n}{r} \binom{n+r-1}{r-1} \widehat{{c}}_1(\overline{E}) +
a \left( \sum_{\substack{\alpha_1 + \cdots + \alpha_r = n, \\
\alpha_1 \geq 0, \ldots, \alpha_r \geq 0}}
\log \left( \frac{n!}{\alpha_1 ! \cdots \alpha_r !} \right) \right)}$.
\item
If $X$ is regular, then
\begin{multline*}
\widehat{\operatorname{ch}}_2 \left( \operatorname{Sym}^n(\overline{E}) \right) =
\binom{n+r}{r+1} \widehat{\operatorname{ch}}_2(\overline{E}) +
\frac{1}{2} \binom{n+r-1}{r+1} \widehat{{c}}_1(\overline{E})^2 \\
+ a \left(
\frac{n}{r}
\sum_{\substack{\alpha_1 + \cdots + \alpha_r = n, \\
\alpha_1 \geq 0, \ldots, \alpha_r \geq 0}}
\log \left( \frac{n!}{\alpha_1 ! \cdots \alpha_r !} \right)
c_1(\overline{E}) \right).
\end{multline*}
\end{enumerate}
\end{Proposition}
{\sl Proof.}\quad
In \cite{SoVan}, C. Soul\'{e} gives similar formulae in implicit forms.
We follow his idea to calculate them.
\medskip
(1)
First of all, we fix notation.
We set
\[
S_{r, n} = \{
(\alpha_1, \ldots, \alpha_r) \in
({\mathbb{Z}}_{+})^r \mid \alpha_1 + \cdots + \alpha_r = n \},
\]
where ${\mathbb{Z}}_{+} = \{ x \in {\mathbb{Z}} | x \geq 0 \}$.
For $I = (\alpha_1, \ldots, \alpha_r) \in S_{r, n}$ and
rational sections $s_1, \ldots, s_r$ of $E$,
we denote
$s_1^{\alpha_1} \cdots s_r^{\alpha_r}$ by $s^I$ and
$\alpha_1 ! \cdots \alpha_r !$ by $I !$.
Let $s_1, \ldots, s_r$ be independent
rational sections of $E$.
Then, $\{ s^I \}_{I \in S_{r,n}}$
forms independent rational sections of $\operatorname{Sym}^n(E)$.
First, let us see that
\addtocounter{Claim}{1}
\begin{equation}
\label{eqn:1:prop:chern:class:sym:power}
\operatorname{div} \left( \bigwedge_{I \in S_{r,n}} s^I \right)
= \frac{n}{r} \binom{n+r-1}{r-1}
\operatorname{div} ( s_1 \wedge \cdots \wedge s_r ).
\end{equation}
This is a local question.
So let $x \in X$ and $\{ \omega_1, \ldots, \omega_r \}$
be a local basis of $E$ around $x$.
We set $s_i = \sum_{j=1}^r a_{ij}\omega_j$.
Then,
$s_1 \wedge \cdots \wedge s_r = \det(a_{ij})
\omega_1 \wedge \cdots \wedge \omega_r$.
Let $K$ be a rational function field of $X$.
Since the characteristic of $K$ is zero,
any $1$-dimensional representation of
$\operatorname{GL}_r(K)$ is a power of the determinant.
Thus, there is an integer $N$ with
\[
\bigwedge_{I \in S_{r,n}} s^I =
\det(a_{ij})^N \bigwedge_{I \in S_{r,n}} \omega^I.
\]
Here, by an easy calculation, we can see that
\[
N = \frac{n}{r} \binom{n+r-1}{r-1}.
\]
Thus, we get
(\ref{eqn:1:prop:chern:class:sym:power}).
Next, let us see that
\addtocounter{Claim}{1}
\begin{multline}
\label{eqn:2:prop:chern:class:sym:power}
- \log \det \left(
\operatorname{Sym}^n(h)(s^I, s^J)
\right)_{I,J \in S_{r,n}} = \\
- \frac{n}{r} \binom{n+r-1}{r-1}
\log \det (h(s_i, s_j))_{i,j} +
\sum_{I \in S_{r,n}}
\log \left( \frac{n !}{I !} \right).
\end{multline}
Let $x \in X({\mathbb{C}})$ and $\{ e_1, \ldots, e_r \}$ an
orthonormal basis of $E \otimes \kappa(x)$.
We set $s_i = \sum_{i=1}^r b_{ij}e_j$.
Moreover, we set
$s^I = \sum_{J \in S_{r,n}} b_{IJ}e^J$.
Then, in the same way as before,
$\det (b_{IJ}) = \det (b_{ij})^N$.
Further, since
\[
\operatorname{Sym}^n(h)(s^I, s^J) = \sum_{I', J' \in S_{r,n}}
b_{II'}\operatorname{Sym}^n(h)(e^{I'}, e^{J'}) \overline{b_{J'J}},
\]
we have
\begin{align*}
\det \left( \operatorname{Sym}^n(h)(s^I, s^J) \right)_{I,J \in S_{r,n}} & =
|\det(b_{IJ})|^2
\det \left( \operatorname{Sym}^n(h)(e^I, e^J) \right)_{I,J \in S_{r,n}} \\
& =
|\det(b_{ij})|^{2N} \prod_{I \in S_{r,n}} \frac{I !}{n !}.
\end{align*}
Thus, we get (\ref{eqn:2:prop:chern:class:sym:power}).
Therefore, combining (\ref{eqn:1:prop:chern:class:sym:power})
and (\ref{eqn:2:prop:chern:class:sym:power}),
we obtain (1).
\medskip
(2)
First, we recall an elementary fact.
Let $\Phi \in {\mathbb{R}}[X_1, \ldots, X_r]$ be a symmetric homogeneous
polynomial, and $M_r({\mathbb{C}})$ the algebra of complex $r \times r$
matrices.
Then, there is a unique polynomial map
$\underline{\Phi} : M_r({\mathbb{C}}) \to {\mathbb{C}}$ such that
$\underline{\Phi}$ is invariant under conjugation by $\operatorname{GL}_r({\mathbb{C}})$
and its value on a diagonal matrix
$\operatorname{diag}(\lambda_1, \ldots, \lambda_r)$ is equal to
$\Phi(\lambda_1, \ldots, \lambda_r)$.
Let us consider the natural homomorphism
\[
\rho_{r,n} : \operatorname{Aut}_{{\mathbb{C}}}({\mathbb{C}}^r) \to \operatorname{Aut}_{{\mathbb{C}}}(\operatorname{Sym}^n({\mathbb{C}}^r))
\]
as complex Lie groups, which induces a homomorphism
\[
\gamma_{r, n} = d(\rho_{r,n})_{\operatorname{id}}
: \operatorname{End}_{{\mathbb{C}}}({\mathbb{C}}^r) \to \operatorname{End}_{{\mathbb{C}}}(\operatorname{Sym}^n({\mathbb{C}}^r))
\]
as complex Lie algebras.
Let $\{ e_1, \ldots, e_r \}$ be the standard basis of ${\mathbb{C}}^r$.
Then, $\{ e_I \}_{I \in S_{r,n}}$ forms a basis of
$\operatorname{Sym}^n({\mathbb{C}}^r)$, where $e_I = e_{1}^{\alpha_1} \cdots e_r^{\alpha_r}$
for $I = (\alpha_1, \ldots, \alpha_r)$.
Let us consider the symmetric polynomial
\[
\operatorname{ch}_2^{r,n} = \frac{1}{2}
\sum_{I \in S_{r,n}} X_I^2
\]
in ${\mathbb{R}}[ X_I ]_{I \in S_{r,n}}$.
Then, by the previous remark, using the basis $\{ e_I \}_{I \in S_{r,n}}$,
we have a polynomial map
\[
\underline{\operatorname{ch}_2^{r,n}} : \operatorname{End}_{{\mathbb{C}}}(\operatorname{Sym}^n({\mathbb{C}}^r))
\to {\mathbb{C}}
\]
such that
$\underline{\operatorname{ch}_2^{r,n}}$ is
invariant under conjugation by $\operatorname{Aut}_{{\mathbb{C}}}(\operatorname{Sym}^n({\mathbb{C}}^r))$
and
\[
\underline{\operatorname{ch}_2^{r,n}}
\left( \operatorname{diag}(\lambda_I)_{I \in S_{r, n}} \right) =
\operatorname{ch}_2^{r,n}(\ldots, \lambda_I, \ldots).
\]
Here we consider a polynomial map given by
\[
\begin{CD}
\theta_{r, n} :
\operatorname{End}_{{\mathbb{C}}}({\mathbb{C}}^r) @>{\gamma_{r,n}}>>
\operatorname{End}_{{\mathbb{C}}}(\operatorname{Sym}^n({\mathbb{C}}^r))
@>{\underline{\operatorname{ch}_2^{r,n}}}>> {\mathbb{C}}.
\end{CD}
\]
Since $\gamma_{r,n}(P A P^{-1}) =
\rho_{r,n}(P)\gamma_{r,n}(A)\rho_{r,n}(P)^{-1}$ for
all $A \in \operatorname{End}_{{\mathbb{C}}}({\mathbb{C}}^r)$ and $P \in \operatorname{Aut}_{{\mathbb{C}}}({\mathbb{C}}^r)$,
$\theta_{r,n}$ is invariant under conjugation by
$\operatorname{Aut}_{{\mathbb{C}}}({\mathbb{C}}^r)$.
Let us calculate
\[
\theta_{r,n}(\operatorname{diag}(\lambda_1, \ldots, \lambda_r)).
\]
First of all,
\[
\gamma_{r,n}(\operatorname{diag}(\lambda_1, \ldots, \lambda_r)) =
\operatorname{diag}\left( \ldots,
\left( \alpha_1 \lambda_1 + \cdots + \alpha_r \lambda_r \right),
\ldots \right)_{(\alpha_1, \ldots, \alpha_r) \in S_{r,n}}.
\]
Thus,
\[
\theta_{r,n}(\operatorname{diag}(\lambda_1, \ldots, \lambda_r)) =
\frac{1}{2}
\sum_{(\alpha_1, \ldots, \alpha_r) \in S_{r,n}}
\left( \alpha_1 \lambda_1 + \cdots + \alpha_r \lambda_r \right)^2.
\]
On the other hand, by easy calculations,
we can see that
\[
\sum_{(\alpha_1, \ldots, \alpha_r) \in S_{r,n}}
\left( \alpha_1 \lambda_1 + \cdots + \alpha_r \lambda_r \right)^2 =
\binom{n+r}{r+1}\left( \lambda_1^2 + \cdots + \lambda_r^2 \right) +
\binom{n+r-1}{r+1}\left( \lambda_1 + \cdots + \lambda_r \right)^2.
\]
Therefore, we get
\[
\theta_{r,n}(\operatorname{diag}(\lambda_1, \ldots, \lambda_r)) =
\frac{1}{2} \binom{n+r}{r+1}\left( \lambda_1^2 + \cdots + \lambda_r^2 \right) +
\frac{1}{2} \binom{n+r-1}{r+1}\left( \lambda_1 + \cdots + \lambda_r \right)^2.
\]
Hence,
\addtocounter{Claim}{1}
\begin{equation}
\label{eqn:1:(2):prop:chern:class:sym:power}
\theta_{r,n} = \binom{n+r}{r+1} \underline{\operatorname{ch}_2} +
\frac{1}{2} \binom{n+r-1}{r+1} \underline{(c_1)^2},
\end{equation}
where ${\displaystyle \operatorname{ch}_2(X_1, \ldots, X_r) =
\frac{1}{2}(X_1^2 + \cdots + X_r^2)}$ and
$c_1(X_1, \ldots, X_r) = X_1 + \cdots + X_r$.
Let $M$ be a complex manifold and $\overline{F} = (F, h_F)$
a Hermitian vector bundle of rank $r$ on $M$.
Let $K_{\overline{F}}$ be the curvature form of $\overline{F}$, and
$K_{\operatorname{Sym}^n(\overline{F})}$ the curvature form of $\operatorname{Sym}^n(\overline{F})$.
Then,
\[
K_{\operatorname{Sym}^n(\overline{F})} =
\left( \gamma_{r, n} \otimes \operatorname{id}_{A^{1,1}(M)} \right)
(K_{\overline{F}}).
\]
Thus, by (\ref{eqn:1:(2):prop:chern:class:sym:power}),
\addtocounter{Claim}{1}
\begin{equation}
\label{eqn:2:(2):prop:chern:class:sym:power}
\operatorname{ch}_2 \left( \operatorname{Sym}^n(F,h_F) \right) =
\binom{n+r}{r+1} \operatorname{ch}_2(F,h_F) +
\frac{1}{2} \binom{n+r-1}{r+1} {c}_1(F,h_F)^2.
\end{equation}
Now let $\overline{E} = (E, h)$ be a Hermitian vector bundle on
a regular arithmetic variety $X$.
Let $h'$ be another Hermitian metric of $E$. Then, using the definition
of Bott-Chern secondary characteristic classes and
(\ref{eqn:2:(2):prop:chern:class:sym:power}),
\begin{multline*}
\widehat{\operatorname{ch}}_2 \left( \operatorname{Sym}^n(E, h) \right) -
\widehat{\operatorname{ch}}_2 \left( \operatorname{Sym}^n(E, h') \right) = \\
a \left(
\binom{n+r}{r+1} \widetilde{\operatorname{ch}_2}(E, h, h') +
\frac{1}{2} \binom{n+r-1}{r+1} \widetilde{{c}_1^2}(E, h, h')
\right).
\end{multline*}
Thus,
\[
\widehat{\operatorname{ch}}_2 \left( \operatorname{Sym}^n(E, h) \right) -
\binom{n+r}{r+1} \widehat{\operatorname{ch}}_2(E, h) -
\frac{1}{2} \binom{n+r-1}{r+1} \widehat{{c}}_1(E, h)^2
\]
does not depend on the choice of the metric $h$.
Therefore, in order to show (2),
by using splitting principle \cite[3.3.2]{GSCh},
we may assume that
\[
(E, h) = \overline{L}_1 \oplus \cdots \oplus \overline{L}_r,
\]
where $\overline{L}_i = (L_i, h_i)$'s
are Hermitian line bundles.
Then,
\[
\operatorname{Sym}^n(\overline{E}) =
\bigoplus_{\substack{\alpha_1 + \cdots + \alpha_r = n, \\
\alpha_1 \geq 0, \ldots, \alpha_r \geq 0}}
\overline{L}_1^{\otimes \alpha_1} \otimes \cdots \otimes
\overline{L}_r^{\otimes \alpha_r} \otimes
\left( {\mathcal{O}}_X, \frac{\alpha_1 ! \cdots \alpha_r !}{n!} h_{can} \right).
\]
Therefore, $\widehat{\operatorname{ch}}_2 \left( \operatorname{Sym}^n(\overline{E}) \right)$
is equal to
\[
\sum_{\substack{\alpha_1 + \cdots + \alpha_r = n, \\
\alpha_1 \geq 0, \ldots, \alpha_r \geq 0}}
\left\{
\widehat{\operatorname{ch}}_2 \left( \overline{L}_1^{\otimes \alpha_1} \otimes \cdots \otimes
\overline{L}_r^{\otimes \alpha_r} \right)
-\log \left( \frac{\alpha_1 ! \cdots \alpha_r !}{n!} \right)
a\left( {c}_1
\left( \overline{L}_1^{\otimes \alpha_1} \otimes \cdots \otimes
\overline{L}_r^{\otimes \alpha_r} \right) \right)
\right\}.
\]
On the other hand, since
\begin{multline*}
\sum_{(\alpha_1, \cdots, \alpha_r) \in S_{r,n}}
\log \left( \frac{n!}{\alpha_1 ! \cdots \alpha_r !} \right)
\left( \alpha_1 X_1 + \cdots + \alpha_r X_r \right) \\
= \left(
\frac{n}{r}
\sum_{(\alpha_1, \cdots, \alpha_r) \in S_{r,n}}
\log \left( \frac{n!}{\alpha_1 ! \cdots \alpha_r !} \right)
\right)(X_1 + \cdots + X_r),
\end{multline*}
we have
{\allowdisplaybreaks
\begin{align*}
\widehat{\operatorname{ch}}_2 \left( \operatorname{Sym}^n(\overline{E}) \right)
= & \binom{n+r}{r+1} \widehat{\operatorname{ch}}_2(\overline{E}) +
\frac{1}{2} \binom{n+r-1}{r+1} \widehat{{c}}_1(\overline{E})^2 \\
& + \sum_{(\alpha_1, \cdots, \alpha_r) \in S_{r,n}}
\log \left( \frac{n!}{\alpha_1 ! \cdots \alpha_r !} \right)
a \left( \alpha_1 {c}_1(\overline{L_1}) + \cdots +
\alpha_r{c}_1(\overline{L_r}) \right) \\
= & \binom{n+r}{r+1} \widehat{\operatorname{ch}}_2(\overline{E}) +
\frac{1}{2} \binom{n+r-1}{r+1} \widehat{{c}}_1(\overline{E})^2 \\
& \qquad
+ \left(
\frac{n}{r}
\sum_{(\alpha_1, \cdots, \alpha_r) \in S_{r,n}}
\log \left( \frac{n!}{\alpha_1 ! \cdots \alpha_r !} \right)
\right)
a ( {c}_1(\overline{E})).
\end{align*}
}
Thus, we get (2).
\QED
\subsection{Arithmetic Chern classes of
$\overline{E} \otimes \overline{E}^{\vee}$}
\setcounter{Theorem}{0}
Here, let us consider arithmetic Chern classes of
$\overline{E} \otimes \overline{E}^{\vee}$.
\begin{Proposition}
\label{prop:ch2:end}
Let $X$ be a regular arithmetic variety and
$(E, h)$ a Hermitian vector bundle of rank $r$ on $X$.
Then,
\[
\widehat{\operatorname{ch}}_2(E \otimes E^{\vee}, h \otimes h^{\vee})
= 2r \widehat{\operatorname{ch}}_2(E, h) - \widehat{{c}}_1(E, h)^2
= (r-1) \widehat{{c}}_1(E, h)^2 - 2r \widehat{{c}}_2(E, h).
\]
\end{Proposition}
{\sl Proof.}\quad
Since $\widehat{\operatorname{ch}}_i(E^{\vee}, h^{\vee}) = (-1)^i
\widehat{\operatorname{ch}}_i(E, h)$ and
$\widehat{\operatorname{ch}}(E \otimes E^{\vee}, h \otimes h^{\vee})
= \widehat{\operatorname{ch}}(E, h) \cdot \widehat{\operatorname{ch}}(E^{\vee}, h^{\vee})$,
we have
\begin{align*}
\widehat{\operatorname{ch}}_2(E \otimes E^{\vee}, h \otimes h^{\vee}) & =
r \widehat{\operatorname{ch}}_2(E, h) + \widehat{{c}}_1(E, h) \cdot
\widehat{{c}}_1(E^{\vee}, h^{\vee}) + r \widehat{\operatorname{ch}}_2(E^{\vee}, h^{\vee}) \\
& = 2r \widehat{\operatorname{ch}}_2(E, h) - \widehat{{c}}_1(E, h)^2.
\end{align*}
The last assertion is derived from
the fact
\[
\widehat{\operatorname{ch}}_2(E, h) = \frac{1}{2} \widehat{{c}}_1(E,h)^2
- \widehat{{c}}_2(E, h).
\]
\QED
\section{The proof of the relative Bogomolov's inequality in the arithmetic case}
\renewcommand{\theTheorem}{\arabic{section}.\arabic{subsection}}
The purpose of this section is to give
the proof of the following theorem.
\addtocounter{subsection}{1}
\begin{Theorem}[Relative Bogomolov's inequality in the arithmetic case]
\label{thm:relative:Bogomolov:inequality:arithmetic:case}
Let $f : X \to Y$ be a projective morphism of regular arithmetic varieties
such that every fiber of $f_{{\mathbb{C}}} : X({\mathbb{C}}) \to Y({\mathbb{C}})$ is a reduced
and connected curve with only ordinary double singularities.
Let $(E, h)$ be a Hermitian vector bundle of rank $r$ on $X$,
and $y$ a closed point of $Y_{{\mathbb{Q}}}$.
If $f$ is smooth over $y$ and $\rest{E}{X_{\bar{y}}}$ is
semi-stable, then
\[
\widehat{\operatorname{dis}}_{X/Y}(E, h) = f_* \left( 2r \widehat{{c}}_2(E, h) - (r-1)\widehat{{c}}_1(E, h)^2 \right)
\]
is weakly positive at $y$ with respect to any subsets
$S$ of $Y({\mathbb{C}})$ with the following
properties: \textup{(1)} $S$ is finite, and
\textup{(2)} $f_{{\mathbb{C}}}^{-1}(z)$ is smooth and $\rest{E_{{\mathbb{C}}}}{f_{{\mathbb{C}}}^{-1}(z)}$ is
poly-stable for all $z \in S$.
\end{Theorem}
\renewcommand{\theTheorem}{\arabic{section}.\arabic{subsection}.\arabic{Theorem}}
\subsection{Sketch of the proof of the relative Bogomolov's inequality}
The proof of the relative Bogomolov's inequality is very long, so that
for reader's convenience,
we would like to give a rough sketch of
the proof of it.
\bigskip
{\bf Step 1.}\quad
Using the Donaldson's Lagrangian, we reduce to the case where
the Hermitian metric $h$ of $E$ along $f_{{\mathbb{C}}}^{-1}(z)$ is
Einstein-Hermitian for each $z \in S$.
\medskip
{\bf Step 2.}\quad
We set
\[
\overline{F}_n = \operatorname{Sym}^n \left(
\operatorname{\mathcal{E}\textsl{nd}}(\overline{E}) \otimes f^*(\overline{H}) \right) \otimes
\overline{A} \otimes f^*(\overline{H}),
\]
where $\overline{A}$ is a Hermitian line bundle
on $X$ and $\overline{H}$ is a Hermitian line bundle on $Y$.
Later we will specify these $\overline{A}$ and $\overline{H}$.
By virtue of the arithmetic Riemann-Roch for stable curves
(cf. Theorem~\ref{thm:arith:Riemann:Roch:stable:curves}) and
formulae of arithmetic Chern classes for symmetric powers
(cf. \S\ref{subsec:formula:chern:sym:power}), we can see that
\[
\frac{1}{(r^2 + 1)!} \widehat{\operatorname{dis}}_{X/Y}(\overline{E}) =
- \lim_{n \to \infty} \frac{\widehat{{c}}_1(\det Rf_*(F_n), h_n)}{n^{r^2+1}},
\]
where $h_n$ is a generalized metric of $\det Rf_*(F_n)$
such that $\widehat{{c}}_1(\det Rf_*(F_n), h_n)
\in \widehat{\operatorname{CH}}_{L^1}^1(Y)$ and
$h_n$ coincides with the Quillen metric $h_Q^{\overline{F}_n}$
at each $z \in S$.
\medskip
{\bf Step 3.}\quad
We assume that $A$ is very ample and $A \otimes \omega_{X/Y}^{-1}$
is ample.
We choose an arithmetic variety $B \subset X$
such that $B \in |A^{\otimes 2}|$, $B \to Y$ is \'{e}tale over $y$,
and $B({\mathbb{C}}) \to Y({\mathbb{C}})$ is \'{e}tale over each $z \in S$.
(Exactly speaking, $B$ is not realized as an element of $|A^{\otimes 2}|$.
For simplicity, we assume it.)
We set $\overline{G}_n = \rest{\overline{F}_n}{B}$ and
$g = \rest{f}{B}$.
Here we suppose that
$g_*(\rest{\operatorname{\mathcal{E}\textsl{nd}}(\overline{E})}{B}) \otimes \overline{H}$ and
$g_*(\rest{\overline{A}}{B}) \otimes \overline{H}$
are generated by small sections at $y$
with respect to $S$.
Applying the Riemann-Roch formula for generically finite morphisms
(cf. Theorem~\ref{thm:arith:Riemann:Roch:gen:finite:morphism}),
we can find a generalized metric $g_n$ of
$\det g_*(G_n)$ such that
$g_n$ is equal to the Quillen metric
of $\overline{G}_n$ at each $z \in S$,
$\widehat{{c}}_1(\det g_*(G_n), g_n) \in \widehat{\operatorname{CH}}_{L^1}^1(Y)$, and
\[
\lim_{n \to \infty} \frac{\widehat{{c}}_1(\det g_*(G_n), g_n)}{n^{r^2+1}}
= 0.
\]
Let us consider the exact sequence:
\[
0 \to f_*(F_n) \to g_*(G_n) \to R^1f_*(F_n \otimes A^{\otimes -2})
\]
induced by $0 \to F_n \otimes A^{\otimes -2} \to F_n \to G_n \to 0$.
Let $Q_n$ be the image of
\[
g_*(G_n) \to R^1f_*(F_n \otimes A^{\otimes -2}).
\]
The natural $L^2$-metric of $g_*(G_n)$
around $z$ induces the quotient metric $\tilde{q}_n$
of $Q_n$ around $z$ for each $z \in S$.
Thus, we can find a $C^{\infty}$ metric $q_n$ of $\det Q_n$
such that $q_n$ is equal to $\det \tilde{q}_n$ at each $z \in S$.
Since
\[
\det Rf_*(F_n) = \det g_*(G_n) \otimes (\det Q_n)^{\otimes -1} \otimes
\left( \det R^1f_*(F_n) \right)^{\otimes -1},
\]
we have the generalized metric $t_n$ of $\det R^1f_*(F_n)$ such that
\[
(\det Rf_*(F_n), h_n) =
(\det g_*(G_n), g_n) \otimes (\det Q_n, q_n)^{\otimes -1} \otimes
(\det R^1f_*(F_n), t_n)^{\otimes -1}.
\]
{\bf Step 4.}\quad
We set $
a_n = \max_{z \in S} \{ \log t_n(s_n,s_n)(z) \}$,
where $s_n$ is the canonical section of
$\det R^1f_*(F_n)$.
In this step, we will show that
$\widehat{{c}}_1(\det Q_n, q_n)$ is semi-ample at $y$ with respect to $S$ and
$a_n \leq O(n^{r^2}\log(n))$.
The semi-ampleness of $\widehat{{c}}_1(\det Q_n, q_n)$
at $y$ is derived from
Proposition~\ref{prop:find:small:section} and the fact
that $g_*(\rest{\operatorname{\mathcal{E}\textsl{nd}}(\overline{E})}{B}) \otimes \overline{H}$ and
$g_*(\rest{\overline{A}}{B}) \otimes \overline{H}$
are generated by small sections at $y$ with respect to $S$.
The estimation of $a_n$ involves asymptotic behavior of analytic torsion
(cf. Corollary~\ref{cor:asymp:analytic:torsion})
and a comparison of sup-norm with $L^2$-norm
(cf. Lemma~\ref{lem:comparison:sup:L2}).
{\bf Step 5.}\quad
Thus, using the last equation in Step 3,
we can get a decomposition
\[
-\frac{\widehat{{c}}_1(\det Rf_*(F_n), h_n)}{n^{r^2+1}} =
\alpha_n + \beta_n
\]
such that $\alpha_n$ is semi-ample at $y$ with respect to $S$ and
$\lim_{n \to \infty} \beta_n = 0$.
\subsection{Preliminaries}
\setcounter{Theorem}{0}
First of all, we will prepare three lemmas for the proof
of the relative Bogomolov's inequality.
\begin{Lemma}
\label{lem:comparison:sup:L2}
Let $M$ be a $d$-dimensional compact K\"{a}hler manifold,
$\overline{E} = (E, h)$ a flat Hermitian vector bundle
of rank $r$ on $M$, and $\overline{V} = (V, k)$
a Hermitian line bundle.
Then, there is a constant $c$ such that,
for any $n > 0$ and any $s \in H^0(M, \operatorname{Sym}^n(E) \otimes V)$,
\[
\Vert s \Vert_{\sup} \leq c n^{d+r-1} \Vert s \Vert_{L^2}.
\]
\end{Lemma}
{\sl Proof.}\quad
Let $f : P = \operatorname{Proj} \left(\bigoplus_{i \geq 0} \operatorname{Sym}^i(E) \right) \to M$
be the projective bundle of $E$, and
$L = {\mathcal{O}}_P(1)$ the tautological line bundle of $E$ on $P$.
Let $h_L$ be the quotient metric of $L$ induced by
the surjective homomorphism $f^*(E) \to L$ and
the Hermitian metric $f^*(h)$ of $f^*(E)$.
Let $\Omega_M$ be a K\"{a}hler form of $M$.
Since $\overline{E}$ is flat,
$c_1(L, h_L)$ is positive semi-definite of rank $r-1$.
Thus, $f^*(\Omega_M) + c_1(L, h_L)$ gives rise to
a fundamental $2$-form $\Omega_P$ on $P$.
Moreover, by virtue of the flatness of $\overline{E}$,
we have $c_1(L, h_L)^r = 0$.
Thus,
\[
\Omega_P^{d+r-1} = \binom{d+r-1}{d} f^*(\Omega_M^d) \wedge c_1(L, h_L)^{r-1}.
\]
By \cite[Lemma~30]{GSRR}, there is a constant $c$ such that
\[
\Vert s' \Vert_{\sup} \leq c n^{d+r-1} \Vert s' \Vert_{L^2}
\]
for any $n > 0$ and any $s' \in H^0(P, L^{\otimes n} \otimes f^*(V))$,
where ${\displaystyle \Vert s' \Vert_{L^2} = \int_{P} |s'|^2 \Omega_P^{d+r-1}}$.
We denote a homomorphism
\[
f^*(\operatorname{Sym}^n(E)) \otimes f^*(V) \to L^{\otimes n} \otimes f^*(V)
\]
by $\alpha_n$.
As in the proof of \cite[(44)]{GSRR}, we can see that,
for any $s \in H^0(M, \operatorname{Sym}^n(E) \otimes V)$,
\[
|s|^2 = \binom{n+r-1}{r-1}
\int_{P \to M} | \alpha_n(s) |^2 c_1(L, h_L)^{r-1}.
\]
Thus,
\[
|s|^2 \leq \binom{n+r-1}{r-1}
\int_{P \to M} \Vert \alpha(s) \Vert_{\sup}^2 c_1(L, h_L)^{r-1}
= \binom{n+r-1}{r-1} \Vert \alpha(s) \Vert_{\sup}^2.
\]
Therefore, we get
\[
\Vert s \Vert_{\sup}^2 \leq \binom{n+r-1}{r-1} \Vert \alpha_n(s) \Vert_{\sup}^2
\]
for all $s \in H^0(M, \operatorname{Sym}^n(E) \otimes V)$.
On the other hand,
{\allowdisplaybreaks
\begin{align*}
\Vert \alpha_n(s) \Vert_{L^2}^2 & =
\int_{P} |\alpha_n(s)|^2 \Omega_P^{r} \\
& =
\binom{d+r-1}{d} \int_M \int_{P \to M} |\alpha_n(s)|^2 f^*(\Omega_M^d) \wedge c_1(L, h_L)^{r-1} \\
& =
\binom{d+r-1}{d} \int_M \Omega_M^d \int_{P \to M} |\alpha_n(s)|^2 c_1(L, h_L)^{r-1} \\
& =
\binom{d+r-1}{d} \binom{n+r-1}{r-1}^{-1} \int_M |s|^2 \Omega_M^d \\
& =
\binom{d+r-1}{d} \binom{n+r-1}{r-1}^{-1} \Vert s \Vert_{L^2}^2.
\end{align*}
}
Therefore,
\begin{align*}
\Vert s \Vert_{\sup}^2
& \leq \binom{n+r-1}{r-1} \Vert \alpha_n(s) \Vert_{\sup}^2 \\
& \leq \binom{n+r-1}{r-1} c^2 n^{2(d+r-1)} \Vert \alpha_n(s) \Vert_{L^2}^2 \\
& = \binom{d+r-1}{d} c^2 n^{2(d+r-1)} \Vert s \Vert_{L^2}^2.
\end{align*}
Thus, we get our lemma.
\QED
Here we recall Einstein-Hermitian metrics of vector bundles.
Let $M$ be a $d$-dimensional
compact K\"{a}hler manifold with a K\"{a}hler form $\Omega_M$,
and $E$ a vector bundle on $M$.
We say $E$ is {\em stable} (resp. {\em semistable})
{\em with respect to $\Omega_M$}
if, for any subsheaves $F$ of $E$
with $0 \subsetneq F \subsetneq E$,
\[
\frac{1}{\operatorname{rk} F} \int_M c_1(F) \wedge \Omega_M^{d-1}
<
\frac{1}{\operatorname{rk} E} \int_M c_1(E) \wedge \Omega_M^{d-1}.
\]
\[
\left( \text{resp.}\quad
\frac{1}{\operatorname{rk} F} \int_M c_1(F) \wedge \Omega_M^{d-1}
\leq
\frac{1}{\operatorname{rk} E} \int_M c_1(E) \wedge \Omega_M^{d-1}. \right)
\]
Moreover, $E$ is said to be {\em poly-stable with respect to $\Omega_M$}
if $E$ is semistable with respect to $\Omega_M$ and
$E$ has a decomposition $E = E_1 \oplus \cdots \oplus E_s$ of vector bundles
such that each $E_i$ is stable with respect to $\Omega_M$.
Let $h$ be a Hermitian metric of $E$.
We say $h$ is {\em Einstein-Hermitian with respect to $\Omega_M$}
if there is a constant $\rho$
such that
$K(E, h) \wedge \Omega_M^{d-1} = \rho \Omega_M^d \otimes \operatorname{id}_E$,
where $K(E, h)$ is the curvature form given by $(E, h)$ and
$\operatorname{id}_E$ is the identity map in $\operatorname{\mathcal{H}\textsl{om}}(E, E)$.
The Kobayashi-Hitchin correspondence tells us that
$E$ has an Einstein-Hermitian metric with respect to $\Omega_M$ if and only if
$E$ is poly-stable with respect to $\Omega_M$.
\begin{Lemma}
\label{lem:sum:EH:metric}
Let $M$ be a compact K\"{a}hler manifold with
a K\"{a}hler form $\Omega_M$, and $E$ a poly-stable vector
bundle with respect to $\Omega_M$ on $M$.
If $h$ and $h'$ are Einstein-Hermitian metrics of $E$
with respect to $\Omega_M$, then so is $h+h'$.
\end{Lemma}
{\sl Proof.}\quad
Let $E = E_1 \oplus \cdots \oplus E_s$ be a decomposition into
stable vector bundles.
If we set $h_i = \rest{h}{E_i}$ and
$h'_i = \rest{h'}{E_i}$ for each $i$, then $h_i$ and $h'_i$
are Einstein-Hermitian metrics of $E_i$ and we have the following
orthogonal decompositions:
\[
(E, h) = \bigoplus_{i=1}^s (E_i, h_i)
\quad\text{and}\quad
(E, h') = \bigoplus_{i=1}^s (E_i, h'_i)
\]
(cf. \cite[Chater~IV, \S~3]{Ko}).
Thus, we may assume that $E$ is stable.
In this case, by virtue of the uniqueness of Einstein-Hermitian metric,
there is a positive constant $c$ with $h' = ch$.
Thus, $h + h' = (1+c)h$. Hence $h+h'$ is Einstein-Hermitian.
\QED
\begin{Lemma}
\label{lem:polystable:complex:conjugation}
Let $C$ be a compact Riemann surface.
Considering $C$ as a projective variety over ${\mathbb{C}}$,
let $\overline{C} = C \otimes_{{\mathbb{C}}} {\mathbb{C}}$ be the tensor
product via the complex conjugation.
Let $E$ be a vector bundle on $C$, and
$\overline{E} = E \otimes_{{\mathbb{C}}} {\mathbb{C}}$ on $\overline{C}$.
Then, $E$ is poly-stable on $C$ if and only if $\overline{E}$
is poly-stable on $\overline{C}$.
\end{Lemma}
{\sl Proof.}\quad
This is an easy consequence of the fact that
if $F$ is a vector bundle on $C$, then
$\deg(F) = \deg(\overline{F})$.
\QED
\subsection{Complete proof of the relative Bogomolov's inequality}
\setcounter{Theorem}{0}
\renewcommand{\theClaim}{\arabic{section}.\arabic{subsection}.\arabic{Claim}}
\renewcommand{\theequation}{\arabic{section}.\arabic{subsection}.\arabic{Claim}}
Let us start the complete proof of the relative Bogomolov's inequality.
Considering $S \cup F_{\infty}(S)$ instead of $S$,
we may assume that $F_{\infty}(S) = S$
by virtue of Lemma~\ref{lem:polystable:complex:conjugation}.
For each $z \in S$, let $\Omega_z$ be the K\"{a}hler form induced
by the metric of $\overline{\omega}_{X/Y}$ along $f_{{\mathbb{C}}}^{-1}(z)$.
Since $\rest{E_{{\mathbb{C}}}}{f_{{\mathbb{C}}}^{-1}(z)}$ is poly-stable for all $z \in S$,
there is a $C^{\infty}$ Hermitian metric $h'$ of $E_{{\mathbb{C}}}$ such that
$\rest{h'}{f_{{\mathbb{C}}}^{-1}(z)}$ is Einstein-Hermitian with respect to $\Omega_z$
for all $z \in S$.
It is easy to see that $\rest{\overline{F_{\infty}^*(h')}}{f_{{\mathbb{C}}}^{-1}(z)}$
is Einstein-Hermitian with respect to $\Omega_z$
for all $z \in S$. Thus,
if $h'$ is not invariant under $F_{\infty}$,
then, considering $h' + \overline{F_{\infty}^*(h')}$,
we may assume that $h'$ is invariant under $F_{\infty}$.
For, by Lemma~\ref{lem:sum:EH:metric},
$h' + \overline{F_{\infty}^*(h')}$
is Einstein-Hermitian with respect to $\Omega_z$ on $f_{{\mathbb{C}}}^{-1}(z)$
for each $z \in S$.
Here we claim:
\begin{Claim}
\label{claim:assume:Einstein:Hermitian}
There is a $\gamma \in L^1_{\operatorname{loc}}(Y({\mathbb{C}}))$ such that
$a(\gamma) \in \widehat{\operatorname{CH}}_{L^1}^1(Y;S)$ and
$\gamma(z) \geq 0$ for each $z \in S$, and
\[
\widehat{\operatorname{dis}}_{X/Y}(E, h) = \widehat{\operatorname{dis}}_{X/Y}(E, h') + a(\gamma).
\]
\end{Claim}
{\sl Proof.}\quad
We set ${\displaystyle \phi = \sqrt[r]{\det(h')/\det(h)}}$.
Then, it is easy to see that
$\widehat{\operatorname{dis}}_{X/Y}(E, \phi h) = \widehat{\operatorname{dis}}_{X/Y}(E, h)$.
Thus, we may assume that $\det(h) = \det(h')$.
Then, we have
\[
\widehat{\operatorname{dis}}_{X/Y}(E, h) - \widehat{\operatorname{dis}}_{X/Y}(E, h') =
a \left( - f_* (2r \widetilde{\operatorname{ch}_2}(E, h, h')) \right).
\]
Hence if we set $\gamma = - f_* (2r \widetilde{\operatorname{ch}_2}(E, h, h'))$,
then $a(\gamma) \in \widehat{\operatorname{CH}}_{L^1}^1(Y;S)$.
On the other hand, by \cite[(ii) of Corollary~1.30]{BGSAT},
$- f_*(\widetilde{\operatorname{ch}_2}(E, h, h'))(z)$ is
nothing more than Donaldson's Lagrangian (for details,
see \cite[\S6]{MoBG}).
Thus, we get $\gamma(z) \geq 0$ for each $z \in S$.
\QED
\medskip
By the above claim, we may assume that
$\rest{h}{f_{{\mathbb{C}}}^{-1}(z)}$ is Einstein-Hermitian for each $z \in S$.
Let $\overline{A} = (A, h_A)$ be a Hermitian line bundle
on $X$ such that $A$ is very ample, and
$A \otimes \omega_{X/Y}^{\otimes -1}$ is ample.
If we take a general member $M'$ of $|A_{{\mathbb{Q}}}^{\otimes 2}|$,
then, by Bertini's theorem (cf. \cite[Theorem~6.10]{JB}),
$M'$ is smooth over ${\mathbb{Q}}$,
and $M' \to Y_{{\mathbb{Q}}}$ is \'{e}tale over $y$.
Note that if $Z$ is an algebraic set of ${\mathbb{P}}^N_{{\mathbb{C}}}$,
$U$ is a non-empty Zariski open set of ${\mathbb{P}}^N_{{\mathbb{Q}}}$, and
$U({\mathbb{Q}}) \subseteq Z({\mathbb{C}})$, then $Z = {\mathbb{P}}^N_{{\mathbb{C}}}$.
Hence, we may assume that $M'({\mathbb{C}}) \to Y({\mathbb{C}})$ is
\'{e}tale over $z$ for all $z \in S$.
Let $M' = M'_1 + \cdots + M'_{l_1} + M'_{l_1 + 1} + \cdots + M'_{l_2}$
be the decomposition of $M'$ into irreducible components
(actually, the decomposition into connected components
because $M'$ is smooth over ${\mathbb{Q}}$) such that $f_{{\mathbb{Q}}}(M'_i) = Y_{{\mathbb{Q}}}$
for $1 \leq i \leq l_1$ and $f_{{\mathbb{Q}}}(M'_j) \subsetneq Y_{{\mathbb{Q}}}$
for $l_1 + 1 \leq j \leq l_2$.
Let $M_i$ ($i=1, \ldots, l_1$) be the closure of $M'_i$ in $X$.
We set $M = M_1 + \cdots + M_{l_1}$ and
$B = M_1 \coprod \cdots \coprod M_{l_1}$ (disjoint union).
Then, there is a line bundle $L$ on $X$ with $M \in | A^{\otimes 2} \otimes L |$.
Note that $\rest{L}{X_y} \simeq {\mathcal{O}}_{X_y}$ and $
\rest{L_{{\mathbb{C}}}}{f_{{\mathbb{C}}}^{-1}(z)} \simeq {\mathcal{O}}_{f_{{\mathbb{C}}}^{-1}(z)}$ for all $z \in S$
because
$y \not\in \bigcup_{j=l_1+1}^{l_2} f_{{\mathbb{Q}}}(M'_j)$ and
$z \not\in \bigcup_{j=l_1+1}^{l_2} f_{{\mathbb{C}}}(M'_j({\mathbb{C}}))$.
We denote the morphism $B \to M \to X$ by $\iota$, and
the morphism
$B \overset{\iota}{\longrightarrow} X \overset{f}{\longrightarrow} Y$
by $g$.
We remark that the morphism $B \to M$ is an isomorphism over ${\mathbb{Q}}$.
Further, we set
\[
\overline{F} = \operatorname{\mathcal{E}\textsl{nd}}(E, h) = (E \otimes E^{\vee}, h \otimes h^{\vee}).
\]
Then, $h \otimes h^{\vee}$ is a flat metric along $f_{{\mathbb{C}}}^{-1}(z)$
for each $z \in S$
because $h \otimes h^{\vee}$ is Einstein-Hermitian and
$\deg \left( E \otimes E^{\vee} \right) = 0$ along $f_{{\mathbb{C}}}^{-1}(z)$.
We choose a Hermitian line bundle $\overline{H} = (H, h_H)$ on $Y$
such that $g_*(\iota^*(A)) \otimes H$ and
$g_*(\iota^*(F)) \otimes H$ are generated by small sections at $y$
with respect to $S$.
Moreover, we set
\[
\overline{F}_n = \operatorname{Sym}^n \left(
\overline{F} \otimes f^*(\overline{H}) \right) \otimes
\overline{A} \otimes f^*(\overline{H})
= \left( \operatorname{Sym}^n \left( F \otimes f^*(H) \right) \otimes A \otimes f^*(H),
k_n \right).
\]
\begin{Claim}
\label{claim:terms:right:R:R:formla}
There are $Z_0, \ldots, Z_{r^2} \in \widehat{\operatorname{CH}}_{L^1}^1(Y;S)_{{\mathbb{Q}}}$ and
$\beta \in L^1_{\operatorname{loc}}(Y({\mathbb{C}}))$ such that
$a(\beta) \in \widehat{\operatorname{CH}}_{L^1}^1(Y;S)$,
and
\[
f_* \left( \widehat{\operatorname{ch}}_2(\overline{F}_n) -
\frac{1}{2}
\widehat{{c}}_1(\overline{F}_n) \cdot \widehat{{c}}_1(\overline{\omega}_{X/Y})
\right) =
\frac{n^{r^2 + 1}}{(r^2 + 1) !}
f_* (\widehat{\operatorname{ch}}_2(\overline{F})) +
\sum_{i=0}^{r^2} Z_i n^i + a(b_n \beta),
\]
where
${\displaystyle
b_n = \sum_{\substack{\alpha_1 + \cdots + \alpha_{r^2} = n, \\
\alpha_1 \geq 0, \ldots, \alpha_{r^2} \geq 0}}
\log \left( \frac{n!}{\alpha_1 ! \cdots \alpha_{r^2} !} \right)}$.
\end{Claim}
{\sl Proof.}\quad
Since $\operatorname{Sym}^n(\overline{F} \otimes f^*(\overline{H})) \otimes \overline{A}
\otimes f^*(\overline{H})$
is isometric to
$\operatorname{Sym}^n(\overline{F}) \otimes f^*(\overline{H})^{\otimes (n+1)} \otimes
\overline{A}$,
\begin{multline*}
\widehat{\operatorname{ch}}_2(\overline{F}_n) =
\widehat{\operatorname{ch}}_2(\operatorname{Sym}^n(\overline{F})) +
\widehat{{c}}_1(\operatorname{Sym}^n(\overline{F})) \cdot
\widehat{{c}}_1(f^*(\overline{H})^{\otimes (n+1)} \otimes \overline{A}) \\
+\binom{n+r^2-1}{r^2-1}
\widehat{\operatorname{ch}}_2(f^*(\overline{H})^{\otimes (n+1)} \otimes \overline{A}).
\end{multline*}
Here since $\det(\overline{F}) = \overline{{\mathcal{O}}}_X$,
by Proposition~\ref{prop:chern:class:sym:power},
\[
\widehat{{c}}_1(\operatorname{Sym}^n(\overline{F})) = a(b_n)
\quad\text{and}\quad
\widehat{\operatorname{ch}}_2(\operatorname{Sym}^n(\overline{F})) = \binom{n+r^2}{r^2+1}
\widehat{\operatorname{ch}}_2(\overline{F}).
\]
Thus, by Proposition~\ref{prop:projection:formula:line:bundle},
\begin{align*}
f_* \left( \widehat{{c}}_1(\operatorname{Sym}^n(\overline{F})) \cdot
\widehat{{c}}_1(f^*(\overline{H})^{\otimes (n+1)} \otimes \overline{A}) \right)
& = f_* \left(
b_n a \left((n+1) f^*(c_1(\overline{H})) + c_1(\overline{A}) \right) \right) \\
& = a \left( b_n f_*(c_1(\overline{A})) \right).
\end{align*}
On the other hand, using the projection formula
(cf. Proposition~\ref{prop:projection:formula:line:bundle}),
\begin{align*}
f_* \left(
\widehat{\operatorname{ch}}_2(f^*(\overline{H})^{\otimes (n+1)} \otimes \overline{A}) \right)
& = \frac{1}{2} f_* \left[ \left(
(n+1) \widehat{{c}}_1(f^*(\overline{H})) + \widehat{{c}}_1(\overline{A})
\right)^2 \right] \\
& = \frac{1}{2} f_* \left[
(n+1)^2 \widehat{{c}}_1(f^*(\overline{H}))^2 +
2(n+1) \widehat{{c}}_1(f^*(\overline{H})) \cdot \widehat{{c}}_1(\overline{A}) +
\widehat{{c}}_1(\overline{A})^2 \right] \\
& = (n+1) \deg_f(A) \widehat{{c}}_1(\overline{H}) +
\frac{1}{2} f_* \left( \widehat{{c}}_1(\overline{A})^2 \right),
\end{align*}
where $\deg_f(A)$ is the degree of $A$ on the generic fiber of $f$.
Therefore, we have
\begin{multline*}
f_* \widehat{\operatorname{ch}}_2(\overline{F}_n) =
\binom{n+r^2}{r^2+1} f_* \widehat{\operatorname{ch}}_2(\overline{F}) + \\
\binom{n+r^2-1}{r^2-1} \left(
(n+1) \deg_f(A) \widehat{{c}}_1(\overline{H}) +
\frac{1}{2} f_* \left( \widehat{{c}}_1(\overline{A})^2 \right)
\right) +
a \left( b_n f_*(c_1(\overline{A})) \right).
\end{multline*}
Thus, there are $Z'_0, \ldots, Z'_{r^2} \in \widehat{\operatorname{CH}}^1(Y;S)_{{\mathbb{Q}}}$
such that
\addtocounter{Claim}{1}
\begin{equation}
\label{eqn:ch2:Fn}
f_* \widehat{\operatorname{ch}}_2(\overline{F}_n) =
\frac{n^{r^2+1}}{(r^2+1)!} f_* \widehat{\operatorname{ch}}_2(\overline{F}) +
\sum_{i=0}^{r^2} Z'_i n^i +
a \left( b_n f_*(c_1(\overline{A})) \right).
\end{equation}
Further, since
$\widehat{{c}}_1(\overline{F}_n) \cdot \widehat{{c}}_1(\overline{\omega}_{X/Y})$
is equal to
\[
\left(
\widehat{{c}}_1(\operatorname{Sym}^n(\overline{F})) + \binom{n+r^2-1}{r^2-1}
((n+1) \widehat{{c}}_1(f^*(\overline{H})) + \widehat{{c}}_1(\overline{A}) )
\right) \cdot
\widehat{{c}}_1(\overline{\omega}_{X/Y}),
\]
we have
\begin{align*}
f_* \left(
\widehat{{c}}_1(\overline{F}_n) \cdot \widehat{{c}}_1(\overline{\omega}_{X/Y})
\right)
& = a \left( b_n f_*(c_1(\overline{\omega}_{X/Y})) \right) + \\
& \qquad
\binom{n+r^2-1}{r^2-1} \left(
(n+1) (2g-2) \widehat{{c}}_1(\overline{H}) +
f_* \left( \widehat{{c}}_1(\overline{A}) \cdot
\widehat{{c}}_1(\overline{\omega}_{X/Y}) \right)
\right).
\end{align*}
Thus, there are $Z''_0, \ldots, Z''_{r^2} \in \widehat{\operatorname{CH}}^1(Y;S)_{{\mathbb{Q}}}$
such that
\addtocounter{Claim}{1}
\begin{equation}
\label{eqn:c1:Fn:c1:w}
f_* \left(
\widehat{{c}}_1(\overline{F}_n) \cdot \widehat{{c}}_1(\overline{\omega}_{X/Y})
\right) =
\sum_{i=0}^{r^2} Z''_i n^i +
a \left( b_n f_*(c_1(\overline{\omega}_{X/Y})) \right).
\end{equation}
Thus, combining (\ref{eqn:ch2:Fn}) and
(\ref{eqn:c1:Fn:c1:w}), we get our claim.
\QED
Let $h_{X/Y}$ be a $C^{\infty}$ Hermitian metric of
$\det Rf_* {\mathcal{O}}_X$ over $Y({\mathbb{C}})$
such that $h_{X/Y}$ is invariant under $F_{\infty}$.
Then, since the Quillen metric $h^{\overline{{\mathcal{O}}_X}}_Q$
of $\det Rf_* {\mathcal{O}}_X$ is a generalized metric,
there is a real valued $\phi \in L^1_{\operatorname{loc}}(Y({\mathbb{C}}))$ such that
$h^{\overline{{\mathcal{O}}_X}}_Q = e^{\phi} h_{X/Y}$ and
$F_{\infty}^*(\phi) = \phi \ (\operatorname{a.e.})$.
Adding a suitable real valued $C^{\infty}$ function $\phi'$ with
$F_{\infty}^*(\phi') = \phi'$ to $\phi$
(replace $h_{X/Y}$ by $e^{-\phi'}h_{X/Y}$ accordingly),
we may assume that $\phi(z) = 0$ for
all $z \in S$.
Here, we set
${\displaystyle h_n = \exp \left( - \binom{n+r^2-1}{r^2-1} \phi \right)
h^{\overline{F}_n}_Q}$.
Then, $h_n$ is a generalized metric of $\det Rf_* F_n$ with
$F_{\infty}^*(h_n) = \overline{h}_n \ (\operatorname{a.e.})$.
Moreover,
\begin{multline*}
\widehat{{c}}_1 \left( \det Rf_* F_n, h_n \right) -
\binom{n+r^2-1}{r^2-1} \widehat{{c}}_1 \left( \det Rf_* {\mathcal{O}}_X, h_{X/Y} \right) \\
= \widehat{{c}}_1 \left( \det Rf_* F_n, h^{\overline{F}_n}_Q \right) -
\binom{n+r^2-1}{r^2-1} \widehat{{c}}_1
\left( \det Rf_* {\mathcal{O}}_X, h^{\overline{{\mathcal{O}}_X}}_Q \right).
\end{multline*}
Here, since
\[
\widehat{{c}}_1 \left( \det Rf_* F_n, h^{\overline{F}_n}_Q \right) -
\binom{n+r^2-1}{r^2-1} \widehat{{c}}_1
\left( \det Rf_* {\mathcal{O}}_X, h^{\overline{{\mathcal{O}}_X}}_Q \right)
\in \widehat{\operatorname{CH}}_{L^1}^1(Y;S)_{{\mathbb{Q}}}
\]
by Theorem~\ref{thm:arith:Riemann:Roch:stable:curves}
and $\widehat{{c}}_1(\det Rf_* {\mathcal{O}}_X, h_{X/Y}) \in \widehat{\operatorname{CH}}^1(Y;S)$,
we have
\[
\widehat{{c}}_1 \left( \det Rf_* F_n, h_n \right) \in \widehat{\operatorname{CH}}_{L^1}^1(Y;S)_{{\mathbb{Q}}}.
\]
Further, by the arithmetic Riemann-Roch theorem for
stable curves (cf. Theorem~\ref{thm:arith:Riemann:Roch:stable:curves}),
\begin{multline*}
\widehat{{c}}_1 \left( \det Rf_*(F_n), h_n \right) -
\binom{n+r^2-1}{r^2-1} \widehat{{c}}_1
\left( \det Rf_*({\mathcal{O}}_X), h_{X/Y} \right) \\
= f_* \left( \widehat{\operatorname{ch}}_2(\overline{F}_n) -
\frac{1}{2}
\widehat{{c}}_1 (\overline{F}_n) \cdot \widehat{{c}}_1 (\overline{\omega}_{X/Y})
\right).
\end{multline*}
Therefore,
by Claim~\ref{claim:terms:right:R:R:formla},
there are $W_0, \ldots, W_{r^2} \in \widehat{\operatorname{CH}}_{L^1}^1(Y;S)_{{\mathbb{Q}}}$ and
$\beta \in L^1_{\operatorname{loc}}(Y({\mathbb{C}}))$ such that
$a(\beta) \in \widehat{\operatorname{CH}}_{L^1}^1(Y;S)$, and
\addtocounter{Claim}{1}
\begin{equation}
\label{eqn:1:proof:arith:BG:inq}
\widehat{{c}}_1 \left( \det Rf_*(F_n), h_n \right) =
\frac{n^{r^2 + 1}}{(r^2 + 1) !}
f_* (\widehat{\operatorname{ch}}_2(\overline{F})) +
\sum_{i=0}^{r^2} W_i n^i + a(b_n \beta).
\end{equation}
\begin{Claim}
\label{claim:dis:lim:c1:Fn}
${\displaystyle
\frac{1}{(r^2+1)!} \widehat{\operatorname{dis}}_{X/Y}(\overline{E}) =
-\lim_{n \to \infty}
\frac{\widehat{{c}}_1 \left( \det Rf_*(F_n), h_n \right)}{n^{r^2+1}}
}$ in $\widehat{\operatorname{CH}}_{L^1}^1(Y;S)_{{\mathbb{Q}}}$.
\end{Claim}
{\sl Proof.}\quad
By virtue of Proposition~\ref{prop:ch2:end},
$f_*(\widehat{\operatorname{ch}}_2(\overline{F})) = -\widehat{\operatorname{dis}}_{X/Y}(\overline{E})$.
Thus, by (\ref{eqn:1:proof:arith:BG:inq}),
it is sufficient to show that $0 \leq b_n \leq O(n^{r^2})$.
It is well known that
\[
\frac{\log(\theta_1) + \cdots + \log(\theta_N)}{N} \leq
\log \left( \frac{\theta_1 + \cdots + \theta_N}{N} \right)
\]
for positive numbers $\theta_1, \ldots, \theta_N$.
Thus, noting
${\displaystyle
\sum_{\substack{\alpha_1 + \cdots + \alpha_{r^2} = n, \\
\alpha_1 \geq 0, \ldots, \alpha_{r^2} \geq 0}}
\frac{n!}{\alpha_1 ! \cdots \alpha_{r^2} !} = (r^2)^n}$,
we have
\[
0 \leq
\sum_{\substack{\alpha_1 + \cdots + \alpha_{r^2} = n, \\
\alpha_1 \geq 0, \ldots, \alpha_{r^2} \geq 0}}
\log \left( \frac{n!}{\alpha_1 ! \cdots \alpha_{r^2} !} \right)
\leq
\binom{n+r^2-1}{r^2-1} \log \left(
\frac{ (r^2)^n }{\binom{n+r^2-1}{r^2-1}}
\right)
\leq O(n^{r^2}).
\]
\QED
We set $\overline{G}_n = \iota^*(\overline{F}_n)$.
Then, by Theorem~\ref{thm:arith:Riemann:Roch:gen:finite:morphism},
\[
\widehat{{c}}_1 \left( \det Rg_*(G_n), h_Q^{\overline{G}_n} \right) -
\binom{n+r^2-1}{r^2 -1}
\widehat{{c}}_1 \left( \det Rg_*({\mathcal{O}}_{B}),
h_Q^{\overline{{\mathcal{O}}}_{B}} \right) \in \widehat{\operatorname{CH}}_{L^1}^1(Y;S)
\]
and
\[
\widehat{{c}}_1 \left( \det Rg_*(G_n), h_Q^{\overline{G}_n} \right) -
\binom{n+r^2-1}{r^2 -1}
\widehat{{c}}_1 \left( \det Rg_*({\mathcal{O}}_{B}),
h_Q^{\overline{{\mathcal{O}}}_{B}}
\right)
= g_* \left( \widehat{{c}}_1 (\overline{G}_n) \right).
\]
As before, we can take a $C^{\infty}$ Hermitian metric $h_{B/Y}$
of $\det Rg_*({\mathcal{O}}_{B})$ over $Y({\mathbb{C}})$ and
a real valued $\varphi \in L^1_{\operatorname{loc}}(Y({\mathbb{C}}))$ such that
$h_Q^{\overline{{\mathcal{O}}}_{B}} = e^{\varphi} h_{B/Y}$,
$F_{\infty}^*(h_{B/Y}) = \overline{h}_{B/Y}$,
$F_{\infty}^*(\varphi) = \varphi \ (\operatorname{a.e.})$,
and $\varphi(z) = 0$ for all $z \in S$. We set
\[
g_n = \exp \left( - \binom{n+r^2-1}{r^2-1} \varphi \right)
h^{\overline{G}_n}_Q.
\]
Then,
\[
\widehat{{c}}_1 \left( \det Rg_*(G_n), g_n \right) -
\binom{n+r^2-1}{r^2 -1}
\widehat{{c}}_1 \left( \det Rg_*({\mathcal{O}}_{B}), h_{B/Y} \right)
= g_* \left( \widehat{{c}}_1 (\overline{G}_n) \right)
\]
and $\widehat{{c}}_1 \left( \det Rg_*(G_n), g_n \right)
\in \widehat{\operatorname{CH}}_{L^1}^1(Y;S)$.
Moreover, in the same as in Claim~\ref{claim:terms:right:R:R:formla},
we can see that
\[
g_* \left( \widehat{{c}}_1 (\overline{G}_n) \right) =
a(\deg(g)b_n) + \binom{n+r^2-1}{r^2-1}
\left( (n+1) g_* \widehat{{c}}_1(g^*(\overline{H})) +
g_* \widehat{{c}}_1(\iota^*(\overline{A})) \right).
\]
Thus, there are $W'_0, \ldots, W'_{r^2} \in
\widehat{\operatorname{CH}}_{L^1}^1(Y;S)_{{\mathbb{Q}}}$ such that
\[
\widehat{{c}}_1 \left( \det Rg_*(G_n), g_n \right)
= \sum_{i=0}^{r^2} W'_i n^i + a(b_n \deg(g)).
\]
Therefore, we have
\addtocounter{Claim}{1}
\begin{equation}
\label{eqn:2:proof:arith:BG:inq}
\lim_{n \to \infty}
\frac{\widehat{{c}}_1 \left( \det Rg_*(G_n), g_n \right)}{n^{r^2+1}}
= 0
\end{equation}
in $\widehat{\operatorname{CH}}_{L^1}^1(Y;S)_{{\mathbb{Q}}}$.
Let us consider an exact sequence:
\[
0 \to F_n \otimes A^{\otimes -2} \otimes L^{\otimes -1} \to F_n \to \rest{F_n}{M} \to 0.
\]
Since $F$ is semi-stable and of degree $0$
along $X_y$ and $\rest{L}{X_y} = {\mathcal{O}}_{X_y}$,
we have
\[
f_*(F_n \otimes A^{\otimes -2}\otimes L^{\otimes -1}) = 0
\]
on $Y$. Thus, the above exact sequence gives rise to
\[
0 \to f_*(F_n) \to (\rest{f}{M})_*(\rest{F_n}{M})
\to R^1f_*(F_n \otimes A^{\otimes -2}\otimes L^{\otimes -1})
\to R^1 f_*(F_n).
\]
Let $Q_n$ be the cokernel of
\[
f_*(F_n) \to (\rest{f}{M})_*(\rest{F_n}{M}) \to g_*(G_n).
\]
Let $U$ be the maximal Zariski open set of $Y$ such that
$f$ is smooth over $U$ and $g$ is \'{e}tale over $U$.
Moreover, let $U_n$ be the maximal Zariski open set of $Y$
such that
\[
\begin{cases}
\text{(a) $U_n \subset U$, } \\
\text{(b) $(\rest{f}{M})_*(\rest{F_n}{M})$ coincides with $g_*(G_n)$ over $U_n$,} \\
\text{(c) $R^1f_*(F_n) = 0$ over $U_n$, and} \\
\text{(d) $f_*(F_n)$, $g_*(G_n)$ and $Q_n$ are
locally free over $U_n$.}
\end{cases}
\]
Then, $y \in (U_n)_{{\mathbb{Q}}}$ and $S \subseteq U_n({\mathbb{C}})$.
For, since $A \otimes \omega_{X/Y}^{-1}$ is ample on $X_y$ and
$E$ is semi-stable on $X_y$,
we can see that $R^1 f_*(F_n) = 0$ around $y$, which implies that
$f_*(F_n)$ is locally free around $y$.
Further, since $f_*(F_n)$ and $(\rest{f}{M})_*(\rest{F_n}{M})$ are free at $y$,
$R^1 f_*(F_n) = 0$ around $y$, and
$(\rest{f}{M})_*(\rest{F_n}{M})$ coincides with $g_*(G_n)$ around $y$,
we can easily check that $Q_n$ is free at $y$. Thus,
$y \in (U_n)_{{\mathbb{Q}}}$.
In the same way, we can see that $S \subseteq U_n({\mathbb{C}})$.
Next let us consider a metric of $\det Q_n$.
$g_*(G_n)$ has the Hermitian metric
$(\rest{f}{M})_*\left( \rest{k_n}{M} \right)$ over $U_n({\mathbb{C}})$,
where $k_n$ is the Hermitian metric of $\overline{F}_n$.
Let $\tilde{q}_n$ be the quotient metric of $Q_n$
over $U_n({\mathbb{C}})$ induced by
$(\rest{f}{M})_*\left( \rest{k_n}{M} \right)$. Let $q_n$ be a $C^{\infty}$
Hermitian metrics of
$\det Q_n$ over $Y({\mathbb{C}})$ such that $F_{\infty}^*(q_n) = q_n$ and
$q_n(z) = \det \tilde{q}_n(z)$ for all $z \in S$.
(If $q_n$ is not invariant under $F_{\infty}$, then consider
$(1/2)\left(q_n + \overline{F_{\infty}^*(q_n)}\right)$.)
Here since
$\det Rf_*(F_n) \simeq \det f_*(F_n) \otimes \left(\det R^1 f_*(F_n)\right)^{-1}$
and
$\det f_*(F_n) \simeq \det g_*(G_n) \otimes (\det Q_n)^{-1}$,
we have
\[
\det Rf_*(F_n) \simeq \det g_*(G_n) \otimes (\det Q_n)^{-1} \otimes
\left(\det R^1 f_*(F_n)\right)^{-1}.
\]
Further, we have generalized metrics
$h_n$, $g_n$ and $q_n$ of
$\det Rf_*(F_n)$, $\det g_*(G_n)$ and $\det Q_n$.
Thus, there is a generalized metric $t_n$
of $\det R^1 f_*(F_n)$ such that
the above is an isometry.
As in the proof of Proposition~\ref{prop:find:small:section},
let us construct a section of $\det R^1f_*(F_n)$.
First, we fix a locally free sheaf $P_n$ on $Y$ and
a surjective homomorphism $P_n \to R^1 f_*(F_n)$.
Let $P'_n$ be the kernel of $P_n \to R^1 f_*(F_n)$.
Then, $P'_n$ is a torsion free sheaf and has the same rank
as $P_n$ because $R^1 f_*(F_n)$ is a torsion sheaf.
Noting that $\left( \bigwedge^{\operatorname{rk} P'_n} P'_n \right)^{*}$
is an invertible sheaf on $Y$,
we can identify $\det R^1 f_*(F_n)$ with
\[
\bigwedge^{\operatorname{rk} P_n} P_n \otimes
\left( \bigwedge^{\operatorname{rk} P'_n} P'_n \right)^{*}.
\]
Moreover, the homomorphism
$\bigwedge^{\operatorname{rk} P'_n} P'_n \to \bigwedge^{\operatorname{rk} P_n} P_n$
induced by $P'_n \hookrightarrow P_n$
gives rise to
a non-zero section $s_n$ of $\det R^1 f_*(F_n)$.
Note that $s_n(y) \not= 0$ and $s_n(z) \not= 0$
for all $z \in S$ because $R^1 f_*(F_n) = 0$ at $y$ and $z$.
Here we set
\[
a_n = \max_{z \in S} \{ \log t_n(s_n,s_n)(z) \}.
\]
By our construction, we have
\[
\widehat{{c}}_1(\det R^1 f_*(F_n), e^{-a_n}t_n) \in \widehat{\operatorname{CH}}_{L^1}^1(Y;S).
\]
and an isometry
\addtocounter{Claim}{1}
\begin{multline}
\label{eqn:isometry:det:bundle}
(\det Rf_*(F_n), h_n) \simeq \\
(\det g_*(G_n), g_n) \otimes (\det Q_n, q_n)^{-1} \otimes
(\det R^1 f_*(F_n), e^{-a_n}t_n)^{-1} \otimes ({\mathcal{O}}_Y, e^{-a_n}h_{can}).
\end{multline}
Here we claim:
\begin{Claim}
\label{claim:gen:small:sec:Q:n}
$(\det Q_n, q_n)$ is generated by small sections at $y$ with respect to $S$.
\end{Claim}
{\sl Proof.}\quad
First of all,
\[
g_*\left( \iota^*(F) \otimes g^*(H) \right) =
g_*(\iota^*(F)) \otimes H
\quad\text{and}\quad
g_*\left( \iota^*(A) \otimes g^*(H) \right) =
g_*(\iota^*(A)) \otimes H
\]
are generated by small section at $y$ with respect to $S$.
Thus, by (2) and (3) of Proposition~\ref{prop:find:small:section},
\[
g_*(G_n) = g_* \left(
\operatorname{Sym}^n (\iota^*(F) \otimes g^*(H)) \otimes \iota^*(A) \otimes g^*(H)
\right)
\]
is generated by small sections at $y$ with respect to $S$.
Thus, by (1) of Proposition~\ref{prop:find:small:section},
$(Q_n, \tilde{q}_n)$ is generated by small sections at $y$ with
respect to $S$.
Hence, by (4) of Proposition~\ref{prop:find:small:section},
$(\det Q_n, q_n)$ is generated by small sections at $y$
with respect to $S$ because
$q_n(z) = \det \tilde{q}_n (z)$ for all $z \in S$.
\QED
Next we claim:
\begin{Claim}
\label{claim:estimate:sequences:an:bn}
$a_n \leq O(n^{r^2} \log(n))$.
\end{Claim}
{\sl Proof.}\quad
It is sufficient to show that
$\log t_n(s_n,s_n)(z) \leq O(n^{r^2}\log(n))$
for each $z \in S$.
Let $\{ e_1, \ldots, e_{l_n} \}$
be an orthonormal basis of
$g_*(G_n) \otimes \kappa(z)$ with respect to
$g_*(\rest{k_n}{B})(z)$ such that
$\{ e_1, \ldots, e_{m_n} \}$ forms a basis
of $f_*(F_n) \otimes \kappa(z)$.
Then, $e_1 \wedge \cdots \wedge e_{m_n}$,
$e_1 \wedge \cdots \wedge e_{l_n}$ and
$\bar{e}_{m_n + 1} \wedge \cdots \wedge \bar{e}_{l_n}$
form bases of
$\det (f_*(F_n)) \otimes \kappa(z)$,
$\det (g_*(G_n)) \otimes \kappa(z)$, and
$\det (Q_n) \otimes \kappa(z)$ respectively, and
$(e_1 \wedge \cdots \wedge e_{m_n}) \otimes
(\bar{e}_{m_n + 1} \wedge \cdots \wedge \bar{e}_{l_n}) =
e_1 \wedge \cdots \wedge e_{l_n}$,
where $\bar{e}_{m_n + 1}, \ldots, \bar{e}_{l_n}$
are images of $e_{m_n + 1}, \ldots, e_{l_n}$ in
$Q_n \otimes \kappa(z)$.
Then,
\[
\left| (e_1 \wedge \cdots \wedge e_{m_n}) \otimes s_n^{\otimes -1}
\right|_{h_n}^2(z) =
\frac{|e_1 \wedge \cdots \wedge e_{l_n}|_{g_n}^2(z)}{
|\bar{e}_{m_n + 1} \wedge \cdots \wedge \bar{e}_{l_n}|_{q_n}^2(z)
|s_n|_{t_n}^2(z)} = |s_n|_{t_n}^{-2}(z),
\]
where $|a|_{\lambda} = \sqrt{\lambda(a,a)}$ for
$\lambda = h_n, g_n, q_n, t_n$.
Moreover, let $\Omega_{z}$ be the K\"{a}hler form
induced by the metric of $\overline{\omega}_{X/Y}$ along $f_{{\mathbb{C}}}^{-1}(z)$.
Then, there is a Hermitian metric $v_n$ of $H^0(f_{{\mathbb{C}}}^{-1}(z), F_n)$
defined by
\[
v_n(s, s')
= \int_{f_{{\mathbb{C}}}^{-1}(z)} k_n(s, s') \Omega_{z}.
\]
Here $R^1 f_*(F_n) = 0$ at $z$. Thus,
$(\det R^1 f_*(F_n))_z$ is canonically isomorphic to ${\mathcal{O}}_{Y({\mathbb{C}}), z}$.
Since $(P'_n)_z = (P_n)_z$, under the above isomorphism,
$s_n$ goes to the determinant of
$(P_n)_z \overset{\operatorname{id}}{\longrightarrow} (P_n)_z$,
namely $1 \in {\mathcal{O}}_{Y({\mathbb{C}}), z}$.
Hence, by the definition of Quillen metric,
\[
\left| (e_1 \wedge \cdots \wedge e_{m_n}) \otimes s_n^{\otimes -1}
\right|_{h_n}^2(z)
= \det( v_n(e_i, e_j) )
\exp \left( -T \left( \rest{\overline{F}_n}{f_{{\mathbb{C}}}^{-1}(z)}
\right) \right).
\]
Therefore,
\[
\log |s_n|_{t_n}^2(z) =
T \left( \rest{\overline{F}_n}{f_{{\mathbb{C}}}^{-1}(z)} \right) -
\log \det( v_n(e_i, e_j) ).
\]
By Corollary~\ref{cor:asymp:analytic:torsion},
\[
T \left( \rest{\overline{F}_n}{f_{{\mathbb{C}}}^{-1}(z)} \right)
\leq O(n^{r^2} \log(n)).
\]
Thus, in order to get our claim,
it is sufficient to show that
\[
- \log \det( v_n(e_i, e_j) ) \leq O(n^{r^2-1}\log(n)).
\]
Let $s$ be an arbitrary section of $H^0(f_{{\mathbb{C}}}^{-1}(z), F_n)$.
Then, by Lemma~\ref{lem:comparison:sup:L2},
\[
g_*\left( \rest{k_n}{B} \right)(s, s) =
\sum_{x \in g_{{\mathbb{C}}}^{-1}(z)} |s|_{k_n}^2(x) \leq
\deg(g) \sup_{x \in f_{{\mathbb{C}}}^{-1}(z)} \{ |s|_{k_n}^2(x) \} \leq
\deg(g) c^2 n^{2r^2} \Vert s \Vert^2_{L^2}
\]
for some constant $c$ independent of $n$.
Thus, by \cite[Lemma~3.4]{MoBG} and our choice of $e_i$'s,
\[
1 = \det \left( g_*\left(\rest{k_n}{B}\right)(e_i, e_j) \right)
\leq
\left( \deg(g) c^2 n^{2r^2} \right)^{\dim_{{\mathbb{C}}} H^0(f_{{\mathbb{C}}}^{-1}(z), F_n)}
\det \left( v_n(e_i, e_j) \right).
\]
Using Riemann-Roch theorem, we can easily see that
\[
\dim_{{\mathbb{C}}} H^0(f_{{\mathbb{C}}}^{-1}(z), F_n) \leq O(n^{r^2-1}).
\]
Thus, we have
\[
-\log \det( v_n(e_i, e_j) ) \leq O(n^{r^2-1}\log(n)).
\]
Hence, we obtain our claim.
\QED
Let us go back to the proof of our theorem.
By the isometry (\ref{eqn:isometry:det:bundle}), we get
\begin{align*}
-\widehat{{c}}_1(\det Rf_*(F_n), h_n) & =
-\widehat{{c}}_1( \det g_*(G_n), g_n) + \widehat{{c}}_1(\det Q_n, q_n)
+ \widehat{{c}}_1(\det R^1 f_*(F_n), e^{-a_n}t_n) - a(a_n) \\
& = \left[
\widehat{{c}}_1(\det Q_n, q_n) + \widehat{{c}}_1(\det R^1 f_*(F_n),
e^{-a_n}t_n) + a\left( \max \{ -a_n, 0 \} \right) \right]
\\ & \qquad\qquad
+ \left[ -\widehat{{c}}_1( \det g_*(G_n), g_n) +
a (\min \{ -a_n, 0 \}) \right].
\end{align*}
Here we set
\[
\begin{cases}
{\displaystyle
\alpha_n = \frac{(r^2+1)!}{n^{r^2+1}} \left[
\widehat{{c}}_1(\det Q_n, q_n) + \widehat{{c}}_1(\det R^1 f_*(F_n), e^{-a_n}t_n)
+ a\left( \max \{ -a_n, 0 \} \right)
\right],} \\
{} \\
{\displaystyle
\beta_n = \frac{(r^2+1)!}{n^{r^2+1}} \left[
-\widehat{{c}}_1( \det g_*(G_n), g_n) +
a (\min \{ -a_n, 0 \})
\right].}
\end{cases}
\]
Then,
\[
\frac{-(r^2+1)!\widehat{{c}}_1(\det Rf_*(F_n), h_n)}{n^{r^2+1}} =
\alpha_n + \beta_n.
\]
By (\ref{eqn:2:proof:arith:BG:inq})
and Claim~\ref{claim:estimate:sequences:an:bn},
${\displaystyle \lim_{n \to \infty} \beta_n = 0}$
in $\widehat{\operatorname{CH}}_{L^1}^1(Y;S)_{{\mathbb{Q}}}$.
Therefore, by Claim~\ref{claim:dis:lim:c1:Fn},
\[
\widehat{\operatorname{dis}}_{X/Y}(\overline{E}) =
\lim_{n \to \infty}
\frac{-(r^2+1)! \widehat{{c}}_1 \left( \det Rf_*(F_n), h_n \right)}
{n^{r^2+1}} = \lim_{n\to\infty} (\alpha_n + \beta_n) =
\lim_{n \to \infty} \alpha_n
\]
in $\widehat{\operatorname{CH}}_{L^1}^1(Y;S)_{{\mathbb{Q}}}$.
On the other hand, it is obvious that
\[
\widehat{{c}}_1(\det R^1 f_*(F_n), e^{-a_n}t_n)
\quad\text{and}\quad
a\left( \max \{ -a_n, 0 \} \right)
\]
is semi-ample at $y$ with respect to $S$.
By Claim~\ref{claim:gen:small:sec:Q:n},
$\widehat{{c}}_1(\det Q_n, q_n)$ is semi-ample at $y$ with respect to $S$.
Thus, $\alpha_n$ is semi-ample at $y$ with respect to $S$.
Hence we get our theorem.
\QED
\renewcommand{\theClaim}{\arabic{section}.\arabic{subsection}.\arabic{Theorem}.\arabic{Claim}}
\renewcommand{\theequation}{\arabic{section}.\arabic{subsection}.\arabic{Theorem}.\arabic{Claim}}
\section{Preliminaries for Cornalba-Harris-Bost's inequality}
\label{section:Bost:type:inequality:preparation}
This section is a preparatory one for the next section,
where we will prove the relative Cornalba-Harris-Bost's inequality
(cf. Theorem~\ref{thm:semistability:imply:average:semi-ampleness}).
Moreover, in the next section, we will see how
the relative Bogomolov's inequality
(Theorem~\ref{thm:relative:Bogomolov:inequality:arithmetic:case})
and
the relative Cornalba-Harris-Bost's inequality
(Theorem~\ref{thm:semistability:imply:average:semi-ampleness})
are related
(cf. Proposition~\ref{prop:Bogomolov:to:Bost}).
\subsection{Normalized Green forms}
\label{subsec:normalized:Green:form}
\setcounter{Theorem}{0}
Let $Y$ be a smooth quasi-projective variety over ${\mathbb{C}}$,
$\overline{E}=(E,h)$ a Hermitian vector bundle of rank $r$ on $Y$.
Let $\pi : {\mathbb{P}}(E) \to Y$ be the canonical morphism,
where ${\mathbb{P}}(E) = \operatorname{Proj} (\bigoplus_{i \ge 0} \operatorname{Sym}^{i}(E^{\lor}))$.
We equip the canonical quotient bundle ${\mathcal{O}}_{E}(1)$ on ${\mathbb{P}}(E)$
with the quotient metric via
$\pi ^* (E^{\vee}) \to {\mathcal{O}}_{E}(1)$.
We will denote this Hermitian line bundle by $\overline{{\mathcal{O}}_{E}(1)}$.
Furthermore, let $\Omega = {c}_1(\overline{{\mathcal{O}}_{E}(1)})$ be
the first Chern form.
The purpose of this subsection is that, for
every cycle $X \subset {\mathbb{P}}(E)$
whose all irreducible components map surjectively to $Y$,
we give a Green form $g_X$
such that on a general fiber,
it is an $\Omega$-normalized
Green current in the sense of \cite[2.3.2]{BGS}.
Let $X$ be a cycle of codimension $p$ on ${\mathbb{P}}(E)$ such that
every irreducible component of $X$ maps surjectively to $Y$.
An $L^1$-form $g_X$ on ${\mathbb{P}}(E)$
satisfying the following conditions is called
an {\em $\Omega$-normalized Green form},
(or simply a {\em normalized Green form} when
no confusion is likely).
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item
There are $d$-closed $L^1$-forms $\gamma_i$ of type $(p-i, p-i)$
on $Y$ ($i=0, \ldots, p$) with
\[
dd^c([g_X]) + \delta_X
= \sum_{i=0}^{p} \left[ \pi^*(\gamma_i) \wedge \Omega^i \right].
\]
\item
$\pi_*(g_X \wedge \Omega^{r-p}) = 0$.
\end{enumerate}
Note that $\gamma_p$ is the degree of $X$ along a general fiber of $\pi$.
Let $X = \sum_i a_i X_i$ be the irreducible decomposition
of $X$ as cycles.
Let $\tilde{X}_i \to X_i$ be a desingularization of $X_i$, and
$\tilde{f}_i : \tilde{X}_i \to Y$ the induced morphism.
The main result of this subsection is the following.
\begin{Proposition}
\label{prop:normalized:Green:form}
With notation as above,
there exists an $\Omega$-normalized Green form $g_X$ on ${\mathbb{P}}(E)$
satisfying the following property.
If $y \in Y$ and
$\tilde{f}_i$ is smooth over $y$ for every $i$,
then there is an open set $U$ containing $y$ such that
$\gamma_0, \ldots, \gamma_p$ are $C^{\infty}$ on $U$ and that
$\rest{g_X}{\pi^{-1}(U)}$ is a Green form of logarithmic type for $X_U$,
where $\gamma_0, \ldots, \gamma_p$ are $L^1$-forms
in the definition of $\Omega$-normalized Green form.
\end{Proposition}
To prove the above proposition,
let us begin with the following two lemmas.
\begin{Lemma}
\label{lemma:auxiliary:green:form}
There exist a Green form $g$ of logarithmic type along $X$, and
$d$-closed $C^{\infty}$ forms $\beta_i$ of type $(p-i, p-i)$
on $Y$ \textup{(}$i=0, \ldots, p$\textup{)} such that
\[
dd^c([g]) + \delta_X
= \sum_{i=0}^{p} \left[ \pi^*(\beta_i) \wedge \Omega^i \right].
\]
\end{Lemma}
{\sl Proof.}\quad
We divide the proof into three steps.
{\bf Step 1.} : The case where $Y$ is projective.
Let $g_1$ be a Green form of logarithmic type along $X$ such that
\[
dd^c([g_1]) + \delta_X
= [\omega]
\]
where $\omega$ is a smooth form on ${\mathbb{P}}(E)$.
Then, we can find a smooth form $\eta$ on ${\mathbb{P}}(E)$ of the form
\[
\eta = \sum_{i=0}^{p} \pi^*(\beta_i) \wedge \Omega^i
\]
which represents the same cohomology class as $\omega$,
where $\beta_i$ is a $d$-closed $C^{\infty}$-form of type
$(p-i, p-i)$ on $Y$.
Since $\omega - \eta$ is $d$-exact $(p,p)$-form,
by the $dd^c$-lemma, there is a smooth $(p-1,p-1)$-form $\phi$ with
$\omega - \eta = dd^c(\phi)$.
Thus, if we set $g = g_1 - \phi$,
then $g$ is of logarithmic type along $X$ and
\[
dd^c([g]) + \delta_X = dd^c([g_1]) - dd^c(\phi) + \delta_X = [\eta].
\]
\medskip
{\bf Step 2.} : Let $h'$ be another Hermitian metric of $E$, and
$\Omega'$ the Chern form of ${\mathcal{O}}_{E}(1)$ arising from $h'$.
In this step, we will prove that
if the lemma holds for $h'$, then so does it for $h$.
By our assumption,
there exist a Green form $g'$ of logarithmic type along $X$, and
$d$-closed $C^{\infty}$ forms $\beta'_i$ ($i=0, \ldots, p$)
of type $(p-i, p-i)$
on $Y$ such that
\[
dd^c([g']) + \delta_X
= \sum_{i=0}^{p} \left[ \pi^*(\beta'_i) \wedge {\Omega'}^i \right].
\]
On the other hand, there is a real $C^{\infty}$-function $a$ on
${\mathbb{P}}(E)$ with $\Omega' - \Omega = dd^c(a)$.
Here note that if $v$ is a $\partial$ and $\overline{\partial}$-closed
form on ${\mathbb{P}}(E)$, then
$dd^c(v \wedge a) = v \wedge dd^c(a)$.
Thus, it is easy to see that there is a
$C^{\infty}$ form $\theta$ on ${\mathbb{P}}(E)$ such that
\[
\sum_{i=1}^p \pi^*(\beta'_i) \wedge {\Omega'}^i =
dd^c(\theta) + \sum_{i=1}^p \pi^*(\beta'_i) \wedge {\Omega}^i.
\]
Therefore, if we set $g = g' - \theta$ and $\beta_i = \beta'_i$,
then we have our assertion for $h$.
\medskip
{\bf Step 3.} : General case.
Using Hironaka's resolution \cite{Hiro},
there is a smooth projective variety
$Y'$ over ${\mathbb{C}}$ such that $Y$ is an open set of $Y'$.
Moreover, using \cite[Exercise~5.15 in Chapter~II]{Hartshorne},
there is a coherent sheaf $E'$ on $Y'$ with
$\rest{E'}{Y} = E$.
Further, taking a birational modification along $Y' \setminus Y$
if necessary, we may assume that $E'$ is locally free.
Let $h'$ be a Hermitian metric of $E'$ over $Y'$.
Since ${\mathbb{P}}(E)$ is an Zariski open set of ${\mathbb{P}}(E')$,
let $X'$ be the closure of $X$ in ${\mathbb{P}}(E')$.
Then, by Step~1, our assertion holds for
$(E', h')$ and $X'$.
Thus, so does it for $(E, \rest{h'}{Y})$ and $X$.
Therefore, by Step~2, we can conclude our lemma.
\QED
\begin{Lemma}
\label{lemma:push:g:is:L1}
Let $g$ be a Green form of logarithmic type along $X$
and $\omega$ a $C^{\infty}$-form with $dd^c([g]) + \delta_X = [\omega]$.
If we set $\varsigma = \pi_* (g \wedge \Omega^{r-p})$, then
$\varsigma \in L^1_{loc}(Y)$ and
$dd^c([\varsigma]) \in L^1_{loc}(\Omega_Y^{1,1})$.
Moreover, if $y \in Y$ and $\tilde{f}_i$ is smooth over $y$
for every $i$,
then $\varsigma$ is $C^{\infty}$ around $y$.
\end{Lemma}
{\sl Proof.}\quad
By Proposition~\ref{prop:push:forward:B:pq},
$\varsigma$ is an $L^1$-function on $Y$ and
\begin{align*}
dd^c([\varsigma])
& = dd^c( \pi_* ([g \wedge \Omega^{r-p}]))
= \pi_* dd^c([g \wedge \Omega^{r-p}]) \\
& = \pi_* dd^c([g]) \wedge \Omega^{r-p}
= \pi_* ([\omega] \wedge \Omega^{r-p})
- \pi_* (\delta_X \wedge \Omega^{r-p}) \\
& = \pi_* [\omega \wedge \Omega^{r-p}]
- \sum_i a_i \pi_* (\delta_{X_i} \wedge \Omega^{r-p}) \\
& = \pi_* [\omega \wedge \Omega^{r-p}]
- \sum_i a_i (\tilde{f}_i)_* [\tilde{f}_i^* (\Omega^{r-p})].
\end{align*}
Thus, $dd^c([\varsigma]) \in L^1_{loc}(\Omega_Y^{1,1})$.
Moreover, if $y \in Y$ and $\tilde{f}_i$ is smooth over $y$
for every $i$, then, by the above formula,
$dd^c([\varsigma])$ is $C^{\infty}$ around $y$.
Thus, by virtue of
\cite[(i) of Theorem~1.2.2]{GSArInt},
$\varsigma$ is $C^{\infty}$ around $y$.
\QED
Let us start the proof of Proposition~\ref{prop:normalized:Green:form}.
Let $g$ be a Green form constructed in
Lemma~\ref{lemma:auxiliary:green:form}.
Then, there are $d$-closed $\beta_i$'s with
$\beta_i \in A^{p-i,p-i} (Y)$ and
\[
dd^c([g]) + \delta_X
= \sum_{i=0}^{p} \left[ \pi^*(\beta_i) \wedge \Omega^i \right].
\]
If we set
$\varsigma = \pi_* (g \wedge \Omega^{r-p})$,
then by Lemma~\ref{lemma:push:g:is:L1},
$\varsigma$ is locally an $L^1$-form. We put
\[
g_X = g - \pi^*(\varsigma) \Omega^{p-1},
\]
which is clearly locally an $L^1$-form on ${\mathbb{P}}(E)$.
We will show that $g_X$ satisfies the conditions (i) and (ii).
Using $\int_{{\mathbb{P}}(E) \to Y} \Omega^{r-1} = 1$, (ii) can be readily checked.
Moreover,
\begin{align*}
dd^c([g_X]) + \delta_X
& = \sum_{i=0}^{p} \left[ \pi^*(\beta_i) \wedge \Omega^i \right]
- dd^c [\pi^*(\varsigma) \Omega^{p-1}] \\
& = \beta_p \Omega^p +
\pi^*([\beta_{p-1}] - dd^c([\varsigma])) \wedge \Omega^{p-1} +
\sum_{i=0}^{p-2} \left[ \pi^*(\beta_i) \wedge \Omega^i \right].
\end{align*}
The remaining assertion is easily derived from
Lemma~\ref{lemma:push:g:is:L1}.
\QED
\begin{Remark}
\label{rem:norm:Green:general:fiber}
Let $y$ be a point of $Y$ such that
$\tilde{f}_i$ is smooth over $y$ for every $i$.
Then, by Proposition~\ref{prop:normalized:Green:form},
on the fiber $\pi^{-1}(y)$,
$\rest{g_X}{\pi^{-1}(y)}$ is a Green form of logarithmic type along
$X_{y}$. Moreover,
\[
dd^c ([\rest{g_X}{\pi^{-1}(y)}]) + \delta_{X_y} = \deg (X_y)
[ \rest{\Omega^p}{\pi^{-1}(y)} ]
\]
and
\[
\int_{\pi^{-1}(y)} \left( \rest{g_X}{\pi^{-1}(y)} \right)
\left( \rest{\Omega^{r-p}}{\pi^{-1}(y)} \right) = 0.
\]
Thus, $\rest{g_X}{\pi^{-1}(y)}$ is a
$\Omega$-normalized Green form on $\pi^{-1}(y)$, and it is also
a $\Omega$-normalized Green current in the sense of \cite[2.3.2]{BGS}.
\end{Remark}
\subsection{Associated Hermitian vector bundles}
\label{subsec:associated:herm:vb}
\renewcommand{\theClaim}{\arabic{section}.\arabic{subsection}.\arabic{Theorem}}
\renewcommand{\theequation}{\arabic{section}.\arabic{subsection}.\arabic{Theorem}}
\setcounter{Theorem}{0}
Let $\operatorname{\mathbf{GL}}_r = \operatorname{Spec} {\mathbb{Z}} [X_{11},X_{12},\cdots,X_{rr}]_{\det(X_{ij})}$
be the general linear group of rank $r$ and
$\operatorname{\mathbf{SL}}_r = \operatorname{Spec} {\mathbb{Z}} [X_{11},X_{12},\cdots,X_{rr}]/(\det(X_{ij})-1)$
be the special linear group of rank $r$.
Let $\rho : \operatorname{\mathbf{GL}}_r \to \operatorname{\mathbf{GL}}_R$ be a morphism of group schemes.
First, we note that
\[
\rho({\mathbb{C}}) (\overline{A}) = \overline{\rho({\mathbb{C}}) (A)},
\]
where $\rho({\mathbb{C}}) : \operatorname{\mathbf{GL}}_r({\mathbb{C}}) \to \operatorname{\mathbf{GL}}_R({\mathbb{C}})$ is the induced morphism
and $A \in \operatorname{\mathbf{GL}}_r({\mathbb{C}})$.
Indeed, the above equality is nothing but the associativity of the map
\[
\operatorname{Spec} {\mathbb{C}} \overset{-}{\longrightarrow} \operatorname{Spec} {\mathbb{C}}
\overset{A}{\longrightarrow} \operatorname{\mathbf{GL}}_r
\overset{\rho}{\longrightarrow} \operatorname{\mathbf{GL}}_R.
\]
Next, we consider the following condition for $\rho$;
\addtocounter{Theorem}{1}
\begin{equation}
\label{eqn:condition}
\rho({}^t A) = {}^t \rho (A) \qquad \text{for any $A \in \operatorname{\mathbf{GL}}_r$}.
\end{equation}
In the group scheme language,
this condition means $\rho$ commutes with the transposed morphism.
Let $\operatorname{U}_r({\mathbb{C}})
= \{A \in \operatorname{\mathbf{GL}}_r({\mathbb{C}}) \mid {}^t A \cdot \overline{A} = I_r \}$
be the unitary group of rank $r$.
If a group morphism $\rho : \operatorname{\mathbf{GL}}_r \to \operatorname{\mathbf{GL}}_R$
commutes with the transposed morphism, then
\begin{align*}
I_R & = \rho({\mathbb{C}}) (I_r) = \rho({\mathbb{C}}) ({}^t A \cdot \overline{A}) \\
& = \rho({\mathbb{C}})({}^t A) \cdot \rho({\mathbb{C}})(\overline{A})
= {}^t \rho({\mathbb{C}})(A) \cdot \overline{\rho({\mathbb{C}})(A)},
\end{align*}
namely, $\rho({\mathbb{C}})$ maps $\operatorname{U}_r({\mathbb{C}})$ into $\operatorname{U}_R({\mathbb{C}})$.
Let $k$ be an integer.
A morphism $\rho : \operatorname{\mathbf{GL}}_r \to \operatorname{\mathbf{GL}}_R$ of group schemes is said to be
{\em of degree $k$} if
\[
\rho(t I_r) = t^k I_R \qquad \text{for any $t$}.
\]
In the group scheme language,
this means that the diagram
\begin{equation*}
\begin{CD}
\operatorname{\mathbf{GL}}_1 @>{\lambda_r}>> \operatorname{\mathbf{GL}}_r \\
@V{\alpha}VV @VV{\rho}V \\
\operatorname{\mathbf{GL}}_1 @>{\lambda_R}>> \operatorname{\mathbf{GL}}_R
\end{CD}
\end{equation*}
commutes, where $\lambda_r$ and $\lambda_R$ are given by
$t \mapsto \operatorname{diag}(t,t,\cdots,t)$ and $\alpha$ is given by $t \mapsto t^k$.
\medskip
Let $Y$ be an arithmetic variety,
$\overline{E} = (E,h)$ a Hermitian vector bundle of rank $r$ on $Y$ and
$\rho : \operatorname{\mathbf{GL}}_r \to \operatorname{\mathbf{GL}}_R$ be
a morphism of group schemes satisfying
commutativity with the transposed morphism.
In the following, we will show that we can naturally construct
a Hermitian vector bundle
$\overline{E}^{\rho} = (E^{\rho},h^{\rho})$,
which we will call the
{\em associated Hermitian vector bundle} with respect to
$\overline{E}$ and $\rho$.
First, we construct $E^{\rho}$.
Let $\{ Y_{\alpha} \}$ be an affine open covering such that
$\phi_{\alpha}: E \vert_{Y_{\alpha}}
\overset{\sim}{\longrightarrow} {\mathcal{O}}_{Y_{\alpha}}^{\oplus r}$
gives a local trivialization.
On $Y_{\alpha} \cap Y_{\beta}$, we set the transition function
$g_{\alpha\beta} = \phi_{\alpha} \cdot \phi_{\beta}^{-1}$,
which can be seen as an element of
$\operatorname{\mathbf{GL}}_r(\Gamma({\mathcal{O}}_{Y_{\alpha} \cap Y_{\beta}}))$.
Then we define the associated vector bundle $E^{\rho}$
as the vector bundle of rank $R$ on $Y$
with the transition functions
$\rho(\Gamma({\mathcal{O}}_{Y_{\alpha} \cap Y_{\beta}})) (g_{\alpha\beta})$;
\[
E^{\rho} = \coprod_{\alpha} {\mathcal{O}}_{Y_{\alpha}}^{\oplus R} / \sim.
\]
Next, we define metric on $E^{\rho}$.
Let $h^{\alpha}$ be the Hermitian metric
on ${\mathcal{O}}_{Y_{\alpha}}^{\oplus r}$ over $Y_{\alpha}$
such that
$\phi_{\alpha}: E \vert_{Y_{\alpha}}
\overset{\sim}{\longrightarrow} {\mathcal{O}}_{Y_{\alpha}}^{\oplus r}$
becomes isometry over $Y_{\alpha}({\mathbb{C}})$. Let
\begin{gather*}
e_1^{\alpha} = {}^t (1,0,\cdots,0), \cdots, e_r^{\alpha} = {}^t (0,\cdots,0,1)
\in \Gamma({\mathcal{O}}_{Y_{\alpha}}^{\oplus r}), \\
f_1^{\alpha} = {}^t (1,0,\cdots,0), \cdots, f_R^{\alpha} = {}^t (0,\cdots,0,1)
\in \Gamma({\mathcal{O}}_{Y_{\alpha}}^{\oplus R})
\end{gather*}
be the standard local frames of
${\mathcal{O}}_{Y_{\alpha}}^{\oplus r}$ and ${\mathcal{O}}_{Y_{\alpha}}^{\oplus R}$.
We set
\[
H_{\alpha} = (h^{\alpha}(e_{i}^{\alpha},e_{j}^{\alpha}))_{1 \leq i,j \leq r}.
\]
Then $H_{\alpha}$ is a $C^{\infty}$-map over $Y_{\alpha}({\mathbb{C}})$
and, for each point $y$ in $Y_{\alpha}({\mathbb{C}})$,
$H_{\alpha}(y)$ is a positive definite Hermitian matrix.
Let $\rho(C^{\infty}(Y_{\alpha}({\mathbb{C}}))) :
\operatorname{\mathbf{GL}}_r(C^{\infty}(Y_{\alpha}({\mathbb{C}}))) \to \operatorname{\mathbf{GL}}_R(C^{\infty}(Y_{\alpha}({\mathbb{C}})))$
be the induced map.
\addtocounter{Theorem}{1}
\begin{Claim}
$\rho(C^{\infty}(Y_{\alpha}({\mathbb{C}}))) (H_{\alpha})$
is a $C^{\infty}$-map over $Y_{\alpha}({\mathbb{C}})$ and,
for each point $y$ in $Y_{\alpha}({\mathbb{C}})$,
$\rho(C^{\infty}(Y_{\alpha}({\mathbb{C}})))(H_{\alpha})(y)$
is a positive definite Hermitian matrix.
\end{Claim}
{\sl Proof.}\quad
The first assertion is obvious. For the second one, we note that
there is a matrix $A \in \operatorname{\mathbf{GL}}_r({\mathbb{C}})$
such that ${}^t A \cdot \overline{A} = H_{\alpha}(y)$.
Then it is easy to see that $\rho(C^{\infty}(Y_{\alpha}({\mathbb{C}})))(H_{\alpha})(y)$
is a positive definite Hermitian matrix by using \eqref{eqn:condition}.
\QED
Now we define a metric $h^{\rho_{\alpha}}$
on ${\mathcal{O}}_{Y_{\alpha}}^{\oplus R}$ over $Y_{\alpha}$ by
\[
h^{\rho_{\alpha}}(f_{k}^{\alpha},f_{l}^{\alpha})
= \rho(C^{\infty}(Y_{\alpha}({\mathbb{C}}))) (H_{\alpha})_{kl}
\]
for $1 \leq k,l \leq R$.
\addtocounter{Theorem}{1}
\begin{Claim}
$\{h^{\rho_{\alpha}}\}_{\alpha}$ glue together
to form a Hermitian metric on $E^{\rho}$.
\end{Claim}
{\sl Proof.}\quad
Let $s_{\alpha} = {}^t (s_1^{\alpha},\cdots,s_R^{\alpha})
\in \Gamma({\mathcal{O}}_{Y_{\alpha}}^{\oplus R} \vert_{Y_{\alpha} \cap Y_{\beta}})$
and $s_{\beta} = {}^t (s_1^{\beta},\cdots,s_R^{\beta})
\in \Gamma({\mathcal{O}}_{Y_{\beta}}^{\oplus R} \vert_{Y_{\alpha} \cap Y_{\beta}})$.
Then they give the same section of
$E^{\rho} \vert_{Y_{\alpha} \cap Y_{\beta}}$ if
${}^t (s_1^{\alpha},\cdots,s_R^{\alpha}) =
\rho(g_{\alpha\beta}) {}^t (s_1^{\beta},\cdots,s_R^{\beta})$.
In this case, we write $s_{\alpha} \sim s_{\beta}$.
Now we take $s_{\alpha} \sim s_{\beta}$ and $t_{\alpha} \sim t_{\beta}$.
Then by a straightforward calculation using \eqref{eqn:condition} and
$H_{\beta} = {}^t g_{\alpha\beta} H_{\alpha} \overline{g_{\alpha\beta}}$,
we get $h^{\rho_{\alpha}}(s_{\alpha},t_{\alpha}) =
h^{\rho_{\beta}}(s_{\beta},t_{\beta})$ on $Y_{\alpha} \cap Y_{\beta}$.
\QED
\begin{Remark}
Let $\operatorname{id}_r : \operatorname{\mathbf{GL}}_r \to \operatorname{\mathbf{GL}}_r$ be the identity morphism,
$\rho_1 = (\operatorname{id}_r)^{\otimes k}$,
$\rho_2 = \operatorname{Sym}^k(\operatorname{id}_r)$,
and $\rho_3 = \bigwedge^k(\operatorname{id}_r)$.
Further, let $\rho_4 : \operatorname{\mathbf{GL}}_r \to \operatorname{\mathbf{GL}}_r$ be
the group homomorphism given by $A \mapsto {}^t A^{-1}$.
Then $\rho_1$, $\rho_2$, $\rho_3$ and $\rho_4$ are
of degree $k$, $k$, $k$ and $-1$, respectively.
Let $(E,h)$ be a Hermitian vector bundle of rank $r$. Then
the associated vector bundles are
$(E^{\otimes k}, h^{\otimes k})$, $(\operatorname{Sym}^k(E), h^{\rho_2})$,
$(\bigwedge^k(E), h^{\rho_3})$ and $(E^{\lor}, h^{\lor})$.
Note, for example, that
$h^{\rho_2}$ is not the quotient metric $h_{quot}$ given by
$E^{\otimes k} \to \operatorname{Sym}^k(E)$; Indeed, for a locally orthogonal basis
$e_1,\cdots,e_r$ of $\overline{E}$ and $\alpha_1,\cdots,\alpha_r \in {\mathbb{Z}}$,
$h^{\rho_2}(e_1^{\alpha_1}\cdots e_r^{\alpha_r},
e_1^{\alpha_1}\cdots e_r^{\alpha_r}) = 1$, while
$h_{quot}(e_1^{\alpha_1}\cdots e_r^{\alpha_r},
e_1^{\alpha_1}\cdots e_r^{\alpha_r})
= {\alpha_1 !\cdots\alpha_r !}/ r !$.
\end{Remark}
\renewcommand{\theClaim}{\arabic{section}.\arabic{subsection}.\arabic{Theorem}.\arabic{Claim}}
\renewcommand{\theequation}{\arabic{section}.\arabic{subsection}.\arabic{Theorem}.\arabic{Claim}}
\subsection{Chow forms and their metrics}
\label{subsec:Chow:forms:and:their:metrics}
\renewcommand{\theClaim}{\arabic{section}.\arabic{subsection}.\arabic{Theorem}}
\renewcommand{\theequation}{\arabic{section}.\arabic{subsection}.\arabic{Theorem}}
\setcounter{Theorem}{0}
Let $Y$ be a regular arithmetic variety, and
$\overline{E}=(E,h)$ a Hermitian vector bundle of rank $r$ on $Y$.
Let $\rho : \operatorname{\mathbf{GL}}_r \to \operatorname{\mathbf{GL}}_R$ be
a group scheme morphism of degree $k$
commuting with the transposed morphism
and
$\overline{E}^{\rho}=(E^{\rho},h^{\rho})$
the associated Hermitian bundle of rank $R$.
We give the quotient metric on ${\mathcal{O}}_{E^{\rho}}(1)$
via $\pi^*({E^{\rho}}^{\vee}) \to {\mathcal{O}}_{E^{\rho}}(1)$.
We denote this Hermitian line bundle by
$\overline{{\mathcal{O}}_{E^{\rho}}(1)}$. Further,
let $\Omega_{\rho} = {c}_1(\overline{{\mathcal{O}}_{E^{\rho}}(1)})$
be the first Chern form.
Let $X$ be an effective cycle in ${\mathbb{P}} (E^{\rho})$
such that $X$ is flat over $Y$
with the relative dimension $d$
and degree $\delta$ on the generic fiber.
Let $g_X$ be a $\Omega_{\rho}$-normalized Green form for $X$
and we set $\widehat{X}= (X, g_X)$.
Then
$\widehat{X} \in \widehat{Z}^{R-1-d}_{L^1}({\mathbb{P}}(E^{\rho}))$.
Thus $\widehat{{c}}_1(\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1}\cdot\widehat{X}$
belongs to $\widehat{\operatorname{CH}}^{R}_{L^1}({\mathbb{P}} (E^{\rho}))_{{\mathbb{Q}}}$. Hence,
\[
\pi_*\left(
\widehat{{c}}_1(\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1}\cdot\widehat{X} \right)
\in \widehat{\operatorname{CH}}^{1}_{L^1}(Y)_{{\mathbb{Q}}}.
\]
Let us consider elementary properties of
$\pi_*\left(
\widehat{{c}}_1(\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1}\cdot\widehat{X} \right)$.
\begin{Proposition}
\label{prop:when:Bost:divisor:smooth}
Let $X = \sum_{k=1}^l a_k X_k$ be the irreducible decomposition
of $X$ as cycles.
Let $\phi_k : \tilde{X}_k \to X_k$
be a generic resolution of singularities of $X_k$
for each $k$,
i.e.,
$\phi_k$ is a proper birational morphism
such that $(\tilde{X}_k)_{{\mathbb{Q}}}$ is smooth over ${\mathbb{Q}}$.
Let
$i_k : X_k \hookrightarrow {\mathbb{P}}(E^{\rho})$ be the inclusion map and
$j_k : \tilde{X}_k \to {\mathbb{P}}(E^{\rho})$ the composition map $i_k \cdot \phi_k$.
Also we let
$f_k : X_k \to Y$ be the composition map $\pi \cdot i_k$ and
$\tilde{f}_k : \tilde{X}_k \to Y$ the composition map $\pi \cdot j_k$.
Let $Y_0$ be the maximal open set of $Y$ such that
$\tilde{f}_k$ is smooth over there for every $k$.
Then, we have the following.
\begin{enumerate}
\renewcommand{\labelenumi}{(\arabic{enumi})}
\item
${\displaystyle
\pi_* \left(\widehat{{c}}_1(\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1}
\cdot \widehat{X} \right) =
\sum_{k=1}^l a_k \tilde{f}_k{}_*
(\widehat{{c}}_1(j_k^* \overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1}).
}$ \\
In particular,
$\pi_*\left(
\widehat{{c}}_1(\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1}\cdot\widehat{X} \right)$
is independent of
the choice of normalized Green forms $g_X$ for $X$, and
$\pi_* \left(\widehat{{c}}_1(\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1}
\cdot \widehat{X} \right)
\in \widehat{\operatorname{CH}}_{L^1}^1(Y; Y_0({\mathbb{C}}))$.
\item
Let $y$ be a closed point of $(Y_0)_{{\mathbb{Q}}}$, and
$\Gamma'$ the closure of $\{ y \}$ in $Y$.
Here we choose $g_X$ as in
Proposition~\textup{\ref{prop:normalized:Green:form}}.
Then, there is a representative $(Z, g_Z)$ of
$\widehat{{c}}_1(\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \cdot \widehat{X}$
such that $\pi^{-1}(\Gamma')$ and $Z$ intersect properly, and
$\rest{g_Z}{\pi^{-1}(z)}$ is locally integrable for each
$z \in O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}(y)$.
\end{enumerate}
\end{Proposition}
{\sl Proof.}\quad
We may assume that $X$ is reduced and irreducible, so that
we will omit index $k$ in the following.
(1)
Let $g_X$ be a $\Omega_{\rho}$-normalized Green form for $X$.
Then, by virtue of Proposition~\ref{prop:formula:restriction:intersection},
\[
\widehat{{c}}_1(\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \cdot \widehat{X}
= j_* \left( \widehat{{c}}_1(j^* \overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \right)
+ a(\Omega_{\rho}^{d+1} \wedge [g_X]).
\]
Therefore, since $\pi_* (g_X \wedge \Omega_{\rho}^{d+1}) = 0$ by
the definition of $g_X$,
we get
\[
\pi_* \left(\widehat{{c}}_1(\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1}
\cdot \widehat{X} \right) =
\pi_* j_* \left( \widehat{{c}}_1(j^* \overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \right)
= \tilde{f}_* \left( \widehat{{c}}_1(j^* \overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \right).
\]
\medskip
(2)
First of all, we need the following lemma.
\begin{Lemma}
\label{lem:intersect:proper}
Let $T$ be a quasi-projective integral scheme over ${\mathbb{Z}}$,
$L_1, \ldots, L_n$ line bundles on $T$, and
$\Gamma$ a cycle on $T$.
Then, there is a cycle $Z$ on $T$ such that
$Z$ is rationally equivalent to $c_1(L_1) \cdots c_1(L_n)$,
and that $Z$ and $\Gamma$ intersect properly.
\end{Lemma}
{\sl Proof.}\quad
We prove this lemma by induction on $n$.
First, let us consider the case $n=1$.
Let $\Gamma = \sum_{i=1}^r a_i \Gamma_i$ be the irreducible
decomposition as cycles.
Let $\gamma_i$ be a closed point of
$\Gamma_i \setminus \bigcup_{j \not= i} \Gamma_j$, and
$m_i$ the maximal ideal at $\gamma_i$.
Let $H$ be an ample line bundle on $X$.
Choose a sufficiently large integer $N$ such that
\[
H^1(T, H^{\otimes N} \otimes m_1 \otimes \cdots \otimes m_r) =
H^1(T, H^{\otimes N} \otimes L_1 \otimes m_1 \otimes \cdots \otimes m_r)
= 0.
\]
Then, the natural homomorphisms
\[
H^0(T, H^{\otimes N}) \to \bigoplus_{i=1}^r H^{\otimes N} \otimes \kappa(\gamma_i)
\quad\text{and}\quad
H^0(T, H^{\otimes N} \otimes L_1) \to \bigoplus_{i=1}^r
H^{\otimes N} \otimes L_1 \otimes \kappa(\gamma_i)
\]
are surjective.
Thus, there are sections $s_1 \in H^0(T, H^{\otimes N})$ and
$s_2 \in H^0(T, H^{\otimes N} \otimes L_1)$ such that
$s_1(\gamma_i) \not= 0$ and $s_2(\gamma_i) \not= 0$ for all $i$.
Then, $\operatorname{div}(s_2) - \operatorname{div}(s_1) \sim c_1(L_1)$, and
$\operatorname{div}(s_2) - \operatorname{div}(s_1)$ and $\Gamma$ intersect properly.
Next we assume $n > 1$. Then, by hypothesis of induction,
there is a cycle $Z'$ such that
$Z' \sim c_1(L_1) \cdots c_1(L_{n-1})$, and
$Z'$ and $\Gamma$ intersect properly.
Let $Z' = \sum_{j} b_j T_j$ be the decomposition as cycles.
We set $\Gamma_j = (T_j \cap \operatorname{Supp}(\Gamma))_{red}$.
Then, using the case $n=1$,
there is a cycle $Z_j$ such that $Z_j \sim c_1(\rest{L_n}{T_j})$, and
$Z_j$ and $\Gamma_j$ intersect properly.
Thus, if we set $Z = \sum_j b_j Z_j$, then
$Z \sim c_1(L_1) \cdots c_1(L_n)$, and
$Z$ and $\Gamma$ intersect properly.
\QED
Let us go back to the proof of (2) of
Proposition~\ref{prop:when:Bost:divisor:smooth}.
By virtue of Lemma~\ref{lem:intersect:proper},
there is a cycle $Z$ on $X$ such that
$Z \sim c_1\left(i^* {\mathcal{O}}_{E^{\rho}}(1) \right)^{d+1}$, and that $Z$ and
$f^{-1}(\Gamma')$ intersect properly.
Then, $Z \sim c_1({\mathcal{O}}_{E^{\rho}}(1))^{d+1} \cdot X$, and
$Z$ and $\pi^{-1}(\Gamma')$ intersect properly.
Let $\phi_X$ be a Green form of logarithmic type for $X$.
Then, since
\[
\widehat{{c}}_1(\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \cdot (X, \phi_X)
\in \widehat{\operatorname{CH}}^{R}({\mathbb{P}}(E^{\rho})),
\]
there is a Green form $\phi_Z$ of logarithmic type for $Z$ such that
$(Z, \phi_Z)$ is a representative of
$\widehat{{c}}_1(\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \cdot (X, \phi_X)$.
Thus, if we set
\[
g_Z = \phi_Z + c_1(\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \wedge (g_X - \phi_X),
\]
then $(Z, g_Z)$ is a representative of
$\widehat{{c}}_1(\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \cdot \widehat{X}$.
Since $Z$ and $\pi^{-1}(\Gamma')$ intersect properly and
$g_X$ has the property in Proposition~\ref{prop:normalized:Green:form},
we can easily see that
$g_Z$ is locally integrable along $\pi^{-1}(z)$ for each
$z \in O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}(y)$.
\QED
Here we recall some elementary results of Chow forms.
Details can be found in \cite{Bo}.
Consider the incidence subscheme $\Gamma$ in the product
\[
{\mathbb{P}}(E^{\rho}) \times_{Y} {\mathbb{P}}(E^{\rho}{}^{\lor})^{d+1}
= {\mathbb{P}}(E^{\rho}) \times_{Y} {\mathbb{P}}(E^{\rho}{}^{\lor})
\times_{Y} \cdots \times_{Y} {\mathbb{P}}(E^{\rho}{}^{\lor}).
\]
Let $\imath : \Gamma \to {\mathbb{P}}(E^{\rho})$ and $\jmath :
\Gamma \to {\mathbb{P}}(E^{\rho}{}^{\lor})^{d+1}$
be projection maps. The Chow divisor $\operatorname{Ch}(X)$ of $X$ is defined by
\[
\operatorname{Ch}(X) = \jmath_* \imath^* (X).
\]
The following facts are well-known:
\begin{enumerate}
\item $\operatorname{Ch}(X)$ is an effective cycle of codimension $1$
in ${\mathbb{P}}(E^{\rho}{}^{\lor})^{d+1}$;
\item $\operatorname{Ch}(X)$ is flat over $Y$;
\item For any $y \in Y$,
$\operatorname{Ch}(X)_y$ is a divisor of type $(\delta,\delta,\cdots,\delta)$
in ${\mathbb{P}}(E^{\rho}{}^{\lor})^{d+1}_{y}$.
\end{enumerate}
Let $p : {\mathbb{P}}(E^{\rho}{}^{\lor})^{d+1} \to Y$ be the canonical morphism,
and $p_i : {\mathbb{P}}(E^{\rho}{}^{\lor})^{d+1}
\to {\mathbb{P}}(E^{\rho}{}^{\lor})$ the projection to the $i$-th factor.
Then, by the above properties,
there is a line bundle $L$ on $Y$ and a section $\Phi_{X}$ of
\[
H^0 \left({\mathbb{P}}(E^{\rho}{}^{\lor})^{d+1},
p^*(L)\otimes\bigotimes_{i=1}^{d+1}p_i^*
{\mathcal{O}}_{E^{\rho}{}^{\lor}}(\delta) \right)
\]
such that $\operatorname{div}(\Phi_{X}) = \operatorname{Ch}(X)$. Since
\[
p_* \left(
p^* (L) \otimes \bigotimes_{i=1}^{d+1} p_i^*
{\mathcal{O}}_{E^{\rho}{}^{\lor}}(\delta)
\right)
=
L \otimes (\operatorname{Sym}^{\delta}(E^{\rho}))^{{}\otimes {d+1}},
\]
we may view $\Phi_{X}$ as an element of
$H^0(Y,L \otimes (\operatorname{Sym}^{\delta}(E^{\rho}))^{{}\otimes {d+1}})$.
We say $\Phi_{X}$ is a {\em Chow form} of $X$.
Clearly $\Phi_{X}$ is uniquely determined up to $H^0(Y,{\mathcal{O}}_{Y}^{\times})$.
As in \cite[Proposition 1.2 and its remark]{Bo}, we have
\[
{c}_1(L)
= \pi_* \left({c}_{1}({\mathcal{O}}_{E^{\rho}}(1))^{d+1}\cdot {X}\right).
\]
We give a generalized metric $h_{L}$ on $L$ so that
$\overline{L} = (L,h_L)$ satisfies the equality
\addtocounter{Theorem}{1}
\begin{equation}
\label{eqn:metric:L}
\widehat{{c}}_1(\overline{L})
= \pi_* \left(\widehat{{c}}_{1}(\overline{{\mathcal{O}}_{E^{\rho}}(1)})^
{d+1}\cdot\widehat{X}\right)
\end{equation}
in $\widehat{\operatorname{CH}}^1_{L^1}(Y)$.
Note that we can also give a metric $L$ by the equation
\[
{\mathcal{O}}_{{\mathbb{P}}(E^{\rho}{}^{\lor})^{d+1}}(\operatorname{Ch}(X))
= p^*(L)\otimes\bigotimes_{i=1}^{d+1}p_i^* {\mathcal{O}}_{E^{\lor}}(\delta)
\]
and by suitably metrizing other terms,
as is implicitly done in \cite[1.5]{Zh}.
We do not however pursue this here.
\renewcommand{\theClaim}{\arabic{section}.\arabic{subsection}.\arabic{Theorem}.\arabic{Claim}}
\renewcommand{\theequation}{\arabic{section}.\arabic{subsection}.\arabic{Theorem}.\arabic{Claim}}
\subsection{Restriction of Chow forms on fibers}
\label{subsec:pullback}
\setcounter{Theorem}{0}
In this section we will consider the restriction of Chow forms
on fibers.
Let $Y,\overline{E},\rho,X$
be as in \S \ref{subsec:Chow:forms:and:their:metrics}.
Let $y$ be a closed point of $Y_{{\mathbb{Q}}}$.
Let $\Gamma'$ be the closure of $\{ y \}$ in $Y$, and
$\Gamma$ the normalization of $\Gamma'$.
Let $f : \Gamma \to Y$ be the natural morphism.
We set $E_{\Gamma} = f^*(E)$ and
$\overline{E_{\Gamma}} = (E_{\Gamma},f^*(h))$.
Also we put $(E^{\rho})_{\Gamma} = f^*(E^{\rho})$ and
$\overline{(E^{\rho})_{\Gamma}} = ((E^{\rho})_{\Gamma}, f^*(h^{\rho}))$.
Then $(\overline{E_{\Gamma}})^{\rho}$ is equal to
$\overline{(E^{\rho})_{\Gamma}}$, so that
we denote $(E^{\rho})_{\Gamma}$ by $E^{\rho}_{\Gamma}$, and
$\overline{(E^{\rho})_{\Gamma}}$ by $\overline{E^{\rho}_{\Gamma}}$.
Considering the following fiber product
\begin{equation*}
\begin{CD}
{\mathbb{P}}(E^{\rho}_{\Gamma}) @>{f '}>> {\mathbb{P}}(E^{\rho}) \\
@VV{\pi'}V @VV{\pi}V \\
\Gamma @>{f}>> Y
\end{CD}
\end{equation*}
we set $X_{\Gamma} = {f'}^*(X)$.
Then $X_{\Gamma}$ is flat over $\Gamma$
with the relative dimension $d$
and the degree $\delta$ on the generic fiber.
For this quadruplet $(\Gamma,\overline{E_{\Gamma}},\rho,X_{\Gamma})$
in place of the quadruplet $(Y,\overline{E},\rho,X)$,
we can define in the same way the Hermitian line bundle
$\overline{{\mathcal{O}}_{E^{\rho}_{\Gamma}}(1)}$
on ${\mathbb{P}}(E^{\rho}_{\Gamma})$,
an arithmetic $L^1$-divisor
$\widehat{X_{\Gamma}} = (X_{\Gamma}, g_{X_{\Gamma}})$
on ${\mathbb{P}}(E^{\rho}_{\Gamma})$
and the arithmetic $L^1$-divisor
$\pi '_* \left(\widehat{{c}}_{1}(
\overline{{\mathcal{O}}_{E^{\rho}_{\Gamma}}(1)})^{d+1}\cdot
\widehat{X_{\Gamma}}\right)$
on $\Gamma$.
Further, we have
$\overline{L_{\Gamma}} = (L_{\Gamma}, h_{\Gamma})$
satisfying
\[
\widehat{{c}}_1(\overline{L_{\Gamma}})
= \pi '_* \left(\widehat{{c}}_{1}(\overline{{\mathcal{O}}_{E^{\rho}_{\Gamma}}(1)})^
{d+1}\cdot\widehat{X_{\Gamma}}\right)
\]
in $\widehat{\operatorname{CH}}_{L^1}^1(\Gamma)$.
We also have $\operatorname{Ch}(X_{\Gamma})$. Moreover,
letting $p_i ' : {\mathbb{P}}((E^{\rho}_{\Gamma})^{\lor})^{d+1}
\to {\mathbb{P}}((E^{\rho}_{\Gamma})^{\lor})$
be the $i$-th projection,
there is a section $\Phi_{X_{\Gamma}}$ of
\[
H^0 \left( {\mathbb{P}}((E^{\rho}_{\Gamma})^{\lor})^{d+1},
{p '}^*(L_{\Gamma})\otimes
\bigotimes_{i=1}^{d+1} {p_i'}^*
{\mathcal{O}}_{(E^{\rho}_{\Gamma})^{\lor}}(\delta) \right)
=
H^0 \left(\Gamma,
L_{\Gamma} \otimes (\operatorname{Sym}^{\delta}
((E^{\rho}_{\Gamma})^{\lor}))^{\otimes d+1} \right),
\]
such that $\operatorname{div}(\Phi_{X_{\Gamma}}) = \operatorname{Ch}(X_{\Gamma})$.
Let us consider the following fiber product,
\begin{equation*}
\begin{CD}
{\mathbb{P}}((E^{\rho}_{\Gamma})^{\lor})^{d+1}
@>{g '} >> {\mathbb{P}}((E^{\rho})^{\lor})^{d+1} \\
@VV{p '}V
@VV{p}V \\
\Gamma @>{f}>> Y
\end{CD}
\end{equation*}
Then, we have the following proposition.
\begin{Proposition}
\label{proposition:pullback}
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item
${g'}^* \operatorname{Ch}(X) = \operatorname{Ch}(X_{\Gamma})$.
Moreover, we can choose $\Phi_{X_{\Gamma}}$ to be $f ^* \Phi_X$.
\item
Let $X_1,\cdots,X_l$
be the irreducible components of $X_{red}$.
Assume that, for every $1 \leq i \leq l$,
there is a generic resolution of singularities
$\phi_i : \tilde{X_i} \to X_i$
such that the induced map
$\tilde{X_i} \to Y$
is smooth over $y$ for every $i$.
Then the equality
\begin{align*}
f^* \pi_* \left(\widehat{{c}}_{1}(
\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \cdot \widehat{X} \right)
& = \pi '_* f'{}^* \left(\widehat{{c}}_{1}(
\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \cdot \widehat{X} \right) \\
& = \pi '_* \left(\widehat{{c}}_{1}(
\overline{{\mathcal{O}}_{E^{\rho}_{\Gamma}}(1)})^{d+1}
\cdot \widehat{X_{\Gamma}} \right).
\end{align*}
holds in $\widehat{\operatorname{CH}}_{L^1}^1(\Gamma)$.
In other words, $f^* (L,h_L) = (L_{\Gamma}, h_{L_{\Gamma}})$.
\end{enumerate}
\end{Proposition}
{\sl Proof.}\quad
(i) If $f$ is flat,
then this follows from the base change theorem.
In the case $f$ is not flat, we refer to the remark \cite[4.3.2(i)]{BGS},
or we can easily see this using Appendix~A.
(ii)
We take $g_X$ as in Proposition~\ref{prop:normalized:Green:form}.
Let $\alpha = \widehat{{c}}_{1}(
\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \cdot \widehat{X}
\in \widehat{\operatorname{CH}}_{L^1}^1(Y)$.
By Proposition~\ref{prop:when:Bost:divisor:smooth},
we can take a representative
$(Z,g_Z)$ of $\alpha$ such that
$Z$ and $\pi^{-1}(\Gamma')$ intersect properly, and
$g_Z$ is locally integrable along $\pi^{-1}(w)$ for all
$w \in O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}(y)$.
Now we have
\begin{align*}
f^* \pi_*(\alpha)
& = f^* (\pi_*Z, [\pi_* g_Z]) \\
& = \left( f^* \pi_*Z,
\sum_{w \in O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}(y)}
\left( \int_{\pi^{-1} (w)} g_Z \right) \cdot w \right).
\end{align*}
On the other hand, we have
\begin{equation*}
\pi '_* f'{}^*(\alpha)
= \left( \pi '_* f'{}^* Z,
\sum_{w \in O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}(y)}
\left( \int_{\pi^{-1} (w)} g_Z \right) \cdot w \right).
\end{equation*}
Moreover, by Appendix A, $f^* \pi_*Z$ is equal to $\pi '_* f'{}^* Z$
as a cycle. Thus we have proven the first equality.
Now we will prove the second equality.
Let $\phi_X$ be a Green form of logarithmic type for $X$.
Since
\[
{f'}^* : \bigoplus_{i \geq 0} \widehat{\operatorname{CH}}^i({\mathbb{P}}(E^{\rho})) \to
\bigoplus_{i \geq 0} \widehat{\operatorname{CH}}^i({\mathbb{P}}(E^{\rho}_{\Gamma}))
\]
is a homomorphism of rings (cf. \cite[5) of Theorem in 4.4.3]{GSArInt}).
Thus,
\[
{f'}^* \left(\widehat{{c}}_{1}(
\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \cdot (X, \phi_X) \right) =
{f'}^* \widehat{{c}}_{1}(
\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \cdot {f'}^*(X, \phi_X).
\]
Therefore, since we take $g_X$ as in
Proposition~\ref{prop:normalized:Green:form}, we can see
\begin{align*}
{f'}^* \left(\widehat{{c}}_{1}(
\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \cdot \widehat{X} \right) & =
{f'}^* \left(\widehat{{c}}_{1}(
\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \cdot ((X, \phi_X) + (0, g_X - \phi_X)) \right) \\
& =
{f'}^* \left(\widehat{{c}}_{1}(
\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \cdot (X, \phi_X) \right) +
a \left(
{f'}^*(c_{1}(\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \wedge (g_X - \phi_X))\right) \\
& =
{f'}^* \widehat{{c}}_{1}(
\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \cdot
{f'}^*(X, \phi_X) +
a \left(
{f'}^* (c_{1}(\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \wedge (g_X - \phi_X)) \right)\\
& =
{f'}^* \widehat{{c}}_{1}(
\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1} \cdot
{f'}^*(X, g_X)
\end{align*}
Moreover, as pointed out in Remark~\ref{rem:norm:Green:general:fiber},
${f'}^* g_X$ is a normalized Green form for ${f'}^* X$.
Thus we have got the second equality.
\QED
\renewcommand{\theClaim}{\arabic{section}.\arabic{subsection}.\arabic{Theorem}.\arabic{Claim}}
\renewcommand{\theequation}{\arabic{section}.\arabic{subsection}.\arabic{Theorem}.\arabic{Claim}}
\subsection{Chow stability and field extensions}
\label{subsec:Chow:stability:field:ext}
\setcounter{Theorem}{0}
Let $\rho : \operatorname{\mathbf{GL}}_r \to \operatorname{\mathbf{GL}}_R$ be
a group scheme morphism of degree $k$
commuting with the transposed morphism.
Let $S$ be a ring (commutative, with the multiplicative identity).
For a positive integer $\delta$ and $d$, we consider
$\operatorname{Sym}^{\delta} (S^R)^{\otimes d+1}$.
Then through the induced group homomorphism
$\rho(S) : \operatorname{\mathbf{GL}}_r(S) \to \operatorname{\mathbf{GL}}_R(S)$,
$\operatorname{\mathbf{GL}}_r(S)$ (or $\operatorname{\mathbf{SL}}_r(S)$)
acts on
$\operatorname{Sym}^{\delta} (S^R)^{\otimes d+1}$.
\begin{Proposition}
\label{prop:stability:and:det}
Let $K$ be an infinite field
and $L$ an extension field of K.
Let $P$ be a homogeneous polynomial of degree $e$ on
$\operatorname{Sym}^{\delta} (L^R)^{\otimes d+1}$, i.e.,
$P \in \operatorname{Sym}^e(\operatorname{Sym}^{\delta} (L^R)^{\otimes d+1}{}^{\lor})$.
Then we have the following.
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item
$P$ is $\operatorname{\mathbf{SL}}_r(K)$-invariant
if and only if
$P$ is $\operatorname{\mathbf{SL}}_r(L)$-invariant.
\item
If P is $\operatorname{\mathbf{SL}}_r(K)$-invariant, then
\[
P(v^{\sigma}) ^r = (\det \sigma) ^{ek(d+1)\delta} P(v)^r
\]
in $L$
for all $ v \in \operatorname{Sym}^{\delta} (L^R)^{\otimes d+1}$
and $\sigma \in \operatorname{\mathbf{GL}}_r(L)$.
\end{enumerate}
\end{Proposition}
\proof
(i) We only need to prove the `only if' part.
Let $S_L(P) = \{ \sigma \in \operatorname{\mathbf{SL}}_r(L) \mid P^{\sigma} = P \}$
be the stabilizer of $P$ in $\operatorname{\mathbf{SL}}_r(L)$.
$S_L(P)$ is a closed algebraic set of $\operatorname{\mathbf{SL}}_r(L)$
and contains $\operatorname{\mathbf{SL}}_r(K)$.
Since $\operatorname{\mathbf{SL}}_r(K)$ is Zariski dense in $\operatorname{\mathbf{SL}}_r(L)$,
$S_L(P)$ must coincide with $\operatorname{\mathbf{SL}}_r(L)$.
(ii) Let $M$ be an extension field
of $L$ such that
it has an $r$-th root $\xi$ of $\det \sigma$.
If $\sigma '$ is defined by $\sigma = \xi\sigma '$,
then $\sigma ' \in \operatorname{\mathbf{SL}}_r(M)$.
Since $P$ is $\operatorname{\mathbf{SL}}_r(M)$-invariant by (i),
we find
\begin{align*}
P(v^{\sigma})^r & = P\left( (\rho({\mathbb{C}})(\xi\sigma ')) \cdot v\right) ^r
= P\left( (\xi^k \rho({\mathbb{C}})(\sigma ')) \cdot v \right) ^r \\
& = \xi^{rek(d+1)\delta} P(v)^r
= (\det \sigma) ^{ek(d+1)\delta} P(v)^r.
\end{align*}
in $M$.
Hence the equality holds in $L$ because both sides belong to $L$.
\QED
\begin{Remark}
More strongly, we can show that,
for any integral domain $S$ of characteristic zero,
if $P \in \operatorname{Sym}^e(\operatorname{Sym}^{\delta} (S^R)^{\otimes d+1}{}^{\lor})$
is $\operatorname{\mathbf{SL}}_r({\mathbb{Z}})$-invariant,
then $P$ is $\operatorname{\mathbf{SL}}_r(S)$-invariant.
\end{Remark}
Now let $Y$, $\overline{E}$, $\rho$ and $X$
be as in \S \ref{subsec:Chow:forms:and:their:metrics}.
Recall that for a closed point $y$ of $Y_{{\mathbb{Q}}}$,
$\operatorname{Ch}(X)_y$ is a divisor of type $(\delta,\delta,\cdots,\delta)$
in ${\mathbb{P}}(E^{\rho}{}^{\lor})^{d+1}_{y}$.
Hence the Chow form restricted on $y$, i.e.,
$\Phi_X \vert_y = \Phi_{X_y}$ is an element of
$\operatorname{Sym}^{\delta}(K^R)^{d+1}$.
We say that $X_y$ is {\em Chow semi-stable}
if $\Phi_{X_y} \in \operatorname{Sym}^{\delta}(K^R)^{d+1}$
is semi-stable under the action of $\operatorname{\mathbf{SL}}_r(K)$, where
$K$ is the residue field of $y$.
\begin{Lemma}
\label{lemma:stability:generators:over:Z}
There are a positive integer $e$
and
$\operatorname{\mathbf{SL}}_r({\mathbb{Q}})$-invariant homogeneous polynomials
$P_1,\cdots,P_l \in \operatorname{Sym}^{e}(\operatorname{Sym}^{\delta}({\mathbb{Z}}^R)^{d+1}{}^{\lor})$,
which depend only on $\rho$, $d$ and $\delta$,
with the following property.
For any closed points $y \in Y_{{\mathbb{Q}}}$,
if $X_y$ is Chow semistable, then
there is a $P_i$ such that
$P_i (\Phi_{X_y}) \neq 0$.
\end{Lemma}
\proof
$\operatorname{\mathbf{SL}}_r({\mathbb{Q}})$
acts linearly on $\operatorname{Sym}^{\delta}({\mathbb{Q}}^R)^{d+1}$.
Since $\operatorname{\mathbf{SL}}_r({\mathbb{Q}})$ is a reductive group,
we can take $\operatorname{\mathbf{SL}}_r({\mathbb{Q}})$-invariant
homogeneous polynomials $Q_1,\cdots,Q_l$
such that they form generators of the algebra of
$\operatorname{\mathbf{SL}}_r({\mathbb{Q}})$ invariant polynomials
on $\operatorname{Sym}^{\delta}({\mathbb{Q}}^R)^{d+1}$.
By clearing the denominators,
we may assume that $Q_1,\cdots,Q_l$ is defined over ${\mathbb{Z}}$.
Let $e_i$ be the degree of $Q_i$ for $i = 1,\cdots,l$.
We take a positive integer $e$ such that $e_i \vert e$
for $i = 1,\cdots,l$.
We set $P_i = Q_i^{e/e_i}$.
Let us check that $P_i$'s have the desired property.
Since $X_y$ is Chow semistable, there is a
$\operatorname{\mathbf{SL}}_r(K)$-invariant homogeneous
polynomial $F$ on $\operatorname{Sym}^{\delta}(K^R)^{d+1}$
with $F(\Phi_{X_y}) \not= 0$, where
$K$ is the residue field of $y$.
Let us choose $\alpha_1, \ldots, \alpha_n \in K$ and
homogeneous polynomials $F_1, \ldots, F_n$
over ${\mathbb{Q}}$ such that
$F = \alpha_1 F_1 + \cdots + \alpha_n F_n$ and that
$\alpha_1, \ldots, \alpha_n$ are linearly independent over ${\mathbb{Q}}$.
Here, for $\sigma \in \operatorname{\mathbf{SL}}_r({\mathbb{Q}})$,
\[
F^{\sigma} = \alpha_1 F_1^{\sigma} + \cdots + \alpha_n F_{n}^{\sigma}
\]
and $F_{i}^{\sigma}$'s are homogeneous polynomials
over ${\mathbb{Q}}$. Thus, we can see that $F_i$'s are $\operatorname{\mathbf{SL}}_r({\mathbb{Q}})$-invariant.
Moreover, since
\[
F(\Phi_{X_y}) = \alpha_1 F_1(\Phi_{X_y}) + \cdots + \alpha_n F_{n}(\Phi_{X_y}),
\]
there is $F_i$ with $F_i(\Phi_{X_y}) \not= 0$.
On the other hand, $F_i$ is an element of ${\mathbb{Q}}[Q_1, \ldots, Q_l]$.
Thus, we can find $Q_j$ with $Q_j(\Phi_{X_y}) \not= 0$, namely
$P_j(\Phi_{X_y}) \not= 0$.
\QED
\medskip
\section{Semi-stability and positiveness in a relative case}
\label{section:semistability:positiveness:relative}
\subsection{Cornalba-Harris-Bost's inequality in a relative case}
\label{subsec:CHB:inequality}
\setcounter{Theorem}{0}
Let $Y$ be an arithmetic variety and
$\overline{E} = (E,h)$ a Hermitian vector bundle of rank $r$ on $Y$.
Let $\rho : \operatorname{\mathbf{GL}}_r \to \operatorname{\mathbf{GL}}_R$
be a group scheme morphism of degree $k$
commuting with the transposed morphism.
Before we prove the relative Cornalba-Harris-Bost's inequality,
we need three lemmas.
\begin{Lemma}
\label{lemma:P:to:sheafhom}
Let $L$ be a line bundle on $Y$.
Let $P \in \operatorname{Sym}^e(\operatorname{Sym}^{\delta}({\mathbb{Z}}^R)^{d+1}{}^{\lor}) \backslash \{0\}$
and suppose that $P$ is $\operatorname{\mathbf{SL}}_r({\mathbb{Q}})$-invariant.
Then there is a polynomial map of sheaves
\[
\varphi_P : L \otimes \operatorname{Sym}^{\delta} (E^{\rho})^{\otimes d+1}
\to L^{\otimes er} \otimes (\det E)^{\otimes ek(d+1)\delta}
\]
given by $P^r$,
namely,
$\varphi_P$ is locally defined by the evaluation in terms of $P^r$.
\end{Lemma}
{\sl Proof.}\quad
Let $U$ be a Zariski open set, and
$\phi : \rest{E}{U}
\overset{\sim}{\longrightarrow} {\mathcal{O}}_{U}^{\oplus n}$
and
$\psi : \rest{L}{U}
\overset{\sim}{\longrightarrow} {\mathcal{O}}_{U}$
local trivializations of $E$ and $L$ respectively.
Then, by the construction of $E^{\rho}$, we have
\[
\phi_{\rho,\delta,d}:
\rest{\left( \operatorname{Sym}^{\delta} \left(E^{\rho}\right)^{\otimes d+1} \right)}{U}
\overset{\sim}{\longrightarrow}
\operatorname{Sym}^{\delta} \left({\mathcal{O}}_{U}^{\oplus R}\right)^{\otimes d+1}.
\]
Thus we get
\[
\psi \otimes \phi_{\rho,\delta,d}:
\rest{\left( L \otimes \operatorname{Sym}^{\delta} \left(E^{\rho}\right)^{\otimes d+1}
\right)}{U}
\overset{\sim}{\longrightarrow}
\operatorname{Sym}^{\delta} \left({\mathcal{O}}_{U}^{\oplus R}\right)^{\otimes d+1}.
\]
Here, we define
\[
\rest{\varphi_P}{U} :
\rest{\left( L \otimes \operatorname{Sym}^{\delta} (E^{\rho})^{\otimes d+1} \right)}{U}
\to
\rest{\left( L^{\otimes er} \otimes (\det E)^{\otimes ek(d+1)\delta} \right)}{U}
\]
such that the following diagram is commutative.
\[
\begin{CD}
\rest{\left( L \otimes \operatorname{Sym}^{\delta} \left(E^{\rho}\right)^{\otimes d+1} \right)}{U}
@>{\psi \otimes \phi_{\rho,\delta,d}}>>
\operatorname{Sym}^{\delta} \left({\mathcal{O}}_{U}^{\oplus R}\right)^{\otimes d+1} \\
@V{\rest{\varphi_P}{U}}VV @VV{P^r}V \\
\rest{\left( L^{\otimes er} \otimes (\det E)^{\otimes ek(d+1)\delta} \right)}{U}
@>{\psi^{er} \otimes \det(\phi)^{ek(d+1)\delta}}>>
{\mathcal{O}}_U,
\end{CD}
\]
where $P^r$ is the map given by the evaluation in terms of the polynomial $P^r$.
In order to see that the local $\rest{\varphi_P}{U}$
glues together on $Y$,
it is sufficient to show that
$\rest{\varphi_P}{U}$ does not depend on the choice of
local trivializations $\phi$ and $\psi$.
Let $\phi' : \rest{E}{U}
\overset{\sim}{\longrightarrow} {\mathcal{O}}_{U}^{\oplus n}$
and
$\psi' : \rest{L}{U}
\overset{\sim}{\longrightarrow} {\mathcal{O}}_{U}$
be another local trivializations.
In the same way, we have the following commutative diagram.
\[
\begin{CD}
\rest{\left( L \otimes \operatorname{Sym}^{\delta} \left(E^{\rho}\right)^{\otimes d+1} \right)}{U}
@>{\psi' \otimes \phi'_{\rho,\delta,d}}>>
\operatorname{Sym}^{\delta} \left({\mathcal{O}}_{U}^{\oplus R}\right)^{\otimes d+1} \\
@V{\rest{\varphi'_P}{U}}VV @VV{P^r}V \\
\rest{\left( L^{\otimes er} \otimes (\det E)^{\otimes ek(d+1)\delta} \right)}{U}
@>{{\psi'}^{er} \otimes \det(\phi')^{ek(d+1)\delta}}>>
{\mathcal{O}}_U
\end{CD}
\]
We set the transition functions
$g = \phi \cdot(\phi')^{-1}$ and
$h = \psi \cdot(\psi')^{-1}$.
Then by a straightforward calculation using
(ii) of Proposition~\ref{prop:stability:and:det},
we get, on $U$,
\[
P^r \cdot (\psi \otimes \phi_{\rho,\delta,d})
= h^{re}\det(g)^{ek (d+1)\delta}
P^r \cdot (\psi' \otimes \phi'_{\rho,\delta,d}),
\]
which implies
\[
\left({\psi}^{er} \otimes \det(\phi)^{ek(d+1)\delta}\right) \cdot
\left(\rest{\varphi_P}{U}\right) =
h^{re}\det(g)^{ek (d+1)\delta}
\left({\psi'}^{er} \otimes \det(\phi')^{ek(d+1)\delta}\right) \cdot
\left(\rest{\varphi'_P}{U}\right).
\]
Here note that
\[
h^{re}\det(g)^{ek (d+1)\delta}
=
\left({\psi}^{er} \otimes \det(\phi)^{ek(d+1)\delta}\right) \cdot
\left({\psi'}^{er} \otimes \det(\phi')^{ek(d+1)\delta}\right)^{-1}.
\]
Thus, we obtain $\rest{\varphi_P}{U} = \rest{\varphi'_P}{U}$.
\QED
Suppose now $L$ is given a generalized metric $h_L$.
Since both sides of
\[
\varphi_P : L \otimes \operatorname{Sym}^{\delta} (E^{\rho})^{\otimes d+1}
\to L^{\otimes er} \otimes (\det E)^{\otimes ek(d+1)\delta}
\]
in the lemma above are then equipped with metrics,
we can consider the norm of $\varphi_P$.
Before evaluating the norm of $\varphi_P$,
we define the norm of $P$ as follows;
We first define the metric $\Vert \cdot \Vert_{can}$
on $\operatorname{Sym}^{\delta}({\mathbb{C}}^{n})^{\otimes d+1}$
induced from the usual Hermitian metric on ${\mathbb{C}}$;
We then define $|\!|\!| P |\!|\!|$ by
\[
|\!|\!| P |\!|\!| =
\sup_{v \in \operatorname{Sym}^{\delta}({\mathbb{C}}^{n})^{\otimes d+1} \setminus \{0\}}
\frac{\vert P(v) \vert}{\Vert v \Vert_{can}^e},
\]
where $P$ is regarded as an element of
$\operatorname{Sym}^e((\operatorname{Sym}^{\delta}({\mathbb{C}}^{m}))^{\otimes d+1})^{\lor})$.
\begin{Lemma}
\label{lemma:norm:of:morphism:P}
For any section $s \in H^0(Y,L \otimes (\operatorname{Sym}^{\delta}(E^{\rho}))^{\otimes d+1})$
and any complex point $y \in Y({\mathbb{C}})$
around which $h_L \otimes (\operatorname{Sym}^{\delta}(h^{\rho}))^{\otimes d+1}$ is
$C^{\infty}$,
we have
\[
\Vert \varphi_P(s) \Vert (y) \le
|\!|\!| P |\!|\!|^r \, \Vert s \Vert^{er} (y).
\]
\end{Lemma}
\proof
By choosing bases, $\overline{E}(y)$ and in $\overline{L}(y)$ are isometric to
${\mathbb{C}}^n$ and ${\mathbb{C}}$ with the canonical metrics, respectively.
Then, with respect to these bases,
$\overline{E}^{\rho}$ is by its construction isomorphic to
${\mathbb{C}}^R$ with the canonical metric.
Recalling that
$\varphi_P$ is given by the evaluation by $P^r$
once we fix local trivializations of $E$ and $L$,
the desired inequality follows from the definition of
$|\!|\!| P |\!|\!|$.
\QED
Now let $X$ be an effective cycle in ${\mathbb{P}}(E^{\rho})$
such that $X$ is flat over $Y$
with the relative dimension $d$ and the degree $\delta$ on the generic fiber.
In \S\ref{subsec:Chow:forms:and:their:metrics}
we constructed a Chow form $\Phi_X$ of $X$,
which is an element of
$H^0(Y, L \otimes (\operatorname{Sym}^{\delta}(E^{\rho}))^{{}\otimes {d+1}})$.
Recall that
$L$ is given a generalized metric by \eqref{eqn:metric:L}.
For each irreducible component $X_i$ of $X_{red}$,
let $\tilde{X_i} \to X_i$ be
a generic resolution of singularities of $X_i$.
Moreover, let $Y_0$ be the maximal open set of $Y$ such that
the induced morphism $\tilde{X}_i \to Y$ is smooth over $Y_0$
for every $i$.
Further, we fix terminologies.
Let $T$ be a quasi-projective scheme over ${\mathbb{Z}}$,
$t$ a closed point of $T_{{\mathbb{Q}}}$, and $K$ the residue field
of $t$. By abuse use of notation, let $t : \operatorname{Spec}(K) \to T$
be the induced morphism by $t$.
We say $t$ is {\em extensible in $T$} if $t : \operatorname{Spec}(K) \to T$
extends to $\tilde{t} : \operatorname{Spec}(O_K) \to T$, where
$O_K$ is the ring of integers in $K$.
Note that if $T$ is projective over ${\mathbb{Z}}$, then
every closed point of $T_{{\mathbb{Q}}}$ is extensible in $T$.
Let $V$ be a set, $\phi$ a non-negative function on $V$, and
$S$ a finite subset of $V$.
We define the geometric mean $\operatorname{g.\!m.}(\phi; S)$ of
$\phi$ over $S$ to be
\[
\operatorname{g.\!m.}(\phi; S) = \left( \prod_{s \in S} \phi(s) \right)^{1/\#(S)}.
\]
We will evaluate the norm of $\Phi_X$.
\begin{Lemma}
\label{lemma:evaluation:of:norm:of:Phi}
There is a constant $c_1 (R,d,\delta)$
depending only on $R,d$ and $\delta$
with the following property.
For any closed points $y$ of $(Y_0)_{{\mathbb{Q}}}$
with $y$ extensible in $Y$,
\[
\operatorname{g.\!m.}\left(
\Vert \Phi_{X} \Vert_{
\overline{L}\otimes (\operatorname{Sym}^{\delta}(\overline{E}^{\rho}))^{{}\otimes {d+1}}}
;\ O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{{\mathbb{Q}}})}(y) \right)
\leq c_1 (R,d,\delta).
\]
\end{Lemma}
{\sl Proof.}\quad
Let $K$ be the residue field of $y$.
Let $\Gamma$ be the normalization of the closure of $\{ y \}$ in $Y$.
Then, since $y$ is extensible in $Y$,
$\Gamma = \operatorname{Spec}(O_K)$.
Thus, by virtue of Proposition~\ref{proposition:pullback},
we may assume $Y = \operatorname{Spec}(O_K)$.
In this case, the estimate of the Chow form was already given
in \cite[Proposition~1.3]{Bo} and \cite[4.3]{BGS}.
Indeed if we let $k_L$ be the metric on $L$ such that
\[
\Vert \Phi_{X} \Vert_
{(L,k_L) \otimes (\operatorname{Sym}^{\delta}(\overline{E}^{\rho}))^{{}\otimes {d+1}}} (w)
= 1
\]
for every $w \in O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{{\mathbb{Q}}})}(y)$,
then
$\widehat\deg (L,h_L) = h_{\overline{{\mathcal{O}}_E(1)}}(X)$
and $\widehat\deg (L,k_L) = h_{Herm}(\operatorname{Ch}(X))$, in the notation of \cite{BGS}.
\QED
Now we will state a relative case of Cornalba-Harris-Bost's inequality.
\begin{Theorem}
\label{thm:semistability:imply:average:semi-ampleness}
Let $Y$ be a regular arithmetic variety,
$\overline{E} = (E,h)$ a Hermitian vector bundle of rank $r$ on $Y$,
$\rho : \operatorname{\mathbf{GL}}_r \to \operatorname{\mathbf{GL}}_R$ a group scheme morphism of degree $k$
commuting with the transposed morphism.
Let $X$ be an effective cycle in ${\mathbb{P}} (E^{\rho})$
such that $X$ is flat over $Y$
with the relative dimension $d$
and degree $\delta$ on the generic fiber.
Let $X_1,\ldots,X_l$
be the irreducible components of $X_{red}$, and
$\tilde{X}_i \to X_i$ a generic resolution of singularities of $X_i$.
Let $Y_0$ be the maximal open set of $Y$ such that
the induced morphism $\tilde{X}_i \to Y$ is smooth over $Y_0$ for
every $i$.
Let $(B, h_B)$ be a line bundle equipped with
a generalized metric on $Y$ given by the equality:
\[
\widehat{{c}}_1(B, h_B) =
r \pi_* \left(\widehat{{c}}_{1}(\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1}
\cdot\widehat{X} \right)
+ k \delta (d+1) \widehat{{c}}_{1}(\overline{E}).
\]
Then, $h_B$ is $C^{\infty}$ over $Y_0$. Moreover,
there are a positive integer
$e=e(\rho,d,\delta)$, a positive integer $l=l(\rho,d,\delta)$,
a positive constant $C=C(\rho,d,\delta)$, and
sections $s_1, \ldots, s_l \in H^0(Y, B^{\otimes e})$
with the following properties.
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item
$e$, $l$, and $C$ depend only on $\rho$, $d$, and $\delta$.
\item
For a closed point $y$ of $Y_{{\mathbb{Q}}}$, if $X_y$ is Chow semistable,
then $s_i(y) \not= 0$ for some $i$.
\item
For all $i$ and all closed points $y$ of $(Y_0)_{{\mathbb{Q}}}$
with $y$ extensible in $Y$,
\[
\operatorname{g.\!m.}\left( \left( h_B^{\otimes e} \right)(s_i, s_i);\
O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}(y)\right)
\leq C.
\]
\end{enumerate}
In particular, if we set
\[
\beta =
e \left(
r \pi_* \left(\widehat{{c}}_{1}(\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1}
\cdot\widehat{X} \right)
+ k \delta (d+1) \widehat{{c}}_{1}(\overline{E})\right)
+ a (\log C),
\]
then, for any closed point $y \in (Y_0)_{{\mathbb{Q}}}$ with
$X_y$ Chow semistable,
there is a representative $(D, g)$ of $\beta$
such that $D$ is effective, $y \not\in D$, and that
\[
\sum_{w \in O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}(z)} g(w) \geq 0
\]
for all $z \in (Y_0)_{{\mathbb{Q}}}$ with $z$ extensible in $Y$.
\end{Theorem}
Note that if $\rho$ is the identity morphism,
then, by the proof below,
$C(\rho,d,\delta)$ is depending only on $r,d,\delta$. \\
{\sl Proof.}\quad
First of all, by
Proposition~\ref{prop:when:Bost:divisor:smooth},
\[
r \pi_* \left(\widehat{{c}}_{1}(\overline{{\mathcal{O}}_{E^{\rho}}(1)})^{d+1}
\cdot\widehat{X} \right)
+ k \delta (d+1) \widehat{{c}}_{1}(\overline{E})
\in \widehat{\operatorname{CH}}_{L^1}^1(Y; Y_0({\mathbb{C}})).
\]
Thus, $h_B$ is $C^{\infty}$ over $Y_0({\mathbb{C}})$.
By Lemma~\ref{lemma:stability:generators:over:Z},
there are a positive integer $e$ and
$\operatorname{\mathbf{SL}}_r({\mathbb{Q}})$-invariant homogeneous polynomials
$P_1,\cdots,P_l \in \operatorname{Sym}^{e}(\operatorname{Sym}^{\delta}({\mathbb{Z}}^R)^{d+1}{}^{\lor})$
depending only on $\rho$, $d$ and $\delta$ such that
if $X_y$ is Chow semistable for a closed point $y$ of $Y_{{\mathbb{Q}}}$,
then $P_i(\Phi_{X_{y}}) \ne 0$
for some $P_i$.
For later use,
we put
$c_2 (\rho,d,\delta)
= \max \{ |\!|\!| P_1 |\!|\!|, \cdots,
|\!|\!| P_l |\!|\!| \}$,
which is a constant depending only on $\rho$, $d$ and $\delta$.
Recall that the Chow form $\Phi_X$ is an element of
$H^0(Y,L \otimes (\operatorname{Sym}^{\delta}(E^{\rho}))^{{}\otimes {d+1}})$
and
by Lemma~\ref{lemma:P:to:sheafhom}
$P_i$ induces a polynomial map of sheaves
\[
\varphi_{P_i} : L \otimes \operatorname{Sym}^{\delta} (E^{\rho})^{\otimes d+1}
\to L^{\otimes er} \otimes (\det E)^{ek(d+1)\delta}.
\]
Hence we have
\[
\varphi_{P_i} (\Phi_{X})
\in H^0 \left(Y,L^{\otimes e r}
\otimes (\det E)^{e k(d+1)\delta} \right)
= H^0 (Y, B^{\otimes e})
\]
by \eqref{eqn:metric:L}.
Here we set $s_i = \varphi_{P_i} (\Phi_{X})$.
Then, the property (ii) is obvious
by the construction of $\varphi_{P_i}$
and (i) of Proposition~\ref{proposition:pullback}.
Now we will evaluate
$\Vert s_i \Vert$.
Let $y$ be a closed point of $(Y_0)_{{\mathbb{Q}}}$ with
$y$ extensible in $Y$.
Combining
Lemma~\ref{lemma:norm:of:morphism:P} and
Lemma~\ref{lemma:evaluation:of:norm:of:Phi},
we have
\begin{align*}
\operatorname{g.\!m.}\left( \Vert s_i \Vert ;\ O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{{\mathbb{Q}}})}(y)
\right) & \leq
\operatorname{g.\!m.}\left( |\!|\!| P_i |\!|\!|^r \, \Vert \Phi_{X} \Vert^{e r};\
O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{{\mathbb{Q}}})}(y) \right) \\
& \leq c_2(\rho,d,\delta) ^{r} c_1 (R,d,\delta) ^{e r}.
\end{align*}
Now we put
\[
C(\rho,d,\delta)
= c_1(R,d,\delta)^{2r} c_2(\rho,d,\delta)^{2e r},
\]
which is a positive constant depending only on $\rho$, $d$ and $\delta$.
Thus, we get (iii).
\QED
\begin{Remark}
\label{rem:geom:analog:Cornalba-Harris-Bost}
Here let us consider the geometric analogue of
Theorem~\ref{thm:semistability:imply:average:semi-ampleness}.
Let $Y$ be an algebraic variety over an algebraically closed
field $k$, $E$ a vector bundle of rank $r$,
$\rho : \operatorname{\mathbf{GL}}_r \to \operatorname{\mathbf{GL}}_R$ a group scheme morphism of degree $l$
commuting with the transposed morphism.
Let $X$ be an effective cycle in ${\mathbb{P}} (E^{\rho})$
such that $X$ is flat over $Y$
with the relative dimension $d$
and degree $\delta$ on the generic fiber.
Here we set
\[
b_{X/Y}(E, \rho) =
r \pi_* \left(c_{1}({\mathcal{O}}_{E^{\rho}}(1))^{d+1} \cdot X \right)
+ l \delta (d+1) c_{1}(E),
\]
which is a divisor on $Y$.
In the same way as in the proof of
Theorem~\ref{thm:semistability:imply:average:semi-ampleness}
(actually, this case is much easier than
the arithmetic case),
we can show the following.
\begin{quote}
There is a positive integer $e$ depending only on $\rho$, $d$,
and $\delta$ such that,
if $X_y$ is Chow semi-stable for some $y \in Y$,
then
\[
H^0(Y, {\mathcal{O}}_Y(e b_{X/Y}(E, \rho))) \otimes {\mathcal{O}}_Y \to
{\mathcal{O}}_Y(e b_{X/Y}(E, \rho))
\]
is surjective at $y$.
\end{quote}
This gives a refinement of \cite[Theorem~3.2]{Bo}.
\end{Remark}
\subsection{Relationship of two theorems}
\label{section:Bogomolov:to:Bost}
\setcounter{Theorem}{0}
In this subsection we will see some relationship between
the relative Bogomolov's inequality
(Theorem~\ref{thm:relative:Bogomolov:inequality:arithmetic:case})
and
the relative Cornalba-Harris-Bost's inequality
(Theorem~\ref{thm:semistability:imply:average:semi-ampleness}).
For this purpose, we will first show a more intrinsic version of
Theorem~\ref{thm:semistability:imply:average:semi-ampleness}.
\begin{Proposition}
\label{prop:intrinsic:relbost}
Let $f : X \to Y$ be a flat morphism of
regular projective arithmetic varieties with $\dim f = d$.
Let $L$ be a relatively very ample line bundle
such that $E = f_* (L)$ is a vector bundle of rank $r$ on $Y$.
Let $\eta$ be the generic point of $X$
and $\delta = \deg (L_{\eta}^d)$.
Moreover, let
$i : X \to {\mathbb{P}}(E^{\lor})$ be the embedding over $Y$.
Assume that $E$ is equipped with an Hermitian structure $h$
so that $L$ is also endowed with the Hermitian structure
by $i^* {\mathcal{O}}_{E^{\lor}}(1) \simeq L$.
Let $Y_0$ be the maximal open set of $Y$ such that
$f$ is smooth over $Y_0$.
Then, there is a positive integer
$e(r,d,\delta)$ and a positive constant $C(r,d,\delta)$
depending only on $r,d,\delta$ with the following properties.
If we set
\[
\beta = e(r,d,\delta) \left(
r f_* (\widehat{{c}}_1 (\overline{L})^{d+1})
- \delta (d+1) \widehat{{c}}_{1}(\overline{E})\right)
+ a (\log C(r,d,\delta)),
\]
then, for any closed point $y \in (Y_0)_{{\mathbb{Q}}}$ with
$X_y$ Chow semistable,
there is a representative $(D, g)$ of $\beta$
such that $D$ is effective, $y \not\in D$, and
\[
\sum_{w \in O_{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}(z)} g(w) \geq 0
\]
for all $z \in (Y_0)_{{\mathbb{Q}}}$.
\end{Proposition}
{\sl Proof.}\quad
We identify $X$ with its image by $i$.
Let $\pi : {\mathbb{P}}(E) \to Y$ be the projection.
Then, by Proposition~\ref{prop:when:Bost:divisor:smooth},
we get
\[
\pi_* \left(\widehat{{c}}_{1}(\overline{{\mathcal{O}}_{E^{\lor}}(1)})^{d+1}
\cdot\widehat{X} \right)
=
f_* (\widehat{{c}}_1 (\overline{L})^{d+1})
\]
Thus,
applying Theorem~\ref{thm:semistability:imply:average:semi-ampleness}
for $(Y,E^{\lor},\operatorname{id},X)$,
we get our assertion.
\QED
The following proposition will be derived from
Theorem~\ref{thm:relative:Bogomolov:inequality:arithmetic:case}.
\begin{Proposition}
\label{prop:Bogomolov:to:Bost}
Let $f : X \to Y$ be a projective morphism of regular arithmetic varieties
such that every fiber of $f_{{\mathbb{C}}} : X({\mathbb{C}}) \to Y({\mathbb{C}})$ is a reduced
and connected curve with only ordinary double singularities.
We assume that the genus $g$ of the generic fiber of $f$
is greater than or equal to $1$.
Let $L$ be a line bundle on $X$
such that \textup{(1)}the degree $\delta$ of
$L$ on the generic fiber is greater than or equal to $2g+1$,
\textup{(2)} $E = f_* (L)$ is a vector bundle of rank $r$ on $Y$
\textup{(}actually $r = \delta + 1 - g$\textup{)},
and that \textup{(3)} $f^*(E) \to L$ is surjective.
Assume that $E$ is equipped with an Hermitian structure $h$
so that $L$ is also endowed with the quotient metric
by $f^* (E) \to L$.
Let $Y_0$ be the maximal open set of $Y$ such that
$f$ is smooth over $Y_0$.
Then, for any closed points $y$ of $(Y_0)_{{\mathbb{Q}}}$,
\[
r f_* (\widehat{{c}}_1 (\overline{L})^{2})
- 2 \delta \widehat{{c}}_1 (\overline{E})
\]
is weakly positive at $y$ with respect to
any finite subsets of $Y_0({\mathbb{C}})$.
\end{Proposition}
Note that if the base space is $\operatorname{Spec}(O_K)$, then
the second author showed in \cite[Theorem~1.1]{MorFh}
the above inequality (under weaker assumptions)
using \cite[Corollary~8.9]{MoBG}.
Since we can prove Proposition~\ref{prop:Bogomolov:to:Bost}
in the same way as \cite[Theorem~1.1]{MorFh},
we will only sketch the proof.
{\sl Proof.}\quad
Let $S = \operatorname{Ker} (f^* (E) \to L)$ and $h_S$ the submetric of $S$
induced by $h$.
Then, by \cite{EL},
$S_{z}$ is stable for all $z \in Y_0({\mathbb{C}})$.
Applying Theorem~\ref{thm:relative:Bogomolov:inequality:arithmetic:case}
for $\overline{S} = (S,h_S)$, we obtain that if $y$ is a closed point
of $(Y_0)_{{\mathbb{Q}}}$, then
\[
f_* ( 2 (r-1) \widehat{{c}}_2(\overline{S}) - (r-2) \widehat{{c}}_1(\overline{S})^2)
\]
is weakly positive at $y$ with respect to
any finite subsets of $Y_0({\mathbb{C}})$.
If we set $\rho = \widehat{{c}}_2(f^*\overline{E})
- \widehat{{c}}_2(\overline{S}\oplus \overline{L})$,
then there is $g \in L^1_{loc}(Y({\mathbb{C}}))$ such that
$f_*(\rho) = a(g)$, $g$ is $C^{\infty}$ over $Y_0({\mathbb{C}})$, and
$g > 0$ on $Y_0({\mathbb{C}})$.
Now by a straightforward calculation, we have
\begin{multline*}
f_* ( 2 (r-1) \widehat{{c}}_2(\overline{S}) - (r-2) \widehat{{c}}_1(\overline{S})^2)
+ 2 (r-1) f_*(\rho) \\
= f_* \left( 2 (r-1) \widehat{{c}}_2(f^* \overline{E})
- (r-2) \widehat{{c}}_1(f^* \overline{E})^2 \right)
+ f_* \left( r \widehat{{c}}_1(\overline{L})^2
- 2 \widehat{{c}}_1(f^* \overline{E})
\cdot \widehat{{c}}_1(\overline{L}) \right) \\
= r f_* (\widehat{{c}}_1(\overline{L})^2)
- 2 \delta \widehat{{c}}_1(\overline{E}).
\end{multline*}
\QED
Let us compare
Proposition~\ref{prop:intrinsic:relbost}
with Proposition~\ref{prop:Bogomolov:to:Bost}.
Both of them give some arithmetic positivity
of the same divisor
(although $d=1$ in Proposition~\ref{prop:Bogomolov:to:Bost}),
under the assumption of some semi-stability
(of Chow or of vector bundles).
The former has advantage
since it treats varieties of arbitrary relative dimension.
On the other hand,
the latter has advantage
since it shows that
the anonymous constant in the former
is zero (see also \cite{Zh}).
Moreover, in the complex case,
the counterpart of the relative Bogomolov's inequality
of Theorem~\ref{thm:relative:Bogomolov:inequality:arithmetic:case}
has a wonderful application to the moduli of stable curves
(\cite{MoRB}).
\renewcommand{\thesection}{Appendix \Alph{section}}
\renewcommand{\theTheorem}{\Alph{section}.\arabic{Theorem}}
\renewcommand{\theClaim}{\Alph{section}.\arabic{Theorem}.\arabic{Claim}}
\renewcommand{\theequation}{\Alph{section}.\arabic{Theorem}.\arabic{Claim}}
\setcounter{section}{0}
\section{Commutativity of push-forward and pull-back}
\label{sec::comm:push:pull}
Let $f : X \to Y$ be a smooth proper morphism of regular noetherian schemes, and
$u : Y' \to Y$ a morphism of regular noetherian schemes.
Let $X' = X \times_{Y} Y'$ and
\[
\begin{CD}
X @<{u'}<< X' \\
@V{f}VV @VV{f'}V \\
Y @<{u}<< Y'
\end{CD}
\]
the induced diagram.
Let $Z$ be a cycle of codimension $p$ and $|Z|$ the support of $Z$.
We assume that $\operatorname{codim}_{X'}({u'}^{-1}(|Z|)) \geq p$.
Then, it is easy to see that $\operatorname{codim}_{Y'}(u^{-1}(|f_*(Z)|)) \geq p - d$,
where $d = \dim X - \dim Y$.
Thus, we can define $f'_*({u'}^*(Z))$ and $u^*(f_*(Z))$ as
elements of $Z^{p-d}(Y')$.
It is well known, we believe, that
$f'_*({u'}^*(Z)) = u^*(f_*(Z))$ in $Z^{p-d}(Y')$.
We could not however find any suitable references for the above fact, so that
in this section, we would like to give the proof of it.
\bigskip
Let $X$ be a regular noetherian scheme, and $T$ a closed subscheme of $X$.
We denote by $K'_T(X)$ the Grothendieck group generated by
coherent sheaves $F$ with $\operatorname{Supp}(F) \subseteq T_{red}$ modulo
the following relation:
$[F] = [F'] + [F'']$ if there is an exact sequence
$0 \to F' \to F \to F'' \to 0$.
Let $p$ be a non-negative integer, and
$X^{(p)}$ the set of all points $x$ of $X$ with
$\operatorname{codim}_X \overline{\{ x \}} = p$.
We define $Z^p_T(X)$ to be
\[
Z^p_T(X) = \bigoplus_{ x \in X^{(p)} \cap T} {\mathbb{Z}} \cdot \overline{\{ x \}}.
\]
We assume that $\operatorname{codim}_X T \geq p$. Then, we can define
the natural homomorphism
\[
z^p : K'_T(X) \to Z^p_T(X)
\]
to be
\[
z^p([F]) = \sum_{x \in X^{(p)} \cap T} l_{{\mathcal{O}}_{X, x}}(F_x)
\cdot \overline{\{ x \}},
\]
where $l_{{\mathcal{O}}_{X, x}}(F_x)$ is the length of $F_x$ as
${\mathcal{O}}_{X,x}$-modules.
Note that if $\operatorname{codim}_X T > p$, then $z^p = 0$.
\medskip
Let $f : X \to Y$ be a proper morphism of regular noetherian schemes, and
$T$ a closed subscheme of $X$. Then, we define
the homomorphism $f_* : K'_T(X) \to K'_{f(T)}(Y)$ to be
\[
f_*([F]) = \sum_{i \geq 0} [R^i f_*(F)].
\]
Here we set $d = \dim X - \dim Y$.
Let $p$ be a non-negative integer with $\operatorname{codim}_X T \geq p$ and
$p \geq d$. Then, $\operatorname{codim}_Y f(T) \geq p - d$.
First, let us consider the following proposition.
\begin{Proposition}
\label{prop:comm:push:forward}
With notation as above, the diagram
\[
\begin{CD}
K'_T(X) @>{z^p}>> Z^p_T(X) \\
@V{f_*}VV @VV{f_*}V \\
K'_{f(T)}(Y) @>{z^{p-d}}>> Z^{p-d}_{f(T)}(Y)
\end{CD}
\]
is commutative.
\end{Proposition}
{\sl Proof.}\quad
For a coherent sheaf $F$ on $X$ with
$\operatorname{Supp}(F) \subseteq T_{red}$, there is a filtration
$0 = F_0 \subseteq F_1 \subseteq \cdots \subseteq F_n = F$
with $F_i/F_{i-1} \simeq {\mathcal{O}}_X/P_i$ for some
prime ideal sheaves $P_i$ on $X$.
Then,
\[
\begin{cases}
f_*(z^p([F])) = \sum_{i=1}^n f_*(z^p([{\mathcal{O}}_X/P_i])) \\
z^{p-d}(f_*([F])) = \sum_{i=1}^n z^{p-d}(f_*([{\mathcal{O}}_X/P_i])).
\end{cases}
\]
Thus, we may assume that $F = {\mathcal{O}}_X/P$ for some prime ideal
sheaf $P$ with $\operatorname{Supp}({\mathcal{O}}_X/P) \subseteq T_{red}$ and
$\operatorname{codim}_X(\operatorname{Spec}({\mathcal{O}}_X/P)) = p$.
We set $Z = \operatorname{Spec}({\mathcal{O}}_X/P)$. Then, $z^p([{\mathcal{O}}_X/P]) = Z$.
First, let us consider the case where $\dim f(Z) < \dim Z$.
In this case, $f_*(Z) = 0$. On the other hand,
since $\operatorname{Supp}(R^i f_*({\mathcal{O}}_X/P))\subseteq f(Z)$,
we can see that $z^{p-d}([R^i f_*({\mathcal{O}}_X/P)]) = 0$ for all $i \geq 0$.
Thus, $z^{p-d}(f_*([{\mathcal{O}}/P])) = 0$.
Next, we assume that $\dim f(Z) = \dim Z$.
Then, $Z \to f(Z)$ is generically finite. Thus,
$\operatorname{Supp}(R^i f_*({\mathcal{O}}_X/P))$ is a proper closed subset of $f(Z)$
for each $i \geq 1$. Therefore, we have
\[
z^{p-d}(f_*([{\mathcal{O}}_X/P])) = z^{p-d}([f_*({\mathcal{O}}_X/P)]) = f_*(Z).
\]
\QED
Let $g : Z \to X$ be a morphism of regular noetherian schemes, and
$T$ a closed subscheme of $X$.
Then, we define the homomorphism $g^* : K'_T(X) \to K'_{f^{-1}(T)}(Z)$
to be
\[
g^*([F]) = \sum_{i \geq 0} (-1)^i [L_if^*(F)].
\]
Let $p$ be a non-negative integer with $\operatorname{codim}_X T \geq p$ and
$\operatorname{codim}_Z (g^{-1}(T)) \geq p$.
Here let us consider
the following proposition.
\begin{Proposition}
\label{prop:well:def:pull:back}
Let $F$ and $G$ be coherent sheaves on $X$ with
$\operatorname{Supp}(F), \operatorname{Supp}(G) \subseteq T_{red}$.
If $z^p([F]) = z^p([G])$, then $z^{p}(g^*([F])) =
z^{p}(g^*([G]))$.
\end{Proposition}
{\sl Proof.}\quad
Let
$0 = F_0 \subseteq F_1 \subseteq \cdots \subseteq F_n = F$
and
$0 = G_0 \subseteq G_1 \subseteq \cdots \subseteq G_m = F$
be filtrations of $F$ and $G$ respectively such that
$F_i/F_{i-1} \simeq {\mathcal{O}}_X/P_i$ and
$G_j/G_{j-1} \simeq {\mathcal{O}}_X/Q_j$ for some
prime ideal sheaves $P_i$ and $Q_j$ on $X$.
Then,
\[
\begin{cases}
z^{p}(g^*([F])) = \sum_{i=1}^n z^{p}(g^*([{\mathcal{O}}_X/P_i])) \\
z^{p}(g^*([G])) = \sum_{j=1}^m z^{p}(g^*([{\mathcal{O}}_X/Q_j]))
\end{cases}
\]
Thus, it is sufficient to show that
$z^{p}(g^*([{\mathcal{O}}_X/P])) = 0$ for all prime ideals $P$ with
\[
\text{
$\operatorname{Supp}({\mathcal{O}}_X/P) \subseteq T_{red}$,
$\operatorname{codim}_X (\operatorname{Supp}({\mathcal{O}}_X/P)) > p$ and
$\operatorname{codim}_Z (g^{-1}(\operatorname{Supp}({\mathcal{O}}_X/P))) = p$.
}
\]
This is a consequence of the following lemma.
\QED
\begin{Lemma}
\label{lem:vanshing:tor:alt:sum}
Let $(A, m)$ and $(B, n)$ be regular local rings,
$\phi : A \to B$ a homomorphism of local rings, and
$M$ an $A$-module of finite type.
If $\operatorname{Supp}(M \otimes_A B) = \{ n \}$ and
\[
\operatorname{codim}_{\operatorname{Spec}(B)} (\operatorname{Supp}(M \otimes_A B)) < \operatorname{codim}_{\operatorname{Spec}(A)} (\operatorname{Supp}(M)),
\]
then
\[
\sum_{i \geq 0} (-1)^i l_B(\operatorname{Tor}_i^A(M, B)) = 0.
\]
\end{Lemma}
{\sl Proof.}\quad
We freely use notations in \cite[Chapter~I]{SoAr}.
Let $f : \operatorname{Spec}(B) \to \operatorname{Spec}(A)$ be a morphism induced by
$\phi : A \to B$.
We set $Y = \operatorname{Supp}(M)$ and $q = \operatorname{codim}_{\operatorname{Spec}(A)} (\operatorname{Supp}(M))$.
Let $P_{\cdot} \to M$ be a free resolution of $M$.
Then, $[P_{\cdot}] \in F^q K_0^Y(\operatorname{Spec}(A))$.
Thus, by \cite[(iii) of Theorem~3 in I.3]{SoAr},
\[
[f^*(P_{\cdot})] = [ P_{\cdot} \otimes_A B ]
\in F^q K_0^{\{ n \}}(\operatorname{Spec}(B))_{{\mathbb{Q}}}
\]
because $f^{-1}(Y) = \operatorname{Supp}(M \otimes_A B) = \{ n \}$.
On the other hand, since
\[
q > \operatorname{codim}_{\operatorname{Spec}(B)} (\operatorname{Supp}(M \otimes_A B)) = \dim B,
\]
we have $F^q K_0^{\{ n \}}(\operatorname{Spec}(B))_{{\mathbb{Q}}} = \{ 0 \}$.
Thus, $[ P_{\cdot} \otimes_A B ] = 0$ in $K_0^{\{ n \}}(\operatorname{Spec}(B))$
because
\[
K_0^{\{ n \}}(\operatorname{Spec}(B)) \simeq {\mathbb{Z}}
\]
has no torsion.
This shows us our assertion.
\QED
As a corollary of Proposition~\ref{prop:well:def:pull:back},
we have the following.
\begin{Corollary}
\label{cor:comm:pull:back}
With notation as in Proposition~\textup{\ref{prop:well:def:pull:back}},
\[
\begin{CD}
K'_T(X) @>{z^p}>> Z^p_T(X) \\
@V{g^*}VV @VV{g^*}V \\
K'_{f^{-1}(T)}(Z) @>{z^{p}}>> Z^{p}_{f^{-1}(T)}(Z)
\end{CD}
\]
is commutative. Note that
$g^* : Z^p_T(X) \to Z^{p}_{f^{-1}(T)}(Z)$ is defined by
$g^*(Z) = z^{p}(g^*([{\mathcal{O}}_Z]))$ for each integral cycle $Z$
in $Z^p_T(X)$.
\end{Corollary}
Let $f : X \to Y$ be a flat proper morphism of regular noetherian schemes, and
$u : Y' \to Y$ a morphism of regular noetherian schemes.
Let $X' = X \times_{Y} Y'$ and
\[
\begin{CD}
X @<{u'}<< X' \\
@V{f}VV @VV{f'}V \\
Y @<{u}<< Y'
\end{CD}
\]
the induced diagram.
We assume that $X'$ is regular.
Note that if $f$ is smooth, then $X'$ is regular.
We set $d = \dim X - \dim Y = \dim X' - \dim Y'$.
Let $T$ be a closed subscheme of $X$, and $p$ a non-negative
integer with $\operatorname{codim}_X T \geq p$, $\operatorname{codim}_{X'} ({u'}^{-1}(T)) \geq p$
and $p \geq d$.
Note that $\operatorname{codim}_Y f(T) \geq p-d$ and
$\operatorname{codim}_{Y'} (u^{-1}(f(T))) \geq p - d$ because
$u^{-1}(f(T)) = f'({u'}^{-1}(T))$.
Then, we have the following proposition.
\begin{Proposition}
\label{prop:comm:cycle:push:pull}
The diagram
\[
\begin{CD}
Z^p_T(X) @>{{u'}^*}>> Z^{p}_{{u'}^{-1}(T)}(X') \\
@V{f_*}VV @VV{f'_*}V \\
Z^{p-d}_{f(T)}(Y) @>{u^*}>> Z^{p-d}_{u^{-1}(f(T))}(Y')
\end{CD}
\]
is commutative.
\end{Proposition}
{\sl Proof.}\quad
Since $f$ is flat, by \cite[Proposition 3.1.0 in IV]{SGA6},
for any coherent sheaves $F$ on $X$,
\[
L_{\cdot}u^* \left( R^{\cdot} f_*(F) \right) \overset{\sim}{\longrightarrow}
R^{\cdot}f'_* \left( L_{\cdot}{u'}^*(F)\right),
\]
which shows that the diagram
\[
\begin{CD}
K'_T(X) @>{{u'}^*}>> K'_{{u'}^{-1}(T)}(X') \\
@V{f_*}VV @VV{f'_*}V \\
K'_{f(T)}(Y) @>{u^*}>> K'_{u^{-1}(f(T))}(Y')
\end{CD}
\]
is commutative.
Thus, by virtue of Proposition~\ref{prop:comm:push:forward} and
Corollary~\ref{cor:comm:pull:back},
we can see our proposition.
\QED
\bigskip
|
2000-04-03T15:32:20 | 9710 | alg-geom/9710023 | en | https://arxiv.org/abs/alg-geom/9710023 | [
"alg-geom",
"math.AG"
] | alg-geom/9710023 | Norbert A'Campo | Norbert A'Campo | Real deformations and complex topology of plane curve singularities | 16 pages TeX with 11 fig-xx.eps | Ann. Fac. Sci. Toulouse Math. (6) 8 (1999), no. 1, 5-23 | null | null | null | This is the paper as published. The topology of a complex plane curve
singularity with real branches is deduced from any real deformation having
delta crossings. An example of the computation of the global geometric
monodromy of a polynomial mapping is added.
| [
{
"version": "v1",
"created": "Mon, 20 Oct 1997 00:13:22 GMT"
},
{
"version": "v2",
"created": "Mon, 3 Apr 2000 13:32:16 GMT"
}
] | 2007-05-23T00:00:00 | [
[
"A'Campo",
"Norbert",
""
]
] | alg-geom | \section{1}{Introduction}
The geometric monodromy $T$ of a curve singularity in the complex
plane is a
diffeomorphism of a compact surface with boundary $(F,\partial F)$
inducing
the identity on the boundary, which is well defined up to isotopy
relative to
the boundary. The geometric monodromy of a curve singularity in the
complex
plane determines the local topology of the singularity. As element of
the
mapping class group of the surface $(F,\partial F)$, the diffeomorphism
$T$
can be written as a composition of Dehn twists. In section 3 of this
paper the
geometric monodromy of an isolated plane curve singularity is written
explicitly as a
composition of right Dehn twists. In fact, a global graphical
algorithm
for the construction of the surface $(F,\partial F)$ with a system of
simply closed curves on it is given in section 4, such that the curves
of this system are the vanishing cycles of a real morsification of the
singularity. In section 5, as an illustration, the global geometric
monodromy
of the polynomial $y^4-2y^2x^3+x^6-x^7-4yx^5,$ which has two critical
fibers, is computed. \br
The germ of a curve singularity in $\Bbb C^2$ is a finite union of
parametrized
local branches $b_i:\Bbb C \to \Bbb C^2, 1 \leq i \leq r.$ First observe,
that
without loss of generality for the local topology, we can assume that
the
branches have a real polynomial parameterization. The combinatorial
data used
to describe the geometric monodromy of a curve singularity come from
generic
real polynomial deformations of the parameterizations of the local
branches
$b_{i,t}:\Bbb C \to \Bbb C^2, 1 \leq i \leq r, t \in [0,1]$, such
that:\br
(i) $b_{i,0}=b_i, 1 \leq i \leq r,$\br
(ii) for some $\rho > 0$ the intersection of the union of the branches
with the $\rho-$ball $B$ at the singular point of curve in $\Bbb C^2$ is
a representative of the germ of the curve and $B$ is a Milnor ball for
the
germ,\br
(iii) the images of $b_{i,t},1 \leq i \leq r, t \in [0,1],$ intersect
the
boundary of the ball $B$ transversally,\br
(iv) the union of the images $b_{i,t}(\Bbb R), 1 \leq i \leq r,$ has for
every $t \in (0,1]$ the maximal possible number of double points in the
interior of $B$.\br
Such deformations correspond to real morsifications of the defining
equation
of the singularity and were used to study the local monodromy
in [AC2],[AC3],[G-Z]. Real deformations of singu\-larities of plane
algebraic curves with the maximal possible number of double points in
the real plane were
discovered by Charlotte Angas Scott [S1,S2]. I thank Egbert Brieskorn
for
having drawn my attention on the references [S1],[S2].
In section 6 we start with a connected divide, which defines as
explained in section 3 a classical link. We will construct a map from
the complement of the link of a connected divide to the circle and
prove that this map is a fibration.
This fibration is for a divide of a plane curve singularity a model
for the Milnor fibration of the singularity. The link of most connected
divides are hyperbolic. In a forth coming paper we will study the
geometry of a link of a divide.
We used MAPLE for the drawings of parametrized curves and for the
computation of suitable deformations of the polynomial equations.
Of great help for the investigation of topological changes in
families of polynomial equations is the mathematical software SURF
which
has been developed by Stefan Endrass. I thank Stefan Endrass warmly
for
permitting me to use SURF. Part of this work was done in Toulouse and
I thank the members of the Laboratory \'Emile Picard for their
hospitality.
\par\noindent
\section{2}{\fam\bffam\tenbf Real deformations of plane curve singularities}
Let $f:\Bbb C^2 \to \Bbb C$ be the germ at $0 \in \Bbb C^2$ of an holomorphic
map
with $f(0)=0$ and having an isolated singularity $S$ at $0$. We are
mainly
interested in the study of topological properties of singularities,
therefore we can assume without loss of generality that the germ $f$
is a
product of locally irreducible real polynomials. Having chosen a Milnor
ball
$B(0,\rho)$ for $f$, there exists a real polynomial deformation family
$f_t,t \in [0,1],$ of $f$ such that for all $t$ the $0$-level of $f_t$
is transversal to the boundary of the ball $B(0,\rho)$ and such that
for
all $t \in ]0,1]$ the $0$-level of $f_t$ has $\delta$ transversal
double
points in the interior of the disk $D(0,\rho):=B(0,\rho) \cap \Bbb R^2$,
where
the Milnor number $\mu$ and the number $r$ of local branches of $f$
satisfy
$\mu=2\delta-r+1.$ In particular, the $0$-level of $f_t, t \in ]0,1],$
has
in $D(0,\rho)$ no self tangencies or triple intersections. It is
possible
to choose for $f_t,t \in [0,1],$ a family
of defining equations for the union of the images of
$b_{i,t}, 1 \leq i \leq r.$ The deformation $f_t,t \in [0,1]$ is called
a
real morsification with respect to the Milnor ball $B(0,\rho)$ of $f.$
So, the $0$-level of the restriction of
$f_t,t \in ]0,1],$ to $D(0,\rho)$ is an immersion without
self-tangencies and
having only transversal self-intersections of $r$ copies of an interval
(see [AC2],[AC3],[G-Z]). The $0$-level of the restriction of
$f_t,t \in ]0,1],$ to $D(0,\rho)$ is up to a diffeomorphism independent
of
$t,$ it is called a divide ("partage" in [AC2]) and it is shown that
for
instance the divide determines the homological monodromy group of the
versal deformation of the singularity. Figure 1 represents a divide for
the singularity at $0 \in \Bbb C^2$ of the curve $(y^5-x^3)(x^5-y^3)=0$.
\midinsert
\cline{\epsffile{fig-1.eps}}
\medskip
\centerline{Figure 1: A divide for $(y^5-x^3)(x^5-y^3)=0.$}
\endinsert
\remark{\fam\bffam\tenbf Remark}
The transversal isotopy class of the divide of a singularity with real
branches is not a topological invariant of the singularity. The
singularities of $y^4-2y^2x^3+x^6+x^7$ and $y^4-2y^2x^3+x^6-x^7$ have
congruent but not transversal isotopic divides. The singularities
$(x^2-y^2)(x^2-y^3)(y^2-x^3)$ and $(x^2-3xy+2y^2)(x^2-y^3)(y^2-x^3)$
are topologically equivalent but can not have congruent divides.
The singularity $y^3-x^5$ admits two divides, which give a model for
the smallest possible transition, according to the mod 4 congruence of
V. Arnold [A] and its
celebrated strengthening to a mod 8 congruence of
V. A. Rohlin [R1,R2], of odd ovals to
even ovals for projective real M-curves of even degree. I owe this remark to
Oleg Viro [V].
More precisely, there exist a polynomial family $f_s(x,y), s \in \Bbb R,$ of
polynomials of degree $6,$ having the central symmetry
$f_s(x,y)=-f_{-s}(-x,-y)$ such that the levels
$f_s(x,y)=0,\ s \not=0,$
are divides for the singularity $f_0(x,y)=y^3-x^5.$ Moreover, for $s \in \Bbb R,\
s\not=0,$
the divide $f_s(x,y)=0$ has for regions on which the function has the sign of
the parameter $s.$ Therefor,
at
$s=0$ four regions of $f_s(x,y)=0$ collapse and hence, four ovals of
$f_s(x,y)=(s/2)^E$ collapse and change parity
at $s=0$ if the exponent $E$ is big and odd.
\midinsert
\cline{\epsffile{divide_links.eps} \ \epsffile{divide_rechts.eps}}
\medskip
\centerline{Figure 2a: The divides $f_{\pm 1}(x,y)=0.$}
\endinsert
In Figure $2b$ are drawn the
smoothings of the
divides $f_{-1}(x,y)=0$ and $f_{+1}(x,y)=0$ of
the singularity of $f_0(x,y)=y^3-x^5$ at $0$. Each of the smoothings
$\{f_{\pm1}(x,y)=\pm \epsilon \}\cap D$ consists of four ovals and a
chord, such that the ovals lie
on the positive, respectively on the negative side, of the chord.
Such a family $f_s(x,y)$
is for instance given by:
\midinsert
\cline{\epsffile{links.eps} \ \epsffile{rechts.eps}}
\medskip
\centerline{Figure 2b: Four ovals change parity.}
\endinsert
$$
y^3-x^5
-{{125}\over{8}}\,{s}^{3}{x}^{6}
+({{375}\over{64}}\,{s}^{6}
+{{245}\over{16}}\,{s}^{4}
-{{25}\over{4}}\,{s}^{2}){x}^{5}
+{{75}\over{4}}\,{s}^{2}{x}^{4}y
$$
$$
+({{2695}\over{128}}\,{s}^{7}
+{{21625}\over{256}}\,{s}^{9}
-{{35}\over{8}}\,{s}^{3}
-{{4847}\over{160}}\,{s}^{5}){x}^{4}
-(5\,s
+{{159}\over{4}}\,{s}^{3}
-{{75}\over{4}}\,{s}^{5}){x}^{3}y
$$
$$
-{{15}\over{2}}\,s{x}^{2}{y}^{2}
-({{2703}\over{500}}\,{s}^{6}
+{{1281}\over{32}}\,{s}^{8}
-{{29625}\over{512}}\,{s}^{12}
+{{2345}\over{128}}\,{s}^{10})x^3
$$
$$
-({{17625}\over{256}}\,{s}^{8}
+{{3583}\over{32}}\,{s}^{6}
+{{5793}\over{400}}\,{s}^{4}){x}^{2}y
-({{95}\over{2}}\,{s}^{4}
+{{53}\over{10}}\,{s}^{2})x{y}^{2}
$$
$$
-({{997}\over{4000}}\,{s}^{9}
+{{42875}\over{2048}}\,{s}^{15}
+{{4575}\over{256}}\,{s}^{13}
+{{857}\over{320}}\,{s}^{11}){x}^{2}
$$
$$
-({{177325}\over{2048}}\,{s}^{11}
+{{1803}\over{200}}\,{s}^{7}
+{{35441}\over{512}}\,{s}^{9})xy
-({{6395}\over{128}}\,{s}^{7}
+{{317}\over{40}}\,{s}^{5}){y}^{2}
$$
$$
+({{19871}\over{1280}}\,{s}^{14}
+{{10165}\over{1024}}\,{s}^{16}
-{{59125}\over{4096}}\,{s}^{18}
+{{4171}\over{2000}}\,{s}^{12})x
$$
$$
+({{51025}\over{4096}}\,{s}^{12}
+{{54223}\over{25600}}\,{s}^{10}
-{{153725}\over{16384}}\,{s}^{14})y
$$
\endremark
\remark{\fam\bffam\tenbf Problem}
Classify up to transversal isotopy, i.e. isotopy through immersions
with
only transversal double and triple point crossings, the divides for
an
isolated real plane curve singularity.
\endremark
\goodbreak
\section{3}{Complex topology of plane curve singularities}
In this section we wish to explain how one can read off from the divide
of
a plane curve singularity $S$ the local link $L$, the Milnor fiber and
the
geometric monodromy group of the singularity. In particular, we will
give
the geometric monodromy of the singularity explicitly as a product of
Dehn twists.
Let $P \subset D(0,\rho)$ be the divide of the singularity $f$. For a
tangent
vector $v \in TD(0,\rho)=D(0,\rho) \times\Bbb R^2$ of $D$ at the point
$p\in D(0,\rho)$ let $J(v) \in \Bbb C^2$ be the point $p+iv.$ The Milnor
ball
$B$ can be viewed as
$$
B(0,\rho)=\{J(v) \mid v \in T(D(0,\rho)) \ \text{ and }\ \|J(v)\| \leq
\rho\}.
$$
Observe that
$$L(P):=\{J(v) \mid v \in T(P) \ \text{ and }\ \|J(v)\|=\rho \}$$
is a closed submanifold of dimension one in the boundary of the Milnor
ball $B(0,\rho)$. We call $L(P)$ the link of the divide $P$.
Note further that
$$R(P):=\{J(v) \mid v \in T(P) \ \text{ and }\ \|J(v)\| \leq \rho \}$$
is an immersed surface in $B(0,\rho)$ with boundary $L(P)$ having
only
transversal double point singularities. Let $F(P)$ be the surface
obtained from $R(P)$ by replacing the local links of its singularities
by cylinders. The differential model of those replacements is as
follows:
let $\chi: \Bbb C^2 \to \Bbb R$ be a smooth bump function at $0 \in \Bbb C^2$;
replace the immersed surface
$\{(x,y) \in \Bbb C^2 \mid xy=0\}$ by the smooth surface
$\{(x,y) \in \Bbb C^2 \mid xy=\tau^2\chi(x/\tau,y/\tau)\}$, where $\tau$
is a
sufficiently small positive real number. We call $R(P)$ the singular
and
$F(P)$ the regular ribbon surface of the divide $P.$ The connected,
compact
surface $F(P)$ has genus $g:=\delta-r+1$ and $r$ boundary components.
Note, that $g$ is the number of regions of the divide $P$. A region of
$P$
is a connected component of the complement of $P$ in $D(0,\rho)$,
which
lies in the interior of $D(0,\rho)$. For the example drawn in Figure
$1,$ we
have $r=2,\delta=17,g=16.$
The ribbon surface $R(P)$ carries a natural orientation, since
parametrized by an open subset of the tangent space $T(\Bbb R).$ Hence the
surface $F(P)$ and the link $L(P)$ are also naturally oriented. We
orient $B$ as a submanifold of $-T\Bbb R^2,$ which is the orientation of
$B$ as a submanifold in $\Bbb C^2.$
\proclaim{Theorem 1}
Let $P$ be the divide for an isolated plane curve singularity $S.$ The
submanifold $(F(P),L(P))$ is up to isotopy a model for the Milnor
fiber of the singularity $S$.
\endgroup\bigbreak
\demo{ \fam\bffam\tenbf Proof} Choose $0 < \rho_{-} < \rho$ such that $P \cap
D(0,\rho_{-})$ is
still a divide for the singularity $S.$ Along the divide the singular
level
$F_{t,0}:=\{(x,y) \in B \mid f_t(x,y)=0 \}$ is up to order $1$ tangent
to the
immersed surface $R(P)$. Hence, for
$B_{-}':=\{u+iv \in B(0,\rho_{-}) \mid u,v \in
\Bbb R^2, ||v|| \leq \rho'\}$ with $0 < \rho' << \rho,$ the intersections
$R'(P):=\partial B_{-}' \cap
R(P)$ and $F'_{t,0}:=\partial B_{-}' \cap F_{t,0}$ are
transversal and are regular
collar neighbourhoods of
the divide in $R(P)$ and in $F_{t,0}.$ Therefore the
nonsingular level $F_{t,\eta}:=\{(x,y) \in B(0,\rho) \mid f_t(x,y)=\eta
\}$,
where $\eta \in \Bbb R$ is
sufficiently small, contains in its interior $F'_{t,\eta}:=B_{-}' \cap
F_{t,\eta},$ which is a diffeomorphic copy of the surface with
boundary $F(P)$. Since $F'_{t,\eta}$ and $F_{t,\eta}$ are connected
surfaces both with $r$ boundary components and the
intersection forms on the first homology
are isomorphic, the difference $F_{t,\eta}\setminus F'_{t,\eta}$ is a
union of open collar
tubular neighbourhoods of the boundary components of the surface
$F_{t,\eta}$. So, the surfaces
$F_{t,\eta},F'_{t,\eta}$ and $F(P)$ are diffeomorphic.
We conclude by observing that the nonsingular levels $F_{t,\eta}$
and the Milnor fiber are connected in
the local unfolding through nonsingular levels.
\penalty-100\null\hfill\qed\bigbreak
From this proof it follows also that the local link $L(S)$ of the
singularity
$S$ in
$\partial B$ is
cobordant to the sub\-manifold $\partial F'_{t,0}$ in $\partial
B_{-}'.$ The
cobordism is given by the pair $(B \setminus
\text{int}(B_{-}'),F_{t,\eta}
\setminus \text{int}(F'_{t,\eta})).$ It
is clear, that the pairs $(\partial B,L(P)),$
$(\partial B_{-}',\partial F'_{t,0})$ and
$(\partial B_{-}',\partial F'_{t,\eta})$ are
diffeomorphic.
One can prove even more:
\proclaim{Theorem 2}
Let $P$ be the divide for an isolated plane curve singularity $S.$
The pairs $(\partial B,L(S))$, where $L(S)$ is the local link of the
singularity
$S,$ and $(\partial B,L(P))$ are diffeomorphic.
\endgroup\bigbreak
The proof is given in section 6.
\remark{\fam\bffam\tenbf Remark}
The signed planar Dynkin diagram of the divide determines up to isotopy
the
divide of the singularity. It follows from Theorem 2, that the signed
planar
Dynkin diagram
determines geometrically the topology of the singularity. Using the
theorem
of Burau and Zariski stating that the topological type of a plane curve
singularity is determined by the mutual intersection numbers of the
branches
and the Alexander polynomial of each branch, the authors L. Balke and
R. Kaenders [B-K] have proved that the signed Dynkin diagram,
without its
planar embedding, determines the topology of the singularity.
\endremark
\midinsert
\cline{\epsffile{fig-3.eps}}
\medskip
\centerline{Figure $3:$ The link $P_v.$}
\endinsert
We need a combinatorial description of the surface $F(P).$ For a divide
$P$
we define: A vertex of $P$ is double point of $P$, and an edge of $P$
is the
closure of a connected component of the complement of the vertices in
$P$.
Now we choose an orientation of $\Bbb R^2$, and a small deformation $\bar
f$
of the polynomial $f$ such that the $0$-level of $\bar f$ is the divide
$P.$
We call a region of the divide positive or negative according to the
sign
of $\bar f$. We orient the boundaries of the positive regions such that
the
outer normal and the oriented tangents of the boundary agree in this
order
with the chosen orientation of $\Bbb R^2$. We choose a midpoint on each
edge,
which connects two vertices. The link $P_v$ of a vertex $v$ is the
closure
of the connected component of the complement of the midpoints in $P$
containing the given vertex $v$.
\midinsert
\cline{\epsffile{fig-4.eps}}
\medskip
\centerline{Figure $4:$ A piece of surface $F_v.$}
\endinsert
For each vertex $v$ of $P$ we will construct
a piece of surface $F_v$, such that those pieces glue together and
build
$F(P)$. Let $P_v$ be the link of the vertex $v$. Call $c_v,c_v'$ the
endpoints of the branches of $P_v,$ which are oriented towards $v,$ and
$d_v,d_v'$ the endpoints of the branches of $P_v,$ which are oriented
away
from $v$. Thus, $c_v,c_v',d_v,d_v'$ are midpoints or endpoints of the
divide $P$ (see Figure $3.$) Using an orientation of the divide $P$ we
label
$c_v,c_v'$ such that $c_v'$
comes after $c_v$, and we label $d_v,d_v'$ such that the sector
$c_v,d_v$ is
in a positive region. Then $F_v$ is the surface with boundary and
corners
drawn in Figure $4.$ There are 8 corners and there are 8 boundary
components
in between the corners,
4 of them will get a marking by $c_v,c_v',d_v,d_v'$, which will
determine the gluing with the piece of the next vertex and 4 do not
have a marking. The
gluing of the pieces $F_v$ along the marked boundary components
according to
the gluing scheme given by the divide $P$ yields the surface $F(P)$. On
the
pieces $F_v$ we have drawn oriented curves colored red, white, and
blue.
The white curves are simple closed pairwise disjoint curves. The
surface
$F(P)$ will be oriented such that the curves taken in the order
red-white-blue have nonnegative intersections.
The remaining red curves
glue together and build a red graph on $F(P)$. The remaining blue
curves
build a blue graph. After deleting each contractible component of the
red or
blue graph, each of the remaining components contains a simple closed
red or
blue curve. All together, we have constructed on $F(P)$ a system of
$\mu$
simple closed curves $\delta_1,\delta_2, ...
,\delta_{\mu-1},\delta_\mu$,
which we list by first taking red, then white and finally blue. We
denote
by $n_{+}$ the number of red curves which is also the number of
positive
regions, by $n_{\cdot}$ the number of crossing points and by $n_{-}$
the
number of blue which equals the number of negative regions of the
divide $P$.
Let $D_i$ be the right Dehn twist along the curve $\delta_i.$ A model
for
the right Dehn twist is the linear action $(x,y) \mapsto (x+y,y)$ on
the
cylinder $\{(x,y) \in \Bbb R/\Bbb Z \times \Bbb R \mid 0 \leq y \leq
1\}$ with as orientation the product of the natural orientations of the
factors. A right Dehn twist
around a simply
closed curve $\delta$ on an oriented surface is obtained by embedding
the
model as an oriented bicollar neighbourhood of $\delta$ such that
$\delta$
and $\Bbb R/\Bbb Z \times \{1/2\}$ of the model match.
The local geometric monodromy
of the singularity of $xy=0$ is as diffeomorphism a right Dehn
twist (voir le Th\'eor\`eme Fondamental, page 23 de [L], et page 95 de
[P-S]). Using as in [AC2] a local version of a Theorem of Lefschetz,
one obtains:
\proclaim{Theorem 3}
Let $P$ be the divide for an isolated plane curve singularity $S.$ The
Dehn
twists $D_i$ are generators for the geometric monodromy group of the
unfolding
of the singularity $S$. The product $T:=D_{\mu}D_{\mu-1} ... D_2D_1$ is
the
local geometric monodromy of the singularity $S$.
\endgroup\bigbreak
\section{4}{The singularity $D_5$ and a graphical algorithm in
general}
We will work out the picture for the singularity $D_5$ with the
equation
$x(x^3-y^2)$ and the divide given by the deformation
$(x-s)(x^3+5sx^2-y^2), s \in [0,1],$ which is shown for $s=1$
in Figure $5.$ There are one positive triangular region, one negative
region and three
crossings. By gluing three pieces together, one gets the Milnor fiber
with a system of
vanishing cycles as depicted in Figure $6.$
\midinsert
\cline{\epsffile{fig-5.eps}}
\medskip
\centerline{Figure $5:$ A divide for the singularity $D_5.$}
\endinsert
An easy and fast graphical algorithm of visualizing the Milnor fiber
with a
system of vanishing cycles directly from the divide is as follows:
think the divide
as a road network which has $\delta$ junctions, and replace
every junction by a
roundabout,
which leads you to a new road network with $4\delta$ T-junctions.
Realize now
every road section in between two T-junctions by a strip with a half
twist. Do the same for every road section in between a
T-junction and the boundary of the divide. Altogether you will need
$6\delta+r$ strips. The core line of the four
strips of a roundabout is a white vanishing cycle, the strips
corresponding
to boundary edges and
corners of a positive or negative region have as core line a red or blue
vanishing cycle.
\midinsert
\cline{\epsffile{fig-6.eps}}
\medskip
\centerline{Figure $6:$ Milnor fiber with vanishing cycles for $D_5.$}
\endinsert
In Figure $7$ is worked out the singularity with two
Puiseux pairs and $\mu=16$, where we used the divide from Figure $9.$
\midinsert
\cline{\epsffile{fig-7.eps}}
\medskip
\centerline{Figure $7:$ Milnor fiber with vanishing cycles for
$y^4-2y^2x^3+x^6-x^7-4yx^5.$}
\endinsert
We have drawn
for convenience in
Figure $7$ only one red, white, or blue cycle.
We have also indicated the position of the arc $\alpha,$ which will
play a role in the next section.
\goodbreak
\section{5}{An example of global geometric monodromy}
Let $b:\Bbb C \to \Bbb C^2, b(t):=(t^6+t^7,t^4)$ be the parametrized curve
$C$
having at $b(0)=(0,0)$ the singularity with two essential Puiseux pairs
and
with local link the compound cable knot $(2,3)(2,3).$ The polynomial
$f(x,y):=y^4-2y^2x^3+x^6-x^7-4yx^5$ is the equation
of $C.$ The function $f:\Bbb C^2 \to \Bbb C$ has besides $0$ the only other
critical
value $c:=14130940973168155968/558545864083284007.$
The fiber of $0$ has besides its singularity at $(0,0)$ a nodal
singularity
at $(-8,-4)$, which corresponds to the node $b(-1+i)=b(-1-i)=(-8,-4).$
The
geometric monodromy of the singularity at $(0,0),$ which is up to
isotopy
piecewise of finite order, is described in [AC1]. The fiber of $c$ has
a
nodal singularity at $(1014/343,16807/79092).$ The singularity at
infinity
of the curve $C$ is at the point $(0:1:0)$ and its local equation is
$z^3-2z^2x^4+zx^6-x^7-4zx^5$, whose singularity is topologically
equivalent
to the singularity $u^3-v^7$ with Milnor number $12.$
The function $f$ has no critical values coming from infinity. We aim at
a
description of the global geometric monodromy of the function $f.$
Working with the distance on $\Bbb C^2$ given by $||(x,y)||^2:= |x|^2+
4|y|^2,$
we have that the parametrized curve $b$ is transversal to the spheres
$S_r:=\{(x,y) \in \Bbb C^2 \mid |x|^2+ 4|y|^2=r^2\}$ with center $0 \in
\Bbb C^2$
and radius $r > 0.$ So for $0<r< 8\sqrt{2},$ the intersection
$K_r:=C \cap S_r$ is the local knot in $S_r$ of the singularity at
$0 \in \Bbb C^2$ (see Figure $8$), at $r=8\sqrt{2}$ the knot $K_r$ is
singular with one
transversal crossing, and for $8\sqrt{2} < r$ the knot $K_r$ is the so
called knot at infinity of the curve $C.$ The crossing at the bottom of Fig.
$8$ flips for $r=8\sqrt{2}$ and the knot $K_r, 8\sqrt{2} < r,$
becomes the $(4,7)$ torus knot.
By making one extra total
twist
in a braid presentation of the knot $K_r$ one gets the local knot of
the
singularity at infinity of the projective completion of the curve $C.$
\midinsert
\cline{\epsffile{fig-8.eps}}
\medskip
\centerline{Figure $8$: The torus cable knot $(2,3)(2,3).$}
\endinsert
From the above we get the following partial description of the global
geometric monodromy. The typical regular fiber $F:=f^{-1}(c/2)$ is the
interior of the oriented surface obtained as the union of two pieces
$A$
and $B,$ where $A$ is a surface of genus $8$ with one boundary
component and
$B$ is a cylinder.
The pieces are glued together in the following way:
in each boundary component of $B$ there is an arc, which is glued to an
arc in the boundary of $A.$ The interior of $A$ or $B$ can be thought
of
as a Milnor fiber of the singularity at $0$ or $(-8,-4).$ So, the
geometric
monodromy around $0$ is a diffeomorphism with support in the interior
of
$A$ and $B,$ given for instance by a construction as in Paragraph $2.$
The piece $A$ can be constructed from the divide in Figure $9.$
\midinsert
\leftline{\epsffile{fig-9.eps}}
\medskip
\centerline{Figure $9$: The curve $(x_s(t),y_s(t)),\, s:=1,$ as divide
for the singularity of $C.$}
\endinsert
Clearly, the monodromy in $B$ is a positive Dehn twist around
the simple essential closed curve $\delta_{17}$ in $B,$ whereas the
monodromy in $A$ is a product of positive Dehn twists around a system
$(\delta_1, \dots ,\delta_{16})$ of $16$ red, white or blue curves.
The monodromy around the critical point $c$ is a positive Dehn twist
around
a simple curve $\delta_{18},$ in $F,$ which is the union of two simple
arcs
$\alpha \subset A$ and $\beta \subset B.$ The arcs $\alpha$ and $\beta$
have
their endpoints $p,q \in A \cap B$ in common, and moreover the points
$p$ and
$q$ lie in different components of $A \cap B.$ The arc $\beta$ cuts
the curve
$\delta_{17}$ transversally in one point. The arc $\alpha$ intersects
the
curves $(\delta_1, \dots ,\delta_{16})$ transversally in some way.
For the position of the system $(\delta_{17},\, \beta)$ in B there is
up to
a diffeomorphism of the pair $(B,A \cap B)$ only one possibility. To
obtain
a complete description of the global monodromy it remains to describe
the
position of the system $(\delta_1, \dots ,\delta_{16},\, \alpha)$
in $(A,A \cap B).$
We consider the family with parameter $s$ of parametrized curves with
para\-meter $t$:
$$x_s(t)=T(4,t)/8={t}^{4}-{t}^{2}+1/8,\,\,
y_s(t)=sT(6,t)/32+T(7,t)/64=$$
$$s{t}^{6}+{t}^{7}-3/2\,s{t}^{4}-7/4\,{t}^{5}+{{9}/{16}}\,s{t}^{2}+
{{7}/{8}}\,{t}^{3}-1/32\,s-{{7}/{64}}\,t,$$
where $T(d,t)$ is the Chebychev polynomial of degree $d.$ Let
$$
f_s(x,y):={s}^{4}{x}^{6}-{{3}/{128}}\,{s}^{4}{x}^{4}+{{1}/{1024}}\,{s}
^{4}{x}^{3}-2\,{s}^{2}{y}^{2}{x}^{3}-4\,sy{x}^{5}-
$$
$${x}^{7}+{{9}/{65536}}\,{s}^{4}{x}^{2}-{{3}/{262144}}\,{s}^{4}x+{{3}/{128}}
\,{s}^{2}{y}^{2}x-{{1}/{4096}}\,{s}^{2}{x}^{3}+
$$
$$
{{5}/{64}}\,s
y{x}^{3}+{{7}/{256}}\,{x}^{5}+{{1}/{4194304}}\,{s}^{4}-{{1}/{1024}}\,{s}^{2}{y}^{2}-{{1}/{1024}}\,sy{x}^{2}+
$$
$$
{y}^{4}+
{{3}/{1048576}}\,{s}^{2}x-
{{5}/{16384}}\,syx-{{7}/{
32768}}\,{x}^{3}-{{1}/{8388608}}\,{s}^{2}+
$$
$$
{{1}/{131072}}\,sy-{{1}/{4096}}\,{y}^{2}+{{7}/{16777216}}\,x+{{1}/{134217728}}
$$
be the equation, monic in $y,$ for the curve $(x_s(t),y_s(t),$ whose
real image is for $s=1$ a divide (see Figure $9$) for the singularity
of $C$ at $(0,0).$
The $0-$level of $f_s$ for $s=1$ consists of this divide and an
isolated
minimum not in one of its regions, which corresponds to the minimum of
the
restriction of $f$ to $\Bbb R^2$ at $(-8,-4).$
For $a$ small, we call $\delta_{17,a,s} \subset \{f_s=a\}$
the vanishing cycle of the local minimum of $f_s,s=1,$ which does not
belong
to a region. The curve $(x_s(t),y_s(t)), t \in \Bbb R^2,s=7/24\,\sqrt{2},$
has
$8$ nodes, a cusp at $t=-1/2\,\sqrt{2},$(see Figure $10$) and at
infinity a singularity with
Milnor number 12. We now vary the parameter $s \in
[7/24\,\sqrt{2}-\sigma,1]$
from $1$ to $7/24\,\sqrt{2}-\sigma$ for a very small $\sigma >0.$
\midinsert
\cline{\epsffile{fig-10.eps}}
\medskip
\centerline{Figure $10$: The curves $(x_s(t),y_s(t))$ for $
s:=7/24\,\sqrt{2}.$}
\endinsert
The value of the local minimum, which does not belong to a region,
becomes
smaller and by adjusting the parameter $a$ we can keep the cycle
$\delta_{17,a,s}$ in the new region which emerges from the cusp at
$s=7/24\,\sqrt{2}.$
Since the total Milnor number of $f_s$ is
$(\hbox{\fam0 \tenrm degree}(f_s)-1)(\hbox{\fam0 \tenrm degree}(f_s)-2)-12=18,$
it follows that all its singularities have Milnor number $1$ for
$s=7/24\,\sqrt{2}-\sigma.$ The vanishing cycle of the node, which
appears when deforming the cusp singularity, will be called
$\delta_{18,s}$ and the vanishing cycle in the region of the divide of
Figure $9$, in whose boundary the cusp appeared, will be
called $\delta_{16,s}.$
\midinsert
\cline{\epsffile{fig-11.eps}}
\medskip
\centerline{Figure $11$: A divide, which does not come from a
singularity.}
\endinsert
We label the vanishing cycles on the regular ribbon surface $F_{-}$ of
the divide of Figure $11$ by $\delta_{1}, ... ,\delta_{15}.$ The
cycles
$\delta_{17,a,s},\delta_{18,s},\delta_{16,s}$ deform without changing
their intersection pattern and $\delta_{16,s}$ becomes the cycle
$\delta_{16}$ of the regular ribbon surface $F_{+}$ of the divide of
Figure $9.$ Observe that the regular ribbon surface $F_{-}$ of the
divide
of Figure $11$ is naturally a subset of the regular ribbon surface
$F_{+}$ of the
divide of Figure $9.$ The description of the position of the system
$(\delta_1, \dots ,\delta_{16},\, \alpha)$ in $(A,A \cap B),$ for which
we are looking, is the system $(\delta_1, \dots ,\delta_{16})$ on
$F_{+},$ where the relative cycle $\alpha$ is a simple arc on
$F_{+}-F_{-}$ with endpoints on the boundary of $F_{+}$ and cutting
the cycle $\delta_{16}$ transversally in one point. Observe that
$F_{+}-F_{-}$ is a strip with core $\delta_{16}$ (see Figure $7$).
\goodbreak
\section{6}{Connected divides and fibered knots. Proof of Theorem 2}
In this section we assume, without loss of generality, that a divide
is linear and orthogonal near its crossing points. For a connected
divide $P \subset D(0,\rho),$ let $f_P:D(0,\rho) \to \Bbb R$ be a generic
$C^{\infty}$ function,
such that $P$ is its $0$-level and that each region has exactly one
local maximum or minimum. Such a function exist for a connected divide
and is well defined up to sign and isotopy. In particular, there are no
critical points of saddle type other then the crossing points of the
divide. We assume moreover that the function $f_P$ is quadratic and
euclidean in a neighborhood of its critical points, i.e. for euclidean
coordinates (X,Y) with center at a critical point $c$ of $f_P$ we have
in a neighborhood of $c$ the expression $f_P(X,Y)=f_P(c)+XY$ or
$f_P(X,Y)=f_P(c)+X^2+Y^2.$ Let $\chi:D(0,\rho) \to [0,1]$ be a
$C^{\infty},$ positive function, which evaluates to zero outside of
the neighborhoods where $f_P$ is quadratic and to $1$ in some smaller
neighborhood of the critical points of $f_P.$ Let
$\theta_P:\partial B(0,\rho) \to \Bbb C$ be given
by: $\theta_{P,\eta}(J(v)):=f_P(x)+
i \eta df_P(x)(u)-{1\over 2}\eta^2\chi(x)H_{f_P}(v)$ for
$J(v)=(x,u) \in TD(0,\rho)=D \times \Bbb R^2$ and $\eta \in \Bbb R, \eta >
0.$
Observe that the Hessian $H_{f_P}$ is locally
constant in a the neighborhood of the critical points of $f_P.$
The function $\theta_{P,\eta}$ is
$C^{\infty}$. Let
$\pi_{P,\eta}:\partial B(0,\rho) \setminus L(P) \to S^1$ be defined
by:$\pi_{P,\eta}(J(v)):=\theta_{P,\eta}(J(v))/
|\theta_{P,\eta}(J(v))|.$
\proclaim{Theorem 4}
Let $P \subset D(0,\rho)$ be a divide, such that the system of
immersed curves is connected.
The link $L(P)$ in
$\partial B(0,\rho)=\{J(v) \mid v \in T(D(0,\rho)) \ \text{ and
}\ \|J(v)\| =\rho\}$ is a fibered link. The map
$\pi_P:=\pi_{P,\eta}$ is for $\eta$ sufficiently small, a fibration
of the complement of $L(P)$ over $S^1.$ Moreover the fiber of the
fibration $\pi_P$ is $F(P)$ and the geometric monodromy is the product
of Dehn twist as in Theorem 3.
\endgroup\bigbreak
The map $\pi_P$ is compatible with a regular product
tubular neighborhood of $L(P)$ in $\partial B(0,\rho).$ The map $\pi_P$
is a
submersion, so, since already a fibration near
$L(P),$ it is a fibration by a theorem of Ehresmann. The graphical
algorithm, see Figure $7$, produces in fact, up to a small isotopy of
the image, the projection of the
fiber $\pi_P^{-1}(i)$ on $D(0,\rho).$
This
projection is except above the twist of the strips a submersion.
The proof of Theorem $4$ is given in the forthcoming paper [AC4] on
generic immersions of curves and knots.
\demo{ \fam\bffam\tenbf Proof of Theorem $2$} The oriented fibered links $L(S)$ and $L(P)$
have the same geometric monodromies according to the Theorem 3 and 4.
So, the links
$L(S)$ and $L(P)$ are
diffeomorphic.
\penalty-100\null\hfill\qed\bigbreak
\remark{\fam\bffam\tenbf Remark}
Let $f(x,y)=0$ be a singularity $S,$ such that written in the
canonical coordinates of the charts of the embedded resolution
the branches of the
strict transform have equations of the form $u=a, a \in \Bbb R.$ Let
$f_t(x,y), t \in [0,1]$ be a morsification, with its divide $P$ in
$D(0,\rho),$ obtained by blowing down generic real linear
translates of the strict transforms, as in [AC2]. We strongly
believe that with the use of [B-C1,B-C2] the following
transversallity property can be obtained,
and which we state as a problem:\br
Their exists $\rho'_0>0,$ such that for all $t \in [0,1]$ and for all
$\rho' \in (0,\rho'_0]$ the $0$-levels of
$f_t(x,y)$ in $\Bbb C^2$ meet transversally the boundary of
$$
B(0,\rho,\rho'):=\{(x+iu,y+iv) \in \Bbb C^2 \mid x^2+y^2+u^2+v^2 \leq
\rho^2, u^2+v^2 \leq {\rho'}^2\}.
$$
It is easy to deduce from this transversallity statement an isotopy
between the links $L(S)$ and $L(P).$
\endremark
\remark{\fam\bffam\tenbf Remark}
Bernard Perron has given a proof for the triviallity of the cobordism
from $L(S)$ to $L(P),$ which uses the holomorphic convexity of the
balls $B(0,\rho,\rho')$ of the previous remark [P].
\endremark
\par
\noindent
\Refs
\parskip=0pt
\par
\key{AC1}
Norbert A'Campo,
\it Sur la monodromie des singularit\'es isol\'ees d'hypersurfaces
complexes,
\rm Invent. Math.
\bf 20
\rm (147--170),
1973.
\par
\key{AC2}
Norbert A'Campo,
\it Le Groupe de Monodromie du D\'eploiement des Singularit\'es
Isol\'ees de Courbes Planes I,
\rm Math. Ann.
\bf 213
\rm (1--32),
1975.
\par
\key{AC3}
Norbert A'Campo,
\it Le Groupe de Monodromie du D\'eploiement des Singularit\'es
Isol\'ees de Courbes Planes II,
\rm Actes du Congr\`es International des Math\'e\-ma\-ti\-ciens, Vancouver
\rm (395--404),
1974.
\par
\key{AC4}
Norbert A'Campo,
\it Generic immersions of curves, knots,
monodromy and \"Uber\-schneidungszahl,
\rm Publ. Math. IHES, to appear \br
http://xxx.lanl.gov/abs/math/9803081.
\par
\key{A}
V. Arnold,
\it On the arrangement of the ovals of real plane curves,
involutions
of 4-dimensional smooth manifolds, and the arithmetic of integral
quadratic forms,
\rm Funct. Anal. Appl.
\bf 5
\rm (1--9),
1971.
\par
\key{B-K}
Ludwig Balke and Rainer Kaenders,
\it On certain type of Coxeter-Dynkin diagrams of plane curve
singularities,
\rm Topology
\bf 35
\rm (39--54),
1995.
\par
\key{B-C1}
F. Bruhat, H. Cartan,
\it Sur la structure des sous-ensembles analytiques r\'eels,
\rm C. R. Acad. Sci. Paris
\bf 244
\rm (988--990),
1957.
\par
\key{B-C2}
F. Bruhat, H. Cartan,
\it Sur les composantes irr\'eductibles d'un sous-ensemble
analytique
r\'eel,
\rm C. R. Acad. Sci. Paris
\bf 244
\rm (1123--1126),
1957.
\par
\key{G-Z}
S. M. Gusein-Zade,
\it Matrices d'intersections pour certaines singularit\'es de
fonctions de 2 variables,
\rm Funkcional. Anal. i Prilozen
\bf 8
\rm (11--15),
1974.
\par
\key{L}
S. Lefschetz,
\it L'Analysis Situs et la G\'eom\'etrie Alg\'ebrique,
\rm Collection de Monographies sur la Th\'eorie des Fonctions,
Gauthier- Villars et $C^{ie}$,
\rm Paris,
1924.
\par
\key{M}
J. Milnor,
\it Singular Points on Complex Hypersurfaces,
\rm Ann. of Math. Studies
\bf 61
\fam0 \tenrm Princeton University Press,
\rm Princeton,
1968.
\par
\key{P}
B. Perron,
\it Preuve d'un Th\'eor\`eme de N. A'Campo sur les d\'eformations
r\'eelles des singularit\'es alg\'ebriques complexes planes,
\rm Preprint,
Universit\'e de Bourgogne,
\rm Dijon,
1998.
\par
\key{P-S}
\'Emile Picard et Georges Simart,
\it Th\'eorie des Fonctions Alg\'ebriques de deux variables
ind\'ependantes, Tome I,
Gauthier- Villars et Fils,
\rm Paris,
1897.
\par
\key{R1}
V. A. Rohlin,
\it Congruence modulo 16 in Hilbert's sixteenth problem I,
\rm Funct. Anal. Appl.
\bf 6
\rm (301--306),
1972.
\par
\key{R2}
V. A. Rohlin,
\it Congruence modulo 16 in Hilbert's sixteenth problem II,
\rm Funct. Anal. Appl.
\bf 7
\rm (163--164),
1973.
\par
\key{S1}
Charlotte Angas Scott,
\it On the Higher Singularities of Plane Curves,
\rm Amer. J. Math.
\bf 14
\rm (301--325),
1892.
\par
\key{S2}
Charlotte Angas Scott,
\it The Nature and Effect of Singularities of Plane Algebraic Curves,
\rm Amer. J. Math.
\bf 15
\rm (221--243)
1893.
\par
\key{V}
Oleg Viro,
\it Private communication,
\rm
\bf
\rm
\rm Sapporo,
1990.
\par\endgroup
\bigskip
\address{Universit\"at Basel \br Rheinsprung 21 \br CH-4051 Basel}
\null\firstpagetrue\vskip\bigskipamount
\title{Erratum: \br
Real deformations and complex topology \br
of plane curve singularities}
\shorttitle{Real deformations and complex topology.}
\vskip2\bigskipamount
In Section $5$ the parametrized curve $C$ should be $b(t):=(t^4,t^6+t^7)$
instead of $b(t):=(t^6+t^7,t^4)$ and accordingly $(-8,-4)$ has to
be $(-4,-8)$. We
intersect $C$ with the family of spheres
$S_r:=\{(x,y) \in \Bbb C^2 \mid 4|x|^2+ |y|^2=r^2\}$.
For $0<r< 8\sqrt{2},$ the intersection
$K_r:=C \cap S_r$ is the local knot in $S_r$ of the singularity at
$0 \in \Bbb C^2$, at $r=8\sqrt{2}$ the knot $K_r$ is
singular with one
transversal crossing at $(-8,-4)$, and for $8\sqrt{2} < r$ the
knot $K_r$ is the so
called knot at infinity of the curve $C.$ Fig. $8$ of the text is a knot
projection of $K_r$ for small $r$. It is not possible to obtain
from this projection with only one crossing flip the type of
the knot $K_r$ for $r > 8\sqrt{2}$.
The figure here below is the stereographic
knot projection of $K_r$ for $r = 8\sqrt{2}-1$,
which is not a minimal knot projection.
For $r=8\sqrt{2}$ the crossing at the bottom
flips and the knot $K_r, 8\sqrt{2} < r,$
becomes the $(4,7)$ torus knot.
The knot projection is a braid projection, where
the axis is in the central pentagonal region.
The braid word is
$acabcaAabacabacabacab$ and flips at $r=8\sqrt{2}$ to
$acabcaaabacabacabacab$.
\midinsert
\cline{\epsffile{vor4_7.ps}}
\medskip
\endinsert
\vskip 6.6cm This picture was made with KNOTSCAPE.
\bye
|
1997-10-20T12:49:54 | 9710 | alg-geom/9710025 | en | https://arxiv.org/abs/alg-geom/9710025 | [
"alg-geom",
"math.AG"
] | alg-geom/9710025 | Stephan Endrass | Stephan Endrass | Minimal even sets of nodes | LaTeX 2e, 17 pages | null | null | null | null | We extend some results on even sets of nodes which have been proved for
surfaces up to degree 6 to surfaces up to degree 10. In particular, we give a
formula for the minimal cardinality of a nonempty even set of nodes.
| [
{
"version": "v1",
"created": "Mon, 20 Oct 1997 10:49:54 GMT"
}
] | 2007-05-23T00:00:00 | [
[
"Endrass",
"Stephan",
""
]
] | alg-geom | \section{Setup}\label{sect:setup}
Let $S\subset\Pthree\left(\CC\right)$ be a hypersurface of degree $s$ with $\mu$ ordinary
double points (nodes) as its only singularities. Such a surface will be
called a {\em nodal surface} in the sequel. Denote by
$N=\left\{P_1,\ldots,P_\mu\right\}\subset S$ the set of nodes of $S$.
The maximum number of nodes of a nodal surface of degree $d$ is denoted
classically by $\mu\left(d\right)$. There is a lot of (old) literature
on nodal surfaces and estimates for $\mu\left(d\right)$ (see \cite{endrass}).
For $d=1,2,\ldots,6$
the numbers $\mu\left(d\right)$ are $0,1,4,16,31,65$ and for every
$k\in\left\{0,1,\ldots,\mu\left(d\right)\right\}$ there exists at least one nodal
surface of degree $d$ with exactly $k$ nodes. In the case of cubic
nodal surfaces ($d=3$), this follows from Cayley's and Schl\"afli's
classification of singular cubic surfaces \cite{cayley},
\cite{schlaefli}. For quartic nodal
surfaces ($d=4$) the fact that $\mu\left(4\right)=16$ is due to
Kummer \cite{kummer}, whereas the construction of arbitrary nodal quartics
goes back to Rohn \cite{rohn}. The first quintic nodal surface ($d=5$) with
31 nodes has been constructed by Togliatti in 1940 \cite{togliatti}. In 1971,
Beauville \cite{beauville} showed that this is in fact the maximal number.
The construction of sextic nodal surfaces ($d=6$) with $1,\ldots,64$
nodes has been given by Catanese and Ceresa
\cite{cataneseceresa}. In 1994, Barth \cite{barth}
constructed a sextic nodal surface with 65 nodes. Shortly afterwards,
Jaffe and Ruberman \cite{jafferuberman}
proved that 65 is the maximal number. Both Beauville
and Jaffe/Ruberman use the code of a nodal surface in their proofs.
This code is a $\mathbb{F}_2$ vector space which carries the information
of the low degree contact surfaces of the nodal surface. If a nodal
surface has ``nearly'' $\mu\left(d\right)$ nodes, its code often becomes
accessible.
Let $v\in\mathbb{N}$ and denote
$\delta\left(v\right)=2\left(v/2-\left\lfloor v/2\right\rfloor\right)$.
This number is 0 if v is even and 1 if v is odd.
We want to study surfaces $V\subset\mathbb{P}_3$ of degree $v$
with $S.V=2D$ for a (not necessarily smooth or reduced) curve $D$.
In other words, surfaces $V$ which have contact to $S$ along a curve. Let
$\pi\colon\tilde{\mathbb{P}}_3\rightarrow\mathbb{P}_3$ be the embedded resolution of
all nodes of $S$. Given such a surface $V$, the proper transforms
of $S$ and $V$ are calculated as
\begin{equation*}
%
\tilde{S}=\pi^\ast\negthinspace S-2\sum_{i=1}^\mu E_i
\quad\text{and}\quad
\tilde{V}=\pi^\ast\negthinspace V-\sum_{i=1}^\mu \nu_i E_i,
%
\end{equation*}
where $E_i=\pi^{-1}\left(P_i\right)$ is the exceptional divisor
corresponding to $P_i$ and $\nu_i=\operatorname{mult}\left(V,P_i\right)$ for every
node $P_i\in N$. On the smooth surface $\tilde{S}$ we have
$\tilde{V}\sim_{lin}2\tilde{D}+\sum_{i=1}^\mu\theta_iE_i$, where $\tilde{D}$ is the
proper transform of $D$ and the $\theta_i$'s are nonnegative integers.
Let $H\in\operatorname{Div}\left(\mathbb{P}_3\right)$ be a hyperplane section, then
\begin{equation*}
%
2\tilde{D}\sim_{lin}v\pi^\ast\negthinspace H-
\sum_{i=1}^\mu\left(\nu_i+\theta_i\right)E_i,
%
\end{equation*}
where $\tilde{D} .E_i=\nu_i+\theta_i=\operatorname{mult}\left(D,P_i\right)=\eta_i$. This shows
that in $\operatorname{Pic}\left(\smash{\tilde{S}}\right)$ the divisor class
$\left[\delta\left(v\right)\pi^\ast\negthinspace H+\sum_{\text{$\eta_i$ odd}}E_i\right]$
is divisible by $2$.
This is a remarkable fact, since every $E_i$ is on $\tilde{S}$ a smooth,
rational curve with self intersection $-2$. In particular
$E_i\not\sim_{lin}E_j$ for $i\neq j$.
For any set of nodes $M\subseteq N$, let $E_M=\sum_{P_i\in M}E_i$ be the sum
of exceptional curves corresponding to the nodes in $M$.
\begin{definition}\label{definition:even}
%
A set $M\subseteq N$ of nodes of $M$ is called {\em strictly even},
if the cocycle
class $\operatorname{cl}\left[E_M\right]\in\cohom{2}{\smash{\tilde{S}},\mathbb{Z}}$
is divisible by $2$. $M$ is called {\em weakly even}, if the cocycle
class $\operatorname{cl}\left[\pi^\ast\negthinspace H+E_M\right]\in\cohom{2}{\smash{\tilde{S}},\mathbb{Z}}$
is divisible by $2$. $M$ is called {\em even} if $M$ is
strictly or weakly even.
%
\end{definition}
So the set of nodes $M=\left\{P_i\in N\mid
\text{$\operatorname{mult}\left(D,P_i\right)$ is odd}\right\}$ through which $D$
passes with odd multiplicity is strictly even if
$v$ is even and weakly even if $v$ is odd.
\begin{definition}\label{definition:cut}
%
Let $M\subseteq N$ be an even set of nodes of $S$. If
$V\subset\mathbb{P}_3$ is a surface with $S.V=2D$ and $M$ is the
set of nodes of $S$ through which $D$ passes with odd
multiplicity, we say that {\em $M$ is cut out by $V$ via $D$}.
%
\end{definition}
Conversely, if $M\subseteq N$ is even, consider the linear system
$\mylinsys{v}{-}{M}$ for
$v\in\mathbb{N}$ even if $M$ is strictly even and odd if $M$ is
weakly even.
For $v\gg 0$ this linear
system is nonempty by R.R.~and Serre duality. Then for every $v$ such that
$\mylinsys{v}{-}{M}\neq\emptyset$ and for
every divisor $\overline{D}\in\mylinsys{v}{-}{M}$
we can find a surface $V\subset\mathbb{P}_3$ of degree $v$ which cuts out $M$.
The construction is as follows: $\overline{D}$ is effective, so it
admits a decomposition $\overline{D}=\tilde{D}+\sum_{i=1}^\mu \tau_iE_i$ such
that $\tilde{D}$ is effective and contains no exceptional component and all
the numbers $\tau_i$ are nonnegative. In particular we have
for all $j\in\left\{1,\ldots,\mu\right\}$ that
\begin{equation*}
%
\tilde{D} .E_j=\left(\frac{1}{2}\left(v\pi^\ast\negthinspace H-E_M\right)
-\sum_{i=1}^\mu\tau_i E_i\right).E_j=
\left\{\begin{array}{l@{\quad}l}
2\tau_j & \text{is even if $P_j\not\in M$,}\\
2\tau_j +1 & \text{is odd if $P_j\in M$.}
\end{array}\right.
%
\end{equation*}
But $2\overline{D}\in\left|v\pi^\ast\negthinspace H-E_M\right|$ on the surface $\tilde{S}$, so
$2\overline{D}$ is cut out by a surface
$\overline{V}\in\left|v\pi^\ast\negthinspace H-E_M\right|$ in ${\tilde{\mathbb{P}}}_3$. Let $V=\pi_\ast\left(\overline{V}\right)$
and $D=\pi_\ast\left(\overline{D}\right)=\pi_\ast\left(\smash{\tilde{D}}\right)$,
then by construction
$S.V=2D$ and $\operatorname{mult}\left(D,P_i\right)=\tilde{D} .E_i$ for all $i$.
So $M$ is exactly the set of nodes of $S$ through which $D$ passes
with odd multiplicity.
This shows that $M$ is cut out by $V$ via $D$.
Furthermore we see that only nodal surfaces of even degree do admit weakly
even sets of nodes.
If the surface $V$ cuts out an even set of nodes $M$ on $S$ via $D$, then
in general $D$ is not unique with respect to $M$. The set of these contact
curves is parameterized by the linear system
$L_M=\mylinsys{v}{-}{M}$ which is a projective space
of dimension $\mycohomd{0}{v}{-}{M}-1$.
In particular, if $\mycohomd{0}{v}{-}{M}\geq 1$
then there exists a surface of degree $v$ which cuts out $M$. It is funny
to compute these dimensions, though often not possible.
The canonical divisor of $\tilde{S}$ is
$\smash{K_{\tilde{S}}}\sim_{lin}\left(s-4\right)\pi^\ast\negthinspace H$.
Define $\binom{n}{k}=0$ for $n<k$, then Riemann Roch for the
bundle $\mybundle{v}{-}{w}$ reads as
\begin{equation*}
%
\mychi{v}{-}{w}=\frac{sv}{8}\left(v-2s+8\right)
+\binom{s-1}{3}+1-\frac{\left|w\right|}{4}.
%
\end{equation*}
The symmetric difference of two strictly even sets of nodes is
strictly even again,
so the set $C_S=\left\{M\subseteq N\mid\text{$M$ is strictly even}\right\}$
carries the natural structure of a $\mathbb{F}_2$ vector space sitting inside
$\mathbb{F}_2^\mu$. Hence $C_S$ is a binary linear code, which is called
{\em the code of $S$}. The symmetric difference of two weakly even sets of
nodes is strictly even and the symmetric difference of a strictly even set
and a weakly even
set is weakly even. Thus the set
$\overline{C}_S=\left\{M\subseteq N\mid\text{$M$ is even}\right\}$ is a
binary code of dimension $\dim_{\mathbb{F}_2}\left(C_S\right)\leq
\dim_{\mathbb{F}_2}\left(\overline{C}_S\right)\leq
\dim_{\mathbb{F}_2}\left(C_S\right)+1$ sitting also inside $\mathbb{F}_2^\mu$.
The elements of $\overline{C}_S$ are called {\em words},
and for every word $w\in \overline{C}_S$ its weight $\left|w\right|$ is its
number of nodes. Let $e_1,\ldots,e_\mu,h$ be the canonical basis of
$\mathbb{F}_2^\mu\oplus\mathbb{F}_2$ and consider
\begin{equation*}
%
\begin{array}{ccccc}
%
\mathbb{F}_2^\mu & \overset{j}{\longrightarrow} &
\mathbb{F}_2^\mu\oplus\mathbb{F}_2 & \overset{\lambda}{\longrightarrow} &
\cohom{2}{\smash{\tilde{S}},\mathbb{F}_2} \\
& & e_i & \longmapsto &
\operatorname{cl}\left[E_i\right] \bmod 2 \\
& & h & \longmapsto &
\operatorname{cl}\left[\pi^\ast\negthinspace H\right]\bmod 2
%
\end{array}
%
\end{equation*}
The projection of $\ker\left(\lambda\right)$ onto the first factor
is nothing but $\overline{C}_S$, and $\ker\left(\lambda\circ j\right) = C_S$.
If $s$ is even (resp.~odd),
then $\operatorname{im}\left(\lambda\right)$
(resp.~$\operatorname{im}\left(\lambda\circ j\right)$)
is a total isotropic subspace of $\cohom{2}{\smash{\tilde{S}},\mathbb{F}_2}$ with respect
to the intersection product.
This shows \cite{beauville} that
\begin{align*}
%
\dim_{\mathbb{F}_2}\left(C_S\right) &
\geq\mu -\frac{1}{2}b_2\left(\smash{\tilde{S}}\right),\\
\dim_{\mathbb{F}_2}\left(\overline{C}_S\right) &
\geq\mu +1 -\frac{1}{2}b_2\left(\smash{\tilde{S}}\right)
\quad\text{($s$ even)}.
%
\end{align*}
The weight of every word $w\in C_S$ is divisible by 4. If
$s=\deg S$ is even, then the weight of every word is divisible by
8 \cite{catanese}.
\subsection{Coding theory}
We recall some definitions and facts from coding theory \cite{lint},
\cite{wall}. Let $C\subseteq\mathbb{F}_2^n$ be a linear code and let $e_1,\ldots,e_n$
be the canonical basis of $\mathbb{F}_2^n$. $C$ is called {\em even}
if $2\mid\left|w\right|$ for every $w\in C$ and
{\em doubly even} if $4\mid\left|w\right|$ for every $w\in C$.
The dual code of $C$ is defined as
\begin{equation*}
%
C^\perp = \left\{v\in\mathbb{F}_2^n\mid\left< v,w\right>_{\mathbb{F}_2}=0\ \forall w\in C
\right\}.
%
\end{equation*}
If $C$ is doubly even, then $C\subseteq C^\perp$. Since
$n=\dim_{\mathbb{F}_2}\left(C\right)+\dim_{\mathbb{F}_2}\left(C^\perp\right)$
we also get $2\dim\left(C\right)\leq n$ with equality iff $C$ is
self dual. For $w\in C$ the support of $w$ is the linear subspace
of $\mathbb{F}_2^n$ which is spanned by the ones of $w$, i.e.~
\begin{equation*}
%
\operatorname{supp}\left(w\right) = \operatorname{span}_{\mathbb{F}_2}\left\{e_i\mid
\left<e_i,w\right>=1\right\}.
%
\end{equation*}
The image of the projection $p_w\colon C\rightarrow\operatorname{supp}\left(w\right)$
is called projection of $C$ onto the support of $w$ and denoted by
$C_w$. Assume that $2d\mid\left|v\right|$ for all $v\in C$ for some
$d\in\mathbb{N}$. Since $\left|v+w\right|+2\left|v\cap w\right| =
\left|v\right|+\left|w\right|$ and $p_w\left(v\right)=v\cap w$ we see that
$d\mid v'$ for all $v'\in C_w$.
Now the code $C_S$ of the nodal surface $S$ is always doubly even.
If $s=\deg\left(S\right)$ is even, then $\left(C_S\right)_w$ is doubly even
for all $w\in C_S$.
A $\left[n,k,d\right]$-code is a $k$-dimensional linear code
$C\subseteq \mathbb{F}_2^n$ with $\left|w\right|\geq d$
for all $w\in C\setminus\left\{0\right\}$.
Many methods have been found to give bounds on $k$
for fixed $n$ and $d$. One of the simplest to apply is the
\begin{theorem}\label{theorem:griesmer}
%
(Griesmer bound) For a $\left[n,k,d\right]$ code always
$n\geq \sum_{i=0}^{k-1} \left\lceil d/2^i\right\rceil$.
%
\end{theorem}
\subsection{Examples}
The following examples exhibit the trivial and some of the the well known
cases of even sets of nodes \cite{beauville}.
\begin{example}\label{example:cone}
%
Let $S$ be a quadratic cone and let $P_1$ be its node. Every line
$L\subset S$ runs through $P_1$ and there exists exactly one plane
$H$ with $S.H=2L$. So $\overline{C}_S$ is spanned by $w=\left\{P_1\right\}$
and $\mycohomd{0}{}{-}{w}=2$.
%
\end{example}
\begin{example}\label{example:cubic}
%
Let $S$ be a cubic nodal surface, then $C_S$ can only be
non trivial if $S$ has exactly $\mu\left(3\right)=4$ nodes
$P_1,\ldots,P_4$. But $b_2\left(\smash{\tilde{S}}\right)=7$, so
$\dim_{\mathbb{F}_2}\left(C_S\right)\geq 1$. It follows that
$\dim_{\mathbb{F}_2}\left(C_S\right)=1$ and $C_S$ is spanned by
$w=\left\{P_1,\ldots,P_4\right\}$. But $w$ is
cut out by a quadric: Riemann-Roch on $\tilde{S}$ gives
$\mychi{2}{-}{w}=3$. From Serre duality we get
$\mycohomd{2}{2}{-}{w}=\mycohomd{0}{-4}{+}{w}=0$. One easily checks that
$\mybundle{4}{-}{w}$ is ample, so by Kodaira vanishing also
$\mycohomd{1}{2}{-}{w}=\mycohomd{1}{-4}{+}{w}=0$.
This implies that $\mycohomd{0}{2}{-}{w}=3$, so there exists a two
parameter family of quadric surfaces which cut out $w$.
%
\end{example}
\begin{example}\label{example:quartic}
%
A quartic nodal surface $S$ with $\mu\left(4\right)=16$ nodes is a
Kummer surface. Since $b_2\left(\smash{\tilde{S}}\right)=22$, we have
$\dim_{\mathbb{F}_2}\left(C_S\right)\geq 5$. On the other hand all nonzero
words of $C_S$ must have weight $8$ or $16$. So $C_S$ is a
$\left[16,k,8\right]$ code for some $k\geq 5$. The Griesmer
bound implies $k\leq 5$, so
$C_S$ is a $\left[16,5,8\right]$ code. Every such code has exactly
one word of weight 16 and 30 words of weight 8. Moreover $C_S$
is (up to permutation of columns) spanned by the rows of the
following table.
%
\begin{equation*}
%
\newcommand{\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}}{\smash{\hspace*{0.06cm}\blacksquare\hspace*{0.06cm}}}
\newcommand{\\\hline}{\\\hline}
%
\begin{array}{*{16}{|@{}c@{}}|}\hline
%
\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & & & & & & & \\\hline
\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & & & &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & & & \\\hline
\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & \\\hline
\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& \\\hline
\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}\\\hline
%
\end{array}
%
\end{equation*}
%
\end{example}
\begin{example}\label{example:quintic}
%
A quintic nodal surface $S$ with $\mu\left(5\right)=31$ nodes is
called Togliatti surface. One computes $b_2\left(\smash{\tilde{S}}\right)=53$,
so again $\dim_{\mathbb{F}_2}\left(C_S\right)\geq 5$. By \cite{beauville}, all
even sets of nodes on $S$ have weight $16$ or $20$. So $C_S$
is a $\left[31,k,16\right]$ code for some $k\geq 5$. The Griesmer
bound gives $31\geq 16+8+4+2+1+\left(k-5\right)$, so $k\leq 5$.
This shows that $C_S$ is a $\left[31,5,16\right]$ code. Every such code
has exactly 31 words of weight 16 and no word of weight 20.
Moreover, $C_S$ is (up to a permutation of columns)
spanned by the rows of the following table.
%
\begin{equation*}
%
\newcommand{\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}}{\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}}
%
\begin{array}{*{31}{|@{}c@{}}|}\hline
%
\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&
& & & & & & & & & & & & & & \\\hline
\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & & & & & & & &
\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & & & & & & \\\hline
\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & & & &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & & & &
\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & & & &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & & \\\hline
\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & &
\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& & &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}&\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& \\\hline
\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &
\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}& &\hspace*{0.03cm}\blacksquare\hspace*{0.03cm}\\\hline
%
\end{array}
%
\end{equation*}
%
\end{example}
\begin{example}\label{example:sextic}
%
Let $S$ be a nodal sextic surface with $\mu\left(6\right)=65$ nodes.
Every nonzero word $w\in C_S$ must have weight
$24$, $32$, $40$ or $56$ \cite{jafferuberman}.
We have $b_2\left(\smash{\tilde{S}}\right)=106$,
so $\dim_{\mathbb{F}_2}\left(C_S\right)\geq 12$.
If $C_S$ contains no word of weight $56$, then
$\dim_{\mathbb{F}_2}\left(C_S\right)=12$ \cite{jafferuberman},
\cite{wall}. A short argument runs as follows:
By the Griesmer bound $C_S$ contains a word $w$ of weight $24$.
Clearly $p_w\colon C_S\rightarrow\left(C_S\right)_w$ has trivial
kernel, so $\left(C_S\right)_w$ is a doubly even
$\left[24,\dim_{\mathbb{F}_2}\left(C_S\right),4\right]$ code.
Hence $\dim_{\mathbb{F}_2}\left(C_S\right)\leq 12$.
It is not clear if $C_S$ is unique up to permutation. It is also
not known if any nodal sextic surface can have
even sets of $56$ or $64$ nodes.
%
\end{example}
\subsection{The theorem}
For nodal surfaces of degree $6$, Jaffe and Ruberman proved that the smallest
possible nonzero strictly even sets of nodes are the ones cut out by
quadrics. This seems to be true for nodal surfaces of arbitrary
degree, though we only can prove a few cases. For weakly even sets
of nodes, the corresponding statement is proved.
\begin{definition}\label{definition:eminmax}
%
For $s\in\mathbb{N}$ the \emph{minimal cardinality of an even set of nodes
on a nodal surface of degree $s$} is defined as
%
\begin{align*}
%
e_{min}\left(s\right) &= \min\left\{
\left|w\right|\mid\text{$w\in C_S$, $S$ nodal of degree $s$}
\right\}, \\
\overline{e}_{min}\left(s\right) &= \min\left\{
\left|w\right|\mid\text{$w\in \overline{C}_S$, $S$ nodal of degree $s$}
\right\}.
%
\end{align*}
%
\end{definition}
Our main result is the following
\begin{theorem}\label{theorem:main}
%
\begin{itemize}
%
\item[\romannum{1}] (Strictly even sets of nodes)
Let $s\in\left\{3,4,5,6,7,8,10\right\}$. Then
%
\begin{equation*}
%
e_{min}\left(s\right) =\left\{\begin{array}{cl}
s\left(s-2\right) & \text{if $s$ is even,}\\
\left(s-1\right)^2 & \text{if $s$ is odd.}
\end{array}\right.
%
\end{equation*}
%
Moreover $\left|w\right|=e_{min}\left(s\right)$ if
and only if
$w$ is cut out by a quadric surface.
%
\item[\romannum{2}] (Weakly even sets of nodes)
Let $s\in\left\{2,4,6,8\right\}$. Then
%
\begin{equation*}
%
\overline{e}_{min}\left(s\right)=\frac{s\left(s-1\right)}{2}.
%
\end{equation*}
%
Moreover $\left|w\right|=\overline{e}_{min}\left(s\right)$ if
and only if $w$ is cut out by a plane.
%
\end{itemize}
%
\end{theorem}
A close examination of the proof of theorem \ref{theorem:main} exhibits
that certain weights strictly greater than $e_{min}\left(s\right)$ and
$\overline{e}_{min}\left(s\right)$ cannot appear.
\begin{corollary}\label{corollary:main}
%
For any nodal surface $S$ of degree $s$, there exist no
even sets of nodes with the following weights.
%
\begin{equation*}
%
\begin{array}{|c||c|c|c|c|}\hline
s & 6 & 7 & 8 & 10 \\\hline\hline
\text{weakly even} & 19,23 & & 32,36,\ldots,56 & \\\hline
\text{strictly even} & & 40 & 56 & 88,96,104,112 \\\hline
\end{array}
%
\end{equation*}
%
\end{corollary}
If $w\in\overline{C}_S$ is cut out by a smooth cubic surface, then
$\left|w\right|=3s\left(s-3\right)/2$ \cite{catanese}. The corollary states
that all weights in the open interval
$\left] \overline{e}_{min}\left(s\right),3s\left(s-3\right)/2\right[$ do not
appear for weakly even set of nodes.
In the case of strictly even sets of nodes, the gap is the interval
$\left] e_{min}\left(s\right),2s\left(s-4\right)\right[$. Note that
if $w\in C_S$ is cut out by a smooth quartic surface, then
$\left|w\right|=2s\left(s-4\right)$.
\begin{remark}
%
It follows from example \ref{example:cone} and example
\ref{example:cubic} that the theorem is true for $s=2,3$.
%
\end{remark}
\subsubsection*{Acknowledgments}
%
I would like to thank D.~van Straten for valuable discussions.
%
\section{The formula of Gallarati}
The contact of hypersurfaces in $\mathbb{P}_r$ along a $r-2$ dimensional variety
has been (to our knowledge) studied first by D.~Gallarati
\cite{gallarati}.
He stated the following
\begin{theorem}\label{theorem:gallarati}
%
Let $F_m$, $G_n\subset\mathbb{P}_r$ be hypersurfaces of degree $m$ and $n$
with $F_m.G_n=qC$ for some $r-2$ dimensional variety $C$. Assume
that $F_m$ and $G_n$ have at most double points on $C$. If the
singular locus of $F_m$ on $C$ (resp.~$G_n$ on $C$) is a $r-3$
dimensional variety of degree $t$ (resp.~$s$), then
%
\begin{equation*}
%
q\left(t-s\right) = mn\left(m-n\right).
%
\end{equation*}
%
\end{theorem}
If one allows the surfaces $F_m$ and $G_n$ to have points of
higher multiplicity on $C$, then simple examples show that this number is
dependent on the local geometry. But the philosophy of Gallarati's
theorem is that in the situation of contact of hypersurfaces
the hypersurface of higher degree must have more or harder
singularities on the contact variety than the hypersurface of
lower degree.
We will prove a variant of the above theorem which gives a lower bound
for the size of an even set of nodes. If $S\subset\mathbb{P}_3$ is a nodal surface
recall that for every even set of nodes $w$ on $S$ there exists a surface
$V\subset\mathbb{P}_3$ such that $S.V=2D$ and $w$ is just the set of nodes
of $S$ through which $D$ passes with odd multiplicity. We estimate the
number of nodes through which $D$ passes with multiplicity one.
For a slightly more general setup, let $M$ be a smooth projective threefold
and let $S\subset M$ be a nodal surface. Assume that a surface
$V\subset M$ intersects $S$ as $S.V=rD+D'$, $r\geq 2$, for an
irreducible curve $D$ which is not contained in the support of $D'$.
\begin{definition}\label{definition:smooth}
%
A node $P$ of $S$ is called $D$-smooth if $P\in D$ and $P$ is a
smooth point of $V$.
%
\end{definition}
This definition is justified by the following
\begin{lemma}\label{lemma:smooth}
%
Let $P$ be a node of $S$. If $P$ is $D$-smooth, then
$P$ is a smooth point of $D$. Moreover $r=2$ and
$P\not\in\operatorname{supp}\left(D'\right)$.
%
\end{lemma}
{\noindent\bf Proof:\ } There exists a neighborhood $U$ of $P$ in $M$ which is
biholomorphic to some open neighborhood of the origin $\mathbf{0}\in\mathbb{C}^3$,
so it suffices to prove the lemma for two affine hypersurfaces
$S$, $V\subset\mathbb{C}^3$. We study the intersection with a general
plane through $\mathbf{0}$.
Let $L\cong\mathbb{P}_2$ be the set of all planes $H\subset\mathbb{C}^3$ through
$\mathbf{0}$ and let $T=T_{\mathbf{0}}V\in L$ be the tangent plane to $V$
in $\mathbf{0}$. Then for all $H\in L\setminus\left\{T\right\}$, the
curve $C_H=V.H$ is smooth in $\mathbf{0}$. The set of all planes
$H\in L$ which have contact to the tangent cone $C_{\mathbf{0}}S$ of
$S$ in $\mathbf{0}$ is parameterized by a smooth conic $Q\subset L$.
For all $H\in L\setminus Q$, the curve $F_H=S.H$ has an ordinary
double point in $\mathbf{0}$. While varying $H$ in
$L\setminus\left(Q\cup\left\{T\right\}\right)$, the tangent lines
$T_{\mathbf{0}}C_H$ sweep out $T_{\mathbf{0}}V$, while the tangent lines to
both branches of $F_H$ in $\mathbf{0}$ sweep out $C_{\mathbf{0}}S$.
So there exists a plane $\tilde{H}\in
L\setminus\left(Q\cup\left\{T\right\}\right)$ such that $T_{\mathbf{0}}C_{\tilde{H}}$
is not contained in $C_{\mathbf{0}}S$. Therefore
$C_{\tilde{H}}$ and $F_{\tilde{H}}$ meet transversal in $\mathbf{0}$, hence on $\tilde{H}$
we have local intersection multiplicity
$\left(F_{\tilde{H}}.C_{\tilde{H}}\right)_{\mathbf{0}}=2$. Then of course
\begin{align*}
%
2 &= \left(F_{\tilde{H}}.C_{\tilde{H}}\right)_{\mathbf{0}}
= \left(\left.S\right|_{\tilde{H}}.\left.V\right|_{\tilde{H}}\right)_{\mathbf{0}}
= \left(S.V.\smash{\tilde{H}}\right)_{\mathbf{0}} \\
&= \left(\left(rD+D'\right).H\right)_{\mathbf{0}}
= r\left(D.\smash{\tilde{H}}\right)_{\mathbf{0}}+
\left(D'.\smash{\tilde{H}}\right)_{\mathbf{0}}.
%
\end{align*}
Now $\mathbf{0}\in D$ implies $\left(D.\smash{\tilde{H}}\right)_{\mathbf{0}}\geq 1$. Since
$r\geq 2$ we get $\left(D.\smash{\tilde{H}}\right)_{\mathbf{0}}=1$, $r=2$ and
$\left(D'.\smash{\tilde{H}}\right)_{\mathbf{0}}=0$. This proves the lemma.$\square$\medskip\par
Now we give the lower bound for the number of $D$-smooth nodes of $S$.
\begin{proposition}\label{proposition:irreducible}
%
Assume that $D\not\subseteq\operatorname{sing}\left(V\right)$ and let $\beta$
be the number
of singular points of $V$ on $D$ which are smooth points of $S$.
Then $S$ has at least $D.\left(S-V\right)+\beta$ nodes which are
$D$-smooth.
%
\end{proposition}
{\noindent\bf Proof:\ } To prove the theorem we would like to have everything smooth.
There exists a sequence of blowups (embedded resolution of the
singular locus of $S$, $V$ and $D$)
\begin{equation*}
%
\tilde{M} =M_n\overset{\pi_n }{\longrightarrow}
M_{n-1} \overset{\pi_{n-1}}{\longrightarrow}
\ldots \overset{\pi_2 }{\longrightarrow}
M_1 \overset{\pi_1 }{\longrightarrow}
M_0=M.
%
\end{equation*}
Let $S_i$, $V_i$ and $D_i$ denote the proper transforms of $S$, $V$ and $D$
with respect to $\pi_i\circ\pi_{i-1}\circ\ldots\circ\pi_1$. We can
define divisors $D_i'$ by
$S_i.V_i=rD_i+D_i'$ with $D_i\not\subset\operatorname{supp}\left(D_i'\right)$,
$1\leq i\leq n$. Moreover we can arrange the maps $\pi_i$ in such
a way that the following conditions hold.
\begin{itemize}
%
\item[\romannum{1}] $\pi_1\colon M_1\rightarrow M$
is the blowup of $M$ in all
points which are singular for both $S$ and $V$.
\item[\romannum{2}]
$\pi=\pi_{n-1}\circ\ldots\circ\pi_2\colon M_{n-1}\rightarrow M_1$
is the embedded resolution of the singular locus of $V_1$,
i.e.~every map $\pi_{i+1}$ is a blowup of $M_i$ centered
in a smooth variety $Z_i\subset M_i$ such that $Z_i$
is either a point or a smooth curve, $1\leq i\leq n-2$.
\item[\romannum{3}]
$\pi_n\colon\tilde{M}\rightarrow M_{n-1}$ is the embedded resolution
of the singularities of $S_{n-1}$ and $D_{n-1}$.
%
\end{itemize}
Now one has to keep track of the intersection numbers
$D_i.\left(S_i-V_i\right)$ as $i$ increases. We study each of the
three maps separately.
{\bf\romannum{1}} Let $Z=\left\{P_1,\ldots,P_\alpha\right\}=
\operatorname{sing}\left(S\right)\cap\operatorname{sing}\left(V\right)$, then
$M_1=\operatorname{Blow}_Z M$. Denote by $E_j=\pi_0^{-1}\left(P_j\right)$ the
exceptional divisor corresponding to $P_j$. The proper transforms are
calculated as
$S_1=\pi_1^\ast S-2\sum_{j=1}^\alpha E_j$ and
$V_1=\pi_1^\ast V-\sum_{j=1}^\alpha m_jE_j$,
where $m_j=\operatorname{mult}\left(V,P_j\right)\geq 2$, $1\leq j\leq\alpha$.
Then the intersection number can be estimated as
\begin{equation*}
%
D_1.\left(S_1-V_1\right) = D_1.\pi_1^\ast\left(S-V\right)
+\sum_{j=1}^\alpha\left(m_j-2\right)D_1.E_j
\geq D.\left(S-V\right).
%
\end{equation*}
This is just the information we need, so let us consider the second case.
{\bf\romannum{2}} Every blowup $\pi_{i+1}$ gives rise to an exceptional divisor
$F_{i+1}=\pi_{i+1}^{-1}\left(Z_i\right)$. In the $\left(i+1\right)$-st
step always $S_i$ is smooth in all points of $S_i\cap Z_i$,
whereas $V_i$ is singular in all points of $Z_i$. So the proper transforms
are
\begin{equation*}
%
S_{i+1}=\pi_{i+1}^\ast S_i-n_iF_{i+1}
\quad\text{and}\quad
V_{i+1}=\pi_{i+1}^\ast V_i-p_iF_{i+1}
%
\end{equation*}
where $n_i=\operatorname{mult}\left(S_i,Z_i\right)\in\left\{0,1\right\}$ and
$p_i=\operatorname{mult}\left(S_i,Z_i\right)\geq 2$. So this time the intersection
number in question is just
\begin{align*}
%
D_{i+1}.\left(S_{i+1}\right. &-\left.V_{i+1}\right) =
D_{i+1}.\pi_{i+1}^\ast\left(S_i-V_i\right)+
\left(p_i-n_i\right)D_{i+1}.F_{i+1} \\
&= D_i.\left(S_i-V_i\right)+
\left\{\begin{array}{l@{\quad}l}
\left(p_i-1\right)\operatorname{mult}\left(D_i,Z_i\right) &
\text{if $Z_i$ is a point,}\\
p_i\sum_{P\in Z_i\cap D_i}\operatorname{mult}\left(D_i,P\right) &
\text{if $Z_i$ is a curve,}
\end{array}\right. \\
&\geq D_i.\left(S_i-V_i\right)+ \#\left(Z_i\cap D_i\right).
%
\end{align*}
But every singularity of $V$ on $D$ outside the singular locus
of $S$ counts at least once. So by induction
\begin{equation*}
%
D_{n-1}.\left(S_{n-1}-V_{n-1}\right)\geq
D_1.\left(S_1-V_1\right)+\beta\geq
D.\left(S-V\right)+\beta.
%
\end{equation*}
where $\beta=\#\left(\left(\operatorname{sing}\left(V\right)\cap D\right)\setminus
\operatorname{sing}\left(S\right)\right)$.
{\bf\romannum{3}} As for the third case we note that $V_{n-1}$ is smooth and
$S_{n-1}$ is nodal with $S_{n-1}.V_{n-1}=rD_{n-1}+D_{n-1}'$. Either
$D_{n-1}\cap\operatorname{sing}\left(S_{n-1}\right)=\emptyset$ or
$D_{n-1}$ contains at least one node of $S$. But then $r=2$ by lemma
\ref{lemma:smooth}
and $D_{n-1}$ is smooth in all nodes of $S_{n-1}$. In both cases,
$S_{n-1}$ and $D_{n-1}$ do not have common singularities.
Let $P_{\alpha+1},\ldots,P_{\alpha+\eta}$ be the nodes of $S_{n-1}$ on
$D_{n-1}$ and let
$P_{\alpha+\eta+1},\ldots,P_{\alpha+\eta+\tau}$ be the remaining nodes
of $S_{n-1}$. Moreover let $E_j=\pi_n^{-1}\left(P_j\right)$,
$\alpha+1\leq j\leq \alpha+\eta+\tau$. But the embedded resolution of the
singularities of $D_{n-1}$ on $V_{n-1}$ is the same as on $S_{n-1}$, so
the proper transforms are
\begin{align*}
%
\tilde{S} = S_n &= \pi_n^\ast S_{n-1}-E_D-2\sum_{j=\alpha+1}^{\alpha+\eta+\tau}
E_j, \\
\tilde{V} = V_n &= \pi_n^\ast V_{n-1}-E_D-\sum_{j=\alpha+1}^{\alpha+\eta}E_j-
\sum_{k=\alpha+\eta+1}^{\alpha+\eta+\tau} q_kE_k,
%
\end{align*}
where $q_k=\operatorname{mult}\left(V_k,P_k\right)\in\left\{0,1\right\}$ and $E_D$ is a
sum of exceptional divisors corresponding to the singularities of $D_{n-1}$.
Set $\tilde{D}=D_n$ and calculate
\begin{align*}
%
\tilde{D}.\left(\smash{\tilde{S}-\tilde{V}}\right) &=
\tilde{D}.\pi_n^\ast\left(S_{n-1}-V_{n-1}\right) -
\sum_{j=\alpha+1}^{\alpha+\eta}\tilde{D}.E_j \\
&= D_{n-1}.\left(S_{n-1}-V_{n-1}\right)-\eta \\
&\geq D.\left(S-V\right)-\eta+\beta.
%
\end{align*}
On the other hand the smooth surfaces $\tilde{S}$ and $\tilde{V}$ have contact of order
$r-1\geq 1$ along the smooth curve $\tilde{D}$. So the tangent bundles
$T_{\tilde{S}}$ and $T_{\tilde{V}}$ agree along $\tilde{D}$. This implies that
the normal bundles $N_{\tilde{D}\mid\tilde{S}}$ and $N_{\tilde{D}\mid\tilde{V}}$ coincide, thus
\begin{equation*}
%
\left(\smash{{\tilde{D}}^2}\right)_{\tilde{S}}=
\deg\left(\smash{N_{\tilde{D}\mid\tilde{S}}}\right)=
\deg\left(\smash{N_{\tilde{D}\mid\tilde{V}}}\right)=
\left(\smash{{\tilde{D}}^2}\right)_{\tilde{V}}.
%
\end{equation*}
Now by adjunction formula $\tilde{D}.K_{\tilde{V}}=\tilde{D}.K_{\tilde{S}}$. Using the adjunction
formula again we see that
\begin{equation*}
%
0 = \tilde{D}.K_{\tilde{S}}-\tilde{D}.K_{\tilde{V}}
= \tilde{D}.\left.\left(\smash{K_{\tilde{M}}+\tilde{S}}\right)\right|_{\tilde{S}}-
\tilde{D}.\left.\left(\smash{K_{\tilde{M}}+\tilde{V}}\right)\right|_{\tilde{V}}
= \tilde{D}.\left(\smash{\tilde{S}-\tilde{V}}\right).
%
\end{equation*}
This gives the desired formula $\eta\geq D.\left(S-V\right)+\beta$.
If $V$ is also a nodal surface one can see easily that we have
equality.$\square$\medskip\par
The application to surfaces in $M=\mathbb{P}_3$ gives the following
\begin{corollary}\label{corollary:irreducible}
%
Let a nodal surface $S\subset\mathbb{P}_3$ of degree $s$ and an irreducible
surface $V\subset\mathbb{P}_3$ of degree $v$ intersect as $S.V=2D$ for
curve $D$ on $S$. Assume that $V$ is not singular along
a curve contained in $S$ and let $\beta$ be the number of singular
points of $V$ which are smooth for $S$. If $s>v$, then $D$ is
reduced. Moreover $V$ cuts out an even set of at least
$sv\left(s-v\right)/2+\beta$ nodes on $S$ with equality if $V$
is also nodal.
%
\end{corollary}
{\noindent\bf Proof:\ } Just run proposition \ref{proposition:irreducible}
on every irreducible component of $D$.$\square$\medskip\par
It is possible to extend proposition \ref{proposition:irreducible}
to the case when the surface $V$ is not irreducible, but reduced.
The proof however works different.
\begin{proposition}\label{proposition:reduced}
%
Let $S\subset\mathbb{P}_3$ be a nodal surface, $n\in\mathbb{N}$ and let
$V_1,\ldots,V_n\subset\mathbb{P}_3$ be different
irreducible surfaces of degrees $v_1,\ldots,v_n$ satisfying the
following conditions:
%
\begin{itemize}
%
\item[\romannum{1}] $V_i$ is not singular along a curve
contained in $S$, $1\leq i\leq n$,
\item[\romannum{2}] $v_i=\deg\left(V_i\right)<s$, $1\leq i\leq n$ and
\item[\romannum{3}] $S.\left(V_1+\ldots +V_n\right)=2D$ for a (not
necessarily reduced) divisor $D$ on $S$.
%
\end{itemize}
%
Then the reduced surface $V=V_1+\ldots +V_n$ of degree
$v=v_1+\ldots +v_n$ cuts out an even set of nodes $w\in\overline{C}_S$
of weight $\left|w\right|\geq sv\left(s-v\right)\negthinspace /2$.
%
\end{proposition}
{\noindent\bf Proof:\ } Since $v_i<s$ and $V_i$ is not singular along a curve contained
in $S$ there exist reduced divisors $D_i$ and $R_i$ on $S$ which
do not have a common component such that $S.V_i=2D_i+R_i$,
$1\leq i\leq n$. But
\begin{equation*}
%
S.V=S.\left(V_1+\ldots +V_n\right)=
2\left(D_1+\ldots +D_n\right)+R_1+\ldots +R_n.
%
\end{equation*}
This implies that $R_i\subset\bigcup_{j\neq i}V_j$ and thus $R_i$ has
a decomposition $R_i=\sum_{j\neq i}R_{i,j}$ such that
$R_{i,j}\subset V_i\cap V_j$. Now we count the nodes of $S$ through
which $D$ passes with multiplicity 1. Denote $d_i=\deg\left(D_i\right)$,
$r_i=\deg\left(R_i\right)$ and $r_{i,j}=\deg\left(R_{i,j}\right)$.
By corollary \ref{corollary:irreducible},
$V_i$ contains at least $d_i\left(s-v_i\right)$
nodes of $S$ through which $D_i$ passes with multiplicity 1. All these
nodes lie outside $R_i$. We cannot simply add these numbers:
some nodes might be counted more than once. But every node $P\in w$
on $V_i$ which is counted more than once is contained also
in some $V_j$ for a $j\neq i$, hence in $F_{i,j}=V_i.V_j$.
Let $f_{i,j}=\deg\left(F_{i,j}\right)$. Let $C$ be an irreducible
component of $F_{i,j}$ and let $c=\deg\left(C\right)$.
We have the following possibilities:
\begin{itemize}
%
\item $C\not\subset S$. In this case $C$ contains
at most $cs/2$ nodes of $S$.
\item $C\subset S$ is a component of $R_i$. Here $C$ does not contain
any node that we counted.
\item $C\subset S$ is a component of $D_i$ and $D_j$. Here $V_i$
and $V_j$ have contact to $S$ along $C$
and thus $C$ appears in $F_{i,j}$ with
multiplicity $\geq 2$. Clearly $C$ contains at most
$c\left(s-1\right)$ nodes of $S$.
\item $C\subset S$ is a component of $D_i$, but not of $D_j$.
Then $V_i$ and $V_j$ meet transversal along $C$, so
$S.\left(V_i+V_j\right)=3C+other\ curves$. So there exist
a $k\not\in\left\{i,j\right\}$ such that $C\in F_{i,k}$.
So $C$ appears with multiplicity $\geq 2$ in
$\sum_{j\neq i}F_{i,j}$. Again $C$ contains at most
$c\left(s-1\right)$ nodes of $S$.
%
\end{itemize}
Since every component of $R_{i,j}$ is contained in $F_{i,j}$, this
shows that $F_{i,j}$ contains at most
$\left(f_{i,j}-r_{i,j}\right) s/2$ nodes that we counted. So $V_i$
contains at least
\begin{equation*}
%
d_i\left(s-v_i\right)-\sum_{j\neq i}\left(f_{i,j}-r_{i,j}\right)
%
\end{equation*}
nodes through which $D$ passes with multiplicity 1. This implies
\begin{align*}
%
\left|w\right| &\geq
\sum_{i=1}^n d_i\left(s-v_i\right)
-\frac{s}{2}\sum_{j\neq i}\left(f_{i,j}-r_{i,j}\right)\\
&=\frac{1}{2}\left(\sum_{i=1}^n\left(sv_i-r_i\right)\left(s-v_i\right)
+sr_i -s\sum_{j\neq i}v_iv_j\right)\\
&=\frac{1}{2}\left(s^2\left(v_1+\ldots v_n\right)
-s\left(v_1^2+\ldots +v_n^2+2\sum_{i<j}v_iv_j\right)
+r_1v_1+\ldots +r_nv_n\right)\\
&\geq \frac{sv}{2}\left(s-v\right)
%
\end{align*}
This completes the proof.$\square$\medskip\par
\section{Contact surfaces and quadratic systems}
In this section we apply the previous results to our initial
situation. So let again $S\subset\mathbb{P}_3$ be a nodal surface of degree $s$ and
$V\subset\mathbb{P}_3$ a reduced surface of degree $v$ such that $S.V=2D$ for some
curve $D$. We give a complete analysis of
the situation when $V$ is a plane or a quadric.
Using the notation of the first paragraph, $V$ cuts out an even
set of nodes $w\in\overline{C}_S$. Recall that the linear system
$L_w=\mylinsys{v}{-}{w}$ parameterizes all contact curves of the form
$D'=(1/2)S.V'$ where $V'$ is a surface of degree $v$ which cuts out $w$.
In some cases, $V$ will be the unique surface
of degree $v$ which cuts out $w$.
Now $2D\in\mathbb{P}\left(\cohom{0}{\obundletS{vH}}\right)$ is the restriction of
$V\in\mathbb{P}\left(\cohom{0}{\obundle{\mathbb{P}_3}{vH}}\right)$ to $S$. Consider the
exact sequence
\begin{equation*}
%
0\longrightarrow
\obundle{\mathbb{P}_3}{\left(v-s\right)H}\longrightarrow
\obundle{\mathbb{P}_3}{vH}\longrightarrow
\obundle{S}{vH}\longrightarrow
0.
%
\end{equation*}
Since $\cohom{i}{\obundle{\mathbb{P}_3}{\left(v-s\right)H}}=0$ for $s>v$,
$i=0,1$, the induced map
$\cohom{0}{\obundle{\mathbb{P}_3}{vH}}\rightarrow
\cohom{0}{\obundle{S}{vH}}$ is an isomorphism. So if $v<s$, then $V$ is the
unique surface of degree $v$ cutting out $w$ via $D$ and
$L_w=\mylinsys{v}{-}{w}$ in fact parameterizes the
space of all surfaces of degree $v$ which cut out $w$. This space is
not a linear system, but the quadratic system
\begin{equation*}
%
Q_w=\left\{V'\mid S.V'=2D'\text{\ with\ }D'\in L_w\right\}.
%
\end{equation*}
It is constructed as follows:
If $\mycohomd{0}{v}{-}{w}=n+1\geq 2$ we can
find $n+1$ linearly independent sections
$s_0,\ldots,s_n\in\mycohom{0}{v}{-}{w}$.
Clearly all products $s_is_j\in\cohom{0}{\obundletS{v\pi^\ast\negthinspace H-E_w}}$
for $0\leq i\leq j\leq n$.
So there exist sections $g_{i,j}\in\cohom{0}{\obundle{{\tilde{\mathbb{P}}}_3}{v\pi^\ast\negthinspace H-E_w}}$
over ${\tilde{\mathbb{P}}}_3$ which restrict to $s_is_j$ under the
identification $\obundle{{\tilde{\mathbb{P}}}_3}{v\pi^\ast\negthinspace H-E_w}\otimes \obundles{\tilde{S}}\cong
\obundletS{v\pi^\ast\negthinspace H-E_w}$.
Outside the exceptional locus we can view the $g_{i,j}$ as sections
of $\obundle{\mathbb{P}_3}{vH}$. Since $w$ has codimension $\geq 2$ in
$\mathbb{P}_3$, these sections extend also to $w$. This implies that
\begin{align*}
%
Q_w &= Q\left(g_{i,j}\mid 0\leq i\leq j\leq n\right) \\
&= \left\{\sum_{i=0}^n\lambda_i^2g_{i,i}+
2\sum_{0\leq i<j\leq n}\lambda_i\lambda_j g_{i,j}=0\mid
\left(\lambda_0:\ldots:\lambda_n\right)\in\mathbb{P}_n\right\}.
%
\end{align*}
Therefore the quadratic system $Q_w$ is the image of an embedding of
Veronese type of $\mathbb{P}_n$ into the space
$\mathbb{P}_{\binom{v+3}{3}}$ parameterizing all surfaces of degree $v$.
In general, $Q_w$ will not contain any linear subspace.
The quadratic system $Q_w$ admits a decomposition $Q_w=B_w+F_w$ where
$B_w$ is a reduced surface of degree $b\leq v$ and the base locus of
$F_w$ (if any) consists only of curves and points. If $F_w$ has no
basepoints then $B_w$ cuts out $w$ and so
$\mycohomd{0}{b}{-}{w}=1$.
\begin{definition}\label{definition:stable}
%
An even set of nodes $w\in\overline{C}_S$ is called
%
\begin{equation*}
%
\left.\begin{array}{c}
\text{semi stable}\\\text{stable}\\\text{unstable}
\end{array}\right\}
\quad\text{in degree $v$ if}\quad
\left\{\begin{array}{c}
\text{$F_w$ is basepointfree,}\\
\text{$F_w=\emptyset$,}\\
\text{$F_w$ has basepoints.}
\end{array}\right.
%
\end{equation*}
%
\end{definition}
The base locus of $Q_w$ is
$B\left(Q_w\right)=\left\{g_{i,j}=0\mid 0\leq i\leq j\leq n\right\}$.
It is contained in the discriminant locus
$Z\left(Q_w\right)=\left\{g_{i,i}g_{j,j}=g_{i,j}^2
\mid 0\leq i<j\leq n\right\}$. There is also a Bertini type theorem
for quadratic systems.
\begin{lemma}\label{lemma:bertini}
%
(Bertini for quadratic systems) The general element of $Q_w$
is smooth outside $Z\left(Q_w\right)$.
%
\end{lemma}
{\noindent\bf Proof:\ } The proof runs like the proof of the Bertini theorem in
\cite{griffithsharris}.$\square$\medskip\par
Next we give a different characterization of stability.
\begin{proposition}\label{proposition:stable}
%
Let $w\in\overline{C}_S$. The surface $B_w$ is always reduced and
%
\begin{itemize}
\item[\romannum{1}] $w$ is stable in degree $v$ if and only if
$\mycohomd{0}{v}{-}{w}=1$.
\item[\romannum{2}] $w$ is semi stable in degree $v$ if and only if
$F_w$ contains a square. Then $B_w$ cuts out
$w$ and either every surface in $F_w$ is a square
or the general surface in $F_w$ is reduced.
\item[\romannum{3}] $w$ is unstable in degree $v$ if and only if
$F_w$ contains no square. Then $B_w$ does not
cut out $w$ and the general surface in $Q_w$ is reduced.
%
\end{itemize}
%
\end{proposition}
{\noindent\bf Proof:\ } \romannum{1} follows from the definition. So let $w$ be not stable
in degree $v$. We use induction on $n$.
$n=1$: By construction $\gcd\left(g_{0,0},g_{0,1},g_{1,1}\right)=g$ is
reduced. Let $\overline{g}_{i,j}=g_{i,j}/g$ and let
$\overline{Q}_w=Q\left(\overline{g}_{0,0},\overline{g}_{0,1},\overline{g}_{1,1}\right)$. Now we have two cases.
a) If $\overline{g}_{0,0}\overline{g}_{1,1}=\overline{g}_{0,1}^2$, then $\overline{g}_{0,0}$ and $\overline{g}_{1,1}$
must be squares. So $\overline{g}_{0,0}=a_0^2$, $\overline{g}_{1,1}=a_1^2$ and thus
$\overline{g}_{0,1}=a_0a_1$. Hence $B_{w}=\left\{g=0\right\}$ cuts out $w$. So
by construction the quadratic system
$F_w=\left\{\smash{\left(\lambda_0a_0+\lambda_1a_1\right)^2}
\mid\left(\lambda_0:\lambda_1\right)\in\mathbb{P}_1\right\}$ contains only
squares. Then $F_w$ is free and $w$ is semi stable in degree $v$.
b) $Z\left(\overline{Q}_w\right)= \left\{\overline{g}_{0,0}\overline{g}_{1,1}=\overline{g}_{0,1}^2\right\}$
is a surface. If all surfaces of $Q_w$ are not reduced,
then by lemma \ref{lemma:bertini} all surfaces of
$\overline{Q}_w$ contain a component of $Z\left(Q_w\right)$.
So this component is constant for all surfaces in $\overline{Q}_w$,
which contradicts
$\gcd\left(\overline{g}_{0,0},\overline{g}_{0,1},\overline{g}_{1,1}\right)=1$. So the general surface
in $Q_w$ is reduced. Now assume $B_w$ cuts out $w$. Then by construction
$F_w$ contains squares, so $F_w$ is free and $w$ is semi stable in
degree $v$. Otherwise $B_w$ does not cut out $w$, so $F_w$ must have
basepoints in $w$. Then $w$ is unstable in degree $v$.
$n-1\Rightarrow n$: Again
$\gcd\left(g_{i,j}\mid 0\leq i\leq j\leq n\right)=g$ is reduced. Consider
the quad\-ra\-tic system
$Q=Q\left(g_{i,j}\mid 0\leq i\leq j\leq n-1\right)$. Either
$\gcd\left(g_{i,j}\mid 0\leq i\leq j\leq n-1\right)$ is reduced and
we're done or it's not reduced. For
$\lambda=\left(\lambda_0:\ldots:\lambda_{n-1}\right)\in\mathbb{P}_{n-1}$
let
\begin{equation*}
%
g_\lambda =\sum_{i=0}^{n-1}\lambda_i^2g_{i,i}
+2\sum_{0\leq i<j\leq n-1}\lambda_i\lambda_j g_{i,j}
\quad\text{and}\quad
h_\lambda=\sum_{i=0}^{n-1}\lambda_ig_{i,n}.
%
\end{equation*}
Now consider the quadratic system
\begin{equation*}
%
R_\lambda =\left\{
t^2 g_\lambda+2t\lambda_nh\lambda+\lambda_n^2g_{n,n}=0
\mid\left(t:\lambda_n\right)\in\mathbb{P}_1\right\}.
%
\end{equation*}
While varying $\lambda\in\mathbb{P}_{n-1}$,
$\gcd\left(g_\lambda,h_\lambda,g_{n,n}\right)$ is constant on an open
dense subset, since it contains only factors of $g_{n,n}$. So for
general $\lambda$ $\gcd\left(g_\lambda,h_\lambda,g_{n,n}\right)=
\gcd\left(g_{i,j}\mid 0\leq i\leq j\leq n\right)=g$ is reduced.
By the first part either $g_\lambda g_{n,n}=h_\lambda^2$ for all
$\lambda$, so $g_\lambda$ and $g_{n,n}$ are always squares modulo
$g$. Then $w$ is semi stable in degree $v$. Or the general surface
in $R_\lambda$ and hence in $Q_w$ is reduced. Again either
$B_w$ cuts out $w$ and we're in the semi stable case or $B_w$ does not cut
out $w$. Then $w$ is unstable in degree $v$.$\square$\medskip\par
\begin{corollary}\label{lemma:unique}
%
If $2v<s$ then $w$ is semi stable in degree $v$.
%
\end{corollary}
{\noindent\bf Proof:\ } On $S$ we have
$g_{0,0}g_{1,1}-g_{0,1}^2=s_0^2s_1^2-\left(s_0s_1\right)^2=0$. Let
$S=\left\{f=0\right\}$, then either $g_{0,0}g_{1,1}=g_{0,1}^2$ or
$f\mid g_{0,0}g_{1,1}-g_{0,1}^2$. But the second case implies
$2v\geq s$, so we are in the first case. Then we always run
into case a) in the proof of proposition \ref{proposition:stable}.$\square$\medskip\par
\begin{corollary}
%
If $w$ is semi stable in degree $v$ and $2\deg\left(F_w\right)<s$,
then $F_w$ contains only squares.
%
\end{corollary}
{\noindent\bf Proof:\ } $F_w$ contains a square $W_0=\left\{g_{0,0}=g_0^2=0\right\}$.
Now take any other $W_1=\left\{g_{1,1}=0\right\}\in F_w$ and consider
the quadratic system generated by $g_{0,0}$ and $g_{1,1}$:
$g_{0,0}$, $g_{1,1}$ give rise to sections $s_0^2$, $s_1^2$ over
$\tilde{S}$. Then $s_0s_1$ is the restriction of a section $g_{0,1}$ to
$\tilde{S}$. The quadratic system in question is just
$Q=Q\left(g_{0,0},g_{0,1},g_{1,1}\right)$. But
$g_{0,0}g_{1,1}-g_{0,1}^2$ vanishes on $S$. Since
$\deg\left(g_{0,0}g_{1,1}-g_{0,1}^2\right)=2\deg\left(F_w\right)<s$,
we have $g_{0,0}g_{1,1}=g_0^2g_{1,1}=g_{0,1}^2$. This implies that
also $g_{1,1}$ is a square.$\square$\medskip\par
\begin{proposition}\label{proposition:unstable}
%
If $w$ is unstable in degree $v$, then there exists a surface
$W$ of degree $2v-s$ such that $w$ is cut out by a reduced surface
$V$ of degree $v$ satisfying:
%
\begin{itemize}
%
\item[\romannum{1}] $V$ is not singular on $S$ outside $W$.
\item[\romannum{2}] If $V$ is singular along a curve $C\subset S$,
then $C$ is a curve of triple points of $W$.
%
\end{itemize}
%
\end{proposition}
{\noindent\bf Proof:\ } $w$ is cut out by a reduced surface, so we can assume that
$g_{0,0}$ is square free.
Again $g_{0,0}g_{1,1}-g_{0,1}^2$ vanishes on $S$ and $g_{0,0}$,
$g_{1,1}$ are linearly independent. So there exists a polynomial
$\alpha$ of degree $2v-s$ such that $\alpha f=g_{0,0}g_{1,1}-g_{0,1}^2$.
Let $W=\left\{\alpha =0\right\}$ and let
$V_\lambda=\left\{
\lambda_0^2 g_{0,0}+2\lambda_0\lambda_1 g_{0,1}+\lambda_1^2 g_{1,1}=0
\mid\lambda=\left(\lambda_0:\lambda_1\right)\in\mathbb{P}_1\right\}$.
For every point $P\in\mathbb{P}_3$ we can choose affine coordinates
$\left(z_1,z_2,z_3\right)$ on an affine neighborhood $U$ of $P$.
For any function $h$ on $U$, we identify the total derivative
$Dh$ with the gradient $\nabla h$ and $D^2h$ with the Hesse matrix
$H\left(h\right)$. We find that
\begin{align*}
%
D\left(\alpha f\right) &=
\alpha\nabla f+f\nabla\alpha =
g_{0,0}\nabla g_{1,1}+g_{1,1}\nabla g_{0,0}-
2g_{0,1}\nabla g_{0,1},\\
D^2\left(\alpha f\right) &=
\alpha H\left(f\right)+f H\left(\alpha\right)+
\nabla\alpha{\nabla f}^t+\nabla f{\nabla\alpha}^t\\
&= g_{0,0}H\left(g_{1,1}\right)+
g_{1,1}H\left(g_{0,0}\right)-
2g_{0,1}H\left(g_{0,1}\right)\\
&\hspace*{4ex}+
\nabla g_{0,0}{\nabla g_{1,1}}^t+
\nabla g_{1,1}{\nabla g_{0,0}}^t-
2\nabla g_{0,1}{\nabla g_{0,1}}^t.
%
\end{align*}
Now let $P\in S$. We have to consider two different cases:
a) $P\in\operatorname{sing}\left(S\right)\setminus W$, so $f\left(P\right)=0$,
$\nabla f\left(P\right)=0$ and
$\operatorname{rk}\left(H\left(f\right)\left(P\right)\right)=3$.
If $P$ is a basepoint of $Q$ then
\begin{align*}
%
H\left(\alpha f\right)\left(P\right) &=
\alpha\left(P\right)H\left(f\right)\left(P\right) \\
&= \left(\nabla g_{0,0}{\nabla g_{1,1}}^t+
\nabla g_{1,1}{\nabla g_{0,0}}^t-
2\nabla g_{0,1}{\nabla g_{0,1}}^t\right)\left(P\right).
%
\end{align*}
But $P\not\in W$, so $\alpha\left(P\right))\neq 0$ and
$\operatorname{rk}\left(H\left(\alpha f\right)\left(P\right)\right)=3$. This is only
possible if $\nabla g_{0,0}$, $\nabla g_{1,1}$ and $\nabla g_{0,1}$
are linearly independent in $P$. So every surface $V_\lambda$
is smooth in $P$. If $P$ is not a basepoint of $Q$ then the general
surface $V_\lambda$ will not contain $P$.
b) Let $P\in\operatorname{smooth}\left(S\right)\setminus W$. Here
$f\left(P\right)=0$, $\nabla f\left(P\right)\neq 0$ and
$\alpha\left(P\right)\neq 0$. Then
$\nabla\left(\alpha f\right)\left(P\right)=
\alpha\left(P\right)\nabla f\left(P\right)\neq 0$, so $P$ is not a basepoint
of $Q$. Assume now we have chosen $\lambda$
such that $P\in V_\lambda$. After a permutation
of indices we can assume $\lambda_0=1$, so
$V_\lambda=\left\{
g_{0,0}+2\lambda_1 g_{0,1}+\lambda_1^2 g_{1,1}=0\right\}$.
Since $P$ is not a basepoint we have $g_{1,1}\left(P\right)\neq 0$.
Together with $\left(g_{0,0}g_{1,1}-g_{0,1}^2\right)\left(P\right)=0$
we get $\lambda_1=-\left(g_{0,1}/g_{1,1}\right)\left(P\right)$.
Then
\begin{align*}
%
\nabla &\left(g_{0,0}+2\lambda_1 g_{0,1}+\lambda_1^2
g_{1,1}\right)\left(P\right)=\\
&\hspace*{5ex}=\frac{1}{g_{1,1}\left(P\right)}
\left(g_{1,1}\nabla g_{0,0}-2 g_{0,1}\nabla g_{0,1}
+g_{0,0}\nabla g_{1,1}\right)\left(P\right)\\
&\hspace*{5ex}=\frac{1}{g_{1,1}\left(P\right)}
\alpha\left(P\right)\nabla f\left(P\right).
%
\end{align*}
We see that $P$ is a smooth point of $V_\lambda$,
so together with a) we have proved \romannum{1}.
c) Assume that $V_\lambda$ is singular along a curve
$C_\lambda\subset S$ and let $m_\lambda=\operatorname{mult}\left(V_\lambda,C_\lambda\right)$.
Then $C_\lambda$ is a continuous family of curves and
$m=\min\left\{m_\lambda\mid\lambda\in\mathbb{P}_1\right\}$ is equal
to $m_\lambda$ on an open dense subset of $\mathbb{P}_1$. Now
\romannum{1} says that
$C_\lambda\subseteq S\cap W$ for all $\lambda\in\mathbb{P}_1$.
But $S\cap W$ is itself a curve, so this family is
in fact constant. So let $C=C_{(0:1)}$.
Now $g_{0,0}g_{1,1}-g_{0,1}^2=\alpha f$ vanishes to the $2m$-th order
along $C$ and $\operatorname{mult}\left(S,C\right)=1$, so $\alpha$ vanishes
to the $\left(2m-1\right)$-st order along $C$.$\square$\medskip\par
\begin{corollary}\label{corollary:unstable}
%
Let $w\in\overline{C}_S$.
%
\begin{itemize}
%
\item[\romannum{1}]
If $w$ is unstable in degree $s/2$, then $\left|w\right|=s^3/8$.
\item[\romannum{2}]
If $w$ is unstable in degree $\left(s+1\right)\!/2$
(resp.~$\left(s+2\right)/2$),
then $\left|w\right|\geq s\left(s-1\right)^2/8$
(resp.~$s\left(s-2\right)^2/8$).
%
\end{itemize}
%
\end{corollary}
{\noindent\bf Proof:\ } In the first case $W=\emptyset$. So the general surface in
$Q_w$ is not singular on $S$, hence irreducible. Now apply
corollary \ref{corollary:irreducible}.
In the second case
$\deg\left(W\right)\leq 2$, so $W$ has no triple curve.
Now apply proposition \ref{proposition:reduced}.$\square$\medskip\par
Now here comes our analysis what happens if $V$ is a plane or a quadric.
\begin{proposition}\label{proposition:planequadric}
%
Let $w\in\overline{C}_S$.
%
\begin{itemize}
%
\item[\romannum{1}] If $w$ is cut out by a plane $H$, then
$\left|w\right|=s\left(s-1\right)\!/2$.
Moreover $w$ is stable in degree
$1$ if $s>2$ and unstable in degree $1$ otherwise.
\item[\romannum{2}] If $w$ is cut out by a reduced quadric
$Q$, then
%
\begin{equation*}
%
\left|w\right| = \left\{\begin{array}{c@{\quad}l}
s\left(s-2\right) & \text{if $s$ is even,}\\
\left(s-1\right)^2 & \text{if $s$ is odd.}
\end{array}\right.
%
\end{equation*}
%
Moreover $w$ is stable in degree $2$ if $s>4$ and unstable in
degree $2$ otherwise.
%
\end{itemize}
%
\end{proposition}
{\noindent\bf Proof:\ } \romannum{1} $H$ is smooth, so $\left|w\right|=s\left(s-1\right)/2$ by
corollary \ref{corollary:irreducible}.
If $s>2$ then $2\deg\left(H\right)=2<s$, so
$w$ is semi stable in degree $1$ by lemma \ref{lemma:unique}. But
then $w$ is stable in degree $1$.
In the case $s=2$ example \ref{example:cone} shows
that $w$ is unstable in degree $1$.
\romannum{2} Assume first $Q$ is nodal. Then
$\left|w\right|\in\left\{s\left(s-2\right),s\left(s-2\right)+1\right\}$ by corollary
\ref{corollary:irreducible}.
But $s\left(s-2\right)+1=\left(s-1\right)^2$ and
$4\mid\left|w\right|$ imply the above formula for $\left|w\right|$.
Now let $Q=H_1+H_2$ where $H_1\neq H_2$ are planes and
set $L=H_1\cap H_2$. If $L\subset S$, then
$S.H_i=2D_i+L$ and each $D_i$ contains exactly
$\left(s-1\right)^2\!/2$ nodes which are $D_i$-smooth by proposition
\ref{proposition:irreducible}.
Clearly $L$ cannot contain any node of $w$. So $\left|w\right|=\left(s-1\right)^2$.
If $L\not\subset S$, then $S.H_i=2D_i$. Every
$D_i$ is reduced and contains exactly $s(s-1)/2$ $D_i$-smooth nodes.
In every point $P\in L\cap S$
both $H_1$ and $H_2$ are tangent to $S$, so $P$ is a
node of $S$. Both $H_1$ and $H_2$ have contact to the tangent
cone $C_PS$ of $S$ at $P$. This implies $L\not\subset C_PS$,
hence $\operatorname{mult}\left(S,L;P\right)=2$. Therefore $L$ contains exactly
$s/2$ such nodes and $\left|w\right|=s\left(s-2\right)$.
If $s>4$, then $w$ is stable in degree $2$. Now let $s\leq 4$.
In any case $F_w$ cannot contain a square. So $w$ is unstable
if $\mycohomd{0}{2}{-}{w}>1$. For $s=3$ this follows from
example \ref{example:cubic}. If $s=4$ then we find using Serre duality that
$\mycohomd{2}{2}{-}{w}=\mycohomd{0}{-2}{+}{w}=0$.
Therefore it follows that
$\mycohomd{0}{2}{-}{w}\geq\mychi{2}{-}{w}=2$.$\square$\medskip\par
\section{The proof of theorem \ref{theorem:main}}
This section is devoted entirely to the proof of theorem \ref{theorem:main}.
Let $S$ and $V$ with $S.V=2D$ as in the first section.
\begin{lemma}\label{lemma:equal}
%
\cite{catanese}
Let $w\in \overline{C}_S$ be an even set of nodes.
Let $n\in\mathbb{N}$ be even if $w$ is strictly even and
odd if $w$ is weakly even. Then for all $i\geq 0$
%
\begin{equation*}
%
\cohomd{i}{\obundletS{\left(n\pi^\ast\negthinspace H +E_w\right)\negthinspace /2}} =
\cohomd{i}{\obundletS{\left(n\pi^\ast\negthinspace H -E_w\right)\negthinspace /2}}.
%
\end{equation*}
%
\end{lemma}
\proofwith{ of theorem \ref{theorem:main}}
\romannum{1} By proposition \ref{proposition:planequadric}
the even sets $w\in\overline{C}_S$ cut
out by planes satisfy $\left|w\right|=s\left(s-1\right)\!/2$. We show that no smaller
even sets can occur. The proof also explains the ``gaps'' of
corollary \ref{corollary:main}. So let $w\in\overline{C}_S\setminus\left\{0\right\}$ be weakly even.
$s=4$: Since $\mychi{}{-}{w}=\left(10-\left|w\right|\right)\!/4$ is an integer
we must have
$\left|w\right|\in\left\{2,6,10,14\right\}$. By Serre duality and lemma
\ref{lemma:equal}
$\mycohomd{2}{}{-}{w}=\mycohomd{0}{-}{-}{w}=0$. Now let $\left|w\right|\leq 6$, then
$\mycohomd{0}{}{-}{w}\geq 1$. So $w$ is cut out by a plane, hence $\left|w\right|=6$.
$s=6$: Here
$\mychi{}{-}{w}=\mychi{3}{-}{w}=\left(35-\left|w\right|\right)\!/4$,
so $\left|w\right|\in\left\{3,7,11,\ldots\right\}$. Let $\left|w\right|<27$,
so $\left|w\right|\leq 23$ and $\mychi{}{-}{w}\geq 3$. Following proposition
\ref{proposition:stable},
we see that $w$ is either stable in degree $1$ or no plane cuts out $w$,
hence $\mycohomd{2}{3}{-}{w}=\mycohomd{0}{}{-}{w}\in\left\{0,1\right\}$.
This implies $\mycohomd{0}{3}{-}{w}\geq 2$. Now corollary
\ref{corollary:unstable} tells us
that $w$ is semi stable in degree $3$. But $w$ cannot be stable in degree
$3$, so $w$ is stable in degree $1$. Hence
$w$ is cut out by a plane. It follows that $\left|w\right|=15$ and that there are
no weakly even sets of $19$ and $23$ nodes on a nodal sextic surface.
$s=8$: Now $\mychi{3}{-}{w}=\mychi{5}{-}{w}=21-\left|w\right| /4$, so
$\left|w\right|\in\left\{4,8,12,\ldots\right\}$. Let $\left|w\right|<60$, then
$\left|w\right|\leq 56$ and $\mychi{3}{-}{w}\geq 7$.
Assume that $\mycohomd{0}{3}{-}{w}=\mycohomd{2}{5}{-}{w}\leq 1$. It
follows that $\mycohomd{0}{5}{-}{w}\geq 6$,
hence $w$ is unstable in degree $5$.
So $\left|w\right|\geq 60$ by corollary \ref{corollary:unstable}, contradiction.
This implies that $w$ is semi stable in degree $3$ and stable in degree $1$.
Again $w$ is cut out by a plane, hence $\left|w\right|=28$. Moreover, a nodal octic
surface cannot have weakly even sets of $32, 36, 40,\ldots,56$ nodes.
\romannum{2} This is essentially a copy of the methods of \romannum{1}.
Let $w\in C_S\setminus\left\{0\right\}$ be strictly even.
$s=4$: We have $\left|w\right|\in\left\{8,16\right\}$. If $\left|w\right|=9$ then
$\mychi{2}{-}{w}=2$. But $\mycohomd{2}{2}{-}{w}=\mycohomd{0}{-2}{-}{w}=0$,
so by Serre duality and lemma \ref{lemma:equal}
$\mycohomd{0}{2}{-}{w}\geq 2$ and $w$ is cut out by a quadric.
$s=5$: Now $\left|w\right|\in\left\{4,8,12,\ldots\right\}$ and
$\mychi{2}{-}{w}=5-\left|w\right|/4$. As usual we find that
$\mycohomd{2}{2}{-}{w}=\cohomd{0}{\obundletS{-E_w/2}}=0$. So if
$\left|w\right|\leq 16$ then $\mycohomd{0}{2}{-}{w}\geq 1$ and $w$ is cut out
by a quadric. Then $\left|w\right|=16$ by proposition \ref{proposition:planequadric}.
$s=6$: Here $\left|w\right|\in\left\{8,16,24,\ldots\right\}$ and
$\mychi{2}{-}{w}=8-\left|w\right| /4$. This time we find that
$\mycohomd{2}{2}{-}{w}=\mycohomd{0}{2}{-}{w}$. So if $\left|w\right|\leq 24$, then
$\mycohomd{0}{2}{-}{w}\geq 1$ and $w$ is cut out by a quadric surface.
But then $\left|w\right|=24$.
$s=7$: We modify the proof as follows. One calculates
$\mychi{2}{-}{w}=\mychi{4}{-}{w}=14-\left|w\right|/4$. Let $\left|w\right|<44$, so
$\left|w\right|\leq40$. By proposition \ref{proposition:planequadric}
$\mycohomd{0}{2}{-}{w}\in\left\{0,1\right\}$, so
$\mycohomd{0}{4}{-}{w}\geq 3$. If $w$ is unstable in degree $4$ then
$\left|w\right|\geq42$, contradiction. So $w$ is semi stable in degree $4$ and
stable in degree $2$. Now $w$ is cut out by a quadric and thus
$\left|w\right|=36$. In particular, there is no even set of $40$ nodes on
a nodal septic surface.
$s=8$: Using the same argument as for $s=7$, we get
$\mychi{4}{-}{w}=20-\left|w\right|/4$ and
$\mycohomd{2}{4}{-}{w}=\mycohomd{0}{4}{-}{w}$. Let $\left|w\right|<64$, then
$\left|w\right|\leq56$ and $\mycohomd{0}{4}{-}{w}\geq 3$. Again by corollary
\ref{corollary:unstable}
$w$ cannot be unstable in degree $4$. Hence $w$ is semi stable in degree
$4$ and stable in degree $2$. In particular $\left|w\right|=48$ and there is
no even set of $56$ nodes on a nodal octic surface.
$s=10$: Finally we calculate $\mychi{6}{-}{w}=40-\left|w\right|/4$ and as before
$\mycohomd{2}{6}{-}{w}=\mycohomd{0}{6}{-}{w}$. Let
$\left|w\right|<120$, so $\left|w\right|\leq 112$ and $\mycohomd{0}{6}{-}{w}\geq 6$. As before
$w$ is semi stable in degree $6$. If $w$ was stable in degree $4$ then
$\mycohomd{0}{6}{-}{w}=4$, contradiction. So $w$ is semi stable in degree
$w$ and stable in degree $2$. Again $\left|w\right|=80$, hence there are no
strictly even sets of $88$, $96$, $104$ and $112$ nodes
on a nodal surface of degree $10$.$\square$\medskip\par
\section{Examples revisited}
We want to go a little more into the example of quartics. Many of the
facts stated in this example can be found in \cite{gallarati}.
\begin{example}
%
Let $S$ be a nodal quartic surface and let $w\in C_S$ with
$\left|w\right|=8$. We have seen that $\mycohomd{0}{2}{-}{w}\geq 2$
and that $w$ is unstable in degree $2$. Let $Q_w$ be the
quadratic system of quadrics which cut out $w$. The base locus
of $Q_w$ is contained in the surface $W$ of proposition
\ref{proposition:unstable}. But here $W=\emptyset$, so the
only basepoints of $Q_w$ are the nodes of $S$. It follows from
lemma \ref{lemma:smooth} and Bertini that the general element
in $L_w=\mylinsys{2}{-}{w}$ is a smooth elliptic curve.
In fact $Q_w$ defines an elliptic fibration of
$\tilde{S}$. If $\mycohomd{0}{2}{-}{w}>2$, then we will find two
smooth elliptic curves on $\tilde{S}$ intersecting in at least one point,
contradiction. This shows that $\mycohomd{0}{2}{-}{w}=2$.
Taking into account that $\mydivisor{4}{-}{w}$ is nef and
big on $\tilde{S}$, we can calculate the numbers of the next table.
If $\left|w\right|=16$ one computes
$\mycohomd{i}{4}{-}{w}$ in the same fashion.
%
\begin{equation*}
%
\begin{array}{|c||c|c|c|}\hline
%
\left|w\right|=8 & h^0 & h^1 & h^2 \\\hline\hline
\mydivisor{2}{-}{w} & 2 & 0 & 0 \\\hline
\mydivisor{4}{-}{w} & 8 & 0 & 0 \\\hline
%
\end{array}
\quad\quad
\begin{array}{|c||c|c|c|}\hline
%
\left|w\right|=16 & h^0 & h^1 & h^2 \\\hline\hline
\mydivisor{2}{-}{w} & 0 & 0 & 0 \\\hline
\mydivisor{4}{-}{w} & 6 & 0 & 0 \\\hline
%
\end{array}
%
\end{equation*}
%
Now let $w\in\overline{C}_S$ be weakly even. Here $\mydivisor{3}{-}{w}$ is
big and nef, so we find the following table.
%
\begin{equation*}
%
\begin{array}{|c||c|c|c|}\hline
%
\left|w\right|=6 & h^0 & h^1 & h^2 \\\hline\hline
\mydivisor{ }{-}{w} & 1 & 0 & 0 \\\hline
\mydivisor{3}{-}{w} & 5 & 0 & 0 \\\hline
%
\end{array}
\quad\quad
\begin{array}{|c||c|c|c|}\hline
%
\left|w\right|=10 & h^0 & h^1 & h^2 \\\hline\hline
\mydivisor{ }{-}{w} & 0 & 0 & 0 \\\hline
\mydivisor{3}{-}{w} & 4 & 0 & 0 \\\hline
%
\end{array}
%
\end{equation*}
%
\end{example}
\section{Concluding remarks}
It is very likely that theorem \ref{theorem:main} is true for surfaces
of arbitrary degree. Unfortunately I cannot prove this. The main
obstruction is to exclude the
possibility that an irreducible contact surface is singular along
a curve which is contained in the nodal surface.
In \cite{barth2} Barth gave a construction of nodal surfaces
admitting even sets of nodes. The surfaces are constructed
as degeneracy locus of a generic quadratic form on a
globally generated vector bundle
on $\mathbb{P}_3$.
For convenience we give a list of the strictly even sets of nodes
which have been obtained so far. Note that Barth's construction
gives exactly the gap of corollary \ref{corollary:main}.
\begin{center}
%
\begin{tabular}{l|l}
%
degree & even sets \\\hline
3 & 4 \\
4 & 8,16 \\
5 & 16,20 \\
6 & 24,32,40 \\
8 & 48,64,72,80,\ldots,128 \\
10 & 80,120,128,136,\ldots,208
%
\end{tabular}
%
\end{center}
|
1997-12-04T19:47:18 | 9710 | alg-geom/9710011 | en | https://arxiv.org/abs/alg-geom/9710011 | [
"alg-geom",
"math.AG"
] | alg-geom/9710011 | Andrew Kresch | Andrew Kresch | Canonical rational equivalence of intersections of divisors | LaTeX2e, 14 pages; expanded intro; new first section fixes some
errors | null | null | null | null | We consider the operation of intersecting with a locally principal Cartier
divisor (i.e., a Cartier divisor which is principal on some neighborhood of its
support). We describe this operation explicitly on the level of cycles and
rational equivalences and as a corollary obtain a formula for rational
equivalence between intersections of two locally principal Cartier divisors.
Such canonical rational equivalence applies quite naturally to the setting of
algebraic stacks. We present two applications: (i) a simplification of the
development of Fulton-MacPherson-style intersection theory on Deligne-Mumford
stacks, and (ii) invariance of a key rational equivalence under a certain group
action (which is used in developing the theory of virtual fundamental classes
via intrinsic normal cones).
| [
{
"version": "v1",
"created": "Thu, 9 Oct 1997 00:28:31 GMT"
},
{
"version": "v2",
"created": "Thu, 4 Dec 1997 18:47:17 GMT"
}
] | 2016-08-30T00:00:00 | [
[
"Kresch",
"Andrew",
""
]
] | alg-geom | \section{Introduction}
One way to define an operation in intersection theory is to define
a map on the group of algebraic cycles together with
a map on the group of rational equivalences
which commutes with the boundary operation.
Assuming the maps commute with smooth pullback, the extension of
the operation to the setting of algebraic stacks is automatic.
The goal of the first section of this paper
is to present the operation of intersecting
with a principal Cartier divisor in this light.
We then show how this operation lets us obtain a
rational equivalence which is fundamental to intersection theory.
A one-dimensional family of cycles on
an algebraic variety always admits a unique limiting cycle, but a family of
cycles over the punctured affine plane may yield different limiting
cycles if one approaches the origin from different directions.
An important step in the historical development of intersection theory was
realizing how to prove
that any two such limiting cycles are rationally equivalent.
The results of the first section yield, as a corollary,
a new, explicit formula for this rational equivalence.
Another important rational equivalence in intersection theory is
the one that is used to demonstrate commutativity of Gysin maps
associated to regularly embedded subschemes.
In section 2, we exhibit a two-dimensional family of cycles such
that the cycles we obtain from specializing in two different ways
are precisely the ones we need to show to be rationally equivalent to obtain
the commutativity result.
Our explicit rational equivalence
respects smooth pullback, and hence the generalization to stacks is
automatic.
This simplifies intersection theory on Deligne-Mumford stacks as in \cite{v},
where construction of such a rational equivalence
fills the most difficult section of that important paper.
Since our rational equivalence arises by considering families of cycles
on a larger total space,
we are able to deduce (section 3) that the rational
equivalence is invariant under a certain naturally arising group action.
The key observation is that we can manipulate the situation
so that the group action extends to the total space.
This equivariance result is used, but appears with mistaken proof,
in \cite{bf}, where an important new tool of modern intersection theory ---
the theory of virtual fundamental classes --- is developed.
The author would like to thank S. Bloch, W. Fulton, T. Graber,
and R. Pandharipande for helpful advice and
the organizers and staff of the Mittag-Leffler Institute for
hospitality during the 1996--97 program in algebraic geometry.
\section{Intersection with divisors}
In this section we work exclusively on schemes of finite type
over a fixed base field.
The term variety denotes integral scheme, and by a subvariety we mean an
integral closed subscheme.
We denote by $Z_*X$, $W_*X$, and $A_*X$, respectively, the group of
algebraic cycles, group of rational equivalences, and Chow group
of a scheme $X$.
The boundary map $W_*X\to Z_*X$ is denoted $\partial$.
We refer to \cite{f} for basic definitions and properties from
intersection theory.
Given a Cartier divisor $D$ we denote by $[D]$ the associated
Weil divisor
(it is important to note that the notion of Weil divisor makes
sense on arbitrary varieties, \cite{f} \S 1.2).
If $X$ is a variety then we
denote by $X^1$ the set of subvarieties of
codimension 1.
\begin{defn}
Let $X$ be a variety and let $D$ be a Cartier divisor.
Let $\pi\colon \widehat X\to X$
be the normalization map.
The {\em support} of $D$, denoted $|D|$, is defined to be
$\pi(\bigcup_{\substack{W\in\widehat X^1\\ \mathop{\rm ord}\nolimits_W \pi^*D\ne 0}} W).$
\end{defn}
\begin{rem}
This agrees with the na\"\i{}ve notion of support (the union of
all subvarieties appearing with nonzero coefficient in
$[D]$) when $X$ is normal or when $D$ is effective.
\end{rem}
\begin{rem}
There is yet another notion of support which appears in \cite{f}.
There, the support of a divisor is a piece of data
that must be specified along with the divisor.
Given a Cartier divisor $D$ on a variety $X$, let $Z$ be any
closed subscheme such that away from $Z$ the canonical section
of ${\mathcal O}(D)$ is well-defined and nonvanishing.
Then, \cite{f} defines an intersection operation
$A_k(X)\to A_{k-1}(Z)$.
Unfortunately, the support $|D|$ which we have defined
is not generally a support in this sense.
Hence in the definition below we require that our divisors be
specified by defining functions which are regular away from
their supports.
\end{rem}
We shall denote by $|D|^0$ the set of irreducible components of $|D|$.
\begin{defn}
\label{pdivisor}
Let $X$ be a variety.
A {\em $P$-divisor} on $X$ is a tuple $(U,U',x)$ such that
\begin{itemize}
\vspace{-12pt}
\item[(i)] $U$ and $U'$ are nonempty open subschemes of $X$ such
that $U\cup U'=X$;
\item[(ii)] $x\in k(U)^*$;
\item[(iii)] $x|_{U\cap U'}\in {\mathcal O}^*(U\cap U')$; and
\item[(iv)] the data $(x\in k(U)^*, 1\in k(U')^*)$ specifies
a Cartier divisor $D$ such that $|D|=X\setminus U'$.
\end{itemize}
\end{defn}
By abuse of terminology, we call $D$ a $P$-divisor
if $D$ is the Cartier divisor associated to a
$P$-divisor as in (iv).
Given a $P$-divisor as above, we call
$x$ the {\em local defining function}.
A $P$-divisor may be pulled back via a morphism of varieties
provided that the image of the morphism is not
contained in the support of the underlying Cartier divisor.
\begin{exas}
\begin{itemize}
\item[] \hspace{-30pt} (i)\hspace{5pt}Let $X$ be a normal variety.
Let $x\in k(X)^*$ specify a principal Cartier divisor $D$.
Then $(X, X\setminus |D|, x)$ is a $P$-divisor.
\item[(ii)] Let $X$ be a variety.
Every effective principal Cartier divisor is a $P$-divisor.
\item[(iii)] Let $X$ be a variety, and let $\pi\colon X\to \mathbb P^1$ be
a dominant morphism.
Then the fiber of $\pi$ over $\{0\}$ is a $P$-divisor.
\end{itemize}
\end{exas}
The operation of intersecting with a Cartier divisor is
generally defined only on the level of rational equivalence classes of cycles.
When $V\subset |D|$, we have
$D\cdot[V]=c_1({\mathcal O}(D)|_V)\mathbin{\raise.4pt\hbox{$\scriptstyle\cap$}}[V]$, and
there is generally no way to pick canonically
a cycle representing this first Chern class.
The exception is when ${\mathcal O}(D)|_{|D|}$ is trivial, or in our terminology,
$D$ is a $P$-divisor.
Then, we may define a cycle-level intersection operation
(see \cite{f}, Remark 2.3).
\begin{defn}
Let $X$ be a variety, and let $D$ be a $P$-divisor on $X$.
The cycle-level intersection operation
$$D\cdot {-}\colon Z_k(X)\to Z_{k-1}(|D|)$$
is given by
$$D\cdot [V] = \begin{cases}
{}[D|_V] & \text{if $V\not\subset |D|$}; \\
0& \text{if $V\subset |D|$}.
\end{cases}$$
\end{defn}
The claim that this map passes to rational equivalence and hence gives an
intersection operation
$D\cdot{}\colon A_k(X)\to A_{k-1}(|D|)$
is proved in \cite{f}, but not in a way that makes it easy to see
how $D\cdot\alpha$ is to be rationally equivalent to zero if
$\alpha$ is a cycle that is rationally equivalent to zero.
Following the program set out in the introduction, we would like
to demonstrate this fact by giving an explicit map on rational equivalences
which commutes with the boundary operation.
\begin{defn}
Let $X$ be a variety, and let $D$ be a $P$-divisor on $X$ with
local defining function $x$.
Say $V$ is a subvariety of $X$ with normalization
$\pi\colon \widehat V\to V$,
and suppose $y\in k(V)^*$.
We define the intersection operation on the level of rational equivalences
$$D\cdot{-}\colon W_k(X)\to W_{k-1}(|D|)$$
by
\begin{equation}
\label{maponrat}
D\cdot y = \begin{cases}
{}\pi_*\bigl(\bigoplus_{W\in |\pi^*D|^0} (y^{\mathop{\rm ord}\nolimits_W x} / x^{\mathop{\rm ord}\nolimits_W y}) |_W
\bigr) &
\text{if $V\not\subset |D|$}; \\
0& \text{if $V\subset |D|$}.
\end{cases}
\end{equation}
Here, $\pi_*\colon W_*\widehat V\to W_*V$ is pushforward of
rational equivalence.
\end{defn}
\begin{rem}
This definition explains why we a require the definition of a $P$-divisor
to include more
data than just that of the underlying Cartier divisor.
The map (\ref{maponrat}) actually depends on the choice of defining function.
\end{rem}
\begin{pr}
Let $X$ be a variety and let $D$ be a $P$-divisor on $X$.
Then the diagram
$$
\xymatrix{
W_k(X) \ar[r]^(.43){D\cdot{}}\ar[d]_\partial & W_{k-1}(|D|) \ar[d]_\partial \\
Z_k(X) \ar[r]^(.43){D\cdot{}} & Z_{k-1}(|D|)
}
$$
commutes.
\end{pr}
This follows easily from
\begin{pr}
Let $X$ be a normal variety and let $x$ and $y$ be rational functions
with associated principal Cartier divisors $D$ and $E$.
For $V\in X^1$ set $a_V=\mathop{\rm ord}\nolimits_V x$ and $b_V=\mathop{\rm ord}\nolimits_V y$.
Then
\begin{align}
\sum_{V\in X^1} \partial(y^{a_V}/x^{b_V}|_V)&=0; \label{eqone} \\
\partial(D\cdot y) &= D\cdot(\partial\, y); \label{eqtwo} \\
D\cdot[E] - E\cdot[D] &= \sum_{V\in |D|^0\cap |E|^0}
\partial(y^{a_V}/ x^{b_V}|_V). \label{eqthree}
\end{align}
\end{pr}
\begin{proof}
If we split the sum in (\ref{eqone}) into a sum over $V\in |D|^0$ and
a sum over $V\not\in |D|^0$ we obtain (\ref{eqtwo}).
Similarly if we split away the terms with $V\in |D|^0\cap |E|^0$
we obtain (\ref{eqthree}) from (\ref{eqone}).
So, for a fixed variety $X$ and fixed divisors $D$ and $E$, the
three assertions are equivalent.
Now, we get (\ref{eqone}) as a consequence of the tame symbol in $K$-theory,
cf.\ \cite{q} \S7,
or by the following elementary geometric argument.
We quickly reduce to the case where
$D$ and $E$ are effective.
Then, when $D$ and $E$ meet properly,
(\ref{eqthree}) follows from \cite{f}, Theorem 2.4, case 1.
An induction on {\em excess of intersection}
$$\varepsilon(D,E)=\max_{V\in X^1} a_V\cdot b_V$$
completes the proof: if we denote the normalized
blow-up along the ideal $(x,y)$ by $\sigma\colon X'\to X$
and denote the exceptional divisor by $Z$ then we may write
$\sigma^*D = Z + D'$ and $\sigma^*E = Z + E'$,
and now $|D'|\cap |E'|=\emptyset$
and
$\max(\varepsilon(D',Z), \varepsilon(E',Z))<
\varepsilon(D,E)$
(assuming $D$ and $E$ do not meet properly), cf.\ \cite{f}, Lemma 2.4.
The result pushes forward.
\end{proof}
\begin{cor}
\label{canrat}
Let $D$ and $E$ be $P$-divisors on a variety $X$,
with respective local defining functions $x$ and $y$.
Let $\pi\colon\widehat X\to X$ be the normalization map.
Then
$$D\cdot[E] - E\cdot[D] = \partial\, \omega$$
where $\omega\in W_*(|D|\cap |E|)$ is given by
$$\omega = \sum_{V\in |\pi^*D|^0\cap |\pi^*E|^0}
\pi_*(y^{\mathop{\rm ord}\nolimits_Vx}/ x^{\mathop{\rm ord}\nolimits_Vy}|_V).$$
\end{cor}
\section{Application to intersection theory on stacks}
All stacks (and schemes) in this section are algebraic stacks of Artin type,
\cite{a}, \cite{l}, which are locally of finite type over the base field.
The notion of $P$-divsor on a stack makes sense (it is as in
Definition \ref{pdivisor} with ``open subscheme'' replaced by
``open substack,''
where by ``Cartier divisor'' in part (iv) of the definition we
mean a global section of the sheaf ${\mathcal K}^*/{\mathcal O}^*$ for the
Zariski topology, and where
normalization, order along a substack of codimension 1,
and support of a Cartier divisor are well defined on stacks because
they all respect smooth pullback and hence can be defined locally).
Since an Artin stack possesses
a smooth cover by a scheme, the operation of intersecting
with a $P$-divisor on a stack comes for free
once we know that this operation on schemes commutes with
smooth pullback.
Also for free we get Corollary \ref{canrat} in the setting of stacks:
the formation of $\omega$ from $X$, $D$, and $E$ commutes with smooth
pullback.
\begin{pr}
Let $X$ be a variety, let $Y$ be a scheme,
and let $f\colon Y\to X$ be a smooth morphism.
Let $D$ be a $P$-divisor on $X$.
Then $f^*\smallcirc D\cdot{} = (f^*D)\cdot{}\smallcirc f^*$,
both as maps on cycles and as maps on rational equivalences.
\end{pr}
We now turn to an application of Corollary \ref{canrat} to
intersection theory on Deligne-Mumford stacks
(where a reasonable intersection theory exists, cf.\ \cite{g}, \cite{v}).
Central to intersection theory on schemes is the Gysin map corresponding
to a regularly embedded subscheme, since the diagonal of
a smooth scheme is a regular embedding and this way we obtain
an intersection product on smooth schemes.
The diagonal morphism for a smooth Deligne-Mumford stack is not generally
an embedding, but it is representable and unramified.
\begin{lm}
\label{unram}
Let $f\colon F\to G$ be a representable morphism of Artin stacks.
Then $f$ is unramified if and only if there exists
a commutative diagram
$$
\xymatrix{
U\ar[r]^g\ar[d] & V \ar[d] \\
F\ar[r]^f & G
}
$$
such that the vertical maps are smooth surjective,
$g$ is a closed immersion of schemes,
and the induced morphism $U\to F\times_GV$ is \'etale.
\end{lm}
\begin{proof}
This is \cite{v}, Lemma 1.19.
Because this is such a basic fact about properties of
morphisms in algebraic geometry, we present an elementary proof
in the Appendix.
\end{proof}
To describe a representable morphism,
we use the terminology {\em local immersion} as a synonym for
{\em unramified} and call $f$ above a
{\em regular local immersion} if moreover $g$ is a regular
embedding of schemes.
Since formation of normal cone is of a local nature, an obvious
patching construction produces the normal cone $C_XY$ to a local
immersion $X\to Y$; the cone is a bundle in case
$X\to Y$ is a regular local immersion.
To get Fulton-MacPherson-style intersection theory on Deligne-Mumford stacks
we clearly need to have Gysin maps for regular local immersions.
In \cite{v}, the author supplies this needed Gysin map by giving
a (long, difficult) proof of
the stack analogue of \cite{f}, Theorem 6.4, namely
\begin{pr}
\label{bigrat}
Let $X\to Y$ and $Y'\to Y$ be local immersions of Artin stacks.
Then
$[C_{X\times_YC_{Y'}Y}C_{Y'}Y]=[C_{C_XY\times_YY'}C_XY]$
in $A_*(C_XY\times_YC_{Y'}Y)$.
\end{pr}
\begin{rem}
Though our focus is on applications to intersection theory on
Deligne-Mumford stacks, we continue to make use of constructions which
behave well locally with respect to smooth pullback, and hence
our results are valid in the more general setting of Artin stacks.
\end{rem}
\begin{rem}
Given a stack $X$ which is only locally of finite type over a base field,
we must take $Z_*X$ to be the group of {\em locally finite} formal
linear combinations of integral closed substacks.
More intrinsically, $Z_*X$ is the group of global sections of the
sheaf for the smooth topology $\mathcal Z_*$
which associates to a stack of finite type
the free abelian group on integral closed substacks.
Similarly, $W_*X$ is the group of global sections of sheaf $\mathcal W_*$.
As always, $A_*X$ is defined to be $Z_*X/\partial W_*X$.
\end{rem}
The methods of the last section allow us to supply a new, simpler proof of
this proposition.
\begin{proof}
Recall that given a closed immersion $X\to Y$ there are associated
spaces
\begin{align*}
M_XY &= \mathop{{\rm B}\ell}\nolimits_{X\times\{0\}}Y\times\mathbb P^1, \\
M^\circ_XY &= M_XY \setminus \mathop{{\rm B}\ell}\nolimits_{X\times\{0\}}Y\times\{0\},
\end{align*}
cf.\ \cite{f} \S 5.1.
Given a locally closed immersion, say with
$U$ is an open subscheme of $Y$
and $X$ a closed subscheme of $U$, then
$M^\circ_XY:=M^\circ_XU\amalg_{U\times\mathbb A^1} Y\times\mathbb A^1$ makes sense
and is independent of the choice of $U$.
This lets us define $M^\circ_FG$ when $F\to G$ is a local
immersion of stacks, as follows.
Assume we have a diagram as in the statement of Lemma \ref{unram},
and set $R=U\times_FU$ and $S=V\times_GV$.
There are projections $q_1, q_2\colon S\to G$.
Define $s_i\colon M^\circ_RS\to M^\circ_UV$ ($i=1,2$) to be the composite
$M^\circ_RS\to M^\circ_{U\times_GV}S\to M^\circ_UV$,
where the first map is induced by the open immersion $R\to U\times_GV$
and the second, by pullback via $q_i$.
Then $[M^\circ_RS\rightrightarrows M^\circ_UV]$ is the smooth
groupoid presentation of a stack which we denote $M^\circ_FG$.
We have, by descent, a morphism $M^\circ_FG\to\mathbb P^1$, which is flat and
has as general fiber a copy of $G$ and as special fiber the
normal cone $C_FG$.
In the situation at hand, this construction gives
$$(s\times t)\colon M^\circ_XY\times_Y M^\circ_{Y'}Y\to \mathbb P^1\times\mathbb P^1,$$
and hence a pair of $P$-divisors,
$D$ (corresponding to $s$)
and $E$ (corresponding to $t$).
We note that
$(s\times t)^{-1}(\{0\}\times\{0\})=C_XY\times_YC_{Y'}Y$.
Since the restriction of $s\times t$ to
$\mathbb P^1\times\mathbb P^1\setminus \{0\}\times\{0\}$ is flat, we have
\begin{align*}
{}[D] &= [C_XY\times_Y M^\circ_{Y'}Y] \mod Z_*(C_XY\times_YC_{Y'}Y), \\
{}[E] &= [M^\circ_XY\times_Y C_{Y'}Y] \mod Z_*(C_XY\times_YC_{Y'}Y).
\end{align*}
We examine the fiber of $s\times t$ over $\mathbb P^1\times\{0\}$ more closely.
The fiber square
$$\xymatrix{
i^*C_{Y'}Y \ar[r] \ar[d] & C_{Y'}Y \ar[d] \\
X \ar[r]^i & Y
}$$
gives rise to a closed immersion $f$ making
$$\xymatrix@C=2pt{
M^\circ_{i^*C_{Y'}Y}C_{Y'}Y \ar[rr]^(.48)f\ar[dr]_h &&
M^\circ_XY \times_Y C_{Y'}Y \ar[dl]^g \\
& {\mathbb P^1}
}$$
commute (where $g$ is first projection followed by $s$).
Since $f$ is an isomorphism away from the fiber over 0,
we see in fact that
$$[E] = [M^\circ_{i^*C_{Y'}Y}C_{Y'}Y] \mod Z_*(C_XY\times_YC_{Y'}Y),$$
and since $h$ is flat we find
$$D\cdot [E] = [C_{i^*C_{Y'}Y}C_{Y'}Y].$$
Similarly, if $j$ denotes the map $Y'\to Y$ then
$$E\cdot [D] = [C_{j^*C_XY}C_XY]$$
and so the rational equivalence $\omega\in W_*(C_XY\times_YC_{Y'}Y)$ of
Corollary \ref{canrat} satisfies
$$\partial\,\omega = [C_{X\times_YC_{Y'}Y}C_{Y'}Y] -
[C_{C_XY\times_YY'}C_XY]. \qed $$
\renewcommand{\qed}{}\end{proof}
\begin{rem}
The map $M^\circ_FG\to G$ associated to a local immersion of
stacks is not generally separated, though this should cause
the reader no concern, since intersection theory is valid
even on non-separated schemes and stacks.
In fact, even those operations of \cite{v} which require a
so-called finite parametrization may be carried out
on arbitrary Deligne-Mumford stacks which are of finite type
over a field (no such operations show up in this paper).
This is so thanks to the proof, \cite{l} (10.1), that
every Deligne-Mumford stack of finite type over a field
possesses a finite parametrization, i.e.,
admits a finite surjective map from a scheme.
\end{rem}
\begin{rem}
The reader who wishes greater generality may see easily that
all results in this section are valid in the setting of
Artin stacks which are locally of finite type over an
excellent Dedekind domain.
\end{rem}
\section{Equivariance for tangent bundle action}
We continue to work with stacks which are locally of finite type
over some base field.
A special case of Proposition \ref{bigrat} is when
$i\colon X\to Y$ is a local immersion of smooth Deligne-Mumford stacks.
Suppose $j\colon Y'\to Y$ is a local immersion,
with $Y'$ an arbitrary Deligne-Mumford stack.
Recall that the local immersion $j$ gives rise
to a natural group action of $j^*T_Y$ on $C_{Y'}Y$.
In short, the action is given locally (say $Y$ is an affine scheme and
$Y'$ is the closed subscheme given by the ideal $I$) by
considering the action of $T_Y|_{Y'}$ on $\mathop{\rm Spec}\nolimits \mathop{\rm Sym}\nolimits (I/I^2)$
induced by the map $I/I^2\to \Omega^1_Y$ and proving
(\cite{bf}, Lemma 3.2) that the
normal cone $\mathop{\rm Spec}\nolimits \bigoplus I^k/I^{k+1}$ is invariant under
the group action.
If we let $N_XY$ be the normal bundle to $X$ in $Y$ and denote simply
by $N$ its pullback to $X':=X\times_YY'$,
then $C_XY\times_YC_{Y'}Y$ is identified with $N\times_{X'} i^*C_{Y'}Y$.
Viewing $T_{Y'}$ as a subbundle of $j^*T_Y$,
we have the natural action of $T_{Y'}|_{X'}$ on
$i^*C_{Y'}Y$.
This plus the trivial action on $N$ gives an action
of $T_{Y'}|_{X'}$ on $N\times_{X'}i^*C_{Y'}Y$.
\begin{thm}
The rational equivalence between
$[C_{i^*C_{Y'}Y}C_{Y'}Y]$ and $[N\times_{X'}C_{X'}Y']$
produced in the proof of Proposition \ref{bigrat}
is invariant under the action of
$T_{Y'}|_{X'}$ on $N\times_{X'}C_{X'}Y'$
described above.
\end{thm}
As a consequence, the rational equivalence
descends to a rational equivalence on
the stack quotient $[N\times_{X'}i^*C_{Y'}Y\,/\,T_{Y'}|_{X'}]$.
This fact is exploited in \cite{bf}
(Lemma 5.9, where the authors invoke the incorrect stronger
claim appearing in Proposition 3.5 that the rational equivalence
is equivariant for the bigger group $T_Y|_{X'}$).
\begin{proof}
The question is local, so we may assume $Y$ is an irreducible
scheme, smooth and of finite type over the base field,
$X$ is an smooth irreducible closed subscheme of $Y$,
and $Y'$ is a closed subscheme of $Y$.
If $X\subset Y'$ then the group action is trivial and there is
nothing to prove, so we assume the contrary.
\begin{lm}
Let $Y$ be a smooth irreducible scheme of finite type over
a field $k$, of dimension $n$, let $X$ be a
smooth irreducible closed subscheme of $Y$ of codimension $d$, and
let $Y'$ be a closed subscheme of $Y$ such that $X\not\subset Y'$.
Let $x$ be a closed point of $Y'\cap X$.
Then, after suitable base change by a finite separable extension of
the base field, and after shrinking $Y$ to a neighborhood of $x$ in $Y$,
there exists an \'etale map $f\colon Y\to \mathbb A^n$
such that $X$ maps into a linear subspace of $\mathbb A^n$ of codimension $d$
and such that $Y'\to f(Y')$ is \'etale.
\end{lm}
\begin{proof}
We may assume $x$ is a $k$-valued point,
and moreover that $Y$ sits in $\mathbb A^l$ with $X=\mathbb A^{l-d}\cap Y$
(for suitable $l$).
We may take $x$ to be the origin of $\mathbb A^l$.
We consider as candidates for $f$ all linear functions mapping
the flag $\mathbb A^{l-d}\subset \mathbb A^l$ into the flag $\mathbb A^{n-d}\subset \mathbb A^n$.
Those $f$ with $f_*\colon T_{x,Y}\to T_{f(x),\mathbb A^n}$ surjective
form an open subscheme $U$ of $\mathbb A^{nl-dl+d^2}$.
Define locally closed subschemes $V_1$ and $V_2$ of $Y\times U$ by
$$V_1=\{(y,f)\in (Y'\cap X\setminus\{x\})\times U\,|\,
f(y)=0\}$$
and
$$V_2=\{(y,f)\in (Y'\setminus X)\times U\,|\,f(y)=0\},$$
and let $pr_2\colon Y\times U\to U$ be projection.
A dimension count using the fact that $X\not\subset Y'$
gives $\dim(V_1) < \dim(U)$ and $\dim(V_2) < \dim(U)$,
and hence
$U\setminus \bigl( \overline{pr_2(V_1)}\cup \overline{pr_2(V_2)} \bigr)$
is nonempty.
\end{proof}
Since the rational equivalence of the proof of
Proposition \ref{bigrat} commutes with \'etale base change,
we are reduced by the Lemma to the case where
$Y=\mathbb A^n$ and $X=\mathbb A^m$ (as a linear subspace of $\mathbb A^n$).
Now we need the
\begin{keyobs}
Assume $Y=\mathbb A^n$ and $Y'$ is a closed subscheme of $Y$.
Identify $T_Y$, as a group scheme over $Y$,
with the additive group $\mathbb A^n$.
Then there is a group action of $\mathbb A^n$ on $\widetilde M^\circ_{Y'}Y$
(which we define to be the fiber of $M^\circ_{Y'}Y\to\mathbb P^1$ over $\mathbb A^1$)
which restricts to the natural action of $T_Y$
on $C_{Y'}Y$.
\end{keyobs}
Indeed, we let $\mathbb A^n$ act on $Y\times\mathbb A^1$ by
$$(a_1,\ldots,a_n)\cdot(x_1,\ldots,x_n,t)=(x_1+ta_1,\ldots,x_n+ta_n).$$
By the universal property of blowing up, this extends uniquely to an
action of $\mathbb A^n$ on $\widetilde M^\circ_{Y'}Y$.
If $Y'$ is given by the ideal $(f_1,\ldots,f_k)$, and if
we view $\widetilde M^\circ_{Y'}Y$ as the closure of
the graph of
$(f_1/t,\ldots,f_k/t)\colon Y\times(\mathbb A^1\setminus\{0\})\to
\mathbb A^k=\mathop{\rm Spec}\nolimits k[z_1,\ldots,z_k]$, then
the action is given coordinatewise by
$${\mathbf a}=(a_1,\ldots,a_n)\colon z_i\mapsto z_i+
(f_i({\mathbf x} + t{\mathbf a}) - f_i({\mathbf x}))/t,$$
so at $t=0$ we recover $z_i\mapsto z_i+D_{\mathbf a}f_i({\mathbf x})$.
This is the natural action of $T_Y$ on $C_{Y'}Y$.
Concluding the proof of equivariance, we observe that
$\widetilde M^\circ_{\mathbb A^m}\mathbb A^n$ fits into the fiber diagram
$$
\xymatrix{
{\widetilde M^\circ_{\mathbb A^m}\mathbb A^n\times_{\mathbb A^n}\widetilde M^\circ_{Y'}\mathbb A^n}
\ar[r] \ar[d] & {\widetilde M^\circ_{Y'}\mathbb A^n} \ar[d] \\
{\widetilde M^\circ_{\mathbb A^m}\mathbb A^n} \ar[r] \ar[d] &
{\mathbb A^n} \ar[d] \\
{\widetilde M^\circ_{\{0\}}\mathbb A^{n-m}} \ar[r] &
{\mathbb A^{n-m}}
}$$
and now the action from the Key Observation of $\mathbb A^m\subset \mathbb A^n$ on
$\widetilde M^\circ_{Y'}Y$, plus the trivial action of $\mathbb A^m$ on
$M^\circ_{\{0\}}\mathbb A^{n-m}$, combine to give a group action
of $\mathbb A^m$ on $\widetilde M^\circ_XY\times_Y\widetilde M^\circ_{Y'}Y$.
The function
$\widetilde M^\circ_XY\times_Y\widetilde M^\circ_{Y'}Y\to \mathbb A^1\times\mathbb A^1$
which is used in Corollary \ref{canrat} is invariant for
this $\mathbb A^m$-action.
Since the rational equivalence of the proof of
Proposition \ref{bigrat} is compatible with smooth pullback,
we get the desired equivariance result.
\end{proof}
\section{Appendix: unramified morphisms}
We give an elementary algebraic proof of the following fact.
\begin{lm}
Let $S\to T$ be an unramified morphism of affine schemes which are of
finite type over a base field $k$.
Then there exists a commutative diagram of affine schemes
$$\xymatrix{
U \ar[r]^g \ar[d] & V \ar[d] \\
S \ar[r]^f & T
}$$
such that the vertical maps are \'etale surjective
and such that $g$ is a closed immersion.
\end{lm}
This fact plus the local nature of the property of being unramified
gives us Lemma \ref{unram}.
\begin{proof}
Say $S=\mathop{\rm Spec}\nolimits A$, $T=\mathop{\rm Spec}\nolimits B$, and $f$ is given algebraically
by $f^*\colon B\to A$.
Recall that for $f$ to be unramified means that
for every maximal ideal ${\mathfrak p}$ of $A$ with
${\mathfrak q}=f({\mathfrak p})$, we have
$f^*({\mathfrak q})\cdot A_{\mathfrak p}={\mathfrak p}A_{\mathfrak p}$,
and the induced field extension
$B/{\mathfrak q}\to A/{\mathfrak p}$ is separable.
{\em Case 1:} The induced field extension
$B/{\mathfrak q}\to A/{\mathfrak p}$ is an isomorphism.
Then, if $x_1,\ldots,x_n$ are generators of $A$ as a $k$-algebra,
we may write
$$x_i=f^*(t_i)+w_i$$
with $t_i\in B$ and $w_i\in {\mathfrak p}$, for each $i$.
Since $f$ is unramified, we have
$$w_i=\sum_{j=1}^{m_i} \frac{f^*(y_{ij})p_{ij}}{q_i}$$
for some $y_{ij}\in{\mathfrak q}$, $p_{ij}\in A$, and
$q_i\in A\setminus{\mathfrak p}$.
Choose representative polynomials $P_{ij}$ and $Q_i$ in
$k[X_1,\ldots,X_n]$ such that
$P_{ij}(x_1,\ldots,x_n)=p_{ij}$ and
$Q_i(x_1,\ldots,x_n)=q_i$.
Let
\begin{eqnarray*}
\lefteqn{
V=\mathop{\rm Spec}\nolimits B[X_1,\ldots, X_n]\bigm/\big(\,X_1Q_1-t_1Q_1-\sum_{j=1}^{m_1}
y_{1j}P_{1j}\,,\ \ldots,}\hspace{130pt} \\
& & X_nQ_n-t_nQ_n-\sum_{j=1}^{m_n}y_{nj}P_{nj}\,\big),
\end{eqnarray*}
and define
$g\colon S\to V$ by $B\stackrel{f^*}\rightarrow A$ and $X_i\mapsto x_i$,
and let $\varphi\colon V\to T$ be given by inclusion of $B$.
Then $g$ is a closed immersion, and by the Jacobian criterion
$\varphi$ is \'etale in some neighborhood of $g({\mathfrak p})$.
{\em Case 2:} The field extension
$B/{\mathfrak q}\to A/{\mathfrak p}$ is separable.
Let $k'$ be the maximal subfield of $A/{\mathfrak p}$ which is
separable over $k$,
and make the \'etale base change
$\mathop{\rm Spec}\nolimits k'\to \mathop{\rm Spec}\nolimits k$ to get
$f'\colon S'\to T'$.
Now $S'$ has an $A/{\mathfrak p}$-valued point which maps to
${\mathfrak p}\in S$, and since $k'$ together with
$B/{\mathfrak q}$ generates all of $A/{\mathfrak p}$
we are now in the situation of Case 1.
\end{proof}
|
1998-01-07T19:12:19 | 9710 | alg-geom/9710032 | en | https://arxiv.org/abs/alg-geom/9710032 | [
"alg-geom",
"hep-th",
"math.AG"
] | alg-geom/9710032 | Sergey Barannikov | Sergey Barannikov, Maxim Kontsevich | Frobenius Manifolds and Formality of Lie Algebras of Polyvector Fields | 12 pages, AMS-TeX; typos and a sign corrected, appendix added.
Submitted to IMRN | International Mathematics Research Notices, Volume 1998, Issue 4,
Pages 201-215 | 10.1155/S1073792898000166 | null | null | We construct a generalization of the variations of Hodge structures on
Calabi-Yau manifolds. It gives a Mirror partner for the theory of genus=0
Gromov-Witten invariants
| [
{
"version": "v1",
"created": "Tue, 28 Oct 1997 23:19:11 GMT"
},
{
"version": "v2",
"created": "Wed, 7 Jan 1998 18:12:18 GMT"
}
] | 2023-02-21T00:00:00 | [
[
"Barannikov",
"Sergey",
""
],
[
"Kontsevich",
"Maxim",
""
]
] | alg-geom | \section{1. Frobenius manifolds}
Remind the definition of formal Frobenius (super) manifold as given in [D],
[M],
[KM]. Let $\bold H$ be a finite-dimensional $\bold Z_2$-graded
vector space over
$\C$.\footnote{One can use an
arbitrary field of characteristic zero instead of $\C$ everywhere} It
is convenient to choose
some set of coordinates $x_{\bold H}=\{x^a\}$ which defines
the basis $\{\p_a:=\p/\p x^a\}$
of vector fields. One of the given coordinates is distinguished and
is denoted by $x_0$.
Let $A^c_{ab}\in \C [[x_{\bold H}]]$ be a formal power series
representing 3-tensor field,
$g_{ab}$ be a nondegenerate symmetric pairing on $\bold H$.
To simplify notations in superscripts we replace
$\text{deg}\, (x^a)$ by $\bar a$.
One can use the $A_{ab}^c$ in order to define a structure of
$\C [[x_{\bold H}]]$-algebra
on $\bold H\otimes\C [[x_{\bold H}]]$, the (super)space of all
continuous derivations of
$\C[[x_{\bold H}]]$,
with multiplication denoted by $\circ$:
$$ \p_a\circ\p_b:=\sum_c A_{ab}^c\p_c$$
One can use $g_{ab}$ to define the symmetric $\C[[x_{\bold H}]]$-pairing
on $\bold H\otimes\C[[x_{\bold H}]]$:
$$\langle \p_a,\p_b\rangle:=g_{ab}$$
These data define the structure
of formal Frobenius manifold on $\bold H$ iff the following equations hold:
\item{(1)} (Commutativity/Associativity)
$$\forall a,b,c \,\,\,\,\,\, A^c_{ba}=(-1)^{\bar a\bar b}A_{ab}^c\eqno (1a)$$
$$\forall a,b,c,d\,\,\,\,\,\,\sum_{e}A^e_{ab} A_{ec}^d
=(-1)^{\bar a(\bar b+\bar c)}\sum_{e}A_{bc}^e
A_{ea}^d\eqno (1b)$$
equivalently, $A^c_{ab}$ are structure constants of a supercommutative
associative \break $\C[[x_{\bold H}]]$-algebra
\item {(2)}(Invariance) Put $A_{abc}=\sum_e A^e_{ab}g_{ec}$ $$\forall a,b,c
\,\,\,\,\,\,A_{abc}=
(-1)^{\bar a(\bar b+\bar c)}A_{bca}\,\,\,\, ,$$ equivalently,
the pairing $g_{ab}$ is invariant with respect to the multiplication $\circ$
defined by
$A_{ab}^c$.
\item {(3)} (Identity) $$\forall a,b \,\,\,\,\,\,A_{0a}^b=\delta^b_a$$
equivalently
$\p_0$ is an identity of
the algebra $\bold H\otimes\C[[x_{\bold H}]]$
\item {(4)} (Potential) $$\forall a,b,c,d\,\,\,\,\,\, \p_d
A^c_{ab}=(-1)^{\bar a\bar d}\p_a
A^c_{db}\,\,\,\, ,$$ which implies, assuming (1a) and (2), that the
series $A_{abc}$ are the third derivatives of a single power series
$\Phi\in \bold H\otimes
\C[[x_{\bold H}]]$
$$A_{abc}=\p_a\p_b\p_c\Phi$$
\section{2. Moduli space via deformation functor}
The material presented in this section is standard (see [K2] and references
therein).
Let us remind the definition of the
functor ${\Def_\gtg}$
associated with a differential graded Lie algebra $\goth g$ .
It acts from the category of Artin algebras to the category of
sets.
Let $\goth A$ be an Artin algebra with the maximal ideal denoted by $\gt m$.
Define the set $${\Def_\gtg}(\gt A):=\{d\g+{[\g,\g]\over 2}=0 |\g
\in (\gt g\otimes\gt m)^1\}/\Gamma_{\gt A}^0$$
where the quotient is taken modulo action
of the group $\Gamma_{\gt A}^0$ corresponding to the nilpotent Lie algebra
$(\gt g\otimes\gt m)^0$. The action of the group can be described via the
infinitesimal action of its
Lie algebra: $$\alpha\in \gt g\otimes\gt m \to \dot \g=d\alpha +[\g,\alpha]$$
Sometimes functor $\Def_\gtg$ is represented by
some topological algebra
$\Cal O_{\Cal M_\gtg}$ (projective limit of Artin algebras) in the sense that
the functor
$\Def_\gtg$ is equivalent to the functor
$\text{Hom}_{continuous}(\Cal O_{\Cal M_\gtg},\,\cdot\,)$.
For example, $H^0(\gtg)=0$ is a sufficient condition for this.
If $\Def_\gtg$ is representable then one can associate formal moduli space to
$\gtg$ by defining
the ``algebra of functions''
on the formal moduli space to be the \break algebra $\Cal O_{\Cal M_\gtg}$.
We will need the $\Z$-graded extension of the functor $\Def_\gtg$.
The definition
of $\Def^{\,\Z}_\gtg$ is obtained from the definition of
$\Def_\gtg$ via inserting
$\Z-$graded Artin algebras instead of the usual ones everywhere.
A sufficient and probably necessary condition for the functor
$\Def^{\,\Z}_\gtg$ to be representable is that $\gtg$ must be
quasi-isomorphic to an abelian graded Lie algebra.
We will see in \S 2.1 that this is the case for $\gtg=\bt$.
Hence one can associate formal (graded) moduli space \footnote{We will
omit the superscript $\Z$ where it does not seem to lead to a confusion.}
$\Cal M_\bt$
to the Lie algebra $\gtg$.
\subhead {2.1 Extended moduli space of complex structure}
\endsubhead
Let $M$ be a connected compact complex manifold of dimension $n$,
with vanishing
$1$-st
Chern class $c_1(T_M)=0\in \text{Pic}(M)$. We assume that there exists a
K\"ahler metric on $M$, although we will not fix it. By Yau's theorem there
exists a Calabi-Yau metric on $M$.
It follows from the condition $c_1(T_M)=0$ there exists an everywhere
nonvanishing holomorphic volume form $\Omega\in \Gamma(X,\Lambda^n T_M^*)$.
It is defined up to a multiplication by a constant. Let us fix a choice of
$\om$.
It induces isomorphism
of cohomology groups: $H^q(M,\Lambda^pT_M)\simeq H^q(M,\Omega^{n-p})$;
$\g\mapsto \gamma\vdash\om$.
Define differential $\dv$ of degree $-1$ on $\bt$ by the formula :
$$(\dv\g)\vdash\om=\p(\g\vdash\om)$$
The operator $\dv$ satisfies the following identity (Tian-Todorov lemma) :
$$ [\g_1,\g_2]=(-1)^{\text{deg}\g_1+1}(\dv(\g_1\wedge\g_2) -
(\dv \g_1)\wedge\g_2 -
(-1)^{\text{deg}\g_1+1}\g_1\wedge\dv\g_2)\eqno (2.1)$$
where $\text{deg}\g=p+q-1$ for $\g\in\Gamma(M,\Lambda^q\bar
T^*_M\otimes\Lambda^q
T_M)$.
Denote by $\bold H$ the total homology space of
$\dv$ acting on $\bt[1]$. Let $\{\dl_a\}$ denote a graded basis in the
vector space $\oplus_{p,q}H^q(M,\Lambda^pT_M)$,
$\dl_0=1\in H^0(M,\Lambda^0T_M)$ .
Let us redefine the degree
of $\dl_a$ as follows
$$|\dl_a|:=p+q-2\,\,\,\,\,\text{for}\, \,\,\,\,\,\dl_a\in
H^q(M,\Lambda^pT_M)$$
Then $\{\dl_a\}$ form a graded basis in $\bold H$.
Denote
by $\{t^a\}, t^a\in \bold H^*,\,\text{deg}\,t^a=-|\dl_a|$ the basis
dual to $\{\dl_a\}$. Denote by
$\C[[t_{\bold H}]]$ the algebra of formal power series on
$\Z$-graded vector space $\bold H$.
\proclaim{Lemma 2.1} The functor $\Def^{\,\Z}_\bt$ associated with
$\bt$ is canonically
equivalent to the functor represented by the
algebra $\C[[t_{\bold H}]]$.
\endproclaim
\proof
It follows from (2.1) that the maps $$(\bt,\db)\gets(\Ker\dv,\db)\to (\bold
H[-1],d=0)\eqno (2.2)$$ are morphisms of differential graded Lie
algebras. Then the
$\p\db$-lemma, which says that
$$\Ker\db\cap \Ker\dv\cap(\im\dv\oplus\im\db)=\im\dv\circ\db,\eqno (2.3) $$
shows that these
morphisms are
quasi-isomorphisms (this argument is standard in the theory of minimal models,
see [DGMS]).
Hence (see e.g. theorem in \S 4.4 of [K2]) the deformation
functors associated with the three differential graded Lie algebras are
canonically equivalent. The deformation functor associated with trivial algebra
$(\bold H[-1],d=0)$ is represented by the algebra $\C[[t_{\bold H}]]$.
\endproof
\corollary 2.2 The moduli space $\Cal M_\bt$ associated to
$\bt$ is smooth. The
dimension of $\Cal M_\bt$ is equal to
$\sum_{p,q}\text{dim }H^q(M,\Lambda^pT_M)$ of the dimension of the space
of first order deformations associated with $\bt$.\endproclaim
\Remark The Formality theorem proven in [K2] implies
that the differential graded Lie algebra controlling the
$A_\infty$-deformations
of $\Cal D^bCoh(M)$ is quasi-isomorphic to $\bt$. Here we have proved that
$\bt$ is quasi-isomorphic to an abelian graded Lie algebra. Therefore,
the two differential graded Lie algebras are formal,
i.e. quasi-isomorphic to their cohomology Lie algebras
endowed with zero differential.\endremark
\corollary 2.3 There exists a solution to the Maurer-Cartan equation
$$\db{\hat\g}(t)+{[{\hat\g}(t),{\hat\g}(t)]\over
2}=0\eqno (2.4)$$
in formal power series with values in $\bt$
$${\hat\g}(t)=\sum_a\hat\g_at^a+{1\over
{2!}}\sum_{a_1,a_2}\hat\g_{a_1a_2}t^{a_1}t^{a_2} +\ldots\in (\bt\otimes\bold
C[[t_{\bold H}]])^1$$ such that
the cohomology classes $[\hat\g_a]$ form a basis of cohomology of the
complex
$(\bt,\db)$\endproclaim
\Remark The deformations of the complex structure are controlled by the
differential graded Lie algebra
$$\bt_{(0)}:=\bigoplus_k \bt_{(0)}^k, \,\,\bt_{(0)}^k=
\Gamma(M,\Lambda^{k}\overline{T}^*_M\otimes T_M)$$ The
meaning of this is that
the completion of the algebra of functions on the moduli space of complex
structures on $M$ represents $\Def_{\bt_{(0)}}$ (or $\Def_{\bt_{(0)}}^{\,\Z}$
restricted to the category of Artin algebras concentrated in degree
$0$). The natural embeddings $\bt_{(0)}\hookrightarrow \bt$ induces
embedding of the corresponding moduli spaces. In terms of the solutions
to Maurer-Cartan equation the deformations of complex
structure are singled out
by the condition $\g(t)\in
\Gamma(M,\Lambda^1\overline{T}^*_M\otimes\Lambda^1T_M)$.\endremark
\Remark Similar thickening of the moduli space of complex structures were
considered by Z\.~Ran in [R].\endremark
\section{3. Algebra structure on the tangent sheaf of the moduli space }
Let $R$ denotes a $\Z$-graded Artin algebra over $\C$, $\g^R
\in (\bt\otimes R)^1$ denotes
a solution to the Maurer-Cartan equation (2.4).
The linear extension of the wedge product gives a structure of graded
commutative algebra on $\bt\otimes R[-1]$. Let $\g^R$ be a solution to the
Maurer-Cartan equation in $(\bt\otimes R)^1$. It defines a differential
$\db_{\g^R}=\db+\{\g^R,\cdot\}$ on $\bt\otimes R[1]$. Denote the cohomology of
$\db_{\g^R}$ by $T_{\g^R}$.
The space of first order variations of $\g^R$ modulo gauge equivalence is
identified with $T_{\g^R}$. Geometrically one can think of $\g^R$ as
a morphism
from the formal variety corresponding to algebra $R$ to the formal moduli
space. An element of $T_{\g^R}$ corresponds to a section of the
preimage of the
tangent sheaf.
Note that
$\db_{\g^R}$ acts as a differentiation of the (super)commutative $R-$algebra
$t\otimes R[-1]$.
Therefore $T_{\g^R}[-2]$ inherits the structure of (super)commutative
associative
algebra over $R$. This structure is functorial with respect to the morphisms
$\phi_*: T_{\g^{R_1}}\to T_{\g^{R_2}}$ induced by homomorphisms
$\phi : R_1\to R_2$.
Let ${\hat\g}(t)\in (\bt\widehat\otimes\C[[t_{\bold H}]])^1$ be a
solution to the
Maurer-Cartan satisfying the condition of corollary 2.3.
It follows from this condition
that the $\C[[t_{\bold H}]]$-module $T_{{\hat\g}(t)}$ is freely generated
by the classes of partial
derivatives $[\p_a{\hat\g}(t)]$.
Therefore we have
\proclaim{Proposition 3.1} There exists formal power series $A^c_{ab}(t)\in
\bt\widehat\otimes\C[[t_{\bold H}]]$ satisfying
$$ \p_a{\hat\g}\wedge\p_b{\hat\g}=\sum_c A^c_{ab}\p_c{\hat\g}\,\,\text{mod}
\,\db_{{\hat\g}(t)}$$
The series $A_{ab}^c(t)$ are the structure constants of the commutative
associative \break $\C[[t_{\bold H}]]$@-algebra
structure on $\bold H\otimes\C[[t_{\bold H}]][-2]$.
\endproclaim
\qed
\Remark Note that on the tangent space at zero this algebra structure
is given by the obvious multiplication on $\oplus_{p,q}H^q(M,\Lambda^pT_M)$.
This is "Mirror dual" to the ordinary multiplication on
$\oplus_{p,q}H^q(M,\om^p_{\widetilde M})$.
\endremark
\section{4. Integral}
Introduce linear functional on $\bt$
$$\int \g:=\int_M (\g\vdash\om)\wedge \om$$
\claim 4.1 It satisfies the following identyties:
$$\aligned\int\db\g_1\wedge\g_2=(-1)^{\text{deg}\g_1}\int\g_1\wedge\db\g_2\\
\int\dv\g_1\wedge\g_2=(-1)^{\text{deg}\g_1+1}\int\g_1\wedge\dv\g_2
\endaligned\eqno (4.1)$$
for $\g_i\in \Gamma(M,\Lambda^{q_i}\overline T^*_M\otimes\Lambda^{p_i}T_M),
\,\, i=1,2$ where $\text{deg}\,\g_i=p_i+q_i-1$ .\endproclaim\qed
\section{5. Metric on $\Cal{T_M}$}
There exists a natural metric (i.e. a nondegenerate (super)symmetric
$\Cal O_{M_\bt}$-linear pairing) on the sheaf $\Cal T_{\Cal M_\bt}$.
In terms of a solution to the Maurer-Cartan equation $\g^R
\in (\bt\otimes R)^1$ it
means that there exists an $R$-linear graded symmetric pairing on $T_{\g^R}$,
which is functorial with respect to $R$. Here $T_{\g^R}$
denotes the cohomology of $\db_{\g^R}$ defined in \S 3.
The pairing is defined by the formula
$$\langle h_1,h_2\rangle:=\int h_1\wedge h_2\,\,\,\text{for}\,\,h_1,h_2\in
T_{\g^R} $$
where we assumed for simplicity that $\g^R\in \Ker\dv\otimes R$.
It follows from (2.2) (see also lemma 6.1) that such a choice of $\g^R$ in the
given class of gauge equivalence is always possible.
\claim 5.1 The pairing is compatible with the algebra
structure.\endproclaim
\noindent{\qed}
\section{6. Flat coordinates on moduli space.}
Another ingredient in the definition of Frobenius structure is the choice
of affine structure on the moduli space associated with $\bt$.
The lemma 2.1 identifies $\Cal M_{\bt}$ with the moduli space associated
with trivial algebra $(\bold H[-1],d=0)$. The latter moduli space is
the affine space $\bold H$. The affine coordinates $\{t_a\}$ on $\bold H$
give coordinates on $\Cal M_{\bt}$. This choice of coordinates on the
moduli space
corresponds to a specific
choice of a universal solution to the Maurer-Cartan equation over
$\C[[t_{\bold H}]]$.
\proclaim{Lemma 6.1} There exists a solution to the Maurer-Cartan equation in
formal power series with values in $\bt$
$$\db{\hat\g}(t)+{[{\hat\g}(t),{\hat\g}(t)]\over
2}=0,\,{\hat\g}(t)=\sum_a\hat\g_at^a+{1\over
{2!}}\sum_{a_1,a_2}\hat\g_{a_1a_2}t^{a_1}t^{a_2} +\ldots\in
(\bt\widehat\otimes\C[[t_{\bold H}]])^1$$ such that
\roster
\item (Universality) the cohomology classes $[\hat\g_a]$ form a basis of
cohomology
of the
complex
$(\bt,\db)$
\item (Flat coordinates) $\hat\g_a\in\Ker\dv,\,\,\hat\g_{a_1\ldots a_k}\in
\im\dv\,\,\,
\text{for}\,\,\, k\geq 2$
\item (Flat identity) $\p_0\hat\g(t)=\bold 1$, where $\p_0$ is the coordinate
vector field
corresponding to $[\bold 1]\in \bold H[-1]$
\endroster
\endproclaim
\proof The theorem of \S 4.4 in [K2] shows that there exists $L_\infty$
morphism $f$
homotopy inverse to the natural morphism $(\Ker\dv,\db)\to \bold H[-1]$
(for the definition of $L_\infty$-morphism see \S 4.3 in [K2]).
Put $\Delta(t)=\sum_a(\Delta_a[-1]) t^a$ , where $\Delta_a[-1]$ denotes the
element $\dl_a$ having degree shifted by one. Then
$${\g}(t)=\sum_n{1\over{n!}}f_n(\Delta(t)\wedge\dots\wedge\Delta(t))$$
satisfies
the conditions $(1)-(2)$. To fulfill the condition $(3)$ $\g(t)$ must be
improved slightly. Define the differential graded Lie algebra
$\widetilde{\Ker}$
as follows
\roster
\item $\widetilde{\Ker}_i=\Ker \dv\subset \bt_i\,\text{for}\,i\geq 0$
\item $\widetilde{\Ker}_{-1}=\im \dv\subset \bt_{-1}$
\endroster
Note that the algebra $\Ker \dv$ is the sum of the algebra
$\widetilde{\Ker}$ and trivial algebra of constants $\C\otimes\bold 1[-1]$.
Let
$\tilde f$ be a homotopy inverse to the natural quasi-isomorphism
$\widetilde{\Ker}\to \bold H[-1]_{\geq 0}$. Put $\tilde\Delta(t)=\sum_{a\neq
0}(\Delta_a[-1]) t^a$. Then
$$\hat{\g}(t)=\bold 1t_0 +\sum_n{1\over{n!}}\tilde
f_n(\tilde\Delta(t)\wedge\dots\wedge\tilde\Delta(t))$$ satisfies all
the conditions.
\endproof
\Remark Any two formal power series satisfying conditions of lemma 6.1 are
equivalent under the natural action of the group associated with the \break Lie
algebra
$(\widetilde\Ker \dv\widehat\otimes\C[[t_{\bold H}]])^0$.
\endremark
\Remark It is possible to write down an explicit formula for the components of
the morphism $f$ in terms of Green functions of the Laplace operator acting
on differential forms on $M$.
\endremark
\Remark After the identification of the moduli space $\Cal M_\bt$ with
$\bold H$, provided by lemma 2.1, the complex structure moduli space
corresponds to the \break subspace $\bold H^1(M,\Lambda^1T_M)$.
In the case of classical moduli space of the complex structures on $M$
the analogous lemma was proved in [T]. The coordinates arising on
the classical moduli
space of complex structures coincide with so called
"special" coordinates of [BCOV].\endremark
\remark{Notation}
Denote $${\hat\g}(t)=\sum_a\hat\g_at^a+{1\over
{2!}}\sum_{a_1,a_2}\hat\g_{a_1a_2}t^{a_1}t^{a_2} +\ldots\,\in (\bt\otimes\bold
C[[t_{\bold H}]])^1$$
a solution to the Maurer-Cartan equation satisfying conditions of lemma 6.1.
\endremark
The parameters of a miniversal solution to the Maurer-Cartan equation
over $\C[[t_{\bold H}]]$ serve as coordinates on the moduli space. The
specific choice of coordinates corresponding to the solution to the
Maurer-Cartan equation satisfying conditions (1)-(2) of lemma 6.1 corresponds
to choice
of coordinates on moduli space that are flat with
respect to the natural (holomorphic) metric $g_{ab}$.
\claim 6.2 The power series $\langle
\p_a{\hat\g}(t),\p_b{\hat\g}(t)\rangle\in\C[[t_{\bold H}]]$ has only
constant term in the power series expansion at $t=0$.\endproclaim
\proof $\langle x,y\rangle=0$
for $x\in \Ker\dv, y\in \im\dv$.\endproof
\remark{Notation}
Denote $g_{ab}:=\langle \p_a{\hat\g}(t),\p_b{\hat\g}(t)\rangle$.
\endremark
Thus we have constructed all the ingredients of the Frobenius structure on $\Cal
M_{\bt}$:
the tensors $A^c_{ab}(t)$, $g_{ab}$ and the coordinates $\{t_a\}$ that are flat
with
respect
to $g_{ab}$.
\Remark The 3-tensor $A_{ab}^c(t)$ on $\Cal M_\bt$ does not depend on the choice
of $\om$. The 2-tensor $g_{ab}$ is multiplied by $\lambda^2$ when $\om$ is
replaced by
$\lambda\om$\endremark
\claim 6.3 The structure constants satisfy $A_{0a}^b=\delta^b_a$.\endproclaim
\proof It follows from the condition (3) imposed on $\hat\g(t)$\endproof
We have checked that the tensors $A^c_{ab}(t),\,g_{ab}$ have the properties
(1)-(3)
from the definition of the Frobenius structure. It remains to us to check the
\break property (4).
\section{ 7. Flat connection and periods}
Let ${\hat\g}(t)\in (\bt\otimes\bold
C[[t_{\bold H}]])^1$ be
a solution to the Maurer-Cartan equation satisfying conditions (1)-(2) of lemma
6.1.
Then the formula
$$\om(t):=e^{\hat\g(t)}\vdash\om $$ defines a closed form of mixed degree
depending formally on $t\in \bold H$.
For $t\in \bold H^{-1,1}$ $\hat\g(t)\in \Gamma(M,\overline T\otimes T^*)$
represents a
deformation of complex
structure. Then $\om(t)$ is
a holomorphic $n$-form in the complex structure corresponding to $t\in \bold
H^{-1,1}$,
where $n=\text{dim}_\C\, M$.
Let $\{p^a\}$ denote the set of sections of
$\Cal T^*_{\bold H}$ that form a framing dual to $\{\p_a\}$.
Define a (formal) connection on $\Cal T^*_{\bold H}$ by the covariant
derivatives:
$$\nabla_{\p_a}(p^c):=\sum_b A_{ab}^c p^b \eqno (7.1)$$
Strictly speaking this covariant derivatives are formal power series
sections of $\Cal T^*_{\bold H}$.
Let us put
$$\Pi_{ai}={\p\over\p t^a}\int_{\Gamma_i} \om(t) \eqno
(7.2)$$
where $\{\Gamma_i\}$ form a basis in $\bold H_*(M,\C)$. In particular
$$\Pi_{0i}=\int_{\Gamma_i} \om(t)$$
if $\hat\g(t)$ satisfies the condition (3) of lemma 6.1.
\lemma 7.1 The periods
$\Pi_{i}=\sum_{a}\Pi_{ai} p^a$ are flat sections of $\nabla$\endproclaim
\noindent{\it Proof.\,}Let $\p_\ta=\sum_a \ta^a \p_a$ be an even constant vector
field, i.e. $\ta^a$ are even constants for even $\p_a$ and odd for odd $\p_a$.
It is enough to prove that
$$\p_\ta\p_\ta\int_{\Gamma_i}\om(t)=\sum_c A^c_{\ta\ta}\p_c \int_{\Gamma_i}
\om(t)$$
where $A^c_{\ta\ta}$ are the algebra structure constants defined via
$\sum_a\ta^a\p_a
\circ\sum_a\ta^a\p_a=\sum_c
A^c_{\ta\ta}\p_c$ (see \S 1).
Note that the operators $\dv$ and $\db_{\hat\g(t)}$ acting on
$\bt\widehat\otimes\C[[t_{\bold H}]]$
satisfy a version of $\p\db$-lemma :
$$\Ker
\dv\cap\Ker\db_{\hat\g(t)}\cap(\im\dv\oplus\im\db_{\hat\g(t)})=
\im\dv\circ\db_{\hat\g(t)}\eqno(7.3)$$
Equivalently, there exists decomposition of
$\bt\widehat\otimes\C[[t_{\bold H}]]=X_0\oplus X_1\oplus
X_2\oplus X_3\oplus Y$ into direct sum of
graded vector spaces, such that
the only nonzero components of $\db_{\hat\g(t)}$, $\dv$ are isomorphisms
$\db_{\hat\g(t)}:X_0\mapsto X_1,\,\,\,X_2\mapsto X_3;\,\,\,\,\,
\dv:X_0\mapsto X_2,\,\,\,X_1\mapsto X_3$.
Differentiating twice the Maurer-Cartan equation (2.4)
with respect to $\p_\ta$ and using (2.1) one obtaines
$$\dv (\p_{\ta}\hat\g(t)\wedge\p_{\ta}\hat\g(t)-\sum_c
A_{\ta\ta}^c\p_c\hat\g(t))=-\db_{\hat\g(t)}\p_{\ta}\p_{\ta}\hat\g(t) \eqno
(7.4)$$
It follows from $\p\db$-lemma for $\dv$, $\db_{\hat\g(t)}$ and the equation
(7.4)
that there exist formal power series
$\alpha_{\ta}(t)\in \bt\widehat\otimes
\C[[t_{\bold H}]]$ such that
$$\eqalign{&\p_{\ta}\hat\g(t)\wedge\p_{\ta}\hat\g(t)-\sum_c
A_{\ta\ta}^c\p_c\hat\g(t)=\db_{\hat\g(t)}\alpha_{\ta}(t),\cr &
\p_{\ta}\p_{\ta}\hat\g(t)=\dv(\alpha_{\ta}(t)) \cr}$$
Therefore
$$\p_{\ta}\p_{\ta}\om(t)=\sum_c
A_{\ta\ta}^c{\p_c}\om(t) +
d(\alpha_{\ta}e^{\hat\g(t)}\vdash\om)$$\qed
It follows from the condition (1) imposed on $\hat\g(t)$ that $\Pi_i$ form
a (formal) framing of $\Cal T^*_{\bold H}$.
\corollary 7.2 The connection $\nabla$ is flat\endproclaim
\claim 7.3 The structure constants $A^c_{ab}$ satisfy the potentiality condition
(4)
in flat coordinates.\endproclaim
\proof
If one puts symbolically $\nabla=\nabla_0 + A$ then the flatness of $\nabla$
implies that
$$\nabla_0 A+{1\over 2}[A,A]=0$$
Notice that
associativity and commutativity of the algebra defined by $A^c_{ab}$ imply
that $$[A,A]=0 .$$
Therefore
$$\nabla_0(A)=0. $$
\endproof
We have completed the proof of the fact that $A^c_{ab}(t)$ and $g_{ab}$
define the Frobenius structure on $\Cal M_\bt$ in the flat coordinates
$\{t_a\}$.
\Remark In fact one can write an explicit formula for the potential of the
Frobenius structure. Let us put ${\hat\g}(t)=\sum_a\hat\g_a t^a+\dv\alpha(t),
\alpha(t)\in (\bt\widehat\otimes t_{\bold H}^2\C[[t_{\bold H}]])^0$. Put
$$\Phi=\int
-{1\over 2}\db\alpha\wedge\dv\alpha+{1\over
6}{\hat\g}\wedge{\hat\g}\wedge{\hat\g}$$
Then one checks easily that $A_{abc}=\p_a\p_b\p_c\Phi$ (see Appendix).
In the case $\text{dim}_\C M=3,\, \hat \g \in
\bt^{-1,1}=\Gamma(M,\overline{T}^*_M\otimes T_M)$
this formula gives the critical value of so called Kodaira-Spencer Lagrangian of
[BCOV].
\endremark
\Remark Define differential Batalin-Vilkovisky algebra as $\Z_2$-graded
commutative associative
algebra $A$ equipped with odd differentiation $\db,\,\, \db^2=0$ and odd
differential operator $\dv$ of order $\leq 2$ such that
$\dv^2=0,\,\,\,\dv\db+\db\dv=0,\,\,\,\dv(1)=0$. One can use the formula (2.1)
to define the structure of $\Z_2$-graded
Lie algebra on $\Pi A$. Assume that the operators $\db,\dv$ satisfy
$\p\db$-lemma (2.3).
Assume in addition that $A$ is equipped with a linear functional
$\int:A\to \C$
satisfying (4.1) such that the metric defined as in \S 5 is nodegenerate. Then
the same construction as above produces the Frobenius
structure
on the $\Z_2$-graded moduli space $\Cal M_{\Pi A}$. One can define
the tensor product of two such Batalin-Vilkovisky algebras.
Operator $\dv$ on $A_1\otimes A_2$ is given by
$\dv_1\otimes 1+1\otimes\dv_2$. Also, $\db$ on $A_1\otimes A_2$ is
$\db_1\otimes 1+1\otimes\db_2$.
It is naturally to expect that the Frobenius manifold corresponding to
$A_1\otimes A_2$
is equal to the tensor product of Frobenius manifolds corresponding to
$A_1,A_2$, defined in [KM] in terms of the corresponding algebras over operad
$\{H_*(\overline
M_{0,n+1})\}$.
\endremark
\section{8. Scaling transformations}
The vector field $E=\sum_{a}-{1\over 2}|\dl_a|t^a\p_a$ generates the
scaling symmetry on $\bold H$.
\proclaim{Proposition 8.1} ${\Cal Lie}_EA_{abc}=({1\over
2}(|\dl_a|+|\dl_b|+|\dl_c|)+3-\text{dim}_\C M)A_{abc}$
\endproclaim
\proof $A_{abc}=\int_M \p_{a}{\hat\g}\wedge\p_{b}{\hat\g}\wedge\p_{c}{\hat\g}$.
Note that
$\int \g\neq 0$ implies that $\g\in \bt_{2n-1}$. The proposition follows from
the grading condition on $\hat\g(t)$.
\endproof
\corollary 8.2 ${\Cal Lie}_EA_{ab}^c=({1\over
2}(|\dl_a|+|\dl_b|-|\dl_c|)+1)A_{ab}^c$
\endproclaim
Note that the vector field $E$ is conformal with respect to the metric
$g_{ab}$.
Therefore
the proposition 8.~1 shows that $E$ is the Euler vector field of the Frobenius
structure on $\bold H$ (see [M]). Such a vector field is defined uniquely up to
a multiplication by a constant.
The simplest invariant of
Frobenius manifolds is the spectrum
of the operator $[E,\cdot]$ acting on infinitesimal generators of translations
and the weight of the
tensor $A_{ab}^c$ under the ${\Cal Lie}_E$-action.
Usually
the normalization of $E$ is chosen so that $[E,\p_0]=1$.
In our case the spectrum of $[E,\cdot]$ is equal to $$\bigcup_d \{1-d/2\}
\,\,\text{\eightrm with multiplicity}\sum_{q-p=d-n}\text{dim}H^q(M,\Omega^p)$$
Note that this spectrum and the weight of $A_{ab}^c$ coincide identically
with the corresponding quantities of the Frobenius structure arising from the
Gromov-Witten invariants of the dual Calabi-Yau manifold $\widetilde M$.
\section {9.Further developments.}
Conjecturally the constructed Frobenius manifold is related to the Gromov-Witten
invariants of
$\widetilde M$
in the following way. One can rephrase the present construction
in purely algebraic terms using \v Cech instead of Dolbeault realization of the
simplicial
graded Lie algebra $\Lambda^*\Cal T_M$. The only
additional ``antiholomorphic'' ingredient that is used is the choice of a
filtration
on $H^*(M,\C)$ complementary to the Hodge filtration. The Frobenius structure,
arising from the limiting weight filtration corresponding to a point with
maximal
unipotent
monodromy on moduli space of complex structures on M,
coincides conjecturally with
Frobenius structure on $H^*(\widetilde M,\C)$, obtained from the
Gromov-Witten invariants.
Our construction of Frobenius manifold is a particular case of
a more general construction. Other cases of this construction include
the Frobenius manifold structure on the moduli space of singularities of
analytic
functions found by K.\,Saito, the Frobenius manifold structure on the
moduli space of ``exponents of algebraic functions''. The latter case is a
Mirror Symmetry
partner to the structure arising from Gromov-Witten invariants on Fano
varieties.
All these cases seem to be related with yet undiscovered generalization of
theory of
Hodge structures.
We hope to return to this in the next paper.
\specialhead Appendix \endspecialhead
Let ${\hat\g}(t)=\sum_a\hat\g_a t^a+\dv\alpha(t), \alpha(t)\in
({\bold t}\otimes t_{\bold H}^2{\bold C}[[t_{\bold H}]])^1$ be
a solution to Maurer-Cartan equation satisfying conditions (1)-(2) of lemma
6\.1.
Put $$\Phi=\int -{1\over 2}\db\alpha\wedge\dv\alpha+{1\over
6}{\hat\g}\wedge{\hat\g}\wedge{\hat\g}$$
\proclaim{Proposition } $A_{abc}=\p_a\p_b\p_c\Phi$
\endproclaim
\proof. Let $\p_{\ta}=\sum_a \ta_a\p_a$ be an even constant vector
field in $\bold H$. It is enough to prove
that $\int
(\p_{\ta}{\hat\g}\wedge\p_{\ta}{\hat\g}\wedge{\p_{\ta}\hat\g})=
\p^3_{\ta\ta\ta}\Phi$. Let us differentiate the terms in $\Phi$
$$\eqalign{&\p^3_{\ta\ta\ta}({\hat\g}\wedge\hat\g\wedge\hat\g)=
18\p^2_{\ta\ta}{\hat\g}
\wedge\p_{\ta}{\hat\g}\wedge{\hat\g}+
3\p^3_{\ta\ta\ta}{\hat\g}\wedge{\hat\g}\wedge\hat\g+
6\p_{\ta}{\hat\g}\wedge\p_{\ta}{\hat\g}
\wedge\p_{\ta}{\hat\g} \cr}$$
$$\eqalign{&\p^3_{\ta\ta\ta}(\db\alpha\wedge\dv\alpha)=
\db\alpha\wedge(\p^3_{\ta\ta\ta}\dv\alpha)+
(\p^3_{\ta\ta\ta}\db\alpha)\wedge\dv\alpha
+3(\p_{\ta}\db\alpha)\wedge(\p^2_{\ta\ta}\dv\alpha)+\cr
&+3(\p^2_{\ta\ta}\db\alpha)\wedge(\p_{\ta}\dv\alpha)\cr}\eqno(*)$$
Notice that
$$\eqalign{&\int(\p^3_{\ta\ta\ta}\db\alpha)\wedge\dv\alpha=
(-1)^{\text{deg}\db\alpha+1}\int(\p^3_{\ta\ta\ta}\dv\db\alpha)\wedge\alpha=
\int(\p^3_{\ta\ta\ta}\dv\db\alpha)\wedge\alpha=\cr
&=-\int(\p^3_{\ta\ta\ta}\db\dv\alpha)\wedge\alpha=
-(-1)^{\text{deg}\dv\alpha}\int(\p^3_{\ta\ta\ta}\dv\alpha)\wedge\db\alpha=
\int(\p^3_{\ta\ta\ta}\dv\alpha)\wedge\db\alpha=\cr
&=(-1)^{(\text{deg}\dv\alpha+1)(\text{deg}\db\alpha+1)}\int
\db\alpha\wedge(\p^3_{\ta\ta\ta}\dv\alpha)=
\int\db\alpha\wedge(\p^3_{\ta\ta\ta}\dv\alpha)\cr}$$
\noindent Hence, the first two terms in (*) give the same contribution.
\noindent We have $$\eqalign{&\int\db\alpha\wedge(\p^3_{\ta\ta\ta}\dv\alpha)=
(-1)^{\text{deg}\db\alpha+1}\int\dv\db\alpha\wedge(\p^3_{\ta\ta\ta}\alpha)
=\int\dv\db\alpha\wedge(\p^3_{\ta\ta\ta}\alpha)=\cr
&=-\int\db\dv\alpha\wedge(\p^3_{\ta\ta\ta}\alpha)=
-\int\db\hat\g\wedge(\p^3_{\ta\ta\ta}\alpha)=
{1\over 2}\int[\hat\g,\hat\g]\wedge(\p^3_{\ta\ta\ta}\alpha)=\cr
&={1\over 2}\int\dv(\hat\g\wedge\hat\g)\wedge(\p^3_{\ta\ta\ta}\alpha)=
(-1)^{(\text{deg}(\hat\g\wedge\hat\g)+1)}{1\over
2}\int(\hat\g\wedge\hat\g)\wedge(\p^3_{\ta\ta\ta}\dv\alpha)=\cr
&={1\over 2}\int(\hat\g\wedge\hat\g)\wedge\p^3_{\ta\ta\ta}\hat\g\cr}$$
\noindent Similarly,$$\eqalign{
\int(\p_{\ta}\db\alpha)\wedge(\p^2_{\ta\ta}\dv\alpha)&={1\over 2}\int
\p_{\ta}(\hat\g\wedge\hat\g)\wedge\p^2_{\ta\ta}\hat\g \cr
\int(\p^2_{\ta\ta}\db\alpha)\wedge(\p_{\ta}\dv\alpha)&={1\over 2}\int
(\p^2_{\ta\ta}\hat\g)\wedge\p_{\ta}(\hat\g\wedge\hat\g) \cr
}$$
\endproof
\Refs
\widestnumber\key{DGMS}
\ref\key BCOV\by M.Bershadsky, S.Cecotti, H.Ooguri, C.Vafa \paper
Kodaira-Spencer theory of gravity and exact results for quantum string
amplitudes \jour Comm.Math.Phys.\vol 164\yr 1994\pages 311--428\endref
\ref\key DGMS \by P.Deligne, Ph.Griffiths, J.Morgan, D.Sullivan\paper Real
homotopy
theory of K\"ahler \break manifolds\jour Inventiones Math.\vol 29\yr 1975\pages
245--274
\endref
\ref\key D\by B.Dubrovin \paper\nofrills Geometry of 2d topological field
theories;\inbook LNM 1620\publ Springer\yr 1996\break\pages 120@-348\endref
\ref\key K1 \by M.Kontsevich \paper Homological algebra of Mirror Symmetry
\jour Proccedings of the
International Congress of Mathematicians\vol I \pages 120-139\yr 1994 \publ
Birkh\"auser \publaddr Z\"urich\endref
\ref\key K2\bysame\paper Deformation quantization of Poisson manifolds, \rm I
\jour q-alg/9709040\endref
\ref\key KM \by M.Kontsevich, Yu.I.Manin \paper Gromov-Witten classes,
quantum cohomology, and enumerative geometry\jour Comm.Math.Phys.\vol 164
\yr 1994 \pages 525--562\endref
\ref\key M \by Yu.I.Manin \paper Frobenius manifolds, quantum cohomology and
moduli spaces {\rm I,II,III} \break \paperinfo Preprint MPI 96-113\publ
Max-Planck-Institut f\"ur Mathematik \endref
\ref\key T \by A.Todorov\paper
The Weil-Petersson geometry of the moduli space of $su(n\geq 3)$
(Calabi-Yau) manifolds \rm I
\jour Comm.Math.Phys.\vol 126 \yr 1989\endref
\ref\key R \by Z.Ran\paper\nofrills Thickening Calabi-Yau moduli spaces;\inbook
{\sl in} Mirror Symmetry \rm II\eds B.R.Greene, S.Yau\publ AMS/IP International
Press
\bookinfo Studies in advanced mathematics\yr 1997 \pages 393--400
\endref
\endRefs
\enddocument
\end
|
1998-03-12T13:54:01 | 9710 | alg-geom/9710008 | en | https://arxiv.org/abs/alg-geom/9710008 | [
"alg-geom",
"math.AG"
] | alg-geom/9710008 | Wolfgang Ebeling | Wolfgang Ebeling and Sabir M. Gusein-Zade | On the index of a vector field at an isolated singularity | AMS-LaTeX, 11 p. with 1 fig.; remarks, definition, and references
added to Section 1 | null | null | Dept. of Math., University of Hannover, Preprint No. 285 | null | We consider manifolds with isolated singularities, i.e., topological spaces
which are manifolds (say, $C^\infty$--) outside discrete subsets (sets of
singular points). For (germs of) manifolds with, so called, cone--like
singularities, a notion of the index of an isolated singular point of a vector
field is introduced. There is given a formula for the index of a gradient
vector field on a (real) isolated complete intersection singularity. The
formula is in terms of signatures of certain quadratic forms on the
corresponding spaces of thimbles.
| [
{
"version": "v1",
"created": "Tue, 7 Oct 1997 07:51:44 GMT"
},
{
"version": "v2",
"created": "Thu, 12 Mar 1998 12:53:58 GMT"
}
] | 2007-05-23T00:00:00 | [
[
"Ebeling",
"Wolfgang",
""
],
[
"Gusein-Zade",
"Sabir M.",
""
]
] | alg-geom | \section*{Introduction}\label{sec0}
An isolated singular point of a vector field on ${\Bbb{R}}^n$ or on an
$n$-dimensional smooth manifold has a natural integer invariant~---
the index. The formula of Eisenbud, Levine and Khimshiashvili (\cite{EL},
\cite{Kh}) expresses the index of an (algebraically) isolated
singular point of a vector field as the signature of a quadratic form
on a local algebra associated with the singular point. For a
singular point of a gradient vector field there is a formula in terms
of signatures of certain quadratic forms defined by the action of the
complex conjugation on the corresponding Milnor lattice (\cite{GZ},
\cite{V}).
We define a generalisation of the notion of the index of an isolated
singular point of a vector field on a manifold with
isolated cone-like singularities in such a way that the Poincar\'e--Hopf
theorem (the sum of indices of singular points of a vector field on a
closed manifold is equal to its Euler characteristic) holds. It seems
that this notion (though very natural) cannot be found in the
literature in an explicit form. In particular, the index is defined for
vector fields on a germ of a real algebraic variety with an isolated
singularity. We give a generalisation of a formula from
\cite{GZ} for the gradient vector field on an isolated complete
intersection singularity. For that we define (in a somewhat formal way)
the notion of the variation operator for a complete intersection
singularity. We show that this operator is an invariant of the
singularity.
\section{Basic definitions}\label{sec1}
A {\em manifold with isolated singularities} is a topological
space $M$ which has the structure of a smooth (say, $C^\infty$--) manifold
outside of a discrete set $S$ (the {\em set of singular points} of $M$).
A {\em diffeomorphism} between two such manifolds is a homeomorphism
which sends the set of singular points onto the set of singular points and
is a diffeomorphism outside of them. We say that $M$ has a {\em
cone-like singularity} at a (singular) point $P\in S$ if there exists a
neighbourhood of the point $P$ diffeomorphic to the cone over a smooth
manifold $W_P$ ($W_P$ is called the {\em link} of the point $P$). In
what follows we assume all manifolds to have only cone-like
singularities. A (smooth or continuous) {\em vector field} on a
manifold $M$ with isolated singularities is a (smooth or continuous) vector
field on the set $M\setminus S$ of regular points of $M$. The {\em set
of singular points} $S_X$ of a vector field $X$ on a (singular) manifold
$M$ is the union of the set of usual singular points of $X$ on $M\setminus
S$ (i.e., points at which $X$ tends to zero) and of the set $S$ of singular
points of $M$ itself.
For an isolated {\em usual} singular point $P$ of a vector field $X$
there is defined its index $\mbox{ind}_PX$ (the degree of the map
$X/\Vert X\Vert:\partial B\to S^{n-1}$ of the boundary of a small ball $B$
centred at the point $P$ in a coordinate neighbourhood of $P$;
$n=\mbox{dim}\,M$). If the manifold $M$ is closed and has no singularities
($S=\emptyset$) and the vector field $X$ on $M$ has only isolated
singularities, then
\begin{equation}\label{eq1}
\sum_{P\in S_X}\mbox{ind}_PX=\chi(M)
\end{equation}
($\chi(M)$ is the Euler characteristic of $M$).
Let $(M, P)$ be a cone-like singularity (i.e., a germ of a manifold
with such a singular point) and let $X$ be a vector field defined on an
open neighbourhood $U$ of the point $P$. Suppose that $X$ has no singular
points on $U\setminus\{P\}$. Let $V$ be a closed cone--like neighbourhood of
$P$ in $U$ ($V\cong CW_P$, $V\subset U$). On the cone $CW_P=(I\times
W_P)/(\{0\}\times W_P)$ ($I=[0, 1]$) there is defined a natural vector field
$\partial/\partial t$ ($t$ is the coordinate on $I$). Let $X_{rad}$ be the
corresponding vector field on $V$. Let $\widetilde X$ be a smooth vector
field on $U$ which coincides with $X$ near the boundary $\partial U$ of
the neighbourhood $U$ and with $X_{rad}$ on $V$ and has only isolated
singular points.
\begin{definition} The {\em index} $\mbox{ind}_PX$ of the vector field $X$
at the point $P$ is equal to
$$1+\sum_{Q\in S_{\widetilde X}\setminus\{P\}}\mbox{ind}_Q\widetilde X$$
(the sum is over all singular points $Q$ of $\widetilde X$ except $P$ itself).
\end{definition}
For a cone-like singularity at a point $P\in S$, the link $W_P$ and thus
the cone structure of a neighbourhood are, generally speaking, not
well-defined (cones over different manifolds may be {\em locally}
diffeomorphic). However it is not difficult to show
that the index $\mbox{ind}_PX$ does not depend on the choice of a cone
structure on a neighbourhood and on the choice of the vector field
$\widetilde X$.
\begin{example} The index of the ``radial" vector field $X_{rad}$ is equal
to $1$. The index of the vector field $(-X_{rad})$ is equal to $1-\chi(W_P)$
where $W_P$ is the link of the singular point $P$.
\end{example}
\begin{proposition}\label{prop1}
For a vector field $X$ with isolated singular points on a closed manifold $M$
with isolated singularities, the relation {\rm(\ref{eq1})} holds.
\end{proposition}
\begin{definition} One says that a singular point $P$ of a manifold $M$
(locally diffeomorphic to the cone $CW_P$ over a manifold $W_P$) is
{\em smoothable} if $W_P$ is the boundary of a smooth compact manifold.
\end{definition}
The class of smoothable singularities includes, in particular, the class of
(real) isolated complete intersection singularities. For such a singularity,
there is a distinguished cone--like structure on its neighbourhood.
Let $(M, P)$ be a smoothable singularity (i.e., a germ of a manifold
with such a singular point) and let $X$ be a vector field on $(M, P)$
with an isolated singular point at $P$. Let $V=CW_P$ be a closed
cone--like neighbourhood of the point $P$; $X$ is supposed to have no singular
points on $V\setminus\{P\}$. Let the link $W_P$ of the point $P$ be the
boundary of a compact manifold $\widetilde V_P$. Using a smoothing one can
consider the union
$\widetilde V_P\cup_{W_P}(W_P\times[1/2, 1])$ of $\widetilde V_P$ and
$W_P\times[1/2, 1]\subset CW_P$ with the natural identification of
$\partial\widetilde V_P=W_P$ with $W_P\times\{1/2\}$ as a smooth manifold
(with the boundary $W_P\times\{1\}$). The restriction of the vector field
$X$ to $W_P\times[1/2, 1]\subset CW_P$ can be extended to a smooth vector
field $\widetilde X$ on $\widetilde V_P\cup_{W_P}(W_P\times[1/2, 1])$ with
isolated singular points.
\begin{proposition}\label{prop2}
The index $\mbox{ind}_PX$ of the vector field $X$ at the
point $P$ is equal to
$$\sum_{Q\in S_{\widetilde X}}\mbox{ind}_Q\widetilde X -
\chi(\widetilde V_P)+1$$
(the sum is over all singular points of $\widetilde X$ on $\widetilde
V_P$).
\end{proposition}
\begin{remark} In \cite{S}, \cite{GSV} there was defined a notion of the
index of a vector field at an isolated singular point of a complex variety
(satisfying some conditions). That definition does not coincide with the
one given here. These definitions differ by the Euler characteristic of the
smoothing of the singularity of the variety. One can say that the
index of \cite{S}, \cite{GSV} depends on the Euler characteristic of
a smoothing and thus is well-defined only for a singularity with
well-defined topological type of a smoothing (at least with well-defined
Euler characteristic of it). It is valid, e.g., for {\em complex}
isolated complete intersection singularities.
A closely related notion has been discussed in \cite{BG}. That
notion can be considered as a relative version of the index defined here.
After a
previous version of this paper had been submitted and put on the Duke
preprint server
as alg-geom/9710008, the authors' attention was drawn to the preprint
\cite{ASV},
where a somewhat more general notion is defined, which coincides with the index
considered here for real analytic varieties with isolated singularities.
\end{remark}
A generic (smooth or continuous) vector field
on a (singular) analytic variety has zeroes only at isolated points. Thus
it is desirable to have a definition of the index of such
a point. One can use the following definition.
Let $(V,0)\subset ({\Bbb{R}}^N, 0)$ be a germ of a real algebraic variety
and let $X$ be a continuous vector field on $(V,0)$ (i.e., the
restriction of a continuous vector field on $({\Bbb{R}}^N, 0)$ tangent to
$V$ at each point) which has an isolated zero at the origin (in $V$).
Let ${\EuScript S}=\{\Xi\}$ be a semianalytic Whitney stratification of $V$
such that its only zero-dimensional stratum $\Xi^0$ consists of the origin.
Let $\Xi$ be a stratum of the stratification ${\EuScript S}$ and let $Q$ be a point
of $\Xi$.
A neighbourhood of the point $Q$ in $V$ is diffeomorphic to the direct
product of a linear space ${\Bbb{R}}^k$ (the dimension $k$ of which is equal to the
dimension of the stratum $\Xi$) and the cone $CW_Q$ over a compact
singular analytic variety $W_Q$. (A diffeomorphism between two
stratified spaces is a homeomorphism which is a diffeomorphism on
each stratum.) In particular a neighbourhood $U(0)$ of the origin is
diffeomorphic to the cone $CW_0$ over a singular variety $W_0$.
It is not difficult to show that there exists a
(continuous) vector field $\widetilde X$ on $(V, 0)$
such that:
\begin{enumerate}
\item the vector field $\widetilde X$ is defined on the neighbourhood
$U(0)\cong CW_0$ of the origin;
\item $\widetilde X$ coincides with the vector field $X$ in a
neighbourhood of the base $\{1\}\times W_0$ of the cone $CW_0$;
\item the vector field $\widetilde X$ has only a finite number of zeroes;
\item each point $Q\in U(0)$ with $\widetilde X(Q)=0$ has a neighbourhood
diffeomorphic to $({\Bbb{R}}^k,0)\times CW_Q$ in which $\widetilde X(y,z)$
($y\in {\Bbb{R}}^k$, $z\in CW_q$)
is of the form $Y(y)+Z_{rad}(z)$, where $Y$ is a germ of a vector
field on $({\Bbb{R}}^k,0)$ with an isolated singular point at the origin,
$Z_{rad}$ is the radial vector field on the cone $CW_Q$.
\end{enumerate}
Let $S_{\widetilde X}$ be the set of zeroes of the vector field
$\widetilde X$ (including the origin). For a point $Q\in S_{\widetilde X}$,
let $\widetilde{ind}(Q):=ind_0 Y$, where $Y$ is the vector field on $({\Bbb{R}}^k, 0)$
described above. We define $ind(0)$ to be equal to $1$ (in this case
$k=0$).
\begin{definition}
$ind_{(V,0)}X=\sum\limits_{Q\in S_{\widetilde X}}
\widetilde{ind}(Q)$.
\end{definition}
\section{On the topology of isolated complete intersection
singularities}\label{sec2}
Let $(V,0) \subset ({\Bbb{C}}^{n+p},0)$ be an $(n-1)$-dimensional isolated complete
intersection singularity (abbreviated {\em icis} in the sequel) defined by a
germ of an analytic mapping
$$f=(f_1, \ldots , f_{p+1}): ({\Bbb{C}}^{n+p},0) \to ({\Bbb{C}}^{p+1},0).$$
(We use somewhat strange notations for the dimension and the number of
equations in order to be consistent with the notations in
Section~\ref{sec3}.) For $\delta > 0$, let
$B_\delta$ be the ball of radius
$\delta$ around the origin in ${\Bbb{C}}^{n+p}$. For $\delta > 0$ small enough and
for a generic $t \in
{\Bbb{C}}^{p+1}$ with $0< \| t \| << \delta$, the set $$V_t = f^{-1}(t) \cap
B_\delta$$ is a manifold with boundary and is called a {\em Milnor fibre} of
the {\em icis}
$(V,0)$ (or of the germ $f$). The diffeomorphism type of $V_t$ does not depend
on $t$. The manifold $V_t$ is homotopy equivalent to the bouquet of $\mu$
spheres of dimension $(n-1)$, where $\mu$ is the Milnor number of the
{\em icis}
$(V,0)$.
For $t \in {\Bbb{C}}^{p+1}$, $0 \leq i \leq p$ we define
\begin{eqnarray*}
(V^{(i)},0) & := & (\{ x \in B_\delta: f_1(x)= \ldots = f_{p-i+1}(x) =
0 \},0), \\
V^{(i)}_t & := & \{ x \in B_\delta: f_j(x)=t_j, 1 \leq j \leq p-i+1 \}
\end{eqnarray*}
and we set
$(V^{(p+1)}, 0) := ({\Bbb{C}}^{n+p}, 0)$.
We assume that $(f_1, \ldots , f_{p+1})$
is a system of functions such that for $0\leq i \leq p $ the germ
$(V^{(i)},0)$ is an $(n+i-1)$-dimensional {\em icis}. For any $t \in
{\Bbb{C}}^{p+1}$ with $0 < | t_1 | << |t_2| << \ldots << |t_{p+1} | << \delta$, the
set $V^{(i)}_t$ is the Milnor fibre of the {\em icis} $(V^{(i)},0)$. Here the
condition $0 < | t_1 | << |t_2| << \ldots << |t_{p+1} |$ means that
$t_1, \ldots ,t_{p+1}$ have to be chosen in such a way that for each $i$,
$1 \leq i \leq p+1$, all the critical values of the function $f_i$ on
$V^{(p-i+2)}_t$ are contained in the disc of radius $|t_i|$ around $0$.
We put
\begin{eqnarray*}
\hat{H}^{(i)} & := & H_{n+i}(V^{(i+1)}_t,V^{(i)}_t)
\quad \mbox{for} \ 0 \leq i \leq p-1 , \\
\hat{H}^{(p)} & := & H_{n+p}(B_\delta, V^{(p)}_t).
\end{eqnarray*}
We have short exact sequences (cf.\ \cite{Eb}, \cite{AGV}):
$$
\begin{array}{ccccccccc}
0 & \rightarrow & H_n(V'_t) & \rightarrow & \hat{H} &
\rightarrow & H_{n-1}(V_t) & \rightarrow & 0 \\
0 & \rightarrow & H_{n+1}(V^{(2)}_t) & \rightarrow &
\hat{H}' & \rightarrow & H_n(V'_t) & \rightarrow & 0 \\
& & \vdots & & \vdots & & \vdots & & \\
0 & \rightarrow & H_{n+p-1}(V^{(p)}_t) & \rightarrow &
\hat{H}^{(p-1)} &
\rightarrow & H_{n+p-2}(V^{(p-1)}_t) & \rightarrow & 0 \\
& & 0 & \rightarrow & \hat{H}^{(p)} & \rightarrow
& H_{n+p-1}(V^{(p)}_t) & \rightarrow & 0
\end{array}
$$
They give rise to a long exact sequence (cf.\ \cite[p.~163]{AGV})
$$ 0 \rightarrow \hat{H}^{(p)} \rightarrow \hat{H}^{(p-1)} \rightarrow \ldots
\rightarrow \hat{H}' \rightarrow \hat{H} \rightarrow H_{n-1}(V_t)
\rightarrow 0 .$$
Each of the modules in this sequence is a free ${\Bbb{Z}}$-module of finite
rank. Let
$\nu_i:= {\rm rank}\, \hat{H}^{(i)}$. Then
$$\mu = {\rm rank}\, H_{n-1}(V_t) = \sum_{i=0}^{p} (-1)^i \nu_i.$$
On each of the modules we have an intersection form
$\langle\cdot,\,\cdot\rangle$ defined as
in \cite{Eb}. On the module $H_{n+i-1}(V^{(i)}_t)$ ($0\le i\le p$) it is
the usual intersection form; on the module $\hat{H}^{(i)}$ it is the
pullback of the intersection form by the natural (boundary) homomorphism
$\hat{H}^{(i)}\to H_{n+i-1}(V^{(i)}_t)$. The form on
$H_{n-1}(V_t)$ is symmetric if $n$ is odd and skew-symmetric if
$n$ is even. The form on $\hat{H}^{(i)}$ is symmetric if $n+i$ is odd and
skew-symmetric if $n+i$ is even.
Denote by $\hat{H}^\ast$ the dual module of $\hat{H}=H_n(V'_t,V_t)$ and
let $(\cdot,\,\cdot) : \hat{H}^\ast \times \hat{H} \to {\Bbb{Z}}$ be the
Kronecker pairing. We want to define a variation operator
${\rm Var}: \hat{H}^\ast \to
\hat{H}$ or rather its inverse ${\rm Var}^{-1} : \hat{H} \to \hat{H}^\ast$. For this
purpose we need the notion of a distinguished basis of thimbles.
Let $\tilde{f}_{p+1}:V'_t\to {\Bbb{C}}$ be a generic perturbation of the
restriction of the function $f_{p+1}$ to $V'_t$ which has only non-degenerate
critical points with different critical values $z_1, \ldots , z_\nu$
($\nu=\nu_0$). Let $z_0$ be a non-critical value of $\tilde{f}_{p+1}$ with
$\| z_0 \| > \| z_j\|$ for $j=1, \ldots , \nu$. The level set
$\{ x\in V'_t : \tilde{f}_{p+1}(x) = z_0 \}$ is diffeomorphic to the Milnor
fibre $V_t$ of the {\em icis} $(V,0)$. Let $u_j$, $j=1, \ldots , \nu$, be
non-self-intersecting paths connecting the critical values $z_j$ with the
non-critical value $z_0$ in such a way that they lie inside the disc $D_{\| z_0
\|} = \{ z \in {\Bbb{C}} : \| z \| \leq \| z_0 \| \}$ and every two of them intersect
each other only at the point $z_0$. We suppose that the paths $u_j$ (and
correspondingly the critical values $z_j$) are numbered clockwise according to
the order in which they arrive at $z_0$ starting from the boundary of the disc
$D_{\| z_0 \|}$. Each path $u_j$ defines up to orientation a thimble
$\hat{\delta}_j$ in the relative homology group $\hat{H}$. The system $\{ \hat{\delta}_1 ,
\ldots , \hat{\delta}_\nu \}$ is a basis of $\hat{H}$. A basis obtained in this way is
called {\em distinguished}. The self-intersection number of a thimble $\hat{\delta}$ is
equal to
$$\langle \hat{\delta}, \hat{\delta} \rangle = (-1)^{n(n-1)/2}(1+(-1)^{n-1}).$$
The {\em Picard-Lefschetz transformation} $h_{\hat{\delta}}:\hat{H} \to \hat{H}$
corresponding to the thimble
$\hat{\delta}$ is given by (cf.\ \cite{Eb})
$$h_{\hat{\delta}} (y) = y + (-1)^{n(n+1)/2}\langle y,\hat{\delta} \rangle \hat{\delta} \quad {\rm for}
\ y \in \hat{H}.$$
Going once around the disc $D_{\| z_0 \|}$ in the positive direction
(counterclockwise) along the boundary induces an automorphism of $\hat{H}$,
the {\em (classical) monodromy operator} $h_\ast$. If $\{ \hat{\delta}_1,
\ldots, \hat{\delta}_\nu\}$ is a distinguished basis of $\hat{H}$, then the
monodromy operator is given by
$$h_\ast = h_{\hat{\delta}_1} \circ h_{\hat{\delta}_2} \circ \cdots \circ h_{\hat{\delta}_\nu}.$$
\begin{definition} Let $\{ \hat{\delta}_1, \ldots , \hat{\delta}_\nu \}$ be a
distinguished basis of thimbles of $\hat{H}$ and let $\{ \nabla_1, \ldots ,
\nabla_\nu \}$ be the corresponding dual basis of $\hat{H}^\ast$. The
linear operator ${\rm Var}^{-1} : \hat{H} \to \hat{H}^\ast$ (inverse of the {\em
variation operator}) is defined by
$$ {\rm Var}^{-1}(\hat{\delta}_i) = (-1)^{n(n+1)/2} \nabla_i - \sum_{j < i} \langle \hat{\delta}_i,
\hat{\delta}_j \rangle \nabla_j.$$
\end{definition}
\begin{proposition}
The definition of the operator ${\rm Var}^{-1}$ does not depend on the choice of the
distinguished basis.
\end{proposition}
\begin{proof}
Any two distinguished bases of thimbles can be transformed into each
other by the braid group transformations $\alpha_j$, $j=1, \ldots ,
\nu-1$, and by changes of orientations (see, e.g., \cite{AGV},
\cite{Eb}). Here the operation $\alpha_j$ is defined as follows:
$$ \alpha_j (\hat{\delta}_1, \ldots , \hat{\delta}_\nu) = (\hat{\delta}'_1, \ldots ,
\hat{\delta}'_\nu)$$ where $\hat{\delta}'_j = h_{\hat{\delta}_j}(\hat{\delta}_{j+1})= \hat{\delta}_{j+1} +
(-1)^{n(n+1)/2}\langle \hat{\delta}_{j+1},\hat{\delta}_{j} \rangle \hat{\delta}_j$,
$\hat{\delta}'_{j+1} = \hat{\delta}_j$, and
$\hat{\delta}'_i = \hat{\delta}_i$ for $i \neq j, j+1$.
It is easily seen that the definition of ${\rm Var}^{-1}$ is invariant under a change of
orientation. Therefore it suffices to show that the definition of ${\rm Var}^{-1}$ is
invariant under the transformation $\alpha_j$. One easily computes:
\begin{eqnarray*}
\langle \hat{\delta}'_r ,\hat{\delta}'_s \rangle & = & \langle \hat{\delta}_r, \hat{\delta}_s \rangle \quad
\mbox{for} \ 1 \leq r,s \leq \nu, \ r,s \neq j,j+1, \\
\langle \hat{\delta}'_j ,\hat{\delta}'_{j+1} \rangle & = & -\langle \hat{\delta}_j, \hat{\delta}_{j+1}
\rangle , \\
\langle \hat{\delta}'_r ,\hat{\delta}'_j \rangle & = & \langle \hat{\delta}_r,
\hat{\delta}_{j+1} \rangle +(-1)^{n(n+1)/2}\langle \hat{\delta}_{j+1}, \hat{\delta}_j \rangle \langle
\hat{\delta}_r, \hat{\delta}_j \rangle \ \mbox{ for} \ r \neq j,j+1, \\
\langle \hat{\delta}'_r ,\hat{\delta}'_{j+1} \rangle & = & \langle \hat{\delta}_r, \hat{\delta}_j \rangle
\quad \mbox{for} \ r \neq j,j+1.
\end{eqnarray*}
Let $(\nabla'_1, \ldots , \nabla'_\nu)$ be the dual basis corresponding
to $(\hat{\delta}'_1, \ldots ,\hat{\delta}'_\nu)$.
Then
\begin{eqnarray*}
\nabla_{j+1} & = & \nabla'_j \\
\nabla_{j} & = & \nabla'_{j+1} + (-1)^{n(n+1)/2} \langle \hat{\delta}_{j+1} ,
\hat{\delta}_j \rangle \nabla'_{j}.
\end{eqnarray*}
One has
\begin{eqnarray*}
{\rm Var}^{-1}(\hat{\delta}'_j) & = &
{\rm Var}^{-1}(\hat{\delta}_{j+1}) + (-1)^{n(n+1)/2} \langle \hat{\delta}_{j+1} , \hat{\delta}_j
\rangle {\rm Var}^{-1}(\hat{\delta}_j) \\
& = & (-1)^{n(n+1)/2} \nabla_{j+1} - \sum_{k<j+1} \langle \hat{\delta}_{j+1} ,
\hat{\delta}_k \rangle \nabla_k \\
& & + (-1)^{n(n+1)/2}\langle \hat{\delta}_{j+1} , \hat{\delta}_j \rangle
((-1)^{n(n+1)/2}\nabla_j - \sum_{k<j} \langle \hat{\delta}_j , \hat{\delta}_k \rangle
\nabla_k ) \\
& = & (-1)^{n(n+1)/2}\nabla_{j+1} \\
& & - \sum_{k<j} (\langle \hat{\delta}_{j+1} ,
\hat{\delta}_k \rangle + (-1)^{n(n+1)/2} \langle \hat{\delta}_{j+1} , \hat{\delta}_j \rangle
\langle \hat{\delta}_j , \hat{\delta}_k \rangle ) \nabla_k \\
& = & (-1)^{n(n+1)/2}\nabla'_j - \sum_{k<j} \langle \hat{\delta}'_j ,\hat{\delta}'_k
\rangle \nabla'_k,
\end{eqnarray*}
\begin{eqnarray*}
{\rm Var}^{-1}(\hat{\delta}'_{j+1}) & = &
(-1)^{n(n+1)/2}\nabla_j - \sum_{k<j} \langle \hat{\delta}_j , \hat{\delta}_k
\rangle \nabla_k \\
& = & (-1)^{n(n+1)/2}(\nabla_j - (-1)^{n(n+1)/2}\langle \hat{\delta}_{j+1} ,
\hat{\delta}_j \rangle \nabla_{j+1} ) \\
& & + \langle \hat{\delta}_{j+1} ,
\hat{\delta}_j \rangle \nabla_{j+1} - \sum_{k<j} \langle \hat{\delta}_j , \hat{\delta}_k
\rangle \nabla_k \\
& = & (-1)^{n(n+1)/2}\nabla'_{j+1} - \sum_{k<j+1} \langle \hat{\delta}'_{j+1}
, \hat{\delta}'_k \rangle \nabla'_k .
\end{eqnarray*}
For $i>j+1$ we have
\begin{eqnarray*}
{\rm Var}^{-1}(\hat{\delta}'_i) & = &
(-1)^{n(n+1)/2}\nabla_i - \sum_{k<i} \langle \hat{\delta}_i , \hat{\delta}_k
\rangle \nabla_k \\
& = & (-1)^{n(n+1)/2}\nabla'_i - \hspace{-1.5mm}
\sum\limits_{\substack{k<i \\ k \neq j,j+1}} \langle \hat{\delta}'_i
, \hat{\delta}'_k \rangle \nabla'_k -\langle
\hat{\delta}_i , \hat{\delta}_{j+1} \rangle \nabla_{j+1} - \langle \hat{\delta}_i, \hat{\delta}_j
\rangle \nabla_j \\
& = & (-1)^{n(n+1)/2}\nabla'_i - \hspace{-1.5mm}
\sum\limits_{\substack{k<i \\ k \neq j,j+1}} \! \! \langle \hat{\delta}'_i ,
\hat{\delta}'_k
\rangle \nabla'_k \\ & & - \langle \hat{\delta}'_i, \hat{\delta}'_j \rangle \nabla'_j +
(-1)^{n(n+1)/2}\langle \hat{\delta}_{j+1} , \hat{\delta}_j \rangle \langle \hat{\delta}_i ,
\hat{\delta}_j \rangle \nabla'_j \\
& & - \langle \hat{\delta}'_i , \hat{\delta}'_{j+1} \rangle
\nabla'_{j+1} - (-1)^{n(n+1)/2}\langle \hat{\delta}_{j+1} , \hat{\delta}_j \rangle
\langle \hat{\delta}'_i , \hat{\delta}'_{j+1} \rangle \nabla'_j \\
& = & (-1)^{n(n+1)/2}\nabla'_i - \sum_{k<i} \langle
\hat{\delta}'_i , \hat{\delta}'_k \rangle \nabla'_k
\end{eqnarray*}
The corresponding formula for $i<j$ is obvious.
\end{proof}
\begin{remark}
There is an interesting problem to give an invariant
(topological) definition of the variation operator.
\end{remark}
Let $S: \hat{H} \to \hat{H}^\ast$ be the mapping defined by the
intersection form on $\hat{H}$: $(Sx, y) =\langle x , y \rangle$,
$x,\,y\in \hat{H}$. The mapping ${\rm Var}^{-1}$ is defined
in such a way that one has the equality $S = -{\rm Var}^{-1} + (-1)^n ({\rm Var}^{-1})^T$
where $({\rm Var}^{-1})^T$ means the transpose operator $({\rm Var}^{-1})^T :
\hat{H}^{\ast\ast}= \hat{H} \to \hat{H}^\ast$.
\begin{remark}
We emphasize that the intersection number $\langle \hat{\delta}_i , \hat{\delta}_j
\rangle$ is the entry of the matrix of the operator $S$ with {\em
column} index $i$ and {\em row} index $j$: see the remark in
\cite[p.~45]{AGV}.
\end{remark}
\begin{remark}
The operator $V: \hat{H}\to \hat{H}^\ast$ defined in
\cite[p.~18]{Eb} differs from ${\rm Var}^{-1}$ by sign.
\end{remark}
\begin{proposition}
The monodromy operator $h_\ast$ and the operator ${\rm Var}^{-1}$ are related by the
following formula:
$$h_\ast = (-1)^n {\rm Var} ({\rm Var}^{-1})^T. $$
\end{proposition}
\begin{proof}
This can be computed directly; see also \cite[Chap.~V, \S 6,
Exercice 3]{B}.
\end{proof}
In the same way, for each $i$ with $1 \leq i \leq p$ an operator
${\rm Var}^{-1}_i : \hat{H}^{(i)} \to (\hat{H}^{(i)})^\ast$ is defined.
\section{The index of the gradient vector field on an isolated
complete intersection singularity}\label{sec3}
Let $(V', 0)=\{f_1=f_2=\ldots=f_p=0\}\subset({\Bbb{C}}^{n+p}, 0)$ be a real
$n$-dimensional
{\em icis} (it means that the function germs $f_i:({\Bbb{C}}^{n+p},
0)\to({\Bbb{C}}, 0)$ are real). We assume that its real part $V'\cap{\Bbb{R}}^{n+p}$
does not coincide with the origin (and thus is $n$-dimensional). Let
$g=f_{p+1}:({\Bbb{C}}^{n+p}, 0)\to({\Bbb{C}}, 0)$ be a germ of a real analytic function
such that its restriction to $V'\setminus\{0\}$ has no critical points. A
Riemannian metric on ${\Bbb{R}}^{n+p}$ determines the gradient vector field
$X=\mbox{grad}\,g$ of the restriction of the function $g$ to
$(V'\cap{\Bbb{R}}^{n+p})\setminus\{0\}$. This vector field has no singular
points on a punctured neighbourhood of the origin in
$V'\cap{\Bbb{R}}^{n+p}$. Since the space of Riemannian metrics is connected,
the index \,$\mbox{ind}_0\,X$ of the gradient vector field doesn't
depend on the choice of a metric. In the case $p=0$ (and thus
$V'={\Bbb{C}}^{n}$) the index of the gradient vector field of a function
germ $g$ can be expressed in terms of the action of the complex
conjugation on the Milnor lattice of the singularity $g$ (\cite{GZ},
\cite{V}). We give a generalisation of such a formula for {\em icis}.
Let $0<\varepsilon_1\ll\varepsilon_2\ll\ldots\ll\varepsilon_{p+1}$ be real and small enough, let
$s=(s_1,\,\ldots,\, s_{p+1})$ with $s_i=\pm 1$. For $0\le i\le p$, let
$\hat{H}^{(i)}=\hat{H}^{(i)}_{s\varepsilon}$ be the corresponding space of thimbles:
$\hat{H}^{(i)}=H_{n+i}(V^{(i+1)}_{s\varepsilon}, V^{(i)}_{s\varepsilon})$ for $0\le i\le
p-1$ ($s\varepsilon=(s_1\varepsilon_1, s_2\varepsilon_2, \ldots, s_{p+1}\varepsilon_{p+1}))$; see
Section~\ref{sec2} for $i=p$. Let $\sigma^{(i)}_s$ be the action of the
complex conjugation on the space $\hat{H}^{(i)}$, let
${\rm Var}^{-1}_i:\hat{H}^{(i)}\to (\hat{H}^{(i)})^\ast$
be the inverse of the corresponding variation operator. The operator
${\rm Var}^{-1}_i\sigma^{(i)}_s$ acts from the space $\hat{H}^{(i)}$ to its dual
$(\hat{H}^{(i)})^\ast$ and thus defines a bilinear form on $\hat{H}^{(i)}$.
\begin{theorem}\label{theo1}
The bilinear forms ${\rm Var}^{-1}_i\sigma^{(i)}_s$ are symmetric and
non-degener\-ate, and we have
\begin{eqnarray}\label{eq2}
{\rm ind}_0\,{\rm grad}\,g & = & s_{p+1}^{n}(-1)^{\frac{n(n+1)}{2}}
{\rm sgn}\,{\rm Var}^{-1}_{}\sigma^{}_s \nonumber \\
& & +\sum\limits_{i=1}^p(-1)^{\frac{(n+i)(n+i+1)}{2}}{\rm
sgn}\,{\rm Var}^{-1}_{i}\sigma^{(i)}_s.
\end{eqnarray}
\end{theorem}
\begin{corollary}
The right-hand side of the equation {\rm (\ref{eq2})} does not depend on
$s=(s_1,\,\ldots,\, s_{p+1})$.
\end{corollary}
\begin{proof} Let us consider the restriction of the function $f_i$ to the
manifold $V^{(p-i+2)}_{s\varepsilon}$. It may have degenerate critical points. Let
$\widetilde f_i:V^{(p-i+2)}_{s\varepsilon}\to{\Bbb{C}}$ be its real morsification (i.e., a
perturbation of $f_i$ which is a Morse function on $V^{(p-i+2)}_{s\varepsilon}$ and
maps its real part $V^{(p-i+2)}_{s\varepsilon}\cap{\Bbb{R}}^{n+p}$ to ${\Bbb{R}}$). For
$c\in{\Bbb{R}}$, let $M_c^{(i)}=\{x\in
V^{(p-i+2)}_{s\varepsilon}\cap{\Bbb{R}}^{n+p}:\widetilde f_i\le c\}$. The topological
space $M_c^{(i)}$ is homotopy equivalent to
$V^{(p-i+2)}_{s\varepsilon}\cap{\Bbb{R}}^{n+p}$ or to
$V^{(p-i+1)}_{(s_1\varepsilon_1,\ldots, s_{i-1}\varepsilon_{i-1}, -\varepsilon_i)}\cap{\Bbb{R}}^{n+p}$
for $c$ greater than or less than all the critical values of $\widetilde f_i$
respectively. The standard arguments of Morse theory give
$$
\chi(V^{(p-i+2)}_{s\varepsilon}\cap{\Bbb{R}}^{n+p})=\chi(V^{(p-i+1)}_{(s_1\varepsilon_1,\ldots,
s_{i-1}\varepsilon_{i-1}, -\varepsilon_i)}\cap{\Bbb{R}}^{n+p}) + \sum\limits_{Q\in
S_{{\rm grad}\,\widetilde f_i}}{\rm ind}_Q\,{\rm grad}\,\widetilde f_i.
$$
Applying the same reasonings to the function $-\widetilde f_i$ one has
\begin{eqnarray*}
\chi(V^{(p-i+2)}_{s\varepsilon}\cap{\Bbb{R}}^{n+p}) & =
& \chi(V^{(p-i+1)}_{(s_1\varepsilon_1,\ldots, s_{i-1}\varepsilon_{i-1},
\varepsilon_i)}\cap{\Bbb{R}}^{n+p}) \\
& & + (-1)^{n+p-i+1}\sum\limits_{Q\in
S_{{\rm grad}\,\widetilde f_i}}{\rm ind}_Q\,{\rm grad}\,\widetilde f_i.
\end{eqnarray*}
Thus
$$
\chi(V^{(p-i+2)}_{s\varepsilon}\cap{\Bbb{R}}^{n+p})=\chi(V^{(p-i+1)}_{s\varepsilon}\cap{\Bbb{R}}^{n+p}) +
(-s_i)^{n+p-i+1}
\sum\limits_{Q\in
S_{{\rm grad}\,\widetilde f_i}}{\rm ind}_Q\,{\rm grad}\,\widetilde f_i.
$$
(cf.\ \cite[Lemma in \S 2]{A}). From Proposition 1 one has
\begin{eqnarray*}
\lefteqn{{\rm ind}_{0}{\rm grad}\, g} \\
& & =\hspace{-3mm}\sum\limits_{Q\in
S_{{\rm grad}\,\widetilde f_{p+1}}}\hspace{-3mm}{\rm ind}_Q{\rm
grad}\,\widetilde
f_{p+1} - \chi(V'_{s\varepsilon}\cap{\Bbb{R}}^{n+p}) + 1 \\
& & =\hspace{-3mm}\sum\limits_{ Q\in S_{{\rm grad}\,\widetilde
f_{p+1}}}\hspace{-4.5mm}{\rm ind}_Q{\rm grad}\,\widetilde f_{p+1}
+ (-s_p)^{(n+1)}\hspace{-3mm}\sum\limits_{Q\in S_{{\rm grad}\,\widetilde
f_{p}}}\hspace{-3.5mm}{\rm ind}_Q{\rm grad}\,\widetilde f_{p}
- \chi(V''_{s\varepsilon}\cap{\Bbb{R}}^{n+p}) + 1 \\
& & = \ldots =\\
& & = \hspace{-3mm}\sum\limits_{Q\in S_{{\rm grad}\,\widetilde
f_{p+1}}} \hspace{-3mm}{\rm ind}_Q{\rm grad}\,\widetilde f_{p+1}
+ \sum\limits_{i=1}^p(-s_{p-i+1})^{(n+i)}\hspace{-3mm}\sum\limits_{Q\in
S_{{\rm grad}\,\widetilde f_{p-i+1}}}\hspace{-3mm}{\rm ind}_Q{\rm
grad}\,\widetilde
f_{p-i+1}.
\end{eqnarray*}
Now Theorem~\ref{theo1} follows from the following statement.
\end{proof}
\begin{theorem}\label{theo2}
$$
\sum\limits_{Q\in S_{{\rm grad}\,\widetilde f_{p-i+1}}}
{\rm ind}_Q{\rm grad}\,\widetilde f_{p-i+1}
=(s_{p-i+1})^{n+i}(-1)^{\frac{(n+i)(n+i+1)}{2}}{\rm
sgn}\,{\rm Var}^{-1}_i\sigma^{(i)}_s.
$$
\end{theorem}
\begin{proof}
Let us suppose that $s_{p-i+1} =1$. The case $s_{p-i+1}=-1$ can be
reduced to this by multiplying the function $f_{p-i+1}$ (and the function
$\widetilde{f}_{p-i+1}$) by $(-1)$. Let $\widetilde{s} = (s_1, \ldots ,
s_{p-i}, -1, s_{p-i+2}, \ldots, s_{p+1})$. Without loss of generality we
can suppose that all critical values of the function
$\widetilde{f}_{p-i+1}$ lie inside the circle
$\{z: \| z \|\le\frac{\varepsilon_{p-i+1}}{2} \}$ and have different real parts
(except, of course, values at complex conjugate points). Let us identify
the space
$\hat{H}^{(i)}_{\widetilde{s}}$ with the space $\hat{H}^{(i)}_s$ using a
path which connects $-\varepsilon_{p-i+1}$ with $+\varepsilon_{p-i+1}$ in the upper
half plane outside the circle
$\{z: \| z\| < \frac{\varepsilon_{p-i+1}}{2} \}$ (e.g., the half circle
$\{z: \| z\| = \varepsilon_{p-i+1}\}$). This identification permits to consider
$\sigma_s^{(i)}$ and
$\sigma_{\widetilde{s}}^{(i)}$ as operators on the space $\hat{H}^{(i)}
=\hat{H}^{(i)}_{s\varepsilon}$. Just as in \cite{GZ} the classical monodromy
operator $h_{\ast}^{(i)}: \hat{H}^{(i)} \to \hat{H}^{(i)}$ can be
represented in the form
\begin{equation}\label{eq3}
h_{\ast}^{(i)} = \sigma^{(i)}_s \sigma^{(i)}_{\widetilde{s}}.
\end{equation}
A distinguished basis of the space $\hat{H}^{(i)}$ is defined by a
system of paths connecting the critical values of the function
$\widetilde{f}_{p-i+1}$ with the non-critical value $\varepsilon_{p-i+1}$. Let
us choose the following system of paths (cf.\ Fig.~\ref{fig1}).
\begin{figure}
\vspace{2cm}
\centering
\unitlength1cm
\begin{picture}(9,5)
\put(0,0){\includegraphics{Fig1.eps}}
\put(0.7,2.7){$-\varepsilon_{p-i+1}$}
\put(8.4,2.7){$\varepsilon_{p-i+1}$}
\end{picture}
\caption{The choice of paths}\label{fig1}
\end{figure}
The paths from real critical values go vertically upwards
up to the boundary of the circle $\{ z: \| z \| \leq
\frac{\varepsilon_{p-i+1}}{2} \}$. The paths from complex conjugate critical
values go vertically (upwards or downwards to the real axis and then go
vertically upwards to the boundary of the circle $\{ z: \| z \| \leq
\frac{\varepsilon_{p-i+1}}{2} \}$ avoiding from the right side a neighbourhood
of the critical value with positive imaginary part. From the boundary of
the circle $\{ z: \| z \| \leq \frac{\varepsilon_{p-i+1}}{2} \}$ all the paths
go to the non-critical value $\varepsilon_{p-i+1}$ in the upper half plane (see
Fig.~\ref{fig1}). The cycles are ordered in the usual way which in this
case means that they follow each other in the order of decreasing real
parts of the corresponding critical values; the vanishing cycle
corresponding to the critical value with negative imaginary part
precedes that with the positive one.
In the sequel we shall consider the matrices of the operators
$\sigma^{(i)}_s$, $\sigma^{(i)}_{\widetilde{s}}$, ${\rm Var}^{-1}_i$, etc.\ as block
matrices with blocks of size $1 \times 1$, $1 \times 2$, $2 \times 1$, and $2
\times 2$ corresponding to real critical values and to pairs of complex
conjugate critical values of the function $\widetilde{f}_{p-i+1}$. The
matrix of the operator $\sigma_s^{(i)}$ is an upper triangular block
matrix. Its diagonal entry corresponding to a real critical value is
equal to
$(-1)^m$ where $m$ is the Morse index of the critical point. A diagonal
block of size $2 \times 2$ corresponding to a pair of complex conjugate
critical values is equal to
$\left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)$. The matrix of
the operator
$\sigma_{\widetilde{s}}^{(i)}$ is a lower triangular block matrix (we do
not need a precise description of its diagonal blocks). The matrix of the
operator
${\rm Var}^{-1}_i$ is upper triangular with diagonal entries equal to
$(-1)^{(n+i)(n+i+1)/2}$
(the dual of the space $\hat{H}^{(i)}$ is
endowed with the basis dual to the one of $\hat{H}^{(i)}$). One has
$h_{\ast}^{(i)} = \sigma_s^{(i)} \sigma_{\widetilde{s}}^{(i)} =
(-1)^{n+i} {\rm Var}_i ({\rm Var}^{-1}_i)^T$. Thus
${\rm Var}^{-1}_i \sigma_s^{(i)} = (-1)^{n+i} ({\rm Var}^{-1}_i)^T \sigma_{\widetilde{s}}^{(i)}$.
The matrices ${\rm Var}^{-1}_i\sigma_s^{(i)}$ and $({\rm Var}^{-1}_i)^T
\sigma_{\widetilde{s}}^{(i)}$
are upper triangular and lower triangular respectively. Thus the
matrix ${\rm Var}^{-1}_i \sigma_s^{(i)}$ is in fact block diagonal with
the diagonal entry
$(-1)^{((n+i)(n+i+1)/2) +m}$
corresponding to a real
critical point of the function $\widetilde{f}_{p-i+1}$ ($m$ is the Morse
index) and with the diagonal block of the form
$$(-1)^{\frac{(n+i)(n+i+1)}{2}} \left( \begin{array}{cc} a & 1 \\ 1 & 0
\end{array} \right) $$
corresponding to a pair of complex conjugate critical points (up to a sign
$a$ is the intersection number of the corresponding cycles). This description
implies Theorem~\ref{theo2}.
\end{proof}
The formula~(\ref{eq2}) expresses the index of a gradient vector
field in terms of bilinear forms on the spaces of thimbles. Actually
each second summand of it can be expressed in terms of bilinear forms on
the corresponding spaces of vanishing cycles. Let $\Sigma_s^{(i)}$ be the
quadratic form on the space $H_{n+i-1}(V_{s\varepsilon}^{(i)})$ of vanishing
cycles defined by
$\Sigma_s^{(i)}(x,y)=\langle\sigma_s^{(i)}x,y\rangle$. As above, let
$\widetilde{s} = (s_1,\linebreak[0] \ldots,\linebreak[0]
s_{p-i},\linebreak[0] -1,\linebreak[0] s_{p-i+2},
\ldots, s_{p+1})$.
\begin{theorem}\label{theo3} For $n+i$ odd
$$
\sum\limits_{Q\in S_{{\rm grad}\,\widetilde f_{p-i+1}}}
{\rm ind}_Q{\rm grad}\,\widetilde f_{p-i+1}
=s_{p-i+1}(-1)^{\frac{n+i+1}{2}}({\rm
sgn}\,\Sigma_{\widetilde s}^{(i)}-{\rm sgn}\,\Sigma_s^{(i)})/2.
$$
\end{theorem}
\begin{proof}
It is essentially the same as in \cite{GZ}. One has to notice that the
kernel of the natural (boundary) homomorphism
$\hat{H}^{(i)}_{s\varepsilon}\to H_{n+i-1}(V_{s\varepsilon}^{(i)})$ is contained in the
kernel of the quadratic form $\Sigma_s^{(i)}$ and thus
\ ${\rm sgn}\,\Sigma_{\widetilde s}^{(i)}$ coincides with the signature
of the form $\langle\sigma_s^{(i)}\cdot,\cdot\rangle$ on the space
$\hat{H}^{(i)}_{s\varepsilon}$.
\end{proof}
\begin{remark} For $n+i$ even, it is not possible to express the number
$$\sum\limits_{Q\in S_{{\rm grad}\,\widetilde f_{p-i+1}}}
{\rm ind}_Q{\rm grad}\,\widetilde f_{p-i+1}$$ in terms of invariants
defined by the space of vanishing cycles. It can be understood from the
following example. Let $n=2$, $p=1$, $f_1=x_1^2+x_2^2-x_3^2$, $f_2=x_3$,
$i=0$. The discussed sum is different for $s_1=1$ and for $s_1=-1$
(i.e., for $t_1=s_1\varepsilon_1$ positive or negative). However the line
$\ell=\{t_1=0\}$ is not in the bifurcation set for vanishing cycles: it
doesn't lie in the discriminant of the map $(f_1, f_2):({\Bbb{C}}^3,
0)\to({\Bbb{C}}^2, 0)$. On the other hand the discriminant of the map
$f_1:({\Bbb{C}}^3, 0)\to({\Bbb{C}}, 0)$ coincides with the origin $0\in{\Bbb{C}}$ and
thus the line $\ell$ is in the bifurcation set for thimbles.
\end{remark}
\begin{remark}
It seems that a relation between complex conjugation
and the monodromy operator (similar to (\ref{eq3})) was first used in
\cite{A'C}. In \cite{GZ} it was written in an explicit way. In
\cite{MP} a similar relation was used to find some properties of
the Euler characteristics of links of complete intersection varieties.
This paper partially inspired our work.
\end{remark}
|
1997-10-07T22:55:30 | 9710 | alg-geom/9710009 | en | https://arxiv.org/abs/alg-geom/9710009 | [
"alg-geom",
"math.AG"
] | alg-geom/9710009 | Sandra DiRocco | Gian Mario Besana, Sandra Di Rocco | On The Projective Normality of Smooth Surfaces of degree nine | 22 pages, AmsLatex, see home pages
http://www.emunix.emich.edu/~gbesana/ and
http://www.math.kth.se/~sandra/Welcome | null | null | null | null | We investigate the projective normality of smooth, linearly normal surfaces
of degree 9. All non projectively normal surfaces which are not scrolls over a
curve are classified. Results on the projective normality of surface scrolls
are also given. One of the reasons that brought us to look at this question is
our desire to find examples for a long standing problem in adjunction theory.
Andreatta followed by a generalization by Ein and Lazarsfeld posed the problem
of classifying smooth n-dimensional varieties (X,L) polarized with a very ample
line bundle L, such that the adjoint linear system |H| = |K + (n-1)L| gives an
embedding which is not projectively normal. After a detailed check of the non
projectively normal surfaces found in this work no examples were found except
possibly a blow up of an elliptic P^1-bundle whose existence is uncertain.
| [
{
"version": "v1",
"created": "Tue, 7 Oct 1997 20:55:29 GMT"
}
] | 2007-05-23T00:00:00 | [
[
"Besana",
"Gian Mario",
""
],
[
"Di Rocco",
"Sandra",
""
]
] | alg-geom | \section{introduction}
Smooth projective varieties of small degree have been classified over
the years
and thoroughly studied, e.g. \cite{Io1}, \cite{Io2}, \cite{ok2},
\cite{ok8},
\cite {Ale}, \cite{ADS}. A variety $X~\subset~\Pin{n}$ is {\it
projectively normal} if
the
maps
$H^0(\Pin{n},\oofp{n}{k}) \to H^0(X, \oof{X}{k})$ are surjective for all
$k
\ge 1$ or
in other words if hypersurfaces of degree $k$ cut complete linear
systems
on $X$ for every $k\ge 1.$ In
\cite{Alibaba} the projective normality of varieties of degree $d \le 8$
of
any dimension was
investigated. This work is concerned with the projective normality of
smooth
projective surfaces, embedded by the complete linear system associated
with a
very ample line bundle
$L$ of degree
$d=9.$ Such surfaces are either embedded in $\Pin{4}$ or have sectional
genus $g \le 7.$ Therefore they are completely classified in
\cite{au-ra} and
\cite{LiAq}.
One of the reasons that brought us to look at this question is our
desire
to find examples for a long standing problem in adjunction theory.
Andreatta
\cite{ce}, followed by a generalization by Ein and Lazarsfeld
\cite{EL},
posed the problem of classifying smooth $n$-dimensional varieties $(X,
\cal{L})$
polarized
with a very ample line
bundle $\cal{L}$, such that the adjoint linear system $|L| = |K + (n-1)
\cal{L}|$ gives an
embedding
which is not projectively normal. Andreatta and Ballico \cite{an-ba1}
gave examples of
surfaces $(S, \cal{L})$ with the above behavior, where $d=\deg{S} = 10$
under the
adjoint embedding. Alzati, Bertolini and the first author in
\cite{Alibaba} found no
example with
$d\le 8.$ After a detailed check of the non projectively normal
surfaces
found in this work no examples were found except possibly a blow
up of an elliptic $\Pin{1}$-bundle whose existence is uncertain. See
section
\ref{K+L} for details.
Our findings concerning the projective normality of surfaces of degree
nine are
collected in the the following theorem (see
\brref{notation} for notation):
\begin{theo}
\label{thetheorem} Let $S$ be a smooth surface embedded by the complete
linear system associated with a very ample line
bundle
$L$ as a surface of degree $9$ and sectional genus $g$ in $\Pin{N}. $
Assume $(S, L)$
is not a scroll over a curve. Then
$(S, L)$ fails to be projectively normal if and only if it belongs to
the
following list:
\begin{center}
\begin{tabular}{|c|c|c|c|} \hline
$\Pin{N} $ & $g$& $S$& $L$\\ \hline\hline
$\Pin{5}$ & $4$ & $Bl_3X$ where $X$ is a $\Pin{1}$-bundle & \\
& & over an elliptic
curve,
$e=0$&
$2\frak{C_0}+3\frak{f}-\sum_iE_i$
\\
\hline
$\Pin{5}$ & $5$ & Rational conic bundle $S=Bl_{15} \bold{F_e}$ , $0\le e
\le 5$ &
$2 \frak{C_0} + (6+e)\frak{f} -
\sum_iE_i $
\\
\hline
$\Pin{4}$ & $6$ &$Bl_{10}(\Pin{2})$&$3p^*(\oofp{2}{1})-\sum_i4E_i$ \\
\hline
$\Pin{4}$ & $7$ &$Bl_{15}(\Pin{2})$&$9p^*(\oofp{2}{1})-\sum_1^63E_i-$\\
& & &$\sum_7^92E_j-\sum_{10}^{15}E_k $\\ \hline
$\Pin{4}$ & $6$ & Projection of an Enriques surface & \\
& & of degree 10 in ${\bf P}^5$& cf. \cite{au-ra} \\ \hline
$\Pin{4}$ & $7$ & Minimal elliptic surface & cf. \cite{au-ra} \\ \hline
$\Pin{4}$ & $8$ & Minimal surface of general type& cf. \cite{au-ra}\\
\hline
\end{tabular}
\end{center}
\end{theo}
The projective normality of surfaces which are scrolls over a curve of
genus $g,$
not included in the above theorem, was also investigated. Results are
collected in
Proposition \ref{scrollprop}. We were not able to prove or disprove the
projective
normality of scrolls over trigonal curves of genus $3,4,5.$
The authors would like to thank Ciro Ciliberto, Antonio Lanteri and
Andrew J.
Sommese for many helpful conversations and W. Chach\'{o}lski for his
insight and
patience.
\section{Background material}
\subsection{NOTATION}
\label{notation}
Throughout this article $S$ denotes a smooth connected projective
surface
defined over the complex field {\bf C}. Its
structure sheaf is denoted by
${\cal O_S}$ and the canonical sheaf of holomorphic $2$-forms on $S$ is
denoted by
$K_S$.
For any coherent sheaf $\Im$ on $S$, $h^i(\Im )$ is the complex
dimension of
$H^i(S,\Im)$ and
$\chi=\chi(\cal{O}_S)=\chi(S)=\sum_i(-1)^ih^i(\cal{O}_S).$
Let $L$ be a line bundle on $S.$ If $L$ is ample the pair $(S, L)$ is
called
a {\it
polarized surface}.
The following notation is used:\\
$|L|$, the complete linear system associated with L;\\
$d = L^2,$ the degree of $L$;\\
$g=g(S, L)$, the {\it sectional genus} of $(S, L)$, defined by
$2g-2=L\cdot
(K_S+L).$ If $C\in |L|$ is an irreducible and reduced element then
$g=g(C)$
is the
arithmetic genus of $C$;\\
$\Delta (S, L) = \Delta = 2+L^2-h^0(L)$ the Delta genus of $(S, L)$;\\
$\bold{F_e}$, the Hirzebruch surface of invariant $e$ ;\\
$E^*$ the dual of a vector bundle $E$.\\
Cartier divisors, their associated line bundles and the invertible
sheaves
of their
holomorphic sections are used with no distinction. Given two divisors
$L$
and $M$
we denote linear equivalence by
$L \sim M$ and numerical equivalence by $L \equiv M.$
The blow up of a surface $X$ at $n$ points is denoted by $p: S=Bl_nX \to
X.$
When $X$ is a
$\Pin{1}$-bundle over a curve with fundamental section $C_0$ and generic
fibre $f$ it is
$Num(X) ={\Bbb Z}[C_0]
\oplus
{\Bbb Z}[f]$ and
the following shorthand is used: $\frak{C_0}=p^*(C_0)$ and $\frak{f}=
p^*(f).$
A polarized surface $(S, L)$
is a {\em scroll}
or a {\em conic bundle} over a curve $C$ if there exists a surjective
morphism $p: S
\to C$ with connected fibers and an ample line bundle
$H$ on
$C$ such that, respectively, $ K_S + 2L = p^*(H)$ or $ K_S + L =
p^*(H).$
If $(S, L)$ is a
scroll then $S$ is a $\Pin{1}$-bundle over $C$ and $L \cdot f = 1$ for
every fibre $f.$
In section
\ref{genere6} the notion of {\it reduction} of a smooth polarized
surface
is shortly
used. The best reference is
\cite{BESO}.
\subsection{\small CASTELNUOVO BOUND}
Let $C\subset \Pin{N}$, then by Castelnuovo's lemma
\begin{equation}
\label{cast}
g(C)\leq\left[ \frac{d-2}{N-2}\right ](d-N+1-
(\left[ \frac{d-2}{N-2}\right ]-1)\frac{N-2}{2})
\end{equation}
where$[x]$ denotes the greatest integer $\leq x$.
\subsection{\small PROJECTIVE NORMALITY}
Let $S$ be a surface embedded in $\Pin{N}.$ $S$ is said to be {\it
k-normal} if the map
$$H^0({\cal O}_{\Pin{N}}(k))\longrightarrow H^0({\cal O}_S(k))$$ is
surjective .
$S$ is said to be {\it projectively normal} if it is $ k$-normal for
every $k\geq 1$.
An ample line bundle $L$ on $S$ is {\it normally generated} if
$S^kH^0(L)\to H^0(L^k)$ is surjective for every $k\geq 1$. If $L$ is
normally generated then it is very ample and $S$, embedded in $\Pin{N}$
via
$|L|$ is projectively normal. A polarized surface $(S, L)$ is said
to be projectively normal if $L$ is very ample and $S$ is
projectively normal under the embedding given by $L.$
A polarized surface $(S, L)$ has a {\it ladder} if there exists an
irreducible and
reduced element
$C \in |L|.$ The ladder is said to be
{\it regular} if $H^0(S, L)\to H^0(C, \restrict{L}{C})$ is onto. If
$L$ is very ample $(S, L)$ clearly has a ladder.
We recall the following general result due to Fujita:
\begin{theo}[\cite{fu}]
\label{fujitatheo}
Let $(S, L)$ be a polarized surface with a ladder. Assume
$g(L)\geq\Delta$. Then \\ i) The ladder is regular if $d\geq 2\Delta
-1$;\\
ii) L is normally generated, $g=\Delta$ and $H^1(S, tL)= 0$ for any t,
if $d\geq 2\Delta +1$.
\end{theo}
\subsection{\small $k$ - REGULARITY}
\label{kreg}
A good reference is \cite{mu1}.
A coherent sheaf ${\cal F}$ over $\Pin{n}$ is {\em k-regular} if
$h^i({\cal F}(k-i))=0$ for
all $i >0.$ If ${\cal F}$ is $k$-regular then it is $k+1$-regular. If
$X\subset \Pin{n} $ is an irreducible variety such that ${\cal I}(X)$ is
$k$-regular
then the homogeneous ideal $I_X=\oplus H^0(\iof{X}{t})$ is generated in
degree $\le
k.$ This fact implies that if ${\cal I}_X$ is $k$-regular then $X$
cannot
be embedded
with a $t\ge( k+1)$- secant line.
\subsection{\small CLIFFORD INDEX}
Good references are \cite{mart}, \cite{GL}.
Let $C$ be a projective curve and
$H$ be any line bundle on $C$. The Clifford index of $H$ is defined as
follows:
$$cl(H)=d-2(h^0(H)-1).$$
The Clifford index of the curve is $cl(C)=\text {min}\{cl(H) |
h^0(H)\geq 2 \text{ and }h^1(H)\geq 2 \}$.
$ H$ {\it contributes }to the Clifford index of $C$ if $h^0(H)\geq 2
\text{ and }h^1(H)\geq 2$ and $H$ {\it computes}
the Clifford index of $C$ if $cl(C)=cl(L)$. For a general curve $C$
it is $cl(C)=\left [\frac{g-1}{2}\right ]$
and in any case $cl(C)\leq\left [\frac{g-1}{2}\right ]$. By
Clifford's theorem a special line bundle $L$ on $C$ has $cl(L)\geq
0$ and
the equality holds if and only if $C$ is hyperelliptic and $L$ is a
multiple of the unique
$g^1_2$.\\ If
$cl(C)=1$ then $C$ is either a plane quintic curve or a trigonal curve.
The following results dealing with the projective normality of curves
and relating it to the Clifford index are listed for the convenience of
the reader.
\begin{theo}[\cite{GL}]
\label{glcliff}
Let L be a very ample line bundle on a smooth irreducible complex
projective
curve C with:
$$deg(L)\geq 2g+1-2h^1(L)-cl(C)$$
then $L$ is normally generated.
\end{theo}
In the case of hyperelliptic curves, because there are no special
very-ample line
bundles, the following is true.
\begin{prop}[\cite{la-ma}] \label{hyper}
A hyperelliptic curve of genus $g$ has no normally generated line
bundles
of degree
$\leq 2g$.
\end{prop}
\begin{lemma}[ \cite{la-ma}]
\label{2norm}
Let $L$ be a base point free line bundle of degree $d\geq g+1$ on a
curve
of genus $g$. Then $L$ is normally generated if and only if the natural
map $H^0(L)\otimes H^0(L)\to H^0(2L)$ is onto.
\end{lemma}
It follows that a polarized curve $(C, L)$ with
$d\geq g+1$ and $L$ very ample is projectively normal if and only if it
is 2-normal.
The projective normality of a polarized surface $(S, L)$ will be often
established by investigating the property
for a general hyperplane section. The main tools used are the following
results.
\begin{theo}[\cite{fu}]
\label{fujitatheo2}
Let $(S, L)\supset
(C,\restrict{L}{C})$ be a polarized surface with a ladder. If
the ladder is regular and $\restrict{L}{C}$ is normally generated then
$L$ is
normally generated.
\end{theo}
\begin{lemma}[\cite{Alibaba}]
\label{besanaignorans}
Let $(S, L)\supset
(C,\restrict{L}{C})$ be a polarized surface with a regular ladder.
Assume $h^1(L)=0$
and $\Delta=g.$
Then $L$ is normally generated if and
only if $\restrict{L}{C}$ is normally generated.
\end{lemma}
\subsection{\small SURFACES EMBEDDED IN QUADRIC CONES}
\label{qcones}
As Lemma \ref{2norm} suggests, the hyperplane section technique
will often reduce the projective normality of a surface to its
$2$-normality. It is useful then to recall the detailed investigation
of surfaces in $\Pin{5}$ contained in singular quadrics, done in
\cite{Alibabaquad}.
Let $\Gamma$ be a four dimensional quadric cone in
$\Pin{5}$ and let $\sigma:\Gamma^*\longrightarrow\Gamma$ be the blow up of $\Gamma$ along the
vertex, with exceptional divisor $T$. Suppose $S\subset\Gamma$ is a smooth
surface and let
$S'$ be the strict transform of $S$ in $\Gamma^*$ under $\sigma$.
The Chern classes of $S'$ and $\Gamma^*$ satisfy the following
standard relation:
\begin{equation}
\label{DPF}
\restrict{c_2(\Gamma^*)}{S'}=S'S'+\restrict{c_1(\Gamma^*)}{{S'}}c_1(S')-
K_{S'}^2+c_2(S')
\end{equation}
If $rank(\Gamma)=5$, \ $\Gamma$ is a cone with vertex a point $P$ over a smooth
quadric $Q' \subset \Pin{4}.$
Following \cite{Alibabaquad} let
\begin{xalignat}{2}
C(W):&=& \text{ the cone over the cycle $W \subset Q'$ with vertex $V$}
&
\notag
\\
\sigma:& \Gamma^* \to \Gamma &\text{ the blow up map}& \notag \\
H_{Q'}: &=& \text {the hyperplane section of $Q'$}& \notag \\
l_{Q'}:&=& \text {the generator of $A_1(Q')$}& \notag \\
p_{Q'}:&=& \text {the generator of $A_0(Q')$ }&\notag
\end{xalignat}
According to the above notation it is
\begin{gather}
Pic(Q') = <H_{Q'}> \notag \\
H^2_{Q'} = 2 l_{Q'} \notag\\
A_0(Q') =<p_{Q'}> \notag
\end{gather}
Further, let
\begin{gather}
Z = \sigma^{-1}(C(H_{Q'})) \notag \\
S = \sigma^{-1}(C( l_{Q'})) \notag \\
F = \sigma^{-1}(C(p_{Q'})), \notag
\end{gather}
and denote by $\overline{H}$ the cycle $H_{Q'}$ in $\Gamma^*$ and
by $\overline{l}$ the
cycle $l_{Q'}$ in $\Gamma^* .$
The Chow Rings of $\Gamma^*$ are then given by:
\begin{gather}
Pic(\Gamma^*) = <Z, \tau>\\
A_2(\Gamma^*) = <\overline{H}, S>\\
A_1(\Gamma^*) = < \overline{l}, F>.
\end{gather}
and it is
\begin{xalignat}{3}
c_1(\Gamma^*) &= 2(\tau + Z) & c_2(\Gamma^*) & = Z^2 + 6\tau Z
&T&=\tau-Z
\end{xalignat}
\begin{xalignat}{2}
T&=\tau-Z& S'&=\alpha \overline{H} +\beta X
\end{xalignat}
The intersection table is then the following:
\vskip .5 cm
\setlength{\unitlength}{ .4cm}
\begin{center}
\begin{picture}(13,13)
\put(1,0){\line(0,1){13}}
\put(5,0){\line(0,1){13}}
\put(9,0){\line(0,1){13}}
\put(13,0){\line(0,1){13}}
\put(0,0){\line(1,0){13}}
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\put(1.5,10.5){$\overline{H}$}
\put(3.5,.5){$0$}
\put(3.5,2.5){$1$}
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\put(3.5,8.5){$2X$}
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\put(7.5,4.5){$0$}
\put(7.5,6.5){$1$}
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\put(7.5,10.5){$\overline{l}$}
\put(6.5,2){0}
\put(9.5,10.5){$1$}
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\put(11.5,8.5){$0$}
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\end{picture}
\end{center}
If $rank(\Gamma)=4$, \ $\Gamma$ is a cone with vertex a line $r$ over a smooth
quadric
surface
$Q\subset \Pin{3}.$ Following
\cite{Alibabaquad} let
$\tau$ be the tautological divisor on $\Gamma^*$ and
Let $C(W)$ denote the cone with vertex
$r$ over the cycle $W\subset \overline{Q}.$ Let
\begin{xalignat}{3}
Pic \ \overline{Q} : &= <\ell_1, \ell_2> & A_0(\overline{Q}) : &= <p> &Q
:
&= \sigma^{-1}(\overline{Q}) \notag \\
P_1 :&= \sigma^{-1}(C( \ell_1)) & p_1: &=\tau \cdot P_1 & \overline{\ell}_1:&=
\tau \cdot
p_1 \notag \\
P_2 :&= \sigma^{-1}(C( \ell_2)) & p_2: &=\tau \cdot P_2 & \overline{\ell}_2:&=
\tau \cdot
p2 \notag \\
F :&= \sigma^{-1}(C(p))& \ell:&= \tau \cdot F & & \notag
\end{xalignat}
With the above notation it is:
\begin{xalignat}{3}
Pic (\Gamma^*) &= <\tau, P_1, P_2> & A_2(\Gamma^*) &= < Q,p_1, p_2, F> &
A_1(\Gamma^*) &= < \overline{\ell}_1, \overline{\ell}_2, \ell> \notag
\end{xalignat}
It is also $T=\tau-P_1-P_2$, $c_1(\Gamma^*)=4\tau-T$,
$c_2(\Gamma^*)=3Q+4p_1+4p_2.$
Because $S'$ is an effective cycle it is
$S'=\alpha Q+\beta p_1+\gamma p_2+\delta F$ with
$\alpha\geq 0$,
$\alpha +\beta\geq 0$, $\alpha+ \gamma \ge 0$ and $deg(S)=\tau\cdot
\tau\cdot
S'=2\alpha+\beta+\gamma+\delta.$
With the above notation we have the
following intersection table:
\vskip .5cm
\setlength{\unitlength}{ .7cm}
\begin{center}
\begin{picture}(11,11)
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\multiput(.2,8.2)(2,2){2}{$P_1$}
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\multiput(.2,3.2)(7,7){2}{$F$}
\multiput(.2,2.2)(8,8){2}{$\overline{\ell_1}$}
\multiput(.2,1.2)(9,9){2}{$\overline{\ell_2}$}
\multiput(.2,.2)(10,10){2}{$\ell$}
\multiput(1.2,8.2)(1,1){2}{$p_1$}
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\multiput(1.2,5.2)(4,4){2}{$\overline{\ell_1}$}
\multiput(1.2,4.2)(5,5){2}{$\overline{\ell_2}$}
\multiput(1.2,3.2)(1,1){3}{$\ell$}
\multiput(5.2,7.2)(1,1){3}{$\ell$}
\multiput(3.2,6.2)(1,1){2}{$\overline{\ell_1}$}
\multiput(2.2,5.2)(3,0){2}{$0$}
\multiput(2.2,6.2)(2,2){2}{$\overline{\ell_1}$}
\multiput(2.2,7.2)(1,1){2}{$F$}
\multiput(2.2,8.2)(3,0){2}{$0$}
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\put(.2,10.2){$\cdot$}
\end{picture}
\end{center}
where $\star = \overline{\ell_1} + \overline{\ell_2}$ and the empty
spaces
are intended to
be $0.$
\section{Surfaces in $\Pin{4}$}
Throughout this section $S$ will be a surface embedded in
$\Pin{4}$ by a very ample
line bundle $L,$ non degenerate of degree $9,$ with sectional genus
$g.$
Surfaces of degree $9$ in $\Pin{4}$ have been completely classified by
Aure and Ranestad in \cite{au-ra}. The investigation of the
projective normality of such surfaces is essentially contained in
their work. For completeness we present the global picture in this
section.
\begin{theo}[ \cite{au-ra}]\label{class}
Let $(S, L)$ be as above and $\chi=\chi(\cal {O}_S)$ then $S$ is a regular
surface with $K^2=6\chi-5g+23$, where:
\begin{itemize}
\item[1)] $g=6$, $\chi=1$ and $S$ is
rational or the projection of an Enriques surface of degree $10$ in
$\Pin{5}$ with center of projection on the surface;
\item[2)] $g=7$ and $\chi =1$ and $S$ is rational, or $\chi =2$ and $S$
is a minimal elliptic surface;
\item[3)] $g=8$ and $\chi =2$ and $S$ is a
K3-surface with $5$ $(-1)$-lines, or $\chi =3$ and $S$ is a minimal
surface of general type;
\item[4)] $g=9$, $\chi=4$ and $S$ is linked $(3,4)$
to a cubic scroll;
\item[5)] $g=10$, $\chi=6$ and $S$ is a complete
intersection $(3,3)$;
\item[6)] $g=12$, $\chi = 9$ and $S$ is linked to a
plane.
\end{itemize}
Moreover if $g\geq 7$ then $S$ is contained in at least two
quartic surfaces.
\end{theo}
\begin{prop}$(S, L)$ as above is projectively normal if and only if:
\begin{itemize}
\item [a)] $g=8$, $\chi =2$ and $S$ is a K3-surface with $5$
$(-1)$-lines.
\item[b)] $g=9$, $\chi=4$ and $S$ is linked $(3,4)$ to a cubic scroll
\item[c)] $g=10$, $\chi=5$ and S is a complete intersection
(3.3);
\item[d)] $g=12$, $\chi = 9$ and $S$ is linked to a plane.
\end{itemize}
\end{prop}
\begin{pf} Let us examine the surfaces in Theorem \ref{class};\\ Let
$C\in|L|$ be a
generic smooth element. Since all the surfaces are regular we always
have
$h^0(\restrict{L}{C})=4$. Note that for $g\leq 9$
$d(2\restrict{L}{C})<2g-2$, then
$h^1(2\restrict{L}{C})=0$. If $g=6$ then $h^1(\restrict{L}{C})=0$ and
thus
$h^1(L)=0$. Because
$h^0(2\restrict{L}{C})=h^1(2\restrict{L}{C})+19-6=13$ the following
exact
sequence
\begin{equation}
\label{duelle}
0\longrightarrow L\longrightarrow 2L\longrightarrow 2\restrict{L}{C}\longrightarrow 0
\end{equation}
gives $h^0(2L)=h^0(L)+h^0(2\restrict{L}{C})=18.$ Thus the map
$H^0({\cal O}_{{\bf P}^4}(2))\longrightarrow H^0({\cal O_S}(2))$ cannot be
surjective,
being $h^0({\cal O}
_{{\bf P}^4}(2))=15.$ This means that $S$ is not $2$-normal and
therefore it is not
projectively normal. \\ If $g=7$ then $h^0(2\restrict{L}{C})=12$,
$h^1(\restrict{L}{C})=1$ and
$h^1(2L)=0$ by
\cite[2.10]{au-ra}. Therefore from \brref{duelle} and the regularity of
S it follows that
$16\leq h^0(\restrict{L}{C})\leq 17$, which implies that $S$ is not
projectively
normal, as above.
If $g=8$ then $h^0(2\restrict{L}{C}) =11$, $h^1(L)\leq 1$ and
$h^1(2L)=0$ by
\cite[2.11]{au-ra}. For degree reasons $S$ cannot be contained in any
quadric hypersurface being contained in at least a quartic. Therefore
$S$
is $2$-normal if and only if $H^0(L)=15$.
>From \brref{duelle} we get
$ h^0(2L)\leq 15,16$ respectively if $\chi=2,3$. If $\chi=3$ $S$ is not
projectively normal as above.
If $\chi=2$ $S$ is $2$-normal and therefore projectively normal by
Lemma
\ref{2norm}.
If $g=9$ $(S, L)$ is projectively normal by linkage, see
\cite[2.13]{au-ra}.
If $g=10$ $S$ is a complete intersection and thus projectively normal.
If $g=12$ $S$ is linked to a plane and therefore it is arithmetically
Cohen-Macaulay by linkage, which implies $S$ projectively normal.
\end{pf}
\section{Surfaces embedded in $\Pin{N} ,$ $N\ge 5.$}
Let $ S$ be a smooth surface, let $L$ be a very ample line bundle on
$S$
and let
$${\cal S}_g =\{(S, L) \text{ as above}
\ | L^2 = 9,
h^0(L)\ge 6, g(S, L)=g \text{ and} (S, L) \text{ is not a scroll}\}.$$
If
$(S, L)\in{\cal S}_g$,
by Castelnuovo's Lemma
$g\leq 7.$
Let $\cal{S} =
\bigcup_{g=0}^7
\cal{S}_g.$
In the following lemmata a few preliminary results are collected.
\begin{lemma}
Let $(S, L)\in{\cal S}$ and let $C\in|L|$ be a smooth generic element.
Then
\begin{itemize}
\item[a)] if $h^1(\restrict{L}{C})=0$ then
$g(L)\leq5$;
\item[b)] if $h^1(\restrict{L}{C})=1$ then $g(L)=7$ or $6$;
\item[c)] if $h^1(\restrict{L}{C})\geq 2$ then $g(L)=7$;
\end{itemize}\label{h1LC}
\end{lemma}
\begin{pf}
a) For $g\geq 1$ we have $h^0(\restrict{L}{C})\geq 3$. If
$h^0(\restrict{L}{C})=3, 4$ then
$h^0(L)\le 4, 5$ respectively, which is a contradiction. Thus
$h^0(\restrict{L}{C})
\ge 5$ i.e.
$5\leq h^0(\restrict{L}{C})=10-g$, i.e.
$g\leq 5.$\\
b) Since $h^0(K_C-\restrict{L}{C})=1$ then $d\leq 2g-2$. Moreover being
$d$ odd it
is $d\leq 2g-3$
c) If $h^1(\restrict{L}{C})\geq 2$ then $K_S|_C$ is a special divisor on
$C$ and it contributes to $cl(C),$ thus
$cl(C)\geq 0$,
i.e \\ $0\leq d(K_C-\restrict{L}{C})-2h^0(K_C-\restrict{L}{C})+2\leq
2g-13$
, from which we get
$g\geq 7$.
\end{pf}
\begin{lemma}\label{nonP41}
Let $(S, L) \in \cal S$ . Then $(S, L)$ is
projectively normal or
\begin{itemize}
\item[a)] $(S, L) \subset \Pin{5}$, $g=4$, $S$ is $\Pin{1}$-bundle over an
elliptic curve,
$e=-1$, $L=3C_0.$
\item[b)] $(S, L) \subset \Pin{5}$ is a conic bundle over an elliptic
curve,
$g=4$.
\item[c)] $(S, L) \subset \Pin{5}$ $g = 5,6.$
\end{itemize}
\end{lemma}
\begin{pf}
In our hypothesis it is $\Delta(S, L)=11-h^0(L)$. If codim$ (S) =1$ then
$S$ is
projectively normal. Because $S$ is not embedded in $\Pin{4}$, we can
assume
codim$ (S) \ge 3$ i.e. $h^0(L) \geq 6$, i.e $\Delta(S)\leq 5$.
First assume $\Delta < 5.$ It is $9=d\geq 2\Delta + 1$, then if
$g\geq\Delta$
$(S, L)$ is projectively normal by Theorem \ref{fujitatheo}.
Because $g=0$ implies $\Delta = 0$ it follows that $(S, L)$ is
projectively
normal if
$\Delta=0,1.$
Moreover if $g=1$ then $(S, L)$ is an elliptic scroll which is impossible.
Therefore $(S, L)$ is projectively
normal if
$\Delta = 2.$
Let now $\Delta=3.$ By
\cite{Io1} it is $g=3$ and therefore $(S, L)$ is projectively normal
Let now $\Delta= 4.$ By \cite{Io4} Theorem 3, $(S, L)$ is projectively
normal
unless,
possibly, if it is a
scroll over a curve of genus $g=2,$ which is impossible.
Let now $\Delta(S)=5$, i.e $h^0(L)=6$.
If $g(L)=7$ then $S$ is a
Castelnuovo Surface, see \cite{Har1}, and thus
projectively normal.
Let $g=2,3.$ Simple cohomological computations, using the
classification
given in \cite{Io1} show that there are no such surfaces in $\Pin{5}.$
Let $g=4$ and let $C \in |L|$ be a generic hyperplane section. By Lemma
\ref{h1LC}
it is $h^1(\restrict{L}{C})=0$ and therefore $h^0(\restrict{L}{C})=6$ by
Riemann Roch. This shows that
$q(S) \ne 0.$ By \cite{LiAq} and \cite{Io4} the only possible cases are
b)
and c) in the
statement .
\end{pf}
\subsection{SECTIONAL GENUS $g=4$ }
\label{genusfour}
In this subsection the projective normality of pairs $(S, L)\in{
\cal S}_4$ is studied. By Lemma \ref{nonP41} and \cite{LiAq} we have to
investigate
the following cases :
\begin{itemize}
\item[Case 1.] $(S, L) \subset \Pin{5}$ is a $\Pin{1}$-bundle over an
elliptic curve, $e=-1$, $L\simeq3C_0.$
\item[Case 2.] $(S, L) \subset \Pin{5}$ is the blow up $p: S \to X$ of a
$\Pin{1}$-bundle over an elliptic curve at $3$ points, $e=-1$,
$L\equiv 2\frak{C_0}+2\frak{f}-\sum_1^3E_i$ where $\frak{C_0} =
p^*(C_0)$ and
$\frak{f}=p^*{f}.$
\item[Case 3.]$(S, L) \subset \Pin{5}$ is the blow up $p: S \to X$ of a
$\Pin{1}$-bundle over an elliptic curve at $3$ points, $e=0$,
$L\equiv 2\frak{C_0}+3\frak{f}-\sum_1^3E_i$ where $\frak{C_0} =
p^*(C_0)$ and
$\frak{f}=p^*{f}.$
\end{itemize}
\begin{lemma}
Let $(S, L) \in \cal{S}_4$ be as in \brref{genusfour} Cases 1,2,3 above.
The following
are equivalent:
\begin{enumerate}
\item[i)]$(S, L)$ is $2$- normal;
\item[ii)]$\cal{I}_S$ is $3$-regular;
\item[iii)] $(S, L)$ is projectively normal;
\item[iv)] $h^0(\cal{I}_S(2)) = 0,$ i.e. $S$ is not contained
in any quadric hypersurface.
\end{enumerate}
\label{g4noquad}
\end{lemma}
\begin{pf}
Assume $(S, L)$ is $2$-normal, i.e. $h^1(\cal{I}_S(2)) = 0.$ Since $p_g(S)
=
0$ and
$h^1(L)=0$ in all the cases under consideration, it is not hard to check
that $\cal{I}_S$
is
$3$-regular. By
\cite[pg. 99]{mu1} it follows that
$h^1(\cal{I}_S(k)) = 0$ for all $k \ge 2$ and thus $(S, L)$ is
projectively
normal.
Therefore i), ii) and iii) are equivalent. Since $h^0(\oofp{5}{2}) =
21$ and
$h^0(\oof{S}{2}) = 21,$
$S$ is $2$- normal if and only if $h^0(\cal{I}_S(2)) = 0.$
\end{pf}
\begin{rem}
$(S, L)$ as in \brref{genusfour} Case 3 is a congruence of lines of
$\Pin{3}$ of bi-degree
$(3,6),$ cf.
\cite{gr} and thus not projectively normal by Lemma \ref{g4noquad}.
\end{rem}
\begin{rem}
$(S, L)$ as in \brref{genusfour} Case 1 was shown to be projectively
normal by
Homma \cite{Ho2}. Following an idea due to Sommese we offer below a
different
proof.
\end{rem}
\begin{prop}
\label{propdiandrew}
Let $(S, L)$ be as in \brref{genusfour} Case 1, then $(S, L)$ is
projectively
normal.
\end{prop}
\begin{pf}
Lemma \ref{g4noquad} shows that it is enough to show that $(S, L)$ is
$2$-normal.
Let $E$ be an elliptic curve with fixed origin $O.$ It was shown in
\cite{be-so2} that
$S$ can be viewed as the quotient of $X=E\times E$ under the involution
$ \iota:
X\to X$ given by $ \iota(x,y) = (y,x).$ Let $q: X \to S$ be the quotient
map
and $p_i: X \to E,$ $i=1,2$ be the projections onto the factors. One can
see that $
q^*(C_0) = p_1^*(\oof{E}{P}) \otimes p_2^*(\oof{E}{P}),$ where $P$ is a
point on $E.$ Let
$L_i = p^*_i(\oof{E}{P}).$ It is
$$ H^0(X, q^*(L)) = H^0(X, q^*(3C_0)) = H^0(3L_1) \otimes H^0(3L_2),$$
and therefore $h^0(q^*(L)) = 9.$
Let $H^0(X, q^*L)^{\iota}=\{ \sigma \in H^0(X,q^*L) \text{ such that }
\sigma \iota =
\sigma \}$, i.e. the subspace of global holomorphic sections of
$q^*(L)$
which are
invariant under
$\iota.$ Notice that there is a natural isomorphism $H^0(S, L) \simeq
H^0(X,q^*(L))^{\iota}.$
Because $deg (3L_i) \ge 2g(E) + 1$ the map $H^0(3L_i) \otimes H^0(3L_i)
\to
H^0(2(3L_i))$ is surjective. It follows that $$H^0(q^*(L)) \otimes
H^0(q^*(L)) \to
H^0(q^*(2L))$$ is surjective.
To conclude it is enough to show that
\begin{equation}
\label{2normiota}
H^0(q^*(L))^{\iota} \otimes H^0(q^*(L))^{\iota} \to
H^0(q^*(2L))^{\iota}
\end{equation}
is surjective.
Let $\alpha_1, \alpha _2, \alpha_3$ be a base for $H^0(E, \oof{E}{3P})$
and let
$a_i=p_1^*(\alpha_i)$ and $b_i=p_2^*(\alpha_i)$ be bases for $H^0(3L_1)$
and
$H^0(3L_2)$ respectively.
With the above notation $\{ a_i \otimes b_j\}$ form a base for
$H^0(q^*(L)).$
Notice that an element $a_i \otimes b_j$ with $i \ne j$ cannot be
$\iota$-invariant.
To see this assume $a_i \otimes b_j$ were $\iota$-invariant and let $z
\in
E$ be a point
such that
$\alpha_i(z) = 0$ and
$\alpha_j(z) \ne 0.$ Then
$(a_i \otimes b_j)(z,y) = (a_i \otimes b_j)(y,z)$ implies
$\alpha_i(y)\alpha_j(z) = 0$ for
every
$y,$ which is impossible. Simple direct checks show that
$$
a_1 \otimes b_1, \ a_2 \otimes b_2, \ a_3 \otimes b_3, a_1 \otimes b_2 +
\
a_2 \otimes
b_1, a_1 \otimes b_3 + \ a_3 \otimes
b_1, a_2 \otimes b_3 + \ a_3 \otimes
b_2
$$
form a basis for $H^0(q^*L)^{\iota}$ and that $$t_1= a_1 \otimes
b_2,t_2=
a_1 \otimes
b_3,t_3= a_2
\otimes b_3$$
span a $3$-dimensional subspace $V$ such that $ H^0(q^*(L)) =
H^0(q^*(L))^{\iota}
\oplus V.$
To conclude the proof of the surjectivity of (\ref{2normiota}) it is
enough
to show
that elements of the form $v \otimes s$ or $s \otimes v$ where $v \in V$
and $s \in
H^0(q^*L)^{\iota}$ , and elements of the form $v_1 \otimes v_2$ where
$v_i
\in V$,
cannot be $\iota$-invariant.
Let $v= p_1^*(\gamma_1) \otimes p_2^*(\gamma_2)$ and let $s=
p_1^*(\sigma_1)
\otimes p_2^*(\sigma_2).$ For every $(x,y) \in X$ it is
\begin{gather}
\label{onesidevs}
(v \otimes s)(x,y) = \gamma_1(x) \gamma_2(y)\sigma_1(x) \sigma_2(y) =
\gamma_1(x) \gamma_2(y) \sigma_1(y) \sigma_2(x)\\
\label{othersidevs}
(v \otimes s)(y,x) = \gamma_1(y) \gamma_2(x) \sigma_1(y) \sigma_2(x).
\end{gather}
Equating \brref{onesidevs} and \brref{othersidevs} shows that $v$ is
invariant, a
contradiction. The same argument takes care of the case $s \otimes v.$
Let now $v_1 = \sum c_i t_i$ and $v_2 = \sum d_i t_i,$ where the
$t_i's$
are as above
and assume that $v_1 \otimes v_2$ is invariant. Since $\oof{E}{3P}$ is
generated by
global sections there is a point $z \in E$ such that $\alpha_1(z) \ne
0,$ while
$\alpha_2(z) = \alpha_3(z) = 0.$
The fact that $(v_1 \otimes v_2)(z,y) =
(v_1 \otimes
v_2)(y, z)$ for every $y$ implies$$(c_1 + d_1)\alpha_1(z) \alpha_2(y) =
0$$
for every $y,$ which gives $c_1 = -d_1.$ Repeating the argument
permuting the
indices it follows that $v_2 = - v_1.$
It is then enough to show that $v_1 \otimes (-v_1)$ is not
$\iota$-invariant. If it
were it would follow that $(v_1(x, y) - v _1(y, x))(v_1(x, y) + v_1(y,
x))
= 0$ for all
$x,y
\in E.$ Because $v_1$ is not invariant and it is not zero everywhere,
this is a
contradiction.
\end{pf}
\begin{lemma}\label{tipidicurve}
Let $(S, L)$ be as in \brref{genusfour} Case 2 . Then
\begin{enumerate}
\item[i)] If $r$ is a line contained
in $S,$ then
$r=E_i$ or $r=\frak{f}-E_i$, $i=1,2,3.$
\item[ii)] If $C\subset S$ is a reduced irreducible cubic with $C^2 \ge
0$
then $C$ is
a curve whose numerical class is $C \equiv \frak{C_0} + \frak{f} -
\sum_1^3 E_i$
\item[iii)] If $C\subset S$ is a reduced irreducible quartic with $C^2
\ge 0$ then $C$
is a curve whose numerical class is one of the following:
\begin{enumerate}
\item[a)] $C\equiv \frak{C_0}$
\item[b)] $C \equiv \frak{C_0} +\frak{f} - E_i -E_j $
\end{enumerate}
\end{enumerate}
\end{lemma}
\begin{pf}
Let $r=a\frak{C_0}+b\frak{f}-\sum_1^3a_iE_i$ be a line in $S$. Then
$L \cdot r=4a+2b-\sum_1^3a_i=1$ and
$0=2g(r)=2+a(a-1)+2b(a-1)-\sum_1^3a_i(a_1-1)$. It follows that
\begin{gather}
\sum_1^3a_i^2=a^2+3a+2ab+1\geq\frac{1}{3}(4a+2b-1)^2 \notag \\
\text{i.e.} \ \ \ \ \ 4b(b-1)+13a(a-1)+10ab-4a-2\leq 0
\label{diseqretta}
\end{gather}
Since $r$ is an irreducible smooth curve either $r=E_i$ or $r$ is the
strict
transform of an irreducible curve on the $\Pin{1}$-bundle and the
following cases
can occur:
\begin{itemize}
\item $a=0$ and $b=1$, that gives us the fibers through the points blown
up, i.e. $r=\frak{f}-E_i$ for $i=1,2,3$.
\item $a=1$ and $b\geq 0$ , for which \brref{diseqretta} would imply
$b=0$,
$\sum_1^3a_i =3$ and $\sum a_i^2=5$, i.e. $r=\frak{C_0}-2E_i-E_j$ for
$i,j=1,2,3$. But this
would imply the existence of an irreducible curve in $|C_0|$ passing
through a
point
$P_i$ with multiplicity $2,$ that would imply $C_0 \cdot f=2$, where
$f$
is the fiber
through $P_i$, which is a contradiction.
\item $a\geq 2$ and $b\geq -\frac{a}{2}$. Let $a=2+h$ with $h\geq 0.$
>From \brref{diseqretta} it follows that
$4b^2+16+8h^2+27h+5h(h+2b)+8(2b+h)\leq 0$ and therefore
$ 4b^2+8h^2+17h\leq 0$, which is impossible unless $b=0$, $a=2$ which
contradicts \brref{diseqretta}.
\end{itemize}
Cases ii) and iii) follow from similar computations and Castelnuovo's
bound
on the
arithmetic genus of curves.
\end{pf}
Techniques found in \cite{Alibabaquad} and a detailed analysis of the
geometry of
hyperplane sections will be used to deal with Case 2.
\begin{prop}
\label{selincones} Let $(S, L) \in \cal{S}_4$ be as in Case 2. Then
$(S, L)$ is
projectively normal.
\end{prop}
\begin{pf}
By contradiction assume $(S, L)$ is not projectively normal. Lemma
\ref{g4noquad}
then implies that
$(S, L)$ must be contained in a quadric hypersurface $\Gamma.$ From
\cite{ar-so2}
it follows that $\Gamma$ must be singular.
\begin{case} $rk(\Gamma)=5.$
\end{case} Let $P$ be the vertex of the quadric cone $\Gamma.$ Following
the
notation of subsection
\ref{qcones} let $S'=\alpha \overline{H} +X.$ If $P$ is not contained
in
$S$, then $S=S',$
$deg(S)=2\alpha$, which is impossible. We can assume $P\in S$. Then
$S'=4\overline{H}+X$ because $deg(S')=\tau^2 S'$
and $E^2=(\restrict{T}{S'})^2=-1.$ Moreover $c_1(\Gamma^*)|_{S'} \cdot
c_1(S')=2(2L-E)(-K_S'-E)$, since
$\tau|_ {S'}=\sigma^*(L)$.
Plugging the above obtained values into \brref{DPF} a contradiction
is
reached.
\begin{case} $rk(\Gamma)=4$
\end{case}
Let $r$ be the line vertex of the cone.
>From $deg(S')=9$ we have
\begin{gather}
9=2\alpha+\beta+\gamma+\delta
\end{gather}
If $r\subset S$ then by Lemma \ref{tipidicurve} $r=E_i$ or
$r=\frak{f}-E_i$. Notice that in this case $S'= S.$ Let $T|_{S'}=\lambda
r,$ since
$T|_{S'}\cdot\tau|_{S'}=
\delta$ and $(T|_{S'})^2=-\delta^2$ we have $\lambda=\delta$ and
$\beta+\gamma=\delta-
\delta^2.$ Moreover $(S')^2=2(\alpha +\beta)(\alpha +\gamma)+2\alpha
\gamma$,
$c_2(\Gamma^*)|_{S'}=14\alpha+7(\beta +
\gamma)+3\delta$, $c_1(\Gamma^*)|_{S'}=4\sigma^*(L)-\delta r$. But from
$9=2\alpha+\beta+\gamma+
\delta$ the only possible values are
$(\alpha,\delta,\beta+\gamma)=(3,1,2)$
which give a
contradiction in \brref{DPF}.
If $S\cap r= \emptyset$ then $9=2\alpha$, since $T|_{S'}=0,$ which is a
contradiction.
If $S\cap r=\{P_1,...,P_k\}$, let $\mu_j$ be the multiplicity of
intersection at $P_j$ and
let $s=\sum\mu_j$. Then $(T|_{S'})^2=-\sum \mu_j=-s.$ If any of the
$\mu_j's$ is
strictly greater then $1,$ $S'$ acquires a singularity of type $A_{\mu_j
-
1}$ at a
point of $\overline{E_j},$ where $\overline{E_j}$ are the
exceptional divisors of $\restrict{\sigma}{S'}.$ Notice that $(\tau
T)S'=0$
gives
$S'=\alpha Q+\beta p_1+\gamma p_2$, with $\beta+\gamma=s.$
Moreover it is $\alpha\geq 2$ because $\alpha$ can be viewed as the
degree
of the
generically finite rational map $\psi: S\longrightarrow Q$ induced by the
projection
from the
vertex of
$\Gamma$, where $S$ is not birational to $Q$. Thus the only possible values
are
$(\alpha, s)=(4,1),(3,3),(2,5)$. Let us assume at first that $\mu_j =
1$
for all $j,$ so
that $S'$ is smooth.
It is $(S')^2=2\alpha(\alpha+h) +2\beta\gamma$,
$c_2(\Gamma^*)|_{S'}=63$,
$c_1(\Gamma^*)|_{S'}=4\sigma^*(L)-\sum\overline{E_j}.$ Using the admissible
values for
$\alpha$ and $s$ we get a contradiction in \brref{DPF}.
Therefore for at least one $j$ it is $\mu_j \ge 2$ and $(\alpha,s) =
(4,1)$ does
not occur.
Let $\Pi\subset \Pin{3}$ be a general 2-plane tangent to $Q.$ Then $\Pi
\cap Q = \ell_1
\cup
\ell_2$ where $\ell_i$ is a line in $\Pi.$ Cutting $S$ with
the hyperplane spanned by $\Pi$ and $r$ we get a degree nine divisor
$D\in |L|$
which must be reducible as $D=D_1 \cup D_2$ where $\psi(D_i) =\ell_i.$
Moving $\Pi$ along $\ell_i$ we can see that $D_j$ moves at least in a
pencil. Therefore $h^0(D_i)\ge 2$ for $i=1,2.$ Moreover the above
argument shows
that $D_i$ is spanned away from $S\cap r.$ In particular $D_i$ cannot
have
a fixed
component, therefore $D_i^2 \ge 0.$ Let
$d_i = L
\cdot D_i.$ Then
$(d_1,d_2) = (1,8), (2,7), (3,6),(4,5).$
Lemma \ref{tipidicurve} shows that $S$
contains only a finite number of lines, therefore the first case cannot
happen.
In the
second case, moving $\Pi$ along $\ell_2$, $S$ could be given a conic
bundle
structure
over $\Pin{1}$ which is not possible.
When $(d_1,d_2) = (3,6)$ notice that $D_1$ must be reduced and
irreducible
because $S$ contains only a finite number of lines. Therefore $D_1\equiv
\frak{C_0}
+
\frak{f} - \sum_i E_i$ as in Lemma
\ref{tipidicurve} ii). When $s=3$, $\psi $ is a generically $3:1$ map
while
when $s=5$
$\psi
$ is a generically $2:1$ map. Therefore there is always at least a point
$P\in S\cap
r$ such that
$P
\not \in D_1.$
pertanto
meno con
Because $h^0(X, C_0 + f)=3$ and $C_0 + f$ is spanned,
$h^0(D_1) \le 2.$ Since $|D_1|$ must be at least a pencil, it is
$h^0(D_1)
= 2.$ This
shows that the complete linear system $|D_1|$ is obtained by moving
$\Pi$ along
$\ell_2.$ A member
of
$|D_1|$ passing through $P$
can then be found, contradiction.
Let now $(d_1, d_2) = (4,5).$ Assume $D_1$ reduced and irreducible. Then
$D_1$
must be as in Lemma \ref{tipidicurve} iii). Because $h^0(\frak{C_0}) =1$
it is
$D_1\equiv \frak{C_0} + \frak{f} - E_i -E_j.$ We claim that $D_1$ is
then a
smooth
elliptic quartic embedded in
$\Pin{3}.$ To see this notice that every element of $|C_0 + f|$ on $X$
is smooth
with the only exception of one curve, reducible as the union $C_0 \cup
f.$
Moreover notice that the same argument used above shows that $h^0(D_1) =
2$
and $|D_1|$
is obtained by moving $\Pi$ along $\ell_2.$ Because
$\psi(D_1) =\ell_1$, for degree reasons
$D_1$ must go through at least one point in $S\cap r.$ Because $\mu_j\ge
2$
for at
least one $j$ and $h^0(D_1)=2$ we can always assume that $D_1$ has a
$(k\ge
3)$-secant line.
It is known (see \cite{Io2}) that the ideal of such quartics in
$\Pin{3}$
is generated
by quadrics and therefore they cannot have $(k \ge 3)$-secant lines.
Let $D_1$ now be reducible or non reduced. $D_1$ cannot be reducible
with
lines as
components since $S$ contains only a finite number of lines. A simple
numerical check shows that the only smooth conics on $S$ have numerical
class
$\frak{f}$. Therefore we can assume $D_1 \equiv 2\frak{f.}$ As it was
pointed out above $D_1$
must pass through at least a point $P \in S \cap r$ but this contradicts
$\frak{f}^2 =
0.$
\begin{case} $rk(\Gamma)=3$
\end{case}
Assume $S \subset \Gamma$ where $\Gamma$ is a quadric cone with $rk
\Gamma
= 3$ and vertex $ V \simeq \Pin{2}$ over a smooth conic
$\gamma\subset\Sigma
\simeq \Pin{2}.$ Let $\psi: S -->\gamma$ be the rational linear
projection from $V.$
Let $\ell_1, \ell_2$ and $\ell_3$ be distinct lines in $\Sigma$ such
that
$\gamma \cap
\ell_1 =\{P_1, P_2\}$,\ \ $\gamma \cap \ell_2
=\{P_2, P_3\}$, and $\gamma \cap \ell_3
=\{P_1, P_3\}$, where $P_i \neq P_j.$ Let $D_i$ be the hyperplane
sections
of $S$
given by the hyperplanes spanned by $V$ and $\ell_i.$ Notice that $D_i$
must be
reducible.
Assume that $D_i$ has no components contained in $V$ for at least one
$i$, say
$i=1.$ Then
$D_1\sim A+B$ where $\psi(A)=P_1$ and $\psi(B)=P_2.$ Let $L\cdot A =
a$ and
$L \cdot B=b.$ It follows that $D_2 \sim A'+B$ and $D_3 \sim A'+A$.
Notice
that $A
\sim A'$ and so $L\cdot A'=a.$ This leads to the contradiction
$2a=L\cdot
D_3=9.$
It follows that for any hyperplane section $D$ obtained
with the hyperplane spanned by a line $\ell \subset \Sigma$ and $V$ it
must
be $D
\sim 2C + F$ where $F\subset V$ and no component of $C$ is contained in
$V.$
Because $|C|$ is at least a pencil and it cannot clearly have fixed
components, it must
be $C^2\ge 0.$
Since $S$ contains only a finite number of lines and it is not a
rational
conic bundle
it is $L\cdot C = 3,4.$ If $L\cdot C =3$ $C$ must be irreducible and
therefore as in
Lemma \ref{tipidicurve}, i.e. $C\equiv \frak{C_0}+\frak{f} - \sum_i
E_i.$
It follows that
$F \equiv E_1+E_2+E_3$ which is impossible because these three lines are
disjoint.
If $L\cdot C=4$ and $C$ is reduced and irreducible then $C \equiv
\frak{C_0} +
\frak{f} - E_i-E_j$ as in Lemma \ref{tipidicurve} iii) (the case
$C\equiv
\frak{C_0}$ cannot happen since
$h^0(\frak{C_0})=1$). Then
$F \equiv E_i+E_j - E_k$ which is impossible. If $C$ is reducible or non
reduced then
$C \equiv 2\frak{f}$ and $F\equiv 2\frak{C_0}-2\frak{f} -\sum_iE_i $
which
is not
effective because
$C_0$ is ample on $X$ and $C_0\cdot(2C_0-2f) =0.$
\end{pf}
\subsection{SECTIONAL GENUS $g =5$}
In this section we will study the projective normality of pairs
$(S, L)\in{
\cal S}_5.$ From Lemma \ref{nonP41} it follows that either $(S, L)$ is
known to
be projectively normal or $S\subset
\Pin{5}$ and hence
$\Delta(S, L)=5.$
Since $g=\Delta$ and $d\geq 2\Delta-1$ the ladder
is regular and therefore $(S, L)$ is projectively normal if
$(C,\restrict{L}{C})$ is
projectively
normal.
\begin{lemma}
\label{equivalent}
Let $(S, L)\in{\cal S}_5$ then if $h^1(L)=0$ the following statements
are equivalent:
\begin{itemize}
\item[1)] $(S, L)$ is projectively normal
\item[2)] $(C,\restrict{L}{C})$ is projectively normal
\item[3)] $S$ is contained in exactly one quadric hypersurface in
$\Pin{5}$.
\end{itemize}
\end{lemma}
\begin{pf} Since $h^1(L)=0,$ $(S, L)$ is projectively normal if and only
if
$(C, \restrict{L}{C})$
is projectively normal by Lemma \ref{besanaignorans}. Moreover $(C,
\restrict{L}{C})$ is
projectively normal if and only if it is 2-normal, by Lemma \ref{2norm}.
Consider the exact sequence:
$$0\longrightarrow{\cal I}_S(1)\longrightarrow{\cal I}_S(2)\longrightarrow{\cal I}_C(2)\longrightarrow 0.$$
It is $h^1({\cal I}_S(1))=0$ since $L$ is linearly normal and $h^0({\cal
I}_S(1))=0$ since
$S$ is non degenerate. It follows that $h^0({\cal I}_S(2))=h^0({\cal
I}_C(2)).$
Because $h^0({\cal O}_{\Pin{4}}(2))=15$ and $h^0(2\restrict{L}{C})=14$
we
have that
$(C, \restrict{L}{C})$ is
projectively normal if and only if $h^0({\cal I}_C(2))=1$ and therefore
$(S, L)$ is
projectively normal if and only if $S$ is contained in exactly one
quadric.
\end{pf}
\begin{lemma} \label{trighyper}
Let $(S, L) \in \cal{S}_5.$ Then either $(S, L)$ is projectively normal or
it has a
hyperelliptic or trigonal section $C \in |L|.$
\end{lemma}
\begin{pf}
Since
$g(C)=5$ then
$cl(C)\le 2.$ If $cl(C)=2$ then by Theorem \ref{glcliff} $(C,
\restrict{L}{C})$ is
projectively normal and thus $(S, L)$ is projectively normal by regularity
of
the ladder.
\end{pf}
\begin{lemma}
\label{cbhyper} Let $(S, L)\in{\cal S_5}$ with a hyperelliptic section
$C\in |L|.$ Then $S$ is
a rational conic bundle, not projectively normal.
\end{lemma}
\begin{pf} Surfaces with hyperelliptic sections are classified in
\cite{so-v}.
By degree considerations the only possible case is a rational conic
bundle.
Since
$h^1(L)=0$
Proposition \ref{hyper} and Lemma \ref{equivalent} imply that $(S, L)$ is
not
projectively normal.
\end{pf}
\begin{lemma}
\label{cbtrig}
There are no conic bundles $(S, L)$ with a trigonal
section $C\in |L|$ in ${\cal S}_5$ .
\end{lemma}
\begin{pf} If $S$ is a conic bundle with trigonal section then
\cite{fa} Lemma 1.1 gives
$g=2q+2$ which is impossible because \cite{LiAq} gives $q\leq 1.$
\end{pf}
\begin{theo}
\label{g5theo}
Let $(S, L)\in{\cal S}_5.$ Then $(S, L)$ fails to be projectively
normal if and
only if it is
\begin{itemize}
\item[1)] A rational conic bundle;
\item[2)] $(S, L)=(Bl_{12}\bold{ F_1}, 3\frak{C_0}-5\frak{f}-12p)$ with
trigonal section $C \in |L|$
and \\$\restrict{L}{C}=K_C-g^1_3+D$.
\end{itemize}
\end{theo}
\begin{pf}
>From \cite{LiAq}, Lemma \ref{trighyper}, Lemma
\ref{cbhyper}
and Lemma \ref{cbtrig}, the following cases are left to investigate:
\begin{center}
\begin{tabular}{|l|l|l|l|r|} \hline
Case &$S$ & $L$& existence\\ \hline\hline
1 & $Bl_{10}\Pin{2}$ & $7p^*(\cal O_{\Pin{2}}(1))-10\sum_1^{10} 2E_i$ &
Yes \\ \hline
2 & $Bl_{12}\Pin{2}$ & $6p^*(\cal O_{\Pin{2}}(1))-\sum_1^5
2E_i-\sum_6^{12} E_j$ &Yes\\ \hline
3 & $Bl_{10}\bold {F_e}$, $e=0,1,2$ &
$4\frak{C_0}+(2e+5)\frak{f}-\sum_1^7
2E_i-\sum_8^{
10} E_j$ & Yes\\ \hline
4 & $Bl_{12} \bold{ F_1}$ & $3\frak{C_0}+5\frak{f}-\sum_1^{12}E_i$ & ?
\\ \hline
\end{tabular}
\end{center}
where the hyperplane section is trigonal.
In case 2 $(S, L)$ admits a first reduction $(S',L')$ with $d'=16$. By
\cite{fa} $(S, L)$ cannot have trigonal section and therefore it is
projectively normal.
In case 3 it is $K_S^2=-2$ so that $K_S(K_S+L)=-3$. Then by
\cite{bri-la}
Theorem 2.1
$(S, L)$ cannot have trigonal section thus it is projectively normal.
Case 1 is a congruence of lines of $\Pin{3}$ of bi-degree $(3,6)$
studied in detail
in \cite{ar-so2}.
In particular if ${\cal I}_S^*$ is the ideal of $S$ in the grassmanian
$G(1,3)$ of
lines of $\Pin{3},$ it is $h^0({\cal I}_S^*(2))=0$. From:
$$ 0\longrightarrow {\cal I}_G\longrightarrow{\cal I}_S\longrightarrow{\cal I}_S^*\longrightarrow 0$$
recalling that $G\in|{\cal O}_{\Pin{5}}(2)|$ we get $h^0({\cal
I}_S(2))=1$
and therefore
$(S, L)$ is projectively normal by Lemma \ref{equivalent}.
By \cite{GL} Corollary 1.6 a trigonal curve of genus 5 and degree 9 in
$\Pin{4}$ fails to
be projectively normal if and only if it is embedded via a line bundle
$\restrict{L}{C}=K_C-g^1_3+D$ where
$D$ is an effective divisor of degree 4. Notice that this means that $
C$
is embedded in
$\Pin{4}$ with a foursecant line.
\end{pf}
\begin{rem} (ADDED IN PROOF)
\label{bali}
After this work was completed the first author and A. Alzati proved in
in \cite{bali}
that there exist no surfaces as in Theorem \ref{g5theo} Case 2).
Therefore
this case does not appear in the table of Theorem 1.1.
\end{rem}
\subsection{SECTIONAL GENUS $g= 6$}
\label{genere6}
In this subsection we will study the projective normality of pairs
$(S, L)\in{
\cal S}_6$. Notice that by Lemma \ref{nonP41} $L$ embeds $S$ in
$\Pin{5}$
and the
ladder is regular.
\begin{lemma}
\label{2normg6}Let $C$ be a curve of genus $6$ embedded in $\Pin{4}$ by
the
complete linear system associated with a very ample line bundle
$\restrict{L}{C}$ of
degree
$9$. Then
$C$ is
$2$-normal.
\end{lemma}
\begin{pf} Consider the exact sequence: $$0\longrightarrow {\cal I}_C\otimes{\cal
O}_{
\Pin{4}}(2)\to{\cal O}_{\Pin{4}}(2)\to{\cal O}_C(2)\longrightarrow 0$$
Since $h^0({\cal O}_{\Pin{4}}(2))=15$ and $h^0({\cal
O}_C(2))=h^0(2\restrict{L}{C})=18+1-6=13$ it is $h^0(\iof{C}{2}) \ge 2$
with the
map
$H^0({\cal O}_{\Pin{4}}(2))\longrightarrow H^0({\cal O}_C(2))$ surjective if and
only
if $h^0(
{\cal I}_C(2))=2.$ Assume $h^0(\iof{C}{2})\ge 3,$ and let $Q_i$ for $i
=1,2,3$ be
three linearly independent quadric hypersurfaces containing $C.$ Because
deg$C
=9$ and $C$ is non degenerate it must be $dim(\cap_iQ_i) = 2.$ Let
$\frak{S} =
\cap_iQ_i$ then $h^0(\frak{S}) \ge 3.$ But $\frak{S}$ is a complete
intersection
$(2,2)$ in $\Pin{4}$ and it is easy to see that $h^0(\frak{S}) = 2$,
contradiction.
\end{pf}
\begin{cor}\label{gen6} Let $(S, L) \in \cal{S}_6.$ Then $(S,L)$ is
projectively normal.
\end{cor}
\begin{pf} Let $C\in|L|$ be a generic section. From Lemma \ref{2normg6}
and
Lemma
\ref{2norm} it follows that $(C,\restrict{L}{C})$ is projectively
normal.
Since the
ladder is regular
this implies that $(S, L)$ is projectively normal.
\end{pf}
\section{Results on Scrolls}
A $n$-dimensional polarized variety $(X, L)$ is said to be a scroll over a
smooth curve
$C$ of genus $g$ if there is a vector bundle \map{\pi}{E}{C} of rank $r
=
rk\,E =
n+1$ such that $(X, L) \simeq ({\Bbb P}(E), \oof{{\Bbb P}(E)}{1}).$
Recall that given a vector bundle $E$ over a curve $C$,
$\mu (E)$ and $\mu^- (E)$ of
$E$ are defined as ( see \cite{bu} for details) $$\mu(E) = \frac{deg
E}{rk E} =
\frac{d}{r}.$$
$$\mu^-(E)=min\{\mu(Q)|E\rightarrow Q\rightarrow 0\}$$
$E$ will be called {\em very
ample} to signify that the tautological line bundle $\taut{E}$ is a very
ample line bundle on
${\Bbb P}(E).$
From \cite[Th 5.1.A]{bu} and general properties of projectivized bundles
it follows that:
\begin{prop}[\cite{bu}]\label{buscroll} Let $(X, L)$ be a scroll. If
$\mu^-(E)>2g$ then
$(X, L)$ is projectively normal.
\end{prop}
The following Lemma is essentially due to Ionescu \cite{fa-li9} :
\begin{lemma}[Ionescu]\label{scrolls}
Let $(X,L)$ be an n-dimensional scroll over a hyperelliptic curve $C$ of
genus $g$
with $L$ very ample. Then
$\Delta=ng.$
\end{lemma}
\begin{pf}
a) Let $X=\Proj{E}$, $L=\taut{E}$ and $\pi:X\longrightarrow
C$.\\ By the Riemann-Roch theorem and the fact that $\pi_*({\cal
O}_X(1))=E$ it follows that:
$$h^0(L)=h^0(C,E)=h^1(C,E)+d-n(g-1)$$
Thus it is enough to show that $h^1(C,E)=0$.\\
By Serre duality $h^1(C,E)=h^0(C,K_C\bigotimes E^{*})$. Assume
$h^1(C,E)\neq 0.$ A non trivial section $\sigma\in
h^0(C, K_C\bigotimes E^*)$ gives the following
surjection:
\begin{equation}
\label{sigmasur}
(K_C\bigotimes E^*)^*\longrightarrow{\cal O}_C(-D)\longrightarrow 0
\end{equation} where
$D$ is the divisor on $C$ associated to $\sigma$. Tensoring
\brref{sigmasur} with $K_C$ we obtain
$$E\longrightarrow K_C-D\longrightarrow 0.$$ Because $\taut{E}$ is very ample,
$K_C-D$ is very ample on C. Moreover $K_C-D$ is a special line bundle on
$C$ because $h^1(K_C-D)=h^0(D)>0$. This is impossible because $C$ is
hyperelliptic.
\end{pf}
\begin{lemma}
\label{scrollg2}
Let $(S,L)$ be a two-dimensional scroll over a curve of genus 2 and
degree 9 in
${\Bbb P}^{6}$ with
$L$ very ample. Then
$X={\Bbb P}(E)$ with $E$ stable and $(X, L)$ is projectively normal.
\end{lemma}
\begin{pf} Let $S={\Bbb P}(E)$ where $E$ is a rank $2$ vector bundle of
degree $9$ over a smooth curve
of genus $2$. If $E$ is stable then $\mu^-(E)=\mu(E)=\frac{9}{2}>4$ and
so by Proposition \ref{buscroll} $(S,L)=({\Bbb P}(E),{\cal O}_{{\Bbb
P}(E)})$ is projectively normal.
Assume now $E$ non stable. Then there exists a line bundle $Q$ with
deg($Q)\leq 4$ such that $E\to Q\to 0$. This contradicts the very
ampleness
of $Q$
as a quotient of a very ample $E.$
\end{pf}
\begin{prop}
\label{scrollprop}
Let $(S, L)$ be a scroll of degree $d=9$ over a smooth curve $C$ of genus
$g.$ Then
$(S, L)$ is projectively normal unless possibly if $C$ is trigonal, $3\le
g
\le 5$ and
$S\subset \Pin{5}.$
\end{prop}
\begin{pf}
Following the proof of Lemma \ref{nonP41} if $\Delta \ge 2$ and $g=1$
then
$(S, L)$ is an elliptic scroll (see \cite{fu})
and therefore projectively normal by
\cite{Alibaba} or \cite{Ho1},\cite{Ho2}.
If $\Delta =4$ and $g=2$ then $(S, L) $ is projectively normal by Lemma
\ref{scrollg2}.
Let $\Delta=5.$ If $g=6$ then $(S, L)$ is projectively normal by
\ref{gen6}.
If $g=5$
by Theorem \ref{fujitatheo}, \ref{fujitatheo2}, \ref{glcliff}
$(S, L)$ is projectively normal unless $cl(C)\le 1.$ If $g=3,4$ then it is
always $cl(C)\le 1.$
By Lemma \ref{scrolls} $C$ must be trigonal.
\end{pf}
\section{An Adjunction Theoretic Problem}
\label{K+L}
The question of finding examples for the problem posed by Andreatta, Ein
and
Lazarsfeld (see introduction) is addressed below.
\begin{cor}
Let $(S, L)$ be a surface polarized with a very ample line bundle of
degree $d=9$
such that the embedding given by $|L|$ is not projectively normal. Then
there
does not exist a very ample line bundle $\cal{L}$ such that $L= K_S +
\cal{L}$ unless
$(S, L)$ is the blow up of an elliptic $\Pin{1}$-bundle as in the first
case of Theorem
\ref{thetheorem}.
\end{cor}
\begin{pf}
Let $(S, L)$ be as in the Table of Theorem \ref{thetheorem}, not as in the
first case. Assume $L=K +
\cal{L}$ with $\cal{L}$ very ample. Computing $\cal{L}^2$ and
$g(\cal{L})$ and
using \cite{LiAq} lead to a contradiction
in every case.
Similarly a contradiction is reached if $(S, L)$ is a scroll over a curve
of
genus $3,4,5.$
\end{pf}
\begin{rem}
The existence of an example of a surface as in case 1 of Theorem
\ref{thetheorem}
where
$L=K+\cal{L}$ with $\cal{L}$ very ample is a very delicate question.
Let $E$ be an indecomposable rank $2$ vector bundle over an elliptic
curve
$\cal{C}$ with
$c_1(E) =0$ and let
$X =\Proj{E}.$ Let $C_0$ be the fundamental section, let $M$ be any line
bundle
whose numerical class is $2C_0 + f$ and let $p: S=Bl_3X
\to X$ be the blow up
of $X$ at three points $P_i$ $i=1,..,3.$ Using the same notation for the
blow up
introduced in subsection
\ref{notation} consider a line bundle
$L\equiv 2\frak{C_0} + 3\frak{f} -
\sum_iE_i.$
Notice that $L\equiv K_S + \cal{L}$ where $\cal {L}\equiv 4\frak{C_0} +
3\frak{f} -
\sum_i2E_i.$
Moreover $\cal{L} \equiv K_S + H,$ $H^2 = 9,$ $H \equiv 3T$ where $T=
p^*(M) -
\sum_1^3 E_i.$ Recent results of Yokoyama and Fujita \cite{fuyo},
\cite{Yoko}) show that the $P_i's $ can be chosen generally enough to
have
$T$ ample but not effective.
Reider's theorem then shows that
$\cal{L}$ is very ample if it is possible to choose the $P_i's$ such
that for
\underline{every} line bundle $M$ whose numerical class is $2C_0 + f$
it is
$|p^*(M)-\sum_i E_i|= \emptyset.$
\end{rem}
|
1997-10-11T00:32:03 | 9710 | alg-geom/9710013 | en | https://arxiv.org/abs/alg-geom/9710013 | [
"alg-geom",
"math.AG"
] | alg-geom/9710013 | Ilya Zakharevich | Ilya Zakharevich | Quasi-algebraic geometry of curves I. Riemann-Roch theorem and Jacobian | 99 pages, AmsLaTeX | null | null | null | null | We discuss an analogue of Riemann-Roch theorem for curves with an infinite
number of handles. We represent such a curve X by its Shottki model, which is
an open subset U of CP^{1} with infinite union of circles as a boundary. An
appropriate bundle on X is \omega^{1/2} \otimes L, L being a bundle with (say)
constants as gluing conditions on the circles. An admissible section of an
appropriate bundle on X is a holomorphic half-form on U with given gluing
conditions and H^{1/2}-smoothness condition.
We study the restrictions on the mutual position of the circles and the
gluing constants which guarantee the finite dimension of the space of
appropriate sections of admissible bundles, and make the Riemann-Roch theorem
hold. The resulting Jacobian variety is described as an infinite-dimension
analogue of a torus.
| [
{
"version": "v1",
"created": "Fri, 10 Oct 1997 22:32:01 GMT"
}
] | 2007-05-23T00:00:00 | [
[
"Zakharevich",
"Ilya",
""
]
] | alg-geom | \section{Introduction }
The need to carve out a set of curves of infinite genus for which
``most'' theorems of algebraic geometry are true comes out from the
following observations:
\begin{enumerate}
\item
The existence of algebro-geometric description of solutions of
infinite-dimensional integrable systems;
\item
The ability to describe the series of perturbation theory for string
amplitudes as integrals over moduli spaces;
\item
The hope that the union of compactifications of moduli spaces may
have a simpler geometry than the moduli spaces themselves.
\end{enumerate}
Different approaches which would result in different sets of curves
are possible (as in papers of Feldman, Kn\"orrer and Trubowitz
cf.~\cite{FelKnoTru96Inf}), thus we first motivate our choice of
tools
(Shottki model, $ H^{1/2} $-topology, capacities and half-forms) as (probably) a
best one to fulfill the expectations of the above origins of the theory.
Until Section~\ref{s0.10} we discuss how the above topics motivate the choice
of the Shottki model as a way to describe a curve of infinite genus.
After this (up to section~\ref{s0.40}) we describe motivations for the
choice of half-forms, $ H^{1/2} $-topology, and generalized-Sobolev-spaces (or
capacities) to describe admissible sections on a given curve. In the
remaining part of the introduction we do a walk-through the methods and
results one can find in this paper, as well as some historic remarks.
\subsection{Integrable systems and algebro-geometric solutions } A great
break-through in the first topic came with the work \cite{McKTru76Hil}. A
hyperelliptic curve of genus $ g $ with real branching points and a divisor of
degree $ g $ on this curve allows one to construct a so-called
{\em algebro-geometric\/} solution of KdV equation
\begin{equation}
u_{xx x}+12uu_{x}-4u_{t}=0.
\notag\end{equation}
Such solutions are called $ g $-{\em gap potentials}. McKean and Trubowitz studied
what substitutes algebraic geometry for solutions of KdV which are {\em not\/}
finite gap potentials. To such a solution they associated a curve (i.e.,
a complex manifold of dimension 1) which was hyperelliptic of {\em infinite
genus}, i.e., had infinitely many branching points, and some substitution
for the notion of a divisor of degree $ g $. It was shown that for the curves
which are related to KdV equation one can construct a well-behaved
analogue of $ 1 $-dimensional algebraic geometry. One should consider this
analogue as a generalization of algebraic geometry to hyperelliptic
curves of infinite genus (in fact only to some special curves of this
type).
Note that other integrable systems lead to different classes of
curves which appear in algebro-geometric solutions for the systems. Thus
one may expect that infinite-dimensional integrable systems may
lead to generalizations of algebraic geometry to different classes of
curves of infinite genus.
For this approach one of the richest systems is so called KP system
\begin{equation}
\left(u_{x xx}+12uu_{x}-4u_{t}\right)_{x} +3u_{yy}=0.
\notag\end{equation}
To describe an algebro-geometric solution of a KP system one starts with
an arbitrary algebraic curve, and an arbitrary linear bundle on this
curve. Thus to generalize the approach of \cite{McKTru76Hil} to the KP
equation, one needs to study curves of infinite genus of {\em generic\/} form (as
opposed to a hyperelliptic curve) and bundles on them.
Suppose for a minute that we have such a generalization, i.e., a
collection of curves and bundles on them. Call the members of these
families {\em admissible}. We assume that to any admissible curve and a bundle
on it we can associate some solution of KP equation. Let us investigate
what can we deduce about this collection from the known properties of KP
system (the properties below are applicable at least to algebro-geometric
solutions).
The dynamics of KP leaves the curve the same, but changes the
bundle. Thus the collection of admissible bundles should be reach enough
to include all the bundles obtained by time-flow of KP. In fact KP can be
generalized to include an infinite collection of commuting flows (with
different time variables), and they (taken together) can transform any
bundle to any other one (at least in finite-genus case). Thus we should
expect that we need our collection to include {\em all\/} the possible bundles,
thus the whole Jacobian.
The dynamics of KP is described in terms of locally affine structure
on the Jacobian, hence one should be able to describe the Jacobian as a
quotient of a vector space by a lattice. Algebraic geometry identifies
the Jacobian with a quotient of the space of global holomorphic forms by
the forms with integer periods. Thus one needs something similar to this
description.
There is an alternative description of a solution of KP equation in
terms of so called $ \tau $-{\em functions}. The relation of algebro-geometric
description with the description in terms of $ \tau $-functions needs a
construction of a Laurent series of a meromorphic global section of an
admissible bundle. This ceases to be trivial in infinite-genus case,
since already the description in \cite{McKTru76Hil} shows that one may need to
apply this operation at the {\em infinity\/} of the curve, i.e., at the points
added to the curve to compactify it. These points {\em are not\/} smooth points
of the curve, the curve is not even a topological manifold near these
points.
Finally, to get somewhat {\em explicit\/} description of solutions of KP as
functions of several variables, one needs a way to explicitly describe
global sections of linear bundles. Moreover, the bundles we need to
consider should have a finite number (preferably one!) of independent
global sections. A tool to construct such bundles in the case of finite
genus is the Riemann--Roch theorem, which gives an estimate on the
dimension of sections of the given bundles, and this estimate is precise
in the case of bundles in generic position. These sections are described
in terms of $ \theta $-functions, thus we will also need to describe $ \theta $-functions.
Collecting all this together, we see that we need to describe curves
$ X $ which can be equipped with a linear bundle $ {\cal L} $ such that the space of
global sections is finite-dimensional. One should be able to define what
does it mean that two bundles $ {\cal L} $ and $ {\cal L}' $ are equivalent, describe the
equivalence classes of bundles in terms of global holomorphic $ 1 $-forms,
give a local description of sections of the bundles, and give a global
description of sections of the bundles in terms of $ \theta $-functions.
In this paper we do not complete this program. However, we describe
all the ingredients but the last one.
\subsection{Universal Grassmannian }\label{s0.4}\myLabel{s0.4}\relax The other two origins, the string theory
and geometry of moduli spaces, come into play if we consider the
algebro-geometric methods of solving integrable systems in the other
direction, as a way to find information of algebro-geometric type from
solutions of integrable systems (similar to solution of Shottki problem
in \cite{Shio86Char}).
The key idea is that in the investigated cases the set of
algebro-geometric solutions is {\em dense\/} in the set of all solutions, thus
the set of the solutions is a {\em completion\/} of the set of algebro-geometric
solutions. Since the the set of algebro-geometric solutions is a moduli
space of appropriate structures, and the set of solutions is a linear
space (due to the possibility to solve the Cauchy problem), we see that
the moduli space has a completion which has a topology of a linear space.
While the moduli spaces carry remarkable measures \cite{BeiMan86Mum} used
in the integrals of the string theory, this linear space has a symplectic
structure, so one may expect that this simplectic structure may have a
relationship to the measure on the subsets corresponding to
algebro-geometric solutions.
To carry out this program one needs to investigate how an arbitrary
solution may be approximated by algebro-geometric solutions. For KdV
equation this question was answered by \cite{McKTru76Hil} (for a fixed
spectrum), ameliorated to describe the inclusion of the subset of finite
gap potentials up to a diffeomorphism in \cite{ZakhFinGap2,ZakhFinGap3}, up
to a symplectomorphism in \cite{BatBloGuil95Sym}. The answer is that the above
inclusion is isomorphic to inclusion of trigonometric polynomials into
the space of all functions (for a very wide range of spaces of
functions). Since trigonometric polynomials fill finite-dimensional
coordinate subspaces in spaces of functions (using Fourier coefficients
as coordinates), we see that this picture is very similar to one of a
divisor with normal crossings.
Much simpler problem is how an algebro-geometric solution may be
approximated by (simpler) algebro-geometric solutions. A remarkable fact
is that such problems for different classes of integrable systems may be
solved via a uniform approach: compactification of moduli spaces via
{\em Universal Grassmannian}.
The reason is that the standard way to associate a solution of an
integrable system to an algebro-geometric data comes from consideration
of the mapping to {\em Universal Grassmannian}. The Universal Grassmannian
gives a convenient way to say that the space of sections of a linear
bundle on one curve is close to the space of sections on another linear
bundle on another curve.
Consider an algebraic curve $ X $ with a fixed point $ P\in X $ and a local
coordinate system $ x $ in a neighborhood of $ P $ which maps $ P $ to $ 0\in{\Bbb C} $. Let $ {\cal L} $ be
a linear bundle on $ X $, fix a trivialization of this bundle in a
neighborhood of $ P $. One can associate a Laurent series $ {\frak l}\left(\varphi\right) $ to any section
$ \varphi $ of $ {\cal L} $ in a punctured neighborhood of $ P $: using a coordinate system on $ X $
and the trivialization of $ {\cal L} $, one can write this section as a function of
$ z\approx0 $, $ z\not=0 $. Let $ V $ be the space of meromorphic sections of $ {\cal L} $ which are
holomorphic outside of $ P $. (Abusing divisor notations, one can write $ V $ as
$ \Gamma\left(X,{\cal L}\left(\infty\cdot P\right)\right)=\bigcup_{k}\Gamma\left(X,{\cal L}\left(k\cdot P\right)\right) $.) Then $ {\frak l}\left(V\right) $ is a subspace of the space $ {\frak L} $ of
Laurent series. Let the universal Grassmannian $ \operatorname{Gr}\left({\frak L}\right) $ be the Grassmannian
of subspaces of $ {\frak L} $. Then $ {\frak l}\left(V\right)\in\operatorname{Gr}\left(V\right) $ depends on $ X $, $ {\cal L} $, the coordinate system
near $ P $, and on the trivialization of $ {\cal L} $ near $ P $.
On $ {\frak L} $ there are natural actions of the group of formal
diffeomorphisms\footnote{I.e., invertible $ \infty $-jets of mappings $ {\Bbb C} \to {\Bbb C} $ with $ 0\in{\Bbb C} $ being a fixed point.} of $ \left({\Bbb C},0\right) $ and the group of multiplication by invertible
Taylor series. Let $ G $ be the semidirect product of these groups. The
action of $ G $ on $ {\frak L} $ corresponds (via $ {\frak l} $) to changes of coordinate system on
$ X $, and a change of the trivialization of $ {\cal L} $. We see that to a triple $ \left(X,P,{\cal L}\right) $
we can naturally associate a point $ {\frak l}\left(X,P,{\cal L}\right)\in\operatorname{Gr}\left({\frak L}\right)/G $.
Note that while only the mapping to $ \operatorname{Gr}\left({\frak L}\right)/G $ is invariantly defined,
one can get a canonical lifting to $ \operatorname{Gr}\left({\frak L}\right) $ as far as $ X\not={\Bbb C}P^{1} $. The description
below does not behave well w.r.t. deformations of the curve/bundle, but
can be easily modified to do so. Given a bundle $ {\cal L} $, we can find $ k\in{\Bbb Z} $ such
that $ {\cal L}\left(k\cdot P\right) $ has only one independent section, and $ {\cal L}\left(\left(k+1\right)\cdot P\right) $ has another
one. The ratio $ z $ of these two sections identifies a neighborhood of $ P $
with a neighborhood of $ \infty\in{\Bbb C}P^{1} $, and this identification is defined up to an
affine transformation. Now the only section of $ {\cal L}\left(k\cdot P\right) $ gives a local
identification of $ {\cal L} $ with $ {\cal O}\left(-k\cdot P\right) $. Together with a coordinate system in a
neighborhood of $ P $ it gives a local section of $ {\cal L} $ defined up to
multiplication by a constant.
What remains is to pick up a coordinate system from those which
differ by an affine transformation. If $ X\not={\Bbb C}P^{1} $, then for some $ l>k+1 $ there
is a section of $ {\cal L}\left(l\cdot P\right) $ of the form $ z^{l}+p_{l-1}\left(z\right)+az^{-m}+bz^{-m-1}+O\left(z^{-m-2}\right) $ with
$ a\not=0 $, $ p_{l-1} $ being a polynomial of degree $ l-1 $, and $ m>0 $. Taking the minimal
possible $ l $, we see that the condition that $ a=1 $, $ b=0 $ picks up a coordinate
system out of the above class, unique up to multiplication by a root of
degree $ l+m $ of 1. (If $ X={\Bbb C}P^{1} $, then of course there is no canonically
defined coordinate system, since $ \left(X,P,{\cal L}\right) $ has automorphisms.) Now using
this coordinate system and the corresponding trivialization of $ {\cal L} $ one gets
a canonically defined image of $ \left(X,P,{\cal L}\right) $ in $ \operatorname{Gr}\left({\frak L}\right) $.
Let the {\em Teichmuller\/}--{\em Jacoby\/} space $ {\cal N}_{g,1,d} $ be the moduli space (i.e.,
the ``set'' of equivalence classes) of triples $ \left(X,P,{\cal L}\right) $ with $ g\left(X\right)=g $, and
$ \deg \left({\cal L}\right)=d $. What is of primary importance to us is the fact that $ {\frak l}|_{{\cal N}_{g,1,d}} $
is an injection (even in the case when $ X $ may have double points). Indeed,
\begin{enumerate}
\item
Meromorphic functions on $ X $ may be described as ratios of elements in
$ {\frak l}\left(X,P,{\cal L}\right) $, thus normalization $ \operatorname{Norm}\left(X\right) $ of $ X $ may be reconstructed basing on
$ {\frak l}\left(X,P,{\cal L}\right) $;
\item
The point $ P $ can be reconstructed since we know all the meromorphic
functions on $ X $, and know the order of pole at $ P $;
\item
To reconstruct the lifting $ \bar{{\cal L}} $ of $ {\cal L} $ to $ \operatorname{Norm}\left(X\right) $ note that if $ \varphi\in V $, then
the divisor (of zeros) of $ \varphi $ can be reconstructed as poles of
functions in $ {\frak l}\left(\varphi\right)^{-1}{\frak l}\left(V\right) $.
\item
Finally, to describe the gluings one needs to perform to get $ X $ from
$ \operatorname{Norm}\left(X\right) $ it is sufficient to consider a complement to $ \Gamma\left({\cal L}\left(\infty\cdot P\right)\right) $ in
$ \Gamma\left(\bar{{\cal L}}\left(\infty\cdot P\right)\right) $.
\end{enumerate}
As a corollary, the mapping $ {\frak l} $ defines an inclusion of $ {\cal N}_{g,1,d} $ into
$ \operatorname{Gr}\left({\frak L}\right)/G $. Since the image has a natural structure of a topological space,
this inclusion defines some natural {\em compactification\/} $ \bar{{\cal N}}_{g,1,d} $ of the
Teichm\"uller--Jacoby space. Indeed, consider the closure of $ {\frak l}\left({\cal N}_{g,1,d}\right) $ in
$ \operatorname{Gr}\left({\frak L}\right)/G $. As we will see it shortly, this closure is an image of a compact
smooth manifold\footnote{In fact we will see that already the {\em normalization\/} of the image is
smooth.}. The description of this compactification is very similar
to the description of Deligne--Mumford compactification, it is carried
out by adding to $ {\cal N}_{g,1,d} $ so called {\em semistable\/} objects. We will see that
these object are non-smooth curves and sheaves on them.
The mapping to Universal Grassmannian is a way to collect all the
moduli spaces for different genera into ``one big heap''. It is relevant to
curves of infinite genus since on a back yard of this heap one expects to
find moduli spaces of curves of infinite genus.
\subsection{Semistable elliptic curves }\label{s0.5}\myLabel{s0.5}\relax In this section we consider the case
$ g=1 $, where one can calculate the mapping to the Universal Grassmannian
explicitly. We will see that in this case the closure of the image is
smooth\footnote{As we will see it later, $ g=1 $ is last case when one does not need
any normalization of the image.}.
Fix an elliptic curve $ E $. Then the set of classes of equivalence of
linear bundles of degree 0 (i.e., the Jacobian variety) is isomorphic to
the curve itself (after we fix a point $ P $ on the curve). Given any point
$ Q\in E $, the divisor $ Q-P $ determines a linear bundle $ {\cal O}\left(Q-P\right) $ on $ E $, and any
bundle of degree 0 is isomorphic to exactly one bundle of this form.
In the case of $ g=1 $ it is not instructive to consider the
Teichm\"uller--Jacoby manifold {\em literally}, since it is tainted by the fact
that an involution $ \sigma $ of an elliptic curve $ E $ sends any bundle $ {\cal L} $ of degree
0 to $ {\cal L}^{-1} $, thus the moduli space of bundles of degree 0 on an elliptic
curve fixed {\em up to an isomorphism\/} is $ E/\sigma $, i.e., a rational curve. To fix
the problem one can add some harmless discrete parameter which would
prohibit $ \sigma $ to be an automorphism, say, consider collections $ \left(X,P,{\cal L},\alpha\right) $, $ \alpha $
being a homology class modulo $ {\Bbb Z}/3Z $. However, knowing that we can fix this
problem, we are going to ignore it whatsoever, since it can also be
avoided by considering curves close to the given one, what we are going
to do anyway.
For every elliptic curve $ E $ there is a unique number $ j\in{\Bbb C} $ such that $ E $
is isomorphic to the curve
\begin{equation}
y^{2}=\left(500j-1\right)x^{3}-15jx-j.
\notag\end{equation}
The number $ j $ is called the $ j $-{\em invariant\/} of the curve. The moduli
(Teichm\"uller) space of curves of genus 1 may be identified with $ {\Bbb C} $ via
$ j $-invariant. The Teichm\"uller--Jacoby space is fibered over the
Teichm\"uller space with the fiber being the Jacobian, i.e., the elliptic
curve itself (i.e., in the standard notations this space coincides with
$ {\cal M}_{1,2} $). To compactify the Teichm\"uller space one adds a point with $ j=\infty $,
obtaining $ {\Bbb C}P^{1} $ as the compactified Teichm\"uller space.
To visualize the compactification of the Teichm\"uller space it is
more convenient to consider the family of elliptic curves $ X_{\varepsilon} $ given by
$ y^{2}=x^{2}-x^{3}-\varepsilon $. When $ \varepsilon \to 0 $ (so $ j=\frac{1}{15^{3}\varepsilon\left(4-27\varepsilon\right)} \to \infty $) one gets a rational
curve with a double point. Below we discuss how the corresponding
compactification of Teichm\"uller--Jacoby space looks like. We will see
that instead of adding one point, we need to add a rational curve with a
selfintersection. The compactified Teichm\"uller--Jacoby space maps to the
compactified Teichm\"uller space, with the fiber over $ \infty $ being the above
singular curve.
Consider an arbitrary elliptic curve $ E $ with large a $ j $-invariant, and
a bundle of degree 0 on $ E $. One can identify $ E $ with $ X_{\varepsilon} $ for an appropriate
small $ \varepsilon $. Fix a point $ P_{\varepsilon} $ on $ X_{\varepsilon} $, then for any bundle $ {\cal L} $ of degree 0 on $ X_{\varepsilon} $
one can identify $ {\cal L} $ with $ {\cal O}\left(Q_{\varepsilon}-P_{\varepsilon}\right) $, $ Q_{\varepsilon} $ being an appropriate point on $ X_{\varepsilon} $.
Fix $ P_{\varepsilon} $ to be the point of $ X $ which has $ y $-coordinate being 0, and is close
to $ \left(1,0\right)=P_{0} $. Suppose that $ {\cal L}\not={\cal O} $, thus $ Q_{\varepsilon}\not=P_{\varepsilon} $.
Let us describe meromorphic sections of $ {\cal L}={\cal O}\left(Q_{\varepsilon}-P_{\varepsilon}\right) $ with the only
pole being at $ P_{\varepsilon} $. It is the same as to describe sections of $ {\cal O}\left(Q_{\varepsilon}+kP_{\varepsilon}\right) $,
$ k\gg0 $, i.e., meromorphic functions on $ X_{\varepsilon} $ with a (possible) simple pole at
$ Q_{\varepsilon} $, and any pole at $ P_{\varepsilon} $. These sections are uniquely determined by their
singular part at $ P_{\varepsilon} $ (since there is no meromorphic function on an
elliptic curve $ X_{\varepsilon} $ which has only a simple pole at $ Q_{\varepsilon} $). Thus to describe
the space of these sections, it is sufficient to describe functions $ f_{k} $ on $ X_{\varepsilon} $
which have a pole of prescribed order $ k $ at $ P_{\varepsilon} $ and possibly an additional
pole at $ Q_{\varepsilon} $. (Note that $ f_{0}\equiv 1 $ should be considered as a section of $ {\cal O}\left(Q_{\varepsilon}-P_{\varepsilon}\right) $
with a pole at $ P_{\varepsilon} $, since holomorphic sections of $ {\cal O}\left(Q_{\varepsilon}-P_{\varepsilon}\right) $ are identified
with functions having a zero at $ P_{\varepsilon} $.)
Note that $ y $ is a meromorphic function on $ X_{\varepsilon} $ with a triple pole at
infinity of $ X_{\varepsilon} $, $ x $ has a double pole at infinity of $ X_{\varepsilon} $. Let $ P_{\varepsilon} $ have
coordinates $ \left(a_{\varepsilon},0\right) $, the line through $ P_{\varepsilon} $ and $ Q_{\varepsilon} $ have the equation
$ c_{\varepsilon}x+y=d_{\varepsilon} $, the third point $ R_{\varepsilon} $ of intersection of this line and $ X_{\varepsilon} $ has
$ x $-coordinate $ b_{\varepsilon} $. If $ k=2k_{1} $ is even, one can take $ f_{k}=\frac{1}{\left(x-a_{\varepsilon}\right)^{k_{1}}} $. If
$ k=2k_{1}-1 $, and $ k_{1}>1 $, then $ yf_{k+1} $ has no pole at infinity, thus can be
considered as $ f_{k} $. What remains is to describe $ f_{1} $. Note that $ \frac{1}{c_{\varepsilon}x+y-d_{\varepsilon}} $
has a pole at $ P_{\varepsilon} $, $ Q_{\varepsilon} $, and at $ R_{\varepsilon} $. Moreover, it has a zero of third order
at infinity. Multiplying this function by $ x-b_{\varepsilon} $, we kill the pole at $ R_{\varepsilon} $,
thus get $ f_{1}=\frac{x-b_{\varepsilon}}{c_{\varepsilon}x+y-d_{\varepsilon}} $.
Let us investigate how the space spanned by $ f_{i} $, $ i\geq0 $, depends on $ X_{\varepsilon} $
and $ Q_{\varepsilon} $. First of all, $ \varepsilon $ depends smoothly on $ Q_{\varepsilon} $ (if coordinates of $ Q_{\varepsilon} $ are
$ q_{1},q_{2} $, then $ \varepsilon=q_{2}^{2}-q_{1}^{2}-q_{1}^{3} $). From this moment on, we may denote $ Q_{\varepsilon} $ by just
$ Q $, since $ \varepsilon $ is a function of $ Q $. Second, $ x $ depends smoothly on $ y $ and $ \varepsilon $ in a
neighborhood of $ P_{\varepsilon} $ on $ X_{\varepsilon} $, {\em including\/} the case $ \varepsilon=0 $. Thus $ P_{\varepsilon} $ depends
smoothly on $ Q $, {\em including\/} cases $ \varepsilon=0 $ and $ Q=\left(0,0\right)\in{\Bbb C}^{2} $. Hence $ c_{\varepsilon} $, $ d_{\varepsilon} $ depend
smoothly on $ Q $. Substituting $ y=-c_{\varepsilon}x+d_{\varepsilon} $ into $ y^{2}=x^{2}-x^{3}-\varepsilon $, we get a cubic
equation in $ x $, which has $ q_{1} $ as a root, and two well-separated roots at $ a_{\varepsilon} $
and $ b_{\varepsilon} $. In particular, $ b_{\varepsilon} $ depends smoothly on $ Q $. Note that one can
consider $ y $ as a local coordinate near $ P_{\varepsilon} $, thus the functions $ f_{k} $ can be
written as function of $ y $ if $ y $ is small. We see that all the functions
$ f_{k}\left(y\right) $ depend smoothly on $ \left(q_{1},q_{2}\right) $ (at least as far as $ q_{1} $, $ q_{2} $ are small, $ y $
is small).
If $ q_{2}^{2}-q_{1}^{2}-q_{1}^{3}\not=0 $, the described above space $ V_{q_{1},q_{2}}=\left<f_{i}\left(y\right)\right>_{i\geq0}\subset{\frak L} $ is
the image of $ \left(X_{\varepsilon},P_{\varepsilon},{\cal L}\right) $ in the Universal Grassmannian, i.e., it is in
$ {\frak l}\left({\cal N}_{1,1,0}\right) $. We see that the closure of $ {\frak l}\left({\cal N}_{1,1,0}\right) $ contains at least the
subspaces $ V_{q_{1},q_{2}} $ with $ q_{2}^{2}-q_{1}^{2}-q_{1}^{3}=0 $. What remains is to give a
description of these subspaces as spaces of sections of sheaves on
non-smooth curves. We will see that if $ \left(q_{1},q_{2}\right)\not=\left(0,0\right) $, it is sufficient to
consider {\em linear bundles\/} on non-smooth curves, but if $ \left(q_{1},q_{2}\right)=\left(0,0\right) $, a
consideration of a sheaf is unavoidable.
It is obvious {\em which\/} non-smooth curve we need to consider: the limit
$ X_{0} $ of $ X_{\varepsilon} $ when $ \varepsilon \to $ 0, with the equation $ y^{2}=x^{2}-x^{3} $. There is a
parameterization mapping $ {\Bbb C}P^{1} \to X_{0} $, $ \lambda \mapsto \left(x,y\right) $ if $ y=\lambda x $. It is a bijection
outside of $ \lambda=\pm1 $, these two points are both mapped to (0,0). Thus $ X_{0} $ is a
rational curve with two points glued together. To describe a linear
bundle $ {\cal L} $ on $ X_{0} $ one needs to describe its lifting to $ {\Bbb C}P^{1} $, and the
identification of two fibers over $ \lambda=\pm1 $. If the lifting has degree 0, it
is a trivial bundle, thus one can describe the gluing by a number $ \theta\in{\Bbb C}^{*} $.
Denote this bundle $ {\cal L}_{\theta} $, it has a global holomorphic section iff $ \theta=1 $.
Note that if $ P_{0}=\left(1,0\right)\in X_{0} $, $ Q\in X_{0} $, $ Q\not=\left(0,0\right) $, then the mapping $ f\left(\lambda\right) \to
\frac{\lambda-\lambda_{0}}{\lambda}f\left(\lambda\right) $ gives an isomorphism of $ {\cal O}\left(Q-P_{0}\right) $ to $ {\cal L}_{\theta} $ if $ \theta=\frac{1-\lambda_{0}}{1+\lambda_{0}} $, $ \lambda_{0} $
being the coordinate of $ Q $. The defined above functions $ f_{k}\left(y\right) $ form a basis
of meromorphic sections of $ {\cal O}\left(Q-P_{0}\right) $, thus we see that for $ Q\in X_{0} $ the defined
above subspace $ V_{q_{1},q_{2}}\subset{\frak L} $ may be interpreted as $ {\frak l}\left(X_{0},P_{0},{\cal L}_{\theta}\right) $.
Now investigate what happens if $ Q \to \left(0,0\right) $ along $ X_{0} $, i.e., $ \theta \to 0 $ or
$ \theta \to \infty $. Instead of $ y $, consider the coordinate $ \lambda $ in a neighborhood of $ P_{0} $
on $ X_{0} $. In this coordinate the space $ V\left(X_{0},P_{0},{\cal L}_{\theta}\right) $ is the set of polynomials
$ p\left(\mu\right) $ in $ \mu=\lambda^{-1} $ which satisfy $ p\left(1\right)=\theta p\left(-1\right) $. Denote this subspace of $ {\Bbb C}\left[\mu\right] $ by
$ V_{\theta} $. Obviously, the subspace $ V_{\theta} $ depends smoothly on $ \theta\in{\Bbb C}P^{1} $, $ V_{0} $ consists of
polynomials with a zero at 1, $ V_{\infty} $ consists of polynomials with a zero at $ -1 $.
Moreover, images of $ V_{0} $ and $ V_{\infty} $ in $ \operatorname{Gr}\left({\frak L}\right)/G $ coincide, since $ V_{\infty}=\frac{\mu+1}{\mu-1}V_{0} $,
and $ \frac{\mu+1}{\mu-1} $ is smooth near $ \mu=\infty $, therefore $ f\left(\mu\right) \to \frac{\mu+1}{\mu-1}f\left(\mu\right) $
corresponds to an element of $ G $.
Returning back to the representation of $ V_{q_{1},q_{2}} $, $ q_{1}=q_{2}=0 $, in terms of
$ f_{k} $, we see that $ f_{1}=x/y=\lambda^{-1} $, $ f_{2}=\left(x-1\right)^{-1}=-\lambda^{-2} $, thus $ f_{2k}=\pm\lambda^{-2k} $; $ y=\lambda\left(1-\lambda^{2}\right) $,
thus $ f_{2k+1}=\frac{1-\lambda^{2}}{\lambda^{2k-1}}=\lambda^{-2k-1}-\lambda^{-2k+1} $. We see that this space is $ {\Bbb C}\left[\lambda^{-1}\right] $.
Hence the limit of spaces of sections of $ {\cal O}\left(Q-P\right) $ when $ Q \to \left(0,0\right) $ is
isomorphic to the space of meromorphic functions on the normalization $ {\Bbb C}P^{1} $
of $ X_{0} $ with the only pole at $ \lambda=0 $, i.e., with $ {\frak l}\left({\Bbb C}P^{1},0,{\cal O}\right) $. If we want to
describe this space in the same terms as we described the mapping $ {\frak l} $, we
should consider sections of $ {\cal O} $ on $ {\Bbb C}P^{1} $ as sections of the direct image $ \pi_{*}{\cal O} $
on $ X_{0} $, $ \pi $ being the projection of $ {\Bbb C}P^{1} $ to $ X_{0} $. Since $ \pi_{*}{\cal O} $ is no longer a
sheaf of sections of a linear bundle, we conclude that we {\em need\/} to
consider sheaves instead of bundles.
The only thing which remains to prove is that the completion of the
image $ {\frak l}\left({\cal N}_{1,1,0}\right) $ is smooth. We had already shown the part of the
completion we consider here is an image of a neighborhood of 0 in $ {\Bbb C}^{2} $.
What remains to prove is that the mapping $ {\Bbb C}^{2} \xrightarrow[]{V_{\bullet}} \operatorname{Gr}\left({\frak L}\right)/G $ has a
non-degenerate derivative at (0,0). It is sufficient to pick up a section
of the projection $ \operatorname{Gr}\left({\frak L}\right) \to \operatorname{Gr}\left({\frak L}\right)/G $, and show that a lifting of $ V_{\bullet} $ to this
section has a non-degenerate derivative at (0,0). Again, it is sufficient to
restrict this mapping to $ X_{0} $ and show that the derivatives along two
branches of $ X $ at (0,0) are independent.
One branch of $ X_{0} $ gives a family of subspaces $ V_{\theta} $, $ \theta\approx0 $, $ V_{\theta}\subset{\Bbb C}\left[\lambda^{-1}\right] $. It
is a simple but tedious calculation to show that the lifting defined in
Section~\ref{s0.4} results in a smooth lifting of $ V_{\bullet} $, and that the smooth
liftings resulting from two branches are non-degenerate, and have
different derivatives indeed.
We see that to describe the elements in the completion of $ {\frak l}\left({\cal N}_{1,1,0}\right) $
in a way similar to the description of $ {\frak l} $ one needs to consider both
degenerated curves and degenerated bundles (i.e., sheaves) on these
curves. Using the fact that the space of sections of $ \pi_{*}{\cal O} $ is close to the
set of sections of $ {\cal O}\left(Q\right) $ if $ Q $ is close to the double point (0,0) of $ X_{0} $, it
is reasonable to abuse notations and write $ \pi_{*}{\cal O} $ as $ {\cal O}\left(O\right) $, here $ O=\left(0,0\right) $.
\subsection{Compactified moduli spaces } We start with a description of elements of
the would-be compactified moduli spaces. As we have seen it in Section
~\ref{s0.5}, we need to allow the curves to have double points, and allow a
generalization of a notion of bundle. We call such bundles {\em non-smooth\/}
bundles, in fact they are sheaves which are (locally) direct images of
bundles on the normalization\footnote{Recall that in the case $ \dim =1 $ {\em normalization\/} coincides with ungluing
double points and straightening out cusps.} of the curve.
\begin{definition} Let $ Y $ be a connected curve with only singularities $ \operatorname{Sing}\left(Y\right) $
being double points, let $ \widetilde{Y} $ be the normalization of $ Y $. Define
$ g\left(Y\right)=g\left(\widetilde{Y}\right)+\operatorname{card}\left(\operatorname{Sing}\left(Y\right)\right) $, for a subset $ S_{0}\subset\operatorname{Sing}\left(Y\right) $ let $ Y_{S_{0}} $ be the curve
obtained by ungluing points in $ S_{0} $. A {\em smooth linear bundle\/} $ {\cal L} $ on $ Y $ is a
linear bundle $ \widetilde{{\cal L}} $ on $ \widetilde{Y} $ with a fixed identification of fibers at points of $ \widetilde{Y} $
over the same point on $ Y $. Define {\em degree\/} of a smooth linear bundle as the
$ \deg \left(\widetilde{{\cal L}}\right) $. A {\em section\/} of $ {\cal L} $ is a section of $ \widetilde{{\cal L}} $ compatible with identifications
of fibers.
A (not necessarily smooth) {\em linear bundle\/} $ {\cal L} $ on $ Y $ is a collection
$ \left(S_{0},\bar{{\cal L}}\right) $ of $ S_{0}\subset\operatorname{Sing}\left(Y\right) $ and a smooth linear bundle $ \bar{{\cal L}} $ on $ Y_{S_{0}} $. A {\em section\/} of $ {\cal L} $
is a section of $ \bar{{\cal L}} $. Let $ \deg \left({\cal L}\right)=\deg \left(\bar{{\cal L}}\right)+\operatorname{card}\left(S_{0}\right) $, call $ S_{0} $ the set of {\em double
points\/} of $ {\cal L} $. Let $ P\subset\operatorname{Smooth}\left(Y\right) $ be a finite collection of points. Call the
collection $ \left(Y,P,{\cal L}\right) $ {\em semistable\/} if $ \left(Y,P,{\cal L}\right) $ has no infinitesimal
automorphisms, i.e., if any rational connected component of $ \widetilde{Y} $ has at
least three points in $ P $ or over double points of $ Y $, and if $ Y $ is a
(smooth) elliptic curve, then either $ P\not=\varnothing $, or $ \deg {\cal L}\not=0 $. \end{definition}
In other words, for a ``non-smooth linear bundle'' $ {\cal L} $ the double points
of the curve are broken into two subsets: for one (smooth points of $ {\cal L} $) we
fix identifications of fibers of the lifting to normalizations, for
another one (as we have seen in Section 0.5, one should consider them as
{\em poles\/} of $ {\cal L} $) local sections of $ {\cal L} $ may have ``different'' values at two
branches of $ Y $ which meat at a given point.
Note that the smooth structures on the Teichm\"uller space and on the
Jacobian of a smooth curve provide an atlas on the set of equivalence
classes of smooth curves/bundles. The explicit deformation which we are
going to describe now will provide an atlas in a neighborhood of a class
of a semistable curve.
Consider a curve $ Y $ as in the definition, let $ Q\in Y $ be a double point,
and $ y_{1} $, $ y_{2} $ be coordinates on the branches of $ Y $ near $ Q $, so $ Y $ is locally
isomorphic to $ y_{1}y_{2}=0 $. Let $ {\cal L} $ be a linear bundle over $ Y $ with a fixed
trivialization near $ Q $. Suppose that $ Q $ is a double point of $ {\cal L} $, thus local
sections of $ {\cal L} $ can be written as (unrelated) functions $ f_{1}\left(y_{1}\right) $, $ f_{2}\left(y_{2}\right) $.
Note that trivialization of $ {\cal L} $ induces identification of fibers of $ \widetilde{{\cal L}} $ over
$ Q $. Let $ \overset{\,\,{}_\circ}{{\cal L}} $ be the bundle on $ Y $ (smooth near $ Q $) obtained by addition of
this identification to $ {\cal L} $. Extend the bundle $ \overset{\,\,{}_\circ}{{\cal L}} $ from $ Y $ to $ Y\cup\left\{\left(y_{1},y_{2}\right) |
|y_{1}|,|y_{2}|\ll 1\right\} $ trivially using the trivialization of $ {\cal L} $ near $ Q $, denote this
extension as $ \bar{{\cal L}} $.
Define a deformation $ \left(Y_{\varepsilon_{1},\varepsilon_{2}},{\cal L}_{\varepsilon_{1},\varepsilon_{2}}\right) $ of $ \left(Y,{\cal L}\right) $ (here $ \varepsilon_{1},\varepsilon_{2}\in{\Bbb C} $ are
fixed small numbers) by the following recipe:
\begin{enumerate}
\item
if $ \varepsilon_{1}\varepsilon_{2}=0 $, then $ Y_{\varepsilon_{1},\varepsilon_{2}}=Y $;
\item
if $ \varepsilon_{1}=\varepsilon_{2}=0 $, then $ {\cal L}_{\varepsilon_{1},\varepsilon_{2}}={\cal L} $;
\item
if $ \varepsilon_{1}\not=0 $, but $ \varepsilon_{2}=0 $, then $ {\cal L}_{\varepsilon_{1},\varepsilon_{2}} $ is the bundle $ \overset{\,\,{}_\circ}{{\cal L}}\left(P_{\varepsilon_{1}}\right) $, here $ P_{\varepsilon_{1}} $ is the
point on $ Y $ with $ y_{1}=\varepsilon_{1} $; similarly for $ \varepsilon_{1}=0 $, but $ \varepsilon_{2}\not=0 $;
\item
If $ \varepsilon_{1}\not=0 $, $ \varepsilon_{2}\not=0 $, then $ Y_{\varepsilon_{1},\varepsilon_{2}} $ is obtained by gluing the hyperbola
$ y_{1}y_{2}=\varepsilon_{1}\varepsilon_{2} $ to $ Y $ (with a small neighborhood of $ Q $ removed) via coordinate
projections, and $ {\cal L}_{\varepsilon_{1},\varepsilon_{2}}=\overset{\,\,{}_\circ}{{\cal L}}\left(P_{\varepsilon_{1},\varepsilon_{2}}\right) $, here $ \overset{\,\,{}_\circ}{{\cal L}}_{\varepsilon_{1},\varepsilon_{2}} $ is the restriction of $ \bar{{\cal L}} $ to
$ Y_{\varepsilon_{1},\varepsilon_{2}} $, the point $ P_{\varepsilon_{1},\varepsilon_{2}} $ is the point on hyperbola with coordinates
$ \left(\varepsilon_{1},\varepsilon_{2}\right) $.
\end{enumerate}
If $ Y $ had some marked points $ \left\{P_{i}\right\}\subset\operatorname{Smooth}\left(Y\right) $, then $ Y_{\varepsilon_{1},\varepsilon_{2}} $ has the same
marked points (correctly defined since $ Y_{\varepsilon_{1},\varepsilon_{2}} $ is identified with $ Y $
outside a small neighborhood of $ Q $).
Note that if $ P\in\operatorname{Smooth}\left(Y\right) $, then $ {\frak l}\left(Y,P,{\cal L}\right) $ is well-defined in $ \operatorname{Gr}\left({\frak L}\right)/G $.
\begin{lemma} $ g\left(Y_{\varepsilon_{1},\varepsilon_{2}}\right)=g\left(Y\right) $, $ \deg {\cal L}_{\varepsilon_{1},\varepsilon_{2}}=\deg {\cal L} $. If $ \left(Y,P,{\cal L}\right) $ is semistable, then
$ \left(Y_{\varepsilon_{1},\varepsilon_{2}},P,{\cal L}_{\varepsilon_{1},\varepsilon_{2}}\right) $ is semistable too. If $ P\in\operatorname{Smooth}\left(Y\right) $, then
$ {\frak l}\left(Y_{\varepsilon_{1},\varepsilon_{2}},P,{\cal L}_{\varepsilon_{1},\varepsilon_{2}}\right)\in\operatorname{Gr}\left({\frak L}\right)/G $ depends smoothly on $ \varepsilon_{1},\varepsilon_{2} $. \end{lemma}
Similarly, if $ Q $ is a double point of $ Y $, but not a double point of $ {\cal L} $,
define a one-parametric deformation $ \left(Y_{\varepsilon},{\cal L}_{\varepsilon}\right) $ by gluing to $ Y $ a hyperbola
$ y_{1}y_{2}=\varepsilon $ without changing $ {\cal L} $ (far from $ Q $) and the trivialization of $ {\cal L} $. A
statement similar to the above lemma continues to be true.
Let $ \bar{{\cal N}}_{g,1,d} $ be the set of equivalence classes of semistable
collections $ \left(Y,P,{\cal L}\right) $, let $ \bar{{\cal N}}_{g,1,d}^{\left(m,k\right)}\subset\bar{{\cal N}}_{g,1,d} $ be the subset of $ \bar{{\cal N}}_{g,1,d} $
consisting of curves with exactly $ m $ double points and bundles with
exactly $ k $ double points $ \left(k\leq m\right) $. Note that $ \bar{{\cal N}}_{g,1,d}^{\left(0,0\right)}={\cal N}_{g,1,d} $, and the
complement to $ {\cal N}_{g,1,d} $ in $ \bar{{\cal N}}_{g,1,d} $ is a disjoint union of $ \bar{{\cal N}}_{g,1,d}^{\left(m,k\right)} $. Call
these subsets {\em strata\/} of $ \bar{{\cal N}}_{g,1,d} $. Moreover, note that each $ \bar{{\cal N}}_{g,1,d}^{\left(m,k\right)} $
carries a natural smooth structure (inherited from the smooth structures
on Teichm\"uller spaces and Jacobians). (In fact the above smooth
structures can be refined to structures of orbifolds, but we ignore this
refinement here.)
Let $ \left(Y,y_{0},{\cal L}\right)\in\bar{{\cal N}}_{g,1,d}^{\left(m,k\right)} $, $ \varepsilon\in{\Bbb C}^{m+k} $ is small. Taking coordinate systems
in neighborhoods of double points, and trivializations of $ {\cal L} $ in these
neighborhoods, we obtain a deformation $ \left(Y_{\varepsilon},P,{\cal L}_{\varepsilon}\right)\in\bar{{\cal N}}_{g,1,d} $. Since
$ \bar{{\cal N}}_{g,1,d}^{\left(m,k\right)} $ has a natural smooth structure, we see that {\em a piece\/} of $ \bar{{\cal N}}_{g,1,d} $
is fibered over $ \bar{{\cal N}}_{g,1,d}^{\left(m,k\right)} $ with fibers being small balls in $ {\Bbb C}^{m+k} $.
Define a structure of an manifold on $ \bar{{\cal N}}_{g,1,d} $ using the above
fibration as an atlas in a neighborhood of a point in $ \bar{{\cal N}}_{g,1,d}^{\left(m,k\right)} $.
\begin{proposition} $ \bar{{\cal N}}_{g,1,d} $ is a smooth compact manifold (orbifold). $ {\frak l}\left(\bar{{\cal N}}_{g,1,d}\right) $
coincides with the closure of $ {\frak l}\left({\cal N}_{g,1,d}\right) $. \end{proposition}
\begin{proposition} \label{prop0.33}\myLabel{prop0.33}\relax $ {\frak l}\left(X,P,{\cal L}\right)={\frak l}\left(X',P',{\cal L}'\right) $ iff $ \left(X,P,{\cal L}\right) $ becomes isomorphic
to $ \left(X',P',{\cal L}'\right) $ after ungluing of double points of $ {\cal L} $ and $ {\cal L}' $. \end{proposition}
This means that $ \bar{{\cal N}}_{g,1,d} $ is a natural smooth compactification of
$ {\cal N}_{g,1,d} $. Moreover, the explicit coordinates in a neighborhood of a stratum
show that the {\em boundary\/} $ \bar{{\cal N}}_{g,1,d}\smallsetminus{\cal N}_{g,1,d} $ is a divisor with normal
intersections. As it follows from Proposition~\ref{prop0.33}, the image
$ {\frak l}\left(\bar{{\cal N}}_{g,1,d}\right) $ can be obtained from $ \bar{{\cal N}}_{g,1,d} $ by contracting smooth
submanifolds into points, thus the normalization of the image coincides
with $ \bar{{\cal N}}_{g,1,d} $.
\subsection{Adjacency of moduli spaces and Shottki model }\label{s0.8}\myLabel{s0.8}\relax We have seen that
the consideration of the mapping to the universal Grassmannian leads to a
remarkable compactification of Teichm\"uller--Jacoby space. Moreover, one
can use this compactification to define inclusions of compactified
Teichm\"uller--Jacoby spaces for different $ g $, $ d $ one into another. These
inclusions are going to be compatible with the mapping $ {\frak l} $ to Universal
Grassmannian, and may be constructed studying the properties of the
mapping $ {\frak l} $.
Consider a curve $ X $ with a bundle $ {\cal L} $ (we allow $ X $ and $ {\cal L} $ to have double
points). Given two smooth points $ Q_{1} $, $ Q_{2} $ on $ X $ one can construct a curve
$ X_{Q_{1}Q_{2}} $ obtained by gluing $ Q_{1} $ with $ Q_{2} $. This curve has one more double
point, and genus $ g\left(X\right)+1 $. One can consider the bundle $ {\cal L}_{Q_{1}Q_{2}} $ on $ X_{Q_{1}Q_{2}} $ which
has an extra double point at $ Q_{1}\sim Q_{2} $, otherwise coincides with $ {\cal L} $. Note that
sections of $ {\cal L} $ can be considered as sections of $ {\cal L}_{Q_{1}Q_{2}} $ and visa versa,
hence $ {\frak l}\left(X,P,{\cal L}\right) = {\frak l}\left(X_{Q_{1}Q_{2}},P,{\cal L}_{Q_{1}Q_{2}}\right) $ for any $ P $.
Using this remark one can easily include the compactification of one
moduli space into the boundary of another one. Indeed, one can generalize
the construction of $ \bar{{\cal N}}_{g,1,d} $ to the case of $ n $ marked points $ \left\{P_{i}\right\} $ (instead
of one $ P $), the only change being that one needs to consider $ \operatorname{Gr}\left({\frak L}^{n}\right) $
instead of $ \operatorname{Gr}\left({\frak L}\right) $. Now associate an element of $ \bar{{\cal N}}_{g+l,n-2l,d+l} $ to an element
$ \left(Y,\left\{P_{i}\right\},{\cal L}\right)\in\bar{{\cal N}}_{g,n,d} $ as follows: take first $ 2l $ points out of $ \left\{P_{i}\right\} $, and glue
them pairwise. Since we do not glue fibers of $ {\cal L} $ at these points, the
resulting double points on the resulting curve $ Y_{1} $ are double points of a
bundle $ {\cal L}_{1} $. Clearly, $ \left(Y_{1},\left\{P_{2l+i}\right\},{\cal L}_{1}\right)\in\bar{{\cal N}}_{g+l,n-2l,d+l} $. Thus we obtain a
mapping $ \bar{{\cal N}}_{g,n,d} \to \bar{{\cal N}}_{g+l,n-2l,d+l} $. Note that this correspondence does not
change the space of global sections of $ {\cal L} $, thus the image under the map $ {\frak l} $.
Since we are especially interested in $ \bar{{\cal N}}_{g,1,d} $, let us modify the
above construction to get a mapping $ \bar{{\cal N}}_{g,1,d} \to \bar{{\cal N}}_{g+l,1,d+l} $. To do this it
is sufficient to define a mapping $ \bar{{\cal N}}_{g,1,d} \to \bar{{\cal N}}_{g,1+2l,d} $. We will define such a
mapping for every {\em most degenerate\/} curve, which is a semistable curve with
the class on a stratum of $ \dim =0 $ on the moduli space. The mapping will be
given by {\em gluing\/} this curve to the given curve. Since there are finitely
many most degenerate curves (they are enumerated by appropriately colored
trees), we will obtain a finite number of mappings $ \bar{{\cal N}}_{g,1,d} \to \bar{{\cal N}}_{g,1+2l,d} $.
A most degenerate curve $ \left(Z,\left\{z_{j}\right\}\right) $ of genus 0 with $ 2l+2 $ marked points
$ z_{j} $, $ 0\leq j\leq2l+1 $ has $ 2l-1 $ double points, and its normalization has $ 2l $
connected components. Any bundle of degree 0 over $ Z $ is trivial. Given a
curve $ \left(Y,P_{0},{\cal L}\right) $, glue $ Z $ to $ Y $ by identifying $ P_{0} $ and $ z_{0} $ and identify $ {\cal L}|_{P_{0}} $
with $ {\cal O}_{Z}|_{z_{0}} $ arbitrarily, denote the resulting bundle $ {\cal L}' $. The resulting
collection $ \left(Y\cup Z,\left\{z_{j}\right\},{\cal L}'\right) $ is obviously an element of $ \bar{{\cal N}}_{g,1+2l,d} $, and
gluing together $ z_{j} $, $ 1\leq j\leq2l $, we obtain an element of $ \bar{{\cal N}}_{g+l,1,d+l} $.
We see that $ {\frak l}\left(X,P,{\cal L}\right) $ can be approximated by $ {\frak l}\left(X',P',{\cal L}'\right) $ (here $ X $
and $ X' $ are smooth curves) with $ g\left(X'\right)<g\left(X\right) $ if $ \left(X,P,{\cal L}\right) $ is close to a
subset of codimension 2 which consists of semistable curves/bundles such
that a bundle has a double point.
To visualize better the above surgery we improve the description in
two ways. First, let us revisit the process of $ \varepsilon $-deformation.
We glue a hyperbola $ y_{1}y_{2}=\varepsilon $ to the coordinate lines $ y_{1}=0 $, $ y_{2}=0 $ via
the coordinate projections. One can momentarily see that this is
equivalent to gluing together two regions $ \left\{|y_{1}|\geq\sqrt{|\varepsilon|}\right\} $, $ \left\{|y_{2}|\geq\sqrt{|\varepsilon|}\right\} $
along the boundary via $ y_{2}=\varepsilon/y_{1} $. Given a bundle $ {\cal L} $ with trivializations
near $ Q_{1} $ and $ Q_{2} $ one obtains a bundle $ \overset{\,\,{}_\circ}{{\cal L}} $ on the resulting curves (its
sections are sections of $ {\cal L} $ outside of the disks with compatible
restrictions to the disk boundaries). The deformed bundle is $ \overset{\,\,{}_\circ}{{\cal L}}\left(Q\right) $, here $ Q $
is close to the circle $ |y_{1}|=|y_{2}|=\sqrt{|\varepsilon|} $ on the deformed curve.
There is an alternative description of the deformed curve. Above we
were removing two disks of radii $ \varepsilon_{1} $ and $ \varepsilon_{2} $ near points $ y_{1} $, $ y_{2} $ and gluing
the boundaries. The radii of disks were variable, so it was hard to
visualize the picture. Remove instead bigger disks of (fixed) radii $ \delta_{1} $,
$ \delta_{2} $. What remains is to glue two annuli $ \left\{\varepsilon_{i}<|z|<\delta_{i}\right\} $, $ i=1,2 $, to the
resulting manifold with a boundary, and glue inner boundaries of annuli
together. Each annulus is conformally equivalent to a cylinder with ratio
length/radius being $ \log \frac{\delta_{i}}{\varepsilon_{i}} $. Thus we need to glue in a cylinder
$ S^{1}\times\left(0,L\right) $ of {\em conformal length\/} $ L=\log \frac{\delta_{1}\delta_{2}}{\varepsilon_{1}\varepsilon_{2}} $. If $ \varepsilon_{1,2} $ decrease, it is
equivalent to gluing longer and longer {\em handles\/} between circles of radii
$ \delta_{1,2} $. Note also that we modify $ {\cal L} $ by adding a pole inside this handle.
Second, note that the above gluing mapping $ y_{2}=\varepsilon/y_{1} $ is defined not
only on the boundary of the above regions, but in the regions themselves.
Thus continuing the identification of boundaries, one can identify the
region $ \left\{|y_{2}|\geq\sqrt{|\varepsilon|}\right\} $ with $ \left\{|y_{1}|\leq\sqrt{|\varepsilon|}\right\} $. If a part of curve corresponds to
a subset of $ \left\{|y_{2}|\geq\sqrt{|\varepsilon|}\right\} $, then it is identified with a subset of
$ \left\{|y_{1}|\leq\sqrt{|\varepsilon|}\right\} $. Suppose that the curve we deform has rational smooth
components. If one component $ Y_{i} $ contains $ l_{i} $ double points, then to
perform the above deformation we remove $ l_{i} $ disks of radii $ \varepsilon_{ik} $, $ k=1,2 $,
around these points. What remains is a sphere without $ l_{i} $ disks, and we
need to glue several such spheres together identifying boundaries by
fraction-linear mappings. Say, if $ i\not=j $ and $ Y_{j} $ is glued to $ Y_{i} $ via gluing
$ \varphi_{i} $, then $ \varphi\left(Y_{j}\right) $ is a subset of a removed disk for $ Y_{i} $. Consider now the
union $ Y_{i}\cup\varphi_{i}\left(Y_{j}\right) $. It is again a sphere with several disks removed, and the
boundaries of this disks are identified with the boundaries of other
components (or different circles on $ Y_{i}\cup\varphi_{i}\left(Y_{j}\right) $) via fraction-linear
mappings.
Continuing this process as long as we can, the result is {\em one\/} sphere
with several disks removed, and boundaries of these disks are identified
pairwise via fraction-linear mappings. This is so called {\em Shottki model\/} of
the curve, and we see that the analogue of Deligne--Mumford
compactification we consider here leads naturally to this model of
deformed curves. One can do a similar thing in the case when the initial
curve contains non-rational components as well. In this case one should
restrict attention to the ``connected part'' of the curve which consists of
rational components.
We conclude that in the simplest case the above process of
deformation may be described as this (for one particular choice of
degenerate rational curve with 3 marked points): Take two points $ Q_{1} $, $ Q_{2} $
which are close to the marked point $ P $ and much closer to each other than
to $ P $. Now remove two non-intersecting disks around $ Q_{1} $ and $ Q_{2} $, and glue a
very long handle between boundaries of these disks. Here a disk which
contains $ P $, $ Q_{1} $ and $ Q_{2} $ plays the r\^ole of the ``old'' marked point, we assume
that $ z_{0} $ and $ z_{4} $ (using the notations from the beginning of this section)
are on the same smooth component of the degenerate rational curve.
\subsection{Second compactification } The previous section shows that compactified
moduli spaces are included one into another, thus one can consider the
direct limit, i.e., the {\em union\/} of these spaces. The mapping $ {\frak l} $ identifies
this union $ {\frak N}_{d-g}=\bigcup_{l}\bar{{\cal N}}_{g+l,1,d+l} $, with a subset of the Universal
Grassmannian, and (at last!) we have the ingredients necessary for the
discussion what the closure of this subset may look like.
Consider an element $ \left(Y,P,{\cal L}\right) $ of $ \bar{{\cal N}}_{g+l,1,d+l} $ for a very big $ l $. Suppose
that it is close to an element $ \left(Y',P',{\cal L}'\right) $ of $ \bar{{\cal N}}_{g,1,d} $. To get $ Y $ we glue a
most degenerate rational curve to $ Y' $, the result is a curve with $ 2l $
components and $ 2l+2 $ marked points. Next we glue marked points pairwise,
and $ \varepsilon $-deform the resulting curve at the double points. However, the
double points are naturally broken into two categories: double points on
the attached rational curve, and doubled marked points. Let us change the
order of gluing and deformations: glue in the degenerate curve, $ \varepsilon $-deform
the double points on this curve, then glue marked points together and
$ \varepsilon $-deforming them. Note that the former double points are not double
points of the linear bundle, but the latter ones are double points for
the bundle.
Deformation of double points on a degenerate rational curve leads
to a Shottki model for some curve, i.e., a sphere with several disks
removed. However, since the genus is 0, there is no removed disks at all,
thus we get a sphere with $ 2l+2 $ marked points. The condition that the
parameters of deformation are small is translated into the fact that
double ratios of marked points on the sphere are very big, i.e., that
{\em conformally\/} the centers are {\em well separated}. Deform now the double point
where the degenerated curve is glued to $ Y' $. This leads to the part of the
above sphere being identified (by a fraction-linear mapping) with a disk
around $ P' $, and the marked points go to points of this disk. We conclude
that this part of deformation corresponds to picking up a collection of
points which are close to $ P' $ and conformally well separated.
The second part of the deformation is the removing of small disks
around the marked points, and gluing together the boundaries of these
disks (or, what is the same, gluing long handles to boundaries of bigger
fixed disks). In the most important case $ g=0 $, thus the description of a
neighborhood of a point on $ {\frak N}_{k} $ is related to studying curves of genus $ g $
obtained by gluing together $ 2g $ small conformally well separated disks on
$ {\Bbb C}P^{1} $. Note that the notion of being {\em well separated\/} depends on the
combinatorics of degenerated curves we glue in to get the limit point of
$ {\frak N}_{k} $.
Since we expect that $ {\frak l}\left({\frak N}_{k}\right) $ is dense in the space of solutions of an
integrable system, one can approximate a point in this space by a
sequence of curves and bundles. Assuming the best case scenario, we can
obtain a term of this sequence by a small deformation of the previous
term (here a small deformation is taken in the sense of algebraic
geometry, i.e., one takes a couple of points on the curve, glues them
together and deforms the double point slightly). We see that such a point
of the space can be described as a {\em curve of potentially infinite\/} genus,
i.e., a sequence of the curves where the next one is obtained by gluing
long handles to the previous one.
In this paper we show that one can consider a {\em curve of actually
infinite\/} genus instead. Such a curve is obtained in the same manner as
the above sequence of finite-genus curves: one takes an infinite
collection of ``well-separated'' points on $ {\Bbb C}P^{1} $, removes a disk of a small
radius around each point, and glues the boundaries pairwise.\footnote{The exact meaning of the above terms ``well-separated collection'' and
circles being ``of small radii'' is the main topic of this paper. The
answers vary a little bit depending on what problem one considers, for
the list of results see Section~\ref{s0.120}.} We show that
under suitable conditions the standard theorems of algebraic geometry
hold for the resulting curves as well. This shows that the {\em completed\/}
moduli space $ \bar{{\frak N}}_{-1} $ of such curves$ + $bundles may be important in algebraic
geometry. {\em If\/} this moduli space coincides with the whole space of
solutions (i.e., the best case scenario has place indeed), one can see
that the completed moduli space has a very simple topology. Indeed, by
Cauchy--Kovalevskaya theorem the space of solutions is identified with
the space of initial data for the solutions, which is a topological
linear space.
One can see that under assumption that the moduli space described in
this paper coincides with the set of solutions of integrable systems, the
study of compactified moduli spaces simplifies a lot by a transition to
the case of infinite genus. This simplification may suggest additional
approaches to the problem of studying the moduli spaces in finite genus
as well.
Unfortunately, at this moment it is unclear whether the best case
scenario is applicable to the moduli space as a whole. As we noted above,
the results of \cite{McKTru76Hil} show that this is true for the real part of
the hyperelliptic subset of the moduli space.
\subsection{Growth conditions and divisors }\label{s0.10}\myLabel{s0.10}\relax The main target of this
paper is to describe how to quantify the conditions on the collection of
circles to be ``conformally well separated'' and ``of small radii'' so that
the main theorems of algebraic geometry are still valid for the resulting
curves of infinite genus. However, before we discuss the conditions on
the nonlinear data (curves and bundles), we should discuss the
simpler conditions on the linear data (sections of above bundles).
The basic theorem of $ 1 $-dimensional algebraic geometry states that
any linear bundle on a compact analytic curve has a finite-dimensional
space of global sections. The moment we try to drop the compactness
condition this theorem breaks, since any non-compact analytic curve is a
Stein manifold, thus any sheaf on it has a giant space of global
sections.
One way to fix this situation is to consider growth conditions. Let
$ \bar{X} $ be a compact curve with a linear bundle $ \bar{{\cal L}} $. Suppose that $ \bar{X} $ and $ \bar{{\cal L}} $ are
provided with Hermitian metrics. Pick up a point $ \infty\in\bar{X} $, let $ X=\bar{X}\smallsetminus\left\{\infty\right\} $, $ {\cal L}=\bar{{\cal L}}|_{X} $.
Then bounded sections of $ {\cal L} $ can be uniquely extended to analytic sections
of $ \bar{{\cal L}} $, thus $ {\cal L} $ has a finite-dimensional space of bounded sections.
Similarly, if we consider sections of $ {\cal L} $ with magnitude going to 0 when $ x
\to \infty $, it is the same as to consider sections of $ \bar{{\cal L}}\left(-1\cdot\infty\right) $ on $ \bar{X} $, here $ -1\cdot\infty $
is a divisor on $ \bar{X} $.
Going in a different direction, consider $ L_{2} $-sections of $ {\cal L} $. It is
easy to see that this is also equivalent to consideration of sections of
$ \bar{{\cal L}} $. If we consider a different metric on $ X $, say, $ \frac{dx^{2}}{\operatorname{dist}\left(x,\infty\right)^{2\alpha}} $, then
to consider holomorphic on $ X L_{2} $-sections is the same as to consider
$ \bar{{\cal L}}\left(\left[\alpha\right]\cdot\infty\right) $. Similarly, different choice of metric on $ {\cal L} $ will lead to
different shift of $ \bar{{\cal L}} $ by a divisor at $ \infty $.
We see that different growth conditions applied to sections of $ {\cal L} $
lead to different ``effective'' continuations of $ {\cal L} $ to $ \bar{X} $. Obviously, the
situation becomes more complicated when $ \bar{X}\smallsetminus X $ consists of more than
one point. If $ \bar{X}\smallsetminus X $ is discrete, different possible choices of growth
conditions lead to a lattice which is isomorphic to the lattice of
divisors on $ \bar{X} $ concentrated on $ \bar{X}\smallsetminus X $. Situation goes out of control if $ \bar{X}\smallsetminus X $
is a ``massive'' set, or $ \bar{X} $ is not a smooth manifold at all. In such cases
different choices of metrics on $ X $ and $ {\cal L} $ (or some other data controlling
the growth of sections) form a very complicated lattice. However, there
is a way to select ``good'' elements of this lattice.
\subsection{Riemann--Roch and growth control }\label{s0.20}\myLabel{s0.20}\relax The Riemann--Roch theorem says
that on a compact analytic curve there is a relationship between a {\em degree
of\/} $ {\cal L} $, which is an easily calculable geometric characteristics of $ {\cal L} $, and
the dimensions of spaces of global sections of $ {\cal L} $ and $ \omega\otimes{\cal L}^{-1} $, $ \omega $ being the
linear bundle of holomorphic forms:
\begin{equation}
\dim \Gamma\left({\cal L}\right)-\dim \Gamma\left(\omega\otimes{\cal L}^{-1}\right) = \deg {\cal L}-g+1.
\notag\end{equation}
Let us show how this theorem might be applied to picking up ``correct''
growth conditions on non-compact analytic curves.
Consider some fixed growth conditions for $ X $. They select a new sense
for the functor $ \Gamma\left(X,\bullet\right) $. Suppose again that $ \bar{X}\smallsetminus X $ is discrete, $ {\cal L}=\bar{{\cal L}}|_{X} $. We
have seen above that consideration of these growth conditions is
equivalent to consideration of $ \bar{{\cal L}}\left(D\right) $, here $ D $ is some divisor at infinity,
i.e., $ \Gamma\left(X,{\cal L}\right)=\Gamma\left(\bar{X},\bar{{\cal L}}\left(D\right)\right) $. One should expect that $ D $ does not depend on $ {\cal L} $ (at
least when $ {\cal L} $ does not change a lot). Suppose that there is a way to
determine $ \deg \bar{{\cal L}} $ basing on $ {\cal L} $, say, we describe bundles by
divisors on $ X $.
If the dimensions of sections of linear bundles on $ X $ with given
growth conditions satisfy the Riemann--Roch theorem, then an easy
calculation shows that $ 2D\sim0 $. In particular, suppose that $ {\cal L} $ was a
restriction of a bundle $ \bar{{\cal L}} $ of half-forms on $ \bar{X} $, i.e., $ \bar{{\cal L}}\otimes\bar{{\cal L}}\simeq\omega $. Then we see
that the bundle on $ \bar{X} $ which corresponds to $ \bar{{\cal L}}|_{X} $ with given growth
conditions is $ \bar{{\cal L}}\left(D\right) $. Note that $ \bar{{\cal L}}\left(D\right)\otimes\bar{{\cal L}}\left(D\right)=\omega $. In particular, the growth
conditions which satisfy Riemann--Roch theorem ``preserve'' the set of
bundles of half-forms (or $ \theta $-{\em characteristics\/}): the sections on $ X $ of one
$ \theta $-characteristic $ \bar{{\cal L}} $ which satisfy the growth conditions can be naturally
identified with sections on $ \bar{X} $ of some other $ \theta $-characteristic $ \bar{{\cal L}}\left(D\right) $.
Suppose now that the growth conditions on sections of $ {\cal L} $ are picked
up in some invariant way---whatever it means. Because of that we expect
that $ D $ is a linear combination of points on $ \bar{X}\smallsetminus X $ with the same
coefficients. Since $ \deg D=0 $, thus $ D $ is 0.
We see that Riemann--Roch provides a selection criterion for picking
up growth conditions which ``do not add'' points at infinity to a given
divisor. By analogy, one can apply the same criterion in cases when $ \bar{X}\smallsetminus X $
is massive or $ \bar{X} $ does not exist: If there is an invariant way to describe
the growth conditions, and the Riemann--Roch theorem holds for a class of
bundles, then one may describe the part at infinity of the divisor of a
section of such a bundle: a section has no ``poles'' at infinity if it
satisfies the growth conditions.
\subsection{Handles, Sobolev spaces, and representations of $ \protect \operatorname{SL}_{2}\left({\Bbb C}\right) $ }\label{s0.30}\myLabel{s0.30}\relax Let us
apply heuristics from the previous section to the case of a curve $ X $ of
infinite genus, i.e., a Riemannian sphere with an infinite number of
disks removed, and infinite number of handles glued in along the cut
lines (or just any surface glued in from ``pants'' in such a way that the
graph of gluing is of infinite genus). This is not a compact Riemannian
surface, so one needs to consider ``infinities'' of this surface, and fix
some growth conditions near these infinities.
Moreover, if the genus is infinite, one needs to be especially
careful with Riemann--Roch theorem, since the right-hand side contains
genus $ g\left(X\right) $ (i.e., number of handles) of the curve, which is not a number
any more, but infinity. To save the theorem, let us consider the quantity
$ \deg {\cal L}-g+1 $ as a unity. One way to do it is to consider some fixed bundle $ {\cal M} $
on $ X $ which is ``naturally constructed'' and satisfies the condition
$ \deg {\cal M}=g-1 $ for finite-genus $ X $, and consider sections of $ {\cal L}\otimes{\cal M} $ instead of
sections $ {\cal L} $. Then the Riemann--Roch theorem may be rewritten as
\begin{equation}
\dim \Gamma\left({\cal L}\otimes{\cal M}\right)-\dim \Gamma\left(\omega\otimes{\cal M}^{-1}\otimes{\cal L}^{-1}\right) = \deg {\cal L}.
\notag\end{equation}
Note that $ \deg \omega\otimes{\cal M}^{-1}=\deg {\cal M} $, so $ {\cal N}=\omega\otimes{\cal M}^{\otimes-2} $ is a ``naturally constructed'' bundle
of degree 0. There are $ 2^{g} $ different square roots of a given bundle of
degree 0, and fixing one solution $ {\cal R} $ to $ {\cal R}^{2}=\omega\otimes{\cal M}^{\otimes-2} $ we may change $ {\cal M} $ to
$ {\cal M}\otimes{\cal R}^{-1} $. After such a change the formula simplifies to
\begin{equation}
\dim \Gamma\left({\cal L}\otimes{\cal M}\right)-\dim \Gamma\left({\cal L}\otimes{\cal M}^{-1}\right) = \deg {\cal L},
\notag\end{equation}
which has an additional advantage of being symmetric w.r.t. $ {\cal L} \mapsto {\cal L}^{-1} $.
Moreover, $ {\cal M} $ is now a solution of $ {\cal M}^{2}=\omega $.
We come to the following heuristic: to consider Riemann--Roch
theorem for a curve $ X $ with infinitely many handles one should fix a
bundle of half-forms $ {\cal M}=\omega^{1/2} $ on $ X $, consider a linear bundle $ {\cal L} $ of finite
degree, and apply appropriate growth conditions to sections of $ {\cal L}\otimes{\cal M} $. We
expect that the choice of growth conditions is very restricted, since any
possible change is equivalent to consideration of different square root
of $ \omega $ (see Section~\ref{s0.20}).
One should expect that thus obtained growth conditions are in some
way ``invariant'' (since so rigid), thus geometrically defined. It is still
meaningful to apply these conditions in the case $ {\cal L}={\cal O} $, when $ {\cal L}\otimes{\cal M} $ is the
sheaf of half-forms. We come to the following conclusions:
Conditions of boundness (i.e., $ L_{\infty} $-topology) cannot be applied (since
there is no geometrically-defined norm on the fibers of the bundle of
half-forms), as well as $ L_{2} $-type restrictions (since a square of a
holomorphic half-form is a half-form on $ X_{{\Bbb R}} $, thus cannot be invariantly
integrated). One can easily see that the fourth degree of a half form $ \alpha $
(more precise, $ \alpha^{2}\otimes\bar{\alpha}^{2} $) is a top-degree-form on $ X_{{\Bbb R}} $, thus a good candidate
is the condition of integrability of fourth degree, i.e., $ L_{4} $-topology.
It turns out that generic Banach spaces (like $ L_{4} $) are not convenient
for cut-and-glue operations we are going to perform, but there is a way
to get a Hilbert structure instead of the Banach one. Note that $ s $-th
derivative of $ \alpha $-form changes as a $ \left(\alpha+s\right) $-form under coordinate
transformations (in the main term), thus its square is a top-degree form
on $ X_{{\Bbb R}} $ (in the main term) if $ \alpha+s=1 $. We conclude that the notion that
$ 1/2 ${\em -th derivative of a half-form is square-integrable\/} has a chance to be
geometrizable. This notion leads us to consideration of $ H^{1/2} $-Sobolev
spaces with values in (holomorphic) half-forms. (We discuss the basics of
the theory of Sobolev spaces in Section~\ref{h2}) In fact taking different $ s $
one gets an entire hierarchy of Banach spaces $ W_{\frac{2}{s+1/2}}^{s} $ which
interpolate between $ L_{4} $ and $ H^{1/2} $. However, only one of these spaces is a
Hilbert one, and having a Hilbert space is very convenient for our method
of divide and conquer.
As we will see it in Section~\ref{s2.70}, the spaces of functions we
obtain in such a way are closest possible analogues of Hardy spaces. Note
also that they are very small modifications of a particular
representation of $ \operatorname{SL}_{2}\left({\Bbb C}\right) $ from a supplementary series (see Section
~\ref{s2.60}). Indeed, the supplementary series is realized in $ s $-forms on
$ \left({\Bbb C}P^{1}\right)_{{\Bbb R}} $, and the Hilbert structure in these spaces is equivalent to
$ H^{1-2s} $-structure. Taking $ s=1/4 $, we see that the $ H^{1/2} $-structure on sections
of $ \omega^{1/4}\otimes\bar{\omega}^{1/4} $ may be defined\footnote{Recall that $ H^{s} $-structure on sections of bundles on manifolds is defined
only up to equivalence of topologies.} to be invariant w.r.t. fraction-linear
transformations. One should compare this result with the above heuristic
about $ H^{1/2} $-structure on sections of $ \omega^{1/2} $ being ``invariant in the main
term'' w.r.t. diffeomorphisms (we use an additional heuristic that
$ \omega^{1/4}\otimes\bar{\omega}^{1/4} $ is ``close'' to $ \omega^{1/2} $).
\subsection{Thick infinity }\label{s0.40}\myLabel{s0.40}\relax We conclude that a working definition of {\em admissible\/}
sections on a sphere with some disks removed is that the section is a
half-form on the domain which can be extended as a $ H^{1/2} $-section of $ \omega^{1/2} $
to the whole sphere (though a usual heuristic about $ H^{1/2} $ is that it is a
smoothness class, it restricts the growth as well).
The usual definition of $ H^{s} $ on a manifold with a boundary (see
Section~\ref{s2.20}) allows one to work with function within the interior of
the manifold only. In our case the smooth part of the boundary (union of
circles) is not closed (if the number of circles is infinite), so one
loses the info about what happens on the {\em dust}, i.e., in the accumulation
points of the disks. This has some very inconvenient consequences. Let $ S $
be the sphere with interiors of disks removed, $ V $ be the closure of $ {\Bbb C}P^{1}\smallsetminus S $.
Then the sections we consider are elements of $ H^{1/2}\left({\Bbb C}P^{1}\smallsetminus V\right) $. The condition
of such a $ \varphi $ being holomorphic can be written as $ \bar{\partial}\varphi=0\in H^{-1/2}\left({\Bbb C}P^{1}\smallsetminus V\right) $ (since
taking derivative moves one notch down on Sobolev scale). However, if the
dust $ S\cap V $ is massive enough, there are functions of smoothness $ H^{-1/2} $ with
support in $ S\cap V $, thus there functions $ \varphi $ of smoothness $ H^{1/2} $ such that $ \bar{\partial}\varphi $
has support on the dust only. Since $ \bar{\partial}\varphi=0\in H^{-1/2}\left({\Bbb C}P^{1}\smallsetminus V\right) $, such functions
look like holomorphic ones, but in fact the dust is the part of infinity
on our curve of infinite genus, thus one should consider the divisor of
these functions as having components at infinity (since $ \bar{\partial}\varphi $ is not zero
there). This ruins our preparations to count the divisor at infinity
correctly.
One possible way out of this deadlock is to prohibit collections of
circles with a massive dust, say, restrict the Hausdorff dimension of the
dust to be smaller than 1. We pick up a different approach: we massage
the definition of the Sobolev space for a subset of a manifold in such a
way that it now takes into account the behaviour on the boundary even if
the boundary is massive. These are so called {\em generalized\/} Sobolev spaces.
As a corollary, we can consider pretty monstrous curves of infinite
genus.
For example, one can take a Serpinsky carpet\footnote{Which is a two-dimensional analogue of the Cantor set, obtained by
repeated removing of a smaller rectangle inside a bigger one (or removing
a triangle with vertices on the sides of a bigger triangle).} on a complex plane,
and take one disk inside each thrown away triangle/rectangle. Gluing the
boundaries of disk pairwise gives a curve we can deal with (if radii of
disks decrease quick enough). In particular, the dust can have a positive
measure (in fact in our theory the only condition on the dust is that it
is nowhere dense, so can be a set of accumulation points).
One can see that smooth points on the resulting curve have
infinitely many connected components, and it is the dust which keeps
these components together. The components are tubes, and the boundaries
of these tubes are glued to the dust, which is connected. We see that
points of the dust need to be considered as legitimate points on the
curve of infinite genus, and the curve is {\em not a topological manifold\/} in
neighborhoods of these points.
The striking fact is that it is possible to strengthen the
restriction on the radii of the circles in such a way that any global
holomorphic function has an {\em asymptotic\/} Taylor expansion near any point of
the dust (see Section~\ref{s4.95})! Thus the dust would consist of points
which one has a full right to call {\em smooth points}.
\subsection{Fight with $ H^{1/2} $ } While consideration of half-forms of smoothness $ H^{1/2} $
{\em enormously\/} simplifies the work with infinities on the curve, it is one of
the worst choices when we consider gluing conditions on sections on the
glued together circles on the complex sphere. The reason is that Sobolev
spaces with half-integer indices do not satisfy a lot of properties of
other Sobolev spaces when restrictions to hypersurfaces is considered
(this is the case when one usually considers Besov spaces instead of
Sobolev ones).
To fight with this, we define a notion of {\em mollified restriction\/} to a
hypersurface. Applying this restriction to the case of one removed disk
$ \left\{|z|>1\right\} $ one can momentarily see that the space $ H^{1/2} $ we consider coincides
with the Hardy space $ {\cal H}^{2} $ for the circle $ \left\{|z|=1\right\}! $ We see that the space of
``global holomorphic functions'' we consider is a generalization of the
Hardy space to the case when the boundary consists of many circles. What
is more, the Hardy space $ {\cal H}^{\infty} $ (of multiplicators in $ {\cal H}^{2} $) also plays an
important r\^ole when we consider equivalence of bundles in Section~\ref{s8.7}.
\subsection{Main results }\label{s0.120}\myLabel{s0.120}\relax Since technical details take a lot of place
in the
course of discussion, we make a concise list of main results (without
discussing what the different conditions on the curve/bundle mean).
In Section~\ref{s35.40} we prove (an analogue of) Riemann--Roch theorem
for $ \bar{\partial}\colon {\cal O} \to \bar{\omega} $ in assumption that the matrix $ {\bold M}_{2}=\left(e^{-l_{ij}}-\delta_{ij}\right) $ (formed basing
on conformal distances $ l_{ij} $ between disks) gives a compact mapping $ l_{2} \to
l_{2} $. In Section~\ref{s7.90} we show that the mapping $ \bar{\partial}\colon {\cal O} \to \bar{\omega} $ has a maximal
rank compatible with its index. In Section~\ref{s9.20} we show an analogues
result for $ \bar{\partial}\colon \omega \to \omega\otimes\bar{\omega} $ under an additional assumption that disks have a
{\em thickening\/} (in fact we show that the mapping of taking $ A $-periods of
global homomorphic forms is an isomorphisms). In Section~\ref{s9.40} we
describe the image of the $ \left(A,B\right) $-period mapping. In all these cases the
mappings in question have an infinite index, so we massage these mappings
to get a mapping of index 0 (by extending the domain or target spaces)
which is in fact an isomorphism.
In Section~\ref{s7.30} we show that the space of global holomorphic
sections of $ \omega^{1/2}\otimes{\cal L} $ is finite-dimensional provided the matrix
$ {\bold M}_{1}=\left(a_{i}e^{-l_{ij}/2}-a_{i}\delta_{ij}\right) $: give a compact mapping $ l_{2} \to l_{2} $. Here $ \left(a_{i}\right) $ is some
sequence associated to the bundle $ {\cal L} $ (it consists of norms of gluing
cocycle for $ {\cal L} $). In the section~\ref{s7.40} we show that the duality
\begin{equation}
\operatorname{Coker}\left(\bar{\partial}\colon \omega^{1/2}\otimes{\cal L} \to \omega^{1/2}\otimes{\cal L}\otimes\bar{\omega}\right) \simeq \operatorname{Ker}\left(\bar{\partial}\colon \omega^{1/2}\otimes{\cal L}^{-1} \to \omega^{1/2}\otimes{\cal L}^{-1}\otimes\bar{\omega}\right)
\notag\end{equation}
holds as far as $ {\bold M}_{1} $ gives a bounded operator $ l_{2} \to l_{2} $.
In Section~\ref{s5.60} we show that the above mappings satisfy the
Riemann--Roch theorem as far as both $ \left(a_{i}e^{-l_{ij}/2}-a_{i}\delta_{ij}\right) $ and
$ \left(a'_{i}e^{-l_{ij}/2}-a'_{i}\delta_{ij}\right) $ give compact operators $ l_{2} \to l_{2} $ (here $ a'_{i} $ is the
sequence associated to $ {\cal L}^{-1} $). In the same section we show that these
mappings are Fredholm under an additional condition that $ \left(a_{i}\right) $, $ \left(a_{i}'\right) $ are
bounded.
In Section~\ref{s4.50} we show that the above conditions give no
restrictions on the {\em dust\/} of the corresponding M\"obius group. In Section
~\ref{s7.70} we discuss heuristics which show that our version of the
Riemann--Roch theorem is close to the strongest possible.
In Section~\ref{s8.40} we show that an existence of so called
{\em Hilbert\/}--{\em Schmidt\/} bundle on a curve shows that $ \omega^{1/2} $ is also
Hilbert--Schmidt. In Section~\ref{s8.50} we show that the set of
Hilbert--Schmidt bundles is convex (in the sense of multiplication of
bundles), thus it is possible to define a {\em group\/} of {\em strong
Hilbert\/}--{\em Schmidt\/} bundles. In Section~\ref{s8.7} we define the notion of two
bundles being {\em equivalent}, and define the Jacobian. In Section~\ref{s8.80} we
define divisors, and in Section~\ref{s8.90} the mapping to the Universal
Grassmannian (using asymptotic series at points of the dust defined in
Section~\ref{s4.95}).
In Section~\ref{s7.90} we show that if $ \sum_{i\not=j}e^{-l_{ij}}<\infty $, then any admissible
bundle of degree 0 can be described by locally-constant cocycles (and
some amplifications of this result). In Section~\ref{s9.60} we show that the
period matrix is symmetric, and the imaginary part is positive. In
Section~\ref{s9.70} we show that the Jacobian coincides with the quotient of
the space of possible periods by the lattice of {\em integer\/} periods, as far
as $ \sum_{i\not=j}e^{-l_{ij}}<\infty $. Finally, in Section~\ref{s9.80} we show how to modify the
notion of Hodge structure to describe the curves we study in this paper.
\subsection{Main tools } We already noted that to deal with massive infinities we
need the notion of generalized Sobolev space (introduced in Section
~\ref{s2.40}), and to deal with gluing conditions along the boundaries of
disks we need mollified restriction/extension mappings (defined in
Section~\ref{s5.31}).
We discuss the relationship of the ``invariant'' growth conditions (we
need them to have a good enumeration of divisor at infinity, see Section
~\ref{s0.20}) with the supplementary series of representations of $ \operatorname{SL}_{2}\left({\Bbb C}\right) $ in
Section~\ref{s2.60}. In Section~\ref{s2.70} we demonstrate that the classical Hardy
space $ {\cal H}^{2} $ is a particular case of spaces we consider here (for ``curves of
genus 1/2'', when the fundamental domain is a disk). In Section~\ref{s8.7} we
show that the space of multiplicators (needed for defining equivalence of
bundles) gives $ {\cal H}^{\infty} $ in the particular case of a disk as a fundamental
domain.
To handle divide-and-conquer strategy of dealing with curves of
infinite genus we consider {\em almost perpendicular\/} families of subspaces
(studied in Section~\ref{s5.61}). To study duality we employ spaces of
{\em strong sections\/} of bundles (see Section~\ref{s6.50}), to study $ \bar{\partial} $ as a
Fredholm operator we employ spaces of {\em weak sections\/} (see Section
~\ref{s5.61}). (In most important cases these spaces coincide.)
To define $ B $-periods on a curve we apply a method of averaging
(Section~\ref{s9.41}) of paths connecting two sides of an $ A $-cycle.
\subsection{Historic remarks } In the spring of 1981 Yu.~I.~Manin was overwhelmed
by unusual amount of sophomore students who asked him to be their
undergraduate advisor. Contrary to his habits of the time, Yuri Ivanovich
presented a list of topics he thought would be interesting and
instructive for us to work on. One of these topics was a generalization
of algebro-geometric description of solutions of KP to a curve of
infinite genus represented by its Shottki model. (It was R.~Ismailov who
started to work on this topic.)
Five years later, when integrable systems reached the peak of their
popularity, we obtained first results on the Riemann--Roch theorem which
were similar to the results of this paper. The curves were represented in
the way that was in fashion that time: as collections of pants glued
along boundaries. The restrictions on the curves/bundles were
jaw-breaking and very $ \operatorname{ad} $-hockish. Since Manin's suggestions were
completely forgotten already, there was nothing similar to the Shottki
model approach. However, half-forms and $ H^{1/2} $-topology were already
present at that time.
When in the end of the eighties the first variant of this paper was
taking shape, it became clear that the conditions on curves/sheaves
simplify {\em enormously\/} if one restricts attention to gluings of boundary
circles which are fraction-linear, and gluing conditions for the bundles
are given by locally constant cocycles (relative to half-forms). To our
great surprise, with these restrictions the pants would glue together
into the Shottki model of the curve (see Section~\ref{s0.8}), and sheaves
glue into the bundle of half-forms on $ {\Bbb C}P^{1} $. This finished a complete
circle returning the settings back into context of the question of
Manin's of 1981!
Note that though the importance of Shottki model, of half-forms and
of $ H^{1/2} $-growth conditions was clear long time ago, it was the
formalization of the mollified restriction mapping which emphasized the
capacity of infinity and the parallelism with the usual study of Hardy
spaces $ {\cal H}^{2} $ and $ {\cal H}^{\infty} $. This formalization (together with the rest of this
paper) would not appear if not the fruitful discussions with A.~Tyurina
in spring of 1996.
During the last several years Feldman, Kn\"orrer and Trubowitz made a
major breakthrough using an unrelated approach (cf.~\cite{FelKnoTru96Inf}).
The author is most grateful to I.~M.~Gelfand, A.~Givental,
A.~Goncharov, D.~Kazhdan, M.~Kontsevich, Yu.~I.~Manin, H.~McKean,
V.~Serganova, A.~Tyurina and members of A.~Morozov's seminar for
discussions which directed many approaches applied here. It was the
patient work of V.~Serganova which improved readability of the most
obscure pieces of this paper.
This work was partially supported by NSF grant and Sloan
foundation fellowship.
\section{Geometry of half-forms }
\subsection{Half-forms and holomorphic half-forms } Consider an oriented real
manifold $ M $. For $ \alpha\in{\Bbb C} $ consider a (complex) linear bundle $ \Omega_{M}^{\alpha} $ with
transition functions $ \left(\det \frac{\partial Y}{\partial X}\right)^{\alpha} $ if $ Y=Y\left(X\right) $. We will write a section of
this bundle as $ \varphi\left(X\right)dX^{\alpha} $, so $ \varphi\left(X\right)dX^{\alpha}=\psi\left(Y\right)dY^{\alpha} $ if $ Y=Y\left(X\right) $ and
$ \varphi\left(X\right)\left(\det \frac{\partial Y}{\partial X}\right)^{\alpha}=\psi\left(Y\right) $. When $ M $ is clear from context, we will denote $ \Omega_{M}^{\alpha} $
by $ \Omega^{\alpha} $.
In what follows we will be most interested in $ \Omega^{1/2} $. Note that
$ \Omega^{1/2}\otimes\Omega^{1/2} $ is canonically isomorphic to the bundle of top de Rham forms.
For any two global sections $ \varphi $, $ \psi $ of $ \Omega^{1/2} $ the product $ \varphi\bar{\psi} $ is a top de Rham
form, thus the pairing
\begin{equation}
\left(\varphi,\psi\right) \mapsto \int\varphi\bar{\psi}
\notag\end{equation}
gives a structure of pre-Hilbert space on $ \Gamma\left(M,\Omega^{1/2}\right) $. In particular, a
notion of $ L_{2} $-section of $ \Omega^{1/2} $ is canonically defined.
If $ M $ is a complex-analytic manifold, the power $ \left(\det \frac{\partial Y}{\partial X}\right)^{\alpha} $ is not
uniquely defined (unless $ \alpha\in{\Bbb Z} $), thus $ \Omega^{\alpha} $ is not canonically defined. We use
an $ \operatorname{ad} $ hoc definition of an analogue of $ \Omega^{1/2} $ as follows:
Denote by $ \omega $ the top holomorphic de Rham bundle for $ M $. Consider a
linear bundle $ l $ with an isomorphism $ i\colon l\otimes l\simeq\omega $. We call sections of such a
bundle {\em holomorphic half-forms}, and will denote $ l = \omega^{1/2} $. In what follows
we will consider at most one bundle of half-forms on a given manifold.
Note that any local diffeomorphism $ M \to N $ induces a bundle of
half-forms on $ M $ given a bundle of half-forms on $ N $.
\subsection{Half-forms on complex curves }\label{s1.20}\myLabel{s1.20}\relax In what follows we are most
interested
in complex curves. On a compact complex curve of genus $ g $ there are $ 2^{g} $
different bundles of half-forms. If $ g=0 $, then there is a uniquely defined
bundle of half-forms, isomorphic to $ {\cal O}\left(-1\right) $.
To pick up an isomorphism of $ {\cal O}\left(-1\right)^{\otimes2}={\cal O}\left(-2\right) $ with $ \omega $, one must fix an
element $ v\in\Lambda^{2}V^{*} $, here $ V $ is $ 2 $-dimensional space, projectivization of which
is $ {\Bbb C}P^{1} $. Indeed, let $ {\bold e} $ be the Euler vector field on $ V $, $ {\bold e} = z_{1}\frac{\partial}{\partial z_{1}} +
z_{2}\frac{\partial}{\partial z_{2}} $. Consider $ v $ as a constant $ 2 $-form on $ V $, then $ {\bold e}\lrcorner v $ is a $ 1 $-form on
$ V $ which induces a global section of $ \omega\otimes{\cal O}\left(2\right) $ on $ PV. $ Multiplication on this
section gives an isomorphism of $ {\cal O}\left(-2\right) $ and $ \omega $.
This shows that there should be an action of $ \operatorname{SL}\left(2,{\Bbb C}\right) $ on the bundle
of half-forms, compatible with the action of $ \operatorname{SL}\left(2,{\Bbb C}\right) $ on $ {\Bbb C}P^{1} $. Indeed, in
affine coordinate $ z $ on $ {\Bbb C}P^{1}\smallsetminus\left\{\infty\right\} $, a half-form may be written as $ \varphi\left(z\right)dz^{1/2} $.
Here if domain of $ \varphi $ contains $ \infty $, then $ \varphi\left(\infty\right)=0 $. To write an action of
$ m=\left(
\begin{matrix}
a & b \\ c & d
\end{matrix}
\right)\in\operatorname{SL}\left(2,{\Bbb C}\right) $, note that the derivative of the corresponding action
$ z'=\displaystyle\frac{az+b}{cz+d} $ on $ {\Bbb C}P^{1} $ has a canonical square root $ \frac{1}{cz+d} $. Thus
$ \varphi\left(z'\right)dz'{}^{1/2}\sim\varphi\left(z'\right)\frac{1}{cz+d}dz^{1/2} $.
Note that it is the restriction of this action on $ \operatorname{SL}\left(2,{\Bbb R}\right) $ what is
used to define automorphic forms on $ \left\{\operatorname{Im} z>0\right\} $. Note also that this action
cannot be pushed down to an action of $ \operatorname{PGL}\left(2,{\Bbb C}\right) $.
\subsection{Restriction onto $ S^{1} $ } Consider now a annulus $ B=\left\{r<|z|<R\right\} $ on a complex
plane. Obviously, there are only two different bundles of half-forms on
$ B $. The inclusion of $ B $ into $ {\Bbb C}P^{1} $ induces a bundle $ \omega^{1/2} $ of half-forms on $ B $.
Another bundle of half-forms is $ \omega^{1/2}\otimes\mu $, here $ \mu $ is a locally constant
sheaf with monodromy $ -1 $ on $ B $, obviously $ \mu^{2} $ is the constant sheaf $ \underline{{\Bbb C}}_{ } $. Note
that the holomorphic form $ \frac{dz}{iz} $ on $ B $ has no square root in $ \omega^{1/2} $, but
has two in $ \omega^{1/2}\otimes\mu $, one may write these square roots symbolically as
$ \pm\frac{\left(dz\right)^{1/2}}{i^{1/2}z^{1/2}} $.
Consider now a restriction of $ \omega^{1/2}\otimes\mu $ onto $ S^{1}=\left\{|z|=1\right\}\subset B $ (we assume
that $ r<1<R $). Consider a coordinate $ t = \operatorname{Im} \ln z $ on $ S^{1} $, and identify
$ \frac{dz^{1/2}}{i^{1/2}z^{1/2}}\in\Gamma\left(S^{1},\omega^{1/2}\otimes\mu\right) $ with $ dt^{1/2}\in\Gamma\left(S^{1},\Omega_{S^{1}}^{1/2}\right) $. A simple
calculation shows that this identification gives an isomorphism of
$ \omega^{1/2}|_{S^{1}} $ with $ \Omega_{S^{1}}^{1/2}\otimes\mu $, which is preserved by real-analytic
diffeomorphisms of $ S^{1} $ with a given lift to $ \mu $.
In general, an immersion of $ S^{1} $ in a complex curve $ X $ with a bundle of
half-forms $ \omega $ induces an isomorphism of $ \omega^{1/2}|_{S^{1}} $ either with $ \Omega_{S^{1}}^{1/2}\otimes\mu $, or with
$ \Omega_{S^{1}}^{1/2} $. One may call immersions of the first type $ A $-cycles, of the second
one $ B $-cycles. Until Section~\ref{h9} the immersions we consider are going to be
$ A $-cycles, thus we will always have a factor $ \mu $.
Note that the above isomorphism is defined up to multiplication by
$ \pm1 $, and both $ \Omega_{S^{1}}^{1/2}\otimes\mu $ and $ \Omega_{S^{1}}^{1/2} $ have natural pre-Hilbert structures,
thus $ \omega^{1/2}|_{S^{1}} $ has a natural pre-Hilbert structure as well.
\section{Sobolev spaces }\label{h2}\myLabel{h2}\relax
In what follows $ {\cal S} $ denotes the space of rapidly decreasing smooth
functions, $ {\cal D} $ denotes the space of smooth functions with compact support,
and $ {\cal S}' $ and $ {\cal D}' $ are the dual spaces. $ l_{2} $ denotes the Hilbert space of
square-integrable sequences, and $ L_{2} $ denotes the space of square-integrable
functions. The
symbol $ \bigoplus V_{i} $ denotes the space of sequences with finite number of non-zero
terms. If $ V_{i} $ are Hilbert spaces, $ \bigoplus_{l_{2}}V_{i} $ denotes the space of
square-integrable sequences $ \left(v_{i}\right) $ with $ v_{i}\in V_{i} $.
The material of this section is mostly standard, however, note that
we also discuss some topics which are not parts of standard curriculum of
Sobolev spaces: in Section~\ref{s2.40} we introduce {\em generalized\/} Sobolev
spaces, in Sections~\ref{s2.60} and~\ref{s2.70} we give descriptions of
supplementary series of representations of $ \operatorname{SL}\left(2,{\Bbb C}\right) $ and of Hardy space in
terms of Sobolev spaces. In addition to the results which are covered in
this section, in Section~\ref{s2.10} we introduce {\em mollifications\/} of
mappings of restriction to a submanifold (in the case $ s=1/2 $), and of
mapping of extension from submanifold (in the case $ s=0 $), both in the
cases when the non-mollified mappings are not continuous.
\subsection{Euclidean case }\label{s2.02}\myLabel{s2.02}\relax Consider a vector space $ {\Bbb R}^{n} $ with the standard
Euclidean structure. Consider $ s\in{\Bbb R} $ and a norm $ \|\bullet\|_{s} $ on the set $ {\cal S} $ of rapidly
decreasing $ C^{\infty} $-functions:
\begin{equation}
\|f\left(x\right)\|_{s} = \int\left(1+|\xi|^{2}\right)^{s}|\widehat{f}\left(\xi\right)|^{2}d\xi,
\notag\end{equation}
here $ \widehat{f}\left(\xi\right) $ is the Fourier transform of $ f $. By definition, {\em the Sobolev space\/}
$ H^{s}\left({\Bbb R}^{n}\right) $ is the completion of $ {\cal S} $ w.r.t. this norm. It is naturally
isomorphic to the set of $ L_{2} $-functions $ g\left(\xi\right) $ w.r.t. the measure
$ \left(1+|\xi|^{2}\right)^{s/2}d\xi $. Since the inverse Fourier transform of $ g\left(\xi\right) $ is a
well-defined generalized function on $ {\Bbb R}^{n} $, there is a natural inclusion
$ H^{s}\left({\Bbb R}^{n}\right)\hookrightarrow{\cal S}' $, compatible with the inclusion $ {\cal S}\hookrightarrow H^{s}\left({\Bbb R}^{n}\right) $.
The pairing
\begin{equation}
\left(f,g\right) \mapsto \int f\bar{g}\,dx
\notag\end{equation}
extends to a pairing $ H^{s}\left({\Bbb R}^{n}\right)\otimes H^{-s}\left({\Bbb R}^{n}\right) \to {\Bbb C} $ which is a pairing of Hilbert
spaces. This pairing is compatible with the pairing between $ {\cal S} $ and $ {\cal S}' $.
Let $ D $ be a closed subset of $ {\Bbb R}^{n} $, $ U $ be an open subset of $ {\Bbb R}^{n} $. Define a
Hilbert subspace
\begin{equation}
\overset{\,\,{}_\circ}{H}^{s}\left(D\right)=H^{s}\left({\Bbb R}^{n}\right) \cap \left\{f\in{\cal S}' \mid \operatorname{Supp} f\subset D\right\},
\notag\end{equation}
and a quotient Hilbert space
\begin{equation}
H^{s}\left(U\right)=H^{s}\left({\Bbb R}^{n}\right)/\overset{\,\,{}_\circ}{H}\left({\Bbb R}^{n}\smallsetminus U\right).
\notag\end{equation}
The following properties of Sobolev spaces are most important in
analysis:
\begin{enumerate}
\item
If $ D $ is compact, $ V $, $ U $ are open, $ V\subset D\subset U\subset{\Bbb R}^{n} $, and $ \varphi\colon U \to \varphi\left(U\right) $ is a
diffeomorphism, then $ \varphi^{*} $ induces invertible bounded operators $ \overset{\,\,{}_\circ}{H}^{s}\left(\varphi\left(D\right)\right) \to
\overset{\,\,{}_\circ}{H}^{s}\left(D\right) $, $ H^{s}\left(V\right) \to H^{s}\left(V\right) $.
\item
If $ D $ is compact, then any differential operator $ A $ of degree $ d $ with
smooth coefficients gives a bounded operators $ \overset{\,\,{}_\circ}{H}^{s}\left(D\right) \to \overset{\,\,{}_\circ}{H}^{s-d}\left(D\right) $, $ H^{s}\left(V\right) \to
H^{s-d}\left(V\right) $.
\item
In the same way a pseudodifferential operator of degree $ d $
gives a bounded operators $ \overset{\,\,{}_\circ}{H}^{s}\left(D\right) \to H^{s-d}\left(U\right) $.
\item
The mapping of restriction of smooth functions onto $ {\Bbb R}^{n-k}\subset{\Bbb R}^{n} $ extends
to a bounded operator
\begin{equation}
r\colon H^{s}\left({\Bbb R}^{n}\right) \to H^{s-\frac{k}{2}}\left({\Bbb R}^{n-k}\right)\text{ if }s>\frac{k}{2}.
\notag\end{equation}
\item
Dually, extension-by-$ \delta $-function of generalized functions on $ {\Bbb R}^{n-k} $ to
generalized functions on $ {\Bbb R}^{n} $ gives a bounded operator $ e\colon H^{s}\left({\Bbb R}^{n-k}\right) \to
H^{s-\frac{k}{2}}\left({\Bbb R}^{n}\right) $ if $ s<0 $.
\item
If $ D $ is compact and has a smooth boundary, then $ \overset{\,\,{}_\circ}{H}^{s}\left(D\right)^{\perp}\subset H^{-s}\left({\Bbb R}^{n}\right) $
coincides with $ \overset{\,\,{}_\circ}{H}^{-s}\left(\overline{{\Bbb R}^{n}\smallsetminus D}\right) $.
\end{enumerate}
\subsection{Sobolev spaces on manifolds }\label{s2.20}\myLabel{s2.20}\relax If $ M $ is a paracompact $ C^{\infty} $-manifold, then
$ H^{s}\left(M\right) $ can be defined as a subspace of generalized functions on $ M $
which consists of functions $ f $ such that for any $ U\subset M $ and $ \varphi\colon \widetilde{U} \overset{\sim}\to \widetilde{V}\subset{\Bbb R}^{n} $ the
restriction $ f|_{\widetilde{U}} $ satisfies $ \left(\varphi^{-1}\right)^{*}f\in H^{s}\left(V\right) $ (here $ \widetilde{U} $ is a neighborhood of $ \bar{U} $,
$ V=\varphi\left(U\right) $).
An alternative definition is that for an appropriate partition of
unity $ \left(U_{\alpha},\sigma_{\alpha}\right) $, $ \operatorname{Supp} \sigma_{\alpha}\subset\subset U_{\alpha} $, $ \sum_{\alpha}\sigma_{\alpha}=1 $, the products $ \sigma_{\alpha}f $ satisfy $ \left(\varphi_{\alpha}^{-1}\right)^{*}\left(\sigma_{\alpha}f\right)
\in \overset{\,\,{}_\circ}{H}\left(V_{\alpha}\right) $, here $ \varphi_{a}\colon U_{\alpha} \overset{\sim}\to V_{\alpha}\subset{\Bbb R}^{n} $.
In the same way one may define $ H^{s} $-sections of $ C^{\infty} $-vector bundles
over $ M $. We denote the space of $ H^{s} $-section of a bundle $ E $ by $ H^{s}\left(M,E\right) $.
Obviously, if $ M $ is compact, one may define a structure of Hilbert
space on $ H^{s}\left(M\right) $, but this structure is not uniquely defined. However, the
corresponding topology on $ H^{s}\left(M\right) $ is well-defined. For a general manifold $ M $
one defines a topology on $ H^{s}\left(M\right) $ as an inverse limit w.r.t. topologies on
$ H^{s}\left(U\right) $, $ U $ being open subsets of $ M $ diffeomorphic to bounded open subsets of
$ {\Bbb R}^{n} $.
If $ D $ is a closed subset of $ M $, and $ U $ is an open subset, define
$ \overset{\,\,{}_\circ}{H}^{s}\left(D\right)\subset H^{s}\left(M\right) $ as subspace of function with support in $ D $, and $ H^{s}\left(U\right) $ as
$ H^{s}\left(M\right)/\overset{\,\,{}_\circ}{H}^{s}\left(M\smallsetminus U\right) $.
Note that if $ M={\Bbb R}^{n} $, the above definition produces a different space
$ H_{\text{loc}}^{s}\left({\Bbb R}^{n}\right) $ of functions on $ {\Bbb R}^{n} $ than the space $ H^{s}\left({\Bbb R}^{n}\right) $ defined in Section
~\ref{s2.02}. A generalized function $ f $ on $ {\Bbb R}^{n} $ is in $ H_{\text{loc}}^{s}\left({\Bbb R}^{n}\right) $ if in any bounded
domain it is equal to a function from $ H^{s}\left({\Bbb R}^{n}\right) $.
The properties of Sobolev spaces $ H^{s}\left({\Bbb R}^{n}\right) $ have direct analogues for
$ H^{s}\left(M\right) $, thus diffeomorphisms of manifolds and differential operators act
on $ H^{s}\left(M\right) $, and one can restrict/extend Sobolev sections to/from
submanifolds. In particular, the existence of a parametrix for an elliptic
differential operator shows that
on a compact manifold $ M $ an elliptic operator $ A $ of degree $ d $ gives a Fredholm
operator $ H^{s}\left(M\right) \to H^{s-d}\left(M\right) $ (with suitable changes if $ A $ acts in vector
bundles).
Note that on compact manifolds one can define the Sobolev spaces
using arbitrary elliptic (pseudo)differential operators:
\begin{proposition} Consider a compact manifold $ M $ with a metric and a
positive self-adjoint elliptic operator $ A $ of degree $ d $. Then the $ s $-Sobolev
norm is equivalent to the norm
\begin{equation}
\|f\| = \int_{M}|\left(1+A\right)^{s/d}f|^{2}d\mu.
\notag\end{equation}
\end{proposition}
We will also use the following statement:
\begin{proposition} Consider two linear bundles $ {\cal L}_{1} $ and $ {\cal L}_{2} $ on a compact
manifold $ M $ such that $ {\cal L}_{1}\otimes{\cal L}_{2}\simeq\Omega^{\text{top}}\left(M\right) $. Then the pairing $ \int\alpha\beta $ between smooth
sections of $ {\cal L}_{1} $ and $ {\cal L}_{2} $ can be extended to non-degenerate pairing between
Hilbert spaces $ H^{s}\left(M,{\cal L}_{1}\right) $ and $ H^{-s}\left(M,{\cal L}_{2}\right) $ for an arbitrary $ s\in{\Bbb R} $.
\end{proposition}
\subsection{Capacity } Consider a manifold $ M $. We say that $ s $-{\em capacity\/} of a closed subset
$ S\subset M $ is
non-zero, if there is a non-zero function $ f\in H^{s}\left(M\right) $ such that $ \operatorname{Supp} f\subset S $. We
say that $ S $ has a {\em capacity dimension\/} $ \geq d $, if $ s $-capacity of $ S $ is non-zero
for $ s=-\frac{\dim M - d}{2} $.
Since $ \delta $-function of a point belongs to $ H^{s}\left(M\right) $ with $ s<-\frac{\dim M}{2} $, a
capacity dimension of a point is $ \geq d $ if $ d<0 $. It is clear that this
estimate cannot be improved. In the same
way the capacity dimension of a submanifold $ S $ is $ \geq d $ iff $ d<\dim S $.
For a general subset $ S $ the same is true if one considers
Hausdorff dimension \cite{Hor83Dis}.
\subsection{Generalized Sobolev spaces }\label{s2.40}\myLabel{s2.40}\relax In Section~\ref{s5.10} we consider the space
of $ H^{1/2} $-half-forms which are holomorphic outside of a closure $ \bar{U} $ of a union $ U $
of disks in $ {\Bbb C}P^{1} $. We will see that this space is too big for our
purposes if $ \bar{U} $
is ``much bigger'' than $ U $. We will need to
consider half-forms which ``are holomorphic'' in $ \bar{U}\smallsetminus U $ as well as in $ {\Bbb C}P^{1}\smallsetminus\bar{U} $.
To have this we need the half-forms to be {\em defined\/} in $ {\Bbb C}P^{1}\smallsetminus U $ instead
of $ {\Bbb C}P^{1}\smallsetminus\bar{U} $. The problem with this is that $ U $ is not closed, so the usual
definition of $ H^{1/2} $-half-forms as of a quotient-space does not work. This
shows the need for the following
\begin{definition} Consider a subset $ U $ of the manifold $ M $. Let $ \overset{\,\,{}_\circ} H^{s}\left(U\right) $ denotes
the closure of the subspace $ L\subset H^{s}\left(M\right) $
\begin{equation}
L=\left\{f\in H^{s}\left(M\right) \mid \operatorname{Supp} f\subset U\right\}.
\notag\end{equation}
Consider $ V\subset M $. Let $ H^{s}\left(V\right)=H^{s}\left(M\right)/\overset{\,\,{}_\circ} H^{s}\left(M\smallsetminus V\right) $. \end{definition}
Note that we do not require that the subset $ U $ is closed, and $ V $ is
open. If they are, then we get the standard definitions of $ \overset{\,\,{}_\circ}{H}^{s} $ and $ H^{s} $.
Obviously, $ \overset{\,\,{}_\circ} H^{s}\left(U\right)\subset\overset{\,\,{}_\circ} H^{s}\left(\bar{U}\right) $,
but this inclusion may be proper, as the following construction shows\footnote{The construction can be simplified, but in the current form it is an
example of domains we are going to deal with.}.
Consider a disjoint family of closed subsets $ V_{i}\subset M $, Let $ {\cal V}=\overline{\bigcup V_{i}}\smallsetminus
\bigcup V_{i} $. Suppose that
\begin{enumerate}
\item
The natural mapping $ \bigoplus_{l_{2}}\overset{\,\,{}_\circ} H^{s}\left(V_{i}\right) \xrightarrow[]{\iota} H^{s}\left(M\right) $ is a (continuous)
injection\footnote{I.e., the image is closed, and the mapping is an isomorphism on the
image.}.
\item
The $ s $-capacity of $ {\cal V} $ is positive.
\end{enumerate}
The first condition insures that the space $ \overset{\,\,{}_\circ} H^{s}\left(\bigcup V_{i}\right) $ is the image of
the mapping $ \iota $. Hence any non-zero function $ f\in\overset{\,\,{}_\circ} H^{s}\left(\bigcup V_{i}\right) $ satisfies the
condition $ \operatorname{Supp} f \cap \bigcup V_{i}\not=\varnothing $.
On the other hand, the second condition shows that there is a
non-zero function $ f\in H^{s}\left(M\right) $ such that $ \operatorname{Supp} f\subset{\cal V} $. Obviously,
$ f\in\overset{\,\,{}_\circ} H^{s}\left(\overline{\bigcup V_{i}}\right) $, but $ f\notin\overset{\,\,{}_\circ} H^{s}\left(\bigcup V_{i}\right) $.
In Section~\ref{s4.50} we show how to construct a family of disks
$ V_{i} $ which satisfy the first condition. The centers of these disks may be
an arbitrary prescribed locally discrete set. Moreover, one can easily find an
appropriate set of centers such that the corresponding set $ {\cal V} $ does not
depend on radii and coincides with an arbitrary given closed set $ {\cal V}_{0} $ with
empty interior.
In particular, if there exist a closed set $ {\cal V} $ with an empty interior
and non-zero $ s $-capacity, then one can construct a subset $ U $ of $ {\Bbb C}P^{1} $ such that
the inclusion $ \overset{\,\,{}_\circ} H^{s}\left(U\right)\subset\overset{\,\,{}_\circ} H^{s}\left(\bar{U}\right) $ is proper. Taking appropriate Cantor sets,
one gets that this is possible with any $ s<0 $.
On the other hand, if $ U $ has smooth boundary, then $ \overset{\,\,{}_\circ} H^{s}\left(U\right)=\overset{\,\,{}_\circ} H^{s}\left(\bar{U}\right) $.
\subsection{Rescaling on $ {\Bbb R}^{n} $ and $ {\Bbb C}^{n} $ }\label{s2.50}\myLabel{s2.50}\relax Consider an $ H^{s} $-function $ f\left(x\right) $ on $ {\Bbb R}^{n} $ with
compact
support. The Fourier transform $ \widehat{f}\left(\xi\right) $ is real-analytic, thus the integral
\begin{equation}
\int|\xi|^{2s}|\widehat{f}\left(\xi\right)|^{2}d\xi,
\label{equ3.7}\end{equation}\myLabel{equ3.7,}\relax
can be defined in the sense of generalized functions (i.e., analytic
continuation in $ s $) near $ \xi=0 $. Since $ f\in H^{s} $, the integral
converges near $ \infty $. Moreover, if $ s>-\frac{n}{2} $, the integral converges near $ \xi=0 $,
thus its value is a limit of Riemann sums, hence is non-negative.
In any case the integral defines a quadratic form on $ \overset{\,\,{}_\circ}{H}^{s}\left(D\right) $ for any
compact $ D $.
\begin{nwthrmi} If $ s>-\frac{n}{2} $, the integral defines a norm on $ \overset{\,\,{}_\circ}{H}^{s}\left(D\right) $, and this form
is equivalent to the Hilbert norm. \end{nwthrmi}
The advantage of the norm~\eqref{equ3.7} is the fact that it is covariant
with respect to dilatations. Define a mapping $ {\frak D}_{a}^{s}\colon {\cal S} \to {\cal S} $ by
\begin{equation}
\left({\frak D}_{a}^{s}f\right)\left(x\right) = a^{s}f\left(ax\right),\qquad a,s\in{\Bbb R},\quad a\not=0.
\notag\end{equation}
Then $ {\frak D}_{a}^{-s+\frac{n}{2}} $ is an isometry w.r.t. the norm~\eqref{equ3.7}. Note also that
that $ {\frak D}_{a}^{tn} $ is a natural action of a dilatation on sections of the linear
bundle $ \Omega^{t} $.
Combining this with the previous statement, we conclude that any
similarity transform\footnote{I.e., a composition of translations, rotations and dilatations.} $ {\bold T} $ acts as a uniformly bounded operator in the space
of $ H^{s} $-sections of $ \Omega^{\alpha} $
\begin{equation}
{\bold T}^{*}\colon \overset{\,\,{}_\circ}{H}^{s}\left({\bold T}D,\Omega^{\alpha}\right) \to \overset{\,\,{}_\circ}{H}^{s}\left(D,\Omega^{\alpha}\right),\qquad \alpha = \frac{1}{2}-\frac{s}{n},
\notag\end{equation}
as far as both $ D $ and $ {\bold T}D $ remain in the same disk $ \left\{|x|<R\right\} $, and $ R $ is fixed.
If we consider $ {\Bbb C}^{n}={\Bbb R}^{2n} $ and the linear bundle $ \omega^{1/2} $ over $ {\Bbb C}^{n} $, then the
action of dilatations\footnote{In fact one needs to consider $ 2 $-sheeted covering of the group of
similarity transforms.} on this bundle is the same\footnote{Strictly speaking, differs on a multiplication by a constant of
magnitude 1.} as on $ \Omega_{{\Bbb R}^{2n}}^{1/4} $. We
conclude that similarity transforms act as uniformly bounded transformations
\begin{equation}
\overset{\,\,{}_\circ}{H}^{n/2}\left({\bold T}D,\omega^{1/2}\right) \to \overset{\,\,{}_\circ}{H}^{n/2}\left(D,\omega^{1/2}\right)
\notag\end{equation}
as far as both $ D $ and $ {\bold T}D $ remain in the same disk $ \left\{|z|<R\right\} $, and $ R $ is fixed.
\subsection{$ \protect \operatorname{PGL}\left(2,{\Bbb C}\right) $-invariant realization of $ H^{1/2}\left({\Bbb C}P^{1}\right) $. }\label{s2.60}\myLabel{s2.60}\relax
Consider double ratio
\begin{equation}
\left(a:b:c:d\right) = \frac{a-b}{b-d}:\frac{a-c}{c-d},\qquad a,b,c,d\in{\Bbb C}P^{1}.
\notag\end{equation}
Double ratio $ \left(z_{1}:z_{1}+\delta z_{1}:z_{2}:z_{2}+\delta z_{2}\right) $ gives a section $ \rho $ of the bundle
$ T^{*}{\Bbb C}P^{1}\boxtimes T^{*}{\Bbb C}P^{1} $ over $ {\Bbb C}P^{1}\times{\Bbb C}P^{1} $.
In affine coordinates it can be written as $ \frac{1}{\left(z_{1}-z_{2}\right)^{2}}dz_{1}\,dz_{2} $. By
construction this section is invariant with respect to the action of
$ \operatorname{PGL}\left(2,{\Bbb C}\right) $.
Consider $ K_{s}=\frac{\rho^{s} \bar{\rho}^{s}}{\Gamma\left(-2s+1\right)} $. It is a section of $ \Omega^{s}\left({\Bbb C}P_{{\Bbb R}}^{1}\right) \boxtimes \Omega^{s}\left({\Bbb C}P_{{\Bbb R}}^{1}\right) $,
which in local coordinates looks like
$ K_{s}\left(z_{1},z_{2}\right)=\frac{1}{|z_{1}-z_{2}|^{4s}\Gamma\left(-2s+1\right)}dZ_{1}^{s}\,dZ_{2}^{s} $ and has no singularity outside
of $ z_{1}=z_{2} $. Here $ dZ=dx\wedge dy $ if $ z=x+iy. $ The operator with kernel $ K_{s} $ defines a
pairing
\begin{equation}
\left(\alpha,\beta\right)_{s}=\int_{{\Bbb C}P^{1}\times{\Bbb C}P^{1}}K_{s}\left(z_{1},z_{2}\right)\alpha\left(z_{1}\right)\bar{\beta}\left(z_{1}\right)
\notag\end{equation}
on the space $ \Gamma\left(\Omega^{1-s}\left({\Bbb C}P_{{\Bbb R}}^{1}\right)\right) $ if $ s<\frac{1}{2} $, this pairing depends on $ s $
analytically \cite{GelShil58Gen}, and may be continued to an arbitrary $ s $.
\begin{proposition} $ \left(,\right)_{s} $ is a positive-definite Hermitian form on
$ \Gamma\left(\Omega^{1-s}\left({\Bbb C}P_{{\Bbb R}}^{1}\right)\right) $, if $ \left|s-\frac{1}{2}\right|<\frac{1}{2} $. It is equivalent to the $ H^{-1+2s} $-norm
on this space. \end{proposition}
\begin{proof} For $ s=\frac{1}{2} $ we get a standard pairing on $ \Omega^{1/2} $ (since $ K_{1/2} $ is a
$ \delta $-function), so it is sufficient to consider $ s\not=\frac{1}{2} $. Take an affine
coordinate system $ z $ on $ {\Bbb C}P^{1}\smallsetminus\left\{\infty\right\} $. A smooth section of $ \Omega^{t} $ near $ \infty $ is
represented by a smooth $ t $-form $ f\left(z\right)dZ^{t} $ with an asymptotic
\begin{equation}
|z|^{-4t}g\left(1/z\right)dZ^{t},
\notag\end{equation}
near $ \infty $. Here $ g\left(w\right) $ is a smooth function of $ x $ and $ y $, $ w=x+yi. $ Thus the
Fourier transform $ \widehat{f}\left(\zeta\right) $ of $ f $ is rapidly decreasing, smooth outside of $ \zeta=0 $,
and has an asymptotic expansion
\begin{equation}
h\left(\zeta\right)+|\zeta|^{2-4t}h_{1}\left(\zeta\right)+O\left(|\zeta|^{N}\right)
\notag\end{equation}
near 0, here $ N>0 $ is an arbitrary integer, $ h $ and $ h_{1} $ are smooth functions,
and $ t\not=\frac{1}{2} $.
The pairing $ \left(,\right)_{s} $ written in terms of $ \widehat{\alpha} $, $ \widehat{\beta} $ is the pairing
\begin{equation}
\operatorname{const}\cdot\int\widehat{\alpha}\left(\zeta\right)\bar{\widehat{\beta}}\left(\zeta\right)|\zeta|^{4s-2}d\zeta d\bar{\zeta}
\label{equ3.74}\end{equation}\myLabel{equ3.74,}\relax
(in the sense of generalized functions) and we see
that for $ t=1-s $ and $ \left|s-\frac{1}{2}\right|<\frac{1}{2} $ the integral converges, thus the
pairing is positive.
On the other hand, the local equivalence of the norm~\eqref{equ3.74} with
$ H^{2s-1} $ is already proved in the previous section (cf. Equation~\eqref{equ3.7}). \end{proof}
\begin{remark} Note that what we got is a complementary series of unitary
representations of $ \operatorname{SL}\left(2,{\Bbb C}\right) $. \end{remark}
\begin{remark} \label{rem2.60}\myLabel{rem2.60}\relax The defined pairings are compatible with duality $ \int\alpha\beta $
between $ \Omega^{s} $ and $ \Omega^{1-s} $. Note also that in the case $ s=0 $ the pairing has $ \operatorname{rk}=1 $,
dually, in the case $ s=1 $ the pairing becomes $ -\left(\Delta\alpha,\beta\right) $, which has constants
in the null-space. \end{remark}
Put $ s=\frac{1}{4} $. We get a $ \operatorname{PGL}\left(2,{\Bbb C}\right) $-invariant implementation\footnote{I.e., a Hilbert structure which is equivalent to the Sobolev Hilbert
structure which is in turn defined up to equivalence (compare with
Section~\ref{s2.20}).} of the space
$ H^{-1/2}\left(\Omega^{3/4}\right) $. Note now that $ \Omega^{3/4}=\omega^{3/4}\otimes\bar{\omega}^{3/4} $ differs not so much from the
space $ \omega^{1/2}\otimes\bar{\omega} $ we are most interested in. The difference $ \omega^{1/4}\otimes\bar{\omega}^{-1/4} $ is a
bundle with transitions functions $ \left(\frac{D}{\bar{D}}\right)^{1/4} $ of magnitude 1 (here
$ D=dz/dw $). In particular, a choice of affine coordinate $ z $ on $ {\Bbb C}P^{1} $ gives an
identification of $ \omega^{1/2}\otimes\bar{\omega} $ with $ \Omega^{3/4} $ (outside of infinity). This
identification commutes with natural actions of $ \operatorname{Aff}\left({\Bbb C}\right) $ on $ \Omega^{3/4} $ and
$ \omega^{1/2}\otimes\bar{\omega} $ (up to multiplication by a constant of magnitude 1). Here $ \operatorname{Aff}\left({\Bbb C}\right) $
is the group of affine transformations on $ {\Bbb C} $ (in fact to get an action on
$ \omega^{1/2}\otimes\bar{\omega} $ one needs to consider $ 2 $-covering of $ \operatorname{Aff}\left({\Bbb C}\right) $).
This explains the almost-invariance of $ H^{-1/2}\left({\Bbb C},\omega^{1/2}\otimes\bar{\omega}\right) $ with respect
to affine transformations, discussed in Section~\ref{s2.50}. Indeed, consider
a disk $ K\subset{\Bbb C}P^{1} $. On $ {\Bbb C}P^{1}\smallsetminus K $ multiplication by $ dz^{1/4}d\bar{z}^{-1/4} $ gives an isomorphism
of the sheaves $ \omega^{1/2}\otimes\bar{\omega} $ and $ \Omega^{3/4} $. Taking $ \infty $ as a center of $ K $, we obtain the
results of Section~\ref{s2.50} in the case $ n=1 $.
In what follows the following result is sufficient for us to show
conformal invariance of the objects we introduce:
\begin{amplification} \label{amp2.65}\myLabel{amp2.65}\relax Fix a metric on $ {\Bbb C}P^{1} $ and $ \varepsilon>0 $. Consider $ \varphi\in\operatorname{SL}\left(2,{\Bbb C}\right) $, let
$ A\subset{\Bbb C}P^{1} $, $ B=\varphi\left(A\right) $. If both $ {\Bbb C}P^{1}\smallsetminus A $ and $ {\Bbb C}P^{1}\smallsetminus B $ contain disks of radius $ \varepsilon $ with
centers at $ c_{A} $, $ c_{B} $, then the norm of $ \varphi^{*}\colon \overset{\,\,{}_\circ}{H}^{-1/2}\left(B,\omega^{1/2}\otimes\bar{\omega}\right) \to
\overset{\,\,{}_\circ}{H}^{-1/2}\left(A,\omega^{1/2}\otimes\bar{\omega}\right) $ is bounded by
\begin{equation}
C\left(\varepsilon\right)\left(\frac{\operatorname{diam}\left(A\right)}{\operatorname{dist}\left(\varphi^{-1}\left(c_{B}\right),A\right)}\right)^{1/2}.
\notag\end{equation}
By duality, the same bound is valid for action of $ \varphi^{*} $ in $ H^{1/2}\left(\bullet,\omega^{1/2}\right) $.
\end{amplification}
\begin{proof} Indeed, $ SU\left(2\right) $ is compact, thus acts on $ H^{1/2}\left({\Bbb C}P^{1},\omega^{1/2}\otimes\bar{\omega}\right) $ by
uniformly bounded operators. Thus it is sufficient to consider the case when
a disk $ K $ of radius $ \varepsilon $ is fixed (say, if $ c_{A}=c_{B}=\infty $, thus $ |z| > 1/\varepsilon $), and
$ A\cap K=B\cap K=\varnothing $.
Let $ \alpha $, $ \beta $ be sections of $ \omega^{1/4}\otimes\bar{\omega}^{-1/4} $ and $ \omega^{-1/4}\otimes\bar{\omega}^{1/4} $ which coincide
with $ \bar{d}z^{1/4}d\bar{z}^{-1/4} $ and $ dz^{-1/4}d\bar{z}^{1/4} $ inside $ {\Bbb C}P^{1}\smallsetminus K $ correspondingly, and have a
compact support. Then $ \varphi^{*}|_{A}f $, $ f\in\omega^{1/2}\otimes\bar{\omega} $, is equal to
\begin{equation}
\mu_{\varphi}\beta\varphi_{3/4}^{*}|_{A}\left(\alpha f\right),\qquad \mu_{\varphi}=\left(\frac{\varphi'}{|\varphi'|}\right)^{1/2}
\notag\end{equation}
(here $ \varphi_{3/4}^{*} $ acts on $ \Omega^{3/4} $). On the other hand, multiplications by $ \alpha $ and $ \beta $
are bounded operators, and $ \varphi_{3/4}^{*} $ is unitary, so what remains to prove is
a bound on operator of multiplication by $ \mu_{\varphi} $ in $ H^{-1/2} $ or in $ H^{1/2} $. It is
sufficient to consider $ H^{1/2} $.
Let us apply interpolation theorem now: the norm of multiplication
by $ \mu_{\varphi} $ in $ H^{1/2} $ is no more than geometric mean of norms in $ H^{0}=L_{2} $ and in $ H^{1} $.
Since $ |\mu_{\varphi}|=1 $, it is unitary in $ H^{0} $, so it is sufficient to estimate the norm
in $ H^{1} $, which is bounded by $ \operatorname{const}\cdot\max |\mu_{\varphi}'| $, i.e., by
$ \operatorname{const}\cdot\operatorname{dist}\left(\varphi^{-1}\left(\infty\right),A\right)^{-1} $.
To get the estimate in the theorem, note that we have a freedom of
rescaling $ A $, thus may assume that $ \operatorname{diam}\left(A\right)=1 $. \end{proof}
\subsection{Hardy space }\label{s2.70}\myLabel{s2.70}\relax Let $ K = \left\{z \mid |z|\leq1\right\} $. The Hardy space $ {\cal H} $ is the
subspace of $ L_{2}\left(\partial K\right) $ consisting of functions with Fourier coefficients $ \left(a_{n}\right) $
which vanish for $ n<0 $. For a function $ f\in{\cal H} $, $ f=\sum a_{k}z^{k} $, $ |z|=1 $, let $ c_{f}=\sum a_{k}z^{k} $ be
defined for $ |z|<1 $. The latter series converges, and defines a holomorphic
function inside $ K $.
\begin{lemma} The mapping $ f \mapsto c_{f} $ is an injection $ L_{2}\left(\partial K\right) \to H^{1/2}\left(K\right) $. The
image of this injection coincides with
\begin{equation}
\operatorname{Ker}\left(H^{1/2}\left(K\right) \xrightarrow[]{\bar{\partial}} H^{-1/2}\left(K\right)\right).
\notag\end{equation}
\end{lemma}
\begin{proof} It is sufficient to prove that $ \|z^{k}\|_{H^{1/2}\left(K\right)} $ is bounded from
above and from below when $ k\in{\Bbb N} $. Consider a concentric disk $ K_{1} $ of radius
$ R<1 $. Since $ \|z^{k}\|_{H^{1/2}\left(K_{1}\right)}\leq\|z^{k}\|_{H^{1}\left(K_{1}\right)}=O\left(kR^{k}\right) $, it is $ o\left(1\right) $, thus it is
sufficient to consider the norm of $ z^{k} $ in a narrow annulus with external
boundary $ \partial K $.
In turn, taking coordinate $ \log z=a+ib $, $ b\in{\Bbb R}/2\pi{\Bbb Z} $, it is sufficient to
consider $ H^{1/2} $-norm of $ e^{ka} $ on the half-line $ a\leq0 $, more precise, it is
sufficient to consider $ L_{2} $-norm of $ \left(\frac{d}{da}+k\right)^{1/2}e^{ka} $, $ a\leq0 $. Indeed, we need to
show that there is a continuation of $ e^{k\left(a+ib\right)} $ outside of $ a\leq0 $ such that
the $ H^{1/2} $-norm of this continuation is bounded, and that the $ H^{1/2} $-norm of
any such continuation is bounded from below. One can suppose that the
continuation is $ e^{ik b}\varphi\left(a\right) $, so one needs to estimate $ \int\left(1+k^{2}+\alpha^{2}\right)|\widehat{\varphi}\left(\alpha\right)|^{2}d\alpha $.
The operator $ \left(\frac{d}{da}+k\right)^{1/2} $ maps this norm to a norm equivalent to
$ L_{2} $-norm, moreover, it sends functions with support in $ a\geq0 $ into itself
(since $ k\geq0 $). Thus it defines an invertible operator $ H^{1/2}\left({\Bbb R}_{\leq0}\right) \to
H^{0}\left({\Bbb R}_{\leq0}\right)=L_{2}\left({\Bbb R}_{\leq0}\right) $.
This means that $ \left(\frac{d}{da}+k\right)^{1/2}e^{ka}|_{a\leq0}=\left(\frac{d}{da}+k\right)^{1/2}\varphi\left(a\right)|_{a\leq0} $, thus we
should not care about the choice of continuation $ \varphi $. Moreover, since
$ \left(\frac{d}{da}+k\right)^{1/2} $ does not move support to the left,
$ \left(\frac{d}{da}+k\right)^{1/2}e^{ka}=C\left(k\right)e^{ka} $, $ a\leq0 $. Since $ \left(\frac{d}{da}+k\right)^{1/2} $ is an operator of
convolution with $ \frac{a_{+}^{-3/2}e^{-ka}}{\Gamma\left(-1/2\right)} $ (in the sense of generalized
functions, $ x_{+}^{s} $ is 0 if $ x<0 $, $ x^{s} $ if $ x>0 $ and $ s>0 $, and is defined by analytic
continuation from the region $ s>0 $, where the above description defines it
completely \cite{GelShil58Gen}), to estimate $ C\left(k\right) $ one needs to calculate
\begin{align} -\int_{0}^{\infty}x_{+}^{-3/2}e^{-kx}e^{-kx}dx & \buildrel{\text{def}}\over{=} -\int_{0}^{\infty}x^{-3/2}\left(e^{-2kx}-1\right)dx
\notag\\
& = k^{1/2}\int_{0}^{\infty}\left(kx\right)^{-1/2}\frac{1-e^{-2kx}}{kx}d\left(kx\right) = k^{1/2}C.
\notag\end{align}
Thus we need to show that $ k^{1/2}\|e^{ka}|_{a\leq0}\|_{L_{2}} $ is bounded from above and from
below, which is obvious. \end{proof}
\begin{remark} It is clear that if $ f\in{\cal H}\cap C^{\infty} $, then $ f=c_{f}|_{\partial K} $. Moreover, if $ f\cdot\delta_{\partial K} $ is
the continuation of $ f $ to $ {\Bbb C} $ by $ \delta $-function, then $ c_{f}=\operatorname{const}\cdot\bar{\partial}^{-1}\left(f\cdot\delta_{\partial K}\right) $ (see
Section~\ref{s3.30} for description of $ \bar{\partial}^{-1} $). Several following sections are
dedicated to defining something similar in the case $ f\in{\cal H} $. \end{remark}
\begin{remark} \label{rem2.10}\myLabel{rem2.10}\relax In Section~\ref{h4} we will construct a generalization of the
following modification of this result: instead of $ K $ we consider an
isomorphic domain $ {\Bbb C}P^{1}\smallsetminus K $. Instead of functions on $ {\Bbb C}P^{1} $ and $ \partial K $ we consider
sections of $ \omega^{1/2} $ and $ \Omega^{1/2}\otimes\mu $ correspondingly. \end{remark}
\section{Cauchy kernel for $ \omega^{1/2} $ }
\subsection{$ \bar{\partial} $ on piecewise-analytic functions }\label{s3.05}\myLabel{s3.05}\relax Let $ M $ be a complex curve
with a
linear bundle $ {\cal L} $. Consider a domain $ D\subset M $ with a smooth boundary $ \partial D=\gamma $. Let
$ f\left(z\right) $, $ z\in\bar{D} $, be an analytic section of $ {\cal L} $ in $ D $ which continues as a smooth
section to $ \bar{D} $. Extend $ f $ outside of $ D $ as 0.
Consider $ \bar{\partial}f $. It is a section of $ {\cal L}\otimes\bar{\omega} $, which vanishes on $ M\smallsetminus\gamma $. Since $ f $
has a jump of the first kind along $ \gamma $, and $ \bar{\partial} $ is a differential operator of
the first order, $ \bar{\partial}f $ has at most a $ \delta $-function singularity along $ \gamma $. One may
locally write
\begin{equation}
\bar{\partial}f = g\left(z\right)\cdot\delta_{h\left(z\right)=0}.
\notag\end{equation}
Here $ g\left(z\right) $ is a smooth section of $ {\cal L}\otimes\bar{\omega} $, $ h $ is a local equation of $ \gamma $.
Let us calculate $ g|_{\gamma} $ in terms of $ f $. First of all, if one changes the
equation $ h=0 $ by a different equation $ hh_{0}=0 $ (here $ h_{0} $ has no zeros close to
the point in question), the coefficient at $ g\left(z\right) $ changes to
\begin{equation}
\delta_{hh_{0}=0}=\frac{1}{h_{0}}\delta_{h=0},
\notag\end{equation}
since $ \delta $-function has homogeneity degree $ -1 $. Thus $ g\left(z\right)|_{\gamma} $ should be
multiplied by $ h_{0}|_{\gamma} $ to preserve the same value of the product. This shows
that to get an invariant description of $ g\left(z\right)|_{\gamma} $ one needs to write
\begin{equation}
g\left(z\right)|_{\gamma} = G\left(z\right)\otimes dh|_{\gamma}.
\notag\end{equation}
Here $ dh|_{\gamma} $ is considered as a section of the conormal bundle $ N^{*}\gamma $ to $ \gamma $,
thus $ G\left(z\right) $ should be a section of $ \left({\cal L}\otimes\bar{\omega}\right)|_{\gamma}\otimes\left(N^{*}\gamma\right)^{*} = \left({\cal L}\otimes\bar{\omega}\right)|_{\gamma}\otimes N\gamma $.
Now $ G\left(z\right) $ does not depend on the parameterization of a neighborhood
of $ \gamma $, so it should be directly expressible in terms of $ f\left(z\right) $.
\begin{lemma} $ \bar{\omega}|_{\gamma} $ is canonically isomorphic to $ \left(N^{*}\gamma\right)\otimes{\Bbb C} $. \end{lemma}
\begin{proof} Indeed, $ \bar{\omega} $ is defined as a quotient of $ \left(T^{*}M_{{\Bbb R}}\right)\otimes{\Bbb C} $ by holomorphic
forms, moreover, $ \bar{\omega}_{{\Bbb R}} $ is an isomorphic
image of $ T^{*}M_{{\Bbb R}} $. Since $ N^{*}\gamma\subset T^{*}M_{{\Bbb R}} $, there is a natural non-zero mapping
$ N^{*}\gamma \to \bar{\omega}_{{\Bbb R}} $, which gives the required isomorphism after complexification. \end{proof}
As a corollary, $ \left({\cal L}\otimes\bar{\omega}\right)|_{\gamma}\otimes N\gamma \simeq {\cal L}|_{\gamma} $. Now the following fact is obvious
from calculations in local coordinates:
\begin{proposition} $ G\left(z\right) = -f\left(z\right)|_{\gamma} $. \end{proposition}
In particular, we see that one can write
\begin{equation}
\bar{\partial}f=-e\circ r\left(f\right)
\label{equ3.02}\end{equation}\myLabel{equ3.02,}\relax
in terms of operators $ r $ of restriction and $ e $ of extension-by-$ \delta $-function.
Thus we can consider $ -e\circ r $ as an ``approximation'' to $ \bar{\partial} $ which is good on
functions which are analytic far from $ \gamma $. This approximation is exact on
functions with a jump of the first kind.
Yet another way to treat this identity is to write it as
\begin{equation}
-\bar{\partial}^{-1}\circ e\circ r\left(F\right) = \vartheta_{D}F.
\notag\end{equation}
Here $ F $ is a function which is holomorphic in a neighbourhood of $ \bar{D} $, $ \vartheta_{D} $ is
a function which is 1 on $ D $ and 0 outside of $ D $. We give a mollified
version of this statement in Section~\ref{s4.35}.
\subsection{Self-duality of $ \bar{\partial} $ } Consider a mapping $ \bar{\partial}\colon \omega^{1/2} \to \omega^{1/2}\otimes\bar{\omega} $. Note that the
spaces of sections of these bundles are dual w.r.t. the pairing
\begin{equation}
\left(\varphi,\psi\right) = \int_{M}\varphi\psi.
\label{equ3.20}\end{equation}\myLabel{equ3.20,}\relax
\begin{lemma} $ \bar{\partial} $ is skew-symmetric w.r.t. the above pairing,
\begin{equation}
\left(\varphi_{1},\bar{\partial}\varphi_{2}\right)+\left(\varphi_{2},\bar{\partial}\varphi_{1}\right)=0.
\notag\end{equation}
\end{lemma}
\begin{proof} It is sufficient to show that $ \left(\varphi_{1},\bar{\partial}\varphi_{1}\right)=0 $, i.e., to study
\begin{equation}
\int_{M}\varphi\bar{\partial}\varphi = \frac{1}{2}\int_{M}\bar{\partial}\varphi^{2} = \frac{1}{2}\int_{M}d\varphi^{2}
\notag\end{equation}
which obviously vanishes. \end{proof}
Note that this supports the heuristic that the bundle $ \omega^{1/2} $ is ``the
best one'' of the powers of $ \omega $.
Used literally, the above considerations were applicable to the
space of smooth sections of the bundles in question. On the other hand,
the pairing~\eqref{equ3.20} extends to a continuous pairing between $ H^{1/2}\left(M,\omega^{1/2}\right) $
and $ H^{-1/2}\left(M,\omega^{1/2}\otimes\bar{\omega}\right) $, and the operator $ \bar{\partial} $ maps one of these spaces to
another. Using the facts from Section~\ref{s2.20} We get a
\begin{proposition} The operator $ \bar{\partial} $ gives a canonically defined
Fredholm symplectic form on $ H^{1/2}\left(M,\omega^{1/2}\right) $. This form is non-degenerate if
$ M={\Bbb C}P^{1} $. \end{proposition}
Here we call a bilinear form $ \alpha $ on a Hilbert space $ {\cal H} $ a {\em Fredholm form},
if the corresponding operator $ \alpha\colon H \to H^{*}=H $ is Fredholm.
Recall that in the case $ M={\Bbb C}P^{1} $ the space $ H^{1/2}\left(M,\omega^{1/2}\right) $ is the ``best''
of Sobolev spaces for $ \omega^{1/2} $, since it allows the action of $ \operatorname{SL}\left(2,{\Bbb C}\right) $ (at
least ``in small'') by bounded operators. Here we see another ``nice''
property of this particular Sobolev space: there is a canonically defined
invertible pairing on this space (invertible in the sense that the
corresponding operator is bounded).
\begin{remark} If we could construct a self-adjoint operator which enjoys the
above properties of the operator $ \bar{\partial} $, i.e., is $ \operatorname{SL}\left(2,{\Bbb C}\right) $-invariant and
Fredholm, one would be (almost) able
to define an invariant Sobolev space structure on $ H^{1/2}\left(M,\omega^{1/2}\right) $. The only
thing missing would be positive definiteness. \end{remark}
\subsection{The kernel of $ \bar{\partial}^{-1} $ }\label{s3.30}\myLabel{s3.30}\relax Suppose $ M={\Bbb C}P^{1} $. In this case the operator $ \bar{\partial}:
{\cal D}'\left(\omega^{1/2}\right) \to {\cal D}'\left(\omega^{1/2}\otimes\bar{\omega}\right) $
is invertible, thus the operator $ \bar{\partial}^{-1} $ is canonically defined. Consider a
kernel $ K\left(x,y\right) $ of this operator,
\begin{equation}
\left(\bar{\partial}^{-1}f\right)\left(x\right) = \int K\left(x,y\right)f\left(y\right).
\notag\end{equation}
Obviously, $ K\left(x,y\right) $ is a section of $ \omega^{1/2} \boxtimes \omega^{1/2} $.
To calculate $ K\left(x,y\right) $ consider a $ \delta $-section $ \delta_{z_{0}} $ of $ \omega^{1/2} $ at $ z_{0} $ (defined
up to a scalar multiple). Then $ K\left(\bullet,z_{0}\right)=\operatorname{const}\cdot\bar{\partial}^{-1}\delta_{z_{0}} $ is a holomorphic
section of $ \omega^{1/2} $ on $ {\Bbb C}P^{1}\smallsetminus\left\{z_{0}\right\} $. It is easy to see that it has a simple pole
at $ z=z_{0} $, thus has no zeros (since $ \omega^{1/2}\simeq{\cal O}\left(-1\right) $). We conclude that $ K\left(x,y\right) $
has a simple pole at $ x=y $. There is only one such a section of $ \omega^{1/2} \boxtimes
\omega^{1/2} $. Now one may easily recognize the kernel for $ \bar{\partial}^{-1} $ in the Cauchy
formula
\begin{equation}
\bar{\partial}^{-1}\delta_{z=z_{0}}dz^{1/2}d\bar{z} = \frac{1}{2\pi i} \frac{1}{z-z_{0}}dz^{1/2}
\notag\end{equation}
The right-hand side $ f\left(z,z_{0}\right) $ has no pole at $ \infty $, and is uniquely determined
by this condition and the equation
\begin{equation}
\bar{\partial}f=\delta_{z=z_{0}}dz^{1/2}d\bar{z}
\notag\end{equation}
in $ {\Bbb C}P^{1}\smallsetminus\left\{\infty\right\} $. We conclude that
\begin{equation}
K\left(x,y\right) = \frac{1}{2\pi i} \frac{1}{x-y}dx^{1/2}dy^{1/2}.
\notag\end{equation}
\subsection{Cauchy kernel and $ L_{2} $ } Consider a smooth embedded curve $ \gamma\hookrightarrow{\Bbb C}P^{1} $ (we do
not suppose that $ \gamma $ is compact). Any smooth function $ f $ on $ \gamma $ with a
compact support may be extended as a generalized $ \delta $-function $ e\left(f\right) $ on $ {\Bbb C}P^{1} $
with a support on $ \gamma $, same for half-forms on $ \gamma $. Applying $ \bar{\partial}^{-1} $ to the
result, we obtain a form on $ {\Bbb C}P^{1} $ which is holomorphic outside of $ \gamma $.
Here we discuss when this mapping $ \bar{\partial}^{-1}\circ e $ allows $ f\in L_{2}\left(\gamma\right) $ instead of
$ f\in{\cal D}\left(\gamma\right) $.
\begin{remark} In what follows the main example of $ \gamma $ is a disjoint union of
infinite number of circles.
\end{remark}
Fix a point $ z\in{\Bbb C}P^{1} $ and suppose that some neighborhood of $ z $ does not
intersect $ \gamma $. Then for a half-form $ f $
\begin{align} \frac{\bar{\partial}^{-1}\circ e\left(f\right)}{dz^{1/2}d\bar{z}}|_{z_{0}} & \buildrel{\text{def}}\over{=} \left< \bar{\partial}^{-1}\circ e\left(f\right), \delta_{z_{0}}dz^{1/2}\right>
\notag\\
& = -\left< e\left(f\right), \bar{\partial}^{-1}\left(\delta_{z_{0}}dz^{1/2}\right)\right>
\notag\\
& =-\left< e\left(f\right), K\left(\bullet,z_{0}\right)dz^{-1/2} \right> = \int_{C}f\left(x\right)K\left(x,z_{0}\right)dz^{-1/2}.
\notag\end{align}
We conclude that the linear functional of calculating $ \bar{\partial}^{-1}\circ e\left(f\right) $ at $ z_{0} $ is
given by the kernel $ K\left(\bullet,z_{0}\right)|_{\bullet\in C} $. However, since $ K\left(\bullet,z_{0}\right) $ is bounded
outside of a neighborhood of $ z_{0} $, we see that
\begin{proposition} \label{prop4.12}\myLabel{prop4.12}\relax The mapping $ \bar{\partial}^{-1}\circ e $ extends to a mapping from $ L_{2}\left(\gamma\right) $ to
the space $ \operatorname{Hol}\left({\Bbb C}P^{1}\smallsetminus\bar{\gamma}\right) $ of forms holomorphic outside of $ \gamma $ iff the length of
$ \gamma $ is finite. \end{proposition}
\section{Toy theory }\label{h35}\myLabel{h35}\relax
In this section we study an example which shows the main framework
of our approach without any of the complications related to the necessity
to consider $ H^{1/2} $-spaces. This is an example related to dealing with
{\em partial period mapping\/} $ \Gamma\left(M,\omega\right) \to {\Bbb C}^{g} $ of taking periods of global
holomorphic forms along $ A $-cycles on the Riemann surface. In fact, we
start from studying a related space of global holomorphic functions which
are allowed to have constant jumps along $ A $-cycles.
\subsection{Toy global space } Consider a complex curve $ M $ and the mapping $ \partial\bar{\partial}\colon \Omega^{0} \to
\Omega^{1} $ (let us recall that $ \Omega^{\text{top}}=\Omega^{1} $ in our notations for fractional forms).
The integral $ -\int_{M}if\cdot\partial\bar{\partial} \bar{f} $ defines a sesquilinear form on functions with
compact support, it is Hermitian, and the only functions which are in the
null-space of this form are constants, since the integral is equal to
$ \int_{M}i\partial f\cdot\bar{\partial} \bar{f}\geq0 $.
This integral defines a pre-Hilbert structure on global sections of
$ \Omega^{0} $ modulo constants, this structure is compatible with the
Sobolev $ H^{1} $-topology.
Moreover, if $ M={\Bbb C}P^{1} $, this structure is $ \operatorname{PGL}\left(2,{\Bbb C}\right) $-invariant, and coincides
with the structure of supplementary series of representations of $ \operatorname{SL}\left(2,{\Bbb C}\right) $
(in notations of Section~\ref{s2.60} it is the case $ s=1 $, compare with Remark
~\ref{rem2.60}). As a result, we obtain a $ \operatorname{PGL}\left(2,{\Bbb C}\right) $-invariant realization of
Hilbert space $ H^{1}\left({\Bbb C}P^{1}\right)/\operatorname{const} $.
The dual space is the subspace $ H_{\int=0}^{-1}\left({\Bbb C}P^{1},\Omega^{1}\right) $ of $ H^{-1}\left({\Bbb C}P^{1},\Omega^{1}\right) $
consisting of forms with integral 0, thus the latter space also carries
an $ \operatorname{PGL}\left(2,{\Bbb C}\right) $-invariant Hilbert structure. To describe it, note that Remark
~\ref{rem2.60} is still applicable, thus in the limit $ s \to 0 $ the pre-Hilbert
structure of Section~\ref{s2.60} becomes a form of $ \operatorname{rk}=1 $ with $ H_{\int=0}^{-1}\left({\Bbb C}P^{1},\Omega^{1}\right) $ in
the null-space. As a corollary, $ \frac{d}{ds}|_{s=0}\left(\alpha,\beta\right)_{s} $ gives a correctly
defined positive form on $ H_{\int=0}^{-1}\left({\Bbb C}P^{1},\Omega^{1}\right) $ which should be dual to the
pairing on $ H^{1}\left({\Bbb C}P^{1}\right)/\operatorname{const} $.
Taking the $ s $-derivative of the kernel $ K_{s} $ from Section~\ref{s2.60}, we get
$ \log |z_{1}-z_{2}| $ (when restricted on $ {\Bbb C}\subset{\Bbb C}P^{1} $), which is indeed scaling-invariant
up to addition of a constant. (Note that the constant is irrelevant,
since the forms $ \alpha $ and $ \beta $ we are going to pair $ K $ with have vanishing
integral.) What the above argument shows is that the continuation of this
kernel to $ {\Bbb C}P^{1} $ is invariant w.r.t. fraction-linear mappings.
\subsection{Almost-perpendicularity } The description of the pairing on
$ H_{\int=0}^{-1}\left({\Bbb C}P^{1},\Omega^{1}\right) $ immediately implies
\begin{proposition} Consider two disjoint disks $ K_{1} $, $ K_{2} $ on $ {\Bbb C}P^{1} $ of conformal
distance $ l $. Let $ H_{1,2} = \overset{\,\,{}_\circ}{H}_{\int=0}^{-1}\left(K_{1,2},\Omega^{1}\right)\subset H_{\int=0}^{-1}\left({\Bbb C}P^{1},\Omega^{1}\right) $ . Let $ P_{l} $ be the
orthogonal projector from one subspace to another. Then
\begin{equation}
\|P_{l}\| \sim e^{-l}.
\notag\end{equation}
The equivalence means that the quotient of two sides remains bounded and
separated from 0 when $ l $ varies. \end{proposition}
\begin{proof} Since the natural norm on $ H_{\int=0}^{-1}\left(\Omega^{1}\right) $ is invariant with respect
to $ \operatorname{PGL} $, the angle between the subspaces $ H_{1,2} $ depends on $ l $ only. Note that
it is sufficient to prove the statement in the case $ l>\varepsilon $ for some fixed $ \varepsilon>0 $.
Now proceed as in the proof of Proposition~\ref{prop3.170}. The only change is
that the kernel is now $ \log |L+z_{1}-z_{2}| $, and on forms with integral 0 it is
equivalent to $ \left(\log |L+z_{1}-z_{2}|-\log L-\operatorname{Re}\frac{z_{1}-z_{2}}{L}\right) $, which is $ L^{-2} $ times an
operator of rank 1, plus much smaller operator. \end{proof}
\begin{remark} Note that norm of this projector is much smaller than the
norm of the projector from Proposition~\ref{prop3.170} (if $ l $ is big enough). \end{remark}
\begin{corollary} \label{cor35.20}\myLabel{cor35.20}\relax Consider a family of disjoint disks $ \left\{K_{i}\right\} $ in $ {\Bbb C}P^{1} $ with
pairwise conformal distances $ l_{ij} $, $ i\not=j $. Let $ l_{i i}=0 $. If the matrix $ \left(e^{-l_{ij}}\right) $
gives a bounded operator $ l_{2} \to l_{2} $, then the natural extension-by-0 mapping
\begin{equation}
\bigoplus_{l_{2}}\overset{\,\,{}_\circ}{H}_{\int=0}^{-1}\left(K_{i},\Omega^{1}\right) \to H_{\int=0}^{-1}\left({\Bbb C}P^{1},\Omega^{1}\right)
\notag\end{equation}
is a continuous injection, and, dually, the natural restriction mapping
\begin{equation}
H^{1}\left({\Bbb C}P^{1}\right)/\operatorname{const} \to \bigoplus_{l_{2}}H^{1}\left(K_{i}\right)/\operatorname{const}
\notag\end{equation}
is a continuous surjection. \end{corollary}
\subsection{Topology on the boundary }\label{s35.20}\myLabel{s35.20}\relax Consider a disk $ K\subset{\Bbb C}P^{1} $. Given a
function $ f $ on $ \partial K $, one can consider its decomposition into a sum $ f_{+}+f_{-} $ of
functions which can be analytically extended into/outside of $ K $. Such a
decomposition
exists if $ f $ is in $ L_{2} $, and the summands are uniquely defined up to
addition of a constant.
If $ f\in H^{1/2}\left(\partial K\right) $, then $ f_{+} $ and $ f_{-} $ are $ H^{1} $-functions on $ K $ and on $ {\Bbb C}P^{1}\smallsetminus K $
(compare with Section~\ref{s2.70}), if $ f\in H^{1/2}\left(\partial K\right)/\operatorname{const} $, then $ f_{+}\in H^{1}\left(K\right)/\operatorname{const} $,
$ f_{-}\in H^{1}\left({\Bbb C}P^{1}\smallsetminus K\right)/\operatorname{const} $, and $ f_{\pm} $ are uniquely defined by $ f $. Thus $ H^{1} $-norms of $ f_{\pm} $
are correctly defined. In the same way as in Section~\ref{s2.70} one can prove
\begin{lemma} \label{lm35.30}\myLabel{lm35.30}\relax The described above extension mapping
\begin{equation}
H^{1/2}\left(\partial K\right) \to H^{1}\left(K\right)/\operatorname{const} \oplus H^{1}\left({\Bbb C}P^{1}\smallsetminus K\right)/\operatorname{const}
\notag\end{equation}
is a continuous injection. \end{lemma}
Since the Hilbert structure on the right-hand side can be made
$ \operatorname{PGL}\left(2,{\Bbb C}\right) $-invariant, the extension mapping defines the $ \operatorname{PGL}\left(2,{\Bbb C}\right) $-invariant
realization of the Hilbert structure on $ H^{1/2}\left(\partial K\right) $. The property of
invariance can be described in the following way: let $ \varphi $ be a
fraction-linear mapping, $ \varphi\left(K\right)=K' $. Then $ \varphi^{*} $ defines a unitary operator
\begin{equation}
\varphi^{*}\colon H^{1/2}\left(\partial K'\right) \to H^{1/2}\left(\partial K\right).
\notag\end{equation}
\begin{remark} \label{rem35.35}\myLabel{rem35.35}\relax In the same way as we defined a $ \operatorname{PGL}\left(2,{\Bbb C}\right) $-invariant
pairing on sections of $ \Omega^{1-s} $ on $ {\Bbb C}P^{1} $ (in Section~\ref{s2.60}), one can define a
$ \operatorname{PGL}\left(2,{\Bbb R}\right) $-invariant pairing on section of $ \Omega_{{\Bbb R}P^{1}}^{1-s} $ on $ {\Bbb R}P^{1}=S^{1} $. In this way
one gets so called supplementary series of representations of $ \operatorname{SL}\left(2,{\Bbb R}\right) $. By
expressing the pairing in appropriate coordinate systems on $ {\Bbb R}P^{1}=S^{1} $ one
can easily see that the described above $ \operatorname{SL}\left(2,{\Bbb R}\right) $-invariant Hilbert
structure coincides with one of these structures. \end{remark}
\begin{definition} The representation $ f=f_{+}+f_{-} $ decomposes $ H^{1/2}\left(\partial K\right)/\operatorname{const} $ into a
direct sum of two subspaces which we denote $ H_{+}^{1/2}\left(\partial K\right)/\operatorname{const} $ and
$ H_{-}^{1/2}\left(\partial K\right)/\operatorname{const} $. \end{definition}
Combining the above description of $ H^{1/2}\left(\partial K\right) $ with results of the
previous section, one obtains
\begin{corollary} In the conditions of Corollary~\ref{cor35.20} the natural
restriction mapping
\begin{equation}
H^{1}\left({\Bbb C}P^{1}\right)/\operatorname{const} \to \bigoplus_{l_{2}}H^{1/2}\left(K_{i}\right)/\operatorname{const}
\notag\end{equation}
is a continuous surjection. \end{corollary}
\subsection{Space of holomorphic functions }\label{s35.30}\myLabel{s35.30}\relax Consider a family $ \left\{K_{i}\right\} $ of
disjoint disks in $ {\Bbb C}P^{1} $.
\begin{definition} We say that a generalized function $ f $ on $ {\Bbb C}P^{1} $ is
$ H^{1} $-{\em holomorphic in\/} $ {\Bbb C}P^{1}\smallsetminus\bigcup K_{i} $ if $ f\in H^{1}\left({\Bbb C}P^{1}\smallsetminus\bigcup K_{i}\right) $, and
$ \bar{\partial}f=0\in H^{0}\left({\Bbb C}P^{1}\smallsetminus\bigcup K_{i},\bar{\omega}\right)= L_{2}\left({\Bbb C}P^{1}\smallsetminus\bigcup K_{i},\bar{\omega}\right) $. Denote the
the space of $ H^{1} $-holomorphic functions in $ {\Bbb C}P^{1}\smallsetminus\bigcup K_{i} $ by $ {\cal H}^{\left(1\right)} $.
\end{definition}
Note that the Sobolev spaces in this definition are generalized
ones. Note also that the Hilbert norm on $ L_{2}\left({\Bbb C}P^{1}\smallsetminus\bigcup K_{i},\bar{\omega}\right) $ is canonically
defined by $ \|\alpha\|^{2}=-i\int\bar{\alpha}\alpha $.
\begin{theorem} \label{th35.15}\myLabel{th35.15}\relax Suppose that the conformal distance between $ \partial K_{i} $ and $ \partial K_{j} $
is $ l_{ij} $, and the matrix $ \left(e^{-l_{ij}}\right) $ gives a bounded operator $ l_{2} \to l_{2} $.
Consider the mapping of taking the boundary value:
\begin{equation}
b\colon {\cal H}^{\left(1\right)}/\operatorname{const} \to \bigoplus_{l_{2}}H^{1/2}\left(\partial K_{i}\right)/\operatorname{const}.
\notag\end{equation}
Let $ b_{-} $ be the component of this mapping going into $ \bigoplus_{l_{2}}H_{-}^{1/2}\left(\partial K_{i}\right)/\operatorname{const} $.
Then $ b_{-} $ is a bounded invertible mapping. \end{theorem}
\begin{proof} We already know that $ b_{-} $ is bounded. To show that one can
reproduce a function by the component $ b_{-} $ of its restriction to a boundary
consider a function $ f\in{\cal H}^{\left(1\right)} $. By definition, it is a restriction of some
function $ g\in H^{1}\left({\Bbb C}P^{1}\right) $, and this function $ g $ is defined up to a function with
support in $ \bigcup K_{i} $. By the condition of $ f $ being holomorphic, $ \bar{\partial}g $ is an
element of $ L_{2}\left(\bigcup K_{i},\bar{\omega}\right)=\bigoplus_{l_{2}}L_{2}\left(K_{i},\bar{\omega}\right) $, and $ \bar{\partial}g $ is defined up to addition of
$ \bar{\partial}\overset{\,\,{}_\circ}{H}^{1}\left(\bigcup K_{i}\right) $. On the other hand, the topology on $ H^{1}\left(X\right) $ (here $ X $ is a manifold
with coordinates $ x_{k} $) can be defined by the norm $ \|f\|_{L_{2}}^{2}+\sum\|\partial_{x_{k}}f\|_{L_{2}}^{2} $, thus
$ \overset{\,\,{}_\circ}{H}^{1}\left(\bigcup K_{i}\right) = \bigoplus_{l_{2}}\overset{\,\,{}_\circ}{H}^{1}\left(K_{i}\right) $.
We conclude that $ \bar{\partial}g $ is a canonically defined element of
\begin{equation}
\bigoplus_{l_{2}}L_{2}\left(K_{i},\bar{\omega}\right)/\bar{\partial}\overset{\,\,{}_\circ}{H}^{1}\left(K_{i}\right).
\notag\end{equation}
\begin{lemma} Consider a disk $ K\subset{\Bbb C}P^{1} $. Then there exists a canonical
isomorphism
\begin{equation}
L_{2}\left(K,\bar{\omega}\right)/\bar{\partial}\overset{\,\,{}_\circ}{H}^{1}\left(K\right) \simeq H_{-}^{1/2}\left(\partial K\right)/\operatorname{const}.
\notag\end{equation}
\end{lemma}
\begin{proof} Let us start with a left-to-right mapping. Let
$ \alpha\in L_{2}\left(K,\bar{\omega}\right)/\bar{\partial}\overset{\,\,{}_\circ}{H}^{1}\left(K\right) $. Since $ \bar{\partial} $ is an elliptic operator without cokernel and
with $ 1 $-dimensional null-space, $ \bar{\partial}^{-1}\alpha $ is an $ H^{1} $-function defined up to
addition of a constant and addition of an element of $ \overset{\,\,{}_\circ}{H}^{1}\left(K\right) $. Thus the
restriction of $ f $ to $ \partial K $ is defined up to a constant, so it is an element
of $ H^{1/2}\left(\partial K\right)/\operatorname{const} $. Moreover, it is in $ H_{-}^{1/2}\left(\partial K\right)/\operatorname{const} $ since $ \bar{\partial}^{-1}\alpha $ is
holomorphic outside $ K $.
To get right-to-left mapping note that any element of
$ H_{-}^{1/2}\left(\partial K\right)/\operatorname{const} $ can be (by definition) continued to a holomorphic outside
of $ K $ function $ f $, and this function is of class $ H^{1} $. Thus to this
continuation one can apply the arguments which precede the lemma.
\end{proof}
To finish the proof of the theorem note that the knowledge of $ r_{-}\left(f\right) $
allows one to construct $ \bar{\partial}f $ (by lemma), so the application of $ \bar{\partial}^{-1} $
reconstructs $ f $.\end{proof}
\subsection{Toy Riemann--Roch theorem }\label{s35.40}\myLabel{s35.40}\relax Consider a family of disjoint disks
$ K_{i} $, $ i\in I $, and the corresponding space of $ H^{1} $-holomorphic functions. Suppose
that $ I $ has an involution ' which interchanges two halves of $ I=I_{+}\amalg I_{+}' $. Fix
automorphisms $ \varphi_{i} $ of $ {\Bbb C}P^{1} $, $ i\in I $, such that $ \varphi_{i'}=\varphi_{i}^{-1} $, and $ \varphi_{i}\left(\partial K_{i}\right) $ is $ \partial K_{i'} $
with inverted orientation. Note that if the set $ I $ is finite, then after
gluing $ \partial K_{i} $ via $ \varphi_{i} $ one gets a curve of genus $ \operatorname{card}\left(I_{+}\right)=\operatorname{card}\left(I\right)/2 $.
Associate to a function $ f\in{\cal H}^{\left(1\right)} $ the jump of its boundary value after
such a gluing:
\begin{equation}
{\cal J}\colon f \mapsto \left(f|_{\partial K_{j}}-\varphi_{j}^{*}\left(f|_{\partial K_{j'}}\right)\right),\quad j\in I_{+}.
\label{equ4.101}\end{equation}\myLabel{equ4.101,}\relax
\begin{theorem} \label{th35.45}\myLabel{th35.45}\relax Suppose that the conformal distance between $ \partial K_{i} $ and $ \partial K_{j} $
is $ l_{ij} $, and the matrix $ \left(e^{-l_{ij}}-\delta_{ij}\right) $ gives a compact operator $ l_{2} \to l_{2} $.
Then the mapping
\begin{equation}
{\cal J}\colon {\cal H}^{\left(1\right)}/\operatorname{const} \to \bigoplus\Sb l_{2} \\ j\in I_{+}\endSb H^{1/2}\left(\partial K_{j}\right)/\operatorname{const}
\notag\end{equation}
is a Fredholm operator of index 0. \end{theorem}
\begin{proof} Consider the composition $ \widetilde{{\cal J}}={\cal J}\circ b_{-}^{-1} $. Let us show that $ \widetilde{{\cal J}} $ is
Fredholm of index 0, this would immediately imply the statement of the
theorem. The mapping $ \varphi_{j}^{*} $ interchanges $ \pm $-components of $ H^{1/2}\left(\partial K_{j}\right)/\operatorname{const} $ and
$ H^{1/2}\left(\partial K_{j'}\right)/\operatorname{const} $, thus one can identify
\begin{equation}
H^{1/2}\left(\partial K_{j}\right)/\operatorname{const} =H_{-}^{1/2}\left(\partial K_{j}\right)/\operatorname{const} \oplus H_{+}^{1/2}\left(\partial K_{j}\right)/\operatorname{const}
\notag\end{equation}
with $ H_{-}^{1/2}\left(\partial K_{j}\right)/\operatorname{const} \oplus H_{-}^{1/2}\left(\partial K_{j'}\right)/\operatorname{const} $. Denote the composition of $ \widetilde{{\cal J}} $ and
this identification by $ {\cal K}\colon \bigoplus_{l_{2}}H_{-}^{1/2}\left(\partial K_{j}\right)/\operatorname{const} \to \bigoplus_{l_{2}}H_{-}^{1/2}\left(\partial K_{j}\right)/\operatorname{const} $.
For $ f\in{\cal H}^{\left(1\right)} $ denote by $ f_{j\pm} $ the $ \pm $-components of $ f|_{\partial K_{j}} $. One can easily see
that $ {\cal K}\left(\left(f_{j-}\right)_{j\in I}\right)=\left(f_{j-}-\varphi_{j}^{*}\left(f_{j'+}\right)\right)_{j\in I} $, in other words, $ {\cal K}=\operatorname{id}-\varphi^{*}\circ b_{+}\circ b_{-}^{-1} $,
here $ b_{+}=b-b_{-} $. Since $ \varphi^{*}=\bigoplus_{l_{2}}\varphi_{j}^{*} $ is an isometry, to prove the theorem it is
enough to show that $ b_{+}\circ b_{-}^{-1} $ is compact. Now we investigate how to
reconstruct $ H_{+}^{1/2} $-components of restriction to a boundary via
$ H_{-}^{1/2} $-components.
Consider the operator with the Cauchy kernel $ \left(y-x\right)^{-1}dy $, $ x,y\in{\Bbb C}P^{1} $.
Restrict this kernel to $ \bigcup\partial K_{i} $. One gets an operator which sends functions
on $ \bigcup\partial K_{i} $ to functions on $ \bigcup\partial K_{i} $. Put zeros instead of the diagonal terms
(which send functions on $ \partial K_{i} $ to functions on $ \partial K_{i} $), and call the resulting
operator $ {\bold K} $. Obviously, the functions in the image can be holomorphically
extended to $ K_{i} $. Moreover, if a function was non-zero on $ \partial K_{i} $ only, and
could be analytically continued inside $ K_{i} $, then this function is in the
zero-space of $ {\bold K} $.
Thus the operator $ {\bold K} $ maps $ \bigoplus H_{-}^{1/2}\left(\partial K_{i}\right)/\operatorname{const} $ to $ \bigoplus H_{+}^{1/2}\left(\partial K_{i}\right)/\operatorname{const} $.
Moreover, as the proof of Theorem~\ref{th35.15} shows, $ b_{+}\circ b_{-}^{-1}= \frac{{\bold K}}{2\pi i} $, since
the pseudodifferential operator $ \bar{\partial}^{-1} $ has $ \frac{dy}{2\pi i\left(x-y\right)} $ as the null-space.
We conclude that the image $ V_{1}=b\left({\cal H}^{\left(1\right)}/\operatorname{const}\right) $ in $ \bigoplus H^{1/2}\left(\partial K_{i}\right)/\operatorname{const} $ is
a graph of operator $ \frac{{\bold K}}{2\pi i}\colon \bigoplus H_{-}^{1/2}\left(\partial K_{i}\right)/\operatorname{const} \to \bigoplus H_{+}^{1/2}\left(\partial K_{i}\right)/\operatorname{const} $. On
the other hand, consider the subspace $ V_{2} $ of $ \bigoplus H^{1/2}\left(\partial K_{i}\right)/\operatorname{const} $ consisting
of functions which are in the null-space of $ {\cal J} $, i.e., the subspace given
by the condition
\begin{equation}
f|_{\partial K_{j}}=\varphi_{j}^{*}\left(f|_{\partial K_{j'}}\right),\qquad j\in I_{+}.
\notag\end{equation}
Since $ \varphi_{j}^{*} $ interchanges $ H_{-}^{1/2}\left(\partial K_{j}\right)/\operatorname{const} $ and $ H_{+}^{1/2}\left(\partial K_{j'}\right)/\operatorname{const} $ and is
unitary, we conclude that the subspace $ V_{2} $ is a graph of a unitary
mapping $ \bigoplus\varphi_{i}^{*}|_{H_{+}^{1/2}\left(\partial K_{i}\right)} $.
Thus we are in conditions of the abstract Riemann--Roch theorem
(Theorem~\ref{th6.50}), which finishes the proof. \end{proof}
\begin{remark} This theorem is an infinite-genus variant of Riemann--Roch
theorem for the case of the bundle $ {\cal O} $ on an algebraic curve $ M $. Indeed, the
latter theorem says that the mapping $ \bar{\partial}\colon H^{s}\left(M,{\cal O}\right) \to H^{s-1}\left(M,\bar{\omega}\right) $
between Sobolev spaces is a Fredholm mapping of index $ 1-g\left(M\right) $. The theory
of elliptic operators says that in a case of smooth compact $ M $ the value
of $ s $ is irrelevant, but in our context we are forced to use the value
$ s=1 $.
The relation of the mapping $ \bar{\partial} $ to the mapping $ {\cal J} $ is the standard (in
mathematical physics) trick of {\em reduction to boundary}. We do cuts in the
curve $ M $ to obtain a region $ S={\Bbb C}P^{1}\smallsetminus\bigcup K_{i} $. Instead of $ H^{s}\left(M,{\cal O}\right) $ we consider
subspace $ V $ of $ H^{s}\left(S,{\cal O}\right) $ which consists of functions which satisfy gluing
conditions on the boundary. These gluing conditions are similar to $ {\cal J}f=0 $,
but are applied to functions (instead of functions modulo constants),
thus there is an extra condition per cut, total $ g $ extra conditions. Let
\begin{equation}
{\cal J}_{H^{1}}\colon H^{1}\left(S,{\cal O}\right)/\operatorname{const} \to \bigoplus_{l_{2}}H^{1/2}\left(\partial K_{j}\right)/\operatorname{const}
\notag\end{equation}
be defined by the formula~\eqref{equ4.101}. Thus $ V/\operatorname{const} $ is a subspace of $ \operatorname{Ker}{\cal J}_{H^{1}} $
of codimension $ g-1 $. Now the translation of the classical Riemann--Roch
theorem is that the mapping $ \bar{\partial}\colon \operatorname{Ker}{\cal J}_{H^{1}} \to H^{s-1}\left(S,\bar{\omega}\right) $ is of index 0. (Note
that the choice $ s=1 $ insures that there should be no gluing conditions for
elements of $ H^{s-1}\left(M,\bar{\omega}\right) $.)
Now to solve the equation $ \bar{\partial}f=h $, $ {\cal J}_{H^{1}}f=0 $ we reduce it to a boundary
value problem: we use the fact that $ \bar{\partial} $ is a surjection on $ {\Bbb C}P^{1} $, thus one
can immediately find a function $ f_{0} $ such that $ \bar{\partial}f_{0}=h $. Thus $ f_{1}=f-f_{0} $ should
satisfy $ \bar{\partial}f_{1}=0 $, $ {\cal J}_{H^{1}}f_{1}=-{\cal J}_{H^{1}}f_{0} $. This shows that the index of the mapping $ {\cal J} $
is 0 in the algebraic case $ g<\infty $. We see that Theorem~\ref{th35.45} is indeed an
infinite-dimensional analogue of Riemann--Roch theorem for the bundle $ {\cal O} $.
\end{remark}
\begin{remark} It is possible to modify Theorem~\ref{th35.45} to make it applicable
to deformations of the bundle $ {\cal O} $ as well (see Section~\ref{s5.30} for details).
Since to describe Jacobian we need the special case $ {\cal O} $ only, we do not
pursue this venue here. \end{remark}
\begin{remark} For the particular case of the bundle $ {\cal O} $ it is possible to prove
a much stronger result than Riemann--Roch theorem: that $ {\cal J} $ is an
isomorphism. However, since this result is not true out of context of toy
theory, we postpone its proof until Section~\ref{s7.90}, when it is needed for
description of Jacobian. \end{remark}
\section{$ H^{1/2} $-theory }
As we have seen in Section~\ref{h35}, Riemann--Roch theorem can be
easily proven given appropriate ingredients, such as {\em almost
perpendicularity\/} of functions with far-separated supports, ability to
{\em restrict\/} a global section to the boundary, ability to {\em invert\/} $ \bar{\partial} $ basing on
boundary values, and an ability to glue a function provided we know the
values inside and outside the given curve with appropriate compatibility
conditions along the curve. However, when one tries to apply the same
technique to the topology we are most interested in, i.e.,
$ H^{1/2} $-topology, significant difficulties arise.
\subsection{In an ideal world } In what follows we use the (complex) case $ n=1 $ of
Section~\ref{s2.50}. We have seen that the space $ H^{1/2}\left(M,\omega^{1/2}\right) $ should have a
special significance in studying the holomorphic half-forms. Moreover,
the results of Section~\ref{s2.70} suggest that one would be able to describe
holomorphic elements of $ H^{1/2}\left(D,\omega^{1/2}\right) $, $ D\subset M $, by their restrictions on $ \partial D $.
If our world were the ideal world, then the following properties would be
satisfied:
\begin{enumerate}
\item
The action of $ \operatorname{SL}\left(2,{\Bbb C}\right) $ on $ H^{1/2}\left({\Bbb C}P^{1},\omega^{1/2}\right) $ would be an action by
uniformly bounded operators;
\item
For an embedded curve $ \gamma \to {\Bbb C}P^{1} $ the restriction mappings
$ r\colon H^{s}\left({\Bbb C}P^{1},\omega^{1/2}\right) \to H^{s-1/2}\left(\gamma,\Omega^{1/2}\otimes\mu\right) $ (defined for $ s>\frac{1}{2} $) would be defined
for $ s=\frac{1}{2} $ as well;
\item
For the same curve the mapping $ e\colon H^{s}\left(\gamma,\Omega^{1/2}\otimes\mu\right) \to H^{s-1/2}\left({\Bbb C}P^{1},\omega^{1/2}\otimes\bar{\omega}\right) $ of
continuation by $ \delta $-function (defined for $ s<0 $) would be defined for $ s=0 $ as
well.
\item
If a curve $ \gamma $ divides a domain $ S $ into two parts $ S_{1} $, $ S_{2} $, then the
natural restriction mapping $ H^{1/2}\left(S\right) \to H^{1/2}\left(S_{1}\right)\oplus H^{1/2}\left(S_{2}\right) $ would be an
isomorphism.
\end{enumerate}
To make a long story short, in the ideal world the main technical
tools of this paper would behave in a civilized manner, which would
simplify the exposition a lot.
Identification of the Hardy space with the subspace of holomorphic
functions would be provided by the restriction $ r $. The composition
\begin{equation}
H^{1/2}\left({\Bbb C}P^{1},\omega^{1/2}\right) \xrightarrow[]{r} H^{0}\left(\gamma,\Omega^{1/2}\otimes\mu\right) \xrightarrow[]{e} H^{-1/2}\left({\Bbb C}P^{1},\omega^{1/2}\otimes\bar{\omega}\right)
\notag\end{equation}
would be a ``model'' of $ \bar{\partial} $-operator\footnote{In the sense of Section~\ref{s3.05}, i.e., it would coincide with $ -\bar{\partial} $ on
piecewise holomorphic functions.}
\begin{equation}
H^{1/2}\left({\Bbb C}P^{1},\omega^{1/2}\right) \xrightarrow[]{\bar{\partial}} H^{-1/2}\left({\Bbb C}P^{1},\omega^{1/2}\otimes\bar{\omega}\right),
\notag\end{equation}
so that the composition (Cauchy formula)
\begin{equation}
H^{1/2}\left({\Bbb C}P^{1},\omega^{1/2}\right) \xrightarrow[]{r} H^{0}\left(\gamma,\Omega^{1/2}\otimes\mu\right) \xrightarrow[]{e} H^{-1/2}\left({\Bbb C}P^{1},\omega^{1/2}\otimes\bar{\omega}\right)\xrightarrow[]{\bar{\partial}^{-1}} H^{1/2}\left({\Bbb C}P^{1},\omega^{1/2}\right)
\notag\end{equation}
would be continuous. By construction the functions in the image of this
operator are holomorphic outside of $ \gamma $, and the operator $ r $ gives
an injection
of this subspace into $ H^{0}\left(\gamma,\Omega^{1/2}\otimes\mu\right) $. Now one may consider
this as a Cauchy formula, since the composition is an identity operator
on the subspace of holomorphic functions in the domain $ D $ bounded by $ \gamma $.
All these operators would be canonically defined by $ D $, and
compatible with projective mappings $ {\Bbb C}P^{1} \to {\Bbb C}P^{1} $, $ D \to D_{1} $. Thus one would
be able to consider the image of $ r $ on subspace of holomorphic forms
in $ H^{1/2}\left(D,\omega^{1/2}\right) $ as a ``model'' of this space.
The last property in the list would allow us to glue together the
forms which are provided by different means on pieces $ S_{1} $ and $ S_{2} $.
Since we are confined to the current world, the above program will
not work, thus we need some workarounds against above three non-facts. We
will consider a Riemannian structure on $ {\Bbb C}P^{1} $, and will need to control the
size of domains in question, in order for results of Section~\ref{s2.50} to be
applicable. We will also need mollifications of operators $ r $ and $ e $, and
will need some restrictions on what we can glue together.
\subsection{Mollification }\label{s2.10}\myLabel{s2.10}\relax Here we introduce a mapping
\begin{equation}
H^{0}\left(S^{1},\Omega^{1/2}\otimes\mu\right) \xrightarrow[]{\widetilde{e}} H^{-1/2}\left(S^{1}\times\left(-\varepsilon,\varepsilon\right),\omega^{1/2}\right)
\notag\end{equation}
which a closest existing analogue for the (non-existing) mapping $ e $ from
the previous section. It will be an invertible mapping onto its image,
and it will depend on additional parameters $ a_{n} $, $ n\in{\Bbb Z} $. (Later we will need
some particular choice of parameters $ \left(a_{n}\right) $, appropriate for the elliptic
operator $ \bar{\partial} $ we study.)
Fix $ \varepsilon>0 $ and a sequence $ \left(a_{n}\right)_{n\in{\Bbb Z}} $ such that $ 0<a<a_{n}<A $ for fixed
constants $ a $ and $ A $. Fix a smooth function $ \sigma\left(y\right) $, $ y\in\left(-\varepsilon,\varepsilon\right) $, with a
compact support and integral 1. Now map a half-form $ e^{2\pi i kx}dx^{1/2} $,
$ x\in S^{1}={\Bbb R}/{\Bbb Z} $, $ k\in{\Bbb Z}+\frac{1}{2} $, into
\begin{equation}
a_{k}e^{2\pi i kx}\frac{\sigma\left(|k|y\right)}{|k|}dz^{1/2},\qquad x\in S^{1}\text{, }y\in\left(-\varepsilon,\varepsilon\right),\quad z=x+iy,
\notag\end{equation}
and continue this mapping linearly to $ L_{2}\left(S^{1},\Omega^{1/2}\otimes\mu\right)=H^{0}\left(S^{1},\Omega^{1/2}\otimes\mu\right) $.
\begin{lemma} This mapping is an injection, i.e., an invertible mapping onto its
image. \end{lemma}
The dual mapping $ H^{1/2}\left(S^{1}\times\left(-\varepsilon,\varepsilon\right),\omega^{1/2}\right) \xrightarrow[]{\widetilde{r}} H^{0}\left(S^{1},\Omega^{1/2}\otimes\mu\right) $ given by
\begin{align} f\left(x,y\right)dz^{1/2} \mapsto g\left(x\right) & = \sum_{k}g_{k}e^{2\pi kx}dx^{1/2},
\notag\\
g_{k} & = \int f\left(x,y\right)e^{-2\pi kx}\frac{\sigma\left(|k|y\right)}{|k|}dx\,dy
\notag\end{align}
has a similar property: it is a surjection, i.e., an invertible mapping
from the quotient by its null-space. It is a close analogue of the
(non-existing) mapping $ r $ from the previous section.
Suppose $ a_{k}=1 $ for any $ k $. If $ \varepsilon \to $ 0, then in weak topology for
mappings $ {\cal D} \to {\cal D}' $ the constructed mappings converge to the mappings of
extension as $ \delta $-function and restriction, but on Sobolev spaces the norms
of these mappings go to $ \infty $,
\begin{remark} In what follows we will use this mapping with following
modifications: we suppose that $ \operatorname{Supp}\sigma\subset\left[0,\varepsilon\right] $, thus we map a half-form on $ S^{1} $
to a half-form concentrated on a small collar to the {\em right\/} of $ S^{1} $ in
$ S^{1}\times\left[-\varepsilon,\varepsilon\right] $. In fact $ S^{1}\times\left[-\varepsilon,\varepsilon\right] $ will be identified with a
annulus
$ \left\{R\,e^{-2\pi\varepsilon}<|z-z_{0}|<R\,e^{2\pi\varepsilon}\right\} $ in $ {\Bbb C} $, and the mapping would send half-forms on
$ \left\{|z|=R\right\} $ to half-forms concentrated on the {\em outside\/} collar.
Similarly, the dual mapping
\begin{equation}
H^{1/2}\left(R\,e^{-2\pi\varepsilon}<|z-z_{0}|<R\,e^{2\pi\varepsilon}\right) \xrightarrow[]{\widetilde{r}} H^{0}\left(\left\{|z|=R\right\}\right)
\notag\end{equation}
will depend only on value of $ f\left(z\right) $ on the outside collar. \end{remark}
\subsection{Mollification suitable for $ \bar{\partial} $ }\label{s5.31}\myLabel{s5.31}\relax Here we are going to fix the
values for the coefficients $ \left(a_{n}\right) $ from Section~\ref{s2.10} which are most
suitable for the operator $ \bar{\partial} $.
\begin{proposition} \label{prop4.15}\myLabel{prop4.15}\relax Fix $ \varepsilon>0 $. Let $ \gamma= \left\{z \mid |z|=1\right\} $, $ \widetilde{K} = \left\{z \mid |z|<e^{2\pi\varepsilon}\right\} $. For
appropriate $ A $ and $ a $ there exists a sequence $ \left(a_{n}\right) $, $ 0<a<a_{n}<A $, such that
the corresponding restriction mapping $ \widetilde{r}\colon H^{1/2}\left(\widetilde{K},\omega^{1/2}\right) \to L_{2}\left(\gamma,\Omega^{1/2}\otimes\mu\right) $
coincides with the usual restriction on half-forms which are holomorphic
between $ \gamma $ and $ \widetilde{K} $. The numbers $ a_{n} $, thus the operator $ \widetilde{r} $, are uniquely
determined.
\end{proposition}
Dually,
\begin{proposition} \label{prop4.16}\myLabel{prop4.16}\relax Let $ U=K\cup\left({\Bbb C}P^{1}\smallsetminus\widetilde{K}\right) $. Consider the Cauchy kernel
restricted to $ \widetilde{K} $. It gives two operators: an operator $ \overset{\,\,{}_\circ} H^{-1/2}\left(\widetilde{K}\right) \xrightarrow[]{\bar{\partial}^{-1}}
H^{1/2}\left(U\right) $, and an operator $ C^{\infty}\left(\gamma\right) \xrightarrow[]{\bar{\partial}^{-1}} {\cal D}'\left(U\right) $. Consider the mollification $ \widetilde{e} $
of the extension mapping $ e $ corresponding to a sequence $ \left(a_{n}\right) $. With
appropriate choice of the sequence $ \left(a_{n}\right) $ the composition
\begin{equation}
C^{\infty}\left(\gamma\right)\hookrightarrow L_{2}\left(\gamma\right) \xrightarrow[]{\widetilde{e}} \overset{\,\,{}_\circ} H^{-1/2}\left(\widetilde{K}\right) \xrightarrow[]{\bar{\partial}^{-1}} H^{1/2}\left(U\right) \hookrightarrow {\cal D}'\left(U\right)
\notag\end{equation}
coincides with the mapping $ C^{\infty}\left(\gamma\right) \xrightarrow[]{\bar{\partial}^{-1}} {\cal D}'\left(U\right) $. The numbers $ a_{n} $, thus the
operator $ \widetilde{e} $, are uniquely determined by the above condition.
\end{proposition}
\begin{remark} Note that the mapping $ \widetilde{r} $ is a left inverse to the mapping $ f \mapsto
c_{f} $ from Section~\ref{s2.70}. \end{remark}
\subsection{$ \protect \widetilde{e}\circ\protect \widetilde{r} $ as an approximation to $ \bar{\partial} $ }\label{s2.25}\myLabel{s2.25}\relax Consider a circle $ \gamma $ in $ {\Bbb C} $. In what
follows we consider particular mollifications of the $ \delta $-inclusion $ C^{\infty}\left(\gamma\right)
\xrightarrow[]{e} {\cal D}'\left({\Bbb C}\right) $ and restriction $ {\cal D}\left({\Bbb C}\right) \xrightarrow[]{r} C^{\infty}\left(S^{1}\right) $, which correspond to the only
sequences $ \left(a_{n}\right) $ which satisfy the conditions of Propositions~\ref{prop4.15},
~\ref{prop4.16}. The notations $ \widetilde{e}_{\gamma} $ and $ \widetilde{r}_{\gamma} $ are reserved for these two mappings,
we may denote them $ \widetilde{e} $ and $ \widetilde{r} $ if the circle to apply them for is clear from
context.
A central tool in the following discussion is the mollification of
identity~\eqref{equ3.02}:
\begin{proposition} Consider a circle $ \gamma $ or radius $ \rho $ which bounds a disk $ K $.
Let $ \widetilde{K} $ be a concentric disk of radius $ \rho e^{2\pi\varepsilon} $. Let $ F $ be a holomorphic
half-form in $ \widetilde{K}\smallsetminus K $. Let
\begin{equation}
G=\bar{\partial}^{-1}\circ\widetilde{e}_{\gamma}\circ\widetilde{r}_{\gamma}\left(F\right).
\notag\end{equation}
If $ F $ is a holomorphic half-form in $ {\Bbb C}P^{1}\smallsetminus K $, then $ G $ coincides with $ F $ outside
of $ \widetilde{K} $, and is 0 in $ U $. If $ F $ is a holomorphic half-form in $ \widetilde{K} $, then $ G $
coincides with $ -F $ inside of $ K $, and is 0 outside of $ \widetilde{K} $. \end{proposition}
\begin{amplification} \label{amp4.21}\myLabel{amp4.21}\relax Fix $ \varepsilon,R>0 $. Consider disks $ K_{\rho}=\left\{|z|<\rho\right\} $,
$ K'_{\rho}=\left\{|z|<\rho e^{2\pi\varepsilon}\right\} $, $ \widetilde{K}_{\rho}=\left\{|z|<\rho e^{4\pi\varepsilon}\right\} $ in $ {\Bbb C} $. Let $ \gamma=\partial K_{\rho} $. There exists a
mapping
\begin{equation}
\lambda_{\rho}\colon H^{1/2}\left(\widetilde{K}_{\rho},\omega^{1/2}\right) \to H^{1/2}\left(\widetilde{K}_{\rho},\omega^{1/2}\right)
\notag\end{equation}
such that
\begin{enumerate}
\item
For $ 0<\rho<R $ the mapping $ \lambda_{\rho} $ is continuous with the norm uniformly
bounded by a constant depending on $ \varepsilon $ and $ R $ only;
\item
$ \lambda_{\rho}\left(f\right)|_{K_{\rho}}=0 $ for any $ f\in H^{1/2}\left(\widetilde{K}_{\rho}\right) $;
\item
$ \left(f-\lambda_{\rho}\left(f\right)\right)|_{\widetilde{K}_{\rho}\smallsetminus K'_{\rho}}=0 $ for any $ f\in H^{1/2}\left(\widetilde{K}_{\rho}\right) $;
\item
if $ f $ is holomorphic in $ \widetilde{K}_{\rho}\smallsetminus K_{\rho} $, then $ \bar{\partial}\lambda_{\rho}\left(f\right) = \left(\widetilde{e}_{\gamma}\circ\widetilde{r}_{\gamma}\right)\left(f\right) $.
\end{enumerate}
\end{amplification}
\begin{proof} During the proof we abuse notations and do not mention the
bundle $ \omega^{1/2} $ in notations for Sobolev spaces.
Note that for a half-form $ f $ which is holomorphic outside of $ K $ all
the statements of the amplification are true if we take
$ \lambda_{\rho}\left(f\right)=\left(\bar{\partial}^{-1}\circ\widetilde{e}\circ\widetilde{r}\right)\left(f\right) $. If $ f $ is holomorphic inside of $ \widetilde{K} $, then
$ \lambda_{\rho}\left(f\right)=f-\left(\bar{\partial}^{-1}\circ\widetilde{e}\circ\widetilde{r}\right)\left(f\right) $ works. This uniquely defines $ \lambda_{\rho}\left(f\right) $ if $ f $ is
holomorphic inside of $ \widetilde{K}_{\rho}\smallsetminus K_{\rho} $. What we need is to adjust this formula to
the case of non-holomorphic half-forms.
Since $ R $ is fixed, we can consider the norm from Section~\ref{s2.50}
instead of the equivalent Sobolev $ H^{1/2} $-norm. Since for the former norm
the Sobolev spaces $ H^{1/2}\left(\widetilde{K}_{\rho}\right) $ with different $ \rho $ are naturally isomorphic, it
is enough to consider $ \rho=1 $.
Note that for a holomorphic $ f $ the image $ \lambda_{\rho}\left(f\right) $ depends on $ f|_{\widetilde{K}_{\rho}\smallsetminus K_{\rho}} $
only. We are going to define $ \lambda_{\rho} $ in general case such that it satisfies
the same condition. Thus $ \lambda_{\rho} $ is a mapping $ H^{1/2}\left(\widetilde{K}_{\rho}\smallsetminus K_{\rho}\right) \to H^{1/2}\left(\widetilde{K}_{\rho}\right) $ such
that a half-form in the image is 0 inside $ K_{\rho} $. We may substitute
conformally equivalent domain $ S^{1}\times\left(-2\varepsilon,2\varepsilon\right) $ instead of $ \widetilde{K}_{\rho}\smallsetminus K_{\rho} $.
Now $ H^{1/2}\left(S^{1}\times\left(-2\varepsilon,2\varepsilon\right)\right) $ is an orthogonal sum of subspaces $ L_{n} $,
$ n\in{\Bbb Z}+\frac{1}{2} $, spanned by half-forms of the form $ \varphi\left(y\right)e^{2\pi i\quad nx}dz^{1/2} $,
$ \left(x,y\right)\in S^{1}\times\left(-2\varepsilon,2\varepsilon\right) $, $ z=x+iy $, thus it is enough to construct uniformly
bounded mappings in these subspaces. For a given $ \lambda $ let $ \lambda^{\left(n\right)} $ be defined as
\begin{equation}
\lambda\left(\varphi\left(y\right)e^{2\pi i nx}dz^{1/2}\right)=\lambda^{\left(n\right)}\left(\varphi\left(y\right)\right)e^{2\pi i nx}dz^{1/2}.
\notag\end{equation}
Note that the conditions on $ \lambda^{\left(n\right)} $ are: $ \lambda^{\left(n\right)}\left(\varphi\left(y\right)\right) $ vanishes if $ y<0 $,
$ \lambda^{\left(n\right)}\left(\varphi\left(y\right)\right)=\varphi\left(y\right) $ if $ y>\varepsilon $, and $ \lambda^{\left(n\right)}\left(e^{ny}\right) $ has a prescribed value
(obtained basing on the last condition of the amplification).
Define $ \widetilde{\sigma}_{n}\left(y\right) $ via
\begin{equation}
\lambda^{\left(n\right)}\left(e^{ny}\right) = \widetilde{\sigma}_{n}\left(y\right)e^{ny}.
\notag\end{equation}
Obviously, $ \widetilde{\sigma}_{n}\left(y\right)=0 $ if $ y<0 $, $ \widetilde{\sigma}_{n}\left(y\right)=1 $ if $ y>\varepsilon $. The function $ \widetilde{\sigma}_{n}\left(y\right) $ is
uniquely determined by the last condition of the theorem.
After all these remarks {\em define\/} $ \lambda|_{L_{n}} $ by
\begin{equation}
\varphi\left(y\right)e^{2\pi i nx}dz^{1/2} \to \sigma_{n}\left(y\right)\varphi\left(y\right)e^{2\pi i nx}dz^{1/2}.
\notag\end{equation}
This mapping satisfies all the conditions of the amplification, with a
possible exception of uniform boundness of these mappings for different
$ n $. To prove the boundness, note that
\begin{equation}
\widetilde{\sigma}_{n}\left(y\right)=\widetilde{\sigma}_{1}\left(ny\right),
\notag\end{equation}
and the mappings
\begin{equation}
L_{1} \xrightarrow[]{m_{n}} L_{n}\colon \varphi\left(y\right)e^{2\pi i x} \mapsto n^{-1/2}\varphi\left(ny\right)e^{2\pi i nx}
\notag\end{equation}
are uniformly bounded together with their inverse mappings. \end{proof}
\subsection{Mollification for $ L_{2}\left(\Omega^{1/2}\otimes\mu\right) $ }\label{s4.35}\myLabel{s4.35}\relax Consider a family of circles $ \gamma_{i} $ on
$ {\Bbb C}P^{1} $.
Fix a projective isomorphism of $ \gamma_{i} $ with $ \left\{|z|=1\right\} $ for every $ i $. This
isomorphism identifies an annulus $ \left\{e^{-2\pi\varepsilon}<|z|<e^{2\pi\varepsilon}\right\} $ with some neighborhood $ U_{i} $
of $ \gamma_{i} $.
The $ \bar{\partial} $-adjusted mollification of $ e $ gives a mapping
\begin{equation}
\widetilde{e}_{i}\colon L_{2}\left(\gamma_{i},\Omega^{1/2}\otimes\mu\right) \to H^{-1/2}\left({\Bbb C}P^{1},\omega^{1/2}\otimes\bar{\omega}\right).
\notag\end{equation}
This mapping is a injection of topological vector spaces. In the
following sections we discuss under which conditions $ \sum_{i}\widetilde{e}_{i} $ is an
injection. Here $ \sum_{i}\widetilde{e}_{i} $ is considered as a mapping from $ L_{2}\left(\coprod\gamma_{i},\Omega^{1/2}\otimes\mu\right) $ to
$ H^{-1/2}\left({\Bbb C}P^{1},\omega^{1/2}\otimes\bar{\omega}\right) $.
This injection is going to be a central tool in Section~\ref{s5.10},
which allows one to identify the space of ``admissible'' holomorphic
half-forms with the space of their boundary values. To provide the
criterion for this mapping to be an injection, we need to introduce a
notion of ``almost perpendicular'' subspaces, to study relative position of
Sobolev subspaces corresponding to different domains, and to investigate
the relative position of domains inside $ {\Bbb C}P^{1} $.
First, we suppose that $ \varepsilon $ is small enough and the subsets $ U_{i} $ do not
intersect. Then it is clear that $ \sum_{i}\widetilde{e}_{i} $ has no null-vectors. Obviously, $ \sum_{i}\widetilde{e}_{i} $
were an injection if the images of $ \widetilde{e}_{i} $ were perpendicular for
different $ i $.
We are going to investigate when these images are {\em almost perpendicular\/}
for big $ i $. In other words, when the sum of images is direct in the sense
of Hilbert topology (i.e., it is an orthogonal sum after an appropriate
change of the Hilbert norm to an equivalent one).
\subsection{Almost perpendicular subspaces }\label{s5.61}\myLabel{s5.61}\relax
\begin{lemma} \label{lm4.41}\myLabel{lm4.41}\relax Consider a Hilbert space $ H $ and a family of subspaces $ H_{i} $, $ i\in{\Bbb N} $.
Denote by $ a_{ij} $ the orthogonal projector $ H_{i} \to H_{j} $. Let
$ A=\left(a_{ij}\right) $ be
the matrix of an operator $ {\bold A}\colon \bigoplus H_{i} \to \coprod H_{i} $, $ \coprod H_{i} $ being the space of
arbitrary sequences $ \left(h_{i}\right) $, $ h_{i}\in H_{i} $. Then the natural mapping $ \bigoplus H_{i} \to H $
extends to a Fredholm mapping
\begin{equation}
\bigoplus_{l_{2}}H_{i} \xrightarrow[]{i} \operatorname{Im} i\subset H,\qquad \left(h_{i}\right) \mapsto \sum h_{i},
\notag\end{equation}
onto its image iff {\bf A }induces a Fredholm mapping $ \bigoplus_{l_{2}}H_{i} \to \bigoplus_{l_{2}}H_{i} $. \end{lemma}
\begin{proof} Consider $ {\bold A} $ as $ i^{*}\circ i $. \end{proof}
\begin{corollary} Suppose that the conditions of the previous lemma the matrix $ \alpha =
\left(\|a_{ij}\| - \delta_{ij}\right) $ corresponds to a compact mapping $ l_{2} \to l_{2} $. Then the mapping
$ \oplus_{l_{2}}H_{i} \xrightarrow[]{i} H $ is a Fredholm operator onto its image. In particular, this is
true if
\begin{equation}
\sum_{i\not=j}\|a_{ij}\|^{2}<\infty.
\notag\end{equation}
Moreover, if there is a number $ C $ and a family of unit vectors $ v_{i}\in V_{i} $ such
that
\begin{equation}
\|a_{ij}\| < C|\left(v_{i},v_{j}\right)|,
\notag\end{equation}
then the mapping $ i $ is continuous iff $ \alpha $ provides a bounded mapping $ l_{2} \to
l_{2} $.
\end{corollary}
\subsection{Conformal distance } In the applications we consider the curve $ \gamma $ is a
union of countably many connected components $ \gamma=\coprod\gamma_{i} $, and all the
components $ \gamma_{i} $ but a finite number are circles. Let $ \widetilde{e}_{i}=\widetilde{e}_{\gamma_{i}} $ be the mollified
extension mapping, and
\begin{equation}
H_{i}=\operatorname{Im} \widetilde{e}_{i}.
\notag\end{equation}
If $ H_{i} $ satisfy the conditions of Lemma~\ref{lm4.41}, then the mapping $ \sum_{i}\widetilde{e}_{i} $ from
Section~\ref{s4.35} is a Fredholm mapping onto its image. So
the next thing to study is how to calculate the norm of the projection
\begin{equation}
\operatorname{Im} \widetilde{e}_{i} \to \operatorname{Im} \widetilde{e}_{j}.
\label{equ4.33}\end{equation}\myLabel{equ4.33,}\relax
The subspace $ \operatorname{Im} \widetilde{e}_{i} $ lies inside a subspace $ \overset{\,\,{}_\circ}{H}\left(\widetilde{K}_{i}\right) $, $ \widetilde{K}_{i} $ being
any disk which contains an appropriate neighborhood of $ \gamma_{i} $. Thus to
majorate the norm of the projection~\eqref{equ4.33} we start with
the case of two disjoint disks $ \widetilde{K}_{i} $, $ \widetilde{K}_{j} $ on $ {\Bbb C}P^{1} $, and consider the
orthogonal projection from $ \overset{\,\,{}_\circ}{H}^{-1/2}\left(\widetilde{K}_{i}\right) $ to $ \overset{\,\,{}_\circ}{H}^{-1/2}\left(\widetilde{K}_{j}\right) $ inside $ H^{-1/2}\left({\Bbb C}P^{1}\right) $. As
we will see, the norm of this projection can be majorated by a number
which depends only on some kind of {\em distance\/} between $ \widetilde{K}_{i} $ and $ \widetilde{K}_{j} $.
Two disjoint simple curves $ \gamma_{1} $, $ \gamma_{2} $ on $ {\Bbb C}P^{1} $ bound a tube $ U $,
which is
conformally equivalent to exactly one tube $ S^{1}\times\left[0,l\right] $, $ l>0 $. Here $ S^{1} $ is
a circle of circumference $ 2\pi $.
\begin{definition} We call the number $ l $ the {\em conformal distance\/} between $ \gamma_{1} $ and
$ \gamma_{2} $. \end{definition}
\begin{proposition} Consider three simple curves $ \gamma_{1} $, $ \gamma_{2} $, $ \gamma_{3} $ such that $ \gamma_{2} $
separates $ \gamma_{1} $ and $ \gamma_{3} $. Then the conformal distance $ l\left(\gamma_{1},\gamma_{3}\right) \geq
l\left(\gamma_{1},\gamma_{2}\right) +l\left(\gamma_{2},\gamma_{3}\right) $. \end{proposition}
\begin{proof} Conformal distance between $ \gamma $ and $ \gamma' $ is $ \geq l $ iff there exists a
function $ \varphi $ defined between $ \gamma $ and $ \gamma' $ such that $ \varphi|_{\gamma}=0 $, $ \varphi|_{\gamma'}=l $, and
the ``energy'' $ \int\partial\varphi\bar{\partial}\varphi \leq l $. Combining two such functions, one defined between
$ \gamma_{1} $ and $ \gamma_{2} $, another between $ \gamma_{2} $ and $ \gamma_{3} $, we obtain the statement. \end{proof}
\begin{lemma} The only $ \operatorname{PGL}_{2}\left({\Bbb C}\right) $-invariant of a couple of disjoint disks on $ {\Bbb C}P^{1} $
is the conformal distance. \end{lemma}
\subsection{Subspaces of $ \Omega^{3/4} $ }\label{s3.8}\myLabel{s3.8}\relax Consider two disjoint disks $ K_{1} $, $ K_{2} $ on $ {\Bbb C}P^{1} $ of
conformal
distance $ l $. Since the natural norm on $ H^{-1/2}\left(\Omega^{3/4}\right) $ (see Section~\ref{s2.60}) is
invariant with respect
to $ \operatorname{PGL} $, the angle between the subspaces $ \overset{\,\,{}_\circ}{H}^{-1/2}\left(K_{1},\Omega^{3/4}\right) $ and
$ \overset{\,\,{}_\circ}{H}^{-1/2}\left(K_{2},\Omega^{3/4}\right) $ depends on $ l $ only. Let $ P_{l} $ be the orthogonal projector from
one subspace to another.
\begin{proposition} \label{prop3.170}\myLabel{prop3.170}\relax Fix $ \varepsilon>0 $. If $ l\geq\varepsilon $, then
\begin{equation}
\|P_{l}\| \sim e^{-l/2}.
\notag\end{equation}
The equivalence means that the quotient of two sides remains bounded and
separated from 0. \end{proposition}
\begin{proof} The norm $ \|P_{l}\| $ is a smooth function of $ l $, thus one needs to
prove only the asymptotic when $ l \to \infty $. One may represent two disks of
conformal distance $ \approx l\gg 1 $ as $ |z|\leq1 $ and $ |z-e^{l/2}|\leq1 $. Let $ L=e^{l/2} $. Consider the
coordinate $ z_{1}=z $ in the first disk, and $ z_{2}=z-L $ in the second one. What we
need to prove is that the kernel $ K_{1}\left(z_{1},z_{2}\right)=\frac{1}{|z_{1}-z_{2}+L|} $ in $ \left\{\left(z_{1},z_{2}\right) |
|z_{1}|,|z_{2}|\leq1\right\} $ gives an operator of the norm $ \sim1/L $.
Since the radii are fixed now, we can consider functions instead of
$ 3/4 $-forms. Since the norm of operator with the kernel $ K_{0}=\frac{1}{L} $ is
$ O\left(\frac{1}{L}\right) $, it is enough to prove that the kernel $ K_{2}\left(z_{1},z_{2}\right)=\frac{1}{|z_{1}-z_{2}+L|} -
\frac{1}{L} $ corresponds to an operator of norm $ o\left(1/L\right) $. Let us estimate
Hilbert--Schmidt norm of this operator. It is equal to the
$ H^{1/2}\otimes_{l_{2}}H^{1/2} $-norm of $ K_{2} $. On the other hand, the last norm is bounded by
$ H^{1} $-norm, which is obviously $ O\left(L^{-2}\right) $.\end{proof}
\begin{remark} \label{rem3.75}\myLabel{rem3.75}\relax Note that if $ 0<s<1 $, then a similar statement is true in
$ H^{1-2s}\left(\Omega^{s}\right) $. We will need only the following statement:
\begin{equation}
\|P_{l}\| = O\left(e^{-l/2}\right)\text{ if }0<s<\frac{3}{4},
\notag\end{equation}
and we will use it in the case $ s=\frac{1}{4} $ only. \end{remark}
\begin{corollary} \label{cor3.80}\myLabel{cor3.80}\relax Consider a family of disjoint closed disks $ K_{i}\subset{\Bbb C}P^{1} $,
$ i\in{\Bbb N} $, with conformal distance between $ K_{i} $ and $ K_{j} $ being $ l_{ij} $. Put $ l_{i i}=0 $. Let
$ {\cal L} $ be a linear bundle on $ {\Bbb C}P^{1} $. Suppose that the closure
of $ \bigcup K_{i} $ does not coincide with $ {\Bbb C}P^{1} $. Suppose that $ \inf _{i\not=j}l_{ij}>0 $, and that the
matrix $ \left(e^{-l_{ij}/2}\right) $ corresponds to a bounded operator $ l_{2} \to l_{2} $. Then the
inclusions $ \overset{\,\,{}_\circ}{H}^{-1/2}\left(K_{i},{\cal L}\right)\hookrightarrow H^{-1/2}\left({\Bbb C}P^{1},{\cal L}\right) $ extend to a continuous injection
\begin{equation}
\bigoplus_{l_{2}}\overset{\,\,{}_\circ}{H}^{-1/2}\left(K_{i},{\cal L}\right)\hookrightarrow H^{-1/2}\left({\Bbb C}P^{1},{\cal L}\right).
\notag\end{equation}
Dually, the restrictions $ H^{1/2}\left({\Bbb C}P^{1},{\cal L}\right) \to H^{1/2}\left(K_{i},{\cal L}\right) $ extend to a continuous
surjection
\begin{equation}
H^{1/2}\left({\Bbb C}P^{1},{\cal L}\right) \to \bigoplus_{l_{2}}H^{1/2}\left(K_{i},{\cal L}\right).
\notag\end{equation}
\end{corollary}
\begin{proof} If $ {\cal L}=\Omega^{3/4} $ in the first part of the theorem, or $ {\cal L}=\Omega^{1/4} $ in the
second one, then everything is proved. Otherwise let $ U $ be an open subset
of $ {\Bbb C}P^{1}\smallsetminus\overline{\bigcup K_{i}} $. An isomorphism of $ {\cal L} $ and $ \Omega^{3/4} $ (or $ \Omega^{1/4} $) on $ {\Bbb C}P^{1}\smallsetminus U $ proves
the rest. \end{proof}
Using the results of Remark~\ref{rem3.75} we obtain the following
statement:
\begin{proposition} In the conditions of Corollary~\ref{cor3.80}
\begin{equation}
\bigoplus_{l_{2}}\overset{\,\,{}_\circ}{H}^{1/2}\left(K_{i},{\cal L}\right) \to H^{1/2}\left({\Bbb C}P^{1},{\cal L}\right).
\notag\end{equation}
(components being inclusions) is an isomorphism to its image. The image
of this mapping is $ \overset{\,\,{}_\circ}{H}^{1/2}\left(\bigcup_{i}K_{i},{\cal L}\right) $. \end{proposition}
\section{Generalized Hardy space }\label{h4}\myLabel{h4}\relax
\subsection{Hilbert space of holomorphic half-forms }\label{s5.10}\myLabel{s5.10}\relax
In this section we are going to construct a Hilbert space (or at
least a space with Hilbert topology) which models global holomorphic
half-forms on a curve which is not necessarily compact.
Consider a family of disjoint closed disks $ K_{i}\subset{\Bbb C}P^{1} $, $ i\in I $.
Let $ \varepsilon>0 $, $ \widetilde{K}_{i} $ be the concentric disk to $ K_{i} $ of radius $ e^{2\varepsilon}\cdot\operatorname{radius}\left(K_{i}\right) $.
Suppose that the disks $ \widetilde{K}_{i} $ do not intersect, and denote the conformal distance
between $ \widetilde{K}_{i} $ and $ \widetilde{K}_{j} $ by $ l_{ij} $. Put $ l_{i i}=0 $. Suppose that
\begin{nwthrmii} Say that the collection of disks is {\em well-separated\/} if
\begin{enumerate}
\item
$ \inf _{i\not=j}l_{ij}>0 $;
\item
$ \overline{\bigcup\widetilde{K}_{i}}\not={\Bbb C}P^{1} $; and
\item
the matrix $ \left(e^{-l_{ij}/2}\right) $ gives a bounded operator $ l_{2} \to l_{2} $;
\end{enumerate}
\end{nwthrmii}
Note that the interior of the closure $ \overline{{\Bbb C}P^{1}\smallsetminus\bigcup K_{i}} $ is non-empty by
the above conditions, and
\begin{proposition} \label{prop5.16}\myLabel{prop5.16}\relax If disks $ K_{i} $ are well-separated, then $ \sum_{i}\operatorname{radius}\left(K_{i}\right)<\infty $,
thus Cauchy kernel defines a mapping from $ L_{2}\left(\bigcup\partial K_{i},\Omega^{1/2}\otimes\mu\right) $ to the space
of analytic functions in $ {\Bbb C}P^{1}\smallsetminus\overline{\bigcup K_{i}} $. \end{proposition}
\begin{proof} Indeed, since rows of $ \left(e^{-l_{ij}/2}\right) $ are in $ l_{2} $, $ \sum_{j} e^{-l_{ij}} <\infty $. On the
other hand, $ e^{-l_{ij}} $ is approximately proportional to radius of $ K_{j} $ when $ i $ is
fixed. \end{proof}
Moreover, if we slightly decrease the value of $ \varepsilon $, then the first two
conditions on the family $ \left\{K_{i}\right\} $ will be automatically satisfied, so only
the last condition is the new one.
\begin{definition} The {\em generalized Hardy space\/} $ {\cal H}\left({\Bbb C}P^{1}, \left\{K_{i}\right\}\right) $ is the space
\begin{equation}
\left\{f\left(z\right)\in H^{1/2}\left({\Bbb C}P^{1}\smallsetminus\bigcup_{i}K_{i}, \omega^{1/2}\right) \mid \bar{\partial}f=0\in H^{-1/2}\left({\Bbb C}P^{1}\smallsetminus\bigcup_{i}K_{i},\omega^{1/2}\otimes\bar{\omega}\right)\right\}.
\notag\end{equation}
\end{definition}
Note that elements of $ {\cal H}\left({\Bbb C}P^{1}, \left\{K_{i}\right\}\right) $ are holomorphic half-forms in
$ {\Bbb C}P^{1}\smallsetminus\overline{\bigcup\widetilde{K}_{i}} $. Note also that the Sobolev space in the definition is a
generalized Sobolev space, since $ {\Bbb C}P^{1}\smallsetminus\bigcup_{i}K_{i} $ is not open if the set $ I $ is
infinite.
\begin{remark} Note that if $ I $ consists of one element, then $ {\cal H} $ is the usual
Hardy space from Section~\ref{s2.70} with modifications outlined in Remark
~\ref{rem2.10}. \end{remark}
In Section~\ref{s4.90} we introduce a slightly weaker condition on
domains $ K_{i} $, and will use it instead of Condition A. Note that we are going
to mention the space $ {\cal H} $ only in cases when the Condition A is satisfied.
\subsection{Hilbert operator }\label{s5.20}\myLabel{s5.20}\relax A section of $ \Omega^{1/2}\otimes\mu $ on a circle can be
decomposed into the Fourier series
\begin{equation}
f\left(t\right)dt^{1/2}=\sum_{n\in{\Bbb Z}+1/2}a_{n}e^{2\pi i nt}dt^{1/2}
\notag\end{equation}
(half-integers appear because of the factor $ \mu $), hence it can be written
as a sum of a component $ f_{+} $ which can be holomorphically extended to a
section of $ \omega^{1/2} $ inside the circle, and a component $ f_{-} $ which may be
extended outside of the circle. Moreover, norms of $ f_{\pm} $ are bounded by the
norm of $ f $, and the decomposition is unique.
We denote this decomposition $ L_{2}^{+}\left(\gamma,\Omega^{1/2}\otimes\mu\right) \oplus L_{2}^{-}\left(\gamma,\Omega^{1/2}\otimes\mu\right) $.
\begin{proposition} Consider the operator $ {\bold K}_{+} $ with Cauchy kernel acting in the
space $ \bigoplus_{l_{2}}L_{2}\left(\partial K_{i},\Omega^{1/2}\otimes\mu\right) $. Let $ {\bold K} $ be the operator $ \widetilde{{\bold K}} $ with diagonal blocks
$ L_{2}\left(\partial K_{i},\Omega^{1/2}\otimes\mu\right) \to L_{2}\left(\partial K_{i},\Omega^{1/2}\otimes\mu\right) $ removed. If the disks $ K_{i} $ are
well-separated, then $ {\bold K} $ is bounded. If we decompose
\begin{equation}
L_{2}\left(\partial K_{i},\Omega^{1/2}\otimes\mu\right) = L_{2}^{+}\left(\partial K_{i},\Omega^{1/2}\otimes\mu\right) \oplus L_{2}^{-}\left(\partial K_{i},\Omega^{1/2}\otimes\mu\right),
\notag\end{equation}
then the only nonzero blocks of $ {\bold K} $ act from $ L_{2}^{-}\left(\partial K_{i},\Omega^{1/2}\otimes\mu\right) $ to
$ L_{2}^{+}\left(\partial K_{j},\Omega^{1/2}\otimes\mu\right) $, $ i\not=j $. \end{proposition}
This statement is an immediate corollary of the first part of
\begin{lemma} \label{lm5.05}\myLabel{lm5.05}\relax Consider a Hilbert space $ H=\bigoplus_{l_{2}}H_{i} $ and a linear operator
$ A\colon H \to H $ with bounded blocks $ A_{ij}\colon H_{j} \to H_{i} $. Let $ a_{ij}=\|A_{ij}\| $. Suppose that
the matrix $ \left(a_{ij}\right) $ gives a bounded operator $ \alpha\colon l_{2} \to l_{2} $. Then
\begin{enumerate}
\item
$ A $ is bounded, $ \|A\|\leq\|\alpha\| $;
\item
$ A $ is compact if $ \alpha $ is compact and each of operators $ A_{ij} $ is compact.
\end{enumerate}
\end{lemma}
\begin{proof} To bound the operator $ A $ it is enough to bound $ |\left(x, Ay\right)| $ for
$ |x|=|y|=1 $, $ x,y\in H $. On the other hand, decomposition of $ H $ gives $ x=\left(x_{i}\right) $,
$ y=\left(y_{i}\right) $, and $ |x|^{2}=\sum|x_{i}|^{2} $, $ |y|^{2}=\sum|y_{i}|^{2} $. This gives
\begin{equation}
|\left(x,Ay\right)| = \left|\sum_{ij}\left(x_{i},A_{ij}y_{j}\right)\right| \leq \sum_{ij}|\left(x_{i},A_{ij}y_{j}\right)| \leq \sum_{ij}\|A_{ij}\||x_{i}||y_{j}|
\notag\end{equation}
and the latter sum is just $ \left(\xi,\alpha\eta\right) $ with $ \xi=\left(|x_{i}|\right)\in l_{2} $, similarly for $ \eta $.
To prove the second part it is enough to show that we can
approximate $ A $ (in norm) by compact operators. Since $ \alpha $ is
compact, we can approximate it (in norm) by a matrix with finite number
of non-zero elements. The relation of norm of $ A $ and norm of $ \alpha $ shows that
$ A $ can be approximated by an operator $ A' $ which has the same blocks as $ A $
in finite number of places, all the rest is 0.
Since blocks of $ A $ are compact, $ A' $ is compact as well. We conclude
that $ A $ can be approximated with arbitrary precision by a compact
operator, thus $ A $ is compact.\end{proof}
\begin{corollary} \label{cor4.35}\myLabel{cor4.35}\relax Any block-row and block-column of $ {\bold K} $ gives a compact
operator. \end{corollary}
\subsection{Boundary map } Here we consider which of the facts from Section~\ref{s2.70}
have sense for the generalized Hardy space as well.
Fix $ i\in I $, let $ S_{i}=\partial K_{i} $. Consider the mollified restriction
mapping
\begin{equation}
H^{1/2}\left(\widetilde{K}_{i}\smallsetminus K_{i},\omega^{1/2}\right) \xrightarrow[]{\widetilde{r}_{i}} L_{2}\left(S_{i},\Omega^{1/2}\otimes\mu\right).
\notag\end{equation}
Since any element $ f $ of the generalized Hardy space is $ H^{1/2} $ in $ \widetilde{K}_{i}\smallsetminus K_{i} $ (and
holomorphic inside this annulus), the restriction $ \widetilde{r}_{i}\left(f\right) =f|_{S_{i}} $ of this
half-form on $ S_{i} $ is an $ L_{2} $-section of $ \Omega^{1/2}\left(S_{i}\right)\otimes\mu $. In the rest of this
section we are going to abuse notations and denote $ \widetilde{r}_{i}\left(f\right) $ as $ f|_{S_{i}} $.
\begin{theorem} \label{th4.40}\myLabel{th4.40}\relax Suppose that the disks $ K_{i} $ are well-separated. Then the
mappings $ \widetilde{r}_{i} $ taken together provide a mapping
\begin{equation}
{\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) \xrightarrow[]{\widetilde{r}} \displaystyle\coprod L_{2}\left(S_{i},\Omega^{1/2}\otimes\mu\right),
\notag\end{equation}
which is fact is a continuous mapping
\begin{equation}
{\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) \xrightarrow[]{\widetilde{r}} \bigoplus_{l_{2}}L_{2}\left(S_{i},\Omega^{1/2}\otimes\mu\right).
\notag\end{equation}
The image of $ \widetilde{r} $ is closed. Moreover, the mapping $ \widetilde{r} $ is invertible
onto its image.
\end{theorem}
\begin{proof} In fact, all the main ingredients for the proof of this theorem
are already here. An element $ f\in{\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $ is by definition a
restriction of some
$ H^{1/2} $-section $ g $ of $ \omega^{1/2} $ on the whole sphere $ {\Bbb C}P^{1} $. The restrictions of $ g $ to
$ \widetilde{K}_{i} $ give an element of $ \bigoplus_{l_{2}}H^{1/2}\left(\widetilde{K}_{i},\omega^{1/2}\right) $ by Corollary~\ref{cor3.80}. On the
other hand, given $ f $, the restriction of $ g $ on $ \widetilde{K}_{i} $ is defined up to a
section with support on $ K_{i} $. Hence the mollified restriction on $ S_{i} $ is
correctly defined, and it has a norm majorated by some multiple of the
norm of the restriction on $ \widetilde{K}_{i} $. Hence the restriction mapping $ \widetilde{r} $ is bounded
indeed.
To show that $ \operatorname{Im} \widetilde{r} $ is closed, let us construct a left inverse $ l $ this
operator. Then $ \widetilde{r}l $ is going to be a projection on $ \operatorname{Im} \widetilde{r} $, which will prove
the closeness.
Consider the mapping $ \lambda_{i} $ from Section~\ref{s2.25}, associated to the
disks $ K_{i} $, $ \widetilde{K}_{i} $. Then $ g - \lambda_{i}\left(g\right) $
\begin{enumerate}
\item
vanishes outside of $ \widetilde{K}_{i} $;
\item
coincides with $ g $ inside of $ K_{i} $;
\item
has a norm bounded by $ C\cdot\|g|_{\widetilde{K}_{i}}\| $;
\item
depends on values of $ g $ outside of $ K $ only.
\end{enumerate}
Combining all this together, we get
\begin{equation}
F = g+\sum_{i}\left(\lambda_{i}\left(g\right)-g\right)
\notag\end{equation}
which is a half-form of the norm bounded by
\begin{equation}
\|g\|+C'\cdot\left(\sum\|g|_{\widetilde{K}_{i}}\|^{2}\right)^{1/2}= O\left(\|g\|\right).
\notag\end{equation}
The half-form $ F $ is equal to $ g $ (thus to $ f $) outside of $ \bigcup\widetilde{K}_{i} $, and is equal to
0 inside all $ K_{i} $. Moreover, since $ f $ is holomorphic, one can calculate $ \bar{\partial}F|_{\widetilde{K}_{i}} $ as
$ \widetilde{e}_{i}\left(f|_{\partial K_{i}}\right) $.
\begin{proposition} Let $ f $ be a half-form from the generalized Hardy space.
There exists a half-form $ F\in H^{1/2}\left({\Bbb C}P^{1}\right) $ such that:
\begin{enumerate}
\item
$ \|F\| < C\cdot\|f\| $;
\item
$ f=F $ outside of $ \overline{\bigcup\widetilde{K}_{i}} $;
\item
$ F=0 $ inside of $ K_{i} $ for any $ i $;
\item
$ \bar{\partial}F = \sum\widetilde{e}_{i}\left(f|_{\partial K_{i}}\right) $.
\end{enumerate}
\end{proposition}
\begin{proof} The only part which needs proof is the last one. First of all,
the sum in the right-hand side obviously converges in $ H^{-1/2} $, since the
disks $ \widetilde{K}_{i} $ are separated far enough, and norms of the restrictions of $ f $
onto $ \partial K_{i} $ form a sequence in $ l_{2} $. The same arguments show that the
right-hand side is contained in the generalized Sobolev subspace
$ \overset{\,\,{}_\circ}{H}^{-1/2}\left(\bigcup\widetilde{K}_{i}\right) $. Moreover, $ \bar{\partial}\left(F-g\right) $ is contained in the same space, and $ \bar{\partial}f $ is
there by the definition of the generalized Hardy space. Since $ g $ coincides
with $ f $ outside of $ \bigcup K_{i} $, $ \bar{\partial}F\in\overset{\,\,{}_\circ}{H}^{-1/2}\left(\bigcup\widetilde{K}_{i}\right) $.
Thus we know that the difference of the right-hand side and
left-hand side is contained in $ \overset{\,\,{}_\circ}{H}^{-1/2}\left(\bigcup\widetilde{K}_{i}\right) $ and is 0 inside any $ \widetilde{K}_{i} $. On the
other hand, since $ \widetilde{K}_{i} $ are well-separated, $ \overset{\,\,{}_\circ}{H}^{-1/2}\left(\bigcup\widetilde{K}_{i}\right) = \bigoplus_{l_{2}}\overset{\,\,{}_\circ}{H}^{-1/2}\left(\widetilde{K}_{i}\right) $,
what finishes the proof.\end{proof}
\begin{remark} It is obvious that $ f \mapsto F $ is a continuous linear mapping
$ {\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) \to H^{1/2}\left({\Bbb C}P^{1},\omega^{1/2}\right) $. Moreover, it is an injection. To show
this, one needs only to prove that $ f|_{\bigcup\widetilde{K}_{i}} $ is determined by $ F $ (and bounded
by the norm of $ F $). Since the disks $ \widetilde{K}_{i} $ are well-separated, it is enough
to show this for one particular disk $ \widetilde{K}_{i} $.
Since $ \widetilde{e}_{i} $ is an injection, $ \bar{\partial}F $ determines $ \widetilde{r}_{i}\left(f\right) $, thus $ \lambda_{i}\left(g\right) $ (by
construction of $ \lambda $), thus $ \left(F-f\right)|_{\widetilde{K}_{i}} $. We obtained
\end{remark}
\begin{corollary} The relation $ f \mapsto F $ is an injection $ {\cal H}\left({\Bbb C}P^{1}, \left\{K_{i}\right\}\right) \to
H^{1/2}\left({\Bbb C}P^{1},\omega^{1/2}\right) $. Since $ \bar{\partial}\colon H^{1/2}\left({\Bbb C}P^{1},\omega^{1/2}\right) \to H^{-1/2}\left({\Bbb C}P^{1},\omega^{1/2}\otimes\bar{\omega}\right) $ is an
isomorphism, the relation $ f \mapsto \bar{\partial}F $ gives an injection
$ {\cal H}\left({\Bbb C}P^{1}, \left\{K_{i}\right\}\right) \to H^{-1/2}\left({\Bbb C}P^{1},\omega^{1/2}\otimes\bar{\omega}\right) $. Moreover, the last mapping may be
pushed through $ {\cal H}\left({\Bbb C}P^{1}, \left\{K_{i}\right\}\right) \to \bigoplus_{l_{2}}H^{-1/2}\left(\widetilde{K}_{i},\omega^{1/2}\otimes\bar{\omega}\right) $. \end{corollary}
Since the operator $ \bar{\partial} $ has no null-space on $ H^{1/2}\left({\Bbb C}P^{1}, \omega^{1/2}\right) $, we obtain
\begin{corollary} $ \bar{\partial}^{-1}\left(\sum\widetilde{e}_{i}\left(f|_{\partial K_{i}}\right)\right) $ equals $ f $ outside of $ \bigcup\widetilde{K}_{i} $, i.e.,
modulo $ \overset{\,\,{}_\circ}{H}^{1/2}\left(\bigcup\widetilde{K}_{i}\right) $. \end{corollary}
Since $ \bar{\partial}F $ is determined by $ \widetilde{r}\left(f\right) $, we found a left inverse to the mapping
$ \widetilde{r} $, thus it is an injection and the image is closed. This finishes the
proof of Theorem~\ref{th4.40}. {}\end{proof}
The next step is to describe the image of the operator in question.
Let $ \gamma_{i}=\partial K_{i} $, $ \gamma=\bigcup\gamma_{i} $. We claim that an element of the image of the mapping
$ \widetilde{r} $ is uniquely determined by the minus-components, and any collection of
minus-components of bounded norm is possible.
\begin{proposition} \label{prop5.28}\myLabel{prop5.28}\relax Consider a decomposition of the space $ L_{2}\left(\gamma_{i}, \Omega^{1/2}\right) =
L_{2}^{+}\left(\gamma_{i}\right) \oplus L_{2}^{-}\left(\gamma_{i}\right) $ into subspaces of half-forms which may be holomorphically
extended
inside the circle and outside the circle. The image of $ \widetilde{r} $ consists of
sequences $ \left(f_{i}^{\pm}\right) $ such that
\begin{equation}
f_{i}^{+}=\sum_{j\not=i}{\bold K}f_{j}^{-}.
\notag\end{equation}
Here $ {\bold K} $ is the Hilbert operator, i.e., the operator with Cauchy kernel. \end{proposition}
\begin{proof} First of all, note that the proof of Theorem~\ref{th4.40}
together with Propositions~\ref{prop4.12} and~\ref{prop5.16} shows that knowing
$ \widetilde{r}\left(f\right) $ one can write $ f $ by an explicit formula outside of $ \overline{\bigcup\widetilde{K}_{i}} $. On the
other hand, it is easy to see that if we throw away the restriction that
we want our operators to be continuous in $ H^{s} $-topology, one can
reconstruct $ f $ outside of $ \bigcup\bar{K}_{i} $.
Indeed, since the curve $ \gamma $ has a finite length, restriction of any
smooth section $ \alpha $ of $ \omega^{1/2} $ to $ \gamma $ is in $ L_{2}\left(\gamma\right) $, thus $ \int\alpha|_{S}\widetilde{r}\left(f\right) $ is correctly
defined. By duality, this means that extension $ e\left(\widetilde{r}\left(f\right)\right) $ of $ \widetilde{r}\left(f\right) $ to $ {\Bbb C}P^{1} $ by
$ \delta $-function is a correctly defined generalized section of $ \omega^{1/2}\otimes\bar{\omega} $. (Note
that we consider $ e $, not $ \widetilde{e}! $)
\begin{lemma} Half-form $ f $ coincides with $ \bar{\partial}^{-1}e\left(\widetilde{r}\left(f\right)\right) $ outside of $ \bigcup\bar{K}_{i} $. \end{lemma}
\begin{proof} Indeed, by the construction of $ \widetilde{e}_{i} $, $ \bar{\partial}^{-1} \widetilde{e}_{i}\left(\widetilde{r}_{i}\left(f\right)\right) $ coincides with
$ \bar{\partial}^{-1}e\left(\widetilde{r}_{i}\left(f\right)\right) $ outside of $ \widetilde{K}_{i}\smallsetminus K_{i} $, thus $ \bar{\partial}^{-1}e\left(\widetilde{r}\left(f\right)\right) $ coincides with $ f $ outside of
$ \bigcup\widetilde{K}_{i} $. On the other hand, $ \bar{\partial}^{-1}e\left(\widetilde{r}\left(f\right)\right) $ is holomorphic outside of $ \bar{\gamma} $, thus
this equality can be extended up to $ \gamma $. \end{proof}
Consider now $ f=\bar{\partial}^{-1}e\left(\widetilde{r}\left(f\right)\right) $ near $ \gamma_{i} $. Breaking $ \widetilde{r}\left(f\right) $ into two
components, $ \left(\widetilde{r}\left(f\right)-\widetilde{r}_{i}\left(f\right)\right) $ and $ \widetilde{r}_{i}\left(f\right) $, we conclude that
\begin{equation}
f=\bar{\partial}^{-1}e\left(\widetilde{r}\left(f\right)-\widetilde{r}_{i}\left(f\right)\right) + \bar{\partial}^{-1}e\left(\widetilde{r}_{i}\left(f\right)\right).
\notag\end{equation}
The first summand is holomorphic inside $ \widetilde{K}_{i} $ and coincides with $ \sum_{j\not=i}{\bold K}f_{j}^{-} $,
since $ {\bold K} $ kills $ f_{j}^{+} $. The second summand is holomorphic outside of $ \partial K_{i} $, so
its $ + $-part vanishes on $ \partial K_{i} $, which finishes the proof of Proposition
~\ref{prop5.28}. {}\end{proof}
\subsection{Gluing conditions }\label{s5.30}\myLabel{s5.30}\relax We continue using notations of Section
~\ref{s5.10}. Suppose that the set $ I $ has an involution $ '\colon I\to I $ which
interchanges two subsets $ I_{+} $ and $ I_{+}' $, $ I=I_{+}\amalg I_{+}' $. Thus all the disks $ K_{i} $ are
divided into pairs $ \left(K_{i},K_{i'}\right) $, $ i\in I_{+} $. Fix a fraction-linear identification $ \varphi_{i} $
of $ \partial K_{i'} $ and $ \partial K_{i} $ which reverses the orientation of the circles, $ \varphi_{i'}=\varphi_{i}^{-1} $.
Let $ \varphi_{i} $ identifies the boundary of the disk $ \widetilde{K}_{i'} $ with the boundary of $ \overset{\,\,{}_\circ}{K}_{i} $, and
$ \partial\widetilde{K}_{i} $ with $ \partial\overset{\,\,{}_\circ}{K}_{i'} $, thus $ \overset{\,\,{}_\circ}{K}_{i}\subset K_{i}\subset\widetilde{K}_{i} $.
Let $ R_{i} $ be the annulus between $ \widetilde{K}_{i} $ and $ \overset{\,\,{}_\circ}{K}_{i} $. The mapping $ \varphi_{i} $ identifies
$ R_{i} $ with $ R_{i'} $. Let $ S $ be the part of $ {\Bbb C}P^{1} $ which lies outside of all the disks
$ \overset{\,\,{}_\circ}{K}_{i} $. Glue the annuli $ R_{i}\subset S $ with $ R_{i'}\subset S $ using $ \varphi_{i} $.
\begin{definition} A {\em model space\/} $ \bar{M} $ is the set obtained from $ S $ by identifying the
annuli $ R_{i} $ and $ R_{i'} $ using $ \varphi_{i} $. \end{definition}
Note that $ \bar{M} $ consists of two parts: a smooth manifold $ M $ which is the
image of $ S\smallsetminus\overline{\bigcup\overset{\,\,{}_\circ}{K}_{\bullet}} $, and the rest, which one should consider as ``infinity''
$ M_{\infty} $ of the manifold $ M $ (compare with Section~\ref{s0.10}). Unfortunately, the
topology on $ \bar{M} $ in neighborhood of infinity is not suitable for studying
the Riemann--Roch theorem, so we will not consider it in this paper.
Note that Section~\ref{s4.95} suggests a different topology on $ \bar{M} $. We will
use some features of this topology when we discuss a mapping into
universal Grassmannian.
\subsection{Strong sections and duality }\label{s6.50}\myLabel{s6.50}\relax In the notations of the previous
section
consider now holomorphic functions $ \psi_{i} $ defined in $ R_{i} $. Suppose that $ \psi_{i} $ are
nowhere 0, and $ \psi_{i'}\circ\varphi_{i}=\psi_{i}^{-1} $.
Define $ {\cal L} $ to be a sheaf on $ \bar{M} $ associated with gluing conditions $ \psi_{i} $,
i.e., for $ U\subset\bar{M} $ the section of $ {\cal L} $ on $ U $ is a function $ f $ on $ \widetilde{U}\subset{\Bbb C}P^{1} $ such that
$ f\left(\varphi_{i}\left(x\right)\right)=\psi_{i}\left(x\right)f\left(x\right) $ whenever both sides have sense (here $ \widetilde{U} $ is an
appropriate covering subset of $ {\Bbb C}P^{1} $).
Let $ {\cal L}^{-1} $ be the sheaf associated with gluing conditions $ \psi_{i}^{-1} $.
Similarly define the tensor product of two sheaves defined via gluing
conditions.
The identifications $ \varphi_{i} $ fixes an identification $ \varphi_{i}^{*} $ of half-forms on
$ R_{i} $ and on $ R_{i'} $ up to a sign. Choose this sign for all $ i\in I_{+}. $\footnote{This corresponds to picking a representative of $ \varphi_{i} $ in $ 2 $-sheet cover
$ \operatorname{SL}\left(2,{\Bbb C}\right) \to \operatorname{PGL}\left(2,{\Bbb C}\right) $.} Let $ \omega^{1/2}\otimes{\cal L} $ be
the sheaf on $ \bar{M} $ consisting of half-forms on $ S $ such that
\begin{equation}
\varphi_{i}^{*}\left(\alpha|_{R_{i'}}\right) = \psi_{i}\cdot\alpha|_{R_{i}},\qquad i\in I.
\label{equ5.31}\end{equation}\myLabel{equ5.31,}\relax
Similarly define $ \omega^{1/2}\otimes{\cal L}\otimes\bar{\omega} $.
To write an analogue of~\eqref{equ5.31} for sections in Sobolev spaces,
consider a subannulus $ \overset{\,\,{}_\circ}{R}_{i}\subset R_{i} $ such that $ \overset{\,\,{}_\circ}{R}_{i'}=\varphi_{i}\left(\overset{\,\,{}_\circ}{R}_{i}\right) $. We suppose that one can
find numbers $ C,D>0 $ and a sequence $ \left(s_{i}\right) $, $ i\in I $, $ |s_{i}|+C+D<1 $, such that the
annulus $ \overset{\,\,{}_\circ}{R}_{i} $ is described as $ \left(\left(s_{i}-C\right)\varepsilon,\left(s_{i}+C\right)\varepsilon\right)\times S^{1} $ in the conformal
coordinate system such that $ R_{i} $ is $ \left(-\varepsilon,\varepsilon\right)\times S^{1} $. Let $ K'_{i} $ be the disk bounded
by the inner boundary of $ \overset{\,\,{}_\circ}{R}_{i} $.
Consider a subset $ \overset{\,\,{}_\circ}{S}\subset S $, $ \overset{\,\,{}_\circ}{S}={\Bbb C}P^{1}\smallsetminus\bigcup_{i}K'_{i} $. One obtains the same manifold $ \bar{M} $
by gluing $ \overset{\,\,{}_\circ}{R}_{i}\subset\overset{\,\,{}_\circ}{S} $ as by gluing $ R\subset S $, but in what follows it will be more
convenient to have a choice of annuli $ \overset{\,\,{}_\circ}{R}_{i} $, and have them separated from
boundary of $ R_{i} $.
\begin{definition} Let $ H^{s}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) $ be the subspace of $ H^{s}\left({\Bbb C}P^{1}\smallsetminus\bigcup_{i}K'_{i},\omega^{1/2}\right) $
consisting of sections which satisfy the gluing conditions~\eqref{equ5.31},
similarly define $ H^{s}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\otimes\bar{\omega}\right) $. \end{definition}
Dually,
\begin{definition} Let $ \overset{\,\,{}_\circ}{H}^{s}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) $ be the quotient of $ \overset{\,\,{}_\circ}{H}^{s}\left({\Bbb C}P^{1}\smallsetminus\bigcup_{i}K'_{i}, \omega^{1/2}\right) $ by
the subspace of $ \overset{\,\,{}_\circ}{H}^{s}\left(\bigcup_{i}\overset{\,\,{}_\circ}{R}_{i}, \omega^{1/2}\right) $ consisting of sections which satisfy the
gluing conditions
\begin{equation}
\varphi_{i}^{*}\left(\alpha|_{\overset{\,\,{}_\circ}{R}_{i'}}\right) = -\psi_{i}\cdot\alpha|_{\overset{\,\,{}_\circ}{R}_{i}},\qquad i\in I,
\label{equ5.32}\end{equation}\myLabel{equ5.32,}\relax
similarly define $ \overset{\,\,{}_\circ}{H}^{s}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\otimes\bar{\omega}\right) $. \end{definition}
Indeed, these definitions are dual due to
\begin{lemma} \label{lm5.22}\myLabel{lm5.22}\relax If the disks $ K_{i} $ are well-separated, the spaces
$ H^{1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) $ and $ \overset{\,\,{}_\circ}{H}^{-1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}^{-1}\otimes\bar{\omega}\right) $ are mutually dual w.r.t. the
pairing $ \int\alpha\beta $. \end{lemma}
\begin{proof} The spaces $ H^{s}\left({\Bbb C}P^{1}\smallsetminus\bigcup_{i}K'_{i},\omega^{1/2}\otimes{\cal L}\right) $ and $ \overset{\,\,{}_\circ}{H}^{-s}\left({\Bbb C}P^{1}\smallsetminus\bigcup_{i}K'_{i},\omega^{1/2}\otimes{\cal L}^{-1}\otimes\bar{\omega}\right) $
are mutually dual w.r.t. this pairing by definition. What remains to
prove is that the orthogonal complement to the subspace of
$ H^{s}\left({\Bbb C}P^{1}\smallsetminus\bigcup_{i}K'_{i},\omega^{1/2}\otimes{\cal L}\otimes\bar{\omega}\right) $ given by~\eqref{equ5.31} is given by
\begin{equation}
\varphi_{i}^{*}\left(\alpha|_{R_{i'}}\right) = -\psi_{i}^{-1}\cdot\alpha|_{R_{i}},\qquad i\in I,
\label{equ5.33}\end{equation}\myLabel{equ5.33,}\relax
in $ \overset{\,\,{}_\circ}{H}^{-s}\left({\Bbb C}P^{1}\smallsetminus\bigcup_{i}K'_{i},\omega^{1/2}\otimes{\cal L}^{-1}\otimes\bar{\omega}\right) $, provided $ s=1/2 $. This statement if obvious
for any fixed $ i\in I $.
On the other hand, solutions to~\eqref{equ5.33} form a direct sum over $ i $, and
by the second part of Corollary~\ref{cor3.80}, solutions to~\eqref{equ5.31} form a
``direct'' intersection (i.e., an intersection of subspaces with almost
orthogonal complements).\end{proof}
\begin{lemma} \label{lm5.25}\myLabel{lm5.25}\relax Suppose that the disks $ K_{i} $ are well-separated. The natural
mapping
\begin{equation}
\overset{\,\,{}_\circ}{H}^{-1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\otimes\bar{\omega}\right) \to H^{-1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\otimes\bar{\omega}\right)
\label{equ5.36}\end{equation}\myLabel{equ5.36,}\relax
is an isomorphism. Dually,
\begin{equation}
\overset{\,\,{}_\circ}{H}^{1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) \to H^{1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right)
\notag\end{equation}
is an isomorphism. \end{lemma}
\begin{proof} We may suppose that all the rings $ \overset{\,\,{}_\circ}{R}_{i} $ have conformal
distance between boundaries greater than $ 2C\varepsilon $, $ C>0 $. Consider a function
$ \sigma\left(x\right) $, $ x\in\left(-C\varepsilon,C\varepsilon\right) $, such that $ \sigma\left(x\right)=0 $ near the left end, $ \sigma\left(x\right)+\sigma\left(1-x\right)=1 $. This
gives a cut-off function in all the rings $ \overset{\,\,{}_\circ}{R}_{i} $, and we may extend it by 0
into all $ K_{i}' $, and by 1 into $ S $. We obtain a function on $ {\Bbb C}P^{1} $.
By the results of Section~\ref{s3.8} and Amplification~\ref{amp2.65} the
multiplication by this function is a bounded operator in $ H^{1/2} $. Clearly,
this operator provides an inverse to~\eqref{equ5.36}. \end{proof}
The operator $ \bar{\partial} $ gives mappings
\begin{align} \overset{\,\,{}_\circ}{H}^{s}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) & \to \overset{\,\,{}_\circ}{H}^{s-1}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\otimes\bar{\omega}\right),
\notag\\
H^{s}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) & \to H^{s-1}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\otimes\bar{\omega}\right),
\notag\end{align}
which by the previous lemma induce mappings
\begin{align} \overset{\,\,{}_\circ}{H}^{1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) & \xrightarrow[]{\bar{\partial}} H^{-1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\otimes\bar{\omega}\right),
\notag\\
H^{1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) & \xrightarrow[]{\bar{\partial}} \overset{\,\,{}_\circ}{H}^{-1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\otimes\bar{\omega}\right)
\notag\end{align}
(obviously, it does not matter which of isomorphisms of Lemma~\ref{lm5.25} we
use to obtain these mappings).
\begin{definition} Define the space $ \Gamma_{\text{strong}}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) $ of {\em strong global
holomorphic sections\/} of $ \omega^{1/2}\otimes{\cal L} $ as
\begin{equation}
\Gamma_{\text{strong}}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) = \operatorname{Ker} \left(H^{1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) \xrightarrow[]{\bar{\partial}} H^{-1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\otimes\bar{\omega}\right)\right).
\notag\end{equation}
\end{definition}
Note that the conditions on strong global sections $ \alpha $ are: they are
holomorphic sections of $ \omega^{1/2}\otimes{\cal L}|_{M} $ (since $ \bar{\partial}\alpha $ vanishes on $ M $), they do not
grow very quick near $ M_{\infty} $ (since they belong to $ H^{1/2} $), and they have no
residue on $ M_{\infty} $ (since $ \bar{\partial}\alpha $ vanishes on $ M_{\infty} $).
\subsection{Weak sections }\label{s6.60}\myLabel{s6.60}\relax The space $ \Gamma_{\text{strong}} $ from the last section is
good
for studying the duality conditions, but it is not suitable for for
description of global sections via boundary conditions.
\begin{definition} Define the space $ \Gamma_{\text{weak}}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) $ of {\em weak global holomorphic
sections\/} of $ \omega^{1/2}\otimes{\cal L} $ as forms from the generalized Hardy space
$ \alpha\in{\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $ such that the (mollified) restrictions on the circles
$ \partial K_{i} $ satisfy the gluing conditions. In other words,
\begin{equation}
\Gamma_{\text{weak}}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) = \left\{\alpha\in{\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) \mid \psi_{i}\widetilde{r}_{i}\left(\alpha\right)=\varphi_{i}^{*}\left(\widetilde{r}_{i'}\left(\alpha\right)\right) \right\}.
\notag\end{equation}
\end{definition}
There is a natural mapping
\begin{equation}
\Gamma_{\text{strong}} \to \Gamma_{\text{weak}}.
\notag\end{equation}
\begin{theorem} \label{th5.31}\myLabel{th5.31}\relax Suppose that the disks $ K_{i} $ are well-separated, and for
some $ A>1 $ either $ |\psi_{i}\left(z\right)|<A $, or $ |\psi_{i}\left(z\right)|>1/A $ for any $ i $ and $ z\in R_{i} $. Then the
above mapping is an isomorphism for an appropriate choice of annuli $ \overset{\,\,{}_\circ}{R}_{i} $. \end{theorem}
\begin{proof} Indeed, any weak section $ \alpha $ is a holomorphic form inside
$ {\Bbb C}P^{1}\smallsetminus\overline{\bigcup_{i}K_{i}} $. Consider $ \psi_{i}^{-1}\cdot\varphi_{i}^{*}\left(\alpha\right) $, it is a holomorphic form inside
$ K_{i}\smallsetminus\overset{\,\,{}_\circ}{K}_{i} $. Since $ \alpha $ satisfies gluing conditions, $ \alpha|_{\widetilde{K}_{i}\smallsetminus K_{i}} $ has the same Laurent
coefficients as $ \psi_{i}^{-1}\cdot\varphi_{i}^{*}\left(\alpha\right) $ (we used compatibility of $ \widetilde{r} $ with $ r $ on
holomorphic forms), thus these two forms are restrictions of the same
holomorphic form defined inside $ \widetilde{K}_{i}\smallsetminus\overset{\,\,{}_\circ}{K}_{i} $. We see that $ \alpha $ can be extended (as
a holomorphic form) into $ {\Bbb C}P^{1}\smallsetminus\overline{\bigcup_{i}\overset{\,\,{}_\circ}{K}_{i}} $, and this holomorphic form
satisfies the gluing conditions~\eqref{equ5.31}.
What remains to prove is that we can bound the norm of this
extension. On the other hand, this is a local statement, since one can
represent $ \alpha=\alpha_{1}+\alpha_{2} $, and $ \alpha=0 $ inside an appropriate circle concentric with
$ \widetilde{K}_{i} $, $ \alpha_{2}=0 $ outside of $ \bigcup\widetilde{K}_{i} $. Thus the only thing we need to prove is that
inside $ R_{i} $ the form $ \alpha $ can be extended across $ \partial K_{i} $ without increasing its
norm too much.
Take $ C=\frac{1}{4} $, let $ \varepsilon_{i}=C $ if $ |\psi_{i}\left(z\right)|<A $, $ \varepsilon_{i}=-C $ otherwise. With this
choice of $ \overset{\,\,{}_\circ}{R}_{i} $ we know that the $ H^{1/2} $-norms of $ \alpha|_{\widetilde{K}_{i}\smallsetminus K_{i}} $ and $ \alpha|_{K_{i}\smallsetminus K'_{i}} $ are
bounded by the norms of $ \alpha|_{\widetilde{K}_{i}\smallsetminus K_{i}} $ and $ \alpha|_{\widetilde{K}_{i'}\smallsetminus K_{i'}} $. Now the theorem becomes a
corollary of the following
\begin{lemma} Fix two numbers $ A>a>0 $. Consider an annulus $ R $ with concentric
boundaries and conformal distance between boundaries between $ A $ and $ a $.
Let $ V_{R} $ be the space of holomorphic half-forms in $ R $ which belong to
$ H^{1/2}\left(R\right) $. Then the mapping of taking boundary value
\begin{equation}
b\colon H^{1/2}\left(R,\omega^{1/2}\right) \to L_{2}\left(\partial R,\Omega^{1/2}\right)\colon \alpha \mapsto \alpha|_{\partial R}
\notag\end{equation}
is an invertible mapping to its (closed) image, and the norms of this
mapping and its inverse are bounded by numbers depending on $ A $ and $ a $
only.
\end{lemma}
This lemma is a variation of what we did in Section~\ref{s2.70}, with a
disk substituted with an annulus. It can be proven in the same way as the
case of a disk.
This finishes the proof of the theorem. \end{proof}
\begin{remark} To complement the notion of weak holomorphic sections, let us
define the spaces of ``weak'' $ H^{\pm1/2} $-section in such a way that
\begin{equation}
\Gamma_{\text{weak}}\left(\bar{M}, \omega^{1/2}\otimes{\cal L}\right) = \operatorname{Ker}\left(H_{\text{weak}}^{1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) \xrightarrow[]{\bar{\partial}} H_{\text{weak}}^{-1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\otimes\bar{\omega}\right) \right):
\notag\end{equation}
\end{remark}
\begin{definition} Let $ H_{\text{weak}}^{1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) $ be the subspace of $ H^{1/2}\left({\Bbb C}P^{1}\smallsetminus\bigcup_{i}K_{i},\omega^{1/2}\right) $
consisting of sections $ \alpha $ which satisfy the gluing conditions:
\begin{equation}
\psi_{i}\widetilde{r}_{i}\left(\alpha\right)=\varphi_{i}^{*}\left(\widetilde{r}_{i'}\left(\alpha\right)\right),\qquad i\in I.
\notag\end{equation}
Let $ H_{\text{weak}}^{-1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\otimes\bar{\omega}\right) = H^{-1/2}\left({\Bbb C}P^{1}\smallsetminus\bigcup_{i}K_{i},\omega^{1/2}\otimes\bar{\omega}\right) $.
\end{definition}
\subsection{Finite-degree bundles }\label{s4.90}\myLabel{s4.90}\relax Let $ d_{i} $ be the {\em index\/} of $ \psi_{i} $, i.e., the
degree of the mapping $ \arg \psi_{i}\colon K_{i} \to S^{1} $. We say that the collection of
gluing data $ \left\{\psi_{i}\right\} $ is of {\em finite degree\/} if $ d_{i}=0 $ for all but the finite
number of indices $ i\in I $. The collection $ \left\{\psi_{i}\right\} $ is {\em semibounded\/} if for some
fixed number $ C $ and any $ i $ either $ |\psi_{i}\left(z\right)| $ or $ |\psi\left(z\right)|_{i}^{-1} $ is bounded by $ C $ if
$ z\in R_{i} $. The {\em degree\/} of the finite-degree collection $ \left\{\psi_{i}\right\} $ is the sum
\begin{equation}
\sum_{i\in I_{+}}d_{i} = \frac{1}{2}\sum_{i\in I}d_{i}.
\notag\end{equation}
\begin{definition} The {\em degree\/} of a bundle $ {\cal L} $ defined by gluing conditions $ \psi_{i} $ is
the degree of the collection $ \left\{\psi_{i}\right\} $. \end{definition}
\begin{amplification} In what follows we are going to use the following
generalization of these constructions: we allow a substitution of a
finite number of simply-connected domains with smooth boundaries instead
of disks $ K_{i} $. (For such an $ i $ one should substitute any bigger domain
instead of $ \widetilde{K}_{i} $.) However, we still require that the identifications $ \varphi_{i} $ are
fraction-linear. \end{amplification}
\subsection{Stratification of infinity }\label{s4.95}\myLabel{s4.95}\relax Consider a smooth curve $ \gamma $ on $ {\Bbb C}P^{1} $ of
finite length. Suppose that a metric on $ {\Bbb C}P^{1} $ is fixed, and $ z\notin\gamma $. Let
$ \rho_{k}\left(z,\gamma\right)=\|\operatorname{dist}\left(z,y\right)^{-k}\|_{L_{2}\left(\gamma\right)} $, here $ y\in\gamma $.
\begin{lemma} $ \rho_{k}\left(x,\gamma\right) $ is a semicontinuous function of $ x $, thus
\begin{equation}
D_{k,R}=\left\{z\in{\Bbb C}P^{1} \mid \rho_{k}\left(x,\gamma\right)\leq R \right\}
\notag\end{equation}
is a compact subset of $ {\Bbb C}P^{1}\smallsetminus\gamma $. \end{lemma}
Let $ D_{k}=\bigcup_{R\in{\Bbb R}}D_{k,R} $. It is a subset of $ {\Bbb C}P^{1}\smallsetminus\gamma $, moreover, $ {\Bbb C}P^{1}\smallsetminus\bar{\gamma}\subset D_{k} $.
Let $ \overset{\,\,{}_\circ}{D}_{k,R}=D_{k,R}\cap\left({\Bbb C}P^{1}\smallsetminus\bar{\gamma}\right) $, $ \overset{\,\,{}_\circ}{D}_{k}=\bigcup_{R\in{\Bbb R}}\overline{\overset{\,\,{}_\circ}{D}_{k,R}} $. Since length of $ \gamma $ is finite,
$ \overset{\,\,{}_\circ}{D}_{0}={\Bbb C}P^{1} $.
\begin{definition} Define a filtration of $ {\cal K}=\bar{\gamma}\smallsetminus\gamma $ by $ {\cal K}^{\left(k\right)}={\cal K}\cap\overset{\,\,{}_\circ}{D}_{k+1} $, $ k\geq-1 $. Let
$ {\cal K}^{\left(\infty\right)}=\bigcap{\cal K}^{\left(k\right)} $. \end{definition}
Suppose that $ \gamma $ is the boundary of a well-separated family of
circles, $ \gamma=\bigcup\partial K_{i} $.
\begin{theorem} Let $ z_{0}\in{\cal K}^{\left(k\right)} $, $ z $ be a coordinate system near $ z_{0} $, $ f\in{\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $.
Then $ f $ has an asymptotic decomposition
\begin{equation}
f\left(z\right)dz^{-1/2} = f_{0}+f_{1}\left(z-z_{0}\right)+f_{2}\left(z-z_{0}\right)^{2}+\dots +f_{k}\left(z-z_{0}\right)^{k}+o\left(z^{k}\right)
\notag\end{equation}
when $ z \to z_{0} $ along $ \overset{\,\,{}_\circ}{D}_{k,R} $ for an appropriate $ R\gg0 $. \end{theorem}
\begin{proof} The half-form $ f $ is holomorphic inside $ {\Bbb C}P^{1}\smallsetminus\overline{\bigcup K_{i}} $. The Cauchy
formula show that inside $ \overset{\,\,{}_\circ}{D}_{k,R} $ the derivatives $ f^{\left(l\right)} $, $ l\leq k $, are bounded.
Moreover, these derivatives are given by some integrals along $ \gamma $, and
these integrals remain well-defined in $ D_{k,R} $ as well. Let $ f_{l} $ be the values
of these integrals in $ z_{0}\in D_{k,R_{1}} $ (here $ R_{1}\gg0 $).
Since $ z_{0}\in\overset{\,\,{}_\circ}{D}_{k+1} $, it is in the closure of $ \overset{\,\,{}_\circ}{D}_{k+1,R} $ for an appropriate $ R $. It
is easy to see that $ f^{\left(k\right)}\left(z\right) $ has a limit $ f_{k} $ when $ z \to z_{0} $ along $ \overset{\,\,{}_\circ}{D}_{k+1,R} $. Same
is true for $ f^{\left(l\right)}\left(z\right) $, $ l\leq k $.
Consider the integral for
\begin{equation}
\frac{f\left(z\right)dz^{-1/2} - \left(f_{0}+f_{1}\left(z-z_{0}\right)+f_{2}\left(z-z_{0}\right)^{2}+\dots +f_{k}\left(z-z_{0}\right)^{k}\right)}{\left(z-z_{0}\right)^{k}}.
\notag\end{equation}
It is
\begin{equation}
\int_{\gamma}K_{k}\left(z,z_{0},\zeta\right)f\left(\zeta\right),
\notag\end{equation}
here
\begin{align} K\left(z,\zeta\right) & =\frac{d\zeta^{1/2}}{\zeta-z},
\notag\\
K_{k}\left(z,z_{0},\zeta\right) & =\frac{K\left(z,\zeta\right)-\sum_{l=0}^{k}\frac{d^{k}K}{dz^{k}}|_{z=z_{0}}\frac{\left(z-z_{0}\right)^{k}}{k!}}{\left(z-z_{0}\right)^{k}}.
\notag\\
& = \frac{z-z_{0}}{\left(\zeta-z_{0}\right)^{k+1}}K\left(z,\zeta\right) = \frac{1}{\left(\zeta-z_{0}\right)^{k}}\left(K\left(z,\zeta\right)-K\left(z_{0},\zeta\right)\right).
\notag\end{align}
We need to show that this integral goes to 0 when $ z \to z_{0} $ along $ \overset{\,\,{}_\circ}{D}_{k+1,R} $.
Consider a function $ \rho\left(r\right) $ such that $ \lim _{r\to0}\rho\left(r\right)=0 $, $ \lim _{r\to0}\rho\left(r\right)/r=\infty $. Here
$ \gamma_{r} $ is the intersection of $ \gamma $ with the disk of radius $ r $ about $ z_{0} $, instead
of $ \widetilde{r}_{\gamma}\left(f\right)\in L_{2}\left(\gamma\right) $ we write just $ f $.
Break the integral into three parts: two (which we do not want to
separate yet) along $ \gamma_{\rho\left(|z-z_{0}|\right)} $, the other along $ \gamma\smallsetminus\gamma_{\rho\left(|z-z_{0}|\right)} $. Since
\begin{equation}
K_{k}= \frac{z-z_{0}}{\zeta-z_{0}}\frac{K\left(z,\zeta\right)}{\left(\zeta-z_{0}\right)^{k}} = o\left(\frac{K\left(z_{0},\zeta\right)}{|\zeta-z_{0}|^{k}}\right)
\notag\end{equation}
along the second part of $ \gamma $ if $ |z-z_{0}| \to $ 0, and since
\begin{equation}
\frac{K\left(z_{0},\zeta\right)}{|z_{0}-\zeta|^{k}}\in L_{2}\left(\gamma\right)
\notag\end{equation}
as a function of $ \zeta $, it is enough to show that the first two part of the
integral go to 0. Since $ \lim _{r\to0} \int_{\gamma_{\rho\left(r\right)}}\left|f\left(\zeta\right)\right|^{2}=0 $, we need to show only
that the $ L_{2} $-norm of $ K_{k}\left(z,z_{0},\zeta\right) $ is bounded when $ \zeta\in\gamma_{\rho\left(|z-z_{0}|\right)} $.
Subdivide $ \gamma_{\rho\left(|z-z_{0}|\right)} $ once more: into $ \gamma_{\varepsilon|z-z_{0}|} $ and
$ \gamma_{\rho\left(|z-z_{0}|\right)}\smallsetminus\gamma_{\varepsilon|z-z_{0}|} $. Here $ \varepsilon\ll1 $. Since on the second part
$ K\left(z_{0},\zeta\right)=O\left(K\left(z,\zeta\right)\right) $, we see that $ K_{k}\left(z,z_{0},\zeta\right) = O\left(\frac{K\left(z,\zeta\right)}{\left(\zeta-z\right)^{k}}\right) $, thus $ K_{k} $
has a bounded $ L_{2} $-norm. We conclude that the integral along the second
part goes to 0 when $ z \to z_{0} $. On the first part $ K\left(z,\zeta\right)=O\left(K\left(z_{0},\zeta\right)\right) $, thus
\begin{equation}
K_{k}\left(z,z_{0},\zeta\right) = \frac{1}{\left(\zeta-z_{0}\right)^{k}}\left(K\left(z,\zeta\right)-K\left(z_{0},\zeta\right)\right) =
O\left(\frac{K\left(z_{0},\zeta\right)}{\left(\zeta-z_{0}\right)^{k}}\right)=O\left(\frac{1}{\left(z-z_{0}\right)^{k+1}}\right),
\notag\end{equation}
so it has a bounded $ L_{2} $-norm as well. \end{proof}
\begin{definition} Consider a model $ \bar{M} $ of a curve. Say that a point $ z\in M_{\infty} $ is of
smoothness $ C^{k} $, if $ z\in{\cal K}^{\left(k\right)} $, here $ {\cal K}=M_{\infty} $. \end{definition}
\begin{remark} Note that the stratification we used is related to the
following inclusion of $ M $ into $ L_{2}\left(\gamma,\Omega^{1/2}\otimes\mu\right) $:
\begin{equation}
z \mapsto \widetilde{r}_{\gamma}\left(K_{z}\right),
\notag\end{equation}
here $ K_{z}\left(\zeta\right)=\frac{d\zeta^{1/2}}{\zeta-z} $ is the Cauchy kernel. In other words,
\begin{equation}
z \mapsto \frac{d\zeta^{1/2}}{\zeta-z}|_{\gamma}
\notag\end{equation}
after a choice of coordinate $ \zeta $ on $ {\Bbb C}P^{1} $. The points of smoothness $ C^{0} $
correspond to limit points of this inclusion, the points of
smoothness $ C^{k} $ correspond to limit points of $ k $-jets continuation of this
mapping. \end{remark}
\section{Riemann--Roch theorems }
\subsection{Abstract Riemann--Roch theorem } We say that two vector subspaces $ V_{1} $,
$ V_{2} $ of a topological vector space $ H $ {\em satisfy the Riemann\/}--{\em Roch theorem\/} if
$ \dim V_{1}\cap V_{2}<\infty $, and $ \operatorname{codim}\left(\overline{V_{1}+V_{2}}\right)<\infty $. We call the number
\begin{equation}
\dim V_{1}\cap V_{2} - \operatorname{codim}\left(\overline{V_{1}+V_{2}}\right)
\notag\end{equation}
the {\em index\/} of two subspaces. If $ V_{1}+V_{2}=\overline{V_{1}+V_{2}} $, we say that $ V_{1},V_{2} $ satisfy
the {\em strong form\/} of the theorem.
\begin{remark} Note that if $ V_{1} $, $ V_{2} $ satisfy the strong form of the theorem,
then the natural mapping $ V_{1} \to V/V_{2} $ is a Fredholm mapping with the index
being the index of $ V_{1} $, $ V_{2} $. If $ V_{1} $, $ V_{2} $ satisfy the weak form of the
theorem, then this mapping is a continuous mapping $ p $ with $ \dim \operatorname{Ker} p - \dim
\operatorname{Coker} p $ being the index of $ V_{1},V_{2} $. Here $ \operatorname{Coker} $ is the quotient by the
closure of the image. \end{remark}
Consider a direct sum of two Hilbert spaces $ H=H_{1}\oplus H_{2} $. Consider two
closed vector subspaces $ V_{1},V_{2}\subset H $ such that the projection of $ V_{i} $ on $ H_{i} $
has no
null-space and a dense image. This means that one can consider $ V_{1} $ as a graph
of a mapping $ A_{1}\colon H_{1} \to H_{2} $, similarly $ V_{2} $ is a graph of $ A_{2}\colon H_{2} \to H_{1} $.
Mappings $ A_{1,2} $ are closed, but not necessarily bounded.
\begin{lemma}[abstract finiteness] If $ A_{2} $ is bounded, and $ A_{1}\circ A_{2} $ is compact,
then $ V_{1}\cap V_{2} $ is finite dimensional. \end{lemma}
\begin{proof} The projection of $ V_{1}\cap V_{2} $ to $ H_{2} $ is a subspace of $ \operatorname{Ker}\left(A_{1}\circ A_{2}-\boldsymbol1\right) $,
thus is finite-dimensional. \end{proof}
\begin{proposition}[strong form] If $ A_{1} $, $ A_{2} $ are bounded, and $ A_{1}\circ A_{2} $
is compact, then $ V_{1} $ and $ V_{2} $ satisfy the strong form of Riemann--Roch
theorem with index 0. \end{proposition}
\begin{proof} The projection of $ V_{1}\cap V_{2} $ to $ H_{2} $ is $ \operatorname{Ker}\left(A_{1}\circ A_{2}-\boldsymbol1\right) $, which implies
the statement about $ \dim V_{1}\cap V_{2} $. The statement
about $ \operatorname{codim}\overline{V_{1}+V_{2}} $ follows from the fact that the orthogonal complements
to $ V_{1} $ and $ V_{2} $ satisfy the same conditions as $ V_{2} $ and $ V_{1} $ with linear
mappings being $ -A_{1}^{*} $, $ -A_{2}^{*} $.
To show that $ \dim V_{1}\cap V_{2} = \operatorname{codim}\left(\overline{V_{1}+V_{2}}\right) $ note that $ \operatorname{Ker}\left(A_{1}\circ A_{2}-\boldsymbol1\right) $ is
dual to $ \operatorname{Coker}\left(A_{2}^{*}\circ A_{1}^{*}-\boldsymbol1\right) $. \end{proof}
\begin{proposition}[weak form] If $ A_{2} $ is bounded, and both
$ A_{1}\circ A_{2} $ and $ A_{1}^{*}\circ A_{2}^{*} $ are compact,
then $ V_{1} $ and $ V_{2} $ satisfy the Riemann--Roch theorem with
index 0. \end{proposition}
\begin{proof} We already know that $ \dim V_{1}\cap V_{2} $ and $ \operatorname{codim} \overline{V_{1}+V_{2}} $ are finite.
The only thing to prove is that
\begin{equation}
\dim \operatorname{Ker}\left(A_{1}\circ A_{2}-\boldsymbol1\right) = \dim \operatorname{Coker}\left(A_{1}^{*}\circ A_{2}^{*}-\boldsymbol1\right).
\notag\end{equation}
Obviously, $ \left(A_{1}^{*}\circ A_{2}^{*}\right)^{*} $ is the closure of $ A_{2}\circ A_{1} $, thus
\begin{equation}
A_{2}\left(\operatorname{Ker}\left(A_{1}\circ A_{2}-\boldsymbol1\right)\right) \subset \operatorname{Ker}\left(\left(A_{1}^{*}\circ A_{2}^{*}\right)^{*}-\boldsymbol1\right),
\notag\end{equation}
hence
\begin{equation}
\dim \operatorname{Ker}\left(A_{1}\circ A_{2}-\boldsymbol1\right) \leq \dim \operatorname{Ker}\left(\left(A_{1}^{*}\circ A_{2}^{*}\right)^{*}-\boldsymbol1\right) = \dim \operatorname{Coker}\left(A_{1}^{*}\circ A_{2}^{*}-\boldsymbol1\right).
\notag\end{equation}
Application of the same argument to the dual operators shows the opposite
unequality. \end{proof}
We say that two vector subspaces $ V $, $ V' $ of a vector space $ H $ are
{\em comparable}, if $ V\cap V' $ is of finite codimension in both $ V $ and $ V' $. The
{\em relative dimension\/} $ \operatorname{reldim}\left(V,V'\right) $ is $ \operatorname{codim}\left(V\cap V'\subset V\right)-\operatorname{codim}\left(V\cap V'\subset V'\right) $. The
following theorem is a direct corollary of the above statement:
\begin{theorem}[Riemann--Roch theorem] \label{th6.50}\myLabel{th6.50}\relax Consider two
vector subspaces $ V_{1,2}\subset H $ of a Hilbert space $ H=H_{1}\oplus H_{2} $. Suppose that $ V_{1} $ is
comparable with the graph of a closed mapping $ A_{1}\colon H_{1} \to H_{2} $ and the
relative dimension of $ V_{1} $ and this graph is $ d_{1} $. Suppose $ V_{2} $ is comparable
with the graph of a closed mapping $ A_{2}\colon H_{2} \to H_{1} $ and the relative
dimension of $ V_{2} $ and this graph is $ d_{2} $.
\begin{enumerate}
\item
{\bf(weak form) }If $ A_{2} $ is bounded, and both $ A_{1}\circ A_{2} $ and $ A_{1}^{*}\circ A_{2}^{*} $ are
compact, then $ V_{1} $ and $ V_{2} $ satisfy the Riemann--Roch theorem with the index
being $ d_{1}+d_{2} $.
\item
{\bf(strong form) }If both $ A_{1} $ and $ A_{2} $ are bounded, and $ A_{1}\circ A_{2} $ is compact,
then $ V_{1} $ and $ V_{2} $ satisfy the strong form of Riemann--Roch theorem with the
index being $ d_{1}+d_{2} $.
\end{enumerate}
\end{theorem}
\subsection{Riemann problem }\label{s5.5}\myLabel{s5.5}\relax Consider a holomorphic function $ \psi\left(z\right) $ defined
in a annulus $ U=\left\{z \mid 1-\varepsilon\leq|z|\leq1+\varepsilon\right\} $, let $ S^{1}=\left\{z \mid |z|=1\right\} $. Suppose that $ \psi\left(z\right) $ is
nowhere 0, and consider the subspace $ V_{\psi} $ in $ L_{2}\left(S^{1},\omega^{1/2}\right)\oplus L_{2}\left(S^{1},\omega^{1/2}\right) $ consisting
of pairs of the form $ \left(f\left(z\right),\psi\left(z\right)f\left(z\right)\right) $. The Hilbert space $ L_{2}\left(S^{1},\omega^{1/2}\right) $ is a
direct sum of subspaces $ L_{2}^{\pm}\left(S^{1},\omega^{1/2}\right) $ consisting of forms which can be
holomorphically continued into two regions $ S^{1} $ divides $ {\Bbb C}P^{1} $ into.
Let $ p $ be the projection of $ V_{\psi} $ to
\begin{equation}
L_{2}^{+}\left(S^{1},\omega^{1/2}\right)\oplus L_{2}^{-}\left(S^{1},\omega^{1/2}\right)\subset L_{2}\left(S^{1},\omega^{1/2}\right)\oplus L_{2}\left(S^{1},\omega^{1/2}\right).
\notag\end{equation}
\begin{lemma} If $ \operatorname{ind}\psi=0 $, then $ p $ is invertible. \end{lemma}
\begin{proof} Consider a linear bundle $ {\cal L} $ over $ {\Bbb C}P^{1} $ with isomorphisms to
$ \omega^{1/2} $ over $ U^{+}=\left\{|z|<1+\varepsilon\right\} $ and $ U^{-}=\left\{|z|>1-\varepsilon\right\} $, and the gluing data being $ l^{+}=\psi l^{-} $.
Since $ \deg {\cal L}=\deg \omega^{1/2}+\operatorname{ind}\psi $, and a linear bundle over $ {\Bbb C}P^{1} $ is determined by
its degree up to an isomorphism, we see that $ {\cal L}\simeq\omega^{1/2} $ if $ \operatorname{ind}\psi=0 $.
Since $ \operatorname{Ker} p $ consists of global sections of $ {\cal L} $, $ \operatorname{Ker} p=\left\{0\right\} $. Similarly,
consideration of orthogonal complement to $ V_{\psi} $ shows that $ \operatorname{Coker} p=\left\{0\right\} $. This
finishes the proof, since the operator is obviously Fredholm. \end{proof}
Consider now another linear bundle $ {\cal L}' $ with trivializations over $ U^{+} $
and over $ U^{-} $ with gluing data $ l^{+}=\psi l^{-} $. The same arguments as above show
that $ {\cal L}' $ is trivial, thus it has a (unique up to multiplication by a
constant) global section. This means that there are functions $ l^{\pm} $ defined
on $ U^{\pm} $ such that $ l^{+}=\psi l^{-} $. Since this global section has no zeros, $ l^{\pm} $ have
no zeros inside the domain of definition, thus $ \psi=\left(l^{+}\right)^{-1}l^{-} $. We see that
any function $ \psi $ such that $ \operatorname{ind}\psi=0 $ can be represented as a product $ \psi=\psi_{+}\psi_{-} $ of
a parts $ \psi_{+} $, $ \psi_{-} $ which can be holomorphically extended inside/outside a
circle without zeros.
Let as write the mapping $ p^{-1} $ in terms of $ \psi_{\pm} $. For any $ L_{2} $-section $ \omega $ of
$ \Omega^{1/2}\otimes\mu $ on $ \left\{|z|=1\right\} $ let $ \omega=\omega_{+}+\omega_{-} $ be the (unique) decomposition of $ \omega $ into a
sum of forms which can holomorphically extended inside/outside of the
unit circle. Given $ \omega_{+} $ and $ \omega_{-}' $ we want to find $ \omega_{-} $ and $ \omega'_{+} $ from the
equality
\begin{equation}
\psi\left(\omega_{+}+\omega_{-}\right) = \omega'_{+}+\omega'_{-},\qquad \text{or\qquad }\psi_{-}\left(\omega_{+}+\omega_{-}\right) = \psi_{+}^{-1}\left(\omega'_{+}+\omega'_{-}\right).
\notag\end{equation}
Taking the $ + $-part we see that $ \left(\psi_{-}\omega_{+}\right)_{+} = \psi_{+}^{-1}\omega'_{+}+\left(\psi_{+}^{-1}\omega'_{-}\right)_{+} $, thus
\begin{equation}
\omega'_{+} = \psi_{+}\left(\psi_{-}\omega_{+}\right)_{+} - \psi_{+}\left(\psi_{+}^{-1}\omega'_{-}\right)_{+} = \psi\omega_{+} - \psi_{+}\left(\psi_{-}\omega_{+}\right)_{-} - \psi_{+}\left(\psi_{+}^{-1}\omega'_{-}\right)_{+},
\notag\end{equation}
similarly $ \left(\psi_{-}\omega_{+}\right)_{-}+\psi_{-}\omega_{-} = \left(\psi_{+}^{-1}\omega'_{-}\right)_{-} $, thus
\begin{equation}
\omega_{-} = \psi_{-}^{-1}\left(\psi_{+}^{-1}\omega'_{-}\right)_{-}-\psi_{-}^{-1}\left(\psi_{-}\omega_{+}\right)_{-} = \psi^{-1}\omega'_{-} - \psi_{-}^{-1}\left(\psi_{+}^{-1}\omega'_{-}\right)_{+} - \psi_{-}^{-1}\left(\psi_{-}\omega_{+}\right)_{-}.
\notag\end{equation}
These two formulae express $ \omega_{-} $ and $ \omega'_{+} $ in terms of $ \omega'_{-} $ and $ \omega_{+} $, thus give
an inverse mapping to $ p $.
Let $ |\psi|_{++}=\frac{\max |\psi_{+}|}{\min |\psi_{+}|} $, $ |\psi|_{--}=\frac{\max |\psi_{-}|}{\min |\psi_{-}|} $, $ |\psi|_{+-}=\max
|\psi_{+}| \max |\psi_{-}| $, $ |\psi|_{-+}=\max |\psi_{+}|^{-1} \max |\psi_{-}|^{-1} $,
$ |\psi|_{0}=\max \left(|\psi|_{++},|\psi|_{--},|\psi|_{+-},|\psi|_{-+}\right) $. Then the norm of $ p^{-1} $ is bounded by
$ C\cdot|\psi|_{0} $.
\begin{definition} \label{def5.155}\myLabel{def5.155}\relax The {\em Riemann norm\/} $ \|\psi\|_{{\bold R}} $ of $ \psi $ is the norm of the
operator $ p^{-1} $. \end{definition}
The following lemma is a corollary of the fact that one can find
factorization $ \psi=\psi_{+}\psi_{-} $ using integral operators applied to $ \log \psi $:
\begin{lemma} \label{lm5.160}\myLabel{lm5.160}\relax Let $ \log \psi\left(z\right) $ is defined using any branch of logarithm.
Then
\begin{equation}
\|\psi\|_{{\bold R}}< C \exp C \max _{z\in U} |\log \psi\left(z\right)|
\notag\end{equation}
for an appropriate $ C $ (which depends on $ \varepsilon $ only). \end{lemma}
Let $ \Pi_{\psi} $ be the composition of $ p^{-1} $ with the projection of
$ L_{2}\left(S^{1},\omega^{1/2}\right)\oplus L_{2}\left(S^{1},\omega^{1/2}\right) $ to $ L_{2}^{-}\left(S^{1},\omega^{1/2}\right)\oplus L_{2}^{+}\left(S^{1},\omega^{1/2}\right) $,
\begin{equation}
\Pi_{\psi}\colon L_{2}^{+}\left(S^{1},\omega^{1/2}\right)\oplus L_{2}^{-}\left(S^{1},\omega^{1/2}\right) \to L_{2}^{-}\left(S^{1},\omega^{1/2}\right)\oplus L_{2}^{+}\left(S^{1},\omega^{1/2}\right).
\notag\end{equation}
In other words, $ \left(\left(f_{+},g_{+}\right),\left(f_{-},g_{-}\right)\right) $ lies on the graph of $ \Pi_{\psi} $ if
$ g_{+}+g_{-}=\psi\left(f_{+}+f_{-}\right) $.
We see that the norm of $ \Pi_{\psi} $ is bounded by $ C\cdot|\psi|_{0} $. Moreover, $ \Pi_{\psi} $ can
be written as a sum
\begin{equation}
\Pi_{\psi} = \left(
\begin{matrix}
0 & \psi^{-1} \\ \psi & 0
\end{matrix}
\right) + k.
\label{equ5.52}\end{equation}\myLabel{equ5.52,}\relax
Obviously, $ k $ is compact (as any Hankel operator). Indeed, components of $ k $
look like $ f_{+} \mapsto \left(mf_{+}\right)_{-} $. The Hilbert operator $ f=f_{+}+f_{-} \buildrel{H}\over{\mapsto} f_{+}-f_{-} $
is a pseudodifferential operator of degree 0, thus components of $ K $ may
be written as $ \left[m,H\right] $, thus are pseudodifferential operators of degree $ -1 $,
thus compact.
\begin{lemma} \label{lm5.60}\myLabel{lm5.60}\relax Consider an invertible function $ \psi $ defined in a
neighborhood of $ |z|=1 $ and having $ \operatorname{ind}=k $. Let $ V_{\psi}\subset H=L_{2}\left(S^{1}\right)\oplus L_{2}\left(S^{1}\right) $ be
\begin{equation}
V_{\psi}=\left\{\left(f_{1},f_{2}\right) \mid f_{2}=\psi f_{1}\right\}.
\notag\end{equation}
Define $ H^{\pm}=L_{2}\left(S^{1}\right)^{\pm}\oplus L_{2}\left(S^{1}\right)^{\mp} $. Then for an appropriate bounded operator $ \pi\colon H^{+}
\to H^{-} $ the $ \operatorname{graph}\left(\pi\right) $ is compatible with $ V_{\psi} $, and $ \operatorname{reldim}\left(H_{2},\operatorname{graph}\left(\pi\right)\right)=k $. \end{lemma}
\begin{proof} Let $ \psi\left(z\right)=z^{k}\psi_{0}\left(z\right) $. The function $ \psi_{0} $ has $ \operatorname{ind}=0 $, thus the
corresponding subspace $ V_{\psi_{0}} $ is the graph of $ \Pi_{\psi_{0}} $. Let
$ H_{0}^{\pm}=L_{2}\left(S^{1}\right)^{\pm}\oplus z^{k}L_{2}\left(S^{1}\right)^{\mp} $. We see that $ V_{\psi} $ is a graph of a bounded mapping $ H_{0}^{+}
\to H_{0}^{-} $.
Since $ H_{0}^{+} $ is compatible with $ H_{0} $ of relative dimension $ k $, we momentarily
obtain the required statement about $ V_{\psi} $. \end{proof}
\begin{remark} Note that $ \|\pi\| $ may be bounded in the same way as in Lemma
~\ref{lm5.160}, but neither this result, nor~\eqref{equ5.52} are going to be needed in
what follows. \end{remark}
\subsection{Finiteness theorem }\label{s7.30}\myLabel{s7.30}\relax Consider a family of disks and
gluing conditions $ K_{\bullet} $, $ \varphi_{\bullet} $, $ \psi_{\bullet} $ from Section~\ref{s5.30}. Let
$ {\cal H}^{\pm}=\bigoplus_{l_{2}}L_{2}^{\pm}\left(\partial K_{i},\Omega^{1/2}\otimes\mu\right) $, $ i\in I $. Then the operator with Cauchy kernel defines
a Hilbert mapping $ {\bold K}\colon {\cal H}^{-} \to {\cal H}^{+} $ (see Section~\ref{s5.20}). This mapping depends
on the circles $ \partial K_{i} $ only, not on the gluing conditions $ \varphi_{\bullet} $, $ \psi_{\bullet} $.
Since the Hilbert structure on $ L_{2}\left(\partial K_{i},\Omega^{1/2}\otimes\mu\right) $ is invariant w.r.t.
fraction-linear mappings (as is decomposition into $ \pm $-parts), the
identification $ \varphi_{i}\colon \partial K_{i'} \to \partial K_{i} $ gives an isomorphism of Hilbert spaces
$ L_{2}\left(\partial K_{i},\Omega^{1/2}\otimes\mu\right) $ and $ L_{2}\left(\partial K_{i'},\Omega^{1/2}\otimes\mu\right) $ which interchanges $ + $-part and $ - $-part.
Suppose that $ \operatorname{ind}\psi_{i}=0 $. Then the operator $ \Pi_{\psi_{i}} $ from Section~\ref{s5.5}
together with an identification given by $ \varphi_{i} $ gives a mapping $ \Pi_{\varphi_{i},\psi_{i}} $
\begin{equation}
L_{2}^{+}\left(\partial K_{i},\Omega^{1/2}\otimes\mu\right)\oplus L_{2}^{+}\left(\partial K_{i'},\Omega^{1/2}\otimes\mu\right) \to L_{2}^{-}\left(\partial K_{i},\Omega^{1/2}\otimes\mu\right)\oplus L_{2}^{-}\left(\partial K_{i'},\Omega^{1/2}\otimes\mu\right),
\notag\end{equation}
the graph of this mapping consists of half-forms on $ \partial K_{i} $ and
$ \partial K_{i'} $ which differ by multiplication by $ \psi_{i} $. In other words, $ \left(\left(f_{+},g_{+}\right),\left(f_{-},g_{-}\right)\right) $
lies on this graph if
\begin{equation}
\left(g_{+}+g_{-}\right)\left(\varphi_{i}^{-1}\left(z\right)\right)=\psi_{i}\left(z\right)\left(f_{+}+f_{-}\right)\left(z\right),\qquad z\in\partial K_{i}.
\notag\end{equation}
If $ \operatorname{ind}\psi_{i}\not=0 $, instead of $ \Pi_{\varphi_{i},\psi_{i}} $ consider an arbitrary operator
between the same spaces such that the graph of $ P $ is comparable with the
set of half-forms which differ by multiplication by $ \psi_{i} $ (see Lemma
~\ref{lm5.60}). The
only fact important in what follows is that this is a bounded operator
(see Corollary~\ref{cor4.35}).
Let $ \Pi=\Pi_{\left\{\psi\right\}}=\bigoplus_{i\in I_{+}}\Pi_{\varphi_{i},\psi_{i}}\colon {\cal H}^{+} \to {\cal H}^{-} $. Similarly, let $ \Pi_{\left\{\psi^{-1}\right\}} $ corresponds
to gluing data $ \varphi_{i} $, $ \psi_{i}^{-1} $. Since $ \Pi $ consists of bounded diagonal blocks, it
is a closed operator.
Let $ \bar{M} $ be a curve determined by gluing conditions $ \varphi_{\bullet} $, $ {\cal L} $ be a bundle
on $ \bar{M} $ determined by gluing conditions $ \psi_{\bullet} $.
\begin{theorem} Suppose that the disks $ K_{i} $ are well separated, $ {\cal L} $ is a
finite-degree bundle, and $ \Pi_{\left\{\psi\right\}}\circ{\bold K} $ is compact. Then both
$ \Gamma_{\text{strong}}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) $ and $ \Gamma_{\text{weak}}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) $ are finite-dimensional. \end{theorem}
\begin{proof} We give only a sketch of a proof, since the details are the
same as in the case of Riemann--Roch theorem (see Section~\ref{s5.60}). Since
$ \Gamma_{\text{strong}} $ is identified with a subspace of $ \Gamma_{\text{weak}} $, it is enough to show that
$ \dim \Gamma_{\text{weak}}<\infty $. On the other hand, $ \Gamma_{\text{weak}} $ is defined as an intersection of the
generalized Hardy space $ {\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $ with the subspace of forms which
satisfy the
gluing conditions. We are going to reduce the statement of the theorem
to the abstract finiteness theorem.
To do this, note that Proposition~\ref{prop5.28} identifies $ {\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $
with the graph of $ {\bold K}\colon {\cal H}^{-} \to {\cal H}^{+} $, thus the only thing to note is that fact
that the subspace of forms which satisfy the gluing conditions is
compatible with the graph of $ \Pi_{\left\{\psi\right\}} $. \end{proof}
\subsection{Duality }\label{s7.40}\myLabel{s7.40}\relax Consider the curve $ \bar{M} $ and a sheaf $ {\cal L} $ from the previous
section and the dual sheaf $ {\cal L}^{-1} $ with inverse gluing conditions $ \left\{\psi_{i}^{-1}\right\} $.
\begin{theorem} \label{th5.70}\myLabel{th5.70}\relax Suppose that the disks $ K_{i} $ are well-separated, and $ {\cal L} $
is of finite degree. The mappings
\begin{equation}
H^{1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) \xrightarrow[]{\bar{\partial}} H^{-1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\otimes\bar{\omega}\right)
\notag\end{equation}
and
\begin{equation}
H^{1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}^{-1}\right) \xrightarrow[]{-\bar{\partial}} H^{-1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}^{-1}\otimes\bar{\omega}\right)
\notag\end{equation}
are mutually dual, thus dimension of null-space of one mapping is equal
to the dimension of cokernel of another one. \end{theorem}
\begin{proof} This is a direct corollary of Lemmas~\ref{lm5.22} and~\ref{lm5.25}. \end{proof}
Using Theorems~\ref{th5.70} and~\ref{th5.31} together with selfduality of $ {\bold K} $
and the fact that $ \Pi_{\psi}^{t}=\Pi_{\psi^{-1}} $, we obtain
\begin{corollary} Suppose that the disks $ K_{i} $ are well separated, and $ {\cal L} $ is
finite-degree and semibounded, and both $ \Pi_{\left\{\psi\right\}}\circ{\bold K} $ and $ \Pi_{\left\{\psi^{-1}\right\}}\circ{\bold K} $ are
compact. Then the bounded operator
\begin{equation}
H_{\text{weak}}^{1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) \xrightarrow[]{\bar{\partial}} H_{\text{weak}}^{-1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\otimes\bar{\omega}\right)
\notag\end{equation}
has finite-dimensional null-space and cokernel. \end{corollary}
\subsection{Riemann--Roch for curves }\label{s5.60}\myLabel{s5.60}\relax Now we are ready to state
\begin{theorem}[Riemann--Roch] \label{th5.15}\myLabel{th5.15}\relax Suppose that the disks $ K_{i} $ are
well-separated, and $ \psi_{i} $ is a finite-degree family.
\begin{enumerate}
\item
If both $ \Pi_{\left\{\psi\right\}}\circ{\bold K} $ and $ \Pi_{\left\{\psi^{-1}\right\}}\circ{\bold K} $ are compact operators then the mapping
\begin{equation}
H_{\text{weak}}^{1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) \xrightarrow[]{\bar{\partial}} H_{\text{weak}}^{-1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\otimes\bar{\omega}\right)
\notag\end{equation}
has finite-dimensional null-space and cokernel, and
\begin{equation}
\dim \operatorname{Ker} \bar{\partial} - \dim \operatorname{Coker} \bar{\partial} = \sum_{i\in I_{+}} \operatorname{ind} \psi_{i}.
\notag\end{equation}
\item
If $ \Pi_{\left\{\psi\right\}} $ is bounded, and $ {\bold K} $ is compact, then the mapping is Fredholm
of index $ \sum_{i\in I_{+}} \operatorname{ind} \psi_{i} $.
\end{enumerate}
\end{theorem}
\begin{proof} The only thing we need to do is to describe the null-space and
the image of $ \bar{\partial} $. We are going to reduce this description to the abstract
Riemann--Roch theorem. First of all, the null-space consists of
half-forms in the generalized Hardy space, so $ \widetilde{r} $ maps it injectively to a
subspace of $ \bigoplus_{l_{2}}L_{2}\left(\partial K_{i},\Omega^{1/2}\otimes\mu\right) $. On the other hand, the image of $ \operatorname{Ker}\bar{\partial} $ in
$ \bigoplus_{l_{2}}L_{2}\left(\partial K_{i},\Omega^{1/2}\otimes\mu\right) $ is described as intersection of the image of the
generalized Hardy space and a subspace in $ \bigoplus_{l_{2}}L_{2}\left(\partial K_{i},\Omega^{1/2}\otimes\mu\right) $ consisting
of half-forms which satisfy the gluing conditions.
The first condition can be written as
\begin{equation}
f_{i}^{+}=\sum_{j\not=i}{\bold K}_{ij}f_{j}^{-},
\notag\end{equation}
here $ f_{i}^{\pm} $ are $ \pm $-parts of the component of $ f $ in $ L_{2}\left(\partial K_{i},\Omega^{1/2}\otimes\mu\right) $. The second
condition is
\begin{equation}
f'_{i'}=\psi_{i}f_{i}.
\notag\end{equation}
Here $ f\left(t\right)'=f\left(-t\right) $, and we suppose that we use compatible parameterizations
of $ \partial K_{i} $ and of $ \partial K_{i'} $, i.e., such that $ \varphi_{i} $ send parameter $ t $ on $ \partial K_{i} $ to
parameter $ -t $ on $ \partial K_{i'} $.
Consider the decomposition $ {\cal H}={\cal H}^{+}\oplus{\cal H}^{-} $. Vectors which satisfy the
first condition are in the graph of operator $ {\bold K}\colon {\cal H}^{-} \to {\cal H}^{+} $. Consider the
vector space of vectors $ {\cal H}_{2} $ which satisfy the second conditions. It is a
direct sum over $ i\in I_{+} $ of subspaces of
\begin{equation}
L_{2}\left(\partial K_{i},\Omega^{1/2}\otimes\mu\right)\oplus L_{2}\left(\partial K_{i'},\Omega^{1/2}\otimes\mu\right),
\notag\end{equation}
and all the components but a finite number have $ \psi $ with $ \operatorname{ind}=0 $, thus are
described as graphs of $ \Pi_{\varphi_{i},\psi_{i}} $. As a corollary, we conclude that this
subspace is compatible with the graph of the mapping $ \Pi $.
To finish the description of the null-space the only thing which remains
to prove is to show that the relative dimension of $ {\cal H}_{2} $ and $ \operatorname{graph}\left(\Pi\right) $ is
$ \sum_{i\in I_{+}}\operatorname{ind}\psi_{i} $. However, because of decomposition of $ {\cal H}_{2} $ into a direct sum it
follows from the corresponding fact for each component, i.e., from
Lemma~\ref{lm5.60}.
To describe the image, consider
\begin{equation}
\alpha\in H_{\text{weak}}^{-1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\otimes\bar{\omega}\right) = H^{-1/2}\left({\Bbb C}P^{1}\smallsetminus\bigcup K_{i},\omega^{1/2}\otimes\bar{\omega}\right).
\notag\end{equation}
Since $ \bar{\partial} $ is an isomorphism
\begin{equation}
H^{1/2}\left({\Bbb C}P^{1},\omega^{1/2}\right) \to H^{-1/2}\left({\Bbb C}P^{1},\omega^{1/2}\otimes\bar{\omega}\right),
\notag\end{equation}
the element $ \bar{\partial}^{-1}\alpha\in H^{1/2}\left({\Bbb C}P^{1},\omega^{1/2}\right) $ is defined up to addition of an
element of $ \bar{\partial}^{-1} \overset{\,\,{}_\circ}{H}^{-1/2}\left(\bigcup K_{i},\omega^{1/2}\otimes\bar{\omega}\right) $. On the other hand, the latter space is
$ {\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $, i.e., $ \bar{\partial}^{-1} $ gives a correctly defined isomorphism
\begin{equation}
H^{-1/2}\left({\Bbb C}P^{1}\smallsetminus\bigcup K_{i},\omega^{1/2}\otimes\bar{\omega}\right) \to H^{1/2}\left({\Bbb C}P^{1},\omega^{1/2}\right)/{\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right),
\notag\end{equation}
hence $ \bar{\partial}^{-1}\alpha $ is an element of the latter space. Thus
\begin{equation}
\alpha\in\operatorname{Im}\left(H_{\text{weak}}^{1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) \xrightarrow[]{\bar{\partial}} H_{\text{weak}}^{-1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\otimes\bar{\omega}\right)\right)
\notag\end{equation}
is equivalent to
\begin{equation}
\partial^{-1}\alpha\in {\cal H}_{2}/{\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) \buildrel{\text{def}}\over{=}\left({\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right)+{\cal H}_{2}\right)/{\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right)\text{ .}
\notag\end{equation}
In particular, if $ {\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right)+{\cal H}_{2} $ is closed, then $ \operatorname{Im} \bar{\partial} $ is closed, and in
any case the codimension of the closure of $ \operatorname{Im}\bar{\partial} $ is equal to codimension of
the closure of $ {\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right)\oplus{\cal H}_{2} $. \end{proof}
From Lemma~\ref{lm5.160} we momentarily obtain
\begin{corollary} Suppose that $ {\bold K} $ is compact, and $ \psi_{i}=\exp \Phi_{i} $ for all $ i $ but a
finite number. If for an appropriate $ C $ and any $ i $
\begin{equation}
|\Phi_{i}| < C,
\notag\end{equation}
then the operator
\begin{equation}
H_{\text{weak}}^{1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\right) \xrightarrow[]{\bar{\partial}} H_{\text{weak}}^{-1/2}\left(\bar{M},\omega^{1/2}\otimes{\cal L}\otimes\bar{\omega}\right)
\notag\end{equation}
is Fredholm. \end{corollary}
\begin{definition} Call a pair $ \left(\bar{M},{\cal L}\right) $ {\em admissible\/} if $ \bar{M} $ is given by a family of
well-separated disks, $ {\cal L} $ is of finite degree, and both $ \Pi_{\left\{\psi\right\}}\circ{\bold K} $ and $ \Pi_{\left\{\psi^{-1}\right\}}\circ{\bold K} $
are compact. Call $ \bar{M} $ {\em admissible\/} if $ \left(\bar{M},\boldsymbol1\right) $ is admissible. Here {\bf1 }is a bundle
over $ \bar{M} $ with $ \psi_{i}\equiv 1 $. \end{definition}
\subsection{Criterion of admissibility of a curve }\label{s4.50}\myLabel{s4.50}\relax Recall that in Section
~\ref{s5.10} we considered a matrix $ \left(e^{-l_{ij}/2}\right) $ which was supposed to give a
bounded operator in $ l_{2} $.
\begin{proposition} If the matrix $ \left(e^{-l_{ij}/2}-\delta_{ij}\right) $ gives a compact operator $ l_{2} \to
l_{2} $, then $ \bar{M} $ is admissible. \end{proposition}
\begin{proof} Indeed, for the bundle $ {\cal L}=\boldsymbol1 $ the operator $ \Pi $ is an isometry, thus
we need to show that $ {\bold K} $ is compact, which is a corollary of Lemma~\ref{lm5.05}. \end{proof}
\begin{corollary} If $ \sum_{i\not=j}e^{-l_{ij}}<\infty $, then $ \bar{M} $ is admissible. \end{corollary}
\begin{corollary} Fix a locally discrete subset $ I\subset{\Bbb C}P^{1} $ (i.e., for any point $ i $
of $ I $ there is a punctured neighborhood of $ i $ which does not intersect
$ I $). Then there exists a family of disks $ K_{i} $, $ i\in I $ with $ \operatorname{center}\left(K_{i}\right)=i $ such
that for any involution ' and any gluing data $ \varphi_{i} $ the corresponding curve $ \bar{M} $
is admissible. Moreover, one can chose $ K_{i} $ in such a way that $ M_{\infty} $ consists
of points of smoothness $ C^{\infty} $ (see Section~\ref{s4.95}). \end{corollary}
\begin{corollary} For an arbitrary nowhere dense subset $ N $ of $ {\Bbb C}P^{1} $ there exists
an admissible curve $ \bar{M} $ such that $ M_{\infty}=N $. Moreover, it is possible to make
every point of $ N $ to be of smoothness $ C^{\infty} $. \end{corollary}
\begin{remark} While the above statements are obvious, note that construction
of examples and counterexamples may be simplified a lot by an additional
restriction:
\begin{equation}
\operatorname{dist}\left(i,j\right) \geq \varepsilon\cdot\operatorname{dist}\left(i,N\right)\qquad \text{for any }i,j\in I\text{, }i\not=j.
\notag\end{equation}
Here $ \varepsilon\ll1 $. To construct such a family $ I $ for a given nowhere dense set $ N $,
let $ N_{k}=\left\{z \mid 2^{-k-1}\leq\operatorname{dist}\left(z,N\right)\leq2^{-k}\right\} $. Fix $ k $, and let $ \delta=2^{-k} $. Consider a $ \delta/8 $-net
for $ N_{k} $.
By removing some points from this net one can obtain $ \delta/4 $-net such
that it does not have two points closer than $ \delta/8 $. This net is necessarily
finite.
Now consider the union of these finite sets over $ k\in{\Bbb N} $, and again
remove net points from $ N_{k+1} $ which are closer than $ \delta/8 $ to net points
in $ N_{k} $. One obviously obtains a set $ I $ with required properties.
Now to chose the radius of $ K_{i} $ denote by $ n_{k} $ the number of points in
$ I\cap N_{k} $. Let $ \operatorname{radius}\left(K_{i}\right)=\frac{f\left(k\right)}{n_{k}} $ if $ i\in N_{k} $, here $ f\left(k\right) $ is rapidly decreasing
function. Picking appropriate $ f\left(k\right) $, one obtains disks which satisfy the
given above requirements. \end{remark}
The following property of admissibility is obvious:
\begin{nwthrmiii} If we change a finite number of contours $ \partial K_{i} $ and/or a finite
numbers of identifications $ \varphi_{i} $, this does not change the admissibility of
the resulting curve. \end{nwthrmiii}
\subsection{Filling the gap }\label{s7.70}\myLabel{s7.70}\relax This section contains heuristics only, so
anyone
interested exceptionally in exact results should proceed directly to
Section~\ref{h8}.
Consider once more the criterion of admissibility of a curve. It
says that if the operator $ {\bold K} $ is compact, the curve is admissible (together
with any bundle which is defined by gluing functions $ \psi_{i} $ with bounded
Riemann norm $ |\psi|_{{\bold R}} $). On the other hand, to obtain this result we study the
generalized Hardy space, which is correctly defined if $ {\bold K} $ is bounded.
Thus the tools we use leave a gap between the objects for which the
analysis is applicable (i.e., $ {\bold K} $ is bounded), and objects for which we get
the admissibility (i.e., $ {\bold K} $ is compact). How can we use the existence of
this gap?
We propose to consider this gap as a confirmation that the
Riemann--Roch theorem we obtained is {\em almost unimprovable}. Indeed, we know
that the finiteness condition can be written as $ \dim \operatorname{Ker}\left(\Pi\circ{\bold K}-1\right)<\infty $. On the
other hand, the invertible operator $ \Pi $ depends on the family $ \left(\psi_{i}\right) $ which
has very high degree of freedom (even if we consider the strong form of
Riemann--Roch theorem, so $ \Pi $ is required to be bounded), and one should
expect that the condition
\begin{equation}
\dim \operatorname{Ker}\left(\Pi\circ{\bold K}-1\right)<\infty\text{ for every choice of }\psi_{i}\text{ with }|\psi_{\bullet}|_{{\bold R}}<\infty
\notag\end{equation}
should be very close to the condition
\begin{equation}
{\bold K}\text{ has discrete spectrum near }|\lambda|=1.
\label{equ4.88}\end{equation}\myLabel{equ4.88,}\relax
In other words, the pairs $ \left(\Pi,{\bold K}\right) $ with an invertible $ \Pi $, bounded $ {\bold K} $, and
{\em infinite-dimensional\/} null-space of $ \Pi{\bold K}-1 $ are plentiful (at least if we
drop ``geometric'' conditions on $ \Pi $ and $ {\bold K} $, and consider abstract operators),
and they form a ``natural boundary'' of the set of curves for which
Riemann--Roch theorem has a chance to be true. This natural boundary is
quite close to the boundary of the set of compact operators, which is
another confirmation of our thesis.
Note that~\eqref{equ4.88} may lead to a different description of possible
$ {\bold K} $, like $ \frac{{\bold K}}{1+{\bold K}^{2}} $ being compact. It is unclear, however, whether the above
boundary separates the set of compact operators as a connected component
of the set of operators $ {\bold K} $ with compact $ \frac{{\bold K}}{1+{\bold K}^{2}} $.
\subsection{Moduli space } The description of a complex curve by disks $ \left\{K_{i}\right\} $,
involution ' and gluings $ \varphi_{i} $ leaves a feeling of being incomplete, since
in the case of finite genus it is enough to provide just gluings $ \varphi_{i} $ which
generate a subgroup of $ \operatorname{SL}\left(2,{\Bbb C}\right) $. To describe the quotient by this subgroup
one can take any fundamental domain for the subgroup, and different
choices of the fundamental domain result in the same geometric data.
To get a similar description in the case of infinite genus, note
that in Section~\ref{s5.10} instead of the restriction that $ \widetilde{K}_{i} $ {\em is\/} a concentric
with $ K_{i} $ disk of radius $ e^{2\varepsilon}\cdot\operatorname{radius}\left(K_{i}\right) $ one can require that $ \widetilde{K}_{i} $ {\em contains\/}
such a circle. This leads to the construction of strong sections of
$ \omega^{1/2}\otimes{\cal L} $ in the same way as in Section~\ref{s6.50}, and Theorem~\ref{th5.31} can be
refined as
\begin{amplification} Fix a metric on $ {\Bbb C}P^{1} $ and $ \varepsilon>0 $. Consider a family of
elements $ \varphi_{i} $ of $ \operatorname{SL}\left(2,{\Bbb C}\right) $. Suppose that there exists a family of disjoint
domains $ \widetilde{K}_{i}\subset{\Bbb C}P^{1} $, $ i\in I $, which satisfy the following properties:
\begin{enumerate}
\item
All $ \widetilde{K}_{i} $ but a finite number are disks;
\item
Let $ K'_{i} $ be a concentric with $ \widetilde{K}_{i} $ disk of radius $ \left(1-\varepsilon\right)\operatorname{radius}\left(\widetilde{K}_{i}\right) $ (or
any domain in $ \widetilde{K}_{i} $ if $ \widetilde{K}_{i} $ is not a disk). Suppose that for an involution ':
$ I\to $I one has $ \varphi_{i'}=\varphi_{i}^{-1} $, $ \varphi_{i}\left(K'_{i}\right)\cup K'_{i'}={\Bbb C}P^{1} $;
\item
Let $ \psi_{i} $ be holomorphic functions defined inside $ \widetilde{K}_{i}\cap\varphi_{i}^{-1}\left(\widetilde{K}_{i'}\right) $ which
satisfy $ \psi_{i'}\circ\varphi_{i}=\psi_{i}^{-1} $, and $ \operatorname{ind}\psi_{i}=0 $ for all but a finite number of $ i\in I $.
\item
Suppose that the pairwise conformal distances $ l_{ij} $ between $ \widetilde{K}_{i} $, $ \widetilde{K}_{j} $
satisfy the condition that the matrix $ \left(e^{-l_{ij}/2}-\delta_{ij}\right) $ gives a compact
mapping $ l_{2} \to l_{2} $, and all the functions $ \psi_{i} $ are bounded taken together.
\end{enumerate}
Let $ \bar{M} $ be a curve given by gluing $ S=\bigcap\varphi_{i}^{-1}\left(K'_{i}\right) $ together via $ \varphi_{i} $, $ {\cal L} $ be
a bundle on $ \bar{M} $ given by cocycle $ \psi_{i} $. Define strong sections of $ \omega^{1/2}\otimes{\cal L} $
associated with family $ \left\{\widetilde{K}_{i}\right\} $ as forms $ \alpha $ in $ H^{1/2}\left(\bigcap\varphi_{i}^{-1}\left(K'_{i}\right),\omega^{1/2}\right) $ which
satisfy $ \varphi_{i}^{*}\left(\alpha\right)=\psi_{i}\cdot\alpha $ whenever both sides have sense.
Let $ I=I_{+}\amalg I_{+}' $. For any choice of circles $ \gamma_{i} $ in $ K_{i}'\cap\varphi_{i}^{-1}\left(K'_{i'}\right) $, $ i\in I_{+} $, let
$ K_{i} $ be the disk bounded by $ \gamma_{i} $ inside $ K'_{i} $ (with appropriate modifications
if $ K_{i} $ is not a disk). Let $ K_{i'}={\Bbb C}P^{1}\smallsetminus\varphi_{i}\left(K_{i}\right) $, $ i\in I_{+} $. Then the space of strong
sections of $ \omega^{1/2}\otimes{\cal L} $ coincides with the set of weak sections associated to
the family $ \left\{K_{i}\right\} $.
In particular, the Riemann--Roch theorem (in the strong form) is
valid for strong sections associated to the family $ \left\{\widetilde{K}_{i}\right\} $. \end{amplification}
The proof of this statement is a corollary of the proof of
Theorem~\ref{th5.31}.
We see that at least for bundles described by bounded cocycles one
does not need to specify {\em precisely\/} the circles which cut the curve, one
can vary them in wide ranges (which depend on the Kleinian group) without
any change to the geometric data. One possible objection to usability of
the above theorem is that for different choices $ \left\{\gamma_{i}^{\left(1\right)}\right\} $, $ \left\{\gamma_{i}^{\left(2\right)}\right\} $ of the
circles the cocycles $ \psi_{i} $ which describe the bundle need to be defined
everywhere between the circles for the theorem to be applicable. However,
in Section~\ref{s7.90} we will see that any bounded bundle of degree 0 may be
described by constant cocycles, thus this objection becomes void. (If the
degree is not 0, one can take all the cocycles to be constant except one
pair described by rational functions.)
Call a Kleinian group {\em admissible\/} if it has generators $ \varphi_{i} $, $ i\in I_{+} $, which
satisfy the conditions of the amplification. The following conjecture
would show that the curve (with a fixed family of $ A $-cycles) is
completely described by the corresponding Kleinian group, and the
restriction on distances between $ \widetilde{K}_{i} $ is in fact the restriction on the
Kleinian group.
Consider an admissible Kleinian group. A {\em fundamental family\/} is
a family of domains $ \widetilde{K}_{\bullet} $ which satisfies the conditions of amplification. If
$ \widetilde{K}_{i}^{\left(1\right)}\subset\widetilde{K}_{i}^{\left(2\right)} $, $ i\in I $, then call these families {\em equivalent}, and continue this
relation by transitivity. The amplification shows that equivalent
families lead to the same spaces of sections of bundles.
\begin{conjecture} Any two fundamental families for an admissible
Kleinian group are equivalent. \end{conjecture}
Let the {\em fine moduli space\/} be the set of admissible Kleinian groups
up to conjugation in $ \operatorname{PGL}\left(2,{\Bbb C}\right) $.
\begin{conjecture} Consider a complex curve $ M $. Let $ \gamma_{i}^{\left(1\right)} $, $ i\in I^{\left(1\right)} $, be a
disjoint family of smooth embedded cycles in $ M $ such that $ M\smallsetminus\bigcup\gamma_{i} $ is
conformally equivalent to a fundamental domain of an admissible Kleinian
group. Let $ \gamma_{i}^{\left(2\right)} $ be a different family which satisfies same conditions.
Then all the cycles $ \gamma_{i}^{\left(1\right)} $ except a finite number are homotopic to cycles
in $ \gamma_{i}^{\left(2\right)} $. \end{conjecture}
\begin{conjecture} Two families of cuts from the previous conjecture lead to
the same set of $ C^{n} $-points at infinity (see Section~\ref{s4.95}) for any $ n\geq0 $. \end{conjecture}
\begin{definition} Suppose that families $ \left\{K_{i}^{\left(1\right)},',\varphi_{i}^{\left(1\right)}\right\} $ and $ \left\{K_{i}^{\left(2\right)},',\varphi_{i}^{\left(2\right)}\right\} $
have all the accumulation points of class $ C^{0} $. Say that these families
{\em describe the same curve\/} if the set of finite points of the corresponding
curves are equivalent as complex manifolds, and the equivalence continues
by continuity to the points at infinity. Let the {\em moduli space\/} be the set
of equivalence classes of such families up to relationship that they
describe the same curve. \end{definition}
Note that it is not reasonable to drop the consideration of points
at infinity.
\begin{example} Indeed, consider a curve $ \bar{M} $ with the Serpinsky carpet as the
set $ M_{\infty} $ of accumulation points of disks. The Serpinsky carpet breaks $ {\Bbb C}P^{1} $
into a union of triangles. Suppose that each triangle contains exactly
one disk $ K_{i} $. Then the smooth points on $ M $ form a disjoint union of tubes,
so the only invariant is the conformal lengths of these tubes. This is
one parameter per handle, much smaller than three parameters per handle
as one would expect to have from finite-genus theory. The remaining
parameters must be contained in the data for gluing the boundary of each
smooth component to the set of points at infinity. \end{example}
\section{Set of admissible bundles }\label{h8}\myLabel{h8}\relax
Here we investigate the structure of the set $ {\frak L} $ of admissible bundles
$ {\cal L} $ over the given curve (described by the model space $ \bar{M} $, as in Section
~\ref{s5.30}). The mapping $ {\cal L} \mapsto \left(\Pi_{\left\{\psi\right\}}\circ{\bold K},\Pi_{\left\{\psi^{-1}\right\}}\circ{\bold K} \right) $ into a pair of compact
operators allows one to consider the topology on $ {\frak L} $ induced by the
operator topology on the space of compact operators.
Thus $ {\frak L} $ is a topological space. We will describe some remarkable
subsets of $ {\frak L} $ and algebraic structures on these subsets.
\subsection{Exceptional indices } Since operators $ \Pi_{\varphi_{i},\psi_{i}} $ are bounded, and row-blocks
and column-blocks of the operator $ {\bold K} $ are compact, we can conclude that
instead of compactness of $ \Pi{\bold K} $ one can require compactness of restriction
of $ \Pi{\bold K} $ to the direct sum of all but a finite number of $ L_{2}\left(\partial K_{i},\Omega^{1}\otimes\mu\right) $. We
call the indices of excluded contours {\em exceptional indices}.
In particular, we can include all the non-circular contours and all
contours $ \partial K_{i} $ such that $ \operatorname{ind}\psi_{i}\not=0 $ in the set of exceptional indices. Thus
whenever we discuss admissibility conditions we can suppose that $ \operatorname{ind} \psi_{i}=0 $
and all the $ K_{i} $ are disks.
The following facts are obvious:
\begin{proposition} If we change a finite number of functions $ \psi_{i} $, this does
not change the admissibility of the resulting bundle. If we multiply
functions $ \psi_{i} $ by constants $ c_{i} $ with $ |c_{i}| $, $ |c_{i}^{-1}| $ being bounded,
this does not change the admissibility of the bundle. \end{proposition}
\subsection{Hilbert--Schmidt bundles } In practice the condition of admissibility
is very hard to use directly, since there is no practically useful
criterion of compactness which is necessary and sufficient. The closest
simple-to-check approximation is the Hilbert--Schmidt condition.
\begin{definition} We say that a curve with a bundle $ \left(\bar{M},{\cal L}\right) $ is {\em Hilbert\/}--{\em Schmidt\/}
if both operators $ \Pi_{\left\{\psi\right\}}{\bold K} $ and $ \Pi_{\left\{\psi^{-1}\right\}}{\bold K} $ are Hilbert--Schmidt operators. \end{definition}
\begin{remark} Note that since the Hilbert structure on the sections of
the bundle of half-forms on a curve is canonically defined, so is the
notion of Hilbert--Schmidt operator. \end{remark}
\subsection{Involution } Consider an admissible bundle $ {\cal L} $ and the dual bundle $ {\cal L}^{-1} $
(defined by inverse gluing conditions). The following statement is a
direct corollary of definitions:
\begin{proposition} If $ {\cal L} $ is admissible, so is $ {\cal L}^{-1} $. If $ {\cal L} $ is Hilbert--Schmidt,
so is $ {\cal L}^{-1} $. \end{proposition}
\subsection{Hilbert--Schmidt criterion }\label{s8.40}\myLabel{s8.40}\relax From the description of a
solution of the Riemann problem one can easily get the
following criterion:
\begin{theorem} Consider a family of disks $ K_{i} $, $ i\in I $, with an involution ' on I
and gluing data $ \varphi_{i} $, $ \psi_{i} $.
\begin{enumerate}
\item
If
\begin{equation}
\sum_{i\not=j}\left(\left|\psi_{i}\right|_{{\bold R}}^{2}+\left|\psi_{i}^{-1}\right|_{{\bold R}}^{2}\right)e^{-l_{ij}} < \infty
\notag\end{equation}
then $ {\cal L} $ is Hilbert--Schmidt (here $ ||_{{\bold R}} $ is the Riemann norm, see Definition
~\ref{def5.155}).
\item
Suppose that all but a finite
number of functions $ \psi_{i} $ are constant. Then the corresponding curve $ M $ with a
bundle $ {\cal L} $ is Hilbert--Schmidt iff
\begin{equation}
\sum_{i\not=j}\left(\left|\psi_{i}\right|^{2}+\left|\psi_{i}\right|^{-2}\right)e^{-l_{ij}} < \infty
\notag\end{equation}
(here the indices $ i $ for which $ \psi_{i} $ is not constant are excluded from
summation).
\end{enumerate}
Here $ l_{ij} $ is the conformal distance between $ \partial K_{i} $ and $ \partial K_{j} $.
\end{theorem}
This theorem follows immediately from the following
\begin{lemma} \label{lm6.30}\myLabel{lm6.30}\relax Let $ k_{ij} $, $ i,j\in I $ be the Hilbert--Schmidt norm of the block of $ {\bold K} $
which maps $ L_{2}^{-}\left(\partial K_{i},\omega^{1/2}\otimes\mu\right) \to L_{2}^{+}\left(\partial K_{j},\omega^{1/2}\otimes\mu\right) $. Then
\begin{equation}
k_{ij} = \sum\Sb s>0 \\ s\in{\Bbb Z}+1/2\endSb e^{-sl_{ij}} = O\left(e^{-l_{ij}/2}\right).
\notag\end{equation}
\end{lemma}
\begin{proof} It is enough to prove that the characteristic numbers of the
block of $ {\bold K} $ which is the mapping
\begin{equation}
L_{2}^{-}\left(\partial K_{i},\omega^{1/2}\otimes\mu\right) \to L_{2}^{+}\left(\partial K_{j},\omega^{1/2}\otimes\mu\right)
\notag\end{equation}
are $ e^{-sl_{ij}} $ for $ s>0 $, $ s\in{\Bbb Z}+\frac{1}{2} $. Since $ {\bold K} $ is invariant with respect to
fraction-linear transformations, we can assume that $ \partial K_{i} $ and $ \partial K_{j} $ bound a
tube $ S^{1}\times\left(0,l_{ij}\right) $ of circumference $ 2\pi $ and length $ l_{ij} $. Then characteristic
vectors of $ {\bold K} $ correspond to holomorphic $ 1/2 $-forms
\begin{equation}
e^{i sx}e^{-sy}dz^{1/2},\qquad z=x+iy,\quad \left(x,y\right)\in S^{1}\times\left(0,l_{ij}\right),
\notag\end{equation}
the restrictions on $ y=0 $ being the characteristic vectors themselves, the
restrictions on $ y=l_{ij} $ being their images, which are $ e^{-sl_{ij}} $ times smaller.
Now the condition that the restriction on $ y=0 $ is a section of $ \Omega\otimes\mu $ gives
the condition that $ s\in{\Bbb Z}+\frac{1}{2} $, the condition that this restriction is in
$ - $-part of $ L_{2} $ gives $ s>0 $.\end{proof}
\begin{corollary}
\begin{enumerate}
\item
A curve $ \bar{M} $ is Hilbert--Schmidt iff
\begin{equation}
\sum_{i\not=j}e^{-l_{ij}} < \infty;
\notag\end{equation}
\item
If a curve $ \bar{M} $ allows a Hilbert--Schmidt bundle, it is
Hilbert--Schmidt itself.
\end{enumerate}
\end{corollary}
\begin{proof} Indeed, only the second part requires proof, and it follows
from
\begin{equation}
\sum_{ij}k_{ij}^{2} \leq \frac{1}{2}\sum_{ij}\left(|\psi_{i}|^{2}+|\psi_{i}|^{-2}\right)k_{ij}^{2}.
\notag\end{equation}
\end{proof}
\subsection{$ \protect \log $-convexity }\label{s8.50}\myLabel{s8.50}\relax We were not able to prove the following
\begin{conjecture} Let $ {\cal L} $ and $ {\cal M} $ are two bundles on $ \bar{M} $ given by gluing
conditions. Suppose that both $ {\cal M}\otimes{\cal L} $ and $ {\cal M}\otimes{\cal L}^{-1} $ are admissible. Then $ {\cal M} $ is
admissible as well. \end{conjecture}
However, the following statement is true:
\begin{proposition} \label{prop6.70}\myLabel{prop6.70}\relax Let $ {\cal L} $ and $ {\cal M} $ are two bundles on $ \bar{M} $ given by
gluing
conditions with functions $ \psi_{i} $ and $ \gamma_{i} $, and all but a finite number of these
functions are constant. Suppose that both $ {\cal M}\otimes{\cal L} $ and $ {\cal M}\otimes{\cal L}^{-1} $ are
Hilbert--Schmidt. Then $ {\cal M} $ is Hilbert--Schmidt as well. \end{proposition}
\begin{proof} We may suppose that all $ \psi_{i} $ and $ \gamma_{i} $ are constant. Then in
notations of Lemma~\ref{lm6.30} we know that
\begin{gather} \sum_{ij}\left(\left|\psi_{i}\gamma_{i}\right|^{2}+\left|\psi_{i}\gamma_{i}\right|^{-2}\right)k_{ij}^{2} < \infty,
\notag\\
\sum_{ij}\left(\left|\psi_{i}\gamma_{i}^{-1}\right|^{2}+\left|\psi_{i}\gamma_{i}^{-1}\right|^{-2}\right)k_{ij}^{2} < \infty,
\notag\end{gather}
and want to prove that
\begin{equation}
\sum_{ij}\left(\left|\psi_{i}\right|^{2}+\left|\psi_{i}\right|^{-2}\right)k_{ij}^{2} < \infty.
\notag\end{equation}
However, this is an obvious corollary of relation between geometric
mean and arithmetic mean. \end{proof}
\subsection{Types of admissible bundles } In what follows we are going to study
Hilbert--Schmidt bundles, thus we may assume that the curve is
Hilbert--Schmidt itself.
\begin{definition}
\begin{enumerate}
\item
Call the bundle $ \omega\otimes{\cal L} $ on a curve $ \bar{M} $ {\em real\/} if all the gluing
functions $ \psi_{i} $ for this bundle are constants of magnitude 1.
\item
Call the bundle $ \omega\otimes{\cal L} $ on a curve $ \bar{M} $ {\em bounded\/} if all the gluing
functions $ \psi_{i} $ for this bundle taken together are bounded in $ ||_{{\bold R}} $-norm.
\item
Call the bundle $ \omega\otimes{\cal L} $ on a curve $ \bar{M} $ {\em strongly Hilbert\/}--{\em Schmidt\/} if
the bundle $ \omega\otimes{\cal L}^{n} $ is Hilbert--Schmidt for any $ n\in{\Bbb Z} $.
\end{enumerate}
\end{definition}
\begin{lemma} If the bundles $ \omega\otimes{\cal L} $ and $ \omega\otimes{\cal M} $ on a curve $ \bar{M} $ are strongly
Hilbert--Schmidt, then $ \omega\otimes{\cal L}\otimes{\cal M} $ is also strongly Hilbert--Schmidt. \end{lemma}
\begin{proof} Since $ \omega\otimes{\cal L}^{2n} $ and $ \omega\otimes{\cal M}^{2n} $ are Hilbert--Schmidt, such is $ \omega\otimes{\cal L}^{n}\otimes{\cal M}^{n} $
by the $ \log $-convexity. \end{proof}
\begin{lemma} Any real bundle is bounded. Any bounded bundle is strongly
Hilbert--Schmidt. \end{lemma}
\begin{proposition} Consider an admissible curve $ \bar{M} $ and a bundle $ {\cal L} $ defined by
gluing conditions $ \psi_{i} $. Suppose that for any $ N>0 $
\begin{equation}
\sum_{i\not=j}\left(\max _{R_{i}} \left|\psi_{i}\right|^{N}+\max _{R_{i}}\left|\psi_{i}\right|^{-N}\right)e^{-l_{ij}} < \infty.
\notag\end{equation}
Then $ {\cal L} $ is strongly Hilbert--Schmidt. \end{proposition}
\begin{proof} Indeed, this is a direct corollary of Lemma~\ref{lm5.160}. \end{proof}
Topology on $ {\frak L} $ defines a topology on the set of bounded bundles.
Note that the latter topology is very easy to describe. Fix a number $ M>1 $,
and consider the subset of bundles with $ |\psi_{i}|_{{\bold R}}<M $, $ i\in I $. Then the topology
on this subset is the topology of direct product. This topology is
important for the description of Jacobian in Section~\ref{s9.70}.
\subsection{Multiplicators, equivalence and Jacobians }\label{s8.7}\myLabel{s8.7}\relax Consider what can play
a r\^ole of a mapping $ {\frak m}\colon {\cal L}_{1} \to {\cal L}_{2} $ between two bundles on $ \bar{M} $ defined by
gluing conditions. Inside the ``smooth'' part of the curve such mapping
should be a multiplication by a section $ a $ of a holomorphic bundle, thus
$ {\frak m} $ is determined by a function $ a $ which is holomorphic inside $ {\Bbb C}P^{1}\smallsetminus\overline{\bigcup K_{i}} $.
We start with discussing heuristics for the properties of the
function $ a $. On one hand, $ {\frak m} $ should send holomorphic sections to
holomorphic one. If $ \bar{\partial}a $ is defined in a neighborhood of the infinity $ M_{\infty} $,
and is not 0, then by Leibniz rule $ {\frak m} $ will not send holomorphic sections
of $ {\cal L}_{1} $ to holomorphic section of $ {\cal L}_{2} $.
On the other hand, $ {\frak m} $ should send $ H^{1/2} $-sections to $ H^{1/2} $-sections.
Since $ 1\in H^{1/2} $, $ a\in H^{1/2} $. Moreover, if $ {\frak m} $ is bounded, multiplication by $ a $
should send $ L_{2} $-sections of $ \Omega^{1/2}\left(\bigcup_{i}\partial K_{i}\right)\otimes\mu $ to $ L_{2} $-sections. Since the only
multiplicators in $ L_{2} $ are essentially bounded functions, we conclude that
the restriction that $ a $ is bounded on the domain of definition and is in
$ H^{1/2} $ looks like a particularly good candidate.
\begin{proposition} \label{prop8.140}\myLabel{prop8.140}\relax Consider a function $ a $ which is holomorphic
inside
$ {\Bbb C}P^{1}\smallsetminus\overline{\bigcup K_{i}} $. Identify $ {\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $ with a subspace of the space of
analytic functions on $ {\Bbb C}P^{1}\smallsetminus\overline{\bigcup K_{i}} $. Suppose that the disks $ K_{i} $ are well
separated. Multiplication by $ a $ preserves $ {\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $ if and only if $ a $
is a restriction of an element $ \widetilde{a}\in H^{1/2}\left({\Bbb C}P^{1}\smallsetminus\bigcup K_{i}\right) $ such that
$ \bar{\partial}\widetilde{a}=0\in H^{-1/2}\left({\Bbb C}P^{1}\smallsetminus\bigcup K_{i}\right) $ and the restriction $ \widetilde{r}\left(\widetilde{a}\right)\in L_{2}\left(\bigcup\partial K_{i}\right) $ of $ \widetilde{a} $ to the
boundary is essentially bounded. \end{proposition}
\begin{proof} Indeed, we may suppose that $ \infty\in K_{i} $ for some $ i\in I $. Let us show the
``only if'' part first. Considering $ a\cdot dz^{1/2} $ we see that if $ a $ is a
multiplicator in $ {\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $, then $ \widetilde{a} $ satisfying the first two conditions
of the proposition exists. Since the operator of multiplication by $ a $ is
automatically bounded in $ {\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $, we see that the boundary value of
$ \widetilde{a} $ is essentially bounded by the norm $ M $ of this operator.
Indeed, if $ \varepsilon>0 $ and the essential supremum of $ \widetilde{r}\left(\widetilde{a}\right) $ is bigger than
$ M/\left(1-\varepsilon\right) $, then there is an arc in $ \bigcup\partial K_{i} $ such that $ |\widetilde{r}\left(\widetilde{a}\right)|>M/\left(1-\varepsilon\right) $ on a
subset of this arc of relative measure greater than $ 1-\varepsilon $. Taking
$ \|a\cdot\frac{dz^{1/2}}{z-z_{0}}\|_{L_{2}\left(\bigcup\partial K_{i}\right)} $ with $ z_{0} $ close to this arc and inside $ \bigcup K_{i} $, we
obtain a contradiction.
The proof of the ``if'' part consists of three parts. First, let us
show that if $ \alpha\in{\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $, then $ a\alpha $ is a generalized half-form correctly
defined up to addition of a half-form with support in $ \bigcup\partial K_{i} $ (here we say
that the support of a generalized function $ \beta $ is in a set $ U $---not
necessarily closed---if it is a weak limit of generalized functions $ \beta_{n} $
such that $ \operatorname{Supp}\beta_{n}\subset U $). Indeed, for any manifold $ M $ the formula
$ \left<\alpha\beta,\varphi\right>\buildrel{\text{def}}\over{=}\left<\alpha,\varphi\beta\right> $, $ \varphi\in{\cal D}\left(M\right) $, $ \alpha\in H^{s}\left(M\right) $, $ \beta\in H^{-s}\left(M,\Omega^{\text{top}}\right) $, shows that there is
a natural pairing $ \left(\alpha,\beta\right) \mapsto \alpha\beta $ of $ H^{s}\left(M\right) $ with $ H^{-s}\left(M,\Omega^{\text{top}}\right) $ with values in
$ {\cal D}'\left(M\right) $.
This pairing is weakly bicontinuous, moreover, for any
smooth vector field $ v\in\operatorname{Vect}\left(M\right) $ the Leibniz identity
\begin{equation}
v\left(\alpha\beta\right)=\left(v\alpha\right)\beta+\alpha\left(v\beta\right)
\notag\end{equation}
holds if $ \alpha\in H^{s}\left(M\right) $, $ \beta\in H^{1-s}\left(M,\Omega^{\text{top}}\right) $. Since both $ a $ and $ \alpha $ are of smoothness
$ H^{1/2} $, we see that $ a\alpha $ is indeed a generalized function defined with the
described above ambiguity (take $ s=1/2 $, and make appropriate changes to
adjust the above discussion to half-forms). Moreover, $ \bar{\partial}\left(a\alpha\right) =\left(\bar{\partial}a\right)\alpha +
a\left(\bar{\partial}\alpha\right) $, thus $ \bar{\partial}\left(a\alpha\right)=0 $ as a generalized function defined up to addition
of a function with support in $ \bigcup\partial K_{i} $. What remains to prove is that
$ \bar{\partial}\left(a\alpha\right)\in H^{-1/2}\left({\Bbb C}P^{1},\omega^{1/2}\otimes\bar{\omega}\right) $ after an appropriate choice of continuation of $ a\alpha $
to $ {\Bbb C}P^{1} $.
As a second step, fix $ i\in I $ and show that $ a\alpha $ has an appropriate
extension into $ K_{i} $. Consider restriction of $ a\alpha $ to a small collar outside
$ \partial K_{i} $.
\begin{lemma} Consider two concentric circles $ K\subset\widetilde{K}\subset{\Bbb C} $, and a holomorphic
half-form $ \beta $ in $ \widetilde{K}\smallsetminus K $. Consider a $ L_{2} $-half-form $ B $ on $ \partial K $, and suppose that
Laurent coefficients of $ \beta $ coincide with Fourier coefficients of $ B $. Then
$ \beta $ has an $ H^{1/2} $-continuation into $ K $ with the $ H^{1/2} $-norm being $ O\left(\|B\|_{L_{2}}\right) $. \end{lemma}
\begin{proof} Indeed, we know that positive Fourier coefficients of $ B $ are
$ O\left(e^{-\varepsilon k}\right) $, and negative are in $ l_{2} $. Consider positive and
negative parts of $ \beta $ separately. The positive part automatically continues
into $ K $, and the bound is a corollary of results of Section~\ref{s2.70}.
Consider now the negative part. We can suppose that $ K=\left\{z \mid |z|<1 \right\} $,
and $ \beta $ is $ \sum B_{k}z^{-k}dz^{1/2} $, $ \left(B_{k}\right)\in l_{2} $. Extend $ \beta $ into $ K $ as $ \sum B_{k}\bar{z}^{k}dz^{1/2} $. Let us
show that this extension satisfies the conditions of the lemma.
It is sufficient to show that the $ H^{-1/2} $-norm of $ \bar{\partial}\beta=\sum kB_{k}\bar{z}^{k-1}\vartheta_{K_{1}}dz^{1/2}d\bar{z} $ is
bounded (here $ \vartheta_{K_{1}} $ is the characteristic function of $ K_{1} $). Since different
components of this form are perpendicular in $ H^{-1/2} $, it is sufficient to bound
the norm of one component. To do this it is sufficient to apply the methods
of Section~\ref{s2.70}. {}\end{proof}
Application of this lemma to $ a\alpha $ shows that $ a\alpha $ may be extended into
every disk $ K_{i} $, moreover, that after these extensions the norms
$ \|\bar{\partial}\left(a\alpha\right)|_{K_{i}}\|_{H^{-1/2}} $ form a sequence in $ l_{2} $. Indeed, $ \widetilde{r}_{i}\left(a\right) $ is a bounded
function on $ \partial K_{i} $, thus $ \widetilde{r}_{i}\left(a\right)\widetilde{r}_{i}\left(\alpha\right) $ has its $ L_{2} $-norm bounded by $ L_{2} $-norm of
$ \widetilde{r}_{i}\left(\alpha\right) $, and the latter norms (for different $ i $) form a sequence in $ l_{2} $.
Third, we need to show that the above extensions can be glued
together to an extension to $ {\Bbb C}P^{1} $. Consider an arbitrary
generalized-function-extension of $ a\alpha $ to $ {\Bbb C}P^{1} $ and a generalized function
$ \bar{\partial}\left(a\alpha\right) $. It is a generalized function with support in the disjoint union
$ \bigcup K_{i} $, thus the components $ \bar{\partial}\left(a\alpha\right)|_{K_{i}} $ are well defined generalized forms $ b_{i} $,
$ \operatorname{Supp} b_{i}\subset K_{i} $. On the other hand, above we constructed an $ H^{1/2} $-extension of
$ a\alpha $ to $ K_{i} $, let $ \overset{\,\,{}_\circ}{b}_{i} $ be $ \bar{\partial}\left(a\alpha\right) $ obtained from this extension.
\begin{lemma} $ b_{i}-\overset{\,\,{}_\circ}{b}_{i}=\bar{\partial}\beta_{i} $, $ \beta_{i} $ being a generalized function with support in $ K_{i} $. \end{lemma}
This lemma is an obvious corollary of the fact that
restriction of $ \bar{\partial}^{-1}\left(b_{i}-\overset{\,\,{}_\circ}{b}_{i}\right) $ to a collar around $ \partial K_{i} $ can be holomorphically
extended inside of $ K_{i} $. Since both series $ \sum b_{i} $ and $ \sum\overset{\,\,{}_\circ}{b}_{i} $ converge
(one in $ {\cal D}' $, another in $ H^{-1/2} $), we conclude that $ \sum\left(b_{i}-\overset{\,\,{}_\circ}{b}_{i}\right) $ converges in $ {\cal D}' $,
thus $ \sum\beta_{i} $ converges in $ {\cal D}' $.
We conclude that $ \beta=a\alpha-\sum\beta_{i} $ is a generalized function such that
\begin{enumerate}
\item
$ \bar{\partial}\beta $ has support in $ \bigcup K_{i} $;
\item
$ \beta $ coincides with $ a\alpha $ inside $ {\Bbb C}P^{1}\smallsetminus\overline{\bigcup K_{i}} $;
\item
$ \bar{\partial}\beta $ is in $ H^{-1/2} $.
\end{enumerate}
Last condition implies $ \beta\in H^{1/2} $, which shows that $ a\alpha $ can be extended to
an element of $ H^{1/2}\left({\Bbb C}P^{1},\omega^{1/2}\right) $, thus $ a\alpha\in{\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $. This finishes the
proof of Proposition~\ref{prop8.140}. {}\end{proof}
\begin{definition} Call a bounded operator $ {\cal M}\colon {\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) \to {\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $ a
{\em multiplicator}, if for some point $ z_{0}\in{\Bbb C}P^{1}\smallsetminus\overline{\bigcup K_{i}} $ there is a formal power
series $ \nu $ at $ z_{0} $ such that for any $ f\in{\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $ formal power series $ {\cal M}f $
and $ \nu f $ coincide. \end{definition}
(This is just a formal way to say that $ {\cal M} $ is a bounded operator of
multiplication by a holomorphic function.)
\begin{definition} Let $ {\cal H}^{\infty}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right)\subset H^{1/2}\left({\Bbb C}P^{1}\smallsetminus\bigcup K_{i}\right) $ consists of functions $ f $ such
that the restriction $ \widetilde{r}\left(f\right) $ to $ \bigcup\partial K_{i} $ (which is automatically in $ L_{2}\left(\bigcup\partial K_{i}\right) $)
is essentially bounded. Define a norm on this space by taking the
essential maximum of $ |\widetilde{r}\left(f\right)| $. \end{definition}
\begin{amplification} Multiplicators form an algebra. The set of multiplicators
coincides with $ {\cal H}^{\infty}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $. One can choose a norm in $ {\cal H}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $ in such
a way that the operator norm of any multiplicator coincides with the
$ {\cal H}^{\infty} $-norm. \end{amplification}
\begin{proof} The only statement which needs a proof is the last one, and the
norm in question is the norm induced by the inclusion into
$ \bigoplus_{l_{2}}L_{2}\left(\partial K_{i},\Omega^{1/2}\otimes\mu\right) $. \end{proof}
\begin{corollary} Let $ z_{0}\in{\Bbb C}P^{1}\smallsetminus\overline{\bigcup K_{i}} $, $ f\in{\cal H}^{\infty}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $. Then $ |f\left(z_{0}\right)|\leq \|f\|_{{\cal H}^{\infty}} $. \end{corollary}
\begin{proof} One can suppose that $ \infty\in K_{i} $ for some $ i\in I $, so that $ dz^{1/2}\in{\cal H} $. Since
$ L_{2} $-norm on $ {\Bbb C}P^{1}\smallsetminus\overline{\bigcup K_{i}} $ is majorated by $ H^{1/2} $-norm, we see that
$ \|f^{n}dz^{1/2}\|_{L_{2}}\leq C\cdot\|f\|_{{\cal H}^{\infty}}^{n} $, $ n\in{\Bbb N} $. Hence $ |f\left(z_{0}\right)|^{n}\leq C\cdot\|f\|_{{\cal H}^{\infty}}^{n} $, thus $ |f\left(z_{0}\right)|\leq\|f\|_{{\cal H}^{\infty}} $.
\end{proof}
Last conditions on $ {\frak m} $ is that it should preserve the gluing conditions.
In particular, if $ {\cal L}_{1} $ is defined by gluing conditions $ \psi_{i} $, and $ {\cal L}_{2} $ by $ \xi_{i} $,
then
\begin{equation}
\xi_{i}=\frac{a\circ\varphi_{i}}{a}\psi_{i}.
\label{equ7.65}\end{equation}\myLabel{equ7.65,}\relax
\begin{definition} Say that linear bundles $ {\cal L}_{1} $, $ {\cal L}_{2} $ defined by gluing conditions
$ \left\{\psi_{i}\right\} $ and $ \left\{\xi_{i}\right\} $ are {\em bounded-equivalent\/} if there exists a function
$ a\in{\cal H}^{\infty}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $ such that $ \xi_{i}=\frac{a\circ\varphi_{i}}{a}\psi_{i} $ and $ a^{-1}\in{\cal H}^{\infty}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $. \end{definition}
\begin{remark} It is obvious that a bundle which is bounded-equivalent to a
bounded bundle is bounded itself. Moreover, if a bundle is
bounded-equivalent to a (strongly) Hilbert--Schmidt bundle, it is
(strongly) Hilbert--Schmidt itself. \end{remark}
\begin{definition} The {\em Jacobian\/} is the set of equivalence classes of admissible
bundles of degree 0. The {\em bounded Jacobian\/} is the subset of Jacobian which
consists of classes of bounded bundles, similarly for {\em(strongly)
Hilbert\/}--{\em Schmidt Jacobian}, and {\em real Jacobian}. {\em Constant Jacobian\/} is formed
from classes of bundles defined by constant gluing functions, similarly
one can define different flavors of constant Jacobians. \end{definition}
Multiplication by an appropriate rational function with zeros and
poles inside $ \bigcup\overset{\,\,{}_\circ}{K}_{i} $ shows that
\begin{proposition} Any admissible bundle of degree 0 is bounded-equivalent
to a bundle with all the gluing functions $ \psi_{i} $ of index 0. \end{proposition}
\subsection{Divisors }\label{s8.80}\myLabel{s8.80}\relax Consider a model $ \left({\Bbb C}P^{1}, \left\{K_{\bullet}\right\}, \left\{\varphi_{\bullet}\right\}\right) $ of a curve $ \bar{M} $,
and a
rational function $ a $ on $ {\Bbb C}P^{1} $ with a divisor $ D $. Suppose that the part of $ D $
inside $ \bigcup R_{i} $ is invariant w.r.t. $ \varphi_{\bullet} $, and $ D $ does not intersect with the
infinity $ M_{\infty}\subset{\Bbb C}P^{1} $ of the curve $ M $. If the bundle $ {\cal L} $ with gluing data $ \left\{\psi_{i}\right\} $ is
admissible, so is the bundle $ L\left(D\right) $ given by the gluing data
\begin{equation}
\xi_{i}=\frac{a\circ\varphi_{i}}{a}\psi_{i}.
\notag\end{equation}
Obviously, $ \deg {\cal L}\left(D\right)=\deg {\cal L}+\deg ' D $, if $ \deg ' D $ is the degree of the part of $ D $
inside $ M\smallsetminus\bigcup R_{i} $ plus half the degree of the part of $ D $ inside $ \bigcup R_{i} $.
Moreover, if we change the part of $ D $ inside $ \bigcup\overset{\,\,{}_\circ}{K}_{i} $, the bundle $ {\cal L}\left(D\right) $ will
change to a bounded-equivalent one.
In particular, to any finite subset $ D_{0} $ of $ M $ (with integer
multiplicities fixed for any point of $ D_{0} $) we associate a transformation $ {\cal L}
\mapsto {\cal L}\left(D_{0}\right) $ where the right-hand side is defined up to equivalence. Note
that if we fix a point $ Z\in\bigcup\overset{\,\,{}_\circ}{K}_{i} $, then we can complete any divisor $ D_{0} $ on $ M $
to a divisor on $ {\Bbb C}P^{1} $ of degree 0 by adding some multiple of $ Z $, thus one
can define $ {\cal L}\left(D_{0}\right) $ uniquely. Moreover, if $ D_{0} $ is of degree 0, then $ {\cal L}\left(D_{0}\right) $
does not depend on the choice of $ Z $.
Consider now two sequences of points $ \left(x_{k}\right),\left(y_{k}\right)\subset{\Bbb C}P^{1} $. Let
$ a_{k}\left(z\right)=\frac{z-x_{k}}{z-y_{k}} $, and $ a\left(z\right)=\prod_{k}a_{k}\left(z\right) $. If $ x_{k} $ is sufficiently close to $ y_{k} $,
and both these points are in appropriate neighborhood of $ K_{k} $,
then the infinite product converges and defines a bounded function on
$ {\Bbb C}P^{1}\smallsetminus\overline{\bigcup\overset{\,\,{}_\circ}{K}_{i}} $. We conclude that it is possible to consider also some
``infinite'' divisors (of finite degree) on $ M $.
Consider now a finite divisor $ D\subset{\Bbb C}P^{1} $ such that $ D\cap M_{\infty} $ consists of one
point $ z_{0}\in M_{\infty} $ (with some multiplicity). Changing $ D $ to $ D-\left(\deg D\right)\cdot Z $, we can
consider $ D $ as a divisor of a rational function $ a $, thus $ {\cal L}\left(D\right) $ is correctly
defined by gluing conditions conditions~\eqref{equ7.65}. For $ {\cal L}\left(D\right) $ to be
admissible for any such $ D $ and any admissible $ {\cal L} $ it is sufficient that for
some constant $ C $
\begin{equation}
\operatorname{dist}\left(z_{0},\operatorname{center}\left(K_{i'}\right)\right) \leq C\cdot\operatorname{dist}\left(z_{0},\operatorname{center}\left(K_{i}\right)\right),\qquad i\in I
\notag\end{equation}
(since the disks $ K_{i} $ are well-separated). On the other hand, suppose that
all the bundles with gluing functions
\begin{equation}
\Psi_{i}=\left(\frac{\operatorname{dist}\left(z_{0},\operatorname{center}\left(K_{i'}\right)\right)}{\operatorname{dist}\left(z_{0},\operatorname{center}\left(K_{i}\right)\right)}\right)^{N},\qquad n\in{\Bbb Z},
\notag\end{equation}
are Hilbert--Schmidt. Then by $ \log $-convexity the bundle $ L\left(D\right) $ is strongly
Hilbert--Schmidt if $ {\cal L} $ is strongly Hilbert--Schmidt.
\begin{definition} Say that the point $ z_{0}\in M_{\infty} $ is {\em bounded\/} if for any $ N\in{\Bbb Z} $ the sheaf
with constant gluing functions
\begin{equation}
\Psi_{i}=\left(\frac{\operatorname{dist}\left(z_{0},\operatorname{center}\left(K_{i'}\right)\right)}{\operatorname{dist}\left(z_{0},\operatorname{center}\left(K_{i}\right)\right)}\right)^{N},\qquad n\in{\Bbb Z}\text{, }i\in I,
\notag\end{equation}
is Hilbert--Schmidt. \end{definition}
We see that if the point $ z_{0}\in M_{\infty} $ is bounded, then the function
$ a_{N}\left(z\right)=\left(\frac{z-z_{0}}{z-Z}\right)^{N} $ defines a strongly Hilbert--Schmidt sheaf of degree
1. In particular, one can consider finite divisors on $ \bar{M} $ which consist of
points of $ M $ and bounded points on $ M_{\infty} $. Similarly, one can also consider
some infinite divisors on $ \bar{M} $.
\subsection{Universal Grassmannian and bundles }\label{s8.90}\myLabel{s8.90}\relax
\begin{definition} Let $ V $ is the vector space of sequences $ \left(a_{k}\right) $, $ k\in{\Bbb Z} $, such that
$ a_{k}=0 $ for $ k\ll0 $. The vector space $ V $ carries a natural topology of inductive
limit of projective limits. Let
\begin{equation}
V_{+}=\left\{\left(a_{k}\right) \mid a_{k}=0\text{ if }k<0\right\},\qquad V_{-}=\left\{\left(a_{k}\right)\in V \mid a_{k}=0\text{ if }k\geq0\right\}.
\notag\end{equation}
Say that the vector subspace $ W\subset V $ is {\em admissible\/} if $ W\cap V_{-} $ is
finite-dimensional, and $ W+V_{-} $ is
closed and has a finite codimension in $ V $. Let {\em universal Grassmannian\/} $ {\cal G} $ be
the set of admissible vector subspaces with natural topology. \end{definition}
Consider an admissible pair $ \left(\bar{M},{\cal L}\right) $ and a smooth point $ z_{0}\in M $. Pick up a
coordinate system $ z $ in neighborhood of $ z_{0} $, and a half-form $ f $ defined in
the same neighborhood. Now any global section of $ {\cal L} $ may be written as
$ g\left(z\right)f\left(z\right) $ with $ g\left(z\right) $ being a holomorphic function which is correctly
defined in the neighborhood of $ z_{0} $.
Call $ f $ a {\em local section\/} of $ {\cal L} $ near $ z_{0} $. If $ {\cal L}' $ is an equivalent to $ {\cal L} $
bundle, and the equivalence is given by multiplication by $ a $, we obtain a
local section $ af $ of $ {\cal L}' $. Say that local sections $ f $ and $ af $ are {\em equivalent}.
Consider now a bundle $ {\cal L}\left(k\cdot z_{0}\right) $, defined using a point $ Z\in\overset{\,\,{}_\circ}{K}_{i} $ (as in the
previous section). A section $ h\left(\zeta\right) $, $ \zeta\in{\Bbb C}P^{1} $, of this bundle can be
identified (via multiplication by $ \left(\frac{\zeta-Z}{\zeta-z_{0}}\right)^{k} $) with a ``meromorphic
section'' of $ {\cal L} $, i.e., one can write it as $ g\left(z\right)f\left(z\right) $, $ g\left(z\right) $ being a
meromorphic function correctly defined in the neighborhood of $ z_{0} $.
Moreover, $ g\left(z\right) $ has poles only at $ z_{0} $. One can momentarily see that $ g\left(z\right) $
does not depend on the choice of the point $ Z $, more precise, for a
different choice of $ Z $ there is a different choice of $ h $ which gives the
same $ g\left(z\right) $.
Associate to any such function $ g\left(z\right) $ the sequence of its Laurent
coefficients. Consider this sequence as an element of $ V $. Let $ W $ be the
vector subspace of $ V $ spanned by all possible functions $ g\left(z\right) $ for
bundles $ {\cal L}\left(k\cdot z_{0}\right) $, $ k\in{\Bbb Z} $. The Riemann--Roch theorem momentarily implies that
$ W $ is admissible. Indeed, the condition that $ W\cap V_{-} $ is
finite-dimensional means that $ {\cal L} $ has a finite-dimensional space of
global sections. The condition on $ W+V_{-} $ is implied by the following fact:
\begin{lemma} Consider a admissible semibounded bundle $ {\cal L} $ on a curve $ \bar{M} $, and a
point $ z_{0}\in M $. Then
\begin{enumerate}
\item
for big enough $ k $ the bundle $ {\cal L}\left(-k\cdot z_{0}\right) $ has no sections.
\item
for big enough $ k $ the bundle $ {\cal L}\left(\left(k+1\right)\cdot z_{0}\right) $ has one more section than
$ {\cal L}\left(k\cdot z_{0}\right) $.
\end{enumerate}
\end{lemma}
\begin{proof} The second statement is a corollary of the first one, of
Riemann--Roch theorem and duality. The first one is obvious, since the
global sections of $ {\cal L}\left(-k\cdot z_{0}\right) $ are naturally identified with global
sections of $ {\cal L} $ which have a zero of $ k $-th order at $ z_{0} $. \end{proof}
We obtained
\begin{proposition} To each admissible pair $ \left(\bar{M},{\cal L}\right) $ with a fixed smooth point
$ z_{0}\in M $, a coordinate system near $ z_{0} $, and a local section $ f $ of $ {\cal L} $ one can
associate a point $ W\in{\cal G} $. If we change $ {\cal L} $ and $ f $ to an equivalent bundle with
a local section, $ W $ does not change. \end{proposition}
One can generalize this proposition to some points at infinity. If
$ z_{0}\in M_{\infty} $, define a {\em local section\/} of $ {\cal L} $ near $ z_{0} $ in the same way as above, i.e.,
as a non-zero section of $ \omega^{1/2} $ near $ z_{0} $. In what follows we use only the
$ \infty $-jet of this section.
\begin{amplification} Consider an admissible pair $ \left(\bar{M},{\cal L}\right) $ with a fixed point at
infinity $ z_{0}\in M_{\infty} $, a coordinate system near $ z_{0} $, and a local section $ f $ of $ {\cal L} $.
Suppose that $ {\cal L} $ is Hilbert--Schmidt, and $ z_{0} $ is bounded and of class $ C^{\infty} $. To
this data we can associate a point $ W\in{\cal G} $. If we change $ {\cal L} $ and $ f $ to an
equivalent bundle with a local section, $ W $ does not change. \end{amplification}
\begin{proof} Since $ z_{0} $ is bounded, and $ {\cal L} $ is Hilbert--Schmidt, $ {\cal L}\left(k\cdot z_{0}\right) $ is
Hilbert--Schmidt too, thus the Riemann--Roch theorem is applicable. Since
$ z_{0} $ is of class $ C^{\infty} $, any section of $ {\cal L}\left(k\cdot z_{0}\right) $ has a (formal) Taylor series
near $ z_{0} $, thus the corresponding section of $ {\cal L} $ has a (formal) Laurent
series.
What remains to be proved is the fact that equivalence between
bundles can be pushed to infinity points of class $ C^{\infty} $. Note that one can
associate an element of the generalized Hardy space to any bounded
function $ a $. Indeed, suppose that $ \zeta=\infty\in K_{i} $, then $ a\cdot d\zeta^{1/2} $ is an element of
$ {\cal H} $. In particular, $ a $ has an asymptotic expansion near $ z_{0} $, thus
multiplication by $ a $ maps (formal) Laurent series at $ z_{0} $ to themselves. \end{proof}
\section{Structure of Jacobian }\label{h9}\myLabel{h9}\relax
\subsection{Constant Jacobian }\label{s7.90}\myLabel{s7.90}\relax Consider the involution $ '\colon I\to I $. It defines a
transposition matrix $ t=\left(t_{ij}\right) $, $ i,j\in I $, $ t_{ij}=\delta_{ij'} $. We use the results of toy
theory (see Section~\ref{h35}) to obtain the following
\begin{theorem} \label{th7.25}\myLabel{th7.25}\relax Let $ l_{ij} $ be pairwise conformal distances between disks
$ K_{i} $. Suppose that the matrix $ {\cal R}=\left(e^{-l_{ij}}-\delta_{ij}\right) $ defines a compact mapping $ l_{2} \to
l_{2} $ and a bounded mapping $ l_{\infty} \to l_{\infty} $, and that for some $ N>0 $ the matrix $ \left(t{\cal R}\right)^{N} $
defines a mapping $ l_{\infty} \to l_{2} $. Then for any admissible curve obtained by
gluing circles $ \partial K_{i} $ the constant bounded Jacobian coincides with the
bounded Jacobian. \end{theorem}
\begin{proof} Indeed, to show this we need to show that for any cycle $ \left\{\psi_{i}\right\}_{i\in I} $
with $ \deg \psi_{i}=0 $ and $ |\psi_{i}|<C $ one can find a bounded collection of constants $ c_{i} $
such that $ c_{i}\psi_{i} =\frac{a\circ\varphi_{i}}{a} $ for some $ a $ such that $ a,a^{-1}\in{\cal H}^{\infty}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $.
Taking logarithms, we see that it is sufficient to show that the mapping
\begin{equation}
{\cal J}_{b}\colon f+C \mapsto \left(f|_{\partial K_{j}}-\varphi^{*}\left(f|_{\partial K_{j'}}\right)+C_{j}\right)_{j\in I_{+}}
\notag\end{equation}
from bounded analytic functions (modulo constants) to functions on
boundary (modulo
constants) with bounded $ \pm $-parts is surjective. In Section~\ref{s35.40} we have
seen that (given the first condition of the theorem) a similar mapping
\begin{equation}
{\cal J}\colon {\cal H}^{\left(1\right)}/\operatorname{const} \to \bigoplus\Sb l_{2} \\ j\in I_{+}\endSb H^{1/2}\left(\partial K_{j}\right)/\operatorname{const}
\notag\end{equation}
is a bounded mapping of index 0. We are going to prove the
surjectivity by using a combination of following lemmas:
\begin{lemma} \label{lm8.31}\myLabel{lm8.31}\relax If $ \left(e^{-l_{ij}}-\delta_{ij}\right) $ gives a compact operator $ l_{2} \to l_{2} $, then the
mapping $ {\cal J} $ is a bijection. \end{lemma}
\begin{proof} Since we know the index of $ {\cal J} $, it is sufficient to show that
$ \operatorname{Ker}{\cal J}=0 $. Let $ f\in{\cal H}^{\left(1\right)}/\operatorname{const} $ and $ {\cal J}f=0 $. We are going to show that $ \|f\|_{H^{1}}=0 $.
Since $ \partial $ is elliptic, and $ \bar{\partial}f=0 $, it is sufficient to show that $ \|\partial f\|_{L_{2}}=0 $,
i.e., that $ \int\partial f\wedge\bar{\partial}\bar{f}=0 $, the integral is taken over $ {\Bbb C}P^{1}\smallsetminus\bigcup K_{i} $. Denote the
domain of integration by $ S $. Note that the above integral is well defined,
since $ \partial f\in H^{0}\left(S\right)=L_{2}\left(S\right) $.
First of all, since $ \wedge $-product of any two sections of $ \omega $ is 0, one can
add $ \partial\bar{f} $ to $ \bar{\partial}\bar{f} $ without changing the integral, thus it is sufficient to show
that $ \int_{S}\partial f\wedge d\bar{f} = $ 0. Additionally, since $ \bar{\partial}f=0 $ on $ S $, one can change $ \partial f $ to $ df $
without changing the integral, thus it is sufficient to show that
$ \int_{S}df\wedge d\bar{f}=0 $. Second, use the duality identity $ \int_{S}d\alpha=\int_{\partial S}\alpha $. Applying it
(at first formally), we get
\begin{align} \int_{S}df\wedge d\bar{f} & = \int_{S}d\left(f\,d\bar{f}\right) = \int_{\partial S}f\,d\bar{f} = -\sum_{i\in I}\int_{\partial K_{i}}f\,d\bar{f}
\notag\\
& = -\sum_{i\in I_{+}}\left(\int_{\partial K_{i}}f\,d\bar{f} + \int_{\partial K_{i'}}f\,d\bar{f}\right).
\notag\end{align}
We want to show that $ \int_{\partial K_{i}}f\,d\bar{f} + \int_{\partial K_{i'}}f\,d\bar{f}=0 $. Indeed,
$ \int_{\partial K_{i'}}f\,d\bar{f}=-\int_{\partial K_{i}}\varphi_{i'}^{*}\left(f\,d\bar{f}\right) $, the sign appears since $ \varphi_{i} $ changes the
orientations of $ \partial K_{\bullet} $. Now the above equality becomes obvious, since
$ f\circ\varphi_{i}=f+C_{i} $ when both sides are defined, thus the sum of the integrals is
$ -C_{i}\int_{\partial K_{i}}d\bar{f}=0 $.
What remains to be proved is that one can indeed apply the formula
$ \int_{S}d\alpha=\int_{\partial S}\alpha $ in our situation, when $ \alpha $ is not smooth, and $ S $ has non-smooth
boundary. Consider an $ H^{1} $-extension $ g $ of $ f $ to $ {\Bbb C}P^{1} $. Since $ dg\wedge d\bar{g}\in L_{1}\left(\Omega^{\text{top}}\right) $,
the integral $ \int_{S}df\wedge d\bar{f} $ can be represented as
\begin{equation}
\int_{{\Bbb C}P^{1}}dg\wedge d\bar{g} - \int_{\bigcup K_{i}}dg\wedge d\bar{g} = \int_{{\Bbb C}P^{1}}dg\wedge d\bar{g} - \sum\int_{K_{i}}dg\wedge d\bar{g}.
\notag\end{equation}
It remains to prove that $ \int_{{\Bbb C}P^{1}}dg\wedge d\bar{g} = $ 0, and $ \int_{K_{i}}dg\wedge d\bar{g} = \int_{\partial K_{i}}g\,d\bar{g}. $ Note
that in both identities the boundary is already smooth (it is empty in
the first one!).
Let us prove that if $ M $ is a two-dimension manifold, $ S $ is a
compact subset of $ M $ with a smooth boundary, and $ g_{1},g_{2}\in H^{1}\left(M\right) $, then
\begin{equation}
\int_{S}dg_{1}\wedge dg_{2}=\int_{\partial S}g_{1}\,dg_{2}.
\notag\end{equation}
Here we understand the right-hand side as a natural pairing between
$ g_{1}|_{\partial S}\in H^{1/2}\left(\partial S\right) $ and $ d\left(g_{2}|_{\partial S}\right)\in H^{-1/2}\left(\partial S,\Omega_{\partial S}^{\text{top}}\right) $. However, both sides define
bounded bilinear functionals on $ H^{1}\left(M\right) $, thus it is sufficient to check
them on a dense subset $ C^{\infty}\left(M\right) $, where they are true due to de Rham theory.
\end{proof}
To continue the proof of the theorem, note that the operator $ {\cal J} $ is
related to the operator $ {\bold K} $ in the following way: identify $ {\cal H}^{\left(1\right)}/\operatorname{const} $ with
$ \bigoplus\Sb l_{2} \\ i\in I\endSb H_{-}^{1/2}\left(\partial K_{i}\right)/\operatorname{const} $. Let $ \Pi_{i} $ be the identification of $ H_{+}^{1/2}\left(\partial K_{i}\right) $ with
$ H_{-}^{1/2}\left(\partial K_{i'}\right) $ via $ \varphi^{*} $, let $ \Pi^{\left(1\right)} $ be the direct sum of such identifications.
As we have seen it in Section~\ref{s35.40}, the image of $ {\cal J} $ coincides with the
image of $ \Pi^{\left(1\right)}\circ{\bold K}-1 $ (which is a mapping
\begin{equation}
\bigoplus_{l_{2}}H_{-}^{1/2}\left(\partial K_{i}\right)/\operatorname{const} \to \bigoplus_{l_{2}}H_{-}^{1/2}\left(\partial K_{i}\right)/\operatorname{const},
\notag\end{equation}
but we identify $ \bigoplus_{l_{2}}H_{-}^{1/2}\left(\partial K_{i}\right)/\operatorname{const} $ with $ \bigoplus\Sb l_{2} \\ j\in I_{+}\endSb H^{1/2}\left(\partial K_{j}\right)/\operatorname{const} $ via $ \varphi_{\bullet}^{*} $).
\begin{lemma} \label{lm8.33}\myLabel{lm8.33}\relax
\begin{enumerate}
\item
Consider two disjoint disks $ K_{1} $, $ K_{2} $ on $ {\Bbb C}P^{1} $ of conformal distance $ l $.
The operator $ {\bold K} $ with smooth kernel $ \frac{dy}{y-x} $ defines bounded operators
$ H^{1/2}\left(\partial K_{1}\right) \to L_{\infty}\left(\partial K_{2}\right) $, $ L_{\infty}\left(\partial K_{1}\right) \to L_{\infty}\left(\partial K_{2}\right) $, $ H^{1/2}\left(\partial K_{1}\right) \to H^{1/2}\left(\partial K_{2}\right) $, $ L_{\infty}\left(\partial K_{1}\right)
\to H^{1/2}\left(\partial K_{2}\right) $ with the norms being $ O\left(e^{-l}\right) $.
\item
Consider a family of disjoint disks $ \left\{K_{i}\right\} $ on $ {\Bbb C}P^{1} $ with conformal
distances $ l_{ij} $, and suppose that the matrix $ {\cal R}=\left(e^{-l_{ij}}-\delta_{ij}\right) $ defines bounded
mappings $ l_{2} \to l_{2} $ and $ l_{\infty} \to l_{\infty} $, and for some $ N>0 $ the matrix $ \left(t{\cal R}\right)^{N} $ defines
a mapping $ l_{\infty} \to l_{2} $. Then the operator $ {\bold K} $ gives bounded mappings
$ \bigoplus_{l_{2}}H^{1/2}\left(\partial K_{i}\right) \to \bigoplus_{l_{2}}H^{1/2}\left(\partial K_{i}\right) $ and $ L_{\infty}\left(\bigcup\partial K_{i}\right) \to L_{\infty}\left(\bigcup\partial K_{i}\right) $, moreover, the
operator $ \left(\Pi^{\left(1\right)}{\bold K}\right)^{N} $ gives a bounded mapping $ L_{\infty}\left(\bigcup\partial K_{i}\right) \to \bigoplus_{l_{2}}H^{1/2}\left(\partial K_{i}\right) $.
\item
In the conditions of the previous part of the lemma let
$ f_{-}\in L_{\infty}\left(\bigcup\partial K_{i}\right) $. Suppose that $ + $-parts of $ f_{-} $ on all the $ \partial K_{i} $ vanish. Then $ {\bold K}f_{-} $
is a bounded analytic function.
\end{enumerate}
\end{lemma}
Now we are able to invert the mapping $ {\cal J}_{b} $. Consider a fixed function
$ f $ on $ \bigcup\partial K_{i} $ with bounded $ + $-part and bounded $ - $-part. If $ f-f\circ\varphi_{i} $ is a
constant for any $ i\in I $, then $ + $-part of $ f|_{\partial K_{i}} $ can be reconstructed basing on
-part of $ f|_{\partial K_{i'}} $, thus one can identify $ f $ (modulo constants) with the
collection of $ - $-parts of $ f $ on $ \bigcup_{i\in I}\partial K_{i} $ (modulo constants).
The restrictions on $ l_{ij} $ show that the radii of disks $ K_{i} $ for a
sequence from $ l_{1} $. Hence the Cauchy kernel allows one to construct an
analytic function $ \widetilde{f} $ on $ S $ such that the $ - $-parts of $ \widetilde{f}|_{\partial K_{i}} $ and $ f|_{\partial K_{i}} $
coincide (modulo constants). Moreover, $ {\cal J}_{b}\widetilde{f} = \left(\Pi^{\left(1\right)}\circ{\bold K}-1\right)\left(f\right) $. Since $ f $ is
bounded, it is in $ L_{2} $, thus $ \widetilde{f}\in H^{1/2} $. Since $ \widetilde{f}|_{\bigcup\partial K_{i}} $ is bounded, $ \widetilde{f} $ is a
multiplicator.
We see that $ \Pi^{\left(1\right)}\circ{\bold K}\left(f\right) $ is a well-defined bounded function on
$ \bigcup_{j\in I_{+}}\partial K_{j} $, moreover, $ \left(\Pi^{\left(1\right)}\circ{\bold K}\right)^{N}\left(f\right)\in\bigoplus_{l_{2}}H^{1/2}\left(\partial K_{j}\right) $. In particular,
$ \left(\Pi^{\left(1\right)}\circ{\bold K}\right)^{N}\left(f\right)={\cal J}F $ for some function $ F\in{\cal H}^{\left(1\right)} $. To show that $ f $ is in the image
of $ \Pi^{\left(1\right)}\circ{\bold K}-1 $, it remains to prove is that $ f-\left(\Pi^{\left(1\right)}\circ{\bold K}\right)^{N}\left(f\right) $ is in the image
of $ \Pi^{\left(1\right)}\circ{\bold K}-1 $, what is obvious.
This finishes the proof of Theorem~\ref{th7.25}. {}\end{proof}
To demonstrate a particular case in which the assumptions of Theorem
~\ref{th7.25} hold, consider
\begin{lemma} Suppose that the matrix $ \left(a_{ij}\right) $ gives a Hilbert--Schmidt operator
$ l_{2} \to l_{2} $, i.e., $ \sum|a_{ij}|^{2}<\infty $. Then the matrix $ \left(a_{ij}^{2}\right) $ gives a bounded
operator $ l_{\infty} \to l_{2} $.
If $ a_{ij}>0 $ for any $ i,j\in I $, and the matrix $ \left(a_{ij}\right) $ gives a compact mapping
$ l_{2} \to l_{2} $, then $ \left(a_{ij}^{2}\right) $ also gives a compact mapping $ l_{2} \to l_{2} $. \end{lemma}
\begin{corollary} If $ \bar{M} $ is a Hilbert--Schmidt curve, then the bounded Jacobian
of $ \bar{M} $ coincides with the constant bounded Jacobian. \end{corollary}
In Section~\ref{s9.70} we show that under suitable restrictions on $ \bar{M} $ the
above Jacobians are in $ 1 $-to-1 correspondence with quotients of
topological vector spaces by $ {\Bbb Z} $-lattices. On the other hand, this
statement is obvious when applied to real Jacobian, since it is a direct
product of circles $ |\psi_{i}|=1 $. To construct an isomorphism of this product
with a quotient by a lattice, one can take the Hilbert space $ l_{2} $ with the
standard basis $ e_{i} $, and the lattice spanned by $ \alpha_{i}e_{i} $, with $ \alpha_{i} \to $ 0.
The description of the real part of Jacobian was first obtained in
\cite{McKTru76Hil} in the case of a real hyperelliptic curve of a special form.
\subsection{The partial period mapping }\label{s9.20}\myLabel{s9.20}\relax Consider a theory similar to one
discussed in Section~\ref{s35.30}, but related to holomorphic $ 1 $-forms instead
of holomorphic functions. Consider a family $ \left\{K_{i}\right\} $ of disjoint disks in
$ {\Bbb C}P^{1} $.
\begin{definition} We say that a generalized-function section $ \alpha $ of the linear
bundle $ \omega $ on $ {\Bbb C}P^{1} $ is $ H^{0} $-{\em holomorphic in\/} $ {\Bbb C}P^{1}\smallsetminus\bigcup K_{i} $ if $ \alpha\in H^{0}\left({\Bbb C}P^{1}\smallsetminus\bigcup K_{i},\omega\right) =
L_{2}\left({\Bbb C}P^{1}\smallsetminus\bigcup K_{i},\omega\right) $, and $ \bar{\partial}\alpha=0\in H^{-1}\left({\Bbb C}P^{1}\smallsetminus\bigcup K_{i},\omega\otimes\bar{\omega}\right) $. Denote the the space of
$ H^{0} $-holomorphic forms in $ {\Bbb C}P^{1}\smallsetminus\bigcup K_{i} $ by $ {\cal H}^{\left(0\right)} $. \end{definition}
Since one can extend a $ H^{0} $-holomorphic form $ \alpha $ into any disk $ K_{i} $ by 0
without increasing $ H^{0} $-norm,
$ \alpha $ has a canonical extension to $ {\Bbb C}P^{1} $, and $ \bar{\partial}\alpha $ is concentrated on $ \bigcup\partial K_{i} $.
Moreover, if $ K_{i} $ has a neighborhood which does not intersect with other
disks, one can define a {\em boundary value\/} $ \alpha|_{\partial K_{i}} $ of $ \alpha $ on $ \partial K_{i} $, which is a
element of $ H^{-1/2}\left(\partial K_{i},\Omega_{\partial K_{i}}^{1}\right) $ (note that if $ \gamma $ is a curve in $ {\Bbb C}P^{1} $, then
$ \omega|_{\gamma}\simeq\Omega_{\gamma}^{1} $; one can get the $ H^{-1/2} $-restriction on the smoothness of the
boundary value in the same way as in Section~\ref{s2.70}). Now $ \bar{\partial}\alpha $ can be
described as the extension of $ -\alpha|_{\partial K_{i}} $ from $ \partial K_{i} $ to $ {\Bbb C}P^{1} $ by $ \delta $-function (see
~\eqref{equ3.02}). (Such an extension is a correctly defined continuous mapping
$ H^{-1/2}\left(\partial K_{i},\Omega_{\partial K_{i}}^{1}\right) \to H^{-1}\left({\Bbb C}P^{1},\omega\otimes\bar{\omega}\right) $.)
Since the norm in $ H^{1}\left({\Bbb C}P^{1}\right) $ can be described by a local (non-invariant)
formula $ \|\alpha\|_{l_{2}}^{2}+\|\alpha_{,x}\|_{l_{2}}^{2}+\|\alpha_{,y}\|_{l_{2}}^{2} $ (here $ x+iy $ is the local coordinate on
$ {\Bbb C}P^{1} $), functions with non-intersecting support are orthogonal in $ H^{1} $.
Dually, if $ U_{i} $ are pairwise non-intersecting, then the natural mapping
\begin{equation}
H^{-1}\left({\Bbb C}P^{1},\omega\otimes\bar{\omega}\right) \to \bigoplus_{l_{2}}H^{-1}\left(U_{i},\omega\otimes\bar{\omega}\right)
\notag\end{equation}
is a continuous epimorphism.
\begin{definition} Fix $ \varepsilon>0 $. Say that disks $ \left\{K_{i}\right\} $ have a {\em thickening\/} $ \left\{U_{i}\right\} $ if $ K_{i} $
have non-intersecting neighborhoods $ U_{i} $ such that a pair $ K_{i}\subset U_{i} $ is
conformally equivalent to $ \left\{|z|<1\right\}\subset\left\{|z|<1+\varepsilon\right\} $. Say that $ \left\{K_{i}\right\} $ have a {\em uniform
thickening\/} if the neighborhood $ U_{i} $ can be picked up to be concentric to $ K_{i} $
circles (assume that a metric on $ {\Bbb C}P^{1} $ is fixed). \end{definition}
(Note that the existence of uniform thickening does not depend on
the metric on $ {\Bbb C}P^{1} $.)
In particular, if $ \left\{K_{i}\right\} $ have a thickening $ \left\{U_{i}\right\} $, then the above
arguments show that $ \left(\|\alpha|_{\partial K_{i}}\|_{H^{-1/2}}\right)_{i\in I}\in l_{2} $. Note also that $ H^{-1/2}\left(\partial K_{i},\Omega_{\partial K_{i}}^{1}\right) $
is dual to $ H^{1/2}\left(\partial K_{i}\right) $, thus the subspace $ H_{\int=0}^{-1/2}\left(\partial K_{i},\Omega_{\partial K_{i}}^{1}\right) $ of forms with
vanishing integral has an $ \operatorname{PGL}\left(2,{\Bbb C}\right) $-invariant Hilbert structure (in the
same sense as in Section~\ref{s35.20}).
\begin{definition} Call the mapping $ \alpha \mapsto \left(\int_{\partial K_{i}}\alpha\right)_{i\in I} $, $ H^{0}\left({\Bbb C}P^{1}\smallsetminus\bigcup K_{i},\omega\right) \to l_{2} $ the
{\em integration mapping}. Fix an involution $ '\colon I\to I $ which interchanges two
parts of $ I $, $ I=I_{+}\amalg I_{+}' $, and fraction-linear identifications $ \varphi_{i}\colon \partial K_{i} \to \partial K_{i} $
(with the same conditions as in Section~\ref{s35.40}), then a {\em global
holomorphic form\/} $ \alpha $ is an element of $ {\cal H}^{\left(0\right)} $ such that $ \varphi_{i}^{*}\left(\alpha|_{\partial K_{i'}}\right)=\alpha|_{\partial K_{i}} $.
Define the {\em partial period mapping\/} $ P $ by restriction of the integration
mapping to global holomorphic forms, and taking integrals only for $ i\in I_{+} $.
Let $ \bar{M} $ be the curve obtained by gluing $ \partial K_{i} $ together via $ \varphi_{i} $. Denote by
$ \Gamma\left(\bar{M},\omega\right) $ the space of global holomorphic forms on $ \bar{M} $, i.e., the space of
global holomorphic forms compatible with $ \varphi_{\bullet} $. \end{definition}
The significant difference of this case and the case of Section
~\ref{s35.30} is that the mapping $ \bar{\partial}\colon H^{0}\left({\Bbb C}P^{1},\omega\right) \to H^{-1}\left({\Bbb C}P^{1},\omega\otimes\bar{\omega}\right) $ has no
null-space, but has $ 1 $-dimensional cokernel, thus it is not easy to
reconstruct $ \alpha $ basing on $ \bar{\partial}\alpha $ by local formulae. However, it is easy to
prove
\begin{lemma} \label{lm9.90}\myLabel{lm9.90}\relax Suppose that $ \left\{K_{i}\right\} $ has a thickening. Then the partial period
mapping $ P $ satisfies the identity
\begin{equation}
\dim \operatorname{Ker} P = \dim \operatorname{Ker} {\cal J}\colon {\cal H}^{\left(1\right)}/\operatorname{const} \to \bigoplus_{l_{2}}H^{1/2}\left(\partial K_{j}\right)/\operatorname{const}.
\notag\end{equation}
\end{lemma}
\begin{proof} We claim that $ \partial $ gives a mapping from one null-space to another
one. Let $ \alpha\in\operatorname{Ker} P $. Then $ \alpha|_{\partial K_{i}}\in H^{-1/2}\left(\partial K_{i},\Omega_{\partial K_{i}}^{1}\right) $ has an antiderivative
$ f_{i}\in H^{1/2}\left(\partial K_{i}\right) $, and the harmonic extension of $ f_{i} $ into $ K_{i} $ has a bounded
$ H^{1} $-norm. Since $ \alpha $ is closed in $ U_{i}\smallsetminus K_{i} $, it has an antiderivative. Since $ \partial $ is
elliptic, the antiderivative has smoothness $ H^{1} $. It is
possible to pick up the constant in such a way that two antiderivatives
coincide on $ \partial K_{i} $. Taking de Rham differential of resulting $ H^{1} $-function, we
see that one can extend $ \alpha $ into $ K_{i} $ preserving the closeness, and the
related increase of the norm of $ \alpha $ is bounded by $ \|\alpha|_{\partial K_{i}}\|_{H^{-1/2}} $.
Repeating this operation for all the $ K_{i} $, we obtain a closed
extension of $ \alpha $ to $ {\Bbb C}P^{1} $ with the $ H^{0} $-norm bounded by $ \|\alpha\|_{H^{0}} $ (here we use
locality of $ L_{2} $-norm). Thus $ \alpha $ has a $ H^{1} $-antiderivative $ f $ on $ {\Bbb C}P^{1}\smallsetminus\bigcup K_{i} $, thus
$ \alpha=df $, hence $ \alpha=\partial f $. Obviously, $ f\in\operatorname{Ker} {\cal J} $.
On the other hand, if $ f\in\operatorname{Ker} {\cal J} $, then $ \partial f\in\operatorname{Ker} P $.\end{proof}
\begin{remark} As we have seen in Section~\ref{s7.90}, in assumptions of Theorem
~\ref{th35.45} the dimension in the lemma is 0, thus $ P $ defines an inclusion of
global holomorphic forms into $ l_{2} $. Note that we do not claim that this
inclusion is a monomorphism---estimates in Section~\ref{s9.12} show that it is
not if $ \operatorname{card}\left(I\right)=\infty $. \end{remark}
\subsection{$ A $-Periods }\label{s9.12}\myLabel{s9.12}\relax To describe the image of the integration mapping,
consider the mapping $ \bar{\partial}^{-1}\colon H_{\int=0}^{-1}\left({\Bbb C}P^{1},\omega\otimes\bar{\omega}\right) \to H^{0}\left({\Bbb C}P^{1},\omega\right) $. It is a
continuous elliptic operator. To make the formulae localizable, fix a top
form $ \beta_{0} $ with integral 1, and extend $ \bar{\partial}^{-1} $ to any form on $ {\Bbb C}P^{1} $ by $ \bar{\partial}^{-1}\beta\buildrel{\text{def}}\over{=}
\bar{\partial}^{-1}\left(\beta-\beta_{0}\int_{{\Bbb C}P^{1}}\beta\right) $. This operator is still elliptic, but is not an
isomorphism any more.
Let $ \alpha\in{\cal H}^{\left(0\right)} $, consider $ \alpha|_{\partial K_{i}} $. In contrast with the cases of Section
~\ref{s35.20} and Proposition~\ref{prop5.28} the $ - $-components of $ \left(\alpha|_{\partial K_{i}}\right)_{i\in I} $ are not
arbitrary. To describe possible values of $ \left(\alpha|_{\partial K_{i}}\right)_{i\in I} $, break $ \alpha|_{\partial K_{i}} $ into
two parts, one $ \alpha_{i}^{\left(0\right)} $ with integral 0, another one $ \alpha_{i}^{\left(1\right)} $ proportional to
some fixed $ 1 $-form $ \mu_{i} $ on $ \partial K_{i} $. Use the same letters for
$ \delta $-function-extensions of these forms to $ {\Bbb C}P^{1} $. Suppose that $ 1 $-forms $ \mu_{i} $ (one
per disk boundary) are normalized to have integral 1 and have uniformly
bounded $ H^{-1/2} $-norms when $ i $ varies, and that disks $ K_{i} $ have a thickening.
Then $ \sum\alpha_{i}^{\left(1\right)} $ converges in $ \bigoplus_{l_{2}}H^{-1/2}\left(\partial K_{i}\right) $, same for $ \sum\alpha_{i}^{\left(0\right)} $. Note that if
conditions of Theorem~\ref{th35.45} hold, then $ \alpha^{\left(0\right)}=\bar{\partial}^{-1}\left(\sum\alpha_{i}^{\left(0\right)}\right)\in H^{0}\left({\Bbb C}P^{1},\omega\right) $.
Since $ \alpha=\bar{\partial}^{-1}\sum_{i}\alpha|_{\partial K_{i}} $, $ \alpha^{\left(1\right)}=\bar{\partial}^{-1}\left(\sum\alpha_{i}^{\left(1 \right)}\right)=\alpha-\bar{\partial}^{-1}\left(\sum\alpha_{i}^{\left(0\right)}\right) $ is also in $ H^{0}\left({\Bbb C}P^{1},\omega\right) $.
We see that any form $ \alpha\in{\cal H}^{\left(0\right)} $ can be represented as a sum $ \alpha^{\left(0\right)}+\alpha^{\left(1\right)} $,
$ \alpha^{\left(0\right)},\alpha^{\left(1\right)}\in{\cal H}^{\left(0\right)} $ such that integrals of $ \alpha^{\left(0\right)} $ around each disk $ K_{i} $ vanishes,
and $ \alpha^{\left(1\right)} $ is a linear combination of $ 1 $-forms $ \bar{\partial}^{-1}\mu_{i} $, $ i\in I $. Consider the
subspace $ {\cal H}_{0}^{\left(0\right)}\subset{\cal H}^{\left(0\right)} $ consisting of $ 1 $-forms which satisfy $ \int_{\partial K_{i}}\alpha=0 $ for any
$ i\in I $, and the subspace $ {\cal H}_{1}^{\left(0\right)} $ of forms satisfying $ \int_{\partial K_{i}}\alpha=\int_{\partial K_{i}'}\alpha $ for any $ i\in I $.
Let $ {\frak c}={\cal H}^{\left(0\right)}/{\cal H}_{0}^{\left(0\right)} $, $ {\frak c}''={\cal H}_{1}^{\left(0\right)}/{\cal H}_{0}^{\left(0\right)} $. The decompositon $ \alpha=\alpha^{\left(0\right)}+\alpha^{\left(1\right)} $ gives a
splitting of $ {\cal H}^{\left(0\right)} $ into a direct sum of $ {\cal H}_{0}^{\left(0\right)} $ and the span of $ \bar{\partial}^{-1}\mu_{i} $, $ i\in I $.
As a corollary, the images of forms $ \bar{\partial}^{-1}\mu_{i} $ form a basis in $ {\frak c} $, and the Gram
matrix of the pairing in $ {\frak c} $ can be calculated as $ \left(\bar{\partial}^{-1}\mu_{i},\bar{\partial}^{-1}\mu_{j}\right)_{L_{2}\left({\Bbb C}P^{1}\right)} $.
Similarly, projections of $ \bar{\partial}^{-1}\mu_{i}-\bar{\partial}^{-1}\mu_{i'} $, $ i\in I_{+} $, form a basis in $ {\frak c}'' $, and one
can easily calculate the Gram matrix of this basis. Let us estimate
elements of these Gram matrices.
We may assume that $ \infty\in K_{i_{0}} $ for some $ i_{0}\in I $, and that $ \operatorname{Supp}\beta_{0}\subset K_{i_{0}} $. Then
the operator $ \bar{\partial}^{-1} $ restricted to $ {\Bbb C}P^{1}\smallsetminus K_{i_{0}} $ has $ \frac{dx}{y-x} $ as a kernel, and we
may suppose that $ \mu_{i} $ is $ d\vartheta_{i}/2\pi $, $ \vartheta_{i} $ being the natural angle coordinate on
$ \partial K_{i}\subset{\Bbb C} $. Let us calculate the Gram matrix $ \left(G_{ij}\right) $ for $ \bar{\partial}^{-1}d\vartheta_{i}\in{\cal H}^{\left(0\right)} $. If the
radius of $ K_{i} $ is $ r_{i} $, and the distance between centers of $ K_{i} $ and $ K_{j} $ is $ d_{ij} $,
then $ G_{ij} $ can be estimated as
\begin{equation}
\int\Sb r_{i}<|z|<C \\ |z-d_{ij}|>r_{j}\endSb\frac{1}{z\overline{\left(z-d_{ij}\right)}}dx\,dy
\notag\end{equation}
here $ C\gg0 $ (and depends on $ \beta_{0} $), $ z=x+iy. $ Note that the integral over $ |z|>C/2 $
is greater than a constant which does not depends on $ d_{ij} $ and $ r_{i},r_{j} $, thus
this integral is bounded from below. If $ i=j $, then it behaves as $ \log \frac{1}{r} $.
If $ i\not=j $, then the part outside the disk of radius $ 2d_{ij} $ is $ \sim\log \frac{1}{d_{ij}} $, and
the part inside this disk is scaling-invariant. Thus to estimate the
second part one may assume $ d_{ij}=1 $. However, the integral converges
absolutely inside the whole disk $ |z|<2 $, thus this part is bounded.
Thus we have estimates for the elements of the Gram matrix\footnote{Note that these estimates may be not sufficient to describe the Hilbert
structure on $ {\frak c} $ up to equivalence.}.
Knowledge of this Gram matrix gives a complete description of the
subspace $ {\cal B} $ of $ H^{-1}\left({\Bbb C}P^{1},\omega\otimes\bar{\omega}\right) $ spanned by extensions-by-$ \delta $-function of forms
on $ \partial K_{i} $, $ i\in I $.
Similarly one can estimate the elements of the Gram matrix for $ {\frak c}'' $.
Since elements of $ {\frak c}'' $ correspond to $ 2 $-forms with integral 0, this Gram
matrix does not change when we apply a conformal transformation to disks
$ K_{i} $. The diagonal elements are $ \frac{l_{i i'}}{\pi} $, and the off-diagonal ones are $ C
\frac{1}{\pi}\log |\lambda\left(c_{i},c_{i'},c_{j},c_{j'}\right)|+O\left(\sum\varepsilon_{k}r_{k}\right) $, here $ \lambda $ is the double ratio, $ c_{\bullet} $ and
$ r_{\bullet} $ are the center and radius of $ K_{\bullet} $. The constants $ \varepsilon_{\bullet} $ can be calculated as
$ c_{i}=\frac{c_{j}-c_{j'}}{\left(c_{i}-c_{j}\right)\left(c_{i}-c_{j'}\right)} $.
\begin{remark} Note that we have estimates for the elements of this Gram
matrix. It is easy to check that if the matrix $ \left(r_{i}/d_{ij}\right)_{i,j\in I} $ gives a
compact operator $ l_{2} \to l_{2} $, these estimates allow one to reconstruct the
Hilbert norm up to equivalence, thus to describe the space $ {\frak c}'' $ completely.
\end{remark}
Note that the diagonal entries of the Gram matrix for $ {\frak c} $ are not
bounded. Since any element in the image of the integration mapping has a
finite norm w.r.t. this matrix, this shows, in particular, that the image
of the integration mapping does not coincide with $ l_{2} $. Similarly, the Gram
matrix for $ {\frak c}'' $ contains arbitrarily big elements, thus $ {\frak c}'' $ also differs
from $ l_{2} $, thus image of partial period mapping differs from $ l_{2} $. As we will
see it in Section~\ref{s9.40}, this image coincides with the space $ {\frak c}' $ defined
below.
So far the spaces we consider depended on the relative position of
disks $ K_{i} $ only. Now suppose that a fraction-linear orientation-changing
identification $ \varphi_{i} $ of $ \partial K_{i} $ and $ \partial K_{i'} $ is fixed, and consider the subspace $ {\cal B}' $
of $ {\cal B} $ which is spanned by extensions-by-$ \delta $-function of forms on $ \partial K_{i} $, $ i\in I $,
which are preserved by the identifications $ \varphi_{i} $. Let $ {\cal B}_{0} $ be the subspace of
$ {\cal B} $ spanned by extensions-by-$ \delta $-function of forms on $ \partial K_{i} $, $ i\in I $, with integral
0 along each $ \partial K_{i} $, let $ {\cal B}_{0}'={\cal B}'\cap{\cal B}_{0} $.
It is clear that $ {\cal B}/{\cal B}_{0}\simeq{\frak c} $, Let us describe $ {\cal B}'/{\cal B}_{0}' $. Obviously, the
image of $ {\cal B}'/{\cal B}_{0}' $ in $ {\frak c} $ is a subspace of $ {\frak c}'' $, however, if the identifications
$ \varphi_{i} $ ``squeeze'' the circles $ \partial K_{i} $ too much, this image may be a proper
subspace of $ {\frak c}'' $.
Indeed, the dual statement to Remark~\ref{rem35.35} shows that one can
define a $ \operatorname{PGL}\left(2,{\Bbb R}\right) $-invariant norm on $ H_{\int=0}^{-1/2}\left(S^{1},\Omega^{\text{top}}\right) $ which is equivalent
to Hilbert norm. However, there is no $ \operatorname{PGL}\left(2,{\Bbb R}\right) $-invariant norm on the
whole space $ H^{-1/2}\left(S^{1},\Omega^{\text{top}}\right) $. For $ g\in\operatorname{PGL}\left(2,{\Bbb R}\right) $ consider how $ g $ changes the
Sobolev norm on $ H^{-1/2}\left(S^{1},\Omega^{\text{top}}\right) $. It is enough to estimate $ N\left(g\right)=\|g^{*}\left(\beta\right)\| $, $ \beta $
being an arbitrary form with non-zero integral. It is clear that
different choices of Sobolev norm on $ H^{-1/2}\left(S^{1},\Omega^{\text{top}}\right) $\footnote{Recall that it is defined up to equivalence only.} and different
choices of $ \beta $ would change $ N\left(g\right) $ to something of the form $ CN\left(g\right)+O\left(1\right) $ only.
Since $ \operatorname{PGL}\left(2,{\Bbb R}\right) $ has a compact subgroup $ U\left(1\right) $ of dimension 1, it is enough
to estimate $ N\left(g\right) $ on double classes $ U\left(1\right)\backslash\operatorname{PGL}\left(2,{\Bbb R}\right)/U\left(1\right) $ only, thus one can
assume $ g=\operatorname{diag}\left(\lambda,1\right) $, $ \lambda\geq1 $.
In turn, it is sufficient to estimate $ H^{-1/2} $-norm of $ \frac{\lambda dx}{1+\lambda^{2}x^{2}} $ on
$ {\Bbb R} $, and one can do it explicitly since the Fourier transfrom can be easily
calculated, it is proportional to $ e^{-\lambda|\xi|} $. Thus $ N\left(g\right)=C \log \lambda + O\left(1\right) $. Using
this, one can easily obtain
\begin{proposition} Consider the image $ {\frak c}' $ of $ {\cal B}'/{\cal B}_{0}' $ in $ {\frak c}={\cal B}/{\cal B}_{0} $. Consider $ \varphi_{i} $ as
elements of $ \operatorname{SL}\left(2,{\Bbb R}\right)\subset{\Bbb R}^{4} $. The space $ {\frak c}' $ consists of sequences $ \left(p_{i}\right)\in{\frak c}'' $ which
satisfy $ \left(p_{i}\log \|\varphi_{i}\|\right)\in l_{2} $. \end{proposition}
We see that $ {\frak c}' $ can be described by Gram matrix
$ G'_{ij}=\delta_{ij}\log \|\varphi_{i}\|+G_{ij}'' $, $ G'' $ being the Gram matrix for $ {\frak c}'' $.
\subsection{$ B $-periods }\label{s9.41}\myLabel{s9.41}\relax What we defined in Section~\ref{s9.20} was integration
of global holomorphic forms along $ A $-cycles. Finite-genus theory shows
that it is important to study additional integrals along $ B $-cycles. They
should depend on the choice
of cuts on the Riemann surface, and we are not in the conditions when one
can easily proceed with such cuts.
Since we have only $ H^{0} $-smoothness of global holomorphic forms, one
cannot invent a priori bounds on integrals of these forms along arbitrary
curves. Indeed, the ``infinity'' $ M_{\infty}\subset{\Bbb C}P^{1} $ can break the complex sphere into
infinitely many
connected components (see the example in Section~\ref{s0.40}), thus one cannot
assume that the cuts do not
intersect $ M_{\infty} $. Moreover, even if there is only one connected component, it
is not clear how to make infinitely many cuts in $ {\Bbb C}P^{1} $ which would not
intersect each other. In fact it {\em is\/} possible to make such cuts, but in
general the lengths of these cuts form a quickly increasing sequence.
However, if one has a strip $ \left(-\varepsilon,\varepsilon\right)\times\left(a,b\right) $ embedded into the complex
curve, then the value of $ \int_{a}^{b} $ averaged along $ \left(-\varepsilon,\varepsilon\right) $ is well defined and
may be bounded as $ O\left(\varepsilon^{-1/2}\right) $. Indeed, this average is $ L_{2} $-pairing
\begin{equation}
\frac{1}{2\varepsilon}\int\alpha\wedge dx
\notag\end{equation}
with $ dx/\varepsilon $, here $ \left(x,y\right) $ are coordinates on the strip $ -\varepsilon<x<\varepsilon $.
Thus we are not going to define the $ B $-periods as integrals over
curves, but as some averaged integrals. On the level of would-be homology
of the surface we will pair the form with cycles on the surface which are
not ``geometric'' cycles, but linear combinations of them.
Consider the involution ' and gluings $ \varphi_{i} $. Suppose that $ 0\in K_{i_{0}} $,
$ \infty\in K_{i_{0}'} $, and that the disks $ \left\{K_{i}\right\} $ have a uniform thickening.
Parameterize
the set of rays going from 0 to $ \infty $ by the angle $ \vartheta $, and deform each ray
slightly in such a way that it avoids all the circles $ K_{i} $. We require
that deformations of two rays which differ by the angle $ \Delta\vartheta $ do not become
closer than $ \varepsilon\cdot\Delta\vartheta $ (outside of $ K_{i_{0}} $). Say, let $ \widetilde{K}_{i} $ be the concentric with $ K_{i} $
disk of radius $ \left(1+2\varepsilon\right)\operatorname{radius}\left(K_{i}\right) $, and deform the ray $ {\cal R} $ inside $ \widetilde{K}_{i} $ so that
it moves along an appropriate arc going between $ K_{i} $ and $ \widetilde{K}_{i} $. If
the
$ \operatorname{dist}\left({\cal R},\operatorname{center}\left(K_{i}\right)\right)/\operatorname{radius}\left(\widetilde{K}_{i}\right)=\rho\leq1 $, one can take the radius of the arc to
be $ \left(1+\varepsilon+\varepsilon\rho\right)\operatorname{radius}\left(K_{i}\right) $.
In fact we need to choose whether the arcs are going to leave the
disk $ K_{i} $
on the right or on the left. We use the following algorithm: let $ O_{i} $ be
the fixed point of $ \varphi_{i} $ inside $ K_{i} $. If the ray $ {\cal R} $ leaves $ O_{i} $ on the right, let
the deformation leave $ K_{i} $ on the right, otherwise leave it on the left.
(This choice is going to be important in the proof of Proposition
~\ref{prop8.33}. Note that the choice of the direction of the turnout does not
depend on the metric on $ {\Bbb C}P^{1} $.)
Now we assume that 0 and $ \infty $ are fixed points of $ \varphi_{i_{0}} $. As a last
correction, in a neighborhood of $ K_{i_{0}'} $ change a ray $ {\cal R} $ to a logarithmic
spiral so that the intersection of $ {\cal R} $ with $ \partial K_{i_{0}} $ and intersection of $ {\cal R} $ with
$ \partial K_{i_{0}'} $ are glued together by $ \varphi_{i_{0}} $. Say, if $ \partial K_{i_{0}'} = \left\{z \mid |z|=R\right\} $, take the
part of a spiral $ d\vartheta = C\,dr/\varepsilon r $ inside $ \left\{z \mid e^{-\varepsilon}R <|z|<R\right\} $, here $ C=\operatorname{Arg} \varphi_{i_{0}} $ is
defined up to addition of a multiple of $ 2\pi $. Now each deformed ray
represents a closed curve after the gluing by $ \varphi_{i_{0}} $ is performed.
Under these conditions the averaged over $ \vartheta\in\left[0,2\pi\right] $ integral over the
deformed rays (between $ \partial K_{i_{0}} $ and $ \partial K_{i_{0}'} $) is correctly defined. It
represents a combination of cycles, thus is a cycle itself. Denote this
linear functional on $ {\cal H}^{\left(0\right)}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $ by $ {\cal Q}_{i_{0}} $.
Restricting this linear functional to global holomorphic forms, we
call the integral along this cycle $ B $-{\em period\/} of the global holomorphic
form {\em from\/} $ \partial K_{i_{0}} $ {\em to\/} $ \partial K_{i_{0}'} $. The above description shows that
\begin{lemma} $ B $-period of $ \alpha $ from $ \partial K_{i} $ to $ \partial K_{i'} $ is bounded by $ C\cdot l_{i i'}\|\alpha\|_{L_{2}} $, here
$ l_{i i'} $ is the conformal distance between $ \partial K_{i_{0}} $ and $ \partial K_{i_{0}'} $. The constant $ C $
depends on $ \varepsilon $ only. \end{lemma}
\begin{definition} Associate to $ \alpha\in\Gamma\left(\bar{M},\omega\right) $ the sequence $ \left(q_{j}\right)_{j\in I_{+}} $, $ q_{j} $ being the
$ B $-period of $ \alpha $ from $ \partial K_{j} $ to $ \partial K_{j'} $. Denote this mapping by $ Q $. \end{definition}
\begin{remark} Note that in the case of finite genus one chooses the
$ B $-cycles to be non-intersecting. This assures that the matrix of periods
is symmetric. The above construction takes average of
different cycles connecting given points, thus one can get an impression
that the resulting matrix of periods will have much worse properties than
in the standard settings.
However, the properties turn out to be exactly the same, due to our
choice of direction of turnouts, and the following surprising result: \end{remark}
\begin{proposition} Consider 4 different points $ x_{0},x_{1},y_{0},y_{1}\in{\Bbb C}P^{1} $. Parameterize
the set of (circular) arcs connecting $ x_{0} $ with $ x_{1} $ by the angle at $ x_{0} $, and
do the same with arcs connecting $ y_{0} $ with $ y_{1} $. This provides a measure on
the set of arcs connecting $ x_{0} $ with $ x_{1} $, same for $ y_{0} $ and $ y_{1} $. Then the
average index of intersection of an arc $ x_{0}x_{1} $ with an arc $ y_{0}y_{1} $ is 0. \end{proposition}
\subsection{Space of periods }\label{s9.40}\myLabel{s9.40}\relax In Section~\ref{s9.12} we have shown that under
mild assumption the space $ {\cal H}^{\left(0\right)} $ (which, loosely speaking, consists of
holomorphic forms with jumps along the cuts on the curve) is a sum of
two components: forms with integrals 0 along $ A $-cycles (which can be
described by locale data on each cut), and some explicitely defined
Hilbert subspace $ {\frak c} $ of ``small'' dimension (two basis vectors per each cut
on the curve). Only the elements of $ {\cal H}^{\left(0\right)} $ which have equal integrals along
two sides of the cut have a chance to correspond to a global holomorphic
form on $ \bar{M} $, thus only subspace $ {\frak c}'\subset{\frak c} $ is interesting for us.
\begin{theorem} Suppose that the matrix $ \left(e^{-l_{ij}}-\delta_{ij}\right) $ defines a compact
operator $ l_{2} \to l_{2} $, and the disks $ K_{i} $ have a thickening. Then the partial
period mapping $ P $ sends global holomorphic forms to elements of $ {\frak c}' $.
Moreover, it is an isomorphism onto $ {\frak c}' $. \end{theorem}
\begin{proof} The first statement is an immediate corollary of the
description of the elements of $ {\cal H}^{\left(0\right)} $ via the space $ {\cal B}' $ in Section~\ref{s9.12}.
We start the proof of the second one by showing that $ P $ is a component of
a Fredholm operator of index 0.
In the notations of Section~\ref{s9.12}, consider the mapping $ {\cal H}_{1}^{\left(0\right)} \xrightarrow[]{\pi}
{\frak c}'' $. Let $ {\cal H}_{2}^{\left(0\right)}=\pi^{-1}\left({\frak c}'\right) $. Then the mapping
\begin{equation}
{\cal J}_{\omega}\colon {\cal H}_{2}^{\left(0\right)} \to \bigoplus\Sb l_{2} \\ i\in I_{+}\endSb H_{\int=0}^{-1/2}\left(\partial K_{i},\Omega_{\partial K_{i}}^{1}\right)\colon \alpha \mapsto \left(\alpha|_{\partial K_{i}}-\varphi^{*}\left(\alpha|_{\partial K_{i'}}\right)\right).
\notag\end{equation}
is continuous. The same arguments as in the proof of Theorem~\ref{th35.45}
show that $ {\cal J}_{\omega}|_{{\cal H}_{0}^{\left(0\right)}} $ is a Fredholm operator of index 0. Combining this
mapping with the projection $ {\cal H}_{1}^{\left(0\right)} \xrightarrow[]{\pi} {\cal H}_{2}^{\left(0\right)}/{\cal H}_{0}^{\left(0\right)}={\frak c}' $, we see that $ \alpha \mapsto
\left({\cal J}_{\omega}\left(\alpha\right),\pi\left(\alpha\right)\right) $ is a Fredholm mapping of index 0. If $ \alpha $ is in the null-space
of this mapping, then $ \alpha $ is a non-trivial global holomorphic form with
vanishing $ A $-periods, thus $ \alpha\in\operatorname{Ker} P $, and $ \alpha=0 $. Hence this mapping is an
isomorphism. Since $ P=\pi|_{\operatorname{Ker}{\cal J}_{\omega}} $, it is an isomorphism as well. \end{proof}
Let $ \alpha_{i}=\bar{\partial}^{-1}\beta_{i} $. The theorem says that for any element $ \left(c_{i}\right)\in{\frak c}' $ one can
find an element $ \alpha\in{\cal H}^{\left(0\right)} $ such that integrals of $ \alpha $ along $ \partial K_{i} $ are 0 for any
$ i\in I $, and $ \sum c_{i}\alpha_{i}+\alpha $ is a global holomorphic form (which automatically has
$ A $-periods $ c_{i} $). Moreover, $ \|\alpha\|_{{\cal H}^{\left(0\right)}}\leq C\cdot\|\left(c_{i}\right)\|_{{\frak c}'} $.
\begin{proposition} \label{prop9.42}\myLabel{prop9.42}\relax Let $ \beta\in{\cal H}^{\left(0\right)} $, and integrals of $ \beta $ along $ \partial K_{i} $ are 0 for any
$ i\in I $, let $ \left(c_{i}\right)\in{\frak c}' $. Then $ \left|\sum_{j\in I_{+}}c_{j}{\cal Q}_{j}\left(\beta\right)\right| \leq C\|\beta\|_{{\cal H}^{\left(0\right)}}\cdot\|\left(c_{i}\right)\|_{{\frak c}'} $ for an
appropriate $ C $ which does not depend on $ \beta $ and $ \left(c_{i}\right) $. \end{proposition}
\begin{proof} Suppose that $ \beta $ can be holomorphically continued into all disks
$ K_{i} $ except $ K_{i_{0}} $. Then $ \beta=\partial f $, and $ f $ is holomorphic outside $ K_{i_{0}} $. Let
$ j,j'\not=i_{0} $. The
construction of $ {\cal Q}_{j} $ shows that one can calculate $ {\cal Q}_{j}\left(\beta\right) $ as $ f\left(y_{1}\right)-f\left(y_{0}\right) $, $ y_{0} $,
$ y_{1} $ being two fixed points of $ \varphi_{j} $. Let $ \Psi_{j} $ be the fraction-linear function
with a zero and a pole at $ y_{0} $ and $ y_{1} $. One can momentarily see that
$ {\cal Q}_{j}\left(\beta\right)=\int_{\partial K_{i_{0}}}\beta\log \Psi_{j} $. Similarly, $ {\cal Q}_{i_{0}}\left(\beta\right)=0 $.
Similar statements are true for forms $ \beta $ such that they can be
holomorphically continued into all the disks $ K_{i} $ except a finite number.
Since any form $ \beta $ which satisfies conditions of the proposition can be
approximated by such forms, we see that it is sufficient to show that
$ \sum_{j\not=i,i'}c_{j}\log \Psi_{j} $ converges in $ \bigoplus_{l_{2}}H^{1/2}\left(\partial K_{i}\right)/\operatorname{const} $, or that $ \sum_{j\not=i,i'}c_{j}d\Psi_{j}/\Psi_{j} $
converges in $ \bigoplus H^{-1/2}\left(\partial K_{i},\Omega^{1}\right) $. In turn, it is sufficient to show
convergence of $ \sum_{j\not=i,i'}c_{j}d\Psi_{j}/\Psi_{j} $ in $ \bigoplus_{l_{2}}H^{0}\left(K_{i},\omega\right)=L_{2}\left(\bigcup K_{i},\omega\right) $. On the other
hand, if $ \sigma_{j} $ is 1 in $ K_{j} $ and $ K_{j'} $, and 0 otherwise, then
$ \bar{\partial}\sum_{j\not=i,i'}c_{j}\sigma_{j}d\Psi_{j}/\Psi_{j} =\sum_{j\not=i,i'}c_{j}\beta_{j}\in H^{-1} $, hence $ \sum c_{j}\sigma_{j}d\Psi_{j}/\Psi_{j} $ converges in $ L_{2} $. \end{proof}
\begin{proposition} \label{prop9.25}\myLabel{prop9.25}\relax Let $ \left(d_{i}\right),\left(c_{i}\right)\in{\frak c}' $, and $ \alpha\in{\cal H}^{\left(0\right)} $ such that integrals of
$ \alpha $ along $ \partial K_{i} $ are 0 for any $ i\in I $, and $ \alpha_{c}=\sum c_{i}\alpha_{i}+\alpha $ is a global holomorphic
form on $ \bar{M} $ (automatically with $ A $-periods $ c_{i} $). Then $ \sum d_{j}{\cal Q}_{j}\left(\alpha_{c}\right)
\leq C\|\left(c_{i}\right)\|_{{\frak c}'}\cdot\|\left(d_{i}\right)\|_{{\frak c}'} $. \end{proposition}
\begin{proof} It is sufficient to show that $ \sum d_{j}{\cal Q}_{j}\left(\sum c_{i}\alpha_{i}\right)\leq C_{1}\|\left(c_{i}\right)\|_{{\frak c}'}\cdot\|\left(d_{i}\right)\|_{{\frak c}'} $. In
the notations of the previous proposition $ \alpha_{i}-\alpha_{i'}=d\Psi_{i}/\Psi_{i} $ outside of $ K_{i} $ and
$ K_{i'} $. One can assume that only a finite number of $ c_{i} $ and $ d_{i} $ is non-zero.
Then $ \sum_{i\in I}c_{i}\alpha_{i} $ is $ \sum_{i\in I_{+}}c_{i}d\Psi_{i}/\Psi_{i} $.
Now it should be obvious that $ {\cal Q}_{j}\left(\alpha_{i}\right)=\int\sigma_{i}\sigma_{j}\alpha_{i}\wedge\alpha_{j} $, which finishes the
proof. \end{proof}
\subsection{Period matrix }\label{s9.60}\myLabel{s9.60}\relax Since $ P $ is an isomorphism $ \Gamma\left(\bar{M},\omega\right) \to {\frak c}' $, one can
consider
the mapping $ \Omega=Q\circ P^{-1} $. Write this mapping using coordinate ``basis'' in $ {\frak c}' $:
\begin{definition} Let $ \widetilde{\alpha}_{j} $, $ j\in I_{+} $, be the global holomorphic form on $ \bar{M} $ such that
$ P\left(\widetilde{\alpha}_{i}\right) $ has 1 on $ j $-th position, $ -1 $ at $ j' $-position, 0 at the other
positions. Let $ \Omega_{ij} $ be the $ B $-period of $ \widetilde{\alpha}_{j} $ from $ \partial K_{i} $ to $ \partial K_{i'} $, $ i\in I_{+} $. \end{definition}
\begin{proposition} \label{prop8.33}\myLabel{prop8.33}\relax Let $ \alpha\in\Gamma\left(\bar{M},\omega\right) $ and the sequence $ \left(p_{j}\right)=P\left(\alpha\right) $ has only a
finite number of non-zero elements. Let $ Q\left(\alpha\right)=\left(q_{j}\right) $. Then $ \|\alpha\|_{L_{2}}^{2}=
i\sum_{I_{+}}\left(\bar{p}_{j}q_{j}-p_{j}\bar{q}_{j}\right) $. \end{proposition}
\begin{proof} Consider a representative of $ \alpha $ in $ {\cal H}^{\left(0\right)}\left({\Bbb C}P^{1},\left\{K_{i}\right\}\right) $. Since all the
$ A $-periods of $ \alpha $ but a finite number are 0, $ \alpha $ can be extended (without
changing the norm too much) into all the disks but a finite number
preserving the closeness. Consider the remaining disks. Since the
integral
of $ \alpha $ around $ K_{i} $ is opposite to the integral around $ K_{i'} $, we see that
if we
connect $ K_{i} $ and $ K_{i'} $ by a cut, then the integral of $ \alpha $ around the resulting
hole is 0. Make smooth cuts which connect the remaining disks pairwise
(according to $ ' $) and do not intersect. One can suppose that two ends
of the cut---one on $ \partial K_{i} $, another on $ \partial K_{i'} $---are identified by $ \varphi_{i} $.
After the cuts are performed, on the resulting domain $ 1 $-form $ \alpha $ is
closed, and the integral along any component of (piecewise-smooth)
boundary is 0. Thus one can write $ \alpha = df $, $ f $ being a function of
smoothness $ H^{1} $. The restriction of $ f $ to any smooth curve is well-defined,
and is of smoothness $ H^{1/2} $. When one goes from one side of the cut to
another one along $ \partial K_{i} $, $ f $ grows by $ p_{i} $. Let $ \overset{\,\,{}_\circ}{q}_{i} $ be the change of $ f $ when one
goes from $ \partial K_{i} $ to $ \partial K_{i'} $ along the cut (choosing the side of the cut so that
the direction is counterclockwise).
Note that $ f $ is holomorphic near $ \partial K_{i} $ and $ \partial K_{i'} $ thus the {\em value\/} of $ f $ at
points is well-defined, thus the change of $ f $ along the cut is
well-defined. (In generic point $ z $ of the cut $ f\left(z\right) $ is not correctly
defined, since $ f $ is only of smoothness $ H^{1/2} $.)
Let $ \gamma_{i} $ be the part of the boundary of the domain consisting of the
circles $ \partial K_{i} $, $ \partial K_{i'} $ and both sides of the cut which connects
them.
Now take into account that $ \|\alpha\|_{L_{2}}^{2}=i\int\alpha\wedge\bar{\alpha}=i\int\partial f\wedge\bar{\partial}\bar{f} $. Proceeding as in
Section~\ref{s7.90}, we see that the only change to the arguments is that
instead of taking integrals along $ \partial K_{i}\cup\partial K_{i'} $, one needs to take some
integrals along $ \gamma_{i} $. As there, the integral along $ \partial K_{i}\cup\partial K_{i'} $ vanishes, so
what remains is
\begin{equation}
\|\alpha\|_{L_{2}}^{2}=-i\sum_{i}\int_{\gamma_{i}}f\,d\bar{f}
\notag\end{equation}
summation being over $ i $ such that $ p_{i}\not=0 $. The cycle $ \gamma_{i} $ consists of 4 parts:
two going around $ \partial K_{i} $ and $ \partial K_{i'} $, another two going along sides of the cut.
On the first two parts $ d\bar{f} $ are identified via $ \varphi_{i} $ (with opposite signs),
and $ f $ differs by $ \overset{\,\,{}_\circ}{q}_{i} $, thus the total integral is $ -\bar{p}_{i}\overset{\,\,{}_\circ}{q}_{i} $. On the second two
$ d\bar{f} $ coincide (but the orientation is opposite), and $ f $ differs by $ p_{i} $, thus
the total integral is $ p_{i}\overline{\overset{\,\,{}_\circ}{q}_{i}} $.
What remains to prove is that we may substitute $ q_{i} $ instead of $ \overset{\,\,{}_\circ}{q}_{i} $. To
do this one needs to investigate the relationship between $ q_{i} $ and $ \overset{\,\,{}_\circ}{q} $.
First, one can describe $ \overset{\,\,{}_\circ}{q}_{i} $ as an integral of $ df $ along one side of the
cut. Indeed, though $ d\left(f|_{\gamma_{i}}\right) $ is of smoothness $ H^{-1/2} $ on each part of $ \gamma_{i} $, it
is actually analytic near circles $ \partial K_{i} $, $ \partial K_{i'} $, thus the pairing with the
fundamental cycle of the interval (which has jumps at the ends of the
interval!) is well-defined.
Second, one can suppose that the cuts are in fact piecewise-smooth
(as far as non-smooth points are in $ {\Bbb C}P^{1}\smallsetminus\overline{\bigcup K_{i}} $), and consist of arcs of
circles. Let us recall that $ q_{i} $ is the average integral of $ \alpha $ along ``rays''
which connect 0 and $ \infty $ (after an appropriate choice of coordinate system).
Here ``rays'' are curves with differ from rays inside circles $ \widetilde{K}_{j} $ only, and
consist of arcs of circles. What is more, one does not need to deform a
ray into a ``ray'' inside the circle $ K_{j} $ as far as $ p_{j}=0 $, since a closed
continuation of $ \alpha $ inside $ K_{j} $ is already fixed, thus the deformation does
not change the value of the integral along the ``ray''. Note that each
ray intersects cuts along a finite number of points (except a finite
number of rays which may contain whole pieces of cuts), thus the same is
true for ``rays''.
We see that the integral of $ \alpha $ along each ``ray'' is now well defined
(recall that in general setting only the average was well-defined),
moreover, it is easy to calculate this integral using representation
$ \alpha=df. $ The integral along a ``ray'' $ {\cal R} $ is equal to the change of $ f $ on
the ends minus the jumps of $ f $ at the finite number of points where ``ray''
intersects cuts, thus it is
\begin{equation}
f\left(\operatorname{end}\left({\cal R}\right)\right)-f\left(\operatorname{start}\left({\cal R}\right)\right)-\sum_{j}n_{ij}p_{j},\qquad n_{ij}\in{\Bbb Z}\text{, }i,j\in I.
\notag\end{equation}
The change of $ f $ is equal to $ \overset{\,\,{}_\circ}{q}_{i} $, thus the average value of the integral is
$ \overset{\,\,{}_\circ}{q}_{i} $ minus sum of some real multiples of $ p_{j} $:
\begin{equation}
q_{i}=\overset{\,\,{}_\circ}{q}_{i}-\sum_{j}\nu_{ij}p_{j},\qquad \nu_{ij}\in{\Bbb R}\text{, }i,j\in I,
\notag\end{equation}
and $ \nu_{ij} $ are averaged values of $ n_{ij} $.
Finally, use our choice of direction of turnout around the disk $ K_{i} $. It
insures that the following fact is true:
\begin{lemma} $ \nu_{ij}=\nu_{ji} $ if $ i\not=i $. \end{lemma}
\begin{proof} Extend the cut between $ \partial K_{j} $ and $ \partial K_{j'} $ to fixed points of $ \varphi_{j} $ (one
inside each of $ K_{j} $ and $ K_{j'} $) along
straight intervals. Call the resulting curve $ \gamma'_{j} $. Then $ 2\pi\nu_{ij} $ is the
change of $ \operatorname{Arg} z $ along the curve $ \gamma_{j}' $ (we again suppose that 0 and $ \infty $ are
fixed points of $ \varphi_{i} $). Similarly, $ \pi\nu_{ji} $ is the change of $ \operatorname{Arg}\Psi_{j}\left(z\right) $ along $ \gamma'_{i} $,
here $ \Psi_{j}\left(z\right) $ is a fraction-linear function with a pole and a zero at fixed
points of $ \varphi_{j} $.
We need to show that $ \operatorname{Im} \int_{\gamma_{j}'}\frac{dz}{z}+\int_{\gamma_{i}'}\frac{d\Psi_{j}\left(z\right)}{\Psi_{j}\left(z\right)}=0 $. However,
\begin{equation}
\int_{\gamma_{j}'}\frac{dz}{z}=\frac{1}{2\pi i} \int_{\gamma_{j}'}\frac{dz}{z}\operatorname{Jump}\left(\log \Psi_{j}\left(z\right)\right) =\frac{1}{2\pi i}\int_{\gamma_{j}''}\log \Psi_{j}\left(z\right) d \log z
\notag\end{equation}
here we take an arbitrary branch of $ \log \Psi\left(z\right) $ defined outside of $ \gamma_{j}' $, and
$ \gamma_{j}'' $ is a loop around $ \gamma_{j}' $. Now the identity is obvious, since $ \gamma_{j}'' $ is
homotopic to $ -\gamma_{i}'' $. \end{proof}
The lemma implies that plugging in $ \overset{\,\,{}_\circ}{q}_{i} $ instead of $ q_{i} $ into the formula
of the proposition gives the same value, which finishes the proof of
Proposition~\ref{prop8.33}. {} \end{proof}
\begin{proposition} \label{prop8.35}\myLabel{prop8.35}\relax Let $ \alpha,\beta\in\Gamma\left(\bar{M},\omega\right) $ and the sequences $ \left(p_{j}\right)=P\left(\alpha\right) $, $ \left(p_{j}'\right)=P\left(\beta\right) $
have only a finite number of non-zero elements. Let $ Q\left(\alpha\right)=\left(q_{j}\right) $,
$ Q\left(\beta\right)=\left(q_{j}'\right) $. Then $ \sum_{I_{+}}\left(p_{j}q'_{j}-p'_{j}q_{j}\right) = $ 0. \end{proposition}
\begin{proof} The proof of Proposition~\ref{prop8.33} with minor changes is
applicable, the integral to consider is $ 0=\int\alpha\beta $. \end{proof}
The following statement is an immediate corollary of Proposition
~\ref{prop9.25}:
\begin{proposition} In the conditions of the previous proposition
$ \sum_{I_{+}}p_{j}q'_{j}\leq C\|\alpha\|\cdot\|\beta\| $. \end{proposition}
\begin{corollary} The period matrix $ \Omega_{ij} $ is symmetric, $ \operatorname{Im}\Omega_{ij} $ is a positive
real symmetric matrix, and the matrix $ \Omega_{ij} $ defines a bounded symmetric
form on the Hilbert space with the pairing given by the Gram matrix
$ \operatorname{Im}\Omega_{ij} $. \end{corollary}
\subsection{Bounded Jacobian as a torus }\label{s9.70}\myLabel{s9.70}\relax In Section~\ref{s7.90} we have seen that
(under mild assumptions) the Jacobian coincides with the constant
Jacobian. Given a topological space $ S $ and a set $ I $, let $ S_{l_{\infty}}^{I}\buildrel{\text{def}}\over{=}\bigcup_{K}K^{I} $,
here $ K $ runs over compact subsets of $ S $. Obviously, there is a surjection
from $ \left({\Bbb C}^{*}\right)_{l_{\infty}}^{I_{+}} $ to the constant bounded Jacobian, which sends a sequence
$ \left(\psi_{i}\right)_{i\in I} $ such that $ \psi_{i'}=\psi_{i}^{-1} $ into a corresponding line bundle. Here we are
going to show that the kernel of this surjection is a lattice in
$ \left({\Bbb C}^{*}\right)_{l_{\infty}}^{I_{+}} $, as in finite-genus case. To avoid defining a lattice in
$ \left({\Bbb C}^{*}\right)_{l_{\infty}}^{I_{+}} $, consider $ \left({\Bbb C}^{*}\right)_{l_{\infty}}^{I_{+}} $ as the set of coordinate-wise exponents of
$ {\Bbb C}_{l_{\infty}}^{I_{+}} $. We obtain a mapping from $ {\Bbb C}_{l_{\infty}}^{I_{+}} $ to the bounded Jacobian which sends
$ \left(2\pi i{\Bbb Z}\right)_{l_{\infty}}^{I_{+}} $ to the origin in the Jacobian.
We are going to prove that there is a lattice $ L\subset{\Bbb C}^{I_{+}} $ which goes to
the origin. First, construct generators of this lattice:
\begin{proposition} \label{prop9.72}\myLabel{prop9.72}\relax Suppose that the matrix $ \left(e^{-l_{ij}}-\delta_{ij}\right) $ gives a compact
operator $ l_{2} \to l_{2} $ and a compact operator $ l_{1} \to l_{1} $. Let $ j\in I_{+} $. There exists
a global holomorphic form $ \alpha^{\left(j\right)} $ on $ \bar{M} $ such that the $ A $-periods of $ \alpha^{\left(j\right)} $
vanish except for the $ j $-th one, which is equal to $ 2\pi i $. Then $ a_{j}=\exp \int\alpha^{\left(j\right)} $
is an element of $ {\cal H}^{\infty} $. The corresponding cocycle $ \psi_{kj}=a_{j}\circ\varphi_{k}/a_{j} $ is (locally)
constant, and coincides with $ \exp 2\pi i\Omega_{kj} $. \end{proposition}
\begin{proof} Existence of $ \alpha^{\left(j\right)} $ is a corollary of results of Section~\ref{s9.40}.
The only statement we need to prove is that $ a_{j}\in{\cal H}^{\infty} $. In turn, it is
sufficient to prove that $ \operatorname{Im}\Omega_{kj} $ is bounded (for a fixed $ j $). On the other
hand, $ \operatorname{Im}\Omega_{kj} $ consists of two parts which correspond to decomposition
$ \alpha^{\left(j\right)}=2\pi i\alpha_{j}+o^{\left(j\right)} $, here $ \alpha_{j} $ is defined as in Section~\ref{s9.40}, and $ o^{\left(j\right)}\in{\cal H}^{\left(0\right)} $
and has $ A $-periods 0. The description of the cycle for $ B $-period shows
that first part of $ \operatorname{Im}\Omega_{kj} $ is bounded by $ \log |l_{jj'}| $, thus it is sufficient
to estimate the second part $ \operatorname{Im}{\cal Q}_{k}\left(o^{\left(j\right)}\right) $.
This estimate follows from the following lemma:
\begin{lemma} Suppose that the matrix $ \left(e^{-l_{ij}}-\delta_{ij}\right) $ gives a compact operator
$ l_{2} \to l_{2} $ and a compact operator $ l_{1} \to l_{1} $.
\begin{enumerate}
\item
Let $ \alpha\in H^{-1/2}\left(\partial K_{i},\Omega^{\text{top}}\right) $. Denote by the same letter
extension-by-$ \delta $-function of $ \alpha $ to $ {\Bbb C}P^{1} $. Then $ \bar{\partial}^{-1}\alpha\in{\cal H}^{\left(0\right)} $, and
$ {\cal Q}_{k}\left(\bar{\partial}^{-1}\alpha\right)=O\left(\|\alpha\|_{H^{-1/2}}\right) $ uniformly in $ k\not=i $.
\item
The sequence $ \left(\|o^{\left(j\right)}|_{\partial K_{k}}\|_{H^{-1/2}}\right)_{k\in I}\in l_{1} $.
\end{enumerate}
\end{lemma}
\begin{proof} The first part follows from the explicit construction of cycles
for $ B $-periods. Prove the second part.
Since sum of radii of disks $ K_{i} $ is finite, one can estimate that the
sequence $ \left(\|\alpha_{j}|_{\partial K_{k}}\|_{H^{-1/2}}\right)_{k\in I}\in l_{1} $. Denote the space of such $ 1 $-forms on
$ \bigcup_{i}\partial K_{i} $ by $ {\cal H}_{1} $. On the other hand, $ o^{\left(j\right)} $ is uniquely determined by the
conditions $ {\cal J}_{\omega}o^{\left(j\right)}=-2\pi i{\cal J}_{\omega}\alpha_{j} $ and $ o^{\left(j\right)}\in{\cal H}_{0}^{\left(0\right)} $, here $ {\cal J}_{\omega} $ is the operator from
Section~\ref{s9.40}. What remains to prove is that $ {\cal J}_{\omega} $ sends the subspace
$ {\cal H}_{0}^{\left(0\right)}\cap{\cal H}_{1} $ onto itself. We know that $ {\cal J}_{\omega} $ sends $ {\cal H}_{0}^{\left(0\right)} $ onto itself.
However, from the
restrictions on $ l_{ij} $ one immediately obtains that $ {\cal J}_{\omega} $ gives a Fredholm
operatorar $ {\cal H}_{1} \to {\cal H}_{1} $ of index 0. Since $ {\cal J}_{\omega} $ has no kernel, this operator is
an isomorphism. \end{proof}
This finishes the proof of Proposition~\ref{prop9.72}. {}\end{proof}
\begin{definition} Call a multiplicator $ a\in{\cal H}^{\infty} $ {\em lattice-like}, if $ a^{-1}\in{\cal H}^{\infty} $
and $ \psi_{k}=a^{-1}\cdot\left(a\circ\varphi_{k}\right) $ is a constant function on $ \partial K_{k} $ for every $ k\in I $. \end{definition}
Now to each $ j\in I_{+} $ we associated a lattice-like multiplicator $ a_{j} $,
which induces a cocycle $ \left(\psi_{k}\right)_{k\in I_{+}} $, $ \psi_{k}=\exp 2\pi i\Omega_{kj} $. The bundle given by this
cocycle is isomorphic to a trivial one (via $ a_{j} $). Since $ \operatorname{Im}\Omega_{kj} $ gives a
positive Hermitian form (defined at least on real sequences of finite
length), columns of $ \operatorname{Im}\Omega_{kj} $ generate the subspace which is
dense in the set of real sequences (with topology of direct product),
which shows in conditions of Proposition~\ref{prop9.72}
\begin{proposition} Consider the space $ {\Bbb C}_{l_{\infty}}^{I_{+}} $ of bounded sequences $ \left(\Psi_{k}\right) $ with
topology induced from the direct product $ {\Bbb C}^{I_{+}} $. Consider the subgroup $ L $
generated by rows $ \left(\Omega_{kj}\right) $, $ j\in I_{+} $, and $ {\Bbb Z}_{l_{\infty}}^{I_{+}} $. This subgroup is a lattice,
i.e., its $ {\Bbb R} $-span is dense in $ {\Bbb C}_{l_{\infty}}^{I_{+}} $. \end{proposition}
To finish the description of bounded Jacobian, it remains to prove
that any lattice-like multiplicator $ a $ is a product of powers of $ a_{j} $,
$ j\in I_{+} $. Recall that the index of function $ a $ is the change of $ \frac{\operatorname{Arg} a}{2\pi i} $
along a contour. Note that the {\em index\/} of $ a_{j} $ around $ \partial K_{k} $ is $ \pm1 $ if $ k=j,j' $,
and is 0 otherwise. Thus one reconstruct the needed powers of $ a_{j} $ by
taking indices of $ a $ around $ \partial K_{j} $, $ j\in I_{+} $. The only things we need to prove
is that a lattice-like multiplicator $ a $ with all indices around $ \partial K_{j} $ being
0 is constant, and only a finite number of indices of a lattice-like
multiplicator is non-zero. The first statement is a direct corollary of
Lemma~\ref{lm8.31} and the following
\begin{lemma} \label{lm9.91}\myLabel{lm9.91}\relax Suppose that the conformal distances $ l_{ij} $ satisfy the
condition that $ \left(e^{-l_{ij}}-\delta_{ij}\right) $ gives a bounded operator $ l_{\infty} \to l_{2} $. Then any
lattice-like multiplicator is of class $ {\cal H}^{\left(1\right)} $. \end{lemma}
Recall that the space $ {\cal H}^{\left(1\right)} $ was defined in Section~\ref{s35.30}.
\begin{proof} Since $ a\,dz^{1/2}\in{\cal H}\left({\Bbb C}P^{1},\left\{K_{j}\right\}\right) $, Theorem~\ref{th4.40} shows that $ a $ can be
reconstructed from its restriction to $ \bigcup_{j\in I}\partial K_{j} $ using Cauchy formula. In
particular, $ + $-part of $ \partial a $ on $ \partial K_{j} $ is given by an integral along $ \bigcup_{k\not=j}\partial K_{k} $.
The restrictions on $ l_{ij} $ guarantie that the $ + $-parts of $ \partial a $ on the circles
$ \partial K_{j} $ is in $ \oplus_{l_{2}}H^{-1/2}\left(\partial K_{j}\right) $ modulo constants. Since $ a^{-1}\cdot\left(a\circ\varphi_{k}\right) $ is constant
and bounded, $ - $-parts of $ \partial a $ are also in this space. Since the integral of
$ \partial a $ along each circle vanishes, arguments similar to ones in Section
~\ref{s35.30} show that $ \partial a\in L_{2}\left({\Bbb C}P^{1}\smallsetminus\bigcup K_{i},\omega\right) $, thus $ a\in{\cal H}^{\left(1\right)} $. \end{proof}
Iteration of the procedure used in the proof of the lemma shows that
\begin{amplification} \label{amp9.94}\myLabel{amp9.94}\relax Conclusions of Lemma~\ref{lm9.91} remain true if
$ \left(e^{-l_{ij}}-\delta_{ij}\right) $ gives a bounded operator $ l_{\infty} \to l_{\infty} $, and some power of this
operator sends $ l_{\infty} $ to $ l_{2} $. \end{amplification}
To prove that only a finite number of indices of a lattice-like
multiplicator $ a $ is non-zero, it is enough to show that $ a^{-1}\partial a\in L_{2}\left(U,\omega\right) $, $ U $
being homotopic to $ {\Bbb C}P^{1}\smallsetminus\bigcup K_{i} $. Indeed, indices of $ a $ are proportional to
periods of $ a^{-1}\partial a $, and periods of $ L_{2} $-form form a sequence in $ l_{2} $.
In turn, since $ a^{-1} $ is a multiplicator, thus is bounded, this follows
from the fact that $ \partial a\in L_{2}\left(U,\omega\right) $. We obtain
\begin{theorem} Consider a family of non-intersecting disks $ K_{i}\subset{\Bbb C}P^{1} $ with
pairwise conformal distances. Let $ {\cal R} $ be the matrix $ \left(e^{-l_{ij}}-\delta_{ij}\right) $. Suppose
that $ {\cal R} $ gives a compact mapping $ l_{2} \to l_{2} $ and a bounded mapping $ l_{\infty} \to l_{\infty} $,
and that some power of $ {\cal R} $ gives a bounded mapping $ l_{\infty} \to l_{2} $. If disks $ K_{i} $
have a uniform thickening, then the bounded Jacobian coincides with the
quotient of $ {\Bbb C}_{l_{\infty}}^{I_{+}} $ by the lattice generated by $ {\Bbb Z}_{l_{\infty}}^{I_{+}} $ and rows of the
period matrix $ \left(\Omega_{ij}\right) $. \end{theorem}
\begin{remark} Formally speaking, we defined the Jacobians in the case when
$ \left(e^{-l_{ij}/2}-\delta_{ij}\right) $ gives a compact operator $ l_{2} \to l_{2} $, thus the statement of
theorem is abuse of notations. However, it is easy to define all the
ingredients needed for the definition of the bounded Jacobian as
far as the conditions of Amplification~\ref{amp9.94} are satisfied. \end{remark}
Let us recall that the conditions of the theorem are automatically
satisfied for any Hilbert--Schmidt curve, thus we get a complete
description of the bounded Jacobian in this case. It is similar to the
finite-genus case, where Jacobian is a quotient of a finite-dimensional
complex vector space by a lattice.
Moreover, note that the topology on the constant bounded Jacobian is
inherited from the topology of direct product on $ {\Bbb C}^{I_{+}} $, thus the above
description allows one to reconstruct the topology on the bounded
Jacobian as well.
\subsection{Rigged Hodge structure }\label{s9.80}\myLabel{s9.80}\relax Let us wrap the results of the previous
section into the familiar form of Hodge structures. In fact the resulted
structure will be a hybrid of a Hodge structure and a structure of a
rigged topological vector space.
Let $ I_{+} $ be an arbitrary set. Let $ H_{{\Bbb Z}}^{A} $, $ H_{{\Bbb R}}^{A} $, $ H_{{\Bbb C}}^{A} $ be the spaces of
sequences $ \left(p_{i}\right)_{i\in I_{+}} $ with only a finite number of non-zero terms (with
integer/real/complex terms), $ H_{{\Bbb Z}}^{B} $, $ H_{{\Bbb R}}^{B} $, $ H_{{\Bbb C}}^{B} $ be the spaces of sequences
$ \left(q_{i}\right)_{i\in I_{+}} $ without any restriction on growth, and $ H_{{\Bbb Z}}= H_{{\Bbb Z}}^{A}\oplus H_{{\Bbb Z}}^{B} $, $ H_{{\Bbb R}}=H_{{\Bbb R}}^{A}\oplus H_{{\Bbb R}}^{B} $,
$ H_{{\Bbb C}}=H_{{\Bbb C}}^{A}\oplus H_{{\Bbb C}}^{B} $. One can define an operation of complex conjugation on the
spaces $ H_{{\Bbb C}}^{\bullet} $, this operation leaves $ H_{{\Bbb R}}^{\bullet} $ fixed. The spaces $ H_{{\Bbb Z}} $, $ H_{{\Bbb R}} $, $ H_{{\Bbb C}} $ have a
natural symplectic structure
\begin{equation}
\left[\left(\left(p_{i}\right),\left(q_{i}\right)\right),\left(\left(p'_{i}\right),\left(q_{i}'\right)\right)\right] = \sum_{i}\left(p_{i}q_{i}'-p'_{i}q_{i}\right)
\notag\end{equation}
such that the components $ H_{\bullet}^{A,B} $ are Lagrangian and mutually dual. Fix an
arbitrary mapping $ \Omega\colon H_{{\Bbb C}}^{A} \to H_{{\Bbb C}}^{B} $. Let $ \Omega_{1}\colon H_{{\Bbb C}}^{A} \to H_{{\Bbb C}}^{A}\oplus H_{{\Bbb C}}^{B}=H_{{\Bbb C}} $ be $ \operatorname{id}\oplus\Omega $.
Denote $ \operatorname{Im}\Omega_{1}\subset H_{{\Bbb C}} $ by $ H^{1,0} $. Suppose that (compare Proposition~\ref{prop8.35}) $ H^{1,0} $
is Lagrangian. Moreover, suppose (compare Proposition~\ref{prop8.33}) that if
$ \alpha\in H^{1,0} $, then $ \operatorname{Im} \left[\alpha,\bar{\alpha}\right]\gg0 $ in the following sense:
\begin{enumerate}
\item
$ \operatorname{Im} \left[\alpha,\bar{\alpha}\right]\geq0 $ (here $ \alpha \mapsto \bar{\alpha} $ is the complex conjugation on $ H_{{\Bbb C}} $);
\item
the equality is achieved only if $ \alpha=0 $;
\item
the completion $ {\frak h}^{1,0} $ of $ H^{1,0} $ w.r.t. the norm $ \|\alpha\|^{2}=\operatorname{Im} \left[\alpha,\bar{\alpha}\right] $ has no
vectors of length 0.
\end{enumerate}
Let $ H^{0,1} = \left\{\alpha\in H_{{\Bbb C}} \mid \bar{\alpha}\in H^{1,0}\right\} $. The mapping $ \Omega_{1} $ gives an inclusion $ i $ of $ H_{{\Bbb C}}^{A} $
into $ {\frak h}^{1,0} $.
\begin{definition} A {\em rigged Hodge structure\/} is a subspace $ H^{1,0}\subset H_{{\Bbb C}} $ which
satisfies the above conditions, and such that the projection $ p\colon H^{1,0} \to
H_{{\Bbb C}}^{B} $ can be continuously extended to a mapping $ p\colon {\frak h}^{1,0} \to H_{{\Bbb C}}^{B} $. \end{definition}
Note the mappings $ H_{{\Bbb C}}^{A} \xrightarrow[]{i} {\frak h}^{1,0} \xrightarrow[]{p} H_{{\Bbb C}}^{B} $ equip $ {\frak h}^{1,0} $ with a structure
of a rigged topological vector space \cite{GelVil64Gen}. It is clear that $ H^{1,0} $
is a rigged Hodge structure iff $ \Omega $ is symmetric, $ \operatorname{Im}\Omega>0 $, $ -C\cdot\operatorname{Im}\Omega<\operatorname{Re}\Omega<C\cdot\operatorname{Im}\Omega $.
From now on suppose that $ H^{1,0} $ is a rigged Hodge structure. Let $ {\frak h}^{0,1} $
be the complexly conjugated space to $ {\frak h}^{1,0} $, $ {\frak h}_{{\Bbb C}}={\frak h}^{1,0}\oplus{\frak h}^{0,1} $. The vector space
$ {\frak h}_{{\Bbb C}} $ has a natural operation of complex conjugation $ \left(\alpha,\alpha'\right) \mapsto \left(\bar{\alpha}',\bar{\alpha}\right) $, let
$ {\frak h}_{{\Bbb R}} $ be the subspace $ \left(\alpha,\bar{\alpha}\right) $ of fixed points of this complex conjugation.
There is a natural extension of the projection $ {\frak h}^{1,0} \to H_{{\Bbb C}}^{B} $ to $ {\frak h}_{{\Bbb C}} \to H_{{\Bbb C}}^{B} $.
This mapping is compatible with the complex conjugation, thus induces a
mapping $ {\frak h}_{{\Bbb R}} \to H_{{\Bbb R}}^{B} $. Let $ {\frak h}_{{\Bbb C}}^{A} $ be the kernel of the mapping $ {\frak h}_{{\Bbb C}} \to H_{{\Bbb C}}^{B} $,
similarly for $ {\frak h}_{{\Bbb R}}^{A} $.
Let $ \bar{\Omega}_{1}\colon H_{{\Bbb C}}^{A} \to {\frak h}^{0,1} $ be the complex conjugate to the mapping $ \Omega_{1}\colon H_{{\Bbb C}}^{A}
\to {\frak h}^{1,0} $, $ \bar{\Omega}_{1}v=\overline{\Omega_{1}\bar{v}} $. Define a pre-Hilbert structure on $ H_{{\Bbb C}}^{A} $ via
$ \|v\|_{A}^{2}=\|\Omega_{1}v\|_{{\frak h}^{1,0}} $. Then $ \Omega $ extends to a mapping from Hilbert completion
$ \left(H_{{\Bbb C}}^{A}\right)^{\text{compl}} $ of $ H_{{\Bbb C}}^{A} $ to $ {\frak h}^{1,0} $, same for $ \bar{\Omega}_{1} $ and $ {\frak h}^{0,1} $. Let $ {\frak h}_{{\Bbb C}}^{B} $ be the image of
this completion w.r.t. $ \Omega_{1}-\bar{\Omega}_{1} $. Identify $ {\frak h}_{{\Bbb C}}^{B} $ with a subspace of $ H_{{\Bbb C}}^{B} $ via the
projection $ {\frak h}_{{\Bbb C}} \to H_{{\Bbb C}}^{B} $.
The restriction on $ \Omega $ immediately imply
\begin{proposition} $ {\frak h}_{{\Bbb C}} $ carries a natural symplectic structure, the
corresponding mapping $ {\frak h}_{{\Bbb C}} \to {\frak h}_{{\Bbb C}}^{*} $ is invertible, and $ {\frak h}_{{\Bbb C}} $ is a direct sum of
Lagrangiann subspaces $ {\frak h}_{{\Bbb C}}^{A} $ and $ {\frak h}_{{\Bbb C}}^{B} $. \end{proposition}
Since the subspace $ {\frak h}_{{\Bbb C}}^{B} $ is stable w.r.t. complex conjugation, one can
define $ {\frak h}_{{\Bbb R}}^{B} $. Similarly, define $ {\frak h}_{{\Bbb Z}}^{A} $ as the image of $ H_{{\Bbb Z}}^{A} $ in $ {\frak h}_{{\Bbb C}}^{A} $. Note that
$ {\frak h}_{{\Bbb C}}^{A} $ and $ {\frak h}_{{\Bbb C}}^{B} $ are mutually dual, so $ {\frak h}_{{\Bbb Z}}^{A} $ is identified with a lattice in
$ {\frak h}_{{\Bbb C}}^{B} $. Say that $ v\in{\frak h}_{{\Bbb Z}}^{B}\subset{\frak h}_{{\Bbb C}}^{B} $ if $ \left(v,w\right)=0 $ for any $ w\in{\frak h}_{{\Bbb Z}}^{A} $.
\begin{proposition} The subset $ {\frak h}_{{\Bbb Z}}^{B} $ generates $ {\frak h}_{{\Bbb C}}^{B} $. \end{proposition}
\begin{proof} Let $ {\bold e}_{j} $, $ j\in I_{+} $, be the natural basis of $ H_{{\Bbb Z}}^{A} $. Let $ V_{j} $ be a subspace
of $ {\frak h}_{{\Bbb C}}^{A} $ generated by $ {\bold e}_{k} $, $ k\not=j $. If $ V_{j}={\frak h}_{{\Bbb C}}^{A} $, then $ {\bold e}_{j} $ is a linear combination
of $ {\bold e}_{k} $, $ k\not=j $, which contradicts $ \operatorname{Im}\Omega \gg $ 0. Appropriate multiple of a normal
vector to $ V_{i} $ is in $ {\frak h}_{{\Bbb Z}}^{B} $. It is obvious that these elements generate $ {\frak h}_{{\Bbb C}}^{B} $. \end{proof}
\begin{definition} Say that the rigged Hopf structure is {\em integer\/} if $ {\frak h}_{{\Bbb Z}}^{A} $ is
closed in $ {\frak h}_{{\Bbb C}}^{A} $. \end{definition}
The condition of being integer is a restriction from below on $ \operatorname{Im}\Omega $,
say, it prohibits $ \operatorname{Im}\Omega=\operatorname{diag}\left(\lambda_{i}\right) $, $ \lambda_{i}\in l_{2} $.
We obtained 4 subspaces $ {\frak h}_{{\Bbb C}}^{A} $, $ {\frak h}_{{\Bbb C}}^{B} $, $ {\frak h}^{1,0} $ and $ {\frak h}^{0,1} $ of $ {\frak h}_{{\Bbb C}} $. Note that
$ {\frak h}_{{\Bbb R}}^{A} $, $ {\frak h}_{{\Bbb R}}^{B} $ are generated by $ {\Bbb Z} $-lattices $ {\frak h}_{{\Bbb Z}}^{A} $, $ {\frak h}_{{\Bbb Z}}^{B} $ in them. If we consider $ {\frak h}_{{\Bbb C}} $
and $ {\frak h}^{0,1} $ as real vector spaces, then $ {\frak h}_{{\Bbb C}} ={\frak h}_{{\Bbb R}}\oplus{\frak h}^{0,1} $. Note that $ {\frak h}^{1,0} $ is
naturally identified with $ {\frak h}_{{\Bbb C}}/{\frak h}^{0,1} $, thus $ {\frak h}^{1,0}\simeq{\frak h}_{{\Bbb R}} $. Let $ {\frak L} $ be the image of
$ {\frak h}_{{\Bbb Z}}^{A}\oplus{\frak h}_{{\Bbb Z}}^{B}\subset{\frak h}_{{\Bbb R}}\subset{\frak h}_{{\Bbb C}} $ in $ {\frak h}_{{\Bbb C}}/{\frak h}^{0,1} $, identify $ {\frak L} $ with a subgroup of $ {\frak h}^{1,0} $. It is clear
that $ {\frak L} $ is a lattice in $ {\frak h}^{1,0}\simeq{\frak h}_{{\Bbb R}} $.
Start from these 4 subspaces of $ {\frak h}_{{\Bbb C}} $, the complex conjugation and
symplectic structure on $ {\frak h}_{{\Bbb C}} $, and lattices in the real parts of the first
two spaces. Try to reconstruct the initial rigged Hodge structure.
Suppose that $ {\frak h}_{{\Bbb C}}^{A} $, $ {\frak h}_{{\Bbb C}}^{B} $ are stable w.r.t. the complex conjugation, and that
$ {\frak h}^{1,0} $, $ {\frak h}^{0,1} $ are interchanged by complex conjugation. Suppose that the last
two subspaces project isomorphically on any subspace of the first two
(along the other one). Thus $ {\frak h}^{1,0} $ and $ {\frak h}^{0,1} $ are graphs of invertible
mappings $ \Omega_{1},\bar{\Omega}_{1}\colon {\frak h}_{{\Bbb C}}^{A} \to {\frak h}_{{\Bbb C}}^{B} $, these mappings are mutually complex
conjugate.
Now the symplectic structure on $ {\frak h}_{{\Bbb C}} $ identifies $ {\frak h}_{{\Bbb Z}}^{B} $ with a subset of
the dual lattice to $ {\frak h}_{{\Bbb Z}}^{A} $. This reconstructs the groups $ H_{{\Bbb Z}}^{A} $, $ H_{{\Bbb Z}}^{B} $, together
with $ \Omega_{1} $ they allow to reconstruct the mapping $ \Omega $ from $ H_{{\Bbb C}}^{A}=H_{{\Bbb Z}}^{A}\otimes{\Bbb C} $ to
$ H_{{\Bbb C}}^{B}=H_{{\Bbb Z}}^{B}\otimes{\Bbb C} $. The positivity of $ \operatorname{Im}\Omega $ is translated into positivity of $ \left(\alpha,\bar{\alpha}\right) $,
$ \alpha\in{\frak h}^{1,0} $, and the condition $ {\frak h}^{1,0}\oplus{\frak h}^{0,1}={\frak h} $.
Now the results of this chapter imply
\begin{proposition} Consider a family of disks $ K_{i}\subset{\Bbb C}P^{1} $, $ i\in I $, with pairwise
conformal distances $ l_{ij} $. Let $ {\cal R}=\left(e^{-l_{ij}}-\delta_{ij}\right) $. Suppose that $ {\cal R} $ gives a
compact mapping $ l_{2} \to l_{2} $ and a bounded mapping $ l_{\infty} \to l_{\infty} $, and that some
power of $ {\cal R} $ gives a bounded mapping $ l_{\infty} \to l_{2} $. Consider an involution $ ':
I\to $I, $ I=I_{+}\coprod I'_{+} $, glue boundaries of disks pairwise using fraction-linear
mappings $ \varphi_{i}=\varphi_{i'}^{-1} $, let $ \left(\Omega_{jk}\right) $, $ j,k\in I_{+} $, be the matrix of periods of the
resulting curve $ \bar{M} $. Then $ \Omega_{jk} $ gives rise to a rigged Hodge structure. \end{proposition}
Note that it is natural to call the space $ {\frak h}_{{\Bbb C}} $ of this rigged Hodge
structure {\em the first cohomology space\/} of the curve $ \bar{M} $. The space $ {\frak h}^{1,0} $ can
be identified with $ \Gamma\left(\bar{M},\omega\right) $. As in finite-genus case, the quotient $ {\frak h}^{1,0}/{\frak L} $
is identified with the (bounded) Jacobian of the curve.
|
1997-11-14T15:20:36 | 9710 | alg-geom/9710019 | en | https://arxiv.org/abs/alg-geom/9710019 | [
"alg-geom",
"math.AG"
] | alg-geom/9710019 | Barbara Russo | B. Russo and M. Teixidor i Bigas | On a conjecture of Lange | 13 pages, amslatex, deleted the result of irreducibility in theorems
0.2 and 0.3 | null | null | null | null | Let C be a projective smooth curve of genus g> 1. Let E be a vector bundle of
rank r on C. For each integer r'<r, associate to E the invariant
s_{r'}(E)=r'deg(E)-rdeg(E') where E'is a subbundle of E of rank r' and maximal
degree. For every r', one can stratify the moduli space of stable vector
bundles according to the value of the invariant. Lange's conjecture says that
this strata are non-empty and of the right dimension if s_{r'}>0. The
conjecture has recently been solved thanks to work of Lange- Narasimhan,
Lange-Brambila-Paz, Ballico and the authors. The purpose of this paper is to
give a simpler proof of the result valid without further assumptions. The
method of proof provides additional information on the geometry of the strata.
We can prove that each strata (which is irreducible) is contained in the
closure of the following one. We also show the unicity of the maximal subbundle
when s\le r'(r-r')(g-1). Our methods can be used to study twisted Brill-Noether
loci and to give a new proof of Hirschowitz Theorem about the non-speciality of
the tensor product of generic vector bundles.
| [
{
"version": "v1",
"created": "Thu, 16 Oct 1997 10:00:20 GMT"
},
{
"version": "v2",
"created": "Fri, 14 Nov 1997 14:20:37 GMT"
}
] | 2007-05-23T00:00:00 | [
[
"Russo",
"B.",
""
],
[
"Bigas",
"M. Teixidor i",
""
]
] | alg-geom | \section*{Introduction}
Let $C$ be a projective non-singular curve of genus $g\ge 2$ .
Let $E$ be a vector bundle of rank $r$ and degree $d$.
Fix a positive integer $r'<r$. Define
$$s_{r'}(E)=r'd-r\max _{E'}\{ degE'|rk E'=r', E'\subset E\} $$
Notice that $E$ is stable if and only if $s_{r'}(E)>0$ for every $r'<r$.
On the other
hand, for a generic stable $E$
$$r'(r-r')(g-1)\le s_{r'}(E)<r'(r-r')(g-1)+r$$
(cf [L] Satz 2.2 p.452 and [Hi] Th.4.4).
One can then stratify the moduli space
$U(r,d)$ of vector bundles of rank $r$ and degree $d$ according
to the value of $s$. Define
$$U_{r',s}(r,d)=\{ E\in U(r,d)|s_{r'}(E)=s\} $$
We want to study this stratification. A vector bundle $E\in U_{r',s}(r,d)$
can be written in an exact sequence
$$0\rightarrow E'\rightarrow E\rightarrow E''\rightarrow 0$$
with $E',E''$ vector bundles of ranks $r',r''$ and degrees
$d',d''$ satisfying $r=r'+r'', d=d'+d'', r'd-rd'=r'd''-r''d'=s$.
Note that the condition $s>0$ is equivalent to the inequality of slopes
$\mu (E')<\mu (E'')$. One expects that when this condition
is satisfied, a generic such extension will yield a stable $E$.
We call this statement Lange's conjecture (cf. [L]).
The conjecture is now solved and a great deal is known about the geometry
of the strata : the rank two case is treated in [L,N], the case
$s\leq \min (r',r'')(g-1)$ in [B,B,R]. In [T1], the result is proved for the
generic curve and for every curve if $E$ is assumed to be only semistable.
This apparently implies the result also for $E$ stable (cf.[B]).
In [B,L], a proof is provided for $g\ge (r+1)/2$.
The purpose of this paper is to give a simpler proof of the result
valid without further assumptions. The method is somehow the converse of
the one used by Brambila-Paz and Lange in [B,L]. They start with the
most general $E$ in $U(r,d)$ and then show that a suitable
transformation of $E$ gives a new bundle with smaller $s$.
They need to check then that such an $E$ is in fact stable.
Here, we start with an $E$ with the smallest possible $s$ and
produce an $E'$ with larger $s$. Stability then comes for free
because $E'$ is more general than $E$. The drawback is that one needs
to prove existence of stable vector bundles with small $s$ but this
is surprisingly easy.
Our method of proof provides additional information on the
geometry of the strata. We can prove that $U_{r',s}(r,d)$ is
contained in the closure of $U_{r',s+r}(r,d)$ as well as the unicity
of the subbundle (see also [T2])
Our results can be stated in the following
\begin{Thm}
\label{Theorem}
Assume that $0<s\le r'(r-r')(g-1) ,s\equiv r'd(r)$.
Write $d'={r'd-s\over r}$. If $g\ge 2$,
then $U_{r's}(r,d)$ is non-empty, irreducible of dimension
$$dim U_{r's}(r,d)= r^2(g-1)+1+s-r'(r-r')(g-1)$$
Moreover, a generic
$E\in U_{r',s}(r,d)$ can be written in an exact sequence
$$0\rightarrow E'\rightarrow E\rightarrow E''\rightarrow 0$$
with both $E',E''$ stable and $E'$ is the unique subbundle
of $E$ of rank $r'$ and degree $d'$.
\end{Thm}
\begin{Thm}
\label{Theorem2}
If $s\ge r'(r-r')(g-1)$, every stable vector bundle
has subbundles of rank $r'$ and degree $d'$. Denote by
$$A_{r',d'}(E)= \{ E'|rk E'=r',deg E'=d',
E'\subset E ,E' {\rm saturated}\} .$$
Then, for generic $E$, $A_{r',d'}(E)$ has dimension
$$dimA_{r',d'}(E) =r'(r-r')(g-1)-s.$$
\end{Thm}
These results and our methods of proof can be used to study
twisted Brill-Noether loci. We can show the following
\begin{Thm}
\label{BrillNoether}
(twisted Brill-Noether for one section).
Let $E$ be a generic vector bundle of rank $r_E$ and degree $d_E$.
Consider the twisted Brill-Noether loci $W^0_{r_F,d_F}(E)$. This is defined
as the
subset of the moduli space $U(r_F,d_F)$ consisting
of those $F$ such that $h^0(F^*\otimes E)\ge 1$.
Then the dimension of $W^0_{r_F,d_F}(E)$ is the expected dimension
given by the Brill-Noether number
$$ \rho ^0_{r_F,d_F}(E)=r_F(r_F-r_E)(g-1)+r_Fd_E-r_Ed_F$$
if this number is positive and is empty otherwise.
Moreover, when non-empty
its generic elements considered as maps $F\rightarrow E$
have maximal rank.
\end{Thm}
We also include a proof of Hischowitz's Theorem that states that
the tensor product of two generic vector bundles is non-special.
\bigskip
Acnowledgments: The first author was partially supported by MURST
GNSAGA of CNR (Italy), Max-Planck Institut of Bonn and
the University of Trento. The second author is visiting the
Mathematics Department of the University of Cambridge.
This
collaboration started during the Europroj meeting ``Vector Bundles and
Equations'' in Madrid. Both authors are members of the Europroj group
VBAC and received support from Europroj
and AGE to attend this conference.
\section{Existence and dimensionality}
In this section we prove the existence of extensions with central
term stable and we compute the dimension of the set of vector bundles
that fit in such exact sequences. We need several preliminary results.
\begin{Lem}
\label{h^0=0}
Let $E$ be a stable vector bundle. Assume that we have an
exact sequence
$$0\rightarrow E'\rightarrow E\rightarrow E''\rightarrow 0.$$
Then $h^0(E^{''*}\otimes E')=0$.
\end{Lem}
\begin{pf}
A non-zero map $E''\rightarrow E'$ induces an endomorphism of $E$
that is not an homothethy. This is impossible if $E$ is stable.
\end{pf}
\begin{Thm}[Hirschowitz]
\label{Hirschowitz}
The tensor product of two generic vector bundles is not special.
\end{Thm}
This result was stated and proved in [Hi], 4.6. As this is ,
unfortunately, still unpublished, we provide an alternative proof
below.
\begin{pf} We shall denote by $r_G,d_G$ the rank and degree of
a given sheaf say $G$.
By Serre duality, it is enough to show that if $E,F$ are generic
vector bundles, then $h^0(F^*\otimes E)>0$ implies
$\chi (F^*\otimes E)>0$.
Assume $h^0(F^*\otimes E)\not= 0$. Then, there is a non-zero
map $F\rightarrow E$. Denote by $F'$ its kernel, $I$ its image,
$E''$ its cokernel. Let $T$ be the torsion subsheaf of $E''$ and
$\bar E=E''/T$.
We then have the following exact sequences of sheaves
$$0\rightarrow F'\rightarrow F\rightarrow I\rightarrow 0$$
$$ \begin{array}{ccccccccc}
& & & & & & 0& & \\
& & & & & & \downarrow & & \\
& &0 & & & & T & & \\
& &\downarrow & & & & \downarrow & & \\
0&\rightarrow & I&\rightarrow &E&\rightarrow &E''&\rightarrow &0\\
& &\downarrow & &\downarrow & &\downarrow & & \\
0&\rightarrow &\bar I&\rightarrow &E&\rightarrow &\bar E&\rightarrow &0\\
& &\downarrow & & & &\downarrow & & \\
& &T & & & &0 & & \\
& &\downarrow & & & & & & \\
& & 0& & & & & & \\
\end{array}$$
As $T$ is a torsion sheaf, $I$ is determined by $\bar I$, the support
of $T$ and for every point in the support a map from the fiber
of $\bar I$ at the point to the basefield.
Hence,
$$dim\{ \bar I \}\le dim \{ I\} +r_I degT$$
As any vector bundle can be deformed to a stable vector bundle,
(cf.[N,R]Prop.2.6), $F',\bar I, \bar E$ depend at most on
$r_{F'}^2(g-1)+1, r_{\bar E}^2(g-1)+1, r_{\bar I}^2(g-1)+1$
moduli respectively.
From \ref{h^0=0} and the stability of $E,F$,
$h^0(I^*\otimes F)=0, h^0(\bar E^*\otimes \bar I)=0$.
Notice that $F$ is determined by $F',I$ and an extension class
in $H^1(I^*\otimes F')$ up to homotethy. Similarly,
$E$ is determined by $\bar I,\bar E$ and an extension class
in $H^1(\bar E^*\otimes \bar I)$ up to homotethy.
From the genericity of the pair $E,F$, we find
$$r_F^2(g-1)+1+r_E^2(g-1)+1=dimU(r_F,d_F)+dimU(r_E,d_E)\le$$
$$\le dimU(r_{F'},d_{F'})+dim U(r_I,d_I)+dim U(r_{\bar E},d_{\bar E})
+r_IdegT+$$
$$+h^1(I^*\otimes F')-1+h^1(\bar E^*\otimes \bar I)-1$$
$$\le (r_{F'}^2+r_I^2+r_{\bar E^2}+r_{F'}r_I+r_Ir_{\bar E})(g-1)
+1+r_{F'}d_I-r_Id_{F'}+r_Id_{\bar E}-r_{\bar E}d_I+r_IdegT$$
This condition can be written as
$$(*)r_{F'}d_I-r_Id_{F'}+r_Id_{\bar E}-r_{\bar E}d_I+r_IdegT-
r_I(r_I+r_{\bar E}+r_{F'})(g-1)-1\ge 0$$
From the genericity of $E,F$ and [L] Satz 2.2, we obtain
$$\mu (I)-\mu(F')\ge g-1, \mu(\bar E)-\mu (\bar I)\ge g-1$$
Hence
$$\mu (\bar E)-\mu (F')\ge 2(g-1)+degT/r_I$$
Equivalently
$$r_{F'}d_{\bar E}-r_{\bar E}d_{F'}\ge 2r_{F'}r_{\bar E}(g-1)+
{r_{F'}r_{\bar E}\over r_I}deg T$$
Adding (*) and this last inequality, we find
$$\chi (F^*\otimes E)\ge 1+r_{F'}r_{\bar E}(g-1)+{r_{F'}r_{\bar E}
\over r_I}deg T+r_{F'}degT\ge 1$$
\end{pf}
\begin{Lem}
\label{cotasubf}
Denote by $V_{r',s}(r,d)$ the set of stable $E$ that can be
written in an exact sequence of vector bundles
$$0\rightarrow E'\rightarrow E\rightarrow E''\rightarrow 0$$
with $rk E'=r', deg E'=d'$. Assume that $V_{r's}(r,d)$ is non-empty.
If $s\le r'(r-r')(g-1)$,
then the generic such $E$ has only a finite number of subbundles
of rank $r'$ and degree $d'$ such that $ r'd-rd'=s$. If $s\ge r'(r-r')(g-1)$,
then the dimension of the space of subbundles of rank $r'$ and degree
$d'$ of the generic $E$ is at most
$s-r'(r-r')(g-1)$
\end{Lem}
\begin{pf}
Several proofs of this fact appear in the literature .
We sketch a proof here for the convenience of the reader.
The set of subbundles of rank $r'$ and degree $d'$ of $E$ is parametrised
by the quotient scheme of $E$ of the corresponding rank and degree.
The tangent space to this quotient scheme at the point corresponding
to a bundle $E$ with subbundle $E'$ and quotient $E''$ is
$H^0(E^{'*}\otimes E'')$. As $E$ is generic, we can assume $E',E''$
generic. Then, from \ref{Hirschowitz}, $E^{'*}\otimes E''$ is non-special.
Hence, if $s\le r'(r-r')(g-1) ,h^0(E^{'*}\otimes E'') =0$ while if
$s\ge r'(r-r')(g-1)$, then $ h^1(E^{'*}\otimes E'')=0$ and so
$h^0(E^{'*}\otimes E'')=s-r'(r-r')(g-1)$.
\end{pf}
\begin{Prop}
\label{Irr}
With the notations of \ref{cotasubf}, if $V_{r',s}(r,d)$
is non-empty, then, it is irreducible
and the generic $E\in V_{r',s}(r,d)$ can be written in an
exact sequence as above with $E',E''$ stable. Moreover,
$dimV_{r',s}(r,d)= \min [r^2(g-1)+1, r^2(g-1)+1+s-r'(r-r')(g-1)]$
\end{Prop}
\begin{pf}
This proof appears in [T]. We give a sketch here for the convenience
of the reader .
Consider an extension
$$0\rightarrow E'_0\rightarrow E_0 \rightarrow E''_0\rightarrow 0$$
with $E_0$ stable. From [N,R] Prop.2.6, there are irreducible families of
vector bundles ${\cal M}', {\cal M}''$ containing $E'_0, E''_0$ respectively
and whose generic member is stable. Consider the universal family
of extensions ${\bf P}$ of an $E''\in {\cal M''}$ by an $E'\in {\cal M'}$.
Consider the open subset $U\subset {\cal M'}\times {\cal M''}$ consisting
of those pairs $(E',E'')$ such that $h^0(E^{''*}\otimes E')=0$.
As $\mu (E')<\mu (E'')$, $U$ contains all pairs in which both
$E',E''$ are stable.
From \ref{h^0=0}, $(E'_0,E''_0)\in U$. As $h^1(E^{''*}\otimes E')$
is constant on $U$, the inverse image ${\bf P}(U)$ of $U$ in
${\bf P}$ is irreducible.
This proves that the given extension can be deformed to an extension
with both $E',E''$ stable. By the stability of $E_0$, the generic
central term in an extension in ${\bf P}(U)$ is stable.
Consider the canonical rational map
$\pi :{\bf P}(U)\rightarrow U(r,d)$. By definition $V_{r',d'}(r,d)$ is the
image of this map. Hence, it is irreducible. The dimension of ${\bf P}$
can be computed as
$$dim{\bf P}=dim{\cal M}+dim{\cal M}'+h^1(E^{''*}\otimes E')-1=
(r^2-r'r'')(g-1)+1+s$$
From \ref{cotasubf}, the fibers of $\pi $ have dimension
$\max [0,s-r'(r-r')(g-1)]$. Hence, the result follows.
\end{pf}
\begin{Prop}
\label{Irrsubf}
Assume $s>r'(r-r')(g-1)$ and $E$ is a generic stable vector bundle.
If $A_{r'd'}(E)$ is non-empty, then it has
dimension $s-r'(r-r')(g-1)$.
\end{Prop}
\begin{pf}
With the notations in the proof of
\ref{Irr}, $A_{r',d'}(r,d)$ are the fibers of $\pi$.
Its dimension has been computed already.
\end{pf}
\begin{Def}
Let $E$ be a vector bundle. A vector bundle $\tilde E$
is called an elementary transformation of $E$ if there is an exact
sequence
$$0\rightarrow \tilde E \rightarrow E \rightarrow {\bf C}_P \rightarrow 0$$
Here ${\bf C}_P$ denotes the skyscraper sheaf isomorphic to the
base field with support on the point $P$.
A vector bundle $\bar E$ is called a dual elementary
transformation of $E$ if $E$ is an elementary
transformation of $\bar E$. Equivalently, the dual of $\bar E$
is an elementary transformation of the dual of $E$ or equivalently
there is an exact sequence
$$\bar E \rightarrow E(Q)\rightarrow {\bf C}_Q^{r-1}\rightarrow 0.$$
\end{Def}
\begin{Lem}
\label{TE}
Let
$$0\rightarrow E' \rightarrow E\rightarrow E''\rightarrow 0$$
be an exact sequence of vector bundles. Then, for a generic elementary
transformation $\tilde E$ of $E$, we have an exact sequence
of vector bundles
$$0\rightarrow \tilde E' \rightarrow \tilde E\rightarrow E''\rightarrow 0$$
where $\tilde E'$ is a generic elementary transformation of
$E$.
\end{Lem}
\begin{pf} There is an injective map $0\rightarrow E'_P \rightarrow E_P$.
The elementary transformation depends on the choice of a map
$E_P\rightarrow {\bf C}_P\rightarrow 0$. If this map is generic,
it induces a non-zero map $E'_P\rightarrow {\bf C}_P$.
Hence, we have a diagram
$$\begin{array}{ccccccccc}
& &0 & &0 & & & & \\
& &\downarrow & &\downarrow & & & & \\
0&\rightarrow& \tilde E'&\rightarrow &\tilde E&
\rightarrow &\tilde E''&\rightarrow &0\\
& &\downarrow & &\downarrow & & \downarrow & & \\
0&\rightarrow& E'&\rightarrow & E&
\rightarrow & E''&\rightarrow &0\\
& &\downarrow & &\downarrow & & & & \\
& &{\bf C}_P&\rightarrow & {\bf C}_P& & & & \\
& &\downarrow & &\downarrow & & & & \\
& & 0 & & 0 & & & & \\
\end{array}$$
This proves the statement.
\end{pf}
\begin{Lem}
\label{TE*} Let
$$0\rightarrow E' \rightarrow E\rightarrow E''\rightarrow 0$$
be an exact sequence of vector bundles. Then, for a generic dual
elementary transformation $\bar E$ of $E$, we have an exact sequence
of vector bundles
$$0\rightarrow E' \rightarrow \bar E\rightarrow \bar E''\rightarrow 0$$
where $\bar E''$ is a generic dual elementary transformation of
$E''$.
\end{Lem}
\begin{pf}:
Dualise the proof above
\end{pf}
\begin{Prop}
\label{exist}
Let
$$0\rightarrow E' \rightarrow E\rightarrow E''\rightarrow 0$$
be an exact sequence of vector bundles. Assume that $E$ is stable.
Then, there exists an exact sequence of vector bundles
$$0\rightarrow \hat E' \rightarrow \hat E \rightarrow \hat E''\rightarrow 0$$
satisfying
\begin{description}
\item[i)] $deg \hat E'=deg E'-1, deg \hat E=deg E$\medskip
\item[ii)] $\hat E$ is stable.
\end{description}
\end{Prop}
\begin{pf}
Take first an elementary transformation
of the exact sequence
based at a point $P$. Take next a dual elementary transformation
based at a point $Q$. From the two Lemmas above, $deg \hat E'=deg E+1$.
We now construct a family of these transformations which contains
$E$ as one of its members: let the point $Q$ vary until it coincides
with $P$. Then, with a suitable choice of the dual transformation,
one can go back to $E$. The existence of this family of
vector bundles together with the stability of $E$, implies the stability
of the generic $\hat E$.
\end{pf}
\begin{Cor}
\label{inclusio}
If $U_{r's}(r,d)$ is non empty,
then it is contained in the closure of $U_{r', s+r}(r,d)$.
\end{Cor}
\begin{pf}
Take $E\in U_{r',s}(r,d)$ and consider an exact sequence
$$0\rightarrow E'\rightarrow E\rightarrow E''\rightarrow 0$$
with $E'$ of rank $r'$ and maximal degree $d'$.
In the proof above, we construct a family with special
member $E$ and generic member $\tilde E$ that fits in an exact sequence
$$0\rightarrow \tilde E'\rightarrow \tilde E\rightarrow \tilde E''
\rightarrow 0$$
with $deg(\tilde E')=d'+1$.
Hence, this $\tilde E\in V_{r',s+r}(r,d)$.
From \ref{Irr}, $V_{r',s+r}(r,d)$ is irreducible and from the
dimensionality statement in \ref{Irr}, $V_{r',s+r}(r,d)\not\subseteq
V_{r',s-kr}(r,d), k\ge 0$.
Hence the generic element in $V_{r',s+r}(r,d)$
is in $U_{r',s+r}(r,s)$.
\end{pf}
\begin{Prop}
\label{spetita}
Let $C$ be a projective non-singular curve of
genus $g\ge 2$. Consider an exact sequence of vector bundles on $C$
$$0\rightarrow E'\rightarrow E\rightarrow E''\rightarrow 0.$$
Denote by $r',r'',r,d',d'',d$ the ranks and degrees of $E',E'',E$.
Assume that $E',E''$ are generic stable vector bundles
of their ranks and degrees.
If $0<r'd-rd'\le r$, then, the generic such $E$ is stable.
\end{Prop}
\begin{pf} Assume that $E$ is not stable. Let $F$ be a subbundle
of $E$ such that $\mu (F)\ge \mu (E)$. Up to replacing $F$ by a subbundle
of smaller rank or by its saturation, we can assume $F$ stable and
$E/F$ without torsion. As $E'$ is stable and
$\mu (E')<\mu (E)$, $F$ gives rise to a non-zero map
$\phi :F\rightarrow E''$. Denote by $F'$ its kernel, $F''$
its image.
Denote by $r_{F'}, r_{F},
r_{F''}, d_{F'}, d_{F}, d_{F''}$ the ranks and degrees of the bundles
$F',F,F''$.
{\it Claim 1.} $ r_{F''}=r''$.
Proof of Claim 1: Assume $ r_{F''}<r''$.
By the genericity of $E''$ this implies
$r_{F''}d''-r''d_{F''}\ge r_{F''}(r''-r_{F''})(g-1)$ (cf.[L] Satz 2.2).
Equivalently
$$\mu (F'')\le \mu (E'')-(1-(r_{F''}/r''))(g-1)$$
By the initial assumption $r'd-rd'\le r$,
$$(*)\mu (E'')\le \mu (E) +1/r''.$$
As $F$ is a destabilizing subbundle,
$$\mu (E)\le \mu (F)$$
and from the stability of $F$
$$\mu (F) \le \mu (F'')$$
with equality if and only if $F=F''$.
Notice that
$$1/r''-(1-(r_{F''}/r''))(g-1)\le 1/r''-(1-(r_{F''}/r''))\le 0$$
With equalities if and only if $g=2, r_{F''}=r''-1$.
Puting together all of the above inequalities, we find that they are all
equalities. This proves Claim 1 except in the case when all of the following
properties are satisfied:
\begin{description}
\item[i)] $g=2, r_{F''}=r''-1$
\item[ii)]$\mu(E)=\mu(F)$, $F'=0$ and $F''=F$ is a subsheaf of
$E''$
\item[iii)] $ (r''-1)d''-r''d_F=r''-1$
\end{description}
We shall see at the end of the proof that this situation does not
correspond to a generic $E$. This will finish the proof of Claim 1.
{\it Claim 2.} $F'=0$
Proof of Claim 2.
Note that $E^*$ satisfies the hypothesis in \ref{spetita} . If $F$ is a
maximal destabilising subbundle of $E$ and we write $G=E/F$,
then $G^*$ is a maximal destabilising subbundle of $E^*$. Then, Claim 2
follows from Claim 1.
From now on, we assume that $F$ is a subbundle of $E''$ and $r_F=r''$
As $F$ is a destabilising subbundle, $\mu (F)\ge \mu (E)$.
As $F$ is a subbundle of $E''$, $\mu (F)\le \mu (E'')$.
Using (*) and $r_F=r''$, we obtain either $d_F=d''$ or
$d_{F}= d''-1$ and $\mu (F)=\mu (E)=\mu (E'')-1/r''$.
If $F= E''$, the sequence splits and the extension
is not generic. If
$d_F=d''-1$, we have an exact sequence
$$0\rightarrow F\rightarrow E''\rightarrow T\rightarrow 0$$
where $T$ is a torsion sheaf of degree one supported at
one point, say $P$. Then, $F$ is determined by the choice of
$P$ and a map from $E_P$ to the base field defined up to homothety.
Therefore the
number of moduli for such $F$ is at most $r''$.
Consider the pull-back diagram
$$\begin{array}{ccccccccc} 0& \rightarrow &E'&\rightarrow &E&\rightarrow &E''&
\rightarrow &0\cr
& &\uparrow & &\uparrow & &\uparrow & & \cr
0& \rightarrow &E'&\rightarrow &E\times _{E''}F&\rightarrow &F&
\rightarrow &0\cr
\end{array} $$
As $F$ is a subsheaf of $E$, the bottom row splits. Hence the top row
corresponds to an element in the kernel of the map
$$ H^1(E^{''*}\otimes E')\rightarrow H^1(F^*\otimes E')\rightarrow 0.$$
This kernel has dimension at most $h^0(T\otimes E')=deg(T)\times rk(E')=r'$.
Therefore the dimension of the subspace of $ H^1(E^{''*}\otimes E')$ that
may correspond to unstable extensions
is at most
$$dim\{ F \} +r'\le r''+r'=r.$$
On the other hand,
$$ h^1(E^{''*}\otimes E')=r'd-rd'+r'(r-r')(g-1)=
r'(r-r')(g-1)+r$$
where the last equality comes from the condition $ \mu (E)=\mu (E'')-1/r''$ .
It is then clear that the generic extension is stable.
We now prove that the special situation at the end of Claim 1 does not occur:
Notice that from condition iii) and the stability of $E''$, $E''$
does not have a subbundle of rank $r''-1$ and degree higher than
$d_F$. Hence, we have a pull-back diagram as above but $E/F=L$ is a line
bundle. From condition iii) and \ref{cotasubf},
there is only a finite number of possible
$F$ for a given $E''$. From the genericity of $E''$, both $F$ and $L$ are
generic (and depend only on $E''$ and not on $E'$). Hence, it is enough
to show that the canonical map
$$ H^1(E^{''*}\otimes E')\rightarrow H^1(F^*\otimes E')$$
is non-zero. As this map is surjective, this is equivalent to
$ H^1(F^*\otimes E')\not= 0$. From Riemann-Roch
$ h^1(F^*\otimes E')\ge r'(r''-1)+r''(r''-1)(\mu (F)-\mu (E'))>0$
where the last inequality comes from ii) $\mu (E')<\mu (E)$ and
$r_F=r''-1>0$.
\end{pf}
\begin{pf}. We now prove \ref{Theorem} except for the unicity of the
subbundle that we
postpone to next section.
From \ref{cotasubf} , \ref{Irr} and \ref{inclusio} ,
it is enough to prove the non-emptiness
of $U_{r's}(r,d)$. Take now any positive $s$. Write $s=ar+\bar s, \
0<\bar s\le r$. Then, $U_{r',\bar s}(r,d)$ is non empty
by \ref{spetita}. Applying
\ref{exist} $a$-times, we obtain the non-emptiness of $V_{r',s}(r,d)$.
From the definitions of $V_{r',s}(r,d),U_{r',s}(r,d)$, a generic element of
$V_{r',s}(r,d)$ belongs to an $U_{r',\tilde s}(r,d)$ for some
$\tilde s\le s$. In order to prove the non-emptiness of
$U_{r',s}(r,d)$, it is enough to see that
$V_{r',s}(r,d)\not\subset V_{r',\tilde s}(r,d), \tilde s<s$.
From the dimensionality statement in Lemma 5,
this is true.
\end{pf}
\begin{pf} The proof of \ref{Theorem2} is similar to the proof
of \ref{Theorem}: use
\ref{spetita}, \ref{exist} and the dimensionality statement in
\ref{Irr}.
\end{pf}
\section{Brill-Noether for twisted bundles and unicity of subbundles}
In this section we prove the result that we stated in the introduction
about twisted Brill-Noether Theory. The corresponding result
for the untwisted case (i.e. $E={\cal O}$) is well-known (cf[S] Th IV 2.1).
We then use this result to show the unicity of the Lange subbundle.
\begin{pf} (of \ref{BrillNoether})
Assume that $ W^0_{r_F,d_F}(E)$ is non-empty.
Consider an element $F$ in this set. This gives rise to
a non-zero map $F\rightarrow E$. Denote by $F'$ its kernel,
$F''$ its image. Then $F''$ is a subbundle of $E$. Moreover, we have an
exact sequence
$$0\rightarrow F'\rightarrow F\rightarrow F''\rightarrow 0.$$
Assume first $r_{F''}<r_E$. From \ref{Theorem}, the set of
saturated subbundles
of $E$ of rank $r_{F''}$ and degree $d_{F''}$ has
dimension
$$dim\{ F''\} = r_{F''}(r_{F''}-r_{E})(g-1)+r_{F''}d_{E}-r_{E}d_{F''}$$
if this number is positive and is empty otherwise. The set of
non-saturated subbundles has dimension smaller than this number.
Consider then the case in which $r_{F''}=r_E$. Then, the quotient
$E/F''$ is torsion. The choice of $F''$ depends on the choice of the support
of this quotient and for each point $P$ on the support the choice of
a map (up to homothety) to the base field $E_P\rightarrow {\bf C}$ .
Hence,
$dim\{ F''\} \le r_E(d_E-d_{F''})$. This coincides with the bound above
for the case $r_{F''}=r_E.$
Any family of vector bundles can be embedded in a family with
generic member stable(cf [NR] Prop.2.6). Hence,
$F'$ varies in a parameter space of dimension at most
$$dim \{ F' \} =r_{F'}^2(g-1)+1.$$
From \ref{h^0=0}, $h^0(F*{''*}\otimes F')=0$. Using Riemann-Roch,
$$h^1(F^{''*}\otimes F')=
r_{F'}r_{F''}(g-1)+r_{F'}d_{F''}-r_{F''}d_{F'}.$$
The choice of $F$ depends on the choice of the pair $F',F''$ and
the class of the extension up to scalar.
Therefore, the dimension of all possible $F$ is bounded by
$$dim \{ F\} \le [r_{F'}^2+r_{F''}^2-r_{F''}r_E+r_{F'}r_{F''}](g-1)+
r_{F'}d_{F''}-r_{F''}d_{F'}+r_{F''}d_E-r_Ed_{F''}=$$
$$=\rho ^0_{r_F,d_F}(E)-[r_{F'}d_{E''}-d_{F'}r_{E''}-
r_{F'}r_{E''}(g-1)]$$
where $E''$ denotes the quotient of $E$ by $F''$
Let us check that
$$(*)[r_{F'}d_{E''}-d_{F'}r_{E''}-r_{F'}r_{E''}(g-1)]\ge 0.$$
By the genericity of $E$, if $F''$ exists, then
$\mu (E'')-\mu (F'')\ge g-1$ (cf Prop. 2.4)
By the stability of $F$, $\mu (F')<\mu (F'')$. Hence,
$\mu (E'')-\mu (F')\ge (g-1)$. This is equivalent to the inequality
(*) and proves the upper bound for the dimension.
Notice also that (*) vanishes if and only if either $F'=0$
or $F''=E$. In both these cases, the map has maximal rank.
It only remains to prove existence in case the Brill-Noether
number is positive. If $r_F\not= r_E$, this is equivalent to the existence of
a stable subbundle or quotient of $E$ and is contained in \ref{Theorem2}.
If $r_F=r_E$, one needs to check that the generic elementary transformation
of a stable vector bundle is again stable. This is well known.
\end{pf}
\begin{Prop}
Let $E$ be a stable vector bundle obtained as a generic extension
$$0\rightarrow E'\rightarrow E\rightarrow E''\rightarrow 0.$$
Assume that $0<s=r'd-rd'\le r'(r-r')(g-1)$.
Then the only subbundle of $E$ of rank $r'$
and degree $d'$ is $E'$.
\end{Prop}
\begin{pf}
Assume that there were another subbundle $ F'$ of rank $r'$
and degree $d'$. Denote by $F''$ the quotient sheaf $E/F'$.
Claim: If $E$ is general, both $F',F''$ are generic vector bundles
of the given ranks and degrees (i.e. as $E$ varies, $F',F''$ vary
in an open dense subset of the corresponding moduli spaces).
Proof of the claim: If $F''$ had torsion, then $E$ would have a subbundle
of higher degree and from \ref{Theorem} it could not be general.
If $F'$ or $F''$ were not general or were not stable, then
they would move in varieties of dimension strictly smaller than
those parametrising $E',E''$. From \ref{h^0=0}, $h^1(E^{''*}\otimes E')=
h^1(F^{''*}\otimes F')$. From \ref{cotasubf}, every $E$ appears in
at most a finite number of extensions of an $E''$ by an $E'$.
Hence, $E$ could not be general. This proves the claim.
We obtain non-zero maps $ F'\rightarrow E''$ and
$E'\rightarrow F''$. From the genericity of
$E', E''$ and \ref{BrillNoether} , the dimension of the sets of
these $ F',F''$ is at most
$$dim\{ F'\} =r'(r'-r'')(g-1)+r'd''-r''d'=r'(r'-r'')(g-1)+s$$
$$dim \{ F''\} =r''(r''-r')(g-1)+r'd''-r''d'=r''(r''-r')(g-1)+s$$
and both these numbers are positive.
From \ref{h^0=0} and the stability of $E$, $h^0(F^{''*}\otimes F')=0$.
Hence, from Riemmann-Roch
$$h^1(F^{''*}\otimes F')=r'r''(g-1)-[r''d'-r'd'']=
r'r''(g-1)+s$$
We obtain then a bound for the dimension of the set of $E$ for which
there is an exact sequence
$$0\rightarrow F'\rightarrow E\rightarrow F''\rightarrow 0$$
given by
$$dim \{ E\} \le (r^{'2}+r^{''2}-r'r'')(g-1)+3s-1$$
On the other hand, from \ref{Irr},
$$dim \{ E\} =(r^{'2}+r^{''2}+r'r'')(g-1)+s+1$$
It follows then that
$2r'r''(g-1)\le 2s-2$. This contradicts our assumption on $s$.
\end{pf}
|
1997-10-29T20:17:16 | 9710 | alg-geom/9710033 | en | https://arxiv.org/abs/alg-geom/9710033 | [
"alg-geom",
"math.AG"
] | alg-geom/9710033 | Gian Mario Besana | Alberto Alzati and Gian Mario Besana | On the k-normality of some projective manifolds | AMS-LaTeX, 20 pages, to appear in Collect. Math. special volume in
memory of F. Serrano | null | null | null | null | A long standing conjecture, known to us as the Eisenbud Goto conjecture,
states that an n-dimensional variety embedded with degree $d$ in the $N$-
dimensional projective space is $(d-(N-n)+1)$-regular in the sense of
Castelnuovo-Mumford.
In this work the conjecture is proved for all smooth varieties $X$ embedded
by the complete linear system associated with a very ample line bundle $L$ such
that $\Delta (X,L) \le 5$ where $\Delta (X,L) = \dim{X} + \deg{X} -h^0(L).$ As
a by-product of the proof of the above result the projective normality of a
class of surfaces of degree nine in $\Pin{5}$ which was left as an open
question in a previous work of the second author and S. Di Rocco
alg-geom/9710009 is established. The projective normality of scrolls $X
=\Proj{E}$ over a curve of genus 2 embedded by the complete linear system
associated with the tautological line bundle assumed to be very ample is
investigated. Building on the work of Homma and Purnaprajna and Gallego
alg-geom/9511013, criteria for the projective normality of three-dimensional
quadric bundles over elliptic curves are given, improving some results due to
D. Butler.
| [
{
"version": "v1",
"created": "Wed, 29 Oct 1997 19:17:16 GMT"
}
] | 2007-05-23T00:00:00 | [
[
"Alzati",
"Alberto",
""
],
[
"Besana",
"Gian Mario",
""
]
] | alg-geom | \section{Introduction}
A complex projective variety $X\subset \Pin{N}$ is $k$-regular in the
sense of
Castelnuovo-Mumford if
$h^i(\iof{X}{k-i}) = 0$ for all
$i\ge 1$ where $\iofo{X}$ is the ideal sheaf of $X.$ If
$X$ is $k$-regular then the minimal generators of its homogeneous ideal
have
degree less than or equal to $k.$
A long standing conjecture, known to us as the Eisenbud Goto conjecture,
states
that an $n$-dimensional variety $X \subset \Pin{N}$ of degree $\deg X =
d$ is
$(d-(N-n)+1)$-regular. Gruson Lazarsfeld and Peskine \cite{GLP}
established the
conjecture for curves, Lazarsfeld \cite{Laz1} for smooth surfaces and
Ran
\cite{Ran1} for threefolds with high enough codimension. A nice
historical account
of the conjecture and further results can be found in \cite{Kw}.
In section \ref{deltacongettura} the conjecture is proved for all
smooth
varieties $X$ embedded by the complete linear system associated with a
very
ample line bundle $L$ such that $\Delta (X, L) \le 5$ where $\Delta (X, L) =
\dim{X} +
\deg{X} -h^0(L).$
Notice also that in recent times computer algebra systems like
Macaulay have made possible the
explicit construction and study of examples of algebraic varieties
starting from
minimal generators of the homogeneous ideal of the variety. A priori
information
on the $k$-regularity of a variety is therefore useful for these
constructions.
Strictly related to the notion of $k$-regularity is the notion of
$k$-normality of a
projective variety. A variety $X \subset \Pin{N}$ is $k$-normal if
hypersurfaces of
degree $k$ cut a complete linear system on $X$ or, equivalently,
if $h^1(\iof{X}{k})=0.$ If $X$ is $k$-regular it is clearly
$(k-1)$-normal. $X$ is said to
be projectively normal if it is $k$-normal for all $k\ge 1.$
As a by-product of the proof of the above result the projective
normality of a class of surfaces of degree nine in $\Pin{5}$ which was
left as an open
question in \cite{gisa} is established in Lemma \ref{blowupF1}. The non
existence of a
class of scrolls of degree $10$, left as an open problem in
\cite{fa-li10}, is also
established in Remark \ref{scr10}.
In section \ref{g2scrolls} we deal with the projective normality of
scrolls $X
=\Proj{E}$ over a curve of genus $2$ embedded by the complete linear
system
associated with the tautological line bundle $\taut{E},$ assumed to be
very ample.
Two-dimensional such scrolls are shown to be always projectively normal
except
for a class $S$ of non
$2$-normal surfaces of degree eight in
$\Pin{5}$ studied in detail in \cite{alibaba}. Three-dimensional scrolls
$X=\Proj{E}$
of degree
$\deg X \ge 13$ are then shown to be projectively normal if and only if
$E$ does not
admit a quotient
$E
\to
\cal{E}
\to 0$ where $P(\cal{E})$ belongs to the class $S$ of non quadratically
normal
surfaces mentioned above.
In section \ref{ellipticpkbundles}, building on the work of Homma
\cite{Ho1},\cite{Ho2} and Purnaprajna and Gallego \cite{pu-ga}, criteria
for
the projective normality of three-dimensional quadric bundles over
elliptic curves
are given, improving some results contained in \cite{bu}.
\section{General Results and Preliminaries}
\label{prelimsec}
\subsection{Notation}
\label{notation}
The notation used in this work is mostly standard from Algebraic
Geometry. Good references are \cite{H} and \cite{gh}.
The ground field is always the field $\Bbb{ C}$ of complex
numbers. Unless otherwise stated all varieties are supposed to be
projective.
$\Bbb{P}^{N}$ denotes the N-dimensional complex projective space.
Given a projective n-dimensional variety $X$, ${\cal O}_X$
denotes its structure sheaf and $Pic(X)$ denotes the group of line
bundles over $X.$ Line bundles, vector bundles and Cartier
divisors are denoted by capital letters as $L, M,\cal{M} \dots.$ Locally
free sheaves of
rank one, line bundles and Cartier divisors are used interchangeably as
customary.
Let $L, M \in Pic(X)$, let $E$ be a vector bundle of rank $r$ on $X$,
let
$\cal{F}$ be a coherent sheaf on $X$ and let $Y\subset X$ be a
subvariety of $X.$
Then the following notation is used:
\begin{enumerate}
\item[ ] $LM$ the intersection of divisors $L$ and $M$
\item[ ] $L^{n}$ the degree of $L,$
\item[ ] $|L|$ the complete linear system of effective divisors
associated with $L$,
\item[ ]$L_Y$ or $\restrict{L}{Y}$ the restriction of $L$ to $Y,$
\item[ ] $L \sim M$ linear equivalence of divisors
\item[ ] $L \equiv M$ numerical equivalence of divisors
\item[ ] Num$(X)$ the group of line bundles on $X$ modulo numerical
equivalence
\item[ ] $\Bbb{P}(E)$ the projectivized bundle of $E,$ see \cite{H}
\item[ ] $H^i(X, \cal{F})$ the $i^{th}$ cohomology vector space with
coefficients in ${\cal
F},$
\item[ ] $h^i(X,\cal{F})$ the dimension of $H^i(X, \cal{F}),$ here and
immediately
above $X$ is sometimes omitted when no confusion arises.
\end{enumerate}
If $C$ denotes a smooth projective curve of genus $ g$, and $E$ a vector
bundle
over $C$ of deg $E= c_1(E)= d$ and rk$E=r$, we need the following
standard
definitions:
\begin{enumerate}
\item[ ] $E$ is $\it normalized$ if $h^0(E)\ne 0$ and $h^0(E \otimes
\cal{L})=0$
for any invertible sheaf $\cal{L}$ over $C$ with deg$\cal{L}<0$.
\item[ ] $E$ has slope $\mu(E) = \frac{d}{r}$.
\item[ ] $E$ is $\it semistable$ if and only if for every proper
subbundle $S$,
$\mu(S) \leq \mu(E)$. It is $\it stable$ if and only if the inequality
is strict.
\item[ ] The Harder-Narasimhan filtration of $E$ is the
unique filtration:
$$0=E_0\subset E_1\subset ....\subset E_s=E$$
such that
$\frac{E_i}{E_{i-1}}$ is semistable for all $i$, and $\mu_i(E)=\mu
(\frac{E_i}{E_{i-1}})$ is
a strictly decreasing function of $\it i$.
\end{enumerate}
A few definitions from \cite{bu} needed in the sequel are recalled.
Let $0=E_0 \subset E_1 \subset ....\subset E_s=E$ be the
Harder-Narasimhan
filtration of a vector bundle $E$ over $C$. Then
\begin{enumerate}
\item[]$\mu^-(E)=\mu_s(E)=\mu (\frac{E_s}{E_{s-1}})$
\item[]$\mu^+(E)=\mu_1(E)=\mu (E_1)$
\item[]or alternatively
\item[]$\mu^+(E)= \text{ max }\{\mu(S) |0 \to S \to E \}$
\item[]$\mu^-(E)= \text{ min }\{\mu(Q) |E \to Q \to 0 \}$.
\end{enumerate}
It is also $\mu^+(E) \geq \mu(E) \geq \mu^-(E)$ with equality if and
only if $E$ is
semistable.
In particular if $C$ is an elliptic curve, an indecomposable vector
bundle $E$ on $C$ is
semistable and hence $\mu(E) = \mu^-(E)=\mu^+(E)$.
The following definitions are standard in the theory of polarized
varieties. A good
reference is \cite{fu}. A {\em polarized variety } is a pair $(X, L)$
where $X$
is a smooth projective n-dimensional variety and $L$ is an
ample line bundle on $X$. Its {\em sectional genus}, denoted
$g(X, L)$, is defined by $2g(X, L) - 2 = (K_X +
(n-1)L) L^{ n-1}$. Given any $n$-dimensional polarized
variety $(X, L)$ its $\Delta${\em - genus } is defined by $\Delta (X, L) =
dim (X) + L^{ n} - h^0(X, L).$
A polarized variety $(X, L)$ has a {\em ladder} if there
exists a sequence of reduced and irreducible subvarieties $X = X_n
\supset X_{n-1} \dots \supset X_1$ of $X$ where $X_j\in|L_{j+1}| =
|\restrict{L}{X_{j+1}}|.$ Each $(X_j, L_j)$ is called a {\em rung} of
the ladder. If $L$ is generated by global sections $(X, L)$ has a
ladder. A rung $(X_j,
L_j)$ is {\em regular} if
$H^0(X_{j+1},\restrict{L}{X_{j+1}})
\to H^0(X_j,\restrict{L}{X_j})$ is onto. The ladder is regular if all
the
rungs are
regular. If the ladder is regular $\Delta (X_j,L_j) = \Delta(X,L)$ for
all $1\le j \le n.$
A variety $X \subset \Pin{N}$ is {\it k-normal} for some $k \in \Bbb{Z}$
if
$H^0(\Pin{N},
\oofp{N}{k})
\to H^0(\oof{X}{k})$ is onto. Equivalently, if $\iofo{X}$ is the ideal
sheaf of $X,$ $X$ is
$k$-normal if
$h^1(\iof{X}{k}) = 0.$
$X$ is {\it projectively normal} if it is $k$-normal for all $k \ge 1.$
A polarized pair
$(X, L)$ with $L$ very ample is called $k$-normal or projectively normal
if $X$ is
$k$-normal or p.n. in the embedding given by $|L|.$ A polarized variety
$(X, L)$ with
$L$ very ample is always $1$-normal (linearly normal).
A line bundle $L$ on
$X$ is {\em normally (or simply) generated} if the graded algebra
$G(X, L) = \bigoplus_{t\ge 0}H^0(X,tL)$ is generated by $H^0(X,L).$
$L$
is very ample and normally generated if and only if $(X, L)$ is p.n.
A variety $X\subset \Pin{N}$ is $k$-regular, in the sense of
Castelnuovo-Mumford, if
for all
$i\ge 1$ it is $h^i(\iof{X}{k-i})=0.$ A polarized pair $(X, L)$ with $L$
very ample is
$k$-regular if
$X$ is $k$-regular in the embedding given by $|L|.$
If $X$ is $k$-regular then it is $(k+1)$-regular.
\medskip
\subsection{General Results}
\medskip
Let $C$ be a smooth projective curve of genus $g$, $E$ a vector
bundle of rank $n$, with $n \ge 2$, over $C$ and $\pi : X =\Bbb{P}(E)
\to C $ the projectivized bundle
associated to $E$ with the natural projection $\pi$.
Denote
with $\cal{T} = \taut{E} $ the tautological sheaf and with $\frak{F}_P=
\pi^*\cal {O}_{C}(P) $ the line bundle associated with the fiber over
$P\in C.$ Let
$T$ and
$F$ denote the numerical classes respectively of $\cal T$ and
$\frak{F}_P$.
In this work we refer to a polarized variety $(X,\cal{T})$ as a {\it scroll}
over a curve $C$ if
there is a vector bundle
$E$ over $C$ such that $(X,\cal{T})= \scroll{E}$ and $\cal{T}$ is very ample.
\begin{rem}
\label{leray}
Let $\pi : \scroll{E} \to C$ be a $n$-dimensional projectivized bundle
over a curve
$C.$ From Leray's Spectral sequence and standard facts about higher
direct image sheaves (see for example
\cite{H} pg. 253) it follows that
\begin{gather}
H^1(\tautof{E}{t})=H^1(C, S^tE) \text{ for } t\ge 0 \notag\\
H^i(\tautof{E}{t}) = 0 \text {
for } i \ge 2 \text { and } t>-n. \notag
\end{gather}
\end{rem}
Let $D\sim a\cal{T} + \pi^*B$, with $ a\in \Bbb{Z}$, $B\in Pic(C)$
and $\deg B = b$, then $ D \equiv aT+bF .$ Moreover
$\pi_*(\oof{\Proj{E}}{D}) =
S^{a}(E)
\otimes
\cal {O}_{C}(B) $ and hence $\mu^-(\pi_*(\oof{\Proj{E}}{D})=a\mu^-(E)
+b$ (see
\cite{bu}).
Regarding the ampleness, the global generation, and
the normal generation of $D$, a few known criteria useful in the sequel
are listed
here:
\begin{theo}[Miyaoka \cite{Miyao3}]
\label{miyaoteo}
Let $E$ be a vector bundle over a smooth
projective curve $C$ of genus $g$, and $X =\Bbb{P}(E)$ . If $D\equiv
aT+bF$ is a line
bundle over $X$, then $D$ is ample if and only if $a>0$ and $b+a
\mu^-(E) >0$.
\end {theo}
\begin{lemma}
\label{buongg}
(see e.g. \cite{bu}, Lemma 1.12)
Let $E$ be a vector bundle over $C$ of genus $g$.
\begin{itemize}
\item[i)]if $\mu^-(E) > 2g-2$ then $h^1(C, E)=0$
\item[ii)]if $\mu^-(E) > 2g-1$ then $E$ is generated by global sections.
\end{itemize}
\end{lemma}
\begin{lemma}[Butler \cite{bu} Theorem 5.1A]
\label{criteriodelbutler}
Let $E$ be a vector bundle on a
smooth projective curve of genus $g$ and let $D\equiv aT + bF$ be a
divisor on $X
=\Bbb{P}(E).$ If
\begin{equation}
\label{condizionedelbutler}
b+a\mu^-(E) > 2g.
\end{equation}
then $D$ is normally generated.
\end{lemma}
A few basic facts on the {\it Clifford Index} of a curve are recalled.
Good references
are \cite{mart} and \cite{GL}. Let $C$ be a projective curve and
$L$ be any line bundle on $C$. The Clifford index of $L$ is defined as
follows:
$$cl(L)=deg(L) -2(h^0(L)-1).$$
The Clifford index of the curve is $cl(C)=\text {min}\{cl(L) |
h^0(L)\geq 2 \text{ and }h^1(L)\geq 2 \}$.
For a general curve $C$
it is $cl(C)=\left [\frac{g-1}{2}\right ]$
and in any case $cl(C)\leq\left [\frac{g-1}{2}\right ]$. By
Clifford's theorem a special line bundle $L$ on $C$ has $cl(L)\geq
0$ and
the equality holds if and only if $C$ is hyperelliptic and $L$ is a
multiple of the unique
$g^1_2$.\\ If
$cl(C)=1$ then $C$ is either a plane quintic curve or a trigonal curve.
\begin{theo}[\cite{GL}]
\label{glcliff}
Let L be a very ample line bundle on a smooth irreducible complex
projective curve
$C.$ If
$$\deg(L)\geq 2g+1-2h^1(L)-cl(C)$$
then $(C, L)$ is projectively normal.
\end{theo}
\section{The Eisenbud Goto conjecture for low values of $\Delta.$}
\label{deltacongettura}
Let $X \subset \Pin{N}$ be an $n$ dimensional projective variety of
degree $d$. A long
standing conjecture, known to us as the {\it Eisenbud Goto} conjecture,
states that
$X$ should be $(d-(N-n) +1)$-regular, i.e.
$(degree-codimension+1)$-regular.
Many authors worked on the conjecture for low values of the dimension
and
codimension of X. A nice historic account is found in \cite{Kw}. Some of
their results
are collected in the following Theorem.
\begin{theo}[\cite{GLP}, \cite{Laz1}]
\label{lazran}
If $X\subset \Pin{N}$ is any smooth curve or any smooth surface then
$X$ is
$(d-c+1)$-regular where $d =\deg{(X)}$ and $c = \text{codimension} (X)$.
\end{theo}
In this section we would like to offer a proof of the conjecture for
linearly normal smooth varieties with low $\Delta$-genus. Let
$(X, L)$ be a polarized variety with
$L$ very ample. The above conjecture can be restated for the embedding
given by
$|L|$ in terms of
$\Delta$-genus as follows:
{\bf Conjecture } {\it Let $(X, L)$ be a polarized variety with $L$ very
ample. Then $(X, L)$
is
$(\Delta + 2)$-regular.}
\begin{rem}
\label{ipers}
It is straightforward to check that hypersurfaces of degree $d$ are
always
$d$-regular and not $(d-1)$-regular. This shows that the conjecture is
indeed
sharp. On the other hand there are varieties $X\subset \Pin{N}$ which
are
$k$-regular for $k < d-c+1.$ This motivates Definition \ref{extremal}.
\end{rem}
\begin{rem}
It is a classical adjunction theoretic results that given $(X, L)$ with
$L$ very ample, $K_X + tL$ is globally generated, and in particular
$h^0(K_X + tL) \ne 0,$ for
$ t
\ge n$ unless
$t=n$ and $(X, L) = (\Pin{n}, \oofp{n}{1}).$ This fact, Remark \ref{ipers}
and the sequence $0 \to \iofo{X}\to {\cal O}_{\Pin{N}} \to {\cal O}_X \to 0$
suitably twisted show that no linearly normal non degenerate
$n$-dimensional variety $X \subset \Pin{N}$ can be $k$-regular for
$k \le 1.$ Therefore in what follows we will always assume $k \ge 2$
when dealing with $k$-regularity.
\end{rem}
\begin{dfntn}
\label{extremal} Let $X \subset
\Pin{N}$ be a $n$-dimensional variety of degree
$d.$Let
$$r(X)= Min \{k \in \Bbb{Z} | X \text{ is } k-\text{regular}\}.$$ A
variety $X$ is
{\bf extremal} if
$r (X) = d-(N-n)-1.$
A polarized variety $(X, L)$ with $L$ very ample is {\bf extremal} if it
is extremal in
the embedding given by $|L|,$ i.e. if $r(X,L) = Min
\{k \in \Bbb{Z} | (X, L) \text{ is } k-\text{regular}\} = \Delta + 2.$
\end{dfntn}
In what follows we will prove the above conjecture for all
linearly normal manifolds with
$\Delta
\le 5$ obtaining along the way the value of $r (X, L)$ for most of the
same
manifolds.
\begin{lemma}
\label{hyperplanesec}
Let $X \subset \Pin{N}$ be a smooth n-dimensional variety and let $Y
\subset
\Pin{N-1}$ be a generic hyperplane section.
\begin{itemize}
\item[i)] If $X$ is $k$-regular then $Y$ is $k$-regular
\item[ii)] If $Y$ is $k$-regular and $X$ is $(k-1)$-normal then $X$ is
$k$-regular.
\item[iii)] If $X$ is $(r(Y)-1)$-normal then $r(X)=r(Y)$
\end{itemize}
\end{lemma}
\begin{pf}
The exact sequence
\begin{equation}
\label{ideali}
0 \to \iof{X}{k-i} \to \iof{X}{k-i+1} \to\iof{Y}{k-i+1} \to 0.
\end{equation}
immediately gives i). To see ii) consider again sequence \brref{ideali}.
The $k$-regularity of $Y$ gives
$h^{i-1}(\iof{Y}{k-i+1})=0$ for all
$i\ge 2.$ Since $k$ regularity implies $k+1$-regularity it is
$h^i(\iof{Y}{k-i+1})=0$ for all $i \ge 1.$ Therefore
$h^i(\iof{X}{k-i})=h^i(\iof{X}{k-i+1})$ for all $i\ge 2$ from
\brref{ideali} and
iteratively
$h^i(\iof{X}{k-i})=h^i(\iof{X}{k-i+t}$ for all $i\ge 2$ and for all
$t\ge 1.$ Letting $t$
grow, Serre's vanishing theorem gives $h^i(\iof{X}{k-i+t}=0$ for all
$i\ge 2$ and all
$t\ge 1$ and thus
$h^i(\iof{X}{k-i})=0$ for all $i\ge 2.$ Because $X$ is assumed
$(k-1)$-normal it is
$h^1(\iof{X}{k-1})=0$ which concludes the proof of $ii).$
Now $iii)$ follows immediately from $i)$ and $ii).$
\end{pf}
\begin{lemma}
\label{hyperplanesecpol}
Let $(X, L)$ be a polarized variety with $L$ very ample. Let $Y \in |L|$
be a generic
element and assume
$H^0(X, L)\to H^0(Y, \restrict{L}{Y})$ is onto. Then $i), ii),iii)$ as
in Lemma
\ref{hyperplanesec} hold if we replace $X$ by $(X, L)$ and $Y$ by $(Y,
\restrict{L}{Y}).$
\end{lemma}
\begin{pf}
Let $h^0(L)=N+1.$ The surjectivity condition on the restriction map
between global
sections of
$L$ and
$\restrict{L}{Y}$ guarantees that $|\restrict{L}{Y}|$ embedds $Y$ as a
linearly normal
manifold in $\Pin{N-1}$ , therefore the same proof as in Lemma
\ref{hyperplanesec} applies.
\end{pf}
\begin{rem}
\label{liftingpn}
Let $(X, L)$ be a polarized variety with $L$ very ample. Let $Y \in |L|$
be a generic element and assume
$H^0(X, L)\to H^0(Y, \restrict{L}{Y})$ is onto. Then \cite{fu}
Corollary 2.5 shows
that if $(Y, \restrict{L}{Y})$ is projectively normal, so is $(X, L).$
Therefore when the
ladder is regular and $Y$ is p.n. Lemma \ref{hyperplanesecpol} gives $r
(X, L) = r (Y,
\restrict{L}{Y}).$
\end{rem}
\begin{lemma}
\label{rofpncurves}
Let $(C, L)$ be a projectively normal curve with $g\ge 1.$
Then $r (C, L) = Min \{ t \ge 3 |
h^1((t-2) L) =0\}.$
\end{lemma}
\begin{pf}
Let $h^0(L)=N+1$ so that $C\subset \Pin{N}.$ It is
$h^1(\iof{C}{k-1})=0$ for all
$k\ge 2$ because of the projective normality assumption. The sequence
$$0 \to \iof{C}{k-i} \to \oofp{N}{k-i} \to (k-i) L \to 0$$
easily gives $h^i(\iof{C}{k-i})=0$ for all $i \ge 3$ and $k \ge 2.$
The same sequence
gives $h^2(\iof{C}{k-2})=h^1((k-2) L)$ and since
$h^1({\cal O}_C) = g \ge 1$ it is $r (C, L) = Min \{ t \ge 3 | h^1((t-2) L)
=0\}.$
\end{pf}
In order to apply the above lemmata in one occasion
the projective normality of a particular class of
surfaces of degree nine needs to be established. The
following Lemma also improves \cite{gisa}. Here $\Bbb{F}_1$ denotes the
Hirzebruch
rational ruled surface of invariant $e=1,$ $\pi : Bl_{t}S \to S$
denotes the blow up of
a surface $S$ at $t$ points, $E_i$ are the exceptional divisors of the
blow up,
$\frak{C}_0=\pi^*(C_0)$ denotes the pull back of the line bundle
associated with the
fundamental section of
$\Bbb{F}_1$ and
$\frak{f}=\pi^*(f)$ the pull back of the one associated with any fibre
$f$ of the
natural projection
$p:
\Bbb{F}_1
\to \Pin{1}.$
\begin{lemma}
\label{blowupF1}
Let $(S, L)=(Bl_{12} {\Bbb F}_1, 3{\frak C_0} + 5{\frak f} - \sum_i E_i)$.
Then $(S, L)$ is projectively normal.
\end{lemma}
\begin{pf}
The projective normality of linearly normal degree nine surfaces
was studied in \cite{gisa}. Let $(S, L)$ be a surface of degree $9$ and
sectional genus
$5,$ embedded in $\Pin{5}.$
The surface under consideration was
established to be projectively normal unless its generic curve
section $C$ is trigonal and $\restrict{L}{C}=K_C-M+D$ where $M$ is a
divisor in the
$g^1_3$ and $D$ is a divisor of degree $4$ giving a foursecant line for
$C.$
Therefore if $S$ were not p.n. it would admit an infinite number of
$k\ge 4$-secant lines. On the other hand a careful study of the
embedding shows
that
$S$ contains only a finite number of lines and that the only lines
with
self intersection $\ge -1$ are the $12$ exceptional divisors $E_i.$
Thus the formulas
contained in
\cite{LeBz1} can be used. A straightforward calculation using
\cite{LeBz1} shows that $S$ cannot have a infinite number of $k\ge
4$-secants,
contradiction.
\end{pf}
\begin{theo}
\label{alafujita}
Let $(X, L)$ be a $n$ dimensional polarized pair, $n\ge 2,$ with a ladder.
Assume $g=g (X, L) \ge \Delta (X, L) = \Delta$ and $d=L^n \ge 2 \Delta + 1.$
Then :
\begin{itemize}
\item[i)] The
curve section $(C,\restrict{L}{C})$ is $k$-regular if and only if $(X, L)$ is
$k$-regular and $r (X, L)= r
(C,\restrict{L}{C}).$
\item[ii)] Either $\Delta=0,1$ and $(X, L)$ is extremal or $\Delta\ge 2$
and $r (X, L)=3.$
\end{itemize}
\end{theo}
\begin{pf}
From \cite{fu} Theorem
(3.5) and from the fact that a normally genereated ample line bundle is
automatically
very ample it follows that
$L$ is very ample, $g=\Delta$, the ladder is regular and every rung of
the ladder is
projectively normal. Therefore Lemma \ref{hyperplanesecpol} immediately
gives
$i).$
Then $(X, L)$ is extremal if and only if the curve section $(C,\restrict{L}{C})$ is
such. Extremal linearly normal curves were classified in \cite{GLP} and
they are
either rational or elliptic normal curves. Therefore
$(X, L)$ is extremal if and only if $\Delta=g=0,1.$
Now assume $\Delta \ge 2$ and thus $(X, L)$ not extremal. The curve
section $(C,\restrict{L}{C})$ is
embedded in $\Pin{M}$where $M= d - \Delta.$ Since
$h^M(\oofp{M}{k-M})=h^0(\oofp{M}{-1-k})=0$ for all $k \ge 0$ the
sequence $0 \to
\iof{C}{k-i} \to \oofp{M}{k-i} \to \oof{C}{k-i} \to 0$ shows that
$h^i(\iof{C}{k-i}) =
h^{i-1}(\oof{C}{k-i})$ for all $i \ge 2$ and all $k\ge 0.$
Therefore, because $h^1({\cal O}_C) = g \ge 2,$ it must be $r (X, L) \ge 3.$
If $i\ge 3$ then clearly $h^{i-1}(\oof{C}{3-i}) =0$ and thus
$h^i(\iof{C}{3-i})=0.$
It is also $h^2(\iof{C}{1}) = h^1(\oof{C}{1}) = h^1(\restrict{L}{C})=0$
because
$g=\Delta$ and $d\ge 2 \Delta+1 > 2g-2.$
Since every rung of the ladder is projectively normal, in
particular $h^1(\iof{C}{2}) =0$ and thus $(C,\restrict{L}{C})$ is $3$-regular. We can
conclude
that $r
(X, L)=r
(C,\restrict{L}{C}) = 3.$
\end{pf}
\begin{prop}
\label{elscr}
Let $(X,\cal{T}) = (\Proj{E}, \taut{E}) $ be a scroll over an elliptic curve.
Then
$r (X,\cal{T}) = 3.$
\end{prop}
\begin{pf}
Because $h^2(\iofo{X})=h^1({\cal O}_X)=1$ it is $r (X, L) \ge 3.$
We need to show that $h^i(\iof{X}{3-i})=0$ for all $i\ge 1.$ Notice that
$|\cal{T}|$ embeds $X$ into $\Pin{N}$ as a variety of degree $d$ where
$N=d-1.$ Let $i=1.$ It is known, cf. \cite{bu} and \cite{alibaba},
that
elliptic scrolls are projectively normal, so
$ h^1(\iof{X}{2}) = 0.$
Let $i=2.$ From Remark \ref{leray} it is $h^2(\iof{X}{1}) =
h^1(\oof{X}{1}) = h^1(C, E).$ Because $E$ is very ample it is
$\mu^-(E)
>0$ which, by Lemma \ref{buongg} implies $h^1(C, E) = 0.$
For
$i=N$ it is
$h^N( \oofp{N}{3-N}) = h^0(\oofp{N}{-4})=0.$
Therefore it follows that
$h^i(\iof{X}{3-i}) = h^{i-1}(\oof{X}{3-i})$ for all $i\ge 3.$
Remark \ref{leray} gives
$h^{i-1}(\oof{X}{3-i}) = 0$ for $3\le i \le n+1$ while clearly
$h^{i-1}(\oof{X}{3-i})=0$ for $i>n + 1$ since $n=\dim X.$
Therefore
$h^i(\iof{X}{3-i}) = 0$ for all $i\ge 3.$
\end{pf}
\begin{lemma}
\label{diseqonscr}
Let $(\Proj{E},\taut{E})$
be a n-dimensional scroll over a curve of genus $g \ge 2.$ If
$\deg (E) >2g-2$ then
$\Delta
\ge 2n+g-3.$
\end{lemma}
\begin{pf}
Because $d=\deg(det E)=\deg(E) > 2g-2$, it is $h^0(det E) = 1 + d - g$
by Riemann
Roch. Combining this with the inequality $h^0(det E) \ge h^0(E) + r-2$
found in
\cite{Io-To}, it follows that $h^0(E) \le d - n + 3-g$ and therefore
$\Delta \ge 2n + g
-3.$
\end{pf}
\begin{rem}
\label{scr10}
Notice that the above Lemma \ref{diseqonscr} rules out the existence of
scrolls of
degree
$10$ over a curve of genus $g=3$ left as an open possibility in
\cite{fa-li10}.
\end{rem}
We can now prove the main theorems of this section. For $\Delta \le 3$
we
establish the conjecture and give the value of $r (X, L)$ for all pairs.
For $\Delta=4,5$
we establish the conjecture and collect in a remark the known values of
$r (X, L).$
\begin{theo}
\label{Delta+2thm}
Let $(X, L)$ be a n-dimensional polarized pair with $X$ smooth, $L$ very
ample and
$\Delta
\le 5.$ Then $(X, L)$ is $\Delta + 2$-regular.
\end{theo}
\begin{pf}
Because of Theorem \ref{lazran} and Remark \ref{ipers}, the blanket
hypothesis
$n\ge 3$ and $codim X \ge 2$ will be in place throughout this proof.
\begin{case}
$\Delta \le 1$
\end{case}
If $\Delta = 0$ then $(X, L)$ is extremal by Theorem \ref{alafujita}.
Assume
$\Delta=1$, because
$g=0$ implies
$\Delta=0,$ see \cite{fu} Prop. (3.4), it is $g \ge1.$ Because $(X, L)$
is not a
hypersurface it is
$d\ge 3$ and again Theorem
\ref{alafujita} gives $(X, L)$ extremal.
\begin{case}
$\Delta=2$
\end{case}
If $g\le 1$ then
$(X, L)$ must be a two dimensional elliptic scroll, see \cite{fu} Theorem
(10.2).
Proposition
\ref{elscr} gives
$r=3.$ Let $g\ge 2.$ Because $X$ is not a hypersurface it is
$h^0(L) \ge n+3.$ This implies $\Delta\le d-3$ i.e. $d\ge 5$ and then
Theorem
\ref{alafujita} gives $r=3.$
\begin{case}
$\Delta=3$
\end{case}
From
\cite{BEL2} it follows that complete intersections of type
$(2,3)$ have $r=4.$ Following \cite{Io1}
Theorem 4.8 and section 7, it follows from Theorem \ref{alafujita} and
Proposition
\ref{elscr} that the only varieties left to investigate are Bordiga
threefolds scrolls in $\Pin{5}.$ They have the following resolution
with
$N=5.$
\begin{equation}
\label{bordigares}
0 \to \oofp{N}{-4}^{\oplus 3} \to \oofp{N}{-3}^{\oplus 4} \to \iofo{X}
\to 0.
\end{equation}
Equalities $h^{N-1}(\iof{X}{3-N}) =
h^0({\cal O}_{\Pin{N}})=1$ and
$h^i(\iof{X}{3-i})=0$ for all $i \ge 1$ are straightforward to see,
therefore $r=3.$
\begin{case}
$\Delta=4$
\end{case}
Varieties with
$\Delta=4$ are classified. Let us follow the list of varieties
given in \cite{Io4} Theorem 3. Threefolds in $\Pin{5}$ with $d=7$,
$g=5$ or
$g=6$ have respective resolutions as in \brref{D4g5res} and
\brref{D4g6res}
with $N=5.$
\begin{equation}
\label{D4g5res}
0 \to \oofp{N}{-5} \oplus \oofp{N}{-4} \to \oofp{N}{-3}^{\oplus 3} \to
\iofo{X}
\to 0.
\end{equation}
\begin{equation}
\label{D4g6res}
0 \to \oofp{N}{-5}^{ \oplus 2} \to \oofp{N}{-2} \oplus
\oofp{N}{-4}^{\oplus 2} \to
\iofo{X}
\to 0.
\end{equation}
The resolutions
\brref{D4g5res} and \brref{D4g6res} quickly show that
$r=4.$
Complete intersections of type
$(2,2,2)$ have
$r=4$ by
\cite{BEL2}.
Scrolls over a genus $2$ curve must be two-dimensional while elliptic
scrolls
are taken care of by Proposition
\ref{elscr}.
Let now $q=0$ and $g=4.$ If $d\ge 9$ Theorem \ref{alafujita} gives
$r=3.$ On the
other hand since $\Delta=4$ and the codimension must be at least two, it
follows that
$d\ge 7.$ Let us now compare the varieties under consideration with the
lists of
manifolds of degree $7$ and $8$ given in \cite{Io1} and \cite{Io2}.
If $d=8$ then
$X\subset
\Pin{6}$ is a threefold scroll over the quadric surface. Since
$q=0$ the ladder is regular. Consider the curve section
$(C,\restrict{L}{C}).$
Such a $(C, \restrict{L}{C})$ is known to be non hyperelliptic (see
\cite{Io2}) and thus Theorem \ref{glcliff} gives $(C,
\restrict{L}{C})$ p.n. Since
$d =8 > 2g-2 = 6$ it is $h^1(\restrict{L}{C})=0$ and thus $r(C,
\restrict{L}{C})=3$ by lemma \ref{rofpncurves}. Because the ladder is
regular
$r (X, L)=3$ by Remark
\ref{liftingpn}.
If $d=7$ then $X \subset \Pin{5}$ is Palatini's scroll
over the cubic surface. A resolution for $\iofo{X}$ is found in
\cite{BSS3}:
$$0 \to \oofp{5}{-4}^{\oplus 4} \to \Omega^1(-2) \to \iofo{X} \to 0.$$
A simple cohomological calculation gives $r=4.$
\begin{case}
$\Delta=5$
\end{case}
Theorem \ref{alafujita} takes care of cases with $g\ge 5$ and $d\ge 11.$
Manifolds
with degree $d\le 10$ were classified by various authors and we will
examine them
later in the proof. Let us now assume $d \ge 11$ and $g\le 4.$
Because $\Delta=5$ and elliptic scrolls are dealt with in Proposition
\ref{elscr}, it
must be
$g \ge 2.$ Varieties of low sectional genus were classified in
\cite{Io1}. Let us follow the lists given there. If $g=2$ scrolls over a
curve are the only
manifolds to be considered. On the other hand such scrolls of genus $2$
have
$\Delta=2n$ (cf.
\cite{gisa}) so there are no manifolds to examine. If $g=3$ scrolls
over curves are
ruled out by Lemma \ref{diseqonscr} and scroll over $\Pin{2},$ having
$q=0,$ are
ruled out by \cite{Io1} Theorem 4.8 iv). If $g=4$ scrolls over curves
are again ruled
out by Lemma \ref{diseqonscr}. Using standard numerical relations (see
for
example \cite{fa-li9} ( 0.14)) one sees that there are no hyperquadric
fibrations of
dimension $n\ge 3,$ $g=4,$ $\Delta=5$ over
$\Pin{1}$ or over an elliptic curve. Let now $(X, L)$ be a threefold which
is a scroll over
a surface
$(Y, \cal{L})$ with $q(Y)=0,$ $g (X, L)=4.$ Because
$h^1({\cal O}_X)=q(Y)=0,$ recalling that a general hyperplane section of
$X$ is
birational to $Y$ and thus regular, the ladder is regular and then
$\Delta
(X, L) =
\Delta(C,
\restrict{L}{C}) = 4$ by Riemann Roch.
Let us now consider the cases with $d\le 10$ by looking at the
classification
found in \cite{Io2}, \cite{fa-li9}, \cite{fa-li10}. The first non
trivial case occurs with
$d=8.$ $(X, L)$ is a threefold in $\Pin{5},$ admitting a fibration over
$\Pin{1}$ with
generic fibers complete intersections of type $(2,2)$ in $\Pin{4}.$ A
resolution of the
ideal of this variety can be found in \cite{BSS3}. A standard
cohomological calculation
shows that $r (X, L) = 4.$
Let now $d=9.$ From \cite{fa-li9} all varieties to be considered are
threefolds in
$\Pin{6}$ with
$g=5,6,7 \ge \Delta$ and $d=9\ge 2\Delta -1.$
Thus the ladder is regular, see \cite{fu} Theorem 3.5. Let $(S,\restrict{L}{S})$
be the surface section and let $(C,\restrict{L}{C})$ be the curve section. The
projective
normality of linearly normal surfaces of degree nine was studied in
\cite{gisa}. Comparing the list given there with \cite{fa-li9} and
using Lemma
\ref{blowupF1} $(S,\restrict{L}{S})$ is seen to be projectively normal. Remark
\ref{liftingpn}
then gives $r (X, L)= r (S,\restrict{L}{S})
= r (C,\restrict{L}{C}).$
Let now $g=5.$ Then $h^1(tL_C)=0$ for all $t\ge 1$ and from the
structural
sequence of $C$ in $\Pin{4}$ it is easy to see that $ r (C,\restrict{L}{C}) = 3$ if and
only if
$(C,\restrict{L}{C})$ is $2$-normal. On the other hand \cite{fa-li9} shows that in this
case
$h^1({\cal O}_S)=0$ and since $h^1(L_C)=0$ it must be $h^1(L_S)=
0$ and thus $ h^2(\iof{S}{1}) = 0.$ Now the $2$-normality of
$(S,\restrict{L}{S})$ implies the
$2$-normality of $C$ as can be seen from $0 \to \iof{S}{1} \to
\iof{S}{2} \to
\iof{C}{2} \to 0$ and therefore $r (X, L) = 3.$
Let now $g=6.$ First notice that since $h^0(L_C)=5$ it is $h^1(L_C)=1$
and thus
$0 \to \iof{C}{1} \to \oofp{4}{1} \to L_C \to 0$ shows that
$h^2(\iof{C}{1})=h^1(L_C) = 1$ i.e. $(C,\restrict{L}{C})$ cannot be $3$-regular.
Consider the
sequence
\begin{equation}
\label{powersofLS}
0 \to tL_S \to (t+1)\restrict{L}{S} \to (t+1) \restrict{L}{C} \to 0
\end{equation}
for all $t \ge 1.$ Because $\deg{(t+1)\restrict{L}{C}} = 9(t+1) > 2g-2$
it is
$h^1((t+1)\restrict{L}{C}) =0$ for all $t \ge 1.$ Therefore the above
sequence gives
$h^2(t\restrict{L}{S}) =h^2((t+1)\restrict{L}{S})=0$ for all
$t\ge 1$ and thus $h^2(t\restrict{L}{S}) =0$ for all $t \ge 1$ by
Serre's Theorem.
From
\cite{fa-li9} we know that
$q(S)=0$ and
$p_g(S) = 1.$ Thus the sequence $0 \to {\cal O}_S \to \restrict{L}{S}
\to\restrict{L}{C}\to 0$ gives
$h^1(\restrict{L}{S}) = h^2(\restrict{L}{S}) = 0.$ Then the sequence
\brref{powersofLS} for
$t=1$ gives
$h^1(2\restrict{L}{S}) =0.$ The sequence $0 \to \iof{S}{2} \to
\oofp{5}{2} \to 2\restrict{L}{S}
\to 0$ gives
$h^2(\iof{s}{2})= h^1(2L_S) =0.$ Then the sequence $0 \to \iof{S}{2} \to
\iof{S}{3}\to
\iof{C}{3} \to 0$, recalling that $(S,\restrict{L}{S})$ is projectively normal, gives
$h^1(\iof{C}{3})=h^2(\iof{S}{2}) =0$, i.e. $(C,\restrict{L}{C})$ is $3$-normal. The
structure sequence
for $C$ in $\Pin{4}$ then easily shows that $(C,\restrict{L}{C})$ is $4$-regular and
thus $r (X, L) =
4.$
Let now $g=7.$ Noticing that $h^1(\restrict{L}{C})=2$ and recalling from
\cite{fa-li9}
that in this case $q(S) =0$ and $p_g(S) = 2$, the same argument as above
shows that
$(C,\restrict{L}{C})$ is not $3$-regular but it is $4$-regular thus $r (X, L) = 4.$
Let now $d=10.$ From \cite{fa-li10} we see that $h^1({\cal O}_X)=0$ and
therefore the
ladder of these manifolds is regular. Following the list given in
\cite{fa-li10} let $X$ be
a sectional genus $6$, codimension $4$ Mukai manifold of dimension
$3$ or $4.$
The curve section $(C,\restrict{L}{C})$ is then a canonical curve in $\Pin{5}$ and as
such it is
projectively normal. Because $h^1(K_C) =1,$ $h^1(2K_C) = 0$ and the
ladder is regular, it follows from Lemma
\ref{rofpncurves} and Remark
\ref{liftingpn} that
$r (X, L) = r (C,\restrict{L}{C})=4.$
Let now $X$ be any of the remaining threefolds of degree $10$ in
$\Pin{7},$ all of
which have
$g=5,$ according to \cite{fa-li10}. Let
$(C,\restrict{L}{C})$ be a generic curve section. From the classification of manifolds
with
hyperelliptic section (see
\cite{BESO}) it follows that either $X$ is a hyperquadric fibration over
$\Pin{1}$ or
$C$ is not hyperelliptic. In the latter case it is $cl(C)\ge 1$ and
therefore Theorem
\ref{glcliff} gives the projective normality of $C.$
Because $g=5$ it is $h^1(\restrict{L}{C})=0$ and then
Lemma \ref{rofpncurves}, the regularity of the ladder and Remark
\ref {liftingpn} give $r (X, L) = r (C,\restrict{L}{C})= 3.$
Let $(X, L)
\stackrel{\pi}{\to}\Pin{1}$ now be a hyperquadric fibration. Consider
$ W=
\Proj{\oofp{1}{1,1,1,1}}$ and let $\cal{T}={\cal O}_W(1).$ From
\cite{fa-li10} it follows that $X \in|2\cal{T} + \pi^*(\oofp{1}{2})|$
and $L=
\restrict{\cal{T}}{X}.$ The
higher vanishings $h^i(\iof{X}{k-i})=0$ for $i\ge 2$ required for the
$k$-regularity of
$X$ are easily obtained for all $k \ge 3$ from the sequences
\begin{gather}
0\to \iof{X}{k-i}\to \oofp{7}{k-i} \to \oof{X}{k-i} \to 0\notag \\
0\to (k-2-i)\cal{T} +\pi^*(\oofp{1}{-2}) \to (k-i)\cal{T} \to
\oof{X}{k-i} \to 0\notag
\end{gather}
recalling Remark \ref{leray}.
Notice that $|\cal{T}|$
embeds
$W$ in $\Pin{7}$ and the embedding is projectively normal, i.e.
$H^0(\oofp{7}{k})
\to H^0(W, \oof{W}{k})$ is onto for all $k\ge 1.$
Therefore $X$ is $k$-normal
in the embedding given by $\restrict{\cal{T}}{X},$ for some $k,$ if and
only if
$H^0(W,\oof{W}{k}) \to H^0(X,\oof{X}{k})$ is surjective and this
happens if
and only if $H^1(W, (k-2)\cal{T} +\pi^*(\oofp{1}{-2})=0.$
It is $H^1(W, (k-2)\cal{T}
+\pi^*(\oofp{1}{-2})) = H^1(\Pin{1},
\oofp{1}{-2} \otimes S^{k-2}\oofp{1}{1,1,1,1}).$ \\ Combining Lemma
\ref{buongg}
and the fact that
$\mu^-(\oofp{1}{-2}
\otimes S^{k-2}\oofp{1}{1,1,1,1}) = k-4$ it is $H^1(\Pin{1},
\oofp{1}{-2} \otimes S^{k-2}\oofp{1}{1,1,1,1})=0$ for all $k \ge 3.$ On
the other
hand $H^1(\Pin{1},\oofp{1}{-2}) =1$ so $r (X, L) =3.$
\end{pf}
\begin{cor}
Let $(X, L)$ be a $n$-dimensional polarized pair with $X$ smooth, $L$
very ample and
$\Delta
\le 3.$ Then
\begin{itemize}
\item[i)] $(X, L)$ is extremal if and only if it is either a hypersurface
or $\Delta=0,1.$
\item[ii)] If $\Delta=2$ then $r (X, L) = 3.$
\item[iii)] If $\Delta=3$ then $r (X, L) = 3$ unless $(X, L)$ is a complete
intersection of
type $(2,3)$ or a curve of genus $3$ embedded in $\Pin{3}$ as a curve of
type
$(2,4)$ on a smooth quadric hypersurface. In both these cases $r (X, L) =
4. $
\end{itemize}
\end{cor}
\begin{pf}
From the proof of Theorem \ref{Delta+2thm} there are only curves and
surfaces
with $\Delta=3$ to consider.
If $X$ is a curve, since $c\ge 2$, it must be $g\ge 3$ and $d
\ge 6$. If
$d\ge 7$ Theorem \ref{alafujita} gives $r=3.$ If $d=6$ then $X \subset
\Pin{3}$ and
\cite{Io1} section 7 gives three possible types for $X.$ $X$ is linked
to a twisted
cubic by two cubic hypersurfaces, $X$ is of type $(2,4)$ on a smooth
quadric or $X$
is a complete intersection of type $(2,3).$ In the first case
$\iofo{X}$ has a resolution
as in
\brref{bordigares} for
$N=3$
and therefore $r=3.$
In the second case $X$ is not $2$-normal and therefore $r\ge 4.$ By
Theorem
\ref{lazran} $r \le 5.$ By \cite{GLP} $X$ cannot be extremal, therefore
$r=4.$ From
\cite{BEL2} it follows that complete intersections of type
$(2,3)$ have $r=4.$
Assume $n=2.$ As above complete intersections of type $(2,3)$ have
$r=4.$ Following
\cite{Io1} Theorem 4.8 and section 7, it follows from Theorem
\ref{alafujita} and
Proposition
\ref{elscr} that the only varieties left to investigate are Bordiga
surfaces in
$\Pin{4}.$ They have resolutions as
in \brref{bordigares} with $N=4.$ It is
straightforward to check $r=3.$
\end{pf}
\begin{cor}
Let $(X, L)$ be a $n$-dimensional polarized pair with $n \ge 3,$ $X$
smooth, $L$ very
ample,
$\Delta=4$ and $(X, L)$ not a hypersurface. Then $r (X, L) =3$ unless $(X, L)$
is
a complete intersection of type $(2,2,2)$ or any threefold in $\Pin{5}$
of degree $7$
in which cases
$r(X, L) =4.$
\end{cor}
\begin{pf}
Immediate from the proof of Theorem \ref{Delta+2thm}.
\end{pf}
\begin{cor}
Let $(X, L)$ be a $n$-dimensional polarized pair with $n \ge 3,$ $X$
smooth, $L$ very
ample,
$\Delta=5$ and $(X, L)$ not a hypersurface. Then $r (X, L) =3$ unless $(X, L)$
is
in the following list, in
which cases
$r(X, L) =4.$
\begin{itemize}
\item[i)] $(X, L) \subset \Pin{5}$ is a threefold of degree $8$ fibered
over $\Pin{1}$ with
generic fibres complete intersections of type $(2,2)$ (see
\cite{BSS3}).
\item[ii)] $(X, L) \subset \Pin{6}$ is a threefold of degree $9$, $g=6$,
obtained by
blowing up a point on a Fano manifold in $\Pin{7}$, (see \cite{fa-li9}).
\item[iii)] $(X, L) \subset \Pin{6}$ is a threefold of degree $9$, $g=7$,
obtained by
a cubic section of a cone over the Segre embedding of $\Pin{1} \times
\Pin{2}
\subset \Pin{5}$, (see
\cite{fa-li9}).
\item[iv)] $(X, L) \subset \Pin{n+4}$ is a Mukai manifold of degree $10,$
$n=3,4$, $g=6$,
(see \cite{fa-li10}).
\end{itemize}
\end{cor}
\begin{pf}
Immediate from the proof of Theorem \ref{Delta+2thm}.
\end{pf}
\section{Scrolls over curves of genus two}
\label{g2scrolls}
Let $(X,\cal{T})= (\Proj{E}, \taut{E})$ be an $n$-dimensional scroll
over a curve $C$ of genus
$2.$
From
\cite{gisa} (Lemma 5.2) it follows that $\Delta(X,\cal{T}) = 2n$ and
$h^1(\cal{T})=0.$ These
facts will be used without further mention throughout this section.
The same
conclusions can also be drawn from Lemma \ref{buongg} and the following
lemma:
\begin{lemma}
\label{muscrollg2}
Let $E$ be a rank $r$ very ample vector bundle over a genus $2$ curve.
Then
$\mu^-(E) > 3$ and $h^1(C, S^t(E))=0$ for $t\ge 1.$
\end{lemma}
\begin{pf}
By induction on $r.$ If $r=1$ then $E$ is semistable and very ample,
therefore
$\mu^-(E) = \mu(E) \ge 5.$
Let now $r\ge 2$ and assume $\mu^-(E) >3$ for every very ample vector
bundle of
rank up to
$r-1.$ From \cite{Io-To} it is $c_1(E) \ge 3r+1.$ If $E$ is semistable
then $\mu^-(E) =
\mu(E) =\frac{d}{r}\ge 3 + \frac{1}{r}>3.$
Let now $E$ be non semistable. Then there is a
quotient bundle $E \to Q\to 0$ such that $rk(Q) < rk(E)$ and $\mu(Q) =
\mu^-(E).$
Being a quotient of a very ample bundle on a curve, $Q$ is also very
ample and by
induction $\mu^-(E) = \mu(Q) \ge \mu^-(Q) >3.$
Because $\mu^-(S^t(E))=t \mu^-(E) > 3t \ge 3$ it is $h^1(S^t(E))=0$ from
Lemma
\ref{buongg}.
\end{pf}
\begin{prop}
\label{pnscrg2surf} Let $(X,\cal{T})$ be a surface scroll over a curve of genus
$g=2$ with degree
$T^2=d.$ Then
$(X,\cal{T})$ is projectively normal unless $d=8.$ In this case $X$ is as in
\cite{Io2} (4.2).
\end{prop}
\begin{pf}
The projective normality of such scrolls up to degree $8$ was studied
in
\cite{alibaba} where the non projectively normal surfaces in the
statement can be
found. Let us assume $d \ge 9.$ If $E$ is semistable then $\mu^-(E) =
\mu(E) = d/2 >4$
and therefore $(X,\cal{T})$ is p.n. by Lemma \ref{criteriodelbutler}.
Let now $E$ be non semistable. Then $E$ admits a Harder Narasimhan
filtration of
the form $0\to D\to E$ where $D$ is a line bundle. Let now
$Q$ be the quotient line bundle $0\to D \to E\to Q \to 0.$ From the
definition of
$\mu^-$ it is
$\mu^-(E)=\mu(Q) = \deg Q.$ Because $E$ is very ample so must be $Q.$ A
line
bundle $Q$ on a curve of genus $2$ is very ample if and only if $\deg Q
\ge 5.$ Thus
$\mu^-(E) >4$ and $(X,\cal{T})$ is p.n. by Lemma \ref{criteriodelbutler}.
\end{pf}
The above non projectively normal surface scrolls are such because they
are not
$2$-normal, (see \cite{alibaba}). Indeed the next Proposition and Lemma
\ref{muscrollg2} show that
$2$-normality is equivalent to projective normality for scrolls of genus
$2,$
extending a result found in
\cite{pu-ga}.
\begin{prop}
\label{pn2n}
Let $(X,\cal{T})=(\Proj{E},\taut{E})$ be a scroll over a smooth curve $C$ of
genus $g$ such
that
$\mu^-(E)>2g-2.$
Then $(X,\cal{T})$ is projectively normal if and only if it is $2$-normal.
\end{prop}
\begin{pf}
If $(X,\cal{T})$ is p.n. it is obviously $2$-normal. \\ To see the converse
let $n=dim X=rk E$ and let $\pi:X\to C$ be the natural projection.
Reasoning as in \cite{pu-ga} Lemma 1.4, projective normality of
$(X,\cal{T})$ follows from the surjectivity of the maps
\begin{equation}
\label{ontomaps}
H^0((j-1)\cal{T}) \otimes H^0(\cal{T}) \to
H^0(j\cal{T}) \text{ \ \ \ for all \ }
j \ge 2.
\end{equation}
This in turns follows, according to \cite{mu} Theorem 2, from the
vanishing of
$$H^i(X,(j-1-i)\cal{T})=0 \text{\ \ for all } n\ge i\ge 1 \text{ \
\ and for all } j \ge
2.$$ Because $i\le n$ and $j\ge 2$ it is $j-1-i>-n$ and therefore
Remark \ref{leray} shows that $(X,\cal{T})$ is p.n. if
$H^1(X,(j-2)\cal{T})=H^1(C,
S^{j-2}E)=0$ for all $j\ge 2.$
The hypothesis $\mu^-(E)>2g-2$ implies $\mu^-(S^{j-2}E) =
(j-2)\mu^-(E)>2g-2$
for all $j\ge 3.$ From Lemma \ref{buongg} it follows that
$H^1(X,(j-2)\cal{T})=0$ for
all
$j\ge 3.$ This gives all necessary surjectivity in \brref{ontomaps} but
for $j=2.$
Thus $(X,\cal{T})$ is p.n. if
$H^0(\cal{T}) \otimes H^0(\cal{T}) \to H^0(2\cal{T})$ is onto, i.e. if
$(X,\cal{T})$ is $2$-normal.
\end{pf}
\begin{cor}
A scroll $(X,\cal{T})$ over a curve of genus $2$ is p.n. if and only if it is
$2$ normal.
\end{cor}
\begin{pf}
Immediate from Proposition \ref{pn2n} and Lemma \ref{muscrollg2}
\end{pf}
Results on threefold scrolls are collected in the following proposition.
\begin{prop}
Let $(X,\cal{T})=(\Proj{E}, \taut{E})$ be a threefold scroll over a curve of
genus $2.$ Let $d
\ne 12.$ Then
$(X,\cal{T})$ fails to be projectively normal if and only if one of the
following cases occur
\begin{enumerate}
\item[i)] $d=11.$
\item[ii)] $d\ge 13$, $E$ is not semistable and it admits a quotient $E
\to \cal{E}\to 0$ of
rank two and degree eight.
\end{enumerate}
\end{prop}
\begin{pf}
It is known, see \cite{Io-To} or \cite{Io2} and \cite{fa-li9},
\cite{fa-li10},
that there do not
exist threefold scrolls of genus two and
$d\le 10.$ So assume $d\ge 11.$ Because $h^1(\cal{T})=0$ it is
$h^0(\cal{T}) = d-3.$ A simple
computation shows that
$h^0(X,\oof{X}{2}) = 4d-6 > h^0(\oofp{d-4}{2}) = \frac{(d-2)(d-3)}{2}$
if $d \le 11,$ so
that degree $11$ scrolls cannot be $2$-normal.
Assume $d\ge 13.$ If $E$ is semistable then $\mu^-(E) = \mu(E) = d/3 >
4$ and thus
$(X,\cal{T})$ is p.n. by Lemma \ref{criteriodelbutler}.
Let now $E$ be not semistable. Assume $E$ does not admit a degree $8$
and rank $2$
quotient. All quotients $E \to Q \to 0$ must be very ample and thus it
must be $rank
Q = 1$ and
$\deg Q \ge 5$ or $rank Q =2$ and $\deg Q \ge 9.$ Therefore for all $Q$
it is $\mu(Q)
>4$ and thus $\mu^-(E) >4$ and then $(X,\cal{T})$ is p.n. by Lemma \ref{criteriodelbutler}.
Let now $E$ be not semistable with a quotient $E\to \cal{E} \to 0$ of
degree $8$ and
rank $2.$ Notice that $(\Projcal{E},\tautcal{E})$ is one of the non
$2$-normal
surfaces of degree eight embedded in $\Pin{5}$ studied in
\cite{alibaba}.
Let $D$ be the line bundle of degree $d -8$ such that
\begin{equation}
\label{quotient}
0\to D \to E \to
\cal{E}
\to 0.
\end{equation}
Since $d - 8 > 2$ it is $h^1(D) =0$ and thus $H^0(E) = H^0(\cal{E})
\oplus H^0(D).$
Therefore
\begin{equation}
\label{essedueE}
S^2(H^0(E))=S^2(H^0(\cal{E})) \bigoplus H^0(\cal{E}) \otimes
H^0(D)\bigoplus
S^2(H^0(D))
\end{equation}
Consider the sequence obtained by tensoring \brref{quotient} with $D:$
\begin{equation}
\label{tensorD}
0 \to D \otimes D \to E \otimes D \to \cal{E} \otimes D \to 0.
\end{equation}
Because $\deg(D\otimes D) =2(d-8) > 2$ and $\mu^-(\cal{E} \otimes
D)=\mu^-(\cal{E})
+ \mu^-(D) = d-4 > 2$ it follows that $h^1(D\otimes D) = h^1( \cal{E}
\otimes D)=0$ and
thus $H^0(E \otimes D) = H^0(D\otimes D) \oplus H^0(\cal{E} \otimes D)$
and
$h^1(E \otimes D) = 0.$
Considering now the exact sequence
\begin{equation}
\label{simm}
0 \to D \otimes E \to S^2(E) \to S^2(\cal{E}) \to 0
\end{equation}
it follows that
\begin{equation}
\label{hzero}
H^0(S^2(E)) =
H^0(S^2(\cal{E})) \bigoplus H^0(\cal{E} \otimes D)\bigoplus H^0(D
\otimes D).
\end{equation}
Putting together \brref{hzero} and \brref{essedueE}
it follows that the map $\phi: S^2(H^0(E)) \to H^0(S^2(E))$ decomposes
as
\begin{gather}[S^2(H^0(\cal{E}))
\stackrel{\alpha}{\rightarrow}H^0(S^2(\cal{E}))] \notag \\ \bigoplus
\notag \\
[H^0(\cal{E})
\otimes H^0(D) \stackrel{\beta}{\rightarrow} H^0(\cal{E}\otimes D)]
\bigoplus [S^2(H^0(D)) \stackrel{\gamma}{\rightarrow} H^0(D \otimes D)].
\notag
\end{gather}
It was proven in \cite{alibaba} that $\alpha$ is not surjective,
therefore $\phi$
cannot be surjective, i.e. $(X,\cal{T})$ cannot be $2$-normal.
\end{pf}
\begin{rem}
The existence of degree $11$ and $12$ threefold scrolls over curves of
genus $2$
is an open problem. If a degree $12$ such scroll
$(X,\cal{T})=(\Proj{E},\taut{E})$ exists then it is not difficult to see that
$E$ must be
semistable. If it were not semistable then there would be a
destabilizing subbundle\
$\cal{F}$ either of rank $1$ such that
$ \deg \cal{F} > 4$ or of rank $2$ such that $ \deg
\cal{F} > 8.$
In both cases the resulting quotient $0 \to \cal{F} \to E \to Q \to 0$
could
not be very ample for degree reasons, which is a contradiction.
\end{rem}
\begin{rem} Let $(X,\cal{T})=(\Proj{E},\taut{E})$ again be a $3$ dimensional
scroll over a
curve of genus $2.$ If $(X, L)$ is projectively normal, recalling that
$h^1(E) =0,$
the same argument used in Proposition \ref{elscr} gives $r (X, L) = 3.$
If $(X, L)$ is not p.n. , notice that if
$d\ge 13$ it follows that $h^0(\cal{T}) \ge 10$ and thus $(X, \cal{T})$
is $(\Delta +
2)$-regular, i.e. $8$-regular from \cite{Ran1}.
When $d=11,12$ it is easy to check that $h^i(\iof{X}{8-i}) = 0$ for all
$i\ge 2$ while
we were not able to establish the $7$-normality of these manifolds.
\end{rem}
\section{ $\Pin{r}$ bundles over an elliptic curve }
\label{ellipticpkbundles}
Throughout this section let $E$ be a vector bundle of rank $r$ and
degree $d$ over an
elliptic curve
$C$. Let
$(X,\cal{T}) =
\scroll{E}$ and let $D$ be a divisor on $X$ numerically equivalent to
$aT + bF.$
Assume $D$ is very ample. The projective normality of the embedding
given
by $D$ was studied by Homma
\cite{Ho1},
\cite{Ho2} when $r=2,$ and in a more general setting by Butler
\cite{bu} (See also
\cite{alibaba}). In this section the case of $a=2$ and $r=3$ is
addressed and Butler's results are improved in some cases.
\begin{lemma}
\label{mupiudec}
Let $E = \bigoplus_i E_i$ be a decomposable vector bundle over an
elliptic curve. Then
$$\mu^+(E) = max_i \{{\mu(E_i)}\}.$$
\end{lemma}
\begin{pf}
This is essentially \cite{alibaba2} Lemma 2.8, reinterpreted from the
point of view
of $\mu^+$ instead of $\mu^-.$
\end {pf}
\begin{lemma}
\label{mupiu}
Let $(X,\cal{T})$ be as above and let $M_{s}$ be a divisor on $X$ whose
numerical class is
$M_{s}\equiv T+sF. $ Let
$m=min\{ t
\in \Bbb{Z}| h^0(M_t)> 0.\}$ Then $m=-[\mu^+(E)]$ and there exists a
smooth $S \equiv T+mF.$
\end{lemma}
\begin{pf}
From \cite{bu} it follows that for any vector bundle $\cal{G}$ over an
elliptic
curve $\mu^+(\cal{G}) <0$ implies $h^0(\cal{G}) = 0.$
For simplicity of notation let $m^*=-[\mu^+(E)].$ We need to show that
$m=m^*.$ Let
$\cal{L}_t$ be a line bundle on $C$ with degree $t.$ If $t < m^*,$ then
$t=m^*- x$ for some integer
$x\ge 1.$ Then
$\mu^+(E \otimes\cal{L}_t )= \mu^+(E) + t = \mu^+(E) +m^*- x < 1-x \le
0$
Therefore
$ h^0(M_t) = 0$ if $t< m^*$ and thus
\begin{equation} \label{mandm1} m^*\le m.\end{equation}
Let now $E$ be indecomposable and thus semistable. Because $\mu(E)
=\mu^+(E)$ and
$-m^*
\le
\mu^+(E)$ it is $d+rm^* \ge 0.$ If $d +rm^*>0$ then $
h^0(M_{m^*}) >0.$ If $d + rm^*= 0$ then, as in \cite{At}, a line
bundle
$\cal{L}_{m^*}$ of degree $m^*$ can be found by a suitable twist of
degree zero, such
that
$h^0(E \otimes \cal{L}_{m^*}) =1.$
Let now $E = \bigoplus_i E_i$ be decomposable. Then $h^0(E) =
\bigoplus_ih^0(E_i).$
Let $E_{\hat{i}}$ be one of the components such that $\mu^+(E) =
\mu(E_{\hat{i}}).$
As $E_{\hat{i}}$ is indecomposable it follows from above that there
exists a line
bundle
$\cal{L}_{m^*}$ of degree $m^*$ such that
$h^0(E_{\hat{i}}
\otimes
\cal{L}_{m^*})>0.$
Therefore
$h^0(M_{m^*}) > 0$ and thus $m\le m^*$ which combined with
\brref{mandm1} gives $m=m^*.$
If $S\equiv T+m^*F$ is singular
it must be reducible as
$S'
\cup (m^*-t) F$ where
$S'\equiv T+tF$ with $t<m^*$ which is not possible because of the
minimality of
$m^*=m.$ Therefore there must be a smooth $S\equiv T +m^*F.$
\end{pf}
\begin{lemma}
\label{D2npn}
Let $(X,\cal{T})$ and $D$ be as above with $r=3,$ $a \ge 2,$ and $D$ very
ample. If
\begin{itemize}
\item[i)] there exists an ample smooth surface $S \equiv T+x F$ for some
$x \in
\Bbb Z;$
\item[ii)] $(a-1)\mu^-(E) + b - x>1$
\end{itemize}
then
the embedding of $X$ given by $D$
is projectively normal if and only if it is
$2$-normal.
\end{lemma}
\begin{pf}
If the embedding is p.n. it is obviously $2$-normal. As in Lemma
\ref{pn2n} the
projective normality follows from the surjectivity of the maps
\begin{equation}
\label{ontomaps2}
H^0((j-1)D) \otimes H^0(D) \to
H^0(jD) \text{ \ \ \ for all \ }
j \ge 2.
\end{equation}
Assume $j\ge 4.$
Surjectivity in \brref{ontomaps2} follows, according to \cite{mu}, from
the
vanishing of
$$H^i(X,(j-1-i)D)=0 \text{\ \ for all } 3 \ge i\ge 1 \text { \ \
and for all } j \ge 4.$$
Notice that $R^q\pi_*((j-1-i)D)=R^q\pi_*(a(j-1-i) \cal{T}) \otimes
\cal{M}_{i,j}$ where
$\cal{M}_{i,j}$ is a line bundle on $C$ of degree $(j-1-i)b.$ Notice
also, e.g. \cite{H}, that
$R^q\pi_*(a(j-1-i)\cal{T})=0$ unless
$q=0$ and $j-1-i\ge 0,$ or $q=2$ and $a(j-1-i)\le -3.$ Since $a\ge 2,$
$i\le 3$ and
$j\ge 4,$ the last inequality is never satisfied . For $j \ge 4$
Leray's spectral sequence shows that it is enough to show
$H^1(X,(j-2)D)=H^1(C, S^{a(j-2)}E\otimes \cal{M}_{1,j})=0$ which is
guaranteed by
$D$ being ample. This gives all necessary surjectivity in
\brref{ontomaps2} but
for
$j=2,3.$
If $j=2$ the map $H^0(D) \otimes H^0(D) \to H^0(2D)$ is onto by
assumption , being
the embedding $2$-normal.
Assume now $j=3.$
Let $S \equiv T+x F$ be the smooth element whose existence is given by
assumption $i).$ Ampleness of $D$ gives $a \mu^-(E) + b > 0.$ Combining
this with
condition $ii)$ it follows from Lemma \ref{buongg} that $H^1(tD-S) =0$
for $t=1,2,3.$
In particular, following Homma, the commutative diagram below is
obtained :
\begin{alignat}{5}
0 \to H^0(D-S)&\otimes H^0(2D)& &\to& H^0(D)&\otimes H^0(2D) &&\to&
H^0(S,\restrict{D}{S})
&\otimes H^0(2D)\to 0 \notag \\
&\downarrow \alpha& & & &\downarrow \beta & & & &\downarrow
\gamma\\
0 \longrightarrow H^0(3&D-S)& &\longrightarrow& H^0(3&D)&
&\longrightarrow& H^0(S,&\restrict{3D}{S}) \to 0 \notag
\end{alignat}
The surjectivity of $\beta$ will follow from the surjectivity of
$\alpha$ and $\gamma.$
From \cite{Ho1} and \cite{Ho2} it follows that $\restrict{D}{S}$ is
normally generated.
Because
$H^0(2D)
\to H^0(\restrict{2D}{S})$ is surjective from above, it follows that
$\gamma$ is onto.
Lemma \ref{buongg} and condition $ii)$ give $D-S\equiv (a-1)T + (b-x)F$
generated
by global sections. Using this fact and noticing that $H^1(D+S)=0$ being
$D$ very
ample and
$S$ ample, it is straightforward to check that $H^i(2D - i (D-S)) = 0$
for all $i \ge 1.$
Therefore by
\cite{mu} $\alpha$ is onto.
\end{pf}
\begin{prop}
\label{homma}
Let $(X,\cal{T})$ and $D \equiv 2T + bF$ be as above. If
\begin{itemize}
\item[i)] there exists an ample smooth divisor $Y \equiv T+x F$ for some
$x \in
\Bbb Z;$
\item[ii)] $\mu^-(E) + b - x>1$
\end{itemize}
then $|D|$ gives a $2$-normal embedding of $X$ if $|\restrict {D}{Y}|$
gives a
$2$-normal embedding of $Y$.
\end{prop}
\begin{pf}
The proof proceeds along the same lines of the case $j=3$ in the proof
of Lemma
\ref{D2npn}. Let
$Y \equiv T+x F$ be the smooth element whose existence is given by
assumption
$i).$ Ampleness of $D$ gives $2 \mu^-(E) + b > 0.$ Combining this with
condition $ii)$ it
follows that $H^1(tD-Y) =0$ for $t=1,2.$ In particular the following
commutative diagram is obtained :
\begin{alignat}{5}
0 \to H^0(D-Y)&\otimes H^0(D)& &\to& H^0(D)&\otimes H^0(D) &&\to&
H^0(Y,\restrict{D}{Y})
&\otimes H^0(D)\to 0 \notag \\
&\downarrow \alpha& & & &\downarrow \beta & & & &\downarrow
\gamma\\
0 \longrightarrow H^0(2&D-Y)& &\longrightarrow& H^0(2&D)&
&\longrightarrow& H^0(Y,&\restrict{2D}{Y}) \to 0. \notag
\end{alignat}
The surjectivity of $\beta$ will follow from the surjectivity of
$\alpha$ and $\gamma.$
Because $H^0(D) \to
H^0(\restrict{D}{Y})$ is onto from above and $H^0(\restrict{D}{Y})
\otimes
H^0(\restrict{D}{Y}) \to H^0(\restrict{2D}{Y})$ is onto by assumption
it follows that
$\gamma$ is onto.
Condition $ii)$ is equivalent to $D-S\equiv T + (b-x)F$ being generated by
global
sections (see Lemma \ref{buongg} and \cite{alibaba2} Lemma 2.9). Using
this fact and
noticing that
$H^1(Y)=0$ being
$Y$ ample, it is straightforward to check that $H^i(D - i(D-Y)) = 0$ for
all $i \ge 1.$
Therefore by
\cite{mu} $\alpha$ is onto.
\end{pf}
\begin{cor}
\label{hommacor}
Let $(X,\cal{T})$ and $D$ be as above with $r=3$ and $a=2.$ If
\begin{itemize}
\item[i)]$\mu^-(E) > [\mu^+(E)]$
\item[ii)] $\mu^-(E) + b >1-[\mu^+(E)]$
\end{itemize}
then $|D|$ gives a projectively normal embedding.
\end{cor}
\begin{pf}
Let $x=-[\mu^+(E)]$, notice that condition i) and Theorem
\ref{miyaoteo}
give ampleness of $T+xF$. Now combine Lemma
\ref{mupiu}, Lemma
\ref{D2npn}, Proposition
\ref{homma} and the fact that a very ample line bundle on a $2$
dimensional scroll
over an elliptic curve is always normally generated by \cite{Ho1} and
\cite{Ho2}.
\end{pf}
\begin{rem}
Let $E$ be an indecomposable vector bundle of rank $r=3$ and degree
$d\equiv 1 (3).$
For simplicity let us assume that $E$ has been normalized, so $d=1.$
Since $E$ is
indecomposable it is semistable and $\mu^-(E) = \mu ^+(E) = \mu(E) =
1/3.$ The
hypothesis of Corollary \ref{hommacor} are satisfied for $D\equiv 2T +
F.$
and such a $D$ is very ample from \cite{alibaba2} Theorem 4.5. Therefore
$|D|$ gives
a projectively normal embedding. Notice that Butler's results
\cite{bu}, see Lemma
\ref{criteriodelbutler}, were not able to establish the normal
generation of such a $D.$
\end{rem}
\begin{rem}
It is straightforward to check that for a divisor $D$ as in Corollary
\ref{hommacor} it is always $h^0(D) \ge 10$ and therefore the embedding
given by
$|D|$ satisfies the Eisenbud Goto conjecture .
\end{rem}
|
1998-12-14T16:34:24 | 9710 | alg-geom/9710031 | en | https://arxiv.org/abs/alg-geom/9710031 | [
"alg-geom",
"math.AG",
"math.QA",
"q-alg"
] | alg-geom/9710031 | Gregor Masbaum | Jorgen Ellegaard Andersen and Gregor Masbaum | Involutions on Moduli Spaces and Refinements of the Verlinde Formula | 33 pages, Latex, minor modifications, to appear in Mathematische
Annalen | null | null | MSRI 1997-102 | null | The moduli space $M$ of semi-stable rank 2 bundles with trivial determinant
over a complex curve carries involutions naturally associated to 2-torsion
points on the Jacobian of the curve. For every lift of a 2-torsion point to a
4-torsion point, we define a lift of the involution to the determinant line
bundle $\L$. We obtain an explicit presentation of the group generated by these
lifts in terms of the order 4 Weil pairing. This is related to the triple
intersections of the components of the fixed point sets in $M$, which we also
determine completely using the order 4 Weil pairing. The lifted involutions act
on the spaces of holomorphic sections of powers of $\L$, whose dimensions are
given by the Verlinde formula. We compute the characters of these vector spaces
as representations of the group generated by our lifts, and we obtain an
explicit isomorphism (as group representations) with the
combinatorial-topological TQFT-vector spaces of [BHMV]. As an application, we
describe a `brick decomposition', with explicit dimension formulas, of the
Verlinde vector spaces. We also obtain similar results in the twisted (i.e.,
degree one) case.
| [
{
"version": "v1",
"created": "Tue, 28 Oct 1997 09:39:59 GMT"
},
{
"version": "v2",
"created": "Mon, 14 Dec 1998 15:34:18 GMT"
}
] | 2007-05-23T00:00:00 | [
[
"Andersen",
"Jorgen Ellegaard",
""
],
[
"Masbaum",
"Gregor",
""
]
] | alg-geom | \section*{Introduction and Motivation.} The celebrated Verlinde
formula
\cite{Ve}
gives the
dimension of certain vector spaces of so-called `conformal blocks' appearing
in conformal field theory. In this paper, we will take the point of
view of algebraic geometry and think of the conformal blocks
as
holomorphic sections of powers of the determinant line bundle over
moduli spaces of
semi-stable bundles with fixed determinant over a complete
non-singular complex curve
$\Sigma$. According to Atiyah
\cite{At} and Witten
\cite{Wi}, the spaces of holomorphic sections should also fit into
$2+1$-dimensional
`Topological Quantum Field Theories' (TQFT). This geometric construction has
been studied quite a lot (see
{\em e.g.} \cite{ADW,MS,RSW,Hi,CLM,Th,Th2,Sz,BSz,BL,KNR,Fa}).
In \cite{BHMV}, a combinatorial-topological construction of
TQFT-functors was described, based on a particularly simple
construction of
the
Witten-Reshetikhin-Turaev $3$-manifold invariant \cite{RT} in the
$SU(2)$-case through
the Kauffman bracket. In that paper one also constructed certain
involutions on the TQFT-vector spaces which were then used to
decompose the
vector spaces into direct summands. These involutions are associated
to
simple closed
curves on the underlying smooth surface of $\Sigma$, and they generate a
kind
of Heisenberg group
presented in terms of the $mod $ $4$ intersection form. These ideas
were
developed
further in \cite{BM} to construct
spin-refined TQFT's.
The starting point for the present paper was the idea that the involutions of
\cite{BHMV} should correspond on the algebraic-geometric side to the
involutions on moduli space naturally associated to $2$-torsion
points on the Jacobian of $\Sigma$. More precisely, the $2$-torsion
points should correspond to the $mod $ $2$ homology classes of the
simple
closed
curves. Note that the involutions on the spaces of holomorphic
sections require a choice of lift to the determinant line bundle
${\mathcal{L}}$. It is easy to see that these lifts generate a central
extension of the group of $2$-torsion
points whose alternating form is given
by the order $2$ Weil pairing. It follows that this extension is
indeed
abstractly isomorphic
to the one that appeared in
\cite{BHMV}. The ambiguity in the choice of lift is reflected on the
topological side by the choice of a simple closed curve within its $mod
$ $2$ homology class.
One of our motivations in this paper is to establish a more precise
correspondance
between the two theories. The key idea is to define, for every lift of
a $2$-torsion
point ${\alpha}$ to a
$4$-torsion point $a$, an involution $\rho_a$ on ${\mathcal{L}}$ which covers the
action of ${\alpha}$ on the moduli space $M$. The sign of this lift
$\rho_a$ is fixed by requiring it to act as the
identity over a certain component of the fixed point set of the involution
acting on $M$; this component is simply the one containing the class
of the semi-stable bundle $L_a\oplus L_a^{-1}$ (see section
\ref{prel} for more details). We will see in section
\ref{taugamma} that the action of this
lift
$\rho_a$ on holomorphic sections corresponds precisely to
the involution $\tau_{\gamma}$ which is
associated in the \cite{BHMV}-theory to a simple closed curve ${\gamma}$
whose homology class is Poincar\'e dual to $a$. In this way, we
obtain an
explicit isomorphism
(as group representations) with
the TQFT-vector spaces of \cite{BHMV}. We believe this constitutes a
nice confirmation that there should be a natural correspondence
between
the two theories.
The main work in this paper is, however, algebraic-geometric in
nature and consists of a detailed study of our lifts $\rho_a$ and
their action on the Verlinde vector spaces. Our main results are as
follows (see section \ref{smr} for more
complete statements). We show in theorem \ref{1.1} that our lifts
satisfy $$\rho_a \,\rho_b\,=\,\lambda_4(a,b) \,\rho_{a+b}, $$ where
$\lambda_4$ is the order $4$ Weil
pairing (which is the algebraic-geometric analogue of the {\em mod}
$4$ intersection form). Along the way, we also determine in theorem
\ref{evencasei} the
triple intersections of the components of the fixed point sets of
the
action on the various
moduli spaces. The {\em r\^ole} played by the order $4$ Weil
pairing in this context doesn't seem to have been observed before.
We then compute in theorem \ref{1.2} the trace of the induced
involutions
$\rho_a^{\otimes
k}$ on the spaces of holomorphic sections of the $k$-th
tensor power of ${\mathcal{L}}$. This determines the characters of these vector
spaces as representations of the group generated by our lifts. We
also
obtain similar results in the
twisted ({\em i.e.}, degree one) case, where the situation is somewhat
simpler, as only the order $2$ Weil pairing is needed.
A corollary of our results
is a `brick decomposition' of the
spaces of holomorphic sections, the structure of which depends on
the value of the level $k$ modulo $4$. This is, of course, analogous to
the decomposition in
\cite{BHMV}, but we will derive it, as well as explicit
dimension formulas, directly from the
character of the representation. At low levels, similar
decompositions have appeared previously in the work of Beauville
\cite{Be2} (see also Laszlo \cite{L}) for $k=2$, and of van
Geemen
and Previato \cite{vGP1,vGP2}, Oxbury and Pauly
\cite{OP}, and Pauly \cite{P} for $k=4$ (in our notation). Their
approach is based
on the
relationship
with abelian theta-functions and seems quite different
from ours.
This paper is organized as follows. After giving the basic
definitions in section \ref{prel}, we state our main results in
section \ref{smr}. The relationship with the \cite{BHMV}-theory is
discussed in more detail in section \ref{taugamma}, and the `brick
decompositions' are
described in section
\ref{3}. The
remainder of the paper is devoted to the proofs. (See remark
\ref{logstruct} for the logical structure of the proofs.) The reader
interested only in the algebraic geometry may
skip section \ref{taugamma}, and no familiarity with
\cite{BHMV} is necessary to understand the results and their proofs.
{\small \vskip 8pt\noindent{\bf Acknowledgment.}
We would like to thank Ch. Sorger for discussions on this project.}
\section{Basic definitions and notation.}\label{prel}
Let $\Sigma$ be a complete, non-singular curve over the complex
numbers
of genus
$g\geq 2$. Let $M_d$ be the moduli space of
semi-stable bundles of rank $2$ and degree $d\in {\mathbb Z}$ on $\Sigma$.
There is a natural
algebraic action of the degree zero Picard group
$\mathop{\fam0 Pic}\nolimits_0(\Sigma)$ on $M_d$ gotten by tensoring. In this paper, we will use
the standard
identification of $\mathop{\fam0 Pic}\nolimits_0(\Sigma)$ with the Jacobian
$J(\Sigma)$ and speak of an action of $J(\Sigma)$ on $M_d$.
Fix a point
$p\in \Sigma$ and let $[p]$ be the associated line bundle. We
have the determinant morphism $\det : M_d \rightarrow \mathop{\fam0 Pic}\nolimits_d(\Sigma)$. We
put $M=\det^{-1}({\mathcal{O}}_\Sigma)$ and $M'=\det^{-1}([p])$.
As it is explained in \cite{DN} there is a
natural construction of a determinant line bundle ${\mathcal{L}}_d$ over
$M_d$. We will denote the restriction of
${\mathcal{L}}_0$ to
$M$ by ${\mathcal{L}}$ and the restriction of ${\mathcal{L}}_1$ to
$M'$ by ${\mathcal{L}}'$. These two determinant bundles are generators
respectively of the Picard group of $M$ and of $M'$ (see
\cite{DN}).
Let $k\geq 1$ be an integer called the {\em level}. Put
$$Z_k(\Sigma) = H^0(M,{\mathcal{L}}^k)$$
$$Z'_k(\Sigma) = \left\{\begin{array}{ccl}
H^0(M',{{\mathcal{L}}'}^{k/2}) & \ {\rm if} \ k\equiv 0 \ {\rm mod} \ 2\\
0& \ {\rm if} \ k\equiv 1 \ {\rm mod} \ 2\end{array}\right.$$
It is by now well-known that the dimensions of these vector spaces are
given by the
celebrated Verlinde formulas (to be recalled in section \ref{3}).
Following Thaddeus \cite{Th}, we will refer to $Z'_k(\Sigma)$
as the {\em twisted case}, and to $Z_k(\Sigma)$ as the {\em untwisted
case.}
Let $J^{(r)}$ be the subgroup of order $r$ points on $J(\Sigma)=\mathop{\fam0 Pic}\nolimits_0(\Sigma)$. We denote by
$L_{\alpha}$ (resp. $L_a$) the line bundle on
$\Sigma$ corresponding to ${\alpha}\in
J^{(2)}$ (resp. $a\in J^{(4)}$). Note that the group $J^{(r)}$ is identified
with $H^1(\Sigma;\mu_r)$, where $\mu_r\subset {\mathbb C}$ is the group of
$r$-th roots of unity.
The action of ${\alpha}\in J^{(2)}$ on $M_d$ is given by tensoring with
$L_{\alpha}$. Since $L_{\alpha}^{\otimes 2}\cong {\mathcal{O}}_\Sigma$, this preserves $M$ and
$M'$. By abuse of notation, the automorphisms of $M$ and
$M'$ induced by ${\alpha}\inJ^{(2)}$ will again be denoted by ${\alpha}$.
\vskip 8pt\noindent{\bf The lifts $\rho_a$ and $\rho'_{\alpha}$.} By a lift of ${\alpha}$ to
${\mathcal{L}}$ we mean an invertible bundle map from ${\mathcal{L}}$ to itself covering
${\alpha}$. For a lift to exist it suffices that ${\alpha}^*{\mathcal{L}}\cong {\mathcal{L}}$. This
is the case for every ${\alpha}\in
J^{(2)}$, since the Picard
group of $M$ is isomorphic to ${\mathbb Z}$, and ${\mathcal{L}}$ is ample, so that ${\alpha}$
must act trivially on the Picard group. Therefore the
action of any
${\alpha}\in
J^{(2)}$ can be lifted to ${\mathcal{L}}$, and also to ${\mathcal{L}}'$, for the same
reason. Since
the only algebraic
functions on $M$ and $M'$ are the constant ones, it
follows that we can actually choose involutive lifts of each element of
$J^{(2)}$. Any two involutive lifts of ${\alpha}$ agree up to sign.
To fix the signs of the lifts of ${\alpha}$ to ${\mathcal{L}}$ and ${\mathcal{L}}'$, we use the
fact that the sign can be read off in the fiber over a fixed point of
${\alpha}$. Given $0\neq {\alpha}\in J^{(2)}$, we
use the notation $|X|_\alpha$ for the fixed point variety of the
automorphism induced by
${\alpha}$ on the various moduli spaces $X$.
It is well-known that $|M'|_{\alpha}$ is isomorphic to the {\em Prym variety
$P_{\alpha}$} associated to ${\alpha}$, and that $|M|_{\alpha}$ is isomorphic to the
disjoint union of two
copies of the {\em Kummer variety} $P_{\alpha}/{\langle} \pm 1{\rangle}$. This result is
due to Narasimhan and Ramanan \cite{NR}. In particular, $|M'|_{\alpha}$ is
connected and
non-empty, and $|M|_{\alpha}$ has two
components. For any $a\in J^{(4)}$ such that $2a={\alpha}$, we denote by
$|M|_a^+$
the component of $|M|_{\alpha}$ containing
the S-equivalence class of the semistable bundle $L_a\oplus
L^{-1}_a$. (Note that this is indeed fixed under tensoring with
$L_{\alpha}$, since $L_a\otimes L_{\alpha}\cong L_a^{-1}$.) The other component
of $|M|_\alpha$ is denoted by $|M|_a^-$.
\begin{definition}\label{def11}
For $0\neq {\alpha}\in J^{(2)}$, we define $\rho'_\alpha$ to be the involutive lift
to ${\mathcal{L}}'$ of $\alpha$ acting
on $M'$ such that $\rho'_\alpha$ acts by {\em minus} the identity on
the restriction of ${\mathcal{L}}'$
to $|M'|_\alpha$.
\end{definition}
\begin{definition}\label{def12} For $a\in J^{(4)}$ such that $2a={\alpha}\neq 0$, we define
$\rho_a$ to be the involutive lift to ${\mathcal{L}}$ of $\alpha$ acting
on $M$ such that
$\rho_a$ acts by the identity on the restriction of ${\mathcal{L}}$
to the component $|M|^+_a$ of $|M|_\alpha$ specified by $a$.
\end{definition}
It will be convenient to extend this definition by letting
$\rho'_\alpha$ (resp. $\rho_a$) be the identity if ${\alpha}=0$
(resp. $2a=0$).
\vskip 8pt\noindent{\bf Note.} According to Theorem F in \cite{DN}, we have that
${{\mathcal{L}}'}^{-2} \cong K$,
where $K$
is the canonical bundle of $M'$. The natural action of ${\alpha}$ on $K$
coincides with the one induced by both possible involutive lifts $\pm
\rho'_\alpha$. This is because ${\alpha}$ acts as the identity on the fiber
of $K$
over the fixed point set $|M'|_\alpha$, since $|M'|_\alpha$ has even
codimension (see section \ref{fpv}). Similar comments apply in
the untwisted case.
\vskip 8pt
\noindent{\bf The Weil pairing.} (See {\em e.g.} \cite{Ho}.\footnote{We thank
A. Beauville for pointing out this reference.}) Let ${\mathcal M}(\Sigma)$ be the
field of meromorphic
functions on $\Sigma$. The divisor of $f\in {\mathcal M}(\Sigma)$ is denoted by
$(f)$. As already mentioned, we consider $J^{(r)}$ to
be the $r$-torsion points on the group
$\mathop{\fam0 Pic}\nolimits_0(\Sigma)$ which is naturally identified with
$\mathop{\fam0 Div}\nolimits_0(\Sigma)/\mathop{\fam0 Div}\nolimits^{pr}(\Sigma)$. (Here, $\mathop{\fam0 Div}\nolimits_d(\Sigma)$ is the group of degree
$d$ divisors, and
$\mathop{\fam0 Div}\nolimits^{pr}(\Sigma)$ are the principal divisors.) If $D=\sum n_j x_j$ is a
divisor and $f\in
{\mathcal M}(\Sigma)$ a meromorphic function, we put $f(D)=\prod
f(x_j)^{n_j}$. The Weil pairing
$$\lambda_r :
J^{(r)}\times J^{(r)} \rightarrow \mu_r$$ is defined as follows. Given $a,b\in
J^{(r)}$, represent them by divisors
$D_a,D_b$ with disjoint support, and pick $f,g\in {\mathcal M}(\Sigma)$ such that
$(f)=r D_a$ and $(g)=r D_b$. Then
\begin{equation}
\lambda_r(a,b)=\frac{g(D_a)}{f(D_b)}.\nonumber
\end{equation} The Weil pairing is antisymmetric and
non-degenerate. The fact that it takes values in $\mu_r$ follows from
Weil reciprocity (see \cite{GH}, p. 242).
\section{Statement of the main results.}\label{smr}
Let ${\mathcal G}(J^{(2)},{\mathcal{L}})$ be the group of automorphisms
of the determinant line bundle ${\mathcal{L}}$ covering the action of $J^{(2)}$ on
$M$. Since $\alpha^*({\mathcal{L}})\approx {\mathcal{L}}$ for every $\alpha\in
J^{(2)}$, the group ${\mathcal G}(J^{(2)},{\mathcal{L}})$ is a central extension
\begin{equation} {\mathbb C}^* \ \longrightarrow \ {\mathcal G}(J^{(2)},{\mathcal{L}})\
\longrightarrow \ J^{(2)} \label{tildE}.
\end{equation} The same holds for the group ${\mathcal G}(J^{(2)},{\mathcal{L}}')$ of automorphisms
of ${\mathcal{L}}'$ covering the action of $J^{(2)}$ on
$M'$.
\vskip 8pt\noindent{\bf Notation.} Let ${\mathcal{E}}\subset{\mathcal G}(J^{(2)},{\mathcal{L}})$ be the subgroup generated by the
involutions $\rho_a$ ($a \in J^{(4)}$), and let ${\mathcal{E}}'\subset{\mathcal G}(J^{(2)},{\mathcal{L}}')$ be
the subgroup generated by the involutions $\rho'_\alpha$ (${\alpha}\in J^{(2)}$). (See
definitions \ref{def12} and \ref{def11}.)
\vskip 8pt
Our first main result gives a presentation of the groups ${\mathcal{E}}$ and
${\mathcal{E}}'$, as follows.
\begin{theorem}\label{1.1}
The involutions $\rho_a$ and $\rho'_\alpha$ satisfy the following
relations:
\begin{equation} \rho_a \,\rho_b\,=\, \lambda_4(a,b)\, \rho_{a+b}\label{rhoa}
\end{equation}
\begin{equation} \rho'_\alpha\,
\rho'_\beta\,=\, \lambda_2({\alpha},{\beta})\, \rho'_{\alpha+\beta}\label{rhoprima}
\end{equation}
\end{theorem}
It follows that the group ${\mathcal{E}}$ is a central extension
\begin{equation} \mu_4 \ \longrightarrow \ {\mathcal{E}} \ \longrightarrow \
J^{(2)}.\label{tildE2}
\end{equation}
This extension is non-trivial, since the associated alternating form on $J^{(2)}$ is the order $2$
Weil pairing $\lambda_2$. Indeed, this form is
given by the commutator
$$ c({\alpha},{\beta})=\rho_a \rho_b\rho_a^{-1}
\rho_b^{-1}=(\rho_a \rho_b)^2=\lambda_4(a,b)^2=\lambda_2({\alpha},{\beta}).$$ The group ${\mathcal{E}}'$ is a
trivial extension of $J^{(2)}$, {\em i.e.}, it is isomorphic to $\mu_2
\times J^{(2)}$.
\begin{remarks} {\em (i) The group ${\mathcal G}(J^{(2)},{\mathcal{L}})$ is known as the {\em
Heisenberg group}. The fact that its alternating form is the order $2$
Weil pairing $\lambda_2$ is well-known. For example, it follows already from Beauville's isomorphism of $Z_1(\Sigma)$ with the space of abelian theta-functions
\cite{Be0}. This fact is also very easy to see from our point of view
(see remark \ref{coc}). Note that the alternating form determines the extension
(\ref{tildE}) (but not the extension (\ref{tildE2})) up to
isomorphism.
(ii) If the alternating form is known, one knows {\em a priori} that
our lifts $\rho_a$ satisfy $\rho_a\rho_b=\pm
\rho_{a+b}$ if $\lambda_2({\alpha},{\beta})=1$, and $\rho_a\rho_b=\pm i
\rho_{a+b}$ if
$\lambda_2({\alpha},{\beta})=-1$, where $2a={\alpha}$ and $2b={\beta}$. (This is because
the lifts $\rho_a,\rho_b,\rho_{a+b}$ are involutions.) But the sign
of the prefactors $\pm 1$ and $\pm i$ in these relations is, of
course, not determined by the
alternating form. The contribution of theorem
\ref{1.1} is to show that with our `geometric' choice of lifts in terms of
components of the fixed point sets, the
prefactors $\pm 1$ and $\pm i$ are given by the order $4$ Weil
pairing. Similar remarks apply in the twisted case.
}\end{remarks}
\noindent{\bf Note.} In the literature, the Heisenberg
group ${\mathcal G}(J^{(2)},{\mathcal{L}})$ is often described explicitly in terms of a `theta-structure' (see {\em
e.g.} \cite{Be2}), that is, ${\mathcal G}(J^{(2)},{\mathcal{L}})$ is described as a
certain group structure on the set ${\mathbb C}^*\times ({\mathbb Z}/2)^g\times
({\mathbb Z}/2)^g$. A theta-structure allows one to write the extensions
(\ref{tildE}) and (\ref{tildE2}) as push-outs of an extension of $J^{(2)}$ by
$\mu_2$. But this `reduction' to an extension by $\mu_2$ is not
canonical, as it depends on the choice of theta-structure (which comes down,
essentially, to the choice of a symplectic basis of $J^{(2)}\approx
H^1(\Sigma,{\mathbb Z}/2)$).\footnote{The {\em existence} of such a `reduction'
follows already from the fact that the alternating
form $c({\alpha},{\beta})=\lambda_2({\alpha},{\beta})$ takes values in $\mu_2$. But there are many choices for
this `reduction'.} {}From our point of view, such a choice is
neither necessary nor useful, as it would break the symmetry of our
description of the group
${\mathcal{E}}$, which is defined completely intrinsically in terms of the
involutions
$\rho_a$. Therefore we won't use theta-structures in this paper.
\vskip 8pt
Theorem \ref{1.1} is related to the triple intersections of the components of
the fixed point set on $M$. In fact, a key step in the proof is the
following result which does not seem to have
been observed before.
\begin{theorem}\label{evencasei} Assume that ${\alpha}$ and ${\beta}$ are
distinct non-zero elements of $J^{(2)}$ such
that $\lambda_2({\alpha},{\beta})=1\in\mu_2$. Let $a,b\in J^{(4)}$ such that $2a={\alpha}$ and
$2b={\beta}$. Then
$$ |M|^+_a \cap |M|^+_b \cap |M|^+_{a + b} \neq \emptyset \
\Leftrightarrow\ \lambda_4(a,b)=1\in\mu_4$$
$$ |M|^+_a \cap |M|^+_b \cap |M|^-_{a + b} \neq \emptyset \
\Leftrightarrow\ \lambda_4(a,b)=-1\in\mu_4$$
\end{theorem}
\noindent{\bf Note.} Given theorem \ref{evencasei}, theorem \ref{1.1} in
the case $\lambda_4(a,b)=1$
follows immediately. Indeed, if the
triple intersection $|M|_a^+\cap|M|_b^+\cap |M|_{a+b}^+$ is non-empty,
it follows from the definition of the lifts that $\rho_{a+b}=\rho_a
\rho_b$ (since $\rho_a \rho_b
\rho_{a+b}$ must be a constant, and this constant can be computed in
the fiber of ${\mathcal{L}}$ over a triple intersection point.) The proof in
the general case, however, requires some further arguments. The most
interesting case is when $\lambda_4(a,b)=\pm i$. In this situation,
the fixed point sets $|M|_{\alpha}$, $|M|_{\beta}$, $|M|_{{\alpha}+{\beta}}$ don't
intersect, but there is a
${\mathbb P}^1$
intersecting each one of the six components $|M|_a^+$, $|M|_a^-$,
$|M|_b^+$, $|M|_b^-$, $|M|_{a+b}^+$, $|M|_{a+b}^-$, in a point. See section
\ref{Geoinvest}.
\vskip 8pt
Our next result describes $Z_k(\Sigma)$ (resp. $Z'_k(\Sigma)$) as
representations of the group ${\mathcal{E}}$ (resp. ${\mathcal{E}}'$). (Here, ${\mathcal{E}}$ acts on
$Z_k(\Sigma)$ {\em via} the natural action
of $\rho_a^{\otimes k}$ on ${\mathcal{L}}^{\otimes k}$, and similarly for
${\mathcal{E}}'$.) This is based on the following result obtained by
applying the Lefschetz-Riemann-Roch formula. We assume $\alpha\in
J^{(2)}$ is non-zero, and $2a={\alpha}$.
\begin{theorem}\label{1.2} One has
$$ \mathop{\fam0 Tr}\nolimits(\rho_a^{\otimes k}) =
\frac{1+(-1)^k}{2}\left(\frac{k+2}{2}\right)^{g-1}$$
and (for even $k$)
$$\mathop{\fam0 Tr}\nolimits({\rho'_\alpha}^{\otimes k/2}) = (-1)^{k/2}\left(
\frac{k+2}{2}\right)^{g-1}$$
\end{theorem}
\noindent{\bf Note.} In the twisted case and for
levels divisible by $4$, this result is due to Pantev \cite{Pa}. In
the untwisted case and for
levels divisible by $4$, it is also contained in
Beauville's recent paper
\cite{Be3}. Our computation was done independently of his.
\vskip 8pt
The characters of $Z_k(\Sigma)$ and $Z'_k(\Sigma)$ (as representations of
the groups ${\mathcal{E}}$ and ${\mathcal{E}}'$, respectively) are determined by the
formulas in theorem \ref{1.2} together with the trace of the
identity, given by the Verlinde formulas. Therefore the above result
determines these representations up to isomorphism.
A remarkable consequence of this is the following theorem, which was
actually the main motivation for the present
paper.
\begin{theorem}\label{2.4} The spaces $Z_k(\Sigma)$ and $Z'_k(\Sigma)$
are isomorphic, as representations of the groups ${\mathcal{E}}$ and ${\mathcal{E}}'$,
respectively, to the
TQFT-vector spaces constructed in
\cite{BHMV}.
\end{theorem}
This will be discussed in more detail in section
\ref{taugamma}.
\begin{corollary} \label{2.5} If the level $k$ is even, the spaces $Z_k(\Sigma)$
and $Z'_k(\Sigma)$ are decomposed as direct sums of isotypic components
(called `bricks' in what follows)
for the action of ${\mathcal{E}}$ and ${\mathcal{E}}'$. If $k\equiv 0$ mod $4$, the bricks
are indexed by characters of $J^{(2)}$. If $k\equiv 2$ mod $4$, the bricks
are indexed by $\theta$-characteristics on the curve $\Sigma$ (or,
equivalently, by spin structures on $\Sigma$). If the level is odd,
$Z_k(\Sigma)$ is isomorphic, as representation of the group ${\mathcal{E}}$, to
a direct sum of copies of $Z_1(\Sigma)$ or to a direct sum of copies of the
conjugate representation, $\overline{Z_1(\Sigma)}$, according to
the parity of $(k-1)/2$.
\end{corollary}
\begin{remarks}{\em (i) Note that formula (\ref{rhoa}) implies
\begin{equation} \rho_a^{\otimes k} \,\rho_b^{\otimes k}\,=
\,\lambda_4(a,b)^k\,\rho_{a+b}^{\otimes k}. \label{rhoak}
\end{equation} Hence the action
of ${\mathcal{E}}$ on
$Z_k(\Sigma)$ factors through an action of ${\mathcal{E}}'$ if $k$ is even, and
furthermore through
an action of $J^{(2)}$ if $k\equiv 0$ mod $4$. This explains why the
bricks are indexed by characters of $J^{(2)}$ if $k\equiv 0$ mod
$4$. If $k\equiv 2$ mod $4$, the index set are the characters of
${\mathcal{E}}'$ which do not
factor through $J^{(2)}$; such characters can be identified with
$\theta$-characteristics.
(ii) The dimensions of the bricks are the same for all non-trivial
characters of
$J^{(2)}$ in the case $k\equiv 0$ mod $4$, and depend only on the
parity of the
$\theta$-characteristic in the case $k\equiv 2$ mod $4$. In section
\ref{3}, we will give explicit formulas for their dimensions.}
\end{remarks}
\noindent{\bf Note.} Our brick decomposition can be viewed as a
generalisation of an old result of Beauville's \cite{Be2} (see also
Laszlo \cite{L}) at level $2$. Beauville constructed bases of $Z_2(\Sigma)$ (resp.
$Z'_2(\Sigma)$) whose basis elements are indexed by even (resp. odd)
$\theta$-characteristics. This corresponds from our point of view to
the fact that the bricks in level
$2$ are zero, if the $\theta$-characteristic has the `wrong'
parity, and one-dimensional otherwise. Note, however, that the situation in
level $2$ is very special; in general, the bricks are non-zero for
both parities. We would like to mention also
the work of van Geemen and Previato \cite{vGP1,vGP2}, Oxbury and Pauly
\cite{OP}, and Pauly \cite{P}, whose work contains a description of
the bricks in level $4$.
\begin{remark}\label{logstruct}{\em The proofs of our results are
organized as follows. After a description of the fixed
point varieties in section \ref{fpv}, theorem
\ref{evencasei} concerning their triple intersections is proved in section \ref{triple}. As already
observed, theorem \ref{evencasei} implies part of theorem \ref{1.1}; the remainder of the
proof of \ref{1.1} is given in section \ref{Geoinvest}, using some
results about the Hecke correspondence proved in section \ref{Heckecorr}.
Finally, theorem
\ref{1.2} is proved in section \ref{tracecomp}. Both section
\ref{taugamma} (where theorem \ref{2.4} is explained) and section
\ref{3} (where the brick decompositions in
corollary \ref{2.5} are derived)
require only the statements of theorem \ref{1.1} and theorem
\ref{1.2}, but not their proofs.
}\end{remark}
\section{Relationship with the \cite{BHMV}-theory.}\label{taugamma}
In
\cite{BHMV}, the Kauffman bracket (at a
$2p$-th root of unity called $A$) was used to give a combinatorial-topological
construction of TQFT-functors
$V_p$ on a certain $2+1$-dimensional cobordism category. It is
expected that
these functors for $p=2k+4$
correspond (in some natural sense) to
Witten's ones for $SU(2)$ and level $k$. For example, the
dimensions of the vector space $V_p(\Sigma)\otimes {\mathbb C}$ \footnote{By
definition, $V_p(\Sigma)$ is a module over a
certain abstract cyclotomic ring $k_{p}$. By $V_p(\Sigma)\otimes {\mathbb C}$, we mean the
vector space obtained by extending coefficients from $k_p$ to ${\mathbb C}$;
this depends on a
choice of a $2p$-th root of unity $A$ in ${\mathbb C}$.} associated to a closed
oriented
surface $\Sigma$ is also given by the Verlinde
formula (\cite{BHMV}, cor. 1.16).
The aim of the present section is to explain the following more
precise statement of theorem \ref{2.4}. (As already mentioned in the
introduction, the reader interested only in
the algebraic geometry may proceed directly to section \ref{3}.)
\begin{theorem}\label{2.1} Let $p=2k+4$. For the `right' choice of
$2p$-th root of unity $A$, the vector spaces $V_p(\Sigma)\otimes {\mathbb C}$
and
$Z_k(\Sigma)$ are isomorphic as representations of ${\mathcal{E}}$
(this group is denoted by $\Gamma(\Sigma)$ in \cite{BHMV}). This
isomorphism sends the involution $\rho_a^{\otimes k}$ acting on
$Z_k(\Sigma)$ to the involution $\tau_\gamma$ acting on $V_p(\Sigma)\otimes
{\mathbb C}$, where ${\gamma}$ is a simple closed curve on $\Sigma$ representing the
Poincar\'e dual of $a$. A similar statement holds in the twisted
case.
\end{theorem}
\noindent{\bf Note.}
For
simplicity of exposition, a few technical details related to framing
issues will be
suppressed from the discussion here. The reader is also referred to the
survey article \cite{MV}.
\vskip 8pt
By definition, elements of $V_p(\Sigma)$
are represented by linear combinations of compact oriented
$3$-manifolds
$M$ with boundary
equal to $\Sigma$. (No complex structure on $\Sigma$ is needed here.) The
$3$-manifolds may also contain colored links or, more generally,
colored trivalent graphs. In the following, we
assume $p$ is even and put $p=2k+4$. Then a color is
just an element of $\{0,1,2,\ldots,k\}$.
In the \cite{BHMV}-theory, one has involutions $\tau_\gamma$ of
$V_p(\Sigma)$ associated to unoriented simple closed curves $\gamma$ on
$\Sigma$. They are defined as follows. Consider a
vector represented by some $(M,L)$, where $M$ is a $3$-manifold with
boundary $\Sigma$, and $L$ stands for
some colored link inside $M$. Then the action of $\tau_\gamma$
consists of adding to the link $L$ already present in $M$, an
additional component consisting of the curve
$\gamma$ pushed slightly inside $M$ in a neighborhood of $\Sigma=\partial
M$, where $\gamma$ is colored by $k$ (the last
color). It was shown in
section 7 of \cite{BHMV} that (because of the relations which hold in
$V_p(\Sigma)$) this endomorphism $\tau_\gamma$ is an
involution.
Following \cite{BHMV}, p. 917, the groups generated by these
involutions
acting on
$V_p(\Sigma)$ can be described as follows. Let $\Gamma(\Sigma)$ be the
group with one generator $[a]$ for each $a\in H_1(\Sigma;{\mathbb Z}/4)$ plus one
additional generator $u$, and the following relations: the element $u$
is central, $u^4=1$, $[a]^2=1$ for all $a\in H_1(\Sigma;{\mathbb Z}/4)$, and
\begin{equation}
[a]\,[b]\, = \,
u^{a\cdot b} \,[a+b]\label{uu}
\end{equation}
for all $a,b\in H_1(\Sigma;{\mathbb Z}/4)$. Here, $a\cdot
b\in {\mathbb Z}/4$
denotes the {\em mod} $4$ intersection form determined by the orientation
of $\Sigma$.
If a {\em mod} $2$ class $\alpha\in H_1(\Sigma;{\mathbb Z}/2)$ is given, every lift
of $\alpha$
to a {\em mod} $4$ class $a\in H_1(\Sigma;{\mathbb Z}/4)$ determines an element $[a]\in
\Gamma(\Sigma)$. However, up to multiplication by $u^2$, this element
$[a]$ depends only on
$\alpha$. (Indeed, if $a_1,a_2$ are two such lifts, then $[a_1]=u^{2
a_2\cdot x}[a_2]$
where $a_1-a_2=2x$.) From this it follows
easily that the group
$\Gamma(\Sigma)$ is a central extension of $H_1(\Sigma;{\mathbb Z}/2)$ by
${\mathbb Z}/4$. (It can be viewed as some kind of `reduced' Heisenberg group
associated to twice the intersection form.)
Here is the relationship of the group $\Gamma(\Sigma)$ with the involutions
$\tau_\gamma$. Given $\alpha \in H_1(\Sigma;{\mathbb Z}/2)$, a lift of $\alpha$ to
an element of
$\Gamma(\Sigma)$ can also be specified by an unoriented simple closed
curve $\gamma$ representing $\alpha$, as follows. The curve
$\gamma$ determines an
element $a \in H_1(\Sigma;{\mathbb Z}/4)$ up to sign, and the induced element
$[a]\in \Gamma(\Sigma)$ is well-defined,
since $a\cdot a=0$ and hence $[-a]=[a]$.
It is clear from the geometric description of
the $\tau_\gamma$'s given above that the commutation properties of
these involutions are related to the
intersection properties of the associated curves; for example two
such
involutions
obviously commute if the corresponding simple closed curves on $\Sigma$
don't intersect. The fact that the
$\tau_\gamma$'s satisfy precisely the relations in $\Gamma(\Sigma)$ is
shown in prop. 7.5
of \cite{BHMV}. Moreover, it is shown there that the
central
element $u$ acts on $V_{2k+4}(\Sigma)$ as multiplication by
$(-1)^{k+1}A^{(k+2)^2}$.
Now let us identify ${\mathbb Z}/4=\mu_4$ such that $1\in {\mathbb Z}/4$ corresponds to
$i\in\mu_4$. Also, let us use Poincar\'e duality to identify $
J^{(4)}(\Sigma)=H^1(\Sigma;\mu_4)$ with $H_1(\Sigma;{\mathbb Z}/4)$. By a well-known folk
theorem, there exists a
sign $\varepsilon =\pm 1$ such that the Weil pairing is
related to the intersection form as follows: \footnote{The value of
$\varepsilon$ should be known, but we have been
unable to locate it in the
literature.} $$
\lambda_4(a,b)=(\varepsilon i)^{a\cdot b}.$$ We now choose the
primitive root of unity $A$ of order $2p=4k+8$ such that
$(-1)^{k+1}A^{(k+2)^2}=(\varepsilon i)^k$. For instance, we may choose
$A=-\varepsilon
e^{2i\pi/(4k+8)}$. Comparing (\ref{rhoa}) and
(\ref{uu}), we see that the assignment
$$\tau_\gamma \mapsto \rho_a^{\otimes k}$$ (where the curve $\gamma$
represents the Poincar\'e dual of $a$) defines an isomorphism
from the image of $\Gamma(\Sigma)$ in the general linear
group of $V_{2k+4}(\Sigma)\otimes {\mathbb C}$ to the image of the group ${\mathcal{E}}$ in
the general linear
group of $Z_k(\Sigma)$. (In particular, for $k=1$ we have an isomorphism
$\Gamma(\Sigma)\cong {\mathcal{E}}$.)
Using the
dimension formulas given in section 7 of \cite{BHMV}, one can check
that the traces of the involutions $\tau_\gamma$ acting on
$V_p(\Sigma)$ coincide precisely with the traces computed in our theorem
\ref{1.2}. This verification will be omitted.
Since representations of finite groups are determined by their
characters, this proves theorem \ref{2.1} in the untwisted case.
The twisted
case, where $Z_k(\Sigma)$ is replaced with $Z'_k(\Sigma)$, is similar. Here
we must replace $V_p(\Sigma)$ with $V_p(\Sigma')$, where $\Sigma'$ is $\Sigma$
with one puncture
colored by $k$ (the last color). (We maintain the convention
$p=2k+4$.) Then we have again an isomorphism of representations (the
group corresponding to ${\mathcal{E}}'$ is called $\Gamma'(\Sigma)$ in
\cite{BHMV}). The dimension of
$V_p(\Sigma')$ is given by the `twisted Verlinde formula', see
\cite{BHMV}, Remark 5.11. This space is zero if $k$ is odd, and this
is why we define $Z'_k(\Sigma)$ to be zero for odd $k$. (See also Thaddeus
\cite{Th}.) (N.b., the vector
spaces $V_p$
are also defined for surfaces with colored punctures; they are called
`surfaces with colored structure' in \cite{BHMV}. The theory of the involutions
$\tau_\gamma$ works the same for these, with the {\em caveat} that the
curves $\gamma$ must avoid the punctures. We expect analogues of our
results to hold in the case of general colored punctures, using moduli spaces
of parabolic bundles.)
\vskip 8pt
\noindent{\bf Remark.} Although it has been known for some time that $V_p(\Sigma)$ and
$Z_k(\Sigma)$ have the same dimensions, it seems, to the best of our
knowledge, that a {\em natural} isomorphism between the two
theories is still missing. Of course, the fact that these two spaces
are isomorphic also as representations (of canonically isomorphic groups) gives further evidence that there should be a
natural isomorphism between the two theories.
\section{Brick decomposition and dimension formulas.}\label{3}
In this section, we describe $Z_k(\Sigma)$ and $Z'_k(\Sigma)$ as
representations of the groups ${\mathcal{E}}$ and ${\mathcal{E}}'$, respectively. We also
give explicit dimension formulas. The results of this section are
immediate consequences of the isomorphism of theorem \ref{2.1} and the
computations in \cite{BHMV} and \cite{BM}. In order to make this
paper self-contained, we will, however, derive them directly from
theorems \ref{1.1} and \ref{1.2}.
We first recall the Verlinde formula and its twisted counterpart. Put $d_g(k)=dim\ Z_k(\Sigma_g)$ and $d'_g(k)=dim\
Z'_k(\Sigma_g)$, where the subscript $g$ indicates the genus of the curve $\Sigma_g$. \footnote{These numbers are denoted by $d_g(p)$
and $\widehat d_g(p)$ in \cite{BHMV,MV,BM}, where $p=2k+4$.} Then one has
$$d_g(k)=\left(\frac{k+2}{2}\right)^{g-1}\
\sum_{j=1}^{k+1} \ \left(
\sin \frac{ \pi j}{k+2}\right)^{2-2g}$$
$$d'_g(k)=\left(\frac{k+2}{2}\right)^{g-1}\
\sum_{j=1}^{k+1} \ (-1)^{j+1}\left(
\sin \frac{ \pi j}{ k+2}\right)^{2-2g}$$
\noindent{\bf The case $k \equiv 0$ mod $4$.}
In view of formula
(\ref{rhoak}) in section \ref{smr}, the action of the group ${\mathcal{E}}$
on $Z_k(\Sigma_g)$ factors through an action of
$J^{(2)}(\Sigma_g)=H^1(\Sigma_g;\mu_2)$. We
have a direct sum decomposition
$$Z_k(\Sigma_g)=\bigoplus_{h} Z_k(\Sigma_g;h)$$
where $h$ runs through the characters of $J^{(2)}$, and $Z_k(\Sigma_g;h)$
is the subspace of $Z_k(\Sigma_g)$ where ${\alpha}$ acts as multiplication by
$h({\alpha})\in \mu_2={\pm 1}$, for all $\alpha\in J^{(2)}$. We will refer
to $Z_k(\Sigma_g;h)$ as the {\em brick} associated to $h$.
By theorem \ref{1.2}, the character of the representation $Z_k(\Sigma_g)$
takes the same value on all non-trivial elements of $J^{(2)}$, and is
therefore invariant under the automorphism group of $J^{(2)}$. It follows
that the dimension of the brick $Z_k(\Sigma_g;h)$ is the
same for all non-trivial characters $h$ of the group, since the
automorphism group acts transitively on the set of non-trivial
characters. We denote this dimension by $d_g^{(1)}(k)$,
and we put $d_g^{(0)}(k)=dim \
Z_k(\Sigma_g;0)$, where $0$ denotes the trivial character. (Thus, the
space $Z_k(\Sigma_g;0)$ is the fixed point set of the action of the
group $J^{(2)}$ on $Z_k(\Sigma_g)$.) These numbers can be computed from the following two formulas:
$$d_g(k)=d_g^{(0)}(k) +(2^{2g}-1)\, d_g^{(1)}(k)$$
$$d_g^{(0)}(k)-d_g^{(1)}(k)=tr(\alpha)=
\left(\frac{k+2}{2}\right)^{g-1}, \ \ \alpha\neq 0$$ For example, one
has $$d_g^{(0)}(k)=\frac{1}{2^{2g}} \left(d_g(k) + (2^{2g}-1) \left(\frac{k+2}{2}\right)^{g-1}\right)$$
Similarly, $Z'_k(\Sigma_g)$ is the direct sum of bricks
$Z'_k(\Sigma_g;h)$,
and the dimensions ${d'_g}^{(0)}(k)=dim \ Z'_k(\Sigma_g;0)$ and ${d'_g}^{(1)}(k)=dim \ Z'_k(\Sigma_g;h)$ for $h\neq 0$, can be computed as before (just replace $d_g(k)$ with $d'_g(k)$ and $d_g^{(\varepsilon)}(k)$ with ${d'_g}^{(\varepsilon)}(k)$ in the above).
\vskip 8pt\noindent{\bf Example: The case $k=4$.} The numbers $d_g^{(1)}(4)$ and
${d'_g}^{(1)}(4)$ are equal to $(3^{g-1}+1)/2$ and $(3^{g-1}-1)/2$,
respectively, and the numbers $d_g^{(0)}(4)$ and ${d'_g}^{(0)}(4)$
are obtained by adding $3^{g-1}$. These numbers have appeared in Oxbury and Pauly
\cite{OP} and Pauly \cite{P}.
\vskip 8pt
\noindent{\bf The case $k\equiv 2$ mod $4$.}\nopagebreak
In this case, the action of
${\mathcal{E}}$ factors through an action of the group ${\mathcal{E}}'$ but not through an
action of $J^{(2)}$. Indeed, the involutions $\rho_a^{\otimes 2}$ acting on ${\mathcal{L}}^2$ depend only on
$\alpha$, and satisfy the same relations as the
$\rho'_\alpha$'s (see formulas (\ref{rhoprima}) and
(\ref{rhoak}) in section \ref{smr}, and note that $\lambda_4(a,b)^2=\lambda_2({\alpha},{\beta})$). Therefore one has a direct sum
decomposition $$Z_k(\Sigma_g)=\bigoplus_{q} Z_k(\Sigma_g;q)$$ where $q$ runs
through the characters of the group ${\mathcal{E}}'$ which do not factor through
$J^{(2)}$, {\em i.e.}, such that $q$ takes the value $-1$ on the central
element $-1\in\mu_2\subset {\mathcal{E}}'$. Such characters are in $1$-to-$1$ correspondence with
functions $q\colon J^{(2)}\rightarrow \mu_2$ such that
$$q(\alpha+\beta)\,=\,q(\alpha)\,q(\beta)\, \lambda_2(\alpha,\beta).$$ In
other words, $q$ runs through the set of quadratic forms on
$J^{(2)}\cong H^1(\Sigma_g;{\mathbb Z}/2)$
inducing the Weil pairing, {\em i.e.}, the {\em mod} $2$ intersection form. It
is well-known that such quadratic forms correspond to spin
structures, or equivalently, $\theta$-characteristics, on $\Sigma$. (See
Atiyah \cite{At2}, Johnson \cite{Jo}.)
Let ${{\mathcal G}}\cong Sp(2g;{\mathbb Z}/2)$ be the group of automorphisms of $J^{(2)}$ preserving the
order $2$
Weil pairing $\lambda_2$. The group ${\mathcal G}$ acts on ${\mathcal{E}}'$. As in the case
$k\equiv 0$ mod $4$, it follows
from theorem \ref{1.2} that the character of
the representation $Z_k(\Sigma_g)$ is invariant under the
action of ${\mathcal G}$. It is well-known that the induced action of ${\mathcal G}$ on
quadratic forms has two orbits which are characterized by the {\em Arf
invariant}, {\em i.e.}, the action is transitive on the set of forms $q$ with the
same Arf invariant $\mathop{\fam0 Arf}\nolimits(q)\in {\mathbb Z}/2$. (The Arf invariant of the quadratic form
corresponds to the parity of the $\theta$-characteristic.) This shows
that the
dimension of the brick $Z_k(\Sigma_g;q)$ depends only on $\mathop{\fam0 Arf}\nolimits(q)$. Put $d_g^{(\varepsilon)}(k)=dim \ Z_k(\Sigma_g;q_\varepsilon)$, where $q_\varepsilon$ has Arf invariant $\varepsilon\in {\mathbb Z}/2$. These dimensions can be computed from the following two formulas:
$$d_g(k)=2^{g-1}(2^g+1)\,d_g^{(0)}(k) +2^{g-1}(2^g-1)\, d_g^{(1)}(k)$$
$$2^{g-1}(d_g^{(0)}(k)-d_g^{(1)}(k))=tr(\rho_a^{\otimes k})=
\left(\frac{k+2}{2}\right)^{g-1}, \ \ \alpha\neq 0$$
The first formula follows from the fact that the number of quadratic
forms with zero Arf invariant is equal to $2^{g-1}(2^g+1)$. The second
formula follows from the fact\footnote{A nice way to think about this
is to observe that there is a natural bijection between quadratic
forms $q$ with fixed Arf invariant and such that $q({\alpha})=-1$, and
quadratic forms with arbitrary Arf invariant on the
$2g-2$-dimensional space ${\langle} {\alpha}{\rangle} ^\bot/{\langle} {\alpha}{\rangle}$.} that for $\alpha\neq 0$, one has
$$\sharp \{q\,|\,q(\alpha)=-1,\ \mathop{\fam0 Arf}\nolimits(q)=0\}=2^{2g-2}=\sharp \{q\,|\, q(\alpha)=-1,\ \mathop{\fam0 Arf}\nolimits(q)=1\}$$
Similarly, $Z'_k(\Sigma_g)$ is the direct sum of bricks $Z'_k(\Sigma_g;q)$,
and the dimensions ${d'_g}^{(\varepsilon)}(k)=dim \
Z'_k(\Sigma_g;q_\varepsilon)$ can be computed as before (just replace
$d_g(k)$ with $d'_g(k)$ and $d_g^{(\varepsilon)}(k)$ with
${d'_g}^{(\varepsilon)}(k)$ in the above, but notice that
$tr({\rho'_{\alpha}}^{\otimes k/2})$ is now equal to $-\left((k+2)/2\right)^{g-1}$.)
Here are explicit formulas for the dimensions. They are equivalent to the
formulas on p. 264 of \cite{BM} \footnote{Warning: Note that $k$ does not
denote the level in \cite{BM}; one has $p=8k$ in \cite{BM} while in
the present paper $p=2k+4$.}.
\begin{eqnarray} d_g^{(\varepsilon)}(k)&=&{\frac{1}{2^{2g}}} \left(d_g(k)+\left(\frac{k+2}{2}\right)^{g-1} \left((-1)^{\varepsilon} 2^g-1\right)\right)\nonumber\\
{d'_g}^{(\varepsilon)}(k)&=& {\frac{1}{2^{2g}}}\left(d'_g(k) +\left(\frac{k+2}{2}\right)^{g-1} \left(1-(-1)^{\varepsilon} 2^g\right)\right)\nonumber
\end{eqnarray}
\vskip 8pt\noindent{\bf Example: The case $k=2$.} The numbers $d_g^{(0)}(2)$ and
${d'_g}^{(1)}(2)$ are equal to $1$, and the numbers $d_g^{(1)}(2)$
and ${d'_g}^{(0)}(2)$ are zero. Therefore $d_g(2)=2^{g-1}(2^g+1)$ and
$d'_g(2)=2^{g-1}(2^g-1)$ (see Beauville \cite{Be2}).
\vskip 8pt\noindent{\bf Note.} In \cite{BM}, a ${\mathbb Z}/2$-graded TQFT-functor is
constructed on a cobordism category of surfaces equipped with spin
structures (and other things). Given a connected surface $\Sigma$ with a
spin structure $\sigma$, the even (resp. odd) part of this functor is
isomorphic to $Z_k(\Sigma_g;q_\sigma)$ (resp. $Z'_k(\Sigma_g;q_\sigma)$),
where $q_\sigma$ is the quadratic form corresponding to $\sigma$.
\begin{remark}{\em The action of the symplectic group ${\mathcal G}\cong Sp(2g;{\mathbb Z}/2)$
permuting the bricks has the following geometric interpretation. Recall that all elements of ${\mathcal G}$ can be represented by
diffeomorphisms of $\Sigma$. In the \cite{BHMV}-theory, one has, more or less by
definition, a
natural
action of a
certain extended mapping class group on $V_{2k+4}(\Sigma)$. On the
geometric side, there is also a (projective-linear) action
of the mapping class group of $\Sigma$ on $Z_k(\Sigma)$; this action is constructed using
Hitchin's projectively-flat
connection
\cite{Hi}. It is, of course, expected that $V_{2k+4}(\Sigma)\otimes {\mathbb C}$ and $Z_k(\Sigma)$ are isomorphic as
representations of the extended mapping class group. In any case, it
is easy to see in both theories that the
action of
a diffeomorphism
$f$ takes the brick associated to a character $h$ (resp. a quadratic
form $q$) to the brick associated to $f^*(h)$ (resp.
$f^*(q)$). On the geometric side, the main reason for this is that Hitchin's
connection is (projectively) invariant under the actions of both
the mapping class group and the group ${\mathcal{E}}$.
}\end{remark}
\noindent{\bf The odd-level case.}
\begin{proposition} If $k\equiv 1$ mod $2$,
$Z_k(\Sigma)$ is isomorphic, as representation of the group ${\mathcal{E}}$, to
a direct sum of copies of $Z_1(\Sigma)$, if $k\equiv 1$ mod $4$, and to a direct sum of
copies of the
conjugate representation, $\overline{Z_1(\Sigma)}$, if $k\equiv 3$ mod
$4$.
\end{proposition}
\noindent {\bf Proof.} Note that the character
of the representation $Z_1(\Sigma)$ is the function $\chi\colon {\mathcal{E}}\rightarrow {\mathbb C}$
which is zero on all group elements not in the central subgroup $\mu_4\subset
{\mathcal{E}}$, while the trace of a central element $\lambda\in \mu_4$
is $\chi(\lambda)=2^g\,\lambda$. The character of the conjugate
representation, $\overline{Z_1(\Sigma)}$, is of course the conjugate
character $\overline{\chi}$. Now it follows from theorems \ref{1.1}
and \ref{1.2} and formula (\ref{rhoak}) that
the character of $Z_k(\Sigma)$ is a multiple of $\chi$ if $k\equiv 1$ mod $4$, and a multiple of $\overline{\chi}$, if $k\equiv 3$ mod
$4$. It is however well-known that $\chi$ is an irreducible
character, and this proves the proposition.
\vskip 8pt
\noindent{\bf Note.} This result corresponds, {\em via} the isomorphism
$Z_k(\Sigma)\cong V_{2k+4}(\Sigma) \otimes {\mathbb C}$, to theorems 1.5
and 1.6 of
\cite{BHMV}. It is shown there that (for odd $k$)
$$V_{2k+4}(\Sigma)\cong V_2'(\Sigma)\otimes
V_{k+2}(\Sigma)$$ as representations of the group ${\mathcal{E}}$, where
$V_2'(\Sigma)$ and $V_{k+2}(\Sigma)$ are defined in
\cite{BHMV}. Moreover,
the group ${\mathcal{E}}$ acts trivially on $V_{k+2}(\Sigma)$, and
$V_2'(\Sigma_g)$ is isomorphic to $V_6(\Sigma)$, and hence to $Z_1(\Sigma)$
or to $\overline{Z_1(\Sigma)}$, after a change of
coefficients. It would be interesting to have an algebro-geometric
interpretation of the space $V_{k+2}(\Sigma)$, which in the
\cite{BHMV}-theory can be
interpreted as a
$SO(3)$-TQFT vector space. Here, the name $SO(3)$-TQFT just means that the allowed colors in
this TQFT are even, or in other words, are $SU(2)$-representations
which lift to $SO(3)$.
\section{The fixed point varieties.}\label{fpv}
We need to analyze the action of
$J^{(2)}$ on $M$ and $M'$ in order to obtain information about our
lifts. In this section, we describe the various fixed
point varieties, mainly following Narasimhan and Ramanan
\cite{NR}. We also discuss
when two lifts of ${\alpha}\in J^{(2)}$ to $J^{(4)}$ determine the same component of
the fixed point set $|M|_{\alpha}$.
\vskip 8pt\noindent{\bf Notation.} Throughout this paper, we denote by
$L_{\alpha}$ (resp. $L_a$) the line bundle on
$\Sigma$ corresponding to ${\alpha}\in
J^{(2)}$ (resp. $a\in J^{(4)}$).
\vskip 8pt
Let ${\alpha}\inJ^{(2)}$ be nonzero. Let $\pi_{\alpha} : \Sigma^\alpha \rightarrow \Sigma$ be the 2-sheeted unramified covering of
$\Sigma$ corresponding to
$\alpha$, and let $\phi_{\alpha}$ be the covering transformation of $\pi_{\alpha}$.
Using the line bundle $L_{\alpha}$, we can explicitly construct $\Sigma^{\alpha}$ as
$$\Sigma^{\alpha} = \left\{ \xi \in L_{\alpha} | \xi\otimes\xi = 1\right\}$$
using an isomorphism $L^2_{\alpha}\cong {\mathcal{O}}_\Sigma$. The involution $\phi_{\alpha}$ is then
induced by multiplication by $-1$ on $L_{\alpha}$.
Given a line bundle $L$ over $\Sigma^{\alpha}$, the push-down $E={\pi_{\a*}}(L)$ can be
obtained by descending $L\oplus
\phi_{\alpha}^*(L)$ (which is naturally an equivariant bundle) to $\Sigma$:
\begin{equation}
E=(L\oplus \phi_{\alpha}^*(L)){\big/}\langle \phi_{\alpha}\rangle.\nonumber
\end{equation} Here, the
natural involution of $L\oplus \phi_{\alpha}^*(L)$ covering $\phi_{\alpha}$ is
again denoted by $\phi_{\alpha}$. The
fundamental observation is that $E\otimes L_{\alpha}$ is
isomorphic to $E$. This follows formally from the pull-push formula
${\pi_{\a*}}(L)\otimes L'\cong {\pi_{\a*}}(L\otimes \pi_{\alpha}^*L') $ and the fact
that $L_{\alpha}$ pulls back to the trivial bundle on $\Sigma^{\alpha}$.
\begin{remark} \label{fundobs} {\em We will later need the following
explicit isomorphism from $E={\pi_{\a*}}(L)$ to $E\otimes L_{\alpha}$. Note that $\pi_{\alpha}^*L_{\alpha}$, as an equivariant bundle, is
isomorphic to ${\mathcal{O}}_{\Sigma^{\alpha}}^-$, that is, the trivial line bundle $\Sigma^{\alpha}\times
{\mathbb C}$, but with non-trivial action, given by $(x,z)\mapsto (\phi_{\alpha}(x),
-z)$. Therefore $$E\otimes L_{\alpha}=(L\oplus
\phi_{\alpha}^*(L)){\big/}\langle -\phi_{\alpha}\rangle.$$ It follows that the diagonal automorphism
$1\oplus(-1)$ of $L\oplus \phi_{\alpha}^*(L)$ descends to an isomorphism from $E$
to $E\otimes L_{\alpha}$.}\end{remark}
Let $\mathop{\fam0 Nm}\nolimits_{\alpha}\colon \mathop{\fam0 Pic}\nolimits(\Sigma^\alpha) \rightarrow
\mathop{\fam0 Pic}\nolimits(\Sigma)$ be the classical albanese or norm map
induced by the following map on divisors. If $D= \sum n_j x_j\in \mathop{\fam0 Div}\nolimits(\Sigma^{\alpha})$ then
$\mathop{\fam0 Nm}\nolimits_{\alpha}(D) = \sum n_j \pi_{\alpha}(x_j)\in \mathop{\fam0 Div}\nolimits(\Sigma)$.
The {\em Prym variety} $P_\alpha$ associated to $\alpha$ is by definition the
connected component of $\mathop{\fam0 Nm}\nolimits_{\alpha}^{-1}({\mathcal{O}}_\Sigma)$ containing
${\mathcal{O}}_{\Sigma^\alpha}$. It is a principally polarized abelian variety
of dimension $g-1$. The quotient variety
$P_\alpha/\langle\pm 1\rangle$ is called the {\em Kummer variety.}
We define $\theta_{\alpha} : \mathop{\fam0 Pic}\nolimits(\Sigma^\alpha) \rightarrow
\mathop{\fam0 Pic}\nolimits(\Sigma)$ by $$\theta_{{\alpha}}(L) = \det({\pi_{\a*}}(L)) =\mathop{\fam0 Nm}\nolimits_{\alpha}(L)\otimes
L_\alpha$$
(see \cite{NR} for the second equality). Note that
$\phi_{\alpha}$ acts on
$\mathop{\fam0 Pic}\nolimits(\Sigma^\alpha)$ by sending $L$ to $\phi_{\alpha}^*(L)$.
\begin{proposition}[Narasimhan and Ramanan \cite{NR}]\label{6.1} (i) The map $L\mapsto (\pi_{\alpha})_*(L)$
induces isomorphisms
\begin{equation}\label{thetu}
\theta_{\alpha}^{-1}({\mathcal{O}}_\Sigma)/{\langle}\phi_{\alpha} {\rangle} \,\mapright\sim \,|M|_{\alpha}
\end{equation}
\begin{equation}\label{thett}\theta_{\alpha}^{-1}([p])/{\langle}\phi_{\alpha}{\rangle} \,\mapright\sim \,|M'|_{\alpha}
\end{equation}
(ii) Moreover, $\theta_{\alpha}^{-1}({\mathcal{O}}_\Sigma)/{\langle}\phi_{\alpha}{\rangle}$ is isomorphic to two
copies of $P_\alpha/{\langle}\pm 1{\rangle}$, while $\theta_{\alpha}^{-1}([p])/{\langle}\phi_{\alpha}{\rangle}$ is
isomorphic to $P_{\alpha}$.
\end{proposition}
\begin{remark}{\em In the twisted case,
${\pi_{\a*}}\colon \theta_{\alpha}^{-1}([p]) \rightarrow |M'|_{\alpha}$ is a double covering,
with covering transformation $\phi_{\alpha}$. In the untwisted case, the
same holds on the open subvariety of $\theta_{\alpha}^{-1}({\mathcal{O}}_\Sigma)$
where $\phi_{\alpha}$ acts freely, while the fixed
points of $\phi_{\alpha}$ are sent by the map ${\pi_{\a*}}$ bijectively to the points in $|M|_\alpha$
represented by semi-stable but not stable bundles. (See remark
\ref{check} below.)
}\end{remark}
For later use, we need the following more explicit
description of $\theta_{\alpha}^{-1}({\mathcal{O}}_\Sigma)$ and $\theta_{\alpha}^{-1}([p])$. For $d=0,1$, define $ \Phi^d_{\alpha} : \mathop{\fam0 Pic}\nolimits_d(\Sigma^{\alpha}) \rightarrow \mathop{\fam0 Pic}\nolimits_0(\Sigma^{\alpha})=J(\Sigma^{\alpha})$
by
\begin{equation} \Phi^d_{\alpha}(L) = L\otimes\phi^*_{\alpha} L^{-1}.\nonumber
\end{equation} The following two properties (\ref{imPhi1}) and
(\ref{imPhi2}) of the maps $
\Phi^d_{\alpha}$ are elementary facts from the classical theory of
line bundles on curves, see {\em e.g.} Appendix B in \cite{ACGH}.
First, one has the disjoint union
\begin{equation} \mathop{\fam0 Nm}\nolimits_{\alpha}^{-1}({\mathcal{O}}_\Sigma)=\mathop{\fam0 Im}\nolimits\Phi^0_{\alpha} \cup \mathop{\fam0 Im}\nolimits \Phi^1_{\alpha}
\label{imPhi1}
\end{equation} and the Prym
variety $P_{\alpha}$ is equal to the component $\mathop{\fam0 Im}\nolimits\Phi^0_{\alpha}$.
Second, note that $\mathop{\fam0 Nm}\nolimits_{\alpha}(\pi_{\alpha}^*(L_{\beta}))=L_{\beta}^{\otimes 2}={\mathcal{O}}_\Sigma$ for all
$\beta\in J^{(2)}$. Moreover, one has
\begin{equation}
\pi_{\alpha}^*L_{\beta} \in \mathop{\fam0 Im}\nolimits \Phi^d_{\alpha} \ \Leftrightarrow\
\lambda_2({\alpha},{\beta})=(-1)^d. \label{imPhi2}
\end{equation}
Here, $\lambda_2 : J^{(2)}\times J^{(2)} \rightarrow \mu_2$ is the order $2$ Weil
pairing.
Now pick $a\in J^{(4)}$ such that
$2a={\alpha}$, and ${\beta}\inJ^{(2)}$ such that $\lambda_2({\alpha},{\beta})=-1$. Note that
$a'=a+{\beta}$ is another element of $ J^{(4)}$ such that $2a'={\alpha}$. Also, pick
a point
${p_\a}\in\pi_{\alpha}^{-1}(p)\subset \Sigma^{\alpha}$. Note that
$\theta_{\alpha}^{-1}({\mathcal{O}}_\Sigma)$ and $\theta_{\alpha}^{-1}([p])$ are both isomorphic
to $\mathop{\fam0 Nm}\nolimits_{\alpha}^{-1}({\mathcal{O}}_\Sigma)$. {}From (\ref{imPhi1}) and
(\ref{imPhi2}), we have the following description of
$\theta_{\alpha}^{-1}({\mathcal{O}}_\Sigma)$ and $\theta_{\alpha}^{-1}([p])$ as disjoint
unions:
\begin{eqnarray} \theta_{\alpha}^{-1}({\mathcal{O}}_\Sigma)&=&\pi_{\alpha}^*L_{a}\otimes
\mathop{\fam0 Im}\nolimits\Phi^0_{\alpha} \,\cup\, \pi_{\alpha}^*L_{a'} \otimes \mathop{\fam0 Im}\nolimits \Phi^0_{\alpha} \label{untw1}\\
\theta_{\alpha}^{-1}({\mathcal{O}}_\Sigma)&=&\pi_{\alpha}^*L_{a}\otimes
\mathop{\fam0 Im}\nolimits\Phi^0_{\alpha} \,\cup\, \pi_{\alpha}^*L_{a} \otimes \mathop{\fam0 Im}\nolimits \Phi^1_{\alpha} \label{untw2}\\
\theta_{\alpha}^{-1}([p])&=&\pi_{\alpha}^*L_{a}\otimes[p_{\alpha}]\otimes
\mathop{\fam0 Im}\nolimits\Phi^0_{\alpha} \,\cup\, \pi_{\alpha}^*L_{a} \otimes [\phi_{\alpha}(p_{\alpha})]\otimes\mathop{\fam0 Im}\nolimits
\Phi^0_{\alpha} \label{tw}
\end{eqnarray}
\begin{remark} {\em It follows from (\ref{untw2}) and (\ref{tw}) that the action of
$\phi_{\alpha}$ preserves the two components of $\theta_{\alpha}^{-1}({\mathcal{O}}_\Sigma)$, while it
exchanges the two components of $\theta_{\alpha}^{-1}([p])$. By (\ref{imPhi1}), the action of
$\phi_{\alpha}$ on
$\mathop{\fam0 Pic}\nolimits(\Sigma^\alpha)$ restricts to
multiplication by $-1$ (that is, the map $L\mapsto L^{-1}$) on
$\mathop{\fam0 Nm}\nolimits_{\alpha}^{-1}({\mathcal{O}}_\Sigma)$. This proves \ref{6.1}(ii).
}\end{remark}
We can now describe
when two lifts of ${\alpha}\in J^{(2)}$ to $J^{(4)}$ determine the same component of
the fixed point set $|M|_{\alpha}$. Recall that $|M|_a^+$ was defined to be
the component of $|M|_{\alpha}$ containing the S-equivalence
class of the semi-stable bundle $L_a\oplus L^{-1}_a$.
\begin{proposition}\label{4.6} (i) Let $a_1,a_2 \in J^{(4)}$ such that
$2a_1=2a_2={\alpha}$. Then $$|M|_{a_1}^+
= |M|_{a_2}^+ \ \Leftrightarrow\ \lambda_2(a_1-a_2,{\alpha})=1\in\mu_2.$$
(ii) For ${\beta}\in J^{(2)}$, the action of
${\beta}$ on $|M|_{\alpha}$ interchanges the two components of $|M|_{\alpha}$ if
and only if $\lambda_2({\alpha},{\beta})=-1$.
\end{proposition}
\noindent {\bf Proof.} Note that ${\pi_{\a*}}(\pi_{\alpha}^*
(L_{a_i}))\cong L_{a_i}\oplus L_{a_i}\otimes L_{\alpha}\cong L_{a_i}\oplus L_{a_i}^{-1}$. Therefore $ |M|^+_{a_i}$ is the
component
$\pi_{{\alpha}*}(\pi_{\alpha}^*L_{{a_i}}\otimes \mathop{\fam0 Im}\nolimits\Phi^0_{\alpha})$, and (i) follows
from formula (\ref{untw1}). Now part (ii) follows from (\ref{imPhi2}), since
the action of ${\beta}$ on $|M|_{\alpha}$ lifts to tensoring with $\pi_{\alpha}^*L_{\beta}$ on $ \theta_{\alpha}^{-1}
({\mathcal{O}}_\Sigma)$. This completes the proof.
\vskip 8pt\noindent{\bf Note.} Translating the `multiplicative' notation of the
Weil pairing into the `additive' notation in section
\ref{taugamma}, the condition $\lambda_2(a_1-a_2,{\alpha})=1\in\mu_2$
becomes the condition $((a_1-a_2)/2)\cdot {\alpha}=0\in {\mathbb Z}/2$. Thus,
prop. \ref{4.6}(i) implies that $|M|_{a_1}^+
= |M|_{a_2}^+$ if and only if $[a_1]=[a_2]$, where $[a_i]$ is the lift
of ${\alpha}$ to the group $\Gamma(\Sigma)$ defined in section
\ref{taugamma}.
\begin{remark}\label{check}{\em Let $M^{sing}$ denote the set of points of $M$
represented by semistable, but not stable, bundles. A semi-stable
bundle $E$ with $\mathop{\fam0 Gr}\nolimits(E) \cong L_1\oplus L_2$ represents a point in
$M^{sing}$ if and only if $L_2\cong L_1^{-1}$, and this point lies
in $|M|_{\alpha}$ if and only if $L_1\otimes L_{\alpha}\cong L_2$. This shows
that $M^{sing}\cap |M|_{\alpha}$ is precisely the set of points represented
by bundles of the form $L_a\oplus L_a^{-1}$ with $a\in J^{(4)}$ and
$2a={\alpha}$.
Since $L_a\oplus L_a^{-1}\cong{\pi_{\a*}}(\pi_{\alpha}^*(L_a))$, and $\pi_{\alpha}^*(\{L\in \mathop{\fam0 Pic}\nolimits_0(\Sigma) | L^2 \cong L_{\alpha}\})$ is
precisely the fixed point set of $\phi_{\alpha}$ on
$\theta_{\alpha}^{-1}({\mathcal{O}}_\Sigma)$, we see that $\pi_{{\alpha}*}$ induces a bijection from
that fixed point set to $M^{sing}\cap |M|_{\alpha}$, as asserted above.
}\end{remark}
\section{Intersections and triple intersections.} \label{triple}
\vskip 8pt
In this section, we describe how the various fixed point varieties
intersect. The triple intersections will be used in the proof of theorem \ref{1.1}.
\vskip 8pt\noindent{\bf Note.} The
intersection properties of the fixed point sets in relation to the order $2$
Weil pairing are well-known; they are used for example in van Geemen and
Previato \cite{vGP1}. On the other hand, the
relationship of the triple intersections of their individual {\em components} (in the
untwisted case) with the order $4$ Weil pairing seems to be new.
\vskip 8pt
\begin{proposition}\label{6.5}
For $\alpha,\beta\in J^{(2)}$ non-zero distinct elements, we have that
$$|M|_\alpha\cap |M|_\beta \neq
\emptyset\ \Leftrightarrow\ \lambda_2(\alpha,\beta) = 1\in\mu_2$$
$$|M'|_\alpha\cap |M'|_\beta \neq
\emptyset\ \Leftrightarrow\ \lambda_2(\alpha,\beta) = -1\in\mu_2$$
\end{proposition}
\noindent {\bf Proof.}
The morphism $\pi_{{\alpha}*}\colon \mathop{\fam0 Pic}\nolimits_d(\Sigma^\alpha)\rightarrow |M_d|_\alpha$
is surjective and $J^{(2)}$-equivariant. (Here, ${\beta}\in J^{(2)}$ acts on
$\mathop{\fam0 Pic}\nolimits_d(\Sigma^\alpha)$ by $L\mapsto L\otimes \pi_{\alpha}^*L_{\beta}$.) Hence we get the following
description of the intersection
$$|M_d|_\alpha \cap |M_d|_\beta = {\pi_{\a*}}( \{L\in \mathop{\fam0 Pic}\nolimits_d(\Sigma^\alpha) | L\otimes
\pi_{\alpha}^*L_\beta \cong \phi_{\alpha}^* L\mbox{ or } L\}).$$
Hence we see that
$$|M_d|_\alpha \cap |M_d|_\beta \neq \emptyset$$
if and only if
$$\pi_{\alpha}^*L_{\beta} \in \mathop{\fam0 Im}\nolimits \Phi^d_{\alpha}.$$
{}From (\ref{imPhi2}), this is the case if and only if
$\lambda_2({\alpha},{\beta})=(-1)^d $. Using the action of $J(\Sigma)$ on
$|M_d|_\alpha \cap |M_d|_\beta$,
the results for $|M|_\alpha\cap |M|_\beta$ and $|M'|_\alpha\cap
|M'|_\beta$ follow from this.
\begin{proposition}\label{trans} If $\lambda_2({\alpha},{\beta})=1$, the quotient group $J^{(2)}/{\langle}{\alpha},{\beta}{\rangle}$
acts simply transitively on
$|M|_{\alpha}\cap |M|_{\beta}$. In particular, this
intersection has $2^{2g-2}$
elements. If $\lambda_2({\alpha},{\beta})=-1$, the same holds for $|M'|_{\alpha}\cap |M'|_{\beta}$.
\end{proposition}
\noindent {\bf Proof.} Put $I_{{\alpha},{\beta}}=\{L\in \theta_{\alpha}^{-1}({\mathcal{O}}_\Sigma)\,|\,
\pi_{\alpha}^*(L_{\beta})\cong L\otimes \phi_{\alpha}
^*L^{-1}\}$. Then ${\pi_{\a*}}\colon I_{{\alpha},{\beta}}\rightarrow |M|_{\alpha}\cap |M|_{\beta}$ is a
double covering. Note that $J^{(2)}/{\langle} {\alpha}{\rangle}$ acts simply
transitively
on $I_{{\alpha},{\beta}}$. The
involution $\phi_{\alpha}$ is on $ I_{{\alpha},{\beta}}$ the same as tensoring with
$\pi_{\alpha}^*L_{\beta}$,
in other words, the action of ${\beta}$. This proves the result in the
untwisted case. The twisted case is proved similarly.
\vskip 8pt
\noindent{\bf Note.} In view of remark \ref{check}, this description
shows that $|M|_{\alpha}\cap |M|_{\beta}$ is contained in the stable part of
$M$.
\vskip 8pt
\begin{proposition}\label{count} $|M|_{\alpha}\cap |M|_{\beta}$ is the disjoint union of the sets
$|M|_a^\varepsilon\cap |M|_b^\mu $,
where $\varepsilon=\pm$ and $\mu=\pm$, each of
which sets has
$2^{2g-4}$ elements.
\end{proposition}
\noindent {\bf Proof.} Recall from proposition \ref{4.6}(ii) that the action of $\gamma\in
J^{(2)}$ exchanges the components of $|M|_{\alpha}$ if and only if
$\lambda_2({\alpha},{\gamma})=-1$. Thus, the result follows by exploiting the fact
that for every choice of signs $\varepsilon=\pm 1$ and $\mu=\pm 1$, there exists ${\gamma}$
such that $\lambda_2({\alpha},{\gamma}) = \varepsilon$ and
$\lambda_2({\beta},{\gamma})= \mu$.
\vskip 8pt
We now turn to the triple intersections. The first observation is the
following easy lemma.
\begin{lemma} Let ${\alpha},{\beta},{\gamma}$ be
distinct non-zero elements of $J^{(2)}$ such that the triple
intersection $|M|_\alpha \cap |M|_\beta
\cap |M|_{\gamma}$ is non-empty. Then ${\gamma}={\alpha}+{\beta}$. The same holds for the triple intersections
in the twisted case.
\end{lemma}
\noindent {\bf Proof.} Indeed, it
follows from the description in prop. \ref{trans} that the triple
intersection can only be non-empty if $\pi_{\alpha}^*(L_{\beta})=\pi_{\alpha}^*(L_{\gamma})$
which implies ${\gamma}={\alpha}+{\beta}$.
\vskip 8pt\noindent{\bf Note.} Since the group $J^{(2)}$ is commutative,
we have $$|M|_\alpha \cap |M|_\beta = |M|_\alpha \cap |M|_\beta\cap
|M|_{\alpha + \beta},$$ and similarly
in the twisted case.
\vskip 8pt
In the untwisted case, the fixed point sets have two components each, and
we may ask about the triple intersections of the individual
components. Our answer was already stated in theorem \ref{evencasei}. We will prove it in the following
equivalent form.
\begin{theorem}\label{6.8} Assume that ${\alpha}$ and ${\beta}$ are
distinct non-zero elements of $J^{(2)}$ such
that $\lambda_2({\alpha},{\beta})=1\in\mu_2$. Let $a,b\in J^{(4)}$ such that $2a={\alpha}$ and
$2b={\beta}$. Let $\varepsilon,\mu,\nu=\pm 1$ be three signs. Then
\begin{equation}
|M|^\varepsilon_a \cap |M|^\mu_b \cap |M|^\nu_{a + b} \neq \emptyset \
\Leftrightarrow\ \lambda_4(a,b)=\varepsilon\mu\nu\nonumber
\end{equation}\end{theorem}
\noindent{\bf Note.} It follows that if $|M|^\varepsilon_a \cap |M|^\mu_b
\cap |M|^\nu_{a + b}$ is non-empty, then it is equal to all of
$|M|^\varepsilon_a \cap |M|^\mu_b$. This fact can of course be seen
directly using prop. \ref{count}. The important information in the
theorem is that it tells us when $|M|^\varepsilon_a \cap |M|^\mu_b$
intersects $|M|^+_{a + b}$ and when $|M|^-_{a + b}$.
\vskip 8pt
\noindent {\bf Proof.} To simplify notation, we put ${\gamma}={\alpha}+{\beta}$ and $c=a+b$.
Let $E$ represent a point in $|M|_\alpha \cap |M|_\beta \cap
|M|_{\gamma}$. The description of $|M|_\alpha$ in section \ref{fpv} tells us that
there exists ${\mathcal{L}}_a\in
\pi_{\alpha}^*(L_a)\otimes \mathop{\fam0 Im}\nolimits \Phi_{\alpha}^d \subset \mathop{\fam0 Pic}\nolimits_0(\Sigma^{\alpha})$ such that $E\cong{\pi_{\a*}}{\mathcal{L}}_a$;
moreover $E$ lies in $|M|_a^+$ if and only $d$ is even. (See formula
(\ref{untw2}).) Similarly we
have $E\cong{\pi_{\b*}}{\mathcal{L}}_b\cong{\pi_{\g*}}{\mathcal{L}}_c$, where ${\mathcal{L}}_b\in
\pi_{\beta}^*(L_b)\otimes \mathop{\fam0 Im}\nolimits \Phi_{\beta}^{d'}\subset \mathop{\fam0 Pic}\nolimits_0(\Sigma^{\beta})$ and ${\mathcal{L}}_c\in
\pi_{\gamma}^*(L_c)\otimes \mathop{\fam0 Im}\nolimits \Phi_{\gamma}^{d''}\subset \mathop{\fam0 Pic}\nolimits_0(\Sigma^{\gamma})$. Thus, the theorem is
equivalent to the following lemma.
\begin{lemma}\label{6.9} One has $\lambda_4(a,b)=(-1)^{d+d'+d''}$.
\end{lemma}
The proof of this lemma will occupy the remainder of this section.
There is a curve $\widetilde{\S}$ naturally double covering $\Sigma^{\alpha}$, $\Sigma^{\beta}$
and $\Sigma^{{\gamma}}$:
$$\widetilde{\S} = \left\{ (\xi,\eta)\in L_{\alpha}\oplus L_{\beta}\,|\, \xi^2=1=\eta^2\right\}.$$
The projections onto the two factors induce projections
$\pi^{\alpha} \colon \widetilde{\S} \rightarrow \Sigma^{\alpha}$ and $\pi^{\beta} \colon \widetilde{\S} \rightarrow \Sigma^{\beta}.$
The bilinear map $L_{\alpha}\oplus L_{\beta} \rightarrow L_{\alpha}\otimes L_{\beta}$ induces the projection
$\pi^{{\gamma}} \colon \widetilde{\S} \rightarrow \Sigma^{{\gamma}}.$
$$\begin{array}{lll} &\widetilde{\S}&\\
{}^{\textstyle\pi^{\alpha}}\hskip -5pt\swarrow &\downarrow\pi^{\beta}&\searrow^{\textstyle \pi^{\gamma}}\\
\hskip -10pt\Sigma^{\alpha} & \Sigma^{\beta} & \ \ \ \ \Sigma^{\gamma}\\
{}_{\textstyle\pi_{\alpha}}\hskip -5pt\searrow&\downarrow\pi_{\beta}&\swarrow_{\textstyle \pi_{\gamma}}\\
&\Sigma&
\end{array}$$
The deck-transformations of the
coverings $\pi^{\alpha}, \pi^{\beta}, \pi^{{\gamma}}$ will be denoted respectively by $\phi^{\alpha},
\phi^{\beta},\phi^{\gamma}$. Note that $\phi^{\alpha}$ (resp. $
\phi^{\beta}$) is induced by multiplication by $-1$ in the fibers of
$L_{\beta}$ (resp. $L_{\alpha}$), and that $\phi^{\gamma}=\phi^{\alpha}\circ
\phi^{\beta}$. We denote the projection
$\widetilde{\S}\rightarrow \Sigma$ by $\tilde \pi$, so that $$\tilde \pi=\pi_{\alpha}\circ
\pi^{\alpha} =\pi_{\beta}\circ
\pi^{\beta}=\pi_{{\gamma}}\circ
\pi^{{\gamma}}.$$ Notice also that the involution $\phi_{\alpha}$ of $\Sigma^{\alpha}$ is
covered by both $\phi^{\beta}$ and $\phi^{\gamma}$ (but of course not by
$\phi^{\alpha}$, since $\Sigma^{\alpha}=\widetilde{\S}/{\langle}\phi^{\alpha}{\rangle}$). Similar comments apply
to $\phi_{\beta}$ and $\phi_{\gamma}$.
\begin{lemma}\label{6.10} One has $\pi^{{\alpha}*}({\mathcal{L}}_a)\cong \pi^{{\beta}*}({\mathcal{L}}_b)\cong\pi^{{\gamma}*}({\mathcal{L}}_c)$.
\end{lemma}
\noindent {\bf Proof.} Since $E\cong \pi_{{\alpha}*}({\mathcal{L}}_a)$ lies in $|M|_{\alpha}\cap|M|_{\beta}$, we
have $\phi_{\alpha}^*({\mathcal{L}}_a)\cong {\mathcal{L}}_a\otimes \pi_{\alpha}^*(L_{\beta})$ (see the proof
of prop. \ref{trans}). Since $\pi^{{\alpha}*}\pi_{\alpha}^*(L_{\beta})=\tilde\pi^*(L_{\beta})$
is trivial, it follows that $$\tilde
\pi^*(E)\cong\tilde
\pi^*{\pi_{\a*}}({\mathcal{L}}_a)\cong\pi^{{\alpha}*}({\mathcal{L}}_a\oplus \phi_{\alpha}^*({\mathcal{L}}_a))\cong\pi^{{\alpha}*}({\mathcal{L}}_a)\oplus\pi^{{\alpha}*}({\mathcal{L}}_a).$$ Similarly $\tilde
\pi^*(E)\cong
\pi^{{\beta}*}({\mathcal{L}}_b)\oplus\pi^{{\beta}*}({\mathcal{L}}_b)\cong
\pi^{{\gamma}*}({\mathcal{L}}_c)\oplus\pi^{{\gamma}*}({\mathcal{L}}_c)$. Since line bundles are simple,
the lemma follows.
\vskip 8pt
We now turn to the computation of the Weil pairing
$\lambda_4(a,b)$. Represent $a,b\in J^{(4)}$ by divisors
$D_a,D_b\in \mathop{\fam0 Div}\nolimits_0(\Sigma)$ with disjoint support, and put $D_c=D_a+D_b$. Pick $D\in Div_d(\Sigma^{\alpha})$ (resp. $D'\in
Div_{d'}(\Sigma^{\beta})$, resp. $D''\in Div_{d''}(\Sigma^{\gamma})$) such that
$\pi_{\alpha}^*(D_a) +(1-\phi_{\alpha}^*)(D)$ (resp. $\pi_{\beta}^*(D_b)
+(1-\phi_{\beta}^*)(D')$, resp. $\pi_{\gamma}^*(D_c)
+(1-\phi_{\gamma}^*)(D'')$) represents ${\mathcal{L}}_a$ (resp. ${\mathcal{L}}_b$, resp. ${\mathcal{L}}_c$).
Pulling everything up to
$\widetilde{\S}$, we get divisors
$$F_a=\tilde\pi^{*}(D_a)+\pi^{{\alpha}*}(1-\phi_{\alpha}^*)(D)$$
$$F_b=\tilde\pi^{*}(D_b)+\pi^{{\beta}*}(1-\phi_{\beta}^*)(D')$$
$$F_c=\tilde\pi^{*}(D_c)+\pi^{{\gamma}*}(1-\phi_{\gamma}^*)(D'')$$
such that $F_a$ represents $\pi^{{\alpha}*}({\mathcal{L}}_a)$, $F_b$ represents
$\pi^{{\beta}*}({\mathcal{L}}_b)$, and $F_c$ represents $\pi^{{\gamma}*}({\mathcal{L}}_c)$. Since these
three bundles are isomorphic by
lemma \ref{6.10}, there exist meromorphic functions $h_1,h_2\in
{\mathcal M}(\widetilde{\S})$ such that $$(h_2)+F_a\,=\,F_c\,=\, (h_1)+F_b.$$
Let $\mathop{\fam0 Nm}\nolimits^{\alpha}\colon {\mathcal M}(\widetilde{\S})\rightarrow {\mathcal M}(\Sigma^{\alpha})$ be the norm map on meromorphic
functions associated to
the covering $\pi^{\alpha}$. The norm maps associated to the various other
coverings will similarly be denoted by
$\mathop{\fam0 Nm}\nolimits^{\beta},\mathop{\fam0 Nm}\nolimits^{\gamma},\mathop{\fam0 Nm}\nolimits_{\alpha},\mathop{\fam0 Nm}\nolimits_{\beta},\mathop{\fam0 Nm}\nolimits_{\gamma},$ and $\widetilde \mathop{\fam0 Nm}\nolimits$.
\begin{lemma}\label{6.12} (i) Define $f,g\in {\mathcal M}(\Sigma)$ by $f=\widetilde \mathop{\fam0 Nm}\nolimits(h_1)$,
$g=\widetilde \mathop{\fam0 Nm}\nolimits(h_2)$. Then $$(f)=4 \,D_a, \ \ (g)=4\, D_b.$$
(ii) Define $f_{\alpha}=\mathop{\fam0 Nm}\nolimits^{\alpha}(h_1)\in {\mathcal M}(\Sigma^{\alpha})$, $f_{\beta}=\mathop{\fam0 Nm}\nolimits^{\beta}(h_2)\in {\mathcal M}(\Sigma^{\beta})$, $f_{\gamma}=\mathop{\fam0 Nm}\nolimits^{\gamma}(h_1/h_2)\in
{\mathcal M}(\Sigma^{\gamma})$. Then $$f_{\alpha}\circ \phi_{\alpha}=-f_{\alpha}, \ f_{\beta}\circ
\phi_{\beta}=-f_{\beta}, \ \ f_{\gamma}\circ \phi_{\gamma}=-f_{\gamma}.$$
\end{lemma}
\noindent {\bf Proof.} Using that $\phi^{\alpha}$ covers $\phi_{\beta}$ and
$\phi_{\gamma}$, one computes that
$$\pi^{{\alpha}*}((f_{\alpha}))=(h_1)+\phi^{{\alpha}*}(h_1)=2\,\tilde\pi^*(D_a).$$ It
follows that $(f_{\alpha})=2\pi_{\alpha}^*(D_a)$ and hence
$(f)=(\mathop{\fam0 Nm}\nolimits_{\alpha}(f_{\alpha}))=4D_a$, as asserted. This also shows that the
divisor $(f_{\alpha})$ is $\phi_{\alpha}$-invariant. Therefore one has $f_{\alpha}\circ
\phi_{\alpha}=\pm f_{\alpha}$. But $f_{\alpha}$ itself cannot be $\phi_{\alpha}$-invariant,
because if it were, it would descend to a function $h\in{\mathcal M}(\Sigma)$ such
that $(h)=2D_a$, which is impossible since $a$ has order
$4$. Therefore $f_{\alpha}\circ \phi_{\alpha}=-f_{\alpha}$. The other assertions of the
lemma are proved similarly.
\vskip 8pt
By lemma \ref{6.12}(i), we can compute the Weil pairing
$\lambda_4(a,b)$ using the functions $f$ and $g$ (see the definition
in section \ref{prel}). Note that by lemma \ref{6.12}(ii), we have that
$$h_1(\pi^{{\alpha}*}(1-\phi_{\alpha}^*)(D))=f_{\alpha}((1-\phi_{\alpha}^*)(D))=\frac{f_{\alpha}(D)}{(f_{\alpha}\circ
\phi_{\alpha} )(D)}=(-1)^{\deg D}.$$
Thus
\begin{eqnarray*}\label{calc}
\lambda_4(a,b)&=&\frac{g(D_a)}{f(D_b)} =\frac{f(-D_b)}{g(-D_a)}=\frac{h_1(-\tilde\pi^*(D_b))}{h_2(-\tilde\pi^*(D_a))} \\
&=&\frac{h_1(-\tilde\pi^*(D_b) +(h_2))}{h_2(-\tilde\pi^*(D_a) +(h_1))} \\
&=&\frac{h_1(\pi^{{\gamma}*}(1-\phi_{\gamma}^*)(D'') - \pi^{{\alpha}*}(1-\phi_{\alpha}^*)(D))}
{h_2(\pi^{{\gamma}*}(1-\phi_{\gamma}^*)(D'') -
\pi^{{\beta}*}(1-\phi_{\beta}^*)(D'))}\\
&=&f_{\gamma}((1-\phi_{\gamma}^*)(D'')) \frac
{f_{\beta}((1-\phi_{\beta}^*)(D'))}{f_{\alpha}((1-\phi_{\alpha}^*)(D))}\\
&=&(-1)^{\deg(D) +\deg (D')+\deg(D'')}=(-1)^{d+d'+d''}
\end{eqnarray*}
where we have used Weil reciprocity in the fourth equality. This proves lemma
\ref{6.9} and hence theorem \ref{6.8}.
\section{The action of $J^{(2)}$ on the Hecke correspondence.}\label{Heckecorr}
We will make use of the Hecke correspondence in our analysis of the
involutions in section \ref{Geoinvest}. This is a pair of morphisms
\begin{center}
\begin{picture}(120,80)
\put(55,60){${\mathcal P}$}
\put(0,10){$M$}
\put(100,10){$M'$}
\put(50,55){\vector(-1,-1){30}}
\put(65,55){\vector(1,-1){30}}
\put(20,45){$q$}
\put(90,45){$q'$}
\end{picture}
\end{center}
which allows one to `transfer' information from $M$ to $M'$. In this section, we describe the fixed point varieties $|{\mathcal P}|_{\alpha}$ of the action
of the various ${\alpha}\inJ^{(2)}$ on ${\mathcal P}$.
\vskip 8pt \noindent{\bf Notation.} Given a bundle $E$ over $\Sigma$, we denote by
$E_x$ the fiber of $E$ at the point $x\in\Sigma$. Also, for a bundle $E$
representing
a point in the
moduli spaces $M$ or
$M'$, we use the notation
$[E]$ for that point.
\vskip 8pt
We briefly review the construction of ${\mathcal P}$. (See {\em e.g.} Bertram and
Szenes \cite{BSz}.) Let
${\mathcal{U}}$ be a Poincar\'{e} bundle over $\Sigma\times M'$. Thus, if
$[E']\in M'$, the
restriction of ${\mathcal{U}}$ to $\Sigma\times \{[E']\}$ is isomorphic to $E'$. We can uniquely
fix ${\mathcal{U}}$ by requiring that $\det({\mathcal{U}}|_{\{p\}\times M'})$ is an ample generator of
$\mathop{\fam0 Pic}\nolimits(M')$.
We put ${\mathcal P} = {\mathbb P}({\mathcal{U}}|_{\{p\}\times M'})$ and let $ q' : {\mathcal P} \rightarrow M'$ be
the projection. Note that $q'$ is a ${\mathbb P}^1$-fibration and for $[E']\in
M'$, the fiber $(q')^{-1}([E'])$ is isomorphic to the projective space
${\mathbb P}(E'_p)$. In fact, ${\mathcal P}$ can be viewed as the moduli space of pairs
$(E', {\mathcal F})$ where $E'$ is a stable rank $2$ bundle with $\det(E')=[p]$,
and ${\mathcal F}\subset E'_p$ is a one-dimensional subspace, {\em i.e.}, ${\mathcal P}$ is a moduli space of
semi-stable parabolic bundles. We will refer to
${\mathcal F}$ as a {\em line} in $E'_p$. Points in ${\mathcal P}$ will be denoted as
$[(E',{\mathcal F})]$, and we have $q'([(E',{\mathcal F})])=[E']$.
The map $q$ is obtained by the operation of elementary modification at $p$. This means that we have $q([(E',{\mathcal F})])=[E]$ if and
only if there is a short exact sequence (of sheaves)
$$0\rightarrow E\rightarrow E'\mapright{\lambda}{\mathbb C}_p\rightarrow 0$$ such that
$\ker_p(\lambda)={\mathcal F}\subset E'_p$. Here, ${\mathbb C}_p$ is the skyscraper sheaf
at $p$.
The group $J^{(2)}$ acts naturally on ${\mathcal P}$. The action of ${\alpha}\in J^{(2)}$ on
${\mathcal P}$ sends $[(E',{\mathcal F})]$ to $[(E'\otimes L_{\alpha},{\mathcal F}\otimes L_{\alpha})]$. The
morphisms $q$ and $q'$ are $J^{(2)}$-equivariant.
Let ${\alpha}\in J^{(2)}$ be non-zero. We now describe the fixed point variety
$|{\mathcal P}|_{\alpha}$. Recall that ${\pi_{\a*}}\colon \theta_{\alpha}^{-1}([p]) \rightarrow
|M'|_{\alpha}$
is a double covering, where
$\theta_{\alpha}^{-1}([p])\subset \mathop{\fam0 Pic}\nolimits_1(\Sigma^{\alpha})$ consists of two `translates'
of
the Prym variety $P_{\alpha}$.
Let $p_\alpha\in\Sigma^\alpha$ be such that $\pi_{\alpha}(p_\alpha) =
p$. If $L\in \mathop{\fam0 Pic}\nolimits_1(\Sigma^\alpha)$, the projection gives a canonical isomorphism
$$(L\oplus \phi_{\alpha}^*(L))_{p_\a} \mapright \sim ({\pi_{\a*}}(L))_p.$$
\begin{proposition} \label{fixPa} We have an isomorphism $$j_{{p_\a}}\colon
\theta_{\alpha}^{-1}([p]) \,\mapright\sim\, |{\mathcal P}|_{\alpha}$$ defined
by $j_{{p_\a}}(L)=[({\pi_{\a*}}(L),(L\oplus 0)_{p_a})]$.
\end{proposition}
\noindent {\bf Proof.} Put $E'={\pi_{\a*}}(L)$. It is clear that $|{\mathcal P}|_{\alpha}\subset (q')^{-1}
(|M'|_{\alpha})$. Therefore the only question is which lines in $E'_p$
correspond to fixed points of ${\alpha}$ acting on ${\mathbb P}(E'_p)\cong
(q')^{-1}([E'])$.
Let $\psi\colon E'\mapright\sim
E'\otimes L_{\alpha}$ be the isomorphism described in remark
\ref{fundobs}. It is covered by the diagonal automorphism
$\tilde\psi=1\oplus(-1)$ of $L\oplus \phi_{\alpha}^*(L)$. Since $E'$ is stable, it is simple,
hence any other isomorphism is a
non-zero multiple of $\psi$. Therefore $(E',{\mathcal F})$ represents a point in
$|{\mathcal P}|_{\alpha}$ if and only if $\psi({\mathcal F})={\mathcal F}\otimes L_{\alpha}$. Letting $\widetilde
{\mathcal F}$ denote the line in $(L\oplus \phi_{\alpha}^*(L))_{p_\a}$ projecting down to
${\mathcal F}\subset E'_p$, this condition is equivalent to $\tilde\psi(\widetilde
{\mathcal F})=\widetilde
{\mathcal F}$. The only lines in $(L\oplus \phi_{\alpha}^*(L))_{p_\a}$ that $\tilde\psi$ preserves are $(L\oplus 0)_{p_\a}$
and $(0\oplus \phi_{\alpha}^*(L))_{p_\a}$. The first line defines the
point $j_{p_\a}(L)$ in ${\mathcal P}$, and the second line defines the point
$j_{p_\a}(\phi_{\alpha}^*(L))$. This shows that $j_{p_\a}$ is bijective. It is clearly an
algebraic morphism, and since its
domain is smooth, this shows $j_{p_\a}$ is an isomorphism.
\begin{remark}\label{qeq} {\em One has $q'\circ j_{p_\a}={\pi_{\a*}}$. In other words,
$j_{p_\a}$ is an isomorphism of coverings over the identity of $|M'|_{\alpha}$. Note also that
$j_{\phi_{\alpha}({p_\a})}=j_{p_\a} \circ \phi_{\alpha}^*$.}\end{remark}
\noindent{\bf Notation.} Given $a\inJ^{(4)}$ such that $2a={\alpha}$, we denote by
$|{\mathcal P}|_a^+$ the component of $|{\mathcal P}|_{\alpha}$ containing the point
$j_{p_\a}(\pi_{\alpha}^*(L_a)\otimes [p_{\alpha}])$. (See formula (\ref{tw}).) Note
that this point, and hence the definition of the component
$|{\mathcal P}|_a^+$,
depends only on $a$, not
on the choice of $p_{\alpha}$.
\begin{proposition}\label{q-+} One has $q(|{\mathcal P}|_a^+)=|M|_a^-$.
\end{proposition}
\noindent {\bf Proof.} Put $L=\pi_{\alpha}^*(L_a)$ and $L'=L\otimes [p_{\alpha}]$. Then
$E={\pi_{\a*}}(L)$ represents a point in $|M|_a^+$, and $E'={\pi_{\a*}}(L')$
represents a point in $|M'|_{\alpha}$. The short exact sequence of sheaves
$$0\rightarrow L\rightarrow L'\rightarrow {\mathbb C}_{p_\a}\rightarrow 0$$ induces the short exact sequence
\begin{equation} 0\rightarrow E\rightarrow E'\,\mapright{\lambda}\,{\mathbb C}_p\rightarrow 0.
\label{ses}
\end{equation} We need to determine the
line ${\mathcal F}=\ker_p (\lambda)\subset E'_p$. Pulling (\ref{ses}) back to
$\Sigma^{\alpha}$ and restricting to the fiber at $p_{\alpha}$, the map $\lambda$
becomes $$(L'\oplus \phi_{\alpha}^*(L'))_{p_\a}\rightarrow ({\mathbb C}_{p_\a} \oplus
\phi_{\alpha}^*({\mathbb C}_{p_\a}))_{p_\a} ={\mathbb C}_{p_\a} \oplus 0.$$ This shows that ${\mathcal F}=\ker_p
(\lambda)$ is the projection of the line $(0\oplus
\phi_{\alpha}^*(L'))_{p_\a}$. Hence $$[(E',
{\mathcal F})]=j_{p_\a}(\phi_{\alpha}^*(L'))=j_{\phi_{\alpha}(p_{\alpha})}(L').$$ Since $j_{p_\a}(L')\in
|{\mathcal P}|_a^+$, this shows that $[(E',
{\mathcal F})]$ lies
in $|{\mathcal P}|_a^-$ (see remark \ref{qeq}). Recalling that $q([(E', {\mathcal F})]=[E]$, it follows that
$q(|{\mathcal P}|_a^-)=|M|_a^+ $, and also that $q(|{\mathcal P}|_a^+)=|M|_a^-$.
\vskip 8pt
The following observation will be used in section \ref{tracecomp}.
\begin{proposition}\label{qtriv} Let $\nu$ be the relative cotangent
sheaf of $q'\colon {\mathcal P}\rightarrow M'$. Then the restriction of $\nu$ to
$|{\mathcal P}|_{\alpha}$ is numerically trivial.
\end{proposition}
\noindent {\bf Proof.} Note that $\nu|_{|{\mathcal P}|_{\alpha}}$ is the dual of the normal bundle,
$N$, say, of
the inclusion $|{\mathcal P}|_{\alpha}\subset (q')^{-1}(|M'|_{\alpha})$. By
prop. \ref{fixPa}, it suffices to show
that $j_{p_\a}^*(N) $ is numerically trivial. Let $\Lambda$ be a
Poincar\'e bundle over
$\mathop{\fam0 Pic}\nolimits_1(\Sigma^{\alpha})\times \Sigma^{\alpha}$. For $x\in \Sigma^{\alpha}$, let $\Lambda_{x}$ denote its
restriction to $\theta_{\alpha}^{-1}([p])\times \{x\}$. We
have a commutative diagram
$$\begin{array}{cccc}
{\mathbb P}( \Lambda_{p_\alpha} \oplus \Lambda_{\phi_{\alpha}({p_\a})} )
&\mapright{\Pi_{{\alpha}*}} & (q')^{-1}(|M'|_{\alpha})&\ \subset \ {\mathcal P} \\
\downarrow & & \downarrow \, q' &\\
\theta_{\alpha}^{-1}([p]) & \mapright{{\pi_{\a*}}} & |M'|_{\alpha}&
\end{array}$$ Here, $\Pi_{{\alpha}*}$ is the obvious map covering ${\pi_{\a*}}$. (A
point in ${\mathbb P}( \Lambda_{p_\alpha} \oplus \Lambda_{\phi_{\alpha}({p_\a})} )$ is
a point $[L]\in \theta_{\alpha}^{-1}([p])$ together with a line in $
L_{{p_\a}}\oplus L_{\phi_{\alpha}({p_\a})}=(L\oplus \phi_{\alpha}^*(L))_{p_\a}$. This is
sent by
$\Pi_{{\alpha}*}$ to the point represented by ${\pi_{\a*}}(L)$ and the induced line
in $({\pi_{\a*}}(L))_p$.)
Let $s_{{p_\a}}$ be the section of the fibration on
the left defined by $s_{p_\a}(L)=L_{{p_\a}}\oplus 0$, for $[L]\in
\theta_{\alpha}^{-1}([p])$. Then $j_{p_\a}=\Pi_{{\alpha}*}\circ s_{{p_\a}}$, and hence
the inclusion $|{\mathcal P}|_{\alpha}\subset (q')^{-1}(|M'|_{\alpha})$ corresponds to the
inclusion of the image of $s_{p_\a}$ in ${\mathbb P}( \Lambda_{p_\alpha} \oplus
\Lambda_{\phi_{\alpha}({p_\a})} )$. This shows
$$j_{p_\a}^*(N) \cong \Lambda_{p_\a}^*\otimes \Lambda_{\phi_{\alpha}({p_\a})}.$$
Tensoring $\Lambda$ by the pull-back of a bundle over $\mathop{\fam0 Pic}\nolimits_1(\Sigma^{\alpha})$
if necessary, we may assume $\Lambda_{p_\a}$ is trivial. Hence
$j_{p_\a}^*(N)$ is numerically
trivial, proving the proposition.
\section{Investigation of the involutions $\rho_a$ and $\rho'_{\alpha}$.}
\label{Geoinvest}
Let $a\inJ^{(4)}$ such that $2a={\alpha}\neq0$. Recall that the involution
$\rho_a$ is the lift of ${\alpha}$ to ${\mathcal{L}}$ which acts as the identity over
the fixed point component $|M|_a^+$.
\begin{proposition}\label{rhominus} The involution $\rho_a$ acts as minus the identity over
the fixed point component $|M|_a^-$.
\end{proposition}
\noindent {\bf Proof.} Let
$E'$ represent a point $[E']$ in $|M'|_\alpha$. Now
$\alpha$ acts on the fiber of $q'$ over $[E']$ and from our description of
$|{\mathcal P}|_\alpha$, we have that
$\alpha$ has exactly two fixed points on $(q')^{-1}([E'])$. Consider now
$q^*{\mathcal{L}}|_{(q')^{-1}([E'])}$ with its lift of $\alpha$ induced by
$\rho_a$. Let $s_1,s_2$ be the signs by which $\rho_a$ acts over the
two fixed points. By lemma 2.1 in
\cite{BSz} we have that
$q^*{\mathcal{L}}|_{(q')^{-1}([E'])}$ is isomorphic to ${\mathcal{O}}(1)$ over
$(q')^{-1}([E'])\cong {\mathbb P}(E'_p)$. From this we conclude that $s_1s_2 = -1$,
proving the proposition.
\begin{remark}\label{coc} {\em At this point, it follows easily that the
alternating form of the extension ${\mathcal{E}}$ generated by the involutions $\rho_a$
is equal to the order $2$ Weil pairing $\lambda_2$. Indeed, recall
that the alternating form is defined by the commutator pairing
$c({\alpha},{\beta})=\rho_a\rho_b\rho_a^{-1}\rho_b^{-1}$ where $a$ is a lift of
$\alpha$ and $b$ is a
lift of $\beta$. Assume first that $\lambda_2(\alpha, \beta)=1$.
Let us then evaluate $\rho_a\rho_b\rho_a^{-1}\rho_b^{-1}$ in a point in
$|M|^+_a$. Recalling from \ref{4.6}(ii) that $\beta$ preserves this component of $|M|_\alpha$,
we get that
$$c(\alpha,\beta) = 1 \,\rho_b \,(1)^{-1}\, \rho_b^{-1} = 1.$$
If however $\lambda_2(\alpha, \beta)=-1$, then $\beta$ exchanges the two
components, and $$c(\alpha,\beta) = 1 \,\rho_b \,(-1)^{-1}\, \rho_b^{-1} =
-1.$$
Thus $c=\lambda_2$, as asserted.}
\end{remark}
\begin{theorem}\label{8.4} We have
$\rho'_\alpha
\rho'_\beta=\lambda_2({\alpha},{\beta})\rho'_{\alpha+\beta}$.
\end{theorem}
\noindent {\bf Proof.} We may assume none of the classes ${\alpha}$, ${\beta}$, ${\alpha}+{\beta}$, is zero, the
result being obvious otherwise. Consider first the case
$\lambda_2({\alpha},{\beta})=-1$. Then by prop. \ref{6.5} we have that the triple
intersection $
|M'|_\alpha \cap |M'|_\beta \cap |M'|_{\alpha + \beta}$ is non-empty.
Since by definition $\rho'_\alpha$ acts as minus the identity on the
fiber over $|M'|_\alpha$, it
follows that $$\rho'_\alpha \rho'_\beta=-\rho'_{\alpha +
\beta}, $$ proving the result in this case.
Now consider the case
$\lambda_2({\alpha},{\beta})=1$. Pick $a,b\in J^{(4)}$ such that $2a={\alpha}$ and
$2b={\beta}$, and consider the involutions $\rho_a^{\otimes 2}$,
$\rho_b^{\otimes 2}$, and $\rho_{a+b}^{\otimes 2}$, acting on
${\mathcal{L}}^2$. In fact, those involutions depend only on ${\alpha}$ and ${\beta}$, and
not on the choice of $a$ and $b$. By prop. \ref{6.5} we have that the triple
intersection $
|M|_\alpha \cap |M|_\beta \cap |M|_{\alpha + \beta}$ is
non-empty. Note that $\rho_a^{\otimes 2}$ acts as the identity over
both components of $|M|_{\alpha}$. Hence
\begin{equation}\label{rhoH} \rho_a^{\otimes 2} \rho_b^{\otimes 2}=\rho_{a+b}^{\otimes 2}.
\end{equation} Now consider the Hecke correspondence. From Corollary
2.2 in \cite{BSz} (see also Lemma 10.3 in \cite{BLS}) we have that the
canonical bundle $K_{{\mathcal P}}$ of ${\mathcal P}$ satisfies
\begin{equation}\label{Hecke}
K_{{\mathcal P}} \cong (q')^*({{\mathcal{L}}'}^{-1})\otimes q^*({\mathcal{L}}^{-2}).
\end{equation}
From Proposition
\ref{fixPa} we see that $|{\mathcal P}|_\alpha$ has odd codimension, hence $\alpha$ acts by
$-1$ on the restriction of
$K_{\mathcal P}$ to
$|{\mathcal P}|_\alpha$. Our lifts $\rho_a^{\otimes 2}$ and $\rho'_\alpha$ thus
make the isomorphism (\ref{Hecke}) a $J^{(2)}$-equivariant
isomorphism. The action of $J^{(2)}$ on
$K_{{\mathcal P}}$ is obviously a group action. This enables us to compare the lift $\rho'_\alpha$ acting on ${{\mathcal{L}}'}$
and the lift $\rho_a^{\otimes 2}$ acting on ${\mathcal{L}}^{2}$. Thus
(\ref{rhoH})
implies $$\rho'_\alpha \rho'_\beta=\rho'_{\alpha +
\beta},$$ proving the result in the case $\lambda_2({\alpha},{\beta})=1$. This
completes the proof.
\vskip 8pt\noindent{\bf Note.} The equivariance of the isomorphism (\ref{Hecke})
is the reason why we defined $\rho'_\alpha$ to be the lift which acts
as {\em minus} the identity over the fixed point set.
\begin{theorem}\label{8.5} We have $ \rho_a \rho_b=\lambda_4(a,b)\rho_{a+b}$.
\end{theorem}
\noindent {\bf Proof.} We first deal with the case where $\lambda_4(a,b)=\pm 1$, or,
equivalently,
$\lambda_2({\alpha},{\beta})=1$. If ${\alpha}={\beta}=0$, there is nothing to show. If
${\alpha}={\beta}\neq 0$, then $\rho_a$ and $\rho_b$ are lifts of the same class,
hence $\rho_a=\pm \rho_b$. By prop. \ref{rhominus}, we have $\rho_a=\rho_b$ if and only if $a$
and $b$ define the same component of $|M|_{\alpha}$, which in turn is
equivalent, by prop. \ref{4.6}(i), to $\lambda_2(b-a,{\alpha})=1$. But if ${\alpha}={\beta}$ then
$\lambda_2(b-a,{\alpha})=\lambda_4(a,b)$ and $\rho_{a+b}$ is the
identity (by definition). This proves the result in the case ${\alpha}={\beta}$. Finally, if
${\alpha},{\beta},$ and ${\alpha}+{\beta}$ are all three non-zero, the triple intersection
$ |M|_\alpha \cap
|M|_\beta \cap |M|_{\alpha + \beta}$ is non-empty, and we can compute $\rho_a \rho_b
\rho_{a+b}$ in the fiber over an intersection point. By theorem
\ref{evencasei} and prop. \ref{rhominus}, it follows that $\rho_a \rho_b
\rho_{a+b}=\lambda_4(a,b)$, completing the proof in the case
$\lambda_4(a,b)=\pm 1$.
The remainder of this section is devoted to the proof in the case where $\lambda_4(a,b)=\pm
i.$ We
will again use the notations ${\gamma}={\alpha}+{\beta}$ and $c=a+b$.
Let the bundle
$E'$ represent a point $[E']$ in $|M'|_\alpha\cap |M'|_{\beta}\cap
|M'|_{\gamma}$. The three involutions ${\alpha},{\beta},{\gamma}$ induce involutions on
$(q')^{-1}([E'])\subset {\mathcal P}$; recall that $(q')^{-1}([E'])$ is
identified with the
projective space ${\mathbb P}(E'_p)$. Each of these involutions has two fixed
points on ${\mathbb P}(E'_p)$; these are precisely the intersection points of
${\mathbb P}(E'_p)$ with
the fixed point varieties $|{\mathcal P}|_{\alpha},|{\mathcal P}|_{\beta}$, and $|{\mathcal P}|_{\gamma}$. Note that
$\rho_a$ acts as $\mp 1$ on the fiber of $q^*{\mathcal{L}}$ at the intersection point
of ${\mathbb P}(E'_p)$ with
the component $|{\mathcal P}|_a^{\pm}$, since
$q(|{\mathcal P}|_a^+)=|M|_a^-$ by prop. \ref{q-+}. Let
${\mathcal F}_a^{\pm}$, ${\mathcal F}_b^{\pm}$, ${\mathcal F}_c^{\pm}$ be the lines in $E'_p$
corresponding to the intersection points of ${\mathbb P}(E'_p)$ with the components $|{\mathcal P}|_a^{\pm},|{\mathcal P}|_b^{\pm}$,
and $|{\mathcal P}|_c^{\pm}$. As already used in the proof
of prop. \ref{rhominus}, the restriction of $q^*{\mathcal{L}}$ to ${\mathbb P}(E'_p)$ is
the bundle ${\mathcal{O}}(1)$. It will be convenient to transfer the calculation
to the tautological bundle ${\mathcal{O}}(-1)$, whose fiber over a point
represented by a line ${\mathcal F}$ is that line. Of course, ${\mathcal{O}}(-1)$ is the restriction of
$q^*{\mathcal{L}}^{-1}$ to ${\mathbb P}(E'_p)$. For $a\inJ^{(4)}$, let us denote
by
$\hat\rho_a$ the involution
$\rho_a^{\otimes(-1)}$ acting on ${\mathcal{L}}^{-1}$. Then $\hat\rho_a$ acts as $\mp 1$ on the line
${\mathcal F}_a^{\pm}$, and similarly for $\hat\rho_b$ and $\hat\rho_c$ on the lines
${\mathcal F}_b^{\pm}$ and ${\mathcal F}_c^{\pm}$.
It follows easily from this description (or from
the computation of the alternating form in remark \ref{coc}) that one has
$\hat\rho_a\hat\rho_b=\varepsilon \hat\rho_c$ where $\varepsilon\in\{\pm i\}$. (In
fact, the involutions $\hat\rho_a,
\hat\rho_b$ generate a quaternion subgroup $Q_8\subset Sl_2({\mathbb C})$,
covering the commutative subgroup ${\mathbb Z}/2\times {\mathbb Z}/2 \subset PSl_2({\mathbb C})$
generated by ${\alpha},{\beta}$.) Of
course, the sign of $\varepsilon$ is determined by the relative position of the
six lines. The following lemma computes this relative position in terms of the Weil pairing
$\lambda_4(a,b)$.
\begin{lemma}\label{ml} Let $\lambda=\lambda_4(b,a)\in\{\pm i\}$. There is an isomorphism of
$E'_p$ with ${\mathbb C}^2$ sending the six lines ${\mathcal F}_a^{+}$, ${\mathcal F}_a^{-}$,
${\mathcal F}_b^{+}$, ${\mathcal F}_b^{-}$, ${\mathcal F}_c^{+}$, ${\mathcal F}_c^{-}$, to the lines
generated by the vectors
$$\left(\begin{array}{c} 1\\0 \end{array}\right), \
\left(\begin{array}{c} 0\\1 \end{array}\right), \
\left(\begin{array}{c} 1\\1 \end{array}\right), \
\left(\begin{array}{c} 1\\-1\end{array}\right), \
\left(\begin{array}{c} 1\\-\lambda \end{array}\right), \
\left(\begin{array}{c} 1\\ \lambda \end{array}\right) \ .
$$
\end{lemma}
The proof will be given later. Assuming lemma \ref{ml} for the moment,
we see that $\hat\rho_a,
\hat\rho_b$ and $\hat\rho_c$ correspond to the matrices
$$T_a=\left( \begin{array}{cc} -1& 0 \\ 0 & 1 \end{array}\right), \ \
T_b=\left( \begin{array}{cc} 0& -1 \\ -1 & 0 \end{array}\right), \ \
T_c=\left( \begin{array}{cc} 0& -\lambda \\ \lambda & 0
\end{array}\right).$$
Note that $T_aT_b=\lambda T_c$, and hence
$\hat\rho_a\hat\rho_b=\lambda \hat
\rho_c$ and $\rho_a \rho_b=\lambda^{-1}\rho_c$. Thus the remaining case of theorem \ref{8.5} follows directly
from lemma \ref{ml}.
\vskip 8pt
Now let us prove lemma \ref{ml}. The proof uses again the coverings $\Sigma^{\alpha}$,
$\Sigma^{\beta}$, $\Sigma^{\gamma}$, and their common covering $\widetilde{\S}$ (see section \ref{triple}).
Choose a point $\tilde p\in\tilde\pi^{-1}(p)\subset \widetilde{\S}$ and put
\begin{equation}
p_{\alpha}=\pi^{\alpha}(\tilde p), \ p_{\beta}=\pi^{\beta}(\tilde p), \ p_{\gamma}=\pi^{\gamma}(\tilde
p).
\label{palpha}
\end{equation}
Since $[E']\in |M'|_\alpha\cap |M'|_{\beta}\cap
|M'|_{\gamma}$, there exist line bundles ${\mathcal{L}}_a$ over $\Sigma^{\alpha}$, ${\mathcal{L}}_b$ over
$\Sigma^{\beta}$, and ${\mathcal{L}}_c$ over
$\Sigma^{\gamma}$, such that $
E'\cong
{\pi_{\a*}}({\mathcal{L}}_a)\cong
{\pi_{\b*}}({\mathcal{L}}_b)\cong
{\pi_{\g*}}({\mathcal{L}}_c)$. We can fix ${\mathcal{L}}_a$ (resp. ${\mathcal{L}}_b$, resp. ${\mathcal{L}}_c$) uniquely up to isomorphism by requiring
that ${\mathcal{L}}_a\in
\pi_{\alpha}^*(L_a)\otimes[p_{\alpha}]\otimes \mathop{\fam0 Im}\nolimits\Phi_{\alpha}^0$ (resp. ${\mathcal{L}}_b\in
\pi_{\beta}^*(L_b)\otimes[p_{\beta}]\otimes \mathop{\fam0 Im}\nolimits\Phi_{\beta}^0$, resp. ${\mathcal{L}}_c\in
\pi_{\gamma}^*(L_c)\otimes[p_{\gamma}]\otimes \mathop{\fam0 Im}\nolimits\Phi_{\gamma}^0$) (see formula (\ref{tw})).
Let us denote the bundle $\pi^{{\gamma}*}({\mathcal{L}}_c)$ on $\widetilde{\S}$ by $L$. Being a
pull-back bundle, $L$ has a canonical involution, $C$, say, covering the involution $\phi^{\gamma}$ on
$\widetilde{\S}$, and such that $L/{\langle} C{\rangle}$ is the bundle ${\mathcal{L}}_c$ on
$\Sigma^{\gamma}=\widetilde{\S}/{\langle}\phi^{\gamma}{\rangle}$. Proceeding as in lemma \ref{6.10}, it is easy to check that the
bundles $\pi^{{\alpha}*}({\mathcal{L}}_a)$ and $\pi^{{\beta}*}({\mathcal{L}}_b)$ are isomorphic to $L=\pi^{{\gamma}*}({\mathcal{L}}_c)$. Therefore $L$ also has canonical involutions $A$
and $B$, covering $\phi^{\alpha}$ and $\phi^{\beta}$, respectively, such that
$L/{\langle} A{\rangle}\cong {\mathcal{L}}_a$ and $L/{\langle} B{\rangle}\cong {\mathcal{L}}_b$.
\begin{lemma}\label{ABC} One has $AB=\lambda_4(b,a)C$.
\end{lemma}
\noindent {\bf Proof.} As in the proof of lemma \ref{6.9}, represent
$\pi^{{\alpha}*}({\mathcal{L}}_a)$, $\pi^{{\beta}*}({\mathcal{L}}_b)$, and $L=\pi^{{\gamma}*}({\mathcal{L}}_c)$, by
divisors $F_a$, $F_b$, $F_c$, respectively, such that
$$F_a=\tilde\pi^{*}(D_a)+\pi^{{\alpha}*}(p_{\alpha}+(1-\phi_{\alpha}^*)(D))$$
$$F_b=\tilde\pi^{*}(D_b)+\pi^{{\beta}*}(p_{\beta}+(1-\phi_{\beta}^*)(D'))$$
$$F_c=\tilde\pi^{*}(D_c)+\pi^{{\gamma}*}(p_{\gamma}+(1-\phi_{\gamma}^*)(D''))$$
where $D_a,D_b\in \mathop{\fam0 Div}\nolimits_0(\Sigma)$ represent $a,b\in J^{(4)}$, $D_c=D_a+D_b$, $D\in Div_0(\Sigma^{\alpha})$, $D'\in
Div_{0}(\Sigma^{\beta})$, and $D''\in Div_{0}(\Sigma^{\gamma})$. As before, since the three
bundles are isomorphic, there exist meromorphic functions $h_1,h_2\in
{\mathcal M}(\widetilde{\S})$ such that $$(h_2)+F_a\,=\,F_c\,=\, (h_1)+F_b.$$
The action of our involutions $A,B$ and $C$ on $L$ can be
described on local sections as
follows. Since $L={\mathcal{O}}(F_c)$, a local section over some open set $U\subset
\widetilde{\S}$ is just a meromorphic function $s$ on $U$ such that $(s)+F_c|_U\geq 0$. The action of $C$ is
simply given by $$ C\ : s \mapsto s\circ \phi^{\gamma},$$ since the divisor
$F_c$ was pulled back from $\Sigma^{\gamma}$, and $C$ is the canonical
involution of the pull-back bundle.
The involution $A$ is nothing but the
canonical involution of the pull-back bundle $\pi^{{\alpha}*}({\mathcal{L}}_a)={\mathcal{O}}(F_a)$,
conjugated by an
isomorphism with $L={\mathcal{O}}(F_c)$. Since $(h_2)=F_c-F_a$, multiplication
by $h_2$ gives such
an isomorphism ${\mathcal{O}}(F_c)\,\mapright\sim \,{\mathcal{O}}(F_a)$. Therefore the action
of $A$ on local sections of $L={\mathcal{O}}(F_c)$ is
$$ A\ : s \mapsto ((s h_2)\circ \phi^{\alpha})h_2^{-1}=(s\circ
\phi^{\alpha})\,k_A= (s\
k_A^{-1})\circ \phi^{\alpha}$$
where we have put $k_A=(h_2\circ \phi^{\alpha})/h_2$. (N.b., one may think about this as
follows: $s\circ
\phi^{\alpha}$ is a local section of $\phi^{{\alpha}*}L={\mathcal{O}}(\phi^{{\alpha}*}F_c)$, and
multiplication by $k_A$ describes an isomorphism
${\mathcal{O}}(\phi^{{\alpha}*}F_c)\,\mapright\sim \,{\mathcal{O}}(F_c)$.)
Similarly, $B$ acts on local sections of
${\mathcal{O}}(F_c)$ as $$ B\ : s \mapsto ((s h_1)\circ \phi^{\beta})h_1^{-1}=(s\circ
\phi^{\beta})\,k_B= (s\
k_B^{-1})\circ \phi^{\beta}$$
where $k_B=(h_1\circ \phi^{\beta})/h_1$.
Put \begin{equation} \lambda= \frac{k_B}{k_A} =
\frac{h_2}{h_1} \,\frac {h_1\circ \phi^{\beta}}{h_2\circ
\phi^{\alpha}}\label{la1}
\end{equation} Note that $\lambda$ is a constant, since
$(k_B)=(k_A)$. Since $AB$ acts on
local sections by
$$AB\ : s\mapsto (((s\circ
\phi^{\beta})\,k_B) \circ \phi^{\alpha})\, k_A=((s\circ
\phi^{\beta})\,k_B\,k_A^{-1})\circ \phi^{\alpha} =\lambda \, s\circ \phi^{\gamma},$$
we have $AB=\lambda C$.
Now let us show that $\lambda=\lambda_4(b,a)$. As in section
\ref{triple},
we use the functions $f,g\in {\mathcal M}(\Sigma)$ defined by $f=\widetilde
\mathop{\fam0 Nm}\nolimits(h_1)$ and $g=\widetilde \mathop{\fam0 Nm}\nolimits(h_2)$. A computation shows that the
statements of lemma \ref{6.12} hold word for word. We can therefore
compute $\lambda_4(a,b)=g(D_a)/f(D_b)$ exactly as before. Note that
the divisors $D,D',D''$ have degree zero, so that the terms
involving
the functions $f_{\alpha},f_{\beta}, f_{\gamma}$ are now equal to $1$. We thus obtain
\begin{eqnarray*}
\lambda_4(a,b)&=&\frac{h_1(\pi^{{\gamma}*}(p_{\gamma})-\pi^{{\alpha}*}(p_{\alpha}))}
{h_2(\pi^{{\gamma}*}(p_{\gamma})-\pi^{{\beta}*}(p_{\beta}))}
=\frac{h_1(\phi^{\gamma}(\tilde p)-\phi^{\alpha}(\tilde p))}{h_2(\phi^{\gamma}(\tilde p)-\phi^{\beta}(\tilde
p))} \\
&=&
\frac{h_1}{h_2} \,\frac {h_2\circ \phi^{\alpha}}{h_1\circ
\phi^{\beta}} \,(\phi^{\gamma}(\tilde p))
\end{eqnarray*}
where we have used (\ref{palpha}) in the last but one step.
Comparing this with formula (\ref{la1}), we have
$\lambda=\lambda_4(a,b)^{-1}=\lambda_4(b,a)$, proving lemma \ref{ABC}.
We return to the proof of lemma \ref{ml}. Recall that
\begin{equation}
E'\cong
{\pi_{\a*}}({\mathcal{L}}_a)=({\mathcal{L}}_a\oplus\phi_{\alpha}^*{\mathcal{L}}_a)/{\langle}\phi_{\alpha}{\rangle}. \label{uniqueflag}
\end{equation}
Since ${\mathcal{L}}_a\in
\pi_{\alpha}^*(L_a)\otimes[p_{\alpha}]\otimes \mathop{\fam0 Im}\nolimits\Phi_{\alpha}^0$, we have
$$j_{p_\a}({\mathcal{L}}_a)\in |{\mathcal P}|_a^+,$$ where $j_{p_\a}$ is the isomorphism of
prop. \ref{fixPa}. Thus, the line ${\mathcal F}_a^+\subset
E'_p$ (representing the unique intersection point in ${\mathbb P}(E'_p)\cap
|{\mathcal P}|_a^+$) is the image of the line $({\mathcal{L}}_a\oplus 0)_{{p_\a}}\subset
({\mathcal{L}}_a\oplus\phi_{\alpha}^*{\mathcal{L}}_a)_{{p_\a}}$ under the
natural projection. Also, the line ${\mathcal F}_a^-$ is the image of $(0\oplus
\phi_{\alpha}^*({\mathcal{L}}_a))_{{p_\a}}$. Note that since $E'$ is stable, the isomorphism
(\ref{uniqueflag}) is unique up to scalar multiples, hence the lines
${\mathcal F}_a^{\pm}$ are
well-determined by this description.
Similarly, using the isomorphism of $E'$ with ${\pi_{\b*}}({\mathcal{L}}_b)$ and with
${\pi_{\g*}}({\mathcal{L}}_c)$, the lines ${\mathcal F}_b^+$ and ${\mathcal F}_b^-$ correspond to the projections of
the lines $({\mathcal{L}}_b\oplus 0)_{{p_\b}}$ and $(0\oplus
\phi_{\beta}^*({\mathcal{L}}_b))_{{p_\b}}$, and the lines ${\mathcal F}_c^+$ and ${\mathcal F}_c^-$ correspond to the projections of
the lines $({\mathcal{L}}_c\oplus 0)_{{p_\g}}$ and $(0\oplus
\phi_{\gamma}^*({\mathcal{L}}_c))_{{p_\g}}$.
Let us now understand the relative position of the six lines in
$E'_p$.
Note that since $AB=-BA$, the involution $B$ of $L$ covering the involution
$\phi^{\beta}$ on $\widetilde{\S}$ induces a map
$$L/{\langle} -A{\rangle}\,\mapright{}\,L/{\langle} A{\rangle}$$ covering the involution $\phi_{\alpha}$ on
$\Sigma^{\alpha}$. We may choose isomorphisms ${\mathcal{L}}_a\cong L/{\langle} A{\rangle}$ and
$\phi_{\alpha}^*{\mathcal{L}}_a\cong L/{\langle} -A{\rangle}$ such that this map becomes the canonical
map $\phi_{\alpha}^*{\mathcal{L}}_a \rightarrow {\mathcal{L}}_a$ covering $\phi_{\alpha}$. Therefore
${\pi_{\a*}}({\mathcal{L}}_a)$ is isomorphic to the bundle $E_a$ defined by
$$E_a\ = \ L\oplus L {\Big/} \bigl<
\left( \begin{array}{cc} A& 0 \\ 0 & -A \end{array}\right),
\left( \begin{array}{cc} 0& B \\ B & 0 \end{array}\right)\bigr>.$$
(Notice that the two matrices commute, so that $E_a$ is indeed a
well-defined bundle on $\Sigma=\widetilde{\S}/\langle\phi^{\alpha},\phi^{\beta}{\rangle}$.) Moreover, the lines
${\mathcal F}_a^+$ and ${\mathcal F}_a^-$ in $E'_p$ correspond, {\em via} an isomorphism
$E'\cong E_a$ (which is unique up to scalar multiples), to the
images of the lines $(L\oplus 0)_{\tilde p}$
and $(0\oplus L)_{\tilde p}$ under the projection from
$L\oplus L$ to $E_a$. Here, we have used that $\pi^{\alpha}(\tilde p)={p_\a}$
by our choice of $p_{\alpha}$ in (\ref{palpha}).
Similarly, ${\pi_{\b*}}({\mathcal{L}}_b)$ is isomorphic to the bundle $E_b$ defined by
$$E_b\ = \ L\oplus L {\Big/} \bigl<
\left( \begin{array}{cc} B& 0 \\ 0 & -B \end{array}\right),
\left( \begin{array}{cc} 0& A \\ A & 0 \end{array}\right)\bigr>,$$ and
since $\pi^{\beta}(\tilde p)={p_\b}$,
the lines ${\mathcal F}_b^+$ and ${\mathcal F}_b^-$ in $E'_p$ correspond to the
images of the same lines $(L\oplus 0)_{\tilde p}$
and $(0\oplus L)_{\tilde p}$, but now projected from
$L\oplus L$ to $E_b$.
Lastly, ${\pi_{\g*}}({\mathcal{L}}_c)$ is isomorphic to the bundle $E_c$ defined by
$$E_c\ = \ L\oplus L {\Big/} \bigl<
\left( \begin{array}{cc} C& 0 \\ 0 & -C \end{array}\right),
\left( \begin{array}{cc} 0& A \\ A & 0 \end{array}\right)\bigr>, $$
and as before, since $\pi^{\gamma}(\tilde p)={p_\g}$, the lines ${\mathcal F}_c^+$ and ${\mathcal F}_c^-$ in $E'_p$ correspond to the
images of the lines $(L\oplus 0)_{\tilde p}$
and $(0\oplus L)_{\tilde p}$ under the projection from
$L\oplus L$ to $E_c$.
The three bundles $E_a$,$E_b$, $E_c$ are all isomorphic to
$E'$. In fact, we have isomorphisms
$\psi_{X}\colon E_b\mapright\sim E_a$ and $\psi_{Y}\colon
E_c\mapright\sim
E_a$ induced by the endomorphisms of $L\oplus L$ defined by the
matrices $$X=\left( \begin{array}{cc} 1& 1 \\ 1 & -1
\end{array}\right), \ \ \ Y=\left( \begin{array}{cc} 1& 1 \\
-\lambda & \lambda \end{array}\right).$$
The verification, which uses that $AB=\lambda C$ by lemma \ref{ABC},
is left to the reader.
Let us identify the fiber $(L\oplus
L)_{\tilde p}$ in the obvious way with ${\mathbb C}\oplus {\mathbb C}$ and consider the
isomorphism $$E'_p\,\mapright\sim\, (E_a)_p
\stackrel{\sim}\longleftarrow (L\oplus
L)_{\tilde p} = {\mathbb C}\oplus {\mathbb C},$$ where the first map is induced by an
isomorphism $E'\cong E_a$ and the second map is the projection.
Then the six lines ${\mathcal F}_a^{+}$, ${\mathcal F}_a^{-}$,
${\mathcal F}_b^{+}$, ${\mathcal F}_b^{-}$, ${\mathcal F}_c^{+}$, ${\mathcal F}_c^{-}$, correspond to the lines
in ${\mathbb C}\oplus {\mathbb C}$ generated by the vectors
$$\left(\begin{array}{c} 1\\0 \end{array}\right), \
\left(\begin{array}{c} 0\\1 \end{array}\right), \
X\left(\begin{array}{c} 1\\0 \end{array}\right) , \
X\left(\begin{array}{c} 0\\1 \end{array}\right), \
Y\left(\begin{array}{c} 1\\0 \end{array}\right) , \
Y\left(\begin{array}{c} 0\\1 \end{array}\right) \ .
$$
These vectors are precisely the ones in the statement of lemma
\ref{ml}.
This proves lemma \ref{ml}, and completes the proof of theorem
\ref{8.5}.
\section{The trace computation.}\label{tracecomp}
In this section, we prove theorem \ref{1.2} by computing the trace of ${\rho'_\alpha}^{\otimes k/2}$
and $\rho_a^{\otimes k}$ using the Lefschetz-Riemann-Roch fixed point
formula. In the twisted case, the computation is rather straightforward,
since $M'$ is smooth, and the relevant cohomology classes
on the fixed
point set are given in \cite{NR}.
This computation has been done in a different context by Pantev \cite{Pa}. We
repeat the calculation here and in the process we correct a misprint
in his formula. In the untwisted case, the moduli space and the fixed point
sets are not smooth. We circumvent this problem by transferring the
computation to ${\mathcal P}$, using some results of \cite{BSz}.
\vskip 8pt\noindent{\bf Note.} Beauville \cite{Be3} has recently
computed the traces in the untwisted case in a different way by transferring the calculation to
$M'$. \footnote{Our computation was
done independently of his.} Beauville
considers, more generally, rank $r$ bundles, and his formula agrees with ours in the
case $r=2$. He does not, however, choose lifts to the line bundle
${\mathcal{L}}$, and his result (for $r=2$) concerns only the case
$k\equiv 0$ mod $4$, where one has a group action of $J^{(2)}$.
\begin{proposition}
The trace of the involution ${\rho'_\alpha}^{\otimes k/2}$ is given by
$$\mathop{\fam0 Tr}\nolimits( {\rho'_\alpha}^{\otimes k/2})= (-1)^{k/2}\left( \frac{k+2}{2}\right)^{g-1}.$$
\end{proposition}
\noindent {\bf Proof.} The Lefschetz-Riemann-Roch fixed point formula \cite{AS} states that
\[ \mathop{\fam0 Tr}\nolimits({\rho'_\alpha}^{\otimes k/2}) = {\widetilde {Ch}}({{\mathcal{L}}'}|_{|M'|_\alpha})^{k/2}\,
{\widetilde {Ch}}(\lambda_{-1}N_\alpha)^{-1}\,\mathop{\fam0 Td}\nolimits(|M'|_{\alpha})\cap [|M'|_\alpha].\]
Here $\mathop{\fam0 Td}\nolimits(|M'|_{\alpha})$ is the Todd class,
$N_\alpha$ is the conormal bundle of $|M'|_\alpha$, $\lambda_{t}$ is
the operation defined by $\lambda_t E= \sum
t^i \Lambda^i E$, and $${\widetilde {Ch}}(E)=Ch(E_+ -E_-),$$ where $Ch$ is the Chern
character, and, for any
${\mathbb Z}/2$-equivariant bundle $E$ over the fixed point set $|M'|_\alpha$,
$E_+$ and $E_-$ are the $\pm 1$-eigenbundles.
Since the fixed point set $|M'|_\alpha$ is isomorphic to the Prym
variety $P_\alpha$, its Todd class is $1$.
The cohomology class ${\widetilde {Ch}}(\lambda_{-1}N_\alpha)$ was computed in Proposition
4.2 in \cite{NR}. Let $\Theta$ be the restriction to $P_\alpha$ of the
principal polarization of $J_0(\Sigma^\alpha)$. The result of Narasimhan and Ramanan
then states that
\begin{equation}
{\widetilde {Ch}}(\lambda_{-1}N_\alpha) = 2^{2(g-1)}e^{-2\Theta}. \label{Thet2}
\end{equation}
By the construction of ${{\mathcal{L}}'}$ in \cite{DN}, we have the following relation
between
$\Theta$ and
$Ch({{\mathcal{L}}'}|_{|M'|_\alpha})$.
\begin{equation}
Ch({{\mathcal{L}}'}|_{|M'|_\alpha}) = e^{2\Theta}. \label{Thet1}
\end{equation} Note that ${\widetilde {Ch}}({{\mathcal{L}}'}|_{|M'|_\alpha})=-
Ch({{\mathcal{L}}'}|_{|M'|_\alpha})$, by our definition of $\rho'_{\alpha}$.
Hence we get that
\[ \mathop{\fam0 Tr}\nolimits({\rho'_\alpha}^{\otimes k/2})= (-1)^{k/2}2^{-2(g-1)} e^{(k+2)\Theta}\cap
[P_\alpha].\] Using Corollary 4.16 in \cite{NR}, which states that
\begin{equation}
\Theta^{g-1}\cap [P_\alpha] = (g-1)!2^{g-1}, \label{Thet3}
\end{equation}
the result follows.
\begin{proposition}
The trace of the involution $\rho_a^{\otimes k}$ is given by
$$\mathop{\fam0 Tr}\nolimits(\rho_a^{\otimes k}) = \frac{1+(-1)^k}{2}\left(
\frac{k+2}{2}\right)^{g-1}.$$
\end{proposition}
\noindent {\bf Proof.}
The morphism $q$ induces a $J^{(2)}$-equivariant morphism
$$q^* : H^0(M,{\mathcal{L}}^k) \rightarrow H^0({\mathcal P},q^*{\mathcal{L}}^k).$$
According to \cite{BSz} we have that this morphism is an isomorphism and that
$$H^i({\mathcal P},q^*{\mathcal{L}}^k) = 0,$$
for $i>0$. Hence we just need to apply the Lefschetz-Riemann-Roch
fixed point theorem to $({\mathcal P},q^*{\mathcal{L}}^k)$. The fixed point set $|{\mathcal P}|_{\alpha}$
has two components, $|{\mathcal P}|_a^+$ and $|{\mathcal P}|_a^-$, each of which is
isomorphic to the Prym variety $P_{\alpha}$, and hence has trivial Todd
class. In order to understand $q^*{\mathcal{L}}|_{|{\mathcal P}|_{\alpha}}$ and the conormal
bundle $N(|{\mathcal P}|_\alpha)$, consider the following exact sequence
$$ 0 \rightarrow (q')^*T^*_{M'} \rightarrow T^*_{\mathcal P} \rightarrow \nu \rightarrow 0,$$
where $\nu$ is the relative cotangent sheaf of $q'$. We
conclude that
$$K_{\mathcal P} \cong \nu \otimes (q')^*K_{M'}\cong \nu \otimes (q')^*
({\mathcal{L}}')^{-2},$$ since $K_{M'} \cong ({\mathcal{L}}')^{-2}$ \cite{DN}.
By cor. 2.2 in \cite{BSz} (see equation (\ref{Hecke}) in the proof of
theorem \ref{8.4}), we get that
$$q^*{\mathcal{L}}^2 \cong (q')^*{{\mathcal{L}}'} \otimes \nu^{-1}.$$
Now recall from prop. \ref{qtriv} that the line bundle $\nu|_{|{\mathcal P}|_{\alpha}}$ is
numerically trivial. It follows that
$$Ch(q^*{\mathcal{L}}|_{|{\mathcal P}|_a^{\pm}})=\bigl(Ch((q')^*{{\mathcal{L}}'}|_{|{\mathcal P}|_a^{\pm}})\bigr)^{1/2}=e^\Theta$$
where in the last equality we have used formula (\ref{Thet1}), after
identifying both $|{\mathcal P}|_a^+$ and
$|{\mathcal P}|_a^-$ with the Prym variety $P_{\alpha}$. Next, observe that $\alpha$
acts as $-1$ on
$\nu|_{|{\mathcal P}|_{\alpha}}$, since one has an exact
sequence (of bundles over $|{\mathcal P}|_{\alpha}$)
\begin{equation}
0\rightarrow (q')^*N_{\alpha} \rightarrow N(|{\mathcal P}|_\alpha)\rightarrow \nu|_{|{\mathcal P}|_\alpha}\rightarrow
0.\label{exse}
\end{equation} where $N_{\alpha}=N(|M'|_\alpha)$ as before. Therefore
$${\widetilde {Ch}}(\lambda_{-1}\nu|_{|{\mathcal P}|_a^{\pm}})={\widetilde {Ch}}(1-\nu|_{|{\mathcal P}|_a^{\pm}})=Ch(1+1)=2,$$
and
the exact sequence (\ref{exse})
gives us that $${\widetilde {Ch}}(\lambda_{-1}(N(|{\mathcal P}|_a^{\pm})) )=
2\, {\widetilde {Ch}}(\lambda_{-1}(N_{\alpha}))=2\cdot
2^{2(g-1)}e^{-2\Theta}, $$ where we have used formula (\ref{Thet2}) in
the last equality, after again identifying $|{\mathcal P}|_a^+$ and
$|{\mathcal P}|_a^-$ with $P_{\alpha}$.
Now recall that $\rho_a$ acts with opposite signs on the restriction
of $q^*{\mathcal{L}}$ to the two components $|{\mathcal P}|_a^+$ and
$|{\mathcal P}|_a^-$. In fact, it acts as $\mp 1$ over $|{\mathcal P}|_a^{\pm
}$, by prop. \ref{q-+}. Therefore $${\widetilde {Ch}}(q^*{\mathcal{L}}|_{|{\mathcal P}|_a^{\pm }})=\mp
Ch(q^*{\mathcal{L}}|_{|{\mathcal P}|_a^{\pm }})=\mp e^\Theta.$$ Putting everything together, the fixed point formula gives
$$\mathop{\fam0 Tr}\nolimits(\rho_a^{\otimes k}) = (1 + (-1)^k) e^{k\Theta} \,
2^{-1} 2^{-2(g-1)} e^{2\Theta} \cap
[P_\alpha].$$
The proposition now follows as in the twisted case from formula
(\ref{Thet3}).
|
1997-10-15T19:38:35 | 9710 | alg-geom/9710018 | en | https://arxiv.org/abs/alg-geom/9710018 | [
"alg-geom",
"math.AG"
] | alg-geom/9710018 | Sandra DiRocco | Sandra Di Rocco | Generation of $k$-jets on Toric Varieties | 14M25, 14J60, 14C20(14C25, 14E25), 17 pages, AmsLatex, see home page
http://www.math.kth.se/~sandra/Welcome | null | null | null | null | In this notes we study $k$-jet ample line bundles $L$ on a non singular toric
variety $X$, i.e. line bundles with global sections having arbitrarily
prescribed $k$-jets at a finite number of points. We introduce the notion of an
associated $k$-convex $\D$-support function, $\psi_L$, requiring that the
polyhedra $P_L$ has edges of length at least $k$. This translates to the
property that the intersection of $L$ with the invariant curves, associated to
every edge, is $\geq k$. We also state an equivalent criterion in terms of a
bound of the Seshadri constant $\e(L,x)$. More precisely we prove the
equivalence of the following: (1) $L$ is $k$-jet ample; (2) $L\cdot C\geq k$,
for any invariant curve $C$; (3) $\psi_L$ is $k$-convex; (4) the Seshadri
constant $\e(L,x)\geq k$ for each $x\in X$.
| [
{
"version": "v1",
"created": "Wed, 15 Oct 1997 17:38:35 GMT"
}
] | 2007-05-23T00:00:00 | [
[
"Di Rocco",
"Sandra",
""
]
] | alg-geom | \section*{Introduction}
The notion of $k$-jet ampleness has been introduced by Demailly to describe line bundles $L$ whose global sections can have arbitrarily prescribed $k$-jets at every single point $x\in X$, see \cite{Dem}. Beltrametti and Sommese generalized it by considering $k$-jets supported on a finite number of points.
\vskip6pt
{\em A line bundle $L$ is said to be $k$-jet ample on $X$ if for any collection of $r$ points, $(x_1,\cdots,x_r)$, and any $r$-ple of positive integers $(k_1,\cdots,k_r)$, with $\sum k_i=k+1$ the natural map
$$H^0(X,L)\to H^0(L\otimes{\cal O}_X/{\frak m}_{x_1}^{k_1}\otimes \cdots \otimes {\frak m}_{x_r}^{k_r})$$
is surjective}.
\vskip6pt
Notice that $0$-jet ampleness is equivalent to being spanned by global sections and $1$-jet ampleness is equivalent to being very ample.
During the last years many results on $k$-jet ample line bundles on surfaces have been established, \cite{BeSok, EiLa, BaSz, BaDRSz}. Up to our knowledge the problem is still quite open for higher dimensional varieties, besides few cases like $\pn{n}$ and Fano varieties \cite{BeSok, BeDRSo}.
In \cite{Cox1} D. Cox has introduced ``homogeneous coordinates" on a toric variety $X(\Delta)$. For the points invariant under the torus action the situation looks similar to the projective space case. Using this system of local coordinates we give a description of the fibers of the $k$-jet bundle $J_k(L)$ on fixed points, see section \ref{kjet}.
According to Oda and Demazure a line bundle $L$ on $X(\Delta)$ is generated by global sections (respectively very ample) if the $\Delta$-support function $\psi_L$ is {\em convex (respectively strictly convex)}. This suggests to use a ``higher convexity" property for $\psi_L$ in the cases $k\geq 2$.\\
In section \ref{kconvex} we introduce the notion of a {\em $k$-convex $\Delta$-support function}, which for $k=0,1$ agrees with being convex or strongly convex.\\
The $\Delta$-support function $\psi_L$ is $k$-convex if the polyhedra $P_L$, associated to $L$, has edges of length at least $k$. This translates to the property that the intersection of $L$ with the invariant curves, associated to every edge, is $\geq k$, which is a generalization of the toric Nakai criterion for ample line bundle.
A key step in the proof is the reduction to the case where the considered points are invariant under the torus action. We are grateful to T. Ekedahl for suggesting to use the Borel's fixed point theorem, and for pointing out the sufficiency of the reduction argument.
Our result states that in order to check the $k$-jet ampleness of a line bundle $L$ it is enough to have a bound on the intersection $L\cdot C$, for all the invariant curves $C$. This can be applied to the study of ``local positivity". In section \ref{applications} we report a series of results on blow-ups and higher adjoint bundles, which in the toric case can be shown by means of a direct checking on intersections. We also state an equivalent criterion for $k$-jet ampleness in terms of a bound of the Seshadri constant $\epsilon(L,x)$. This can be thought as a toric version of the Seshadri criterion for ample line bundles, generalized to $k$-jet ampleness.
In this paper we prove:\\
{\em Let $L$ be a line bundle on a non singular toric variety $X(\Delta)$, then the following statements are equivalent:
\begin{itemize}
\item $L$ is $k$-jet ample;
\item $L\cdot C\geq k$, for any $T$-invariant curve $C$, [Proposition \ref{inter}];
\item $\psi_L$ is $k$-convex, [Theorem \ref{main}];
\item the Seshadri constant $\epsilon(L,x)\geq k$ for each $x\in X$, [Proposition \ref{sesh}].
\end{itemize}}
\vskip6pt
It is a pleasure to acknowledge valuable discussions with T. Ekedahl, M. Boij, D. Laksov and D. Cox. This research was initiated during the author's stay at Mittag-Leffler Institute and KTH and reached a final stage at the Max Planck Institute. To all those institutions we owe thanks for their support.
\section*{ Notation}
We will use standard notation in Algebraic Geometry. The groundfield will always be the field of complex numbers.\\
By abuse of terminology the words line bundle and Cartier divisor will be used with no distinction, as well as the multiplicative and additive structure.\\
For basic notions on toric varieties we refer to \cite{Fu, Oda, Ew} and for a nice survey on the resent progress on toric geometry we refer to \cite{Cox2}.\\
When not stated $X$ will always denote a smooth $n$-dimensional toric variety and $L$ a line bundle on it.\\
\section{Toric Varieties}
Let $N$ be an $n$-dimensional lattice and $\Delta=\cup \sigma_i$ be a complete and regular fan, meaning:
\begin{itemize}
\item $supp(\Delta)=N_{\Bbb R}=N\otimes {\Bbb R}$ and
\item for every $r$-dimensional cone $\sigma\in\Delta$, there exists a $\Bbb Z$-basis of $N_{\Bbb R}$, $\{\rho_1,\cdots,\rho_n\}$, such that the subset $\{\rho_1,\cdots,\rho_r\}$ spans $\sigma$.
\end{itemize}
We will denote by $X=X(\Delta)$ the associated non singular $n$-dimensional toric variety and by $\Delta(t)$ the set of $t$-dimensional cones in $\Delta$.\\
Let $M=Hom_{\Bbb Z}(N,{\Bbb Z})$ be the dual lattice so that $X$ is obtained by gluing together the affine toric varieties $X_{\sigma}=Spec({\Bbb C}[\check{\sigma}\cap M])$, where $\check{\sigma}=\{v\in M_{\Bbb R}:\langle v,\sigma\rangle\geq 0\}$, and $\sigma\in\Delta$.\\
Each $m\in M$ can be viewed as a rational function $\chi^m:T=N\otimes{\Bbb C}^*\to \Bbb C^*.$\\
There is a $1-1$ correspondence between $r$-dimensional cones $\tau\in\Delta(r)$ and $T$-invariant codimension $r$ subvarieties of $X$, which will be denoted by $V(\tau)$.\\
Let $D_i=V(\rho_i)$ be the $T$-invariant divisors corresponding to the one dimensional cones $\rho_i\in\Delta(1)$. The set $\{D_i\}_{\rho_i\in\Delta(1)}$ form a set of generators for the Picard group of $X$ and thus every line bundle $L$ can be written in terms of those principal divisors:
$$L=\sum_{\rho_i\in\Delta(1)} a_i D_i$$
We will denote by $P_L$ the associated convex polyhedra:
$$P_L=\{m\in M_{\Bbb R}:\langle m, \rho_i\rangle\geq -a_i\}$$
This gives a nice way of expressing the global sections of $L$:
$$H^0(X,L)=\bigoplus_{m\in P_L\cap M}{\Bbb C}{\cal X}^m$$
Recently David Cox \cite{Cox1}, has introduced the notion of homogeneous coordinates on a toric variety.
There is a $1-1$ correspondence between $T$-invariant principal divisors $D_i$ and linear monomials ${\cal X}_i$ on $X$.
The polynomial ring is then defined as:
$$S={\Bbb C}[{\cal X}_i:\rho_i\in\Delta(1)]$$
and the grading is given by the group of divisors modulo rational equivalence, $Pic(X)$, i.e. two rationally equivalent divisors $D$ and $E$ are associated to monomials ${\cal X}_D$ and ${\cal X}_E$ of the same degree.\\
Considering the exact sequence:
$$0\to M\to \oplus_{\rho_i\in\Delta(1)}{\Bbb Z}\cdot D_i\to Pic(X)\to 0$$
we associate to each $m\in M$ a divisor $\sum \langle m,\rho_i\rangle D_i=div(\chi^m)$.\\
The global sections of $L$ are generated by the monomials of the form $$(\Pi_i {\cal X}_i^{\langle m,\rho_i\rangle+a_i})_{m\in P_D\cap M}$$
The notion of $\Delta$-support function will be use constantly throughout this paper:
\begin{definition}\cite[2.1]{Oda} A real valued function
$f:\cup_i\sigma_i\to\bf R$ is a $\Delta$-linear support function if it is
$\Bbb Z$-valued on $N\cap(\cup_i\sigma_i)$ and it is linear on each
$\sigma_i$.
\end{definition} This means that for each $\sigma$ there exists $m_\sigma\in M$
such that
$f(n)=\langle m_\sigma,n\rangle$ for $n\in\sigma$
and $\langle m_\sigma,n\rangle=\langle m_\tau,n\rangle$ when $\tau$ is a face of $\sigma$.
To each divisor $L$ we associate a $\Delta$-support function $\psi_L$ defined by:
$$\psi_L(\rho_i):=-a_i$$
\section{$k$-jet Bundles}\label{kjet}
Let $\Delta$ be the diagonal in $X\times X$ and $p:X\times X\to X$ the projection onto the first factor. The $k$-th jet-bundle associated to $L$ is the vector bundle associated to the sheaf:
$$p^*L/p^*L\otimes {\cal I}_{\Delta}^{k+1}$$
Where ${\cal I}_{\Delta}$ is the ideal sheaf of $\Delta$. It is a vector bundle of rank ${k+n}\choose n$ whose fiber is
$$J_k(L)_x=L_x\otimes{\cal O}_X/m_x^{k+1}$$
For details on jet bundles we refer the reader to \cite[Ch.I]{KuSp}.
There are natural maps (defined on the sheaf level):
$$i_k:L\to J_k(L)$$
sending the germ of a section $s$ at a point $x\in X$ to its $k$-th jet.
More specifically for $s\in H^0(X,L)$ $i_k(s(x))\in \bigoplus_1^{{k+n}\choose n}\Bbb C$ is the ${k+n}\choose n$-ple determined by the coefficients of the terms of degree up to $k$, in the Taylor expansion of $s$ around $x$.\\
So if $(x_1,\cdots,x_n)$ are local coordinates around $x_0=(0,0,\cdots,0)$ and $s=\sum c_{i_1,\cdots,i_r}\prod x_i^{i_j}$ then
$$i_k(s(x_0))=(\cdots,\frac{\partial s}{\pr{x_1}{t_1}\cdots\pr{x_r}{t_r}},\cdots)|_{x=x_0}=(\cdots,(\text{{\Tiny constant}})\cdot c_{t_1,\cdots,t_r},\cdots) $$
where $t_1+\cdots+t_r\leq k$. For example $i_1(s(x))$ consists in the constant and linear term.
The following definition formalizes the property for a linear series $|L|$ on $X$ to generate $k$-jets on one or more points of $X$. When more points are considered $L$ is said to generate ``simultaneous jets" at those points.
\begin{definition}
Let ${\cal Z}=\{x_1,\cdots,x_r\}$ be a finite collection of distinct points on $X$. $L$ is said to be $k$-jet ample on ${\cal Z}$ (or equivalently the series $|L|$ is said to generate all $k$-jets on ${\cal Z}$) if for any $r$-ple of positive integers $(k_1,\cdots,k_r)$, such that $\sum_1^r k_i=k+1$ the map:
$$H^0(X,L)\to H^0(L\otimes {\cal O}_X/{\frak m}_{x_1}^{k_1}\otimes\cdots\otimes {\frak m}_{x_r}^{k_r})$$ is surjective, where ${\frak m}_{x}^k$ is $k$-th tensor power of the maximal ideal sheaf ${\frak m}_x$.
$L$ is $k$-jet ample on $X$ if it is $k$-jet ample on each such ${\cal Z}$ in $X$.
\end{definition}
Clearly from the definition:
\begin{itemize}
\item We can rewrite the map above as:
$$\psi_{{\cal Z}}^{k_1,\cdots,k_r}:H^0(X,L)\to \bigoplus_1^r (J_{k_i-1}(L))_{x_i}$$
defined by $\psi_{{\cal Z}}^{k_1,\cdots,k_r}(s)=(i_{k_1-1}(s(x_1)),\cdots,i_{k_r-1}(s(x_1)))$.
We say then that $L$ is $k$-jet ample on $X$ if the map $\psi_{{\cal Z}}^{k_1,\cdots,k_r}$ is surjective for any ${\cal Z}$ and any $(k_1,\cdots,k_r)\in\Bbb Z^r_+$, such that $\sum k_i=k+1$.
\item If $L$ is $0$-jet ample then $L$ is generated by its global sections;
\item if $L$ is $1$-jet ample then using the sections in $H^0(X,L)$ we can define an embedding $i:X\to \pn{N}$ and thus $L$ is very ample.
\end{itemize}
Using the homogeneous coordinates introduced in the previous section the $k$-jets at the $T$-invariant points $x(\sigma)=V(\sigma)$ can be better described in terms of the polyhedra associated to $L$.\\
Let $\sigma=\langle \rho_1,\cdots,\rho_n\rangle$, where $\sigma\in\Delta(n)$ and $\rho_i\in\Delta(1)$ are the one dimensional cones generating $\sigma$.
The point $x(\sigma)$, lies on the intersection of the divisors $D_i$, $i=1,\cdots,n$: $x(\sigma)\in\cap_1^n({\cal X}_i=0)$. Then the maximal ideal is generated by the linear monomials in ${\cal X}_i$:
$${\frak m}_{x(\sigma)}=\langle {\cal X}_1,\cdots,{\cal X}_n\rangle$$
and thus
$${\frak m}_{x(\sigma)}^{k+1}=\langle \Pi_{\rho_i\subset\sigma}{\cal X}_i^{t_i}|t_1+\cdots+t_n=k+1\rangle$$
i.e. the generators are the monomials of ``degree$=k+1$" in the variables ${\cal X}_1,\cdots,{\cal X}_n$ ( here by degree we mean the sum of the powers of the variables, i.e. the usual one). \\
Each ${{\cal X}}^m$, generator of $H^0(X,L)$, can be written in the local coordinates $({\cal X}_1,\cdots,{\cal X}_n)$ as follows. Fix $\{\rho_1,\cdots,\rho_n\}$ as basis of $N$ and let $\{m_1,\cdots,m_n\}$ the dual basis. In this coordinate system $m=\sum \langle m,\rho_i\rangle m_i$ and the germ of ${{\cal X}}^m$ at $x(\sigma)$ is:
$${{\cal X}}^m|_{x(\sigma)}=\prod_{i=1}^n {\cal X}_i^{\langle m,\rho_i\rangle+a_i}$$
Taking its $k$-th jet means ``killing" all the monomials of degree $\geq k+1$ in the variables ${\cal X}_1,\cdots{\cal X}_n$:
$$ i_k({{\cal X}}^m(x(\sigma)))=(\cdots, \frac{\partial{{\cal X}}^m}{\pr{x_1}{t_1}\cdots\pr{x_r}{t_r}},\cdots)|_{x=x(\sigma)}$$
\ex\label{delpezzo} Let $N={\Bbb Z}^2$ and $\Delta$ be the $2$-dimensional fan composed by the following $6$ cones, and their edges:\\
$\sigma_1=\langle (0,1),(1,1)\rangle,\,\sigma_2=\langle (1,1),(1,0)\rangle,\,\sigma_3=\langle (1,0),(0,-1)\rangle\\
\sigma_4=\langle (0,-1),(-1,-1)\rangle,\,\sigma_5=\langle (-1,-1),(-1,0)\rangle,\,\sigma_6=\langle (-1,0),(0,-1)\rangle\\$
$X(\Delta)$ is the equivariant blow up of $\pn{2}$ in the $3$ fixed points, i.e. a Del Pezzo surface of degree $6$.\\
Let $L=D_1+D_2+D_3+D_4+D_5+D_6=-K_{X(\Delta)}$, where the $D_i's$ are associated to the edges in the order given above. Let $\sigma=\langle (0,1),(1,1)\rangle$ and let $\{m_1,m_2\}$ be the basis dual to $\{(0,1),(1,1)\}$.\\
In this basis $P_L$ is the convex hull of the points $$\{(0,1),(1,1),(1,0),(-1,0),(-1,-1),(-1,0)\}$$
and thus the generators of $H^0(X,L)$ are $$\{1,{\cal X}_1,{\cal X}_2,{\cal X}_1{\cal X}_2,{\cal X}_1^2{\cal X}_2,{\cal X}_1{\cal X}_2^2,{\cal X}_1^2{\cal X}_2^2\}$$
Moreover ${\frak m}_{x(\sigma)}^2=\langle {\cal X}_1{\cal X}_2,{\cal X}_1^2,{\cal X}_2^2\rangle$ and then
$$J_1(L)_{x(\sigma)}={\Bbb C}\oplus {\Bbb C}{\cal X}_1\oplus {\cal X}_2$$
\ex\label{pn1} Let $X=\pn{n}$, then $\Delta$ is the fan composed by $(n+1)$ $n$-dimensional cones spanned by the $(n+1)$ edges
\begin{itemize}
\item $\rho_i=(0\cdots,0,\underbrace{1}_{i-th},0,\cdots,0)$ for $i=1,\cdots n$
\item $\rho_{n+1}=(-1,\cdots,-1)=-rho_1-\cdots -\rho_n$
\end{itemize}
Let $D_1,\cdots,D_{n+1}$ be the associated $T$-invariant principal divisors and let $L=D_1+\cdots+D_k={\cal O}_{\pn{n}}(k)$.\\
recall that the Picard group is generated by one principal divisor $D_i$ and that $D_i\equiv D_j$ for $i\neq j$. So we can think of $L$ as
$$L=t_1D_1+\cdots+t_nD_n;\,\,t_1+\cdots+t_n=k$$
Let $\sigma=\langle\rho_1,\cdots\rho_n\rangle$, and fix the basis $\{\rho_1,\cdots,\rho_n\}$ with dual $\{m_1,\cdots,m_n\}$. In this basis the polyhedra $P_L$ is the convex hall of the $(n+1)$ points
$$\{ (-1,\cdots,-1),(k,-1,\cdots,-1),\cdots,(-1,\cdots,-1,k,-1,\cdots,-1),\cdots,(-1,\cdots,k)\}$$
Then for any decomposition $t_1,\cdots,t_n$ of positive integers such that $\sum_1^n(t_1+1)=k$ the lattice point $m=\sum _1^n t_1m_1\in P_L$ and any lattice point $m\in P_L$ can be written in this form. The situation stays the same if we consider another $\sigma\in\Delta$. It follows that
$$J_k(L)_{x(\sigma)}=H^0(X,L)=\bigoplus_{t_1+\cdots+t_r=k}{\Bbb C}\prod_1^r{\cal X}_1^{t_i}$$
In particular if $L={\cal O}(1)$ then $J_1(L)$ is trivial. This is in fact a characterization of the projective space, cf.\cite{So}.
\section{ $k$-convex functions}\label{kconvex}
In order to study positivity properties of line bundles Demazure and Oda introduced the definition of convex and strictly convex $\Delta$-support function.
\begin{theorem}\cite[Th. 2.13]{Oda} A line bundle $L$ on $X$ is globally generated ( i.e. $0$-jet ample) if and only if $\psi_L$ is convex and it is very ample (i.e. $1$-jet ample) if and only if $\psi_L$ is strictly convex.
\end{theorem}
A natural way of generalizing such a criterion to higher jets is to introduce a definition of ``higher convexity".
\begin{definition} Let $\psi$ be a $\Delta$-linear support function with
$\psi(v)=\langle m_\sigma, v\rangle$ for each $v\subset\sigma\in\Delta$.
we will say that $\psi$ is {\em $k$-convex} if for any
$\sigma\in\Delta$ and $v\not\subset\sigma$ $$\langle m_\sigma,v\rangle\geq \psi(v)+k
$$
\end{definition}
Confronting the notion of convex and strictly convex function, see \cite{Oda}, it is clear that
\begin{itemize}
\item $\psi$ is $0$-convex if and only if it is convex;
\item $\psi$ is $1$-convex if and only if it is strongly convex.
\end{itemize}
\rem\label{additive}It is clear from the definition that:
\begin{itemize}
\item If $\psi$ is $k$ convex then it is $t$-convex for any $t\leq k$;
\item If $\psi_1$ is $t_1$-convex and $\psi_2$ is $t_2$-convex, then $(\psi_1+\psi_2)$ is $(t_1+t_2)$-convex.
\end{itemize}
\vskip10pt
The meaning of convexity and strong convexity of a $\Delta$-support function associated to a line bundle $L$ is quite clear at least at the fixed points $x(\sigma)\in\Delta$. If $\psi_L$ is convex then:
$${{\cal X}}^{m_{\sigma}}(x(\sigma))=\prod_1^n {\cal X}_i^{a_1-a_i}\neq 0$$
If $\sigma=\langle \rho_1,\cdots,\rho_n\rangle$ and $\sigma'=\langle \rho_0,\rho_2,\cdots,\rho_n\rangle$ then
$${{\cal X}}^{m_{\sigma'}}(x(\sigma))=\prod_2^n {\cal X}_i^{a_1-a_i}{\cal X}_1^{\langle m_{\sigma'},\rho_1\rangle+a_1}=0$$
in the case $\langle m_{\sigma'},\rho_1\rangle+a_1>0$, i.e. $\psi_L$ strictly convex.
In other words if $\psi_L$ is convex then for each invariant point there is a non vanishing section, and $\psi_L$ strictly convex implies that different invariant points can be separated.\\
The notion of $k$-convexity generalizes the above property to more points with possible multiplicities.
More geometrically a $\Delta$-support function $\psi$ is $k$-convex if
for each $\sigma\in\Delta$ the graph of the defining linear function
$\langle m_{\sigma},\,\,\rangle$ is ``very" high compared to the graph of $\psi$.\\
Recall that if $\psi_L$ is convex then the polyhedra $P_L$ is the convex hull of the points $m_{\sigma}$ in $M_{\Bbb R}$. If $\psi_L$ is strictly convex then there is a correspondence between the faces in $\Delta$ and the set of non empty faces of $P_L$ (cf. \cite[2.12]{Oda}). Any face $F\subset P_L$ corresponds to
$$F^*=\{n\in N_{\Bbb R}|<m,n>=\psi_L(n),\text{ for any }m\in F\}\in\Delta$$
and any cone $\sigma\in\Delta$ corresponds to
$$\sigma^*=\{m\in M_{\Bbb R}|<m,n>=\psi_L(n),\text{ for any }n\in \sigma\}\subset P_L$$ Then $\psi_L$ being $k$-convex means that the length of the edges of the polyhedra, corresponding to $\tau=\sigma_i\cap\sigma_j$, are bigger or equal to $k$. This implies that $P_L$ is ``big enough" to choose points in it, corresponding to sections with an arbitrary prescribed jet.
\begin{lemma}\label{lenght}
Let $P_L$ be the polyhedra associated to $L$ and assume $\psi_L$ is $k$-convex, then
\begin{enumerate}
\item For each $\tau=\sigma_i\cap\sigma_j\in\Delta(n-1)$ the length of the associated edge $\tau^*$ is\\ $l(\tau^*)=|m_{\sigma_i}-m_{\sigma_j}|=l_{i,j}\geq k$;
\item Let $\sigma_1,\cdots,\sigma_r$ be the $n$-dimensional cones in $\Delta$. For each partition \\
$(t_1^1,\cdots,t_n^1,t_1^2,\cdots,t_n^2,\cdots,t_n^r)$ where $t_i^j\geq 0$, $\sum_{j=1}^nt_j^i=k_i-1$ and $\sum_1^rk_i=k+1$, and for any $\sigma_i\in\Delta(n)$ we can find $m\in P_L$ such that
\begin{itemize}
\item $\langle m, \rho_l\rangle=-a_l+t_l^i$ for all $\rho_j\subset\sigma_i$
\item $\langle m, \rho_l\rangle\geq-a_l+ t_l^j$ for all $\rho_j\not\subset\sigma_i$ with equality only if $t_l=0$
\end{itemize}
\end{enumerate}
\end{lemma}
\begin{pf} of (1). Let $\sigma_i=\langle \rho_1,\cdots,\rho_n\rangle$ and $\sigma_j=\langle \rho_2,\cdots,\rho_{n+1}\rangle$. Then using the basis $\{m_1,\cdots,m_n\}$ dual to $\{\rho_1,\cdots,\rho_n\}$
$m_{\sigma_i}=(-a_1,\cdots,-a_n)$ and $m_{\sigma_j}=(-a_1+l_{i,j},\cdots,-a_n)$, where $l_{i,j}\geq k$ since $\psi_L$ is $k$-convex.\end{pf}
\begin{pf} of (2). Fix $\sigma=\langle \rho_1,\cdots,\rho_n\rangle$ for simplicity of notation and let \\$\sigma_i=\langle \rho_1,\cdots,\check{\rho_i},\cdots,\rho_n,\overline{\rho_i}\rangle$ the $n$-cone so that $\sigma\cap\sigma_i=\tau_i=\langle \rho_1,\cdots,\check{\rho_i},\cdots,\rho_n\rangle$. By (1) the edge $\tau_i^*$ has lenght at least $k$. Choose the $t_i$-th lattice point next to $m_{\sigma}$ traveling on $\tau^*$ towards $m_{\sigma_i}$, i.e
$$\overline{m_i}=(-a_1,\cdots,-a_i+t_i,\cdots,-a_n)=m_{\sigma}+(\frac {t_i}{l_i})(m_{\sigma_i}-m_{\sigma_i})$$ in the basis dual to $\{\rho_1,\cdots,\rho_n\}$, where $\langle m_{\sigma_i},\rho_i\rangle=-a_i+l_i\geq -a_i+k$ by hypothesis. Traveling on the $n$ edges next to $m_{\sigma}$ we get
$$m=m_{\sigma}+\sum_1^n(\frac {t_i}{l_i})(m_{\sigma_i}-m_{\sigma})$$
Rewriting it in the form
$$m=(1-\sum_1^n(\frac {t_i}{l_i}))m_{\sigma}+\sum_1^n(\frac {t_i}{l_i})m_{\sigma_i}$$
It is clear that $m$ is a convex combination of $\{m_{\sigma},m_{\sigma_1},\cdots,m_{\sigma_n}\}$, since \\$0\leq \sum_1^n(\frac {t_i}{l_i})\leq \frac{1}{k}\sum_1^n t_i\leq 1$ and therefore $m\in P_L=Conv(m_{\sigma})_{\sigma\in\Delta(n)}$. Moreover
\begin{itemize}
\item $\langle m,\rho_i\rangle=-a_i+t_i$ for $i=1,\cdots,n$
\item if $\rho_l\not\subset\sigma$ then
\begin{align*}
\langle m,\rho_l\rangle&=(1-\sum (\frac{t_i}{l_i}))\langle m_{\sigma},\rho_l\rangle+\sum [(\frac{t_i}{l_i})\langle m_{\sigma_i},\rho_l\rangle]\\
&> (1-\sum (\frac{t_i}{l_i}))(-a_l+k)+\sum [(\frac{t_i}{l_i})(-a_l)]\\
&=-a_l-k(\sum \frac{t_i}{l_i})+k\geq -a_l+(k-\sum t_i)
\end{align*}
If $k_j=0$ for all $j\neq i$ then $\sum t_i=k_i-1=k$ and thus $\langle m,\rho_l\rangle\geq -a_l.$ Otherwise $k-\sum t_i<t_l$ and thus $\langle m,\rho_l\rangle>-a_l+t_l.$
\end{itemize}
\end{pf}
\vskip10pt
\section{The Main Result}
Let us first formulate an equivalent criterion for $\psi_L$ to be $k$-convex in terms of the intersections of the divisor $L$ with the $T$-invariant rational curves associated to each $\tau\in\Delta(n-1)$.\\
In fact the polyhedra $P_L$ having edges of length at least $k$ translates to the restriction of $L$ to each curve, corresponding to such edges, being at least $k$.\\ This is in a way a generalization of the ``toric Nakai criterion", cfr. \cite[Th. 2.18]{Oda}.
\begin{proposition}\label{inter} Let $L$ be a line bundle on a smooth $n$-dimensional toric variety $X$. Then $\psi_L$ is $k$-convex if and only if the restriction $L|_{V(\tau)}$ has degree $\geq k$, for every $\tau\in\Delta(n-1)$.\\
Equivalently if and only if $L_{V(\tau)}={\cal O}_{\pn{1}}(a)$ with $a\geq k$ for every $\tau\in\Delta(n-1)$.
\end{proposition}
\proof Let $\tau=\sigma_0\cap\sigma_1$ and assume $\sigma_i=\langle \tau, n_i\rangle$ for $i=0,1$. Then, since we are assuming $X$ to be non singular, there exists a $\Bbb Z$-basis
$\langle n_i, n_2,\cdots,n_n\rangle$ and $(n-1)$ integers $(s_2,\cdots,s_n)$ such that:
$$n_0+n_1-\sum_2^{n-2}s_in_i=0$$
Write $L=-\sum_i \psi_L(n_i)D_i$ where $D_1, D_0$ are the principal divisors associated to the edges $n_0, n_2$ and $D_i$ are the ones associated to $n_i$, $i=2,\cdots,n$.
then
\begin{itemize}
\item $D_1\cdot V(\tau)=D_0\cdot V(\tau)=1$
\item $D_i\cdot V(\tau)=-s_i$ for $i=2,\cdots,n$
\item $D_j\cdot V(\tau)=0$ otherwise
\end{itemize}
\[\begin{array}{ll}
L\cdot V(\tau)&=\sum (-\psi_L(n_i))D_i\cdot V(\tau)=\\
&=-\psi_L(n_0)-\psi_L(n_1)+\sum_2^n \psi_L(n_i)s_i\\
&=\langle m_{\sigma_1}, n_0\rangle -\psi_L(n_0)
\end{array}\]
It follows that $L\cdot V(\tau)\geq k$ for all $\tau\in\Delta(n-1)$ if and only if for any $\sigma\in\Delta$ and $\langle\rho_j, \sigma\cap\sigma'\rangle=\sigma'$ the inequality
$$\langle m_{\sigma}, \rho_j\rangle -\psi_L(\rho_j)\geq k$$
In other words $L\cdot V(\tau)\geq k$ if and only if the support function $\psi_L$ is $k$-convex.\qed
The relation between $k$-convex $\Delta$-support functions and $k$-very ampleness is given by the following:
\begin{theorem}\label{main} A line bundle $L$ generates $k$-jets on $X$ if and only if the $\Delta$-support function $\psi_L$ is $k$-convex.
\end{theorem}
We need the following reduction step:\\
\noindent{\rm C\sc laim} {\em If $L$ is $k$-jet ample on any $r$-ple of fixed points $\{x(\sigma_1),\cdots,x(\sigma_r)\}$, $\sigma_i\neq \sigma_j$, then it is $k$-jet ample on $X$.}
\begin{pf}{\rm \sc of the claim.} To every $(x_1,\cdots,x_{k+1})\in X^{k+1}$ we can associate ${\cal Z}=(x_1,\cdots,x_r)$, a collection of $r\leq k+1$ distinct points of $X$, and $(k_1,\cdots,k_r)$, an $r$-ple of positive integers such that $\sum k_i=k+1$, simply counting the multeplicities of each $x_i$:
$$(\underbrace{x_1,\cdots,x_1}_{k_1},\underbrace{x_2,\cdots,x_2}_{k_2},\cdots,\underbrace{x_r,\cdots,x_r}_{k_r})=[{\cal Z}=(x_1,\cdots,x_r),(k_1,\cdots,k_r)]$$
Then to each $\underline{x}\in X^{k+1}$ we can associate the map:
$$\psi_{\underline{x}}=\psi_{{\cal Z}}^{k_1,\cdots,k_r}:H^0(X,L)\to \bigoplus_1^r (J_{k_{i-1}}(L))_{x_i} $$
Let $C=\{\underline{x}\in X^{k+1}\text{ such that }coker(\psi_{\underline{x}})\neq 0\}$. Since $\psi_{\underline{x}}$ is an equivariant map, $C$ inherits the torus action from $X$, i.e. it is an invariant closed subvariety of $X^{k+1}$, and hence proper. If $L$ is not $k$-jet ample on $X$, then $\psi_{\underline{x}}$ is not surjective for some $\underline{x}=[{\cal Z},(k_1,\cdots,k_r)]$, which means $C\neq\emptyset$. But this implies $C^{T}\neq\emptyset$, where $C^{T}$ is the set of the fixed points in $C$. To see this one can apply Borel's fixed point theorem ( see \cite[21.1]{Hum}) or more directly observe that $C$ is a lower dimensional toric variety and thus it must contain fixed points. It follows that there exists $\underline{x}\in X^{k+1}$, fixed by the torus action, for which $\psi_{\underline{x}}$ is not surjective. Such $\underline{x}$ must have all the components fixed so it is of the form $(x(\sigma_1),\cdots,x(\sigma_r))$, which is a contradiction.\end{pf}
\begin{pf}{\sc of the theorem.}
If $L$ is a $k$-jet ample line bundle then the restriction $L|_{V(\tau)}$ to every $\tau\in\Delta(n-1)$ is $k$-jet ample, i.e. $L_{V(\tau)}={\cal O}_{\pn{1}}(a)$ with $a\geq k$ for every $\tau\in\Delta(n-1)$.
Proposition \ref{inter} then implies that $\psi_L$ is $k$-convex.\\
Assume now that $\psi_L$ is $k$-convex. By the reduction step it suffices to prove that the map $\psi_{{\cal Z}}^{(k_1,\cdots,k_r)}$ is surjective for each ${\cal Z}=\{x(\sigma_1),\cdots,x(\sigma_r)\}$, with $k_1+\cdots+k_r=k+1$. This follows immediately from Lemma \ref{lenght}. For each $k_i$ and for each partition $t_1^i+\cdots+t_n^i=k_{i}-1$ we can choose $m\in P_L$ such that
\begin{itemize}
\item ${\cal X}^{m}=\prod_{\rho_j\subset\sigma_i}{\cal X}_j^{t_j^i}$ around $x(\sigma_i)$ and
\item ${\cal X}^{m}=\prod_{\rho_j\subset\sigma_l}{\cal X}_j^{t_j^l+c_j^l}$ around $x(\sigma_l)\neq x(\sigma_i)$, with $c_j^l>o$ for some $j$ and for any partition $\sum t_j^l\leq k_l$
\end{itemize}
This means that
$$(i_{k_i}( {\cal X}^{m}(x(\sigma_1)),\cdots,i_{k_r}( {\cal X}^{m}(x(\sigma_r)))=(0,\cdots,1,0,\cdots,0)$$
the non zero term corresponding to $\frac{\partial {\cal X}^{m}}{\pr{x_1}{t_1}\cdots\pr{x_r}{t_r}}|_{x=x(\sigma_i)}$. Enough to prove the surjectivity.\end{pf}
\rem\label{add}From \ref{additive} it follows immediately that:
\begin{itemize}
\item if $L$ is $k$-jet ample then it is $t$-jet ample for any $t\leq k$;
\item If $L$ is $k$-jet ample and $E$ is $t$-jet ample then the line bundle $E\otimes L$ is $(k+t)$-jet ample. In fact $\psi_{E\otimes L}=\psi_L+\psi_E$.
\end{itemize}
It is worth observing that in the toric case the notion of $k$-jet ampleness is equivalent to the notion of $k$-very ampleness (which is weaker in general).
For basic properties of $k$-very ample line bundles we refer to \cite{BeSoB}.
\begin{definition}
$L$ is said to be $k$-very ample if for every zeroscheme $({\cal Z},{\cal O}_{{\cal Z}})$ of length $h^0({\cal O}_{{\cal Z}})=k+1$ the map $H^0(X,L)\to H^0({\cal Z},L\otimes{\cal O}_{{\cal Z}})$ is onto.
\end{definition}
\begin{proposition}
A line bundle $L$ on a smooth toric variety $X$ is $k$-very ample if and only if it is $k$-jet ample.
\end{proposition}
\proof If $L$ is $k$-jet ample then it is $k$-very ample ( see \cite[Prop. 2.2]{BeSok}).\\
If $L$ is $k$-very ample then the degree of $L$ restricted to any irreducible curve must be $\geq k$, i.e. $L\cdot V(\tau)\geq k$ for all $\tau\in\Delta(n-1)$ and thus it is $k$-jet ample by \ref{inter}. \qed
\section{Examples}
In this section we work out few examples, for which the $k$-jet ampleness has been studied otherwise, to convince the reader that this is in fact the right way to formulate the result.
\ex\label{pn}{\sc The projective space $\pn{n}$.} Notation as in \ref{pn1}.
Let $L=t_1D_1+\cdots+t_{n+1}D_{n+1}\cong (t_1+\cdots+t_{n+1})D_1$. By \ref{inter} $L$ is $k$-jet ample if and only if $V({\sigma_i})\cdot L\geq k$ where $\sigma_i$ is the n-dimensional cone $\langle \rho_1,\cdots\check{\rho_i},\cdots,\rho_{n+1}\rangle$ and $D_i$ is the divisor associated to the edge $\rho_i$. Since $\rho_{n+1}+\rho_i+\sum_{j\neq i}\rho_j=0$
$$V({\sigma_i})\cdot L=t_1+\cdot+t_{n+1}\geq k$$
In other words ${\cal O}_{\pn{N}}(a)$ is $k$-jet ample if and only if it is $k$-very ample if and only if $a\geq k$, as proven in \cite{BeSok}.
\vskip10pt
\ex\label{fn}{\sc The Hirzebruch surface ${\Bbb F}_n$.}
Let $\{e_1,e_2\}$ be the standard basis for ${\Bbb R}^2$. The Hirzebruch surface ${\Bbb F}_n$ is the toric surface associated to the fan $\Delta$ spanned by the following $2$-cones:
$$\sigma_1=\langle e_1,e_2\rangle,\,\sigma_2=\langle e_2,-e_1\rangle,\,\sigma_3=\langle -e_2,-e_1+ne_2\rangle,\,\sigma_4=\langle -e_1+ne_2,e_2\rangle$$
Let $D_1,\cdots,D_4$ be the divisors associated respectively to $e_2,e_1,-e_2,-e_1+ne_2$. Then
we have the following intersection matrix:
\[\begin{pmatrix}D_1^2&D_1\cdot D_2&D_1\cdot D_3&D_1\cdot D_4\cr
D_2\cdot D_1&D_2^2&D_2\cdot D_3&D_2\cdot D_4\cr
D_3\cdot D_1&D_2\cdot D_3&D_3^2&D_3\cdot D_4\cr
D_4\cdot D_1&D_2\cdot D_4&D_4\cdot D_3&D_4^2\cr
\end{pmatrix}=
\begin{pmatrix}-n&1&0&1\cr
1&0&1&0\cr
0&1&n&1\cr
1&0&1&0\cr
\end{pmatrix}\]
Recall that $D_3\equiv D_1+nD_2$ and $D_2\equiv D_4$.
Let $L=a_1D_1+\cdots+a_2D_4=(a_1+a_3)D_1+(a_4+a_2+na_3)D_2$. Then by \ref{inter} $L$ is $k$-jet ample if and only if it is $k$-very ample if and only if
\begin{itemize}
\item $L\cdot D_1=a_4-na_1+a_2\geq k$
\item $L\cdot D_2=a_1+a_3\geq k$
\item $L\cdot D_3=a_2+na_3+a_4\geq k$
\item $L\cdot D_4=a_1+a_2\geq k$
\end{itemize}
which of course is equivalent to saying that $L=aE_o+bf$, where $E_0$ is the section of minimal selfintersection $-n$ ( i.e. $D_1$) and $f$ the general fiber of the projection onto $\pn{1}$ (i.e. $D_2$), is $k$-jet ample if and only if
\begin{itemize}
\item $a=a_1+a_3\geq k$
\item $-an+b=-na_1-na_3+a_4+na_3+a_2=a_4-na_1+a_2\geq k$
\end{itemize}
This conditions have been given by Beltrametti-Sommese in \cite{BeSok}, using a decomposition argument.
\vskip10pt
\ex{\sc Del Pezzo surfaces.}
The toric Del Pezzo surfaces are $\pn{2}$, ${\Bbb F}_1$ and the equivariant blow up of $\pn{2}$ in $2$ or $3$ points.
The most interesting one is the last one.
Let $S$ be the equivariant blow up of $\pn{2}$ in the $3$ invariant points as described in \ref{delpezzo}.
The principal divisors $D_1,\cdots,D_6$ are the $6$ $(-1)$-curves on the surfaces, i.e. the three exceptional divisors and the pull back of the three lines passing through two of the $3$ points blown up.\\
Proposition \ref{inter} says that $L$ is $k$-jet ample if and only if it is $k$-very ample if and only if the intersection with all the $(-1)$-curves on the surface is $\geq k$.\\
This criterion has been given for $k$-very ampleness in \cite{DR} using a generalization of Reider's theorem.\\
Note that the equivalence between $k$-jet ampleness and $k$-very ampleness is not always true for Del Pezzo surfaces. In fact it is not hard to see that if $S$ is the blow up of $\pn{2}$ in $7$ points in general position ( so it is not toric), then the line bundle $L=-2K_S$ is $2$-very ample but it is not $2$-jet ample.
\section{ Local positivity applications}\label{applications}
In this section we report some nice applications of $k$-jet ampleness to the study of ``local positivity" of line bundles. Most of it is a survey on well known results. We think it is interesting to show how these results can be established rather easily in the case of toric varieties.
\vskip10pt
\noindent{\sc Blow ups}. A $k$-jet ample line bundle carries its positivity along blow ups at a finite number of points. The following property has been proved by Beltrametti and Sommese in \cite{BeSo96} and it has a very ``visible" proof in the toric case. We refer to \cite{Oda} for notation and definition of equivariant blow ups.
\begin{proposition}\label{blowup} Let $p:X(\Delta')\to X(\Delta)$ be the equivariant blow up of $X(\Delta)$ at $r$ points, $x_1,\cdots,x_r$, and let $L$ be a $k$-jet ample line bundle on $X(\Delta)$. Then $p^*(L)-\sum\epsilon_iE_i$ is min$(k-\sum\epsilon_i,\epsilon_1,\cdots,\epsilon_r)$-jet ample on $X(\Delta)$, where $E_i$'s are the exceptional divisors.
\end{proposition}
\begin{pf}Use induction on $r$. Assume the number of edges in $\Delta$ is $N$. \\
If $r=1$, let $x=x(\sigma)$, $\sigma=\langle \rho_1,\cdots,\rho_n\rangle$ and $E_1=E$ be the divisor associated to the edge $\overline{\rho}=\rho_1+\cdots+\rho_n$. In the new fan $\Delta'$ there are $n$ new $n$-cones $\sigma_i=\langle \overline{\rho},\rho_1,\cdots,\check{\rho_i},\cdots,\rho_n\rangle$. Moreover let $\overline{D_i}$ be the divisors in $Pic(X(\Delta'))$ corresponding to the edges $\rho_i$, then
\begin{itemize}
\item $\overline{D_i}=p^*(D_i)-E$ for $i=1,...,n$;
\item $\overline{D_i}=D_i$ for $i=n+1,...,N$.
\end{itemize}
If $L=\sum a_i D_i$ then
$${\cal H}=p^*(L)-\epsilon E=\sum_1^N a_ip^*(D_i)-\epsilon E=\sum_1^N a_i\overline{D_i}-(\epsilon+\sum_i^n a_i)E$$
Let $\tau\in\Delta'(n-1)$. If $\tau\in\Delta(n-1)$ then clearly ${\cal H}\cdot V(\tau)=L\cdot V(\tau)\geq k$. If $\tau\in\Delta'-\Delta$ then it is one of the following:
\begin{itemize}
\item[(a)] $\sigma_i\cap\sigma_j=\langle \overline{\rho},\rho_1,\cdots,\check{\rho_i},\check{\rho_j},\cdots,\rho_n\rangle$
\item[(b)] $\sigma_i\cap\langle\rho_{n+1},\rho_1,\cdots,\check{\rho_i},\cdots,\rho_n\rangle$
\end{itemize}
Following the lines of \ref{inter}:\\
In case (a), since $\rho_i+\rho_j-\overline{\rho}+\sum_{l\neq i,j}\rho_l=0$
$${\cal H}\cdot V(\tau)=-a_1-a_2+\epsilon+\sum_1^n a_l+
\sum_{l\neq i,j}a_l=\epsilon$$
In case (b), assume $\rho_{n+1}+\rho_i-\sum_{j\neq i}s_j\rho_j=0$, then $\rho_n+1+\overline{\rho}-\sum_{j\neq i}(s_j-1)\rho_j=0$ and
$${\cal H}\cdot V(\tau)=-a_{n+1}-\epsilon-\sum_1^n a_j+\sum_{j\neq i}s_j+\sum_{j\neq i}a_i=L\cdot V(\tau')-\epsilon$$
where $\tau'=\langle \rho_1,\cdots,\check{\rho_i},\cdots,\rho_n,\rho_{n+1}\rangle\cap\sigma\in\Delta(n-1)$\\
If $r>1$, iterating this process, after $r$ blow-ups ${\cal H}=p_{r}^*(L')-\epsilon_rE$, where \\$p^*_r:X(\Delta')\to X(\Delta_{r-1})$ is the $r$-th blow up map. By induction $L'$ is min$(k-\sum_1^{r-1}\epsilon_1,\epsilon_r,\cdots,\epsilon_{r-1})$-jet ample on $X(\Delta_{r-1})$. Clearely from what done before
$${\cal H}\cdot V(\tau)\geq min(L'\cdot V(\tau')-\epsilon_r,\epsilon_r)\geq min(k-\sum_1^n\epsilon_i,\epsilon_1,\cdots,\epsilon_r)$$ for any $\tau\in \Delta'(n-1)$.\end{pf}
\vskip10pt
\noindent{\sc Toric Seshadri criterion}. An ample line bundle on a smooth projective variety, $X$, is characterized by the positive value of its Seshadri constant at each point.\\
Let $L$ be a nef line bundle on $X$. For every irreducible curve $C\subset X$, $m_x(C)$ denotes the multiplicity of $C$ at the point $x\in C$ and
$$m(C)=\sup_{x\in C}\{m_x(C)\}$$.
\begin{theorem}(Seshadri \cite[7.1]{Ha}) A line bundle $L$ on $X$ is ample if and only if there exists $\epsilon>0$ such that $L\cdot C\geq \epsilon\cdot m(C)$ for every irreducible curve $C\subset X$.
\end{theorem}
As for the Nakai criterion we can generalize the Seshadri criterion to $k$-jet ampleness on toric varieties.\\
Let us first reformulate the Seshadri's theorem in the ``modern languige" of Seshadri constants.\\
For a nef line bundle $L$ the Seshadri constant of $L$ at a point $x\in X$ is the real number
$$\epsilon(L,x)=\inf_{x\in C}\frac{L\cdot C}{m_x(C)}=\sup_{\epsilon}\{\epsilon\in {\Bbb R}|p^*(L)-\epsilon L\text{ is nef }\}$$
where the inf is taken over all the irreducible curves containing $x$ and $p$ is the blow-up map of $X$ at $x$.
Then one can immediately see that The Seshadri criterion says that $L$ is ample if and only if $\epsilon(L,x)>0$ for every $x\in X$.\\
Demailly showed that the Seshadri constant is a measure of the highest degree jets that can be generated by the global sections of $L$:
\begin{proposition}\label{dem}\cite{Dem}
Let $s(L,x)$ be the largest integer such that $|L|$ generates $s$-jets at $x$. Then
$$\epsilon(L,x)= \limsup_{n\to\infty}\frac{s(nL,x)}{n}$$
\end{proposition}
Since every ample line bundle on a toric variety is very ample the toric Seshadri criterion says that $L$ is a ample if and only if $\epsilon(L,x)\geq 1$ for every $x\in X(\Delta)$. More generally:
\begin{proposition}\label{ses}
A line bundle $L$ on a non singular toric variety $X(\Delta)$ is $k$-jet ample if and only if there exists an $\epsilon\geq k$ such that
$$L\cdot V(\tau)\geq \epsilon \cdot m(V(\tau))$$ for every invariant curve $V(\tau)$.\end{proposition}
\begin{pf} Assume $L$ is $k$-jet ample, where $k$ is the biggest integer such that the property is true. Then by \ref{dem} $\epsilon(L,x)=k$, since $nL$ is $(nk)$-jet ample by \ref{add}. It follows that $\frac{L\cdot V(\tau)}{m(V(\tau))}\geq k$ for every $\tau\in\Delta$.\\
Assume now that $L\cdot V(\tau)\geq k\cdot m(V(\tau))$ for any $\tau\in\Delta$. Let $\tau=\sigma_i\cap\sigma_j$, we have to prove that $L\cdot V(\tau)\geq k$.
Consider the invariant point $x=V(\sigma_i)\in V(\tau)$, clearely $m_{x}(V(\tau))=1$. Then
$L\cdot V(\tau)\geq k\cdot m(V(\tau))\geq k\cdot m_x(V(\tau))\geq k$.
\end{pf}
\begin{corollary}\label{sesh} A line bundle $L$ on a non singular toric variety is $k$-jet ample if and only if $\epsilon(L,x)\geq k$ for every $x\in X$.\end{corollary}
\begin{pf} If $L$ is $k$-jet ample then $\epsilon(L,x)=k$ by \ref{dem}.\\
If $\epsilon(L,x)\geq k$ for every $x\in X$ then in particular $L\cdot V(\tau)\geq k \cdot m(V(\tau))$ for every invariant curve $V(\tau)$ and thus $L$ is $k$-jet ample by \ref{ses}.\end{pf}
\vskip10pt
\noindent{\sc Higher adjoint series}. By use of bounds on the Seshadry constant of $L$ at a sufficiently general point Ein and Lazarsfeld showed that:
\begin{proposition}\cite[5.14]{Laz} Let $L$ be an ample line bundle on a smooth surface $S$. Then the adjoint series $|K_S+(k+3)L|$ generates $k$-jets at a sufficiently general point $x\in S$.\end{proposition}
The fact that for a line bundle on a toric variety being ample is equivalent to being very ample implies a simple generalization.
\begin{proposition} Let $S$ be a non singular toric surface and let $L$ an ample line bundle on it such that $L^2>1$. Then
\begin{itemize}
\item[(a)] $|K_S+(k+2)L|$ generates $k$-jets at every point $x\in S$;
\item[(b)] $|K_S+(2k+2)|$ generates $k$-jets on $S$.
\end{itemize}
\end{proposition}
\begin{pf}(a) By the Nakai toric criterium $L$ is in fact very ample and by \ref{add} $(k+2)L$ is $(k+2)$-jet ample. Let $x\in S$, using the long exact sequence:
$$\to H^0(K_S+(k+2)L)\to H^0((K_S+(k+2)L)/{\frak m}_x^{k+1}))\to H^1((K_S+(k+2)L)\otimes {\frak m}_x^{k+1})\to$$
the vanishing of $H^1((K_S+(k+2)L)\otimes {\frak m}_x^{k+1})$ would imply the result. Let $p:\overline{S}\to S$ be the blow up of $S$ at $x$ with $E=p^{-1}(x)$, then by Leray spectral sequence and Serre duality
$$H^1((K_S+(k+2)L)\otimes {\frak m}_x^{k+1})=H^1(K_{\overline{S}}+[p^*((k+2)L)-(k+2)E])$$
Kawamata vanishing theorem applies since $p^*((k+2)L)-(k+2)E$ is nef and big and thus $K_S+(k+2)L$ is $k$-jet ample at any point $x\in S$.\\
(b) Consider now simoultaneus jets supported on $\{x_1,\cdots x_r\}\in S$ and $(k_1,\cdots,k_r)\in{\Bbb Z}_+^r$ such that $\sum k_1=k+1$. Let $p:\overline{S}\to S$ the blow up of $S$ at $x_1,\cdots, x_r$ with $E_i=p^{-1}(x_i)$. Using the same exact sequence as above it suffices to prove that:
$$H^1((K_S+(2k+2)L)\otimes({\frak m}_{x_1}^{k_1}\otimes\cdots\otimes {\frak m}_{x_r}^{k_r}))=H^1(K_{\overline{S}}+[p^*((2k+2)L)-\sum (k_i+1)E_i])=0$$
Since $(2k+2)L$ is $(2k+2)$-jet ample and $\sum(k_i+1)\leq 2k+2$, $p^*((2k+2)L)-\sum (k_i+1)E_i$ is spanned by \ref{blowup}. Moreover
$$(p^*((2k+2)L)-\sum (k_i+1)E_i)^2>(2k+2-\sum(k_1+1))(2k+2+\sum(k_1+1))\geq 0$$
Then Kawamata vanishing theorem applies to gine the nedeed vanishing.\end{pf}
\vskip10pt
\noindent{\sc The $k$-reduction}. In the case of surfaces we can make some further remarks about the ``$k$-reduction" process (see \cite{BeSo}).
Let $S$ be a non singular toric surface and $L$ a $k$-very ample line bundle on it. Since $k$-jet very ampleness and $k$-very ampleness are equivalent we will use freely the property of being $k$-very ample according to the criterion given in \ref{main}.
Assume the adjoint bundle is not $k$-very ample i.e., the surface contains $(-1)$-curves whose intersection with
$L$ is exactly $k$. If the $k$-adjoint bundle
$kK_S+ L$ is nef we can contract down those curves and get the
$k$-reduction $(S',L')$.
Notice that if $L=-kK_{S}$ then $-K_{S}$ is ample and hence $S$ is a toric
Del Pezzo surface. We can then compute directly the nefness of the $k$-adjoint bundle obtaining the same result as in \cite{BeSo}. Notation as in \ref{pn},\ref{fn}.
\begin{proposition} Let $L\neq -kK_{S}$ be a $k$-very ample line bundle on
$S$. Then $kK_{S}+L$ is nef unless:
\begin{itemize}
\item $S=\pn{2}$ and $L=aD_1$ with $a<3k$ ;
\item $S={\Bbb F}_r$ and $L=aD_1+bD_2$ with $a< 2k$.
\end{itemize}
\end{proposition}
\begin{pf}Let $L=\sum a_iD_i$ and $D_i^2=-s_i$, then
$$(kK_S+L)\cdot D_i=L\cdot D_i+k(s_i-2)\geq 0\text{ if }s_i\geq 1$$
Recall that $S$ is isomorphic to $\pn{2}$, ${\Bbb F}_n$ or their equivariant blow up in a finite number of points. If $S$ is minimal then intersecting $L$ with the basic generators of $Pic(S)$ and imposing at least one intersection to be less then $k$ gives the cases in the statement. Assume now $S$ not minimal.
If $S=Bl_r(\pn{2})$ (i.e. the blow up of $\pn{2}$ in $r$ points) let $D_1,D_j,D_l$ be the divisors associated to the edges $(0,1),(1,0),(-1,-1)$ respectively. If $r\geq 2$, for each $D_i$ generator of $Pic(S)$, the corresponding weight $D_i^2=-s_i\leq -1$ unless possibly only one among $(s_1,s_j,s_l)$, say $s_1=-1$. But in this case $D_1\equiv D_2+\sum_1^rD_i$, the $D_i$'s being the exceptional divisors, $L\cdot D_1\geq (r+1)k$ and
$$(kK_S+L)D_1\geq (r+1)k-3k\geq 0$$
If $r=1$ then $S\cong {\Bbb F}_1$.\\
If $S=Bl_r({\Bbb F}_n)$, let $D_l,D_j,D_3,D_h$ be the divisors corresponding to the edges $(0,1),\\
(1,0),(0,-1),(-1,n)$ respectively. We can assume the weights $s_i\geq 1 $, unless possibly $s_3=-n+s<1$, since $Bl_1{\Bbb F}_0\cong Bl_2(\pn{2})$ and $D_j\equiv -D_h$. But in this case $D_3\equiv D_l+nD_j+\sum_1^{r-s}D_i$, $L\cdot D_3\geq (n+1+r-s)k$ and
$$(kK_S+L)D_3\geq (n+r+1-s)k-(n-s+2)k=(r-1)k\geq 0$$\end{pf}
\small
|
1997-10-12T07:02:17 | 9710 | alg-geom/9710015 | en | https://arxiv.org/abs/alg-geom/9710015 | [
"alg-geom",
"math.AG"
] | alg-geom/9710015 | null | Zvezdelina E. Stankova-Frenkel | Moduli of Trigonal Curves | 69 pages, 34 figures, Latex2e | null | null | null | null | We study the moduli of trigonal curves. We establish the exact upper bound of
${36(g+1)}/(5g+1)$ for the slope of trigonal fibrations. Here, the slope of any
fibration $X\to B$ of stable curves with smooth general member is the ratio
$\delta_B/\lambda_B$ of the restrictions of the boundary class $\delta$ and the
Hodge class $\lambda$ on the moduli space $\bar{\mathfrak{M}}_g$ to the base
$B$. We associate to a trigonal family $X$ a canonical rank two vector bundle
$V$, and show that for Bogomolov-semistable $V$ the slope satisfies the
stronger inequality ${\delta_B}/{\lambda_B}\leq 7+{6}/{g}$. We further describe
the rational Picard group of the {trigonal} locus $\bar{\mathfrak T}_g$ in the
moduli space $\bar{\mathfrak{M}}_g$ of genus $g$ curves. In the even genus
case, we interpret the above Bogomolov semistability condition in terms of the
so-called Maroni divisor in $\bar{\mathfrak T}_g$.
| [
{
"version": "v1",
"created": "Sun, 12 Oct 1997 05:02:10 GMT"
}
] | 2007-05-23T00:00:00 | [
[
"Stankova-Frenkel",
"Zvezdelina E.",
""
]
] | alg-geom | \section*{\hspace*{1.9mm}1. Introduction}
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\label{introduction}
In this paper $\overline{\mathfrak M}_g$ denotes the
Deligne-Mumford compactification of the moduli space
of smooth curves over $\mathbb{C}$ of genus $g\geq 2$.
The boundary locus $\Delta$ of
$\overline{\mathfrak M}_g$ consists of nodal curves with finite automorphism
groups, which together with the smooth curves are referred to as {\it
stable} curves. The locus of hyperelliptic curves will be denoted by
$\overline{\mathfrak{I}}_g$, and the closure of the locus of trigonal curves
will be denoted by $\overline{\mathfrak{T}}_g$.
\medskip The main objects of our study will be families
of genus $g$ stable curves, whose general members
are smooth. Associated to any such {\it flat and proper}
family $f\!:\!X\!\rightarrow\! B$
are three basic invariants $\lambda|_B$, $\delta|_B$ and $\kappa|_B$.
We define these in Section~\ref{definition}
as divisors on $B$, but
for most purposes one can think of them as integers by considering their
respective degrees. The invariant $\delta|_B$ counts, with appropriate
multiplicities, the number of singular fibers of $X$.
The self-intersection of the relative dualizing sheaf $\omega_f$ on
$X$ defines $\kappa|_B$, and its pushforward to $B$ is a rank $g$
locally free sheaf, whose determinant is $\lambda|_B$.
\smallskip
The basic relation $12\lambda|_B=\delta|_B+\kappa|_B$ and the positivity of
the three invariants for non-isotrivial families force the {\it slope}
$\displaystyle{\frac{\delta|_B}{\lambda|_B}}$ of $X/\!_{\displaystyle{B}}$
to fall into the interval $[0,12)$ (cf.~Sect.~\ref{slope-non-isotrivial}).
In fact, Cornalba-Harris and Xiao establish for this slope
an exact upper bound of $8+4/g$, which is achieved only for certain
hyperelliptic families (cf.~Theorem~\ref{CHX}).
However, if the base curve $B$ passes through
a {\it general}
point of $\overline{\mathfrak{M}}_g$, Mumford-Harris-Eisenbud
give the better bound of $6+\on{o}(1/g)$ (cf.~Theorem~\ref{generalbound1}).
The families violating this inequality are
entirely contained in the closure $\overline{\mathcal{D}}_k$
of the locus of $k$-sheeted covers of ${\mathbf P}^1$, for
a suitably chosen $k$. In particular, for $k=2$ we recover the
{hyperelliptic} locus $\overline{\mathfrak{I}}_g$,
for $k=3$ - the {trigonal} locus $\overline{\mathfrak{T}}_g$, etc.
Therefore, higher than the above
``generic'' ratio can be obtained only for families with special
linear series, such as $g^1_2$, $g^1_3$, etc.
These observations clearly raise the following
\medskip
\noindent{\bf Question.} {\it According to the possession of special
linear series, is there a stratification of $\overline{\mathfrak M}_g$
which would give successively smaller slopes $\delta/\lambda$? What
would be the successive upper bounds with respect to such a stratification?}
\smallskip
The following result, whose proof will be given in the paper,
answers this question for an exact upper bound for
families with linear series $g^1_3$.
\medskip
\noindent{\bf Theorem I.}
{\it If $f\!:\!X{\rightarrow} B$ is a trigonal nonisotrivial family
with smooth general member, then the slope of $X/\!_{\displaystyle{B}}$
satisfies:
\begin{equation*}
\frac{\delta|_B}{\lambda|_B}\leq \frac{36(g+1)}{5g+1}\cdot
\end{equation*}
Equality is achieved if and only if all fibers are irreducible,
$X$ is a triple cover of a ruled surface $Y$ over $B$,
and a certain divisor class $\eta$ on $X$ is numerically zero.}
\smallskip
To understand the importance of this result and the above question,
consider Mumford's alternative description
of the basic invariants (cf.~ Sect.~\ref{linebundles}):
$\lambda|_B$, $\delta|_B$ and $\kappa|_B$ are restrictions of certain
rational divisor classes $\lambda, \delta, \kappa\in
\on{Pic}_{\mathbb{Q}}\overline{\mathfrak{M}}_g$. Specifically,
$\delta=\delta_0+\cdots+\delta_{[g/2]}$, where $\delta_i$ the class
of the boundary divisor $\Delta_i$ of $\overline{\mathfrak{M}}_g$,
and $\on{Pic}_{\mathbb{Q}}\overline{\mathfrak{M}}_g$
is freely generated by the Hodge class $\lambda$ and the boundary
classes $\delta_i$ for $g\geq 3$ (cf.~\cite{Ha2}).
Thus, our question about a stratification of $\overline{\mathfrak{M}}_g$
translates into a question about
the relations among the fundamental classes of various
subvarieties defined by geometric conditions in $\overline{\mathfrak{M}}_g$ .
Moreover, such a stratification would provide a link between
the {\it global} invariant $\lambda$ (the degree of the Hodge bundle on
$\overline{\mathfrak M}_g$) and the {\it locally defined} invariant $\delta$ of
the singularities of our families. In the process of estimating the
ratio $\delta / \lambda$ we hope to understand the geometry of interesting
loci in $\overline{\mathfrak M}_g$, and describe their rational Picard groups.
\smallskip
Such a program for the hyperelliptic locus $\overline{\mathfrak{I}}_g$
is completed by Cornalba-Harris (cf.~Theorem~\ref{theoremCHPic}),
who exhibit generators and relations
for $\on{Pic}_{\mathbb{Q}}{\overline{\mathfrak{I}}_g}$.
The typical examples of families with maximal ratio of $8+{4}/{g}$ are
constructed as blow-ups of pencils of hyperelliptic curves,
embedded in the same ruled surface.
\smallskip
Similar examples for
{trigonal families} yield the slope $7+{6}/{g}$, but as Theorem I
shows, this ratio is {\it not} an upper bound. This happens because of
an ``extra'' {\it Maroni} locus in $\overline{\mathfrak{T}}_g$ (cf.~
Sect.~12).
While a general trigonal curve embeds in ${\mathbf F}_0={\mathbf P}^1\times
{\mathbf P}^1$ or in the blow-up ${\mathbf F}_1$
of ${{\mathbf P}^2}$ at a point, the remaining
trigonal curves embed in other rational
ruled surfaces and comprise a closed subset in
$\overline{\mathfrak{T}}_g$, called the Maroni locus. The proof of
Theorem II, stated below, implies that
all trigonal families achieving the maximal bound lie entirely in the
Maroni locus, and moreover, their members
are embedded in ruled surfaces ``as far as possible
from the generic'' ruled surfaces ${\mathbf F}_0$ and ${\mathbf F}_1$.
\smallskip
The ratio $7+{6}/{g}$, though not the ``correct'' maximum,
plays a significant role in understanding the geometry of the trigonal
locus, and in describing its rational Picard group. In particular, in
a linear relation is established between the Hodge class,
the boundary classes on $\overline{\mathfrak{T}}_g$, and
a canonically defined vector bundle $V$ of rank 2 on a ruled surface
$\widehat{Y}$ (cf.~Sect.~9):
\medskip
\noindent{\bf Theorem II.}
{\it Let $\delta_0$ denote the boundary class in $\overline
{\mathfrak{T}}_g$ corresponding to irreducible singular curves, and let
$\delta_{k,i}$ be the remaining boundary classes.
For any trigonal non-isotrivial family with general smooth member, we have
\begin{equation*}
(7g+6)\lambda|_B=g\delta_0|_B+\sum_{k,i} \widetilde{c}_{k,i}\delta_{k,i}|_B
+\frac{g-3}{2}(4c_2(V)-c_1^2(V)),
\end{equation*}
where $\widetilde{c}_{k,i}$
is a quadratic polynomial in $i$ with linear coefficients
in $g$, and it is determined by the geometry of $\delta_{k,i}$.}
\smallskip
For example, $\widetilde{c}_{1,i}=3(i+2)(g-i)/2$
corresponds to the boundary divisor $\Delta{\mathfrak{T}}_{1,i}$,
whose general member is the join in three points of two trigonal curves
of genera $i$ and $g-i-2$, respectively (cf.~Fig.~\ref{Delta-k,i}).
\medskip
Recall that the vector bundle $V$ is called {\it Bogomolov semistable}
if its Chern classes satisfy $4c_2(V)\geq c_1^2(V)$ (cf.~\cite{Bo,Re}).
We show in Section~9 the following
\medskip
\noindent{\bf Theorem III.}
{\it For any trigonal nonisotrivial family $X\rightarrow B$
with general smooth member, if $V$ is Bogomolov semistable, then
\begin{equation*}
\frac{\delta|_B}{\lambda|_B}\leq 7+\frac{6}{g}\cdot
\end{equation*}}
\medskip
In the even genus case, the Maroni locus is in fact a divisor on
$\overline{\mathfrak{T}}_g$, whose class we denote by $\mu$.
We further recognize the ``Bogomolov quantity''
$4c_2(V)-c_1^2(V)$ as counting roughly four times the number of
Maroni fibers in $X$, and deduce
\medskip
\noindent{\bf Theorem IV.}
{\it For even $g$, $\on{Pic}_{\mathbb{Q}}\overline{\mathfrak{T}}_g$ is
freely generated by all boundary divisors $\delta_0$ and $\delta_{k,i}$, and
the Maroni divisor $\mu$. The class of the Hodge bundle on
$\overline{\mathfrak{T}}_g$ is expressed in terms of these generators as
the following linear combination:
\begin{equation*}
(7g+6)\lambda|_{\overline{\mathfrak{T}}_g}=g\delta_0+
\sum_{k,i}\widehat{c}_{k,j}\delta_{k,i}+2(g-3){\mu}.
\end{equation*}}
Consequently, the condition $\eta\equiv 0$ in Theorem I can be interpreted
as a relation among the number of irreducible singular curves and the
``Maroni'' fibers in our family: $(g+2)\delta_0|_B=-72(g+1)\mu|_B$, and hence
maximal slope families are entirely contained in the Maroni locus of
$\overline{\mathfrak{T}}_g$ (cf.~Theorem~\ref{maximalmaroni}).
The stated theorems complete the program for the trigonal locus
$\overline{\mathfrak{T}}_g$, which was outlined earlier in this section.
\smallskip
An important interpretation of these results can be traced
back to \cite{MHE}, where it is shown that the moduli space
$\mathfrak{M}_g$ is of {\it general type}. The $k$-gonal locus
$\overline{\mathcal{D}}_k$ is realized in terms of the generating classes as:
$[\overline{\mathcal{D}}_k]=a\lambda-b\delta-\mathcal{E}$ for some $a,b>0$,
and an effective boundary combination $\mathcal{E}$.
Restricting to a general curve $B\subset \overline{\mathfrak{M}}_g$, we have
$\overline{\mathcal{D}}_k|_B> 0$, and hence
$a\lambda|_B-b\delta|_B>0$. Because of
Seshadri's criterion for ampleness of line bundles,
in effect, we are asking for all positive numbers $a$ and $b$ such that
the linear combination $a\lambda-b\delta$ is ample on $\overline{\mathfrak{M}}_g$.
The smaller the ratio $a/b$ is, the stronger result we obtain.
In other words, we are aiming at a maximal bound of $\delta /\lambda$,
when we think of these classes as restricted to any curve $B\subset
\overline{\mathfrak{M}}_g$.
In view of this, part of this paper can be described as looking for all {\it
ample} divisors on $\overline{\mathfrak{T}}_g$ of the form
$a\lambda-b\delta$ with $a,b>0$. Theorem I then gives the necessary
condition $(5g+1)a\geq 36(g+1)b$ (compare with \cite{M2,MHE,CH}).
Some of the results can be applied to the study of
the discriminant loci of a certain type of triple
covers of surfaces.
\smallskip
The methods and ideas for the trigonal case are
in principal extendable to more general families of $k$-gonal curves.
We refer the reader to Sect.~13
for a general maximal bound for tetragonal curves (for
$g$ odd), and conjectures for the maximal
and general bounds for any $d$-gonal and other families of stable curves.
\bigskip
\begin{center}{\sc Acknowledgments}\end{center}
\medskip
This paper is based on my Ph.D. thesis at Harvard University.
Joe Harris, my advisor, introduced me to the problem of finding
a stratification of the moduli space with respect to a descending sequence
of slopes of one--parameter families. I am very grateful to him for
his advice and support throughout my work on the present thesis.
I would like to thank Fedor Bogomolov,
David Eisenbud, Benedict Gross, Brendan Hassett,
David Mumford, Tony Pantev and Emma Previato for the helpful discussions
that I have had with them at different stages of the project, as well as
Kazuhiro Konno for providing me with his recent results on trigonal
families. A source of inspiration and endless moral support has been my
husband, Edward Frenkel, to whom goes my gratitude and love.
\bigskip
\section*{\hspace*{1.9mm}2. Preliminaries}
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\subsection{Definition of $\lambda|_B,\,\,\delta|_B$ and
$\kappa|_B$ in Pic$B$}
\label{definition} Let $f:X\rightarrow B$ be a flat proper
one-parameter family of stable curves of genus $g$, where
$B$ is a smooth projective curve. Assume in addition that the
general member of $X$ is {\it smooth} (cf.~Fig.~\ref{family}).
\medskip
\begin{figure}[h]
$$\psdraw{family}{1.5in}{1.5in}$$
\hspace*{3mm}
\caption{Trigonal family $f\!:\!X\!\rightarrow\! B$}
\label{family}
\end{figure}
Let $\omega_f=\omega_X\otimes f^*\omega_B^{-1}$ be the relative dualizing
sheaf of $f$. Its pushforward $f_*(\omega_f)$ is a locally free sheaf
on $B$ of rank $g$, and we set
\[\lambda|_B=\lambda_X:=\wedge^gf_*(\omega_f)\in \on{Pic}B\]
to be its determinant. The sheaf $f_*(\omega_f)$ is known as the ``Hodge
bundle'' on $B$, and $\lambda|_B$ - as the ``Hodge class'' of $B$.
In a similar way, we set $\kappa|_B$ to be the self-intersection of $\omega_f$:
\[\kappa|_B=\kappa_X:=f_*(c_1^2(\omega_f))\in\on{Pic}B.\]
\bigskip
The definition of $\delta|_B$, on the other hand, is local and requires
some notation. Let $q$ be any singular point of a fiber $X_b$,
$b\in B$. Since the general fiber of $X$ is smooth, the total space
of $X$ near $q$ is given locally analytically by $xy=t^{m_q}$,
where $x$ and $y$ are local parameters on $X_b$, $t$ is a local parameter
on $B$ near $b$, and $m_q\geq 1$.
(This follows from the one-dimensional versal
deformation space of the nodal singularity at $q$.) For any other
point $q$ of $X$ we set $m_q=0$.
Now we can define
\[\delta|_B=\delta_X:=f_*(\sum_{q\in X}m_qq)\in \on{Pic}B.\]
By abuse of notation, we shall use the same letters for the line bundles
$\lambda|_B, \,\,\kappa|_B$ and $\delta|_B$ and for their respective degrees,
e.g. $\lambda|_B=\on{deg}\lambda|_B$.
\medskip\noindent{\bf Remark 2.1.} It is possible to define the three basic
invariants for a wider variety of families. In particular, dropping the
assumption of smoothness of the general fiber roughly means that the base
curve $B$ is contained entirely in the boundary locus of
$\overline{\mathfrak M}_g$. Since such families are not discussed in
our paper, we shall not give here these definitions. The existence,
however, of such invariants for any one-parameter family of stable curves
will follow from the description of $\lambda,\,\,\delta$ and $\kappa$ as
``global'' classes in $\on{Pic}_{\mathbb Q}\overline{\mathfrak M}_g$ (cf.
Sect.~\ref{linebundles}).
\medskip\noindent{\bf Remark 2.2.} It is also possible to consider families
whose special members are not stable, e.g. cuspidal, tacnodal and other
types of singular curves. One reduces to the above cases by applying
{\it semistable reduction} (cf.~\cite{FM}).
\subsection{The line bundles $\lambda,\,\,\delta$ and $\kappa$ in
Pic$_{\mathbb Q}\overline{\mathfrak M}_g$}
\label{linebundles} Another way to
interpret the classes $\lambda|_B,\,\,\delta|_B$ and $\kappa|_B$ is to
think of them as rational divisor classes on $\overline{\mathfrak{M}}_g$. In fact,
Mumford shows that such invariants, defined for {\it
any} proper flat family $X\rightarrow S$ and natural under base change,
induce line bundles in Pic$_{\mathbb Q}\overline{\mathfrak M}_g$. Here follows
a rough sketch of the argument (cf.~\cite{M2}).
\smallskip
Consider $\on{Hilb}^{p(x)}_r$, the Hilbert scheme
parametrizing all closed subschemes of ${\mathbf P}^r$ with Hilbert polynomial
$p(x)=dx-g+1$ for some $d=2n(g-1)\gg 0$ and $r=d-g$. Let
$\mathcal H\subset \on{Hilb}^{p(x)}_r$ be the locally closed smooth subscheme
of $n$-canonical stable curves of genus $g$. Then $\overline{\mathfrak M}_g$ is
the GIT-quotient of $\mathcal H$ by ${\on{PGL}}_r$. Let
\[\rho:\mathcal H\rightarrow \overline{\mathfrak M}_g=\mathcal H/{\on{PGL}}_r\]
be the natural surjection, and let $(\on{Pic}\mathcal H)^{{\on{PGL}}_r}$ be
the subgroup of isomorphism classes of line bundles on $\mathcal H$ invariant
under the action of ${\on{PGL}}_r$.
Consider also $\on {Pic}_{\on{fun}}\overline{\mathfrak M}_g$,
the group of line bundles on the {\it moduli functor}. An element $L$ of
$\on {Pic}_{\on{fun}}\overline{\mathfrak M}_g$ consists of the following
data: for any proper flat family $f:X\rightarrow S$ of stable curves
a line bundle $L_S$ on $S$ natural under base change. Two such elements
are declared isomorphic under the obvious compatibility conditions.
Naturally, a line bundle on $\overline{\mathfrak M}_g$ gives rise by
pull-back to a line bundle on the moduli functor. In fact,
this map is an inclusion with a torsion cokernel, and
$\on{Pic}_{\on{fun}}\overline{\mathfrak M}_g$ is torsion free and isomorphic
to $(\on{Pic}\mathcal H)^{{\on{PGL}}_r}$:
\[\on{Pic}\overline{\mathfrak M}_g\stackrel{\rho^*}{\hookrightarrow}
\on{Pic}_{\on{fun}}\overline{\mathfrak M}_g\cong (\on{Pic}\mathcal H)^{{\on{PGL}}_r}.\]
Hence we may regard all these groups as sublattices of
$\on{Pic}_{\mathbb Q}\overline{\mathfrak M}_g$. In particular,
\[\on{Pic}_{\on{fun}}\overline{\mathfrak M}_g\otimes {\mathbb Q}\cong
\on{Pic}_{\mathbb Q}\overline{\mathfrak M}_g,\]
and any line bundle on the moduli functor can be thought of as a rational
class on $\overline{\mathfrak M}_g$. These identifications allow us to make
the following
\medskip\noindent{\bf Definition 2.1.}
In $\on{Pic}_{\mathbb Q} \overline{\mathfrak M}_g$ we define the line bundles
$\lambda,\,\,\kappa$ and $\delta$ by
\[\lambda=\det\pi_*(\omega_{\mathcal {C}/\mathcal H}),\,\, \kappa=\pi_*c_1(\omega
_{\mathcal {C}/\mathcal H})^2,\,\,
\delta=\mathcal{O}_{\mathcal H}(\Delta\mathcal H),\]
where ${\mathcal{C}}\subset \mathcal H\times {\mathbb P}^r$ is the universal curve
over $\mathcal H$, $\pi:\mathcal C \rightarrow \mathcal H$ is the projection map,
$\omega_{\mathcal{C}/\mathcal H}$ is the relative dualizing sheaf of $\pi$, and
$\Delta\mathcal H\subset \mathcal H$ is the divisor of singular curves on $\mathcal H$.
\medskip
As defined, $\lambda,\kappa$ and $\delta$ lie in
$\on{Pic}_{\mathbb Q}\overline{\mathfrak M}_g$, and as such they are only
{\it rational} Cartier divisors on $\overline{\mathfrak M}_g$. In
\cite{MHE} one can find examples where $\lambda$ does {\it not}
descend to a line bundle on $\overline{\mathfrak M}_g$.
On the other hand, it is obvious from which divisor on $\overline{\mathfrak M}_g$
our $\delta$ comes: $\delta=\mathcal O_{\overline{\mathfrak M}_g}(\Delta)$, where
$\Delta$ denotes the divisor on
$\overline{\mathfrak M}_g$ comprised of all singular stable curves.
Again, due to singularities of the total space of $\overline{\mathfrak
M}_g$, $\Delta$ is only a {\it rational} Cartier divisor.
In fact, the only locus of $\overline{\mathfrak{M}}_g$ on which
$\lambda,\,\,\delta$ and $\kappa$ are necessarily {\it integer} divisor classes
is $(\overline{\mathfrak{M}}_g)_0$ - the locus of automorphism-free curves.
\smallskip
We can further define the {\it boundary} classes $\delta_0,
\delta_1,...,\delta_{[\frac{g}{2}]}$ in $\on{Pic} _{\mathbb Q}\overline
{\mathfrak M}_g$. Let $\Delta_i$ be the $\mathbb Q$--Cartier divisor on $\overline
{\mathfrak M}_g$ whose general member is an irreducible nodal curve with
a single node (if $i=0$), or the join of two irreducible
smooth curves of genera $i$ and $g-i$ intersecting
transversally in one point (if $i>0$). Setting $\delta_i={\mathcal O}_{\overline
{\mathfrak M}_g}(\Delta_i)$, we have $\Delta=\sum_i\Delta_i$ and
$\delta=\sum_i\delta_i$.
\smallskip As the following result of Harer \cite{Ha1,Ha2} suggests that,
in order to describe the geometry of the moduli space
$\overline{\mathfrak{M}}_g$, it will be useful to study
the divisor classes defined above, and to understand the relations
between them.
\begin{thm}[Harer] The Hodge class $\lambda$ and
the boundary classes $\delta_0,\delta_1,...,\delta_{[\frac{g}{2}]}$ generate
$\on{Pic}_{\mathbb Q} \overline{\mathfrak M}_g$, and for $g\geq 3$ they are
linearly independent.
\end{thm}
It is easy to recognize the restrictions of $\lambda,\,\,\delta$
and $\kappa$ to a curve $B$ in $\overline{\mathfrak{M}}_g$ as the previously
defined $\lambda|_B,\,\,\delta|_B$ and $\kappa|_B$. For example,
the restriction of $\delta$ to the base curve $B$ counts,
with appropriate multiplicities, the number of intersections of
$B$ with the boundary components $\Delta_i$ of $\overline{\mathfrak{M}}_g$.
As a final remark, applying Grothendieck Riemann-Roch Theorem (GRR) to the map
$\pi:\mathcal C \rightarrow \mathcal H$ and the sheaf $\omega_{\mathcal{C}/\mathcal H}$,
implies the basic relation:
\begin{equation}
12\lambda=\kappa+\delta.
\label{GRR}
\end{equation}
\subsection{Slope of non-isotrivial families}
\label{slope-non-isotrivial} Let $f:X\rightarrow B$
be our family of stable curves with a smooth general member. By definition,
$\delta_B\geq 0$. Further, all
locally free quotients of the Hodge bundle $f_*(\omega_f)$
have non-negative degrees \cite{key13}. If $X$ is a
non-isotrivial family, then $\lambda|_B>0$, and since
the relative canonical divisor $K_{X/B}$ is nef,
$\kappa|_B>0$ \cite{key32}. In particular, we can divide by $\lambda|_B$.
\medskip
\noindent{\bf Definition 2.2.} The {\it slope} of a non-isotrivial family
$f:X\rightarrow B$ of stable curves with a smooth general member
is the ratio
\[\on{slope}(X/\!_{\displaystyle{B}}):=\frac{\delta|_B}{\lambda|_B}
\cdot\]
Suppose we make a base change $B_1\rightarrow B$ of degree $d$,
and set $X_1=X\times_{B}B_1$ to be the pull-back of
our family over the new base $B_1$ (cf.~Fig.~\ref{basechange}).
Then the three invariants on $B$ will
pull-back to the corresponding invariants on $B_1$, and
their degrees will be multipied by $d$,
e.g. $\lambda|_{B_1}=d\lambda|_B$, etc. In particular, the
slope of $X/_{\displaystyle{B}}$ will be preserved.
\setlength{\unitlength}{10mm}
\begin{figure}[t]
\begin{picture}(1.8,1.8)(-0.2,3.9)
\put(0,4){$B_1\stackrel{d}{\longrightarrow} B$}
\put(0,5.1){$X_1\,\,{\longrightarrow}\,X$}
\multiput(0.2,5)(1.25,0){2}{\vector(0,-1){0.6}}
\end{picture}
\hspace*{3mm}
\vspace*{-1mm}
\caption{Base change}
\label{basechange}
\end{figure}
\smallskip
In view of (\ref{GRR}), restricting to the base curve $B$ yields
\begin{equation}
12\lambda|_B=\kappa|_B+\delta|_B.
\end{equation}
From the positivity conditions above, we deduce that
$0\leq \on{slope}(X/\!_{\displaystyle{B}})<12.$
\subsection{Statement of the problem and what is known}
\label{statement}
It is natural to ask whether we can find a better estimate for the slope of
$X$. The first fundamental result in this direction is the following
\begin{thm}[Cornalba-Harris, Xiao] Let $f:X\rightarrow B$ be a nonisotrivial
family with smooth general member. Then the slope of the family satisfies:
\begin{equation}
\frac{\delta|_B}{\lambda|_B}\leq 8+\frac{4}{g}\cdot
\label{8+4/g}
\end{equation}
Equality holds if and only if the general fiber of $f$ is hyperelliptic,
and all singular fibers are irreducible.
\label{CHX}
\end{thm}
Note that the upper bound is achieved only for hyperelliptic families.
Such families are of very special type since
the hyperelliptic locus $\overline{\mathfrak I}_g$ has codimension $g-2$ in
$\overline{\mathfrak M}_g$.
On the other hand, if the base curve $B$ is
general enough, a much better estimate can be shown (cf.~\cite{MHE}):
\begin{thm}[Mumford-Harris-Eisenbud] If $B$ passes through a general
point $[C]\in \overline{\mathfrak{M}}_g$, then
\vspace*{-4mm}
\begin{equation}
\frac{\delta|_B}{\lambda|_B}\leq 6+\on{o}(\frac{1}{g})\cdot
\label{6+o(1/g)}
\end{equation}
\label{generalbound1}
\end{thm}
For example, when $g$ is odd, we can set $k=(g+1)/2$ and
define the divisor $\overline{\mathcal{D}}_k$ in ${\overline{\mathfrak M}}_g$ as
the closure of the $k$-sheeted covers of ${\mathbf P}^1$:
\[\overline{\mathcal{D}}_k=\overline{\{C\in{\mathfrak{M}}_g\,\,|\,\,C\,\,
\on{has}\,\,{g}^1_k\}}.\]
\begin{figure}[h]
$$\psdraw{general}{1.5in}{1.3in}$$
\caption{General curve $B\not \subset \overline{\mathcal{D}}_k$}
\label{generalcurve}
\end{figure}
If our family is not entirely contained in $\overline{\mathcal{D}}_k$, or
equivalently, if $B$ passes through a point $[C]\not\in\overline{\mathcal{D}}_k$
(cf.~Fig.~\ref{generalcurve}),
\begin{equation}
\frac{\delta|_B}{\lambda|_B}\leq 6+\frac{12}{g+1}\cdot
\label{6+12/(g+1)}
\end{equation}
\smallskip\noindent
Higher than the ``generic'' ratio can be obtained, therefore, only
for a very special type of families: those entirely contained in
$\overline{\mathcal{D}}_k$, and hence possessing ${g}^1_2$, ${g}^1_3$, etc.
\subsubsection{The rational Picard group of the hyperelliptic locus
$\overline{\mathfrak{I}}_g$}
\label{rationalhyper}
In proving the maximal bound $8+4/g$, Cornalba-Harris also
describe $\on{Pic}_{\mathbb Q}\overline{\mathfrak I}_g$
by exhibiting generators and relations (cf.~\cite{CH}).
Here we briefly discuss their result.
\smallskip
Recall the irreducible divisors $\Delta_i$ on $\overline{\mathfrak M}_g$.
For $i=1,...,[g/2]$, $\Delta_i$ cuts out an irreducible divisor
on $\overline{\mathfrak I}_g$, while the intersection
$\Delta_0\cap\overline{\mathfrak I}_g$ breaks
up into several components:
\[\Delta_0\cap\overline{\mathfrak I}_g=\Xi_0\cap\Xi_1\cap\cdots
\cap\Xi_{[\frac{g-1}{2}]}.\]
Set $\xi_i=\mathcal O_{\overline{\mathfrak I}_g}(\Xi_i)$ for the class of $\Xi_i$ in
$\overline{\mathfrak I}_g$, and retain the symbols $\lambda$ and $\delta_i$
for their corresponding restrictions to
$\on{Pic}_{\mathbb Q}\overline{\mathfrak I}_g$. Thus,
$\delta_i:=\mathcal O_{\overline{\mathfrak I}_g}(\Delta_i\cap\overline{\mathfrak I}_g)$
for all $i$. Finally note that the class $\delta_0$ is realised in
$\on{Pic}_{\mathbb Q}\overline{\mathfrak I}_g$ as the sum
\[\delta_0=\xi_0+2\xi_1+\cdots+2\xi_{[\frac{g-1}{2}]}.\]
The coefficient $2$ roughly means that
$\Delta_0$ is {\it double} along $\Xi_i$, for $i>0$.
\begin{thm}[Cornalba-Harris] The classes
$\xi_0,\cdots,\xi_{[\frac{g-1}{2}]}$ and $\delta_1,\cdots,\delta_{[\frac{g}
{2}]}$ freely generate
$\on{Pic}_{\mathbb Q}{\overline{\mathfrak I}_g}$. The Hodge class
$\lambda\in \on{Pic}_{\mathbb Q}{\overline{\mathfrak I}_g}$ is expressed
in terms of these generators as the following linear combination:
\begin{equation}
(8g+4)\lambda=g\xi_0+\sum_{i=1}^{[(g-1)/2]}2(i+1)(g-i)\xi_i
+\sum_{j=1}^{[g/2]}4j(g-j)\delta_j.
\label{CHPic}
\end{equation}
\label{theoremCHPic}
\end{thm}
For a specific family $f:X\rightarrow B$ of hyperelliptic
stable curves this relation reads:
\[(8g+4)\lambda|_B=g\xi_{0}|_B+\sum_{i=1}^{[(g-1)/2]}2(i+1)(g-i)\xi_{i}|_B
+\sum_{j=1}^{[g/2]}4j(g-j)\delta_{j}|_B\]
\[\Rightarrow\,\, (8+4/g)\lambda|_B\geq \xi_{0}|_B+\sum_i2\xi_{i}|_B+
\sum_j2\delta_{j}|_B=\delta|_B.\]
This yields the desired $8+4/g$
inequality for the slope of a hyperelliptic family,
and shows that the maximum can be obtained exactly when all
$\xi_1,\cdots,\xi_{[\frac{g-1}{2}]},\delta_1,\cdots,\delta_{[\frac{g}{2}]}$
vanish on $B$. In other words, the singular fibers of $X$ belong only to
the boundary divisor $\Xi_0$, and hence are irreducible. In Appendix
we review the description of the divisors $\Xi_i$ via admissible covers, and
give an alternative proof of Theorem~\ref{theoremCHPic}.
\subsubsection{Example of a hyperelliptic family with maximal slope}
\label{example} We present here a typical example in which the
upper bound $8+4/g$ is achieved, and show how to
calculate explicitly the basic invariants $\lambda|_B$ and $\delta|_B$
for this family.
\medskip
\noindent{\bf Example 2.1.} Consider a pencil $\mathcal{P}$ of hyperelliptic
curves of genus $g$
on ${\mathbf P}^1\!\times \!{\mathbf P}^1$. Because of genus considerations,
its members must be of type $(2,g+1)$.
Our family \newline
$f\!:\!X\!\rightarrow \!{\mathbf P}^1$ will be obtained by blowing-up
${\mathbf P}^1\!\times\! {\mathbf P}^1$ at the $4(g+1)$ base points of the pencil
in order to separate its members (cf.~Fig.~\ref{ratio8+4/g}). Hence,
$\chi(X)=\chi({\mathbf P}^1\!\times\! {\mathbf P}^1)+4(g+1)$ for the
corresponding topological Euler characteristics. Riemann-Hurwitz formula for
the map $f$ gives a second relation:
$\chi(X)=\chi({\mathbf P}^1)\chi(X_b)+\delta|_B$, where ${\mathbf P}^1$
is the base $B$ and $X_b$ is the general
fiber of $X$. Putting together, $\delta|_B=8g+4.$
\begin{figure}[t]
\begin{picture}(1,3)(0,3.1)
\put(-0.7,5.5){$X\,\,\hookrightarrow\,\,{\mathbf P}^1\!
\!\times\! {\mathbf P}^1\!\!\times\!{\mathbf P}^1$}
\put(-0.5,5.4){\vector(1,-1){1}}
\put(1.7,5.4){\vector(-1,-1){1}}
\put(0,3.9){${\mathbf P}^1\!\!\times\! {\mathbf P}^1$}
\put(0.4,2.7){${\mathbf P}^1$}
\put(0.6,3.8){\vector(0,-1){0.6}}
\end{picture}
\caption{Ratio $8+4/g$}
\label{ratio8+4/g}
\end{figure}
The total space
of $X$ is a divisor on ${\mathbf P}^1\!\!\times\! {\mathbf P}^1\!\!\times\!
{\mathbf P}^1$ of type
$(2,g+1,1)$, and the map $f\!:\!X\!\rightarrow \!{\mathbf P}^1$ is the
restriction to $X$ of the third projection $\pi_3\!:\!{\mathbf P}^1\!\!\times\!
{\mathbf P}^1\!\!\times\!{\mathbf P^1}\!\rightarrow \!{\mathbf P}^1$.
Using standard methods,
we compute $h^0( (f_*(\omega_f))(-2))=0$. From the positivity of all free
quotients of the Hodge bundle on ${\mathbf P}^1$, $f_*(\omega_f)$
splits as a direct sum $\bigoplus_{i=1}^g
{\mathcal O}_{{\mathbf P}^1}(a_i)$ for some $a_i>0$.
Then, for $f_*(\omega_f)(-2)
=\bigoplus_i\mathcal{O}_{\mathbf P^1}(a_i-2)$
to have no sections, all $a_i$'s must be at most $1$. Finally,
\begin{equation*}
f_*(\omega_f)=\bigoplus_{i=1}^g{\mathcal O}_{{\mathbf P}^1}(+1),\,\,\lambda|_B=g,
\,\,\on{and}\,\,\frac{\delta|_B}{\lambda|_B}=8+\frac{4}{g}
\cdot
\end{equation*}
\subsubsection{The Trigonal Locus $\overline{\mathfrak{T}}_g$}
\label{trigonallocus} In a similar vein
as in the above example, we consider pencils of trigonal curves on ruled
surfaces, and obtain the slope $7+6/g$. It is somewhat reasonable to expect
that this is the maximal ratio. Recall that a bundle $\mathcal{E}$ on a curve
$B$ is {\it semistable} if for any proper subbundle $\mathcal{F}$, we have
$q(\mathcal{F})\leq q(\mathcal{E})$, where $q$ is
the quotient of the degree and the rank
of the corresponding bundle. Following Xiao's approach in the proof of
Theorem~\ref{CHX}, Konno shows
that for non-hyperelliptic fibrations of genus $g$ with semistable
Hodge bundle $f_*\omega_{f}$ (cf.~\cite{HN,key5}):
\begin{equation}
\frac{\delta|_B}{\lambda|_B}\leq 7+\frac{6}{g}\cdot
\label{7+6/g}
\end{equation}
As for any trigonal families, he establishes the inequality (cf.~\cite{key6}):
\begin{equation}
\frac{\delta|_B}{\lambda|_B}\leq
\frac{22g+26}{3g+1}\sim 7\frac{1}{3}+\on{o}
(\frac{1}{g})\cdot
\end{equation}
Examples of trigonal families achieving this ratio were not found, which
suggested that this bound might be too big. On the other hand, in trying to
disprove the smaller bound $7+6/g$, we naturally arrived at counterexamples
pointing to a third intermediate ratio (cf.~Theorem~\ref{maximal bound2}):
\begin{equation}
\frac{36(g+1)}{5g+1}\sim 7\frac{1}{5}+\on{o}
(\frac{1}{g})\cdot
\end{equation}
The difference between the last two estimates may seem negligible, but this
would not be so when both $\lambda|_B$ and $\delta|_B$ become large and
we attempt to bound $\lambda|_B$ from below by $\delta|_B$. What is more
important, the second ratio is in fact {\it exact}, and we give
equivalent conditions for it to be achieved (cf.~Sect.~\ref{whenmaximal},
\ref{Maroni-maximal}). This maximal bound confirms Chen's result
for genus $g=4$ in \cite{Chen}.
\smallskip
The reader may ask why the ``generic'' examples for the maximum in the
hyperelliptic case fail to provide also the maximum in the trigonal
case. As we noted in the Introduction, the answer is closely related to
the so-called {\it Maroni} locus in $\overline{\mathfrak{T}}_g$.
More precisely, if ${\mathbf F}_k={\mathbf P}({\mathcal O}_{{\mathbf P}^1}\oplus
{\mathcal O}_{{\mathbf P}^1}(k))$ denotes the corresponding rational ruled surface,
a general curve $C$ embeds in ${\mathbf F}_0$ is
$g$ is even, and in ${\mathbf F}_1$ if $g$ is odd. The Maroni locus consists
of those curves that embed in ${\mathbf F}_k$ with $k\geq 2$. The number
$k/2$ is referred to as the {\it Maroni invariant} of $C$.
In these terms, the examples of pencils of trigonal curves on ${\mathbf F}_0$
and ${\mathbf F}_1$ have the lowest possible constant Maroni invariant,
and we shall see that the maximum bound can be obtained only for families
entirely contained in the Maroni locus, and having very high Maroni
invariants.
\medskip
The ``semistable'' bound $7+6/g$ appears in Theorem~\ref{7+6/g Bogomolov2},
where we give instead a sufficient {\it Bogomolov-semistability} condition
$4c_2(V)-c_1^2(V)\geq 0$ for a canonically associated to $X$ vector bundle
$V$ of rank 2. The rational Picard group of $\overline{\mathfrak{T}}_g$
is described in terms of generators and relations in
Section~\ref{generators}, providing thus in the trigonal case
an analog of Theorem~\ref{theoremCHPic}. Note the apparent similarity
of the coefficients $\widetilde{c}_{k,i}$ of the trigonal boundary classes
and the coefficients of the hyperelliptic boundary classes.
This is not coincidental. In fact, the $\widetilde{c}_{k,i}$'s are
in a sense the ``smallest'' coefficients that could have been
associated to the corresponding classes $\delta_{k,i}$
(cf.~Fig.~\ref{Delta-k,i}): they are symmetric with respect
to the two genera of the components in the general member of $\delta_{k,i}$.
A crucial role in the proof of Theorem~\ref{Pic trigonal} is played
by the interpretation of the above Bogomolov semistability condition
in terms of the Maroni locus it $\overline{\mathfrak{T}}_g$ (cf.~Sect.~\ref
{interpretation}).
\medskip
\subsection{The idea of the proof}
\label{idea}
Let $f:X\rightarrow B$ be a family of stable curves, whose general member
$X_b$ is a smooth trigonal curve.
By definition, $X_b$ is a triple cover of ${\mathbf P}^1$. We would like
to study how this triple cover varies as $X_b$ moves in the family $X$.
Thus, it would be desirable to represent $X$, by analogy with
$X_b$, as a triple cover of a ruled
surface $Y$, comprised by the image lines ${\mathbf P}^1$. Unfortunately, due
to existence of hyperelliptic and other
special singular fibers, this is not always possible.
\setlength{\unitlength}{10mm}
\begin{figure}[h]
\begin{picture}(3,4.9)(-1,2)
\put(0,4){$\widetilde{X}\,\stackrel{\widetilde{\phi}}
{\longrightarrow}\, \widetilde{Y}$}
\put(0,5.1){$\widehat{X}\,\stackrel{\widehat{\phi}}{\longrightarrow}\,
\widehat{Y}$}
\multiput(0.2,3.85)(-1.3,-0.7){2}{\vector(1,-1){0.5}}
\put(1.4,3.85){\vector(-1,-1){0.5}}
\multiput(0.5,3)(-0.5,0.85){2}{\vector(-2,-1){0.8}}
\put(0.6,2.9){$\widetilde{B}$}
\put(-1.3,3.3){$X$}
\put(-0.7,2.3){$B$}
\multiput(0.1,5)(1.5,0){2}{\vector(0,-1){0.6}}
\put(1.6,6.2){\vector(0,-1){0.6}}
\put(1.35,6.3){${\mathbf P}V$}
\put(0.2,5.6){\vector(2,1){1.2}}
\end{picture}
\vspace*{-3mm}
\caption{Basic construction}
\label{Basic construction idea}
\end{figure}
\subsubsection{The basic construction} The ``closest''
model of such a triple cover can be obtained after a finite number of
birational transformations on $X$, and a possible base change over the base
$B$. This way we construct a {\it
quasi-admissible} cover $\widetilde{\phi}:{\widetilde{X}}
\rightarrow {\widetilde{Y}}$ over a new base
${\widetilde{B}}$ (cf.~Prop.~\ref{propquasi}). Here ${\widetilde{Y}}$ is a
{\it birationally} ruled surface over $\widetilde{B}$ with reduced, but non
necessarily irreducible, special fibers: $\widetilde{Y}$ allows for
{\it pointed stable} rational fibers, i.e. trees of ${\mathbf P}^1$'s with
points marked in a certain (stable) way.
The map $\widetilde{\phi}$ expresses any fiber
$\widetilde{X}_ b$
as a triple quasi-admissible cover of the corresponding {\it pointed
stable} rational curve $\widetilde{Y}_b$. To calculate
effectively our invariants $\lambda,\delta$ and $\kappa$, we need that
$\widetilde{\phi}$ be {\it flat},
which could force a few additional blow-ups on $\widetilde{X}$
and $\widetilde{Y}$. We end up with a flat proper triple
cover $\widehat{\phi}:
\widehat{X}\rightarrow \widehat{Y}$, where certain fibers of
$\widehat{X}$ and $\widehat{Y}$
are allowed to be {\it non-reduced}: these are the scheme-theoretic
preimages under the blow-ups on $\widetilde{X}$ and $\widetilde{Y}$.
We call such covers $\widehat{\phi}$ {\it effective}.
\smallskip\quad
We observe next that any smooth trigonal curve $C$ can
be naturally embedded in a ruled surface ${\mathbf F}_k$ over $B$. If
$\alpha:C\rightarrow {{\mathbf P}^1}$ is the corresponding triple cover, there
is an exact sequence of locally free sheaves on ${{\mathbf P}^1}$:
\begin{equation*}
0\rightarrow {V}\rightarrow {\alpha}_*{\mathcal O}_{C}\stackrel
{\on{tr}}{\rightarrow}{\mathcal O}_{{\mathbf P}^1}\rightarrow 0.
\end{equation*}
The projectivization $\mathbf P V$ of the rank 2 vector bundle $V$ is the
ruled surface ${\mathbf F}_k$.
\smallskip
This construction can be extended as $C$ moves in the effective cover
$\widehat{\phi}:\widehat{X}\rightarrow \widehat{Y}$. The flatness of
$\widehat{\phi}$ forces the pushforward ${\phi}_*{\mathcal O}_{\widehat{X}}$
to be a locally free sheaf of rank 3 on $\widehat{Y}$, and the finiteness
of $\widehat{\phi}$ ensures the existence of a {\it trace map}
$\on{tr}:{\phi}_*{\mathcal O}_{\widehat{X}}\rightarrow {\mathcal O}_{\widehat{Y}}$.
Again, the kernel $V$ of $\on{tr}$
is the desired rank 2 vector bundle on $\widehat{Y}$, in
whose projectivization, ${\mathbf P}V$, we embed $\widehat{X}$
(cf.~Fig.~\ref{Basic construction idea}).
\subsubsection{Chow Rings Calculations}
We can now use the relations in the Chow rings of ${\mathbb{A}}({\mathbf P}V)$,
$\mathbb{A}\widehat{Y}$ and $\mathbb{A}\widehat{X}$ to calculate the invariants
$\lambda_{\widehat{X}}$ and $\delta_{\widehat{X}}$,
appropriately defined for the new family
$\widehat{X}\rightarrow {\widetilde{B}}$ of semistable and occasionally
non-reduced fibers. Then, of course, we translate
$\lambda_{\widehat{X}}$ and $\delta_{\widehat{X}}$ into $\lambda_{{X}}$
and $\delta_{{X}}$ with the necessary adjustments from the
birational transformations on $X$ and the base change on $B$.
We compare the resulting expressions to obtain a relation among
$\lambda_X$ and $\delta_X$.
\subsubsection{Boundary of the Trigonal Locus}
As we vary the base curve $B$ inside $\overline{\mathfrak{T}}_g$,
we actually obtain a relation among the restrictions of $\lambda$ and $\delta$
in $\on{Pic}_{\mathbb{Q}}\overline{\mathfrak{T}}_g$, rather than just
among $\lambda|_B=\!\lambda_X$ and $\delta|_B=\!\delta_X$ in
$\on{Pic}B$.
\smallskip
{\it In terms of what} have we thus represented and linked
$\lambda|_{\overline{\mathfrak{T}}_g}$ and $\delta|_{\overline
{\mathfrak{T}}_g}$?
To answer this question, we need first to understand the
boundary divisors of the trigonal locus $\overline{\mathfrak{T}}_g$.
As we shall see, there
are seven types of such divisors, denoted by
$\Delta{\mathfrak{T}}_{0}$ and $\Delta{\mathfrak{T}}_{k,i}$ for $k=1,...,6$.
Each type is
determined by the specific geometry of its general member. For example,
$\Delta{\mathfrak{T}}_0$ is the closure of all irreducible trigonal curves with
one node, while $\Delta{\mathfrak{T}}_{2,i}$ corresponds to joins in two points
of a trigonal and a hyperelliptic curve with genera $i$ and $g-1-i$,
respectively (cf.~Fig.~\ref{Delta-k,i}).
Naturally, we derive an expression for the restriction of
the divisor class $\delta\in\on{Pic}_{\mathbb{Q}}\overline{\mathfrak{M}}_g$ to
$\overline{\mathfrak{T}}_g$:
\begin{equation*}
\delta|_{\displaystyle{\overline{\mathfrak{T}}_g}}=\delta_0+
\sum_{i=1}^{\scriptscriptstyle{[(g-2)/2]}}3\delta_{1,i}
+\sum_{i=1}^{\scriptscriptstyle{g-2}}2\delta_{2,i}
+\sum_{i=1}^{\scriptscriptstyle{[g/2]}}\delta_{3,i}
+\sum_{i=1}^{\scriptscriptstyle{[(g-1)/2]}}3\delta_{4,i}+
\sum_{i=1}^{\scriptscriptstyle{g-1}}\delta_{5,i}
+\sum_{i=1}^{\scriptscriptstyle{[g/2]}}\delta_{6,i}.
\label{delta}
\end{equation*}
Here $\delta_0$ and $\delta_{k,i}$ are the divisor classes of
$\Delta{\mathfrak{T}}_0$ and
$\Delta{\mathfrak{T}}_{k,i}$ in $\on{Pic}_{\mathbb{Q}}\overline{\mathfrak{T}}_g$.
\subsubsection{Relations among $\lambda$ and $\delta$}
For a fixed family $X\rightarrow B$ with a smooth trigonal general member,
we establish a relation among the Hodge class $\lambda|_B$, the
boundary classes $\delta_{k,i}|_B$, and the Bogomolov quantity
$4c_2(V)-c_1^2(V)$ for the associated vector bundle $V$:
\begin{equation}
(7g+6)\lambda|_B=g\delta_0|_B+\sum_{k,i}\widetilde{c}_
{k,i}\delta_{k,i}|_B+\frac{g-3}{2}(4c_2(V)-c_1^2(V)).
\label{tobelifted}
\end{equation}
The polynomial coefficients $\widetilde{c}_{k,i}$ are
comparatively larger than the corresponding coefficients of the
boundary divisors in the expression for
$\delta|_{\displaystyle{\overline{\mathfrak{T}}_g}}$.
As a result, we rewrite (\ref{tobelifted}) as
\begin{equation}
(7g+6)\lambda|_B=g\delta|_B+\mathcal{E}|_B+\frac{g-3}{2}(4c_2(V)-c_1^2(V)),
\label{E-argument}
\end{equation}
where $\mathcal{E}$ is an effective combination of the boundary classes on
$\overline{\mathfrak{T}}_g$. In particular, if $V$ is Bogomolov semistable,
the slope satisfies (cf.~Theorem~\ref{7+6/g Bogomolov2}):
\begin{equation}
\on{slope}(X/_{\displaystyle{B}})\leq 7+\frac{6}{g}.
\label{idea7+6/g}
\end{equation}
Further, we describe $\on{Pic}_{\mathbb{Q}}\overline{\mathfrak{T}}_g$
as generated freely by the restriction $\lambda|_{\overline{\mathfrak{T}}_g}$
and the boundary classes of $\overline{\mathfrak{T}}_g$. In the even genus
$g$ case, we can replace $\lambda|_{\overline{\mathfrak{T}}_g}$ by
a geometrically defined class $\mu$, corresponding to the so-called
Maroni divisor in $\overline{\mathfrak{T}}_g$. This, of course, means
that the Hodge class $\lambda|_{\overline{\mathfrak{T}}_g}$ must be
some linear combination of the boundary classes and $\mu$. The Bogomolov
quantity is interpreted as
\[4c_2(V)-c_1^2(V)=4\mu|_B+0\cdot \delta_0|_B+\sum_{k,i}\alpha_{k,i}
\delta_{k,i}|_B,\]
which in turn ``lifts'' (\ref{tobelifted}) to the wanted relation in
$\on{Pic}_{\mathbb{Q}}\overline{\mathfrak{T}}_g$:
\begin{equation*}
(7g+6)\lambda|_{\overline{\mathfrak{T}}_g}=g\delta_0+
\sum_{k,i}\widehat{c}_{k,i}\delta_{k,i}+2(g-3){\mu}.
\end{equation*}
We have not yet computed explicitly all coefficients $\widehat{c}_{k,i}$.
In the cases which we have completed ($\Delta_{0}\mathfrak{T}_g$
and $\Delta_{1,i}\mathfrak{T}_g$), these coefficients
turn out again sufficiently large so that
we can repeat the argument in (\ref{E-argument}).
Thus, if $X$ has
at least one non-Maroni fiber, and its singular fibers belong to
$\Delta_{0}\mathfrak{T}_g\cup\Delta_{1,i}\mathfrak{T}_g$,
then $\mu|_B\geq 0$, and hence the stronger bound of (\ref{idea7+6/g}) holds
(cf.~Prop.~\ref{Maroni inequality} and Conj.~\ref{Maroni-conj}).
\subsubsection{Maximal Bound}
Since the Bogomolov semistability condition
$4c_1^2(V)-c_2(V)\geq 0$ is not always satisfied, the above discussion
shows that $7+6/g$ is {\it not} the maximal bound for the
slope of trigonal families, Therefore, we need another,
more subtle, estimate. The expressions for $\lambda|_B$ and $\delta|_B$
suggest that any maximal
bound would be equivalent to an inequality involving $c_1^2(V)$,
$c_2(V)$, and possibly some other invariants. We construct a specific
divisor class $\eta$ on $\widetilde{X}$, for which the {\it Hodge Index}
theorem implies $\eta^2\leq 0$, and we translate this into
$9c_2(V)-2c_1^2(V)\geq 0$ (cf.~Prop.~\ref{genindex}).
We notice that the only reasonable
way to replace Bogomolov's condition $4c_1^2(V)-c_2(V)\geq 0$
by the newly found inequality is by subtracting the following quantities:
\begin{equation*}
36(g+1)\lambda|_B-(5g+1)\delta|_B= \mathcal{E}^{\prime}|_B+(g-3)\big(9c_2(V)-
2c_1^2(V)\big),
\label{maximum1}
\end{equation*}
so that the ``left-over'' linear combination of boundary divisors
$\mathcal{E}^{\prime}$ is again effective (cf.~Theorem~\ref{maximal relation2}).
Hence, we conclude that for {\it
all} trigonal families:
\[\on{slope}(X/_{\displaystyle{B}})\leq \frac{36(g+1)}{5g+1}
\cdot\]
\subsection{The organization of the paper}
\label{organization}
The presentation of the {\it Basic Construction} is done in several stages.
Fig.~\ref{stages} shows schematically the connection between the
three types of covers, admissible, quasi-admissible and effective, in
relation to the original family $X\rightarrow B$ of stable curves.
\vspace*{4mm}
\begin{figure}[h]
$$\psdraw{stages}{1.6in}{1.35in}$$
\caption{Types of covers}
\label{stages}
\end{figure}
We start in Section~\ref{hurwitz} by introducing a compactification
$\overline{\mathcal{H}}_{d,g}$ of the Hurwitz scheme, parametrizing {\it
admissible} $d$-uple covers of stable pointed rational curves. Using its
coarse moduli properties, we show in Section ~\ref{admissible} the existence
of admissible covers of surfaces $X^a\rightarrow Y^a$ associated to the
original family $f\!:\!X\!\rightarrow \!B$. Next we modify these covers to
{\it quasi-admissible} covers $\widetilde{\phi}:\widetilde{X}
\rightarrow \widetilde{Y}$ (cf.~Prop.~\ref{propquasi}),
and further to {\it effective} covers $\widehat{\phi}:\widehat{X}\rightarrow
\widehat{Y}$ in order to resolve the technical difficulties arising from the
non-flatness of $\widetilde{\phi}$ (cf.~Sect.~\ref{effectivecovers}).
\bigskip
We devote Section~4
to the study of the boundary components of the trigonal
locus $\overline{\mathfrak{T}}_g$ inside the moduli space
$\overline{\mathfrak{M}}_g$, and express the restriction
$\Delta|_{\overline{\mathfrak{T}}_g}$ as a linear combination of the
boundary divisors (cf.~Prop.~\ref{divisorrel}).
In Section~6
we complete the Basic Construction by embedding the effective
cover $\widehat{X}$ in a rank 1 projective bundle ${\mathbf P}V$ over
$\widehat{Y}$.
\medskip
For convenience of the reader, the proofs of the maximal $36(g+1)/(5g+1)$
and the semistable $7+6/g$ bounds are presented first in the special,
but fundamental case when the original family
$f\!:\!X\!\rightarrow \!B$
is already an effective triple cover of a ruled surface
$Y$ (cf.~Sect.~7). The discussion results in finding the
coefficients of $\delta_0$ in two different expressions of
$\lambda|_{\overline{\mathfrak{T}}_g}$, but, as it turns out, the knowledge of
these coefficients is enough to determine the desired two bounds. We refer
to this as the {\it global} calculation.
The Hodge Index Theorem and Nakai-Moishezon criterion
on $X$ complete the global calculation in Sect.~\ref{indextheorem}.
A discussion of maximal bound examples can be found in
Section~\ref{whenmaximal}.
\medskip
The {\it local} calculations in Sections~8-10
compute the contributions of the other boundary \vspace*{-1mm}classes
$\delta_{k,i}$, and express $\lambda|_{\overline{\mathfrak{T}}_g}$ in terms
of these contributions and the Chern classes of the rank 2 vector bundle
$V$ on $\widehat{Y}$. For clearer exposition, the proofs of the
two bounds are shown first for a {\it general} base curve $B$ (i.e.
$B$ intersects transversally the boundary components in general points),
and then in Section~11 the results are extended to {\it any}
base curve $B$. We develop the necessary notation and techniques for the
local calculations in Section ~\ref{conventions}.
\medskip
Section~12 discusses the relation between the Bogomolov
semistability condition and the Maroni locus, and describes the structure of
$\on{Pic}_{\mathbb{Q}}\overline{\mathfrak{T}}_g$. In Section~\ref{Maroni-maximal}
we give another interpretation of the conditions for the maximal bound.
\medskip
We present further results and conjectures for $d$-gonal families in
Section~13. In the Appendix, we
give another proof of the $8+4/g$ bound in
the hyperelliptic case and show an application of the
maximal trigonal bound to the study of the discriminant
locus of certain triple covers.
\bigskip
\section*{\hspace*{1.9mm}3. Quasi-Admissible Covers of Surfaces}
\setcounter{section}{3}
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\setcounter{prop}{0}
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\label{quasi-admissible}
We first review briefly
the theory of admissible covers. For more details, we refer
the reader to \cite{MHE,HM}.
\subsection{The Hurwitz scheme $\overline{\mathcal H}_{d,g}$}
\label{hurwitz}
Let ${\mathcal H}_{d,g}$ be the {\it small Hurwitz scheme}
parametrizing the pairs
$(C,\phi)$, where $C$ is a smooth curve of genus $g$ and $\phi:C\rightarrow
{\mathbf P}^1$ is a cover of degree $d$, simply branched over $b=2d+2g-2$
distinct points. Since $C\in {\mathfrak M}_g$, there is a
natural map ${\mathcal H}_{d,g}\rightarrow {\mathfrak M}_g$, whose
image contains an open dense subset of ${\mathfrak M}_g$. The theory of
admissible covers provides the commutative diagram in
Fig.~\ref{Hurwitz figure}.
\begin{figure}[h]
\begin{picture}(5,3.5)(-0.8,2.2)
\put(0,4){${\mathcal H}_{d,g}\hookrightarrow \overline{\mathcal H}_{d,g}$}
\put(0.4,3.85){\vector(1,-1){0.9}}
\put(1.9,3.85){\vector(1,-1){0.9}}
\put(0.4,4.2){\vector(1,1){0.9}}
\put(1.9,4.2){\vector(1,1){0.9}}
\put(2.4,4.5){${pr}_1$}
\put(2.4,3.5){${pr}_2$}
\put(1.1,2.5){${\mathfrak P}_{0,b}\hookrightarrow
\overline{\mathfrak P}_{0,b}$}
\put(1.1,5.2){${\mathfrak M}_g\hookrightarrow \overline{\mathfrak M}_g$}
\end{picture}
\caption{Hurwitz scheme}
\label{Hurwitz figure}
\end{figure}
There ${\mathfrak P}_{0,b}$ (resp. $\overline{\mathfrak P}_{0,b}$)
is the moduli space of $m$-pointed ${\mathbf P}^1$'s (resp. of stable
$m$-pointed rational curves), and
$\overline{\mathcal H}_{d,g}$ is a compactification of the Hurwirz
scheme. The points of $\overline{\mathcal H}_{d,g}$ correspond to triples
$(C,(P;p_1,...,p_m),\phi)$,
where $C$ is a connected reduced nodal curve of genus $g$,
$(P;p_1,...,p_m)$ is a stable
$m$-pointed rational curve, and $\phi:C\rightarrow P$ is a
so-called {\it admissible cover}.
\medskip
\noindent{\bf Definition 3.1.} Given the curves $C$ and $P$ as above, an
{\it admissible cover} $\phi:C\rightarrow P$ is a regular map
satisfying the following conditions:
\smallskip
(A1) $\phi^{-1}(P_{\on{sm}})=C_{\on{sm}}$ and $\phi:C_{\on{sm}}
\rightarrow P_{\on{sm}}$
is simply branched over the distinct points $p_1,...,p_b\in P_{\on{sm}}$;
(A2) for every $q\in C_{\on{sing}}$ lying over a node $p\in P$, the two
branches through $q$ map with the same ramification index
to the two branches through $p$.
\begin{figure}[h]
$$\psdraw{nonstable}{1in}{1in}$$
\caption{Admissible model}
\label{nonstable}
\end{figure}
\smallskip
Note that $C$ is not necessarily a stable curve, but contracting its
destabilizing rational chains yields the corresponding stable
curve $pr_1(C)\in \overline{\mathfrak M}_g$. In such a case, we say that
$C$ is the ``admissible model'' for $pr_1(C)$ (cf.~Fig.~\ref{nonstable}).
Harris-Mumford have shown that
the compactification $\overline{\mathcal H}_{d,g}$ is in fact
a {\it coarse moduli space} for the admissible covers $\phi:C\rightarrow P$.
\begin{figure}[h]
$$\psdraw{admfamilies}{3.4in}{1.4in}$$
\hspace*{4.5mm}\vspace*{-5mm}
\caption{Admissible family}
\label{admfamilies}
\end{figure}
\subsection{Local properties of admissible covers}
\label{localproperties}
When we vary the admissible covers of curves in families, the
local structure of the corresponding total spaces becomes apparent.
Let $\phi:\mathcal C\rightarrow \mathcal P$ be a proper flat family (over a scheme
$\mathcal B$) of admissible covers of curves (cf.~Fig.~\ref{admfamilies}).
Assume that $\phi$ is \'{e}tale
everywhere except over the nodes of the fibers of $\mathcal P/_
{\textstyle{\mathcal B}}$,
and except over some sections $\sigma_i:\mathcal B\rightarrow \mathcal C$ and their
images $\omega_i:\mathcal B\rightarrow \mathcal P$: there $\phi$ is simply branched
along $\sigma_i$ over $\omega_i$ for all $i$. If $q\in {\mathcal C}_b$ is a point
lying above a node $p\in {\mathcal P}_b$ for some $b\in \mathcal B$, then
$\mathcal C_b$ has a node at $q$, and locally analytically we can describe
$\mathcal C,\mathcal P$ and $\phi$ near $q$ and $p$ by:
\[\left\{\begin{array}{lll}
\mathcal C: & xy=a, &x,y\,\,\on{generate}\,\,\widehat{\mathfrak m}_{q,\mathcal
C_b},\,\, a\in \widehat{\mathcal O}_{b,\mathcal B},\\
\mathcal P: & uv=a^n, &u,v\,\,\on{generate}\,\,\widehat{\mathfrak m}_{p,\mathcal
P_b},\\
\phi: & u=x^n,v=u^n.
\end{array}\right.\]
\smallskip
One can see that $n$ is the index of ramification of $\phi$ at $q$,
and that fiberwise $\mathcal C_b\rightarrow \mathcal P_b$ is
an admissible cover (of curves). From now on, by
{\it admissible covers} we mean, more generally,
families $\mathcal C\rightarrow \mathcal P$ over $\mathcal B$ with the above description.
\smallskip
The local properties of the admissible cover $\phi:\mathcal C\rightarrow \mathcal P$
over the nodes in $\mathcal P_b$ forces singularities on the total spaces
of $\mathcal C$ and $\mathcal P$. Since we will be interested only in the cases when
the base $\mathcal B$ is a smooth projective curve $B$ and the general
fiber of $\mathcal C$
is smooth, we can always pick a generator $t$ for $\widehat{\mathcal O}_{b,B}$,
and express $a=t^l$ for some $l\in {\mathbb N}$.
\begin{figure}[h]
$$\psdraw{singular}{1.2in}{1.2in}$$
\caption{Singularity of $\mathcal C$}
\label{singular}
\end{figure}
\noindent{\bf Example 3.1.} Let the triple
admissible cover $\phi\!:\!\mathcal C\!\rightarrow\!\mathcal P$
contain the fiber $\mathcal C_b$ as in Fig.~\ref{singular}. At $q$,
${\mathcal C}$ is given by
$\,\,xy=t^{l}$, and at $p$, $\mathcal P$ is given by
$uv=t^{2l}$, where $u=x^2,\,\,v=y^2$. This
forces at $r$ the local equation
$xy=t^{2l}$ ($u=x,\,\,v=y$).
Even if $\mathcal C$ is smooth at $q$ ($l=1$), $\mathcal C$
and $\mathcal P$ will be singular at $r$ and $p$, respectively
($xy=t^2,\,\,uv=t^2$). Compare this with the non-flat cover of ramification
index 1 in Fig.~\ref{mult4}.
\smallskip
Recall that a rational double point $s$ on a surface $S$ is of type
$A_{l-1}$ if locally analytically $S$ is given at $s$ by the
equation $xy=t^l$. Thus, $r$ and $p$ above are rational double points on
$\mathcal C$ and $\mathcal P$, respectively, of type $A_{l-1}$.
\medskip
\noindent{\bf Remark 3.1.}
In the sequel, we use the fact that the projection
$pr_1\!:\!\overline{\mathcal H}_{d,g}\!\rightarrow
\!\overline{\mathfrak P}_{0,b}$ is a {\it finite} map. From the weak valuative
criterion for properness, this means that given a family of admissible
covers $\phi:{\mathcal C}^*\rightarrow {\mathcal P}^*$ over the punctured disc
${\on {Spec}}\,{\mathbb C}((t))$, there is some $n\in {\mathbb N}$ for which
$\phi$ extends to a family $\phi_n:{\mathcal C}_n\rightarrow {\mathcal P}_n$
of admissible covers
over ${\on {Spec}}\,{\mathbb C}[[t^{1/n}]]$. In particular, if the base for
the admissible cover $\phi:X^*\rightarrow Y^*$ is an open set $B^*$ of a smooth
projective curve $B$, modulo a finite base change, we can extend this
to a family of admissible covers $X^a\rightarrow Y^a$ over the whole curve $B$.
\subsection{Admissible covers of surfaces}
\label{admissible}
Consider a family $f:X\rightarrow B$ of stable curves of genus $g$, whose
general member is smooth and $d$-gonal. Let
$\psi:B\rightarrow {\overline{\mathfrak M}_g}$ be
the canonical map, and let $\overline B$ denote
the fiber product $B\times_{\overline{\mathfrak M}_g}
\overline{\mathcal H}_{d,g}$.
\begin{figure}
\begin{picture}(5,4)(-0.7,1.9)
\put(-2,4){$\overline{B}_0\subset
\overline B\stackrel{\eta}{\longrightarrow}
\overline{\mathcal H}_{d,g}$}
\multiput(-0.8,3.85)(1.4,0){2}{\vector(0,-1){0.9}}
\put(-1,2.5){$B\stackrel{\psi}{\longrightarrow}
\,\overline{\mathfrak M}_{g}$}
\put(-2.5,3.3){$X$}
\put(0.7,3.35){$\scriptstyle{pr_1}$}
\put(-3.5,4.8){$\overline{X}$}
\multiput(-3.2,4.7)(1.5,-0.8){2}{\vector(2,-3){0.7}}
\multiput(-2.1,3.4)(-1,1.5){2}{\vector(3,-2){1.1}}
\put(3,3.7){$B^*\subset B\stackrel{\eta}{\longrightarrow}
\overline{\mathcal H}_{d,g}$}
\multiput(4.2,5.05)(1.4,-1.5){2}{\vector(0,-1){0.9}}
\put(3.2,5.05){\vector(0,-1){0.9}}
\put(5.4,2.2){$\overline{\mathfrak M}_{g}$}
\put(5.7,3.1){$\scriptstyle{pr_1}$}
\put(3,5.2){$X^*\subset X$}
\put(4.3,3.6){\vector(1,-1){1}}
\put(4.8,3.1){$\scriptstyle{\psi}$}
\end{picture}
\caption{$\eta:\overline{B}\rightarrow\overline{\mathcal {H}}_{d,g}$
\hspace*{10mm}{\sc Figure 12.} Simply branched $C$\hspace*{-20mm}}
\label{map eta}
\end{figure}
\addtocounter{figure}{1}
If the general member of $X$ has infinitely many ${g}_d^1$'s,
the variety $\overline B$ will have dimension $\geq 2$.
We can resolve this by considering an intersection of
the appropriate number of hyperplane sections of
$\overline B$, and picking a one-dimensional
component $\overline B_0$ dominating $B$. The curve
$\overline B_0$ might be
singular, but by normalizing it and pulling $X$
over it, we get another family of stable curves (cf.~Fig.~\ref{map eta}):
\[\overline X=X\times_B(\overline B_0)^{\on{norm}}
\rightarrow (\overline B_0)^{\on{norm}}.\]
Since the two
families have the same basic invariants, we can replace
the original with the new one, and assume
the existence of a map $\eta:B\rightarrow
\overline{\mathcal H}_{d,g}$ compatible with $\psi:
B\rightarrow \overline{\mathfrak M}_g$. In other
words, $\eta$ associates to every fiber $C$ of $X$ a
specific $g^1_d$ on $C$ or, possibly, a $g^1_d$
on an admissible model $C^a$ of $C$.
\smallskip
Let $B^*$ be the open subset of
$B$ over which {\it all} fibers are smooth and $d$-gonal. For simplicity,
assume for now that all the fibers over $B^*$ can be
represented as admissible covers of ${\mathbf P}^1$ via the chosen
$g^1_d$'s, i.e. they are {\it simply branched} covers of ${\mathbf P}^1$
over $m$ distinct points of ${\mathbf P}^1$.
Denote by $X^*$ the restriction of $X$ over $B^*$ (cf.~Fig.12).
\smallskip
The map $\eta:B^*\rightarrow {\mathcal H}_{d,g}$ induces
a section
\[\sigma:B^*\rightarrow {\on{Pic}}^d(X^*/B^*),\] where
${\on{Pic}}^d(X^*/B^*)$ is the {\it relative degree $d$ Picard variety} of
$X^*$ over $B^*$. ${\on{Pic}}^d(X^*/B^*)$ parametrizes
the line bundles on $X^*$ of relative degree $d$.
The image $\sigma(B^*)\subset {\on{Pic}}^d(X^*/B^*)$ is a class of line bundles
on $X^*$ whose fiberwise restrictions are the chosen $g^1_d$'s.
Let $\mathcal L$ be a representative of this class, and let
$Y^*$ be the ruled surface ${\mathbf P}((f_*{\mathcal L})^{\widehat
{\phantom{n}}})$ over $B^*$.
The map $\phi:X^*\rightarrow Y^*$ induced by $\mathcal L$
defines an admissible cover over $B^*$, as shown in Fig.~\ref{construction
Y*}.
\setlength{\unitlength}{10mm}
\begin{figure}[h]
\begin{picture}(5,2.2)(-0.3,2.6)
\put(0,4){$X^*\stackrel{\phi}{\longrightarrow} Y^*={\mathbf P}((f_*{\mathcal L})^
{\widehat{\phantom{n}}})$}
\put(0.2,3.85){\vector(1,-1){0.5}}
\put(1.4,3.85){\vector(-1,-1){0.5}}
\put(0.6,2.9){$B^*$}
\put(0.05,3.4){$f$}
\put(1.3,3.4){$h$}
\end{picture}
\caption{ Construction of $Y^*$}
\label{construction Y*}
\end{figure}
\bigskip
From Remark 3.1, $\phi$ extends to a family of admissible covers
${\phi}^a:{X}^a\rightarrow {Y}^a$ over the whole base $B$.
Since ${X}^a$ and $X$ are
isomorphic over $B^*$, they are birational to each other.
In other words, the fibers $C$ of $X$, over which $\mathcal L$ does not
extend to the base-point free linear series $g^1_d=\sigma_1(b)$, are
modified by blow-ups and blow-downs so as to arrive at their admissible
models in ${X}^a$. We have thus proved the following
\begin{lem} Let $f:X\rightarrow B$ be a family of stable curves,
whose general member over an open subset $B^*\subset B$
is a smooth $d$-uple admissible cover of ${\mathbf P}^1$.
Then, modulo a finite base change, there exists an admissible cover
of surfaces ${X}^a\rightarrow {Y}^a$
over $B$ such that ${X}^a$ is obtained from $X$ by a finite number of
birational transformations performed on the fibers over $B-B^*$.
\label{quasicov}
\end{lem}
\subsection{Quasi-admissible covers}
\label{quasi-covers}
In case the general member of $X$ is {\it not} an admissible cover
of ${\mathbf P}^1$, e.g. it is trigonal with a total point of ramification,
we have to modify the above construction. To start with, we cannot
expect to obtain an {\it admissible} cover $X^*\rightarrow Y^*$, even
modulo a finite base change. This leads us to consider a different kind
of covers, which we call {\it quasi-admissible}.
\medskip
\noindent{\bf Definition 3.2.} A {\it quasi-admissible cover}
$\widetilde{\phi}:
C\rightarrow P$ of a nodal curve $C$ over a semistable pointed rational
curve $P$
is a regular map which behaves like an admissible cover over the singular
locus of $P$, i.e. for any $q\in C$ lying over a node $p\in P$
the two branches through $q$ map with the same ramification index
to the two branches through $p$.
\smallskip
\begin{figure}[h]
$$\vspace*{5mm}\psdraw{quasi}{4.5in}{0.5in}$$
\vspace*{-6mm}
\caption{Quasi-admissible covers over $\mathbf P^1$}
\label{quasicovers}
\end{figure}
Quasi-admissible covers differ from admissible covers in allowing
more diverse behavior of $C$ over $P_{\on{sm}}$, e.g. having singularities,
higher ramification points and multiple simple ramification points.
Fig.~\ref{quasicovers}
displays several degree 3 quasi-admissible covers over ${\mathbf P}^1$:
However, any quasi-admissible cover can be obtained
from an admissible cover $\phi^a\!:\!C^a\!\rightarrow\! P^a$
by simultaneous contractions of components in $P^a$
and their (rational) inverses on $C^a$.
\medskip
\noindent{\bf Definition 3.3.} A {\it minimal} quasi-admissible cover
$\widetilde{\phi}:C\rightarrow P$ is minimal with respect to the number of
components of $P$. In other words, one cannot apply more simultaneous
contractions on $C\rightarrow P$ and end with another quasi-admissible cover.
\smallskip
\noindent{\bf Example 3.2.}
A smooth trigonal curve $C$ with a total point of ramification $q$ is a
minimal quasi-admissible cover of $P={\mathbf P}^1$. Blowing up $q$ on $C$
and $p=\widetilde{\phi}(q)\in P$, gives an admissible cover $C^a=C\cup
C_1\rightarrow P\cup P_1$, where $C_1\cong {\mathbf P}^1$ maps
three-to-one onto $P_1\cong {\mathbf P}^1$ with a total
point of ramification $q=C_1\cap C$ (cf.~Fig.~\ref{quasi/adm}).
\bigskip
\begin{figure}[h]
$$\psdraw{quasiadm}{2.4in}{0.8in}$$
\caption{Quasi virsus admissible covers}
\label{quasi/adm}
\end{figure}
\medskip
The motivation for using {\it minimal} quasi-admissible covers,
instead of just admissible or quasi-admissible covers, is that the former are
the closest covers to the original families $X\rightarrow B$ of stable
curves, and calculations
on them will yield the best possible estimate for the ratio
$\delta_X/\lambda_X $ (cf.~Fig.~\ref{stages}).
\subsubsection{Quasi-admissible covers for families with
higher ramification sections}
Now let us consider the remaining case of
a family $X\rightarrow B$, whose general member over
$B^*$ is smooth and $d$-gonal, but {\it not} an admissible cover of
${\mathbf P}^1$. After a possible base change, we still have the map
(cf.~Fig.~{12})
\[\eta:B\longrightarrow {\overline {\mathcal H}}_{d,g}.\]
It associates to every fiber $C$ a $g^1_d$ on its admissible
model $C^a$. Let $C^a\rightarrow P^a$ be the
corresponding admissible cover. Since $C$ itself is $d$-gonal, and
by assumption it does not possess
a $g^1_e$ with $e<d$, $C$ must be a $d$-uple cover of some component of
$P^a$. In particular, the $g^1_d$ on
$C^a$ restricts to a $g^1_d$ on $C$. Thus, in effect, $\eta$
gives again a section $\sigma:B^*\rightarrow {\on{Pic}}^d(X^*/B^*)$.
As before, we obtain a degree $d$ finite map $\phi:X^*\rightarrow Y^*$ to
the ruled surface ${\mathbf P}((f_*{\mathcal L})^{\widehat{\phantom{n}}})$
over $B^*$. Note that
this is a family of {\it minimal quasi-admissible} covers.
\medskip
We extend $\phi$ over the curve $B$ as follows.
For simplicity, assume that $d=3$. Let $R$ be the
ramification divisor of $\phi$ in $X^*$. By hypothesis,
there is a component $R_0$ of $R$ which passes through
total ramification points and dominates $B^*$.
Letting $\overline{R}_0$
be the closure of $R_0$ in $X$, we can normalize it and pull the family $X$
over it. So we may assume that $\overline{R}_0$ is a section of $X\rightarrow
B$. If there are some other components $R_1,R_2,...,R_l$ of the
ramification divisor $R$ passing through higher ramification points,
we repeat the same procedure for them, until we ``straighten out'' all
$\overline{R}_i$'s into sections of $X\rightarrow B$.
Let $E_i=\phi(R_i)$ be the corresponding sections of $Y^*$ over $B^*$.
We can shrink $B^*$ in order to exclude any fibers with isolated
higher ramification points.
\smallskip
Consider a fiber $C$ in $X^*$. Let $\{r_i=C\cap R_i\}$ be its
total ramification points, and let $\{p_i=\phi(r_i)\}$ be their images
on $P=\phi(C)$ in $Y^*$. It is clear that blowing-up all $r_i$'s
and $p_i$'s will give an admissible triple cover $C^a=
\on {Bl}_{\{r_i\}}(C)\rightarrow P^a=\on {Bl}_{\{p_i\}}(P)$.
The $g^1_d$, giving
this cover, is the original one assigned by $\eta:B^*\rightarrow
{\overline{\mathcal H}}_{d,g}$. We globalize this construction by
blowing-up the sections ${R}_i$ on $X^*$ and $E_i$ on $Y^*$.
Similarly as above, we obtain a triple admissible cover of surfaces
$\phi^*:\on{Bl}_{\cup R_i}(X^*)\rightarrow \on{Bl}_{\cup E_i}
(Y^*)$ over $B^*$. The properness of $pr_1:
\overline{\mathcal H}_{d,g}\rightarrow \overline{\mathfrak M}_g$ allows us
to extend this to an admissible cover ${\phi}^a:
\overline{\on{Bl}_{\cup R_i}(X^*)}
\rightarrow\overline{\on{Bl}_{\cup E_i}(Y^*)}$ over $B$ (cf.~
Fig.~\ref{blowing up}).
\begin{figure}[h]\hspace*{-30mm}
\begin{picture}(3,5)(4.3,-0.5)
\put(2,3.7){$\overline{{\mathcal R}_i}\subset
\overline{\on{Bl}_{\cup R_i}(X^*)}\stackrel{{\phi}^a}
{\longrightarrow} \overline{\on{Bl}_{\cup E_i}(Y^*)}\supset
\overline{{\mathcal E}_i}$}
\put(2,2.2){${{\mathcal R}_i}\subset {\on{Bl}_{\cup R_i}(X^*)}
\stackrel{\phi^*}{\longrightarrow}
{\on{Bl}_{\cup E_i}(Y^*)}\supset{{\mathcal E}_i}$}
\put(2,0.7){$R_i\hspace{0.5mm}\subset \hspace{6.6mm}
X^*\hspace{6mm}\stackrel{\phi}{\longrightarrow}\hspace{7.3mm}Y^*
\hspace*{5.6mm}\supset E_i$}
\multiput(2.2,3.5)(6.1,0){2}{\vector(0,-1){0.9}}
\multiput(3.9,3.5)(2.8,0){2}{\vector(0,-1){0.9}}
\multiput(2.2,2)(6.1,0){2}{\vector(0,-1){0.9}}
\multiput(3.9,2)(2.8,0){2}{\vector(0,-1){0.9}}
\end{picture}
\vspace*{-10mm}
\caption{Blowing up ${R}_i$ and ${E}_i$ \hspace*{15mm}
{\sc Figure 17.} Over $B^*$\hspace*{-10mm}}
\label{blowing up}
\hspace*{80mm}\begin{picture}(3,0)(6.7,-2.3)
\put(7.8,2.2){${X}^q\stackrel{\phi^q}{\longrightarrow} {Y}^q$}
\multiput(8,2)(1.4,0){2}{\vector(0,-1){0.9}}
\multiput(8.1,1.5)(1.4,0){2}{$\wr$}
\put(7.8,0.7){$X^*\longrightarrow Y^*$}
\put(8,0.5){\vector(1,-1){0.5}}
\put(9.4,0.5){\vector(-1,-1){0.5}}
\put(8.5,-0.4){$B^*$}
\end{picture}
\label{over B*}
\end{figure}
\addtocounter{figure}{1}
\smallskip
Denote by ${\mathcal R}_i$
the component of $\on {Bl}_{\cup R_i}(X^*)$, obtained by blowing up
${R}_i\subset X^*$,
and let $\overline{{\mathcal R}_i}$ be its closure in
$\overline{\on{Bl}_{\cup R_i}(X^*)}$.
Define similarly ${\mathcal E}_i\subset \on{Bl}_{\cup E_i}(Y^*)$ and
$\overline{\mathcal E}_i
\subset \overline{\on{Bl}_{\cup E_i}(Y^*)}$. The admissible cover
$\phi^a$ maps
$\overline{\mathcal R}_i$ to $\overline{\mathcal E}_i$, so that after removing
all the $\overline{\mathcal R}_i$'s and $\overline{\mathcal E}_i$'s we still have
a triple cover
\[{\phi}^q:{X}^q=\overline{\on{Bl}_{\cup R _i}
(X^*)}-\cup\overline{\mathcal R_i}\longrightarrow
{Y}^q=\overline{\on{Bl}_{\{{ E_i}\}}
(Y^*)}-\cup\overline{\mathcal E_i}.\]
Note that ${X}^q\cong X$ and ${Y}^q\cong Y$ over the open set
$B^*$, and that ${Y}^q$ is a birationally ruled surface over $B$
(cf.~Fig.~17).
Finally, note that from the quasi-admissible cover
${\phi}^q:{X}^q\rightarrow {Y}^q$ we obtain a family
$\widetilde{\phi}:\widetilde{X}\rightarrow
\widetilde{Y}$ of {\it minimal}
quasi-admissible covers: simply contract the unnecessary
rational components in the fibers of ${X}^q$ and
${Y}^q$, and observe that the triple map ${\phi}^q$ restricts to the
corresponding triple map $\widetilde{\phi}$.
\smallskip This completes the construction of minimal quasi-admissible
covers for any family $X\rightarrow B$ with general smooth trigonal member.
The cases $d>3$ are only notationally more difficult. One has to keep
track of the possibly different higher multiplicities in $C$ and multiple
double points in $C$ over the same $p\in P$. The construction
of an admissible cover ${X}^a\rightarrow {Y}^a$
goes through with minimal modifications. We combine the results of this
section in the following
\begin{prop}
Let $f:X\rightarrow B$ be a family of stable curves,
whose general member over an open subset $B^*\subset B$
is smooth and $d$-gonal.
Then, modulo a finite base change, there exists a minimal quasi-admissible
cover of surfaces $\widetilde{X}\rightarrow \widetilde{Y}$
over $B$ such that $\widetilde{X}$ is obtained from $X$ by a finite number of
birational transformations performed on the fibers over
$B-B^*$.
\label{propquasi}
\end{prop}
\medskip
\section*{\hspace*{1.9mm}4. The Boundary $\Delta{\mathfrak{T}}_g$
of the Trigonal Locus $\overline{\mathfrak{T}}_g$}
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\label{boundarycomponents}
\subsection{Description and notation for the boundary of
$\overline{\mathfrak{T}}_g$}
\label{description} In this section we shall see that
there are {\it seven types} of boundary divisors of
$\overline{\mathfrak{T}}_g$, each denoted by $\Delta{\mathfrak{T}}_{k,i}$
for $k=0,1,...,6$. The second index $i$ is determined in the following
way. Let $C=C_1\cup C_2$ be the
general member of $\Delta{\mathfrak{T}}_{k,i}$, where $C_1$ and $C_2$ are smooth curves.
If $C_1$ and $C_2$ are both trigonal or both hyperelliptic, then we set
$i$ to be the smaller of the two genera $p(C_1)$ or $p(C_2)$. If, say, $C_1$ is
a trigonal, but $C_2$ is hyperelliptic, then we set $i$ to be
genus of the trigonal component $C_1$.
The only exception to this rule occurs
when $C$ is irreducible (and hence of genus $g$ with exactly one
node). We denote this boundary component by $\Delta{\mathfrak{T}}_0$.
\smallskip
When we view a general member $C$ roughly as a triple
cover of ${\mathbf P}^1$'s in the Hurwitz scheme (consider the pull-back
$pr_1[C]\in\overline{\mathcal{H}}_{3,g}$), then it may or may not be
ramified. If there is no ramification, then $C$ lies in one of the first
four types of trigonal boundary divisors $\Delta{\mathfrak{T}}_{k,i}$,
$k=0,1,2,3$. Ramification index 1 characterizes the general members of
$\Delta{\mathfrak{T}}_{4,i}$ and $\Delta{\mathfrak{T}}_{5,i}$, and in case of
$\Delta{\mathfrak{T}}_{6,i}$ the ramification index is 2
(cf.~Fig.~\ref{Delta-k,i}).
\smallskip
There is an alternative description of the boundary components
$\Delta{\mathfrak{T}}_{k,i}$'s of $\overline{\mathfrak{T}}_g$.
\vspace*{-1mm}If one such $\Delta{\mathfrak{T}}_{k,i}$ lies in the
restriction $\Delta_0\big|_{\displaystyle{\overline{\mathfrak{T}}_g}}$
of the divisor $\Delta_0$ in $\overline{\mathfrak{M}}_g$,
\vspace*{-1mm}then $\Delta{\mathfrak{T}}_{k,i}$ is one of $\Delta{\mathfrak{T}}_0,\,\,
\Delta{\mathfrak{T}}_{1,i},\,\,\Delta{\mathfrak{T}}_{2,i},$
or $\Delta{\mathfrak{T}}_{4,i}$. The partial normalization of their general
members $C$ is still connected, i.e. $C$ is either irreducible,
or the join of two smooth curves meeting in at least two
points. Correspondingly, for the general member $C$ of the remaining
three types of boundary components,
$\Delta{\mathfrak{T}}_{3,i},\,\,\Delta{\mathfrak{T}}_{5,i}$ and $\Delta{\mathfrak{T}}_{6,i}$, the irreducible
components of $C$ intersect transversely in exactly one point, so that the
normalization of $C$ is disconnected.
\bigskip
\begin{figure}[h]
$$\psdraw{boundary}{4.5in}{1in}$$
\begin{picture}(6,1)(2.7,-1.4)
\put(-0.7,2.45){$\Delta{\mathfrak{T}}_{0} \hspace{19mm}\Delta{\mathfrak{T}}_{1,i}
\hspace{25mm}\Delta{\mathfrak{T}}_{2,i}
\hspace{23mm}\Delta{\mathfrak{T}}_{3,i}$}
\put(2.8,1.2){$\scriptstyle{i=1,2,...,}\left[\frac{g-2}{2}\right]\hspace{17mm}
\scriptstyle{i=1,2,..., g-2}\hspace{17mm}\scriptstyle{i=1,2,
...,} \left[\frac{g}{2}\right]$}
\put(-0.2,0.5){$\Delta{\mathfrak{T}}_{4,i}\hspace{31mm}\Delta{\mathfrak{T}}_{5,i}
\hspace{31mm}\Delta{\mathfrak{T}}_{6,i}$}
\put(01.1,-0.6){$\scriptstyle{i=1,2,...,\left[\frac{g-1}{2}\right]\hspace{24mm}
i=1,2,...,g-1 \hspace{22mm}i=1,2,...,\left[\frac{g}{2}\right]}$}
\end{picture}
\vspace*{-10mm}
\caption{Boundary Components $\Delta\mathfrak{T}_{k,i}$
of $\overline{\mathfrak{T}}_g$}
\label{Delta-k,i}
\end{figure}
\begin{prop} The boundary divisors of $\overline{\mathfrak{T}}_g$ can
be grouped in seven types: $\Delta{\mathfrak{T}}_{0}$ and $\Delta{\mathfrak{T}}_{k,i}$
for $k=1,...,6$. Their general members and range of index $i$ are shown in
Fig.~\ref{Delta-k,i}. The boundary of $\overline{\mathfrak{T}}_g$ consists of
$\Delta{\mathfrak{T}}_{0}$, $\Delta{\mathfrak{T}}_{k,i}$, and the codimension 2
component $\overline{\mathfrak{I}}_g$ of hyperelliptic curves.
\label{boundary}
\end{prop}
\medskip
Consider the projection map
$pr_1:\overline{\mathcal{H}}_{3,g}\rightarrow \overline{\mathfrak{M}}_g$, whose
image is the trigonal locus $\overline{\mathfrak{T}}_g$. Thus, the
inverse image of
each boundary divisor $\Delta{\mathfrak{T}}_{k,i}$ will be a boundary
divisor $\Delta{\mathcal{H}}_{k,i}$ in $\overline{\mathcal{H}}_{3,g}$.
The converse, however, is not always true, i.e. certain boundary divisors
of $\overline{\mathcal{H}}_{3,g}$ contract under $pr_1$ to smaller subschemes of
$\overline{\mathfrak{T}}_g$, e.g. the hyperelliptic locus $\overline{\mathfrak{I}}_g$.
With the description of the Hurwitz scheme
$\overline{\mathcal{H}}_{3,g}$, given in Section~3,
it is easier to determine first $\overline{\mathcal{H}}_{3,g}$'s boundary
divisors. Thus, we postpone the proof of Proposition~\ref{boundary} until
the end of the next subsection.
\subsubsection{The Boundary of $\overline{\mathcal{H}}_{3,g}$.}
\label{admissibleaboundary}
\begin{prop} The boundary divisors of $\overline{\mathcal{H}}_{3,g}$ can
be grouped in six types:
$\Delta{\mathcal{H}}_{k,i}$
for $k=1,...,6$. Their general members and range of index $i$ are shown
in Fig.~\ref{admissible-k,i}.
\begin{figure}[h]
\bigskip
$$\psdraw{admissible}{4.5in}{1in}$$
\begin{picture}(5,1)(3,-2)
\put(-0.9,1.85){$\Delta{\mathcal{H}}_{1,i}
\hspace{33mm}\Delta{\mathcal{H}}_{2,i}
\hspace{32mm}\Delta{\mathcal{H}}_{3,i}$}
\put(0.3,0.7){$\scriptstyle{i=1,2,...,}\left[\frac{g-2}{2}\right]\hspace{29mm}
\scriptstyle{i=1,..., g-1}\hspace{24mm}\scriptstyle{i=0,1,
...,} \left[\frac{g}{2}\right]$}
\put(-0.9,0.15){$\Delta{\mathcal{H}}_{4,i}\hspace{33mm}\Delta{\mathcal{H}}_{5,i}
\hspace{32mm}\Delta{\mathcal{H}}_{6,i}$}
\put(0.3,-1.2){$\scriptstyle{i=1,2,...,\left[\frac{g-1}{2}\right]\hspace{29mm}
i=1,2,...,g-1 \hspace{23mm}i=1,2,...,\left[\frac{g}{2}\right]}$}
\end{picture}
\vspace*{-5mm}
\caption{Boundary Components of $\overline{\mathcal{H}}_{3,g}$}
\label{admissible-k,i}
\end{figure}
\label{boundary2}
\end{prop}
\begin{proof}
A general member $A$ of the boundary $\Delta{\mathcal{H}}$ is a
triple admissible cover of a chain of {\it two} ${\mathbf P}^1$.
(From the dimension calculations that follow it will become clear that
an admissible cover of a chain of three or more $\mathbf P^1$'s will
generate a subscheme in $\overline{\mathcal{H}}_{3,g}$ of codimension $\geq 2$.)
Note that {\it
three} connected components of $A$ over one ${\mathbf P}^1$ means that they are
all smooth ${\mathbf P}^1$'s themselves, and hence they can all be
contracted simultaneously, leaving us with a smooth trigonal curve, or
with a hyperelliptic curve with an attached ${{\mathbf P}}^1$, neither of which
cases by dimension count corresponds to a {\it general} member of
a boundary component $\Delta{\mathcal{H}}_{k,i}$. Considering all
combinations of one or two connected components of $A$ over each
${\mathbf P}^1$, we generate a list of the possible general members of
the boundary divisors $\Delta{\mathcal{H}}_{k,i}$.
To see which of these are indeed of codimension 1 in
$\overline{\mathcal{H}}_{3,g}$, we do the following calculation. First we
note that, for
a fixed set of $2i+4$ ramification points in ${\mathbf P}^1$, there are finitely
many covers of degree $3$ and genus $i$, that is,
\[\on{dim}\overline{\mathfrak{T}}_i=2i+4-3=2i+1.\]
Substracting $3$ takes
into account the projectively equivalent triples of points on ${\mathbf P}^1$.
In particular, $\on{dim}\overline{\mathfrak{T}}_g=2g+1$.
A similar agrument (with $2i+2$ ramification points) shows that for
the hyperelliptic locus:
\[\on{dim}\overline{\mathfrak{I}}_i=2i+2-3=2i-1.\]
These computations are valid for $i>0$, whereas $0=
\on{dim}\overline{\mathfrak{T}}_i=\on{dim}\overline{\mathfrak{I}}_i$.
\smallskip
Thus, to compute the dimensions of the six types of subschemes of
$\overline{\mathcal{H}}_{3,g}$, one adds the corresponding dimensions of
$\overline{\mathfrak{T}}_i$ and $\overline{\mathfrak{I}}_j$, making
the necessary adjustments for the choice of intersection points on
the components of each curve $A$. For example, when $i>0$
the dimension of the subscheme with general member $A$,
shown in Fig.~\ref{admissible-k,i}, is
\[\on{dim}\overline{\mathfrak{T}}_i+\on{dim}\overline{\mathfrak{T}}_{g-i-2}+1+1=2g.\]
The final 1's account for the choice of triples of points in the
$g^1_3$'s on each component. We conclude that for
$i=1,2,...,[(g-2)/2]$ the join at three points of two trigonal curves,
one of genus $i$ and the other of genus $g-i-2$, is the general member of
a boundary component of $\overline{\mathcal{H}}_{3,g}$. We denote it by
$\Delta{\mathcal{H}}_{1,i}$. The range of $i$ stops at
$[(g-2)/2]$ for symmetry considerations. When $i=0$, the corresponding
subscheme has a smaller dimension of $2g-2$ and hence no boundary divisor
is generated by such curves.
\smallskip
As another example, consider the fifth sketch in Fig.~\ref{admissible-k,i}.
It corresponds to the join at one point
of a trigonal curve $C_1$ of genus $i$, a hyperelliptic curve $C_2$
of genus $g-i$, and an attached ${\mathbf P}^1$ to $C_2$ to make the whole curve a
triple cover. Note that $C_1$ and $C_2$ intersect transversally at a point
$q$, but when presented as covers of ${\mathbf P}^1$ they both have
ramifications at $q$ of index 1. On all such curves $C_1$ and $C_2$
the total number of ramification points over ${\mathbf P}^1$ is finite, and
hence their choice does not affect the dimension of our subscheme. Thus,
\[\on{dim}\overline{\mathfrak{T}}_i+\on{dim}\overline{\mathfrak{I}}_{g-i}=2i+1+2(g-i)=
2g.\]
Therefore, this subscheme is in fact a divisor in
$\overline{\mathcal{H}}_{3,g}$, which we denote by $\Delta
{\mathcal{H}}_{5,i}$. The cases of $i=0$ or $i=g$ lead to
contractions of unstable rational components ($C_1$ or $C_2$), and do not
yield the necessary dimension of $2g$. Hence, $i=1,2,...,g-1$.
\smallskip
In the case of $\Delta{\mathcal{H}}_{6,i}$, the two components $C_1$ and
$C_2$ meet transversally in one point $q$, but both have ramification of
index $2$ at $q$ as triple covers of ${\mathbf P}^1$. Smooth trigonal curves of
genus $i$ with such high ramification form a codimension 1 subscheme of the
trigonal locus ${\mathfrak{T}}_i$, hence the dimension of
$\Delta{\mathcal{H}}_{6,i}$ is
\[\on{\dim}\overline{\mathfrak{T}}_i-1+\on{dim}\overline{\mathfrak{T}}_{g-i}-1=
(2i+1)-1+(2(g-i)+1)-1=2g.\]
Thus, $\Delta{\mathcal{H}}_{6,i}$ is a boundary divisor in
$\overline{\mathcal{H}}_{3,g}$ for $i=1,2,...,[g/2]$. The case of $i=0$ yields
dimension $2g-1$, and hence we disregard it.
\smallskip
The remaining cases are treated similarly. We conclude
that $\overline{\mathcal{H}}_{3,g}$ has
six types of boundary divisors, $\Delta{\mathcal{H}}_{k,i}$,
whose general members and range of indices
are indicated in Fig.~\ref{admissible-k,i}. \end{proof}
\subsubsection{Boundary of $\overline{\mathfrak{T}}_g$.
Proof of Proposition~\ref{boundary}}
\label{trigonalboundary}
Having described the boundary of $\overline{\mathcal{H}}_{3,g}$, it remains to
check which of the divisors $\Delta\mathcal{H}_{k,i}$ preserve their
dimension under the map $pr_1$ and hence map into divisors of $\overline
{\mathfrak{T}}_g$. The only ``surprises'' can be expected where $pr_1$
contracts unstable ${\mathbf P}^1$, such as in $\Delta{\mathcal{H}}_{2,i}$,
$\Delta{\mathcal{H}}_{3,i}$, and $\Delta{\mathcal{H}}_{5,i}$. In fact, only
$\Delta{\mathcal{H}}_{2,g-1}$ and $\Delta{\mathcal{H}}_{3,0}$ diverge from
the common pattern; in all other cases, we set $\Delta{\mathfrak{T}}_{k,i}:=
pr_1\left(\Delta{\mathcal{H}}_{k,i}\right)$
to be the corresponding boundary divisor in $\overline{\mathfrak{T}}_g$.
\smallskip
The map $pr_1$ contracts the three rational components of
the general member of $\Delta{\mathcal{H}}_{3,0}$, leaving only
a smooth hyperelliptic curve of genus $g$. Thus, the image
$pr_1\left(\Delta{\mathcal{H}}_{3,0}\right)$ is the hyperelliptic locus
$\overline{\mathfrak{I}}_g$, which is of dimension $2g-1$. Hence
$\Delta{\mathcal{H}}_{3,0}$ does not yield a divisor in
$\overline{\mathfrak{T}}_g$, but a boundary component of codimension 2.
\smallskip
Finally we consider $\Delta{\mathcal{H}}_{2,g-1}$. After we contract
its two rational components, we arrive at an {\it irreducible nodal}
trigonal curve with exactly one node. The dimension of the subscheme
of such curves is
\[\on{dim}\overline{\mathfrak{T}}_{g-1}+1=2(g-1)+1+1=2g,\]
where the final $1$ indicates the choice of a triple of points on
a smooth trigonal curve (belonging to the $g^1_3$), two of which
will be identified as a node. Correspondingly, we obtain another divisor
in $\overline{\mathfrak{T}}_g$, which we denote by $\Delta{\mathfrak{T}}_0.$
\qed
\subsection{Multiplicities of the boundary divisors $\Delta{\mathfrak{T}}_{k,i}$
in the restriction $\delta|_{\overline{\mathfrak{T}}_g}$}
\label{multiplicities}
By abuse of
notation, we will denote by $\delta_0$ and $\delta_{k,i}$
the classes in $\on{Pic}_{\mathbb{Q}}\overline{\mathfrak{T}}_g$
of $\Delta{\mathfrak{T}}_0$ and $\Delta{\mathfrak{T}}_{k,i}$, respectively.
\begin{prop} The divisor class $\delta\in\on{Pic}_{\mathbb{Q}}
\overline{\mathfrak{M}}_g$
restricts to $\overline{\mathfrak{T}}_g$ as the following linear combination
of the boundary classes in $\overline{\mathfrak{T}}_g$:
\begin{equation}
\delta|_{\displaystyle{\overline{\mathfrak{T}}_g}}=\delta_0+
\sum_{i=1}^{\scriptscriptstyle{[(g-2)/2]}}3\delta_{1,i}
+\sum_{i=1}^{\scriptscriptstyle{g-2}}2\delta_{2,i}
+\sum_{i=1}^{\scriptscriptstyle{[g/2]}}\delta_{3,i}
+\sum_{i=1}^{\scriptscriptstyle{[(g-1)/2]}}3\delta_{4,i}+
\sum_{i=1}^{\scriptscriptstyle{g-1}}\delta_{5,i}
+\sum_{i=1}^{\scriptscriptstyle{[g/2]}}\delta_{6,i}.
\label{divisorrel}
\end{equation}
\end{prop}
\noindent{\it Proof.} Let us rewrite equation (\ref{divisorrel}) in the form
\[\delta|_{\displaystyle{\overline{\mathfrak{T}}_g}}=(\on{mult}_{\delta}
\delta_0)\delta_0+\sum_{k,i}(\on{mult}_{\delta}\delta_{k,i})\delta_{k,i},\]
and call $\on{mult}_{\delta}\delta_{k,i}$ the {\it multiplicity} of
$\delta_{k,i}$ in $\delta|_{\overline{\mathfrak{T}}_g}$.
This linear relation simply counts the contribution
of each singular curve of a specific boundary type in $\Delta\mathfrak{T}_g$
to the degree of
$\delta$. Recall that for any trigonal family $f:X\rightarrow B$:
\[\on{deg}\delta|_B=\sum_{q\in X}m_q.\]
Here $m_q$ denotes the local analytic multiplicity of the total space of
$X$ nearby $q$ measured by the equation $xy=t^{m_q}$, where
$x$ and $y$ are local parameters on the singular fiber $X_b$, and
$t$ is a local parameter on $B$ near $b=f(q)$.
\smallskip
For each boundary class $\Delta{\mathfrak{T}}_{k,i}$
of $\overline{\mathfrak{T}}_g$, we consider its general member
$C\!=\!C_1\cup C_2$,
and a base curve $B$ in $\overline{\mathfrak{T}}_g$ which intersects
transversally $\Delta{\mathfrak{T}}_{k,i}$ in $[C]$. In the
corresponding one-parameter trigonal family $f:X\rightarrow B$,
we must find the sum of the multiplicities $m_q$ of the singularities
of $C$. Thus, \[\on{mult}_{\delta}\delta_{k,i}=\sum_{\,\,q\in
C_{\on{sing}}}\!\!m_q.\]
For most of the divisors classes, this sum is actually quite straight forward.
For example, the general member $[C]\in \Delta{\mathfrak{T}}_{3,i}$
is the join of two smooth hyperelliptic curves $C_1$ and $C_2$, which
intersect transversally in one point $q$. The family $X$,
constructed as above, will be given locally analytically nearby $q$ by
$xy=t$, and hence $\on{mult}_{\delta}\delta_{k,i}=m_q=1$.
A similar situation occurs in the cases of $\Delta\mathfrak{T}_0,
\Delta\mathfrak{T}_{5,i}$ and $\Delta\mathfrak{T}_{6,i}$: there is one point
of transversal intersection (or one node) forcing
\[\on{mult}_{\delta}\delta_0=
\on{mult}_{\delta}\delta_{k,i}=1\,\,\on{for}\,\, k=3,5,6.\]
\smallskip
In the cases of $\Delta\mathfrak{T}_{2,i}$ and $\Delta\mathfrak{T}_{1,i}$ there
are correspondingly two or three points of transversal intersection,
forcing \[\on{mult}_{\delta}\delta_{2,i}=2\,\,\on{and}\,\,
\on{mult}_{\delta}\delta_{1,i}=3.\] This can be also interpreted
by the fact that $\Delta\mathfrak{T}_{2,i}$ and
$\Delta\mathfrak{T}_{1,i}$ lie entirely in the divisor
$\Delta_0$ in $\overline{\mathfrak{M}}_g$ with,
$\Delta_0$ being {\it double} along $\Delta\mathfrak{T}_{2,i}$ and
{\it triple} along $\Delta\mathfrak{T}_{1,i}$.
\medskip
A slightly more complex situation occurs in the case of
$\Delta\mathfrak{T}_{4,i}$. The general member $C$ consists of two curves
$C_1$ and $C_2$, meeting transversally in two points $q$ and $r$
(see Fig.~\ref{mult4}). But,
as in an admissible triple cover of two ${\mathbf P}^1$'s, the points $q$ and
$r$ behave differently: at one of them, say $r$, the triple cover
is {\it not} ramified, while at $q$ there is ramification of index $1$.
In the local analytic rings of $p,q$ and $r$
the generators of $\widehat{\mathcal{O}}_{Y,p}$ map into the squares of
the generators of $\widehat{\mathcal{O}}_{X,q}$: $u\mapsto x^2, v\mapsto y^2$, and
of course, $t\mapsto t$, so that the local equation of $Y$ near $p$ is
$uv=t^2$, and that of $X$ near $q$ is $xy=t$. But since the triple cover
is a local isomorphism of $\widehat{\mathcal{O}}_{Y,p}$ into $\widehat{\mathcal{O}}
_{X,r}$, the total space of $X$ near $r$ is given locally analytically
by $zw=t^2$ ($u\mapsto z, v\mapsto w, t\mapsto t$).
Therefore, $m_q=1$, but $m_r=2$, and
\[\on{mult}_{\delta_0}\delta_{4,i}=m_q+m_r=3.\,\,\,\qed\]
\begin{figure}
$$\psdraw{mult4}{2.8in}{1.3in}$$
\caption{The multiplicity mult$_{\delta_0}\delta_{4,i}$}
\label{mult4}
\end{figure}
\subsection{The hyperelliptic locus $\overline{\mathfrak{I}}_g$ inside
$\overline{\mathfrak{T}}_g$}
\label{hyperelliptic locus}
Although the relations proved in this paper
will be valid on the Picard group $\on{Pic}_{\mathbb
Q}\overline{\mathfrak{T}}_g$, it will be interesting to check what happens
with the hyperelliptic curves inside the trigonal locus
$\overline{\mathfrak{T}}_g$.
We noted that $\overline{\mathfrak{I}}_g$ is the
only boundary component of $\overline{\mathfrak{T}}_g$ of
codimension 2. It is obtained as the image $pr_1(\Delta{\mathcal H}_{3,0})$.
By blowing up a point on a smooth hyperelliptic
curve $C$, we add a $\mathbf P^1$--component to $C$ to make it a triple cover
$C^{\prime}$ (cf.~Fig.~\ref{smoothhyper}).
It terms of the quasi-admissible covers, such $C^{\prime}$
behaves exactly as an irreducible singular trigonal curve in
$\Delta{\mathfrak{T}}_0$. However, $C$
does not contribute to the invariant $\delta|_B$, as defined in
Section~\ref{definition}. In fact, in a certain sense,
it even decreases $\delta|_B$.
To simplify the exposition, we shall postpone the discussion
of families with hyperelliptic fibers until Section~11, where
we will explain the behavior of trigonal families with finitely many
hyperelliptic fibers in terms of the exceptional divisor $\Delta{\mathcal H}_{3,0}$
of the projection $pr_1$.
A similar phenomenon occurs with the boundary component
$\Delta\mathfrak{T}_{1,0}=pr_1(\Delta\mathcal{H}_{1,0})$, but it does not
make sense to exclude its members from our discussion, since they behave
exactly as members of the boundary divisor $\Delta\mathfrak{T}_{1,i}$ for
$i\geq 1$.
\subsection{The invariants $\mu(C)$}
\label{The invariants} In the transition from the original
family $X\rightarrow B$ to the minimal quasi-admissible family
$\widetilde{X}\rightarrow \widetilde{Y}$ over $\widetilde{B}$, certain
changes occur in the calculation of the basic invariants. To start with,
it is easy to redefine
$\lambda_{\widetilde{X}},\kappa_{\widetilde{X}}$ and
$\delta_{\widetilde{X}}$ for $\widetilde{X}\rightarrow \widetilde{B}$:
simply use the corresponding definitions from Section~\ref{definition}.
Since we are interested in the slope of the family, which
is invariant under base change, we may assume that $\widetilde{B}:=B$
and that $X$ is the pull-back over the new base $\widetilde{B}$.
Now the difference between $X$ and $\widetilde{X}$
is reduced to several ``quasi-admissible'' blow-ups on $X$.
\smallskip
Blowing up smooth or rational double points
on a surface does not affect its structure sheaf. Therefore,
the degrees of the Hodge bundles on the two surfaces $X$ and $\widetilde{X}$
will be the same: $\lambda_{\widetilde{X}}=\lambda_X$.
On the other hand, blowing up a smooth point on a surface
decreases the square of its dualizing sheaf by 1, while there is no effect when
blowing up a rational double point. Each type of singular fibers $C$ in $X$
requires apriori
different quasi-admissible modifications (or no modifications at all),
and thus decreases $\kappa_X$ by some nonnegative integer, denoted by
$\mu(C)$:
\begin{equation}
\kappa_X=\kappa_{\widetilde{X}}+\sum_{C}\mu(C).
\end{equation}
Thus, $\mu(C)$ counts the number
of ``smooth blow-ups'' on $C$, which are needed to obtain
the minimal
quasi-admissible cover $\widetilde{C}\rightarrow C$
within the surface quasi-admissible cover $\widetilde{\phi}:
\widetilde{X}\rightarrow \widetilde{Y}$.
\smallskip In the following Lemma, we compute the invariants $\mu(C)$
only for the general members of the boundary $\Delta{\mathfrak{T}}_g$
(cf.~Fig.~\ref{Delta-k,i}). The remaining, more special, singular curves
in $\Delta{\mathfrak{T}}_g$ will be linear combinations of these $\mu(C)$'s
(cf.~Sect.~11).
\begin{lem} If $\mu_{k,i}$ denotes the invariant $\mu(C)$ for a general
curve $C\in\Delta\mathfrak{T}_{k,i}$, then
\begin{eqnarray*}
\on{(a)}&&\mu_0=\mu_{1,i}=\mu_{4,i}=\mu_{6,i}=0;\\
\on{(b)}&&\mu_{2,i}=1;\\
\on{(c)}&&\mu_{3,i}=\mu_{5,i}=2.
\end{eqnarray*}
\label{mu(C)}\vspace*{-5mm}
\end{lem}
\noindent{\it Proof.} The general members of the boundary
$\Delta{\mathcal H}$ are in fact the minimal quasi-admissible covers associated to
the general members of the boundary $\Delta{\mathfrak{T}}$, except for
$\Delta_0$ which has $\mu_0=0$. Thus, we trace the blow-ups necessary to
transform the curves in Fig.~\ref{Delta-k,i} to the curves in
Fig.~\ref{admissible-k,i}. For example, no blow-ups are needed in the case
of $\Delta_{1,i}$, so that $\mu_{1,i}=0$, while we need 2 blow-ups in
the case of $\Delta_{3,i}$, and hence $\mu_{3,i}=2$.
\smallskip
The only interesting situation occurs for $\Delta_{5,i}$. Apparently,
there is only {\it one} added component $\mathbf P ^1$ to the original
$C\in\Delta_{3,i}$, but the lemma states that $\mu_{3,i}=2$. The difference
comes from the fact that near the intersection $r=C\cup\mathbf P^1$
the surface $\widetilde{X}$ has equation $xy=t^2$, i.e. $r$ is a rational
double point on $\widetilde{X}$ of type $A_1$ (a similar situation occurred
in Fig.~\ref{mult4}). To obtain such a point $r$ in place of a smooth
point $r_1$ on $X$, we first blow up $r_1$, and then on the
obtained exceptional divisor we blow up another point $r_2$, so as to end
with a {\it chain of two} $\mathbf P^1$'s (cf.~Fig.~\ref{mu-5,i}). Finally,
we blow down the first $\mathbf P^1$, and develop the required rational
double point $r$. As a result, we have two ``smooth'' and one
``singular'' blow-ups, which implies $\mu_{5,i}=2.$ \qed
\bigskip
\begin{figure}[h]
$$\psdraw{mu}{3in}{1in}$$
\caption{Quasi-admissible blow-ups on $\Delta_{5,i}$}
\label{mu-5,i}
\end{figure}
\section*{\hspace*{1.9mm}5. Effective Covers}
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\label{effectivecovers}
In this section we construct the final type of triple covers in
the Basic Construction. These will not be necessary for the global
calculation in Section~7, so the reader may wish to skip
this more technical part on a first reading, and assume in
Section~6 that all covers are flat.
\subsection{Construction of
effective covers $\widehat{X}\rightarrow\widehat{Y}$}
\label{constructioneffective} Consider a quasi-admissible cover
$\widetilde{\phi}:\widetilde{X}\rightarrow \widetilde{Y}$, as given in
Prop.~\ref{propquasi}. In order to use the fact that the pushforward
$\widetilde{\phi}_*{\mathcal{O}_{\widetilde{X}}}$
is locally free on $\widetilde{Y}$, we need to assure that the map
$\widetilde{\phi}$ is {\it flat}. Unfortunately, there are certain points
on $\widetilde{X}$ where this fails to be true: exactly where the fibers
of $\widetilde{X}$ are ramified as triple covers of the corresponding
fibers of $\widetilde{Y}$. The situation can be resolved by several further
blow-ups.
\smallskip
Namely, we work locally analytically near the points in $\widehat{X}$
of ramifications index 1 or 2, and consider correspondingly two cases.
\subsubsection{Case of ramification index 1}
\label{caseram1}
This case involves
members of the boundary divisors $\Delta{\mathfrak{T}}_{4,i}$ and
$\Delta{\mathfrak{T}}_{5,i}$. Let $q$ be the point of ramification in the
fiber of $\widetilde{X}$ over the point $p$ in the fiber of $\widetilde{Y}$
(cf.~Fig.~\ref{ram}).
We use the pull-back
of the map $\widetilde{\phi}$ to study the embedding of the completion of
the local ring of $p$ into that of $q$:
\[\widehat{\mathcal{O}}_{\widetilde{Y}\!,p}=\mathbb{C}[[u,v,t]]
\big/_{\displaystyle{(uv-t^2)}}
\stackrel{\widetilde{\phi}^{\#}}
{\hookrightarrow}\widehat{\mathcal{O}}_{\widetilde{X}\!,q}=
\mathbb{C}[[x,y,t]]\big/_{\displaystyle{(xy-t)}}.\]
\begin{figure}[t]
$$\psdraw{ram}{2.7in}{1.2in}$$
\caption{Non-flat cover of ramification index 1}
\label{ram}
\end{figure}
\noindent Therefore, as an $\widehat{\mathcal{O}}_{\widetilde{Y}\!,p}$-module,
\[\widehat{\mathcal{O}}_{\widetilde{X}\!,q}= \widehat{\mathcal{O}}_{\widetilde{Y}\!,p}+
\widehat{\mathcal{O}}_{\widetilde{Y}\!,p}x+
\widehat{\mathcal{O}}_{\widetilde{Y}\!,p}y.\]
However, this is not a locally-free
$\widehat{\mathcal{O}}_{\widetilde{Y},p}$-module: for instance, one
relation among the generators is $(v-t)x+(u-t)y=0$.
\smallskip
Alternatively, the fiber of $\phi$ over $p$ is supported at $q$, but
it is of degree 3 rather than 2,
which would have been necessary for the flatness of a degree $2$ map.
Indeed, as $\mathbb{C}-$vector spaces:
\[\widehat{\mathcal{O}}_{\widetilde{X}\!,q}\otimes_{\widehat{\mathcal{O}}_
{\widetilde{Y}\!,p}}
\on{Spec}k(p)\cong \widehat{\mathcal{O}}_{\widetilde{X}\!,q}\big/_{\displaystyle
{\widehat{\mathfrak{m}}_{Y\!,p}\widehat{\mathcal{O}}_{\widetilde{X}\!,q}}}\cong
\mathbb{C}[[x,y]]\big/_{\displaystyle{(x^2,y^2,xy)}}=\mathbb{C}\oplus
\mathbb{C}x\oplus\mathbb{C}y.\]
In Fig.~\ref{ram} one can visually observe the two distinct tangent
directions at $q$ making it a {\it fat} point of degree $3$.
\medskip
We conclude that $\widetilde{\phi}$ is indeed
non-flat at $q$. To resolve this, we blow-up $\widetilde{Y}$ at $p$
and $\widetilde{X}$ at $q$, denoting the new surfaces by $\widehat{Y}$ and
$\widehat{X}$. It is easy to see that they fit into the following coming
diagram:
\smallskip
\begin{figure}[h]
$$\psdraw{blow1}{3.5in}{1.4in}$$
\caption{Resolving the case of ram. index 1}
\label{resolve1}
\end{figure}
\smallskip
In order to keep the map $\widehat{\phi}:\widehat{X}\rightarrow \widehat{Y}$
of degree 3, we need to blow-up one further point on $\widetilde{X}$:
if the inverse image of $p$ is $\{q,r\}$ we blow-up $r$, and thus we add
the necessary component to $\widehat{X}$ to make it a triple cover of
$\widehat{Y}$ (cf.~Fig~\ref{coef2.fig}).
\medskip
\subsubsection{Case of ramification index 2}
\label{caseram2}
The only boundary component,
where ramification index 2 occurs, is $\Delta{\mathfrak{T}}_{6,i}$.
Similarly as above, $\widetilde{\phi}:
\widetilde{X}\rightarrow \widetilde{Y}$ is non-flat at $q$. Indeed,
$\widehat{\mathcal{O}}_{\widetilde{X}\!,q}$ is generated as an
$\widehat{\mathcal{O}}_{\widetilde{Y}\!,p}$-module
by $1,x,y,x^2,y^2$, but not-freely
due to the relation $u\cdot x+v\cdot y-t\cdot x^2-t\cdot y^2=0$. To
resolve the apparent non-flatness of $\widetilde{\phi}$, we can blow-up
once $\widetilde{X}$ and $\widetilde{Y}$ at
$q$ and $p$, but this would not be sufficient. In fact, we must make
further blows-ups on each surface, as Fig.~\ref{resolve2} suggests:
two more on $\widetilde{X}$ and one more on $\widetilde{Y}$.
\medskip
\begin{figure}[h]
$$\psdraw{blow2}{5.3in}{1.4in}$$
\caption{Resolving the case of ram. index 2}
\label{resolve2}
\end{figure}
\medskip
In both cases of ramification index 1 or 2,
the new map $\widehat{\phi}:\widehat{X}\rightarrow
\widehat{Y}$ is obtained from $\widetilde{\phi}$ by a base change, and hence
$\widehat{\phi}$
is {\it proper} and {\it finite}, and by construction, also a {\it flat}
morphism. We call such covers {\it effective}.
\medskip
The above considerations combined with Prop.~\ref{quasicov}
imply the existence of effective covers for our families of trigonal curves:
\begin{prop} Let $X/\!_{\displaystyle{B}}$ be a family of
trigonal curves with smooth general member. After several blow-ups (and
possibly modulo a base change) we can associate to it an effective cover
$\widehat{\phi}:\widehat{X}\rightarrow \widehat{Y}$.
\label{effexist}
\end{prop}
Here $\widehat{Y}$ is a birationally ruled surface over $B$.
If the base curve $B$ is {\it not} tangent to the boundary divisors $\Delta
{\mathfrak{T}}_{k,i}$, then the resulting surfaces $\widehat{X}$ and
$\widehat{Y}$ will have smooth total spaces. If, moreover, $B$ intersects
the $\Delta_{k,i}$'s only in their general points (as given in
Fig.~\ref{Delta-k,i}), then the special fibers of $\widehat{Y}$ and
$\widehat{X}$ are easy to describe
(cf.~Fig.~\ref{coef1.fig}-\ref{coef3.fig}). For example, $\widehat{Y}$'s
special fibers are either
chains of two or three reduced projective lines, or chains of five
smooth rational curves with non-reduced middle component of
multiplicity two. The special fibers of $\widehat{X}$ can also contain
nonreduced components (of multiplicity 2 or 3), and this occurs only in the
ramification cases discussed above ($\Delta{\mathfrak{T}}_{k,i}$ for $k=4,5,6$).
\subsection{Change of $\lambda_X,\kappa_X$ and $\delta_X$ in
the effective covers}
\label{change}
This is an analog to the discussion in Section~\ref{The invariants}.
After the necessary base changes
we again identify, without loss of generality,
the new base curve $\widetilde{B}$ with $B$,
and the pull-back of $X$ over $\widetilde{B}$ with $X$, and we redefine
the basic invariants $\lambda_{\widehat{X}}$ and $\kappa_{\widehat{X}}$
for the effective family $\widehat{X}$ over $\widetilde{B}$.
(It doesn't
make sense to define directly $\delta_{\widehat{X}}$, because of the
nonreduced fiber components in $\widehat{X}$. We could, of course, set
$\delta_{\widehat{X}}=12\lambda_{\widehat{X}}-\kappa_{\widehat{X}}$,
but we will not need this in the sequel.) Now the original $X$ and
the effective $\widehat{X}$ differ by ``quasi-admissible'' and ``effective''
blow-ups. The connections between the invariants of $X$, $\widetilde{X}$ and
$\widehat{X}$ are expresssed by the following
\begin{lem} With the above notation,
\begin{eqnarray*}
\on{(a)}&\!\!\!\!&\displaystyle{\lambda_X}=\lambda_{\widetilde{X}}=
\lambda_{\widehat{X}};\\
\on{(b)}&\!\!\!\! &\kappa_X=\kappa_{\widetilde{X}}+\sum_{C}\mu(C);\\
\on{(c)}&\!\!\!\! &\displaystyle{\kappa_{\widetilde{X}}=
\kappa_{\widehat{X}}+\sum_{\on{ram}1}1+\sum_{\on{ram}2}3}.
\end{eqnarray*}
\label{changeinv}\vspace*{-10mm}
\end{lem}
\begin{proof} In view of Lemma~\ref{mu(C)}, the first and the
second statements are obvious. Obtaining a flat cover
$\widehat{X}\rightarrow \widehat{Y}$ requires blowing up on $\widetilde{X}$
one smooth point for each
ramification index 1, and three smooth points for each ramification
index 2. Hence the relation between $\kappa_{\widehat{X}}$ and
$\kappa_{\widetilde{X}}.$ \end{proof}
\section*{\hspace*{1.9mm}6.
Embedding $\widehat{X}$ in a Projective Bundle over $\widehat{Y}$}
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\label{embedding}
Given the effective degree $3$ map $\widehat{\phi}:\widehat{X}\rightarrow
\widehat{Y}$, our next step is to embed
$\widehat{X}$ into a projective bundle $\mathbf P V$ of rank $1$ over
the birationally ruled surface $\widehat{Y}$. We shall consider a degree
$3$ morphism $\widehat{\phi}$, but the same
discussion is valid for any degree $d$.
\subsection{Trace map}
\label{tracemap} Since $\widehat{\phi}$ is flat and finite,
the pushforward $\widehat{\phi}_*(\mathcal O_{\widehat{X}})$
is a locally free sheaf on $\widehat{Y}$ of rank $3$.
Define the {\it trace} map
\[\on{tr}:\widehat{\phi}
_*(\mathcal O_{\widehat{X}})\rightarrow \mathcal O_{\widehat{Y}}\]
as follows. The finite field extension $K(\widehat{X})$ of
$K(\widehat{Y})$ induces the {\it algebraic} trace map
$\on{tr^\#}:K(\widehat{X})\rightarrow K(\widehat{Y})$, defined by
$\on{tr^\#}(a)=\textstyle{\frac{1}{3}}(a_1+a_2+a_3)$. Here
the $a_i$'s are the conjugates of $a$ over $K(\widehat{Y})$
in an algebraic closure of $K(\widehat{X})$. The restriction
$\on{tr^\#}{|}_{K(\widehat{Y})}=\on{id}_{K(\widehat{Y})}$.
Over an affine open $U={\on {Spec}}\,A\subset
\widehat{Y}$ and its affine inverse $\widehat{\phi}^{-1}(U)=
{\on {Spec}}\,B\subset
\widehat{X}$, $B$ is the integral closure of $A$ in its field of fractions
$K(\widehat{X})$. Therefore,
the trace map restricts to the $A$-module homomorphism
$\on{tr^\#}:B\rightarrow A$. We have a commutative diagram:
\medskip
\begin{figure}[h]
\vspace*{2mm}
\begin{picture}(4,2.5)(4,0)
\put(7.8,2.2){$B\hspace{1mm}\hookrightarrow \hspace{1mm}K(\widehat{X})$}
\multiput(8,1.2)(1.6,0){2}{\vector(0,1){0.9}}
\put(7.8,0.7){$A\hspace{1mm}\hookrightarrow \hspace{1mm}K(\widehat{Y})$}
\put(7.8,1.65){\oval(0.35,0.9)[l]}
\put(7.8,1.2){\vector(1,0){0.1}}
\put(7.8,2.1){\line(1,0){0.1}}
\put(9.8,1.65){\oval(0.35,0.9)[r]}
\put(9.8,1.2){\vector(-1,0){0.1}}
\put(9.8,2.1){\line(-1,0){0.1}}
\multiput(7,1.5)(3.1,0){2}{$\on{tr}^{\#}$}
\put(3.2,0.7){$U=\on{Spec}\,A$}
\put(2.3,2.2){$\widehat{\phi}^{-1}(U)=\on{Spec}\,B$}
\put(3.7,1.5){$\big\downarrow$}
\put(3.7,1.4){$\big\downarrow$}
\put(3.3,1.5){$\widehat{\phi}$}
\end{picture}
\vspace*{-6mm}
\caption{The trace map}
\end{figure}
The so-defined local maps $\on{tr}:\widehat{\phi}_*\mathcal O_{\on {Spec}\,B}
\twoheadrightarrow
\mathcal O_{\on {Spec}\,A}$ patch up to give a global trace map
$\on{tr}:\widehat{\phi}_*\mathcal O_{\widehat{X}}
\twoheadrightarrow \mathcal O_{\widehat{Y}}.$
Let $\mathcal V$ be the kernel of $\on{tr}$:
\begin{equation}
0\rightarrow {\mathcal V}\rightarrow {\widehat{\phi}}
_*{\mathcal O}_{\widehat X}\stackrel
{\on{tr}}{\rightarrow}{\mathcal O}_{\widehat Y}\rightarrow 0.
\label{splitting}
\end{equation}
Note that $\mathcal V$ is locally free of rank $2$.
The natural inclusion
$\mathcal O_{\widehat{Y}}\hookrightarrow \widehat{\phi}
_*\mathcal O_{\widehat{X}}$, composed
with $\on{tr}$, is the identity on $\mathcal O_{\widehat{Y}}$, hence the
exact sequence splits:
\begin{equation}
{\widehat{\phi}}_*{\mathcal O}_{\widehat X}={\mathcal O}_{\widehat Y}\oplus {\mathcal V}.
\label{directsum}
\end{equation}
\smallskip
\subsubsection{Geometric interpretation of the trace map}
It is useful to interpret the trace map geometrically in terms of the
corresponding vector bundles $\widehat{\phi}_*{O_{\widehat X}}$, $O_{\widehat Y}$ and
$V$ associated to the sheaves
$\widehat{\phi}_*{\mathcal O}_{\widehat X}$, ${\mathcal O}_{\widehat Y}$ and $\mathcal V$.
We again localize over affine opens,
and if necessary, we shrink $U=\on{Spec}\,A$ so that $\widehat{\phi}_*
\mathcal O_{\widehat X}$ becomes a {\it free} $\mathcal O_{\widehat Y}$-module.
\smallskip
Let $p$ be a closed point in $\on{Spec}\,A$ with maximal ideal
$\mathfrak{p} \subset A$, having three distinct preimages $q,r,s\in\on{Spec}\,B$
with maximal ideals $\mathfrak{q,r,s}\subset B$.
Since $B$ is a free $A$-module, the
quotient $B/{\mathfrak{p}}B$ is a 3-dim'l algebra over the ground
field $\on{k}(p)=A/{\mathfrak{p}}$, i.e. a 3-dim'l vector space over ${\mathbb C}$.
On the other hand, from $\mathfrak{qrs}=\mathfrak{q}\cap
\mathfrak{r}\cap \mathfrak{s}$ and the Chinese Remainder Theorem, it is clear that
$B/{\mathfrak{p}}B\cong B/{\mathfrak{q}} \oplus B/{\mathfrak{r}} \oplus
B/{\mathfrak{s}}\cong {\mathbb C}\overline{f}_q\oplus{\mathbb
C}\overline{f}_r\oplus{\mathbb C}\overline{f}_s.$
The generators $\overline{f}_q,\overline{f}_r,\overline{f}_s$ are
chosen as usual: $\overline{f}_q$, for instance,
is the residue in $\on{k}(q)$
of a function $f_q\in B$ such that $f_q\equiv 1(\on{mod}\mathfrak{q}),\,\,
f_q\equiv 0(\on{mod}\mathfrak{r,s})$.
\smallskip
In the Groethendieck style, the vector bundle
over $\widehat Y$ associated to $\widehat{\phi}_*\mathcal O_{\widehat X}$ is
$\on{ Spec}\on{S}(B_A)$, where $\on{S}(B_A)$ is the symmetric algebra
of $B$ over $A$. Its fiber over $p$ is the pull-back
$\on{ Spec}\on{S}(B_A)\times _{\on{Spec\,A}}\on{Spec\,k}(p) =
\on{ Spec}(\on{S}(B_A)\times_A A/\mathfrak {p})=\on{Spec}\on{S(B}/{\mathfrak{p}}B).$
We prefer to work with the dual $\widehat{\phi}_*{O_{\widehat X}}$
of this bundle, and the same goes for
projectivizations: we projectivize the 1-dim'l subspaces of
$\widehat{\phi}_*{O_{\widehat X}}$ rather than its 1-dim'l quotients.
In view of this convention, the fiber of the bundle
$\widehat{\phi}_*{O_{\widehat X}}$ is canonically identified as
\[(\widehat{\phi}_*{O_{\widehat X}})_p=B/{\mathfrak p}B\cong {\mathbb C}\overline{f}_q
\oplus{\mathbb
C}\overline{f}_r\oplus{\mathbb C}\overline{f}_s \cong {\bf A}^3_{\mathbb C}.\]
The generators $\overline{f}_q,\overline{f}_r,\overline{f}_s$
span three lines in ${\bf A}^3_{\mathbb C}$, which can be
canonically described: the line $\Lambda_q=
{\mathbb C}\overline{f}_q$, for example, corresponds
to all functions regular at $q,r$ and $s$, and vanishing at $r$ and $s$.
\smallskip
Similarly, the vector bundle $O_{\widehat Y}$ associated to
$\mathcal O_{\widehat Y}$ has fiber $(O_{\widehat Y})_p=A/{\mathfrak {p}}
\cong {\mathbb C}\overline{h}_p$, where $h_p$ is a
function near $p$ having residue $h_p(p)=1$ in $\on{k}(p)$. The trace
map $\on{tr}:\widehat{\phi}_*\mathcal O_{\widehat{X}}
\twoheadrightarrow \mathcal O_{\widehat{Y}}$ translates fiberwise in terms
of the vector bundles $\widehat{\phi}_*{O_{\widehat X}}$ and
$O_{\widehat Y}$ as:
\[\on{tr}_p:{\mathbb C}\overline{f}_q\oplus{\mathbb
C}\overline{f}_r\oplus{\mathbb C}\overline{f}_s \rightarrow
{\mathbb C}\overline{h}_p,\,\,\overline{f}\mapsto \frac{1}{3}
(f(q)+f(r)+f(s)).\]
Finally, the locally free subsheaf ${\mathcal V}=\on{Ker(tr)}\subset
\widehat{\phi}_*{\mathcal
O}_{\widehat X}$ is associated to a vector bundle $V$ with fiber
$V_p=\{\overline{f}\,\,|\,\,f(q)+f(r)+f(s)=0\}
\subset (\widehat{\phi}_*{O_{\widehat X}})_q.$
Equivalently, from the direct sum (\ref{directsum}),
$V_p=(\widehat{\phi}_*{O_{\widehat X}})_p \big{/}_{\textstyle{\Lambda}}$, where the line
$\Lambda=\{\overline{f}\,\,|\,\,f(q)=f(r)=f(s)\}$
corresponds to pull-backs of functions regular at $p$.
\begin{figure}[h]
$$\psdraw{embedding}{3in}{1.88in}$$
\caption{Geometric interpretation of $tr$}
\label{geometric}
\end{figure}
Since the four lines $\Lambda_q,\Lambda_r,\Lambda_s$
and $\Lambda$ are in general position in the fiber $(\widehat{\phi}
_*{O_{\widehat X}})_p$, modding out by $\Lambda$ yields three
distinct lines in the fiber $V_p$ (cf.~Fig.~\ref{geometric}).
Therefore, projectivizing $V_p$ naturally induces three distinct points
$Q,R,S$ in the fiber ${\mathbf P}^1$ of ${\mathbf P}V$.
Going the other way around the diagram, we first projectivize
$(\widehat{\phi}_*{O_{\widehat X}})_q\cong{\bf A}^3$,
and then we project from the point
$[\Lambda]$ onto the fiber of ${\mathbf P}V$. In other words, $\pi_{[\Lambda]}
:{\mathbf P}^2\dashrightarrow {\mathbf P}^1$ is well-defined at
$[\Lambda_q],[\Lambda_r]$ and $[\Lambda_s]$.
\medskip This completes the interpretation of the trace map in the case
of three distinct preimages $q,r,s$ in $\widehat{X}$.
In case of only two or one preimage of $p$ in $\widehat{X}$, one modifies
correspondingly the above interpretation.
\subsection{$\widehat{X}$ embeds naturally in
$\mathbf P V$ over $\widehat{Y}$}
\label{naturally}
We construct the map $i:\widehat{X}\hookrightarrow \mathbf P V$ via the use of
an invertible relative dualizing sheaf $\omega_{\widehat X/ \widehat Y}$.
Its existence imposes a mild technical condition on the schemes $\widehat X$
and $\widehat Y$: we assume that they are Gorenstein, i.e. Cohen-Macaulay
with invertible dualizing sheaves $\omega_{\widehat X/{\mathbb C}}$ and
$\omega_{\widehat Y/{\mathbb C}}$.
In our situation this will be sufficient. As we noted in
Section~\ref{constructioneffective}, when the base curve $B$ is {\it not}
tangent to the boundary divisors $\Delta{\mathfrak{T}}_{k,i}$, then
$\widehat{X}$ and $\widehat{Y}$ are smooth surfaces. The remaining
cases are ``local'' base changes of these, and the construction
carries over.
\begin{prop}
Let $\widehat{\phi}:\widehat X\rightarrow \widehat Y$ be a flat and finite
degree $d$ morphism of Gorenstein schemes, with $\widehat Y$ integral. Then
$\widehat{\phi}$ factors through a natural embedding of $\widehat X$ into
the projective bundle $\mathbf P V$, followed by the projection $\pi:
\mathbf P V\rightarrow \widehat Y$ (cf.~Fig.~\ref{basicconstruction}).
\label{propembedding}
\end{prop}
For easier referencing in the sequel,
we have kept the notation $\widehat X$ and $\widehat Y$,
but the statement is true for {\it any} schemes
satisfying the Gorenstein condition. For another proof of
Prop.~\ref{propembedding}, see \cite{embedding}.
\medskip
\noindent{\it Proof of Prop.~\ref{propembedding}}.
Here we construct the map $i:\widehat X\rightarrow \mathbf P V$,
give the proof of its regularity, and point out how to show its injectivity.
\begin{figure}[h]
\begin{picture}(6,2.5)(-2.3,2.5)
\put(-1.7,4){$\mathbf P({\mathcal O}_{\widehat Y})\stackrel
{\on{tr}^{\#}}{\hookrightarrow}\mathbf P({\widehat{\phi}}_*{\mathcal O}_{\widehat X})
\stackrel{\rho}{\dashrightarrow} \mathbf P V$}
\put(0.95,3.05){\vector(0,1){0.8}}
\put(0.75,2.6){$\widehat X$}
\put(0.6,3.35){$\psi$}
\put(1.1,2.95){\vector(2,1){1.85}}
\put(1.7,3.35){$i$}
\end{picture}
\caption{Embedding $i:\widehat X\hookrightarrow \mathbf P V$}
\label{construction of i}
\end{figure}
\subsubsection{Construction of the embedding map}
According to Prop. II.7.12
\cite{Hartshorne}, to give
a morphism $\psi:\widehat X\rightarrow {\mathbf P}(\widehat{\phi}
_*(\mathcal O_{\widehat X}))$ over $\widehat Y$ is equivalent to give an
invertible sheaf $\mathcal L$ on $\widehat Y$ and a surjective map of sheaves
$\widehat{\phi}^*(\widehat{\phi}_*(\mathcal O_{\widehat X}){\textstyle
{\widehat{\phantom{n}}}})\twoheadrightarrow \mathcal L$. Recall from {\it
relative Serre-duality} that $(\widehat{\phi}_*\mathcal O_{\widehat X}){\textstyle
{\widehat{\phantom{n}}}}\cong \widehat{\phi}
_*\omega_{\widehat X/\widehat Y}$, and let
$\mathcal L=\omega_{\widehat X/\widehat Y}$. The natural morphism
\[\sigma:\widehat{\phi}^*\widehat{\phi}
_*\omega_{\widehat X/\widehat Y}\rightarrow\omega_{\widehat X/\widehat Y}\]
is {\it surjective}. This is in fact true for any quasicoherent sheaf
$\mathcal F$ on $\widehat X$. Indeed, restricting to the
affine open sets $\widehat{\phi}:\on{Spec}\,B\rightarrow \on{Spec}\,A$, we have
$\mathcal F=M^{\sim}$ for some finitely generated $B$-module $M$, and
$\widehat{\phi}^*\widehat{\phi}
_*{\mathcal F}=\widehat{\phi}^*(M_A)^{\sim}=(M_A\otimes_A B)^{\sim}.$
The surjective $B$-module homomorphism $M_A\otimes_A B \twoheadrightarrow M$,
given by $m\otimes b \mapsto b\circ m$, induces
$\widehat{\phi}^*\widehat{\phi}_*{\mathcal F}\twoheadrightarrow \mathcal F$.
Thus, the above map $\sigma$ naturally defines a morphism
$\psi:\widehat X\rightarrow {\mathbf P}(\widehat{\phi}_*(\mathcal O_
{\widehat X}))$ over $\widehat Y$. Projectivizing
$0\rightarrow {\mathcal O}_{\widehat Y} \rightarrow {\widehat{\phi}}
_*{\mathcal O}_{\widehat X}\rightarrow {\mathcal V} \rightarrow 0$
gives a sequence of projective bundles, as in Fig.~\ref{construction of i}.
The map $\rho$ is undefined exactly on the image of $\on{tr}
^{\#}$. Composing $\rho$ with the map $\psi$ yields the
map $i:\widehat{X}\dasharrow \mathbf P V$, which we claim is
a regular map.
\subsubsection{Regularity and injectivity of $i$.}
To see regularity, we show that the restriction
of $\sigma|_{\widehat{\phi}^*(\mathcal V{{\widehat{\phantom{n}}}})}
:{\widehat{\phi}^*(\mathcal V{{\widehat{\phantom{n}}}})}
\rightarrow \omega_{\widehat X/\widehat Y}$ is also
surjective. Indeed, we again work locally, and let $B=A\oplus C$
be the decomposition of $B$ via the trace map as a free $A$-module, where
$C=A\cdot b_1\oplus A\cdot b_2$ with $\on{tr}b_1=\on{tr}b_2=0$.
Let $\omega_{\widehat X/\widehat Y}=M^{\sim}$ for some finitely
generated $B$-module $M$. Recall that
$\widehat{\phi}_*\omega_{\widehat X/\widehat Y}\cong
(\widehat{\phi}_*\mathcal O_{\widehat X}){\textstyle{\widehat{\phantom{n}}}}$,
so that as $A$-modules: $M\cong (B_A){{\widehat{\phantom{n}}}}=\on{Hom}_A
(B,A)$, and $B$ acts on $M$ by
\[(b\circ f)(x)=f(bx)\,\,\,\on{for}\,\,\,f\in \on{Hom}_A(B,A),\,x\in B.\]
Naturally, the sheaf $\mathcal V=C^{\sim}$, and $\widehat{\phi}^*
(\mathcal V{{\widehat{\phantom{n}}}})=(\on{Hom}_A(C,A)\otimes_A B)^{\sim}$,
where we think of $f\in \on{Hom}_A(C,A)$ as an element
of $\on{Hom}_A(B,A)$ by extending it via $f(1)=0$. Our statement is
equivalent to showing that the $B$-module homomorphism
\[\sigma:\on{Hom}_A(C,A)\otimes_A B \rightarrow \big{(}\on{Hom}_A(B,A)
\big{)_B},\,\,f\otimes b \mapsto b\circ f,\]
is surjective. In fact, it suffices to show that the trace map is in the
image of $\sigma$, i.e. to find $c_1,c_2\in B$ such that
\begin{equation}
\on{tr}\equiv c_1\circ {\pi_1}+c_2\circ {\pi_2}.
\label{trace equation}
\end{equation}
Here $\pi_j:B\rightarrow A$
gives the $b_j$-coordinate of $b\in B$, $j=1,2$. Set $c_1=b_1-\pi_1(b_1^2)$ and
$c_2=-\pi_1(b_1b_2)$. Evaluating both sides of (\ref{trace equation})
at $1,b_1$ and $b_2$ yields the same result, hence the identity is
established, and $\sigma|_{\widehat{\phi}^*(\mathcal V{{\widehat{\phantom{n}}}})}
:{\widehat{\phi}^*(\mathcal V{{\widehat{\phantom{n}}}})}
\rightarrow \omega_{\widehat X/\widehat Y}$ is surjective.
\smallskip We have shown that the composition $\rho\circ \psi=
i:\widehat{X}\dasharrow \mathbf P V$ is a regular map on $\widehat{X}$.
Alternatively, one could employ the geometric interpretation
of the trace map. A {\it general} point $p\in {\widehat Y}$ has three preimages
$q,r,s$ in $\widehat X$, each of which defines canonically a distinct point
$[\Lambda_q],[\Lambda_r]$ or $[\Lambda_s]$ in the fiber
of ${\mathbf P}(\widehat{\phi}_*{\mathcal O}_{\widehat X})$. As we pointed above, the
projection $\pi_{[\Lambda]}
:{\mathbf P}^2\dashrightarrow {\mathbf P}^1$ is well-defined at
$[\Lambda_q],[\Lambda_r]$ and $[\Lambda_s]$. But $\pi_{[\Lambda]}$ is
precisely the fiberwise restriction of $\mathbf P({\widehat{\phi}}_*
{\mathcal O}_{\widehat X})
\stackrel{\rho}{\dashrightarrow} \mathbf P V$, which shows again that the
composition $i=\rho\circ\psi:\widehat{X}\dasharrow \mathbf P V$ is regular
on an open set of $\widehat X$. One makes the necessary modifications in
the cases of fewer preimages of $p$ in $\widehat{X}$.
Finally, one can show, using similar methods (either algebraically or
geometrically), that the map $i$ is also an embedding. \qed
\medskip
\noindent{\bf Remark 6.1.} Since the
general fiber $C$ of $\widehat{X}$ is a smooth trigonal curve,
the restriction $i|_{\displaystyle{C}}$ embeds $C$ in a ruled surface ${\mathbf
F}_k$ over the corresponding fiber $F_{\widehat{Y}}=
\mathbf P^1$ of $\widehat{Y}$, where ${\mathbf F}_k=\mathbf P(V|_{F_{\widehat{Y}}})$.
In Section~\ref{Maroniinvariant}
we will describe how the ruled surface ${\mathbf F}_k$ varies
as the fiber $C$ varies in $\widehat{X}$.
\bigskip
\section*{\hspace*{1.9mm}7.
Global Calculation on a Triple Cover $X\rightarrow Y$}
\setcounter{section}{7}
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\label{triplecover}
In this section we consider the simplest case of effective covers, namely,
when the original family $X$ is itself a triple cover of
a {\it ruled surface} $Y$ over the base curve $B$. This happens exactly when
all fibers of $X$ are irreducible, and the
linear system of $g^1_3$'s on the
smooth fibers extends over the singular fibers to base-point free
line bundles of degree 3 with two linearly independent sections. As we saw in
Section~\ref{quasi-admissible}, we can patch together
all these $g^1_3$'s in a line bundle ${\mathcal L}$ on the total
space of $X$. Thus, $X$ will map to ${\mathbf P}(H^0(X,{\mathcal
L})^{\widehat{\phantom{n}}})$ via
$\phi_{\mathcal L}$, and this map will factor through our
ruled surface $Y$ over $B$:
\begin{figure}[h]
$$\psdraw{triple}{2.7in}{1.31in}$$
\caption{Basic triple cover case.}
\label{fig.triplecover}
\end{figure}
Equivalently, we can describe such a
family $X\rightarrow B$ via the classification of the boundary components
of the trigonal locus in Section~4:
in $\overline{\mathfrak T}_g$ the base
curve $B$ meets only the boundary component
$\Delta{\mathfrak{T}}_0$ of irreducible curves ($\delta_0|_B>0$), and
there are no hyperelliptic fibers in $X$
($B\cap \overline{\mathfrak I}_g=\emptyset$).
\subsection{Global versus local calculation}
\label{global} As it will turn out, the
calculation of the slope $\delta_X /\lambda_X$ in this basic case yields
the actual maximal bound $\frac{36(g+1)}{5g+1}$: any addition of
singular fibers belonging to
other boundary components of $\overline{\mathfrak T}_g$ will
only decrease the ratio. Henceforth, we distinguish among two types of
calculation: {\it global} and {\it local}. The {\it global}
calculation refers to finding the
coefficients of $\delta_0$ and
the Hodge bundle $\lambda|_{\overline{\mathfrak T}_g}$
in a relation in $\on {Pic}_{\mathbb Q}\overline{\mathfrak T}_g$ involving {\it all}
boundary classes. The {\it local}
calculation provides the remaining coefficients by considering {\it local
invariants} of each individual boundary class (cf.~Sect.~8).
\subsection{The basic construction}
\label{basic} For the remainder of this section, we consider a family
$X\rightarrow B$ such that, as above, $X$ is a triple cover of the
corresponding ruled surface $Y$, and we carry out the proposed global
calculation.
\unitlength 0.11in
\begin{figure}[h]
\begin{picture}(10,9)(25,21)
\put(26.5,24){$X$}
\put(28.4,24.2){\vector(1,0){3.1}}
\put(32,24){$Y$}
\put(29,20){$B$}
\put(27.3,23.5){\vector(1,-1){2}}
\put(32.3,23.5){\vector(-1,-1){2}}
\put(26.6,22){$f$}
\put(31.75,22){$h$}
\put(29.45,24.7){$\phi$}
\put(27.3,25.4){\vector(1,1){2}}
\put(30.3,27.4){\vector(1,-1){2}}
\put(29,27.8){${\mathbf P}V$}
\put(32,26.4){$\pi$}
\put(27.4,26.4){$i$}
\end{picture}
\caption{Triple Case}
\label{basicconstruction}
\end{figure}
Recall that
the pushforward of the structure sheaf ${\mathcal O}_X$ to $Y$ is a locally
free sheaf of rank $3$. In the exact sequence:
\[0\rightarrow E\rightarrow
{\phi}_*{\mathcal O}_X\stackrel {\on{tr}}{\rightarrow}{\mathcal O}_Y\rightarrow 0,\]
the kernel of the trace map $\on{tr}$ is a vector
bundle $E$ on $Y$ of rank $2$,
and $X$ naturally
embeds in the rank 1 projective bundle
${\mathbf P}V$ over $Y$, where $V=E\,\widehat{\phantom{n}}$.
Any rank 2 vector bundle $E$ has the same
projectivizations as its dual bundle $V$ since $E\cong \bigwedge^2E\otimes V$,
where $\bigwedge^2E$ is a line bundle. For easier notation, in the trigonal
case we use the dual $V$ instead of $E$ from Section~\ref{embedding}.
A basis for $\on{Pic}Y$ can be chosen by letting $F_Y$ be the
fiber of $Y$, and $B^{\prime}$ be any section of
$Y\rightarrow B$. Hence $\on{Pic}Y={\mathbb Z}B^{\prime}\oplus {\mathbb Z}F_Y$.
We normalize by replacing $B^{\prime}$ with
the ${\mathbb Q}$-linear combination
$B_0=B^{\prime}-\displaystyle{\frac{(B^{\prime})^2}{2}}{F_Y}$, and
provide a $\mathbb{Q}$-basis for $\on{Pic}_{\mathbb Q}Y$:
\begin{equation}
\on{Pic}_{\mathbb Q}Y={\mathbb Q}B_0\oplus {\mathbb Q}F_Y\,\,\,\on{with}\,\,\,
B_0^2=F_Y^2=0\,\,\,\on{and}\,\,\,B_0\cdot F_Y=1.
\label{normalize}
\end{equation}
Let $\zeta$ denote the class of the
hyperplane line bundle ${\mathcal O}_{{\mathbf P}V}(+1)$
on ${\mathbf P}V$, and let $c_1(V)$ and $c_2(V)$ be the Chern classes
of $V$ on $Y$. The Chow ring $A({\mathbf P}V)$ is
generated as a $\pi^*(A(Y))$-module by $\zeta$
with the only relation:
\begin{equation}
\zeta^2+\pi^*c_1(V)\zeta+\pi^*c_2(V)=0.
\label{zeta-relation}
\end{equation}
In particular, for the Picard groups:
\begin{equation}
\on {Pic}_{\mathbb Q}({\mathbf P}V)=\pi^*(\on{Pic}
_{\mathbb Q}Y)\oplus {\mathbb Q}\zeta.
\label{Q-basis}
\end{equation}
\smallskip
\noindent As a divisor on ${\mathbf P}V$, the surface
$X$ meets the fiber $F_{\pi}$ of $\pi$
generically in three points ($X$ maps three-to-one onto $Y$).
Thus in the Chow ring $A({\mathbf P}V)$ we have $[X]\cdot [F_{\pi}]=3$, which
simply means that $X$ can be expressed as
\[X\sim 3\zeta + \pi^*D\]
for some divisor $D$ on $Y$ (see (\ref{Q-basis})).
With respect to the chosen basis for $\on{Pic}_{\mathbb Q}Y$:
\begin{equation}
D\sim aB_0+bF_Y\,\,\on{and}\,\,
c_1(V)\sim cB_0+dF_Y
\label{define D,c1(V)}
\end{equation}
for some $a,b,c,d\in {\mathbb Z}$.
Note that $\deg(D|_{B_0})=b$ and $\deg(c_1(V)|_{B_0})=d$.
\subsection{Relation among the divisor classes $D$ and $c_1(V)$}
\label{relation}
It is evident that the divisors $D$ and $c_1(V)$ cannot be
independent on ruled surface
$Y$ since both are canonically determined by the surface $X$.
The relation is in fact quite straightforward.
\begin{lem} With the above notation for the triple cover
$\phi:X\rightarrow Y$, we have $D=2c_1(V)$ in $\on{Pic}Y$.
\label{D=2c_1(V)}
\end{lem}
\begin{proof} We start with the standard exact sequence for the divisor
$X$ on ${\mathbf P}V$:
\begin{equation}
0\rightarrow \mathcal{O}_{{\mathbf P}V}(-X)\rightarrow \mathcal{O}_{{\mathbf P}V}
\rightarrow \mathcal{O}_X\rightarrow 0.
\label{X-divisorsequence}
\end{equation}
Pushing to $Y$ yields:
\begin{equation}
0 \!\rightarrow \!\pi_*\mathcal{O}_{{\mathbf P}V}(-X) \!\rightarrow\!
\pi_*\mathcal{O}_{{\mathbf P}V}\!\rightarrow\! \pi_*\mathcal{O}_X \!\rightarrow \!
R^1\pi_*\mathcal{O}_{{\mathbf P}V}(-X) \!\rightarrow \!R^1
\pi_*\mathcal{O}_{{\mathbf P}V} \!\rightarrow \cdots
\label{pushforward}
\end{equation}
\noindent It is easy to show that $R^1\pi_*\mathcal{O}_{{\mathbf P}V}=0$ and
$\pi_*\mathcal{O}_{{\mathbf P}V}(-X)=0$. This follows from
Grauert's theorem \cite{Hartshorne}:
$h^1(\mathcal{O}_{{\mathbf P}V}|_{F_{\pi}})=h^1(\mathcal
O_{{\mathbf P}^1})=0,$ and
\[h^0(\mathcal{O}_{{\mathbf P}V}(-X)|_{F_{\pi}})=h^0(\mathcal{O}_{{\mathbf P}V}
(-3\zeta-\pi^*D)|_{F_{\pi}})=h^0(\mathcal{O}_{{\mathbf P}^1}(-3))=0.\]
Furthermore, $\pi_*\mathcal{O}_{{\mathbf P}V}=
\mathcal{O}_Y$ and $\pi_*\mathcal{O}_X=\phi_*\mathcal{O}_X$, so that
(\ref{pushforward}) becomes
\begin{equation}
0\rightarrow \mathcal{O}_Y\rightarrow \phi_*\mathcal{O}_X\rightarrow
R^1\pi_*\mathcal{O}_{{\mathbf P}V}(-X)\rightarrow 0.
\label{remainingnonzero}
\end{equation}
From relative Serre-duality, and using the first Chern class of the
relative dualizing sheaf, $c_1(\omega_{\pi})$ (cf.~(\ref{omega-pi})):
\[R^1\pi_*\mathcal{O}_{{\mathbf P}V}(-X) \cong \big(\pi_*(\omega_{\pi}\otimes
\mathcal {O}_{{\mathbf P}V}(X))\big)\widehat{\phantom{t}}=
\big(\pi_*\mathcal{O}_{{\mathbf P}V}(\zeta+\pi^*D-\pi^*c_1(V))\big)
\widehat{\phantom{t}}.\]
\noindent
Since $\pi_*\mathcal{O}_{{\mathbf P}V}(\zeta)=V\widehat{\phantom{t}}$ (cf.~
~\cite{BPV}),
\begin{equation}
R^1\pi_*\mathcal{O}_{{\mathbf P}V}(-X) \cong \big[V\widehat{\phantom{t}}
\otimes \mathcal{O}_Y(D-c_1(V))\big]\widehat{\phantom{t}}.
\label{tranformedsequence}
\end{equation}
Finally, combining (\ref{tranformedsequence}) with (\ref{remainingnonzero})
and $\phi_*\mathcal{O}_X/\mathcal{O}_Y=V\widehat{\phantom{t}}$, we arrive at
\[V\cong V\widehat{\phantom{t}}\otimes \mathcal{O}_Y(D-c_1(V))\,\,\Rightarrow\,\,
\mathcal{O}_Y(D-c_1(V))\cong \bigwedge ^2 V\cong\mathcal{O}_Y(c_1(V)),\]
and hence $D=2c_1(V)$ in $\on{Pic}Y. \,$ \end{proof}
\subsection{Global calculation of $\lambda_X$ and $\kappa_X$.}
\label{globalcalculation}
In the following proposition we express $\lambda_X$ and
$\kappa_X$ in terms of
$\deg(c_1(V)|_{B_0})=d$ and the Chern class polynomial
$c_1^2(V)-4c_2(V)$, both of which are
independent of the choice of the vector bundle $V$.
Indeed, if we twist
$V$ by a line bundle $M$ on $Y$ and set $V^{\prime}=V\otimes M$, then
\[c_1(V^{\prime})=c_1(V)+2c_1(M),\,\,
c_2(V^{\prime})=c_2(V)+c_1(V)c_1(M)+c_1(M)^2,\]
\[\Rightarrow\,\,c_1(V^{\prime})^2-4c_2(V^{\prime})=c_1(V)^2-4c_2(V).\]
Recall the notation of (\ref{define D,c1(V)}).
In order to make $d$ also invariant, we use $b=2d$ from
Lemma~\ref{D=2c_1(V)} and write $d=2b-3d$.
Now, if we replace ${\mathbf P}V$ with its isomorphic
${\mathbf P}V^{\prime}$ (cf.~Fig.~\ref{invariance}), and set $\zeta^{\prime}=i^*
\zeta\otimes(\pi^{\prime})^*M^{-1}$ to be the new hyperplane bundle,
then in $\on{Pic}({\mathbf P}V)$:
$X\sim 3\xi^{\prime}+(\pi^{\prime})^*D^{\prime}$ with $D^{\prime}\sim
D+3c_1(M)$. Hence
\[2D^{\prime}-3c_1(V^{\prime})=2D+6c_1(M)-3c_1(V)-6c_1(M)=2D-3c_1(V),\]
and equating their degrees on $B_0$, we obtain $2b^{\prime}-3d^{\prime}=2b-3d$.
\unitlength 0.11in
\begin{figure}[h]
\begin{picture}(10,7)(25,20.5)
\put(26,24){${\mathbf P}V^{\prime}$}
\put(28.4,24.2){\vector(1,0){3.1}}
\put(32,24){${\mathbf P}V$}
\put(29,20){$Y$}
\put(27.3,23.5){\vector(1,-1){2}}
\put(32.3,23.5){\vector(-1,-1){2}}
\put(26.5,22){$\pi^{\prime}$}
\put(31.75,22){$\pi$}
\put(29.45,24.7){$i$}
\end{picture}
\caption{$V^{\prime}=V\otimes M$}
\label{invariance}
\end{figure}
In other words, the following formulas for $\lambda_X$ and $\kappa_X$
would be valid for any vector bundle $V^{\prime}$ in place of the
canonically defined $V$, as long as the diagram of the basic construction
(cf.~Fig.~\ref{basicconstruction}) is satisfied, and as long as we adjust the
degree $d=\deg(c_1(V)|_{B_0})$ by its invariant form $2b-3d=
2\deg(D|_{B_0})-3\deg(c_1(V)|_{B_0})$.
\begin{prop}
Let $\phi:X\rightarrow Y$ be a degree 3 map from the original family
$X$ of trigonal curves to the ruled surface $Y$ over $B$. The
invariants $\lambda_X$ and $\kappa_X$ are given by the formulas:
\begin{eqnarray*}
\lambda_X&=&\displaystyle{\frac{g}{2}\deg\big(c_1(V)|_{B_0}\big)+\frac{1}{4}
\big(c_1(V)^2-4c_2(V)\big)},\\
\kappa_X&=&\displaystyle{\frac{5g-6}{2}\deg\big(c_1(V)|_{B_0}\big)+\frac{3}{4}
\big(c_1(V)^2-4c_2(V)\big)}.
\end{eqnarray*}
\label{lambda_X,kappa_X}\vspace*{-6mm}
\end{prop}
\noindent We defer the proof of Prop.~\ref{lambda_X,kappa_X}
to Subsections 7.4.2-3.
\subsubsection{Notation and Basic Tools.}
\label{notation} The proof of Prop.~\ref{lambda_X,kappa_X}
consists of two calculations in the Chow ring
of ${\mathbf P}V$; one uses versions of Riemann-Roch theorem on $X$ and ${\mathbf P}V$,
and the other uses the adjunction formula on ${\mathbf P}V$ for the
divisor $X$. Here we discuss these statements and set up the necessary
notation.
\medskip
In order to work in ${\mathbb A}({{\mathbf P}}V)$, we express the Chern classes of
${\mathbf P}V$ in terms of the hyperplane class $\zeta$ and the Chern classes
of $Y$. We first recall that
$\pi_*\mathcal{O}_{{\mathbf P}V}(+1)\cong V\widehat{\phantom{t}}$.
In the Euler sequence on ${{\mathbf P}V}$:
\begin{equation}
0\rightarrow \mathcal{O}_{{\mathbf P}V} \rightarrow \mathcal{O}_{{\mathbf P}V}(+1)\otimes
\pi^*V\rightarrow \mathcal{T}_{\pi} \rightarrow 0,
\label{Eulersequence}
\end{equation}
we compare the Chern polynomials $c_t(\mathcal{O}_{{\mathbf P}V}(+1)\otimes
\pi^*V)=c_t(\mathcal{T}_{\pi})$, and obtain:
\begin{eqnarray}
K_{{\mathbf P}V}&\!\!\!=&\!\!\!\!-2\zeta-\pi^*c_1(V)+\pi^*K_Y,
\label{K_PV}\\
c_1(\omega_{\pi})&\!\!\!=&\!\!\!\!-2\zeta-\pi^*c_1(V),
\label{omega-pi}\label{omega_pi}\\
c_2({\mathbf P}V)&\!\!\!=&\!\!\!\!-2\zeta\pi^*K_Y+\pi^*\big(c_1(V)K_Y+c_2(Y)\big).
\label{c_2(PV)}
\end{eqnarray}
Here $\mathcal{T}_{\pi}$ and $\omega_{\pi}$ are correspondingly the relative
tangent and the
relative dualizing sheaves of $\pi$, while $K_{{\mathbf P}V}$ is the
class of the canonical sheaf on ${\mathbf P}V$. On the ruled surface $Y$ over
the curve $B$ of genus $g_{\scriptscriptstyle B}$ we similarly have
\begin{eqnarray}
\hspace{9mm}K_Y&\!\!\!=&\!\!\!\!-2B_0+h^*(K_B)
\equiv -2B_0+(2g_{\scriptscriptstyle B}-2)F_Y
\label{K_Yglobal},\\
\hspace{9mm}c_2(Y)&\!\!\!=&\!\!4(1-g_{\scriptscriptstyle B}).
\label{c_2(Y)global}
\end{eqnarray}
\medskip
Now let $C$ be the general fiber of $X$, i.e. a smooth trigonal curve of
genus $g$. Assuming the Basic construction for the triple cover
$X\rightarrow Y$ (cf.~Fig.~\ref{basicconstruction}),
we have the following lemmas.
\begin{lem}If $\chi(\mathcal{E})$ denotes the holomorphic Euler characteristic
of any sheaf $\mathcal{E}$, then the invariant $\lambda_X$ is expressible as
$\lambda_X=\chi({\mathcal O}_X)-\chi({\mathcal O}_B)\cdot
\chi({\mathcal O}_C).$
\label{Euler}
\end{lem}
\noindent{\it Proof.} From
Grothendieck-Riemann-Roch theorem for the map $f:X\rightarrow B$,
\[\on{ch}(f_{!}\mathcal{O}_X).\on{td}\mathcal{T}_B=
f_*(\on{ch}\mathcal{O}_X.\on{td}\mathcal {T}_X),\]
where $\mathcal{T}_X$ and $\mathcal{T}_B$ are the corresponding tangent sheaves.
Since the fibers of $f$ are one-dimensional, $f_{!}\mathcal{O}_X=
f_*\mathcal{O}_X-R^1\!f_*\mathcal{O}_X=\mathcal{O}_B-(f_*\omega_f)\widehat{\phantom{t}}$.
Substituting:
\[\big(1-g+c_1(f_*\omega_f)\big).\big(1-\frac{1}{2}K_B\big)=
f_*\big(1-\frac{1}{2}K_X+\frac{1}{12}(K^2_X+c_2(X))\big),\]
\[\Rightarrow\,\,
c_1(f_*\omega_f)=\frac{1}{12}f_*(K^2_X+c_2(X))-\frac{g-1}{2}K_B,\]
\[\,\,\,\,\,\,\,\,\,\,\Rightarrow\,\,
\lambda_X=\on{deg}(f_*\omega_f)=\chi(\mathcal{O}_X)-\chi(\mathcal{O}_B)\cdot
\chi(\mathcal{O}_C).\,\,\,\qed\]
\medskip
Note the similarity between this formula and the formula for $\delta_B$
in Example 2.1. Both quantities are expressed as differences of the
Euler characteristic (holomorphic or topological)
on the total space of $X$ and the product of the
corresponding Euler characteristics on the base $B$ and the general fiber $C$.
Lemma~\ref{Euler} suggests that in order to calculate $\lambda_X$,
we must have control over $\chi(\mathcal{O}_X)$.
\begin{lem} In the Chow ring of ${\mathbf P}V$:
\[\chi({\mathcal O}_X)=\frac{1}{12}X\big[\big(X+K_{{\mathbf P}V}\big)
\big(2X+K_{{\mathbf P}V}\big)+
c_2({\mathbf P}V)\big]\]
\label{holomorphicEuler}\vspace*{-7mm}
\end{lem}
\begin{proof} From the standard exact sequence
(\ref{X-divisorsequence}) for the divisor $X$ on ${\mathbf P}V$ we have
$\chi({\mathcal O}_X)=\chi({\mathcal O}_{{\mathbf P}V})-\chi({\mathcal O}_{\mathbf
{P}V}(-X))$. On the other hand, Hirzebruch-Riemann-Roch claims that
for any sheaf $\mathcal E$ on ${\mathbf P}V$:
$\chi(\mathcal{E})=\on{deg}\big(\on{ch}(\mathcal{E}).\on{td}\mathcal{T}_{{\mathbf P}V}
\big)_3$. Applying this to
the line bundles $\mathcal{O}_{{\mathbf P}V}$ and ${\mathcal O}_{{\mathbf P}V}(-X)$,
and subtracting the results completes the proof of the lemma. \end{proof}
\medskip
The reader may have noticed that all quantities discussed in the above
lemmas are elements
of the third graded piece ${\mathbb A}^3({\mathbf P}V)\otimes \mathbb{Q}$ of the Chow ring
${\mathbb A}({\mathbf P}V)\otimes \mathbb{Q}$. Hence they are cubic polynomials in the
class $\zeta$, whose coefficients are appropriate products of pull-backs
from ${\mathbb A}(Y)\otimes \mathbb{Q}$.
The higher degrees $\zeta^3$ and $\zeta^2$ can be decreased using the
basic relation (\ref{zeta-relation}), while $\zeta$ itself can be
altogether eliminated by noticing that for any $\vartheta\in {\mathbb A}^2(Y)$:
\begin{equation}
\zeta.\pi^*(\on{point})=\zeta.F_{\pi}=1\,\,
\Rightarrow\,\,\zeta . \pi^*\vartheta=\on{deg}\vartheta.
\label{trivial}
\end{equation}
It is also useful to remember the trivial fact that for any divisors $D_i$
on $Y$, $\on{dim}Y=2$ implies
$D_1.D_2.D_3=0=D_i.c_2(V)$.
\smallskip
\begin{figure}[h]
$$\psdraw{intersection}{1.8in}{1.8in}$$
\caption{Intersection of $X$ and $\pi^*F_Y$}
\label{intersection}
\end{figure}
\begin{lem}[Adjunction formula] The canonical bundle $\omega_Z$
of a smooth divisor $Z$ on the smooth variety $T$ can be expressed as
$\omega_Z\cong \omega_T\otimes \mathcal{O}_T(Z)\otimes
\mathcal{O}_Z$. Consequently,
\[K_X^2=\big(K_{{\mathbf P}V}+X\big)^2X\,\,\,{and}\,\,\,
g+2=\on{deg}c_1(V)|_{F_Y}.\]
\label{adjunction}\vspace*{-4mm}
\end{lem}
\noindent{\it Proof.} For the general statement of the adjunction formula
see \cite{Hartshorne}. The expression for $K_X^2$ is a straightforward
application to the divisor $X$ on ${\mathbf P}V$:
$K_X=\big(K_{{\mathbf P}V}+X\big)\big|_X$ is being squared in ${\mathbb A}({\mathbf P}V)$.
As for the genus $g$ of the
general member $C$ of our family, we consider a general fiber
$F_Y$ of $Y$ (cf.~Fig.~\ref{intersection}).
Its pullback $\pi^*F_Y$ is a rational ruled surface $\mathbf F$
over $F_Y$, embedded in the 3-fold ${\mathbf P}V$. The intersection of $\mathbf F$
with the surface $X$ is the trigonal fiber $C=X\cdot\pi^*F_Y=
(3\zeta+2\pi^*c_1(V))|_{\pi^*F_Y}$.
From the adjunction formulas
for $C\subset \pi^*F_Y$ and $\pi^*F_Y\subset {\mathbf P}V$:
\begin{eqnarray*}
2g-2&\!\!=&\!\!(K_{\pi^*F_Y}+C)\cdot C=\big((K_{{\mathbf P}V}+\pi^*F_Y)\big|_{\pi^*F_Y}
+X\big|_{\pi^*F_Y}\big)\cdot X\big|_{\pi^*F_Y}\\
&\!\!=&\!\!\big(\zeta+\pi^*c_1(V)+\pi^*K_Y+\pi^*F_Y\big)\big(
3\zeta+2\pi^*c_1(V)\big)\cdot \pi^*F_Y\\
&\!\!=&\!\!(2c_1(V)+3K_Y)\cdot
F_Y=2\on{deg}c_1(V)\big|_{F_Y}-6. \,\,\,\qed
\end{eqnarray*}
\subsubsection{Global Calculation of $\lambda_X$.}
\label{globallambda}
We substitute in Lemma~\ref{holomorphicEuler}
the expressions (\ref{K_PV}--\ref{c_2(PV)}) for $X,\,\,
K_{{\mathbf P}V}$ and $c_2(V)$, as well as the identity $D=2c_1(V)$:
\[\chi(\mathcal{O}_X)=\frac{3\xi+2\pi^*c_1(V)}{12}\big[
\big(\xi+\pi^*c_1(V)+\pi^*K_Y\big)
\big(4\xi+3\pi^*c_1(V)+\pi^*K_Y\big)\]
\[-2\xi\pi^*K_Y+\pi^* c_1(V)\pi^*K_Y+\pi^*c_2(Y)\big].\]
Applying the necessary reductions, we arrive at:
\[\chi({\mathcal O}_X)=\frac{1}{2}\big(c_1^2(V)-2c_2(V)\big)+\frac{1}{2}c_1(V)K_Y+
\frac{1}{4}\big(K_Y^2+c_2(Y)\big).\]
We expect our formula for $\lambda_X$ to be independent of the base curve
$B$. The contribution of $g_{\scriptscriptstyle B}$ in $\chi(\mathcal{O}_X)$ can be
written as: $(g_{\scriptscriptstyle B}-1)\on{deg}c_1(V)\big|_{F_Y}+\chi(\mathcal{O}_Y)=
(g_{\scriptscriptstyle B}-1)(g-1)$,
but this is precisely the adjustment $\chi(\mathcal{O}_B)\chi(\mathcal{O}_C)$
given in Lemma~\ref{Euler}. Thus,
\[\lambda_X=\frac{1}{2}\big(c_1^2(V)-2c_2(V)\big)-\on{deg}c_1(V)\big|_{B_0}.\]
It remains to notice that $c_1^2(V)=2\on{deg}c_1(V)|_{F_Y}
\on{deg}c_1(V)|_{B_0}=2(g+2)\on{deg}c_1(V)|_{B_0}$
and rewrite $\lambda_X$ in the form
\[\lambda_X=\frac{1}{4}\big
(c_1^2(V)-4c_2(V)\big)+\frac{g}{2}\on{deg}c_1(V)\big|_{B_0}.\,\,\,\qed\]
\subsubsection{Global Calculation of $\kappa_X$.}
\label{globalkappa} Since $\omega_f=
\omega_X\otimes \omega_B^{-1}$,
\begin{equation}
\kappa_X=(K_X-\pi^*K_B)^2=K_X^2-8(g_{\scriptscriptstyle B}-1)(g-1).
\label{kappa}
\end{equation}
\noindent From Lemma~\ref{adjunction} we calculate
\begin{eqnarray*}
K_X^2&\!\!=\!\!\!&(K_{{\mathbf P}V}+X)^2X=\big(\xi+\pi^*c_1(V)+\pi^*K_Y\big)^2
(3\xi+2\pi^*c_1(V))\\
&\!\!=\!\!\!\!&2c_1^2(V)-3c_2(V)+4c_1(V)K_Y+3K_Y^2.
\end{eqnarray*}
\noindent
We calculate the contribution of $g_{\scriptscriptstyle B}$
in $K_X^2$: $8(g_{\scriptscriptstyle B}-1)\on{deg}c_1(V)|_{F_Y}+
24(1-g_{\scriptscriptstyle B})=8(g_{\scriptscriptstyle B}-1)(g-1)$,
which is exactly the necessary adjustment for $\kappa_X$
in (\ref{kappa}). Therefore,
\begin{eqnarray*}
\kappa_X&\!\!\!=\!\!\!&2c_1^2(V)-3c_2(V)-8\on{deg}c_1(V)\big|_{B_0}\\
&\!\!\!=\!\!\!&\frac{3}{4}\big(c_1^2(V)-4c_2(V)\big)+\frac{5}{2}
\on{deg}c_1(V)\big|_{B_0}\on{deg}c_1(V)\big|_{F_Y}-8\on{deg}c_1(V)\big|_{B_0}\\
&\!\!\!=\!\!\!&\frac{3}{4}(c_1(V)^2-4c_2(V))+\frac{5g-6}{2}
\on{deg}c_1(V)\big|_{B_0}.\,\,\,\qed
\end{eqnarray*}
\subsection{Index theorem on the surface $X$.}
\label{indextheorem} Now that we have completed
the proof of Prop.~\ref{lambda_X,kappa_X}, we notice that any
bound on the ratio $\delta_X/\lambda_X$ would be equivalent to some
inequality involving the genus $g$ and the two invariants
discussed earlier: $\on{deg}c_1(V)|_{B_0}$ and the quantity
$c_1(V)^2\!-\!4c_2(V)$. This inequality
should be a fairly general one, since the only relevant information in our
situation is that $X$ is a triple cover of a ruled surface $Y$.
One way of obtaining such general inequalities in ${\mathbb A}^2(X)\otimes
\mathbb{Q}$
is via
\begin{thm}[Hodge Index] Let $H$ be an ample
divisor on the smooth surface $X$, and let $\eta$ be a divisor on $X$,
numerically not equivalent to 0. If $\eta \cdot H=0$, then $\eta^2<0$.
\label{Index}
\end{thm}
The question here, of course, is how to find suitable divisors $H$ and
$\eta$ that would yield our result for the maximal slope bound.
For that, we make use of the triple cover
$\phi:X\rightarrow Y$. If $H$ is any {\it ample} divisor on $Y$, then
its pullback $\phi^*H$ to $X$ is also ample. This follows from
\begin{thm}[Nakai-Moishezon Criterion] A divisor $A$ on the smooth surface
$X$ is ample if and only if $A^2>0$ and $A\cdot C>0$ for all irreducible
curves $C$ in $X$.
\label{Nakai}
\end{thm}
\noindent
Since $H$ is ample itself, $(\phi^*H)^2=3H^2>0$ and $(\phi^*H)\!\cdot\!
C=H\!\cdot\!\phi_*(C)>0$
for any curve $C$ on $X$, so that $\phi^*H$ is also ample on $X$.
Now, if we find a divisor $\eta$ on $X$ such that $\eta\cdot\phi^*\on{Pic}
Y=0$, we will have assured that $\eta\cdot\phi^*H=0$, and then
the Index theorem
will assert $\eta^2\leq 0$. As $X$ is a divisor itself on ${\mathbf P}V$,
its Picard group naturally contains the restriction of $\on{Pic}{\mathbf P}V$ to
$X$. We look for $\eta$ inside this subgroup, and for our purposes we may write
it in the form $\eta=\big(\zeta+\pi^*C_1\big)\big|_X$
for some $C_1\in \on{Pic}_{\mathbb Q}Y$. Let $C$ be any divisor class
$\on{Pic}_{\mathbb {Q}}Y$. We compute
\[\eta\cdot \phi^*C=\big(\zeta+\pi^*C_1\big)\big(3\zeta+2\pi^*c_1(V)\big)
\pi^*C=C(3C_1-c_1(V)).\]
We want this to be zero for all $C$, so we naturally take
$C_1={\displaystyle\frac{1}{3}}c_1(V)\in\on{Pic}_{\mathbb {Q}}Y$.
We summarize the above discussion in
\begin{lem}[Index Theorem on $X$] The divisor class
$\eta=\big(\zeta+\frac{1}{3}\pi^*c_1(V)\big)\big|_X$ on $X$
satisfies $\eta\cdot \phi^*\on{Pic}(Y)=0$. In particular, for
an ample divisor $H$ on $Y$, the pullback $\phi^*H$ is also ample on $X$
and $\eta \cdot \phi^*H=0$. Consequently, $\eta^2\leq 0$ with equality
if and only if $\eta$ is numerically equivalent to $0$ on $X$.
\label{Eta}
\end{lem}
We have shown that
\begin{equation}
0\geq3\eta^2=3\big(\zeta+\frac{1}{3}\pi^*c_1(V)\big)^2
\big(3\zeta+2\pi^*c_1(V)\big)=2c_1^2(V)-9c_2(V),
\label{eta}
\end{equation}
or equivalently,
\begin{equation}
2(g+2)\on{deg}c_1(V)\big|_{B_0}-9\big(c_1^2(V)-4c_2(V)\big)\geq 0.
\label{indexinequality}
\end{equation}
We are now ready to find a maximal bound for the slope of $X$. Recall the
formulas for $\lambda_X$ and $\kappa_X$ (cf.~Prop.~\ref{lambda_X,kappa_X}),
and write
\[\delta_X=12\lambda_X-\kappa_X=\displaystyle{
\frac{7g+6}{2}\on{deg}c_1(V)\big|_{B_0}
+\frac{9}{4}\big(c_1^2(V)-4c_2(V)\big)}.\]
\medskip
In view of the type of bound for the ratio $\delta_X/\lambda_X$, which
we aim to achieve, we have to eliminate any extra terms and use inequality
(\ref{eta}). Our only choice is to subtract
\begin{eqnarray*}
36(g+1)\lambda_X-(5g+1)\delta_X&\!\!\!=\!\!\!&
\frac{1}{2}\big(36(g+1)g-(5g+1)(7g+6)\big)
\on{deg}c_1(V)\big|_{B_0}+\\
&& +\frac{1}{4}\big(9(g+1)-9(5g+1)\big)\big(c_1^2(V)-4c_2(V)\big)\\
&\!\!\!=\!\!\!&
\frac{1}{2}(g^2-g-6)\on{deg}c_1(V)\big|_{B_0}-\frac{9}{4}(g-3)
\big(c_1^2(V)-4c_2(V)\big)\\
&\!\!\!=\!\!\!&\frac{g-3}{4}\big[2(g+2)\on{deg}c_1(V)\big|_{B_0}-
9\big(c_1^2(V)-4c_2(V)\big)\big]\\
&\!\!\!=\!\!\!&(g-3)(9c_2(V)-2c_1^2(V))\geq 0
\end{eqnarray*}
\smallskip
As a result, we establish an exact maximal bound for the slopes of our
triple covers:
\begin{thm}[Main Theorem in Triple Cover Case]
Given a triple cover \newline$\phi:X\!\rightarrow \!Y$ satisfying in
the Basic construction, the slope of $X$ satisfies
\[\frac{\delta_X}{\lambda_X}\leq \frac{36(g+1)}{5g+1}\cdot\]
Equality is achieved if and only if $g=3$, or $g>3$ and $\eta\equiv 0$ on $X$.
\label{maintheorem}
\end{thm}
\subsection{When is the maximal bound achieved?}
\label{whenmaximal}
\subsubsection{The branch divisor of $\phi$}
From GRR, applied to $\phi:X\rightarrow Y$ and the sheaf $\mathcal{O}_X$, we
obtain a description of $c_1(V)$:
\[\on{ch}(\phi_{!}\mathcal{O}_X).\on{td}\mathcal{T}_Y=
\phi_*(\on{ch}\mathcal{O}_X.\on{td}\mathcal {T}_X),\]
\[\on{ch}(\phi_*\mathcal{O}_X)\big(1-\frac{1}{2}K_Y+\frac{1}{12}(K_Y^2+c_2(Y))\
\big)=\phi_*\big(1-\frac{1}{2}K_X+\frac{1}{12}(K_X^2+c_2(X)\big)\]
\[\Rightarrow c_1(\phi_*\mathcal{O}_X)=-\frac{1}{2}\big(\phi_*K_X-3K_Y).\]
For the {\it ramification} divisor $R$ on $X$ we know
$K_X=\phi^*K_Y+R$, so that
$\phi_*K_X=3K_Y+\phi_*R$. Hence $c_1(V)=-c_1(\phi_*\mathcal{O}_X)=
\frac{1}{2}\phi_* R$. In other words, from Lemma~\ref{D=2c_1(V)}
we conclude that $c_1(V)$ is half of the {\it branch}
divisor $D$ on $Y$.
On the other hand, we can rewrite the condition $\eta\equiv 0$
in the following way:
\[0 \equiv 3\eta=\big(3\zeta+\pi^*c_1(V)\big)\big|_X=
\big(X-\pi^*c_1(V)\big)\big|_X=
c_1\big(\mathcal{O}_{{\mathbf P}V}(X)\big|_X\big)-\pi^*c_1(V)\big|_X\]
\[\Leftrightarrow\,\,c_1\big(\mathcal{O}_{{\mathbf P}V}(X)\big|_X\big)\equiv
\frac{1}{2}\phi^*D.\]
The self-intersection of $X$ on ${\mathbf P}V$ satisfies (cf.~
\cite{self-intersection})
\[i^*i_*(1_X)=c_1(\mathcal{N}_{X/{\mathbf P}V})\,\,\Rightarrow\,\,
X\cdot X=i_*c_1(\mathcal{N}_{X/{\mathbf P}V}).\]
In particular, our condition $\eta\equiv 0$ can be expressed as
$\displaystyle
{c_1(\mathcal{N}_{X/{\mathbf P}V})\equiv \displaystyle{\frac{1}{2}}\phi^*D}$.
\subsubsection{Examples of the maximal bound}
Constructing examples of families achieving the maximal bound is not
so easy, considering that the condition $\eta\equiv 0$ is not
useful in practice. Instead, we start from the Basic construction and
attempt to find a ruled surface $Y$ and a rank 2 vector bundle $V$ on it
satisfying the equality in (\ref{indexinequality}), as well as
the ``genus condition'' given in Lemma~\ref{adjunction}. The former will
ensure the maximal ratio $\delta/\lambda = 36(g+1)/(5g+1)$, while the
latter will imply that the fibers of our family are indeed of genus $g$.
The remaining question is what linear series $3\zeta+\phi^*D$
has an irreducible member with at most rational double points
as singularities, which would serve as the total space of our family $X$.
\medskip
It is hard to work with the canonically defined bundle $V=
\phi_*(\mathcal{O}_X)/\mathcal{O}_Y$, since not every vector bundle $W$ of rank $2$
on $Y$ is of this form for some surface $X$. But any $W$ is
congruent to some $V$ after a twist by an appropriate line bundle $M$:
$V=W\otimes M$, and ${\mathbf P}V\cong {\mathbf P}W$.
So, it seems reasonable to start with $W$ rather than $V$,
and use the invariant forms of our required equalities
(cf.~Sect.~\ref{globalcalculation}). This means
replacing the degrees of $c_1(V)$ on $B_0$ and $F_Y$ by the corresponding
invariant
degrees of $2D-3c_1(V)$. Thus, we need for some divisor $\widehat{D}$ on $Y$:
\begin{equation}
2(g+2)(2\on{deg}\widehat{D}\big|_{B_0}-3\on{deg}c_1(W)\big|_{B_0})
=9\big(c_1^2(W)-4c_2(W)\big),
\label{condition1}
\end{equation}
\begin{equation}
g+2=2\on{deg}\widehat{D}\big|_{F_Y}-3\on{deg}c_1(W)\big|_{F_Y}.
\label{condition2}
\end{equation}
For a general fiber $F_Y$ of $Y$ consider the rational ruled surface
(cf.~Fig.~\ref{intersection}):
\[{\mathbf F}_e=\pi^*F_Y={\mathbf P}(W|_{F_Y})=
{\mathbf P}(\mathcal{O}_{{\mathbf P}^1}\oplus
\mathcal{O}_{{\mathbf P}^1}(e)),\,\,\,\on{with}\,\,
e\geq 0,\]
Let $S^{\prime}$ be the section in ${\mathbf F}_e$ with
self-intersection $(S^{\prime})^2=-e$, and let $F_{\pi}$ be the fiber of
${\mathbf F}_e$ (in terms of the map $\pi:{\mathbf P}V\rightarrow Y$,
$F_{\pi}=\pi^*(\on{pt})$). Since a general
fiber $C$ of our family is embedded in ${\mathbf F}_e$,
the linear system \[|C|=\big|3S^{\prime}+\frac{g+2+3e}{2}F_{\pi}\big|\]
has an irreducible nonsingular member. Equivalently,
$C\cdot S^{\prime}\geq 0$, i.e.
\begin{equation}e\leq (g+2)/3\,\,\,\on{and}\,\,\,
e\equiv g(\on{mod}2)
\label{e-conditions}
\end{equation}
(compare with Lemma~\ref{gentrig}).
This forces three types of extremal examples
according to the remainders $g(\on{mod}3)$.
\bigskip
\noindent{\bf Example 7.1 ($g\equiv 0(\on{mod}3)$).} Let $g=3e$ for
some $e\in \mathbb{N}$. Set the base curve $B={\mathbf P}^1$, and the ruled surface
\[Y={\mathbf P}(\mathcal{O}_B\oplus \mathcal{O}_B(6))={\mathbf F}_6.\]
Let $B^{\prime}$ be the section in $Y$ with smallest
self-intersection: $(B^{\prime})^2=-6$, thus
$B_0=B^{\prime}+3F_Y$ with $B_0^2=0$. Let $Q=B^{\prime}+6F_Y$, and
we choose two divisors $\widehat{D}$ and $E$ on $Y$ as follows:
\[\widehat{D}=(g+1)Q\,\,\,\on{and}\,\,\,E=eB^{\prime}+2(g+1)F_Y.\]
For the vector bundle $W$ on $Y$ we set $W=\mathcal{O}_Y\oplus
\mathcal{O}_Y(E)$ so that $c_1(W)=E$ and $c_2(W)=0$.
We claim that the linear system $L=|3\zeta+\pi^*\widehat{D}|$ on
the 3-fold ${\mathbf P}W$ contains an irreducible smooth member,
which we set to be our surface $X$ with maximal ratio $\delta/\lambda$.
\begin{figure}[h]
$$\psdraw{maximal1}{2.6in}{1.2in}$$
\caption{Example with $g\equiv 0$(mod$3$)}
\label{maximal1.fig}
\end{figure}
Indeed, it is trivial to check conditions
(\ref{condition1}--20). Further,
for {\it any} fiber $F_Y$ of $Y$:
\[\pi^*F_Y={\mathbf P}\big(\mathcal{O}_{{\mathbf P}^1}\oplus \mathcal{O}_{{\mathbf P}^1}
(E\cdot F_Y)\big)={\mathbf P}\big(\mathcal{O}_{{\mathbf P}^1}\oplus \mathcal{O}_{{\mathbf P}^1}
(e)\big)={\mathbf F}_e,\]
so that $e=g/3$ satisfies the required conditions (\ref{e-conditions}).
\medskip
The only nontrivial fact is the existence of the wanted member $X$ in the
linear system $L$ on ${\mathbf P}W$. Consider two sections $\Sigma_0$
and $\Sigma_1$ of ${\mathbf P}W$ corresponding to the subbundles
$\mathcal{O}_Y$ and $\mathcal{O}_Y(E)$ of $W$, respectively:
$\Sigma_0\in |\zeta|,\,\,\,\Sigma_1\in |\zeta+\pi^*(E)|,$ so that
$\Sigma_1\sim \Sigma_0+E$ (cf.~Fig.~\ref{maximal1.fig}).
Note that $\Sigma_0\cdot \Sigma_1=0$ and
$\Sigma_0\cdot L=\Sigma_0\cdot \pi^*B^{\prime}$.
In other words, if $G=\pi^*B^{\prime}$ is the ruled surface
over $B^{\prime}$, then $\Sigma_0$ intersects every irreducible
member of $L$ in the curve $R=\Sigma_0\cap G$. On the other hand,
if a member of $L$ meets $\Sigma_0$ in a point outside $R$, then this
member contains entirely $\Sigma_0$. Thus, $L$ does not distinguish
the points on $\Sigma_0$, and $R$ is in the base locus of $L$.
Similarly, the restriction $L|_G=|3\Sigma_0|_G|=|3R|$
has exactly one section on $G$, namely, $3R$. Again it
follows that $L$ does not distinguish the points on $G$.
\smallskip
Away from the closed subset $Z=\Sigma_0\cup G$,
the linear system $L$ is in fact very
ample. This can be checked by showing directly that $L$ separates
points and tangent vectors on ${\mathbf P}W-Z$.
Therefore, $L$ defines a
rational map \[\phi_L:{\mathbf P}W \rightarrow {\mathbf P}(H^0(L)^{\widehat{
\phantom{o}}})={\mathbf P}^N.\]
The map $\phi_L$ is regular on ${\mathbf P}W-R$, embeds ${\mathbf P}W-Z$, and
contracts $\Sigma_0-R$ and $G-R$ to two distinct
points $p$ and $q$ in ${\mathbf P}^N$. By Bertini's theorem
(cf.~\cite{Hartshorne}), the {\it general} member of $L$ is
{\it smooth} away from the base locus $R$.
Let $H$ be a general hyperplane in ${\mathbf P}^N$
not passing through $p$ and $q$. Pulling $H$ back to ${\mathbf P}W$
yields a member $X$
of $L$ not containing $\Sigma_0$ or $G$, and hence
irreducible.
\medskip
It remains to show that the total space of $X$ has at most
finitely many double point singularities along the curve $R$.
Since the member $3\Sigma_1+G\in|L|$ is smooth along $R$, then
the general member of $|L|$ must be smooth along $R$. Hence our surface $X$
has, in fact, a smooth total space. This concludes the construction
of our maximal bound family of trigonal curves.
\bigskip
\noindent{\bf Example 7.2 ($g\equiv 1(\on{mod}3)$).} Set $g=3e-2$ for
$e\in{\mathbb N}$. Then $e$ satisfies the requirements of our construction:
$e=(g+2)/3$ and $e\equiv g (\on{mod}2)$. For the ruled surface $Y$ we
choose $Y=\mathbf P^1\times\mathbf P^1$. Let $E$ and $\widehat{D}$ be
the following divisors on $Y$:
$E=eB_0+fF_Y\,\,\on{and}\,\,\widehat{D}=3E,$
where $f\in{\mathbb N}$. The vector bundle $W$ on $Y$ is then defined by
$W=\mathcal{O}_Y\oplus\mathcal{O}_Y(E)$.
Finally, we indentify the total space of the surface $X$
with an irreducible smooth member of the linear system
$L=|3\zeta+\pi^*\widehat{D}|$ on the 3-fold $\mathbf P W$.
\smallskip
The verification of this construction is similar to the previous example.
Here $L$ is very ample everywhere on $\mathbf P W$ except on the section
$\Sigma_0$, which is contracted to a point under the map $\phi_{L}$.
This example, in somewhat different context, is shown in \cite{small}.
\medskip
\noindent{\bf Remark 7.1} The case of $g\equiv 2(\on{mod}3)$ is
complicated by the fact that we cannot take $e=(g+1)/3$, for then
$e\not \equiv g(\on{mod}2)$. For example, if $g=5$, then the only
possibility is $e=1$. In the notation of Section~12,
all trigonal curves have lowest Maroni invariant of $1$, and there is
no Maroni locus. For now,
in this case we have not been able to construct a trigonal
family with singular general member, whose ratio is $36(g+1)/(5g+1)$.
\bigskip
\section*{\hspace*{1.9mm}8.
Local Calculation of $\lambda,\delta$ and $\kappa$ in
the General Case}
\setcounter{section}{8}
\setcounter{subsection}{0}
\setcounter{subsubsection}{0}
\setcounter{lem}{0}
\setcounter{thm}{0}
\setcounter{prop}{0}
\setcounter{defn}{0}
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\label{generalcase}
\subsection{Notation and conventions}
\label{Conventions}
In this section we consider the general case of a trigonal
family $X\rightarrow B$. For convenience of notation, we shall assume
that the base curve $B$ intersects transversally and in general points
the boundary divisors of $\overline{\mathfrak{T}}_g$
(cf.~Fig.~\ref{Delta-k,i}). We will call such a base
curve {\it general}, and use this definition throughout
Section~8-10. Since
we work in the rational Picard group of $\overline{\mathfrak{M}}_g$, all arguments
and statements in the remaining cases are shown similarly in
Sect.~11. From Prop.~6.1,
we may assume that modulo a base change, our family $X\!\rightarrow\!
B$ fits in the following commutative diagram:
\setlength{\unitlength}{10mm}
\begin{figure}[h]
\begin{picture}(3,4.3)(-1,2)
\put(0,3.9){$\widetilde{X}\,\stackrel{\phi}{\longrightarrow}
\widetilde{Y}$}
\put(0,5.1){$\widehat{X}\,\stackrel{\widehat{\phi}}{\longrightarrow}
\widehat{Y}$}
\multiput(0.2,5)(0,-1.2){3}{\vector(0,-1){0.6}}
\put(1.45,3.8){\vector(-1,-2){1}}
\put(0,2.8){$X$}
\put(0,1.6){$B$}
\put(1.55,5){\vector(0,-1){0.6}}
\put(1.55,6.2){\vector(0,-1){0.6}}
\put(1.65,5.8){$\pi$}
\put(1.3,6.3){${\mathbf P}V$}
\put(0.2,5.6){\vector(2,1){1.2}}
\end{picture}
\caption{General base $B$}
\label{general B}\vspace*{-4mm}
\end{figure}
\subsubsection{Relations in ${\rm Pic}_{\mathbb{Q}}\widehat{Y}$ and
${\rm Pic}_{\mathbb{Q}}{\mathbf P}V$}
\label{relations}
The special fibers of of $\widehat{X}$ and of the birationally ruled
surface $\widehat{Y}$ over $B$ are described in
Fig.~\ref{coef1.fig}--\ref{coef3.fig}. Since each such fiber in $\widehat{Y}$
is a {\it chain} $T$ of rational components, we can fix one of the end
components to be the {\it root} $R$. We keep the
notation $E^-$ ($E^+$, respectively) for the ancestor (descendants,
respectively)
of a component $E$ in $T$. We also fix a general fiber $F_{\widehat{Y}}\cong
{{\mathbf P}}^1$ of $\widehat{Y}$, and a section $B_{\widehat{Y}}$, which is
the pullback
of the corresponding section $B_0$ in $\widetilde{Y}$ (cf.~(\ref{normalize})).
The rational Picard group of $\widehat{Y}$ is generated by $F_{\widehat{Y}}$,
$B_{\widehat{Y}}$ and all non-root components $E$ of the special
fibers of $\widehat{Y}$:
\[\on{Pic}_{\mathbb{Q}}\widehat{Y}=\mathbb{Q}
B_{\widehat{Y}}\bigoplus \mathbb{Q}F_{\widehat{Y}}\!\!\!\bigoplus
_{E-\on{not}\,\on{root}}\!\!\!\mathbb{Q}E.\]
The intersection numbers of these generators are as follows:
$B_{\widehat{Y}}^2=0=F_{\widehat{Y}}^2,\,\,\,
B_{\widehat{Y}}\cdot F_{\widehat{Y}}=1,\,\,\on{and}\,\,
E\cdot B_{\widehat{Y}}=E\cdot F_{\widehat{Y}}=0.$
\vspace*{-3mm}
\setlength{\unitlength}{7mm}
\begin{figure}[h]
\begin{picture}(3,2.3)(12.6,4.5)
\put(6,5.3){\line(1,0){2}}
\multiput(5,4.6)(0,-1.7){2}{\line(2,1){1.6}}
\multiput(6,3.6)(0,-0.1){2}{\line(1,0){2}}
\put(9.1,4.2){\line(-2,-1){1.6}}
\multiput(4.9,5)(0,-1.8){2}{$E^-$}
\multiput(6.8,5.45)(0,-1.7){2}{$E$}
\put(8.7,4.3){$E^+$}
\end{picture}
\vspace*{9mm}
\caption{$m_{E}$\hspace*{100mm}}
\label{m E}
\begin{picture}(7,4)(3,-2.4)
\put(4.45,2.9){\line(2,1){1.6}}
\put(5.5,3.6){\line(1,0){3}}
\put(6.5,3.4){\line(-1,1){1.2}}
\put(7.5,3.4){\line(1,1){1.2}}
\put(4.5,3.2){$E^-$}
\put(6.8,2.9){$E$}
\put(5.1,4.75){$E^+$}
\put(5.9,4.85){$E^+$}
\put(7.6,4.85){$E^+$}
\put(8.4,4.75){$E^+$}
\put(6.75,3.4){\line(-1,2){0.65}}
\put(7.25,3.4){\line(1,2){0.65}}
\end{picture}\vspace*{-38mm}
\caption{$E^2$}
\label{E^2}
\begin{picture}(3,4)(-1,-1.9)
\multiput(6,5.3)(0,-0.1){2}{\line(1,0){2}}
\multiput(5,4.6)(0,-1.65){2}{\line(2,1){1.6}}
\put(6,3.6){\line(1,0){2}}
\multiput(9.1,4.2)(0,1.8){2}{\line(-2,-1){1.6}}
\put(9.1,4.3){\line(-2,-1){1.6}}
\multiput(4.8,5)(0,-1.7){2}{$R$}
\multiput(6.8,5.45)(0,-1.7){2}{$E^-$}
\multiput(8.7,4.3)(0,1.8){2}{$E$}
\end{picture}
\vspace*{-37.5mm}
\caption{$\theta_E$\hspace*{-90mm}}
\label{theta_E}
\end{figure}
We also set $m_{\!\stackrel{\phantom{.}}{E}}=E\cdot E^-$ (cf.~Fig.~\ref{m E}):
\[m_{\!\stackrel{\phantom{.}}{E}}=\left\{\begin{array}{l}
0\,\,\on{if}\,\,E=R\,\,\on{root},\\
1\,\,\on{if}\,\,E\,\,\on{and}\,\,E^-\,\,\on{reduced},\\
2\,\,\on{if}\,\,E\,\,\on{or}\,\,E^-\,\,\on{nonreduced}.
\end{array}\right.\]
In this notation, due to the fact that $E\cdot T=E\cdot
F_{\widehat{Y}}=0$, the self-intersection of any $E$ is computed by
(cf.~Fig.~\ref{E^2}):
\[E^2=-\sum_{\!\stackrel{\scriptstyle{E^{\prime}\not= E}}
{E^{\prime}\cap E\not= \emptyset}} E\cdot
E^{\prime}=-\sum_{\!\stackrel{\scriptstyle{E^{\prime}=E^+}}
{\on{or}\,\,E^{\prime}=E}}m(E^{\prime})\]
\smallskip
In order to express the dualizing sheaf $K_{\widehat{Y}}$
in terms of the above generators of $\on{Pic}_{\mathbb{Q}}\widehat{Y}$,
for each component $E$ in $\widehat{Y}$ we denote by $\theta_E$ the length
of the path $\stackrel{\longrightarrow}
{RE}$, omitting any nonreduced components except for
$E$ itself. For example, in the two cases in Fig.~\ref{theta_E}
we have $\theta_E=1$ and $\theta_E=2$. Note that $\theta_R=0$.
Considering the ``effective'' blow-ups on $\widetilde{Y}$,
necessary to construct $\widehat{Y}$, we immediately obtain the following
identities (compare with (\ref{K_PV}) and (\ref{K_Yglobal})).
\begin{lem} In $\on{Pic}_{\mathbb{Q}}\widehat{Y}$
and $\on{Pic}_{\mathbb{Q}}{\mathbf P}V$:
\begin{eqnarray*}
\on{(a)}&\!\!\!\!&\!\!
\displaystyle{K_{\widehat{Y}}
\equiv -2B_{\widehat{Y}}+(2g_B-2)F_{\widehat{Y}}+\sum_E \theta_EE},\\
\vspace*{-2mm}
\on{(b)}&\!\!\!\!&\!\!K_{{\mathbf P}V}\equiv -2\zeta -\pi^*
c_1(V)+\pi^*K_{\widehat{Y}},\\
\on{(c)}&\!\!\!\!&\!\!K_{{\mathbf P}V/\!_{\scriptstyle{B}}}
\equiv -2\zeta -\pi^*c_1(V)-2\pi^*B_{\widehat{Y}}+\sum_E \theta_E\pi^*E.
\end{eqnarray*}
\label{Kdivisors}\vspace*{-8mm}
\end{lem}
The hyperplane section $\zeta$ of ${\mathbf P}V$ and
the rank 2 vector bundle $V$ on $\widehat{Y}$ are defined similarly
as in Section~7.
Thus, in $\on{Pic}_{\mathbb{Q}}{\mathbf P}V$ we have $\widehat{X}\sim 3\zeta +\pi^*D$
for a certain divisor $D$ on $\widehat{Y}$. By analogy with
Lemma~\ref{D=2c_1(V)}, one shows that $D\equiv 2c_1(V)$ in
$\on{Pic}_{\mathbb{Q}}Y$, so that
\begin{equation}
\widehat{X}\equiv 3\zeta+ 2\pi^*c_1(V).
\label{genX}
\end{equation}
Using the above notation for $\on{Pic}_{\mathbb{Q}}\widehat{Y}$ we can write
for some half-integers $c,d,\gamma_{\!\stackrel{\phantom{.}}{E}}$:
\begin{equation}
\displaystyle{c_1(V)\equiv cB_{\widehat{Y}}+dF_{\widehat{Y}}+
\sum_{E}\gamma_{\!\stackrel{\phantom{.}}{E}}E}.
\label{genc_1(V)}
\end{equation}
Here we can assume that $\gamma_{\!\stackrel{\phantom{.}}{R}}=0$
by replacing $R$ with a linear combination of the remaining components $E$
in its chain $T$ (compare with (\ref{define D,c1(V)})).
Finally, we need the top Chern classes of $\widehat{Y}$ and ${\mathbf P}V$
in terms of intersections of known divisors and other known invariants
of the two surfaces (compare with (\ref{c_2(PV)}) and (\ref{c_2(Y)global})).
\begin{lem} In the Chow rings $\mathbb{A}(\widehat{Y})$ and
$\mathbb{A}({\mathbf P}V)$ the following equalities are true:
\begin{eqnarray*}
\on{(a)}&\!\!&\!\!\!\!c_2(\widehat{Y})=c_2(Y)+\sum_{E\not =R} 1=4(1-g_B)+\sum_
{E\not = R} 1, \\
\on{(b)}&\!\!&\!\!\!\!c_2({\mathbf P}V)=
c_2(\widehat{Y})-\pi^*K_{\widehat{Y}}(2\zeta+\pi^*c_1(V)),\\
\on{(c)}&\!\!&\!\!\!\!\displaystyle{
c_2({\mathbf P}V/\!_{\displaystyle{B}})=
-\pi^*K_{\widehat{Y}/B}(2\zeta+\pi^*c_1(V))+\sum_{E\not =R} 1}.
\end{eqnarray*}
\end{lem}
\label{conventions}
\subsubsection{A technical lemma}
\label{technicallemma} In the sequel,
we will work with several functions defined on the set of components
$\{E\}$ in $\widehat{Y}$. For easier calculations, to
any such function $f$ we associate the {\it difference function} $F$
by setting $F_{\!\stackrel{\phantom{.}}{E}}:=
f_{\!\stackrel{\phantom{.}}{E}}-f_{\!\stackrel{\phantom{.}}
{E^-}}$ for all $E$. Since $R^-$ does not exist, we define
$f_{R^-}=0$ for all roots $R$ in $\widehat{Y}$.
\begin{lem}For any functions $f$ and $h$ defined on the
set of components $\{E\}$ in $\widehat{Y}$, the following identity
holds true:
\[\sum_E f_{\!\stackrel{\phantom{.}}{E}}E \cdot \sum_E
h_{\!\stackrel{\phantom{.}}{E}}E=-\sum_E
(m\!\cdot\! F\!\cdot\! H)_{\!\stackrel{\phantom{.}}{E}}.\]
\label{technical}\vspace*{-7mm}
\end{lem}
\noindent{\it Proof.} We rewrite the lefthand side as
$\displaystyle{\sum_{E_1\not= E_2}f_{\!\stackrel{\phantom{.}}{E_1}}
h_{\!\stackrel{\phantom{.}}{E_2}}E_1E_2+\sum_E f_{\!\stackrel{\phantom{.}}{E}}h_{\!\stackrel{\phantom{.}}{E}}E^2=}$
\[=\sum_{E}\left(f_{\!\stackrel{\phantom{.}}{E^-}}
h_{\!\stackrel{\phantom{.}}{E}}+f_{\!\stackrel{\phantom{.}}{E}}
h_{\!\stackrel{\phantom{.}}{E^-}}\right)m_{\!\stackrel{\phantom{.}}{E}}-
\sum_E
\left(f_{\!\stackrel{\phantom{.}}{E}}h_{\!\stackrel{\phantom{.}}{E}}+
f_{\!\stackrel{\phantom{.}}{E^-}}
h_{\!\stackrel{\phantom{.}}{E^-}}\right)m_{\!\stackrel{\phantom{.}}{E}}=\]
\[=\sum_E\left(f_{\!\stackrel{\phantom{.}}{E^-}}-
f_{\!\stackrel{\phantom{.}}{E}}\right)\left(h_{\!\stackrel{\phantom{.}}{E}}
-h_{\!\stackrel{\phantom{.}}{E^-}}\right)m_{\!\stackrel{\phantom{.}}{E}}
=\sum_E(m\!\cdot\! F\!\cdot\! H)
_{\!\stackrel{\phantom{.}}{E}}.\,\,\,\qed\]
We have noted that all three functions $m,\theta$ and $\gamma$ are zero on the
roots $R$ in $\widehat{Y}$. Since we shall be working specifically with
these three functions, it makes sense to restrict from now on all sums
$\sum_E$ only to the non-roots $E$ in $\widehat{Y}$.
With this in mind, in every application
of Lemma~\ref{technical} one must check that
the corresponding functions $f$ and $h$ have the same property:
$f_R=0=h_R$, so that we can restrict the sums
in Lemma~\ref{technical} also to all {\it non-roots}
$E$ in $\widehat{Y}$. In fact, in all cases this verification
will be obvious as $f$ and
$h$ will be, for the most part, linear combinations of $\theta$ and $\gamma$.
\smallskip
\noindent{\bf Example 8.1.} From expression (\ref{genc_1(V)}) for $c_1(V)$
as a divisor on $\widehat{Y}$, and Lemma~\ref{technical}:
\begin{equation} c_1^2(V)=2cd+
\sum_{E}\gamma_{\!\stackrel{\phantom{.}}{E}}E\cdot
\sum_{E}\gamma_{\!\stackrel{\phantom{.}}{E}}E=
2cd-\sum_{E}m_{\!\stackrel{\phantom{.}}{E}}
\Gamma^2_{\!\stackrel{\phantom{.}}{E}}.
\label{c^2_1(V)}
\end{equation}
\subsection{Computation of the invariants $\lambda_{\widehat{X}},
\kappa_{\widehat{X}}$ and $\delta$}
\label{computation}
The following proposition~\ref{hatlambda} is a generalization
of the corresponding statement in Section~7 (cf.~
Prop.~\ref{lambda_X,kappa_X}). We set $\Gamma_{
\!\stackrel{\phantom{.}}{E}}=\gamma_{\!\stackrel
{\phantom{.}}{E}}-\gamma_{\!\stackrel{\phantom{.}}{E^-}}$ and
$\Theta_{\!\stackrel{\phantom{.}}{E}}
=\theta_{\!\stackrel{\phantom{.}}{E}}
-\theta_{\!\stackrel{\phantom{.}}{E^-}}$ to be the difference functions of
$\gamma$ and $\theta$.
\begin{prop}
The degrees of the invariants $\lambda_{\widehat{X}}$ and
$\kappa_{\widehat{X}}$
on $\widehat{X}$ are given by
\[\lambda_{\widehat{X}}=d(g+1)-c_2(V)-\frac{1}{4}\sum_E
\left\{m_{\!\stackrel{\phantom{.}}{E}}\cdot\left(2\Gamma^2+2\Gamma\cdot \Theta
+\Theta^2\right)_{\!\stackrel{\phantom{.}}{E}}-1\right\},\]
\vspace*{-6mm}\[\,\,\,\,\kappa_{\widehat{X}}=4dg-3c_2(V)-\sum_Em_{\!\stackrel{\phantom{.}}{E}}\left
(2\Gamma^2+4\Gamma\Theta+3\Theta^2\right)_{\!\stackrel{\phantom{.}}{E}}.\]
\label{hatlambda}\vspace*{-6mm}
\end{prop}
\begin{proof}
One starts with the Euler characteristic formula
$\lambda_{\widehat{X}}=\chi(\mathcal{O}_{\widehat{X}})-
\chi(\mathcal{O}_C)\cdot \chi(\mathcal{O}_B),$
or the adjunction formula
$\kappa_{\widehat{X}}=\big(\widehat{X}+K_{{\mathbf P}V/\!_{\scriptstyle{B}}})
^2\widehat{X}.$
The rest of the proof is a straight forward calculation, which uses the
equalities given in~\ref{conventions}, and
is substantially simplified by Lemma~\ref{technical}. \end{proof}
\begin{cor} The degree $\delta$ on the original family $X$ is given by
\[\delta=4d(2g+3)-9c_2(V)-\!\sum_T\mu(T)-\!\sum_{\on{ram}1} 1-\!
\sum_{\on{ram}2}
3-\!\sum_E\left\{m_{\!\stackrel{\phantom{.}}{E}}
\left(4\Gamma^2+2\Gamma\Theta\right)_{\!\stackrel{\phantom{.}}{E}}-3\right\}\]
\label{hatdelta}\vspace*{-4mm}
\end{cor}
Here $\mu(T)$ stands for the quasi-admissible contribution to
$\kappa_{\widehat{X}}$ of the preimage
$C=\widehat{\phi}^*T$ in $\widehat{X}$, as defined in Lemma~\ref{mu(C)}.
\smallskip
\begin{proof} Since $\lambda=\lambda_{\widehat{X}}$,
$\kappa=\kappa_{\widehat{X}}+\sum_T \mu(T)+
\sum_{\on{ram}1} 1+\sum_{\on{ram}2}3$,
and $\delta=12\lambda-\kappa$, the statement immediately
follows from Prop.~\ref{hatlambda}.\end{proof}
\subsection{The arithmetic genus $p_{{E}}$,
and the invariants $\Gamma_{{E^{\prime}}}$ and $\Theta_{{E^{\prime}}}$}
\label{arithmetic}For a component $E$ in a special fiber $T$ of
$\widehat{Y}$, we define $T(E)$ to be the subtree of $T$ generated
by the component $E$. In other
words, $T(E)$ is the union of all components $E^{\prime}\in T$ such that
$E^{\prime}\geq E$ (cf. Fig.~\ref{subtree}). For simplicity, we set
$p_{\!\stackrel{\phantom{.}}{E}}:=p_a\big(\phi^*(T(E))\big)$
to be the arithmetic genus of the inverse image $\phi^*(T(E))$ in
$\widehat{X}$. It can be easily computed via the following
analog of Lemma~\ref{adjunction},
where $T$ consisted of a single component $E=R$.
\begin{figure}[h]
$$\psdraw{ex5}{1.2in}{1.2in}$$
\caption{Subtree $T(E)$}
\label{subtree}
\end{figure}
\begin{lem} For a general base curve $B$ and for any non-root component
$E\in T$:
\begin{equation}
\displaystyle{p_{\!\stackrel{\phantom{.}}{E}}=-m_{\!\stackrel
{\phantom{.}}{E}}\left(\Gamma_E+\frac{3(\Theta_E+1)}{2}\right)+1}.
\label{arithmetic equation}
\end{equation}
\label{arithgenus}\vspace*{-6mm}
\end{lem}
\noindent{\it Proof.} From the adjunction formula for the divisor
$\phi^*(T(E))$ in $\widehat{X}$:
\[2p_{\!\stackrel{\phantom{.}}{E}}-2=\left(K_{\widehat{X}}+\phi^*(T(E))\right)\phi^*(T(E))=
\left((K_{{\mathbf P}V}+\widehat{X})|_{\widehat{X}}+\sum_{E^{\prime}}
\delta_{E^{\prime}}\widehat{\phi}^*E^{\prime}\right)\sum_{E^{\prime}}
\delta_{E^{\prime}}\widehat{\phi}^*E^{\prime}.\]
Here $\delta_{E^{\prime}}=0$ if $E^{\prime}<E$, and $\delta_{E^{\prime}}=1$
otherwise. Thus, the sums above are effectively taken
over all $E^{\prime}\geq E$. Substituting the expressions
for $K_{{\mathbf P}V}$ and $\widehat{X}$ as divisors in ${{\mathbf P}V}$
from Lemma~\ref{Kdivisors} and (\ref{genX}), we arrive at
\[2p_{\!\stackrel{\phantom{.}}{E}}-2=\sum_{E^{\prime}}\left(2
\gamma_{\!\stackrel{\phantom{.}}{E^{\prime}}}+
3\theta_{\!\stackrel{\phantom{.}}{E^{\prime}}}+
3\delta_{\!\stackrel{\phantom{.}}{E^{\prime}}}
\right)E^{\prime} \sum_{E^{\prime}}
\delta_{\!\stackrel{\phantom{.}}{E^{\prime}}}E^{\prime}.\]
Set $\Delta_{\!\stackrel{\phantom{.}}{E}}
=\delta_{\!\stackrel{\phantom{.}}{E}}
-\delta_{\!\stackrel{\phantom{.}}{E^-}}$, i.e.
$\Delta_{\!\stackrel{\phantom{.}}{E^{\prime}}}=1$
only if $E^{\prime}=E$; otherwise,
$\Delta_{\!\stackrel{\phantom{.}}{E^{\prime}}}=0$.
By Lemma~\ref{technical},
\[2p_{\!\stackrel{\phantom{.}}{E}}-2=-\sum_{E^{\prime}} m_
{\!\stackrel{\phantom{.}}{E^{\prime}}}
\left(2\Gamma_{\!\stackrel{\phantom{.}}{E^{\prime}}}+
3\Theta_{\!\stackrel{\phantom{.}}{E^{\prime}}}+3
\Delta_{\!\stackrel{\phantom{.}}{E^{\prime}}}\right)\Delta_{\!\stackrel{\phantom{.}}{E^{\prime}}}\,\,
\Rightarrow\,\,2p_{\!\stackrel{\phantom{.}}{E}}-2=
-m_{\!\stackrel{\phantom{.}}{E}}\left(2\Gamma_{\!\stackrel{\phantom{.}}{E}}
+3\Theta_{\!\stackrel{\phantom{.}}{E}}+3\right).\,\,\,\qed\]
\smallskip
Now we can easily compute the invariants
$m_{\!\stackrel{\phantom{.}}{E}}$,
$\Theta_{\!\stackrel{\phantom{.}}{E^{\prime}}}$ and
$\Gamma_{\!\stackrel{\phantom{.}}{E^{\prime}}}$, appearing in the formulas
for $\lambda_{X}$ and $\kappa_{X}$.
\begin{cor} There are three possibilities for the triple
$(m_{\!\stackrel{\phantom{.}}{E}},
\Theta_{\!\stackrel{\phantom{.}}{E^{\prime}}},
\Gamma_{\!\stackrel{\phantom{.}}{E^{\prime}}})$, depending on
whether the components $E$ and $E^-$ of $T$ are reduced:
\begin{eqnarray*}
\on{(a)}\,\on{if}\,E,E^-\,\on{reduced},\,\on{then}&\!\!\!&\!\!\!\!
m_{\!\stackrel{\phantom{.}}{E}}=1,\,\,\Theta_{\!\stackrel{\phantom{.}}{E}}
=1,\,\,\Gamma_{\!\stackrel{\phantom{.}}{E}}=
-(p_{\!\stackrel{\phantom{.}}{E}}+2).\\
\on{(b)}\, \on{if}\,E\,\,\on{nonreduced}, \,\on{then}&\!\!\!&\!\!\!\!
m_{\!\stackrel{\phantom{.}}{E}}=2,\,\,\Theta_{\!\stackrel{\phantom{.}}{E}}
=1,\,\,\Gamma_{\!\stackrel{\phantom{.}}{E}}=
-({p_{\!\stackrel{\phantom{.}}{E}}+5})/{2}.\\
\on{(c)}\,\on{if}\,E^-\!\on{nonreduced}, \,\on{then}&\!\!\!&\!\!\!\!
m_{\!\stackrel{\phantom{.}}{E}}=2,\,\,\Theta_{\!\stackrel{\phantom{.}}{E}}
=0,\,\,\Gamma_{\!\stackrel{\phantom{.}}{E}}=
-({p_{\!\stackrel{\phantom{.}}{E}}+2})/{2}.
\end{eqnarray*}
\label{constants}\vspace*{-5mm}
\end{cor}
\begin{proof} Note that for the list all possible
special fibers $T$ of $\widehat{Y}$, each component $E$ fits in exactly one
of the three cases above (cf.~Fig.~\ref{coef1.fig}--\ref{coef3.fig}).
The proof of the statement is
immediate from the definitions of $m_{\!\stackrel{\phantom{.}}{E}}$ and
$\Theta_{\!\stackrel{\phantom{.}}{E^{\prime}}}$, and
from Lemma~\ref{arithgenus}. \end{proof}
\bigskip\section*{\hspace*{1.9mm}9.
The Bogomolov Condition $4c_2-c_1^2$ and the $7+6/g$ Bound in
$\overline{\mathfrak{T}}_g$}
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\label{Bogomolov1}
With the conventions of Section~8, we state
the main proposition of the section.
\begin{prop} There exists an effective $\mathbb Q$-linear combination
$\mathcal{E}$ of boundary divisors $\Delta{\mathfrak{T}}_{k,i}$, not containing
$\Delta{\mathfrak{T}}_0$, such that
for a general base curve $B$ in $\overline{\mathfrak{T}}_g$:
\[(7g+6)\lambda|_B=g\delta|_B+\mathcal{E}|_B+\frac{g-3}{2}\left(4c_2(V)-c_1^2(V)
\right).\]
\label{bogomolov1}\vspace*{-10mm}
\end{prop}
For a shorthand notation, we denote by $\mathfrak{S}$
the difference
\[\mathfrak{S}:=(7g+6)\lambda|_B-g\delta|_B -\frac{g-3}{2}\left(
4c_2(V)-c_1^2(V)\right).\]
Using the results of the previous section, we can write:
\begin{eqnarray*}
\mathfrak{S}&=&
-\frac{1}{4}\sum_E\left\{m_{\!\stackrel{\phantom{.}}{E}}\left(6\Gamma^2+
6(g+2)\Gamma\Theta+(7g+6)\Theta^2\right)_{\!\stackrel{\phantom{.}}{E}}+5g-6
\right\}\\
&&+\sum_T g\mu(T)+\sum_{\on{ram}1} g +\sum_{\on{ram}3} 3g.
\end{eqnarray*}
We defer the
proof of Prop.~\ref{bogomolov1} until the end of this section, when
all of the terms in this sum will be computed.
\subsection{Grouping the contributions of each $\Delta{\mathfrak{T}}_{k,i}$ in
$\mathfrak{S}$}
\label{Grouping}
Substituting the results of Corollary~\ref{constants}
in the expression for $\mathfrak{S}$, we eliminate
$m_{\!\stackrel{\phantom{.}}{E}}$,
$\Theta_{\!\stackrel{\phantom{.}}{E^{\prime}}}$ and
$\Gamma_{\!\stackrel{\phantom{.}}{E^{\prime}}}$:
\begin{eqnarray*}
\mathfrak{S}&\!\!=\!\!&\sum_T g\mu(T)+\sum_{\on{ram}1} g+\sum_{\on{ram}2} 3g+
\frac{1}{4}\sum_{E,E^-\on{red}}\!\!\left(6(2+p_{\!\stackrel{\phantom{.}}{E}})
(g-p_{\!\stackrel{\phantom{.}}{E}})-12g\right)\\
&\!\!-\!\!
&\frac{1}{4}\sum_{E^-\on{nonred}}\!\!\!\!\!\left(3(p_{\!\stackrel{\phantom{.}}{E}}+2)^2+5g-6\right)+\frac{1}{4}\sum_{E\on{nonred}}\!\!\left(3(p_{\!\stackrel{\phantom{.}}{E}}+5)(2g-p_{\!\stackrel{\phantom{.}}{E}}-1)-19g+6\right).
\end{eqnarray*}
For each chain $T$ in $\widehat{Y}$,
the inverse image $\widehat{\phi}^*(T)$ in $\widehat{X}$ is a member
(or a blow-up of a member) of exactly one boundary divisor
$\Delta{\mathfrak{T}}_{k,i}$. Consequently,
to find the contribution to $\mathfrak{S}$ of a specific $\Delta{\mathfrak{T}}_{k,i}$,
we calculate the sum in $\mathfrak{S}$ corresponding to all types of
special fibers $\widehat{\phi}^*(T)$.
\begin{figure}[h]
$$\hspace*{-7mm}\psdraw{coef1}{4.5in}{1.3in}$$
\caption{Coefficients with no ramification}
\label{coef1.fig}
\end{figure}
\subsubsection{Contributions of $\Delta{\mathfrak{T}}_{1,i},
\Delta{\mathfrak{T}}_{2,i}$ and $\Delta{\mathfrak{T}}_{3,i}$}
\label{contribution1}
Fig.~\ref{coef1.fig} presents the
special fibers corresponding to the boundary divisors
$\Delta{\mathfrak{T}}_{1,i},\,\,\Delta{\mathfrak{T}}_{2,i},\,\,\Delta{\mathfrak{T}}_{3,i}$. In each of these cases,
there is only one component $E$ in $T$ besides the root
$R=E^-$. Thus, the subchain $T(E)$ in $T$ is trivial -- it consists
only of $E$. Its inverse image $\widehat{\phi}^*(E)$ is connected
for $\Delta{\mathfrak{T}}_{1,i}$, and consists of two connected curves for
$\Delta{\mathfrak{T}}_{2,i}$ and $\Delta{\mathfrak{T}}_{3,i}$. Setting the genus of the inverse
image of $R$ to be $i$, it is easy to see that the genus $p_{\!\stackrel{\phantom{.}}{E}}$ of
$\phi^*(E)$ is $g-i-2$ in the first two cases, and $g-i-1$ in the
third case.(The total genus of the original fiber of $X$,
drawn in full lines, must be $g$.)
Finally, counting the number of ``quasi-admissible''
blow-ups (drawn by dashed lines), we see that $\mu(T)=0$ for $\Delta{\mathfrak{T}}_{1,i}$,
$\mu(T)=1$ for $\Delta{\mathfrak{T}}_{2,i}$, and $\mu(T)=2$ for
$\Delta{\mathfrak{T}}_{3,i}$ (cf.~Lemma~\ref{mu(C)}).
Note that there are no ramification modifications.
The contribution of each such fiber $\widehat{\phi}^*T$
to the sum $\mathfrak{S}$ is only
one summand of the first type ($E,E^-$reduced), plus the
quasi-admissible adjustment $g\mu(T)$. If $\widehat{\phi}^*T$
corresponds to the boundary divisor $\Delta{\mathfrak{T}}_{k,i}$, we denote this
contribution by $c_{k,i}$. In conclusion,
\[c_{k,i}=
\frac{1}{4}\big(6(2+p_{\!\stackrel{\phantom{.}}{E}})(g-p_{\!\stackrel
{\phantom{.}}{E}})-12g\big)+g\mu(T)\,\,\Rightarrow\,\,
c_{k,i}=\frac{3}{2}(i+2)(g-i)-(4-k)g,\,\,k=1,2,3.\]
\subsubsection{Contributions of $\Delta{\mathfrak{T}}_{4,i}$ and
$\Delta{\mathfrak{T}}_{5,i}$: ramification index 1}
\label{contribution2}
In each of these cases,
the fiber $T$ of $\widehat{Y}$ consists of two rational curves $E_1$ and $E_2$,
and the root $R=E_1^-$ (cf.~Fig.~\ref{coef2.fig}). There are
no nonreduced components in $T$, so the contribution to $\mathfrak{S}$
consists of two summands of the first type ($E,E^-$ nonreduced),
plus a quasi-admissible adjustment of $\mu(T)=1$ for $\Delta{\mathfrak{T}}_{5,i}$,
and a ramification adjustment of $g$ in both cases:
\[c_{k,i}=\frac{1}{4}\sum_{j=1,2}
\big(6(2+p_{\!\stackrel{\phantom{.}}{E_j}})(g-p_
{\!\stackrel{\phantom{.}}{E_j}})
-12g\big)+g\mu(T)+g\,\,\on{for}\,\,k=4,5.\]
\begin{figure}[h]
$$\psdraw{coef2}{3.25in}{2in}$$
\caption{Coefficients for ramification index 1}
\label{coef2.fig}
\end{figure}
The arithmetic genus of the nonreduced component of $\widehat{X}$ is
$-2$, and its intersection number with each of the neighboring components is
2. Setting $p_a(\widehat{\phi}^*R)=i$ forces $p_a(\widehat{\phi}^*E_2)=
g-i-1$. Hence, $p_{\!\stackrel{\phantom{.}}{E_1}}=g-i-1$ and
$p_{\!\stackrel{\phantom{.}}{E_2}}=g-i-2$.
Substituting:
\[c_{k,i}=3(g-i)(i+1)-\frac{7g-3}{2}+g\mu(T),\]
\[\vspace*{-5mm}
c_{4,i}=3(i+1)(g-i)-\frac{7g-3}{2},\,c_{5,i}=3(i+1)(g-i)-\frac{7g-3}{2}+2g.\]
\subsubsection{Contribution of $\Delta{\mathfrak{T}}_{6,i}$: ramification
index 2}
\label{contribution3}
It remains to consider the case of ramification index 2.
Here there are four components $E$ besides the root $R$ in the special
fiber $T\subset \widehat{Y}$. Consequently, there
are four summands in $\mathfrak{S}$ corresponding to the $E_i$'s: $E_1$ and
$E_4$ yield summands of the first type ($E,E^-$ reduced),
$E_2$ yields a summand of the second type ($E$ nonreduced),
and $E_3$ yields a summand of the third type ($E^-$ nonreduced).
\begin{figure}[h]
$$\psdraw{coef3}{3.5in}{1.5in}$$
\caption{Coefficients for ramification index 2}
\label{coef3.fig}
\end{figure}
Since $\mu(T)=0$, and the ramification adjustment is $3g$, we obtain for the
contribution of $\Delta{\mathfrak{T}}_{6,i}$ to $\mathfrak{S}$ the following expression:
\begin{eqnarray*}
c_{6,i}\!\!&\!\!=\!\!&\!\!\frac{1}{4}\big(6(2+
p_{\!\stackrel{\phantom{.}}{E_1}})(g-
p_{\!\stackrel{\phantom{.}}{E_1}})-12g\big)+
\frac{1}{4}\big(6(2+p_{\!\stackrel{\phantom{.}}{E_4}})(g-
p_{\!\stackrel{\phantom{.}}{E_4}})-12g\big)+\\
\!\!&\!\!+\!\!&\!\!\frac{1}{4}\left(3(
p_{\!\stackrel{\phantom{.}}{E_2}}+5)(2g-
p_{\!\stackrel{\phantom{.}}{E_2}}-1)-19g+6\right)
-\frac{1}{4}\left(3(p_{\!\stackrel{\phantom{.}}{E_3}}+2)^2+5g-6\right)+3g.
\end{eqnarray*}
The arithmetic genera of the components in $\widehat{X}$ are denoted in
the Fig.~\ref{coef3.fig}. It is easy to see that
$p_{\!\stackrel{\phantom{.}}{E_4}}
=i$, $p_{\!\stackrel{\phantom{.}}{E_3}}=i-3$,
$p_{\!\stackrel{\phantom{.}}{E_2}}=i-2$,
$p_{\!\stackrel{\phantom{.}}{E_1}}=i-2$. Finally,
\[c_{6,i}=\frac{9}{2}i(g-i)-\frac{3}{2}(g-1).\]
\subsection{Proof of Proposition~\ref{bogomolov1}}
\label{Proof}
In the above discussion
we calculated the contributions of the boundary divisors
$\Delta{\mathfrak{T}}_{k,i}$ to the sum $\mathfrak{S}$, so that
$\mathfrak{S}=\sum_{k,i}c_{k,i}$ with $k=1,...,6$, and the
corresponding limits for the index $i$ (cf.~Prop.~\ref{boundary}).
It is now clear what the divisor
$\mathcal{E}$ should be. We set
$\mathcal{E}:=\sum_{k,i}c_{k,i}\Delta{\mathfrak{T}}_{k,i}$, and thus,
$\mathfrak{S}=\mathcal{E}|_B$,
\[\Rightarrow\,\,\,
(7g+6)\lambda|_B=g\delta|_B+\mathcal{E}|_B+\frac{g-3}{2}(4c_2(V)-c_1^2(V)).\]
Using the restrictions on the index $i$ for each type of boundary divisor
$\Delta{\mathfrak{T}}_{k,i}$, one can easily deduce that all coefficients $c_{k,i}> 0$.
For instance, when $i=1,...,[g/2]$:
\[c_{6,i}=\frac{9}{2}i(g-i)-\frac{3}{2}(g-1)>\frac{9}{2}1\cdot
(g-1)-\frac{3}{2}(g-1)=3(g-1)>0.\]
In other words, $\mathcal{E}$ is an effective rational linear
combination of boundary divisors in $\overline{\mathfrak{T}}_g$, which by
construction does not contain $\Delta{\mathfrak{T}}_0.\,\,\,\qed$
\subsection{The slope bound $7+6/g$ and a relation
restricted to the base curve $B$}
\label{slopebound}
Recall that a vector bundle $V$ of rank 2 is {\it Bogomolov
semistable} if $4c_2(V)\geq c^2_1(V)$.
\begin{prop}[$7+6/g$ bound]
For a general base curve $B$, if the canonically associated vector bundle $V$
is Bogomolov semistable, then the slope of $X/\!_{\displaystyle{B}}$ is
bounded by
\[\frac{\delta|_B}{\lambda|_B}\leq 7+\frac{6}{g}\cdot\]
\label{7+6/g Bogomolov}
\end{prop}
\vspace*{-5mm}\begin{proof} The statement follows directly from
Prop.~\ref{bogomolov1}. Indeed, since $\mathcal{E}$ is effective, then
$\mathcal{E}|_B\geq 0$. By hypothesis, $4c_2(V)-c^2_1(V)\geq
0$, and $g\geq 3$. Hence, $(7g+6)\lambda|_B\geq g\delta|_B.$ \end{proof}
\begin{cor} For a general base curve $B$
the following relation holds true:
\begin{eqnarray*}
\!(7g+6)\lambda|_B&\!\!\!=\!\!\!&g\delta_0|_B+
\frac{g-3}{2}\left(4c_2(V)-c_1^2(V)\right)\\
&\!\!\!+\!\!\!\!\!\!&
\sum_{i=1}^{[(g-2)/2]}\frac{3}{2}(i+2)(g-i)\delta_{1,i}|_B+
\sum_{i=1}^{g-2}\frac{3}{2}(i+2)(g-i)\delta_{2,i}|_B\\
&\!\!\!+&\sum_{i=1}^{[g/2]}\frac{3}{2}(i+1)(g-i+1)\delta
_{3,i}|_B+\sum_{i=1}^{[(g-1)/2]}
\big(3(i+1)(g-i)-\frac{g-3}{2}\big)\delta_{4,i}|_B\\
&\!\!\!+&\sum_{i=1}^{g-1}\big(3(i+1)(g-i)-\frac{g-3}{2}\big)
\delta_{5,i}|_B+\sum_{i=1}^{[g/2]}
\big(\frac{9}{2}i(g-i)-{\frac{g-3}{2}}\big)\delta_{6,i}|_B.
\end{eqnarray*}
\label{analog1}\vspace*{-7mm}
\end{cor}
\noindent{\it Proof.} This is an immediate consequence of the established
relation in Prop.~\ref{bogomolov1}. We replace $\delta$ by the
linear combination (\ref{divisorrel}) of the boundary
classes of $\overline{\mathfrak{T}}_g$, and write
\[(7g+6)\lambda=g\delta_0|_B+\sum_{k,i}\widetilde{c}_{k,i}\delta_{k,i}|_B
+\frac{g-3}{2}(4c_2(V)-c_1^2(V)),\]
for some new coefficients $\widetilde{c}_{k,i}$. Recall that
$\on{mult}_{\delta}(\delta_{k,i})$ denotes the {\it multiplicity}
of $\delta_{k,i}$ in $\delta$, so that
$\widetilde{c}_{k,i}=c_{k,i}+\on{mult}_{\delta}(\delta_{k,i})g$.
For example, the coefficient of $\delta_{1,i}$ is
\[\widetilde{c}_{1,i}=\left\{\frac{3}{2}(i+2)(g-i)-3g\right\}+3g=
\frac{3}{2}(i+2)(g-i),\]
or the coefficient of $\delta_{5,i}$ is
\[\widetilde{c}_{5,i}=\left\{3(i+2)(g-i)-\frac{7g-3}{2}+2g\right\}+g=
3(i+1)(g-i)-\frac{g-3}{2}.\,\,\,\qed\]
\medskip\section*{10. Generalized Index Theorem and Upper Bound}
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\label{Index1}
\begin{prop}[Index Theorem on $\widehat{X}$] For a general base curve $B$ and
for the rank 2 vector bundle $V$ on $\widehat{Y}$, we have
$9c_2(V)-2c_1^2(V)\geq 0.$
\label{genindex}
\end{prop}
\begin{proof}
The proof is identical to that of
Theorem~\ref{indextheorem}. One considers the divisor $\eta$ on
$\widehat{X}$ defined by
\[\eta:=\left(\zeta+\frac{1}{3}\pi^*c_1(V)\right)\big|_
{\widehat{X}},\]
and shows that $\eta$ kills the pullback of any divisor on $\widehat{Y}$.
In particular, $\eta$ kills an ample divisor on $\widehat{X}$.
By the index theorem on $\widehat{X}$, $\eta^2
\leq 0$. From expression (\ref{genX}),
this can be also written as $9c_2(V)-2c_1^2(V)\geq 0.$ \end{proof}
\medskip
As in Section~7, the index theorem on $\widehat{X}$ suggests to
replace the Bogomolov difference $4c_2(V)-c^2_1(V)$ by another
linear combination of $c_2(V)$ and $c^2_1(V)$, which would behave in a more
``predictable'' way, namely, by $9c_2(V)-2c_1^2(V)$. In the process of doing
so, the only way to eliminate the unnecessary global
terms $d$ and $c$ from a relation among $\lambda|_B$ and $\delta|_B$
is to subtract:
$36(g+1)\lambda|_B-(5g+1)\delta|_B.$
\begin{prop} For a general base curve $B$ and an effective rational
combination $\mathcal{E}^{\prime}$ of the
boundary divisors $\Delta{\mathfrak{T}}_{k,i}$, not containing $\Delta{\mathfrak{T}}_0$, we have:
\[36(g+1)\lambda|_B=(5g+1)\delta|_B+\mathcal{E}^{\prime}|_B+(g-3)
\big(9c_2(V)-2c_1^2(V)\big).\]
\label{indexrelation}\vspace*{-5mm}
\end{prop}
Note the apparent similarity between this relation and
Prop.~\ref{bogomolov1}. One may use the latter to prove the former, but
the calculations are not simpler than if one starts from scratch. We will
show a sketch of this proof, leaving the details to the reader, and referring
to the proof of Prop.~\ref{bogomolov1} for comparison.
\begin{proof} We denote by $\mathfrak{S}^{\prime}$ the difference
\[\mathfrak{S}^{\prime}:=
36(g+1)\lambda|_B-(5g+1)\delta|_B -(g-3)\left(9c_2(V)-2c_1^2(V)\right).\]
Substituting for $\delta|_B,\lambda|_B$ and $c_1^2(V)$ the
corresponding identities from Prop.~\ref{hatlambda} and
Example 8.1, and recalling that $c=g+2$ (cf.~Lemma~\ref{adjunction}),
we write $\mathfrak{S}^{\prime}$ as
\begin{eqnarray*}
\mathfrak{S}^{\prime}&=
&-\sum_E\left\{m_{\!\stackrel{\phantom{.}}{E}}\left(8\Gamma^2+
8(g+2)\Gamma\Theta+9(g+1)\Theta^2\right)_{\!\stackrel{\phantom{.}}{E}}+6(g-1)
\right\}\\
&&+\,(5g+1)\left(\sum_T \mu(T)+\sum_{\on{ram}1}1 +\sum_{\on{ram}3} 3\right).
\end{eqnarray*}
As in Lemma~\ref{bogomolov1}, we group the above summands for every
special fiber in $\widehat{X}$, and correspondingly, for every
chain $T$ in $\widehat{Y}$. Recall Corollary~\ref{constants},
and the computations of the arithmetic genera
$p_{\!\stackrel{\phantom{.}}{E}}$ in the previous section:
\begin{eqnarray*}
\mathfrak{S}^{\prime}
&\!\!=\!\!&(5g+1)\big(\sum_T \mu(T)+\sum_{\on{ram}1} 1+\sum_{\on{ram}2}
3\big)+\!\!\sum_{E,E^-\on{red}}\!\!\left(8(p_{\!\stackrel{\phantom{.}}{E}}+2)
(g-p_{\!\stackrel{\phantom{.}}{E}})-3(5g+1)\right)\\
&\!\!-\!\!\!\!&\sum_{E^-\on{nonred}}\!\!\!\!\!\left(4
(p_{\!\stackrel{\phantom{.}}{E}}+2)^2+6(g-1)\right)\,+
\sum_{E\on{nonred}}\!\!\left(4(p_{\!\stackrel{\phantom{.}}{E}}+5)
(2g-1-p_{\!\stackrel{\phantom{.}}{E}})-12(g-1)\right).
\end{eqnarray*}
With this at hand, it is not hard to calculate the
contributions $d_{k,i}$ of each boundary component $\Delta{\mathfrak{T}}_{k,i}$
to the sum $\mathfrak{S}^{\prime}$:
\[\begin{array}{|l|l|}\hline
\!d_{1,i}\stackrel{\phantom{l}}{=}8(i+2)(g-i)\phantom{+1}\!-3(5g+1)\!&
\!d_{4,i}=16(i+1)(g-i)-2(g-3)-3(5g+1)\\
\!d_{2,i}\stackrel{\phantom{l}}{=}8(i+2)(g-i)\phantom{+1}-\!2(5g+1)\!&
\!d_{5,i}\stackrel{\phantom{l}}{=}16(i+1)(g-i)-2(g-3)-\phantom{3}(5g+1)\\
\!d_{3,i}\stackrel{\phantom{l}}{=}8(i+1)(g-i+1)-(5g+1)\!&
\!d_{6,i}\stackrel{\phantom{l}}{=}24i(g-i)-(5g+1).\phantom{\big)}\\\hline
\end{array}\]
\label{d_{k,i}table}
Let $\mathcal{E}^{\prime}=\sum_{k,i}d_{k,i}\Delta{\mathfrak{T}}_{k,i}$. Then
$\mathfrak{S}^{\prime}=\mathcal{E}^{\prime}|_B$, and the desired relation would be
established if $\mathcal{E}^{\prime}$ is effective. Given the
restrictions on the indices $i$ of the coefficients $d_{k,i}$ in
Prop.~\ref{boundary}, one
easily shows that all $d_{k,i}>0.$ \end{proof}
\begin{prop}[Maximal Bound]
For a general base curve $B$, the slope satisfies:
\[\frac{\delta}{\lambda}\leq \frac{36(g+1)}{5g+1},\]
with equality if and only all fibers of $X$ are irreducible
curves, and either $g=3$ or
the divisor $\eta$ on the total space of ${X}$ is numerically zero.
\label{genmaximal}
\end{prop}
\begin{proof} From the Index Theorem on $\widehat{X}$, it follows
that $9c_2(V)-2c_1^2(V)\geq 0$. Since $\mathcal{E}^{\prime}$ is
effective, $\mathcal{E}^{\prime}|_B\geq 0$. Then Prop.~\ref{indexrelation}
implies $36(g+1)\lambda|_B\geq(5g+1)\delta|_B$, with equality exactly
when $9c_2(V)-2c_1^2(V)=0$ and $\mathcal{E}^{\prime}|_B=0$. The latter
means that $B\cap \Delta{\mathfrak{T}}_{k,i}=\emptyset$ because all coefficients
$d_{k,i}$ of $\mathcal{E}^{\prime}$ are strictly positive. In other words,
the family $\widehat{X}$ has only {\it irreducible} fibers
($B\cap\Delta{\mathfrak{T}}_{0}\not = \emptyset$).
This takes us back to Section~7,
where we presented the global calculation
on the triple cover $X\rightarrow Y$. There we concluded that
the {\it index condition} $9c_2(V)-2c_1^2(V)=0$ was equivalent to
$\eta\equiv 0$ on $X$($=\widehat{X}$), or the genus $g=3$. \end{proof}
\begin{cor} For a general base curve $B$,
\begin{eqnarray*}\vspace*{-1mm}
\!36(g+1)\lambda|_B&\!\!\!\!=\!\!\!\!&(5g+1)\delta_0|_B+
(g-3)\left(9c_2(V)-2c_1^2(V)\right)\\
&\!\!\!\!+\!\!\!\!\!&\sum_{i=0}^{[(g-2)/2]}8(i+2)(g-i)\delta_{1,i}|_B+
\sum_{i=1}^{g-2}8(i+2)(g-i)\delta_{2,i}|_B\\
&\!\!\!\!+\!\!\!\!\!&\sum_{i=1}^{[g/2]}8(i+1)(g-i+1)\delta_{3,i}|_B+
\!\!\!\sum_{i=1}^{[(g-1)/2]}\!\!\!\!\!
\big(16(i+1)(g-i)-2(g-3)\big)\delta_{4,i}|_B\\
&\!\!\!\!+\!
\!\!\!&\sum_{i=1}^{g-1}\big(16(i+1)(g-i)-2(g-3)\big)\delta_{5,i}|_B+
\sum_{i=1}^{[g/2]}24i(g-i)\delta_{6,i}|_B.
\end{eqnarray*}
\label{analog2}
\end{cor}
\label{page analog2}
\noindent{\it Proof.} We only need to substitute the known expressions
for the divisors $\mathcal{E}^{\prime}$ and $\delta$ into
Prop.~\ref{indexrelation}:
\[36(g+1)\lambda|_B=(5g+1)\delta_0|_B+\sum_{k,i}\big((5g+1)\on{mult}_{\delta}
(\delta_{k,i})+d_{k,i}\big)+(g-3)\big(9c_2(V)-2c_1^2(V)\big).\]
The rest is a simple calculation. For example, the total coefficient
$\widetilde{d}_{3,i}$ of $\delta_{3,i}$ equals
\begin{eqnarray*}
d_{3,i}+(5g+1)\on{mult}_{\delta}(\delta_{3,i})&=&\{8(i+1)(g-i+1)-(5g+1)\}+
(5g+1)\cdot 1\\
&=&8(i+1)(g-i+1).\,\,\,\qed
\end{eqnarray*}
\medskip
\section*{11. Extension to an Arbitrary Base $B$}
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\label{arbitrary}
We extend now the results of
Sect.~8-10 to arbitrary nonisotrivial
families $X\!\rightarrow \!B$ with smooth trigonal general member. The
essential case is when $B$ is {\it not} tangent to the boundary
$\Delta{\mathfrak{T}}$, from which the remaining cases easily follows.
\subsection{The base curve $B$ not tangent to $\Delta\mathfrak{T}$}
\label{nontangentB}
We now drop the hypothesis of the base curve $B$
intersecting the boundary divisors in general points. Instead, for now we only
assume that the base curve $B$ is not tangent
the boundary $\Delta{\mathfrak{T}}$.
This means that all special fibers of $X$ locally look like the
general ones (cf.~Fig.~\ref{coef1.fig}--\ref{coef3.fig}). Therefore,
from the quasiadmissible cover
$\widetilde{X}\rightarrow \widetilde{Y}$ we can construct an effective cover
$\widehat{\phi}:\widehat{X}\rightarrow \widehat{Y}$ of {\it smooth}
surfaces $\widehat{X}$ and $\widehat{Y}$.
(The {\it smoothness} indicates
that $B$ is {\it not} tangent to any $\Delta{\mathfrak{T}}_{k,i}$. Otherwise,
there would be a higher local multiplicity $xy=t^n$ near a node
of a special fiber $C_{X}$, $n> 1$. Hence $\widehat{X}$ would be
obtained locally via a base change from a smooth surface, but $\widehat{X}$
would have a singular total space.)
Now the special fibers of $\widehat{Y}$ are {\it trees} $T$ (rather than
just chains) of reduced smooth rational
curves with occasional nonreduced rational components of multiplicity 2.
The latter occur again exactly for each singular point in
$C_{\widetilde{X}}$ of
ramification index 2 under the quasiadmissible cover $\widetilde{\phi}:
\widetilde{X}\rightarrow\widetilde{Y}$ (cf.~Fig.~\ref{coef3.fig}).
\smallskip
The notation and conventions from Sections
\ref{conventions} are also valid here. In particular, for any tree $T$,
we fix one of its end (nonreduced) components to be its root $R$, and we define
as before the functions $m,\theta,\gamma$ on the components $E$ of $T$.
Moreover, since Lemma
\ref{technical} can be applied also for any tree $T$, the calculations of
$\lambda_{\widehat{X}},\kappa_{\widehat{X}}$ and $\delta$ in
Prop.~\ref{hatlambda}
and Cor.~\ref{hatdelta} go through without any modifications.
\smallskip Finally, we wish to extend all results of
Sections~8-10 over the new base $B$.
The only difference arises in the final calculation of the
coefficients $c_{k,i}$ and $d_{k,i}$. The fiber $C_X$ in $X$, corresponding to
a tree $T$, may now lie
in the intersection of {\it several} boundary divisors
$\Delta{\mathfrak{T}}_{k,i}$. Such a trigonal curve $C_X$ is called a {\it special
boundary} curve.
Accordingly, its contribution $c_{\!\stackrel{\phantom{.}}{T}}$ to
$\mathfrak{S}$ (or $d_{\!\stackrel{\phantom{.}}{T}}$ to $\mathfrak{S}^{\prime}$)
will be distributed among these divisors
$\Delta{\mathfrak{T}}_{k,i}$'s, rather than
just yielding a single coefficient $c_{k,i}$ (or $d_{k,i}$) as before.
\begin{figure}[h]
$$\psdraw{arbitrary}{1.6in}{1.5in}$$
\caption{Moving $B$}
\label{arbitrary1}
\end{figure}
This can be easily resolved. The idea is to replace any special singular
fiber in $\widehat{X}$ by a suitable combination of {\it general} fibers,
without changing the sums $\mathfrak{S}$ and $\mathfrak{S}^{\prime}$. We can
imagine this as ``moving'' the base curve $B$ in $\overline{\mathfrak{T}}_g$
{\it away from} the special singular locus of $\overline{\mathfrak{T}}_g$,
and replacing it with a {\it general} base curve $B^{\prime}$, as defined
in Section~8. For example, in Fig.~\ref{arbitrary1}
the base $B$ passes through a point $p$ in the intersection of
two boundary divisors $\Delta\mathfrak{T}_{k,i}$. Two new general points
$p_1$ and $p_2$, each lying on a single $\Delta\mathfrak{T}_{k,i}$, replace the
special point $p$, and thus $B$ moves to a {\it general}
curve $B^{\prime}$.
\begin{lem} Let $C_X$ be a special boundary curve in
$\overline{\mathfrak{T}}_g$. Denote by
$\alpha_{k,i}$ the degree of the point $[C_X]$ in the intersection
$\Delta{\mathfrak{T}}_{k,i}\cdot B$. Then the contributions of
$T=\widehat{\phi}(C_{\widehat{X}})$ to
$\mathfrak{S}$ and to $\mathfrak{S}^{\prime}$ are
$c_{\!\stackrel{\phantom{.}}{T}}=\sum_{k,i}\alpha_{k,i}c_{k,i}$ and
$d_{\!\stackrel{\phantom{.}}{T}}=\sum_{k,i}\alpha_{k,i}d_{k,i}$, respectively.
\label{contributions d_Tc_T}
\end{lem}
\noindent{\it Proof:} Rewrite $\mathfrak{S}$ and
$\mathfrak{S}^{\prime}$ as sums over the non-root components $E$ of the special
trees $T$:
\begin{eqnarray*}
\mathfrak{S}&\!\!=\!\!&
\sum_{E,E^-\on{red}}F_1(p_{\!\stackrel{\phantom{.}}{E}})+
\sum_{E^-\on{nonred}}\!\!\!\!\!F_2(p_{\!\stackrel{\phantom{.}}{E}})
+\sum_{E\on{nonred}}\!\!F_3(p_{\!\stackrel{\phantom{.}}{E}})
+gH,\\
\mathfrak{S}^{\prime}&\!\!=\!\!&
\sum_{E,E^-\on{red}}G_1(p_{\!\stackrel{\phantom{.}}{E}})+
\sum_{E^-\on{nonred}}\!\!\!\!\!G_2(p_{\!\stackrel{\phantom{.}}{E}})
+\sum_{E\on{nonred}}\!\!G_3(p_{\!\stackrel{\phantom{.}}{E}})+(5g+1)H,
\end{eqnarray*}
where $H=\sum_T\mu(T)+\sum_{\on{ram}1}1+\sum_{\on{ram}2}3$ is the
quasi-admissible and effective adjustment, and the functions $F_i$ and
$G_j$ are quadratic polynomials in $p_{\!\stackrel{\phantom{.}}{E}}$
with linear coefficients in $g$.
Recall that in these sums each non-root component $E$ appears
exactly once, and $p_{\!\stackrel{\phantom{.}}{E}}$ is the arithmetic
genus of the inverse image $\widehat{\phi}^*(T(E))$ of the
subtree $T(E)$ generated by $E$.
\smallskip
There is a simple way to recognize the boundary divisors
$\Delta{\mathfrak{T}}_{k,i}$ in which
a special trigonal fiber $C_X$ lies. Consider the
corresponding ``effective'' fiber $C_{\widehat{X}}=\widehat{\phi}^*T$
in $\widehat{X}$. For any non-root component $E$ in $T$ there are
two possibilities: either $\widehat{\phi}^*E$ and
$\widehat{\phi}^*{E^-}$ are both reduced, or
$E$ is part of a chain of length 3 or 5, constructed to
resolve ramifications in the quasi-admissible fiber $C_{\widetilde{X}}$.
\smallskip
\begin{figure}[h]
$$\psdraw{notchain}{3.8in}{1.8in}$$ \vspace*{-2mm}
\caption{$E\not\subset$ chain $\rightarrow\,\,
\alpha_{1,i},\alpha_{2,i},\alpha_{3,i}$}
\label{notchain}
\end{figure}
\subsubsection{Contributions to the degrees
$\alpha_{1,i},\alpha_{2,i},\alpha_{3,i}$}
Consider the first situation, and denote by $C^{\prime}$ the preimage
$\widehat{\phi}^*E\cup \widehat{\phi}^*{E^-}$ in $\widehat{X}$. Thus,
$C^{\prime}$ corresponds
to a general member of $\Delta{\mathfrak{T}}_{1,i},\Delta{\mathfrak{T}}_{2,i},
\Delta{\mathfrak{T}}_{3,i}$, possibly of lower genus (cf.~Fig.~\ref{notchain}).
As part of the fiber $C_{\widehat{X}}$, the curve $C^{\prime}$ is
represented for simplicity by the triple intersection of two {\it smooth}
trigonal curves (in $\Delta{\mathfrak{T}}_{1,i}$), but it could have
corresponded to any general member of $\Delta{\mathfrak{T}}_{2,i}$
or $\Delta{\mathfrak{T}}_{3,i}$. The solid box
encompasses the preimage $\widehat{\phi}^*T(E)$, and the dashed box
encompasses the preimage of the rest, $\widehat{\phi}^*\big(T-T(E)\big)$.
Each of these boxes represents a limit of a quasi-admissible
curve, $C_1$ or $C_2$,
which is naturally a triple cover of ${\mathbf P}^1$. Thus, we can
{\it ``smoothen''}
each box to such a curve $C_i$. As a result we obtain
a quasiadmissible curve $C_1\cup C_2$ of total genus $g$, which
corresponds to a general member
of $\Delta{\mathfrak{T}}_{1,i},\Delta{\mathfrak{T}}_{2,i}$ or $\Delta{\mathfrak{T}}_{3,i}$. Depending on
which divisor $\Delta{\mathfrak{T}}_{k,i}$ is evoked, there is a
corresponding contribution of $1$ to the
coefficient $\alpha_{k,i}$: $[C_X]\in\Delta{\mathfrak{T}}_{k,i}$.
\smallskip
Note that the arithmetic genus of $C_2$ is the previously
defined $p_{\!\stackrel{\phantom{.}}{E}}$. The contribution of $E$
to $\mathfrak{S}$ is $F_1(p_{\!\stackrel{\phantom{.}}{E}})$ plus the possible
quasi-admissible adjustment in $\mu(T)$ needed to obtain $\widetilde
{\phi}^*(E\cup E^-)$.
In view of the above ``smoothening'', this can be thought of as
the contribution of $C_2$ in the effective curve $C_1\cup C_2$, and by
Prop.~\ref{bogomolov1} this is exactly the coefficient $c_{k,i}$.
The same argument works in the case of $\mathfrak{S}^{\prime}$ from
Prop.~\ref{indexrelation}. We conclude that $\alpha_{k,i}$ (for $k=1,2,3$)
equals the number of $c_{k,i}$'s and $d_{k,i}$'s in $\mathfrak{S}$ and
$\mathfrak{S}^{\prime}$, respectively.
\subsubsection{Contributions to the degrees
$\alpha_{4,i},\alpha_{5,i},\alpha_{6,i}$} We treat analogously
the remaining case
when the component $E$ is part of a chain of length 3 or 5.
Here, however, one must consider {\it simultaneously
all} the components $E$ of $T$ participating in such a chain, and take a
quasi-admissible limit only {\it over the reduced} curves in $C_{\widehat{X}}$.
In Fig.~\ref{chain} one can see all three ramification cases, or
equivalently, the boundary divisors $\Delta{\mathfrak{T}}_{4,i},\Delta{\mathfrak{T}}_{5,i}$
and $\Delta{\mathfrak{T}}_{6,i}$. For simplicity, we have again depicted the
reduced components in $\widehat{X}$ by smooth trigonal curves,
which may not always be true for every tree $T$: they could, for instance,
be singular or reducible, but they will keep the ramification index 1 or 2
at the appropriate points.
\begin{figure}[h]
$$\psdraw{chain}{5.5in}{2in}$$
\caption{$E\subset$ chain $\rightarrow\,\,
\alpha_{4,i},\alpha_{5,i},\alpha_{6,i}$}
\label{chain}
\end{figure}
\newline
To see how $c_{k,i}$ and $d_{k,i}$ are obtained,
let us calculate, for example, the contributions of $E_1,E_2,E_3$ and
$E_4$ in the case of $\Delta{\mathfrak{T}}_{6,i}$. The inverse images in
$\widehat{X}$ of $T-T(E_1)$ and $T(E_4)$ are marked by dashed and
solid boxes, respectively.
We {\it ``smoothen''} each box by a smooth trigonal curve, $C_1$ or $C_2$, and
keep the inverse images of $E_1$,$E_2$ and $E_3$. Thus, we obtain
a general member ${C}^{\prime\prime}$
of $\Delta{\mathfrak{T}}_{6,i}$. The arithmetic genera, necessary to calculate the
contribution of ${C}^{\prime\prime}$ to $\mathfrak{S}$, are given from
right to left by: \[p_a(C_2)=p_{\!\stackrel{\phantom{.}}{E_4}},\,\,
p_{\!\stackrel{\phantom{.}}{E_3}}=p_{\!\stackrel{\phantom{.}}{E_4}}\!\!-3,\,\,
p_{\!\stackrel{\phantom{.}}{E_2}}=p_{\!\stackrel{\phantom{.}}{E_4}}\!\!-2,\,\,
p_{\!\stackrel{\phantom{.}}{E_1}}=p_{\!\stackrel{\phantom{.}}{E_4}}\!\!-2.\]
As in the proof of Prop.~\ref{bogomolov1}, we substitute these
in the sum $\mathfrak{S}$, and for $i=p_{\!\stackrel{\phantom{.}}{E_4}}$ we obtain
\[F_1(E_1)+F_1(E_4)+F_2(E_2)+F_3(E_3)+3g=\frac{9}{2}
p_{\!\stackrel{\phantom{.}}{E_4}}(g-p_{\!\stackrel{\phantom{.}}{E_4}})-
\frac{3}{2}(g-1)=c_{6,i}.\]
Combining all of the above
results, we conclude that the contributions of any tree $T$ to the sums
$\mathfrak{S}$ and $\mathfrak{S}^{\prime}$ are
$c_{\!\stackrel{\phantom{.}}{T}}=\sum_{k,i}\alpha_{k,i}c_{k,i}\,\,\on{and}\,\,
d_{\!\stackrel{\phantom{.}}{T}}=\sum_{k,i}\alpha_{k,i}d_{k,i}.$\qed
\smallskip
This allows us to extend all results of
Sect.~\ref{Bogomolov1}--\ref{Index1} to the case of a base curve $B$
meeting transversally the boundary $\Delta{\mathfrak{T}}_g$.
\subsection{Extension to an arbitrary base $B$, not contained in
$\Delta{\mathfrak{T}}_g$}
\label{extension}
If the base curve $B$ happens to be {\it tangent} to a boundary divisor
$\Delta\mathfrak{T}_{k,i}$ at a point $[C_{X}]$, then over some node $p$ of the
corresponding tree $T=\widehat{\phi}(C_{\widehat{X}})$ {\it all}
local analytic multiplicities $m_q$ (cf.~Sect.~\ref{definition})
will be multiplied by the degree
of tangency of $B$ and $\Delta\mathfrak{T}_{k,i}$. Fig.~\ref{Local
multiplicities} presents a few examples of possible fibers in $\widetilde{X}$:
\begin{figure}[h]
$$\psdraw{tangent}{5.4in}{1.2in}$$
\caption{Local multiplicities}
\label{Local multiplicities}
\end{figure}
\smallskip
In the nonramification cases of $\Delta\mathfrak{T}_{1,i},\Delta\mathfrak{T}_{2,i}$ and
$\Delta\mathfrak{T}_{3,i}$,
this would force rational double points as singularities on
the total spaces of $\widehat{X}$ and $\widehat{Y}$,
whereas in the ramification cases of
$\Delta\mathfrak{T}_{4,i},\Delta\mathfrak{T}_{5,i}$ and
$\Delta\mathfrak{T}_{6,i}$, one may arrive at surfaces $\widehat{X}$
and $\widehat{Y}$, nonnormal over some nonreduced fibers.
But in both cases, one can roughly view the
corresponding fibers as being obtained by a base change from the general or
special fibers of
Sect.~8 and Sect. ~11.1. Alternatively,
one can go through the arguments of the paper for the new surfaces
$\widehat{X}$ and $\widehat{Y}$ (normalizing, if necessary), and
notice that all formulas (e.g. Euler characteristic formula for
$\lambda$, Index theorem on $\widehat{X}$, adjunction formula in $\mathbf P V$,
etc.) hold for surfaces with double point singularities.
\smallskip
Thus, in effect, one may replace a given singular fiber $C_X$ by
a bunch of general boundary curves $C$, following
the procedure described in Section~\ref{nontangentB}. Furthermore, if
some of these general curves $C$ are ``multiple'' (i.e. $B$ is tangent to
$\Delta\mathfrak{T}_{k,i}$ at $[C]$), one may in turn replace each $C$
by several ``transversal'' general boundary curves, and refer to the
statements in Sections~8.3 and 9.3. The only notational
difference in this approach will appear in the definition of the invariants
$m,\theta$ and $\gamma$ from Sect.~8: now we have to allow
for them to be {\it rational}, rather than integral, due to possible
rational intersections $E\cdot E^-$. This will be
``compensated'' in the final calculations, which will take into account the
multiplicity of the corresponding fibers, and roughly speaking,
will ``multiply back'' our invariants $\delta,\lambda$ and $\kappa$
by what they were divided by in the beginning of the calculations.
\bigskip
This concludes the proof of our results for all families of stable curves
$X\rightarrow B$ with general smooth trigonal members.
\subsection{Statements of the results for any family $X\rightarrow B$}
\label{results}
In the following list of results,
Theorems~\ref{7+6/g relation2} and \ref{maximal relation2} can be viewed as
local trigonal analogs of Cornalba-Harris's relation in the Picard group of the
hyperelliptic locus $\overline{\mathfrak{I}}_g$ (cf.~Theorem~\ref{CHPic}).
Similarly, Theorem~\ref{maximal bound2} is the analog of the $8+4/g$
maximal bound in the hyperelliptic case (cf.~Theorem~\ref{CHX}).
\begin{thm}[$7+6/g$ relation]
For any family $X\rightarrow B$ of stable curves with smooth
trigonal general member, if
$V$ is the canonically associated to $X$ vector bundle of rank 2, then
the following relation holds true
\[(7g+6)\lambda|_B=g\delta|_B+\mathcal{E}|_B+\frac{g-3}{2}\left(4c_2(V)-c_1^2(V)
\right),\]
where $\mathcal{E}$ is an effective rational linear combination of
boundary components of $\overline{\mathfrak{T}}_g$,
not containing $\Delta{\mathfrak{T}}_0$. In particular,
\[(7g+6)\lambda|_B=g\delta_0|_B+\sum_{k,i}\widetilde{c}_{k,i}
\delta_{k,i}|_B+\frac{g-3}{2}\left(4c_2(V)-c_1^2(V)\right),\]
where $\widetilde{c}_{k,i}$ is
quadratic polynomial in $i$ with linear coefficients in $g$, and it is
determined by the geometry of $\delta_{k,i}$ (cf.~p.~\pageref{analog1}).
\label{7+6/g relation2}
\end{thm}
\begin{thm}[$7+6/g$ bound]
For any nonisotrivial family $X\rightarrow B$ of stable curves with smooth
trigonal general member, if the canonically associated to $X$
vector bundle $V$ is Bogomolov semistable, then the slope of
$X/\!_{\displaystyle{B}}$ is bounded from above by
\[\frac{\delta}{\lambda}\leq 7+\frac{6}{g}\cdot\vspace*{-5mm}\]
\label{7+6/g Bogomolov2}
\end{thm}
\begin{thm}[Index relation]
For any family $X\rightarrow B$ of stable curves with smooth
trigonal general member,
if $V$ is the canonically associated to $X$ vector bundle of rank 2, then
the following relation holds true
\[36(g+1)\lambda|_B=(5g+1)\delta|_B+\mathcal{E}^{\prime}|_B+(g-3)
\big(9c_2(V)-2c_1^2(V)\big),\]
where $\mathcal{E}^{\prime}$ is an effective rational combination of the
boundary divisors $\Delta{\mathfrak{T}}_{k,i}$, not containing
$\Delta{\mathfrak{T}}_0$. In particular,
\[36(g+1)\lambda|_B=(5g+1)\delta_0|_B+\sum_{k,i}\widetilde{d}_{k,i}
\delta_{k,i}|_B+(g-3)\left(9c_2(V)-2c_1^2(V)\right),\]
where $\widetilde{d}_{k,i}$ is
quadratic polynomial in $i$ with linear coefficients in $g$, and it is
determined by the geometry of $\delta_{k,i}$ (cf.~p.~\pageref{page analog2}).
\label{maximal relation2}
\end{thm}
\begin{thm}[Maximal bound]
For any nonisotrivial family $X\rightarrow B$ of stable curves
with smooth trigonal general member,
the slope of $X/\!_{\displaystyle{B}}$ satisfies:
\[\frac{\delta}{\lambda}\leq \frac{36(g+1)}{5g+1},\]
with equality if and only all fibers of $X$ are irreducible
curves, and either $g=3$ or
the divisor $\eta$ on the total space of ${X}$ is numerically zero.
\label{maximal bound2}
\end{thm}
\label{list of theorems}
\subsection{What happens with the hyperelliptic locus
$\overline{\mathfrak{I}}_g$}
As we promised in Section~\ref{hyperelliptic locus}, we consider the
contribution of the hyperelliptic locus to the above theorems.
For any hyperelliptic curve $C$, we need to blow up a point on $C$
before it starts ``behaving'' like a trigonal curve in the quasi-admissible
and effective covers. Below we have shown what happens to a smooth
or general singular hyperelliptic curve (cf.~Fig.~\ref{hyperboundary}
for the admissible classification of the boundary locus $\Delta\mathfrak{I}_g$).
\begin{figure}[h]
$$\psdraw{smoothhyper}{1in}{1in}$$
\caption{$\overline{\mathfrak{I}}_g\cap\Delta\mathfrak{T}_0$}
\label{smoothhyper}
\end{figure}
\subsubsection{Smooth hyperelliptic curves}
We blow up $C$ at a point, and thus add a smooth rational component $\mathbf P^1$
to make it a triple cover $C^{\prime}$ (cf.~Fig.~\ref{smoothhyper}).
The quasi-admissible adjustment of $C$ is $\mu(C^{\prime})=1$. From here
on, $C$ will behave essentially like a smooth trigonal curve.
Therefore, in all relations $C$ is going to contribute $g$ or $(5g+1)$,
depending on what $\delta$ is multiplied by.
\subsubsection{Singular hyperelliptic curves in $\Delta\mathfrak{T}_{2,i}$
and $\Delta\mathfrak{T}_{5,i}$}
The necessary effective and quasi-admissible modifications are shown in
Fig.~\ref{singularhyper}--47.
In the first case, there are two hyperelliptic components intersecting
transversally in two points. For the quasi-admissible cover, we need
two ``smooth'' blow-ups, which makes $\mu=2$. From now on, this curve
will behave like a element of $\Delta\mathfrak{T}_{2,i}$, where $\mu_{2,i}=1$.
Thus, the coefficient
in, say, the maximal bound relation will be: $\widetilde{d}_{2,i}+(5g+1)$,
due to the extra blow-up in $\mu$.
\begin{figure}[h]
$$\psdraw{singularhyper}{5in}{1.1in}$$
\caption{$\overline{\mathfrak{I}}_g\cap\Delta\mathfrak{T}_{2,i}$\hspace{22mm}
{\sc Figure 47.} $\overline{\mathfrak{I}}_g\cap\Delta\mathfrak{T}_{5,i}$
\hspace*{25mm}}
\label{singularhyper}
\end{figure}
\addtocounter{figure}{1}
In the second case, two hyperelliptic components meet transversally
in one point, but have a ramification index 1 at this point when viewed as
double covers. Fig.46 presents first the quasi-admissible modification:
as in the case of $\Delta\mathfrak{T}_{5,i}$, the local
analytic multiplicity between
the two rational components is $2$, which means that we must have made
three ``smooth'' blow-ups and one ``singular'' blow-down. As a result,
$\mu=3$. From here on, this curve behaves exactly as a general member of
$\Delta\mathfrak{T}_{5,i}$.
Recall that $\mu_{5,i}=2$, and the extra $1$ in the hyperelliptic case
accounts for the one extra blow-up. Therefore, the coefficient
of this fiber $C$, say, in the maximal bound relation, will be
$\widetilde{d}_{5,i}+(5g+1)$.
\smallskip
We conclude that a base curve $B$, passing through the hyperelliptic locus,
will contribute in the results listed in Section~\ref{results}
roughly $g$, or $(5g+1)$, times the number of elements in
$B\cap\overline{\mathfrak{I}}_g$. We cannot write the latter in the form
of a scheme-theoretic intersection, since $\overline{\mathfrak{I}}_g$ is of
a larger codimension in $\overline{\mathfrak{T}}_g$.
\label{hypercalculations}
\smallskip
One can explain these extra summands in the expressions for $\lambda$ in
the following way. Recall the projection map $pr_1:\overline{\mathcal{H}}_{3,g}
\rightarrow \overline{\mathfrak{T}}_g$. The exceptional locus of $pr_1$
is the admissible boundary divisor $\Delta{\mathcal{H}}_{3,0}$, which is
blown down to the codimension 2 hyperelliptic locus $\overline{\mathfrak{I}}_g$
inside $\overline{\mathfrak{T}}_g$. For calculation purposes, it will
be easier to work instead with the space of minimal quasi-admissible
covers $\overline{\mathcal Q}_{3,g}$, which replaces $\overline{\mathcal{H}}_{3,g}$.
The same situation of a blow-down
occurs, where the exceptional divisor in $\overline{\mathcal Q}_{3,g}$ consists
of reducible curves $C^{\prime}$, as shown in Fig.~\ref{smoothhyper}.
\begin{figure}[h]
$$\psdraw{exceptional}{1.3in}{1.8in}$$
\caption{$B\cap \overline{\mathfrak{I}}_g$}
\label{exceptional.fig}
\end{figure}
Let $D$ be the linear combination of divisors in $\overline{\mathfrak{T}}_g$
given by the restriction $\Delta|_{\overline{\mathfrak{T}}_g}$, and consider
a curve $B\subset \overline{\mathfrak{T}}_g$, intersecting the hyperelliptic
locus in finitely many points.
By abuse of notation, we
denote by $pr_1$ the projection from $\overline{\mathcal Q}_{3,g}$ to
$\overline{\mathfrak{T}}_g$. Then for the intersection $D\cdot B$ we have:
\[D\cdot B=pr_1^*(D)\cdot pr_1^*(B)=pr_1^*(D)\cdot(\overline{B}+\sum
E_j),\]
where $\overline{B}$ is the proper transform of $B$, and the $E_j$'s
are the corresponding exceptional curves above $B$. Note that each $E_j$ is
in fact a line $\mathbf P^1$ representing all possible quasi-admissible covers,
arising from
a hyperelliptic curve $[C]\in B\cap \overline{\mathfrak{I}}_g$. From
Fig.~\ref{smoothhyper}, these are the blow-ups of $C$ at a point, one for
each involution pair $\{p_1,p_2\}\in g^1_2$, and that is \vspace*{3mm}why
$E_j\cong\mathbf P^1$.
The extra summands on p.~\pageref{list of theorems}, induced by
the base curve $B$, are result of the extra intersections $pr_1^*(D)\cdot E_j$
from above. Indeed, the relations, as they stand, compute only
$pr_1^*(D)\cdot\overline{B}$, the component corresponding to families
with general smooth members. From the calculations
on p.~\pageref{hypercalculations}, we expect that each $pr_1^*(D)\cdot E_j=1$,
and this will account for the extra $1$ apprearing in all $\mu$'s.
\smallskip
To verify this, we
only needs to show $\delta|_{E_j}=1$. Since we cannot pick out canonically
one point $p_i$ from each hyperelliptic pair $\{p_1,p_2\}$ on $C$, and
thus construct a family of blow-ups at $p_i$ of $C$ over
$E_j\cong \mathbf P^1$, we make a base change of degree two $C\rightarrow E_j$.
\begin{figure}[h]
$$\psdraw{deltahyper}{1.1in}{1.3in}$$
\caption{$\delta|_{C}=2$}
\label{deltahyper.fig}
\end{figure}
We construct a family over $C$, corresponding to {\it all} blow-ups
of $C$ at point $p\in C$. This is simply the products $C\times C$ and
$\mathbf P^1\times C$, identified at two sections $S_i$: $S_1$ is the diagonal on
$C\times C$, and $S_2$ is a trivial section of $\mathbf P^1\times C$ over $C$
(cf.~Fig.~\ref{deltahyper.fig}).
From \cite{CH}, for the base curve $C$ of this family,
the degree $\delta|_C$ is computed as
\[\delta|_C=\delta_{C\times C}+\delta_{\mathbf P^1\times C}+S_1^2+S_2^2=
0+0+2+0=2.\]
Taking into account the base \vspace*{2mm}change $C\rightarrow E_j$,
$\delta|_{E_j}=1$.
\smallskip Finally, if we allow for our families to have finitely many
hyperelliptic fibers, we adjust the relation in ~\ref{7+6/g relation2}
by $g\Delta{\mathcal H}_{3,0}\cdot B$, and the relation in
~\ref{maximal relation2} by $(5g+1)\Delta{\mathcal H}_{3,0}\cdot B$. The
two bounds in Theorems~\ref{7+6/g Bogomolov2}-\ref{maximal bound2} are
unaffected by the above discussion.
\setcounter{section}{12}
\bigskip\section*{12. Interpretation of the Bogomolov Index $4c_2-c_1^2$
via the Maroni Divisor}
\setcounter{subsection}{0}
\setcounter{subsubsection}{0}
\setcounter{lem}{0}
\setcounter{thm}{0}
\setcounter{prop}{0}
\setcounter{defn}{0}
\setcounter{cor}{0}
\setcounter{conj}{0}
\setcounter{claim}{0}
\setcounter{remark}{0}
\setcounter{equation}{0}
\label{Bog-Maroni}
\subsection{The Maroni invariant of trigonal curves}
\label{Maroniinvariant}
For any smooth trigonal curve $C$, consider the triple cover
$f:C\rightarrow {{\mathbf P}^1}$. The pushforward $f_*(\mathcal{O}_{C})$,
as we noted before, is a locally free sheaf of rank 3 on ${{\mathbf P}^1}$, and
hence decomposes into a direct sum of three invertible sheaves on
${{\mathbf P}}^1$:
\[f_*(\mathcal{O}_{C})=\mathcal{O}_{{\mathbf P}^1}\oplus \mathcal{O}_{{\mathbf P}^1}(a)\oplus
\mathcal{O}_{{\mathbf P}^1}(b).\]
The first summand is trivial due to the split exact sequence
\[0\rightarrow {V}\rightarrow {\alpha}_*{\mathcal O}_{C}\stackrel
{\on{tr}}{\rightarrow}{\mathcal O}_{{\mathbf P}^1}\rightarrow 0,\]
where $V=\mathcal{O}_{{\mathbf P}^1}(a)\oplus\mathcal{O}_{{\mathbf P}^1}(b)$.
From GRR, $a+b=g+2$. We have observed in Section~6
that $C$ embeds in the rational
ruled surface ${\mathbf P}V={\mathbf F}_k$, for $k=|b-a|$.
\medskip
\noindent{\bf Definition 12.1.} The {\it Maroni invariant} of an
irreducible trigonal curve
$C$ is the difference $|b-a|$. The {\it Maroni locus}
in $\overline{\mathfrak{T}}_g$ is the closure of the set of curves
with Maroni invariants $\geq 2$ (cf.~[Ma]).
\begin{lem} For a general trigonal
curve $C$ the vector bundle $V$ is {\it balanced}, i.e.
the integers $a$ and $b$ are equal or 1 apart according to
$g(\on{mod}2)$.
\label{gentrig}
\end{lem}
\begin{proof} Let $a\leq b$.
The statement follows from a dimension count of
the linear system $L=|3B_0+\frac{g+2}{2}F|$
on the ruled surface ${\mathbf F}_{b-a}={\mathbf F}_k$.
Indeed, all trigonal curves with Maroni invariant $(b-a)/2$ are elements
of $L$. If $p:{\mathbf F}_k\rightarrow {\mathbf P}^1$ is the projection map,
the projective dimension of $L$ equals
\[r(L)=h^0\big(p_*\mathcal{O}_{{\mathbf F}_k}(3B_0+\textstyle{\frac{g+2}{2}}
F)\big)-1.\]
Denoting by $\widetilde{B}=B_0-\frac{k}{2}F$ the section of
${\mathbf F}_k$ with smallest self-intersection of $-k$, we have
$p_*\mathcal{O}_{{\mathbf F}_k}(\widetilde{B})\cong \mathcal{O}_{{\mathbf P}^1}\oplus
\mathcal{O}_{{\mathbf P}^1}(-k)$. The necessary pushforward from above is:
\begin{equation*}
p_*\mathcal{O}_{{\mathbf F}_k}(3\widetilde{B}+\textstyle{\frac{g+2+3k}{2}}F)=
\on{Sym}^3(\mathcal{O}_{{\mathbf P}^1}\!\!\oplus\mathcal{O}_{{\mathbf P}^1}(-k))
\otimes \mathcal{O}_{{\mathbf P}^1}({\textstyle{\frac{g+2+3k}{2}}})=
\!\!\!\!\displaystyle{\bigoplus_{j=\pm 1,\pm 3}}
\!\!\!\!\mathcal{O}_{{\mathbf P}^1}({\textstyle{\frac{g+2+jk}{2}}}).
\end{equation*}
Since an irreducible trigonal curve $C$ lies in $L$, we have
$C\cdot \widetilde{B}\geq 0$, hence $g+2-3k\geq 0$ and $g\equiv k
(\on{mod}2)$. Evaluating the sections of this sum of sheaves, we obtain
$r(L)=2g+7.$
The ruled surface ${\mathbf F}_k$ has automorphisms, inducing automorphisms
of the linear system $L$. We need to mod out these in order to obtain
the dimension of the space of trigonal curves embedded in ${\mathbf F}_k$.
The group $\on{Aut}{\mathbf F}_k$ is a product (not necessarily direct)
of the base automorphisms $\on{Aut}{\mathbf P}^1=\on{PGL}_2$, and
the projective automorphisms of the vector bundle $V$. The latter is an open
set (up to projectivity)
of the homomorphisms of $V$ into $V$, and hence has the same dimension as:
\[\on{Hom}(V,V)\cong H^0(V\otimes V\,\,\widehat{\phantom{n}})=
H^0\big(\mathcal{O}_{{\mathbf P}^1}(-k)\oplus\mathcal{O}_{{\mathbf P}^1}\oplus
\mathcal{O}_{{\mathbf P}^1}\oplus\mathcal{O}_{{\mathbf P}^1}(k)\big).\]
For $k>0$, $\on{dim}\on{Aut}V=k+3$, while for $k=0$,
$\on{dim}\on{Aut}V=4$. We conclude that the dimension of
the set of trigonal curves with Maroni invariant $k/2$ is
\[r(L)-\on{dim}\on{Aut}{\mathbf F}_k=
\left\{\begin{array}{l} 2g+1\,\,\on{if}\,\,k=0,\\
2g+2-k\,\,\on{if}\,\,k>0.
\end{array}\right.\]
\medskip When $k=0$ or $k=1$, this space corresponds to an open dense set of
$\overline{\mathfrak{T}}_g$.
For an even $g$ a general trigonal curve has Maroni invariant $0$
and therefore embeds in ${\mathbf F}_0=
{\mathbf P}^1\times{\mathbf P}^1$, while for an odd $g$
a general trigonal curve has Maroni invariant $1$ and embeds in
${\mathbf F}_1=\on{Bl}_{\on{pt}}(\mathbf P^2)$.
In both cases, the vector bundle $V$ is balanced. \end{proof}
\begin{cor} For $g$ even, the Maroni locus is a divisor in
$\overline{\mathfrak{T}}_g$ whose general member embeds in ${\mathbf F}_2$.
For $g$ odd, the Maroni locus has codimension 2 in $\overline{\mathfrak{T}}_g$
and its general member embeds in ${\mathbf F}_3$.
\label{maronilocus}
\end{cor}
\noindent{\bf Remark 12.1.} It will be useful to identify precisely the
group of authomorphisms of the linear system $L$ for $k=0,1$. We have
$\on{Aut}(\mathbf P^1\!\times \mathbf P^1)\cong PGL_2\times PGL_2\times
{\mathbb Z}/2{\mathbb Z}$. The last factor comes from the
exchange of the fiber and the base of $\mathbf P^1\times \mathbf P^1$ and it
is relevant only for $g=4$: then $L=|3B_0+3F|$. Otherwise,
for any even $g>4$:
\[\on{Aut}L\cong PGL_2\times PGL_2.\]
When $g$ is odd, the ruled surface ${\mathbf F}_1$ can be thought of as the
blow-up of $\mathbf P^2$ at the point $q=[0,0,1]$. Any automorphism of
$\on{Bl}_q{\mathbf P^2}$ carries the exceptional divisor $E_q$ of $\mathbf F_1$
to itself, and hence is induced by an automorphism of the plane preserving
the point $q$. The group of such automorphisms of $\mathbf P^2$
is the subgroup of $PGL_3$ corresponding to matrices:
\[\left(\begin{array}{ccc} a_{11} & a_{12} & 0\\
a_{21} & a_{22} & 0\\
a_{31} & a_{32} & a_{33}
\end{array}\right).\]
Taking into account the discriminant of these matrices, we easily
identify for odd $g$:
\[\on{Aut}L\cong \mathbf A^2\times GL_2.\]
Note that all of the above groups $\on{Aut}L$ have dimension $6$,
which was claimed already in Lemma~\ref{gentrig}.
\subsection{Generators of Pic$_{\mathbb{Q}}\overline{\mathfrak{T}}_g$}
\label{generators}
\begin{prop} The rational Picard group of
$\overline{\mathfrak{T}}_g$, $\on{Pic}_{\mathbb{Q}}\overline{\mathfrak{T}}_g$,
is freely generated by the boundary classes
$\delta_0$, $\delta_{k,i}$, and one additional class, which for even genus
$g$ coincides with the Maroni class $\mu$.
\label{genPic}
\end{prop}
\begin{proof} Since a general trigonal curve $C$ embeds in
the ruled surface ${\mathbf F}_k$ ($k=0,1$),
$C$ is a member of the linear system $L=|3B_0+
\frac{g+2}{2}F|$ on ${\mathbf F}_k$.
Let $U$ be the open set inside ${\mathbf P}L\cong
\mathbf P^{2g+7}$ corresponding to the {\it smooth trigonal} members of $L$.
The surjection \[{\mathbb Z}=\on{Pic}\mathbf P^{2g+7}\twoheadrightarrow
\on{Pic}U \]
has a nontrivial kernel, because the set of singular trigonal curves in
${\mathbf F}_k$ is a divisor in ${\mathbf P}L$.
Hence $\on{Pic}U={\mathbb Z}/n{\mathbb Z}$ for some integer $n\!>\!0$, and
$\on{Pic}_{\mathbb{Q}}U\!=\!0$.
\medskip
The image of the natural
projection map $p:U\rightarrow \overline{{\mathfrak{T}}}_g$
is the open dense set $W$ of smooth trigonal curves
with lowest Marone invariant of $0$ or $1$.
Let $F$ denote the fiber of $p$. From Remark 12.1,
\[F\cong\left\{\begin{array}{l}
PGL_2\times PGL_2\,\,\on{if}\,\,g-\on{even},g>4;\\
PGL_2\times PGL_2\times {\mathbb Z}/2{\mathbb Z}\,\,\on{if}\,\,g=4;\\
\on{Aut}L\cong {\mathbf A}^2\times GL_2\,\,\on{if}\,\,g-\on{odd}.
\end{array}\right.\]
Leray spectral sequence or other methods (cf.~~\cite{Gr-Ha,Milne}) yield:
\[H^1(W,f_*{\mathcal O}^*_U)\hookrightarrow
H^1(U,{\mathcal O}^*_U).\]
Pushing the exponential sequence on $U$ to $W$:
\[0\rightarrow {\mathbb Z}\rightarrow {\mathcal O}_U \rightarrow {\mathcal O}^*_U
\rightarrow 0\,\,
\Rightarrow \,\,0\rightarrow {\mathbb Z}\rightarrow {\mathcal O}_{W}
\rightarrow f_*{\mathcal O}^*_U\rightarrow R^1\!\!f_*{\mathbb Z}.\]
Combining with the exponential sequence on $W$:
\[0\rightarrow {\mathcal O}^*_{W}
\rightarrow f_*{\mathcal O}_U^* \rightarrow R^1\!\!f_*{\mathbb Z}\,\,
\Rightarrow\,\,H^1(W,{\mathcal O}^*_{W})
\stackrel{p^*}{\rightarrow}H^1(U,{\mathcal O}_X^*),\]
with $\on{ker}p^*\subset H^0(W,R^1\!\!f_*{\mathbb Z})\subset H^1(F,{\mathbb Z})$.
For even $g$, $H^1(F,{\mathbb Z})$ is torsion (a direct sum of copies of
${\mathbb Z}/2{\mathbb Z}$), but for odd $g$ it is isomorphic to ${\mathbb Z}$.
\smallskip
Hence, for even $g$ we have the natural embedding
$p^*:\on{Pic}_{\mathbb Q}W\hookrightarrow \on{Pic}_{\mathbb Q}U$,
and in view of $\on{Pic}_{\mathbb{Q}}U=0$, it follows that $\on{Pic}_{\mathbb{Q}}
W=0$. The complement of $W$ in $\overline{\mathfrak{T}}_g$ is
the union of the boundary of $\overline{\mathfrak{T}}_g$ and
the Maroni divisor. Therefore, $\delta_0$, $\delta_{k,i}$ and $\mu$
generate $\on{Pic}_{\mathbb Q}\overline{\mathfrak{T}}_g$.
Since the class of the Hodge bundle $\lambda$ is {\it not} a linear
combination of the boundary classes (cf.~p.~\pageref{list of theorems}),
the boundary divisors are {\it not} sufficient to generate
the rational Picard group of $\overline{\mathfrak{T}}_g$, and
$\mu$ must be linearly independent of them. We conclude that
$\delta_0$, $\delta_{k,i}$, and $\mu$ generate freely
$\on{Pic}_{\mathbb Q}\overline{\mathfrak{T}}_g$ for even genus $g$.
\smallskip
For $g$-odd,
$p^*:\on{Pic}_{\mathbb{Q}}W\rightarrow \on{Pic}_{\mathbb{Q}}U$
is either an inclusion, or has a kernel with one generator. Since
the Maroni locus for $g$-odd is not a divisor, an inclusion would
imply as above that $\lambda$ is a linear combination of the boundary
classes, which is not true. Hence, $\on{ker}p^*={\mathbb Q}$ and
$\on{Pic}_{\mathbb{Q}}W$ is generated freely by the boundary
classes $\delta_0$ and $\delta_{k,i}$, and one additional class. \end{proof}
\subsection{The Bogomolov condition and the Maroni divisor}
\label{interpretation}
\begin{prop} For even genus $g$ and a base curve $B$,
not contained in $\Delta{\mathfrak{T}}_g$:
\[(7g+6)\lambda=g\delta_0+
\sum_{k,i}\widehat{c}_{k,i}\delta_{k,i}+2(g-3)\mu,\]
where $\widehat{c}_{k,i}$ are certain polynomial coefficients computed
similarly as $\widetilde{c}_{k,i}$. (cf.~p.~\pageref{analog1})
\label{Maroni2}
\end{prop}
\begin{proof} We set $g=2(m-1)$.
Let us consider for now only families with irreducible trigonal
fibers, i.e. the base curve $B$ intersects only
the boundary component $\Delta{\mathfrak{T}}_0$.
\smallskip
\noindent{\it Case 1.}
If $B$ does not intersect the Maroni divisor $\mu$, then the Maroni
invariant of the fibers in $X$ is constant, and equal to $0$.
The fibers $C$ of $X$ embed in the projectivization
${\mathbf P}(V|_{F_Y})\cong {\mathbf P}^1\times {\mathbf P}^1$. Since $\on{deg}V|_{F_Y}
=g+2$ and $V$ is balanced, the restriction of $V$ to the fiber $F_Y$ on
the ruled surface $Y$ is
\[V|_{F_Y}=\mathcal{O}_{{\mathbf P}^1}(m)\oplus
\mathcal{O}_{{\mathbf P}^1}(m).\]
Moreover, $V|_{F_Y}$ does not jump as $F_Y$ moves, so that
$V$ can be written as:
\[V\cong h^*M\otimes\mathcal{O}_{Y}\left(mB_0\right)\]
for some vector bundle $M$ of rank 2 on $B$. But the Bogomolov
index $4c_2(V)-c_1^2(V)$ is independent of twisting $V$ by line
bundles, in particular, by $\mathcal{O}_{Y}\left(mB_0\right)$, so that
\[4c_2(V)-c_1^2(V)=4c_2(h^*M)-c_1^2(h^*M)=
4c_2(M)-c_1^2(M)=0.\]
The last equality follows from $c_2(M)=0=c_1^2(M)$ for any
bundle on the curve $B$. We conclude that $4c_2(V)-c_1^2(V)=4\mu|_B=0$.
\medskip
\begin{figure}[h]
$$\psdraw{marone1}{1.5in}{1.3in}$$
\caption{$B\cap \mu$ in $\overline{\mathfrak{T}}_g$}
\label{intersectBandmu}
\end{figure}
{\it Case 2.} Now let $B$ intersect the Maroni divisor $\mu$
in {\it finitely} many points. Assume also that
these points are {\it general} in $\mu$, i.e. they correspond to
trigonal curves $C$ embeddable in the ruled surface ${\mathbf F}_2$.
We twist $V$ by a line bundle $M=
\mathcal{O}_{Y}\left(mB_0\right)$, and set $\widetilde{V}=V\otimes M$, so that
$\on{deg}\widetilde{V}|_{F_Y}=0$ and
\[\,\,\,\,\left\{\begin{array}{l}
\widetilde{V}|_{F_Y}=
\mathcal{O}_{{\mathbf P}^1}\oplus \mathcal{O}_{{\mathbf P}^1}\,\,\,
\phantom{(-1)(1)}\on{when}\,\,F_Y\,\,\on{is\,\,generic,}\\
\widetilde{V}|_{F_Y}=\mathcal{O}_{{\mathbf P}^1}(-1)\oplus
\mathcal{O}_{{\mathbf P}^1}(1)\,\,\,\on{when}\,\,F_Y\,\,\on{is\,\,special}.
\end{array}\right.\]
Then $\widetilde{V}$ is the middle term of a short exact sequence on $Y$
\[0\rightarrow \widetilde{V}^{\prime} \rightarrow \widetilde{V} \rightarrow
\mathcal{I} \rightarrow 0,\]
where $\widetilde{V}^{\prime}=h^*(h_*\widetilde{V})$ is a vector bundle of
rank 2 on $Y$. In the notation of \cite{Br}, let $W$ be the
sum of the special fibers of $Y$, and let $Z$ be the union of certain
isolated points on each member of $W$, so that
$\mathcal{I}=\mathcal{I}_{Z\subset W}$ is the ideal sheaf of $Z$ inside $W$.
Note that the number of the special fibers, which comprise $W$,
equals $\on{deg}Z=\mu|_B$.
We can now compute the Chern classes of $\widetilde{V}$:
\[\left\{\begin{array}{l}
c_1(\widetilde{V})=c_1(\widetilde{V}^{\prime})+W=\on{a\,\,sum\,\,of\,\,
fibers\,\,of\,\,Y},\\
c_2(\widetilde{V})=c_2(\widetilde{V}^{\prime})+\on{deg}Z=\on{deg}Z.
\end{array}\right.\]
The last equality follows from the fact that $\widetilde{V}^{\prime}$
is the pull-back of a bundle on the curve $B$, hence of zero higher
Chern classes. We conclude that $c_1^2(\widetilde{V})=0$, and
\[4c_2({V})-c_1^2({V})=
4c_2(\widetilde{V})-c_1^2(\widetilde{V})=4\on{deg}Z=4\mu|_B.\]
Putting the above two cases together, we have for any family with
irreducible trigonal members, not entirely contained in the Maroni locus:
\begin{equation}
4c_2({V})-c_1^2({V})=4\mu|_B.
\end{equation}
Prop.~\ref{genPic} then implies that $\lambda$ is a linear
combination of the boundary and the Maroni class:
\[(7g+6)\lambda|_{\overline{\mathfrak{T}}_g}=g\delta_0+\sum_{k,i}
\widehat{c}_{k,i}\delta_{k,i}+2(g-3)\mu,\]
where the coefficients $\widehat{c}_{k,i}$ are computed in a similar way, or by
direct computation with families of singular trigonal curves
(cf.~\cite{CH}). \end{proof}
\medskip
We can combine the above results in the following
\begin{thm} For even $g$, $\on{Pic}_{\mathbb{Q}}\overline{\mathfrak{T}}_g$ is
freely generated by all boundary classes $\delta_0$ and $\delta_{k,i}$, and
the Maroni class $\mu$. The class of the Hodge bundle on
$\overline{\mathfrak{T}}_g$ is expressed in terms of these generators as
the following linear combination:
\begin{equation*}
(7g+6)\lambda|_{\overline{\mathfrak{T}}_g}=g\delta_0+
\sum_{k,i}\widehat{c}_{k,i}\delta_{k,i}+2(g-3){\mu}.
\end{equation*}
\label{Pic trigonal}\vspace*{-5mm}
\end{thm}
\noindent{\bf Remark 12.2.} Note that the coefficients
$\widehat{c}_{k,i}$ depend on the specific decriptions of the Maroni
curves that appear in the boundary divisors $\Delta{\mathfrak{T}}_{k,i}$,
and they are {\it not} always equal to the corresponding
coefficients $\widetilde{c}_{k,i}$ in Theorem \ref{7+6/g relation2}.
Indeed, in the above Proposition, we have shown that
\begin{equation}
4c_2({V})-c_1^2({V})=4\mu|_B+
\sum_{k,i}\alpha_{k,i}\delta_{k,i},
\label{alpha-coef}
\end{equation}
for some $\alpha_{k,i}$, which may be non-zero. Hence,
$\widehat{c}_{k,i}=\widetilde{c}_{k,i}+\frac{g-3}{2}\alpha_{k,i}$.
\smallskip
For example, consider the case of $\Delta{\mathfrak{T}}_{1,i}$, and let
$C=C_1\cup C_2$ be a general member of it. If $C$ is also Maroni, then
there exists a family $X\rightarrow B$, whose general fiber is
an irreducible Maroni curve, and one of whose special fibers is our $C$.
We can assume, modulo a base change and certain blow-ups not affecting
$C$, that this family fits in the basic construction diagram (cf.~
Fig.~\ref{general B}). Let ${\mathbf R}_1$ and ${\mathbf R}_2$ be
the two ruled surfaces in which $C_1$ and $C_2$ are embedded, and
let $E_1$ and $E_2$ be the projections of $C_1$ and $C_2$ in the
birationally ruled surface $\widehat{Y}$. Then $F=E_1+E_2$ is
a special fiber of $\widehat{Y}$, with self-intersections $E_1^2=E_2^2=-1$.
\smallskip
Now, the general member of $X$, being Maroni, is embedded in
a ruled surface $\mathbf F_2$ with a section $L$ of self-intersection $-2$.
The union of such $L$'s forms a surface in the 3-fold $\mathbf PV$, whose
closure we denote by $S$. Evidently, $S\cong \widehat{Y}$, at least
outside their special fibers. Let
$S$ intersect ${\mathbf R}_1$ and ${\mathbf R}_2$ in curves $L_1$ and
$L_2$ (over $E_1$ and $E_2$).
We claim that at least one of ${\mathbf R}_1$ and ${\mathbf R}_2$
is {\it not} isomorphic to $\mathbf F_0=\mathbf P^1\times \mathbf P^1$.
It will suffice to show that $L_1$ or $L_2$ has negative self-intersection.
\smallskip
Indeed, suppose to the contrary that $L_m^2\geq 0$ in ${\mathbf R}_m$
($m=1,2$). Note that $S\cdot \mathbf R_m=L_m$ in $\mathbf PV$, so that
\[L_m^2=S|_{\mathbf R_m}\cdot S|_{\mathbf R_m}= S^2\cdot \mathbf R_m\,\,
\Rightarrow\,\,S^2(\mathbf R_1+\mathbf R_2)\geq 0.\]
On the other hand, $\mathbf R_1+\mathbf R_2$ is the fiber of the
projection $\mathbf PV\rightarrow \widehat{Y}$, and as such it is
linearly equivalent to the general fiber $\mathbf F_2$. Hence
\[0\leq S^2\cdot \mathbf F_2=S|_{\mathbf F_2}\cdot S|_{\mathbf F_2}=L^2=-2,\]
a contradiction.
We conclude that if $C=C_1\cup C_2$ is a Maroni curve of boundary type
$\Delta{\mathfrak{T}}_{1,i}$, then either $C_1$ or $C_2$ (or both)
is embedded in a ruled surface $\mathbf F_k$ with $k\geq 1$. This
already distinguishes the cases of odd and even genus $i$.
\smallskip
When $i=g(C_1)$ is even (and hence $j=g(C_2)=g-j-2$ is also even),
the general member of $\Delta{\mathfrak{T}}_{1,i}$ is embedded in a join
of two $\mathbf F_0$'s (each $C_m\subset \mathbf F_0$), and hence
it is {\it not} Maroni. Based on this observation, one can easily find
the coefficient $\alpha_{1,i}$ for $i$-even. To do this, consider
the birationally ruled surface $Y$ which is the blow-up of
$\mathbf F_0$ at one point. Let again the two components of the
special fiber of $Y$ be $E_1$ and $E_2$, and projectivize the trivial vector
bundle $V=\mathcal{O}_Y\oplus \mathcal{O}_Y$: ${\mathbf P}V=Y\times
\mathbf P^1$. By taking an appropriate linear system in ${\mathbf P}V$,
one obtains a family of trigonal curves $X$, whose fibers are all
irreducible and embedded in $\mathbf F_0$, except for a special
reducible curve $C$ over $E_1\cup E_2$ of the specified above type.
Hence none of $X$'s members are Maroni, and so $\mu|_B=0$. Further,
$4c_2(V)-c_1^2(V)=0$, and $\delta_{1,i}|_B=1$, so that
equation~(\ref{alpha-coef}) implies $\alpha_{1,i}=0$, and
hence $\widehat{c}_{k,i}=\widetilde{c}_{k,i}$ for $i$-even.
\smallskip
The situation is quite different when the genus $i$ is odd. Then both
components of the general member $C$ of $\Delta{\mathfrak{T}}_{1,i}$
are embedded in $\mathbf F_1$'s, and hence $C$ is potentially Maroni.
One can take further the above general argument of
intersection theory on ${\mathbf P}V$, and show that the
curves $L_1$ and $L_2$ are in fact both
sections of negative self-intersection $-1$ in these $\mathbf F_1$'s:
consider the product $S\cdot X \cdot {\mathbf F}_2$ and
its variation over the special fiber of $\widehat{Y}$. But we know
that $L_1$ and $L_2$ intersect, as the fiber of $S$ over
$\widehat{Y}$ is connected.
Thus, the curve $C$ would be Maroni if and only if the two corresponding
ruled surfaces $\mathbf F_1$ are glued along one of their fibers so that
their negative sections intersect on that fiber. (This decsription
can be alternatively derived by considering the degenerations of
the $g^1_3$'s on the irreducible Maroni curves.)
To find $\alpha_{1,i}$ in this case, we construct a similar example
as above, only changing $V$ to $\mathcal O_Y\oplus \mathcal O_Y(E_1)$.
This, while keeping the general fiber embedded in $\mathbf F_0$, has the
effect of embedding the special one in a ``Maroni'' gluing of two
$\mathbf F_1$'s. We have $4c_2(V)-c_1^2(V)=-E_1^2=1$, $\mu|_B=1$,
and $\delta_{1,i}|_B=1$, so that equation~(\ref{alpha-coef}) implies
$\alpha_{1,i}=-3$ for $i$-odd, and hence
$\widehat{c}_{k,i}=\widetilde{c}_{k,i}-3/2(g-3)$.
\bigskip
\begin{figure}[t]
$$\psdraw{maroni}{1.5in}{1.5in}$$
\caption{Maroni curves in $\Delta_{1,i}\mathfrak{T}_g$, $i$-odd}
\label{Maroni-boundary}
\end{figure}
One can similarly compute the remaining coefficients
$\alpha_{k,i}$, by first figuring out which boundary curves in
$\Delta_{k,i}\mathfrak{T}_g$ are Maroni, then constructing
an appropriate vector bundle $V$, and finally using
equation~(\ref{alpha-coef}) to compute $\alpha_{k,i}$, and
hence $\widehat{c}_{k,i}$. \qed
\begin{prop} For $g$-even, if the base curve $B$ is not entirely
contained in the Maroni divisor, and the singular members of
$X$ belong only to $\Delta_0\mathfrak{T}_g
\cup \Delta_{1,i}\mathfrak{T}_g$, then the slope of the family $X/_B$
satisfies:
\[\frac{\delta}{\lambda}\leq 7+\frac{6}{g}.\]
\label{Maroni inequality}\vspace*{-5mm}
\end{prop}
\begin{conj} For $g$-even, if the base curve $B$ is not entirely
contained in the Maroni divisor, then the slope of the family $X/_B$
satisfies:
\[\frac{\delta}{\lambda}\leq 7+\frac{6}{g}.\]
\label{Maroni-conj}
\end{conj}
\subsection{The Maroni divisor and the maximal bound}
\label{Maroni-maximal}
Even though for odd genus $g$ the Maroni locus is not large enough
to be a divisor in $\overline{\mathfrak{T}}_g$, we can define a
{\it generalized Maroni} divisor class by extending the relation from
the $g$-even case.
\medskip
\noindent{\bf Definition 12.2.} For any genus, we define the
{\it generalized Maroni} class $\mu$
in $\on{Pic}_{\mathbb{Q}}\overline{\mathfrak{T}}_g$ by
\[\mu:=\frac{1}{2(g-3)}\big\{(7g+6)\lambda-g\delta_0-
\sum_{k,i}\widehat{c}_{k,i}\delta_{3,i}\big\}.\]
\begin{thm} The maximal bound $36(g+1)/(5g+1)$ is attained
for a trigonal family of curves $X\rightarrow B$ if and only
all fibers of $X$ are irreducible and
\[\delta_0|_B=-\frac{72(g+1)}{g+2}\mu|_B\]
\label{maximalmaroni}\vspace*{-5mm}
\end{thm}
\begin{proof} The fact that $X$ must have only irreducible fibers
in order to attain the maximum bound is already known from
Theorem~\ref{genmaximal}. This means $\delta_{k,i}|_B=0$ for all
$k,i$. Then, Theorem~\ref{bogomolov1} implies:
\begin{equation}
(7g+6)\lambda|_B=g\delta_0|_B+\frac{g-3}{2}\mu|_B.
\end{equation}
Assume that the maximal bound is attained, i.e.
$36(g+1)\lambda|_B=(5g+1)\delta_0|_B$. Substituting for
$\lambda|_B$ in the above equation, yields the desired equality.
The converse follows similarly. \end{proof}
\medskip
\noindent{\bf Remark 12.3.} In the $g$-even case, this equality has a
specific meaning. Since the Maroni class $\mu$ corresponds to an
effective divisor on $\overline{\mathfrak{T}}_g$, the equality (and hence
the maximal bound) is achieved only for base curves $B$ entirely contained
in the Maroni divisor, so that the restriction $\mu|_B$ can be negative.
In fact, in all found examples, the base $B$ is contained in a very small
subloci of the Maroni loci, defined by the highest possible Maroni invariant.
\medskip
\noindent{\bf Remark 12.4.} Theorem~\ref{Pic trigonal}
and Prop.~\ref{maximalmaroni}
do not have analogs in the hyperelliptic case: there is no additional
Maroni divisor to generate $\on{Pic}_{\mathbb Q}\overline{\mathfrak{I}}_g$
together with the boundary $\Delta\mathfrak{I}_g$.
\medskip
\noindent{\bf Remark 12.5.} When $g=3$, there is no Maroni locus in
$\overline{\mathfrak{T}}_3$ either. Indeed, since an irreducible
trigonal curve of genus $3$ embeds
only in ruled surfaces ${\mathbf F}_k$ with $k$-odd and
$k\leq (g+2)/3=5/3$, then {\it all} irreducible trigonal curves embed in
${\mathbf F}_1$, and correspondingly they all have the lowest possible
Maroni invariant $k=1$. However, $\on{Pic}_{\mathbb Q}\overline{\mathfrak{T}}_3$
is not generated by the boundary classes of $\overline{\mathfrak{T}}_3$: as
Prop.~\ref{genPic} asserts, in the odd genus case there is always one
additional generating class.
\smallskip
On the other hand, the results on p.~\pageref{list of theorems}
yield apriori {\it two} relations among $\lambda$ and the $\delta_{k,i}$'s.
This would have been a contradiction to the {\it freeness} of the
generators above, unless these two relations are the same. This is in fact
what happens:
\[9\lambda=\delta_0+3\delta_{2,1}+3\delta_{3,1}+4\delta_{4,1}+4\delta_{5,1}
+3\delta_{5,2}+3\delta_{6,1},\]
as restricted to any base curve $B\not\subset\Delta\overline{\mathfrak{T}}_3$.
Note the convenient disappearance of the ``extra''
$(g-3)$--summands in the coefficients of $\delta_{4,i},\delta_{5,i},
\delta_{6,i}$).
Then the maximal and the semistable ratios
both equal $9$, and are attained for families with
irreducible trigonal members.
\bigskip\section*{13. Further Results and Conjectures}
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\label{furtherresults}
\subsection{Results and conjectures for $d$-gonal families, $d\geq 4$}
I have carried out some preliminary research in the $d$-gonal case, and
while the methods and ideas for the trigonal case are
in principle extendable, this appears to be a substantially more
subtle and complex problem.
More precisely, let $\overline{\mathcal{D}}_d$ be the closure in
$\overline{\mathfrak{M}}_g$ of the stable curves expressible as $d$-sheeted covers
of ${\mathbf P}^1$. One possible goal is
to complete the program of describing generators and relations
for the rational Picard groups $\on{Pic}_{\mathbb{Q}}\overline{\mathcal{D}}_d$,
and to find the exact maximal bounds for the slopes of $d$-gonal
families.
\smallskip
For example, I have obtained the following bound for the slope of a
general tetragonal family with smooth general member (for odd genus $g$):
\[\frac{\delta}{\lambda}\leq 6\frac{2}{3}+\frac{64}{3(3g+1)}=
\frac{4(5g+7)}{3g+1}.\]
I have also conjectured formulas for the maximal
and general bounds for any $d$-gonal and other families
of stable curves. Entering these formulas are the {\it Clifford index} of
curves, {\it Bogomolov semistability} conditions for higher rank bundles,
and some new geometrically described loci in $\overline{\mathcal{D}}_d$.
Generalizing the idea of the Maroni locus in the trigonal case,
these loci are characterized, for example, in the tetragonal case by the
dimensions of the multiples of the $g^1_4$-series. In particular,
there will be another generator of $\on{Pic}_{\mathbb Q}\overline{\mathfrak{T}}_4$
besides the boundary and Maroni divisors.
\smallskip
In the following I present some of these conjectures
on the upper bounds for $\overline{\mathcal{D}}_d$.
We start by comparing all known maximal and general
bounds functions of the genus $g$:
\[\begin{array}{|c|c|c|c|c|c|}
\hline\hline
\stackrel{\vspace*{1mm}}{\on{locus\,\, in \,\,}\overline{\mathfrak{M}}_g}
&\on{bound}& g=1& g=2& g=3& g=5\\
\hline\hline\vspace*{1mm}
\on{general}\,\overline{\mathfrak{M}}_g&
\stackrel{\vspace*{1mm}}{\displaystyle{ 6+\frac{12}{g+1\vspace*{1mm}}}}
& 12 & 10 &9 &8\\
\hline\vspace*{1mm}
\on{hyperelliptic}\,\overline{\mathcal{H}}_g=\overline{\mathcal{D}}_2&
\stackrel{\vspace*{1mm}}{\displaystyle{8+ \frac{4}{g\vspace*{1mm}}}}
& 12 & 10 & - &-\\
\hline\vspace*{1mm}
\on{trigonal}\,\overline{\mathcal{T}}_g=\overline{\mathcal{D}}_3
& \stackrel{\vspace*{1mm}}{\displaystyle{ \frac{36(g+1)}{5g+1
\vspace*{1mm}}}}
& 12 & - & 9 &- \\
\hline\vspace*{1mm}
\on{ gen. tetragonal=\overline{\mathcal{D}}_4}
&\stackrel{\vspace*{1mm}}{\displaystyle{
\frac{4(5g+7)}{3g+1\vspace*{1mm}}}}& 12 & - & -&8\\
\hline
\end{array}\]
\medskip
The pattern appearing in this table is clear: the general bound
$6+\displaystyle{12/(g+1)}$
coincides with each of the other bounds exactly twice
for some special values of the genus $g$. Evidently,
$g=1$ is one of these special values, yielding 12 everywhere. (I
owe this observation to Benedict Gross.)
Let $g_d$ be the other genus $g$ for which the
general formula in $\overline{\mathfrak{M}}_g$ and the maximal formula for
$\overline{\mathcal{D}}_d$ coincide, i.e. $g_2=2$, $g_3=3$, $g_5=5$.
We notice that for these genera $g_d$
the moduli spaces $\overline{\mathfrak{M}}_2,\overline{\mathfrak{M}}_3$
and $\overline{\mathfrak{M}}_5$ consist only of
hyperelliptic, trigonal or tetragonal curves, respectively.
In general, {\it Brill-Noether} theory (cf.~\cite{ACGH}) asserts
that for complete linear series
$g^r_d=g^1_d$ the expected dimension of the variety of $g^1_d$'s on
a smooth curve of genus $g$ is $\rho=g-(r+1)(g-d+r)=2(d-1)-g,$ and hence
the smallest genus $g$ for which
$\overline{\mathfrak{M}}_g=\overline{\mathcal{D}}_d\supsetneq
\overline{\mathcal{D}}_{d-1}$
is $g=2d-3$. Thus we set $g_d=2d-3$ for $d\geq 3$ and $g_2=2$. Note that
this coincides with the previously found $g_3=3$ and $g_5=5$.
\begin{conj} If $\mathcal{F}_d(g)$ is an exact upper bound for the slopes of
families of stable curves with smooth $d$-gonal general member (locus
$\overline{\mathcal{D}}_d$), then
\begin{eqnarray*}
&&(a)\,\,\mathcal{F}_d(1)=12.\\
&&(b)\,\,\mathcal{F}_d(g_d)=6+\displaystyle{\frac{12}{g_d+1}}\cdot
\end{eqnarray*}
\label{conj2}\vspace*{-5mm}
\end{conj}
It is reasonable to expect that the upper bounds for $\overline{\mathcal{D}}_d$
will be ratios of linear functions of the genus $g$:
$\mathcal{F}_d(g)=(Ag+B)/(Cg+D)$.
Conjecture~\ref{conj2} then estimates the difference between $\mathcal{F}_d(g)$
and the general bound for $\overline{\mathfrak{M}}_g$ up to a factor
$f_d=D/C$.
\begin{conj} The exact upper bounds $\mathcal{F}_d(g)$ are given by
\[\mathcal{F}_d(g)=6+\frac{12}{g+1}+6\frac{(1-f_d)(g-g_d)(g-1)}{(g+f_d)(g_d+1)(g+1)},\]\vspace*{-3mm}
or equivalently, \vspace*{-3mm}
\[\mathcal{F}_d(g)=6+\frac{6}{g+f_d}\left(1+f_d+\frac{1-f_d}{g_d+1}(g-1)\right).\]
\end{conj}
I have a conjecture on how to determine the remaining
factor $f_d$, which seems to be closely related to the coefficients
of the linear expression in [EMH]
for the divisor $\overline{\mathcal{D}}_{\frac{g+1}{2}}$
in terms of the Hodge bundle $\lambda$ and
the boundary classes $\delta_i$ on $\overline{\mathfrak{M}}_g$.
These conjectures are supported by the work of Cornalba-Harris on the
{ hyperelliptic locus} $\overline{\mathcal{H}}_g=\overline{\mathcal{D}}_2$,
by the results of this paper
on the { trigonal locus} $\overline{\mathcal{T}}_g=\overline{\mathcal{D}}_3$,
and by partial results on the tetragonal locus $\overline{\mathcal{D}}_4$.
\smallskip In view of Remark 12.5, the equality between the maximal
and semistable trigonal bounds for $g=3$ suggests that a similar situation
might occur for other $d$-gonal families. It is reasonable to expect
two or more ``semistable'' bounds, depending on the number of extra
generators in $\on{Pic}_{\mathbb Q}{\overline{\mathcal D}}_d$.
\smallskip One of these ``semistable'' bounds relates to
families obtained as blow-ups of pencils of $d$-gonal curves on
a ruled surface ${\mathbf F}_k$. Example 2.1 yields the maximal bound
$8+4/g$ for hyperelliptic families (no extra generator besides
the boundary classes), and a similar example in the trigonal case
yields the $7+6/g$ semistable bound (one extra generator, the
Maroni locus). We generalize this to any $d$-gonal family of
curves embedded in an arbitrary ruled surface ${\mathbf F}_k$. Invariably, the
slope of $X/\!_{\displaystyle{B}}$ is:
\begin{equation*}
\frac{\delta|_B}{\lambda|_B}=\left(6+\frac{2}{d-1}\right)+\frac{2d}{g}\cdot
\end{equation*}
\medskip
\begin{conj}
Let $X$ be a family of $d$-gonal curves of genus $g$ whose
base $B$ is not contained in a certain codimension 1 closed subset of
$\overline{\mathcal D}_d$. Then the slope of $X/\!_{\displaystyle{B}}$ satisfies:
\begin{equation*}
\frac{\delta|_B}{\lambda|_B}\leq \left(6+\frac{2}{d-1}\right)+\frac{2d}{g}\cdot
\end{equation*}
\label{clifford} \vspace*{-5mm}
\end{conj}
Conjectures~\ref{clifford}--4
are modifications of earlier conjectures of Joe Harris.
\subsection{A look at families with special $g^r_d$'s, $r\geq 2$}
The discussion so far was primarily
concerned with the loci $\overline{\mathcal{D}}_d\subset
\overline{\mathfrak{M}}_g$ corresponding
to linear series $g^1_d$. But all of our problems are well-defined
and quite interesting to solve for curves with series $g^r_d$ of dimension
$r>1$. Equivalently, we consider the loci $\overline{\mathcal{D}}^r_d$
of curves mapping with degree $d$ to ${\mathbf P}^r$, $r\geq 1$.
\medskip
\noindent{\bf Definition 13.1.} The {\it Clifford index} $\mathfrak {c}$
of a smooth curve $C$ is defined as
\[\mathfrak{c}=\on{min}_L\left\{\on{deg} {L} -2\on{dim}{L}\right\}\]
where $L$ runs over all effective special linear series ${L}$
on $C$.
\medskip
Clifford's theorem implies
${\mathfrak{c}}\geq 0$, with equality if and only if $C$ is
hyperelliptic, i.e. ${L}=g^1_2$ (cf.~\cite{ACGH}).
On the other hand, ${\mathfrak{c}}=1$
means that there exists a $g^r_d$ on $C$ with $d-2r=1$. From Marten's
Theorem, $\on{dim}W^r_d(C)\leq d-2r-1=0$, where $W^r_d$ is the
variety parametrizing complete linear series on $C$ of degree $d$
and dimension at least $r$. Therefore, we must have
$\on{dim}W^r_d=0$. But then Mumford's theorem
asserts that $C$ is either trigonal, or
bi-elliptic, or a smooth plane quintic. The bi-elliptic case would mean
that $W^r_d$ consists of $g^2_6$'s, which contradicts the dimension of
$\on{dim}W^r_d$. In short, $\mathfrak{c}=1$
if and only if $C$ is not hyperelliptic and possesses a $g^1_3$ or a $g^2_5$.
\smallskip
Thus, according to the Clifford index,
the first case with $r\geq 2$ is the space of plane quintics. Consider a
general pencil of such, and blow up the plane at its 25 base
points. The resulting family $X=\on{Bl}_{25}{\mathbf P^2}\rightarrow\mathbf P^1$
is easily seen to have slope $8=7+6/g$, which corresponds to
the bound in Conjecture~\ref{clifford} with $d-2$ replaced by the Clifford
index $\mathfrak{c}=1$. Finally, note that for a $d$-gonal curve
$C$ of genus $g$, by definition $\mathfrak{c}\leq d-2$, so that when
$g\gg d$ we may generalize to:
\begin{conj} For a general family $X\rightarrow B$ of genus $g$ stable curves
whose general member has Clifford index $\mathfrak{c}$ and whose base
$B$ is a general curve in $\overline{\mathcal D}^r_d$,
the slope of $X/\!_{\displaystyle{B}}$ satisfies:
\begin{equation*}
\frac{\delta_X}{\lambda_X}\leq\left(6+\frac{2}{\mathfrak{c}+1}\right)+
\frac{2\mathfrak{c}+4}{g}\,\,\,\on{for}\,\,\mathfrak{c}<\!<g\cdot
\label{clifford1}
\end{equation*}
\end{conj}
\noindent{\bf Remark 13.1.} It is worth noting that the stratification
of $\overline{\mathfrak{M}}_g$, for which we asked
in the Introduction, is not obtained
via the Clifford index $\mathfrak{c}$. For example, Xiao constructs families
of bi--elliptic curves $C$ with slope $8$ (cf.~\cite{Xiao}),
which is between the hyperelliptic and the trigonal maximal bounds.
Since $C$ has a $g^1_4$ as bi--elliptic, this already exceeds the
conjectured maximal bounds for the tetragonal case. This shows
that in some of the above
conjectures we have to exclude the subset of
bi--elliptic curves from the tetragonal locus $\overline{\mathcal D}_4$, and that
similar modifications might be necessary for the
other loci $\overline{\mathcal D}_d$. More precisely, it seems plausible
that the stratification of $\overline{\mathfrak{M}}_g$ according to
successively lower slope bounds is related not just to the existence
of a specific linear series $g^r_d$, but also to the number, dimension and
description of the irreducible components of corresponding varieties $W^r_d$.
\subsection{Other methods via the moduli space
$\overline{\mathcal{M}}_{g,n}({\mathbf P}^r,d)$}
The approach in the $g^1_d$-cases is based on a modification
of the Harris-Mumford's [EHM] {\it Hurwitz scheme
of admissible covers}, which parametrized the
$d$-uple covers of stable pointed rational curves.
However, in the more general situation for linear series with larger
dimensions $r>1$, such a compactification via
admissible covers does not exist, so we have to look for a
different solution.
\smallskip
Consider moduli spaces of stable maps
$\overline{\mathcal{M}}_{g,n}({\mathbf P}^r,d)$. They parametrize {\it stable}
maps $(C,p_1,p_2,...,p_n;\mu)$, where $C$ is a projective, connected
nodal curve of arithmetic genus $g$, the $p_i$'s are marked
distinct nonsingular points on $C$, and the map
$\mu:C\rightarrow{\mathbf P}^r$ has image $\mu_*([C])=d[\on{line}]$ and
satisfies certain stability conditions (cf.~\cite{K,KM}). The space
$\overline{\mathcal{M}}_{g,n}({\mathbf P}^r,d)$ seems to be the right
compactification which we need in order to extend our results to
families with $g^r_d$-series on the fibers: the moduli space
of stable maps is somewhat more ``sensitive'' in describing our loci
$\overline{\mathcal{D}}^r_d$ in terms of their geometry.
\smallskip
Going back to the $g^1_d$-problems, one can also see the combinatorial
flavor that stands in the background of these questions.
It is probably not coincidental that the spaces
$\overline{\mathcal{M}}_{g,n}({\mathbf P}^r,d)$ are also combinatorially
defined and give rise to many enumerative problems. It will be useful
to understand better the loci $\overline{\mathcal{D}}^r_d$ via their
connection with the Kontsevich spaces
$\overline{\mathcal{M}}_{g,n}({\mathbf P}^r,d)$, and ultimately to solve the
remaining questions on $\on{Pic}_{\mathbb{Q}}\overline{\mathcal{D}}^r_d$
for any $d,r$, as well as related interesting enumerative problems that
will inevitably arise from such considerations.
\section*{14. Appendix: The Hyperelliptic Locus $\overline{\mathfrak{I}}_g$}
\setcounter{section}{14}
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\label{hyperelliptic}
In this section we give a proof of Theorems~\ref{theoremCHPic} and~\ref{CHX},
following the same ideas and methods as in the trigonal case. We refer
the reader to previous sections for a detailed proof of certain
statements.
\subsection{Boundary locus of $\overline{\mathfrak{I}}_g$}
\label{hyperellipticboundary}
Cornalba-Harris
describe the boundary of $\overline{\mathfrak{I}}_g$ as consisting of
several boundary components, whose general members and indexing are shown
in Fig.~\ref{hyperboundary} (cf.~\cite{CH}). The restriction of the divisor class $\delta$ to $\overline{\mathfrak{I}}_g$
is the following linear combination:
\begin{equation}
\delta\big|_{\overline{\mathfrak{I}}_g}=\delta_0+2\sum_{i=1}^
{[(g-1)/2]}\xi_i+
\sum_{j=1}^{\left[g/2\right]}\delta_j,
\label{boundaryrel}
\end{equation}
where $\xi_i$ and $\delta_i$ are the classes in
$\on{Pic}_{\mathbb{Q}}\overline{\mathfrak{I}}_g$ of the boundary
divisors $\Xi_i$ and $\Delta_j$.
\newpage
\begin{figure}[h]
$$\psdraw{hyper1}{4.5in}{0.9in}$$
\caption{Boundary of the hyperelliptic locus $\overline{\mathfrak{I}}_g$}
\label{hyperboundary}
\end{figure}
\vspace*{-8mm}$$\Xi_0;\,\,\Xi_i,\,{\scriptstyle {i=1,...,[(g-1)/2]}}
;\,\,\Delta_j,\,{\scriptstyle{j=1,...,[g/2]}}$$
\subsection{Effective covers and embedding for hyperelliptic families}
\label{embeddinghyperelliptic}
In the case of a hyperelliptic family $f:X\rightarrow B$, a minimal
quasi-admissible cover coincides with the original family $X$, because no
blow-ups are necessary to perform on the fibers of $X$: these are
already quasi-admissible double covers. Thus, we have a
degree 2 map $\phi=\widetilde{\phi}:X\rightarrow Y$ for some birationally
ruled surface $Y$ over $B$. As for an effective cover
$\widehat{\phi}:\widehat{X}\rightarrow \widehat{Y}$, only the boundary
divisors $\Delta_i$ require blow-ups (cf.~Fig.~\ref{hyperboundary}).
This is analogous to the ``ramification index 1'' discussion in
Fig.~\ref{ram}--\ref{resolve1}. Thus, while in $\widehat{X}$ the special
fibers may have occasional nonreduced rational components of multiplicity
2, the fibers of $\widehat{Y}$ are always
trees of reduced smooth ${\mathbf P}^1$'s.
\medskip
In the case of a smooth hyperelliptic curve $C$, we consider the
natural double sheeted map $f:C\rightarrow {\mathbf P}^1$. The pushforward
$f_*{\mathcal{O}_C}$ is a rank 2 vector bundle on ${\mathbf P}^1$, which fits into the
short exact sequence
\begin{equation*}
0\rightarrow {\mathcal{O}_{{\mathbf P}^1}(g+1)}\rightarrow {f}_*{\mathcal O}_{C}\stackrel
{\on{tr}}{\rightarrow}{\mathcal O}_{{\mathbf P}^1}\rightarrow 0.
\end{equation*}
We can embed $C$ in the rational ruled surface
${\mathbf P}((f_*\mathcal{O}_C)\,\,\hat{})$.
We generalize this construction
to the effective cover $\widehat{\phi}:\widehat{X}\rightarrow \widehat{Y}$
by setting $V:=({\phi}_*{\mathcal O}_{X})\,\,\hat{}$. For some line
bundle $E$ on $\widehat{Y}$:
\begin{equation*}
0\rightarrow {E}\rightarrow {\widehat{\phi}}_*{\mathcal O}_{\widehat{X}}\stackrel
{\on{tr}}{\rightarrow}{\mathcal O}_{\widehat{Y}}\rightarrow 0.
\end{equation*}
Then $\widehat{X}$ naturally
embeds in the threefold ${\mathbf P}V$. Let $\pi:{\mathbf P}V\rightarrow
\widehat{Y}$ be the corresponding projection map.
\subsection{The invariants $\lambda,\delta$ and $\kappa$}
\label{Hyperinvariants}
As a divisor in ${\mathbf P}V$, $\widehat{X}\equiv 2\zeta+\pi^*D$,
for some divisor $D$ on $\widehat{Y}$. From the adjunction
formula, $g=\on{deg}c_1(V)|_{F_{\widehat{Y}}}-1=c-1$, where
$c_1(V)=cB_0+dF_Y$.
The arithmetic genus of the inverse image $\widehat{\phi}^*T(E)$ is
given by \[p_{\!\stackrel{\phantom{.}}{E}}=-m_{\!\stackrel{\phantom{.}}{E}}
\left(\Gamma_{\!\stackrel{\phantom{.}}{E}}+
\Theta_{\!\stackrel{\phantom{.}}{E}}\right).\]
It turns out that these are
the only differences between the set-up of the hyperelliptic and the
trigonal case. The definitions of the functions $m,\theta$ and $\gamma$,
as well as the formulas for $c_1(V), K_{{\mathbf P}V}, c_2({\mathbf P}V)$ and
the congruence $D\equiv 2c_1(V)$ are valid without any modifications.
\smallskip
As in the trigonal case, it will be sufficient to consider only the
cases when the base curve $B$ intersects {\it transversally}
the boundary divisors of $\overline{\mathfrak{I}}_g$.
But then for all non-root components $E$ in $\widehat{Y}$:
\[m_{\!\stackrel{\phantom{.}}{E}}=1=\Theta_{\!\stackrel{\phantom{.}}{E}}\,\,
\on{and}\,\,
\Gamma_{\!\stackrel{\phantom{.}}{E}}=-(p_{\!\stackrel{\phantom{.}}{E}}+1).\]
We can now easily calculate the invariants on $X$.
\begin{prop} For any family $f:X\rightarrow B$ of hyperelliptic curves
with smooth general member and a base curve $B$ intersecting transversally
the boundary of $\overline{\mathfrak{I}}_g$:\
\begin{eqnarray*}
\lambda_X&\!\!=\!\!&dg+\frac{1}{2}\sum_{E\not =R}\Gamma_
{\!\stackrel{\phantom{.}}{E}}
(\Gamma_{\!\stackrel{\phantom{.}}{E}}+1),\\
\kappa_X&\!\!=\!\!&4d(g-1)-2\sum_{E\not
=R}(\Gamma_{\!\stackrel{\phantom{.}}{E}}+1)^2+\sum_{\on{ram}1}1,\\
\delta_X&\!\!=\!\!&4d(2g+1)+2\sum_{E\not = R}
(\Gamma_{\!\stackrel{\phantom{.}}{E}}+1)(1-2\Gamma_
{\!\stackrel{\phantom{.}}{E}})+\sum_{\on{ram}1}1.
\end{eqnarray*}
\label{hyperinvariants}
\end{prop}
With this, we are ready to show the linear relations among $\lambda|_B$ and
the boundary restrictions $\delta_i|_B$ and $\xi_i|_B$. It is
evident that in order to cancel the ``global'' term $d$, one
must subtract $(8g+4)\lambda_X|_B-g\delta|_B$, which is the main
idea of the next theorem.
\begin{thm} There exists an effective linear combination $\mathcal{E}_h$ of
the boundary divisors of $\overline{\mathfrak{I}}_g$, not containing
$\Xi_0$, such that for any family $f:X\rightarrow B$ of hyperelliptic curves
with smooth general member:
\[(8g+4)\lambda_X|_B=g\delta|_B+\mathcal{E}_h|_B\]
\label{hyperrelation}\vspace*{-8mm}
\end{thm}
\begin{proof} We consider the difference
\begin{eqnarray*}
\mathfrak{S}_h&\!\!=\!\!&(8g+4)\lambda_X|_B-g\delta|_B=
2\sum_{E\not = R}(1+\Gamma_{\!\stackrel{\phantom{.}}{E}})(g+\Gamma_
{\!\stackrel{\phantom{.}}{E}})+\sum_{\on{ram}1}g\\
&\!\!=\!\!&2\sum_{E\not = R}p_{\!\stackrel{\phantom{.}}{E}}(g-1+p_
{\!\stackrel{\phantom{.}}{E}})+\sum_{\on{ram}1}g.
\end{eqnarray*}
In the hyperelliptic case, as opposed to the trigonal case, there
is only {\it one type} of non-root components $E$, namely, such that
both $E$ and $E^-$ are reduced. That is why there is just one type of summands
in $\mathfrak{S}_h$.
\smallskip
As in Section~\ref{arbitrary}, it is sufficient to calculate the
above sum for general members of $\Xi_{i}$ and
$\Delta_i$, as described in Prop.~\ref{Delta-k,i}, i.e. for a {\it general}
base curve $B$.
\subsubsection{Contribution of the boundary divisors $\Xi_{i}$}
\label{hypercontribution1}
This case is analogous to the case of $\Delta_{3,i}$ (cf.~
Subsection~\ref{contribution1}). The arithmetic genus
$p_{\!\stackrel{\phantom{.}}{E}}=g-i-1$, and the corresponding summand in
$\mathfrak{S}_h$ is
\[e_i=2p_{\!\stackrel{\phantom{.}}{E}}(g-1+p_
{\!\stackrel{\phantom{.}}{E}})
=2i(g-i-1)>0,\]
where $i=1,...,[(g-1)/2]$.
\subsubsection{Contribution of the boundary divisors $\Delta_{j}$}
\label{hypercontribution2}
Compare this with the contribution of $\Delta_{5,j}$ (subsection
\ref{contribution2}). There are two non-root components $E_1$ and
$E_2$ in the special fiber of $\widehat{Y}$ ($E_1^-=R$),
whose invariants are $p_{\!\stackrel{\phantom{.}}{E_1}}=g-j-1$ and
$p_{\!\stackrel{\phantom{.}}{E_2}}=g-j$. With the
ramification adjustment of $g$, the contribution of $\Delta_j$
to the sum $\mathfrak{S}_h$ is
\begin{equation*}
f_j=
2p_{\!\stackrel{\phantom{.}}{E_1}}(g-1+p_{\!\stackrel{\phantom{.}}{E}})+
p_{\!\stackrel{\phantom{.}}{E_2}}(g-1+p_{\!\stackrel{\phantom{.}}{E}})+g
=4j(g-j)-g>0,
\end{equation*}
where $j=1,...,[g/2]$.
\medskip
Finally, for the appropriate indices $i$ and $j$
we set $\displaystyle{\mathcal{E}_h:=\sum_{i>0}e_i\Xi_i+\sum_{j>0}f_j\Delta_j.}$
This is an effective combination of
boundary divisors in $\overline{\mathfrak{I}}_g$, not containing $\Delta_0$
by construction, and satisfying $\mathfrak{S}_h=\mathcal{E}_h|_B$. \end{proof}
\medskip
Theorem~\ref{hyperrelation} implies immediately the following
\begin{cor}
Let $f:X\rightarrow B$ be a nonisotrivial
family with smooth general member. Then the slope of the family satisfies:
\begin{equation}
\frac{\delta|_B}{\lambda|_B}\leq 8+\frac{4}{g}.
\label{second8+4/g}
\end{equation}
Equality holds if and only if the general fiber of $f$ is hyperelliptic,
and all singular fibers are irreducible.
\end{cor}
It is now straightforward to prove the fundamental
relation in $\on{Pic}_{\mathbb{Q}}\overline{\mathfrak{I}}_g$, shown first
in \cite{CH}. In Theorem~\ref{hyperrelation},
we add to the coefficients $e_i$ and $f_j$
the corresponding multiplicities $\on{mult}_{\delta}\xi_i$ and
$\on{mult}_{\delta}\delta_j$:
\[\widetilde{e}_i=e_i+2\cdot g=2(i+1)(g-i),\,\,\,
\widetilde{f}_j=f_j+1\cdot g=4j(g-j).\] Using the fact that
$\on{Pic}_{\mathbb{Q}}\overline{\mathfrak{I}}_g$ is generated freely
by the boundary classes $\xi_i$ and $\delta_j$ (see \cite{CH}),
we obtain
\[(8g+4)\lambda=g\delta_0+\sum_{i>0}\widetilde{e_i}\xi_i+
\sum_{j>0}\widetilde{f_j}\delta_j.\]
\begin{thm} In the Picard group of the hyperelliptic locus,
$\on{Pic}_{\mathbb{Q}}\overline{\mathfrak{I}}_g$, the class of the Hodge bundle
$\lambda$ is expressible in terms of the boundary divisor classes of
$\overline{\mathfrak{I}}_g$ as:
\begin{equation*}
(8g+4)\lambda=g\xi_0+\sum_{i=1}^{[(g-1)/2]}2(i+1)(g-i)\xi_i
+\sum_{j=1}^{[g/2]}4j(g-j)\delta_j.
\end{equation*}
\label{CHPic2}
\end{thm}
|
1997-10-23T05:14:17 | 9710 | alg-geom/9710026 | en | https://arxiv.org/abs/alg-geom/9710026 | [
"alg-geom",
"math.AG"
] | alg-geom/9710026 | Dmitry Kaledin | D. Kaledin | Hyperkaehler structures on total spaces of holomorphic cotangent bundles | 100 pages, LaTeX2e | null | null | null | null | Let $M$ be a Kaehler manifold, and consider the total space $T^*M$ of the
cotangent bundle to $M$. We show that in the formal neighborhood of the zero
section $M \subset T^*M$ the space $T^*M$ admits a canonical hyperkaehler
structure, compatible with the complex and holomorphic symplectic structures on
$T^*M$. The associated hyperkaehler metric $h$ coincides with the given Kaehler
metric on the zero section $M \subset T^*M$. Moreover, $h$ is invariant under
the canonical circle action on $T^*M$ by dilatations along the fibers of $T^*M$
over $M$. We show that a hyperkaehler structure with these properties is
unique. When the Kaehler metric on $M$ is real-analytic, we show that this
formal hyperkaehler structure can be extended to an open neighborhood of the
zero section. We also prove a hyperkaehler analog of the Darboux-Weinstein
Theorem. To prove these results, we use the machinery of $R$-Hodge structures,
following Deligne and Simpson.
| [
{
"version": "v1",
"created": "Thu, 23 Oct 1997 03:14:16 GMT"
}
] | 2007-05-23T00:00:00 | [
[
"Kaledin",
"D.",
""
]
] | alg-geom | \section*{Introduction}
A hyperk\"ahler manifold is by definition a Riemannian manifold $M$
equipped with two anti-commuting almost complex structures $I$, $J$
parallel with respect to the Levi-Civita connection. Hyperk\"ahler
manifolds were introduced by Calabi in \cite{C}. Since then they
have become the topic of much research. We refer the reader to
\cite{Bes} and to \cite{HKLR} for excellent overviews of the
subject.
Let $M$ be a hyperk\"ahler manifold. The almost complex structures
$I$ and $J$ generate an action of the quaternion algebra ${\Bbb H}$ in the
tangent bundle $\Theta(M)$ to the manifold $M$. This action is
parallel with respect to the Levi-Civita connection. Every
quaternion $h \in {\Bbb H}$ with $h^2 = -1$, in particular, the product $K
= IJ \in {\Bbb H}$, defines by means of the ${\Bbb H}$-action an almost complex
structure $M_h$ on $M$. This almost complex structure is also
parallel, hence integrable and K\"ahler. Thus every hyperk\"ahler
manifold $M$ is canonically K\"ahler, and in many different
ways. For the sake of convenience, we will consider $M$ as a
K\"ahler manifold by means of the complex structure $M_I$, unless
indicated otherwise.
One of the basic facts about hyperk\"ahler manifolds is that the
K\"ahler manifold $M_I$ underlying a hyperk\"ahler manifold $M$ is
canonically holomorphically symplectic. To see this, let $\omega_J$,
$\omega_K$ be the K\"ahler forms for the complex structures $M_J$,
$M_K$ on the manifold $M$, and consider the $2$-form $\Omega =
\omega_J + \sqrt{-1}\omega_K$ on $M$. It is easy to check that the
form $\Omega$ is of Hodge type $(2,0)$ for the complex structure
$M_I$ on $M$. Since it is obviously non-degenerate and closed, it is
holomorphic, and the K\"ahler manifold $M_I$ equipped with the form
$\Omega$ is a holomorphically symplectic manifold.
It is natural to ask whether every holomorphically symplectic
manifold $\langle M,\Omega\rangle$ underlies a hyperk\"ahler
structure on $M$, and if so, then how many such hyperk\"ahler
structures are there. Note that if such a hyperk\"ahler structure
exists, it is completely defined by the K\"ahler metric $h$ on
$M$. Indeed, the K\"ahler forms $\omega_J$ and $\omega_K$ are by
definition the real and imaginary parts of the form $\Omega$, and
the forms $\omega_J$ and $\omega_K$ together with the metric define
the complex structures $J$ and $K$ on $M$ and, consequently, the
whole ${\Bbb H}$-action in the tangent bundle $\Theta(M)$. For the sake of
simplicity, we will call a metric $h$ on a holomorphically
symplectic manifold $\langle M,\Omega\rangle$ hyperk\"ahler if the
Riemannian manifold $\langle M, h \rangle$ with the quaternionic
action associated to the pair $\langle \Omega, h \rangle$ is a
hyperk\"ahler manifold.
It is known (see, e.g., \cite{Beauv}) that if the holomorphically
symplectic manifold $M$ is compact, for example, if $M$ is a
$K3$-surface, then every K\"ahler class in $H^{1,1}(M)$ contains a
unique hyperk\"ahler metric. This is, in fact, a consequence of the
famous Calabi-Yau Theorem, which provides the canonical Ricci-flat
metric on $M$ with the given cohomology class. This Ricci-flat
metric turns out to be hyperk\"ahler. Thus in the compact case
holomorphically symplectic and hyperk\"ahler manifolds are
essentially the same.
The situation is completely different in the general case. For
example, all holomorphically symplectic structures on the formal
neighborhood of the origin $0 \in \C^{2n}$ in the $2n$-dimensional
complex vector space $\C^{2n}$ are isomorphic by the appropriate
version of the Darboux Theorem. On the other hand, hyperk\"ahler
structures on this formal neighborhood form an infinite-dimensional
family (see, e.g., \cite{HKLR}, where there is a construction of a
smaller, but still infinite-dimensional family of hyperk\"ahler
metrics defined on the whole $\C^{2n}$). Thus, to obtain meaningful
results, it seems necessary to restrict our attention to
holomorphically symplectic manifolds belonging to some special
class.
The simplest class of non-compact holomorphically symplectic
manifolds is formed by total spaces $T^*M$ to the cotangent bundle
to complex manifolds $M$. In fact, the first examples of
hyperk\"ahler manifolds given by Calabi in \cite{C} were of this
type, with $M$ being a K\"ahler manifold of constant holomorphic
sectional curvature (for example, a complex projective space). It
has been conjectured for some time that every total space $T^*M$ of
the cotangent bundle to a K\"ahler manifold admits a hyperk\"ahler
structure. The goal of this paper is to prove that this is indeed
the case, if one agrees to consider only an open neighborhood $U
\subset T^*M$ of the zero section $M \subset T^*M$. Our main result
is the following.
\begin{thm}\label{th.1}
Let $M$ be a complex manifold equipped with a K\"ahler metric. The
metric on $M$ extends to a hyperk\"ahler metric $h$ defined in the
formal neighborhood of the zero section $M \subset T^*M$ in the
total space $T^*M$ to the holomorphic cotangent bundle to $M$. The
extended metric $h$ is invariant under the action of the group
$U(1)$ on $T^*M$ given by dilatations along the fibers of the
canonical projection $\rho:T^*M \to M$. Moreover, every other
$U(1)$-invariant hyperk\"ahler metric on the holomorphically
symplectic manifold $T^*M$ becomes equal to $h$ after a holomorphic
symplectic $U(1)$-equivariant automorphism of $T^*M$ over $M$.
Finally, if the K\"ahler metric on $M$ is real-analytic, then the
formal hyperk\"ahler metric $h$ converges to a real-analytic metric
in an open neighborhood $U \subset T^*M$ of the zero section $M
\subset T^*M$.
\end{thm}
Many of the examples of hyperk\"ahler metrics obtained by
Theorem~\ref{th.1} are already known. (See, e.g., \cite{K1},
\cite{K2}, \cite{Nak}, \cite{H}, \cite{BG}, \cite{Sw}.) In these
examples $M$ is usually a generalized flag manifold or a homogeneous
space of some kind. On the other hand, very little is known for
manifolds of general type. In particular, it seems that even for
curves of genus $g \geq 2$ Theorem~\ref{th.1} is new.
We would like to stress the importance of the $U(1)$-invariance
condition on the metric in the formulation of
Theorem~\ref{th.1}. This condition for a total space $T^*M$ of a
cotangent bundle is equivalent to a more general compatibility
condition between a $U(1)$-action and a hyperk\"ahler structure on a
smooth manifold introduced by Hitchin (see, e.g., \cite{H}). Thus
Theorem~\ref{th.1} can be also regarded as answering a question of
Hitchin's in \cite{H}, namely, whether every K\"ahler manifold can
be embedded as the sub-manifold of $U(1)$-fixed points in a
$U(1)$-equivariant hyperk\"ahler manifold. On the other hand, it is
this $U(1)$-invariance that guarantees the uniqueness of the metric
$h$ claimed in Theorem~\ref{th.1}.
We also prove a version of Theorem~\ref{th.1} ``without the
metrics''. The K\"ahler metric on $M$ in this theorem is replaced
with a holomorphic connection $\nabla$ on the cotangent bundle to
$M$ without torsion and $(2,0)$-curvature. We call such connections
{\em K\"ah\-le\-ri\-an}. The total space of the cotangent bundle $T^*M$ is
replaced with the total space ${\overline{T}M}$ of the complex-conjugate to the
tangent bundle to $M$. (Note that {\em a priori} there is no complex
structure on ${\overline{T}M}$, but the $U(1)$-action by dilatations on this
space is well-defined.) The analog of the notion of a hyperk\"ahler
manifold ``without the metric'' is the notion if a hypercomplex
manifold (see, e.g., \cite{Bo}). We define a version of Hitchin's
condition on the $U(1)$-action for hypercomplex manifolds and prove
the following.
\begin{thm}\label{th.2}
Let $M$ be a complex manifold, and let ${\overline{T}M}$ be the total space of
the complex-conjugate to the tangent bundle to $M$ equipped with an
action of the group $U(1)$ by dilatation along the fibers of the
projection ${\overline{T}M} \to M$. There exists a natural bijection between the
set of all K\"ah\-le\-ri\-an connections on the cotangent bundle to $M$
and the set of all isomorphism classes of $U(1)$-equivariant
hypercomplex structures on the formal neighborhood of the zero
section $M \subset {\overline{T}M}$ in ${\overline{T}M}$ such that the projection $\rho:{\overline{T}M}
\to M$ is holomorphic. If the K\"ah\-le\-ri\-an connection on $M$ is
real-analytic, the corresponding hypercomplex structure is defined
in an open neighborhood $U \subset {\overline{T}M}$ of the zero section.
\end{thm}
Our main technical tool in this paper is the relation between
$U(1)$\--equi\-va\-ri\-ant hyperk\"ahler manifolds and the theory of
$\R$-Hodge structures discovered by Deligne and Simpson (see
\cite{De2}, \cite{Simpson}). To emphasize this relation, we use the
name {\em Hodge manifolds} for the hypercomplex manifolds equipped
with a compatible $U(1)$-action.
It must be noted that many examples of hyperk\"ahler manifolds
equipped with a compatible $U(1)$-action are already known. Such
are, for example, many of the manifolds constructred by the
so-called hyperk\"ahler redution from flat hyperk\"ahler spaces (see
\cite{H} and \cite{HKLR}). An important class of such manifolds is
formed by the so-called quiver varieties, studied by Nakajima
(\cite{Nak}). On the other hand, the moduli spaces $\M$ of flat
connections on a complex manifold $M$, studied by Hitchin
(\cite{Hcurves}) when $M$ is a curve and by Simpson (\cite{Smoduli}
in the general case, also belong to the class of Hodge manifolds, as
Simpspn has shown in \cite{Simpson}. Some parts of our theory,
especially the uniqueness statement of Theorem~\ref{th.1}, can be
applied to these known examples.
\medskip
We now give a brief outline of the paper. Sections 1-3 are
preliminary and included mostly to fix notation and
terminology. Most of the facts in these sections are well-known.
\begin{itemize}
\item In Section 1 we have collected the necessary facts from linear
algebra about quaternionic vector spaces and $\R$-Hodge
structures. Everything is standard, with an exception, perhaps, of
the notion of {\em weakly Hodge map}, which we introduce in
Subsection~\ref{w.H.sub}.
\item We begin Section 2 with introducing a technical notion of a
{\em Hodge bundle} on a smooth manifold equipped with a
$U(1)$-action. This notion will be heavily used throughout the
paper. Then we switch to our main subject, namely, various
differential-geomteric objects related to a quaternion action in the
tangent bundle. The rest of Section 2 deals with almost quaternionic
manifolds and the compatibility conditions between an almost
quaternionic structure and a $U(1)$-action on a smooth manifold $M$.
\item In Section 3 we describe various integrability conditions on
an almost quaternionic structure. In particular, we recall the
definition of a hypercomplex manifold and introduce
$U(1)$-equivariant hypercomplex manifolds under the name of {\em
Hodge manifolds}. We then rewrite the definition of a Hodge manifold
in the more convenient language of Hodge bundles, to be used
throughout the rest of the paper. Finally, in
Subsection~\ref{polarization} we discuss metrics on hypercomplex and
Hodge manifolds. We recall the definition of a hyperk\"ahler
manifold and define the notion of a {\em polarization} of a Hodge
manifold. A polarized Hodge manifold is the same as a hyperk\"ahler
manifold equipped with a $U(1)$-action compatible with the
hyperk\"ahler structure of the sense of Hitchin, \cite{H}.
\item The main part of the paper begins in Section 4. We start with
arbitrary Hodge manifolds and prove that in a neighborhood of the
subset $M^{U(1)}$ of ``regular'' $U(1)$-fixed points every such
manifold $M$ is canonically isomorphic to an open neighborhood of
the zero section in a total space $\overline{T}M^{U(1)}$ of the
tangent bundle to the fixed point set. A fixed point $m \in M$ is
``regular'' if the group $U(1)$ acts on the tangent space $T_mM$
with weights $0$ and $1$. We call this canonical isomorphism {\em
the linearization} of the regular Hodge manifold.
The linearization construction can be considered as a hyperk\"ahler
analog of the Darboux-Weinstein Theorem in the symplectic
geometry. Apart from the cotangent bundles, it can be applied to the
Hitchin-Simpson moduli space $\M$ of flat connections on a K\"ahler
manifold $X$. The regular points in this case correspond to stable
flat connections such that the underlying holomorphic bundle is also
stable. The linearization construction provides a canonical
embedding of the subspace $\M^{reg} \subset \M$ of regular points
into the total space $T^*\M_0$ of the cotangent bundle to the space
$\M_0$ of stable holomorphic bundles on $X$. The unicity statement
of Theorem~\ref{th.1} guarantees that the hyperk\"ahler metric on
$\M^{reg}$ provided by the Simpson theory is the same as the
canonical metric constructed in Theorem~\ref{th.1}.
\item Starting with Section 5, we deal only with total spaces ${\overline{T}M}$
of the complex-conjugate to tangent bundles to complex manifolds
$M$. In Section 5 we describe Hodge manifolds structures on ${\overline{T}M}$ in
terms of certain connections on the locally trivial fibration ${\overline{T}M}
\to M$. ``Connection'' here is understood as a choice of the
subbundle of horizontal vectors, regardless of the vector bundle
structure on the fibration ${\overline{T}M} \to M$. We establish a
correspondence between Hodge manifold structures on ${\overline{T}M}$ and
connections on ${\overline{T}M}$ over $M$ of certain type, which we call {\em
Hodge connection}.
\item In Section 6 we restrict our attention to the formal
neighborhood of the zero section $M \subset {\overline{T}M}$. We introduce the
appropriate ``formal'' versions of all the definitions and then
establish a correspondence between formal Hodge connections on ${\overline{T}M}$
over $M$ and certain differential operators on the manifold $M$
itself, which we call {\em extended connections}. We also introduce
a certain canonical algebra bundle $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ on the complex
manifold $M$, which we call {\em the Weil algebra} of the manifold
$M$. Extended connections give rise to natural derivations of the
Weil algebra.
\item Before we can proceed with classification of extended
connections on the manifold $M$ and therefore of regular Hodge
manifolds, we need to derive some linear-algebraic facts on the Weil
algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$. This is the subject of Section 7. We begin
with introducing a certain version of the de Rham complex of a
smooth complex manifold, which we call {\em the total de Rham
complex}. Then we combine it the material of Section 6 to define the
so-called {\em total Weil algebra} of the manifold $M$ and establish
some of its properties. Section 7 is the most technical part of the
whole paper. The reader is advised to skip reading it until needed.
\item Section 8 is the main section of the paper. In this section we
prove, using the technical results of Section 7, that extended
connections on $M$ are in a natural one-to-one correspondence with
K\"ah\-le\-ri\-an connections on the cotangent bundle to $M$
(Theorem~\ref{kal=ext}). This proves the formal part of
Theorem~\ref{th.2}.
\item In Section 9 we deal with polarizations. After some
preliminaries, we use Theorem~\ref{th.2} to deduce the formal part
of Theorem~\ref{th.1} (see Theorem~\ref{metrics}).
\item Finally, in Section~\ref{convergence} we study the convergence
of our formal series and prove Theorem~\ref{th.1} and
Theorem~\ref{th.2} in the real-analytic case.
\item We have also attached a brief Appendix, where we sketch a more
conceptual approach to some of the linear algebra used in the paper,
in particular, to our notion of a weakly Hodge map. This approach
also allows us to describe a simple and conceptual proof of
Proposition~\ref{ac}, the most technical of the facts proved in
Section 7. The Appendix is mostly based on results of Deligne and
Simpson (\cite{De2}, \cite{Simpson}).
\end{itemize}
\noindent
{\bf Acknowledgment.} I would like to thank A. Beilinson,
D. Kazhdan, A. Levin, L. Posicelsky, A. Shen and A. Tyurin for
stimulating discussions. I am especially grateful to my friends
Misha Verbitsky and Tony Pantev for innumerable discussions,
constant interest and help, without which this paper most probably
would not have been written. I would also like to mention here how
much I have benefited from a course on moduli spaces and Hodge
theory given by Carlos Simpson at MIT in the Fall of 1993. On a
different note, I would like to express my dearest gratitude to
Julie Lynch, formerly at International Press in Cambridge, and also
to the George Soros's Foundation and to CRDF for providing me with a
source of income during the preparation of this paper.
\tableofcontents
\section{Preliminary facts from linear algebra}
\subsection{Quaternionic vector spaces}
\refstepcounter{subsubsection Throughout the paper denote by ${\Bbb H}$ the $\R$-algebra of quaternions.
\noindent {\bf Definition.\ } A {\em quaternionic vector space} is a finite-dimensional left module
over the algebra ${\Bbb H}$.
Let $V$ be a quaternionic vector space. Every algebra embedding
$I:\C \to {\Bbb H}$ defines by restriction an action of $\C$ on
$V$. Denote the corresponding complex vector space by $V_I$.
Fix once and for all an algebra embedding $I:\C \to {\Bbb H}$ and call the
complex structure $V_I$ {\em the preferred complex structure} on
$V$.
\refstepcounter{subsubsection \label{u.acts.on.h}
Let the group $\C^*$ act on the algebra ${\Bbb H}$ by conjugation by
$I(\C^*)$. Since $I(\R^*) \subset I(\C^*)$ lies in the center of the
algebra ${\Bbb H}$, this action factors through the map $N:\C^* \to
\C^*/\R^* \cong U(1)$ from $\C^*$ to the one-dimensional unitary group
defined by $N(a) = a^{-1}\overline{a}$. Call this action {\em the
standard action} of $U(1)$ on ${\Bbb H}$.
The standard action commutes with the multiplication and leaves
invariant the subalgebra $I(\C)$. Therefore it extends to an action
of the complex algebraic group $\C^* \supset U(1)$ on the algebra
${\Bbb H}$ considered as a right complex vector space over $I(\C)$. Call
this action {\em the standard action} of $\C^*$ on ${\Bbb H}$.
\refstepcounter{subsubsection \noindent {\bf Definition.\ } An {\em equivariant} quaternionic vector space is a quaternionic
vector space $V$ equipped with an action of the group $U(1)$ such that the
action map
$$
{\Bbb H} \otimes_\R V \to V
$$
is $U(1)$-equivariant.
The $U(1)$-action on $V$ extends to an action of $\C^*$ on the complex
vector space $V_I$. The action map ${\Bbb H} \otimes_\R V \to V$ factors through a
map
$$
\Mult:{\Bbb H} \otimes_\C V_I \to V_I
$$
of complex vector spaces. This map is $\C^*$-equivariant if and only if $V$
is an equivariant quaternionic vector space.
\refstepcounter{subsubsection The category of complex algebraic representations $V$ of the
group $\C^*$ is equivalent to the category of graded vector spaces $V
= \oplus V^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. We will say that a representation $W$ is {\em of
weight $i$} if $W = W^i$, that is, if an element $z \in \C^*$ acts
on $W$ by multiplication by $z^k$. For every representation $W$ we
will denote by $W(k)$ the representation corresponding to the
grading
$$
W(k)^i = W^{k+i}.
$$
\refstepcounter{subsubsection \label{h=c+c}
The algebra ${\Bbb H}$ considered as a complex vector space by means of right
multiplication by $I(\C)$ decomposes ${\Bbb H} = I(\C) \oplus \overline{\C}$ with respect
to the standard $\C^*$-action. The first summand is of weight $0$, and the
second is of weight $1$. This decomposition is compatible with the left
$I(\C)$-actions as well and induces for every complex vector space $W$ a
decomposition
$$
{\Bbb H} \otimes_\C W = W \oplus \overline{\C} \otimes_\C W.
$$
If $W$ is equipped with an $\C^*$-action, the second summand is canonically
isomorphic to $\overline{W}(1)$, where $\overline{W}$ is the vector space
complex-conjugate to $W$.
\refstepcounter{subsubsection Let $V$ be an equivariant quaternionic vector space. The action map
$$
\Mult:{\Bbb H} \otimes_\C V_I \cong V_I \oplus \overline{\C} \otimes V_I \to V_I
$$
decomposes $\Mult = {\sf id} \oplus j$ for a certain map $j:\overline{V_I}(1)
\to V_I$. The map $j$ satisfies $j \circ \overline{j} = -{\sf id}$, and we
obviously have the following.
\begin{lemma}\label{explicit.eqvs}
The correspondence $V \longmapsto \langle V_I, j \rangle$ is an
equivalence of categories bet\-ween the category of equivariant
quaternionic vector spaces and the category of pairs $\langle W, j
\rangle$ of a graded complex vector space $W$ and a map
$j:\overline{W^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}} \to W^{1-{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ satisfying $j \circ
\overline{j} = -{\sf id}$.
\end{lemma}
\refstepcounter{subsubsection We will also need a particular class of equivariant quaternionic
vector spaces which we will call regular.
\noindent {\bf Definition.\ } An equivariant quaternionic vector space $V$ is called {\em regular}
if every irreducible $\C^*$-subrepresentation of $V_I$ is either trivial or
of weight $1$.
\begin{lemma}\label{regular.quaternionic}
Let $V$ be a regular $\C^*$-equivariant quaternionic vector space and let
$V^0_I \subset V_I$ be the subspace of $\C^*$-invariant vectors. Then the
action map
$$
V^0_I \oplus \overline{V^0_I} \cong {\Bbb H} \otimes_\C V^0_I \to V_I
$$
is invertible.
\end{lemma}
\proof Let $V^1_I \subset V_I$ be the weight $1$ subspace with respect to
the $ gm$-action. Since $V$ is regular, $V_I = V^0_I \oplus V^1_I$. On the
other hand, $j:\overline{V_I} \to V_I$ interchanges $V^0_I$ and
$V^1_I$. Therefore $V^1_I \cong \overline{V^0_I}$ and we are done.
\hfill \ensuremath{\square}\par
Thus every regular equivariant quaternionic vector space is a sum of
several copies of the algebra ${\Bbb H}$ itself. The corresponding Hodge structure
has Hodge numbers $h^{1,0} = h^{0,1}$, $h^{p,q} = 0$ otherwise.
\subsection{The complementary complex structure}\label{complementary}
\refstepcounter{subsubsection
Let $J:\C \to {\Bbb H}$ be another algebra embedding. Say that embeddings $I$ and
$J$ are {\em complementary} if
$$
J(\sqrt{-1}) I(\sqrt{-1}) = - I(\sqrt{-1}) J(\sqrt{-1}).
$$
Let $V$ be an equivariant quaternionic vector space. The standard
$U(1)$-action on ${\Bbb H}$ induces an action of $U(1)$ on the set of all algebra
embeddings $\C \to {\Bbb H}$. On the subset of embeddings complementary to $I$
this action is transitive. Therefore for every two embeddings $J_1,J_2:\C
\to {\Bbb H}$ complementary to $I$ the complex vector spaces $V_{J_1}$ and
$V_{J_2}$ are canonically isomorphic. We will from now on choose for
convenience an algebra embedding $J:\C \to {\Bbb H}$ complementary to $I$ and
speak of {\em the complementary complex structure} $V_J$ on $V$; however,
nothing depends on this choice.
\refstepcounter{subsubsection For every equivariant quaternionic vector space $V$ the
complementary embedding $J:\C \to {\Bbb H}$ induces an isomorphism
$$
J \otimes {\sf id}:\C \otimes_\R V \to {\Bbb H} \otimes_{I(\C)} V_I
$$
of complex vector spaces. Let $\Mult:{\Bbb H} \otimes_{I(\C)} V_I \to V_I$, $\Mult:\C
\otimes_\R V \to V_J$ be the action maps. Then there exists a unique
isomorphism $H:V_J \to V_I$ of complex vector spaces such that the diagram
$$
\begin{CD}
\C \otimes_\R V @>{J \otimes {\sf id}}>> {\Bbb H} \otimes_{I(\C)} V_I\\
@V{\Mult}VV @V{\Mult}VV \\
V_J @>{H}>> V_I
\end{CD}
$$
is commutative. Call the map $H:V_J \to V_I$ {\em the standard
isomorphism} between the complementary and the preferred complex structures
on the equivariant quaternionic vector space $V$.
\refstepcounter{subsubsection Note that both $V_I$ and $V_J$ are canonically isomorphic to $V$ as
real vector spaces; therefore the map $H:V_J \to V_I$ is in fact an
automorphism of the real vector space $V$. Up to a constant this
automorphism is given by the action of the element $I(\sqrt{-1}) +
J(\sqrt{-1}) \in {\Bbb H}$ on the ${\Bbb H}$-module $V$.
\subsection{$\protect\R$-Hodge structures}
\refstepcounter{subsubsection Recall that {\em a pure $\R$-Hodge structure $W$ of weight
$i$} is a pair of a graded complex vector space $W = \oplus_{p+q=i}
W^{p,q}$ and a {\em real structure} map $\overline{\ }:\overline{W^{p,q}}
\to W^{q,p}$ satisfying $\overline{\ } \circ \overline{\ } = {\sf id}$. The bigrading
$W^{p,q}$ is called {\em the Hodge type bigrading}. The dimensions
$h^{p,q} = \dim_\C W^{p,q}$ are called {\em the Hodge numbers} of
the pure $\R$-Hodge structure $W$. Maps between pure Hodge
structures are by definition maps of their underlying complex vector
spaces compatible with the bigrading and the real structure maps.
\refstepcounter{subsubsection Recall also that for every $k$ the {\em Hodge-Tate} pure $\R$-Hodge
structure $\R(k)$ of weight $-2k$ is by definition the
$1$-dimensional complex vector space with complex conjugation equal
to $(-1)^k$ times the usual one, and with Hodge bigrading
$$
\R(k)^{p,q} = \begin{cases} \R(k), \quad p=q=-k,\\ 0, \quad
\text{otherwise}. \end{cases}
$$
For a pure $\R$-Hodge structure $V$ denote, as usual, by $V(k)$ the
tensor product $V(k) = V \otimes \R(k)$.
\refstepcounter{subsubsection \label{w.1}
We will also need special notation for another common $\R$-Hodge
structure, which we now introduce. Note that for every complex $V$
be a complex vector space the complex vector space $V \otimes_\R \C$
carries a canonical $\R$-Hodge structure of weight $1$ with Hodge
bigrading given by
$$
V^{1,0} = V \subset V \otimes_\R \C \qquad\qquad V^{0,1} =
\overline{V} \otimes_\R \C.
$$
In particular, $\C \otimes_\R \C$ carries a natural $\R$-Hodge
structure of weight $1$ with Hodge numbers $h^{1,0} = h^{0,1} =
1$. Denote this Hodge structure by $\W_1$.
\refstepcounter{subsubsection Let $\langle W, \overline{\ } \rangle$ be a pure Hodge structure, and
denote by $W_\R \subset W$ the $\R$-vector subspace of elements
preserved by $\overline{\ }$. Define the {\em Weil operator} $C:W \to W$ by
$$
C = (\sqrt{-1})^{p-q}:W^{p,q} \to W^{q,p}.
$$
The operator $C:W \to W$ preserves the $\R$-Hodge structure, in
particular, the subspace $W_\R \subset W$. On pure $\R$-Hodge
structures of weight $0$ the Weil operator $C$ corresponds to the
action of $-1 \in U(1) \subset \C^*$ in the corresponding
representation.
\refstepcounter{subsubsection \label{Weil}
For a pure Hodge structure $W$ of weight $i$ let
$$
j = C \circ \overline{\ }:\overline{W^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},1-{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}} \to W^{1-{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}.
$$
If $i$ is odd, in particular, if $i=1$, then $j \circ \overline{j} =
-{\sf id}$. Together with Lemma~\ref{explicit.eqvs} this gives the following.
\begin{lemma}\label{eqvs.hodge}
The category of equivariant quaternionic vector spaces is equivalent to the
category of pure $\R$-Hodge structures of weight $1$.
\end{lemma}
\refstepcounter{subsubsection Let $V$ be an equivariant quaternionic vector space, and let
$\langle W, \overline{\ } \rangle$ be $\R$-Hodge structure of weight $1$
corresponding to $V$ under the equivalence of
Lemma~\ref{eqvs.hodge}. By definition the complex vector space $W$
is canonically isomorphic to the complex vector space $V_I$ with the
preferred complex structure on $V$. It will be more convenient for
us to identify $W$ with the complementary complex vector space $V_J$
by means of the standard isomorphism $H:V_J \to V_I$. The
multiplication map
$$
\Mult:V_I \otimes_\R \C \cong V \otimes_\C {\Bbb H} \to V_J
$$
is then a map of $\R$-Hodge structures. The complex conjugation $\overline{\ }:W \to
\overline{W}$ is given by
\begin{equation}\label{i.conj}
\overline{\ } = C \circ H \circ j \circ H^{-1} = C \circ i:V_J \to \overline{V_J},
\end{equation}
where $i:V_J \to \overline{V_J}$ is the action of the element
$I(\sqrt{-1}) \subset {\Bbb H}$.
\subsection{Weakly Hodge maps}\label{w.H.sub}
\refstepcounter{subsubsection Recall that the category of pure $\R$-Hodge structures of weight $i$
is equivalent to the category of pairs $\langle V, F^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\rangle$ of a
real vector space $V$ and a decreasing filtration $F^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ on the complex
vector space $V_\C = V \otimes_\R \C$ satisfying
$$
V_\C = \bigoplus_{p+q=i}F^pV_\C \cap \overline{F^qV_\C}.
$$
The filtration $F^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is called {\em the Hodge filtration}.
The Hodge type grading and the Hodge filtration are related by $V^{p,q} =
F^pV_\C \cap \overline{F^qV_\C}$ and $F^pV_\C = \oplus_{k \geq
p}V^{k,i-k}$.
\refstepcounter{subsubsection \label{weakly.hodge} Let $\langle V,F^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\rangle$ and
$\langle W,F^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \rangle$ be pure $\R$-Hodge structures of weights
$n$ and $m$ respectively. Usually maps of pure Hodge structures are
required to preserve the weights, so that $\Hom(V,W)=0$ unless
$n=m$. In this paper we will need the following weaker notion of
maps between pure $\R$-Hodge structures.
\noindent {\bf Definition.\ } An $\R$-linear map $f:V \to W$ is said to be {\em weakly
Hodge} if it preserves the Hodge filtrations.
Equivalently, the complexified map $f:V_\C \to W_\C$ must decompose
\begin{equation}\label{H.type}
f = \sum_{0 \leq p \leq m-n}f^{p,m-n-p},
\end{equation}
where the map $f^{p,m-n-p}:V_\C \to W_\C$ is of Hodge type
$(p,m-n-p)$. Note that this condition is indeed weaker than the
usual definition of a map of Hodge structures: a weakly Hodge map
$f:V \to W$ can be non-trivial if $m$ is strictly greater than
$n$. If $m < n$, then $f$ must be trivial, and if $m=n$, then weakly
Hodge maps from $V$ to $W$ are the same as usual maps of $\R$-Hodge
structures.
\refstepcounter{subsubsection We will denote by ${{\cal W}{\cal H}odge}$ the category of pure $\R$-Hodge
structures of arbitrary weight with weakly Hodge maps as
morphisms. For every integer $n$ let ${{\cal W}{\cal H}odge}_n$ be the full
subcategory in ${{\cal W}{\cal H}odge}$ consisting of pure $\R$-Hodge structures
of weight $n$, and let ${{\cal W}{\cal H}odge}_{\geq n}$ be the full subcategory
of $\R$-Hodge structures of weight not less than $n$. Since weakly
Hodge maps between $\R$-Hodge structures of the same weight are the
same as usual maps of $\R$-Hodge structures, the category
${{\cal W}{\cal H}odge}_n$ is the usual category of pure $\R$-Hodge structures of
weight $n$.
\refstepcounter{subsubsection \label{w.k} Let $\W_1 = \C \otimes_\R \C$ be the pure Hodge
structure of weight $1$ with Hodge numbers $h^{1,0}=h^{0,1}=1$, as
in \ref{w.1}. The diagonal embedding $\C \to \C \otimes_\R \C$
considered as a map $w_1:\R \to \W_1$ from the trivial pure
$\R$-Hodge structure $\R$ of weight $0$ to $\W_1$ is obviously
weakly Hodge. It decomposes $w_1 = w_1^{1,0} + w_1^{0,1}$ as in
\eqref{H.type}, and the components $w_1^{1,0}:\C \to \W_1^{1,0}$ and
$w_1^{0,1}:\C \to \W_1^{0,1}$ are isomorphisms of complex vectors
spaces. Moreover, for every pure $\R$-Hodge structure $V$ of weight
$n$ the map $w_1$ induces a weakly Hodge map $w_1:V \to \W_1 \otimes
V$, and the components $w_1^{1,0}:V_\C \to V_\C \otimes \W_1^{1,0}$
and $w_1^{0,1}:V_\C \to V_\C \otimes \W_1^{0,1}$ are again
isomorphisms of complex vector spaces.
More generally, for every $k \geq 0$ let $\W_k = S^k\W_1$ be the
$k$-th symmetric power of the Hodge structure $\W_1$. The space
$\W_k$ is a pure $\R$-Hodge structure of weight $k$, with Hodge
numbers $h^{k,0} = h^{k-1,1} =\ldots = h^{0,k} = 1$. Let $w_k:\R \to
\W_k$ be the $k$-th symmetric power of the map $w_1:\R \to
\W_1$. For every pure $\R$-Hodge structure $V$ of weight $n$ the map
$w_k$ induces a weakly Hodge map $w_k:V \to \W_k \otimes V$, and the
components
$$
w_k^{p,q}:V_\C \to V_\C \otimes \W_k^{p,k-p}, \qquad 0 \leq p \leq k
$$
are isomorphisms of complex vector spaces.
\refstepcounter{subsubsection \label{w.k.uni}
The map $w_k$ is a universal weakly Hodge map from a pure $\R$-Hodge
structures $V$ of weight $n$ to a pure $\R$-Hodge structure of
weight $n+k$. More precisely, every weakly Hodge map $f:V \to V'$
from $V$ to a pure $\R$-Hodge structure $V'$ of weight $n+k$ factors
uniquely through $w_k:V \to \W_k \otimes V$ by means of a map
$f':\W_k \otimes V \to V'$ preserving the pure $\R$-Hodge
structures. Indeed, $V_\C \otimes \W_k = \bigoplus_{0 \leq p \leq k}
V_\C \otimes \W_k^{p,k-p}$, and the maps $w_k^{p,k-p}:V_\C \to V_\C
\otimes \W_k^{p,k-p}$ are invertible. Hence to obtain the desired
factorization it is necessary and sufficient to set
$$
f' = f^{p,k-p} \circ \left(w_k^{p,k-p}\right)^{-1}:V_\C \otimes
\W_k^{p,k-p} \to V_\C \to \V'_\C,
$$
where $f = \sum_{0 \leq p \leq k}f^{p,k-p}$ is the Hodge type decomposition
\eqref{H.type}.
\refstepcounter{subsubsection It will be convenient to reformulate the universal properties
of the maps $w_k$ as follows. By definition $\W_k = S^k\W_1$,
therefore the dual $\R$-Hodge structures equal $\W_k^* = S^k\W_1^*$,
and for every $n,k \geq 0$ we have a canonical projection
${\sf can}:\W_n^* \otimes \W_k^* \to \W_{n+k}^*$. For every pure
$\R$-Hodge structure $V$ of weight $k \geq 0$ let $\Gamma(V) = V
\otimes \W_k^*$.
\begin{lemma}\label{g.ex}
The correspondence $V \mapsto \Gamma(V)$ extends to a functor
$$
\Gamma:{{\cal W}{\cal H}odge}_{\geq 0} \to {{\cal W}{\cal H}odge}_0
$$
adjoint on the right to the canonical embedding ${{\cal W}{\cal H}odge}_0
\hookrightarrow {{\cal W}{\cal H}odge}_{\geq 0}$.
\end{lemma}
\proof Consider a weakly Hodge map $f:V_n
\to V_{n+k}$ from $\R$-Hodge structure $V_n$ of weight $n$ to a pure
$\R$-Hodge structure $V_{n+k}$ of weight $n+k$. By the universal
property the map $f$ factors through the canonical map $w_k:V_n \to
V_n \otimes \W_k$ by means of a map $f_k:V_n \otimes \W_k \to
V_{n+k}$. Let $f_k':V_k \to V_{n+k} \otimes \W_k^*$ be the adjoint
map, and let
\begin{align*}
\Gamma(f) = {\sf can} \circ f_k':&\Gamma(V_n) = V_n \otimes \W_k^* \to
V_{n+k} \otimes \W_n^* \otimes \W_k^* \to\\
\to &\Gamma(V_{n+k}) = V \otimes \W_{n+k}^*.
\end{align*}
This defines the desired functor $\Gamma:{{\cal W}{\cal H}odge}_{\geq 0} \to
{{\cal W}{\cal H}odge}_0$. The adjointness is obvious.
\hfill \ensuremath{\square}\par
\noindent {\bf Remark.\ } See Appendix for a more geometric description of the functor
$\Gamma:{{\cal W}{\cal H}odge}_{\geq 0} \to {{\cal W}{\cal H}odge}_0$.
\refstepcounter{subsubsection \label{l.r}
The complex vector space $\W_1 = \C \otimes_\R \C$ is equipped with
a canonical skew-symmetric trace pairing $\W_1 \otimes_\C \W_1 \to
\C$. Let $\gamma:\C \to \W_1^*$ be the map dual to $w_1:\R \to \W_1$
under this pairing. The map $\gamma$ is not weakly Hodge, but it
decomposes $\gamma = \gamma^{-1,0} + \gamma^{0,-1}$ with respect to
the Hodge type grading. Denote $\gamma_l = \gamma^{-1,0}$, $\gamma_r
= \gamma^{0,-1}$. For every $0 \leq p \leq k$ the symmetric powers
of the maps $\gamma_l$, $\gamma_r$ give canonical complex-linear
embeddings
$$
\gamma_l, \gamma_r:\W_p^* \to \W_k^*.
$$
\refstepcounter{subsubsection The map $\gamma_l$ if of Hodge type $(p-k,0)$, while
$\gamma_r$ is of Hodge type $(0, p-k)$, and the maps $\gamma_l$,
$\gamma_r$ are complex conjugate to each other. Moreover, they are
each compatible with the natural maps ${\sf can}:\W_p^* \otimes \W_q^*
\to \W_{p+q}^*$ in the sense that ${\sf can} \circ (\gamma_l \otimes
\gamma_l) = \gamma_l \circ {\sf can}$ and ${\sf can} \circ (\gamma_r \otimes
\gamma_r) = \gamma_r \circ {\sf can}$. For every $p$, $q$, $k$ such that
$p+q \geq k$ we have a short exact sequence
\begin{equation}\label{p+q}
\begin{CD}
0 @>>> \W_{p+q-k} @>{\gamma_r - \gamma_l}>> \W_p^* \oplus \W_q^*
@>{\gamma_l + \gamma_l}>> \W_k^* @>>> 0
\end{CD}
\end{equation}
of complex vector spaces. We will need this exact sequence in
\ref{gamma.use}.
\refstepcounter{subsubsection \label{gamma.tensor}
The functor $\Gamma$ is, in general, not a tensor functor. However,
the canonical maps ${\sf can}:\W_n^* \otimes \W_k^* \to \W_{n+k}^*$
define for every two pure $\R$-Hodge structures $V_1$, $V_2$ of
non-negative weight a surjective map
$$
\Gamma(V_1) \otimes \Gamma(V_2) \to \Gamma(V_1 \otimes V_2).
$$
These maps are functorial in $V_1$ and $V_2$ and commute with the
associativity and the commutativity morphisms. Moreover, for every
algebra $\A$ in the tensor category ${{\cal W}{\cal H}odge}_{\geq 0}$ they turn
$\Gamma(\A)$ into an algebra in ${{\cal W}{\cal H}odge}_0$.
\subsection{Polarizations}
\refstepcounter{subsubsection \label{hyperherm}
Consider a quaternionic vector space $V$, and let $h$ be a Euclidean metric
on $V$.
\noindent {\bf Definition.\ } The metric $h$ is called {\em Qua\-ter\-nionic\--Her\-mi\-tian} if
for any algebra embedding $I:\C \to {\Bbb H}$ the metric $h$ is the real part of
an Hermitian metric on the complex vector space $V_I$.
Equivalently, a metric is Quaternionic-Hermitian if it is invariant under the
action of the group $SU(2) \subset {\Bbb H}$ of unitary quaternions.
\refstepcounter{subsubsection \label{hermhodge}
Assume that the quaternionic vector space $V$ is equivariant. Say
that a metric $h$ on $V$ is {\em Hermitian-Hodge} if it is
Qua\-ter\-nionic-\-Her\-mi\-tian and, in addition, invariant under the
$U(1)$-action on $V$.
Let $V_I$ be the vector space $V$ with the preferred complex structure $I$,
and let
$$
V_I = \bigoplus V^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}
$$
and $j:\overline{V^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}} \to V^{1-{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ be as in
Lemma~\ref{explicit.eqvs}. The metric $h$ is Hermitian-Hodge if and only if
\begin{enumerate}
\item it is the real part of an Hermitian metric on $V_I$,
\item $h(V^p,V^q) = 0$ whenever $p \neq q$, and
\item $h(j(a),b) = - h(a,j(b))$ for every $a,b \in V$.
\end{enumerate}
\refstepcounter{subsubsection Recall that a {\em polarization} $S$ on a pure $\R$-Hodge
structure $W$ of weight $i$ is a bilinear form $S:W \otimes W \to
\R(-i)$ which is a morphism of pure Hodge structures and satisfies
\begin{align*}
S(a,b) &= (-1)^{i}S(b,a)\\
S(a,Ca) &> 0
\end{align*}
for every $a,b \in W$. (Here $C:W \to W$ is the Weil operator.)
\refstepcounter{subsubsection \label{pol}
Let $V$ be an equivariant quaternionic vector space equipped with an
Euclidean metric $h$, and let $\langle W, \overline{\ } \rangle$ be the pure
$\R$-Hodge structure of weight $1$ associated to $V$ by
Lemma~\ref{eqvs.hodge}. Recall that $W = V_J$ as a complex vector
space. Assume that $h$ extends to an Hermitian metric $h_J$ on
$V_J$, and let $S:W \otimes W \to \R(-1)$ be the bilinear form
defined by
$$
S(a,b) = h(a,C\overline{b}), \qquad a,b \in W_\R \subset W.
$$
The form $S$ is a polarization if and only if the metric $h$ is
Hermitian-Hodge. This gives a one-to-one correspondence between the set of
Hermitian-Hodge metrics on $V$ and the set of polarizations on the Hodge
structure $W$.
\refstepcounter{subsubsection Let $W^*$ be the Hodge structure of weight $-1$ dual to $W$.
The sets of polarizations on $W$ and on $W^*$ are, of course, in a
natural one-to-one correspondence. It will be more convenient for us
to identify the set of Hermitian-Hodge metrics on $V$ with the set
of polarizations on $W^*$ rather then on $W$.
Assume that the metric $h$ on the equivariant quaternionic vector
space $V$ is Hermitian-Hodge, and let $S \in \Lambda^2(W) \subset
\Lambda^2(V \otimes \C)$ be the corresponding polarization. Extend
$h$ to an Hermitian metric $h_I$ on the complex vector space $V$
with the preferred complex structure $V_I$, and let
$$
\omega_I \in V_I \otimes \overline{V_I} \subset \Lambda^2(V
\otimes_\R \C)
$$
be the imaginary part of the corresponding Hermitian metric on the
dual space $V_I^*$. Let $i:V_J \to \overline{V_J}$ the action of the
element $I(\sqrt{-1}) \in {\Bbb H}$. By \eqref{i.conj} we have
$$
\omega_I(a,b) = h(a,i(b)) = h(a,C\overline{b}) = S(a,b)
$$
for every $a,b \in V_J \subset V \otimes \C$. Since $\omega_I$ is
real, and $V \otimes \C = V_J \oplus \overline{V_J}$, the $2$-forms
$\omega_I$ and $S$ are related by
\begin{equation}\label{omega.and.Omega}
\omega_I = \frac{1}{2}(S + \nu(S)),
\end{equation}
where $\nu:V \otimes \C \to \overline{V \otimes \C}$ is the usual
complex conjugation.
\section{Hodge bundles and quaternionic manifolds}\label{hbqm.section}
\subsection{Hodge bundles}\label{hb.sub}
\refstepcounter{subsubsection Throughout the rest of the paper, our main tool in studying
hyperk\"ahler structures on smooth manifolds will be the equivalence
between equivariant quaternionic vector spaces and pure $\R$-Hodge
structures of weight $1$, established in Lemma~\ref{explicit.eqvs}.
In order to use it, we will need to generalize this equivalence to
the case of vector bundles over a smooth manifold $M$, rather than
just vector spaces. We will also need to consider manifolds equipped
with a smooth action of the group $U(1)$, and we would like
our generalization to take this $U(1)$-action into account.
Such a generalization requires, among other things, an appropriate
notion of a vector bundle equipped with a pure $\R$-Hodge structure.
We introduce and study one version of such a notion in this section,
under the name of a Hodge bundle (see
Definition~\ref{hodge.bundles}).
\refstepcounter{subsubsection Let $M$ be a smooth manifold equipped with a smooth
$U(1)$-action (or {\em a $U(1)$-manifold}, to simplify the
terminology), and let $\iota:M \to M$ be the action of the element
$-1 \in U(1) \subset \C^*$.
\noindent {\bf Definition.\ } \label{hodge.bundles}
An {\em Hodge bundle of weight $k$} on $M$ is a pair
$\langle\E,\overline{\ }\rangle$ of a $U(1)$-equi\-va\-ri\-ant complex
vector bundle $\E$ on $M$ and a $U(1)$-equivariant bundle map
$\overline{\ }:\overline{\iota^*\E}(k) \to \E$ satisfying $\overline{\ } \circ
\iota^*\overline{\ } = {\sf id}$.
Hodge bundles of weight $k$ over $M$ form a tensor $\R$-linear
additive category, denoted by ${{\cal W}{\cal H}odge}_k(M)$.
\refstepcounter{subsubsection\label{w.hodge}
Let $\E$, $\F$ be two Hodge bundles on $M$ of weights $m$ and
$n$. Say that a bundle map. or, more generally, a differential operator
$f:\E \to \F$ is {\em weakly Hodge} if
\begin{enumerate}
\item $f = \overline{\iota^*f}$, and
\item there exists a decomposition $f = \sum_{0 \leq n-m} f_i$ with
$f_i$ being of degree $i$ with respect to he $U(1)$-equivariant
structure. (In particular, $f=0$ unless $n \geq m$, and we always
have $f_k = \overline{\iota^* f_{n-m-k}}$.)
\end{enumerate}
Denote by ${{\cal W}{\cal H}odge}(M)$ the category of Hodge bundles of arbitrary
weight on $M$, with weakly Hodge bundle maps as morphisms. For every
$i$ the category ${{\cal W}{\cal H}odge}_i(M)$ is a full subcategory in
${{\cal W}{\cal H}odge}(M)$. Introduce also the category ${{\cal W}{\cal H}odge}^\D(M)$ with
the same objects as ${{\cal W}{\cal H}odge}(M)$ but with weakly Hodge
differential operators as morphisms. Both the categories
${{\cal W}{\cal H}odge}(M)$ and ${{\cal W}{\cal H}odge}^\D(M)$ are additive $\R$-linear tensor
categories.
\refstepcounter{subsubsection\label{H-type}
For a weakly Hodge map $f:\E \to \F$ call the canonical
decomposition
$$
f = \sum_{0 \leq i \leq m-n} f_i
$$
{\em the $H$-type decomposition}.
\label{gamma.m}
For a Hodge bundle $\E$ on $M$ of non-negative weight $k$ let
$\Gamma(\E) = \E \otimes \W_k^*$, where $\W_k$ is the canonical pure
$\R$-Hodges structure introduced in \ref{w.k}. The universal
properties of the Hodge structures $\W_k$ and Lemma~\ref{g.ex}
generalize immediately to Hodge bundles. In particular, $\Gamma$
extends to a functor $\Gamma:{{\cal W}{\cal H}odge}_{\geq 0}(M) \to
{{\cal W}{\cal H}odge}_0(M)$ adjoint on the right to the canonical embedding.
\refstepcounter{subsubsection \label{u.trivial}
If the $U(1)$-action on $M$ is trivial, then Hodge bundles of weight
$i$ are the same as real vector bundles $\E$ equipped with a Hodge
type grading $\E = \oplus_{p+q=i} \E^{p,q}$ on the complexification
$\E_\C = \E \otimes_\R \C$. In particular, if $M={\operatorname{pt}}$ is a single
point, then ${{\cal W}{\cal H}odge}(M) \cong {{\cal W}{\cal H}odge}^\D(M)$ is the category of
pure $\R$-Hodge structures. Weakly Hodge bundle map are then the
same as weakly Hodge maps of $\R$-Hodge structures considered in
\ref{weakly.hodge}. (Thus the notion of a Hodge bundle is indeed a
generalization of the notion of a pure $\R$-Hodge structure.)
\refstepcounter{subsubsection The categories of Hodge bundles are functorial in $M$, namely, for
every smooth map $f:M_1 \to M_2$ of smooth $U(1)$-manifolds $M_1$, $M_2$
there exist pull-back functors
\begin{align*}
f^*:{{\cal W}{\cal H}odge}(M_1) &\to {{\cal W}{\cal H}odge}(M_1) \\
f^*:{{\cal W}{\cal H}odge}^\D(M_1) &\to {{\cal W}{\cal H}odge}^\D(M_1).
\end{align*}
In particular, let $M$ be a smooth $U(1)$-manifold and let $\pi:M \to {\operatorname{pt}}$
be the canonical projection. Then every $\R$-Hodge structure $V$ of weight
$i$ defines a constant Hodge bundle $\pi^*V$ on $M$, which we denote for
simplicity by the same letter $V$. Thus the trivial bundle $\R =
\Lambda^0(M) = \pi^*\R(0)$ has a natural structure of a Hodge bundle of
weight $0$.
\refstepcounter{subsubsection \label{de.Rham}
To give a first example of Hodge bundles and weakly Hodge maps,
consider a $U(1)$-manifold $M$ equipped with a $U(1)$-invariant
almost complex structure $M_I$. Let $\Lambda^i(M,\C) = \oplus
\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},i-{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(M_I)$ be the usual Hodge type decomposition
of the bundles $\Lambda^i(M,\C)$ of complex valued differential
forms on $M$. The complex vector bundles $\Lambda^{p,q}(M_I)$ are
naturally $U(1)$-equivariant. Let
$$
\overline{\ }:\Lambda^{p,q}(M_I) \to \iota^*\overline{\Lambda^{q,p}(M_I)}
$$
be the usual complex conjugation, and introduce a $U(1)$-equivariant
structure on $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ by setting
$$
\Lambda^i(M,\C) = \bigoplus_{0 \leq j \leq i} \Lambda^{j,i-j}(M)(j).
$$
The bundle $\Lambda^i(M,\C)$ with these $U(1)$-equivariant structure
and complex conjugation is a Hodge bundle of weight $i$ on $M$. The
de Rham differential $d_M$ is weakly Hodge, and the $H$-type
decomposition for $d_M$ is in this case the usual Hodge type
decomposition $d = \partial + \bar\partial$.
\refstepcounter{subsubsection \noindent {\bf Remark.\ } Definition~\ref{hodge.bundles} is somewhat technical. It can be
heuristically rephrased as follows. For a complex vector bundle $\E$ on
$M$ the space of smooth global section $C^\infty(M,\E)$ is a module over the
algebra $C^\infty(M,\C)$ of smooth $\C$-valued functions on $M$, and the
bundle $\E$ is completely defined by the module $C^\infty(M,\E)$. The
$U(1)$-action on $M$ induces a representation of $U(1)$ on the algebra
$C^\infty(M,\C)$. Let $\nu:C^\infty(M,\C) \to C^\infty(M,\C)$ be
composition of the complex conjugation and the map $\iota^*:C^\infty(M,\C)
\to C^\infty(M,\C)$. The map $\nu$ is an anti-complex involution; together
with the $U(1)$-action it defines a pure $\R$-Hodge structure of weight $0$
on the algebra $C^\infty(M,\C)$. Giving a weight $i$ Hodge bundle structure
on $\E$ is then equivalent to giving a weight $i$ pure $\R$-Hodge structure
on the module $C^\infty(M,\E)$ such that the multiplication map
$$
C^\infty(M,\C) \otimes C^\infty(M,\E) \to C^\infty(M,\E)
$$
is a map of $\R$-Hodge structures.
\subsection{Equivariant quaternionic manifolds}
\refstepcounter{subsubsection We now turn to our main subject, namely, various
dif\-feren\-ti\-al\--ge\-o\-met\-ric structures on smooth manifolds
associated to actions of the algebra ${\Bbb H}$ of quaternions.
\refstepcounter{subsubsection\noindent {\bf Definition.\ } A smooth manifold $M$ is called {\em quaternionic} if it
is equipped with a smooth action of the algebra ${\Bbb H}$ on its
cotangent bundle $\Lambda^1(M)$.
Let $M$ be a quaternionic manifold. Every algebra embedding $J:\C \to {\Bbb H}$
defines by restriction an almost complex structure on the manifold
$M$. Denote this almost complex structure by $M_J$.
\refstepcounter{subsubsection Assume that the manifold $M$ is equipped with a smooth action of the
group $U(1)$, and consider the standard action of $U(1)$ on the vector
space ${\Bbb H}$. Call the quaternionic structure and the $U(1)$-action on $M$
{\em compatible} if the action map
$$
{\Bbb H} \otimes_\R \Lambda^1(M) \to \Lambda^1(M)
$$
is $U(1)$-equivariant.
Equivalently, the quaternionic structure and the $U(1)$-action are
compatible if the action preserves the almost complex structure $M_I$, and
the action map
$$
{\Bbb H} \otimes_\C \Lambda^{1,0}(M_I) \to \Lambda^{1,0}(M_I)
$$
is $U(1)$-equivariant.
\refstepcounter{subsubsection \noindent {\bf Definition.\ } A quaternionic manifold $M$ equipped with a compatible smooth
$U(1)$-action is called {\em an equivariant quaternionic manifold}.
For a $U(1)$-equivariant complex vector bundle $\E$ on $M$ denote by
$\E(k)$ the bundle $\E$ with $U(1)$-equivariant structure twisted by the
$1$-dimensional representation of weight $k$. Lemma~\ref{explicit.eqvs}
immediately gives the following.
\begin{lemma}\label{explicit.qm}
The category of quaternionic manifolds is equivalent to the category of
pairs $\langle M_I, j \rangle$ of an almost complex manifold $M_I$ and a
$\C$-linear $U(1)$-equivariant smooth map $j:\Lambda^{0,1}(M_I)(1) \to
\Lambda^{0,1}(M_I)$ satisfying $j \circ \overline{j} = -{\sf id}$.
\end{lemma}
\subsection{Quaternionic manifolds and Hodge bundles}
\refstepcounter{subsubsection Let $M$ be a smooth $U(1)$-manifold. Recall that we have
introduced in Subsection~\ref{hb.sub} a notion of a Hodge bundle on
$M$. Hodge bundles arise naturally in the study of quaternionic
structures on $M$ in the following way. Define a {\em quaternionic
bundle} on $M$ as a real vector bundle $\E$ equipped with a left
action of the algebra ${\Bbb H}$, and let $\Bun(M,{\Bbb H})$ be the category of
smooth quaternionic vector bundles on the manifold $M$. Let also
$\Bun^{U(1)}(M,{\Bbb H})$ be the category of smooth quaternionic bundles
$\E$ on $M$ equipped with a $U(1)$-equivariant structure such that
the ${\Bbb H}$-action map ${\Bbb H} \to {{\cal E}\!nd\:}(\E)$ is
$U(1)$-equivariant. Lemma~\ref{eqvs.hodge} immediately generalizes
to give the following.
\begin{lemma}\label{eqb.hodge}
The category $\Bun^{U(1)}(M,{\Bbb H})$ is equivalent to the category of Hodge
bundles of weight $1$ on $M$.
\end{lemma}
\refstepcounter{subsubsection Note that if the $U(1)$-manifold $M$ is equipped with an almost complex
structure, then the decomposition ${\Bbb H} = \overline{\C} \oplus I(\C)$ (see
\ref{h=c+c}) induces an isomorphism
$$
{\sf can}:{\Bbb H} \otimes_{I(\C)} \Lambda^{0,1} \cong \Lambda^{1,0}(M) \oplus
\Lambda^{0,1}(M) \cong \Lambda^1(M,\C).
$$
The weight $1$ Hodge bundle structure on $\Lambda^1(M,\C)$ corresponding to
the natural quaternionic structure on ${\Bbb H} \otimes_{I(\C)} \Lambda^{0,1}(M)$
is the same as the one considered in \ref{de.Rham}.
\refstepcounter{subsubsection Assume now that the smooth $U(1)$-manifold $M$ is equipped
with a compatible quaternionic structure, and let $M_I$ be the
preferred almost complex structure on $M$. Since $M_I$ is preserved
by the $U(1)$-action on $M$, the complex vector bundle
$\Lambda^{0,1}(M_I)$ of $(0,1)$-forms on $M_I$ is naturally
$U(1)$-equivariant.
The quaternionic structure on $\Lambda^1(M)$ induces by
Lemma~\ref{eqb.hodge} a weight-$1$ Hodge bundle structure on
$\Lambda^{0,1}(M_I)$. The corresponding $U(1)$-action on
$\Lambda^{0,1}(M_I)$ is induced by the action on $M_I$, and the real
structure map
$$
\overline{\ }:\Lambda^{1,0}(M_I)(1) \to \Lambda^{0,1}(M_I)
$$
is given by $\overline{\ } = \sqrt{-1} \left( \iota^* \circ j \right)$. (Here $j$
is induced by quaternionic structure, as in Lemma~\ref{explicit.qm}).
\refstepcounter{subsubsection \label{lambda_j=lambda_i}
Let $M_J$ be the complementary almost complex structure on the
equi\-va\-ri\-ant quaternionic manifold $M$. Recall that in
\ref{complementary} we have defined for every equivariant
quaternionic vector space $V$ the standard isomorphism $H:V_J \to
V_I$. This construction can be immediately generalized to give a
complex bundle isomorphism
$$
H:\Lambda^{0,1}(M_J) \to \Lambda^{0,1}(M_I).
$$
Let $P:\Lambda^1(M,\C) \to \Lambda^{0,1}(M_J)$ be the natural projection,
and let $\Mult:{\Bbb H} \otimes_{I(\C)} \Lambda^{0,1}(M_I) \to
\Lambda^{0,1}(M_I)$ be the action map. By definition the diagram
$$
\begin{CD}
\Lambda^1(M,\C) @>{{\sf can}}>> {\Bbb H} \otimes_{I(\C)} \Lambda^{0,1}(M_I) \\
@V{P}VV @V{\Mult}VV \\
\Lambda^{0,1}(M_J) @>{H}>> \Lambda^{0,1}(M_I)
\end{CD}
$$
is commutative. Since the map $\Mult$ is compatible with the Hodge bundle
structures, so is the projection $P$.
\noindent {\bf Remark.\ } This may seems paradoxical, since the complex conjugation
$\overline{\ }$ on $\Lambda^1(M,\C)$ does not preserve $\Ker P =
\Lambda^{0,1}(M_J)$. However, under our definition of a Hodge bundle
the conjugation on $\Lambda^1(M,\C)$ is $\iota^* \circ \overline{\ }$ rather
than $\overline{\ }$. Both $\overline{\ }$ and $\iota^*$ interchange
$\Lambda^{1,0}(M_J)$ and $\Lambda^{0,1}(M_J)$.
\refstepcounter{subsubsection The standard isomorphism $H:\Lambda^{0,1}(M_J) \to
\Lambda^{0,1}(M_I)$ does not commute with the Dolbeault differentials. They
are, however, related by means of the Hodge bundle structure on
$\Lambda^{0,1}(M_I)$. Namely, we have the following.
\begin{lemma}\label{d=d_0+d_0}
The Dolbeault differential $D:\Lambda^0(M,\C) \to \Lambda^{0,1}(M_I)$ for
the almost complex structure $M_J$ is weakly Hodge. The $U(1)$-invariant
component $D_0$ in the $H$-type decomposition $D = D_0 + \overline{D_0}$ of
the map $D$ coincides with the Dolbeault differential for the almost complex
structure $M_I$.
\end{lemma}
\proof The differential $D$ is the composition $D = P \circ d_M$ of the de
Rham differential $d_M:\Lambda^0(M,\C) \to \Lambda^1(M,\C)$ with the
canonical projection $P$. Since both $P$ and $d_M$ are weakly Hodge, so is
$D$. The rest follows from the construction of the standard isomorphism
$H$.
\hfill \ensuremath{\square}\par
\subsection{Holonomic derivations}
\refstepcounter{subsubsection Let $M$ be a smooth $U(1)$-manifold. In order to give a
Hodge-theoretic description of the set of all equivariant quaternionic
structures on $M$, it is convenient to work not with various complex
structures on $M$, but with associated Dolbeault differentials. To do this,
recall the following universal property of the cotangent bundle
$\Lambda^1(M)$.
\begin{lemma}\label{universal}
Let $M$ be a smooth manifold, and let $\E$ be a complex vector bundle on
$M$. Every differential operator $\partial:\Lambda^0(M,\C) \to \E$ which is a
derivation with respect to the algebra structure on $\Lambda^0(M,\C)$
factors uniquely through the de Rham differential $d_M:\Lambda^0(M,\C) \to
\Lambda^1(M,\C)$ by means of a bundle map $P:\Lambda^1(M,\C) \to \E$.
\end{lemma}
\refstepcounter{subsubsection \label{holonomic}
We first use this universal property to describe almost complex structures.
Let $M$ be a smooth manifold equipped with a complex vector bundle $\E$.
\noindent {\bf Definition.\ } A derivation $D:\Lambda^0(M,\C) \to \E$ is called {\em holonomic} if
the associated bundle map $P:\Lambda^1(M,\C) \to \E$ induces an isomorphism
of the subbundle $\Lambda^1(M,\R) \subset \Lambda^1(M,\C)$ of real
$1$-forms with the real vector bundle underlying $\E$.
By Lemma~\ref{universal} the correspondence
$$
M_I \mapsto \left\langle \Lambda^{0,1}(M_I), \bar\partial \right\rangle
$$
identifies the set of all almost complex structures $M_I$ on $M$
with the set of all pairs $\langle \E, D \rangle$ of
a complex vector bundle $\E$ and a holonomic derivation $D:\Lambda^0(M,\C)
\to \E$.
\refstepcounter{subsubsection Assume now that the smooth manifold $M$ is equipped with smooth
action of the group $U(1)$. Then we have the following.
\begin{lemma}\label{qm.hodge}
Let $\E$ be a weight $1$ Hodge bundle on the smooth $U(1)$-manifold
$M$, and let
$$
D:\Lambda^0(M,\C) \to \E
$$
be a weakly Hodge holonomic derivation. There exists a unique
$U(1)$\--equi\-va\-ri\-ant quaternionic structure on $M$ such that
$\E \cong \Lambda^{0,1}(M_J)$ and $D$ is the Dolbeault differential
for the complementary almost complex structures $M_J$ on $M$.
\end{lemma}
\proof Since the derivation $M$ is holonomic, it induces an almost
complex structure $M_J$ on $M$. To construct an almost complex
structure $M_I$ complementary to $M_J$, consider the $H$-type
decomposition $D = D_0 + \overline{D_0}$ of the derivation
$D:\Lambda^0(M,\C) \to \E$ (defined in \ref{H-type}).
The map $D_0$ is also a derivation. Moreover, it is holonomic. Indeed, by
dimension count it is enough to prove that the associated bundle map
$P:\Lambda^1(M,\R) \to \E$ is injective. Since the bundle $\Lambda^1(M,\R)$
is generated by exact $1$-forms, it is enough to prove that any real valued
function $f$ on $M$ with $D_0f = 0$ is constant. However, since $D$ is
weakly Hodge,
$$
Df = D_0 f + \overline{D_0} f = D_0 f + \overline{ D_0 \overline{f} } = D_0
f + \overline{ D_0 f},
$$
hence $D_0f = 0$ if and only if $Df=0$. Since $D$ is holonomic, $f$ is indeed
constant.
The derivation $D_0$, being holonomic, is the Dolbeault differential
for an almost complex structure $M_I$ on $M$. Since $D_0$ is by
definition $U(1)$-equivariant, the almost complex structure $M_I$ is
$U(1)$-invariant. Moreover, $\E \cong \Lambda^{0,1}(M_I)$ as
$U(1)$-equivariant complex vector bundles. By Lemma~\ref{eqb.hodge}
the weight $1$ Hodge bundle structure on $\E$ induces an equivariant
quaternionic bundle structure on $\E$ and, in turn, a structure of an
equivariant quaternionic manifold on $M$. The almost complex structure
$M_I$ coincides by definition with the preferred almost complex
structure.
It remains to notice that by Lemma~\ref{d=d_0+d_0} the Dolbeault differential
$\bar\partial_J$ for the complementary almost complex structure on $M$ indeed equals
$D = D_0 + \overline{D_0}$.
\hfill \ensuremath{\square}\par
Together with Lemma~\ref{d=d_0+d_0}, this shows that the set of
equivariant quaternionic structures on the $U(1)$-manifold $M$ is
naturally bijective to the set of pairs $\langle \E, D \rangle$ of a
weight $1$ Hodge bundle $\E$ on $M$ and a weakly Hodge holonomic
derivation $D:\Lambda^0(M,\C) \to \E$.
\section{Hodge manifolds}
\subsection{Integrability}
\refstepcounter{subsubsection There exists a notion of integrability for quaternionic manifolds
analogous to that for the almost complex ones.
\noindent {\bf Definition.\ } A quaternionic manifold $M$ is called {\em hypercomplex} if for two
complementary algebra embeddings $I,J:\C \to {\Bbb H}$ the almost complex
structures $M_I,M_J$ are integrable.
In fact, if $M$ is hypercomplex, then $M_I$ is integrable for any algebra
embedding $I:\C \to {\Bbb H}$. For a proof see, e.g., \cite{K}.
\refstepcounter{subsubsection When a quaternionic manifold $M$ is equipped with a compatible
$U(1)$-action, there exist a natural choice for a pair of almost complex
structures on $M$, namely, the preferred and the complementary one.
\noindent {\bf Definition.\ } An equivariant quaternionic manifold $M$ is called a {\em Hodge
manifold} if both the preferred and the equivariant almost complex
structures $M_I$, $M_J$ are integrable.
Hodge manifolds are the main object of study in this paper.
\refstepcounter{subsubsection \label{standard}
There exists a simple Hodge-theoretic description of
Hodge manifolds based on Lemma~\ref{qm.hodge}. To give it (see
Proposition~\ref{explicit.hodge}), consider an equivariant quaternionic
manifold $M$, and let $M_J$ and $M_I$ be the complementary and the
preferred complex structures on $M$. The weight $1$ Hodge bundle structure
on $\Lambda^{0,1}(M_J)$ induces a weight $i$ Hodge bundle structure on the
bundle $\Lambda^{0,i}(M_J)$ of $(0,i)$-forms on $M_J$. The standard
identification $H:\Lambda^{0,1}(M_J) \to \Lambda^{0,1}(M_I)$ given in
\ref{lambda_j=lambda_i} extends uniquely to an algebra isomorphism
$H:\Lambda^{0,i}(M_J) \to \Lambda^{0,i}(M_I)$.
Let $D:\Lambda^{0{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(M_J) \to \Lambda^{0,{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M_J)$ be the Dolbeault
differential for the almost complex manifold $M_J$.
\begin{lemma}\label{yet.another.lemma}
The equivariant quaternionic manifold $M$ is Hodge if and only if
the following holds.
\begin{enumerate}
\item $M_J$ is integrable, that is, $D \circ D = 0$, and
\item the differential $D:\Lambda^{0,i}(M_J) \to \Lambda^{0,i+1}(M_J)$
is weakly Hodge for every $i \geq 0$.
\end{enumerate}
\end{lemma}
\proof Assume first that the conditions \thetag{i}, \thetag{ii}
hold. Condition \thetag{i} means that the complementary almost complex
structure $M_J$ is integrable. By \thetag{ii} the map $D$ is weakly Hodge.
Let $D = D_0 + \overline{D_0}$ be the $H$-type decomposition. The
map $D_0$ is an algebra derivation of
$\Lambda^{0,{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(M_I)$. Moreover, by Lemma~\ref{d=d_0+d_0} the map
$D_0:\Lambda^0(M,\C) \to \Lambda^{0,1}(M_J)$ is the Dolbeault
differential $\bar\partial_I$ for the preferred almost complex structure
$M_I$ on $M$. (Or, more precisely, is identified with $\bar\partial_I$
under the standard isomorphism $H$.) But the Dolbeault differential
admits at most one extension to a derivation of the algebra
$\Lambda^{0,{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(M_J)$. Therefore $D_0 = \bar\partial_I$ everywhere.
The composition $D_0 \circ D_0$ is the $(2,0)$-component in the
$H$-type decomposition of the map $D \circ D$. Since $D \circ D = 0$,
$$
D_0 \circ D_0 = \bar\partial_I \circ \bar\partial_I = 0.
$$
Therefore the preferred complex structure $M_I$ is also integrable, and the
manifold $M$ is indeed Hodge.
Assume now that $M$ is Hodge. The canonical projection $P:\Lambda^1(M,\C)
\to \Lambda^{0,1}(M_J)$ extends then to a multiplicative projection
$$
P:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \Lambda^{0,{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(M_J)
$$
from the de Rham complex of the complex manifold $M_I$ to the Dolbeault
complex of the complex manifold $M_J$.
The map $P$ is surjective and weakly Hodge, moreover, it
commutes with the differentials. Since the de Rham
differential preserves the pre-Hodge structures, so does the Dolbeault
differential $D$.
\hfill \ensuremath{\square}\par
\refstepcounter{subsubsection Lemma~\ref{yet.another.lemma} and Lemma~\ref{qm.hodge} together
immediately give the following.
\begin{prop}\label{explicit.hodge}
The category of Hodge manifolds is equivalent to the category of triples
$\langle M, \E, D \rangle$ of a smooth $U(1)$-manifold $M$, a weight $1$
Hodge bundle $\E$ on $M$, and a weakly Hodge algebra derivation
$$
D = D^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(\E) \to \Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(\E)
$$
such that $D \circ D = 0$, and the first component
$$
D^0:\Lambda^0(M,\C) = \Lambda^0(\E) \to \E = \Lambda^1(\E)
$$
is holonomic in the sense of \ref{holonomic}.
\end{prop}
\subsection{The de Rham complex of a Hodge manifold}
\refstepcounter{subsubsection Let $M$ be a Hodge manifold. In this subsection we study in some
detail the de Rham complex $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ of the manifold $M$, in
order to obtain information necessary for the discussion of metrics on $M$
given in the Subsection~\ref{polarization}. The reader is advised to skip
this subsection until needed.
\refstepcounter{subsubsection Let $\Lambda^{0,{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(M_J)$ be the Dolbeault complex for the
complementary complex structure $M_J$ on $M$. By
Proposition~\ref{explicit.hodge} the complex vector bundle
$\Lambda^{0,i}(M_J)$ is a Hodge bundle of weight $i$ on $M$, and the
Dolbeault differential $D:\Lambda^{0,{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(M_J) \to
\Lambda^{0,{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M_J)$ is weakly Hodge. Therefore $D$ admits an $H$-type
decomposition $D = D_0 + \overline{D_0}$.
\refstepcounter{subsubsection Consider the de Rham complex $\Lambda^i(M,\C)$ of the smooth
manifold $M$. Let $\Lambda^i(M,\C) = \oplus_{p+q}
\Lambda^{p,q}(M_J)$ be the Hodge type decomposition for the
complementary complex structure $M_J$ on $M$, and let
$\nu:\Lambda^{p,q}(M_J) \to \overline{\Lambda^{q,p}(M_J)}$ be the
usual complex conjugation. Denote also
$$
f^\nu = \nu \circ f \circ \nu^{-1}
$$
for any map $f:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$.
Let $d_M:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C)$ be the de
Rham differential, and let $d_M = D + D^\nu$ be the Hodge type
decomposition for the complementary complex structure $M_J$ on $M$.
Since the Dolbeault differential, in turn, equals $D = D_0 +
\overline{D_0}$, we have
$$
d_M = D_0 + \overline{D_0} + D_0^\nu + \overline{D_0}^\nu.
$$
\refstepcounter{subsubsection \label{6_I}
Let $\bar\partial_I:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C)$ be the
Dolbeault differential for the preferred complex structure $M_I$ on $M$. As
shown in the proof of Lemma~\ref{yet.another.lemma}, the $(0,1)$-component
of the differential $\bar\partial_I$ with respect to the complex structure $M_J$
equals $D_0$. Therefore the $(1,0)$-component of the complex-conjugate map
$\partial_I = \bar\partial_I^\nu$ equals $D_0^\nu$. Since $d_M = \bar\partial_I + \partial_I$, we
have
\begin{align*}
\bar\partial_I &= D_0 + \overline{D_0}^\nu\\
\partial_I &= \overline{D_0} + D_0^\nu
\end{align*}
\refstepcounter{subsubsection \label{6_I^H}
The standard isomorphism $H:\Lambda^{0,1}(M_J) \to \Lambda^{0,1}(M_I)$
introduced in \ref{standard} extends uniquely to a bigraded algebra
isomorphism $H:\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(M_J) \to
\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(M_I)$. By definition of the map $H$, on
$\Lambda^0(M,\C)$ we have
\begin{align}\label{identities}
\begin{split}
\bar\partial_I &= H \circ D_0 \circ H^{-1}\\
\partial_I &= H \circ D_0^\nu \circ H^{-1}\\
d_M &= \partial_I + \bar\partial_I = H \circ (D_0 + D_0^\nu) H^{-1}.
\end{split}
\end{align}
The right hand side of the last identity is the algebra derivation
of the de Rham complex $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$. Therefore, by
Lemma~\ref{universal} it holds not only on $\Lambda^0(M,\C)$, but on
the whole $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$. The Hodge type decomposition for
the preferred complex structure $M_I$ then shows that the first two
identities also hold on the whole de Rham complex
$\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$.
\refstepcounter{subsubsection Let now $\xi = I(\sqrt{-1}):\Lambda^{0,1}(M_J) \to
\Lambda^{1,0}(M_J)$ be the operator corresponding to the preferred
almost complex structure $M_I$ on $M$. Let also $\xi = 0$ on
$\Lambda^0(M,\C)$ and $\Lambda^{1,0}(M_J)$, and extend $\xi$ to a
derivation $\xi:\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(M_J) \to
\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}-1,{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M_J)$ by the Leibnitz rule. We finish this
subsection with the following simple fact.
\begin{lemma}\label{xi.lemma}
On $\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},0}(M_J) \subset \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ we have
\begin{align}\label{xi.eq}
\begin{split}
\xi \circ D_0 + D_0 \circ \xi &= \overline{D}_0^\nu\\
\xi \circ \overline{D}_0 + \overline{D}_0 \circ \xi &= -D_0^\nu.
\end{split}
\end{align}
\end{lemma}
\proof It is easy to check that both identities hold on
$\Lambda^0(M,\C)$. But both sides of these identities are algebra
derivations of $\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},0}(M_J)$, and the right hand sides
are holonomic in the sense of \ref{holonomic}. Therefore by
Lemma~\ref{universal} both identities hold on the whole
$\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},0}(M_J)$.
\hfill \ensuremath{\square}\par
\subsection{Polarized Hodge manifolds}\label{polarization}
\refstepcounter{subsubsection Let $M$ be a quaternionic manifold. A Riemannian metric $h$ on $M$
is called {\em Qua\-ter\-ni\-onic-\-Her\-mi\-ti\-an} if for every point $m
\in M$ the induces metric $h_m$ on the tangent bundle $T_mM$ is
Qua\-ter\-ni\-onic\--Her\-mi\-ti\-an in the sense of \ref{hyperherm}.
\noindent {\bf Definition.\ } A {\em hyperk\"ahler manifold} is a hypercomplex manifold $M$
equipped with a Quaternionic-Hermitian metric $h$ which is K\"ahler for
both integrable almost complex structures $M_I$, $M_J$ on $M$.
\noindent {\bf Remark.\ } In the usual definition (see, e.g., \cite{Bes}) the integrability of
the almost complex structures $M_I$, $M_J$ is omitted, since it is
automatically implied by the K\"ahler condition.
\refstepcounter{subsubsection Let $M$ be a Hodge manifold equipped with a Riemannian metric $h$.
The metric $h$ is called {\em Hermitian-Hodge} if it is
Quaternionic-Hermitian and, in addition, invariant under the $U(1)$-action
on $M$.
\noindent {\bf Definition.\ } Say that the manifold $M$ is {\em polarized} by the Hermitian-Hodge
metric $h$ if $h$ is not only Quaternionic-Hermitian, but also hyperk\"ahler.
\refstepcounter{subsubsection \label{positive} Let $M$ be a Hodge manifold. By
Proposition~\ref{explicit.hodge} the holomorphic cotangent bundle
$\Lambda^{1,0}(M_J)$ for the complementary complex structure $M_J$ on $M$
is a Hodge bundle of weight $1$. The holomorphic tangent bundle
$\Theta(M_J)$ is therefore a Hodge bundle of weight $-1$. By \ref{pol} the
set of all Hermitian-Hodge metrics $h$ on $M$ is in natural one-to-one
correspondence with the set of all polarizations on the Hodge bundle
$\Theta(M_J)$.
Since $\theta(M)$ is of odd weight, its polarizations are
skew-symmetric as bilinear forms and correspond therefore to smooth
$(2,0)$-forms on the complex manifold $M_J$. A $(2,0)$-form $\Omega$
defines a polarization on $\Theta(M_J)$ if and only if
$\Omega \in C^\infty(M,\Lambda^{2,0}(M_J))$
considered as a map
$$
\Omega:\R(-1) \to \Lambda^{2,0}(M_J)
$$
is a map of weight $2$ Hodge bundles, and for an arbitrary smooth
section $\chi \in C^\infty(M,\Theta(M_J))$ we have
\begin{equation}\label{P}
\Omega(\chi,\overline{\iota^*(\chi)}) > 0.
\end{equation}
\refstepcounter{subsubsection Assume that the Hodge manifold $M$ is equipped with an
Hermitian-Hodge metric $h$. Let $\Omega \in
C^\infty(M,\Lambda^{2,0}(M_J))$ be the corresponding polarization,
and let $\omega_I \in C^\infty(M,\Lambda^{1,1}(M_I))$ be the
$(1,1)$-form on the complex manifold $M_I$ associated to the
Hermitian metric $h$. Either one of the forms $\Omega$, $\omega_I$
completely defines the metric $h$, and by \eqref{omega.and.Omega} we
have
$$
\Omega + \nu(\Omega) = \omega_I,
$$
where $\nu:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ is the complex
conjugation.
\begin{lemma}\label{pol.hm}
The Hermitian-Hodge metric $h$ polarizes $M$ if and only if the
corresponding $(2,0)$-form $\Omega$ on $M_J$ is holomorphic, that is,
$$
D\Omega = 0,
$$
where $D$ is the Dolbeault differential for complementary complex structure
$M_J$.
\end{lemma}
\proof Let $\omega_I,\omega_J \in \Lambda^2(M,\C)$ be the K\"ahler forms
for the metric $h$ and complex structures $M_I$, $M_J$ on $M$. The metric
$h$ is hyperk\"ahler, hence polarizes $M$, if and only if $d_M\omega_I =
d_M\omega_J = 0$.
Let $D = D_0 + \overline{D_0}$ be the $H$-type decomposition and
let $H:\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(M_J) \to \Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(M_J)$ be the
standard algebra identification introduced in \ref{6_I^H}.
By definition $H(\omega_I) = \omega_J$. Moreover, by
\eqref{identities} $H^{-1} \circ d_M \circ H = D_0 +
\overline{D_0}^\nu$, hence
$$
H(d_M\omega_J) = D_0 \omega_I + \overline{D_0}^\nu\omega_I,
$$
and the metric $h$ is hyperk\"ahler if and only if
\begin{equation}\label{K}
d_M\omega_I = (D_0+\overline{D_0}^\nu)\omega_I = 0
\end{equation}
But $2\omega_I = \Omega + \nu(\Omega)$. Since $\Omega$ is of Hodge
type $(2,0)$ with respect to the complementary complex structure
$M_J$, \eqref{K} is equivalent to
$$
\overline{D_0} \Omega = D_0 \Omega = \overline{D_0}^\nu \Omega =
D_0^\nu \Omega = 0.
$$
Moreover, let $\xi$ be as in Lemma~\ref{xi.lemma}. Then $\xi(\Omega)
= 0$, and by \eqref{xi.eq} $D_0\Omega = \overline{D}_0\Omega = 0$
implies that $D_0^\nu\Omega = \overline{D}_0^\nu\Omega = 0$ as well.
It remains to notice that since the metric $h$ is Hermitian-Hodge,
$\Omega$ is of $H$-type $(1,1)$ as a section of the weight $2$ Hodge
bundle $\Lambda^{2,0}(M_J)$. Therefore $D\Omega = 0$ implies
$\overline{D_0}\Omega = D_0\Omega = 0$, and this proves the lemma.
\hfill \ensuremath{\square}\par
\noindent {\bf Remark.\ } This statement is wrong for general hyperk\"ahler manifolds
(eve\-ry\-thing in the given proof carries over, except for the
implication $D\Omega=0 \Rightarrow
D_0\Omega=\overline{D_0}\Omega=\overline{D_0}^\nu \Omega = D_0^\nu
\Omega = 0$, which depends substantially on the $U(1)$-action). To
describe general hyperk\"ahler metrics in holomorphic terms, one has
to introduce the so-called {\em twistor spaces} (see, e.g.,
\cite{HKLR}).
\section{Regular Hodge manifolds}
\subsection{Regular stable points}
\refstepcounter{subsubsection Let $M$ be a smooth manifold equipped with a smooth
$U(1)$-action with differential $\phi_M$ (thus $\phi_M$ is a smooth
vector field on $M$). Since the group $U(1)$ is compact, the subset
$M^{U(1)} \subset M$ of points fixed under $U(1)$ is a smooth
submanifold.
Let $m \in M^{U(1)} \subset M$ be a point fixed under $U(1)$. Consider the
representation of $U(1)$ on the tangent space $T_m$ to $M$ at $m$. Call
the fixed point $m$ {\em regular} if every irreducible subrepresentation of
$T_m$ is either trivial or isomorphic to the representation on $\C$ given
by embedding $U(1) \subset \C^*$. (Here $\C$ is considered as a
$2$-dimensional real vector space.) Regular fixed points form a union of
connected component of the smooth submanifold $M^{U(1)} \subset M$.
\refstepcounter{subsubsection Assume that $M$ is equipped with a complex structure preserved by
the $U(1)$-action. Call a point $m \in M$ {\em stable} if for any $t \in
\R, t \geq 0$ there exists $\exp(\sqrt{-1}t\phi_M)m$, and the limit
$$
m_0 \in M, m_0 = \lim_{t \to +\infty} \exp(\sqrt{-1}t\phi_M)m
$$
also exists.
\enlargethispage{10mm}
\refstepcounter{subsubsection For every stable point $m \in M$ the limit $m_0$ is obviously fixed
under $U(1)$. Call a point $m \in M$ {\em regular stable} if it is stable
and the limit $m_0 \in M^{U(1)}$ is a regular fixed point.
Denote by $M^{reg} \subset M$ the subset of all regular stable points. The
subset $M^{reg}$ is open in $M$.
\noindent {\bf Example.\ } Let $Y$ be a complex manifold with a holomorphic bundle $\E$ and let
$E$ be the total space of $\E$. Let $\C^*$ act on $M$ by dilatation along the
fibers. Then every point $e \in E$ is regular stable.
\refstepcounter{subsubsection Let $M$ be a Hodge manifold. Recall that the $U(1)$-action on $M$
preserves the preferred complex structure $M_I$.
\noindent {\bf Definition.\ } A Hodge manifold $M$ is called {\em regular} if $M_I^{reg} = M_I$.
\subsection{Linearization of regular Hodge manifolds}
\refstepcounter{subsubsection Consider a regular Hodge manifold $M$. Let $\Delta \subset
\C$ be the unit disk equipped with the multiplicative semigroup
structure. The group $U(1) \subset \Delta$ is embedded into $\Delta$
as the boundary circle.
\begin{lemma}\label{regular}
The action $a:U(1) \times M \to M$ extends uniquely to a holomorphic
action $\tilde{a}:\Delta \times M_I \to M_I$. Moreover, for every $x
\in \Delta \setminus \{0\}$ the action map $\wt{a}(x):M_I \to M_I$
is an open embedding.
\end{lemma}
\proof Since $M$ is regular, the exponential flow $\exp(it\phi_M)$
of the differential $\phi_M$ of the action is defined for all
positive $t \in \R$. Therefore $a:U(1) \times M \to M$ extends
uniquely to a holomorphic action
$$
\tilde{a}:\Delta^* \times M_I \to M_I,
$$
where $\Delta^* = \Delta \backslash \{0\}$ is the punctured
disk. Moreover, the exponential flow converges as $t \to +\infty$,
therefore $\tilde{a}$ extends to $\Delta \times M_I$
continuously. Since this extension is holomorphic on a dense open
subset, it is holomorphic everywhere. This proves the first claim.
To prove the second claim, consider the subset $\wt{\Delta} \subset
\Delta^*$ of points $x \in \Delta$ such that $\wt{a}(x)$ is
injective and \'etale. The subset $\wt{\Delta}$ is closed under
multiplication and contains the unit circle $U(1) \subset
\Delta^*$. Therefore to prove that $\wt{\Delta} = \Delta^*$, it
suffices to prove that $\wt{\Delta}$ contains the interval $]0,1]
\subset \Delta^*$.
By definition we have $\wt{a}(h) = \exp(-\sqrt{-1}\log h \phi_M)$
for every $h \in ]0,1] \subset \Delta^*$. Thus we have to prove that
if for some $t \in \R, t \geq 0$ and for two points $m_1,m_2 \in M$
we have
$$
\exp(\sqrt{-1}t\phi_M)(m_1) = \exp(\sqrt{-1}t\phi_M)(m_2),
$$
then $m_1 = m_2$. Let $m_1$, $m_2$ be such two points and let
$$
t = \inf\{t \in \R, t \geq 0, \exp(\sqrt{-1}t\phi_M)(m_1) =
\exp(\sqrt{-1}t\phi_M)(m_2)\}.
$$
If the point $m_0 = \exp(\sqrt{-1}t\phi_M)(m_1) =
\exp(\sqrt{-1}t\phi_M)(m_2) \in M$ is not $U(1)$-invariant, then it
is a regular point for the vector field $\sqrt{-1}\phi_M$, and by
the theory of ordinary differential equations we have $t = 0$ and
$m_1 = m_2 = m_0$.
Assume therefore that $m_0 \in M^{U(1)}$ is $U(1)$-invariant. Since
the group $U(1)$ is compact, the vector field $\sqrt{-1}\phi_M$ has
only a simple zero at $m_0 \subset M^{U(1)} \subset M$. Therefore
$m_0 = \exp(\sqrt{-1}t\phi_M)m_1$ implies that the point $m_1 \in M$
also is $U(1)$-invariant, and the same is true for the point $m_2
\in M$. But $\wt{a}(\exp t)$ acts by identity on $M^{U(1)} \subset
M$. Therefore in this case we also have $m_1=m_2=m_0$.
\hfill \ensuremath{\square}\par
\refstepcounter{subsubsection Let $V = M_I^{U(1)} \subset M_I$ be the submanifold of fixed points
of the $U(1)$ action. Since the action preserves the complex structure on
$M_I$, the submanifold $V$ is complex.
\begin{lemma}
There exists a unique $U(1)$-invariant holomorphic map
$$
\rho_M:M_I \to V
$$
such that $\rho_M|_{V} = {\sf id}$.
\end{lemma}
\proof For every point $m \in M$ we must have $\rho_M(m) =
\displaystyle\lim_{t \to +\infty} \exp(i t \phi_M)$, which proves uniqueness.
To prove that $\rho_M$ thus defined is indeed holomorphic, notice that the
diagram
$$
\begin{CD}
M_I @>{0 \times {\sf id}}>> \Delta \times M\\
@V{\rho_M}VV @VV{\tilde{a}}V\\
V @>>> M_I
\end{CD}
$$
is commutative. Since the action $\tilde{a}:\Delta \times M_I \to M_I$ is
holomorphic, so is the map $\rho_M$.
\hfill \ensuremath{\square}\par
\refstepcounter{subsubsection Call the canonical map $\rho_M:M_I \to M_I^{U(1)}$ the {\em canonical
projection} of the regular Hodge manifold $M$ onto the submanifold $V \subset
M$ of fixed points.
\begin{lemma}
The canonical projection $\rho_M:M \to M^{U(1)}$ is submersive, that
is, for every point $m \in M$ the differential $d\rho_M:T_mM \to
T_{\rho(m)}M^{U(1)}$ of the map $\rho_M$ at $m$ is surjective.
\end{lemma}
\proof Since $\rho_M|_{M^{U(1)}} = {\sf id}$, the differential $d\rho_M$
is surjective at points $m \in V \subset M$. Therefore it is
surjective on an open neighborhood $U \supset V$ of $V$ in $M$. For
any point $m \in M$ there exists a point $x \in \Delta$ such that $x
{\:\raisebox{3pt}{\text{\circle*{1.5}}}} m \in U$. Since $\rho_M$ is $\Delta$-invariant, this implies that
$d\rho_M$ is surjective everywhere on $M$.
\hfill \ensuremath{\square}\par
\refstepcounter{subsubsection Let $\Theta(M/V)$ be the relative tangent bundle of the holomorphic
map $\rho:M \to V$. Let $\Theta(M)$ and $\Theta(V)$ be the tangent bundles
of $M$ and $V$ and consider the canonical exact sequence of complex bundles
$$
0 \longrightarrow \Theta(M/V) \longrightarrow \Theta(M) \overset{d\rho_M}{\longrightarrow}
\rho^*\Theta(V) \longrightarrow 0,
$$
where $d\rho_M$ is the differential of the projection $\rho_M:M \to
V$.
The quaternionic structure on $M$ defines a $\C$-linear map
$j:\Theta(M) \to \overline{\Theta}(M)$. Restricting to $\Theta(M/V)$ and
composing with $d\rho_M$, we obtain a $\C$-linear map $j:\Theta(M/V)
\to \rho^*\overline{\Theta}(V)$.
\refstepcounter{subsubsection \label{overline.T}
Let ${\overline{T}V}$ be the total space of the bundle $\overline{\Theta}(V)$
complex-conjugate to the tangent bundle $\Theta(V)$, and let
$\rho:{\overline{T}V} \to V$ be the projection. Let the group $U(1)$ act on
${\overline{T}V}$ by dilatation along the fibers of the projection $\rho$.
Since the canonical projection $\rho_M:M \to V$ is $U(1)$-invariant,
the differential $\phi_M$ of the $U(1)$-action defines a section
$$
\phi_M \in C^\infty(M,\Theta(V/M)).
$$
The section $j(\phi_M) \in C^\infty(M,\rho_M^*\overline{\Theta}(V))$
defines a map $\Lin_M:M \to {\overline{T}V}$ such that $\Lin_M \circ \rho =
\rho_M:M \to V$. Call the map $\Lin_M$ {\em the linearization} of the
regular Hodge manifold $M$.
\begin{prop}\label{linearization}
The linearization map $\Lin_M$ is a $U(1)$-equivariant open embedding.
\end{prop}
\proof The map $j:\Theta(M/V) \to \rho^*\overline{\Theta}(V)$ is of
degree $1$ with respect to the $U(1)$-action, while the section
$\phi_M \in C^\infty(M,\Theta(M/V))$ is $U(1)$-invariant. Therefore
the map $\Lin_M$ is $U(1)$-equivariant.
Consider the differential $d\Lin_M:T_m(M) \to T_m({\overline{T}V})$ at a point $m \in V
\subset M$. We have
$$
d\Lin_M = d\rho_M \oplus d\rho_M \circ j:T_m(M) \to T_m(V) \oplus
\overline{T}_m(V)
$$
with respect to the decomposition $T_m({\overline{T}V}) = T_m(V) \oplus
\overline{T}(V)$. The tangent space $T_m$ is a regular quaternionic vector
space. Therefore the map $d\Lin_M$ is bijective at $m$ by
Lemma~\ref{regular.quaternionic}. Since $\Lin_M$ is bijective on $V$, this
implies that $\Lin_M$ is an open embedding on an open neighborhood $U
\subset M$ of the submanifold $V \subset M$.
To finish the proof of proposition, it suffices prove that the
linearization map $\Lin_M:M_I \to {\overline{T}V}$ is injective and \'etale on
the whole $M_I$. To prove injectivity, consider arbitrary two points
$m_1,m_2 \in M_I$ such that $\Lin_M(m_1)=\Lin_M(m_2)$. There exists
a point $x \in \Delta \setminus \{0\}$ such that $x {\:\raisebox{3pt}{\text{\circle*{1.5}}}} m_1, x
{\:\raisebox{3pt}{\text{\circle*{1.5}}}} m_2 \in U$. The map $\Lin_M$ is $U(1)$-equivariant and
holomorphic, therefore it is $\Delta$-equivariant, and we have
$$
\Lin_M(x {\:\raisebox{3pt}{\text{\circle*{1.5}}}} m_1) = x {\:\raisebox{3pt}{\text{\circle*{1.5}}}} \Lin_M(m_1) = x {\:\raisebox{3pt}{\text{\circle*{1.5}}}} \Lin_M(m_2) =
\Lin_M(x {\:\raisebox{3pt}{\text{\circle*{1.5}}}} m_2).
$$
Since the map $\Lin_M:U \to {\overline{T}V}$ is injective, this implies that $x
{\:\raisebox{3pt}{\text{\circle*{1.5}}}} m_1 = x {\:\raisebox{3pt}{\text{\circle*{1.5}}}} m_2$. By Lemma~\ref{regular} the action map
$x:M_I \to M_I$ is injective. Therefore this is possible only if
$m_1 = m_2$, which proves injectivity.
To prove that the linearization map is \'etale, note that by
Lemma~\ref{regular} the action map $x:M_I \to M_I$ is not only
injective, but also \'etale. Since $\Lin_M$ is \"etale on $U$, so is
the composition $\Lin_M \circ x:M_I \to U \to {\overline{T}V}$ is \'etale. Since
$\Lin_M \circ x = x \circ \Lin_M$, the map $\Lin_M:M_I \to {\overline{T}V}$ is
\'etale at the point $m_1 \in M_I$.
Thus the linearization map is also injective and \'etale on the
whole of $M_I$. Hence it is indeed an open embedding, which proves
the propostion.
\hfill \ensuremath{\square}\par
\subsection{Linear Hodge manifold structures}
\refstepcounter{subsubsection By Proposition~\ref{linearization} every regular Hodge manifold $M$
admits a canonical open embedding $\Lin_M:M \to {\overline{T}V}$ into the total space
${\overline{T}V}$ of the (complex-conjugate) tangent bundle to its fixed points
submanifold $V \subset M$. This embedding induces a Hodge manifold
structure on a neighborhood of the zero section $V \subset {\overline{T}V}$.
In order to use the linearization construction, we will need a
characterization of all Hodge manifold structures on neighborhoods of $V
\subset {\overline{T}V}$ obtained in this way (see \ref{lin.def}). It is convenient to
begin with an invariant characterization of the linearization map $\Lin_M:M
\to {\overline{T}V}$.
\refstepcounter{subsubsection \label{tau}
Let $V$ be an arbitrary complex manifold, let ${\overline{T}V}$ be the total space
of the complex-conjugate to the tangent bundle $\Theta(V)$ to
$V$, and let $\rho:{\overline{T}V} \to V$ be the canonical projection. Contraction with
the tautological section of the bundle $\rho^*\overline{\Theta(V)}$ defines
for every $p$ a bundle map
$$
\tau:\rho^*\Lambda^{p+1}(V,\C) \to \rho^*\Lambda^{p}(V,\C),
$$
which we call {\em the tautological map}. In particular, the
induced map
$$
\tau:C^\infty(V,\Lambda^{0,1}(V)) \to C^\infty({\overline{T}V},\C)
$$
identifies the space $C^\infty(V,\Lambda^{0,1}(V))$ of smooth $(0,1)$-forms
on $V$ with the subspace in $C^\infty({\overline{T}V},\C)$ of function linear along the
fibers of the projection ${\overline{T}V} \to V$.
\refstepcounter{subsubsection Let now $M$ be a Hodge manifold. Let $V \subset M_I$ be the
complex submanifold of $U(1)$-fixed points, and let $\rho_M:M \to V$
be the canonical projection. Assume that $M$ is equipped with a
smooth $U(1)$-equivariant map $f:M \to {\overline{T}V}$ such that $\rho_M = \rho
\circ f$. Let $\bar\partial_I$ be the Dolbeault differential for the
preferred complex structure $M_I$ on $M$, and let $\phi \in
\Theta(M/V)$ be the differential of the $U(1)$-action on $M$. Let
also $j:\Lambda^{0,1}(M_I) \to \Lambda^{1,0}(M_I)$ be the map
induced by the quaternionic structure on $M$.
\begin{lemma}\label{lin.char}
The map $f:M \to {\overline{T}V}$ coincides with the linearization map if and only if
for every $(0,1)$-form $\alpha \in C^\infty(V,\Lambda^{0,1}(V))$ we have
\begin{equation}\label{L}
f^*\tau(\alpha) = \langle \phi, j(\rho_M^*\alpha) \rangle.
\end{equation}
Moreover, if $f = \Lin_M$, then we have
\begin{equation}\label{LL}
f^*\tau(\beta) = \langle \phi, j(f^*\beta) \rangle
\end{equation}
for every smooth section $\beta \in C^\infty({\overline{T}V},\rho^*\Lambda^1(V,\C))$.
\end{lemma}
\proof Since functions on ${\overline{T}V}$ linear along the fibers separate points,
the correspondence
$$
f^* \circ \tau:C^\infty(V,\Lambda^{0,1}(V)) \to C^\infty(M,\C)
$$
characterizes the map $f$ uniquely, which proves the ``only if''
part of the first claim. Since by assumption $\rho_M = \rho \circ
f$, the equality \eqref{L} is a particular case of \eqref{LL} with
$\beta = \rho^*\alpha$. Therefore the ``if'' part of the first claim
follows from the second claim, which is a rewriting of the
definition of the linearization map $\Lin_M:M \to {\overline{T}V}$ (see
\ref{overline.T}).
\hfill \ensuremath{\square}\par
\refstepcounter{subsubsection Let now $\Lin_M:M \to {\overline{T}V}$ be the linearization map for the
regular Hodge manifold $M$. Denote by $U \subset {\overline{T}V}$ the image of
$\Lin_M$. The subset $U \subset {\overline{T}V}$ is open and
$U(1)$-invariant. In addition, the isomorphism $\Lin_M:M \to U$
induces a regular Hodge manifold structure on $U$.
Donote by $\Lin_U$ the linearization map for the regular Hodge
manifold $U$. Lemma~\ref{lin.char} implies the following.
\begin{corr}\label{lin.lin}
We have $\Lin_M \circ \Lin_U = \Lin_M$, thus the linearization map
$\Lin_U:U \to {\overline{T}V}$ coincides with the given embedding $U \hookrightarrow {\overline{T}V}$.
\end{corr}
\proof Let $\alpha \in C^\infty(V,\Lambda^{0,1}(V)$ be a
$(0,1)$-form on $V$. By Lemma~\ref{lin.char} we have
$\Lin_U^*\tau(\alpha) = \langle \phi_U, j_U(\rho_U^*\alpha)\rangle$,
and it suffices to prove that
$$
\Lin_M^*(\Lin_U^*(\tau(\alpha))) = \langle \phi_M, j_M(\rho_M^*\alpha)
\rangle.
$$
By definition we have $\rho_M = \rho_U \circ \Lin_M$. Moreover, the
map $\Lin_M$ is $U(1)$-equivariant, therefore it sends $\phi_M$ to
$\phi_U$. Finally, by definition it commutes with the quaternionic
structure map $j$. Therefore
$$
\Lin_M^*(\Lin_U^*(\tau(\alpha))) = \Lin_M^*(\langle \phi_U,
j_U(\rho_U^*\alpha) \rangle)= \langle \phi_M, j_M(\rho_M^*\alpha)
\rangle,
$$
which proves the corollary.
\hfill \ensuremath{\square}\par
\refstepcounter{subsubsection \noindent {\bf Definition.\ } \label{lin.def}
Let $U \subset {\overline{T}V}$ be an open $U(1)$-invariant neighborhood of the zero
section $V \subset {\overline{T}V}$. A Hodge manifold structure on ${\overline{T}V}$ is called {\em
linear} if the associated linearization map $\Lin_U:U \to {\overline{T}V}$ coincides
with the given embedding $U \hookrightarrow {\overline{T}V}$.
By Corollary~\ref{lin.lin} every Hodge manifold structure on a subset $U
\subset {\overline{T}V}$ obtained by the linearization construction is linear.
\refstepcounter{subsubsection We finish this section with the following simple observation, which
we will need in the next section.
\begin{lemma}\label{aux2}
Keep the notations of Lemma~\ref{lin.char}. Moreover, assume given a
subspace $\A \subset C^\infty({\overline{T}V},\rho^*\Lambda^1(V,\C))$ such that
the image of $\A$ under the restriction map
$$
\Res:C^\infty({\overline{T}V},\rho^*\Lambda^1(V,\C)) \to
C^\infty(V,\Lambda^1(V,\C))
$$
onto the zero section $V \subset {\overline{T}V}$ is the whole space
$C^\infty(V,\Lambda^1(V,\C))$. If \eqref{LL} holds for every section
$\beta \in \A$, then it holds for every smooh section
$$
\beta \in C^\infty({\overline{T}V},\rho^*\Lambda^1(V,\C)).
$$
\end{lemma}
\proof By assumptions sections $\beta \in \A$ generate the
restriction of the bundle $\rho^*\Lambda^1(V,\C)$ onto the zero
section $V \subset {\overline{T}V}$. Therefore there exists an open neighborhood
$U \subset {\overline{T}V}$ of the zero section $V \subset {\overline{T}V}$ such that the
$C^\infty(U,\C)$-submodule
$$
C^\infty(U,\C) {\:\raisebox{3pt}{\text{\circle*{1.5}}}} \A \subset C^\infty(U,\rho^*\Lambda^1(V,\C))
$$
is dense in the space $C^\infty(U,\rho^*\Lambda^1(V,\C))$ of smooth
sections of the pull-back bundle $\rho^*\Lambda^1(V,\C)$. Since
\eqref{LL} is continuous and linear with respect to multiplication
by smooth functions, it holds for all sections $\beta \in
C^\infty(U,\rho^*\Lambda^1(V,\C))$. Since it is also compatible with
the natural unit disc action on $M$ and ${\overline{T}V}$, it holds for all
sections $\beta \in C^\infty({\overline{T}V},\rho^*\Lambda^1(V,\C))$ as well.
\hfill \ensuremath{\square}\par
\section{Tangent bundles as Hodge manifolds}\label{section.5}
\subsection{Hodge connections}
\refstepcounter{subsubsection The linearization construction reduces the study of arbitrary
regular Hodge manifolds to the study of linear Hodge manifold
structures on a neighborhood $U \subset {\overline{T}V}$ of the zero section $V
\subset {\overline{T}V}$ of the total space of the complex conjugate to the
tangent bundle of a complex manifold $V$. In this section we use the
theory of Hodge bundles developed in Subsection~\ref{hb.sub} in
order to describe Hodge manifold structures on $U$ in terms of
connections on the locally trivial fibration $U \to V$ of a certain
type, which we call {\em Hodge connections} (see
\ref{hodge.con}). It is this description, given in
Proposition~\ref{equiv}, which we will use in the latter part of the
paper to classify all such Hodge manifold structures.
\refstepcounter{subsubsection \label{conn}
We begin with some preliminary facts about connections on locally
trivial fibrations. Let $f:X \to Y$ be an arbitrary smooth map of
smooth manifolds $X$ and $Y$. Assume that the map $f$ is submersive,
that is, the codifferential $\delta_f:f^*\Lambda^1(Y) \to
\Lambda^1(X)$ is an injective bundle map. Recall that a {\em
connection} on $f$ is by definition a splitting $\Theta:\Lambda^1(X)
\to f^*\Lambda^1(Y)$ of the canonical embedding $\delta_f$.
Let $d_X$ be the de Rham differential on the smooth manifold
$X$. Every connection $\Theta$ on $f:X \to Y$ defines an algebra
derivation
$$
D = \Theta \circ d_X:\Lambda^0(X) \to f^*\Lambda^1(Y),
$$
satisfying
\begin{equation}\label{conn.eq}
D \rho^* h = \rho^* d_Y h
\end{equation}
for every smooth function $h \in C^\infty(Y,\R)$. Vice versa, by the
universal property of the cotangent bundle (Lemma~\ref{universal})
every algebra derivation $D:\Lambda^0(X) \to \Lambda^1(Y)$
satisfying \eqref{conn.eq} comes from a unique connection $\Theta$
on $f$.
\refstepcounter{subsubsection Recall also that a connection $\Theta$ is called {\em flat}
if the associated derivation $D$ extends to an algebra
derivation
$$
D:f^*\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(Y) \to f^*\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(Y)
$$
so that $D \circ D = 0$. The splitting $\Theta:\Lambda^1(X) \to
f^*\Lambda^1(Y)$ extends in this case to an algebra map
$$
\Theta:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(X) \to f^*\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(Y)
$$
compatible with the de Rham differential $d_X:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(X) \to
\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(X)$.
\refstepcounter{subsubsection We will need a slight generalization of the notion of connection.
\noindent {\bf Definition.\ } Let $f:X \to Y$ be a smooth submersive morphism of complex
manifolds. A {\em $\C$-valued connection} $\Theta$ on $f$ is a
splitting $\Theta:\Lambda^1(Y,\C) \to f^*\Lambda^1(X,\C)$ of the
codifferential map $\delta f:f^*\Lambda^1(Y,\C) \to \Lambda^1(X,\C)$
of complex vector bundles. A $\C$-valued connection $\Theta$ is
called {\em flat} if the associated algebra derivation
$$
D = \Theta \circ d_X:\Lambda^0(X,\C) \to f^*\Lambda^1(Y,\C)
$$
extends to an algebra derivation
$$
D:f^*\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(Y,\C) \to f^*\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(Y,\C)
$$
satisfying $D \circ D = 0$.
As in \ref{conn}, every derivation $D:\Lambda^0(X,\C) \to
f^*\Lambda^1(Y,\C)$ satisfying \eqref{conn.eq} comes from a unique
$\C$-valued connection $\Theta$ on $f:X \to Y$.
\noindent {\bf Remark.\ } By definition for every flat connection on $f:X \to Y$ the
subbundle of horizontal vectors in the the tangent bundle
$\Theta(X)$ is an involutive distribution. By Frobenius Theorem this
implies that the connection defines locally a trivialization of the
fibration $f$.
This is no longer true for flat $\C$-valued connections: the
subbundle of horizontal vectors in $\Theta(X) \otimes \C$ is only
defined over $\C$, and the Frobenius Theorem does not apply. One can
try to correct this by replacing the splitting
$\Theta:\Lambda^1(X,\C) \to f^*\Lambda^1(Y,\C)$ with its real part
$\Re\Theta:\Lambda^1(X) \to \Lambda^1(Y)$, but this real part is, in
general, no longer flat.
\refstepcounter{subsubsection For every $\C$-valued connection $\Theta:\Lambda^1(X,\C) \to
f^*\Lambda^1(Y,\C)$ on a fibration $f:X \to Y$ the kernel
$\Ker\Theta \subset \Lambda^1(X,\C)$ is canonically isomorphic to
the quotient $\Lambda^1(X,\C)/\delta_f(f^*\Lambda^1(Y,\C))$, and the
composition
$$
R = \Theta \circ d_X:\Lambda^1(X,\C)/\delta_f(f^*\Lambda^1(Y,\C) \cong
\Ker\Theta \to f^*\Lambda^2(Y,\C)
$$
is in fact a bundle map. This map is called {\em the curvature} of
the $\C$-valued connection $\Theta$. The connection $\Theta$ is flat
if and only if its curvature $R$ vanishes.
\refstepcounter{subsubsection \label{conn.setup}
Let now $M$ be a complex manifold, and let $U \subset {\overline{T}M}$ be an
open neighborhood of the zero section $M \subset {\overline{T}M}$ in the total
space ${\overline{T}M}$ of the complex-conjugate to the tangent bundle to
$M$. Let $\rho:U \to M$ be the natural projection. Assume that $U$
is invariant with respect to the natural action of the unit disc
$\Delta \subset \C$ on ${\overline{T}M}$.
\refstepcounter{subsubsection Since $M$ is complex, by \ref{de.Rham} the bundle
$\Lambda^1(M,\C)$ is equipped with a Hodge bundle structure of
weight $1$. The pullback bundle $\rho^*\Lambda^1(M,\C)$ is then also
equipped with a weight $1$ Hodge bundle structure.
Our description of the Hodge manifold structures on the subset $U
\in {\overline{T}M}$ is based on the following notion.
\noindent {\bf Definition.\ } \label{hodge.con}
A {\em Hodge connection} on the pair $\langle M, U\rangle$ is
a $\C$-valued connection on $\rho:U \to M$ such that the associated
derivation
$$
D:\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)
$$
is weakly Hodge in the sense of \ref{w.hodge}. A Hodge connection is
called {\em flat} if it extends to a weakly Hodge derivation
$$
D:\rho^*\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \rho^*\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C)
$$
satisfying $D \circ D = 0$.
\refstepcounter{subsubsection Assume given a flat Hodge connection $D:\Lambda^0(U,\C) \to
\rho^*\Lambda^1(M,\C)$ on the pair $\langle U,M \rangle$, and assume
in addition that the derivation $D$ is holonomic in the sense of
\ref{holonomic}. Then the pair $\langle D,
\rho^*\Lambda^1(M,\C)\rangle$ defines by
Proposition~\ref{explicit.hodge} a Hodge manifold structure on $U$.
It turns out that every Hodge manifold structure on $U$ can be
obtained in this way. Namely, we have the following.
\begin{prop}\label{equiv}
There correspondence $D \mapsto \langle \rho^*\Lambda^1(M,\C), D
\rangle$ is a bijection between the set of all flat Hodge
connections $D$ on the pair $\langle U, M\rangle$ such that
$D:\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)$ is holonomic in the
sense of \ref{holonomic}, and the set of all Hodge manifold
structures on the $U(1)$-manifold $U$ such that the projection
$\rho:U_I \to M$ is holomorphic for the preferred complex structure
$U_I$ on $U$.
\end{prop}
\refstepcounter{subsubsection The crucial part of the proof of Proposition~\ref{equiv} is
the following observation.
\begin{lemma}\label{ident}
Assume given a Hodge manifold structure on the $U(1)$-manifold $U
\subset {\overline{T}M}$. Let $\delta_\rho:\rho^*\Lambda^1(M,\C) \to
\Lambda^1(U,\C)$ be the codifferential of the projection $\rho:U \to
M$, and let $P:\Lambda^1(U,\C) \to \Lambda^{0,1}(U_J)$ be the
canonical projection. The bundle mao given by the composition
$$
P \circ \delta_\rho:\rho^*\Lambda^1(M,\C) \to \Lambda^{0,1}(U_J)
$$
is an isomorphism of complex vector bundles.
\end{lemma}
\proof Since the bundles $\rho^*\Lambda^1(M,\C)$ and
$\Lambda^{0,1}(U_J)$ are of the same rank, and the maps
$\delta_\rho:\rho^*\Lambda^1(M,\C) \to \Lambda^1(U,\C)$ and $P \circ
\delta_\rho$ are equivariant with respect to the action of the unit
disc on $U$, it suffices to prove the claim on $M \subset U$. Let $m
\in M$ be an arbitrary point, and let $V = T^*_m{\overline{T}M}$ be the
cotangent bundle at $m$ to the Hodge manifold $U \subset {\overline{T}M}$. Let
also $V^0 \subset V$ be the subspace of $U(1)$-invariant vectors in
$V$.
The space $V$ is an equivariant quaternionic vector space. Moreover,
the fibers of the bundles $\rho^*\Lambda^1(M,\C)$ and
$\Lambda^{0,1}(U_J)$ at the point $m$ are complex vector spaces, and
we have canonical identifications
\begin{align*}
\rho^*\Lambda^1(M,\C)|_m &\cong V^0_I \oplus \overline{V^0_I},\\
\Lambda^{0,1}(U_J)|_m &\cong V_J.
\end{align*}
Under these identifications the map $P \circ \delta_\rho$ at the point
$m$ coincides with the action map $V^0_I \oplus \overline{V^0_I} \to
V_J$, which is invertible by Lemma~\ref{regular.quaternionic}.
\hfill \ensuremath{\square}\par
\refstepcounter{subsubsection By Proposition~\ref{explicit.hodge} every Hodge manifold
structure on $U$ is given by a pair $\langle \E, D\rangle$
of a Hodge bundle $\E$ on $U$ of weight $1$ and a holonomic
derivation $D:\Lambda^0(U,\C) \to \E$. Lemma~\ref{ident} gives an
isomorphism $\E \cong \rho^*\Lambda^1(M,\C)$, so that $D$ becomes a
flat $\C$-valued connection on $U$ over $M$. To prove
Proposition~\ref{equiv} it suffices now to prove the following.
\begin{lemma}
The complex vector bundle isomorphism
$$
P \circ \delta_\rho:\rho^*\Lambda^1(M,\C) \to \Lambda^{0,1}(U_J)
$$
associated to a Hodge manifold structure on $U$ is compatible with
the Hodge bundle structures if and only if the projection $\rho:U_I
\to M$ is holomorphic for the preferred complex structure $U_I$ on
$U$.
\end{lemma}
\proof The preferred complex structure $U_I$ induces a Hodge bundle
structure of weight $1$ on $\Lambda^1(U,\C)$ by \ref{de.Rham}, and
the canonical projection $P:\Lambda^1(U,\C) \to \Lambda^{0,1}(U_J)$
is compatible with the Hodge bundle structures by
\ref{lambda_j=lambda_i}. If the projection $\rho:U_I \to M$ is
holomorphic, then the codifferential
$\delta_\rho:\rho^*\Lambda^1(M,\C) \to \Lambda^1(U,\C)$ sends the
subbundles $\rho^*\Lambda^{1,0}(M),\rho^*\Lambda^{0,1}(M) \subset
\rho^*\Lambda^1(M,\C)$ into, respectively, the subbundles
$\Lambda^{1,0}(U_I),\Lambda^{0,1}(U_I) \subset
\Lambda^1(U,\C)$. Therefore the map $\delta_\rho:\rho^*\Lambda^1(M,\C)
\to \Lambda^1(U,\C)$ is compatible with the Hodge bundle structures,
which implies the ``if'' part of the lemma.
To prove the ``only if'' part, assume that $P \circ \delta_\rho$ is a
Hodge bundle isomorphism. Since the complex conjugation
$\nu:\overline{\Lambda^{0,1}(U_I)} \to \Lambda^{1,0}(U_J)$ is
compatible with the Hodge bundle structures, the projection
$\overline{P}:\Lambda^1(U,\C) \to \Lambda^{1,0}(U_J)$ and the
composition $\overline{P} \circ \delta_\rho:\rho^*\Lambda^1(M,\C) \to
\Lambda^{1,0}(U_J)$ are also compatible with the Hodge bundle
structures. Therefore the map
$$
P \oplus \overline{P}:\Lambda^1(U,\C) \to \Lambda^{1,0}(U_J) \oplus
\Lambda^{0,1}(U_J)
$$
is a Hodge bundle isomorphism, and the composition
$$
\delta_\rho \circ (P \oplus \overline{P}):\rho^*\Lambda^1(M,\C) \to
\Lambda^1(U,\C)
$$
is a Hodge bundle map. Therefore the codifferential
$\delta_\rho:\rho^*\Lambda^1(M,\C) \to \Lambda^1(U,\C)$ is compatible
with the Hodge bundle structures. This means precisely that the
projection $\rho:U_I \to M$ is holomorphic, which finishes the proof
of the lemma and of Proposition~\ref{equiv}.
\hfill \ensuremath{\square}\par
\subsection{The relative de Rham complex of $U$ over
$M$}\label{relative.de.rham.sub}
\refstepcounter{subsubsection Keep the notation of the last subsection. To use
Proposition~\ref{equiv} in the study of Hodge manifold structures on
the open subset $U \subset {\overline{T}M}$, we will need a way to check whether
a given Hodge connection on the pair $\langle U,M \rangle$ is
holonomic in the sense of \ref{holonomic}. We will also need to
rewrite the linearity condition \ref{lin.def} for a Hodge manifold
structure on $U$ in terms of the associated Hodge connection $D$. To
do this, we will use the so-called {\em relative de Rham complex} of
$U$ over $M$. For the convenience of the reader, and to fix
notation, we recall here its definition and main properties.
\refstepcounter{subsubsection Since the projection $\rho:U \to M$ is submersive, the
codifferential
$$
\delta_\rho:\rho^*\Lambda^1(M,\C) \to \Lambda^1(U,\C)
$$
is injective. The {\em relative cotangent bundle} $\Lambda^1(U/M,C)$
is by definition the quotient bundle
$$
\Lambda^1(U/M,\C) = \Lambda^1(U,\C)/\delta_\rho(\rho^*\Lambda^1(M,\C)).
$$
Let $\pi:\Lambda^1(U,\C) \to \Lambda^1(U/M,\C)$ be the natural
projection. We have by definition the short exact sequence
\begin{equation}\label{ex.seq}
\begin{CD}
0 @>>> \rho^*\Lambda^1(M,\C) @>\delta_\rho>> \Lambda^1(U,\C) @>\pi>>
\Lambda^1(U/M,\C) @>>> 0
\end{CD}
\end{equation}
of complex vector bundles on $U$.
\refstepcounter{subsubsection The composition $d^r = \pi \circ d_U$ of the de Rham differential
$d_U$ with the projection $\pi$ is an algebra derivation
$$
d^r:\Lambda^0(U,\C) \to \Lambda^1(U/M,\C),
$$
called {\em the relative de Rham differential}. It is a first order
differential operator, and $d^rf = 0$ if and only if the smooth
function $f:U \to \C$ factors through the projection $\rho:U \to
M$.
Let $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U/M,\C)$ be the exterior algebra of the bundle
$\Lambda^1(U/M,\C)$. The projection $\pi$ extends to an algebra
map
$$
\pi:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U,\C) \to \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U/M,\C).
$$
The differential $d^r$ extends to an algebra derivation
$$
d^r:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U/M,\C) \to \Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(U/M,\C)
$$
satisfying $d^r \circ d^r = 0$, and we have $\pi \circ d_U = d^r
\circ \pi$. The differential graded algebra $\langle
\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U/M,\C),d^r\rangle$ is called {\em the relative de
Rham complex} of $U$ over $M$.
\refstepcounter{subsubsection Since the relative de Rham differential $d^r$ is linear with
respect to multiplication by functions of the form $\rho^*f$ with $f
\in C^\infty(M,\C)$, it extends canonically to an operator
$$
d^r:\rho^*\Lambda^i(M,\C) \otimes \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U/M,\C) \to
\rho^*\Lambda^i(M,\C) \otimes \Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(U/M,\C).
$$
The two-step filtration $\rho^*\Lambda^1(M,\C) \subset
\Lambda^1(U,\C)$ induces a filtration on the de Rham complex
$\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U,\C)$, and the $i$-th associated graded quotient
of this filtration is isomorphic to the complex $\langle
\rho^*\Lambda^i(M,\C) \otimes \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U/M,\C), d^r \rangle$.
\refstepcounter{subsubsection Since $U \subset {\overline{T}M}$ lies in the total space of the
complex-conjugate to the tangent bundle to $M$, we have a canonical
algebra isomorphism
$$
{\sf can}:\overline{\rho^*\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)} \to \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U/M,\C).
$$
Let $\tau:C^\infty(M,\Lambda^1(M,\C)) \to C^\infty(U,\C)$ be
the tautological map sending a $1$-form to the corresponding linear
function on ${\overline{T}M}$, as in \ref{tau}. Then for every smooth
$1$-form $\alpha \in C^\infty(M,\Lambda^1(M,\C))$ we have
\begin{equation}\label{can.and.tau}
{\sf can}(\rho^*\alpha) = d^r\tau(\alpha).
\end{equation}
\refstepcounter{subsubsection The complex vector bundle $\Lambda^1(U/M,\C)$ has a natural
real structure, and it is naturally $U(1)$-equivariant. Moreover,
the decomposition $\Lambda^1(M,\C) = \Lambda^{1,0}(M) \oplus
\Lambda^{0,1}(M)$ induces a decomposition
$$
\Lambda^1(U/M,\C) = {\sf can}(\Lambda^{1,0}(M)) \oplus
{\sf can}(\Lambda^{0,1}(M)).
$$
This allows to define, as in \ref{de.Rham}, a canonical Hodge bundle
structure of weight $1$ on $\Lambda^1(U/M,\C)$. It gives rise to a
Hodge bundle structure on $\Lambda^i(U/M,\C)$ of wieght $i$, and the
relative de Rham differential $d^r:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U/M,\C) \to
\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(U/M,\C)$ is weakly Hodge.
\refstepcounter{subsubsection The canonical isomorphism
${\sf can}:\overline{\rho^*\Lambda^1(M,\C)} \to \Lambda^1(U/M,\C)$ is not
compatible with the Hodge bundle structures. The reason for this is
that the real structure on the Hodge bundles $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U/M,\C)$
is, by definition \ref{de.Rham}, twisted by $\iota^*$, where
$\iota:{\overline{T}M} \to {\overline{T}M}$ is the action of $-1 \in U(1) \subset
\C$. Therefore, while ${\sf can}$ is $U(1)$-equivariant, it is not
real. To correct this, introduce an involution
$\zeta:\Lambda^1(M,\C) \to \Lambda^1(M,\C)$ by
\begin{equation}\label{zeta}
\zeta = \begin{cases} {\sf id} &\text{ on }\Lambda^{1,0}(M) \subset
\Lambda^1(M,\C) \\
-{\sf id} &\text{ on }\Lambda^{0,1}(M) \subset \Lambda^1(M,\C)
\end{cases}
\end{equation}
and set
\begin{equation}\label{eta}
\eta = {\sf can} \circ
\rho^*\overline{\zeta}:\rho^*\overline{\Lambda^1(M,\C)} \to
\rho^*\overline{\Lambda^1(M,\C)} \to \Lambda^1(U/M,\C)
\end{equation}
Unlike ${\sf can}$, the map $\eta$ preserves the Hodge bundle structures.
It will also be convenient to twist the tautological map
$\tau:\rho^*\Lambda^1(M,\C) \to \Lambda^0(U,\C)$ by the involution
$\zeta$. Namely, define a map $\sigma:\rho^*\Lambda^1(M,\C) \to
\Lambda^0(U,\C)$ by
\begin{equation}\label{sigma1}
\sigma = \tau \circ \rho^*\overline{\zeta}:
\rho^*\overline{\Lambda^1(M,\C)} \to
\rho^*\overline{\Lambda^1(M,\C)} \to \Lambda^0(U/M,\C)
\end{equation}
By \eqref{can.and.tau} the twisted tautological map $\sigma$ and the
canonical map $\eta$ satisfy
\begin{equation}\label{eta.and.sigma}
\eta(\rho^*\alpha) = d^r\sigma(\alpha)
\end{equation}
for every smooth $1$-form $\alpha \in C^\infty(M,\Lambda^1(M,\C))$.
\refstepcounter{subsubsection Let $\phi \in \Theta(U)$ be the differential of the canonical
$U(1)$-action on $U \subset {\overline{T}M}$. The vector field $\phi$ is real
and tangent to the fibers of the projection $\rho:U \to
M$. Therefore the contraction with $\phi$ defines an algebra
derivation
\begin{align*}
\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(U/M,\C) &\to \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U/M,\C)\\
\alpha &\mapsto \langle \phi, \alpha \rangle
\end{align*}
The following lemma gives a relation between this derivation, the
canonical weakly Hodge map $\eta:\rho^*\Lambda^1(M,\C) \to
\Lambda^1(U/M,\C)$ given by \eqref{eta}, and the tautological map
$\tau:\rho^*\Lambda^1(M,\C) \to \Lambda^0(U,\C)$.
\begin{lemma}\label{phi.and.tau}
For every smooth section $\alpha \in
C^\infty(U,\rho^*\Lambda^1(M,\C))$ we have
$$
\sqrt{-1}\tau(\alpha) = \langle \phi, \eta(\alpha) \rangle \in
C^\infty(U,\C).
$$
\end{lemma}
\proof Since the equality that we are to prove is linear with
respect to multiplication by smooth functions on $U$, we may assume
that the section $\alpha$ is the pull-back of a smooth $1$-form
$\alpha \in C^\infty(M,\Lambda^1(M,\C))$. The Lie derivative
$\LL_\phi:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U,\C) \to \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U,\C)$ with respect
to the vector field $\phi$ is compatible with the projection
$\pi:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U,\C) \to \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U/M,\C)$ and defines
therefore an algebra derivation $\LL_\phi:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U/M,\C) \to
\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U/M,\C)$. The Cartan homotopy formula gives
\begin{equation}\label{eqqq.1}
\LL_\phi \tau(\alpha) = \langle \phi, d^r\tau(\alpha) \rangle.
\end{equation}
The function $\tau(\alpha)$ on ${\overline{T}M}$ is by definition $\R$-linear
along the fibers of the projection $\rho:{\overline{T}M} \to M$. The subspace
$\tau(C^\infty(M,\Lambda^1(M,\C))) \subset C^\infty(U,\C)$ of such
functions decomposes as
$$
\tau(C^\infty(M,\Lambda^1(M,\C))) =
\tau(C^\infty(M,\Lambda^{1,0}(M))) \oplus
\tau(C^\infty(M,\Lambda^{0,1}(M,\C))),
$$
and the group $U(1)$ acts on the components with weight $1$ and
$-1$. Therefore the derivative $\LL_\phi$ of the $U(1)$-action acts
on the components by multiplication with $\sqrt{-1}$ and
$-\sqrt{-1}$. By definition of the involution $\zeta$ (see
\eqref{zeta}) this can be written as
\begin{equation}\label{eqqq.2}
\LL_\phi \tau(\alpha) = \sqrt{-1}\tau(\zeta(\alpha)).
\end{equation}
On the other hand, by \eqref{can.and.tau} and the definition of the
map $\eta$ we have
\begin{equation}\label{eqqq.3}
d^r\tau(\alpha) = {\sf can}(\alpha) = \eta(\zeta(\alpha)).
\end{equation}
Substituting \eqref{eqqq.2} and \eqref{eqqq.3} into \eqref{eqqq.1}
gives
$$
\sqrt{-1}\tau(\zeta(\alpha)) = \langle \phi, \eta(\zeta(\alpha)),
$$
which is equivalent to the claim of the lemma.
\hfill \ensuremath{\square}\par
\subsection{Holonomic Hodge connections}
\refstepcounter{subsubsection We will now describe a convenient way to check whether a
given Hodge connection $D$ on the pair $\langle U,M \rangle$ is
holonomic in the sense of \ref{holonomic}. To do this, we proceed as
follows.
Consider the restriction $\Lambda^1(U,\C)|_M$ of the bundle
$\Lambda^1(U,\C)$ to the zero section $M \subset U \subset {\overline{T}M}$, and
let
$$
\Res:\Lambda^1(U,\C)|_M \to \Lambda^1(M,\C)
$$
be the restriction map. The kernel of the map $\Res$ coincides with
the conormal bundle to the zero section $M \subset U$, which we
denote by $S^1(M,\C)$. The map $\Res$ splits the restriction of
exact sequence \eqref{ex.seq} onto the zero section $M \subset U$,
and we have the direct sum decomposition
\begin{equation}\label{hor.vert}
\Lambda^1(U,\C)|_M = S^1(M,\C) \oplus \Lambda^1(M,\C).
\end{equation}
\refstepcounter{subsubsection\label{S1}
The $U(1)$-action on $U \subset {\overline{T}M}$ leaves the zero section $M
\subset U$ invariant and defines therefore a $U(1)$-action on the
conormal bundle $S^1(M,\C)$. Together with the usual real structure
twisted by the action of the map $\iota:{\overline{T}M} \to {\overline{T}M}$, this defines a
Hodge bundle structure of weight $0$ on the bundle $S^1(M,\C)$.
Note that the automorphism $\iota:{\overline{T}M} \to {\overline{T}M}$ acts as $-{\sf id}$ on the
Hodge bundle $S^1(M,\C)$, so that the real structure on $S^1(M,\C)$
is minus the usual one. Moreover, as a complex vector bundle the
conormal bundle $S^1(M,\C)$ to $M \subset {\overline{T}M}$ is canonically
isomorphic to the cotangent bundle $\Lambda^1(M,\C)$. The Hodge type
grading on $S^1(M,\C)$ is given in terms of this isomorphism by
$$
S^1(M,\C) = S^{1,-1}(M) \oplus S^{-1,1}(M) \cong \Lambda^{1,0}(M)
\oplus \Lambda^{0,1}(M) = \Lambda^1(M,\C).
$$
\refstepcounter{subsubsection Let
$$
C_{lin}^\infty(U,\C) = \tau(C^\infty(M,\Lambda^1(M,\C))) \subset
C^\infty(U,\C)
$$
be the subspace of smooth functions linear along the fibers of the
canonical projection $\rho:U \subset {\overline{T}M} \to M$. The relative de
Rham differential defines an isomorphism
\begin{equation}\label{iso}
d^r:C_{lin}^\infty(U,\C) \to C^\infty(M,S^1(M,\C)).
\end{equation}
This isomorphism is compatible with the canonical Hodge structures of weight
$0$ on both spaces, and it is linear with respect to multiplication
by smooth functions $f \in C^\infty(M,\C)$.
\refstepcounter{subsubsection \label{pr.part}
Let now $D:\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)$ be a Hodge
connection on the pair $\langle U,M \rangle$, and let
$\Theta:\Lambda^1(U,\C) \to \rho^*\Lambda^(M,\C)$ be the
corresponding bundles map. Since $D$ is a $\C$-valued connection,
the restriction $\Theta|_M$ onto the zero section $M \subset M$
decomposes as
\begin{equation}\label{hor.vert.1}
\Theta = D_0 \oplus {\sf id}:S^1(M,\C) \oplus \Lambda^1(M,\C) \to
\Lambda^1(M,\C)
\end{equation}
with respect to the direct sum decomposition \eqref{hor.vert} for a
certain bundle map $D_0:S^1(M,\C) \to \Lambda^1(M,\C)$.
\noindent {\bf Definition.\ } The bundle map $D_0:S^1(M,\C) \to \Lambda^1(M,\C)$ is called
the {\em principal part} of the Hodge connection $D$.
\refstepcounter{subsubsection Consider the map $D_0:C^\infty(M,S^1(M,\C)) \to
C^\infty(M,\Lambda^1(M,\C))$ on the spaces of smooth sections
induced by the principal part $D_0$ of a Hodge connection $D$.
Under the isomorphism \eqref{iso} this map coincides with the
restriction of the composition
$$
\Res \circ D:C^\infty(U,\C) \to C^\infty(U,\rho^*\Lambda^1(M,\C))
\to C^\infty(M,\Lambda^1(M,\C))
$$
onto the subspace $C^\infty_{lin}(U,\C) \subset
C^\infty(U,\C)$. Each of the maps $\Res$, $D$ is weakly Hodge, so
that this composition also is weakly Hodge. Since the isomorphism
\eqref{iso} is compatible with the Hodge bundle structures, this
implies that the principal part $D_0$ of the Hodge connection $D$ is
a weakly Hodge bundle map. In particular, it is purely imaginary
with respect to the usual real structure on the conormal bundle
$S^1(M,\C)$.
\refstepcounter{subsubsection We can now formulate the main result of this subsection.
\begin{lemma}\label{aux1}
A Hodge connection $D:\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)$ on
the pair $\langle U,M \rangle$ is holonomic in the sense of
\ref{holonomic} on an open neighborhood $U_0 \subset U$ of the zero
section $M \subset U$ if and only if its principal part
$D_0:S^1(M,\C) \to \Lambda^1(M,\C)$ is a complex vector bundle
isomorphism.
\end{lemma}
\proof By definition the derivation $D:\Lambda^0(U,\C) \to
\rho^*\Lambda^1(M,\C)$ is holonomic in the sense of \ref{holonomic}
if and only if the corresponding map
$$
\Theta:\Lambda^1(U,\R) \to \rho^*\Lambda^1(M,\C)
$$
is an isomorphism of real vector bundles. This is an open condition.
Therefore the derivation $D$ is holonomic on an open neighborhood
$U_0 \supset M$ of the zero section $M \subset U$ if and only if the
map $\Theta$ is an isomorphism on the zero section $M \subset U$
itself.
According to \eqref{hor.vert.1}, the restriction $\Theta|_M$
decomposes as $\Theta_M = D_0 + {\sf id}$, and the principal part
$D_0:S^1(M,\C) \to \Lambda^1(M,\C)$ of the Hodge connection $D$ is
purely imaginary with respect to the usual real structure on
$\Lambda^1(U,\C)|_M$, while the identity map
${\sf id}:\rho^*\Lambda^1(M,\C) \to \rho^*\Lambda^1(M,\C)$ is, of course,
real. Therefore $\Theta_M$ is an isomorphism if and only is $D_0$ is
an isomorphism, which proves the lemma.
\hfill \ensuremath{\square}\par
\subsection{Hodge connections and linearity}\label{hodge.lin.subsec}
\refstepcounter{subsubsection \label{aux}
Assume now given a Hodge manifold structure on the subset $U \subset
{\overline{T}M}$, and let $D:\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)$ be the
associated Hodge connection on the pair $\langle U, M\rangle$ given
by Proposition~\ref{equiv}. We now proceed to rewrite the linearity
condition \ref{lin.def} in terms of the Hodge connection $D$.
Let $j:\Lambda^1(U,\C) \to \overline{\Lambda^1(U,\C)}$ be the
canonical map defined by the quaternionic structure on $U$, and let
$\iota^*:\Lambda^1(U,\C) \to \iota^*\Lambda^1(U,\C)$ be the action
of the canonical involution $\iota:U \to U$. Let also
$D^\iota:\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)$ be the operator
$\iota^*$-conjugate to the Hodge connection $D$.
We begin with the following identity.
\begin{lemma}\label{aux.lemma}
For every smooth function $f \in C^\infty(U,\C)$ we have
$$
d^rf = \frac{\sqrt{-1}}{2}\pi(j(\delta_\rho(D-D^\iota)(f))),
$$
where $\pi:\Lambda^1(U,\C) \to \Lambda^1(U/M,\C)$ is the canonical
projection, and
$$
\delta_\rho:\rho^*\Lambda^1(M,\C) \to \Lambda^1(U,\C)
$$
is the codifferential of the projection $\rho:U \to M$.
\end{lemma}
\proof By definition of the Hodge connection $D$ the Dolbeault
derivative $\bar\partial_Jf$ coincides with the $(0,1)$-component of the
$1$-form $\delta_\rho(Df) \in \Lambda^1(U,\C)$ with respect to the
complementary complex structure $U_J$ on $U$. Therefore
$$
\bar\partial_J f = \frac{1}{2} \delta_\rho(Df) + \frac{\sqrt{-1}}{2} j(\delta_\rho(Df)).
$$
Applying the complex conjugation $\nu:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U,\C) \to
\overline{\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U,\C)}$ to this equation, we get
\begin{align*}
\begin{split}
\partial_J f &= \nu\left(\frac{1}{2} \delta_\rho(D\nu(f)) + \frac{\sqrt{-1}}{2}
j(\delta_\rho(D\nu(f))))\right) = \\
&= \frac{1}{2} \nu(\delta_\rho(D\nu(f))) - \frac{\sqrt{-1}}{2} j(\nu(\delta_\rho(D\nu(f)))).
\end{split}
\end{align*}
Since the map $\delta_\rho \circ D:\Lambda^0(U,\C) \to
\rho^*\Lambda^1(M,\C)$ is weakly Hodge, we have
$$
\delta_\rho(D(\iota^*\nu(f))) = \iota^*\nu(\delta_\rho(Df)).
$$
Therefore $\nu(\delta_\rho(D(f))) = \delta_\rho(D^\iota(\nu(f)))$, and
we have
$$
\partial_J f = \frac{1}{2} \delta_\rho(D^\iota f) - \frac{\sqrt{-1}}{2} j(\delta_\rho(D^\iota
f)).
$$
Thus the de Rham derivative $d_Uf$ equals
$$
d_Uf = \partial_Jf + \bar\partial_Jf = \frac{1}{2} \delta_\rho((D+D^\iota)f) + \frac{\sqrt{-1}}{2}
j(\delta_\rho((D-D^\iota)f)).
$$
Now, by definition $\delta_\rho \circ \pi = 0$. Therefore
$$
d^rf = \pi(d_Uf) = \frac{\sqrt{-1}}{2}\pi(j(\delta_\rho((D-D^\iota)f))),
$$
which is the claim of the lemma.
\hfill \ensuremath{\square}\par
\refstepcounter{subsubsection We will also need the following fact. It can be derived
directly from Lemma~\ref{aux.lemma}, but it is more convenient to
use Lemma~\ref{aux1} and the fact that the Hodge connection
$D:\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)$ is holonomic.
\begin{lemma}\label{aux3}
In the notation of Lemma~\ref{aux.lemma}, let
$$
\A = \delta_\rho\left((D-D^\iota)\left(C_{lin}^\infty(U,\C)\right)\right)
\subset C^\infty(U, \rho^*\Lambda^1(M,\C))
$$
be the subspace of sections $\alpha \in
C^\infty(U,\rho^*\Lambda^1(M,\C))$ of the form $\alpha =
\delta_\rho((D-D^\iota)f)$, where $f \in C^\infty(U,\C)$ lies in the
subspace $C_{lin}^\infty(U,\C) \subset C^\infty(U,\C)$ of smooth
functions on $U$ linear along the fibers of the projection $\rho:U
\to M$. The restriction $\Res(\A) \subset
C^\infty(M,\Lambda^1(M,\C))$ of the subspace $\A$ onto the zero
section $M \subset U$ is the whole space
$C^\infty(M,\Lambda^1(M,\C))$.
\end{lemma}
\proof Let $D_0 = \Res \circ D:C^\infty_{lin}(U,\C) \to
C^\infty(M,\Lambda^1(M,\C))$ be the principal part of the Hodge
connection $D$ in the sense of Definition~\ref{pr.part}. Since the
canonical automorphism $\iota:{\overline{T}M} \to {\overline{T}M}$ acts as $-{\sf id}$ on
$C_{lin}^\infty(U,\C)$, we have $D_0^\iota = - D_0$. Therefore
\begin{multline*}
\Res(\A) = \Res \circ (D - D^\iota)\left(C_{lin}^\infty(U,\C)\right) = \\
=(D_0-D^\iota_0)\left(C_{lin}^\infty(U,\C)\right) =
D_0\left(C_{lin}^\infty(U,\C)\right).
\end{multline*}
Since the Hodge connection $D$ is holonomic, this space coincides
with the whole $C^\infty(M,\Lambda^1(M,\C))$ by Lemma~\ref{aux1}.
\hfill \ensuremath{\square}\par
\refstepcounter{subsubsection We now apply Lemma~\ref{aux.lemma} to prove the following criterion
for the linearity of the Hodge manifold structure on $U$ defined by
the Hodge connection $D:\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)$.
\begin{lemma}\label{explicit.lin}
The Hodge manifold structure on $U \subset {\overline{T}M}$ corresponding to a
Hodge connection $D:\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)$ is
linear in the sense of \ref{lin.def} if and only if for every smooth
function $f \in C^\infty(U,\C)$ linear along the fibers of the
projection $\rho:U \subset {\overline{T}M} \to M$ we have
\begin{equation}\label{expl.lin}
f = \frac{1}{2}\sigma\left((D-D^\iota)f\right),
\end{equation}
where $\sigma:\rho^*\Lambda^1(M,\C) \to \Lambda^0(U,\C)$ is the
twisted tautological map introduced in \eqref{sigma1}, and
$D^\iota:\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)$ is the operator
$\iota^*$-conjugate to $D$, as in \ref{aux}.
\end{lemma}
\proof By Lemma~\ref{lin.char} the Hodge manifold structure on $U$
is linear if and only if for every $\alpha \in
C^\infty(U,\rho^*\Lambda^1(M,\C))$ we have
\begin{equation}\label{eee.to.prove}
\langle \phi, j(\alpha) \rangle = \tau(\alpha),
\end{equation}
where $\phi$ is the differential of the $U(1)$-action on $U$,
$j:\Lambda^1(U,\C) \to \overline{\Lambda^1(U,\C)}$ is the operator
given by the quaternionic structure on $U$, and
$\tau:\rho^*\Lambda^1(M,\C) \to \Lambda^0(U,\C)$ is the tautological
map sending a $1$-form on $M$ to the corresponding linear function
on ${\overline{T}M}$, as in \ref{tau}. Moreover, by Lemma~\ref{aux3} and
Lemma~\ref{aux2} the equality \eqref{eee.to.prove} holds for all
smooth sections $\alpha \in C^\infty(U,\rho^*\Lambda^1(M,\C))$ if
and only if it holds for sections of the form
\begin{equation}\label{special}
\alpha = \frac{\sqrt{-1}}{2}\delta_\rho((D-D^\iota)f),
\end{equation}
where $f \in C_{lin}^\infty(U,\C) \subset C^\infty(U,\C)$ is linear
along the fibers of $\rho:U \to M$.
Let now $f \in C^\infty(U,\C)$ be a smooth function on $U$ linear
along the fibers of $\rho:U \to M$, and let $\alpha$ be as in
\eqref{special}. Since $\phi$ is a vertical vector field on $U$ over
$M$, we have $\langle \phi, j(\alpha) \rangle = \langle \phi,
\pi(j(\alpha))\rangle$, where $\pi:\Lambda^1(U,\C) \to
\Lambda^1(U/M,\C)$ is the canonical projection. By
Lemma~\ref{aux.lemma}
\begin{equation}\label{thelema}
\langle \phi, j(\alpha) \rangle = \langle \phi, \pi(j(\alpha))
\rangle = \langle \phi, d^rf \rangle.
\end{equation}
Since the function $f$ is linear along the fibers of $\rho:U \to M$,
we can assume that $f = \sigma(\beta)$ for a smooth $1$-form $\beta
\in C^\infty(M,\Lambda^1(M,\C)$. Then by \eqref{eta.and.sigma} and
by Lemma~\ref{phi.and.tau} the right hand side of \eqref{thelema} is
equal to
$$
\langle \phi, d^rf \rangle = \langle \phi, d^r(\sigma(\beta))
\rangle = \langle \phi, \eta(\beta) \rangle = \sqrt{-1}
\tau(\beta).
$$
Therefore, \eqref{eee.to.prove} is equivalent to
\begin{equation}\label{horus}
\sqrt{-1}\tau(\beta) =
\tau\left(\frac{\sqrt{-1}}{2}\delta_\rho((D-D^\iota)\sigma(\beta))\right).
\end{equation}
But we have $\tau = \sigma \circ \zeta$, where
$\zeta:\rho^*\Lambda^1(M,\C) \to \rho^*\Lambda^1(M,\C)$ is the
invulution introduced in \eqref{zeta}. In particular, the map
$\zeta$ is invertible, so that \eqref{horus} is in turn equivalent
to
$$
\sigma(\beta) = \frac{1}{2}\sigma(\delta_\rho((D-D^\iota)\sigma(\beta))),
$$
or, substituting back $f = \sigma(\beta)$, to
$$
f = \frac{1}{2}\sigma(\delta_\rho((D-D^\iota)f)),
$$
which is exactly the condition \eqref{expl.lin}.
\hfill \ensuremath{\square}\par
\refstepcounter{subsubsection \noindent {\bf Definition.\ } \label{hodge.conn.lin}
A Hodge connection $D$ on the pair $\langle U, M \rangle$ is called
{\em linear} if it satisfies the condition~\ref{expl.lin}.
We can now formulate and prove the following more useful version of
Proposition~\ref{equiv}.
\begin{prop}\label{equiv.bis}
Every linear Hodge connection $D\!:\!\Lambda^0(U,\C) \to
\rho^*\Lambda^1(M,\C)$ on the pair $\langle U, M\rangle$ defines a
linear Hodge manifold structure on an open neighborhood $V \subset
U$ of the zero section $M \subset U$, and the canonical projection
$\rho:V_I \to M$ is holomorphic for the preferred complex structure
$V_I$ on $V$. Vice versa, every such linear Hodge manifold structure
on $U$ comes from a unique linear Hodge connection $D$ on the pair
$\langle U,M \rangle$.
\end{prop}
\proof By Proposition~\ref{equiv} and Lemma~\ref{explicit.lin}, to
prove this proposition suffices to prove that if a Hodge connection
$D:\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)$ is linear, then it is
holonomic in the sense of \ref{holonomic} on a open neighborhood $V
\subset U$ of the zero section $M \subset U$. Lemma~\ref{aux1}
reduces this to proving that the principal part $D_0:S^1(M,\C) \to
\Lambda^1(M,\C)$ of a linear Hodge connection $D:\Lambda^0(U,\C) \to
\rho^*\Lambda^1(M,\C)$ is a bundle isomorphism.
Let $D:\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)$ be such
connection. By \eqref{expl.lin} we have
$$
\frac{1}{2} \sigma \circ (D_0 - D_0^\iota) = {\sf id}:S^1(M,\C) \to
\Lambda^1(M,\C) \to S^1(M,\C).
$$
Since $\sigma:\Lambda^1(M,\C) \to S^1(M,\C)$ is a bundle
isomorphism, so is the bundle map $D_0 - D_0^\iota:S^1(M,\C) \to
\Lambda^1(M,\C)$. As in the proof of Lemma~\ref{aux3}, we have $D_0
= - D_0^\iota$. Thus $D_0 = \frac{1}{2}(D_0-D_0^\iota):S^1(M,\C) \to
\Lambda^1(M,\C)$ also is a bundle isomorphism, which proves the
proposition.
\hfill \ensuremath{\square}\par
\section{Formal completions}\label{formal.section}
\subsection{Formal Hodge manifolds}
\refstepcounter{subsubsection Proposition~\ref{equiv.bis} reduces the study of arbitrary
regular Hodge manifolds to the study of connections of a certain
type on a neighborhood $U \subset {\overline{T}M}$ of the zero section $M
\subset {\overline{T}M}$ in the total space ${\overline{T}M}$ of the tangent bundle to a
complex manifold $M$. To obtain further information we will now
restrict our attention to the {\em formal} neighborhood of this zero
section. This section contains the appropriate definitions. We study
the convergence of our formal series in Section~\ref{convergence}.
\refstepcounter{subsubsection Let $X$ be a smooth manifold and let $\Bun(X)$ be the category of
smooth real vector bundles over $X$. Let also $\Diff(X)$ be the
category with the same objects as $\Bun(X)$ but with differential operators
as morphisms.
Consider a closed submanifold $Z \subset X$. For every two real vector
bundles $\E$ and $\F$ on $X$ the vector space $\Hom(\E,\F)$ of bundle maps
from $\E$ to $\F$ is naturally a module over the ring $C^\infty(X)$ of
smooth functions on $X$. Let $\J_Z \subset C^\infty(X)$ be the ideal of
functions that vanish on $Z$ and let $\Hom_Z(\E,\F)$ be the $\J_Z$-adic
completion of the $C^\infty(X)$-module $\Hom(\E,\F)$.
For any three bundles $\E,\F,\G$ the composition map
$$
\Mult:\Hom(\E,\F) \otimes \Hom(\F,\G) \to \Hom(\E,\G)
$$
is $C^\infty(X)$-linear, hence extends to a map
$$
\Mult:\Hom_Z(\E,\F) \otimes \Hom_Z(\F,\G) \to \Hom_Z(\E,\G).
$$
Let $\Bun_Z(X)$ be the category with the same objects as $\Bun(X)$
and for every two objects $\E$, $\F \in \Ob\Bun(X)$ with
$\Hom_Z(\E,\F)$ as the space of maps between $\F$ anf $\F$. The
category $\Bun_Z(X)$, as well as $\Bun(X)$, is an additive tensor
category.
\refstepcounter{subsubsection The space of differential operators $\Diff(\E,\F)$ is also a
$C^\infty(X)$ module, say, by left multiplication. Let $\Diff_Z(\E,\F)$ be
its $\J_Z$-completion. The composition maps in $\Diff(X)$ are no longer
$C^\infty(X)$-linear. However, they still are compatible with the
$\J_Z$-adic topology, hence extend to completions. Let $\Diff_Z(X)$ be the
category with the same objects as $\Bun(X)$ and with $\Diff_Z(\E,\F)$ as
the space of maps between two objects $\E,\F \in \Ob \Bun(X)$.
By construction we have canonical {\em $Z$-adic completion functors}
$$
\Bun(X) \to \Bun_Z(X) \text{ and } \Diff(X) \to \Diff_Z(X).
$$
Call the categories $\Bun_Z(X)$ and $\Diff_Z(X)$ {\em the $Z$-adic
completions} of the categories $\Bun(X)$ and $\Diff(X)$.
\refstepcounter{subsubsection When the manifold $X$ is equipped with a smooth action of
compact Lie group $G$ fixing the submanifold $Z$, the completion
construction extends to the categories of $G$-equivariant bundles on
$M$. When $G = U(1)$, the categories ${{\cal W}{\cal H}odge}(X)$ and
${{\cal W}{\cal H}odge}^\D(X)$ defined in \ref{w.hodge} also admit canonical
completions, denoted by ${{\cal W}{\cal H}odge}_Z(X)$ and ${{\cal W}{\cal H}odge}_Z^\D(X)$.
\refstepcounter{subsubsection Assume now that the manifold $X$ is equipped with a smooth
$U(1)$-action fixing the smooth submanifold $Z \subset X$.
\noindent {\bf Definition.\ } A {\em formal quaternionic structure} on $X$ along the
submanifold $Z \subset X$ is given by an algebra map
$$
\Mult: {\Bbb H} \to {{\cal E}\!nd\:}_{\Bun_Z(X)} \left(\Lambda^1(X)\right)
$$
from the algebra ${\Bbb H}$ to the algebra ${{\cal E}\!nd\:}_{\Bun_Z(X)}
\left(\Lambda^1(X)\right)$ of endomorphisms of the cotangent bundle
$\Lambda^1(X)$ in the category $\Bun_Z(X)$.
A formal quaternionic structure is called {\em equivariant} if the
map $\Mult$ is equivariant with respect to the natural $U(1)$-action
on both sides.
\refstepcounter{subsubsection Note that Lemma~\ref{universal} still holds in the situation
of formal completions. Consequently, everything in
Section~\ref{hbqm.section} carries over word-by-word to the case of
formal quaternionic structures. In particular, by
Lemma~\ref{qm.hodge} giving a formal equivariant quaternionic
structure on $X$ along $Z$ is equivalent to giving a pair $\langle
\E, D \rangle$ of a Hodge bundle $\E$ on $X$ and a holonomic algebra
derivation $D:\Lambda^0(X) \to \E$ in ${{\cal W}{\cal H}odge}_Z^\D(X)$.
\refstepcounter{subsubsection The most convenient way to define Hodge manifold structures
on $X$ in a formal neighborhood of $Z$ is by means of the Dolbeault
complex, as in Proposition~\ref{explicit.hodge}.
\noindent {\bf Definition.\ } A {\em formal Hodge manifold structure} on $X$ along $Z$ is a pair of
a pre-Hodge bundle $\E \in \Ob {{\cal W}{\cal H}odge}_Z(X)$ of weight $1$ and an
algebra derivation $D^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\E \to \Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}\E$ in
${{\cal W}{\cal H}odge}^\D_Z(X)$ such that $D^0:\Lambda^0(\E) \to \E$ is
holonomic and $D^0 \circ D^1 = 0$.
\refstepcounter{subsubsection Let $U \subset X$ be an open subset containing $Z \subset
X$. For every Hodge manifold structure on $U$ the $Z$-adic
completion functor defines a formal Hodge manifold structure on $X$
along $Z$. Call it {\em the $Z$-adic completion} of the given
structure on $U$.
\noindent {\bf Remark.\ } Note that a Hodge manifold structure on $U$ is completely
defined by the preferred and the complementary complex structures
$U_I$, $U_J$, hence always real-analytic by the Newlander-Nirenberg
Theorem. Therefore, if two Hodge manifold structures on $U$ have
the same completion, they coincide on every connected component of
$U$ intersecting $Z$.
\subsection{Formal Hodge manifold structures on tangent bundles}
\refstepcounter{subsubsection Let now $M$ be a complex manifold, and let ${\overline{T}M}$ be the total
space of the complex-conjugate to the tangent bundle to $M$ equipped
with an action of $U(1)$by dilatation along the fibers of the
projection $\rho:{\overline{T}M} \to M$. All the discussion above applies to the
case $X = {\overline{T}M}$, $Z = M \subset {\overline{T}M}$. Moreover, the linearity
condition in the form given in Lemma~\ref{lin.char} generalizes
immediately to the formal case.
\noindent {\bf Definition.\ } A formal Hodge manifold structure on ${\overline{T}M}$ along $M$ is called
{\em linear} if for every smooth $(0,1)$-form $\alpha \in
C^\infty(M,\Lambda^{0,1}(M))$ we have
$$
\tau(\alpha) = \langle \phi, j(\rho^*) \rangle \in
C^\infty_M({\overline{T}M},\C),
$$
where $j$ is the map induced by the formal quaternionic structure on
${\overline{T}M}$ and $\phi$ and $\tau$ are as in Lemma~\ref{lin.char}.
\refstepcounter{subsubsection As in the non-formal case, linear Hodge manifold structures
on ${\overline{T}M}$ along $M \subset {\overline{T}M}$ can be described in terms of
differential operators of certain type.
\noindent {\bf Definition.\ } \label{formal.hodge.con}
A {\em formal Hodge connection} on ${\overline{T}M}$ along $M \subset {\overline{T}M}$ is an
algebra derivation
$$
D:\Lambda^0({\overline{T}M},\C) \to \rho^*\Lambda^1(M,\C)
$$
in ${{\cal W}{\cal H}odge}_M^\D({\overline{T}M})$ such that for every smooth function $f \in
C^\infty(M,\C)$ we have $D\rho^*=\rho^*d_Mf$, as in
\eqref{conn.eq}. A formal Hodge connection is called {\em flat} if
it extends to an algebra derivation
$$
D:\rho^*\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C)
$$
in ${{\cal W}{\cal H}odge}_M^\D({\overline{T}M})$ such that $D \circ D = 0$. A formal Hodge
connection is called {\em linear} if it satisfies the condition
\eqref{expl.lin} of Lemma~\ref{explicit.lin}, that is, for every
function $f \in C^\infty_{lin}({\overline{T}M},\C)$ linear along the fibers of
the projection $\rho:{\overline{T}M} \to M$ we have
$$
f = \frac{1}{2}\sigma\left((D-D^\iota)f\right),
$$
where $\sigma:\rho^*\Lambda^1(M,\C) \to \Lambda^0({\overline{T}M},\C)$ is the
twisted tautological map introduced in \eqref{sigma1}, the
automorphism $\iota:{\overline{T}M} \to {\overline{T}M}$ is the multiplication by $-1 \in
\C$ on every fiber of the projection $\rho:{\overline{T}M} \to M$, and
$D^\iota:\Lambda^0({\overline{T}M},\C) \to \rho^*\Lambda^1(M,\C)$ is the
operator $\iota^*$-conjugate to $D$, as in \ref{aux}.
The discussion in Section~\ref{section.5} generalizes immediately to
the formal case and gives the following.
\begin{lemma}\label{form.hdg}
Linear formal Hodge manifold structures on ${\overline{T}M}$ along the zero
section $M \subset {\overline{T}M}$ are in a natural one-to-one correspondence
with linear flat formal Hodge connections on ${\overline{T}M}$ along $M$.
\end{lemma}
\subsection{The Weil algebra}
\refstepcounter{subsubsection Let, as before, $M$ be a complex manifold and let ${\overline{T}M}$ be
the total space of the complex conjugate to its tangent bundle, as
in \ref{overline.T}. In the remaining part of this section we give
a description of the set of all formal Hodge connections on ${\overline{T}M}$
along $M$ in terms of certain differential operators on $M$ rather
than on ${\overline{T}M}$. We call such operators {\em extended connections} on
$M$ (see \ref{ext.con} for the definition). Together with a
complete classification of extended connections given in the next
Section, this description provides a full classification of regular
Hodge manifolds ``in the formal neighborhood of the subset of
$U(1)$-fixed points''.
\refstepcounter{subsubsection Before we define extended connections in
Subsection~\ref{ext.con.subsec}), we need to introduce a certain
algebra bundle in ${{\cal W}{\cal H}odge}(M)$ which we call {\em the Weil
algebra}. We begin with some preliminary facts.
Recall (see, e.g., \cite{Del}) that every additive category $\A$
admits a canonical completion $\displaystyle\operatornamewithlimits{Lim}_{\longleftarrow}\A$ with respect to filtered
projective limits. The category $\displaystyle\operatornamewithlimits{Lim}_{\longleftarrow}\A$ is also additive, and it
is tensor if $\A$ was tensor. Objects of the canonical completion
$\displaystyle\operatornamewithlimits{Lim}_{\longleftarrow}\A$ are called {\em pro-objects in $\A$}.
\refstepcounter{subsubsection\label{rho_*}
Let $\rho:{\overline{T}M} \to M$ be the canonical projection. Extend the
pullback functor $\rho^*:\Bun(M) \to \Bun({\overline{T}M})$ to a functor
$$
\rho^*:\Bun(M) \to \Bun_M({\overline{T}M})
$$
to the $M$-adic completion $\Bun_M({\overline{T}M})$. The functor $\rho^*$
admits a right adjoint direct image functor
$$
\rho_*:\Bun_M({\overline{T}M}) \to \displaystyle\operatornamewithlimits{Lim}_{\longleftarrow}\Bun(M).
$$
Moreover, the functor $\rho_*$ extends to a functor
$$
\rho_*:\Diff_M({\overline{T}M}) \to \displaystyle\operatornamewithlimits{Lim}_{\longleftarrow}\Diff(M).
$$
Denote by $\B^0(M,\C) = \rho_*\Lambda^0({\overline{T}M})$ the direct image under
the projection $\rho:{\overline{T}M} \to M$ of the trivial bundle
$\Lambda^0({\overline{T}M})$ on ${\overline{T}M}$.
The compact Lie group $U(1)$ acts on ${\overline{T}M}$ by dilatation along the
fibers, and the functor $\rho_*:\Diff_M({\overline{T}M}) \to \displaystyle\operatornamewithlimits{Lim}_{\longleftarrow}\Diff(M)$
obviously extends to a functor $\rho_*:{{\cal W}{\cal H}odge}^\D_M({\overline{T}M}) \to
\displaystyle\operatornamewithlimits{Lim}_{\longleftarrow}{{\cal W}{\cal H}odge}(M)$. The restriction of $\rho_*$ to the subcategory
${{\cal W}{\cal H}odge}_M({\overline{T}M}) \subset {{\cal W}{\cal H}odge}^\D_M({\overline{T}M})$ is adjoint on the
right to the pullback functor $\rho^*:{{\cal W}{\cal H}odge}(M) \to
{{\cal W}{\cal H}odge}_M({\overline{T}M})$.
\refstepcounter{subsubsection The constant bundle $\Lambda^0({\overline{T}M})$ is canonically a Hodge
bundle of weight $0$. Therefore $\B^0(M,\C) = \rho_*\Lambda^0(M,\C)$
is also a Hodge bundle of weight $0$. Moreover, it is a commutative
algebra bundle in $\displaystyle\operatornamewithlimits{Lim}_{\longleftarrow}{{\cal W}{\cal H}odge}_0(M)$. Let $S^1(M,\C)$ be the
conormal bundle to the zero section $M \subset {\overline{T}M}$ equipped with a
Hodge bundle structure of weight $0$ as in \ref{S1}, and denote by
$S^i(M,\C)$ the $i$-th symmetric power of the Hodge bundle
$S^1(M,\C)$. Then the algebra bundle $\B^0(M,\C)$ in
$\displaystyle\operatornamewithlimits{Lim}_{\longleftarrow}{{\cal W}{\cal H}odge}_0(M)$ is canonically isomorphic
$$
\B^0(M,\C) \cong \widehat{S}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)
$$
to the completion $\widehat{S}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ of the symmetric algebra
$S^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ of the Hodge bundle $S^1(M,\C)$ with respect to the
augmentation ideal $S^{>0}(M,\C)$.
Since the $U(1)$-action on $M$ is trivial, the category
${{\cal W}{\cal H}odge}(M)$ of Hodge bundles on $M$ is equivalent to the category
of pairs $\langle \E, \overline{\ } \rangle$ of a complex bundle $\E$
equipped with a Hodge type grading
$$
\E = \bigoplus_{p,q} \E^{p,q}
$$
and a real structure $\overline{\ }:\E^{p,q} \to \overline{\E^{q,p}}$. The
Hodge type grading on $\B^0(M,\C)$ is induced by the Hodge type
grading $S^1(M,\C) = S^{1,-1}(M) \oplus S^{-1,1}(M)$ on the
generators subbundle $S^1(M,\C) \subset \B^0(M,\C)$, which was
described in \ref{S1}.
\noindent {\bf Remark.\ } The complex vector bundle $S^1(M,\C)$ is canonically isomorphic
to the cotangent bundle $\Lambda^1(M,\C)$. However, the Hodge bundle
structures on these two bundles are different (in fact, they have
different weights).
\refstepcounter{subsubsection \label{Weil.defn}
Consider the pro-bundles
$$
\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) = \rho_*\rho^*\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)
$$
on $M$. The direct sum $\oplus\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ is a graded algebra in
$\displaystyle\operatornamewithlimits{Lim}_{\longleftarrow}\Bun(M,\C)$. Moreover, since for every $i \geq 0$ the bundle
$\Lambda^i(M,\C)$ is a Hodge bundle of weight $i$ (see
\ref{de.Rham}), $\B^i(M,\C)$ is also a Hodge bundle of weight
$i$. Denote by
$$
\B^i(M,\C) = \bigoplus_{p+q=i} \B^{p,q}(M,\C)
$$
the Hodge type bigrading on $\B^i(M,\C)$.
The Hodge bundle structures on $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ are compatible with
the multiplication. By the projection formula
$$
\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \cong \B^0(M,\C) \otimes \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C),
$$
and this isomorphism is compatible with the Hodge bundle structures on
both sides.
\noindent {\bf Definition.\ } Call the algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ in $\displaystyle\operatornamewithlimits{Lim}_{\longleftarrow}{{\cal W}{\cal H}odge}(M)$ {\em
the Weil algebra} of the complex manifold $M$.
\refstepcounter{subsubsection \label{iota.Weil}
The canonical involution $\iota:{\overline{T}M} \to {\overline{T}M}$ induces an algebra
involution $\iota^*:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$. It acts on
generators as follows
$$
\iota^* = -{\sf id}:S^1(M,\C) \to S^1(M,\C) \qquad
\iota^* = {\sf id}:\Lambda^1(M,\C) \to \Lambda^1(M,\C).
$$
For every operator $N:\B^p(M,\C) \to \B^q(M,\C)$, $p$ and $q$
arbitrary, we will denote by
$$
N^\iota = \iota^* \circ N \circ \iota^*:\B^p(M,\C) \to
\B^q(M,\C)
$$
the operator $\iota^*$-conjugate to $N$.
\refstepcounter{subsubsection \label{sigma}
The twisted tautological map $\sigma:\rho^*\Lambda^1(M,\C) \to
\Lambda^0({\overline{T}M},\C)$ introduced in \ref{sigma1} induces via the
functor $\rho_*$ a map $\sigma:\B^1(M,\C) \to \B^0(M,\C)$. Extend
this map to a derivation
$$
\sigma:\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C) \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)
$$
by setting $\sigma = 0$ on $S^1(M,\C) \subset \B^0(M,\C)$. The
derivation $\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C) \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ is not weakly
Hodge. However, it is real with respect to the real structure on the
Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$.
\refstepcounter{subsubsection \label{C.Weil}
By definition of the twisted tautological map (\ref{sigma1},
\ref{tau}), the derivation $\sigma:\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C) \to
\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ maps the subbundle $\Lambda^1(M,\C) \subset
\B^1(M,\C)$ to the subbundle $S^1(M,\C) \subset \B^0(M,\C)$ and
defines a complex vector bundle isomorphism $\sigma:\Lambda^1(M,\C)
\to S^1(M,\C)$. To describe this isomorphism explicitly, recall that
sections of the bundle $\B^0(M,\C)$ are the same as formal germs
along $M \subset {\overline{T}M}$ of smooth functions on the manifold ${\overline{T}M}$. The
sections of the subbundle $S^1(M,\C) \subset \B^0(M,\C)$ form the
subspace of functions linear along the fibers of the canonical
projection $\rho:{\overline{T}M} \to M$. The isomorphism $\sigma:\Lambda^1(M,\C)
\to S^1(M,\C)$ induces an isomorphism between the space of smooth
$1$-forms on the manifold $M$ and the space of smooth functions on
${\overline{T}M}$ linear long the fibers of $\rho:{\overline{T}M} \to M$. This isomorphism
coincides with the tautological one on the subbundle $\Lambda^{1,0}
\subset \Lambda^1(M,\C)$, and it is minus the tautological
isomorphism on the subbundle $\Lambda^{0,1} \subset
\Lambda^1(M,\C)$.
Denote by
$$
C = \sigma^{-1}:S^1(M,\C) \to \Lambda^1(M,\C)
$$
the bundle isomorphism inverse to $\sigma$. Note that the complex
vector bundle isomorphism $\sigma:\Lambda^1(M,\C) \to S^1(M,\C)$ is
real. Moreover, it sends the subbundle $\Lambda^{1,0}(M) \subset
\Lambda^1(M,\C)$ to $S^{1,-1}(M) \subset S^1(M,\C)$, and it sends
$\Lambda^{0,1}(M)$ to $S^{-1,1}(M)$. Therefore the inverse
isomorphism $C:S^1(M,\C) \to \Lambda^1(M,\C)$ is weakly Hodge. It
coincides with the tautological isomorphism on the subbundle
$S^{1,-1} \subset S^1(M,\C)$, and it equals minus the tautological
isomorphism on the subbundle $S^{-1,1} \subset S^1(M,\C)$.
\subsection{Extended connections}\label{ext.con.subsec}
\refstepcounter{subsubsection We are now ready to introduce the extended connections. Keep
the notation of the last subsection.
\noindent {\bf Definition.\ } \label{ext.con}
An {\em extended connection} on a complex manifold $M$ is a
differential operator $D:S^1(M,\C) \to \B^1(M,\C)$ which is weakly
Hodge in the sense of \ref{w.hodge} and satisfies
\begin{equation}\label{e.c}
D(fa) = fDa + a \otimes df
\end{equation}
for any smooth function $f$ and a local section $a$ of the pro-bundle
$\B^0(M,\C)$.
\refstepcounter{subsubsection \label{red}
Let $D$ be an extended connection on the manifold $M$.
By \ref{Weil.defn} we have canonical bundle isomorphisms
$$
\B^1(M,\C) \cong \B^0(M,\C) \otimes \Lambda^1(M,\C) \cong \bigoplus_{i
\geq 0} S^i(M,\C) \otimes \Lambda^1(M,\C).
$$
Therefore the operator $D:S^1 \to \B^1$ decomposes
\begin{equation}\label{aug.con}
D = \sum_{p \geq 0}D_p, \quad D_p:S^1(M,\C) \to S^i(M,\C) \otimes
\Lambda^1(M,\C).
\end{equation}
By \eqref{e.c} all the components $D_p$ except for the $D_1$ are
weakly Hodge bundle maps on $M$, while
$$
D_1:S^1(M,\C) \to S^1(M,\C) \otimes \Lambda^1(M,\C)
$$
is a connection in the usual sense on the Hodge bundle $S^1(M,\C)$.
\noindent {\bf Definition.\ } The weakly Hodge bundle map $D_0:S^1(M,\C) \to
\Lambda^1(M,\C)$ is called {\em the principal part} of the extended
connection $D$ on $M$. The connection $D_1$ is called {\em the
reduction} of the extended connection $D$.
\refstepcounter{subsubsection Extended connection on $M$ are related to formal Hodge
connections on the total space ${\overline{T}M}$ of the complex-conjugate to the
tangent bundle to $M$ by means of the direct image functor
$$
\rho_*:{{\cal W}{\cal H}odge}^\D_M({\overline{T}M}) \to \displaystyle\operatornamewithlimits{Lim}_{\longleftarrow}{{\cal W}{\cal H}odge}^\D(M).
$$
Namely, let $D:\Lambda^0(M,\C) \to \rho^*\Lambda^1(M,\C)$ be a
formal Hodge connection on ${\overline{T}M}$ along $M$ in the sense of
\ref{formal.hodge.con}. The restriction of the operator
$$
\rho_*D:\B^0(M,\C) \to \B^1(M,\C)
$$
to the generators subbundle $S^1(M,\C) \subset \B^0(M,\C)$ is then
an extended connection on $M$ in the sense of \ref{ext.con}. The
principal part $D_0:S^1(M,\C) \to \Lambda^1(M,\C)$ of the Hodge
connection $D$ in the sense of \ref{pr.part} coincides with the
principal part of the extended connection $\rho_*D$.
\refstepcounter{subsubsection We now write down the counterparts of the flatness and
linearity conditions on a Hodge connection on ${\overline{T}M}$ for the
associated extended connection on $M$. We begin with the linearity
condition \ref{hodge.conn.lin}. Let $D:S^1(M,\C) \to \B^1(M,\C)$ be
an extended connection on $M$, let $\sigma:\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C) \to
\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ be the algebra derivation introduced in \ref{sigma},
and let
$$
D^\iota:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C)
$$
be the operator $\iota^*$-conjugate to $D$ as in \ref{iota.Weil}.
\noindent {\bf Definition.\ } \label{lin.ext.con}
An extended connection $D$ is called {\em linear} if for
every local section $f$ of the bundle $S^1(M,\C)$
we have
$$
f = \frac{1}{2} \sigma((D-D^\iota)f).
$$
This is, of course, the literal rewriting of
Definition~\ref{hodge.conn.lin}. In particular, a formal Hodge
connection $D$ on ${\overline{T}M}$ is linear if and only if so is the extended
connection $\rho_*D$ on $M$.
\refstepcounter{subsubsection \label{deriv}
Next we rewrite the flatness condition \ref{hodge.con}. Again, let
$$
D:S^1(M,\C) \to \B^1(M,\C)
$$
be an extended connection on $M$. Since the algebra pro-bundle
$\B^0(M,\C)$ is freely generated by the subbundle $S^1(M,\C) \subset
\B^1(M,\C)$, by \eqref{e.c} the operator $D:S^1(M,\C) \to
\B^1(M,\C)$ extends uniquely to an algebra derivation
$$
D:\B^0(M,\C) \to \B^1(M,\C).
$$
Moreover, we can extend this derivation even further to a
derivation of the Weil algebra
$$
D:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C)
$$
by setting
\begin{equation}\label{D=d}
D(f \otimes \alpha) = Df \otimes \alpha + f \otimes d\alpha
\end{equation}
for any smooth section $f \in C^\infty(M,\B^0(M,\C))$ and any smooth
form $\alpha \in C^\infty(M,\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C))$. We will call this
extension {\em the derivation, associated to the extended connection
$D$}.
Vice versa, the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ is generated by the
subbundles
$$
S^1(M,\C),\Lambda^1(M,\C) \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C).
$$
Moreover, for every algebra derivation $D:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to
\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C)$ the condition \eqref{D=d} completely defines the
restriction of $D$ to the generator subbundle $\Lambda^1(M,\C)
\subset \B^1(M,\C)$. Therefore an algebra derivation
$D:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ satisfying \eqref{D=d} is
completely determined by its restriction to the generators subbundle
$S^1(M,\C) \to \B^1(M,\C)$. If the derivation $D$ is weakly Hodge,
then this restriction is an extended connection on $M$.
\refstepcounter{subsubsection \noindent {\bf Definition.\ } \label{flat.ext.con}
The extended connection $D$ is called {\em flat} if the associated
derivation satisfies $D \circ D = 0$.
If a formal Hodge connection $D$ on ${\overline{T}M}$ is flat in the sense of
\ref{hodge.con}, then we have a derivation
$D:\rho^*\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \rho^*\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C)$ such
that $D \circ D = 0$. The associated derivation
$\rho_*D:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C)$ satisfies
\eqref{D=d}. Therefore the extended connection $\rho_*D:S^1(M,\C)
\to \B^1(M,\C)$ is also flat.
\refstepcounter{subsubsection It turns out that one can completely recover a Hodge
connection $D$ on ${\overline{T}M}$ from the corresponding extended connection
$\rho_*D$ on $M$. More precisely, we have the following.
\begin{lemma}
The correspondence $D \mapsto \rho_*D$ is a bijection between the
set of formal Hodge connections on ${\overline{T}M}$ along $M \subset {\overline{T}M}$ and
the set of extended connections on $M$. A connection $D$ is flat,
resp. linear if and only if $\rho_*D$ is flat,
resp. linear.
\end{lemma}
\proof To prove the first claim of the lemma, it suffices to prove
that every extended connection on $M$ comes from a unique formal
Hodge connection on the pair $\langle {\overline{T}M},M\rangle$. In general, the
functor $\rho_*$ is not fully faithful on the category $\Diff(M)$,
in other words, it does not induce an isomorphism on the spaces of
differential operators between vector bundles on ${\overline{T}M}$. However, for
every complex vector bundle $\E$ on ${\overline{T}M}$ the functor $\rho_*$ does
induce an isomorphism
$$
\rho_*:\Der_M(\Lambda^0(M,\C),\E) \cong
\Der_{\B^0(M,\C)}(\B^0(M,\C),\rho_*\E)
$$
between the space of {\em derivations} from $\Lambda^0(M,\C)$ to
$\F$ completed along $M \subset {\overline{T}M}$ and the space of derivations
from the algebra $\B^0(M,\C) = \rho_*\Lambda^0(M,\C)$ to the
$\B^0(M,\C)$-module $\rho_*\E$. Therefore every derivation
$$
D':\B^0(M,\C) \to \B^1(M,\C) = \rho_*\rho^*\Lambda^1(M,\C)
$$
must be of the form $D'=\rho_*D$ for some derivation
$$
D:\Lambda^0({\overline{T}M},\C) \to \rho^*\Lambda^1(M,\C)
$$
It is easy to check that $D$ is a Hodge connection if and only if
$D'=\rho_*D$ is weakly Hodge and satisfies \eqref{e.c}. By
\ref{deriv} the space of all such derivations $D':\B^0(M,\C) \to
\B^1(M,\C)$ coincides with the space of all extended connections on
$M$, which proves the first claim of the lemma.
Analogously, for every extended connection $D' = \rho_*D$ on $M$,
the canonical extension of the operator $D'$ to an algebra
derivation $D':\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C)$ constructed in
\ref{deriv} must be of the form $\rho_*D$ for a certain weakly Hodge
differential operator $D:\rho^*\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to
\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C)$. If the extended connection $D'$ is flat,
then $D' \circ D' = 0$. Therefore $D \circ D = 0$, which means that
the Hodge connection $D$ is flat. Vice versa, if the Hodge
connection $D$ is flat, then it extends to a weakly Hodge derivation
$D:\rho^*\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C)$ so that $D
\circ D = 0$. The equality $D \circ D = 0$ implies, in particular,
that the operator $\rho_*D$ vanishes on the sections of the form
$$
Df = df \in C^\infty(M,\Lambda^1(M,\C)) \subset
C^\infty(M,\B^1(M,\C)),
$$
where $f \in C^\infty(M,\C)$ is a smooth function on $M$. Therefore
$\rho_*D$ coincides with the de Rham differential on the subbundle
$\Lambda^1(M,\C) \subset \B^1(M,\C)$. Hence by \ref{deriv} it is
equal to the canonical derivation $D':\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to
\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C)$. Since $D \circ D = 0$, we have $D' \circ D' =
0$, which means that the extended connection $D'$ is flat.
Finally, the equivalence of the linearity conditions on the Hodge
connection $D$ and on the extended connection $D' = \rho_*D$ is
trivial and has already been noted in \ref{lin.ext.con}.
\hfill \ensuremath{\square}\par
This lemma together with Lemma~\ref{form.hdg} reduces the
classification of linear formal Hodge manifold structures on ${\overline{T}M}$
along the zero section $M \subset {\overline{T}M}$ to the classification of
extended connections on the manifold $M$ itself.
\section{Preliminaries on the Weil algebra}\label{Weil.section}
\subsection{The total de Rham complex}\label{de.rham.sub}
\refstepcounter{subsubsection Before we proceed further in the study of extended
connections on a complex manifold $M$, we need to establish some
linear-algebraic facts on the structure of the Weil algebra
$\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ defined in \ref{Weil.defn}. We also need to
introduce an auxiliary Hodge bundle algebra on $M$ which we call
{\em the total Weil algebra}. This is the subject of this
section. Most of the facts here are of a technical nature, and the
reader is advised to skip this section until needed.
\refstepcounter{subsubsection We begin with introducing and studying a version of the de
Rham complex of a complex manifold $M$ which we call {\em the total
de Rham complex}. Let $M$ be a smooth complex
$U(1)$-manifold. Recall that by \ref{de.Rham} the de Rham complex
$\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ of the complex manifold $M$ is canonically a
Hodge bundle algebra on $M$. Let $\Lambda_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M) =
\Gamma(\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C))$ be the weight $0$ Hodge bundle obtained
by applying the functor $\Gamma$ defined in \ref{gamma.m} to the de
Rham algebra $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$. By \ref{gamma.tensor} the bundle
$\Lambda_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M)$ carries a canonical algebra structure. By
\ref{de.Rham} the de Rham differential $d_M$ is weakly
Hodge. Therefore it induces an algebra derivation
$d_M:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M) \to \Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot}(M)$ which is
compatible with the Hodge bundle structure and satisfies $d_M \circ
d_M = 0$.
\noindent {\bf Definition.\ } The weight $0$ Hodge bundle algebra $\Lambda_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M)$ is
called {\em the total de Rham complex} of the complex manifold $M$.
\refstepcounter{subsubsection By definition
$$
\Lambda^i_{tot}(M) = \Gamma(\Lambda^i(M,\C)) = \Lambda^i(M,\C) \otimes \W^*_i,
$$
where $\W^*_i = S^i\W^*_1$ is the symmetric power of the $\R$-Hodge
structure $\W^*_1$, as in \ref{w.k}. To describe the structure of
the algebra $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M)$, we will use the following
well-known general fact. (For the sake of completeness, we have
included a sketch of its proof, see \ref{gen.symm.proof}.)
\begin{lemma}\label{symm}
Let $A$, $B$ be two objects in an arbitrary $\Q$-linear symmetric
tensor category $\A$, and let $\CC^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} = S^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(A \otimes B)$ be
the sum of symmetric powers of the object $A \otimes B$. Note that
the object $\CC^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is naturally a commutative algebra in $\A$ in
the obvious sense. Let also $\wt{\CC}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} = \bigoplus_k S^kA
\otimes S^kB$ with the obvious commutative algebra structure. The
isomorphism $\CC^1 \cong \wt{\CC}^1 \cong A \otimes B$ extends to a
surjective algebra map $\CC^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \wt{\CC}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$, and its
kernel $\J^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \subset \CC^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is the ideal generated by the
subobject $\J^2 = \Lambda^2(A) \otimes \Lambda^2(B) \subset S^2(A
\otimes B)$.
\end{lemma}
\refstepcounter{subsubsection The category of complexes of Hodge bundles on $M$ is
obviously $\Q$-linear and tensor. Applying
Lemma~\ref{symm} to $A = \W^*_1$, $B = \Lambda^1(M,\C)
[1]$ immediately gives the following.
\begin{lemma}\label{total.rel}
The total de Rham complex $\Lambda_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M)$ of the complex
manifold $M$ is generated by its first component $\Lambda^1_{tot}(M)$,
and the kernel of the canonical surjective algebra map
$$
\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(\Lambda_{tot}^1(M)) \to \Lambda_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M)
$$
from the exterior algebra of the bundle $\Lambda^1_{tot}(M)$ to
$\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M)$ is the ideal generated by the subbundle
$$
\Lambda^2\W_1 \otimes S^2(\Lambda^1(M,\C)) \subset S^2(\Lambda^1_{tot}(M)).
$$
\end{lemma}
\refstepcounter{subsubsection \label{S}
We can describe the Hodge bundle $\Lambda^1_{tot}(M)$ more explicitly
in the following way. By definition, as a $U(1)$-equivariant complex
vector bundle it equals
$$
\Lambda^1_{tot}(M) = \Lambda^1(M,\C) \otimes \W_1^* = \left(
\Lambda^{1,0}(M)(1) \oplus \Lambda^{0,1}(M)(0)\right) \otimes \left(\C(0)
\oplus \C(-1)\right),
$$
where $\Lambda^{p,q}(M)(i)$ is the $U(1)$-equivariant bundle
$\Lambda^{p,q}(M)$ tensored with the $1$-dimensional representation
of weight $i$, and $\C(i)$ is the constant $U(1)$-bundle
corresponding to the representation of weight $i$. If we denote
\begin{align*}
S^1(M,\C) &= \Lambda^{1,0}(M)(1) \oplus \Lambda^{0,1}(M)(-1) \subset
\Lambda^1_{tot}(M),\\
\Lambda^1_{ll}(M) &= \Lambda^{1,0}(M) \subset \Lambda^1_{tot}(M),\\
\Lambda^1_{rr}(M) &= \Lambda^{0,1}(M) \subset \Lambda^1_{tot}(M),
\end{align*}
then we have
$$
\Lambda^1_{tot}(M) = S^1(M,\C) \oplus \Lambda^1_{ll}(M) \oplus
\Lambda^1_{rr}(M).
$$
The complex conjugation $\overline{\ }:\Lambda^1_{tot}(M) \to
\iota^*\overline{\Lambda^1_{tot}(M)}$ preserves the subbundle
$$
S^1(M,\C) \subset \Lambda^1_{tot}(M,\C)
$$
and interchanges $\Lambda^1_{ll}(M)$ and $\Lambda^1_{rr}(M)$.
\refstepcounter{subsubsection\label{S.Hodge.type}
If the $U(1)$-action on the manifold $M$ is trivial, then Hodge
bundles are the same as bigraded complex vector bundles with a real
structure. In this case the Hodge bigrading on the Hodge bundle
$\Lambda^1_{tot}(M,\C)$ is given by
\begin{align*}
\left(\Lambda^1_{tot}(M)\right)^{1,-1} &= S^{1,-1}(M,\C) =
\Lambda^{1,0}(M)(1),\\
\left(\Lambda^1_{tot}(M)\right)^{-1,1} &= S^{-1,1}(M,\C) =
\Lambda^{0,1}(M)(-1),\\
\left(\Lambda^1_{tot}(M)\right)^{0,0} &= \Lambda^1_{ll}(M) \oplus
\Lambda^1_{rr}(M) = \Lambda^1(M,\C).
\end{align*}
Under these identifications, the real structure on
$\Lambda^1_{tot}(M,\C)$ is minus the one induced by the usual real
structure on the complex vector bundle $\Lambda^1(M,\C)$.
\noindent {\bf Remark.\ } The Hodge bundle $S^1(M,\C)$ is canonically isomorphic to the
conormal bundle to the zero section $M \subset {\overline{T}M}$, which we have
described in \ref{S1}.
\refstepcounter{subsubsection \label{gamma.use}
Recall now that we have defined in \ref{l.r} canonical embeddings
$\gamma_l,\gamma_r:\W_p^* \to \W_k^*$ for every $0 \leq p \leq
k$. Since $\W_0^* = \C$, for every $p,q \geq 0$ we have by
\eqref{p+q} a short exact sequence
\begin{equation}\label{cap.cup}
\begin{CD}
0 @>>> \C @>>> \W_p^* \oplus \W_q^* @>{\gamma_l \oplus \gamma_r}>>
\W_{p+q}^* @>>> 0
\end{CD}
\end{equation}
of complex vector spaces. Recall also that the embeddings
$\gamma_l$, $\gamma_r$ are compatible with the natural maps
${\sf can}:\W_p^* \otimes \W_q^* \to \W_{p+q}^*$. Therefore the
subbundles defined by
\begin{align*}
\Lambda^k_l(M) &= \bigoplus_{0 \leq p \leq k} \gamma_l(\W_p^*) \otimes
\Lambda^{p,k-p}(M) \subset \Lambda^k_{tot}(M) = \bigoplus_{0 \leq p \leq k} \W_k^*
\otimes \Lambda^{p,k-p}(M)\\
\Lambda^k_r(M) &= \bigoplus_{0 \leq p \leq k} \gamma_r(\W_p^*) \otimes
\Lambda^{k-p,p}(M) \subset \Lambda^k_{tot}(M) = \bigoplus_{0 \leq p \leq k} \W_k^*
\otimes \Lambda^{k-p,p}(M)
\end{align*}
are actually subalgebras in the total de Rham complex
$\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M)$.
\refstepcounter{subsubsection \label{l.r.rel}
To describe the algebras $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_l(M)$ and
$\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_r(M)$ explicitly, note that we obviously have
$\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M) = \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_l(M) +
\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_r(M)$. Moreover, in the notation of \ref{S} we have
\begin{align*}
\Lambda^1_l(M) &= S^1(M,\C) \oplus \Lambda^1_{ll}(M) \subset \Lambda^1_{tot}(M),\\
\Lambda^1_r(M) &= S^1(M,\C) \oplus \Lambda^1_{rr}(M) \subset \Lambda^1_{tot}(M).
\end{align*}
By Lemma~\ref{symm}, the algebra
$$
\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_l(M) = \left(\bigoplus_p \W_p^* \otimes
\Lambda^{p,0}(M)\right) \otimes \left( \bigoplus_q \Lambda^{0,q}(M) \right)
$$
is the subalgebra in the total de Rham complex $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M)$
generated by $\Lambda^1_l(M)$, and the ideal of relations is
generated by the subbundle
$$
S^2(\Lambda^{1,0}(M)) \otimes \Lambda^2(\W_1^*) \subset
\Lambda^2(\Lambda^1_l(M)).
$$
Analogously, the subalgebra $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_r(M) \subset
\Lambda^1_{tot}(M)$ is generated by $\Lambda^1_r(M)$, and the relations
are generated by
$$
S^2(\Lambda^{0,1}(M)) \otimes \Lambda^2(\W_1^*) \subset
\Lambda^2(\Lambda^1_r(M)).
$$
\refstepcounter{subsubsection We will also need to consider the ideals in these algebras
defined by
\begin{align*}
\Lambda^k_{ll}(M) &= \bigoplus_{1 \leq p \leq k} \gamma_l(\W_p^*) \otimes
\Lambda^{p,k-p}(M) \subset \Lambda^k_l(M)\\
\Lambda^k_{rr}(M) &= \bigoplus_{1 \leq p \leq k} \gamma_r(\W_p^*) \otimes
\Lambda^{k-p,p}(M) \subset \Lambda^k_r(M)
\end{align*}
The ideal $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{ll}(M) \subset \Lambda^1_l(M)$ is
generated by the subbundle $\Lambda^1_{ll}(M) \subset
\Lambda^1_l(M)$, and the ideal $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{rr}(M) \subset
\Lambda^1_r(M)$ is generated by the subbundle $\Lambda^1_{rr}(M)
\subset \Lambda^1_r(M)$.
\refstepcounter{subsubsection \label{left.right}
Denote by $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_o(M) = \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_l(M) \cap
\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_r(M) \subset \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M)$ the intersection of
the subalgebras $\Lambda_l^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M)$ and $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_r(M)$.
Unlike either of these subalgebras, the subalgebra
$\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_o(M) \subset \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M)$ is compatible with
the weight $0$ Hodge bundle structure on the total de Rham complex.
By \eqref{cap.cup} we have a short exact sequence
\begin{equation}\label{shrt}
\begin{CD}
0 @>>> \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) @>>> \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_l(M) \oplus \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_r(M)
@>>> \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M) @>>> 0
\end{CD}
\end{equation}
of complex vector bundles on $M$. Therefore the algebra
$\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_o(M)$ is isomorphic, as a complex bundle algebra, to
the usual de Rham complex $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$. As a Hodge bundle
algebra it is canonically isomorphic to the exterior algebra of the
Hodge bundle $S^1(M,\C)$ of weight $0$ on the manifold $M$.
Finally, note that the short exact sequence \eqref{shrt} induces a
direct sum decomposition
$$
\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M) \cong \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{ll}(M) \oplus
\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_o(M) \oplus \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{rr}(M).
$$
\refstepcounter{subsubsection \noindent {\bf Remark.\ } The total de Rham complex $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M)$ is
related to Simpson's theory of Higgs bundles (see \cite{Shiggs}) in
the following way. Recall that Simpson has proved that every
(sufficiently stable) complex bundle $\E$ on a compact complex
manifold $M$ equipped with a flat connection $\nabla$ admits a
unique Hermitian metric $h$ such that $\nabla$ and the $1$-form
$\theta = \nabla - \nabla^h \in C^\infty(M,\Lambda^1({{\cal E}\!nd\:}\E))$
satisfy the so-called {\em harmonicity condition}. He also has shown
that this condition is equivalent to the vanishing of a certain
curvature-like tensor $R \in \Lambda^2(M,{{\cal E}\!nd\:}\E)$ which he
associated canonically to every pair $\langle \nabla, \theta
\rangle$.
Recall that flat bundles $\langle \E, \nabla \rangle$ on the
manifold $M$ are in one-to-one correspondence with free differential
graded modules $\E \otimes \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ over the de Rham
complex $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$. It turns out that complex bundles
$\E$ equipped with a flat connection $\nabla$ and a $1$-form $\theta
\in C^\infty(M, \Lambda^1({{\cal E}\!nd\:}\E))$ such that Simpson's tensor $R$
vanishes are in natural one-to-one correspondence with free
differential graded modules $\E \otimes \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M)$ over the
total de Rham complex $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M)$. Moreover, a pair
$\langle\theta,\nabla\rangle$ comes from a variation of pure
$\R$-Hodge structure on $\E$ if and only if there exists a Hodge
bundle structure on $\E$ such that the product Hodge bundle
structure on the free module $\E \otimes \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M)$ is
compatible with the differential.
\refstepcounter{subsubsection\label{gen.symm.proof}
\proof[Proof of Lemma~\ref{symm}]
For every $k \geq 0$ let $G = \Sigma_k \times \Sigma_k$ be the
product of two copies of the symmetric group $\Sigma_k$ on $k$
letters. Let $\V_k$ be the $\Q$-representation of $G_k$ induced from
the trivial representation of the diagonal subgroup $\Sigma_k
\subset G_k$. The representation $\V_k$ decomposes as
$$
\V_k = \bigoplus_V V \boxtimes V,
$$
where the sum is over the set of irreducible representations $V$ of
$\Sigma_k$. We obviously have
$$
\CC^k = \Hom_{G_k}\left(\V_k, A^{\otimes k} \otimes B^{\otimes
k}\right) = \bigoplus_V \Hom_{\Sigma_k}\left(V,A^{\otimes k}\right)
\otimes \Hom_{\Sigma_k}\left(V,A^{\otimes k}\right) .
$$
Let $\J^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \subset \CC^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ be the ideal generated by
$\Lambda^2A \otimes \Lambda^2B \subset S^2(A \otimes B)$. It is easy
to see that
$$
\J^k = \sum_{1 \leq l \leq k-1} \bigoplus_V
\Hom_{\Sigma_k}\left(V,A^{\otimes k}\right) \otimes
\Hom_{\Sigma_k}\left(V,A^{\otimes k}\right) \subset \CC^k,
$$
where the first sum is taken over the set of $k-1$ subgroups
$\Sigma_2 \subset \Sigma_k$, the $l$-th one transposing the $l$-th
and the $l+1$-th letter, while the second sum is taken over all
irreducible constituents $V$ of the representation of $\Sigma_k$
induced from the sign representation of the corresponding $\Sigma_2
\subset \Sigma_k$. Now, there is obviously only one irreducible
representation of $\Sigma_k$ which is not encountered as an index in
this double sum, namely, the trivial one. Hence $\CC^k / \J^k = S^kA
\otimes S^kB$, which proves the lemma.
\hfill \ensuremath{\square}\par
\subsection{The total Weil algebra}\label{t.W.sub}
\refstepcounter{subsubsection \label{S.and.Lambda}
Assume from now on that the $U(1)$-action on the complex manifold
$M$ is trivial. We now turn to studying the Weil algebra of the
manifold $M$. Let $S^1(M,\C) = S^{1,-1}(M,\C) \oplus S^{-1,1}(M,\C)$
be the weight $0$ Hodge bundle on $M$ introduced in \ref{S1}. To
simplify notation, denote
\begin{align*}
S^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} &= \widehat{S}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(S^1(M,\C))\\
\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} &= \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C),
\end{align*}
where $\widehat{S}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is the completed symmetric power, and let
$$
\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} = \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) = S^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}
$$
be the Weil algebra of the complex manifold $M$ introduced in
\ref{Weil.defn}. Recall that the algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ carries a
natural Hodge bundle structure. In particular, it is equipped with a
Hodge type bigrading $\B^i = \sum_{p+q=i} \B^{p,q}$.
\refstepcounter{subsubsection \label{aug}
We now introduce a different bigrading on the Weil algebra
$\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. The commutative algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is freely generated
by the subbundles
$$
S^1 = S^{1,-1} \oplus S^{-1,1} \subset \B^0 \quad\text{ and }\quad
\Lambda^1 = \Lambda^{1,0} \oplus \Lambda^{0,1} \subset \B^1,
$$
therefore to define a multiplicative bigrading on the algebra
$\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ it suffices to assign degrees to these generator
subbundles $S^{1,-1},S^{-1,1},\Lambda^{1,0},\Lambda^{0,1} \subset
\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$.
\noindent {\bf Definition.\ } The {\em augmentation bigrading} on $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is the
multiplicative bigrading defined by setting
\begin{align*}
\deg S^{1,-1} &= \deg \Lambda^{1,0} = ( 1, 0 )\\
\deg S^{-1,1} &= \deg \Lambda^{0,1} = ( 0, 1 )
\end{align*}
on generators $S^{1,-1},S^{-1,1},\Lambda^{1,0},\Lambda^{0,1} \subset
\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$.
We will denote by $\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}_{p,q}$ the component of the Weil
algebra of augmentation bidegree $(p,q)$. For any linear map
$a:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ we will denote by $a = \sum_{p,q}a_{p,q}$
its decomposition with respect to the augmentation bidegree.
It will also be useful to consider a coarser {\em augmentation
grading} on $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$, defined by $\deg\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{p,q} = p + q$. We
will denote by $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_k = \bigoplus_{p+q=k}\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{p,q}$ the
component of $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ of augmentation degree $k$.
\refstepcounter{subsubsection Note that the Hodge bidegree and the augmentation bidegree
are, in general, independent. Moreover, the complex conjugation
$\overline{\ }:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \overline{\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ sends $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{p,q}$ to
$\overline{\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{q,p}}$. Therefore the augmentation bidegree
components $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{p,q} \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ are not Hodge
subbundles. However, the coarser augmentation grading is compatible
with the Hodge structures, and the augmentation degree $k$-component
$\B^i_k \subset \B^i$ carries a natural Hodge bundle structure of
weight $i$. Moreover, the sum $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{p,q} + \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{q,p}
\subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is also a Hodge subbundle.
\refstepcounter{subsubsection \label{total.Weil}
We now introduce an auxiliary weight $0$ Hodge algebra bundle on
$M$, called the total Weil algebra. Recall that we have defined in
\ref{gamma.m} a functor $\Gamma:{{\cal W}{\cal H}odge}_{\geq 0}(M) \to
{{\cal W}{\cal H}odge}_0(M)$ adjoint on the right to the canonical
embedding. Consider the Hodge bundle $\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} = \Gamma(\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}})$
of weight $0$ on $M$. By \ref{gamma.tensor} the multiplication on
$\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ induces an algebra structure on $\Gamma(\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}})$.
\noindent {\bf Definition.\ } The Hodge algebra bundle $\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ of weight $0$ is called
{\em the total Weil algebra} of the complex manifold $M$.
\noindent {\bf Remark.\ } For a more conceptual description of the functor $\Gamma$ and
the total Weil algebra, see Appendix.
\refstepcounter{subsubsection By definition of the functor $\Gamma$ we have $\B_{tot}^k = \B^k
\otimes \W_k^* = S^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \Lambda^k \otimes \W_k^* = S^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}
\otimes \Lambda^k_{tot}$, where $\Lambda^k_{tot} = \Lambda^k \otimes
\W_{\:\raisebox{3pt}{\text{\circle*{1.5}}}}^* = \Gamma(\Lambda^k)$ is the total de Rham complex
introduced in Subsection~\ref{de.rham.sub}. We have also introduced
in Subsection~\ref{de.rham.sub} Hodge bundle subalgebras
$\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_o, \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_l, \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_r \subset
\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ in the total de Rham complex $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ and
ideals $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{ll} \subset \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_l$,
$\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{rr} \subset \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_r$ in the algebras
$\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_l$, $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_r$. Let
\begin{align*}
\B_o^k &= S^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \Lambda_o^k \subset \B_{tot}^k\\
\B_l^k &= S^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \Lambda_l^k \subset \B_{tot}^k\\
\B_r^k &= S^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \Lambda_r^k \subset \B_{tot}^k
\end{align*}
be the associated subalgebras in the total Weil algebra $\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$
and let
\begin{align*}
\B_{ll}^k &= S^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \Lambda_{ll}^k \subset \B_l^k\\
\B_{rr}^k &= S^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \Lambda_{rr}^k \subset \B_r^k
\end{align*}
be the corresponding ideals in the Hodge bundle algebras
$\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_l$, $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_r$.
By \ref{left.right} we have bundle isomorphisms $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} =
\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_l + \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_r$ and $\Lambda_o^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} =
\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_l \cap \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_r$, and the direct sum
decomposition $\Lambda_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \cong \Lambda_{ll}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \oplus
\Lambda_o^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \oplus \Lambda_{rr}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. Therefore we also have
\begin{align}\label{drct}
\begin{split}
\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} &= \B_l^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} + \B_r^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} = \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{ll} \oplus
\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_o \oplus \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{rr}\\
\B_o^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} &= \B_l^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \bigcap \B_r^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}
\end{split}
\end{align}
Moreover, the algebra $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_o$ is isomorphic to the usual
de Rham complex $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$, therefore the subalgebra
$\B_o^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ is isomorphic to the usual Weil
algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. These isomorphisms are {\em not} weakly Hodge.
\refstepcounter{subsubsection The total Weil algebra carries a canonical weight $0$ Hodge
bundle structure, and we will denote the corresponding Hodge type
grading by upper indices: $\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} =
\oplus_p\left(\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\right)^{p,-p}$. The augmentation bigrading
on the Weil algebra introduced in \ref{aug} extends to a bigrading
of the total Weil algebra, which we will denote by lower indices. In
general, both these grading and the direct sum decomposition
\eqref{drct} are independent, so that, in general, for every $i \geq
0$ we have a decomposition
$$
\B_{tot}^i = \bigoplus_{n,p,q} \left(\B_{ll}^i\right)_{p,q}^{n,-n}
\oplus \left(\B_o^i\right)_{p,q}^{n,-n} \oplus
\left(\B_{rr}^i\right)_{p,q}^{n,-n}.
$$
We would like to note, however, that some terms in this
decomposition vanish when $i = 0,1$. Namely, we have the following
fact.
\begin{lemma}\label{total.aug}
Let $n,k$ be arbitrary integers such that $k \geq 0$.
\begin{enumerate}
\item If $n+k$ is odd, then $\left(\B^0_{tot}\right)_k^{n,-n} = 0$.
\item If $n+k$ is even, then $\left(\B^1_{ll}\right)_k^{n,-n} =
\left(\B^1_{rr}\right)_k^{n,-n} = 0$, while if $n+k$ is odd, then
$\left(\B^1_o\right)_k^{n,-n} = 0$.
\end{enumerate}
\end{lemma}
\proof
\begin{enumerate}
\item The bundle $\B^0_{tot}$ by definition coincides with $\B^0$, and
it is generated by the subbundles $S^{1,-1},S^{-1,1} \subset
\B^0$. Both these subbundles have augmentation degree $1$ and Hodge
degree $\pm 1$, so that the sum $n+k$ of the Hodge degree with the
augmentation degree is even. Since both gradings are multiplicative,
for all non-zero components $\B^{n,-n}_k \subset \B^0$ the sum $n+k$
must also be even.
\item By definition we have $\B^1_{tot} = \B^0 \otimes \Lambda^1_{tot}$. The
subbundle $\Lambda^1_{tot} \subset \B^1_{tot}$ has augmentation degree $1$,
and it decomposes
$$
\Lambda^1_{tot} = \Lambda^1_o \oplus \Lambda^1_{ll} \oplus
\Lambda^1_{rr}.
$$
By \ref{S} we have $\Lambda^1_o \cong S^1 = S^{1,-1} \oplus
S^{-1,1}$ as Hodge bundles, so that the Hodge degrees on
$\Lambda^1_o \subset \Lambda^1_{tot}$ are odd. On the other hand, the
subbundles $\Lambda^1_{ll},\Lambda^1_{rr} \subset \Lambda^1_{tot}$ are
by \ref{S.Hodge.type} of Hodge bidegree $(0,0)$. Therefore the sum
$n+k$ of the Hodge and the augmentation degrees is even for
$\Lambda^1_o$ and odd for $\Lambda^1_{ll}$ and
$\Lambda^1_{rr}$. Together with \thetag{i} this proves the claim.
\hfill \ensuremath{\square}\par
\end{enumerate}
\subsection{Derivations of the Weil algebra}
\refstepcounter{subsubsection We will now introduce certain canonical derivations of the
Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ which will play an important part in
the rest of the paper. First of all, to simplify notation, for any
two linear maps $a,b$ let
$$
\{ a,b \} = a \circ b + b \circ a
$$
be their anticommutator, and for any linear map $a:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to
\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+i}$ let $a = \sum_{p+q=i} a^{p,q}$ be the Hodge type
decomposition. The following fact is well-known, but we have
included a proof for the sake of completeness.
\begin{lemma}
For every two odd derivations $P,Q$ of a
graded-commutative algebra $\A$, their anticommutator $\{P,Q\}$ is
an even derivation of the algebra $\A$.
\end{lemma}
\proof Indeed, for every $a,b \in \A$ we have
\begin{align*}
\{P,Q\}(ab) &= P(Q(ab)) + Q(P(ab)) \\
& = P(Q(a)b + (-1)^{\deg a}aQ(b)) +
Q(P(a)b + (-1)^{\deg a}aP(b)) \\
& = P(Q(a))b + (-1)^{\deg Q(a)}Q(a)P(b)
+ (-1)^{\deg a}P(a)Q(b) \\
&\quad + aP(Q(b)) + Q(P(a))b + (-1)^{\deg P(a)}P(a)Q(b) \\
&\quad + (-1)^{\deg a}Q(a)P(b) + aQ(P(b)) \\
& = P(Q(a))b - (-1)^{\deg a}Q(a)P(b)
+ (-1)^{\deg a}P(a)Q(b) \\
&\quad + aP(Q(b)) + Q(P(a))b - (-1)^{\deg a}P(a)Q(b) \\
&\quad + (-1)^{\deg a}Q(a)P(b) + aQ(P(b)) \\
& = P(Q(a))b + aP(Q(b)) + Q(P(a))b + a Q(P(b)) \\
& = \{P,Q\}(a)b + a\{P,Q\}(b).
\end{align*}
\hfill \ensuremath{\square}\par
\refstepcounter{subsubsection \label{C.and.sigma}
Let $C:S^1 \to \Lambda^1$ be the canonical weakly Hodge map
introduced in \ref{C.Weil}. Extend $C$ to an algebra derivation
$C:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ by setting $C = 0$ on $\Lambda^1
\subset \B^1$. By \ref{C.Weil} the derivation $C$ is weakly Hodge.
The composition
$$
C \circ C = \frac{1}{2}\{C,C\}:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+2}
$$
is also an algebra derivation, and it obviously vanishes on
generators $S^1,\Lambda^1 \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. Therefore $C \circ C =
0$ everywhere.
Let also $\sigma:\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ be the derivation
introduced in \ref{sigma}. The derivation $\sigma$ is not weakly
Hodge; however, it is real and admits a decomposition $\sigma =
\sigma^{-1,0} + \sigma^{0,-1}$ into components of Hodge types
$(-1,0)$ and $(0,-1)$. Both these components are algebra derivations
of the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. We obviously have $\sigma \circ
\sigma = \sigma^{-1,0} \circ \sigma^{-1,0} = \sigma^{0,-1} \circ
\sigma^{0,-1} = 0$ on generators $S^1,\Lambda^1 \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$,
and, therefore, on the whole Weil algebra.
\noindent {\bf Remark.\ } Up to a sign the derivations $C,\sigma$ and their Hodge
bidegree components coincide with the so-called {\em Koszul
differentials} on the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} = S^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes
\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$.
\refstepcounter{subsubsection \label{total.C}
The derivation $C:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ is by definition weakly
Hodge. Applying the functor $\Gamma$ to it, we obtain a derivation
$C:\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B_{tot}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ of the total Weil algebra
$\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ preserving the weight $0$ Hodge bundle structure on
$\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. The canonical identification $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \cong
\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_o \subset \B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is compatible with the derivation
$C:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot}$. Moreover, by \ref{C.and.sigma}
this derivation satisfies $C \circ C = 0:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} \to
\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+2}_{tot}$. Therefore the total Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$
equipped with the derivation $C$ is a complex of Hodge bundles of
weight $0$.
The crucial linear algebraic property of the total Weil algebra
$\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ of the manifold $M$ which will allow us to classify
flat extended connections on $M$ is the following.
\begin{prop}\label{ac}
Consider the subbundle
\begin{equation}\label{sbcmp}
\bigoplus_{p,q \geq 1}\left(\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}\right)_{p,q} \subset
\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}
\end{equation}
of the total Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ consisting of the components
of augmentation bidegrees $(p,q)$ with $p,q \geq 1$. This subbundle
equipped with the differential
$C:\left(\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}\right)_{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}} \to
\left(\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot}\right)_{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ is an acyclic complex of
Hodge bundles of weight $0$ on $M$.
\end{prop}
\refstepcounter{subsubsection We sketch a more or less simple and conceptual proof of
Proposition~\ref{ac} in the Appendix. However, in order to be able
to study in Section~\ref{convergence} the analytic properties of our
formal constructions, we will need an explicit contracting homotopy
for the complex \eqref{sbcmp}, which we now introduce.
The restriction of the derivation $C:\B^0_{tot} \to \B^1_{tot}$ to the
subbundle $S^1 \subset \B_0 \cong \B^0_{tot}$ induces a Hodge bundle
isomorphism
$$
C:S^1 \to \Lambda^1_o \subset \Lambda^1_{tot} \subset \B^1_{tot}.
$$
Define a map $\sigma_{tot}:\Lambda^1_{tot} \to S^1$ by
\begin{equation}\label{sigma.c.eq}
\sigma_{tot} =
\begin{cases}
0 \quad &\text{ on }\quad\Lambda^1_{ll},\Lambda^1_{rr} \subset \Lambda^1_{tot},\\
C^{-1} \quad &\text{ on }\quad\Lambda^1_o \subset \Lambda^1_{tot}.
\end{cases}
\end{equation}
The map $\sigma_{tot}:\Lambda^1_{tot} \to S^1$ preserves the Hodge bundle
structures of weight $0$ on both sides. Moreover, its restriction to
the subbundle $\Lambda^1 \cong \Lambda^1_o \subset \Lambda^1_{tot}$
coincides with the canonical map $\sigma:\Lambda^1 \to S^1$
introduced in \ref{C.and.sigma}.
\refstepcounter{subsubsection \label{sigma.l}
Unfortunately, unlike $\sigma:\Lambda^1 \to S^1$, the map
$\sigma_{tot}:\Lambda^1_{tot} \to S^1$ does {\em not} admit an extension to
a derivation $\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot} \to \B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ of the total Weil
algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$. We will extend it to a {\em bundle map}
$\sigma_{tot}:\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ in a somewhat roundabout
way. To do this, define a map $\sigma_l:\Lambda^1_l \to S^0 \subset
\B^0$ by
$$
\sigma_l =
\begin{cases}
0 \quad &\text{ on } \quad\Lambda_{ll}^1 \subset \Lambda_l^1 \quad
\text{ and on }\quad \left(\Lambda^1_o\right)^{-1,1} \subset
\Lambda^1_{tot}, \\
C^{-1} \quad &\text{ on } \quad\left(\Lambda^1_o\right)^{1,-1} \subset
\Lambda^1_{tot}.
\end{cases}
$$
and set $\sigma_l = 0$ on $S^1$. By \ref{gamma.use} we have
$$
\Lambda^1_l = \Lambda^{1,0} \oplus \left( \Lambda^{0,1} \otimes
\W_1^*\right).
$$
The map $\sigma_l:\Lambda^1_l \to S^1$ vanishes on the second
summand in this direct sum, and it equals $C^{-1}:\Lambda^{1,0} \to
S^{1,-1} \subset S^1$ on the first summand. The restriction of the map
$\sigma_l$ to the subbundle
$$
\Lambda^{1,0} \oplus \Lambda^{0,1} = \Lambda^1 \cong \Lambda^1_o \subset
\Lambda^1_l
$$
vanishes on $\Lambda^{0,1}$ and equals $C^{-1}$ on
$\Lambda^{1,0}$. Thus it is equal to the Hodge type-$(0,-1)$
component $\sigma^{0,-1}:\Lambda^1 \to S^1$ of the canonical map
$\sigma:\Lambda^1 \to S^1$.
\refstepcounter{subsubsection By \ref{l.r.rel} the algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_l$ is generated by the
bundles $S^1$ and $\Lambda^1_l$, and the ideal of relations is
generated by the subbundle
\begin{equation}\label{rell}
S^2\left(\Lambda^{0,1}\right) \otimes \Lambda^2\left(\W_1^*\right)
\subset \Lambda^2\left(\Lambda^1_l\right).
\end{equation}
Since the map $\sigma_l:\Lambda^1_l \to S^1$ vanishes on
$\Lambda^{0,1} \otimes \W_1^* \subset \Lambda^1_l$, it extends to an
algebra derivation $\sigma_l:\B_l^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_l$. The
restriction of the derivation $\sigma_l$ to the subalgebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}
\cong \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_0 \subset \B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ coincides with the
$(0,-1)$-component $\sigma^{0,-1}$ of the derivation
$\sigma:\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$.
Analogously, the $(-1,0)$-component $\sigma^{-1,0}$ of the
derivation $\sigma:\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ extends to an algebra
derivation $\sigma_r:\B_r^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1} \to \B_r^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ of the
subalgebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_r \subset \B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. By definition, the
derivation $\sigma_l$ preserves the decomposition $\B_l^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} =
\B_{ll} \oplus \B_o^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$, while the derivation $\sigma_r$
preserves the decomposition $\B_r^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} = \B_{rr}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \oplus
\B_o^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. Both these derivations vanish on $\B_{tot}^0$, therefore
both are maps of $\B_{tot}^0$-modules. In addition, the compositions
$\sigma_l \circ \sigma_l$ and $\sigma_r \circ \sigma_r$ vanish on
generator and, therefore, vanish identically.
\refstepcounter{subsubsection \label{sigma.c}
Extend both $\sigma_l$ and $\sigma_r$ to the whole $\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ by
setting
\begin{equation}\label{nol}
\sigma_l = 0 \text{ on }\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_r \qquad \sigma_r = 0 \text{ on
}\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_l,
\end{equation}
and let
$$
\sigma_{tot} = \sigma_l + \sigma_r:\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot} \to \B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}.
$$
On $\Lambda^1_{tot} \subset \B^1_{tot}$ this is the same map as in
\eqref{sigma.c.eq}. The bundle map $\sigma_{tot}:\B_{tot}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1} \to
\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ preserves the direct sum decomposition \eqref{drct}, and
its restriction to $\B_o^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ coincides with
the derivation $\sigma$. Note that neither of the maps $\sigma_l$,
$\sigma_r$, $\sigma_{tot}$ is a derivation of the total Weil algebra
$\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. However, all these maps are linear with respect to the
$\B_{tot}^0$-module structure on $\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ and preserve the
decomposition \eqref{drct}. The map $\sigma_{tot}:\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot} \to
\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ is equal to $\sigma_l$ on $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{ll} \subset
\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$, to $\sigma_r$ on $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{rr}$ and to
$\sigma:\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ on $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \cong \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_o
\subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$. Since $\sigma_l \circ \sigma_l = \sigma_r \circ
\sigma_r = \sigma \circ \sigma = 0$, we have $\sigma_{tot} \circ
\sigma_{tot} = 0$.
\refstepcounter{subsubsection The commutator
$$
h = \{C, \sigma_{tot}\}:\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}
$$
of the maps $C$ and $\sigma_{tot}$ also preserves the decomposition
\eqref{drct}, and we have the following.
\begin{lemma}\label{h.acts}
The map $h$ acts as multiplication by $p$ on
$\left(\B_{ll}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\right)_{p,q}$, as multiplication by $q$ on
$\left(\B_{rr}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\right)_{p,q}$ and as multiplication by $(p+q)$
on $\left(\B_o^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\right)_{p,q}$.
\end{lemma}
\proof It suffices to prove the claim separately on each term in the
decomposition \eqref{drct}. By definition $\sigma_{tot} = \sigma_l +
\sigma_r$, and $h = h_l + h_r$, where $h_l = \{\sigma_l,C\}$ and
$h_r = \{\sigma_r,C\}$. Moreover, $h_l$ vanishes on $\B_{rr}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$
and $h_r$ vanishes on $\B_{ll}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. Therefore it suffices to
prove that $h_l = p{\sf id}$ on $\left(\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_l\right)_{p,q}$ and that
$h_r = q{\sf id}$ on $\left(\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_r\right)_{p,q}$. The proofs of these
two identities are completely symmetrical, and we will only give a
proof for $h_l$.
The algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_l$ is generated by the subbundles $S^1 \subset
\B^0_l$ and $\Lambda^1_l \subset \B^1_l$. The augmentation bidegree
decomposition of $S^1$ is by definition given by
$$
S^1_{1,0} = S^{1,-1} \qquad\qquad S^1_{0,1} = S^{-1,1},
$$
while the augmentation bidegree decomposition of $\Lambda^1_l$ is
given by
$$
\left(\Lambda^1_l\right)_{1,0} = \Lambda^{1,0} \qquad
\left(\Lambda^1_l\right)_{0,1} = \Lambda^{0,1} \otimes \W_1^*.
$$
By the definition of the map $\sigma_l:\Lambda^1_l \to S^1$ (see
\ref{sigma.l}) we have $h_l = \{C,\sigma_l\} = {\sf id}$ on
$\Lambda^{1,0}$ and $S^{1,-1}$, and $h_l = 0$ on $\Lambda^{0,1}
\otimes \W_1^*$ and on $S^{-1,1}$. Therefore for every $p,q \geq 0$
we have $h_l=p{\sf id}$ on the generator subbundles $S^1_{p,q}$ and on
$\left(\Lambda^1_l\right)_{p,q}$. Since the map $h_l$ is a
derivation and the augmentation bidegree is multiplicative, the same
holds on the whole algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_l =
\oplus\left(\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_l\right)_{p,q}$.
\hfill \ensuremath{\square}\par
Lemma~\ref{h.acts} shows that the map $\sigma_{tot}$ is a homotopy,
contracting the subcomplex \eqref{sbcmp} in the total Weil algebra
$\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$, which immediately implies Proposition~\ref{ac}.
\noindent {\bf Remark.\ } In fact, in our classification of flat extended connections
given in Section~\ref{main.section} it will be more convenient for
us to use Lemma~\ref{h.acts} directly rather than refer to
Proposition~\ref{ac}.
\refstepcounter{subsubsection We finish this section with the following corollary of
Lemma~\ref{total.aug} and Lemma~\ref{h.acts}, which we will need in
Section~\ref{convergence}.
\begin{lemma}\label{h.on.b1}
Let $n=\pm 1$. If the integer $k \geq 1$ is odd, then the map
$h:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ acts on $\left(\B_{tot}^1\right)^{n,-n}_k$
by multiplication by $k$. If $k = 2m \geq 1$ is even, then the
endomorphism $h:\left(\B_{tot}^1\right)^{n,-n}_k \to
\left(\B_{tot}^1\right)^{n,-n}_k$ is diagonalizable, and its only
eigenvalues are $m$ and $m-1$.
\end{lemma}
\proof If $k$ is odd, then $\left(\B_{tot}^1\right)^{n,-n}_k =
\left(\B_o^1\right)^{n,-n}_k$ by Lemma~\ref{total.aug}, and
Lemma~\ref{h.acts} immediately implies the claim. Assume that the
integer $k = 2m$ is even. By Lemma~\ref{total.aug} we have
$$
\left(\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\right)^{n,-n}_k =
\left(\B_{ll}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\right)^{n,-n}_k \oplus
\left(\B_{rr}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\right)^{n,-n}_k = \B^{n,-n}_{k-1} \otimes
\left(\Lambda^1_{ll} \oplus \Lambda^1_{rr}\right).
$$
The bundle $\B^0$ is generated by subbundles $S^{1,-1}$ and
$S^{-1,1}$. The first of these subbundles has augmentation bidegree
$(1,0)$, while the second one has augmentation bidegree
$(0,1)$. Therefore for every augmentation bidegree component
$\B^{n,-n}_{p,q} \subset \B^{n,-n}_{k-1}$ we have $p - q = n$ and
$p+q=k-1$. This implies that $\B^{n,-n}_{k-1} = \B^{n,-n}_{p,q}$
with $p=m-(1-n)/2$ and $q=m-(1+n)/2$.
By definition the augmentation bidegrees of the bundles
$\Lambda^1_{ll}$ and $\Lambda^1_{rr}$ are, respectively, $(0,1)$ and
$(1,0)$. Lemma~\ref{h.acts} shows that the only eigenvalue of the
map $h$ on $\left(\B^1_{ll}\right)_{k+1}$ is $p=(m-(1-n)/2))$, while
its only eigenvalue on $\left(\B^1_{rr}\right)_{k+1}$ is
$q=(m-(1+n)/2)$. Since $n = \pm 1$, one of these numbers equals $m$
and the other one equals $m-1$.
\hfill \ensuremath{\square}\par
\section{Classification of flat extended connections}\label{main.section}
\subsection{K\"ah\-le\-ri\-an connections}
\refstepcounter{subsubsection Let $M$ be a complex manifold. In
Section~\ref{formal.section} we have shown that formal Hodge
manifold structures on the tangent bundle ${\overline{T}M}$ are in one-to-one
correspondence with linear flat extended connections on the manifold
$M$ (see \ref{ext.con}--\ref{flat.ext.con} for the definitions). It
turns out that flat linear extended connections on $M$ are, in turn,
in natural one-to-one correspondence with differential operators of
a much simpler type, namely, connections on the cotangent bundle
$\Lambda^{1,0}(M)$ satisfying certain vanishing conditions
(Theorem~\ref{kal=ext}). We call such connections {\em
K\"ah\-le\-ri\-an}. In this section we use the results of
Section~\ref{Weil.section} establish the correspondence between
extended connections on $M$ and K\"ah\-le\-ri\-an connections on
$\Lambda^{1,0}(M)$.
\refstepcounter{subsubsection We first give the definition of K\"ah\-le\-ri\-an
connections. Assume that the manifold $M$ is equipped with a
connection
$$
\nabla:\Lambda^1(M) \to \Lambda^1(M) \otimes \Lambda^1(M)
$$
on its cotangent bundle $\Lambda^1(M)$. Let
\begin{align*}
T &= \Alt \circ \nabla - d_M:\Lambda^1(M) \to \Lambda^2(M) \\
R &= \Alt \nabla \circ \nabla:\Lambda^1(M) \to \Lambda^1(M) \otimes
\Lambda^2(M)
\end{align*}
be its torsion and curvature, and let $R = R^{2,0} + R^{1,1} +
R^{0,2}$ be the decomposition of the curvature according to the
Hodge type.
\noindent {\bf Definition.\ } The connection $\nabla$ is called {\em K\"ah\-le\-ri\-an} if
\begin{align*}
T &= 0 \tag{i}\\
R^{2,0} &= R^{0,2} = 0 \tag{ii}
\end{align*}
\noindent {\bf Example.\ } The Levi-Civita connection on a K\"ahler manifold is K\"ah\-le\-ri\-an.
\noindent {\bf Remark.\ } The condition $T=0$ implies, in particular, that the component
$$
\nabla^{0,1}:\Lambda^{1,0}(M) \to \Lambda^{1,1}(M)
$$
of the connection $\nabla$ coincides with the Dolbeault
differential. Therefore a K\"ah\-le\-ri\-an connection is always
holomorphic.
\refstepcounter{subsubsection Recall that in \ref{red} we have associated to any extended
connection $D$ on $M$ a connection $\nabla$ on the cotangent bundle
$\Lambda^1(M,\C)$ called the reduction of $D$. We can now formulate
the main result of this section.
\begin{theorem}\label{kal=ext}
\begin{enumerate}
\item If an extended connection $D$ on $M$ is flat and linear, then
its reduction $\nabla$ is K\"ah\-le\-ri\-an.
\item Every K\"ah\-le\-ri\-an connection $\nabla$ on $\Lambda^1(M,\C)$ is
the reduction of a unique linear flat extended connection $D$ on
$M$.
\end{enumerate}
\end{theorem}
The rest of this section is taken up with the proof of
Theorem~\ref{kal=ext}. To make it more accessible, we first give an
informal outline. The actual proof starts with
Subsection~\ref{pf.first.sub}, and it is independent from the rest
of this subsection.
\refstepcounter{subsubsection Assume given a K\"ah\-le\-ri\-an connection $\nabla$ on the
manifold $M$. To prove Theorem~\ref{kal=ext}, we have to construct a
flat linear extended connection $D$ on $M$ with reduction $\nabla$.
Every extended connection decomposes into a series $D = \sum_{k \geq
0}D_k$ as in \eqref{aug.con}, and, since $\nabla$ is the reduction
of $D$, we must have $D_1 = \nabla$. We begin by checking in
Lemma~\ref{lin.aug} that if $D$ is linear, then $D_0 = C$, where
$C:S^1(M,\C) \to \Lambda^1(M,\C)$ is as in \ref{C.Weil}. The sum $C
+ \nabla$ is already a linear extended connection on $M$. By
\ref{deriv} it extends to a derivation $D_{\leq 1}$ of the Weil
algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ of the manifold $M$, but this derivation
does not necessarily satisfy $D_{\leq 1} \circ D_{\leq 1} = 0$, thus
the extended connection $D_{\leq 1}$ is not necessarily flat.
We have to show that one can add the ``correction terms'' $D_k, k
\geq 2$ to $D_{\leq 1}$ so that $D = \sum_k D_k$ satisfies all the
conditions of Theorem~\ref{kal=ext}. To do this, we introduce in
\ref{red.Weil} a certain quotient $\wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ of the Weil
algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$, called the reduced Weil algebra. The
reduced Weil algebra ia defined in such a way that for every
extended connection $D$ the associated derivation $D:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)
\to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C)$ preserves the kernel of the surjection
$\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$, thus inducing a derivation
$\wt{D}:\wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \wt{\B}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C)$. Moreover, the algebra
$\wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ has the following two properties:
\begin{enumerate}
\item \label{first.prop} The derivation $\wt{D}:\wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to
\wt{\B}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C)$ satisfies $\wt{D} \circ \wt{D} = 0$ if and only if
the connection $D_1$ is K\"ah\-le\-ri\-an.
\item \label{second.prop} Let $\wt{D}$ be the weakly Hodge derivation
of the quotient algebra $\wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ induced by an arbitrary
linear extended connection $D_{\leq 1}$ and such that $\wt{D} \circ \wt{D}
= 0$. Then the derivation $\wt{D}$ lifts uniquely to a weakly Hodge
derivation $D$ of the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ such that $D
\circ D = 0$, and the derivation also $D$ comes from a linear
extended connection on $M$ (see Proposition~\ref{main} for a precise
formulation of this statement).
\end{enumerate}
\refstepcounter{subsubsection The property \ref{first.prop} is relatively easy to check,
and we do it in the end of the proof, in
Subsection~\ref{pf.last.sub}. The rest is taken up with establishing
the property \ref{second.prop}. The actual proof of this statement
is contained in Proposition~\ref{main}, and
Subsection~\ref{pf.first.sub} contains the necessary preliminaries.
Recall that we have introduced in \ref{aug} a new grading on the
Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$, called the augmentation grading, so
that the component $D_k$ in the decomposition $D = \sum_kD_k$ is of
augmentation degree $k$. In order to lift $\wt{D}$ to a derivation $D$
so that $D \circ D = 0$, we begin with the given lifting $D_{\leq
1}$ and then add components $D_k, k \geq 2$, one by one, so that on
each step for $D_{\leq k} = D_{\leq 1} + \sum_{2 \leq p \leq k}$ the
composition $D_{\leq k} \circ D_{\leq k}$ is zero in augmentation
degrees from $0$ to $k$. In order to do it, we must find for each
$k$ a solution to the equation
\begin{equation}\label{tslv}
D_0 \circ D_k = - R_k,
\end{equation}
where $R_k$ is the component of augmentation degree $k$ in the
composition $D_{\leq k-1} \circ D_{\leq k-1}$. This solution must be
weakly Hodge, and the extended connection $D_{\leq k} = D_{\leq k-1}
+ D_k$ must be linear.
We prove in Lemma~\ref{lin.aug} that since $D_{\leq 0}$ is linear,
we may assume that $D_0 = C$. In addition, since $\wt{D} \circ \wt{D} =
0$, we may assume by induction that the image of $R_k$ lies in the
kernel $\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ of the quotient map $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to
\wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$.
\refstepcounter{subsubsection In order to analyze weakly Hodge maps from $S^1(M,\C)$ to the
Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$, we apply the functor
$\Gamma:{{\cal W}{\cal H}odge}_{\geq 0}(M) \to {{\cal W}{\cal H}odge}_0(M)$ constructed in
\ref{gamma.m} to the bundle $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ to obtain the total
Weil algebra $\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) = \Gamma(\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C))$ of weight
$0$, which we studied in Subsection~\ref{t.W.sub}. The Hodge bundle
$S^1(M,\C)$ on the manifold $M$ is of weight $0$, and, by the
universal property of the functor $\Gamma$, weakly Hodge maps from
$S^1(M,\C)$ to $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ are in one-to-one correspondence
with Hodge bundle maps from $S^1(M,\C)$ to the total Weil algebra
$\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M,\C)$. The canonical map $C:S^1(,\C) \to \B^1(M,\C)$
extends to a derivation $C:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M,\C) \to
\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot}(M,\C)$. Moreover, the weakly Hodge map $R_k:S^1(M,\C)
\to \B^2(M,\C)$ defines a Hodge bundle map $R_k^{tot}:S^1(M,\C) \to
\B^2_{tot}(M,\C)$, and solving \eqref{tslv} is equivalent to finding a
Hodge bundle map $D_k:S^1(M,\C) \to \B^1_{tot}(M,\C)$ such that
\begin{equation}\label{tslvc}
C \circ D_k = - R_k.
\end{equation}
\refstepcounter{subsubsection Recall that by \ref{C.and.sigma} the derivation
$$
C:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M,\C) \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot}(M,\C)
$$
satisfies $C \circ C = 0$, so that the total Weil algebra
$\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M,\C)$ becomes a complex with differential $C$. The
crucial part of the proof of Theorem~\ref{kal=ext} consists in
noticing that the subcomplex $\I_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) =
\Gamma(\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)) \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M,\C)$ of the total Weil
algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M,\C)$ corresponding to the kernel
$\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ of the quotient map
$\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ is canonically contractible.
This statement is analogous to Proposition~\ref{ac}, and we prove it
in the same way. Namely, we check that the subcomplex $\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}
\subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ is preserved by the bundle map
$\sigma_{tot}:\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ constructed in
\ref{sigma.c}, and that the anticommutator $h =
\{\sigma_{tot},C\}:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M,\C) \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M,\C)$ is invertible
on the subcomplex $\I_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M,\C)$
(Corollary~\ref{h.inv} of Lemma~\ref{h.acts}). We also check that $C
\circ R_k^{tot} = 0$, which implies that the Hodge bundle map
\begin{equation}\label{sltn}
D_k = -h^{-1} \circ \sigma_{tot} \circ R_k^{tot}:S^1(M,\C) \to \B^1_{tot}(M,\C)
\end{equation}
provides a solution to the equation \eqref{tslvc}.
\refstepcounter{subsubsection To establish the property \ref{second.prop}, we have to
insure additionally that the extended connection $D = D_{\leq k}$ is
linear, and and we have to show that the solution $D_k$ of
\eqref{tslvc} with this property is unique. This turns out to be
pretty straightforward. We show in Lemma~\ref{lin.aug} that $D_{\leq
k}$ is linear if and only if
\begin{equation}\label{lnr}
\sigma_{tot} \circ D_k = 0.
\end{equation}
Moreover, we show that the homotopy $\sigma_{tot}:\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot}(M,\C)
\to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M,\C)$ satisfies $\sigma_{tot} \circ \sigma_{tot} =
0$. Therefore the solution $D_k$ to \eqref{tslvc} given by
\eqref{sltn} satisfies \eqref{lnr} automatically.
The uniqueness of such a solution $D_k$ follows from the
invertibility of $h = C \circ \sigma_{tot} + \sigma_{tot} \circ C$. Indeed,
for every two solutions $D_k,D_k'$ to \eqref{tslv}, both satisfying
\eqref{lnr}, their difference $P = D_k - D'_k$ satisfies $C \circ P
= 0$. If, in addition, both $D_k$ and $D'_k$ were to satisfy
\eqref{lnr}, we would have had $\sigma_{tot} \circ P = 0$. Therefore $h
\circ P = 0$, and $P$ has to vanish.
\refstepcounter{subsubsection These are the main ideas of the proof of
Theorem~\ref{kal=ext}. The proof itself begins in the next
subsection, and it is organized as follows. In
Subsection~\ref{pf.first.sub} we express the linearity condition on
an extended connection $D$ in terms of the associated derivation
$D_{tot}:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_C \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot}$ of the total Weil algebra
$\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ of the manifold $M$. After that, we introduce in
Subsection~\ref{pf.third.sub} the reduced Weil algebra
$\wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ and prove Proposition~\ref{main}, thus reducing
Theorem~\ref{kal=ext} to a statement about derivations of the
reduced Weil algebra. Finally, in Subsection~\ref{pf.last.sub} we
prove this statement.
\noindent {\bf Remark.\ } In the Appendix we give, following Deligne and Simpson, a
more geomteric description of the functor $\Gamma:{{\cal W}{\cal H}odge}_{\geq 0}
\to {{\cal W}{\cal H}odge}$ and of the total Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}(M,\C)$,
which allows to give a simpler and more conceptual proof for the key
parts of Theorem~\ref{kal=ext}.
\subsection{Linearity and the total Weil algebra}\label{pf.first.sub}
\refstepcounter{subsubsection Assume given an extended connection $D:S^1 \to \B^1$ on the
manifold $M$, and extend it to a derivation $D:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to
\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ of the Weil algebra as in \ref{deriv}. Let $D =
\sum_{k\geq 0} D_k$ be the augmentation degree decomposition. The
derivation $D$ is weakly Hodge and defines therefore a derivation $D
= \sum_{k\geq 0} D_k:\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B_{tot}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ of the total
Weil algebra $\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$.
Before we begin the proof of Theorem~\ref{kal=ext}, we give the
following rewriting of the linearity condition \ref{lin.ext.con} on
the extended connection $D$ in terms of the total Weil algebra.
\begin{lemma}\label{lin.aug}
The extended connection $D$ is linear if and only if $D_0=C$ and
$\sigma_{tot} \circ D_k = 0$ on $S^1 \subset \B_{tot}^0$ for every $k \geq
0$.
\end{lemma}
\proof Indeed, by Lemma~\ref{total.aug} for odd integers $k$ and $n
= \pm 1$ the subbundle $\left(\B_o^1\right)^{n,-n}_{k+1} \subset
\B_{tot}^1$ vanishes. Therefore the map $D_k:S^1 \to \B_{tot}^1$ factors
through $\B_{ll}^1 \oplus \B_{rr}^1$. Since by definition
(\ref{sigma.c}) we have $\sigma_{tot}=0$ on both $\B_{ll}^1$ and
$\B_{rr}^1$, for odd $k$ we have $\sigma_{tot} \circ D_k = 0$ on $S^1$
regardless of the extended connection $D$. On the other hand, for
even $k$ we have $\left(\B_{tot}^1\right)^{n,-n}_{k+1} =
\left(\B_o^1\right)^{n,-n}_{k+1}$. Therefore on $S^1$ we have
$\sigma_{tot} \circ D_k = \sigma \circ D_k$ (where $\sigma:\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}
\to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is as in \ref{C.and.sigma}). Moreover, since
$\sigma:\Lambda^1 \to S^1$ is an isomorphism, $D_0 = C$ is
equivalent to $\sigma \circ D_0 = {\sf id}:S^1 \to \Lambda^1 \to
S^1$. Therefore the condition of the lemma is equivalent to the
following
\begin{equation}\label{tprv}
\sigma \circ D_k =
\begin{cases}
{\sf id}, \quad &\text{ for } k=0\\
0, \quad &\text{ for even integers } k > 0.
\end{cases}
\end{equation}
Let now $\iota^*:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ be the operator given by the
action of the canonical involution $\iota:{\overline{T}M} \to {\overline{T}M}$, as in
\ref{iota.Weil}, and let $D^\iota = \sum_{k \geq 0} D_k^\iota =
\iota^* \circ D \circ(\iota^*)^{-1}$ be the operator
$\iota^*$-conjugate to the derivation $D$. The operator $\iota^*$
acts as $-{\sf id}$ on $S^1 \subset \B^0$ and as ${\sf id}$ on $\Lambda^1
\subset \B^1$. Since it is an algebra automorphism, it acts as
$(-1)^{i+k}$ on $\B_k^i \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. Therefore $D_k^\iota =
(-1)^{k+1}D_k$, and \eqref{tprv} is equivalent to
$$
\sigma \circ \frac{1}{2}(D - D^\iota) = {\sf id}:S^1 \to \B^1 \to S^1,
$$
which is precisely the definition of a linear extended connection.
\hfill \ensuremath{\square}\par
\subsection{The reduced Weil algebra}\label{pf.third.sub}
\refstepcounter{subsubsection We now begin the proof of Theorem~\ref{kal=ext}. Our first
step is to reduce the classification of linear flat extended
connections $D:S^1 \to \B^1$ on the manifold $M$ to the study of
derivations of a certain quotient $\wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ of the Weil algebra
$\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. We introduce this quotient in this subsection under the
name of reduced Weil algebra. We then show that every extended
connection $D$ on $M$ induces a derivation $\wt{D}:\wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to
\wt{\B}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ of the reduced Weil algebra, and that a linear flat
extended connection $D$ on $M$ is completely defined by the
derivation $\wt{D}$.
\refstepcounter{subsubsection By Lemma~\ref{h.acts} the anticommutator $h =
\{C,\sigma_{tot}\}:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ of the canonical bundle
endomorphisms $C,\sigma_{tot}$ of the total Weil algebra $\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is
invertible on every component $\left(\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}\right)_{p,q}$ of
augmentation bidegree $(p,q)$ with $p,q \geq 1$.
The direct sum $\oplus_{p,q \geq 1}\left(\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\right)_{p,q}$ is
an ideal in the total Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$, and it is obtained
by applying the functor $\Gamma$ to the ideal $\oplus_{p,q \geq
1}\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{p,q}$ in the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. For technical
reasons, it will be more convenient for us to consider the smaller
subbundle
$$
\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} = \bigoplus_{p \geq 2,q \geq 1}\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{p,q} +
\bigoplus_{p \geq 1, q \geq 2} \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{p,q} \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}
$$
of the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. The subbundle $\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is a Hodge
subbundle in $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$, and it is an ideal with respect to the
multiplication in $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$.
\refstepcounter{subsubsection \noindent {\bf Definition.\ } \label{red.Weil}
The reduced Weil algebra $\wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} = \wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ of the
manifold $M$ is the quotient
$$
\wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} = \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} / \I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}
$$
of the full Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ by the ideal $\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$.
The reduced Weil algebra decomposes as
$$
\wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} = \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{1,1} \oplus \bigoplus_{p \geq 0}\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{p,0}
\oplus \bigoplus_{q \geq 0}\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{0,q}
$$
with respect to the augmentation bigrading on the Weil algebra
$\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. The two summands on the right are equal to
\begin{align*}
\bigoplus_{p \geq 0}\B_{p,0}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} &= \bigoplus_{p geq
0}S^p\left(S^{1,-1}\right) \otimes \Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},0}, \\
\bigoplus_{q \geq 0}\B_{0,q}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} &= \bigoplus_{q geq
0}S^q\left(S^{-1,1}\right) \otimes \Lambda^{0,{\:\raisebox{3pt}{\text{\circle*{1.5}}}}},
\end{align*}
\refstepcounter{subsubsection Since $\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is a Hodge subbundle, the reduced Weil
algebra carries a canonical Hodge bundle structure compatible with
the multiplication. It also obviously inherits the augmentation
bigrading, and defines an ideal $\I_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} = \Gamma(\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}})
\subset \B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ in the total Weil algebra
$\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. Lemma~\ref{h.acts} immediately implies the following
fact.
\begin{corr}\label{h.inv}
The map $h=\{C,\sigma_{tot}\}:\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is invertible
on $\I_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \subset \B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$.
\end{corr}
\refstepcounter{subsubsection \label{wD}
Let now $D:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ be the derivation associated to
the extended connection $D$ as in \ref{deriv}. The derivation $D$
does not increase the augmentation bidegree, it preserves the ideal
$\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ and defines therefore a weakly Hodge
derivation of the reduced Weil algebra $\wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$, which we denote
by $\wt{D}$. If the extended connection $D$ is flat, then the
derivation $\wt{D}$ satisfies $\wt{D} \circ \wt{D} = 0$.
We now prove that every derivation $\wt{D}:\wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \wt{\B}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$
of this type comes from a linear flat extended connection $D$, and
that the connection $D$ is completely defined by $\wt{D}$. More
precisely, we have the following.
\begin{prop}\label{main}
Let $D:S^1 \to \B^1$ be a linear but not necessarily flat extended
connection on $M$, and let $\wt{D}:\wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \wt{\B}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ be the
associated weakly Hodge derivation of the reduced Weil algebra
$\wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. Assume that $\wt{D} \circ \wt{D} = 0$.
There exists a unique weakly Hodge bundle map $P:S^1 \to \I^1$ such
that the extended connection $D' = D + P:S^1 \to \B^1$ is linear
and flat.
\end{prop}
\refstepcounter{subsubsection
\proof Assume given a linear extended connection $D$ satisfying the
condition of Proposition~\ref{main}. To prove the proposition, we
have to construct a weakly Hodge map $P:S^1 \to \I^1$ such that the
extended connection $D+P$ is linear and flat. We do it by induction
on the augmentation degree, that is, we construct one-by-one the
terms $P_k$ in the augmentation degree decomposition $P = \sum_k
P_k$. The identity $\wt{D} \circ \wt{D} = 0$ is the base of the induction,
and the induction step is given by applying the following lemma to
$D + \sum_{i=2}^k P_k$, for each $k \geq 1$ in turn.
\begin{lemma}\label{main.ind}
Assume given a linear extended connection $D:S^1 \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$
on $M$ and let $D:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ also denote the
associated derivation. Assume also that the composition $D \circ
D:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+2}$ maps $S^1$ into $\I^2_{>k} =
\oplus_{p>k}\I^2_p$.
There exists a unique weakly Hodge bundle map $P_k:S^1 \to
\I^1_{k+1}$ such that the extended connection $D' = D + P_k:S^1 \to
\B^1$ is linear, and for the associated derivation $D':\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to
\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ the composition $D' \circ D'$ maps $S^1$ into
$\I^2_{>k+1} = \oplus_{p>k+1}\I^2_p$.
\end{lemma}
\proof Let $D:\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ be the derivation of the
total Weil algebra associated to the extended connection $D$, and
let
$$
R:\left(\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\right)_{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to
\left(\B_{tot}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+2}\right)_{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+k}
$$
be the component of augmentation degree $k$ of the composition $D
\circ D:S^1 \to \B^2_{tot}$. Note that by \ref{deriv} the map $R$
vanishes on the subbundle $\Lambda^1_{tot} \subset
\left(\B^1_{tot}\right)_1$. Moreover, the composition $C \circ
R:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+3}$ of the map $R$ with the canonical
derivation $C:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot}$ vanishes on the
subbundle $S^1 \subset \B^0_{tot}$. Indeed, since $C$ maps $S^1$ into
$\Lambda^1_{tot}$, the composition $C \circ R$ is equal to the
commutator $[C,R]:S^1 \to \B^3_{tot}$. Since the extended connection
$D$ is by assumption linear, we have $D_0=C$, and
\begin{align*}
C \circ R = [C,R] &= \sum_{0 \leq p \leq k}[C,D_p \circ D_{k-p}] =\\
&= [C,\{C,D_k\}] + \sum_{1 \leq p \leq k-1}[C,D_p \circ D_{k-p}].
\end{align*}
Since $C \circ C = 0$, the first term in the right hand side
vanishes. Let $\Theta = \sum_{1 \leq p \leq k-1}D_p:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} \to
\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot}$. Then the second term is the component of
augmentation degree $k$ in the commutator $[C,\Theta \circ
\Theta]:S^1 \to \B^3_{tot}$. By assumption $\{D,D\}=0$ in augmentation
degrees $<k$. Therefore we have $\{C,\Theta\} = - \{\Theta,\Theta\}$
in augmentation degrees $< k$. Since $\Theta$ increases the
augmentation degree, this implies that in augmentation degree $k$
$$
[C,\Theta \circ \Theta] = \{C,\Theta\} \circ \Theta - \Theta \circ
\{C,\Theta\} = [\Theta,\{\Theta,\Theta\}],
$$
which vanishes tautologically.
The set of all weakly Hodge maps $P:S^1 \to \I_k^1$ coincides with
the set of all maps $P:S^1 \to \left(\I_{tot}^1\right)_k$ preserving the
Hodge bundle structures. Let $P$ be such a map, and let
$D':\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B_{tot}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ be the derivation associated to
the extended connection $D' = D + P$.
Since the extended connection $D$ is by assumption linear, by
Lemma~\ref{lin.aug} the extended connection $D'$ is linear if and
only if $\sigma_{tot} \circ P = 0$. Moreover, since the augmentation
degree-$0$ component of the derivation $D$ equals $C$, the
augmentation degree-$k$ component $Q:S^1 \to \B^2_{tot}$ in the
composition $D' \circ D'$ is equal to
$$
Q = R + \{C,(D'-D)\}.
$$
By definition $D'-D:\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B_{tot}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ equals $P$ on
$S^1 \subset \B_{tot}^0$ and vanishes on $\Lambda^1_{tot} \subset
\B_{tot}^1$. Since $C$ maps $S^1$ into $\Lambda^1_{tot} \subset \B_{tot}^1$, we
have $Q = R + C \circ P$. Thus, a map $P$ satisfies the condition of
the lemma if and only if
$$
\begin{cases}
C \circ P = -R\\
\sigma_{tot} \circ P = 0
\end{cases}
$$
To prove that such a map $P$ is unique, note that these equations imply
$$
h \circ P = (\sigma_{tot} \circ C + C \circ \sigma_{tot}) \circ P = \sigma_{tot}
\circ R,
$$
and $h$ is invertible by Corollary~\ref{h.inv}. To prove that such a
map $P$ exists, define $P$ by
$$
P = -h^{-1} \circ \sigma_{tot} \circ R:S^1 \to \I^1_{k+1}.
$$
The map $h = \{C,\sigma_{tot}\}$ and its inverse $h^{-1}$ commute with
$C$ and with $\sigma_{tot}$. Since $\sigma_{tot} \circ \sigma_{tot} = C \circ
C0$, we have $\sigma_{tot} \circ P = h^{-1} \circ \sigma_{tot} \circ
\sigma_{tot} \circ R = 0$. On the other hand, $C \circ R = 0$. Therefore
\begin{align*}
C \circ P &= - C \circ h^{-1} \circ \sigma_C \circ R = - h^{-1} \circ
C \circ \sigma_{tot} \circ R = \\
&= h^{-1} \circ \sigma_{tot} \circ C \circ R - h^{-1} \circ h \circ R = -R.
\end{align*}
This finishes the proof of the lemma and of Proposition~\ref{main}.
\hfill \ensuremath{\square}\par
\subsection{Reduction of extended connections}\label{pf.last.sub}
\refstepcounter{subsubsection We now complete the proof of Theorem~\ref{kal=ext}. First we
will need to identify explicitly the low Hodge bidegree components
of the reduced Weil algebra $\wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. The following is easily
checked by direct inspection.
\begin{lemma}
We have
\begin{align*}
&\wt{\B}^{2,-1} \oplus \wt{\B}^{1,0} \oplus \wt{\B}^{0,1} \oplus \wt{\B}^{-1,2} =
\Lambda^1 \oplus \left(S^1 \otimes \Lambda^1\right) \subset \wt{\B}^1\\
&\wt{\B}^{3,-2} \oplus \wt{\B}^{2,-1} \oplus \wt{\B}^{1,1} \oplus \wt{\B}^{-1,2}
\oplus \wt{\B}^{-2,3} = \\
&\qquad\qquad\qquad\qquad\qquad\qquad = \left(S^{1,-1} \otimes
\Lambda^{2,0}\right) \oplus \left(S^{-1,1} \otimes
\Lambda^{0,2}\right) \oplus \Lambda^2 \subset \wt{\B}^2
\end{align*}
\end{lemma}
\refstepcounter{subsubsection Let now $\nabla:S^1 \to S^1 \otimes \Lambda^1$ be an
arbitrary real connection on the bundle $S^1$. The operator
$$
D = C + \nabla:S^1 \to \Lambda^1 \oplus \left(\Lambda^1 \otimes S^1\right)
\subset \B^1
$$
is then automatically weakly Hodge and defines therefore an extended
connection on $M$. This connection is linear by
Lemma~\ref{lin.aug}. Extend $D$ to a derivation $D:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to
\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ as in \ref{deriv}, and let $\wt{D}:\wt{\B}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to
\wt{\B}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ be the associated derivation of the reduced Weil
algebra.
\begin{lemma}\label{kal=red}
The derivation $\wt{D}$ satisfies $\wt{D} \circ \wt{D} = 0$ if and only if
the connection $\nabla$ is K\"ah\-le\-ri\-an.
\end{lemma}
\proof Indeed, the operator $\wt{D} \circ \wt{D}$ is weakly Hodge, hence
factors through a bundle map
\begin{multline*}
\wt{D} \circ \wt{D}: S^1 \to \wt{\B}^{3,-1} \oplus \wt{\B}^{2,0} \oplus \wt{\B}^{1,1}
\oplus \wt{\B}^{0,2} \oplus \wt{\B}^{-1,3} = \\
= \left(S^{1,-1} \otimes \Lambda^{2,0}\right) \oplus \left(S^{-1,1}
\otimes \Lambda^{0,2}\right) \oplus \Lambda^2 \subset \B^2.
\end{multline*}
By definition we have
$$
\wt{D} \circ \wt{D} = (C + \nabla) \circ (C + \nabla) = \{C, \nabla\} + \{
\nabla, \nabla \}.
$$
An easy inspection shows that the sum is direct, and the first
summand equals
$$
\{C, \nabla\} = T \circ C:S^1 \to \Lambda^2,
$$
where $T$ is the torsion of the connection $\nabla$, while the
second summand equals
$$
\{\nabla, \nabla\} = R^{2,0} \oplus R^{0,2}: S^{1,-1} \oplus
S^{-1,1} \to \left(S^{1,-1} \otimes \Lambda^{2,0}\right) \oplus
\left(S^{-1,1} \otimes \Lambda^{0,2}\right),
$$
where $R^{2,0}$, $R^{0,2}$ are the Hodge type components of the
curvature of the connection $\nabla$. Hence $\wt{D} \circ \wt{D} = 0$ if
and only if $R^{2,0} = R^{0,2} = T = 0$, which proves the lemma and
finishes the proof of Theorem~\ref{kal=ext}.
\hfill \ensuremath{\square}\par
\refstepcounter{subsubsection We finish this section with the following corollary of
Theorem~\ref{kal=ext} which gives an explicit expression for
the augmentation degree-$2$ component $D_2$ of a flat linear
extended connection $D$ on the manifold $M$. We will need this
expression in Section~\ref{metrics.section}.
\begin{corr}\label{D.2}
Let $D = \sum_{k \geq 0}D_k:S^1 \to \B^1$ be a flat linear extended
connection on $M$, so that $D_0 = C$ and $D_1$ is a K\"ah\-le\-ri\-an
connection on $M$. We have
$$
D_2 = \frac{1}{3}\sigma \circ R,
$$
where $\sigma:\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is the canonical derivation
introduced in \ref{sigma}, and $R = D_1 \circ D_1:S^1 \to S^1
\otimes \Lambda^{1,1} \subset \B^2$ is the curvature of the
K\"ah\-le\-ri\-an connection $D_1$.
\end{corr}
\proof Extend the connection $D$ to a derivation $D_C = \sum_{k \geq
0}D^{tot}_k:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot}$ of the total Weil algebra. By
the construction used in the proof of Lemma~\ref{main.ind} we have
$D^{tot}_2 = h^{-1} \circ \sigma_{tot} \circ R_{tot}:S^1 \to
\left(\B^1_{tot}\right)_3$, where $h:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ is as in
Lemma~\ref{h.acts}, the map $\sigma_{tot}:\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$
is the canonical map constructed in \ref{sigma.c}, and $R^{tot}:S^1 \to
\left(\B^2_{tot}\right)_3$ is the square $R_{tot} = D^{tot}_1 \circ D^{tot}_1$ of
the derivation $D^{tot}_1:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot}$. By
Lemma~\ref{h.on.b1} the map $h$ acts on $\left(\B_{tot}^1\right)_3$ by
multiplication by $3$. Moreover, it is easy to check that
$$
\left(B^2_{tot}\right)_3 = \left(S^1 \otimes \Lambda^2\right) \oplus
\left(S^{-1,1} \otimes \Lambda^{2,0}\right)\oplus
\left(S^{1,-1} \otimes \Lambda^{0,2}\right),
$$
and the map $R^{tot}:S^1 \to \left(B^2_{tot}\right)_3$ sends $S^1$ into the
first summand in this decomposition and coincides with the curvature
$R:S^1 \to S^1 \otimes \Lambda^2 \subset \left(B^2_{tot}\right)_3$ of
the K\"ah\-le\-ri\-an connection $D_1$. Therefore $\sigma_{tot} \circ R^{tot} =
\sigma \circ R$, which proves the claim.
\hfill \ensuremath{\square}\par
\section{Metrics}\label{metrics.section}
\subsection{Hyperk\"ahler metrics on Hodge manifolds}
\refstepcounter{subsubsection Let $M$ be a complex manifold equipped with a K\"ah\-le\-ri\-an
connection $\nabla$, and consider the associated linear formal Hodge
manifold structure on the tangent bundle ${\overline{T}M}$. In this section we
construct a natural bijection between the set of all polarizations
on the Hodge manifold ${\overline{T}M}$ in the sense of
Subsection~\ref{polarization} and the set of all K\"ahler metrics on
$M$ compatible with the given connection $\nabla$.
\refstepcounter{subsubsection \label{restr}
Let $h$ be a hyperk\"ahler metric on ${\overline{T}M}$, or, more
generally, a formal germ of such a metric in the neighborhood of the
zero section $M \subset {\overline{T}M}$. Assume that the metric $h$ is
compatible with the given hypercomplex structure and Hermitian-Hodge
in the sense of \ref{hermhodge}, and let $\omega_I$ be the K\"ahler
form associated to $h$ in the preferred complex structure ${\overline{T}M}_I$ on
${\overline{T}M}$.
Let $h_M$ be the restriction of the metric $h$ to the zero section
$M \subset {\overline{T}M}$, and let $\omega \in C^\infty(M,\Lambda^{1,1}(M))$
be the associated real $(1,1)$-form on the complex manifold
$M$. Since the embedding $M \subset {\overline{T}M}_I$ is holomorphic,
the form $\omega$ is the restriction
onto $M$ of the form $\omega_I$. In particular, it is closed, and
the metric $h_M$ is therefore K\"ahler.
\refstepcounter{subsubsection The main result of this section is the following.
\begin{theorem}\label{metrics}
Restriction onto the zero section $M \subset {\overline{T}M}$ defines a
one-to-one correspondence between
\begin{enumerate}
\item K\"ahler metrics on $M$ compatible with the K\"ah\-le\-ri\-an
connection $\nabla$, and
\item formal germs in the neighborhood on $M \subset {\overline{T}M}$ of
Hermitian-Hodge hyperk\"ahler metrics on ${\overline{T}M}$ compatible with the
given formal Hodge manifold structure.
\end{enumerate}
\end{theorem}
\noindent
The rest of this section is devoted to the proof of this theorem.
\refstepcounter{subsubsection In order to prove Theorem~\ref{metrics}, we reformulate it in
terms of polarizations rather than metrics. Recall (see
\ref{positive}) that a {\em polarization} of the formal Hodge
manifold ${\overline{T}M}$ is by definition a $(2,0)$-form $\Omega \in
C^\infty_M({\overline{T}M},\Lambda^{2,0}({\overline{T}M}_J))$ for the complementary complex
structure ${\overline{T}M}_J$ which is holomorphic, real and of $H$-type $(1,1)$
with respect to the canonical Hodge bundle structure on
$\Lambda^{2,0}({\overline{T}M}_J)$, and satisfies a certain positivity condition
\eqref{P}.
\refstepcounter{subsubsection By Lemma~\ref{pol.hm} Hermitian-Hodge hyperk\"ahler metrics
on ${\overline{T}M}$ are in one-to-one correspondence with polarizations. Let
$h$ be a metric on $M$, and let $\omega_I$ and $\omega$ be the
K\"ahler forms for $h$ on ${\overline{T}M}_I$ and on $M \subset {\overline{T}M}_I$.
The corresponding polarization $\Omega \in
C^\infty({\overline{T}M},\Lambda^{2,0}({\overline{T}M}_J))$ satisfies by
\eqref{omega.and.Omega}
\begin{equation}\label{pol2kal}
\omega_I = \frac{1}{2}\left(\Omega + \nu(\Omega)\right) \in \Lambda^2({\overline{T}M},\C),
\end{equation}
where $\nu:\Lambda^2({\overline{T}M},\C) \to \overline{\Lambda^2({\overline{T}M},\C)}$ is
the usual complex conjugation.
\refstepcounter{subsubsection Let $\rho:{\overline{T}M} \to M$ be the natural projection, and let
$$
\Res:\rho_*\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},0}({\overline{T}M}_J) \to \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)
$$
be the map given by the restriction onto the zero section $M \subset
{\overline{T}M}$. Both bundles are naturally Hodge bundles of the same weight on
$M$ in the sense of \ref{hodge.bundles}, and the bundle map $\Res$
preserves the Hodge bundle structures. Since $\Omega$ is of $H$-type
$(1,1)$, the form $\Res\Omega \in C^\infty(M,\Lambda^2(M,\C))$ is
real and of Hodge type $(1,1)$. By \eqref{pol2kal}
\begin{multline*}
\Res\Omega = \frac{1}{2}\left(\Res\Omega + \overline{\Res\Omega}\right) =
\frac{1}{2}(\Omega+\nu(\Omega))|_{M \subset {\overline{T}M}} = \\
= \omega_I|_{M \subset
{\overline{T}M}} = \omega \in \Lambda^{1,1}(M).
\end{multline*}
Therefore to prove Theorem~\ref{metrics}, it suffices to prove the
following.
\begin{itemize}
\item \label{to.prove} For every polarization $\Omega$ of the formal
Hodge manifold ${\overline{T}M}$ the restriction $\omega = \Res\Omega \in
C^\infty(\Lambda^{1,1}(M))$ is compatible with the connection
$\nabla$, that is, $\nabla\omega=0$. Vice versa, every real positive
$(1,1)$-form $\omega \in C^\infty(\Lambda^{1,1}(M))$ satisfying
$\nabla\omega=0$ extends to a polarization $\Omega$ of ${\overline{T}M}$, and
such an extension is unique.
\end{itemize}
This is what we will actually prove.
\subsection{Preliminaries}\label{drm.mod}
\refstepcounter{subsubsection\label{descr} We begin with introducing a convenient model for
the holomorphic de Rham algebra $\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},0}({\overline{T}M}_J)$ of the
complex manifold ${\overline{T}M}_J$, which would be independent of the Hodge
manifold structure on ${\overline{T}M}$. To construct such a model, consider
the relative de Rham complex $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}({\overline{T}M}/M,\C)$ of ${\overline{T}M}$ over
$M$ (see \ref{relative.de.rham.sub} for a reminder of its definition
and main properties). Let $\pi:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}({\overline{T}M},\C) \to
\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}({\overline{T}M}/M,\C)$ be the canonical projection. Recall that
the bundle $\Lambda^i({\overline{T}M}/M,\C)$ of relative $i$-forms on ${\overline{T}M}$ over
$M$ carries a natural structure of a Hodge bundle of weight
$i$. Moreover, we have introduced in \eqref{eta} a Hodge bundle
isomorphism
$$
\eta:\rho^*\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}({\overline{T}M}/M,\C)
$$
between $\Lambda^i({\overline{T}M}/M,\C)$ and the pullback
$\rho^*\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ of the bundle of $\C$-valued $i$-forms
on $M$.
\begin{lemma}\label{ident.bis}
\begin{enumerate}
\item The projection $\pi$ induces an algebra isomorphism
$$
\pi:\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},0}({\overline{T}M}_J) \to \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}({\overline{T}M}/M,\C)
$$
compatible with the natural Hodge bundle structures.
\item Let $\alpha \in C^\infty_M({\overline{T}M},\Lambda^{i,0}({\overline{T}M}_J))$ be a
smooth $(i,0)$-form on ${\overline{T}M}_J$, and consider the smooth $i$-form
$$
\beta = \eta^{-1}\pi(\alpha) \in C^\infty_M({\overline{T}M},\Lambda^i({\overline{T}M},\C))
$$
on ${\overline{T}M}$. The forms $\alpha$ and $\beta$ have the same restriction
to the zero section $M \subset {\overline{T}M}$.
\end{enumerate}
\end{lemma}
\proof Since $\eta$, $\pi$ and the restriction map are compatible
with the algebra structure on $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$, it suffices to
prove both claims for $\Lambda^1(M,\C)$. For every bundle $\E$ on
${\overline{T}M}$ denote by $\E|_{M \subset {\overline{T}M}}$ the restriction of $\E$ to the
zero section $M \subset {\overline{T}M}$. Consider the bundle map
$$
\chi = \eta \circ \Res:\Lambda^{1,0}({\overline{T}M}_J)|_{M \subset {\overline{T}M}} \to
\Lambda^1(M,\C) \to \Lambda^1({\overline{T}M}/M,\C)|_{M \subset {\overline{T}M}}.
$$
The second claim of the lemma is then equivalent to the identity
$\chi = \pi$.\! Moreover, note that the contraction with
the canonical vector field $\phi$ on ${\overline{T}M}$ defines an injective map
$i_\phi:C^\infty(M,\Lambda^1({\overline{T}M}/M,\C)|_M) \to
C^\infty({\overline{T}M},\C)$. Therefore it suffices to prove that $i_\phi \circ
\chi = i_\phi \circ \pi$.
Every smooth section $s$ of the bundle $\Lambda^{1,0}({\overline{T}M}_J)|_{M \subset {\overline{T}M}}$
is of the form
$$
s = (\rho^*\alpha + \sqrt{-1}j\rho^*\alpha)|_{M \subset {\overline{T}M}},
$$
where $\alpha \in C^\infty(M,\Lambda^1(M,\C))$ is a smooth $1$-form
on $M$, and $j:\Lambda^1({\overline{T}M},\C) \to \overline{\Lambda^1({\overline{T}M},\C)}$
is the map induced by the quaternionic structure on ${\overline{T}M}$. For such
a section $s$ we have $\Res(s) = \alpha$, and by \eqref{phi.and.tau}
$i_\phi(\chi(s)) = \sqrt{-1}\tau(\alpha)$, where
$\tau:C^\infty(M,\Lambda^1(M,\C)) \to C^\infty({\overline{T}M},\C)$ is the
tautological map introduced in \ref{tau}. On the other hand, since $\pi
\circ \rho^* = 0$, we have
$$
i_\phi(\pi(s)) = i_\phi(\pi(\sqrt{-1}j\rho^*\alpha)) =
i_\phi(\sqrt{-1}j\rho^*\alpha).
$$
Since the Hodge manifold structure on ${\overline{T}M}$ is linear, this equals
$$
i_\phi(\pi(s)) = \sqrt{-1}i_\phi(j(\rho^*\alpha)) =
\sqrt{-1}\tau(\alpha) = \chi(s),
$$
which proves the second claim of the lemma. Moreover, it shows that
the restriction of the map $\pi$ to the zero section $M \subset {\overline{T}M}$
is an isomorphism. As in the proof of Lemma~\ref{ident}, this
implies that the map $\pi$ is an isomorphism on the whole ${\overline{T}M}$,
which proves the first claim and finishes the proof of the lemma.
\hfill \ensuremath{\square}\par
\refstepcounter{subsubsection Lemma~\ref{ident.bis}~\thetag{i} allows to define the bundle
isomorphism
$$
\pi^{-1} \circ \eta:\rho^*\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to
\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}({\overline{T}M}/M,\C) \to \Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},0}({\overline{T}M}_J),
$$
between $\rho^*\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ and $\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},0}({\overline{T}M}_J)$,
and it induces an isomorphism
$$
\rho_*(\eta \circ \pi^{-1}):\rho_*\rho^*\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \cong
\rho_*\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},0}({\overline{T}M}_J)
$$
between the direct images of these bundles under the canonical
projection $\rho:{\overline{T}M} \to M$.
On the other hand, by adjunction we have the canonical embedding
$$
\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \hookrightarrow \rho_*\rho^*\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C),
$$
and by the projection formula it extends to an isomorphism
$$
\rho_*\rho^*\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \cong \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \otimes
\B^0,
$$
where $\B^0 = \rho_*\Lambda^0({\overline{T}M},\C)$ is the $0$-th component of
the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ of $M$. All these isomorphisms are
compatible with the Hodge bundle structures and with the
multiplication.
\refstepcounter{subsubsection \label{ident.punkt}
It will be convenient to denote the image $\rho_*(\eta \circ
\pi^{-1})\left(\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)\right) \subset
\rho_*\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},0}({\overline{T}M}_J)$ by $L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ or, to simplify
the notation, by $L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. (The algebra $L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ is, of
course, canonically isomorphic to $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$.) We then
have the identification
\begin{equation}\label{ident.zero}
L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \B^0 \cong \rho_*\rho^*\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \cong
\rho_*\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},0}({\overline{T}M}_J).
\end{equation}
This identification is independent of the Hodge manifold structure
on ${\overline{T}M}$. Moreover, by Lemma~\ref{ident.bis}~\thetag{ii} the
restriction map $\Res:\rho_*\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},0}({\overline{T}M}_J) \to
\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ is identified under \eqref{ident.zero} with the
canonical projection $L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \B^0 \to L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes
\B^0_0 \cong L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$.
By Lemma~\ref{ident} we also have the identification
$\rho_*\Lambda^{0,{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}({\overline{T}M}_J) \cong \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. Therefore
\eqref{ident.zero} extends to an algebra isomorphism
\begin{equation}\label{ident.formula}
\rho_*\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}({\overline{T}M}_J) \cong
\rho_*\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}({\overline{T}M}/M,\C) \otimes \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \cong
L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}.
\end{equation}
This isomorphism is also compatible with the Hodge bundle structures
on both sides.
\subsection{The Dolbeault differential on $\protect{\overline{T}M}_J$}
\refstepcounter{subsubsection Our next goal is to express the Dolbeault differential
$\bar\partial_J$ of the complex manifold ${\overline{T}M}_J$ in terms of the model for
the de Rham complex $\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}({\overline{T}M}_J)$ given by
\eqref{ident.formula}. For every $k \geq 0$ denote by
$$
D:L^k \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to L^k \otimes \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}.
$$
the differential operator induced by
$\bar\partial_J:\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}({\overline{T}M}_J) \to
\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}({\overline{T}M}_J)$ under \eqref{ident.formula}. The
operator $D$ is weakly Hodge. It satisfies the Leibnitz rule with
respect to the algebra structure on $L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$, and
we have $D \circ D = 0$. By definition for $k=0$ it coincides with
the derivation $D:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ defined by the Hodge
manifold structure on ${\overline{T}M}$. For $k > 0$ the complex $\langle L^k
\otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}, D \rangle$ is a free differential graded module
over the Weil algebra $\langle\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}},D\rangle$.
\refstepcounter{subsubsection \label{dr.L}
The relative de Rham differential $d^r$ (see
Subsection~\ref{relative.de.rham.sub}) induces under the isomorphism
\eqref{ident.formula} an algebra derivation
$$
d^r:L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1} \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}.
$$
The derivation $d^r$ also is weakly Hodge, and we have the
following.
\begin{lemma}\label{D.dr}
The derivations $D$ and $d^r$ commute, that is,
$$
\{D,d^r\} = 0:L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1} \otimes
\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}.
$$
\end{lemma}
\proof The operator $\{D,d^r\}$ satisfies the Leibnitz rule, so it
suffices to prove that it vanishes on $\B^0$, $\B^1$ and $L^1
\otimes \B^0$. Moreover, the $\B^0$-modules $\B^1$ and $L^1 \otimes
\B^0$ are generated, respectively, by local sections of the form $Df$ and
$d^rf$, $f \in \B^0$. Since $\{D,d^r\}$ commutes with $D$ and $d^r$,
it suffices to prove that it vanishes on $\B^0$. Finally,
$\{D,d^r\}$ is continuous in the adic topology on $\B^0$. Since the
subspace
$$
\{fg|f,g \in \B^0, Df = d^rg = 0\} \subset \B^0
$$
is dense in this topology, it suffices to prove that for a local
section $f \in \B^0$ we have $\{D,d^r\}f=0$ if either $d^rf=0$ of
$Df=0$.
It is easy to see that for every local section $\alpha \in \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$
we have $d^r\alpha = 0$ if and only if $\alpha \in \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_0$ is of
augmentation degree $0$. By definition the derivation $D$ preserves
the component $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_0 \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ of augmentation degree
$0$ in $\B^0$. Therefore $d^rf=0$ implies $d^rDf=0$ and consequently
$\{D,d^r\}f=0$. This handles the case $d^rf=0$. To finish the proof,
assume given a local section $f \in \B^0$ such that $Df=0$. Such a
section by definition comes from a germ at $M \subset {\overline{T}M}$ of a
holomorphic function $\wh{f}$ on ${\overline{T}M}_J$. Since $\wh{f}$ is
holomorphic, we have $\bar\partial_J\wh{f}=0$ and $d\wh{f} =
\partial_J\wh{f}$. Therefore $d^rf = \pi(d\wh{f}) = \pi(\partial_Jf)$, and
$$
Dd^rf = \pi(\bar\partial_J\partial_J\wh{f}) = -\pi(\partial_J\bar\partial_J\wh{f}) = 0,
$$
which, again, implies $\{D,d^r\}f = 0$.
\hfill \ensuremath{\square}\par
\refstepcounter{subsubsection Let now $\nabla=D_1:S^1 \to S^1 \otimes \Lambda^1$ be the
reduction of the extended connection $D$. It induces a connection on
the bundle $L^1 \cong S^1$, and this connection extends by the
Leibnitz rule to a connection on the exterior algebra $L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ of
the bundle $L^1$, which we will also denote by $\nabla$.
Denote by $R = \nabla \circ \nabla:L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes
\Lambda^2$ the curvature of the connection $\nabla$. Since $\nabla
\circ \nabla = \frac{1}{2}\{\nabla,\nabla\}$, the operator $R$ also
satisfies the Leibnitz rule with respect to the multiplication in
$L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$.
\refstepcounter{subsubsection Introduce the augmentation grading on the bundle $L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}
\otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ by setting $\deg L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} = 0$. The derivation
$D:L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$
obviously does not increase the augmentation degree, and we have the
decomposition $D = \sum_{k \geq 0}D_k$. On the other hand, the
derivation $d^r$ preserves the augmentation degree. Therefore
Lemma~\ref{D.dr} implies that for every $k \geq 0$ we have
$\{D_k,d^r\} = 0$. This in turn implies that $D_0 = 0$ on $L^p$ for
$p > 0$, and therefore $D_0 = {\sf id} \otimes C:L^p \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to
L^p \otimes \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$. Moreover, this allows to identify
explicitly the components $D_1$ and $D_2$ of the derivation
$D:L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \B^1$. Namely, we have the following.
\begin{lemma}\label{D.1.2}
We have
\begin{align*}
D_1 &= \nabla:L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \B^1_1 = L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes
\Lambda^1\\
D_2 &= \frac{1}{3}\sigma \circ R:L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \B^1_2,
\end{align*}
where $\sigma = {\sf id} \otimes \sigma:L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1} \to
L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is as in \ref{C.and.sigma}.
\end{lemma}
\proof Since both sides of these identities satisfy the Leibnitz
rule with respect to the multiplication in $L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$, it suffices to
prove them for $L^1$. But $d^r:\B^0 \to L^1 \otimes \B^0$ restricted
to $S^1 \subset \B^0$ becomes an isomorphism $d^r:S^1 \to
L^1$. Since $\{D_1,d^r\} = \{D_2,d^r\} = 0$, it suffices to prove
the identities with $L^1$ replaced with $S^1$. The first one then
becomes the definition of $\nabla$, and the second one is
Corollary~\ref{D.2}. \hfill \ensuremath{\square}\par
\subsection{The proof of Theorem~\ref{metrics}}
\refstepcounter{subsubsection We can now prove Theorem~\ref{metrics} in the form
\ref{to.prove}. We begin with the following corollary of
Lemma~\ref{D.1.2}.
\begin{corr}\label{red.corr}
Let $\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ be the ideal introduced in
\ref{red.Weil}. An arbitrary smooth section $\alpha \in
C^\infty(M,L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}})$ satisfies
\begin{equation}\label{rrd}
D\alpha \in C^\infty(M, L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \I^1) \subset C^\infty(M,
L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \B^1)
\end{equation}
if and only if $\nabla \alpha = 0$.
\end{corr}
\proof Again, both the identity \eqref{rrd} and the equality
$\nabla\alpha$ are compatible with the Leibnitz rule with respect to
the multiplication in $\alpha$. Therefore it suffices to prove that
they are equivalent for every $\alpha \in L^1$. By definition of the
ideal $\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ the equality \eqref{rrd} holds if and only if
$D_1\alpha = D_2\alpha = 0$. By Lemma~\ref{D.1.2} this is equivalent
to $\nabla\alpha = \sigma \circ R(\alpha) = 0$. But since $R =
\nabla \circ \nabla$, $\nabla\alpha = 0$ implies $\sigma \circ
R(\alpha) = 0$, which proves the claim.
\hfill \ensuremath{\square}\par
\refstepcounter{subsubsection Let now $\Omega \in C^\infty(M, \rho_*\Lambda^{2,0}({\overline{T}M}_J)
\cong L^2 \otimes \B^0$ be a polarization of the Hodge manifold
${\overline{T}M}_J$, so that $\Omega$ is of Hodge type $(1,1)$ and
$D\Omega=0$. Let $\omega = \Res\Omega \in
C^\infty(M,\Lambda^{1,1}(M,\C))$ be its restriction, and let $\Omega
= \sum_{k \geq 0}\Omega_k$ be its augmentation degree decomposition.
As noted in \ref{ident.punkt}, the restriction map
$\Res:\rho_*\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},0}({\overline{T}M}_J) \to \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ is
identified under the isomorphism \eqref{ident.formula} with the
projection $\LL^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \B^0 \to L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ onto the component of
augmentation degree $0$. Therefore $\omega = \Omega_0$. Since the
augmentation degree-$1$ component $\left(L^2 \otimes
B^0_{tot}\right)^{1,1}_1=0$ and $D\Omega = 0$, we have $\nabla\omega =
D_1\Omega_0 = 0$, which proves the first claim of
Theorem~\ref{metrics}.
\refstepcounter{subsubsection To prove the second claim of the theorem, let $\omega$ be a
K\"ahler form on $M$ compatible with the connection $\nabla$, so
that $\nabla\omega = 0$. We have to show that there exists a unique
section $\Omega = \sum_{k \geq 0}\Omega_k \in C^\infty(M, L^2
\otimes \B^0)$ which is of Hodge type $(1,1)$ and satisfies
$D\Omega=0$ and $\Omega_0 = \omega$. As in the proof of
Theorem~\ref{kal=ext}, we will use induction on $k$. Since $\Omega$
is of Hodge type $(1,1)$, we must have $\Omega_1 = 0$, and by
Corollary~\ref{red.corr} we have $D(\Omega_0 + \Omega_1) \in
C^\infty(M, L^2 \otimes \I^1)$, which gives the base of our
induction. The induction step is given by applying the following
proposition to $\sum_{0 \geq p \geq k}\Omega_k$ for each $k \geq 1$
in turn.
\begin{prop}\label{metrics.ind}
Assume given integers $p,q,k; p,q \geq 0, k \geq 1$ and assume given
a section $\alpha \in C^\infty(M, L^{p+q} \otimes \B^0)$ of Hodge
type $(p,q)$ such that
$$
D\alpha \in C^\infty(M, L^{p+q} \otimes \I^1_{\geq k}),
$$
where $\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{\geq k} = \oplus_{m \geq k}\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_m$. Then there
exists a unique section $\beta \in C^\infty(M, L^{p+q} \otimes
\B_k)$ of the same Hodge type $(p,q)$ and such that $D(\alpha +
\beta) \in C^\infty(M, L^{p+q} \otimes \I^1_{\geq k+1})$.
\end{prop}
\proof Let $\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ be the total Weil algebra introduced in
\ref{total.Weil}, and consider the free module $L^{p+q} \otimes
\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ over $\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ generated by the Hodge bundle
$L^{p+q}$. This module carries a canonical Hodge bundle structure of
weight $p+q$. Consider the maps $C:\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B_{tot}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$,
$\sigma_{tot}:\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1} \to \B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ introduced in \ref{total.C}
and \ref{sigma.c}, and let $C = {\sf id} \otimes C,\sigma_{tot} = {\sf id} \otimes
\sigma_{tot}:L^{p+q} \otimes \B_{tot} \to L^{p+q} \otimes \B_{tot}$ be the
associated endomorphisms of the free module $L^{p+q} \otimes
\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$.
The maps $C$ and $\sigma_{tot}$ preserve the Hodge bundle structure.
The commutator $h = \{C, \sigma_{tot}\}: \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ is
invertible on $\I_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \subset \B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ by
Corollary~\ref{h.inv} and acts as $k{\sf id}$ on $\B^0_k \subset
\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$. Therefore the endomorphism
$$
h = {\sf id} \otimes h = \{C,\sigma_{tot}\}:L^{p+q} \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} \to
L^{p+q} \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}
$$
is invertible on $L^{p+q} \otimes \I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ and acts as $k{\sf id}$ on
$L^{p+q} \otimes \B^0_k$.
Since the derivation $D:L^{p+q} \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to L^{p+q} \otimes
\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ is weakly Hodge, it induces a map $D^{tot}:L^{p+q} \otimes
\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to L^{p+q} \otimes \B_{tot}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$, and $D\alpha \in
L^{p+q} \otimes \I^1_{\geq k}$ if and only if the same holds for
$D^{tot}\alpha$. To prove uniqueness, note that $D_0 = C$ is injective
on $L^{p+q} \otimes \B^0_k$. If there are two sections
$\beta,\beta'$ satisfying the conditions of the proposition, then
$D_0(\beta-\beta')=0$, hence $\beta = \beta'$.
To prove existence, let $\gamma = (D^{tot}\alpha)_k$ be the component of
the section $D^{tot}\alpha$ of augmentation degree $k$. Since $D^{tot} \circ
D^{tot}=0$, we have $C\gamma = D^{tot}_0\gamma = 0$ and $C \circ \sigma_{tot}
\gamma = h \gamma$. Let $\beta = - \frac{1}{k} \circ
\sigma_{tot}(\gamma)$. The section $\beta$ is of Hodge type $(p,q)$ and
of augmentation degree $k$. Moreover, it satisfies
$$
D^{tot}_0\beta = C\beta = -Ch^{-1}\sigma_{tot}(\gamma) = -h^{-1} \circ C
\circ \sigma_{tot} \gamma = -h^{-1} \circ h\gamma = -\gamma.
$$
Therefore $D_{tot}(\alpha+\beta)$ is indeed a section of $L^{p+q}
\otimes (I^1_{tot})_{\geq k+1}$, which proves the proposition and
finishes the proof of Theorem~\ref{metrics}.
\hfill \ensuremath{\square}\par
\subsection{The cotangent bundle}
\refstepcounter{subsubsection For every K\"ahler manifold $M$ Theorem~\ref{metrics}
provides a canonical formal hyperk\"ahler structure on the total
space ${\overline{T}M}$ of the complex-conjugate to the tangent bundle to
$M$. In particular, we have a closed holomorphic $2$-form $\Omega_I$
for the preferred complex structure ${\overline{T}M}_I$ on $M$.
Let $T^*M$ be the total space to the cotangent bundle to $M$
equipped with the canonical holomorphic symplectic form $\Omega$. To
obtain a hyperk\"ahler metric of the formal neighborhood of the zero
section $M \subset T^*M$, one can apply an appropriate version of
the Darboux Theorem, which gives a local symplectic isomorphism
$\kappa:{\overline{T}M} \to T^*M$ in a neighborhood of the zero
section. However, this theorem is not quite standard in the
holomorphic and formal situations. For the sake of completeness, we
finish this section with a sketch of a construction of such an
isomorphism $\kappa:{\overline{T}M} \to T^*M$ which can be used to obtain a
hyperk\"ahler metric on $T^*M$ rather than on ${\overline{T}M}$.
\refstepcounter{subsubsection We begin with the following preliminary fact on the
holomorphic de Rham complex of the manifold ${\overline{T}M}_I$.
\begin{lemma}\label{de.rham.exact}
Assume given either a formal Hodge manifold structure on the
$U(1)$-manifold ${\overline{T}M}$ along the zero section $M \subset {\overline{T}M}$, or an
actual Hodge manifold structure on an open neighborhood $U \subset
{\overline{T}M}$ of the zero section.
\begin{enumerate}
\item For every point $m \in M$ there exists an open neighborhood $U
\subset {\overline{T}M}_I$ such that the spaces $\Omega^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U)$ of holomorphic
forms on the complex manifold ${\overline{T}M}_I$ (formally completed along $M
\subset {\overline{T}M}$ if necessary) equipped with the holomorphic de Rham
differential $\partial_I:\Omega^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U) \to \Omega^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(U)$ form an
exact complex.
\item If the subset $U \subset {\overline{T}M}$ is invariant under the
$U(1)$-action on ${\overline{T}M}$, then the same is true for the subspaces
$\Omega^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_k(U) \subset \Omega^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U)$ of forms of weight $k$
with respect to the $U(1)$-action.
\item Assume further that the canonical projection $\rho:{\overline{T}M}_I \to
M$ is holomorphic for the preferred complex structure ${\overline{T}M}_I$ on
${\overline{T}M}$. Then both these claims hold for the spaces
$\Omega^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U/M)$ of relative holomorphic forms on $U$ over $M$.
\end{enumerate}
\end{lemma}
\proof The claim \thetag{i} is standard. To prove \thetag{ii}, note
that, both in the formal and in the analytic situation, the spaces
$\Omega^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U)$ are equipped with a natural topology. Both this
topology and the $U(1)$-action are preserved by the holomorphic
Dolbeault differential $\partial_I$.
The subspaces $\Omega^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{fin}(U) \subset \Omega^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(U)$ of
$U(1)$-finite vectors are dense in the natural topology. Therefore
the complex $\langle \Omega^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{fin}(U),\partial_I\rangle$ is also
exact. Since the group $U(1)$ is compact, we have
$$
\Omega^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{fin}(U) = \bigoplus_k\Omega^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_k(U),
$$
which proves \thetag{ii}. The claim \thetag{iii} is, again,
standard.
\hfill \ensuremath{\square}\par
\refstepcounter{subsubsection We can now formulate and prove the main result of this
subsection.
\begin{prop}
Assume given a formal polarized Hodge manifold structure on the
manifold ${\overline{T}M}$ along the zero section $M \subset {\overline{T}M}$ such that the
canonical projection $\rho:{\overline{T}M}_I \to M$ is holomorphic for the
preferred complex structure ${\overline{T}M}_I$ on ${\overline{T}M}$. Let $\Omega_I$ be the
associated formal holomorphic $2$-form on ${\overline{T}M}_I$. Let $T^*M$ be the
total space of the cotangent bundle to the manifold $M$ equipped
with a canonical holomorphic symplectic form $\Omega$. There exists
a unique $U(1)$-equivariant holomorphic map $\kappa:{\overline{T}M}_I \to T^*M$,
defined in a formal neighborhood of the zero section, which commutes
with the canonical projections onto $M$ and satisfies $\Omega_I =
\kappa^*\Omega$. Moreover, if the polarized Hodge manifold structure
on ${\overline{T}M}_I$ is defined in an open neighborhood $U \subset {\overline{T}M}$ of the
zero section $M \subset {\overline{T}M}$, then the map $\kappa$ is also defined
in a (possibly smaller) open neighborhood of the zero section.
\end{prop}
\proof By virtue of the uniqueness, the claim is local on $M$, so
that we can assume that the whole $M$ is contained in a
$U(1)$-invariant neighborhood $U \subset {\overline{T}M}_I$ satisfying the
conditions of Lemma~\ref{de.rham.exact}. Holomorphic maps $\kappa:U
\to T^*M$ which commute with the canonical projections onto $M$ are
in a natural one-to-one correspondence with holomorphic sections
$\alpha$ of the bundle $\rho^*\Lambda^{1,0}(M)$ on ${\overline{T}M}_I$. Such a
map $\kappa$ is $U(1)$-equivariant if and only if the corresponding
$1$-form $\alpha \in \Omega^1(U)$ is of weight $1$ with respect to
the $U(1)$-action. Moreover, it satisfies $\kappa^*\Omega=\Omega_I$
if and only if $\partial_I\alpha = \Omega_I$. Therefore to prove the
formal resp. analytic parts of the proposition it suffices to prove
that there exists a unique holomorphic formal resp. analytic section
$\alpha \in C^\infty(U,\rho^*\Lambda^{1,0}(M)$ which is of weight
$1$ with respect to the $U(1)$-action and satisfies
$\partial_I\alpha=\Omega_I$.
The proof of this fact is the same in the formal and in the analytic
situations. By definition of the polarized Hodge manifold the
$2$-form $\Omega_I \in \Omega^2(U)$ is of weight $1$ with respect to
the $U(1)$-action. Therefore by
Lemma~\ref{de.rham.exact}~\thetag{ii} there exists a holomorphic
$1$-form $\alpha \in \Omega^1(U)$ of weight $1$ with respect to the
$U(1)$-action and such that $\partial_I\alpha = \Omega_I$. Moreover, the
image of the form $\Omega_I$ under the canonical projection
$\Omega^2(U) \to \Omega^2(U/M)$ is zero. Therefore by
Lemma~\ref{de.rham.exact}~\thetag{iii} we can arrange so that the
image of the form $\alpha$ under the projection $\Omega^1(U) \to
\Omega^1(U/M)$ is also zero, so that $\alpha$ is in fact a section
of the bundle $\rho^*\Lambda^{1,0}(M)$. This proves the existence
part. To prove uniqueness, note that every two such $1$-forms must
differ by a form of the type $\partial_if$ for a certain holomorphic
function $f \in \Omega^0(U)$. Moreover, by
Lemma~\ref{de.rham.exact}~\thetag{ii} we can assume that the
function $f$ is of weight $1$ with respect to the $U(1)$-action. On
the other hand, by Lemma~\ref{de.rham.exact}~\thetag{iii} we can
assume that the function $f$ is constant along the fibers of the
canonical projection $\rho:{\overline{T}M}_I \to M$. Therefore we have $f=0$
identically on the whole $U$.
\hfill \ensuremath{\square}\par
\section{Convergence}\label{convergence}
\subsection{Preliminaries}
\refstepcounter{subsubsection Let $M$ be a complex manifold. By Theorem~\ref{kal=ext} every
K\"ah\-le\-ri\-an connection $\nabla:\Lambda^1(M,\C) \to \Lambda^1(M,\C)
\otimes \Lambda^1(M,\C)$ on the cotangent bundle $\Lambda^1(M,\C)$
to the manifold $M$ defines a flat linear extended connection
$D:S^1(M,\C) \to \B^1(M,\C)$ on $M$ and therefore a formal Hodge
connection $D$ on the total space ${\overline{T}M}$ of the complex-conjugate to
the tangent bundle to $M$. By Proposition~\ref{equiv.bis}, this
formal Hodge connection defines, in turn, a formal Hodge manifold
structure on ${\overline{T}M}$ in the formal neighborhood of the zero section $M
\subset {\overline{T}M}$.
In this section we show that if the K\"ah\-le\-ri\-an connection $\nabla$
is real-analytic, then the corresponding formal Hodge manifold
structure on ${\overline{T}M}$ is the completion of an actual Hodge manifold
structure on an open neighborhood $U \subset {\overline{T}M}$ of the zero
section $M \subset {\overline{T}M}$. We also show that if the connection
$\nabla$ comes from a K\"ahler metric $\omega$ on $M$, then the
corresponding polarization $\Omega$ of the formal Hodge manifold
${\overline{T}M}$ defined in Theorem~\ref{metrics} converges in a neighborhood
$U' \subset U \subset {\overline{T}M}$ of the zero section $M \subset {\overline{T}M}$ to a
polarization of the Hodge manifold structure on $U'$. Here is the
precise formulation of these results.
\begin{theorem}\label{converge}
Let $M$ be a complex manifold equipped with a real-analytic
K\"ah\-le\-ri\-an connection $\nabla:\Lambda^1(M,\C) \to \Lambda^1(M,\C)
\otimes \Lambda^1(M,\C)$ on its cotangent bundle
$\Lambda^1(M,\C)$. There exists an open neighborhood $U \subset {\overline{T}M}$
of the zero section $M \subset {\overline{T}M}$ in the total space ${\overline{T}M}$ of the
complex-conjugate to the tangent bundle to $M$ and a Hodge manifold
structure on $U \subset {\overline{T}M}$ such that its completion along the zero
section $M \subset {\overline{T}M}$ defines a linear flat extended connection
$D$ on $M$ with reduction $\nabla$.
Moreover, assume that $M$ is equipped with a K\"ahler metric
$\omega$ such that $\nabla\omega=0$, and let $\Omega \in
C^\infty_M({\overline{T}M},\Lambda^2({\overline{T}M},\C))$ be the formal polarization of the
Hodge manifold structure on $U \subset {\overline{T}M}$ along $M \subset
{\overline{T}M}$. Then there exists an open neighborhood $U' \subset U$ of $M
\subset U$ such that $\Omega \in C^\infty(U',\Lambda^2({\overline{T}M},\C))
\subset C^\infty_M({\overline{T}M},\Lambda^2({\overline{T}M},\C))$.
\end{theorem}
\refstepcounter{subsubsection \label{formal.Weil}
We begin with some preliminary observations. First of all, the
question is local on $M$, therefore we may assume that $M$ is an
open neighborhood of $0$ in the complex vector space $V = \C^n$. Fix
once and for all a real structure and an Hermitian metrics on the
vector space $V$, so that it is isomorphic to its dual $V \cong
V^*$.
The subspace $\J \subset C^\infty(M,\C)$ of functions vanishing at
$0 \in M$ is an ideal in the algebra $C^\infty(M,\C)$, and $\J$-adic
topology on $C^\infty(M,\C)$ extends canonically to the de Rham
algebra $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ of the manifold $M$ and, further, to
the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ of $M$ introduced in
\ref{Weil.defn}. Instead of working with bundle algebra
$\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ on $M$, it will be convenient for us to consider
the vector space
$$
\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} = C^\infty_\J(M,B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)),
$$
which is by definition the $\J$-adic completion of the space
$C^\infty(M,\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C))$ of global sections of the Weil
algebra. This vector space is canonically a (pro-)algebra over
$\C$. Moreover, the Hodge bundle structure on $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$
induces an $\R$-Hodge structure on the algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$.
\refstepcounter{subsubsection The $\J$-adic completion $C^\infty_\J(M,\C)$ of the space of
smooth functions on $M$ is canonically isomorphic to the completion
$\wh{S}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(V)$ of the symmetric algebra of the vector space $V
\cong V^*$. The cotangent bundle $\Lambda^1(M,\C)$ is isomorphic to
the trivial bundle $\V$ with fiber $V$ over $M$, and the completed de Rham
algebra $C^\infty_\J(M,\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C))$ is isomorphic to the
product
$$
C^\infty_\J(M,\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)) \cong S^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(V) \otimes
\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(V).
$$
This is a free graded-commutative algebra generated by two copies of
the vector space $V$, which we denote by $V_1 = V \subset
\Lambda^1(V)$ and by $V_2 = V \subset S^1(V)$. It is convenient to
choose the trivialization $\Lambda^1(M,\C) \cong \V$ in such a way
that the de Rham differential $d_M:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to
\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C)$ induces an identity map $d_M:V_2 \to V_1
\subset C^\infty_\J(M,\Lambda^1(M,\C))$.
\refstepcounter{subsubsection The complex vector bundle $S^1(M,\C)$ on $M$ is also
isomorphic to the trivial bundle $\V$. Choose a trivialization
$S^1(M,\C) \cong \V$ in such a way that the canonical map
$C:S^1(M,\C) \to \Lambda^1(M,\C)$ is the identity map. Denote by
$$
V_3 = V \subset C^\infty(M,S^1(M,\C)) \subset \B^0
$$
the subset of constant sections in $S^1(M,\C) \cong \V$. Then the
Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ becomes isomorphic to the product
$$
\B^i \cong \wh{S}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(V_2 \oplus V_3) \otimes \Lambda^i(V_1)
$$
of the completed symmetric algebra $\wh{S}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(V_2 \oplus V_3)$ of
the sum $V_2 \oplus V_3$ of two copies of the vector space $V$ and
the exterior algebra $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(V_1)$ of the third copy of the
vector space $V$.
\refstepcounter{subsubsection Recall that we have introduced in \ref{aug} a grading on the
Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ which we call {\em the augmentation
grading}. It induces a grading on the the Weil algebra
$\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. The augmentation grading on $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is
multiplicative, and it is obtained by assigning degree $1$ to the
generator subspaces $V_1,V_3 \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ and degree $0$ to the
generator subspace $V_2 \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. As in \ref{aug}, we will
denote the augmentation grading on $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ by lower indices.
We will now introduce yet another grading on the algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$
which we will call {\em the total grading}. It is by definition the
multiplicative grading obtained by assigning degree $1$ to {\em all}
the generators $V_1,V_2,V_3 \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ of the Weil algebra
$\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ We will denote by $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{k,n} \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ the
component of augmentation degree $k$ and total degree $n$. Note that
by definition $n,k \geq 0$ and, moreover, $n \geq k$.
\noindent {\bf Remark.\ } In \ref{aug} we have also defined a finer {\em augmentation
bigrading} on the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ \, and it this
bigrading that was denoted by double lower indices throughout
Section~\ref{main.section}. We will now longer need the augmentation
bigrading, so there is no danger of confusion.
\refstepcounter{subsubsection The trivialization of the cotangent bundle to $M$ defines an
isomorphism ${\overline{T}M} \cong M \times V$ and a constant Hodge connection
on the pair $\langle {\overline{T}M},M \rangle$. The corresponding extended
connection $D^{const}:S^1(M,\C) \to \B^1(M,\C)$ is the sum of the
trivial connection
$$
\nabla^{const}_1:S^1(M,\C) \to S^1(M,\C) \otimes
\Lambda^1(M,\C) \subset \B^1(M,\C)
$$
on $S^1(M,\C) \cong \V$ and the canonical isomorphism
$$
C = {\sf id}:S^1(M,\C) \to \Lambda^1(M,\C) \subset \B^1(M,\C).
$$
The derivation $D^{const}:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ of the Weil
algebra associated to the extended connection $D^{const}$ by
\ref{deriv} is equal to
\begin{align*}
D^{const} &= C = {\sf id}:V_3 \to V_1\\
D^{const} &= d_M = {\sf id}:V_2 \to V_1\\
D^{const} &= d_M = 0 \text{ on }V_1
\end{align*}
on the generator spaces $V_1,V_2,V_3 \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. In
particular, the derivation $D^{const}$ preserves the total degree.
\refstepcounter{subsubsection Let now $D:S^1(M,\C) \to \B^1(M,\C)$ be the an arbitrary
linear extended connection on the manifold $M$, and let
$$
D = \sum_{k \geq 0}D_k:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}
$$
be the derivation of the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ associated to the
extended connection $D$ by \ref{deriv}. The derivation $D$ admits a
finer decomposition
$$
D = \sum_{k,n \geq 0} D_{k,n}:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}
$$
according to both the augmentation and the total degree on
$\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. The summand $D_{k,n}$ by definition raises the
augmentation degree by $k$ and the total degree by $n$.
\refstepcounter{subsubsection Since the extended connection $D:S^1(M,\C) \to \B^1(M,\C)$ is
linear, its component $D_0:S^1(M,\C) \to \Lambda^1(M,\C)$ of
augmentation degree $0$ coincides with the canonical isomorphism
$C:S^1(M,\C) \to \Lambda^1(M,\C)$. Therefore the restriction of the
derivation $D:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ to the generator subspace
$V_3 \subset S^1(M,\C) \subset \B^0$ satisfies
$$
D_0 = C = D^{const}_0 = D^{const}_{0,0}:V_3 \to V_1 \subset \B^1.
$$
In particular, all the components $D_{0,n}$ except for $D_{0,0}$
vanish on the subspace $V_3 \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$.
The restriction of the derivation $D$ to the subspace
$\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ by definition coincides
with the de Rham differential $d_M:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to
\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C)$. Therefore on the generator subspaces
$V_1,V_2 \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ we have $D = d_M = D^{const}$. In
particular, all the components $D_{k,n}$ except for $D_{1,0}$ vanish
on the subspaces $V_1,V_2 \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$.
\refstepcounter{subsubsection The fixed Hermitian metric on the generator spaces $V_1 = V_2
= V_3 = V$ extends uniquely to a metric on the whole Weil algebra
such that the multiplication map $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to
\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is an isometry. We call this metric {\em the standard
metric} on $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. We finish our preliminary observations with
the following fact which we will use to deduce
Theorem~\ref{converge} from estimates on the components $D_{n,k}$ of
the derivation $D:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$.
\begin{lemma}\label{est=conv}
Let $D = \sum_{n,k}D_{n,k}:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ be a
derivation associated to an extended connection $D$ on the manifold
$M$. Consider the norms $\|D_{k,n}\|$ of the restrictions
$D_{k,n}:V_3 \to \B^1_{n+1,k+1}$ of the derivations
$D_{k,n}:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ to the generator subspace $V_3
\subset \B^0$ taken with respect to the standard metric on the Weil
algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$. If for certain constants $C,\varepsilon > 0$ and for
every natural $n \geq k \geq 0$ we have
\begin{equation}\label{est}
\|D_{k,n}\| < C\varepsilon^n,
\end{equation}
then the formal Hodge connection on ${\overline{T}M}$ along $M$ associated to
$D$ converges to an actual real-analytic Hodge connection on the
open ball of radius $\varepsilon$ in ${\overline{T}M}$ with center at $0 \in M \subset
{\overline{T}M}$. Conversely, if the extended connection $D$ comes from a
real-analytic Hodge connection on an open neighborhood $U \subset
{\overline{T}M}$, and if the Taylor series for this Hodge connection converge in
the closed ball of radius $\varepsilon$ with center at $0 \in M \subset
{\overline{T}M}$, then there exists a constant $C>0$ such that \eqref{est} holds
for every $n,k \geq 0$.
\end{lemma}
\proof The constant Hodge connection $D^{const}$ is obviously defined
on the whole ${\overline{T}M}$, and every other formal Hodge connection on ${\overline{T}M}$
is of the form
$$
D = D^{const} + d^r \circ \Theta:\Lambda^0({\overline{T}M},\C) \to
\rho^*\Lambda^1(M,\C),
$$
where $d^r:\Lambda^0({\overline{T}M},\C) \to \Lambda^1({\overline{T}M}/M,\C)$ is the
relative de Rham differential, and $\Theta \in
C^\infty_M({\overline{T}M},\Lambda^1({\overline{T}M}/M,\C) \otimes \rho^*\Lambda^1(M,\C))$
is a certain relative $1$-form on the formal neighborhood of $M
\subset {\overline{T}M}$ with values in the bundle $\rho^*\Lambda^1(M,\C)$. Both
bundles $\Lambda^1({\overline{T}M}/M,\C)$ and $\rho^*\Lambda^1(M,\C)$ are
canonically isomorphic to the trivial bundle $\V$ with fiber $V$ on
${\overline{T}M}$. Therefore we can treat the $1$-form $\Theta$ as a formal germ
of a $\End(V)$-valued function on ${\overline{T}M}$ along $M$. The Hodge
connection $D$ converges on a subset $U \subset {\overline{T}M}$ if and only if
this formal germ comes from a real-analytic $\End(V)$-valued
function $\Theta$ on $U$.
The space of all formal Taylor series for $\End(V)$-valued functions
on ${\overline{T}M}$ at $0 \in M \subset {\overline{T}M}$ is by definition equal to $\End(V)
\otimes \B^0$. Moreover, for every $n \geq 0$ the component
$\Theta_n \in \B^0_n \otimes \End(V) = \Hom(V,\B^0_n \otimes V)$ of
total degree $n$ of the formal power series for the function
$\Theta$ at $0 \in {\overline{T}M}$ is equal to the derivation
$$
\sum_{0 \leq k \leq n}D_{k,n}:V=V_3 \to V=V_1 \otimes \bigoplus_{0 \leq k
\leq n}\B^0_{k,n}.
$$
Every point $x \in {\overline{T}M}$ defines the ``evaluation at $x$'' map
$$
\ev_x:C^\infty({\overline{T}M},\End(V)) \to \C,
$$
and the formal Taylor series for $\Theta \in \End(V) \otimes \B^0$
converges at the point $x \in {\overline{T}M}$ if and only if the series
$$
\Theta(x) = \sum_{n \geq 0}\ev_x(\Theta_n) \in \End(V)
$$
converges. But we have
$$
\|\ev_x(\Theta_n)\| = \left\|\sum_{0 \leq k \leq n}D_{k,n}\right\||x|^n,
$$
where $|x|$ is the distance from the point $x$ to $0 \in {\overline{T}M}$. Now
the application of standard criteria of convergence finishes the
proof of the lemma.
\hfill \ensuremath{\square}\par
\subsection{Combinatorics}
\refstepcounter{subsubsection We now derive some purely combinatorial facts needed to
obtain estimates for the components $D_{k,n}$ of the extended
connection $D$. First, let $a_n$ be the Catalan numbers, that is,
the numbers defined by the recurrence relation
$$
a_n = \sum_{1 \leq k \leq n-1} a_k a_{n-k}
$$
and the initial conditions $a_1 = 1$, $a_n = 0$ for $n \leq 0$. As
is well-known, the generating function $f(z) = \sum_{k \geq
0}a_kz^k$ for the Catalan numbers satisfies the equation $f(z) =
f(z)^2 + z$ and equals therefore
$$
f(z) = \frac{1}{2} - \sqrt{\frac{1}{4} - z}.
$$
The Taylor series for this function at $z=0$ converges for $4|z|
< 1$, which implies that
$$
a_k < C(4+\varepsilon)^k
$$
for some positive constant $C > 0$ and every $\varepsilon > 0$.
\refstepcounter{subsubsection \label{b.n.k}
We will need a more complicated sequence of integers, numbered by
two natural indices, which we denote by $b_{k,n}$. The sequence
$b_{k,n}$ is defined by the recurrence relation
$$
b_{k,n} = \sum_{p,q;1 \leq p \leq k-1} \frac{q+1}{k} b_{p,q}b_{k-p,n-q}
$$
and the initial conditions
$$
\begin{cases}
b_{k,n} &= 0 \quad \text{ for } \quad k \leq 0,\\
b_{k,n} &= 0 \quad \text{ for } \quad k = 1, n < 0,\\
b_{k,n} &= 1 \quad \text{ for } \quad k = 1, n \geq 0,
\end{cases}
$$
which imply, in particular, that if $n < 0$, then $b_{k,n} = 0$ for
every $k$. For every $k \geq 1$ let $g_k(z) = \sum_{n \geq
0}b_{k,n}z^n$ be the generating function for the numbers
$b_{k,n}$. The recurrence relations on $b_{k,n}$ give
\begin{align*}
g_k(z) &= \frac{1}{k}\sum_{1 \leq p \leq
k-1}g_{k-p}(z)\left(1+z\frac{\partial}{\partial z}\right)(g_p(z)) \\
&=\frac{1}{2k}\sum_{1 \leq p \leq k-1}\left(2+z\frac{\partial}{\partial
z}\right)(g_p(z)g_{k-p}(z)),
\end{align*}
and the initial conditions give
$$
g_1(z) = \frac{1}{1-z}.
$$
\refstepcounter{subsubsection Say that a formal series $f(z)$ in the variable $z$ is {\em
non-negative} if all the terms in the series are non-negative real
numbers. The sum and product of two non-negative series and the
derivative of a non-negative series is also obviously
non-negative. For two formal series $s(z)$, $t(z)$ write $s(z) \ll
t(z)$ if the difference $t(z) - s(z)$ is a non-negative power
series.
Our main estimate for the generating functions $g_k(z)$ is the
following.
\begin{lemma}
For every $k \geq 1$ we have
$$
g_k(z) \ll a_k\frac{1}{(1-z)^{2k-1}},
$$
where $a_k$ are the Catalan numbers.
\end{lemma}
\proof Use induction on $k$. For $k=1$ we have $g_1(z) =
\frac{1}{1-z}$ and $a_1 = 1$, which gives the base for
induction. Assume that the claim is proved for all $p < k$. Since
all the $g_n(z)$ are non-negative power series, this implies that for
every $p$, $1 \leq p \leq k-1$ we have
$$
g_p(z)g_{k-p}(z) \ll
a_pa_{k-p}\frac{1}{(1-z)^{2p-1}}\frac{1}{(1-z)^{2k-2p-1}} =
a_pa_{k-p}\frac{1}{(1-z)^{2k-2}}.
$$
Therefore
\begin{align*}
\left(2+z\frac{\partial}{\partial z}\right)(g_p(z)g_{k-p}(z)) &\ll
a_pa_{k-p}\left(2+z\frac{\partial}{\partial z}\right)\frac{1}{(1-z)^{2k-2}} \\
&= a_pa_{k-p}\left(\frac{2}{(1-z)^{2k-2}} +
\frac{(2k-2)z}{(1-z)^{2k-1}} \right) \\
&= a_pa_{k-p}\left(\frac{2k-2}{(1-z)^{2k-1}} -
\frac{2k-4}{(1-z)^{2k-2}}\right) \\
&\ll (2k-2)a_pa_{k-p}\frac{1}{(1-z)^{2k-1}}.
\end{align*}
Hence
\begin{align*}
g_k(z) &= \frac{1}{2k}\sum_{1 \leq p \leq k-1}\left(2+z\frac{\partial}{\partial
z}\right)(g_p(z)g_{k-p}(z)) \\
&\ll \frac{2k-2}{2k}\sum_{1 \leq p \leq
k-1}a_pa_{k-p}\frac{1}{(1-z)^{2k-1}} \\
&\ll \frac{1}{(1-z)^{2k-1}}\sum_{1 \leq p \leq k-1}a_pa_{k-p} =
a_k\frac{1}{(1-z)^{2k-1}},
\end{align*}
which proves the lemma.
\hfill \ensuremath{\square}\par
\refstepcounter{subsubsection This estimate yields the following estimate for the numbers
$b_{k,n}$.
\begin{corr}\label{combin}
The power series
$$
g(z) = \sum_{k,n} b_{k,n}z^{n+k} = \sum_{k \geq 1}g_k(z)z^k
$$
converges for $z < 3-\sqrt{8}$. Consequently, for every $C_2$ such
that $(3-\sqrt{8})C_2 > 1$ there exists a positive constant $C > 0$
such that
$$
b_{n,k} < CC_2^{n+k}
$$
for every $n$ and $k$. (One can take, for example, $C_2 = 6$.)
\end{corr}
\proof Indeed, we have
\begin{equation}\label{g(z)}
g(z) \ll \sum_{k \geq 1}a_kz^k\frac{1}{(1-z)^{2k-1}} =
(1-z)f\left(\frac{z}{(1-z)^2}\right),
\end{equation}
where $f(z) = \frac{1}{2} - \sqrt{\frac{1}{4}-z}$ is the generating
function for the Catalan numbers. Therefore
$$
g(z) \ll (1-z)\left(\frac{1}{2} - \sqrt{\frac{1}{4} -
\frac{z}{(1-z)^2}}\right),
$$
and the right hand side converges absolutely when
$$
\frac{|z|}{(1-z)^2} < \frac{1}{4}.
$$
Since $3 - \sqrt{8}$ is the root of the quadratic equation $(1-z)^2
= 4z$, this inequality holds for every $z$ such that $|z| < 3 -
\sqrt{8}$.
\hfill \ensuremath{\square}\par
\refstepcounter{subsubsection \label{b.n.k.m}
To study polarizations of Hodge manifold structures on ${\overline{T}M}$, we
will need yet another recursive sequence of integers, which we
denote by $b_{k,n}^m$. This sequence is defined by the recurrence
relation
$$
b^m_{k,n} = \sum_{p,q;1 \leq p \leq k-1} \frac{q+m(k-p)}{k}
b^m_{p,q}b_{k-p,n-q}
$$
and the initial conditions
$$
\begin{cases}
b^m_{k,n} &= 0 \quad \text{ unless } \quad k,m \leq 0,\\
b^m_{k,n} &= 0 \quad \text{ for } \quad k = 1, n < 0,\\
b^m_{k,n} &= 1 \quad \text{ for } \quad k = 1, n \geq 0.
\end{cases}
$$
\refstepcounter{subsubsection \label{c.k.n}
To estimate the numbers $b^m_{k,n}$, consider the auxiliary sequence
$c_{k,n}$ defined by setting
$$
c_{k,n} = \sum_{p,q;1 \leq p \leq k-1}c_{p.q}b_{k-p,n-q} \qquad k
\geq 2,
$$
and $c_{k,n} = b_{k,n}$ for $k \leq 1$. The generating series $c(z)
= \sum_{k,n \geq 0}c_{k,n}z^{n+k}$ satisfies
$$
c(z) = c(z)g(z) + \frac{z}{1-z},
$$
so that we have $c(z) = \frac{z}{(1-z)(1-g(z))}$, which is
non-singular when $|z|, |g(z)| < 1$ and $g(z)$ is non-singular.
By \eqref{g(z)} the latter inequality holds if
$$
\left|(1-z)\left(\frac{1}{2} - \sqrt{\frac{1}{4} -
\frac{z}{(1-z)^2}}\right)\right| < 1,
$$
which holds in the whole disc where $g(z)$ converges, that is, for
$|z| < 3-\sqrt{8}$. Therefore, as in Corollary~\ref{combin}, we have
\begin{equation}\label{cnk}
c_{k,n} < C6^{n+k}
\end{equation}
for some positive constant $C$.
\refstepcounter{subsubsection We can now estimate the numbers $b^m_{k,n}$.
\begin{lemma}\label{combin.2}
For every $m,k,n$ we have
\begin{equation}\label{bnkm.indu}
b^m_{k,n} \leq (2m)^{k-1}c_{k,n}b_{k,n},
\end{equation}
where $c_{k,n}$ are as in \ref{c.k.n} and $b_{k,n}$ are the
numbers introduced in \ref{b.n.k}. Consequently, we have
$$
b^m_{k,n} < C(72m)^{n+k+m}
$$
for some positive constant $C > 0$.
\end{lemma}
\proof Use induction on $k$. The case $k=1$ follows from the initial
conditions. Assume the estimate \eqref{bnkm.indu} proved for all
$b^m_{p,q}$ with $p < k$. Note that by the recurrence relations we
have $b_{p,q} \leq b_{k,n}$ and $c_{p,q} \leq c_{k,n}$ whenever $p <
k$. Therefore
\begin{align*}
b^m_{k,n} &= \sum_{p,q;1 \leq p \leq k-1} \frac{q+m(k-p)}{k}
b^m_{p,q}b_{k-p,n-q} \\
&\leq \sum_{p,q;1 \leq p \leq k-1} \frac{q}{k}b^m_{p,q}b_{k-p,n-q} +
\sum_{p,q;1 \leq p \leq k-1} b^m_{p,q}b_{k-p,n-q} \\
&\leq \sum_{p,q;1 \leq p \leq k-1}2^{p-1} c_{p,q} \frac{q}{k}
b_{p,q}b_{k-p,n-q} + \sum_{p,q;1 \leq p \leq k-1} 2^{p-1} b_{p,q}
c_{p,q} b_{k-p,n-q} \\
&\leq 2^{k-2} c_{k,n}\sum_{p,q;1 \leq p \leq k-1} \frac{q+1}{k}
b_{p,q} b_{k-p,n-q} \\
&\quad + 2^{k-2}b_{k,n}\sum_{p,q;1 \leq p \leq k-1}
c_{p,q} b_{k-p,n-q} \\
&= 2^{k-2}c_{k,n}b_{k,n} + 2^{k-2}c_{k,n}b_{k,n} = 2^{k-1}c_{k,n}b_{k,n},
\end{align*}
which proves \eqref{bnkm.indu} for $b^m_{k,n}$. The second estimate of
the lemma now follows from \eqref{cnk} and Corollary~\ref{combin}.
\hfill \ensuremath{\square}\par
\subsection{The main estimate}
\refstepcounter{subsubsection \label{norms}
Let now $D = \sum_{k,n}D_{n,k}:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ be a
derivation of the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ associated to a flat
linear extended connection on $M$. Consider the restriction
$D_{k,n}:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{p,q} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}{p+k,q+n}$ of the derivation
$D_{k,n}$ to the component $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{p,q}\subset\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ of
augmentation degree $p$ and total degree $q$. Since both
$\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{p,q}$ and $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}{p+k,q+n}$ are finite-dimensional
vector spaces, the norm of this restriction with respect to the
standard metric on $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is well-defined. Denote this norm by
$\|D_{k,n}\|_{p,q}$.
By Lemma~\ref{est=conv} the convergence of the Hodge manifold
structure on ${\overline{T}M}$ corresponding to the extended connection $D$ is
related to the growth of the norms $\|D_{k,n}\|_{1,1}$. Our main
estimate on the norms $\|D_{k,n}\|_{1,1}$ is the following.
\begin{prop}\label{estimate}
Assume that there exist a positive constant $C_0$ such that for
every $n$ the norms $\|D_{1,n}\|_{1,1}$ and $\|D_{1,n}\|_{0,1}$
satisfy
$$
\|D_{1,n}\|_{1,1}, \|D_{1,n}\|_{1,0} < C_0^n.
$$
Then there exists a positive constant $C_1$ such that for every
$n,k$ the norm $\|D_{k,n}\|_{1,1}$ satisfies
$$
\|D_{k,n}\|_{1,1} < C_1^n.
$$
\end{prop}
\refstepcounter{subsubsection \label{metr}
In order to prove Proposition~\ref{estimate}, we need some
preliminary facts. Recall that we have introduced in
\ref{total.Weil} the total Weil algebra $\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ of the
manifold $M$, and let
$$
\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} = C^\infty_\J(M,\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C))
$$
be the algebra of its smooth sections completed at $0 \subset M$. By
definition for every $k \geq 0$ we have $\B_{tot}^k = \B^k \otimes
\W_k^*$, where $\W_k$ is the $\R$-Hodge structure of weight $k$
universal for weakly Hodge maps, as in \ref{w.k.uni}. There exists a
unique Hermitian metric on $\W_k$ such that all the Hodge components
$\W^{p,q} \subset \W_k$ are orthogonal and all the Hodge degree
components $w_k^{p,q}$ of the universal weakly Hodge map $w_k:\R(0)
\to \W_k$ are isometries. This metric defines a canonical Hermitian
metric on $\W_k^*$.
\noindent {\bf Definition.\ } The {\em standard metric} on the total Weil algebra
$\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is the product of the canonical metric and the standard
metric on $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$.
\refstepcounter{subsubsection By Lemma~\ref{total.rel} the total Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$
is generated by the subspaces $V_2,V_3 \subset \B^0 = \B^0_{tot}$ and
the subspace $V_1 \otimes \W_1^* \subset \B^1_{tot}$, which we denote by
$V_1^{tot}$. The ideal of relations for the algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ is the
ideal in $S^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(V_2 \oplus V_3)\otimes\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(V_1^{tot})$
generated by $S^2(V_1)\otimes\Lambda^2(\W_1^*) \subset
\Lambda^2(V_1^{tot})$.
The direct sum decomposition \eqref{drct} induces a direct sum
decomposition
$$
V^{tot}_1 = V^{ll}_1 \oplus \V^o_1 \oplus V^{rr}_1
$$
of the generator subspace $V^{tot}_1 \subset \B^1_{tot}$. The subspaces
$V^o_1 \subset V^{tot}_1$ and $V^{ll}_1 \oplus V^{rr}_1 \subset V^{tot}_1$
are both isomorphic to the vector space $V_1$. More precisely, the
universal weakly Hodge map $w_1:\R(0) \to \W_1$ defines a projection
$$
P:V^{tot}_1 = V_1 \otimes \W_1^* \to V_1,
$$
and the restriction of the projection $P$ to either of the subspaces
$V^o_1,V^{ll}_1 \oplus V^{rr}_1 \subset V^{tot}_1$ is an
isomorphism. Moreover, either of these restrictions is an isometry
with respect to the standard metrics.
\refstepcounter{subsubsection \label{P2}
The multiplication in $\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is not an isometry with respect
to this metric. However, for every $b_1,b_2 \subset \B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ we
have the inequality
$$
\|b_1b_2\| \leq \|b_1\|{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\|b_2\|.
$$
Moreover, this inequality becomes an equality when $b_1 \subset
\B^0_{tot}$. In particular, if we extend the map $P:V^{tot}_1 \to V_1$ to a
$\B^0$-module map
$$
P:\B^1_{tot} \to \B^1,
$$
then the restriction of the map $P$ to either of the subspaces
$\B^1_o, \B^1_{ll}\oplus\B^1_{rr} \subset \B^1_{tot}$ is an isometry
with respect to the standard metric. Therefore the norm of the
projection $P:\B^1_{tot} \to \B^1$ is at most $2$.
\refstepcounter{subsubsection The total and augmentation gradings on the Weil algebra
$\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ extend to gradings on the total Weil algebra
$\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$, also denoted by lower indices. The extended connection
$D$ on $M$ induces a derivation $D^{tot} =
\sum_{n,k}D_{k,n}^{tot}:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot}$. As in
\ref{norms}, denote by $\|D^{tot}_{k,n}\|_{p,q}$ the norm of the map
$D^{tot}_{k,n}:\left(\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}\right)_{p,q} \to
\left(\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}\right)_{p+k,q+n}$ with respect to the standard
metric on $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$. The derivations $D^{tot}_{k,n}$ are related to
$D_{k,n}$ by
$$
D_{k,n} = P \circ D^{tot}_{k,n}:\B^0 \to \B^1,
$$
and we have the following.
\begin{lemma}\label{total.ne.total}
For every $k$, $n$ and $p=0,1$ we have
$$
\|D_{k,n}\|_{p,1} = \|D^{tot}_{k,n}\|_{p,1}.
$$
\end{lemma}
\proof By definition we have $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{0,1} \oplus \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{1,1} =
V_1 \oplus V_2 \oplus V_3$. Moreover, the derivation $D_{k,n}$
vanishes on $V_1$, hence $D^{tot}_{k,n}$ vanishes on $V_1^{tot}$. Therefore
it suffices to compare their norms on $V_2 \oplus V_3 \subset \B^0 =
\B^0_{tot}$.
Since on $S^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(V_2) \subset \B^0$ the derivation $D_{k,n}$
coincides with the de Rham differential, the derivation $D^{tot}_{k,n}$
maps the subspace $V_2$ into $\B^1_{ll}\oplus\B^1_{rr}$. Moreover,
by Lemma~\ref{total.aug} the derivation $D_{k,n}^{tot}$ maps $V_3$
either into $\B_o^1$ or into $\B^1_{ll} \oplus \B^1_{rr}$, depending
on the parity of the number $k$. Since $D_{k,n} = P \circ D^{tot}_{k,n}$
and the map $P:\B^1_{tot} \to \B^1$ is an isometry on both $\B^1_o
\subset \B^1_{tot}$ and $\B^1_{ll}\oplus\B^1_{rr} \subset \B^1_{tot}$, we
have $\|D_{k,n}\| = \|D^{tot}_{k,n}\|$ on both $V_2 \subset \B^0$ and
$V_3 \subset \B^0$, which proves the lemma.
\hfill \ensuremath{\square}\par
This lemma allows to replace the derivations $D_{k,n}$ in
Proposition~\ref{estimate} with associated derivations $D_{k,n}^{tot}$ of
the total Weil algebra $\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$.
\refstepcounter{subsubsection Since the extended connection $D$ is linear and flat, the
construction used in the proof of Lemma~\ref{main.ind} shows that
\begin{equation}\label{indu}
D^{tot}_k = h^{-1} \circ \sigma_{tot} \circ \sum_{1 \leq p \leq k-1}D_p^{tot}
\circ D^{tot}_{k-p}:V_3 \to \left(\B^1_{tot}\right)_{k+1},
\end{equation}
where $\sigma_{tot}:\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ is the canonical map
constructed in \ref{sigma.c}, and $h:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} \to \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ is
as in Lemma~\ref{h.acts}. Both $\sigma_{tot}$ and $h$ preserve the
augmentation degree. In order to obtain estimates on
$\|D_{k,n}\|_{1,1} = \|D_{k,n}^{tot}\|_{1,1}$, we have to estimate the
norms $\|h^{-1}\|$ and $\|\sigma_{tot}\|$ of the restrictions of maps
$h^{-1}$ and $\sigma_{tot}$ on the subspace
$\left(\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\right)_{k+1} \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$.
By Lemma~\ref{h.on.b1} the map $h:\left(\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\right)_{k+1} \to
\left(\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\right)_{k+1}$ is diagonalizable, with eigenvalues
$k+1$ if $k$ is even and $(k+1)/2,(k-1)/2$ if $k$ is odd. Since $m
\geq 2$, in any case on $\left(\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\right)_{k+1}$ we have
\begin{equation}\label{h.est}
\|h^{-1}\| < \frac{3}{k}.
\end{equation}
\refstepcounter{subsubsection \label{sigma.est.punkt}
To estimate $\sigma_{tot}:\B^2_{tot} \to \B^2_{tot}$, recall that, as noted in
\ref{sigma.c}, the map $\sigma_{tot}$ is a map of $\B^0$-modules. The
space $\B^2_{tot} = \B^0 \otimes \left(\B^2_{tot}\right)_2$ is a free
$\B^0$-module generated by a finite-dimensional vector space
$\left(\B^2_{tot}\right)_2$. The map $\sigma_{tot}$ preserves the
augmentation degree, hence it maps $\left(\B^2_{tot}\right)_2$ into the
finite-dimensional vector space $\left(\B^1_{tot}\right)_2$. Therefore
there exists a constant $K$ such that
\begin{equation}\label{sigma.est}
\|\sigma_{tot}\| \leq K
\end{equation}
on $\left(\B^2_{tot}\right)_2$. Since we have
$\|b_1b_2\|=\|b_1\|{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\|b_2\|$ for every $b_1 \in \B^0$, $b_2 \in
\B^2_{tot}$, and the map $\sigma_{tot}$ is $\B^0$-linear, the estimate
\eqref{sigma.est} holds on the whole $\B^2_{tot} = \B^0 \otimes
\left(\B^2_{tot}\right)_2$. We can assume, in addition, that $K \geq 1$.
\noindent {\bf Remark.\ } In fact $K = 2$, but we will not need this.
\refstepcounter{subsubsection Let now $b_{k,n}$ be the numbers defined recursively in
\ref{b.n.k}. Our estimate for $\|D^{tot}_{k,n}\|_{p,q}$ is the
following.
\begin{lemma}\label{est.indu}
In the assumptions and notations of Proposition~\ref{estimate}, we
have
$$
\|D^{tot}_{k,n}\|_{p,q} < q(3K)^{k-1} C_0^nb_{k,n}
$$
for every $k,n$.
\end{lemma}
\proof Use induction on $k$. The base of induction is the case
$k=1$, when the inequality holds by assumption. Assume that for some
$k$ we have proved the inequality for all
$\|D^{tot}_{m,n}\|_{p,q}$ with $m < k$, and fix a number $n \geq k$.
Consider first the restriction of $D^{tot}_{k,n}$ onto the generator
subspace $V_3 \subset \B^0$. Taking into account the total degree,
we can rewrite \eqref{indu} as
$$
D^{tot}_{k,n} = h^{-1} \circ \sigma_{tot} \circ \sum_{1 \leq p \leq k-1}\sum_q
D^{tot}_{k-p,n-q} \circ D^{tot}_{p,q}:V_3 \to \left(\B^1_{tot}\right)_{k+1}.
$$
Therefore the norm of the map $D^{tot}_{k,n}:V_3 \to \B_{tot}^1$ satisfies
$$
\|D^{tot}_{k,n}\} \leq \|h^{-1}\| {\:\raisebox{3pt}{\text{\circle*{1.5}}}} \|\sigma_{tot}\| {\:\raisebox{3pt}{\text{\circle*{1.5}}}} \sum_{1 \leq p
\leq k-1}\sum_q\|D^{tot}_{k-p,n-q}\|_{p+1,q+1} {\:\raisebox{3pt}{\text{\circle*{1.5}}}} \|D^{tot}_{p,q}\|_{1,1}.
$$
Substituting into this the estimates \eqref{h.est},
\eqref{sigma.est} and the inductive assumption, we get
$$
\|D^{tot}_{k,n}\| < \frac{3}{k}K\sum_{1 \leq p \leq
k-1}\sum_q(q+1)(3K)^{k-2}C_0^nb_{k-p,n-q}b_{p,q} = (3K)^{k-1}C_0^nb_{k,n}.
$$
Since by definition $D^{tot}_{k,n}$ vanishes on $V_2 \subset \B^0$ and
on $V^{tot}_1 \subset \B^1_{tot}$, this proves that
$$
\|D^{tot}_{k,n}\|_{p,1} < (3K)^{k-1}C_0^nb_{k,n}
$$
when $p=0,1$. Since the map $D^{tot}_{k,n}:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot} \to
\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}_{tot}$ is a derivation, the Leibnitz rule and the triangle
inequality show that for every $p$, $q$
$$
\|D^{tot}_{k,n}\|_{p,q} < q(3K)^{k-1}C_0^nb_{k,n},
$$
which proves the lemma.
\hfill \ensuremath{\square}\par
\refstepcounter{subsubsection \proof[Proof of Proposition~\ref{estimate}]
By Lemma~\ref{total.ne.total} we have $\|D_{k,n}\|_{p,1} =
\|D^{tot}_{k,n}\|_{p,1}$, and by Lemma~\ref{est.indu} we have
$$
\|D_{k,n}\|_{p,1} = \|D^{tot}_{k,n}\|_{p,1} < (3K)^{k-1}C_0^nb_{n,k}.
$$
Since $k \leq n$ and $K \geq 1$, this estimate together with
Corollary~\ref{combin} implies that
$$
\|D_{k,n}\|_{p,1} < C(3K)^{k-1}C_0^n6^{2n} < C(108KC_0)^n
$$
for some positive constant $C > 0$, which proves the proposition.
\hfill \ensuremath{\square}\par
\refstepcounter{subsubsection Proposition~\ref{estimate} gives estimates for the derivation
$D:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ or, equivalently, for the Dolbeault
differential
$$
\bar\partial_J:\Lambda^{0,{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}({\overline{T}M}_J) \to \Lambda^{0,{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}({\overline{T}M}_J)
$$
for the complementary complex structure ${\overline{T}M}_J$ on ${\overline{T}M}$ associated
to the extended connection $D$ on $M$. To prove the second part of
Theorem~\ref{converge}, we will need to obtain estimates on the
Dolbeault differential $\bar\partial_J:\Lambda^{p,0}({\overline{T}M}_J) \to
\Lambda^{p,1}({\overline{T}M}_J)$ with $p > 0$. To do this, we use the model for
the de Rham complex $\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}({\overline{T}M}_J)$ constructed in
Subsection~\ref{drm.mod}.
Recall that in \ref{ident.punkt} we have identified the direct image
$\rho_*\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}({\overline{T}M}_J)$ of the de Rham algebra of the
manifold ${\overline{T}M}$ with the free module over the Weil algebra
$\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ generated by a graded algebra bundle
$L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ on $M$. The Dolbeault differential $\bar\partial_J$ for
the complementary complex structure ${\overline{T}M}_J$ induces an algebra
derivation $D:L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to
L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \otimes \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C)$, so that the free module
$\rho_*\Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}({\overline{T}M}_J) \cong L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \otimes
\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ becomes a differential graded module over the Weil
algebra.
\refstepcounter{subsubsection The algebra bundle $L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ is isomorphic
to the de Rham algebra $\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$. In particular, the
bundle $L^1(M,\C)$ is isomorphic to the trivial bundle $\V$ with
fiber $V$ over $M$. By \ref{dr.L} the relative de Rham differential
$$
d^r:\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}({\overline{T}M}/M,\C) \to \Lambda^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}({\overline{T}M}/M,\C)
$$
induces a derivation
$$
d^r:L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \to
L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(M,\C) \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C),
$$
and we can choose the trivialization $L^1(M,\C) \cong \V$ in such a
way that $d^r$ identifies the generator subspace $V_3 \subset \B^0$
with the subspace of constant sections of $\V \cong L^1(M,\C)
\subset L^1(M,\C) \otimes \B^0(M,\C)$.
\refstepcounter{subsubsection \label{LLL}
Denote by
$$
\LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}} = C^\infty_\J(L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \otimes
\B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C))
$$
the $\J$-adic completion of the space of smooth sections of the
algebra bundle $L^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C) \otimes \B_{tot}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(M,\C)$ on $M$. The
space $\LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ is a bigraded algebra equipped with the
derivations $d^r:\LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}} \to \LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1,{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$,
$D^{tot}:\LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}} \to \LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$, which commute by
Lemma~\ref{D.dr}. The algebra $\LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ is the free
graded-commutative algebra generated by the subspaces $V_1,V_2,V_3
\subset \LL^{0,{\:\raisebox{3pt}{\text{\circle*{1.5}}}}} = \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ and the subspace $V = d^r(V_3)
\subset \LL^{1,0}$, which we denote by $V_4$.
\refstepcounter{subsubsection As in \ref{metr}, the given metric on the generator subspaces
$V_1=V_2=V_3=V_4=V$ extends uniquely to a multiplicative metric on
the algebra $\LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$, which we call {\em the standard
metric}. For every $k > 0$, introduce the total and augmentation
gradings on the free $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$-module $\LL^{k,{\:\raisebox{3pt}{\text{\circle*{1.5}}}}} =
\Lambda^k(V_4) \otimes \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ by setting $\deg \Lambda^k(V_4) =
(0,0)$. Let $D = D_{k,n}$ be the decomposition of the derivation
$D:\LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}} \to \LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ with respect to the
total and the augmentation degrees. Denote by $\|D_{k,n}\|^p_q$ the
norm with respect to the standard metric of the restriction of the
derivation $D_{k,n}:\LL^{p,0} \to \LL^{p,1}$ to the component in
$\LL^{p,0}$ of total degree $q$.
\refstepcounter{subsubsection Let now $\LL_{tot}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}} = \Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(V_4) \otimes
\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ be the product of the exterior algebra
$\Lambda^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(V_4)$ with the total Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$. We
have the canonical identification $\LL^{p,q}_{tot} = \LL^{p,q} \otimes
\W_q^*$, and the canonical projection $P:\LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}_{tot} \to
\LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$, identical on $\LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},0}_{tot} = \LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},0}$.
As in \ref{P2}, the norm of the projection $P$ on $\LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},1}_{tot}$
is at most $2$.
The derivation $D:\LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},0} \to \LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},1}$ induces a
derivation $D^{tot}:\LL_{tot}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},0} \to \LL_{tot}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},1}$, related to $D$
by $D = P \circ D^{tot}$. The gradings and the metric on
$\LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ extend to $\LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}_{tot}$, in particular,
we have the decomposition $D^{tot} = \sum_{k,n}D^{tot}_{k,n}$ with respect
to the total and the augmentation degrees. Denote by
$\|D_{k,n}^{tot}\|^p_q$ the norm with respect to the standard metric of
the restriction of the derivation $D^{tot}_{k,n}:\LL_{tot}^{p,0} \to
\LL_C^{p,1}$ to the component in $\LL^{p,0}$ of total degree
$q$. Since $\|P\|\leq 2$ on $\LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},1}_{tot}$, we have
\begin{equation}\label{DcL}
\|D_{k,n}^{tot}\|^p_q \leq 2{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\|D_{k,n}\|^p_q.
\end{equation}
\refstepcounter{subsubsection The estimate on the norms $\|D^{tot}_{k,n}\|^\LL_{p,q}$ that we
will need is the following.
\begin{lemma}\label{D2est}
In the notation of Lemma~\ref{est.indu}, we have
$$
\|D^{tot}_{k,n}\|^p_q < 2(q+pk)(3K)^{k-1} C_0^nb_{k,n},
$$
for every $p,q,k,n$.
\end{lemma}
\proof Since $D_{k,n}$ satisfies the Leibnitz rule, it suffices to
prove the estimate for the restriction of the derivation $D^{tot}_{k,n}$
to the generator subspaces $V_2,V_3,V_4 \subset \LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},0}$. By
\eqref{DcL} it suffices to prove that on $V_2,V_3,V_4$ we have
$$
\|D_{k,n}\|^p_q < (q+pk)(3K)^{k-1} C_0^nb_{k,n}.
$$
On the generator subspaces $V_2,V_3 \subset \B^0$ we have $p=0$, and
this equality is the claim of Lemma~\ref{est.indu}. Therefore it
suffices to consider the restriction of the derivation $D_{k,n}$ to
the subspace $V_4 = d^r(V_3) \subset \LL^{1,0}$. We have $d^r \circ
D_{k,n} = D_{k,n} \circ d^r:V_3 \to \LL^{1,1}$. Moreover, $d^r =
{\sf id}:V_3 \to V_4$ is an isometry. Since the operator $d^r:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}
\to \LL^{1,{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ satisfies the Leibnitz rule and vanishes on the
generators $V_1,V_2 \subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$, the norm $\|d^r\|_k$ of its
restriction to the subspace $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_k$ of augmentation degree $k$
does not exceed $k$. Therefore
\begin{align*}
\|D_{k,n}\|^1_0 &= \left\|D_{k,n}|_{V_4}\right\| =
\left\|D_{k,n}\circ d^r|_{V_3}\right\| = \left\|d^r \circ
D_{k,n}|_{V_3}\right\| \\
&\leq \left\|D_{k,n}|_{V_3}\right|{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\left\|d^r_{\B^1_k}\right\| < k
(3K)^{k-1} C_0^nb_{k,n},
\end{align*}
which proves the lemma.
\hfill \ensuremath{\square}\par
\subsection{The proof of Theorem~\ref{converge}}
\refstepcounter{subsubsection We can now prove Theorem~\ref{converge}. Let $D:S^1(M,\C) \to
\B^1(M,\C)$ be a flat linear extended connection on $M$. Assume that
its reduction $D_1=\nabla:S^1(M,\C) \to S^1(M,\C) \otimes
\Lambda^1(M,\C)$ is a real-analytic connection on the bundle
$S^1(M,\C)$.
The operator $D_1:S^1(M,\C) \to S^1(M,\C) \otimes \Lambda^1(M,\C)
\subset \B^1(M,\C)$ considered as an extended connection on $M$
defines a Hodge connection $D_1:\Lambda^0({\overline{T}M},\C) \to
\rho^*\Lambda^1(M,\C)$ on the pair $\langle {\overline{T}M},M \rangle$, and this
Hodge connection is also real-analytic. Assume further that the
Taylor series at $0 \subset M \subset {\overline{T}M}$ for the Hodge connection
$D_1$ converge in the closed ball of radius $\varepsilon>0$.
\refstepcounter{subsubsection Let $D = \sum_{n,k}D_{k,n}:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to \B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ be the
derivation of the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ associated to the extended
connection $D$. Applying Lemma~\ref{est=conv} to the Hodge
connection $D_1$ proves that there exists a constant $C>0$ such that
for every $n \geq 0$ the norm $\|D_{1,n}\|_{1,1}$ of the restriction
of the derivation $D_{1,n}$ to the generator subspace $V_3 =
\B^0_{1,1} \subset \B^0$ satisfies
$$
\|D_{1,n}\|_{1,1} < CC_0^n,
$$
where $C_0 = 1/\varepsilon$.
By definition the derivation $D_{1,n}$ vanishes on the generator
subspace $V_1 \subset \B^1$. If $n > 0$, then it also vanishes on
the generator subspace $V_2 = \B^0_{0,1} \subset \B^0$. If $n=0$,
then its restriction to $V_2 = \B^0_{0,1} \subset \B^0$ is the
identity isomorphism $D_{1,0}={\sf id}:V_2 \to V_1$. In any case, we have
$\|D_{1,n}\|_{0,1} \leq 1$. Increasing if necessary the constant
$C_0$, we can assume that for any $n$ and for $p=0,1$ we have
$$
\|D_{1,n}\|_{p,1} < C_0^n.
$$
\refstepcounter{subsubsection We can now apply our main estimate,
Proposition~\ref{estimate}. It shows that there exists a constant
$C_1 > 0$ such that for every $k,n$ we have
$$
\|D_{k,n}\|_{1,1} < C_1^n.
$$
Together with Lemma~\ref{est=conv} this estimate implies that the
formal Hodge connection $D$ on ${\overline{T}M}$ along $M \subset {\overline{T}M}$
corresponding to the extended connection $D$ converges to a
real-analytic Hodge connection on an open neighborhood $U \subset
{\overline{T}M}$ of the zero section $M \subset {\overline{T}M}$. This in turn implies the
first claim of Theorem~\ref{converge}.
\refstepcounter{subsubsection To prove the second claim of Theorem~\ref{converge}, assume
that the manifold $M$ is equipped with a K\"ahler form $\omega$
compatible with the K\"ah\-le\-ri\-an connection $\nabla$, so that
$\nabla\omega=0$. The differential operator $\nabla:\Lambda^{1,1}(M)
\to \Lambda^{1,1}(M) \otimes \Lambda^1(M,\C)$ is elliptic and
real-analytic. Since $\nabla\omega=0$, the K\"ahler form $\omega$ is
also real-analytic.
For every $p,q \geq 0$ we have introduced in \ref{LLL} the space
$\LL^{p,q}$, which coincides with the space of formal germs at $0
\in M \subset {\overline{T}M}$ of smooth forms on ${\overline{T}M}$ of type $(p,q)$ with
respect to the complementary complex structure ${\overline{T}M}_J$. The spaces
$\LL^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}},{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ carry the total and the augmentation
gradings. Consider $\omega$ as an element of the vector space
$\LL^{2,0}_0$, and let $\omega = \sum_n\omega_n$ be the total degree
decomposition. The decomposition $\omega = \sum_n \omega_n$ is the
Taylor series decomposition for the form $\omega$ at $0 \in M$. Since
the form $\omega$ is real-analytic, there exists a constant $C_2$
such that
\begin{equation}\label{omega.est}
\|\omega_n\| < C_2^n
\end{equation}
for every $n$.
\refstepcounter{subsubsection Let $\Omega = \sum_k \Omega_k = \sum_{k,n} \Omega_{k,n}
\subset \LL^{2,0}$ be the formal polarization of the Hodge manifold
${\overline{T}M}$ at $M \subset {\overline{T}M}$ corresponding to the K\"ahler form $\omega$
by Theorem~\ref{metrics}. By definition we have $\omega =
\Omega_0$. Moreover, by construction used in the proof of
Proposition~\ref{metrics.ind} we have
\begin{equation}\label{Omega.rec}
\Omega_k = -\frac{1}{k}\sum_{1 \leq p \leq
k-1}\sigma_{tot}(D^{tot}_{k-p}\Omega_p),
\end{equation}
where $D^{tot}:\LL^{2,0} \to \LL^{2,1}$ is the derivation associated to
the extended connection $D$ on $M$ and $\sigma_{tot}:\LL^{2,{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1} \to
\LL^{2,{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ is the extension to $\LL^{2,{\:\raisebox{3pt}{\text{\circle*{1.5}}}}} = L^2_0 \otimes
\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ of the canonical endomorphism of the total Weil algebra
$\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{tot}$ constructed in the proof of
Proposition~\ref{metrics.ind}.
\refstepcounter{subsubsection The map $\sigma_{tot}:\LL^{2,1} \to \LL^{2,0}$ is a map of
$\B^0_{tot}$-modules. Therefore, as in \ref{sigma.est.punkt}, there
exists a constant $K_1 > 0$ such that
\begin{equation}\label{K1}
\|\sigma_{tot}\| < K_1
\end{equation}
on $\LL^{2,1}$. We can assume that $K_1 > 3K$, where $K$ is as in
\eqref{sigma.est}. Together with the recursive formula
\eqref{Omega.rec}, this estimate implies the following estimate on
the norms $\|\Omega_{k,n}\|$ of the components $\Omega_{k,n}$ of the
formal polarization $\Omega$ taken with respect to the standard
metric.
\begin{lemma}
For every $k,n$ we have
$$
\|\Omega_{k,n}\| < (2K_1)^{k-1}C^nb^2_{k,n},
$$
where $C = \max(C_0,C_2)$ is the bigger of the constants $C_0$,
$C_2$, and $b^2_{k,n}$ are the numbers defined recursively in
\ref{b.n.k.m}.
\end{lemma}
\proof Use induction on $k$. The base of the induction is the case
$k=1$, where the estimate holds by \eqref{omega.est}. Assume the
estimate proved for all $\Omega_{p,n}$ with $p < k$, and fix a
number $n$. By \eqref{Omega.rec} we have
$$
\Omega_{k,n} = -\frac{1}{k}\sum_{1 \leq p \leq
k-1}\sum_q\sigma_{tot}(D^{tot}_{k-p,n-q}\Omega_{p,q}).
$$
Substituting the estimate \eqref{K1} together with the inductive
assumption and the estimate on $\|D^{tot}_{k-p,n-q}\|^2_q$ obtained in
Lemma~\ref{D2est}, we get
\begin{align*}
\|\Omega_{k,n}\| &< \frac{1}{k}\sum_{1 \leq p \leq k-1}\sum_q
\|\sigma_{tot}\|{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\|D^{tot}_{k-p,n-q}\|^2_q{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\|\Omega_{p,q}\| \\
&< \frac{1}{k}\sum_{1 \leq p \leq k-1}\sum_qK_1 {\:\raisebox{3pt}{\text{\circle*{1.5}}}}
2(q+2(k-p))(3K)^{k-p-1}C_0^{n-q}b_{k-p,n-q} \\
&\qquad\qquad\qquad\qquad{\:\raisebox{3pt}{\text{\circle*{1.5}}}} (2K_1)^{p-1}C^qb_{p,q} \\
&< \frac{1}{k}(2K_1)^{k-1}C^n\sum_{1 \leq p \leq
k-1}\sum_q(q+2(k-p))b_{k-p,n-q}b^2_{p,q} \\
&= (2K_1)^{k-1}C^nb^2_{k,n},
\end{align*}
which proves the lemma.
\hfill \ensuremath{\square}\par
\refstepcounter{subsubsection This estimate immediately implies the last claim of
Theorem~\ref{converge}. Indeed, together with Lemma~\ref{combin.2}
it implies that for every $k,n$
$$
\|\Omega_{k,n}\| < (C_3)^n
$$
for some constant $C_3>0$. But $\Omega = \sum_n\sum_{0 \leq k \leq
n}\Omega_{k,n}$ is the Taylor series decomposition for the formal
polarization $\Omega$ at $0 \subset M$. The standard convergence
criterion shows that this series converges in an open ball of radius
$1/C_3 > 0$. Therefore the polarization $\Omega$ is indeed
real-analytic in a neighborhood of $0 \subset M \subset {\overline{T}M}$.
\hfill \ensuremath{\square}\par
\section*{Appendix}
\refstepcounter{section}
\refstepcounter{subsection}
\renewcommand{\thesection}{A}
\addcontentsline{toc}{section}{Appendix}
\refstepcounter{subsubsection In this appendix we describe a well-known Borel-Weyl type
localization construction for quaternionic vector spaces (see,
e.g. \cite{HKLR}) which provides a different and somewhat more
geometric approach to many facts in the theory of Hodge
manifolds. In particular, we establish, following Deligne and
Simpson (\cite{De2}, \cite{Simpson}), a relation between Hodge
manifolds and the theory of mixed $\R$-Hodge structures. For the
sake of simplicity, we consider only Hodge manifold structures on
the formal neighborhood of $0 \in \R^{4n}$ instead of actual Hodge
manifolds, as in Section~\ref{convergence}. To save the space all
proofs are either omitted or only sketched.
\refstepcounter{subsubsection Let $\SB$ be the Severi-Brauer variety associated to the
algebra ${\Bbb H}$, that is, the real algebraic variety of minimal right
ideals in ${\Bbb H}$. The variety $\SB$ is a twisted $\R$-form of the
complex projective line ${\C P}^1$.
For every algebra map $I:\C \to {\Bbb H}$ let the algebra ${\Bbb H} \otimes_\R
\C$ act on the $2$-dimensional complex vector space ${\Bbb H}_I$ by left
multiplication, and let $\widehat{I} \subset {\Bbb H} \otimes_\R \C$ be
the annihilator of the subspace $I(\C) \subset {\Bbb H}_I$ with respect to
this action. The subspace $\widehat{I}$ is a minimal right ideal in
${\Bbb H} \otimes_\R \C$. Therefore it defines a $\C$-valued point
$\widehat{I} \subset \SB(\C)$ of the real algebraic variety
$\SB$. This establishes a bijection between the set $\SB(\C)$ and
the set of algebra maps from $\C$ to ${\Bbb H}$.
\refstepcounter{subsubsection Let $\Shv(\SB)$ be the category of flat coherent sheaves on
$\SB$. Say that a sheaf $\E \in \Ob\Shv(\SB)$ is {\em of weight $p$}
if the sheaf $\E \otimes \C$ on ${\C P}^1 = \SB \otimes \C$ is a sum of
several copies of the sheaf ${\cal O}(p)$.
Consider a quaternionic vector space $V$. Let $\I \in {\Bbb H}
\otimes {\cal O}_\SB$ be the tautological minimal left ideal in the
algebra sheaf ${\Bbb H} \otimes {\cal O}_\SB$, and let $\loc{V} \in
\Ob\Shv(\SB)$ be the sheaf defined by
$$
\loc{V} = V \otimes {\cal O}_\SB / \I {\:\raisebox{3pt}{\text{\circle*{1.5}}}} V \otimes {\cal O}_\SB.
$$
The correspondence $V \mapsto \loc{V}$ defines a functor from
quaternionic vector spaces to $\Shv(\SB)$. It is easy to check that
this functor is a full embedding, and its essential image is the
subcategory of sheaves of weight $1$. Call $\loc{V}$ {\em the
localization} of the quaternionic vector space $V$. For every
algebra map $\I:\C \to {\Bbb H}$ the fiber $\loc{V}|_{\widehat{I}}$ of the
localization $\loc{V}$ over the point $\widehat{I} \subset \SB(\C)$
corresponding to the map $i:\C \to {\Bbb H}$ is canonically isomorphic to
the real vector space $V$ with the complex structure $V_I$.
\refstepcounter{subsubsection The compact Lie group $U(1)$ carries a canonical structure of
a real algebraic group. Fix an algebra embedding $I:\C \to {\Bbb H}$ and
let the group $U(1)$ act on the algebra ${\Bbb H}$ as in
\ref{u.acts.on.h}. This action is algebraic and induces therefore an
algebraic action of the group $U(1)$ on the Severi-Brauer variety
$\SB$. The point $\widehat{I}:\Spec\C \subset \SB$ is preserved by
the $U(1)$-action. The action of the group $U(1)$ on the complement
$\SB \setminus \widehat{I}(\Spec\C) \subset \SB$ is free, so that
the variety $\SB$ consists of two $U(1)$-orbits. The corresponding
orbits of the complexified group $\C^* = U(1) \times \Spec\C$ on the
complexification $\SB \times \Spec\C \cong {\C P}$ are the pair of
points $0,\infty \subset {\C P}$ and the open complement ${\C P} \setminus
\{0,\infty\} \cong \C^* \subset {\C P}$.
Let $\Shv^{U(1)}(\SB)$ be the category of $U(1)$-equivariant flat
coherent sheaves on the variety $\SB$. The localization construction
immediately extends to give the equivalence $V \mapsto \loc{V}$
between the category of equivariant quaternionic vector spaces and
the full subcategory in $\Shv^{U(1)}(\SB)$ consisting of sheaves of
weight $1$. For an equivariant quaternionic vector space $V$, the
fibers of the sheaf $\loc{V}$ over the point $\widehat{I} \subset
\SB(\C)$ and over the complement $\SB \setminus
\widehat{I}(\Spec\C)$ are isomorphic to the space $V$ equipped,
respectively, with the preferred and the complementary complex
structures $V_I$ and $V_J$.
\refstepcounter{subsubsection The category of $U(1)$-equivariant flat coherent sheaves on
the variety $\SB$ admits the following beautiful description, due to
Deligne.
\begin{lemma}[ (\cite{De2},\cite{Simpson})]\label{DS}
\begin{enumerate}
\item For every integer $n$ the full subcategory
$$
\Shv^{U(1)}_n(\SB) \subset \Shv^{U(1)}(\SB)
$$
of sheaves of weight $n$ is equivalent to the category of pure
$\R$-Hodge structures of weight $n$.
\item The category of pairs $\langle \E, W_{\:\raisebox{3pt}{\text{\circle*{1.5}}}}\rangle$ of a flat
$U(1)$-equivariant sheaf
$$
\E \in \Ob\Shv^{U(1)}(\SB)
$$
and an increasing filtration $W_{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ on $\E$ such that for every
integer $n$
\begin{equation}\label{hg}
W_n\E/W_{n-1}\E \quad \text{ is a sheaf of weight } \quad n \quad
\text{ on } \quad \SB
\end{equation}
is equivalent to the category of mixed $\R$-Hodge structures. (In
particular, it is abelian.)
\end{enumerate}
\end{lemma}
\refstepcounter{subsubsection For every pure $\R$-Hodge structure $V$ call the
corresponding $U(1)$-e\-qui\-va\-ri\-ant flat coherent sheaf on the
variety $\SB$ {\em the localization} of $V$ and denote it by
$\loc{V}$. For the trivial $\R$-Hodge structure $\R(0)$ of weight
$0$ the sheaf $\loc{\R(0)}$ coincides with the structure sheaf
${\cal O}$ on $\SB$. If $V$, $W$ are two pure $\R$-Hodge structures,
then the space $\Hom(\loc{V},\loc{W})$ of $U(1)$-equivariant maps
between the corresponding sheaves coincides with the space of weakly
Hodge maps from $V$ to $W$ in the sense of Subsection~\ref{w.H.sub}.
For every pure $\R$-Hodge structure $V$ the space
$\Gamma(\SB,\loc{V})$ of the global sections of the sheaf $\loc{V}$
is equipped with an action of the group $U(1)$ and carries therefore
a canonical $\R$-Hodge structure of weight $0$. This $\R$-Hodge
structure is the same as the universal $\R$-Hodge structure
$\Gamma(V)$ of weight $0$ constructed in Lemma~\ref{g.ex}. This
explains our notation for the functor $\Gamma:{{\cal W}{\cal H}odge}_{\geq 0} \to
{{\cal W}{\cal H}odge}_0$.
\refstepcounter{subsubsection Assume given a complex vector space $V$ and let $M$ be the
formal neighborhood of $0 \in V$. Let $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ be the Weil algebra
of the manifold $M$, as in \ref{formal.Weil}. For every $n \geq 0$
the vector space $\B^n$ is equipped with an $\R$-Hodge structure of
weight $n$, so that we can consider the localization
$\loc{\B^n}$. The sheaf $\oplus\loc{\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ is a commutative
algebra in the tensor category $\Shv^{U(1)}(\SB)$. We will call {\em
the localized Weil algebra}.
The augmentation grading on $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ defined in \ref{aug} is
compatible with the $\R$-Hodge structures. Therefore it defines an
augmentation grading on the localized Weil algebra
$\loc{\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$. The finer augmentation bigrading on $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$
does not define a bigrading on $\loc{\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$. However, it does
define a bigrading on the complexified algebra $\loc{\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}
\otimes \C$ of $\C^*$-equivariant sheaves on the manifold $\SB
\otimes \C \cong {\C P}$.
\refstepcounter{subsubsection Assume now given a flat extended connection $D:\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \to
\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}$ on $M$. Since the derivation $D$ is weakly Hodge, it
corresponds to a derivation $D:\loc{\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}} \to
\loc{\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}}$ of the localized Weil algebra $\loc{\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$.
It is easy to check that the complex $\langle \loc{\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}, D
\rangle$ is acyclic in all degrees but $0$. Denote by $\HH^0$ the
$0$-th cohomology sheaf $H^0(\loc{\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}})$. The sheaf $\HH^0$
carries a canonical algebra structure. Moreover, while the
derivation $D$ does not preserve the augmentation grading on
$\loc{\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$, it preserves the decreasing {\em augmentation
filtration} $\left(\loc{\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}\right)_{\geq {\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$. Therefore
we have a canonical decreasing filtration on the algebra $\HH^0$,
which we also call the augmentation filtration.
\refstepcounter{subsubsection It turns out that the associated graded quotient algebra $\gr \HH^0$
with respect to the augmentation filtration does not depend on the
extended connection $D$. To describe it, introduce the $\R$-Hodge
structure $W$ of weight $-1$ by setting $W = V$ as a real vector
space and
\begin{align}\label{W.dfn}
\begin{split}
W^{-1,0} &= V \subset V \otimes_\R \C,\\
W^{0,-1} &= \overline{V} \subset V \otimes_\R \C.
\end{split}
\end{align}
The $k$-th graded piece $\gr_k \HH^0$ with respect to the
augmentation filtration is then isomorphic to the symmetric power
$S^k(\loc{W})$ of the localization $\loc{W}$ of the $\R$-Hodge
structure $W$. In particular, it is a sheaf of weight $-n$, so that
up to a change of numbering the augmentation filtration on $\HH^0$
satisfies the condition \eqref{hg}. The extension data between these
graded pieces depend on the extended connection $D$. The whole
associated graded algebra $\gr \HH^0$ is isomorphic to the completed
symmetric algebra $\widehat{S}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(\loc{W})$.
\refstepcounter{subsubsection Using standard deformation theory, one can show that the
algebra map $\HH^0 \to \loc{\B^0}$ is the universal map from the
algebra $\HH^0$ to a complete commutative pro-algebra in the tensor
category of $U(1)$-equivariant flat coherent sheaves of weight $0$
on $\SB$. Moreover, the localized Weil algebra $\loc{\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$
coincides with the relative de Rham complex of $\loc{\B^0}$ over
$\HH^0$. Therefore one can recover, up to an isomorphism, the whole
algebra $\loc{\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ and, consequently, the extended connection
$D$, solely from the algebra $\HH^0$ in $\Shv^{U(1)}(\SB)$. Together
with Lemma~\ref{DS}~\thetag{ii} this gives the following, due also
to Deligne (in a different form).
\begin{prop}\label{DS.prop}
The correspondence $\langle \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}, D \rangle \mapsto \HH^0$ is a
bijection between the set of all isomorphism classes of flat
extended connections on $M$ and the set of all algebras $\HH^0$ in
the tensor category of mixed $\R$-Hodge structures equipped with an
isomorphism $\gr^W_{-1}\HH^0 \cong W$ between the $-1$-th associated
graded piece of the weight filtration on $\HH^0$ and the pure
$\R$-Hodge structure $W$ defined in \eqref{W.dfn} which induces for
every $n \geq 0$ an isomorphism $\gr^W_{-n}\HH^0 \cong S^nW$.
\end{prop}
\noindent {\bf Remark.\ } The scheme $\Spec\HH^0$ over $\SB$ coincides with the so-called
twistor space of the manifold ${\overline{T}M}$ with the hypercomplex structure
given by the extended connection $D$ (see \cite{HKLR} for the
definition). Deligne's and Simpson's (\cite{De2}, \cite{Simpson})
approach differs from ours in that they use the language of twistor
spaces to describe the relation between $U(1)$-equivariant
hypercomplex manifolds and mixed $\R$-Hodge structures. Since this
requires some additional machinery, we have avoided introducing
twistor spaces in this paper.
\refstepcounter{subsubsection We will now try to use the localization construction to
eludicate some of the complicated linear algebra used in
Section~\ref{main.section} to prove our main theorem. As we have
already noted, the category ${{\cal W}{\cal H}odge}$ of pure $\R$-Hodge
structures with weakly Hodge maps as morphisms is identified by
localization with the category $\Shv^{U(1)}(\SB)$ of
$U(1)$-equivariant flat coherent sheaves on $\SB$. Moreover, the
functor $\Gamma:{{\cal W}{\cal H}odge}_{\geq 0} \to {{\cal W}{\cal H}odge}_0$ introduced in
Lemma~\ref{g.ex} is simply the functor of global sections
$\Gamma(\SB,{\:\raisebox{3pt}{\text{\circle*{1.5}}}})$.
\refstepcounter{subsubsection Consider the localized Weil algebra $\loc{\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ with the
derivation $C:\loc{\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}} \to \loc{\B^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}}$ associated to
the canonical weakly Hodge derivation introduced in
\ref{C.and.sigma}. The differential graded algebra $\langle
\loc{\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}},C \rangle$ is canonically an algebra over the
completed symmetric algebra $\widehat{S}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(V)$ generated by the
constant sheaf on $\SB$ with the fiber $V$. Moreover, it is a free
commutative algebra generated by the complex
\begin{equation}\label{cmplx}
V \longrightarrow V(1)
\end{equation}
placed in degrees $0$ and $1$, where $V(1)$ is the
$U(1)$-equivariant sheaf of weight $1$ on $\SB$ corresponding to the
$\R$-Hodge structure given by $V(1)^{1,0} = V$ and $V(1)^{0,1} =
\overline{V}$.
\refstepcounter{subsubsection The homology sheaves of the complex \eqref{cmplx} are
non-trivial only in degree $1$. This non-trivial homology sheaf is a
skyscraper sheaf concentrated in the point $\widehat{I}(\Spec\C)
\subset \SB$ with fiber $V$. The associated sheaf on the
complexification $\SB \otimes \C \cong {\C P}$ splits into the sum of
skyscraper sheaf with fiber $V$ concentrated at $0 \in {\C P}$ and the
skyscraper sheaf with fiber $\overline{V}$ concentrated at $\infty
\in {\C P}$. This splitting cooresponds to the splitting of the complex
\eqref{cmplx} itself into the components of augmentation bidegrees
$(1,0)$ and $(0,1)$.
\refstepcounter{subsubsection Let now $\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} = \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}_{\geq 0,\geq 0}$ be the sum of
the components in the Weil algebra $\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ of augmentation
bidegree greater or equal than $(1,1)$. The subspace $\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}
\subset \B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is compatible with the $\R$-Hodge structure. The
crucial point in the proof of Theorem~\ref{kal=ext} is
Proposition~\ref{ac}, which claim the acyclycity of the complex
$\langle \Gamma(\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}), C \rangle$. It is this fact that becomes
almost obvious from the point of view of the localization
construction. To show it, we first prove the following.
\begin{lemma}
The complex $\langle\loc{\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}},C\rangle$ of $U(1)$-equivariant
sheaves on $\SB$ is acyclic.
\end{lemma}
\proof It suffices to prove that the complex $\loc{I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}} \otimes
\C$ of sheaves on $\SB \otimes \C \cong {\C P}$ is acyclic. To prove
it, let $p,q \geq 1$ be arbitrary integer, and consider the
component $\left(\loc{B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}\right)_{p,q}$
of augmentation bidegree $(p,q)$ in the localized Weil algebra
$\loc{\B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$. By definition we have
$$
\left(\loc{B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}\right)_{p,q} = \left(\loc{B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}\right)_{p,0}
\otimes \left(\loc{B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}\right)_{0,q}
$$
Since the complex $\left(\loc{B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}\right)_{p,0} =
S^p\left(\loc{B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}\right)_{1,0}$ has homology concentrated at $0
\in {\C P}$, while the complex $\left(\loc{B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}\right)_{0,q} =
S^q\left(\loc{B^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}\right)_{0,1}$ has homology concentrated at
$\infty \in {\C P}$, their product is indeed acyclic.
\hfill \ensuremath{\square}\par
Now, we have $\Gamma(\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}) = \Gamma(\SB,\loc{\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}})$, and
the functor $\Gamma(\SB,{\:\raisebox{3pt}{\text{\circle*{1.5}}}})$ is exact on the full subcategory in
$\Shv^{U(1)}(\SB)$ consisting of sheaves of positive
weight. Therefore the complex $\langle \Gamma(\I^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}),C \rangle$
is also acyclic, which gives an alternative proof of
Proposition~\ref{ac}.
\refstepcounter{subsubsection We would like to finish the paper with the following
observation. Proposition~\ref{DS.prop} can be extended to the
following claim.
\begin{prop}
Let $M$ be a complex manifold. There exists a naturla bijection
between the set of isomorphism classes of germs of Hodge manifold
structures on ${\overline{T}M}$ in the neighborhood of the zero section $M
\subset {\overline{T}M}$ and the set of multiplicative filtrations $F^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ on
the sheaf ${\cal O}_\R(M) \otimes \C$ of $\C$-valued real-analytic
functions on $M$ satisfying the following condition:
\begin{itemize}
\item For every point $m \in M$ let $\widehat{{\cal O}}_m$ be the
formal completion of the local ring ${\cal O}_m$ of germs of
real-analytic functions on $M$ in a neighborhood of $m$ with respect
to the maximal ideal. Consider the filtration $F^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ on
$\widehat{{\cal O}}_m \otimes \C$ induced by the filtration $f^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$
on the sheaf ${\cal O}_\R(M) \otimes \C$, and for every $k \geq 0$ let
$W_{-k}\widehat{{\cal O}}_m \subset \widehat{{\cal O}}_m$ be the $k$-th
power of the maximal ideal in $\widehat{{\cal O}}_m$. Then the triple
$\langle \widehat{{\cal O}}_m, F^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}, W^{\:\raisebox{3pt}{\text{\circle*{1.5}}}} \rangle$ is a mixed
$\R$-Hodge structure. (In particular, $F^k = 0$ for $k > 0$.)
\end{itemize}
\end{prop}
If the Hodge manifold structure on ${\overline{T}M}$ is such that the projection
$\rho:{\overline{T}M}_I \to M$ is holomorphic for the preferred complex
structure ${\overline{T}M}_I$ on ${\overline{T}M}$, then it is easy to see that the first
non-trivial piece $F^0{\cal O}_\R(M) \otimes \C$ of the filtration
$F^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ on the sheaf ${\cal O}_\R(M) \otimes \C$ coincides with the
subsheaf ${\cal O}(M) \subset {\cal O}_\R(M) \otimes \C$ of holomorphic
functions on $M$. Moreover, since the filtration $F^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ is
multiplicative, it is completely defined by the subsheaf
$F^{-1}{\cal O}_\R(M) \otimes \C \subset {\cal O}_\R(M) \otimes \C$. It
would be very interesting to find an explicit description of this
subsheaf in terms of the K\"ah\-le\-ri\-an connection $\nabla$ on $M$
which corresponds to the Hodge manifold structure on ${\overline{T}M}$.
|
1994-09-06T14:14:54 | 9406 | alg-geom/9406004 | en | https://arxiv.org/abs/alg-geom/9406004 | [
"alg-geom",
"math.AG"
] | alg-geom/9406004 | Fumiharu Kato | Fumiharu Kato | Log Smooth Deformation Theory | 29 pages, Latex version 2.09, Kyoto-Math 94-07 | null | null | null | null | This paper gives a foundation of log smooth deformation theory. We study the
infinitesimal liftings of log smooth morphisms and show that the log smooth
deformation functor has a representable hull. This deformation theory gives,
for example, the following two types of deformations: (1) relative deformations
of a certain kind of a pair of an algebraic variety and a divisor of it, and
(2) global smoothings of normal crossing varieties. The former is a
generalization of the relative deformation theory introduced by Makio, and the
latter coincides with the logarithmic deformation theory introduced by Kawamata
and Namikawa.
| [
{
"version": "v1",
"created": "Wed, 15 Jun 1994 08:42:17 GMT"
},
{
"version": "v2",
"created": "Mon, 4 Jul 1994 03:21:09 GMT"
},
{
"version": "v3",
"created": "Tue, 6 Sep 1994 08:56:10 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Kato",
"Fumiharu",
""
]
] | alg-geom | \section{Introduction}
In this article, we formulate and develop the theory of
{\it log smooth deformation}. Here, log smoothness (more precisely,
logarithmic smoothness) is a concept in {\it log geometry} which is
a generalization of ``usual'' smoothness of morphisms of algebraic
varieties. Log geometry is a beautiful geometric theory which
succesfully generalizes and unifies the scheme theory and the theory
of torus embeddings. This theory was first planned and founded by
Fontaine and Illusie, based on their idea of {\it log structures} on
schemes, and further developed by Kato \cite {Kat1}.
Recently, the importance of log
geometry comes to be recognized by many geometers and applied to
various fields of algebraic and arithmetic geometry. One of such
applications can be seen in the recent work of Steenbrink \cite {Ste1}.
In the present paper, we attempt to apply log geometry to extend the usual
smooth deformation theory by using the concept of log smoothness.
Log smoothness is one of the most important concepts in log geometry,
and is a log geometric generalization of usual smoothness. For example,
varieties with toric singularities or normal crossing varieties may
become log smooth over certain logarithmic points. Kato \cite {Kat1} showed
that any log smooth morphism is written \'{e}tale locally by the composition
of a usual smooth morphism and a morphism induced by a homomorphism
of monoids which essentially determines the log structures
(Theorem \ref{lisse}). On the other hand, log
smoothness is described by means of {\it log differentials} and
{\it log derivations} similarly to usual smoothness by means of
differentials and derivations. Hence if we consider the log smooth
deformation by analogy with the usual smooth deformation, it is
expected that
the first order deformation is controled by the sheaf of
log derivations. This intuition motivated this work and we shall see
later that this is, in fact, the case.
In the present paper, we construct
log smooth deformation functor by the concept of infinitesimal log smooth
lifting. The goal of this paper is to show that this functor has a
representable hull in the sense of Schlessinger
\cite {Sch1}, under certain conditions on the underlying schemes
(Theorem \ref{hull}). At the end of this paper,
we give two examples of our log smooth
deformation theory, which are summarized as follows:
\vspace{3mm}
1. {\sc Deformations with divisors} (\S \ref{exam1}): Let $X$ be a
variety over a field $k$. Assume that the variety $X$ is covered
by \'{e}tale open sets
which are smooth over affine torus embeddings, and there exists a divisor
$D$ of $X$ which is the union of the closures of
codimension 1 torus orbits. Then, there exists a log
structure ${\cal M}$ on $X$ such that the log scheme $(X, {\cal M})$ is log smooth
over $k$ with trivial
log structure. (The converse is also true in a certain excellent category
of log schemes.)
In this case, our log smooth deformation is a deformation of the
pair $(X,D)$. If $X$ itself is smooth and $D$ is a smooth divisor of $X$,
our deformation coincides with the relative deformation studied by Makio
\cite {Mak1}.
\vspace{3mm}
2. {\sc Smoothings of normal crossing varieties} (\S \ref{exam2}):
If a connected scheme of finite type $X$ over a field $k$ is,
\'{e}tale locally, isomorphic to an affine normal crossing variety
$\mathop{\rm Spec}\nolimits k[z_1,\ldots,z_n]/(z_1\cdots z_d)$, then we call $X$ a
normal crossing variety over $k$. If $X$ is $d$--semistable
(cf.\ \cite {Fri1}), there
exists a log structure ${\cal M}$ on $X$ such that $(X,{\cal M})$ is log smooth over a
standard log point $(\mathop{\rm Spec}\nolimits k,{\bf N})$ (Theorem \ref{dss}). Then, our log
smooth deformation is nothing but a smoothing of $X$. If the singular
locus of $X$ is connected, our deformation theory coincides
with the one introduced by Kawamata and Namikawa \cite {K-N1}.
\vspace{3mm}
The composition of this paper is as follows. We recall some basic notions of
log geometry in the next section, and review the definition and basic
properties of log smoothness in section 3. In section 4, we study the
characterization of log smoothness by means of the theory of torus
embeddings according to Illusie \cite {Ill1} and Kato \cite {Kat1}. In
section 5, we recall the definitions and basic properties of log derivations
and log differentials. In section 6 and section 7, we give proofs of
theorems stated in section 4. Section 8 is devoted to the formulation of
log smooth deformation theory. This section is the main section of this
present paper. We prove the existance of a representable hull of the
log smooth deformation functor in section 9. In section 10 and section 11,
we give two examples of log smooth deformation. For the reader's convenience,
in section 12, we give a proof of the result of Kawamata and Namikawa
\cite {K-N1} which is relevant to our log smooth deformation.
The author would like to express his thanks to Professors Kazuya Kato and
Yoshinori Namikawa for
valuable suggestions and advice. The author is also very grateful to
Professor Luc Illusie for valuable advice on this paper.
\vspace{3mm}
{\sc Convention}. We assume that all monoids are commutative and have
neutral elements. A homomorphism of monoids is assumed to preserve
neutral elements.
We write the binary operations of all monoids
multiplicatively except in the case of ${\bf N}$ (the monoid of non--negative
integers), ${\bf Z}$, etc., which
we write additively.
All sheaves on schemes are considered
with respect to the \'{e}tale topology.
\section{Fine saturated log schemes}
In this and subsequent sections, we use the terminology of log geometry
basically as in \cite {Kat1}.
Let $X$ be a scheme.
We view the structure sheaf $\O_X$ of $X$ as a sheaf of monoids under
multiplication.
\begin{dfn}{\rm (cf.\ \cite [\S 1]{Kat1})
A {\it pre--log structure} on $X$ is a homomorphism ${\cal M}\rightarrow\O_X$ of
sheaves of monoids where ${\cal M}$ is a sheaf of monoids on $X$.
A pre--log structure $\alpha:{\cal M}\rightarrow\O_X$ is said to be a
{\it log structure} on $X$ if $\alpha$ induces an isomorphism
$$
\alpha:\alpha^{-1}({\cal O}^\times_X)\stackrel{\sim}{\longrightarrow}{\cal O}^\times_X.
$$}
\end{dfn}
\noindent
Given a pre--log structure $\alpha:{\cal M}\rightarrow\O_X$, we can
construct the {\it associated log structure}
$\alpha^{\rm a}:{\cal M}^{\rm a}\rightarrow\O_X$
functorially by
\begin{equation}\label{asslog1}
{\cal M}^{\rm a}=({\cal M}\oplus{\cal O}^\times_X)/\P
\end{equation}
and
$$
\alpha^{\rm a}(x,u)=u\cdot\alpha(x)
$$
for $(x,u)\in{\cal M}^{\rm a}$, where $\P$ is the submonoid defined by
$$
\P=\{(x,\alpha(x)^{-1})\: |\: x\in\alpha^{-1}({\cal O}^\times_X)\}.
$$
Here, in general, the quotient $M/P$ of a monoid $M$ with respect
to a submonoid
$P$ is the coset space $M/\sim$ with induced monoid structure,
where the equivalence relation $\sim$ is defined by
$$
x\sim y\Leftrightarrow xp=yq \,\mbox{ for some $p,q\in P$}.
$$
${\cal M}^{\rm a}$ has a universal mapping property:
if $\beta:{\cal N}\rightarrow\O_X$ is a log structure on $X$ and
$\varphi:{\cal M}\rightarrow{\cal N}$ is a homomorphism of sheaves of monoids such that
$\alpha=\beta\circ\varphi$, then there exists a unique lifting
$\varphi^{\rm a}:{\cal M}^{\rm a}\rightarrow{\cal N}$.
Note that the monoid ${\cal M}^{\rm a}$ defined by (\ref{asslog1})
is the push--out of the diagram
$$
{\cal M}\supset
\alpha^{-1}({\cal O}^\times_X)\stackrel{\alpha}{\longrightarrow}{\cal O}^\times_X
$$
in the category of monoids, and the homomorphism $\alpha^{\rm a}$ is
induced by
$\alpha$ and the inclusion ${\cal O}^\times_X\hookrightarrow\O_X$. We sometimes denote
the monoid ${\cal M}^{\rm a}$ by ${\cal M}\oplus_{\alpha^{-1}({\cal O}^\times_X)}{\cal O}^\times_X$.
Note that we have the natural isomorphism
\begin{equation}\label{basic1}
{\cal M}/\alpha^{-1}({\cal O}^\times_X)\stackrel{\sim}{\longrightarrow}
{\cal M}^{\rm a}/{\cal O}^\times_X.
\end{equation}
\begin{dfn}{\rm
By a {\it log scheme}, we mean a pair $(X,{\cal M})$ with a scheme $X$ and
a log structure ${\cal M}$ on $X$. A {\it morphism} of log schemes
$f:(X,{\cal M})\rightarrow(Y,{\cal N})$ is a pair $f=(f,\varphi)$ where
$f:X\rightarrow Y$ is a morphism of schemes and
$\varphi:f^{-1}{\cal N}\rightarrow{\cal M}$ is a homomorphism of sheaves of monoids
such that the diagram
$$
\begin{array}{ccc}
f^{-1}{\cal N}&\stackrel{\varphi}{\longrightarrow}&{\cal M}\\
\vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow\\
f^{-1}\O_Y&\longrightarrow&\O_X
\end{array}
$$
commutes. }
\end{dfn}
\begin{dfn}\label{logequiv}{\rm
Let $\alpha:{\cal M}\rightarrow\O_X$ and $\alpha':{\cal M}'\rightarrow\O_X$ be log
structures on a scheme $X$. These log structures are said to be
{\it equivalent} if there exists an isomorphism
$\varphi:{\cal M}\stackrel{\sim}{\rightarrow}{\cal M}'$ such that
$\alpha=\alpha'\circ\varphi$, i.e., there exists an isomorphism of log schemes
$(X,{\cal M})\stackrel{\sim}{\rightarrow}(X,{\cal M}')$ whose underlying morphism of
schemes is the identity ${\rm id}_X$.
Let $\beta:{\cal N}\rightarrow\O_Y$ and $\beta':{\cal N}'\rightarrow\O_Y$ be log
structures on a scheme $Y$. Let $f:(X,{\cal M})\rightarrow(Y,{\cal N})$ and
$f':(X,{\cal M}')\rightarrow(Y,{\cal N}')$ be morphisms of log schemes. Then $f$ and $f'$
are said to be {\it equivalent} if there exist isomorphisms
$\varphi:{\cal M}\stackrel{\sim}{\rightarrow}{\cal M}'$ and
$\psi:{\cal N}\stackrel{\sim}{\rightarrow}{\cal N}'$ such that
$\alpha=\alpha'\circ\varphi$, $\beta=\beta'\circ\psi$ and the diagram
$$
\begin{array}{ccc}
{\cal M}&\stackrel{\varphi}{\longrightarrow}&{\cal M}'\\
\vphantom{\bigg|}\Big\uparrow&&\vphantom{\bigg|}\Big\uparrow\\
f^{-1}{\cal N}&\underrel{\longrightarrow}{f^{-1}\psi}&f^{-1}{\cal N}'
\end{array}
$$
commutes. }
\end{dfn}
\noindent
We denote the category of log schemes by ${\bf LSch}$. For $(S,\L)\in\mathop{\rm Obj}\nolimits({\bf LSch})$, we
denote the category of log schemes over $(S,\L)$ by ${\bf LSch}_{(S,\L)}$.
The following examples play important roles in the sequel.
\begin{exa}\label{trilog}{\rm
On any scheme $X$, we can define a log structure by the inclusion
${\cal O}^\times_X\hookrightarrow\O_X$, called the {\it trivial} log structure.
Thus, we have an inclusion functor from the category of schemes to that
of log schemes. We often denote the log scheme $(X, {\cal O}^\times_X\hookrightarrow\O_X)$
simply by $X$.
}
\end{exa}
\begin{exa}\label{canlog}{\rm
Let $A$ be a commutative ring. For a monoid $P$ , we can define a log structure
canonically on the scheme $\mathop{\rm Spec}\nolimits A[P]$, where $A[P]$ denotes the monoid ring
of $P$ over $A$, as the log structure associated to the natural homomorphism,
$$
P\stackrel{\alpha}{\longrightarrow}A[P].
$$
This log structure is called the {\it canonical log structure} on
$\mathop{\rm Spec}\nolimits A[P]$.
Thus we obtain a log scheme which we denote simply by $(\mathop{\rm Spec}\nolimits A[P], P)$.
A monoid homomorphism $P\rightarrow Q$
induces a morphism $(\mathop{\rm Spec}\nolimits A[Q], Q)\rightarrow(\mathop{\rm Spec}\nolimits A[P], P)$ of log
schemes. Thus, we have a contravariant functor from the category of monoids
to ${\bf LSch}_{\mathop{\rm Spec}\nolimits A}$.
}
\end {exa}
\begin{exa}\label{torlog}{\rm
Let $\Sigma$ be a fan on $N_{\bf R}={\bf R}^d$, $N={\bf Z}^d$, and $X_{\Sigma}$
the toric variety determined by the fan $\Sigma$ over a commutative
ring $A$. Then, we get an induced log structure on the scheme
$X_{\Sigma}$ by gluing the log structures associated to the
homomorphisms
$$
M\cap\sigma^{\vee}\longrightarrow A[M\cap\sigma^{\vee}],
$$
for each cone $\sigma$ in $\Sigma$, where $M=\mathop{\rm Hom}\nolimits_{{\bf Z}}(N, {\bf Z})$. Thus, a toric
variety $X_{\Sigma}$ is naturally viewed as a log scheme over $\mathop{\rm Spec}\nolimits A$,
which we denote by $(X_{\Sigma}, \Sigma)$.
}
\end{exa}
Next, we define important subcategories of ${\bf LSch}$. These subcategories are
closely related with {\it charts} defined as follows.
\begin{dfn}{\rm
Let $(X,{\cal M})\in\mathop{\rm Obj}\nolimits({\bf LSch})$. A {\it chart} of ${\cal M}$ is a homomorphism
$P\rightarrow{\cal M}$ from the constant sheaf of a monoid $P$ which induces
an isomorphism from the associated log structure $P^{\rm a}$ to ${\cal M}$. }
\end{dfn}
\begin{dfn}{\rm
Let $f:(X,{\cal M})\rightarrow(Y,{\cal N})$ be a morphism in ${\bf LSch}$. A {\it chart} of
$f$ is a triple $(P\rightarrow{\cal M},Q\rightarrow{\cal N},Q\rightarrow P)$, where
$P\rightarrow{\cal M}$ and $Q\rightarrow{\cal N}$ are charts of ${\cal M}$ and ${\cal N}$,
respectively, and $Q\rightarrow P$ is a homomorphism for which the diagram
$$
\begin{array}{ccc}
Q&\longrightarrow&P\\
\vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow\\
f^{-1}{\cal N}&\longrightarrow&{\cal M}
\end{array}
$$
is commutative. }
\end{dfn}
\begin{dfn}{\rm (cf.\ \cite [\S 2]{Kat1})
A log structure ${\cal M}\rightarrow\O_X$ on a scheme $X$
is said to be {\it fine} if ${\cal M}$ has \'{e}tale
locally a chart $P\rightarrow{\cal M}$ with $P$ a finitely generated integral
monoid. Here, in general, a monoid $M$ is said to be
{\it finitely generated} if there exists a surjective homomorphism
${\bf N}^n\rightarrow M$ for some $n$, and a monoid $M$ is
said to be {\it integral} if $M\rightarrow\gp{M}$ is injective, where
$\gp{M}$ denotes the Grothendieck group associated with $M$.
A log scheme $(X,{\cal M})$ with a fine log structure ${\cal M}\rightarrow\O_X$ is called
a {\it fine} log scheme. }
\end{dfn}
\noindent
We denote the category of fine log schemes by ${\bf LSch}^{\rm f}$. Similarly, we denote the
category of fine log schemes over $(S,\L)\in\mathop{\rm Obj}\nolimits({\bf LSch}^{\rm f})$ by ${\bf LSch}^{\rm f}_{(S,\L)}$,
The category ${\bf LSch}^{\rm f}$ (resp. ${\bf LSch}^{\rm f}_{(S,\L)}$) is a full subcategory of ${\bf LSch}$
(resp. ${\bf LSch}_{(S,\L)}$). Both ${\bf LSch}$ and ${\bf LSch}^{\rm f}$ have fiber products.
But the inclusion functor ${\bf LSch}^{\rm f}\hookrightarrow{\bf LSch}$ does not preserve fiber
products (cf. Lemma \ref{fpro}). The inclusion functor ${\bf LSch}^{\rm f}\hookrightarrow
{\bf LSch}$ has a right adjoint ${\bf LSch}\rightarrow{\bf LSch}^{\rm f}$ \cite[(2.7)]{Kat1}. Then, the
fiber product of a diagram $(X,{\cal M})\rightarrow(Z,\P)\leftarrow(Y,{\cal N})$ in
${\bf LSch}^{\rm f}$ is the image of that in ${\bf LSch}$ by this adjoint functor. Note that the
underlying scheme of the fiber product of
$(X,{\cal M})\rightarrow(Z,\P)\leftarrow(Y,{\cal N})$ in ${\bf LSch}$ is $X\times_{Z}Y$, but
this is not always the case in ${\bf LSch}^{\rm f}$.
\vspace{3mm}
Next, we introduce more excellent subcategory of ${\bf LSch}$.
\begin{dfn}\label{satin}{\rm
Let $M$ be a monoid and $P$ a submonoid of $M$.
The monoid $P$ is said to be
{\it saturated} in $M$ if $x\in M$ and $x^n\in P$ for some positive
integer $n$ imply
$x\in P$. An integral monoid $N$ is said to be
{\it saturated} if $N$ is saturated in $\gp{N}$. }
\end{dfn}
\begin{exa}{\rm
Put $M={\bf N}$ and $P=l\cdot M$ for an integer $l>1$.
Then $P$ is saturated but not
saturated in $M$. }
\end{exa}
\begin{dfn}{\rm
A fine log scheme $(X,{\cal M})\in\mathop{\rm Obj}\nolimits ({\bf LSch}^{\rm f})$ is said to be {\it saturated} if
the log structure ${\cal M}$ is a sheaf of saturated monoids. }
\end{dfn}
\noindent
We denote the category of fine saturated log
schemes by ${\bf LSch}^{\rm fs}$. Similarly, we denote the category of fine saturated
log schemes over $(S,\L)\in\mathop{\rm Obj}\nolimits({\bf LSch}^{\rm fs})$ by ${\bf LSch}^{\rm fs}_{(S,\L)}$.
The category ${\bf LSch}^{\rm fs}$ (resp. ${\bf LSch}^{\rm fs}_{(S,\L)}$) is a full subcategory of ${\bf LSch}^{\rm f}$
(resp. ${\bf LSch}^{\rm f}_{(S,\L)}$).
The following lemma is an easy consequence of
\cite [Lemma (2.10)]{Kat1}.
\begin{lem}
Let $f:(X,{\cal M})\rightarrow(Y,{\cal N})$ be a morphism in ${\bf LSch}^{\rm fs}$, and $Q\rightarrow{\cal N}$
a chart of ${\cal N}$, where $Q$ is a finitely generated integral saturated
monoid. Then there exists \'{e}tale locally a chart
$(P\rightarrow{\cal M},Q\rightarrow{\cal N},Q\rightarrow P)$
of $f$ extending $Q\rightarrow{\cal N}$ such that the monoid $P$ is also finitely
generated, integral and saturated.
\end{lem}
\begin{lem}
The inclusion functor ${\bf LSch}^{\rm fs}\lhook\joinrel\longrightarrow{\bf LSch}^{\rm f}$ has a right adjoint.
\end{lem}
\noindent{\sc Proof.}\hspace{2mm}
Let $M$ be an integral monoid. Define
$$
\sat{M}=\{x\in\gp{M}\: |
\: x^n\in M \;\mbox{for some positive integer $n$}\}.
$$
Then $\sat{M}$ is an integral saturated monoid. For any
integral saturated monoid $N$ and homomorphism $M\rightarrow N$, there
exists a unique lifting $\sat{M} \rightarrow N$. In this sense, $\sat{M}$
is the universal saturated monoid associated with $M$. Let $(X,{\cal M})$ be a
fine log scheme. Then we have \'{e}tale locally a chart $P\rightarrow{\cal M}$.
This chart defines a morphism $X\rightarrow\mathop{\rm Spec}\nolimits {\bf Z}[P]$ \'{e}tale locally.
Let $X'=X\times_{\mathop{\rm Spec}\nolimits {\bf Z}[P]}\mathop{\rm Spec}\nolimits {\bf Z}[\sat{P}]$. Then $X'\rightarrow
\mathop{\rm Spec}\nolimits {\bf Z}[\sat{P}]$ induces a log structure ${\cal M}'$
by the associated log structure of
$\sat{P}\rightarrow{\bf Z}[\sat{P}]\rightarrow\O_{X'}$, and $(X',{\cal M}')$ is a fine
saturated log scheme. This procedure defines
a functor ${\bf LSch}^{\rm f}\rightarrow{\bf LSch}^{\rm fs}$. It is easy to see that this functor is the
right adjoint of the inclusion functor ${\bf LSch}^{\rm fs} \hookrightarrow{\bf LSch}^{\rm f}$. $\Box$
\begin{cor}
${\bf LSch}^{\rm fs}$ has fiber products. More precisely, the fiber product of
morphisms $(X,{\cal M})\rightarrow(Z,\P)\leftarrow(Y,{\cal N})$ in ${\bf LSch}^{\rm fs}$ is the
image of that in ${\bf LSch}^{\rm f}$ by the right adjoint functor of
${\bf LSch}^{\rm fs}\lhook\joinrel\longrightarrow{\bf LSch}^{\rm f}$.
\end{cor}
\section{Log smooth morphisms}
In this section, we review the definition and basic properties of log
smoothness (cf. \cite {Kat1}).
\begin{dfn}\label{pback}{\rm
Let $f:X\rightarrow Y$ be a morphism of schemes, and ${\cal N}$ a log structure
on $Y$. Then the {\it pull--back} of ${\cal N}$, denoted by $f^{*}{\cal N}$, is the log
structure on $X$ associated with the pre--log structure
$f^{-1}{\cal N}\rightarrow f^{-1}\O_Y\rightarrow\O_X$. A morphism of log schemes
$f:(X,{\cal M})\rightarrow(Y,{\cal N})$ is said to be {\it strict} if the induced
homomorphism $f^{*}{\cal N}\rightarrow{\cal M}$ is an isomorphism.
A morphism of log schemes
$f:(X,{\cal M})\rightarrow(Y,{\cal N})$ is said to be an {\it exact closed immersion} if
it is strict and
$f:X\rightarrow Y$ is a closed immersion in the ususal sense. }
\end{dfn}
\noindent
Exact closed immersions are stable under base change in ${\bf LSch}^{\rm f}$
\cite [(4.6)]{Kat1}.
\begin{lem}\label{basic2}
Let $\alpha:{\cal M}\rightarrow\O_X$ and $\alpha':{\cal M}'\rightarrow\O_X$ be fine log
strctures on a scheme $X$ with a homomorphism $\varphi:{\cal M}\rightarrow{\cal M}'$ of
monoids such that $\alpha=\alpha'\circ\varphi$. Then, $\varphi$ is an
isomorphism if and only if $\varphi\ {\rm mod}\ {\cal O}^\times_X:{\cal M}/{\cal O}^\times_X
\rightarrow{\cal M}'/{\cal O}^\times_X$ is an isomorphism.
\end{lem}
\noindent
The proof is straightforward.
\begin{lem}\label{basic3}
Let $f:(X,{\cal M})\rightarrow(Y,{\cal N})$ be a morphism of log schemes. Then, we have
the natural isomorphism
$$
f^{-1}({\cal N}/{\cal O}^\times_Y)\stackrel{\sim}{\longrightarrow}f^{*}{\cal N}/{\cal O}^\times_X.
$$
In particular, $f$ is strict if and only if the induced morphism
$$
f^{-1}({\cal N}/{\cal O}^\times_Y)\stackrel{\sim}{\longrightarrow}{\cal M}/{\cal O}^\times_X.
$$
\end{lem}
\noindent{\sc Proof.}\hspace{2mm}
The first part is easy to see. For the second part, apply (\ref {basic1}) and
Lemma \ref{basic2}.
$\Box$
\begin{lem}{\rm (cf. \cite [(1.7)]{Kaj1})}\label{fpro}
Let
\begin{equation}\label{fpro1}
(X,{\cal M})\rightarrow(Z,\P)\leftarrow(Y,{\cal N})
\end{equation}
be morphisms in ${\bf LSch}^{\rm fs}$.
If $(Y,{\cal N})\rightarrow(Z,\P)$ is strict,
then the fiber product of {\rm (\ref{fpro1})} in ${\bf LSch}^{\rm fs}$ is isomorphic to
that in ${\bf LSch}$. In particular, the underlying scheme of the fiber product of
{\rm (\ref{fpro1})} in ${\bf LSch}^{\rm fs}$ is isomorphic to $X\times_{Z}Y$.
\end{lem}
\noindent{\sc Proof.}\hspace{2mm}
Let $P\rightarrow\P$ be a chart of $\P$, where $P$ is a finitely generated
integral saturated monoid. Since $(Y,{\cal N})\rightarrow(Z,\P)$ is strict,
$\rightarrow\P\rightarrow{\cal N}$ is a chart of ${\cal N}$
by Lemma \ref{basic2}, Lemma \ref{basic3}, and (\ref{basic1}).
Take a chart
$$
\begin{array}{ccc}
P&\longrightarrow&M\\
\vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow\\
\P&\longrightarrow&{\cal M}
\end{array}
$$
of $(X,{\cal M})\rightarrow(Z,\P)$ extending $P\rightarrow\P$. Set $W=X\times_{Z}Y$.
There exists an induced homomorphism $M\rightarrow\O_W$. Define a log
structure on $W$ by this homomorphism. Then this
log scheme $(W,M\rightarrow\O_W)$ is the fiber product of (\ref{fpro1}) in
${\bf LSch}$. Since the associated log structure of $M\rightarrow\O_W$ is fine and
saturated, $(W,M)$ is, indeed, the fiber product of (\ref{fpro1}) in ${\bf LSch}^{\rm fs}$.
$\Box$
\begin{dfn}\label{defthick}{\rm
The exact closed immersion $t:(T',\L')\rightarrow(T,\L)$
is said to be a {\it thickening of order $\leq n$}, if
${\cal I}=\mathop{\rm Ker}\nolimits(\O_T\rightarrow\O_{T'})$ is a nilpotent ideal such that
${\cal I}^{n+1}=0$. }
\end{dfn}
\begin{lem}\label{thick}{\rm (cf.\ \cite {Ill1}).}
Let $(T,\L)$ and $(T',\L')$ be fine log schemes.
If $(t,\theta):(T',\L')\rightarrow(T,\L)$ is a thickening of order 1,
there exists a commutative diagram with exact rows:
$$
\begin{array}{ccccccccc}
1&\rightarrow&1+{\cal I}&\hookrightarrow&t^{-1}\L&
\stackrel{\theta}{\rightarrow}&\L'&\rightarrow&1\\
&&\parallel&&\cap&&\cap\\
1&\rightarrow&1+{\cal I}&\rightarrow&t^{-1}\gp{\L}&
\underrel{\rightarrow}{\gp{\theta}}&\gp{\L'}&\rightarrow&1\rlap{,}
\end{array}
$$
where ${\cal I}=\mathop{\rm Ker}\nolimits(\O_T\rightarrow\O_{T'})$, such that the right square of this
commutative diagram is cartesian.
\end{lem}
\noindent
The proof is straightforward.
Note that the multiplicative monoid $1+{\cal I}$ is identified with the additive
monoid ${\cal I}$ by $1+x\mapsto x$ since ${\cal I}^2=0$.
\begin{dfn}{\rm (cf.\ \cite [(3.3)]{Kat1})
Let $f:(X,{\cal M})\rightarrow(Y,{\cal N})$ be a morphism in ${\bf LSch}^{\rm f}$. $f$ is said to be
{\it log smooth} if the following conditions are satisfied:
\begin{enumerate}
\item $f$ is locally of finite presentation,
\item for any commutative diagram
$$
\begin{array}{ccc}
(T',\L')&\stackrel{s'}{\longrightarrow}&(X,{\cal M})\\
\llap{$t$}\vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow\rlap{$f$}\\
(T,\L)&\underrel{\longrightarrow}{s}&(Y,{\cal N})
\end{array}
$$
in ${\bf LSch}^{\rm f}$,
where $t$ is a thickening of order 1, there exists \'{e}tale locally
a morphism $g:(T,\L)\rightarrow(X,{\cal M})$ such that $s'=g\circ t$ and
$s=f\circ g$.
\end{enumerate}}
\end{dfn}
\noindent
The proofs of the following two propositions are straightforward and are
left to the reader.
\begin{pro}\label{usulisse}
Let $f:(X,{\cal M})\rightarrow(Y,{\cal N})$ be a morphism in ${\bf LSch}^{\rm f}$.
If $f$ is strict,
then $f$ is log smooth if and only if $f$ is smooth in the usual sense.
\end{pro}
\begin{pro}\label{bextlisse}
For $(S,\L)\in\mathop{\rm Obj}\nolimits({\bf LSch}^{\rm f})$ and $(X,{\cal M}),(Y,{\cal N})\in\mathop{\rm Obj}\nolimits({\bf LSch}^{\rm f}_{(S,\L)})$,
let $f:(X,{\cal M})\rightarrow(Y,{\cal N})$ be a morphism in ${\bf LSch}^{\rm f}_{(S,\L)}$.
Assume that $f$ is log smooth. If $(S',\L')$ is a log scheme over $(S,\L)$,
then the induced morphism
$$
(X,{\cal M})\times_{(S,\L)}(S',\L')\rightarrow(Y,{\cal N})\times_{(S,\L)}(S',\L')
$$
is also log smooth.
\end{pro}
\section{Toroidal characterization of log smoothness}
The following theorem is due to Kato \cite{Kat1},
and we prove it in \S \ref{prf1} for the reader's convenience.
\begin{thm}\label{lisse}{\rm (\cite [(3.5)]{Kat1})}
Let $f:(X,{\cal M})\rightarrow(Y,{\cal N})$ be a morphism in ${\bf LSch}^{\rm f}$. and $Q\rightarrow{\cal N}$
a chart of ${\cal N}$, where $Q$ is a finitely generated integral
monoid. Then the following conditions are equivalent.
\begin{description}
\item[{\rm 1.}] $f$ is log smooth.
\item[{\rm 2.}] There exists \'{e}tale locally a chart
$(P\rightarrow{\cal M},Q\rightarrow{\cal N},Q\rightarrow P)$ of $f$ extending
$Q\rightarrow{\cal N}$, where $P$ is a finitely generated integral
monoid, such that
\begin{description}
\item[{\rm (a)}] $\mathop{\rm Ker}\nolimits(\gp{Q}\rightarrow\gp{P})$ and the torsion part of
$\mathop{\rm Coker}\nolimits(\gp{Q}\rightarrow\gp{P})$ are finite groups of orders invertible
on $X$,
\item[{\rm (b)}] $X\rightarrow Y\times_{\mathop{\rm Spec}\nolimits {\bf Z}[Q]}\mathop{\rm Spec}\nolimits {\bf Z}[P]$ is
smooth (in
the usual sense).
\end{description}
\end{description}
Moreover, if $f:(X,{\cal M})\rightarrow(Y,{\cal N})$ is a log smooth
morphism in ${\bf LSch}^{\rm fs}$ and
$Q\rightarrow{\cal N}$ is a chart of ${\cal N}$ such that $Q$ is finitely generated,
integral and saturated,
then there exists a chart
$(P\rightarrow{\cal M},Q\rightarrow{\cal N},Q\rightarrow P)$ of $f$ as above
with $P$ also saturated.
\end{thm}
\begin{rem}\label{lisserem}{\rm
The proof of Theorem \ref{lisse} in \S \ref{prf1} shows that we can require
in the condition 2. (a) that $\gp{Q}\rightarrow\gp{P}$ is injective without
changing the conclusion.
}
\end{rem}
We give some important examples of log smooth morphisms in the following.
Let $k$ be a field.
\begin{dfn}\label{logpt}{\rm
A log structure on $\mathop{\rm Spec}\nolimits k$ is called a log structure of a {\it logarithmic
point} if it is equivalent (Definition \ref{logequiv})
to the associated log structure of
$\alpha:Q\rightarrow k$, where $Q$ is a monoid having no
invertible element other than 1 and $\alpha$ is a homomorphism defined by
$$
\alpha(x)=\left\{
\begin{array}{ll}
1&\mbox{if $x=1$,}\\
0&\mbox{otherwise.}
\end{array}
\right.
$$
Note that this log structure is equivalent to
$Q\oplus k^{\times}\rightarrow k$.
We denote the log scheme obtained in this way by $(\mathop{\rm Spec}\nolimits k,Q)$.
The log scheme $(\mathop{\rm Spec}\nolimits k,Q)$ is called a
{\it logarithmic point}.
Especially, if $Q={\bf N}$, the logarithmic point $(\mathop{\rm Spec}\nolimits k,{\bf N})$ is said to be the
{\it standard log point}. }
\end{dfn}
\noindent
If $k$ is algebraically closed, any log structure on $\mathop{\rm Spec}\nolimits k$ is equivalent
to a log structure of logarithmic point (cf.\ \cite {Ill1}).
Note that if we set
$Q=\{1\}$, then the log structure of the logarithmic point induced by $Q$ is
the trivial log structure (Example \ref{trilog}).
\begin{exa}\label{toroex}{\rm
Let $P$ be a submonoid of a group $M={\bf Z}^d$ such that $\gp{P}=M$ and that
$P$ is
saturated. Let $Q$ be a submonoid of $P$, which is saturated but not
necessarily saturated in $P$.
We assume the following:
\begin{enumerate}
\item the monoid $Q$ has no invertible element other than 1,
\item the order of the torsion part of $M/\gp{Q}$ is invertible
in $k$.
\end{enumerate}
Let $R={\bf Z}[1/N]$ where $N$ is the order of the torsion part of
$M/\gp{Q}$.
The latter assumption implies, by Theorem \ref{lisse}, that
$(\mathop{\rm Spec}\nolimits R[P],P)\rightarrow(\mathop{\rm Spec}\nolimits R[Q],Q)$ (see Example \ref{canlog})
is log smooth. Define $\mathop{\rm Spec}\nolimits k\rightarrow\mathop{\rm Spec}\nolimits R[Q]$ by
$\alpha:Q\rightarrow k$ as in Definition \ref{logpt}. Let $X$ be
a scheme over $k$ which is smooth over $\mathop{\rm Spec}\nolimits k\times_{\mathop{\rm Spec}\nolimits R[Q]}\mathop{\rm Spec}\nolimits R[P]$.
Then we have a diagram
$$
\begin{array}{ccc}
X\\
\vphantom{\bigg|}\Big\downarrow\\
\mathop{\rm Spec}\nolimits k\times_{\mathop{\rm Spec}\nolimits R[Q]}\mathop{\rm Spec}\nolimits R[P]&\longrightarrow&\mathop{\rm Spec}\nolimits R[P]\\
\vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow\\
\mathop{\rm Spec}\nolimits k&\longrightarrow&\mathop{\rm Spec}\nolimits R[Q]\rlap{.}
\end{array}
$$
Define a log structure ${\cal M}$ on $X$ by the pull--back of the canonical log
structure on $\mathop{\rm Spec}\nolimits R[P]$.
Then we have a morphism
$$
f:(X,{\cal M})\longrightarrow(\mathop{\rm Spec}\nolimits k,Q)
$$
of fine saturated log schemes. This morphsim $f$
is log smooth by Proposition \ref{usulisse} and
Proposition \ref{bextlisse}.
We denote this log scheme $(X,{\cal M})$ simply by $(X,P)$. }
\end{exa}
\begin{exa}\label{toroidal}{\rm (Toric varieties.)
In this and the following examples, we use the notation appearing in
Example \ref{toroex}. Let $\sigma$ be a cone in $N_{{\bf R}}={\bf R}^d$ and
$\sigma^{\vee}$ be its dual cone in $M_{{\bf R}}={\bf R}^d$. Set
$P=M\cap\sigma^{\vee}$ and $Q=\{0\}\subset P$. Then,
$\mathop{\rm Spec}\nolimits k\times_{\mathop{\rm Spec}\nolimits {\bf Z}[Q]}\mathop{\rm Spec}\nolimits {\bf Z}[P]$ is $k$--isomorphic to $\mathop{\rm Spec}\nolimits k[P]$
which is nothing but an affine toric variety.
Let $X\rightarrow\mathop{\rm Spec}\nolimits k[P]$ be a smooth morphism.
Then $(X,P)\rightarrow\mathop{\rm Spec}\nolimits k$ is log smooth. }
\end{exa}
\begin{exa}\label{ssreduc}{\rm (Variety with normal crossings.)
Let $\sigma$ be the cone in $M_{{\bf R}}={\bf R}^d$ generated by $e_1,\ldots,e_d$, where
$e_i=(0,\ldots,0,1,0,\ldots 0)\,\mbox{($1$ at the $i$--th entry)}$,
$1\leq i\leq d$. Let $\tau$ be the subcone generated by
$a_1e_1+\cdots+a_de_d$ with positive integers $a_j$ for $j=1,\ldots,d$.
We assume that $\mbox{\rm GCD}(a_1,\ldots,a_d)(=N)$
is invertible in $k$. Set $R={\bf Z}[1/N]$.
Then, by setting $P=M\cap\sigma$ and $Q=M\cap\tau$, we see that
$\mathop{\rm Spec}\nolimits k\times_{\mathop{\rm Spec}\nolimits R[Q]}\mathop{\rm Spec}\nolimits R[P]$ is $k$--isomorphic to
$\mathop{\rm Spec}\nolimits k[z_1,\ldots,z_d]/(z_1^{a_1}\cdots z_d^{a_d})$ and $f$ is induced by
$$
\begin{array}{ccl}
{\bf N}^d&\longrightarrow&k[z_1,\ldots,z_d]/(z_1^{a_1}\cdots z_d^{a_d})\\
\llap{$\varphi$}\vphantom{\bigg|}\Big\uparrow&&\vphantom{\bigg|}\Big\uparrow\\
{\bf N}&\longrightarrow&k,
\end{array}
$$
where the morphism in the first row is defined by $e_i\mapsto z_i,\,
(1\leq i\leq d)$, and $\varphi$ is defined by
$\varphi(1)=a_1e_1+\cdots+a_de_d$.
Let $X\rightarrow \mathop{\rm Spec}\nolimits k[z_1,\ldots,z_d]/(z_1^{a_1}\cdots z_d^{a_d})$
be a smooth morphism.
Then, $(X,{\bf N}^d)\rightarrow(\mathop{\rm Spec}\nolimits k,{\bf N})$ is log smooth. }
\end{exa}
The following theorem is an application of Theorem \ref{lisse}.
We prove it in \S \ref{prf2}.
\begin{thm}\label{toroch}
Let $X$ be an algebraic scheme over a field $k$, and ${\cal M}\rightarrow\O_X$ a
fine saturated
log structure on $X$. Then, the log scheme $(X,{\cal M})$ is log smooth over
$\mathop{\rm Spec}\nolimits k$ with trivial log structure if and only if there exist an open
\'{e}tale covering ${\cal U}=\{U_i\}_{i\in I}$ of $X$
and a divisor $D$ of $X$
such that:
\begin{description}
\item[{\rm 1.}] there exists a smooth morphism
$$
h_i:U_i\longrightarrow V_i
$$
where $V_i$ is a affine toric variety over $k$ for each $i\in I$,
\item[{\rm 2.}] the divisor $U_i\cap D$ of $U_i$ is the pull--back of the
union of the closure of
codimension 1 torus orbits of $V_i$ by $h_i$ for each $i\in I$,
\item[{\rm 3.}] the log structure ${\cal M}\rightarrow\O_X$ is equivalent to the
log structure $\O_X\cap j_{\ast}\O^{\times}_{X-D}\hookrightarrow\O_X$ where
$j:X-D\hookrightarrow X$ is the inclusion.
\end{description}
\end{thm}
\begin{cor}\label{toroch1}
Let $X$ be a smooth algebraic variety over a field $k$, and ${\cal M}\rightarrow
\O_X$ a fine saturated log structure on $X$. Then, the log scheme
$(X,{\cal M})$ is log smooth over $\mathop{\rm Spec}\nolimits k$ with trivial log structure if and only
if there exists a reduced normal crossing divisor $D$ of $X$ such that
the log structure ${\cal M}\rightarrow\O_X$ is equivalent to the log structure
$\O_X\cap j_{\ast}\O^{\times}_{X-D}\hookrightarrow\O_X$ where
$j:X-D\hookrightarrow X$ is the inclusion.
\end{cor}
\section{Log differentials and log derivations}
In this section, we are going to discuss the log differentials and
log derivations. These objects are closely related with log smoothness,
and play important roles in the sequel.
To begin with, we introduce a
useful notation which we often use in the sequel. Let $(X,{\cal M})$ be a
log scheme. If we like to omit writing the log structure ${\cal M}$, we write
this log scheme by $\underline{X}$ to distinguish from the underlying scheme $X$.
\begin{dfn}\label{logder}
{\rm (cf.\ \cite {Kat1}, in different notation)
Let $\underline{X}=(X,{\cal M})$ and $\underline{Y}=(Y,{\cal N})$ be fine log schemes,
and $f=(f,\varphi):\underline{X}\rightarrow\underline{Y}$ a morphism, where
$\varphi:f^{-1}{\cal N}\rightarrow{\cal M}$ is a homomorphism of sheaves of monoids.
\begin{enumerate}
\item Let ${\cal E}$ be an $\O_X$--module.
The sheaf of {\it log derivations} $\mathop{{\cal D}er}\nolimits_{\underline{Y}}(\underline{X},{\cal E})$
of $\underline{X}$ to ${\cal E}$ over $\underline{Y}$
is the sheaf of germs of couples $(D,D{\rm log} )$, where
$D\in\mathop{{\cal D}er}\nolimits_Y(X,{\cal E})$ and $D{\rm log} :{\cal M}\rightarrow{\cal E}$,
such that the following conditions are satisfied:
\begin{description}
\item[{\rm (a)}] $D{\rm log} (ab)=D{\rm log} (a)+D{\rm log} (b),\mbox{ for}\,a,b\in{\cal M}$,
\item[{\rm (b)}] $\alpha(a)D{\rm log} (a)=D(\alpha(a)),\mbox{ for}\,a\in{\cal M}$,
\item[{\rm (c)}] $D{\rm log} (\varphi(c))=0,\mbox{ for}\,c\in f^{-1}{\cal N}$.
\end{description}
\item The sheaf of {\it log differentials} of $\underline{X}$ over $\underline{Y}$ is the
$\O_X$--module defined by
$$
\Omega^1_{\underline{X}/\underline{Y}}=
[\Omega^1_{X/Y}\oplus(\O_X\otimes_{{\bf Z}}\gp{{\cal M}})]/{\cal K},
$$
where ${\cal K}$ is the $\O_X$--submodule generated by
$$
(d\alpha(a),0)-(0,\alpha(a)\otimes a)\;\mbox{and}\;
(0,1\otimes\varphi(b)),
$$
for all $a\in{\cal M}$, $b\in f^{-1}{\cal N}$.
\end{enumerate}}
\end{dfn}
\noindent
These are coherent $\O_X$--modules if $Y$ is locally noetherian and $X$
locally of finite type over $Y$ (cf.\ \cite {Ill1}).
The proofs of the following three propositions are found in
\cite [\S 3]{Kat1}.
\begin{pro}
Let $\underline{X}$, $\underline{Y}$, $f$, and ${\cal E}$ be the same as in Definition \ref{logder}.
Then there is a natural isomorphism
$$
\mathop{{\cal H}om}\nolimits_{\O_X}(\Omega^1_{\underline{X}/\underline{Y}},{\cal E})
\stackrel{\sim}{\longrightarrow}
\mathop{{\cal D}er}\nolimits_{\underline{Y}}(\underline{X},{\cal E}),
$$
by $u\mapsto(u\circ d,u\circ d{\rm log} )$,
where $d$ and $d{\rm log}$ are defined by
$$
d:\O_X\rightarrow\Omega^1_{X/Y}\rightarrow\Omega^1_{\underline{X}/\underline{Y}}
$$
and
$$
d{\rm log} :{\cal M}\rightarrow\O_X\otimes_{{\bf Z}}\gp{{\cal M}}\rightarrow\Omega^1_{\underline{X}/\underline{Y}}.
$$
\end{pro}
\begin{pro}\label{genbun}
Let $\underline{X}\stackrel{f}{\rightarrow}\underline{Y}\stackrel{g}{\rightarrow}\underline{Z}$ be morphisms
of fine log schemes.
\begin{enumerate}
\item[{\rm 1.}] There exists an exact sequence
$$
f^{*}\Omega^1_{\underline{Y}/\underline{Z}}\rightarrow\Omega^1_{\underline{X}/\underline{Z}}
\rightarrow\Omega^1_{\underline{X}/\underline{Y}}\rightarrow 0.
$$
\item[{\rm 2.}] If $f$ is log smooth, then
\begin{equation}\label{diffseq}
0\rightarrow f^{*}\Omega^1_{\underline{Y}/\underline{Z}}\rightarrow\Omega^1_{\underline{X}/\underline{Z}}
\rightarrow\Omega^1_{\underline{X}/\underline{Y}}\rightarrow 0
\end{equation}
is exact.
\item[{\rm 3.}] If $g\circ f$ is log smooth and {\rm (\ref{diffseq})}
is exact and splits locally, then $f$ is log smooth.
\end{enumerate}
\end{pro}
\begin{pro}\label{bungen}
If $f:\underline{X}\rightarrow\underline{Y}$ is log smooth, then $\Omega^1_{\underline{X}/\underline{Y}}$ is a locally
free $\O_X$--module of finite type.
\end{pro}
\begin{exa}\label{fan1}{\rm (cf.\ \cite [Chap. 3, \S(3.1)]{Oda1})
Let $X_{\Sigma}$ be a toric variety over a field $k$
determined by a fan $\Sigma$ on $N_{{\bf R}}$ with
$N={\bf Z}^d$. Consider the log scheme $(X_{\Sigma},\Sigma)$
(Example \ref{torlog}) over $\mathop{\rm Spec}\nolimits k$.
Then we have isomorphisms of $O_X$--modules
$$
\mathop{{\cal D}er}\nolimits_k(\underline{X},\O_X)\cong\O_X\otimes_{{\bf Z}}N\; \; {\rm and}\; \;
\Omega^1_{\underline{X}/k}\cong\O_X\otimes_{{\bf Z}}M,
$$
where $M=\mathop{\rm Hom}\nolimits_{{\bf Z}}(N,{\bf Z})$. }
\end{exa}
\begin{exa}{\rm
For $X=\mathop{\rm Spec}\nolimits k[z_1,\ldots,z_{n}]/(z_1\cdots z_l)$,
let $f:(X,{\cal M})\rightarrow(\mathop{\rm Spec}\nolimits k,{\bf N}\rightarrow k)$ be the log smooth
morphism defined in Example \ref{ssreduc}. Then $\mathop{{\cal D}er}\nolimits_{\underline{k}}(\underline{X},\O_X)$
is a free $\O_X$--module generated by
$$
z_1\frac{\partial}{\partial z_1},\ldots,z_l\frac{\partial}{\partial z_l},
\frac{\partial}{\partial z_{l+1}},\ldots,\frac{\partial}{\partial z_n}
$$
with a relation
$$
z_1\frac{\partial}{\partial z_1}+\cdots+z_l\frac{\partial}{\partial z_l}=0.
$$
The sheaf $\Omega^1_{\underline{X}/\underline{k}}$ is a free $\O_X$--module generated by the
{\it logarithmic differentials}:
$$
\frac{dz_1}{z_1},\ldots,\frac{dz_l}{z_l},dz_{l+1},\ldots,dz_n
$$
with a relation
$$
\frac{dz_1}{z_1}+\cdots+\frac{dz_l}{z_l}=0.
$$
In the complex analytic case, the sheaf $\Omega^1_{\underline{X}/\underline{k}}$ is nothing but the
sheaf of {\it relative logarithmic differentials} introduced in, for
example, \cite [\S 3]{Fri1}, and \cite [\S 2]{K-N1}. }
\end{exa}
\section{The proof of Theorem 4.1}\label{prf1}
In this section, we give a proof of Theorem \ref{lisse} due to
Kato \cite {Kat1}.
Before proving the general case,
we prove the following proposition.
\begin{pro}\label{canlisse}
Let $A$ be a commutative ring and $h:Q\rightarrow P$ a homomorphism of
finitely generated integral monoids. The homomorphism $h$ induces the
morphism of log schemes
$$
f:\underline{X}=(\mathop{\rm Spec}\nolimits A[P],P)\longrightarrow\underline{Y}=(\mathop{\rm Spec}\nolimits A[Q],Q).
$$
We set $K=\mathop{\rm Ker}\nolimits(\gp{h}:\gp{Q}\rightarrow\gp{P})$ and
$C=\mathop{\rm Coker}\nolimits(\gp{h}:\gp{Q}\rightarrow\gp{P})$, and denote the torsion part of $C$
by $\tor{C}$. If both $K$ and $\tor{C}$ are finite groups of order invertible
in $A$, then $f$ is log smooth.
\end{pro}
\noindent{\sc Proof.}\hspace{2mm}
Suppose we have a commutative diagram
$$
\begin{array}{ccccc}
(T',\L')&\stackrel{s'}{\longrightarrow}&\underline{X}&=&(\mathop{\rm Spec}\nolimits A[P],P)\\
\llap{$t$}\vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow\rlap{$f$}\\
(T,\L)&\underrel{\longrightarrow}{s}&\underline{Y}&=&(\mathop{\rm Spec}\nolimits A[Q],Q)
\end{array}
$$
in ${\bf LSch}^{\rm f}$, where the morphism $t$ is a thickening of order 1.
Since we may work \'{e}tale locally, we may assume that $T$ is affine.
Set
$$
{\cal I}=\mathop{\rm Ker}\nolimits(\O_T\rightarrow \O_{T'}).
$$
Since the morphism $t$ is a thickening of order 1, by Lemma \ref{thick},
we have the following commutative diagram with exact rows:
$$
\begin{array}{ccccccccc}
1&\longrightarrow&1+{\cal I}&\lhook\joinrel\longrightarrow&\L&\stackrel{t^*}
{\longrightarrow}&
\L'&\longrightarrow&1\\
&&\parallel&&\cap&&\cap\\
1&\longrightarrow&1+{\cal I}&\longrightarrow&\gp{\L}&
\underrel{\longrightarrow}{\gp{(t^*)}}&\gp{\L'}&
\longrightarrow&1\rlap{.}
\end{array}
$$
\noindent
Note that the right square of the above commutative diagram is cartesian.
\vspace{3mm}
First, consider the following commutative diagram with exact rows:
$$
\begin{array}{ccccccccccc}
1&\longrightarrow&K&\longrightarrow&\gp{Q}
&\stackrel{\gp{h}}{\longrightarrow}&\gp{P}&\longrightarrow&C
&\longrightarrow&1\\
&&\vphantom{\bigg|}\Big\downarrow\rlap{$u$}&&\vphantom{\bigg|}\Big\downarrow\rlap{$v$}&&
\vphantom{\bigg|}\Big\downarrow\rlap{$w$}\\
1&\longrightarrow&1+{\cal I}&\longrightarrow&\gp{\L}
&\underrel{\longrightarrow}{\gp{(t^*)}}&\gp{\L'}&\longrightarrow
&1\rlap{.}
\end{array}
$$
The multiplicative monoid $1+{\cal I}$ is isomorphic to the additive monoid ${\cal I}$
by $1+x\mapsto x$ since ${\cal I}^2=0$. If the order of $K$ is invertible in $A$,
then we have $u=1$, and hence there exists a morphism $a':R\rightarrow
\gp{\L}$ with $R=\mbox{\rm Image}\,(\gp{h}:\gp{Q}\rightarrow\gp{P})$ such
that $a'\circ\gp{h}=v$ and $\gp{(t^*)}\circ a'=w$.
\vspace{3mm}
Next, we consider the following commutative diagram with exact rows:
$$
\begin{array}{ccccccccccc}
&&1&\longrightarrow&R&\stackrel{i}{\longrightarrow}&\gp{P}
&\longrightarrow&C&\longrightarrow&1\\
&&&&\llap{$a'$}\vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow\rlap{$w$}\\
0&\longrightarrow&{\cal I}&\longrightarrow&\gp{\L}
&\underrel{\longrightarrow}{\gp{(t^*)}}&\gp{\L'}&\longrightarrow
&1\rlap{.}
\end{array}
$$
We shall show that there exists a homomorphism $a'':\gp{P}\rightarrow\gp{\L}$
such that $a''\circ t=a'$ and $\gp{(t^*)}\circ a''=w$. The obstruction of
existence of $a''$ lies in $\mbox{\rm Ext}^1(C,{\cal I})$. In general, if a
positive integer $n$ is invertible in $A$ then we have
$\mbox{\rm Ext}^1({\bf Z}/n{\bf Z},{\cal I})=0$. Combining this
with $\mbox{\rm Ext}^1({\bf Z},{\cal I})=0$,
we have $\mbox{\rm Ext}^1(C,{\cal I})=0$ since the order of the torsion part of $C$
is invertible in $A$. Hence a homomorphism $a''$ exists.
Since the diagram
$$
\begin{array}{ccc}
\L&\stackrel{t^*}{\longrightarrow}&\L'\\
\cap&&\cap\\
\gp{\L}&\underrel{\longrightarrow}{\gp{(t^*)}}&\gp{\L'}\rlap{.}
\end{array}
$$
is cartesian, we found a
homomorphism
$$
a:P\longrightarrow\L
$$
such that $t^*\circ a=(s')^*$ and $a\circ h=s^*$.
Using this $a$, we can construct a
morphism of log schemes
$$
g:(T,\L)\longrightarrow\underline{X}=(\mathop{\rm Spec}\nolimits A[P],P)
$$
such that $g\circ t=s'$ and $s\circ g=f$. $\Box$
\vspace{3mm}
Now, let us prove Theorem 4.1. First, we prove the implication
$2\Rightarrow 1$. Let $R={\bf Z}[1/(N_1\cdot N_2)]$ where $N_1$ is the order of
$\mathop{\rm Ker}\nolimits(\gp{Q}\rightarrow\gp{P})$ and $N_2$ is the order of the torsion part
of $\mathop{\rm Coker}\nolimits(\gp{Q}\rightarrow\gp{P})$. By the assumption (a), we have
$$
Y\times_{\mathop{\rm Spec}\nolimits {\bf Z}[Q]}\mathop{\rm Spec}\nolimits {\bf Z}[P]\cong Y\times_{\mathop{\rm Spec}\nolimits R[Q]}\mathop{\rm Spec}\nolimits R[P].
$$
Since $X\rightarrow Y\times_{\mathop{\rm Spec}\nolimits R[Q]}\mathop{\rm Spec}\nolimits R[P]$ is smooth by (b),
$f$ is log smooth due to Proposition \ref{usulisse},
Proposition \ref{bextlisse} and Proposition \ref{canlisse}.
\vspace{3mm}
Next, let us prove the converse. Assume the morphism
$f$ is log smooth. Then, the sheaf $\Omega^1_{\underline{X}/\underline{Y}}$ is a locally free
$\O_X$--module of finite type (Proposition \ref{bungen}). Take any point
$x\in X$. We denote by $\bar{x}$ a separable closure of $x$.
\begin{ste}{\rm
Consider the morphism of $\O_X$--modules
$$
1\otimesd{\rm log}:\O_X\otimes_{{\bf Z}}\gp{{\cal M}}\longrightarrow
\Omega^1_{\underline{X}/\underline{Y}},
$$
which is surjective by the definition of $\Omega^1_{\underline{X}/\underline{Y}}$. Then we can take
elements $t_1,\ldots,t_r\in{\cal M}_{\bar{x}}$ such that the system
$\{d{\rm log} t_i\}_{1\leq i\leq r}$ is a $\O_{X,\bar{x}}$--base of
$\Omega^1_{\underline{X}/\underline{Y},\bar{x}}$.
Consider the homomorphism $\psi:{\bf N}^r\rightarrow{\cal M}_{\bar{x}}$ defined by
$$
{\bf N}^r\ni(n_1,\ldots,n_r)\mapsto t_1^{n_1}\cdots t_r^{n_r}\in{\cal M}_{\bar{x}}.
$$
Combining this $\psi$ with the homomorphism $Q\rightarrow f^{-1}({\cal N})_{\bar{x}}
\rightarrow{\cal M}_{\bar{x}}$, we have a homomorphism $\varphi:H={\bf N}^r\oplus
Q\rightarrow{\cal M}_{\bar{x}}$.
}
\end{ste}
\begin{ste}{\rm
Let $k(\bar{x})$ denote the residue field at $\bar{x}$. We have a homomorphism
\begin{equation}\label{bunchan}
k(\bar{x})\otimes_{{\bf Z}}{\bf Z}^r\longrightarrow k(\bar{x})\otimes_{{\bf Z}}
\mathop{\rm Coker}\nolimits(f^{-1}(\gp{{\cal N}}/{\cal O}^\times_Y)_{\bar{x}}\rightarrow\gp{{\cal M}}_{\bar{x}}/
{\cal O}^\times_{X,\bar{x}})
\end{equation}
by $k(\bar{x})\otimes_{{\bf Z}}\gp{\psi}:k(\bar{x})\otimes_{{\bf Z}}{\bf Z}^r\rightarrow
k(\bar{x})\otimes_{{\bf Z}}\gp{{\cal M}}_{\bar{x}}$ and canonical projectiones
$\gp{{\cal M}}_{\bar{x}}\rightarrow\gp{{\cal M}}_{\bar{x}}/{\cal O}^\times_{X,\bar{x}}\rightarrow
\mathop{\rm Coker}\nolimits(f^{-1}(\gp{{\cal N}}/{\cal O}^\times_Y)_{\bar{x}}\rightarrow\gp{{\cal M}}_{\bar{x}}/
{\cal O}^\times_{X,\bar{x}})$.
We claim that this morphism (\ref{bunchan}) is surjective.
In fact, this morphism coincides with the composite morphism
$$
k(\bar{x})\otimes_{{\bf Z}}{\bf Z}^r\rightarrow k(\bar{x})\otimes_{\O_{X,\bar{x}}}
\Omega^1_{\underline{X}/\underline{Y},\bar{x}}\rightarrow k(\bar{x})\otimes_{{\bf Z}}
\mathop{\rm Coker}\nolimits(f^{-1}(\gp{{\cal N}}/{\cal O}^\times_Y)_{\bar{x}}\rightarrow\gp{{\cal M}}_{\bar{x}}/
{\cal O}^\times_{X,\bar{x}}),
$$
where the first morphism is induced by $d{\rm log}\circ\psi$ and the second one
by the canonical projection, and these morphisms are clearly surjective.
Hence the morphism (\ref{bunchan}) is surjective.
On the other hand, the homomorphism
$$
\gp{Q}\longrightarrow f^{-1}({\cal N}/{\cal O}^\times_Y)_{\bar{x}}
$$
is surjective since $Q\rightarrow{\cal N}$ is a chart of ${\cal N}$. Hence, the
homomorphism
$$
k(\bar{x})\otimes_{{\bf Z}}\gp{Q}\longrightarrow k(\bar{x})\otimes_{{\bf Z}}
f^{-1}({\cal N}/{\cal O}^\times_Y)_{\bar{x}}
$$
is surjective, and then, the homomorphism
$$
1\otimes_{{\bf Z}}\gp{\varphi}:k(\bar{x})\otimes_{Z}\gp{H}\longrightarrow
k(\bar{x})\otimes_{{\bf Z}}({\cal M}_{\bar{x}}/{\cal O}^\times_{X,\bar{x}})
$$
is surjective.
This shows that the cokernel $C=\mathop{\rm Coker}\nolimits(\gp{\varphi}:\gp{H}\rightarrow
\gp{{\cal M}}_{\bar{x}}/{\cal O}^\times_{X,\bar{x}})$ is annihilated by an integer $N$
invertible in $\O_{X,\bar{x}}$.
}
\end{ste}
\begin{ste}{\rm
Take elements $a_1,\ldots,a_d\in\gp{{\cal M}}_{\bar{x}}$ which generates $C$. Then
we can write $a_i^n=u_i\varphi(b_i)$ for $u_i\in{\cal O}^\times_{X,\bar{x}}$ and
$b_i\in\gp{H}$, for $i=1,\ldots,d$. Since ${\cal O}^\times_{X,\bar{x}}$ is
$N$--divisible, we can write $u_i=v_i^N$ for $v_i\in{\cal O}^\times_{X,\bar{x}}$, for
$i=1,\ldots,d$, and hence we may suppose $a_i^N=\varphi(b_i)$,
replacing $a_i$ by $a_i/v_i$, for $i=1,\ldots,d$. Let $G$ be the push--out
of the diagram
$$
\gp{H}\longleftarrow{\bf Z}^d\longrightarrow{\bf Z}^d,
$$
where ${\bf Z}^d\rightarrow\gp{H}$ is defined by $e_i\mapsto b_i$,
and ${\bf Z}^d\rightarrow{\bf Z}^d$ is defined by $e_i\mapsto
Ne_i$ for $i=1,\ldots,d$. Then $\gp{\varphi}:
\gp{H}\rightarrow\gp{{\cal M}}_{\bar{x}}$ and ${\bf Z}^d\rightarrow\gp{{\cal M}}_{\bar{x}}$,
defined by $e_i\mapsto a_i$ for $i=1,\ldots,d$, induce the homomorphism
$$
\phi:G\longrightarrow\gp{{\cal M}}_{\bar{x}}
$$
which maps $G$ surjectively onto $\gp{{\cal M}}_{\bar{x}}/{\cal O}^\times_{X,\bar{x}}$.
Define $P=\phi^{-1}({\cal M}_{\bar{x}})$, then $P$ defines a chart of ${\cal M}$ on some
neighborhood of $\bar{x}$ (\cite [Lemma 2.10]{Kat1}).
If ${\cal M}$ is saturated, $P$ is also saturated.
There exists an induced map $Q\rightarrow P$ which defines a chart of $f$ on
some neighborhood of $\bar{x}$.
Since $\gp{H}\rightarrow\gp{P}$ is injective, $\gp{Q}\rightarrow\gp{P}$ is
injective. The cokernel $\mathop{\rm Coker}\nolimits(\gp{H}\rightarrow\gp{P})$ is annihilated
by $N$, hence $\tor{\mathop{\rm Coker}\nolimits(\gp{Q}\rightarrow\gp{P})}$ is finite and
annihilated by $N$.
}
\end{ste}
\begin{ste}{\rm
Set $X'=Y\times_{\mathop{\rm Spec}\nolimits {\bf Z}[Q]}\mathop{\rm Spec}\nolimits {\bf Z}[P]$ and $g:X\rightarrow X'$. We need to
show that the morphism $g$ is smooth in the usual sense.
Since $\underline{X}$ has the log structure induced by $g$ from $\underline{X}'=(X',P)$, it suffice
to show that $g$ is log smooth (Proposition \ref{usulisse}). Since
$k(\bar{x})\otimes_{{\bf Z}}(\gp{P}/\gp{Q})\cong k(\bar{x})\otimes_{{\bf Z}}{\bf Z}^d\cong
k(\bar{x})\otimes_{\O_{X,\bar{x}}}\Omega^1_{\underline{X}/\underline{Y},\bar{x}}$ and
$\Omega^1_{\underline{X}/\underline{Y}}$ is locally free, we have $\Omega^1_{\underline{X}/\underline{Y}}\cong
\O_X\otimes_{{\bf Z}}(\gp{P}/\gp{Q})$ on some neighborhood of $\bar{x}$.
On the other hand, by direct calculations, one sees that
$\Omega^1_{\underline{X}'/\underline{Y}}\cong\O_{X'}\otimes_{{\bf Z}[P]}\Omega^1_{{\bf Z}[P]/{\bf Z}[Q]}\cong
\O_{X'}\otimes_{{\bf Z}}(\gp{P}/\gp{Q})$. Hence we have $g^*\Omega^1_{\underline{X}'/\underline{Y}}
\cong\Omega^1_{\underline{X}/\underline{Y}}$. This implies $g$ is log smooth due to Proposition
\ref{genbun} (in fact, $g$ is {\it log \'{e}tale} (cf.\ \cite{Kat1})).
}
\end{ste}
\noindent
This completes the proof of the theorem. $\Box$
\section{The proof of Theorem 4.7}\label{prf2}
In this section, we give a proof of Theorem \ref{toroch}.
If $V=\mathop{\rm Spec}\nolimits k[P]$ is an affine toric variety, then it is easy to
see that the log structure associated to $P\rightarrow k[P]$ is equivalent
to the log structure $\O_X\cap j_{\ast}\O^{\times}_{V-D}\hookrightarrow
\O_X$ where $D$ is the union of the closure of
codimention 1 torus orbits of $V$ and
$j:V-D\hookrightarrow V$ is the inclusion. Hence, the ``if'' part of
Theorem \ref{toroch} is easy to see. Let us prove the converse.
Let $(X,{\cal M})$ be as in the assumption of Theorem \ref{toroch} and
$f:(X,{\cal M})\rightarrow\mathop{\rm Spec}\nolimits k$ the structure morphism.
The key--lemma is the following.
\begin{lem}\label{keylem1}
We can take \'{e}tale locally a chart $P\rightarrow{\cal M}$ of ${\cal M}$
such that
\begin{description}
\item[{\rm 1.}] the chart $(P\rightarrow{\cal M},1\rightarrow k^{\times},
1\rightarrow P)$ of $f$ satisfies the conditions {\rm (a)} and {\rm (b)} in
Theorem \ref{lisse},
\item[{\rm 2.}] $P$ is a finitely generated integral saturated monoid, and
has no torsion element.
\end{description}
Here, by a torsion element, we mean an element $x\neq 1$ such that
$x^n=1$ for some positive integer $n$.
\end{lem}
First, we are going to show that the theorem follows from the above lemma.
Since the monoid $P$ has no torsion element, $P$ is the saturated submonoid of
a finitely generated free abelian group $\gp{P}$. Hence, $X$ is \'{e}tale
locally smooth over affine toric varieties, and the log structure ${\cal M}$ on $X$
is \'{e}tale locally equivalent to the pull--back of the log structure
induced by the union of the closure of
codimension 1 torus orbits. Since, these log structure
glue to the log structure ${\cal M}$ on $X$, the pull--back of the union
of the closure of codimension 1
torus orbits glue to a divisor on $X$. In fact,
this divisor is the compliment of the
largest open subset $U$ such that ${\cal M}|_U$ is trivial with the reduced scheme
structure.
Hence our assertion is proved.
\vspace{3mm}
Now, we are going to prove Lemma \ref{keylem1}. We may work \'{e}tale locally.
Take a chart $(P\rightarrow{\cal M},1\rightarrow k^{\times},1\rightarrow P)$ of
$f$ as in Theorem \ref{lisse}. We may assume that $P$ is saturated.
Define
$$
\tor{P}=\{x\in P\: |\: x^n=1\,(\mbox{for some $n$})\}.
$$
$\tor{P}$ is a subgroup in $P$.
Take a decomposition $\gp{P}=\fr{G}\oplus\tor{G}$ of the finitely
generated abelian group $\gp{P}$, where
$\fr{G}$ (resp.\ $\tor{G}$) is a free (resp.\ torsion) subgroup of
$\gp{P}$. Then we have the equalities $\tor{P}=P\cap\tor{G}=\tor{G}$ since
$P$ is saturated. Define a submonoid $\fr{P}$ by $\fr{P}=P\cap\fr{G}$.
\begin{cla}
$P=\fr{P}\oplus\tor{P}$.
\end{cla}
\noindent{\sc Proof.}\hspace{2mm}
Take $x\in P$. Decompose $x=yz$ in $\gp{P}$ such that $y\in\fr{G}$ and
$z\in\tor{G}=\tor{P}$. Since $y^n=(xz^{-1})^n=x^n\in P$ for a large $n$,
we have $y\in P$. Hence $y\in\fr{P}$.
$\Box$
\vspace{3mm}\noindent
Define $\fr{\alpha}:\fr{P}\rightarrow\O_X$ by
$\fr{P}\hookrightarrow P\stackrel{\alpha}{\rightarrow}\O_X$.
\begin{cla}
The homomorphism
$\fr{\alpha}:\fr{P}\rightarrow\O_X$ defines a log structure
equivalent to ${\cal M}$.
\end{cla}
\noindent{\sc Proof.}\hspace{2mm}
If $x\in\tor{P}$, then $\alpha(x)\in{\cal O}^\times_X$ since
$\alpha(x)^n=1$ for a large $n$.
Hence $\alpha(\tor{P})\subset{\cal O}^\times_X$. This implies that the associated log
structure of $\fr{P}$ is equivalent to that of $P$. $\Box$
\vspace{3mm}\noindent
Hence, the morphism $f$ is equivalent to the morphsim induced by the diagram
$$
\begin{array}{ccc}
\fr{P}&\stackrel{\fr{\alpha}}{\longrightarrow}&\O_X\\
\llap{$\fr{\varphi}$}\vphantom{\bigg|}\Big\uparrow&&\vphantom{\bigg|}\Big\uparrow\\
1&\underrel{\longrightarrow}{\lambda}&k,
\end{array}
$$
Then we have to check the conditions (a) and (b) in
Theorem \ref{lisse}. The condition (a) is easy to verify.
Let us check the condition (b). We need to show that the morphism
$$
X\longrightarrow\mathop{\rm Spec}\nolimits k[\fr{P}]
$$
induced by $X\rightarrow\mathop{\rm Spec}\nolimits{\bf Z}[P]\rightarrow\mathop{\rm Spec}\nolimits{\bf Z}[\fr{P}]$ is smooth.
\begin{cla}
The morphism
\begin{equation}\label{sepext}
\mathop{\rm Spec}\nolimits k[P]\longrightarrow
\mathop{\rm Spec}\nolimits k[\fr{P}]
\end{equation}
induced by $\fr{P}\hookrightarrow P$ is \'{e}tale.
\end{cla}
\noindent{\sc Proof.}\hspace{2mm}
Since $P=\fr{P}\oplus\tor{P}$, we have
$k[P]=k[\fr{P}]\otimes_{k}k[\tor{P}]$.
Since every element
in $\tor{P}$ is roots of 1, and the order of $\tor{P}$ is
invertible in $k$, the morphism
$$
k\lhook\joinrel\longrightarrow k[\tor{P}]
$$
is a finite separable extension of the field $k$.
This shows that the morphism (\ref{sepext}) is \'{e}tale. $\Box$
\vspace{3mm}\noindent
Now we have proved Lemma \ref{keylem1}, and hence, Theorem \ref{toroch}.
\section{Formulation of log smooth deformation}
{}From now on, we fix the following notation. Let $k$ be a field and
$Q$ a finitely generated integral saturated monoid having no
invertible element other than 1.
Then we have a logarithmic point (Definition \ref{logpt}) $\underline{k}=(\mathop{\rm Spec}\nolimits k, Q)$.
Let $f=(f,\varphi):\underline{X}=(X,{\cal M})\rightarrow\underline{k}=(\mathop{\rm Spec}\nolimits k,Q)$ be
a log smooth morphism in ${\bf LSch}^{\rm fs}$.
\vspace{3mm}
Let $\Lambda$ be a complete noetherian local ring with the residue field $k$.
For example, $\Lambda=k$ or $\Lambda=\mbox{the Witt vector ring of $k$}$
when $k$ is perfect.
We denote, by $\Lambda[[Q]]$, the completion of the monoid ring
$\Lambda[Q]$ along
the maximal ideal $\mu+\Lambda[Q\setminus\{1\}]$ where $\mu$ denotes the
maximal ideal of $\Lambda$. The completion $\Lambda[[Q]]$ is a complete local
$\Lambda$--algebra and is noetherian since $Q$ is finitely generated.
If the monoid $Q$ is isomorphic to ${\bf N}$ then the ring $\Lambda[[Q]]$ is
isomorphic to $\Lambda[[t]]$ as local
$\Lambda$--algebras.
Let ${\cal C}_{\Lambda[[Q]]}$ be the category of artinian local $\Lambda[[Q]]$--algebras with
the residue field $k$, and $\widehat{{\cal C}}_{\Lambda[[Q]]}$ be the category of pro--objects
of ${\cal C}_{\Lambda[[Q]]}$ (cf.\ \cite {Sch1}).
For $A\in\mathop{\rm Obj}\nolimits(\widehat{{\cal C}}_{\Lambda[[Q]]})$, we define a log structure on the scheme $\mathop{\rm Spec}\nolimits A$ by
the associated log structure
$$
Q\oplus A^{\times}\longrightarrow A
$$
of the homomorphism $Q\rightarrow\Lambda[[Q]]\rightarrow A$.
We denote, by $(\mathop{\rm Spec}\nolimits A, Q)$, the log scheme obtained in this way.
\begin{dfn}{\rm
For $A\in\mathop{\rm Obj}\nolimits({\cal C}_{\Lambda[[Q]]})$, a {\it log smooth lifting} of
$f:(X,{\cal M})\rightarrow(\mathop{\rm Spec}\nolimits k,Q)$ on $A$ is a morphism
$\widetilde{f}:(\widetilde{X},\widetilde{{\cal M}})\rightarrow(\mathop{\rm Spec}\nolimits A,Q)$
in ${\bf LSch}^{\rm fs}$ together with a cartesian diagram
$$
\begin{array}{ccc}
(X,{\cal M})&\longrightarrow&(\widetilde{X},\widetilde{\M})\\
\llap{$f$}\vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow\rlap{$\widetilde{f}$}\\
(\mathop{\rm Spec}\nolimits k,Q)&\longrightarrow&(\mathop{\rm Spec}\nolimits A,Q)
\end{array}
$$
in ${\bf LSch}^{\rm fs}$.
Two liftings are said to be {\it isomorphic} if they are isomorphic in
${\bf LSch}^{\rm fs}_{(\mathop{\rm Spec}\nolimits A,Q)}$. }
\end{dfn}
\noindent
Note that $(\mathop{\rm Spec}\nolimits k,Q)\rightarrow(\mathop{\rm Spec}\nolimits A,Q)$
is an exact closed immersion, and hence,
the above diagram is cartesian in ${\bf LSch}^{\rm fs}$ if and only if so is in ${\bf LSch}$
(Lemma \ref{fpro}).
In particular, the underlying morphisms of log smooth liftings are
(not necessarily flat) liftings in the usual sense.
Moreover, since exact closed immersions are stable under base changes,
$(X,{\cal M})\rightarrow(\widetilde{X},\widetilde{\M})$ is also an exact closed immersion.
If either $Q=\{1\}$ or $Q={\bf N}$, the underlying morphisms of any
log smooth liftings of $f$ are flat since these morphisms of log schemes are
{\it integral} (cf.\ \cite {Kat1}).
Hence, in this case, the underlying morphisms
of log smooth liftings of $f$ are flat liftings of $f$.
\vspace{3mm}
Take a local chart $(P\rightarrow {\cal M}, Q\rightarrow Q\oplus k^{\times},
Q\rightarrow P)$ of $f$
extending the given $Q\rightarrow k$ as in Theorem \ref{lisse} such that
$\gp{Q}\rightarrow\gp{P}$ is injective (Remark \ref{lisserem}). Then, $f$
factors throught $\mathop{\rm Spec}\nolimits k\times_{\mathop{\rm Spec}\nolimits {\bf Z}[Q]}\mathop{\rm Spec}\nolimits {\bf Z}[P]$ by the smooth
morphism $X\rightarrow\mathop{\rm Spec}\nolimits k\times_{\mathop{\rm Spec}\nolimits {\bf Z}[Q]}\mathop{\rm Spec}\nolimits {\bf Z}[P]$ and the
natural projection \'{e}tale locally. For $A\in\mathop{\rm Obj}\nolimits({\cal C}_{\Lambda[[Q]]})$, a smooth lifting
\begin{equation}\label{loclift}
\widetilde{X}\longrightarrow\mathop{\rm Spec}\nolimits A\times_{\mathop{\rm Spec}\nolimits {\bf Z}[Q]}\mathop{\rm Spec}\nolimits {\bf Z}[P]
\end{equation}
of $X\rightarrow\mathop{\rm Spec}\nolimits k\times_{\mathop{\rm Spec}\nolimits {\bf Z}[Q]}\mathop{\rm Spec}\nolimits {\bf Z}[P]$, with the naturally
induced log structure, gives a local log smooth lifting of $f$. Note that
this local lifting $(\widetilde{X},\widetilde{\M})\rightarrow
(\mathop{\rm Spec}\nolimits A,Q)$ is log smooth.
Conversely, suppose $\widetilde{f}:(\widetilde{X},\widetilde{\M})\rightarrow
(\mathop{\rm Spec}\nolimits A,Q)$ is a local log smooth lifting of $f$ on $A$.
\begin{lem}\label{liftlem1}
The local chart $(P\rightarrow {\cal M}, Q\rightarrow Q\oplus k^{\times},
Q\rightarrow P)$ of $f$
lifts to the local chart
$(P\rightarrow\widetilde{\M}, Q\rightarrow Q\oplus A^{\times}, Q\rightarrow P)$ of $\widetilde{f}$.
\end{lem}
\noindent{\sc Proof.}\hspace{2mm}
The proof is done by the induction with respect to the length of
$A$. Take $A'\in\mathop{\rm Obj}\nolimits({\cal C}_{\Lambda[[Q]]})$ with surjective morphism
$A\rightarrow A'$ such that $I=\mathop{\rm Ker}\nolimits(A\rightarrow A')\neq 0$ and $I^2=0$.
Let $\widetilde{f}':(\widetilde{X}',\widetilde{\M}')\rightarrow
(\mathop{\rm Spec}\nolimits A',Q)$ be a pull--back of $\widetilde{f}$. Then, $\widetilde{f}'$ is a
log smooth lifting of $f$ to $A'$. By the induction, we have the lifted
local chart $(P\rightarrow\widetilde{\M}', Q\rightarrow Q\oplus A'^{\times},
Q\rightarrow P)$
of $\widetilde{f}'$. Since $(\widetilde{X}',\widetilde{\M}')\rightarrow
(\widetilde{X},\widetilde{\M})$ is a thickening of order 1, by Lemma
\ref{thick}, we have the
following commutative diagram with exact rows:
$$
\begin{array}{ccccccccc}
0&\rightarrow&{\cal I}&\hookrightarrow&\widetilde{\M}&
\rightarrow&\widetilde{\M}'&\rightarrow&1\\
&&\parallel&&\cap&&\cap\\
0&\rightarrow&{\cal I}&\rightarrow&\gp{\widetilde{\M}}&
\rightarrow&\gp{\widetilde{\M}'}&\rightarrow&1\rlap{,}
\end{array}
$$
where ${\cal I}=\mathop{\rm Ker}\nolimits(\O_{\widetilde{X}}\rightarrow\O_{\widetilde{X}'})$.
The right square of this diagram is cartesian. Consider the
following commutative diagram of abelian groups with exact rows and columns:
$$
\begin{array}{ccccccc}
\mathop{\rm Hom}\nolimits(\gp{P},{\cal I})&\rightarrow&\mathop{\rm Hom}\nolimits(\gp{P},\gp{\widetilde{\M}})&\rightarrow&
\mathop{\rm Hom}\nolimits(\gp{P},\gp{\widetilde{\M}'})&\rightarrow&\mathop{\rm Ext}\nolimits^1(\gp{P},{\cal I})\\
\vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow\\
\mathop{\rm Hom}\nolimits(\gp{Q},{\cal I})&\rightarrow&\mathop{\rm Hom}\nolimits(\gp{Q},\gp{\widetilde{\M}})&\rightarrow&
\mathop{\rm Hom}\nolimits(\gp{Q},\gp{\widetilde{\M}'})\\
\vphantom{\bigg|}\Big\downarrow\\
\mathop{\rm Ext}\nolimits^1(C,{\cal I})\rlap{,}
\end{array}
$$
where $C=\mathop{\rm Coker}\nolimits(\gp{Q}\hookrightarrow\gp{P})$. By our assumption, we have
$\mathop{\rm Ext}\nolimits^1(\gp{P},{\cal I})=0$ and $\mathop{\rm Ext}\nolimits^1(C,{\cal I})=0$, since the order of the
torsion part of each $\gp{P}$ and $C$ is invertible in $A$.
Then, by an easy diagram
chasing, we can show that the given morphism $\gp{P}\rightarrow
\gp{\widetilde{\M}'}$ can be lifted to a morphism $\gp{P}\rightarrow
\gp{\widetilde{\M}}$ such that
$$
\begin{array}{ccc}
\gp{P}&\longrightarrow&\gp{\widetilde{\M}}\\
\vphantom{\bigg|}\Big\uparrow&&\vphantom{\bigg|}\Big\uparrow\\
\gp{Q}&\longrightarrow&\gp{Q}\oplus A^{\times}
\end{array}
$$
is commutative. Then, we have the morphism $P\rightarrow\widetilde{\M}$ and
we see that the diagram
$$
\begin{array}{ccc}
\P&\longrightarrow&\widetilde{\M}\\
\vphantom{\bigg|}\Big\uparrow&&\vphantom{\bigg|}\Big\uparrow\\
Q&\longrightarrow&Q\oplus A^{\times}
\end{array}
$$
is commutative. Since $\widetilde{\M}/{\cal O}^\times_{\widetilde{X}}
\stackrel{\sim}{\rightarrow}\widetilde{\M}'/{\cal O}^\times_{\widetilde{X}'}$, we can
easily show that the morphism $P\rightarrow\widetilde{\M}$ defines a chart
(Lemma \ref {basic3}).
$\Box$
\vspace{3mm}
\noindent
Then, $\widetilde{f}$ is factors throught $\mathop{\rm Spec}\nolimits A\times_{\mathop{\rm Spec}\nolimits {\bf Z}[Q]}\mathop{\rm Spec}\nolimits {\bf Z}[P]$
by the induced morphism
$\widetilde{X}\rightarrow\mathop{\rm Spec}\nolimits A\times_{\mathop{\rm Spec}\nolimits {\bf Z}[Q]}
\mathop{\rm Spec}\nolimits {\bf Z}[P]$ and the natural projection,
and we have the following commutative diagram
$$
\begin{array}{ccc}
X&\longrightarrow&\widetilde{X}\\
\vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow\\
\mathop{\rm Spec}\nolimits k\times_{\mathop{\rm Spec}\nolimits {\bf Z}[Q]}\mathop{\rm Spec}\nolimits {\bf Z}[P]&\longrightarrow&
\mathop{\rm Spec}\nolimits A\times_{\mathop{\rm Spec}\nolimits {\bf Z}[Q]}\mathop{\rm Spec}\nolimits {\bf Z}[P]\\
\vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow\\
\mathop{\rm Spec}\nolimits k&\longrightarrow&\mathop{\rm Spec}\nolimits A\rlap{,}
\end{array}
$$
such that the each square is cartesian. Hence,
$\widetilde{X}\rightarrow\mathop{\rm Spec}\nolimits A\times_{\mathop{\rm Spec}\nolimits {\bf Z}[Q]}
\mathop{\rm Spec}\nolimits {\bf Z}[P]$ is smooth since it is a smooth lifting of the smooth morphism
$X\rightarrow\mathop{\rm Spec}\nolimits k\times_{\mathop{\rm Spec}\nolimits {\bf Z}[Q]}\mathop{\rm Spec}\nolimits {\bf Z}[P]$, and is unique
up to isomorphisms by the classical theory. Therefore, we
have proved the following proposition.
\begin{pro}\label{liftloc}{\rm (cf. \cite [(3.14)]{Kat1})}
For $A\in{\cal C}_{\Lambda[[Q]]}$, a log smooth lifting of
$f:(X,{\cal M})\rightarrow(\mathop{\rm Spec}\nolimits k,Q)$ on $A$
exists \'{e}tale locally, and is unique up to isomorphisms. In particular,
log smooth liftings are log smooth.
\end{pro}
Let $\widetilde{f}:(\widetilde{X},\widetilde{\M})\rightarrow(\mathop{\rm Spec}\nolimits A,Q)$ be a log smooth lifting of $f$ to
$A$, and $u:A'\rightarrow A$ a surjective homomorphism in ${\cal C}_{\Lambda[[Q]]}$ such that
$I^2=0$ where $I=\mathop{\rm Ker}\nolimits (u)$. Suppose $\widetilde{f}':(\widetilde{X}',\widetilde{\M}')\rightarrow(\mathop{\rm Spec}\nolimits A',Q)$
is a log smooth lifting of $f$ to $A'$ which is also a lifting of $\widetilde{f}$.
Let $(P\rightarrow\widetilde{\M}', Q\rightarrow Q\oplus A'^{\times}, Q\rightarrow P)$ be
a local chart of $\widetilde{f}'$ which is a lifting of $(P\rightarrow{\cal M}, Q\rightarrow
Q\oplus k^{\times}, Q\rightarrow P)$. Define a local chart $(P\rightarrow\widetilde{\M},
Q\rightarrow Q\oplus A^{\times}, Q\rightarrow P)$ of $\widetilde{f}$ by $P\rightarrow{\cal M}'
\rightarrow{\cal M}$ and $Q\rightarrow Q\oplus A'^{\times}\rightarrow Q\oplus
A^{\times}$. An automorphism $\Theta:(\widetilde{X}',\widetilde{\M}')\stackrel{\sim}{\rightarrow}
(\widetilde{X}',\widetilde{\M}')$ over $(\mathop{\rm Spec}\nolimits A',Q)$ which is identity on $(\widetilde{X},\widetilde{\M})$ induces
an automorphism $\theta:\gp{\widetilde{\M}'}\stackrel{\sim}{\rightarrow}\gp{\widetilde{\M}'}$.
Consider the diagram
$$
\begin{array}{ccccccccc}
&&&&\gp{P}&=&\gp{P}\\
&&&&\llap{$\alpha'$}\vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow\rlap{$\alpha$}\\
1&\rightarrow&1+{\cal I}&\rightarrow&\gp{\widetilde{\M}'}&\rightarrow&\gp{\widetilde{\M}}&
\rightarrow&1.
\end{array}
$$
For $a\in\gp{P}$, the element
$\alpha'(a)\cdot[\theta\circ\alpha'(a)]^{-1}$ is in
$1+{\cal I}$. Then, we have a morphism $\Delta:\gp{P}\rightarrow{\cal I}=I\cdot\O_{\widetilde{X}'}
\cong I\otimes_{A}\O_{\widetilde{X}}$ by
$\Delta(a)=\alpha'(a)\cdot[\theta\circ\alpha'(a)]^{-1}-1$.
The morphism $\Delta$
lifts to the morphism $\Delta:\gp{P}/\gp{Q}\rightarrow I\otimes_{A}\O_{\widetilde{X}}$
and defines a morphism of $\O_{\widetilde{X}}$--modules
$$
\O_{\widetilde{X}}\otimes_{{\bf Z}}(\gp{P}/\gp{Q})\rightarrow I\otimes_{A}\O_{\widetilde{X}}.
$$
Since $\Omega^1_{\underline{\widetilde{X}}/\underline{A}}\cong\O_{\widetilde{X}}\otimes_{{\bf Z}}(\gp{P}/\gp{Q})$ \'{e}tale
locally, this defines a local section of
$$
\mathop{{\cal H}om}\nolimits_{\O_{\widetilde{X}}}(\Omega^1_{\underline{\widetilde{X}}/\underline{A}},I\otimes_{A}\O_{\widetilde{X}})\cong
\mathop{{\cal D}er}\nolimits(\underline{\widetilde{X}},\O_{\widetilde{X}})\otimes_{A}I.
$$
Conversely, for a local section
$(D,D{\rm log})\in\mathop{{\cal D}er}\nolimits(\underline{\widetilde{X}},\O_{\widetilde{X}})\otimes_{A}I$,
$D$ induces an automorphism of $O_{\widetilde{X}'}$ and $D{\rm log}$ induces an
automorphism of $\widetilde{\M}'$, and then, indues an automorphism of $(\widetilde{X}',\widetilde{\M}')$.
By this, applying the arguement in SGA I \cite{Gro1} Expos\'{e} 3, we get
the following proposition.
\begin{pro}\label{liftauto}{\rm (cf. \cite [(3.14)]{Kat1})}
Let $\widetilde{f}:(\widetilde{X},\widetilde{\M})\rightarrow(\mathop{\rm Spec}\nolimits A,Q)$ be a
log smooth lifting of $f$ to $A$, and $u:A'\rightarrow A$ a surjective
homomorphism in ${\cal C}_{\Lambda[[Q]]}$ such that $I^2=0$ where $I=\mathop{\rm Ker}\nolimits (u)$
(i.e., $(\mathop{\rm Spec}\nolimits A,Q)\rightarrow(\mathop{\rm Spec}\nolimits A',Q)$ is a thickening of order
$\leq 1$).
\begin{enumerate}
\item The sheaf of germs of lifting automorphisms of $\widetilde{X}$ to
$A'$ is
$$
\mathop{{\cal D}er}\nolimits_{\underline{A}}(\underline{\widetilde{X}},\O_{\widetilde{X}})\otimes_A I.
$$
\item The set of isomorphism classes of log smooth liftings of
$\widetilde{X}$ to $A'$ is isomorphic to
$$
\mbox{\rm H}^1(\widetilde{X},\mathop{{\cal D}er}\nolimits_{\underline{A}}(\underline{\widetilde{X}},\O_{\widetilde{X}}))\otimes_A I.
$$
\item The lifting obstructions of $\widetilde{X}$ to $A'$ are in
$$
\mbox{\rm H}^2(\widetilde{X},\mathop{{\cal D}er}\nolimits_{\underline{A}}(\underline{\widetilde{X}},\O_{\widetilde{X}}))\otimes_A I.
$$
\end{enumerate}
\end{pro}
Define the {\it log smooth deformation functor} ${\bf LD}={\bf LD}_{\underline{X}/\underline{k}}$ by
$$
{\bf LD}_{\underline{X}/\underline{k}}(A)=\{\mbox{isomorphism class of log smooth lifting of
$f:\underline{X}\rightarrow\underline{k}$ on $A$}\}
$$
for $A\in\mathop{\rm Obj}\nolimits({\cal C}_{\Lambda[[Q]]}$).
This is a covariant functor from ${\cal C}_{\Lambda[[Q]]}$ to $\mbox{\bf Ens}$, the category of sets,
such that ${\bf LD}_{\underline{X}/\underline{k}}(k)$
consists of one point.
We shall prove the following theorem in the next section.
\begin{thm}\label{hull}
If the underlying scheme $X$ is proper over $k$,
then the log deformation functor ${\bf LD}_{\underline{X}/\underline{k}}$ has a
representable hull {\rm (cf.\ \cite {Sch1})}.
\end{thm}
\section{The proof of Theorem 8.5}
We are going to prove Theorem \ref{hull} by checking M. Schlessinger's
criterion (\cite [Theorem 2.11]{Sch1}) for ${\bf LD}$. Let
$u_1:A_1\rightarrow A_0$ and $u_2:A_2\rightarrow A_0$
be morphisms in ${\cal C}_{\Lambda[[Q]]}$. Consider the map
\begin{equation}\label{desant}
{\bf LD}(A_1\times_{A_0}A_2)\longrightarrow{\bf LD}(A_1)\times_{{\bf LD}(A_0)}{\bf LD}(A_2).
\end{equation}
Then we shall check the following conditions.
\begin{description}
\item[(H1)] The map (\ref{desant}) is a surjection whenever
$u_2:A_2\rightarrow A_0$ is a surjection.
\item[(H2)] The map (\ref{desant}) is a bijection when $A_0=k$ and
$A_2=k[\epsilon]$, where $k[\epsilon]=k[E]/(E^2)$.
\item[(H3)] $\mbox{dim}_k(t_{{\bf LD}})<\infty$,
where $t_{{\bf LD}}={\bf LD}(k[\epsilon])$.
\end{description}
By Proposition \ref{liftauto}, we have
$$
t_{{\bf LD}}\cong\mbox{\rm H}^1(X,\mathop{{\cal D}er}\nolimits_{\underline{k}}(\underline{X},\O_X)).
$$
Our assumption implies that $t_{{\bf LD}}$ is finite dimensional since
$\mathop{{\cal D}er}\nolimits_{\underline{k}}(\underline{X},\O_X)$ is a coherent $\O_X$--module.
Hence we need to check (H1) and (H2).
Set $B=A_1\times_{A_0}A_2$. Let $v_i:B\rightarrow A_i$ be the natural map
for $i=1,2$. We denote the morphisms of schemes associated
to $u_i$ and $v_i$ also by $u_i:\mathop{\rm Spec}\nolimits A_0\rightarrow\mathop{\rm Spec}\nolimits A_i$ and
$v_i:\mathop{\rm Spec}\nolimits A_i\rightarrow\mathop{\rm Spec}\nolimits B$ for $i=1,2$, respectively.
\vspace{3mm}
\noindent{\sc Proof of (H1).}\hspace{2mm}
Suppose the homomorphism $u_2:A_2\rightarrow A_0$ is surjective.
Take an element
$(\eta_1,\eta_2)\in{\bf LD}(A_1)\times_{{\bf LD}(A_0)}{\bf LD}(A_2)$ where $\eta_i$ is
an isomorphism class of a log smooth lifting
$f_i:(X_i,{\cal M}_i)\rightarrow(\mathop{\rm Spec}\nolimits A_i,Q)$ for each $i=1,2$.
The equality
${\bf LD}(u_1)(\eta_1)={\bf LD}(u_2)(\eta_2)(=\eta_0)$ implies that there exists an
isomorphism
$(u_2)^*(X_2,{\cal M}_2)\stackrel{\sim}{\rightarrow}(u_1)^*(X_1,{\cal M}_1)$
over $(\mathop{\rm Spec}\nolimits A_0,Q)$.
Here, $(u_i)^*(X_i,{\cal M}_i)$ is the pull--back of $(X_i,{\cal M}_i)$ by
$u_i:\mathop{\rm Spec}\nolimits A_0\rightarrow\mathop{\rm Spec}\nolimits A_i$ for $i=1,2$.
Set $(X_0,{\cal M}_0)=(u_1)^*(X_1,{\cal M}_1)$. We denote the induced morphism of
log schemes $(X_0, {\cal M}_0)\rightarrow(X_i, {\cal M}_i)$ by $\underline{u}_i$ for
$i=1,2$. Then we have the following commutative diagram:
$$
\begin{array}{ccccc}
(X_1,{\cal M}_1)&\stackrel{\underline{u}_1}{\longleftarrow}&(X_0,{\cal M}_0)&
\stackrel{\underline{u}_2}{\longrightarrow}&(X_2,{\cal M}_2)\\
\llap{$f_1$}\vphantom{\bigg|}\Big\downarrow&&\llap{$f_0$}\vphantom{\bigg|}\Big\downarrow&&
\vphantom{\bigg|}\Big\downarrow\rlap{$f_2$}\\
(\mathop{\rm Spec}\nolimits A_1,Q)&\stackrel{u_1}{\longleftarrow}&(\mathop{\rm Spec}\nolimits A_0,Q)&
\stackrel{u_2}{\longrightarrow}&(\mathop{\rm Spec}\nolimits A_2,Q).
\end{array}
$$
We have to find an element $\xi\in{\bf LD}(B)$, which represents a lifting of $f$
to $B$, such that ${\bf LD}(v_i)(\xi)=\eta_i$ for $i=1,2$. Consider a scheme
$Z=(|X|,\O_{X_1}\times_{\O_{X_0}}\O_{X_2})$ over $\mathop{\rm Spec}\nolimits B$. Define a log
structure on $Z$ by the natural homomorphism
$$
{\cal N}={\cal M}_1\times_{{\cal M}_0}{\cal M}_2\longrightarrow
\O_Z=\O_{X_1}\times_{\O_{X_0}}\O_{X_2}.
$$
It is easy to verify that this homomorphism is a log structure. Since the
diagram
$$
\begin{array}{ccc}
{\cal N}&\longrightarrow&\O_Z\\
\vphantom{\bigg|}\Big\uparrow&&\vphantom{\bigg|}\Big\uparrow\\
\llap{$Q\cong$\hspace{1mm}}Q\times_{Q}Q&\longrightarrow&B
\end{array}
$$
is commutative, we have a morphism $g:(Z,{\cal N})\rightarrow(\mathop{\rm Spec}\nolimits B,Q)$ of log
schemes. By the construction, we have the morphism
$v_i:(X_i,{\cal M}_i)\rightarrow (Z,{\cal N})$ for $i=1,2$ such that the diagram
$$
\begin{array}{ccc}
(X_1,{\cal M}_1)&\stackrel{\underline{v}_1}{\longrightarrow}&(Z,{\cal N})\\
\llap{$\underline{u}_1$}\vphantom{\bigg|}\Big\uparrow&&\vphantom{\bigg|}\Big\uparrow\rlap{$\underline{v}_2$}\\
(X_0,{\cal M}_0)&\underrel{\longrightarrow}{\underline{u}_2}&(X_2,{\cal M}_2)
\end{array}
$$
is commutative. Since $u_2:A_2\rightarrow A_0$ is surjective, the underlying
morphism $X_1\rightarrow Z$ of $\underline{v}_1$ is a closed immersion in the
classical sense. We have to show that the morphism $\underline{v}_1$ is an
exact closed immersion. Take a local chart $(P\rightarrow{\cal M}, Q\rightarrow
Q\oplus k^{\times},
Q\rightarrow P)$ of $f$ as in Theorem \ref{lisse} such that $\gp{Q}
\rightarrow\gp{P}$ is injective. By Lemma \ref{liftlem1}, this local chart
lifts to the local chart of $f_i$ for each $i=0,1,2$. Since
$u_2:A_2\rightarrow A_0$ is surjective, we have the following isomorphism
$$
{\cal N}/{\cal O}^\times_Z\stackrel{\sim}{\rightarrow}
({\cal M}_1/{\cal O}^\times_{X_1})\times_{({\cal M}_0/{\cal O}^\times_{X_0})}({\cal M}_2/{\cal O}^\times_{X_2}).
$$
By this, one sees that $P\cong P\times_{P}P\rightarrow{\cal N}$ is a local chart
of ${\cal N}$. This shows that $(Z,{\cal N})$ is a fine saturated log scheme, and
$\underline{v}_1$ is an exact closed immersion.
Then, the exact closed immersion $(X,{\cal M})\rightarrow (X_1,{\cal M}_1)\rightarrow
(Z,{\cal N})$ gives $g$ a structure of log smooth lifting of $f$ to $(\mathop{\rm Spec}\nolimits B,Q)$.
Hence, $g$ represents an element $\xi\in{\bf LD}(B)$.
It is easy to verify that ${\bf LD}(v_i)(\xi)=\eta_i$ for $i=1,2$ since the morphism
$f_1$,$f_2$ and $g$ have the common local chart.
Thus (H1) is now proved.
$\Box$
\vspace{3mm}
\noindent{\sc Proof of (H2).}\hspace{2mm}
We continue to use the same notation as above. First, we prepare the following
lemma.
\begin{lem}\label{basic4}
Let $g':(Z',{\cal N}')\rightarrow (\mathop{\rm Spec}\nolimits B,Q)$ be a log smooth lifting of $f$ with
a commutative diagram
$$
\begin{array}{ccc}
(X_1,{\cal M}_1)&\longrightarrow&(Z',{\cal N}')\\
\llap{$\underline{u}_1$}\vphantom{\bigg|}\Big\uparrow&&\vphantom{\bigg|}\Big\uparrow\\
(X_0,{\cal M}_0)&\underrel{\longrightarrow}{\underline{u}_2}&(X_2,{\cal M}_2)
\end{array}
$$
of liftings such that $(v_i)^{*}(Z',{\cal N}')\stackrel{\sim}{\leftarrow}
(X_i,{\cal M}_i)$ over $(\mathop{\rm Spec}\nolimits A_i,Q)$ for $i=1,2$. Then, the natural morphism
$(Z,{\cal N})\rightarrow(Z',{\cal N}')$ is an isomorphism.
\end{lem}
\noindent{\sc Proof.}\hspace{2mm}
We may work \'{e}tale locally. By Lemma \ref{liftlem1}, the local chart
$(P\rightarrow{\cal M}, Q\rightarrow Q\oplus k^{\times}, Q\rightarrow P)$
of $f$ lifts to a local
chart of $g'$. Take a local chart $(P\rightarrow{\cal N}, Q\rightarrow Q\oplus
B^{\times},
Q\rightarrow P)$ of $g$ by $P\rightarrow{\cal N}'\rightarrow{\cal N}$.
Then, the schemes $Z$ and $Z'$ are smooth liftings of
$X\rightarrow\mathop{\rm Spec}\nolimits k\times_{\mathop{\rm Spec}\nolimits {\bf Z}[Q]}\mathop{\rm Spec}\nolimits {\bf Z}[P]$ to
$\mathop{\rm Spec}\nolimits B\times_{\mathop{\rm Spec}\nolimits {\bf Z}[Q]}\mathop{\rm Spec}\nolimits {\bf Z}[P]$. Hence, we have only to show that
the natural morphism $Z\rightarrow Z'$ of underlying schemes is an
isomorphism. But this follows from the classical theory
\cite [Corollary 3.6]{Sch1} since each $X_i$ is a smooth lifting of
$X\rightarrow\mathop{\rm Spec}\nolimits k\times_{\mathop{\rm Spec}\nolimits {\bf Z}[Q]}\mathop{\rm Spec}\nolimits {\bf Z}[P]$ to
$\mathop{\rm Spec}\nolimits A_i\times_{\mathop{\rm Spec}\nolimits {\bf Z}[Q]}\mathop{\rm Spec}\nolimits {\bf Z}[P]$ for $i=0,1,2$.
$\Box$
\vspace{3mm}
\noindent
Let $g':(Z',{\cal N}')\rightarrow (\mathop{\rm Spec}\nolimits B,Q)$ be a log smooth lifting of $f$
which represents a class $\xi'\in{\bf LD}(B)$. Suppose that the class $\xi'$ is
mapped to $(\eta_1,\eta_2)$ by (\ref {desant}). Then,
$$
(X_0,{\cal M}_0)\stackrel{\sim}{\rightarrow}
(v_1\circ u_1)_{*}(Z',{\cal N}')\cong (v_2\circ u_2)_{*}(Z',{\cal N}')
\stackrel{\sim}{\leftarrow}(X_0,{\cal M}_0)
$$
defines an automorphism $\theta$ of the lifting $(X_0,{\cal M}_0)$.
If this automorphism
$\theta$ lifts to an automorphism $\theta'$ of the lifting $(X_1,{\cal M}_1)$
such that $\theta'\circ \underline{u}_1=\underline{u}_1\circ\theta'$, then,
replacing $(X_1,{\cal M}_1)\rightarrow(Z',{\cal N}')$ by
$(X_1,{\cal M}_1)\stackrel{\theta'}{\rightarrow}(X_1,{\cal M}_1)\rightarrow(Z',{\cal N}')$,
we have a commutative diagram as in Lemma \ref{basic4}, and then, we have
$\xi=\xi'$. Now if $A=k$ (so that $(X_0,{\cal M}_0)=(X,{\cal M})$, $\theta={\rm id}$),
$\theta'$ exists.
Thus (H2) is proved.
$\Box$
\vspace{3mm}\noindent
Thus we have proved Theorem \ref{hull}.
\section{Example 1: Log smooth deformation over trivial base}
\label{exam1}
As we have seen in Theorem \ref{toroch}, any log scheme $(X,{\cal M})$ which is
log smooth over $\mathop{\rm Spec}\nolimits k$ with the trivial log structure is smooth over
an affine torus embedding \'{e}tale locally.
\begin{exa}{\rm (Usual smooth deformations.)
Let $X$ be a smooth algebraic variety over a field $k$.
Then $X$ with the trivial
log structure is log smooth over $\mathop{\rm Spec}\nolimits k$
(this is the case $D=0$ in Corollary \ref{toroch1}), and our log smooth
deformation of $X$ is nothing but the usual smooth deformation of $X$. }
\end{exa}
\begin{exa}{\rm (Generalized relative deformations.)
Let $X$ be an algebraic variety over a field $k$.
Assume that there exists a fine saturated log
structure ${\cal M}$ on $X$ such that $f:(X,{\cal M})\rightarrow\mathop{\rm Spec}\nolimits k$ is log
smooth. Then, by Theorem \ref{toroch}, $X$ is covered by \'{e}tale open
sets which are smooth over affine toric varieties, and the log structure
${\cal M}\rightarrow\O_X$ is equivalent to the log structure defined by
\begin{equation}\label{logdiv}
{\cal M}=j_{\ast}{\cal O}^\times_{X-D}\cap\O_X
\end{equation}
for some divisor $D$ of $X$,
where $j$ is the inclusion $X-D\hookrightarrow X$.
In this situation, our log smooth deformation
of $f$ is the deformation of the pair $(X,D)$.
If $X$ is smooth over $k$, then $D$ is a reduced normal crossing
divisor (Corollary \ref{toroch1}).
Assume $X$ is smooth over $k$, then we have the exact sequence
$$
0\rightarrow\mathop{{\cal D}er}\nolimits_k(\underline{X},\O_X)\rightarrow\mathop{{\cal D}er}\nolimits_k(X,\O_X)(=\Theta_X)
\rightarrow{\cal N}\rightarrow 0,
$$
where ${\cal N}$ is an $\O_X$--module locally written by
$$
({\cal N}_{D_1|X}\otimes\O_{D_1})\oplus\cdots\oplus
({\cal N}_{D_d|X}\otimes\O_{D_d}),
$$
where $D_1,\ldots,D_d$ are local components of $D$, and ${\cal N}_{D_i|X}$
is the normal bundle of $D_i$ in $X$ for $i=1,\ldots,d$.
Then, we have an exact sequence
$$
\mbox{\rm H}^0(D,{\cal N})\rightarrow t_{{\bf LD}}\rightarrow
\mbox{\rm H}^1(X,\Theta_X)\rightarrow
\mbox{\rm H}^1(D,{\cal N}).
$$
In this sequence, $\mbox{\rm H}^0(D,{\cal N})$ is viewed as the set of
isomorphism classes of locally trivial deformations of $D$ in $X$, and
$\mbox{\rm H}^1(D,{\cal N})$ is viewed as a set of obstructions of
deformations
of $D$ in $X$. Hence this sequence explains
the relation between the log smooth
deformation and the usual smooth deformation. Note that, if $D$ is a
smooth divisor on $X$, the log smooth deformation is nothing but the
{\it relative deformation} of the pair $(X,D)$ studied by Makio
\cite {Mak1}. }
\end{exa}
\begin{exa}{\rm (Toric varieties.)
Let $X_{\Sigma}$ be a complete toric variety
over a field $k$ defined by a fan $\Sigma$ in $N_{{\bf R}}$,
and consider the log scheme $\underline{X}=(X_{\Sigma},\Sigma)$
(Example \ref{torlog}) over $\mathop{\rm Spec}\nolimits k$. We have seen in Example \ref{fan1} that
$$
\mathop{{\cal D}er}\nolimits_k(\underline{X},\O_X)\cong\O_X\otimes_{{\bf Z}}N,
$$
and hence is a globally free $\O_X$--module. Since
$\mbox{\rm H}^1(X,\mathop{{\cal D}er}\nolimits_k(\underline{X},\O_X))=0$, any toric varieties are
infinitesimally rigid with respect to our log smooth deformation.
Note that toric varieties without log structures are not necessarily rigid
with respect to the usual smooth deformation. }
\end{exa}
\section{Example 2: Smoothings of normal crossing varieties}
\label{exam2}
Let $k$ be a field. A {\it normal crossing variety} over $k$ is a seperated,
connected, and geometrically reduced scheme $X$ of finite type over $k$ which
is covered by an \'{e}tale open covering $\{X_\lambda\}_{\lambda\in\Lambda}$ such that
each $X_{\lambda}$ is isomorphic to $\mathop{\rm Spec}\nolimits k[z_1,\ldots,z_n]/(z_1\cdots z_{d_\lambda})$
over $k$ where $n-1={\rm dim}_k X$. Let $(\mathop{\rm Spec}\nolimits k,{\bf N})$ be a standard log point
(Definition \ref{logpt}).
\begin{dfn}{\rm (cf. \cite[(2.6)]{Kaj1})\
A log structure $\alpha:{\cal M}\rightarrow\O_X$ on a normal crossing variety $X$
is called a log structure of {\it semistable type} over $(\mathop{\rm Spec}\nolimits k,{\bf N})$ if
the following conditions are satisfied:
\begin{enumerate}
\item there exists an \'{e}tale open covering $\{X_\lambda\}_{\lambda\in\Lambda}$ of $X$
such that, for each $\lambda\in\Lambda$, $X_\lambda$ is isomorphic to
$\mathop{\rm Spec}\nolimits k[z_1,\ldots,z_n]/(z_1\cdots z_{d_\lambda})$ over $k$ and the log
structure ${\cal M}_\lambda={\cal M}|_{X_\lambda}$ has a chart $\beta_{\lambda}:{\bf N}^{d_\lambda}
\rightarrow{\cal M}_{\lambda}$ such that $\alpha\circ\beta_{\lambda}(e_i)=z_i$ for
$i=1,\ldots, d_{\lambda}$, where $e_i=(0,\ldots,0,1,0,\ldots,0)$ ($1$ at
the $i$--th entry),
\item there exists a morphism $f:(X,{\cal M})\rightarrow (\mathop{\rm Spec}\nolimits k,{\bf N})$ which has
a local chart $(\beta_{\lambda}:{\bf N}^{d_\lambda}\rightarrow{\cal M},
{\bf N}\rightarrow{\bf N}\oplus
k^{\times}, \varphi_{\lambda}:{\bf N}\rightarrow{\bf N}^{d_{\lambda}})$ where
$\varphi_{\lambda}$ is the diagonal homomorphism, for each $\lambda\in\Lambda$.
\end{enumerate}
}
\end{dfn}
\vspace{3mm}
\noindent
The morphism $f:(X,{\cal M})\rightarrow (\mathop{\rm Spec}\nolimits k,{\bf N})$ defined as above is called a
{\it logarithmic semistable reduction}. Note that a logarithmic semistable
reduction is log smooth (Example \ref{ssreduc}).
Let $f:(X,{\cal M})\rightarrow(\mathop{\rm Spec}\nolimits k,{\bf N})$ be a logarithmic semistable reduction.
Then, \'{e}tale locally, $X$ is isomorphic to
$k[z_1,\ldots,z_n]/(z_1\cdots z_d)$, and $f$ is induced by
the diagram
$$
\begin{array}{ccl}
{\bf N}^d&\longrightarrow&k[z_1,\ldots,z_n]/(z_1\cdots z_d)\\
\llap{$\varphi$}\vphantom{\bigg|}\Big\uparrow&&\vphantom{\bigg|}\Big\uparrow\\
{\bf N}&\longrightarrow&k.
\end{array}
$$
For $A\in\mathop{\rm Obj}\nolimits({\cal C}_{\Lambda[[{\bf N}]]})$, the log structure on $A$ is
defined by $\gamma:{\bf N}\rightarrow A$, $\gamma(1)=\pi\in m_A$, where $m_A$
is the maximal ideal of $A$. Then the log smooth lifting of $f$ on $A$
is locally equivalent to the morphism induced by the diagram
$$
\begin{array}{ccl}
{\bf N}^d&\longrightarrow&A[z_1,\ldots,z_n]/(z_1\cdots z_d-\pi)\\
\llap{$\varphi$}\vphantom{\bigg|}\Big\uparrow&&\vphantom{\bigg|}\Big\uparrow\\
{\bf N}&\underrel{\longrightarrow}{\gamma}&A.
\end{array}
$$
Hence, our log smooth deformation carries out the smoothing of the normal
crossing variety $X$.
\vspace{3mm}
Now we assume that the singular locus $X_{\mbox{sing}}=D$ of
$X$ is connected. Then our deformation theory coincides with the
{\it logarithmic deformation theory of normal crossing varieties} introduced
by Kawamata and Namikawa \cite {K-N1} in the complex analytic
situation. The following theorem is essentially due to Kawamata and
Namikawa \cite {K-N1}, and we prove it in the next section.
\begin{thm}\label{dss}
Let $X$ be a normal crossing variety over $k$. Then, $X$ has a
log structure of semistable type over $(\mathop{\rm Spec}\nolimits k,{\bf N})$
if and only if $X$ is $d$--semistable, i.e.,
${\cal E}xt^1(\Omega^1_X,\O_X)\cong\O_D$ {\rm (cf.\ \cite {Fri1})}.
\end{thm}
\section{The proof of Theorem 11.2}
Let $X$ be a normal crossing variety over a field $k$ of dimension $n$.
Then the scheme $X$ is covered by an \'{e}tale open covering
$\{X_{\lambda}\}_{\lambda\in\Lambda}$ such that each $X_{\lambda}$ is isomorphic to the
divisor in the $(n+1)$--affine space over $k$, $V_\lambda=\mathop{\rm Spec}\nolimits k[z^{\ssd{\la}}_0,\ldots,
z^{\ssd{\la}}_n]\cong \mbox{\bf A}^{(n+1)}_k$, defined by $z^{\ssd{\la}}_0\cdots z^{\ssd{\la}}_{d_\lambda}=0$
for some $0\leq d_\lambda \leq n$.
We call this covering $\U=\{X_\la\hookrightarrow V_\la\}$ a {\it coordinate covering} of $X$.
If $\U=\{X_\la\hookrightarrow V_\la\}$ is a coordinate covering, the {\it singular locus} of
$X$, denoted by $D$, is the closed subscheme of $X$ defined locally by
the equations $z^{\ssd{\la}}_0\cdotsz^{\ssd{\la}}_{i-1}\cdotz^{\ssd{\la}}_{i+1}\cdotsz^{\ssd{\la}}_{d_\lambda}=0$ for
$0\leq i \leq d_\lambda$ in $V_\lambda$.
We write the defining ideals of $X_\lambda$ and $D$
in $V_\lambda$ as
$$
\begin{array}{ccl}
{\cal I}_{X_\lambda} &=& (z^{\ssd{\la}}_0\cdots z^{\ssd{\la}}_{d_\lambda}),\vspace{2mm} \\
{\cal I}_{D_\lambda} &=& (z^{\ssd{\la}}_0\cdotsz^{\ssd{\la}}_{i-1}\cdotz^{\ssd{\la}}_{i+1}\cdots
z^{\ssd{\la}}_{d_\lambda}\mid 0\leq i \leq d_\lambda).
\end{array}
$$
These are ideals in $\O_{V_\lambda}$. In the sequel,
we fix these notation and conventions, sometimes omitting the
index $\lambda$.
\vspace{3mm}
Let us describe ${\cal T}^1_X={\cal E}xt^1_{\O_X}(\Omega^1_X, \O_X)$
locally. Consider the following exact sequence,
$$
0\rightarrow{\cal I}_X/{\cal I}^2_X\rightarrow\Omega^1_V\otimes\O_X
\rightarrow\Omega^1_X\rightarrow 0.
$$
Here, we omitted the index $\lambda$. Its dual is
$$
0\rightarrow{\cal T}^0_X\rightarrow\Theta_V\otimes\O_X
\rightarrow{\cal N}_{X\mid V}.
$$
Then, we have an isomorphism ${\cal T}^1_X\cong
\mathop{\rm Coker}\nolimits(\Theta_V\otimes\O_X\rightarrow{\cal N}_{X\mid V})$.
\begin{lem}\label{locT1}
${\cal T}^1_X$ is an invertible $\O_D$--module. More precisely,
we have an isomorphism of $\O_D$--modules
$$
{\cal T}^1_X\cong({\cal I}_X/{\cal I}^2_X)^\vee\otimes\O_D.
$$
\end{lem}
\noindent{\sc Proof.}\hspace{2mm}
First, note that
${\cal I}_X/{\cal I}^2_X=\O_X\cdot d(z_0\cdots z_d)$, and
$\Omega^1_V\otimes\O_X=\oplus^n_{i=1}\O_X\cdot dz_i$.
Here, we omited the index $\lambda$. Let us denote
the inclusion ${\cal I}_X/{\cal I}^2_X\hookrightarrow\Omega^1_V\otimes\O_X$ by
$\iota$.
For each $f\in\Theta_V\otimes\O_X=
{\cal H}om_{\O_X}(\Omega^1_V\otimes\O_X, \O_X)$,
set $f_i=f(dz_i)\in\O_X$. Then, $\iota^* f(d(z_0\cdots z_d))=
\sum^d_{i=0}f_i z_0\cdots z_{i-1}\cdot z_{i+1}\cdots z_d\in{\cal I}_D\otimes\O_X$.
Hence, $\mbox{\rm Image}\,\iota^*\subseteq{\cal H}om_{\O_X}({\cal I}_X/{\cal I}^2_X, {\cal I}_D\otimes\O_X)$.
Easy to see the converse. Then, we have $\mbox{\rm Image}\,\iota^*=
{\cal H}om_{\O_X}({\cal I}_X/{\cal I}^2_X, {\cal I}_D\otimes\O_X)$.
Hence, we get
${\cal T}^1_X\cong\mathop{\rm Coker}\nolimits\iota^*\cong{\cal H}om_{\O_X}({\cal I}_X/{\cal I}^2_X, \O_D)
\cong({\cal I}_X/{\cal I}^2_X)^\vee\otimes\O_D$.
$\Box$
\vspace{3mm}
Let $X$ be a reduced normal crossing variety over a field $k$
and $\U=\{X_\la\hookrightarrow V_\la\}$ a
coordinate covering of $X$.
A system $\{(\zeta^{\ssd{\la}}_0,\ldots,\zeta^{\ssd{\la}}_n)\}_{\lambda\in\Lambda}$, where $\zeta^{\ssd{\la}}_i\in
{\rm H}^0(X_\lambda, \O_X)$ for $\lambda\in\Lambda$ and $i=0,\ldots,n$,
is said to be a {\it log system} on $X$ with respect to $\U=\{X_\la\hookrightarrow V_\la\}$ if
the following conditions are satisfied:
\begin{enumerate}
\item for $0\leq i\leq d_\lambda$, we have
$
\zeta^{\ssd{\la}}_i=u^{\ssd{\lambda}}_i\cdotz^{\ssd{\la}}_i
$
for some $u^{\ssd{\lambda}}_i\in{\rm H}^0(X_\lambda, \O^\times_X)$,
\item for $d_\lambda <j\leq n$, $\zeta^{\ssd{\la}}_j$ is invertible
on $X_\lambda$.
\end{enumerate}
Then we have the following lemma.
\begin{lem}\label{ltrns}
On each $X_{\lambda\mu}=X_\lambda\cap X_\mu\neq\emptyset$,
there exists a transition relation such as
\begin{equation}\label{trans}
\zeta^{\ssd{\la}}_i=u^{\ssd{\la\mu}}_i\zeta^{\ssd{\mu}}_{\sigma^{\ssd{\la\mu}}(i)}\ (0\leq i\leq n),
\end{equation}
for some invertible function $u^{\ssd{\la\mu}}_i$ on $X_{\lambda\mu}$ and a permutation
$\sigma^{\ssd{\la\mu}}$.
{\rm (As is seen in the following proof, these $u^{\ssd{\la\mu}}_i$'s
and $\sigma^{\ssd{\la\mu}}$'s are {\it not} unique. )}
\end{lem}
\noindent{\sc Proof.}\hspace{2mm}
Set $E^{\ssd{\lambda}}=\{i\mid(\zeta^{\ssd{\la}}_i
\mid_{X_{\lambda\mu}})\not\in{\rm H}^0(X_{\lambda\mu}, \O^\times_X)\}$,
and set $E^{\ssd{\mu}}$ similarly.
There is a one to one correspondence between elements in
$E^{\ssd{\lambda}}$ and components of $X_{\lambda\mu}$ by
$i\leftrightarrow\{\zeta^{\ssd{\la}}_i=0\}$.
This implies that there exists a bijection
$\sigma: E^{\ssd{\lambda}}\stackrel{\sim}{\rightarrow}E^{\ssd{\mu}}$
such that $\{\zeta^{\ssd{\la}}_i=0\}=\{\zeta^{\ssd{\mu}}_{\sigma(i)}=0\}$.
Hence, for $i\in E^{\ssd{\lambda}}$, we can write $\zeta^{\ssd{\la}}_i
=u^{\ssd{\la\mu}}_i\cdot\zeta^{\ssd{\mu}}_{\sigma(i)}$ for some
$u^{\ssd{\la\mu}}_i\in{\rm H}^0(X_{\lambda\mu}, \O^\times_X)$.
On the other hand, if $i$ is not in $E^{\ssd{\lambda}}$,
we can write as (\ref{trans}) for these $i$'s,
since $\zeta^{\ssd{\la}}_i$ is invertible on $X_{\lambda\mu}$
(note that the number of
these $i$'s coincides with that of those $j$'s which are not in
$E^{\ssd{\mu}}$).
$\Box$
\begin{pro}\label{cridss}{\rm (cf.\ \cite {K-N1})}
The following conditions are equivalent.
\begin{enumerate}
\item $X$ is $d$--semistable.
\item There exist a log system $\{(\zeta^{\ssd{\la}}_0,\ldots,\zeta^{\ssd{\la}}_n)\}$
on $X$ and its transition system $\{(u^{\ssd{\la\mu}}_i, \sigma^{\ssd{\la\mu}})\}$
as in Lemma \ref{ltrns} such that the equality
\begin{equation}\label{cdss}
u^{\ssd{\la\mu}}_0\cdotsu^{\ssd{\la\mu}}_n=1
\end{equation}
holds on each $X_\lambda\cap X_\mu\neq\emptyset$.
\end{enumerate}
\end{pro}
\noindent{\sc Proof.}\hspace{2mm}
($1\Rightarrow 2$) Assume that the normal crossing variety
$X$ is $d$--semistable. Due to Lemma \ref{locT1}, the invertible
$\O_X$--module
$({\cal I}_X/{\cal I}^2_X)\otimes\O_D\cong{\cal I}_X/{\cal I}_X{\cal I}_D$ is trivial, i.e.,
$$
{\cal I}_X/{\cal I}_X{\cal I}_D\quad\cong\quad\O_D.
$$
Let us denote the natural projection
${\cal I}_X/{\cal I}^2_X\rightarrow{\cal I}_X/{\cal I}_X{\cal I}_D$
by $p$. The sheaf ${\cal I}_X/{\cal I}_X{\cal I}_D$ is a free $\O_D$-module and
$p(z_0\cdots z_d)$ is a $\O_D$-free base of it. By the above isomorphism,
the global section $1\in\O_D$ correspondes to $p(v\cdot z_0\cdots z_d)$,
for some invertible function $v$ on $X$.
Set
$$
\zeta_i=\left\{
\begin{array}{ll}
v\cdot z_0&(i=0), \\
z_i&(1\leq i\leq d), \\
1&(d<i\leq n).
\end{array}
\right.
$$
Then, the system $\{(\zeta^{\ssd{\la}}_0,\ldots,\zeta^{\ssd{\la}}_n)\}$ is a log system
on $X$.
Due to Lemma \ref{ltrns}, we can take a transition system
$\{(u^{\ssd{\la\mu}}_i, \sigma^{\ssd{\la\mu}})\}$ such that the transition relation
(\ref{trans}) holds.
On each $X_{\lambda\mu}\neq\emptyset$, we have $p_\lambda(\zeta^{\ssd{\la}}_0\cdots\zeta^{\ssd{\la}}_n)=
p_\mu(\zeta^{\ssd{\mu}}_0\cdots\zeta^{\ssd{\mu}}_n)$.
Then, we have $u^{\ssd{\la\mu}}_0\cdotsu^{\ssd{\la\mu}}_n p_\mu(\zeta^{\ssd{\mu}}_0\cdots\zeta^{\ssd{\mu}}_n)=
p_\mu(\zeta^{\ssd{\mu}}_0\cdots\zeta^{\ssd{\mu}}_n)$.
Since, $p_\mu(v^{\ssd{\mu}}z^{\ssd{\mu}}_0\cdotsz^{\ssd{\mu}}_{d_{\mu}})$
is a $\O_D$--free base of ${\cal I}_X/{\cal I}_X{\cal I}_D$, we have
$u^{\ssd{\la\mu}}_0\cdotsu^{\ssd{\la\mu}}_n=1$ on $D$.
Hence, we can write
$$
u^{\ssd{\la\mu}}_0\cdotsu^{\ssd{\la\mu}}_n=1+\sum^n_{j=0}a_j
z^{\ssd{\mu}}_0\cdotsz^{\ssd{\mu}}_{j-1}\cdotz^{\ssd{\mu}}_{j+1}\cdotsz^{\ssd{\mu}}_n.
$$
Replacing every $u^{\ssd{\la\mu}}_j$ with
$$
u^{\ssd{\la\mu}}_j-a_jz^{\ssd{\mu}}_0\cdotsz^{\ssd{\mu}}_{j-1}\cdotz^{\ssd{\mu}}_{j+1}\cdotsz^{\ssd{\mu}}_n
(u^{\ssd{\la\mu}}_0\cdotsu^{\ssd{\la\mu}}_{j-1}\cdotu^{\ssd{\la\mu}}_{j+1}\cdotsu^{\ssd{\la\mu}}_n)^{-1},
$$
we get $u^{\ssd{\la\mu}}_0\cdotsu^{\ssd{\la\mu}}_n=1$ on $X_{\lambda\mu}$ as desired.
\vspace{3mm}
($2\Rightarrow 1$) The local section
$\zeta^{\ssd{\la}}_0\otimes\cdots\otimes\zeta^{\ssd{\la}}_n$ is a local generator
of $({\cal I}_X/{\cal I}^2_X)\otimes\O_D$.
By (\ref{cdss}), these local
generators glue to a global section. Hence, the invertible
$\O_D$--module $({\cal I}_X/{\cal I}^2_X)\otimes\O_D$
is trivial, and its dual ${\cal T}^1_X$ is also trivial.
$\Box$
\vspace{3mm}
Let $\U=\{X_\la\hookrightarrow V_\la\}$ be a coordinate covering on $X$, and take a log system
$\{(\zeta^{\ssd{\la}}_0,\ldots,\zeta^{\ssd{\la}}_n)\}$ with respect to $\U=\{X_\la\hookrightarrow V_\la\}$. On each $X_\lambda$,
consider a pre--log structure $\alpha_\lambda:{\bf N}^{n+1}\longrightarrow\O_{X_\lambda}$,
defined by $\alpha_\lambda(e_i)=\zeta^{\ssd{\la}}_i$ for $0\leq i\leq n$.
Then, the associate log structure of this pre--log structure is
equivalent to the log structure ${\cal M}_\lambda$ of semistable type
on the normal crossing variety $X_\lambda$.
We write this log structure, according to \S 1, by
\begin{equation}\label{loclog}
{\cal M}_\lambda={\bf N}^{n+1}\oplus_{\alpha^{-1}_\lambda(\O^\times_{X_\lambda})}
\O^\times_{X_\lambda}
\stackrel{{\bar{\alpha}}_\lambda}{\longrightarrow}\O_{X_\lambda},
\end{equation}
and ${\bar{\alpha}}_\lambda(e_i, u)=u\cdot\zeta^{\ssd{\la}}_i$,
for $0\leq i\leq n$.
Due to Lemma \ref{ltrns}, on each $X_{\lambda\mu}=X_\lambda\cap X_\mu\neq\emptyset$,
there exists a non canonical isomorphism
$$
\begin{array}{ccc}
{\bf N}^{n+1}
\oplus_{\alpha^{-1}_\lambda(\O^\times_{X_{\lambda\mu}})}
\O^\times_{X_{\lambda\mu}} &
\stackrel{\phi_{\lambda\mu}}{\longrightarrow} &
{\bf N}^{n+1}
\oplus_{\alpha^{-1}_\mu(\O^\times_{X_{\lambda\mu}})}
\O^\times_{X_{\lambda\mu}} \\
\vphantom{\bigg|}\Big\downarrow & & \vphantom{\bigg|}\Big\downarrow \\
\O_{X_{\lambda\mu}} & = & \O_{X_{\lambda\mu}}
\end{array}
$$
defined by
\begin{equation}\label{isom}
\phi_{\lambda\mu}(e_i, u)=(e_{\sigma^{\ssd{\la\mu}}(i)}, u\cdotu^{\ssd{\la\mu}}_i),
\end{equation}
for $0\leq i\leq n$.
\begin{lem}\label{patch}
Any isomorphism $\phi_{\lambda\mu}$ which makes the above diagram commute
is written in the form of (\ref{isom}).
\end{lem}
\noindent{\sc Proof.}\hspace{2mm}
Changing indicies suitably, we may assume that
\begin{itemize}
\item for $0\leq i\leq d$, $\alpha_\lambda(e_i)$ and
$\alpha_\mu(e_i)$ are not invertible on $X_{\lambda\mu}$,
\item for $d< i\leq n$, $\alpha_\lambda(e_i)$ and
$\alpha_\mu(e_i)$ are invertible $X_{\lambda\mu}$.
\end{itemize}
Set $\phi_{\lambda\mu}(e_i, 1)=(\sum_j a_i^j e_j, u^{\ssd{\la\mu}}_i)$,
for $0\leq i\leq n$.
\vspace{3mm}
\noindent{\sc Case}\vspace{2mm} $0\leq i\leq d\:$:
Since ${\bar{\alpha}}_\mu\circ\phi_{\lambda\mu}={\bar{\alpha}}_\lambda$,
the matrix $(a_i^j)_{0\leq i,j\leq d}$ is a permutation matrix
(Note that $A,B\in{\rm M}_d({\bf N})$ and $AB=1$ implies that $A$ and $B$ are
permutation matrices).
Then we may assume $a_i^j=\delta_i^j\quad(0\leq i,j\leq d)$.
Hence, we can write $\phi_{\lambda\mu}(e_i, 1)=(e_i, u^{\ssd{\la\mu}}_i)\cdot(b_i, 1)$,
where $\alpha_\mu(b_i)\in{\rm H}^0(X_{\lambda\mu}, \O^\times_X)$.
Since we have the equality
$$
(e_i, u^{\ssd{\la\mu}}_i\cdot\alpha_\mu(b_i))\cdot(b_i, \alpha_\mu(b_i)^{-1})
=(e_i, u^{\ssd{\la\mu}}_i)\cdot(b_i, 1),
$$
we get $\phi_{\lambda\mu}(e_i, 1)=(e_i, u^{\ssd{\la\nu}}_i\cdot\alpha_\mu(b_i))$
in the quotient monoid
$\mbox{\rm N}^{n+1}\oplus_{\alpha^{-1}_\mu(\O^\times_{X_{\lambda\mu}})}
\O^\times_{X_{\lambda\mu}}$.
\vspace{3mm}
\noindent{\sc Case}\vspace{2mm} $d<i\leq n\:$: Since
${\bar{\alpha}}_\mu\circ\phi_{\lambda\mu}(e_i, 1)$ is invertible,
we have $a_i^j=0$,
for $d<i\leq n$ and $0\leq j\leq d$.
This implies $\phi_{\lambda\mu}(e_i, 1)=(c_i, u^{\ssd{\la\mu}}_i)$,
where $\alpha_\mu(c_i)\in{\rm H}^0(X_{\lambda\mu}, \O^\times_X)$.
Since
$$
(c_i, u^{\ssd{\la\mu}}_i)\cdot(e_i, \alpha_\mu(e_i)^{-1})=
(e_i, \alpha_\mu(c_i)u^{\ssd{\la\mu}}_i\alpha_\mu(e_i)^{-1})\cdot
(c_i, \alpha_\mu(c_i)^{-1}),
$$
we get $\phi_{\lambda\mu}(e_i, 1)=
(e_i, \alpha_\mu(c_i)\cdotu^{\ssd{\la\nu}}_i\cdot\alpha_\mu(e_i)^{-1})$
in the quotient monoid
$\mbox{\rm N}^{n+1}\oplus_{\alpha^{-1}_\mu(\O^\times_{X_{\lambda\mu}})}
\O^\times_{X_{\lambda\mu}}$.
\vspace{3mm}
Hence, for $0\leq i\leq n$, we have the equality $\phi_{\lambda\mu}(e_i, 1)=
(e_i, \widetilde{u^{\ssd{\la\mu}}_i})$,
for some invertible function $\widetilde{u^{\ssd{\la\mu}}_i}$ on $X_{\lambda\mu}$,
in the quotient monoid
$\mbox{\rm N}^{n+1}\oplus_{\alpha^{-1}_\mu(\O^\times_{X_{\lambda\mu}})}
\O^\times_{X_{\lambda\mu}}$.
Combining this with $\phi_{\lambda\mu}(0, u)=(0, u)$, we have the desired
result.
$\Box$
\vspace{3mm}
\setcounter{ste}{0}
Now, let us prove Theorem 11.2.
Assume that $X$ is $d$--semistable. Take a coordinate covering $\U=\{X_\la\hookrightarrow V_\la\}$,
a log system $\{(\zeta^{\ssd{\la}}_0,\ldots,\zeta^{\ssd{\la}}_n)\}$ and a transition system
$\{(u^{\ssd{\la\mu}}_i, \sigma^{\ssd{\la\mu}})\}$ such that (\ref{cdss}) holds.
Then, each $X_\lambda$ has a log structure by (\ref{loclog}).
Moreover, there exists a system of isomorphisms
$\{\phi_{\lambda\mu}\}$ defined by (\ref{isom}). Now we are going to show
that $\{{\cal M}_\lambda\}$ glues to a log structure ${\cal M}$ on $X$ of the desired type.
\begin{ste}{\rm
Let us prove that
$\phi_{\mu\nu}\circ\phi_{\lambda\mu}=\phi_{\lambda\nu}$ on each
$X_{\lambda\mu\nu}=X_\lambda\cap X_\mu\cap X_\nu\neq\emptyset$, i.e.,
$\{\phi_{\lambda\mu}\}$ satisfies the 1-cocycle condition.
Set
\begin{eqnarray*}
\phi_{\mu\nu}\circ\phi_{\lambda\mu}(e_i, 1) & = &
(e_{\tau\circ\sigma(i)}, v_{\sigma(i)}u_i) \\
\phi_{\lambda\nu}(e_i, 1) & = & (e_{\rho(i)}, w_i),
\end{eqnarray*}
where we put $\sigma=\sigma^{\ssd{\la\mu}}$, $\tau=\sigma^{\ssd{\mu\nu}}$, $\rho=\sigma^{\ssd{\la\nu}}$,
$u=u^{\ssd{\la\mu}}$, $v=u^{\ssd{\mu\nu}}$, and $w=u^{\ssd{\la\nu}}$.
Changing indicies suitably, we may assume
\begin{itemize}
\item for $0\leq i\leq d$, $\zeta^{\ssd{\la}}_i\mid_{X_{\lambda\mu\nu}}$
is not invertible on $X_{\lambda\mu\nu}$,
\item for $d<i\leq n$, $\zeta^{\ssd{\la}}_i\mid_{X_{\lambda\mu\nu}}$
is invertible on $X_{\lambda\mu\nu}$.
\end{itemize}
Since ${\bar{\alpha}}_\nu\circ\phi_{\mu\nu}\circ\phi_{\lambda\mu}=
{\bar{\alpha}}_\lambda={\bar{\alpha}}_\nu\circ\phi_{\lambda\nu}$, we have
\begin{equation}\label{str1}
v_{\sigma(i)}u_i\zeta^{\ssd{\nu}}_{\tau\circ\sigma(i)}=\zeta^{\ssd{\la}}_i=
w_i\zeta^{\ssd{\nu}}_{\rho(i)}\quad(0\leq i\leq n).
\end{equation}
\vspace{3mm}
\noindent
{\sc Case}\hspace{2mm} $d<i\leq n\:$: Since the equality (\ref{str1})
holds, both
$\zeta^{\ssd{\nu}}_{\tau\circ\sigma(i)}$ and $\zeta^{\ssd{\nu}}_{\rho(i)}$ are invertible
on $X_{\lambda\mu\nu}$ and we have
$$
(e_{\tau\circ\sigma(i)}, v_{\sigma(i)}u_i)\cdot
(e_{\rho(i)}, (\zeta^{\ssd{\nu}}_{\rho(i)})^{-1})=
(e_{\rho(i)}, w_i)\cdot
(e_{\tau\circ\sigma(i)}, (\zeta^{\ssd{\nu}}_{\tau\circ\sigma(i)})^{-1}).
$$
Hence, we have $(e_{\tau\circ\sigma(i)}, v_{\sigma(i)}u_i)=
(e_{\rho(i)}, w_i)$ in the quotient monoid
${\bf N}^{n+1}
\oplus_{\alpha^{-1}_\nu(\O^\times_{X_{\lambda\mu\nu}})}
\O^\times_{X_{\lambda\mu\nu}}$.
\vspace{3mm}
\noindent
{\sc Case}\hspace{2mm} $0\leq i\leq d\:$: Since the components
$\{\zeta^{\ssd{\nu}}_{\tau\circ\sigma(i)}=0\}$ and $\{\zeta^{\ssd{\nu}}_{\rho(i)}=0\}$
of $X_{\lambda\mu\nu}$ coincides due to (\ref{str1}),
we have $\tau\circ\sigma(i)=\rho(i)$.
Hence, we have $\zeta^{\ssd{\nu}}_{\rho(i)}=w_i^{-1}v_{\sigma(i)}u_i\zeta^{\ssd{\nu}}_{\rho(i)}$.
This imples
\begin{equation}\label{str2}
w_i^{-1}v_{\sigma(i)}u_i=1+
a_iz^{\ssd{\nu}}_{\rho(0)}\cdotsz^{\ssd{\nu}}_{\rho(i-1)}\cdotz^{\ssd{\nu}}_{\rho(i+1)}
\cdotsz^{\ssd{\nu}}_{\rho(d)},
\end{equation}
for some $a_i\in\O_{X_{\lambda\mu\nu}}$.
Since
\begin{eqnarray*}
\rho(\{d+1,\ldots,n\})&=&\tau\circ\sigma(\{d+1,\ldots,n\})\\
&=&\{j\mid(\zeta^{\ssd{\nu}}_j\mid_{X_{\lambda\mu\nu}})\in
{\rm H}^0(X_{\lambda\mu\nu}, \O^\times_X)\},
\end{eqnarray*}
and $\zeta^{\ssd{\nu}}_{\rho(j)}=w_j^{-1}v_{\sigma(j)}u_j
\zeta^{\ssd{\nu}}_{\tau\circ\sigma(j)}$ for $d<j\leq n$, we have
$$
\prod_{d<j\leq n}w_j^{-1}v_{\sigma(j)}u_j=1.
$$
On the other hand, by our assumptions
$u_1\cdots u_n=1$, etc., we have
$$
\prod_{0\leq i\leq n}w_j^{-1}v_{\sigma(j)}u_j=1.
$$
Hence, we get the equality
$$
\prod_{0\leq i\leq d}w_j^{-1}v_{\sigma(j)}u_j=1.
$$
By this and (\ref{str2}), we have
$$
\sum_{0\leq i\leq d}a_iz^{\ssd{\nu}}_{\rho(0)}\cdots
z^{\ssd{\nu}}_{\rho(i-1)}\cdotz^{\ssd{\nu}}_{\rho(i+1)}\cdots
z^{\ssd{\nu}}_{\rho(d)}=0,
$$
and consequently we get $a_i=0$, that is, $w_i^{-1}v_{\sigma(i)}u_i=1$.
Hence, also in this case, we have
$(e_{\tau\circ\sigma(i)}, v_{\sigma(i)}u_i)=(e_{\rho(i)}, w_i)$.
Therefore, we have proved that $\phi_{\mu\nu}\circ\phi_{\lambda\mu}=
\phi_{\lambda\nu}$ holds and $\{{\cal M}_\lambda\}$ glues to a log structure ${\cal M}$ on $X$.
}
\end{ste}
\begin{ste}{\rm The system of morphisms
$\{f_\lambda\}$,
where $f_\lambda:(X_\lambda, {\cal M}_\lambda)\rightarrow
(\mathop{\rm Spec}\nolimits k, {\bf N})$ is defined as in Definition \ref{canlog},
glues to the morphism $f$ if and only if
the following diagram commutes:
$$
\begin{array}{ccc}
{\bf N}^{n+1}\oplus_{\alpha^{-1}_\lambda(\O^\times_{X_{\lambda\mu}})}
\O^\times_{X_{\lambda\mu}} & \stackrel{\phi_{\lambda\mu}}{\longrightarrow} &
{\bf N}^{n+1}\oplus_{\alpha^{-1}_\mu(\O^\times_{X_{\lambda\mu}})}
\O^\times_{X_{\lambda\mu}} \\
\vphantom{\bigg|}\Big\uparrow & & \vphantom{\bigg|}\Big\uparrow \\
{\bf N}\oplus k^\times
& = & {\bf N}\oplus k^\times,
\end{array}
$$
where ${\bf N}\oplus k^\times\rightarrow k$ is the associated log
structure of ${\bf N}\rightarrow k$, and the homomorphism
${\bf N}\oplus k\rightarrow
{\bf N}^{n+1}\oplus_{\alpha^{-1}_\lambda
(\O^\times_{X_{\lambda\mu}})}\O^\times_{X_{\lambda\mu}}$
is defined by $(1, 1)\mapsto (e_0+\cdots+e_n, 1)$.
The commutativity of this diagram is equivalent to the equality
\begin{eqnarray*}
(e_0+\cdots+e_n, 1) & = & \phi_{\lambda\mu}(e_0+\cdots+e_n, 1) \\
& = & (e_0+\cdots+e_n, u^{\ssd{\la\mu}}_0\cdotsu^{\ssd{\la\mu}}_n).
\end{eqnarray*}
It is easy to see that this is equivalent to $u^{\ssd{\la\mu}}_0\cdotsu^{\ssd{\la\mu}}_n=1$.
Hence, $\{f_\lambda\}$ glues to the morphism
$f$ due to Proposition \ref{cridss}.
Thus, we have proved that ${\cal M}$ is the log structure on $X$ of the desired
type.
}
\end{ste}
\begin{ste}{\rm Let us prove the converse.
Assume that $X$ has a log structure of semistavle type.
Then, by Lemma \ref{patch}, Step 2 above and
Proposition \ref{cridss},
it is easy to show that $X$ is $d$--semistable.
}
\end{ste}
Thus, the proof of Theorem 11.2 is completed.
|
1995-11-21T05:59:22 | 9406 | alg-geom/9406006 | en | https://arxiv.org/abs/alg-geom/9406006 | [
"alg-geom",
"math.AG"
] | alg-geom/9406006 | Dr. Yakov Karpishpan | Yakov Karpishpan | Infinite-dimensional Lie algebras and the period map for curves | 36 pages, LaTeX. This corrects a reference in the earlier version | null | null | null | null | We compute higher-order differentials of the period map for curves and show
how they factor through the corresponding higher Kodaira-Spencer classes. Our
approach is based on the infinitesimal equivariance of the period map, due to
Arbarello and De Concini \cite{AD}.
| [
{
"version": "v1",
"created": "Thu, 23 Jun 1994 16:11:25 GMT"
},
{
"version": "v2",
"created": "Wed, 29 Jun 1994 18:29:35 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Karpishpan",
"Yakov",
""
]
] | alg-geom | \section{Notation and some preliminaries}
\label{sect:notat}
The results discussed in this paper owe their explicitness to a
very concrete object, the field of Laurent power series
$$
\H={\bf C}((z))={\bf C}[[z]][z^{-1}]\ .
$$
Most of the time we will regard it merely as an
infinite-dimensional vector space. It has several distinguished
subspaces:
$$
\H_+={\bf C}[[z]]\ \ \ {\rm and} \ \ \
\H_-= \mbox{the span of negative powers of}\ z\ .
$$
Thus $\H=\H_+\oplus\H_-$ . Also,
$$
\H_+'=z\H_+, \ \ \ {\rm and} \ \ \ \H'=\H'_+\oplus \H_-\ .
$$
\begin{Def}
$<f,g>={\rm Res}_{z=0}fdg$\ .
\end{Def}
This is a symplectic form on $\H$, non-degenerate on $\H'$.
\begin{Def}
$${\bf sp}(\H')=\{\alpha\in
\mbox{\rm End}(\H')|<\alpha(x),y>+<\alpha(y),x>=0\ \ \mbox{\rm for all}\
x,y\in\H'\} \ .$$
\end{Def}
\noindent{\bf Facts:} (a) $\H$ (and, hence, $\H'$) is a
topological vector space with the $z$-adic topology.
(b) ${\bf sp}(\H')$ is isomorphic to the completion
$\widehat{S^2}(\H')$ of $S^2(\H')$ --- the symmetric square of
$\H'$, where $S^2(\H')$ embeds in ${\bf sp}(\H')$ via
$$
hk \mapsto \{x\mapsto <h,x>k+<k,x>h\}\ .
$$
Finally, $\d=\H\frac{d}{dz}$ will denote the {\em Witt} Lie
algebra of formal vector fields on a punctured disc. Its central
extension is the more famous Virasoro algebra.
We will also use $\d_+=\H_+\frac{d}{dz}$.
The Lie algebras $\d$ and ${\bf sp}(\H')$ will appear in the
setting of the following
\begin{Def}
\label{def:inf-hom}
A Lie algebra $L$ {\em acts by vector fields} on
a manifold $M$ if there is a homomorphism (or
anti-homomorphism) of Lie algebras
$$
L\longrightarrow \Gamma(M,\Theta_M)\ .
$$
If the composed map to the tangent space of $M$ at a point $x$
$$
L\longrightarrow \Gamma(M,\Theta_M)\longrightarrow T_xM
$$
is surjective for each $x\in M$, $M$ is called {\em
infinitesimally homogeneous}, and one says that $L$ provides an
{\em infinitesimal uniformization} for $M$.
\end{Def}
We will write $\Omega^1_X$ and $\omega_X$ interchangeably when $X$
is a curve. In turn, ``curve" will mean a complex algebraic curve.
We will consider the classical topology and the analytic structure
on $X$ only when dealing with the cohomology of $X$ with
coefficients in ${\bf Z}$ or ${\bf C}$, and when using the exponential
sequence in the proof of Lemma \ref{lemma:Lambda}, where we write
$X^{an}$.
We will also use the following notation: if ${\bf g}$ is a Lie
algebra, then ${\cal U}{\bf g}$ is its universal enveloping
algebra, and $\overline{\cal U}{\bf g}:={\cal U}{\bf g}/{\bf C}$.
${\cal U}^{(k)}{\bf g}$ (respectively, $\overline{\cal U}^{(k)}{\bf
g}$) will denote the elements of order$\leq k$ in the natural
filtration of ${\cal U}{\bf g}$ (respectively, $\overline{\cal
U}{\bf g}$).
Finally, we will use the fact that Lie algebras of endomorphisms of
Hodge structures, with or without polarization, carry a Hodge
structure of their own, always of weight 0. E.g. if $H=\oplus
H^{p,q}$ is a HS and ${\bf g}=\mbox{\rm End}(H)$, then the Hodge
decomposition of ${\bf g}$ is
$$
{\bf g}=\oplus{\bf g}^{-k,k}\ ,
$$
where ${\bf g}^{-k,k}=\{f\in\mbox{\rm End}(H)\ |\ f(H^{p,q})\subset
H^{p-k,q+k}\ \ \forall p,q\}$.
\section{Infinitesimal uniformization of moduli of curves}
Let $X$ be a complete curve of genus $g\geq 2$, $p$ --- a point
on $X$, and
$$
u:\widehat{\cal O}_{X,p}\stackrel{\cong}{\longrightarrow}\H_+
$$
--- a formal local coordinate at $p$; $u$ extends to an
isomorphism of fields of fractions, also defining the
obvious monomorphisms
$$
\Theta_{X,p}\hookrightarrow \H_+\frac{d}{dz}\subset\d\ , \ \ \
\Omega^1_{X,p}\hookrightarrow \H_+dz , \ \ \
\Omega_X^1(*p)_p\hookrightarrow \H dz , \ \ \ {\rm etc.},
$$
all of which will also be denoted by $u$. As any point on a
complete curve, $p$ is an ample divisor on $X$. Therefore,
$X-p$ is an affine open set in $X$. Choose an affine
neighborhood $V$ of $p$ in $X$. Then $\{(X-p), V\}$ is an affine
covering of $X$, suitable for computing the \v{C}ech cohomology
of $X$ with coefficients in a coherent sheaf. Thus we have an
exact sequence
\begin{eqnarray}
\label{seq:basic}
& & \\ & &
0\rightarrow\Gamma(X-p,\Theta_X)\oplus\Gamma(V,\Theta_X)
\stackrel{\delta}{\rightarrow}\Gamma(V-p,\Theta_X)
\stackrel{\pi}{\rightarrow}H^1(X,\Theta_X)\rightarrow 0\ .
\nonumber
\end{eqnarray}
Exactness on the left is a consequence of $H^0(X,\Theta_X)=0$,
which, in turn, follows from the assumption $g\geq 2$.
The sheaf $\Theta_X$ is filtered by the subsheaves
$\Theta_X(-ip)$ of vector fields vanishing at $p$ to an order
$\geq i$ \ ($i\geq 0$). This induces a decreasing filtration
$P^i$ on spaces of sections over $V$, and hence over $X-p$ and
$V-p$. The \v{C}ech differential $\delta$ is strictly compatible
with $P^{\bullet}$, and so is the projection $\pi$, once
$H^1(X,\Theta_X)$ receives the induced filtration $P^{\bullet}$
from $\Gamma(V-p, \Theta_X)$. Therefore, the sequence
(\ref{seq:basic}) remains exact when reduced modulo $P^i$.
\begin{Lemma} The maps
$$
u_i:\Gamma(V-p, \Theta_X)/P^i\longrightarrow\d/z^i\d_+
$$
induced via the identification $u$ are isomorphisms for each
$i>0$.
\end{Lemma}
\ \\ \noindent {\bf Proof.\ \ } Suppose the points $q_1,\ldots,q_m$ constitute the complement of $V$ in
$X$, and let $Q$ be the effective divisor $q_1+\ldots+q_m$. Since $Q$ is ample,
for $N$ sufficiently large $H^1(X,\Theta_X(NQ-ip))$ will vanish. We may also
assume $deg{\cal L}\ge
q g-1$. With this choice of $N$, set ${\cal L}=\Theta_X(NQ-ip)$. Then the
Riemann-Roch Theorem gives
$$
H^0(X,{\cal L}(kp))=\deg {\cal L}+k+1-g
$$
for each $k\geq 1$, which means that for each $k$ there exists a section of
$\Theta_V(-ip)$, regular on $V-p$ and with a pole of order exactly $k$ at $p$.
Thisimplies the surjectivity of $u_i$.
Now,
$$u:\Gamma(V-p,{\cal O}_X)\longrightarrow\H$$
is injective, since any regular function on $V-p$ is
completely determined by its Laurent expansion. And
$$u^{-1}(z^i\H_+)=\Gamma(V-p,{\cal O}_X(-ip))\ ,$$
implying that the $u_i$'s are injective too.
\ $\displaystyle\Box$\\ \ \par
\begin{Cor}
$\llim\Gamma(V-p,\Theta_X)/P^i=\d$ .
\end{Cor}
\begin{Lemma}
\label{lemma:surj}
Passing to the inverse limit in the exact sequence obtained
from (\ref{seq:basic}) by reduction $\bmod P^i$ produces an exact
sequence
\begin{equation}
0\rightarrow
u(\Gamma(X,\Theta_X(*p)))\oplus\d_+\longrightarrow\d\longrightarrow
H^1(X,\Theta_X)\rightarrow 0
\label{seq:final}
\end{equation}
\end{Lemma}
\ \\ \noindent {\bf Proof.\ \ } First we note that
$\Gamma(X-p,\Theta_X)=\Gamma(X,\Theta_X(*p))$ by \cite{Gro}. Also,
$P^i\Gamma(X,\Theta_X(*p))=0$ for all $i>0$, since $X$ supports no
non-zero global regular vector fields by virtue of the assumption
$g\geq 2$. Hence
$\Gamma(X,\Theta_X(*p))^{\wedge}=\Gamma(X,\Theta_X(*p))$.
Second,
$$
\llim H^1(X,\Theta_X)/P^i=H^1(X,\Theta_X)\ ,
$$
because for all sufficiently large $i$
$$
H^1(X,\Theta_X)/P^i=H^1(X,\Theta_X)\ ;
$$
this is simply a consequence of $H^1(X,\Theta_X)$ being
finite-dimensional. Finally, inverse limits preserve the
exactness of (\ref{seq:basic}) $\bmod P^i$, because the directed
system
$$
\{A_i=(\Gamma(X-p,\Theta_X)\oplus\Gamma(V,\Theta_X))/P^i\}
$$
satisfies the Mittag-Leffler condition (see \cite{L}, III,
Prop. (9.3)):
{\em For each $n$, the decreasing sequence of images of natural
maps $\varphi_{mn}:A_m\rightarrow A_n\ \ (m\geq n)$ stabilizes.}
\noindent This is trivially so since all $\varphi_{mn}$ are
surjective in our situation. \ $\displaystyle\Box$\\ \ \par
Assume now that $X$ moves in a flat family
\begin{eqnarray}
{\cal X} & \supset & X_t \nonumber\\
\pi\ \downarrow & & \downarrow \\
S & \ni & t\ \ ,\nonumber
\label{family:flat}
\end{eqnarray}
with a section ${\bf p}:S\rightarrow{\cal X}$ and a local
coordinate
$$
{\bf u}:\widehat{\cal O}_{{\cal X},{\bf p}}
\stackrel{\sim}{\longrightarrow}\Gamma(S,{\cal O}_S)
\otimes\H_+
$$
on $\cal X$ along $\bf p$, so that the restriction of $\bf u$
to $X_t$ provides a local formal coordinate $u_t$ near $p_t$.
For each $t\in S$ one has an analogue of
(\ref{seq:final}). In particular, for each $t$ there is a
surjection$$
\d\longrightarrow H^1(X_t,\Theta_{X_t}) \ ;
$$
these glue together into a map
\begin{equation}
\d \longrightarrow\Gamma(S,R^1\pi_{*}\Theta_{{\cal X}/S}) \ .
\label{map:rel}
\end{equation}
Assume further that $S$ is a disc centered at $0$ in ${\bf C}^{3g-3}$,
and the family
\begin{equation}
\begin{array}{ccc}
{\cal X} & \supset & X \\
\pi\ \downarrow & & \downarrow \\
S & \ni & 0
\end{array}
\label{deform:X}
\end{equation}
is a miniversal deformation of the curve $X$. Then the
Kodaira-Spencer map of the family,
$$
\kappa:\Theta_S\longrightarrow R^1\pi_{*}\Theta_{{\cal X}/S}\ ,
$$
isan isomorphism. Composing its inverse with the map
in (\ref{map:rel}) yields a linear map
\begin{equation}
\lambda:\d\longrightarrow\Gamma(S,\Theta_S)\ .
\label{map:Lie}
\end{equation}
\begin{Lemma}
\label{lemma:act}
The map $\lambda$ in (\ref{map:Lie}) is an
anti-homomorphism of Lie algebras.
\end{Lemma}
\ \\ \noindent {\bf Proof.\ \ }
$\cal X$ admits an acyclic Stein covering ${\cal W}=\{W_0,W_1\}$
with $W_0\cong S\times V$ and $W_1={\cal X}-{\bf p}\cong S\times
{X-{\bf p}}$. It follows from the proof of the previous lemma that
the map $\lambda$ fits in the commutative diagram
$$
\begin{array}{ccccc}
\Gamma(S,{\cal O}_S)\otimes\d & \hookleftarrow & \d &
\stackrel{\lambda}{\longrightarrow} & \Gamma(S,\Theta_S)\\ & & \\
j\uparrow\cong & && &
\cong\downarrow\kappa \\ & && & \\
\Gamma(W_0\cap W_1,\Theta_{{\cal X}/S})^{\wedge} & &
\stackrel{\textstyle c}{\,\longrightarrow\hspace{-12pt} & & \Gamma(S,R^1\pi_{*}\Theta_{{\cal
X}/S}) \end{array}\ ,
$$
where $ ^{\wedge}$ indicates completion with respect to the
filtration by the order of vanishing along $\bf p$, and $j$ is the
isomorphism given by taking Laurent expansionsof relative vector
fields on $W_0\cap W_1$ along $\bf p$ via $\bf u$.
We begin by reviewing the
definition of $\kappa$. The Kodaira-Spencer map
$\kappa$ is the connecting morphism in the direct-image
sequence of the short exact sequence of ${\cal O}_{\cal X}$
modules
\begin{equation}
0\rightarrow\Theta_{{\cal X}/S}\longrightarrow\Theta_{\cal X}
\longrightarrow\pi^*\Theta_S\rightarrow 0\ \ .
\label{seq:KS}
\end{equation}
This contains an exact subsequence of $\pi^{-1}{\cal
O}_S$-modules
\begin{equation}
0\rightarrow\Theta_{{\cal X}/S}\longrightarrow
\widetilde{\Theta}_{\cal X}
\longrightarrow\pi^{-1}\Theta_S\rightarrow 0\ \ .
\label{subseq:KS}
\end{equation}
whose direct-image sequence also has $\kappa$ as a connecting
morphism (see \cite{BS} and also \cite{EV}). Furthermore,
(\ref{subseq:KS}) is an exact sequence of sheaves of Lie
algebras. The ${\bf C}$-linear brackets on $\widetilde{\Theta}_{\cal
X}$ and $\pi^{-1}\Theta_S$ are inherited from $\Theta_{\cal X}$
and $\Theta_S$, respectively. The bracket on $\Theta_{{\cal
X}/S}$ is even $\pi^{-1}{\cal O}_S$-linear.
We are ready to prove the lemma. Take any $\zeta,\xi\in\d$, and
let $Z=\lambda(\zeta),\ \Xi=\lambda(\xi)$. We wish to show that
$[\zeta,\xi]=-[Z,\Xi]$, where the first bracket istaken in
the Witt Lie algebra $\d$, and the second is in
$\Gamma(S,\Theta_S)$. The elements $\zeta$ and $\xi$ of $\d$,
which we identify with their pre-images under $j$, may be taken as
Kodaira-Spencer representatives of $Z$ and $\Xi$. Lift $Z$
to some sections of $\widetilde{\Theta}_{\cal X}$, $\zeta_0\in$ on
$W_0$ and $\zeta_1\in$ on $W_2$, and similarly for $\Xi$: $\xi_0\in
\Gamma(W_0,\widetilde{\Theta}_{\cal X})$, and $\xi_1\in
\Gamma(W_1,\widetilde{\Theta}_{\cal X})$. Then $\zeta_1-\zeta_0$
and $\xi_1-\xi_0$, with all terms restricted to $W_{01}=W_0\cap
W_1$, also give KS representatives for $Z$ and $\Xi$. In
particular,
$$
\zeta=\zeta_1-\zeta_0+\delta\theta\ ,
$$
and
$$
\xi=\xi_1-\xi_0+\delta\eta\ ,
$$
where $\theta$ and $\eta$ are some elements of $\check{C}^0({\cal
W},\Theta_{{\cal X}/S})^{\wedge}$. Then $[Z,\Xi]$ admits as its KS
representative the following expression, all terms of which are
restricted to $W_{01}$:
\begin{eqnarray}
\label{brackets}
\lefteqn{[\zeta_1,\xi_1]-[\zeta_0,\xi_0]=}\\
& =& [\zeta_1,\xi_1]-
[\zeta_1-\zeta+\delta\theta,\xi_1-\xi+\delta\eta] \nonumber\\
& =&
-[\zeta,\xi]+[\zeta_1,\xi]+[\zeta,\xi_1]+[\delta\theta,\xi_0]+
[\zeta_0,\delta\eta] \ .\nonumber
\end{eqnarray}
The Lie bracket of a section of
$\widetilde{\Theta}_{\cal X}$ with that of $\Theta_{{\cal X}/S}$
is again a section of $\Theta_{{\cal X}/S}$, which implies that
the last two terms in (\ref{brackets}) are in
$\delta\check{C}^0({\cal W},\Theta_{{\cal X}/S})$. We may assume
that $\bf u$ is induced by an isomorphism $u:\widehat{\cal O}_{X,p}
\rightarrow\H_+$ viathe identification $W_{01}\cong
S\times\{V-p\}$. The identification allows us to label some
vector fields on $W_{01}$ as horizontal or vertical. By
construction, $\zeta$ and $\xi$ are vertical and constant in the
horizontal direction. The fields $\zeta_1$ and $\xi_1$, on the
other hand, may be chosen to be horizontal and constant in the
vertical direction. Then $[\zeta_1,\xi]=[\zeta,\xi_1]=0$. Collecting
what is left of (\ref{brackets}), we conclude that $-[\zeta,\xi]$
is a Kodaira-Spencer representative for $[Z,\Xi]$,
which proves the lemma. \ $\displaystyle\Box$\\ \ \par
Recallingdefinition \ref{def:inf-hom}, we may summarize lemmas
\ref{lemma:act} and \ref{lemma:surj} in the following theorem,
due to\cite{BMS,Ko,BS}, cf. \cite{N,TUY}.
\begin{Thm}
\label{thm:S:infhom}
For any curve $X$ of genus $g\geq 2$ the Witt Lie algebra $\d$
acts by vector fields on the base $S$ of a miniversal deformation
of$X$, making $S$ infinitesimally homogeneous.
\end{Thm}
\refstepcounter{Thm Theaction above clearly
depends on the choice of a point $p_t$ on each curve $X_t$, as
well as on a formal parameter $u_t$ at $p_t$. For our purposes
all these choices are equally good. More canonically, one may
consider the moduli space of triples $(X,p,u)$, encompassing
all possible choices of $p$ and $u$ on each $X$. The action of
$\d$ extends to such ``dressed" moduli spaces $\hat{\cal M}_g$,
making them also infinitesimally homogeneous. We will not need
these constructions, since the questions we study are local on
$\hat{\cal M}_g$.
\newpage
\section{Infinitesimal uniformization of period domains of weight
one}
By definition, a {\em Hodge structure of weight one} consists of a
lattice $\Lambda\cong{\bf Z}^{2g}$ and a decomposition of its
complexification $H=\Lambda\otimes{\bf C}$, $H=H^{1,0}\oplus H^{0,1}$,
such that $H^{1,0}=\overline{H^{0,1}}$. The {\em Hodge filtration}
$F^{\bullet}$ on $H$ is given by $F^0=H$, $F^1=H^{1,0}$,
$F^0=0$. The HS $(\Lambda,H,F^{\bullet})$ is {\em principally
polarized} if $\Lambda$ is equipped with a unimodular symplectic
form $Q(\ ,\ )$ such that $(u,v)=Q(\bar{u},v)$ is a
{\em positive-definite} Hermitian form on $H^{1,0}$ (and on
$H^{0,1}$).
The data $(\Lambda,H,F^{\bullet},Q)$ defines {\em a
principally-polarized abelian variety} $A=H^{0,1}/i(\Lambda)$,
where $i$ denotes the composition of the inclusion
$\Lambda\rightarrow\Lambda\otimes{\bf C}=H$ with the projection
$H=H^{1,0}\oplus H^{0,1}\rightarrow H^{0,1}$.
As is well-known, the space ${\bf D}$ of all Hodge structures
$(H,F^{\bullet})$ with a given lattice $\Lambda$ and polarization
$Q$ (={\em the period domain}) can be identified with the Siegel
upper half-space ${\bf H}_g$ of complex symmetric $g\times g$ matrices
whose imaginary parts are positive-definite. The moduli space of
principally-polarized abelian varieties of dimension $g$, ${\cal
A}_g$, is a quotient of ${\bf H}_g$ by the action of $Sp(2g,{\bf Z})$.
Note that ${\bf D}$ is a homogeneous space for the group$Sp(2g,{\bf R})$.
We wish to present ${\bf D}$ locally as an infinitesimally homogeneous
space for ${\bf sp}(\H')$.
\begin{Def}
An {\em extended Hodge structure} (of weight one) is a triple
$(Z,K_0,\Lambda)$, where $Z$ is a maximal isotropic subspace of
$\H'$ (with respect to the symplectic form $<\ ,\ >$), $K_0$ is a
codimension $g$ subspace of $Z$, and $\Lambda$ is a rank $2g$
lattice in $K^{\perp}_0/K_0$, subject to several conditions.
\end{Def}
First of all, $Z\cap\H'_+=0$. This implies the splittings
$\H'=Z\oplus\H'_+$ and
$$
H:=K^{\perp}_0/K_0=H^{1,0}\oplus H^{0,1}\ ,
$$
where $H^{0,1}=Z/K_0$, and $H^{1,0}=K^{\perp}_0\cap \H'_+$.
Let $Q$ be the bilinear form induced on $H$ by $\frac{1}{2\pi i}<\
,\ >$ on $\H'$. The
remaining conditions state that $H=\Lambda\otimes{\bf C}$, defining a
real structure on $H$, that $H^{1,0}=\overline{H^{0,1}}$ with
respect to this structure, and that $Q$ is unimodular on $\Lambda$.
Thus $(\Lambda, H, H^{1,0},H^{0,1},Q)$ is a principally-polarized
HS of weight one.
Arbarello and De Concini introduced an extended version of the
Siegel upper half-space, $\widehat{{\bf H}}_g$, on which $Sp(2g,{\bf Z})$
acts transitively, and the quotient manifold $\widehat{\cal A}_g$
parameterizes extended Hodge structures. The latter may also be
regarded as``extended abelian varieties" in view of the
following commutative diagram:
\begin{equation}
\begin{array}{ccc}
\widehat{{\bf H}}_g & \longrightarrow & \widehat{\cal A}_g \\
&& \\
\downarrow & \swarrow & \downarrow \\
&& \\
{\bf H}_g & \longrightarrow & {\cal A}_g
\end{array}
\label{diag:extended}
\end{equation}
The horizontal maps are quotients with respect to the
$Sp(2g,{\bf Z})$-action. All spaces are manifolds (the top two are
infinite dimensional), except ${\cal A}_g$, which is a
$V$-manifold. Note that all maps in the upper triangle are smooth.
\begin{Prop}[\cite{AD}]
$\widehat{\cal A}_g$ is an infinitesimally
homogeneous space for ${\bf sp}(\H')$.
\label{prop:A:infhom}
\end{Prop}
Obviously, this also makes ${\bf D}={\bf H}_g$ {\em locally} infinitesimally
homogeneous for ${\bf sp}(\H')$. Let us work out the surjection
$$
{\bf sp}(\H')\longrightarrow T_H{{\bf D}}
$$
explicitly.
At any point $H=H^{1,0}\oplus H^{0,1}$ of ${\bf D}$,
$$
T_H{\bf D}=\mbox{\rm Hom}^{(s)}(H^{1,0}, H^{0,1})=S^2H^{0,1}\ .
$$
Suppose $H$ comes from an extended HS $(Z,K_0,\Lambda)$. Then any
$\alpha\in\mbox{\rm End}(\H')$ induces a map
\begin{equation}
H^{1,0}=K_0^{\perp}\cap \H'_+\longrightarrow\H' \ .
\label{sect:of:proj}
\end{equation}
We use the formulas $\H'=Z\oplus\H'_+$ and $K_0\cap \H'_+=0$ to observe
that
$$
H^{0,1}\cong Gr_F^{0} H= Z/K_0\cong \H'/K_0+\H'_+\ .
$$
Then (\ref{sect:of:proj}), composed with the natural projection
$$
\H'\longrightarrow\H'/K_0+\H'_+\ ,
$$
yields an element $a\in\mbox{\rm Hom}(H^{1,0},H^{0,1})$.
When $\alpha\in{\bf sp}(\H')$,
$$<\alpha(x),y>=-<x,\alpha(y)>\ ,$$
i.e. $<x,\alpha(y)>=<y,\alpha(x)>$ for all $x,y\in\H'$. Hence
$$
Q(x,a(y))=Q(y,a(x))
$$
for all $x,y\in H^{1,0}$, which means $a$ is {\em symmetric}:
$$
a\in\mbox{\rm Hom}^{(s)}(H^{1,0},H^{0,1})=S^2H^{0,1}\ .
$$
For reasons that will be clear later, we prefer $-a\in S^2H^{0,1}$.
Thus $\alpha\mapsto -a$ indeed defines a map
\begin{equation}
\label{res:rho}
\rho:{\bf sp}(\H')\longrightarrow T_H{\bf D}=
\mbox{\rm Hom}^{(s)}(H^{1,0},H^{0,1})=S^2H^{0,1}\ .
\end{equation}
\refstepcounter{Thm
\label{rho}
For further use we record that the above construction presents
the uniformizing map (\ref{res:rho}) as a restriction of a more
broadly defined map
\begin{equation}
\mbox{\rm End}(\H')\longrightarrow
\mbox{\rm Hom}(H^{1,0},H^{0,1})\ .
\end{equation}
Both maps will be denoted $\rho$.
\refstepcounter{Thm In view of (\ref{diag:extended}), Proposition
\ref{prop:A:infhom} implies that a sufficiently small open set $U$
in ${\bf D}$ is an infinitesimally homogeneous space under the action of
${\bf sp}(\H')$. However, the action is not unique --- it depends
on the choice of a lift from $U$ to $\widehat{\cal A}_g$.
\section{The extended period map}
Let $X$ be a complete smooth curve, $p$ --- a point on $X$, and
$u:\widehat{\cal O}_{X,p}\stackrel{\cong}{\longrightarrow}\H_+$
--- a formal local parameter at $p$. In this section we review
Arbarello and De Concini's construction associating an extended HS
$(Z,K_0,\Lambda)$ to the data $(X,p,u)$. When the triple $(X,p,u)$
varies in a flat family over some base $S$, this construction
defines ``an extended period map"
$$
\widehat{\Phi}: S\longrightarrow\widehat{\cal A}_g\ ,
$$
such that the usual period map $\Phi: S\longrightarrow {\bf D}$
naturally factors through $\widehat{\Phi}$.
\begin{Def}
$K_0:=u(\Gamma(X-p,{\cal O}_X))\cap\H'$\ .
\end{Def}
\noindent This is the same as putting $K_0=u(\Gamma(X-p,{\cal
O}_X))/{\bf C}$.
Note that $\Gamma(X-p,{\cal O}_X)=\Gamma(X,{\cal O}_X(*p))$ by a
theorem of Grothendieck \cite{Gro}, and that
$$
u:\Gamma(X-p,{\cal O}_X)\longrightarrow\H
$$
is injective. There are no non-constant regular functions on
$X$, and so $K_0\cap\H'_+=0$.
\begin{Lemma}
$H^1(X,{\cal O}_X)\cong \H/\H_++K_0$.
\label{H1:O}
\end{Lemma}
\ \\ \noindent {\bf Proof.\ \ }
This follows from the exact sequence
$$
0\rightarrow\Gamma(V,{\cal O}_X)\oplus\Gamma(X-p,{\cal O}_X)\longrightarrow
\Gamma(V-p,{\cal O}_X)\longrightarrow H^1(X,{\cal O}_X)\rightarrow 0
$$
by completion with respect to the order-of-vanishing filtration $P^{\bullet}$
as in (\ref{lemma:surj}). \ $\displaystyle\Box$\\ \ \par
Furthermore, $K_0$ is an isotropic
subspace of $\H'$, i.e. $K_0$ is contained in $K_0^{\perp}$, the
orthogonal complement of$K_0$ in $\H'$ with respect to the
symplectic form $<\ ,\ >$. We can be more specific about $K_0^{\perp}$.
\begin{Def}
$\Omega:=\{f\in\H'\ |\ df\in u(\Gamma(X-p,\Omega^1_X))\}$.
\end{Def}
We have $\Omega\cong\Gamma(X-p,\Omega^1_X)=\Gamma(X,\Omega^1_X(*p))$.
Now, Grothendieck's Algebraic De Rham Theorem \cite{Gro}, coupled wirh the
injectivity of the map $d:\H'\rightarrow\H dz$ and of $u$, gives
\begin{equation}
\Omega/K_0\cong
\frac{\Gamma(X,\Omega^1_X(*p))}{d\,\Gamma(X,{\cal O}_X(*p))} \cong
H^1(X-p,{\bf C})=H^1(X,{\bf C})\ .
\label{Groth:DR}
\end{equation}
\begin{Lemma}
$K_0^{\perp}=\Omega$.
\label{K:perp}
\end{Lemma}
\ \\ \noindent {\bf Proof.\ \ }
If $f\in K_0$ and $g\in\Omega$, then $fdg$ is the Laurent expansion of a
globally defined one-form on $X$ with poles only at $p$. Then ${\rm Res}_0
fdg=0$, i.e. $<K_0,\Omega>=0$, and so $\Omega\subseteq K_0^{\perp}$. The
well-known duality theorem of Serr
e \cite{S} implies that the residue pairing induces a duality between
$H^0(X,\Omega_X^1)$ and $H^1(X,{\cal O}_X)$. The first of these groups is
isomorphic to $\Omega\cap\H'_+$, the second --- to
$$
\frac{\H}{\H_++K_0}=\frac{\Omega+\H}{\H_++K_0}\cong\frac{\Omega}{\Omega\cap(\H_++K_0)}= \frac{\Omega}{\Omega\cap\H_++K_0}\ .
$$
This implies that the residue pairing on $\Omega/K_0$ is non-degenerate.
Coupled with the earlier statements that $\Omega\subseteq K_0^{\perp}$ and
$K_0\subset\Omega^{\perp}$, we have $\Omega^{\perp}=K_0$ and
$(K_0^{\perp})^{\perp}\subseteq \Omega^{\perp}
$, which means that $(K_0^{\perp})^{\perp}=K_0$.
This, in turn, says that $<\ ,\ >$ is non-degenerate on $K_0^{\perp}/K_0$.
However, the pairing is 0 on $K_0^{\perp}\cap\H_+$ (since it is 0 on all of
$\H_+$), and on
$$
\frac{K_0^{\perp}}{(K_0^{\perp}\cap\H_+)+K_0}\cong
\frac{K_0^{\perp}+\H}{\H_++K_0}=\frac{\H}{\H_++K_0}\cong H^1(X,{\cal O}_X)\ .
$$
Then $K_0^{\perp}\cap\H_+$ must be dual to
$$
\frac{K_0^{\perp}}{(K_0^{\perp}\cap\H_+)+K_0}\cong H^1(X,{\cal O}_X)
$$
under the residue pairing on $K_0^{\perp}/K_0$, which implies
$K_0^{\perp}=\Omega$. \ $\displaystyle\Box$\\ \ \par
\begin{Cor}
$K_0^{\perp}/K_0\cong H^1(X,{\bf C})$.
\label{EHS:HS}
\end{Cor}
At this point we make the observation that the Laurent expansion
via $u$ at $p$ can be made well-defined not only for regular functions on a
punctured neighborhood of $p$, but also for sections of ${\cal
O}_X/{\bf Z}$:
\begin{Def}
$K:=u(\Gamma(X-p,{\cal O}_X/{\bf Z}))\cap \H'$\ .
\end{Def}
Of course, by means of the exponential map, $\Gamma(X-p,{\cal
O}_X/{\bf Z})$ may be regarded as a subspace of $\Gamma(X-p,{\cal
O}^*_{X^{an}})$.
In other words, $K$ consists of those $f\in\H'$ for which
$e^f$ lies in $u(\Gamma(X-p,{\cal O}^*_{X^{an}}))$. Obviously,
$K_0\subset K$.
Since the exterior derivative $d$ of a constant function is 0, $d$
is well-defined on ${\cal O}_X/{\bf Z}$, and (\ref{K:perp}) implies
that $K\subset K_0^{\perp}$.
\Def $\Lambda:=K/K_0$\ .
\begin{Lemma}
\label{lemma:Lambda}
The isomorphism (\ref{EHS:HS}):
$K_0^{\perp}/K_0\stackrel{\simeq}{\longrightarrow}H^1(X,{\bf C})$ maps
$\Lambda$ onto $H^1(X,{\bf Z})$; in particular,
$K_0^{\perp}/K_0\cong\Lambda\otimes{\bf C}$.
\end{Lemma}
\ \\ \noindent {\bf Proof.\ \ } The starting point in identifying $H^1(X,{\bf Z})$ with
$\Lambda$ is the exponential sequence (on $X^{an}$, of course)
\begin{equation}
\begin{array}{rcccccl}
& & & & & {\cal O}^*_{X^{an}} & \\
& & & & & e\,\uparrow\,\cong & \\
0\longrightarrow & {\bf Z} & \longrightarrow & {\cal O}_{X^{an}}
& \longrightarrow& {\cal O}_{X^{an}}/{\bf Z} & \longrightarrow 1\
\end{array}
\label{seq:exp}
\end{equation}
and its cohomology sequence
\begin{equation}\textstyle
\begin{array}{rcl}
H^1(X^{an},{\cal O}^*_{X^{an}})\ & & {\bf Z}\\
e\,\uparrow\,\cong\ \ \ \ \ \ \ \ \ \ & & \ \| \\
H^1(X^{an},{\bf Z}) \hookrightarrow
H^1(X^{an},{\cal O}_{X^{an}}) \rightarrow
H^1(X^{an}, {\cal O}_{X^{an}}/{\bf Z}) & \makebox[0pt]{$\,\,\longrightarrow\hspace{-12pt$} &
H^2(X^{an},{\bf Z})
\end{array}
\label{seq:exp:coho}
\end{equation}
But we also have an algebraic partial analogue of (\ref{seq:exp})
on $X$:
$$
0\longrightarrow{\bf Z}\longrightarrow{\cal O}_X\longrightarrow {\cal
O}_X/{\bf Z}\longrightarrow 1\ ,
$$
with the cohomology sequence
$$
\begin{array}{ccccccc}
0 & & & & & & 0 \\
\| & & & & & & \| \\
H^1(X,{\bf Z}) & \longrightarrow &
H^1(X,{\cal O}_X) & \longrightarrow & H^1(X,{\cal O}_X/{\bf Z}) &
\longrightarrow & H^2(X,{\bf Z})
\end{array}
$$
mapping functorially to (\ref{seq:exp:coho}):
$$
\begin{array}{rcccl}
0\longrightarrow H^1(X^{an},{\bf Z}) \longrightarrow &
H^1(X^{an},{\cal O}_{X^{an}}) & \longrightarrow&
{\rm Pic}^0(X) & \longrightarrow 0\\
& \cong\ \uparrow & & \uparrow & \\
& H^1(X,{\cal O}_X) & \stackrel{\cong}{\longrightarrow} &
H^1(X,{\cal O}_X/{\bf Z}) & \ .
\end{array}
$$
The commutativity of the square implies that the right
verical arrow is surjective.
We also have the commutative ladder with exact columns
$$
\begin{array}{ccc}
0 & & \\
\uparrow & & \\
H^1(X,{\cal O}_X) & \stackrel{\cong}{\longrightarrow} &
H^1(X,{\cal O}_X/{\bf Z})\\
\uparrow & & \uparrow \\
\Gamma(V-p,{\cal O}_X) & \longrightarrow &
\Gamma(V-p,{\cal O}_X/{\bf Z})\\
\uparrow & & \uparrow \\
\Gamma(V,{\cal O}_X)\oplus\Gamma(X-p,{\cal O}_X) & \longrightarrow
& \Gamma(V,{\cal O}_X/{\bf Z})\oplus\Gamma(X-p,{\cal O}_X/{\bf Z}) \\
\uparrow & & \uparrow \\
0 & & 0
\end{array}
$$
Again we note that the upper right vertical arrow must be
surjective.
Splicing the two diagrams, and completing with respect to the
order-of-vanishing filtration $P^{\bullet}$ as in (\ref{lemma:surj}),
we get
\begin{equation}
\begin{array}{rcccl}
& 0 & & 0 & \\
& \uparrow & & \uparrow & \\
0\longrightarrow H^1(X^{an},{\bf Z}) \longrightarrow &
H^1(X^{an},{\cal O}_{X^{an}}) & \longrightarrow&
{\rm Pic}^0(X) & \longrightarrow 0\\
& \uparrow & & \uparrow & \\
& \H & = & \H & \\
& \uparrow & & \uparrow & \\
& \H_++K_0 & \longrightarrow & \H_++K & \\
& \uparrow & & \uparrow & \\
& 0 & & 0 &
\end{array}
\label{ladder:ABC}
\end{equation}
The lemma now follows by simple homological algebra. Consider the
vertical ladder in the above diagram as a short exact sequence of
three complexes
$$
0\longrightarrow A^{\bullet}\longrightarrow B^{\bullet}
\longrightarrow C^{\bullet}\longrightarrow 0\ .
$$
Then $H^0(C^{\bullet})=H^1(X^{an},{\bf Z})$, $H^1(A^{\bullet})=K/K_0$,
and the connecting map in the corresponding
cohomology sequence is precisely the sought-after isomorphism
\begin{equation}
H^1(X^{an},{\bf Z})\stackrel{\cong}{\longrightarrow}\Lambda=K/K_0\ .
\label{sought-after}
\end{equation}
It remains to show that this isomorphism is induced by theone in
(\ref{EHS:HS}). The map (\ref{sought-after}) factors through the monomorphism
$$
H^1(X^{an},{\bf Z})\longrightarrow H^1(X^{an},{\cal O}_{X^{an},})\ ,
$$
which, in turn, factors through $H^1(X^{an},{\bf C})$. And the vertical sequences in
(\ref{ladder:ABC}) may be amended as in the proof of Lemma \ref{K:perp}. Then
we arrive at the following variant of (\ref{ladder:ABC}):
$$
\begin{array}{rccl}\textstyle
& H^1(X^{an},{\bf C}) & 0 & \ \ \ \ \ 0 \\
& \nearrow\ \ \uparrow\ \ \searrow & \uparrow &\ \ \ \ \ \uparrow \\
H^1(X^{an},{\bf Z}) & \longrightarrow &
H^1(X^{an},{\cal O}_{X^{an}}) & \rightarrow
{\rm Pic}^0(X)\rightarrow 0\\
& | & \uparrow & \ \ \ \ \ \uparrow \\
& \ \ \ K_0^{\perp}\ = & K_0^{\perp} & = \ K_0^{\perp} \\
& \uparrow & \uparrow & \ \ \ \ \ \uparrow \\
& \ \ \ K_0\ \rightarrow & (K_0^{\perp} \cap \H_+)+K_0 & \rightarrow
(K_0^{\perp} \cap \H_+)+K \rightarrow\Lambda\\
& \uparrow & \uparrow & \ \ \ \ \ \uparrow \\
& 0 & 0 & \ \ \ \ \ 0
\end{array}
$$
With this diagram it is easy to trace the map $H^1(X^{an},{\bf Z}) \longrightarrow
\Lambda$ and see that it fits in the commutative square
$$
\begin{array}{ccc}
H^1(X^{an},{\bf C}) & \longrightarrow & K_0^{\perp}/K_0\\
\uparrow & & \uparrow\\
H^1(X^{an},{\bf Z}) & \longrightarrow & \Lambda
\end{array}
$$
with natural inclusions as the vertical arrows. \ $\displaystyle\Box$\\ \ \par
\begin{Prop}
The isomorphism in (\ref{EHS:HS}) is {\em symplectic}, identifying
${\frac{1}{2\pi i}<\ ,\ >}$ on $\Lambda$ with the intersection form $Q(\ ,\ )$
on $H^1(X,{\bf Z})$.
\end{Prop}
\ \\ \noindent {\bf Proof.\ \ } The above proposition is established in \cite{AD}, following \cite{SW}, by
reasoning
similar to that in the proof of the Riemann reciprocity laws. Alternatively, we
can identify the residue pairing with the cup product
$$
H^0(X,\Omega_X)\otimes H^1(X,{\cal O}_X)\longrightarrow H^1(X,\Omega_X^1)\cong
H^2(X,{\bf C})\cong {\bf C}\ ,
$$
as Serre suggests in \cite{S}, and then relate the cup product to the
intersection pairing. \ $\displaystyle\Box$\\ \ \par
We now complete the identifications above to include the Hodge
structure. First, $U:=K_0^{\perp}\cap\H'_+$ is easily seen to be
mapped onto
$$
H^0(X,\Omega_X^1)=H^{1,0}(X)=F^1H^1(X,{\bf C})
$$
by the isomorphism (\ref{EHS:HS}). Let $\overline{U}$ be the
complex conjugate of $U$ with respect to the real structure which
$\Lambda$ defines on $K_0^{\perp}/K_0=\Lambda\otimes{\bf C}$. Then
(\ref{EHS:HS}) identifies $\overline{U}$ with $H^{0,1}(X)$.
Finally, let $Z\subset K_0^{\perp}$ to be the pre-image of
$\overline{U}$ with respect to the projection
$$
K_0^{\perp}\,\longrightarrow\hspace{-12pt K_0^{\perp}/K_0\ .
$$
It is easy to see that $Z\cap\H'_+=0$,
$\H'=Z\oplus\H'_+$, and that $Z$ is a maximal isotropic subspace
of $\H'$.
To summarize, we have constructed an extended HS $(Z,K_0,\Lambda)$
out of the data $(X,p,u)$.
\section{Infinitesimal equivariance of the period map}
We will work with a miniversal deformation $\pi:{\cal
X}\rightarrow S$ of a complete smooth curve $X$ of genus $g\geq
2$, as in (\ref{deform:X}), with a sufficiently small contractible
open Stein manifold $S$ as its base.
It was shown in Theorem \ref{thm:S:infhom} that $S$ is an
infinitesimally homogeneous space for $\d$. Consider the usual
and the extended period maps on $S$:
\begin{eqnarray}
S & \stackrel{\widehat{\Phi}}{\longrightarrow} & \widehat{\cal
A}_g\nonumber\\
& &\nonumber\\
\Phi\downarrow & j\nearrow & \downsurj \\
& &\nonumber\\
U & \subset & {\bf D}={\bf H}_g\ .\nonumber
\label{maps:period}
\end{eqnarray}
Let $U$ be a neighborhood of $\Phi(0)$ in ${\bf D}$ containing the
image of $S$; we assume that $U$ is small enough to admit lifts to
$\widehat{\cal A}_g$. Choose the lift $j$ making the diagram
commutative (i.e. $j\circ\Phi=\widehat{\Phi}$ on $S$). This makes
$U$ an infinitesimally homogeneous space for ${\bf sp}(\H')$.
\Def $\varphi: \d \longrightarrow {\bf sp}(\H')$ is the
Lie-algebra homomorphism given by
$$
f\frac{d}{dz}\longmapsto\{g\mapsto fg'\ \ \ \forall g\in \H'\}\ .
$$
Using the identification ${\bf sp}(\H')\cong\widehat{S^2}(\H')$
(see Section \ref{sect:notat}), the map $\varphi$ may also be
written as
\begin{eqnarray*}
\varphi:\d & \longrightarrow & \widehat{S^2}(\H')\\
z^{k+1}\frac{d}{dz} & \longmapsto &
\frac{1}{2}\sum_{j\in{\bf Z}-\{0\}} z^{-j}z^{j+k}\ .
\end{eqnarray*}
We note that $\varphi$ is an irreducible representation of the
Witt algebra on $\H'$, described in \cite{KR}, (1.2), where it is
denoted $V'_{0,0}$.
The following is an adaptation of a theorem of Arbarello and De
Concini \cite{AD}.
\begin{Thm}
\label{thm:equi}
The period map $\Phi: S\longrightarrow U\subset{\bf D}$
is infinitesimally equivariant, i.e. there exists a commutative
diagram
\begin{equation}
\label{phi:induce:dPhi}
\begin{array}{ccc}
\d & \stackrel{\varphi}{\longrightarrow} & {\bf sp}(\H') \\
\downarrow & & \downarrow \\
\Gamma(S,\Theta_S) & \stackrel{d\Phi}{\longrightarrow} &
\Gamma(U,\Theta_{{\bf D}}) \ .
\end{array}
\end{equation}
The vertical arrows are Lie algebra
anti-homomorphisms, while the horizontal ones are Lie algebra
homomorphisms. The vertical arrows induce surjections onto $T_tS$
(respectively, $T_{H}{\bf D}$) for any point $t\in S$ (respectively,
$H\in U\subset{\bf D}$). \end{Thm}
\refstepcounter{Thm The vertical arrows are not unique.
\newpage
\section{The second differential of the period map}
\label{second:diff}
We continue with a miniversal deformation(\ref{deform:X}) of
$X$. Theorem \ref{thm:equi} allows one to calculate the various
differentials of the period map. We
begin by specializingdiagram (\ref{phi:induce:dPhi}) to $0\in S$:
\begin{equation}
\label{diag:equi:at_0}
\begin{array}{ccc}
\d & \stackrel{\varphi}{\longrightarrow} & {\bf sp}(\H') \\
\downsurj & & \downsurj \\
T_0S & \stackrel{d_0\Phi}{\longrightarrow} & T_{\Phi(0)}{\bf D}
\end{array}
\end{equation}
A well-known theorem of Griffiths \cite{Gri} factors $d_0\Phi$ as
\begin{equation}
\label{diag:Griffiths}
\begin{array}{ccl}
T_0S & \stackrel{d_0\Phi}{\longrightarrow} & T_{\Phi(0)}{\bf D} \\
\kappa\downarrow\cong & & \ || \\
H^1(\Theta_X) & \stackrel{\nu}{\longrightarrow} &
\mbox{\rm Hom}^{(s)}(H^0(\omega_X),H^1({\cal O}_X))\ ,
\end{array}
\end{equation}
where $\kappa$ is the Kodaira-Spencer isomorphism, and
$\nu$ is the map defined by the cup-product pairing
\begin{equation}
\label{cup}
H^1(\Theta_X)\otimes
H^0(\omega_X)\stackrel{\smile}{\longrightarrow} H^1({\cal O}_X)\ ,
\end{equation}
itself induced by the contraction pairing of sheaves
$\Theta_X\otimes\omega_X\stackrel{\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,}{\rightarrow}{\cal O}_X$.
Splicing (\ref{diag:equi:at_0}) and (\ref{diag:Griffiths}) yields
\begin{equation}
\label{diag:post-Griffiths}
\begin{array}{ccl}
\d & \stackrel{\varphi}{\longrightarrow} & {\bf sp}(\H')\\
\downsurj & & \ \downsurj{\scriptstyle \rho} \\
H^1(\Theta_X) & \stackrel{\nu}{\longrightarrow} &
\mbox{\rm Hom}^{(s)}(H^0(\omega_X),H^1({\cal O}_X))\ ,
\end{array}
\end{equation}
which we want to work out explicitly. As before, let $V$ be an
affine open set in $X$ containing $p$, so that $X-p$ and $V$ form
an affine covering of $X$. Let $\xi$ be a vector field on $V-p$
with $u(\xi)=f(z)\frac{d}{dz}$ in $\d$. Let $\omega$ be a global
holomorphic 1-form on $X$ with $u(\omega|_{V-p})=dg$ for some
$g\in\H'$. Then the cup-product pairing (\ref{cup}) gives
$$
[\xi]\smile[\omega]=[-\xi\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,(\omega|_{V-p})]\in H^1({\cal
O}_X)\ .
$$
We observe that the minus sign is built into $\rho$ (see
(\ref{rho})), and that
$$
u(\xi\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,(\omega|_{V-p}))=f\frac{d}{dz}\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,
dg=fg'=\varphi(f\frac{d}{dz})g\ ,
$$
which is how (\ref{diag:post-Griffiths}) and, indeed, the theorem
of Arbarello and De Concini (\ref{thm:equi}) is proved.
We would like to work out an equally explicit realization of the
second differential of $\Phi$ (higher-order cases are
similar). Our starting point is again Theorem \ref{thm:equi}. We
simply pass from Lie algebras to their (reduced) enveloping
algebras to obtain a commutative diagram
\begin{equation}
\label{diag:U:2}
\begin{array}{lcl}
\overline{\cal U}^{(2)}\d & \stackrel{\varphi^{(2)}}{\longrightarrow} & \overline{\cal U}^{(2)}{\bf sp}(\H')
\\ \downarrow & & \downarrow \\
\Gamma(S,\Theta^{(2)}_S) & \stackrel{d^2\Phi}{\longrightarrow} &
\Gamma(U,\Theta^{(2)}_{{\bf D}})\ ,
\end{array}
\end{equation}
where $\Theta^{(2)}={\cal D}^{(2)}/{\cal O}$ stands for the
second-order tangent sheaf, and $\overline{\cal U}^{(2)}$ is the notation introduced
in Section \ref{sect:notat}.
Again, to be precise, the maps emanating from the upper-left
corner reverse the order of products, while the remaining maps are
the second-degree parts of filtered ring homomorphisms.
Restricting to $0\in S$, we obtain
\begin{equation}
\label{equi:at_0:two}
\begin{array}{ccl}
\overline{\cal U}^{(2)}\d & \stackrel{\varphi^{(2)}}{\longrightarrow} &
\overline{\cal U}^{(2)}{\bf sp}(\H')\\
\lambda^{(2)}\downsurj\& & \ \downsurj\rho^{(2)} \\
T_0^{(2)}S & \stackrel{d^2_0\Phi}{\longrightarrow} &
T_0^{(2)}{\bf D}\ .
\end{array}
\end{equation}
\begin{Prop}
\label{split}
The second tangent space of the period domain ${\bf D}$ at the
point corresponding to a HS $(H,F^{\bullet})$ admits a canonical
splitting $$
T_F^{(2)}{\bf D}=T_F{\bf D}\oplus S^2T_F{\bf D}\ .
$$
\end{Prop}
\ \\ \noindent {\bf Proof.\ \ }
Let ${\bf g}=\mbox{\rm End}(H)$ ($={\bf gl}(2g,{\bf C})$), and ${\bf
s}={\bf sp}(H)$ (symplectic with respect to the polarization on
$H$). Then ${\bf D}$ is infinitesimally homogeneous under the action of
$\bf s$, and
$$
T_F{\bf D}\cong{\bf s}^{-1,1}\cong\mbox{\rm Hom}^{(s)}(F^1,H/F^1)\ .
$$
We also have a natural surjection $\overline{\cal U}^{(2)}{\bf s}\,\longrightarrow\hspace{-12pt T_F^{(2)}{\bf D}$.
Its restriction to $\overline{\cal U}^{(2)}{\bf s}^{-1,1}$ is an isomorphism by reason
of dimension. But ${\bf s}^{-1,1}$ is an abelian Lie algebra, i.e.
$$
\overline{\cal U}^{(2)}{\bf s}^{-1,1}={\bf s}^{-1,1}\oplus S^2{\bf s}^{-1,1}\ .$$
\ $\displaystyle\Box$\\ \ \par
In view
of Proposition \ref{split},
$d^2_0\Phi:T_0^{(2)}S\rightarrow T_{\Phi(0)}^{(2)}{\bf D}$ breaks up
into a direct sum of two components:
{\em the symbol map}
$$
\begin{array}{ccl}
T_0^{(2)}S & \stackrel{\sigma}{\longrightarrow} & S^2{\bf
s}^{-1,1}\\
(\Upsilon+\sum_iZ_i\Xi_i)|_0 & \longmapsto &
d_0\Phi(Z|_0)\otimes d_0\Phi(\Xi|_0)\ \ \ (\mbox{\rm order does
not matter here}),
\end{array}
$$
where $\Upsilon,Z_i,\Xi_i\in\Gamma(S,\Theta_S)$, and
{\em the linear part}
$$
\ell: T_0^{(2)}S\longrightarrow{\bf s}^{-1,1}\ .
$$
It is the linear part that is really interesting. A typical
second-order tangent vector to $S$ at 0,
$(\Upsilon+\sum_iZ_i\Xi_i)|_0$, is sent by $\ell$ to
$$
d_0\Phi(\Upsilon)+\sum_i\ell((Z_i\Xi_i)|_0)\ .
$$
Thus, it suffices to understand $\ell((Z\Xi)|_0)$ for
$Z,\Xi\in\Gamma(S,\Theta_S)$.
By surjectivity of $\lambda^{(2)}$ in
(\ref{equi:at_0:two}), we may assume that the vector fields $Z$
and $\Xi$ on $S$ are the images, respectively, of
some $f_1\frac{d}{dz}$ and $f_2\frac{d}{dz}$ in $\d$, under the map
$\d\rightarrow\Gamma(S,\Theta_S)$, whose restriction$\lambda$ is.
Then (\ref{equi:at_0:two}) implies that
\begin{equation}
\label{vw}
d_0^2\Phi((Z\Xi)|_0)=\rho^{(2)}\circ\varphi^{(2)}(f_1\frac{d}{dz}
f_2\frac{d}{dz})=\rho^{(2)}(vw) \,
\end{equation}
where $v=\varphi(f_1\frac{d}{dz})$ and
$w=\varphi(f_2\frac{d}{dz})$.
Now, the map$\rho^{(2)}$ in (\ref{equi:at_0:two})
is not induced by $\rho:{\bf sp}(\H')\rightarrow{\bf s}^{-1,1}$,
which was a restriction of the map, also denoted $\rho$ in
(\ref{rho}),
$$
\mbox{\rm End}(\H')\longrightarrow{\bf g}^{-1,1}\ .
$$
In fact, the maps $\rho$ are not even Lie algebra morphisms.
Nevertheless, there is a way to reduce$\rho^{(2)}$ to
$\rho$. This will require a more detailed understanding of the
infinitesimal action of ${\bf sp}(\H')$; in fact, we need to work
out how the group $Sp(\H')\subset\mbox{\rm Aut}(\H')$ acts on a neighborhood
of a point in $\widehat{\cal A}_g$.
Since any element in the group $\mbox{\rm Aut}(\H')$ may be written as
$I+\alpha$, where $\alpha\in\mbox{\rm End}(\H')$, we have the following map
from $\mbox{\rm Aut}(\H')$ to $\mbox{\rm Aut}(H)$:
$$
A\longmapsto I-\rho(\alpha), \ {\rm where}\ I+\alpha=A^{-1}\ .
$$
This map will be denoted $R$. So
\begin{equation}
R(A)=I-\rho(A^{-1}-I)\ .
\label{R}
\end{equation}
\noindent {\bf Caution:} $R$ is not a group homomorphism.
Let $(H,F^{\bullet}_t)$ be the Hodge structure corresponding to a
point in $U$ near $\Phi(0)$. The HS $(H,F^{\bullet}_t)$ comes from
an extended HS $(Z_t,K_{0,t},\Lambda_t)$. The assumption that $U$ is
small and infinitesimally homogeneous under ${\bf sp}(\H')$ implies
that there exists $A_t\in Sp(\H')$ such that $K_{0,t}$ and $Z_t$
are images under $A_t$ of $K_0$ and $Z$, respectively (we refer to
the components of the extended HS corresponding to $\Phi(0)$).
\begin{Lemma}
\label{how:Sp:acts}
In this situation $F_t^1=R(A_t)F^1$.
\end{Lemma}
\ \\ \noindent {\bf Proof.\ \ }
We wish to compare the Hodge structures
$$
(H=K_0^{\perp}/K_0,F^1=K_0^{\perp}\cap\H'_+)
$$
and
$$
(H_t=K_{0,t}^{\perp}/K_{0,t},F_t^1=K_{0,t}^{\perp}\cap\H'_+)\ .
$$
To do so, we identify $H_t$ with $H$ by $A_t^{-1}$. Then the
comparison involves two subspaces of $H$:
$F^1=K_0^{\perp}\cap\H'_+$ and
$$
A_t^{-1}F_t^1=A_t^{-1}(K_{0,t}^{\perp})\cap A_t^{-1}(\H'_+)=
K_0^{\perp}\cap A_t^{-1}(\H'_+)\ .
$$
We regard $U$ as a subset of the Grassmannian
$$
Grass(F^1,H)=\mbox{\rm Aut}(H)/\{A\,|\,A(F^1)\subseteq F^1\}\ .
$$
Any element of $\mbox{\rm Aut}(H)$ may be written as $I+T$ for some
$T\in{\bf g}$, and if$I+T\in\{A\,|\,A(F^1)\subseteq F^1\}$,
then $T\in{\bf g}^{0,0}$. If some $I+T\in\mbox{\rm Aut}(H)$ moves
$F^1$ to $A_t^{-1}F^1_t$, then so does $I+T^{-1,1}$, where the
subscript refers to the $(-1,1)$-component of $T$ under the direct
sum decomposition ${\bf g}={\bf g}^{-1,1}\oplus{\bf g}^{0,0}\oplus{\bf
g}^{1,-1}$.
Thus we only need to find the map
$$
T^{-1,1}:H^{1,0}=\H'_+\cap K_0^{\perp}\longrightarrow
\H'/\H'_++K_0\cong H^{0,1}
$$
which measures deviation of $A_t^{-1}F^1_t$ from $F^1$. The
above formulas for $F^1$ and $A_t^{-1}F^1_t$ show that $T^{-1,1}$
is induced by
$$
A_t^{-1}:\H'_+\longrightarrow \H'/\H'_+\ .
$$
But if $A_t^{-1}=I+\alpha$, then
$$
\alpha:\H'_+\longrightarrow \H'/\H'_+
$$
induces the same $T^{-1,1}$. Recalling the definition of $\rho$
(\ref{rho}), this says that $T^{-1,1}=-\rho(\alpha)$. It remains
to consult the definition of $R$ (\ref{R}) and to abuse notation
by putting $F_t^1=A_t^{-1}F_t^1$. \ $\displaystyle\Box$\\ \ \par
We are now able to establish the principal formula relating the
two infinitesimal uniformizations of $U$ on the second-order level.
\begin{Lemma}
\label{two:unifs}
Let $v,w\in\mbox{\rm End}(\H')$. Then
$$
\rho^{(2)}(vw)=\rho(v)\rho(w)-\rho(w\circ v)\ ,
$$
where $w\circ v$ denotes the composition law in $\mbox{\rm End}(\H')$.
\end{Lemma}
\ \\ \noindent {\bf Proof.\ \ }
Let $V=\rho(v)$, $W=\rho(w)$. $V$ and $W$ are vectors in
$T_{\Phi(0)}{\bf D}\cong {\bf s}^{-1,1}$. For any $Z\in{\bf s}^{-1,1}$
we will write $\widetilde{Z}$ to denote the vector field on $U$
correspondingto $Z$ under the Lie algebra homomorphism
$$
{\bf s}^{-1,1}\longrightarrow\Gamma(U,\Theta_{{\bf D}})\ .
$$
In particular, $\widetilde{V}|_{\Phi(0)}=V$,
$\widetilde{W}|_{\Phi(0)}=W$. Let $f$ be any smooth function on
$U$. Then
\begin{eqnarray*}
\lefteqn{\rho^{(2)}(vw)f=} & & \\
& = & \frac{\partial^2}{\partial t\partial s}|_0
f\{R(\exp tv\circ\exp sw)\Phi(0)\}\\
& = & \frac{\partial^2}{\partial t\partial s}|_0
f\{[I-\rho(\exp(-sw)\circ\exp(-tv) -I)]\Phi(0)\}\\
& = & \frac{\partial^2}{\partial t\partial s}|_0
f\{[I-\rho(-sw-tv+tsw\circ v+\ldots)]\Phi(0)\}\\
& = & \frac{\partial^2}{\partial t\partial s}|_0
f\{(I+tV+sW-ts\rho(w\circ v)+\ldots)\Phi(0)\}\\
& = & \frac{d}{dt}|_0\left\{\frac{\partial}{\partial s}|_0
f\{[(I+tV+o(t))+s(W-t\rho(w\circ v)+o(t))+o(s)]\Phi(0)\}
\right\}\\
& = & \frac{d}{dt}|_0\left\{
[(\widetilde{W}-t(\rho(w\circ v))^{\sim}+o(t))f]
\{(I+tV+o(t))\Phi(0)\}\right\}\\
& = & \frac{d}{dt}|_0\biggl\{
[\widetilde{W}f]\{(I+tV+\ldots)\Phi(0)\}-
t[(\rho(w\circ v))^{\sim}f]\{(I+tV+\ldots)\Phi(0)\}+\\
& & +o(t) \biggr\}\\
& = & V\widetilde{W}f-\rho(w\circ v)f\ .
\end{eqnarray*}\ $\displaystyle\Box$\\ \ \par
Observe that $\rho$ takes
its values in ${\bf g}^{-1,1}$, which is an abelian Lie algebra.
Thus the lemma gives a splitting of
$$
\rho^{(2)}:\overline{\cal U}^{(2)}\mbox{\rm End}(\H')\longrightarrow\overline{\cal U}^{(2)}{\bf g}^{-1,1}={\bf
g}^{-1,1}
\oplus S^2{\bf g}^{-1,1}\ :
$$
$V\widetilde{W}=\rho(v)\rho(w)$ ispurely quadratic (=the symbol
part), and $-\rho(w\circ v)$ is the linear part. Going back to
(\ref{vw}),this implies that $\ell((Z\Xi)|_0)=-\rho(w\circ v)$,
which proves the following
\begin{Thm}
\label{Thm:main}
If $Z,\Xi\in \Gamma(S,\Theta_S)$ lift to $f_1\frac{d}{dz},
f_2\frac{d}{dz}\in\d$, then
the linear part of $d^2_0\Phi$,
$$
\ell:T_0^{(2)}S\longrightarrow T_{\Phi(0)}{\bf D}={\bf s}^{-1,1}\ ,
$$
sends
$(Z\Xi)|_0$ to the negative of the image under
$\rho:\mbox{\rm End}(\H')\rightarrow{\bf g}^{-1,1}$ of the composition in
$\mbox{\rm End}(\H')$ of $\varphi(f_1\frac{d}{dz})$ and
$\varphi(f_2\frac{d}{dz})$, in reverse order:
\begin{equation}
g\longmapsto f_2f'_1g'+f_1f_2g''\ .
\label{prescription}
\end{equation}
\end{Thm}
\refstepcounter{Thm A composition (in $\mbox{\rm End}(\H')$) of two elements of ${\bf
sp}(\H')$ need not be in ${\bf sp}(\H')$. In particular, it is not
a priori obvious that the image of $\ell$ is in
$$
\mbox{\rm Hom}^{(s)}(H^0(\omega_X),H^1({\cal
O}_X))\cong{\bf s}^{-1,1}\ .
$$
There is a better-known object which carries part of the
information contained in the linear part $\ell$ of the period
map's second differential. It is {\em the second fundamental form
of the VHS} of \cite{CGGH}, the map
$$
{\rm II}: T_0^{(2)}S/T_0S=S^2T_0S\longrightarrow
T_{\Phi(0)}{\bf D}/im\,(d_0\Phi)
$$
induced by $\ell$.
\begin{Thm}
\label{Thm:II}
The prescription (\ref{prescription}) for computing $\ell$ gives
a formula for {\rm II}, which coincides with that in \cite{K1}, $\S 6$:
\begin{equation}
\label{II}
{\rm II}(Z\otimes\Xi)=
\{\omega\mapsto\xi\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\pounds_\zeta\omega\}\bmod im\,(d_0\Phi)
\end{equation}
for any$Z,\Xi\in T_0S$ with KS representatives
$\zeta,\xi\in\Gamma(V-p,\Theta_X)$, and $\omega\in H^0(X,\omega_X)$.
\end{Thm}
\ \\ \noindent {\bf Proof.\ \ } Recall that a choice of a point $p$ on the curve $X$ and a local
parameter $z$ near $p$ allows one to represent
$\omega\in H^0(X,\omega_X)$ by some $g\in\H'$ with $dg=\omega$ near
$p$. The vectors $Z$ and $\Xi$ are the images under $\rho$ of
some $f_1\frac{d}{dz}$ and $f_2\frac{d}{dz}$ in $\d$,
i.e. $f_1\frac{d}{dz}$ and $f_2\frac{d}{dz}$ are the
Laurent expansions at $p$ of $\zeta$ and $\xi$, respectively.
Working out (\ref{II}) in terms of $z$, using the formula for the
Lie derivative
$$ \pounds_\zeta\omega=d\zeta\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\omega+\zeta\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,
d\omega\ ,
$$
easily yields (\ref{prescription}) ($\bmod\
im\,(d_0\Phi)$),as was already done, in fact, in \cite{K1}, $\S
6$. \ $\displaystyle\Box$\\ \ \par
\refstepcounter{Thm
Thus the prescription for ${\rm II}$ given in \cite{K1} turns out to be
well-defined for second-order differential operators on $S$ and
not just their symbols, and the values given by that prescription
are not merely equivalence classes modulo $im\,(d_0\Phi)$.
\newpage
\section{Relation with the second Kodaira-Spencer class}
\label{rel:withKS2}
As explained at the beginning of the previous section, the first
differential of the period map is given bycup product with the
(first) Kodaira-Spencer class $\kappa=\kappa_1$ of the
deformation. In \cite{K2} we have shown that ${\rm II}$ depends only on
the {\em second} Kodaira-Spencer class $\kappa_2$ (more precisely,
on $\kappa_2\ \bmod\ im\,(\kappa_1)$) introduced recently in
\cite{BG}, \cite{EV} and \cite{R1}. In this section we will
explain, in the case of curves, how the full second differential
$$
d^2_0\Phi: T_0^{(2)}S\longrightarrow T_{\Phi(0)}^{(2)}{\bf D}
$$
factors through the second KS mapping
$$
\kappa_2: T_0^{(2)}S\longrightarrow {\bf T}_X^{(2)}\ .
$$
Let us recall first the construction of ${\bf T}_X^{(2)}$, the space of
second-order deformations of $X$. Our reference is \cite{R1} or
\cite{K2}.
Let $X_2$ denote the symmetric product of the curve $X$ with
itself;write
$$
g: X\times X\longrightarrow X_2
$$
for the obvious projection map, and $i:X\hookrightarrow X_2$ for the
inclusion of the diagonal. Then ${\bf T}_X^{(2)}={\bf H}^1(X_2,{\cal
K}^{\bullet})$, where ${\cal K}^{\bullet}$ is the sheaf complex on
$X_2$
$$
\begin{array}{ccc}
{\scriptstyle -1} & & {\scriptstyle 0}\\
(g_*(\Theta_X^{\makebox[0pt][l]{$\scriptstyle\times$ 2}))^- & \stackrel{[\ ,\
]}{\longrightarrow} & i_*\Theta_X\ .
\end{array}
$$
Here $\makebox[0pt][l]{$\times$$ stands for the exterior tensor product on
$X\times X$, $(\ \ )^-$ denotes anti-invariants of the
${\bf Z}/2{\bf Z}$-action, and the differential is the restriction to the
diagonal followed by the Lie bracket of vector fields.
In practice it seems easier to do the following. Letting
$C^{\bullet}$ denote the \v{C}ech cochain complex
$\check{C}^{\bullet}({\cal U},\Theta_X)$ of $\Theta_X$ with
respect to an affine covering $\cal U$ of $X$, one may compute
${\bf T}_X^{(2)}$ as the cohomology of the simple complex associated to
the double complex
$$
\begin{array}{cccc}
{\scriptstyle 2} & (C^1\otimes C^1)^{(s)} & & \\
& 1\otimes\delta\uparrow -\delta\otimes 1 & & \\
{\scriptstyle 1} & (C^0\otimes C^1 + C^1\otimes C^0)^- &
\stackrel{[\ ,\ ]}{\longrightarrow} & C^1 \\
& 1\otimes\delta\uparrow +\delta\otimes 1 & & \uparrow\delta \\
{\scriptstyle 0} & (C^0\otimes C^0)^- & \stackrel{[\ ,\
]}{\longrightarrow} & C^0 \\
& {\scriptstyle -1} & & {\scriptstyle 0}
\end{array}
$$
The superscripts $^{(s)}$ and $^-$ denote the invariants and the
anti-invariants, respectively, of the ${\bf Z}/2$-action.
Working with ${\cal U}=\{X-p,V\}$ and using Laurent expansions at
$p$, we may follow the proof of Lemma \ref{lemma:surj} and replace
$C^1$ with $\d$ and $C^0$ with the completion of its image in
$\d\oplus\d$. The resulting bicomplex still computes ${\bf T}_X^{(2)}$.
In particular, ${\bf T}_X^{(2)}$ is a quotient of
$\d\oplus(\d\otimes\d)^{(s)}$.
\begin{Lemma}
\label{ks2:rep}
Assume the vector fields $Z$ and $\Xi$ on $S$ are the images
of $\zeta,\xi\in\d$ under the infinitesimal uniformization map
$\lambda: \d\longrightarrow\Gamma(S,\Theta_S)$. Then
$$
\frac{1}{2}([\xi,\zeta]+\zeta\otimes\xi+\xi\otimes\zeta)\in
\d\oplus(\d\otimes\d)^{(s)}
$$
is a representative for $\kappa_2((Z\Xi)|_0)\in {\bf T}_X^{(2)}$.
\end{Lemma}
\ \\ \noindent {\bf Proof.\ \ } The Kodaira-Spencer maps are compatible with the symbol map
${\bf T}_X^{(2)}\longrightarrow S^2{\bf T}_X^1$ in the sense that there is a
commutative diagram
$$
\begin{array}{ccc}
T_0^{(2)}S & \stackrel{\kappa_2}{\longrightarrow} & {\bf T}_X^{(2)}\\
\downarrow && \downarrow\\
S^2T_0S & \stackrel{\kappa_1^2}{\longrightarrow} & S^2{\bf T}_X^1\ .
\end{array}
$$
Thus it is natural to look for a representative of
$\kappa_2((Z\Xi)|_0)$ of the form
$$
\theta+\frac{1}{2}(\zeta\otimes\xi+\xi\otimes\zeta)\in
\d\oplus(\d\otimes\d)^{(s)}
$$
for some $\theta\in \d$. The construction of $\kappa_2$ as the
connecting morphism in a certain long exact sequence
(\cite{EV,R1,R2}), presented more explicitly in \cite{K2}, offers
the following way to determine $\theta$. Working with a covering
${\cal W}=\{W_0,W_1\}$ of $\cal X$ as in the proof of Lemma
\ref{lemma:act}, and using the subsheaf $\widetilde{\Theta}_{\cal
X}$ of $\Theta_{\cal X}$ introduced in (\ref{subseq:KS}), let
$\zeta_0,\zeta_1$ be lifts of $Z$ to $\Gamma(W_0,
\widetilde{\Theta}_{\cal X})$, $\Gamma(W_1,
\widetilde{\Theta}_{\cal X})$. Write $\widetilde{\zeta}$ to denote
$\zeta_0+\zeta_1$ viewed as a cochain in $\check{C}^0({\cal W},
\widetilde{\Theta}_{\cal X})$. A slight modification of the proof
of Prop. 2 in \cite{K2} shows that $\theta$ should be cohomologous
(in $\check{C}^1({\cal W},\Theta_{{\cal X}/S})$) to
$$
\frac{1}{2}([\widetilde{\zeta},\xi]+[\xi,\widetilde{\zeta}])=
\frac{1}{2}([\zeta_0,\xi]+[\xi,\zeta_1])=
\frac{1}{2}[\xi,\zeta_1-\zeta_0]\ .
$$
But $\zeta_1-\zeta_0$ is cohomologous to $\zeta$. Hence we can take
$\theta=\frac{1}{2}[\xi,\zeta]$. \ $\displaystyle\Box$\\ \ \par
In (\ref{split}) we explained how $T_{\Phi(0)}^{(2)}{\bf D}$
splits into $T_{\Phi(0)}{\bf D}\oplus S^2T_{\Phi(0)}{\bf D}$, with the
second differential of the period map breaking up accordingly:
$$
d_0^2\Phi=\ell\oplus \sigma\ .
$$
The symbol part factors through the square of the first KS class:
$$
\begin{array}{ccrcl}
T_0^{(2)}S && \stackrel{\sigma}{\longrightarrow} & &
S^2T_{\Phi(0)}{\bf D} \\
& & & & \\
\downsurj & & {\scriptstyle (d_0\Phi)^2}\nearrow & & \ \ \uparrow
\nu_1^2\\
& & & & \\
S^2T_0S & & \stackrel{\kappa_1^2}{\longrightarrow} & &
S^2{\bf T}_X=S^2\mbox{\rm Hom}^{(s)}(H^0(\omega_X),H^1({\cal O}_X))\ .
\end{array}
$$
This diagram may be directly obtained from (\ref{diag:Griffiths})
and carries no additional information.
Now to the linear part $\ell$ of $d_0^2\Phi$. ``Recall" the
canonical bijection $b$ given by the composition of the obvious
maps
$$
b: \d\oplus(\d\otimes\d)^{(s)}\hookrightarrow
\d\oplus(\d\otimes\d) \,\longrightarrow\hspace{-12pt \overline{\cal U}^{(2)}\d\ .
$$
\begin{Lemma}
\label{lemma:canon.bij}
The canonical bijection $b$ fits in the commutative
square
$$
\begin{array}{ccc}
\overline{\cal U}^{(2)}\d & \stackrel{b}{\longleftarrow} &
\d\oplus(\d\otimes\d)^{(s)}\\
\downsurj & & \downsurj \\
T_0^{(2)}S & \stackrel{\kappa_2}{\longrightarrow} & {\bf T}_X^{(2)}
\end{array}
$$
with bijective horizontal arrows, and surjective vertical ones.
\end{Lemma}
\ \\ \noindent {\bf Proof.\ \ } It suffices to show that if $\zeta,\xi\in\d$ lift the vector
fields $Z$ and $\Xi$ on $S$, then $b^{-1}(\zeta\xi)$ lifts
$\kappa_2((\Xi Z)|_0)$ under the projection
$$
\d\oplus(\d\otimes\d)^{(s)}\,\longrightarrow\hspace{-12pt {\bf T}_X^{(2)}\ .
$$
In other words, we
must verify that $b^{-1}(\zeta\xi)$ is a KS representative for
$(\Xi Z)|_0\in T_0^{(2)}S$. From the definition of $b$ it easily
follows that
$$
b^{-1}(\zeta\xi)=
\frac{1}{2}([\xi,\zeta]+\zeta\otimes\xi+\xi\otimes\zeta)\ .
$$
This, together with Lemma \ref{ks2:rep}, implies our statement. \ $\displaystyle\Box$\\ \ \par
\begin{Def}
We define $\nu_2:{\bf T}_X^{(2)}\longrightarrow T_{\Phi(0)}{\bf D}=
\mbox{\rm Hom}^{(s)}(H^0(\omega_X),H^1({\cal O}_X))$ as the composition
$$
{\bf T}_X^{(2)}\stackrel{\kappa_2^{-1}}{\longrightarrow}T_0^{(2)}S
\stackrel{\ell}{\longrightarrow}T_{\Phi(0)}{\bf D}\ .
$$
\end{Def}
Thus we have a commutative triangle
\begin{equation}
\begin{array}{ccccc}
T_0^{(2)}S & & \stackrel{\ell}{\longrightarrow} & &
T_{\Phi(0)}{\bf D}\\
& & & & \\
& \searrow\kappa_2 & & \nu_2\nearrow & \\
& & & & \\
& & {\bf T}_X^{(2)} & & \ .
\end{array}
\end{equation}
\begin{Thm}
\label{Interpret:coho}
$\nu_2:{\bf T}_X^{(2)}={\bf H}^1({\cal
K}^{\bullet})\longrightarrow\mbox{\rm Hom}(H^0(\omega_X),H^1({\cal O}_X))$
is induced by the pairing
\begin{equation}
{\bf H}^1({\cal K}^{\bullet})\otimes H^0(\omega_X)\longrightarrow
H^1({\cal O}_X)\ ,
\label{pairing:coho}
\end{equation}
defined on the \v{C}ech cochain level by the coupling
\begin{equation}
\label{pairing:cochain}
\begin{array}{cclcl}
(\check{C}^1(\Theta_X))^{\otimes 2} \oplus
\check{C}^1(\Theta_X) & \times\ & \check{C}^0(\omega_X) &
\longrightarrow & \check{C}^1({\cal O}_X) \\
(\zeta\otimes\xi+\upsilon) &
\times & \omega & \longmapsto &
\xi\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\pounds_{\zeta}\omega-\upsilon\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\omega\ .
\end{array}
\end{equation}
\end{Thm}
\ \\ \noindent {\bf Proof.\ \ } It suffices to study the effect of
$\nu_2=\ell\circ\kappa_2^{-1}$ on an element
$x$ of ${\bf T}_X^{(2)}$ represented by
$$
\frac{1}{2}(\zeta\otimes\xi+\xi\otimes\zeta)+\upsilon\in
(\d\otimes\d)^{(s)}\oplus \d\ .
$$
According to Lemma \ref{ks2:rep},
$$
\kappa_2^{-1}(x)=(Z\Xi+\Upsilon-\frac{1}{2}[Z,\Xi])|_0\ ,
$$
where $Z,\Xi$ and $\Upsilon$ are the images of $\zeta,\xi$ and
$\upsilon$, respectively, under the uniformization map $\lambda:
\d\longrightarrow\Gamma(S,\Theta_S)$. Note that
$\lambda([\zeta,\xi])=-[Z,\Xi]$.
As explained in the previous section,
$\ell((Z\Xi+\Upsilon-\frac{1}{2}[Z,\Xi])|_0)$ is a map
$H^0(\omega_X)\longrightarrow H^1({\cal O}_X)$ given by
\begin{equation}
\omega\longmapsto\xi\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\pounds_{\zeta}\omega
-(\upsilon-\frac{1}{2}[\zeta,\xi])\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\omega\ .
\label{pre-Cartan}
\end{equation}
However, Cartan's identity gives
$$
\pounds_{\zeta}(\xi\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\omega)-\xi\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\pounds_{\zeta}\omega=
[\zeta,\xi]\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\omega\ ,
$$
and on a curve $\pounds_{\zeta}(\xi\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\omega)=
\zeta\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\pounds_{\xi}\omega$. Hence the right-hand side of
(\ref{pre-Cartan}) equals
$$
\frac{1}{2}(\xi\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\pounds_{\zeta}\omega+
\zeta\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\pounds_{\xi}\omega)-\upsilon\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\omega\ .
$$
\ $\displaystyle\Box$\\ \ \par
\begin{Cor}
\label{II:factors}
The second fundamental form of the VHS, {\rm II}, factors through
$S^2{\bf T}_X$.
\end{Cor}
\noindent This fact was already proved in complete generality
(for a deformation of any compact K\"{a}hler manifold) in
\cite{K2}, using Archimedean cohomology.
We conclude with a diagram summarizing the relationships between
some of the maps discussed in this section:
\begin{equation}\begin{array}{ccccc}
& & \ell & & \\
& & & & \\
T_0^{(2)}S & \stackrel{\kappa_2}{\longrightarrow} &
{\bf T}^{(2)}_X & \stackrel{\nu_2}{\longrightarrow} &
T_{\Phi(0)}{\bf D} \\
\downsurj && \downsurj & & \downsurj\\
S^2T_0S & \stackrel{\kappa_1^2}{\longrightarrow} &
S^2{\bf T}_X &
\stackrel{\nu_2/\,im\,\nu_1}{\longrightarrow} & T_{\Phi(0)}{\bf D}/\,im\,\nu_1 \\
& & & & \\
& & {\rm II} & &
\end{array}
\label{relationships}
\end{equation}
\section{The higher-order case}
We have the following analogues of the results in sections
\ref{second:diff} and \ref{rel:withKS2}. The proofs, which are
notationally cumbersome transcriptions of the $n=2$ case, are omitted.
\begin{Prop}
The $n^{th}$ tangent space of the period domain ${\bf D}$ at a point
corresponding to a HS $(H,F^{\bullet})$ admits a canonical
splitting
$$
T_F^{(n)}{\bf D}=T_F{\bf D}\oplus S^2T_F{\bf D} \oplus\ldots\oplus S^nT_F{\bf D}\ .
$$
\end{Prop}
\noindent The $n^{th}$ differential of the period map splits
accordingly:
$$
d_0^n\Phi=\ell_1^{(n)}+\ldots+\ell_n^{(n)}\ .
$$
E.g. what we called $\ell$ and $\sigma$ earlier are $\ell_1^{(2)}$
and $\ell_2^{(2)}$, respectively.
Thus it suffices to describe the $k^{th}$ component of $d_0^n\Phi$,
$$
\ell_k^{(n)}: T_0^{(n)}{\bf D}\longrightarrow S^k T_{\Phi(0)}{\bf D}\ .
$$
\begin{Thm}
If $Z_1,\ldots,Z_n\in\Gamma(S,\Theta_S)$ lift to
$\zeta_1,\ldots,\zeta_n\in\d$, then
\newcounter{bean}
\begin{list}{\rm\alph{bean})}{\usecounter{bean}}
\item $\ell_1^{(n)}$ sends $(Z_1\ldots Z_n)|_0$ to $(-1)^{n-1}$ times the image
under
$\rho:\mbox{\rm End}(\H')\longrightarrow{\bf g}^{-1,1}$ of the composition
in $\mbox{\rm End}(\H')$ of
$\varphi(\zeta_1),\ldots,\varphi(\zeta_n)$ in reverse
order;
\item $\ell_k^{(n)}$ is the sum, over all partitions of $k$, of the
symmetrized tensor products
$$
\overline{\bigotimes}_{\sum_i p_i=k}\ell_1^{(p_i)}\ ;
$$
\item $d_0^{n}\Phi$, as well as each $\ell_k^{(n)}$, factors
through ${\bf T}_X^{(n)}$.
\end{list}
\label{thm:main:higher}
\end{Thm}
We may add to (a) that in terms of the covering $\{V,X-p\}$ of $X$
as above, $\ell_1^{(n)}((Z_1\ldots Z_n)|_0)$ can be also described as
follows: it is a map
$$
H^0(X,\omega_X)\longrightarrow H^1(X,{\cal O}_X)
$$
sending the class represented by a form $\omega$ on $V$ to the class
represented by the function
$$
(-1)^n\zeta_n\,\rule{.1in}{.5pt}\rule{.5pt}{1.5mm}\,\,\pounds_{\zeta_{n-1}}\ldots \pounds_{\zeta_1}
\omega
$$
on $V-p$. Here we assume that the lifts $\zeta_i\in
\d$ of $Z_i\in\Gamma(S,\Theta_S)$ converge and define regular vector
fields on
$V-p$.
\newpage
|
1994-06-17T17:26:53 | 9406 | alg-geom/9406005 | en | https://arxiv.org/abs/alg-geom/9406005 | [
"alg-geom",
"math.AG"
] | alg-geom/9406005 | Charles Walter | Charles H. Walter | Pfaffian Subschemes | 26 pages, AMS-LaTeX | null | null | null | null | A subscheme $X\subset \Bbb P^{n+3}$ of codimension $3$ is {\em Pfaffian} if
it is the degeneracy locus of a skew-symmetric map $f:\cal{E}\spcheck(-t) @>>>
\cal{E}$ with $\cal{E}$ a locally free sheaf of odd rank on $\Bbb P^{n+3}$. It
is shown that a codimension $3$ subscheme $X\subset\Bbb P^{n+3}$ is Pfaffian if
and only if it is locally Gorenstein, subcanonical (i.e.\ $\omega_X\cong\cal
O_X(l)$ for some integer $l$), and the following parity condition holds: if
$n\equiv 0\pmod{4}$ and $l$ is even, then $\chi (\cal O_X (l/2))$ is also even.
The paper includes a modern version of the Horrocks correspondence, stated in
the language of derived categories. A local analogue of the main theorem is
also proved.
| [
{
"version": "v1",
"created": "Fri, 17 Jun 1994 15:28:12 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Walter",
"Charles H.",
""
]
] | alg-geom | \subsection{Outline of the Paper}
In the first section we review the proof of the local version of
Theorem \ref{main} given by Buchsbaum and Eisenbud (\cite{BE} Theorem
2.1). We show that their proof will work for us if we can replace
their minimal projective resolution by a locally free resolution of
$\cal O_X$ which satisfies two properties (Proposition
\ref{conditions}). The rest of the paper is devoted to finding a
locally free resolution of $\cal O_X$ which satisfies these
properties.
Our main tool for constructing this locally free resolution is the
Horrocks correspondence of \cite{H}. In the second section of the
paper, we give a modern description of this correspondence using
derived categories. This point of view is not identical to Horrocks',
so we have felt it prudent to include a full proof of Horrocks'
principal result (Theorem \ref{Horrocks}) from this point of view.
However, the derived categories viewpoint is useful because it permits
us to further develop Horrocks' ideas so as to obtain a method for
transfering a portion of the cohomology of the coherent sheaf $\cal
O_X$ to a locally free sheaf in a controlled way (Proposition
\ref{functorial}). This is critical for our construction.
In the third section we apply the Horrocks correspondence to construct
a particular locally free resolution of the form (\ref{resol}). The
basic idea is to cut in half the cohomology of the subscheme $X$ by
using truncations of $\bold R\Gamma_*(\cal I_X)$. Our results on the
Horrocks correspondence then permit us to find a vector bundle $\cal
F_1$ whose intermediate cohomology is one of the halves of the
cohomology of $\cal O_X$. Moreover, there is a natural morphism from
this $\cal F_1$ to $\cal I_X$. This more or less gives the right half
of the resolution, and the left half comes from the conventional
methods of the Serre correspondence. We then show that if the
cohomology of $\cal O_X$ was cut in half properly (viz.\ if the
subcomplex carries an ``isotropic'' half of the cohomology), then the
resolution is self-dual in a very strong way: i.e.\ any chain map from
the resolution to its dual which extends the identity on $\cal{O}_X$
is necessarily an isomorphism of complexes. This is one of the
properties required of the locally free resolution in order to make
the Buchsbaum-Eisenbud proof work.
In the fourth section we show that our locally free resolution of
$\cal O_X$ can be endowed with a commutative differential graded
algebra structure. This is a matter of calculating the obstruction to
the lifting of a certain map. This is the second property required of
the locally free resolution in order for the Buchsbaum-Eisenbud proof
to work. This will complete the proof of Theorem \ref{main}.
In the fifth section we consider Theorem \ref{main} in characteristic
$2$. Essentially, certain lemmas in the fourth section fail in
characteristic $2$ and must be replaced by analogues which are
slightly different.
In the sixth section we consider the results for regular local rings.
Theorem \ref{main} concerning projective spaces has an obvious
analogue (Theorem \ref{punc:spec}) for the punctured spectrum of a
regular local ring. We show that this analogue is equivalent to
Theorem \ref{RLR}.
\begin{ack}
The author would like to thank R.~M.~Mir\`o-Roig who brought the
problem to his attention and with whom he had several discussions
concerning it. The paper was written in the context of the Space
Curves group of Europroj.
\end{ack}
\section{The Buchsbaum-Eisenbud Proof}
In this section we review Buchsbaum and Eisenbud's proof of the local
version of Theorem \ref{main}. In particular, we describe the two
conditions that a locally free resolution of $\cal O_X$ must satisfy
in order for their proof to show that a subcanonical subscheme
$X\subset \Bbb{P}^{n+3}$ is Pfaffian (Proposition \ref{conditions}).
\begin{theorem}[\cite{BE} Theorem 2.1]
\label{local}
Let $R$ be a regular local ring and $I$ an ideal of $R$ of height $3$
such that $R/I$ is a Gorenstein ring. Then $I$ has a minimal
projective resolution of the form
\[
0 @>>> R @>{g\spcheck}>> F\spcheck @>f>> F @>g>> R @>>> R/I
\]
such that $F$ of odd rank $2p+1$, the map $f$ is skew-symmetric, and
$g$ is composed of the Pfaffians of order $2p$ of $f$.
\end{theorem}
\begin{pf*}{Sketch of Buchsbaum and Eisenbud's proof of Theorem
\ref{local}}
One considers a minimal projective resolution of $R/I$. Since $R/I$
is Gorenstein, it is of the form
\[
\bold{P}^*: \qquad\quad 0 @>>> R @>{d_3}>> F_2 @>{d_2}>> F_1 @>{d_1}>>
R
\]
We now seek to find a way of identifying $F_2\cong F_1\spcheck$ so
that $d_2$ becomes skew-symmetric.
The first step is to endow $\bold{P}^*$ with the structure of a
commutative associative differential graded algebra (\cite{BE}
pp.~451--453). To define the multiplication, they define
$S_2(\bold{P}^*) = (\bold{P}^* \otimes \bold{P}^*)/M^*$ where $M^*$ is
the graded submodule of $\bold{P}^*\otimes\bold{P}^*$ generated by
\[
\{a\otimes b-(-1)^{(\deg a)(\deg b)}b\otimes a \mid a,b \text{
homogeneous elements of }\bold{P}^*\}.
\]
Using universal properties of projective modules, they then construct
a map of complexes $\Phi : S_2(\bold{P}^*) @>>> \bold{P}^*$ which
extends the multiplication $R/I \otimes R/I$ and which is the identity
on the subcomplex $R\otimes\bold{P}^* \subset S_2(\bold{P}^*)$. This
makes $\bold{P}^*$ into a commutative differential graded algebra.
The associativity of this algebra follows from the fact that it of
length $3$, i.e. $P_n = 0$ for $n\geq 4$.
The next step (p.~455) is to note that the multiplication $F_i \otimes
F_{3-i} @>>> F_3 = R$ induces maps $s_i: F_i @>>> F_{3-i}\spcheck$ and
a commutative diagram:
\begin{equation}
\label{dual}
\begin{CD}
\bold{P}^*: & \qquad & 0 @>>> R @>{d_3}>> F_2 @>{d_2}>> F_1
@>{d_1}>> R \\
&&&& @| @VV{s_2}V @VV{s_1}V @| \\
(\bold{P}^*)\spcheck : && 0 @>>> R @>{d_1\spcheck}>> F_1\spcheck
@>{-d_2\spcheck}>> F_2\spcheck @>{d_3\spcheck}>> R
\end{CD}
\end{equation}
This map of complexes is an extension of the Gorenstein duality
isomorphism $R/I \cong \omega _{R/I} = \operatorname{Ext} ^3_R(R/I,R)$ to the
minimal projective resolutions of $R/I$ and $\omega _{R/I}$. Since
any map between minimal projective resolutions which extends an
isomorphism in degree $0$ must be an isomorphism, it follows that the
$s_i$ are all isomorphisms.
We can therefore use the identification $s_2 : F_2\cong F_1\spcheck$.
A very simple computation (p.~465) shows that with this
identification, the commutativity and associativity of the
differential graded algebra structure on $\bold{P}^*$ imply the
skew-symmetry of $d_2$. In particular $d_2$ must have even rank (say
$2p$), and $F_2$ must have odd rank $2p+1$. The identification of
$d_1$ and $d_3$ with the vectors of Pfaffians of order $2p$ of $d_2$
is a lengthy but unproblematic computation (pp.~458--464).
\end{pf*}
Now let $X$ be a locally Gorenstein subcanonical subscheme of
codimension $3$ in $\Bbb{P}^{n+3}$ with $\omega_X\cong\cal O_X (l)$. We wish
to repeat the proof we have just sketched only with $\bold{P}^*$
replaced by a locally free resolution of $\cal O_X$:
\begin{equation}
\label{P}
\cal{P}^*: \qquad\quad 0 @>>> \cal{L} @>{d_3}>> \cal{F}_2
@>{d_2}>> \cal{F}_1 @>{d_1}>> \cal O_{\Bbb{P}^{n+3}}
\end{equation}
where we will write $\cal{L}$ in place of $\omega_{\Bbb{P}^{n+3}}(-l)$ in order
to simplify our diagrams.
A careful reading yields only two places where the fact that
$\bold{P}^*$ is a minimal projective resolution of $R/I$ was used in a
way that does not immediately carry over to the locally free
resolution $\cal{P}^*$. The first place was in the definition of the
map $\Phi : S_2(\bold{P}^*) @>>> \bold{P}^*$ which made $\bold{P}^*$
into a commutative differential graded algebra. Therefore we will
need to show directly the existence of a map of complexes
\[
\begin{CD}
S_2(\cal{P}^*): & \quad & \dotsb & \:\longrightarrow\: & \cal{L}
\oplus [\cal{F}_2\otimes \cal{F}_1] &
\:\stackrel{\sigma}{\longrightarrow}\: &
\cal{F}_2 \oplus \Lambda^2 \cal{F}_1 & \:\longrightarrow\: &
\cal{F}_1 & \:\longrightarrow\: & \cal O_{\Bbb{P}^{n+3}} \\
&&&&@VV{\phi _3}V @VV{\phi _2}V @| @| \\
\cal{P}^*: && 0 & \:\longrightarrow\: & \cal{L} &
\:\stackrel{d_3}{\longrightarrow}\: & \cal{F}_2 &
\:\stackrel{d_2}{\longrightarrow}\: & \cal{F}_1 &
\:\stackrel{d_1}{\longrightarrow}\: & \cal O_{\Bbb{P}^{n+3}}
\end{CD}
\]
The critical problem in defining the morphism of complexes is the
following. Let $\psi : \Lambda^2 \cal{F}_1 @>>> \ker (d_1)$ be
defined by $\psi (a\wedge b)=d_1(a)b-d_1(b)a$. We then must lift
\begin{equation}
\label{liftdiag}
\begin{CD}
&&&&&& \Lambda^2 \cal{F}_1 \\
&&&&&& @VV{\psi}V \\
0 @>>> \cal{L} @>>> \cal{F}_2 @>>> \ker (d_1) @>>> 0
\end{CD}
\end{equation}
to a $\phi\in\operatorname{Hom} (\Lambda ^2\cal{F}_1,\cal{F}_2)$. Once that is
done, the rest of the chain map follows. For one may define $\phi _2
= (1_{\cal{F}_2},\phi )$. Then
\[
\phi _2\circ\sigma (\cal{L}\oplus [\cal{F}_2\otimes \cal{F}_1])
\subset \ker (d_2) = \cal{L}.
\]
So $\phi _2\circ\sigma$ factors through $\cal{L}$, allowing one to
define $\phi _3$. Thus one can put a commutative associative
differential graded algebra structure on $\cal{P}^*$ provided $\psi$
can be lifted. The obstruction to lifting $\psi$ lies in $\operatorname{Ext}
^1(\Lambda^2 \cal{F}_1,\cal{L}) \cong H^{n+2}(\Lambda^2
\cal{F}_1(l))^*$.
Once we have the commutative differential graded algebra structure on
$\cal{P}^*$, we may use it to define maps $s_i : \cal{F}_i @>>>
\cal{F}_{3-i}\spcheck \otimes \cal{L}$ and a commutative diagram
analogous to (\ref{dual}):
\begin{equation}
\label{dual2}
\begin{CD}
\cal{P}^*: & \qquad & 0 @>>> \cal{L} @>{d_3}>> \cal{F}_2 @>{d_2}>>
\cal{F}_1 @>{d_1}>> \cal O_{\Bbb{P}^{n+3}} \\
&&&& @| @VV{s_2}V @VV{s_1}V @| \\
(\cal{P}^*)\spcheck : &&0 @>>> \cal{L} @>{d_1\spcheck}>>
\cal{F}_1\spcheck\otimes\cal{L} @>{-d_2\spcheck}>>
\cal{F}_2\spcheck\otimes\cal{L} @>{d_3\spcheck}>> \cal O_{\Bbb{P}^{n+3}}
\end{CD}
\end{equation}
The vertical maps extend the isomorphism $\cal O_X\cong \omega_X(-l)=
\cal{E}xt^3(\cal O_X,\cal{L})$. We now run into the second problem
with locally free resolutions. Namely, a morphism of locally free
resolutions which extends an isomorphism in degree $0$ is not
automatically an isomorphism between the resolutions. But we reach
the conclusion:
\begin{proposition}
\label{conditions}
Suppose $X$ is a locally Gorenstein subcanonical subscheme of
codimension $3$ in $\Bbb{P}^{n+3}$ with $\omega_X\cong\cal O_X (l)$. Then $X$
will be a Pfaffian scheme if $\cal O_X$ has a locally free resolution
$\cal{P}^*$ as in \rom{(}\ref{P}\rom{)} satisfying the following two
conditions:
\rom(a\rom) Any morphism of complexes $\cal{P}^* @>>>
(\cal{P}^*)\spcheck$ as in \rom{(}\ref{dual2}\rom{)} which extends the
identity of $\cal O_X$ is an isomorphism of complexes, and
\rom(b\rom) The morphism $\psi$ of \rom(\ref{liftdiag}\rom) lifts to a
map $\phi\in\operatorname{Hom} (\Lambda ^2\cal{F}_1,\cal{F}_2)$.
\end{proposition}
We will now construct locally free resolutions $\cal{P}^*$ satisfying
the conditions of the proposition. Our method involves the Horrocks
correspondence.
\section{The Horrocks Correspondence}
In this section we give a modern description of the Horrocks
correspondence of \cite{H} using derived categories. We include a
full proof of the principal properties of the correspondence from this
point of view (Theorem \ref{Horrocks}). Taking advantage of the
greater flexibility of the derived category viewpoint, we develop a
technique which allows us to transfer a prescribed portion of the
cohomology of $\cal O_X$ to prescribed parts of a locally free
resolution (Proposition \ref{functorial}).
\subsection{Notation and Generalities}
We first recall some generalities about complexes. If $\frak{A}$ is
an abelian category, let $C(\frak{A})$ (resp.\ $K(\frak{A})$,
$D(\frak{A})$) denote the category (resp.\ homotopy category, derived
category) of complexes of objects of $\frak{A}$, and let
$C^b(\frak{A})$, $C^-(\frak{A})$, $C^+(\frak{A})$, etc., denote the
corresponding complexes of bounded (resp.\ bounded above, bounded
below) complexes of objects of $\frak{A}$. When speaking of
complexes, we will generally reserve the word ``isomorphism'' for
isomorphisms in $C(\frak A)$. Isomorphisms in $K(\frak A)$ (resp.\
$D(\frak A)$) are referred to as homotopy equivalences (resp.\
quasi-isomorphisms).
If $r$ is an integer, then any complex $C^*$ of objects of $\frak{A}$
has two {\em canonical truncations} at $r$ and a {\em naive
truncation}:
\begin{align*}
\begin{CD}
\tau _{\leq r}(C^*): & \qquad & \cdots & \:\rightarrow\: & C^{r-2} &
\:\rightarrow\: & C^{r-1} & \:\rightarrow\: &
\ker(\delta^r) & \:\rightarrow\: & 0 & \:\rightarrow\: & 0 &
\:\rightarrow\: & \cdots , \\
\tau _{> r}(C^*): & \qquad & \cdots & \:\rightarrow\: & 0 &
\:\rightarrow\: & 0 & \:\rightarrow\: & C^r/\ker(\delta^r) &
\:\rightarrow\: & C^{r+1} & \:\rightarrow\: & C^{r+2} &
\:\rightarrow\: & \cdots . \\
\sigma _{\geq r}(C^*): & \qquad & \cdots & \:\rightarrow\: & 0 &
\:\rightarrow\: & 0 & \:\rightarrow\: & C^r & \:\rightarrow\: &
C^{r+1} & \:\rightarrow\: & C^{r+2} & \:\rightarrow\: & \cdots .
\end{CD}
\end{align*}
All the truncations are functorial in $C(\frak{A})$. The canonical
truncations are functorial in $K(\frak{A})$ and $D(\frak{A})$ as well.
We will often find it more convenient to write $\tau_{<r+1}$ instead
of $\tau_{\leq r}$.
Suppose now that $\frak{A}$ has enough projectives. Every bounded
above complex $C^*$ of objects in $\frak{A}$ admits a {\em projective
resolution}, i.e.\ a quasi-isomorphism $P^* @>>> C^*$ with $P^*$ a
complex of projectives (\cite{Ha} Proposition I.4.6). The projective
resolution of a complex is unique up to homotopy equivalence. If
$C^*$ and $E^*$ are bounded above complexes of objects in $\frak{A}$,
and if $P^* @>>> C^*$ is a projective resolution of $C^*$, then there
is a natural isomorphism $\operatorname{Hom}_{D^-(\frak{A})}(C^*,E^*) \cong
\operatorname{Hom}_{K^-(\frak{A})}(P^*,E^*)$. In particular if $\frak{P}$ denotes
the full subcategory of projective objects of $\frak{A}$, then the
natural functor $K^-(\frak{P}) @>>> D^-(\frak{A})$ is an equivalence
of categories (\cite{Ha} Proposition I.4.7). This can be refined to
the following statement:
\begin{lemma}
\label{category}
Let $\frak{A}$ be an abelian category with enough projectives, and let
$\frak{P}$ be the full subcategory of projective objects of
$\frak{A}$. Suppose $A\subset D^-(\frak{A})$ and $P\subset
K^-(\frak{P})$ are full subcategories such that $\operatorname{ob}(P)\subset \operatorname{ob}(A)$
and every object of $A$ has a projective resolution belonging to $P$.
Then the natural functor $P @>>> A$ is an equivalence of categories.
\end{lemma}
Let $S=k[X_0,\dots,X_N]$ be the homogeneous coordinate ring of $\Bbb{P}^N$,
and let $\frak{m}=(X_0,\dots,X_N)$ be its irrelevant ideal. Let
$\operatorname{Mod}_{\mit{S},\operatorname{gr}}$ be the category of graded $S$-modules. Then $\operatorname{Mod}_{\mit{S},\operatorname{gr}}$ has
enough projectives, namely the free modules. We will call a complex
$P^*$ of projectives in $\operatorname{Mod}_{\mit{S},\operatorname{gr}}$ a {\em minimal} if all its objects
$P^i$ are free of finite rank and its differential $\delta^*$
satisfies $\delta^i(P^i) \subset \frak m P^{i+1}$ for all $i$. If
$C^*$ is a bounded above complex of objects in $\operatorname{Mod}_{\mit{S},\operatorname{gr}}$ whose
cohomology modules $H^i(C^*)$ are all finitely generated, then $C^*$
has a {\em minimal projective resolution}, i.e.\ a projective
resolution by a minimal complex of projectives. The next lemma, which
is a well known consequence of Nakayama's lemma, says that minimal
projective resolutions are unique up to isomorphism and not merely up
to homotopy equivalence:
\begin{lemma}
\label{NAK}
Let $\phi: P^* @>>> Q^*$ be a homotopy equivalence between {\em
minimal} complexes of free graded $S$-modules of finite rank. Then
$\phi$ is an isomorphism.
\end{lemma}
Let $\operatorname{Mod_{\cal O}}$ be the category of sheaves of $\cal{O}_{\Bbb{P}^N}$-modules.
For $\cal{E}$ a sheaf of $\cal{O}_{\Bbb{P}^N}$-modules, let $\Gamma
_*(\cal{E})=\bigoplus _{t\in\Bbb{Z}} \Gamma(\cal{E}(t))$. Then
$\Gamma_*$ defines a left exact functor from $\operatorname{Mod_{\cal O}}$ to $\operatorname{Mod}_{\mit{S},\operatorname{gr}}$. It
has a right derived functor $\bold{R}\Gamma_*: D^b(\operatorname{Mod_{\cal O}}) @>>>
D^b(\operatorname{Mod}_{\mit{S},\operatorname{gr}})$ whose cohomology functors we denote $H^i_*(\cal{E}) =
\bigoplus _{t\in\Bbb{Z}}H^i(\cal{E}(t))$. The functor $\Gamma_*$ has
an exact left adjoint $\widetilde{\ \ }$, the functor of associated
sheaves.
Let $\Gamma_{\frak m}: \operatorname{Mod}_{\mit{S},\operatorname{gr}} @>>> \operatorname{Mod}_{\mit{S},\operatorname{gr}}$ be the functor
associating to a graded $S$-module $M$ the maximal submodule
$\Gamma_{\frak m}(M) \subset M$ supported at the origin $0$ of $\Bbb
A^{N+1}$. This functor is also left exact and has a right derived
functor $\bold R\Gamma_{\frak m}: D^b(\operatorname{Mod}_{\mit{S},\operatorname{gr}}) @>>> D^b(\operatorname{Mod}_{\mit{S},\operatorname{gr}})$.
Its cohomology functors are denoted $H^i_{\frak m}$.
\begin{lemma}
\label{bounds}
Let $P^*$ be a bounded complex of free graded $S$-modules of finite
rank where $S=k[X_0,\dots,X_N]$. If $P^*$ is minimal, then
\begin{align*}
\max\{i\mid P^i\neq 0\} = & \max\{i\mid H^i(P^*)\neq 0\},\\
\min\{i\mid P^i\neq 0\} = & \min\{i\mid H^i_{\frak m}(P^*)\neq 0\}
-N-1.
\end{align*}
\end{lemma}
\begin{pf}
The assertion about maxima is a simple and well-known application of
the minimality condition and Nakayama's Lemma. The assertion about
minima, which is essentially the Auslander-Buchsbaum theorem, reduces
to the assertion about maxima by Serre duality.
\end{pf}
\subsection{The Horrocks Correspondence}
We now begin to describe the components of the Horrocks
correspondence. Let $\frak{B}$ be the full subcategory of $\operatorname{Mod_{\cal O}}$ of
locally free sheaves of finite rank, and let $\frak{Z}$ denote the
full category of $D^b(\operatorname{Mod}_{\mit{S},\operatorname{gr}})$ of complexes $C^*$ such that
$H^i(C^*)$ is of finite length for $0<i<N$ and $H^i(C^*)$ vanishes for
all other $i$.
The Horrocks correspondence consists of a functor $\zeta: \frak{B}
@>>> \frak{Z}$ and a map $\cal{H}: \operatorname{ob}(\frak{Z}) @>>> \operatorname{ob}(\frak{B})$
in the opposite direction. The functor $\zeta$ is simply
$\tau_{>0}\tau_{< N}\bold{R}\Gamma _*$. For $\cal{E}$ a vector bundle
on $\Bbb{P}^N$, the cohomology of $\zeta(\cal{E})$ is of course:
\[
H^i(\zeta(\cal{E}))=
\begin{cases}
H^i_*(\cal{E}) & \text{if }0<i<N,\\
0 & \text{otherwise.}
\end{cases}
\]
Since $\cal{E}$ is locally free of finite rank, $H^i_*(\cal{E})$ is of
finite length for $0<i<N$. So $\zeta(\cal{E})\in\operatorname{ob}(\frak{Z})$.
We now define $\cal{H}$. Any $C^*\in\operatorname{ob}(\frak{Z})$ has a minimal
projective resolution $P^* @>>> C^*$. We define $\cal{H}(C^*)$ to be
the kernel of the differential $\widetilde\delta^0:\widetilde P^0 @>>>
\widetilde P^1$. Then $\cal{H}(C^*)$ is a vector bundle because it
fits into an exact complex of vector bundles
\begin{equation}
\label{Hcomp}
\dotsb @>>> 0 @>>> \cal{H}(C^*) @>>> \widetilde P^0 @>>> \widetilde
P^1 @>>> \dotsb @>>> \widetilde P^{N-1} @>>> 0 @>>> \dotsb .
\end{equation}
Note that $\cal{H}(C^*)$ is well-defined up to isomorphism because the
minimal projective resolution $P^*$ of $C^*$ is unique up to
isomorphism because of Lemma \ref{NAK}. However, $\cal{H}$ is not a
functor.
The principal results of Horrocks' paper \cite{H} can be described in
the following way:
\begin{theorem}[Horrocks]
\label{Horrocks}
Let $\frak{B}$ be the category of locally free sheaves of finite rank
on $\Bbb{P}^N$, and let $\frak{Z}$ be the full subcategory of $D^b(\operatorname{Mod}_{\mit{S},\operatorname{gr}})$
of complexes $C^*$ such that $H^i(C^*)$ is of finite length if
$0<i<N$, and $H^i(C^*)=0$ for all other $i$. Let $\zeta =
\tau_{>0}\tau_{< N}\bold{R}\Gamma _*: \frak{B} @>>> \frak{Z}$, and let
$\cal{H}:\operatorname{ob}(\frak{Z}) @>>> \operatorname{ob}(\frak{B})$ be the map defined as in
\rom(\ref{Hcomp}\rom) above.
\rom(a\rom) If $\cal{E}\in\operatorname{ob}(\frak{B})$, then $\cal{E}
\cong \cal{H}\zeta(\cal{E}) \oplus \bigoplus _i
\cal{O}_{\Bbb{P}^N}(n_i)$ for some integers $n_i$.
\rom(b\rom) If $C^*\in\operatorname{ob}(\frak{Z})$, then $\zeta\cal{H}(C^*)\simeq
C^*$.
\rom(c\rom) If $\cal{E},\cal{F}\in\operatorname{ob}(\frak{B})$, then
$\operatorname{Hom}_{\frak{Z}}(\zeta(\cal{E}),\zeta(\cal{F}))\cong
\operatorname{Hom}(\cal{E},\cal{F})/\operatorname{Hom}_{\Phi}(\cal{E},\cal{F})$ where
$\operatorname{Hom}_{\Phi}(\cal{E},\cal{F})$ is the set of all morphisms which
factor through a direct sum of line bundles.
\end{theorem}
The theorem may be read as saying the following. Call two vector
bundles $\cal{E}$ and $\cal{F}$ {\em stably equivalent} if there exist
sets of integers $\{n_i\}$ and $\{m_j\}$ such that $\cal{E} \oplus
\bigoplus_i\cal{O}_{\Bbb{P}^N}(n_i) \cong \cal{F}\oplus \bigoplus_j
\cal{O}_{\Bbb{P}^N}(m_j)$. Then the theorem says that $\zeta$ and $\cal{H}$
induce a one-to-one correspondence between stable equivalence classes
of vector bundles on $\Bbb{P}^N$ and quasi-isomorphism classes of complexes
in $\frak{Z}$.
For Horrocks' proof of the theorem, see \cite{H} Lemma 7.1 and Theorem
7.2 and the discussion between them. However, Horrocks' definition of
the category $\frak{Z}$ and the functor $\zeta$ are different from
ours, and demonstrating the equivalence of the definitions is somewhat
tedious. So instead of referring the reader to Horrocks' paper, we
give a new proof. The first step is the following lemma:
\begin{lemma}
\label{tauresol}
\rom(a\rom) Suppose
\[
P^*:\qquad \dotsb @>>> 0 @>>> P^0 @>>> P^1 @>>> \dotsb @>>>
P^{N-1} @>>> 0 @>>> \dotsb
\]
is a complex of free graded $S$-modules of finite rank such that
$H^i(P^*)$ is a module of finite length for $0<i<N$. Let $\cal{E}=
H^0(P^*)\sptilde$. Then $P^*$ is quasi-isomorphic to
$\tau_{<N}\bold{R}\Gamma_*(\cal{E})$.
\rom(b\rom) Conversely, if $\cal{E}$ is a vector bundle on $\Bbb{P}^N$, then
the minimal projective resolution of
$\tau_{<N}\bold{R}\Gamma_*(\cal{E})$ is of the above form.
\end{lemma}
\begin{pf}
(a) Note that the complex $\widetilde{P}^*$ of coherent sheaves on
$\Bbb{P}^N$ has vanishing cohomology in degrees different from $0$. So it
is quasi-isomorphic to $H^0(\widetilde{P}^*) = \cal{E}$. Hence the
triangle of functors of \cite{W} Proposition 1.1:
\[
\bold{R}\Gamma_{\frak m} @>>> \operatorname{Id} @>>> \bold{R}\Gamma_*\circ\sptilde
@>>> \bold{R}\Gamma_{\frak m}[1],
\]
when applied to $P^*$, yields a triangle
\begin{equation}
\label{triangle}
\bold{R}\Gamma_{\frak m}(P^*) @>>> P^* @>\beta>>
\bold{R}\Gamma_*(\cal{E}) @>>> \bold{R}\Gamma_{\frak m}(P^*)[1].
\end{equation}
By Lemma \ref{bounds}, we have $H^i_{\frak m}(P^*)=0$ for $i \leq N$.
So $H^i(\beta): H^i(P^*) @>>> H^i_*(\cal{E})$ is an isomorphism for
$i<N$. Therefore $\beta$ induces a quasi-isomorphism of $P^*$ onto
$\tau_{<N}\bold{R}\Gamma_* (\cal{E})$.
(b) Conversely, if $\cal{E}$ is a vector bundle on $\Bbb{P}^N$, then
$H^i_*(\cal{E})$ is finitely generated for $i<N$. Hence
$\tau_{<N}\bold{R}\Gamma_*(\cal{E})$ has a minimal projective
resolution $P^*$. For $0<i<N$ the module $H^i(P^*) = H^i_*(\cal E)$
is of finite length because $\cal E$ is locally free. By construction
$H^i(P^*) = H^i(\tau_{<N}\bold R\Gamma_*(\cal E)) = 0$ for $i \geq N$.
So we have $P^i = 0$ for $i\geq N$ by Lemma \ref{bounds}. Looking
again at the triangle (\ref{triangle}), we see by the construction of
$P^*$ that $H^i(\beta)$ is an isomorphism for $i<N$ and an injection
for $i=N$. So $H^i_{\frak m}(P^*) = 0$ for $i\leq N$. So by Lemma
\ref{bounds} we see that $P^i=0$ for $i\leq -1$. Thus $P^*$ has the
form asserted by the lemma.
\end{pf}
We now wish to functorialize the previous lemma. Let $B\subset
K^b(\operatorname{Mod}_{\mit{S},\operatorname{gr}})$ be the full subcategory of complexes of the form
\begin{equation}
\label{rightresol}
\dotsb @>>> 0 @>>> P^0 @>>> P^1 @>>> \dotsb @>>> P^{N-1} @>>>
0 @>>> \dotsb
\end{equation}
such that the $P^i$ are free of finite rank for all $i$, the modules
$H^i(P^*)$ are of finite length for $0<i<N$ and the differentials
satisfy $\delta^i(P^i)\subset \frak{m}P^{i+1}$ for all $i$. For any
vector bundle $\cal E$ on $\Bbb{P}^N$ we now define $P^*(\cal E)$ as the
minimal projective resolution of $\tau_{<N}\bold R\Gamma_* (\cal E)$.
By Lemma \ref{tauresol}, $P^*(\cal E)$ is always an object of $B$.
\begin{lemma}
\label{Bequiv}
The functor $P^*: \frak{B} @>>> B$ which associates to an $\cal{E}
\in\operatorname{ob}(\frak{B})$ the minimal projective resolution of $\tau_{<N}
\bold{R}\Gamma_*(\cal{E})$ is an equivalence of categories with
inverse given by $C^*\mapsto H^0(C^*)\sptilde$.
\end{lemma}
\begin{pf}
Since the functor $\tau_{<N}\bold R\Gamma_*: \frak B @>>>
D^-(\operatorname{Mod}_{\mit{S},\operatorname{gr}})$ has a left inverse $H^0(-)\sptilde$, it induces an
equivalence between $\frak B$ and the full subcategory $A\subset
D^-(\operatorname{Mod}_{\mit{S},\operatorname{gr}})$ of complexes quasi-isomorphic to complexes in the image
of $\tau_{<N}\bold R\Gamma_*$. But by Lemma \ref{tauresol}, the full
subcategory $B\subset K^-(\operatorname{Mod}_{\mit{S},\operatorname{gr}})$ has the properties that
$\operatorname{ob}(B)\subset\operatorname{ob}(A)$ and that the minimal projective resolution of
every object of $A$ belongs to $B$. Hence the natural functor $B @>>>
A$ is also an equivalence of categories by Lemma \ref{category}.
Since $P^*$ is exactly the composition of the equivalence
$\tau_{<N}\bold R\Gamma_*: \frak B @>>> A$ with the inverse of the
equivalence $B @>>> A$, it is an equivalence. The inverse of $P^*$
remains the same as that of $\tau_{<N} \bold R\Gamma_*$, namely
$H^0(-)\sptilde$.
\end{pf}
Now the graded module associated to a vector bundle $\cal E$ on $\Bbb{P}^N$
has a minimal projective resolution:
\[
0 @>>> Q^{-(N-1)} @>>> \dotsb @>>> Q^{-1} @>>> Q^0 @>>> \Gamma_*(\cal
E)
\]
For any $\cal E$ we now define the following complexes in addition to
the $P^*(\cal E)$ defined above. First we set:
\[
Q^*(\cal E):\qquad \dotsb @>>> 0 @>>> Q^{-(N-1)} @>>> \dotsb @>>>
Q^{-1} @>>> Q^0 @>>> 0 @>>> \dotsb.
\]
We then let $R^*(\cal E)$ be the natural concatenation of $Q^*(\cal
E)$ with $P^*(\cal E)$ induced by the composition $Q^0
\twoheadrightarrow \Gamma_*(\cal E) \hookrightarrow P^0$:
\[
R^*(\cal E):\qquad\dotsb @>>> 0 @>>> Q^{-(N-1)} @>>> \dotsb @>>> Q^0
@>>> P^0 @>>> \dotsb @>>> P^{N-1} @>>> 0 @>>> \dotsb
\]
Thus $R^i(\cal E) = P^i(\cal E)$ for $i\geq 0$, and $R^i(\cal E) =
Q^{i+1}(\cal E)$ for $i<0$. Note that although the projective
complexes $P^*(\cal E)$ and $Q^*(\cal E)$ are minimal, $R^*(\cal E)$
may not be minimal, because there may be a direct factor of $Q^0(\cal
E)$ which is mapped isomorphically onto a direct factor of $P^0(\cal
E)$. However, one may write $R^*(\cal E)$ as the direct sum of a
minimal complex of projectives $R^*_{\min}(\cal E)$
\begin{align*}
R^*_{\min}(\cal E): \qquad \dotsb @>>> Q^{-2} @>>> Q^{-1} @>>>
Q^0_{\min} & @>>> P^0_{\min} @>>> P^1 @>>> P^2 @>>> \dotsb\\
\intertext{and of an exact complex of projectives}
\stepcounter{equation}\tag{\theequation}\label{L:module}
\dotsb @>>> 0 @>>> L & @>\operatorname{Id}>> L @>>> 0 @>>> \dotsb.
\end{align*}
The complexes $Q^*(\cal E)$, $R^*(\cal E)$, and $R^*_{\min}(\cal E)$
are all functorial (in the homotopy category) in $\cal E$. Moreover,
we may use the identification between the categories $\frak B$ and $B$
to define complexes $Q^*(P^*)$, $R^*(P^*)$, and $R^*_{\min}(P^*)$ for
$P^*$ in $B$. Namely, $Q^*(P^*)$ is the minimal projective resolution
of $H^0(P^*)$, $R^*(P^*)$ is the concatenation of $Q^*(P^*)$ with
$P^*$, etc.
We now define a homotopy category of complexes of type $R^*_{\min}$.
More formally, let $Z\subset K^b(\operatorname{Mod}_{\mit{S},\operatorname{gr}})$ be the full subcategory of
minimal complexes of projective modules of finite rank of the form
\begin{equation}
\label{R:complex}
\dotsb @>>> 0 @>>> R^{-N} @>>> \dotsb @>>> R^{-1} @>>> R^0 @>>> \dotsb
@>>> R^{N-1} @>>> 0 @>>> \dotsb
\end{equation}
such that the cohomology modules $H^i(R^*)$ are of finite length for
$0<i<N$ and vanish for all other $i$.
We need one more lemma before proving Theorem \ref{Horrocks}.
\begin{lemma}
\label{Zequiv}
The natural functor $Z @>>> \frak{Z}$ is an equivalence of categories.
\end{lemma}
\begin{pf}
Let $R^*$ be the minimal projective resolution of an object $C^*$ of
$\frak Z$. Since $H^i(R^*)= H^i(C^*)=0$ for $i\geq N$, we have
$R^i=0$ for $i\geq N$ by Lemma \ref{bounds}. Moreover, all the
$H^i(C^*)$ are of finite length, so $H^i_{\frak m}(C^*) = H^i(C^*)$
for all $i$. In particular, $H^i_{\frak m}(R^*)=H^i_{\frak m}(C^*)
=0$ for $i\leq 0$. So $R^i=0$ for $i\leq -N-1$ by Lemma \ref{bounds}.
Thus the minimal projective resolution of any object of $\frak Z$ is
in $Z$. The lemma now follows from Lemma \ref{category}.
\end{pf}
\begin{pf*}{Proof of Theorem \ref{Horrocks}}
Lemma \ref{Bequiv} permits us to identify a vector bundle $\cal E$
with the complex $P^*(\cal E)$ of $B$. Since $P^*(\cal E)$ is already
quasi-isomorphic to $\tau_{<N}\bold R\Gamma_*(\cal E)$, the complex
$\zeta(\cal E) = \tau_{>0}\tau_{<N} \bold R\Gamma_*(\cal E)$ is
quasi-isomorphic to the complex
\[
\dotsb @>>> 0 @>>> \Gamma_*(\cal E) @>>> P^0 @>>> P^1 @>>> \dotsb @>>>
P^{N-1} @>>> 0 @>>> \dotsb
\]
and hence to the complexes $R^*(\cal E)$ and $R^*_{\min}(\cal E)$.
Hence the object $\zeta(\cal E)$ in $\frak Z$ is quasi-isomorphic to
the object $R^*_{\min}(\cal E)$ of $Z$. Hence after identifying
$\frak B$ and $\frak Z$ with $B$ and $Z$ by Lemmas \ref{Bequiv} and
\ref{Zequiv}, the functor $\zeta$ may be identified with the functor
from $B$ to $Z$ which associates to any complex $P^*$ in $B$ the
corresponding complex $R^*_{\min}$ as described earlier.
Similarly, given any object $C^*$ of $\frak Z$ with minimal projective
resolution $R^*$, the definitions say that $P^*(\cal H(C^*))=
\sigma_{\geq 0}(R^*)$, the naive truncation. Thus the map $\cal H:
\operatorname{ob}(\frak Z) @>>> \operatorname{ob}(\frak B)$ may be identified with $\sigma_{\geq
0}: \operatorname{ob}(Z) @>>> \operatorname{ob}(B)$. Note that since all objects of $Z$ and $B$
are minimal complexes of projective modules, homotopy equivalence
classes of objects of $Z$ and $B$ coincide with isomorphism classes.
Hence the map $\sigma_{\geq 0}: \operatorname{ob}(Z) @>>> \operatorname{ob}(B)$ preserves homotopy
equivalence. Since $Z$ and $B$ are subcategories of the homotopy
category, this means that $\sigma_{\geq 0}$ is well-defined on objects
of $Z$. However, $\sigma_{\geq 0}$ and hence $\cal H$ are not
well-defined on morphisms of $Z$.
(a) The above identifications now say if $\cal E\in\operatorname{ob}(\frak B)$, then
$\cal H\zeta(\cal E)$ is the object of $\frak B$ corresponding to the
complex $\sigma_{\geq 0}(R^*_{\min}(\cal E))$:
\[
\sigma_{\geq 0}(R^*_{\min}(\cal E)): \qquad \dotsb @>>> 0 @>>>
P^0_{\min} @>\mu>> P^1 @>>> \dotsb @>>> P^{N-1} @>>> 0 @>>> \dotsb .
\]
By Lemma \ref{Bequiv}, the sheaf $\cal H\zeta(\cal E)$ is
$\ker(\mu)\sptilde$. So $\cal E = \cal H\zeta(\cal
E)\oplus\widetilde{L}$ where $L$ is the projective module of
(\ref{L:module}). Since $\widetilde{L}$ is now a direct sum of line
bundles, (a) follows.
(b) If $C^*$ is an object of $\frak Z$ with minimal projective
resolution $R^*$ in $Z$ of the form (\ref{R:complex}), then the above
computations identify $\cal H(C^*)$ in $\frak B$ with $P^*(\cal
H(C^*)) = \sigma_{\geq 0}(R^*)$ in $B$. Thus $\zeta\cal H(C^*)$
becomes identified with $R^*_{\min}(\cal H(C^*))$ which is just $R^*$
again. Since $R^*$ is quasi-isomorphic to $C^*$, we have $\zeta\cal
H(C^*)\simeq C^*$ as desired.
(c) After identifying $\frak B$ with $B$ and $\frak Z$ with $Z$,
assertion (c) becomes the statement: For any pair of objects $E^*$ and
$F^*$ in $B$, the natural map
\begin{equation}
\label{B:to:Z}
\operatorname{Hom}_B(E^*,F^*) @>>> \operatorname{Hom}_Z(R^*_{\min}(E^*),R^*_{\min}(F^*))
\end{equation}
is surjective and its kernel is the subspace of morphisms which factor
through an object of $B$ of the form
\begin{equation}
\label{triv:comp}
\dotsb @>>> 0 @>>> L @>>> 0 @>>> \dotsb
\end{equation}
with $L$ a free graded $S$-module of finite rank appearing in degree
$0$.
We first prove surjectivity. Suppose $\phi\in \operatorname{Hom}_Z
(R^*_{\min}(E^*), R^*_{\min}(F^*))$. Since $Z$ is a homotopy
category, $\phi$ is actually a homotopy equivalence class of maps in
$C(\operatorname{Mod}_{\mit{S},\operatorname{gr}})$. So we may choose a chain map $f$ in the class $\phi$.
Then $f$ may be extended to a chain map $\overline f: R^*(E^*) @>>>
R^*(F^*)$ by defining it to be $0$ on the exact factor of the type
(\ref{L:module}). Then $\sigma_{\geq 0}\overline f$ maps $E^*$ to
$F^*$, and its homotopy class in $B$ has image $\phi$ in $Z$. This
proves surjectivity.
We now compute the kernel of (\ref{B:to:Z}). First if $\alpha\in
\operatorname{Hom}_B(E^*,F^*)$ factors through a complex $L^*$ of the form
(\ref{triv:comp}), then $R^*_{\min}(\alpha)$ factors through
$R^*_{\min}(L^*)=0$ and so vanishes. So the kernel of (\ref{B:to:Z})
contains all morphisms which factor through complexes of the form
(\ref{triv:comp}).
Conversely, suppose $\alpha$ is in the kernel of (\ref{B:to:Z}).
Since $\alpha$ is a morphism in $B$, it is a homotopy class of chain
maps from which we may choose a member $\beta$. We may complete
$\beta$ to a chain map $\rho: R^*(E^*) @>>> R^*(F^*)$.
\[
\begin{CD}
R^*(E^*) & \qquad & \dotsb & \:\rightarrow\: & 0 & \:\rightarrow\: &
\overline E^{-N} & \:\rightarrow\: & \dotsb & \:\rightarrow\: &
\overline E^{-1} & \:\rightarrow\: & E^0 & \:\rightarrow\: & \dotsb &
\:\rightarrow\: & E^{N-1} & \:\rightarrow\: & 0 & \:\rightarrow\: &
\dotsb \\
@VV{\rho}V && @VVV @VVV && @VVV @VV{\beta}V && @VV{\beta}V @VVV \\
R^*(F^*) & \qquad & \dotsb & \:\rightarrow\: & 0 & \:\rightarrow\: &
\overline F^{-N} & \:\rightarrow\: & \dotsb & \:\rightarrow\: &
\overline F^{-1} & \:\rightarrow\: & F^0 & \:\rightarrow\: & \dotsb &
\:\rightarrow\: & F^{N-1} & \:\rightarrow\: & 0 & \:\rightarrow\: &
\dotsb
\end{CD}
\]
The homotopy class of $\rho$ is the image of $\alpha$ under $R^*$ and
so must vanish by hypothesis. (Note that $R^*$ and $R^*_{\min}$ are
homotopy equivalent.) Thus $\rho$ is homotopic to $0$. Thus if we
write $\delta^i$ for the differentials of $R^*(E^*)$, and $\epsilon^i$
for the differentials of $R^*(F^*)$, then there is a chain homotopy $h
= (h^i)$ such that $\rho^i = h^{i+1}\delta^i + \epsilon^{i-1}h^i$ for
all $i$. Now restrict $h$ to a chain homotopy $\widehat{h} =
(\widehat{h}^i)$ with $\widehat{h}^i: E^i @>>> F^{i-1}$ defined by
defined by $\widehat{h}^i = h^i$ for all $i\geq 1$, and $\widehat{h}^i
= 0$ for all $i\leq 0$. Then $\beta$ is homotopic to a morphism whose
components are
\[
\beta^i-(\widehat{h}^{i+1}\delta^i + \epsilon^{i-1}\widehat{h}^i)=
\begin{cases}
\rho^i-(h^{i+1}\delta^i+\epsilon^{i-1}h^i) = 0 & \text{if }i\geq1,\\
\rho^0-h^1\delta^0 = \epsilon^{-1}h^0 & \text{if }i=0,\\
0 & \text{if }i\leq -1.
\end{cases}
\]
Hence the homotopy class $\alpha$ of $\beta$ factors through the complex
\[
\dotsb @>>> 0 @>>> \overline F^{-1} @>>> 0 @>>> \dotsb
\]
of type (\ref{triv:comp}). So the kernel of (\ref{B:to:Z}) is as
asserted. This completes the proof of the theorem.
\end{pf*}
We will use three further results concerning the Horrocks
correspondence. The first will permit us to use the Horrocks
correspondence to constuct locally free resolutions of coherent
sheaves.
\begin{proposition}
\label{functorial}
Let $\cal{Q}$ be a quasi-coherent sheaf on $\Bbb{P}^N$, let
$C^*\in\operatorname{ob}(\frak{Z})$, and let $\beta : C^* @>>> \tau_{>0}\tau_{<N}
\bold{R}\Gamma _*(\cal{Q})$ be a morphism in $D^b(\operatorname{Mod}_{\mit{S},\operatorname{gr}})$. Then there
exists a morphism of quasi-coherent sheaves $\widetilde{\beta} :
\cal{H}(C^*) @>>> \cal{Q}$ such that $\beta =\tau_{>0}\tau_{<N}
\bold{R}\Gamma _*(\widetilde{\beta})$. In particular, the induced
morphisms $H^i_*(\cal{H}(C^*)) @>>> H^i_*(\cal{Q})$ are the same as
$H^i(\beta)$ for $1\leq i\leq N-1$.
\end{proposition}
\begin{pf}
Let $R^*$ be a minimal projective resolution of $C^*$, and let
$\cal{I}^*$ be an injective resolution of $\cal{Q}$. Then $\beta$ may
be identified with an actual chain map
\[
\begin{CD}
\dotsb & \:\rightarrow\: & R^{-2} & \:\rightarrow\: & R^{-1} &
\:\rightarrow\: & R^0 & \:\stackrel{\lambda}{\longrightarrow}\: & R^1 &
\:\rightarrow\: & \dotsb & \:\rightarrow\: & R^{N-2} & \:\rightarrow\:
& R^{N-1} & \:\rightarrow\: & 0 \\
&& @VVV @VVV @VVV @VVV && @VVV @VVV \\
\dotsb & \:\rightarrow\: & 0 & \:\rightarrow\: & \Gamma_*(\cal{Q}) &
\:\rightarrow\: & \Gamma_*(\cal{I}^0) & \:\stackrel{\mu}{\longrightarrow}\: &
\Gamma_*(\cal{I}^1) & \:\rightarrow\: & \dotsb & \:\rightarrow\: &
\Gamma_*(\cal{I}^{N-2}) & \:\rightarrow\: & \ker(\delta^{N-1}) &
\:\rightarrow\: & 0
\end{CD}
\]
Thus $\beta$ induces a morphism $\widetilde{\beta}$ from $\cal{H}(C^*)
= \ker(\lambda)\sptilde$ to $\cal{Q} = \ker(\mu)\sptilde$.
We now need to calculate $\bold{R}\Gamma_*(\widetilde{\beta})$.
Consider the complex
\[
P^*: \qquad \dotsb @>>> 0 @>>> R^0 @>>> R^1 @>>> \dotsb @>>> R^{N-1}
@>>> 0 @>>> \dotsb .
\]
The previous diagram induces a new commutative diagram
\[
\begin{CD}
\widetilde{P}^*&: & \qquad\qquad & \dotsb & \:\rightarrow\: & 0 &
\:\rightarrow\: & \widetilde R^0 & \:\rightarrow\: & \widetilde R^1 &
\:\rightarrow\: & \dotsb & \:\rightarrow\: & \widetilde R^{N-2} &
\:\rightarrow\: & \widetilde R^{N-1} & \:\rightarrow\: & 0 &
\:\rightarrow\: & 0 & \:\rightarrow\: & \dotsb \\
@VV{\overline \beta}V &&& @VVV @VVV @VVV && @VVV @VVV @VVV @VVV\\
\cal{I}^*&: && \dotsb & \:\rightarrow\: & 0 & \:\rightarrow\: &
\cal{I}^0 & \:\rightarrow\: & \cal{I}^1 & \:\rightarrow\: & \dotsb &
\:\rightarrow\: & \cal{I}^{N-2} & \:\rightarrow\: & \cal{I}^{N-1} &
\:\rightarrow\: & \cal{I}^N & \:\rightarrow\: & 0 & \:\rightarrow\: &
\dotsb
\end{CD}
\]
between resolutions of $\cal{H}(C^*)$ and $\cal{Q}$ extending
$\widetilde \beta$. Let $\gamma: P^* @>>> J^*$ be an injective
resolution of $P^*$. Then $\overline\beta$ factors through
$\widetilde\gamma$ as $\widetilde{P}^* @>>> \widetilde{J}^* @>>>
\cal{I}^*$. Applying $\Gamma_*$ now gives a factorization
\begin{equation}
\label{factorize}
P^* @>>> \Gamma_*(\widetilde{J}^*) @>>> \Gamma_*(\cal{I}^*).
\end{equation}
Now $\overline\beta$ is a map between resolutions of $\cal{H}(C^*)$
and $\cal{Q}$, respectively, which extends $\widetilde{\beta}:
\cal{H}(C^*) @>>> \cal{Q}$, while $\widetilde\gamma$ is a
quasi-isomorphism. So the map $\widetilde{J}^* @>>> \cal{I}^*$ is a
map between injective resolutions of $\cal{H}(C^*)$ and $\cal{Q}$
extending $\widetilde\beta$. So by definition, the second arrow of
(\ref{factorize}) is $\bold{R}\Gamma_*(\widetilde{\beta}) :
\bold{R}\Gamma_*(\cal{H}(C^*)) @>>> \bold{R}\Gamma_*(\cal{Q})$. On
the other hand, the proof of Lemma \ref{tauresol}(a) shows that the
first arrow of (\ref{factorize}) can be identified with the truncation
$\tau_{<N}(\bold{R}\Gamma_*(\cal{H}(C^*))) @>>>
\bold{R}\Gamma_*(\cal{H}(C^*))$ because it induces isomorphisms
$H^i(P^*) \cong H^i(\Gamma_*(\widetilde{J}^*)) = H^i_*(\cal{H}(C^*))$
for $i < N$. Hence $\Gamma_*(\overline\beta): P^* @>>>
\Gamma_*(\cal{I}^*)$ can be identified with the composition of the
truncation $\tau_{<N}(\bold{R}\Gamma_*(\cal{H}(C^*))) @>>>
\bold{R}\Gamma_*(\cal{H}(C^*))$ with $\bold{R}\Gamma_*
(\widetilde{\beta})$. Thus $\tau_{<N}\bold{R}\Gamma_*
(\widetilde{\beta})$ may be identified with the diagram
\[
\begin{CD}
\dotsb & \:\rightarrow\: & 0 &
\:\rightarrow\: & R^0 & \:\rightarrow\: & R^1 &
\:\rightarrow\: & \dotsb & \:\rightarrow\: & R^{N-2} & \:\rightarrow\:
& R^{N-1} & \:\rightarrow\: & 0 & \:\rightarrow\: & \dotsb \\
&& @VVV @VVV @VVV && @VVV @VVV @VVV\\
\dotsb & \:\rightarrow\: & 0 &
\:\rightarrow\: & \Gamma_*(\cal{I}^0) & \:\rightarrow\: &
\Gamma_*(\cal{I}^1) & \:\rightarrow\: & \dotsb & \:\rightarrow\: &
\Gamma_*(\cal{I}^{N-2}) & \:\rightarrow\: & \ker(\delta^{N-1}) &
\:\rightarrow\: & 0 & \:\rightarrow\: & \dotsb
\end{CD}
\]
induced by $\beta$. Truncating on the left, we reach a diagram
equivalent to the first diagram of the proof of the proposition. So
$\beta = \tau_{>0}\tau_{<N} \bold{R}\Gamma _*(\widetilde{\beta})$.
\end{pf}
We now need two homological criteria for maps of vector bundles to be
isomorphisms.
\begin{lemma}
\label{isom}
Let $\cal E$ and $\cal F$ be vector bundles on $\Bbb{P}^N$ with neither
containing a line bundle as a direct factor. If $\alpha: \cal E @>>>
\cal F$ is a map such that $H^i_*(\alpha): H^i_*(\cal E) @>>>
H^i_*(\cal F)$ is an isomorphism for $0<i<N$, then $\alpha$ is an
isomorphism.
\end{lemma}
\begin{pf}
We use the notation of the proof of Theorem \ref{Horrocks}. Let
$E^*=P^*(\cal E)$ and $F^*=P^*(\cal F)$, and let $\overline\alpha: E^*
@>>> F^*$ be the map induced by $\alpha$. The hypothesis $\cal E =
\cal H\zeta(\cal E)$ implies that $E^*$ is homotopy equivalent to
$\sigma_{\geq 0}R^*_{\min}(E^*)$, or equivalently that $R^*(E^*)$ is a
minimal complex of projectives. Similarly, $R^*(F^*)$ is a minimal
complex of projectives. The hypothesis on $\alpha$ implies that
$\zeta(\alpha): \zeta(\cal E) @>>> \zeta(\cal F)$ is a
quasi-isomorphism. This in turn translates into
$R^*_{\min}(\overline\alpha)$ being a homotopy equivalence. But
because of the earlier hypotheses, this means that
$R^*(\overline\alpha): R^*(E^*) @>>> R^*(F^*)$ is a homotopy
equivalence between the minimal complexes of projectives. Hence by
Lemma \ref{NAK} $R^*(\overline\alpha)$ is actually an isomorphism of
complexes. So its naive truncation $\sigma_{\geq 0}
R^*(\overline\alpha) = \overline\alpha$ is also an isomorphism.
Therefore $\alpha$ is an isomorphism.
\end{pf}
We will also need a slight generalization of the previous lemma.
\begin{lemma}
\label{second:isom}
Let $\cal E = \cal H\zeta(\cal E)\oplus\bigoplus\cal O_{\Bbb{P}^N}(n_i)$ and
$\cal F$ be vector bundles on $\Bbb{P}^N$, and let $\cal Q$ be a coherent
sheaf on $\Bbb{P}^N$. Suppose that there exist morphisms $\alpha: \cal E
@>>> \cal F$ and $\beta: \cal F @>>> \cal Q$ such that
\rom(i\rom) $H^i_*(\alpha): H^i_*(\cal E) @>>> H^i_*(\cal F)$ is an
isomorphism for $0<i<N$,
\rom(ii\rom) $\beta\alpha$ takes the generators of the factors
$S(n_i)$ of $\Gamma_*(\cal E)$ onto a minimal set of generators of the
module $\overline Q :=\Gamma_*(\cal Q)/\beta\alpha(\Gamma_*(\cal
H\zeta(\cal E)))$,
\rom(iii\rom) $\cal E$ and $\cal F$ have the same rank.
Then $\alpha$ is an isomorphism.
\end{lemma}
\begin{pf}
Write $\cal F = \cal H\zeta(\cal F)\oplus\bigoplus\cal O_{\Bbb{P}^N}(m_j)$.
The splittings of $\cal E$ and of $\cal F$ into direct factors are not
canonical. But choosing such splittings gives an injection $\cal
H\zeta (\cal E) \hookrightarrow \cal E$ and a projection $\cal F
\twoheadrightarrow \cal H\zeta (\cal F)$. Then the composition
\[
\overline{\alpha}:\quad \cal H\zeta (\cal E) @>>> \cal E @>{\alpha}>>
\cal F @>>> \cal H\zeta (\cal F)
\]
is, like $\alpha$, an isomorphism on $H^i_*$ for $0<i<N$. So
$\overline \alpha$ is an isomorphism by Lemma \ref{isom}. Hence by
identifying $\cal H\zeta(\cal F)$ with $\alpha (\cal H\zeta(\cal E))
\subset \cal F$, we see that $\alpha$ induces a morphism of diagrams
\begin{equation}
\label{split}
\begin{CD}
0 @>>> \cal H\zeta(\cal E) @>>> \cal E @>>> \bigoplus \cal
O_{\Bbb{P}^N}(n_i) @>>> 0\\
&& @| @VV{\alpha}V @VV{\alpha_1}V \\
0 @>>> \cal H\zeta(\cal F) @>>> \cal F @>>> \bigoplus \cal
O_{\Bbb{P}^N}(m_j) @>>> 0
\end{CD}
\end{equation}
The morphisms $\alpha$ and $\beta$ therefore induce maps
\[
\bigoplus S(n_i) @>{\Gamma_*(\alpha_1)}>> \bigoplus S(m_j)
@>{\overline \beta}>> \overline Q = \Gamma_*(\cal Q)/ \beta\alpha
(\Gamma_*(\cal H\zeta(\cal E))).
\]
The composition is a surjection corresponding to a minimal set of
generators of $\overline Q$ by hypothesis (ii). Hence the righthand
map $\overline \beta$ must be a surjection corresponding to a set of
generators of $\overline Q$. However, the two free modules have the
same rank by hypothesis (iii). Hence $\overline \beta$ also
corresponds to a minimal set of generators, and $\Gamma_*(\alpha_1)$
must be an isomorphism. So returning to diagram (\ref{split}),
$\alpha_1$ and hence $\alpha$ are isomorphisms.
\end{pf}
\section{The Self-Dual Resolution}
Let $X\subset \Bbb{P}^{n+3}$ be a locally Gorenstein subcanonical subscheme of
equicodimension $3$ satisfying the parity condition. In this section
we use the Horrocks correspondence and especially Proposition
\ref{functorial} to construct a locally free resolution of $\cal O_X$.
We then use Lemma \ref{second:isom} to show that the resolution
satisfies condition (a) of Proposition
\ref{conditions}.
In the course of the construction we will need a more refined variant
of the canonical truncation. Namely, suppose $D^*$ is a complex of
objects in an abelian category with differentials $\delta^i: D^i @>>>
D^{i+1}$. Suppose $r$ is an integer, and $W\subset H^r(D^*)$ a
subobject. Then $W$ may be pulled back to a $\overline W$ satisfying
\[
\operatorname{im}(\delta^{r-1})\subset \overline W \subset \ker(\delta^r) \subset
D^r
\]
We then define:
\[
\tau_{\leq r,W}(D^*): \qquad \dotsb @>>> D^{r-2} @>>> D^{r-1}
@>>> \overline{W} @>>> 0 @>>> 0 @>>> \dotsb \quad .
\]
The cohomology of this complex is given by
\[
H^i(\tau_{\leq r,W}(D^*)) =
\begin{cases}
H^i(D^*) & \text{if }i<r,\\ W & \text{if }i=r,\\0 & \text{if }i>r.
\end{cases}\\
\]
We will also use the following conventions. If $\cal{E}$ is a
coherent sheaf on $\Bbb{P}^N$ and $\alpha,\beta\in\Bbb{Q}$ are not both
integers, then we define $H^{\alpha}(\cal{E}(\beta)) = 0$. Also if
$D^*$ is a complex and $\alpha\in\Bbb Q$, we define $\tau_{\leq
\alpha}(D^*) = \tau_{\leq [\alpha]}(D^*)$.
\subsection{Definition of the Locally Free Resolution}
Suppose $X\subset\Bbb{P}^{n+3}$ is a locally Gorenstein subscheme of
equidimension $n>0$ such that $\omega_X\cong\cal O_X(l)$ for some
integer $l$ and such that $h^{n/2}(\cal O_X(l/2))$ is even.
Let $\nu = n/2$ and $l' = l/2$. By hypothesis $H^{\nu}(\cal O_X(l'))$
is an even-dimensional vector space (zero if $n$ or $l$ is odd)
equipped with a nondegenerate $(-1)^{\nu}$-symmetric bilinear form
\[
H^{\nu}(\cal O_X(l')) \times H^{\nu}(\cal O_X(l')) @>>> H^n(\cal
O_X(l)) \cong k .
\]
Let $U\subset H^{\nu}(\cal O_X(l'))$ be an isotropic subspace of
maximal dimension $h^{\nu}(\cal O_X(l'))/2$. Let
\begin{equation}
\label{w}
W = U \oplus \bigoplus_{t>l'} H^{\nu}(\cal O_X(t)) \subset
H^{\nu}_*(\cal O_X)
\end{equation}
We begin the construction of the locally free resolution with the
short exact sequence
\begin{equation}
\label{ideal}
0 @>>> \cal I_X @>>> \cal O_{\Bbb{P}^{n+3}} @>>> \cal O_X @>>> 0.
\end{equation}
Since $H^i_*(\cal O_X)\cong H^{i+1}_*(\cal I_X)$ for $0<i<n+2$, we
have $W\subset H^{\nu+1}_*(\cal I_X)$.
Now since $X$ is locally Cohen-Macaulay of equidimension $n$, the
modules $H^i_*(\cal I_X)$ are of finite length for $0<i<n+1$. Hence
the truncated complex $C^*_X = \tau_{>0} \tau_{\leq\nu+1,W}
\bold{R}\Gamma_*(\cal I_X)$ has cohomology modules $H^i(C^*_X)$ of
finite length for $0<i\leq \nu+1$, while $H^i(C^*_X)=0$ for all other
$i$. Hence $C^*_X$ is in $\frak Z$.
The definition of $C^*_X$ as a truncation means that it is endowed
with a natural map $\beta: C^*_X @>>> \tau_{>0}\tau_{<n+3}
\bold{R}\Gamma_*(\cal I_X)$. By Proposition \ref{functorial} this map
induces a morphism $\widetilde{\beta}: \cal{H}(C^*_X) @>>> \cal I_X$.
Let $Q$ be the cokernel
\[
H^0_*(\cal{H}(C^*_X)) @>{H^0_*(\widetilde{\beta})}>> H^0_*(\cal I_X)
@>>> Q @>>> 0.
\]
Let $d_1,\dots,d_r$ be the degrees of a minimal set of generators of
$Q$. These generators lift to $H^0_*(\cal I_X)$, allowing us to
define a surjection
\begin{equation}
\label{gamma}
\gamma: \cal{F}_1 := \cal{H}(C^*_X)\oplus\bigoplus\cal O_{\Bbb{P}^{n+3}}(-d_i)
\twoheadrightarrow \cal I_X .
\end{equation}
By construction, $\cal F_1$ is locally free.
Let $\cal K = \ker(\gamma)$. We may then attach the short exact
sequence $0 @>>> \cal K @>>> \cal F_1 @>>> \cal I_X @>>> 0$ to the
short exact sequence (\ref{ideal}) to get an exact sequence
\begin{equation}
\label{partial}
0 @>>> \cal K @>>> \cal F_1 @>>> \cal O_{\Bbb{P}^{n+3}} @>>> \cal O_X @>>> 0.
\end{equation}
The construction described above leads immediately to the following
conclusions about the cohomology of $\cal F_1$ and about the induced
morphisms $H^i_*(\gamma): H^i_*(\cal F_1) @>>> H^i_*(\cal I_X)$ (cf.\
Proposition \ref{functorial}).
\begin{itemize}
\item $H^i_*(\gamma)$ is surjective (resp.\ an isomorphism) for $i=0$
(resp.\ $0<i<\nu+1$).
\item $H^{\nu+1}_*(\gamma): H^{\nu+1}_*(\cal F_1) \cong W
\hookrightarrow H^{\nu+1}_*(\cal I_X)$ is injective.
\item $H^i_*(\cal F_1)=0$ for $\nu+1<i<n+3$.
\end{itemize}
One may now draw the following conclusions about the cohomology of
$\cal K$.
\begin{itemize}
\item $H^i_*(\cal K) = 0$ for $0<i<\nu+2$.
\item $H^{\nu+2}_*(\cal K)\cong H^{\nu}_*(\cal O_X)/W$.
\item $H^i_*(\cal K) \cong H^{i-2}_*(\cal O_X)$ for $\nu+2<i<n+3$.
\end{itemize}
To finish the definition of the locally free resolution, consider the
isomorphisms
\[
\operatorname{Ext}^1(\cal K,\omega_{\Bbb{P}^{n+3}}(-l)) \cong H^{n+2}(\cal K(l))^* \cong
H^n(\cal O_X(l))^* \cong H^0(\cal O_X).
\]
The extension class corresponding to $1\in H^0(\cal O_X)$ gives a
short exact sequence
\begin{equation}
\label{extension}
0 @>>> \omega_{\Bbb{P}^{n+3}}(-l) @>>> \cal F_2 @>>> \cal K @>>> 0
\end{equation}
which we may attach to (\ref{partial}) to get a complex of the type
(\ref{P}) resolving $\cal O_X$
\begin{equation}
\label{free:resol}
\cal P^*:\qquad 0 @>>> \omega_{\Bbb{P}^{n+3}}(-l) @>>> \cal{F}_2 @>>> \cal{F}_1
@>>> \cal O_{\Bbb{P}^{n+3}} .
\end{equation}
\begin{lemma}
\label{duality}
The sheaves $\cal F_1$ and $\cal F_2$ in the resolution
\rom{(\ref{free:resol})} satisfy $H^i_*(\cal F_2) \cong \left(
H^{n+3-i}_*(\cal F_1) \right)^*(l)$ for $0<i<n+3$.
\end{lemma}
\begin{pf}
If $0<i<\nu+2$, then $H^i_*(\cal F_2) \cong H^i_*(\cal K) = 0$ and
$H^{n+3-i}_*(\cal F_1) = 0$. So the lemma holds for these values of
$i$.
If $i=\nu+2$, then $H^{\nu+2}_*(\cal F_2) \cong H^{\nu+2}_*(\cal K)
\cong H^{\nu}_*(\cal O_X)/W$, while $H^{\nu+1}_*(\cal F_1) \cong W$.
However, the submodule $W\subset H^{\nu}_*(\cal O_X)$ has been
constructed so that it is an isotropic submodule with respect to the
perfect pairing of Serre duality
\[
H^{\nu}_*(\cal O_X)\times H^{\nu}_*(\cal O_X) @>>> H^n_*(\cal
O_X) @>{\operatorname{tr}}>> k(-l).
\]
Moreover the length of $W$ is half the length of $H^{\nu}_*(\cal
O_X)$. Hence $W = W^{\perp}$, and the duality isomorphism
$H^{\nu}_*(\cal O_X) \cong \left( H^{\nu}_*(\cal O_X) \right)^*(l)$
carries the submodule $W$ onto $\left(H^{\nu}_*(\cal O_X)/W
\right)^*(l)$.
If $\nu+2 < i < n+2$, then $H^i_*(\cal F_2) \cong H^i_*(\cal K) \cong
H^{i-2}_*(\cal O_X)$, while $H^{n+3-i}_*(\cal F_1) \cong
H^{n+3-i}_*(\cal I_X) \cong H^{n+2-i}_*(\cal O_X)$. The asserted
duality is then simply the Serre duality pairing
\[
H^{i-2}_*(\cal O_X)\times H^{n+2-i}_*(\cal O_X) @>>> H^n_*(\cal
O_X) @>{\operatorname{tr}}>> k(-l).
\]
Finally if $i=n+2$, we have an exact sequence
\[
0 @>>> H^{n+2}_*(\cal F_2) @>>> H^{n+2}_*(\cal K) @>>>
H^{n+3}_*(\omega_{\Bbb{P}^{n+3}}(-l)).
\]
Now $H^{n+2}_*(\cal K) \cong H^{n}_*(\cal O_X)$. Moreover, the fact
that the extension class defining $\cal F_2$ corresponded under the
Serre duality identifications to $1\in H^0_*(\cal O_X) \cong \left(
H^{n+2}_*(\cal K) \right)^*(l)$ implies that the last exact sequence
dualizes to
\[
H^0_*(\cal O_{\Bbb{P}^{n+3}}) @>1>> H^0_*(\cal O_X) @>>> \left( H^{n+2}_*(\cal
F_2) \right)^*(l) @>>> 0.
\]
Hence $\left( H^{n+2}_*(\cal F_2) \right)^*(l) \cong H^1_*(\cal I_X)
\cong H^1_*(\cal F_1)$. Dualizing now gives the last of the asserted
isomorphisms.
\end{pf}
\begin{corollary}
The coherent sheaf $\cal F_2$ in the resolution
\rom{(\ref{free:resol})} is locally free.
\end{corollary}
\begin{pf}
Since $\cal F_1$ is locally free, $H^i_*(\cal F_1)$ is of finite
length for $0<i<n+3$. So by the lemma, $H^i_*(\cal F_2)$ is also of
finite length for $0<i<n+3$. But this implies that $\cal F_2$ is
locally free.
\end{pf}
\begin{proposition}
\label{cond:a}
The locally free resolution \rom{(\ref{free:resol})} satisfies
condition \rom(a\rom) of Proposition \ref{conditions}.
\end{proposition}
\begin{pf}
We write $\cal L=\omega_{\Bbb{P}^{n+3}}(-l)$. We have to show that if there is
a commutative diagram
\begin{equation}
\label{self:dual}
\begin{CD}
0 @>>> \cal{L} @>{d_3}>> \cal{F}_2 @>{d_2}>>
\cal{F}_1 @>{d_1}>> \cal O_{\Bbb{P}^{n+3}}\\
&& @| @VV{s_2}V @VV{s_1}V @| \\
0 @>>> \cal{L} @>{d_1\spcheck}>>
\cal{F}_1\spcheck\otimes\cal{L} @>{-d_2\spcheck}>>
\cal{F}_2\spcheck\otimes\cal{L} @>{d_3\spcheck}>> \cal O_{\Bbb{P}^{n+3}}
\end{CD}
\end{equation}
such that the vertical maps extend the identity on $\cal O_X$, then
$s_1$ and $s_2$ are isomorphisms.
By exactness, the image of $d_3\spcheck$ is $\cal I_X$. We will show
that $s_1$ is an isomorphism by applying Lemma \ref{second:isom} to
the composition
\[
\cal F_1 @>{s_1}>> \cal F_2\spcheck\otimes \cal L \twoheadrightarrow
\cal I_X.
\]
Note that this composition is exactly the surjection $\gamma: \cal F_2
\twoheadrightarrow \cal I_X$ of (\ref{gamma}). Hence the composition
\[
H^i_*(\cal F_1) @>{H^i_*(s_1)}>> H^i_*(\cal F_2\spcheck\otimes\cal L) @>>>
H^i_*(\cal I_X)
\]
is injective for $0<i<n+3$. A fortiori, $H^i_*(s_1)$ is also
injective for $0<i<n+3$.
However, by Serre duality $H^i_*(\cal F_2\spcheck\otimes \cal L)
\cong \left( H^{n+3-i}_*(\cal F_2) \right)^*(l)$ for all $i$. So by
Lemma \ref{duality}, we have $H^i_*(\cal F_2\spcheck\otimes \cal L)
\cong H^i_*(\cal F_1)$ for $0<i<n+3$. Hence for each $0<i<n+3$, the
morphism $H^i_*(s_1)$ is an injection of modules of the same finite
length. Hence $H^i_*(s_1)$ is an isomorphism for $0<i<n+3$. Thus
condition (i) of Lemma \ref{second:isom} holds.
Condition (ii) of Lemma \ref{second:isom} holds because of the method
of construction of $\cal F_1$ and of the surjection $\gamma$ in
(\ref{gamma}). Finally exactness in the resolution implies that $\cal
F_1$ and $\cal F_2$ have the same rank. Hence $\cal F_1$ and $\cal
F_2\spcheck\otimes\cal L$ also have the same rank, which is condition
(iii) of Lemma \ref{second:isom}. Hence all three conditions of Lemma
\ref{second:isom} hold, and we may conclude that $s_1$ is an
isomorphism.
The map $s_2$ must now also be an isomorphism by the five-lemma. This
completes the proof of the proposition.
\end{pf}
\section{The Differential Graded Algebra Structure}
In this section we finish the proof of Theorem \ref{main} by showing
that the locally free resolution (\ref{free:resol}) defined in the
previous section satisfies condition (b) of Proposition
\ref{conditions}. That is to say, we show that the locally free
resolution (\ref{free:resol}) admits a commutative, associative
differential graded algebra structure.
Throughout this section we assume that the characteristic is not $2$.
We recall what needs to be proven. In the previous section we defined
a locally free resolution (\ref{free:resol}) of $\cal O_X$
\[
0 @>>> \cal L @>{d_3}>> \cal F_2 @>{d_2}>> \cal F_1 @>{d_1}>> \cal
O_{\Bbb{P}^{n+3}} @>>> \cal O_X.
\]
Let $\cal K=\ker(d_1)$. We then had a morphism $\psi: \Lambda^2\cal
F_1 @>>> \cal K$ defined by $\psi(a\wedge b) = d_1(a)b-d_1(b)a$. We
also have a long exact sequence
\[
\cdots @>>> \operatorname{Hom}(\Lambda^2\cal F_1,\cal F_2) @>>> \operatorname{Hom}(\Lambda^2\cal
F_1,\cal K) @>>> \operatorname{Ext}^1(\Lambda^2\cal F_1,\cal L) @>>> \cdots.
\]
According to diagram (\ref{liftdiag}), the problem is to lift $\psi\in
\operatorname{Hom}(\Lambda^2\cal F_1,\cal K)$ to a $\phi\in\operatorname{Hom}(\Lambda^2\cal
F_1,\cal F_2)$. The obstruction to doing this is simply the image of
$\psi$ in
\[
\operatorname{Ext}^1(\Lambda^2\cal F_1,\cal L) = H^{n+2}(\Lambda^2\cal F_1(l))^*.
\]
Our first goal will therefore be to compute $H^{n+2}(\Lambda^2\cal
F_1(l))$. We begin by considering a complex of locally free sheaves
on $\Bbb{P}^N$.
\begin{equation}
\label{G}
\cal G^*: \qquad 0 @>>> \cal G^0 @>>> \cal G^1 @>>> \cdots @>>> \cal
G^r @>>> 0.
\end{equation}
There is an involution
\begin{align*}
T: \quad \cal G^*\otimes\cal G^* \ & @>>> \ \cal G^*\otimes\cal G^*\\
a\otimes b \ & \mapsto \ (-1)^{(\deg a)(\deg b)}b\otimes a
\end{align*}
interchanging the factors of $\cal G^*\otimes\cal G^*$. Since the
characteristic is not $2$, the complex $\cal G^*\otimes\cal G^*$
splits into a direct sum of subcomplexes on which $T$ acts as
multiplication by $\pm 1$, viz.\ $\cal G^*\otimes\cal G^* = S_2(\cal
G^*) \oplus \Lambda^2(\cal G^*)$. The complex $\Lambda^2(\cal G^*)$
is of the form
\begin{equation}
\label{Lambda:length}
\Lambda^2(\cal G^*):\qquad 0 @>>> \cal H^0 @>>> \cal H^1 @>>> \cdots
@>>> \cal H^{2r} @>>> 0
\end{equation}
where (cf.\ \cite{BE} p.\ 452)
\begin{equation}
\label{Lambda:formula}
\cal H^i \cong \bigoplus_{q<i/2} \left( \cal G^q\otimes \cal G^{i-q}
\right) \oplus
\begin{cases}
0 &\text{if $i$ is odd,}\\
\Lambda^2(\cal G^{i/2}) &\text{if }i\equiv 0\pmod 4,\\
S_2(\cal G^{i/2}) &\text{if }i\equiv 2\pmod 4.
\end{cases}
\end{equation}
\begin{lemma}
\label{Lambda}
Suppose $\cal G^*$ is a complex of locally free sheaves on $\Bbb{P}^N$ as in
\rom{(\ref{G})} which is exact except in degree $0$. Let $\cal E =
H^0(\cal G^*)$. Then $\Lambda^2(\cal G^*)$ is an exact sequence of
locally free sheaves which is exact except in degree $0$, and
$H^0(\Lambda^2(\cal G^*)) = \Lambda^2\cal E$.
\end{lemma}
\begin{pf}
The standard spectral sequences of the double complex $\cal
G^*\otimes\cal G^*$ degenerate to show that the simple complex $\cal
G^*\otimes \cal G^*$ is exact except in degree $0$, and $H^0(\cal
G^*\otimes\cal G^*) = \cal E\otimes\cal E$. Thus the augmented
complex $0 @>>> \cal E\otimes\cal E @>>> \cal G^*\otimes\cal G^*$ is
exact, and consequently its direct factor $0 @>>> \Lambda^2\cal E @>>>
\Lambda^2(\cal G^*)$ is also exact.
\end{pf}
\begin{lemma}
\label{max:cohom}
Suppose $\cal E$ is a locally free sheaf on $\Bbb{P}^N$. Let $r<N/2$ be an
integer. Suppose that $H^i_*(\cal E)=0$ for $r<i<N$. Then
\rom(a\rom) $H^i_*(\Lambda^2\cal E)=0$ for $2r<i<N$,
\rom(b\rom) $H^{2r}_*(\Lambda^2\cal E)\cong S_2 (H^r_*(\cal E))$ if
$r$ is odd, and $H^{2r}_*(\Lambda^2\cal E)\cong \Lambda^2 (H^r_*(\cal
E))$ if $r$ is even.
\rom(c\rom) If $H^r(\cal E(t))=0$ for $t<q$ for some integer $q$, then
$H^{2r}(\Lambda^2\cal E(t))=0$ for $t<2q$, while $H^{2r}(\Lambda^2\cal
E(2q)) \cong S_2 (H^r(\cal E(q)))$ if $r$ is odd, and $H^{2r}
(\Lambda^2 \cal E(2q)) \cong \Lambda^2 (H^r(\cal E(q)))$ if $r$ is
even.
\end{lemma}
\begin{pf}
By Lemma \ref{tauresol}(b), the minimal projective resolution of $P^*$
of the truncation $\tau_{<N}\bold R\Gamma_*(\cal E)$ is a complex of
free graded $S$-modules such that $P^i=0$ unless $0\leq i\leq N-1$.
Indeed, since $H^i(\tau_{<N}\bold R\Gamma_*(\cal E))=0$ for all $i>r$,
Lemma \ref{bounds} indicates that $P^i=0$ unless $0\leq i\leq r$,
i.e.\ $P^*$ is of the form
\[
P^*: \qquad 0 @>>> P^0 @>>> \cdots @>>> P^{r-1} @>>> P^r @>>> 0 .
\]
We now consider the complex of free graded $S$-modules
\[
\Lambda^2(P^*):\qquad 0 @>>> \Lambda^2 P^0 @>>> \cdots @>>> P^{r-1}
\otimes P^r @>>> T_2 (P^r) @>>> 0.
\]
where $T_2(P^r) = \Lambda^2(P^r)$ if $r$ is even, and $T_2(P^r) =
S_2(P^r)$ if $r$ is odd (cf.\ \eqref{Lambda:length} and
\eqref{Lambda:formula}). According to Lemma \ref{tauresol}, the
complex of sheaves $\widetilde P^*$ associated to $P^*$ is exact
except in degree $0$ where the homology is $\cal E$. So Lemma
\ref{Lambda} implies that the complex of sheaves $\Lambda^2(\widetilde
P^*)$ is also exact except in degree $0$ where the homology is
$\Lambda^2\cal E$. The complex $\Lambda^2(P^*)$ of graded $S$-modules
therefore has homology of finite length except in degree $0$.
Moreover, the complex $\Lambda^2(P^*)$ vanishes except in degrees
between $0$ and $2r<N$, and the coefficients of its differentials lie
in $\frak m$ because it those of $P^*$ and therefore $P^*\otimes P^*$
do. It now follows from Lemma \ref{tauresol}(a) that $\Lambda^2(P^*)$
is the minimal projective resolution of $\tau_{<N}\bold
R\Gamma_*(\Lambda^2 \cal E)$.
Therefore $H^i_*(\Lambda^2\cal E)\cong H^i(\Lambda^2(P^*))$ for all
$i<N$. In particular, since $\Lambda^2(P^*)$ is concentrated in
degrees between $0$ and $2r$ by \eqref{Lambda:length}, we see that
$H^i_*(\Lambda^2\cal E)=0$ for $2r<i<N$. This is part (a) of the
lemma.
For (b) note that $H^r_*(\cal E)$ and $H^{2r}_*(\Lambda^2\cal E)$ has
respective presentations
\[
\begin{CD}
P^{r-1} & \:\overset{\delta}{\longrightarrow}\: & P^r &
\:\rightarrow\: & H^r_*(\cal E) & \:\rightarrow\: & 0 ,\\
P^{r-1}\otimes P^r & \:\overset{\delta_1}{\longrightarrow}\: &
T_2(P^r) & \:\rightarrow\: & H^{2r}_*(\Lambda^2\cal E) &
\:\rightarrow\: & 0,
\end{CD}
\]
where $\delta_1(e\otimes f) = \delta(e)f\in T_2 (P^r)$. But since the
presentation of $T_2(H^r_*(\cal E))$ is of exactly this form, we see
that $H^{2r}_*(\Lambda^2\cal E) \cong T_2(H^r_*(\cal E))$, as asserted
by the lemma.
For (c) write $H=H^r(\cal E(q))$. The hypothesis that $H^r(\cal
E(t))=0$ for $t<q$ implies that
\(
P^r = \left( H\otimes_k S(-q)\right) \oplus F
\)
with $F=\bigoplus S(-n_i)$ for some $n_i>q$. Then
\(
T_2(P^r) = \left( T_2 H \otimes_k S(-2q) \right) \oplus G
\)
with $G = \left( H\otimes_k F(-q) \right) \oplus T_2 F = \bigoplus
S(-m_j)$ for some $m_j>2q$. Since the presentation of
$H^{2r}_*(\Lambda^2 \cal E)$ given above has the property that no
direct factor of $P^{r-1}\otimes P^r$ is mapped surjectively onto a
factor of $T_2(P^r)$, it now follows that $H^{2r}(\Lambda^2\cal
E(t))=0$ for $t<2q$, and $H^{2r}(\Lambda^2\cal E(2q)) \cong T_2 H$.
\end{pf}
\begin{corollary}
\label{Lambda:cohom}
Let $n$, $l$, and $X\subset \Bbb{P}^{n+3}$ be as in Theorem \ref{main}.
Suppose that $U\subset H^{n/2}(\cal O_X(l/2))$ is the maximal
isotropic subspace defined in \eqref{w}, and that $\cal F_1$ is the
locally free sheaf defined in \eqref{gamma}. Then
\[
H^{n+2}(\Lambda^2\cal F_1(l)) \cong \begin{cases} 0 &\text{if $n$ or
$l$ is odd,}\\ S_2 U &\text{if $l$ is even, and }n\equiv 0\pmod 4,\\
\Lambda^2 U &\text{if $l$ is even, and }n\equiv 2\pmod 4.
\end{cases}
\]
\end{corollary}
\begin{pf}
If $n$ is odd, then $H^i_*(\cal F_1)=0$ for $(n+1)/2 < i < n+3$. So
Lemma \ref{max:cohom}(a) applies with $r=(n+1)/2$. Therefore
$H^i_*(\Lambda^2\cal F_1) = 0$ for $n+1<i<n+3$, i.e.\
$H^{n+2}(\Lambda^2\cal F_1(t)) = 0$ for all $t$.
If $n$ is even but $l$ is odd, then Lemma \ref{max:cohom}(c) applies
with $r=(n+2)/2$ and $q=(l+1)/2$. Then $H^{n+2}(\Lambda^2\cal
F_1(t))=0$ for all $t < l+1$.
If $l$ and $n$ are even, then Lemma \ref{max:cohom}(c) applies with
$r=(n+2)/2$ and $q=l/2$. Since $H^{(n+2)/2}(\cal F_1(l/2))\cong U$,
it follows that $H^{n+2}(\Lambda^2\cal F_1(l))\cong \Lambda^2 U$ if
$r$ is even, and $H^{n+2}(\Lambda^2\cal F_1(l))\cong S_2 U$ if $r$ is
odd. The corollary follows.
\end{pf}
\begin{lemma}
\label{cond:b}
If $\cal F_1$ is the locally free sheaf defined in \eqref{gamma}, then
the image of the map $\psi$ of \eqref{liftdiag} in
$\operatorname{Ext}^1(\Lambda^2\cal F_1,\cal L) \cong H^{n+2}(\Lambda^2\cal
F_1(l))^*$ vanishes.
\end{lemma}
\begin{pf}
If $n$ or $l$ is odd, then $H^{n+2}(\Lambda^2\cal F_1(l)) = 0$
according to Corollary \ref{Lambda:cohom}, so the image of $\psi$ is
evidently zero.
If $n$ and $l$ are even, then we claim that the image of $\psi$ in
$H^{n+1}(\Lambda^2\cal F_1(l))^*$ is the map
\[
\bigl\{ S_2 U \text{ or } \Lambda^2 U \bigr\} @>>> k
\]
which is the restriction to $U$ of the pairing $H^{n/2}(\cal O_X(l/2))
\times H^{n/2}(\cal O_X(l/2)) @>>> k$ of \eqref{pairing}. Since $U$
was chosen isotropic, this map vanishes.
In order to prove the claim, we consider the diagonal $i: \Bbb{P}^{n+3} =
\Delta\subset\Bbb{P}^{n+3}\times\Bbb{P}^{n+3}$. Then there is a natural inclusion
$i(X)\subset X\times X$ which corresponds to a restriction map
\begin{equation}
\label{rest:diag}
\cal O_{X\times X} @>>> i_* \cal O_X.
\end{equation}
This map is essentially the multiplication $\cal O_X\otimes\cal O_X
@>>> \cal O_X$. In any case applying $\bold R\Gamma_*$ to
\eqref{rest:diag} gives the cup product map
\begin{equation}
\label{cup:product}
\bold R\Gamma_*(\cal O_X)\otimes_k \bold R\Gamma_*(\cal O_X) @>>>
\bold R\Gamma_*(\cal O_X).
\end{equation}
Now consider the ``resolution'' of $\cal O_X$ given in \eqref{partial}
\[
\cal K^*:\qquad 0 @>>> \cal K @>>> \cal F_1 @>{d_1}>> \cal O_{\Bbb{P}^{n+3}}
@>>> 0.
\]
The complex $\cal K^*$ is quasi-isomorphic to $\cal O_X$. Hence the
restriction to the diagonal map \eqref{rest:diag} corresponds to a
morphism in the derived category
\[
p_1^*\cal K^*\otimes p_2^*\cal K^* @>>> i_*\cal K^*.
\]
In fact this morphism in the derived category is represented by an
actual map of complexes of sheaves
\begin{equation}
\label{FFK}
\begin{CD}
\dotsb & \:\rightarrow\: & p_1^*\cal K \oplus (p_1^*\cal F_1 \otimes
p_2^*\cal F_1) \oplus p_2^*\cal K & \:\rightarrow\: & p_1^*\cal F_1
\oplus p_2^*\cal F_1 & \:\rightarrow\: & \cal O_{\Bbb P\times\Bbb P} &
\:\rightarrow\: & 0\\
&& @VVV @VVV @VVV \\
0 & \:\rightarrow\: & i_*\cal K & \:\rightarrow\: & i_*\cal F_1 &
\:\rightarrow\: & i_*\cal O_{\Bbb P} & \:\rightarrow\: & 0
\end{CD}
\end{equation}
All the vertical maps are straightforward restrictions to the diagonal
except for the component $p_1^*\cal F_1 \otimes p_2^*\cal F_1 @>>>
i_*(\cal K)$ which is defined (like $\psi$ of
\eqref{liftdiag}) by noting that the composition
\begin{equation}
\label{FFK2}
\begin{CD}
p_1^*\cal F_1 \otimes p_2^*\cal F_1 & \:\rightarrow\: p_1^*\cal F_1
\oplus p_2^*\cal F_1 \:\rightarrow\: & i_* \cal F_1 \\
p_1^*(a)\otimes p_2^*(b) & \mapsto & i_*\left(d_1(a)b-d_1(b)a\right)
\end{CD}
\end{equation}
is contained in the kernel of $i_*\cal F_1 @>>> i_*\cal O_{\Bbb P}$.
Now since $\cal K^*$ is quasi-isomorphic to $\cal O_X$, if we apply
$\bold R\Gamma_*$ to \eqref{FFK} we get a morphism $\bold
R\Gamma_*(p_1^*\cal K\otimes p_2^*\cal K) @>>> \bold R\Gamma_*(i_*\cal
K^*)$ in $D^b_{\grssmod}$ which is quasi-isomorphic to
\eqref{cup:product}. In particular, the maps of hypercohomology are
quasi-isomorphic to the cup product
\[
H^n_*(p_1^*\cal O_X\otimes p_2^*\cal O_X) \cong \bigoplus_i H^i_*(\cal
O_X)\otimes_k H^{n-i}_*(\cal O_X) @>>> H^n_*(\cal O_X).
\]
The hypercohomology $H^n_*(\cal K^*)\cong H^n_*(\cal O_X)$ is of
course the same as the $H^n$ of the total complex of the double
complex
\[
0 @>>> \bold R\Gamma_*(\cal K) @>>> \bold R\Gamma_*(\cal F_1) @>>>
\bold R\Gamma_*(\cal O_{\Bbb P}) @>>> 0.
\]
According to the calculations at the beginning of the previous
section, this $H^n_*$ is all attributable to $\cal K$, i.e.\ the
truncation $\cal K^* @>>> \cal K[2]$ induces an isomorphism
$H^n_*(\cal O_X) \cong H^n_*(\cal K^*) \cong H^{n+2}_*(\cal K)$.
Similarly, the hypercohomology $H^n_*(p_1^*\cal K^*\otimes p_2^*\cal
K^*)$ is the same as the $H^n$ of the total complex of $\bold
R\Gamma_*$ of the first row of \eqref{FFK}. The submodule $W\otimes_k W
\subset H^n_*(p_1^*\cal O_X\otimes p_2^*\cal O_X)$ is
attributable as the $H^{n+2}_*$ of the factor $p_1^*\cal F_1\otimes
p_2^*\cal F_1$ in the first row of \eqref{FFK}. Therefore $H^{n+2}_*$
of the vertical map $p_1^*\cal F_1 \otimes p_2^*\cal F_1 @>>> \cal
K$ is simply the cup product map $W\otimes_k W @>>> H^n_*(\cal O_X)$.
Now the fact that $i_*(\cal K)$ is supported on $\Delta$, plus the
symmetry of the product map imply that the vertical map of \eqref{FFK}
factors as
\begin{equation}
\label{factorization}
\begin{CD}
p_1^*\cal F_1\otimes p_2^*\cal F_1 & \:\rightarrow\: & i_*(\cal
F_1\otimes \cal F_1) & \:\rightarrow\: & i_*(\Lambda^2\cal F_1)
& \:\overset{i_*(\psi)}{\longrightarrow}\: & i_*(\cal K).\\
p_1^*(a)\otimes p_2^*(b) & \mapsto & i_*(a\otimes b) & \mapsto &
i_*(a\wedge b) & \mapsto & d_1(a)b-d_1(b)a
\end{CD}
\end{equation}
We wish to calculate $H^{n+2}_*$ of the above morphisms. Let
\[
P^*:\qquad 0 @>>> P^0 @>>> \dotsb @>>> P^{(n+2)/2} @>>> 0
\]
be a minimal projective resolution of $\tau_{<n+3}\bold R\Gamma_*(\cal
F_1)$ (cf.\ Lemma \ref{tauresol}). Then if one applies
$\tau_{<n+3}\bold R\Gamma_*$ to the first two morphisms of
\eqref{factorization}, one gets the natural maps
\[
P^*\otimes_k P^* @>>> P^*\otimes_S P^* @>>> \Lambda^2(P^*)
\]
(cf.\ the proof of Lemma \ref{max:cohom}). All three complexes are
supported in degrees between $0$ and $n+2$, and applying $H^{n+2}$
gives surjections
\[
W\otimes_k W \twoheadrightarrow W\otimes_S W \twoheadrightarrow
\bigl\{ S_2 W \text{ or } \Lambda^2 W \bigr\}.
\]
It therefore follows that $H^{n+2}_*(\psi): H^{n+2}_*(\Lambda^2\cal
F_1) @>>> H^{n+2}_*(\cal K)$ is isomorphic to the cup product map
$\bigl\{ S_2 U \text{ or } \Lambda^2 U \bigr\} @>>> H^n_*(\cal O_X)$.
In particular, in degree $l$ the morphism $H^{n+2}(\cal F_1(l)) @>>>
H^{n+2}(\cal K(l))$ is the same as $\bigl\{ S_2 U \text{ or }
\Lambda^2 U \bigr\} @>>> H^n(\cal O_X(l))$.
We now have to consider the extension of \eqref{extension}
\[
0 @>>> \omega_{\Bbb P}(-l) @>>> \cal F_2 @>>> \cal K @>>> 0.
\]
(Recall $\cal L = \omega_{\Bbb P}(-l)$.) In the associated long
exact sequence of cohomology
\[
\dotsb @>>> H^{n+2}(\cal F_2(l)) @>>> H^{n+2}(\cal K(l)) @>\operatorname{tr}>>
H^{n+3}(\omega_{\Bbb{P}^{n+3}}) \cong k,
\]
the differential is the element of $H^{n+2}(\cal K(l))^* \cong
\operatorname{Ext}^1(\cal K(l),\omega_{\Bbb{P}^{n+3}})$ corresponding to the extension class.
So by construction the differential is the trace map $\operatorname{tr}\in
H^{n+2}(\cal K(l))^*\cong H^n(\cal O_X(l))^*$ which corresponds under
Serre duality to $1\in H^0(\cal O_X)$.
Now the image of $\psi\in \operatorname{Ext}^1(\Lambda^2\cal F_1(l),\omega_{\Bbb P})
\cong H^{n+2}(\Lambda^2\cal F_1(l))^*$ is exactly the composition
\[
H^{n+2}(\Lambda^2\cal F_1(l)) @>{H^{n+2}(\psi)}>> H^{n+2}(\cal K(l))
@>\operatorname{tr}>> H^{n+3}(\omega_{\Bbb{P}^{n+3}}) \cong k.
\]
By our previous calculations, this is the composition of the cup
product map $\bigl\{ S_2 U \text{ or } \Lambda^2 U \bigr\} @>>>
H^n(\cal O_X(l))$ with the trace map $H^n(\cal O_X(l)) @>>> k$.
Therefore this composition is the restriction to $S_2 U$ or $\Lambda^2
U$ of the Serre duality pairing $H^{n/2}(\cal O_X(l/2))\otimes
H^{n/2}(\cal O_X(l/2)) @>>> k$. This is what was claimed at the
beginning of the proof of the lemma. Since $U$ was chosen isotropic,
this composition vanishes, i.e.\ the image of $\psi$ in $\operatorname{Ext}^1(\cal
K,\cal L)$ vanishes.
\end{pf}
\begin{pf*}{Proof of Theorem \ref{main}}
According to Proposition \ref{conditions}, in order to prove Theorem
\ref{main} it suffices to find a locally free resolution
\[
0 @>>> \cal L @>>> \cal F_2 @>>> \cal F_1 @>>> \cal O_{\Bbb{P}^{n+3}} @>>> \cal O_X
\]
which satisfies two conditions. But the locally free resolution
defined in \eqref{free:resol} was shown to satisfy the first of these
conditions was shown in Proposition \ref{cond:a}. Moreover, this
resolution was just shown to satisfy the second condition in Lemma
\ref{cond:b}. Hence Theorem \ref{main} holds.
\end{pf*}
\section{Characteristic $2$ Computations}
In the introduction, we asserted that Theorem \ref{main} also holds in
characteristic $2$ provided the phrase ``$n\equiv 0\pmod 4$'' in the
parity condition is replaced by the phrase ``$n$ is even.'' In this
section we justify that assertion by proving analogues of Lemmas
\ref{Lambda} and \ref{max:cohom} and Corollary \ref{Lambda:cohom} in
characteristic $2$. These were the only steps in the proof of
Theorem \ref{main} where we used the assumption that the
characteristic is not $2$.
Throughout this section we assume that the characteristic is $2$.
We recall certain simple facts from modular representation theory.
Let $R$ be a commutative algebra over a field of characteristic $2$,
and let $V$ be a free $R$-module. Let $t\in\operatorname{End}(V\otimes V)$ be the
endomorphism $t(a\otimes b) = a\otimes b - b\otimes a$. Set $D_2 V
=\ker(t)$, and $\Lambda^2 V =\operatorname{im}(t)$, and $S_2 V = \operatorname{coker}(t)$. Since
$t^2=0$ in characteristic $2$, there are inclusions
\[
0 \subset \Lambda^2 V \subset D_2 V \subset V\otimes V
\]
and corresponding surjection of quotients of $V\otimes V$
\[
V\otimes V \twoheadrightarrow S_2 V \twoheadrightarrow \Lambda^2 V
@>>> 0.
\]
The subquotient $D_2 V /\Lambda^2 V$ is $F(V)$, the Frobenius pullback
of $V$. It is a free module of the same rank as $V$. This $F(V)$ is
also the kernel of the surjection $S_2 V \twoheadrightarrow \Lambda^2
V$.
Note that the natural map from $S_2 V $ to $V\otimes V$ given by $xy
\mapsto x\otimes y + y\otimes x$ is not injective because any $x^2
\mapsto 0$. The map is the composition $S_2 V \twoheadrightarrow
\Lambda^2 V \hookrightarrow V\otimes V$ with kernel $F(V)$.
The operations $D_2$, $\Lambda^2$, $S_2$, and $F$ are all functorial.
Therefore we may define $D_2\cal E$, $\Lambda^2\cal E$, $S_2\cal E$,
and $F(\cal E)$ for any locally free sheaf $\cal E$ on any scheme $X$
over a field of characteristic $2$.
In some ways $F$ has better properties than the others. If $M =
(m_{ij}): R^n @>>> R^m$ is a morphism of free $R$-modules, then $F(M)
= (m_{ij}^2): R^n @>>> R^m$. One may use this formula together with
the Buchsbaum-Eisenbud exactness criterion \cite{BE:exact} to show
that $F$ of an exact sequence of locally free sheaves is exact. As
a result of this we get the following lemma.
\begin{lemma}
\label{frob}
Let $\cal E$ be a locally free sheaf on $\Bbb{P}^N$. If for some integer
$r$ one has $H^i_*(\cal E)=0$ for $r<i<N$, then $H^i_*(F(\cal E))=0$
for $r<i<N$ also.
\end{lemma}
\begin{pf}
Let $P^*$ be the minimal projective resolution of $\tau_{<N}\bold
R\Gamma_*(\cal E)$. By Lemmas \ref{tauresol} and \ref{bounds}, $P^*$
has the form
\[
P^*:\qquad 0 @>>> P^0 @>>> P^1 @>>> \dotsb @>>> P^r @>>> 0.
\]
Moreover, $P^*$ is exact except in degree $0$ away from the irrelevant
ideal $\frak m\subset S$, and $H^0(\widetilde P^*)=\cal E$.
The functoriality and exactness of $F$ now imply that
\[
F(P^*):\qquad 0 @>>> F(P^0) @>>> F(P^1) @>>> \dotsb @>>> F(P^r) @>>> 0
\]
is exact except in degree in degree $0$ away from the irrelevant
ideal, and has $H^0(F(\widetilde P^*))=F(\cal E)$. Applying Lemma
\ref{tauresol} again, we conclude that $F(P^*)$ is the minimal projective
resolution of $\tau_{<N}\bold R\Gamma_*(F(\cal E))$.
So if $r<i<N$, then $H^i_*(F(\cal E)) = H^i(F(P^*))=0$ since $F(P^*)$
vanishes in degrees greater than $r$.
\end{pf}
\begin{corollary}
\label{l:s}
Let $\cal E$ be a locally free sheaf on $\Bbb{P}^N$ over a field of
characteristic $2$ such that $H^{N-1}_*(\Lambda^2\cal E)=0$.
Suppose $r<N$ is an integer such that $H^i_*(\cal E)=0$ for $r<i<N$.
Then $H^i_*(S_2\cal E) \cong H^i_*(\Lambda^2\cal E)$ for $r<i<N$.
\end{corollary}
\begin{pf}
We consider the exact sequence $0 @>>> F(\cal E) @>>> S_2\cal E @>>>
\Lambda^2\cal E @>>> 0$ and the associated long exact sequence
\[
\dotsb @>>> H^i_*(F(\cal E)) @>>> H^i_*(S_2\cal E) @>>>
H^i_*(\Lambda^2\cal E) @>>> H^{i+1}_*(F(\cal E)) @>>> \dotsb
\]
The hypothesis $H^i_*(\cal E)=0$ for $r<i<N$ implies also
$H^i_*(F(\cal E))=0$ for $r<i<N$ by Lemma \ref{frob}. Hence the long
exact sequence implies that $H^i_*(S_2\cal E)\cong
H^i_*(\Lambda^2\cal E)$ for $r<i<N-1$ and that $H^{N-1}_*(S_2\cal
E) \hookrightarrow H^{N-1}_*(\Lambda^2\cal E)$ is injective. But
by hypothesis $H^{N-1}_*(\Lambda^2\cal E)=0$, so $H^{N-1}_*(S_2\cal
E)=0$ as well.
\end{pf}
We now prove the analogue of Lemma \ref{max:cohom}.
\begin{lemma}
\label{char:two}
Suppose $\cal E$ is a locally free sheaf on $\Bbb{P}^N$ over a field of
characteristic $2$. Let $0 < r <N/2$ be an integer such that
$H^i_*(\cal E)=0$ for $r<i<N$. Then
\rom(a\rom) $H^i_*(\Lambda^2\cal E)=0$ for $2r<i<N$,
\rom(b\rom) $H^{2r}_*(\Lambda^2\cal E)\cong S_2 (H^r_*(\cal E))$.
\rom(c\rom) If $H^r(\cal E(t))=0$ for $t<q$ for some integer $q$, then
$H^{2r}(\Lambda^2\cal E(t))=0$ for $t<2q$, while $H^{2r}(\Lambda^2\cal
E(2q)) \cong S_2 (H^r(\cal E(q)))$.
\end{lemma}
\begin{pf}
Let $P^*$ be the minimal projective resolution of $\tau_{<N}\bold
R\Gamma_*(\cal E)$
\[
P^*:\qquad 0 @>>> P^0 @>{\delta^0}>> P^1 @>{\delta^1}>> P^2 @>>>
\dotsb @>>> P^r @>>> 0
\]
Let $\cal P = \widetilde P^0$, and let $\cal F = \widetilde{\ker(
\delta^1)}$. Then we have an exact sequence $0 @>>> \cal E @>>> \cal
P @>>> \cal F @>>> 0$ such that $\cal P$ is a direct sum of line
bundles, $H^i_*(\cal F) = H^{i+1}_*(\cal E)$ for $0<i<N-1$, and
$H^{N-1}_*(\cal F)=0$.
It is easy to see that there is a natural exact complex
\begin{equation}
\label{lambda:s}
0 @>>> \Lambda^2 \cal E @>>> \Lambda^2 \cal P @>>> \cal P\otimes
\cal F @>>> S_2\cal F @>>> 0.
\end{equation}
We now prove parts (a) and (b) of the lemma by induction on $r$. If
$r=1$, then $\cal F = \widetilde P^1$ is a direct sum of line bundles,
and the complex \eqref{lambda:s} is just the augmented complex
\[
0 @>>> \Lambda^2 \cal E @>>> \Lambda^2(\widetilde P^*)
\]
which is still exact in this case. So we may conclude just as in
Lemma \ref{max:cohom} that $H^i_*(\Lambda^2\cal E)=0$ for $2<i<N$,
and that $H^2_*(\Lambda^2\cal E) = S_2(H^1_*(\cal E))$.
If $r>1$, then $H^i_*(\cal F)=0$ for $r-1<i<N$. So by induction
$H^i_*(\Lambda^2\cal F)=0$ for $2r-2<i<N$, and also $H^{2r-2}_*
(\Lambda^2\cal F)) \cong S_2(H^{r-1}_*(\cal F)) \cong S_2(H^r_*(\cal
E))$. It now follows from Corollary \ref{l:s} that $H^i_*(S_2\cal
F)\cong H^i_*(\Lambda^2\cal F)=0$ for $r-1 < i < N$. So in particular
$H^i_*(S_2\cal F)=0$ for $2r-2<i<N$ and that $H^{2r-2}_*(S_2\cal
F)\cong S_2(H^r_*(\cal E))$. Now since $\cal P$ is a direct sum of
line bundles, we have $H^i_*(\Lambda^2\cal P)=0$ for $0<i<N$, and
$H^i_*(\cal P\otimes\cal F)=0$ for $r-1<i<N$. So if we break up
\eqref{lambda:s} into short exact sequences and take its graded
cohomology, we can deduce that $H^i_*(\Lambda^2\cal E)) \cong
H^{i-2}_* (S_2\cal F)$ for $r+1<i<N$. Since $r+1<2r$, this gives
parts (a) and (b) of the lemma.
Part (c) of the lemma follows from part (b) by the same argument as in
Lemma \ref{max:cohom}.
\end{pf}
We have the following corollary in analogy with Corollary
\ref{Lambda:cohom}.
\begin{corollary}
\label{S2U}
Let $n$, $l$, and $X\subset \Bbb{P}^{n+3}_k$ be as in Theorem \ref{main} with
$k$ a field of characteristic $2$. Suppose that $U\subset
H^{n/2}(\cal O_X(l/2))$ is the maximal isotropic subspace defined in
\eqref{w}, and that $\cal F_1$ is the locally free sheaf defined in
\eqref{gamma}. Then
\[
H^{n+2}(\Lambda^2\cal F_1(l)) \cong \begin{cases} 0 &\text{if $n$ or
$l$ is odd,}\\ S_2 U &\text{if $n$ and $l$ are even.} \end{cases}
\]
\end{corollary}
The proof of Lemma \ref{cond:b} in characteristic $2$ is essentially
the same as in the previous section, only with Lemma \ref{char:two}
and Corollary \ref{S2U} replacing their analogues, and with $T_2 =
S_2$ always. Hence Theorem \ref{main} also holds in characteristic
$2$ as long as one treats all even $n$ the same.
\section{The Local Version of the Main Theorem}
In this section we consider Theorem \ref{RLR}, the local version of
our main result. We state a variant version which is clearly a local
analogue of Theorem \ref{main} with an identical proof, and then show
that this variant version is equivalent to Theorem \ref{RLR}.
Let $(R,\frak m,k)$ be a regular local ring, and let $U = \operatorname{Spec}(R) -
\{\frak m\}$ be the punctured spectrum of $R$. We say that a closed
subscheme $Y\subset U$ of pure codimension $3$ is {\em Pfaffian} if
$O_X$ has a locally free resolution on $U$
\[
0 @>>> \cal{O}_U @>h>> \cal{E}\spcheck @>f>> \cal{E} @>g>> \cal{O}_U
@>>> \cal O_X
\]
where $\cal E$ is a locally free $\cal O_U$-module of odd rank $2p+1$,
$f$ is skew-symmetric, and $g$ and $h=g\spcheck$ are given locally by
the Pfaffians of order $2p$ of $f$. The following theorem is the
obvious local analogue of Theorem \ref{main}.
\begin{theorem}
\label{punc:spec}
Let $(R,\frak m,k)$ be a regular local ring of dimension $n+4>4$ with
residue field not of characteristic $2$, and let $U=\operatorname{Spec}(R)-\{\frak
m\}$. Let $X\subset U$ be a closed subscheme of pure codimension $3$.
Then $X$ is Pfaffian if and only if the following three conditions
hold:
\rom(a\rom) $X$ is locally Gorenstein,
\rom(b\rom) $\omega_X\cong \cal O_X$, and
\rom(c\rom) if $n\equiv 0\pmod 4$, then $H^{n/2}(\cal O_X)$ is of even
length.
\end{theorem}
Theorem \ref{punc:spec} may be proven in exactly the same manner as
Theorem \ref{main}. All results concerning the Buchsbaum-Eisenbud
proof, the Horrocks correspondence, Serre/local duality, the
cohomology of $H^{n+2}(\Lambda^2\cal F_1)$ work identically for graded
modules over polynomial rings over $k$ and for modules over regular
local $k$-algebras. There is only one point which is in any way more
subtle in the local case. Namely, if $n$ is even, then one has a
Matlis duality pairing of $R$-modules of even finite length
\[
H^{n/2}(\cal O_X) \times H^{n/2}(\cal O_X) @>>> k.
\]
This pairing is perfect in the sense that for any submodule $M\subset
H^{n/2}(\cal O_X)$ one has
\[
\operatorname{length}(M)+\operatorname{length}(M^\perp) = \operatorname{length}(H^{n/2}(\cal O_X)).
\]
In order to be able to define $C^*_X$ and $\cal F_1$ as in
\eqref{gamma} one must choose an isotropic submodule $W$ of
length equal to half that of $H^{n/2}(\cal O_X)$. But it is not
difficult to show that this is possible.
We now compare Theorems \ref{RLR} and \ref{punc:spec}. First of all,
$E=\Gamma(\cal E)$ gives a bijective correspondence between locally
free sheaves $\cal E$ on $U$ and reflexive $R$-modules $E$ such that
$E_{\frak p}$ is a free $R_{\frak p}$-module for all prime ideals
$\frak p\neq \frak m$. There is also bijective correspondence
betweenclosed subschemes $X\subset U$ of pure codimension $3$ and
unmixed ideals $I\subset R$ of height $3$ given by $I=\Gamma(\cal
I_X)$. Hence an ideal $I$ is Pfaffian in the sense of Theorem
\ref{RLR} if and only if the corresponding subscheme $X\subset U$ is
Pfaffian in the sense of Theorem \ref{punc:spec}.
The three conditions (a), (b), and (c) of the two theorems also
correspond. In the case of (a) this is obvious. For (b) note that
$\omega_{R/I}\cong \Gamma(\omega_X)$ since for all $\frak p\in U$ one
has $\omega_{R/I,\frak p} = \operatorname{Ext}^3_{R_{\frak p}}((R/I)_{\frak
p},R_{\frak p}) = \omega_{X,\frak p}$, and $\omega_{R/I}$ is
saturated. Similarly $(R/I)^{\operatorname{sat}} \cong \Gamma(\cal O_X)$. This
gives the equivalence of the two conditions (b).
As for the conditions (c), first note that the dimension $n$ in
Theorem \ref{RLR} corresponds to $n+4$ in Theorem \ref{punc:spec}.
But if one uses $n$ as in the latter theorem, one has
\[
H^{n/2}(U,\cal O_X) \cong H^{(n+2)/2}(U,\cal I_X) \cong
H^{(n+4)/2}_{\frak m}(I).
\]
Hence the two conditions (c) correspond.
Therefore the two theorems \ref{RLR} and \ref{punc:spec} are
equivalent, as claimed.
In equicharacteristic $2$ the computations of Section 5 remain true in
the local case. So Theorems \ref{RLR} and \ref{punc:spec} are true in
equicharacteristic $2$ provided one changes the phrase ``$n\equiv
0\pmod 4$'' in the parity condition to ``$n$ is even.'' If $R$ is a
regular local ring with residue field of characteristic $2$ and
quotient field of characteristic $0$, a different set of calculations
is needed. These are unfortunately somewhat involved, and we do not
reproduce them here.
|
1993-05-20T18:42:25 | 9305 | alg-geom/9305011 | en | https://arxiv.org/abs/alg-geom/9305011 | [
"alg-geom",
"math.AG"
] | alg-geom/9305011 | Rita Pardini | Rita Pardini, Francesca Tovena | On the fundamental group of an abelian cover | 17 pages Latex Version 2.09 | null | null | null | null | We study the behaviour of the topological fundamental group under totally
ramified abelian covers (a special case of abelian Galois covers) of complex
projective varieties of dimension at least 2.
| [
{
"version": "v1",
"created": "Thu, 20 May 1993 17:36:21 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Pardini",
"Rita",
""
],
[
"Tovena",
"Francesca",
""
]
] | alg-geom | \section{Introduction.}
\hspace{6 mm}This work generalizes a result of Catanese and the second
author, who analyze in \cite{kn:Cato} the fundamental group of a
special type of covering ${f:Y\rightarrow
X}$, with Galois group $({\bf Z}/m{\bf
Z})^{2}$, of a complex smooth projective surface $X$, the so-called
"$m$-th root extraction" of a divisor $D$ on $X$.
By means of standard topological methods, the fundamental group
$\pi_{1}(Y)$ can be described in that case
as a central extension of the group $\pi_{1}(X)$, as follows:
\begin{equation}
\label{ext1}
0\longrightarrow {\bf Z}/r{\bf Z}\longrightarrow \pi_{1}(Y)
\longrightarrow \pi_{1}(X) \longrightarrow 1,
\end{equation}
$r$ being a divisor of $m$ which depends only on the divisibility of
$\pi^{*}(D)$ in $H^2(\tilde{X},{\bf Z})$, where
${\pi:\tilde{X}\rightarrow X}$ is the universal covering of $X$.
The main result of \cite{kn:Cato} (see Thm.2.16) is that
the group cohomology class corresponding to the extension
(\ref{ext1}) can be explicitly computed in terms of the first
Chern class of $D$.
This is an instance of a more general philosophy: in principle,
it should be possible to recover all the information
about an abelian cover ${f:Y\to X}$ from the "building data"
of the cover, i.e., from the Galois group $G$, the
components of the branch locus, the inertia subgroups
and the eigensheaves of ${f_{*}{\cal O}_{Y}}$ under the natural
action of $G$ (see section 2 or \cite{kn:Rita} for more
details).
Actually, the description of the general abelian cover given in
\cite{kn:Rita} enables us to treat (under some mild assumptions on the
components $D_{1},\ldots D_{k}$ of the branch locus) the case of any
totally ramified abelian covering ${f:Y\rightarrow X}$, with $X$ a
complex projective variety of dimension at least $2$ (cf. section 2
for the definition of a totally ramified abelian cover).
Using the same methods as in \cite{kn:Cato}, we show that
$\pi_{1}(Y)$ is a central extension as before:
\begin{equation}
\label{ext2}
0\longrightarrow K\longrightarrow \pi_{1}(Y) \longrightarrow
\pi_{1}(X) \longrightarrow 1\,,
\end{equation}
where $K$ is a finite abelian group which is determined by the
building data of the cover and the cohomology classes of
$\pi^{*}(D_{1}),\ldots \pi^{*}(D_{k})$ (cf. Prop.\ref{top}).
Consistently with the above "philosophy", a statement analogous to
Thm.2.16 of \cite{kn:Cato} actually holds in the general case: our
main result (Thm.\ref{mtgen}, Rem.\ref{dipdaLchi}) can be summarized by
saying that the class of the extension (\ref{ext2}) can be recovered
from the Chern classes of the $D_j$'s and of the eigensheaves of
$f_*{\cal O}_Y$; in some special case, this relation can be described
in a particularly simple way (Thm.\ref{mt}, Cor.\ref{mtcor},
Rem.\ref{rem}). Moreover, one can construct examples of not
homeomorphic varieties realized as covers of a projective variety $X$ with the
same Galois group, branch locus and inertia subgroups (cf.
Rem.\ref{dipdaLchi}).
The idea of the proof is to exploit a natural representation of
$\pi_{1}(Y)$ on a vector bundle on the universal covering
$\tilde{X}$ of $X$ and the spectral sequence describing the
cohomology of a quotient, in order to relate the group
cohomology class of the extension (\ref{ext2}) to
the geometry of the covering. These are basically
the same ingredients as in the proof of \cite{kn:Cato},
but we think that we have reached here a more conceptual
and clearer understanding of the argument.
\noindent
{\bf Acknowledgements:} we wish to express our heartfelt
thanks to Fabrizio Catanese, who suggested that the result
of \cite{kn:Cato} was susceptible of generalization
and encouraged us to investigate this problem.
\section{A brief review of abelian covers.}
\hspace{6 mm}In this section we set the notation and, for the reader's
convenience, we collect here the definitions and the
notions concerning abelian covers that will be needed later. For
further details and proofs, we refer to \cite{kn:Rita},
sections 1 and 2.
Let $X$, $Y$ be complex algebraic varieties of dimension at
least 2, smooth and projective, and let ${f:Y\to X}$
be a finite abelian cover, i.e. a Galois cover with finite
abelian Galois group $G$.
The bundle $f_{*}({\cal O} _{Y})$ splits as a sum of one dimensional
eigensheaves under the action of $G$, so that one has:
\begin {equation}
\label{splitting}
f_{*}({\cal O} _{Y})=
\bigoplus_{\chi \in G^{*}}L_{\chi}^{-1}
={\cal O} _{X}
\oplus ( \bigoplus _{\chi \in G^{*}\setminus \{1\}}
L_{\chi}^{-1})
\end{equation}
where $G^{*}$ denotes the group of characters of G and $G$ acts on
$L_{\chi}^{-1}$ via the character $\chi$.
We warn the reader that the notation here and in the next section
is dual to the one adopted in \cite{kn:Rita}; however this
does not affect the formulas quoted from there.
Under our assumptions, the ramification locus of $f$ is a divisor.
Let $D_{1}, \ldots D_{k}$ be the irreducible components of
the branch locus $D$ and let $R_{j} = f^{-1}(D_{j})$,
${j = 1,} \ldots k$. For every ${j = 1, \ldots k}$, one defines the
{\em inertia subgroup} $G_{j} = \{g\in G | g(y)=y \;
{\rm for \;each}\;y\in R_{j}\}$. Given any point
$y_{0} \in R_{j}$, one obtains a natural representation
of $G_{j}$ on the normal space to $R_{j}$ at $y_{0}$
by taking differentials. The corresponding character, that we
denote by $\psi_{j}$, is independent of the choice
of the point $y_{0} \in R_{j}$. By standard results, the subgroup
$G_{j}$ is cyclic and the character $\psi_{j}$ generates the group
$G_{j}^{*}$ of the characters of $G_{j}$. We denote by $m_{j}$ the
order of $G_{j}$, by $m$ the least common multiple of the $m_{j}$'s
and by $g_{j}$ the generator of $G_{j}$ such that
$\psi_{j}(g_{j}) = exp(\frac{2\pi \sqrt{-1}}{m_{j}})$.
In what follows we will always assume that the cover
${f:Y\rightarrow X}$ is {\em totally ramified}, i.e., that
the subgroups $G_{j}$ generate $G$; then the group of
characters $G^{*}$ injects in
${\bigoplus_{j=1}^{k}G_{j}^{*}}$ and every
${\chi \in G^{*}}$ may be written uniquely as:
\begin{equation}
\label{char0}
\chi = \sum_{j=1}^{k} \;a_{\chi,j}\,\psi_{j},
\hspace{10mm}\,0\leq a_{\chi,j}< m_{j} \;{\rm for\,every}\,j\,.
\end{equation}
In particular, let $\chi_{1}, \ldots \chi_{n} \in G^{*}$
be such that $G^{*}$ is the direct sum of the cyclic
subgroups generated by the $\chi $'s, and let $d_{i}$
be the order of $\chi_{i}$, $i = 1, \ldots n$. Write:
\begin{equation}
\label{char}
\chi_{i} = \sum_{j=1}^{k} \;a_{ij}\,\psi_{j},
\;\;\,\;0\leq a_{ij}< m_{j}\,, \;\;\, i= 1, \ldots n\,.
\end {equation}
Then one has (\cite{kn:Rita}, Prop.2.1):
\begin{equation}
\label{eqstr}
d_{i} L_{\chi_{i}}
\equiv \sum_{j=1}^{k}\frac{d_{i}a_{ij}}{m_{j}}\;D_{j}
\;\;\;\;i= 1, \ldots n\,
\end{equation}
the corresponding isomorphism of line bundles being induced
by multiplication in the ${\cal O}_{X}$-algebra
$f_{*}{\cal O}_{Y}$. More generally, if $\chi=\sum_{i=1}^n
b_{\chi,i}\chi_i$, with $0\leq b_{\chi,i}<d_i
\;\forall\,i$, one has (\cite{kn:Rita}, Prop.2.1):
\begin{equation}\label{Lchigen}
L_{\chi}\equiv \sum_{i=1}^n b_{\chi,i}L_{\chi_{i}} -
\sum_{j=1}^k q^{\chi}_{j} D_j\,.
\end{equation}
where $q^{\chi}_{j}$ is the integral part of the rational
number ${\sum_{i=1}^n \frac{b_{\chi,i}a_{ij}}{m_j}}$, $j=1,\ldots
k$.
Equations (\ref{eqstr}) are the characteristic relations of an
abelian cover. Actually, since $X$ is complete, for assigned $G$,
$D_{j}$, $G_{j}$, $\psi_{j}$, $j = 1, \ldots k$, to each set of line
bundles $L_{\chi_{i}}$, $i = 1, \ldots n$, satisfying (\ref{eqstr})
there corresponds a unique, up to isomorphism, $G$-cover of $X$,
branched on the $D_{j}$'s and such that $G_{j}$ is the inertia
subgroup of $D_{j}$ and $\psi_{j}$ is the corresponding character
(\cite{kn:Rita}, Thm.2.1). Moreover, the cover is actually smooth
under suitable assumptions on the building data.
\section{The fundamental group and the universal covering of $Y$.}
\hspace{6 mm}We keep the notation introduced in the previous section.
\begin{Definition}{\rm (\cite{kn:MM}, pag.218)}
A smooth divisor $\Delta$ on a variety $X$ is called
{\rm flexible}
if there exists a smooth divisor $\Delta ' \equiv \Delta$
such that $\Delta ' \cap \Delta \neq \emptyset$
and $\Delta$ and $\Delta '$ meet transversely.
\end{Definition}
We recall that a flexible divisor on a projective surface is connected
(see \cite{kn:Ca}, Remark 1.5). Hence, by considering a general linear
section, one deduces easily that a flexible divisor on a
projective variety of dimension $\geq 2$ is connected.
\begin{Proposition}
\label{top}
Let $X$, $Y$ be smooth projective varieties over {\bf C} of
dimension $n$ at least 2. Let ${f:Y\to X}$ be a totally ramified
abelian cover branched on irreducible, flexible and ample divisors
$\{D_j\}_{j=1,\ldots k}$. Then:
\begin{list}%
{\alph{alph})}{\usecounter{alph}}
\item
The natural map ${f_{*}:\pi_{1}(Y)\to \pi_{1}(X)}$
is surjective.
\item Let $K = ker(f_{*})$; then $K$ is finite and
\begin{equation}
\label{ext}
0\longrightarrow K\longrightarrow \pi_{1}(Y) \longrightarrow
\pi_{1}(X) \longrightarrow 1.
\end{equation}
is a central group extension.
\item Let ${\pi :\tilde{X}\to X}$ be the universal covering of $X$
and ${\tilde{D} = \pi^{-1}(D)}$;
then $\tilde{D}_{j}=\pi^{-1}(D_{j})$ is connected, j = 1, \ldots k.
Denote by $H^{i}_{c}$ the cohomology with compact supports and by
${\rho : H^{2n-2}_{c}(\tilde{X}) \to H^{2n-2}_{c}(\tilde{D})
\cong \bigoplus ^{k}_{j=1}\,{\bf Z}\tilde{D}_{j}}$ the restriction
map. Finally, let $\sigma$ be the map defined by:
\begin{equation}
\begin{array}{crclc}
\sigma:&H^{2n-2}_{c}(\tilde{D})
\cong \bigoplus ^{k}_{j=1}\,{\bf Z}\tilde{D}_{j}
&\to& \bigoplus_{j=1}^{k}G_{j}&\\
&{\tilde{D}_{j}} &\mapsto &g_{j}&.
\end{array}
\end{equation}
Then ${N = \ker(\bigoplus G_{j} \to G)}$ contains
${\rm Im}(\sigma \circ \rho)$ and $K$ is isomorphic
to the quotient group $N/{\rm Im}(\sigma \circ \rho)$.
\end{list}
\end{Proposition}
{\sc Proof.} a) and the fact that the extension (\ref{ext}) is central
can be proven exactly as in \cite{kn:Ca}, Thm.1.6 and in
\cite{kn:Cato}, Lemma 2.1.
For the proof of c) (that implies that $K$ is finite), we refer the
reader to \cite{kn:Ca}, Prop.1.8 and to \cite{kn:Cato}, proof of
Thm.2.16, Step I. One only has to notice that, by Lefschetz theorem
(cf. \cite{kn:Bott}, Cor. of Thm.1), $\pi_{1}(D_{j})$ surjects onto
$\pi_{1}(X)$, hence $\tilde{D}_{j}=\pi^{-1}(D_{j})$ is connected and
smooth for every $j = 1, \ldots k$.\hfill\qed
\begin{Remark}{\rm
\begin{list}%
{\alph{alph})}{\usecounter{alph}}
\item
{}From Prop.\ref{top}, c), it follows in particular
that the kernel $K$ of the surjection ${f_*:\pi_1(Y)\to \pi_1(X)}$
does not depend on the choice of the solution $L_\chi$ of
(\ref{eqstr}), once $G$, the $g_j$'s and the class of the
$\tilde{D}_j$'s in $H^2(\tilde{X},{\bf Z}/m_j{\bf Z})$, $j=1,\ldots
k$, are fixed.
\item
If $f:Y\to X$ is an abelian cover
as in the hypotheses of Prop.\ref{top}, then
$H^1(Y,{\cal O}_Y)\cong H_1(X,{\cal O}_X)$ by (\ref{splitting}) and the
Kodaira Vanishing Theorem. Moreover, according to Prop.\ref{top}, a)
the map ${f_*:H_1(Y,{\bf Z})\to H_1(X,{\bf Z})}$ is surjective; thus
the map ${f_*: alb(Y)\to alb(X)}$ between the Albanese varieties is an
isomorphism.
\end{list}
}
\end{Remark}
\begin{Proposition}
\label{tilde}
In the same hypotheses as in Prop.\ref{top}, let ${q:\tilde{Y}\to Y}$
be the universal cover of $Y$ and let
${\tilde{f\,}:\tilde{Y}\to \tilde{X}}$ be the map
lifting $f:Y\to X$. Then $\tilde{f\,}$ is a totally ramified
abelian cover of $\tilde{X}$ with group
${\tilde{G} = (\bigoplus_{j=1}^k G_{j})/{\rm Im}(\sigma \circ \rho)}$,
branched on $\tilde{D}$.
\end{Proposition}
{\sc Proof.} By diagram chasing, it is easy to show that
${\pi_{1}(\tilde{X}\setminus \tilde{D})}$ is isomorphic
to the kernel $V$ of the surjection
${\pi_{1}(X\setminus D)\to \pi_{1}(X)}$ induced by the inclusion
${X\setminus D\subset X}$.
Since the $D_{i}$'s are flexible, one proves as in (\cite{kn:Cato},
Lemma 2.1) that $V$ is an abelian group. It follows that
$\tilde{f\,}$, being branched on $\tilde{D}$, is an abelian cover.
Consider now the fiber product $Y'$ of ${f:Y\to X}$ and
${\pi:\tilde{X} \to X}$, with the natural maps
${f':Y'\to\tilde{X}}$ and ${q':Y'\to Y}$; $f'$ is a $G$-cover
ramified on $\tilde{D}$ and $q'$ is unramified. According to
Prop.\ref{top}, b), the universal covering ${q:\tilde{Y}\to Y}$ of
$Y$ factors as ${q=q' \circ q''}$, for a suitable unramified cover
${q'':\tilde{Y} \to Y'}$ with group $K$, giving a commutative diagram
as follows: \begin{equation} \label{diagramma}
\begin{picture}(60,85)(-4,-40)
\put (8,0){\vector(1,0){20}}
\put (9,30){\vector(1,-1){20}}
\put (0,30){\vector(0,-1){19}}
\put (0,-8){\vector(0,-1){19}}
\put (42,-8){\vector(0,-1){19}}
\put (8,-38){\vector(1,0){20}}
\put (-4,-2){$Y'$}
\put (36,-2){$\tilde{X}$}
\put (-3,36){$\tilde{Y}$}
\put (-4,-40){$Y$}
\put (36,-40){$X$}
\put(23,20){${\scriptstyle \tilde{f\,}}$}
\put(13,3){${\scriptstyle f'}$}
\put(-10,20){${\scriptstyle q''}$}
\put(13,-35){${\scriptstyle f}$}
\put(-10,-20){${\scriptstyle q'}$}
\put(44,-20){${\scriptstyle \pi}$}
\end{picture}
\end{equation}
In particular, ${K\cong \pi_{1}(Y')}$ and
${\tilde{f\,}=f'\circ q''}$. Hence, the Galois group $\tilde{G}$ of
$\tilde{f\,}$ is given as an extension:
\begin{equation}
\label{extgrouO}
0\longrightarrow K \longrightarrow\tilde{G}
\longrightarrow G \longrightarrow 0
\end{equation}
Moreover, if one denotes by $\tilde{G}_{j}$ the inertia subgroup of
$\tilde{D}_{j}$
with respect to $\tilde{f\,}$, then $\tilde{G}_{j}$ maps
isomorphically onto $G_{j}$ for every $j=1, \ldots k$.
The isomorphism
${\tilde{G} = (\bigoplus G_{j})/{\rm Im}(\sigma \circ \rho)}$
can be obtained by computing the fundamental group of
$\tilde{Y}$ as in \cite{kn:Cato}, proof of Thm.2.16.
\hfill\qed
\vspace{3mm}
The following lemma will be used in the next section.
\begin{Lemma}
\label{commut}
Consider the subgroups $\pi_{1}(Y)$ and $\tilde{G}$ of
${Aut(\tilde{Y})}$; then one has:
\begin{equation}
\beta g = g\beta\,\hspace{15mm}\forall g\in \tilde{G},
\forall\beta \in \pi_{1}(Y).
\end{equation}
\end{Lemma}
{\sc Proof.} Since the cover $\tilde{f\,}$ is totally ramified, it is
enough to show that all the elements of $\pi_{1}(Y)$ commute with
$g_{j}$, $j=1, \ldots k$.
Let ${\beta \in \pi_{1}(Y)}$ and fix $j=1, \ldots k$.
We remark firstly that ${\beta g_{j}\beta^{-1}}$ is actually an
element of ${\tilde{G}\subset Aut(\tilde{Y})}$. In fact, consider
the classes represented by ${\beta g_{j}}$ and ${g_{j}\beta}$ modulo
$K$: they do coincide as automorphisms of ${Y'\subseteq
Y\times\tilde{X}}$, since the group ${G\times\pi_{1}(X)}$ acts there
via the natural action on the components. So, ${\beta
g_{j}\beta^{-1}g_{j}^{-1}\in K}$ and ${\beta g_{j}\beta^{-1}\in
g_{j}K\subseteq\tilde{G}}$, as desired.
By diagram (\ref{diagramma}), we have
$\tilde{R}_{j}=\tilde{f\,}^{-1}(\tilde{D}_{j})=q^{-1}(R_{j})\;\forall$
$j=1, \ldots k$. Since $R_{j}=f^{-1}(D_{j})$ is ample and connected,
the same argument as in the proof of Lemma \ref{top}, c) shows
that $\tilde{R}_{j}$ is connected. So,
${\beta \tilde{R}_{j}=\tilde{R}_{j}}$ and ${\beta g_{j}\beta^{-1}}$
fixes ${\tilde{R}_{j}}$ pointwise, namely
${\beta g_{j}\beta^{-1}\in \tilde{G}_{j}}$.
Finally, recalling the definition of the character ${\psi_{j}\in
G_{j}^{*}}$ introduced in section 2, one checks immediately that
${\psi_{j}(\beta
g_{j}\beta^{-1})=\psi_{j}(g_{j})}$.
The conclusion now follows from the faithfulness of $\psi_{j}$.
\hfill\qed
\section{Computing the cohomology class of the central extension
$0\to K \to\pi_{1}(Y)\to \pi_{1}(X)\to 1$.}
\hspace{6 mm}We keep the notation and the assumptions introduced
in the previous sections, unless the contrary is explicitly
stated. We need two technical Lemmas in order to state the main result
of this paper.
\begin{Lemma}
\label{Lemma}
Let ${\cal H}$ be a finite abelian group and $\zeta_1$,\ldots
$\zeta_m\,\in\,{\cal H}$ be such that ${\cal H}=\bigoplus_{j=1}^m
<\zeta_j>$ is the direct sum of the cyclic
subgroups generated by $\zeta_j$, j=1, \ldots m; denote
by $h_j$ the order of $\zeta_j$. Let ${p\,\in\,{\bf Z}}$ be a prime
and ${\cal H}_{p}$ be the p-torsion subgroup of ${\cal H}$. Let
$\chi_{1}, \ldots \chi_{t} \in {\cal H}_{p}$ such that
${<\chi_{1}, \ldots \chi_{t}>=\bigoplus_{i=1}^t<\chi_{i}>}$. Finally,
let $d_{i}$ be the order of $\chi_{i}$ and write
${\chi_{i}=\sum_{j=1}^{h} a_{ij}\zeta_{j}}$ with ${0\leq a_{ij}<h_j}$.
Then, $\forall$ $x_{1}, \ldots x_{t} \in {\bf Z}$ and $\forall$
$\gamma \geq 1$, the system:
\begin{equation}
\label{sistlemma} \sum_{j=1}^{m}\frac{d_{i}a_{ij}}{h_{j}}s_{j}\equiv
x_{i}\;\;\; {\rm mod}\,p^{\gamma}\;\;\;\;\;\;\;\;\;i=1,\ldots t
\end{equation}
admits a solution $(s_{1}, \ldots s_{m})\,\in\,{\bf Z}^m$.
\end{Lemma}
{\sc Proof.} We set $c_{ij}=\frac{d_{i}a_{ij}}{h_{j}}$ and, for
$x\,\in\,{\bf Z}$, we denote by $\overline{x}$ the class of $x$ in
${\bf Z}/p{\bf Z}$. We proceed by induction on $\gamma$.
Let $\gamma =1$. We show that
the matrix $(\overline{c}_{ij})$ has rank $t$.
Let $y_{1}, \ldots y_{m} \in {\bf Z}$ and assume that:
\begin{equation}
\sum_{i}\overline{c}_{ij}\overline{y}_{i}=0\;\;\;\;\;\forall\,j=1,
\ldots m\,.
\end{equation}
This implies that:
\begin{equation}
\sum_{i=1}^tc_{ij}y_{i}\equiv 0\;\;\;{\rm
mod}\,p\;\;\;\;\;\;\;\forall\,j=1, \ldots m
\end{equation}
so that:
\begin{equation}
\sum_{i=1}^t\frac{y_{i}d_{i}}{p}\frac{a_{ij}}{h_{j}}\;\in\,
{\bf Z}\;\;\;\;\forall\,j=1, \ldots m\,.
\end{equation}
Recalling that $p$ divides $d_{i}$ $\forall\,i$, we deduce that:
\begin{equation}
\sum_{i=1}^t\left(\frac{y_{i}d_{i}}{p}\right)\,a_{ij}\equiv 0
\;\;\;{\rm mod}\,h_{j}\;\;\;\;\;\;\;\forall\,j=1, \ldots m\,,
\end{equation}
so that $\sum_{i}\frac{y_{i}d_{i}}{p}\chi_{i}$ is the zero element in
${\cal H}$. By the hypothesis on the $\chi_{i}$'s, it
follows that:
\begin{equation}
\frac{y_{i}d_{i}}{p}\equiv 0\;\;\;{\rm mod}\,d_{i}
\end{equation}
and finally:
\begin{equation}
y_{i}\equiv 0\;\;\;{\rm mod}\,p
\end{equation}
showing, as desired, that the rows of the matrix $(\overline{c}_{ij})$
are linearly independent over ${\bf Z}/p{\bf Z}$.
Let now $\gamma > 1$ and assume by inductive hypothesis that ${(s_{1},
\ldots s_{m})\in {\bf Z}^{m}}$ is a solution of the system
(\ref{sistlemma}).
We set $s_{j}'=s_{j}+\delta_{j}p^{\gamma}$ and we look for a suitable
choice of the integers $\delta_{j}$. We have:
\begin{equation}
\begin{array}{ccl}
\sum_{j}c_{ij}s_{j}'&=&\sum_{j}c_{ij}s_{j}+p^{\gamma}
\sum_{j}c_{ij}\delta_{j}\\
&=&x_{i}+p^{\gamma}y_{i}+p^{\gamma}
\sum_{j}c_{ij}\delta_{j}\;\;\;\;\;\exists\,y_{i}\,\in\,{\bf Z},
\end{array}
\end{equation}
so that:
\begin{equation}
\sum_{j}c_{ij}s_{j}'\equiv x_{i}\;\;{\rm mod}\,p^{\gamma +1}
\;\;\;\Longleftrightarrow\;\;\;\sum_{j}c_{ij}\delta_{j} \equiv -y_{i}
\;\;\;{\rm mod}\,p
\end{equation}
and the latter system has a solution, by the case $\gamma = 1$. This
conclude the proof.\hfill\qed
We come back to the study of the cover $f$:
\begin{Lemma}
\label{coeffinK}
Let $A$ be the subgroup of $Pic(X)$ generated by
$D_{1}, \ldots D_{k}$ and $L_{\chi}$, $\chi\in G^{*}$. Then there
exist $M_{1}, \ldots M_{q}\in Pic(X)$ such that
${A=\bigoplus_{l=1}^{q}<M_{l}>}$ and
\begin{equation}
\left(\begin{array}{c}
D_{1}\\
\vdots\\
D_{k}
\end{array}\right)\;\;\equiv\;\;C\,
\left(\begin{array}{c} M_{1}\\
\vdots\\
M_{q}
\end{array}\right)
\end{equation}
where $C=(c_{jl})$ is a matrix with integral coefficients such
that each column $(c_{jl})_{j=1,\ldots k}$ represents an element of
${N=\,{\rm ker}\,(\bigoplus_{j=1}^kG_j\to G)}$.
\end{Lemma}
{\sc Proof.} $A$ is a finitely generated abelian group, so one can
write ${A=F\bigoplus T}$, where $T$ is the torsion part of $A$ and $F$
is free.
Denote by $\{\xi_{l}\}_{l}$ a set of free generators of $F$ and by
$\{\eta_{l}\}_{l}$ a set of generators of $T$ such that
${T=\bigoplus<\eta_{l}>}$ and the order $o(\eta_{l})$ of $\eta_{l}$ is
the power of a prime, $\forall l$. Let finally $\chi_{i}$ be
generators of $G^{*}$ such that $G^{*}=\bigoplus_{i=1}^n<\chi_{i}>$
and the order $o(\chi_{i})$ of $\chi_{i}$ is the power of a prime,
$\forall i$.
One can write:
\begin{eqnarray}
\label{Lchii}L_{\chi_{i}}\equiv
\sum_l\lambda_{il}\eta_{l}+\sum_l\lambda_{il}'\xi_{l}\hspace{10mm}
\forall\,i=1,\ldots n,\\
\label{Djlemma}D_{j}\equiv\sum_l c_{jl}\eta_{l}+\sum_l
c_{jl}'\xi_{l}\hspace{10mm} \forall\,j=1,\ldots k,
\end{eqnarray}
where the coefficients $\lambda_{il}'$ and $c_{jl}'$ are
uniquely determined, whereas $\lambda_{il}$ and $c_{jl}$ are
determined only up to a multiple of $o(\eta_{l})$.
We can apply the analysis of section 2 to the cover $f$. We write
${\chi_i=\sum_{j=1}^{k}a_{ij}\psi_j}$, with ${0\leq
a_{ij}<m_j}$, and we set ${d_{i}=o(\chi_i)}$ as in the previous
Lemma; the equations (\ref{eqstr}) become here:
\begin{equation} \label{eqstrf}
d_{i} L_{\chi_{i}}
\equiv \sum_{j=1}^{k}\frac{d_{i}a_{ij}}{m_{j}}\;D_{j}\;
\hspace{10mm}i=1,\ldots n,
\end{equation}
so that we must have:
\begin{equation}
d_{i}\lambda_{il}'=\sum_{j=1}^k\frac{d_{i}a_{ij}}{m_{j}}\;c_{jl}'
\hspace{10mm}i=1,\ldots n,
\end{equation}
showing that $(c_{jl}')_{j=1,\ldots k}$ represents an element of $N$,
$\forall \,l$: in fact, by duality, ${(t_1,\ldots,t_k)\,\in\,
{\bf Z}^k}$ represents an element of $N$ if and only if it satisfies
the relations:
\begin{equation}\label{relN}
\sum_{j=1}^k\frac{a_{ij}}{m_{j}}t_j\;\in\;{\bf
Z}\hspace{15mm}\forall\,i=1,\ldots n. \end{equation}
For the coefficients of the torsion part, we have:
\begin{equation}
d_{i}\lambda_{il}\,\eta_{l}=\left(
\sum_{j=1}^k\frac{d_{i}a_{ij}}{m_{j}}\;c_{jl}\right)\;\eta_{l}
\end{equation}
so that:
\begin{equation}
\label{cong}
d_{i}\lambda_{il}\equiv
\sum_{j=1}^k\frac{d_{i}a_{ij}}{m_{j}}\;c_{jl}
\;\;\;{\rm mod}\,o(\eta_{l})\;.
\end{equation}
We fix an index $l$. Let $p$ be a prime such that
$o(\eta_l)=p^{\alpha}$. We want to show that, for a suitable
choice of the $c_{jl}$, the following relation holds $\forall\,i=1,
\ldots n$:
\begin{equation}
\label{tesi}
d_{i}\lambda_{il}\equiv
\sum_{j=1}^k\frac{d_{i}a_{ij}}{m_{j}}\;c_{jl}
\;\;\;{\rm mod}\,d_{i}\,.
\end{equation}
Let $\chi_{i}$ be a generator such
that ${d_{i}\equiv 0}$ mod $p$ and set
${d_{i}=p^{\alpha_{i}}}$.
By (\ref{cong}), it is enough to consider the case in which
${\alpha<\alpha_{i}}$.
Setting $c_{jl}''= c_{jl}+p^{\alpha}s_{j}$ and recalling (\ref{cong}),
one has:
\begin{equation}
\begin{array}{ccl}
\sum_{j=1}^k\frac{d_{i}a_{ij}}{m_{j}}\;c_{jl}''&=&
\sum_{j=1}^k\frac{d_{i}a_{ij}}{m_{j}}\;c_{jl}+
p^{\alpha}\sum_{j=1}^k\frac{d_{i}a_{ij}}{m_{j}}\;s_{j}\\
&=&d_{i}\lambda_{il}-p^{\alpha}x_{i}+p^{\alpha}
\sum_{j=1}^k\frac{d_{i}a_{ij}}{m_{j}}\;s_{j}
\end{array}
\end{equation}
for a suitable choice of integers $x_{i}$. One concludes that the
relation (\ref{tesi}) holds if and only if:
\begin{equation}
\sum_{j=1}^k\frac{d_{i}a_{ij}}{m_{j}}\;s_{j}\equiv x_{i}
\;\;\;{\rm mod}\,p^{\alpha_{i}-\alpha}\;.
\end{equation}
Let $\beta = {\rm max}\,\{\alpha_{i}-\alpha\}_{i}$. The system of
congruences:
\begin{equation}
\sum_{j=1}^k\frac{d_{i}a_{ij}}{m_{j}}\;s_{j}\equiv x_{i}
\;\;\;{\rm mod}\,p^{\beta}\;\hspace{10mm}\forall\,i \mbox{ such
that } d_{i}\equiv 0\;\mbox{ mod } p
\end{equation}
admits a solution by Lemma \ref{Lemma}. So, we can assume that the
coefficients $(c_{jl})_{j=1,\ldots k}$ in (\ref{Djlemma}) satisfy
(\ref{tesi}) for every $i$ such that $d_{i}\equiv 0$ mod $p$.
To complete the proof, let $\gamma$ be an
integer $\gg 0$; we can still modify the coefficients as
${c_{jl}''=c_{jl}+ p^{\gamma}t_{j}}$. It is enough to notice that,
setting ${d={\rm lcm}\{d_i\,|\,d_{i}\not\equiv 0\;\mbox{ mod }
p\}}$, then $d$ and $p$ are coprime and the system of
congruences:
\begin{equation}
c_{jl}+ p^{\gamma}t_{j}\equiv 0\hspace{10mm}{\rm mod}\,d
\hspace{15mm}\forall\,j
\end{equation}
admits a solution. So we can assume that $c_{jl}''\equiv 0$ mod $d$,
and the proof is complete. \hfill\qed
\vspace{3mm}
To any decomposition (\ref{decteor}) as in Lemma \ref{coeffinK}, we
associate a cohomology class in $H^2(X,K)$:
\begin{Definition}\label{defxi}
Given a decomposition (\ref{decteor}) as in Lemma \ref{coeffinK},
consider the map:
\begin{equation}
\begin{array}{ccl}
{\bf Z}^{q}&\to &N\\
(x_1,\ldots x_q) &\mapsto&\sum_{l=1}^q\,x_l\underline{c}_l=
(\sum_{l=1}^q\,x_lc_{jl})_{j=1,\ldots k}
\end{array}
\end{equation}
and denote by ${\Theta:{\bf Z}^{q}\to K}$ its composition with
the projection ${N\to K}$ (cf. Prop.\ref{top}, c)). Then, set:
\begin{equation}
\xi=\Theta_*([M_1], \ldots [M_q])\,,
\end{equation}
where ${\Theta_*:H^2(X,{\bf Z}^q)\cong\bigoplus^q H^2(X,{\bf Z})\to
H^2(X,K)}$ is the map induced in cohomology by $\Theta$ and
$[M]$ is the Chern class of a divisor $M$ on $X$.
\end{Definition}
\vspace{3mm}
We briefly recall some facts about quotients by a properly
discontinuous group action (see for instance \cite{kn:Mu}, Appendix to
section 1, \cite{kn:Grot}, ch. 5).
Let $\tilde{X}$ be a simply connected variety, let $\Gamma$
be a group acting properly and discontinuously on
$\tilde{X}$ and let ${p:\tilde{X}\rightarrow X=\tilde{X}/\Gamma}$
be the projection onto the quotient. Consider the following
two functors:
$$
\begin{array}{l}
M\stackrel{F}{\longrightarrow} M^{\Gamma} ,\,{\rm for}\,M
\,{\rm a}\, \Gamma \mbox{-module}\\
{\cal F}\stackrel{H}{\longrightarrow}
H^0 (\tilde{X},p^{*}{\cal F}),\,{\rm for}\,{\cal F}
\,{\rm a\,locally\,constant\,sheaf \, on}\, X\,.
\end{array}
$$
The spectral sequence associated to the functor $F\circ H$
yields in this case the exact sequence of cohomology
group:
\begin{equation}
\label{spseq}
0\longrightarrow H^{2}(\Gamma,H^{0}(\tilde{X},p^{*}{\cal F}))
\longrightarrow H^{2}(X,{\cal F}) \longrightarrow
H^{2}(\tilde{X},p^{*}{\cal F})^{\Gamma}
\end{equation}
that will be used several times in the following and it
is natural with respect to the sheaf maps on $X$.
\begin{Theorem}\label{mtgen}
Let $X$, $Y$ be smooth projective varieties over {\bf C} of dimension
at least 2. Let ${f:Y \rightarrow X}$ be a totally ramified finite
abelian cover branched on a divisor with flexible and ample
components ${\{D_{j}\}_{j=1, \ldots k}}$. According to Prop.\ref{top},
b), the map $f$ induces a central extension: \begin{equation}
\label{extgrouteogen}
0\longrightarrow K \longrightarrow\pi_{1}(Y)
\stackrel{f_{*}}{\longrightarrow} \pi_{1}(X) \longrightarrow 1
\end{equation}
Denote by ${c(f)\, \in \,H^{2}(\pi_{1}(X), K)\subseteq H^{2}(X, K)}$
the cohomology class classifying the extension (\ref{extgrouteogen}).
Let:
\begin{equation} \label{decteor}
\left(\begin{array}{c}
D_{1}\\
\vdots\\
D_{k}
\end{array}\right)\;\;\equiv\;\;C\,
\left(\begin{array}{c} M_{1}\\
\vdots\\
M_{q}
\end{array}\right)
\end{equation}
be a decomposition as in Lemma \ref{coeffinK} and let $\xi\,\in\,
H^{2}(X,K)$ be the class defined in Def.\ref{defxi}.
In this notation, one has:
\begin{equation}
c(f)=\xi;
\end{equation}
in particular, the class $\xi$ does not
depend on the chosen decomposition.
\end{Theorem}
{\sc Proof.} It is enough to show that $\xi$ and $c(f)$ admit
cohomologous representatives. This can be done in three steps.
\vspace{3 mm}
{\sc Step I}: we compute a cocycle representing $c(f)\,\in\,H^2(X,K)$.
We start by choosing suitable trivializations of the line bundles
that appear in the computation.
Set $\Gamma = \pi_{1}(X)$ and $\tilde{\Gamma} = \pi_{1}(Y)$. Let
$\{U_{r}\}$ be a sufficiently fine cover of $X$ such that $\Gamma$
acts transitively on the set of connected components of
$\pi^{-1}(U_{r})$, $\forall$ $r$. If we fix a component $V_{r}$ of
$\pi^{-1}(U_{r})$, then ${\pi^{-1}(U_{r}) = \cup_{\gamma \in
\Gamma}\,\gamma(V_{r})}$; for every ${\gamma \in \Gamma}$ we write:
\begin{equation}
\gamma(V_{r}) = V_{(\gamma, r)}
\end{equation}
and, in particular: $V_{(1, r)}=V_{r}$.
Such a covering has the following properties:
\begin{list}%
{\alph{alph})}{\usecounter{alph}}
\item For every $(r, s)$ such that ${U_{r}\cap U_{s} \neq \emptyset}$,
there exists a unique element ${\beta (r, s) \in \Gamma}$ such that:
\begin{equation}
V_{(1, r)} \cap V_{(\beta (r, s), s)} \neq \emptyset\;.
\end{equation}
\item If ${U_{r}\cap U_{s} \neq \emptyset}$, then $V_{(\gamma, r)}$
and $V_{(\gamma \beta (r, s), s)}$ have nonempty intersection.
\item Since $\pi$ is a local homeomorphism, if
${U_{r}\cap U_{s}\cap U_{t} \neq \emptyset}$, then:
\begin{equation}
\emptyset\;\neq\; V_{(\beta (r, s), s)}\cap V_{(\beta (r, t), t)}.
\end{equation}
Hence the following relation is satisfied for every
${U_{r}\cap U_{s}\cap U_{t} \neq \emptyset}$:
\begin{equation}
\beta (r, t) = \beta (r, s) \beta (s, t)\;.
\end{equation}
In particular: $\beta (s, r) = \beta (r, s)^{-1}$.
\end{list}
\noindent
For later use, we set:
\begin{equation}
V_{(\alpha, r, s)} = \alpha(V_{(1, r)}\cap V_{(\beta (r, s), s)})
= V_{(\alpha, r)}\cap V_{(\alpha\beta (r, s), s)}
\end{equation}
for every $\alpha \in \Gamma$ and for every $(r, s)$ such that
${U_{r}\cap U_{s} \neq \emptyset\,}$.
For every $r$ and for every $j = 1, \ldots k$, we choose a local
generator $w^{j}_{r}$ for ${{\cal O}_{X}(-D_{j})}$
on $U_{r}$ (we ask that $w^{j}_{r}$ is a local equation for $D_j$) and for
every pair $(r, s)$ such that ${U_{r}\cap U_{s} \neq \emptyset}$ we
write:
\begin{equation}\label{coc-Dj}
w^{j}_{r} = k^{j}_{(r, s)}\,w^{j}_{s}\;\;{\rm on}\,U_{r}\cap U_{s}\,.
\end{equation}
Now we apply to ${\tilde{f\,}:\tilde{Y}\rightarrow \tilde{X}}$ the
analysis of section 2 , most of which can be easily extended to the
case of analytic spaces. One has: \begin {equation} \label{split}
\tilde{f\,}_{*}({\cal O} _{\tilde{Y}})=
\bigoplus_{\tilde{\chi} \in \tilde{G}^{*}}L_{\tilde{\chi}}^{-1}
\end{equation}
Each element of the group $\tilde{G}$ can be interpreted as an
automorphism of the sheaf $\tilde{f\,}_{*}({\cal O} _{\tilde{Y}})$. In
particular, by duality, the elements of $K\subseteq \tilde{G}$ are
characterized by the property that they induce the identity on the
subsheaf $L_{\chi}$ for every
${\chi\,\in\,G^*\subseteq\tilde{G}^*}$.
Let $\tilde{\chi}_{1}, \ldots \tilde{\chi}_{h} \in \tilde{G}^{*}$ be
such that $\tilde{G}^{*}$ is isomorphic to the direct sum of the
cyclic subgroups generated by the $\tilde{\chi}_{i}$'s, and let
$\tilde{d_i}$ be the order of $\tilde{\chi}_{i}$, $i = 1, \ldots h$.
Let $\tilde{D_{i}}$ be the inverse image of $D_{i}$ via the universal
covering map ${\pi :\tilde{X}\to X}$, as before. If
${\tilde{\chi}_i=\sum_{j=1}^{k}\,\tilde{a}_{ij}\psi_j}$, with ${0\leq
\tilde{a}_{ij} <m_j}$, the system (\ref{eqstr}) yields in this case:
\begin{equation} \label{eqstr1}
\tilde{d_i} L_{\tilde{\chi}_{i}} \equiv
\sum_{j=1}^{k}\frac{\tilde{d_i}\tilde{a}_{ij}}{m_{j}} \;\tilde{D}
_{j}\;\;\;\;i= 1, \ldots h\,.
\end{equation}
So it is possible to choose local generators $\tilde{z}^{i}_{(\alpha,
r)}$ for $L_{\tilde{\chi}_{i}}^{-1}$ on $V_{(\alpha, r)}$ such
that for every ${\alpha \in \Gamma}$ and for every pair $(r, s)$ with
${U_{r}\cap U_{s} \neq \emptyset}$ one has:
\begin{equation}\label{relforz}
\left(\tilde{z}_{(\alpha, r)}^{i}\right)^{\tilde{d_i}}=
\prod_{j=1}^k\left(w_r^j\right)^{\frac{\tilde{d_i}\tilde{a}_{ij}}{m_j}}
\,.
\end{equation}
Writing:
\begin{equation}
\tilde{z}_{(\alpha, r)}^{i} = \tilde{h}^{i}_{(\alpha, r,s)}\,
\tilde{z}^{i}_{(\alpha \beta (r, s), s)} \;\;\;\; {\rm
on}\,V_{(\alpha, r, s)}
\end{equation}
we have:
\begin{equation}
\label{rela}
(\tilde{h}^{i}_{(\alpha, r,
s)})^{\tilde{d_i}}\;=\;\prod_{j=1}^{k}(k^{j}_{(r, s)})
^{\frac{\tilde{d_i}\tilde{a}_{ij}}{m_{j}}}\;.
\end{equation}
and the cocycle condition for $\tilde{h}^{i}_{(\alpha, r,
s)}$, that will be often used later on, yields
the relation:
\begin{equation}
\label{coch}
1=\tilde{h}^{i}_{(\alpha, r, s)}\tilde{h}^{i}_{(\alpha\beta(r,s), s,
t)} \tilde{h}^{i}_{(\alpha\beta(r,t), t, r)}\;
\hspace{15mm}\forall\,\alpha\,\in\,\Gamma,\forall\,i,\forall\,
r,s,t\mbox{ with }U_r\cap U_s \cap U_t\neq\emptyset. \end{equation}
We observe that the generator $\tilde{z}^{i}_{(\alpha,r)}$ is
determined by (\ref{relforz}) only up to a constant of the form
$exp2\pi \sqrt{-1} \,(u^i_{(\alpha,r)}/\tilde{d_i})$ with
${u^i_{(\alpha,r)}\in {\bf Z}}$. Moreover, according to
(\ref{Lchigen}), every choice of local generators $w^j_r$ for ${\cal
O}_X(-D_j)$ and $\tilde{z}_{(\alpha, r)}^{i}$ for
$L_{\tilde{\chi}_i}^{-1}$ induces a choice of
local generators for $L_{\tilde{\chi}}^{-1}$,
${\forall\,\tilde{\chi}\,\in\,\tilde{G}^*}$, by the rule:
\begin{equation}\label{ztildechi} \tilde{z}_{(\alpha,
r)}^{\tilde{\chi}}=\prod_{i=1}^n\left( \tilde{z}_{(\alpha, r)}^{i}
\right)^{b_{\tilde{\chi},i}}
\prod_{j=1}^k\,(w^j_r)^{-q^{\tilde{\chi}}_{j}}\,\hspace{15mm} \mbox{if
}\,\tilde{\chi}=\sum_{i=1}^h b_{\tilde{\chi},i}\tilde{\chi}_i,\;\;
0\leq b_{\tilde{\chi},i}<\tilde{d_i}\,.
\end{equation}
where $q^{\tilde{\chi}}_j$ denotes the integral part of the real
number $\sum_{i=1}^h b_{\tilde{\chi},i} \,\tilde{a}_{ij}$.
Let now $\chi_1, \ldots \chi_n$ be a set of generators for $G^*$ such
that $G^*$ is the direct sum of the cyclic subgroups generated by the
$\chi_v$'s and the order $d_v$ of $\chi_v$ is a power of a prime
number, $v=1, \ldots n$. We recall that ${G^*\subseteq \tilde{G}^*}$
and,
${\forall\,\chi\,\in\,G^*}$, the corresponding eigensheaf $L_{\chi}$ is
a pullback from $X$. We write
${\chi_v=\sum_{i=1}^n b_{vi}\tilde{\chi}_i\,\in\,\tilde{G}^*}$
(${0\leq b_{vi}<\tilde{d_i}}$) and $q^{\chi_v}_j=q^{v}_j$; the
corresponding local generator for ${L_{\chi_v}^{-1}}$ chosen in
(\ref{ztildechi}) is:
\begin{equation}\label{zG*}
z_{(\alpha, r)}^{v}=\prod_{i=1}^h\left(
\tilde{z}_{(\alpha, r)}^{i}
\right)^{b_{vi}}\prod_{j=1}^k\,(w^j_r)^{-q^{v}_j} \;;
\end{equation}
we show that, for a suitable choice of the $\tilde{z}_{(\alpha,
r)}^{i}$, we can assume that the expression in (\ref{zG*}) is
independent from $\alpha$. In fact, using the characteristic equations
of the cover $f$, one can choose a local base $y^v_r$ of
$L_{\chi_v}^{-1}$ on $V_{(\alpha, r)}$ that does not depend on
$\alpha$ and satisfies the relation:
\begin{equation}\label{eqstrsuX}
\left(y^v_r\right)^{d_v}=\prod_{j=1}^k\left(w^j_r
\right)^{\frac{a_{vj} d_v}{m_j}}\,.
\end{equation}
Since $\sum_{i=1}^h b_{vi}\tilde{a}_{ij}=q^v_jm_j+a_{vj}$
$\forall\,j=1,\ldots k$, the two local generators $y^v_r$ and
$z_{(\alpha, r)}^{v}$ on $V_{(\alpha,r)}$ differ by a $d_v$-th
root of unity, that we denote by $exp( 2 \pi
\sqrt{-1}\,(x^v_{(\alpha,r)}/d_v))$. If we multiply
$\tilde{z}_{(\alpha, r)}^{i}$ by $exp( 2 \pi
\sqrt{-1}(u^i_{(\alpha,r)}/\tilde{d_i}))$, then
${(x^v_{(\alpha,r)}/d_v)}$ becomes
${(x^v_{(\alpha,r)}/d_v)+\sum_{i=1}^h
(b_{vi}/\tilde{d_i})\,u^i_{(\alpha, r)}}$. Hence, we only need to
solve the linear system of congruences, $\forall\,(\alpha, r)$:
\begin{equation} \sum_{i=1}^h\frac{d_v
b_{vi}}{\tilde{d_i}}u^i_{(\alpha, r)}\equiv x^v_{(\alpha,
r)}\;\;\;\mbox{mod}\,d_v\hspace{20 mm}v=1, \ldots n; \end{equation}
since we assume that $d_v$ is a power of a prime number, this system
admits a solution according to the Chinese Remainder's Theorem and
Lemma \ref{Lemma}.
So we can assume that the expression $z_{(\alpha, r)}^{v}$ in
(\ref{zG*}) does not depend on $\alpha$ and it is the pullback of a
local generator of the corresponding eigensheaf on $X$: we write
$z_{r}^{v}=z_{(\alpha, r)}^{v}$. For later use, we define the
corresponding cocycle $h^{v}_{(r,s)}$ ($v=1,\ldots n$) by the rule:
\begin{equation}\label{cochv}
z_{r}^{v}=h^{v}_{(r,s)}z_{s}^{v} \hspace{15 mm}\mbox{on }U_r\cap
U_s
\end{equation}
and we observe that, according to (\ref{eqstrsuX}), the following
relation holds:
\begin{equation}\label{relhv}
(h^{v}_{(r,s)})^{d_v}=\prod_{j=1}^k(k^j_{(r,s)})^{\frac{a_{vj}
d_v}{m_j}}\,\hspace{15mm}\mbox{if
}\chi_v=\sum_{j=1}^k a_{ij}\psi_j, \;\;0\leq a_{ij}<m_j.
\end{equation}
\vspace{3 mm}
In order to compute the class of the extension
(\ref{extgrouteogen}), for every ${\gamma \in \Gamma}$ we choose a
lifting ${\tilde{\gamma} \in \tilde{\Gamma}}$.
By Lemma \ref{commut}, the induced map ${\tilde{\gamma}_{*}:
\tilde{f\,}_{*} {\cal O}_{\tilde {Y}} \to \tilde{f\,}_{*}{\cal
O}_{\tilde {Y}}}$ is a ${\cal O}_{\tilde {X}}$-algebra isomorphism
lifting ${\gamma : \tilde{X} \to \tilde{X}}$; in terms of the chosen
trivializations we may write:
\begin{equation}
\tilde{z}^{i}_{(\alpha ,r)} \stackrel{\tilde{\gamma}_{*}}{\mapsto}
\sigma^{i,\gamma}_{(\alpha,r)}\tilde{z}^{i}_{(\gamma\alpha
,r)}\;\;\;\;\;\; \;\forall\gamma , \alpha \in \Gamma , i=1,\ldots h
\end{equation}
for a suitable choice of a $\tilde{d}_{i}$-th root of unity
$\sigma^{i,\gamma}_{(\alpha,r)}$, $i=1, \ldots h$.
For later use, we write down the transition relations
for the constants $\sigma^{i,\gamma}_{(\alpha,r)}$. Let $s,t$ be such that
${U_{s}\cap U_{t} \neq \emptyset}$ and let $\alpha,\gamma \in\Gamma$;
then, for $i=1,\ldots h$:
\begin{equation}
\label{rel}
\sigma^{i,\gamma}_{(\alpha ,s)}\tilde{h}^{i}_{(\gamma\alpha ,s,t)}=
\sigma^{i,\gamma}_{(\alpha\beta (s,t),t)}\left(\tilde{h}^{i}_{(\alpha
,s,t)}\circ \gamma^{-1}\right)\;\;\;\;\;\;\;{\rm
on}\;V_{(\gamma\alpha, s,t)}.
\end{equation}
We now exploit the action of the chosen elements in $\tilde{\Gamma}$ on
$\tilde{f\,}_{*} {\cal O}_{\tilde {Y}}$ in order to compute the class
${c(f)\in H^{2}(\Gamma ,K)}$
associated to the extension (\ref{extgrouteogen}), and its image
${c(f)\in H^{2}(X,K)}$.
For any given
$\delta,\gamma\in\tilde{\Gamma}$, the action of
$(\widetilde{\delta\gamma})_{*}^{-1}\tilde{\delta}_{*}
\tilde{\gamma}_{*}$ on $L^{-1}_{\tilde{\chi}_{i}}$, $i = 1,
\ldots h$, is described with respect to the chosen trivializations by:
\begin{equation} \label{comm}
\tilde{z}^{i}_{(\alpha ,r)} \mapsto
\left(\sigma^{i,(\delta\gamma)}_{(\alpha,r)}\right)^{-1}
\sigma^{i,\delta}_{(\gamma\alpha ,r)} \sigma^{i,\gamma}_{(\alpha ,r)}
\tilde{z}^{i}_{(\alpha ,r)}\;\;\;\;\;\;\;\;\;\;\forall
r,\;\forall\alpha\in\Gamma, \; i=1,\ldots h.
\end{equation}
Since (\ref{comm}) represents a line bundle automorphism given by a
root of the unity, the expression does not depend on $(\alpha ,r)$ by
the connectedness of $\tilde{X}$: therefore, we may set $\alpha = 1$.
So, the class ${c(f)\in H^{2}(\Gamma
,K)}$ is represented by the cocycle:
\begin{equation}
c(f)(\delta ,\gamma )=\left(
\left(\sigma^{i,(\delta\gamma)}_{(1,r)}\right)^{-1}
\sigma^{i,\delta}_{(\gamma,r)}
\sigma^{i,\gamma}_{(1,r)}\right)_{i=1,\ldots h}\;
\;\;\;\;\;\forall r,
\end{equation}
where an element of $K\subseteq \tilde{G}$ is represented by
its coordinates with respect to the
basis dual to $\{\tilde{\chi}_{1}, \ldots \tilde{\chi}_{h}\}$.
According to (\cite{kn:Mu}, page 23), the class
${c(f)\in H^{2}(X,K)}$ is represented on $V_{(1,r,s)}\cap V_{(1,
r,t)}$ by the cocycle:
\begin{equation}
\label{uffa}
c(f)_{r,s,t}=c(f)(\beta (r,s),\beta (s,t))=
\left(
\left(\sigma^{i,\beta (r,t)}_{(1,p)}\right)^{-1}
\sigma^{i,\beta (r,s)}_{(\beta (s,t),p)}
\sigma^{i,\beta (s,t)}_{(1 ,p)}
\right)_{i=1,\ldots h}\;
\;\;\;\;\;\;\forall p
\end{equation}
for $r,s,t$ such that
${U_{r}\cap U_{s}\cap U_{t}\neq\emptyset}$.
We set $p=t$ and, by the relation (\ref{rel}), we rewrite (\ref{uffa})
as follows:
\begin{equation}\label{coccf2}
c(f)_{r,s,t}=\left(
\left(\sigma^{i,\beta (r,t)}_{(1,p)}\right)^{-1}
\sigma^{i,\beta (t,r)}_{(1,r)}
\sigma^{i,\beta (s,t)}_{(1 ,t)}
\tilde{h}^{i}_{(\beta (r,s),s,t)}
\left( \tilde{h}^{i}_{(1,s,t)}\circ \beta(s,r)\right)^{-1}
\right)_{i=1, \ldots h}\,;
\end{equation}
this shows that $c(f)_{r,s,t}$ differs from the following cocycle (that
we still denote by $c(f)_{r,s,t}$ by abuse of notation):
\begin{equation}\label{coccfnuovo}
c(f)_{r,s,t}=\left(
\tilde{h}^{i}_{(\beta (r,s),s,t)}
\left( \tilde{h}^{i}_{(1,s,t)}\circ \beta(s,r)\right)^{-1}\right)_{i=1, \ldots
h}\, \end{equation}
by the coboundary of the cochain:
\begin{equation}
g_{r,t}=\left(\sigma^{i,\beta(r,t)}_{(1,t)}\right)_{i=1, \ldots h}\,.
\end{equation}
The cochain $g_{r,t}$ actually takes values in $K$: in fact, it is
enough to check $g_{r,t}$ acts trivially on the eigensheaves
corresponding to the chosen generators $\chi_v$ of $G^*$. This
follows easily by the prevous choices since the action of $\beta(r,t)$
on $L_{\chi_v}^{-1}$ is given locally by
${\prod_{i=1}^h\left(\sigma^{i,\beta(r,t)}_{(1,t)}\right)^{b_{vi}}}$.
\vspace{3mm}
{\sc Step II:} we compute a cocycle representing
$\xi\,\in\,H^2(X,K)$.
For every $r$ and for every $l = 1, \ldots q$, we choose a local
generator $y^{l}_{r}$ for ${\cal O}_X(-M_{l})$ on $U_{r}$; if $M_{l}$
has finite order $e$, then we require:
\begin{equation}
\label{tors}
\left(y^{l}_{r}\right)^{e}=1\,.
\end{equation}
We set $m={\rm lcm}\,\{m_j\}_{j=1, \ldots k}$. For every pair of
indices $(r, s)$ such that ${U_{r}\cap U_{s} \neq \emptyset}$ we write:
\begin{equation}
y^{l}_{r} = \mu^{l}_{(r, s)}\,y^{l}_{s}\;\;{\rm on}\,U_{r}\cap U_{s}
\end{equation}
and we choose a $m$-th root $\hat{\mu}^{l}_{(r, s)}$ of
$\mu^{l}_{(r, s)}$ in such a way that
${\hat{\mu}^{l}_{(s, r)}=(\hat{\mu}^{l}_{(r,
s)})^{-1}}$. Then, as in \cite{kn:Cato}, (2.45), one sees that
the image of the class of $-M_{l}$ in $H^2(X,{\bf Z}/m_l{\bf Z})$ is
represented on ${U_{r}\cap U_{s}\cap U_{t}}$ by the cocycle
${\left(\hat{\mu}^{l}_{(r, s)}
\hat{\mu}^{l}_{(s, t)}
(\hat{\mu}^{l}_{(r, t)})^{-1} \right)^{m/m_l}}$, $l=1, \ldots q$. We
conclude that the class ${\xi=\Theta_*([M_1], \ldots [M_q])}$ is
represented on ${U_{r}\cap U_{s}\cap U_{t}}$ by:
\begin{equation}\label{cocc}
\xi_{r,s,t}=\left(\prod_{j=1}^k\prod_{l=1}^{q}\,\left(\hat{\mu}^{l}_{(r,
s)} \hat{\mu}^{l}_{(s, t)}
\hat{\mu}^{l}_{(t, r)}\right)^{-(m/m_j)
c_{jl}\tilde{a}_{ij}}\right)_{i=1, \ldots h}
\end{equation}
\vspace{3mm}
{\sc Step III:} we show that ${\xi=c(f)}$.
We remark that, according to (\ref{decteor}), the cocycle $k^{j}_{(r,
s)}$ in (\ref{coc-Dj}) representing ${{\cal O}_{X}(-D_{j})}$ ($j=1,
\ldots k$) and the cocycles $\mu^{l}_{(r,s)}$ representing $-M_l$
($l=1, \ldots q$) are related as follows:
\begin{equation}
k^{j}_{(r, s)}=\prod_{l=1}^q\left(\mu^{l}_{(r,
s)}\right)^{c_{jl}}\frac{f^j_r}{f^j_s}
\end{equation}
for suitable nowhere vanishing holomorphic functions $f_r$ on
$U_{ r}$. For every $j=1, \ldots k$ and every $r$, we choose a $m$-th
root $\hat{f}^j_r$ of $f^j_r$ on $U_{ r}$; then, the
expression:
\begin{equation}\label{tildekmu}
\hat{k}^j_{(r,s)}=\prod_{l=1}^{q}\,\left(\hat{\mu}^{l}_{(r,
s)}\right)^{c_{jl}(m/m_j)}
\left(\frac{\hat{f\,}^j_r}{\hat{f\,}^j_s}\right)^{(m/m_j)}
\end{equation}
is a $m_j$-th root of the
cocycle $k^j_{(r,s)}$ and, as before, the product
$\prod_{l=1}^{q}\,\left(\hat{\mu}^{l}_{(r, s)} \hat{\mu}^{l}_{(s,
t)} \hat{\mu}^{l}_{(t, r)}\right)^{ c_{jl}(m/m_j)}$ yields a cocycle
representing the image of the class of $-D_j$ in $H^2(X,{\bf
Z}/m_j{\bf Z})$. In this notation, by (\ref{tors}), we rewrite
as follows the cocycle in (\ref{cocc}) representing the class $\xi$:
\begin{equation}\label{cocc1}
\xi_{r,s,t}=\left(\prod_{j=1}^k
\left( \hat{k}^j_{(r,s)}\hat{k}^j_{(s,t)}\hat{k}^j_{(t,r)}
\right)^{-\tilde{a}_{ij}}\right)_{i=1, \ldots, h}\,.
\end{equation}
Let $\epsilon = exp(\frac{2\pi \sqrt{-1}}{m})$. Then, by
(\ref{rela}), one has:
\begin{equation}\label{defq}
\tilde{h}^{i}_{(\alpha, r, s)}\;=\;\prod_{j=1}^{k}(\hat{k}^{j}_{(r,s)})
^{\tilde{a}_{ij}}\;
\epsilon^{-q^{i}_{(\alpha,r,s)}}
\end{equation}
where $q^{i}_{(\alpha,r,s)}$ is an integer, multiple of
$m/\tilde{d_i}$, and (\ref{coccfnuovo}) may be rewritten as:
\begin{equation}
c(f)_{r,s,t} =
(\epsilon^{q^{i}_{(1,s,t)}-q^{i}_{(\beta(r,s),s,t)}})
_{i=1,\ldots h}\;\;.
\end{equation}
{}From the cocycle condition (\ref{coch}) for
$\tilde{h}^{i}_{(\alpha , r, s)}$, it follows:
\begin{equation}
\xi_{r,s,t}=(\epsilon^{-q^{i}_{(\alpha
,r,s)}-q^{i}_{(\alpha\beta (r,s),s,t)}
-q^{i}_{(\alpha\beta(r,t),t,r)}})_{i=1,\ldots h}\;\;\;\forall\, r,s,t,
\forall\,\alpha\in\Gamma\,.
\end{equation}
In particular, for $\alpha = 1$, one gets:
\begin{equation}
\xi_{r,s,t}=(\epsilon^{-q^{i}_{(1,r,s)}-q^{i}_{(\beta
(r,s),s,t)} -q^{i}_{(\beta (r,t),t,r)}})_{i=1,\ldots h}\;\;.
\end{equation}
So, one has:
\begin{equation}
c(f)_{r,s,t} = \xi_{r,s,t}
(\epsilon^{q^{i}_{(1,r,s)}+q^{i}_{(1,s,t)}
+q^{i}_{(\beta(r,t),t,r)}})_{i=1,\ldots h}\;\;.
\end{equation}
By the definition of $q^{i}_{(\alpha,r,s)}$, this equality can
be rewritten as follows:
\begin{equation}\label{cobord}
c(f)_{r,s,t} = \xi_{r,s,t}
(\epsilon^{q^{i}_{(1,r,s)}-q^{i}_{(1,r,t)}
+q^{i}_{(1,s,t)}})_{i=1,\ldots h}\,.
\end{equation}
To complete the proof of the theorem, we show that we can
choose the $m$-th root $\hat{f\,}_j$ of $f_j$ ($j=1, \ldots k$) so
that:
\begin{equation}
\underline{q}_{(1,r,s)}=\left(\epsilon^{q^{i}_{(1,r,s)}}
\right)_{i=1,\ldots h} \mbox{ is an element of } K, \;\forall\,(r,s).
\end{equation}
Let $\chi_v$ one of the chosen generators of $G^*$. According to
(\ref{zG*}), (\ref{relhv}) and (\ref{defq}), the action of
$\underline{q}_{(1,r,s)}$ on $L_{\chi_v}^{-1}$ is given by a $d_v$-th
root of unity, that we denote by $exp({2\pi
\sqrt{-1}\,\frac{x_v}{d_v}})$ (for a suitable integer $x_v$). We want
to show that we can assume that $x_v\equiv 0$ mod $d_v$, $v=1, \ldots
n$.
We observe that:
\begin{equation}\label{cobinK}
exp({2\pi \sqrt{-1}\,\frac{x_v}{d_v}})=
\prod_{i=1}^h \epsilon^{q^{i}_{(1,r,s)}b_{vi}}=
(h^{v}_{(1,r,s)})^{-1}\prod_{j=1}^k(\hat{k}_{(r,s)}^j)^{a_{vj}}
\end{equation}
and we compute the right-hand side of
(\ref{cobinK}). By (\ref{tildekmu}), one must have:
\begin{equation}
\begin{array}{cl}
h^{v}_{(1,r,s)}&=exp({-2\pi
\sqrt{-1}\,\frac{x_v}{d_v}})\prod_{j=1}^k
(\hat{k}^j_{(r,s)})^{a_{vj}}\\
&= exp({-2\pi
\sqrt{-1}\,\frac{x_v}{d_v}}) \prod_{l=1}^q
(\hat{\mu}^l_{(r,s)})^{m\sum_{j=1}^k \frac{c_{jl}a_{vj}}{m_j}}
\prod_{j=1}^k\left(\frac{\hat{f\,}^j_r}{\hat{f\,}^j_s}
\right)^{a_{vj}(m/m_j)}\\
&= exp({-2\pi
\sqrt{-1}\,\frac{x_v}{d_v}}) \prod_{l=1}^q
(\hat{\mu}^l_{(r,s)})^{\frac{m}{d_v}\sum_{j=1}^k
\frac{c_{jl}d_v a_{vj}}{m_j}}
\prod_{j=1}^k\left(\frac{\hat{f\,}^j_r}{\hat{f\,}^j_s}
\right)^{a_{vj}(m/m_j)}. \end{array}
\end{equation}
On the other hand, as in Lemma
\ref{coeffinK}, we write $L_{\chi_v}\equiv \sum_{l=1}^q
\lambda_{vl}M_l$ and we get the following relation form of cocycles on
$V_{(1,r,s)}$: \begin{equation}
h^{v}_{(1,r,s)}=\prod_{l=1}^q
(\mu^l_{(r,s)})^{\lambda_{vl}}
\frac{\varphi^v_{r}}{\varphi^v_{s}}
\end{equation}
for suitable nowhere vanishing holomorphic functions $\varphi^v_{r}$
on $U_{r}$. According to Lemma \ref{Lemma} and to (\ref{tors}), we can
then assume that in the previous equations one has:
\begin{equation}
\prod_{l=1}^q
(\hat{\mu}^l_{(r,s)})^{\left(\frac{m}{d_v}\sum_{j=1}^k
\frac{c_{jl}d_v a_{vj}}{m_j}\right)-d_v \lambda_{vl}}=1
\end{equation}
so that one gets:
\begin{equation}
\label{xv}
exp({2\pi \sqrt{-1}\,\frac{x_v}{d_v}})=
\prod_{j=1}^k\left(\frac{\hat{f\,}^j_r}{\hat{f\,}^j_s}
\right)^{a_{vj}(m/m_j)}\frac{\varphi^v_s}{\varphi^v_r}\,.
\end{equation}
We observe that we may assume that:
\begin{equation}
\varphi^v_r= exp(2\pi \sqrt{-1}\,\frac{t^v_r}{d_v})
\prod_{j=1}^k(\hat{f\,}^j_r)^{a_{vj}(m/m_j)}
\end{equation}
for suitable integers $t^v_r$; hence the equation
(\ref{xv}) gives:
\begin{equation}
\frac{x_v}{d_v}- \frac{t^v_s}{d_v}+\frac{t^v_r}{d_v}\;\in\;{\bf Z}.
\end{equation}
If we replace $\hat{f\,}^j_r$ by $exp(2\pi
\sqrt{-1}\,\frac{s^j_r}{m})\hat{f\,}^j_r$, then
$\frac{t^u_r}{d_v}$ is replaced by
$\frac{t^u_r}{d_v}+\sum_{j=1}^k\frac{a_{vj}s_j}{m_j}$. Therefore, we
need to solve the system:
\begin{equation}
\sum_{j=1}^k\frac{a_{vj}s^j_r}{m_j}\equiv t^v_r \hspace{10mm}
\mbox{mod}\,d_v \hspace{15mm}v=1,\ldots n.
\end{equation}
Since this is possible according to Lemma \ref{Lemma} and the Chinese
Remainder's Theorem, the proof is complete.
\hfill\qed
\begin{Remark}\label{dipdaLchi}
The cohomology class of the extension (\ref{extgrouteogen}) of the
fundamental groups depends on the choice of the solution
$\{L_\chi\}$ of the characteristic relations (\ref{eqstr}) for the
covering $f$. Moreover, covers corresponding to different
solutions $\{L_\chi\}$ may not be homeomorphic.
{\rm This is shown, for instance, by the following class
of examples. Denote by $e_i$ the standard generators of the group
$({\bf Z}/4{\bf Z})^3$ and let $G$ be the quotient of $({\bf Z}/4{\bf
Z})^3$ by the subgroup generated by ${2e_1+2e_2+2e_3}$. Let now $X$
be a smooth projective surface such that ${H^2(\pi_1(X), {\bf Z}/2
{\bf Z})\neq 0}$ and Pic($X$) has a 2-torsion element $\eta$ whose
class in ${H^2(X, {\bf Z}/2{\bf Z})}$ is non zero. Fix a very ample
divisor $H$ on $X$ and choose suitable divisors $D_i$ ($i=1,2,3$) such
that $D_i\equiv 4H$ and the $D_i$'s are in general position. Then
there exists a smooth abelian $G$-cover ${f:Y\to X}$ ramified on the
$D_i$'s ($i=1,2,3$), with inertia subgroup $G_i=<e_i>={\bf Z}/4{\bf
Z}$ and character $\psi_i$ dual to $e_i$, respectively. In fact,
taking the characters $\chi_1=\psi_1+3\psi_3$,
$\chi_2=\psi_2+3\psi_3$, $\chi_3=2\psi_3$ as generators of $G^*$, the
characteristic relations (\ref{eqstr}) of the cover $f$ are:
\begin{equation} \label{eqstrex}
\left\{ \begin{array}{lll}
4L_1&\equiv&D_1+3D_3\\ 4L_2&\equiv&D_2+3D_3\\ 2L_3&\equiv&D_3
\end{array}
\right.
\end{equation}
and admit, in particular, the solution $L_1= L_2=
L_3= 2H$. Under these hypotheses, $L_3$ generates the
subgroup $<D_i,L_\chi>$ ($i=1,2,3,\chi\,\in\,G^*$) of Pic($X$) and the
decomposition $D_i\equiv 2L_3$ has the properties requested in
Prop.\ref{coeffinK}. According to Prop.\ref{top} and Thm.\ref{mtgen},
since the pull back $\tilde{D}_i$ of $D_i$ under the universal cover
$\tilde{X}$ of $X$ is 2-divisible, $\forall\,i$, then the map $f$
induces a central extension of the form:
\begin{equation}
0\to {\bf Z}/2{\bf Z}\to \pi_1(Y)\to \pi_1(X)\to 1
\end{equation}
and the cohomology class of
this extension in $H^2(\pi_1(X), {\bf Z}/2{\bf Z})\subseteq H^2(X,{\bf
Z}/2{\bf Z})$ is the image $\Psi_*([L_3])$ of the Chern class of
$L_3$ under the map induced in cohomology by the standard projection
$\Psi:{\bf Z}\to {\bf Z}/2{\bf Z}$: so, this class is trivial.
Let now $\overline{Y}$ be the $G$-cover of $X$ corresponding to the
solution $\overline{L}_i=2H+\eta$ ($i=1,2,3$) of (\ref{eqstrex});
in this case the cohomology class describing $\pi_1(\overline{Y})$
is given by $\Psi_*([L_3+\eta])$ and, by the hypotheses made, it is
not trivial.
In particular, when $X$ is a
projective variety with $\pi_1(X)={\bf Z}/2{\bf
Z}$, the previous construction yields two non
homeomorphic $G$-covers $Y$, $\overline{Y}$ of $X$, branched on the
same divisor, with the same inertia subgroups and characters, such
that:
\begin{equation} \pi_1(Y)=({\bf Z}/2{\bf
Z})^2\hspace{20mm}\pi_1(\overline{Y})={\bf Z}/4{\bf
Z}\,.
\end{equation}
}
\end{Remark}
\hfill\qed
The following theorem is an attempt to determine to what extent the
class $c(f)$ depends on the choice of the $L_\chi$'s, once the branch
divisor and the covering structure are fixed.
\begin{Theorem} \label{mt}
Same hypotheses and notation as in the statement of Thm.\ref{mtgen}.
Consider the class $c(f)\,\in\,H^{2}(\pi_1(X), K)$ associated to the
central extension (\ref{extgrouteogen}) given by the fundamental
groups and denote by ${i(c(f)) \in
H^{2}(\pi_1(X),\tilde{G})\subseteq H^{2}(X,\tilde{G})}$ its image via
the map induced in cohomology by the inclusion (\ref{extgrouO})
$K\subseteq \tilde{G}$.
Denote by $\Phi$ the group homomorphism defined as follows:
\begin{equation}
\begin{array}{rccl}
\Phi :& {\bf Z}^{k}& \rightarrow &\tilde{G}\\
&(x_{1}, \ldots x_{k})&\rightarrow&
g_{1}^{x_{1}}\cdots g_{k}^{x_{k}}
\end{array}
\end{equation}
and by ${\Phi_{*}: H^{2}(X,{\bf
Z}^{k}) \rightarrow H^{2}(X,\tilde{G})}$ the map induced by $\Phi$
in cohomology.
Then:
\begin{equation}
\label{coc}
i(c(f)) = \Phi_{*}([D_{1}], \ldots [D_{k}])
\end{equation}
where $[\Delta]$ denotes the class of a divisor $\Delta$
on $X$ in ${H^{2}(X,{\bf Z})}$.
\end{Theorem}
\begin{Corollary}\label{mtcor}
Same hypotheses and notation as in Thm.\ref{mtgen}. Assume moreover
that the natural morphism $Hom(\pi_1(X),\tilde{G})\to
Hom(\pi_1(X),G)$, induced by the surjection ${\tilde{G}\to G}$, is
surjective. Then the map ${i:H^2(\pi_1(X),K)\to
H^2(\pi_1(X),\tilde{G})}$ is injective and the class
$\Phi_{*}([D_{1}], \ldots [D_{k}])$ in (\ref{coc}) determines
uniquely the class $c(f)\,\in\,H^2(\pi_1(X),K)$ of the extension
(\ref{extgrouteogen}) of the fundamental groups.
This happens, in particular, if $\pi_1(X)$ is torsion
free or $Hom(\pi_1(X),G)=0$ (e.g., if $\pi_1(X)$ is finite with order
coprime to the order of $G$), or the sequence ${0\to K\to
\tilde{G}\to G\to 0}$ splits.
\end{Corollary}
{\sc Proof of Thm.}\ref{mt}. We keep the notation and the results
in Step I of the proof of Thm.\ref{mtgen}, noticing that the cocycle
$c(f)_{r,s,t}$ in (\ref{coccfnuovo}) also represents the class
$i(c(f))$ in $H^{2}(X,\tilde{G})$.
We want to write down a cocycle representing the class
${\Phi_{*}([D_{1}], \ldots [D_{k}])\in H^{2}(X,\tilde{G})}$
and to show that it represents the same cohomology class as the
cocycle in (\ref{coccfnuovo}).
We consider as before the cocycle $k^{j}_{(r,s)}$ representing
${\cal O}_{X}(-D_{j})$ in the choosen covering ${U_{r}}$ of $X$.
For every
pair of indices $r$, $s$ with ${U_{r}\cap U_{s}\neq\emptyset}$
and for every $j = 1, \ldots k$, we choose a $m_{j}$-th root
$\hat{k}^{j}_{(r,s)}$ of $k^{j}_{(r,s)}$ on ${U_{r}\cap U_{s}}$
in such a way that
${\hat{k}^{j}_{(s,r)}= (\hat{k}^{j}_{(r,s)})^{-1}}$. As before, the
image of the class of $-D_{j}$ in $H^{2}(X,{\bf Z}/m_{j}{\bf Z})$ is
represented on ${U_{r}\cap U_{s}\cap U_{t}}$ by the cocycle
${\hat{k}^{j}_{(r,s)}\hat{k}^{j}_{(s,t)}\hat{k}^{j}_{(t,r)}}$,
${j = 1,} \ldots k$.
Then the class ${-\Phi_{*}([D_{1}], \ldots [D_{k}])\in
H^{2}(X,\tilde{G})}$ is represented on ${U_{r}\cap U_{s}\cap U_{t}}$
by:
\begin{equation} b_{r,s,t} = (\prod^{k}_{j=1}
(\hat{k}^{j}_{(r,s)}\hat{k}^{j}_{(s,t)}\hat{k}^{j}_{(t,r)})
^{\tilde{a}_{ij}})_{i=1,\ldots h}\;\;.
\end{equation}
and we have shown in the equality (\ref{cobord}) in the proof of
Thm.\ref{mtgen}, Step III, that this cocycle represents the same
class then $c(f)_{r,s,t}$ in $H^2(X,\tilde{G})$. \hfill\qed
\begin{Remark}
\label{rem}
{\rm From Thm.\ref{mt} it follows in particular that the class
$i(c(f))\in H^{2}(\Gamma,\tilde{G})$ depends only on the class of the
$D_{j}$'s in $H^{2}(X,{\bf Z}/m_{j}{\bf Z})$ ($j = 1, \ldots k$), once
$G$ and the $g_{j}$'s are fixed. In particular, if $D_j$ is
$m_j$-divisible on $X$ ($\forall\; j=1,\ldots k$), then $i(c(f))=0$.}
\end{Remark}
|
1996-01-22T01:47:12 | 9305 | alg-geom/9305012 | en | https://arxiv.org/abs/alg-geom/9305012 | [
"alg-geom",
"math.AG"
] | alg-geom/9305012 | Claude LeBrun | Claude LeBrun | A Kaehler Structure on the Space of String World-Sheets | 13 pages, LaTeX | null | 10.1088/0264-9381/10/9/006 | null | null | Let (M,g) be an oriented Lorentzian 4-manifold, and consider the space S of
oriented, unparameterized time-like 2-surfaces in M (string world-sheets) with
fixed boundary conditions. Then the infinite-dimensional manifold S carries a
natural complex structure and a compatible (positive-definite) Kaehler metric h
on S determined by the Lorentz metric g. Similar results are proved for other
dimensions and signatures, thus generalizing results of Brylinski regarding
knots in 3-manifolds. Generalizing the framework of Lempert, we also
investigate the precise sense in which S is an infinite-dimensional complex
manifold.
| [
{
"version": "v1",
"created": "Wed, 26 May 1993 18:36:40 GMT"
}
] | 2009-10-22T00:00:00 | [
[
"LeBrun",
"Claude",
""
]
] | alg-geom | \section{Introduction}
Given a collection of circles in
a 4-dimensional oriented Lorentzian space-time, one may consider the
space $\cal S$ of unparameterized oriented time-like compact 2-surfaces with
the given circles as boundary. The main purpose of
the present note is to endow $\cal S$ with the structure
of an infinite-dimensional K\"ahler manifold--- i.e. with
both a complex
structure and a Riemannian metric for which this
complex structure is covariantly constant.
This was motivated by
a construction of Brylinski \cite{bryl}, whereby a K\"ahler
structure is given to
the space of knots in a Riemannian 3-manifold.
In fact, our discussion will be structured so as to apply to
codimension 2 submanifolds of a
space-time of arbitrary
dimension and metrics of arbitrary signature, with the
proviso that we only consider those submanifolds
for which the normal bundle is orientable and has
(positive- or negative-)definite
induced metric; thus Brylinski's construction becomes subsumed as a special
case.
As the reader will therefore see,
complex manifold theory thus comes naturally into play when one
studies codimension 2 submanifolds of a space-time. On the other hand,
complex manifold theory makes a quite different kind of
appearance when one attempts to study the intrinsic
geometry of 2-dimensional manifolds. If some interesting modification
of string theory could be found which invoked
both of these observations simultaneously, one
might hope to thereby explain the puzzling four-dimensionality of the
observed world.
Many of the key technical ideas in the present note are
straightforward generalizations of arguments due to
L\'aszl\'o Lempert
\cite{lemp}, whose lucid study of Brylinski's complex structure
is based on the theory of twistor CR manifolds \cite{leb}.
One of the most striking features of the complex
structures in question is that, while they are formally integrable and
may even admit legions of local holomorphic functions, they
do {\em not} admit enough finite-dimensional complex
submanifolds to be locally modeled on any complex
topological vector space.
This beautifully illustrates the fact, emphasized by Lempert,
that the Newlander-Nirenberg Theorem \cite{nn} fails in
infinite dimensions.
\section{The Space of World-Sheets}
Let $(M, g)$ be a smooth oriented
pseudo-Riemannian n-manifold. We use the term {\em world-sheet} to refer
to a smooth compact
oriented codimension-2
submanifold-with-boundary
$\Sigma^{n-2}\subset M^n$ for which the inner product induced
by $g$ on the conormal bundle
$$\nu^{\ast}_{\Sigma}:=\{ \phi \in T^{\ast}M|_{\Sigma}~~~|~~\phi|_{T\Sigma}\equiv 0
\} $$
of $\Sigma$
is definite at each point. If $g$ is Riemannian, this just means an
oriented submanifold of codimension 2;
on the other hand, if $(M,g)$ is a Lorentzian 4-manifold,
a world-sheet is exactly an oriented time-like 2-surface.
\begin{defn} Let $(M, g)$ be a smooth oriented
pseudo-Riemannian n-manifold, and let $B^{n-3}\subset M^n$ be a smooth
codimension-3 submanifold which is compact, without boundary. We will
then let ${\cal S}_{M,B}$ denote the space of smooth oriented
world-sheets $\Sigma^{n-2}\subset M$ such that
$\partial \Sigma =B$.
\end{defn}
Of course, this space is sometimes empty--- as happens, for example, if $B$
is a single space-like circle in Minkowski 4-space. This said,
${\cal S}_{M,B}$ is automatically a Fr\'echet manifold, and its tangent space at
$\Sigma$ is
$$T_{\Sigma}{\cal S}_{M,B}=\{ v\in \Gamma (\Sigma, C^{\infty}(\nu_{\Sigma}))~~|~~
v|_{\partial \Sigma}\equiv 0\}~ .$$
Indeed, if we choose a tubular neighborhood of $\Sigma$ which is
identified with the normal bundle of an open extension
$\Sigma_{\varepsilon}$ of $\Sigma$ beyond its boundary, every section
of $\nu_{\Sigma}\to \Sigma$ which vanishes on $\partial \Sigma$ is thereby
identified with an imbedded submanifold of $M$, and this
submanifold is still a world-sheet provided the $C^1$ norm of the
section is sufficiently small. This provides ${\cal S}_{M,B}$ with
charts which take values in Fr\'echet spaces, thus giving it the desired
manifold structure.
Since the normal bundle
$\nu_{\Sigma} =(\nu_{\Sigma}^{\ast})^{\ast}=TM/T\Sigma =(T\Sigma)^{\perp}$ of our
world-sheet is of rank 2 and comes equipped with
an orientation as well as a metric induced by
$g$, we may identify $\nu_{\Sigma}$ with a complex line bundle by
taking $J: \nu_{\Sigma}\to\nu_{\Sigma}$, $J^2=-1$ to be rotation by $+90^{\circ}$.
This
then defines an endomorphism ${\cal J}$ of $T{\cal S}$ by
$${\cal J} : T_{\Sigma}{\cal S}_{M,B}\to T_{\Sigma}{\cal S}_{M,B}: v \to J\circ v ~ .$$
Clearly ${\cal J}^2=-1$, so that ${\cal J}$ gives ${\cal S}$ the structure of an almost-complex
Fr\'echet manifold--- i.e. every tangent space of ${\cal S}$ can now be thought of as
a complex Fr\'echet space by defining ${\cal J}$ to be multiplication by
$\sqrt{-1}$. In the next sections, we shall investigate the
integrability properties of this almost-complex structure.
\section{Integrability of the Complex Structure}
Let $(M,g)$ denote, as before, an oriented pseudo-Riemannian manifold.
Let $Gr_2^{+}(M)$
denote the bundle of oriented 2-planes in $T^{\ast}M$ on which
the inner product induced by $g$ is definite. This
smooth ($3n-4$)-dimensional manifold then has a natural CR structure
\cite{leb,ros} of codimension $n-2$. Let us review how this
comes about.
Let $\hat{N}\subset [({\Bbb C}\otimes T^{\ast}M)- T^{\ast}M]$
denote the set of non-real null covectors of $g$, and let $N\subset
{\Bbb P} ({\Bbb C}\otimes T^{\ast}M)$ be its image in the fiber-wise projectivization
of the complexified cotangent bundle. There is then a natural identification
of ${N}$ with $Gr_2^{+}(M)$. Namely, using pairs $u,v\in T_x^{\ast}M$ of
real covectors
satisfying
$\langle u, v\rangle =0$ and $\langle u, u\rangle =\langle v, v\rangle$,
we define a bijection between these two spaces by
$$Gr_2^{+}(M) \ni
\mbox{oriented span}(u,v)\leftrightarrow [u+iv]\in {N}\subset {\Bbb P}
({\Bbb C}\otimes T_x^{\ast}M)$$
which
is independent of the representatives $u$ and $v$. But, letting
$\vartheta=\sum p_jdx^j$ denote the canonical complex-valued 1-form
on the total space of ${\Bbb C}\otimes T^{\ast}M\to M$, and letting $\omega$ be
the restriction
of $d\vartheta$ to $\hat{N}$, the distribution
$$ \hat{D}=\ker (\omega:{\Bbb C}\otimes T\hat{N}\to {\Bbb C}\otimes T^{\ast}\hat{N})$$
is involutive by virtue of the fact that $\omega$ is closed; since
$ \hat{D}$ also contains no non-zero real vectors as a consequence of
the fact that $\hat{N}\cap T^{\ast}M=\emptyset$, $ \hat{D}$ is a CR structure
on $\hat{N}$, the codimension of which can be checked to be
$n-2$. This CR structure is invariant under the natural action
of ${\Bbb C}^{\times}$ on $\hat{N}$ by scalar multiplication, and
thus descends to a CR structure $D$ on $N=Gr_2^{+}(M)$, again of codimension
$n-2$. Moreover, $\vartheta|_{\hat{N}}$ descends to $N$ as a CR
line-bundle-valued 1-form
$$\theta\in \Gamma (N, {\cal E}^{1,0} (L))~,~~ \bar{\partial}_b\theta =0~,
$$
where, letting $T^{1,0}N:=({\Bbb C}\otimes TN)/D$,
$L^{\otimes (n-1)}=\bigwedge^{(2n-3)}T^{1,0}N$,
${\cal E}^{1,0} (L):= C^{\infty} (L\otimes (T^{1,0}N)^{\ast})$,
and $\bar{\partial}_b$ is naturally induced by $d|_D$.
The CR structure $D$ of $N$ may be expressed in the form
$$D=\{ v-iJv~| ~~v\in H\}$$
for a unique rank $2n-2$ sub-bundle
$H$ of the real tangent bundle $TN$ and a unique endomorphism $J$ of $H$
satisfying $J^2=-1$.
In these terms the geometric
meaning of the CR structure of $N$ is
fairly easy to describe. Indeed, if $\varpi : Gr_2^{+}(M)\to M$
is the tautological projection, then
$H_P = (\varpi_{\ast P})^{-1}(P)$ for every oriented definite 2-plane
$P\subset TM$.
On vertical vectors, $J$ acts by the
standard complex structure on the quadric fibers of $N\to M$;
whereas $J$ acts on horizontal vectors by $90^{\circ}$ rotation in the
2-plane $P\subset TM$. This point of view, however,
obscures the fact that $D$ is both involutive
and
unaltered by conformal changes $g\mapsto e^fg$.
A compact $(n-2)$-dimensional submanifold-with-boundary $S\subset N$,
will be called a {\em transverse sheet} if its tangent space
is everywhere tansverse to the CR tangent space of $N$:
$$TN|_S= TY\oplus H|_Y~ .$$ As before,
let $B^{n-3}\subset M$ denote a compact codimension-3
submanifold, and let $\varpi :N\to M$ be the canonical projection.
We will then let
$\hat{{\cal S} }_{N,B}$ denote the set of
transverse sheets $S\subset N$
such that $\varpi$ maps $\partial S$ diffeomorphically onto
$B$. Thus $\hat{{\cal S} }_{N,B}$ is a Fr\'echet manifold whose
tangent space at $S$ is given by
$$T\hat{{\cal S} }_{N,B}|_S=\{ v\in \Gamma (S, C^{\infty}(H|_S))~
|~\varpi_{\ast}(v|_{\partial S})\equiv 0\} ~ ,$$
and hence $J: H\to H$ induces an almost-complex structure
$\hat{\cal J}$ on
$ \hat{{\cal S} }_{N,B}$ by $\hat{\cal J} (v):= J\circ v$.
\begin{propn} The almost-complex structure $\hat{\cal J}$ on the space
$ \hat{{\cal S} }_{N,B}$ of transverse sheets is formally integrable--- i.e.
$$\tau (v,w):= \hat{\cal J}[v,w]
-[v,\hat{\cal J}w] -[\hat{\cal J}v,w]
-\hat{\cal J} [\hat{\cal J}v,\hat{\cal J}w] =0 $$
for all smooth vector fields $v$, $w$ on $\hat{{\cal S} }_{N,B}$.
\end{propn}
\begin{proof}
The Fr\"ohlicher-Nijenhuis torsion $\tau (v,w)$ is tensorial in the sense
that its value at $S$ only depends on the values of $v$ and $w$ at
$S$. Given $v_S,w_S \in \{ v\in \Gamma (S, C^{\infty}(H|_S))~
|~\varpi_{\ast}(v|_{\partial S})\equiv 0\}$, we will now
define
preferred extensions of them as vector fields near $S\in \hat{{\cal S} }_{N,B}$
in such a manner as to simplify the computation of
$\tau (v,w)= \tau (v_S,w_S)$.
To do this, we may first use a partition of unity to
extend $v_S$ and $w_S$ as sections
$\hat{v}, \hat{w}\in \Gamma (N, C^{\infty}(H))$ defined on all of
of $N$ in such a manner that $\hat{v}$ and $\hat{w}$
are tangent to the fibers of $\varpi$ along all of $\varpi^{-1}(B)$.
Now let $U\subset N$ be a tubular neighborhood
of $S$ which is identified with the normal bundle $H$ of
some open extension $S_{\epsilon }$ of $S$, and
let $\hat{U}\subset \hat{{\cal S} }_{N,B}$ be the
set of transverse sheets $S'\subset U$.
We may now define our preferred extensions of
$v$ and $w$ of $v_S$ and $w_S$ on the domain $\hat{U}$ by letting
the values of $v$ and $w$ at $S'\subset U$
be the restrictions of $\hat{v}$ and $\hat{w}$
to ${S'}$. Notice that $[ v,w ]$ is then precisely the vector field
on $\hat{U}$ induced by $[\hat{v}, \hat{w}]$, whereas
$\hat{\cal J}v$ is the vector field induced by $J\hat{v}$.
Since the integrability condition for $(N,D)$ says that
$$J([\hat{v},\hat{w}] -[J\hat{v},J\hat{w}])
=[\hat{v},J\hat{w}] +[J\hat{v},\hat{w}]~ , $$
it therefore follows that
$$\hat{\cal J}[v,w]
-\hat{\cal J} [\hat{\cal J}v,\hat{\cal J}w] =
[v,\hat{\cal J}w] +[\hat{\cal J}v,w] ~ ,$$
so that $\tau (v,w)=0$, as claimed.
\end{proof}
We now observe that there is a canonical imbedding
\begin{eqnarray*} {{\cal S} }_{M,B}&\stackrel{\Psi}{\hookrightarrow} &\hat{{\cal S} }_{N,B}\\
\Sigma &\mapsto & \nu_{\Sigma }
\end{eqnarray*}
obtained by sending a world-sheet to its normal-bundle,
thought of as the image of a section of $Gr_2^+(M)|_{\Sigma }= N|_{\Sigma }$;
thought of in this way, it is easy to see that $\nu_{\Sigma}\subset N$
is a transverse submanifold.
\begin{thm} The imbedding $\Psi$ realizes $({{\cal S} }_{M,B}, {\cal J})$
as a complex submanifold of $(\hat{{\cal S} }_{N,B}, \hat{\cal J})$.
In particular, the almost-complex structure ${\cal J}$
of ${{\cal S} }_{M,B}$ is formally integrable. \label{imb}
\end{thm}
\begin{proof}
The projection $\varpi: N\to M$ induces a map $\hat{\varpi}:
\hat{{\cal S} }_{N,B}\to{{\cal S} }_{M,B}$
which is a left inverse of $\Psi$ and satisfies $\hat{\varpi}_{ *}\hat{\cal J}
={\cal J}\hat{\varpi}_{ *}$. It therefore suffices to show that
the tangent space of the image of
$\Psi$ is $\hat{\cal J}$-invariant.
Now the condition for a transverse sheet $S\subset N$
to be the $\Psi$-lifting of the
world-sheet $\varpi(S)\subset M$ is exactly that $\theta|_S\equiv 0$.
When $S$ satisfies this condition, a connecting field
$v\in\Gamma (S, {\cal E}(H))$
then represents a vector $\hat{v}\in T\hat{\cal S}$ which is
tangent to the image of $\Psi$ iff
\be (v \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm d\theta)|_{TS}+d(v \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm \theta)|_{TS}\equiv 0~ ;\label{leg} \end{equation}
the exterior derivative of $\theta$ may here be calculated in any
local trivialization for the line bundle $L$, since the
left-hand side rescales properly under changes of trivialization so as
define an $L$-valued 1-form on $S$. But since $\theta\in \Gamma (N, {\cal
E}^{1,0}
(L))$ satisfies $\bar{\partial}_b \theta =0$, it follows that
$$(Jv \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm d\theta)|_{TS}+d(Jv \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm \theta)|_{TS}=i(Jv \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm d\theta)|_{TS}+
id(Jv \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm \theta)|_{TS} $$
because $\theta$ and $d\theta$ are of types (1,0) and (2,0), respectively.
The tangent space of the image of $\Psi$
is therefore $\hat{\cal J}$-invariant, and the claim follows.
\end{proof}
\begin{defn} Let $({\Frak X}, {\Frak J})$
be an almost-complex Fr\'echet manifold, and let
$f: U\to {\Bbb C}$ be a differentiable function defined on an
open subset of ${\Frak X}$. We will say that
$f$ is ${\Frak J}$-holomorphic if
$$({\Frak J}v)f=ivf~~\forall v\in TU~.$$
\end{defn}
\begin{defn}
An almost-complex Fr\'echet manifold
$({\Frak X}, {\Frak J})$ is
called {\em weakly integrable} if for each real tangent vector
$w\in T{\Frak X}$ there is a ${\Frak J}$-holomorphic
function $f$ defined on a neighborhood
of the base-point of $w$ such that $wf\neq 0$.
\end{defn}
\begin{thm} Suppose that $(M,g)$ is real-analytic. Then
$(\hat{{\cal S} }_{N,B}, \hat{\cal J})$ is weakly integrable.\label{wint}
\end{thm}
\begin{proof}
If $(M,g)$ is real-analytic, so is the CR manifold $(N,D)$,
and we can therefore
realize $(N,D)$
as a real submanifold of a complex manifold $(2n-3)$-manifold
$\cal N$. This can even be done explicitly by
taking $\cal N$ to be a space of complex null geodesics
for a suitable complexification of $(M,g)$.
Now let $S\subset N\subset {\cal N}$ be any transverse sheet.
Then there is a neighborhood $V\subset {\cal N}$ of $S$ which can
be holomorphically imbedded in some ${\Bbb C}^{\ell}$.
Indeed, let $Y\subset {\cal N}$ be a totally real $(2n-3)$-manifold
containing $S$, let $f: Y\to {\Bbb R}^{\ell}$ be
a smooth imbedding, and let $Y_0$ be a precompact
neighborhood of $S\subset Y$ with smooth boundary.
By \cite{wel},the component functions $f^j|_{Y_0}$ are limits
in the $C^1$ topology
of the restrictions of holomorphic functions. Using such an approximation
of $f$,
we may therefore imbed $Y_0$ as a totally real submanifold
of ${\Bbb C}^{\ell}$ by a map which
extends holomorphically to a neighborhood of
$Y_0$, and this holomorphic extension then automatically
yields a holomorphic imbedding
of some open neighborhood $V\supset S$ in ${\Bbb C}^{\ell}$.
Now suppose that $v$ is a smooth section of $H$ along $S$.
We may express $v$ uniquely as $u+Jw$, where $u$ and $w$ are
tangent to the $Y$. By changing $Y$ if necessary, we can furthermore
assume that $u\not\equiv 0$.
Let $F$ be a smooth function on $Y$ which
vanishes on $S$ and such that the derivative $vF$ is
non-negative and supported near some interior point of $S$;
and let $\varphi$ be a real-valued smooth $(n-2)$-form
on $Y$ whose restriction to $S$ is positive on the
support of $vF$. Set $\psi =F\varphi$. Using
our imbedding of $Y$ in ${\Bbb C}^{\ell}$,
we can express $\psi$ as a family of component functions---
e.g. by arbitrarily declaring that all contractions
of $\psi$ with elements of the normal bundle
$(TY)^{\perp}\subset T{\Bbb C}^{\ell }$ shall vanish.
But, again by \cite{wel}, these component functions are $C^1$-limits
on $Y_0$ of restrictions of
holomorphic functions from a neighborhood of $Y_0\subset {\Bbb C}^{\ell}$.
Thus, by perhaps replacing $V$ with a smaller neighborhood,
there is a holomorphic $(n-2)$-form $\beta$ on $V$ which approximates
$\psi$ well enough that $$\Re e \int_Sv \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm d\beta > \frac{1}{2}
\int_Su \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm d\psi >0$$ and $$\Re e
\int_{\partial S}v \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm \beta > - \frac{1}{2}
\int_Su \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm d\psi~ .$$
Let $\hat{V}:=\{S'\in {\cal S}_{N,B}~|~S'\subset V\}$,
and define $f_{\beta}: \hat{V}\to {\Bbb C}$
by $f_{\beta}(S')=\int_{S'}\beta$. Then $f_{\beta}$
is a holomorphic function on the open set
$\hat{V}\subset {\cal S}_{N,B}$. Indeed, if
$\gamma$ is {\em any} smooth $(n-2)$-form on $V$, and if we set
$f_{\gamma}(S')=\int_{S'}\gamma$, then, for $S'\subset V$,
the derivative of $f_{\gamma}$ in the direction of
$w\in \Gamma (S', C^{\infty}(H))$, $\varpi_*(w)|_{\partial S'}\equiv 0$,
is given by
$$wf_{\gamma}|_S= \int_{S}w \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm d\gamma +
\int_{\partial S'}w \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm\gamma ~ ;$$
for if $w$ is extended to $V$ as a smooth
vector field $\hat{w}$ tangent to the
fibers of $\varpi$ and $S_t$ is obtained by pushing $S'$
along the flow of the vector field $\hat{w}$, then
\begin{eqnarray*} wf_{\gamma}|_{S'}&=&
\left.\frac{d}{dt}\left[\int_{S_t}\gamma\right]\right|_{t=0}
\\&=& \int_{S'}\pounds_{\hat{w}} \gamma
\\&=& \int_{S'}[\hat{w} \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm d\gamma +d(\hat{w} \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm\gamma )]
\\&=& \int_{S'}w \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm d\gamma +\int_{\partial S'}w \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm\gamma ~ .
\end{eqnarray*}
But since $\beta$ is the restriction of a
holomorphic $(n-2)$-form from a region of $\cal N$,
it therefore follows that
\begin{eqnarray*}(\hat{\cal J}w)f|_{S'}&=&
\int_{S'}(Jw ) \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm d\beta +\int_{\partial S'}(Jw) \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm\beta
\\&=& i\int_{S'}w \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm d\beta +i\int_{\partial S'}w \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm\beta
\\&=& iwf|_{S'}~ ,\end{eqnarray*}
showing that the function $f_{\beta}$ induced by $\beta$ is
$\hat{\cal J}$-holomorphic, as claimed.
However, we have also carefully chosen $\beta$ so that the real part
of the expression $\int_{S}v \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm d\beta +\int_{\partial S}v \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm\beta
= vf_{\beta}$ is positive.
For every real tangent vector $v$ on ${\cal S}_{N,B}$,
one can thus find a locally-defined
$\cal J$-holomorphic function whose derivative
is non-trivial in the direction $v$. Hence $\hat{\cal J}$
is weakly integrable, as claimed.
\end{proof}
\begin{cor} If $(M,g)$ is real-analytic, then
$({{\cal S} }_{M,B}, {\cal J})$ is weakly integrable.
\end{cor}
\begin{proof} By
Theorem \ref{imb} and \ref{wint}, $({{\cal S} }_{M,B}, {\cal J})$
can be imbedded
in the weakly integrable almost-complex manifold
$(\hat{{\cal S} }_{N,B}, \hat{\cal J})$.
Since the restriction of a holomorphic function to an almost-complex
submanifold is holomorphic, it follows that $({{\cal S} }_{M,B}, {\cal J})$
is weakly integrable.
\end{proof}
One might instead ask whether $({{\cal S} }_{M,B}, {\cal J})$ is
{\em strongly integrable}--- i.e. locally biholomorphic to a ball in some
complex vector space. The answer is {\bf no}; in contrast to
any strongly integrable almost-complex manifold,
$({{\cal S} }_{M,B}, {\cal J})$ contains very few
finite-dimensional complex submanifolds:
\begin{propn} Suppose that $(M,g)$ is real-analytic.
At a generic point $S\in {{\cal S} }_{M,B}$, a generic $(n-1)$-plane
is not tangent to any $(n-1)$-dimensional ${\cal J}$-complex submanifold.
\end{propn}
\begin{proof}
Let $S\subset M$ be a world-sheet which is {\sl not} real-analytic near
$p\in S$. Let $q\in N=Gr_2^+(M)$ be given by $q=T_pS^{\perp}$, and let
$v_1, \ldots , v_{n-1}\in H_q$ be a set of real vectors such that the
$v_j+iJv_j$ form a basis for $D_q$. Extend these vectors as sections
$\hat{v}_j$ of $H|_{\Psi(S)}$ which satisfy equation (\ref{leg})
along the sheet; this may be done, for example, by first extending each
$v_j$ to just a 1-jet at $p$ satisfying (\ref{leg}) at $p$, projecting this
to a 1-jet via $\varpi$ to yield
a 1-jet of a normal vector field on $S\subset M$, extending this 1-jet as a
section of the normal bundle of $S$, and finally lifting this section
using $\Psi_{*}$. Let $u_1, \ldots , u_n$ be the
elements of $T_S{{\cal S} }_{M,B}$ represented by
$\hat{v}_1, \ldots , \hat{v}_{n-1}$,
and let $P\subset T^{1,0}_S{{\cal S} }_{M,B}$
be spanned by $u_1-i{\cal J}u_1, \ldots , u_n-i{\cal J} u_n$.
Now suppose there were an $\cal J$-holomorphic submanifold
$X\subset {{\cal S} }_{M,B}$
through $S$ with
(1,0)-tangent plane equal to $P$. Then $X$ represents a
family of transverse sheets in $N$ which foliates a neighborhood $U\subset
N$ of $q$;
moreover, because $X$ represents a {\sl holomorphic} family,
the leaf-space projection $\ell : U\to X$ is CR in the sense that
$\ell_*(D)\subset T^{0,1}X$. Since we have assumed that
$(M,g)$ is real-analytic, we may also assume that $U$ has
a real-analytic CR imbedding $U\hookrightarrow {\Bbb C}^{2n-3}$.
Moreover the twistor CR manifold $N$ is automatically
``anticlastic,'' by which I mean that the Levi form
${\cal L}: D\to TN/H: v\mapsto [v,\bar{v}] \bmod H$
is {\sl surjective} at each point of $N$.
This gives rise to a Bochner-Hartogs
extension phenomenon:
every CR function on $U$ extends to a holomorphic function on some
neighborhood of $U\subset {\Bbb C}^{2n-3}$. In particular, every
CR map defined on $U$ must be real-analytic, and this applies in particular
to the leaf-space projection $\ell$. Thus $\Psi(S)$ is real-analytic near
$q$, and $S$ is therefore real-analytic near $p$.
This proves the result by contradiction.
\end{proof}
\section{K\"ahler Structure}
The complex structure $\cal J$ on ${\cal S}$ depends only
on the conformal class $[ g]= \{ e^fg\}$ of our
metric, but we will now specialize by fixing a specific
pseudo-Riemannian metric $g$. Our reason for
doing so is that we thereby induce an
$L^2$-metric on ${\cal S}$. Indeed,
each tangent space
$$T_{\Sigma}{\cal S}_{M,B}=\{ v\in \Gamma (\Sigma, C^{\infty}(\nu_{\Sigma}))~~|~~
v|_{\partial \Sigma}\equiv 0\}~$$ may be equipped with
a positive-definite
inner product by setting
$$h( v, w ) := \int_{\Sigma}g(v, w) ~d\mbox{vol}_{g|_{\Sigma }} .$$
We shall now see that this metric has some quite remarkable
properties.
\begin{thm} The Riemannian metric $h$ on ${\cal S}$ is {\em K\"ahler}
with respect to the
previously-defined complex structure ${\cal J}$.\end{thm}
\begin{proof}
Let
$\Omega$ denote the volume n-form of $g$, and
define a
2-form $\omega$ on ${\cal S}$ by
$$\omega ( v, w ) := \int_{\Sigma}(v\wedge w) \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm \Omega ~ .$$
Obviously, $\omega$ is ${\cal J}$-invariant and
$$h( v, w )=\omega ({\cal J}v, w).$$
We therefore just need to show that $\omega$ is closed.\footnote{
The reader may ask whether it is actually legitimate to
call a Riemannian
manifold K\"ahler when the almost-complex structure in question is
at best weakly integrable. However, formal integrability and the
closure of the K\"ahler form are the only conditions
necessary to insure that
the almost-complex structure tensor is parallel,
even in infinite dimensions.}
To check this,
let us introduce the universal family
\setlength{\unitlength}{1ex}
\begin{center}\begin{picture}(20,17)(0,3)
\put(9.5,17){\makebox(0,0){${\cal F}$}}
\put(1.5,5){\makebox(0,0){${\cal S}$}}
\put(17,5){\makebox(0,0){$M$}}
\put(14.5,12){\makebox(0,0){$p$}}
\put(4.5,12){\makebox(0,0){${\pi }$}}
\put(10.5,15){\vector(2,-3){5.5}}
\put(8,15){\vector(-2,-3){5.5}}
\end{picture}\end{center}
where the
fiber of $\pi$ over $\Sigma \in {\cal S}$ is defined to be $\Sigma \subset M$.
We can then pull $\Omega$
back to ${\cal F}$
to obtain a closed n-form $\alpha=p^{\ast}\Omega$ which
vanishes on the boundary $B=\partial \Sigma$ of every fiber
of $\pi $. But $\omega$ is just obtained from $\alpha$ by integrating
on the fibers of $\pi $:
$$\omega=\pi _{\ast}\alpha~ .$$
Since $\pi _{\ast}$ commutes with $d$ on forms which vanish along the
fiber-wise boundary (cf. \cite{bt}, Prop. 6.14.1), it follows that
$$d\omega=d(\pi _{\ast}\alpha )= \pi _{\ast}d\alpha=
\pi _{\ast}d (p^{\ast}\Omega )=\pi _{\ast}p^{\ast}d\Omega=
\pi _{\ast}p^{\ast}0=0.$$
Thus $h$ is a K\"ahler metric, with K\"ahler form $\omega$.
\end{proof}
To conclude this note, we now observe that $({\cal S}, h)$ is formally
of Hodge type--- but non-compact, of course!
\begin{propn} Modulo a multiplicative constant, the
K\"ahler form $\omega$ of $h$
represents an integer class in cohomology. If,
moreover, $M$ is non-compact, $\omega$ is actualy an exact form, and
its cohomology class thus vanishes.
\end{propn}
\begin{proof}
If $M$ is compact, we may assume that $g$ has total volume 1, so that
its volume form $\Omega$ then represents an element of integer cohomology.
Since $\omega=\pi _{\ast}p^{\ast}\Omega$, its
cohomology class $[\omega ]=\pi _{\ast}p^{\ast}[ \Omega ]$
is therefore integral. If, on the other hand, $M$ is
non-compact, $\Omega =d\Upsilon$ for some
$(n-1)$-form $\Upsilon$, and hence $\omega = d(\pi _{\ast}p^{\ast}\Upsilon)$.
\end{proof}
\bigskip
\noindent
{\bf Acknowledgements.} The author would like to
thank L\'aszl\'o Lempert and
Edward Witten for their suggestions and encouragement.
|
1993-05-17T15:53:31 | 9305 | alg-geom/9305001 | en | https://arxiv.org/abs/alg-geom/9305001 | [
"alg-geom",
"math.AG"
] | alg-geom/9305001 | Kirti | Kirti Joshi | A General Noether-Lefschetz Theorem and applications | 30 pages, in LaTeX. replaced to correct earlier e-mail corruption | null | null | null | null | In this paper we generalize the classical Noether-Lefschetz Theorem to
arbitrary smooth projective threefolds. Let $X$ be a smooth projective
threefold over complex numbers, $L$ a very ample line bundle on $X$. Then we
prove that there is a positive integer $n_0(X,L)$ such that for $n \geq
n_0(X,L)$, the Noether-Lefschetz locus of the linear system $H^0(X,L^n)$ is a
countable union of proper closed subvarieties of $\P(H^0(X,L^n)^*)$ of
codimension at least two. In particular, the {\em general singular member} of
the linear system $H^0(X,L^n)$ is not contained in the Noether-Lefschetz locus.
As an application of our main theorem we prove the following result: Let $X$
be a smooth projective threefold, $L$ a very ample line bundle. Assume that $n$
is very large. Let $S=\P(H^0(X,L^n)^*)$, let $K$ denote the function field of
$S$. Let ${\cal Y}_K$ be the generic hypersurface corresponding to the sections
of $H^0(X,L^n)$. Then we show that the natural map on codimension two cycles
$$ CH^2(X_{\C}) \to CH^({\cal Y}_K) $$ is injective. This is a weaker version
of a conjecture of M. V. Nori, which generalises the Noether-Lefschetz theorem
on codimension one cycles on a smooth projective threefolds to arbitrary
codimension
| [
{
"version": "v1",
"created": "Mon, 3 May 1993 18:08:00 GMT"
},
{
"version": "v2",
"created": "Mon, 17 May 1993 13:53:21 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Joshi",
"Kirti",
""
]
] | alg-geom | \section{Introduction}
In this paper we generalize the classical Noether-Lefschetz
Theorem (see \cite{Lefschetz}, \cite{GH}) to arbitrary smooth
projective threefolds. More specifically, we prove that given any
smooth projective threefold $X$ over complex numbers and a very ample
line bundle $L$ on $X$, there is an integer $n_0(X,L)$ such that if $n
\geq n_0(X,L)$ then the Noether-Lefschetz locus of the linear
system $H^0(X,L^n)$ is a countable union of proper closed subvarieties
of $\P(H^0(X,L^n)^*)$ of codimension at least two. In particular, the
{\em general singular member} of the linear system $H^0(X,L^n)$ is not
contained in the Noether-Lefschetz locus. This generalizes the
results of \cite{GH}. In \cite{Madhav}, we find a
conjecture due to M.~V.~Nori, which generalizes the Noether-Lefschetz
theorem for codimension one cycles on smooth projective threefolds to
higher codimension cycles on arbitrary smooth projective varieties.
As an application of our main theorem we prove a result which can be
thought of as a weaker version of Nori's conjecture for {\em
codimension two cycles} on smooth projective threefolds.
The idea of the proof is borrowed from an elegant paper of
M.~Green (\cite{MarkGreen}) and the work of M.~V.~Nori (see
\cite{Madhav}). In \cite{MarkGreen} it was shown that the
classical Noether-Lefschetz theorem can be reduced to a coherent
cohomology vanishing result and the required vanishing was also proved
for $\P^3$. Though we have used this idea, we prove the required
vanishing by combining the techniques of \cite{Madhav} and of
\cite{MarkGreen}.
The paper is organized as follows. In the next section
(Section \ref{Generalities and notations}) we set up the basic
notations and terminology. Here we have also collected a few facts
which will be used throughout this paper. In Section~\ref{The
Noether-Lefschetz machine} we set up the basic technical machinery. In
this section, we reduce the Infinitesimal Noether-Lefschetz theorem to
certain coherent cohomology vanishing statements. This is the most
crucial part of the paper. Following the unpublished work of N. Mohan
Kumar and V. Srinivas (see \cite{Mohan-Srinivas}), in
Section~\ref{The Noether-Lefschetz Locus}, we show how the
Infinitesimal
Noether-Lefschetz theorem can be used to prove the global
Noether-Lefschetz theorem. The global Noether-Lefschetz theorem is
thus reduced to Infinitesimal Noether-Lefschetz theorems, which in
turn are deduced from coherent cohomology vanishing results. In
Section~\ref{A General Noether-Lefschetz Theorem}, we prove our main
result, Theorem~\ref{main theorem}. By the results of the
Sections~\ref{The Noether-Lefschetz machine} and \ref{The
Noether-Lefschetz Locus}, to prove our main theorem, we are reduced to
proving several cohomology vanishing results. This program is carried
out in Section~\ref{A General Noether-Lefschetz Theorem}. The
technique of the proofs of this section are based on Green's work
(see \cite{MarkGreen}) and a modification of this method due to
Paranjape (see \cite{KP}). Finally, in the last section we give an
application of our main theorem to codimension two cycles on smooth
projective threefolds.
This paper could not have been written without all the help
that we have received from N. Mohan Kumar and Kapil Paranjape. They
have generously shared their ideas and insights on the problems
considered in this paper; moreover N. Mohan Kumar suggested the
problem and also explained to us his unpublished work (with V.
Srinivas) and the work of Green. Kapil Paranjape patiently
explained to us the work of Nori and the subsequent
simplifications of Nori's work due to him. We would like to thank
both N.~Mohan Kumar and Kapil Paranjape for all their help and
encouragement (and also their patience) without which this paper might
not have been written.
We would also like to thank Madhav Nori and V.~Srinivas for
numerous conversations and suggestions.
\section{Generalities and notations}\label{Generalities and
notations}
Let $X/\C$ be a smooth projective threefold, $L$ an ample line
bundle on $X$, spanned by its global sections. For any linear system $
W \subset H^0(X,L) $ we will write $S=\P(W^*)$, and for any point
$s\in S$ we will write $T = \Spec({\cal O}_{\P(W^*)}/m^2_s)$, where $m_s$
is the maximal ideal of the point $s$. The base point free linear
system $H^0(X,L)$ gives a universal family of hypersurfaces with the
parameter space $\P(H^0(X,L)^*)$. Moreover, if $W\subset H^0(X,L)$ is
any sub linear system, then we can restrict the above universal family,
to the linear subspace $\P(W^*) \subset \P(H^0(X,L)^*)$, and get a
universal family of hypersurfaces ${\cal Y}$ parametrised by $\P(W^*)$.
Further, we will also write ${\cal X} = X \times \P(W^*)$. Note that we
have suppressed the dependence of ${\cal X},{\cal Y}$ and $T$ on $W$ and $s$. In
our discussions, it will be clear which $W,s$ the notation refers to,
and there will be no danger of confusion.
When the linear system $W$ is base point free, we can also describe
the universal family as follows. We denote by $M(L,W)$ the locally
free sheaf defined by the exact sequence
\begin{equation}
0 \to M(L,W) \to W \tensor {\cal O}_X \to L \to 0.
\end{equation}
Further as a notational convenience, if $W=H^0(X,L)$, we will write
$M(L) = M(L,W)$. Then ${\cal Y} = \P_X(M(L,W)^*)$. The dual of the exact
sequence above then gives an embedding ${\cal Y} \into {\cal X}$.
For any $S$-scheme $S'$, denote by ${\cal X}_{S'} = {\cal X} \times_{S} S'$. In
particular we have the ``universal infinitesimal deformations''
corresponding to a point $s\in S$, given by ${\cal X}_{T}, {\cal Y}_T$. Note that
for any $S' \to S$, we have an embedding ${\cal Y}_{S'} \into {\cal X}_{S'}$.
We have the projections $p_X : {\cal X} \to X$ and $p_S:{\cal X} \to S$. The
restrictions of these morphisms to ${\cal Y}$ will also be denoted by the
same symbols. Note that $p_S:{\cal Y} \to S$ is smooth over $s\in S$ if the
zero scheme of the section $s\in W$ is a smooth subscheme of $X$. We
will denote by $Y_s$ the zero scheme of $s$.
We also have on ${\cal X}$ (resp. on ${\cal Y}$) the de Rham complex
on ${\cal X}$ (resp. on ${\cal Y}$) denoted by
$\Omega_{\cal X}^{\scriptscriptstyle\bullet}$ (resp. $\Omega_{\cal Y}^{\scriptscriptstyle\bullet}$).
We define a complex
on ${\cal X}$, denoted by $\Omega_{({\cal X},{\cal Y})}^{\scriptscriptstyle\bullet}$ by the exact
sequence
\begin{equation}
0 \to \Omega_{({\cal X},{\cal Y})}^{\scriptscriptstyle\bullet} \to \Omega_{{\cal X}}^{\scriptscriptstyle\bullet} \to
\Omega_{\cal Y}^{\scriptscriptstyle\bullet} \to 0.
\end{equation}
The first two are complexes of locally free sheaves on ${\cal X}$ and the
latter is a complex of locally free sheaves on ${\cal Y}$.
Further as the natural projections ${\cal X} \to X$ and ${\cal Y} \to X$ are
smooth morphisms, we see that we also have a commutative diagram:
\label{main diagram}
\begin{equation}
\Matrix{
& & 0 & & 0 & & 0 & & \cr
& & \da & & \da & & \da & & \cr
0 &\to& p_X^*(\Omega_X^1) \tensor {\cal O}_{\cal X}(-{\cal Y}) &\to& \Omega^1_{({\cal X},{\cal Y})} &\to&
\Omega^1_{({\cal X},{\cal Y})/X}&\to&0\cr
& & \da & & \da & & \da & & \cr
0 &\to& p_X^*(\Omega^1_X) &\to& \Omega^1_{{\cal X}} &\to& \Omega^1_{{\cal X}/X}& \to
&
0\cr
& & \da & & \da & & \da & & \cr
0 &\to& p_X^*(\Omega^1_X)\big|_{\cal Y} &\to& \Omega^1_{{\cal Y}} &\to&
\Omega^1_{{\cal Y}/X}
& \to & 0\cr
& & \da & & \da & & \da & & \cr
& & 0
& & 0 & & 0 & & \cr
}
\end{equation}
Note that as ${\cal Y}$ is a divisor in ${\cal X}$, the sheaves
$\Omega^1_{({\cal X},{\cal Y})}$ and $\Omega^1_{({\cal X},{\cal Y})/X}$ are vector bundles on ${\cal X}$.
\begin{defn}\label{VNL}
We will say the Vanishing Noether-Lefschetz
condition is valid for $(X,L,W,s)$ where $W$ is a
linear system contained in $H^0(X,L)$,
and $s\in W$ if the following
assertion is valid:
$$H^2( X , \Omega^1_{({\cal X}_T,{\cal Y}_T)}\big|_{X\times s} ) = 0.$$
We will symbolically denote this hypothesis by $\mathop{\rm VNL}\nolimits(X,L,W,s)$.
\end{defn}
\begin{defn}
If $X,L$ are as above and $W\subset H^0(X,L)$ is any linear
system we will say that the Infinitesimal Noether-Lefschetz Theorem is
valid at $s\in W$ if in the commutative diagram
$$\Matrix{
\Pic(X) & \to & \Pic({\cal Y}_T) \cr
\searrow & & \swarrow \cr
& \Pic(Y_s) & \cr
}
$$
the equality $\image(\Pic({\cal Y}_T) \to \Pic(Y_s)) = \image(\Pic(X)\to
\Pic(Y_s) ) $ is valid.
We will denote this hypothesis by $\mathop{\rm INL}\nolimits(X,L,W,s)$.
\end{defn}
The key observation which relates these two definitions is the
following proposition, which is just a reformulation of Remark 3.10
of \cite{Madhav}.
\begin{propose}\label{vnl gives inl}
Let $X$ be a smooth projective threefold over complex numbers,
$L$ a very ample line bundle on $X$, $W\subset H^0(X,L)$ a base point
free linear system. If $s\in W$ cuts out a smooth divisor $Y_s$ on
$X$ and if $\mathop{\rm VNL}\nolimits(X,L,W,s)$ is valid then so is $\mathop{\rm INL}\nolimits(X,L,W,s)$.
\end{propose}
\begin{proof}
The morphism $p_S: {\cal X} \to S$ gives rise to an exact sequence
$$ 0 \la p_S^*\Omega^1_S \la \Omega^1_{\cal X} \la \Omega^1_{{\cal X}/S}
\la 0,$$
and a similar exact sequence for the morphism $p_S:{\cal Y} \to S$. Since
$Y_s$ is a smooth divisor on $X$, the morphism $p_S:{\cal Y}\to S$ is smooth
over $s\in S$. Hence we have the following commutative diagram:
\begin{equation}
\Matrix{
& & 0 & & 0 & & 0 & & \cr
& & \da & & \da & & \da & & \cr
0 &\to& p_S^*(\Omega_S^1) \tensor {\cal O}_X(-Y_s) &\to&
\Omega^1_{({\cal X},{\cal Y})}\big|_{X_s} &\to&
\Omega^1_{({\cal X},{\cal Y})/S}\big|_{X_s}&\to&0\cr
& & \da & & \da & & \da & & \cr
0 &\to& p_S^*(\Omega^1_S)\big|_{X_s} &\to&
\Omega^1_{{\cal X}}\big|_{X_s} &\to&
\Omega^1_{{\cal X}/S}\big|_{X_s}& \to & 0\cr
& & \da & & \da & & \da & & \cr
0 &\to& p_S^*(\Omega^1_S)\big|_{Y_s} &\to&
\Omega^1_{{\cal Y}}\big|_{Y_s} &\to&
\Omega^1_{{\cal Y}/S}\big|_{Y_s}& \to & 0\cr
& & \da & & \da & & \da & & \cr
& & 0 & & 0 & & 0 & & \cr
}
\end{equation}
This diagram gives rise to the following cohomology diagram:
\begin{equation}\let\scty=\scriptstyle
\Matrix{
& & \scty{H^1( X , \Omega_X^1)} & \scty{
\mapright{}}
& \scty{H^2( X , p_S^*(\Omega_S^1) \big|_{X_s})} \cr
& & \scty{\mapdown{\beta}} &
& \scty{\mapdown{}} \cr
\scty{H^1(Y_s,\Omega^1_{{\cal Y}}\big|_{Y_s})} & \scty{\mapright{}}
& \scty{H^1(Y_s,\Omega^1_{Y_s})} & \scty{
\mapright{\alpha}}
& \scty{H^2(Y_s,p_S^*(\Omega_S^1)\big|_{Y_s})} \cr
\scty{\mapdown{}} & & \scty{\mapdown{}} &
& \scty{\mapdown{}} \cr
\scty{H^2(X,\Omega^1_{({\cal X},{\cal Y})}\big|_{X_s})} & \scty{\mapright{}}
& \scty{H^2 ( X, \Omega^1_{({\cal X},{\cal Y})/S}\big|_{X_s})}
& \scty{\mapright{}}
& \scty{H^3( X , p_S^*(\Omega^1_S)\tensor {\cal O}_X(-Y_s))} \cr
}
\end{equation}
This diagram is obtained from the previous diagram by taking
the long exact cohomology sequence and noting that we have the
identifications:
\begin{eqnarray*}
\Omega^1_{{\cal X}/S}\big|_{X_s} & \isom & \Omega^1_{X_s} \\
\Omega^1_{{\cal Y}/S}\big|_{Y_s} & \isom & \Omega^1_{Y_s}
\end{eqnarray*}
By hypothesis, the first term on the bottom row (from the
left) is zero (this vanishing is just $\mathop{\rm VNL}\nolimits(X,L,W,s)$). Then it
follows from the injectivity of the bottom row map that $\ker(\alpha)
\subset \image(\beta)$. Further note that one has the trivial
identifications:
\begin{eqnarray*}
H^2(X_s,p_S^*(\Omega^1_S)\big|_{X_s}) & \isom &
H^2(X,{\cal O}_X)\tensor \Omega^1_S\tensor k(s) \\
H^2(Y_s,p_S^*(\Omega^1_S)\big|_{Y_s}) & \isom &
H^2(Y_s,{\cal O}_{Y_s})\tensor \Omega^1_S \tensor
k(s)
,
\end{eqnarray*}
where $k(s)$ denotes the residue field of $s$. Using these
identifications, one notes that the top two rows of this diagram can
be identified with the ``Kodaira-Spencer maps'' for the infinitesimal
deformations ${\cal X}_T$ and ${\cal Y}_T$ respectively (see \cite{CGGH})
(strictly speaking this identification above should be carried
out on $T$).
Since $\ker(\alpha) \subset \image(\beta)$, we see that any
any class $c\in H^1(Y_s,\Omega^1_{Y_s})$ which deforms infinitesimally
is the image of a class in $H^1(X,\Omega^1_X)$. Further as $Y_s$ is
the zero scheme of a section of a very ample line bundle, one has a
surjection $\Pic^0(X) \to \Pic^0(Y_s)$ (this follows easily from
Kodaira vanishing theorem). Moreover, it is easy to see, using the
exponential sequence, that the homomorphism $\Pic(X) \to \Pic(Y_s)$
has torsion free cokernel. So that we see at once that in the
commutative diagram
$$
\Matrix{
\Pic(X) & \to & \Pic({\cal Y}_T) \cr
\searrow & & \swarrow \cr
& \Pic(Y_s) & \cr
}
$$
any line bundle which is in the image of the map $\Pic({\cal Y}_T) \to
\Pic(Y_s)$, comes from $\Pic(X)$. Thus we have proved the proposition.
\end{proof}
This proposition, though a technical assertion, is a crucial result
from the point of view of this paper. In the next section we use this
result to systematically reduce Infinitesimal Noether-Lefschetz
Theorem to a coherent cohomology vanishing which can be checked in practice.
\section{The Noether-Lefschetz machine}\label{The Noether-Lefschetz machine}
The condition $\mathop{\rm VNL}\nolimits(X,L,W,s)$ is now cast into a more
manageable form. The technique which is employed in the proof of the
following proposition is going to be applied repeatedly in different
contexts throughout this section and hence in the subsequent proofs of
this section, we will give the important points and leave the details
to the reader.
\begin{propose}\label{base free and smooth}
Let $(X,L,W,s)$ be such that $W\subset H^0(X,L)$ is a base
point free linear system and $s\in W$ cuts out a smooth divisor on
$X$. Assume that
the following assertions are valid:
\begin{description}
\item[{\rm(\thepropose.1)}] $H^1( X , \Omega^2_X \tensor L ) = 0$,
a
nd
\item[{\rm(\thepropose.2)}] $H^1( X , M(L,W) \tensor K_X \tensor L
) = 0 $.
\end{description}
Then $\mathop{\rm VNL}\nolimits(X,L,W,s)$ is valid.
\end{propose}
\begin{proof}
We closely follow the techniques of proof in \cite{Madhav}.
Indeed, we make repeated application of this method throughout this
section.
Since the morphisms $p_X:{\cal X} \to X$ and $p_X\big|_{\cal Y}: {\cal Y} \to
X$ are smooth we have the following exact sequence of vector bundles
on ${\cal X}$:
\begin{equation}
0 \to p_X^*(\Omega^1_X)\tensor {\cal O}_{\cal X}(-{\cal Y}) \to
\Omega^1_{({\cal X},{\cal Y})} \to \Omega^1_{({\cal X},{\cal Y})/X} \to 0.
\end{equation}
This exact sequence is just the top row of the commutative
diagram (\ref{main diagram}). Further, note that we can restrict this
exact sequence to $X_s$ and get the exact sequence:
\begin{equation}
0 \to p_X^*(\Omega^1_X) \tensor {\cal O}_{\cal X}(-{\cal Y}) \big|_{X_s}
\to \Omega^1_{({\cal X},{\cal Y})} \big|_{X_s} \to
\Omega^1_{({\cal X},{\cal Y})/X}\big|_{X_s} \to 0.
\end{equation}
Thus to prove that the middle term of the above exact sequence
has no $H^2$, it suffices to prove that the extreme terms have no
$H^2$. Now as ${\cal O}_{\cal X}(-{\cal Y})\big|_{X_s} = L^{-1}$,we see that
\begin{equation}
H^2(X , p_X^*(\Omega^1_X)\tensor{\cal O}_{\cal X}(-{\cal Y})\big|_{X_s}) = H^2(X ,
\Omega^1_X\tensor L^{-1}).
\end{equation}
And so the vanishing of this is just the hypothesis
\specialref{base free and smooth}{1}, by Serre duality.
Now we prove that the
second term also has no $H^2$. This is done by the following:
\begin{claim}\label{from Kapils paper}
$$\Omega^1_{({\cal X},{\cal Y})/X}\big|_{X_s} = M(L,W)^* \tensor L^{-1}.$$
\end{claim}
\begin{proof}
This fact is easily verified, it also follows from an
explicit resolution of the sheaves $\Omega^i_{({\cal X},{\cal Y})/X}$ constructed
in \cite{KP}.
\end{proof}
Now the second vanishing is just our hypothesis
\specialref{base free and smooth}{2}, after applying Serre duality.
This proves Proposition \ref{base free and smooth}.
\end{proof}
Before we proceed further we need some notations. If $x\in X$
is any point, we denote by ${\tilde X}_x$ the blowup of $X$ along $x$;
$\pi_x:{\tilde X}_x \to X$ the blowup morphism and $E_x$ the exceptional
divisor. We write $L'_x = \pi_x^*(L)\tensor{\cal O}_{{\tilde X}_x}(-E_x)$.
Frequently, when there is no chance of confusion, we will
suppress the subscript $x$ from the above notations.
Suppose $W\subset H^0(X, m_x \tensor L)=H^0({\tilde X}_x,L'_x)$ is a subspace.
We assume that the linear system $W$ is base point free
on ${\tilde X}$. By Proposition \ref{base free and smooth},
$\mathop{\rm INL}\nolimits({\tilde X},L'_x,W,s)$ is reduced to a vanishing on ${\tilde X}$. The next
proposition reduces the vanishing on ${\tilde X}$ to a coherent cohomology
vanishing on $X$.
\begin{propose}\label{base point and smooth}
Let $(X,L,W,s)$ be as above. Suppose
the following assertions are valid:
\begin{description}
\item[{\rm(\thepropose.1)}] $H^1( X ,
(\pi_x)_*(\Omega^2_{{\tilde X}_x}(-2E_x))\tensor L) =0$, and
\item[{\rm(\thepropose.2)}] \label{b-p-s-2} $H^1( X , (\pi_x)_*
(M(L'_x,W))\tensor K_X \tensor L) = 0$.
\end{description}
Then $\mathop{\rm VNL}\nolimits({\tilde X}_x , L'_x , W, \tilde{s})$ is valid.
\end{propose}
\begin{proof}
The proof is similar to the one given earlier, though it is a
bit more involved. For the purpose of the proof let us use the
following notations: we write ${\cal X} = {\tilde X}_x \times S$, $S=\P(W^*)$,
${\cal Y}=\P_{{\tilde X}_x}(M(L'_x,W)^*)$ and $D=E_x
\times S$. Note that $D$ is a divisor on ${\cal X}$ and that ${\cal O}_{\cal X}(D) =
p_{{\tilde X}_x}^*({\cal O}_{{\tilde X}_x}(E_x))$.
We have the following exact sequence on ${\cal X}$.
$$ 0 \to \Omega^1_{({\cal X},{\cal Y})} \to \Omega^1_{({\cal X},{\cal Y})}(D) \to
\Omega^1_{({\cal X},{\cal Y})} (D) \big|_D \to 0.$$
Now we can restrict this exact sequence to the fibre
${\cal X}_s={\tilde X} \times \{\tilde s\}$ of the projection morphism ${\cal X} \to
S$ over $\tilde s\in S$. Thus we have an exact sequence
$$ 0 \to \Omega^1_{({\cal X},{\cal Y})}\big|_{{\cal X}_s} \to
\Omega^1_{({\cal X},{\cal Y})}(D)\big|_{{\cal X}_s} \to
\Omega^1_{({\cal X},{\cal Y})} (D) \big|_D\big|_{{\cal X}_s} \to 0.$$
Thus to prove that the first term on the left has no $H^2$
(that is to say $\mathop{\rm VNL}\nolimits({\tilde X}_x , L'_x , W, \tilde{s})$), it
suffices to prove the following assertions:
\begin{description}
\item[{\rm(\thepropose.3)}] $H^2({\cal X}_s,
\Omega^1_{({\cal X},{\cal Y})}(D)\big|_{{\cal X}_s} ) = 0$, and
\item[{\rm(\thepropose.4)}] $H^1( {\cal X}_s,
\Omega^1_{({\cal X},{\cal Y})}(D)\big|_{D}\big|_{{\cal X}_s} )= 0$.
\end{description}
These are proved in Lemma \ref{proof of 3.2.3} and Lemma
\ref{proof of 3.2.4}.
\end{proof}
\begin{lemma}\label{proof of 3.2.3}
The hypotheses of Proposition \ref{base point and smooth} imply
\specialref{base point and smooth}{3}.
\end{lemma}
\begin{proof}
First note that $D\big|_{{\cal X}_s} = E_x$ (recall that $D=E_x\times
S$, so this equality is obvious). Now we have the exact sequence
$$ 0 \to p_{{\tilde X}}^*(\Omega^1_{{\tilde X}}) \tensor {\cal O}_X(-{\cal Y}) \to
\Omega^1_{({\cal X}, {\cal Y})} \to \Omega^1_{({\cal X},{\cal Y})/{\tilde X}} \to 0.$$
Twisting this exact sequence by ${\cal O}_{\cal X}(D)$ we get
$$\scriptstyle{ 0 \to p_{{\tilde X}_x}^*(\Omega^1_{{\tilde X}_x}) \tensor {\cal O}_X(-{\cal Y})
\tensor {\cal O}_{\cal X}(D) \to
\Omega^1_{({\cal X}, {\cal Y})} \tensor {\cal O}_{\cal X}(D) \to
\Omega^1_{({\cal X},{\cal Y})/{\tilde X}_x} \tensor {\cal O}_{\cal X}(D) \to 0.}$$
Now restricting to ${\cal X}_s$, we have
$$ 0 \to p_{{\tilde X}}^*(\Omega^1_{{\tilde X}}(E_x)) \tensor {\cal O}_{\cal X}(-{\cal Y})\big|_{{\cal X}_s} \to
\Omega^1_{({\cal X},{\cal Y})}(D)\big|_{{\cal X}_s} \to
\Omega^1_{({\cal X},{\cal Y})/{\tilde X}}(D) \big|_{{\cal X}_s} \to 0,$$
where we have used the fact that ${\cal O}_{\cal X}(D) =
p_{{\tilde X}}^*({\cal O}_{{\tilde X}}(E_x))$ in the term on the left. So the middle term
has no $H^2$ if the extreme terms have no $H^2$. So we have to check
that the hypotheses of the Proposition \ref{base point and smooth}
ensure this.
Note that ${\cal O}_{\cal X}(-{\cal Y}) \big|_{{\cal X}_s} = L'^{-1}_x$, and as
$p^*_{\tilde X}(\Omega^1_{\tilde X} (E_x))\big|_{{\cal X}_s} = \Omega^1_{{\tilde X}}(E_x)$, we
have to prove that the hypotheses of the proposition imply that
\begin{description}
\item[{\rm(\thepropose.5)}] $H^2({\tilde X}, \Omega^1_{\tilde X}(E_x) \tensor
L'^{-1}_x) = 0$, and
\item[{\rm(\thepropose.6)}] $H^2({\tilde X}
,\Omega^1_{({\cal X},{\cal Y})/{\tilde X}}(D)\big|_{{\cal X}_s}) = 0 $.
\end{description}
We will use the Leray spectral sequence to prove that the hypothesis
\specialref{base point and smooth}{1} will imply
\specialref{proof of 3.2.3}{5} and hypothesis
\specialref{base point and smooth}{2} will imply
\specialref{proof of 3.2.3}{6}. This will
prove Lemma \ref{proof of 3.2.3}.
\end{proof}
\begin{sublem}\label{sublemma1}
Hypothesis \specialref{base point and smooth}{1} implies
\specialref{proof of 3.2.3}{5}.
\end{sublem}
\begin{proof}
By Serre duality we are reduced to proving:
$$H^1( {\tilde X},
\Omega^2_{\tilde X}(-E_x) \tensor L'_x ) = 0.$$
Now by Leray spectral sequence for $\pi_x:{\tilde X} \to X$, we
see that the above $H^1$ is vanishes if:
\begin{description}
\item[{\rm(\thepropose.7)}] $H^1(X , (\pi_x)_*(
\Omega^2_{\tilde X}(-E_x) \tensor L'_x)) = 0$, and
\item[{\rm(\thepropose.8)}] $R^1(\pi_x)_*(\Omega^2_{\tilde X}(-E_x)
\tensor L'_x) = 0$.
\end{description}
Note that by the projection formula,
$$(\pi_x)_*(\Omega^2_{\tilde X}(-E_x)
\tensor L'_x) = (\pi_x)_*(\Omega^2_{\tilde X}(-2E_x)) \tensor L.$$
Hence vanishing of \specialref{sublemma1}{7} above is
implied by hypothesis \specialref{base point and smooth}{1}. So
we have to check that $R^1$ is also zero. By projection formula, it
suffices to prove the following:
\begin{equation}\label{whats in a name}
R^1(\pi_x)_*(\Omega^2_{\tilde X}(-2E_x)) = 0.
\end{equation}
Since the fibres of $\pi_x$ have dimension $\leq 2$, $R^2$ satisfies base
change.
Since the $R^2$ is supported on the point $x$, we can compute the
$R^2$ by restricting to $E_x$. Providing we show that $R^2$ is 0, we
get that $R^1$ satisfies base change. Then $R^1$ is also supported on
the point $x$ and can be computed by restricting to $E_x$.
On $E_x$ we have the fundamental exact
sequence
$$ 0 \to {\cal O}_{E_x}(1) \to \Omega^1_{\tilde X}\big|_{E_x} \to
\Omega^1_{E_x} \to 0.$$
An easy calculation shows that this exact sequence splits. So that we have
$$\Omega^1_{\tilde X}\big|_{E_x} = \Omega^1_{E_x} \oplus
{\cal O}_{E_x}(1).$$
So that by taking exteriors, we have
$$\Omega^2_{\tilde X}\big|_{E_x} = \Omega^2_{E_x} \oplus
\Omega^1_{E_x}(1).$$
Hence we have
$$\Omega^2_{\tilde X}(-2E_x)\big|_{E_x} =
\Omega^2_{\tilde X}\big|_{E_x}\tensor {\cal O}_{E_x}(2) =
\Omega^2_{E_x}(2) \oplus \Omega^1_{E_x}(3).$$
Since $E_x \isom \P^2$, we see easily that $H^1$ and $H^2$ of the
latter sheaf vanish.
\end{proof}
\begin{sublem}
Hypothesis \specialref{base point and smooth}{2} implies
\specialref{proof of 3.2.3}{6}
\end{sublem}
\begin{proof}
We want to prove that
$$H^2( {\tilde X} , \Omega^1_{({\cal X},{\cal Y})/{\tilde X}}(D)\big|_{{\cal X}_s}) = 0. $$
Firstly we recall that $\Omega^1_{({\cal X},{\cal Y})/{\tilde X}} \big|_{{\cal X}_s} =
M(L'_x, W)^* \tensor L'^{-1}_x$ (this is just the formula \ref{from
Kapils paper}).
Further ${\cal O}_{\cal X}(D)\big|_{{\tilde X}} = {\cal O}_{\tilde X}(E_x)$. So that we
have to show that
$$ H^2( {\tilde X} , M(L'_x,W)^* \tensor {\cal O}_{\tilde X}(E_x) \tensor
L'^{-1}_x ) = 0.$$
By Serre duality, we have to prove the following vanishing:
$$H^1({\tilde X} , M(L'_x,W) \tensor {\cal O}_{\tilde X}(-E_x) \tensor K_{\tilde X}
\tensor L'_x ) = 0.$$
This is done as before by a Leray spectral sequence argument. We
observe that $ H^1( X , (\pi_x)_*( M(L'_x,W) \tensor {\cal O}_{\tilde X}(-E_x)
\tensor K_{\tilde X} \tensor L'_x)) = 0 $ by the hypothesis \specialref{base
point and smooth}{2}. Thus it suffices to prove that
$$R^1(\pi_x)_*(M(L'_x,W) \tensor {\cal O}_{\tilde X}(-E_x) \tensor K_{\tilde X}
\tensor L'_x) = 0.$$
Using the projection formula and the fact that $K_{\tilde X} = \pi_x^*(K_X)
\tensor {\cal O}_{\tilde X}(2E_x)$, we see that it suffices to prove that
$$R^1(\pi_x)_*(M(L'_x,W)) = 0.$$
As before we will show vanishing of $H^1$ and $H^2$ after
after restricting to $E_x$. This will imply the vanishing of $R^1$ and
$R^2$ as before. Since we have the exact sequence
$$ 0 \to M(L'_x , W) \to W\tensor {\cal O}_{\tilde X} \to L'_x \to 0, $$
restricting this exact sequence to $E_x$, by the universal property
of the tautological bundle ${\cal O}_{E_x}(1)$, we see that we have:
$$M(L'_x,W)\big|_{E_x} = \Omega^1_{E_x}(1) \oplus V' \tensor
{\cal O}_{E_x},$$
for some subspace $V' \subset W$. Now since $E_x\isom\P^2$
one notes that the $H^1$ and $H^2$ of the latter sheaf vanish.
\end{proof}
Finally, to complete the proof of Proposition \ref{base point
and smooth}, it remains to prove \specialref{base point and smooth}{4}:
\begin{lemma}\label{proof of 3.2.4}
$$H^1({\cal X}_s, \Omega^1_{({\cal X},{\cal Y})}(D)\big|_{D}\big|_{{\cal X}_s}) =0.$$
\end{lemma}
\begin{proof}
Observe that as $D=E_x \times S$, we have $D\big|_{{\cal X}_s} =E_x$
and hence $D\big|_{E_x} = {\cal O}_{E_x}(-1)$. We have the
divisor $Z = E_x\times S\cap {\cal Y} = D\cap {\cal Y}$ in $D$.
Note that $D$ is just a product of $\P^2$
and a projective space. Thus the vanishing to be proved is reduced to
a vanishing result on $\P^2$. Further note that by definition $Z =
\P_{E_x}(M(L'_x,W)^*\big|_{E_x})$. We have a natural commutative
diagram
$$\let\scty=\scriptstyle
\Matrix{
& & \scty{0} &&\scty{0} & & \scty{0} & & \cr
& & \scty{\da} &&\scty{\da} & &\scty{\da} & & \cr
\scty{0} & \scty{\to} &\scty{{\cal O}_D(-Z)} &\scty{\to}&\scty{{\cal O}_D} &
\scty{\to} &\scty{{\cal O}_Z} &\scty{\to} & \scty{0}\cr
& & \scty{\da} && \scty{\da} & & \scty{\da} & & \cr
\scty{0} & \scty{\to} & \scty{\Omega^1_{({\cal X},{\cal Y})}(D)\big|_D} &\scty{\to}&
\scty{\Omega^1_{\cal X}(D)\big|_D}
&\scty{\to} &\scty{\Omega^1_{\cal Y}(D)\big|_D} &
\scty{\to} & \scty{0}\cr
& & \scty{\da} && \scty{\da} & & \scty{\da} & & \cr
\scty{0} & \scty{\to} & \scty{\Omega^1_{(D,Z)}\tensor {\cal O}_{E_x}(-1)} &
\scty{\to}&
\scty{\Omega^1_D\tensor
{\cal O}_{E_x}(-1)} &\scty{\to} &\scty{\Omega^1_Z\tensor {\cal O}_{E_x}(-1)} &
\scty{\to} & \scty{0}\cr
& & \scty{\da} && \scty{\da} & & \scty{\da} & & \cr
& &\scty{0} && \scty{0} & & \scty{0} & & \cr
}
$$
Further restricting the right column to ${\cal X}_s = {\tilde X}\times \{s\}$ we see that
the middle term
$H^1({\tilde X} , \Omega^1_{({\cal X},{\cal Y})}(D)\big|_D\big|_{{\cal X}_s}) = 0$
if
\begin{description}
\item[{\rm(\thepropose.9}] $H^1(E_x , {\cal O}_D(-Z)\big|_{{\cal X}_s}) =0$, and
\item[{\rm(\thepropose.10)}] $H^1( E_x , \Omega^1_{(D,Z)}\big|_{E_x}
\tensor
{\cal O}_{E_x}(-1)) = 0$.
\end{description}
Note that as $E_x \isom \P^2$, and as ${\cal O}_D(-Z)\big|_{E_x}$ is a line
bundle on $E_x$, its $H^1$ is trivially zero. So we have to prove
\specialref{proof of 3.2.4}{10}. To do this we proceed as follows: We
have the exact sequence (from the definition)
$$\Matrix{
0 &\to& \Omega^1_{(D,Z)}& \to &\Omega^1_D &\to& \Omega^1_Z& \to& 0\cr
& & & & \Vert & & & &\cr
& & & & \Omega^1_{E_x}\oplus \Omega^1_S & & & &\cr
}
$$
So restricting to $E_x$, as $\tilde{s}$ cuts out a smooth divisor
on ${\tilde X}$, $Z=E_x\cap {\cal Y}$ is a $\P^1$-bundle over $S$
in a Zariski neighbourhood of $\tilde s$.
Then we have the exact sequence
$$
0 \to \Omega^1_{(D,Z)}\big|_{E_x} \to \Omega^1_D\big|_{E_x} \to
\Omega^1_Z \big|_{E_x} \to 0.$$
Now $\Omega^1_D\big|_{E_x}=\Omega^1_{E_x}\oplus
\Omega^1_{S,s}\tensor{\cal O}_{E_x}$, and
similarly we have
$\Omega^1_Z\big|_{E_x} = \Omega^1_{\P^1} \oplus
\Omega^1_{S,s}\tensor{\cal O}_{E_x}$.
This gives an exact sequence
$$\displaylines{
0 \to (\Omega^1_{(\P^2,\P^1)} \oplus \Omega^1_{S,s}\tensor{\cal O}_{E_x})
\tensor {\cal O}_{E_x}(-1) \to
\Omega^1_{\P^2}(-1) \oplus
\Omega^1_{S,s}\tensor {\cal O}_{E_x}(-1)
\to \hfill\cr
\hfill{}\qquad\to \Omega^1_{\P^1}(-1) \oplus
\Omega^1_{S,s} \tensor {\cal O}_{E_x}(-1)\to 0.\hfill\cr
}
$$
Then taking long exact cohomology sequence we have:
$$\displaylines{
H^0(\P^1, \Omega^1_{\P^1}(-1)) \oplus H^0(\P^2,\Omega^1_{S,s}
\tensor {\cal O}_{E_x}(-1)) \to \hfill\cr
\hfill{}\qquad \to H^1( (\Omega^1_{(\P^2,\P^1)}
\oplus \Omega^1_{S,s}\tensor{\cal O}_{E_x})\tensor
{\cal O}_{E_x}(-1)) \to \hfill\cr
\hfill{} \qquad\qquad \to H^1(\Omega^1_{\P^2}(-1))
\oplus H^1(\Omega^1_{S,s}\tensor{\cal O}_{E_x}(-1) ).\hfill\cr
}
$$
And the terms on the extreme vanish so that the middle term vanishes.
Hence we are done. This finishes the proof of Lemma \ref{proof
of 3.2.4}
\end{proof}
For the next proposition, we need some more notation. Let
$$ L''_x = (\pi_x)^*(L)\tensor{\cal O}_{\tilde X}(-2E_x).$$
Observe that image, under $\pi_x$ of smooth sections of
$H^0({\tilde X},L''_x)$, are sections in $H^0(X,L)$ which have a single
ordinary double point at $x$. We now want to reduce the condition
$\mathop{\rm VNL}\nolimits({\tilde X},L''_x,H^0({\tilde X},L''_x),s)$ for smooth $s\in
S=\P(H^0({\tilde X},L''_x)^*)$, to a more manageable form. This is done by
the following.
\begin{propose}\label{base point and ODP}
Assume that $L$ is such that $L''_x$ is very ample on ${\tilde X}$.
Let $s\in W=H^0({\tilde X}, L''_x)$ be a section which cuts out a smooth
divisor on ${\tilde X}$. Assume that the following assertions are valid:
\begin{description}
\item[{\rm(\thepropose.1)}] $H^1(X, L\tensor m_x^3) = 0$,
where $m_x$ is the maximal ideal of $x\in X$.
\item[{\rm(\thepropose.2)}] $H^1( X ,
(\pi_x)_*(\Omega^2_{{\tilde X}_x}(-2E_x)) \tensor L) = 0$, and
\item[{\rm(\thepropose.3)}] $H^1( X , (\pi_x)_*( M(L_x'') )
\tensor K_X \tensor L ) = 0$.
\end{description}
Then $\mathop{\rm VNL}\nolimits({\tilde X},L''_x,H^0({\tilde X},L''_x),s)$ is valid.
\end{propose}
\begin{proof}
Since the linear system $W$ is base point free on ${\tilde X}$, and as
the section $s\in W$ cuts out a smooth divisor on ${\tilde X}$, we see that
Proposition \ref{base free and smooth} can be applied. By this
proposition, it suffices to prove the following assertions:
\begin{description}
\item[{\rm(\thepropose.4)}] $H^1({\tilde X}, \Omega^2_{\tilde X} \tensor L''_x ) = 0$,
and
\item[{\rm(\thepropose.5)}] $H^1( {\tilde X} , M(L''_x,W) \tensor K_{\tilde X} \tensor
L''_x ) =0$.
\end{description}
Thus we have to prove that the hypothesis of the proposition ensure
the vanishing \specialref{base point and ODP}{4} and
\specialref{base point and ODP}{5}. The argument is similar to the
one earlier. Firstly, let us observe that the implication
\specialref{base point and ODP}{2} implies \specialref{base
point and ODP}{4} follows from \ref{sublemma1}. By a Leray
spectral sequence argument, we are reduced to proving
$$R^1(\pi_x)_*(\Omega^2_{\tilde X}(-2E_x)) = 0.$$
But this has been proved during the proof of
Sublemma\ref{sublemma1} as equation (\ref{whats in a name}).
Thus we are done in this case. So it
remains to prove that \specialref{base point and ODP}{1} and
\specialref{base point and ODP}{3} together imply
\specialref{base point and ODP}{5}.
First we check that $R^1(\pi_x)_*(M(L''_x,W)) = 0$. This can be done
by restricting to $E_x$. On $E_x$ we have an exact sequence:
$$ 0 \to M(L''_x,W)\big|_{E_x} \to W\tensor {\cal O}_{E_x} \to L''_x
\big|_{E_x} \to 0.$$
So on noting that $W= H^0({\tilde X},L''_x)=H^0(X,L\tensor m_x^2)$, the
result now follows from \specialref{base point and ODP}{1}
and \specialref{base point and ODP}{3}.
\end{proof}
\section{The Noether-Lefschetz Locus}\label{The Noether-Lefschetz Locus}
For any smooth projective threefold $X$, and $L$ be a very
ample line bundle on $X$, we say smooth member $Y$ of $H^0(X,L)$ lies
in the ``Noether-Lefschetz locus of the linear system $H^0(X,L)$'' if
$\Pic(X)\to \Pic(Y)$ is not a surjection. More generally, if $Y$ is
normal we say $Y$ lies in the ``Noether-Lefschetz locus'' if $\Pic(X)
\to CH^1(Y)$ is not a surjection.
Now we deduce the global Noether-Lefschetz theorem from the
infinitesimal results.
\begin{propose}\label{codim one}
Suppose $X$ is a smooth projective threefold over $\C$ and $L$ is
a very ample line bundle over $X$. Assume that $\mathop{\rm INL}\nolimits(X,L,H^0(X,L),s)$
is valid for all $s\in H^0(X,L)$ which cut out a smooth divisor on
$X$. Then the Noether-Lefschetz locus for the linear system
$H^0(X,L)$ has codimension $\geq 1$ in $\P(H^0(X,L)^*)$.
\end{propose}
\def\bar{K}{\bar{K}}
\begin{proof}
We argue as in \cite{Mohan-Srinivas}. We can assume that the $X,
L$ are defined over a finitely generated field $K/\Q$. Let $\bar{K}$
denote the algebraic closure of $K$. Let $\bar{K}(\eta)$ denote the
rational function field in $\dim H^0(X,L)$ variables over $\bar{K}$. Then
this defines a point $\eta \into S=\P(H^0(X,L)^*)$, which we will call
the $\bar{K}$-generic point of $S$. Let ${\overline\eta} \in S=\P(H^0(X,L)^*)$ be
the corresponding $\bar{K}$-geometric generic point of $S$. Then we claim
that $\Pic(X_{\overline\eta}) \isom \Pic(Y_{\overline\eta})$.
The claim is proved as follows. Suppose that it is not an
isomorphism, then in particular the map is not surjective. Let
$\alpha\in\Pic(Y_{\overline\eta})$ be any line bundle not in the image of the
map. Then this cycle is defined over a finite field extension of the
function field $\bar{K}(\eta)$. Then there exists an \'etale open set $U
\to S$ such that $\alpha$ ``spreads'' out to a line bundle on ${\cal Y}_U$.
By replacing $U$ by a smaller non-empty open subset if necessary, we
can further assume that $\alpha$ is a nontrivial element of
$\Pic({\cal Y}_U)$. Now we restrict $\alpha$ to the fibres of ${\cal Y}_U \to U$.
By shrinking $U$ further we can assume that image of $U$ in $S$ is an
open set which does not meet the discriminant locus of the linear
system $H^0(X,L)$ in $S$. Thus we can now apply the Infinitesimal
Noether-Lefschetz to the fibres over $U$. On the fibres of ${\cal Y}_U \to
U$, as $\mathop{\rm INL}\nolimits(X,L,H^0(X,L),s)$ is valid, we see that the restriction of
$\alpha$ to the fibres over $u\in U$ is zero in $\Pic(Y_u)$ because
infinitesimally there are no extra cycles (note that we have used the
fact that we can identify ${\cal O}_U/m_u^2 \isom {\cal O}_S/m_s^2$, where $s$ is
the image of $u$ under the map $U\to S$). Then by the semi-continuity
theorem we see that the line bundle $\alpha$ on ${\cal Y}_U$ must be
trivial. This proves the claim.
Now it remains to prove that the Noether-Lefschetz locus has
codimension $\geq 1$. We proceed as in \cite{Mohan-Srinivas}. Let
$U$ be the subset of $S$ obtained by removing all the divisors of
$S$, which are defined over $\bar{K}$. Thus we have removed a
countable set of closed subvarieties. If $p\in U$ is a closed point
the map $ \Spec(\C(p)) \into U$ factors through the geometric generic
point ${\overline\eta}$. Further as ${\cal Y}_{\overline\eta}$ is the zero scheme of a section
of an ample line bundle on $X_{\overline\eta}$, we see as $H^1(X_{\overline\eta},
{\cal O}_{X_{\overline\eta}}) \isom H^1(Y_{\overline\eta},{\cal O}_{Y_{\overline\eta}})$. We see that
$\Pic(Y_p)(\bar{K})\isom \Pic(Y_{\overline\eta})(\bar{K})$. This proves the claim.
\end{proof}
An identical argument, with divisors on $S$ which are not
contained in the support of the discriminant locus, we can prove the
following:
\begin{propose}\label{codim two in smooth}
If {}$\mathop{\rm INL}\nolimits(X,L,W,s)$ is valid for all $W$ of codimension one in
$H^0(X,L)$ and $s\in W$ smooth. Then ``smooth part'' of the
Noether-Lefschetz locus has codimension $\geq 2$ in $S$.
\end{propose}
\begin{proof}
Since the proof is almost identical to the one given earlier, we
will only indicate the important point. The idea is to work with
divisor of $D$ which is not contained in the discriminant locus. For
such a divisor, we prove by an argument identical to the one given
above that for the $\bar{K}[A$-geometric generic point ${\overline\eta}_D$ of $D$, we
have an isomorphism $\Pic(X_{{\overline\eta}_D}) \isom \Pic(Y_{{\overline\eta}_D})$. This
is proved as before, except that one has to use $\mathop{\rm INL}\nolimits$ for a
codimension one linear system. For which we use the following Lemma.
Assuming the Lemma for the moment, we complete the proof as follows.
Consider the subset of $D$ obtained by removing all the codimension
one subvarieties of $D$ which are defined over $\bar{K}$. Then the
one argues as in the proof of Proposition~\ref{codim one}.
\end{proof}
\begin{lemma} \label{vnl and inl on the blowup}
Let $(X,L,W,s)$ be such that $W\subset H^0(X,L)$ is linear system
of codimension one with a single base point at $x\in X$. Further
assume that $s$ cuts out a smooth divisor $Y_s$ in $X$. Then we have
$$ \mathop{\rm INL}\nolimits({\tilde X}_x,L'_x,W,\tilde s) \mbox{$\Rightarrow$} \mathop{\rm INL}\nolimits(X,L,W,s).$$
\end{lemma}
\begin{proof}
We use the following commutative diagrams:
$$
\Matrix{
\Pic({\tilde X}_x)&& \mapright{f_2} && \Pic(\tilde{{\cal Y}_T}) \cr
h_2 \searrow && && \swarrow g_2 \cr
&& \Pic({\tilde Y}_s) && \cr
\mapup{\tau_1} && \mapup{\tau_2} && \mapup{\tau_3} \cr
&& \Pic(Y_s) && \cr
h_1 \nearrow && && \nwarrow g_1 \cr
\Pic(X) && \mapright{f_1} && \Pic({\cal Y}_T) \cr
}
$$
And one also has the following commutative diagram:
$$
\Matrix{
0 & \to & \Pic(X) & \mapright{\tau_1} & \Pic({\tilde X}_x) &
\mapright{\tau_1'}& \Z & \to & 0 \cr
& & \mapdown{h_1} & & \mapdown{h_2}&
&\Big\Vert& & \cr
0 & \to & \Pic(Y_s) & \mapright{tau_2}& \Pic({\tilde Y}_s) &
\mapright{\tau_2'} & \Z & \to & 0 \cr
}
$$
As $\mathop{\rm INL}\nolimits({\tilde X}_x,L'_x,W,s)$ is valid, we have
$\image{(h_2)}=\image{(g_2)}$. We want to prove that
$\image{(h_1)}=\image{(g_1)}$. Since the lower triangle commutes, we
have the inclusion $\image{(h_1)} \subset \image{(g_1)}$. So we will
prove the reverse inclusion. Let $\alpha\in \Pic({\cal Y}_T)$, as $\tau_2
g_1 = g_2\tau_3$ we have $\tau_2g_1(\alpha) = g_2\tau_3(\alpha)$. This
implies that there exists $\alpha'\in \Pic({\tilde X}_x)$ such that
$h_2(\alpha') = g_2(\tau_3(\alpha)) = \tau_2(g_1(\alpha))$.
Now chasing the image $g_1(\alpha)\in\Pic(Y_s)$ in the second
diagram, we see that $\tau_2'(h_2(\alpha')) =
\tau_2'(\tau_2(g_1(\alpha))) =0$. Then $\tau_2'(h_2(\alpha')) =
\tau_1'(\alpha') = 0$. So that $\alpha' \in \image(\tau_1)$. So there
is an $\alpha'' \in \Pic(X)$ such that $\alpha' = \tau_1(\alpha'')$.
Now going back to the previous diagram, we see that
$$h_2(\alpha')=h_2(\tau_1(\alpha'')) = \tau_2(h_1(\alpha''))=
\tau_2(g_1(\alpha)).$$
Hence, $\tau_2(h_1(\alpha'')-g_1(\alpha)) =0$. But as $\tau_2$ is
injective, we see that $h_1(\alpha'')=g_1(\alpha)$.
This proves Lemma~\ref{vnl and inl on the blowup}.
\end{proof}
Applying the same technique we can prove:
\begin{propose}\label{codim one in disc.}
If for all $x\in X$, $\mathop{\rm INL}\nolimits({\tilde X},L''_x,H^0({\tilde X},L''_x),s)$ is valid
for all smooth $s\in H^0({\tilde X},L''_x)$, then the intersection of the
Noether-Lefschetz locus of $H^0(X,L)$ with the discriminant locus of
$H^0(X,L)$ has codimension $\geq 2$ in $S=\P(H^0(X,L)^*)$.
\end{propose}
\begin{proof}
Firstly one notes that the hypothesis of the proposition together
with Proposition~\ref{codim one}, implies that the
Noether-Lefschetz locus of $H^0({\tilde X},L''_x)$ is a proper of
codimension $\geq 1$ in $\P(H^0({\tilde X},L''_x)^*)$.
Note that the image of the zero scheme of any smooth $s\in
H^0({\tilde X},L''_x)$ in $X$ is an element of the linear system $H^0(X,L)$
with a single ordinary double point at $x$. Then the intersection of
the Noether-Lefschetz locus with the discriminant locus is clearly of
codimension $\geq 2$. This proves the result.
\end{proof}
\section{A General Noether-Lefschetz Theorem}\label{A General
Noether-Lefschetz Theorem}
In this section we are going to apply the general machinery of the
previous section to prove the following generalization of the
classical Noether-Lefschetz theorem.
\begin{thm}\label{main theorem}
Let $X/\C$ be a smooth projective threefold, $L$ a very ample
line bundle on $X$. Then there exits an integer
$n_0(X,L) > 0$ such that for all $n \geq n_0(X,L)$, the
Noether-Lefschetz locus of the linear system $H^0(X,L^n)$ has
codimension $\geq 2$.
\end{thm}
\begin{proof}
The general machinery for the proof of this result was set up in
the previous section. By the Propositions \ref{base free and
smooth}, \ref{base point and smooth} and \ref{base point and ODP},
and Propositions \ref{codim one}, \ref{codim two in smooth} and
\ref{codim one in disc.} we are reduced to proving the following
assertion:
There exists and $n_0(X,L)$ such that for all $n\geq n_0$ the
following are valid:
\begin{enumerate}
\item $H^1(X,\Omega^2_X\tensor L^n) =0$,
\item for every base point free linear system $W\subset
H^0(X,L^n)$ of codimension at most one, we have
$$H^1(X,M(L^n,W) \tensor K_X \tensor L^n ) =0.$$
\item For all points $x\in X$, for all $n \geq n_0$ we have
$$H^1( X , (\pi_x)_*( \Omega^2_{{\tilde X}_x}(-2E_x)) \tensor
L^n ) = 0, $$
\item for any codimension one linear system $W\subset H^0(X,L^n)$
with a single base point at $x\in X$ the following
holds for all $n \geq n_0$:
$$H^1( X , (\pi_x)_*(M(L'_x,W)) \tensor K_X
\tensor L^n) =0.$$
\item For all $x\in X$ and for all $n\geq n_0$, we have
$$H^1(X , m^3_x \tensor L^n ) =0,$$
\item
$$H^1(X, (\pi_x)_*(\Omega^2_{{\tilde X}_x}(-2E_x)) \tensor
L^n ) = 0;$$
\item for $W= H^0({\tilde X}_x,L''_x)$, we have for all
$x\in X$ and for all $n\geq n_0$:
$$H^1( X , (\pi_x)_*(M(L''_x,W)) \tensor K_X \tensor
L^n ) = 0.$$
\end{enumerate}
By what has transpired so far, these assertions, by the
results of Section~\ref{The Noether-Lefschetz machine}, imply the
infinitesimal Noether-Lefschetz results {\it i.e.\/},\ Propositions \ref{base
free and smooth}, \ref{base point and smooth} and \ref{base point
and ODP}, and finally from these local results by
Propositions \ref{codim one}, \ref{codim two in smooth} and
\ref{codim one in disc.}, we obtain the above global result. This
entire section is devoted to the proof of these seven assertions.
Firstly, note that of these seven assertions, the assertions (1), (3),
(5) and (6) follow immediately from the Semi-continuity Theorem,
Serre's vanishing theorem and from simple Noetherian induction. This
will be left to the reader. The remaining assertions are more
difficult and require more elaborate arguments. We will now prove the
assertions (2), (4) and (7).
\end{proof}
\begin{propose}
There exists a positive integer $n_0$ depending only on $X,L$ such
that for any base point free linear system of codimension at most
one, we have
$$H^1( X , M(L^n,W)\tensor K_X \tensor L^n ) = 0.$$
\end{propose}
\begin{proof}
The idea is to use a method of Mark Green (see
\cite{MarkGreen}) and reduce the result to a regularity computation.
Let $V = H^0(X,L)$, by hypothesis, $X$ embeds in $\P:=\P(V)$.
Let $j:X \into \P$ be the embedding given by the linear system. Write
${\cal I}_X$ for the ideal sheaf of $X$ in $\P$.
Then note that $L = j^*({\cal O}_\P(1))$. We have a surjection:
$$H^0( \P, {\cal O}_\P(n)) \to H^0(X,L^n) \to 0.$$
We write $F=H^0(X,L^n), F'=H^0(\P,{\cal O}_\P(n))$. Then the subspace
$W\subset F$ gives us a subspace $W'\subset F'$. If we choose $n_0 >
\mathop{\rm reg}\nolimits({\cal I}_X)$ then $W'$ is base point free on $\P$. Moreover, one has
$\codim_F'(W') = \codim_F(W)$. Then we have corresponding to the
triple $(\P,{\cal O}_\P(n),W')$ a vector bundle $M':=M({\cal O}_\P(n),W')$ which
is defined by the usual evaluation sequence:
$$ 0 \to M' \to F'\tensor {\cal O}_\P \to {\cal O}_\P(n) \to 0.$$
Write $V' = \ker(W' \to W)$. Now we have the following
commutative diagram of locally free sheaves on $X$:
$$
\Matrix{
& & 0 & & 0 & & & & \cr
& & \da & & \downarrow & & & & \cr
& &V'\tensor{\cal O}_X& = & V'\tensor{\cal O}_X & & & & \cr
& & \da & & \downarrow & & & & \cr
0 & \la & j^*M' & \la & W'\tensor{\cal O}_X & \la & F & \la & 0 \cr
& & \downarrow & & \downarrow & & \parallel & & \cr
0 & \la & M & \la & W\tensor{\cal O}_X & \la & F & \la & 0 \cr
& & \downarrow & & \downarrow & & & & \cr
& & 0 & & 0 & & & & \cr
}
$$
Now any splitting of the middle column gives a (non-canonical)
splitting of the left column. Thus we see that one has a
non-canonical splitting:
$$ j^*(M') = M(L^n,W) \oplus V'\tensor {\cal O}_X.$$
Now we need the following lemma due to Mark Green,
see \cite{MarkGreen}.
\begin{lemma}
Let $W$ be any base point free linear system $W\subset
H^0(\P,{\cal O}_\P(n))$. Then for all $i \geq 1$, and for all $k+i \geq
\codim(W)+1$, we have
$$H^i( \P , M' \tensor {\cal O}_\P(k)) =0,$$
in other words, $M'$ is $(\codim(W)+1)$-regular.
\end{lemma}
Now by projection formula we have $j_*j^* M' =j_*M(L^n,W) \oplus V'
\tensor j_*({\cal O}_X)$. Thus as $j_*(M(L^n,W))$ is a direct summand of $M'
\tensor j_*({\cal O}_X)$, and as the regularity of $M'$ is bounded
independent of $n$, and as the regularity of $j_*({\cal O}_X)$ is a fixed
constant, it follows that the regularity of $j_*M(L^n,W)$ is bounded
independent of $n$. Thus if we assume that $n >
(\codim(W)+1+\mathop{\rm reg}\nolimits(K_X))$, then $H^1(X, M(L^n,W)\tensor K_X \tensor L^n
) =0$. This proves the proposition.
\end{proof}
A slight modification of the same technique gives the
following result.
\begin{propose}
For any codimension one linear system $W\subset H^0(X,L^n)$ with a
single base point $x$, there is an $n_0(X,L)$ such that for all
$n\geq n_0$, we have:
$$H^1( X , (\pi_x)_*(M(L'_x,W)) \tensor K_X \tensor L^n )
=0.$$
\end{propose}
\begin{proof}
We have to modify the argument we gave for the previous proposition.
We need the following variant of the previous lemma.
For $x\in \P$ write $\tilde\P$ for the blowup of $\P$ at $x$,
and write $\pi_x$ for the blowup morphism. Write
$W=H^0(\P,{\cal O}_\P(n)\tensor m_{x,\P})$, where $m_{x,\P}$ is the maximal
ideal of $x$ in $\P$. $W$ gives a natural linear system on $\tilde\P$.
Further, let $M'$ be the vector bundle defined in
Section~\ref{Generalities and notations}, on $\tilde\P$ corresponding
to the linear system $W$. Assume that $n \geq 2$ so that this linear
system is base point free. We have
\begin{lemma}
With the above notations, the coherent sheaf $(\pi_x)_*(M')$ on $\P$
is $1$-regular, {\it i.e.\/},\ for all $i\geq 1$ and for all $k+i \geq 1$ we have
$$H^i(\P, (\pi_x)_*(M') \tensor {\cal O}_\P(k)) = 0.$$
\end{lemma}
\begin{proof}
On $\P$ we have the exact sequence
$$ 0 \to (\pi_x)_*(M') \to W\tensor {\cal O}_\P \to {\cal O}_\P(n) \tensor
m_{x,\P} \to 0.$$
The result now follows from the regularity computation for the sheaf
$m_{x,\P}$.
\end{proof}
\long\def\comment#1\endcomment{}
\comment
Note that we have assumed here that $n$ is large enough, so
that the $W$ spans ${\cal O}_\P(n) \tensor m_{x,\P}$. Now twisting by
${\cal O}_\P(k-i)$ we see that we have
$$\to H^{i-1}({\cal O}(n+k-i) \tensor m_x ) \to H^i( (\pi_x)_*(M')
\tensor {\cal O}(k-i)) \to H^i({\cal O}(k-i)) \tensor W\to$$
Now for $i=1$, $H^0({\cal O}(k-1)) \tensor W \to
H^0({\cal O}(n+k-1)\tensor m_x)$ is a surjection if $k\geq i+1$. And for
$1\leq i \leq \dim\P$, we have $H^i(\P,{\cal O}(k-i) =0$ and for $i=\dim\P$,
$H^i({\cal O}_\P(k-i)) = 0$ if $k-i > -\dim\P-1$ {\it i.e.\/},\ if $k > -1$. So one has
to prove that the first term on the left in the exact sequence above
is zero. This is done as follows: we have the exact sequence
$$ 0 \to m_x \to {\cal O}_\P \to \C_x \to 0,$$
where $\C_x$ is the skyscraper sheaf supported only at $x$. Then
twisting by ${\cal O}(n+k-i)$ we get:
$$ 0 \to {\cal O}(n+k-i)\tensor m_x \to {\cal O}_\P(n+k-i) \to
\C_x\tensor{\cal O}(n+k-i) \to 0.$$
Note the sheaf on the extreme right is still supported at a
point and hence has no cohomology except possibly $H^0$. Hence for
$i\geq 1$ we have $H^i(m_x\tensor{\cal O}(n+k-i)) \isom H^i({\cal O}(n+k-i))$. So
that for $i=\dim\P$, we have $H^i({\cal O}(n+k-i))=0$ if $n+k-i > -\dim\P-1$
{\it i.e.\/},\ if $n+k > -1 $. Thus finally if $k > -1 $ and $k > -1 -n $ then
for $i=\dim\P$, $H^i( (\pi_x)_*(M') \tensor {\cal O}(k) ) = 0 $ if $k+i \geq
1$. Thus we have proved that $(\pi_x)_*(M')$ is $1$-regular.
\end{proof}
\endcomment
Finally, we can now finish the proof of the proposition. We
have an exact sequence
$$ 0 \to \pi_*M' \to {\cal O}_\P \tensor H^0({\cal O}(n)\tensor m_{x,\P})
\to {\cal O}(n) \tensor m_{x,\P} \to 0.$$
Write $K_x = \ker(L^n \tensor m_{x,\P} \tensor {\cal O}_X \to L^n
\tensor m_{x,X})$. Note that $K_x$ is a finite length sheaf, supported
at the point $x$. We assume that $n$ is large enough to ensure that
there is a surjection
$$
\Matrix{
H^0(\P, {\cal O}(n)\tensor m_{x,\P}) & \to & H^0(X, L^n\tensor m_{x,X})& \to& 0\cr
\Vert & & \Vert & & \cr
F' & & F & & \cr
}
$$
Write $W'=\ker(F' \to F)$. Then there is a surjection
$W'\tensor {\cal O}_X \to K_x$. To see this note that $K_x$ is a sheaf
supported at a point, further note that we can assume, by choosing $n$
sufficiently large so that $H^1(X, L^n \tensor I_X ) = 0$, that $F'$
surjects on to $H^0(X, L^n \tensor m_{x,\P} \tensor {\cal O}_X)$. This being
done, we have the required surjection. Note that this choice of $n$
depends only on $X,L$. We have then the following commutative diagram
on $X$
$$\let\sc=\scriptstyle
\Matrix{
& & \sc{0} & & \sc{0} & & \sc{0} & & \cr
& & \sc{\da} & & \sc{\da} & & \sc{\da} & & \cr
\sc{0} & \sc{\to}& \sc{H} &\sc{\to}&\sc{W'\tensor{\cal O}_X} & \sc{\to} & \sc{K_x} &
\sc{\to} & \sc{0}\cr
& & \sc{\da} & & \sc{\da} & & \sc{\da} & & \cr
\sc{0} & \sc{\to}& \sc{j^*M'/ \mathop{\rm Tor}\nolimits^{{\cal O}_X}_1( \C_x, {\cal O}_X)} &\sc{\to}&
\sc{F'\tensor{\cal O}_X} &\sc{\to} & \sc{L^n \tensor m_{x,\P} \tensor
{\cal O}_X} & \sc{\to} & \sc{0}\cr
& & \sc{\da} & & \sc{\da} & & \sc{\da} & & \cr
\sc{0} & \sc{\to}& \sc{M} & \sc{\to}&\sc{F\tensor{\cal O}_X} & \sc{\to} &
\sc{m_{x,X}
\tensor L^n} & \sc{\to} & \sc{0}\cr
& & \sc{\da} & & \sc{\da} & & \sc{\da} & & \cr
& & \sc{0} & & \sc{0} & & \sc{0} & & \cr
}
$$
Note that $\mathop{\rm Tor}\nolimits^{{\cal O}_X}_1(\C_x,{\cal O}_x)$ is supported at $x$ and
consequently is of finite length. Further we note that $K_x$ is
also a sheaf of finite length.
Now we want to compute the regularity of $M$ in terms of
$j^*M'$. On one hand we know that the regularity of $j^*M'$ can be
computed in terms of regularity of $M'$ on $\P$. By the previous lemma,
we know that the regularity of $M'$ on $\P$ is independent of $n$. And
so that regularity of $j^*M'$ depends only on regularity of $M'$ and
on regularity of $j_*{\cal O}_X$. Thus we can choose $n$ to be larger than
the sum of these two numbers.
We note that as the $\mathop{\rm Tor}\nolimits$ term is of finite length, the
regularity of the sheaf $j^*(M')$ is the same as the regularity of the
sheaf $j^*(M')/\mathop{\rm Tor}\nolimits^{{\cal O}_X}_1(\C_x,{\cal O}_X)$ (this is essentially because
the $\mathop{\rm Tor}\nolimits$ has no higher cohomologies, being supported on a single
point). Similarly, as $K_x$ has finite length, the top row of the
diagram says that $F$ and ${\cal O}_X$ have the same regularity. Thus we see
that there is an $n_0$ depending only on $X,L$ such that $M$ is $n_0$
regular. Then the vanishing which is required follows.
\end{proof}
Lastly, the assertion (7) is proved exactly as above. The
point to be observed is the following lemma
\begin{lemma}
Let $W=H^0(\P,{\cal O}(n) \tensor m_x^2 )$, $x\in\P$, define the
coherent sheaf $M''$ on $\P$ by the exact sequence:
$$ 0 \to M'' \to H^0(\P,{\cal O}(n)\tensor m_x^2) \tensor {\cal O}_\P \to
{\cal O}(n) \tensor m_x^2 \to 0.$$
Then $M''$ is $2$-regular on $\P$.
\end{lemma}
\section{Application to codimension two cycles}\label{Applications
to Codimension two cycles}
In this section we give an application of our general
Noether-Lefschetz Theorem to codimension two cycles. The application
which we have, is related to a conjecture of Madhav Nori (see
\cite{Madhav}, \cite{KP}). Let $X$ be a smooth projective variety
and $L$ a very ample line bundle on $X$. Let $S=\P(H^0(X,L^n)^*)$. Let
$K$ be the function field of $S$, and let ${\bar K}$ be its algebraic
closure. Let ${\cal X}=X\times S$ and let ${\cal Y}=\P(M(L^n)^*)\into {\cal X}$ be the
universal hypersurface corresponding to sections of $L^n$. The natural
inclusion $\Spec(K) \into S$ gives rise to a fibre square
$$\Matrix{
{\cal Y}_K & \into & {\cal X}_K \cr
\downarrow & & \downarrow \cr
{\cal Y} & \into & {\cal X} \cr
}
$$
The inclusion ${\cal Y}_K \into {\cal X}_K$ then gives rise to the restriction
map on rational equivalence classes of cycles in codimension $i$:
$$ CH^i({\cal X}_K) \to CH^i({\cal Y}_K).$$
Nori's conjecture in this set up is the following:
\begin{conj}[M. V. Nori]\label{Madhav's conjecture}
If $n$ is sufficiently large then the natural map
$$CH^i({\cal X}_{\bar{K}})\tensor\Q \to CH^i({\cal Y}_{{\bar K}})\tensor\Q$$
is an isomorphism for $i < \dim({\cal Y}_K) $ and an inclusion for
$i=\dim({\cal Y}_K) $.
\end{conj}
Observe that the inclusion of fields $\C \into K$ gives rise
to a morphism ${\cal X}_K \to X_{\C}$ and hence a pull back morphism
$CH^i(X_{\C}) \to CH^i({\cal X}_K)$ on cycle classes. Hence by composition
we get an homomorphism $p_X^* : CH^i(X_{\C}) \to CH^i({\cal Y}_K)$.
We can now prove the following weaker statement:
\begin{thm}\label{codim two cycles}
Let $X$ be a smooth projective threefold, $L$ an ample line
bundle on $X$. If $n$ is sufficiently large then the natural
map (given above):
$$CH^2(X_{\C}) \to CH^2({\cal Y}_K)$$
is an inclusion.
\end{thm}
Before we begin the proof of the theorem, let us note one
immediate consequence of Theorem~\ref{codim two cycles}.
\begin{cor}
Let $X$ be a smooth projective threefold, $L$ a very ample line
bundle on $X$. Then we have
$$ CH^2(X_{\C})\tensor\Q \into CH^2({\cal Y}_{\bar K})\tensor\Q $$
\end{cor}
This is immediate from the fact that the passage from $K$ to
${\bar K}$, annihilates only the torsion in Chow groups.
\begin{proof}
Let $z\in CH^2(X_{\C})$ be a cycle such that $p_X^*(z) = 0$
in $CH^2({\cal Y}_K)$. Then we note that there exists a divisor $D\into S$
such that $z$ is the image of a cycle in $CH^1({\cal Y}_D)$, where ${\cal Y}_D$ is
the pull back of ${\cal Y} \to S$ to $D$ (via the inclusion $D\into S$).
We can find a general pencil $\P^1 \into S$, which meets $D$
transversally. Let $F$ denote the function
field of the $\P^1$ which parametrises the pencil. We have
a commutative diagram in which the rows are complexes:
$$\Matrix{
& & CH^2(X_{\C}) && \cr
& & \mapdown{\tau} & & \cr
CH^1({\cal Y}_D) & \mapright{f} & CH^2({\cal Y}) & \mapright{h} & CH^2({\cal Y}_K) \cr
\mapdown{g} & & \mapdown{g'} & & \cr
CH^1({\cal Y}_{D\cap\P^1}) & \mapright{f'} & CH^2({\cal Y}_{\P^1}) &
\mapright{h'} & CH^2({\cal Y}_F) \cr
}
$$
Since $p_X^*(z) = h(\tau(z)) = 0$, there is a $\psi\in
CH^1({\cal Y}_D)$ such that $f(\psi) = \tau(z)$. Then as the square commutes
and as the bottom row is a complex, we see that the image of $z$ in
$CH^2({\cal Y}_F)$ is also zero. Thus we are reduced to proving the
following stronger assertion.
\end{proof}
\begin{thm}\label{reduction to pencil}
Suppose $n$ is sufficiently large, and let
$S=\P(H^0(X,L^n)^*)$, and let ${\cal Y}_P \into X\times \P^1$ be a general
Lefschetz pencil in $S$. Let $F$ be the function field of the $\P^1$
corresponding to this pencil. Then the natural morphism $CH^2(X_{\C})
\to CH^2({\cal Y}_F)$ is injective.
\end{thm}
\begin{proof}
Since $n$ is sufficiently large, by Theorem~\ref{main
theorem} the Noether-Lefschetz locus is of codimension at least two in
$S$. Therefore we can assume that for every closed point $t\in\P^1$,
we have a surjection $\Pic(X) \to CH^1({\cal Y}_t)$. Moreover by Bertini's
Theorem we can also assume that the base locus of such a pencil is an
irreducible, smooth projective curve $C$.
Let $E\into {\cal Y}_P$ be the exceptional divisor. Observe that
$E=C\times \P^1$. Then it is easy to check that we have an isomorphism
$CH^2({\cal Y}_P) \isom CH^2(X_{\C}) \oplus CH^1(C)$ (see \cite{Fulton}).
Further, we also have the following exact sequence:
$$
\oplus_{t\in\P^1}CH^1({\cal Y}_t) \to CH^2(X)\oplus
CH^1(C) \to CH^2({\cal Y}_F)
$$
and where the direct sum extends over all the closed points of
$\P^1$.
Let $z\in CH^2(X_\C)$ be such that $p_X^*(z)=0$ in
$CH^2({\cal Y}_F)$. Then $p_X^*(z)$ is in the image of the map
$\oplus_{t\in\P^1}CH^1({\cal Y}_t) \to CH^2({\cal Y}_P)$. And hence there are
finitely many closed points $t_1,t_2\ldots,t_m\in \P^1$, and cycles
$s_i \in CH^1({\cal Y}_{t_i})$ such that $p_X^*(z)=\sum_{i}s_i$. By the
choice of our pencil, we see that there are cycles $w_i\in CH^1(X)$
such that $s_i= p_X^*(w_i).{\cal Y}_{t_i}$. But any two fibres of ${\cal Y}_P \to
\P^1$ are rationally equivalent. Hence $p_X^*(z) =
p_X^*(\sum_{i}w_i).{\cal Y}_t$. Writing $w=\sum_{i}w_i$, we can rewrite the
last equality as $p_X^*(z) = p_X^*(w).{\cal Y}_t$.
So to prove that $z =0$, by the projection formula, it
suffices to prove that $w=0$. Assume, if possible, that $w$ is not
zero. Then we have on intersecting with $E$, $p_X^*(z).E =
p_X^*(w).{\cal Y}_t.E$. By the projection formula, $(p_X)_*(p_X^*(z).E) =
z.(p_X)_*(E)$. Since $E$ is the exceptional divisor, we have
$(p_X)_*(E)=0$, an thus $p_X^*(w).{\cal Y}_t.E = p_X^*(z).E =0$. Now the
intersection of ${\cal Y}_t$ with $E$ is $C$, so we see that $p_X^*(w).C
=0$. To contradict this, we need to prove that $ \Pic(X) \to \Pic(C) $
is injective. This is done as follows.
Let $G$ be the Grassmannian of all lines in $S$, and $\tilde{X}
\into X \times G$ be the incidence locus. By choosing a large enough
$n$, we can ensure that the fibres of $p_G:\tilde{X} \to G$ are
irreducible outside a codimension three subset of $G$. This can be
done as follows. Let $R \subset \tilde{X}$ be a subset where the
morphism $p:\tilde{X} \to G$ is not smooth. Then since $p$ is
generically smooth, $R$ is a proper closed subscheme of ${\tilde X}$.
\begin{claim}
$R$ is an irreducible subscheme of $\tilde{X}$.
\end{claim}
\begin{proof}
For simplicity, we write $S=\P^n$. By definition, $(x,L) \in
{\tilde X}$ is in $R$ if and only if the the natural map $T_{\tilde X} \to p^*T_G$
is not surjective. Note that lines in the dual projective space
$\tilde{\P^n}$ correspond to codimension two linear spaces in $\P^n$.
We will use the same notation to denote a line in $\tilde{\P^n}$ and
the corresponding codimension two linear space in $\P^n$. Thus we see
that $p$ is not smooth at $(x,L)$ if and only if the tangent spaces
$T_{X,x}$ and $T_{L,x}$ do not span the tangent space $T_{\P^n,x}$.
One has the following diagram of vector spaces:
$$
\Matrix{
0 & \to & T_{L,x} & \to & T_{\P^n,x} & \to & N_{L/\P^n} & \to
& 0 \cr
& & & & \parallel & & & & \cr
0 & \to & T_{X,x} & \to & T_{\P^n,x} & \to & N_{X/\P^n} & \to
& 0 \cr
}
$$
Then $R$ is the locus of pairs $(x,L)$ where $\coker(T_{L,x}
\to N_{X/\P^n,x}) \neq 0$. So on writing $U=T_{\P^n,x},
E=N_{X/\P^n,x}$, for each $L$, such that $(x,L)\in R$, we have a
codimension $n-2$ subspace $U_L$ of $U$. We are interested in those
which subspaces $U_L$ whose images under the surjection $U\to E$ of
non-zero co-rank. Write $K=\ker(U\to E)$. Clearly the composite map
$U_L \to W$ is not surjective if and only if we have $\dim U\cap K
\geq \dim U -\dim W +1$. Thus we see that the set of such subspaces
$U_L$ is in fact a Schubert subvariety in the Grassmannian of
codimension two subspaces of $U$. It is well know that such varieties
are irreducible.
Now varying $x\in X$ we see that, for every $x\in X$, we have
a surjection of vector spaces $T_{\P^n,x} \to N_{X/\P^n,x}$ and the
set of pairs $(x,L)\in R$ is a codimension two linear subspace
$T_{L,x} \into T_{\P^n,x}$. Furthermore, $(x,L)\in R$ if and only if
the composite map $T_{L,x} \to N_{X/\P^n,x}$ is not surjective. Thus
$R$ is an irreducible variety.
\end{proof}
Now the subscheme $R'$ of ${\tilde X}$ where the fibres of the map
$p:{\tilde X} \to G$ are of dimension two is in fact a subscheme of $R$. It
is in fact a proper subscheme of $R$. This follows from the fact that
since $n$ is large enough, the general singular fibre of $p$ (which a
complete intersection) is irreducible. This is an easy consequence of
Bertini's Theorem. Then the image of $R'$ in $G$ has codimension at
least three.
Now let $Z$ be the open subset of $G$ where the fibres of
$p_G$ are smooth and irreducible. Let $\eta$ be the generic fibre of
$Z$.
We have a complex
$$\oplus_D CH^0(p_G^{-1}(D)) \to \Pic(\tilde{X}) \to
\Pic(\tilde{X}_\eta) $$
where the sum extends over all codimension one subvarieties $D$ in
$G$. By the choice of $n$ as above, $p_G^{-1}(D) $ is irreducible.
Therefore, the image of $CH^0(p_G^{-1}(D)) \to \Pic(\tilde{X})$ is
contained in the image of $\Pic(G) \to \Pic(\tilde{X})$. Note that
$\Pic(\tilde{X}) =\Pic(X) \oplus \Z$. Thus $\Pic(X) \to
\Pic(\tilde{X}_\eta)$ is injective.
Now the following proposition, which seems to be well known
but for which we have been unable to find a convenient reference,
completes the proof.
\begin{propose}\label{specialization}
Let $\pi: \tilde{X} \to X, f:\tilde{X}\to Z$ be such that $f$ is
proper with general fibre irreducible and smooth, and $X,Z$
irreducible, $X$ smooth projective variety over complex numbers. Let
$\eta$ be the generic point of $Z$. Suppose $\Pic(X) \to
\Pic(\tilde{X}_\eta)$ is injective. Then for $s\in Z$ outside a
countable union of proper closed subvarieties of $Z$, the
``specialization map'' $\Pic(X) \to \Pic(\tilde{X}_s)$ is injective.
\end{propose}
\begin{proof}
Recall that the N\'eron-Severi group, $NS(X)
=\Pic(X)/\Pic^0(X)$, is a finitely generated group. Let $\tau \in
NS(X)$ be a numerical class, then it is well-known (see for instance
\cite{Mumford-Curves}) that there exists smooth projective variety
$\Pic^\tau(X)$, which parametrises line bundles on $X$ of numerical
class $\tau$, and a universal line bundle $P_\tau$ on $X \times
\Pic^\tau(X)$ (the ``Poincare bundle'') with the following property:
for any $\alpha\in \Pic^\tau(X)$ the restriction
$P_\tau\big|_{X\times \{\alpha\}}$ is the line bundle $\alpha$ on
$X$. The line bundle $P_\tau$ is unique up to a tensoring with a line
bundle pulled back from $\Pic^\tau(X)$. We have then the following
diagram of morphisms:
\def\mathop{\rm id}\nolimits{\mathop{\rm id}\nolimits}
$$\Matrix{
\tilde{X}\times \Pic^\tau(X) &
\mapright{f'=f\times\mathop{\rm id}\nolimits} & Z \times
\Pic^\tau(X) \cr
\mapdown{\pi\times\mathop{\rm id}\nolimits} & & \cr
X\times \Pic^\tau(X) & & \cr
}
$$
Now recall that $\Pic(X) = \coprod_{\tau\in NS(X)} \Pic^\tau(X)$.
Let $s\in Z$, then the ``specialization map'' $g:\Pic(X) \to
\Pic({\tilde X}_s)$ gives to maps $g_{\tau,s}:\Pic^\tau(X) \to \Pic({\tilde X}_s)$.
Note that $0\in g_{0,s}^{-1}(0)$ for any $s\in Z$. Then to prove
the proposition, it suffices to prove that for $s$ outside a countable
union of proper closed subsets, we have
$$
g_{\tau,s}^{-1}(0) = \cases{\{0\}, &if $\tau = 0$;\cr
\emptyset, & otherwise.\cr
}
$$
Let $P'_\tau = (\pi\times\mathop{\rm id}\nolimits)^*(P_\tau)$. Then we are interested in
the set of points $Z_{\tau}=\{(s,p)\in Z\times \Pic^\tau(X) \big|
P'_\tau\big|_{\tilde{X}_s\times\{p\}}\} = 0$. We claim that $Z_\tau$ is in
fact a closed subset of $Z\times\Pic^\tau(X)$. This is accomplished
by the following well-known:
\begin{schol}
Let $X$ be a smooth projective variety, $S$ any irreducible
smooth variety. Let $X \to S$ be smooth proper morphism, and let
${\cal E}$ be a coherent sheaf on $X$ which is flat over $S$. Then
the set of points $s\in S$ where the $H^0(X_s,{\cal E}_s) \neq 0 $ is
a closed subset of $S$.
\end{schol}
The assertion of the Scholium is clearly local, so we can
assume that the base is affine. Now the result is an easy consequence
of the existence of a Grothendieck complex for ${\cal E}$ (see
\cite{Mumford}), and will be left to the reader.
Now we apply the Scholium to the morphism $f':{\tilde X} \times
\Pic^\tau(X) \to X\times \Pic^\tau(X)$ and with ${\cal E} = P'_\tau$,
and with ${\cal E} = P_\tau^{'-1}$. Now we are done because a line
$L$ bundle on an irreducible variety is trivial if and only if $H^0(L)
\neq 0$ and $H^0(L^{-1})\neq 0$. So the set $Z_\tau$ is closed.
Now the proof breaks up into the above two cases: $\tau=0$ and
$\tau\neq 0$. Let us first dispose of the case $\tau \neq 0$. Since by
the hypothesis of Proposition~\ref{specialization} $\Pic(X) \to
\Pic({\tilde X}_\eta)$ is injective, and since the morphism $f'$ is
generically smooth we see that the subset $Z_\tau$ does not meet
$\{\eta\}\times \Pic^\tau(X)$. Hence, there is a proper closed subset
$D_\tau\subset Z$ such that $Z_\tau \into D_\tau\times \Pic^\tau(X)$.
Hence the image of $Z_\tau$ in $Z$ under the first projection is a
proper closed subset of $Z$.
The argument for $\tau=0$ is almost identical except that at
the generic point, the line bundle $P'_0$ is trivial, and hence at the
generic point, $Z_0$ is contained in a subset $\eta\times\Pic^0(X)
\cup D_0\times\Pic^0(X)$, where $D_0$ is a proper closed subset of
$X$. So to sum up this argument, we have shown that for every class
$\tau$ in $NS(X)$, there is closed subscheme $Z_\tau$ of $Z\times
\Pic^\tau(X)$, such that for $\tau\neq 0$, $Z_\tau\into D_\tau\times
\Pic^\tau(X)$, where $D_\tau\subset Z$ is a proper closed subscheme.
For $\tau=0$, the subscheme $Z_0$ (at the generic point) is contained
in a subscheme of the form $\eta\times\Pic^0(X)\cup D_0\times
\Pic^0(X)$, and again $D_0$ is a proper subscheme of $Z$. Since
$NS(X)$ is finitely generated, it is countable, so by removing a
countable union of closed subsets $D_\tau$ where $\tau\in NS(X)$, we
can ensure that the specialization map is injective.
\end{proof}
\begin{rmk}
In a personal communication, Nori has pointed out to us
that his conjecture is not valid for Chow groups over $\Z$, in
other words one has to work with Chow groups tensor $\Q$. This can
been seen as follows. Suppose $X$ is a smooth projective threefold,
$L$ an ample line bundle on $X$, assume further that $H^1(X,{\cal O}_X) =
0$. Let $n$ be any positive integer. Then the zero scheme $Y$ of a
general section of $H^0(X,L^n)$ has the property that
$H^1({\cal Y}_{K},{\cal O}_{{\cal Y}_K}) =0 $. Then by a well-known result of Roitman
(see \cite{Roitman}), we know that $CH^2({\cal Y}_{\bar K})$ is torsion
free, because the Albanese variety of ${\cal Y}_{\bar K}=0$. So that the
restriction $CH^2(X_{K}) \to CH^2({\cal Y}_{\bar K})$ obliterates all the
torsion in $CH^2(X_{K})$, consequently if $CH^2(X_{K})$ has
non-trivial torsion then the restriction map cannot be injective.
It is possible to write down examples of threefolds which have
the above properties. A well studied example being that of a smooth
general quartic hypersurface in $\P^4$. It is shown in \cite{Bloch}
that group of codimension two cycles algebraically equivalent to zero
modulo rational equivalence, denoted by $A^2(X)$, is isomorphic as a
group to the intermediate Jacobian $J^2(X)$. The latter group is a
complex torus (in fact an Abelian variety) and hence has non-trivial
torsion. Moreover, it is standard that $A^2(X) \subset CH^2(X)$ (see
\cite{Fulton}). A similar example can be found in \cite{Murre}.
\end{rmk}
\end{proof}
\bibliographystyle{plain}
|
1992-06-12T22:14:15 | 9206 | alg-geom/9206005 | en | https://arxiv.org/abs/alg-geom/9206005 | [
"alg-geom",
"math.AG"
] | alg-geom/9206005 | Valery Alexeev | Valery Alexeev | Two Two-dimensional Terminations | 25 pages, 4 figures, LaTeX 2.09 | null | null | null | null | Varieties with log terminal and log canonical singularities are considered in
the Minimal Model Program, see \cite{...} for introduction. In
\cite{shokurov:hyp} it was conjectured that many of the interesting sets,
associated with these varieties have something in common: they satisfy the
ascending chain condition, which means that every increasing chain of elements
terminates. Philosophically, this is the reason why two main hypotheses in the
Minimal Model Program: existence and termination of flips should be true and
are possible to prove.
In this paper we prove that the following two sets satisfy the ascending
chain condition:
1. The set of minimal log discrepancies for $K_X+B$ where $X$ is a surface
with log canonical singularities.
2. The set of groups $(b_1,...b_s)$ such that there is a surface $X$ with log
canonical and numerically trivial $K_X+\sum b_jB_j$. The order on such groups
is defined in a natural way.
| [
{
"version": "v1",
"created": "Fri, 12 Jun 1992 20:13:52 GMT"
}
] | 2015-06-30T00:00:00 | [
[
"Alexeev",
"Valery",
""
]
] | alg-geom | \section{Introduction}
Varieties with log terminal and log canonical singularities are
considered in the Minimal Model Program, see \cite{kmm} for
introduction. In \cite{sh:hyp} it was conjectured that many of the
interesting sets, associated with these varieties have something in
common: they satisfy the ascending chain condition, which means that
every increasing chain of elements terminates (in \cite{sh:hyp} it was
called the upper semi-discontinuaty). Philosophically, this is the
reason why two main hypotheses in the Minimal Model Program: existence
and termination of flips should be true and are possible to prove.
As for the latter, one of the main properties of flips is that log
discrepancies after doing one do not decrease and some of them
actually increase, \cite{sh:old}. Therefore, if one could show that a
set of ``the minimal discrepancies'' satisfies the ascending chain
condition, that would help to prove the termination of flips. The
Shokurov's proof of existence of 3-fold log flips \cite{sh:3f} is
another example of applying the same principle. In fact, to complete
the induction it uses some 1~-~dimensional statement, 2~-~dimensional
analog of which is proved in this paper. For further discussion, see
also
\cite{ag-kol}.
For one of the first examples where the phenomenon is actually proved
let us mention the following
\begin{utv}[\cite{al:fi},\cite{al:tg}]
Let us define the Gorenstein index of an $n$-dimensional Fano variety
$X$ with weak log terminal singularities as the maximal rational
number $r$ such that the anticanonical divisor $-K_X\equiv rH$ with an
ample Cartier divisor $H$. Then a set $$FS_n\cap
[n-2,=+\infty]=\{r(X)|X\; is\; a\; Fano\; variety\; and \;
r(X)>n-2\}$$ satisfies the ascending chain condition and has only the
following limit points: $n-2$ and $n-2+{1\over k}$, $k=1,2,3...$.
\end{utv}
In this paper we prove that the following two sets satisfy the
ascending chain condition:
\begin{num}
\item (Theorems \ref{utv:local_pasc},\ref{utv:local_asc})
The set of minimal log discrepancies for $K_X+B$ where $X$ is a
surface with log canonical singularities.
\item (Theorem \ref{global_asc})
The set of groups $(b_1,...b_s)$ such that there is a surface $X$ with
log canonical and numerically trivial $K_X+\sum b_jB_j$. The order on
such groups is defined in a natural way, see \ref{blessb'}.
\end{num}
\medskip
The proofs heavily use explicit formulae for log discrepancies from
\cite{al:lc}. We do not find it possible to prove them here again.
(This is quite easy anyway).
\begin{askn}
Author would like to thank V.V.Shokurov and J.Koll\'ar for asking the
questions that this paper gives the answers to and for useful
discussions.
\end{askn}
\section{Definitions and recalling}
All varieties in this paper are defined over the algebraically closed
field of characteristic zero. $K_X$ or simply $K$ if the variety $X$
is clear from the context always denote the class of the canonical
divisor.
\subsection{Basics}
\begin{opr}
A {\bf \bfQ-divisor} on a variety $X$ is a formal combination $D=\sum
d_j D_j$ of Weil divisors with rational coefficients.
\end{opr}
\begin{opr}
One says that a \bfQ-divisor $D$ is {\bf \bfQ-Cartier} if some
multiple of it is a Weil divisor with integer coefficients that is a
Cartier divisor.
\end{opr}
\begin{opr}
\label{opr:discr}
Let $f:Y\to X$ be any birational morphism and $F_i$ be exceptional
divisors of this morphism. Consider a divisor of the form $K+B$, where
$B=\sum b_j B_j$ and $0<b_j\le1$. Coefficients $a_i$ in the following
formula $$K_Y+f^{-1}B+\sum F_i = f^*(K+B)+\sum a_i F_i$$ are called
{\bf log discrepancies} of $K+B$. \end{opr}
\begin{opr}
\label{opr:codiscr}
Let $f:Y\to X$ be any birational morphism and $F_i$ be exceptional
divisors of this morphism. Consider a divisor of the form $K+B$, where
$B=\sum b_j B_j$ and $0<b_j\le1$. Coefficients $b_i$ in the following
formula $$K_Y+f^{-1}B+\sum b_i F_i = f^*(K+B)$$ are called {\bf
codiscrepancies} of $K+B$. \end{opr}
\begin{zam}
Evidently there is a simple relation between log discrepancy and
codiscrepancy: $a_i=1-b_i$.
\end{zam}
\begin{opr}
A \bfQ-divisor of the form $K+B$ is said to be {\bf log canonical
(lc)} if
\begin{num}
\item $K+B$ is \bfQ-Cartier
\item there is a resolution of singularities $f:Y\to X$
such that $supp(f^{-1}B)\bigcup F_i$ is a divisor with normal
intersections and all the log discrepancies $a_i\ge0$. \end{num}
\end{opr}
\begin{opr}
A \bfQ-divisor of the form $K+B$ is said to be {\bf log terminal (lt)}
if
\begin{num}
\item $K+B$ is \bfQ-Cartier
\item there is a resolution of singularities
$f:Y\to X$ such that $supp(f^{-1}B)\bigcup F_i$ is a divisor with
normal intersections and all the log discrepancies $a_i>0$. \end{num}
\end{opr}
\subsection{Graphs}
With rare exceptions all the varieties in this paper will be
two-dimensional. No doubt that the case of surfaces is much easier
than that of more-dimensional varieties. One of the reasons for this
is that surface has a natural quadratic form defined by intersection
of curves. Many statements that we need can be formulated in terms of
weighted graphs and become therefore basicly combinatorical problems.
So let us start with a system of curves on a surface that are divided
into two classes: ``internal'', denoted by $F_i$ and ``external'',
denoted by $B_j$.
\begin{opr}
A weighted graph $\Gam$ is the following data:
\begin{num}
\item a ``ground graph'': each vertex $v$ of it corresponds to an ``internal''
curve
$F$ , two different vertices $v_1$ and $v_2$ are connected by wedge of
weight $F_1\cdot F_2$.
\item weights: a vertex $v$ has weight $w=-F^2$
\item genera: a vertex $v$ has genus $p_a(F)$ (arithmetical genus of
the curve)
\item an ``external part'': additional vertices, corresponding to the
``external''
components $B_j$ , connected with vertices $v_i$ if $B_j$ and $F_i$
intersect.
\end{num}
\end{opr}
Vice versa, every weighted graph $\Gam$ corresponds to a system of
curves $\{F_i,B_j\}$.
\begin{opr}
Graph $\Gam$ is said to be {\bf elliptic, parabolic or hyperbolic} if
the corresponding quadratic form $F_i\cdot F_k$ is elliptic, parabolic
or hyperbolic, that is, has the signature $(0,n)$, $(0,n-1)$ or
$(1,n-1)$.
\end{opr}
The following is the basic case when we shall need such graphs: $X$ is
a surface with a divisor $K+B$ and $f:Y\to X$ is a resolution of
singularities of $X$. The curves $F_i$ are exceptional divisors of
$f$ and the curves $B_j$ are strict transforms of the components of
$B$. Note that since a matrix of intersection $(F_i\cdot F_k)$ is
negatively defined, the graph is elliptic and all the weights in this
case are positive integer numbers. Usually we will examine graphs that
correspond to the {\it minimal\/} resolution of singularities.
\begin{opr}
A graph $\Gam$ is said to be {\bf minimal} if it does not contain
internal vertices that have $p_a=0$ and weight 1.
\end{opr}
\begin{opr}
For any graph with a nondegenerate quadratic form $F_i\cdot F_k$ (for
example elliptic or hyperbolic) we define {\bf log discrepancies}
$a_i$ as the solutions of a system of linear equations $$\sum a_i
F_i\cdot F_k=(2p_a(F_k)-2-F^2)+(f^{-1}B+\sum F_i)F_k$$
\end{opr}
\begin{opr}
For any graph with a nondegenerate quadratic form $F_i\cdot F_k$ we
define {\bf codiscrepancies} $b_i$ by the formula $b_i=1-a_i$
\end{opr}
Let us explain the meaning of the two previous definitions. The
formulae above are equivalent to the following: $$(K+\sum b_jB_j+\sum
b_iF_i)F_k=0\quad for \quad any\quad k$$ So if the graph $\Gam$ is an
elliptic graph, corresponding to some birational morphism $f:Y\to X$
we get the previous definitions
\ref{opr:discr}, \ref{opr:codiscr}. Another situation when we shall
use discrepancies and codiscrepancies is the following: $X$ is a
surface with numerically trivial $K+B$, $f:Y\to X$ is some resolution.
Part of the vertices of $\Gam$ correspond to exceptional curves of $f$
and the other part -- to strict transforms of certain curves on $X$.
\begin{opr}
A graph $\Gam$ is said to be {\bf log canonical (lc) or log terminal
(lt)} with respect to $K+B$ if its log discrepancies $a_i\ge0$ or
$a_i>0$ respectively.
\end{opr}
The main object into consideration in this paper is a surface $X$
with a divisor $K+B$ that is lc. So will be the corresponding graphs.
If we ignore the way $B$ meets the ground graph or assume that all the
coefficients of $B_j$ equal 1, then all such graphs are classified in
\cite{kaw:cb} (see also \cite{al:lc}). They are divided into two
classes describing respectively rational and elliptic singularities.
In the case of rational singularities all the genera are equal to 0
(and by this reason will be omited), all the edges are simple (of
weight 1) the ground graphs are those of types $A_n$, $D_n$ and
$E_6,E_7,E_8$. If we fix some number $N$ and consider graphs with
weights $\le N$ then the only infinite series of such graphs are the
following
\begin{picture}(300,150)(-10,0)
\multiput(75,75)(150,0){2}{\begin{picture}(0,0)
\put(0,0){\oval(100,20)}
\put(-7.5,0){\oval(75,10)}
\put(40,0){\circle{10}}
\put(0,25){\vector(0,-1){15}}
\put(-15,-20){\vector(0,1){20}}
\put(-5,30){$m$}
\put(-20,-35){$q$}
\end{picture}}
\multiput(265,50)(0,50){2}{
\put(0,0){\circle{10}}
\put(10,-5){2}
}
\multiput(265,55)(0,25){2}{\put(0,0){\line(0,1){15}}}
\put(85,30){a)}
\put(235,30){b)}
\put(130,10){Figure 1}
\end{picture}
These are typical pictures that we shall use to describe graphs. Long
ovals denote chains of vertices. The numbers $q$ and $m$ denote the
absolute values of determinants of the submatrices of $F_i\cdot F_k$
that include only rows and columns corresponding to the vertices of
the chains. It is very well known (\cite{ri}, comp.\cite{al:lc}) that
any chain is uniquely determined by a pair of coprime numbers $(q,m)$
with $1\le q<m$ and vice versa. In the previous example $q$ and $m$
are any such numbers.
Generally, graphs shall also have external parts that shall be
denoted by crossed vertices.
In the case of elliptic singularities one has ``circles'' of vertices
with $p_a=0$ and a single vertex with $p_a=1$. $B$ is empty and all
the log discrepancies $a_i=0$, codiscrepancies $b_i=1$.
\begin{opr}
{\bf Du Val graph} is an elliptic graphs with all genera = 0, all
weights = 2 and empty external part $B$. It is well known that the
ground graph is then one of the graphs $A_n$, $D_n$, $E_6,E_7$ or
$E_8$.
\end{opr}
\begin{opr}
We say that a graph $\Gam'$ is a subgraph of $\Gam$ if all the
vertices of $\Gam'$ are at the same time vertices of $\Gam$, weights
of vertices and edges of $\Gam'$ and $p_a$ of vertices in $\Gam'$ do
not exceed the corresponding weights and $p_a$ in $\Gam$ and $F_i'\sum
b_j'B_j'\le F_i'\sum b_jB_j $ for the corresponding vertices.
\end{opr}
The following are easy linear algebra statements.
\begin{lem}
\label{a_less1}
Let $\Gam$ be a minimal elliptic graph. Then all the log
discrepancies $a_i\le1$ (codiscrepancies $b_i\ge0$) and if $\Gam$ is
not a Du Val graph then $a_i<1$ ($b_i>0$).
\end{lem}
\par\noindent {\sl Proof:}\enspace Well known.
\qed
\begin{lem}
\label{ag_less_ag'}
Let $\Gam'\subset\Gam$, $\Gam'\neq\Gam$ be two minimal elliptic graphs
and assume that the weights of the vertices are in both graphs the
same. Then for the log discrepancies one has $a_i\le a_i'$ (for
codiscrepancies $b_i\ge b_i'$) and if $\Gam$ is not a Du Val graph
then $a_i< a_i'$ ($b_i> b_i'$). If the weights of $\Gam'$ and $\Gam$
are different then $a_i\le a_i'$ assuming that $\Gam$ is log
canonical.
\end{lem}
\par\noindent {\sl Proof:}\enspace Compare the corresponding systems of linear equations
(see \cite{al:fi}, \cite{al:lc}). \qed
\begin{lem}
\label{ag_greater_ag'}
Let $\Gam'\subset\Gam$, $\Gam'\neq\Gam$ be two graphs such that all
the log discrepancies of $\Gam'$ $a_i\le 1$ (codiscrepancies $b_i\ge
0$) and $v_0$ is a fixed vertex of $\Gam'$. Assume that $\Gam$ is
hyperbolic and that $\Gam-v_0$ is elliptic. Then for the log
discrepancy of $v_0$ one has $a_0\ge a_0' $ ($b_0\le b_0' $) assuming
that $\Gam$ is log canonical.
\end{lem}
\par\noindent {\sl Proof:}\enspace Compare the corresponding systems of linear equations. \qed
\begin{sle}
\label{max_ell}
Let $\Gam$ be a minimal elliptic graph and assume that all the log
discrepancies of $\Gam$ $a_i\ge c>0$. Then weights of the vertices are
bounded from above by $2/c$.
\end{sle}
\par\noindent {\sl Proof:}\enspace Consider a graph $\Gam'$ containing a single vertex of
weight $n$. Then $a'=2/n$.
\qed
\begin{sle}
\label{max_hyp}
Let $\Gam$ be a graph as in \ref{ag_greater_ag'} plus let $v_0$ have
weight 1. Assume that the codiscrepancy of $v_0$ $b_0\ge c>0$. Then
$$\sum_{i\ne0} F_0F_i\le 2+\frac{2}{c}$$
\end{sle}
\par\noindent {\sl Proof:}\enspace Consider a graph $\Gam'$ containing a vertex $v_0$ connected
with $n$ vertices of weight 2. Then $b_0'=2/(n+2)$.
\qed
\subsection{Sequences}
\begin{opr}
Let $X$ be a variety with a log canonical $K+B$. {\bf A log
discrepancy of $\bf K+B$ $ld(K+B)$} is a minimal log discrepancy $a_i$
that appears in \ref{opr:discr} for some birational morphism $f:Y\to
X$.
\end{opr}
It is easy to see that $ld(K+B)$ is well defined and is a nonnegative
rational number.
\begin{opr}
Let $X$ be a surface with a log canonical $K+B$. {\bf A partial log
discrepancy of $\bf K+B$ $pld(K+B)$} is a minimal log discrepancy
$a_i$ that appears in
\ref{opr:discr} for the special birational morphism $h:{\widetilde X}} \def\xtil{{\tilde x}\to X$, where
${\widetilde X}} \def\xtil{{\tilde x}$ is the minimal resolution of singularities.
\end{opr}
\begin{opr}
Let $\xi=\{X^{(n)},K+B^{(n)}| n=1,2...\}$ be a sequence of surfaces.
Then we define $\bf ld(\xi)$ and $\bf pld(\xi)$ as the {\bf sequences}
of real numbers \{$ld({K+B^{(n)}}{ })$\} and \{$pld({K+B^{(n)}}{ })$\} respectively.
\end{opr}
\begin{opr}
\label{LDPLD}
Let $\xi=\{X^{(n)},K+B^{(n)}| n=1,2...\}$ be a sequence of surfaces.
Then we define $\bf LD(\xi)$ and $\bf PLD(\xi)$ as the {\bf subsets}
of real numbers \{$ld({K+B^{(n)}}{ })$\} and \{$pld({K+B^{(n)}}{ })$\} respectively.
\end{opr}
\begin{opr}
We define $ld,pld,LD,PLD$ for graphs in the same way as we have done it for
surfaces.
\end{opr}
\begin{opr}
\label{blessb'}
Let $B=(b_1,b_2...b_s)$ an $B'=(b_1',b_2'...b_t')$ be two groups of
numbers. One says that $B\le B'$ if
\begin{num}
\item $s\geq t$
\item for every $j=1...t$ $b_j\le b_j'$
\end{num}
If, in addition, one of the inequalities in (i) or in (ii) for some
index $j_0$ is strict, one says that $B<B'$.
\end{opr}
\begin{zam}
Because of the part (i) of \ref{blessb'} when considering a
nondecreasing sequence {$B^{(n)}$ } we can always assume, passing to a
subsequence, that the lengths of {$B^{(n)}$ } are in fact the same.
\end{zam}
\subsection{Log Del Pezzo surfaces}
\begin{opr}
A normal surface $X$ is said to be a Del Pezzo surface if $-K$ is an
ample \bfQ-divisor.
\end{opr}
The following is an simple lemma, see \cite{al-nik},\cite{nik} for the
proof which is especially easy if $K$ is lt or lc.
\begin{lem}
\label{Del_Pezzo}
Let $X$ be a log Del Pezzo surface an $h:{\widetilde X}} \def\xtil{{\tilde x}\to X$ be the minimal
resolution of singularities. Then
\begin{num}
\item the Kleiman-Mori cone of effective curves $NE({\widetilde X}} \def\xtil{{\tilde x})$ is generated by
finitely many extremal rays
\item if $X\ne \bfP^2,\bfF_n$ (minimal rational surface) then all the extremal
rays are generated by exceptional curves of $f$ and (-1)-curves.
\end{num}
\end{lem}
\begin{lem}
\label{log_Del_Pezzo}
Let $X$ be a Del Pezzo surface and assume that $K$ is lc. Then $X$ is
one of the following:
\begin{num}
\item a rational surface with rational singularities
\item a generalized cone over a smooth elliptic curve
\end{num}
\end{lem}
\par\noindent {\sl Proof:}\enspace Let $h:{\widetilde X}} \def\xtil{{\tilde x}\to X$ be a minimal desingularization. ${\widetilde X}} \def\xtil{{\tilde x}$ is a
smooth surface and clearly $h^0(NK_{{\widetilde X}} \def\xtil{{\tilde x}}=0$ for any $N>0$, so
${\widetilde X}} \def\xtil{{\tilde x}$ is ruled.
Assume that $X$ has a nonrational singularity. Then by the
classification of log canonical singularities ${\widetilde X}} \def\xtil{{\tilde x}$ contains an
elliptic curve or a circle of rational curves $F_0$ that is disjoint
from other curves, exceptional for $h$. If ${\widetilde X}} \def\xtil{{\tilde x}$ is a locally
trivial $\bfP^1$-bundle then $F_0$ should be an exceptional section of
this bundle and should be smooth. In this case $X$ is a generalized
cone. Otherwise $F_0$ should intersect a curve $E$ with $E^2<0$ that
lies in the fiber of a generically $\bfP^1$-bundle giving the
structure of a ruled surface and such that $E$ is not exceptional for
$h$. By \ref{Del_Pezzo} $E$ is a $(-1)$-curve. The latter is
impossible since $-h^*K=-K_{{\widetilde X}} \def\xtil{{\tilde x}}-F_0-...$ and therefore $-h^*K\cdot
E\le0$.
Now let us assume that $X$ has only rational log canonical
singularities. By the classification again one has
$-h^*K=-K_{{\widetilde X}} \def\xtil{{\tilde x}}-F_0-\sum b_iF_i$, $0\le b_i<1$ and $F_0$ is a
disjoint union of smooth rational curves. Since $-h^*K$ is big, nef,
the Kawamata-Fiehweg vanishing gives
$$h^1({\widetilde X}} \def\xtil{{\tilde x},-F_0)=h^1({\widetilde X}} \def\xtil{{\tilde x},K_{{\widetilde X}} \def\xtil{{\tilde x}}+\sum b_iF_i+(-h^*K))=0$$ and
from the exact sequence $$0\to\cal O_{{\widetilde X}} \def\xtil{{\tilde x}}(-F_0)\to\cal
O_{{\widetilde X}} \def\xtil{{\tilde x}}\to\cal O_{F_0}\to 0$$ one gets $h^1(\cal O_{{\widetilde X}} \def\xtil{{\tilde x}})=0$.
Therefore ${\widetilde X}} \def\xtil{{\tilde x}$ and $X$ are rational surfaces.
\qed
\section{Local case: elliptic log canonical graphs}
{\bf In this section we consider only local situation. $\bfX$ is a
neighbourhood of a surface point $\bfP$ and all the components of
$\bfB$ pass through $\bfP$. }
\begin{utv}[Local boundness]
\label{utv:loc_bound}
Let $X,K+B$ be as above a neighbourhood of a surface point $P$ with lc
$K+B$ and all of $B_j$ pass through $P$. Then $\sum b_j\leq2$.
\end{utv}
\par\noindent {\sl Proof:}\enspace Proved in \cite{ag-kol} for the $n$-dimensional case with
a bound $n$.
\subsection{Minimal resolution}
\begin{utv}[Local partial ascending chain condition]
\label{utv:local_pasc}
Let $\xi=\{X^{(n)},K+B^{(n)}\}$ be a sequence of surfaces such that
\begin{num}
\item $K+B^{(n)}$ is lc
\item $B^{(n)}$ is a nondecreasing sequence
(for example, constant)
\end{num}
Then every increasing subsequence in $pld(\xi)$ terminates.
\noindent If, in the addition, one has
\begin{num}
\setcounter{beam}{2}
\item $B^{(n)}$ is an increasing sequence
\end{num}
then every nondecreasing subsequence in $pld(\xi)$ terminates.
\end{utv}
We prove \ref{utv:local_pasc} in several steps.
\begin{shag}
One can assume that {$K+B^{(n)}$ } and moreover $K$ are lt.
\end{shag}
\par\noindent {\sl Proof:}\enspace Indeed, if in (ii) {$K+B^{(n)}$ } is not lt, then
$pld({K+B^{(n)}}{ })=0$ but we are looking for increasing subsequences of
$pld(\xi)$. In (iii) if {$K+B^{(n)}$ } is not lt, then there exists a partial
resolution $f:Y\to X$ with a single exceptional divisor $F$ such that
the corresponding log discrepancy $a=0$. Then
$$K_Y+F+f^{-1}B=f^*(K+B)$$ and the log adjunction formula for $F$ (see
\cite{sh:3f}) yields $$\sum {k-1+\sum l_j b_j\over k}=2$$ for some
positive integers $k,l_j$. It is easy to see that if {$B^{(n)}$ } are
increasing then the sequence should terminate. \qed
\begin{shag}
By the previous step we can assume that there is a constant $\varepsilon$ so
that for every $n$ $pld({K+B^{(n)}}{ })>\varepsilon$. Then we prove the following
\end{shag}
\begin{lem} All the lt elliptic graphs with $pld({K+B^{(n)}}{ })>\varepsilon$
and $b_j>\varepsilon$ can be described as follows:
\label{fin_many_graphs}
\begin{num}
\item finitely many graphs (that includes the way $B_j$ intersect $F_i$)
\item the graphs given on the next picture, where there are only finitely many
possibilities for the chains of vertices, denoted by ovals and for the
ways $B_j$ meet that vertices
\end{num}
\begin{picture}(320,270)(-10,-30)
\multiput(0,0)(0,100){2}{
\put(5,50){$q_1$}
\put(15,50){\vector(1,0){25}}
\put(65,50){\oval(80,20)}
\put(60,50){\oval(50,10)}
\put(95,50){\circle{5}}
\put(105,50){\line(1,0){12.5}}
\put(122.5,50){\line(1,0){15}}
\multiput(120,50)(20,0){2}{\circle{5}}
\multiput(117.5,55)(20,0){2}{2}
\multiput(142.5,50)(27.5,0){2}{\line(1,0){7.5}}
\multiput(155,50)(5,0){3}{\circle*{1}}
\multiput(35,85)(50,0){2}{\circle{14}}
\multiput(30,90)(50,0){2}{\line(1,-1){10}}
\multiput(30,80)(50,0){2}{\line(1,1){10}}
\multiput(50,80)(10,0){3}{\circle*{1}}
\put(35,78){\line(5,-6){15}}
\put(85,78){\line(-5,-6){15}}
\put(5,20){$m_1$}
\put(15,25){\vector(1,1){15}}
\put(125,20){\vector(-1,1){25}}
\put(125,15){$min$}
\put(110,85){\circle{14}}
\put(105,90){\line(1,-1){10}}
\put(105,80){\line(1,1){10}}
\put(95,52.5){\line(3,5){15}}
}
\multiput(180,30)(0,20){3}{\circle{5}}
\multiput(185,27.5)(0,20){3}{2}
\multiput(180,32.5)(0,20){2}{\line(0,1){15}}
\multiput(180,150)(20,0){2}{\circle{5}}
\multiput(177.5,155)(20,0){2}{2}
\put(182.5,150){\line(1,0){15}}
\put(202.5,150){\line(1,0){12.5}}
\put(255,150){\oval(80,20)}
\put(260,150){\oval(50,10)}
\put(225,150){\circle{5}}
\multiput(235,185)(50,0){2}{\circle{14}}
\multiput(230,190)(50,0){2}{\line(1,-1){10}}
\multiput(230,180)(50,0){2}{\line(1,1){10}}
\multiput(250,180)(10,0){3}{\circle*{1}}
\put(235,178){\line(5,-6){15}}
\put(285,178){\line(-5,-6){15}}
\put(210,185){\circle{14}}
\put(205,190){\line(1,-1){10}}
\put(205,180){\line(1,1){10}}
\put(225,152.5){\line(-3,5){15}}
\put(195,120){\vector(1,1){25}}
\put(175,115){$min$}
\put(305,150){\vector(-1,0){25}}
\put(310,150){$q_2$}
\put(310,120){$m_2$}
\put(305,125){\vector(-1,1){15}}
\put(140,-15){Figure 2}
\end{picture}
Moreover, the log discrepancies of any of suchs graphs satisfy the
following inequality $$pld(K+B)\ge {1-\sum l_jb_j\over m-q}$$
where $\bar b_j=\lim b_j$
and tend
to this number as the chain of 2's gets longer and longer. Here
$l_j=\sum( B_j\cdot F_i) r_i$, where $r_i$ is the determinant of the
short subchain of the ground graph, ``cut off'' by the vertex $v_i$.
\end{lem}
\par\noindent {\sl Proof:}\enspace By \ref{max_ell} the weights of vertices in the graph $\Gam$ are
bounded
by $2/\varepsilon$. Therefore, sacrificing finitely many graphs we can assume
that $\Gam$ is one of the graphs on Fig.1.
First, assume that we are in the case a) of Fig.1, i.e. $\Gam$ is a
chain. Consider the sequence of log discrepancies of vertices in this
chain. By \cite{al:lc} $$a_{i-1}-2a_i+a_{i+1}=(w_i-2)a_i+\sum
b_jB_jF_j\ge(w_i-2+\sum B_jF_i)\varepsilon,$$ therefore the graph of this
function is concave up and, unless $w_i=2$ and all $B_jF_i=0$, it is
not a straight line but is ``very concave up''. Now by \ref{a_less1}
the discrepancies $a_i\le1$. This implies that all the chains are
those on Fig.2 with only finitely many possibilities for the ovals and
with the chains of 2's of an arbitrary length $A$. Also, omiting
finitely many graphs, we can assume that the minimal log discrepancy
is achieved at one of the two vertices, where the arrows point out.
Now we use an explicit formula for the log discrepancies of those
vertices which follows easily from 3.1.8, 3.1.10 of \cite{al:lc}.
Define $\alpha} \def\Alp{\Alpha_1=1-\sum l_j^{(1)}b_j$ for the left part of the chain,
the meaning of $l_j$ being explained in the formulation of the
statement, $\alpha} \def\Alp{\Alpha_2=1-\sum l_j^{(2)}b_j$ be the corresponding
expression for the right par, and let $A$ be the length of the chain
of 2's. Then $$a_1={\alpha} \def\Alp{\Alpha_1(A(m_2-q_2)+m_2)+\alpha} \def\Alp{\Alpha_2q_1\over
A(m_2-q_2)(m_1-q_1)+m_2(m_1-q_1)+q_1(m_2-q_2)}$$ or
$$a_1={{\alpha} \def\Alp{\Alpha_1\over m_1-q_1}(A+{m_2\over m_2-q_2})+ {\alpha} \def\Alp{\Alpha_2\over
m_2-q_2}{q_1\over m_1-q_1}\over A+{m_2\over m_2-q_2}+{q_1\over
m_1-q_1}}$$ with the symmetric expression for $a_2$.
One can note that
\begin{enumerate}
\item if ${\alpha} \def\Alp{\Alpha_1\over m_1-q_1}\le {\alpha} \def\Alp{\Alpha_2\over m_2-q_2}$ then
${\alpha} \def\Alp{\Alpha_1\over m_1-q_1}\le a_1\le {\alpha} \def\Alp{\Alpha_2\over m_2-q_2}$
\item $\lim_{A\to\infty}a_1={\alpha} \def\Alp{\Alpha_1\over m_1-q_1}$
\end{enumerate}
and these two observations complete the proof in the case a) of Fig.1.
The case b) of Fig.1 is handled similarly. Let us mention only that in
the latter case there is only one possible vertex for the minimal log
discrepancy which is given by the formula $$a_1={\alpha} \def\Alp{\Alpha_1\over
m_1-q_1},$$ so this case can be treated formally as a subcase of a)
with $\alpha} \def\Alp{\Alpha_2=0$ and $m_2=q_2$.
\qed
\begin{shag}
The lemma~\ref{fin_many_graphs} implies \ref{utv:local_pasc}.
\end{shag}
\par\noindent {\sl Proof:}\enspace For any fixed graph $\Gam$ if the coefficients of the external part $B$
increase, then by \ref{ag_less_ag'} log discrepancy $pld(K+B)$
decreases. Therefore, we can consider only case (ii) of
\ref{fin_many_graphs}. Passing to a subsequence we can assume that
all the graphs are of the same type and the length of the sequence of
2's increases. But then $$pld({K+B^{(n)}}{ }) \ge{1-\sum l_j\bar b_j\over
m-q}$$ and $$\lim pld({K+B^{(n)}}{ }) = {1-\sum l_j\bar b_j\over m-q}$$ where
$\bar b_j=\lim b_j$, and we are done.
\qed
\begin{sle}\label{corol}
If $B=\emptyset$, then \ref{fin_many_graphs} says that the set of
minimal log discrepancies satisfies the ascending chain condition and
the only limit points are 0 and $1/k$, $k=2,3...$
\end{sle}
\begin{zam}
The statement \ref{corol} is due to V.V.Shokurov (unpublished).
\end{zam}
\subsection{General case}
Later we shall use the local ascending chain condition in the just
proved form, i.e. for the minimal resolution of singularities. But
the minimal resolution of singularities of $X$ is not necessarily a
resolution of singularities for $K+B$, because $supp(f^{-1}B\cup F_i)$
can have nonnormal intersections. Below we prove the statement,
corresponding to \ref{utv:local_pasc} but for $ld(\xi)$ instead of
$pld(\xi)$. We first consider the case when {$X^{(n)}$ } are nonsingular and
then combine our arguments to treat the general situation.
\begin{utv}
\label{nonsingular}
Let $\xi=\{X^{(n)},K+B^{(n)}\}$ be a sequence of {\bf nonsingular}
surfaces such that
\begin{num}
\item $K+B^{(n)}$ is lc
\item $B^{(n)}$ is a nondecreasing sequence
(for example, constant)
\end{num}
Then every increasing subsequence in $ld(\xi)$ terminates.
\noindent If, in the addition, one has
\begin{num}
\setcounter{beam}{2}
\item $B^{(n)}$ is an increasing sequence
\end{num}
then every nondecreasing subsequence in $ld(\xi)$ terminates.
\end{utv}
\par\noindent {\sl Proof:}\enspace
As above, we can assume that that {$K+B^{(n)}$ } are in fact lt.
Now let us find out what happens with a nonsingular surface $X$ with
$K+B$ after a single blow up $f:X\to Y$ at the point $P$, $F$ as
usually denotes the exceptional divisor of $f$. The answer is evident:
\begin{equation}
\label{blowup}
f^*(K+\sum b_jB_j)=K_Y+\sum b_jf^{-1}B_j+(-1+\sum mult_PB_jb_j)F
\end{equation}
and the condition $a>0$ translates to $-1+\sum mult_PB_jb_j<1$. If
$-1+\sum mult_PB_jb_j\le0$, then for any further blowups all the log
discrepancies $a_i\ge a$, so they are irrelevant in finding the
minimal log discrepancy and $K+B$ is lt. However, if this is a
positive number, some negative log discrepancies can appear on the
following steps.
Now let $f:X\to Y$ be a composite of several blow ups. One gets
\begin{equation}
\label{blowups}
f^*(K+\sum b_jB_j)=K_Y+\sum b_jf^{-1}B_j+
\sum(-s_i+\sum t_{ik}b_k)F_i
\end{equation}
with some nonnegative integers $s_i$, $t_{ik}$ and $s_i\le\rho(Y/X)$.
The corresponding log discrepancies are given by $$a_i=1+s_i-\sum
t_{ik}b_k$$ and for fixed $s_i$ and nondecreasing/increasing $b_j$
they evidently form a nonincreasing/decreasing sequences. Note that
there are only finitely many such sequences with $a_i\ge0$. Therefore
\ref{utv:local_pasc} follows from the following lemma.
\begin{lem}
With the assumptions as above, there is a constant $N(\xi)$ so that
for every surface {$X^{(n)}$ } in $\xi$ there exists a birational morphism
$g:{Y^{(n)} }\to{X^{(n)}}{ }$ such that
\begin{num}
\item $\rho({Y^{(n)} }/{X^{(n)}}{ })<N(\xi)$
\item the minimal log discrepancy $ld({K+B^{(n)}}{ })$ is one of the log discrepancies
of $g$.
\end{num}
\end{lem}
\par\noindent {\sl Proof:}\enspace
Let us remind that we are in the local situation, so {$X^{(n)}$ } is a
neighbourhood of a (nonsingular) point $P$. Let $f:{Z^{(n)} }\to{X^{(n)}}{ }$ be a
single blow up at $P$. If in the formula~\ref{blowup} the number
$C=-1+\sum mult_PB_jb_j$ is positive and on {$Y^{(n)}$ } the strict transforms
of $B_j$ intersect at one point and have the same multiplicities as on
{$X^{(n)}$ }, then by the formula~\ref{blowup} on the second blowup
codiscrepancy of the exceptional divisor equals $2C$, after the third
blowup $3C$ and so on (and it should be $\le1$). Since $B^{(n)}$ is
nondecreasing, there exists a constant $\varepsilon(\xi)$ so that for any
$-1+\sum m_jb_j>0$, one also has $-1+\sum m_jb_j>\varepsilon(\xi)$. The
conclusion is that there exists a number $N_1$, depending on $\xi$, so
that after $N_1(\xi)$ blowups the configuration of $B_j$ simplifies in
some way: either the number of curves, passing through the points, or
the multiplicities at those points get smaller; or all the further
blowups are irrelevant in finding the minimal discrepancy.
Let $X^{(n)} {}'$ be $X^{(n)}$ with blown up points,
${K+B^{(n)}}{ }'=f^*({K+B^{(n)}}{ })$. Note that the coefficients of ${B^{(n)}}{ }'$ are still
nonnegative numbers. At the neighbourhood of any point of $X^{(n)}{
}'$\enspace ${B^{(n)}}{ }'$ consists of several curves $B_j+\le 2$ nonsingular
curves $F_i$ with coefficients, given by the formula~\ref{blowups} and
hence, nondecreasing, and from the finite list of possible
combinations. Now we can find the next number $N_2(\xi)$ so that after
$N_2(\xi)$ blowups the configuration of ($B_j$ $+$ $\le 2$ nonsingular
curves) simplifies even further. By induction we get the desired
result.
\qed
\medskip
And finally we prove
\begin{utv}[Local ascending chain condition]
\label{utv:local_asc}
Let $\xi=\{X^{(n)},K+B^{(n)}\}$ be a sequence of surfaces such that
\begin{num}
\item $K+B^{(n)}$ is lc
\item $B^{(n)}$ is a nondecreasing sequence
(for example, constant)
\end{num}
Then every increasing subsequence in $ld(\xi)$ terminates.
\noindent If, in the addition, one has
\begin{num}
\setcounter{beam}{2}
\item $B^{(n)}$ is an increasing sequence
\end{num}
then every nondecreasing subsequence in $ld(\xi)$ terminates.
\end{utv}
\par\noindent {\sl Proof:}\enspace By \ref{fin_many_graphs} all the singularities with $ld({K+B })\ge\varepsilon$
are divided into finite number of series $+$ finite number of graphs. The
latters are taken care by \ref{nonsingular}.
So all we have to do is to consider one of the graphs on Fig.2 with
the chain of 2's that is getting longer and longer. And a simple
calculation shows that for all except finitely many graphs the minimal
log discrepancy is in fact one of log discrepancies of $h:{\widetilde X}} \def\xtil{{\tilde x}\to X$.
\qed
\section{Special hyperbolic log canonical graphs}
\noindent
{\bf Set-up\enspace } In this section $\Gam$ or $(X,K+B)$ always
denote the following:
\begin{opr} We say that a graph $\Gam$ is {\bf special hyperbolic} if
\begin{num}
\item $\Gam$ is hyperbolic and connected
\item all the vertices have $p_a=0$, there is a special vertex
$v_0$ of weight 1, all other vertices $v_i$ have weights $\ge2$
\item $\Gam-v_0$ is elliptic
\item as usually, $\Gam$ may have an external part $B=\sum b_jB_j$
\end{num}
\end{opr}
Such graphs naturally appear when one considers a minimal resolution
of singularities of a Del Pezzo surface $X$ with $\rho(X)=1$ and $B_0$
being a (-1)-curve on the resolution.
\begin{utv}[Local-to-global ascending chain condition]
\label{hyper}
Let $\xi=\{{X^{(n)}}{ },{K+B^{(n)}}{ }\}$ be a sequence of special hyperbolic graphs
with a chosen vertex $B_0$ such that
\begin{num}
\item {$K+B^{(n)}$ } is lc
\item \{{$B^{(n)}$ }\} is an increasing sequence, moreover, $\{b_0^{(n)}\}$ is an
increasing sequence
\item all the log discrepancies $a_i\ge 1-\bar b_0=1-\lim b_0$
\item $K+B^{(n)}$ is numerically trivial
\end{num}
Then $\xi$ terminates.
\end{utv}
\par\noindent {\sl Proof:}\enspace
{\par\noindent{\sl Case 1:}\enspace} $\bar b_0=\lim b_0=1$.
{}From \cite{al:lc} it follows that if $b_0$ is close enough to 1, then
all the singularities (that is, the connected components of
$\Gam-v_0$) and the ways the components of $B$ meet $F_i$ are
exhausted by the following list:
\begin{picture}(300,270)(-30,-30)
\multiput(0,0)(0,100){2}{
\put(20,45){$B_0$}
\put(50,50){\circle{14}}
\put(45,55){\line(1,-1){10}}
\put(45,45){\line(1,1){10}}
\put(130,50){\oval(110,20)}
\put(57,50){\line(1,0){18}}}
\multiput(170,20)(0,30){3}{\circle{10}}
\multiput(170,25)(0,30){2}{\line(0,1){20}}
\multiput(180,17)(0,60){2}{2}
\put(85,150){\circle{5}}
\put(135,150){\oval(80,10)}
\put(115,180){\vector(0,-1){25}}
\put(105,180){$q$}
\put(145,120){\vector(0,1){25}}
\put(150,115){$m$}
\multiput(135,185)(40,0){2}{\circle{14}}
\multiput(130,190)(40,0){2}{\line(1,-1){10}}
\multiput(130,180)(40,0){2}{\line(1,1){10}}
\multiput(150,180)(5,0){3}{\circle*{1}}
\put(135,178){\line(5,-6){15}}
\put(175,178){\line(-5,-6){15}}
\put(110,-15){Figure 3}
\end{picture}
The next step is a formula for the coefficient $b_0$ that follows from
the explicit calculations of \cite{al:lc}:
\begin{equation}
\label{horrible}
b_0={\sum_{s=1}^{N}{{q_s+\alpha} \def\Alp{\Alpha_s}\over m_s}-(N-1)\over
\sum_{s=1}^{N} {q_s\over
m_s}-1}=1-{\frac{(N-2)-\sum_{s=1}^{N}{\frac{\alpha} \def\Alp{\Alpha_s}{m_s}}}
{\sum_{s=1}^{N}{\frac{q_s}{m_s}}-1}}
\end{equation}
with denominator $>0$, where $N$ is a number of connected components
of $\Gam-v_0$. Here $\alpha} \def\Alp{\Alpha_s=1-\sum l_j^sb_j$ as in
\ref{fin_many_graphs}. We consider the second case of the figure~3
formally as a subcase of the first one with $q=m$ and $\alpha} \def\Alp{\Alpha=0$.
Note that by \ref{max_hyp} a number of graphs that $v_0^{(n)}$ is
connected with in the sequence is bounded.
For any fixed $N$ the conditions $\lim b_0=1$ and $b_0<1$ imply
$$ \sum_{s=1}^N{\frac{\alpha} \def\Alp{\Alpha_s}{m_s}}<N-2\quad and\quad
\lim \sum_{s=1}^N{\frac{\alpha} \def\Alp{\Alpha_s}{m_s}}=N-2.$$
We can assume that some of $m_s$ are fixed and others tend to
infinity. For the latters ${\frac{\alpha} \def\Alp{\Alpha_s}{m_s}}\to 0$ and
${\frac{\alpha} \def\Alp{\Alpha_s}{m_s}}>0$. This is so by \ref{fin_many_graphs} (here
it is important again that there is a constant $\varepsilon(\xi)$ so that
$\sum m_jb_j-1>0$ implies $\sum m_jb_j-1>\varepsilon(\xi)$) and by
\ref{utv:loc_bound}. So we can assume that
$\sum_{s=1}^M{\frac{\alpha} \def\Alp{\Alpha_s}{m_s}}<N-2$ and $$\lim
\sum_{s=1}^M{\frac{\alpha} \def\Alp{\Alpha_s}{m_s}} =\lim\sum_{s=1}^M{\frac{1-\sum
l_j^sb_j}{m_s}}=N-2$$ with $m_1...m_M$ being fixed. But this
definitely gives a contradiction. Note that $\sum l_j^sb_j\le 2$ by
\ref{utv:loc_bound}, so there are only finitely many possibilities for
$l_j^s$.
Finally, for $N\ge5$ $$b_0\le1-{\frac{(N-2)-\sum{1\over{m_s}}}{N-1}}
\le 1-{\frac{{N\over 2}-2}{N-1}}\le {7\over 8}$$
\noindent and we are done.
\medskip
{\par\noindent{\sl Case 2:}\enspace} $\bar b_0=\lim b_0<1$.
Since all the log discrepancies $a_i\ge\varepsilon=1-{\bar b}_0$, the only
infinite series of connected components of $\Gam-v_0$ are given by
\ref{fin_many_graphs}. Moreover, for the minimal log discrepancies
there $$\lim\min a_i\le{\frac{1-\sum(B_0F_i)r_ib_0}{m-k}}$$
\noindent and this should be not less than $1-\bar b_0$.
As a conclusion, all the infinite series are given by
\begin{picture}(300,250)(-50,-40)
\multiput(0,0)(0,100){2}{
\put(-5,45){$B_0$}
\put(20,50){\circle{14}}
\put(15,55){\line(1,-1){10}}
\put(15,45){\line(1,1){10}}
\put(27,50){\line(1,0){15.5}}
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\multiput(42.5,55)(20,0){2}{2}
\put(47.5,50){\line(1,0){15}}
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\multiput(80,50)(5,0){3}{\circle*{1}}}
\multiput(105,150)(20,0){2}{\circle{5}}
\multiput(102.5,155)(20,0){2}{2}
\put(107.5,150){\line(1,0){15}}
\put(127.5,150){\line(1,0){17.5}}
\put(190,150){\oval(90,20)}
\put(155,150){\circle{5}}
\put(195,150){\oval(60,10)}
\multiput(155,185)(70,0){2}{\circle{14}}
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\multiput(150,180)(70,0){2}{\line(1,1){10}}
\multiput(180,180)(10,0){3}{\circle*{1}}
\put(155,178){\line(5,-6){15}}
\put(225,178){\line(-5,-6){15}}
\multiput(105,30)(0,20){3}{\circle{5}}
\multiput(110,28.5)(0,20){3}{2}
\multiput(105,32.5)(0,20){2}{\line(0,1){15}}
\put(125,-15){Figure 4}
\end{picture}
Now we would like to use a variant of the formula~\ref{horrible}.
However, $B_0$ can intersect finitely many types of graphs
arbitrarily. Still, for any fixed combination, if $b_j$ increase,
$b_0$ decreases. The situation is exactly the opposite to the one of
elliptic graphs since the signature of the quadratic form is now
$(1,n-1)$ instead of $(0,n)$ and the graph $\Gam-v_0$ is still
elliptic (cf. \ref{max_hyp}).
All the said above implies that for $b_0$ there are only finitely many
possible expressions of the form $$b_0=1-{C_1+\sum
C_2^jb_j-\sum_{s=1}^{N}{\alpha} \def\Alp{\Alpha_s\over m_s}\over
C_3+\sum_{s=1}^{N}{q_s\over m_s}}$$ with fixed
$C_1,C_2^j,C_3,m_s-q_s$, $m_s\to+\infty$, $C_2^j\ge0$ and the
denominator $>0$. Simplifying, $$1-b_0={C_1+\sum
C_2^jb_j-\sum_{s=1}^{N}{\alpha} \def\Alp{\Alpha_s\over m_s}\over
C_3'-\sum_{s=1}^{N}{m_s-q_s\over m_s}}$$
Now $\lim b_0=\bar b_0$ implies $(C_1+\sum C_2^j\bar b_j)/C_3'=1-\bar
b_0$. And, finally, the inequalities ${\alpha} \def\Alp{\Alpha_s\over m_s-q_s}\ge 1-\bar
b_0$ and $C_2^j\ge0$ imply that$1-b_0\le1-\bar b_0$, i.e. $b_0\ge \bar
b_0$ that gives a contradiction.
\qed
\begin{zam}
As the proof shows,
\ref{hyper} is not true without the assumption (iii).
\end{zam}
\section{Global case}
\begin{utv}[Global boundness]
\label{utv:glob_bound}
Let $X$ be a surface with a lc divisor $K+B$ and assume that $f:X\to
Y$ is a contraction of an extremal ray such that $K+B$ if
$f$-nonpositive. Let $B^+=\sum b_j^+B_j^+$ contain all the components
in $B$ that are $f$-positive. Then
\begin{num}
\item if $\dim Y=2$, $\sum b_j^+\le2$
\item if $\dim Y=1$, $\sum b_j^+\le2$
\item if $\dim Y=0$, $\sum b_j^+\le3$
\end{num}
\end{utv}
\par\noindent {\sl Proof:}\enspace
(i) follows from \ref{utv:loc_bound}, because $K_Y+f(B)$ is also lc.
(ii) is clear: if $B^+$ is not empty, then $-K$ should be negative on
the general fiber, so a general fiber $F$ is isomorphic to $\bfP^1$
and $\sum b_j^+\le B^+F\le -KF=2$.
In the case (iii) if $X$ is nonsingular, then $X\simeq {\bfP^2}$ and
the statement is evident. If $X$ is singular, consider a partial
resolution $g:Z\to X$, dominated by the minimal resolution and such
that $\rho(Z)=\rho(X)+1=2$. Then by \ref{Del_Pezzo} there is a second
extremal ray and $g^*(K+B)=K_Z+B_Z$ is nonpositive with respect to
this extremal ray. Since every curve on $Z$ is positive with respect
to at least one of the extremal rays, (iii) with the bound 4 follows
immediately. To get the bound 3 is an easy exercise.
\qed
\begin{zam}
\ref{utv:glob_bound}(iii)
is also proved in \cite{ag-kol} for arbitrary dimension with a bound
$n+1$.
\end{zam}
\begin{utv}[Global ascending chain condition]
\label{global_asc}
Let $\xi=\{X^{(n)},K+B^{(n)}\}$ be a sequence of surfaces such that
\begin{num}
\item $K+B^{(n)}$ is lc
\item $B^{(n)}$ is an increasing sequence
\item $K+B^{(n)}$ is numerically trivial
\end{num}
Then $\xi$ terminates.
\end{utv}
Proof of \ref{global_asc} will be given in several steps.
\setcounter{shag}{0}
\begin{shag}
One can assume that all the surfaces {$X^{(n)}$ } are Del Pezzo surfaces with
$\rho({X^{(n)}}{ })=1$.
\end{shag}
\par\noindent {\sl Proof:}\enspace
We can assume that the lengths of the groups {$B^{(n)}$ } in the sequence $\xi$
are constant and that $b_1$ always increases. Now consider a divisor
$K+B-\varepsilon B_1$ on $X^{(n)}$.
Note here that $B_1$ is $\bfQ$-factorial by the classification of log
canonical singgularities.
It is lc and is not numerically effective
and if $B_1^2\le0$ then $(K+B-\varepsilon B_1)B_1\ge0$. Therefore either
$\rho({X^{(n)}}{ })=1$ and then {$X^{(n)}$ } is a Del Pezzo surface with lc $K+B$ or
there is an extremal ray that does not contract $B_1$. If the
contraction is birational, we make it and repeat the same procedure
again. If it is a fibration, the claim follows from the corresponding
1-dimensional statement.
\qed
\begin{zam} The argument works in the 3-dimensional case as well.
\end{zam}
\begin{shag}
One can assume that there are only finitely many different types of
graphs of singularities that the increasing components of {$B^{(n)}$ } are
passing through.
\end{shag}
\par\noindent {\sl Proof:}\enspace
As usually, we can assume that the groups {$B^{(n)}$ } have the same length.
Now consider the set $PLD(\xi)$. By \ref{utv:local_pasc} this set
satisfies the ascending chain condition and has at least one limit
point. Let $l$ be the minimal limit point of $PLD(\xi)$. Fix the
number $C$ so that all $b_j\ge C$. If the surfaces in $\xi$ contain
singularities that correspond to infinitely many elliptic graphs, then
by \ref{fin_many_graphs} $l\le1-C$. Passing to a subsequence we can
assume that a sequence of minimal log discrepancies, which we shall
denote $\{a^{(n)}_s\}$ is a decreasing sequence and $\lim a^{(n)_s}=l$
(the sequence of codiscrepancies is increasing and $\lim
b_0^{(n)_s}=1-l\ge C$.
Now consider a partial resolution $f:{Y^{(n)} }\to{X^{(n)}}{ }$ which is dominated
by the minimal desingularization and which blows up exactly the curve
$B^{(n)_s}$. Then $$f^*(K+{B^{(n)}}{ })=K_Y+f^{-1}{B^{(n)}}{ }+b^{(n)_s}B^{(n)_s}$$
The surface {$Y^{(n)}$ } has Picard number 2 and by \ref{Del_Pezzo} there is a
second extremal ray, corresponding to a $(-1)$-curve on ${\widetilde Y}} \def\ytil{{\tilde y}={\widetilde X}} \def\xtil{{\tilde x}$.
Let $g:{Y^{(n)} }\to{X'^{(n)}}$ be the contraction of this second extremal ray.
If $g$ is a fibration then restricting of
$K_Y+f^{-1}{B^{(n)}}{ }+b^{(n)_s}B^{(n)_s}$ on the general fibre of $g$ readily
gives a contradiction. Hence, we shall assume that $g$ is a birational
morphism. A divisor $K+{B'^{(n)} }=g_*f^*(k+{B^{(n)}}{ })$ is lc and numerically
trivial, {$B'^{(n)}$ } has either the same number of components as {$B^{(n)}$ } or one
more, and, after passing to a subsequence, $B'^{(n)}$ is an increasing
sequence.
A morphism $g$ can contract one of the components of {$B^{(n)}$ } and we can
assume that it is always, say, $B_0$. However, by \ref{hyper} and
\ref {utv:local_pasc} the sequence $\{b_0^{(n)}\}$ cannot be an
increasing sequence with $\lim b_0^{(n)}\ge1-l$. Therefore, changing the
sequence $\xi=\{X^{(n)}\}$ by a new sequence $\xi'=\{X'^{(n)}\}$, we
are gaining a new component with increasing coefficient that has the
limit $1-l\ge C$. Note that for a new minimal limit point $l'$ of
$PLD(\xi')$ one has $l'\ge l$. This is so because a minimal
desingularization of {$X'^{(n)}$ } is dominated by the minimal
desingularization of {$X^{(n)}$ } and $K+B^{(n)}$, $K+B'^{(n)}$ both are
numerically trivial, so $PLD(\xi')$ is a subset in $PLD(\xi)$.
Repeating the procedure, we get one more component and so on. After
$k$ steps the sum of the coefficients in {$B^{(n)}$ } will be greater than
$kC$. This eventually will get into the contradiction with
\ref{utv:glob_bound}.
\qed
\begin{shag}
One can assume that all the surfaces {$X^{(n)}$ } are isomorphic to each other.
\end{shag}
\par\noindent {\sl Proof:}\enspace
By \ref{log_Del_Pezzo} a surface ${\widetilde X}} \def\xtil{{\tilde x}^{(n)}$ is either a locally
trivial $\bfP^1$-bundle with a section which is a smooth elliptic
curve or a rational surface with rational singularities. In the former
case the statement follows from the 1-dimensional analog by
restricting $B$ to the fiber of the fibration. Now assume we are in
the latter case. By the previous step, there exists a constant
$N(\xi)$ so that for the increasing component $B_1$ of $B$\enspace
$NB_1$ is Cartier. Hence for any curve $D$ on
{$X^{(n)}$ }
$$-KD=\sum b_jB_jD\ge b_1/N\ge C/N$$ Now theorem~2.~3 of \cite{al:fi}
states that for all such surfaces $\rho({\widetilde X}} \def\xtil{{\tilde x})$ is bounded. Therefore
one can get $X^{(n)}$ by blowing up finitely many points from the
minimal rational surface $\bfF_k$. Threfore there are only finitely
many possibilities for the graph of exceptional curves on $X^{(n)}$
except for the fact that one weight $k$ can be arbitrary. Now if
$B_1$ does not lie in the fiber for infinitely many $n$ we prove the
statement restricting a numerically trivial divisor $K+\sum
b_jB_j+\sum b_i F_i$ to the fiber and using \ref{utv:local_pasc}.
Otherwise (recall that $\rho(X)=1$) $B_1$ on $X$ should pass through
the singularity which graph contains the exceptional curve of
$\bfF_k$. By the previous step $k$ is bounded. Hence we can assume
that the surfaces $X^{(n)}$ belong to a bounded family and it is
enough to consider only finitely many of them.
\qed
\begin{zam}
Theorem 2.3 in \cite{al:fi} is stated for log terminal singularities.
But in fact the proof is exactly the same for rational log canonical
singularities.
\end{zam}
\begin{shag}
\ref{global_asc} follows.
\end{shag}
\par\noindent {\sl Proof:}\enspace
Indeed, there are only finitely many possibilities for effective Weil
divisors $B_j$.
\qed
\medskip
The following example shows that \ref{global_asc} is not true without
the assumption~(i).
\begin{pri}
Consider a sequence of surfaces {$X^{(n)}$ } so that ${\widetilde X}} \def\xtil{{\tilde x}^{(n)}=\bfF_n$ and
${B^{(n)}}{ }=(1-1/n)F_1+3/4(F_2+F_3+F_4)$ where $F_1$ is an image of the
infinite section of $\bfF_n$, $F_{2,3,4}$ are fibres. Note that {$K+B^{(n)}$ }
is numerically trivial but is not lc.
\end{pri}
|
1992-06-25T00:15:01 | 9206 | alg-geom/9206009 | en | https://arxiv.org/abs/alg-geom/9206009 | [
"alg-geom",
"math.AG"
] | alg-geom/9206009 | Temporary | G.Mikhalkin | The complex separation and extensions of Rokhlin congruence for curves
on surfaces | 17 pages, LaTeX | null | null | null | null | The subject of this paper is the problem of arrangement of real algebraic
curves on real algebraic surfaces. In this paper we extend Rokhlin,
Kharlamov-Gudkov-Krakhnov and Kharlamov-Marin congruences for curves on
surfaces and give some applications of this extension.
For some pairs consisting of a surface and a curve on this surface (in
particular for M-pairs) we introduce a new structure --- the complex separation
that is separation of the complement of curve into two surfaces. In accordance
with Rokhlin terminology the complex separation is a complex topological
characteristic of real algebraic varieties. The complex separation is similar
to complex orientations introduced by O.Ya.Viro (to the absolute complex
orientation in the case when a curve is empty and to the relative complex
orientation otherwise). In some cases we calculate the complex separation of a
surface (for example in the case when surface is the double branched covering
of another surface along a curve). With the help of these calculations
applications of the extension of Rokhlin congruence gives some new restrictions
for complex orientations of curves on a hyperboloid.
| [
{
"version": "v1",
"created": "Wed, 24 Jun 1992 22:09:25 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Mikhalkin",
"G.",
""
]
] | alg-geom | \section{Introduction}
\subsection{Rokhlin and Kharlamov-Gudkov-Krakhnov congruences
for curves on surfaces}
If $A$ is a real curve of odd degree on real projective surface $B$
then $A$ divides $B$ into two parts $B_+$ where polynomial determining $A$
is non-negative and $B_-$ where this polynomial is non-positive.
V.A.Rokhlin \cite{R} proved the congruence for the Euler characteristic
of $B_-$ under such strong hypotheses that they follow that $B$ is an
M-surface,
$A$ is an M-curve and $B_+$ is contained in a connected component of $B$.
V.M.Kharlamov \cite{Kh}, D.A.Gudkov and A.D.Krakhnov \cite{GK} proved a
relevant
congruence that makes sense sometimes even if $A$ is not M- but (M-1)-curve,
but the hypotheses for $B$ and $B_+$ were not weakened.
\subsection{Description of the paper}
The results of the paper make sense in the case when a pair consisting
of a surface and a curve in this surface is of characteristic type
(for definition see section \ref{not}).
For pairs of characteristic type we introduce a complex separation of the
complement of the curve in the surface that is a new complex topological
characteristic of pairs consisting of a surface and a curve in this surface.
If the curve is empty then the complex separation is a new complex topological
characteristic of surfaces relevant to the complex orientation of surfaces
introduced by O.Viro \cite{V1}.
The main theorem is formulated with the help of the complex separation,
this theorem is a generalization of Rokhlin and Kharlamov-Gudkov-Krakhnov
congruences for curves in surfaces.
The main theorem gives nontrivial restrictions even for curves of odd degree
on some surfaces.
The paper contains also some applications of the main theorem.
We prove a congruence modulo 32 for Euler characteristic of real connected
surface of characteristic type.
We prove some new congruences for curves on a hyperboloid.
We give a direct extension of Rokhlin and Kharlamov-Gudkov-Krakhnov
congruences for curves on projective surfaces, there we avoid both of
the hypotheses that $B$ is an M-surface and $B_+$ is contained in a connected
component of $B$.
We apply the main theorem to the classification of curves of low degrees
on an ellipsoid.
In particular, we get a complete classification of flexible curves
on an ellipsoid of bidegree (3,3) (the notion of flexible curve
is analogous to one introduced by O.Viro \cite{V} for plane case).
One can see that the fact that the theorem can be applied to curves of odd
degree follows that this theorem can not be proved in the Rokhlin approach
using the double covering of the complexification of the surface branched along
the curve (since there is no such a covering for curves of odd degree).
We use the Marin approach \cite{Marin}. All results of this paper apply to
flexible curves as well as to algebraic.
The author is indebted to O.Ya.Viro for his attention to the paper and
consultations.
\section{Notations and the statement of the main theorem}
\label{not}
Let ${\bf C} B$ be a smooth oriented 4-manifold such that its first ${\bf Z}_2$-Betti
number is zero equipped with an involution $conj$ such that the set ${\bf R} B$ of
its fixed points is a surface.
Let ${\bf C} A$ be a smooth surface in ${\bf C} B$ invariant under $conj$ and such that
the intersection of ${\bf C} A$ and ${\bf R} B$ is a curve.
These notations are inspired by algebraic geometry.
It is said that $A$ is of even degree if ${\bf C} A$ is ${\bf Z}_2$-homologous to zero
in ${\bf C} B$ and that $A$ is of odd degree otherwise.
It is said that curve $A$ is of type I if ${\bf R} A$ is ${\bf Z}_2$-homologous to zero
in ${\bf C} A$ and that $A$ is of type II otherwise.
It is said that surface $B$ is of type I$abs$ if ${\bf R} B$ is ${\bf Z}_2$-homologous
to zero in ${\bf C} B$
If $B$ is a real projective plane then it is said that $B$ is of type I$rel$
if ${\bf R} B$ is ${\bf Z}_2$-homologous to a plane section of ${\bf C} B$.
We shall say that pair $(B,a)$ is of characteristic type if the sum of
${\bf R} B$ and ${\bf C} A$ is Poincar\'{e} dual to the second Stiefel-Whitney class
of ${\bf C} B$.
We shall say that surface $B$ is of characteristic type if ${\bf R} B$ is a
characteristic surface in ${\bf C} B$.
Let $b_*$ denote the total ${\bf Z}_2$-Betti number.
It is said that manifold ${\bf C} X$ equipped with involution $conj$ is an
(M-$j$)-manifold if $b_*({\bf R} X)+2j=b_*({\bf C} X)$ where ${\bf R} X$ is the fixed point
set
of $conj$.
One can easily see that Smith theory follows that $j$ is a nonnegative integer
number.
Let $\sigma(M)$ denote the signature of oriented manifold $M$.;
$D_M:H^*(M;{\bf Z}_2)\rightarrow H_*(M;{\bf Z}_2)$ denote Poincar\'{e} duality operator;
$[N]\in H_*(M;{\bf Z}_2)$ denote ${\bf Z}_2$-homology class of submanifold $N\subset M$.
Let $e_A=[{\bf C} A\circ{\bf C} A]_{{\bf C} B}$ denote normal Euler number of ${\bf R} B$ in ${\bf C}
B$.
If $B_\epsilon$ is a surface contained in ${\bf R} B$ and such that
$\partial B_\epsilon={\bf R} A$ that we shall denote by $e_{B_\epsilon}$
the obstruction to extending of line bundle over ${\bf R} A$ and normal to ${\bf R} A$
in ${\bf C} A$ to the line bundle over $B_\epsilon$ normal in ${\bf C} B$ to ${\bf R} B$
evaluated on the twisted fundamental class $[B_\epsilon,\partial B_\epsilon]$
and divided by 2.
One can see that if $(B,A)$ is a nonsingular pair consisting of
an algebraic surface and algebraic curve then
$e_{B_\epsilon}=-\chi(B_\epsilon)$.
Let $\beta(q)$ denote the Brown invariant of ${\bf Z}_4$-valued quadratic form $q$.
\begin{th}
\label{main}
If $(B,A)$ is of characteristic type then there is a natural separation of
${\bf R} B-{\bf R} A$ into surfaces $B_1$ and $B_2$ such that
$\partial B_1=\partial B_2={\bf R} A$
defined by the condition that ${\bf C} A/conj\cup B_j$ is a characteristic surface
in ${\bf C} B/conj (j=1,2)$.
There is a congruence for the Guillou-Marin form $q_j$ on
$H_1({\bf C} A/conj\cup B_j;{\bf Z}_2)$
\begin{displaymath}
e_{B_j}\equiv\frac{e_{{\bf R} B}+\sigma({\bf C} B)}{4}-\frac{e_A}{4}-\beta(q_j)\pmod{8}
\end{displaymath}
\end{th}
\begin{add}
Let $q_j|_{H_1({\bf R} A;{\bf Z}_2)}=0$
\begin{itemize}
\begin{description}
\item[a)] If $A$ is an M-curve then
$e_{B_j}\equiv\frac{e_{{\bf R} B}+\sigma({\bf C} B)}{4}-\frac{e_A}{4}-\beta_j\pmod{8}$
\item[b)] If $A$ is an (M-1)-curve then
$e_{B_j}\equiv\frac{e_{{\bf R} B}+\sigma({\bf C}
B)}{4}-\frac{e_A}{4}-\beta_j\pm1\pmod{8}$
\item[c)] If $A$ is an (M-2)-curve and
$e_{B_j}\equiv\frac{e_{{\bf R} B}+\sigma({\bf C} B)}{4}-\frac{e_A}{4}-\beta_j+4\pmod{8}$
then $A$ is of type I
\item[d)] If $A$ is of type I then
$e_{B_j}\equiv\frac{e_{{\bf R} B}+\sigma({\bf C} B)}{4}-\frac{e_A}{4}\pmod{4}$
\end{description}
\end{itemize}
where $\beta_j$ is the Brown invariant of the restriction
$q_j|_{H_1(B_j;{\bf Z}_2)}$
\end{add}
\subsection{Remark}
Some of components of ${\bf R} A$ can be disorienting loops in ${\bf C} A$
(it is easy to see that number of such components is even).
If $\alpha$ is some 1-dimensional ${\bf Z}_2$-cycle in $B_j$ that is a boundary
of some 2-chain $\beta$ in ${\bf R} B$ containing an even number of
disorienting ${\bf C} A$ components of ${\bf R} A$ then $q_j(\alpha)=0$; if such a number
is odd
then $q_j(\alpha)=2$.
It follows that if $(B,A)$ is of characteristic type then the number
of disorienting ${\bf C} A$ components of ${\bf R} A$ in each component of ${\bf R} B$
is even.
\section{The proof of Theorem 1 and Addendum 1}
\subsection{Calculation of the characteristic class of ${\bf C} B/conj$}
It is not difficult to see the formula for the characteristic classes of
double branched covering:
if $\pi:Y\rightarrow X$ is a double covering branched along $Z$ then
$$w_2(Y)=\pi^*w_2(X)+D^{-1}_Y[Z]$$
Applying this formula to $p:{\bf C} B\rightarrow {\bf C} B/conj$ we get
$$tr(Dw_2({\bf C} B/conj))=Dw_2({\bf C} B)+[{\bf R} B]$$
where $tr:H_2({\bf C} B/conj;{\bf Z}_2)\rightarrow H_2({\bf C} B;{\bf Z}_2)$ is transfer
(i.e. the inverse Hopf homomorphism to $P$).
It is easy to see that transfer can be decomposed as the composition
$$H_2({\bf C} B/conj;{\bf Z}_2)\stackrel{k}{\rightarrow}H_2({\bf C} B/conj,{\bf R} B;{\bf Z}_2)
\stackrel{h}{\rightarrow}H_2({\bf C} B;{\bf Z}_2)$$
where $k$ is an inclusion homomorphism and $h\circ k=tr$.
To prove that $h$ is a monomorphism we use the Smith exact sequence
(see e.g. \cite{W}):
$$H_3({\bf C} B/conj,{\bf R} B;{\bf Z}_2)\stackrel{\gamma_3}{\rightarrow}
H_2({\bf R} B;{\bf Z}_2)\oplus H_2({\bf C} B/conj,{\bf R} B;{\bf Z}_2)\stackrel{\alpha_2}{\rightarrow}
H_2({\bf C} B;{\bf Z}_2)$$
In this sequence the first component of $\gamma_3$ is equal to
the boundary homomorphism $\partial$ of pair $({\bf C} B/conj,{\bf R} B)$;
$\partial$ is a monomorphism since $H_3({\bf C} B/conj;{\bf Z}_2)=0$
(since ${\bf C} B$ and therefore ${\bf C} B/conj$ are simply connected).
It means that no element of type $(0,x)\in H_2({\bf R} B;{\bf Z}_2)\oplus
H_2({\bf C} B/conj,{\bf R} B;{\bf Z}_2), x\neq 0$ is contained in $Im\gamma_3$
and therefore the restriction of $\alpha_2$ to $H_2({\bf C} B/conj,{\bf R} B;{\bf Z}_2)$
is a monomorphism and thus $H$ is a monomorphism.
Now if $[{\bf C} A]=Dw_2({\bf C} B)+[{\bf R} B]$ then
$$Dw_2({\bf C} B/conj)=[{\bf C} A/conj]\in H_2({\bf C} B/conj,{\bf R} B;{\bf Z}_2)$$
The exactness of homology sequence of pair $({\bf C} B/conj,{\bf R} B)$ follows
since $H$ is a monomorphism that there exists a surface $B_1\subset{\bf R} B$
such that $\partial B_1={\bf R} A$ and $W_1=B_1\cup{\bf C} A/conj$ is dual to
$w_2({\bf C} B/conj)$. Let $B_2$ be equal to $Cl({\bf R} B-B_1)$
Surface ${\bf R} B$ is ${\bf Z}_2$-homologous to zero in ${\bf C} B/conj$ since ${\bf C} B$ is
a double covering of ${\bf C} B/conj$ branched along ${\bf R} B$.
It follows that $W_2=B_2\oplus{\bf C} A/conj$ is also a surface dual to
$w_2({\bf C} B/conj)$.
Note that the separation of ${\bf R} B$ into $B_1$ and $B_2$ is unique since
$$dim(ker(in_*:H_2({\bf R} B;{\bf Z}_2)\rightarrow H_2({\bf C} B/conj)))=1$$
as it follows from exactness of the Smith sequence.
Indeed, since $H_3({\bf C} B/conj;{\bf Z}_2)=0$ then this dimension is equal to the
dimension of $H_3({\bf C} B/conj,{\bf R} B;{\bf Z}_2)$;
the dimension of $H_3({\bf C} B/conj,{\bf R} B;{\bf Z}_2)$ is equal to 1 since
$\gamma_4:H_4({\bf C} B/conj,{\bf R} B;{\bf Z}_2)\rightarrow {0}\oplus H_3({\bf C} B/conj,{\bf R} B;
{\bf Z}_2)$ is an isomorphism.
\subsection{Proof of the congruence for $\chi(B_j)$}
\label{prth}
Note that since $W_j,j\in \{1,2\}$ is a characteristic surface in ${\bf C} B/conj$
and ${\bf C} B$ is simply connected
we can apply the Guillou-Marin congruence to pair $({\bf C} B/conj,W_j)$
$$\sigma({\bf C} B/conj)\equiv W_j\circ W_j+2\beta (q_j)\pmod{16}$$
where $q_j:H_1(W_j;{\bf Z}_2)\rightarrow{\bf Z}_4$ is the quadratic form associated
to the embedding of $W_j$ into ${\bf C} B/conj$ (see \cite{GM}).
Similar to the calculations in \cite{Marin} we get that
$$W_j\circ W_j=\frac{e_A}{2}-2\chi(B_j)$$
The Atiyah-Singer-Hirzebruch formula follows that
$$\sigma({\bf C} B/conj)=\frac{\sigma({\bf C} B)-\chi({\bf R} B)}{2}$$
Combining all this we get
$$\chi(B_j)\equiv\frac{e_A}{4}+\frac{\chi({\bf R} B)-\sigma({\bf C} B)}{4}+
\beta(q_j)\pmod{8}$$
Now if $q|_{H_1({\bf R} A;{\bf Z}_2)}=0$ then additivity of the Brown invariant
(see \cite{KV}) follows that
$$\beta(q_j)=\beta(q|_{H_1({\bf C} A/conj;{\bf Z}_2)})+\beta_j$$
It is easy to see that if $A$ is an (M-$j$)-curve then
$rkH_1({\bf C} A/conj;{\bf Z}_2)=j$.
Points a) and b) of the addendum immediately follow from this.
To deduce points c) and d) of the addendum note that
$q|_{H_1({\bf C} A/conj;{\bf Z}_2)}$ is even iff
${\bf C} A/conj$ is an orientable surface iff $A$ is of type I (cf. \cite{Marin},
\cite{KV}).
\section{Some applications of Theorem 1}
\subsection{The case ${\bf C} A$ is empty; congruences for surfaces}
\label{pov}
\subsubsection{}
\label{surface}
If $Dw_2({\bf C} B)=[{\bf R} B]$ then there is defined a complex separation of ${\bf R} B$
into two closed surfaces $B_1$ and $B_2$;
there is defined a ${\bf Z}_4$-quadratic form $q$ on $H_1({\bf R} B;{\bf Z}_2)=
H_1(B_1;{\bf Z}_2)\oplus H_1(B_2;{\bf Z}_2)$ equal to sum of Guillou-Marin forms
of $B_1$ and $B_2$ which are characteristic surfaces in ${\bf C} B/conj$ and
\begin{displaymath}
\chi(B_j)\equiv\frac{\chi({\bf R} B)-\sigma({\bf C} B)}{4}+\beta(q|_{H_1(B_j;{\bf Z}_2)})
\pmod{8}
\end{displaymath}
\subsubsection{}
\label{pusto}
If $Dw_2({\bf C} B)=[{\bf R} B]$ and $B_j$ is empty for some $j$
(that is evidently true if ${\bf R} B$ is connected) then
$$\chi({\bf R} B)\equiv\sigma({\bf C} B)\pmod{32}$$
\subsubsection{}
If $Dw_2({\bf C} B)=[{\bf R} B]$ then $$\chi({\bf R} B)\equiv\sigma({\bf C} B)\pmod{8}$$
{\em\underline{Proof}} It follows from an easy observation that $\chi(B_j)
\equiv\beta(q|_{H_1(B_j;{\bf Z}_2)})\pmod{2}$
\subsubsection{Remark}
According to O.Viro \cite{V} in some cases one can define some more complex
topological characteristics on ${\bf R} B$.
Namely, If $B$ is of type I$abs$ then ${\bf R} B$ possesses two special
reciprocal orientations (so-called semi-orientation) and a special spin
structure.
If $Dw_2({\bf C} B)=[{\bf R} B]$ then ${\bf R} B$ possesses the special $Pin_-$-structure
corresponding to Guillou-Marin form $q_{{\bf C} B}:H_1({\bf R} B;{\bf Z}_2)\rightarrow{\bf Z}_4$
of surface ${\bf R} B$ in ${\bf C} B$.
The complex separation is a new topological characteristic
for surfaces and \ref{surface} may be interpreted as a formula for this
characteristic.
Quadratic form $q$ is not a new complex topological characteristic.
\subsubsection{}
Form $q$ is equal to $q_{{\bf C} B}$ in the case when these forms are defined (i.e.
when
$Dw_2({\bf C} B)=[{\bf R} B]$)
To prove this one can note that the index of a generic membrane in ${\bf C} B$
bounded by curves in ${\bf R} B$ differs from the index of the image of this
membrane in ${\bf C} B/conj$ by number of intersection points of this membrane
and ${\bf R} B$.
\subsubsection{Remark}
If $B$ is a complete intersection in the projective space of hypersurfaces
of degrees $m_j,j=1,\ldots,s$ then the condition that $Dw_2({\bf C} B)=[{\bf R} B]$
is equivalent to the condition that $B$ is of type I$abs$ in the case
when $\sum^{s}_{j=1}m_j\equiv 0\pmod{2}$ and to the condition that $B$ is
of type I$rel$ in the case when $\sum^{s}_{j=1}m_j\equiv 1\pmod{2}$.
\subsection{Calculations for double coverings}
\label{vych}
We see that to apply Theorem 1 one needs to be able to calculate the complex
separation and the corresponding Guillou-Marin form.
We calculate them in some cases in this subsection.
By the semiorientation of manifold $M$ we mean a pair of reciprocal
orientations of $M$ (note that this notion is nontrivial only for non-connected
manifolds).
It is easy to see that two semiorientations determine a separation of $M$;
this separation is a difference of two semiorientations,
namely, two components of $M$ are of the same class of separation iff the
restrictions of the semiorientations on these components are the same.
Let ${\bf C} B$ be the double covering of surface ${\bf C} X$
branched along curve ${\bf C} D$
invariant under the complex conjugation $conj_X$ in ${\bf C} X$.
Suppose ${\bf C} D$ is of type I.
Let ${\bf C} D_+$ be one of two components of ${\bf C} D-{\bf R} D$.
Then ${\bf R} D$ possesses a special semiorientation called the complex
semiorientation (see \cite{R1}).
The invariance of ${\bf C} D$ under $conj_X$ follows that $conj_X$ can be lifted
in two different ways into an involution of ${\bf C} B$.
Let $conj_B$ be one of these two lifts.
Let $X_-=p({\bf R} B))$, $X_+={\bf R} X-int(X_-)$ where $p$ is the covering map.
\subsubsection{}
\label{calor}
Suppose that $Dw_2({\bf C} B)=[{\bf R} B]$.
Then for every component $C$ of $X_+$ the complex separation of $\partial C$
induced from complex separation of ${\bf R} B$ via $p$ (namely, two circles of
$\partial C$ are of the same class of separation iff the lie in the image
under $p$ of the same class of the separation of ${\bf R} B$)
is equal to the difference of the semiorientations on $\partial C$
induced by the complex semiorientation of ${\bf R} D$ and the unique (since $C$ is
connected) semiorientation of $C$.
In particular $C$ is orientable.
{\em\underline{Proof}} Let $\alpha$ and $\beta$ be components of $\partial C$.
Consider two only possible cases: the first case when the semiorientations
induced from ${\bf R} D$ and $C$ are equal on $\alpha$ and $\beta$ and the second
case when they are different (see fig.1, arrows indicate one of two
orientations induced by the complex semiorientation of ${\bf R} D$).
Choose a point $Q_{\alpha}$ in $\alpha$ and $Q_{\beta}$ in $\beta$.
Connect these points by a path $\gamma$ inside $C$ and by a path $\delta$
inside ${\bf C} D_+$ (not visible on the picture) without self-intersection points.
It is easy to see that there exists a disk $F'\subset{\bf C} X$ bounding loop
$\gamma\delta$ and such that the interior of $F'$ does not intersect ${\bf C} D$.
Set $F$ to be equal to $F'\cup conj_X F'$.
Then $p^{-1}(F)$ gives an element of $H_2({\bf C} B)$ (the construction of this
element was suggested by O.Viro \cite{V3}).
Since $p^{-1}(F)$ is invariant under $conj_B$ it gives an element in
$H_2({\bf C} B/conj_B;{\bf Z}_2)$, say $f\in H_2({\bf C} B/conj_B;{\bf Z}_2)$.
Let us calculate the self-intersection number of $f$.
It is easy to see that because of symmetry the self-intersection number of $f$
in ${\bf C} B/conj_B$ is equal to the self-intersection number of $\gamma\delta$
in $C\cup{\bf C} D_+$.
The definition of complex semiorientation follows that the self-intersection
number of $\gamma\delta$ in $C\cup{\bf C} D_+$ (and therefore the self-intersection
number of $f$) is equal to zero in the first case and to one in the second
case.
\subsubsection{Remark}
In the case when ${\bf R} X$ is connected \ref{calor}
completely determines the complex separation of ${\bf R} B$.
\subsubsection{(O.Viro [11])}
\label{vychv}
If $\lambda$ is a loop in ${\bf R} B$ such that $p(\lambda)=\partial(G)$,
where $G\subset{\bf R} X$ then
$$q_{{\bf C} B}(\lambda)\equiv 2\chi(G\cap X_+)\pmod{4}$$
\subsubsection{(O.Viro [11])}
\label{vychq}
Suppose $\gamma$ is a path in $X_-$ connecting points $Q_{\alpha}$ and
$Q_{\beta}$ of components $\alpha$ and $\beta$ of ${\bf R} D$ respectively.
Then $q_{{\bf C} B}(p^{-1}(\gamma))=0$ if the intersection numbers of $\gamma$ with
$\alpha$ and
$\beta$ are of opposite sign (case 1 of fig.1) and
$q_{{\bf C} B}(p^{-1}(\gamma))=2$ otherwise (case 2 of fig.1)
\subsection{New congruences for complex orientations of curves on a
hyperboloid}
In this subsection we apply the results of \ref{pov} and \ref{vych} to
double branched coverings over simplest surfaces of characteristic type,
a plane and a hyperboloid.
To state congruences it is convenient to use the language of integral calculus
based on Euler characteristic developed by O.Viro \cite{V2}.
Let ${\bf R} A$ be a curve of type I in the connected surface ${\bf R} X$.
We equip ${\bf R} A$ with one of two complex orientations and fix $X_{\infty}$
-- one of the components of ${\bf R} X-{\bf R} A$.
If ${\bf R} X-X_{\infty}$ is orientable and ${\bf R} A$ is $R$-homologous to zero
for some ring $R$ of coefficient coefficient
then there is defined function $ind_R:{\bf R} X-{\bf R} A\rightarrow R$ equal to zero
on $X_{\infty}$ and equal to the $R$-linking number with oriented curve ${\bf R} A$
in ${\bf R} X-X_{\infty}$ otherwise.
It is easy to see that $ind_R$ is measurable and defined almost everywhere on
${\bf R} X$ with respect to Euler characteristic.
Evidently, the function ${ind_R}^2:{\bf R} X-{\bf R} A\rightarrow R\otimes R$ does
not depend on
the ambiguity in the choice of one of two complex orientation of ${\bf R} A$.
\subsubsection{}
Consider the case when $X=P^2$, $R={\bf Z}$.
Let $A$ be a plane nonsingular real curve of type I given by polynomial $f_A$
of degree $m=2k$.
Then ${\bf R} A$ is ${\bf Z}_2$-homologous to zero.
Let $X_{\infty}$ be the only nonorientable component of ${\bf R} P^2-{\bf R} A$.
Without loss of generality suppose that $f_A|_{X_{\infty}}<0$.
Define $X_{\pm}$ to be equal to $\{y\in{\bf R} P^2|\pm f_A(y)\ge 0\}$.
Let $p:{\bf C} B\rightarrow{\bf C} P^2$ be the double covering of ${\bf C} P^2$ branched
along ${\bf C} A$ (note that such a covering exists and is unique since $m$ is even
and ${\bf C} P^2$ is simply connected).
Let $conj_B:{\bf C} B\rightarrow{\bf C} B$ be the lift of $conj:{\bf C} P^2\rightarrow{\bf C} P^2$
such that $p({\bf R} B)=X_-$ (as it is usual we denote $Fix(conj_B$ by ${\bf R} B$).
Lemmae 6.6 and 6.7 of \cite{W} immediately follow that $Dw_2({\bf C} B)={\bf R} B$.
Therefore we can apply \ref{surface} and \ref{calor}.
Proposition \ref{calor} follows that $p^{-1}(cl({ind_{{\bf Z}_2}}^{-1}(2+4{\bf Z})))$ is
equal to one of two surfaces of the complex separation of ${\bf R} B$.
Set $B_1$ to be equal to $p^{-1}(cl({ind_{{\bf Z}}}^{-1}_{{\bf Z}_2}(2+4{\bf Z})))$,
note that $B$ is orientable since $ind_{{\bf Z}}|_{X_{\infty}}=0$.
\subsubsection{Lemma}
$$\beta(q|_{H_1(B_1;{\bf Z}_2)})\equiv
4\chi({ind_{{\bf Z}}}^{-1}(3+8{\bf Z})\cup{ind_{{\bf Z}}}^{-1}(-3+8{\bf Z}))
\pmod{8}$$
{\em\underline{Proof}}The boundary of each component $C$ of $p(B_1)$ has one
exterior oval and some interior ovals (with respect to $C$).
We call an interior oval of $C$ $C$-positive if its complex orientation and
the complex orientation of the exterior oval can be extended to some
orientation
of $C$ and we call it $C$-negative otherwise.
Proposition \ref{vychq} follows that $\beta(q|_{H_1(B_1;{\bf Z}_2)})$ is equal to
twice the sum of values of $q$ on on all $C$-negative ovals modulo 8.
Note that $ind_{{\bf Z}}^{-1}(\pm 3+8{\bf Z})$ is just the part of $X_+$ lying inside
odd number of $C$-negative ovals.
The lemma follows now from \ref{vychv}.
\subsubsection{}
The application of \ref{surface} gives that
$$\chi(B_1)\equiv\frac{\chi({\bf R} B)-\sigma({\bf C} B)}{4}+\beta(q|_{H_1(B_1;{\bf Z}_2)})
\pmod{8}$$
Note that $p({\bf R} B)={ind_{{\bf Z}}}^{-1}(2{\bf Z})$, $\chi({ind_{{\bf Z}}}^{-1}(2{\bf Z}))=
1-\chi(1+2{\bf Z})$
and $\sigma({\bf C} B)=2-2k^2$.
We get
$$4\chi({ind_{{\bf Z}}}^{-1}(2+4{\bf Z}))\equiv 1-\chi({ind_{{\bf Z}}}^{-1}(1+2{\bf Z}))-1+k^2+
8\chi({ind_{{\bf Z}}}^{-1}(\pm 3+8{\bf Z}))\pmod{16}$$
We can reformulate this in integral calculus language
$$\int_{{\bf R} P^2}{ind_{{\bf Z}}}^2d\chi\equiv k^2\pmod{16}$$
Thus for projective plane we get nothing new but the reduction modulo 16
of the Rokhlin congruence for complex orientation \cite{R1}
$$\int_{{\bf R} P^2}{ind_{{\bf Z}}}^2d\chi=k^2$$
\subsubsection{}
Consider now the case when $X=P^1\times P^1$.
Let $A$ be a nonsingular real curve of type I in $P^1\times P^1$ of bidegree
$(d,r)$, i.e. the bihomogeneous polynomial $f_A$ determining $A$ is of bidegree
$(d,r)$ where $d$ and $r$ are even numbers.
Let $X_{\infty}$ be a component of ${\bf R} P^2-{\bf R} A$, $X_{\pm}=
\{y\in{\bf R} P^1\times{\bf R} P^1|\pm f_A(y)\ge 0\}$.
Suppose without loss of generality that $f_A|_{X_{\infty}}<0$.
Let $p:{\bf C} B\rightarrow{\bf C} P^2$ be the double covering of ${\bf C} P^2$ branched
along ${\bf C} A$ and let $conj_B:{\bf C} B\rightarrow{\bf C} B$ be the lift of
$conj:{\bf C} P^1\times{\bf C} P^1\rightarrow{\bf C} P^1\times{\bf C} P^1$ such that
$p({\bf R} B)=X_-$.
Nonsingularity of $A$ follows that all components of ${\bf R} A$ non-homologous to
zero are homologous to each other.
Let $e_1,e_2$ form the standard basis of $H_1({\bf R} P^1\times{\bf R} P^1)$ and
let $s,t$ be the coordinates in this basis of a non-homologous to zero
component of ${\bf R} A$ equipped with such an orientation that $s,t\ge 0$.
If all the components of ${\bf R} A$ are homologous to zero then set $s=t=0$.
Then ${\bf R} A$ equipped with the complex orientation produces $l'(se_1+te_2)$
in $H_1({\bf R} P^1\times{\bf R} P^1)$.
Note that $s$ and $t$ are relatively prime and $l'$ is even since both $d$ and
$r$ are even.
\subsubsection{Lemma}
If $l'\equiv 0\pmod{4}$ then $Dw_2({\bf C} B)=[{\bf R} B]$
The proof follows from Lemma 3.1 of \cite{Ma}.
\subsubsection{Lemma}
If $l'\equiv 0\pmod{4}$ then $ind_{{\bf Z}_4}$ is defined
and if $sd+tr\equiv 0\pmod{4}$ then
$$\int_{{\bf R} P^1\times{\bf R} P^1}{ind_{{\bf Z}_4}}^2d\chi\equiv \frac{dr}{2}\equiv
0\pmod{8}$$
{\em\underline{Proof}} Note that the condition that $sd+tr\equiv 0\pmod{4}$
is just equivalent to the orientability of ${\bf R} B$.
Therefore $\beta(q|_{H_1(B_1;{\bf Z}_2)})\equiv 0\pmod{4}$.
Further arguments are similar to the plane case,
we skip them.
\subsubsection{Remark}
The traditional way of proving of formulae of complex orientations for
curve on surfaces (see \cite{Z}) adjusted to the case when the real curve
is only ${\bf Z}_4$-homologous to zero gives only congruence
$$\int_{{\bf R} P^1\times{\bf R} P^1}{ind_{{\bf Z}_4}}^2d\chi\equiv \frac{dr}{2}\pmod{4}$$
The \ref{l=4} shows the worthiness of modulo 4 in this congruence.
If $l'\equiv 0\pmod{8}$ then the traditional way gives that
$$\int_{{\bf R} P^1\times{\bf R} P^1}{ind_{{\bf Z}_8}}^2d\chi\equiv \frac{dr}{2}\pmod{8}$$
\subsubsection{}
\label{l=4}
If $l'\equiv 4\pmod{8}$ and $sd+tr\equiv 2\pmod{4}$
then
$$\int_{{\bf R} P^1\times{\bf R} P^1}{ind_{{\bf Z}_4}}^2d\chi\equiv \frac{dr}{2}+4\pmod{8}$$
{\em\underline{Proof}} Form $q|_{H_1(B_1;{\bf Z}_2)}$ is cobordant to the sum of
a form on an orientable surface and some forms on Klein bottles.
The condition that $l'\equiv 4\pmod{8}$ is equivalent to the condition
that the number of forms on Klein bottles non-cobordant to zero is odd.
Thus
$$\beta(q|_{H_1(B_1;{\bf Z}_2)})\equiv 2\pmod{4}$$
\subsubsection{}
If $l'\equiv 0\pmod{8}$ and $sd+tr\equiv 0\pmod{4}$ then
$$\int_{{\bf R} P^1\times{\bf R} P^1}{ind}_{{\bf Z}_8}^{2}d\chi\equiv \frac{dr}{2}\pmod{16}$$
The proof is similar to the plane case.
\subsubsection{Addendum, new congruences for the Euler characteristic of $B_+$
for curves on a hyperboloid}
\label{b10}
Let $d\equiv r\equiv 0\pmod{2}$ and $\frac{d}{2}t+\frac{r}{2}s+s+t\equiv
1\pmod{2}$.
\begin{itemize}
\begin{description}
\item[a)] If $A$ is an M-curve then $\chi(B_+)\equiv\frac{dr}{2}\pmod{8}$
\item[b)] If $A$ is an (M-1)-curve then $\chi(B_+)\equiv\frac{dr}{2}\pm
1\pmod{8}$
\item[c)] If $A$ is an (M-2)-curve and $\chi(B_+)\equiv\frac{dr}{2}+4\pmod{8}$
then $A$ is of type I
\item[d)] If $A$ is of type I then $\chi(B_+)\equiv 0\pmod{4}$
\end{description}
\end{itemize}
This theorem follows from Theorem 1 and gives some new restriction on the
topology of the arrangement of real nonsingular algebraic curve of even
bidegree on a hyperboloid with non-contractible branches.
Points a) and b) of \ref{b10} in the case when $\frac{d}{2}t+\frac{r}{2}s\equiv
0\pmod{2}, s+t\equiv 1\pmod{2}$ were proved by S.Matsuoka \cite{Ma1} in another
way (using 2-sheeted branched coverings of hyperboloid).
Point d) of \ref{b10} is a corollary of the modification of Rokhlin formula
of complex orientations for modulo 4 case.
\subsection{The case when $A$ is a curve of even degree on projective surface
$B$}
In this subsection we deduce Rokhlin and Kharlamov-Gudkov-Krakhnov congruences
from Theorem 1 and give a direct generalization of these congruences.
Let $B$ be the surface in $P^q$ given by the system of equations
$P_j(x_0,\ldots,x_q)=0,j=1,\ldots,s-1$,
let $A$ be the (M-$k$)-curve given by the system of equations
$P_j(x_0,\ldots,x_q)=0,j=1,\ldots,s-1$,
where $P_j$ are homogeneous polynomials with real coefficients,
$deg P_j=m_j$, $s=q-1$.
Suppose $(B,A)$ is a non-singular pair, ${\bf R} A\neq\emptyset$ and $m_s$ is even.
Denote $B_+=\{x\in{\bf R} B|P_s(x)\geq 0\},B_-=\{x\in{\bf R} B|P_s(x)\leq 0\}$,
$$d=rk(in^{B_+}_*:H_1(B_+;{\bf Z}_2)\rightarrow H_1({\bf R} B;{\bf Z}_2)),
e=rk(in^{{\bf R} A}_*:H_1({\bf R} A;{\bf Z}_2)\rightarrow H_1({\bf R} B;{\bf Z}_2))$$
Set $c$ to be equal to the number of non-contractible in ${\bf R} P^q$ components
of ${\bf R} B$ not intersecting ${\bf R} A$.
Let us reformulate Rokhlin and Kharlamov-Gudkov-Krakhnov congruences for
curve on surfaces in the form convenient for generalization and correcting
the error in \cite{R}.
\subsubsection{(Rokhlin [1], Kharlamov [2], Gudkov-Krakhnov [3])}
\label{RKGK}
Suppose $B$ is an $M$-surface, $e=0$, $B_+$ is contained in one component
of ${\bf R} B$ and in the case when $m_s\equiv 0\pmod{4}$ suppose in addition that
$c=0$.
\begin{itemize}
\begin{description}
\item[a)] If $d+k=0$ then
$$\chi(B_+)\equiv\frac{m_1\ldots m_{s-1}m_s^2}{4}\pmod{8}$$
\item[b)] If $d+k=1$ then
$$\chi(B_+)\equiv\frac{m_1\ldots m_{s-1}m_s^2}{4}\pm 1\pmod{8}$$
\end{description}
\end{itemize}
Indeed, it is easy to see that the hypothesis of \ref{RKGK} without
the condition on $c$ is equivalent to the hypotheses of corresponding theorems
in \cite{R}, \cite{Kh} and \cite{GK},
to reformulate them it is enough to apply the Rokhlin congruence for
M-surfaces.
\subsubsection{Remark (on an error in [1])}
Point 2.3 of \cite{R} contains a miscalculation of characteristic class $x$
of the restriction to $B_-$ of the double covering of ${\bf C} B$ branched along
${\bf C} A$.
In \cite{R} it is claimed that if $m_s\equiv 2\pmod{4}$ then $x=w_1(B_{-}-A)$.
It led to the omission of the condition on $c$ in both Rokhlin and
Kharlamov-Gudkov-Krakhnov congruences.
\subsubsection{Correction of the error in [1]}
If $m_s\equiv 2\pmod{4}$ then $x=in^*\alpha$ where $in$ is the inclusion of
$B_-$
into ${\bf R} P^q$ and $\alpha$ is the only non-zero element of $H^1({\bf R} P^q;{\bf Z}_2)$
{\em\underline{Proof}} Consider $E$ -- the auxiliary surface in $P^q$
given by equation $P_s(x_0,\ldots,x_q)=0$.
Then the construction of the double covering of ${\bf C} P^q$ branched along ${\bf C} E$
in the weighted-homogeneous projective space by equation $\lambda^2=
P_s(x_0,\ldots,x_q)$ follows that the characteristic class of the restriction
of the covering to ${\bf R} P^q-{\bf R} E$ is equal to the restriction of $\alpha$
to ${\bf R} P^q-{\bf R} E$.
Therefore, the characteristic class of the restriction of the covering
to $B_-$ is equal to $in^*\alpha$.
Besides, it is claimed in \cite{R} that in the case of curves on surfaces
(i.e. $n=1$ in notations of \cite{R}) the endomorphism
$\omega:H_*(B_-,A;{\bf Z}_2)\rightarrow H_*(B_-,A;{\bf Z}_2)$ of cap-product with
characteristic class $x$ is trivial.
This is true only if either $m_s\equiv 0\pmod{4}$ or each non-contractible
component of $B_-$ contains at least one component of ${\bf R} A$.
The condition on $C$ in \ref{RKGK} allows to correct proofs of congruences
of \cite{R}, \cite{Kh} and \cite{GK}.
The author though does not know counter-examples to \ref{RKGK} without
condition on $c$ (without the condition on $c$ point a) of \ref{RKGK} is
equivalent to 3.4 of \cite{R}).
\subsubsection{Direct generalization of Rokhlin and Kharlamov-Gudkov-Krakhnov
congruences}
\label{gen}
Suppose that $B$ is of type I$abs$ in the case when $\sum_{j=1}^{s-1}m_j\equiv
0
\pmod{2}$ and that $B$ is of type I$rel$ in the case when
$\sum_{j=1}^{s-1}m_j\equiv 1\pmod{2}$.
Suppose that $m_s$ is even, $e=0$, $B_+$ is contained in one surface of
the complex separation of ${\bf R} B$ and in the case when $m_s\equiv 2\pmod{4}$
suppose in addition that $c=0$.
\begin{itemize}
\begin{description}
\item[a)] If $d+k=0$ then
$$\chi(B_+)\equiv\frac{m_1\ldots m_{s-1}m_s^2}{4}\pmod{8}$$
\item[b)] If $d+k=1$ then
$$\chi(B_+)\equiv\frac{m_1\ldots m_{s-1}m_s^2}{4}\pm 1\pmod{8}$$
\item[c)] If $d+k=2$ and
$$\chi(B_+)\equiv\frac{m_1\ldots m_{s-1}m_s^2}{4}+4\pmod{8}$$
then $A$ is of type I
\item[d)] If $A$ is of type I then
$$\chi(B_+)\equiv\frac{m_1\ldots m_{s-1}m_s^2}{4}\pmod{4}$$
\end{description}
\end{itemize}
\subsubsection{Lemma}
\label{V+=0}
The surface $V_+={\bf C} A/conj\cup B_+$ is ${\bf Z}_2$-homologous to zero in
${\bf C} B/conj$
{\em\underline{Proof}} Consider the diagram
\begin{picture}(200,-100)(-50,0)
\put(-10,0){${\bf C} Y$}\put(60,0){${\bf C}B$}
\put(-20,-40){${\bf C} Y/conj_Y$}\put(60,-40){${\bf C}B/conj .$}
\put(10,4){\vector(1,0){46}}
\put(3,-3){\vector(0,-1){25}}
\put(67,-3){\vector(0,-1){25}}
\put(33,8){$p$}
\end{picture}\\[42pt]
where $p:{\bf C} Y\rightarrow{\bf C} B$ is the double covering over ${\bf C} B$ branched
along ${\bf C} A$ and $conj_Y$ is such a lift of $conj:{\bf C} B\rightarrow{\bf C} B$
that ${\bf R} Y=\{y\in{\bf C} Y|conj_Yy=y\}$ is equal to $p^{-1}(B_-)$.
It is easy to see that this diagram can be expanded to a commutative one
by adding $\phi:{\bf C} Y/conj_Y\rightarrow{\bf C} B/conj$ where $\phi$ is the double
covering map branched along $V_+$.
It follows that $[V_+]=0\in H_2({\bf C} B/conj)$.
\subsubsection{Remark}
The construction of \ref{V+=0} allows to define the separation of ${\bf R} B$
into $B_+$ and $B_-$ for any (not necessarily algebraic) curve
of even degree and invariant with respect to $conj$
in any (not necessarily projective) complex surface ${\bf C} B$ equipped with
almost antiholomorphic involution $conj:{\bf C} B\rightarrow{\bf C} B$ and such that
$H_1({\bf C} B;{\bf Z}_2)=0$.
In this case there is the unique double covering over ${\bf C} B$ branched along
${\bf C} A$ and two possible lifts of $conj:{\bf C} B\rightarrow{\bf C} B$.
The images of the fix point sets of these lifts form the desired
separation.
Note that this separation is different from the complex separation.
Let us give an internal definition of this separation.
Let $x,y$ be points in ${\bf R} B-{\bf R} A$.
Connect these points by path $\gamma$ inside ${\bf C} B-{\bf C} A$.
If the linking number of the loop $\gamma conj\gamma$ and ${\bf C} A$ is equal
to zero then $x$ and $y$ are points of the same surface of the complex
separation, otherwise $x$ and $y$ are points of two different surfaces
of the complex separation.
This remark allows to extend \ref{gen} to the case of flexible curves.
\subsubsection{The proof of 4.4.4}
Note that $Dw_2({\bf C} B)+[{\bf R} B]=0$ since $Dw_2({\bf C} B)\equiv [\infty]
\sum_{j=1}^{s-1}m_j$ where $[\infty]\in H_2({\bf C} B;{\bf Z}_2)$ is the class of
hyperplane section of ${\bf C} B$ and note that $[{\bf C} A]=0$ since $m_s$ is even.
We apply \ref{surface}, let $W$ be the surface of the complex separation
not intersecting $B_+$.
Then \ref{V+=0} follows that $W\cup B_+$ is the surface of the complex
separation of $(B,A)$ since if $W$ is a characteristic surface in
${\bf C} B/conj$ then so is $W_+=W\cup V_+$.
Let us prove now that $q_{W_+}|_{H_1(W;{\bf Z}_2)}$ is equal to $q_W$ where
$q_{W_+}$ and $q_W$ are Guillou-Marin forms of $W_+$ and $W$ in
${\bf C} B/conj$.
Note that $(q_{W_+}-q_W)(x), x\in H_1(W;{\bf Z}_2)$ is equal to the linking number
of $x$ and $V_+$ in ${\bf C} B/conj$ that is equal to the linking number of
$x$ and ${\bf C} A$ in ${\bf C} B$.
It is not difficult to see that the condition on $c$ follows that
the linking number of $x$ and ${\bf C} A$ in ${\bf C} B$ is equal to zero.
Theorem 1 follows that
$$\chi(B_+)+\chi(W)\equiv\frac{e_A}{4}+\frac{\chi({\bf R} B)-\sigma({\bf C} B)}{4}+
\beta(q_W)+\beta(q_{W_+}|_{H_1(W_+;{\bf Z}_2)})\pmod{8}$$
and \ref{surface} follows that
$$\chi(W)\equiv\frac{\chi({\bf R} B)-\sigma({\bf C} B)}{4}+\beta(q_W)\pmod{8}$$
Thus, noting that $e_A=m_1\ldots m_{s-1}m_s^2$ we get that
$$\chi(B_+)\equiv\frac{m_1\ldots m_{s-1}m_s^2}{4}+
\beta(q_{W_+}|_{H_1(V_+;{\bf Z}_2)})\pmod{8}$$
The rest of the proof is similar to that of \ref{prth}.
\subsection{Curves on an ellipsoid}
In the congruences of \ref{gen} we was able to avoid the appearance
of the complex separation with the help of the separation of ${\bf R} B$ into $B_+$
and $B_-$.
For curves of odd degree this does not work since there is no such a
separation.
Although, if ${\bf R} B$ is connected then the complex separation does not provide
the additional information and still can be avoided.
Surfaces $B_1$ and $B_2$ of the complex separation are determined by
the condition that $B_1\cup B_2={\bf R} B$ and $\partial B_1=\partial B_2={\bf R} A$.
The simplest case is the case of curves of odd degree on an ellipsoid.
It is well-known that the complex quadric is isomorphic to ${\bf C} P^1\times{\bf C}
P^1$,
an algebraic curve in quadric is determined by a bihomogeneous polynomial of
bidegree $(d,r)$.
If the curve is real and the quadric is an ellipsoid then $d=r$,
otherwise the curve can not be invariant under the complex conjugation
of an ellipsoid since the complex conjugation of an ellipsoid acts on
$H_2({\bf C} P^1\times{\bf C} P^1={\bf Z}\times{\bf Z}$ in the following way :
$conj_*(a,b)=(-b,-a)$ as it is easy to see considering the behavior of
$conj$ on the generating lines of ${\bf C} P^1\times{\bf C} P^1$.
Thus a real curve on an ellipsoid is the intersection of the ellipsoid and
a surface of degree $d$, this can be regarded as a definition of real curves
on an ellipsoid.
\subsubsection{Theorem}
\label{ell}
Let $A$ be a nonsingular real curve of bidegree $(d,d)$ on ellipsoid $B$.
Suppose that $d$ is odd.
\begin{itemize}
\begin{description}
\item[a)] If $A$ is an M-curve then
$$\chi(B_1)\equiv\chi(B_2)\equiv\frac{d^2+1}{2}\pmod{8}$$
\item[b)] If $A$ is an (M-1)-curve then
$$\chi(B_1)\equiv\frac{d^2+1}{2}\pm 1\pmod{8}$$
$$\chi(B_2)\equiv\frac{d^2+1}{2}\mp 1\pmod{8}$$
\item[c)] If $A$ is an (M-2)-curve and
$$\chi(B_1)\equiv\frac{d^2+1}{2}+4\pmod{8}$$
then $A$ is of type I
\item[d)] If $A$ is of type I then
$$\chi(B_1)\equiv\chi(B_2)\equiv 1\pmod{4}$$
\end{description}
\end{itemize}
{\em\underline{Proof}} $Dw_2({\bf C} B)+[{\bf R} B]+[{\bf C} A]=0$ since an ellipsoid is
of type I$rel$.
Theorem 1 follows \ref{ell} now since
$$\frac{e_A}{4}+\frac{\chi({\bf R} B)-\sigma({\bf C} B)}{4}=\frac{d^2+1}{2}$$
and $\beta_j=0$ because of the triviality of $H_1({\bf R} B;{\bf Z}_2)$.
\subsubsection{Low-degree curves on an ellipsoid}
Consider the application of \ref{ell} to the low-degree curves on an
ellipsoid.
Gudkov and Shustin \cite{GSh} classified real schemes
of curves of bidegree not greater then (4,4) on an ellipsoid.
To prove restrictions for such a classification it was enough to apply
the Harnack inequality and Bezout theorem.
Using the Bezout theorem avoided the extension of such a classification to
the flexible curves.
For the curves of bidegree (4,4) one can avoid the using of the Bezout theorem
using instead the analogue of the strengthened Arnold inequalities for curve
on an ellipsoid.
for the classification of the flexible curves of bidegree (3,3) the old
restrictions does not give the complete system of restrictions,
they do not restrict schemes $2\sqcup 1$$<$$2$$>$, $3\sqcup 1$$<$$1$$>$ and
$2\sqcup 1$$<$$1$$>$ (see \cite{V} for the notations).
Theorem \ref{ell} restricts these schemes and thus completes the classification
of the real schemes of flexible curves of bidegree (3,3) on an ellipsoid.
\subsubsection{Theorem}
The real scheme of a nonsingular flexible curve of bidegree (3,3) on
an ellipsoid is $1$$<1$$<$$1$$>$$>$ or $\alpha\sqcup 1$$<$$\beta$$>,
\alpha>\beta,\alpha+\beta\leq 4$.
Each of this schemes is the real scheme of some flexible (and even algebraic)
curve of bidegree (3,3) on an ellipsoid
\subsubsection{Curves of bidegree (5,5) on an ellipsoid}
Theorem \ref{ell} gives an essential restriction for the M-curves of
bidegree (5,5) on an ellipsoid, \ref{ell} leaves unrestricted 18 possible
schemes of flexible M-curves of bidegree (5,5) and 15 of them can be
realized as algebraic M-curves given by birational transformations of
the appropriate affine curves of degree 6 and another one by Viro
technique of small perturbation from the product of five plane sections
intersecting in two different points.
Thus there are only two schemes of M-curves of bidegree (5,5)
unrestricted and unconstructed, namely, $1\sqcup 1$$<$$6$$>$$\sqcup 1$$<$$8$$>$
and $1\sqcup 1$$<$$5$$>$$\sqcup 1$$<$$9$$>$.
\subsubsection{The case of bidegree (4,4)}
The classification of the real schemes of curves of bidegree (4,4) \cite{GSh}
shows that there is no congruence like \ref{ell} for curves of even degree:
the Euler characteristic of surfaces $B_1$ and $B_2$ for curves of bidegree
(4,4) can be any even number between -8 and 10.
Thus the condition that $Dw_2({\bf C} B)+[{\bf R} B]+[{\bf C} A]=0$ is essential.
\subsubsection{Addendum, the Fiedler congruence for curves on an ellipsoid}
Let $A$ be an M-curve of bidegree $(d,d)$ on ellipsoid $B$.
Suppose that $d$ is even, the Euler characteristic of each component
of $B_1$ is even and $\chi(B_1)\equiv 2\pmod{4}$
then
$$\chi(B_2)\equiv d^2\pmod{16}$$
$$\chi(B_1)\equiv 2-d^2\pmod{16}$$
{\em\underline{Proof}} The formula of complex orientations for curves on
an ellipsoid (see \cite{Z1}) follows that the surface $V$ equal to $B_2\cup
A_+$
is ${\bf Z}_2$-homologous to zero in ${\bf C} B$ where $A_+$ is one of the components
of ${\bf C} A-{\bf R} A$.
Thus $V$ is a characteristic surface in ${\bf C} B$.
Further arguments are similar to that of \cite{F}
\subsection{Curves on cubics}
In this subsection we consider the application of Theorem 1 to
the curves of degree 2 on cubics, surfaces in $P^3$ given by cubic polynomial
(cf.\cite{Mi}) of type I$rel$.
In notations of \ref{gen} we have $q=3$, $s=2$, $m_1=3$, $m_2=2$.
Rokhlin congruence for curves on surfaces gives the complete system
of restrictions for curves of degree 2 on $M$-cubic but for restrictions
for curves of degree 2 on another cubic of type I$rel$ -- cubic diffeomorphic
to the disjoint sum of ${\bf R} P^2$ and $S^2$ we need some new tools.
Theorem 1 suffices for this purpose.
To apply Theorem 1 note that the complex separation of the disjoint cubic
consists of two surfaces -- ${\bf R} P^2$ and $S^2$.
The following is the classification of the real schemes of flexible curves
of degree 2 on a hyperboloid obtained in \cite{Mi}, for more details and
for the classifications on other real cubics see \cite{Mi}.
\subsubsection{Theorem}
Each flexible M-curves of degree 2 on non-connected cubic has one of
the following real schemes.
\begin{itemize}
\begin{description}
\item[a)] $($$<$$3\sqcup 1$$<$$1$$>$$>$$)_{{\bf R} P^2}\sqcup
($$<$$\emptyset$$>$$)_{S^2}$
\item[b)] $($$<$$1$$<$$4$$>$$>$$)_{{\bf R} P^2}\sqcup
($$<$$\emptyset$$>$$)_{S^2}$
\item[c)] $($$<$$\alpha$$>$$)_{{\bf R} P^2}\sqcup
($$<$$5-\alpha$$>$$)_{S^2}, 0\leq\alpha\leq 5$
\end{description}
\end{itemize}
Each of these 8 schemes is the real scheme of some flexible curve of degree 2
on an ellipsoid.
|
1992-08-06T01:30:14 | 9206 | alg-geom/9206008 | en | https://arxiv.org/abs/alg-geom/9206008 | [
"alg-geom",
"math.AG"
] | alg-geom/9206008 | null | Ron Donagi | The fibers of the Prym map | 71 pages, LATEX, (This is a reformatted version. It should print
better than its predecessor.) | null | null | null | null | In this work we use the bigonal, trigonal and tetragonal constructions to
describe the fibers of the Prym map P : R_{g} ---->A_{g-1} inthe cases when it
is dominant, i.e. for g < 7. The most interesting cases are g = 5, where the
fiber is a double cover of the Fano surface of lines on a cubic threefold, and
g=6, where the map is generically finite (of degree 27) with Galois group
WE_{6}, so that the general fiber has the structure of the 27 lines on a cubic
surface. For g > 6, the map is known to be generically injective. The
tetragonal construction gives many counterexamples to injectivity, and we
conjecture that all noninjectivity is due to the tetragonal construction.
| [
{
"version": "v1",
"created": "Wed, 17 Jun 1992 22:18:39 GMT"
},
{
"version": "v2",
"created": "Tue, 23 Jun 1992 19:54:38 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Donagi",
"Ron",
""
]
] | alg-geom | \section{Pryms.}
\subsection{Pryms and parity.}
\ \ \ \ Let $$\pi:\widetilde{C}\rightarrow C$$ be an unramified,
irreducible double cover of a curve $C\in{\cal M}_g$. The genus of
$\widetilde{C}$ is then $2g-1$, and we have the Jacobians $$J:=J(C),
\
\ \ \ \ \ \ \widetilde{J}:=J(\widetilde{C})$$ of dimensions $g,\,
2g-1$
respectively, and the norm homomorphism
$${\rm Nm}:\widetilde{J}\longrightarrow J.$$ Mumford shows [M2] that
$${\rm Ker}({\rm Nm})=P\cup P^-$$ where $P={\cal P}(C,
\widetilde{C})$ is an
abelian subvariety of $\widetilde{J}$, called the Prym variety, and
$P^-$ is its translate by a point of order 2 in $\widetilde{J}$. The
principal polarization on $\widetilde{J}$ induces twice a principal
polarization on the Prym. This appears most naturally when we
consider instead the norm map on line bundles of degree $2g-2$,
$${\rm Nm}:{\rm Pic}^{2g-2}(\widetilde{C})\rightarrow {\rm
Pic}^{2g-2}(C).$$
Let $\omega_{C}\in {\rm Pic}^{2g-2}(C)$ be the canonical bundle of
$C$.
\bigskip
\noindent {\bf Theorem 1.1} (Mumford [M1], [M2])
\begin{list}{{\bf(\arabic{bean})}}{\usecounter{bean}}
\item The two components $P_0, \,P_1$ of ${\rm Nm}^{-1}(\omega_C)$
can be
distinguished by their parity: $$P_i=\{L\in
{\rm Nm}^{-1}(\omega_C) \ \ \ | \ \ \ h^0(L)\equiv i \ \ \ \ \ {\rm
mod.
\ 2}\}, \ \ \ \ \ \ \ \ \ \ i=0,1.$$
\item Riemann's theta divisor $\widetilde{\Theta}'\subset {\rm
Pic}^{2g-2}(\widetilde{C})$ satisfies $$\widetilde{\Theta}'\supset
P_1$$
and $$\widetilde{\Theta}'\cap P_0=2\Xi'$$ where $\Xi'\subset P_0$ is
a
divisor in the principal polarization on $P_0$.
\end{list}
\subsection{Bilinear and quadratic forms.}
\ \ \ \ Let $X\in{\cal A}_g$ be a {\rm PPAV}, and let $Y$ be a torser
(=principal homogeneous space) over $X$. By theta divisor in $Y$
we mean an effective divisor whose translates in $X$ are in the
principal polarization. $X$ acts by translation on the variety
$Y'$ of theta divisors in $Y$, making $Y'$ also into an $X$-torser.
In $X'$ there is a distinguished divisor $$\Theta':=\{\Theta\subset
X|\Theta\ni 0\}\subset X'$$ which turns out to be a theta divisor,
$\Theta'\in X''$. In particular, we have a natural identification
$X''\approx X$ sending $\Theta'$ to $0$. Let $X_2$ be the subgroup
of points of order $2$ in $X$. Inversion on $X$ induces an
involution on $X'$; the invariant subset $X'_2$, consisting of
symmetric theta divisors in $X$, is an $X_2$-torser. Let
$\langle\, ,\, \rangle$
denote the natural ${\bf F}_2$-valued (Weil) pairing on $X_2$. On
$X'_2$ we have an ${\bf F}_2$-valued function
$$q=q_X:X'_2\rightarrow{\bf F}_2$$ sending $\Theta\in X'$ to its
multiplicity at $0\in X$, taken mod. 2.
\bigskip
\noindent {\bf Theorem 1.2} [M1] The function $q_X$ is quadratic.
Its associated bilinear form, on $X_2$, is $\langle \, ,\, \rangle$.
When
$(X,\Theta)$ vary in a family, $q_X(\Theta)$ is locally constant.
When $X$ is a Jacobian $J=J(C)$, these objects have the following
interpretations:
\noindent
\begin{tabbing}
$q(L)$ \= $\approx$ \= $\{ L \in {\rm Pic}^{0}(C) = J \ | \ L^{2} \
\approx
{\cal O}_{C} \} \ \approx \ H^{1}(C,{\bf F}_{2})$ \= \ (semi periods)
\kill
$J'$ \> $\approx$ \> ${\rm Pic}^{g-1}(C) \,$ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \
(use Riemann's theta divisor) \\
$J_{2}$ \> $\approx$ \> $\{ L \in {\rm Pic}^{0}(C) = J \ | \ L^{2} \
\approx
{\cal O}_{C} \} \ \approx \ H^{1}(C,{\bf F}_{2})$ \= \ \ \ \ (semi
periods)
\\
$J'_{2}$ \> $\approx$ \> $\{ L \in {\rm Pic}^{g-1}(C) \ \ \ \ | \
L^{2} \
\approx
{\cal \omega}_{C} \ \} \,$ \ \ \ \ \ \ \ \ \ \ \ \ (theta
characteristics) \\
$q(L)$ \> $\equiv$ \> $h^{0}(C,L)$ mod. 2 \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \
(by Riemann-Kempf)
\end{tabbing}
Explicitly, the theorem says in this case that for $\nu, \sigma\in
J_2$ and $L\in J'_2$:
\bigskip
\noindent ({\bf 1.3}) $\langle \nu,\sigma \rangle \ \ \ \equiv \ \ \
h^0(L)+h^0(L\otimes\nu)+h^0(L\otimes\sigma)+h^0(L\otimes\nu
\otimes\sigma)$ \begin{flushright} mod. 2. \end{flushright}
We note that non-zero elements $\mu\in J_2$ correspond exactly to
irreducible double covers $\pi:\widetilde{C}\rightarrow C$. Let $X$
be
the Prym $P={\cal P}(C,\widetilde{C})$, which we also denote
$P(C,\mu)$, $P(C,\widetilde{C})$, \ $P(\widetilde{C}/C)$ etc. Now
the
divisor $\Xi'\subset P_0$ of Theorem 1.1 gives
a natural identification $$P'\approx P_0\subset\widetilde{J}'.$$ The
pullback $$\pi^*:J\longrightarrow\widetilde{J}$$ sends $J_2$ to
$\widetilde{J}_2$. Since ${\rm Nm}\circ\pi^*=2$, we see that
$$\pi^*(J_2)\subset P_2\cup P^-_2.$$ Let $(\mu)^\perp$ denote the
subgroup of $J_2$ perpendicular to $\mu$ with respect to $\langle \,
, \,
\rangle$.
\bigskip
\noindent {\bf Theorem 1.4}
[M2]\begin{list}{{\bf(\arabic{bean})}}{\usecounter{bean}}
\item For $\tau\in J_2$, $\pi^*\tau\in P_2$ iff
$\tau\in(\mu)^{\perp}$.
\item This gives an exact sequence
$$0\rightarrow(\mu)\rightarrow(\mu)^\perp
\stackrel{\pi^*}{\rightarrow}P_2\rightarrow 0.$$
\item In (2), $\pi^*$ is symplectic, i.e. $$\langle \nu,
\sigma \rangle_J \, = \, \langle \pi^*\nu, \pi^*\sigma\rangle _P, \ \
\ \ \ \ \
\ \
\ \nu, \sigma\in(\mu)^\perp\subset J_2.$$
\end{list}
This equality of bilinear forms can be refined to an equality of
quadratic functions. The identifications $$J'\approx {\rm Pic}^{g-
1}(C), \ \ \ \ \ \ \ \widetilde{J}'\approx {\rm Pic} ^{2g-2}
(\widetilde{C})$$ convert the pullback $$\pi^* :{\rm
Pic}^{g-1}(C)\rightarrow{\rm Pic}^{2g-2}(\widetilde{C})$$ into a map
of
torsers $${\pi^{*}}' : J' \rightarrow \widetilde{J}'$$ over the group
homomorphism $$\pi^{*}:J\rightarrow\widetilde{J}.$$ Let
$${(\mu)^{\perp
}}':=({\pi^{*}}')^{-1}(P'_2).$$ the refinement is:
\bigskip
\noindent {\bf Theorem 1.5}
[D4]\begin{list}{{\bf(\arabic{bean})}}{\usecounter{bean}}
\item ${(\mu)^{\perp}}'$ is contained in $J'_2$ and is a
$(\mu)^\perp$-coset there.
\item ${\pi^{*}}':{(\mu)^{\perp}}'\rightarrow P'_2$ is a map of
torsers over $\pi^*:(\mu)^\perp\rightarrow P_2$.
\item In (2), ${\pi^{*}}'$ is orthogonal, i.e.
$$q_J(\nu)=q_P({\pi^{*}}'\nu), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\nu\in{(\mu)^{\perp}}'.$$
\end{list}
\subsection{The Prym Maps.}
\ \ \ \ Let ${\cal R}_g$ be the moduli space of irreducible double
covers $\pi:\widetilde{C}\rightarrow C$ of non-singular curves
$C\in{\cal M}_g$. Equivalently, ${\cal R}_g$ parametrizes pairs
$(C, \mu)$ with $\mu\in J_2(C)\backslash(0)$, a semiperiod on
$C$. The assignment of the Prym variety to a double cover gives a
morphism $${\cal P}:{\cal R}_g\rightarrow{\cal A}_{g-1}.$$
Let $\iota$ be the involution on $\widetilde{C}$ over $C$. The Abel-
Jacobi map $$\varphi:\widetilde{C}\rightarrow J(\widetilde{C})$$
induces
the Abel-Prym map $$\psi:\widetilde{C}\rightarrow{\rm Ker}({\rm
Nm})$$
$$ \ \ \ \ \ \ \ \ \ \ \ x\longmapsto\varphi(x)-\varphi(\iota x).$$
The image actually lands in the wrong component, $P^-$, but
at least $\psi$ is well-defined up to translation (by a point of
order 2). In particular, its derivative is well-defined; it
factors through $C$, yielding the Prym-canonical map
$$\Psi:C\rightarrow{\bf P}^{g-2}$$ given by the complete linear
system $|\omega_C\otimes\mu|$. Beauville computed the
codifferential of the Prym map:
\bigskip
\noindent {\bf Theorem 1.6} [B1] The codifferential $$d{\cal
P}:T^*_P{\cal A}_{g-1}\rightarrow T^*_{(C,\mu)}{\cal R}_g$$ can be
naturally identified with restriction $$\Psi^*:S^2 H^0(\omega_C
\otimes \mu) \rightarrow H^0(\omega^2_C).$$ In particular,
${\rm Ker}(d{\cal P})$ is given by quadrics through the
Prym-canonical
curve $\Psi(C)\subset{\bf P}^{g-2}$.
Let $\bar{\cal A}_g$ denote a toroidal compactification of ${\cal
A}_g$. Its boundary $\partial\bar{\cal A}_g$ maps to $\bar{\cal
A}_{g-1}$, and the fiber over generic $A\in{\cal
A}_{g-1}\subset\bar{\cal A}_{g-1}$ is the Kummer variety
$K(A):=A/(\pm 1)$. In codimension 1, this picture is independent
of the toroidal compactification used.
Let ${\cal RA}_g$ denote the level moduli space parametrizing
pairs $(A,\mu)$ with $A\in{\cal A}_g \ , \ \mu\in
A_2\backslash(0)$, and let $\overline{\cal RA}_g$ be a toroidal
compactification. In [D3] we noted that its boundary has 3
irreducible components, distinguished by the relation of the
vanishing cycle (mod. 2), $\lambda$, to the semiperiod $\mu$:
\noindent{\bf (1.7)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
$\begin{array}{llll} \partial^{\rm I} \ \ : \!\!\! & \!\!\!\lambda
\!\!\! & \!\! = \!\! & \!\!\! \mu \\ \partial^{\rm II} \ : \!\!\! &
\!\!\! \lambda \!\!\! & \!\! \neq \!\! & \!\!\! \mu, \ { \ }
\!\langle
\lambda,\mu\rangle =0\in{\bf F}_2 \\ \partial^{\rm III}:\langle
\!\!\! &
\!\!\!\lambda \!\!\! & \! , \!\! & \!\!\! \mu\rangle \neq0.
\end{array} $
Let $\bar{\cal M}_g$, $\doublebar{\cal R}_g$ denote the
Deligne-Mumford
stable-curve compactifications of ${\cal M}_g$ and ${\cal R}_g$.At
least in
codimension one, the Jacobi map extends: $$\bar{\cal
M}_g\rightarrow\bar{\cal A}_g \ , \ \doublebar{\cal R}_g\rightarrow
\overline{\cal RA}_g.$$ We use $\partial\bar{\cal M}_g$ \ , \
$\partial^i\doublebar{\cal R}_g$ \ \ \ $(i = {\rm I, \, II, \,
III})$ to
denote
the intersections of $\bar{\cal M}_g$, $\doublebar{\cal R}_g$ with
the
corresponding boundary divisors in $\bar{\cal A}_g$,
$\overline{\cal RA}_g$.
In [B1], Beauville introduced the notion of an allowable double
cover. This leads to the construction ([DS] I, 1.1) of a proper
version of the Prym map, $$\bar{\cal P}:\bar{\cal R}_g\rightarrow
{\cal A}_{g-1}.$$ Roughly, one extends ${\cal P}$ to
$$\doublebar{\cal P}:\doublebar{\cal R}_g\rightarrow\bar{\cal
A}_{g-1},$$ then restricts to the open subset $\bar{\cal
R}_g\subset\doublebar{\cal R}_g$ of covers which are allowable, in
the sense that their Prym is in ${\cal A}_{g-1}$.This condition can
be made
more explicit:
\bigskip
\noindent {\bf Theorem 1.8} [B1] A stable curve $\widetilde{C}$ with
involution $\iota$, quotient $C$, is allowable if and only if all
the fixed points of $\iota$ are nodes of $\widetilde{C}$ where the
branches are not exchanged, and the number of nodes exchanged under
$\iota$ equals the number of irreducible components exchanged under
$\iota$.
We illustrate the possibilities in codimension 1:
\bigskip
\noindent {\bf Examples 1.9}
\begin{list}{{\rm(\Roman{butter})}}{\usecounter{butter}}
\item $X\in{\cal M}_{g-1}, \ p, q\in X, \ \ p\neq q$; let $X_0,
X_1$ be isomorphic copies of $X$. Then $C:=X/(p\sim q)$ is a point
of $\partial\bar{\cal M}_g$. The Wirtinger cover
$$\widetilde{C}:=(X_0\amalg X_1)/(p_0\sim q_1, p_1\sim q_0)$$ gives a
point $$(C,\widetilde{C})\in\partial^{\rm I}\bar{\cal R}_g$$ which is
allowable. The Prym is $${\cal P}(C,\widetilde{C})\approx
J(X)\in{\cal
A}_{g-1}.$$
\item Start with $(\widetilde{X}\rightarrow X)\in{\cal R}_{g-1}$,
choose distinct points $p, q\in X$, let $p_i, q_i(i=0, 1)$ be their
inverse images in $\widetilde{X}$, and set $$C:=X/(p\sim q), \ \ \
\widetilde{C}:=\widetilde{X}/(p_0\sim q_0, p_1\sim q_1).$$ Then
$$(C,\widetilde{C})\in\partial^{\rm II}\doublebar{\cal R}_g$$ is an
unallowable
cover. Its Prym is a ${\bf C}^*$-extension of ${\cal
P}(X,\widetilde{X})$; the extension datum defining this extension is
given by $$\psi(p_0)-\psi(q_0)\in{\cal P}(X,\widetilde{X}),$$ which
is
well defined modulo $\pm 1$ (i.e. in the Kummer), as it should be.
\item $X, p, q$ as before, but now $\widetilde{X}\rightarrow X$ is a
double cover branched at $p, q$; consider Beauville's cover
$$C:=X/(p\sim q), \ \ \
\\widetilde{C}:=\widetilde{X}/(\widetilde{p}\sim\widetilde{q})$$
where
$\widetilde{p},
\widetilde{q}$ are the ramification points in $\widetilde{X}$ above
$p, q$.
Then $(C, \widetilde{C})\in\partial^{\rm III}\bar{\cal R}_g$ is
allowable.
\end{list}
In [M1], Mumford lists all covers $(C, \widetilde{C})\in{\cal R}_g$
whose Pryms are in the Andreotti-Mayer locus (i.e. have theta
divisors singular in codimension 4). A major result in [B1]
(Theorem (4.10)) is the extension of this list to allowable covers
in $\bar{\cal R}_g$. We do not copy Beauville's list here, but we
will refer to it when needed.
\section{Polygonal constructions}
\subsection{The $n$-gonal constructions}
\ \ \ \ Let $$f:C\rightarrow K$$ be a map of non singular algebraic
curves, of
degree $n$, and $$\pi:\widetilde{C}\rightarrow C$$ a branched double
cover. These two determine a $2^n$-sheeted branched cover of $K$,
$$f_*\widetilde{C}\rightarrow K,$$ whose fiber over a
general point $k\in K$
consists of the $2^n$ sections $s$ of $\pi$ over $k$: $$s:f^{-
1}(k)\rightarrow\pi^{-1}f^{-1}(k), \ \ \ \pi\circ s=id.$$ The curve
$f_*\widetilde{C}$ can be realized, for instance, as sitting in ${\rm
Pic}^n(\widetilde{C})$ or $S^n\widetilde{C}$:
\noindent {\bf (2.1)} \ \ \ $f_*\widetilde{C}=\{D\in
S^n\widetilde{C} \ \ | \ \ {\rm Nm}(D)=f^{-1}(k), \ {\rm some} \ k\in
K\}.$
\noindent
(If we think of $\widetilde{C}$ as a local system on an open subset
of $C$,
this
is just the direct image local system on $K$, hence our notation
$f_*\widetilde{C}$.) On $f_*\widetilde{C}$ we have two structures:
an
involution $$\iota:f_*\widetilde{C}\rightarrow f_*\widetilde{C}$$
obtained
by changing all $n$ choices in the section $s$ via the involution
(also denoted $\iota$) of $\widetilde{C}$, and an equivalence
relation
$$f_*\widetilde{C}\rightarrow\widetilde{K}\rightarrow K$$ where
$\widetilde{K}$
is a branched double cover of $K$: two sections $$s_1, s_2:f^{-
1}(k)\rightarrow\pi^{-1}f^{-1}(k)$$ are equivalent if they differ
by an even number of changes.
For even $n$, the involution $\iota$ respects equivalence, so we
have a sequence of maps
\noindent{\bf (2.1.1)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
$f_*\widetilde{C}\rightarrow
f_*\widetilde{C}/\iota\rightarrow\widetilde{K}\rightarrow K$
\noindent of degrees $2, 2^{n-2}, 2$ respectively. For odd $n$ the
equivalence classes are exchanged by $\iota$, so we have instead a
Cartesian diagram:
\begin{equation}
\renewcommand{\theequation}{\bf
{\arabic{section}}.{\arabic{subsection}}.{\arabic{equation}}}
\setcounter{equation}{2}
\begin{diagram}[f ]
\node[2]{f_{*}\widetilde{C}} \arrow{sw} \arrow{se} \\
\node{f_{*}\widetilde{C}/\iota} \arrow{se} \node[2]{\widetilde{K}}
\arrow{sw}
\\
\node[2]{K}
\end{diagram}
\end{equation}
{\bf Remark 2.1.3} In prctice we will often want to allow $C$ to
acquire some
nodes, over which $\pi$ may be etale (as in (1.9 II)) or ramified (as
in
\linebreak (1.9 III)).
We will always consider this as a limiting case of the non-singular
situation,
and interpret the $n$-gonal construction in the limit so as to make
it depend
continuously on the parameters, whenever possible. We will see
various examples
of this below.
\subsection{Orientation}
\ \ \ \ We observe that the branched cover $\widetilde{K}\rightarrow
K$
depends on $f\circ\pi:\widetilde{C}\rightarrow K$, but not on $f,
\pi$
or $C$ directly. More generally, to an $m$-sheeted branched cover
$$g:M\rightarrow K$$ we can associate an $m!$-sheeted branched
cover (the Galois closure of $M$) $$g!:M!\rightarrow K,$$ with an
action of the symmetric group $S_m$; the quotient by the
alternating group $A_m$ gives a branched double cover
$$O(g):O(M)\rightarrow K$$ which we call the orientation cover of
$M$. We say $M$ is orientable (over $K$) if the double cover
$O(M)$ is trivial. One verifies easily that the double cover
$\widetilde{K}\rightarrow K$
(obtained in \S 2.1 from the maps
$\widetilde{C}\stackrel{\pi}{\rightarrow}C\stackrel{f}{\rightarrow}K$
as quotient of $f_*\widetilde{C}$) is the orientation cover
$O(f\circ\pi)$ of $\widetilde{C}$.
\bigskip
\noindent{\bf Corollary 2.2} If $\widetilde{C}$ is orientable over
$K$
then $f_*\widetilde{C}=\widetilde{C}_0\cup\widetilde{C}_1$ is
reducible:
\begin{list}{{\rm(\roman{butter})}}{\usecounter{butter}}
\item For $n$ even, the involution $\iota$ acts on each
$\widetilde{C}_i$ with quotient $C_i$ of degree $2^{n-2}$ over $K, \
\
\ i=0, 1$.
\item For $n$ odd, $\iota$ exchanges $\widetilde{C}_0,
\widetilde{C}_1$.Each
$\widetilde{C}_i$ has degree $2^{n-1}$ over $K$.
\end{list}
\bigskip
\noindent{\bf Lemma 2.3} Branch $(\widetilde{K}/K)=f_*({\rm Branch}
\
(\widetilde{C}/C))$.
This means: if one point of $f^{-1}(k)$ is a branch point of
$\widetilde{C}\rightarrow C$, then $k$ is a branch point of
$\widetilde{K}\rightarrow K$; if two points of $f^{-1}(k)$ are branch
points of $\widetilde{C}\rightarrow C$, then $k$ is not a branch
point
of (the normalization of) $\widetilde{K}\rightarrow K$, but the two
sheets of $\widetilde{K}$ there intersect; etc. In particular, the
ramification behavior of $f:C\rightarrow K$ does not affect the
ramification of $\widetilde{K}$.
\bigskip
\noindent{\bf Corollary 2.4} Let $f:C\rightarrow{\bf P}^1$ be a
branched cover, $\pi:\widetilde{C}\rightarrow C$ an (unramified)
double
cover. Then $\widetilde{C}$ is orientable over ${\bf P}^1$.
(More generally, the conclusion holds whenever $$f_*({\rm Branch}
(\pi))=2D$$ for some divisor $D$ on ${\bf P}^1$, since the
normalization of $O(\widetilde{C})$ is then an unramified double
cover
of the simply connected ${\bf P}^1$, by (2.3). In this situation
we say that $\pi$ has \underline{cancelling ramification.})
\bigskip
\noindent{\bf Remark 2.5} Assume $K={\bf P}^1$ and $\pi$
unramified. The image of $f_*\widetilde{C}$ in ${\rm
Pic}(\widetilde{C})$
is: $$\{L\in{\rm Pic}^n(\widetilde{C}) \ \ | \ \ {\rm Nm}(L)=f^*{\cal
O}_{{\bf P}^1}(1), \ \ \ h^0(L) > 0\}.$$ This is contained in a
translate of $${\rm Nm}^{-1}(\omega_C)=P_0\cup P_1,$$ and the
splitting
(2.2) of $f_*\widetilde{C}$ is ``explained", in this case, by the
splitting (1.1) of ${\rm Ker}({\rm Nm})$, i.e. after translation:
$$\widetilde{C}_i\subset P_i, \ \ \ \ \ i=0, 1,$$ cf. [D1, \S 6],
[B2].
\bigskip
\noindent{\bf Remark 2.6} The splitting of $f_*\widetilde{C}$ can
also
be explained group theoretically. Let $WC_n$ be the group of
signed permutations of $n$ letters, i.e. the subgroup of $S_{2n}$
centralizing a fixed-point-free involution of the $2n$ letters.Let
$WD_n$ be
its subgroup of index 2 consisting of even signed
permutations, i.e. permutations of $n$ letters followed by an even
number of sign changes. (These are the Weyl groups of the Dynkin
diagrams $C_n,D_n$.) Over an arbitrary space $X$, we have
equivalences:
\pagebreak[4]
$$\{ \ \ \ \ \ n{\rm -sheeted \ cover \ } Y\rightarrow
X \ \ \ \ \ \} \ \longleftrightarrow\{ \ \ \ {\rm Representation \
} \pi_1(X)\rightarrow \ \ \ \ \ \ S_n \ \ \}$$ $$ \left\{
\begin{array}{l} n{\rm -sheeted \ cover \ } Y\rightarrow X \\ {\rm
with \ a \ double \ cover \ } \widetilde{Y}\rightarrow Y \end{array}
\right\} \longleftrightarrow \left\{ \begin{array}{l} {\rm
\ Representation \ } \pi_1(X)\rightarrow WC_n \end{array} \right\}
$$ $$ \left\{ \begin{array}{l} n{\rm -sheeted \ cover \ }
Y\rightarrow X \\ {\rm with \ an \ orientable \ double \ \ } \\
{\rm cover \ } \widetilde{Y}\rightarrow Y \ \end{array} \right\}
\longleftrightarrow \left\{ \begin{array}{l} {\rm Representation \
} \pi_1(X)\rightarrow WD_n \end{array} \right\} $$
\medskip
The basic construction of $f_*\widetilde{C}$ then corresponds to the
standard representation $$\rho:WC_n\hookrightarrow S_{2^n}.$$ The
existence of the involution $\iota$ on $f_*\widetilde{C}$ corresponds
to the factoring of $\rho$ through $$WC_{2^{n-1}}\subset S_{2^n}.$$
The restriction $\bar{\rho}$ of $\rho$ to $WD_n$ factors through
$$S_{2^{n-1}}\times S_{2^{n-1}},$$ explaining the splitting when
$\widetilde{C}$ is orientable.
\subsection{The bigonal construction}
\ \ \ \ The case $n=2$ of our construction (``bigonal") takes a
pair of maps of degree 2:
$$\widetilde{C}\stackrel{g}{\rightarrow}C\stackrel{f}{\rightarrow}K$$
and produces another such pair
$$f_*\widetilde{C}\stackrel{g'}{\rightarrow}\widetilde{K}
\stackrel{f'}{\rightarrow} K.$$
Above any given point $k \in K$, the possibilities are:
\begin{list}{(\roman{bean})}{\usecounter{bean}}
\item If $f$, $g$ are etale then so are $f'$ and $g'$.
\item If $f$ is etale and $g$ is branched at one of the two points
$f^{-1}(k)$,
then $f'$ is branched at $k$, and $g'$ is etale there.
\item Vise versa, if $f$ is branched and $g$ is etale then $f'$ is
etale and
$g'$
is branched at one point of $f'^{-1}(k)$.
\item If both $f$ and $g$ are branched over $k$ then so are $f'$,
$g'$.
\item If $f$ is etale and $g$ is branched at both points
$f^{-1}(k)$, then
$\widetilde{K}$ will have a node over $k$, and $g' :
f_{*}\widetilde{C}
\rightarrow \widetilde{K}$ will be a $\partial^{\rm III}$
degeneration, i.e.
will look like (1.9 III).
\item Vice versa, we can extend the bigonal construction by
continuity, as in
(2.1.3), to allow $g : \widetilde{C} \rightarrow C$ to degenerate
to a
$\partial^{\rm III}$-cover. This leads to $f'$ which is etale and
$g'$ which
is branched at both points of $f'^{-1}(k)$.
\end{list}
The following general properties are immediately verified:
\bigskip
\noindent{\bf Lemma
2.7}\begin{list}{{\bf(\arabic{bean})}}{\usecounter{bean}}
\item The bigonal construction is symmetric, i.e. if it takes
$\widetilde{C}\stackrel{g}{\rightarrow}C\stackrel{f}{\rightarrow}K$
to
$\widetilde{C}'\stackrel{g'}{\rightarrow}C'\stackrel{f'}{\rightarrow}
K$
then it takes $\widetilde{C}'\rightarrow C'\rightarrow K$ to
$\widetilde{C}\rightarrow C\rightarrow K$.
\item The bigonal construction exchanges branch loci: $${\rm
Branch}(g')=f_*({\rm Branch}(g)), \ \ \ \ \ {\rm
Branch}(f)=g'_*({\rm Branch}(f')).$$
\end{list}
(As in lemma (2.3), this requires the following convention in case
(vi)
above: the local contribution to Branch($f$) is $2k$, and the
contribution
to Branch($g$) is 0).
The symmetry group of this situation, $WC_2$, is the
dihedral group
of the square: $$WC_2= \ \ \langle r, f \ \ \ | \ \ \
f^2=r^4=(rf)^2=1\rangle.$$
($r=90^\circ$ rotation, $f=$flip around $x$-axis, in the
2-dimensional representation.) It has a non-trivial outer
automorphism (=conjugation by a $45^\circ$ rotation), which
explains why conjugacy classes of representations (of $\pi_1(X)$)
in $WC_2$ come in (bigonally related) pairs.
We list all conjugacy classes of subgroups of $WC_2$ in the
following diagram ($\sim$ denotes conjugate subgroups):
\begin{equation}
\renewcommand{\theequation}{\bf
{\arabic{section}}.{\arabic{equation}}}
\setcounter{equation}{8}
\begin{diagram}[(fr) \sim]
\node[2]{(1)} \arrow{sw,-} \arrow{s,-} \arrow{se,-} \\
\node{(f) \sim (fr^{2})} \arrow{s,-} \node{(r^{2})} \arrow{sw,-}
\arrow{s,-}
\arrow{se,-}
\node{(fr) \sim (fr^{3})} \arrow{s,-} \\
\node{(f,r^{2})} \arrow{se,-} \node{(r)} \arrow{s,-}
\node{(fr,r^{2})}
\arrow{sw,-} \\
\node[2]{WC_{2}}
\end{diagram}
\end{equation}
Correspondingly, we obtain the diagram of curves and maps of degree
2:
\begin{equation}
\renewcommand{\theequation}{\bf
{\arabic{section}}.{\theau}.{\arabic{equation}}}
\setcounter{au}{8}
\setcounter{equation}{1}
\begin{diagram}[\widetilde{C}]
\node[2]{\doubletilde{C}} \arrow{sw} \arrow{s} \arrow{se} \\
\node{\widetilde{C}} \arrow{s} \node{C \times_{K} C'} \arrow{sw}
\arrow{s}
\arrow{se}
\node{\widetilde{C}'} \arrow{s} \\
\node{C} \arrow{se} \node{C''} \arrow{s} \node{C'} \arrow{sw} \\
\node[2]{K}
\end{diagram}
\end{equation}
Here the two sides are bigonally related.
Note that $C'$ is $O(\widetilde{C})$; so if $\widetilde{C}$ is
orientable
(e.g. if $K={\bf P}^1$ and $g$ is unramified) then everything
splits: $$\widetilde{C}'=C_0\amalg C_1\rightarrow K\amalg K=C',$$
$\widetilde{C}$ is Galois over $K$ with group $({\bf Z}/2{\bf Z})^2$
and quotients
\[
\begin{diagram}[C_{0}]
\node[2]{\widetilde{C}} \arrow{sw} \arrow{s} \arrow{se} \\
\node{C_{0}} \arrow{se} \node{C} \arrow{s} \node{C_{1}} \arrow{sw} \\
\node[2]{K}
\end{diagram}
\]
(cf. [M1]), and (2.8.1) simplifies to:
\begin{equation}
\renewcommand{\theequation}{\bf
{\arabic{section}}.{\theau}.{\arabic{equation}}}
\setcounter{au}{8}
\setcounter{equation}{2}
\begin{diagram}[\widetilde{C} {\textstyle \amalg}]
\node[2]{\widetilde{C} {\textstyle \amalg} \widetilde{C}} \arrow{sw}
\arrow{s}
\arrow{se} \\
\node{\widetilde{C}} \arrow{s} \node{C {\textstyle \amalg} C}
\arrow{sw}
\arrow{s} \arrow{se}
\node{C_{0} {\textstyle \amalg} C_{1}} \arrow{s} \\
\node{C} \arrow{se} \node{C} \arrow{s} \node{K {\textstyle \amalg} K}
\arrow{sw} \\
\node[2]{K}
\end{diagram}
\end{equation}
Given an arbitrary branched double cover $\widetilde{C}\rightarrow
C$,
we form its Prym variety
$$P(\widetilde{C}/C):={\rm Ker}^0({\rm
Nm}:J(\widetilde{C})\rightarrow J(C)).$$
It is
an abelian variety (for $C, \widetilde{C}$ non-singular), but in
general not a principally polarized one. Nevertheless, there is a
simple relationship between the bigonally-related Pryms
$P(\widetilde{C}/C)$ and $P(\widetilde{C}'/C'):$ in the case $K={\bf
P}^1$,
Pantazis [P] showed that these abelian varieties are dual to each
other.
\subsection{The trigonal construction.}
\ \ \ \ The case $n=3$ of our construction was discovered by
Recillas [R]. Start with a tower
$$\widetilde{C}\stackrel{\pi}{\rightarrow}C\stackrel{f}{\rightarrow}{
\bf P}^1$$ where $f$ has degree 3, and $\widetilde{C}\rightarrow C$
is
an unramified double cover. By Corollaries (2.4) and (2.2),
$f_*\widetilde{C}$ consists of two copies of a tetragonal curve
$g:X\rightarrow{\bf P}^1$. Since $f$ and $g$ have the same branch
locus by Lemma (2.3), we find from Hurwitz' formula: $${\rm
genus}(X)={\rm genus}(C)-1.$$ All in all, we have constructed a
map: $$T: \left\{ \begin{array}{c} {\rm trigonal \ curves \ } C
{\rm \ of \ } \\ {\rm genus \ } g {\rm \ with \ a \ double \ cover
\ } \widetilde{C} \end{array} \right\} \rightarrow \left\{
\begin{array}{c} {\rm tetragonal \ curves \ } \\ X {\rm \ of \
genus \ } g-1 \end{array} \right\}.$$
We claim that this map is a bijection (except that
sometimes a nonsingular object on one side may correspond to a
singular one on the other): given $g:X\rightarrow{\bf P}^1$, we
recover $\widetilde{C}$ as the relative second symmetric product of
$X$
over ${\bf P}^1$, $$\widetilde{C}:=S^2_{{\bf P}^1}X\rightarrow{\bf
P}^1,$$ whose fiber over $p\in{\bf P}^1$ consists of all unordered
pairs in $g^{-1}(p)$; this has an involution $\iota$
(=complementation of pairs), giving the quotient
$C:=\widetilde{C}/\iota$.
\begin{center}
\begin{tabular}{cccc}
\hspace{1in} & \hspace{1in} & \hspace{1in} & \hspace{1in} \\
\begin{picture}(30,30)(2,1)
\thicklines
\put(2,1){$\circ$} \put(2,25.8){$\circ$}
\put(26,1){$\circ$} \put(26,25.8){${\circ}$}
\end{picture} &
\begin{picture}(30,30)(2,1)
\thicklines
\put(2,1){$\circ$}
\put(7,3.9){\line(1,1){22.4}}
\put(2,25.8){$\circ$}
\put(7,27.9){\line(1,-1){22}}
\put(28,1){$\circ$} \put(28,25.8){${\circ}$}
\end{picture} &
\begin{picture}(30,30)(2,1)
\thicklines
\put(2,1){$\circ$} \put(2,25.8){$\circ$}
\put(4.5,6){\line(0,1){20}}
\put(26,1){$\circ$} \put(28.5,6){\line(0,1){20}}
\put(26,25.8){${\circ}$}
\end{picture} &
\begin{picture}(30,30)(2,1)
\thicklines
\put(2,1){$\circ$} \put(2,25.8){$\circ$} \put(7,28.3){\line(1,0){20}}
\put(7,3.5){\line(1,0){20}}
\put(26,1){$\circ$}
\put(26,25.8){${\circ}$}
\end{picture} \\
& & & \\
$X$ & \multicolumn{3}{c}{$S^{2}_{{\bf P}^{1}}X$ and its involution}
\end{tabular}
\end{center}
In the group-theoretic setup of Remark (2.6), $\bar{\rho}$ induces
an isomorphism $$WD_3\stackrel{\sim}{\rightarrow}S_4.$$ (This is
the standard isomorphism, reflecting the isomorphism of the Dynkin
diagrams $D_3, A_3$.) Recillas' map $T$ then corresponds to
composition of a representation with this isomorphism.
We list a few of the subgroups of $S_4$:
\[
\begin{diagram}[\widetilde{C} {\textstyle \amalg}]
\node[2]{\langle (1) \rangle} \arrow{sw,-} \arrow{se,-} \\
\node{\langle (12) \rangle} \arrow[2]{s,-} \arrow{se,-}
\node[2]{\langle
(12)(34) \rangle}
\arrow{sw,-} \arrow{s,-} \\
\node[2]{\langle (12) , (34) \rangle} \arrow{s,-} \node{K}
\arrow{s,-}
\arrow{sw,-} \\
\node{S_{3}} \arrow{se,-} \node{D} \arrow{s,-} \node{A_{4}}
\arrow{sw,-} \\
\node[2]{S_{4}}
\end{diagram}
\]
$D$: The dihedral group $\langle\!(12), (1324)\!\rangle$ \\
$K=D\cap A_4$: The Klein group $\langle\!(12)(34),
(13)(24)\!\rangle$.
The corresponding curves are:
\[
\begin{diagram}[\widetilde{C} {\textstyle \amalg}]
\node[2]{X !} \arrow{sw,t}{2} \arrow{se,t}{2} \\
\node{Y} \arrow[2]{s,l}{3} \arrow{se,b}{2} \node[2]{Z}
\arrow{sw,b}{2} \arrow{s,r}{2} \\
\node[2]{\widetilde{C}} \arrow{s,r}{2} \node{T !} \arrow{s,r}{3}
\arrow{sw,b}{2} \\
\node{X} \arrow{se,b}{4} \node{C} \arrow{s,r}{3} \node{O}
\arrow{sw,b}{2} \\
\node[2]{{\bf P}^{1}}
\end{diagram}
\]
\noindent $O\approx O(X)\approx O(C)$: The orientation \\
$Y\approx(X\times_{{\bf P}^1}X)$ $\backslash$ (diagonal) \\
$Z\approx\widetilde{C}\times_{{\bf P}^1}O$.
Using either of these constructions, we can easily describe the
behavior of $X, C, \widetilde{C}$ around various types of branch
points. Keeping $X$ non-singular, there are the following five
possible local pictures, cf. [DS, III 1.4].
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|} \cline{2-6}
\multicolumn{1}{c|}{ } & (i) & (ii) & (iii) & (iv) & (v) \\
\cline{2-6} \hline
& \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in}
&
\hspace{.76in} \\
& \hspace{.76in} & \hspace{.76in} & \hspace{.76in} &
\hspace{.76in} &
\hspace{.76in} \\
$X$ & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} &
\hspace{.76in} &
\hspace{.76in} \\
& \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in}
&
\hspace{.76in} \\
& \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in}
&
\hspace{.76in} \\ \hline
& \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in}
&
\hspace{.76in} \\
& \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} &
\hspace{.76in} \\
& \hspace{.76in} & \hspace{.76in} & \hspace{.76in} &
\hspace{.76in} &
\hspace{.76in} \\
$\widetilde{C}$ & \hspace{.76in} & \hspace{.76in} & \hspace{.76in}
&
\hspace{.76in} &
\hspace{.76in} \\
& \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in}
&
\hspace{.76in} \\
& \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in}
&
\hspace{.76in} \\
& \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in}
&
\hspace{.76in} \\ \hline
& \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} &
\hspace{.76in} \\
& \hspace{.76in} & \hspace{.76in} & \hspace{.76in} &
\hspace{.76in} &
\hspace{.76in} \\
$C$ & \hspace{.76in} & \hspace{.76in} & \hspace{.76in} &
\hspace{.76in} &
\hspace{.76in} \\
& \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in}
&
\hspace{.76in} \\
& \hspace{.76in} & \hspace{.76in} & \hspace{.76in} & \hspace{.76in}
&
\hspace{.76in} \\ \hline
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{cc}
\multicolumn{2}{c}{\Large\bf Legend} \\
\begin{tabular}{cl}
\hspace{.35in} & \hspace{1.2in} \\
\hspace{.35in} &unramified sheet \\
\hspace{.35in} & \hspace{1.2in} \\
\hspace{.35in} & \hspace{1.2in} \\
\hspace{.35in} & simple ramification \\
\hspace{.35in} & \hspace{1.2in} \\
\hspace{.35in} & \hspace{1.2in} \\
\hspace{.35in} & node (two unramified \\
\hspace{.35in} & sheets glued together) \\
\hspace{.35in} & \hspace{1.2in} \\
\hspace{.35in} & two ramified sheets\\
\hspace{.35in} & glued together \\
\end{tabular} &
\begin{tabular}{cl}
\hspace{.35in} & \hspace{1.2in} \\
\hspace{.35in} & ramification point \\
\hspace{.35in} & of index 2 \\
\hspace{.35in} & \hspace{1.2in} \\
\hspace{.35in} & \hspace{1.2in} \\
\hspace{.35in} & ramification point \\
\hspace{.35in} & of index 3 \\
\hspace{.35in} & \hspace{1.2in} \\
\hspace{.35in} & \hspace{1.2in} \\
\hspace{.35in} & glueing of two sheets \\
\hspace{.35in} & of different \\
\hspace{.35in} & ramification indices \\
\end{tabular}
\end{tabular}
\end{center}
\begin{list}{{\rm(\roman{butter})}}{\usecounter{butter}}
\item $f, \pi, g$ are \'{e}tale.\item $f$ and $g$ have
simple ramification points, $\pi$ is \'{e}tale.
\item $f$ and $g$ each have a ramification point of index 2, $\pi$
is \'{e}tale.
\item $g$ has two simple ramification points, $\pi$ is a
Beauville cover: \\ $\bar{f}:N\rightarrow{\bf P}^1$ is trigonal,
with a fiber $\{p, q, r\}; \ \bar{\pi}:\widetilde{N}\rightarrow N$ is
branched at $p, q: \ \bar{\pi}^{-1}(p)=\widetilde{p}, \
\bar{\pi}^{-1}(q)=\widetilde{q}$; \ and we have $C=N/(p\sim q), \
\ \ \widetilde{C}=\widetilde{N}/(\widetilde{p}\sim\widetilde{q})$, \
\ and
$\pi:\widetilde{C}\rightarrow C, \ \ \ f:C\rightarrow{\bf P}^1$ are
induced by $\bar{\pi}, \bar{f}$.
\item $g$ has a ramification point of index 3, $\pi$ is Beauville,
$f$ is ramified at one of the two branches of the node of $C$.
\end{list}
\bigskip
Considering first the first three cases, then all five, we
conclude:
\noindent{\bf Theorem 2.9} The trigonal construction gives
isomorphisms $$T^0:{\cal R}^{{\rm
Trig}}_g\stackrel{\sim}{\rightarrow}{\cal M}^{{\rm Tet}, 0}_{g-1}$$
and $$T:\bar{\cal R}^{{\rm
Trig}}_g\stackrel{\sim}{\rightarrow}{\cal M}^{{\rm Tet}}_{g-1},$$
where:
${\cal M}^{{\rm Tet}}_{g-1}$ is the moduli space of (non-singular)
curves of genus $g-1$ with a tetragonal line bundle.
${\cal M}^{{\rm Tet},0}_{g-1}$ is the open subset of tetragonal
curves $X$ with the property that above each point of ${\bf P}^1$
there is at least one etale point of $X$.
${\cal R}^{{\rm Trig}}_g$ is the moduli space of etale double
covers of non-singular curves of genus $g$ with a trigonal bundle.
$\bar{\cal R}^{{\rm Trig}}_g$ is the partial compactification of
${\cal R}^{{\rm Trig}}_g$ using allowable covers in $\bar{\cal
R}_g$ of type $\partial^{\rm III}$ (cf (1.9.III)).
\bigskip
\noindent{\bf Examples 2.10}
\begin{list}{{\rm(\roman{butter})}}{\usecounter{butter}}
\item $\widetilde{C}$ is the trivial cover, $\widetilde{C}=C_0\amalg
C_1$,
iff $X$ is disconnected, \\ $X={\bf P}^1\amalg C$, with
$f=g|_C, \ \ \ id_{{\bf P}^1}=g|_{{\bf P}^1}$.
\item Wirtinger covers $(C_0\amalg C_1) \ / \ (p_0\sim q_1, \
q_0\sim p_1)\rightarrow C/(p\sim q)$, where $\{p, q, r\}$
form a trigonal fiber in $C$, correspond to reducible $X={\bf
P}^1\cup_rC$, the
two components meeting at $r\in C$.
\item $C$ is reducible: $C=H\cup{\bf P}^1$, with $H$
hyperelliptic, and \\ $\widetilde{C}=\widetilde{H}\cup{\bf P}^1$ with
$\widetilde{H}\rightarrow H$ and $\widetilde{{\bf
P}^1}\rightarrow{\bf P}^1$
branched over \\ $B:=H\cap{\bf P}^1$. This situation
corresponds to $g:X\rightarrow{\bf P}^1$ factoring through a
hyperelliptic $H'$. Indeed, such a pair $(C, \widetilde{C})$ is
uniquely determined by the tower $\widetilde{H}\rightarrow
H\rightarrow{\bf P}^1$. The trigonal construction for $C$ is
reduced to the bigonal construction for $H$, which then gives
$X=\widetilde{H}'\rightarrow H'\rightarrow{\bf P}^1$. In particular:
\item $C=H\amalg{\bf P}^1$ is disconnected iff $X=H_0\amalg H_1$ is
disconnected with hyperelliptic pieces, and then
$\widetilde{C}=\widetilde{H}\amalg{\bf P}^1\amalg{\bf P}^1$, where
$\widetilde{H}$ is the Cartesian cover:
$$\widetilde{H}=H_0\times_{{\bf
P}^1}H_1.$$
\end{list}
So far, we have only used the fact that $\widetilde{C}$ is an
orientable double cover of a triple cover. We now use our two
assumptions, that $\pi$ is unramified and that the base $K$ equals
${\bf P}^1$, to obtain an identity of abelian varieties. Namely,
by Remark 2.5 we have a map, natural up to translation.
$$\alpha:X\rightarrow P(\widetilde{C}/C).$$
The result, due to S. Recillas, is:
\bigskip
\noindent{\bf Theorem 2.11} [R] If $X$ is trigonally related to
$(\widetilde{C},C)$, then the above map $\alpha$ induces an
isomorphism
$$\alpha_*:J(X)\stackrel{\sim}{\rightarrow}P(\widetilde{C}/C).$$
\noindent{\bf Proof.}
By naturality of $\alpha$ and irreducibility of ${\cal M}^{{\rm
Tet}}_{g-1}$, it suffices to prove this for any one convenient $X$.
We take $\widetilde{C}\rightarrow C$ to be a Wirtinger cover as in
(2.10)(ii), so $$X={\bf P}^1\cup_rC'.$$ where $p+q+r$ is a trigonal
divisor on $C'$, and $C=C'/(p\sim q)$. We have natural
identifications: $$J(X)\approx J(C')\approx P(\widetilde{C}/C),$$ in
terms of which $\alpha$ becomes the Abel-Jacobi map $\varphi$ on
$C'$, and collapses ${\bf P}^1$ to a point.The induced $\alpha_*$ is
therefore
the identity.
\begin{flushright} QED \end{flushright}
\noindent{\bf Corollary 2.12} All trigonal Pryms are Jacobians,
and all tetragonal Jacobians are Pryms.
\subsection{The tetragonal construction}
\ \ \ \ Consider now a tower $$\widetilde{C}\rightarrow
C\stackrel{f}{\rightarrow}{\bf P}^1$$ where $f$ has degree 4 and
$\widetilde{C}$ is a double cover (unramified) of $C$. The general
construction yields a sequence of maps of degrees 2, 4, 2:
$$f_*\widetilde{C}\rightarrow
f_*\widetilde{C}/\iota\rightarrow\widetilde{\bf
P}^1\rightarrow{\bf P}^1.$$By (2.2) and (2.4) again, $\widetilde{\bf
P}^1$ is
unramified, hence we have
splittings:
\[\begin{array}{rcl}
\widetilde{\bf P}^1 & = & {\bf P}_0^1\amalg{\bf P}_1^1 \\
f_*\widetilde{C} & = & \widetilde{C}_0\amalg\widetilde{C}_1 \\
f_*\widetilde{C}/\iota & = & {C}_0\amalg {C}_1.
\end{array}
\] The tetragonal
construction thus associates to a tower
$$\widetilde{C}\stackrel{2}{\rightarrow}C\stackrel{4}{\rightarrow}{\b
f
P}^1$$ two other towers $$\widetilde{C}_i\rightarrow
C_i\rightarrow{\bf P}^1, \
\ \ \ \ \ \ \ \ \ \ \ i=0, 1$$ of the same type.
\bigskip
\noindent{\bf Lemma 2.13} The tetragonal construction is a
triality, i.e. starting with \ \ \ $\widetilde{C}_0\rightarrow
C_0\rightarrow{\bf P}^1$ \ \ \ it returns \ \ \
$\widetilde{C}\rightarrow C\rightarrow{\bf P}^1$ \ \ \ and \\
$\widetilde{C}_1\rightarrow
C_1\rightarrow{\bf P}^1$.
\ \ \ \ On the group level, the point is this: Our tower
$\widetilde{C}\rightarrow C\rightarrow{\bf P}^1$ corresponds to a
representation (of $\pi_1({\bf P}^1\backslash$ (branch locus)))
in $WD_4$. Now the Dynkin diagram $D_4$ has an automorphism of
order 3:
\vspace{3in}
\noindent This corresponds to an outer automorphism of $WD_4$, of
order 3. Hence representations in $WD_4$ come in packets of three.
The various groups involved are described in some detail in the
proof of Lemma (5.5), below.
\bigskip
\pagebreak[4]
\noindent{\bf Local pictures 2.14} Given the local behavior of $C$
and $\widetilde{C}$ over a point of ${\bf P}^1$, it is quite
straightforward to compute $f_*\widetilde{C}$ and hence
$\widetilde{C}_i,
C_i \ \ (i=0, 1)$ over the same point. Since these local pictures
are needed quite frequently, we record the simplest ones here.
\begin{list}{{\bf(\arabic{bean})}}{\usecounter{bean}}
\item $C, \widetilde{C}$ unramified $\Rightarrow C_i,
\widetilde{C}_i$ are
also unramified.
\item $C$ has one simple ramification point (and two unramified
sheets), $\widetilde{C}\rightarrow C$ unramified $\Rightarrow C_i,
\widetilde{C}_i$ have the same local picture as $C, \widetilde{C}$
respectively.
\item $C$ has two distinct simple ramification points,
$\widetilde{C}\rightarrow C$ unramified $\Rightarrow$ One pair, say
$C_0, \widetilde{C}_0$, has the same local pictures as $C,
\widetilde{C}$,
while the other is a Beauville degeneration: $C_1$ is unramified
but two of its four sheets are glued, $\widetilde{C}_1\rightarrow
C_1$
is ramified over these two sheets (and the ramification points are
glued) while the other sheets are unramified.
\item $C$ is unramified but two of its sheets are glued,
$\widetilde{C}\rightarrow C$ is ramified over these two sheets
$\Rightarrow$ $C_i$ has two distinct ramification points,
$\widetilde{C}_i\rightarrow C_i$ is unramified $(i=0, 1)$. (This is
the same triple as in (3).)
\item $C$ has a simple ramification point and the other two sheets
are glued, $\widetilde{C}$ is ramified over the glued sheets
$\Rightarrow C_i, \widetilde{C}_i$ have the same local pictures as
$C,
\widetilde{C}$.
\item $C$ has a ramification point of index 2 (i.e. 3 of its sheets
are permuted by the local monodromy), $\widetilde{C}\rightarrow C$
unramified $\Rightarrow$ same local picture for
$\widetilde{C}_i\rightarrow C_i$.
\item $C$ has a ramification point of index 3 (all 4 sheets
permuted), $\widetilde{C}\rightarrow C$ unramified $\Rightarrow C_0,
\widetilde{C}_0$ have the same local picture as $C, \widetilde{C}$,
but
$C_1$ has a simple ramification point glued to an unramified point,
so $\widetilde{C}_1$ must be simply ramified over each. (I.e.
$\widetilde{C}_1$ has a point which is simply ramified over ${\bf
P}^1$, glued to a point which has ramification index 3 over ${\bf
P}^1$!)
\end{list}
We note that in examples (3) and (7), the tetragonal construction
applied to $(\widetilde{C}\rightarrow C)\in{\cal RM}_g$ produces an
(allowable) degenerate cover, $(\widetilde{C}_1\rightarrow C_1)\in
\partial^{\rm III}({\cal RM}_g)$.
\begin{center}
\begin{tabular}{ccc}
\begin{tabular}{|c|c|c|} \hline
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline
\end{tabular} &
\begin{tabular}{|c|c|c|} \hline
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline
\end{tabular} &
\begin{tabular}{|c|c|c|} \hline
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline
\end{tabular} \\
(1) & (2) & (3,4)
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ccc}
\begin{tabular}{|c|c|c|} \hline
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline
\end{tabular} &
\begin{tabular}{|c|c|c|} \hline
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline
\end{tabular} &
\begin{tabular}{|c|c|c|} \hline
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline
\end{tabular} \\
(5) & (6) & (7)
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{c}
\begin{tabular}{|c|c|c|} \hline
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in}& \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
$\widetilde{C}$ & $\widetilde{C}_{0}$ & $\widetilde{C}_{1}$ \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
$C$ & $C_{0}$ & $C_{1}$ \\
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\ \hline
\hspace{.32in} & \hspace{.32in} & \hspace{.32in} \\
${\bf P}^{1}$ & ${\bf P}^{1}$ & ${\bf P}^{1}$ \\ \hline
\end{tabular} \\
(pattern)
\end{tabular}
\end{center}
\pagebreak[4]
\noindent{\bf Examples 2.15}
\begin{list}{{\bf(\arabic{bean})}}{\usecounter{bean}}
\item It is perhaps not terribly surprising that the trigonal
construction is a degenerate case of the tetragonal
construction.Start with
\linebreak $\widetilde{C}\rightarrow C$ the split double
cover of the curve $C$ with the tetragonal map
$f:C\stackrel{4}{\rightarrow}{\bf P}^1$. Then
$f_*\widetilde{C}$ splits into 5 components, of degrees 1, 4, 6, 4, 1
respectively over ${\bf P}^1$. The components of degree 4 make up
$\widetilde{C}_1\rightarrow C_1$, which is isomorphic to
$\widetilde{C}\rightarrow C$. The remaining components give
$${\bf
P}^1\amalg\widetilde{T}\amalg{\bf P}^1\rightarrow T\amalg{\bf P}^1$$
where $(\widetilde{T}, T)$ is associated to $C$ by the trigonal
construction. Vice versa, starting with an (unramified) double
cover \\ ${\bf P}^1\amalg\widetilde{T}\amalg{\bf P}^1$ of
$T\amalg{\bf
P}^1$, the tetragonal construction produces \linebreak $C\amalg
C\rightarrow
C$, twice.
\item Let $p+q+r+s$ be a tetragonal divisor on $C$. Then $C/(p\sim
q)$ is still tetragonal. Tacking a node onto the previous example,
we see that the Wirtinger cover $$(C'\amalg C'')/(p'\sim q'',
q'\sim p'')\rightarrow C/(p\sim q)$$ is taken by the tetragonal
construction to :
\noindent $\bullet$ Another Wirtinger Cover, $$(C'\amalg
C'')/(r'\sim s'', s'\sim r'')\rightarrow C/(r\sim s),$$ and to:
\noindent $\bullet$ ${\bf P}^1\cup_{t'}\widetilde{T}\cup_{t''}{\bf
P}^1\rightarrow T\cup_t{\bf P}^1$, where $(\widetilde{T}, T)$ is
associated by the trigonal construction to $C$. (Each copy of
${\bf P}^1$ meets $\widetilde{T}$ or $T$ in the unique point
indicated.
$t\in T$ corresponds to the partition $\{\{p, q\}, \{r, s\}\}$.)
\item We will see in Lemma (3.5) that if $C\rightarrow{\bf P}^1$
factors through a hyperelliptic curve, so do $C_0, C_1$. An
interesting subcase occurs when $C=H^0\cup H^1$ has two
hyperelliptic components, cf. Proposition (3.6).
\item Let $X$ be a non-singular cubic hypersurface in ${\bf P}^4$,
$\ell\subset X$ a line, and $\widetilde{X}$ the blowup of $X$ along
$\ell$, with projection from $\ell$:
$$\pi:\widetilde{X}\rightarrow{\bf
P}^2.$$ This is a conic bundle [CG] whose discriminant is a plane
quintic curve $Q\subset{\bf P}^2$. The set of lines $\ell'\subset
X$ meeting $\ell$ is a double cover $\widetilde{Q}$ of $Q$. Now
choose
a plane $A\subset{\bf P}^4$ meeting $X$ in 3 lines $\ell, \ell',
\ell''$; we get 3 plane quinties $Q, Q', Q''$, with double covers
$\widetilde{Q}, \widetilde{Q}', \widetilde{Q}''$. Note that $\ell',
\ell''$
map to a point $p\in Q$, hence determine a tetragonal map
$f:Q\rightarrow{\bf P}^1$, given by ${\cal O}_Q(1)(-p)$, and
similarly for $Q', Q''$. Our observation is that the 3 objects
$$(\widetilde{Q}, Q, f) \ \ ; \ \ (\widetilde{Q}', Q', f') \ \ ; \ \
(\widetilde{Q}'', Q'', f'')$$ are tetragonally related. Indeed, the
3
maps can be realized simultaneously via the pencil of hyperplanes
$S\subset{\bf P}^4$ containing $A$. Such an $S$ meets $X$ in a
(generally non-singular) cubic surface $Y$. $A$ line in $Y$ (and
not in $A$) which meets $\ell'$, also meets 4 of the 8 lines (in
$Y$, not in $A$) meeting $\ell$, one in each of 4 coplanar pairs.this
gives the
desired injection $\widetilde{Q}'\hookrightarrow
f_*\widetilde{Q}$.
\end{list}
Our main interest in the tetragonal construction stems from:
\bigskip
\noindent{\bf Theorem 2.16} The tetragonal construction commutes
with the Prym map, $$P(\widetilde{C}/C)\approx
P(\widetilde{C}_0/C_0)\approx P(\widetilde{C}_1/C_1).$$
\bigskip
\noindent{\bf Proof}
\ \ \ \ As in Remark (2.5), we have a map
$$\alpha:\widetilde{C}_i\hookrightarrow
f_*\widetilde{C}\rightarrow{\rm
Pic}(\widetilde{C}), \ \ \ \ \ \ \ \ \ \ i=0, 1.$$ The image sits in
a
translate of $P(\widetilde{C}/C)$, so we get induced maps
$$\alpha_*:J(\widetilde{C}_i)\rightarrow P(\widetilde{C}/ C)$$ and
by restriction $$\beta:P(\widetilde{C}_i/ C_i)\rightarrow
P(\widetilde{C}/
C).$$ By Masiewicki's criterion [Ma], $\beta$ will
be an isomorphism if we can show:
\begin{list}{{\rm(\arabic{bean})}}{\usecounter{bean}}
\item The image $\alpha(\widetilde{C}_i)$ of $\widetilde{C}_i$ in
$P(\widetilde{C}/C)$ is symmetric;
\item The fundamental class in $P(\widetilde{C}/C)$ of
$\alpha(\widetilde{C}_i)$ is twice the minimal class, $\frac{2}{(g-
1)!}[\Theta]^{g-1}$.
\end{list}
Now (1) is clear, since the involution on $\widetilde{C}_i$
commutes with $-1$ in $P(\widetilde{C}/C)$. The fundamental class in
(2) can be computed directly, as is done in [B2]. Instead, we find
it here by a degeneration argument: it varies continuously with
$(C, \widetilde{C})\in{\cal RM}^{{\rm Tet}}_g$, which is an
irreducible parameter space, so it suffices to do the computation
for a single $(C, \widetilde{C})$. We take $$C=T\cup_t{\bf P}^1, \
\
\ \widetilde{C}={\bf P}^1\cup_{t'}\widetilde{T}\cup_{t''}{\bf P}^1,$$
as in
Example (2.15)(2). Then $(C_i, \widetilde{C}_i)$ is a Wirtinger
cover,
$i=0, 1$, and the normalization of $C_i$ is the tetragonal curve
$N$ associated to $(T, \widetilde{T})$ by the trigonal construction.
We have identifications $$J(N)\approx P(\widetilde{T}/T)\approx
P(\widetilde{C}/C)$$ (Theorem (2.11)), in terms of which
$\alpha(\widetilde{C}_i)$ consists of the Abel-Jacobi image
$\varphi(N)\subset J(N)$ and of its image under the involution.Thus
the
fundamental class is twice that of $\varphi(N)$, as
required.
(Note: since this argurment works for any double cover
$\widetilde{T}\rightarrow T$, and since any semiperiod on a nearby
\mbox{non-singular} $C$ specializes to a semiperiod on $T\cup_t{\bf
P}^1$
which is supported on $T$, we need only the irreducibility of
${\cal M}^{{\rm Tet}}_g$, instead of ${\cal RM}^{{\rm Tet}}_g$.)
\begin{flushright} QED \end{flushright}
\large
\section{Bielliptic Pryms.}
\ \ \ \ As a first application of the tetragonal construction, we
show that some remarkable coincidences occur among the various loci
in Beauville's list [B1]. The central role here is played byPryms of
bielliptic curves. We see in (3.7), (3.8) that the bielliptic
loci
can be tetragonally related to various other components in
Beauville's
list, and therefore give the same Pryms. As suggested
in [D1], this leads to a complete, short list of the irreducible
components of the Andreotti-Mayer locus in genus $\leq 5$, and
of its intersection with the image of the proper Prym map for
arbitrary $g$. We do not include here the complete analysis of the
Andreotti-Mayer locus itself, since this has already appeared in
[Deb1] and [D5] (together with some corrections to the original
list in [D1]). Nevertheless, we could not resist describing
explicitly the operation of the tetragonal construction on
Beauville's list, as it is such a pretty and straightforward
application of the results of \S 2.
\ \ \ \ We recall Mumford's results on hyperelliptic Pryms. Let
$$f_i:C^i\rightarrow K, \ \ \ \ \ \ i= 0, 1$$ be two ramified
double covers of a curve $K$. The fiber product
$$\widetilde{C}:=C^0\times_KC^1$$ has 3 natural
involutions:$\tau_i(i=0, 1)$,
with quotient $C^i$, and \\
$\tau:=\tau_0\circ\tau_1$, with a new quotient, $C$. This all
fits in a Cartesian diagram:
\[
\begin{diagram}[C]
\node[2]{\widetilde{C}} \arrow{sw,t}{\pi_{1}} \arrow{s,r}{\pi}
\arrow{se,t}{\pi_{0}} \\
\node{C^{0}} \arrow{se,b}{f_{0}} \node{C} \arrow{s,r}{f} \node{C^{1}}
\arrow{sw,b}{f_{1}} \\
\node[2]{K}
\end{diagram}
\]
If the branch loci of $f_0, f_1$ are disjoint, then
$$\pi:\widetilde{C}\rightarrow C$$ is unramified. We say that a
double
cover obtained this way is \linebreak \underline{Cartesian}.
\bigskip
\noindent{\bf Lemma 3.1} Let $f:C\rightarrow K$ be a ramified
double cover. A double cover $$\pi:\widetilde{C}\rightarrow C,$$
given
by a semiperiod $\eta\in J_2(C)$, is Cartesian if and only if \\
$f_*(\eta)=0\in J_2(K)$.
\bigskip
\noindent{\bf Proof:} apply the bigonal construction.
\begin{flushright} QED \end{flushright}
\bigskip
\noindent{\bf Proposition 3.2} [M1]
\begin{list}{{\bf(\arabic{bean})}}{\usecounter{bean}}
\item Any double cover $\widetilde{C}$ of a hyperelliptic $C$ is
Cartesian.
\item Any hyperelliptic Prym is a product of 2 hyperelliptic
Jacobians (one of which may vanish): If $\widetilde{C}$ arises as
$C^0\times_{{\bf P}^1}C^1$ then $$P(\widetilde{C}/C)\approx
J(C^0)\times J(C^1).$$
\end{list}
\bigskip
\noindent{\bf Proof:} (2) follows from (1), (1) follows from lemma
(3.1) with $K = {\bf P}^{1}$. \begin{flushright} QED
\end{flushright}
\ \ \ \ A \underline{bielliptic} curve (aka elliptic-hyperelliptic,
superelliptic, ...) is a branched double cover of an elliptic
curve. In this section we apply the tetragonal construction to
find various identities between bielliptic Pryms and Pryms of
other, usually degenerate, curves. Some of the results extend
to \underline{bihyperelliptic} curves, i.e. branched double
covers of hyperelliptic curves. To warm up, we consider
\underline{Jacobians} of bihyperelliptic curves. Example
(2.10)(iii) can be restated:
\bigskip
{\bf Lemma 3.3} The trigonal construction gives a bijection between:
\noindent
$\bullet \ \ $ Bihyperelliptic, non singular curves $C$:
\[ C \stackrel{f}{\rightarrow} H \stackrel{g}{\rightarrow} {\bf
P}^{1}; \]
$\bullet \ \ $ Reducible trigonal double covers $\widetilde{X}
\rightarrow X$:
\begin{center}
\begin{tabular}{rlccc}
$\begin{diagram}[X]
\node{\widetilde{X}} \arrow{s} \\
\node{X}
\end{diagram}$ &
$\begin{diagram}[X]
\node{=} \\
\node{=}
\end{diagram}$ &
$\begin{diagram}[X]
\node{C'} \arrow{s} \\
\node{H'}
\end{diagram}$ &
$ \begin{diagram}[X]
\node{\cup} \\
\node{\cup}
\end{diagram} $&
$\begin{diagram}[X]
\node{H} \arrow{s} \\
\node{{\bf P}^{1}}
\end{diagram}$
\end{tabular}
\end{center}
where
$X = H' \cup {\bf P}^{1}$ is reducible
$\tau : X \rightarrow {\bf P}^{1}$, the trigonal map, has
degree 2 on $H'$ and 1 \linebreak on ${\bf P}^{1}$.
$\tau(H' \cap {\bf P}^{1})$ = Branch($g$)
$\widetilde{X} \rightarrow X$ is allowable of type $\partial^{\rm
III}$
at each point of $H' \cap {\bf P}^{1}$.
\bigskip
\noindent
We note that $C' \rightarrow H' \rightarrow {\bf P}^{1}$ is
bigonally
related to $C \rightarrow H \rightarrow {\bf P}^{1}$.
\bigskip
\noindent{\bf Corollary 3.4} The Jacobian of a bihyperelliptic
curve $C$,
$$C\stackrel{f}{\rightarrow}H\stackrel{g}{\rightarrow}{\bf P}^1,$$
is isogenous to the product $$J(H)\times P(g_*C, \iota)$$ of a
hyperelliptic Jacobian and a bihyperelliptic (branched) Prym.
\bigskip
\ \ \ \ We move to the Pryms of bihyperelliptic curves. First we
note that this class is closed under the tetragonal construction:
\bigskip
\noindent{\bf Lemma 3.5} \ \ Let $(\widetilde{C}_i, C_i)$ be
tetragonally related to $(\widetilde{C}, C)$, with $C$ non-singular.
If $C\rightarrow{\bf
P}^1$ factors through a (possibly reducible) hyperelliptic $H$, so
do the $C_i$:
$$C_i\stackrel{f_i}{\rightarrow}H_i\stackrel{g_i}{\rightarrow}{\bf
P}^1, \ \ \ \ \ \ i=0, 1.$$
\bigskip
\noindent{\bf Proof.}
The bigonal construction applied to
$$\widetilde{C}\stackrel{\pi}{\rightarrow}C\stackrel{f}{\rightarrow}H
$$
yields $$f_*\widetilde{C}\rightarrow\widetilde{H}\rightarrow H,$$ and
when
applied again to $$\widetilde{H}\rightarrow
H\stackrel{g}{\rightarrow}{\bf P}^1$$ yields
$$g_*\widetilde{H}\rightarrow\widetilde{{\bf P}}^1\rightarrow{\bf
P}^1.$$
Since $\pi$ is unramified, so are $\widetilde{H}\rightarrow H$ and
$\widetilde{\bf P}^1\rightarrow{\bf P}^1$. Hence $\widetilde{\bf
P}^1$
splits: $$\widetilde{\bf P}^1={\bf P}^1_0\amalg{\bf P}^1_1,$$ and
this
splitting climbs its way up the tower:
\[
\begin{array}{rlc}
(g\circ f)_{*}\widetilde{C} & = & \widetilde{C}_{0} {\textstyle
\amalg}
\widetilde{C}_{1} \\
& & \downarrow \\
& & C_{0} {\textstyle \amalg} C_{1} \\
& & \downarrow \\
g_{*}\widetilde{H} & = & {H}_{0} {\textstyle \amalg} {H}_{1} \\
& & \downarrow \\
\widetilde{\bf P}^{1} & = & {\bf P}_{0}^{1} {\textstyle \amalg}
{\bf
P}_{1}^{1} \\
& & \downarrow \\
& & {\bf P}^{1}
\end{array}
\]
\begin{flushright} QED \end{flushright}
\bigskip
\noindent
{\bf Remark 3.5.1} The rational map $f_{i} : C_{i} \rightarrow
H_{i}$ can,
in a couple of cases, fail to be a morphism; this is easily remedied
by
identifying a pair of points in $H_{i}$. Among the local pictures
(2.14), the
ones that
can occur here are (1), (2), (7) and (3) :
\begin{itemize}
\item In cases (1), (2), the hyperelliptic maps $g$, $g_{0}$,
$g_{1}$ are all
unramified at the relevant point, and the $f_{i}$ are morphisms.
\item In case (7), $g$ and $g_{0}$ are ramified, $g_{1}$ is not, $f$
and
$f_{0}$
are (ramified) morphisms, but $f_{1}$ is not, since $C_{1}$ is
singular above a
point where $H_{1}$, as constructed above, is nonsingular. To make
$f_{1}$ into
a morphism, we must glue the two points of $g_{1}^{-1}(k)$.
\item In case (3) we find two possibilities:
\begin{list}{(3\alph{bean})}{\usecounter{bean}}
\item $g$ is etale, $f$ is ramified at both points of $g^{-1}(k)$;
then
$g_{0}$,
$g_{1}$ are also etale, $f_{0}$ is ramified at both points of
$g_{0}^{-1}(k)$,
$C_{1}$ has a node but $f_{1}$ is still a morphism.
\item $g$ is ramified, $f$ is etale; then $g_{0}$ is ramified,
$f_{0}$ is
etale, $g_{1}$
is etale, but the two branches of the node of $C_{1}$ are sent by
$f_{1}$ to
opposite sheets of $H_{1}$, so $f_{1}$ is again not a morphism.
\end{list}
\end{itemize}
\noindent{\bf Proposition 3.6} Let $\widetilde{C}\rightarrow C$ be a
Cartesian double cover of a bihyperelliptic $C$:
$$C\stackrel{f}{\rightarrow}H\stackrel{g}{\rightarrow}{\bf P}^1, \;
\widetilde{C} = C^{0}\times_{H} C^{1}.$$
The tetragonal construction applied to $\widetilde{C}\rightarrow
C\rightarrow{\bf P}^1$ yields:
\noindent $\bullet$ A similar Cartesian tower
$\widetilde{C}_0\rightarrow
C_0\stackrel{f_0}{\longrightarrow}H\stackrel{g_0}{\longrightarrow
}{\bf P}^1$, same $H$.
\noindent $\bullet$ A tower $\widetilde{C}_1\rightarrow
C_1\rightarrow{\bf P}^1$ where:
\begin{tabbing} XXXXX \= \kill
\>$C_1$ is reducible, $C_1=H^0\cup H^1$, \\ \> $H^0, H^1$ are
hyperelliptic, \\ \> $H^0\cap H^1$ maps onto $B:={\rm
Branch}(g)\subset{\bf P}^1$, \\ \>
$\widetilde{C}_1=\widetilde{H}^0\cup\widetilde{H}^1$ is allowable
over
$C_1$, \\
\> $C^{i} \rightarrow H \rightarrow {\bf P}^{1}$ is bigonally related
to
$\widetilde{H}^{i} \rightarrow H^{i} \rightarrow {\bf P}^{1}$, $i$ =
1,2.
\end{tabbing}
Vice versa, the tetragonal construction takes any tower \linebreak
$\widetilde{C}_1\rightarrow C_1\rightarrow{\bf P}^1$ as above to two
Cartesian bihyperelliptic towers $$\widetilde{C}\rightarrow
C\rightarrow H\rightarrow{\bf P}^1 \ \ \ {\rm and} \ \ \
\widetilde{C}_0\rightarrow C_0\rightarrow H\rightarrow{\bf P}^1.$$
The proof is quite straightforward, and we will simply
write down a few of the relationships involved, using the notation of
the
previous
proof:
\noindent $\bullet$ $\widetilde{H}$ splits into two copies of $H$,
by
(3.1).Hence:
\noindent $\bullet$ $g_*\widetilde{H}\approx H\cup {\bf
P}^1\cup {\bf P}^1$, say $H_0\approx H, \ \ H_1 = R^{0} \cup R^{1},
\;
R^{i} \approx {\bf P}^{1}$, $i$ = 0, 1.
\begin{tabbing}
\noindent $\bullet$ \= Let $H^{i}$, $\widetilde{H}^{i}$ be the
inverse image
of
$R^{i}$ in $C_{1}$, $\widetilde{C}_{1}$ respectively. Then \\
\> $\widetilde{H}^{i} \rightarrow H^{i} \rightarrow {\bf P}^{1}$ is
bigonally
related to $C^{i} \rightarrow H \rightarrow {\bf P}^{1}$.
\end{tabbing}
\begin{tabbing}
\noindent $\bullet$ \=The intersection properties of the $H^i$ (or
$\widetilde{H}^i$) can be read off the \\ \>local pictures (2.14.3).
\end{tabbing}
\begin{tabbing}
\noindent $\bullet$ \=Finally, let $\varepsilon:H\to H$ be the
hyperelliptic involution. A cover \\ \>$C^1\to H$ determines a
mirror-image $\varepsilon^*C^1$. The remaining tower \\
\>$\widetilde{C}_0\to C_0\to H\to{\bf P}^1$ is given by the Cartesian
diagram:
\end{tabbing}
\[
\begin{diagram}[C]
\node[2]{\widetilde{C}_{0}} \arrow{sw} \arrow{s} \arrow{se} \\
\node{C^{0}} \arrow{se} \node{C_{0}} \arrow{s}
\node{\varepsilon^{*}C^{1}}
\arrow{sw} \\
\node[2]{H}
\end{diagram}
\]
\begin{flushright} QED \end{flushright}
{\bf Remarks}
\noindent
{\bf (3.6.1)} Since the branch points of $C^{i} \rightarrow H$ map to
the
branch points of $H^{i} \rightarrow {\bf P}^{1}$, we have the
relation
between the genera:
\[ g(H^{i}) = g(C^{i}) - 2\cdot g(H). \]
\noindent
{\bf (3.6.2)} The possible local pictures are exactly the same as in
(3.5.1).
(The use of $C_{0}$, $C_{1}$ in (3.6) is consistent with that of
(2.14).)
\noindent
{\bf (3.6.3)} Another way of proving both lemma (3.5) and proposition
(3.6)
is based on lemma (5.5), which says that the three tetragonal curves
$C, C_{0}, C_{1}$ which are tetragonally related are obtained, via
the
trigonal construction, from one and the same trigonal curve $X$ (with
three distinct double covers). Lemma (3.3) characterizes the possible
curves
$X$, hence proves that the locus of bihyperelliptics is closed under
the
tetragonal construction, lemma (3.5). To complete the proof of
proposition (3.6), one simply needs to characterize the double covers
$\widetilde{X}$ which correspond to Cartesian covers of $C$.
\ \ \ \ For the rest of this section, we specialize to the case
where the hyperelliptic $H$ is an elliptic curve $E$, i.e. $C$ is
bielliptic. First, we write out explicitly the content of
Proposition (3.6) in this case:
\noindent{\bf Corollary 3.7} The Pryms of double covers $\pi
:\widetilde{C}\to C$ where
\noindent $\bullet$ $C$ is bielliptic,
$C\stackrel{f}{\to}E\stackrel{g}{\to}{\bf P}^1$,
\noindent $\bullet$ $\widetilde{C}\to C$ is Cartesian,
$\widetilde{C}=C^0\times_EC^1, \ \ \ C^0$ is of genus $n$,
\noindent are precisely (via the tetragonal construction) the Pryms
of the following allowable double covers $\widetilde{X}\to X$:
\begin{tabbing}
$n=1$: \ \ \=$X$ is obtained from a hyperelliptic curve by
identifying \\ \>two pairs of points, $X=H/(p\sim q, \ \ r\sim s)$.
\\ \\
$n=2$:\>$X=X_0\cup X_1, \ \ X_0$ rational, $X_1$ hyperelliptic,
\\ \>$\#(X_0\cap X_1)\!=\!4$. \\ \\
$n\ge 3$:\>$X=X_0\cup X_1$, \ each $X_i$ hyperelliptic, \
$g(X_0)=n-2$, \\ \>$g(X_1)=g(C)-n-1, \ \ \#(X_0\cap X_1)=4$,
and both \\ \>hyperelliptic maps are restrictions of the same
tetragonal \\ \>map on $X$ (i.e. they agree on $X_0\cap X_1$).
\end{tabbing}
\ \ \ \ Everything here follows directly from the proposition,
except that for $n=1$ we need to use (twice) the following
observation of Beauville. Let $\pi:\widetilde{X}\to X$ be an
allowable
double cover where $$X=Y\cup R, \ \ \ \ \ \ \ \ R
\;\mbox{rational}, \ \ \ \ \ \ \ \ \ Y\cap R=\{a,b\}$$
$$\widetilde{X}=\widetilde{Y}\cup\widetilde{R}, \ \
\widetilde{R}=\pi^{-1}(R)\;\mbox{rational}, \ \
\widetilde{Y}\cap\widetilde{R}=\{\widetilde{a},\widetilde{b}\}$$ and
$\pi$ is
ramified at $\widetilde{a}, \widetilde{b}$, which map to $a,b$.
Construct
a new cover $\widetilde{Z}\to Z$ where
$$\widetilde{Z}:=\widetilde{Y}/(\widetilde{a}\sim\widetilde{b})$$
$$Z:=Y/(a\sim
b).$$ Then this is still allowable, and $$P(\widetilde{Z}/Z)\approx
P(\widetilde{X}/X).$$ (Indeed, there are natural isomorphisms of
generalized Jacobians $$J(\widetilde{Z})\approx J(\widetilde{X}),
\;\;J(Z)\approx J(X)$$ commuting with $\pi_*$ and inducing the
desired isomorphisms.)
\begin{flushright} QED \end{flushright}
\ \ \ \ We are left with the Pryms of non-Cartesian double covers
of bielliptic curves. The result here may be somewhat surprising:
\bigskip
\noindent{\bf Proposition 3.8} Pryms of \underline{non}-Cartesian
double covers of bielliptic curves are precisely the Pryms of
Cartesian covers (of bielliptic curves) with $n(:=g(C_0))=1.$ (The
isomorphism is obtained through a sequence of 2 tetragonal moves.)
\ \ \ \ The point is that if $X=H/(p\sim q, \ \ r\sim s)$ with $H$
hyperelliptic, and $\widetilde{X}\to X$ is an allowable double cover,
then $P(\widetilde{X}/X)$ is the Prym of a Cartesian cover (with
$n=1$)
of a bielliptic curve, as we've just seen; but $X$ has another
$g^1_4$, and applying the tetragonal construction to it yields a
non-Cartesian double cover of a bielliptic curve.
\ \ \ \ The $g^1_4$ is obtained as follows: map $H$ to a conic in
${\bf P}^2$ (by the hyperelliptic map), then project the conic to
${\bf P}^1$ from the unique point $x$ in ${\bf P}^2$ (and not on
the conic) on the intersection of the lines $\overline{pq}$ and
$\overline{rs}$.
\vspace{2in}
\ \ \ \ We should now check that the tetragonal construction yields
a non-Cartesian cover of a bielliptic curve, and that all covers
arise this way. We leave the former to the reader, and do the
latter.
\ \ \ \ Let $\widetilde{C}\to C$ be a non-Cartesian cover of $C$,
which
is bielliptic: $$ C\stackrel{f}{\to}E\stackrel{g}{\to}{\bf P}^1.$$
\noindent Let $(\widetilde{C}_i,C_i)$, \ \ $i=0,1$, be the
tetragonally
related covers. By lemma (3.5), $C_i$ is
bihyperelliptic:$$C_i\stackrel{f_i}{\to}H_i\stackrel{g_i}{\to}{\bf
P}^1.$$ By
the
local pictures (2.14), $$B :=\mbox{Branch}(g)=B_0\amalg B_1, \ \
\ B_i:=\mbox{Branch}(g_i).$$
(As we saw in Remark (3.5.1), the possible pictures are (1), (2),
(7),
(3a) and (3b). Of these, (7) and (3b) contribute to $B$, and each
contributes also to one of the $B_{i}$.)
Since $\#B=4$ \ \ ($E$ is elliptic),
and $\# B_i$ is even and $>0$ (non-Cartesian!), we find $$\# B_i=2,
\;\;\; i=0,1,$$ hence $H_i$ is rational and $C_i$ is hyperelliptic.
Again by the local pictures, $C_i$ will have two nodes, at points
lying over $B_{1-i}$.
\begin{flushright} QED \end{flushright}
\ \ \ \ We observe that the last argument works not only for
bielliptics but also for branched double covers of hyperelliptic
curves of
genus 2, since now $$\#B_0>0, \ \ \#B_1>0, \ \ \#B_0+\#B_1=6, \ \
\#B_i\; \mbox{even}\Rightarrow$$ $$\mbox{either} \ \;\#B_0=2 \
\mbox{or} \ \#B_1=2.$$ However, the resulting hyperelliptic curve
with 4 nodes does not carry other $g^1_4$'s and is not necessarily
related to any other covers.
\ \ \ \ We leave one more corollary of proposition (3.6) to the
reader.
\bigskip
\noindent{\bf Corollary 3.9} Let $K$ be hyperelliptic,
$\widetilde{K}\to K$ a double cover with 2 branch points. Then
$P(\widetilde{K}/K)$ is a hyperelliptic Jacobian.
\noindent(Hint: take both $H$ and $C^0$ in proposition (3.6) to
be rational, show $P(\widetilde{C}_1/C_1)\approx J(C^1)$ and
$C_1=K\cup_{(2\; \mbox{points})}{\bf P}^1, \ K$ hyperelliptic.)
\large
\section{Fibers of ${\cal P}:{\cal R}_6\to{\cal A}_5.$}
\subsection{The structure}
\ \ \ \ We recall the main result of [DS]:
\bigskip
\noindent{\bf Theorem 4.1 [DS]} ${\cal P}:{\cal R}_6\to{\cal A}_5$ is
generically finite, of degree 27.
Recall that ${\cal M}_6^{\rm Tet}$ denotes the moduli space of curves of
genus 6 with a $g^1_4$. The forgetful map ${\cal M}_6^{\rm
Tet}\to{\cal M}_6$
is generically finite, of degree 5 [ACGH]. By base change we get
a corresponding object ${\cal R}_6^{\rm Tet}$, with map
$${\cal R}_6^{\rm Tet}\to{\cal R}_6$$ of degree 5. The tetragonal
construction gives a triality, or (2,2)-corres\-pon\-den\-ce, on
${\cal R}_6^{\rm Tet}$. The image in ${\cal R}_6$ is then a
(10,10)-correspondence:
\medskip
\noindent{\bf (4.1.1)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
${\rm Tet}\subset{\cal R}_6\times{\cal R}_6.$
\bigskip
\noindent{\bf Theorem 4.2} The correspondence Tet induced by the
tetragonal construction on the fiber ${\cal P}^{-1}(A)$, for
generic $A\in{\cal A}_4$, is isomorphic to the incidence correspondence
on the 27 lines on a non-singular cubic surface. The monodromy
group of ${\cal R}_6$ over ${\cal A}_5$ (i.e. the Galois group of its
Galois closure) is the Weyl group $WE_6$, the symmetry group of the
incidence of the 27 lines on the cubic surface.
This was conjectured in [DS] and announced in [D1]. The proof will
be given below. For the symmetry group of the line incidence on a
cubic surface, or other del Pezzo surfaces, we refer to [Dem].
\bigskip
\noindent{\bf (4.3) The blownup map}
Let ${\cal Q}\subset{\cal M}_6$ denote the moduli space of non-singular
plane
quintic curves, ${\cal R\cal Q}$ its inverse image in ${\cal R}_6$. By
Theorem (1.2), it splits:
$${\cal R\cal Q}={\cal R\cal Q}^+ \ \ \cup \ \ {\cal R\cal Q}^-$$ with $(Q,\mu)\in{\cal R\cal Q}^+$
(respectively, ${\cal R\cal Q}^-$) iff $h^0(\mu\otimes{\cal O}_Q (1)$) is
even (respectively, odd). The point is that ${\cal O}_Q(1)$ gives
a uniform choice of theta characteristics over ${\cal Q}$, hence the
spaces of theta characteristics and semiperiods over ${\cal Q}$ are
identified.
Let ${\cal J}$ be the closure in ${\cal A}_5$ of the locus of Jacobians
of
curves, and let ${\C}$ denote the moduli space of non-singular
cubic threefolds. Via the intermediate Jacobian map, we identify
${\C}$ with its image in ${\cal A}_5$.
The Prym map sends ${\cal R\cal Q}^+$ to ${\cal J}$ and ${\cal R\cal Q}^-$ to
${\C}$. Since the fiber dimensions can be positive, it is useful
to consider the blownup Prym map $$\widetilde{\cal
P}:\widetilde{\cal R}_6\to\widetilde{\cal A}_5$$ where ${\cal J},{\C}$ on
the
right are blown up to divisors $\widetilde{\cal J},\widetilde{\C}$,
while
on the left we blow up ${\cal R\cal Q}^+,{\cal R\cal Q}^-$, as well as the locus
${\cal R}_6^{Trig}$ of double covers of trigonal curves. The result is
a morphism which is generically finite over $\widetilde{\cal J}$
and
$\widetilde{\C}$. We recall the geometric description of points of
the
various loci, and
give the map in these geometric terms. This is taken from [CG],
[T] and [DS].
\bigskip
\noindent{\bf (4.3.1)} \ A point of ${\C}$ is given by a non-singular
cubic threefold $X\subset{\bf P}^4$. A point of $\widetilde{\C}$
is
given by a pair $(X,H), \ H\in({\bf P}^4)^*$ a hyperplane.
\bigskip
\noindent{\bf (4.3.2)} \ A point of $\widetilde{{\cal R\cal Q}}$ is given by
($Q,
\mu, L$), or $(Q, \widetilde{Q}, L)$, where $Q\subset{\bf P}^2$ is
a
plane quintic, $L\in({\bf P}^2)^*$ a line, and $\mu$ a semiperiod
on $Q$ (or $\widetilde{Q}$ the corresponding double cover).
\bigskip
\noindent{\bf (4.3.3)} \ The fiber ${\cal
P}^{-1}(J(X))\subset{\cal R\cal Q}^-$ over a cubic threefold $X$ can be
identified with the Fano surface $F(X)$ of lines $\ell\subset
X$.(Projection
from $\ell$ puts a conic bundle structure $\pi: X
--\!\to{\bf P}^2={\bf P}^4/\ell$ on $X$; the corresponding point of
${\cal R\cal Q}^-$ is $(Q,\widetilde{Q})$, where the plane quintic $Q$ is
the
discriminant locus of $\pi$, and its double cover $\widetilde{Q}$
parametrizes lines
$\ell'\in F(X)$ meeting $\ell$.)
\bigskip
\noindent{\bf (4.3.4)} \ The fiber $\widetilde{\cal P}^{-1}(X,H)$
corresponds to the lines $\ell$ in the cubic surface $X\cap H$.For
general
$X,H$, there are 27 of these. The corresponding
objects are of the form $(Q,\widetilde{Q},L)$ where
$(Q,\widetilde{Q})$ are
as above, and $L\subset{\bf P}^2$ is the projection, $L=\pi(H)$.
\bigskip
\noindent{\bf (4.3.5)} \ A point of ${\cal R}_6^{Trig}$ is given by a
curve $T\in{\cal M}_6$ with a trigonal line bundle ${\cal
L}\in\mbox{Pic}^3(T),h^0({\cal L})=2$, and a double cover
$\widetilde{T}\to T$. The fiber of $\widetilde{\cal R}_6^{Trig}$ above
it is
given by the linear system $|\omega_T\otimes{\cal L}^{-2}|$, a
${\bf P}^1$.
\bigskip
\noindent{\bf (4.3.6)} \ A point of ${\cal J}$ is given by the
Jacobian of a curve $C\in{\cal M}_5$. The canonical curve
$\Phi(C)\subset{\bf P}^4$, for general $C$, is the base locus of a
net of quadrics: $$A_p\subset{\bf P}^4, \;\;\;\;\;\;p\in{\bf
P}^2={\bf P}^2(C).$$ A point of $\widetilde{\cal J}$ above $C$ is
then
given by a pair $(C,L)$, where $L$ is a line in ${\bf
P}^2(C)$.(Choosing such a
line is the same as choosing a quartic del Pezzo
surface $$S=S_L=\cap_{p\in L}A_p$$ containing $\Phi(C)$.)
\bigskip
\noindent{\bf (4.3.7)} \ Consider the map
$$\alpha:{\cal M}_5\rightarrow{\cal R\cal Q}^+$$ sending $C\in{\cal M}_5$ to
$\alpha(C)=(Q, \widetilde{Q})$, where: $$Q:=\{p\in{\bf P}^2(C) \ |
\ A_p {\rm \ is \ singular}\}\subset{\bf P}^2(C),$$ and
$\widetilde{Q}$ is the double cover whose fiber over a general
$p\in Q$ corresponds to the two rulings on the rank-4 quadric
$A_p$. This $\alpha$ is a birational isomorphism; its inverse is
the restriction to ${\cal R\cal Q}^+$ of ${\cal P}$.
The fiber $\widetilde{\cal P}^{-1}(C, L)$ over generic $(C,
L)\in\widetilde{\cal J}$ is given by the following 27 objects:
\noindent$\bullet$ The quintic
object$(Q,\widetilde{Q},L)\in\widetilde{{\cal R\cal Q}}^+$,
where$(Q,\widetilde{Q})=\alpha(C)$ and $L$ is the given line in ${\bf
P}^2(C)$.
\noindent$\bullet$ Ten trigonals $T^\varepsilon_i, \ 1\le i\le 5,
\ \ \varepsilon=0, 1$, each with a double cover
$\widetilde{T}^\varepsilon_i$: each of the 5 points $p_i\in Q\cap
L$ determines two $g^1_4$'s on $C$, cut out by the rulings
$R^\varepsilon_i$ on $A_{p_{i}}$, and the $(T^\varepsilon_i,
\widetilde{T}^\varepsilon_i)$ are associated to these by the
trigonal
construction.
\noindent$\bullet$ Sixteen Wirtinger covers $(X_j,
\widetilde{X}_j)\in\partial^I{\cal R}_6$: the quartic del Pezzo
surface
$S_L$ contains 16 lines $\ell_j$ [Dem], each meeting $\Phi(C)$ in
two points, say $p_j, q_j$, and then $$X_j=C/(p_j\sim q_j)$$ and
$\widetilde{X}_j$ is its unique Wirtinger cover (1.9.I).
\bigskip
\noindent{\bf (4.3.8)} \ We observe that the generically finite map
\ \ ${\cal R}_6^{\mbox{Tet}}\to{\cal R}_6$ \ \ has \\ 1-dimensional fibers
over both ${\cal R\cal Q}$ and ${\cal R}^{\mbox{Trig}}_6$. After blowing up
and normalizing, we obtain finite fibers generically over the
exceptional loci. In the limit:
\noindent$\bullet$ Over $(Q,L)$, the 5 $g^1_4$'s correspond to
projections of the plane quintic $Q$ from one of the 5 points
$p_i\in Q\cap L$.
\noindent$\bullet$ Over $(T,D)$, with $T$ a trigonal curve, ${\cal
L}$ the trigonal bundle, and \\ $D\in|\omega_T\otimes{\cal
L}^{-2}|$, four of the $g^1_4$'s are of the form ${\cal L}(q)$ with
$q\in D$ (i.e. they are the trigonal ${\cal L}$ with base point
$q$); the fifth $g^1_4$ is $\omega_T\otimes{\cal L}^{-2}$.
\noindent$\bullet$ Given $X=C/(p\sim q)\in\partial\bar{\cal M}_5$,
there is a pencil $L\subset{\bf P}^2(C)$ of quadrics $A_p, \ p\in
L$, which contain both $\Phi(C)$ and its chord $\overline{pq}$.Among
these
there are 5 quadrics $A_{p_{i}}$ which are singular,
generically of rank 4. Each of these has a single ruling $R_i$
containing a plane containing $\overline{pq}$. These $R_i$ cut the
5 $g^1_4$'s on $X$.
We conclude that the tetragonal correspondence Tet of (4.1.1) lifts
to $$\widetilde{\rm Tet}\subset\widetilde{\cal R}_6 \times
\widetilde{\cal R}_6$$ which is
generically finite, of type (10,10), over each of our special loci.
\bigskip
\noindent {\bf Theorem 4.4 \ Structure of the blownup Prym map.}
Over each of the following loci, the blownup Prym map
$\widetilde{\cal P}$
has the listed monodromy group, and the lifted tetragonal
correspondence $\widetilde{\rm Tet}$ induces the listed structure.
\begin{list}{{\bf(\arabic{bean})}}{\usecounter{bean}}
\item $\widetilde{\C}$: The group is $WE_6$, the structure is that
of
lines on a general non-singular cubic surface.
\item $\widetilde{\cal J}$: The group is $WD_5$, the symmetry
group of
the incidence of lines on a quartic del Pezzo surface, or
stabilizer in $WE_6$ of a line. The structure is that of lines on
a non-singular cubic surface, one of which is marked.
\item ${\cal B}$= the locus of intermediate Jacobians of Clemens'
quartic double solids of genus 5 \ [C1]: The group is $WA_5=S_6$,
the structure is that of lines on a nodal cubic surface.
[Note: ${\cal B}$ is contained in the branch locus of ${\cal P}$ [DS,
V.4] and in fact ([D6], and compare also [SV], [I]) equals the
branch locus. The monodromy along ${\cal B}$ acting on a nearby,
unramified fiber is $({\bf{Z}\rm} \newcommand{\C}{\cal C}/2{\bf{Z}\rm} \newcommand{\C}{\cal C})\times S_6$, or the symmetry group
of a double-six, which is a subgroup of $WE_6$. The group $S_6$
thus occurs as a subquotient of $WE_6$.]
\item (cf. [I]) $\widetilde{\cal P}$ extends naturally to the
boundary $\partial=\partial{\cal A}_5$; the monodromy is $WE_6$ and the
structure is that of lines on a general cubic surface.
\end{list}
We will prove parts (1), (2) and (3) in \S4.2. In the rest of this
section we
show
that theorems (4.1) and (4.2) follow from (4.4).
\bigskip
\noindent{\bf (4.5) Proofs of Theorem (4.2).}
By Theorem (2.16), Tet commutes with ${\cal P}$, therefore
$\widetilde{\rm Tet}$ commutes with $\widetilde{\cal P}$. To
identify
this structure over a generic point, it suffices to do so over any
point over which $\widetilde{\cal P}$ and $\widetilde{\rm Tet}$ are
etale. These conditions hold, e.g., over a generic
$(X,H)\in\widetilde{\C}$, where (4.4.1) identifies the structure.This
implies
that the monodromy is contained in $WE_6$, but we get
all of $WE_6$ already over $\widetilde{\C}$ (by (4.4.1) again), so
we are done.
We can work instead over $\widetilde{\cal J}$: again,
$\widetilde{\cal P}$ and $\widetilde{\rm Tet}$ are etale over generic
$(C, L)\in\widetilde{\cal J}$, and $\widetilde{\rm Tet}$ has the
right
structure there by (4.4.2). This shows $$WD_5\subset
\mbox{Monodromy} \subset WE_6.$$ As there are no intermediate
groups, the monodromy must equal $WD_5$ or $WE_6$.But if it were the
former,
$\widetilde{\cal R}_6$ would be reducible
(since $WD_5$ is the stabilizer in $WE_6$ of one of the 27 lines),
contradiction. \begin{flushright} QED \end{flushright}
\bigskip
\noindent{\bf Remark 4.5.1} \ Along the same lines, we can also
reprove Theorem (4.1). Let $\widetilde{\rm Tet}^i$ denote the $i$-th
iterate of the correspondence $\widetilde{\rm Tet}$. On ${\cal R\cal Q}^-$
we
have:
$$\widetilde{\rm Tet}^2 \mbox{ \ has degree} \ 27, $$
$$\widetilde{\rm Tet}^i=\widetilde{\rm Tet}^2 \;\;\;\mbox{for}\; i\ge
2.$$Since
$\widetilde{\rm Tet}$ is etale there, these properties persist
generically
on $\widetilde{\cal R}_6$. Let $\sim$ be the equivalence relation
generated by $\widetilde{\rm Tet}$. We conclude that $\sim$ has
degree
27, and that $\widetilde{\cal P}$ factors through a proper
quotient: $${\cal
P}':\widetilde{\cal R}_6/\sim\longrightarrow\widetilde{\cal A}_5.$$ We
still need to verify that $\deg ({\cal P}')=1$. There are several
possibilities:
\noindent$\bullet$ We can work over $\widetilde{\cal J}$; as we will
see in (4.7), the fiber of $\widetilde{\cal P}$ there consists of
aunique
$\sim$-equivalence class; so we need to check that ${\cal
P}'$ is unramified at that equivalence class. This reduces to
seeing that $\widetilde{\cal P}$ is unramified at least at one
point of
the fiber; this is trivial at the plane-quintic point. (This
argument avoids some of the detailed computations of the
codifferential on the boundary, [DS, Ch., IV], but is still very
close in spirit to [DS].)
\noindent$\bullet$ We could instead work over any other point of
$\widetilde{\cal A}_5$ over which we know the complete fiber, e.g. over
Andreotti-Mayer points, coming from bielliptic
Pryms, as in \S 3. (This was proposed in [D1], as a way to avoid
the boundary computations.)
\noindent$\bullet$ Izadi [I] applies a similar argument over
boundary points, in $\partial {\cal A}_5$. This lets her reduce the
degree computation over ${\cal A}_5$ to her results on ${\cal A}_4$, cf.
(4.9).
\subsection{Special Fibers}
In this section we exhibit the cubic surface of theorem (4.2)
explicitly over three special loci in ${\cal A}_{5}$. We do not know
how
to do this at the generic point of ${\cal A}_{5}$.
\noindent{\bf (4.6) Cubic threefolds}
{}From (4.3.4) we have an identification of $\widetilde{\cal
P}^{-1}(X,H)$, where $X\subset{\bf P}^4$ is a cubic threefold and
$H\subset{\bf P}^4$ a hyperplane, with the set of lines $\ell$ on
the cubic surface $X\cap H$. For Theorem (4.4.1) we need to check
that two of these, say $\ell, \ell'\in F(X)$, intersect each other
if and only if the corresponding objects $(Q,\widetilde{Q},L),
(Q',\widetilde{Q}',L')$ correspond under $\widetilde{\rm Tet}$. If
the
lines
$\ell,\ell'$ intersect, we are in the situation of (2.15.4), so the
corresponding objects $$(Q,\widetilde{Q},f),
(Q',\widetilde{Q}',f')$$
(notation of (2.15.4)) are tetragonally related. Since $f,f'$ are
both cut out by hyperplanes through the span $A$ of $\ell,\ell'$,
we find points $$p\in Q\cap L, \ \ \ \ p'\in Q'\cap L'$$ (namely,the
projection
of $A$ from $\ell,\ell'$ respectively) such that
$f,f'$ are the projections of $Q$ from $p$ and of $Q'$ from $p'$,
respectively. The description of $\widetilde{\cal R}_6^{\rm Tet}$ in
(4.3.8)
then shows that
$$((Q,\widetilde{Q},L),(Q',\widetilde{Q}',L'))\in\widetilde{\rm
Tet},$$
as required. Since both the line incidence and $\widetilde{\rm Tet}$
are of bidegree (10,10), and we have an inclusion, it must be an
equality.
This shows that $\widetilde{\rm Tet}$ induces on $\widetilde{\cal
P}^{-1}(X,H)$ the structure of line incidence on the cubic surface
$X\cap H$. Fix the ambiant ${\bf P}^4$ and the hyperplane $H$, and
let the cubic $X$ vary. We clearly get all cubic surfaces in $H$
as intersections $X\cap H$; therefore the monodromy group is the
full symmetry group of the line configuration. This completes the
proof of (4.4.1), hence also of Theorem (4.2).
\bigskip
\pagebreak
\noindent{\bf (4.7) Jacobians}
Start with $(C,L)\in\widetilde{\cal J}$. The fiber
$\widetilde{\cal P}^{-
1}(C,L)$ consists of the 27 objects listed in (4.3.7). Each of
these comes with the 5 $g^1_4$'s given in (4.3.8). These give the
correspondence $\widetilde{\rm Tet}$, which we claim is equivalent to
the
line incidence on a cubic surface.
Let $S=S_L$ be the quartic del Pezzo surface determined by $(C,L)$,
as in (4.3.6). Let $S'$ be its blowup at a generic point $r\in S$.
Then $S'$ is a cubic surface; its lines correspond to:
\noindent$\bullet$ $\ell_Q$, the exceptional divisor over $r$.
\noindent$\bullet$ 10 conics through $r$ in $S$; these correspond
naturally to the 10 rulings ${\cal R}^\varepsilon_i$ (as in
(4.3.7)).[Each
${\cal R}^\varepsilon_i$ contains a unique plane through
$r$, which meets $S$ in a conic through $r$.]
\noindent$\bullet$ The 16 lines $\ell_j$ in $S$.
There is thus a natural bijection between the lines of $S'$ and
\linebreak
$\widetilde{\cal P}^{-1}(C,L)$. We need to check that this
correspondence takes incident lines to covers which are
tetragonally related to each other through the $g^1_4$'s of
(4.3.8). To that end, we list the effects of the tetragonal
constructions on our curves. The details are straightforward, and
are omitted.
\bigskip
\noindent{\bf (4.7.1)} \ The quintic $(Q,\widetilde{Q})$, with the
$g_4^1: \ \ {\cal O}_Q(1)(-p_i), \ \ p_i\in Q\cap L$, \ \ is taken
to the two trigonals $$(T_i^\varepsilon,
\widetilde{T}^\varepsilon_i), \;\;\;\;\;\varepsilon=0,1,$$ each
with its unique base-point-free $g_4^1, \ \ \omega_T\otimes{\cal
L}^{-2}$.
\bigskip
\noindent{\bf (4.7.2)} \ The trigonal
$(T^\varepsilon_i,\widetilde{T}^\varepsilon_i)$with its
base-point-free $g^1_4$
goes to $(Q,\widetilde{Q})$ with
${\cal O}_Q(1)(-p_i)$, and to $(T^{1-\varepsilon}_i,
\widetilde{T}^{1-
\varepsilon}_i)$ with the base-point-free $g^1_4$.
Consider $(T^\varepsilon_i,\widetilde{T}^\varepsilon_i)$ with the
$g^1_4 \ \ {\cal L}^\varepsilon_i(p).$ The actual 4-sheeted cover
of ${\bf P}^1$ in this case is reducible, consisting of the
trigonal $T^\varepsilon_i$ together with a copy of ${\bf P}^1$
glued to it at $p$. We are thus precisely in the situation of
Example (2.15.2): both tetragonally related objects are Wirtinger
covers $(X_j,\widetilde{X}_j)$.
\bigskip
\noindent{\bf (4.7.3)} \ A Wirtinger cover $(X_j,\widetilde{X}_j)$
with
the $g^1_4$ cut out by the ruling ${\cal R}^\varepsilon_i$ on the
singular quadric $A_{p_{i}}$, is taken to the trigonal
$(T^\varepsilon_i,\widetilde{T}^\varepsilon_i)$ and to another
Wirtinger cover.
\bigskip
\pagebreak[4]
\noindent{\bf (4.8) Quartic double solids and the branch locus of
${\cal P}$}.
The fiber of ${\cal P}$ over the Jacobian $J(X)\in{\cal B}$ of a
quartic double solid $X$ of genus 5 is described in [DS, V.4],
following ideas of Clemens. It consists of 6 objects
$(C_i,\widetilde{C}_i), \ \ 0\le i\le 5$, each with multiplicity 2,
and
15 objects $(C_{ij},\widetilde{C}_{ij}), \ \ 0\le i<j\le 5$. The
monodromy group $S_6$ permutes the six values of $i$: clearly the
two sets $\{C_i\}$ and $\{C_{ij}\}$ must be separately permuted,
and any permutation of the $C_i$ induces a unique permutation of
the $C_{ij}$. The situation is precisely that of lines on a nodal
cubic surface: the $C_i$ correspond to lines $\ell_i$ through the
node; and the plane through $\ell_i,\ell_j$ meets the cubic
residually in a line $\ell_{i,j}$.
The best way to see the symmetry is to consider Segre's cubic
threefold $Y\subset{\bf P}^4$, image of ${\bf P}^3$ by the linear
system of quadrics through 5 points $p_i$, $1\le i\le 5$, in
general position in ${\bf P}^3$. (cf. [SR] for the details.) $Y$
contains six irreducible, two-dimensional families of lines, which
we call the ``rulings" $R_i, \ \ 0\le i\le 5$: For $1\le i\le 5,
\ \ R_i$ consists of proper transforms of lines through $p_i$;
while $R_0$ parametrizes twisted cubics through $p_1,\cdots,p_5$.$Y$
also
contains 15 planes $\Pi_{ij}, \ \ 0\le i < j\le 5$
\linebreak (= the 5
exceptional divisors and the proper transforms of the 10 planes
$\overline{p_ip_jp_k}$); the ruling $R_i$ is characterized as the
set of lines in ${\bf P}^4$ meeting the 5 planes $\Pi_{ij}, \ \
j\ne i$. The symmetric group $S_6$ acts linearly on ${\bf P}^4$,
preserving $Y$, permuting the $R_i$ and correspondingly the
$\Pi_{ij}$.
The quartic double solids in question are essentially the double
covers $$\zeta: X\to Y$$ branched along the intersection of $Y$
with a quadric $Q\subset{\bf P}^4$. The Prym fiber is obtained as
follows:
\begin{tabbing}
$\bullet$ \=$C_i:= \{$ lines $\ell\in R_i$, tangent to $Q\}$ \\
\> $\widetilde{C}_i:= \{$ irreducible curves $\ell'\subset X$
such that $\zeta(\ell')=\ell\in C_i\}$
\end{tabbing}
Thus $(C_i,\widetilde{C}_i)$ is the discriminant of a conic-bundle
structure on $X$ given by $\zeta^{-1}(R_i)$. The Prym canonical
curve $\Psi(C_i)\subset{\bf P}^4$ is traced by the tangency points
of $\ell$ and $Q$; in particular, $\Psi(C_i)\subset Q$, so
$(C_i,\widetilde{C}_i)$ is a ramification point of ${\cal P}$, by
(1.6).
\noindent$\bullet$ $(C_{ij},\widetilde{C}_{ij})$ is similarly
obtained
as discriminant of a conic bundle structure on $X$ given by
projection from $\Pi_{ij}$, cf. [DS, V4.5].
\bigskip
\noindent{\bf (4.9) Boundary behavior}
In [I], Izadi uses results on the structure of
${\cal P}:{\cal R}_5\to{\cal A}_4$ to find the incidence structure on the
fibers of the compactified map
$\doublebar{\cal P}:\doublebar{\cal R}_6\to\bar{\cal A}_5$
over boundary points of the toroidal compactification $\bar{\cal A}_5$.
The picture is as follows:
\[
\begin{array}{cc}
\begin{diagram}[AA] \node{\begin{array}{r} \; \doublebar{\cal P} \;
\; :
\end{array}}
\end{diagram} &
\begin{diagram}[AA]
\node{\doublebar{\cal R}_{6}}
\arrow{e}
\node{\bar{\cal A}_{5} }
\end{diagram} \\
&
\begin{diagram}[AA]
\node{\cup} \node{\cup}
\end{diagram}
\\
\begin{diagram}[AA]
\node{\begin{array}{r} \partial {\cal P} \; \; : \end{array}} \\
\node{\begin{array}{r} \; {\cal P} \; \; : \end{array} }
\end{diagram} &
\begin{diagram}[AA]
\node{\partial^{\rm II}\doublebar{\cal R}_{6}} \arrow{e}
\arrow{s,l}{\alpha}
\node{\partial\bar{\cal A}_{5} } \arrow{s,r}{\beta} \\
\node{{\cal R}_{5}} \arrow{e}
\node{{\cal A}_{4}}
\end{diagram}
\end{array}
\]
Over general $A\in{\cal A}_4$, the fiber $\beta^{-1}(A)$ is isomorphic
to the Kummer variety $A/(\pm 1)$. Over
$(\widetilde{C},C)\in{\cal R}_5$, the fiber of $\alpha$ is
$S^2\widetilde{C}/\iota$, and $\partial{\cal P}$ becomes (cf. [D3,
(4.6)]) the map
\[
\begin{array}{l}
x+y \mapsto \psi (x) + \psi (y)
\\
\begin{diagram}[AA]
\node{S^{2}\widetilde{C}} \arrow{e} \arrow{s} \node{A} \arrow{s} \\
\node{S^{2}\widetilde{C}/\iota} \arrow{e} \node{A/(\pm 1)}
\end{diagram}
\end{array}
\]
\noindent where $\psi$ is the Abel-Prym map $\widetilde{C}\to A$.All
in all
then, we are considering the map
$$\partial{\cal P}:\cup_{(\widetilde{C},C)\in{\cal P}^{-1}(A)}S^2
\widetilde{C}=:E\longrightarrow A.$$ Theorem (4.1) says that its
degree is 27, and Theorem (4.2) predicts an incidence structure on
its fibers, i.e. a way of associating a cubic surface to each point
$a \in A$.
In \S5 we associate to $A\in{\cal A}_4$ a cubic threefold
$X=\kappa(A)\subset{\bf P}^4$ such that ${\cal P}^{-1}(A)$ is a
double cover of the Fano surface $F(X)$ of lines in $X$. For
generic $a\in A$, we are looking for a cubic surface; it is
reasonable to hope that this should be of the form $H(a)\cap X$,
where $H(a)$ is an appropriate hyperplane in ${\bf P}^4$. We thus
want a map $$H:A\to({\bf P}^4)^*$$ such that $$pr((\partial{\cal
P})^{-1}(a))=\{\mbox{lines in} \ H(a)\cap X\}.$$
\[
\begin{diagram}[A]
\node{E} \arrow[2]{e,t}{\partial {\cal P}} \arrow[2]{s,l}{pr}
\arrow{se}
\node[2]{A}
\arrow[2]{s,r}{H} \\
\node[2]{{\cal P}^{-1}(A)} \arrow{sw} \\
\node{F(X)} \node[2]{({\bf P}^{4})^{*}}
\end{diagram}
\]
Izadi's beautiful observation is that such an $H$ is given by the
linear system $\Gamma_{00}$ (sections of $|2\Theta|$ vanishing to
order $\ge 4$ at 0). The identification of $\Gamma_{00}$ with the
ambiant ${\bf P}^4$ of $X$ uses a construction of Clemens relating
his double solids to $\Gamma_{00}$, and the interpretaton of (a
cover of) $X$ as parametrizing double
solids with intermediate Jacobians isomorphic to $A$, cf. [D6] or
[I].
\section{Fibers of $P:{\cal R}_5\to A_4.$}
\subsection{The general fiber.}
\ \ \ \ Our main result in this section is:
\bigskip
\noindent{\bf Theorem 5.1} For generic $A\in A_4$, the fiber
$\overline{\cal P}^{-1}(A)$ is isomorphic to a double cover of
the Fano
surface $F=F(X)$ of lines on some cubic threefold $X$.
Let ${\cal R}{\cal C}$ denote the inverse image in ${\cal R}A_5$
of
the locus ${\cal C}$ of (intermediate Jacobians $J(X)$ of)
cubic threefolds $X$. We recall from [D4] that it splits into
even
and odd components:
\noindent{\bf (5.1.1)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
${\cal R}{\cal C}={\cal R}{\cal C}^+\amalg {\cal R}{\cal C}^-,$
\noindent distinguished by a parity funciton. This follows from
the existence of a natural theta divisor $\Xi\subset J(X)$,
characterized (cf. [CG]) by having a triple point at $0:\Xi$
translates the parity function $q$ of (1.2), on theta
characteristics, to a parity on semiperiods. More explicitly,
pick
$(Q,\sigma)\in{\cal P}^{-1}(J(X))\subset{\cal R}{\cal Q}^-$;
Mumford's
exact sequence (Theorem (1.4)(2)) says that any $\delta\in
J_2(X)$ is $\pi^*\nu$ for some $\nu\in(\sigma)^\perp\subset
J_2(Q)$. The compatibility result, theorem (1.5), then gives
(cf.
[D4], Proposition (5.1)):
\noindent{\bf (5.1.2)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
$q_X(\delta)=q_Q(\nu)=q_Q(\nu \sigma).$
In case $\delta$ is even, we end up with an isotropic subgroup \\
$(\!\nu,\sigma\!) \ \ \subset \ \ J_2(Q)$, with $\sigma$ odd and
$\nu, \nu\sigma$ even. The Pryms of the latter are therefore
Jacobians of curves:
\noindent{\bf (5.1.3)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
$P(Q,\nu)\approx J(C), \ \ \ P(Q,\nu\sigma)\approx J(C'),$
\noindent and the image of $\sigma$ gives semiperiods $\mu\in
J_2(C),\mu'\in J_2(C')$.
Reversing direction, we can construct an involution
$$\lambda:{\cal
R}_5\longrightarrow {\cal R}_5$$ and a map $$\kappa: {\cal
R}_5\longrightarrow{\cal R}{\cal C}^+,$$ as follows: Start with
$(C,\mu)\in {\cal R}_5$, pick the unique \ \ $(Q,\nu)$ \ \ in \\
${\cal P}^{-1}(C)\cap {\cal R}{\cal Q}^+$, and let
$\sigma,\nu\sigma\in J_2(Q)$ map to $\mu\in J_2(C).$ Then
formula (1.3) reads:
\noindent{\bf (5.1.4)} \ \ \ \ \ \ \ \ \ \ \ \ \ $0\equiv3+{\rm \
even}+q(\sigma)+q(\nu\sigma) \ \ \ \ \ {\rm (mod. \ 2)},$
\noindent so after possibly relabeling, we may assume
$$(Q,\sigma)\in {\cal RQ}^-, \ \ \ (Q,\nu\sigma)\in{\cal RQ}^+$$
so
that there is a well-defined curve $C'\in{\cal M}_5$ and a cubic
threefold $X\in{\cal C}$ such that
$$P(Q,\sigma)\approx J(X)$$ {\bf (5.1.5)}
$$P(Q,\nu\sigma)\approx J(C').$$
\noindent We can thus define $\lambda$ and $\kappa$ by:
$$\lambda(C,\mu):=(C',\mu')$$ {\bf (5.1.6)}
$$\kappa(C,\mu):=(X,\delta),$$
\noindent where $\mu'\in J_2(C'), \ \ \ \delta\in J_2(X)$ are the
images of $\nu\in J_2(Q).$ The precise version of our results is
in terms of $\lambda$ and $\kappa$:
\bigskip
\noindent{\bf Theorem 5.2}
\begin{list}{{\rm(\arabic{bean})}}{\usecounter{bean}}
\item $(C,\mu)$ is related to $\lambda(C,\mu)$ by a sequence of
two
tetragonal constructions. Hence $\lambda$ commutes with the Prym
map:$${\cal P}\circ\lambda={\cal P}, \ \ \
\lambda\circ\lambda=id.$$\item $\kappa$ factors through the Prym map:
$$\kappa:{\cal R}_5\stackrel{{\cal
P}}{\longrightarrow}A_4\stackrel{\chi}{\longrightarrow}{\cal
RC}^+,$$ where $\chi$ is a birational map.
\end{list}
Recall the Abel-Jacobi map [CG], $$AJ:F(X)\longrightarrow
J(X),$$ which is well-defined up to translation in $J(X)$. (It
can
be identified with the Albanese map of the Fano Surface $F(X)$.)A
point
$\delta\in J_2(X)$ determines a double cover of $J(X)$,
hence of $F(X)$.
\bigskip
\noindent{\bf Theorem 5.3} For generic $A\in{\cal A}_4$, set
$$(X,\delta):=\chi(A)=\kappa({\cal P}^{-1}(A))\in {\cal R C}^+.$$
Let $F(X)$ be the Fano surface of $X$, $\widetilde{F(X)}$ its double
cover determined by $\delta$ via the Abel-Jacobi map.
\begin{list}{{\rm(\arabic{bean})}}{\usecounter{bean}}
\item There is a natural isomorphism
$$P^{-1}(A)\approx\widetilde{F(X)}.$$
\item The action of $\lambda$ on the left corresponds to the
sheet
interchange on the right.
\item Two objects $(C,\mu), (C',\mu)\in{\cal P}^{-1}(A)$ are
tetragonally related if and only of the lines $\ell, \ell'\in
F(X)$
which they determine intersect.
\end{list}
\bigskip
\noindent{\bf Remark 5.4} Izadi has recently analyzed the
birational map $\chi$, in [I]. In particular, she shows that
$\chi$ is an isomorphism on an explicitly described, large open
subset of ${\cal A}_4$.
\subsection{Isotropic subgroups.}
\begin{tabbing} X \= \kill
\>By isotropic subgroup of rank $r$ on a curve $C$ we mean an
\\$r$-dimensional
${\bf F}_2$-subspace of $J_2(C)$ on
which the intersection pairing \\ $\langle \ , \ \rangle$ is
identically
zero.Choosing an isotropic subgroup of rank 1 is \\ the same as
choosing
a non-zero semiperiod.
\end{tabbing}
Start with a trigonal curve $T\in {\cal M}_{g+1}$, with a rank-2
isotropic subgroup $W\subset J_2(T)$ whose non-zero elements
we denote $\nu_i, \ \ i=0,1,2.$ The trigonal construction
associates to $(T,\nu_i)$ the tetragonal curve $X_i \in {\cal
M}_g$.Mumford's sequence (1.4)(2) sends $W$ to an isotropic subgroup
of
rank $1$ on $X_i$, whose non-zero element we denote $\mu_i$.
\bigskip
\noindent{\bf Lemma 5.5} The construction above sets up a
bijection between the following data:
\noindent $\bullet$ A trigonal curve $T\in {\cal M}_{g+1}$ with
rank-2 isotropic subgroup.
\noindent $\bullet$ A tetragonally related triple
$(X_i,\mu_i)\in {\cal R}_g, \ \ \ i=0, 1, 2$.
\bigskip
\noindent{\bf Proof.}
We think of $WD_4$ as the group of signed permutations of the 8
objects $\{x^{\pm}_i\}$, $1\leq i\leq 4$. \ Start with a
tetragonal double cover \linebreak $\widetilde{X}_0\longrightarrow
X_0\longrightarrow{\bf P}^1.$ It determines a principal
$WD_4$-bundle over ${\bf P}^1\backslash$(Branch). The original
covers $\widetilde{X}_0, X_0$ are recovered as quotients by the
following subgroups of $WD_4$: $$\widetilde{H}_0:={\rm
Stab}(x^+_1),$$
$$H_0:={\rm Stab}(x^\pm_1),$$
Consider also the subgroup $$G:={\rm Stab}\{\{x^+_1, x^+_2\},
\{x_1^-,x_2^-\}\}.$$ It has index 12 in $WD_4$. Its normalizer
is:$$N(G)={\rm Stab}\{\{x^\pm_1, x^\pm_2\}, \{x^\pm_3,
x^\pm_4\}\},$$ of index 3. The quotient is $$N(G)/G\approx({\bf
Z}/2{\bf Z})^2,$$ so there are 3 intermediate groups
$\widetilde{G}_i,\ \ \ i=0,1,2.$ We single out one of
them:$$\widetilde{G}_0:={\rm Stab}\{x^\pm_1, x^\pm_2\}.$$ The three
subgroups $\widetilde{G}_i$ are not conjugate to each other, but can
be
taken to each other by outer automorphisms of $WD_4$. In fact,
the
action of Out$(WD_4)\approx S_3$ sends $G$, and hence also
$N(G)$,
to conjugate subgroups; it permutes the $\widetilde{G}_i$
transitively,
modulo conjugation; and it also takes $H_0, \widetilde{H}_0$ to
non-conjugate subgroups $H_i, \widetilde{H}_i, \ \ \ i=1,2.$ We
illustrate each of these subgroups as the stabilizer in $WD_4$ of
a corresponding partition of $\left(\begin{array}{llll}x^+_1 &
x^+_2 & x^+_3 & x^+_4 \\ x^-_1 & x^-_2 & x^-_3 & x^-_4
\end{array}
\right)$:
\begin{center}
\begin{tabular}{ccc}
\hspace{1.5in} & \hspace{1.5in} & \hspace{1.5in} \\
\begin{picture}(82,30)(2,1)
\thicklines
\put(2,1){$\circ$} \put(2,25.8){$\circ$}
\put(4.5,6){\line(0,1){20}}
\put(26,1){$\circ$} \put(31,3.5){\line(1,0){20}}
\put(28.5,6){\line(0,1){20}}
\put(26,25.8){${\circ}$} \put(31,28.3){\line(1,0){20}}
\put(50,1){${\circ}$} \put(55,3.5){\line(1,0){20}}
\put(52.5,6){\line(0,1){20}} \put(50,25.8){${\circ}$}
\put(55,28.3){\line(1,0){20}}
\put(74,1){${\circ}$} \put(76.5,6){\line(0,1){20}}
\put(74,25.8){${\circ}$}
\end{picture} &
\begin{picture}(82,30)(2,1)
\thicklines
\put(2,1){$\circ$} \put(2,25.8){$\circ$} \put(7,28.3){\line(1,0){20}}
\put(7,3.5){\line(1,0){20}}
\put(26,1){$\circ$} \put(31,3.5){\line(1,0){20}}
\put(26,25.8){${\circ}$} \put(31,28.3){\line(1,0){20}}
\put(50,1){${\circ}$} \put(55,3.5){\line(1,0){20}}
\put(50,25.8){${\circ}$}
\put(55,28.3){\line(1,0){20}}
\put(74,1){${\circ}$}
\put(74,25.8){${\circ}$}
\end{picture} &
\begin{picture}(82,30)(2,1)
\thicklines
\put(2,1){$\circ$} \put(2,25.8){$\circ$} \put(7,28.3){\line(1,0){20}}
\put(7,3.5){\line(1,0){20}}
\put(26,1){$\circ$} \put(31,3.5){\line(1,0){20}}
\put(26,25.8){${\circ}$} \put(31,28.3){\line(1,0){20}}
\put(50,1){${\circ}$} \put(55,3.9){\line(1,1){22.4}}
\put(50,25.8){${\circ}$}
\put(55,27.9){\line(1,-1){22}}
\put(76,1){${\circ}$}
\put(76,25.8){${\circ}$}
\end{picture} \\
$H_{0}$ & $H_{1}$ & $H_{2}$ \\
& & \\
& & \\
\begin{picture}(82,30)(2,1)
\thicklines
\put(2,1){$\circ$} \put(2,25.8){$\circ$}
\put(7,3.5){\line(1,0){20}}
\put(26,1){$\circ$} \put(31,3.5){\line(1,0){20}}
\put(28.5,6){\line(0,1){20}}
\put(26,25.8){${\circ}$} \put(31,28.3){\line(1,0){20}}
\put(50,1){${\circ}$} \put(55,3.5){\line(1,0){20}}
\put(52.5,6){\line(0,1){20}} \put(50,25.8){${\circ}$}
\put(55,28.3){\line(1,0){20}}
\put(74,1){${\circ}$} \put(76.5,6){\line(0,1){20}}
\put(74,25.8){${\circ}$}
\end{picture} &
\begin{picture}(82,30)(2,1)
\thicklines
\put(2,1){$\circ$} \put(2,25.8){$\circ$} \put(7,28.3){\line(1,0){20}}
\put(26,1){$\circ$}
\put(26,25.8){${\circ}$} \put(31,28.3){\line(1,0){20}}
\put(50,1){${\circ}$}
\put(50,25.8){${\circ}$}
\put(55,28.3){\line(1,0){20}}
\put(74,1){${\circ}$}
\put(74,25.8){${\circ}$}
\end{picture} &
\begin{picture}(82,30)(2,1)
\thicklines
\put(2,1){$\circ$} \put(2,25.8){$\circ$} \put(7,28.3){\line(1,0){20}}
\put(26,1){$\circ$}
\put(26,25.8){${\circ}$} \put(31,28.3){\line(1,0){20}}
\put(50,1){${\circ}$}
\put(50,25.8){${\circ}$}
\put(55,27.9){\line(1,-1){22}}
\put(76,1){${\circ}$}
\put(76,25.8){${\circ}$}
\end{picture} \\
$\widetilde{H}_{0}$ & $\widetilde{H}_{1}$ & $\widetilde{H}_{2}$ \\
& & \\
& & \\
\begin{picture}(82,30)(2,1)
\thicklines
\put(2,1){$\circ$} \put(2,25.8){$\circ$} \put(7,28.3){\line(1,0){20}}
\put(7,3.5){\line(1,0){20}} \put(4.5,6){\line(0,1){20}}
\put(26,1){$\circ$} \put(28.5,6){\line(0,1){20}}
\put(26,25.8){${\circ}$}
\put(50,1){${\circ}$} \put(55,3.5){\line(1,0){20}}
\put(52.5,6){\line(0,1){20}} \put(50,25.8){${\circ}$}
\put(55,28.3){\line(1,0){20}}
\put(74,1){${\circ}$} \put(76.5,6){\line(0,1){20}}
\put(74,25.8){${\circ}$}
\end{picture} & & \\
$N(G)$ & & \\
& & \\
& & \\
\begin{picture}(82,30)(2,1)
\thicklines
\put(2,1){$\circ$} \put(2,25.8){$\circ$} \put(7,28.3){\line(1,0){20}}
\put(7,3.5){\line(1,0){20}} \put(4.5,6){\line(0,1){20}}
\put(26,1){$\circ$} \put(28.5,6){\line(0,1){20}}
\put(26,25.8){${\circ}$}
\put(50,1){${\circ}$}
\put(50,25.8){${\circ}$}
\put(74,1){${\circ}$}
\put(74,25.8){${\circ}$}
\end{picture} &
\begin{picture}(82,30)(2,1)
\thicklines
\put(2,1){$\circ$} \put(2,25.8){$\circ$} \put(7,28.3){\line(1,0){20}}
\put(7,3.5){\line(1,0){20}}
\put(26,1){$\circ$}
\put(26,25.8){${\circ}$}
\put(50,1){${\circ}$} \put(55,3.5){\line(1,0){20}}
\put(50,25.8){${\circ}$}
\put(55,28.3){\line(1,0){20}}
\put(74,1){${\circ}$}
\put(74,25.8){${\circ}$}
\end{picture} &
\begin{picture}(82,30)(2,1)
\thicklines
\put(2,1){$\circ$} \put(2,25.8){$\circ$} \put(7,28.3){\line(1,0){20}}
\put(7,3.5){\line(1,0){20}}
\put(26,1){$\circ$}
\put(26,25.8){${\circ}$}
\put(50,1){${\circ}$} \put(55,3.9){\line(1,1){22.4}}
\put(50,25.8){${\circ}$}
\put(55,27.9){\line(1,-1){22}}
\put(76,1){${\circ}$}
\put(76,25.8){${\circ}$}
\end{picture} \\
$\widetilde{G}_{0}$ & $\widetilde{G}_{1}$ & $\widetilde{G}_{2}$ \\
& & \\
& & \\
\begin{picture}(82,30)(2,1)
\thicklines
\put(2,1){$\circ$} \put(2,25.8){$\circ$} \put(7,28.3){\line(1,0){20}}
\put(7,3.5){\line(1,0){20}}
\put(26,1){$\circ$}
\put(26,25.8){${\circ}$}
\put(50,1){${\circ}$}
\put(50,25.8){${\circ}$}
\put(74,1){${\circ}$}
\put(74,25.8){${\circ}$}
\end{picture} & & \\
$G$& & \\
& & \\
& &
\end{tabular}
\end{center}
Let $X_i, \widetilde{X}_i, T, \stackrel{\approx}{T},\widetilde{T}_i \
\
\ (i=0,1,2)$ be the quotients of the principal $WD_4$-bundle by
the
subgroups $H_i, \widetilde{H}_i, N(G), G, \widetilde{G}_i$
respectively,
compactified to branched covers of ${\bf P}^1$. We see
immediately
that:
\noindent $\bullet$ The trigonal construction takes $X_0\to {\bf
P}^1$ to $\widetilde{T}_0\to T\to{\bf P}^1.$
\noindent $\bullet$ The double cover $\widetilde{X}_0\to X_0$
corresponds via (1.4)(2) to the double cover
$\stackrel{\approx}{T}\to\widetilde{T}_0$.
\noindent $\bullet$ The tetragonal construction acts by outer
antomorphisms,
hence exchanges the three tetragonal double covers
$\widetilde{X}_i\to X_i\to{\bf P}^1.$
Applying the same outer automorphisms, we see that the trigonal
construction also takes $X_i\to{\bf P}^1$ to $\widetilde{T}_i\to T\to
{\bf P}^1, \ i=1,2.$ To a tetragonally related triple
$(\widetilde{X}_i\to X_i \to {\bf P}^1)$ we can thus unambiguously
associate the trigonal \ \ $T\to{\bf P}^1$ \ \ together with the
rank-2, isotropic subgroup corresponding to the covers
$\widetilde{T}_i$. This inverts the construction predecing the
lemma.
\begin{flushright} QED. \end{flushright}
\bigskip
\noindent{\bf Note 5.5.1} The basic fact in the above proof is
that the 3 tetragonals $X_i$ yield the same trigonal $T$. This
can be explained more succinctly: outer automorphisms
take the natural surjection $\alpha_0:WD_4\to\!\to S_4$ to
homomorphisms $\alpha_1,\alpha_2$ which are not conjugate to it.But
the
composition $\beta\circ\alpha_i:WD_4\to\!\to S_3$, where
$\beta:S_4\to\!\to S_3$ is the Klein map, are conjugate to each
other.
\bigskip
\noindent{\bf Construction 5.6} Now let $T\in{\cal M}_{g+1}$ be
a trigonal curve, together with an isotropic subgroup of rank 3,
$$V\subset J_2(T).$$ We think of $V$ as a vector space over
${\bf F}_2$; the projective plane ${\bf P}(V)$ is identified with
$V\backslash(0)$. For each $i\in {\bf P}(V)$, the trigonal
construction gives a tetragonal curve $Y_i\in{\cal M}_g$.Mumford's
sequence
(1.4)(2) gives an isotropic subgroup of rank
2,
$$W_i\subset J_2(Y_i),$$ with a natural identification
$W_i\approx
V/(i).$
Let $U\subset V$ be a rank-2 subgroup, so ${\bf P}(U)\subset{\bf
P}(V)$ is a projective line. Lemma (5.5) shows that the 3
objects $$(Y_i, U/(i))\in{\bf R}_g, \ \ \ \ \ \ \ \ i\in {\bf
P}(U)$$ are
tetragonally related. In particular, they have a
common Prym variety $$P_U\approx{\cal P}(Y_i, U/(i))\in {\cal
A}_{g-1}, \ \ \ \ \ \forall i\in {\bf P}(U).$$Applying (1.4) twice,
we see that
the original rank-3 subgroup
$V$
determines a rank-1 subgroup $$V/U\subset(P_{U})_2,$$
so we let $\mu_{U}\in(P_U)_2$ be its non-zero element.Altogether
then, we have a map $${\bf P}(V)^*\longrightarrow{\cal R
A}_{g-1}$$
$$U\longmapsto(P_{U},\mu_{U}).$$
\bigskip
\noindent {\bf (5.6.1)} Assume now that one of the $Y_i$ happens
to be trigonal. (This can only happen if $g\le 6.$) Whenever
$U\ni i$, we find a tetragonal curve $C_U\in {\cal
M}_{g-1}$ such that $P_U\approx J(C_U)$. Lemma
(5.5), applied to $(Y_i, W_i)$, shows that the 3 objects
$$(C_U,\mu_U)\in{\cal R}_{g-1}, \ \ \ \ \ U\ni i$$ are
tetragonally related, so they have a common Prym variety \\
$A=P_V\in{\cal A}_{g-2}.$
\bigskip
\noindent {\bf (5.6.2)} Assume instead that $g=6$ and that
$P_U$ happens to be a Jacobian $J(C_U)\in{\cal J}_5$, for some
$U\in{\bf P}(V)^*.$ Of the three $Y_i, \ \ i\in U$, we claim two
are trigonal and the third, a plane quintic. Indeed, by (4.7),
the tetragonal triples above $J(C_U)$ consist either of a plane
quintic and two trigonals, as claimed, or of a trigonal and two
Wirtingers. The latter is excluded since the isomorphism
$$J(Y_i)\approx P(T,i)$$ implies that $Y_i$ is non-singular for
each $i\in{\bf P}(V)$.
Assume from now on that $g=6$. Our data consists of:
\noindent $\bullet$ $T\in {\cal M}_7$, trigonal, with $V\subset
J_2(T)$
isotropic of rank 3.
\noindent $\bullet$ For each $i\in {\bf P}(V)$, a curve $Y_i \in
{\cal
M}_6$, with a rank-2 isotropic subgroup $W_i\subset J_2(Y_i)$.
\noindent $\bullet$ For each $U\in {\bf P}(V)^*$, an object
$(P_U,\mu_U)\in
{\cal RA}_5$
\noindent $\bullet$ An abelian variety $A=P_V\in {\cal A}_4.$
We display ${\bf P}(V)$ as a graph with 7 vertices $i\in {\bf
P}(V)$ and 7 edges \\ $U\in{\bf P}(V)^*$, in
(3,3)-correspondence.We write $T$ (or $Q$) on a vertex corresponding
to a
trigonal (or
quintic) curve, and $C$ on an edge corresponding to a Jacobian.We
restate our
observations:
\bigskip
\noindent{\bf (5.6.3)}: Edges through a $T$-vertex are $C$-edges.
\bigskip
\noindent{\bf (5.6.4)}: On a $C$-edge, the vertices are $T,T,Q$.
\bigskip
It follows that only one configuration is possible:
\pagebreak[4]
{ \ }
\vspace{4in}
\centerline{ {\bf Figure 5.7}}
\noindent Thus four of the $Y_i$ are trigonal, the other three
are
quintics, and six of the $P_U$, corresponding to the straight
lines, are Jacobians of curves. Let $U_0\in{\bf P}(V)^*$
correspond to the circle. For $i\in U_0$, \ \ $Y_i$ is a quintic
$Q$. Through $Q$ pass two $C$ edges and $U_0$, and the
semiperiods
corresponding to the $C$-edges are even; by (1.3), the semiperiod
$U_0/(i)$ corresponding to $U_0$ must be \underline{odd}, so
there
is a cubic threefold $X\in{\cal C}$ such that $${\bf
P}_{U_0}\approx J(X).$$ Finally, theorem (1.5), or formula (5.1.2),
shows that the
semiperiod $\delta:=\mu_{U_0}\in J_2(X)$ is \underline{even}.
We observe that the three tetragonally related quintics
correspond
to 3 lines on the cubic threefold which meet each other and thus
form the intersection of $X$ with a (tritangent) plane. We are
thus exactly in the situation of (2.15.4).
\subsection{Proofs.}
\noindent{\bf (5.8)} Theorems (5.1),(5.2) and (5.3) all follow
from the following statements:
\begin{list}{{\rm(\arabic{bean})}}{\usecounter{bean}}
\item $(C,\mu)$ is related to $\lambda(C,\mu)$ by a sequence of
two tetragonal constructions.
\item $\kappa$ is invariant under the tetragonal construction
\item For $(X,\delta)\in {\cal R}{\cal C}^+, \ \ \
\kappa^{-1}(X,\delta)\approx\widetilde{F(X)}$, the isomorphism takes
$\lambda$ to the involution on $\widetilde{F(X)}$ over $F(X)$,
and two objects on the left are tetragonally related iff the
corresponding lines intersect.
\item Any two objects in ${\cal P}^{-1}(A)$, generic $A\in{\cal
A}_4$, are connected by a sequence of (two) tetragonal
constructions.
\end{list}
Indeed, (1) is (5.2)(1); \ (2) and (4) imply the existence of \\
$\chi:{\cal A}_4\longrightarrow{\cal R}{\cal C}^+$ such that
$\kappa=\chi\circ{\cal P}$, while (3) shows that any two objects
in a $\kappa$-fiber are also connected by a sequence of two
tetragonal constructions, so $\chi$ must be birational, giving
(5.2)(2). This gives an isomorphism ${\cal
P}^{-1}(A)\approx\kappa^{-1}(X,\delta)$, so (5.3) follows.
\bigskip
\noindent{\bf (5.9)} We let ${\cal R}^2{\cal Q}^+, {\cal R}^2{\cal
Q}^-$ denote the moduli spaces of plane quintic curves $Q$
together with:
\noindent $\bullet$ A rank-2, isotropic subgroup $W\subset
J_2(Q)$, containing one odd and two even semiperiods, and
\noindent $\bullet$ a marked even (respectively odd) semiperiod in
$W\backslash(0).$
Exchanging the two even semiperiods gives an involution on ${\cal
R}^2{\cal Q}^+$, with quotient ${\cal R}^2{\cal Q}^-$. The
birational map $$\alpha:{\cal M}_5\widetilde{\longrightarrow}{\cal
R}{\cal Q}^+,$$ of (4.3.7), lifts to a birational map
\noindent{\bf (5.9.1)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\
${\cal R}\alpha:{\cal R}_5\widetilde{\longrightarrow}{\cal R}^2{\cal
Q}^+.$
\noindent From the construction of $\lambda$ in (5.1.6) it
follows that the involution on the right hand side corresponds to
$\lambda$ on the left, so we have a commutative diagram:
\begin{equation}
\renewcommand{\theequation}{\bf
{\arabic{section}}.{\theau}.{\arabic{equation}}}
\setcounter{au}{9}
\setcounter{equation}{2}
\begin{diagram}[AA]
\node{{\cal R}_{5}} \arrow{s} \arrow{e,tb}{{\cal R}\alpha}{\sim}
\node{{\cal R}^{2}Q^{+}} \arrow{s,lr}{\pi}{2:1} \\
\node{{\cal R}_{5}/\lambda} \arrow{e,b}{\sim} \node{{\cal
R}^{2}Q^{-}}
\end{diagram}
\end{equation}
Start with $(C,\mu)\in{\cal R}_5$ and any $g^1_4$ on $C$. The
trigonal construction produces a trigonal $Y\in{\cal M}_6$ with
rank-2, isotropic subgroup $W_Y$. On $Y$ we have a natural
$g^1_4$, namely $w_Y\otimes L^{-2}$, where $L$ is the trigonal
bundle; so we bootstrap again, to a trigonal $T\in{\cal M}_7$
with
rank-3 isotropic subgroup $V$. Applying construction (4.6) we
obtain a diagram like (5.7), including an edge for $(C,\mu)$ and
on
it a vertex for $(Q,W_Q):=$ \\ $\pi{\cal R}\alpha(C,\mu)$. But then
$\lambda(C,\mu)$ and $\kappa(C,\mu)$ also appear in the same
diagram, as the two other edges (the line, respectively the
circle) through $Q$! Statement (5.8.1) now follows, since any two
edges of (5.7) which meet in a trigonal vertex are tetragonally
related. (5.8.2) also follows, since any $(C',\mu')$
tetragonally related to
$(C,\mu)$ will appear in the same diagram with $(C,\mu)$ (for the
obvious initial choice of $g^1_4$ on $C$), so they have the same
$\kappa$.
{}From the restriction to ${\cal R}{\cal Q}^-$ of the Prym map we
obtain, by base change:
\begin{equation}
\renewcommand{\theequation}{\bf
{\arabic{section}}.{\theau}.{\arabic{equation}}}
\setcounter{au}{9}
\setcounter{equation}{3}
\begin{diagram}[AA]
\node{{\cal R}^{2}Q^{-}} \arrow{s,l}{\cal RP} \arrow{e}
\node{{\cal R}Q^{-}} \arrow{s,r}{\cal P} \\
\node{{\cal RC}^{+}} \arrow{e} \node{{\cal C}}
\end{diagram}
\end{equation}
Combining with (5.8)(1),(2) and (5.9.2), we find that $\kappa$
factors
\begin{equation}
\renewcommand{\theequation}{\bf
{\arabic{section}}.{\theau}.{\arabic{equation}}}
\setcounter{au}{9}
\setcounter{equation}{4}
\begin{diagram}[AA]
\node{{\cal R}_{5}} \arrow{s} \arrow{e,tb}{{\cal R}\alpha}{\sim}
\node{{\cal R}^{2}Q^{+}} \arrow{s,r}{\pi} \\
\node{{\cal R}_{5}/\lambda} \arrow{e,b}{\sim} \node{{\cal
R}^{2}Q^{-}}
\arrow{s,r}{\cal RP} \\
\node[2]{{\cal RC}^{+}}
\end{diagram}
\end{equation}
We know ${\cal P}^{-1}(X)$ from (4.6), so by (5.9.3):
\noindent{\bf (5.9.5)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ${\cal
RP}^{-1}(X,\delta)\approx{\cal P}^{-1}(X)\approx F(X),$
\noindent and $\kappa^{-1}(X,\delta)$ is a double cover, which by
the following lemma is identified with $\widetilde{F(X)}$. (The
compatibility with $\lambda$ follows from (5.9.4); line incidence
in $F(X)$ corresponds by (4.6) to the tetragonal relation among
the quintics, which by figure (5.7) corresponds, in turn, to
the
tetragonal relation in ${\cal R}_5$, so the proof of (5.8)(3) is
complete.)
\bigskip
\noindent{\bf Lemma 5.10} The Albanese double cover
$\widetilde{F(X)}$ determined by \\ $\delta\in J_2(X)$ is
isomorphic to $\pi^{-1}{\cal R P}^{-1}(X,\delta)$ (notation of
(5.9.4)).
\bigskip
\noindent{\bf Proof.}
The second isomorphism in (5.9.5) sends a line $\ell\in
F(X)$ to the object $(\widetilde{Q}_{\ell},Q_{\ell})\in{\cal
P}^{-1}(X)$, where the curves $\widetilde{Q}_{\ell},Q_{\ell}$
parametrize ordered (respectively, unordered) pairs $\ell',\ell''
\in F(X)$ satisfying: $$\ell + \ell' + \ell''= 0 \ \ \ \ \ \ \ \
\
\ {\rm (sum \ in \ } \ \ \ J(X)).$$
We may of course think of $\widetilde{Q}_{\ell}$ as sitting in
$F(X)$, since $\ell'$ uniquely determines $\ell''$:
$\widetilde{Q}_{\ell}$ is the closure in $F(X)$ of
\noindent{\bf (5.10.1)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
$\{ \ell'\in F(X) \ \ | \ \ \ell'\cap\ell\neq\phi \, , \, \ell' \neq
\ell \}.
$
\medskip
\noindent
The corresponding object of ${\cal R}{\cal P}^{-1}(X,\delta)$ is
$(\stackrel{\approx}{Q}_\ell, \widetilde{Q}_\ell, Q_\ell)$, where
$\stackrel{\approx}{Q}_\ell$ is the inverse image in
$\widetilde{F(X)}$ of $\widetilde{Q}_\ell$ embedded in $F(X)$ via
(5.10.1). Now to specify a point in $\pi^{-1}{\cal R}{\cal
P}^{-1}(X,\delta)$ we need, additionally, a double cover
$\widetilde{Q}_\ell' \to Q_\ell$ satisfying:
\noindent{\bf (5.10.2)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
$\widetilde{Q}_\ell\times_{Q_\ell}\widetilde{Q}_\ell'\approx
\stackrel{\approx}{Q}_\ell \ .$
We need to show that a choice of $\widetilde{\ell}
\in\widetilde{F(X)}$
over $\ell\in F(X)$ determines such a $\widetilde{Q}_\ell'$.Recall
that
$\widetilde{F(X)} \to F(X)$ is obtained by base change,
via
the Albanese map, from the double cover $\widetilde{J(X)} \to
J(X)$ determined by $\delta$. \ $\widetilde{Q}_\ell'$ can
thus be taken to parametrize unordered pairs $\widetilde{\ell}',
\widetilde{\ell}''\in\widetilde{F(X)}$ satisfying: $$\widetilde{\ell}
+ \widetilde{\ell}' + \widetilde{\ell}'' = 0 \ \ \ \ \ \ \ {\rm (sum
\
in} \ \ \ \widetilde{J(X)})\ .$$ The fiber product in (5.10.2)
then parametrizes such ordered pairs, so the required
isomorphism to $\doubletilde{Q}_\ell$ simply sends
$$(\widetilde{\ell}',\widetilde{\ell}'') \mapsto \widetilde{\ell}'.$$
\begin{flushright} Q.E.D. \end{flushright}
Finally, we prove (5.8)(4). Let $\overline{{\cal P}}:
\overline{{\cal R}}_5 \to {\cal A}_4$ be the proper Prym map. By
(5.8)(3) it factors $$\overline{\cal P} = \iota \circ \kappa$$
where $\iota: {\cal R}{\cal C}^+ \to {\cal A}_4$ is a rational
map,
which we are trying to show is birational. It suffices to find
some $A \in{\cal A}_4$ such that:
\begin{list}{{\rm(\arabic{bean})}}{\usecounter{bean}}
\item Any two objects in $\overline{\cal P}^{-1}(A)$ can be
related by a sequence of tetragonal constructions.
\item The differential $d{\cal P}$ is surjective over $A$.
\end{list}
In \S5.4 we see that $(1)$ holds for various examples,
including generic Jacobians $\in{\cal J}_4$: for generic $C \in
{\cal M}_4$, $\overline{\cal P}^{-1}(J(C))$ consists of
Wirtinger covers $\widetilde{C} \to C'$ (with normalilzation $C$) and
of trigonals $T$, and the two types are exchanged by $\lambda$.It is
easier to
check surjectivity of $d{\cal P}$ at the
Wirtingers: by theorem (1.6), this amounts to showing that the
Prym-canonical curve $\Psi(X) \subset {\bf P}^3$ is contained in
no quadrics. By [DS] IV, Propo. 3.4.1, $\Psi(X)$ consists of the
canonical curve $\Phi(C)$ together with an (arbitrarily chosen)
chord. Since $\Phi(C)$ is contained in a unique quadric $Q$,
which does not contain the generic chord, we are done. [Another
argument: it suffices to show that no one quadric contains
$\Psi(T)$ for all trigonal $T$ in ${\cal P}^{-1}(J(C))$.By [DS], III
2.3 we
have $$\cup_T \Psi(T) \ \supset \ \Phi(C) ,$$
so the only possible quadric would be $Q$. Consider the $g^1_4$
on
$C$ given by $\omega_C$(-$p$-$q$), where $p,q \in C$ are such
that the
chord $\overline{\Phi(p),\Phi(q)}$ is not in $Q$. Let $T$ be the
trigonal curve associated to $(C,\omega_C$(-$p$-$q$)), and choose
a plane $A \subset {\bf P}^3$ through $\Phi(p),\Phi(q)$, meeting
$Q$ and $\Phi(C)$ transversally, say $$A \cap \Phi(C) =
\Phi(p+q+\sum^{4}_{i=1}x_i),$$ then by [DS],III 2.1, \ \
$\Psi(T)$
contains the point $$\overline{\Phi(x_1),\Phi(x_2)} \cap
\overline{\Phi(x_3), \Phi(x_4)}$$ which cannot be in
$Q$.]\begin{flushright}
Q.E.D. \end{flushright}
\subsection{Special fibers.}
\begin{tabbing} X \= \kill
\>We want to illustrate the behavior of the Prym map over some \\
special loci in $\overline{\cal A}_4$. The common feature to all
of these examples is that \\ the cubic threefold $X$ given in
Theorem (5.1) acquires a node. We thus \\ begin with a review of
some
results, mostly from [CG], on nodal \\ cubics.
\end{tabbing}
\bigskip
\noindent{\bf (5.11) Nodal cubic threefolds}
There in a natural correspondence between nodal cubic threefolds
$X\subset{\bf P}^4$ and nonhyperelliptic curves $B$ of genus 4.
Either object can
be described by a pair of homogeneous polynomials $F_2, \ F_3$,
of degrees 2 and 3 respectively, in 4 variables $x_1, ..., x_4:X$
has homogeneous equation $0=F_3+x_0F_2$ \ (in ${\bf P}^4$), and
the canonical curve $\Phi(B)$ has equations $F_2=F_3=0$ in ${\bf
P}^3$.
More geometrically, we express the Fano surface $F(X)$ in terms
of $B$. Assume the two $g^1_3$'s on $B$, \ ${\cal L}'$ and
${\cal L}''$, are distinct. They give maps $$\tau', \
\tau'':B\hookrightarrow S^2B$$ sending $r\in B$ to $p+q$ if
$p+q+r$ is a trigonal divisor in $|{\cal L}'|$, \ $|{\cal L}''|$
respectively. We then have the identification
\noindent{\bf (5.11.1)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $F(X)\approx
S^2B/(\tau'(B)\sim\tau''(B))$.
Indeed, we have an embedding $$\tau:B\hookrightarrow F(X),$$
identifying $B$ with the family of lines through the node
$n = (1,0,0,0,0)$. This gives a map $S^2B\rightarrow F(X)$
sending a pair $\ell_{1}, \ell_{2}$ of lines through $n$ to the
residual
intersection with $X$ of the plane $(\ell_1, \ell_2)$.this map
identifies
$\tau'(B)$ with $\tau''(B)$, and induces the
isomorphism (5.11.1).
\bigskip
\noindent{\bf (5.11.2)} A line $\ell\in F(X)$ determines a pair
$(Q,
\widetilde{Q})\in\overline{\cal RQ}^-$, which must be in
$\partial^{\rm II}{\cal RQ}^-$, i.e. for generic $\ell$ we obtain a
nodal quintic $Q$ with \'{e}tale double cover $\widetilde{Q}$. We
can
interpret (5.11.1) in terms of these nodal quintics: Start with
a
divisor $p+q\in S^2B$. Then $\omega_B(-p-q)$ is a $g^1_4$ on
$B$,
so the trigonal construction produces a double cover
$\widetilde{T}\rightarrow T$, where $T\in{\cal M}_5$ comes with a
trigonal bundle ${\cal L}$. The linear system
$|\omega_T\otimes{\cal L}^{-1}|$ maps $T$ to a plane quintic $Q$,
with a single node given by the divisor $|\omega_T\otimes{\cal
L}^{-2}|$ on $T$.
\bigskip
\noindent{\bf (5.11.3)} In the special case that there exists
$r\in B$ such that \\ $p+q=\tau''(r)$, i.e. $p+q+r\in|{\cal
L}''|$ is a trigonal divisor, our $g^1_4$ acquires a base point:
$$\omega_B(-p-q)\approx{\cal L}'(r).$$ As seen in (2.10.ii), the
trigonal construction produces the nodal trigonal curve
$$T:=B/(p'\sim q')$$ with its Wirtinger double cover $\widetilde{T}$,
where $p', q'\in B$ are determined by: $$p'+q'+r\in|{\cal L}'|,$$
i.e. $p'+q'=\tau'(r)$. In this case, the quintic $Q$ is the
projection of $\Phi(B)$ from $\Phi(r)$, with 2 nodes $p\sim q, \
\
p'\sim q'$, and $\widetilde{Q}$ is the reducible double cover with
crossings over both nodes.
\bigskip
\noindent{\bf (5.12) Degenerations in ${\cal RC}^+$.}
We fix our notation as in \S 5.1. Thus we have:
\begin{tabbing}
\=$X\in{\cal C} \ \ \ \ \ \ \ (X, \delta)\in{\cal RC}^+$ \\
\>$(Q,\sigma)\in{\cal RQ}^-, \ \ \ \ \ \ (Q, \nu), \ (Q,
\nu\sigma)\in{\cal RQ}^+$ \\ \>$(C, \mu), \ (C', \mu')\in{\cal
R}_5$ \\ \>$A\in{\cal A}_4$
\end{tabbing}
\noindent and these objects satisfy:
\begin{tabbing}
\= ${\cal P}(Q, \sigma)$ \= $=$ \= $J(X)$ \= , \ \ \ \ \ \ \ \ \
\
\ \ \ \ \ \ \= $\nu, \nu\!$ \= $\!\sigma\mapsto\delta$ \\
\>${\cal
P}(Q, \nu)$ \>= \>$J(C)$ \>, \>$\sigma, \nu\!$
\>$\!\sigma\mapsto\mu$ \\ \>${\cal P}(Q, \nu\sigma)$ \>=
\>$J(C')$
\>, \>$\nu,$ \>$\!\sigma\mapsto\mu'$ \end{tabbing}
\bigskip
\begin{tabbing}
\= ${\cal P}(C, \mu)={\cal P}\!$ \= $\!(C'\!$ \= $, \mu'\!$ \=
$\!)=A$ \\ \>$\lambda(C, \mu)=$ \>$\!(C'\!$ \>$, \mu'\!$ \>$\!)$
\\
\>$\kappa(C, \mu)=$ \>$\!(X$ \>$, \delta\!$ \>$\!).$
\end{tabbing}
Now let $X$ degenerate, acquiring a node, with
$\bar{\varepsilon}\in J_2(X)\backslash(0)$ the vanishing cycle
mod. 2. \ From (5.11) we see that $Q$ also degenerates, with a
vanishing cycle $\varepsilon$ which maps (via. (1.4)) to
$\bar{\varepsilon}$. Lemma (5.9) of [D4] shows that
$\varepsilon$, hence also $\bar{\varepsilon}$, must be even.
There are 3 types of degenerations of $(X, \delta)$,
distinguished
as in (1.7) by the relationship of $\delta, \bar{\varepsilon}$.
(A fourth type, where $Q$ degenerates but $X$ does not, is
explained in (5.13).)The possibilities are summarized below:
\bigskip
\begin{list}{{\rm(\Roman{butter})}}{\usecounter{butter}}
\item If $\bar{\varepsilon}=\delta$ then either
$\varepsilon=\nu$ or $\varepsilon=\nu\sigma$, which gives the same
picture with $C, C'$ exchanged. In case $\varepsilon = \nu$, $(Q,
\nu)$
undergoes
a $\partial^{\rm I}$ degeneration, while $(Q, \nu\sigma)$ is
$\partial^{\rm II}$. (The notation is that of (1.7).) Thus $A$ is a
Jacobian.
\ \ \ The double cover $\widetilde{F(X)}$ is itself a $\partial^I$
cover. In terms of the curve $B$ of (5.11), we have
$$\widetilde{F(X)}=(S^2B)_0\amalg(S^2B)_1 \ \ / \ \
(\tau'(B)_0\sim\tau''(B)_1, \ \ \ \tau''(B)_0\sim\tau'(B)_1).$$This
is clear,
either from the definition of $\widetilde{F(X)}$ viathe Albanese map,
or by
considering the restriction to ${\cal
RP}^{-1}(X, \delta)$ of the double cover $${\cal R}^2{\cal
Q}^+\stackrel{\pi}{\rightarrow}{\cal R}^2{\cal Q}^-$$ of (5.9).
One of the components parametrizes the trigonal objects $(C,
\mu)$, the other parametrizes the nodals $(C', \mu')$.
\item $\sigma$ is always perpendicular to $\varepsilon,
\nu$, and the condition $\langle\bar{\varepsilon}, \delta\rangle=0$
implies
$\langle\varepsilon, \nu\rangle=0$ by (1.4.3). Both $(Q, \nu)$ and
$(Q,
\nu\sigma)$ then give $\partial^{\rm II}$-covers, so $C, C'$ are
nodal. Again by (1.4.3), both $(C, \mu)$ and $(C', \mu')$ are
$\partial^{\rm II}$, so their common Prym $A$ is in
$\partial\bar{\cal
A}_4$.
\ \ \ \ From the Albanese map we see that $\widetilde{F(X)}$ is an
etale cover of $F(X)$. Indeed, $\delta$ comes from a semiperiod
$\delta'$ on $B$, giving a double cover $\widetilde{B}$ with
involution $\iota$; the normalization of $\widetilde{F(X)}$ is then
$S^2\widetilde{B}/\iota$, and $\widetilde{F(X)}$ is obtained by
glueing
above $\tau(B)$.
\item In this case both $(Q, \nu)$ and $(Q, \nu\sigma)$ are
$\partial^{\rm III}$, so $C, C'$ are nonsingular. The node of $Q$
represents a quadric of rank 3 through $\Phi(C)$, so ${\cal L}$
is
cut out by the unique ruling. By the Schottky-Jung relations
[M2],
the vanishing theta null on $C$ descends to one on $A$.
The double cover $\widetilde{F(X)}$ is again a
$\partial^{\rm III}$-cover, in the sense that its normalization is
ramified over $\tau'(B), \tau''(B)$, the sheets being glued.Each of
the
quintics in (5.11.2) gives two points of
$\widetilde{F(X)}$, while the two-nodal quintics (5.11.3) land in the
branch locus of $\pi$ \ (5.9.4).
\end{list}
\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
Degeneration & Degeneration & & & \\
type of & type of & & & \\
\multicolumn{1}{|c|}{$(X,\delta)$} &
\multicolumn{1}{|c|}{$(Q,\sigma,\nu,\nu\sigma)$}
& \multicolumn{1}{|c|}{$(C,\mu)$} & \multicolumn{1}{|c|}{$(C',\mu')$}
&
\multicolumn{1}{|c|}{$A$} \\ \hline \hline
& & & & \\
I : $\bar{\varepsilon} = \delta$ &$\varepsilon = \nu$ &
nonsingular & nodal, $\partial^{\rm I}$ & ${\cal J}_{4}$ \\
& & trigonal & & \\ \hline
& & & & \\
II : $\bar{\varepsilon} \neq \delta$, & $(\varepsilon, \sigma,
\nu)$ &
nodal, $\partial^{\rm II}$& nodal, $\partial^{\rm II}$& $\partial
\bar{\cal
A}_{4}$ \\
\multicolumn{1}{|c|}{$\langle \bar{\varepsilon} , \delta \rangle =
0$} & rank
3 &
& &
\\
& isotropic & & & \\
& subgroup & & & \\ \hline
& & & & \\
III : $\langle \bar{\varepsilon} , \delta \rangle \neq 0$ &
$\langle \varepsilon , \sigma \rangle = 0$ & nonsingular, &
nonsingular, & $\theta_{\rm null}$ \\
&$\langle \varepsilon , \nu \rangle \neq 0$ & has vanishing &
has vanishing & \\
& & thetanull ${\cal L}$, & thetanull ${\cal L}'$, & \\
& & ${\cal L}(\mu)$ even & ${\cal L}'(\mu')$ even & \\ \hline
IV : & $\langle \varepsilon , \nu \rangle = 0$ &
& nonsingular, & \\
nonsingular & $\langle \varepsilon , \sigma \rangle \neq 0$ &
nodal,
$\partial^{\rm II}$ &
has vanishing & ${\cal A}_{4}$ \\
& & & thetanull ${\cal L}'$, & \\
& & & ${\cal L}'(\mu')$ odd & \\ \hline
\end{tabular}
\end{center}
\bigskip
\pagebreak[4]
\noindent{\bf (5.13) Degenerations in ${\cal R}^2{\cal Q}^+$.}
We have just described the universe as seen by a degenerating
cubic threefold. From the point of view of a degenerating plane
quintic, there are a few more possibilities though they lead to
no new components. We retain the notation: $Q, \nu, \sigma,
\varepsilon$ etc.
\noindent 0. \ \ $\varepsilon$ cannot equal $\sigma$, since
$\varepsilon$ is even, $\sigma$ odd.
\noindent I. \ \ $\varepsilon=\nu$ reproduces case I of (5.12), as
does:
\noindent I$'$. \ $\varepsilon=\nu\sigma$.
\begin{tabbing}
\noindent II. \ \=Excluding the above, $\nu, \sigma,
\varepsilon$ generate a subgroup of rank 3. If \\ \>it is
isotropic, we are in case II above.
\end{tabbing}
\noindent III. If $\langle\varepsilon, \sigma\rangle=0$ but
$\langle\varepsilon,
\nu\rangle=\langle\varepsilon, \nu\sigma\rangle\neq0$, we're in case
III.
The only new cases are thus:
\noindent IV. \ \ \ \ \ \ \ \ \ \ \ \ \ $\langle\varepsilon,
\nu \ \ \rangle=0\neq\langle\varepsilon, \sigma\rangle, \ \ {\rm
or}:$
\noindent IV.$'$ \ \ \ \ \ \ \ \ \ \ \ \ \ $\langle\varepsilon,
\nu\sigma\rangle=0\neq\langle\varepsilon, \sigma\rangle,$
which is the same as IV after exchanging $C, C'$.
In case IV, we find:
\noindent $\bullet$ $X$ is non-singular, in fact any $X$ can
arise. What is special is the line $\ell\in F(X)$ corresponding
to $Q:$ it is contained in a plane which is tangent to $X$ along
another line, $\ell'$.
\noindent $\bullet$ $(Q, \nu)$ is a $\partial^{\rm II}$ degeneration,
so
$C$ is nodal, and $(C, \mu)$ is a $\partial^{\rm II}$ degeneration.
\noindent $\bullet$ On the other hand, $(Q, \nu\sigma)$ is
$\partial^{\rm III}$, so $C'$ is non-singular, and has a vanishing
theta null ${\cal L}'$ (corresponding, as before, to the node of
$Q$).
\noindent $\bullet$ This time though, ${\cal L}'(\mu')$ is odd,
so
$A\in{\cal A}_4$ does not inherit a vanishing theta null. In
fact,
any $A\in{\cal A}_4$ arises from a singular quintic with
degeneration of type IV.
So far, we found three loci in $\bar{\cal A}_4$ which are related
to nodal cubics:
\[
\begin{array}{lcl}
{\cal P}\circ \kappa^{-1}(\partial^{\rm I} {\cal RC}^{+}) & \subset &
{\cal
J}_{4} \\
\bar{\cal P}\circ \kappa^{-1}(\partial^{\rm II} {\cal RC}^{+}) &
\subset &
\partial \bar{A}_{4} \\
{\cal P}\circ \kappa^{-1}(\partial^{\rm III} {\cal RC}^{+}) &
\subset &
\theta_{\rm
null}
\end{array}
\]
We are now going to study, one at a time, the fibers of ${\cal P}$
above
generic points in these three loci. We note that related results
have
recently been obtained by Izadi. In a sense, her results are more
precise:
she knows (cf. Remark 5.4) that $\chi$ is an isomorphism on the open
complement ${\cal U}$ of a certain 6-dimensional locus in ${\cal
A}_{4}$.
In [I] she shows that for $A \in {\cal U}$, $\chi (A)$ is singular
if and only
if
\[ A \in {\cal J}_{4} \cup \theta_{\rm null}. \]
Her description of the cubic threefold corresponding to $A \in {\cal
J}_{4}$
complements the one we give below. In general her techiques, based
on
$\Gamma_{00}$, are very different than our degeneration arguments.
\bigskip
\noindent{\bf (5.14) Jacobians}
\bigskip
\noindent{\bf Theorem 5.14} Let $B\in{\cal M}_4$ be a general curve
of genus 4, and let $(X,\delta)=\chi(J(B)).$
\begin{list}{{\rm(\arabic{bean})}}{\usecounter{bean}}
\item $X$ is the nodal cubic threefold corresponding to $B$
(5.11).
\item $(X,\delta)\in\partial^I$, so $\widetilde{F(X)}$ is reducible,
each component is isomorphic to $S^2B.$
\item Let $(Q,\sigma,\nu)$ be the plane quintic with rank-2
isotropic subgroup corresponding to some $\ell\in F(X)$. Then
$Q$
is nodal, with trigonal normalization $T, \ \nu$ is the vanishing
cycle, and \\ $(Q,\sigma)=(Q,\nu\sigma)\in \partial^{\rm II}$.
\item$\bar{\cal P}^{-1}(J(B))$ is isomorphic to $\widetilde{F(X)}$.The
component
corresponding to $\nu$ (respectively $\nu\sigma$)
consists of trigonal curves $T_{p,q}$ (respectively Wirtinger
covers of singular curves $S_{p,q}$), \ \ \ $(p,q)\in S^2B.$
\item The tetragonal construction takes both $S_{p,q}$ and
$T_{p,q}$ to $S_{r,s}$ and $T_{r,s}$ if and only if $p+q+r+s$ is
a
special divisor on $B$. The involution $\lambda$ exchanges
$S_{p,q}, T_{p,q}$.
\item Any two objects in $\bar{\cal P}^{-1}(J(B))$ can be connected
by a sequence of two tetragonal moves (generally, in 10 ways).
\end{list}
\bigskip
\noindent{\bf Proof}
Since at least some of these results are needed for the proof of
(5.8)(4), we do not use Theorem (5.3). For $(p,q)\in S^2B$, we
consider:
\noindent $\bullet$ $\widetilde{T}_{p,q}\to T_{p,q}$, the trigonal
double cover associated by the trigonal construction to $B$ with
the $g^1_4$ given by $\omega_B$(-$p$-$q$).
\noindent $\bullet$ $\widetilde{S}_{p,q}\to S_{p,q}$, the Wirtinger
cover of $S_{p,q}:=B/(p\sim q)$. (When $p=q$, this specializes
to
$B\cup_pR$, where $R$ is a nodal rational curve in which $p$
is a non singular point.)
These objects are clearly in $\bar{\cal P}^{-1}(J(B))$. Beauville's
list ([B1], (4.10)) shows that they exhaust the fiber. This
proves
part (4). Now clearly $\kappa$, as defined in (5.1.6), takes any
of these objects to our $(X,\delta)$; so the analysis in
(5.12)(I)
applies, proving (1)-(3). (Note: this already suffices to
complete the proof of (5.8)(4)!)
Let $r+s+t+u$ be an arbitrary divisor in
$|\omega_B(-p-q)|$.Projection of
$\Phi(B)$ from the chord
$\overline{\Phi(t),\Phi(u)}$
gives (the general) $g^1_4$ on $S_{p,q}$. The tetragonal
construction takes this to the curves $T_{t,u}$ and $S_{r,s}$.(The
situation is
that of (2.15.2).)
On $T_{p,q}$ there are two types of $g^1_4$'s, of the form ${\cal
L}(x)$ and $\omega\otimes{\cal L}^{-1}(-x)$, where ${\cal L}$ is
the trigonal bundle and $x\in T_{p,q}$. Now $x$ corresponds to a
(2,2) partition, say $\{\{r,s\},\{t,u\}\}$, of some divisor in
$|\omega_B(-p-q)|$. The tetragonal construction,
applied to ${\cal L}(x)$, yields the curves $S_{r,s}$ and
$S_{t,u}$; while when applied to $\omega\otimes{\cal
L}^{-1}(-x)$,
it gives $T_{r,s}$ and $T_{t,u}$. Altogether, this proves (5).We
conclude
with:
\bigskip
\noindent{\bf Lemma 5.14.7} Given any $p,q,r,s\in B$, there are
points $t,u\in B$ (in general, 5 such pairs) such that both
$p+q+t+u, r+s+t+u$ are special.
\bigskip
\noindent{\bf Proof}
Let $\alpha, \beta$ be the maps of degree 4 from $B$ to ${\bf
P}^1$ given by \\ $|\omega_B(-p-q)|, |\omega_B(-r-s)|$.
Then $$\alpha\times\beta:B\to{\bf P}^1\times{\bf P}^1$$
exhibits $B$ as a curve of type (4,4) on a non-singular quadric
surface, hence the image has arithmetic genus
$(4-1)^2=9\rangle4=g(B)$,
so there must be (in general, 5) singular points; these give the
desired pairs $(t,u)$. \begin{flushright} QED \end{flushright}
\bigskip
\noindent{\bf (5.15) The Boundary.}
The results in this case were obtained by Clemens [C2].
A general point $A$ of the boundary $\partial\bar{\cal A}_4$ of
a toroidal compactification $\bar{\cal A}_4$ is a ${\bf
C}^*$-extension
of some $A_0\in{\cal A}_3$. The extension data is given by a point
$a$
in the Kummer variety $A_0/(\pm 1)$.
Given $a\in A_0$, consider the curve $$\widetilde{B}=
\widetilde{B}_a:=\Theta\cap\Theta_a\subset A_0$$ (where
$x\in\Theta_a\Leftrightarrow x+a\in\Theta$), and its quotient
$B=B_a$ by the involution $x\mapsto -a-x$. We have
$$(B,\widetilde{B})\in{\cal R}_4$$ and $${\cal P}(B,\widetilde{B})\approx
A_0.$$ The pair $(B,\widetilde{B})$ does not change (up to
isomorphism) when $a$ is replaced by $-a$.
\bigskip
\noindent{\bf Theorem 5.15 ([C2])} Let
$A\in\partial\bar{\cal A}_4$ be the ${\bf C}^*$-extension of
\\ $A_0\in{\cal A}_3$, a generic $P\!P\!A\!V$, determined by $\pm
a\in A_0$. Let $(X, \delta)=\chi(A)$.
\begin{list}{{\bf(\arabic{bean})}}{\usecounter{bean}}
\item $X$ is the nodal cubic threefold corresponding to $B=B_a$.
\item $(X,\delta)\in\partial^{\rm II}$, so $\widetilde{F(X)}$ is the
etale
double cover of $F(X)$ with normalization $S^2\widetilde{B}/\iota$,
as
in (5.12.II).
\item The corresponding quintics $Q$ are nodal; all three of
$\sigma,\nu,\nu\sigma$ are of type $\partial^{\rm II}$.
\item $\doublebar{\cal P}^{-1}(A)$ is isomorphic to $\widetilde{F(X)}$,
and
consists of $\partial^{\rm II}$-covers $(C,\widetilde{C})$ whose
normalizations (at one point) are of the form $(B_b,\widetilde{B}_b)$
for $b=b_1-b_2, \ \ b_1, b_2\in\psi(\widetilde{B})$.
\end{list}
\bigskip
\noindent{\bf Proof}
Clearly $\doublebar{\cal P}^{-1}(A)\subset\partial^{\rm
II}\doublebar{\cal R}_5$, so
consider a pair $(C,\widetilde{C})\in\partial^{\rm II}$, say
$$C=N/(p\sim
q),\;\;\;\widetilde{C}=\widetilde{N}/(p'\sim q', p"\sim q")$$ with
$(N,\widetilde{N})\in\bar{\cal R}_4$. Then
$\doublebar{\cal P}(C,\widetilde{C})$ is a
${\bf C}^*$-extension of $P(N,\widetilde{N})$, with extension data
$$\pm(\psi(p')-\psi(q'))\in{\cal P}(N, \widetilde{N})/(\pm 1).$$ We see
that $\doublebar{\cal P}(C,\widetilde{C})=A$ if and only if
\noindent{\bf (5.15.5)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
$(N,\widetilde{N})\in\bar{\cal P}^{-1}(A_0),$
\noindent and:
\noindent{\bf (5.15.6)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
$\psi(p')-\psi(q')=a, \ \ \ p',q'\in\widetilde{N}.$
\medskip
Now, (5.15.5) says that $(N,\widetilde{N})$ is taken, by its
Abel-Prym map $\psi$, to $(B_b,\widetilde{B}_b)$ for some $b\in A_0$,
and then (5.15.6) translates to: $$a=a_1-a_2, \ \ \ \ \ \ \ \ \ \
\ a_1,a_2\in\Theta\cap\Theta_b$$ which is equivalent to$$b=b_1-b_2, \
\ \ \ \ \
\ \ \ b_1,b_2\in\Theta\cap\Theta_a$$
(take $b_1=a_2+b, \ b_2=a_2)$. This proves (4), and everything
else follows from what we have already seen. \begin{flushright}
QED \end{flushright}
\bigskip
\noindent{\bf (5.16) Theta nulls}
Let $A\in{\cal A}_4$ be a generic $PPAV$ with vanishing thetanull,
and $(C,\widetilde{C})$ a generic element of ${\cal P}^{-1}(A)$. By
[B1], Proposition (7.3), $C$ has a vanishing thetanull. This
implies that the plane quintic $Q$ parametrizing singular
quadrics through $\Phi(C)$ has a node, corresponding to the
thetanull. The corresponding cubic threefold $X$ is thus also
nodal, and we are again in the situation of (5.11.III). I do not
see, however, a more direct way of describing the curve $B$ (or
the cubic $X$) in terms of $A$.
\bigskip
\noindent{\bf (5.17) Pentagons and wheels}.
In [V], Varley exhibits a two dimensional family of double covers
$(C,\widetilde{C})\in{\cal R}_5$ whose Prym is the unique
non-hyperelliptic
$P\!P\!A\!V$ \linebreak $A\in{\cal A}_4$ with 10 vanishing thetanulls.
The
curves
$C$ involved are Humbert curves, and each of these comes with a
distinguished double cover $\widetilde{C}$. As an illustration of our
technique, we work out the fiber of $\doublebar{\cal P}$ over $A$ and the
tetragonal moves on this fiber. This is, of course, a very
special
case of (5.12)(III) or (5.16).
We recall the construction of Humbert curves and their double
covers. Start by marking 5 points $p_1,\cdots,p_5\in{\bf P}^1$.Take
5 copies
$L_i$ of ${\bf P}^1$, and let $E_i$ be the double
cover of $L_i$ branched at the 4 points $p_j, \ \ j\ne i$. Let
\noindent{\bf (5.17.1)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ $A:=\coprod^5_{i=1}L_i, \;\;\;\;\; B:=\coprod^5_{i=1}E_i.$
\noindent The \underline{pentagonal} construction applied to
\noindent{\bf (5.17.2)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ $B\stackrel{g}{\to}A\stackrel{f}{\to}{\bf P}^1$
\noindent($f$ is the forgetful map, of degree 5), yields a
32-sheeted branched cover $f_*B\to{\bf P}^1$ which splits, by
(2.1.1), into 2 copies of the Humbert curve $C$, of degree 16
over ${\bf P}^1$.
Let ${\beta}_I$, $I\subset S:=\{1,\cdots,5\}$, be the
involution of (5.17.2) which fixes $A$ and acts non-trivially on
$E_i, \ \ i\in I$. It induces an involution $\alpha_I$ on
$f_*B$, hence on its quotient $C$. Let $$G:=\{\alpha_I \ \ | \ \
I\subset S\} \ / \ (\alpha_S).$$ Then $C$ is Galois over ${\bf
P}^1$, with group $G\approx({\bf{Z}\rm} \newcommand{\C}{\cal C}/2{\bf{Z}\rm} \newcommand{\C}{\cal C})^4$. Let $G_i, \ \ \ 1\le
i\le 5$, be the image in $G$ of $$\{\alpha_I|i \not\in I, \ \
\#(I)=\;\mbox{even}\}.$$Then $$C/\alpha_i\approx C/G_i\approx E_i,$$
and the
quotient map
$$E_i\approx C/\alpha_i\to C/G_i\approx E_i$$ becomes
multiplication by 2 on $E_i$. In particular, the Humbert curve
$C$ has 5 bielliptic maps $h_i:C\to E_i$. The branch locus of
$h_i$
consists of the 8 points $x\in E_i$ satisfying $g(2x)=p_i$.
For ease of notation, set $E:=E_5, \ \ \ p=p_5\in{\bf P}^1$,
$$C\stackrel{h}{\to}E\stackrel{g}{\to}{\bf P}^1,$$ and
$$\{p^0,p^1\}:=g^{-1}(p)\subset E.$$ Then for $j=0,1, \ \ E$ has
a natural double cover $C^{j}$, branched at the four points
${1\over 2}p^j$ and given by the line bundle ${\cal O}_E(2p^j)$.The
fiber
product
\noindent{\bf (5.17.3)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ $\widetilde{C}:=C^0\times_EC^1$
\noindent gives a Cartesian double cover of $C$.
Replacing $E_5$ by another $E_i$, we get an isomorphic double
cover
$\widetilde{C}$. Here is an invariant description of this cover:
Let $p_{i,j}:=L_i\cap f^{-1}(p_j)\in A$, and consider the curve
$$Q:=A/(p_{i,j}\sim p_{j,i}, \ \ \ i\ne j).$$ Then $Q$ can be
embedded in ${\bf P}^2$ as a pentagon, or completely reducible
plane quintic curve: embed ${\bf P}^1$ as a non-singular conic,
and take $L_i$ to be the tangent line of the conic at $p_i$. We
have two natural branched double covers of $Q$:
\noindent{\bf (5.17.4)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
$\widetilde{Q}_\sigma:=(\coprod^{ \: 5 \ \ \ 1} _{i=1,\varepsilon=0}
\ \ L^\epsilon_i)/(p^0_{i,j}\sim p^1_{j,i}, \ \ \ i\ne j)$
\noindent{\bf (5.17.5)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \
$\widetilde{Q}_\nu:=B/(\widetilde{p}_{i,j}\sim\widetilde{p}_{j,i}, \
\
\ i\ne j),$
\noindent where $\widetilde{p}_{i,j}\in E_i$ is the unique
(ramification) point above $p_{i,j}\in L_i$. We may think of
$\widetilde{Q}_\sigma$ as a "totally $\partial^I$" degeneration, and
of
$\widetilde{Q}_\nu$ as a "totally $\partial^{\rm III}$" degeneration.
We
then find:
\begin{tabbing}
\noindent{\bf (5.17.6)} \=$(Q,\widetilde{Q}_\nu)\in\overline{\cal R\cal Q}^+$
is the quintic double cover corresponding to the \\ \>Humbert
curve $C\in{\cal M}_5$.
\end{tabbing}
\begin{tabbing}
\noindent{\bf (5.17.7)} \=The double cover $\widetilde{Q}_\sigma$ of
$Q$ corresponds, via (1.4.2), to the double \\ \>cover
$\widetilde{C}$ of $C$.
\end{tabbing}
We note that $\widetilde{Q}_\sigma$ is itself an odd cover, so it
corresponds to some (singular) cubic threefold. A moment's
reflection shows that this must be Segre's cubic threefold $Y$ which
we have
already met in (4.8). Indeed, the Fano surface $F(Y)$ consists of the
six rulings $R_i, \ \ \
0\le i\le 5$, plus the 15 dual planes $\Pi_{i,j}^*$ of lines in
$\Pi_{i,j}$ (notation of (4.8)). We see that:
\begin{tabbing}
\noindent{\bf (5.17.8)} \=The discriminant of projection of $Y$
from a line $\ell\in R_i$ is a plane \\ \>pentagon $Q$, with its
double cover $\widetilde{Q}_\sigma$ as above.
\end{tabbing}
The other covers, $\widetilde{Q}_\sigma$, fit together to determine a
point $(Y,\delta)\in\overline{\cal R\C}^+$:
\noindent{\bf (5.17.9)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ $(Y,\delta)=\kappa(C, \widetilde{C}),$
\noindent for any Humbert cover $(C,\widetilde{C})$. The
tetragonal construction takes any $(Q,\widetilde{Q}_\sigma)$ to any
other (in two steps), so we recover Varley's theorem:
\noindent{\bf (5.17.10)} $A:={\cal P}(C, \widetilde{C})\in{\cal A}_4$ is
independent of the Humbert cover $(C,\widetilde{C})$.
But this is not the complete fiber: we have only used one of the
two component types of $F(Y)$. We note:
\noindent{\bf (5.17.11)} The discriminant of projection of $Y$
from a line $\ell\subset\Pi_{ij}$ consists of a conic plus
three lines meeting at a point; the double cover is split.
\vspace{2in}
\begin{center}
\begin{tabular}{cc}
\hspace{2.3in} & \hspace{2.3in} \\
pentagon & wheel
\end{tabular}
\end{center}
Consider a tritangent plane, meeting $Y$ in lines $\ell_i\in R_i,
\ \ \ \ell_j\in R_j$, and $\ell_{ij}\in\Pi_{ij}^*$. It
corresponds to a tetragonal construction involving two pentagons
and a wheel. The other kind of tritangent plane intersects $Y$
in
lines $\ell_{ij}\in\Pi_{ij}^*, \ \ell_{kl}\in\Pi_{kl}^*,
\\ell_{mn}\in\Pi_{mn}^*$, where $\{i,j,k,l,m,n\}=\{0,1,2,3,4,5\}$;
the tetragonal construction then relates three wheels.
\noindent{\bf Theorem 5.18} Let $A\in{\cal A}_4$ be the
non-hyperelliptic $P\!P\!A\!V$ with 10 vanishing thetanulls.
\begin{list}{{\bf(\arabic{bean})}}{\usecounter{bean}}
\item $\chi(A)$ consists of the Segre cubic threefold $Y$, with
its
degenerate semi-period $\delta$ (5.17.9).
\item The corresponding curve $B\in\bar{\cal M}_4$ (5.11) consists of
six ${\bf P}^1$'s:
{ \ }
\vspace{2in}
\item The Fano surface $F(Y)$ consists of the 6 rulings
$R_i \ \ (0\le i\le 5)$ and the 15 dual planes $\Pi_{i,j}^*$.The
plane quintics are pentagons, for $\ell\in R_i$, and wheels, for
$\ell\in\Pi_{ij}^*$, \ all with split covers $\sigma$ \ (5.17.4,
5.17.11). (The $\nu$ covers are branched over all the double
points.)
\item The fiber $\doublebar{\cal P}^{-1}(A)$ is contained in the fixed
locus
of the involution $\lambda:\doublebar{\cal R}_5\to\doublebar{\cal R}_5$
(5.1.6), so
it is a quotient of $F(Y).$
\item $\doublebar{\cal P}^{-1}(A)$ consists of two components:
\begin{itemize}
\item Humbert double covers $\widetilde{C} \rightarrow C$ (5.17.3).
\item Allowable covers $\widetilde{X}_{0} \cup \widetilde{X}_{1}
\rightarrow
X_{0} \cup X_{1}$, where $X_{0}$, $X_{1}$ are elliptic, meeting at
their
4 points of order 2.
\end{itemize}\end{list}
All of this follows from our previous analysis, except (5). The
new, allowable, covers are obtained by applying Corollary (3.7),
with $n=3$, to the Cartesian cover $\widetilde{C}\to C$ in (5.17.3).
It is also easy to see that the plane quintic parametrizing
singular quadrics through the canonical curve $\Phi(X_0\cup X_1)$
is a wheel, and vice versa, that the generalized Prym of any
wheel
(with its $\partial^{\rm III}$-cover) is the generalized Jacobian
$J(X_0\cup X_1)$ of such a curve. Thus every line in $F(Y)$ is
accounted for, so we have the complete fiber
$\doublebar{\cal P}^{-1}(A)$.
\begin{flushright} QED \end{flushright}
\large
\section{Other genera}
\ \ \ \ For $g\le 4$, it is relatively easy to describe the fibers
of ${\cal P}:\bar{\cal R}_g\to{\cal A}_{g-1}$. Indeed, every curve in
${\cal M}_g$ is trigonal, and every $A\in{\cal A}_{g-1}$ is a Jacobian (of a
possibly reducible curve), so the situation is completely
controlled by Recillas' trigonal construction. Similar results can
be obtained, for \linebreak $g\le 3$, by using Masiewicki's criterion
[Ma].
\bigskip
\noindent{\bf (6.1)} \ \underline{$g=1$}. Here $\bar{\cal P}$ sends
$\bar{\cal R}_1\approx{\bf P}^1$ to ${\cal A}_0$ (= a point). Thefibers
of $\bar{\cal P},{\cal P}$ are then ${\bf P}^1, {\bf C}^*$ respectively.
\bigskip
\noindent{\bf (6.2)} \ \underline{$g=2$}. All curves of genus 2 are
hyperelliptic, and all covers are Cartesian (3.2). An element of
${\cal R}_2$ is thus given by 6 points in ${\bf P}^1$, with 4 of them
marked, modulo ${\bf P}GL(2)$; an element $E$ of ${\cal A}_1$ is given
by 4 points of ${\bf P}^1$ modulo ${\bf P}GL(2)$; and ${\cal P}$
forgets the 2 unmarked points. The fiber of ${\cal P}$ is thus
rational; it can be described as $S/G$ where
\[ S := S^{2}\left( {\bf P}^{1} \setminus ({\rm 4 \; points}) \right)
\setminus
({\rm diagonal})\]
and $G \approx ({\bf Z}/2{\bf Z})^{2}$ is the Klein group, whose
action on $S$ is induced from its action on ${\bf P}^{1}$ permuting
the
4 marked points.
We note that $S$ is ${\bf P}^{2}$ minus a conic $C$ and four lines
$L_{i}$ tangent to it. To compactify it we add:
$\bullet$ a $\partial^{\rm I}$ cover for each point of
$C\backslash\cup L_i$,
$\bullet$ a $\partial^{\rm III}$ cover for each point of
$L_i\backslash C$, and
$\bullet$ an "elliptic tail" cover [DS, IV 1.3] for each point in
the exceptional divisor obtained by blowing up one of the points
$L_i\cap C$. (The
limiting double cover obtained is
\[{(E_0\amalg E_1)/\approx} \; \longrightarrow \; {E/\sim}\]
where $\sim$ places a cusp at one of the four marked points
$p_i$ on $E$ and $\approx$ places a
tacnode above it. These curves are unstable, and the family of
elliptic-tail covers gives their stable models, each elliptic tail
being blown down to the cusp.)
The resulting $\overline{S}$ is ${\bf P}^{2}$ with 4 points in
general position blown up, and the compactified fiber is
$\overline{S}/G$,
or ${\bf P}^{2}/G$ with one point blown up.
\bigskip
\noindent{\bf (6.3)} \ \underline{$g=3$}. Fix $A\in{\cal A}_2$. The
Abel-Prym map sends pairs $(C,\tilde{C})\in{\cal P}^{-1}(A)$ to
curves $\psi(\tilde{C})$ in the linear system $|2\Theta|$ on $A$,
uniquely defined modulo translation by the group
$G=A_2\approx({\bf{Z}\rm} \newcommand{\C}{\cal C}/2{\bf{Z}\rm} \newcommand{\C}{\cal C})^4$. The fiber is therefore, birationally,
the quotient ${\bf P}^3/G$. Since some curves in $|2\Theta|$ are
not stable, some blowing up is required to obtain the biregular
model of $\bar{\cal P}^{-1}(A)$. This is carried out in [Ve]. The
quotient ${\bf P}^3/G$ is identified with Siegel's modularquartic
threefold, or the minimal compactification $\bar{\cal A}_2^{(2)}$ of
the moduli space of $P\!P\!A\!V$'s with level-2 structure. To
obtain $\bar{\cal P}^{-1}(A)$, Verra shows that we need to blow
$\bar{\cal A}_2^{(2)}$ up at a point $A'$, corresponding to a level-2
structure on $A$ itself, and along a rational curve. The 2
exceptional divisors then parametrize hyperelliptic and
elliptic-tail covers, respectively.
\bigskip
\noindent{\bf (6.4)} \ \underline{$g=4$}.
As we noted in (5.15), the fiber ${\cal P}^{-1}(A), \ \ A\in{\cal A}_3$,
consists of
covers $(B_a,\tilde{B}_a), \ \ a\in A/(\pm 1):$
$$\tilde{B}_a=\Theta\cap\Theta_a,\;\;\;B_a=\tilde{B}_a/(x\sim(-
a-x)).$$ The fiber is thus (birationally) the Kummer variety
$A/(\pm 1)$.
\bigskip
\noindent{\bf (6.5)} \underline{$g\ge 7$}.
In this case, it was proved in [FS], [K], and [W], that ${\cal P}$ is
generically injective. The results in \S3 show that it is never
injective: on the hyperelliptic loci there are positive-dimensional
fibers, and various coincidences occur on the bielliptic loci. In
[D1] we conjectured:
\bigskip
\noindent{\bf Conjecture 6.5.1} Any two objects in a fiber of
${\cal P}$ are connected by a sequence of tetragonal constructions.
We state this for ${\cal P}$, rather than $\overline{\cal P}$, since
various other phenomena can contribute to non-trivial fibers at
the boundary. For example, all fibers of $\overline{\cal P}$ on
$\partial^{\rm I}$ are two-dimensional. On the other hand, from
the local pictures (2.14) it is clear that the tetragonal
construction can take a nonsingular curve to a singular one. In fact
proposition (3.8) shows that it is possible for two objects in
${\cal R}_g$ to be tetragonally related through an intermediate object of
$\partial{\cal R}_g$, so some care must be taken in clarifying which
class of tetragonal covers should be allowed. The conjecture is
consistent with our results for $g\le 6$. For $g\ge 13$, Debarre
[Deb2] proved it for curves which are neither hyperelliptic,
trigonal, or bielliptic. Naranjo [N] extended this to generic
bielliptics, $g\ge 10$. The following result was communicated to
me by Radionov:
\noindent{\bf Theorem 6.5.2} [Ra] \ \ For $g\ge 7,{\cal R}_g^{\rm Tet}$
is
an irreducible component of the noninjectivity locus of the Prym
map, and for generic $(C,\tilde{C})\in{\cal R}_g^{\rm Tet}$,
${\cal P}^{-1}({\cal P}(C,\tilde{C}))$
consists precisely of three tetragonally related objects.
\pagebreak[4]
\centerline{\bf REFERENCES}
\medskip
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\/,}
\vskip 25pt
\parindent=50pt
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\end{document}
|
1992-06-16T21:10:05 | 9206 | alg-geom/9206006 | fr | https://arxiv.org/abs/alg-geom/9206006 | [
"alg-geom",
"math.AG"
] | alg-geom/9206006 | Jean Francois Mestre | Jean-Francois Mestre | Corps quadratiques dont le 5-rang du groupe des classes est >=3 | 4 pages, LaTeX. (A paraitre dans les Comptes-Rendus de l'Acad. des
Sciences de Paris.) | null | null | null | null | We prove the existence of infinitely many real and imaginary fields whose
5-rank of the class group is >=3.
| [
{
"version": "v1",
"created": "Tue, 16 Jun 1992 19:08:36 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Mestre",
"Jean-Francois",
""
]
] | alg-geom | \section*{Bibliographie\markboth
{REFERENCES}{REFERENCES}}\list
{[\arabic{enumi}]}{\settowidth\labelwidth{[#1]}\leftmargin\labelwidth
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\begin{document}
G\'eom\'etrie alg\'ebrique/ {\it Algebraic geometry.}
\par\medskip
{\large Corps quadratiques dont le $5$-rang du groupe des classes
est $\geq 3$.}
Jean-Fran\c cois Mestre.
{\bf R\'esum\'e.-} Nous prouvons qu'il existe une infinit\'e de corps
quadratiques r\'eels (resp. imaginaires) dont le $5$-rang du groupe
des classes d'id\'eaux est $\geq 3$.
\par\medskip
{\large Quadratic fields whose $5$-rank is $\geq 3$.}
{\bf Abstract.-} We prove the existence of infinitely many real and
imaginary fields whose $5$-rank of the class group is $\geq 3$.
\vspace{5ex}
Soient $K$ un corps de nombres, $\rm Cl\; K$ le groupe des classes d'id\'eaux de
$K$,
et $p$ un nombre premier. Par d\'efinition,
le $p$-rang du groupe des classes de $K$ est la dimension de $\rm Cl\; K/p\rm Cl\; K$
sur $\hbox{\bf F}_p$.
\par\medskip
Nous d\'emontrons ici le th\'eor\`eme suivant:
\begin{theo}
Il existe une infinit\'e de corps quadratiques r\'eels (resp.
imaginaires) dont le $5$-rang du groupe des classes est $\geq 3$.
\end{theo}
L'id\'ee de la d\'emonstration est la suivante: soit $E$ une courbe
elliptique d\'efinie sur $\hbox{\bf Q}$, munie d'un point $P$ d'ordre $5$ rationnel
sur $\hbox{\bf Q}$; si $F$ est la courbe quotient $E/<P>$, notons $\phi:\;\;E\rightarrow F$
l'isog\'enie canonique de $E$ sur $F$.
Le lemme suivant, cons\'equence de \cite{RAYNAUD:schemas} et
du lemme $3$ de \cite{RAYNAUD:jacobienne}, m'a \'et\'e indiqu\'e par M.
Raynaud:
{\sc Lemme.-} {\it
Supposons $E$ semi-stable
en tout nombre premier $p$; soient $K$ un corps quadratique, et
$O_K$ son anneau d'entiers. Si ${\cal E}$
(resp. ${\cal F}$) est le mod\`ele de N\'eron de $E$ (resp. $F$) sur $O_K$,
on a une suite exacte (de sch\'emas en groupes sur $O_K$):
$$0\rightarrow \hbox{\bf Z}/5\hbox{\bf Z} \rightarrow {\cal E} \rightarrow {\cal F}'\rightarrow 0,$$
o\`u $\cal F'$ est un sous-sch\'ema en groupes ouvert de ${\cal F}$ contenant
la composante neutre ${\cal F}^0$ de ${\cal F}$.}
\par\medskip
Par suite, l'image r\'eciproque
par $\phi$ de tout point de ${\cal F}'(O_K)$ engendre une extension
ab\'elienne non ramifi\'ee de degr\'e divisant $5$ de $K$.
Soit $$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$$
une \'equation minimale de Weierstrass de $F$ sur $\hbox{\bf Z}$. Soit $Q=(x,y)$
un point de $F$ tel que $x\in \hbox{\bf Q}$; $Q$ est alors rationnel sur le corps
$K=\hbox{\bf Q}(y)$.
Notons $S$ l'ensemble fini des nombres premiers $p$
tels que le nombre de composantes connexes de la fibre en $p$
du mod\`ele de N\'eron de $F$ sur $\hbox{\bf Z}$ est divisible par $5$.
Supposons que, pour tout nombre premier
$p\in S$, $Q$ ne se r\'eduise pas $\mathop{\;\rm mod}\nolimits p$ en le point singulier de
$F_{/\hbox{\bf F}_p}$. Alors le point $Q$ se prolonge en un
point de ${\cal F}'(O_K)$.
Par suite, si $L=K(\phi^{-1}(Q))$, $L$ est une extension non ramifi\'ee
de $K$. Le th\'eor\`eme d'irr\'eductibilit\'e de Hilbert permet
de montrer que, pour une infinit\'e de tels $x\in \hbox{\bf Q}$, $L/K$ est
de degr\'e $5$ (En fait, comme on le verra plus loin,
on peut donner des crit\`eres effectifs de congruence modulo des nombres
premiers convenables pour assurer que $L/K$ est de degr\'e $5$.)
Nous construisons dans la section suivante une courbe $X$, d\'efinie sur $\hbox{\bf Q}$,
poss\'edant les propri\'et\'es suivantes:
i) Il existe un rev\^etement $\psi$ de degr\'e $2$ de $X$ sur la droite
projective.
ii) Pour $1\leq i\leq 3$, il existe une courbe elliptique $E_i$, semi-stable
sur $\hbox{\bf Z}$, poss\'edant un point $\hbox{\bf Q}$-rationnel $P_i$, et un
rev\^etement ab\'elien $\tau_i$ de groupe de Galois $(\hbox{\bf Z}/2\hbox{\bf Z})^2$ de $X$ sur
la courbe $F_i=E_i/<P_i>$.
Notons $\phi_i$ le morphisme canonique de $E_i$ sur $F_i$.
Nous montrons ensuite que, pour une infinit\'e de nombres rationnels $x$,
si $K=\hbox{\bf Q}(\psi^{-1}(x))$, les trois extensions
$L_i=K(\phi^{-1}(\tau_i(\psi^{-1}(x))))$ de $K$ sont ab\'eliennes de
degr\'e $5$, non ramifi\'ees et ind\'ependantes (i.e. les \'el\'ements
de $\mathop{\rm Hom}\nolimits(\rm Cl\; K,\hbox{\bf Z}/5\hbox{\bf Z})$ dont elles proviennent sont ind\'ependants), ce
qui prouve le th\'eor\`eme.
{\sc Remarque.- } Soit $E$ une courbe elliptique d\'efinie sur $\hbox{\bf Q}$,
munie d'un point rationnel $P$ d'ordre $n$, $n$ entier $\geq 1$. Soit $F$
la courbe quotient $E/<P>$, $\phi:\;\;E\rightarrow F$ le
morphisme canonique, et $y^2=f(x)$ un mod\`ele de Weierstrass
de $F$. Soit $x\in \hbox{\bf Q}$, et $Q$ l'un des deux points de $F$ d'abscisse $x$.
Si $K=\hbox{\bf Q}(\sqrt{f(x)})$,
le th\'eor\`eme de Chevalley-Weil permet de montrer que,
d\`es que la valuation de $x$ est suffisamment n\'egative en chaque
nombre premier o\`u $E$ a mauvaise r\'eduction, l'extension $K(\phi^{-1}(Q))/K$
est non ramifi\'ee. L'hypoth\`ese ``{\it $E$ semi-stable}'' n'est donc
pas n\'ecessaire. N\'eanmoins, dans le cas o\`u elle n'est pas v\'erifi\'ee,
les conditions de congruence sur $x$ sont plus d\'elicates \`a d\'eterminer.
\section{Construction de la courbe $X$}
\subsection{Construction de $E$ et $F$}
La courbe modulaire $X_1(10)$, classifiant les courbes elliptiques
munies d'un point d'ordre $10$, est de genre $0$, et a \'et\'e param\'etr\'ee
par Kubert \cite{KUBERT:universal}: il construit une courbe elliptique $E$
rationnelle sur $\hbox{\bf Q}(f)$, o\`u $f$ est un param\`etre,
poss\'edant un point $P_0$ d'ordre $10$
rationnel sur $\hbox{\bf Q}(f)$.
Si l'on pose $f=(u+1)/2$, et apr\`es un changement de variables, on
trouve comme \'equation de $E$:
$$y^2=(x^2-u(u^2+u-1))(8xu^2+(u^2+1)(u^4-2u^3-6u^2+2u+1)) .$$
Le point $P_0$ d'ordre $10$ est le point d'abscisse
$-{\frac {\left (u^{4}-2\,u^{3}-6\,u^{2}+2\,u+1\right )\left (u^{2}+1
\right )}{8\,u^{2}}}.$
Les formules de V\'elu \cite{VELU:isog} permettent
alors d'obtenir une \'equation de la courbe $F$ quotient de $E$ par
le groupe d'ordre $5$ engendr\'e par $2P_0$;
une \'equation de $F$ est donn\'ee par
$y^2=g_u(x)$, o\`u $$g_u(x)=(x^2-u(u^2+u-1))h_u(x)\;\;\;
{\rm et}\;\;\; h_u(x)=8(u^2+u-1)^2x+(u^2+1)(u^4+22u^3-6u^2-22u+1).$$
De plus, si $u\in \hbox{\bf Q}$, la condition $u\equiv \pm 1 \mathop{\;\rm mod}\nolimits 5$ assure que les
courbes $E$ et $F$ sont semi-stables sur $\hbox{\bf Z}$.
\subsection{Construction de $X$}
Si
$$\left\{\begin{array}{l}
u_1=(t^2+t-1)/(t^2+t+1),\\ u_2=-(t^2+3t+1)/(t^2+t+1),\\
u_3=-(t^2-t-1)/(t^2+t+1)
\end{array}\right.$$
on a
$$u_1(u_1^2+u_1-1)=u_2(u_2^2+u_2-1)=u_3(u_3^2+u_3-1).$$
La courbe $X$, d\'efinie sur $\hbox{\bf Q}(t)$, normalis\'ee de la courbe d'\'equations
$$\left\{\begin{array}{l}
y_1^2=g_{u_1}(x),\\ y_2^2=g_{u_2}(x),\\y_3^2=g_{u_3}(x)
\end{array}\right.$$
est
de genre $5$.
L'application
$\phi:\;\;(x,y_1,y_2,y_3)\mapsto (x,v=y_1/y_2,w=y_1/y_3)$
de $X$ sur la courbe $C$ de genre $0$ et
d'\'equations $$h_{u_1}(x)=v^2h_{u_2}(x)=w^2h_{u_3}(x)$$
est de degr\'e $2$; les quatre points de $C$ de coordonn\'ees $(x,v,w)=(\infty,
\pm u_2/u_1,\pm u_3/u_1)$ sont rationnels sur $\hbox{\bf Q}(t)$
et sont des points de ramification de $\phi$,
donc $C$ est $\hbox{\bf Q}(t)$-isomorphe \`a la
droite projective, et $X$ est hyperelliptique, rev\^etement double de la
droite projective, et poss\`ede quatre points de Weierstrass rationnels
sur $\hbox{\bf Q}(t)$.
Sp\'ecialisons en $t=4$ les formules de la section pr\'ec\'edente.
On trouve $u_1=19/21$, $u_2=-29/21$ et $u_3=-11/21$. Les courbes
$E_i$ et $F_i$, pour $1\leq i\leq 3$, sont donc semi-stables sur $\hbox{\bf Z}$.
Une param\'etrisation de la courbe $C$ est donn\'ee
par
$$v=\frac{29}{19}\frac{53719189282 z^2+26766692861}{ 53719189282 z^2
- 283246634396 z- 26766692861},$$
$$w=\frac{11}{19}\frac{53719189282 z^2+26766692861}{
53719189282 z^2+ 20305766998 z- 26766692861},$$
$$x=\frac{c_4z^4+c_3z^3+c_2z^2+c_1z+c_0}{5167944494559 (4883562662 z+922989409)
(11 z -29) z},$$
avec
$$\begin{array}{ll}
c_0=343898806423252015354080,&c_1=- 411804539876837130626339,\\
c_2=- 642297925780193483509181,&c_3=826467660375890872281118,\\
c_4=1385160622615364964251520.&\end{array}$$
On obtient alors une \'equation hyperelliptique de $X$ en substituant
la fraction rationnelle $x(z)$ ci-dessus dans, par exemple, l'\'equation
de $E_1$, \`a savoir
$$y^2=42(44876601 x-133597561)(9261 x^2-6061).$$
\`A toute valeur rationnelle de $z$
distincte des p\^oles de $x(z)$, on associe ainsi le corps $K=\hbox{\bf Q}(y)$.
\subsection{Conditions sur $z$ pour que le $5$-rang de $\rm Cl\; K$ soit
$\geq 3$}
Si $p$ est un nombre premier, notons $v_p$ la valuation $p$-adique.
Le calcul montre qu'un point de $F_1(K)$
(resp. $F_2(K)$, resp. $F_3(K)$) se prolonge \`a ${\cal F_1}'(O_K)$
(resp. ${\cal F_2}'(O_K)$, resp. ${\cal F_3}'(O_K)$) si et seulement si
son abscisse $x$ v\'erifie $v_{11}(x)\leq -2$, $v_{29}(x)\leq -2$ et
$x\not\equiv 77 \mathop{\;\rm mod}\nolimits 419$ (resp. $v_{11}(x)\leq -2$, $v_{19}(x)\leq -2$,
$x\not\equiv 677 \mathop{\;\rm mod}\nolimits 709$, resp. $v_{19}(x)\leq -2$, $v_{29}(x)\leq -2$,
$x\not\equiv 36 \mathop{\;\rm mod}\nolimits 151$).
Si
\begin{equation}
z\equiv 0 \mathop{\;\rm mod}\nolimits 11.19.29\;\;\;{\rm et}\;\;\;
z\not\equiv \pm 86\mathop{\;\rm mod}\nolimits 419,\end{equation}
les conditions de congruence ci-dessus
sont remplies, et les points correspondants de $F_i(K)$, $i=1,2,3$, se
prolongent en des points de $F'_i(O_K)$.
De plus, soit $l_1=163$, $l_2=701$ et $l_3=1277$; supposons
\begin{equation}
z\equiv 1\mathop{\;\rm mod}\nolimits l_1l_2l_3.\end{equation} Alors:
i) Les id\'eaux $(l_i)$, $i=1,2,3$, se
d\'ecomposent chacun dans $K$ en deux id\'eaux ${\cal P_i}\overline{\cal
P_i}$,
ii) ${\cal P_1}$ (resp. ${\cal P_2}$, resp. ${\cal P_3}$)
est d\'ecompos\'e (resp.
d\'ecompos\'e, resp. inerte) dans $L_1$,
iii) ${\cal P_1}$ (resp. ${\cal P_2}$, resp. ${\cal P_3}$)
est inerte (resp. d\'ecompos\'e, resp.
d\'ecompos\'e) dans $L_2$,
iv) ${\cal P_1}$ (resp. ${\cal P_2}$, resp. ${\cal P_3}$)
est d\'ecompos\'e (resp. inerte,
resp. d\'ecompos\'e) dans $L_3$.
Ceci assure que les extensions $L_i$, $i=1,2,3$, sont ind\'ependantes.
Par suite, d\`es que $z$ v\'erifie les congruences $(1)$ et $(2)$ ci-dessus,
le $5$-rang du groupe des classes de $K$ est $\geq 3$.
Le fait que, lorsque $z$ parcourt une infinit\'e de valeurs
rationnelles, on obtient une infinit\'e de corps quadratiques $K$
provient par exemple du th\'eor\`eme de Faltings (la courbe $X$,
\'etant de genre $>1$, n'a qu'un nombre fini de points rationnels
dans un corps $K$ fix\'e.)
De plus, comme $X$ a trois points de Weierstrass rationnels, donc
r\'eels, on obtient ainsi une infinit\'e de corps quadratiques r\'eels
(resp. imaginaires) dont le $5$-rang du groupe des classes est
$\geq 3$. Par exemple, si $z$ est $>0$ (resp. $<0$) et suffisamment proche
de $0$
(pour la topologie usuelle), $K$ est quadratique imaginaire (resp.
r\'eel).
|
1992-06-16T21:33:05 | 9206 | alg-geom/9206007 | fr | https://arxiv.org/abs/alg-geom/9206007 | [
"alg-geom",
"math.AG"
] | alg-geom/9206007 | Jean Francois Mestre | Jean-Francois Mestre | Rang de courbes elliptiques d'invariant donne | 4 pages. (A paraitre aux Comptes-Rendus de l'Acad. des Sc. de Paris.) | null | null | null | null | We prove that there exist infinitely many elliptic curves over \Q with given
modular invariant, and rank >=2. Furthermore, there exist infinitely many
elliptic curves over $\Q$ with invariant equal at 0 (resp. 1728) and rank >=6
(resp. >=4).
| [
{
"version": "v1",
"created": "Tue, 16 Jun 1992 19:31:34 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Mestre",
"Jean-Francois",
""
]
] | alg-geom | \section*{Bibliographie\markboth
{REFERENCES}{REFERENCES}}\list
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\begin{document}
G\'eom\'etrie alg\'ebrique/ {\it Algebraic geometry.}
\par\medskip
{\large Rang de courbes elliptiques d'invariant donn\'e.}
Jean-Fran\c cois Mestre.
\par\medskip
{\bf R\'esum\'e.-} Nous montrons qu'il existe une infinit\'e de courbes
elliptiques d\'efinies sur $\hbox{\bf Q}$, d'invariant modulaire donn\'e, et de
rang $\geq 2$.
De plus, il existe une infinit\'e de courbes d\'efinies sur $\hbox{\bf Q}$, d'invariant
nul (resp. \'egal \`a $1728$), et de rang $\geq 6$ (resp. $\geq 4$).
\par\medskip
{\large On the rank of elliptic curves with given modular invariant.}
{\bf Abstract.-} We prove that there exist infinitely many elliptic curves over
$\hbox{\bf Q}$
with given modular invariant, and rank $\geq 2$. Furthermore, there exist
infinitely many elliptic curves over $\hbox{\bf Q}$ with invariant equal to $0$
(resp. $1728$),
and rank $\geq 6$ (resp. $\geq 4$).
\vspace{5ex}
Soit $k$ un corps de caract\'eristique nulle, et $t$ une
ind\'etermin\'ee.
Nous prouvons ici les th\'eor\`emes suivants:
\begin{theo} Soit $j$ un \'el\'ement de $k$.
Il existe une courbe elliptique d\'efinie sur $k(t)$,
d'invariant modulaire $j$, qui n'est
pas $k(t)$-isomorphe \`a une courbe elliptique d\'efinie sur $k$,
et qui poss\`ede deux points rationnels sur $k(t)$
lin\'eairement ind\'ependants.
\end{theo}
\begin{theo}
Il existe une courbe
elliptique d\'efinie sur $k(t)$, dont l'invariant modulaire est
\'egal \`a $1728$ (resp. $0$),
qui n'est pas $k(t)$-isomorphe \`a
une courbe d\'efinie sur $k$,
et qui poss\`ede $4$ (resp. $6$) points rationnels sur $k(t)$
lin\'eairement ind\'ependants.
\end{theo}
On en d\'eduit par sp\'ecialisation les corollaires suivants:
\begin{cor} Soit $j$ un \'el\'ement de $\hbox{\bf Q}$. Il existe une
infinit\'e de courbes elliptiques d\'efinies sur $\hbox{\bf Q}$, non
deux \`a deux $\hbox{\bf Q}$-isomorphes, d'invariant modulaire $j$, dont
le rang du groupe de Mordell-Weil est $\geq 2$.
\end{cor}
\begin{cor}
Il existe une infinit\'e de courbes elliptiques d\'efinies
sur $\hbox{\bf Q}$, d'invariant modulaire \'egal \`a $1728$ (resp. $0$),
non deux \`a deux $\hbox{\bf Q}$-isomorphes, dont le rang du groupe de
Mordell-Weil est $\ge 4$ (resp. $\ge 6$).
\end{cor}
\section{D\'emonstration du th\'eor\`eme $1$}
\begin{theo}
Soient $k$ un corps de caract\'eristique nulle, et $E$ et $E'$
deux courbes elliptiques d\'efinies sur $k$. On suppose que les
invariants modulaires $j(E)$ et $j(E')$ ne sont pas simultan\'ement
\'egaux \`a $0$ ou \`a $1728$.
Il existe alors une courbe $C$, rev\^etement quadratique de la droite
projective, d\'efinie sur $k$, et deux morphismes ind\'ependants
$p:\;\;C\rightarrow E$ et $p':\;\;C\rightarrow E'$
d\'efinis sur $k$.
\end{theo}
(On rappelle que deux morphismes $p:\;\;C\rightarrow E$ et $p':\;\;C\rightarrow E'$
sont dits ind\'ependants si les images r\'eciproques par $p^*$ et
$p'^*$ des formes de premi\`ere esp\`ece de $E$ et $E'$ sont
lin\'eairement ind\'ependantes.)
\par\medskip
Soient $y^2=x^3+ax+b$ une \'equation de $E$ et $y^2=x^3+a'x+b'$
une \'equation de $E'$.
L'hypoth\`ese sur $j(E)$ et $j(E')$ implique que
$a=0\rightarrow a'\neq 0$ et
$b=0\rightarrow b'\neq 0$.
Posons $f(x)=x^3+ax+b$ et $g(x)=x^3+a'x+b'$.
Si $u$ est une ind\'etermin\'ee,
l'\'equation (en $x$) $$u^6f(x)=g(u^2x)$$ a pour solution
$x=\phi(u)$, avec
$\phi(u)=-\fracb{b'-u^6b}{u^2(a'-u^4a)}.$
Soit $C$ la courbe d'\'equation
$Y^2=f(\phi(X)).$
Soient
$\rho:\;\;\;C\rightarrow E$ et
$\rho':\;\;\;C\rightarrow E'$ les morphismes donn\'es par
$\rho(X,Y)=(x=\phi(X),y=Y)$ et $\rho'(X,Y)=(x=X^2\phi(X),y=X^3 Y)$.
Si $\omega=\rho^*(dx/y)$ et $\omega'=\rho'^*(dx/y)$, on a
$$\omega/\omega'=
\frac{{ { 3 a { X^{4}} b'} { - 2 { X^{6}} b a'} {- b'a' }}}{{ { X^{3}}
{( { { { X^{6}} b a} { - 3 { X^{2}} b a'}+ { 2 a b'}} )}}},$$
fraction rationnelle en $X$ non constante. Par suite, $\omega$ et $\omega'$
sont ind\'ependantes dans l'espace des formes diff\'erentielles de
premi\`ere esp\`ece de $C$, d'o\`u le th\'eor\`eme.
\par\medskip
{\sc Remarque.} Le calcul montre que le genre de $C$ est
$\leq 10$. Plus pr\'ecis\'ement, si l'invariant modulaire $j(E)$ de $E$
n'est pas \'egal \`a $j(E')$, et si $j(E)$ et $j(E')$ sont distincts
de $0$ et $1728$, le genre de $C$ est \'egal \`a $10$. Si $j(E)=j(E')$,
et distinct de $0$ et $1728$, le genre
de $C$ est \'egal \`a $6$. Si $j(E)=1728$, et $j(E')\neq 0$, le
genre de $C$ vaut $7$. Si $j(E)=0$, et $j(E')\neq 1728$, le genre
de $C$ vaut $8$. Enfin, si $j(E)=0$ et $j(E')=1728$, le genre de $C$
vaut $5$.
\begin{theo}
Soient $k$ un corps de caract\'eristique nulle,
et $j$ un \'el\'ement de $k$.
Il existe une courbe $C$ d\'efinie sur $k$, rev\^etement quadratique de
la droite projective, une courbe elliptique $E$ d\'efinie sur $k$
d'invariant $j$, et deux morphismes ind\'ependants
$p$ et $p'$ de $C$ dans $E$ d\'efinis sur $k$.
\end{theo}
Si $j\in k$, $j\neq 0,1728$, et si $a=b=\fracb{27j}{4(j-1728)}$, la courbe
elliptique $E$, d\'efinie sur $k$, d'\'equation $y^2=x^3+ax+b$ a comme
invariant modulaire $j$.
Le th\'eor\`eme pr\'ec\'edent permet donc de conclure, sauf si
$j=0$ ou $j=1728$.
Or la jacobienne de la courbe de genre $2$, d\'efinie sur
$\hbox{\bf Q}$, d'\'equation $y^2=x^6+1$ est $\hbox{\bf Q}$-isog\`ene
au produit de la courbe elliptique $y^2=x^3+1$, d'invariant modulaire
\'egal \`a $0$, avec elle-m\^eme. D'o\`u le r\'esultat si $j=0$.
De m\^eme, soit $C$ la courbe de genre $2$ d'\'equation
$y^2=(t^2+1)(t^2-2)(2t^2-1).$
Les morphismes $(t,y)\mapsto (t^2,y)$ et $(t,y)\mapsto (1/t^2,y/t^3)$
d\'efinissent deux rev\^etements de $C$ sur la courbe elliptique
d'\'equation $y^2=(x+1)(x-2)(2x-1)$, dont l'invariant modulaire vaut
$1728$. Cela ach\`eve la d\'emonstration du th\'eor\`eme.
\par\medskip
{\sc Remarques.-} Si $E$ est une courbe elliptique d\'efinie sur
$k$, il est parfois possible de trouver une courbe hyperelliptique
d\'efinie sur $k$, de genre $<10$, dont la jacobienne est
$k$-isog\`ene \`a $E\times E\times A$, o\`u $A$ est une vari\'et\'e
ab\'elienne convenable. Par exemple:
1) Soit $E$ une courbe elliptique d\'efinie sur $k$
d'\'equation
$y^2=x^3-ax+b$, o\`u $a$ est non nul et de la forme $\alpha^2+3\beta^2$,
$\alpha$, $\beta \in k$. La conique $x_1^2+x_1x_2+x_2^2=a$ est alors
$k$-isomorphe \`a la droite projective, d'o\`u l'existence
de deux fractions rationnelles $x_1(t)$ et $x_2(t)$ telles
que la fraction rationnelle $f(t)=x_1^3-ax_1+b$ soit \'egale \`a la
fraction rationnelle $x_2^3-ax_2+b$. On en d\'eduit $2$ applications
rationnelles
$(t,y)\mapsto (x_1(t),y)$ et $(t,y)\mapsto (x_2(t),y)$
de la courbe $C$ d'\'equation $y^2=f(t)$ sur $E$. Les fractions
rationnelles $x'_1$ et $x'_2$ n'\'etant pas proportionnelles,
et la courbe $C$ \'etant de genre $3$,
on en d\'eduit que la jacobienne de la courbe
$C$ est $k$-isog\`ene \`a $E\times E\times E_1$,
o\`u $E_1$ est une courbe elliptique d\'efinie sur $k$.
\par\medskip
2) Soient $E_1$ et $E_2$ deux courbes elliptiques, d\'efinies sur
$k$, dont les points d'ordre $2$ appartiennent \`a $k$.
Si $y^2=(x-a)(x-b)(x-c)$ (resp. $y^2=(x-a')(x-b')(x-c')$) est une
\'equation de $E_1$ (resp. $E_2$), quitte \`a permuter les r\^oles
de $a,b,c$,
on peut trouver une application
affine $x\mapsto h(x)=\alpha x+\beta$ telle que $h(a)=a'$, $h(b)=b'$, et
$h(c)\neq c'.$ La jacobienne de la courbe de genre $2$ d'\'equations
$$y^2=(x-a)(x-b)(x-c),\;\;\;z^2=\alpha (x-a)(x-b)(x-h^{-1}(c'))$$
est alors isog\`ene \`a $E_1\times E_2$.
\par\bigskip
Le th\'eor\`eme 1 d\'ecoule ais\'ement du th\'eor\`eme pr\'ec\'edent. En
effet,
si $j\in k$,
d'apr\`es le th\'eor\`eme pr\'ec\'edent,
il existe une courbe
$C$, d\'efinie sur $k$, rev\^etement quadratique de la droite projective,
une courbe elliptique $E$ d\'efinie sur $k$ d'invariant $j$, et
deux morphismes
ind\'ependants $p_1$ et $p_2$ de $C$ sur $E$.
Soit $w$ l'involution
hyperelliptique de $C$; les morphismes $p_1\circ w+p_1$ et
$p_2\circ w+p_2$ de $C$ dans $E$ sont constants, car $w$ agit sur
la jacobienne de $C$ comme $-1$. Par suite, les morphismes
$p'_1=p_1\circ w-p_1$ et $p'_2=p_2\circ w-p_2$ sont ind\'ependants;
si $y^2=f(t)$ est une \'equation de $C$,
et si $E_w$ est la courbe obtenue \`a partir de $E$ par
torsion par $\sqrt{f(t)}$, les points
$P_1=p'_1(t,\sqrt{f(t)})$ et $P_2=p'_2(t,\sqrt{f(t)})$ sont des
points ind\'ependants de $E_w$, rationnels sur $k(t)$.
D'o\`u le th\'eor\`eme $1$.
\section{D\'emonstration du th\'eor\`eme 2}
\subsection{Le cas des courbes d'invariant $j=1728$}
Soit $p(x)=x^4+a_2x^2+a_1x+a_0$ un \'el\'ement de $k[x]$,
dont les racines $x_i$, $1\leq i\leq 4$, appartiennent \`a $k$, et sont de
somme nulle.
La courbe $E$
d'\'equation $x^4+a_2y^2+a_1y+a_0=0$ poss\`ede $4$ points
$k$-rationnels naturels, \`a savoir les points $P_i=(x_i,x_i)$.
Si $a_0=-u^4$, o\`u $u\in k$, $E$ poss\`ede un nouveau point
$k$-rationnel, \`a savoir le point $O=(-u,0)$. Si $a_2(a_1^2-4a_0a_2)\neq 0$,
la courbe $E$ est de genre $1$, et d'invariant modulaire
\'egal \`a $1728$.
Or l'\'equation $a_0=-u^4$ s'\'ecrit $x_1x_2x_3(x_1+x_2+x_3)=u^4.$
Comme me l'a indiqu\'e J.-P. Serre, cette \'equation
a \'et\'e \'etudi\'ee par Euler ([1], p. $660$),
qui a exhib\'e plusieurs courbes unicursales
trac\'ees sur $S$, par exemple la courbe
$$u=1,\;\;\;x_1=t\fracb{2t^2-1}{2t^2+1},\;\;\;
x_2=\fracb{2t^2-1}{2t(2t^2+1)},\;\;\;
x_3=\fracb{4t}{2t^2-1}.$$
Soit donc $x_4=-x_1-x_2-x_3$, o\`u les $x_i$ sont donn\'es par les
formules ci-dessus,
et soit $p=\prod (x-x_i)=x^4+a_2x^2+a_1x+a_0.$
La courbe $E$, d\'efinie sur $k(t)$, d'\'equation
$x^4+a_2y^2+a_1y+a_0$ est de genre $1$; elle est $k(t)$-isomorphe \`a la
courbe elliptique d'\'equation $y^2=x^3+a_2(a_1^2-4a_0a_2)x$.
On v\'erifie que $a_2(a_1^2-4a_0a_2)$
n'est pas une puissance quatri\`eme dans
$k(t)$; par suite, $E$ n'est pas $k(t)$-isomorphe \`a une courbe
d\'efinie sur $k$.
\par\medskip
Pour prouver que les $4$ points $P_i$ sont ind\'ependants,
le point $O$ \'etant choisi comme origine, et d\'emontrer ainsi l'assertion
du th\'eor\`eme $2$ relative aux courbes d'invariant $1728$, il suffit
de v\'erifier que, pour une valeur de $t$, les sp\'ecialisations
des points $P_i$ sont des points ind\'ependants.
Or, pour $t=1$, le calcul, \`a l'aide du logiciel gp, montre que
le d\'eterminant de la matrice des hauteurs des sp\'ecialisations
des points $P_i$ est \'egal \`a $603.61237\ldots$, et est donc non nul.
\subsection{Le cas des courbes d'invariant $0$}
Soit $p\in k[X]$ un polyn\^ome unitaire de degr\'e $6$. Il existe alors
un unique polyn\^ome unitaire $g\in k[X]$, de degr\'e 2, tel que
le polyn\^ome $r=p-g^3$ soit de degr\'e $\leq 3$.
Supposons que les racines
$x_1,\ldots,x_6$ de $p$ soient dans $k$. La courbe $E$ d'\'equation
$r(x)+y^3=0$ contient les $6$ points $k$-rationnels $P_i=(r(x_i),g(x_i))$,
$1\leq i\leq 6$.
De plus, si le discriminant de $r$ est non nul, la courbe $E$ est de
genre $1$ et d'invariant modulaire \'egal \`a $0$.
Si le coefficient de degr\'e $3$ de $r$ est le cube d'un \'el\'ement de $k$,
l'un des points \`a l'infini de $E$ est $k$-rationnel,
et on peut le choisir comme origine $O$ de la courbe elliptique $E$.
Nous allons montrer que, si les $x_i$ sont convenablement choisis, les
points $P_i$ sont alors ind\'ependants.
Sans nuire \`a la g\'en\'eralit\'e du probl\`eme, on peut supposer que
la somme des racines $x_i$ de $p$ est nulle. On peut donc \'ecrire $p$
sous la forme
$p(x)=x^6+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0.$
On a alors $$g(x)=x^2+a_4/3,\;\;\;r(x)=a_3x^3+(a_2-a_4^2/3)x^2+a_1x-a_4^3/27.$$
Le coefficient $a_3$ du polyn\^ome $p$ est homog\`ene de degr\'e $3$ en
les racines $x_i$ de $p$.
L'hypersurface cubique (en les variables $u$ et $x_i$, $1\leq i\leq 5$)
d'\'equation $u^3=a_3$ poss\`ede des sous-vari\'et\'es lin\'eaires
$k$-rationnelles naturelles, par exemple $u=0, x_1=x_2=x_3=-x_4=-x_5$.
Par des manipulations classiques, cela permet d'obtenir des courbes
unicursales trac\'ees sur cette hypersurface. On trouve par exemple
$$\begin{array}{ll}
x_1=-126(35t-19)(14t-13)(t+1),&
x_2=63(-980t^3+3549 t - 3084 t + 1135),\\
x_3=126(35 t - 19) (14 t - 13) (t + 1),&
x_4=63(1127 t^3- 3108 t^2+ 3525 t- 988),\\
x_5=- 113876 t^3+ 265629 t^2- 259980 t + 69103,&
x_6=104615 t^3 - 293412 t^2+ 232197 t - 78364.
\end{array}$$
On obtient ainsi, par la m\'ethode d\'ecrite ci-dessous, une courbe
elliptique $E$, d\'efinie sur $k(t)$, munie de $6$ points $k(t)$-rationnels.
Cette courbe est $k(t)$-isomorphe \`a la courbe $y^2=x^3-16D$, o\`u
$D$ est le discriminant du polyn\^ome $r$.
On v\'erifie que $D$ est
un polyn\^ome irr\'eductible sur $k(t)$, et n'est donc pas une puissance
sixi\`eme. Par suite, $E$ n'est pas $k(t)$-isomorphe \`a une courbe
d\'efinie sur $k$.
Pour prouver que les points $P_i$ sont ind\'ependants,
le point $O$ \'etant choisi comme origine, il suffit de le montrer
pour une valeur convenable de $t$. Or, pour $t=1$, le d\'eterminant
de la matrice des hauteurs normalis\'ees des points $P_i$ vaut
$38462030713.186929\ldots$, et est donc non nul.
\par\medskip
{\sc R\'ef\'erence bibliographique}
\par\medskip
[1] {\sc L. Dickson}, {\it History of the theory of numbers}, vol. 2,
Chelsea $1971$.
\begin{flushright}
UFR de Math\'ematiques,
Universit\'e de Paris VII\\
2 place Jussieu,
75251 Paris Cedex 05.\end{flushright}
\end{document}
|
1996-03-11T06:20:12 | 9602 | alg-geom/9602010 | en | https://arxiv.org/abs/alg-geom/9602010 | [
"alg-geom",
"dg-ga",
"math.AG",
"math.DG"
] | alg-geom/9602010 | Steven Bradlow | Steven Bradlow and Oscar Garcia-Prada | Non-abelian monopoles and vortices | Revised version, in which some minor clarrifications and a number of
unjustly ommitted references have been added. (Apologies to any inadvertantly
offended parties.) To appear in the Proceedings of the 1995 Aarhus Conference
in Geometry and Physics. 25 pages. AMSLaTeX v 2.09 | null | null | null | null | The Seiberg-Witten equations are defined on certain complex line bundles over
smooth oriented four manifolds. When the base manifold is a complex Kahler
surface, the Seiberg-Witten equations are essentially the Abelian vortex
equations. Using known non-abelian generalizations of the vortex equations as a
guide, we explore some non-abelian versions of the Seiberg-Witten equations. We
also make some comments about the differences between the vortex equations that
have previously appeared in the literature and those that emerge as Kahler
versions of Seiberg-witten type equations.
| [
{
"version": "v1",
"created": "Fri, 9 Feb 1996 20:50:43 GMT"
},
{
"version": "v2",
"created": "Fri, 8 Mar 1996 18:22:37 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Bradlow",
"Steven",
""
],
[
"Garcia-Prada",
"Oscar",
""
]
] | alg-geom | \section{Introduction}\label{introduction}
In the short time since their discovery, the Seiberg--Witten equations
have already proved to be a powerful tool in the study of smooth
four-manifolds.
Virtually all the hard-won gains that have been obtained using the heavy
machinery of
Donaldson invariants, can be recovered with a fraction of the effort
if the (SU(2)) anti-self-duality equations are replaced by the
($\mathop{{\fam0 U}}\nolimits(1)$) Seiberg--Witten equations. In addition, the new equations
probe deep features of symplectic structures. They have also been used
to study geometric questions on K\"ahler surfaces. (See \cite{D3} for a
useful survey.)
The impressive success of the original equations has naturally led
to speculation about possible generalizations and other related sets
of equations. The original equations as proposed by Seiberg--Witten
are associated with a Hermitian line bundle, and thus with the
abelian group $\mathop{{\fam0 U}}\nolimits(1)$. One way to generalize the equations is thus
to look for versions based on larger, non-abelian groups. This
means replacing the line bundle with a higher rank complex vector
bundle. Indeed a number of authors have proposed versions of the
equations along these lines. These include, among others,
Okonek and Teleman \cite{OT1,OT2},
Pidstrigach and Tyurin \cite{PT}, Labastida and Mari\~no \cite{LM},
as well as the second author \cite{G5}. Some of these (cf. \cite{PT})
play a key role in attempts to prove the conjecture of Witten \cite{W}
concerning the equivalence of the old Donaldson and the new
Seiberg--Witten\ invariants.
(See also \cite{D3}, and \cite{FL} for more recent progress in
this direction.)
It is striking that no two of the above mentioned authors
consider precisely the same set of equations. One conclusion to be drawn from
this abundance of equations, is
that there is apparently more than one natural way to write down non
abelian versions of the Seiberg-Witten equations. This leads to the
question: Are some versions more reasonable, or more natural, than others?
The material in this paper gives one perspective on this question.
The main idea in our point of view is to exploit the special form of
the Seiberg-Witten equations in the case where the four manifold is
a K\"ahler surface. In this case the original Seiberg--Witten
equations are known to reduce essentially to familiar equations in gauge
theory known as the abelian vortex equations. Looked at from
the opposite direction, the Seiberg--Witten equations serve as
a ''Riemannian version'' of the vortex equations. The key point is
that there are a number of well motivated, natural generalizations
of such vortex equations. All of these are defined on complex
vector bundles over K\"ahler manifolds, and thus in particular over
K\"ahler surfaces. Our guiding principle is
that the generalizations of the Seiberg--Witten equations should provide
''Riemannian versions'' of these vortex-type equations over K\"ahler
surfaces.
In this paper we explore essentially two such non-abelian generalizations.
We also make some remarks concerning a different aspect of the
relation between the vortex and the Seiberg-Witten equations.
This aspect has to do with the parameters which appear in the
vortex equations. In their
original form, these were taken to be real numbers, i.e. constant
functions on the base manifolds. In the versions that emerge from the
Seiberg-Witten equations, the analogous terms turn out to be non-constant
functions (related to the scalar curvature). This has prompted a closer
look at the affected terms in the vortex equation. We discuss various ways
of
incorporating --- and interpreting --- this level of generality in the
analysis of the vortex equations.
In the interests of completeness, we have included a certain amount of
standard background material on the Seiberg-Witten and vortex equations.
\noindent {\bf Acknowledgements.}
The second author wishes to thank the organisers of the Aarhus
Conference in Geometry and Physics, and especially J{\o}rgen Andersen,
for their kind invitation to participate in the Conference,
and to visit the Mathematics Institute in Aarhus,
as well as for their warm hospitality.
\section{The Seiberg--Witten\ monopole equations}\label{sw}
In this section we briefly review the Seiberg--Witten\
equations and the analysis of these equations in the K\"{a}hler\ case.
For more details, see the original papers by Witten \cite{W}
and Kronheimer and Mrowka \cite{KM}, or any recent survey on the subject
(e.g. \cite{D3,G5}).
Let $(X,g)$ be a compact, oriented, Riemannian four-manifold.
To write the Seiberg--Witten\ equations one needs a $\spin^c$-structure on $X$.
This involves the choice of a Hermitian line bundle $L$ on $X$
satisfying that $c_1 (L)\equiv w_2(X)\;\mathop{{\fam0 mod}}\nolimits\;2$.
A $\spin^c$-structure is then a lift of the fibre product of the
$\mathop{{\fam0 SO}}\nolimits(4)$-bundle of orthonormal
frames of $(X,g)$ with the $\mathop{{\fam0 U}}\nolimits(1)$-bundle defined by $L$ to a
$\spin^c(4)$-bundle, according to the short exact sequence
$$
0\longrightarrow{\bf Z}_2\longrightarrow\spin^c(4)\longrightarrow\mathop{{\fam0 SO}}\nolimits(4)\times \mathop{{\fam0 U}}\nolimits(1)\longrightarrow 1.
$$
Using the two fundamental irreducible 2-dimensional representations
of $\spin^c(4)$---the so-called
Spin representations---we can construct the associated vector
bundles of positive and negative spinors $S_L^\pm$.
These are rank 2 Hermitian vector bundles whose determinant is $L$
\cite{H,LaMi}.
The set of $\spin^c$-structures on $X$
is thus parametrised, up to the finite group $H^1(X,{\bf Z}_2)$, by
$$
\spin^c(X)=\{c\in H^2(X,{\bf Z})\;|\; c\equiv w_2(X)\; \mathop{{\fam0 mod}}\nolimits\;2\}.
$$
Let us fix a $\spin^c$-structure $c\in \spin^c(X)$, and let $L=L_c$
be the corresponding Hermitian line bundle, and $S_L^\pm$ the
corresponding spinor bundles.
The Seiberg--Witten\ {\em monopole equations} are equations for a pair $(A,\Psi)$
consisting of a unitary connection on $L$ and a smooth section of ${S_L^+}$.
Using the connection $A$ one has the Dirac operator
$$
D_A:\,\Gamma({S_L^+})\longrightarrow\Gamma({S_L^-}).
$$
The first condition is that $\Psi$ must be in the kernel of the
Dirac operator.
The curvature $F_A\in\Omega^2=\Omega^+\oplus\Omega^-$,
can be decomposed in the self-dual and anti-selfdual parts
$$
F_A=F_A^+ + F_A^-.
$$
Using the spinor $\Psi$ we can consider another self-dual
2-form that we may couple to $F_A^+$ to obtain our second equation.
Let $\mathop{{\fam0 ad}}\nolimits_0{S_L^+}$ be the subbundle of the adjoint bundle of ${S_L^+}$
consistig of the traceless skew-Hermitian endomorphisms --- its fibres
are hence isomorphic to $\gls\glu(2)$.
We have a map
$$
\Omega^0({S_L^+})\longrightarrow\Omega^0(\mathop{{\fam0 ad}}\nolimits_0{S_L^+})
$$
$$
\Psi\mapsto i(\Psi\otimes\Psi^\ast)_0\ ,
$$
where $\Psi^\ast$ is the adjoint of $\Psi$, and the 0 subindex means that
we are taking the trace-free part.
This map is fibrewise modelled on the map ${\bf C}^2\longrightarrow\gls\glu(2)$,
given by $v\mapsto i(v\overline{v}^t)_0$.
One of the basic ingredients that makes the Seiberg--Witten\ equations possible
is the identification between the space of self-dual 2-forms and the
skew-Hermitian automorphism of the positive spin representation
\cite{AHS}.
This is a basic fact in Clifford algebras in dimension four, that takes
place at each point of the manifold, and that can be carried out over
the whole manifold precisely when one has a $\spin^c$-structure.
More specifically, we have the isomorphism
\begin{equation}
\mathop{{\fam0 ad}}\nolimits_0{S_L^+}\cong\Lambda^+\ . \label{iso}
\end{equation}
We can now interpret
$i(\Psi\otimes\Psi^\ast)_0$ as a section of $\Lambda^+$, i.e. as an
element in $\Omega^+$.
The monopole equations consist in the system of equations
\begin{equation}
\left.\begin{array}{l}
D_A \Psi=0\\
F_A^+=i(\Psi\otimes\Psi^\ast)_0
\end{array}\right \}. \label{me}
\end{equation}
In writing the second equation there is an abuse of notation, since we
are not especifying what the isomorphism (\ref{iso}) is. Notice also that
$F_A^+$ is a purely imaginary self-dual 2-form, and hence we are in fact
identifying $\mathop{{\fam0 ad}}\nolimits_0{S_L^+}$ with $i\Lambda^+$.
We shall analyse now the monopole equations in the case in
which $(X,g)$ is K\"{a}hler.
Recall that a K\"{a}hler\ manifold is Spin if and only if there exists
a square root of the canonical bundle $K^{1/2}$ \cite{A,H}.
Moreover the spinor bundles are
$$
S^+=(\Lambda^0\oplus\Lambda^{0,2})\otimes K^{1/2}=K^{1/2}\oplus K^{-1/2}
$$
$$
S^-=\Lambda^{0,1}\otimes K^{1/2}.
$$
In this situation the spinor bundles for the $\spin^c$-structure $c$ are
given by $S_L^\pm=S^\pm\otimes L^{1/2}$ (notice that
$L^{1/2}$ exists since $c_1(L)\equiv 0\;\mathop{{\fam0 mod}}\nolimits\;2$).
Even if $X$ is not Spin, i.e. even if $K^{1/2}$ and
$L^{1/2}$ do not exist, the bundles $S_L^\pm$ do exist. In other words,
there exists a square root of $K\otimes L$. Let us denote
$$
{\hat{L}}=(K\otimes L)^{1/2}.
$$
Then
$$
{S_L^+}={\hat{L}}\oplus \Lambda^{0,2}\otimes {\hat{L}},\;\;{S_L^-}=\Lambda^{0,1}\otimes{\hat{L}}
$$
and
$$
\Gamma({S_L^+})=\Omega^0({\hat{L}})\oplus\Omega^{0,2}({\hat{L}}).
$$
We can write $\Psi$ according to this decomposition as a pair
$\Psi=(\phi,\beta)$. The Dirac operator can be written in this language
as
$$
\overline{\partial}_{\hat{A}} +\overline{\partial}_{\hat{A}}^\ast\;\;:
\Omega^0({\hat{L}})\oplus\Omega^{0,2}({\hat{L}})\longrightarrow\Omega^{0,1}({\hat{L}}),
$$
where $\overline{\partial}_{\hat{A}}$ is the $\overline{\partial}$ operator on ${\hat{L}}$ corresponding to the
connection ${\hat{A}}$ on ${\hat{L}}$ defined by the connection $A$ on $L$ and the
metric connection on $K$ (cf. \cite{H}).
On the other hand recall that
$$
\Lambda^+\otimes{\bf C}=\Lambda^0\omega\oplus\Lambda^{2,0}
\oplus\Lambda^{0,2},
$$
where $\omega$ is the K\"{a}hler\ form.
According to this decomposition, the isomorphism (\ref{iso})
(or rather $\mathop{{\fam0 ad}}\nolimits_0{S_L^+}\cong i\Lambda^+$)
is explicitely given by
\begin{equation}
i(\Psi\otimes\Psi^\ast)_0\mapsto i(|\phi|^2-|\beta|^2)\omega +
\beta\overline{\phi}-\phi\overline{\beta}.
\end{equation}
We may thus write the monopole equations (\ref{me}) as
\begin{equation}
\begin{array}{l}
\overline{\partial}_{\hat{A}}\phi+\overline{\partial}^\ast_{\hat{A}}\beta=0\\
\Lambda F_A=i(|\phi|^2-|\beta|^2) \\
F_A^{2,0}=-\phi \overline{\beta}\\
F_A^{0,2}=\beta\overline{\phi}
\end{array}\label{kme}
\end{equation}
where $\Lambda F_A$ is the contraction of the curvature with the
K\"{a}hler\ form.
It is not difficult to see (cf. \cite{W}) that the solutions to these
equations are such that either $\beta=0$ or $\phi=0$, and moreover it
is not
possible to have irreducible solutions, i.e. solutions with $\Psi\neq 0$,
of both types simultaneously for a fixed $\spin^c$-structure.
We thus have one of the following two situations:
(i)\ $\beta=0$ and the equations reduce to
\begin{equation}
\begin{array}{l}
F_A^{0,2}=0\\
\overline{\partial}_{\hat{A}}\phi=0\\
\Lambda F_A=i|\phi|^2
\end{array}\label{ve1}
\end{equation}
(ii)\ $\phi=0$ and then
\begin{equation}
\begin{array}{l}
F_A^{0,2}=0\\
\overline{\partial}_{\hat{A}}^\ast\beta=0\\
\Lambda F_A=-i|\beta|^2\ .
\end{array} \label{ve2}
\end{equation}
\noindent{\em Remark}.
We have omitted the equation $F_A^{2,0}=0$, since by unitarity of the
connection this is equivalent to $F_A^{0,2}=0$.
Clearly if we have solutions to (\ref{ve1}), from the third equation
in (\ref{ve1}) we
obtain that $\deg L\leq0$, while from (\ref{ve2}) we have $\deg L\geq0$,
where $\deg L$ is the degree of $L$ with respect to the K\"{a}hler\ metric
defined as in (\ref{degree}).
Since we are interested only in irreducible solutions,
obviously these two situations cannot occur simultaneously.
The Hodge star operator interchanges these two
cases, and we can thus concentrate on case (i).
Equations (\ref{ve1}) are essentially the equations known as the
{\em vortex equations}. These are generalisations of the
vortex equations on the Euclidean plane studied by Jaffe and Taubes
\cite{T1,T2,JT}, and have been extensively studied (e.g. in
\cite{B1,B2,G2,G3} ) for compact K\"{a}hler\
manifolds of arbitrary dimension. In that setting, the equations are the
following:
Let $(X,\omega)$ be a compact K\"{a}hler\ manifold
of arbitrary dimension, and
let $(L,h)$ be a Hermitian $C^\infty$ line bundle over $X$.
Let $\tau\in{\bf R}$.
The $\tau$-{\em vortex equations}
\begin{equation}
\left. \begin{array}{l}
F_A^{0,2}=0\\
\overline{\partial}_A\phi=0 \\
\Lambda F_A =\frac{i}{2}(|\phi|^2 -\tau)
\end{array}\right \}, \label{ve}
\end{equation}
are equations for a pair $(A,\phi)$
consisting of a connection on $(L,h)$ and a smooth section of $L$.
The first equation means that $A$ defines a holomorphic
structure on $L$, while the second says that $\phi$ must be holomorphic
with respect to this holomorphic structure.
Coming back to the monopole equations,
we first observe that (\ref{ve1}) can be rewritten in terms of
${\hat{A}}$ only, i.e. not
involving simultaneously $A$ and ${\hat{A}}$. To do this we recall that
${\hat{A}}=(A\otimes a_K)^{1/2}$, where $a_K$ is the metric connection on $K$.
We thus have
\begin{equation}
F_{{\hat{A}}}=\frac{1}{2}(F_A + F_{a_K}), \label{curvature}
\end{equation}
and hence $F_A^{0,2}=0$ is equivalent to $F_{{\hat{A}}}^{0,2}=0$ since
$F_{a_K}^{0,2}=0$.
From (\ref{curvature}), and using that $s=-i\Lambda F_{a_K}$ is the scalar
curvature, we obtain that
$$
\Lambda F_A=2\Lambda F_{\hat{A}}-is,
$$
and hence (\ref{ve1}) is equivalent to
\begin{equation}
\left. \begin{array}{l}
F_{\hat{A}}^{0,2}=0\\
\overline{\partial}_{\hat{A}}\phi=0\\
\Lambda F_{\hat{A}}=\frac{i}{2}(|\phi|^2+ s)
\end{array}
\right\}.\label{ve1'}
\end{equation}
These are the vortex equations on ${\hat{L}}$, but with the parameter
$\tau$ replaced by minus the scalar curvature. As we will
explain in Section \ref{t-vortices}
the existence proofs for the vortex equations can be
easily modified to give an existence
theorem for the equations obtained by replacing the parameter $\tau$ by
a function $t\in C^\infty (X,{\bf R})$ in (\ref{ve}).
However, to compute the Seiberg--Witten\ invariants one can slightly perturb
equations (\ref{me}) in such a way that, when $\beta=0$, equations
(\ref{me}) reduce to the constant function vortex equations, i.e. to
(\ref{ve}) (see e.g. \cite{G5}).
\section{Non-abelian vortex equations}\label{vortices}
As we have seen in the previous section, the Seiberg--Witten\ monopole
equations can be considered as a four-dimensional Riemannian
generalisation of the vortex equations.
This suggests that we may find interesting Seiberg--Witten-type equations
by considering the corresponding analogues of
different equations of vortex-type existing in the literature.
With this objective in mind, in this section we shall
review three different non-abelian generalisations of the vortex
equations described above.
The first one consists in studying the vortex equations on
a Hermitian vector bundle of arbitrary rank. The other two
involve two vector bundles, one of which will actually be a line
bundle in most cases.
Let $(E,H)$ be a Hermitian vector bundle over a compact
K\"{a}hler\ manifold $(X,\omega)$ of complex dimension $n$.
Let $\tau\in{\bf R}$.
The $\tau$-vortex equations
were generalised to this situation in \cite{B2}. As in the line
bundle case, one studies equations
\begin{equation}
\left. \begin{array}{l}
F_A^{0,2}=0\\
\overline{\partial}_A\phi=0 \\
\Lambda F_A =i(\phi\otimes\phi^\ast -\tau{\bf I})
\end{array}\right \} \label{nave}
\end{equation}
for a pair $(A,\phi)$ consisting of a unitary connection on $(E,H)$
and a smooth section of $E$.
By $\phi^\ast$ we denote the adjoint of $\phi$ with respect to $H$, and
${\bf I}\in\mathop{{\fam0 End}}\nolimits E$ is the identity.
Notice that in the abelian equations (\ref{ve}) there is a $1/2$ in the
third equation, while in (\ref{nave}) there is not. This is not essential
since by applying a constant complex gauge transformation we can introduce
an arbitrary positive constant in front of $i\phi\otimes\phi^\ast $.
These vortex equations appear naturally as the equations satisfied by the
minima of the Yang--Mills--Higgs\ functional. This is a functional
defined on the product of the space ${\cal A}$ of unitary connections
on $(E,H)$ and the space of smooth sections $\Omega^0(E)$ by
$$
\mathop{{\fam0 YMH}}\nolimits_\tau(A,\phi)=\|F_A\|^2 +2\|d_A\phi\|^2+
\|\phi\otimes\phi^\ast-\tau{\bf I}\|^2,
$$
where $\|\;\;\|$ denotes the $L^2$-metric.
This is easily seen by rewriting the Yang--Mills--Higgs\ functional --- using
the K\"{a}hler\ identities --- as
\begin{eqnarray}
\mathop{{\fam0 YMH}}\nolimits_\tau(A,\phi) &=& 4\|F_A^{0,2}\|^2+4\|\overline{\partial}_A\phi\|^2+
\|i\Lambda F_A +\phi\otimes\phi^\ast -\tau{\bf I}\|^2 \nonumber \\
& &+4\pi\tau\deg E-
\frac{8\pi^2}{(n-2)!} \int_X\mathop{{\fam0 ch}}\nolimits_2(E)\wedge\omega^{n-2},\nonumber
\end{eqnarray}
where $\deg E$ is the degree of $E$ defined as
\begin{equation}
\deg E=\int_X c_1(E)\wedge \omega^{n-1}\ , \label{degree}
\end{equation}
and $\mathop{{\fam0 ch}}\nolimits_2(E)$ is the second Chern character of $E$, which is represented
in terms of the curvature by
$$
\mathop{{\fam0 ch}}\nolimits_2(E)=-\frac{1}{8\pi^2} \mathop{{\fam0 Tr}}\nolimits(F_A\wedge F_A).
$$
Clearly $\mathop{{\fam0 YMH}}\nolimits_\tau$ achieves its minimum value
$$
4\pi\tau\deg E-
\frac{8\pi^2}{(n-2)!} \int_X\mathop{{\fam0 ch}}\nolimits_2(E)\wedge\omega^{n-2}
$$
if and only if $(A,\phi)$ is a solution to equations (\ref{nave})
(see \cite{B1} for details).
As we will explain in Section \ref{t-vortices}, the vortex equations
also have a symplectic interpretation as moment map equations.
The moment map in question is for a symplectic action of ${\cal G}(E)$, i.e.
the unitary gauge group of $E$, on a certain infinite dimensional
symplectic space.
A natural generalization of these equations is obtained if we regard the
section $\phi$ in (\ref{nave}) as a morphism from
the trivial line bundle to $E$. One can
replace the trivial line bundle by a vector bundle of arbitrary
rank and study equations for connections on both bundles and
a morphism from one to the other.
These are the {\em coupled vortex equations} introduced in \cite{G4}.
Let $(E,H)$ and $(F,K)$ be smooth Hermitian vector bundles over $X$.
Let $A$ and $B$ be unitary connections on $(E,H)$ and $(F,K)$
resp., and let $\phi\in \Omega^0(\mathop{{\fam0 Hom}}\nolimits(F,E))$.
Let $\tau$ and $\tau'$ be real parameters.
The coupled vortex equations are
\begin{equation}
\left. \begin{array}{l}
F_A^{0,2}=0\\
F_B^{0,2}=0\\
\overline{\partial}_{A, B}\phi=0\\
i \Lambda F_A+\phi\phi^\ast=\tau {\bf I}_E\\
i \Lambda F_B-\phi^\ast\phi=\tau'{\bf I}_F
\end{array}\right \}.\label{gcve}
\end{equation}
As in the case of the vortex equations described above, equations
(\ref{gcve})
correspond to the minima of a certain Yang--Mills--Higgs\ functional and are also
moment map equations (cf. \cite{G4}).
The appropriate functional in this case is defined as
$$
\mathop{{\fam0 YMH}}\nolimits_{\tau,\tau'}(A,B,\phi)=\|F_A\|^2 +\|F_B\|^2 +
2\|d_{A\otimes B}\phi\|^2+
\|\phi\phi^\ast-\tau{\bf I}_E\|^2+\|\phi^\ast\phi-\tau'{\bf I}_F\|^2.
$$
The moment map is now for a symplectic action of ${\cal G}(E)\times{\cal G}(F)$, i.e.
for
the product of the unitary groups of $E$ and $F$.
In this paper we will be mostly interested in the case in which $F=L$ is
a line bundle. In this situation the equations can be written as
\begin{equation}
\left. \begin{array}{l}
F_A^{0,2}=0\\
F_B^{0,2}=0\\
\overline{\partial}_{A, B}\phi=0\\
i \Lambda F_A+\phi\otimes\phi^\ast=\tau {\bf I}_E\\
i \Lambda F_B-|\phi|^2=\tau'
\end{array}\right \}.\label{cve}
\end{equation}
It is clear, from taking the trace of the last two equations
in (\ref{cve}) and integrating, that to solve (\ref{cve})
$\tau$ and $\tau'$ must be related by
\begin{equation}
\tau\mathop{{\fam0 rank}}\nolimits E +\tau'=\deg E +\deg L,\label{parameters}
\end{equation}
hence there is only one free parameter.
We shall consider next the {\em framed vortex equations}. The situation
is very similar to the previous one in that it also involves two
vector bundles. In this case, however, the connection on one of the
bundles is fixed. More specifically, let $(E,H)$ and $(F,K)$ be
two Hermitian vector bundles. Let $B$ a fixed Hermitian connection
on $(F,K)$ such that $F_B^{0,2}=0$.
The equations are now for a unitary connection $A$ on $(E,H)$, and
$\phi\in \Omega^0(\mathop{{\fam0 Hom}}\nolimits(F,E))$. As explained in \cite{BDGW}, the
appropriate equations are
\begin{equation}
\left. \begin{array}{l}
F_A^{0,2}=0\\
\overline{\partial}_{A, B}\phi=0\\
i \Lambda F_A+\phi\phi^\ast=\tau {\bf I}_E\\
\end{array}\right \}\label{fve}
\end{equation}
The relation between these equations and the full coupled vortex equations is
perhaps best understood from the symplectic point of view. One sees that the
effect of fixing the data on $F$ is to reduce the symmetry group in the
problem
from ${\cal G}(E)\times{\cal G}(F)$\ to ${\cal G}(E)$. The new equations must thus correspond
to the moment map for the subgroup
${\cal G}(E)\subset{\cal G}(E)\times{\cal G}(F)$. But the moment maps for ${\cal G}(E)$ and for
${\cal G}(E)\times{\cal G}(F)$ are related by a projection from the Lie algebra of
${\cal G}(E)\times{\cal G}(F)$ onto the summand corresponding to the Lie algebra of
${\cal G}(E)$. The effect of this projection is to eliminate the last equations in
(\ref{cve}) (cf.
\cite{BDGW} for more details).
The appropriate moduli space problem corresponds to that of studying
morphisms from a vector bundle with a fixed holomorphic structure
to another vector bundle in which the holomorphic structure is varying.
Such moduli spaces have been studied by Huybrecht and Lehn
\cite{HL1,HL2}, who refer to these objects as {\em framed modules}.
As in the coupled vortex equations, we will be mostly interested in the
case in which $F=L$ is a line bundle.
All the vortex-type equations that we have considered so far involve
one or two real parameters $\tau$ and $\tau'$. As we have seen in
Section \ref{sw}, the study of the Seiberg--Witten\ monopole equations leads to
abelian vortex
equations in which $\tau$ is replaced by a function. The same will
happen in the generalizations of the monopole equations that we are
about to discuss. This will be analysed in detail in Section
\ref{t-vortices}.
\section{Non-abelian monopole equations}\label{monopoles}
Let us go back to the set-up of Section \ref{sw} and let $(X,g)$ be a
compact, oriented, Riemannian, four-dimensional manifold.
Let $c\in \spin^c(X)$ be a fixed $\spin^c$-structure, with corresponding
Hermitian line bundle $L$ and bundles of spinors ${S_L^\pm}$.
Let $(E,H)$ be a Hermitian vector bundle on $X$.
Let $\Psi\in \Gamma (S^+_L\otimes E)$. Using the metrics on $S_L^+$ and
$E$ one has the antilinear identification
$$
S^+_L\otimes E\longrightarrow S^{+*}_L\otimes E^*
$$
$$
\Psi \mapsto \Psi^*.
$$
and hence
$$
\Psi \otimes \Psi^*\in \mathop{{\fam0 End}}\nolimits (S^+_L\otimes E).
$$
We shall also need the map
$$
\mathop{{\fam0 End}}\nolimits (S^+_L\otimes E)\longrightarrow \mathop{{\fam0 End}}\nolimits_0(S^+_L)\otimes \mathop{{\fam0 End}}\nolimits E
$$
$$
\Psi\otimes \Psi^*\mapsto (\Psi\otimes \Psi^*)_0\; ,
$$
as well as the map
$$
\mathop{{\fam0 End}}\nolimits_0(S^+_L)\otimes \mathop{{\fam0 End}}\nolimits E\stackrel{\mathop{{\fam0 Tr}}\nolimits}{\longrightarrow}\mathop{{\fam0 End}}\nolimits_0 (S^+_L)
$$
$$
(\Psi\otimes \Psi^*)_0\mapsto \mathop{{\fam0 Tr}}\nolimits(\Psi\otimes \Psi^*)_0
$$
obtained from the trace map
$$
\mathop{{\fam0 End}}\nolimits_0\,E\longrightarrow \mathop{{\fam0 End}}\nolimits E \stackrel{\mathop{{\fam0 Tr}}\nolimits}{\longrightarrow}{\bf C}\longrightarrow 0.
$$
The endomorphism $(\Psi\otimes \Psi^\ast)_0$ should not be confused
with the completely traceless part of
$\Psi\otimes \Psi^\ast$ --- here we are only
removing the trace corresponding to $S_L^+$.
In this paper we shall consider essentially two different non-abelian
generalizations of the Seiberg--Witten\ equations. While in the first one we will fix
a connection $b$ on $L$
and study equations for a pair $(A,\Psi)$, where $A$
is a unitary connection on $(E,H)$ and $\Psi\in \Gamma(S_L^+\otimes E)$,
in the second one we will allow $b$ to vary as well,
and hence our system of equations will be one for triples $(A,b,\Psi)$.
The first non-abelian version of the Seiberg--Witten\ equations is suggested by the
framed vortex equations:
Let $b$ be a {\em fixed} unitary connection on $L$ and $A$ be a unitary
connection on $E$. Using these two connections and
the Levi--Civita connection one can consider the coupled Dirac operator
$$
D_{A,b}:\; \Gamma({S_L^+}\otimes E)\longrightarrow \Gamma({S_L^-}\otimes E).
$$
and study the equations
\begin{equation}
\left.\begin{array}{l}
D_{A,b}\Psi=0\\
F^+_A=i(\Psi\otimes \Psi^*)_0
\end{array}
\right \}\label{name}
\end{equation}
for the unknowns $A$ and $\Psi\in\Gamma({S_L^+}\otimes E)$.
The Bochner--Weitzenb\"ock formula for $D_{A,b}$ is given by
\begin{equation}
D_{A,b}^\ast D_{A,b}=
\nabla_{A,b}^\ast\nabla_{A,b}+\frac{s}{4} +c(F_{A,b}), \
\label{bw}
\end{equation}
where
$$
F_{A,b}=F_A +\frac{1}{2} F_b\otimes I_E
$$
and $\nabla_{A,b}$ is the connection on $E\otimes S_L$ determined
by the connections $A$ and $b$.
The term $c(F_{A,b})$ in (\ref{bw}) means Clifford multiplication by
$F_{A,b}$. In fact the action of $F_{A,b}$ on
$\Psi\in \Gamma({S_L^+}\otimes E)$ coincides
with the Clifford multiplication with $F_{A,b}^+$
(see \cite{LaMi} for example).
It is thus natural to perturb the second equation in (\ref{name})
by adding the constant self-dual 2-form $\frac{1}{2} F_b^+$,
and consider instead equations
\begin{equation}
\left.\begin{array}{l}
D_{A,b}\Psi=0\\
F^+_{A, b}=i(\Psi\otimes \Psi^*)_0
\end{array}
\right \}. \label{name'}
\end{equation}
These are the equations studied in \cite{OT1}.
The abelian Seiberg--Witten\ equations (\ref{me}) for a $\spin^c$-structure
with line bundle $\tilde L$ can be recovered from (\ref{name})
or (\ref{name'}) by considering the Hermitian bundle
$E=(\tilde L \otimes L^\ast)^{1/2}$.
Notice that this square root exists since
$c_1(\tilde L)\equiv c_1(L)\;\mathop{{\fam0 mod}}\nolimits\; 2$.
Other non-abelian versions of the Seiberg--Witten\ equations that have been considered
include replacing the $\mathop{{\fam0 U}}\nolimits(r)$-bundle $(E,H)$ by an ${SU}(r)$-bundle,
and study equations (\ref{name}) in which
$(\Psi\otimes\Psi^\ast)_0$ is replaced by the completely trace-free part
of $\Psi\otimes\Psi^\ast$. These are the equations studied
in \cite{LM} (see also \cite{OT2})
In other versions, like the one considered in \cite{PT}, one fixes the
connection of $\det E\otimes L$ instead of fixing that of $L$.
In all the versions mentioned above one considers spinors coupled to a
bundle associated the fundamental representation of $\mathop{{\fam0 U}}\nolimits(r)$ or ${SU}(r)$.
Another direction in which the monopole equations can be generalized is
by considering any compact Lie group $G$ and/or an arbitrary
representation. When the manifold is K\"ahler some of these correspond
to the vortex-type equations described in \cite{G1}. These more general
equations will be dealt with somewhere else.
We shall consider next the case in which the connection on $L$ is
also varying.
It is clear that we cannot consider the same equations as in the previous
situation,
since we would not obtain an elliptic complex in the linearization
of the equations --- we need an extra equation.
As the coupled vortex equations (\ref{cve}) suggest, it is natural to
consider the following set of equations for the triple $(A,b,\Psi)$:
\begin{equation}
\left.
\begin{array}{l}
D_{A,b} \Psi =0\\
F^+_A=i(\Psi \otimes\Psi^*)_0\\
F^+_b=2i\mathop{{\fam0 Tr}}\nolimits(\Psi\otimes \Psi^*)_0
\end{array}
\right\}.\label{cme}
\end{equation}
\section{The K\"ahler case}\label{kaehler}
Let now $(X,\omega)$ be a compact K\"{a}hler\ surface.
Let us fix a $\spin^c$-structure $c\in \spin^c(X)$, and let $L=L_c$ be the
corresponding Hermitian line bundle.
As mentioned in Section \ref{sw}, the corresponding spinor bundles for the
$\spin^c$-structure $c$ are given by
$$
{S_L^+}={\hat{L}}\oplus \Lambda^{0,2}\otimes {\hat{L}}\;\;\;\mbox{and}\;\;\;
{S_L^-}=\Lambda^{0,1}\otimes{\hat{L}},
$$
where
$$
{\hat{L}}=(K\otimes L)^{1/2}.
$$
Let $(E,H)$ be a Hermitian vector bundle over $X$.
\begin{equation}
E\otimes{S_L^+}=E\otimes{\hat{L}}\oplus \Lambda^{0,2}\otimes
E\otimes{\hat{L}}\label{E-spinors}
\end{equation}
and hence
$$
\Gamma(E\otimes{S_L^+})=\Omega^0(E\otimes{\hat{L}})\oplus\Omega^{0,2}(E\otimes{\hat{L}}).
$$
We can write $\Psi$ according to this decomposition as a pair
$\Psi=(\phi,\beta)$.
Let $b$ and $A$ be unitary connections on $L$ and $E$, respectively.
Let $a_K$ be the metric connection on $K$. We shall denote
by ${\hat{b}}$ the connection on ${\hat{L}}$ defined by $b$ and $a_K$.
The Dirac operator $D_{A,b}$ can be written in this language as
$$
\overline{\partial}_{A,{\hat{b}}} +\overline{\partial}_{A,{\hat{b}}}^\ast\;:
\Omega^0(E\otimes{\hat{L}})\oplus\Omega^{0,2}(E\otimes{\hat{L}})\longrightarrow
\Omega^{0,1}(E\otimes{\hat{L}}),
$$
where $\overline{\partial}_{A,{\hat{b}}}$ is the $\overline{\partial}$ operator on $E\otimes{\hat{L}}$
corresponding to the connections $A$ and ${\hat{b}}$.
\subsection{Fixed connection on $L$}
We shall perturb equations (\ref{name'}) by a self-dual 2-form $\alpha$
and consider
\begin{equation}
\left.\begin{array}{l}
D_{A,b}\Psi=0\\
F^+_{A, b}=i((\Psi\otimes \Psi^*)_0+\alpha {\bf I})
\end{array}
\right \}. \label{pname}
\end{equation}
We will choose the perturbation to be of K\"{a}hler\ type, that is
$\alpha= -f\omega$, where $f$ is a smooth real function.
Similarly to the abelian case, we can write (\ref{pname}) as
\begin{equation}
\begin{array}{l}
\overline{\partial}_{A,{\hat{b}}}\phi+\overline{\partial}^\ast_{A,{\hat{b}}}\beta=0\\
\Lambda F_{A,b}=i(\phi\otimes \phi^\ast- \Lambda^2
\beta\otimes\beta^\ast-f{\bf I}) \\
F_{A,b}^{2,0}=-\phi\otimes\beta^\ast \\
F_{A,b}^{0,2}=\beta\otimes\phi^\ast\ .
\end{array}\label{kname}
\end{equation}
By $\Lambda^2$ we denote the operation of contracting twice with the
K\"{a}hler\ form.
As in the abelian case, one can see that the solutions to these equations
are such that either $\beta=0$ or $\phi=0$. More precisely
\begin{prop}\label{decoupling}
Let $\overline{f}=\frac{1}{2\pi}\int_X f$.
The only solutions to (\ref{kname}) satisfy either
\noindent (i)\ $\beta=0$,
\begin{equation}
\begin{array}{l}
F_{A,b}^{0,2}=0\\
\overline{\partial}_{A,{\hat{b}}}\phi=0\\
\Lambda F_{A,b}=i(\phi\otimes\phi^\ast-f{\bf I})\ ,
\end{array}\label{nave1}
\end{equation}
then $\mu(E)-1/2\deg L\leq\overline{f}$,
or
\noindent (ii)\ $\phi=0$,
\begin{equation}
\begin{array}{l}
F_{A,b}^{0,2}=0\\
\overline{\partial}_{A,{\hat{b}}}^\ast\beta=0\\
\Lambda F_{A,b}=-i(\Lambda^2\beta\otimes\beta^\ast+f{\bf I})
\end{array}\label{nave2}
\end{equation}
and then $\mu(E)-1/2\deg L\geq\overline{f}$.
\end{prop}
{\em Proof}.
One uses exactly the same method as the one used by Witten \cite{W} in
the abelian case.
Consider the transformation $(A,\phi,\beta)\mapsto (A,-\phi,\beta)$.
Although this is not a symmetry of equations (\ref{kname}), if
$(A,\phi,\beta)$ is a solution $(A,\phi,-\beta)$ must also be.
This is easily seen by considering the functional
$$
\mathop{{\fam0 SW}}\nolimits(A,\Psi)=\|F^+_{A, b}-i((\Psi\otimes \Psi^*)_0+\alpha {\bf I})\|^2
+2\|D_{A,b}\Psi\|^2.
$$
Using the Bochner--Weitzenb\"ock formula (\ref{bw}) and the fact that
on a K\"{a}hler\ manifold the decomposition (\ref{E-spinors}) is parallel
with respect to the connection $\nabla_{A,b}$, we have
\begin{eqnarray}
\mathop{{\fam0 SW}}\nolimits(A,\Psi)& = &\|F_{A,b}^+\|^2+
2\|\nabla_{A,b}\phi\|^2+2\|\nabla_{A,b}\beta\|^2+
\|i((\Psi\otimes \Psi^*)_0+\alpha {\bf I})\|^2 \nonumber\\
& & + \int_X\frac{s}{2}(|\phi|^2 +|\beta|^2)-
2\int_X\langle F_{A,b},i\alpha{\bf I}\rangle.\nonumber
\end{eqnarray}
\noindent{\em Remark}. Notice the analogy between this and the way of rewriting the
Yang--Mills--Higgs\ functional in Section \ref{vortices} using the K\"{a}hler\ identities.
In fact in the K\"{a}hler\ case both things are essentially equivalent.
\hfill$\Box$
Of course the only way in which the two type of solutions
can occur simultaneously is if $\mu(E)=1/2\deg L +\overline{f}$, then
$\Psi=0$ and the equations reduce essentially to the Hermitian--Einstein\ equations.
Since the Hodge operator interchanges the roles of $\phi$ and $\beta$ we
may concentrate in the case $\phi\neq 0$.
We shall write equations
(\ref{nave1}) in a way that we can identify them as the vortex equations
discussed in Section \ref{vortices}. To do this denote by
$$
{\hat{E}}=E\otimes {\hat{L}}\;\;\;\mbox{and}\;\;\; {\hat{A}}=A\otimes {\hat{b}},
$$
where recall that ${\hat{b}}=b\otimes a_K$. We have that
$$
F_{\hat{A}}=F_{A,b}+\frac{1}{2} F_{a_K},
$$
and hence (\ref{nave1}) is equivalent to
\begin{equation}
\begin{array}{l}
F_{\hat{A}}^{0,2}=0\\
\overline{\partial}_{\hat{A}}\phi=0\\
\Lambda F_{\hat{A}}=i(\phi\otimes\phi^\ast+(s/2-f) {\bf I})
\end{array}
\label{nave1'}
\end{equation}
where $s=-i\Lambda F_{a_K}$ is the scalar curvature of $(X,\omega)$, and
we have used that $a_K$ is integrable, i.e. $F_{a_K}^{0,2}=0$.
These are indeed the vortex equations (\ref{nave}) on the bundle ${\hat{E}}$,
with the parameter $\tau$ replaced by the function $t=f-s/2$.
Equations (\ref{nave1}) can also be interpreted as the framed vortex
equations (\ref{fve}). To see this we choose the fixed connection $b$
to be integrable. We then have that $F_{A,b}^{0,2}=F_A^{0,2}$ and
(\ref{nave1}) is equivalent to
\begin{equation}
\begin{array}{l}
F_A^{0,2}=0\\
\overline{\partial}_{A,{\hat{b}}^\ast}\phi=0\\
\Lambda F_A=i(\phi\otimes\phi^\ast- t {\bf I})
\end{array}
\label{nave1''}
\end{equation}
where $t=f+\frac{i}{2}\Lambda F_b$. We thus obtain the framed vortex
equations (\ref{fve}) on $(E,{\hat{L}}^\ast)$ with fixed connection ${\hat{b}}^\ast$
on ${\hat{L}}^\ast$, and parameter $\tau$ replaced by the function $t$.
These two slightly different points of view
in relating (\ref{nave1}) to the vortex equations reflect the close
relation between the usual vortex equations and the framed
vortex equations, as we will explain in Section \ref{t-vortices}.
\subsection{Variable connection on $L$}
We come now to equations (\ref{cme}). In the K\"{a}hler\ situation
these equations can be written as
\begin{equation}
\begin{array}{l}
\overline{\partial}_{A,{\hat{b}}}\phi+\overline{\partial}^\ast_{A,{\hat{b}}}\beta=0\\
\Lambda F_A=i(\phi\otimes \phi^\ast- \Lambda^2
\beta\otimes\beta^\ast) \\
\Lambda F_b=2i(|\phi|^2-|\beta|^2) \\
F_A^{2,0}=-\phi\otimes\beta^\ast\\
F_A^{0,2}=\beta\otimes\phi^\ast\\
F_b^{2,0}=-2\mathop{{\fam0 Tr}}\nolimits(\phi\otimes\beta^\ast)\\
F_b^{0,2}=2\mathop{{\fam0 Tr}}\nolimits(\beta\otimes\phi^\ast) \ . \label{kcme}
\end{array}
\end{equation}
By taking the third equation, subtracting twice the trace of the
second equation in (\ref{kcme}), and integrating we obtain that,
in order to have solutions, we need
$$
\deg E = \frac{1}{2}\deg L.
$$
To avoid this restriction we can perturb, as in the previous case,
the coupled monopole
equations by adding fixed self-dual forms $\alpha, \gamma$, i.e. by
considering
\begin{equation}
\left.
\begin{array}{l}
D_{A,b} \Psi =0\\
F^+_A=i(\Psi \otimes\Psi^*)_0+i\alpha {\bf I}_E\\
F^+_b=2i\mathop{{\fam0 Tr}}\nolimits(\Psi\otimes \Psi^*)_0 +i\gamma \label{pcve}
\end{array}
\right\}.
\end{equation}
In the K\"{a}hler\ case we shall choose
$$
\alpha=-f\omega \;\;\;\;\;\;\;\gamma=2f'\omega,
$$
where $f$ and $f'$ are smooth real functions on $X$.
With this choice of perturbation the second and
third equations in (\ref{kcme}) become respectively
$$
i \Lambda F_A + \phi \otimes \phi^\ast -
\Lambda^2 \beta \otimes\beta^* = f {\bf I}_E
$$
$$
i \Lambda F_b + 2(|\phi|^2 - |\beta|^2) =- 2f'.
$$
A necessary condition for existence of solutions is now
\begin{equation}
\deg E - \frac{1}{2}\deg L = \int_X (rf + f'), \label{ff'}
\end{equation}
where $r = \mathop{{\fam0 rank}}\nolimits E$.
We shall study the more general monopole equations (\ref{pcve}).
\begin{prop}
Let $f,f'\in C^\infty(X,{\bf R})$ be related by (\ref{ff'}) and
denote
$$
\overline{f}=\frac{1}{2\pi}\int_X f \;\;\;\;\mbox{and}\;\;\;\;
\overline{f'}=\frac{1}{2\pi}\int_X f'.
$$
The only solutions to (\ref{kcme}) are such that either
\noindent (i)\ $\beta=0$,
\begin{equation}
\begin{array}{l}
F_A^{0,2}=0\\
F_b^{0,2}=0\\
\overline{\partial}_{A,{\hat{b}}}\phi=0\\
i\Lambda F_A+\phi\otimes\phi^\ast=f{\bf I}_E\\
i\Lambda F_b+2|\phi|^2=-2f'\ ,
\end{array}
\label{cve1}
\end{equation}
then $\mu(E)\leq \overline{f}$, $\deg L\geq 2\overline{f'}$
or
\noindent (ii)\ $\phi=0$,
\begin{equation}
\begin{array}{l}
F_A^{0,2}=0\\
F_b^{0,2}=0\\
\overline{\partial}_{A,{\hat{b}}}^\ast\beta=0\\
i\Lambda F_A-\Lambda^2\beta\otimes\beta^\ast=f {\bf I}_E\\
i\Lambda F_b-2|\beta|^2=-2f'
\end{array}\label{cve2}
\end{equation}
and then $\mu(E)\geq \overline{f}$, $\deg L\leq 2\overline{f'}$.
\end{prop}
{\em Proof}.
This is proved similarly to Proposition \ref{decoupling}, by considering
the functional
$$
\mathop{{\fam0 SW}}\nolimits(A,b,\Psi)=
\| F^+_A-i(\Psi \otimes\Psi^*)_0-i\alpha {\bf I}_E\|^2+
2\|F^+_b-2i\mathop{{\fam0 Tr}}\nolimits(\Psi\otimes \Psi^*)_0 -i\gamma\|^2+
2\| D_{A,b} \Psi\|^2,
$$
and observing that
$$
\langle F_A^+,i(\Psi \otimes\Psi^*)_0\rangle +\frac{1}{2}
\langle F_b^+,2i\mathop{{\fam0 Tr}}\nolimits(\Psi \otimes\Psi^*)_0\rangle =
\langle F_{A,b}^+,i(\Psi \otimes\Psi^*)_0\rangle.
$$
\hfill$\Box$
Again, we can focus on case (i), since by means of the Hodge operator
we can interchange the roles of $\phi$ and $\beta$.
The system of equations (\ref{cve1}) is equivalent to
\begin{equation}
\begin{array}{l}
F_A^{0,2}=0\\
F_{{\hat{b}}^\ast}^{0,2}=0\\
\overline{\partial}_{A,{\hat{b}}^\ast}\phi=0\\
i\Lambda F_A+\phi\otimes\phi^\ast=f{\bf I}_E\\
i\Lambda F_{{\hat{b}}^\ast}-|\phi|^2=(f'-\frac{s}{2})
\end{array}
\label{cve1'}
\end{equation}
where ${\hat{b}} ^\ast$ is the dual connection to ${\hat{b}}$ on ${\hat{L}}^\ast$, which
satisfies that $F_{{\hat{b}}^\ast}=-F_{\hat{b}}$.
We can now identify (\ref{cve1'}) as the coupled vortex equations
(\ref{cve}) on $(E,{\hat{L}}^\ast)$, with the parameters $\tau$ and $\tau'$
replaced by functions.
The existence of solutions to these equations as well as the description
of the moduli space of all solutions will be dealt with in the next
section.
\section{Back to the vortex equations}\label{t-vortices}
We study now the existence of solutions to the monopole
equations in the K\"{a}hler\ case --- equations (\ref{nave1'}),
(\ref{nave1''}) and
(\ref{cve1'}) --- or equivalently the vortex equations in which
the parameters have been replaced by functions.
\subsection{The $t$-vortex equations}
In this section we will examine what happens if the parameter
$\tau$, which appears on the $\tau$-vortex equations (\ref{nave})
is permitted to be a non-constant smooth function, say $t$.
We will refer to the resulting equations as the $t$-vortex equations.
The main result, namely that existence of solutions is
governed entirely by the average value of $t$, has already
been observed by Okonek and Teleman. Here we give a somewhat
different proof than that in \cite{OT1}. We also discuss some
interpretations and implications of the result.
Let $E\rightarrow X$\ be a rank $r$, smooth complex bundle over a
closed K\"{a}hler\ manifold $(X,\omega)$.
It is convenient to look at equations (\ref{nave}) from a
different although equivalent point of view.
Instead of fixing a Hermitian metric $H$ and solving
for $(A,\phi)$ satisfying (\ref{nave}) we fix a
$\overline{\partial}$-operator on $E$, $\overline{\partial}_E$ say, and a section
$\phi\in H^0(X,{\cal E})$, where
${\cal E}$ is the holomorphic bundle determined by $\overline{\partial}_E$,
and solve for a metric $H$ satisfying
\begin{equation}
i\Lambda F_H +\phi\otimes\phi^{*_H}=\tau{\bf I},\label{mve}
\end{equation}
where $F_H$\ is the curvature of the metric connection
determined by $\overline{\partial}_E$\ and $H$. It will be important to explicitly
write $\phi^{*_H}$\ to denote the adjoint of $\phi$ with respect to the
metric $H$.
Equation (\ref{mve}) can be regarded as the defining condition
for a special metric on the holomorphic pair $({\cal E},\phi)$.
In \cite{B2} the first author showed that there is a Hitchin--Kobayashi\ correspondence
between the existence of such metrics and a stability condition for
holomorphic pairs. The appropriate notion of stability is as follows.
\begin{definition} Define the degree of any coherent sheaf
${\cal E}'\subset{\cal E}$ to be
$$
\deg{\cal E}'=\int_X c_1({\cal E}')\wedge \omega^{n-1},
$$
and define the slope of ${\cal E}'$ by
$$
\mu({\cal E}')=\frac{\deg{\cal E}'}{\mathop{{\fam0 rank}}\nolimits {\cal E}'}.
$$
Fix $\tau\in{\bf R}$. The holomorphic pair $({\cal E},\phi)$ is called $\tau$-stable\
if
(1)\ $\mu({\cal E}')<\tau\;\;
\mbox{for every coherent subsheaf}\;\;\;{\cal E}'\subset{\cal E}$, and
(2)\ $\mu({\cal E}/{\cal E}'')>\tau\;\;
\mbox{for every non-trivial coherent subsheaf} \;{\cal E}''\; \mbox{such that}
\; \phi\in H^0(X,{\cal E}'')$.
\end{definition}
The Hitchin--Kobayashi\ correspondence is expressed by the following two propositions.
\begin{prop}[\cite{B2}]\label{easy}
Fix $\tau\in {\bf R}$, let $({\cal E},\phi)$\ be a holomorphic pair, and suppose
that there exists a metric, H, satisfying the $\tau$-vortex equation
(\ref{mve}). Then
either
(1)\ the holomorphic pair $({\cal E},\phi)$\ is $\tau$-stable, or
(2)\ the bundle ${\cal E}$\ splits holomorphically as ${\cal E}={\cal E}_{\phi}\oplus
{\cal E}_{ps}$\ with $\phi\in H^0(X,{\cal E}_{\phi})$, and such that
the holomorphic pair $({\cal E}_{\phi},\phi)$\ is $\tau$-stable, and
${\cal E}_{ps}$\ is polystable with slope equal to $\tau$.
\end{prop}
\begin{prop}[\cite{B2}]\label{hard}
Fix $\tau\in {\bf R}$, let $({\cal E},\phi)$\ be a $\tau$-stable holomorphic
pair.
Then there is a unique smooth metric, H, on $E$\ such that the
$\tau$-vortex equation (\ref{mve})\ is satisfied.
\end{prop}
\noindent{\em Remark}. We are assuming that the K\"{a}hler\ metric is normalized so that
$\mathop{{\fam0 Vol}}\nolimits(X)=2\pi$. Otherwise we need to introduce the factor
$\frac{2\pi}{\mathop{{\fam0 Vol}}\nolimits(X)}$ in the right hand side of (\ref{mve}).
Suppose now that we replace $\tau$\ in equation (\ref{mve}) by a
smooth real valued function $t\in C^{\infty}(X,{\bf R})$ and study
\begin{equation}
i\Lambda F_H +\phi\otimes\phi^{*_H}=t{\bf I}.\label{tmve}
\end{equation}
The question we wish to address is: how does this affect
the Hitchin-Kobayashi correspondence?
One direction is clear: replacing the constant $\tau$\ by the smooth function
$t$\
has absolutely no effect on the proof of Proposition \ref{easy}
(Theorem 2.1.6 in \cite{B2}). The same proof thus yields
\begin{thm}[cf. also Theorem 3.3, \cite{OT1}] Fix a smooth function
$t\in C^{\infty}(X,{\bf R})$.
Let $\overline{t}=\frac{1}{2\pi}\int t$.
If a holomorphic pair $({\cal E},\phi)$\
supports a solution to the $t$-vortex equation (\ref{tmve}), then either
(1)\ the holomorphic pair $({\cal E},\phi)$\ is $\overline t$-stable, or
(2)\ the bundle ${\cal E}$\ splits holomorphically as
${\cal E}={\cal E}_{\phi}\oplus{\cal E}_{ps}$\ with $\phi\in H^0(X,{\cal E}_{\phi})$, and
such that
the holomorphic pair $({\cal E}_{\phi},\phi)$\ is $\overline t$-stable, and
${\cal E}_{ps}$\ is polystable with slope equal to $\overline t$.
\end{thm}
We now consider the converse result, i.e. the analogue of
Proposition \ref{hard} (Theorem 3.1.1 in \cite{B2}). As shown in
\cite{OT1}, the $t$-vortex equation
can be reformulated as an equation with a constant right hand side.
Let $\tau=\overline{t}$. Since
$\int_X(\tau-t)=0$, we can find a smooth function $u$\ such
that $\Delta(u)=\tau-t$. Thus (\ref{tmve}) is equivalent to
\begin{equation}
i\Lambda (F_H+\Delta(u){\bf I})
+\phi\otimes\phi^{*_H}=\tau{\bf I}\ .\label{delta-ve}
\end{equation}
If we define a new metric $K=He^u$,
then (\ref{delta-ve}) becomes
\begin{equation}
i\Lambda F_K
+e^{-u}\phi\otimes\phi^{*_K}=\tau{\bf I}\ .\label{uveOT}
\end{equation}
This is {\it almost} the $\tau$-vortex equation, the only difference being the
prefactor $e^{-u}$\ in the second term. In the analysis of this situation by
Okonek and Teleman, they indicate how the proof (of Theorem 3.1.1) in [B2] can
be modified to accomodate this new wrinkle. (The proof in [B2] employs a
modified Donaldson functional on the space of Hermitian metrics on the bundle
$\cal E$. In [OT1], the authors generalize the functional so that it
accomodates the extra factor of $e^{-u}$, and argue that this has little
effect
on the proof.)
It is interesting to observe that the same result can be achieved without {\it
any modification at all} of the proof in [B2], if one enlarges the catgory in
which the proof is applied. This can be seen as follows. If we set
\begin{equation}
\phi_u=e^{-u/2}\phi\ ,\label{phiu}
\end{equation}
then (\ref{uveOT}) becomes
\begin{equation}
i\Lambda F_K
+\phi_u\otimes\phi_u^{*_K}=\tau{\bf I}\ .\label{uve}
\end{equation}
We thus see that
\begin{lemma} \label{t-tau}
Let $u$\ be given by $\Delta(u)=\tau-t$.
The pair $({\cal E},\phi)$\ admits a metric $H$ satisfying the
$t$-vortex equation if and only if
the pair $({\cal E},\phi_u)$\ admits a metric $K$ satisfying
the $\tau$-vortex equation. The metrics $H$\ and $K$\ are
related by $K=He^u$.
\end{lemma}
It is important to notice that, unless $u$\ is a constant
function, $\phi_u$\ is {\it not} \ a holomorphic section of
${\cal E}$. Indeed $\overline{\partial}_E\phi_u=-\frac{1}{2}\overline{\partial}(u)\phi_u$.
Our key observation is that in Theorem 3.1.1 in \cite{B2}, the
holomorphicity of $\phi$\ is not required either in the
statement or in the proof of the theorem. The proof, and
thus the result remains unchanged if the holomorphic section
$\phi$\ is replaced by a smooth section $\phi_u$\ related to
$\phi$\ by $\phi_u=e^{-u/2}\phi$. The basic reason can be
traced
back to the following simple fact:
\begin{lemma}\label{u-simple}
Let $\phi$, $u$, and $\phi_u$\ be as above.
(1)\ Let ${\cal E}'\subset{\cal E}$\ be any holomorphic
subbundle of ${\cal E}$. Then $\phi$\ is a section of ${\cal E}'$\ if and only if
$\phi_u$\ is a section of ${\cal E}'$.
(2)\ Let $s$\ be any smooth endomorphism of ${\cal E}$. Then
$s\phi=0$\ if and only if $s\phi_u=0$.
\end{lemma}
Notice, for instance, that if we define
$$
\begin{array}{ll}
H^0_u(X,{\cal E})&=e^{-u/2}H^0(X,{\cal E})\\
&=\{e^{-u/2}\phi\in\Omega^0(X,E)\ |\ \phi\in H^0(X,{\cal E})\}\ ,
\end{array}
$$
then the definition of $\tau$-stability can be applied to any pair
$({\cal E},\phi_u)$\ where $\phi_u\in H^0_u(X,{\cal E})$.
\begin{definition} A pair $({\cal E},\phi_u)$, where
$\phi$\ is in
$H^0_u(X,{\cal E})$, will be called a $u$-{\em holomorphic
pair}.
\end{definition}
In view of Lemma \ref{u-simple}, it follows that the $u$-holomorphic
pair $({\cal E},\phi_u)$\ is $\tau$-stable if and only if the
holomorphic pair $({\cal E},\phi)$\ is $\tau$-stable (where $\phi$\
and $\phi_u$\ are related by (\ref{phiu})). Furthermore, without any
alteration whatsoever, the proof of Theorem 3.1.1
in \cite{B2} can be
applied to a $u$-holomorphic pair to prove:
\begin{prop}\label{upair-tvor}
Fix $u\in C^{\infty}(X,{\bf R})$\ and
$\tau\in{\bf R}$. Let $({\cal E},\phi_u)$\ be a $\tau$-stable $u$-holomorphic pair.
Then $E$\ admits a unique smooth metric, say $K$, such that the
$\tau$-vortex equation is satisfied, i.e. such that
$$
i\Lambda F_K +\phi_u\otimes\phi_u^{*_K}=\tau{\bf I}\ .
$$
\end{prop}
Taken together, Lemma \ref{t-tau} and Proposition \ref{upair-tvor} thus
prove
\begin{thm} Fix $\tau=\overline t$\ and suppose that
$({\cal E},\phi)$\ is a $\tau$-stable pair. Then $E$\ supports
a metric satisfying the $t$-vortex equation.
\end{thm}
The above results describe the sense in which the vortex equation is
{\em insensitive} to the precise form of the parameter $t$. This can be
made precise by considering the moduli spaces. Let ${\cal C}$\ be the
space of
holomorphic structures (or, equivalently, integrable $\overline{\partial}_E$-operators)
on $E$, and let
$$
{\cal H}=\{(\overline{\partial}_E,\phi)\in{\cal C}\times\Omega^0(X,E)\ |\
\overline{\partial}_E\phi=0\}\
$$
be the configuration space of holomorphic pairs on $E$.
Let ${\cal V}_t\subset{\cal H}$\ be the set of $t$-{\em vortices}, i.e.
$$
{\cal V}_t=\{(\overline{\partial}_E,\phi)\in{\cal H}\ |\ \mbox{there is a metric satisfying
the $t$-vortex equation}\}\ .
$$
The above results can then be summarized by saying that
(1)\ ${\cal V}_t={\cal V}_{\tau} $\ for all functions $t$\ such that
$\overline t=\tau$, and
(2)\ for generic values of $\tau$, we can identify
${\cal V}_t={\cal H}_{\tau}$\ where ${\cal H}_{\tau}$\ denotes the set of
$\tau$-stable\ holomorphic pairs. In fact, as complex spaces,
${\cal V}_t/{\cal G}^c={\cal V}_{\tau}/{\cal G}^c={\cal B}_{\tau}$, where ${\cal B}_{\tau}$\
is the moduli space of $\tau$-stable holomorphic pairs ---
which has the structure of a variety
(cf. \cite{Be,BD1,BD2,G4,HL1,HL2,Th}).
\noindent{\em Remark}. In the case in which $E=L$ is a line bundle the $\tau$-stability
condition reduces to\
$\deg L<\tau$\ , and the moduli space of $\tau$-stable\ pairs is nothing else but
the space of {\em non-negative divisors} supported by $L$, where by a
non-negative divisor we mean either an effective divisor or
the zero divisor.
Nevertheless, the function $t$\ does carry some information. For example,
the metrics which satisfy the $t$-vortex equation (for fixed $\overline{\partial}_E$\
and $\phi$) do depend on $t$. The following observations shed some light
on the role played by $t$.
As described in \cite{G4}, \cite{BDGW}, the holomorphic pair $({\cal E},\phi)$\
can be identified with the {\em holomorphic triple}
$({\cal E},{\cal O},\phi)$, where ${\cal O}$ is the structure sheaf and $\phi$ is
a morphism ${\cal O}\rightarrow{\cal E}$. From this point of view, the
natural equations to consider are the framed vortex equations
(\ref{fve}).
Coming back to the set-up of Section \ref{vortices}, we want to study
equations (\ref{fve}) for a vector bundle $E$ of arbitrary rank and
$F=L_0$, the trivial line bundle.
As for the usual vortex equations, we will look at (\ref{fve}) as
equations for a metric on $E$.
To do this we fix the holomorphic structure
$\overline{\partial}_{L_0}$ on $L_0$ to be the trivial one i.e.
$(L_0,\overline{\partial}_{L_0})={\cal O}$,
and consider a holomorphic structure $\overline{\partial}_E$ on $E$. Then we take
$\phi:{\cal O}\rightarrow{\cal E}$ to be a holomorphic morphism, where ${\cal E}=(E,\overline{\partial}_E)$.
In contrast with the coupled vortex equations (\ref{cve}) that we will
analyse later, here we need to fix a metric $h$ on $L_0$.
Then solving (\ref{fve})
is equivalent to solving for a metric $H$ on $E$ satisfying
\begin{equation}
i\Lambda F_H+\phi\otimes\phi^\ast=\tau{\bf I}\ .\label{mfve}
\end{equation}
It is important to notice that now $\phi^\ast$ is the adjoint of
$\phi$ with respect to both metrics $H$ and $h$.
The identification between $({\cal E},\phi)$\ and
$({\cal E},{\cal O},\phi)$\ requires a choice of trivializing frame for ${\cal O}$,
say $f$.
If $h(f,f)=e^u$, then (\ref{mfve}) becomes
$$
i\Lambda F_H +e^{-u}\phi\otimes\phi^{*_H}=\tau {\bf I}\ .
$$
Thus we recover the usual $\tau$-vortex equation when we select the
metric on $L_0$\ for which the holomorphic frame of ${\cal O}$\
is also a unitary frame. For other choices of $h$\ we see that we get
essentially equation (\ref{uve}),
or equivalently, the $t$-vortex equation.
{\em From this point of view, the
function $t$\ is determined by the metric on $L_0$.}
The impact of non-constant $t$\ can also be understood from the symplectic
point of view. If we fix a metric, say $H$, on $E$, the induced inner
products
on ${\cal C}$\ and on $\Omega^0(X,E)$\ can be combined to give a symplectic
structure on the configuration space ${\cal H}$. Denoting the symplectic
forms
on ${\cal C}$\ and $\Omega^0(X,E)$\ by $\omega_{H,{\cal C}}$\ and $\omega_{H,0}$\
respectively, we take
$$
\omega_{H,H}=\omega_{H,{\cal C}}+\omega_{H,0}\
$$
as the symplectic form on ${\cal H}$. Let ${\cal G}_H$\ be the unitary gauge
group of $E$\ determined by $H$, and let $\mathop{{\fam0 Lie }}\nolimits {\cal G}_H$\ be its Lie algebra. A
moment map
$\mu:{\cal H}\rightarrow\mathop{{\fam0 Lie }}\nolimits{\cal G}_H^*$\ for the action of ${\cal G}_H$\ on the symplectic
space $({\cal H},\omega_{H,H})$\ is given by
$$
\mu_{H,H}(\overline{\partial}_E,\phi)=\Lambda F_{\overline{\partial}_E,H} -i\phi\otimes\phi^{*_H}\ ,
$$
where we have written $F_{\overline{\partial}_E,H}$ instead of $F_H$ to emphasize that
$F_H$ depends also on the holomorphic structure on $E$.
The $\tau$-vortex equation is thus equivalent to the condition
$\mu_{H,H}(\overline{\partial}_E,\phi)=-i\tau{\bf I}$, and we get an identification of
moduli spaces:
$$
{\cal B}_{\tau}={\cal V}_{\tau}/{\cal G}^c=\mu_{H,H}^{-1}(-i\tau{\bf I})/{\cal G}_H\ .
$$
Replacing $\tau$\ by the non-constant function $t$\ has no effect on the
identification ${\cal V}_{t}/{\cal G}^c=\mu_{H,H}^{-1}(-it{\bf I})/{\cal G}_H$: The
element $it{\bf I}$\
is still a central element in $\mathop{{\fam0 Lie }}\nolimits{\cal G}_H^*$, so the symplectic quotient
at this level is well defined.
(The problem comes in proving that ${\cal V}_t/{\cal G}^c={\cal B}_{\tau}$.)
An alternative point of view makes use of the equivalence between
the equations (\ref{tmve}) and (\ref{uve}). We define
$$
\mu_{H,K}(\overline{\partial}_E,\phi)=\Lambda F_{\overline{\partial}_E,H} -i\phi\otimes\phi^{*_K}\ ,
$$
where $H$\ and $K$\ are metrics on $E$. By the above results, $K=He^u$\
where $\Delta(u)=\tau-t$, then
$$
\mu_{H,H}^{-1}(-it{\bf I})=
\mu_{H,K}^{-1}(-i\tau{\bf I})\ .
$$
The point is that $\mu_{H,K}$\ is also a moment map for the action of
${\cal G}_H$. It arises when the symplectic structure on ${\cal H}$\ is taken to
be
$$
\omega_{H,K}=\omega_{H,{\cal C}}+\omega_{K,0}\ .
$$
{\em From this point of view, the function $t$\ arises from a deformation
of the symplectic structure on ${\cal H}$.}
\subsection{The coupled vortex equations}
Let us consider the set-up in Section \ref{vortices} for the coupled vortex equations\
(\ref{cve}).
As in the previous situation, we want to look at (\ref{cve})
as equations for metrics. In order to do this let us fix holomorphic
structures $\overline{\partial}_E$ and $\overline{\partial}_L$ on $E$ and $L$ respectively. Denote by
${\cal E}$ and ${\cal L}$ the corresponding holomorphic vector bundles. Let
$\phi\in H^0({\cal E}\otimes{\cal L}^\ast)$. Equations (\ref{cve}) are then
equivalent to solving
\begin{equation}
\left. \begin{array}{l}
i \Lambda F_H+\phi\otimes\phi^\ast=\tau {\bf I}_E\\
i \Lambda F_K-|\phi|^2=\tau'
\end{array}\right \}.\label{mcve}
\end{equation}
for metrics $H$ and $K$ on ${\cal E}$ and ${\cal L}$ respectively.
A Hitchin--Kobayashi\ correspondence was proved in \cite{G4}.
The appropriate notion of stability for $({\cal E},{\cal L},\phi)$
can be expressed in terms of the stability of a pair, namely
\begin{definition}\label{st}
The holomorphic triple $({\cal E},{\cal L},\phi)$ is said to be $\tau$-stable if
the holomorphic pair $({\cal E}\otimes{\cal L}^\ast,\phi)$ is $(\tau-\deg L)$-stable.
\end{definition}
\begin{thm}[\cite{G4}]\label{existence-cve}
Let $\tau$ and $\tau'$ be real numbers satisfying
(\ref{parameters}).
Let $({\cal E},{\cal L},\phi)$ a holomorphic triple. Suppose that there exist metrics
$H$ and $K$ satisfying (\ref{mcve}), then either
$({\cal E},{\cal L},\phi)$ is $\tau$-stable\ or
the bundle ${\cal E}$ splits holomorphically as ${\cal E}_\phi\oplus{\cal E}_{ps}$
with $\phi\in H^0(X,{\cal E}_{\phi}\otimes{\cal L}^\ast)$, and such that
$({\cal E}_\phi,{\cal L},\phi)$ is $\tau$-stable\ and ${\cal E}_{ps}$ is polystable with
slope equal to $\tau$.
Conversely, let $({\cal E},{\cal L},\phi)$ be a $\tau$-stable\ triple then there are unique
smooth metrics $H$ and $K$ satisfying the coupled vortex equations
(\ref{mcve}).
\end{thm}
Suppose now that we replace $\tau$ and $\tau'$ in (\ref{mcve}) by
smooth functions $t, t'\in C^\infty(X,{\bf R})$, i.e. we consider
\begin{equation}
\left. \begin{array}{l}
i \Lambda F_H+\phi\otimes\phi^\ast=t {\bf I}_E\\
i \Lambda F_K-|\phi|^2=t'
\end{array}\right \}.\label{tmcve}
\end{equation}
The first thing that we observe is that in order to have solutions
$t$ and $t'$ must satisfy
\begin{equation}
\int_X (r t+t')=\deg E +\deg L\label{t-t'}
\end{equation}
where $r=\mathop{{\fam0 rank}}\nolimits E$.
Next, let us recall the main ideas in the proof of Theorem
\ref{existence-cve}:
The basic fact is that the coupled vortex
equations (\ref{mcve}) are
a dimensional reduction of the Hermitian--Einstein\ equation
for a metric on a certain vector bundle over ${{X\times\bP^1}}$. This bundle ${\cal F}$,
canonically associated to the holomorphic triple $({\cal E},{\cal L},\phi)$,
is an extension on ${{X\times\bP^1}}$ of the form
\begin{equation}
0\longrightarrow p^\ast{\cal E}\longrightarrow {\cal F}\longrightarrow {p^\ast} {\cal L} \otimes q^\ast{\cal O}(2)\longrightarrow 0,
\label{bigbun}
\end{equation}
where $p$ and $q$ are the projections from ${{X\times\bP^1}}$
to $X$ and ${\bf P}^1$ respectively.
This is simply because
$H^1({{X\times\bP^1}}, p^\ast ({\cal E}\otimes{\cal L}^\ast)\otimes q^\ast{\cal O}(2))
\cong H^0(X,{\cal E}\otimes{\cal L}^\ast)$
Let $SU(2)$ act on $X\times {\bf P}^1$,
trivially on $X$, and in the standard way on ${{\bf P}}^1 \cong
SU(2)/U(1)$. This action can be lifted to an action on ${\cal F}$,
trivial on $p^\ast{\cal E}$ and ${p^\ast}{\cal L}$, and standard on
$q^\ast{\cal O}(2)$. The bundle ${\cal F}$ is in this way an $SU(2)$-equivariant
holomorphic vector bundle.
Let $\tau$ and $\tau'$ be related by (\ref{parameters}) and let
\begin{equation}
\sigma=\frac{4\pi}{\tau-\tau'}
\end{equation}
be positive.
Consider the $SU(2)$-invariant K\"{a}hler
metric on $X\times {\bf P}^1$ whose K\"{a}hler form is
$$
\omega_\sigma =p^\ast\omega +\sigma q^\ast\omega_{{\bf P}^1},
$$
where $\omega$ is the K\"{a}hler form on $X$ (normalized such that
$\mathop{{\fam0 Vol}}\nolimits(X)=2\pi$), and $\omega_{{\bf P}^1}$ is
the {\em Fubini-Study} metric with volume 1.
Theorem \ref{existence-cve} is then a consequence of the following two
propositions
and the Hitchin--Kobayashi\ correspondence proved by Donaldson \cite{D1,D2}, and
Uhlenbeck and Yau \cite{UY}).
\begin{prop}[\cite{G4}]\label{dr}
The triple $({\cal E},{\cal L},\phi)$ admits a solution to
(\ref{mcve}) if and only if the vector bundle ${\cal F}$ in (\ref{bigbun})
has a (${SU}(2)$-invariant) Hermitian--Einstein\ metric with respect to $\omega_\sigma$.
\end{prop}
\begin{prop}[\cite{G4}]\label{s-ts}
Suppose that ${\cal E}$ is not isomorphic to ${\cal L}$. Then the triple
$({\cal E},{\cal L},\phi)$
is $\tau$-stable\ if and only if ${\cal F}$ is stable with respect to $\omega_\sigma$.
If ${\cal E}\cong{\cal L}$, then
${\cal F}\cong {p^\ast} {\cal L}\otimes{q^\ast}{\cal O}(1)\oplus{p^\ast} {\cal L}\otimes{q^\ast}{\cal O}(1)$.
\end{prop}
Suppose first that the functions $t$ and $t'$, in addition to satisfying
(\ref{t-t'}), verify that there is a positive constant $\sigma$ so that
\begin{equation}
t-t'=\frac{4\pi}{\sigma}.\label{strong-t-t'}
\end{equation}
The proof of Proposition \ref{dr} then yields
\begin{prop} \label{weakdr}
The triple $({\cal E},{\cal L},\phi)$ admits a solution to
(\ref{tmcve}) if and only if the vector bundle ${\cal F}$ in (\ref{bigbun})
has a (${SU}(2)$-invariant) metric satisfying the weak Hermitian--Einstein\
equation
\begin{equation}
i\Lambda_\sigma F_{\bf H}= t\ {\bf I}
\end{equation}
with respect to the K\"{a}hler\ form
$\omega_\sigma={p^\ast} \omega_X\oplus \sigma{q^\ast}\omega_{{\bf P}^1}$.
\end{prop}
But the existence of a weak Hermitian--Einstein\ metric is in fact equivalent
to the existence of a Hermitian--Einstein\ metric --- as one can see simply by applying a
conformal change to the metric --- and hence equivalent to the
stability of the bundle. We can then combine again Propositions
\ref{weakdr} and \ref{s-ts} to prove the following.
\begin{thm}\label{existence-tcve}
Fix smooth functions $t,t'\in C^\infty(X,{\bf R})$ satisfying (\ref{t-t'})
and (\ref{strong-t-t'}). Let $\overline{t}=\frac{1}{2\pi}\int_X t$ and
$\overline{t'}=\frac{1}{2\pi}\int_X t'$.
Let $({\cal E},{\cal L},\phi)$ a holomorphic triple. Suppose that there exist
metrics $H$ and $K$ satisfying (\ref{tmcve}), then either
$({\cal E},{\cal L},\phi)$ is $\overline{t}$-stable or
the bundle ${\cal E}$ splits holomorphically as ${\cal E}_\phi\oplus{\cal E}_{ps}$
with $\phi\in H^0(X,{\cal E}_{\phi}\otimes{\cal L}^\ast)$, and such that
$({\cal E}_\phi,{\cal L},\phi)$ is $\overline{t}$-stable
and ${\cal E}_{ps}$ is polystable with slope equal to $\overline{t}$.
Conversely, let $({\cal E},{\cal L},\phi)$ be a $\overline{t}$-stable triple then
there are unique smooth metrics $H$ and $K$ satisfying
equations(\ref{tmcve}).
\end{thm}
We will show now that the general coupled vortex equations (\ref{tmcve})
with $t$ and $t'$ satisfying simply (\ref{t-t'})
are also a dimensional reduction, but in this case of a metric on ${\cal F}$
satisfying a certain deformation of the Hermitian--Einstein\ condition. We set, as above,
$$
\sigma=\frac{4\pi}{\overline{t}-\overline{t'}}
$$
where $\overline{t}$ and
$\overline{t'}$ denote the average values of $t$
and $t'$ respectively. Again the proof of Proposition \ref{dr}
yields
\begin{prop} \label{newdr}
The triple $({\cal E},{\cal L},\phi)$ admits a solution to
(\ref{tmcve}) if and only if the vector bundle ${\cal F}$ in (\ref{bigbun})
has a (${SU}(2)$-invariant) metric satisfying the deformed Hermitian--Einstein\
equation
\begin{equation}
i\Lambda_\sigma F_{\bf H}= \overline{t}\ {\bf I} +
\left(\begin{array}{cc}(t-\overline{t}) {\bf I}_1 & 0\\
0&(t'-\overline{t'}){\bf I}_2 \end{array}\right),
\label{bigbun-eqn}
\end{equation}
with respect to the K\"{a}hler\ form
$$
\omega_\sigma={p^\ast} \omega_X\oplus \sigma{q^\ast}\omega_{{\bf P}^1}.
$$
\end{prop}
The deformed Hermitian--Einstein\ equation in this Proposition is similar to the kind
studied in \cite{BG2}. In \cite{BG2} we considered an extension of
holomorphic
vector bundles over a compact K\"{a}hler\ manifold
\begin{equation}
{0\lra \cE_1\lra \cE\lra \cE_2\lra 0}\label{extn}
\end{equation}
and studied metrics $H$ on ${\cal E}$ satisfying the equation
\begin{equation}
i\Lambda F_H=\left(\begin{array}{cc}\tau_1 {\bf I}_1 & 0\\
0&\tau_2 {\bf I}_2 \end{array}\right),\label{ext-eqn}
\end{equation}
where $\tau_1$ and $\tau_2$ are real numbers, related by
$$
\tau_1 r_1 +\tau_2 r_2=\deg {\cal E}.
$$
The reason we can write an equation like (\ref{ext-eqn}) is that the
metric $H$ gives a $C^\infty$ splitting of (\ref{extn}).
We proved an existence theorem for metrics satisfying
(\ref{ext-eqn}) in terms of a notion of stability for the extension
depending on the parameter $\alpha=\tau_1-\tau_2$.
To define this stability condition consider any coherent subsheaf
${\cal E}'\subset{\cal E}$ and write it as a subextension
$$
{0\lra \cE_1'\lra\cE'\lra \cE_2'\lra 0}.
$$
Define the $\alpha$-slope of ${\cal E}'$ as
$$
\mu_\alpha({\cal E}')=\mu({\cal E}')+\alpha\frac{\mathop{{\fam0 rank}}\nolimits{\cal E}_2'}{\mathop{{\fam0 rank}}\nolimits {\cal E}'}.
$$
Then we say that (\ref{extn}) is $\alpha$-stable if and only if for every
non-trivial subsheaf ${\cal E}'\subset{\cal E}$
$$
\mu_\alpha({\cal E}')<\mu_\alpha({\cal E}).
$$
We proved
\begin{thm}[\cite{BG2}]Let $\alpha=\tau_1-\tau_2\leq 0$ and
suppose that (\ref{extn}) is indecomposable (as an extension), then
${\cal E}$ admits a metric satisfying (\ref{ext-eqn}) if and only
(\ref{extn}) is $\alpha$-stable.
\end{thm}
\noindent{\em Remark}. If $\alpha=0$ (\ref{ext-eqn}) reduces to the Hermitian--Einstein\ equation
and the stability condition is the usual stability of the bundle
${\cal E}$.
The deformed Hermitian--Einstein\ equation in Proposition \ref{newdr} differs from
(\ref{ext-eqn}) only in that the constants $\tau_1$ and $\tau_2$
have been replaced by smooth functions, $t_1$ and $t_2$,
satisfying
\begin{equation}
\int(r_1 t_1 +r_2 t_2)=\deg {\cal E}.\label{t1-t2}
\end{equation}
By the same methods used in \cite{BG2} one can readily show one direction of
the
Hitchin-Kobayashi correspondence, namely
\begin{thm}\label{alpha-hk}Let\ $t_1$ and $t_2$ be smooth real functions
satisfying (\ref{t1-t2}) and such that $\alpha=\int(t_1-t_2)\leq 0$. Then
the existence of a metric $H$ on ${\cal E}$ satisfying
\begin{equation}
i\Lambda F_H=\left(\begin{array}{cc}t_1 {\bf I}_1 & 0\\
0&t_2 {\bf I}_2 \end{array}\right),\label{var-extn-eqn}
\end{equation}
implies the $\alpha$-stability of (\ref{extn}).
\end{thm}
It should likewise be possible to adapt the proof of the other
direction of the Hitchin-Kobayashi correspondence.
This will then allow one (by taking $\alpha=0$) to
establish a more general version of Theorem \ref{existence-tcve}, valid
when $t$\ and $t'$\ are smooth functions satisfying just (\ref{t-t'}).
We will discuss this in a future publication.
To describe the moduli space, let ${\cal C}_E$ and ${\cal C}_L$ the sets of
holomorphic structures on $E$ and $L$ respectively. Consider the set
\begin{equation}
{\cal H}(E,L)=
\{(\overline{\partial}_E,\overline{\partial}_L,\phi)\in {\cal C}_E\times{\cal C}_L\times\Omega^0(\mathop{{\fam0 Hom}}\nolimits(L,E))
\;\;|\;\;
\phi\in H^0(X,{\cal E}\otimes{\cal L}^\ast)\} \label{ht}
\end{equation}
of {\em holomorphic triples} on $(E,L)$, where ${\cal E}$ and ${\cal L}$ denote the
holomorphic vector bundles defined by $\overline{\partial}_E$ and $\overline{\partial}_L$
respectively.
Let ${\cal H}_\tau(E,L)\subset{\cal H}(E,L)$ be the set of $\tau$-stable\ holomorphic triples.
This set is invariant under the action of the complex gauge groups
of $E$ and $L$, ${\cal G}^c_E$ and ${\cal G}^c_L$, say. The moduli space of
$\tau$-stable\ triples is defined as
$$
{\cal B}_\tau(E,L)={\cal H}_\tau(E,L)/{\cal G}^c_E\times{\cal G}^c_L.
$$
The set ${\cal B}_\tau(E,L)$, which has naturally the structure of a
variety (cf. \cite{G4}), is closely related to the moduli space
of stable pairs---this is not surprising in view of the definition
\ref{st}.
More precisely, the map
$({\cal E},{\cal L},\phi)\mapsto ({\cal E}\otimes{\cal L}^\ast,\phi)$ exhibits
${\cal B}_\tau(E,L)$ as a $\mathop{{\fam0 Pic}}\nolimits^0$-principal bundle over the moduli
space of ($\tau-\deg L$)-stable pairs on $E\otimes L^\ast$.
The study of the general equations (\ref{gcve}), i.e. the case in which
$F$ is of arbitrary rank, requires the introduction of a new notion of
stability. This was carried out in \cite{BG1}. All the results
explained above should extend appropriately to the higher rank case when one
replaces $\tau$ and $\tau'$ in (\ref{gcve}) by functions $t$ and $t'$.
|
1998-08-29T23:47:41 | 9602 | alg-geom/9602014 | en | https://arxiv.org/abs/alg-geom/9602014 | [
"alg-geom",
"math.AG"
] | alg-geom/9602014 | Alice Silverberg | A. Silverberg and Yu. G. Zarhin | Reduction of abelian varieties | null | null | null | null | null | We study semistable reduction and torsion points of abelian varieties. In
particular, we give necessary and sufficient conditions for an abelian variety
to have semistable reduction. We also study N\'eron models of abelian varieties
with potentially good reduction and torsion points of small order. We study
some invariants that measure the extent to which an abelian variety with
potentially good reduction fails to have good reduction.
| [
{
"version": "v1",
"created": "Mon, 19 Feb 1996 17:09:30 GMT"
},
{
"version": "v2",
"created": "Sat, 29 Aug 1998 21:47:41 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Silverberg",
"A.",
""
],
[
"Zarhin",
"Yu. G.",
""
]
] | alg-geom | \section{Introduction}
In this paper we study the reduction of abelian varieties.
We assume $F$ is a field with a discrete valuation $v$, $X$ is an
abelian variety over $F$, and $n$ is an integer not divisible by
the residue characteristic.
In Part \ref{semistabpart} we give criteria for semistable reduction.
Suppose $n \ge 5$. In Theorem \ref{ssredlem}
we show that $X$ has semistable reduction if and only if
$(\sigma - 1)^2 = 0$ on the $n$-torsion in $X$, for every $\sigma$ in
the absolute inertia group.
In Theorem \ref{ssredconverse} we show (using Theorem \ref{ssredlem})
that $X$ has semistable reduction
if and only if there exists a subgroup of $n$-torsion points such that
the absolute inertia group acts trivially on both it and its orthogonal
complement with respect to the $e_n$-pairing.
We deduce as special cases both Raynaud's criterion (that the abelian
variety have full level $n$ structure for $n \ge 3$; see
Theorem \ref{raynaud}) and the criterion of \cite{semistab} (that the
abelian variety have partial level $n$ structure for $n \ge 5$; see
Theorem \ref{ssred}). We also obtain a (near) converse to the
criterion of \cite{semistab}. The proofs are based on the fundamental
results of Grothendieck on semistable reduction of abelian varieties
(see \cite{SGA}). In \S\ref{except} we allow $n<5$. In
\S\ref{Gsect} we give a measure of potentially good reduction.
We discuss other measures of potentially good reduction in
Part \ref{neronpart}.
In Part \ref{neronpart} we study N\'eron models of abelian
varieties with potentially good reduction and torsion points of
small order. Suppose that the valuation ring is henselian and
the residue field is algebraically closed.
If $X$ has good reduction,
then $X_n \subseteq X(F)$ (this is an immediate corollary of the existence of
N\'eron models; see Lemma \ref{neronlemma} below).
On the other hand, if $X_n \subseteq X(F)$ and $n \ge 3$,
then by virtue of Raynaud's criterion for semistable reduction,
$X$ has good reduction.
Notice that the failure of $X$ to have good reduction is measured by
the dimension $u$ of the unipotent radical of the special fiber of the N\'eron
model of $X$. In particular, $u = 0$ if and only if $X$ has good reduction.
In general, $0 \le u \le \mathrm{dim}(X)$. The equality $u = \mathrm{dim}(X)$
says that $X$ has purely additive reduction.
Another measure of the deviation from good reduction is the (finite) group of
connected components $\Phi$ of the special fiber of the N\'eron model.
If $X$ has good reduction then $\Phi=\{0\}$, but the converse statement
is not true in general.
The aim of \S\ref{main} is to connect explicitly the invariants
$u$ and $\Phi$ with the failure of $X(F)$ to contain all the
$n$-torsion points. This failure can be
measured by the index $[X_n:X_n(F)]$. We assume that at
least ``half'' of the
$n$-torsion points are rational over $F$. More precisely, we assume that
there exists an $F$-rational polarization $\lambda$ on $X$
and a maximal isotropic
(with respect to the pairing $e_{\lambda,n}$ induced from the
Weil $e_n$-pairing by $\lambda$)
subgroup of $X_n$ consisting of $F$-rational points.
If in addition $n \ge 5$, then $X$ has good reduction (see Theorem 7.4
of \cite{semistab}), and therefore $u = 0$, $\Phi=\{0\}$, and $X_n=X_n(F)$.
Therefore, we have to investigate only the cases $n = 2$, $3$, and $4$.
Let $\Phi'$ denote the prime-to-$p$ part of $\Phi$, where $p$ is the
residue characteristic (with $\Phi ' = \Phi$ if $p = 0$).
We show that if $n = 2$ then $\Phi'$ is an elementary abelian
$2$-group and $[X_2:X_2(F)]\#\Phi' = 4^{u}$,
if $n = 3$ then $[X_3:X_3(F)] = 3^{u}$ and
$\Phi' \cong ({\mathbf Z}/3{\mathbf Z})^u$, and
if $n = 4$ then $X_2 \subseteq X(F)$,
$[X_4:X_4(F)] = 4^{u}$, and $\Phi' \cong ({\mathbf Z}/2{\mathbf Z})^{2u}$.
If instead of assuming partial level $n$ structure we assume that
all the points of order $2$ on $X$ are defined over $F$, then
$[X_4:X_4(F)]=4^u$ and $\Phi' \cong ({\mathbf Z}/2{\mathbf Z})^{2u}$.
Earlier work on abelian varieties with potentially good reduction and
on groups of connected components of N\'eron models
has been done by Serre and Tate \cite{Serre-Tate},
Silverman \cite{Silverman},
Lenstra and Oort \cite{LenstraOort},
Lorenzini \cite{Lorenzini}, and Edixhoven \cite{Edixhoven}.
Silverberg would like to thank the IHES and the Bunting
Institute for their hospitality, and the NSA and the Science Scholars
Fellowship Program at the Bunting Institute for financial support.
Zarhin would like to thank the NSF for financial support.
He also would like to thank the organizers of the
NATO/CRM 1998 Summer School
on the Arithmetic and Geometry of Algebraic Cycles for
twelve wonderful days in Banff.
\section{Notation and definitions}
If $F$ is a field, let $F^s$ denote a separable closure.
Suppose that $X$ is an abelian variety defined over
$F$, and $n$ is a positive integer not divisible by the characteristic
of $F$. Let $X^\ast$ denote the dual abelian variety of $X$, let
$X_n$ denote the kernel of multiplication by $n$ in $X(F^s)$, let
$X_n^\ast$ denote the kernel of multiplication by $n$ in $X^\ast(F^s)$,
and let $\boldsymbol \mu _n$ denote the $\mathrm{Gal} (F^s/F)$-module
of $n$-th roots of unity in $F^s$.
The $e_n$-pairing
$$e_n : X_n \times X_n^\ast \to {\boldsymbol \mu}_n$$
is a $\mathrm{Gal}(F^s/F)$-equivariant nondegenerate pairing
(see \S 74 of \cite{WeilAV}).
If $S$ is a subgroup of $X_n$, let
$$S^{\perp_n} =
\{ y \in X_n^\ast : e_n(x,y) = 1 \text{ for every } x \in S \}
\subseteq X_n^\ast.$$
If $\lambda$ is a polarization on $X$, define
$$e_{\lambda ,n} : X_n \times X_n \to {\boldsymbol \mu}_n$$
by $e_{\lambda ,n}(x,y) = e_n(x,\lambda(y))$
(see \S 75 of \cite{WeilAV}). Then
$$\sigma (e_{\lambda ,n}(x_1,x_2)) =
e_{\sigma (\lambda ),n}(\sigma (x_1),\sigma (x_2))$$ for every
$\sigma \in \mathrm{Gal} (F^s/F)$ and $x_1$, $x_2 \in X_n$. If $n$ is relatively prime
to the degree of the polarization $\lambda$, then the pairing
$e_{\lambda ,n}$ is nondegenerate.
If $\ell$ is a prime not equal to the characteristic of $F$,
and $d = \mathrm{dim}(X)$, let
$$\rho_{\ell,X} : \mathrm{Gal}(F^s/F) \to \mathrm{Aut}(T_\ell(X)) \cong \mathrm{M}_{2d}({\mathbf Z}_\ell)$$
denote the $\ell$-adic representation on the Tate module $T_\ell(X)$
of $X$, and let $V_\ell(X) = T_\ell(X) \otimes_{{\mathbf Z}_\ell} {\mathbf Q}_\ell$.
Let $I$ denote the identity matrix in $\mathrm{M}_{2d}({\mathbf Z}_\ell)$.
If $L$ is a Galois extension of $F$,
$v$ is a discrete valuation on $F$, and $w$ is an extension of $v$ to $L$,
let ${\mathcal I}(w/v)$ denote the inertia subgroup at $w$ of $\mathrm{Gal}(L/F)$.
If $X$ is an abelian variety over $F$, let $X_v$ denote the
special fiber of the N\'eron model
of $X$ at $v$ and let $X_v^0$ denote its identity connected
component. Let $a$, $u$, and $t$ denote, respectively, the abelian,
unipotent, and toric ranks of $X_v$. Then $a + u + t = \mathrm{dim}(X)$.
If $p$ ( $\ge 0$) is the residue characteristic of $v$,
let $\Phi '$ denote the prime-to-$p$ part of the group of
connected components of the special
fiber of the N\'eron model of $X$ at $v$ (with $\Phi '$ the full
group of components if $p = 0$).
\begin{defn}
If $v$ is a discrete valuation on a field $F$, we say
the valuation ring is
{\em strictly henselian} if the valuation ring is henselian and
the residue field is algebraically closed.
\end{defn}
\begin{defn}
Suppose $L/F$ is an extension of fields, $w$ is a discrete valuation
on $L$, and $v$ is the restriction of $w$ to $F$. We say that
$w/v$ is {\em unramified} if a uniformizing element of the
valuation ring for $v$
induces a uniformizing element of the valuation ring for $w$
and the residue field extension is separable (see Definition 1
on page 78 of \cite{BLR}).
\end{defn}
\begin{rem}[Remark 5.3 of \cite{semistab}]
\label{ramifiedcyclic}
Suppose $v$ is a discrete valuation on a field $F$, and $m$ is a positive
integer not divisible by the residue characteristic. Then every
degree $m$ Galois extension of $F$ totally
ramified at $v$ is cyclic. If $F(\zeta_m) = F$,
then $F$ has a cyclic extension of degree $m$ which is totally ramified at
$v$. In particular, if the residue characteristic is not $2$ then
$F$ has a quadratic extension which is (totally and tamely) ramified at $v$.
If the valuation ring is henselian and the residue field is separably closed,
then $F= F(\zeta_m)$ and therefore $F$ has a cyclic totally ramified extension
of degree $m$. (See Remark 5.3 of \cite{semistab}.)
Note also that $F$ has no non-trivial unramified extensions if and only if
the valuation ring is henselian and the residue field is separably closed.
\end{rem}
\part{Semistable reduction of abelian varieties}
\label{semistabpart}
\section{Preliminaries}
\begin{defn}
If $k$ is a positive integer, define a finite set
of prime powers $N(k)$ by
$$N(k) = \{\text{prime powers $\ell^m : 0 \le m(\ell - 1) \le k $}\}.$$
\end{defn}
For example, $$N(1) = \{1, 2\}, \quad N(2) = \{1, 2, 3, 4\},$$
$$N(3) = \{1, 2, 3, 4, 8\}, \quad
N(4) = \{1, 2, 3, 4, 5, 8, 9, 16\}.$$
\begin{thm}
\label{quasithm}
Suppose $n$ and $k$ are positive integers, ${\mathcal O}$ is an integral
domain of characteristic zero such that no rational prime which
divides $n$
is a unit in ${\mathcal O}$, $\alpha \in {\mathcal O}$, $\alpha$ has finite
multiplicative order,
and $(\alpha - 1)^k \in n{\mathcal O}$.
If $n \notin N(k)$, then $\alpha = 1$. In particular, if
$(\alpha - 1)^2 \in n{\mathcal O}$ and $n \ge 5$, then $\alpha = 1$.
\end{thm}
\begin{proof}
See Corollary 3.3 of \cite{serrelem}.
\end{proof}
\begin{lem}[Lemma 5.2 of \cite{semistab}]
\label{localglobal}
Suppose that $d$ and $n$ are positive integers, and for each prime $\ell$
which divides
$n$ we have a matrix $A_\ell \in M_{2d}({\mathbf Z}_\ell)$ such that the
characteristic polynomials of the $A_\ell$ have integral coefficients
independent of $\ell$, and such that $(A_\ell - I)^2 \in nM_{2d}({\mathbf Z}_\ell)$.
Then for every eigenvalue $\alpha$ of $A_\ell$, $(\alpha - 1)/\sqrt{n}$
satisfies a monic polynomial with integer coefficients.
\end{lem}
\begin{lem}[Lemma 4.2 of \cite{degree}]
\label{mevals}
Suppose $v$ is a discrete valuation on a field $F$ with residue
characteristic $p \ge 0$, $m$ is a positive integer, $\ell$ is a prime,
$p$ does not divide $m\ell$, $K$ is a degree $m$ extension of $F$
which is totally ramified above $v$, and ${\bar v}$ is an extension of $v$ to
a separable closure $K^s$ of $K$. Suppose that $X$ is an abelian variety over $F$,
and for every $\sigma \in {\mathcal I}({\bar v}/v)$,
all the eigenvalues of $\rho_{\ell,X}(\sigma)$
are $m$-th roots of unity. Then $X$ has
semistable reduction at the extension of $v$ to $K$.
\end{lem}
\section{Criteria for semistable reduction}
\begin{thm}[Galois Criterion for Semistable Reduction]
\label{galcrit}
Suppose $X$ is an abel\-ian variety over a field $F$, $v$ is a discrete
valuation on $F$, $\ell$ is a prime not equal to the residue
characteristic of $v$, ${\bar v}$ is an extension of $v$ to $F^s$,
and ${\mathcal I} = {\mathcal I}({\bar v}/v)$.
Then the following are equivalent:
\begin{enumerate}
\item[(i)] $X$ has semistable reduction at $v$,
\item[(ii)] ${\mathcal I}$ acts unipotently on $T_\ell(X)$; i.e.,
all the eigenvalues of $\rho_{\ell,X}(\sigma)$ are $1$,
for every $\sigma \in {\mathcal I}$,
\item[(iii)] for every $\sigma \in {\mathcal I}$,
$(\rho_{\ell,X}(\sigma) - I)^2 = 0$.
\end{enumerate}
\end{thm}
\begin{proof}
See Proposition 3.5 and Corollaire 3.8 of \cite{SGA}
and Theorem 6 on p.~184 of \cite{BLR}.
\end{proof}
\begin{thm}[Raynaud Criterion for Semistable Reduction]
\label{raynaud}
\hfil Suppose $X$ is an abelian variety over a field $F$ with a
discrete valuation $v$, $m$ is a positive integer not divisible by the
residue characteristic of $v$, and the points of $X_m$ are defined
over an extension of $F$ which is unramified over $v$. If $m \ge 3$, then $X$ has
semistable reduction at $v$.
\end{thm}
\begin{proof}
See Proposition 4.7 of \cite{SGA}.
\end{proof}
\begin{prop}
\label{sslem}
Suppose $X$ is an abelian variety over a field $F$, $v$ is a discrete
valuation on $F$, $n$ is an integer not
divisible by the residue characteristic of $v$,
${\bar v}$ is an extension of $v$ to $F^s$,
and ${\mathcal I} = {\mathcal I}({\bar v}/v)$. Let $S = X_n^{\mathcal I}$, the elements of
$X_n$ on which ${\mathcal I}$ acts as the identity. If
$X$ has semistable reduction at $v$, then
\begin{enumerate}
\item[{(i)}] $(\sigma - 1)^2 X_n = 0$ for every $\sigma \in {\mathcal I}$, and
\item[{(ii)}] ${\mathcal I}$ acts as the identity on $S^{\perp_n}$.
\end{enumerate}
\end{prop}
\begin{proof}
Suppose $X$ has semistable reduction at $v$.
By Theorem \ref{galcrit}, we have (i).
It follows that $\sigma^n = 1$ on $X_n$. Since $n$ is
not divisible by the residue characteristic, $X_n$ is tamely ramified
over $F$. Let ${\mathcal J}$ denote the first ramification group. Then the
action of ${\mathcal I}$ on $X_n$ factors through ${\mathcal I}/{\mathcal J}$. Let $\tau$ denote
a lift to ${\mathcal I}$ of a topological generator of the pro-cyclic group
${\mathcal I}/{\mathcal J}$. Since
$$e_n((\tau - 1)X_n,(X_n^\ast)^{\mathcal I}) = 1,$$
we have
$$\#((X_n^\ast)^{\mathcal I})\#((\tau - 1)X_n) \le \#X_n^\ast.$$
The map from $X_n$ to $(\tau - 1)X_n$ defined by
$y \mapsto (\tau - 1)y$ defines a short exact sequence
$$0 \to S \to X_n \to (\tau - 1)X_n \to 0.$$
Therefore,
$$\#S\#((\tau - 1)X_n) = \#X_n = \#S\#S^{\perp_n}.$$
Similarly,
$$\#((X_n^\ast)^{\mathcal I})\#((\tau - 1)X_n^\ast) = \#X_n^\ast.$$
Therefore,
$$\#S^{\perp_n} = \#((\tau - 1)X_n) \le \#((\tau - 1)X_n^\ast).$$
Since $(\tau - 1)X_n^\ast \subseteq S^{\perp_n}$,
we conclude that
$$S^{\perp_n} = (\tau - 1)X_n^\ast.$$
By (i),
we have $(\tau - 1)^2X_n = 0$. It follows from the natural
$\mathrm{Gal}(F^s/F)$-equivariant isomorphism
$X_n^\ast \cong \mathrm{Hom}(X_n,\boldsymbol \mu_n)$ that $(\tau - 1)^2X_n^\ast = 0$,
and therefore ${\mathcal I}$ acts as the identity on $S^{\perp_n}$.
\end{proof}
\begin{thm}
\label{ssredlem}
Suppose $X$ is an abelian variety over a field $F$, $v$ is a discrete
valuation on $F$, $n$ is an integer not
divisible by the residue characteristic of $v$, $n \ge 5$,
${\bar v}$ is an extension of $v$ to $F^s$,
and ${\mathcal I} = {\mathcal I}({\bar v}/v)$.
Then $X$ has semistable reduction at $v$ if and only if
$(\sigma - 1)^2 X_n = 0$ for every $\sigma \in {\mathcal I}$.
\end{thm}
\begin{proof}
If $X$ has semistable reduction at $v$ then
for every $\sigma \in {\mathcal I}$ we have $(\sigma - 1)^2 X_n = 0$,
by Proposition \ref{sslem}i.
Conversely, suppose $n \ge 5$ and $(\sigma - 1)^2 X_n = 0$
for every $\sigma \in {\mathcal I}$.
Let ${\mathcal I}' \subseteq {\mathcal I}$ be the inertia group for the prime
below ${\bar v}$ in a
finite Galois extension of $F$ over which $X$ has
semistable reduction.
Take $\sigma \in {\mathcal I}$. Then
$\sigma^m \in {\mathcal I}'$ for some $m$.
Let $\ell$ be a prime divisor of $n$.
Theorem \ref{galcrit} implies that
$(\rho_{\ell,X}(\sigma)^m - I)^2 = 0$.
Let $\alpha$ be an eigenvalue of $\rho_{\ell,X}(\sigma)$.
Then $(\alpha^m - 1)^2 = 0$.
Therefore, $\alpha^m = 1$.
By our hypothesis,
$$(\rho_{\ell,X}(\sigma) - I)^2 \in n\mathrm{M}_{2d}({\mathbf Z}_\ell),$$
where $d = \mathrm{dim}(X)$.
By Theorem 4.3 on p.~359 of \cite{SGA},
the characteristic polynomial of $\rho_{\ell,X}(\sigma)$
has integer coefficients which are independent of $\ell$.
By Lemma \ref{localglobal},
$(\alpha - 1)^2 \in n{\bar {\mathbf Z}}$, where ${\bar {\mathbf Z}}$ denotes
the ring of algebraic integers.
Since $n \ge 5$,
by Theorem \ref{quasithm} we have $\alpha = 1$ (i.e.,
${\mathcal I}$ acts unipotently on $T_\ell(X)$).
By Theorem \ref{galcrit}, $X$ has semistable reduction at $v$.
\end{proof}
\begin{thm}
\label{ssredconverse}
Suppose $X$ is an abelian variety over a field $F$, $v$ is a discrete
valuation on $F$, $n$ is an integer not divisible by the residue
characteristic of $v$, $n \ge 5$, ${\bar v}$ is an extension of
$v$ to $F^s$, and ${\mathcal I} = {\mathcal I}({\bar v}/v)$. Then $X$ has semistable
reduction at $v$ if and only if there exists a subgroup $S$ of $X_n$
such that ${\mathcal I}$ acts as the identity on $S$ and on $S^{\perp_n}$.
\end{thm}
\begin{proof}
Suppose there exists a subgroup $S$ as in the statement of the
theorem. The map $x \mapsto (y \mapsto e_n(x,y))$ induces a
$\mathrm{Gal}(F^s/F)$-equivariant isomorphism from $X_n/S$ onto
$\mathrm{Hom}(S^{\perp_n},\boldsymbol \mu_n)$. Suppose $\sigma \in {\mathcal I}$. Then
$\sigma = 1$ on $S^{\perp_n}$ and on $\boldsymbol \mu_n$. Therefore, $\sigma = 1$ on
$X_n/S$. Thus,
$(\sigma - 1)^2X_n \subseteq (\sigma - 1)S = 0$.
By Theorem \ref{ssredlem}, $X$ has semistable reduction at $v$.
Conversely, suppose $X$ has semistable reduction at $v$.
Let $S = X_n^{\mathcal I}$, and apply Proposition \ref{sslem}ii.
\end{proof}
\begin{thm}
\label{ssred}
Suppose $X$ is an abelian variety over a field $F$, $v$ is a discrete
valuation on $F$, $\lambda$ is a polarization on $X$ defined over
an extension of $F$ which is unramified over $v$,
$n$ is a positive integer not
divisible by the residue characteristic of $v$, and $n \ge 5$.
\begin{enumerate}
\item[{(i)}]
If $\widetilde{X}_n$ is a maximal isotropic subgroup of $X_n$ with
respect to $e_{\lambda,n}$, and the points of
$\widetilde{X}_n$ are defined over an extension of $F$ which is
unramified over $v$, then $X$ has semistable reduction at $v$.
\item[{(ii)}] Conversely, if $X$ has semistable reduction at $v$,
and the degree of the polarization $\lambda$ is relatively prime
to $n$, then there exists a maximal isotropic subgroup
of $X_n$ with respect to $e_{\lambda,n}$, all of
whose points are defined over an extension
of $F$ which is unramified over $v$.
\end{enumerate}
\end{thm}
\begin{proof}
Under the hypotheses in (i),
let $S = \widetilde{X}_n$. Then $S^{\perp_n} = \lambda(S)$, and
$X$ has semistable reduction at $v$ by applying
Theorem \ref{ssredconverse}.
Conversely, suppose $X$ has semistable reduction at $v$. Let
${\bar v}$ be an extension of $v$ to $F^s$ and let
${\mathcal I} = {\mathcal I}({\bar v}/v)$. Let $S = X_n^{\mathcal I}$. If $G$ is a subgroup of
$X_n$, let
$$G^{\perp_{\lambda,n}} =
\{ y \in X_n : e_{\lambda,n}(x,y) = 1 \text{ for every } x \in G \}.$$
Since the degree of $\lambda$ is relatively prime to $n$,
$\lambda$ induces an isomorphism between $S^{\perp_{\lambda,n}}$
and $S^{\perp_n}$. Since $\lambda$ is defined over an unramified
extension, ${\mathcal I}$ acts as the identity on $S^{\perp_{\lambda,n}}$
by Proposition \ref{sslem}ii. Therefore,
$S^{\perp_{\lambda,n}} \subseteq S = X_n^{\mathcal I}$.
The pairing $e_{\lambda,n}$ induces a nondegenerate pairing on
$S/S^{\perp_{\lambda,n}}$. Let $H$ be the inverse image
in $S$ (under the natural projection) of a maximal isotropic
subgroup of $S/S^{\perp_{\lambda,n}}$. It is easy to check that
$H$ is a maximal isotropic subgroup of $X_n$ with respect
to $e_{\lambda,n}$, proving (ii).
\end{proof}
\begin{rems}
Raynaud's criterion (Theorem \ref{raynaud}) follows from
Theorem \ref{ssredconverse} by letting
$n = m^2$ and $S = X_m \subset X_{n}$
(since then $S^{\perp_n} = X^\ast_m$, the dual Galois module
of $X_m$, and $n \ge 5$ whenever $m \ge 3$).
The converse of Raynaud's criterion is clearly false,
i.e., semistable reduction does not imply that
the $n$-torsion points are unramified (for $n \ge 3$ and $n$ not
divisible by the residue characteristic), as can be seen, for
example, by
comparing Raynaud's criterion with the N\'eron-Ogg-Shafarevich
criterion for good reduction, and considering an abelian variety
with semistable but not good reduction.
Theorem \ref{ssred}i is Theorem 6.2 of \cite{semistab}.
Similarly, the other results of \cite{semistab} and of \S 3 of
\cite{connected} can readily be generalized to the setting
of Theorem \ref{ssredconverse}.
Theorem \ref{ssred}ii shows that the sufficient condition for
semistability given in Theorem 6.2 of \cite{semistab} comes close
to being a necessary condition.
Note that Theorem \ref{ssred}ii would be false if the condition on
the degree of the polarization were omitted.
\end{rems}
\begin{defn}
Suppose $v$ is a discrete valuation on $F$ of residue characteristic
$p$.
We say $v$ satisfies (*) if at least one of the following
conditions is satisfied:
\begin{enumerate}
\item[(a)] $p \ne 2$,
\item[(b)] the valuation ring
is henselian and the residue field is separably closed.
\end{enumerate}
\end{defn}
The techniques of the above proofs can be extended to prove
the following result. The proof will appear in \cite{etale}.
\begin{thm}
\label{highercoh}
Suppose $X$ is an abelian variety over a field $F$, and $v$ is a discrete
valuation on $F$ of residue characteristic $p \ge 0$.
Suppose $k \in {\mathbf Z}$, and
$0 < k < 2\mathrm{dim}(X)$.
\begin{enumerate}
\item[{(i)}] If either $X$ has semistable reduction at $v$,
or $k$ is even and
$X$ has purely additive reduction at $v$ which becomes
semistable over a quadratic extension of $F$,
then
$$(\sigma - 1)^{k+1}H^k_{\text{\'et}}(X \times_F F^s, {\mathbf Z}_{\ell}) = 0$$
for every $\sigma \in {\mathcal I}$ and every prime $\ell \ne p$,
and
$$(\sigma - 1)^{k+1}H^k_{\text{\'et}}(X \times_F F^s, {\mathbf Z}/n{\mathbf Z}) = 0$$
for every $\sigma \in {\mathcal I}$ and every positive integer $n$ not
divisible by $p$.
\item[{(ii)}] Suppose $n$ is a positive integer not divisible by
$p$, and
$$(\sigma - 1)^{k+1}H^k_{\text{\'et}}(X \times_F F^s, {\mathbf Z}/n{\mathbf Z}) = 0$$
for every $\sigma \in {\mathcal I}$.
Suppose $L$ is a degree $R(k+1,n)$ extension of $F$ which is
totally ramified above $v$, and let $w$ be the extension of
$v$ to $L$.
If $k$ is odd, then $X$ has semistable reduction at $w$.
If $k$ is even and $v$ satisfies (*),
then either $X$ has semistable reduction at $w$, or
$X$ has purely additive reduction at $w$ which becomes
semistable over a quadratic extension of $L$.
\end{enumerate}
\end{thm}
If we restrict to the case where $n \notin N(k+1)$, we obtain
the following result. This result gives necessary and sufficient
conditions for semistable reduction, and also necessary and
sufficient conditions for $X$ to have either semistable reduction
or purely additive reduction which becomes semistable over a
quadratic extension.
\begin{cor}
\label{highercohcor}
Suppose $X$ is an abelian variety over a field $F$, $v$ is a discrete
valuation on $F$ of residue characteristic $p \ge 0$, $k$ and $n$
are positive integers, $\ell$ is a prime number,
$k < 2\mathrm{dim}(X)$, $n$ and $\ell$ are not divisible by $p$,
and $n \notin N(k+1)$.
\begin{enumerate}
\item[{(i)}] Suppose $k$ is odd. Then the following are
equivalent:
\begin{enumerate}
\item[{(a)}] $X$ has semistable reduction at $v$,
\item[{(b)}] for every $\sigma \in {\mathcal I}$,
$$(\sigma - 1)^{k+1}H^k_{\text{\'et}}(X \times_F F^s,
{\mathbf Z}_{\ell}) = 0,$$
\item[{(c)}] for every $\sigma \in {\mathcal I}$,
$$(\sigma - 1)^{k+1}H^k_{\text{\'et}}(X \times_F F^s, {\mathbf Z}/n{\mathbf Z}) = 0.$$
\end{enumerate}
\item[{(ii)}] Suppose $k$ is even and $v$ satisfies (*).
Then the following are equivalent:
\begin{enumerate}
\item[{(a)}] either $X$ has semistable reduction at $v$, or
$X$ has purely additive reduction at $v$ which becomes
semistable over a quadratic extension of $F$,
\item[{(b)}] for every $\sigma \in {\mathcal I}$,
$$(\sigma - 1)^{k+1}H^k_{\text{\'et}}(X \times_F F^s, {\mathbf Z}_{\ell}) =
0,$$
\item[{(c)}] for every $\sigma \in {\mathcal I}$,
$$(\sigma - 1)^{k+1}H^k_{\text{\'et}}(X \times_F F^s, {\mathbf Z}/n{\mathbf Z}) = 0.$$
\end{enumerate}
\end{enumerate}
\end{cor}
\section{Exceptional $n$}
\label{except}
In this section we discuss briefly the ``exceptional'' cases
$n=2,3,4$. For the proofs, and for examples which show the results
are sharp, we refer the reader to \cite{degree}.
First, let us state the following ``one-way"
generalization of Theorem 4.5.
\begin{thm}
\label{oneway}
Suppose $X$ is an abelian variety over a field $F$, $v$ is a discrete
valuation on $F$, and $n$ is an integer greater than $1$ which is not
divisible by the residue characteristic of $v$.
Suppose there exists a subgroup $S$ of $X_n$
such that ${\mathcal I}$ acts as the identity on $S$ and on $S^{\perp_n}$.
Then $X$ has semistable reduction over every degree $R(n)$
extension of $F$ totally ramified above $v$.
\end{thm}
It turns out that the converse statement is not true.
However, the following result gives an ``approximate converse''.
\begin{thm}
\label{bothways}
Suppose $n = 2$, $3$, or $4$, respectively. Suppose
$X$ is an abelian variety over a field $F$, and
$v$ is a discrete valuation on $F$ whose residue characteristic
does not divide $n$.
Suppose
$L$ is an extension of $F$ of degree $4$, $3$, or
$2$, respectively, which
is totally ramified above $v$.
Then the following are equivalent:
\begin{enumerate}
\item[(i)] $X$ has semistable reduction over $L$ above $v$,
\item[(ii)] there exist an abelian variety $Y$ over a finite
extension $K$ of $F$ unramified above $v$, a separable
$K$-isogeny $\varphi : X \to Y$,
and a subgroup
$S$ of $Y_n$ such that
${\mathcal I}$ acts as the identity on $S$ and
on $S^{\perp_n}$.
\end{enumerate}
Further, $\varphi$ can be taken so that
its kernel is killed by $8$, $9$, or $4$, respectively.
If $X$ has potentially good reduction at $v$, then $\varphi$
can be taken so that its kernel is killed by $2$, $3$, or
$2$, respectively.
\end{thm}
In the case of low-dimensional $X$ this result may be improved as follows.
\begin{thm}
\label{ellcor}
In Theorem \ref{bothways},
with $d = \mathrm{dim}(X)$,
$\varphi$ can be taken so that its kernel is killed
by $4$ if $d = 3$ and $n = 2$,
by $3$ if $d =2$ and $n = 3$, and
by $2$ if $d = n = 2$. If $d = 1$, then we can take
$Y = X$ and $\varphi$ the identity map.
\end{thm}
In the case of elliptic curves this implies the following statement.
\begin{cor}
\label{4326cor}
Suppose
$X$ is an elliptic curve over a field $F$, and
$v$ is a discrete valuation on $F$ of residue characteristic
$p \ge 0$.
\begin{enumerate}
\item[(a)] If $p \ne 2$,
then $X$ has semistable reduction above $v$ over a
totally ramified quartic extension of $F$
if and only if
$X$ has an ${\mathcal I}$-invariant point of order $2$.
\item[(b)] If $p \ne 3$,
then $X$ has semistable reduction above $v$ over a totally ramified
cubic extension of $F$
if and only if $X$ has an ${\mathcal I}$-invariant point of order $3$.
\item[(c)]
If $p \ne 2$,
then $X$ has semistable reduction above $v$ over a quadratic
extension of $F$ if and only if
either $X$ has an ${\mathcal I}$-invariant point of order $4$,
or all the points of order $2$ on $X$ are ${\mathcal I}$-invariant.
\item[(d)]
If $p \ne 2$ and $X$ has bad but potentially good reduction
at $v$,
then $X$ has good reduction above $v$ over a quadratic
extension of $F$ if and only if
$X$ has no ${\mathcal I}$-invariant point of order $4$
and all its points of order $2$ are ${\mathcal I}$-invariant.
\item[(e)] Suppose $p$ is not $2$ or $3$. Then the following
are equivalent:
\begin{enumerate}
\item[(i)] $X$ has no ${\mathcal I}$-invariant points of order $2$ or $3$,
\item[(ii)] there does not exist a finite separable extension $L$
of $F$ of degree less than $6$
such that $X$ has semistable reduction at the restriction of
${\bar v}$ to $L$.
\end{enumerate}
\item[(f)] Suppose $p$ is not $2$ or $3$. Then the following
are equivalent:
\begin{enumerate}
\item[(i)] $X$ has no ${\mathcal I}$-invariant points of order $4$ or $3$
and not all the points of order $2$ are ${\mathcal I}$-invariant,
\item[(ii)] there does not exist a finite separable extension $L$
of $F$ of degree less than $4$
such that $X$ has semistable reduction at the restriction
of ${\bar v}$ to $L$.
\end{enumerate}
\end{enumerate}
\end{cor}
In the case of potentially good reduction the following statement
holds true.
\begin{thm}
\label{paddcor}
Suppose
$X$ is an abelian variety over a field $F$,
$v$ is a discrete valuation on $F$ of residue characteristic
$p \ge 0$, and $X$ has purely additive and
potentially good reduction at $v$.
\begin{enumerate}
\item[(a)]
If $p \ne 2$, then $X$ has good reduction above $v$ over a quadratic
extension of $F$ if and only if
there exists a subgroup $S$ of $X_4$ such
that ${\mathcal I}$ acts as the identity on $S$ and on $S^{\perp_4}$.
\item[(b)] If $p \ne 3$,
then $X$ has good reduction above $v$ over a totally ramified cubic
extension of $F$ if and only if
there exists a subgroup $S$ of $X_3$ such
that ${\mathcal I}$ acts as the identity on $S$ and on $S^{\perp_3}$.
\item[(c)] Suppose $p \ne 2$, and
$L/F$ is a degree $4$ extension,
totally ramified above $v$,
which has a quadratic subextension over which $X$ has purely
additive reduction.
Then $X$ has good reduction above $v$ over $L$ if and only if
there exists a subgroup $S$ of $X_2$ such
that ${\mathcal I}$ acts as the identity on $S$ and on $S^{\perp_2}$.
\end{enumerate}
\end{thm}
\section{A measure of potentially good reduction}
\label{Gsect}
Suppose $v$ is a discrete valuation on a field $F$, and
$X$ is an abelian variety over $F$ which has potentially good
reduction at $v$. Let $F_{v}^{nr}$ denote the maximal unramified
extension of the completion of $F$ at $v$, let $L$ denote
the smallest extension of $F_{v}^{nr}$ over which $X$ has good
reduction, and let
$$G_{v,X} = \mathrm{Gal}(L/F_{v}^{nr}).$$
Then $G_{v,X}$ can also be characterized as the inertia group of
the extension $F(X_n)/F$, where $n$ is any integer
greater than $2$ and not divisible by the residue characteristic
of $v$ (see Corollary 2 on p.~497 of \cite{Serre-Tate}).
Clearly, $X$ has good reduction at $v$ if and only if
$G_{v,X} = 1$. The finite group $G_{v,X}$ is a measure of how
far $X$ is from having good reduction at $v$.
If $A$ is a matrix, let $P_A$ denote its characteristic polynomial.
The following result gives constraints on the group $G_{v,X}$.
\begin{thm}
\label{Gthm}
Suppose $v$ is a discrete valuation on a field $F$, and
$X$ is a $d$-dimensional abelian variety over
$F$ which has potentially good reduction at $v$.
Let $G=G_{v,X}$. Suppose $\ell$ is a prime number not
equal to the residue characteristic of $v$.
Then the action of $\mathrm{Gal}(F^s/F)$ on the $\ell$-adic Tate module
$V_\ell(X)$ induces an embedding
$$f : G \hookrightarrow \mathrm{Sp}_{2d}({\mathbf Q}_\ell)$$
which satisfies the following properties.
\begin{enumerate}
\item[(i)]
For every $\sigma \in G$, the coefficients of $P_{f(\sigma)}$
are integers which are independent of $\ell$.
If $X$ has an $F$-polarization of degree not divisible by $\ell$,
then one may choose $f$ so that its image lies in
$\mathrm{Sp}_{2d}({\mathbf Z}_\ell)$.
\item[(ii)]
If either $(\ell,\#G)=1$ or $\ell > d+1$, then there exists
an embedding
$$g : G \hookrightarrow \mathrm{Sp}_{2d}({\mathbf Z}_\ell)$$
such that $P_{g(\sigma)} = P_{f(\sigma)}$
for every $\sigma \in G$.
\item[(iii)]
If $\ell \ge 5$ then there exists an embedding
$$h : G \hookrightarrow \mathrm{Sp}_{2d}(\mathbf{F}_\ell)$$
such that $P_{h(\sigma)} \equiv P_{f(\sigma)} \pmod{\ell}$
for every $\sigma \in G$.
\end{enumerate}
Further, if $\ell \ge 5$ then there exists an embedding
$$G \hookrightarrow \mathrm{Sp}_{2d}({\mathbf Z}_\ell)$$
(which does not necessarily ``preserve'' the characteristic
polynomials obtained from the embedding $f$).
\end{thm}
See \cite{Serre-Tate} for (i), and
see \cite{inertia} for the case $(\ell,\#G)=1$ of
(ii). The remainder of Theorem \ref{Gthm} follows from results
whose proofs will appear elsewhere (along with examples which
show that the results are sharp). Those results apply more
generally to measure how far an abelian variety (not necessarily
with potentially good reduction) is from having semistable reduction.
In some cases, these results apply to more general finite groups
than those obtained as $G_{v,X}$'s.
\part{N\'eron models of abelian varieties with potentially good reduction}
\label{neronpart}
\section{Preliminaries}
In \cite{serrelem}, the following result was obtained as
a corollary of Theorem \ref{quasithm} above.
\begin{prop}[Theorem 6.10a of \cite{serrelem}]
\label{randm}
Suppose $\ell$ is a prime, $m$ and $r$ are positive integers, ${\mathcal O}$ is an
integral domain of characteristic zero with no non-zero infinitely
$\ell$-divisible elements, $\ell{\mathcal O}$ is a maximal ideal of ${\mathcal O}$, $M$ is
a free ${\mathcal O}$-module of finite rank, and $A$ is an endomorphism of $M$
of finite multiplicative order such that
$(A - 1)^{m(\ell - 1)\ell^{r-1}} \in \ell^m\mathrm{End}(M)$.
If $r > 1$, then the torsion
subgroup of $M/(A - 1)M$ is killed by $\ell^{r-1}$.
\end{prop}
\begin{prop}[see Proposition 6.1i and Corollary 7.1 of \cite{semistab}]
\label{pressred}
Suppose $X$ is a $d$-dimensional abelian variety over a field $F$,
$v$ is a discrete valuation on $F$ with residue characteristic
not equal to $2$,
$\lambda$ is a polarization on $X$,
$\widetilde{X}_2$ is a maximal isotropic subgroup of $X_2$ with respect to
$e_{\lambda,2}$, $\lambda$ and the points of
$\widetilde{X}_2$ are defined over an extension of $F$ which is
unramified over $v$,
${\bar v}$ is an extension of $v$ to a separable closure of $F$,
and $\sigma \in {\mathcal I}({\bar v}/v)$.
Then $(\rho_{2,X}(\sigma) - I)^2 \in 2\mathrm{M}_{2d}({\mathbf Z}_2)$,
and $X$ has semistable
reduction above $v$ over every totally ramified Galois
(necessarily cyclic) extension of $F$ of degree $4$.
\end{prop}
Recall that $u$ denotes the unipotent rank of $X_v$, $a$ denotes the
abelian rank, and
$\Phi '$ denotes the prime-to-$p$ part of the group of
connected components of the special
fiber of the N\'eron model of $X$ at $v$, where $p$ is the residue
characteristic of the discrete valuation $v$. If $X$ has potentially
good reduction, then $\mathrm{dim}(X) = a + u$.
\begin{thm}[Theorem 7.5 of \cite{semistab}]
\label{neronmod}
Suppose $v$ is a discrete valuation on a field $F$ with strictly henselian
valuation ring, $X$ is an abelian
variety over $F$ which has potentially good reduction at $v$, and either
\begin{enumerate}
\item[{(a)}] $n = 2$ and the points of $X_2$
are defined over $F$, or
\item[{(b)}] $n = 3$ or $4$, $\lambda$ is a polarization on $X$
defined over $F$, and
the points of a maximal isotropic subgroup of $X_n$ with respect to
$e_{\lambda,n}$ are defined over $F$.
\end{enumerate}
Suppose the residue characteristic $p$ ( $\ge 0$) of $v$ does
not divide $n$.
Then $\Phi ' \cong ({\mathbf Z}/2{\mathbf Z})^{2u}$
if $n = 2$ or $4$, and $\Phi ' \cong ({\mathbf Z}/3{\mathbf Z})^u$ if $n = 3$.
\end{thm}
\begin{lem}
\label{neronlemma}
Suppose
$v$ is a discrete valuation on a field $F$ such that
the valuation ring is strictly henselian.
Suppose $X$ is an abelian
variety over $F$ which has potentially good reduction at $v$,
and
suppose $n$ is a positive integer not divisible by the residue
characteristic of $v$.
Let $\Phi_n$ denote the subgroup of
$X_v/X_v^0$ of points of order dividing $n$.
Then:
\begin{enumerate}
\item[(i)] $(X_v)_n \cong X_n(F)$,
\item[(ii)] $(X_v^0)_n \cong ({\mathbf Z}/n{\mathbf Z})^{2a}$,
\item[(iii)] $\Phi_n \cong (X_v)_n/(X_v^0)_n$, and
\item[(iv)] if $X_n(F) \cong ({\mathbf Z}/n{\mathbf Z})^{b}$,
then $\Phi_n \cong ({\mathbf Z}/n{\mathbf Z})^{b-2a}$.
\end{enumerate}
\end{lem}
\begin{proof}
By Lemma 2 of \cite{Serre-Tate}, the reduction map defines
an isomorphism of $X_n^{\mathcal I}$ onto $(X_v)_n$,
where ${\mathcal I} = {\mathcal I}({\bar v}/v)$ for some extension ${\bar v}$
of $v$ to $F^s$.
Under our hypotheses on $v$, we have
$X_n^{\mathcal I} \cong X_n(F)$. Therefore,
$(X_v)_n \cong X_n(F)$.
As shown in the proof of Lemma 1 of \cite{Serre-Tate},
$(X_v^0)_n \cong ({\mathbf Z}/n{\mathbf Z})^{2a+t}$, where $t$ denotes
the toric rank of $X_v$.
Since $X$ has potentially good reduction at $v$, $t = 0$.
Since $X_v^0$ is $n$-divisible, we have
$\Phi_n \cong (X_v)_n/(X_v^0)_n$. Part (iv) follows
easily from (i), (ii), and (iii).
\end{proof}
\section{N\'eron models}
\label{main}
In Theorem \ref{neron} we generalize Theorem \ref{neronmod}
to the case of partial level $2$ structure. We can recover
Theorem \ref{neronmod}a as a special case.
Recall that $u$ denotes the unipotent rank of $X_v$,
$a$ denotes the abelian rank, and
$\Phi '$ denotes the prime-to-$p$ part of the group of
connected components of the special
fiber of the N\'eron model of $X$ at $v$, where $p$ is the residue
characteristic of $v$ (with $\Phi '$ the full
group of components if $p = 0$).
\begin{thm}
\label{neron}
Suppose $v$ is a discrete valuation on a field $F$, suppose
the valuation ring is strictly henselian, and suppose the residue field
has characteristic $p \ne 2$.
Suppose $(X, \lambda)$ is a $d$-dimensional polarized abelian
variety over $F$, $X$ has potentially good reduction at $v$, and
the points of a maximal isotropic subgroup of $X_2$ with respect to
$e_{\lambda,2}$ are defined over $F$.
Then:
\begin{enumerate}
\item[(i)] $\Phi' \cong ({\mathbf Z}/2{\mathbf Z})^{b-2a} = ({\mathbf Z}/2{\mathbf Z})^{b+2u-2d}$,
where $b$ is defined by $X_2(F) \cong ({\mathbf Z}/2{\mathbf Z})^{b}$,
\item[(ii)] $[X_2 : X_2(F)]\#\Phi' = 2^{2u}$, and
\item[(iii)] $X$ has good reduction at $v$ if and only if
$\Phi' = \{0\}$ and $X_2 \subseteq X(F)$.
\end{enumerate}
\end{thm}
\begin{proof}
Let ${\bar v}$ be an extension of $v$ to a separable closure of $F$,
let ${\mathcal I} = {\mathcal I}({\bar v}/v)$, let $k$ be the residue field of $v$, and
let ${\mathcal J}$ be the first ramification group (i.e., ${\mathcal J}$ is trivial if
$p = 0$ and ${\mathcal J}$ is the pro-$p$-Sylow subgroup of ${\mathcal I}$ if $p > 0$).
Suppose $q$ is a prime not equal to $p$, and let
$\Phi_q$ denote the $q$-part of the
group of connected components of the special
fiber of the N\'eron model of $X$.
Since $X$ has potentially good reduction at $v$, $\rho_{q,X}(\sigma)$
has finite multiplicative order for every $\sigma \in {\mathcal I}$.
Let $\tau$ be a lift to ${\mathcal I}$ of a generator of the pro-cyclic group
${\mathcal I}/{\mathcal J}$. By \S11 of \cite{SGA} (see Lemma 2.1 of \cite{Lorenzini}),
$$\Phi_q \text{ is isomorphic to the torsion subgroup of }
T_q(X)^{\mathcal J}/(\tau - 1)T_q(X)^{\mathcal J}.$$
By Proposition \ref{pressred} and Remark \ref{ramifiedcyclic},
$X$ has semistable reduction (and therefore good reduction) above $v$ over
a totally ramified cyclic Galois extension of $F$ of degree $4$. Therefore
${\mathcal I}$ acts on $T_q(X)$ through a cyclic quotient of order $4$, so
$\rho_{q,X}(\sigma)^4 = I$ for every $\sigma \in {\mathcal I}$. Since $p \ne 2$,
we have $\rho_{q,X}(\sigma) = I$ for every $\sigma \in {\mathcal J}$. Therefore,
$T_q(X)^{\mathcal J} = T_q(X)$. If $q \ne 2$, then
$T_q(X)/(\rho_{q,X}(\tau) - I)T_q(X)$
is torsion-free, so $\Phi_q$ is trivial. Further,
$$\Phi_2 \text{ is isomorphic to the torsion subgroup of }
T_2(X)/(\tau - 1)T_2(X).$$
We have $(\rho_{2,X}(\tau) - I)^2 \in 2\mathrm{M}_{2d}({\mathbf Z}_2)$, by Proposition \ref{pressred}.
By Proposition \ref{randm} with $\ell = 2$, $r = 2$, $m = 1$, and ${\mathcal O} = {\mathbf Z}_2$,
$\Phi_2$ is annihilated by $2$.
Therefore, $\Phi'$ is an elementary abelian $2$-group.
By Lemma \ref{neronlemma}, $\Phi' \cong ({\mathbf Z}/2{\mathbf Z})^{b-2a}$.
Parts (ii) and (iii) follow immediately.
Note that Theorem \ref{neronmod}a is a special case of Theorem
\ref{neron}.
\end{proof}
\begin{thm}
\label{neron3}
Suppose $v$ is a discrete valuation on a field $F$, suppose
the valuation ring is strictly henselian, and suppose the residue field
has characteristic $p \ne 3$.
Suppose $(X, \lambda)$ is a $d$-dimensional polarized abelian
variety over $F$, $X$ has potentially good reduction at $v$, and
the points of a maximal isotropic subgroup of $X_3$ with respect to
$e_{\lambda,3}$ are defined over $F$.
Then:
\begin{enumerate}
\item[(i)] $X_3(F) \cong ({\mathbf Z}/3{\mathbf Z})^{2d-u}$,
\item[(ii)] $X$ has good reduction at $v$ if and only if
$X_3(F) = X_3$, and
\item[(iii)] $X$ has purely additive reduction at $v$ if and only if
$X_3(F) \cong ({\mathbf Z}/3{\mathbf Z})^{d}$.
\end{enumerate}
\end{thm}
\begin{proof}
By Theorem \ref{neronmod},
$\Phi' \cong ({\mathbf Z}/3{\mathbf Z})^u$. Write $X_3(F) \cong ({\mathbf Z}/3{\mathbf Z})^b$.
By Lemma \ref{neronlemma},
$\Phi' \cong ({\mathbf Z}/3{\mathbf Z})^{b-2d+2u}$.
Therefore, $b = 2d - u$, and we obtain the desired result.
\end{proof}
\begin{thm}
\label{neron4}
Suppose $v$ is a discrete valuation on a field $F$ with strictly henselian
valuation ring, $X$ is an abelian variety over $F$ which has potentially
good reduction at $v$, the residue field has characteristic $p \ne 2$,
and either
\begin{enumerate}
\item[{(a)}] the points of $X_2$ are defined over $F$, or
\item[{(b)}] $\lambda$ is a polarization on $X$ defined over $F$, and
the points of a maximal isotropic subgroup of $X_4$ with respect to
$e_{\lambda,4}$ are defined over $F$.
\end{enumerate}
Then
$$X_4(F) \cong ({\mathbf Z}/4{\mathbf Z})^{2a} \times ({\mathbf Z}/2{\mathbf Z})^{2u}.$$
In particular:
\begin{enumerate}
\item[(i)] $X_2 \subseteq X_4(F) \subseteq X_4$,
$[X_4 : X_4(F)] = 2^{2u}$, $[X_4(F) : X_2] = 2^{2a}$,
\item[(ii)] $X$ has good reduction at $v$ if and only if
$X_4(F) = X_4$, and
\item[(iii)] $X$ has purely additive reduction at $v$ if and only if
$X_4(F) = X_2$.
\end{enumerate}
\end{thm}
\begin{proof}
By Theorem \ref{neronmod},
we have $\Phi' \cong ({\mathbf Z}/2{\mathbf Z})^{2u}$. By Lemma \ref{neronlemma},
we have a short exact sequence
$$0 \to ({\mathbf Z}/4{\mathbf Z})^{2a} \to X_4(F) \to ({\mathbf Z}/2{\mathbf Z})^{2u} \to 0.$$
Let $d = \mathrm{dim}(X)$.
Since $X_4(F) \subseteq X_4 \cong ({\mathbf Z}/4{\mathbf Z})^{2d}$,
we conclude that $X_4(F) \cong ({\mathbf Z}/4{\mathbf Z})^{2a} \times ({\mathbf Z}/2{\mathbf Z})^{2u}$.
Note that $X_2 \cong ({\mathbf Z}/2{\mathbf Z})^{2d} = ({\mathbf Z}/2{\mathbf Z})^{2a + 2u}$.
The rest of the result follows immediately.
\end{proof}
As an example, let $X$ be the elliptic curve defined by
the equation $y^2 = x^3 - 9x$, and let $F$ be the maximal unramified
extension of ${\mathbf Q}_3$.
Then $X_2(F) = X_2 = X_4(F)$, $X$ has additive and potentially
good reduction, and $\Phi' \cong ({\mathbf Z}/2{\mathbf Z})^{2}$.
\begin{rems}
If $X$ has a polarization $\lambda$ of odd degree, then $X_2$ is
a maximal isotropic subgroup of $X_4$ with respect to $e_{\lambda,4}$.
As stated in the Introduction,
Theorems \ref{neron3}ii and \ref{neron4}ii are immediate
corollaries of Raynaud's criterion for semistable reduction.
If $X$ has purely additive reduction, then $X_n(F) \cong
\Phi_n$ (see \cite{Lorenzini}).
Suppose $v$ is a discrete valuation on a field $F$,
$X$ is an abelian variety over $F$ with potentially good reduction at $v$,
the valuation ring is strictly henselian, $\ell = 2$ or $3$, and
the residue characteristic is not equal to $\ell$.
Then Theorem 6.1
of \cite{Edixhoven} implies that if $\Phi'$ is an elementary abelian
$\ell$-group,
then $\Phi'$ is a subgroup of $({\mathbf Z}/2{\mathbf Z})^{2u}$ if $\ell = 2$ or of
$({\mathbf Z}/3{\mathbf Z})^{u}$ if $\ell = 3$.
For simplicity of exposition, we do not generalize the results of
\S \ref{main} (or the prerequisite results from \cite{semistab}, or
related results in \S 3 of \cite{connected})
to the setting of Theorem \ref{ssredconverse}, but leave such
generalizations as a straightforward exercise for the reader.
\end{rems}
|
1996-02-06T06:20:19 | 9602 | alg-geom/9602005 | en | https://arxiv.org/abs/alg-geom/9602005 | [
"alg-geom",
"math.AG"
] | alg-geom/9602005 | Frank Sottile | Frank Sottile | Real enumerative geometry and effective algebraic equivalence | 12 pages, LaTeX 2e | J. Pure and Appl. Alg., 117 & 118 (1997) 601--615 | null | null | null | We describe an approach to the question of finding real solutions to problems
of enumerative geometry, in particular the question of whether a problem of
enumerative geometry can have all of its solutions be real. We give some
methods to infer one problem can have all of its solutions be real, given that
a related problem does. These are used to show many Schubert-type enumerative
problems on some flag manifolds can have all of their solutions be real.
| [
{
"version": "v1",
"created": "Tue, 6 Feb 1996 01:03:24 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Sottile",
"Frank",
""
]
] | alg-geom | \section{Introduction}
Determining the common zeroes of a set of polynomials
is further complicated over non-algebraically closed fields such
as the real numbers.
A special case is whether a problem of enumerative geometry can have all
its solutions be real.
We call such a problem {\em fully real}.
Little is known about enumerative geometry from this perspective.
A standard proof
of B\'ezout's Theorem shows the problem of intersecting hypersurfaces in
projective space is fully real.
Khovanskii~\cite{Khovanskii_fewnomials} considers
intersecting hypersurfaces in a torus defined by few monomials
and shows the real zeros are at most a fraction of the
complex zeroes.
Fulton, and more recently, Ronga, Tognoli and Vust~\cite{Ronga_Tognoli_Vust}
have shown the problem of 3264 plane conics tangent to five given
conics is fully real.
The author~\cite{sottile_real_lines} has shown
all problems of enumerating lines incident on linear
subspaces of projective space are fully real.
There are few methods for studying this phenomenon.
We ask:
How can the knowledge that one enumerative problem is fully
real be used to infer that a related problem is fully real?
We give several procedures to accomplish this inference and
examples of their application, lengthening the list of enumerative
problems known to be fully \smallskip real.
We study intersections of any dimension,
not just the zero dimensional intersections of enumerative problems.
Our technique is to deform general intersection
cycles into simpler cycles.
This modification of the classical method of degeneration
was used by Chiavacci and
Escamilla-Castillo~\cite{Chiavacci_Escamilla-Castillo}
to investigate these questions.
Let $\alpha_1,\ldots,\alpha_a$ be cycle classes spanning
the Chow ring of a smooth
variety.
For cycle classes $\beta_1,\ldots,\beta_b$,
there exist integers $c_i$ for $i = 1,\ldots, a$ such that
$$
\beta_1\cdots\beta_b \ =\
\sum_{i=1}^a c_i\cdot \alpha_i.
$$
When the $c_i\geq 0$,
this product formula has a geometric interpretation.
Suppose $Y_1,\ldots,Y_b$ are cycles representing the classes
$\beta_1,\ldots,\beta_b$ which meet generically transversally in a
cycle $Y$.
Then $Y$ is algebraically equivalent to
$Z:=Z_1\cup\cdots\cup Z_a$, where
$Z_i$ has $c_i$ components, each representing the cycle class $\alpha_i$.
This algebraic equivalence is effective if there is a family of cycles
containing both $Y$ and $Z$ whose general member is
a generically transverse intersection of cycles representing the
classes $\beta_1,\ldots,\beta_b$.
If the cycles $Y_1,\ldots,Y_b$ and each component of $Z$ are
defined over ${\Bbb R}$ and both $Y$ and $Z$ are in the same connected
component of the real points of that family,
then the effective algebraic equivalence is real.
Real effective algebraic equivalence
can be used to show an enumerative problem is fully real, or
more generally, to obtain lower bounds on the maximal
number of real solutions.
Suppose the cycles
$Y_1,\ldots,Y_b, W_1,\ldots, W_c$ give an enumerative problem
and the problem obtained by substituting $Z$ for $Y_1,\ldots,Y_b$
has at least $d$ real solutions.
Then there exist real cycles $Y'_1,\ldots,Y'_b$ such that
the original problem (with $Y'_i$ in place of $Y_i$)
has at least $d$ real solutions,
since the number of real solutions is preserved by
small real deformations.
Sections 2 through 5 introduce and develop our basic notions and techniques.
Subsequent sections are devoted to elaborations and
applications of these ideas.
In Section 6, we prove that any enumerative problem on a
flag variety involving five Schubert varieties, three of
which are special Schubert varieties, is fully real.
Given a map $\pi: Y\rightarrow X$ with equidimensional fibres,
real effective algebraic equivalence on $X$ and $Y$ is compared
in Section 7 and used in Sections 8 and 9
to show that many Schubert-type enumerative problems
in two classes of flag varieties are fully real.
A proof of B\'ezout's Theorem in Section 10 suggests another
method for obtaining fully real enumerative problems.
This is applied in Section 11 to show the enumerative problem of
$(n-2)$-planes in
${\Bbb P}^n$ meeting $2n-2$ rational normal curves is fully real.
The author thanks Bernd Sturmfels for encouraging these investigations.
\section{Intersection Problems}
\subsection{Conventions}
Varieties are reduced, complex, and defined over the
real numbers ${\Bbb R}$.
Let $X$ and $Y$ denote smooth
projective varieties and $U$, $V$, and $W$ smooth quasi-projective
varieties.
Equip the real points $X({\Bbb R})$ of $X$
with the classical topology.
Let $A^*X$ be the Chow ring of cycles
modulo algebraic equivalence.
Two subvarieties meet {\em generically transversally}
if they meet transversally along a dense subset of each
component of their intersection.
Such an intersection scheme is reduced at the generic point of each
component, or generically reduced.
A subvariety $\Xi\subset U\times X$ (or $\Xi\rightarrow U$)
with generically reduced equidimensional fibres
over a smooth base
$U$ is a family of {\em multiplicity free cycles on $X$ over $U$.}
All fibres of $\Xi$ over $U$ are algebraically equivalent, and we say
$\Xi\rightarrow U$ {\em represents} that algebraic equivalence class.
\subsection{Chow varieties}\label{sec:Chow}
Positive cycles on $X$ of a fixed dimension and degree are parameterized
by the Chow variety of $X$.
We suppress the dependence on dimension and degree
and write $\mbox{\it Chow}\, X$ for any Chow variety of $X$.
The open Chow variety ${\mbox{\it Chow}}^\circ X$ is the open subset of $\mbox{\it Chow}\, X$
parameterizing multiplicity free cycles on $X$.
There is a tautological family
$\Phi \rightarrow {\mbox{\it Chow}}^\circ X$
of cycles on $X$ with the property
that $\zeta\in{\mbox{\it Chow}}^\circ X$ represents the the fundamental cycle of
the fibre $\Phi_\zeta$.
Let $\Xi\rightarrow U$ be a family of multiplicity
free cycles on $X$.
The association of a point $u$ of $U$ to the fundamental cycle of
the fibre $\Xi_u$ defines the
{\em fibre function} $\phi$, which is algebraic
on a dense open subset $U'$ of $U$.
If $U$ is a curve, then $U=U'$.
\subsection{Proposition}
{\em $\phi(U')$ is dense in the set $\phi(U)$.}\medskip
\noindent{\bf Proof:}
Let $u\in U$ and $C\subset U$ be a smooth curve with $u\in C$ and
$C-\{u\}\subset U'$.
Such a curve is not necessarily closed in
$U$, but is the smooth points of a closed subvariety.
The fibre function $\phi|_C$ of $\Xi|_C \rightarrow C$
is algebraic, hence
$\phi(u)\in \overline{\phi(C-\{u\})} \subset \overline{\phi(U')}$.
\QED\vspace{10pt}
Two families $\Xi\rightarrow U$ and $\Psi\rightarrow V$
of multiplicity free cycles on $X$ are {\em equivalent} if
$\overline{\phi(U)} = \overline{\phi(V)}$, that is, if
they have essentially the same cycles.
Our results remain valid when
one family of cycles is replaced by an equivalent family,
perhaps with the additional assumption that
$\overline{\phi(U({\Bbb R}))} = \overline{\phi(V({\Bbb R}))}$.
The varieties $\mbox{\it Chow}\, X$ and ${\mbox{\it Chow}}^\circ X$ as well as $U'$ and the
morphism $\phi: U' \rightarrow \mbox{\it Chow}\, X$ are defined over
${\Bbb R}$~(\cite{Samuel}, \S I.9).
We use $\phi$ to denote all fibre functions.
Any ambiguity may be resolved by context.
\subsection{Intersection Problems}\label{sec:intersection_problems}
For $1\leq i\leq b$,
let $\Xi_i\rightarrow U_i$ be a family of multiplicity free cycles
on $X$.
Let $U\subset \prod_{i=1}^b U_i$ be the locus where the fibres of the
product family $\prod_{i=1}^b \Xi_i$ meet the (small) diagonal
$\Delta^b_X$ of $X^b$ generically transversally.
Equivalently, $U$ is the locus where fibres of
$\Xi_1,\ldots,\Xi_b$ meet generically transversally in $X$.
If $U$ is nonempty, then
$\Xi_1,\ldots,\Xi_b$ give a {\em (well-posed) intersection problem}.
Given an intersection problem as above, let
$\delta:X\stackrel{\sim}{\longrightarrow}\Delta^b_X\subset X^b$ and set
$\Xi$ to be
$$
\Xi \ :=\ (1_U\times \delta)^* \prod_{i=1}^b \Xi_i \ \subset\ U\times X,
$$
a family of multiplicity
free cycles on $X$ over $U$.
We often suppress the dependence on the original families and write
$\Xi\rightarrow U$ for this intersection problem.
Not all collections of families of cycles give well-posed
intersection problems,
some transversality is needed to guarantee $U$ is nonempty.
When a reductive group acts transitively on $X$, Kleiman's
Transversality Theorem~\cite{Kleiman} has the following consequence.
\subsection{Proposition}\label{prop:transitive_action}
{\em
Suppose a reductive group acts transitively on $X$, $\Xi_1$ is a constant
family, and for
$2\leq i\leq b$, $\Xi_i$ is equivalent to a family of multiplicity free cycles
stable under that action.
Then $\Xi_1,\ldots,\Xi_b$ give a well-posed intersection problem.
}\medskip
Grassmannians and flag varieties have such an action.
For these, we suppose all families of cycles
are stable under that action, and thus
give well-posed intersection problems.
Suppose a reductive group acts on $X$ with a dense open orbit $X'$.
For example, if $X$ is a toric variety, or more generally, a
spherical variety~\cite{Brion_spherical_introduction,%
Knop_spherical_expository,Luna_Vust_Plongements}.
Each family may be stable under that action, but the collection need not
give a well-posed intersection problem as
Kleiman's Theorem~\cite{Kleiman} only guarantees transversality in $X'$.
However, it is often the case that only points of
intersection in $X'$ are desired,
and suitable blow up of $X$ or a different equivariant compactification
of $X'$ exists on which
the corresponding intersection problem is well-posed
(see~\cite{Fulton_introduction_intersection}, \S 1.4
or~\cite{Fulton_intersection}, \S 9 and \S 10.4).
\section{effective algebraic equivalence}
\label{sec:effective_algebraic_equivallences}
Let $\alpha_1,\ldots,\alpha_a$ be distinct
additive generators of $A^*X$,
and for $1\leq i\leq a$, suppose
$\Psi(\alpha_i)\rightarrow V(\alpha_i)$ is a family of multiplicity free
cycles on $X$ representing the cycle class $\alpha_i$.
When $X$ is a Grassmannian or flag variety,
$\alpha_1,\ldots,\alpha_a$ will be the Schubert classes,
and $\Psi(\alpha_i)\rightarrow V(\alpha_i)$ the corresponding families
of Schubert varieties.
A family of multiplicity free cycles
$\Xi\subset U\times X$ has an {\em effective algebraic equivalence}
with {\em witness} $Z\in \overline{\phi(U)}\cap {\mbox{\it Chow}}^\circ X$
if each (necessarily multiplicity free) component of $Z$ is a fibre
of some family $\Psi(\alpha_i)$.
This effective algebraic equivalence is {\em real} if
$Z\in \overline{\phi(U({\Bbb R}))}$ and
each component of $Z$ is a fibre over a real point of some $V(\alpha_i)$.
An intersection problem
$\Xi_1,\ldots,\Xi_b$ has
{\em (real) effective algebraic equivalences}
if its family of intersection cycles $\Xi\rightarrow U$
has (real) effective algebraic equivalences.
\subsection{Products in $A^*X$}
\label{sec:products}
Suppose $\beta_1,\ldots,\beta_b$ are classes from
$\{\alpha_1,\ldots,\alpha_a\}$
and the families $\Psi(\beta_1),\ldots,\Psi(\beta_b)$
give an intersection problem $\Psi\rightarrow V$.
We say $\Psi\rightarrow V$ is given by $\beta_1,\ldots,\beta_b$.
Suppose $\Psi\rightarrow V$ has an
effective algebraic equivalence with witness $Z$.
Fibres of
$\Psi\rightarrow V$ are generically transverse intersections
of fibres of $\Psi(\beta_1),\ldots,\Psi(\beta_b)$, and so have
cycle class $\beta_1\cdots\beta_b$.
As $Z\in \overline{\phi(V)}$, this equals the cycle class of $Z$,
which is $\sum_{i=1}^a c_i \alpha_i$,
where $c_i$ counts the components of $Z$
lying in the family $\Psi(\alpha_i)$.
Thus in $A^*X$, we have
$$
\hspace{2.5in}
\beta_1\cdots\beta_b \ =\ \sum_{i=1}^a c_i \alpha_i.
\hspace{2.2in}
(\ref{sec:products})
$$
To compute products in $A^*X$, classical geometers would try to
understand a generically transverse intersection of degenerate
cycles in special position, as a generic intersection cycle is
typically too difficult to describe.
Effective algebraic equivalence extends this method of degeneration
by also considering
limiting positions of such intersection cycles as the subvarieties
degenerate further, attaining excess intersection.
\subsection{Pieri-type intersection problems}\label{sec:pieri_type}
A Schubert subvariety of a flag variety is determined by a
complete flag ${F\!_{\DOT}}$ and a
coset $w$ of a parabolic subgroup in the symmetric group.
Thus Schubert classes $\sigma_w$ are indexed by these cosets
and families $\Psi_w$ of Schubert varieties have base
${\Bbb F}\ell$, the variety of complete flags.
A {\em special Schubert subvariety} of a Grassmannian
is the locus of planes meeting a fixed linear subspace non-trivially,
or the image of such a subvariety in the dual Grassmannian.
More generally, a special Schubert subvariety of a flag variety
is the pullback of a special Schubert subvariety from
a Grassmannian projection.
If $m$ is the index of a special Schubert class, then the Pieri-type
formulas
of~\cite{Lascoux_Schutzenberger_polynomes_schubert,Sottile_Pieri_Schubert}
show that for any $w$,
there exists a subset $I_{m,w}$ of these cosets
such that
$$
\hspace{2.5in}
\sigma_m\cdot \sigma_w\ =\ \sum_{v\in I_{m,w}}\sigma_v.
\hspace{2.2in}
(\ref{sec:pieri_type})
$$
\subsection{Theorem}\label{thm:pieri_effective_equivalences}
{\em
The intersection problem $\Xi\rightarrow U$ given by the classes
$\sigma_m$ and $\sigma_w$
has real effective algebraic equivalences.
}\medskip
\noindent{\bf Proof:}
The Borel subgroup $B$ of $GL_n{\Bbb C}$ stabilizing a real
complete flag ${F\!_{\DOT}}$
acts on the Chow variety with fixed points the
$B$-stable cycles, which are sums of
Schubert varieties determined by ${F\!_{\DOT}}$.
As Hirschowitz~\cite{Hirschowitz} observed,
$\overline{\phi(U)}$ is $B$-stable, and must
contain a fixed point (\cite{Borel_groups}, III.10.4).
In fact, if ${{F\!_{\DOT}}'}$ is a real flag in linear general position with ${F\!_{\DOT}}$,
then the $B({\Bbb R})$-orbit of $\Omega_m{F\!_{\DOT}}\bigcap \Omega_w{{F\!_{\DOT}}'}$
is a subset of $\phi(U({\Bbb R}))$.
Moreover its closure has a $B({\Bbb R})$-fixed point, as the
proof in~\cite{Borel_groups}
may be adapted to show that complete $B({\Bbb R})$-stable
real analytic sets have fixed points.
Since the coefficients of the sum (\ref{sec:pieri_type}) are
all 1, $\sum_{v\in I_{m,w}}\Omega_w{F\!_{\DOT}}$ is the only $B({\Bbb R})$-stable
cycle in its algebraic equivalence class, and therefore
$$
\sum_{v\in I_{b,w}}\Omega_w{F\!_{\DOT}}\ \in\ \overline{\phi(U({\Bbb R}))}.
\qquad\qquad\qquad\qquad\QED
$$
\section{Fully real enumerative problems}
An {\em enumerative problem} of {\em degree} $d$ is an
intersection problem $\Xi\rightarrow U$
with zero-dimensional fibres of cardinality $d$.
An enumerative problem is {\em fully real} if there exists
$u\in U({\Bbb R})$ with
all points in the fibre $\Xi_u$ real.
In this case, $u = (u_1,\ldots,u_b)$ with $u_i\in U_i({\Bbb R})$
and the cycles $(\Xi_1)_{u_1},\ldots, (\Xi_b)_{u_b}$ meet transversally with
all points of intersection real.
\subsection{Theorem}\label{thm:real_closure}{\em
An enumerative problem $\Xi\rightarrow U$ is fully real if and
only if it has real effective algebraic equivalences.
That is, if and only if there exists a
point $\zeta\in \overline{\phi(U({\Bbb R}))}$
representing distinct real points.
}\medskip
\noindent{\bf Proof:}
The forward implication is a consequence of the definition.
For the reverse,
let $d$ be the degree of $\Xi\rightarrow U$.
Then $\phi:U\rightarrow S^dX$, the Chow variety of effective degree $d$
zero cycles on $X$.
The real points $S^dX({\Bbb R})$ of the Chow variety
represent degree $d$ zero cycles stable
under complex conjugation.
Its dense set of multiplicity free cycles
have an open subset ${\cal M}$ parameterizing cycles of
distinct real points, and $\zeta\in {\cal M}$.
Thus $\phi(U({\Bbb R}))\cap {\cal M}\neq \emptyset$,
which implies $\Xi\rightarrow U$ is fully real.
\QED\vspace{10pt}
The set of witnesses to $\Xi\rightarrow U$ being fully real
contains an open subset $\phi^{-1}({\cal M})\bigcap U({\Bbb R})$.
\section{Curve selection}
Subsequent sections use real effective algebraic
equivalence for one or more families to infer results
about related families.
While intuition supports the claim that the
functions we define between Chow varieties are algebraic
(or at least continuous),
we are unaware of general results verifying this intuition.
An obvious obstruction is that the Chow variety does not represent a functor.
However, weaker claims suffice.
Our tool is the Curve Selection
Lemma~\cite{Benedetti_Risler} of real
semi-algebraic geometry, in the following guise:
\subsection{Curve Selection Lemma}\label{lemma:curve_selection}
{\em Let $V$ be a real variety and $R\subset V({\Bbb R})$ a
semi-algebraic subset.
If $\zeta \in \overline{R}$, then there is a real algebraic map
$f: C\rightarrow V$ with $C$ a smooth curve,
and a point $s$ on a connected arc $S$ of
$C({\Bbb R})$ such that $f(S -\{s\}) \subset R$
and $f(s) = \zeta$.\medskip }
\noindent{\bf Proof:}
By the Curve Selection Lemma
(\cite{Benedetti_Risler}, 2.6.20),
there exists a semi-algebraic function
$g:[0,1]\rightarrow \overline{R}$ with $g(0)=\zeta$,
$g(0,1]\subset R$, and $g$ a real analytic homeomorphism onto its image
in $\overline{R}$.
Let $C^\circ$ be the Zariski closure of $g[0,1]$ in $V$,
and $f:C\rightarrow C^\circ$ its normalization.
Let $S\subset C({\Bbb R})$ be a connected arc of $f^{-1}(g[0,1])$ whose image
contains $g(0)$ and let $s \in S\cap f^{-1}(g(0))$.
\QED
\section{Pieri-type enumerative problems}
\subsection{Theorem}\label{thm:pieri_schubert}
{\em Any enumerative problem in any flag variety involving
five Schubert varieties, three of which are special, is fully real.}
\medskip
This generalizes Theorem 5.2 of~\cite{sottile_explicit_pieri},
the analogous result for Grassmannians.
It requires an additional transversality result.
\subsection{Lemma}\label{lemma:special_transversality}
{\em
Let $(w_1,w_2)$ and $(v_1,v_2)$ be indices of Schubert subvarieties of
a flag variety, with $w_1$ and $w_2$ (respectively $v_1$ and $v_2$)
defining defining Schubert varieties of
the same dimension.
Suppose $m$ is the index of a special Schubert subvariety
such that $(w_1,v_1,m)$ gives an enumerative problem.
If $(w_1,v_1)\neq (w_2,v_2)$, and ${F\!_{\DOT}}, {{F\!_{\DOT}}'}$
are complete flags
in linear general position, then
there is an open set $V$ of the variety
${\Bbb F}\ell$ of
complete flags consisting of flags ${E_{\DOT}}$ such that
$$
\Omega_{w_1}{F\!_{\DOT}} \bigcap \Omega_{v_1}{{F\!_{\DOT}}'} \bigcap \Omega_m {E_{\DOT}}
\qquad\mbox{and}\qquad
\Omega_{w_2}{F\!_{\DOT}} \bigcap \Omega_{v_2}{{F\!_{\DOT}}'} \bigcap \Omega_m {E_{\DOT}}
$$
are transverse intersections which coincide only when empty.
}\medskip
If the three flags are real, then a nonempty intersection
as above is a single real point.
\medskip
\noindent{\bf Proof:}
By Kleiman's Theorem ~\cite{Kleiman},
there is an open subset $U$ of
${\Bbb F}\ell\times{\Bbb F}\ell\times{\Bbb F}\ell$
consisting of triples $({F\!_{\DOT}},{{F\!_{\DOT}}'},{E_{\DOT}})$ such that each
intersection is transverse and so
is either empty or a single point,
by the Pieri-type formulas
of~\cite{Lascoux_Schutzenberger_polynomes_schubert,Sottile_Pieri_Schubert}.
Suppose neither is empty.
Similarly, there is an open subset $V$ of triples for which
$$
\left(\Omega_{w_1}{F\!_{\DOT}} \bigcap \Omega_{w_2}{F\!_{\DOT}} \right)
\bigcap \left(\Omega_{v_1}{{F\!_{\DOT}}'}\bigcap \Omega_{v_2}{{F\!_{\DOT}}'}\right)
\bigcap \Omega_b {E_{\DOT}}
$$
is proper.
Since $(w_1,v_1)\neq (w_2,v_2)$, it is empty.
Thus for triples $({F\!_{\DOT}},{{F\!_{\DOT}}'},{E_{\DOT}})\in U\cap V$,
$$
\Omega_{w_1}{F\!_{\DOT}} \bigcap \Omega_{v_1}{{F\!_{\DOT}}'} \bigcap \Omega_b {E_{\DOT}}\ \neq\
\Omega_{w_2}{F\!_{\DOT}} \bigcap \Omega_{v_2}{{F\!_{\DOT}}'} \bigcap \Omega_b {E_{\DOT}}.
$$
The lemma follows, as $U\cap V$ is stable under the
diagonal action of $GL_n{\Bbb C}$
and the set of pairs $({F\!_{\DOT}},{{F\!_{\DOT}}'})$ in linear
general position is the open $GL_n{\Bbb C}$-orbit
in ${\Bbb F}\ell\times{\Bbb F}\ell$.
\QED
\subsection{Proof of Theorem~\ref{thm:pieri_schubert}:}
Let $\Xi_1,\Xi_2,\Xi_3, \Gamma_1$, and $\Gamma_2$ be families of
Schubert varieties representing the classes
$\sigma_{m_1}, \sigma_{m_2}, \sigma_{m_3}, \sigma_{w_1}$, and $\sigma_{w_2}$.
Suppose $\sigma_{m_1}, \sigma_{m_2}$, and $\sigma_{m_3}$ are special Schubert
classes, and these families give an enumerative problem
$\Xi\rightarrow U$.
By \S \ref{sec:pieri_type}, for each $i=1,2$, the intersection problem
$\Psi_i\rightarrow V_i$ given by the families $\Xi_i$ and $\Gamma_i$
has a real effective algebraic equivalence
with witness
$\sum_{v_i\in I_{m_i,w_i}} \Omega_{v_i}{F\!_{\DOT}}$, for any real flag ${F\!_{\DOT}}$.
Let ${F\!_{\DOT}}$ and ${{F\!_{\DOT}}'}$ be real flags in linear general position
and set
$$
Z_1 \ := \sum_{v_1\in I_{m_1,w_1}} \Omega_{v_1} {F\!_{\DOT}}
\qquad\mbox{and}\qquad
Z_2 \ := \sum_{v_2\in I_{m_2,w_2}} \Omega_{v_2} {{F\!_{\DOT}}'}.
$$
For $i=1,2$,
let $\phi_i$ be the fibre function for $\Psi_i\rightarrow V_i$.
Then $Z_i\in \overline{\phi_i(V_i({\Bbb R}))}$
and by Lemma~\ref{lemma:curve_selection}, there is a
map $f_i : C_i \rightarrow \overline{\phi_i(V_i)}$
with $C_i$ a smooth curve, and a point $s_i$
on a connected arc $S_i$ of
$C_i({\Bbb R})$ such that $f(S_i -\{s_i\})\subset\phi_i(V_i({\Bbb R}))$
and $f_i(s_i) = Z_i$.
Then $f_i^*\Phi\rightarrow C_i$ is a family of multiplicity free cycles,
where $\Phi\rightarrow {\mbox{\it Chow}}^\circ X$ is the tautological family.
Considering pairs of components of $Z_1$ and $Z_2$ separately,
Lemma~\ref{lemma:special_transversality} shows there is a real flag
${E_{\DOT}}$ such that $Z_1\bigcap Z_2 \bigcap \Omega_{m_3} {E_{\DOT}}$
is a transverse intersection all of whose points are real.
Thus $f_1^*\Phi \rightarrow C_1$, $f_2^*\Phi \rightarrow C_2$, and
$\Xi_3\rightarrow {\Bbb F}\ell$ give a well-posed fully real enumerative
problem $\Psi \rightarrow V$, as
$(s_1,s_2,{E_{\DOT}}) \in V({\Bbb R})$.
Let ${\cal M}$ be the open subset of the real points of the
Chow variety parameterizing cycles consisting entirely of real points.
Then $\phi(s_1,s_2,{E_{\DOT}})\in {\cal M}$
and so $\phi^{-1}({\cal M})$ meets
$R:= (S_1-\{s_1\})\times(S_2-\{s_2\})\times \{{E_{\DOT}}\}$.
However, fibres of $\Psi$ over points of
$R$ are fibres of
$\Xi$ over points of $U({\Bbb R})$,
showing $\Xi\rightarrow U$ to be fully real.
\QED
\section{Fibrations}
Suppose $\pi: Y\rightarrow X$ has equidimensional fibres.
If $\Xi\rightarrow U$ is a family of multiplicity free cycles
on $X$ representing the cycle class $\alpha$, its pullback
$\pi^*\Xi := (\pi\times 1_U)^{-1}\Xi\rightarrow U$ is a family of
multiplicity free cycles on
$Y$ representing the cycle class $\pi^*\alpha$.
Suppose $\alpha_1\ldots,\alpha_a$ generate $A^*X$ additively
and $\Psi(\alpha_1),\ldots,\Psi(\alpha_a)$ are
families of cycles representing these generators.
The classes $\pi^*\alpha_1,\ldots,\pi^*\alpha_a$ generate the image of
$A^*X$ in $A^*Y$ and are represented by the families
$\pi^*\Psi(\alpha_1),\ldots,\pi^*\Psi(\alpha_a)$.
Effective algebraic equivalence is preserved by pullbacks:
\subsection{Theorem}\label{thm:pullbacks}
{\em
If $\,\Xi\rightarrow U$ is a family of multiplicity free cycles on
$X$ having effective algebraic equivalences with witness $Z$, then
$\pi^*\Xi\rightarrow U$ is a family of multiplicity free cycles on
$Y$ having effective algebraic equivalences with witness $\pi^{-1}Z$.
Likewise, if $\,\Xi\rightarrow U$ has real effective
algebraic equivalences, then so does $\pi^*\Xi\rightarrow U$.
}\medskip
Associating a cycle $Z$ on $X$ to $\pi^{-1}Z\subset Y$
defines a function $\pi^*: \mbox{\it Chow}\, X \rightarrow \mbox{\it Chow}\, Y$.
If $\phi$ is the fibre function of $\Xi\rightarrow U$, then
$\pi^*\circ \phi$ is the fibre function of $\pi^*\Xi\rightarrow U$.
Letting $W=\phi(U')$ and $R = \phi(U'({\Bbb R}))$, we see that
Theorem~\ref{thm:pullbacks} is a consequence of the following
lemma.
\subsection{Lemma}\label{lemma:pullback}
{\em
Let $W\subset {\mbox{\it Chow}}^\circ X$ be constructible and $V := \overline{W}\cap {\mbox{\it Chow}}^\circ X$.
Then $\pi^*(V)\subset \overline{\pi^*(W)}$ in $\mbox{\it Chow}\, Y$.
Likewise, if $R\subset {\mbox{\it Chow}}^\circ X({\Bbb R})$ is semi-algebraic
and $Q:=\overline{R}\cap {\mbox{\it Chow}}^\circ X({\Bbb R})$, then
$\pi^*(Q)\subset \overline{\pi^*(R)}$ in $\mbox{\it Chow}\, Y({\Bbb R})$.
}\medskip
\noindent{\bf Proof:}
For the first part, let $\zeta\in V$.
We show $\pi^*(\zeta) \in \overline{\pi^*(W)}$.
Let $C^\circ\subset {\mbox{\it Chow}}^\circ X$ be an irreducible curve with
$\zeta\in C^\circ$ and $C^\circ-\{\zeta\} \subset W$.
Let $f: C \rightarrow C^\circ$ be its normalization and let
$s\in f^{-1}(\zeta)$.
Let $\Phi \subset {\mbox{\it Chow}}^\circ X \times X$ be the tautological family.
Then $f^*\Phi$ is a family of multiplicity free cycles on $X$
with fibre function $f$.
Similarly, $\pi^*\circ f$ is the fibre function of the family
$\pi^*(f^*\Phi)$ of multiplicity free cycles on $Y$ over the smooth curve $C$.
As noted in \S \ref{sec:Chow}, this implies $\pi^*\circ f$ is algebraic,
and so
$\pi^*(\zeta) \in \pi^*(f(C)) \subset \overline{\pi^*(W)}$, since
$\pi^*(f(C-\{f^{-1}(\zeta)\}))\subset \pi^*(W)$.
For the second part, suppose $R\subset {\mbox{\it Chow}}^\circ X({\Bbb R})$
and $\zeta \in Q = \overline{R}\bigcap{\mbox{\it Chow}}^\circ X$.
By Lemma~\ref{lemma:curve_selection}, there is a smooth curve $C$,
a connected arc $S\subset C({\Bbb R})$, a point $s\in S$, and
an algebraic map $f: C\rightarrow {\mbox{\it Chow}}^\circ X$ such that
$f(s) = \zeta$ and $f(S-\{s\}) \subset R$.
Arguing as above shows
$\pi^*(\zeta) \in \pi^*(f(S)) \subset \overline{\pi^*(R)}$.
\QED
\section{Schubert-type enumerative problems in
${\Bbb F}\ell_{0,1}{\Bbb P}^n$ are fully real}
The variety ${\Bbb F}\ell_{0,1}{\Bbb P}^n$ of partial
flags $q\in l \subset {\Bbb P}^n$ with
$q$ a point and $l$ a line has projections
$$
p\ :\ {\Bbb F}\ell_{0,1}{\Bbb P}^n \longrightarrow {\Bbb P}^n
\qquad\mbox{and}\qquad
\pi\ :\ {\Bbb F}\ell_{0,1}{\Bbb P}^n\longrightarrow
{\Bbb G}_1{\Bbb P}^n,
$$
where ${\Bbb G}_1{\Bbb P}^n$ is the Grassmannian of lines
in ${\Bbb P}^n$.
A Schubert subvariety of ${\Bbb G}_1{\Bbb P}^n$ is determined by
a partial flag $F\subset P$ of ${\Bbb P}^n$:
$$
\Omega(F,P)\ :=\ \{l \in {\Bbb G}_1{\Bbb P}^n\,|\,
l\cap F \neq \emptyset\ \mbox{\ and }\ l\subset P\}.
$$
If $F$ is a hyperplane of $P$, then $\Omega(F,P) = {\Bbb G}_1P$, the
Grassmannian of lines in $P$.
In addition to $\pi^{-1}\Omega(F,P)$, there is another Schubert subvariety of
${\Bbb F}\ell_{0,1}{\Bbb P}^n$ which projects onto $\Omega(F,P)$ in
${\Bbb G}_1{\Bbb P}^n$:
$$
\widehat{\Omega}(F,P)\ :=\
\{(q,l) \in {\Bbb F}\ell_{0,1}{\Bbb P}^n\,|\,
q\in F \ \mbox{\ and }\ l\subset P\}.
$$
Any Schubert subvariety of ${\Bbb F}\ell_{0,1}{\Bbb P}^n$ is one of
$\Omega(F,P)$ or $\widehat{\Omega}(F,P)$, for suitable $F\subset P$.
The varieties $\widehat{\Omega}(F,P)$ have another description,
which is straightforward to verify:
\subsection{Lemma}\label{lemma:no_hats}
{\em
Let $N,P$ be subspaces of $\,{\Bbb P}^n$.
Then
$$
p^{-1}N \bigcap \pi^{-1}{\Bbb G}_1 P \ =\
\widehat{\Omega}(N\cap P,\,P),
$$
and, if $N$ and $P$ meet properly, this intersection
is generically transverse.
}
\subsection{Corollary}\label{cor:reduction}
{\em
Any Schubert-type enumerative problem on ${\Bbb F}\ell_{0,1}{\Bbb P}^n$
is equivalent to one involving only pullbacks of Schubert subvarieties
of $\,{\Bbb P}^n$ and ${\Bbb G}_1{\Bbb P}^n$.
}\medskip
The next lemma, an exercise in linear algebra,
describes Poincar\'e duality for
Schubert subvarieties of ${\Bbb F}\ell_{0,1}{\Bbb P}^n$.
\subsection{Lemma}\label{lemma:poincare_duality}
{\em
Suppose a linear subspace $N$ meets a partial flag $F\subset P$
properly in ${\Bbb P}^n$.
If $\pi^{-1}\Omega(F,P)$ and $p^{-1}N$ have complimentary
dimension in ${\Bbb F}\ell_{0,1}{\Bbb P}^n$, then their intersection
is empty unless $F$ and $N\cap P$ are points.
In that case, they meet transversally in a single point and
$\pi^{-1}\Omega(F,P)\bigcap p^{-1}N
= (N\cap P,\ \Span{F, N\cap P})$.}
\subsection{Theorem}\label{thm:schubert_fully_real}
{\em
Any Schubert-type enumerative problem in ${\Bbb F}\ell_{0,1}{\Bbb P}^n$
is fully real.
}\medskip
\noindent{\bf Proof:}
By Corollary~\ref{cor:reduction}, it suffices to consider
enumerative problems involving only pullbacks of
Schubert subvarieties of ${\Bbb P}^n$ and ${\Bbb G}_1{\Bbb P}^n$.
Since the intersection of linear subspaces in ${\Bbb P}^n$
is another linear subspace, we may further suppose the enumerative
problem $\Xi\rightarrow U$ is given by families
$p^*\Xi_1,\pi^*\Xi_2,\ldots,\pi^*\Xi_b$, where $\Xi_1$ is the family of
subspaces of a fixed dimension in ${\Bbb P}^n$ and
$\Xi_2,\ldots,\Xi_b$ are families of Schubert subvarieties
of ${\Bbb G}_1{\Bbb P}^n$.
By Theorem~C$'$ of~\cite{sottile_real_lines}, the intersection problem
$\Psi \rightarrow V$ on
${\Bbb G}_1{\Bbb P}^n$ given by $\Xi_2,\ldots,\Xi_b$ has
real effective algebraic equivalences.
Let $Z$ be a witness.
By Theorem~\ref{thm:pullbacks},
$\pi^*\Psi \rightarrow V$ has a real
effective algebraic equivalence with witness $\pi^* Z$.
By Lemma~\ref{lemma:curve_selection},
there is a real algebraic map
$f:C \rightarrow \overline{\phi(W)}\cap {\mbox{\it Chow}}^\circ {\Bbb F}\ell_{0,1}{\Bbb P}^n$
with $C$ a smooth curve, and a point $s$ on
a connected arc $S$ of $ C({\Bbb R})$ such that $f(s) = \pi^{-1}Z$ and
$f(S-\{s\})\subset \phi(V({\Bbb R}))$.
Let $\Phi \rightarrow {\mbox{\it Chow}}^\circ {\Bbb F}\ell_{0,1}{\Bbb P}^n$ be the tautological
family
and consider the family $f^*\Phi \rightarrow C$.
The fibre over $s$ of $f^*\Phi$ is $\pi^{-1}Z$.
Let ${\cal L}$ be the lattice of subspaces of ${\Bbb P}^n$ generated by
the (necessarily real) subspaces defining components of $Z$,
and let $N$ be a real subspace from the family $\Xi_1$ meeting all
subspaces of ${\cal L}$ properly.
By Lemma~\ref{lemma:poincare_duality},
$p^{-1}N\bigcap \pi^{-1}Z$ is transverse with
all points of intersection real.
Thus there is a Zariski open subset $C'$ of $C$ such that fibres of
$f^*\Phi$ over $C'$ meet $p^{-1} N$ transversally.
Then $s\in \overline{(S-\{s\})\bigcap C'({\Bbb R})}$, so there is
a point $t\in S-\{s\}$ such that
$p^{-1}N\bigcap(f^*\Phi)_t$ is transverse and consists
entirely of real points.
But $f(t)\in \phi(V({\Bbb R}))$, so $(f^*\Phi)_t=\Phi_{f(t)}$ is a fibre of
$\pi^*\Psi$ over $V({\Bbb R})$, and hence a
generically transverse intersection of real Schubert varieties
from the families $\pi^*\Xi_2,\ldots,\pi^*\Xi_b$.
Thus $\Xi\rightarrow U$ is fully real.
\QED
\subsection{Effective algebraic equivalence for
${\Bbb F}\ell_{0,1}{\Bbb P}^n$}
Any Schubert-type intersection problem on
${\Bbb F}\ell_{0,1}{\Bbb P}^n$ has real effective algebraic
equivalences.
We give an outline, as a complete analysis is lengthy and involves
no new ideas beyond~\cite{sottile_real_lines}.
By Corollary~\ref{cor:reduction}, it suffices
to consider intersection problems $\Xi\rightarrow U$ given by families
$p^*\Xi_1,\pi^*\Xi_2,\ldots,\pi^*\Xi_b$, where $\Xi_1$ is a family of
subspaces of a fixed dimension in ${\Bbb P}^n$ and
$\Xi_2,\ldots,\Xi_b$ are families of Schubert subvarieties
of ${\Bbb G}_1{\Bbb P}^n$.
In~\cite{sottile_real_lines}, the intersection problem given by
$\Xi_2,\ldots,\Xi_b$ is shown to have real effective algebraic
equivalences with witness $Z$.
Let $\Psi \rightarrow V$ be the intersection problem given by
$p^*\Xi_1$ and the constant family $\pi^{-1}Z$.
Using Theorem~\ref{thm:pullbacks} and Lemma~\ref{lemma:curve_selection}
one may show
$$
\phi(V)\subset \overline{\phi(U)}
\qquad\mbox{and}\qquad
\phi(V({\Bbb R}))\subset \overline{\phi(U({\Bbb R}))}.
$$
It suffices to show $\Psi\rightarrow V$ has real
effective algebraic equivalences.
A proof that $\Psi\rightarrow V$ has real effective algebraic
equivalences mimics the proof of Theorem~E of~\cite{sottile_real_lines},
with the following Lemma playing the role of
Lemma~2.4 of~\cite{sottile_real_lines}.
\subsection{Lemma}\label{lemma:reducible}
{\em Let $F,P,N$, and $H$ be linear subspaces of ${\Bbb P}^n$
and suppose that $H$ is a hyperplane containing neither $P$
nor $N$, $F$ is a proper subspace of $P\cap H$, and $N$ meets $F$,
and hence $P$ properly.
Set $L = N\cap H$.
Then $\pi^{-1}\Omega(F,P)$ and $p^{-1}L$ meet
generically transversally,
$$
\pi^{-1}\Omega(F,P)\bigcap p^{-1}L\ =\
\widehat{\Omega}(N\cap F,\,P) \ +\
\pi^{-1}\Omega(F,P\cap H)\bigcap p^{-1}N,
$$
and the second term is itself an irreducible generically
transverse intersection.}
\medskip
The proof of this statement is almost identical to the proof of
Lemma~2.4 of~\cite{sottile_real_lines}.
\section{Some Schubert-type enumerative problems in
${\Bbb F}\ell_{1,n-2}{\Bbb P}^n$}
The variety ${\Bbb F}\ell_{1,n-2}{\Bbb P}^n$ of partial flags
$l \subset \Lambda \subset {\Bbb P}^n$, where
$l$ is a line and $\Lambda$ an $(n-2)$-plane
has projections
$$
\pi: {\Bbb F}\ell_{1,n-2}{\Bbb P}^n \rightarrow {\Bbb G}_1{\Bbb P}^n
\qquad\mbox{and}\qquad
p: {\Bbb F}\ell_{1,n-2}{\Bbb P}^n\rightarrow {\Bbb G}_{n-2}{\Bbb P}^n,
$$
where ${\Bbb G}_{n-2}{\Bbb P}^n$ is the Grassmannian of $(n-2)$-planes in
${\Bbb P}^n$.
\subsection{Theorem}\label{thm:simple_schubert}
{\em
Any enumerative problem in ${\Bbb F}\ell_{1,n-2}{\Bbb P}^n$
given by pullbacks of Schubert subvarieties of $\,{\Bbb G}_1{\Bbb P}^n$
and ${\Bbb G}_{n-2}{\Bbb P}^n$
is fully real.
}\medskip
\noindent{\bf Proof:}
Suppose $\pi^*\Xi_1,\ldots,\pi^*\Xi_b, p^*\Gamma_1,\ldots,p^*\Gamma_c$
give an enumerative problem on ${\Bbb F}\ell_{1,n-2}{\Bbb P}^n$
where, for $1\leq i\leq b$, $\Xi_i$ is a family of Schubert subvarieties
of ${\Bbb G}_1{\Bbb P}^n$
and for $1\leq j\leq c$, $\Gamma_i$ is a family of Schubert subvarieties
of ${\Bbb G}_{n-2}{\Bbb P}^n$.
By Theorem~\ref{thm:pullbacks} and~\cite{sottile_real_lines},
$\pi^*\Xi_1,\ldots,\pi^*\Xi_b$ give an intersection
problem $\Psi_1\rightarrow V_1$
which has a real algebraic
equivalence with witness $Z_1$.
Identifying ${\Bbb P}^n$ with its dual projective space
gives an isomorphism
${\Bbb G}_{n-2}{\Bbb P}^n
\stackrel{\sim}{\rightarrow} {\Bbb G}_1{\Bbb P}^n$,
mapping Schubert subvarieties to Schubert subvarieties.
It follows that $p^*\Gamma_1,\ldots,p^*\Gamma_c$ give an intersection problem
$\Psi_2\rightarrow V_2$ which has a real algebraic
equivalence with witness $Z_2$.
It suffices to show the enumerative problem
$\Psi\rightarrow V$ given by $\Psi_1$ and $\Psi_2$ is fully real.
Since $Z_1$ and $Z_2$ may be replaced by any translate
by elements of $PGL_{n+1}{\Bbb R}$,
we assume $Z_1$ and $Z_2$ intersect transversally.
Components of $Z_1$ and $Z_2$ are Schubert varieties defined
by real flags.
Moreover, each component of $Z_1$ has complementary dimension to
each component of $Z_2$.
In a flag variety,
Schubert varieties of complimentary dimension
which meet transversally and are defined by real flags
either have empty intersection, or meet in
a single real point.
Thus $Z_1\bigcap Z_2$ consists entirely of real points.
By Lemma~\ref{lemma:curve_selection}, for each $i=1,2$, there is a real
algebraic map $f_i : C_i \rightarrow \overline{\phi(V_i)}$
where $C_i$ is a smooth curve, and a point $s_i$ on a connected arc $S_i$
of $C_i({\Bbb R})$ such that $f(S_i -\{s_i\})\subset\phi(V_i({\Bbb R}))$
and $f_i(s_i) = Z_i$.
The enumerative problem $\Psi'\rightarrow V'$ given by
$f_1^*\Psi_1\rightarrow C_1$ and $f_2^*\Psi_2\rightarrow C_2$
is fully real, as $\Psi'_{(s_1,s_2)} = Z_1\bigcap Z_2$.
Since $(s_1,s_2)\in
\overline{(S_1-\{s_1\})\times(S_2-\{s_2\})\bigcap V'({\Bbb R})}$,
there is a point $(t_1,t_2)\in (S_1-\{s_1\})\times(S_2-\{s_2\})$
such that
$\Psi'_{(t_1,t_2)} = (f_1^*\Psi_1)_{t_1}\bigcap(f_2^*\Psi_2)_{t_2}$
is transverse and consists entirely of real points.
Since $f_i(t_i)\in \phi(V_i({\Bbb R}))$,
we see that $(f_i^*\Psi_i)_{t_i} = (\Psi_i)_{f_i(t_i)}$ is a fibre
of $\Psi_i$ over a point of $V_i({\Bbb R})$.
This shows $\Psi\rightarrow V$ is fully real.
\QED
\section{Powers of Enumerative Problems}
A method to construct a new fully real enumerative problem out of
a given one is illustrated by a proof of B\'ezout's Theorem
in the plane.
We will formalize this method.
\subsection{B\'ezout's Theorem}
{\em
Let $d_1$ and $d_2$ be positive integers.
Then there exist smooth real plane curves $D_1$ and $D_2$
of degrees $d_1$ and $d_2$ meeting transversally in
\medskip$d_1\cdot d_2$ real points.}
\noindent{\bf Proof:}
Two distinct real lines meet in a single real point.
Thus if $D_1$
consists of $d_1$ distinct real lines, $D_2$ of $d_2$,
and if $D_1$ and $D_2$ meet transversally, then $D_1\bigcap D_2$ is
$d_1\cdot d_2$ real points.
The family of real reduced degree $d$ plane
curves has general member a smooth curve and contains
all cycles of $d$ distinct real lines.
Thus the enumerative problem of intersecting reduced curves $D_1$
and $D_2$ of respective degrees $d_1$ and $d_2$ is fully real of
degree $d_1\cdot d_2$.
Moreover, pairs of smooth real curves are dense in the set of pairs
of reduced real curves, showing the enumerative problem of intersecting
two smooth plane curves of respective degrees $d_1$ and $d_2$ is
fully real of
degree $d_1\cdot d_2$.
\QED
\subsection{Powers of intersection problems}
Suppose $\Xi\rightarrow U$ is a family of multiplicity
free cycles on $X$ and $d$ is a positive integer.
If the locus of $d$-tuples
$(u_1,\ldots,u_d)$ such that no two of $\Xi_{u_1},\ldots,\Xi_{u_d}$
share a component is dense in $U^d$, then let
$U^{(d)}$ be an open subset of that locus.
Let $\Xi^{\oplus d}\rightarrow U^{(d)}$ be the family
of multiplicity free cycles whose fibre over
$(u_1,\ldots,u_d)\in U^{(d)}$ is $\sum_{j=1}^d \Xi_{u_j}$.
Suppose $\Xi_1\rightarrow U_1,\ldots,\Xi_b\rightarrow U_b$ are families
of multiplicity free cycles on $X$ giving an
intersection problem $\Xi\rightarrow U$
and $d_1,\ldots,d_b$ is a sequence of positive integers.
Then the families $\Xi_1^{\oplus d_1}\rightarrow U_{1}^{(d_1)},\ldots,
\Xi_b^{\oplus d_b}\rightarrow U_{s}^{(d_b)}$
give a well-posed intersection problem if
general members of the famililies
$\Xi\rightarrow U$ and $\Xi_i\rightarrow U_i$
meet properly, for $1\leq i\leq b$.
When a reductive group $G$ acts transitively
on $X$ and the families of cycles are $G$-stable,
$\Xi_1^{\oplus d_i},\ldots,\Xi_b^{\oplus d_b}$ give an intersection
problem.
Moreover, if $\Xi\rightarrow U$ is fully real, then
so is that intersection problem.
We produce a witness with a particular form.
\subsection{Lemma}\label{lemma:transverse_technicality}
{\em
Suppose $\Xi_1\rightarrow U_1,\ldots,\Xi_b\rightarrow U_b$
give a fully real enumerative problem of degree $d$.
Let $d_1,\ldots,d_b$ be a sequence of positive integers
and suppose that for $1\leq i\leq b$, $V_i$ is $G$-stable subset of
$U_{i}^{(d_i)}$ such that
$\Delta^{d_i}U_i({\Bbb R})\subset \overline{V_i({\Bbb R})}$,
as subsets of $U_i({\Bbb R})^{d_i}$.
Then for $1\leq i\leq b$, there exists $v_i\in V_i({\Bbb R})$ such that
$(\Xi_1^{\oplus d_1})_{v_1},\ldots,(\Xi_b^{\oplus d_b})_{v_b}$
intersect transversally in
$d\cdot d_1\cdots d_b$ real points.}
\medskip
\noindent{\bf Proof:}
The restriction $\Psi_i$ of $\Xi_i^{\oplus d_i}$ to $V_i$ is
$G$-stable.
Thus $\Psi_1,\ldots,\Psi_b$ give a well-posed
enumerative problem $\Psi\rightarrow V$.
We show this is fully real and compute its degree.
Since $\Xi\rightarrow U$ is fully real, there
is an open subset $R$ of points
$u \in U({\Bbb R})$ such that $\Xi_u$ is
$d$ distinct real points.
Since $U({\Bbb R})\subset \prod_{i=1}^b U_i({\Bbb R})$,
for $1\leq i\leq b$ there exists an open subset
$R_i$ of $U_i({\Bbb R})$ such that $\prod_{i=1}^b R_i \subset R$.
Then
$V_i({\Bbb R})\bigcap R^{d_i}\neq \emptyset$, as
$\Delta^{d_i} R_i\subset \Delta^{d_i}U_i({\Bbb R})
\subset \overline{V_i({\Bbb R})}$.
Thus $R' := V({\Bbb R})
\bigcap \prod_{i=1}^b R_i^{d_i}$
is nonempty,
as $V({\Bbb R})$ is dense in
$ \prod_{i=1}^b V_i({\Bbb R})$.
Let $w =(w_{11},\ldots,w_{1d_1},\ldots,w_{b1}\ldots,w_{bd_b})\in R'$.
Here, $w_{ij}\in R_i$ and
$(w_{i1},\ldots,w_{id_i})\in V_i({\Bbb R})$.
If $1\leq j_i\leq d_i$,
then $(w_{1j_1},\ldots,w_{bj_b})\in U({\Bbb R})$.
Furthermore, $\Psi_w = \bigcap_{i=1}^b (\Psi_i)_{(w_{i1},\ldots,w_{id_i})}$
is a transverse intersection, as $R' \subset V$.
Since $(\Psi_i)_{(w_{i1},\ldots,w_{id_i})} =
\sum_{j=1}^{d_i} (\Xi_i)_{w_{ij}}$, we have
$$
\Psi_w \ =\ \bigcap_{i=1}^b \,\sum_{j=1}^{d_i} (\Xi_i)_{w_{ij}}
\ =\ \sum_{\stackrel{\mbox{\scriptsize$j_1,\ldots,j_b$}}{1\leq j_i\leq d_i}}
\ \bigcap_{i=1}^b\,(\Xi_i)_{w_{ij_i}}
\ =\ \sum_{\stackrel{\mbox{\scriptsize$j_1,\ldots,j_b$}}{1\leq j_i\leq d_i}}
\Xi_{(w_{1j_1},\ldots,w_{bj_b})}.
$$
Since this intersection is transverse,
it consists of $d\cdot d_1\cdots d_b$ real points.
\QED
\subsection{Real B\'ezout's Theorem} {\em
Let $d_1,\ldots,d_b$ be positive integers.
Then there exist smooth real hypersurfaces
$H_1,\ldots,H_b$ in ${\Bbb P}^b$ of respective degrees
$d_1,\ldots,d_b$ which intersect transversally
in $d_1\cdots d_b$ real points.
}\medskip
\noindent{\bf Proof:}
Let $\Xi\rightarrow U$ be the family of hyperplanes in ${\Bbb P}^b$.
Since $b$ real hyperplanes in general position meet in a real point,
either simple checking or Lemma~\ref{lemma:transverse_technicality} with
$V:=U^{(d_i)}$ shows that $\Xi^{\oplus d_1},\ldots,\Xi^{\oplus d_b}$
give a fully real enumerative problem of degree $d_1\cdots d_b$.
Note that $\Xi^{\oplus d_i}\rightarrow U^{(d_i)}$ is the family of
hypersurfaces composed of $d_i$ distinct hyperplanes.
Let $W_i\subset {\Bbb P}(\mbox{\em Sym}^{d_i}{\Bbb C}\,^{b+1})$ be the space
of forms of degree $d_i$ with no repeated factors and
$\Gamma_i\rightarrow W_i$ the family of reduced degree $d_i$ hypersurfaces.
Let $W'_i\subset W_i$ be the dense subset of forms determining
smooth hypersurfaces.
Note that $U^{(d_i)}\subset W_i$ and
$\Xi^{\oplus d_i} = \Gamma_i|_{U^{(d_i)}}$.
It follows that $\Gamma_1,\ldots,\Gamma_b$ give a fully real
enumerative problem of degree $d_1\cdots d_b$.
Let $R$ be an open set of witnesses.
Since $U^{(d_i)}({\Bbb R})\subset\overline{W'_i({\Bbb R})}$
and $R$ meets $\prod_{i=1}^sU^{(d_i)}({\Bbb R})$, we see that
$R$ meets $\prod_{i=1}^sW'_i({\Bbb R})$.
That is,
there exist smooth real hypersurfaces
$H_1,\ldots,H_b$ in ${\Bbb P}^b$ of respective degrees
$d_1,\ldots,d_b$ which intersect transversally
in $d_1\cdots d_b$ real points.
\QED
\section{$(n-2)$-planes meeting rational normal curves in
${\Bbb P}^n$}
Let ${\Bbb G}_{n-2}{\Bbb P}^n$ be the Grassmannian of $(n-2)$-planes in
${\Bbb P}^n$, a variety of dimension $2n-2$.
Those $(n-2)$-planes which meet a curve form a hypersurface
in ${\Bbb G}_{n-2}{\Bbb P}^n$.
We synthesize ideas of previous sections to
prove the following theorem.
\subsection{Theorem}\label{thm:rational_normal}
{\em
The enumerative problem of $\,(n-2)$-planes meeting $2n-2$ general
rational normal
curves in ${\Bbb P}^n$ is fully real and has degree
${2n-2\choose n-1}n^{2n-3}$.
}\medskip
\noindent{\bf Proof:}
Identifying ${\Bbb P}^n$ with its dual projective space
gives an isomorphism
${\Bbb G}_{n-2}{\Bbb P}^n
\stackrel{\sim}{\rightarrow} {\Bbb G}_1{\Bbb P}^n$,
mapping Schubert subvarieties to Schubert subvarieties.
By Theorem~C of~\cite{sottile_real_lines}, any enumerative problem
involving Schubert subvarieties of ${\Bbb G}_{n-2}{\Bbb P}^n$ is fully real.
In particular, the enumerative problem given by $2n-2$ copies of the family
$\Xi \rightarrow U$ is fully real,
where $U = {\Bbb G}_1{\Bbb P}^n$ and the fibre of $\Xi$
over $l\in U$ is the Schubert variety $\Omega_l$
of $(n-2)$-planes meeting $l$.
We compute its degree, $d$.
The image of $\Omega_l$
under the isomorphism ${\Bbb G}_{n-2}{\Bbb P}^n
\stackrel{\sim}{\rightarrow} {\Bbb G}_1{\Bbb P}^n$
is the Schubert subvariety of all lines meeting
a fixed $(n-2)$-plane.
Thus $d$ is the number of lines meeting $2n-2$ general $(n-2)$-planes
in ${\Bbb P}^n$.
By Corollary 3.3 of~\cite{sottile_real_lines}, this is the number of
(standard) Young tableaux of shape $(n-1,n-1)$, which is
$\frac{1}{n} {2n-2\choose n-1}$, by the hook length formula
of Frame, Robinson, and Thrall~\cite{FRT}.
Let $e_0,\ldots,e_n$ be real points spanning ${\Bbb P}^n$.
For $1\leq i\leq n$, let $l_i := \Span{e_{i-1},e_i}$.
Then $\Omega_{l_1}+ \cdots+\Omega_{l_n}$
is the fibre of $\Xi^{\oplus n}$ over $(l_1,\ldots,l_n)\in U^{(n)}({\Bbb R})$.
Let $V = PGL_{n+1}{\Bbb C}\cdot (l_1,\ldots,l_n)\subset U^{(n)}$.
For $t\in [0,1]$ and $1\leq i\leq n$, let
$$
l_i(t) \ :=\ \Span{te_{i-1} + (1-t)e_{\overline{i -1}},\,
t e_i + (1-t) e_{\overline{i}}},
$$
where $\overline{j}\in \{0,1\}$ is congruent to $j$ modulo 2.
Let $\gamma(t) := (l_1(t),\ldots,l_n(t))$.
If $t\in (0,1]$, then $\gamma(t)\in V({\Bbb R})$.
Since $\gamma(0) = (l_1,\ldots,l_1)$ and
$\Delta^nU({\Bbb R}) = PGL_{n+1}{\Bbb R}\cdot \gamma(0)$,
it follows that $\Delta^nU({\Bbb R})\subset \overline{V{(\Bbb R})}$.
Then, by Lemma~\ref{lemma:transverse_technicality}, there exist points
$v_1,\ldots,v_{2n-2} \in V({\Bbb R})$ such that
$\Xi^{\oplus n}_{v_1},\ldots,\Xi^{\oplus n}_{v_{2n-2}}$
meet transversally in
${2n-2\choose n-1}n^{2n-3}$ points.
Let $p(m):= n\cdot m +1$, the Hilbert polynomial of a rational normal curve
in ${\Bbb P}^n$.
Let ${\cal H}$ be the open subset of the Hilbert scheme
parameterizing reduced schemes with Hilbert polynomial $p$.
Let $\Psi\subset {\cal H}\times {\Bbb G}_{n-2}{\Bbb P}^n$
be the family of multiplicity free cycles on
${\Bbb G}_{n-2}{\Bbb P}^n$
whose fibre over a curve $C\in {\cal H}$ is the hypersurface
of $(n-2)$-planes meeting $C$.
Note that $p$ is also the Hilbert polynomial of
$l_1\bigcup\cdots\bigcup l_n$.
Let $\lambda \in {\cal H}$ be the point representing
$l_1\bigcup\cdots\bigcup l_n$.
If $V'$ is the $PGL_{n+1}{\Bbb C}$-orbit of $\lambda$ in ${\cal H}$,
then $\Psi|_{V'}\rightarrow V'$ is isomorphic to the family
$\Xi^{\oplus n}\rightarrow V$, under the obvious isomorphism
between $V$ and $V'$.
It follows that the enumerative problem given by $2n-2$
copies of $\Psi\rightarrow {\cal H}$ is fully real.
Let $W$ be the subset of ${\cal H}$ representing rational
normal curves.
We claim $V'({\Bbb R})\subset \overline{W({\Bbb R})}$, from which
it follows that
the enumerative problem of $(n-2)$-planes meeting $2n-2$ rational
normal curves in ${\Bbb P}^n$ is fully real and has degree
${2n-2\choose n-1}n^{2n-3}$.
Let $[x_0,\ldots,x_n]$ be homogeneous coordinates for ${\Bbb P}^n$
dual to the basis $e_0,\ldots,e_n$.
For $t\in {\Bbb C}$, define the ideal ${\cal I}_t$ by
$$
{\cal I}_t \ :=\
( x_ix_j - t x_{i+1}x_{j-1}\, |\, 0\leq i<j\leq n\ \mbox{and}\ j-i\geq 2).
$$
For $t\neq 0$, ${\cal I}_t$ is the ideal of a rational normal curve
and ${\cal I}_0$ is the ideal of $l_1\bigcup\cdots\bigcup l_n$.
This family of ideals is flat.
Let $\varphi:{\Bbb C}\rightarrow {\cal H}$
be the map representing this family.
Then $\varphi({\Bbb R}-\{0\})\subset W({\Bbb R})$.
Noting $\varphi(0)=\lambda$ shows
$\lambda \in \overline{W({\Bbb R})}$.
Since $W({\Bbb R})$ is $PGL_{n+1}{\Bbb R}$-stable,
we conclude that
$V'({\Bbb R})\subset \overline{W({\Bbb R})}$.
\QED
|
1996-02-28T06:20:13 | 9602 | alg-geom/9602018 | en | https://arxiv.org/abs/alg-geom/9602018 | [
"alg-geom",
"math.AG"
] | alg-geom/9602018 | Klaus Altmann | Klaus Altmann | P-Resolutions of Cyclic Quotients from the Toric Viewpoint | 10 pages; LaTeX. dvi and ps file available at
http://www-irm.mathematik.hu-berlin.de/~altmann/PAPER/paper.html | null | null | null | null | P-resolutions of two-dimensional, cyclic quotient singularities have been
introduced to study deformation theory. Those P-resolutions (as well as the
singularities themselves) are toric varieties. In the present paper we give a
straight, elementary description of them just by their defining fans.
| [
{
"version": "v1",
"created": "Tue, 27 Feb 1996 12:50:41 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Altmann",
"Klaus",
""
]
] | alg-geom | \section{Introduction}\label{I}
\neu{I-1}
The break through in deformation theory of (two-dimensional)
quotient singularities $Y$ was Koll\'{a}r/Shepherd-Barron's discovery
of the one-to-one correspondence between so-called P-resolutions, on
the one hand, and components of the versal base space, on the other hand
(cf.\ \cite{KS}, Theorem (3.9)).
It generalizes the fact that all deformations admitting
a simultaneous (RDP-) resolution form one single component, the Artin
component.\\
\par
According to defintion (3.8) in \cite{KS}, P-resolutions are partial
resolutions $\pi:\tilde{Y}\to Y$ such that
\begin{itemize}
\item
the canonical divisor $K_{\tilde{Y}|Y}$ is ample relative to $\pi$ (a
minimality condition) and
\item
$\tilde{Y}$ contains only mild singularities of a certain type (so-called
T-singularities).
\end{itemize}
Despite their definition as those quotient singularities admitting a
$I\!\!\!\!Q$-Gorenstein one-parameter smoothing (\cite{KS}, (3.7)),
there are at least three further descriptions of the class of T-singularities:
An explicit list of their defining group actions on $\,I\!\!\!\!C^2$ (\cite{KS}, (3.10)),
an inductive procedure to construct their resolution graphs
(\cite{KS}, (3.11)), and a characterization using toric language
(\cite{Homog}, (7.3)).\\
The latter one starts with the observation that
affine, two-dimensional toric varieties (given by some rational, polyhedral
cone $\sigma\subseteqI\!\!R^2$) provide exactly the two-dimensional cyclic
quotient singularties. Then, T-singularities come from cones over rational
intervals of integer length placed in height one (i.e.\ contained in the
affine line $(\bullet,1)\subseteqI\!\!R^2$).
\vspace{-2ex}
\begin{center}
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\begin{picture}(155.00,68.00)
\put(0.00,40.00){\line(1,0){138.00}}
\put(10.00,40.00){\circle*{2.00}}
\put(40.00,40.00){\circle*{2.00}}
\put(70.00,40.00){\circle*{2.00}}
\put(100.00,40.00){\circle*{0.00}}
\put(100.00,40.00){\circle*{2.00}}
\put(10.00,10.00){\line(5,2){145.00}}
\put(10.00,10.00){\line(1,2){29.00}}
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\put(100.00,10.00){\circle*{2.00}}
\put(70.00,10.00){\circle*{2.00}}
\put(40.00,10.00){\circle*{2.00}}
\put(10.00,10.00){\circle*{2.00}}
\put(55.00,44.00){\makebox(0,0)[cb]{\footnotesize length 2}}
\put(100,60){\makebox(0,0)[cc]{$\sigma$}}
\end{picture}
\end{center}
If the affine interval is of length $\mu+1$, then the corresponding
T-singularity
will have Milnor number $\mu$ (on the $I\!\!\!\!Q$-Gorenstein one-parameter
smoothing).\\
\par
\neu{I-2}
In \cite{Ch-CQS} and \cite{St-CQS} Christophersen and Stevens gave a
combinatorial
description of all P-resolutions for two-dimensional, cyclic quotient
singularities.
Using an inductive construction method (going through different cyclic
quotients
with step-by-step increasing multiplicity) they have shown that there is a
one-to-one
correspondence between P-resolutions, on the one hand, and certain integer
tuples
$(k_2,\dots,k_{e-1})$ yielding zero if expanded as a (negative) continued
fraction
(cf.\ \zitat{P}{2}), on the other hand.\\
\par
The aim of the present paper is to provide an elementary, direct method for
constructing the
P-resolutions of a cyclic quotient singularity (i.e.\ a two-dimensional toric
variety)
$Y_{\sigma}$. Given a chain $(k_2,\dots,k_{e-1})$ representing zero, we will
give a straight
description of the corresponding polyhedral subdivision of $\sigma$. (In
particular,
the bijection between those
0-chains and P-resolutions will be proved again by a different
method.)\\
\par
\section{Cyclic Quotient Singularities}\label{CQS}
In the following we want to remind the reader of basic notions concerning
continued fractions and cyclic quotients. It should be considered a good
chance to fix notations. References are \cite{Oda} (\S 1.6) or
the first sections in \cite{Ch-CQS} and \cite{St-CQS}, respectively.\\
\par
\neu{CQS-1}
{\bf Definition:}
To integers $c_1,\dots,c_r\inZ\!\!\!Z$ we will assign the continued fraction
$[c_1,\dots,c_r]\inI\!\!\!\!Q$, if the following inductive procedure makes sense
(i.e.\ if no division by 0 occurs):
\begin{itemize}
\item
$[c_r]:=c_r$
\item
$[c_i,\dots,c_r]:= c_i- 1/[c_{i+1},\dots,c_r]$.
\end{itemize}
If $c_i\geq 2$ for $i=1,\dots,r$, then $[c_1,\dots,c_r]$ is always defined
and yields a rational number greater than 1. Moreover, all these numbers may be
represented by those continued fractions in a unique way.\\
\par
\neu{CQS-2}
Let $n\geq 2$ be an integer and $q\in (Z\!\!\!Z/nZ\!\!\!Z)^\ast$ be represented by an
integer
between $0$ and $n$. These data provide a group action of $Z\!\!\!Z/nZ\!\!\!Z$ on $\,I\!\!\!\!C^2$
via the matrix
$\left(\begin{array}{cc}\xi&0\\0&\xi^q\end{array}\right)$
(with $\xi$ a primitive $n$-th
root of unity). The quotient is denoted by $Y(n,q)$.\\
\par
In toric language, $Y(n,q)$ equals the variety $Y_\sigma$ assigned to the
polyhedral cone
$\sigma:=\langle (1,0);(-q,n)\rangle\subseteqI\!\!R^2$. ($Y_\sigma$ is defined as
$\mbox{Spec}\; \,I\!\!\!\!C[\sigma^{\scriptscriptstyle\vee}\capZ\!\!\!Z^2]$ with
\[
\sigma^{\scriptscriptstyle\vee}:=\{r\in(I\!\!R^2)^\ast\,| \;r\geq 0 \mbox{ on } \sigma\}=
\langle [0,1];[n,q]\rangle \subseteq (I\!\!R^2)^\ast \cong I\!\!R^2\,.)
\]
{\bf Notation:} Just to distinguish between $I\!\!R^2$ and its dual
$(I\!\!R^2)^\ast\congI\!\!R^2$,
we will denote these vector spaces by $N_{I\!\!R}$ and $M_{I\!\!R}$, respectively.
(Hence,
$\sigma\subseteq N_{I\!\!R}$ and $\sigma^{\scriptscriptstyle\vee}\subseteq M_{I\!\!R}$.)
Elements of $N_{I\!\!R}\congI\!\!R^2$ are written in paranthesis; elements of
$M_{I\!\!R}\congI\!\!R^2$ are written in brackets. The natural pairing
between $N_{I\!\!R}$ and $M_{I\!\!R}$ is denoted by $\langle\,,\,\rangle$ which should
not be
mixed up with the symbol indicating the generators of a cone. Finally, all
these
remarks apply for the lattices $N\congZ\!\!\!Z^2$ and $M\congZ\!\!\!Z^2$, too.\\
\par
\neu{CQS-3}
Let $n,q$ as before. We may write $n/(n-q)$ and $n/q$ (both are greater than 1)
as continued fractions
\[
n/(n-q) = [a_2,\dots,a_{e-1}] \; \mbox{ and } \; n/q = [b_1,\dots,b_r]
\quad (a_i, b_j\geq 2).
\]
The $a_i$'s and the $b_j$'s are mutually related by Riemenschneider's point
diagram (cf.\ \cite{Riem}).\\
\par
Take the convex hull of $(\sigma^{\scriptscriptstyle\vee}\cap M)\setminus\{0\}$ and denote by
$w^1,w^2,\dots, w^e$ the lattice points on its compact edges. If ordered the
right way, we obtain $w^1=[0,1]$ and $w^e=[n,q]$ for the first and the last
point,
respectively.
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\put(53.00,118.00){\line(-1,-6){11.67}}
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\put(66.00,32.33){\line(1,0){54.00}}
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\put(66.00,32.00){\circle*{3.00}}
\put(121.00,32.00){\circle*{3.00}}
\put(63.00,117.00){\makebox(0,0)[cc]{$w^1$}}
\put(59.00,82.00){\makebox(0,0)[cc]{$w^2$}}
\put(95.00,35.00){\makebox(0,0)[cc]{$\dots$}}
\put(53.00,52.00){\makebox(0,0)[cc]{$w^3$}}
\put(121.00,23.00){\makebox(0,0)[cc]{$w^e$}}
\put(106.00,82.00){\makebox(0,0)[cc]{$\sigma^{\scriptscriptstyle\vee}$}}
\end{picture}
\end{center}
Then, $E:=\{w^1,\dots,w^e\}$ is the minimal generating set (the so-called
Hilbert basis) of the semigroup $\sigma^{\scriptscriptstyle\vee}\cap M$. These point are
related to our first continued fraction by
\[
w^{i-1}+w^{i+1}=a_i\,w^i \quad (i=2,\dots,e-1).
\]
{\bf Remark:}
The surjection $I\!\!N^E\longrightarrow\hspace{-1.5em}\longrightarrow \sigma^{\scriptscriptstyle\vee}\cap M$ provides a minimal
embedding of $Y_{\sigma}$. In particular, $e$ equals its embedding
dimension.\\
\par
In a similar manner we can define $v^0,\dots,v^{r+1}\in\sigma\cap N$
in the original cone; now we
have $v^0=(1,0)$, $v^{r+1}=(-q,n)$, and the relation to the continued
fractions is $v^{j-1}+v^{j+1}=b_j\,v^j$ (for $j=1,\dots,r$).
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\begin{picture}(148.00,146.00)
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\put(67.00,33.00){\line(1,0){81.00}}
\put(144.00,23.00){\makebox(0,0)[cc]{$v^0$}}
\put(68.00,42.00){\makebox(0,0)[cc]{$v^1$}}
\put(40.00,132.00){\makebox(0,0)[cc]{$v^{r+1}$}}
\put(0.00,12.00){\makebox(0,0)[cc]{$0$}}
\put(123.00,101.00){\makebox(0,0)[cc]{$\sigma$}}
\end{picture}
\end{center}
Drawing rays through the origin and each point $v^j$, respectively,
provides a polyhedral subdivision $\Sigma$ of $\sigma$. The corresponding
toric variety $Y_\Sigma$ is a resolution of our singularity
$Y_\sigma$. The numbers $-b_j$ equal the self intersection numbers of the
exceptional divisors; since $b_j\geq 2$, the resolution is the {\em minimal}
one.\\
\par
\section{The Maximal Resolution}\label{MR}
\neu{MR-1}
{\bf Definition:} (\cite{KS}, (3.12)) For a resolution $\pi:\tilde{Y}\to Y$
we may write $K_{\tilde{Y}|Y}:=K_{\tilde{Y}}-\pi^\ast K_Y=
\sum_{j}(\alpha_j-1)E_j$
($E_j$ denote the exceptional divisors, $\alpha_j\inI\!\!\!\!Q$).
Then, $\pi$ will be called {\em maximal}, if it is maximal with respect to the
property $0<\alpha_j<1$.\\
\par
The maximal resolution is uniquely determined and dominates all the
P-resolutions.
Hence, for our purpose, it is
more important than the minimal one. It can be constructed from the minimal
resolution
by sucsessive blowing
up of points $E_i\cap E_j$ with $\alpha_i+\alpha_j\geq 0$ (cf.\ Lemma
(3.13)
and Lemma (3.14) in \cite{KS}).\\
\par
\neu{MR-2}
{\bf Proposition:}
{\em
The maximal resolution of $Y_\sigma$ is toric. It can be obtained by drawing
rays through $0$ and all interior lattice points (i.e.\ $\in N$) of the
triangle
$\Delta:= \mbox{\em conv} \,(0,v^0,v^{r+1})$, respectively.
}\\
\par
{\bf Proof:}
We have to keep track of the rational numbers $\alpha_j$. Hence, we will show
how
they can be ``seen'' in an arbitrary toric resolution of $Y_\sigma$. Let
$\Sigma<\sigma$
be a subdivision generated by one-dimensional rays through the points
$u^0,\dots,u^{s+1}\in\sigma\cap N$. (In particular, $u^0=v^0=(1,0)$ and
$u^{s+1}=v^{r+1}=(-q,n)$; moreover, for the minimal resolution we would
have
$s=r$ and $u^j=v^j$ ($j=0,\dots,r+1$).) Denote by
$c_1,\dots,c_s$ the
integers given by the relations
\[
u^{j-1}+u^{j+1}=c_j\,u^j \qquad (j=1,\dots,s).
\]
(In particular, $c_j =b_j$ for the minimal resolution again.)\\
\par
As usual, the numbers $-c_j$ equal the self intersection numbers of the
exceptional
divisors $E_j$ in $Y_\Sigma$: Indeed, $D:=\sum_i u^i\,E_i$ is a
principal divisor
(if you do not like coefficients $u^i$ from $N$, evaluate them by arbitrary
elements of $M$); hence,
\[
\begin{array}{rcl}
0 \,=\, E_j\cdot D
&=& E_j\cdot (u^{j-1} E_{j-1} + u^j E_j +
u^{j+1} E_{j+1})\\
&=& u^{j-1}+ (E_j)^2\,u^j + u^{j+1}\\
&=& ( c_j + (E_j)^2 )\cdot u^j \qquad\qquad (j=1,\dots,s).
\end{array}
\]
\par
On the other hand, we can use the projection formula to obtain
\[
\begin{array}{rcl}
-2 \,= \, 2\,g(E_j)-2
&=& K_{\tilde{Y}|Y}\cdot E_j + (E_j)^2\\
&=& \sum_i \,(\alpha_i-1) \,(E_i\cdot E_j) + (E_j)^2\\
&=& (\alpha_{j-1}-1) + (\alpha_j -1)\,(E_j)^2 + (\alpha_{j+1}-1) +
(E_j)^2\,,
\vspace{-1ex}
\end{array}
\]
hence
\[
\alpha_{j-1} + \alpha_{j+1} = c_j\, \alpha_j
\qquad (j=1,\dots,s\,;\; \alpha_0,\,\alpha_{s+1}:=1).
\vspace{1ex}
\]
\par
Looking at the definition of the $c_j$'s (via relations among the lattice
points
$u^j$), there has to be some $R\in M_{I\!\!R}$ such that
\[
\alpha_j = \langle u^j,\,R\rangle \qquad\qquad
(j=0,\dots,s+1).
\]
The conditions $\langle u^0,\,R\rangle = \alpha_0 =1$ and
$\langle u^{s+1},\,R\rangle = \alpha_{s+1} =1$ determine $R$ uniquely.
Now, we can see that $\alpha_j$ measures exactly the quotient between the
length of the
line
segment $\overline{0\,u^j}$, on the one hand, and the length of the
$\Delta$-part
of the line through $0$ and $u^j$, on the other hand. In particular,
$\alpha_j<1$ if and only if $u^j$ sits below the line connecting $u^0$ and
$u^{s+1}$.
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\put(108.00,78.00){\makebox(0,0)[lc]{\scriptsize $u^i$ with $\alpha_i>1$}}
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\put(32.00,135.00){\makebox(0,0)[cc]{\scriptsize line $[R=1]$}}
\put(102.00,111.00){\makebox(0,0)[cc]{$\sigma$}}
\end{picture}
\end{center}
This explains how to construct the maximal resolution: Start with the minimal
one
and continue subdividing each small cone $\langle u^j,u^{j+1}\rangle$
into $\langle u^j, u^j+ u^{j+1}\rangle \,\cup \,
\langle u^j + u^{j+1}, u^{j+1}\rangle$ as long as it contains interior
lattice
points below the line $[R=1]$, i.e.\ belonging to $\mbox{int}\, \Delta$.
\hfill$\Box$\\
\par
{\bf Corollary:}
{\em Every P-resolution is toric.}\\
\par
{\bf Proof:} P-resolutions are obtained by blowing down curves in the maximal
resolution.
\hfill$\Box$\\
\par
\neu{MR-3}
{\bf Example:}
We take the example $Y(19,7)$ from \cite{KS}, (3.15). Since
$\sigma=\langle (1,0),\, (-7,19) \rangle$, the interior of $\Delta$ is given by
the
three inequalities
\[
y>0\,,\; 19x+7y>0\,, \mbox{ and}\; 19x+8y<19\;\,
(\mbox{corresponding to } R=[1,\,8/19])\,.
\]
The only primitive (i.e.\ generating rays) lattice points contained in
$\mbox{int}\,\Delta$ are
\[
u^1\!=(0,1)\,,\; u^2\!=(-1,4)\,,\; u^3\!=(-2,7)\,,\; u^4\!=(-1,3)\,,\;
u^5\!=(-5,14)\,,\; u^6\!=(-4,11)\,.
\]
They provide the maximal resolution.
The corresponding $\alpha$'s can be obtained
by taking the scalar product with $R=[1,\,8/19]$, i.e.\
they are $8/19$, $13/19$, $18/19$, $5/19$, $17/19$, and $12/19$.\\
The minimal resolution uses only the rays through $\,u^1=(0,1)$,
$\,u^4=(-1,3)$, and
$\,u^6=(-4,11)$, respectively.\\
\par
\section{P-Resolutions}\label{P}
\neu{P-1}
In this section we will speak about {\em partial} toric resolutions
$\pi:Y_\Sigma\to Y_\sigma$. Nevertheless, we use the same notation as we
did for the maximal resolution: The fan $\Sigma$ subdividing $\sigma$ is
genarated by rays through $u^0,\dots,u^s\in\sigma\cap N$; each ray
$u^j$ corresponds to an exceptional divisor $E_j\subseteq Y_\Sigma$.
However, since $u^{j-1} + u^{j+1}$ need not to be a multiple of
$u^j$, the numbers $c_j$ do not make sense anymore.\\
\par
{\bf Lemma:} (\cite{toricMori}, (4.3))
{\em
For $K:=K_{Y_\Sigma}$ or $K:=K_{Y_\Sigma|Y_\sigma}$ the intersection number
$(E_j\cdot K)$ is positive, zero, or negative, if the line segments
$\overline{u^{j-1} u^j}$ and $\overline{u^j u^{j+1}}$ form a strict
concave, flat, or strict convex ``roof'' over the two cones, respectively.
}
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\put(18.00,66.00){\makebox(0,0)[cc]{$u^{j+1}$}}
\put(46.00,19.00){\makebox(0,0)[cc]{$u^{j-1}$}}
\put(27.00,8.00){\makebox(0,0)[cc]{\footnotesize $(E_j\cdot K)>0$}}
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\put(207.00,8.00){\makebox(0,0)[cc]{\footnotesize $(E_j\cdot K)<0$}}
\put(231.00,60.00){\makebox(0,0)[cc]{$u^j$}}
\put(140.00,47.00){\makebox(0,0)[cc]{$u^j$}}
\put(34.00,42.00){\makebox(0,0)[cc]{$u^j$}}
\put(0.00,14.00){\makebox(0,0)[cc]{$0$}}
\put(90.00,14.00){\makebox(0,0)[cc]{$0$}}
\put(226.00,29.00){\circle*{3.00}}
\put(226.00,54.00){\circle*{3.00}}
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\end{picture}
\end{center}
{\bf Proof:}
Using $K:=K_{Y_\Sigma}= -\sum_{i=0}^{s+1}E_i$
(cf.\ \cite{Oda}, (2.1)) we have
\[
(E_j\cdot K) = - (E_j\cdot E_{j-1}) -(E_j)^2 -
(E_j\cdot E_{j+1})\,.
\]
On the other hand, as in the proof of Proposition \zitat{MR}{2}, we know
that
\[
0 = (E_j\cdot E_{j-1}) \,u^{j-1} + (E_j)^2 \, u^j
+ (E_j\cdot E_{j+1}) \, u^{j+1}\,.
\]
Combining both formulas yields the final result
\[
(E_j\cdot K)\, u^j =
(E_j\cdot E_{j-1}) \,(u^{j-1}-u^j) +
(E_j\cdot E_{j+1}) \,(u^{j+1}-u^j)\,.
\vspace{-3ex}
\]
\hspace*{\fill}$\Box$\\
\par
{\bf Remark:} The previous lemma together with Proposition \zitat{MR}{2}
illustrate again the fact that all P-resolutions (and we just need the
fact that the canonical divisor is relatively ample) are dominated by the
maximal resolution.\\
\par
\neu{P-2}
In \cite{Ch-CQS} Christophersen has defined the set
\[
K_{e-2}:=\{(k_2,\dots,k_{e-1})\in I\!\!N^{e-2}\,|\; [k_2,\dots,k_{e-1}]
\mbox{ is well defined and yields } 0\,\}
\]
of chains representing zero. To every such chain there are assigned
non-negative
integers $q_1,\dots,q_e$ characterized by the following mutually
equivalent properties:
\begin{itemize}
\item
$q_1=0$, $\,q_2=1$, and
$\;q_{i-1} + q_{i+1} = k_i\, q_i
\quad (i=2,\dots,e-1)$;
\item
$q_{e-1}=1$, $\,q_e=0$, and
$\;q_{i-1} + q_{i+1} = k_i\, q_i
\quad (i=2,\dots,e-1)$;
\item
$q_e=0$ and
$\,[k_i,\dots,k_{e-1}]= q_{i-1}/q_i$ with
$\mbox{gcd}(q_{i-1},q_i)=1 \;(i=2,\dots,e-1)$.
\end{itemize}
(The two latter properties do not even use the fact that the continued
fraction $[k_2,\dots,k_{e-1}]$ yields zero.)\\
\par
{\bf Remark:}
The elements of $K_{e-2}$ correspond one-to-one to triangulations of a
(regular)
$(e-1)$-gon with vertices $P_2,\dots,P_{e-1},P_\ast$. Then, the numbers $k_i$
tell
how many triangles are attached to $P_i$. The numbers $q_i$ have an
easy meaning in this language, too.\\
\par
Finally, for a given $Y_\sigma$ with embedding dimension $e$, Christophersen
defines
\[
K(Y_\sigma):=\{(k_2,\dots,k_{e-1})\in K_{e-2}\,|\; k_i\leq a_i\}\,.
\vspace{1ex}
\]
\par
{\bf Theorem:}
{\em
Each P-resolution of $Y_\sigma$ (i.e.\ the corresponding subdivision $\Sigma$
of $\sigma$)
is given by some $\underline{k}\in K(Y_\sigma)$ in the following way:
\vspace{0.5ex}\\
(1) $\Sigma$ is built from the rays that are orthogonal to
$w^i/q_i - w^{i-1}/q_{i-1}\in M_{I\!\!R}$
(for $i=3,\dots,e-1$).
In some sense, if the occuring divisions by zero are interpreted well,
$\Sigma$ may be seen as dual to the Newton boundary generated by
$w^i/q_i\in\sigma^{\scriptscriptstyle\vee}$ ($i=1,\dots,e$).
\vspace{0.5ex}\\
(2) The affine lines $[\langle \bullet,\, w^i\rangle = q_i]$ form the
``roofs'' of the $\Sigma$-cones. In particular, the (possibly degenerate)
cones $\tau^i\in\Sigma$ correspond to the elements $w^1,\dots,w^e\in E$
The ``roof'' over the cone $\tau^i$ has length
$\ell_i:= (a_i-k_i)\,q_i$
(the lattice structure $M\subseteq M_{I\!\!R}$ induces a metric on rational lines).
In particular, $\tau^i$ is degenerated if
and only if $k_i=a_i$. The Milnor number of the T-singularity
$Y_{\tau^i}$
equals $(a_i-k_i-1)$.
}\\
\par
\neu{P-3}
{\bf Proof:}
According to the notation introduced in \zitat{P}{1}, the fan $\Sigma$ consists
of (non-degenerate) cones $\tau^j:=\langle u^{j-1}, u^j \rangle$ with
$j=1,\dots,s+1$. (Except $u^0=(1,0)$ and $u^s=(-q,n)$, their generators
$u^j$
are primitive lattice points (i.e.\ $\in N$) contained in
$\mbox{int}\Delta\subseteq \sigma$.)
\vspace{1ex}\\
{\em Step 1: \quad
For each $\tau^j$ there are $w\in E, d\inI\!\!N$ such that
$\langle u^{j-1}, w \rangle = \langle u^j, w \rangle = d$.}\\
First, it is very clear that there are a primitive lattice point $w\in M$ and
a non-negative number $d\inI\!\!R_{\geq 0}$ admitting the desired properties.
Moreover, since $u^j\in N$, $d$ has to be an integer, and
Reid's Lemma \zitat{P}{1}
tells us that $w\in \sigma^{\scriptscriptstyle\vee}$. It remains to show that $w$ belongs even
to
the Hilbert basis $E\subseteq \sigma^{\scriptscriptstyle\vee}\cap M$.\\
Denote by $\ell$ the length of the line segment $\overline{u^{j-1}u^j}$ on
the
``roof'' line $[\langle \bullet,\,w\rangle =d]$.
Since $\tau^j$ represents a T-singularity, we know from (7.3) of \cite{Homog}
(cf.\ \zitat{I}{1} of the present paper) that $d|\ell$. In particular,
$\overline{u^{j-1}u^j}$ contains the $d$-th multiple $d\cdot u$
of some lattice point $u\in\tau^j\cap M$ (w.l.o.g.\ not belonging to the
boundary
of $\sigma$).
Hence, $\langle u,\, w\rangle =1$ and $u\in \mbox{int}\,\sigma\cap M$,
and this implies $w\in E$.\
\vspace{1ex}\\
{\em Step 2:}
Knowing that each of the cones $\tau^1,\dots,\tau^{s+1}\in\Sigma$ is
assigned
to some element $w\in E$, a
slight adaption of the notation (a renumbering) seems to be very useful:
Let $\tau^i=\langle u^{i-1},\, u^i\rangle$ be the cone assigned to
$w^i\in E$, and denote by $d_i,\, \ell_i$ the height and the length
of its ``roof'' $\overline{u^{i-1}u^i}$, respectively. Some of these cones
might be degenerated, i.e.\ $\ell_i=0$. This it at least true for the
extremal
$\tau^1$ and $\tau^e$ coinciding with the two rays spanning $\sigma$. Here we
have
even $d_1=d_e=0$; in particular $u^0=u^1=(1,0)$ and
$u^{e-1}=u^e=(-q,n)$.\\
Since $d_i|\ell_i$, we may introduce integers $k_i\leq a_i$
yielding
$\ell_i=(a_i-k_i)\,d_i$. For $i=2,\dots,e-1$ they are even
uniquely
determined.
\vspace{1ex}\\
{\em Step 3:}
Using the following three ingrediences
\begin{itemize}
\item[(i)]
$\langle u^{i-1}, \, w^i\rangle =
\langle u^i, \, w^i\rangle = d_i\quad
(i=1,\dots,e)\,$,
\item[(ii)]
$w^{i-1} + w^{i+1} = a_i\, w^i\quad
(i=2,\dots,e-1;\;$ cf.\ \zitat{CQS}{3}), and
\item[(iii)]
$\langle u^i - u^{i-1},\, w^{i-1} \rangle = \ell_i =
(a_i-k_i)\,d_i$ (since $\{w^{i-1},w^i\}$ forms a $Z\!\!\!Z$-basis of
$M$),
\end{itemize}
we obtain
\[
\begin{array}{rcl}
d_{i-1} + d_{i+1}
&=&
(a_i\,d_i + d_{i-1}) -
(a_i\,d_i - d_{i+1})\\
&=&
(a_i\,d_i + \langle u^{i-1}, w^{i-1} \rangle) -
\langle u^i, \, a_i\, w^i- w^{i+1} \rangle\\
&=&
a_i\,d_i + \langle u^{i-1}, w^{i-1} \rangle
- \langle u^i, w^{i-1} \rangle\\
&=&
a_i\,d_i + \langle u^{i-1}-u^i, \,w^{i-1} \rangle\\
&=&
a_i\,d_i - (a_i-k_i)\,d_i
\; = \;
k_i\,d_i\quad (\mbox{for } i=2,\dots,e-1)\,.
\end{array}
\]
In particular, $k_i\geq 0$ (and even $\geq 1$ for $e>3$).
Moreover, since $\{w^{i-1},w^i\}$ forms
a basis of $M$ and $u^{i-1}\in N$ is primitive,
we have $\mbox{gcd}(d_{i-1},d_i)=1$. Hence,
$d_i=q_i$ (both series of integers satisfy the second
of the three properties mentioned in the beginning of \zitat{P}{2}).
Finally, the third of these properties yields
$[k_2,\dots,k_{e-1}]=q_1/q_2 = d_1/d_2 = 0$,
i.e.\ $\underline{k}\in K_{e-2}$.
\vspace{1ex}\\
The reversed direction (i.e.\ the fact that each $K(Y_\sigma)$-element
indeed yields a P-resolution) follows from the above calculations in
a similar manner.
\hfill$\Box$\\
\par
{\bf Remark:}
Subdividing each $\tau^i$ further into $(a_i-k_i)$ equal cones
(with ``roof'' length $q_i$ each) yields the so-called
M-resolution (cf.\ \cite{M}) assigned to a P-resolution. It is defined
to contain only T$_0$-singularities (i.e.\ T-singularities with
Milnor number 0); in exchange, $K_{\tilde{Y}|Y}$ does not need to be
relatively ample anymore. This property is replaced by ``relatively nef''.\\
\par
{\bf Examples:}
(1) The continued fraction $[1,2,2,\dots,2,1]=0$ yields $q_1=q_e=0$
and $q_i=1$ otherwise. In particular, the ``roof'' lines equal
$[\langle \bullet,\,w^i\rangle =1]$ (for $i=2,\dots,e-1$)
describing the RDP-resolution of $Y_\sigma$. The assigned M-resolution
equals the minimal resolution mentioned at the end of \zitat{CQS}{3}.
\vspace{1ex}\\
(2) Let us return to Example \zitat{MR}{3}:
The embedding dimension $e$ of $Y_\sigma$ is $6$, the vector
$(a_2,\dots,a_{e-1})$ equals $(2,3,2,3)$, and, except the trivial RDP element
mentioned in (1),
$K(Y_\sigma)$ contains only $(1,3,1,2)$ and $(2,2,1,3)$.\\
In both cases we already know that $q_1= q_6=0$ and
$q_2=q_5=1$. The remaining values are given by the equation
$q_3/q_4=[k_4,k_5]$, i.e.\ we obtain
$q_3=1$, $q_4=2$ or $q_3=2$, $q_4=3$, respectively.
\vspace{0.5ex}\\
Hence, in case of $(1,3,1,2)$ the fan $\Sigma$ is given by the additional
rays through $(0,1)$ and $(-4,11)$. For $\underline{k}=(2,2,1,3)$ we need the only
one through $(-1,4)$.\\
\par
|
1996-02-05T06:20:34 | 9602 | alg-geom/9602002 | en | https://arxiv.org/abs/alg-geom/9602002 | [
"alg-geom",
"math.AG"
] | alg-geom/9602002 | Israel Vainsencher | Israel Vainsencher | Flatness of families induced by hypersurfaces on flag varieties | 13 pages, LaTeX | null | null | null | null | We answer a question posed by S. Kleiman concerning flatness of the family of
complete quadrics. We also show that any flat family of hypersurfaces on
Grassmann varieties induces a flat family of intersections with the
corresponding flag variety.
| [
{
"version": "v1",
"created": "Fri, 2 Feb 1996 16:58:02 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Vainsencher",
"Israel",
""
]
] | alg-geom | \section*{Introduction}
Let ~${\bf S}$~ be the variety of complete quadrics, ~${\s^{nd}}$~ the open
subset of nondegenerate quadrics and ~${\ff{}}$~ the scheme of
complete flags in ~$\p n$. Let ~$\raise2pt\hbox{$\varphi$}_0:{\s^{nd}}\mbox{${\,\rightarrow\,}$} \hbox{\bf Hilb\hskip1pt}({\ff{}})$~ be the
morphism that assigns to each non\-degen\-erate quadric the locus of its
tangent flags. We prove the following.
{\bf Theorem. } \it ~$\raise2pt\hbox{$\varphi$}_0$~ extends to a morphism
{}~$\raise2pt\hbox{$\varphi$}:{\bf S} \mbox{${\,\rightarrow\,}$} \hbox{\bf Hilb\hskip1pt}(\ff{})$\rm.
This answers affirmatively a question S. Kleiman asked in ([K], p.362).
We first show that ~${{\bf S}}$~ parametrizes a flat family that
restricts, over ~${\s^{nd}}$, to the family of the graphs of the Gauss
map (point $\mapsto$ tangent hyperplane) of nondegenerate quadric
hypersurfaces. The family pertinent to Kleiman's question is obtained
by composing the family of graphs with the appropriate flag bundle
(point $\in$ line $\subset \dots \subset $ hyperplane).
Our proof of flatness for the completed family of graphs relies on
Laksov's description [L] of Semple--Tyrrell's ``standard'' affine open
cover of {\bf S}.
The space of complete conics has recently reappeared as a simple
instance of Kontsevich's spaces of stable maps (cf. [P]). It is also
instrumental for the counting of rational curves on a K3 surface double
cover of the plane (cf. [V]). Complete quadric surfaces play a role
in Narasimhan--Trautmann [NT] study of a compactification of a space of
instanton bundles.
We also show that \em any flat family of hypersurfaces on Grassmann
varieties induces a flat family of subschemes of the corresponding
flag variety \rm (cf. \rf{prop}).
This statement was first obtained as an earlier attempt to answer
Kleiman's question. We observe that for the case of quadric
hypersurfaces the family described in the proposition does not induce
the family of tangent flags. In fact, for conics it yields a double
structure on the graph of the Gauss map.(cf. \S\ref{fim} for details).
\section{The tangent flag to a smooth quadric}
Write $x =(x_1,\dots,x_{n+1})$ (resp. $y =(y_1,\dots,y_{n+1})$) for
the vector of homogeneous coordinates in $\p n$ (resp. $\pd n$ ). Let
{}~$\ff{0,n-1}\subset\p n\times\pd n$~ be the incidence correspondence
``point $\in$ hyperplane''. It is the zeros of the incidence section
{}~$x\cdot{}y$~ of $\mbox{${\cal O}$}_{\p n}(1)\otimes\mbox{${\cal O}$}_{\pd{\,\, n}}(1)$.
Let ~$\mbox{\large$\kappa$}\subset\p n$~ denote a smooth quadric represented by a
symmetric matrix $a$. The Gauss map ~$\gamma:\mbox{\large$\kappa$}\mbox{${\,\rightarrow\,}$}\pd n$~ is given
by $x\mapsto y=x\cdot a$. Hence we have
$$
\gamma^*(\mbox{${\cal O}$}_{\pd{\,\, n}}(1))=\mbox{${\cal O}$}_{\p n}(1)_{|\mbox{\large$\kappa$}}.
$$
The tangent flag ~$\widetilde{\kp}\subset\ff n$~ of ~$\mbox{\large$\kappa$}$~ is equal to the
restriction of the flag bundle
$$
\ff n\mbox{${\,\rightarrow\,}$}\ff{0,n-1}\subset\p n\times\pd n \vspace{-1pt}
$$
over the graph ~${\bf\Gamma}_{\mbox{$\kappa$}}$~ of ~$\gamma$. Consequently,
flatness of the family ~$\{\widetilde{\kp}\}$~ of tangent flags
is equivalent to flatness of the
family of graphs ~$\{{\bf\Gamma}_{\mbox{$\kappa$}}\}$. The latter will be
handled in \S\ref{graphs}.
We proceed to compute the Hilbert polynomial of the graph ${\bf\Gamma}$ of the
Gauss map of a general quadric hypersurface ~$\mbox{\large$\kappa$}\subset\p n$.
\vskip15pt\begin{exx}{\bf{Lemma. }}}\def\el{\rm\end{exx}
Notation as above, the Hilbert polynomial
{}~$\mbox{\raise3pt\hbox{$\chi$}}\big(\mbox{${\cal O}$}_{{\bf\Gamma}}(\mbox{${\cal L}$}^{\otimes t})\big)$~ with respect to
$$
\mbox{${\cal L}$}=\big(\mbox{${\cal O}$}_{\p n}(1)\otimes{}\mbox{${\cal O}$}_{\pd{\, n}}(1)\big)_{|{\bf\Gamma}}
$$
is equal to
$$
{2\,t+n\choose n}-{2(t-1)+n\choose n}.
$$
\label{hilbgen}\el
\vspace{-15pt}\vskip10pt\noindent{\bf Proof.\hskip10pt}
We have $\mbox{${\cal L}$}\cong\mbox{${\cal O}$}_{\p n}(2)_{|\mbox{\large$\kappa$}}$ under the identification ${\bf\Gamma}\cong\mbox{\large$\kappa$}$.
Thus we may compute
$$
\ba{cl}\mbox{\raise3pt\hbox{$\chi$}}(\mbox{${\cal L}$}^{\otimes t})
&= \mbox{\raise3pt\hbox{$\chi$}}\big(\mbox{${\cal O}$}_{\p n}(2t)\big)_{|\mbox{\large$\kappa$}}\\
\noalign{\vskip5pt}
&=\mbox{\raise3pt\hbox{$\chi$}}\big(\mbox{${\cal O}$}_{\p n}(2t)\big) - \mbox{\raise3pt\hbox{$\chi$}}\big(\mbox{${\cal O}$}_{\p n}(2t-2)\big)\\
\noalign{\vskip7pt}
&={2\,t+n\choose n}-{2(t-1)+n\choose n}.
\end{array}
$$\vspace{-30pt}\vskip-7pt\phantom{|}\hfill\mbox{$\Box$}
\section{Hilbert polynomial of loci of rank 1 matrices}
The image of the Segre imbedding $\p n\times\p n\mbox{${\,\rightarrow\,}$}\p N$ is the
variety of matrices of rank one. The image $\Delta$ of the diagonal
$\p n \mbox{${\,\rightarrow\,}$} \p n \times\p n\mbox{${\,\rightarrow\,}$}\p N$ is the subvariety of $symmetric$
matrices of rank one. It's Hilbert polynomial is easily found to be
$$
\hbox{dim\hskip2pt}(H^0(\Delta,\mbox{${\cal O}$}_{\p N}(t))~=~{2t+n\choose n}.
$$
The bi-homogeneous ideal $I_{\Delta}$ of the diagonal is generated by
the 2$\times2$ minors of the matrix
\begin{equation}}\def\ee{\end{equation}\label{2x2}
\left [\ba{cccc}
x_{{1}}&x_{{2}}&\dots&x_{{n+1}}\\\noalign{\medskip}
y_{{1}}&y_{{2}}&\dots&y_{{n+1}}
\end{array}\right].
\ee
Write
$$
S=k[x_1,\dots, x_{{n+1}},y_1,\dots ,y_{{n+1}}]
$$
for the polynomial ring in $2n+2$ variables, and let $S_{i,j}$ denote
the space of bi-hom\-oge\-neous polynomials of bi-degree $(i,j)$. We
have
$$
\hbox{dim\hskip2pt}_k\,S_{t,t}\big/(I_{\Delta})_{t,t}~=~{2t+n\choose n}.
$$
Quite generally, for a closed subscheme $X\subseteq \p m\times\p n$
defined by a bi-homogeneous ideal $I\subseteq S$ we have (cf. [KTB],
p. 189)
$$
H^0\big(X,\mbox{${\cal O}$}_{\p m}(t)\otimes{}\mbox{${\cal O}$}_{\p { n}}(t)_{|X}\big)~=~
S_{t,t}\big/(I)_{t,t}~~\hbox{ for all }~t >>0.
$$
Indeed, the homomorphism
\begin{equation}}\def\ee{\end{equation}\label{rs}\ba{ccc}
R=k[z_{i,j}]\big/\id{z_{i,j}z_{r,s}-z_{r,j}z_{i,s}}&\mbox{${\,\longrightarrow\,}$} &
S\\
z_{i,j}&\mapstochar\longrightarrow& x_iy_j
\end{array}\ee
maps $R_t$ isomorphically onto $S_{t,t}$. Let the bi-homogeneous ideal
$I\subseteq S$ be generated by polynomials of some fixed bidegree
($d,d$). Its inverse image via \rf{rs} generates a homogeneous ideal
$I^\#\subseteq R$.
We have
$$
(R/I^\#)_t\widetilde{\mbox{${\,\longrightarrow\,}$}}S_{t,t}/I_{t,t}.
$$
Now for $t>>0$ we may write
$$
(R/I^\#)_t~=~H^0\big(X,\mbox{${\cal O}$}_{\p N}(t)_{|X}\big)~=~
H^0\big(X,\mbox{${\cal O}$}_{\p m}(t)\otimes{}\mbox{${\cal O}$}_{\p {n}}(t)_{|X}\big).
$$
Let $L(I)$ denote the monomial ideal of initial terms of $I$ with
respect to some bi-graded monomial order. Then we have the equality of
Hilbert functions,
$$
\raise2pt\hbox{$\varphi$}_I(i,j)~=~\raise2pt\hbox{$\varphi$}_{L(I)}(i,j).
$$
This is rather standard: let $f_1,\dots,f_k$ be linearly independent
forms of bidegree $(i,j)$ in $I$. Replacing if needed each $f_\mu$ by
$f_\mu-cf_\nu$ for suitable $c\in k$, we may assume their initial
terms $L\,f_\mu \ne L\,f_\nu $. Hence the initial terms $Lf_1, \dots,
Lf_k$ are linearly independent monomials in $L(I)_{i,j}$. This shows
that $ \raise2pt\hbox{$\varphi$}_I(i,j) \leq \raise2pt\hbox{$\varphi$}_{L(I)}(i,j). $ Conversely, pick monomials
\,$g_1> \dots> g_k$\, in $L(I)_{i,j}$. We have each $g_\mu = Lf_\mu$
for some $f_\mu \in I_{i,j}$. It follows that $f_1, \dots, f_k$ are
linearly independent.
\vskip15pt\begin{exx}{\bf{Lemma. }}}\def\el{\rm\end{exx}\label{k0}
Let ~${\bf\Gamma}_0$~ be the subscheme of $\p n\times\pd n$ defined by the ideal
$$
\id{x_iy_j\mbox{$\,|\,$} 1\leq i<j\leq n+1}+\id{\sum x_iy_i}.
$$Then we have
$$
\raise2pt\hbox{$\varphi$}_{{\bf\Gamma}_0}(t)={2\,t+n\choose n}-{2(t-1)+n\choose n}.
$$
\el
\vskip10pt\noindent{\bf Proof.\hskip10pt}
The whole point is to notice that the $ x_iy_j$ span the ideal of
initial terms of $I_\Delta$ with respect to a suitable order.\footnote
{I'm indebted to P. Gimenez for his precious help on this
matter.} ~In fact, the set of $2\times2$ minors of \rf{2x2} is known
to be a (universal) Gr\"obner basis for $I_\Delta$ (see Sturmfel
[BS]). By the above discussion, we may write
$$
\raise2pt\hbox{$\varphi$}_{L(I_\Delta)}(t)=\raise2pt\hbox{$\varphi$}_{I_\Delta}(t)={2t+n\choose n}.
$$
One checks at once that $\sum x_iy_i$ is a nonzero divisor mod
$L(I_\Delta)$ (see \rf{rmk} (i)). Therefore
$$
\raise2pt\hbox{$\varphi$}_{{\bf\Gamma}_0}(t)=\raise2pt\hbox{$\varphi$}_{L(I_\Delta)}(t)-\raise2pt\hbox{$\varphi$}_{L(I_\Delta)}(t-1).\vspace{-10pt}
$$\phantom{|}\hfill\mbox{$\Box$}
We will deduce flatness for the ``completed'' family of Gauss maps
from the fact that the above Hilbert polynomial at the special point
{}~${\bf\Gamma}_0$~ coincides with the generic one \rf{hilbgen}.
\section{Semple-Tyrrell-Laksov cover}
Let ${\bf U}_n$ denote the group of lower triangular unipotent
($n+1$)-matrices. Thus, ${\bf U}_n$ is isomorphic to the affine space
$\af{n(n+1)/2}$ with coordinate functions ~$u_{i,j},\,1\leq j\leq
i-1,\,i=2\dots n+1$. These are thought of as entries of the matrix,
$$
u~=~\left [\begin {array}{lllll} 1&0&0&\cdots&0\\
\noalign{\medskip}u_{{2,1}}&1&0&\cdots&0\\
\noalign{\medskip}u_{{3,1}}&u_{{3,2}}&1&\cdots&0\\
\noalign{\medskip}\cdots&\cdots&\cdots&\cdots&\cdots\\
u_{{n+1,1}}&u_{{n+1,2}}&u_{{n+1,3}}&u_{{n+1,n}}&1
\end {array}\right ].
$$
Let $d_1,\dots,d_n$ be coordinate functions in $\af n$.
Put
\begin{equation}}\def\ee{\end{equation}\label{d1}
d^{(1)}=\left [\begin {array}{lllll} 1&0&0&\cdots &0\\
\noalign{\medskip}0&d_{{1}}&0&\cdots &0\\
\noalign{\medskip}0&0&d_{{1}}d_{{2}}&\cdots &0\\
\noalign{\medskip}\cdots&\cdots&\cdots&\cdots&\cdots\\
0&0&0&\cdots &d_{{1}}d_{{2}}\cdots d_{{n}}
\end {array}\right ].
\ee
For a matrix $A$ let it's $i$th adjugate be the matrix $\wed i A$ of
all $i\times i$ minors. We denote by $d^{(i)} $ the matrix
obtained from $\wed i d^{(1)}$ by removing the common factor
$d_1^{i-1}d_2^{i-2}\cdots d_{i-1}$. $E.g.,$ for $n=3$ we have
$$\ba{ll}
d^{(1)} &= {\rm diag}(1,\,d_1,\,d_1d_2,\, d_1d_2 d_3)\\
\noalign{\medskip}d^{(2)} &= {\rm
diag}(d_1,\,d_1d_2,\,d_1d_2d_3,\,d_1^2d_2,\,d_1^2d_2d_3,\,d_1^2d_2^2d_3)
\big/(d_1)
\\
&= {\rm diag}(1,\, d_2,\, d_2d_3,\, d_1d_2,\, d_1d_2d_3,\, d_1d_2^2d_3)
\\\noalign{\medskip}
d^{(3)} &=
{\rm diag}(1 ,\, d_3 ,\,d_2d_3 ,\,d_1d_2d_3).
\end{array}$$
The map ~$ {\bf U}_n\times\af n\mbox{${\,\rightarrow\,}$} {\bf S}\subset{
\prod_{i=1}^{i=n}\ps{
S_2(\bigwedge\hskip-8pt\raise6pt\hbox{$^i$}\,\,k^{n+1*})}}$~
defined by sending $(u,\,d)$ to
$$
\big(u\,d^{(1)}\,u^t,\,(\wed2\,u)\,d^{(2)}\,\wed2u^t,\dots,(\wed nu)\,d^{(n)}\,
\wed nu^t\big)
$$
is an isomorphism onto an affine open subset ${\bf S}^0$ of {\bf S}.
The variety of complete quadrics may be covered by translates of
${\bf S}^0$ (cf. Laksov [L]).
Let ${\bf S}^0_d \cong{\bf U}_n\times\af n\!_{d}$ be the
principal open piece defined by $d_1d_2\cdots d_n\ne0$. It maps
isomorphically onto an open subvariety of $\s^{nd}$.
\section{Graph of the Gauss map}\label{graphs}
The variety ~$\s^{nd}$~ of nondegenerate quadrics parametrizes a flat
family of graphs of Gauss maps. For a nondegenerate quadric
represented by a symmetric matrix $a\in\s^{nd}$ the Gauss map is given by
$x\mapsto y=x\cdot a$. We define $\mbox{\hbox{\rm I\hskip-2pt K}}^{nd}\subset\s^{nd}\mbox{$\times$}\p n\mbox{$\times$}\pd
n$ by the bi-homogeneous ideal generated by the incidence relation
{}~$x\cdot y$~ together with the 2$\times$2 minors of the 2$\times(n+1)$
matrix with rows ~$y,\,x\cdot z$,~ where \,$z$\, denotes the generic
symmetric matrix. Clearly ~$\mbox{\hbox{\rm I\hskip-2pt K}}^{nd}\mbox{${\,\rightarrow\,}$}\s^{nd}$ is a map of ${\bf
GL}_{n+1}-$homogeneous spaces.
Now write $a = vc^{(1)}v^t$ with $v\in{\bf U}_n,\, c\in\af n\!_d$
\,($c^{(1)}$\, as in \rf{d1}), and put $x'=xv$, $y'=y(v^{-1})^t$. We
have $y=xa$ iff $y'= x'c^{(1)}$. Let
\begin{equation}}\def\ee{\end{equation}
{\mbox{\hbox{\rm I\hskip-2pt K}}}^0_d ~\subset~ {\bf S}^0_d \times\p n\times\pd n.
\label{g0d}
\ee
be defined by ~$x\cdot y$~ together with the 2$\times$2 minors of the
2$\times(n+1)$ matrix
\begin{equation}}\def\ee{\end{equation}\label{x'd}
\left [\begin {array}{ccccc} x'_{{1}}&d_1x'_{{2}}&d_1d_2x'_{{3}}&\dots&
d_1\cdots d_nx'_{{n+1}}\\
\noalign{\medskip}y'_{{1}}&y'_{{2}}&y'_{{3}}&\dots&y'_{{n+1}}
\end {array}\right ]
\ee
where we put $x'_j=\sum_iu_{ij}x_i$ and likewise $y'_j$ denotes the
$j$th entry of $y(u^{-1})^t$. Thus ${\mbox{\hbox{\rm I\hskip-2pt K}}}^0_d$ is the
total space of the family of Gauss maps parametrized by ${\bf S}^0_d$.
Note that ${\mbox{\hbox{\rm I\hskip-2pt K}}}^0_d \mbox{${\,\rightarrow\,}$} {\bf S}^0_d$ is a smooth quadric
bundle. Its fibre over $(I,(1,\dots,1)) \in {\bf U}_n \times \af
n_{d}$ is equal to the quadric given by \,$\sum x_i^2$\, inside the
``diagonal'' \,$y_1=x_1, \dots,y_{{n+1}}= x_{{n+1}}$\, of ~$\p
n\times\pd n.$
Let
\begin{equation}}\def\ee{\end{equation}\label{gg0} {\mbox{\hbox{\rm I\hskip-2pt K}}}^0 ~\subset~ {\bf S}^0 \times\p n\times\pd n
\ee
be defined by ~$x\cdot y$~ together with the ideal
\begin{equation}}\def\ee{\end{equation}\label{J}
\ba{cl}J=&\langle{}x'_1y'_2-d_1y'_1x'_2,\dots,\,x'_1y'_{n+1}-
d_1\cdots{}d_ny'_1x'_{n+1},\\
&\,\,\,x'_2y'_3-d_2y'_2x'_3,\dots,\,x'_ny'_{n+1}-d_ny'_nx'_{n+1}
\rangle
\end{array}\ee
obtained by cancelling all $d_i$ factors occurring in the above
2$\times$2 minors. We obviously have
{}~$ {\mbox{\hbox{\rm I\hskip-2pt K}}}^0_{\,|\,{\bf S}^0_d} = {\mbox{\hbox{\rm I\hskip-2pt K}}}^0_d$.
We will show that ~$ {\mbox{\hbox{\rm I\hskip-2pt K}}}^0$~ is the scheme theoretic closure of ~$
{\mbox{\hbox{\rm I\hskip-2pt K}}}^0_d$ in ${\bf S}^0 \mbox{$\times$} \p n\mbox{$\times$}\pd n$ (cf. \rf{clos}).
\section{A torus action}
Notation as in \rf{d1}, imbed ~\gm~ in ${\bf GL}_{n+1}$ by sending
$c=(c_1,\dots,c_n)\in\gm$ to $c^{(1)}={\rm diag}(1,\,c_1,\,c_1c_2,\dots)$.
We let ~\gm~ act on ${\bf S}^0$ by
$$
c\cdot (v,b) = (c^{(1)}\, v\, (c^{(1)})^{-1},~(c_1^2b_1, \dots c_n^2b_n)).
$$
This action is compatible with the natural action of ${\bf GL}_{n+1}$
on the space $\ps{S_2(k^{n+1*})}$ of quadrics, $i.e.,$ for a symmetric
matrix ~$a(v,b)\, :=\, v\, b^{(1)}\, v^t$~ as above, we have
$$
\ba{cl}
c^{(1)}\cdot{}a(v,b)&=c^{(1)}\, a(v,b)\,(c^{(1)})^t =
c^{(1)} \,v\,b^{(1)}\,v^t \,(c^{(1)})^t
\\\noalign{\medskip}&=c^{(1)}\,v\,(c^{(1)})^{-1}\,c^{(1)}\,b^{(1)}
\,c^{(1)}\,((c^{(1)})^t)^{-1}\,v^t\, (c^{(1)})^t
\\\noalign{\medskip}&=c^{(1)}\,v\,(c^{(1)})^{-1}\,(c^{(1)})^2\,b^{(1)}
\,((c^{(1)})^t)^{-1}\,v^t\,(c^{(1)})^t
\\\noalign{\medskip}&=a(c\cdot(v,b))\,.
\end{array}$$
It can be also easily checked that ~\gm~ acts compatibly on
${\bf S}^0\times\p n\times\pd n$ and ${\mbox{\hbox{\rm I\hskip-2pt K}}}^0$ is invariant.
Indeed, let $((v,b),x,y)\in{\mbox{\hbox{\rm I\hskip-2pt K}}}^0$. Pick $c\in\gm$.
We have
$$
c\cdot((v,b),x,y)=( (c^{(1)}\, v\, (c^{(1)})^{-1},~(c_1^2b_1, \dots
c_n^2b_n)),~x\,(c^{(1)})^{-1}, \,y\,(c^{(1)})^{t}).
$$
Now $x'=xv$ changes to
$$
x'' ~=~ (x\, (c^{(1)})^{-1})\, (c^{(1)}\, v\, (c^{(1)})^{-1}) ~=~
x\,v\, (c^{(1)})^{-1} ~=~ x'\, (c^{(1)})^{-1}
$$
so that the first row $x'\,b^{(1)}$ in \rf{x'd}
(evaluated at $((v,b),x,y)$)
changes to
$$
x''\,(b^{(1)}\,(c^{(1)})^2) ~=~ x'\, (c^{(1)})^{-1}
\,(b^{(1)}\,(c^{(1)})^2) ~=~ x'\, (b^{(1)}\, c^{(1)}).
$$
Similarly,
$y' ~=~ y\,(v^{-1})^t$ changes to
$$
y'' ~=~ (y\,(c^{(1)})^{t})\,
((c^{(1)}\, v\, (c^{(1)})^{-1})^{-1})^t ~=~
y\,(v^{-1})^t\,(c^{(1)})^{t})
~=~ y'\, c^{(1)}.
$$
Therefore \rf{x'd} changes to the matrix
with rows $x'\, (b^{(1)}\, c^{(1)})$ and $y'\, c^{(1)}$.
Thus evaluation of \rf{J} at $c\cdot((v,b),x,y)$ and at $((v,b),x,y)$
differ only by nonzero multiples.
\vskip15pt\begin{exx}{\bf{Lemma. }}}\def\el{\rm\end{exx}
The orbit of $(I,0)\in{\bf S}^0$ is the unique closed orbit.
\el
\vskip10pt\noindent{\bf Proof.\hskip10pt}
Conjugation of \,$v\in {\bf U}_n$\, by the diagonal matrix \,$c^{(1)}$\,
replaces each entry \,$v_{ij},\,j<i$\, by
$$
\ba{cl}(c^{(1)}\,v\,(c^{(1)})^{-1})_{ij} &=
c^{(1)}_{ii}\,(v\,(c^{(1)})^{-1})_{ij}
= c^{(1)}_{ii}\, v_{ij}\,((c^{(1)})^{-1})_{jj}\\
&= v_{ij}\,c^{(1)}_{ii}/c^{(1)}_{jj}= v_{ij}\,\,c_{i-1}\cdots{}c_{j}.
\end{array}$$
Thus, letting \,$c\ar0$, we see that ($I,0$) is in the orbit closure
{}~$\overline{\gm\cdot(v,b)}$
\vspace{-10pt}\phantom{|}\hfill\mbox{$\Box$}
\section{Proof of the theorem}
\vskip15pt\begin{exx}{\bf{Lemma. }}}\def\el{\rm\end{exx}\label{cc0}
Notation as in \rf{gg0} , the family ~${\mbox{\hbox{\rm I\hskip-2pt K}}}^0\mbox{${\,\rightarrow\,}$}{\bf S}^0$~ is
flat.
\el
\vskip10pt\noindent{\bf Proof.\hskip10pt}
Since ~${\mbox{\hbox{\rm I\hskip-2pt K}}}^0 \mbox{${\,\rightarrow\,}$} {\bf S}^0$~ is equivariant for the ~$\gm-$action, it
suffices to check that the Hilbert polynomial of the fiber over the
representative $(I,0)$ of the unique closed orbit is right, $i.e.,$
coincides with the generic one (cf. Hartshorne [H], thm.9.9, p.261).
Evaluating \rf{J} at $(I,0)$ yields the monomial ideal \rf{2x2}. We
are done by virtue of \rf{hilbgen} and \rf{k0}.\vspace{-25pt}
\phantom{|}\hfill\mbox{$\Box$}
\vskip15pt\begin{exx}{\bf{Lemma. }}}\def\el{\rm\end{exx}
Let $f:X\mbox{${\,\rightarrow\,}$} Y$ be a flat, surjective morphism of schemes. If $U\subseteq Y$
is open and schematically dense in $Y$ then $f^{-1}U$ is open and
schematically dense in $X.$
\el
\vskip10pt\noindent{\bf Proof.\hskip10pt}
We may assume $X,\,Y$ affine. Let $A\subseteq B$ be a flat ring
extension and let $a\in A$ be such that ${\rm Spec}\,A_a$ is
schematically dense in ${\rm Spec}\,A$. This means that every element
in ker\,($A\mbox{${\,\rightarrow\,}$} A_a$) is nilpotent. Flatness implies ker\,($B\mbox{${\,\rightarrow\,}$}
B_a$)=ker\,($A\mbox{${\,\rightarrow\,}$} A_a)\otimes{}B$. Hence ${\rm Spec}\,B_a$ is
schematically dense in ${\rm Spec}\,B$. \vspace{-25pt}
\phantom{|}\hfill\mbox{$\Box$}
\vskip15pt\begin{exx}{\bf{Lemma. }}}\def\el{\rm\end{exx}\label{clos}
Notation as in \rf{gg0} and \rf{g0d}, we have that ~$\mbox{\hbox{\rm I\hskip-2pt K}}^0$~ is
equal to the scheme theoretic closure of ~$\mbox{\hbox{\rm I\hskip-2pt K}}^0_d$.
\el
\vskip10pt\noindent{\bf Proof.\hskip10pt}
In view of \rf{cc0}, we may
apply the previous lemma to ~$\mbox{\hbox{\rm I\hskip-2pt K}}^0\mbox{${\,\rightarrow\,}$}{\bf S}^0\supset{\bf S}^0_d.$
\vspace{-25pt}
\phantom{|}\hfill\mbox{$\Box$}
\vskip15pt\begin{exx}{\bf{Lemma. }}}\def\el{\rm\end{exx}
Let $G$ be an algebraic group and let
$$
\ba{ccc}X^0 &\subset &X \\
\downarrow~&&\downarrow\\
Y^0&\subset &Y
\end{array}$$
be a commutative diagram of maps of ~$G-$varieties. Let
\,$\overline{X}$,~ $\overline{Y}$ denote the closures of
$X^0,\,Y^0.$ If ~$\overline{X}\mbox{${\,\rightarrow\,}$}\overline{Y}$~ is flat over a
neighborhood of a point in each closed orbit then
{}~$\overline{X}\mbox{${\,\rightarrow\,}$}\overline{Y}$~ is flat.
\el
\vskip10pt\noindent{\bf Proof.\hskip10pt} Immediate.
\phantom{|}\hfill\mbox{$\Box$}
We may now finish the proof of the theorem. Let ~$\mbox{\hbox{\rm I\hskip-2pt K}}\subset{\bf S}\mbox{$\times$}\p
n\mbox{$\times$}\pd n$~ be the scheme theoretic closure of ~$\mbox{\hbox{\rm I\hskip-2pt K}}^0$. We have
{}~$\mbox{\hbox{\rm I\hskip-2pt K}}\cap\big({\bf S}^0\mbox{$\times$}\p n\mbox{$\times$}\pd n\big) = \mbox{\hbox{\rm I\hskip-2pt K}}^0$ flat over ~${\bf S}^0$~
by \rf{cc0}. The latter is a neighborhood of a point in the unique closed
orbit of ~${\bf S}$. Now apply the previous lemma to ~$G={\bf GL}_{n+1}$,
$X={\bf S}\mbox{$\times$}\p n\mbox{$\times$}\pd n$, $Y={\bf S}$, $Y^0={\bf S}^{nd}$, $X^0=\mbox{\hbox{\rm I\hskip-2pt K}}^{nd}$.
Finally, since the family of tangent flags is defined by the fibre square,
$$
\ba{ccc}
\widetilde{\mbox{\hbox{\rm I\hskip-2pt K}}}& \mbox{${\,\longrightarrow\,}$} &\ff n \, \mbox{$\times$} \,{\bf S} \\
\downarrow& &\downarrow\\
\mbox{\hbox{\rm I\hskip-2pt K}} & \mbox{${\,\longrightarrow\,}$} &\ff{0,n-1}\,\mbox{$\times$}\,{\bf S}
\end{array}
$$
the composition ~$\widetilde{\mbox{\hbox{\rm I\hskip-2pt K}}}\mbox{${\,\rightarrow\,}$}\mbox{\hbox{\rm I\hskip-2pt K}}\mbox{${\,\rightarrow\,}$}{\bf S}$~ is flat.
\section{Final remarks}\label{fim}
\begin{exx}\em \label{rmk}
(i) The primary decomposition of the monomial ideal in \rf{k0} can be
checked to be given by
$$
\id{x_1,\,x_2,\dots, x_n} \cap \cdots \cap
\id{x_1,\dots, x_i,\,y_{i+2}, \dots, y_{n+1}} \cap \cdots
\cap \id{y_2,\,y_3, \dots, y_{n+1}}.
$$
Thus enlarging it to include the nonzero divisor \,$x\cdot{}y$\,
we see that the special fiber \,${\bf\Gamma}_0$ \, presents no imbedded component.
(ii) The example of $\p n$ acted on by the stabilizer of a point,
blown up at that point might clarify why we were not able
to show directly that the closure of ~$\mbox{\hbox{\rm I\hskip-2pt K}}^{nd}$~ is flat over ~${\bf S}$.
(iii) For $n=1$ we may write the following global equations
for $\mbox{\hbox{\rm I\hskip-2pt K}}$. Let $z,\,w$ be a pair of symmetric 3\vez3 matrices of
independent indeterminates. Then ~$\mbox{\hbox{\rm I\hskip-2pt K}} \subset \p5 \mbox{$\times$} \pd5 \mbox{$\times$} \p2
\mbox{$\times$} \pd2$~ is given by the 2\vez2 minors of the 2\vez3 matrices with
rows $x\cdot z,\,y$ and $x,\,y\cdot w$, in addition to the incidence
relation $x\cdot y$ together with the equation
{}~$3z\cdot w={\rm trace}(z\cdot{}w)I$~for ${\bf S}\subset\p5 \mbox{$\times$} \pd5$.
It would be nice to give a similar description for higher dimension.
(iv) Still assuming $n=1$,
put
$$
{\bf\Gamma}=\{(P,\,\ell,\,\mbox{\large$\kappa$},\,\mbox{\large$\kappa$}')\in\p2\mbox{$\times$}\pd2\mbox{$\times$}\p5\mbox{$\times$}\pd5\mbox{$\,|\,$}
P\in\mbox{\large$\kappa$}\cap\ell,\,\ell\in\mbox{\large$\kappa$}'\}.
$$
It is easy to check that ~${\bf\Gamma}_{|{\bf S}}=\mbox{\hbox{\rm I\hskip-2pt K}}$~ as sets.
Furthermore, ${\bf\Gamma}$ may be endowed with a natural scheme structure such
that ${\bf\Gamma}\mbox{${\,\rightarrow\,}$}\p5\mbox{$\times$}\pd5$ is flat and with Hilbert polynomial of its
fibers equal to $4t$. Thus, ${\bf\Gamma}_{|{\bf S}}\mbox{${\,\rightarrow\,}$}{\bf S}$ is a family of double
structures of genus one on the fibers of $\mbox{\hbox{\rm I\hskip-2pt K}}$.
\end{exx}
In fact, we have the following.
\vskip15pt\begin{exx}{\bf{Proposition. }}}\def\ep{\rm\end{exx}\label{prop}
Any flat family of hypersurfaces on Grassmann varieties induces a
flat family of subschemes of the corresponding flag variety.
\ep
Before considering the general case,
we describe the situation in the projective plane. Thus, let
$$
{\ff2}\subset\p2\times\pd2
$$
be the incidence correspondence ``point ~$\in$~ line''.
Let ~$f_0$~ (resp. ~$f_1$) denote a curve in ~$\p2$~ (resp. ~$\pd2$).
Set
$$
{\bf\Gamma}_{{\underline{f}}}:=(f_0\times f_1)\cap\ff2.
$$
\noindent Then ~${\bf\Gamma}_{{\underline{f}}}$~ is easily seen to be regularly imbedded of codimension
2 in ~${\ff2}$~ (cf. \rf{claim}). Moreover, its Hilbert polynomial
with respect to the ample sheaf ${\cal
O}_{\p2}(1) \otimes {\cal O}_{\pd{\,\,2}}(1)$ restricted to ~${\ff2}$~
depends only on the degrees, say ~$d_0,\,d_1$~ of ~$f_0,f_1$. In fact,
the Koszul complex that resolves the ideal of ~$f_0\times f_1$~ in
$\p2 \times \pd2$~ restricts to a resolution of ~${\bf\Gamma}_{{\underline{f}}}$~ in ~${\ff2}$.
One finds the Hilbert polynomial,
\begin{equation}}\def\ee{\end{equation}
\label{xi}
\xi\hskip-2pt(t)=(d_0+d_1)t- d_0 d_1(d_0+d_1-4)/2 .
\ee
Therefore, as in the final argument for the proof of \rf{cc0},
the parameter space of pairs ~$(f_0,f_1)$, call it ~${\bf T}$
(=$\p{n_0}\times\p{n_1}$~ for suitable ~$n_0,n_1)$,
carries a flat family of curves on ~${\ff2}$.\, Precisely, let
$$
{\bf W}\!_0\subset \p2\times\p{n_0}\hbox{ and }
{\bf W}\!_1\subset \pd2\times\p{n_1}
$$
denote the total spaces of the universal plane curve parametrized by
$\p{n_i}$. Then
$$
{\bf\Gamma}:=({\bf W}\!_0\times\hskip-.38cm\raise-.25cm\hbox{$_{\p2}$}
{\bf W}\!_1)\bigcap{\ff2}\longrightarrow {\bf T}
$$
is a flat family of curves in ~${\ff2}$, with fiber ~${\bf\Gamma}_{{\underline{f}}}$.
\vskip10pt
For the proof of \rf{prop} we
let ~$\gr{r,n}$~ denote the grassmannian of projective subspaces
of dimension \,$r$\, of ~$\p n$.
Recall that the dimension of the variety of complete flags
$\ff n\subset\prod\gr{i,n}$~ is
$$
\hbox{dim\hskip2pt}{\ff n}=1+\cdots+n.
$$
The proposition is an easy consequence of the following.
\vskip15pt\begin{exx}{\bf{Lemma. }}}\def\el{\rm\end{exx} \label{claim}
Let ~$f_0,f_1,\dots,f_n$~ be arbitrary hypersurfaces of
points, lines, \dots, hyperplanes in the appropriate grassmannians of
subspaces of ~$\p{n+1}$. Then the intersection
$$
{\bf\Gamma}_{{\underline{f}}}:= (f_0\times\cdots\times f_n)\cap\ff{n+1}
$$
is of codimension ~$n+1$~ in ~${\ff{n+1}}$.
\ep
\vskip10pt\noindent{\bf Proof.\hskip10pt}
We shall argue by induction on ~$n$.
We may assume all ~$f_i$~ irreducible. Let ~$n=1$. Pick a line ~$h\in
f_1$. Set
$$
h^ {(0)}=\{P\in\p2\mbox{$\,|\,$}{}P\in{}h\}.
$$
The fiber ~$({\bf\Gamma}_{{\underline{f}}})_{h} \simeq
h^ {(0)}\cap f_0$~ is zero dimensional unless ~$h^{(0)}=f_0$. This
occurs for at most one ~$h\in{}f_1$, hence ~${\bf\Gamma}_{{\underline{f}}}$~ is 1--dimensional
(otherwise most of its fibres over ~$f_1$~ would be at least
1--dimensional).
For the inductive step, we set for ~$h\in \pd{n+1}$,
\begin{equation}}\def\ee{\end{equation}\label{hr}
h^ {(r)}=\{g\in\gr{r,n+1}|g\subseteq h\}\simeq\gr{r,n}.
\ee
If the
intersection
$$
f'_r =h^{(r)}\cap f_r
$$
were proper for all ~$r$
and ~$h\in f_n$~ then we would be done by induction. Indeed, we have
$$
({\bf\Gamma}_{{\underline{f}}})_h \simeq(f'_0 \times\cdots\times f'_{n-1})\cap\ff n.
$$
By the induction hypothesis, this is of the right dimension
$$
1+\cdots +n-n =1+\cdots +(n-1).
$$
Since ~$h$~ varies in the ~$n-$dimensional hypersurface ~$f_n$~ of
$\gr{n,n+1}=\pd{n+1}$, we would have
$$
\hbox{dim\hskip2pt} {\bf\Gamma}_{{\underline{f}}}=\big(1+\cdots +(n-1)\big)+n=\big(1+\cdots +(n+1)\big)
-(n+1)
$$
as desired.
However, just as in the case ~$n=1$, it may well happen that the
intersection ~$h^{(r)}\cap f_r$~ be \it not \rm proper for some ~$h,r$.
Thus it remains to be shown that, whenever \hbox{dim\hskip2pt}$({\bf\Gamma}_{{\underline{f}}})_h$~ exceeds the
right dimension, say by ~$\delta$, the hyperplane ~$h$~ is restricted to
vary in a locus of codimension at least ~$\delta$~ in ~$f_n$. This is
taken care of by the lemma below.
\vskip15pt\begin{exx}{\bf{Lemma. }}}\def\el{\rm\end{exx}\label{lema}
Notation as in \rf{hr},
for ~$r=0,\dots,n$~ we have
$$
\hbox{dim\hskip2pt} \{ h\in\pd{n+1}\ |\ h^{(r)}\subseteq\ f_r\} \leq r.
$$
\el
\noindent {\it Proof}. Let ~$\ff{r,n}\subset\pd{n+1}\times\gr{r,n+1}$~ be the
partial flag variety. Form the diagram with natural projections,
$$
\ba{lcr}&\ff{r,n}&\\\noalign{\vskip5pt}
\raise6pt\hbox{$\pi_n$}\! \mbox{\huge$\swarrow$} & & \mbox{\huge$\searrow$}
\raise6pt\hbox{$\pi_r$}~~~~~\\
\noalign{\vskip3pt}\pd{n+1} && ~~\gr{r,n+1}
\end{array}$$
For ~$g_r\in\gr{r,n+1}$, set
$$
g_r^ {(n)}=\{h\in\pd{n+1}\ |\ g_r\subseteq h\}.
$$
We have ~$g_r^ {(n)} \simeq\p{n-r}$~ whence it hits any
subvariety of ~$\pd {n+1}$~ of dimension ~$\geq r+1$. In other
words, for any subvariety ~${\bf Z}\subseteq \pd{n+1}$~ such that
\hbox{dim\hskip2pt} ~${\bf Z}\geq r+1$, we have
$$
\ba{ccl}\pi_r\pi^ {-1}_n{\bf Z} &=&
\{g_r\ |\ \exists\, h\in {\bf Z}\hbox{ s.t. } h\supseteq g_r\}\\
&=& \{g_r\ |\ g_r^{(n)}\cap{\bf Z}\ne\emptyset\}\\
&=&\gr{r,n+1}.
\end{array}$$
The lemma follows by taking {\bf Z}$=
\{ h\in\pd{n+1}\ |\ h^{(r)}\subseteq\ f_r\}$.
Indeed, if \hbox{dim\hskip2pt}{\bf Z}$\geq r+1$, then for all ~$g_r\in\gr{r,n+1}$~ there exists
$h\in{\bf Z}\hbox{ s.t. } h\supseteq g_r$, so ~$g_r\in h^{(r)}\subseteq f_r$,
contradicting that ~$f_r$~ is a hypersurface of ~$\gr{r,n+1}$. \phantom{|}\hfill\mbox{$\Box$}(for
\rf{lema})
Continuing the proof of \rf{claim} we consider the stratification of
$f_n$~ by the condition of improper intersection of ~$f_r$~ with ~$h^
{(r)}$, namely,
$$
\matrix {f_{n,0}&=&\{h\in f_n\ |\ h^{(0)}\subseteq f_0\},&\cr
f_{n,1}&=&\{h\in f_n\ |\ h^{(1)}\subseteq f_1\}&\hskip-.25cm\setminus&
\hskip-.75cm f_{n,0},
\cr
&\vdots&&&\cr
f_{n,n}&=&\{h\in f_n\ |\ h^ {(n)}\subseteq f_n\}&\hskip-.25cm
\setminus&\hskip-.15cm \bigcup\hskip-.45cm\raise-.25cm\hbox{$_{j<n}$}f_{n,j}.}
$$
We will be done if we show
$$
\hbox{dim\hskip2pt}({\bf\Gamma}_{{\underline{f}}})_h\ \leq 1+\cdots+n-
r\quad\forall\ h \in f_{n,r}.
$$
We have already seen that ~$\hbox{dim\hskip2pt}({\bf\Gamma}_{{\underline{f}}})_h=1+\cdots +n-1$~ for ~$h$~ in
$f_{n,n}$. Also, for ~$r=0$, the desired estimate holds because we have
$({\bf\Gamma}_{{\underline{f}}})_h\subseteq (\ff{n+1})_h\simeq\ff n$~ and ~$\hbox{dim\hskip2pt} \ff
n=1+\cdots+n.$~ Let ~$r>0$~ and pick a hyperplane ~$h \in f_{n,r}$. Then
the intersections,
$$
f'_i = h^ {(i)}\cap f_i,
$$
are proper for ~$i=0,\dots,r-1$, whereas for the subsequent index, we
have
$$
h^ {(r)}\cap f_r = h^ {(r)}\simeq \gr{r,n}.\vspace{-10pt}
$$
Thus, we may write,
$$
({\bf\Gamma}_{{\underline{f}}})_h\hookrightarrow
\big(f'_0\times\cdots\times f'_{r-1} \times \gr{r,n
}\times\cdots\times\gr{n-1,n}\big)\bigcap\ff n.
$$
By the induction hypothesis the intersection above is of dimension
dim\,$\ff n-r$~ in view of the following easy
{\bf Remark.\ }\em The validity of \rf{claim} for a given
$n$~ implies properness of the ``partial'' intersection
\vspace{-10pt}
$$
(f_0\times\cdots\times \gr{r,n+1}\times \cdots\times f_n)\cap
{\ff{n+1}},
$$
where one (or more) of the hypersurfaces $f_r\subset\gr{r,n+1}$~ is
replaced by the corresponding full grassmannian\rm.
\phantom{|}\hfill\mbox{$\Box$}(for \rf{claim}) (Feb.2'96)
\vskip15pt
\centerline{\bf REFERENCES} \vskip2pt\parskip2pt\baselineskip13pt
\begin{itemize}}\def\ei{\end{itemize}
\item[]{[H] }R. Hartshorne, {\it Algebraic Geometry},
GTM \# 52 Springer--Verlag (1977).
\item[]{[K] }\ S.L. Kleiman with A. Thorup, {\it ``Intersection theory and
enumerative geometry: A decade in review'', in} Algebraic geometry: Bowdoin
1985, S. Bloch, ed., AMS Proc. of Symp. Pure Math, {\bf 46-2}, p.321-370
(1987).
\item[]{[KT]} S.~L.~Kleiman \& A.~Thorup, {\it Complete bilinear forms},
in Algebraic Geometry, Sundance, 1986, eds. A. Holme and R. Speiser,
pp. 253-320, Lect. Notes in Math. 1311, Springer--Verlag, Berlin,
(1988).
\item[]{[KTB]} $\underline{\hskip2.6cm}$, {\it A Geometric Theory of
the Buchsbaum--Rim Multiplicity},
J. Algebra \bf167\rm, 168-231 (1994).
\item[]{[L] } D. Laksov
{\it Completed quadrics and linear maps}, in Algebraic geometry:
Bowdoin 1985, S. Bloch, ed., AMS Proc. of Symp. Pure Math., {\bf
46-2}, p.371-387 (1987).
\item[]{[NT]} M. S. Narasimhan \& G. Trautmann, \it Compactification of
$M_{\p{\!3}}(0,2)$~ and Poncelet pairs of conics\rm, Pacific J. Math.
\bf145-2\rm, p.255-365
(1990).
\item[]{[P]} R. Pandharipande, {\it ``
Notes On Kontsevich's Compactification Of The Moduli Space Of Maps},
Course notes, Univ. Chicago (1995).
\item[]{[BS]} B. Sturmfels, {\it ``Gr\"obner basis and convex
polytopes''},
Lectures notes at the Holiday Symp. at N. Mexico State Univ.,
Las Cruces (1994).
\item[]{[V]} I. Vainsencher, \it``Conics multitangent to a plane
curve''\rm, in preparation.
\ei
\vskip10pt
\parskip0pt\baselineskip10pt\obeylines
\noindent Departamento de Matem\'atica
\noindent Universidade Federal de Pernambuco
\noindent Cidade Universit\'aria 50670--901 Recife--Pe--Brasil
\noindent email: [email protected]
\enddocument
|
1998-02-11T16:13:54 | 9602 | alg-geom/9602011 | en | https://arxiv.org/abs/alg-geom/9602011 | [
"alg-geom",
"math.AC",
"math.AG"
] | alg-geom/9602011 | Yekutieli Amnon | Amnon Yekutieli | Residues and Differential Operators on Schemes | 35 pages, AMSLaTeX, final version (minor changes), to appear in Duke
Math. J | null | null | null | null | Beilinson Completion Algebras (BCAs) are generalizations of complete local
rings, and have a rich algebraic-analytic structure. These algebras were
introduced in my paper "Traces and Differential Operators over Beilinson
Completion Algebras", Compositio Math. 99 (1995). In the present paper BCAs are
used to give an explicit construction of the Grothendieck residue complex on an
algebraic scheme. This construction reveals new properties of the residue
complex, and in particular its interaction with differential operators.
Applications include: (i) results on the algebraic structure of rings of
differential operators; (ii) an analysis of the niveau spectral sequence of De
Rham homology; (iii) a proof of the contravariance of De Rham homology w.r.t.
etale morphisms; (iv) an algebraic description of the intersection cohomology
D-module of a curve.
| [
{
"version": "v1",
"created": "Wed, 14 Feb 1996 16:47:07 GMT"
},
{
"version": "v2",
"created": "Wed, 11 Feb 1998 15:13:53 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Yekutieli",
"Amnon",
""
]
] | alg-geom | \section{Introduction}
Suppose $X$ is a finite type scheme over a field $k$, with structural
morphism $\pi$. Consider the twisted inverse image functor
$\pi^{!} : \msf{D}^{+}_{\mrm{c}}(k) \rightarrow
\msf{D}^{+}_{\mrm{c}}(X)$
of Grothendieck Duality Theory (see \cite{RD}).
The {\em residue complex} $\mcal{K}^{{\textstyle \cdot}}_{X}$
is defined to be the Cousin complex of $\pi^{!} k$.
It is a bounded complex of quasi-coherent $\mcal{O}_{X}$-modules,
possessing remarkable functorial properties.
In this paper we provide an explicit construction of
$\mcal{K}^{{\textstyle \cdot}}_{X}$. This construction reveals some new properties
of $\mcal{K}^{{\textstyle \cdot}}_{X}$, and also has applications in other
areas of algebraic geometry.
Grothendieck Duality, as developed by Hartshorne in \cite{RD},
is an abstract theory, stated in the language of derived categories.
Even though this abstraction is suitable for many important
applications, often one wants more explicit information.
Thus a significant
amount of work was directed at finding a presentation of
duality in terms of differential forms and residues.
Mostly the focus was on the dualizing sheaf $\omega_{X}$,
in various circumstances. The structure of $\omega_{X}$ as a
coherent $\mcal{O}_{X}$-module and its variance properties are
thoroughly understood by now, thanks to an extended effort
including \cite{Kl}, \cite{KW}, \cite{Li}, \cite{HK1}, \cite{HK2},
\cite{LS} and \cite{HS}.
Regarding an explicit presentation of the full duality theory of
dualizing complexes, there have been some advances in recent years,
notably in the papers \cite{Ye1}, \cite{SY}, \cite{Hu}, \cite{Hg}
and \cite{Sa}.
In this paper we give a totally new construction of the residue
complex $\mcal{K}^{{\textstyle \cdot}}_{X}$, when $k$ is a perfect field of
any characteristic and $X$ is any finite type $k$-scheme.
The main idea is the use of {\em Beilinson Completion Algebras}
(BCAs), which were introduced in \cite{Ye2}.
These algebras are generalizations of complete local rings, and they
carry a mixed algebraic-analytic structure.
A review of BCAs and their properties is included in Section 1,
for the reader's convenience.
Given a point $x \in X$, the complete local ring
$\widehat{\mcal{O}}_{X, x} = \mcal{O}_{X, (x)}$
is a BCA, so according to \cite{Ye2} it has a {\em dual module}
$\mcal{K}(\mcal{O}_{X, (x)})$. This module is a canonical model
for the injective hull of the residue field $k(x)$.
If $(x,y)$ is a saturated chain of points (i.e.\ $y$ is an immediate
specialization of $x$) then there is a BCA
$\mcal{O}_{X, (x,y)}$ and homomorphisms
$\mrm{q} : \mcal{K}(\mcal{O}_{X, (x)}) \rightarrow
\mcal{K}(\mcal{O}_{X, (x,y)})$
and
$\operatorname{Tr} : \mcal{K}(\mcal{O}_{X, (x,y)}) \rightarrow
\mcal{K}(\mcal{O}_{X, (y)})$.
The dual modules $\mcal{K}(-)$ and the homomorphisms
$\mrm{q}$ and $\operatorname{Tr}$ have explicit formulas in terms of differential
forms and coefficient fields. Set
$\delta_{(x,y)} := \operatorname{Tr} \mrm{q} : \mcal{K}(\mcal{O}_{X, (x)}) \rightarrow
\mcal{K}(\mcal{O}_{X, (y)})$.
Define a graded quasi-coherent sheaf $\mcal{K}_{X}^{{\textstyle \cdot}}$ by
\[ \mcal{K}_{X}^{q} := \bigoplus_{\operatorname{dim} \overline{\{x\}} = -q}
\mcal{K}(\mcal{O}_{X, (x)}) \]
and a degree $1$ homomorphism
\[ \delta := (-1)^{q+1} \sum_{(x,y)} \delta_{(x,y)} . \]
It turns out that
$(\mcal{K}_{X}^{{\textstyle \cdot}}, \delta)$ is a residual complex on $X$, and
it is canonically isomorphic to $\pi^{!} k$ in the derived category
$\msf{D}(X)$. Hence it is the residue complex of $X$, as defined in
the first paragraph. The functorial
properties of $\mcal{K}_{X}^{{\textstyle \cdot}}$ w.r.t.\ proper and \'{e}tale
morphisms are obtained directly from
corresponding properties of BCAs, and therefore are reduced to
explicit formulas. All this is worked out in Sections 2 and 3.
An $\mcal{O}_{X}$-module $\mcal{M}$ has a dual complex
$\operatorname{Dual} \mcal{M} := \mcal{H}om^{{\textstyle \cdot}}_{\mcal{O}_{X}}(\mcal{M},
\mcal{K}_{X}^{{\textstyle \cdot}})$.
Suppose
$\mrm{d} : \mcal{M} \rightarrow \mcal{N}$
is a differential operator (DO). In Theorem \ref{thm3.1} we prove
there is a dual operator
$\operatorname{Dual}(\mrm{d}) : \operatorname{Dual} \mcal{N} \rightarrow \operatorname{Dual} \mcal{M}$,
which commutes with $\delta$. The existence of $\operatorname{Dual}(\mrm{d})$
does not follow from
formal considerations of duality theory; it is a consequence of our
particular construction using BCAs
(but cf.\ Remarks \ref{rem3.2} and \ref{rem3.3}).
The construction also provides explicit formulas for
$\operatorname{Dual}(\mrm{d})$ in terms of differential operators and
residues, which are used in the applications in Sections 6 and 7.
Suppose $A$ is a finite type $k$-algebra, and let
$\mcal{D}(A)$ be the ring of differential operators of $A$.
As an immediate application of Theorem \ref{thm3.1} we obtain a
description of the opposite ring $\mcal{D}(A)^{\circ}$,
as the ring of DOs on
$\mcal{K}_{A}^{{\textstyle \cdot}}$ which commute with $\delta$
(Theorem \ref{thm3.2}). In the case of a
Gorenstein algebra it follows that the opposite ring
$\mcal{D}(A)^{\circ}$ is naturally isomorphic to
$\omega_{A} \otimes_{A} \mcal{D}(A) \otimes_{A} \omega_{A}^{-1}$
(Corollary \ref{cor3.6}).
Applying Theorem \ref{thm3.1} to the De Rham complex
$\Omega^{{\textstyle \cdot}}_{X/k}$ we obtain the {\em De Rham-residue complex}
$\mcal{F}_{X}^{{\textstyle \cdot}} = \operatorname{Dual} \Omega^{{\textstyle \cdot}}_{X/k}$.
Up to signs this coincides with El-Zein's complex
$\mcal{K}_{X}^{{\textstyle \cdot},*}$ of \cite{EZ} (Corollary \ref{cor4.3}).
The fundamental class $\mrm{C}_{Z} \in \mcal{F}_{X}^{{\textstyle \cdot}}$,
for a closed subscheme $Z \subset X$, is easily described in this context
(Definition \ref{dfn4.4}).
The construction above works also for a formal scheme
$\mfrak{X}$ which is of formally finite type over $k$, in the sense
of \cite{Ye3}. An example of such a formal scheme is the completion
$\mfrak{X} = Y_{/X}$, where $X$
is a locally closed subset of the finite type $k$-scheme $Y$.
Therefore we get a complex
$\mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}} =
\operatorname{Dual} \widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{X}/k}$.
When $X \subset \mfrak{X}$ is a smooth formal embedding
(see Definition \ref{dfn5.3}) we prove that the cohomology modules
$\mrm{H}^{q}(X, \mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}})$
are independent of $\mfrak{X}$. This is done by analyzing the
$E_{1}$ term of the {\em niveau spectral sequence} converging to
$\mrm{H}^{{\textstyle \cdot}}(X, \mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}})$
(Theorem \ref{thm5.3}). Here we assume $\operatorname{char} k = 0$.
The upshot is that
$\mrm{H}^{q}(X, \mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}}) =
\mrm{H}_{-q}^{\mrm{DR}}(X)$,
the De Rham homology.
There is an advantage in using smooth formal embeddings.
If $U \rightarrow X$ is any \'{e}tale morphism, then there
is an \'{e}tale morphism $\mfrak{U} \rightarrow \mfrak{X}$ s.t.\
$U = \mfrak{U} \times_{\mfrak{X}} X$, so
$U \subset \mfrak{U}$ is a smooth formal embedding. From this we
conclude that
$\mrm{H}_{{\textstyle \cdot}}^{\mrm{DR}}(-)$
is a contravariant functor on $X_{\mrm{et}}$, the small \'{e}tale
site. Previously it was only known that
$\mrm{H}_{{\textstyle \cdot}}^{\mrm{DR}}(-)$ is contravariant for open
immersions (cf.\ \cite{BlO} Example 2.2).
Suppose $X$ is smooth, and let
$\mcal{H}^{p}_{\mrm{DR}}$
be the sheafification of the presheaf
$U \mapsto \mrm{H}^{p}_{\mrm{DR}}(U)$ on $X_{\mrm{Zar}}$.
Bloch-Ogus \cite{BlO} give a flasque resolution of
$\mcal{H}^{p}_{\mrm{DR}}$, the {\em arithmetic resolution}.
It involves the sheaves
$i_{x*} \mrm{H}^{q} \Omega^{{\textstyle \cdot}}_{k(x)/k}$
where $i_{x} : \{x\} \rightarrow X$ is the inclusion map. Our analysis of the
niveau spectral sequence shows that the coboundary operator of this
resolution is a sum of Parshin residues (Corollary \ref{cor5.2}).
Our final application of the new construction of the residue complex
is to describe the {\em intersection cohomology $\mcal{D}$-module}
$\mcal{L}(X, Y)$, when $X$ is an integral curve embedded in a smooth
$n$-dimensional variety $Y$ (see \cite{BrKa}). Again we assume $k$
has characteristic $0$.
In fact we are able to describe all coherent $\mcal{D}_{Y}$-submodules
of $\mcal{H}^{n-1}_{X} \mcal{O}_{Y}$
in terms of the singularities of $X$ (Corollary \ref{cor6.10}).
This description is an algebraic version of Vilonen's work in \cite{Vi},
replacing complex analysis with BCAs and algebraic residues.
It is our hope that a similar description will be found in the general
case, namely $\operatorname{dim} X > 1$. Furthermore, we hope to give in the
future an explicit description of the Cousin complex of
$\operatorname{DR} \mcal{L}(X, Y) = \Omega^{{\textstyle \cdot}}_{Y / k} \otimes
\mcal{L}(X, Y)$.
Note that for $X=Y$ one has
$\mcal{L}(X, Y) = \mcal{O}_{X}$, so this Cousin complex is
nothing but $\mcal{F}^{{\textstyle \cdot}}_{X}$.
\medskip \noindent
{\bf Acknowledgments.}\
I wish to thank J.\ Lipman and S.\ Kleiman for their continued interest
in this work. Thanks also to
P.\ Sastry, R.\ H\"{u}bl, V.\ Hinich, V.\ Berkovich, H.\ Esnault,
K.\ Smith and K.\ Vilonen for helpful conversations, and
thanks to the referee for valuable advice on Sections 4 and 7.
\section{Review of Beilinson Completion Algebras}
Let us begin by reviewing some facts about Topological Local Fields
(TLFs) and Beilinson Completion Algebras (BCAs) from the papers
\cite{Ye1} and \cite{Ye2}.
A semi-topological (ST) ring is a ring $A$, with a linear topology on
its underlying additive group, such that for every $a \in A$ the
multiplication (on either side)
$a : A \rightarrow A$ is continuous.
Let $K$ be a field. We say $K$ is an $n$-dimensional local field
if there is a sequence of complete discrete valuation rings
$\mcal{O}_{1}, \ldots, \mcal{O}_{n}$,
where the fraction field of $\mcal{O}_{1}$ is $K$,
and the residue field of $\mcal{O}_{i}$ is
the fraction field of $\mcal{O}_{i + 1}$.
Fix a perfect field $k$.
A {\em topological local field} of dimension $n$ over $k$ is a
$k$-algebra
$K$ with structures of semi-topological ring and $n$-dimensional local
field, satisfying the following parameterization condition:
there exists an isomorphism of $k$-algebras
$K \cong F((s_{1}, \ldots, s_{n}))$
for some field $F$, finitely generated over $k$,
which respects the two structures.
Here
$F((s_{1}, \ldots, s_{n})) = F((s_{n})) \cdots ((s_{1}))$
is the field of iterated Laurent series, with its inherent topology
and valuation rings ($F$ is discrete).
One should remark that for $n = 1$ we are in the classical situation,
whereas for $n \geq 2$, $F((s_{1}, \ldots, s_{n}))$
is not a topological ring.
TLFs make up a category $\msf{TLF}(k)$,
where a morphism $K \rightarrow L$ is a continuous
$k$-algebra homomorphism which preserves the valuations, and the induced
homomorphism of the last residue fields is finite.
Write $\Omega^{\mrm{sep}, {\textstyle \cdot}}_{K / k}$ for the separated
algebra of differentials; with the parameterization above
$\Omega^{\mrm{sep}, {\textstyle \cdot}}_{K / k} \cong
K \otimes_{F} \Omega^{{\textstyle \cdot}}_{F[\, \underline{s}\, ] / k}$.
Then there is a functorial residue map
$\operatorname{Res}_{L / K} : \Omega^{\mrm{sep}, {\textstyle \cdot}}_{L / k} \rightarrow
\Omega^{\mrm{sep}, {\textstyle \cdot}}_{K / k}$
which is $\Omega^{\mrm{sep}, {\textstyle \cdot}}_{K / k}$-linear and
lowers degree by $\operatorname{dim} L / K$. For instance if
$L = K((t))$ then
\begin{equation} \label{eqn10.1}
\operatorname{Res}_{L / K} \left( \sum_{i} t^{i} \mrm{d} t \wedge \alpha_{i}
\right) =
\alpha_{-1} \in \Omega^{\mrm{sep}, {\textstyle \cdot}}_{K / k} .
\end{equation}
TLFs and residues were initially developed by Parshin and Lomadze,
and the theory was enhanced in \cite{Ye1}.
The notion of a {\em Beilinson completion algebra}
was introduced in \cite{Ye2}. A BCA is a semi-local,
semi-topological $k$-algebra, each of whose residue fields
$A / \mfrak{m}$ is a topological local field. Again there is a
parameterization condition: when $A$ is local,
there should exist a surjection
\[ F((\underline{s})) [[\, \underline{t}\, ]] =
F((\underline{s})) [[ t_{1}, \ldots, t_{m} ]] \twoheadrightarrow A \]
which is strict topologically (i.e.\ $A$ has the quotient topology)
and respects the valuations.
Here $F((\underline{s}))$ is as above and
$F((\underline{s})) [[\, \underline{t}\, ]]$ is the ring of formal power series over
$F((\underline{s}))$.
The notion of BCA is an abstraction of the algebra gotten by
Beilinson's completion, cf.\ Lemma \ref{lem10.1}.
There are two distinguished kinds of homomorphisms between BCAs.
The first kind is a {\em morphism of BCAs} $f : A \rightarrow B$
(see \cite{Ye2} Definition 2.5),
and the category they constitute is denoted $\msf{BCA}(k)$.
A morphism is continuous, respects the valuations on the residue fields,
but in general is not a local homomorphism.
For instance, the homomorphisms
$k \rightarrow k[[ s, t ]] \rightarrow k((s))[[t]] \rightarrow k((s))((t))$ are all morphisms.
$\msf{TLF}(k)$ is a full subcategory of $\msf{BCA}(k)$, consisting
of those BCAs which are fields.
The second kind of homomorphism is an {\em intensification homomorphism}
$u : A \rightarrow \widehat{A}$ (see \cite{Ye2} Definition 3.6).
An intensification is flat, topologically \'{e}tale
(relative to $k$) and unramified (in the appropriate sense).
It can be viewed as a sort of localization or
completion. Here examples are
$k(s)[[t]] \rightarrow k((s))[[t]]$ and
$k(s,t) \rightarrow k(s)((t)) \rightarrow k((s))((t))$.
Suppose $f : A \rightarrow B$ is a morphism of BCAs and
$u : A \rightarrow \widehat{A}$ is an intensification. The Intensification Base
Change Theorem (\cite{Ye2} Theorem 3.8) says there is a BCA
$\widehat{B} = B \otimes_{A}^{(\wedge)} \widehat{A}$, a morphism
$\widehat{f} : \widehat{A} \rightarrow \widehat{B}$ and an intensification
$v : B \rightarrow \widehat{B}$, with
$v f = \widehat{f} u$. These are determined up to isomorphism and
satisfy certain universal properties.
For instance,
$k((s))[[t]] = k(s)[[t]] \otimes_{k(s)}^{(\wedge)} k((s))$.
According to \cite{Ye2} Theorem 6.14, every $A \in \msf{BCA}(k)$
has a dual module $\mcal{K}(A)$.
The module $\mcal{K}(A)$ is a ST $A$-module. Algebraically it is
an injective hull of $A / \mfrak{r}$, where $\mfrak{r}$ is
the Jacobson radical.
$\mcal{K}(A)$ is also a right $\mcal{D}(A)$-module, where
$\mcal{D}(A)$ denotes the ring of continuous differential operators
of $A$ (relative to $k$).
For a ST $A$-module $M$ let
$\operatorname{Dual}_{A} M := \operatorname{Hom}_{A}^{\mrm{cont}}(M, \mcal{K}(A))$.
The dual modules have variance properties w.r.t.\ morphisms and
intensifications.
Given a morphism of BCAs $f : A \rightarrow B$, according to \cite{Ye2}
Theorem 7.4 there is an $A$-linear map
$\operatorname{Tr}_{f} : \mcal{K}(B) \rightarrow \mcal{K}(A)$.
This induces an isomorphism
$\mcal{K}(B) \cong
\operatorname{Hom}_{A}^{\mrm{cont}}(B, \mcal{K}(A))$.
Given an intensification homomorphism $u : A \rightarrow \widehat{A}$,
according to \cite{Ye2} Proposition 7.2 there is an $A$-linear map
$\mrm{q}_{u} : \mcal{K}(A) \rightarrow \mcal{K}(\widehat{A})$.
It induces an isomorphism
$\mcal{K}(\widehat{A}) \cong \widehat{A} \otimes_{A} \mcal{K}(A)$.
Furthermore $\operatorname{Tr}$ and $\mrm{q}$ commute across intensification
base change:
$\operatorname{Tr}_{\widehat{B} / \widehat{A}} \mrm{q}_{B / \widehat{B}} =
\mrm{q}_{\widehat{A} / A} \operatorname{Tr}_{B / A}$.
In case of a TLF $K$, one has
$\mcal{K}(K) = \omega(K) = \Omega^{p, \mrm{sep}}_{K/k}$,
where $p = \operatorname{rank} \Omega^{1, \mrm{sep}}_{K/k}$.
For a morphism of TLFs $f :K \rightarrow L$ one has
$\operatorname{Tr}_{f} = \operatorname{Res}_{f}$, whereas for an intensification
$u : K \rightarrow \widehat{K}$ the homomorphism
$\mrm{q}_{u} : \Omega^{p, \mrm{sep}}_{K/k} \rightarrow
\Omega^{p, \mrm{sep}}_{\widehat{K}/k}$
is the canonical inclusion for a topologically \'{e}tale extension
of fields.
\begin{exa} \label{exa10.1}
Take
$L := k(s,t)$, $\widehat{L} := k(s)((t))$,
$A := k(s)[[t]]$,
$\widehat{A} := k((s))[[t]]$,
$K := k(s)$
and
$\widehat{K} := k((s))$.
The inclusions
$L \rightarrow \widehat{L}$, $K \rightarrow \widehat{K}$ and
$A \rightarrow \widehat{A}$
are intensifications, whereas
$K \rightarrow A \rightarrow \widehat{L}$ and $\widehat{K} \rightarrow \widehat{A}$
are morphisms. Using the isomorphism
$\mcal{K}(A) \cong
\operatorname{Hom}_{K}^{\mrm{cont}}(A, \Omega^{1, \mrm{sep}}_{K / k})$
induced by $\operatorname{Tr}_{A / K}$,
we see that for $\alpha \in \Omega^{2, \mrm{sep}}_{\widehat{L} / k}$
the element
$\operatorname{Tr}_{\widehat{L} / A}(\alpha) \in \mcal{K}(A)$
is represented by the functional
$a \mapsto \operatorname{Res}_{\widehat{L} / K}(a \alpha)$,
$a \in A$.
Also for
$\phi \in \mcal{K}(A)$ the element
$\widehat{\phi} = \mrm{q}_{\widehat{A} / A}(\phi) \in
\mcal{K}(\widehat{A})$
is represented by the unique $\widehat{K}$-linear functional
$\widehat{\phi} : \widehat{A} \rightarrow
\Omega^{1, \mrm{sep}}_{\widehat{K} / k}$
extending $\phi$.
\end{exa}
\begin{rem}
The proof of existence of dual modules with their variance properties
in \cite{Ye2} is straightforward,
using Taylor series expansions, differential operators
and the residue pairing.
\end{rem}
Let $A$ be a noetherian ring and $\mfrak{p}$ a prime ideal.
Consider the exact functor on $A$-modules
$M \mapsto M_{(\mfrak{p})} := \widehat{A}_{\mfrak{p}} \otimes_{A} M$.
For $M$ finitely generated we have
$M_{(\mfrak{p})} \cong
\lim_{\leftarrow i} M_{\mfrak{p}} / \mfrak{p}^{i} M_{\mfrak{p}}$,
and if $M = \lim_{\alpha \rightarrow} M_{\alpha}$, then
$M_{(\mfrak{p})} \cong
\lim_{\alpha \rightarrow} (M _{\alpha})_{(\mfrak{p})}$.
This was generalized by Beilinson (cf.\ \cite{Be}) as follows.
\begin{dfn} \label{dfn10.1}
Let $M$ be an $A$-module and let
$\xi = (\mfrak{p}_{0}, \ldots, \mfrak{p}_{n})$ be a chain of prime
ideals, namely $\mfrak{p}_{i} \subset \mfrak{p}_{i + 1}$.
Define the {\em Beilinson completion} $M_{\xi}$
by recursion on $n$, $n \geq -1$.
\begin{enumerate}
\item If $n = -1$ (i.e.\ $\xi = \emptyset$),
let $M_{\xi} := M$ with the discrete topology.
\item If $n \geq 0$ and $M$ is finitely generated, let
\[ M_{(\mfrak{p}_{0}, \ldots, \mfrak{p}_{n})} :=
\lim_{\leftarrow i}\, (M_{\mfrak{p}_{0}} / \mfrak{p}_{0}^{i}
M_{\mfrak{p}_{0}})_{(\mfrak{p}_{1}, \ldots, \mfrak{p}_{n})} . \]
\item For arbitrary $M$, let $\{ M_{\alpha} \}$ be the set of
finitely generated submodules of $M$, and let
\[ M_{(\mfrak{p}_{0}, \ldots, \mfrak{p}_{n})} :=
\lim_{\alpha \rightarrow}\, (M_{\alpha})_{(\mfrak{p}_{0}, \ldots, \mfrak{p}_{n})}
. \]
\end{enumerate}
\end{dfn}
A chain $\xi = (\mfrak{p}_{0}, \ldots, \mfrak{p}_{n})$
is {\em saturated} if $\mfrak{p}_{i + 1}$ has height $1$ in
$A / \mfrak{p}_{i}$.
\begin{lem} \label{lem10.1}
If $\xi= (\mfrak{p}, \ldots )$ is a saturated chain then
$A_{\xi}$ is a Beilinson completion algebra.
\end{lem}
\begin{proof}
By \cite{Ye1} Corollary 3.3.5,
$A_{\xi}$ is a complete semi-local noetherian ring with
Jacobson radical $\mfrak{p}_{\xi}$, and
$A_{\xi} / \mfrak{p}_{\xi} = K_{\xi}$,
where $K := A_{\mfrak{p}} / \mfrak{p}_{\mfrak{p}}$.
Choose a coefficient field
$\sigma : K \rightarrow \widehat{A}_{\mfrak{p}} = A_{(\mfrak{p})}$.
By \cite{Ye1} Proposition 3.3.6, $K_{\xi}$ is a finite
product of TLFs, and by the proof of \cite{Ye1} Theorem 3.3.8,
$\sigma$ extends to a lifting
$\sigma_{\xi} : K_{\xi} \rightarrow A_{\xi}$. Sending
$t_{1}, \ldots, t_{m}$ to generators of the ideal
$\mfrak{p}$, we get a topologically strict surjection
$K_{\xi} [[ t_{1}, \ldots, t_{m} ]] \twoheadrightarrow A_{\xi}$.
\end{proof}
\section{Construction of the Residue Complex $\mcal{K}^{{\textstyle \cdot}}_{X}$}
Let $k$ be a perfect field, and let $X$ be a scheme of finite type
over $k$. By a {\em chain of points} in $X$ we mean a sequence
$\xi = (x_{0}, \ldots, x_{n})$ of points with
$x_{i + 1} \in \overline{\{x_{i}\}}$.
\begin{dfn}
For any quasi-coherent $\mcal{O}_{X}$-module $\mcal{M}$,
define the Beilinson completion $\mcal{M}_{\xi}$
by taking an affine open neighborhood $U = \operatorname{Spec} A \subset X$
of $x_{n}$, and setting
$\mcal{M}_{\xi} := \Gamma(U, \mcal{M})_{\xi}$
as in Definition \ref{dfn10.1}.
\end{dfn}
These completions first appeared as the local factors of Beilinson's
adeles in \cite{Be}, and were studied in detail in \cite{Ye1}.
According to Lemma \ref{lem10.1}, if
$\xi = (x_{0}, \ldots, x_{n})$ is saturated, i.e.\
$\overline{\{x_{i + 1}\}} \subset \overline{\{x_{i}\}}$
has codimension $1$, then
$\mcal{O}_{X, \xi}$ is a BCA.
We shall be interested in the covertex maps
\[ \begin{array}{rcl}
\partial^{-} : & \mcal{O}_{X,(x_{0})} & \rightarrow \mcal{O}_{X,\xi}
\\
\partial^{+} : & \mcal{O}_{X,(x_{n})} & \rightarrow \mcal{O}_{X,\xi}
\end{array} \]
which arise naturally from the completion process
(cf.\ \cite{Ye1} \S 3.1).
\begin{lem} \label{lem1.1}
$\partial^{+}$ is flat, topologically \'{e}tale relative to $k$, and a
morphism in $\msf{BCA}(k)$.
$\partial^{-}$ is an intensification homomorphism.
\end{lem}
\begin{proof}
By definition
$\partial^{-} = \partial^{1} \circ \cdots \circ \partial^{n}$ and
$\partial^{+} = \partial^{0} \circ \cdots \circ \partial^{0}$, where
$\partial^{i} : \mcal{O}_{X,\partial_{i} \xi} \rightarrow \mcal{O}_{X,\xi}$
is the $i$-th coface operator.
First let us prove that
$\partial^{0} : \mcal{O}_{X,\partial_{0} \xi} \rightarrow \mcal{O}_{X,\xi}$
is a morphism of BCAs. This follows from \cite{Ye1} Theorem 3.3.2 (d),
since we may assume that $X$ is integral with generic point $x_{0}$.
By part (b) of the same theorem,
$\partial^{n} : \mcal{O}_{X,\partial_{n} \xi} \rightarrow
\mcal{O}_{X, \xi}$
is finitely ramified and radically unramified (in the sense of \cite{Ye2}
Definition 3.1).
Now according to \cite{Ye1} Corollary 3.2.8,
$\partial^{i} : \mcal{O}_{X,\partial_{i} \xi} \rightarrow \mcal{O}_{X,\xi}$
is topologically \'{e}tale relative to $k$, for any $i$. We claim
it is also flat. For $i=0$,
$\mcal{O}_{X,\partial_{0} \xi} \rightarrow
(\mcal{O}_{X,\partial_{0} \xi})_{x_{0}}
= (\mcal{O}_{X, x_{0}})_{\partial_{0} \xi}$
is a localization, so it's flat. The map from
$(\mcal{O}_{X, x_{0}})_{\partial_{0} \xi}$ to its
$\mfrak{m}_{x_{0}}$-adic
completion $\mcal{O}_{X, \xi}$ is also flat (these rings are
noetherian). For $i > 0$, by induction on the length of chains,
$\mcal{O}_{X,\partial_{0} \partial_{i} \xi} \rightarrow
\mcal{O}_{X,\partial_{0} \xi}$
is flat, and hence so is
$(\mcal{O}_{X, x_{0}})_{\partial_{0} \partial_{i} \xi} \rightarrow
(\mcal{O}_{X, x_{0}})_{\partial_{0} \xi}$. Now use
\cite{CA} Chapter III \S 5.4 Proposition 4 to conclude that
\[ \mcal{O}_{X,\partial_{i} \xi} =
\lim_{\leftarrow j} (\mcal{O}_{X, x_{0}} / \mfrak{m}_{x_{0}}^{j})_{
\partial_{0} \partial_{i} \xi} \rightarrow
\lim_{\leftarrow j} (\mcal{O}_{X, x_{0}} / \mfrak{m}_{x_{0}}^{j})_{
\partial_{0} \xi} = \mcal{O}_{X, \xi} \]
is flat.
\end{proof}
\begin{exa} \label{1.1}
Take $X = \mbf{A}^{2} := \operatorname{Spec} k[s,t]$,
$x := (0)$,
$y := (t)$ and $z := (s,t)$. Then with
$L := k(x) = \mcal{O}_{X, (x)}$,
$\widehat{L} := k(x)_{(y)} = \mcal{O}_{X, (x, y)}$,
$A := \mcal{O}_{X, (y)}$,
$\widehat{A} := \mcal{O}_{X, (y, z)}$
$K := k(y)$ and
$\widehat{K} := k(y)_{(z)}$
we are exactly in the situation of Example \ref{exa10.1}.
\end{exa}
\begin{dfn}
Given a point $x$ in $X$, let
$\mcal{K}_{X}(x)$ be the skyscraper sheaf with support
$\overline{\{ x \}}$ and group of sections
$\mcal{K}(\mcal{O}_{X,(x)})$.
\end{dfn}
The sheaf $\mcal{K}_{X}(x)$ is a quasi-coherent
$\mcal{O}_{X}$-module,
and is an injective hull of $k(x)$ in the category $\msf{Mod}(X)$ of
$\mcal{O}_{X}$-modules.
\begin{dfn} \label{dfn1.1}
Given a saturated chain $\xi = (x, \ldots, y)$ in $X$, define an
$\mcal{O}_{X}$-linear homomorphism
$\delta_{\xi} : \mcal{K}_{X}(x) \rightarrow \mcal{K}_{X}(y)$, called the
coboundary map along $\xi$, by
\[ \delta_{\xi} : \mcal{K}(\mcal{O}_{X,(x)})
\xrightarrow{ \mrm{q}_{\partial^{-}} }
\mcal{K}(\mcal{O}_{X,\xi}) \xrightarrow{ \operatorname{Tr}_{\partial^{+}} }
\mcal{K}(\mcal{O}_{X,(y)}) . \]
\end{dfn}
Throughout sections 2 and 3 the following convention shall be used.
Let $f: X \rightarrow Y$ be a morphism of schemes, and let $x \in X$,
$y \in Y$ be points. We will
write $x \mid y$ to indicate that $x$ is a closed point in the fiber
$X_{y} := X \times_{Y} \operatorname{Spec} k(y) \cong f^{-1}(y)$. Similarly for
chains: we write
$(x_{0}, \ldots, x_{n}) \mid (y_{0}, \ldots, y_{n})$ if
$x_{i} \mid y_{i}$ for all $i$.
\begin{lem}
Suppose $x \mid y$. Then
$f^{*} : \mcal{O}_{Y,(y)} \rightarrow \mcal{O}_{X,(x)}$ is a
morphism of BCAs. If $f$ is quasi-finite then $f^{*}$ is finite, and
if $f$ is \'{e}tale then $f^{*}$ is an intensification.
\end{lem}
\begin{proof}
Immediate from the definitions.
\end{proof}
\begin{lem} \label{lem1.2}
Suppose $f: X \rightarrow Y$ is a quasi-finite morphism. Let
$\eta = (y_{0}, \ldots, y_{n})$ be a saturated chain in $Y$ and
let $x \in X$, $x \mid y_{n}$. Consider the \textup{(}finite\textup{)}
set of chains in $X$:
\[ \Xi := \{ \xi = (x_{0}, \ldots, x_{n})\ \mid \
\xi \mid \eta \textup{ and } x_{n} = x \}. \]
Then there is a canonical isomorphism of BCAs
\[ \prod_{\xi \in \Xi} \mcal{O}_{X,\xi} \cong
\mcal{O}_{Y,\eta} \otimes_{\mcal{O}_{Y,(y_{n})}}
\mcal{O}_{X,(x)} . \]
\end{lem}
\begin{proof}
Set
$\widehat{X} := \operatorname{Spec} \mcal{O}_{X,(x)}$ and
$\widehat{Y} := \operatorname{Spec} \mcal{O}_{Y,(y_{n})}$, so the induced
morphism
$\widehat{f} : \widehat{X} \rightarrow \widehat{Y}$ is finite.
Let us denote by $\xi, \widehat{\xi}, \widehat{\eta}$ variable saturated
chains in $X, \widehat{X}, \widehat{Y}$ respectively. For any
$\widehat{\eta} \mid \eta$ one has
\begin{equation} \label{eqn1.1}
\prod_{\widehat{\xi} \mid \widehat{\eta}} \mcal{O}_{\widehat{X},
\widehat{\xi}} \cong
\mcal{O}_{\widehat{Y}, \widehat{\eta}}
\otimes_{\mcal{O}_{Y, (y_{n})}} \mcal{O}_{X, (x)},
\end{equation}
by \cite{Ye1} Proposition 3.2.3; note that the completion is defined
on any noetherian scheme. Now from ibid.\ Corollary 3.3.13 one has
$\mcal{O}_{X, \xi} \cong \prod_{\widehat{\xi} \mid \xi}
\mcal{O}_{\widehat{X}, \widehat{\xi}}$, so taking the product over all
$\xi \in \Xi$ and $\widehat{\eta} \mid \eta$ the lemma is proved.
\end{proof}
\begin{dfn} \label{dfn1.2}
Given an \'{e}tale morphism $g : X \rightarrow Y$ and a point $x \in X$, let
$y := g(x)$, so
$g^{*} : \mcal{O}_{Y, (y)} \rightarrow \mcal{O}_{X, (x)}$
is an intensification. Define
\[ \mrm{q}_{g} : \mcal{K}_{Y}(y) \rightarrow g_{*} \mcal{K}_{X}(x) \]
to be the $\mcal{O}_{Y}$-linear homomorphism corresponding to
$\mrm{q}_{g^{*}} : \mcal{K}(\mcal{O}_{Y, (y)}) \rightarrow
\mcal{K}(\mcal{O}_{X, (x)})$
of \cite{Ye2} Proposition 7.2.
\end{dfn}
\begin{prop} \label{prop1.1}
Let $g : X \rightarrow Y$ be an \'{e}tale morphism.
\begin{enumerate}
\rmitem{a} Given a point $y \in Y$, the homomorphism
$1 \otimes \mrm{q}_{g} : g^{*} \mcal{K}_{Y}(y) \rightarrow$ \blnk{3mm} \linebreak
$\bigoplus_{x \mid y}
\mcal{K}_{X}(x)$ is an isomorphism of $\mcal{O}_{X}$-modules.
\rmitem{b} Let
$\eta = (y_{0}, \ldots, y_{n})$ be a saturated chain in $Y$. Then
\[ (1 \otimes \mrm{q}_{g}) \circ g^{*}(\delta_{\eta}) =
(\sum_{\xi \mid \eta} \delta_{\xi}) \circ (1 \otimes \mrm{q}_{g}) \]
as homomorphisms
$g^{*} \mcal{K}_{Y}(y_{0}) \rightarrow \bigoplus_{x_{n} \mid y_{n}}
\mcal{K}_{X}(x_{n})$.
\end{enumerate}
\end{prop}
\begin{proof}
(a)\ Because $\mcal{K}_{Y}(y)$ is an artinian
$\mcal{O}_{Y,y}$-module,
and $g$ is quasi-finite, we find that
\[ g^{*} \mcal{K}_{Y}(y) = \bigoplus_{x \mid y} \mcal{O}_{X, (x)}
\otimes_{\mcal{O}_{Y, (y)}} \mcal{K}_{Y}(y)\ . \]
Now use \cite{Ye2} Proposition 7.2 (i).
\medskip \noindent (b)\
From Lemma \ref{lem1.2} and from \cite{Ye2} Theorem 3.8 we see
that for every $\xi \mid \eta$,
$f^{*} : \mcal{O}_{Y,\eta} \rightarrow \mcal{O}_{X, \xi}$ is both a finite
morphism and an intensification.
By the definition of the coboundary maps, it suffices to verify that the
diagram
\bigskip \noindent
\[ \begin{CD}
\mcal{K}(\mcal{O}_{Y,(y_{0})}) @>{\mrm{q}}>>
\mcal{K}(\mcal{O}_{Y,\eta}) @>{\operatorname{Tr}}>>
\mcal{K}(\mcal{O}_{Y,(y_{n})}) \\
@V{\mrm{q}}VV @V{\mrm{q}}VV @V{\mrm{q}}VV \\
\bigoplus_{x_{0} \mid y_{0}} \mcal{K}(\mcal{O}_{X,(x_{0})})
@>{\mrm{q}}>> \bigoplus_{\xi \mid \eta} \mcal{K}(\mcal{O}_{X,\xi})
@>{\operatorname{Tr}}>>
\bigoplus_{x_{n} \mid y_{n}} \mcal{K}(\mcal{O}_{X,(x_{n})})
\end{CD} \]
\medskip \noindent
commutes. The left square commutes by \cite{Ye2} Proposition 7.2 (iv),
whereas the right square commutes by Lemma \ref{lem1.2} and
\cite{Ye2} Theorem 7.4 (ii).
\end{proof}
\begin{dfn} \label{dfn1.3}
Let $f: X \rightarrow Y$ be a morphism between finite type $k$-schemes,
let $x \in X$ be a point, and let $y := f(x)$. Define an
$\mcal{O}_{Y}$-linear homomorphism
\[ \operatorname{Tr}_{f} : f_{*} \mcal{K}_{X}(x) \rightarrow \mcal{K}_{Y}(y) \]
as follows:
\begin{enumerate}
\rmitem{i} If $x$ is closed in its fiber $X_{y}$, then
$f^{*} : \mcal{O}_{Y,(y)} \rightarrow \mcal{O}_{X,(x)}$ is a morphism in
$\msf{BCA}(k)$. Let $\operatorname{Tr}_{f}$
be the homomorphism corresponding to
$\operatorname{Tr}_{f^{*}} : \mcal{K}(\mcal{O}_{X,(x)}) \rightarrow
\mcal{K}(\mcal{O}_{Y,(y)})$
of \cite{Ye2} Theorem 7.4.
\rmitem{ii} If $x$ is not closed in its fiber, set $\operatorname{Tr}_{f} := 0$.
\end{enumerate}
\end{dfn}
\begin{prop} \label{prop1.2}
Let $f : X \rightarrow Y$ be a finite morphism.
\begin{enumerate}
\rmitem{a} For any $y \in Y$ the homomorphism of
$\mcal{O}_{Y}$-modules
\[ \bigoplus_{x \mid y} f_{*} \mcal{K}_{X}(x) \rightarrow
\mcal{H}om_{\mcal{O}_{Y}}(f_{*} \mcal{O}_{X},
\mcal{K}_{Y}(y)) \]
induced by $\operatorname{Tr}_{f}$ is an isomorphism.
\rmitem{b} Let $\eta = (y_{0}, \ldots, y_{n})$ be a saturated
chain in $Y$. Then
\[ \delta_{\eta} \operatorname{Tr}_{f} =
\operatorname{Tr}_{f} \sum_{\xi \mid \eta} f_{*}(\delta_{\xi}) :
\bigoplus_{x_{0} \mid y_{0}} f_{*} \mcal{K}_{X}(x_{0}) \rightarrow
\mcal{K}_{Y}(y_{n}) . \]
The sums are over saturated chains $\xi = (x_{0}, \ldots, x_{n})$
in $X$.
\end{enumerate}
\end{prop}
\begin{proof}
(a)\
By \cite{Ye1} Proposition 3.2.3 we have
$\prod_{x \mid y} \mcal{O}_{X, (x)} \cong
(f_{*} \mcal{O}_{X})_{(y)}$.
Now use \cite{Ye2} Theorem 7.4 (iv).
\medskip \noindent (b)\
Use the same diagram which appears in the proof of Proposition
\ref{prop1.1},
only reverse the vertical arrows and label them $\operatorname{Tr}_{f}$. Then the
commutativity follows from \cite{Ye2} Theorem 7.4 (i), (ii).
\end{proof}
In \cite{Ye1} \S 4.3 the notion of a system of residue data on a reduced
scheme was introduced.
\begin{prop} \label{prop1.3}
Suppose $X$ is a reduced scheme. Then
$(\{ \mcal{K}_{X}(x) \}, \{ \delta_{\xi} \},$ \linebreak
$\{ \Psi_{\sigma}^{-1} \})$,
where $x$ runs over the points of $X$, $\xi$ runs over the
saturated chains
in $X$, and $\sigma : k(x) \rightarrow \mcal{O}_{X,(x)}$ runs over all
possible coefficient fields, is a system of residue data on $X$.
\end{prop}
\begin{proof}
We must check condition (\dag) of \cite{Ye1} Definition 4.3.10.
So let $\xi = (x, \ldots, y)$ be a saturated chain, and let
$\sigma : k(x) \rightarrow \mcal{O}_{X,(x)}$ and
$\tau : k(y) \rightarrow \mcal{O}_{X,(y)}$ be compatible coefficient fields.
Denote also by $\tau$ the composed morphism
$\partial^{+} \tau : k(y) \rightarrow \mcal{O}_{X,\xi}$.
Then we get a coefficient field
$\sigma_{\xi} : k(\xi) = k(x)_{\xi} \rightarrow \mcal{O}_{X,\xi}$
extending $\sigma$, which is a $k(y)$-algebra map via $\tau$.
Consider the diagram:
\bigskip \noindent
\[ \begin{CD}
\mcal{K}(\mcal{O}_{X,(x)}) @>\mrm{q}>>
\mcal{K}(\mcal{O}_{X,\xi}) @>=>>
\mcal{K}(\mcal{O}_{X,\xi}) @>\operatorname{Tr}>>
\mcal{K}(\mcal{O}_{X,(y)}) \\
@V{\Psi_{\sigma}}VV @V{\Psi_{\sigma_{\xi}}}VV
@V{\Psi_{\tau}}VV @V{\Psi_{\tau}}VV \\
\operatorname{Dual}_{\sigma} \mcal{O}_{X,(x)} @>{\mrm{q}_{\sigma}}>>
\operatorname{Dual}_{\sigma_{\xi}} \mcal{O}_{X,\xi} @>{h}>>
\operatorname{Dual}_{\tau} \mcal{O}_{X,\xi}
@>{\operatorname{Tr}_{\tau}}>>
\operatorname{Dual}_{\tau} \mcal{O}_{X, (y)}
\end{CD} \]
\medskip \noindent
where for a $k(\xi)$-linear homomorphism
$\phi : \mcal{O}_{X,\xi} \rightarrow \omega(k(\xi))$,
$h(\phi) = $ \blnk{3mm} \linebreak
$\operatorname{Res}_{k(\xi) / k(y)} \phi$
(cf.\ \cite{Ye2} Theorem 6.14).
The left square commutes by \cite{Ye2} Proposition 7.2 (iii); the
middle square
commutes by ibid.\ Theorem 6.14 (i); and the right square commutes
by ibid.\ Theorem 7.4 (i), (iii). But going along the bottom
of the diagram we get
$\operatorname{Tr}_{\tau} h \mrm{q}_{\sigma} = \delta_{\xi, \sigma / \tau}$,
as defined in \cite{Ye1} Lemma 4.3.3.
\end{proof}
\begin{lem} \label{lem1.5}
Let $\xi = (x, \ldots, y)$ and $\eta = (y, \ldots, z)$ be saturated
chains in the scheme $X$, and let
$\xi \vee \partial_{0} \eta := (x, \ldots, y, \ldots, z)$ be the
concatenated chain. Then
there is a canonical isomorphism of BCAs
\[ \mcal{O}_{X,\xi \vee \partial_{0} \eta} \cong \mcal{O}_{X,\xi}
\otimes^{(\wedge)}_{\mcal{O}_{X,(y)}} \mcal{O}_{X,\eta} \]
\textup{(}intensification base change\textup{)}.
\end{lem}
\begin{proof}
Choose a coefficient field $\sigma: k(y) \rightarrow \mcal{O}_{X,(y)}$. This
induces a coefficient ring
$\sigma_{\eta}: k(\eta) \rightarrow \mcal{O}_{X,\eta}$, and using \cite{Ye2}
Theorem 3.8 and \cite{Ye1} Theorem 4.1.12 one gets
\[ \mcal{O}_{X,\xi \vee \partial_{0} \eta} \cong
\mcal{O}_{X,\xi} \otimes^{(\wedge)}_{k(y)} k(\eta) \cong
\mcal{O}_{X,\xi} \otimes^{(\wedge)}_{\mcal{O}_{X,(y)}}
\mcal{O}_{X,\eta} . \]
\end{proof}
\begin{lem} \label{lem1.3}
\begin{enumerate}
\item Let $\xi = (x, \ldots, y)$ and
$\eta = (y, \ldots, z)$ be saturated chains and let
$\xi \vee \partial_{0} \eta = (x, \ldots, y, \ldots, z)$. Then
$\delta_{\eta} \delta_{\xi} = \delta_{\xi \vee \partial_{0} \eta}$.
\item Given a point $x \in X$ and an element
$\alpha \in \mcal{K}_{X}(x)$,
for all but finitely
many saturated chains $\xi = (x, \ldots)$ in $X$ one has
$\delta_{\xi}(\alpha) = 0$.
\item \textup{(}Residue Theorem\textup{)} Let $x,z \in X$ be points
s.t.\ $z \in \overline{\{ x \}}$ and
$\operatorname{codim}(\overline{\{ z \}},$ \linebreak
$\overline{\{ x \}}) = 2$.
Then
$\sum_{y}\ \delta_{(x,y,z)} = 0$.
\end{enumerate}
\end{lem}
\begin{proof}
Using Lemma \ref{lem1.5} we see that part
1 is a consequence of the base change property of traces, cf.\
\cite{Ye2} Theorem 7.4 (ii). Assertions
2 and 3 are local, by Proposition \ref{prop1.1}, so we may assume
there is a closed immersion
$f : X \rightarrow \mbf{A}_{k}^{n}$. By Proposition
\ref{prop1.2}, we can replace $X$ with $\mbf{A}_{k}^{n}$, and
so assume
that $X$ is reduced. Now according to Proposition \ref{prop1.3}
and \cite{Ye1} Lemma 4.3.19, both 2 and 3 hold.
\end{proof}
\begin{dfn}
For any integer $q$ define a quasi-coherent sheaf
\[ \mcal{K}_{X}^{q} :=
\bigoplus_{\operatorname{dim} \overline{\{x\}} = -q} \mcal{K}_{X}(x) . \]
By Lemma \ref{lem1.3} there an $\mcal{O}_{X}$-linear homomorphism
\[ \delta := (-1)^{q+1} \sum_{(x,y)} \delta_{(x,y)} :
\mcal{K}_{X}^{q} \rightarrow \mcal{K}_{X}^{q+1} , \]
satisfying $\delta^{2} = 0$. The complex
$(\mcal{K}_{X}^{{\textstyle \cdot}}, \delta)$ is called the {\em Grothendieck
residue complex} of $X$.
\end{dfn}
In Corollary \ref{cor2.2} we will prove that $\mcal{K}_{X}^{{\textstyle \cdot}}$
is canonically isomorphic (in the derived category $\msf{D}(X)$)
to $\pi^{!} k$, where $\pi : X \rightarrow \operatorname{Spec} k$ is the structural
morphism.
\begin{rem}
A heuristic for the negative grading of $\mcal{K}_{X}^{{\textstyle \cdot}}$
and the sign $(-1)^{q+1}$ is that the residue complex is the
``$k$-linear dual'' of a hypothetical ``complex of localizations''
$\cdots \prod \mcal{O}_{X,y} \rightarrow \prod \mcal{O}_{X,x} \rightarrow \cdots$.
Actually, there is a naturally defined complex which is built up from
all localizations and completions: the Beilinson adeles
$\underline{\mbb{A}}^{{\textstyle \cdot}}_{\mrm{red}}(\mcal{O}_{X})$
(cf.\ \cite{Be} and \cite{HY1}).
$\underline{\mbb{A}}^{{\textstyle \cdot}}_{\mrm{red}}(\mcal{O}_{X})$
is a DGA, and $\mcal{K}_{X}^{{\textstyle \cdot}}$ is naturally a right DG-module
over it. See \cite{Ye4}, and cf.\ also Remark \ref{rem5.3}.
\end{rem}
\begin{dfn} \label{eqn1.4}
\begin{enumerate}
\item Let $f:X \rightarrow Y$ be a morphism of schemes. Define a homomorphism of
graded $\mcal{O}_{Y}$-modules
$\operatorname{Tr}_{f} : f_{*} \mcal{K}_{X}^{{\textstyle \cdot}} \rightarrow \mcal{K}_{Y}^{{\textstyle \cdot}}$
by summing the local trace maps of Definition \ref{dfn1.3}.
\item Let $g : U \rightarrow X$ be an \'{e}tale morphism. Define
$\mrm{q}_{g} : \mcal{K}_{X}^{{\textstyle \cdot}} \rightarrow g_{*} \mcal{K}_{U}^{{\textstyle \cdot}}$
by summing all local homomorphisms $\mrm{q}_{g}$
of Definition \ref{dfn1.2}.
\end{enumerate}
\end{dfn}
\begin{thm} \label{thm1.1}
Let $X$ be a $k$-scheme of finite type.
\begin{enumerate}
\rmitem{a} $(\mcal{K}_{X}^{{\textstyle \cdot}}, \delta)$ is a residual complex
on $X$ \textup{(}cf.\ \cite{RD} Chapter \textup{VI \S 1)}.
\rmitem{b} If $g : U \rightarrow X$ is an \'{e}tale morphism, then
$1 \otimes \mrm{q}_{g} : g^{*} \mcal{K}_{X}^{{\textstyle \cdot}} \rightarrow
\mcal{K}_{U}^{{\textstyle \cdot}}$
is an isomorphism of complexes.
\rmitem{c} If $f : X \rightarrow Y$ is a finite morphism, then
$\operatorname{Tr}_{f} : f_{*} \mcal{K}_{X}^{{\textstyle \cdot}} \rightarrow \mcal{K}_{Y}^{{\textstyle \cdot}}$
is a homomorphism of complexes, and the induced map
\[ f_{*} \mcal{K}_{X}^{{\textstyle \cdot}} \rightarrow
\mcal{H}om_{\mcal{O}_{Y}}(f_{*} \mcal{O}_{X},
\mcal{K}_{Y}^{{\textstyle \cdot}}) \]
is an isomorphism of complexes.
\rmitem{d} If $X$ is reduced, then $(\mcal{K}_{X}^{{\textstyle \cdot}}, \delta)$
is canonically isomorphic to the complex constructed in \cite{Ye1} \S
\textup{4.3}.
In particular, if $X$ is smooth irreducible of dimension $n$, there
is a quasi-isomorphism
$\mrm{C}_{X} : \Omega^{n}_{X/k}[n] \rightarrow \mcal{K}^{{\textstyle \cdot}}_{X}$.
\end{enumerate}
\end{thm}
\begin{proof}
Parts (b), (c), (d) are immediate consequences of Propositions
\ref{prop1.1}, \ref{prop1.2} and \ref{prop1.3} here, and \cite{Ye1}
Theorem 4.5.2. (Note that the sign of $\delta$ in \cite{Ye1} is
different.) As for part (a),
clearly $\mcal{K}_{X}^{{\textstyle \cdot}}$ is a direct sum of injective hulls of
all the residue fields in $X$, with multiplicities $1$. It remains to
prove that $\mcal{K}_{X}^{{\textstyle \cdot}}$ has coherent cohomology.
Since this is a local
question, we can assume using part (b) that $X$ is a closed subscheme of
$\mbf{A}_{k}^{n}$. According to parts (c) and (d) of this theorem and
\cite{Ye1} Corollary 4.5.6, $\mcal{K}_{X}^{{\textstyle \cdot}}$ has
coherent cohomology.
\end{proof}
From part (b) of the theorem we get:
\begin{cor}
The presheaf
$U \mapsto \Gamma(U, \mcal{K}_{U}^{{\textstyle \cdot}})$
is a sheaf on $X_{\mrm{et}}$, the small \'{e}tale
site over $X$.
\end{cor}
\begin{dfn} \label{dfn1.6}
For an $\mcal{O}_{X}$-module $\mcal{M}$
define dual complex
\[ \operatorname{Dual}_{X} \mcal{M} :=
\mcal{H}om^{{\textstyle \cdot}}_{\mcal{O}_{X}}(\mcal{M}, \mcal{K}_{X}^{{\textstyle \cdot}}) . \]
\end{dfn}
Observe that since $\mcal{K}_{X}^{{\textstyle \cdot}}$ is complex of injectives the
derived functor
$\operatorname{Dual}_{X} : \msf{D}(X)^{\circ} \rightarrow \msf{D}(X)$
is defined. Moreover, since $\mcal{K}_{X}^{{\textstyle \cdot}}$ is dualizing, the
adjunction morphism
$1 \rightarrow \operatorname{Dual}_{X} \operatorname{Dual}_{X}$
is an isomorphism on
$\msf{D}_{\mrm{c}}^{\mrm{b}}(X)$.
We shall sometimes write $\operatorname{Dual} \mcal{M}$ instead of
$\operatorname{Dual}_{X} \mcal{M}$.
\section{Duality for Proper Morphisms}
In this section we prove that if $f : X \rightarrow Y$ is a proper morphism of
$k$-schemes, then the trace map $\operatorname{Tr}_{f}$
of Definition \ref{dfn1.3} is a homomorphism of complexes, and it induces a
duality in the derived categories.
\begin{prop} \label{prop2.1}
Let $f: X \rightarrow Y$ be a proper morphism between finite type $k$-schemes, and
let $\eta = (y_{0}, \ldots, y_{n})$ be a saturated chain in $Y$. Then there
exists a canonical isomorphism of BCAs
\[ \prod_{\xi \mid \eta} \mcal{O}_{X,\xi} \cong
\prod_{x_{0} \mid y_{0}} \mcal{O}_{X,(x_{0})} \otimes^{(\wedge)}_{
\mcal{O}_{Y,(y_{0})}} \mcal{O}_{Y,\eta}\ , \]
where $\xi = (x_{0}, \ldots, x_{n})$ denotes a variable chain in $X$
lying over $\eta$.
\end{prop}
\begin{proof}
The proof is by induction on $n$. For $n=0$ this is trivial. Assume $n=1$.
Let
$Z := \overline{\{ x_{0} \}}_{\mrm{red}}$,
so $\mcal{O}_{Z, x_{1}}$ is a $1$-dimensional local ring inside
$k(Z) = k(x_{0})$. Considering the integral closure of
$\mcal{O}_{Z, x_{1}}$ we see that
$k((x_{0}, x_{1})) = k(x_{0})_{(x_{1})} =
k(x_{0}) \otimes \mcal{O}_{Z, (x_{1})}$
is the product of the completions of $k(x_{0})$ at all discrete valuations
centered on $x_{1} \in Z$ (cf.\ \cite{Ye1} Theorem 3.3.2).
So by the valuative criterion for properness we get
\begin{equation} \label{eqn2.3}
\prod_{(x_{0}, x_{1}) \mid (y_{0}, y_{1})} k(x_{0})_{(x_{1})} \cong
\prod_{x_{0} \mid y_{0}} k(x_{0}) \otimes_{k(y_{0})} k(y_{0})_{(y_{1})}.
\end{equation}
For $i \geq 1$ the morphism of BCAs
\[ \prod_{x_{0} \mid y_{0}} (\mcal{O}_{X, x_{0}} /
\mfrak{m}_{x_{0}}^{i})
\otimes_{\mcal{O}_{Y, (y_{0})}} \mcal{O}_{Y, (y_{0}, y_{1})} \rightarrow
\prod_{(x_{0}, x_{1}) \mid (y_{0}, y_{1})}
\mcal{O}_{X, (x_{0}, x_{1})} / \mfrak{m}_{(x_{0}, x_{1})}^{i} \]
is bijective, since both sides are flat over
$\mcal{O}_{X, x_{0}} / \mfrak{m}_{x_{0}}^{i}$, and by equation
(\ref{eqn2.3}) (cf.\ \cite{Ye2} Proposition 3.5).
Passing to the inverse limit in $i$ we get an isomorphism of BCAs
\begin{equation} \label{eqn2.4}
\prod_{x_{0} \mid y_{0}} \mcal{O}_{X, (x_{0})} \otimes^{(\wedge)}_{
\mcal{O}_{Y,(y_{0})}} \mcal{O}_{Y,(y_{0}, y_{1})} \cong
\prod_{(x_{0}, x_{1}) \mid (y_{0}, y_{1})} \mcal{O}_{X,(x_{0}, x_{1})}\ .
\end{equation}
Now suppose $n \geq 2$. Then we get
\[ \begin{array}{l}
\prod_{x_{0} \mid y_{0}} \mcal{O}_{X,(x_{0})} \otimes^{(\wedge)}_{
\mcal{O}_{Y,(y_{0})}} \mcal{O}_{Y, \eta} \\[2mm]
\begin{array}{rclc}
\blnk{25mm} & \cong &
\prod_{(x_{0}, x_{1}) \mid (y_{0}, y_{1})} \mcal{O}_{X,(x_{0}, x_{1})}
\otimes^{(\wedge)}_{\mcal{O}_{Y,(y_{0}, y_{1})}} \mcal{O}_{Y, \eta}
& \text{(i)} \\[2mm]
& \cong &
\prod_{(x_{0}, x_{1}) \mid (y_{0}, y_{1})} \mcal{O}_{X,(x_{0}, x_{1})}
\otimes^{(\wedge)}_{\mcal{O}_{Y,(y_{1})}}
\mcal{O}_{Y, \partial_{0} \eta}
& \text{(ii)} \\[2mm]
& \cong &
\prod_{\xi \mid \eta} \mcal{O}_{X,(x_{0}, x_{1})}
\otimes^{(\wedge)}_{\mcal{O}_{X,(x_{1})}}
\mcal{O}_{X, \partial_{0} \xi}
& \text{(iii)} \\[2mm]
& \cong &
\prod_{\xi \mid \eta} \mcal{O}_{X, \xi}
& \text{(iv)}
\end{array}
\end{array} \]
where associativity of intensification base change (\cite{Ye2}
Proposition 3.10)
is used repeatedly; in (i) we use formula (\ref{eqn2.4}); in (ii) we use
Lemma \ref{lem1.5} applied to $\mcal{O}_{Y, \eta}$; in (iii) we use
the induction hypothesis; and (iv) is another application of
Lemma \ref{lem1.5}.
\end{proof}
The next theorem is our version of \cite{RD} Ch.\ VII Theorem 2.1:
\begin{thm} \label{thm2.1}
\textup{(Global Residue Theorem)}\
Let $f : X \rightarrow Y$ be a proper morphism between $k$-schemes of finite type.
Then
$\operatorname{Tr}_{f}: f_{*} \mcal{K}_{X}^{{\textstyle \cdot}} \rightarrow \mcal{K}_{Y}^{{\textstyle \cdot}}$
is a homomorphism of complexes.
\end{thm}
\begin{proof}
Fix a point $x_{0} \in X$, and let $y_{0} := f(x_{0})$. First assume that
$x_{0}$ is closed in its fiber $X_{y_{0}} = f^{-1}(y_{0})$.
Let $y_{1}$ be an immediate specialization of $y_{0}$.
By Proposition \ref{prop2.1} we have
\[ \prod_{x_{1} \mid y_{1}} \mcal{O}_{X, (x_{0}, x_{1})} \cong
\mcal{O}_{X, (x_{0})} \otimes^{(\wedge)}_{\mcal{O}_{Y, (y_{0})}}
\mcal{O}_{Y, (y_{0}, y_{1})}, \]
so just as in Proposition \ref{prop1.2} (b), we get
\[ \delta_{(y_{0}, y_{1})} \operatorname{Tr}_{f} =
\sum_{x_{1} \mid y_{1}} \operatorname{Tr}_{f} \delta_{(x_{0}, x_{1})} :
f_{*} \mcal{K}_{X}(x_{0}) \rightarrow \mcal{K}_{Y}(y_{1}) . \]
Next assume $x_{0}, y_{0}$ are as above, but $x_{0}$ is not closed in
the fiber $X_{y_{0}}$. The only possibility to have an immediate
specialization $x_{1}$ of $x_{0}$ which is closed in its fiber, is if
$x_{1} \in X_{y_{0}}$ and
$Z := \overline{\{ x_{0} \}}_{\mrm{red}} \subset X_{y_{0}}$
is a curve. We have to show that
\begin{equation} \label{eqn2.5}
\sum_{x_{1} \mid y_{0}} \operatorname{Tr}_{f} \delta_{(x_{0}, x_{1})} = 0
: f_{*} \mcal{K}_{X}(x_{0}) \rightarrow \mcal{K}_{Y}(y_{0}) .
\end{equation}
Since
$\mcal{K}_{Z}(x_{0}) \subset \mcal{K}_{X}(x_{0})$
is an essential submodule over $\mcal{O}_{Y, y_{0}}$ it suffices to
check (\ref{eqn2.5}) on $\mcal{K}_{Z}(x_{0})$. Thus we may assume
$X = \overline{\{ x_{0} \}}_{\mrm{red}}$
and
$Y = \overline{\{ y_{0} \}}_{\mrm{red}}$.
After factoring $X \rightarrow Y$ through a suitable finite radiciel morphism
$X \rightarrow \tilde{X}$, and using Proposition \ref{prop1.2}, we may further
assume that $K = k(Y) \rightarrow k(X)$ is separable. Now
\[ \operatorname{Tr}_{f} \delta_{(x_{0}, x_{1})} =
\operatorname{Res}_{k((x_{0}, x_{1})) / K} : \Omega^{n+1}_{k(X) / k} \rightarrow
\Omega^{n}_{K / k} . \]
Since
$\Omega^{n+1}_{k(X) / k} = \Omega^{1}_{k(X) / k} \wedge
\Omega^{n}_{K / k}$,
it suffices to check that
\[ \sum_{x_{1} \in X} \operatorname{Res}_{k((x_{0}, x_{1})) / K} = 0 :
\Omega^{1}_{k(X) / k} \rightarrow K . \]
Let $K'$ be the maximal purely inseparable extension of $K$ in
an algebraic closure, and let
$X' := X \times_{K} K'$. So
\[ k((x_{0}, x_{1})) \otimes_{K} K' \cong
\prod_{(x'_{0}, x'_{1}) \mid (x_{0}, x_{1})} k((x'_{0}, x'_{1})) \]
where $(x'_{0}, x'_{1})$ are chains in $X'$.
According to \cite{Ye1} Lemma 2.4.14 we may assume $k = K = K'$.
Since now $K$ is perfect, we are in the position to use the
well known Residue Theorem for curves (cf.\ \cite{Ye1} Theorem 4.2.15).
\end{proof}
\begin{cor} \label{cor2.1}
Let $f : X \rightarrow Y$ be a morphism between $k$-schemes of finite type, and
let $Z \subset X$ be a closed subscheme which is proper over $Y$. Then
$\operatorname{Tr}_{f}: f_{*} \Gamma_{Z} \mcal{K}_{X}^{{\textstyle \cdot}} \rightarrow
\mcal{K}_{Y}^{{\textstyle \cdot}}$
is a homomorphism of complexes.
\end{cor}
\begin{proof}
Let $\mcal{I} \subset \mcal{O}_{X}$ be the ideal sheaf of $Z$, and
define
$Z_{n} := \mbf{Spec}\, \mcal{O}_{X} / \mcal{I}^{n+1}$,
$n \geq 0$. The trace maps
$\mcal{K}_{Z_{0}}^{{\textstyle \cdot}} \rightarrow \cdots \rightarrow
\mcal{K}_{Z_{n}}^{{\textstyle \cdot}} \rightarrow \cdots \rightarrow \mcal{K}_{X}^{{\textstyle \cdot}}$
of Proposition \ref{prop1.2} induce a filtration by subcomplexes
$\Gamma_{Z} \mcal{K}_{X}^{{\textstyle \cdot}} =
\bigcup_{n=0}^{\infty} \mcal{K}_{Z_{n}}^{{\textstyle \cdot}}$. Now
since each morphism $Z_{n} \rightarrow Y$ is proper,
$\operatorname{Tr}_{f}: f_{*} \mcal{K}_{Z_{n}}^{{\textstyle \cdot}} \rightarrow \mcal{K}_{Y}^{{\textstyle \cdot}}$
is a homomorphism of complexes.
\end{proof}
\begin{thm} \textup{(Duality)}\ \label{thm2.2}
Let $f: X \rightarrow Y$ be a proper morphism between finite type $k$-schemes.
Then for any complex
$\mcal{M}^{{\textstyle \cdot}} \in \mathsf{D}^{\mrm{b}}_{\mrm{c}}(X)$, the
homomorphism
\[ \operatorname{Hom}_{\mathsf{D}(X)} (\mcal{M}^{{\textstyle \cdot}},
\mcal{K}_{X}^{{\textstyle \cdot}})
\rightarrow \operatorname{Hom}_{\mathsf{D}(Y)} (\mrm{R} f_{*} \mcal{M}^{{\textstyle \cdot}},
\mcal{K}_{Y}^{{\textstyle \cdot}}) \]
induced by
$\operatorname{Tr}_{f} : f_{*} \mcal{K}_{X}^{{\textstyle \cdot}} \rightarrow \mcal{K}_{Y}^{{\textstyle \cdot}}$
is an isomorphism.
\end{thm}
\begin{proof}
The proof uses a relative version of Sastry's notion of ``residue
pairs'' and ``pointwise residue pairs'', cf.\ \cite{Ye1} Appendix.
Define a
residue pair relative to $f$ and $\mcal{K}_{Y}^{{\textstyle \cdot}}$, to be a pair
$(\mcal{R}^{{\textstyle \cdot}}, t)$, with $\mcal{R}^{{\textstyle \cdot}}$ a residual
complex on $X$,
and with $t: f_{*} \mcal{R}^{{\textstyle \cdot}} \rightarrow \mcal{K}_{Y}^{{\textstyle \cdot}}$ a
homomorphism of complexes, which represent the functor
$\mcal{M}^{{\textstyle \cdot}} \mapsto \operatorname{Hom}_{\mathsf{D}(Y)} (\mrm{R} f_{*}
\mcal{M}^{{\textstyle \cdot}}, \mcal{K}_{Y}^{{\textstyle \cdot}})$ on
$\mathsf{D}^{\mrm{b}}_{\mrm{c}}(X)$.
Such pairs exist; for instance, we may take $\mcal{R}^{{\textstyle \cdot}}$ to be the
Cousin complex $f^{\triangle} \mcal{K}_{Y}^{{\textstyle \cdot}}$ associated to the
dualizing complex $f^{!} \mcal{K}_{Y}^{{\textstyle \cdot}}$
(cf.\ \cite{RD} ch.\ VII \S 3, or ibid.\ Appendix no.\ 4).
A pointwise residue pair relative to $f$ and $\mcal{K}_{Y}^{{\textstyle \cdot}}$, is
by definition a pair
$(\mcal{R}^{{\textstyle \cdot}}, t)$ as above, but satisfying the condition: for any
closed point $x \in X$, and any coherent $\mcal{O}_{X}$-module $\mcal{M}$
supported on $\{ x \}$, the map
$\operatorname{Hom}_{\mcal{O}_{X}} (\mcal{M}, \mcal{R}^{{\textstyle \cdot}}) \rightarrow
\operatorname{Hom}_{\mcal{O}_{Y}} (f_{*} \mcal{M}, \mcal{K}_{Y}^{{\textstyle \cdot}})$
induced by $t$ is an isomorphism. By the definition of the trace map
$\operatorname{Tr}_{f}$, the pair
$(\mcal{K}_{X}^{{\textstyle \cdot}}, \operatorname{Tr}_{f})$ is a pointwise residue pair.
In fact,
$k \rightarrow \mcal{O}_{Y, (f(x))} \rightarrow \mcal{O}_{X, (x)}$ are morphisms in
$\mathsf{BCA}(k)$, and by \cite{Ye2} Theorem 7.4 (i),(iv) we get
\[ \operatorname{Hom}_{\mcal{O}_{X}} (\mcal{M}, \mcal{K}_{X}^{{\textstyle \cdot}}) \cong
\operatorname{Hom}_{\mcal{O}_{Y}} (f_{*} \mcal{M}, \mcal{K}_{Y}^{{\textstyle \cdot}}) \cong
\operatorname{Hom}_{k} (\mcal{M}_{x}, k). \]
The proof of \cite{Ye1} Appendix Theorem 2 goes through also
in the relative situation: the morphism
$\operatorname{Tr}_{f} : f_{*} \mcal{K}_{X}^{{\textstyle \cdot}} \rightarrow \mcal{K}_{Y}^{{\textstyle \cdot}}$
in $\mathsf{D}(Y)$ corresponds to a morphism
$\zeta: \mcal{K}_{X}^{{\textstyle \cdot}} \rightarrow \mcal{R}^{{\textstyle \cdot}}$ in
$\mathsf{D}(X)$. But since both $\mcal{K}_{X}^{{\textstyle \cdot}}$ and
$\mcal{R}^{{\textstyle \cdot}}$ are residual complexes, $\zeta$ is an actual, unique
homomorphism of complexes (cf.\ \cite{RD} Ch.\ IV Lemma 3.2).
By testing on $\mcal{O}_{X}$-modules
$\mcal{M}$ as above we see that $\zeta$ is indeed an isomorphism of
complexes. So $(\mcal{K}_{X}^{{\textstyle \cdot}}, \operatorname{Tr}_{f})$ is a residue pair.
\end{proof}
Let $\pi: X \rightarrow \operatorname{Spec} k$ be the structural morphism. In \cite{RD}
\S VII.3 we find the twisted inverse image functor
$\pi^{!} : \mathsf{D}^{+}_{\operatorname{c}}(k) \rightarrow \mathsf{D}^{+}_{\operatorname{c}}(X)$.
\begin{cor} \label{cor2.2}
There is a canonical isomorphism
$\zeta_{X} : \mcal{K}_{X}^{{\textstyle \cdot}} \stackrel{\simeq}{\rightarrow} \pi^{!} k$ in
$\mathsf{D}(X)$. It is compatible with proper and \'{e}tale morphisms.
If $\pi$ is proper then
\[ \operatorname{Tr}_{\pi} = \operatorname{Tr}_{\pi}^{\mrm{RD}}
\mrm{R} \pi_{*}(\zeta_{X}) : \pi_{*} \mcal{K}_{X}^{{\textstyle \cdot}} \rightarrow k \]
where
$\operatorname{Tr}_{\pi}^{\mrm{RD}} : \mrm{R} \pi_{*} \pi^{!} k \rightarrow k$
is the trace map of \cite{RD} \S \textup{VII.3}.
\end{cor}
\begin{proof}
The uniqueness of $\zeta_{X}$ follows from considering closed subschemes
$i_{Z} : Z \hookrightarrow X$ finite over $k$. This is because any endomorphism $a$
of $\mcal{K}_{X}^{{\textstyle \cdot}}$ in $\mathsf{D}(X)$
is a global section of $\mcal{O}_{X}$, and $a=1$
iff $i_{Z}^{*}(a) = 1$ for all such $Z$. Existence is proved by covering
$X$ with compactifiable (e.g.\ affine)
open sets and using Theorem \ref{thm2.2}, cf.\
\cite{Ye1} Appendix Theorem 3 and
subsequent Exercise. In particular $\zeta_{X}$ is seen to be compatible
with open immersions. Compatibility with proper morphisms follows from
the transitivity of traces. As for an \'{e}tale morphism $g : U \rightarrow X$,
one has
$g^{*} \mcal{K}_{X}^{{\textstyle \cdot}} \cong \mcal{K}_{U}^{{\textstyle \cdot}}$ by Theorem
\ref{thm1.1} (b), and also
$g^{*} \pi^{!} k = g^{!} \pi^{!} k = (\pi g)^{!} k$. Testing the
isomorphisms on subschemes $Z \subset U$ finite over $k$ shows that
$g^{*}(\zeta_{X}) = \zeta_{U}$.
\end{proof}
\section{Duals of Differential Operators}
Let $X$ be a $k$-scheme of finite type, where $k$ is a perfect field
of any characteristic.
Suppose $\mcal{M}, \mcal{N}$ are $\mcal{O}_{X}$-modules. By a
differential operator (DO)
$D : \mcal{M} \rightarrow \mcal{N}$ over $\mcal{O}_{X}$,
relative to $k$, we mean
in the sense of \cite{EGA} IV \S 16.8.
Thus $D$ has order $\leq 0$ if $D$ is $\mcal{O}_{X}$-linear,
and
$D$ has order $\leq d$ if for all $a \in \mcal{O}_{X}$,
the commutator $[D, a]$ has order $\leq d-1$.
Recall that the dual of an $\mcal{O}_{X}$-module $\mcal{M}$
is
$\operatorname{Dual} \mcal{M} =
\mcal{H}om_{\mcal{O}_{X}}^{{\textstyle \cdot}}(\mcal{M}, \mcal{K}^{{\textstyle \cdot}}_{X})$.
In this section we prove the existence of the dual operator
$\operatorname{Dual}(D)$, in terms of BCAs and residues. This explicit description
of $\operatorname{Dual}(D)$ will be needed for the applications in Sections 5-7.
For direct proofs of existence cf.\ Remarks \ref{rem3.2} and
\ref{rem3.3}.
\begin{thm} \label{thm3.1}
Let $\mcal{M},\mcal{N}$ be two $\mcal{O}_{X}$-modules, and let
$D : \mcal{M} \rightarrow \mcal{N}$ be a differential operator of order
$\leq d$. Then there is a homomorphism of graded sheaves
\[ \operatorname{Dual}(D) :
\operatorname{Dual} \mcal{N} \rightarrow \operatorname{Dual} \mcal{M} \]
having the properties below:
\begin{enumerate}
\rmitem{i} $\operatorname{Dual}(D)$ is a DO of order $\leq d$.
\rmitem{ii} $\operatorname{Dual}(D)$ is a homomorphism of complexes.
\rmitem{iii} Functoriality: if
$E : \mcal{N} \rightarrow \mcal{L}$ is another DO, then
$\operatorname{Dual}(E D) = \operatorname{Dual}(D)\, \operatorname{Dual}(E)$.
\rmitem{iv} If $d = 0$, i.e.\ $D$ is $\mcal{O}_{X}$-linear, then
$\operatorname{Dual}(D)(\phi) = \phi \circ D$ for any
$\phi \in
\mcal{H}om_{\mcal{O}_{X}}^{{\textstyle \cdot}}(\mcal{N}, \mcal{K}^{{\textstyle \cdot}}_{X})$.
\rmitem{v} Adjunction: under the homomorphisms
$\mcal{M} \rightarrow \operatorname{Dual} \operatorname{Dual} \mcal{M}$
and
$\mcal{N} \rightarrow \operatorname{Dual} \operatorname{Dual} \mcal{N}$,
one has
$D \mapsto \operatorname{Dual}(\operatorname{Dual}(D))$.
\end{enumerate}
\end{thm}
\begin{proof}
By \cite{RD} Theorem II.7.8, an $\mcal{O}_{X}$-module $\mcal{M}'$
is noetherian iff there is a surjection
$\bigoplus_{i = 1}^{n} \mcal{O}_{U_{i}} \twoheadrightarrow \mcal{M}'$,
for some open sets $U_{1}, \ldots, U_{n}$. Here $\mcal{O}_{U_{i}}$
is extended by $0$ to a sheaf on $X$.
One has $\mcal{M} = \lim_{\alpha \rightarrow} \mcal{M}_{\alpha}$,
where $\{ \mcal{M}_{\alpha} \}$ is the set of noetherian submodules
of $\mcal{M}$. (We did not assume $\mcal{M},\mcal{N}$ are
quasi-coherent!)
So
\[ \mcal{H}om_{\mcal{O}_{X}}(\mcal{M}, \mcal{K}^{q}_{X})
\cong
\lim_{\leftarrow \alpha}
\mcal{H}om_{\mcal{O}_{X}}(\mcal{M}_{\alpha}, \mcal{K}^{q}_{X}) . \]
Since the sheaf $\mcal{P}^{d}_{X/k}$ of principal parts is coherent,
and
$D : \mcal{M}_{\alpha} \rightarrow \mcal{N}$
induces
\[ \bigoplus_{i = 1}^{n} (\mcal{P}^{d}_{X/k} \otimes \mcal{O}_{U_{i}})
\twoheadrightarrow \mcal{P}^{d}_{X/k} \otimes \mcal{M}_{\alpha} \rightarrow \mcal{N} , \]
we conclude that the module
$\mcal{N}_{\alpha} := \mcal{O}_{X} \cdot D(\mcal{M}_{\alpha}) \subset
\mcal{N}$
is also noetherian. Therefore we may assume that both
$\mcal{M}, \mcal{N}$ are noetherian.
We have
\[ \mcal{H}om^{{\textstyle \cdot}}_{\mcal{O}_{X}}(\mcal{M}, \mcal{K}^{{\textstyle \cdot}}_{X}) =
\bigoplus_{x} \mcal{H}om_{\mcal{O}_{X}}(\mcal{M}, \mcal{K}_{X}(x)) , \]
and
$\mcal{H}om_{\mcal{O}_{X}}(\mcal{M}, \mcal{K}_{X}(x))$
is a constant sheaf with support $\overline{ \{ x \} }$ and module
\[ \mcal{H}om_{\mcal{O}_{X, x}}(\mcal{M}_{x}, \mcal{K}_{X}(x)) =
\operatorname{Hom}_{A}(M, \mcal{K}(A)) = \operatorname{Dual}_{A} M , \]
where
$A := \widehat{\mcal{O}}_{X, x} = \mcal{O}_{X, (x)}$
and
$M := A \otimes \mcal{M}_{x}$.
Note that $M$ is a finitely generated $A$-module.
$D : \mcal{M}_{x} \rightarrow \mcal{N}_{x}$
induces a continuous DO
$D : M \rightarrow N = A \otimes \mcal{N}_{x}$
(for the $\mfrak{m}$-adic topology).
According to \cite{Ye2} Theorem 8.6 there is a continuous DO
\[ \operatorname{Dual}_{A}(D) : \operatorname{Dual}_{A} N \rightarrow \operatorname{Dual}_{A} M . \]
Properties (i), (iii), (iv) and (v) follows directly from \cite{Ye2}
Theorem 8.6 and Corollary 8.8.
As for property (ii), consider any saturated chain
$\xi = (x, \ldots, y)$. Since
$\partial^{-} : \mcal{O}_{X, (x)} \rightarrow \mcal{O}_{X, \xi}$ is an
intensification homomorphism, and since
$\partial^{+} : \mcal{O}_{X, (y)} \rightarrow \mcal{O}_{X, \xi}$
is a morphism in $\mathsf{BCA}(k)$ which is also topologically \'{e}tale,
we see that property (ii) is a consequence of \cite{Ye2}
Thm.\ 8.6 and Cor.\ 8.12.
\end{proof}
Let
$\mcal{D}_{X} := \mcal{D}\textit{iff}_{\mcal{O}_{X} / k}(
\mcal{O}_{X}, \mcal{O}_{X})$
be the sheaf of differential operators on $X$. By definition
$\mcal{O}_{X}$ is a left $\mcal{D}_{X}$-module.
\begin{cor} \label{cor3.1}
If $\mcal{M}$ is a left \textup{(}resp.\ right\textup{)}
$\mcal{D}_{X}$-module, then
$\operatorname{Dual} \mcal{M}$ is a complex of right \textup{(}resp.\
left\textup{)} $\mcal{D}_{X}$-modules. In particular this is true for
$\mcal{K}_{X}^{{\textstyle \cdot}} = \operatorname{Dual} \mcal{O}_{X}$.
\end{cor}
\begin{cor} \label{cor3.2}
Suppose $\mcal{M}^{{\textstyle \cdot}}$ is a complex sheaves, where
each $\mcal{M}^{q}$ is an $\mcal{O}_{X}$-module, and
$\mrm{d} : \mcal{M}^{q} \rightarrow \mcal{M}^{q + 1}$ is a DO.
Then there is a dual complex
$\operatorname{Dual} \mcal{M}^{{\textstyle \cdot}}$.
\end{cor}
Specifically,
$(\operatorname{Dual} \mcal{M}^{{\textstyle \cdot}}, \mrm{D})$
is the simple complex associated to the double complex
$(\operatorname{Dual} \mcal{M}^{{\textstyle \cdot}})^{p, q} :=
\mcal{H}om_{\mcal{O}_{X}}(\mcal{M}^{-p}, \mcal{K}_{X}^{q})$.
The operator is
$\mrm{D} = \mrm{D}' + \mrm{D}''$,
where
\[ \begin{split}
\mrm{D}' & := (-1)^{p+q+1} \operatorname{Dual}(\mrm{d}) :
(\operatorname{Dual} \mcal{M}^{{\textstyle \cdot}})^{p, q} \rightarrow
(\operatorname{Dual} \mcal{M}^{{\textstyle \cdot}})^{p + 1, q} , \\
\mrm{D}'' & := \delta :
(\operatorname{Dual} \mcal{M}^{{\textstyle \cdot}})^{p, q} \rightarrow
(\operatorname{Dual} \mcal{M}^{{\textstyle \cdot}})^{p, q + 1} .
\end{split} \]
It is well known that if $\operatorname{char} k = 0$ and $X$ is smooth
of dimension $n$, then
$\omega_{X} = \Omega^{n}_{X/k}$ is a right $\mcal{D}_{X}$-module.
\begin{prop} \label{prop3.1}
Suppose $\operatorname{char} k = 0$ and $X$ is smooth of dimension $n$. Then
$\mrm{C}_{X} : \Omega^{n}_{X/k} \rightarrow \mcal{K}^{-n}_{X}$
\textup{(}the inclusion\textup{)} is $\mcal{D}_{X}$-linear.
\end{prop}
\begin{proof}
It suffice to prove that any $\partial \in \mcal{T}_{X}$
(the tangent sheaf), which we view as a DO
$\partial : \mcal{O}_{X} \rightarrow \mcal{O}_{X}$,
satisfies
$\operatorname{Dual}(\partial)(\alpha) = - \mrm{L}_{\partial}(\alpha)$,
where $\mrm{L}_{\partial}$
is the Lie derivative, and $\alpha \in \Omega^{n}_{X/k}$.
Localizing at the generic point of $X$ we get
$\partial \in \mcal{D}(k(X))$ and $\alpha \in \omega(k(X))$. Now use
\cite{Ye2} Definition 8.1 and Proposition 4.2.
\end{proof}
\begin{rem} \label{rem3.1}
Proposition \ref{prop3.1} says that in the case
$\operatorname{char} k = 0$ and $X$ smooth, the $\mcal{D}_{X}$-module
structure on $\mcal{K}^{{\textstyle \cdot}}_{X}$ coincides with the standard one,
which is obtained as follows. The quasi-isomorphism
$\mrm{C}_{X} : \Omega^{n}_{X/k} \rightarrow \mcal{K}^{{\textstyle \cdot}}_{X}[-n]$
identifies $\mcal{K}^{{\textstyle \cdot}}_{X}[-n]$ with the Cousin complex
of $\Omega^{n}_{X/k}$, which is computed in the category
$\msf{Ab}(X)$ (cf.\ \cite{Ha} Section I.2).
Since any $D \in \mcal{D}_{X}$ acts $\mbb{Z}$-linearly on
$\Omega^{n}_{X/k}$, it also acts on $\mcal{K}^{{\textstyle \cdot}}_{X}[-n]$.
\end{rem}
\begin{rem} \label{rem3.2}
According to \cite{Sai} there is a direct way to obtain Theorem
\ref{thm3.1} in characteristic $0$. Say $X \subset Y$, with $Y$ smooth.
Then
$\mcal{H}om_{\mcal{O}_{X}}^{{\textstyle \cdot}}(\mcal{M}, \mcal{K}^{{\textstyle \cdot}}_{X})
\cong
\mcal{H}om_{\mcal{O}_{Y}}^{{\textstyle \cdot}}(\mcal{M}, \mcal{K}^{{\textstyle \cdot}}_{Y})$.
Now by \cite{Sai} \S 2.2.3
any DO $D : \mcal{M} \rightarrow \mcal{N}$ of order $\leq d$ can be
viewed as
\[ D \in
\mcal{H}om_{\mcal{O}_{Y}}(\mcal{M}, \mcal{N} \otimes_{\mcal{O}_{Y}}
\mcal{D}^{d}_{Y}) \subset
\mcal{H}om_{\mcal{D}_{Y}}(
\mcal{M} \otimes_{\mcal{O}_{Y}} \mcal{D}_{Y},
\mcal{N} \otimes_{\mcal{O}_{Y}} \mcal{D}_{Y}) \]
(right $\mcal{D}_{Y}$-modules). Since $\mcal{K}^{q}_{Y}$ is a
$\mcal{D}_{Y}$-module (cf.\ Remark \ref{rem3.1}), we get
\[ \mcal{H}om_{\mcal{O}_{Y}}(\mcal{M}, \mcal{K}^{q}_{Y})
\cong
\mcal{H}om_{\mcal{D}_{Y}}(\mcal{M} \otimes_{\mcal{O}_{Y}}
\mcal{D}_{Y}, \mcal{K}^{q}_{Y}) \]
and so we obtain the dual operator $\operatorname{Dual}(D)$.
I thank the referee for pointing out this fact to me.
\end{rem}
\begin{rem} \label{rem3.3}
Suppose $\operatorname{char} k = p > 0$. Then a $k$-linear map
$D : \mcal{M} \rightarrow \mcal{N}$ is a DO over $\mcal{O}_{X}$ iff it is
$\mcal{O}_{X^{(p^{n}/k)}}$-linear, for $n \gg 0$.
Here
$X^{(p / k)} \rightarrow X$ is the Frobenius morphism relative to $k$,
cf.\ \cite{Ye1} Theorem 1.4.9.
Since $\operatorname{Tr}$ induces an isomorphism
\[ \mcal{H}om_{\mcal{O}_{X}}^{{\textstyle \cdot}}(\mcal{M}, \mcal{K}^{{\textstyle \cdot}}_{X})
\cong
\mcal{H}om_{\mcal{O}_{X^{(p^{n}/k)}}}^{{\textstyle \cdot}}(
\mcal{M}, \mcal{K}^{{\textstyle \cdot}}_{X^{(p^{n}/k)}}) \]
we obtain the dual operator $\operatorname{Dual}(D)$.
\end{rem}
Let us finish this section with an application to rings of differential
operators. Given a finitely generated (commutative) $k$-algebra $A$,
denote by $\mcal{D}(A) := \operatorname{Diff}_{A/k}(A,A)$
the ring of differential operators over $A$. Such rings are of interest
for ring theorists (cf.\ \cite{MR} and \cite{HoSt}).
It is well known that if $\operatorname{char} k = 0$ and $A$ is smooth,
then the opposite ring
$\mcal{D}(A)^{\circ} \cong \omega_{A} \otimes_{A} \mcal{D}(A)
\otimes_{A} \omega_{A}^{-1}$,
where $\omega_{A} = \Omega^{n}_{A / k}$.
The next theorem is a vast generalization of this fact.
Given complexes $M^{{\textstyle \cdot}}, N^{{\textstyle \cdot}}$ of $A$-modules let
$\operatorname{Diff}^{{\textstyle \cdot}}_{A / k}(M^{{\textstyle \cdot}}, N^{{\textstyle \cdot}})$
be the complex of $k$-modules which in degree $n$ is
$\prod_{p} \operatorname{Diff}_{A / k}(M^{p}, N^{p+n})$.
Let
$\mcal{K}^{{\textstyle \cdot}}_{A} := \Gamma(X, \mcal{K}^{{\textstyle \cdot}}_{X})$
with
$X := \operatorname{Spec} A$.
By Corollary \ref{cor3.1}, it is a complex of right $\mcal{D}(A)$-modules.
\begin{thm} \label{thm3.2}
There is a natural isomorphism of filtered $k$-algebras
\[ \mcal{D}(A)^{\circ} \cong
\operatorname{H}^{0} \operatorname{Diff}_{A / k}^{{\textstyle \cdot}}(
\mcal{K}_{A}^{{\textstyle \cdot}}, \mcal{K}_{A}^{{\textstyle \cdot}}) . \]
\end{thm}
\begin{proof}
First observe that since DOs preserve support,
$\operatorname{Diff}_{A / k}(\mcal{K}_{A}^{p}, \mcal{K}_{A}^{p-1}) = 0$
for all $p$. This means that every local section
$D \in \operatorname{H}^{0} \operatorname{Diff}_{A / k}^{{\textstyle \cdot}}(\mcal{K}_{A}^{{\textstyle \cdot}},
\mcal{K}_{A}^{{\textstyle \cdot}})$
is a well defined DO
$D : \mcal{K}_{A}^{{\textstyle \cdot}} \rightarrow \mcal{K}_{A}^{{\textstyle \cdot}}$
which commutes with the coboundary $\delta$. Applying
$\operatorname{Dual}$ and taking $0$-th cohomology we obtain a DO
\[ D^{\vee} = \operatorname{H}^{0} \operatorname{Dual}(D) :
\operatorname{H}^{0} \operatorname{Dual} \mcal{K}_{A}^{{\textstyle \cdot}} \rightarrow
\operatorname{H}^{0} \operatorname{Dual} \mcal{K}_{A}^{{\textstyle \cdot}} . \]
But
$\operatorname{H}^{0} \operatorname{Dual} \mcal{K}_{A}^{{\textstyle \cdot}} = A$,
so $D^{\vee} \in \mcal{D}(A)$.
Finally according to Theorem \ref{thm3.1} (v), $D = D^{\vee \vee}$
for
$D \in \operatorname{H}^{0} \operatorname{Diff}_{A / k}^{{\textstyle \cdot}}(\mcal{K}_{A}^{{\textstyle \cdot}},
\mcal{K}_{A}^{{\textstyle \cdot}})$
or $D \in \mcal{D}(A)$.
\end{proof}
Recall that an $n$-dimensional integral domain $A$ is a Gorenstein algebra
iff
$\omega_{A} = \mrm{H}^{-n} \mcal{K}_{A}^{{\textstyle \cdot}} \rightarrow
\mcal{K}_{A}^{{\textstyle \cdot}}[-n]$
is a quasi-isomorphism, and $\omega_{A}$ is invertible.
\begin{cor} \label{cor3.6}
If $A$ is a Gorenstein $k$-algebra,
there is a canonical isomorphism of filtered $k$-algebras
\[ \mcal{D}(A)^{\circ} \cong
\operatorname{Diff}_{A / k}(\omega_{A}, \omega_{A}) \cong
\omega_{A} \otimes_{A} \mcal{D}(A) \otimes_{A} \omega_{A}^{-1} . \]
\end{cor}
\begin{rem}
In \cite{Ho}, the right $\mcal{D}(A)$-module structure on $\omega_{A}$
was exhibited, when $X = \operatorname{Spec} A$ is a curve. Corollary
\ref{cor3.6} was proved there for complete intersection curves.
\end{rem}
\section{The De Rham-Residue Complex}
As before $k$ is a perfect field of any characteristic.
Let $X$ be a $k$-scheme of finite type.
In this section we define a canonical complex on $X$, the De
Rham-residue complex $\mcal{F}^{{\textstyle \cdot}}_{X}$.
As we shall see in Corollary \ref{cor4.3}, $\mcal{F}^{{\textstyle \cdot}}_{X}$
coincides (up to indices and signs) with the double complex
$\mcal{K}^{{\textstyle \cdot}, *}_{X}$ of \cite{EZ}.
According to Theorem \ref{thm3.1}, if $\mcal{M}^{{\textstyle \cdot}}$ is a complex
of sheaves, with each $\mcal{M}^{q}$ an $\mcal{O}_{X}$-module and
$\mrm{d} : \mcal{M}^{q} \rightarrow \mcal{M}^{q+1}$
a DO, then $\operatorname{Dual} \mcal{M}^{{\textstyle \cdot}}$ is a complex of the same kind.
\begin{dfn} \label{dfn4.1}
The {\em De Rham-residue complex} on $X$ is the complex
\[ \mcal{F}_{X}^{{\textstyle \cdot}} :=
\operatorname{Dual} \Omega^{{\textstyle \cdot}}_{X / k} . \]
of Corollary \ref{cor3.2}.
\end{dfn}
Note that the double complex $\mcal{F}_{X}^{{\textstyle \cdot} {\textstyle \cdot}}$ is
concentrated in the third quadrant of the $(p,q)$-plane.
\begin{prop} \label{prop4.2}
$\mcal{F}_{X}^{{\textstyle \cdot}}$ is a right DG module over
$\Omega^{{\textstyle \cdot}}_{X/k}$.
\end{prop}
\begin{proof}
The graded module structure is clear.
It remains to check that
\[ \mrm{D}(\phi \cdot \alpha) = (\mrm{D} \phi) \cdot \alpha +
(-1)^{p+q} \phi \cdot (\mrm{d} \alpha) \]
for $\phi \in \mcal{F}_{X}^{p,q}$ and
$\alpha \in \Omega^{p'}_{X/k}$.
But this is a straightforward computation using Theorem \ref{thm3.1}.
\end{proof}
\begin{prop} \label{prop4.3}
Let $g: U \rightarrow X$ be \'{e}tale. Then there is a homomorphism of
complexes
$\mrm{q}_{g} : \mcal{F}_{X}^{{\textstyle \cdot}} \rightarrow g_{*} \mcal{F}_{U}^{{\textstyle \cdot}}$,
which induces an isomorphism of graded sheaves
$1 \otimes \mrm{q}_{g} : g^{*} \mcal{F}_{X}^{{\textstyle \cdot}} \stackrel{\simeq}{\rightarrow}
\mcal{F}_{U}^{{\textstyle \cdot}}$.
\end{prop}
\begin{proof}
Consider the isomorphisms
$g^{*} \Omega^{{\textstyle \cdot}}_{X/k} \cong \Omega^{{\textstyle \cdot}}_{U/k}$
and
$1 \otimes \mrm{q}_{g} : g^{*} \mcal{K}_{X}^{{\textstyle \cdot}} \rightarrow
\mcal{K}_{U}^{{\textstyle \cdot}}$
of Theorem \ref{thm1.1}. Clearly
$1 \otimes \mrm{q}_{g} :
g^{*} \mcal{F}_{X}^{p,q} \rightarrow \mcal{F}_{U}^{p,q}$
is an isomorphism.
In light of \cite{Ye2} Theorem 8.6 (iv),
$\mrm{q}_{g} : \mcal{F}_{X}^{{\textstyle \cdot}} \rightarrow g_{*} \mcal{F}_{U}^{{\textstyle \cdot}}$
is a homomorphism of complexes.
\end{proof}
Let $f: X \rightarrow Y$ be a morphism of schemes. Define a homomorphism of
graded sheaves
$\operatorname{Tr}_{f} : f_{*} \mcal{F}_{X}^{{\textstyle \cdot}} \rightarrow
\mcal{F}_{Y}^{{\textstyle \cdot}}$
by composing
$f^{*} : \Omega^{{\textstyle \cdot}}_{Y/k} \rightarrow f_{*} \Omega^{{\textstyle \cdot}}_{X/k}$
with
$\operatorname{Tr}_{f} : f_{*} \mcal{K}_{X}^{{\textstyle \cdot}} \rightarrow
\mcal{K}_{Y}^{{\textstyle \cdot}}$
of Definition \ref{dfn1.3}.
\begin{prop} \label{prop4.1}
$\operatorname{Tr}_{f}$ commutes with $\mrm{D}'$. If $f$ is proper then
$\operatorname{Tr}_{f}$ also commutes with $\mrm{D}''$.
\end{prop}
\begin{proof}
Let $y \in Y$ and let $x$ be a closed point in $f^{-1}(y)$. Then
$f^{*}: \mcal{O}_{Y,(y)} \rightarrow \mcal{O}_{X,(x)}$ is a morphism in
$\msf{BCA}(k)$. Applying \cite{Ye2} Cor.\ 8.12 to the DOs
\[ \mrm{d} f^{*} = f^{*} \mrm{d} :
\Omega^{p}_{Y/k, (y)} \rightarrow \Omega^{p+1}_{X/k, (x)} \]
we get a dual homomorphism
\[ \operatorname{Dual}_{f^{*}}(\mrm{d} f^{*}) =
\operatorname{Dual}_{f^{*}}(f^{*} \mrm{d}) : \operatorname{Dual}_{\mcal{O}_{X,(x)}}
\Omega^{p+1}_{X/k, (x)} \rightarrow
\operatorname{Dual}_{\mcal{O}_{Y,(y)}} \Omega^{p}_{Y/k, (y)} , \]
which equals both
$\operatorname{Tr}_{f} \operatorname{Dual}_{X}(\mrm{d})$ and
$\operatorname{Dual}_{Y}(\mrm{d}) \operatorname{Tr}_{f}$.
The commutation of $\mrm{D}''$ with $\operatorname{Tr}_{f}$ in the proper case
is immediate from Thm.\ \ref{thm2.1}.
\end{proof}
Of course if $f : X \rightarrow Y$ is a closed immersion, then $\operatorname{Tr}_{f}$ is
injective, and it identifies $\mcal{F}_{X}^{{\textstyle \cdot}}$ with the
subsheaf
$\mcal{H}om_{\Omega^{{\textstyle \cdot}}_{Y / k}}(\Omega^{{\textstyle \cdot}}_{X / k},
\mcal{F}_{Y}^{{\textstyle \cdot}})$ of
$\mcal{F}_{Y}^{{\textstyle \cdot}}$.
Just as in Corollary \ref{cor2.1} we get:
\begin{cor} \label{cor4.1}
Let $f : X \rightarrow Y$ be a morphism of schemes, and
let $Z \subset X$ be a closed subscheme which is proper over $Y$. Then
the trace map
$\operatorname{Tr}_{f}: f_{*} \Gamma_{Z} \mcal{F}_{X}^{{\textstyle \cdot}} \rightarrow
\mcal{F}_{Y}^{{\textstyle \cdot}}$
is a homomorphism of complexes.
\end{cor}
Suppose $X$ is an integral scheme of dimension $n$. The canonical
homomorphism
\begin{equation}
\operatorname{C}_{X} : \Omega^{n}_{X/k} \rightarrow
\mcal{K}_{X}^{-n} = k(X) \otimes_{\mcal{O}_{X}} \Omega^{n}_{X/k}
\end{equation}
can be viewed as a global section of $\mcal{F}_{X}^{-n, -n}$.
\begin{lem} \label{lem4.3}
Suppose $X$ is an integral scheme. Then
$\mrm{D}' (\operatorname{C}_{X}) = \mrm{D}'' (\operatorname{C}_{X}) = 0$.
\end{lem}
\begin{proof}
By \cite{Ye1} Section 4.5,
$\mrm{D}''(\operatorname{C}_{X}) = \pm \delta (\operatorname{C}_{X}) = 0$.
Next, let $K := k(X)$. Choose $t_{1}, \ldots t_{n} \in K$
such that
$\Omega^{1}_{K / k} = \bigoplus K \cdot \mrm{d} t_{i}$.
Taking products of the $\mrm{d} t_{i}$ as bases of
$\Omega^{n - 1}_{K / k}$ and $\Omega^{n}_{K / k}$, we see from
\cite{Ye2} Theorem 8.6 and Definition 8.1 that
$\operatorname{Dual}_{K}(\operatorname{C}_{X}) = 0$.
\end{proof}
\begin{prop} \label{prop4.4}
If $X$ is smooth irreducible of dimension $n$, then the DG homomorphism
$\Omega^{{\textstyle \cdot}}_{X/k} \rightarrow \mcal{F}^{{\textstyle \cdot}}_{X}[-2n]$,
$\alpha \mapsto \mrm{C}_{X} \cdot \alpha$,
is a quasi-isomorphism.
\end{prop}
\begin{proof}
First note that $\mrm{D}(\mrm{C}_{X}) = 0$, so this is indeed a
DG homomorphism.
Filtering these complexes according to the $p$-degree
we reduce to looking at
$\Omega^{p}_{X/k}[n] \rightarrow \mcal{F}^{p-n, {\textstyle \cdot}}_{X}$.
That is a quasi-isomorphism by Theorem \ref{thm1.1} part d.
\end{proof}
\begin{cor} \label{cor4.3}
The complex $\mcal{F}^{{\textstyle \cdot}}_{X}$ is the same as the complex
$\mcal{K}^{{\textstyle \cdot}, *}_{X}$ of \cite{EZ}, up to signs and indexing.
\end{cor}
\begin{proof}
If $X$ is smooth of dimension $n$ this is because
$\mcal{F}^{{\textstyle \cdot}}_{X} \cong
\Omega^{{\textstyle \cdot}}_{X/k}[n] \otimes \mcal{K}^{{\textstyle \cdot}}_{X}$
is the Cousin complex of
$\bigoplus \Omega^{p}_{X/k}[p]$, and $\mrm{D}'$
is (up to sign) the Cousin functor applied to $\mrm{d}$.
If $X$ is a general scheme embedded in a smooth scheme $Y$,
use Proposition \ref{prop4.1}.
\end{proof}
\begin{dfn} \label{dfn4.4}
Given a scheme $X$, let $X_{1}, \ldots, X_{r}$ be its irreducible
components, with their induced reduced subscheme structures. For each $i$
let $x_{i}$ be the generic point of $X_{i}$, and let $f_{i} : X_{i} \rightarrow X$
be the inclusion morphism. We define the fundamental class $\operatorname{C}_{X}$
by:
\[ \operatorname{C}_{X}:= \sum_{i=1}^{r} \operatorname{length}(\mcal{O}_{X, x_{i}})
\operatorname{Tr}_{f_{i}}(\operatorname{C}_{X_{i}}) \in \mcal{F}_{X}^{{\textstyle \cdot}} . \]
\end{dfn}
The next proposition is easily verified using Propositions \ref{prop4.1}
and \ref{prop4.3}. It should be compared to
\cite{EZ} Theorem III.3.1.
\begin{prop} \label{prop4.7}
For any scheme $X$, the fundamental class
$\operatorname{C}_{X} \in$ \linebreak
$\Gamma(X, \mcal{F}_{X}^{{\textstyle \cdot}})$
is annihilated
by $\mrm{D}'$ and $\mrm{D}''$. If $X$ has pure dimension $n$,
then $\operatorname{C}_{X}$ has bidegree $(-n,-n)$. If $f: X \rightarrow Y$ is a proper,
surjective, generically finite morphism between integral schemes, then
$\operatorname{Tr}_{f}(\operatorname{C}_{X}) = \operatorname{deg}(f) \operatorname{C}_{Y}$.
If $g: U \rightarrow X$ is \'{e}tale, then
$\mrm{C}_{U} = \mrm{q}_{g}(\mrm{C}_{X})$.
\end{prop}
\begin{rem}
In \cite{Ye4} it is shown that $\mcal{F}_{X}^{{\textstyle \cdot}}$ is a right
DG module over the DGA of Beilinson adeles
$\mcal{A}_{X}^{{\textstyle \cdot}} = \underline{\mbb{A}}^{{\textstyle \cdot}}_{\mrm{red}}(
\Omega^{{\textstyle \cdot}}_{X / k})$.
Now let $\mcal{E}$ be a locally free $\mcal{O}_{X}$-module of rank
$r$, and let $Z \subset X$ be the zero locus of a regular section of
$\mcal{E}$. According to the adelic Chern-Weil theory of \cite{HY2}
there is an adelic connection $\nabla$ on $\mcal{E}$ such that
the Chern form
$\mrm{c}_{r}(\mcal{E}; \nabla) \in \mcal{A}_{X}^{2r}$
satisfies
$\mrm{C}_{Z} = \pm \mrm{C}_{X} \cdot \mrm{c}_{r}(\mcal{E}; \nabla)
\in \mcal{F}_{X}^{{\textstyle \cdot}}$.
\end{rem}
\section{De Rham Homology and the Niveau Spectral Sequence}
Let $X$ be a finite type scheme over a field $k$ of characteristic $0$.
In \cite{Ye3} it is shown that if $X \subset \mfrak{X}$ is
a smooth formal embedding (see below) then the De Rham complex
$\widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{X} / k}$ calculates the De Rham
cohomology $\mrm{H}^{{\textstyle \cdot}}_{\mrm{DR}}(X)$.
In this section we will show that the De Rham-residue complex
$\mcal{F}^{{\textstyle \cdot}}_{\mfrak{X}}$ of $\mfrak{X}$ calculates
the De Rham homology $\mrm{H}^{\mrm{DR}}_{{\textstyle \cdot}}(X)$.
This is done by computing the niveau spectral sequence
converging to $\mrm{H}^{{\textstyle \cdot}}(X, \mcal{F}^{{\textstyle \cdot}}_{\mfrak{X}})$
(Theorem \ref{thm5.3}).
We will draw a few conclusions, including
the contravariance of homology w.r.t.\ \'{e}tale morphisms
(Theorem \ref{thm5.1}).
As a reference for algebraic De Rham (co)homology we suggest
\cite{Ha}.
Given a noetherian adic formal scheme $\mfrak{X}$ and a defining ideal
$\mcal{I} \subset \mcal{O}_{\mfrak{X}}$,
let $X_{n}$ be the (usual) noetherian scheme
$(\mfrak{X}, \mcal{O}_{\mfrak{X}} / \mcal{I}^{n+1})$.
Suppose $f : \mfrak{X} \rightarrow \mfrak{Y}$ is a morphism between such
formal schemes, and let
$\mcal{I} \subset \mcal{O}_{\mfrak{X}}$ and
$\mcal{J} \subset \mcal{O}_{\mfrak{Y}}$
be defining ideals such that
$f^{-1} \mcal{J} \cdot \mcal{O}_{\mfrak{X}} \subset \mcal{I}$.
Such ideals are always available.
We get a morphism of (usual) schemes
$f_{0} : X_{0} \rightarrow Y_{0}$.
\begin{dfn} \label{dfn5.2}
A morphism $f : \mfrak{X} \rightarrow \mfrak{Y}$ between (noetherian)
adic formal schemes is called {\em formally finite type}
(resp.\ {\em formally finite} or {\em formally proper})
if the morphism $f_{0} : X_{0} \rightarrow Y_{0}$
is finite type (resp.\ finite or proper).
\end{dfn}
Obviously these notions are independent of the particular defining
ideals chosen.
\begin{exa} \label{exa5.1}
If $X \rightarrow Y$ is a finite type morphism of noetherian schemes,
$X_{0} \subset X$ is a locally closed subset and
$\mfrak{X} = X_{/ X_{0}}$ is the formal completion, then
$\mfrak{X} \rightarrow Y$ is formally finite type.
Such a morphism is called {\em algebraizable}.
\end{exa}
\begin{dfn}
A morphism of formal schemes $\mfrak{X} \rightarrow \mfrak{Y}$ is said to be {\em
formally smooth} (resp.\ {\em formally \'{e}tale}) if, given
a (usual) affine scheme $Z$ and a closed subscheme $Z_{0} \subset Z$
defined by a nilpotent ideal, the map
$\operatorname{Hom}_{\mfrak{Y}}(Z, \mfrak{X})$ \linebreak
$\rightarrow \operatorname{Hom}_{\mfrak{Y}}(Z_{0}, \mfrak{X})$
is surjective (resp.\ bijective).
\end{dfn}
This is the definition of formal smoothness used in \cite{EGA} IV
Section 17.1. We shall also require the next notion.
\begin{dfn}
A morphism $g: \mfrak{X} \rightarrow \mfrak{Y}$ between noetherian formal
sche\-mes is called {\em \'{e}tale} if it is of finite
type (see \cite{EGA} I \S 10.13) and formally \'{e}tale.
\end{dfn}
Note that if $\mfrak{Y}$ is a usual scheme, then so is
$\mfrak{X}$, and $g$ is an \'{e}tale morphism of schemes.
\begin{dfn} \label{dfn5.3}
A {\em smooth formal embedding} of $X$ (over $k$)
is a closed immersion of $X$ into a formal scheme $\mfrak{X}$,
which induces a homeomorphism on the underlying topological spaces, and
such that
$\mfrak{X}$ is of formally finite type and formally smooth over $k$.
\end{dfn}
\begin{exa}
If $X$ is smooth over $Y = \operatorname{Spec} k$ and $X_{0}, \mfrak{X}$
are as in Example \ref{exa5.1}, then $X_{0} \subset \mfrak{X}$
is a smooth formal embedding.
\end{exa}
Let $\xi = (x_{0}, \ldots, x_{q})$ be a saturated chain of points
in the formal scheme $\mfrak{X}$. Choose a defining ideal
$\mcal{I}$, and let $X_{n}$ be as above.
Define the Beilinson completion
$\mcal{O}_{\mfrak{X}, \xi} := \lim_{\leftarrow n}
\mcal{O}_{X_{n}, \xi}$
(which of course is independent of $\mcal{I}$).
\begin{lem} \label{lem5.2}
Let $\mfrak{X}$ be formally finite type over $k$, and let
$\xi$ be a saturated chain in $\mfrak{X}$.
Then $\mcal{O}_{\mfrak{X}, \xi}$ is a BCA over $k$. If
$\mfrak{X} = X_{/ X_{0}}$, then
$\mcal{O}_{\mfrak{X}, \xi} \cong \mcal{O}_{X, \xi}$.
\end{lem}
\begin{proof}
First assume $\mfrak{X} = X_{/ X_{0}}$. Taking $\mcal{I}$ to be
the ideal of $X_{0}$ in $X$, we have
\begin{multline*}
\hspace{1cm}
\mcal{O}_{\mfrak{X}, \xi} =
\lim_{\leftarrow n} (\mcal{O}_{X} / \mcal{I}^{n})_{\xi} \cong
\lim_{\leftarrow m,n} \mcal{O}_{X, \xi} /
(\mcal{I}^{n} \mcal{O}_{X, \xi} + \mfrak{m}_{\xi}^{m}) \\
\cong \lim_{\leftarrow m} \mcal{O}_{X, \xi} / \mfrak{m}_{\xi}^{m} =
\mcal{O}_{X, \xi} . \hspace{1cm}
\end{multline*}
Now by \cite{Ye3} Proposition 1.20 and Lemma 1.1, locally there is a
closed immersion $\mfrak{X} \subset \mfrak{Y}$, with
$\mfrak{Y}$ algebraizable (i.e.\ $\mfrak{Y} = Y_{/ Y_{0}}$).
So there is a surjection
$\mcal{O}_{\mfrak{Y}, \xi} \rightarrow
\mcal{O}_{\mfrak{X}, \xi}$,
and this implies that $\mcal{O}_{\mfrak{X}, \xi}$ is a BCA.
\end{proof}
One can construct the complexes
$\mcal{K}_{\mfrak{X}}^{{\textstyle \cdot}}$ and
$\mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}}$ for a formally finite type
formal scheme $\mfrak{X}$, as follows. Define
$\mcal{K}_{\mfrak{X}}(x) :=
\mcal{K}(\mcal{O}_{\mfrak{X}, (x)})$.
Now let $(x,y)$ be a saturated chain.
Then there is an intensification homomorphism
$\partial^{-}: \mcal{O}_{\mfrak{X}, (x)} \rightarrow
\mcal{O}_{\mfrak{X}, (x,y)}$
and a morphism of BCAs
$\partial^{+}: \mcal{O}_{\mfrak{X}, (y)} \rightarrow
\mcal{O}_{\mfrak{X}, (x,y)}$.
Therefore we get a homomorphism of
$\mcal{O}_{\mfrak{X}}$-modules
$\delta_{(x,y)}: \mcal{K}_{\mfrak{X}}(x) \rightarrow
\mcal{K}_{\mfrak{X}}(y)$. Define a graded sheaf
$\mcal{K}_{\mfrak{X}}^{{\textstyle \cdot}} =
\bigoplus_{x \in \mfrak{X}} \mcal{K}_{\mfrak{X}}(x)$
on $\mfrak{X}$, as in \S 1.
Let
$\widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{X}/k}$ be the complete De Rham
complex on $\mfrak{X}$, and set
$\mcal{F}_{\mfrak{X}}^{p, q} :=
\mcal{H}om_{\mcal{O}_{\mfrak{X}}}(
\widehat{\Omega}^{-p}_{\mfrak{X}/k},
\mcal{K}_{\mfrak{X}}^{q})$.
\begin{prop} \label{prop5.5}
Let $\mfrak{X}$ be a formally finite type formal scheme over $k$.
\begin{enumerate}
\item $\mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}}$
is a complex.
\item If $g : \mfrak{U} \rightarrow \mfrak{X}$ is \'{e}tale, then
there is a homomorphism of complexes
$\mrm{q}_{g} : \mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}} \rightarrow g_{*}
\mcal{F}_{\mfrak{U}}^{{\textstyle \cdot}}$,
which induces an isomorphism of graded sheaves
$1 \otimes \mrm{q}_{g} : g^{*} \mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}} \stackrel{\simeq}{\rightarrow}
\mcal{F}_{\mfrak{U}}^{{\textstyle \cdot}}$.
\item If $f : \mfrak{X} \rightarrow \mfrak{Y}$ is formally proper, then
there is a homomorphism of complexes
$\operatorname{Tr}_{f} : f^{*} \mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}} \rightarrow
\mcal{F}_{\mfrak{Y}}^{{\textstyle \cdot}}$.
\end{enumerate}
\end{prop}
\begin{proof}
1.\ Let $X_{n} \subset \mfrak{X}$ be as before. Then one has
$\mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}} =
\bigcup \mcal{F}_{X_{n}}^{{\textstyle \cdot}}$, so this is a complex.\\
2.\ Take $U_{n} := \mfrak{U} \times_{\mfrak{X}}
X_{n}$; then each $U_{n} \rightarrow X_{n}$ is an \'{e}tale morphism of schemes,
and we can use Proposition \ref{prop4.3}.\\
3.\ Apply Proposition \ref{prop4.1} to $X_{n} \rightarrow Y_{n}$.
\end{proof}
\begin{prop} \label{prop5.1}
Assume $\mfrak{X} = Y_{/X}$ for some smooth irreducible scheme
$Y$ of dimension $n$ and closed set $X \subset Y$. Then there is
a natural isomorphism of complexes
\begin{equation} \label{eqn5.3}
\mcal{F}^{{\textstyle \cdot}}_{\mfrak{X}} \cong
\underline{\Gamma}_{X} \mcal{F}^{{\textstyle \cdot}}_{Y} .
\end{equation}
Hence
$\mcal{F}^{{\textstyle \cdot}}_{\mfrak{X}} \cong
\mrm{R} \underline{\Gamma}_{X} \Omega^{{\textstyle \cdot}}_{Y / k}[2n]$
in the derived category $\mathsf{D}(\mathsf{Ab}(Y))$, and consequently
\[ \mrm{H}^{-q}(X, \mcal{F}^{{\textstyle \cdot}}_{\mfrak{X}}) \cong
\mrm{H}^{2n-q}_{X}(Y, \Omega^{{\textstyle \cdot}}_{Y / k}) =
\mrm{H}^{\mrm{DR}}_{q}(X) . \]
\end{prop}
\begin{proof}
The isomorphism (\ref{eqn5.3}) is immediate from Lemma \ref{lem5.2}.
But according to Proposition \ref{prop4.4}, $\mcal{F}^{{\textstyle \cdot}}_{Y}$
is a flasque resolution of $\Omega^{{\textstyle \cdot}}_{Y/k}[2n]$
in $\mathsf{Ab}(Y)$.
\end{proof}
We need some algebraic results, phrased in the terminology of
\cite{Ye1} \S 1.
Let $K$ be a complete, separated semi-topological (ST) commutative
$k$-algebra, and let
$\underline{t} = (t_{1}, \ldots, t_{n})$ be a sequence of indeterminates.
Let $K[[\, \underline{t}\, ]]$ and $K((\underline{t}))$ be the rings of formal
power series, and of iterated Laurent series, respectively. These
are complete, separated ST $k$-algebras.
Let $T$ be the free $k$-module with basis
$\alpha_{1}, \ldots, \alpha_{n}$ and let
$\bigwedge_{k}^{{\textstyle \cdot}} T$ be the exterior algebra over $k$.
\begin{lem} \label{lem5.1} \textup{(``Poincar\'{e} Lemma'')}\
The DGA homomorphisms
\[ \Omega^{{\textstyle \cdot}, \operatorname{sep}}_{K / k} \rightarrow
\Omega^{{\textstyle \cdot}, \operatorname{sep}}_{K[[\, \underline{t}\, ]] / k} \]
and
\[ \Omega^{{\textstyle \cdot}, \operatorname{sep}}_{K / k} \otimes_{k}
\bigwedge\nolimits_{k}^{{\textstyle \cdot}} T
\rightarrow
\Omega^{{\textstyle \cdot}, \operatorname{sep}}_{K((\underline{t})) / k},\
\alpha_{i} \mapsto \operatorname{dlog} t_{i} \]
are quasi-isomorphisms.
\end{lem}
\begin{proof}
Since
\[ \Omega^{{\textstyle \cdot}, \operatorname{sep}}_{K[[\, \underline{t}\, ]] / k} \cong
K[[\, \underline{t}\, ]] \otimes_{k[\, \underline{t}\, ]}
\Omega^{{\textstyle \cdot}}_{k[\, \underline{t}\, ] / k} \]
the homotopy operator (``integration'') of the Poincar\'{e} Lemma
for the graded polynomial algebra $k[\, \underline{t}\, ]$ works here also.
For $K((t))$ (i.e.\ $n = 1$) we have
\[ \Omega^{{\textstyle \cdot}, \operatorname{sep}}_{K((t)) / k} \cong
\Omega^{{\textstyle \cdot}, \operatorname{sep}}_{K[[t]] / k} \oplus
\Omega^{{\textstyle \cdot}, \operatorname{sep}}_{K[t^{-1}] / k} \wedge \operatorname{dlog} t \]
so we have a quasi-isomorphism. For $n > 1$ use
induction on $n$ and the K\"{u}nneth formula.
\end{proof}
\begin{lem} \label{lem5.4}
Suppose $A$ is a local BCA and
$\sigma, \sigma' : K \rightarrow A$ are two coefficient fields. Then
\[ \mrm{H}(\sigma) = \mrm{H}(\sigma') :
\mrm{H} \Omega^{{\textstyle \cdot}, \operatorname{sep}}_{K/k} \rightarrow
\mrm{H} \Omega^{{\textstyle \cdot}, \operatorname{sep}}_{A/k} . \]
\end{lem}
\begin{proof}
Choosing generators for the maximal ideal of $A$, $\sigma$
induces a surjection of BCAs $\tilde{A} = K [[\, \underline{t}\, ]] \rightarrow A$.
Denote by $\tilde{\sigma} : K \rightarrow \tilde{A}$ the inclusion.
The coefficient field $\sigma'$ lifts to some coefficient field
$\tilde{\sigma}' : K \rightarrow \tilde{A}$. It suffices to show that
$\mrm{H}(\tilde{\sigma}) = \mrm{H}(\tilde{\sigma}')$.
But by Lemma \ref{lem5.1} both of these are bijective, and using the
projection $\tilde{A} \rightarrow K$ we see they are in fact equal.
\end{proof}
Given a saturated chain $\xi = (x, \ldots, y)$ in $X$
and a coefficient field
$\sigma: k(y) \rightarrow \mcal{O}_{X, (y)}$, there is the {\em Parshin residue
map}
\[ \operatorname{Res}_{\xi, \sigma}: \Omega^{{\textstyle \cdot}}_{k(x)/k} \rightarrow
\Omega^{{\textstyle \cdot}}_{k(y)/k} \]
(cf.\ \cite{Ye1} Definition 4.1.3). It is a
map of DG $k$-modules of degree equal to $-(\text{length of } \xi)$.
\begin{prop} \label{prop5.2}
Let $\xi = (x, \ldots, y)$ be a saturated chain in $X$. Then
the map of graded $k$-modules
\[ \operatorname{Res}_{\xi} := \mrm{H}(\operatorname{Res}_{\xi, \sigma}) :
\mrm{H} \Omega^{{\textstyle \cdot}}_{k(x)/k} \rightarrow
\mrm{H} \Omega^{{\textstyle \cdot}}_{k(y)/k} \]
is independent of the coefficient field $\sigma$.
\end{prop}
\begin{proof}
Say $\xi$ has length $n$.
Let $L$ be one of the local factors of $k(\xi) = k(x)_{\xi}$,
so it is an $n$-dimensional topological local field (TLF).
Let $K := \kappa_{n}(L)$, the last residue field of $L$,
which is a finite separable $k(y)$-algebra.
Then $\sigma$ extends uniquely to a morphism of TLFs
$\sigma: K \rightarrow L$, and it is certainly enough to check that
\begin{equation} \label{eqn5.4}
\mrm{H}(\operatorname{Res}_{L/K; \sigma}) :
\mrm{H} \Omega^{{\textstyle \cdot}, \mrm{sep}}_{L/k} \rightarrow
\mrm{H} \Omega^{{\textstyle \cdot}, \mrm{sep}}_{K/k}
\end{equation}
is independent of $\sigma$.
After choosing a regular system of parameters
$\underline{t} = (t_{1}, \ldots, t_{n})$ in $L$ we get
$L \cong K((\underline{t}))$. According to Lemma \ref{lem5.1},
$\mrm{H}(\sigma)$ induces an isomorphism of $k$-algebras
\begin{equation} \label{eqn5.5}
\mrm{H} \Omega^{{\textstyle \cdot}}_{K/k} \otimes_{k}
\bigwedge\nolimits_{k}^{{\textstyle \cdot}} T \cong
\mrm{H} \Omega^{{\textstyle \cdot}, \mrm{sep}}_{L / k} .
\end{equation}
But by Lemma \ref{lem5.4} this isomorphism is independent of $\sigma$.
The map (\ref{eqn5.4}) is
$\mrm{H} \Omega^{{\textstyle \cdot}}_{K/k}$-linear, and
it sends $\bigwedge^{p}_{k} T$ to $0$ if $p < n$, and
$\operatorname{dlog} t_{1} \wedge \cdots \wedge \operatorname{dlog} t_{n}
\mapsto 1$. Hence (\ref{eqn5.4}) is independent of $\sigma$.
\end{proof}
The topological space $X$ has an increasing filtration by families of
supports
$\emptyset = X_{-1} \subset X_{0} \subset X_{1} \subset
\cdots$,
where
\[ X_{q} := \{ Z \subset X \mid Z \text{ is closed and }
\operatorname{dim} Z \leq q \} . \]
We write
$x \in X_{q} / X_{q-1}$ if $\overline{\{x\}} \in X_{q} - X_{q-1}$,
and the set $X_{q} / X_{q-1}$ is called the $q$-skeleton of $X$.
(This notation is in accordance with \cite{BlO}; in \cite{Ye1} $X_{q}$
denotes the $q$-skeleton.)
The {\em niveau filtration} on $\mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}}$ is
$\mrm{N}_{q} \mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}} :=
\underline{\Gamma}_{X_{q}} \mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}}$.
Let us write
$X^{q} / X^{q + 1} := X_{-q} / X_{-q - 1}$
and
$\mrm{N}^{q} := \mrm{N}_{-q}$, so
$\{\mrm{N}^{q} \mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}}\}$
is a decreasing filtration.
\begin{thm} \label{thm5.3}
Suppose $\operatorname{char} k = 0$ and $X \subset \mfrak{X}$ is a smooth
formal embedding. Then in the niveau spectral sequence converging to
$\mrm{H}^{{\textstyle \cdot}}(X, \mcal{F}^{{\textstyle \cdot}}_{\mfrak{X}})$,
the $E_{1}$ term is \textup{(}in the notation of \cite{ML} Chapter
\textup{XI):}
\[ E_{1}^{p,q} =
\mrm{H}^{p+q}_{X^{p} / X^{p + 1}}(X, \mcal{F}^{{\textstyle \cdot}}_{\mfrak{X}})
\cong
\bigoplus_{x \in X^{p} / X^{p + 1}}
\mrm{H}^{q - p} \Omega^{{\textstyle \cdot}}_{k(x)/k} , \]
and the coboundary operator is
$(-1)^{p + 1} \sum_{(x,y)} \operatorname{Res}_{(x,y)}$.
\end{thm}
\begin{proof}
We shall substitute indices
$(p, q) \mapsto (-q, -p)$; this puts us in the first quadrant.
Because $\mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}}$ is a complex of flasque
sheaves, one has
\[ E_{1}^{-q, -p} = \mrm{H}^{-p - q}_{X^{-q} / X^{-q + 1}}(X,
\mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}})
\cong
\bigoplus_{x \in X_{q} / X_{q-1}}
\mrm{H}^{-p} \mcal{H}om^{{\textstyle \cdot}}_{\mcal{O}_{\mfrak{X}}} \left(
\widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{X}/k},
\mcal{K}_{\mfrak{X}}(x) \right) \]
(the operator $\delta$ is trivial on the $q$-skeleton).
Fix a point $x$ of dimension $q$ and let
$B := \mcal{O}_{\mfrak{X}, (x)}$. Then
$\widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{X}/k, (x)} \cong
\Omega^{{\textstyle \cdot}, \mrm{sep}}_{B / k}$
and by definition
\[ \mcal{H}om^{{\textstyle \cdot}}_{\mcal{O}_{\mfrak{X}}} \left(
\widehat{\Omega}^{{\textstyle \cdot}}_{\mfrak{X}/k},
\mcal{K}_{\mfrak{X}}(x) \right)
\cong \operatorname{Dual}_{B} \Omega^{{\textstyle \cdot}, \mrm{sep}}_{B / k} . \]
Choose a coefficient field $\sigma: K = k(x) \rightarrow B$.
By \cite{Ye2} Theorem 8.6 there is an isomorphism of complexes
\[ \Psi_{\sigma} :
\operatorname{Dual}_{B} \Omega^{{\textstyle \cdot}, \mrm{sep}}_{B / k} \stackrel{\simeq}{\rightarrow}
\operatorname{Dual}_{\sigma} \Omega^{{\textstyle \cdot}, \mrm{sep}}_{B/k} =
\operatorname{Hom}^{\mrm{cont}}_{K; \sigma}
(\Omega^{{\textstyle \cdot}, \operatorname{sep}}_{B/k}, \omega(K)) . \]
Here $\omega(K) = \Omega^{q}_{K/k}$ and
the operator on the right is $\operatorname{Dual}_{\sigma}(\mrm{d})$ of
\cite{Ye2} Definition 8.1.
According to \cite{Ye3} \S 3, $k \rightarrow B$ is formally smooth; so $B$ is
a regular local ring, and hence
$B \cong K[[\, \underline{t}\, ]]$.
Put a grading on $\Omega^{{\textstyle \cdot}}_{K [\, \underline{t}\, ] / k}$ by
declaring
$\operatorname{deg} t_{i} = \operatorname{deg} \mrm{d} t_{i} = 1$,
and let
$V_{l} \subset \Omega^{{\textstyle \cdot}}_{K [\, \underline{t}\, ] / k}$
be the homogeneous component of degree $l$.
In particular
$V_{0} = \Omega^{{\textstyle \cdot}}_{K / k}$.
Since $\mrm{d}$ preserves each $V_{l}$, from the definition of
$\operatorname{Dual}_{\sigma}(\mrm{d})$ we see that
\[ \operatorname{Dual}_{\sigma} \Omega^{{\textstyle \cdot}, \mrm{sep}}_{B/k} =
\bigoplus_{l=0}^{\infty} \operatorname{Hom}_{K}(V_{l}, \omega(K)) \]
as complexes. Because the $K$-linear homotopy operator in the proof
of Lemma \ref{lem5.1} also preserves $V_{l}$ we get
$\mrm{H} \operatorname{Hom}_{K}(V_{l}, \omega(K)) = 0$
for $l \neq 0$, and hence
\begin{equation} \label {eqn5.2}
\mrm{H}^{-p} \operatorname{Dual}_{\sigma} \Omega^{{\textstyle \cdot}, \mrm{sep}}_{B/k}
\cong \mrm{H}^{-p} \operatorname{Hom}_{K} (\Omega^{{\textstyle \cdot}}_{K/k}, \omega(K))
\cong \mrm{H}^{q-p} \Omega^{{\textstyle \cdot}}_{K/k}
\end{equation}
(cf.\ proof of Lemma \ref{lem4.3}).
It remains to check that the coboundary maps match up.
Given an immediate specialization $(x,y)$, choose a pair of compatible
coefficient fields
$\sigma: k(x) \rightarrow \mcal{O}_{\mfrak{X}, (x)} = B$ and
$\tau: k(y) \rightarrow \mcal{O}_{\mfrak{X}, (y)} = A$ (cf.\ \cite{Ye1}
Definition 4.1.5). Set
$\widehat{B} := \mcal{O}_{\mfrak{X}, (x,y)}$, so
$f : A \rightarrow \widehat{B}$ is a morphism of BCAs. A cohomology class
$[\phi] \in
\mrm{H}^{-p} \operatorname{Dual}_{B} \Omega^{{\textstyle \cdot}, \mrm{sep}}_{B / k}$
is sent under the isomorphism (\ref{eqn5.2})
to the class $[\beta]$ of some form
$\beta \in \Omega^{q-p}_{k(x)/k}$, such that $\mrm{d} \beta = 0$ and on
$\sigma(\Omega^{{\textstyle \cdot}}_{k(x)/k}) \subset \Omega^{{\textstyle \cdot},
\mrm{sep}}_{B/k}$,
$\phi$ acts like left multiplication by $\beta$. So for
$\alpha \in \Omega^{p}_{k(y)/k}$,
\begin{eqnarray*}
\operatorname{Tr}_{A / k(y)} \operatorname{Tr}_{\widehat{B} / A} \phi f \tau (\alpha)
& = & \operatorname{Res}_{k((x,y)) / k(y); \tau}(\beta \wedge \tau(\alpha)) \\
& = & \operatorname{Res}_{k((x,y)) / k(y); \tau}(\beta) \wedge \alpha .
\end{eqnarray*}
This says that under the isomorphism
\[ \mrm{H}^{-p} \operatorname{Dual}_{A} \Omega^{{\textstyle \cdot}, \mrm{sep}}_{A / k}
\cong \mrm{H}^{q-p-1} \Omega^{{\textstyle \cdot}}_{k(y)/k} , \]
the class $\delta_{(x,y)}([\phi])$ is sent to
$\operatorname{Res}_{(x,y)}([\beta])$.
\end{proof}
\begin{rem} Theorem \ref{thm5.3}, but with
$\operatorname{R} \underline{\Gamma}_{X} \Omega^{{\textstyle \cdot}}_{Y / k}$ instead of
$\mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}}$ (cf.\ Proposition \ref{prop5.1}),
was discovered by Grothendieck
(see \cite{Gr} Footnotes 8,9), and proved by Bloch-Ogus \cite{BlO}.
Our proof is completely different from that in \cite{BlO}, and in
particular we obtain the formula for the coboundary operator as a
sum of Parshin residues. On the other hand the proof in
\cite{BlO} is valid
for a general homology theory (including $l$-adic homology).
Bloch-Ogus went on to prove additional important results, such as the
degeneration of the sheafified spectral sequence
$\mcal{E}^{p,q}_{r}$ at $r=2$, for $X$ smooth.
\end{rem}
The next result is a generalization of \cite{Ha} Theorem II.3.2.
Suppose $X \subset \mfrak{Y}$ is another smooth formal embedding. By a
morphism of embeddings $f : \mfrak{X} \rightarrow \mfrak{Y}$ we mean a
morphism of formal schemes inducing the identity on $X$.
Since $f$ is formally finite, according to Proposition \ref{prop5.5},
$\operatorname{Tr}_{f} : \mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}} \rightarrow
\mcal{F}_{\mfrak{Y}}^{{\textstyle \cdot}}$
is a map of complexes in $\mathsf{Ab}(X)$.
\begin{cor} \label{cor5.5}
Let $f : \mfrak{X} \rightarrow \mfrak{Y}$ be a morphism of embeddings of $X$.
Then
$\operatorname{Tr}_{f} : \Gamma(X, \mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}}) \rightarrow
\Gamma(X, \mcal{F}_{\mfrak{Y}}^{{\textstyle \cdot}})$
is a quasi-isomorphism. If $g : \mfrak{X} \rightarrow \mfrak{Y}$
is another such morphism, then
$\mrm{H}(\operatorname{Tr}_{f}) = \mrm{H}(\operatorname{Tr}_{g})$.
\end{cor}
\begin{proof}
$\operatorname{Tr}_{f}$ induces a map of niveau spectral sequences
$E^{p,q}_{r}(\mfrak{X}) \rightarrow E^{p,q}_{r}(\mfrak{Y})$.
The theorem and its proof imply that these spectral
sequences coincide for $r \geq 1$, hence
$\mrm{H}^{{\textstyle \cdot}}(\operatorname{Tr}_{f})$
is an isomorphism.
The other statement is proved like in \cite{Ye3} Theorem 2.7 (cf.\ next
corollary).
\end{proof}
\begin{cor} \label{cor5.1}
The $k$-module
$\mrm{H}^{q}(X, \mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}})$
is independent of the smooth formal embedding $X \subset \mfrak{X}$.
\end{cor}
\begin{proof}
As shown in \cite{Ye3}, given any two embeddings
$X \subset \mfrak{X}$ and $X \subset \mfrak{Y}$, the completion
of their product along the diagonal
$(\mfrak{X} \times_{k} \mfrak{Y})_{/X}$ is also
a smooth formal embedding of $X$, and it projects to both $\mfrak{X}$
and $\mfrak{Y}$. Therefore by Corollary \ref{cor5.5},
$\mrm{H}^{q}(X, \mcal{F}_{\mfrak{X}}^{{\textstyle \cdot}})$ and
$\mrm{H}^{q}(X, \mcal{F}_{\mfrak{Y}}^{{\textstyle \cdot}})$
are isomorphic. Using triple products we see this isomorphism is
canonical.
\end{proof}
\begin{rem} \label{rem5.2}
We can use Corollary \ref{cor5.1} to {\em define}
$\mrm{H}^{\mrm{DR}}_{{\textstyle \cdot}}(X)$ if some smooth formal embedding exists.
For a definition of $\mrm{H}^{\mrm{DR}}_{{\textstyle \cdot}}(X)$
in general, using a system of local embeddings, see \cite{Ye3}
(cf.\ \cite{Ha} pp.\ 28-29).
\end{rem}
\begin{rem} \label{rem5.3}
In \cite{Ye4} it is shown that $\mcal{F}^{{\textstyle \cdot}}_{\mfrak{X}}$
is naturally a DG module over the adele-De Rham complex
$\mcal{A}^{{\textstyle \cdot}}_{\mfrak{X}} =
\underline{\mathbb{A}}^{{\textstyle \cdot}}_{\mrm{red}}(\widehat{\Omega}^{{\textstyle \cdot}}_{X/k})$,
and this multiplication induces
the cap product of
$\mrm{H}_{\mrm{DR}}^{{\textstyle \cdot}}(X)$ on
$\mrm{H}^{\mrm{DR}}_{{\textstyle \cdot}}(X)$.
\end{rem}
The next result is new (cf.\ \cite{BlO} Example 2.2):
\begin{thm} \label{thm5.1}
De Rham homology $\mrm{H}^{\mrm{DR}}_{{\textstyle \cdot}}(-)$
is contravariant w.r.t.\ \'{e}tale morphisms.
\end{thm}
\begin{proof}
The ``topological invariance of \'{e}tale morphisms'' (see
\cite{Mi} Theorem I.3.23) implies that
the smooth formal embedding $X \subset \mfrak{X}$
induces an ``embedding of \'{e}tale sites''
$X_{\mrm{et}} \subset \mfrak{X}_{\mrm{et}}$.
By this we mean that for every \'{e}tale morphism $U \rightarrow X$ there
is some \'{e}tale morphism $\mfrak{U} \rightarrow \mfrak{X}$, unique up to
isomorphism, s.t.\ $U \cong \mfrak{U} \times_{\mfrak{X}} X$
(see \cite{Ye3}). Then $U \subset \mfrak{U}$ is a smooth formal
embedding. From Proposition \ref{prop5.5} we see there is a complex
of sheaves
$\mcal{F}^{{\textstyle \cdot}}_{\mfrak{X}_{\mrm{et}}}$ on $X_{\mrm{et}}$
with
$\mcal{F}^{{\textstyle \cdot}}_{\mfrak{X}_{\mrm{et}}} |_{U} \cong
\mcal{F}^{{\textstyle \cdot}}_{\mfrak{U}} \cong g^{*}
\mcal{F}^{{\textstyle \cdot}}_{\mfrak{X}}$
for every $g : \mfrak{U} \rightarrow \mfrak{X}$ \'{e}tale
(cf.\ \cite{Mi} Corollary II.1.6). But by Corollary \ref{cor5.1},
$\mrm{H}^{\mrm{DR}}_{{\textstyle \cdot}}(U) =
\mrm{H}^{{\textstyle \cdot}}(U, \mcal{F}^{{\textstyle \cdot}}_{\mfrak{U}})$.
\end{proof}
Say $X$ is smooth irreducible of dimension $n$. Define the sheaf
$\mcal{H}^{p}_{\mrm{DR}}$
on $X_{\mrm{Zar}}$ to be the sheafification of the presheaf
$U \mapsto \mrm{H}^{p}_{\mrm{DR}}(U)$. For any point $x \in X$
let $i_{x} : \{x\} \rightarrow X$ be the inclusion.
Let $x_{0}$ be the generic point, so
$X_{n} / X_{n - 1} = \{ x_{0} \}$.
According to \cite{BlO} there is an exact sequence of sheaves
\[ 0 \rightarrow \mcal{H}^{p}_{\mrm{DR}} \rightarrow
i_{x_{0}\, *} \mrm{H}^{p} \Omega^{{\textstyle \cdot}}_{k(x_{0}) / k}
\rightarrow \cdots \rightarrow
\bigoplus_{x \in X_{q} / X_{q - 1}} i_{x*}
\mrm{H}^{p + q - n} \Omega^{{\textstyle \cdot}}_{k(x)/k}
\rightarrow \cdots \]
called the {\em arithmetic resolution}. Observe that this is a flasque
resolution.
\begin{cor} \label{cor5.2}
The coboundary operator in the arithmetic resolution of
$\mcal{H}^{p}_{\mrm{DR}}$ is
\[ (-1)^{q + 1} \sum_{(x,y)} \operatorname{Res}_{(x,y)} \]
where $\operatorname{Res}_{(x,y)}$ is the Parshin residue of Proposition
\textup{\ref{prop5.2}}.
\end{cor}
\begin{proof}
Take $\mfrak{X} = X$ in Theorem \ref{thm5.3}, and use
\cite{BlO} Theorem 4.2.
\end{proof}
\section{The Intersection Cohomology $\mcal{D}$-module of a Curve}
Suppose $Y$ is an $n$-dimensional smooth algebraic variety over
$\mbb{C}$ and $X$ is a subvariety of codimension $d$. Let
$\mcal{H}_{X}^{d} \mcal{O}_{Y}$
be the sheaf of $d$-th cohomology of $\mcal{O}_{Y}$ with support
in $X$. According to \cite{BrKa}, the holonomic $\mcal{D}_{Y}$-module
$\mcal{H}_{X}^{d} \mcal{O}_{Y}$ has a unique simple coherent
submodule $\mcal{L}(X,Y)$, and the De Rham complex
$\operatorname{DR} \mcal{L}(X,Y) =
\mcal{L}(X,Y) \otimes \Omega^{{\textstyle \cdot}}_{Y^{\mrm{an}}}[n]$
is the middle perversity intersection cohomology sheaf
$\mcal{IC}^{{\textstyle \cdot}}_{X^{\mrm{an}}}$.
Here $Y^{\mrm{an}}$ is the associated complex manifold.
The module $\mcal{L}(X,Y)$ was described explicitly using
complex-analytic methods by Vilonen \cite{Vi} and Barlet-Kashiwara
\cite{BaKa}.
These descriptions show that the fundamental class $\mrm{C}_{X/Y}$ lies
in $\mcal{L}(X,Y) \otimes \Omega^{d}_{Y/k}$, a fact proved earlier
by Kashiwara using the Riemann-Hilbert correspondence and the
decomposition theorem of Beilinson-Bernstein-Deligne-Gabber
(see \cite{Br}).
Now let $k$ be any field of characteristic $0$, $Y$ an $n$-dimensional
smooth variety over $k$, and $X \subset Y$ an integral curve with
arbitrary singularities. In this section we give a description of
$\mcal{L}(X,Y) \subset \mcal{H}_{X}^{n-1} \mcal{O}_{Y}$
in terms of algebraic residues.
As references on $\mcal{D}$-modules we suggest
\cite{Bj} and \cite{Bo} Chapter VI.
Denote by $w$ the generic point of $X$.
Pick any coefficient field
$\sigma : k(w) \rightarrow \widehat{\mcal{O}}_{Y, w} = \mcal{O}_{Y, (w)}$.
As in \cite{Hu} Section 1 there is a residue map
\[ \operatorname{Res}^{\mrm{lc}}_{w, \sigma} :
\mrm{H}_{w}^{n-1} \Omega^{n}_{Y/k} \rightarrow
\Omega^{1}_{k(w) / k} \]
(``lc'' is for local cohomology)
defined as follows. Choose a regular system of parameters
$f_{1}, \ldots, f_{n-1}$ in $\mcal{O}_{Y, w}$,
so that
$\mcal{O}_{Y, (w)} \cong k(w)[[f_{1}, \ldots, f_{n-1}]]$.
Then for a generalized
fraction, with $\alpha \in \Omega^{1}_{k(X) / k}$, we have
\[ \operatorname{Res}^{\mrm{lc}}_{w, \sigma}
\gfrac{\sigma(\alpha) \wedge \mrm{d} f_{1} \wedge \cdots \wedge
\mrm{d} f_{n - 1}}
{f_{1}^{i_{1}} \cdots f_{n - 1}^{i_{n - 1}}} =
\begin{cases}
\alpha & \text{if } (i_{1}, \ldots, i_{n-1}) = (1, \ldots, 1) \\
0 & \text{otherwise} .
\end{cases} \]
Let $\pi : \tilde{X} \rightarrow X$ be the normalization,
and let $\tilde{w}$ be the generic point of $\tilde{X}$.
For any closed point $\tilde{x} \in \tilde{X}$ the residue field
$k(\tilde{x})$ is \'{e}tale over $k$, so it lifts into
$\mcal{O}_{\tilde{X}, (\tilde{x})}$. Hence we get canonical
morphisms of BCAs
$k(\tilde{x}) \rightarrow \mcal{O}_{\tilde{X}, (\tilde{x})} \rightarrow
k(\tilde{w})_{(\tilde{x})}$,
and a residue map
\[ \operatorname{Res}_{(\tilde{w}, \tilde{x})} :
\Omega^{1}_{k(w) / k} \rightarrow
\Omega^{1, \mrm{sep}}_{k(\tilde{w})_{(\tilde{x})} / k} \rightarrow
k(\tilde{w}) . \]
Define
\[ \operatorname{Res}^{\mrm{lc}}_{(\tilde{w}, \tilde{x})} :
\mrm{H}_{w}^{n-1} \Omega^{n}_{Y/k}
\xrightarrow{\operatorname{Res}^{\mrm{lc}}_{w, \sigma}}
\Omega^{1}_{k(w) / k}
\xrightarrow{\operatorname{Res}_{(\tilde{w}, \tilde{x})}} k(\tilde{x}) . \]
We shall see later that
$\operatorname{Res}^{\mrm{lc}}_{(\tilde{w}, \tilde{x})}$
is independent of $\sigma$.
Note that
$\mrm{H}_{w}^{n-1} \Omega^{n}_{Y/k} =
(\mcal{H}_{X}^{n-1} \Omega^{n}_{Y/k})_{w}$.
\begin{thm} \label{thm6.6}
Let $x \in X$ be a closed point and let
$a \in (\mcal{H}_{X}^{n-1} \mcal{O}_{Y})_{x}$.
Then
$a \in \mcal{L}(X, Y)_{x}$ iff
$\operatorname{Res}^{\mrm{lc}}_{(\tilde{w}, \tilde{x})}(a \alpha) = 0$
for all $\alpha \in \Omega^{n}_{Y / k, x}$
and $\tilde{x} \in \pi^{-1}(x)$.
\end{thm}
This is our algebraic counterpart of Vilonen's formula in \cite{Vi}.
The proof of the theorem appears later in this section.
Fix a closed point $x \in X$.
Write $B := \mcal{O}_{Y, (w, x)}$
and
$L := \prod_{\tilde{x} \in \pi^{-1}(x)} k(\tilde{x})$.
\begin{lem}
There is a canonical morphism of BCAs $L \rightarrow B$, and
$B \cong L((g))[[f_{1}, \ldots, f_{n - 1}]]$
for indeterminates $g, f_{1}, \ldots, f_{n - 1}$.
\end{lem}
\begin{proof}
Because $\mcal{O}_{\tilde{X}, (\tilde{x})}$ is a regular local ring
we get
$\mcal{O}_{\tilde{X}, (\tilde{x})} \cong k(\tilde{x})[[g]]$.
It is well known (cf.\ \cite{Ye1} Theorem 3.3.2) that
$k(w)_{(x)} = k(w) \otimes \mcal{O}_{X, (x)}
\cong \prod k(\tilde{w})_{(\tilde{x})}$,
hence $k(w)_{(x)} \cong L((g))$.
Choose a coefficient field $\sigma : k(w) \rightarrow \mcal{O}_{Y, (w)}$.
It extends to a lifting
$\sigma_{(x)} : k(w)_{(x)} \rightarrow \mcal{O}_{Y, (w, x)} = B$
(cf.\ \cite{Ye1} Lemma 3.3.9),
and $L \rightarrow B$ is independent of $\sigma$.
Taking a system of regular parameters
$f_{1}, \ldots, f_{n - 1} \in \mcal{O}_{Y, w}$
we obtain the desired isomorphism.
\end{proof}
The BCA $A := \mcal{O}_{Y, (x)}$ is canonically an algebra over
$K := k(x)$, so
there is a morphism of BCAs
$L \otimes_{K} A \rightarrow B$. Define a homomorphism
\[ T_{x} : \mcal{K}(B) \xrightarrow{\mrm{Tr}}
\mcal{K}(L \otimes_{K} A) \cong
L \otimes_{K} \mcal{K}(A) . \]
Since
$A \rightarrow L \otimes_{K} A \rightarrow B$ are topologically \'{e}tale
(relative to $k$), it follows that $T_{x}$
is a homomorphism of $\mcal{D}(A)$-modules.
Define
\[ V(x) := \operatorname{Coker} \left( K \rightarrow L \right) . \]
Observe that $V(x) = 0$ iff $x$ is either a smooth point or a
geometrically unibranch singularity.
We have
$V(x)^{*} \subset L^{*}$,
where
$(-)^{*} := \operatorname{Hom}_{k}(-, k)$.
The isomorphism $L^{*} \cong L$ induced by $\operatorname{Tr}_{L / k}$
identifies
$V(x)^{*} \cong \operatorname{Ker}(L \xrightarrow{\operatorname{Tr}} K)$.
Since
$\Omega^{n}_{Y / k}[n] \rightarrow \mcal{K}^{{\textstyle \cdot}}_{Y}$
is a quasi-isomorphism we get a short exact sequence
\begin{equation} \label{eqn6.1}
0 \rightarrow (\mcal{H}_{X}^{n-1} \Omega^{n}_{Y/k})_{x} \rightarrow \mcal{K}_{Y}(w)
\xrightarrow{\delta\ } \mcal{K}_{Y}(x) \rightarrow 0 .
\end{equation}
Also we see that
$\mcal{K}(A) = \mcal{K}_{Y}(x) \cong
\mrm{H}^{n}_{x} \Omega^{n}_{Y / k}$.
Now
$\mcal{K}_{Y}(w) = \mcal{K}(\mcal{O}_{Y, (w)}) \subset
\mcal{K}(B)$.
Because the composed map
\[ \mcal{K}_{Y}(w) \xrightarrow{T_{x}}
L \otimes_{K} \mcal{K}_{Y}(x) \xrightarrow{\operatorname{Tr}_{L / K} \otimes 1}
\mcal{K}_{Y}(x) \]
coincides with $\delta$,
and by the sequence (\ref{eqn6.1}), we obtain a homomorphism
of $\mcal{D}_{Y, x}$-modules
\begin{equation} \label{eqn6.11}
T_{x} : (\mcal{H}_{X}^{n-1} \Omega^{n}_{Y/k})_{x} \rightarrow
V(x)^{*} \otimes_{K} \mrm{H}^{n}_{x} \Omega^{n}_{Y / k} .
\end{equation}
\begin{thm} \label{thm6.3}
The homomorphism $T_{x}$
induces a bijection between the lattice of nonzero
$\mcal{D}_{Y, x}$-submodules of
$(\mcal{H}_{X}^{n-1} \Omega^{n}_{Y/k})_{x}$
and the lattice of $k(x)$-submodules of $V(x)^{*}$.
\end{thm}
The proof of the theorem is given later in this section.
In order to globalize we introduce the following notation.
Let $Z$ be the reduced subscheme supported on the singular locus
$X_{\mrm{sing}}$, so $\mcal{O}_{Z} =
\prod_{x \in X_{\mrm{sing}}} k(x)$.
Then $\mcal{V} := \bigoplus_{x \in X_{\mrm{sing}}} V(x)$ and
$\mcal{H}_{Z}^{n} \mcal{O}_{Y} =
\bigoplus_{x \in X_{\mrm{sing}}} \mcal{H}_{ \{x\} }^{n} \mcal{O}_{Y}$
are $\mcal{O}_{Z}$-modules.
Using $\Omega^{n}_{Y / k} \otimes$ to switch between left and right
$\mcal{D}_{Y}$-modules, and identifying $V(x)^{*} \cong V(x)$
by the trace pairing, we see that Theorem \ref{thm6.3}
implies
\begin{cor} \label{cor6.10}
The homomorphism of $\mcal{D}_{Y}$-modules
\[ T := \sum_{x} T_{x} :
\mcal{H}_{X}^{n-1} \mcal{O}_{Y} \rightarrow
(\mcal{H}_{Z}^{n} \mcal{O}_{Y}) \otimes_{\mcal{O}_{Z}} \mcal{V} \]
induces a bijection between the lattice of
nonzero coherent $\mcal{D}_{Y}$-submodules
of $\mcal{H}_{X}^{n-1} \mcal{O}_{Y}$
and the lattice of
$\mcal{O}_{Z}$-submodules of $\mcal{V}$.
\end{cor}
Since $\mcal{H}_{ \{x\} }^{n} \mcal{O}_{Y}$
is a simple $\mcal{D}_{Y}$-submodule, as immediate corollaries we get:
\begin{cor} \label{cor6.1}
$\mcal{H}_{X}^{n-1} \mcal{O}_{Y}$ has a unique simple coherent
$\mcal{D}_{Y}$-submodule
\blnk{2mm} \linebreak
$\mcal{L}(X,Y)$, and the sequence
\begin{equation} \label{eqn6.7}
0 \rightarrow \mcal{L}(X,Y) \rightarrow \mcal{H}_{X}^{n-1} \mcal{O}_{Y}
\xrightarrow{T}
(\mcal{H}_{Z}^{n} \mcal{O}_{Y}) \otimes_{\mcal{O}_{Z}} \mcal{V} \rightarrow 0
\end{equation}
is exact.
\end{cor}
\begin{cor} \label{cor6.2}
$\mcal{H}_{X}^{n-1} \mcal{O}_{Y}$ is a simple coherent
$\mcal{D}_{Y}$-module iff
the singularities of $X$ are all geometrically unibranch.
\end{cor}
According to Proposition \ref{prop4.7} the fundamental class
$\mrm{C}_{X/Y}$ is a double cocycle in
$\mcal{H}om(\Omega^{1}_{Y/k}, \mcal{K}^{-1}_{Y})$,
so it determines a class in
$(\mcal{H}_{X}^{n-1} \mcal{O}_{Y}) \otimes_{\mcal{O}_{Y}}
\Omega^{n-1}_{Y/k}$.
\begin{thm} \label{thm6.2}
$\mrm{C}_{X/Y} \in \mcal{L}(X,Y) \otimes_{\mcal{O}_{Y}}
\Omega^{n-1}_{Y/k}$.
\end{thm}
This of course implies that if $\alpha_{1}, \ldots, \alpha_{n}$
is a local basis of $\Omega^{n-1}_{Y/k}$ and
$\mrm{C}_{X/Y} = \sum a_{i} \otimes \alpha_{i}$, then any nonzero
$a_{i}$ generates $\mcal{L}(X,Y)$ as a $\mcal{D}_{Y}$-module.
The proof of the theorem is given later in this section.
\begin{rem}
As the referee points out, when $k = \mbb{C}$, Corollary \ref{cor6.10}
follows easily from the Riemann-Hilbert correspondence.
In that case we may consider the sheaf $\mcal{V}$ on the analytic space
$X^{\mrm{an}}$, given by
$\mcal{V} := \operatorname{Coker}(\mbb{C}_{X^{\mrm{an}}} \rightarrow
\pi^{\mrm{an}}_{*} \mbb{C}_{\tilde{X}^{\mrm{an}}})$.
Now
$\mcal{IC}_{X^{\mrm{an}}} \cong \pi^{\mrm{an}}_{*}
\mbb{C}_{\tilde{X}^{\mrm{an}}}[1]$.
The triangle
$\mcal{V} \rightarrow \mbb{C}_{X^{\mrm{an}}}[1] \rightarrow$ \linebreak
$\mcal{IC}_{X^{\mrm{an}}} \xrightarrow{+1}$
is an exact sequence in the category of perverse sheaves, and
it is the image of (\ref{eqn6.7})
under the functor
$\operatorname{Sol} = \mrm{R} \mcal{H}om_{\mcal{D}_{Y^{\mrm{an}}}}((-)^{\mrm{an}},
\mcal{O}_{Y^{\mrm{an}}}[n])$.
Nonetheless ours seems to be the first purely algebraic proof
Theorem \ref{thm6.3}
and its corollaries (but cf.\ next remark).
\end{rem}
\begin{rem}
When $Y = \mbf{A}^{2}$ (i.e.\ $X$ is an affine plane curve)
and $k$ is algebraically closed,
Corollary \ref{cor6.2} was partially proved
by S.P.\ Smith \cite{Sm}, using the ring structure of $\mcal{D}(X)$.
Specifically, he proved that if $X$ has unibranch singularities, then
$\mcal{H}_{X}^{1} \mcal{O}_{Y}$ is simple.
\end{rem}
\begin{exa} \label{exa6.1}
Let $X$ be the nodal curve in
$Y = \mbf{A}^{2} = \operatorname{Spec} k [s, t]$
defined by
$f = s^{2} (s+1) - t^{2}$, and let $x$ be the origin.
Take
$r := t / s \in \mcal{O}_{Y, w}$, so
$s = (r+1)(r-1)$.
We see that $\tilde{X} = \operatorname{Spec} k[r]$ and
$r + 1, r - 1$ are regular parameters
at $\tilde{x}_{1}, \tilde{x}_{2}$ respectively on $\tilde{X}$.
For any coefficient field $\sigma$,
\[ \operatorname{Res}^{\mrm{lc}}_{w, \sigma}
\gfrac{\mrm{d} s \wedge \mrm{d} t}{f} =
\operatorname{Res}^{\mrm{lc}}_{w, \sigma}
\gfrac{- \mrm{d} (r + 1) \wedge \mrm{d} f}
{(r + 1)(r - 1) f} = \frac{- \mrm{d} (r + 1)}{(r + 1)(r - 1)} \]
and hence
\[ \operatorname{Res}^{\mrm{lc}}_{(\tilde{w}, \tilde{x}_{1})}
\gfrac{\mrm{d} s \wedge \mrm{d} t}{f} =
\operatorname{Res}_{(\tilde{w}, \tilde{x}_{1})}
\frac{- \mrm{d} (r + 1)}{(r + 1)(r - 1)} = 2 . \]
Likewise
$\operatorname{Res}^{\mrm{lc}}_{(\tilde{w}, \tilde{x}_{2})}
\gfrac{\mrm{d} s \wedge \mrm{d} t}{f} = - 2$.
Therefore
$\gfrac{\mrm{d} s \wedge \mrm{d} t}{f} \notin
\mcal{L}(X, Y) \otimes \Omega^{2}_{Y / k}$.
The fundamental class is
$\mrm{C}_{X / Y} = \gfrac{\mrm{d} f}{f}$, and as generator of
$\mcal{L}(X, Y) \otimes \Omega^{2}_{Y / k}$
we may take
$\gfrac{\mrm{d} s \wedge \mrm{d} f}{f}$.
\end{exa}
Before getting to the proofs we need some general results.
Let $A$ be a BCA over $k$.
The fine topology on an $A$-module $M$ is the quotient topology
w.r.t.\ any surjection $\bigoplus A \twoheadrightarrow M$.
The fine topology on $M$ is $k$-linear, making it a topological
$k$-module (but only a semi-topological (ST) $A$-module).
According to \cite{Ye2} Proposition 2.11.c, $A$ is a Zariski ST ring
(cf.\ ibid.\ Definition 1.7). This means that any finitely
generated $A$-module with the fine topology is separated, and any
homomorphism $M \rightarrow N$ between such modules is topologically
strict. Furthermore if $M$ is finitely generated then it is
complete, so it is a complete linearly topologized $k$-vector space in
the sense of \cite{Ko}.
\begin{lem} \label{lem6.3}
Let $A$ be a BCA.
Suppose $M$ is a countably generated ST $A$-module with the fine
topology. Then $M$ is separated, and any submodule $M' \subset M$
is closed.
\end{lem}
\begin{proof}
Write $M = \bigcup_{i=1}^{\infty} M_{i}$ with $M_{i}$ finitely generated.
Suppose we put the fine topology on $M_{i}$. Then each
$M_{i}$ is separated and $M_{i} \rightarrow M_{i+1}$ is strict.
By \cite{Ye1} Corollary 1.2.6 we have $M \cong \lim_{i \rightarrow} M_{i}$
topologically, so by ibid.\ Proposition 1.1.7, $M$ is separated.
By the same token $M / M'$ is separated too, so $M'$ is closed.
\end{proof}
\begin{prop} \label{prop6.1}
Let $A \rightarrow B$ be a morphism of BCAs, $N$ a finitely generated
$B$-module with the fine topology, and $M \subset N$ a finitely
generated $A$-module. Then
the topology on $M$ induced by $N$ equals the fine $A$-module
topology, and $M$ is closed in $N$.
\end{prop}
\begin{proof}
Since $A$ is a Zariski ST ring we may replace $M$ by any
finitely generated $A$ module $M'$, $M \subset M' \subset N$.
Therefore we can assume $N = B M$ and
$M = \bigoplus_{\mfrak{n} \in \operatorname{Max} B} M \cap N_{\mfrak{n}}$.
So in fact we may assume $A,B$ are both local. Like in the proof of
\cite{Ye2} Theorem 7.4 we may further assume that
$\operatorname{res.dim}(A \rightarrow B) \leq 1$.
Put on $M$ the fine $A$-module topology.
Let $\bar{N}_{i} := N / \mfrak{n}^{i} N$ and
$\bar{M}_{i} := M / (M \cap \mfrak{n}^{i} N)$ with the quotient
topologies.
We claim $\bar{M}_{i} \rightarrow \bar{N}_{i}$ is a strict monomorphism.
This is so because as $A$-modules both have the fine topology,
$\bar{M}_{i}$ is finitely generated and $\bar{N}_{i}$ is countably
generated (cf.\ part 1 in the proof of \cite{Ye1} Theorem 3.2.14).
Just as in part 2 of loc.\ cit.\ we get topological isomorphisms
$M \cong \lim_{\leftarrow i} \bar{M}_{i}$ and
$N \cong \lim_{\leftarrow i} \bar{N}_{i}$, so $M \rightarrow N$ is a strict
monomorphism. But $M$ is complete and $N$ is separated, so $M$ must be
closed.
\end{proof}
Given a topological $k$-module $M$ we set
$M^{*} := \mrm{Hom}_{k}^{\mrm{cont}}(M,k)$
(without a topology).
\begin{lem} \label{lem6.2}
Suppose $M$ is a separated topological $k$-module. Then:
\begin{enumerate}
\item For any subset $S \subset M^{*}$ its perpendicular
$S^{\perp} \subset M$ is a closed submodule.
\item Given a closed submodule $M_{1} \subset M$, one has
$M_{1}^{\perp \perp} = M_{1}$.
\item Suppose $M_{1} \subset M_{2} \subset M$ are closed submodules.
Then there is an exact sequence \textup{(}of untopologized
$k$-modules\textup{)}
\[ 0 \rightarrow M_{2}^{\perp} \rightarrow M_{1}^{\perp} \rightarrow (M_{2} / M_{1})^{*} \rightarrow 0
. \]
\end{enumerate}
\end{lem}
\begin{proof}
See \cite{Ko} Section 10.4, 10.8.
\end{proof}
Let $M,N$ be complete separated topological $k$-modules, and
$\langle -,- \rangle : M \times N \rightarrow k$ a continuous pairing.
We say $\langle -,- \rangle$ is a {\em topological perfect pairing}
if it induces isomorphisms $N \cong M^{*}$ and $M \cong N^{*}$.
\begin{prop} \label{prop6.3}
Assume $k \rightarrow A$ is a morphism of BCAs. Then the residue pairing
$\langle -,- \rangle_{A/k} : A \times \mcal{K}(A) \rightarrow k$,
$\langle a, \phi \rangle_{A/k} = \operatorname{Tr}_{A / k}(a \phi)$,
is a topological perfect pairing.
\end{prop}
\begin{proof}
We may assume $A$ is local. Then
$A = \lim_{\leftarrow i} A / \mfrak{m}^{i}$ and
$\mcal{K}(A) = \lim_{i \rightarrow} \mcal{K}(A / \mfrak{m}^{i})$
topologically.
Let $K \rightarrow A$ be a coefficient field, so both $A / \mfrak{m}^{i}$ and
$\mcal{K}(A / \mfrak{m}^{i}) \cong
\mrm{Hom}_{K}(A / \mfrak{m}^{i}, \omega(K))$
are finite $K$-modules with the fine topology. By \cite{Ye1}
Theorem 2.4.22 the pairing is perfect.
\end{proof}
From here to the end of this section we consider an integral curve $X$
embedded as a closed subscheme in a smooth irreducible $n$-dimensional
variety $Y$.
Fix a closed point $x \in X$, and set
$A := \mcal{O}_{Y, (x)}$ and $K := k(x)$.
Choosing a regular system of parameters at $x$, say
$\underline{t} = (t_{1}, \ldots, t_{n})$, allows us to write
$A = K [[\, \underline{t}\, ]]$. Let
$\mcal{D}(A) := \mrm{Diff}^{\mrm{cont}}_{A/k}(A,A)$.
Since both $K [\, \underline{t}\, ] \rightarrow A$ and $\mcal{O}_{Y,x} \rightarrow A$
are topologically \'{e}tale relative to $k$, we have
\[ \mcal{D}(A) \cong A \otimes_{K}
K [ \textstyle{\frac{\partial}{\partial t_{1}}}, \ldots,
\textstyle{\frac{\partial}{\partial t_{n}}}] \cong
A \otimes_{\mcal{O}_{Y,x}} \mcal{D}_{Y,x} \]
(cf.\ \cite{Ye2} Section 4).
Define $B := \mcal{O}_{Y, (w, x)}$. Since $A \rightarrow B$ is topologically
\'{e}tale relative to $k$, we get a $k$-algebra homomorphism
$\mcal{D}(A) \rightarrow \mcal{D}(B)$. In particular, $B$ and
$\mcal{K}(B)$ are $\mcal{D}(A)$-modules.
Define
$L := \prod_{\tilde{x} \in \pi^{-1}(x)} k(\tilde{x})$
as before.
\begin{lem} \label{lem6.1}
The multiplication map
$A \otimes_{K} L \rightarrow B$ is injective. Its image is a
$\mcal{D}(A)$-submodule of $B$. Any $\mcal{D}(A)$-submodule of
$B$ which is finitely generated over $A$ equals $A \otimes_{K} W$
for some $K$-submodule $W \subset L$.
\end{lem}
\begin{proof}
By \cite{Kz} Proposition 8.9, if $M$ is any $\mcal{D}(A)$-module which is
finitely generated over $A$, then $M = A \otimes_{K} W$, where
$W \subset M$ is the $K$-submodule consisting of all elements killed by
the derivations $\frac{\partial}{\partial t_{i}}$.
Note that $\Omega^{1, \mrm{sep}}_{B / k}$ is free with basis
$\mrm{d} t_{1}, \ldots, \mrm{d} t_{n}$.
Thus it suffices to prove that
\begin{equation} \label{eqn6.5}
L = \{ b \in B\ |\ \ \textstyle{\frac{\partial}{\partial t_{1}}} b =
\cdots = \textstyle{\frac{\partial}{\partial t_{n}}} b = 0 \}
= \mrm{H}^{0} \Omega^{{\textstyle \cdot}, \mrm{sep}}_{B / k} .
\end{equation}
We know that
$B \cong L((g))[[f_{1}, \ldots, f_{n - 1}]]$,
so $B$ is topologically \'{e}tale over the polynomial algebra
$k [g, f_{1}, \ldots, f_{n - 1}]$ (relative to $k$),
and hence
$\mrm{d} g, \mrm{d} f_{1}, \ldots$, \linebreak
$\mrm{d} f_{n - 1}$
is also a basis of $\Omega^{1, \mrm{sep}}_{B / k}$.
It follows that
$\mrm{H}^{0} \Omega^{{\textstyle \cdot}, \mrm{sep}}_{B / k} = L$.
\end{proof}
\begin{proof} (of Theorem \ref{thm6.3})\
Set $\mcal{M} := \mcal{H}_{X}^{n-1} \Omega^{n}_{Y/k}$
and define
$M := A \otimes_{\mcal{O}_{Y, x}} \mcal{M}_{x}$.
Tensoring the exact sequence (\ref{eqn6.1}) with $A$ we get an exact
sequence of $\mcal{D}(A)$-modules
\begin{equation} \label{eqn6.2}
0 \rightarrow M \rightarrow \mcal{K}(B) \xrightarrow{\delta\ } \mcal{K}(A) \rightarrow 0 .
\end{equation}
The proof will use repeatedly the residue pairing
$\langle -,- \rangle_{B/k} : B \times \mcal{K}(B) \rightarrow k$.
By definition of $\delta$ (cf.\ Definition \ref{dfn1.1} and \cite{Ye2}
Section 7) we see that $M = A^{\perp}$.
Consider the closed $k$-submodules
$A \subset A \otimes_{K} L \subset B$ (cf.\ Proposition \ref{prop6.1}).
Applying Lemma \ref{lem6.2} to them, and using
$V(x) = L / K$ and $\mcal{K}(A) \cong A^{*}$,
we get an exact sequence of $\mcal{D}(A)$-modules
\[ 0 \rightarrow (A \otimes_{K} L)^{\perp} \rightarrow M \xrightarrow{T'}
V(x)^{*} \otimes_{K} \mcal{K}(A) \rightarrow 0
. \]
Keeping track of the operations we see that in fact
$T' = T_{x}|_{M}$.
Put the fine $A$-module topology on $M$ and $\mcal{K}(A)$, so
$M \rightarrow V(x)^{*} \otimes_{K} \mcal{K}(A)$ is continuous.
By \cite{Ye1} Proposition 1.1.8, $\mcal{M}_{x} \rightarrow M$ is dense.
Since $\mcal{K}(A)$ is discrete we conclude that
$\mcal{M}_{x} \rightarrow V(x)^{*} \otimes_{K} \mcal{K}(A)$
is a surjection of $\mcal{D}_{Y, x}$-modules.
Thus any $K$-module $W \subset V(x)^{*}$ determines a distinct nonzero
$\mcal{D}_{Y, x}$-module $\mcal{N}_{x} \subset \mcal{M}_{x}$.
Conversely, say
$\mcal{N}_{x} \subset \mcal{M}_{x}$
is a nonzero $\mcal{D}_{Y, x}$-module. On any open set
$U \subset Y$ s.t.\ $U \cap X$ is smooth the module
$\mcal{M}|_{U}$ is a simple coherent $\mcal{D}_{U}$-module (by
Kashiwara's Theorem it corresponds to the $\mcal{D}_{X \cap U}$-module
$\Omega^{1}_{(X \cap U) / k}$).
Therefore the finitely generated $\mcal{D}_{Y, x}$-module $C$ defined
by
\begin{equation} \label{eqn6.4}
0 \rightarrow \mcal{N}_{x} \rightarrow \mcal{M}_{x} \rightarrow C \rightarrow 0
\end{equation}
is supported on $\{x\}$. It follows that $C \cong \mcal{K}(A)^{r}$
for some number $r$.
Tensoring (\ref{eqn6.4}) with $A$ we get an exact sequence
of $\mcal{D}(A)$-modules
\[ 0 \rightarrow N \rightarrow M \rightarrow C \rightarrow 0 \]
with $N \subset M \subset \mcal{K}(B)$.
By faithful flatness of $\mcal{O}_{Y, x} \rightarrow A$ we see that
$\mcal{N}_{x} = \mcal{M}_{x} \cap N$.
We put on $M,N$ the topology induced from $\mcal{K}(B)$, and on
$C$ the quotient topology from $M$.
Now $\mcal{K}(B)$ has the fine $A$-module topology and it is countably
generated over $A$ (cf.\ proof of Proposition \ref{prop6.1}), so by Lemma
\ref{lem6.3} both $M,N$ are closed in $\mcal{K}(B)$.
Using Lemma \ref{lem6.2} and the fact that
$M^{\perp} = A$ we obtain the exact sequence
\[ 0 \rightarrow A \rightarrow N^{\perp} \rightarrow C^{*} \rightarrow 0 , \]
with $N^{\perp} \subset B$.
We do not know what the topology on $C$ is; but it is a ST $A$-module.
Hence the identity map $\mcal{K}(A)^{r} \rightarrow C$ is continuous, and it
induces an $A$-linear injection $C^{*} \rightarrow A^{r}$.
Therefore $C^{*}$, and thus also $N^{\perp}$, are finitely generated over
$A$. According to Lemma \ref{lem6.1},
$N^{\perp} = A \otimes_{K} W$ for some $K$-module $W$,
$K \subset W \subset L$. But $N$ is closed, so $N = (N^{\perp})^{\perp}$.
\end{proof}
\begin{proof} (of Theorem \ref{thm6.2})\
For each $\tilde{x} \in \pi^{-1}(x)$ define a homomorphism
\[ T_{(\tilde{w}, \tilde{x})} : \mcal{K}(B) \xrightarrow{\mrm{Tr}}
\mcal{K}(L \otimes_{K} A) \cong
L \otimes_{K} \mcal{K}(A) \rightarrow
k(\tilde{x}) \otimes_{K} \mcal{K}(A) , \]
so
$T_{x} = \sum T_{(\tilde{w}, \tilde{x})}$.
From the proof of Theorem \ref{thm6.3} we see that
the theorem amounts to the claim that
$T_{\tilde{x}} (\mrm{C}_{X/Y}(\alpha)) = 0$ for every
$\tilde{x}$ and $\alpha \in \Omega^{1}_{Y / k, x}$.
But $\mrm{C}_{X/Y}$ is the image of
$\mrm{C}_{X} \in \mcal{H}om(\Omega^{1}_{X/k},
\mcal{K}^{-1}_{X}(w))$,
so we can reduce our residue calculation to the
curve $\tilde{X}$. In fact it suffices to show that for every
$\alpha \in \Omega^{1}_{X / k, x}$
one has
$\operatorname{Res}_{(\tilde{w}, \tilde{x})} \alpha = 0$.
Since
$\alpha \in \Omega^{1}_{\tilde{X} / k, \tilde{x}}$
this is obvious.
\end{proof}
\begin{proof} (of Theorem \ref{thm6.6})\
According to \cite{SY} Corollary 0.2.11 (or \cite{Hu} Theorem 2.2)
one has
\[ \operatorname{Res}^{\mrm{lc}}_{(\tilde{w}, \tilde{x})} =
(1 \otimes \operatorname{Tr}_{A / K}) T_{(\tilde{w}, \tilde{x})} :
\mrm{H}^{n - 1}_{w} \Omega^{n}_{Y / k} \rightarrow k(\tilde{x}) , \]
which shows that
$\operatorname{Res}^{\mrm{lc}}_{(\tilde{w}, \tilde{x})}$
is independent of $\sigma$. Now use Theorem \ref{thm6.3}.
\end{proof}
\begin{prob}
What is the generalization to $\operatorname{dim} X > 1$?
To be specific, assume $X$ has only an isolated singularity at $x$.
Then we know there is an exact sequence
\[ 0 \rightarrow \mcal{L}(X,Y) \rightarrow \mcal{H}_{X}^{d} \mcal{O}_{Y}
\xrightarrow{T}
\mcal{H}_{ \{x\} }^{n} \mcal{O}_{Y} \otimes_{k(x)} V(x) \rightarrow 0 \]
for some $k(x)$-module $V(x)$.
What is the geometric data determining $V(x)$ and $T$?
Is it true that $T = \sum T_{\xi}$, a sum of ``residues''
along chains $\xi \in \pi^{-1}(x)$, for a suitable
resolution of singularities $\pi : \tilde{X} \rightarrow X$?
\end{prob}
|
1998-11-03T02:02:13 | 9602 | alg-geom/9602023 | en | https://arxiv.org/abs/alg-geom/9602023 | [
"alg-geom",
"math.AG"
] | alg-geom/9602023 | Mitchell Rothstein | Mitchell Rothstein | Sheaves with connection on abelian varieties | 31 pages, AMSLaTeX amsart12. Author's address: [email protected] | null | null | null | null | The Fourier-Mukai transform is lifted to the derived category of sheaves with
connection on abelian varieties. The case of flat connections (D-modules) is
discussed in detail.
| [
{
"version": "v1",
"created": "Wed, 28 Feb 1996 19:15:30 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Rothstein",
"Mitchell",
""
]
] | alg-geom | \section{Introduction}\label{introduction}
\bigskip Let $X$ and $Y$ be abelian varieties over an algebraically closed
field
$k$, dual to one another, and let $\text{Mod}({\mathcal{O}}_X)$ and
$\text{Mod}({\mathcal{O}}_Y)$ be their respective categories of quasicoherent
${\mathcal{O}}$-modules. Mukai proved in \cite{Muk} that the derived categories
$D\text{Mod}({\mathcal{O}}_X)$ and
$D\text{Mod}({\mathcal{O}}_Y)$ are equivalent via a transform now known as the Fourier-Mukai transform,
\begin{equation}
{\mathcal{S}}_1({\mathcal{F}})=\alpha_{2*}({\mathcal{P}}\otimes
\alpha_1^*(-1)^*({\mathcal{F}}))\ ,
\end{equation} where
${\mathcal{P}}$ is the Poincar\'e sheaf and $\alpha_1$ and $\alpha_2$ are the
projections from $X\times Y$ to $X$ and $Y$ respectively. A few years
earlier, Krichever
\cite{K} rediscovered a construction due originally to Burchnall and
Chaundy \cite{BC}, by which the affine coordinate ring of a projective curve
minus a point may be imbedded in the ring of formal differential operators
in one variable. The construction involves the choice of a line bundle on
the curve, and Krichever took the crucial step of asking, in the case of a
smooth curve, how the imbedding varies when the line bundle moves linearly
on the Jacobian. The answer is now well-known, that the imbeddings satisfy
the system of differential equations known as the KP-hierarchy. In fact,
the Krichever construction is an instance of the
Fourier-Mukai transform, with the crucial addition that the transformed sheaf is not only an
${\mathcal{O}}_Y$-module but a ${\mathcal{D}}_Y$-module, where ${\mathcal{D}}_Y$ is the sheaf of
linear differential operators on
$Y$, \cite{N1} \cite{N2} \cite{R}.
This example serves as the inspiration for the present work, which is
concerned with the role of the Fourier-Mukai transform\ in the theory of sheaves on $Y$
equipped with a connection. The main point is that in the derived
category, all sheaves on $Y$ with connection are constructed by the Fourier-Mukai transform\ in
a manner directly generalizing the Krichever construction. The connection
need not be integrable, though the paper focuses mostly on that case.
The basic idea is the following. Set
\begin{equation}
{\mathfrak{g}} = H^1(X, {\mathcal{O}})\ .
\end{equation} Then there is a tautological extension
\begin{equation}\label{taut seq} 0 \longrightarrow {\mathfrak{g}}^* \otimes {\mathcal{O}}
\longrightarrow
{\mathcal{E}}
\overset{\mu}{\longrightarrow} {\mathcal{O}} \longrightarrow 0
\end{equation} given by the extension class
$1 \in \text{End}({\mathfrak{g}}^*) = \text{Ext}^1({\mathcal{O}}, {\mathfrak{g}}^*
\otimes {\mathcal{O}})$. Now let ${\mathcal{F}}$ be any quasicoherent sheaf of
${\mathcal{O}}_X$-modules, and tensor the sequence \eqref{taut seq} with ${\mathcal{F}}$:
\begin{equation}\label{split it!} 0 \longrightarrow {\mathfrak{g}}^* \otimes {\mathcal{F}}
\longrightarrow
{\mathcal{E}} \otimes {\mathcal{F}}
\overset{\mu_{{\mathcal{F}}}}{\longrightarrow} {\mathcal{F}} \longrightarrow 0.
\end{equation} We refer to any splitting of sequence \eqref{split it!} as a
{\it splitting on ${\mathcal{F}}$.} Let $\text{Mod}(\o_X)_{sp}$ denote the category
of pairs
$({\mathcal{F}},\psi)$, where ${\mathcal{F}}$ is a sheaf on $X$ and $\psi:{\mathcal{F}}\longrightarrow {\mathcal{E}}\otimes
{\mathcal{F}}$ is a splitting, with the obvious morphisms.
Let $\text{Mod}(\o_Y)_{cxn}$ denote the category of quasicoherent sheaves on $Y$ equipped
with a connection. In section \ref{mainthm} we use the Fourier-Mukai transform\ to establish
an equivalence of bounded derived categories:
\begin{equation}\label{equiv1} D^b\text{Mod}(\o_X)_{sp}\ \ \leftrightarrow\ \ D^b\text{Mod}(\o_Y)_{cxn}\ .
\end{equation} Note that the extension class of
\eqref{split it!} belongs ${\mathfrak{g}}^*\otimes H^1(X,{\mathcal{E}}nd({\mathcal{F}}))$ and is
therefore a linear map
\begin{equation} H^1(X,{\mathcal{O}})\longrightarrow H^1(X,{\mathcal{E}}nd({\mathcal{F}}))\ .
\end{equation} It is easily seen to be the map on cohomology induced by the
map
${\mathcal{O}} \longrightarrow {\mathcal{E}}nd({\mathcal{F}})$. Thus
\begin{Prop}\label{affine space} Given an ${\mathcal{O}}$-module ${\mathcal{F}}$, there is a
splitting on
${\mathcal{F}}$ if and only if the natural map
\begin{equation}
{\mathcal{O}} \longrightarrow {\mathcal{E}}nd({\mathcal{F}})
\end{equation} induces the $0$-map
\begin{equation}\label{map} H^1(X, {\mathcal{O}})\overset{0}{\longrightarrow} H^1(X,
{\mathcal{E}}nd({\mathcal{F}}))\ \ .
\end{equation} Moreover, if (\ref{map}) holds, then the set of such
splittings is an affine space over \hbox{${\mathfrak{g}}^*\otimes H^0(X,\cal
End({\mathcal{F}}))$.}
\end{Prop} The intuitive idea behind the equivalence \eqref{equiv1} is the
following. Let
$g=\text{dim}(Y)$. Let ${\mathcal{U}}_1,...,{\mathcal{U}}_k$ be an affine open cover of
$X$ and for $i=1,...,g$, let $\{c(i)_{m,n}\}\in Z^1(\{{\mathcal{U}}\},{\mathcal{O}})$ be a
1-cocycle, such that the classes $[c(1)],...,[c(g)]$ form a basis for
$H^1(X,{\mathcal{O}})$. Let $\xi_1,...,\xi_g$ denote this basis, and let
$\omega^1,...,\omega^g$ denote the dual basis for
$H^1(X,{\mathcal{O}})^*$. In light of proposition \ref{affine space}, a splitting
on ${\mathcal{F}}$ amounts to a collection of endomorphisms \hbox{$\psi(i)_n\in
\Gamma({\mathcal{U}}_n,{\mathcal{E}}nd( (-1_X)^*({\mathcal{F}}))$} such that
\begin{equation}
\psi(i)_n-\psi(i)_m=\text{multiplication by}\ -c(i)_{n,m} \ \ .\label{cobdry}
\end{equation} Let ${\mathcal{G}}={\mathcal{S}}_1({\mathcal{F}})$. Then the collection of endomorphisms
$\psi=\{\psi(i)_n\}$ endows ${\mathcal{G}}$ with a connection in the following way. For
each
$n$, there is a connection
$\nabla_n$ relative to ${\mathcal{U}}_n$ on
${\mathcal{P}}\vert_{{\mathcal{U}}_n\times Y}$, such that on the overlaps,
\begin{equation}\label{overlaps}
\nabla_n-\nabla_m=c_{nm}\ \ .
\end{equation} Therefore, one gets a connection relative to $X$ on
$\alpha_1^*(-1_X)^*({\mathcal{F}})\otimes{\mathcal{P}}$ by defining
\begin{equation}\label{relcxn}
\nabla(\phi\otimes\sigma)=\phi\otimes\nabla_n(\sigma)+
\sum_i\omega^i\psi(i)_n(\phi)\otimes
\sigma
\end{equation} for $\phi\in(-1_X)^*({\mathcal{F}})$ and $\sigma\in{\mathcal{P}}$. Now one
applies
$\alpha_{2*}$ to produce a connection on ${\mathcal{G}}$.
In the Krichever construction, $X$ is the Jacobian of a smooth curve $C$
with a base point $P$, and ${\mathcal{F}}$ is
${\mathcal{O}}_C(*P)$, regarded as a sheaf on $X$ by the abel map. The case where
$X$ is an arbitrary abelian variety and ${\mathcal{F}}$ is of the form
${\mathcal{G}}\otimes{\mathcal{O}}_X(*D)$ for a coherent sheaf ${\mathcal{G}}$ and an ample hypersurface
$D\subset X$ has been studied in \cite{N1} and \cite{N2}.
Now consider the curvature tensor. To each object
$({\mathcal{F}},\psi)\in \text{Ob Mod}(\o_X)_{sp}$ we associate a section
\begin{equation} [\psi, \psi]\in
\wedge^2 {\mathfrak{g}}^* \otimes End({\mathcal{F}})\ ,
\end{equation} simply by taking the commutator
$[\psi(i)_n,\psi(j)_n]$, which, by \eqref{cobdry}, is independent of the
chart. Applying the Fourier-Mukai transform to morphisms, one has
\begin{equation}
{\mathcal{S}}_1([{\psi,
\psi}])
\in
\wedge^2
{\mathfrak{g}}^* \otimes End({\mathcal{S}}_1({\mathcal{F}}))\ .
\end{equation} Letting $[\nabla,\nabla]$ denote curvature, one has
(Proposition
\ref{curvature}),
\begin{equation}\label{curvature.intro}
{\mathcal{S}}_1([\psi,\psi])= [\nabla_{\psi},\nabla_{\psi}]\ .
\end{equation} In particular, ${\mathcal{S}}_1$ restricts to a functorial
correspondence
\begin{equation} ({\mathcal{F}},\psi)\ \text{with}\ [\psi,\psi]=0\ \ \rightsquigarrow\
\
{\mathcal{D}}\text{-module structure on}\ {\mathcal{S}}_1({\mathcal{F}})\ .
\end{equation} We prove that this also induces an equivalence of bounded
derived categories (theorem \ref{equiv2}).
The main point regarding the integrable case is the following. Let
\begin{equation} X^{\natural} \overset{\pi}{\longrightarrow} X
\end{equation} denote the ${\mathfrak{g}}^*$-principal bundle associated to the extension
${\mathcal{E}}$. It is known that
$X^{\natural}$ is the moduli space of line bundles on $Y$ equipped with an
integrable connection. For a discussion of $X^{\natural}$ in greater generality, see
\cite{Me}, \cite{Ros}, and \cite{S}. Let
${\mathcal{A}}=\pi_*({\mathcal{O}}_{X^{\natural}})$. Since $\pi$ is an affine morphism, the
category of ${\mathcal{O}}_{X^{\natural}}$-modules is equivalent to category of $\cal
A$-modules. Then the subcategory of $\text{Mod}(\o_X)_{sp}$ whose objects satisfy $[\psi,
\psi]=0$ is precisely $\text{Mod}({\mathcal{A}})=\text{Mod}({\mathcal{O}}_{X^{\natural}})$ (Proposition
\ref{curvature}). Thus we have an equivalence of categories
\footnote{The referee informs us that this equivalence also appears in an
unpublished preprint by Laumon \cite{L}}
\begin{equation}\label{equiv2.intro} D^b\text{Mod}({\mathcal{O}}_{X^{\natural}})\ \
\leftrightarrow\
\ D^b\text{Mod}({\mathcal{D}}_Y)\ .
\end{equation}
The outline of the paper is as follows. In section \ref{mainthm} we prove
the basic equivalence theorem. By way of illustration, section
\ref{Mat's thm} offers a new proof of a theorem of Matsushima on vector
bundles with a connection. In section \ref{integrable case} the equality
\eqref{curvature.intro} is established
and
the equivalence
\eqref{equiv2.intro} is proved. Section \ref{some examples} gives some
examples. In particular, we establish the formula
\begin{equation}
\hat{{\mathcal{A}}} = {\mathcal{D}}_{\{0\}\rightarrow Y}\ .
\end{equation}
Sections \ref{duality} and \ref{char var} contain general results
about coherence, holonomicity and the characteristic variety. Section
\ref{krich} illustrates the theory in the case of the Krichever
construction. In sections \ref{nak} and \ref{PDOs} we refine and extend
several results of Nakayashiki on characteristic varieties of BA-modules
and commuting rings of matrix partial differential operators. This last
topic is a natural setting for the further study of integrable systems;
some brief remarks on this relationship are included at the end. Further
applications to nonlinear partial differential equations will appear in a
future work.
\vskip 12pt
\noindent Acknowledgements: The author wishes to thank Robert Varley for
many valuable discussions.
\section{First equivalence theorem}\label{mainthm}
We adopt the following sign conventions for the Fourier-Mukai transform:
\begin{align}
\text{Mod}({\mathcal{O}}_X)&\overset{{\mathcal{S}}_1}\longrightarrow\text{Mod}({\mathcal{O}}_Y)\notag\\
{\mathcal{S}}_1({\mathcal{F}})&=\alpha_{2*}({\mathcal{P}}\otimes
\alpha_1^*(-1)^*({\mathcal{F}}))\ ,
\end{align}
\begin{align}
\text{Mod}({\mathcal{O}}_Y)&\overset{{\mathcal{S}}_2}\longrightarrow\text{Mod}({\mathcal{O}}_X)\notag\\
{\mathcal{S}}_2({\mathcal{G}})&=\alpha_{1*}({\mathcal{P}}\otimes
\alpha_2^*({\mathcal{G}}))\ .
\end{align}
The fundamental result in \cite{Muk} is
\vskip.1in
\begin{Fund} The derived functors
\begin{align} D^b\text{Mod}({\mathcal{O}}_X)&\overset{R{\mathcal{S}}_1}{\longrightarrow}
D^b\text{Mod}({\mathcal{O}}_Y)\notag\\ D^b\text{Mod}({\mathcal{O}}_Y) &\overset{R\cal
S_2}{\longrightarrow} D^b\text{Mod}({\mathcal{O}}_X)\end{align} are defined, and
\begin{align} R{\mathcal{S}}_1 R{\mathcal{S}}_2 &= T^{-g}\\ R{\mathcal{S}}_2
R{\mathcal{S}}_1 &= T^{-g}\ ,
\end{align} where $T$ is the shift autormophism on the derived category,
\begin{equation} (TF)^n = F^{n+1}.\notag
\end{equation}
\end{Fund}
\begin{pf} \cite[p. 156]{Muk} \end{pf}
The other key result of [Muk] for our purposes is that the Fourier-Mukai transform\ exchanges
tensor product and Pontrjagin product. Letting
$\beta_1$ and $\beta_2$ denote the projections from $Y \times Y$ to
$Y$ and denoting the group law on $Y$ by $m$, the Pontrjagin product is
defined by
\begin{equation}
{\mathcal{F}} * {\mathcal{G}} = m_*(\beta^*_2({\mathcal{F}}) \otimes \beta^*_1({\mathcal{G}}))\ .
\end{equation}
\begin{Funda}
\begin{align}R{\mathcal{S}}_2({\mathcal{G}}_1
\overset{R}{\underset{=}{*}} {\mathcal{G}}_2) &= R{\mathcal{S}}_2({\mathcal{G}}_1)
\overset{L}{\underset{=}{\otimes}} RS_2({\mathcal{G}}_2)\label{exch2} \\
R{\mathcal{S}}_1 ({\mathcal{F}}_1
\overset{L}{\underset{=}{\otimes}} {\mathcal{F}}_2) &= T^g(R{\mathcal{S}}_1({\mathcal{F}}_1)
\overset{R}{\underset{=}{*}} R{\mathcal{S}}_1({\mathcal{F}}_2))\ .\label{exch1}
\end{align}
\end{Funda}
\begin{pf} \cite[p. 160]{Muk} \end{pf}
We want to apply this to the following situation.
Let ${\mathcal{I}} \subset {\mathcal{O}}_Y$ be the ideal
sheaf of $0 \in Y$, and let $k(0)$ be the skyscraper sheaf at the origin
with fiber $k$. Then
$R^0{\mathcal{S}}_2(k(0))={\mathcal{O}}_X$ and $R^i{\mathcal{S}}_2(k(0))=0$ for $i>0$. Thus the
Fourier-Mukai transform\ takes the short exact sequence
\begin{equation}\label{nptext} 0 \longrightarrow {\mathfrak{g}}^* \otimes_k k(0)
\longrightarrow
{\mathcal{O}}/{\mathcal{I}}^2
\longrightarrow k(0) \longrightarrow 0
\end{equation}
to a short exact sequence of vector bundles on $X$, and it is easy to see
that it is precisely the sequence \eqref{taut seq}.
Thus by theorem Mukai 2, if ${\mathcal{F}}
\overset{\psi}{\longrightarrow} {\mathcal{E}} \otimes {\mathcal{F}}$ is a splitting on
${\mathcal{F}}$, $\psi$ induces a morphism
\begin{equation} R{\mathcal{S}}_1({\mathcal{F}}) \overset{{\mathcal{S}}_1(\psi)}{\longrightarrow}
T^{-g}({\mathcal{O}}/{\mathcal{I}}^2) * R{\mathcal{S}}_1({\mathcal{F}})\ .
\end{equation} (We identify objects in an abelian category with
complexes concentrated in degree 0.) If we now take $g^{th}$ cohomology, we
get
\begin{equation}\label{The point}
{\mathcal{S}}_1({\mathcal{F}}) \overset{H^g{\mathcal{S}}_1(\psi)}{\longrightarrow}
{\mathcal{O}}/{\mathcal{I}}^2 *
{\mathcal{S}}_1({\mathcal{F}}).
\end{equation} The point is this. If ${\mathcal{G}}$ is any
${\mathcal{O}}_Y$-module, there is a prolongation sequence
\begin{equation}\label{c} 0 \longrightarrow \Omega^1 \otimes_{{\mathcal{O}}}{\mathcal{G}}
\longrightarrow j({\mathcal{G}})
\overset{\nu_{{\mathcal{G}}}}{\longrightarrow} {\mathcal{G}} \longrightarrow 0,
\end{equation} such that a splitting of $\nu_{{\mathcal{G}}}$ is precisely a
connection on
${\mathcal{G}}$.
As a sheaf of abelian groups,
\begin{equation} j({\mathcal{G}}) = {\mathcal{G}} \oplus (\Omega^1 \otimes_{{\mathcal{O}}} {\mathcal{G}})\ ,
\end{equation} with ${\mathcal{O}}$-module structure
\begin{equation} f (\phi, \omega \otimes \psi) =(f\phi, f \omega \otimes
\psi + df \otimes \phi)\ \ .
\end{equation} Thus a connection on ${\mathcal{G}}$ is a splitting of \eqref{c}.
Since $Y$ is an abelian variety, there is a characterization of
$j({\mathcal{G}})$ in terms of the Pontrjagin product.
\begin{Lem} \label{pont} For any ${\mathcal{O}}_Y$-module ${\mathcal{G}}$,
\begin{equation} j({\mathcal{G}}) = ({\mathcal{O}}/{\mathcal{I}}^2) * {\mathcal{G}}\ .
\end{equation}
\end{Lem}
\begin{pf} Let $Y_1\subset Y\times Y$
denote the first order neighborhood of
the diagonal, and let
$\pi_i: Y_1
\longrightarrow Y$, $i = 1, 2$, denote the two projections. Then
\begin{equation}
j({\mathcal{F}})=\pi_{2*}\pi_1^*({\mathcal{F}})\ .
\end{equation}
(This holds for any variety.)
Let $\tilde{Y} = \text{Spec}({\mathcal{O}}/{\mathcal{I}}^2)$, the first order neighborhood of
$0$ in $Y$. Then $Y_1$ may be identified with
$Y \times \tilde{Y}$ in such a way that $\pi_1$ corresponds to
projection onto the first factor and $\pi_2$ corresponds to the group law,
$\tilde m$. Let $\iota:\tilde Y\to Y$ denote the inclusion map. Then
\begin{align}
{\mathcal{O}}/{\mathcal{I}}^2&*{\mathcal{G}}=
m_*(\beta_2^*\iota_*({\mathcal{O}}_{\tilde Y})\otimes\beta_1^*({\mathcal{G}}))\notag\\
&=
m_*(1\times\iota)_*(\tilde\beta_2^*({\mathcal{O}}_{\tilde Y})\otimes
\tilde\beta_1^*({\mathcal{G}}))\notag\\
&=\tilde m_*
\tilde\beta_1^*({\mathcal{G}}))=j({\mathcal{G}})\ .
\end{align}
\end{pf}
Combining this lemma with the map \eqref{The point}, we see that a
splitting on
${\mathcal{F}}$ induces a splitting of the prolongation sequence of ${\mathcal{S}}_1(\cal
F)$, i.e., a connection on ${\mathcal{S}}_1({\mathcal{F}})$. So we have a functor
\begin{equation}
\text{Mod}(\o_X)_{sp} \overset{S_1}{\longrightarrow}
\text{Mod}(\o_Y)_{cxn}\ .
\end{equation} We will check later that this description is equivalent to
the one given in the introduction.
Conversely, if we apply ${\mathcal{S}}_2$ to a splitting of the prolongation
sequence,
${\mathcal{G}}\overset{\tau}{\longrightarrow} j({\mathcal{G}}) =
{\mathcal{O}}/{\mathcal{I}}^2 * {\mathcal{G}}$, theorem Mukai 2 gives a splitting
\begin{equation}
{\mathcal{S}}_2({\mathcal{G}}) \overset{\psi}{\longrightarrow} {\mathcal{E}} \otimes
{\mathcal{S}}_2({\mathcal{G}})\ .
\end{equation} So we have
\begin{equation}
\text{Mod}(\o_Y)_{cxn} \overset{S_2}{\longrightarrow}
\text{Mod}(\o_X)_{sp}\ .
\end{equation}
The categories $\text{Mod}({\mathcal{O}}_X)_{sp}$ and
$\text{Mod}({\mathcal{O}}_Y)_{cxn}$ are abelian. Moreover, objects in either
$\text{Mod}({\mathcal{O}}_X)_{sp}$ or $\text{Mod}({\mathcal{O}}_Y)_{cxn}$ may be resolved by a \v
Cech resolution with respect to an affine open cover of $X$ or $Y$. Thus
the derived functors
\begin{equation} D^b\text{Mod}({\mathcal{O}}_X)_{sp} \overset{RS_1}{\longrightarrow}
D^b\text{Mod}({\mathcal{O}}_Y)_{cxn}
\end{equation}
\begin{equation} D^b\text{Mod}({\mathcal{O}}_Y)_{cxn} \overset{RS_2}{\longrightarrow}
D^b\text{Mod}({\mathcal{O}}_X)_{sp}
\end{equation} exist.
The main result of this section is
\begin{Thm}\label{equiv}
\begin{align} RS_1 RS_2 &= T^{-g} , \\ RS_2 RS_1 &= T^{-g} .
\end{align}
\end{Thm}
\begin{pf} Let $\zeta$ denote the functor $T^g RS_1 RS_2$. Let
$\underline{for}$ denote the forgetful functor from
$D^b\text{Mod}({\mathcal{O}}_Y)_{cxn}$ to $D^b\text{Mod}({\mathcal{O}}_Y)$.
Then
\begin{equation}\underline{for}\ \zeta = \underline{for}
\end{equation}
by
Mukai's theorem. In particular, for any object
$({\mathcal{F}},\nabla)
\in \text{Ob Mod}(\o_Y)_{cxn}$, $H^i\zeta({\mathcal{F}},\nabla) = 0$ for
$i > 0$. Thus
$\zeta({\mathcal{F}},\nabla)=({\mathcal{F}},\nabla')$ for some new connection $\nabla'$.
Let ${\mathcal{F}} \overset{\tau}{\longrightarrow} j({\mathcal{F}})$ denote the splitting
associated to $\nabla$, and let ${\mathcal{F}}
\overset{\tau'}{\longrightarrow} j({\mathcal{F}})$ denote the splitting associated
to $\nabla'$. Let $ \psi$ denote the corresponding splitting on
${\mathcal{S}}_2({\mathcal{F}})$. Then
$\psi$ is the
$0^{th}$ cohomology of
\begin{equation} R{\mathcal{S}}_2({\mathcal{F}}) \overset{R\cal
S_2(\tau)}{\longrightarrow} {\mathcal{E}}
\overset{L}{\underset{=}{\otimes}} R{\mathcal{S}}_2({\mathcal{F}})\ ,
\end{equation} from which it follows that $\tau'$ is the $0^{th}$ cohomology
of
\begin{equation}
\begin{CD}
{\mathcal{F}} @>{T^gR{\mathcal{S}}_1 R{\mathcal{S}}_2(\tau)}>> j({\mathcal{F}})\ .
\end{CD}
\end{equation} Thus $\tau = \tau'$, again by Mukai's Theorem.
Similarly, if $({\mathcal{G}}, \psi)$ is an object in $\text{Mod}({\mathcal{O}}_X)_{sp}$,
\begin{equation} T^gRS_2 RS_1({\mathcal{G}}, \psi) = ({\mathcal{G}}, \psi)\ .
\end{equation}
The next lemma then completes the proof
of the theorem.
\end{pf}
\begin{Lem} \label{bootstrap} Let $C_1$ and $C_2$ be abelian categories,
and let
\begin{equation} D^bC_1 \underset{F_2}{\overset{F_1}{\rightarrow}}
D^bC_2
\end{equation}
be $\delta$-functors. If $F_1$ and $F_2$ are isomorphic when restricted
to the subcategory $C_1\subset D^bC_1$, then they are isomorphic.
\end{Lem}
\begin{pf} This follows by induction on the cohomological length of an object
in the bounded derived category, using \cite[lemme 12.6, p.104]{Bo} and the
triangle axiom TR3, \cite[p.28]{Bo}, once it is noted that the
constructions used there are functorial.
\end{pf}
\noindent{\bf Remark}\ \ Let
$\gamma_i$, $\gamma_{i,j}$ denote the projections on $X\times Y\times Y$.
The key to Mukai's theorem is the elementary formula
\begin{equation} \label{key}
\gamma^*_{1,2} ({\mathcal{P}}) \otimes \gamma^*_{1,3}({\mathcal{P}}) = (1\times
m)^*({\mathcal{P}})\ .
\end{equation}
This formula also plays a crucial but hidden role in theorem \ref{equiv},
which we would like to make explicit.
Let $\tilde{Y} = \text{Spec}({\mathcal{O}}/{\mathcal{I}}^2)$. Then ${\mathcal{P}}|_{X \times
\tilde{Y}}$ is a line bundle on $X \times \tilde{Y}$, trivial on $X \times
\{0\}$. Set
$\tilde{{\mathcal{P}}} = {\mathcal{P}}|_{X \times \tilde{Y}}$, and let
$\tilde{\alpha}_1: X \times \tilde{Y} \rightarrow X$ be the projection.
Then ${\mathcal{E}} = \tilde{\alpha}_{1*}(\tilde{{\mathcal{P}}})$ and $\mu: {\mathcal{E}} \rightarrow
{\mathcal{O}}$ is the morphism which restricts a section of $\tilde{{\mathcal{P}}}$ to $X \times
\{0\}$.
Let
$\tilde\gamma_i$, $\tilde\gamma_{i,j}$ denote the projections on $X\times
Y\times \tilde Y$. Then we get an infinitesimal form of \ref{key},
\begin{equation}\label{key2}
\tilde{\gamma}^*_{1,2} ({\mathcal{P}}) \otimes
\tilde{\gamma}^*_{1,3}(\tilde{{\mathcal{P}}}) = (1\times \tilde m)^*({\mathcal{P}}) \ .
\end{equation}
If ${\mathcal{G}}$ is a sheaf on $X\times Y$, then a connection on
${\mathcal{G}}$ relative to $X$ is an
isomorphism
\begin{equation}
\tilde{\gamma}_{1,2}^*({\mathcal{G}}) \approx (1\times \tilde m)^*({\mathcal{G}})
\end{equation} restricting to the identity on $X \times Y$. Thus
\eqref{key2} says that
$\tilde{\gamma}^*_{1,3}(\tilde{{\mathcal{P}}})$ is the obstruction to endowing
${\mathcal{P}}$ with a connection relative to $X$. Given
a sheaf ${\mathcal{F}}$ of ${\mathcal{O}}_X$-modules, a splitting on ${\mathcal{F}}$ is
precisely what is needed to cancel this
obstruction. Indeed, a splitting may be regarded as an isomorphism
\begin{equation}
\tilde{\alpha}_1^*({\mathcal{F}})\overset{\psi}{\to}
\tilde{{\mathcal{P}}}\otimes\tilde{\alpha}_1^*({\mathcal{F}})
\end{equation}
restricting to the identity on $X$.
Applying $(-1_X,-1_{\tilde Y})$, we get
\begin{equation}
\tilde\alpha_1^*(-1)^*({\mathcal{F}})\longrightarrow
\tilde{{\mathcal{P}}}\otimes\tilde\alpha_1^*(-1)^*({\mathcal{F}})\ \ .
\end{equation}
If we then apply $\tilde{\gamma}_{1,2}^*$ to
${\mathcal{P}}\otimes\alpha_1^*(-1)^*({\mathcal{F}})$ we find
\begin{align}
\tilde\gamma_{1,2}^*({\mathcal{P}}\otimes&\alpha_1^*(-1)^*({\mathcal{F}}))=
\tilde\gamma_{1,2}^*({\mathcal{P}})\otimes\tilde\gamma_{1,3}^*
\tilde\alpha_1^*(-1)^*({\mathcal{F}})\notag\\ &\approx
\tilde\gamma_{1,2}^*({\mathcal{P}})\otimes\tilde\gamma_{1,3}^*(\tilde{{\mathcal{P}}})
\otimes\tilde\gamma_{1,3}^*\tilde\alpha_1^*(-1)^*({\mathcal{F}})
\notag\\
&=(1\times\tilde
m)^*({\mathcal{P}})\otimes\tilde\gamma_{1,3}^*\tilde\alpha_1^*(-1)^*({\mathcal{F}})\notag\\
&=(1\times\tilde m)^*({\mathcal{P}}\otimes\alpha_1^*(-1)^*({\mathcal{F}}))\ \ ,
\label{cxnalt}
\end{align}
which is a relative connection on ${\mathcal{P}}\otimes\alpha_1^*(-1)^*({\mathcal{F}})$. Then
apply $\alpha_{2*}$ to get a sheaf with connection on $Y$, and this is our
functor $S_1$.
\section{Matsushima's Theorem}\label{Mat's thm}
As an application of theorem
\ref{equiv}, we will give a new proof of Matsushima's Theorem
on the homogeneity of vector bundles admitting a connection \cite{Mat}.
The key is the following lemma, which is of interest in
its own right. Here we assume $char(k)=0$.
\begin{Lem} \label{fin supp} Let ${\mathcal{F}}$ be a coherent ${\mathcal{O}}_X$-module with a
splitting. Then ${\mathcal{F}}$ is finitely supported.
\end{Lem}
\begin{pf} The proof is similar to that of lemma 3.3 in \cite{Muk}.
Assuming $\text{dim(supp}({\mathcal{F}}))>0$, let $C$ be a curve contained in
$\text{supp}({\mathcal{F}})$, and let $\tilde C\overset{\pi}\longrightarrow C$ be its
normalization. Let ${\mathcal{F}}'=\pi^*({\mathcal{F}})$. Then we get a non-zero vector bundle
${\mathcal{F}}''$ on
$\tilde C$ upon taking the quotient of ${\mathcal{F}}'$ by its torsion part. Let
${\mathcal{E}}'$ denote the pullback of ${\mathcal{E}}|_C$ to $\tilde C$. Then any
splitting on ${\mathcal{F}}$ induces a splitting
\begin{equation}
{\mathcal{F}}'\overset{\psi'}\longrightarrow {\mathcal{E}}'\otimes {\mathcal{F}}'\ .
\end{equation}
Since $Tor( {\mathcal{E}}'\otimes {\mathcal{F}}')=
{\mathcal{E}}'\otimes Tor({\mathcal{F}}')$, we get a splitting of the sequence
\begin{equation}\label{can't split}
0 \longrightarrow {\mathfrak{g}}^* \otimes {\mathcal{F}}''
\longrightarrow
{\mathcal{E}}' \otimes {\mathcal{F}}''
\longrightarrow {\mathcal{F}}'' \longrightarrow 0.
\end{equation}
But the extension class of the sequence \eqref{can't split} is the bottom
arrow in the commutative diagram
\begin{equation}
\begin{matrix} H^1(\tilde{C}, {\mathcal{O}}) \\ a\nearrow \qquad\qquad \searrow b
\quad \\ H^1(X, {\mathcal{O}}) \overset{e}{\longrightarrow} H^1(\tilde{C},
{\mathcal{E}}nd({\mathcal{F}}''))
\end{matrix}\ ,
\end{equation}
where $b$ is induced by the natural inclusion ${\mathcal{O}}\overset{\beta}\longrightarrow \cal
End({\mathcal{F}}'')$ and
$a$ is the derivative of the natural map $Pic(X)\to Pic(\tilde C)$. In
particular, $a$ is not the $0$-map. Moreover, in characteristic $0$,
$\beta$ splits by the trace, so $b$ is injective. This is a contradiction,
for now $e\ne 0$ so that \eqref{can't split} does not split. \end{pf}
We then have
\begin{Thm}[Matsushima] Any vector bundle on $Y$ admitting a connection is
homogeneous.
\end{Thm}
\begin{pf} Let ${\mathcal{G}}$ be such a vector bundle. Then the cohomology sheaves
$R^i{\mathcal{S}}_2({\mathcal{G}})$ are ${\mathcal{O}}_X$-coherent and admit splittings. By lemma
\ref{fin supp}, they are all finitely supported. As in
\cite[example 3.2, p. 158]{Muk}, we then have $R^i{\mathcal{S}}_2({\mathcal{G}})=0$ for
$i\ne g$, and ${\mathcal{G}}$ is then the Fourier-Mukai transform\ of a finitely supported sheaf. By
\cite[3.1]{Muk}, ${\mathcal{G}}$ is homogeneous. \end{pf}
\noindent{\bf Remark}\ \ The converse of the statement above is also part
of Matsushima's theorem. That the converse can be proved by the Fourier-Mukai transform\
is already noted in \cite[prop 5.9]{N2}.
\section{Curvature tensor and the integrable case}\label{integrable case}
If $({\mathcal{G}},\nabla)$ is a sheaf with connection on $Y$, then its curvature
tensor is a linear map
\begin{equation}
[\nabla,\nabla]:\wedge^2({\mathfrak{g}})\longrightarrow End({\mathcal{G}})\ \ .
\end{equation}
Before explaining how the curvature can be read off from the
transform of ${\mathcal{G}}$, we want to show that the functor $S_1$ has the
\v Cech description given in the introduction.
Let $\psi_1$ and $\psi_2$ be two splittings on a given
sheaf of ${\mathcal{O}}_X$-modules ${\mathcal{F}}$, and let $\eta=
\psi_1-\psi_2$. By proposition \ref{affine space}, $\eta$ is a map
\begin{equation}
\eta:{\mathfrak{g}}\longrightarrow End({\mathcal{F}})\ .
\end{equation}
Applying the Fourier-Mukai transform\ to $End({\mathcal{F}})$, we get
\begin{equation}
{\mathcal{S}}_1(\eta):{\mathfrak{g}}\longrightarrow End({\mathcal{S}}_1({\mathcal{F}}))\ .
\end{equation}
Denoting the two connections on ${\mathcal{S}}_1({\mathcal{F}})$ by $\nabla_1$ and
$\nabla_2$, it is easy to check
that
\begin{equation}\label{compat}
\nabla_2=
\nabla_1-{\mathcal{S}}_1(\eta)\ .
\end{equation}
Now let ${\mathcal{U}}\overset{\iota}\longrightarrow X$ be an affine open subset,
equipped with a section $\rho\in\Gamma({\mathcal{U}},{\mathcal{E}})$ such that
$\mu(\rho)=1$. Then $(\iota_*({\mathcal{O}}_{{\mathcal{U}}}),\rho)$ is an object in
$\text{Mod}(\o_X)_{sp}$. The corresponding object in $\text{Mod}(\o_Y)_{cxn}$ is a
connection on $\alpha_{2*}({\mathcal{P}}|_{{\mathcal{U}}\times Y})$. Since functions on
${\mathcal{U}}$ act as endomorphisms of $(\iota_*({\mathcal{O}}_{{\mathcal{U}}}),\rho)$, this
connection is linear over $\alpha_{2*}\alpha_1^{-1}({\mathcal{O}}_{{\mathcal{U}}})$. So we in
fact have a connection relative to ${\mathcal{U}}$ on ${\mathcal{P}}|_{{\mathcal{U}}\times Y}$. Call
this connection $\nabla^{\rho}$. If ${\mathcal{F}}$ is any ${\mathcal{O}}_X$-module, $\rho$
induces a splitting on ${\mathcal{F}}|_{{\mathcal{U}}}$. Then if $\psi$ is any other
splitting on ${\mathcal{F}}$, it must take the form
\begin{equation}
\psi(\kern .5em\cdot\kern .5em)=\kern .5em
\cdot
\kern .5em\otimes\rho+
\sum\limits_{i=1}^g\omega^i
\psi(i)^{\rho}(\kern .5em\cdot\kern .5em)\ ,
\end{equation}
for some endomorphisms $\psi(i)^{\rho}$. Again the corresponding sheaf
with connection on $Y$ is the direct image of a sheaf with relative
connection on $X\times Y$, and by \eqref{compat} it has the
form of \eqref{relcxn}.
It is now easy to read off the curvature tensor of
$S_1({\mathcal{F}},\psi)$.
There is an exact sequence
\begin{align}
0 \longrightarrow \wedge^2{\mathfrak{g}}^* \otimes {\mathcal{O}} &\longrightarrow
\wedge^2 {\mathcal{E}} \overset{\mu_2}{\longrightarrow} {\mathfrak{g}}^*
\otimes {\mathcal{O}} \longrightarrow 0,\notag\\
\mu_2(\rho_1 \wedge \rho_2) &= \mu(\rho_1) \rho_2 - \mu(\rho_2) \rho_1.
\end{align}
If we iterate $\psi$ and then skew-symmetrize, we get a map
\begin{equation}
{\mathcal{F}} \overset{[\psi, \psi]}{\longrightarrow} \wedge^2({\mathcal{E}})
\otimes{\mathcal{F}}.
\end{equation} One sees easily that $(\mu_2 \otimes 1) \circ [\psi,
\psi] = 0$, so
\begin{equation}
[\psi,
\psi]
\in \wedge^2 {\mathfrak{g}}^*
\otimes End({\mathcal{F}})\ .\end{equation}
As noted in the introduction,
in terms of the family of endomorphisms $\psi(i)_n$, one
simply has
\begin{equation}\label{bracket}
[\psi, \psi]=\sum \omega^i\wedge\omega^j[\psi(i)_n,\psi(j)_n]\ .
\end{equation}
Applying the functor ${\mathcal{S}}_1$ to morphisms, we get
${\mathcal{S}}_1([\psi,
\psi])
\in
\wedge^2
{\mathfrak{g}}^* \otimes End({\mathcal{S}}_1({\mathcal{F}}))$.
Set $S_1({\mathcal{F}},\psi)=({\mathcal{S}}_1({\mathcal{F}}),\nabla_{\psi})$. Let
$[\nabla,\nabla]$ denote curvature.
\begin{Prop}\label{curvature} $[\nabla_{\psi},\nabla_{\psi}]=\cal
S_1([\psi,\psi])$.
\end{Prop}
\begin{pf} First we claim that the relative connections $\nabla_n$ on
${\mathcal{P}}\vert_{{\mathcal{U}}_n\times Y}$ are integrable. Indeed, from \eqref{overlaps},
the curvature tensor of $\nabla_n$ is independent of the index $n$.
To show
that it is 0, we have only to consider the case where $0 \in {\mathcal{U}}_n$
and note
that all connections on the trivial bundle ${\mathcal{O}}_Y$ are integrable.
Denoting the relative connection on ${\mathcal{P}}\otimes\alpha^*_1(-1_X)^*({\mathcal{F}})$
by
$\nabla=\sum \omega^i \nabla(i)$,
\begin{align}
[\nabla(i), \nabla(j)]&=[\nabla(i)_n\otimes 1 - 1\otimes\psi(i)_n,
\nabla(j)_n\otimes 1 - 1\otimes\psi(j)_n]\notag\\
&=1\otimes[\psi(i)_n,
\psi(j)_n]\ \ .
\end{align}
\end{pf}
Similarly, given $({\mathcal{G}},\nabla)$ in $\text{Ob Mod}(\o_Y)_{cxn}$, we have $\cal
S_2([\nabla,\nabla])\in
\wedge^2({\mathfrak{g}}^*)\otimes End({\mathcal{S}}_2({\mathcal{G}}))$. Setting
$S_2({\mathcal{G}},\nabla)=({\mathcal{S}}_2({\mathcal{G}}),\psi_{\nabla})$,
one proves
\begin{Prop} $[\psi_{\nabla},\psi_{\nabla}]={\mathcal{S}}_2([\nabla,\nabla])$.
\end{Prop}
Turning now to the integrable case,
consider the group extension
\begin{equation}
0 \longrightarrow {\mathfrak{g}}^* \longrightarrow
X^{\natural}
\overset\pi{\longrightarrow} X \longrightarrow 0
\end{equation}
mentioned in the introduction.
The morphism $\pi$ being affine, $X^{\natural}$ is characterized by $\pi_*$ of its
structure sheaf. Define a sheaf
${\mathcal{A}}$ of ${\mathcal{O}}_X$-modules as follows.
For each affine open ${\mathcal{U}} \subset X$ and each
$\rho
\in
{\mathcal{E}}({\mathcal{U}})$ such that $\mu(\rho) = 1$, introduce independent variables
$x^{\rho}_1, \cdots, x^{\rho}_g$. Then
\begin{equation}
{\mathcal{A}}|_{{\mathcal{U}}}=
{\mathcal{O}}|_{{\mathcal{U}}}[x^{\rho}_1, \cdots, x^{\rho}_g]\ .
\end{equation}
Glue these sheaves together by the rule that if
$\tilde{\rho} =\rho + \operatornamewithlimits\sum\limits^g_{i=1}
\omega^i f_i$, then
\begin{equation}
x^{\rho}_i = x^{\tilde{\rho}}_i + f_i
\end{equation} as sections of ${\mathcal{A}}$.
\begin{Def} $X^{\natural}=\text{Spec}({\mathcal{A}})$.
\end{Def}
\begin{Prop}\label{subcat} The full subcategory of
$\text{Mod}(\o_X)_{sp}$ whose objects $({\mathcal{F}}, \psi)$ satisfy $[\psi,
\psi] = 0$ is canonically isomorphic to the category of ${\mathcal{O}}$-modules
on $X^{\natural}$, which is to say the category of ${\mathcal{A}}$-modules on $X$.
\end{Prop}
\begin{pf} Let
$\rho$ be as above. Since ${\mathcal{A}}$ is locally ${\mathcal{O}}[x^{\rho}_1,
\dots, x^{\rho}_g]$, to give an ${\mathcal{A}}$-module structure to an $\cal
O_X$-module ${\mathcal{F}}$ is to choose, for every $\rho$, a set of commuting
endomorphisms $\psi^{\rho}_1,
\dots, \psi^{\rho}_g \in\Gamma({\mathcal{U}},{\mathcal{E}}nd({\mathcal{F}}))$, such that if
\begin{equation}
\tilde{\rho} = \rho + \sum \omega^i f_i\ ,
\end{equation}
then $\psi^{\rho}_i - \psi^{\tilde{\rho}}_i=$
multiplication by
$f_i$. This is clearly the same as lifting ${\mathcal{F}}$ to an object
$({\mathcal{F}}, \psi) \in \text{Ob Mod}(\o_X)_{sp}$ such that $[\psi, \psi] = 0$.
\end{pf}
Now we may restrict our functors $S_1$ and $S_2$ to the subcategories
$\text{Mod}({\mathcal{O}}_{X^{\natural}})\subset\text{Mod}(\o_X)_{sp}$ and
$\text{Mod}({\mathcal{D}}_Y)\subset\text{Mod}(\o_Y)_{cxn}$ respectively, to get functors
\begin{align}
\text{Mod}({\mathcal{O}}_{X^{\natural}})&\overset{S_1}\longrightarrow \text{Mod}(\cal
D_Y)\notag\\
\text{Mod}({\mathcal{D}}_Y)&\overset{S_2}\longrightarrow\text{Mod}({\mathcal{O}}_{X^{\natural}})\ \ .
\end{align}
We then get our second equivalence theorem, whose proof is the same as that
of the first.
\begin{Thm}\label{equiv2}
The derived functors
\begin{equation} D^b\text{Mod}({\mathcal{O}}_{X^{\natural}})
\overset{RS_1}{\longrightarrow} D^b\text{Mod}({\mathcal{D}}_Y)
\end{equation}
\begin{equation} D^b\text{Mod}({\mathcal{D}}_Y)
\overset{RS_2}{\longrightarrow} D^b\text{Mod}({\mathcal{O}}_{X^{\natural}})
\end{equation} exist, and satisfy
\begin{align} RS_1 RS_2 &= T^{-g} , \notag\\ RS_2 RS_1 &= T^{-g} .\notag
\end{align}
\end{Thm}
\section{Some examples}\label{some examples}
\subsection{} Using theorem \ref{equiv2}, one easily recovers the
observation that $X^{\natural}$ is the moduli space of degree-0 line bundles on $Y$
equipped with a connection \cite{Me}.
Indeed, let $\sigma:Z\to X^{\natural}$ be any morphism and let
$\sigma'=\pi\circ\sigma$. Let ${\mathcal{P}}_{\sigma'}$ be the line bundle
on $Z\times Y$ induced by $\sigma'$. Then the ${\mathcal{D}}_Y$-module
$S_1(\sigma_*({\mathcal{O}}_Z))$
is in fact the direct image of a flat connection relative to $Z$ on
${\mathcal{P}}_{\sigma'}$.
In particular, we have the 1:1 correspondence
$$\text{ points of}\ X^{\natural} \leftrightarrow \text{line bundles with
connection on}\ Y\ .$$
\subsection{} Let $J\subset\text{Sym}({\mathfrak{g}})$ be
an ideal. Then $\text{Spec}(\text{Sym}({\mathfrak{g}})/J)$ is a subvariety of the
fiber $\pi^{-1}(0)$, and we may apply $S_1$ to its structure sheaf.
This gives a ${\mathcal{D}}$-module
structure on ${\mathcal{O}}_Y\otimes_k\text{Sym}({\mathfrak{g}})/J$. Writing sections of ${\mathcal{D}}$
in the standard format with functions on the left and invariant
differential operators on the right, make the identification
\begin{equation}
{\mathcal{D}}\simeq{\mathcal{O}}_Y\otimes_k\text{Sym}({\mathfrak{g}})\ .
\end{equation}
Then ${\mathcal{O}}_Y\otimes_kJ$ is a left ideal, and $S_1(
k(0)\otimes_k\text{Sym}({\mathfrak{g}})/J)$ is the quotient ${\mathcal{D}}$-module. In
particular,
\vskip 12pt
\noindent ${\mathcal{D}}_Y$ is the Fourier-Mukai transform\ of the structure sheaf of the subgroup
${\mathfrak{g}}^* =\pi^{-1}(0)\subsetX^{\natural}$.
\subsection{}\label{B itself} Going the other way, we may
ask which ${\mathcal{D}}_Y$-module transforms to ${\mathcal{A}}$. By theorem \ref{equiv2}
this is the same as asking for the Fourier-Mukai transform\ of
${\mathcal{A}}$. Observe first that there is an important filtration $\{\cal
A(m)\}$ on
${\mathcal{A}}$ coming from the local identifications of
${\mathcal{A}}$ with a polymonial algebra over ${\mathcal{O}}$. One has
exact sequences
\begin{equation}\label{filtration}
0 \rightarrow {\mathcal{A}}(m) \rightarrow {\mathcal{A}}(m+1)
\rightarrow Sym^{m+1}({\mathfrak{g}}) \otimes_k {\mathcal{O}}_X \rightarrow 0\ \ .
\end{equation}
We then have
\begin{Lem}
$R^i{\mathcal{S}}_1({\mathcal{A}})=0$ for $i\ne g$, and
$R^g{\mathcal{S}}_1({\mathcal{A}})$ is supported at the origin.\end{Lem}\begin{pf}
By \eqref{filtration} the question reduces to the corresponding assertion
about
${\mathcal{O}}$. The latter is proved in \cite[p.126]{Mum}.\end{pf}
It remains to identify the ${\mathcal{D}}_Y$-module $R^gS_1({\mathcal{A}})$.
If ${\mathcal{G}}$ is an ${\mathcal{O}}_Y$-module, we may form ${\mathcal{D}}
\otimes_{{\mathcal{O}}} {\mathcal{G}}$, using right multiplication by functions as the
${\mathcal{O}}$-module structure on ${\mathcal{D}}$. Then ${\mathcal{D}} \otimes_{{\mathcal{O}}} {\mathcal{G}}$ has a left
${\mathcal{D}}$-module structure. In particular, define
\begin{equation}
{\mathcal{D}}_{\{0\}\rightarrow Y}\overset{def}={\mathcal{D}}\otimes k(0)\ .
\end{equation}
\begin{Prop}\label{fm of B} $R^gS_1({\mathcal{A}}) = {\mathcal{D}}_{\{0\}\rightarrow Y}$.
\end{Prop}
\begin{pf*}{First proof} By theorem \ref{equiv2}, it suffices to prove that
$R^0S_2({\mathcal{D}}_{\{0\}\rightarrow Y}) = {\mathcal{A}}$. Explicitly, $\cal
D_{\{0\}\rightarrow Y}$ is the sheaf of $k$-vector spaces
$\text{Sym}({\mathfrak{g}})(0)$, with the following ${\mathcal{D}}$-module structure.
Given $L \in \text{Sym}({\mathfrak{g}})$ and $\xi \in {\mathfrak{g}}$,
$\xi$ acts on $L$ by $\xi \cdot L = \xi L$. To give the action of
${\mathcal{O}}_Y$, we need only consider $f \in {\mathcal{O}}_{Y,0}$. Expand the
differential operator
$f L$ in the form
$\sum \xi^I g_I$,\ $g_I \in {\mathcal{O}}_{Y, 0}$. Define
$$ :f L: = \sum \xi^I g_I(0)\in\text{Sym}({\mathfrak{g}})\ .
$$ Then $f \cdot L = :f L:$\ .
Let ${\mathcal{F}} = R^0S_2({\mathcal{D}}_{\{0\}\rightarrow Y})$. We want to prove that
${\mathcal{F}}$ is a free rank-1 ${\mathcal{A}}$-module. We have the global section
$1 \in\text{Sym}({\mathfrak{g}}) = H^0(Y, {\mathcal{D}}_{\{0\} \rightarrow
Y})$.
Let ${\mathcal{U}} \subset X$ be an open subset and let
$\sigma$ be a meromorphic section of ${\mathcal{P}}$ on ${\mathcal{U}} \times Y$ such that
$\sigma(0) = 1$, where $\sigma(0) = \sigma|_{{\mathcal{U}}
\times \{0\}}$. Since $\sigma$ is regular on the support of
$\alpha^*_2({\mathcal{D}}_{\{0\}\rightarrow Y})$, there is a well-defined section
$$
\sigma\otimes 1 \in \Gamma({\mathcal{U}} \times Y, {\mathcal{P}} \otimes
\alpha^*_2({\mathcal{D}}_{\{0\}\rightarrow Y})) = \Gamma({\mathcal{U}}, {\mathcal{F}})\ .
$$ If $f$ is a meromorphic function on ${\mathcal{U}} \times Y$, regular on
${\mathcal{U}}
\times \{0\}$, then
$$ f \sigma \otimes 1 = \sigma \otimes :f 1 :\ = f(0) \sigma \otimes 1\ .
$$ Thus $\sigma \otimes 1$ is independent of $\sigma$, and defines a
global section of ${\mathcal{F}}$. Now let
$\rho
\in \Gamma({\mathcal{U}},
{\mathcal{E}})$ satisfy $\mu(\rho) = 1$. Then $\rho$ determines both a relative
connection $\nabla$ on ${\mathcal{P}}|_{{\mathcal{U}} \times Y}$ and a set of sections
$x(1), \dots, x(g) \in \Gamma({\mathcal{U}}, {\mathcal{A}})$, such that
${\mathcal{A}}|_{{\mathcal{U}}} = {\mathcal{O}}[x(1),\dots, x(g)]$. By definition of the
${\mathcal{A}}$-module structure on ${\mathcal{F}}$,
\begin{align} &x(i_1) \dots x(i_k)(\sigma \otimes L)\notag\\ = &x(i_1)
\dots x(i_{k-1})(\nabla_{i_k}(\sigma) \otimes L - \sigma\otimes \xi_{i_k}
L)\ .
\end{align} It now follows by induction on $k$ that $\sigma\otimes 1$
generates ${\mathcal{F}}$ freely as a ${\mathcal{A}}$-module.
\end{pf*}
A second proof will be given in section \ref{duality}.
\section{Coherence, Duality and Holonomicity}\label{duality}
\subsection{} If ${\mathcal{G}}$ is an ${\mathcal{O}}_Y$-module, we may form
${\mathcal{D}}\otimes_{{\mathcal{O}}} {\mathcal{G}}$ using right multiplication by functions as the
${\mathcal{O}}_Y$-module structure on ${\mathcal{D}}$. This leaves us with left multiplication by
elements of ${\mathcal{D}}$ to give a left ${\mathcal{D}}$-module structure. Similarly, if
${\mathcal{F}}$ is an ${\mathcal{O}}_X$-module, we may form the ${\mathcal{A}}$-module
${\mathcal{A}}\otimes{\mathcal{F}}$. Note that
\begin{align}\label{adjoint} Hom_{{\mathcal{D}}}({\mathcal{D}}\otimes_{{\mathcal{O}}} {\mathcal{G}},\ \cdot\
)&=Hom_{{\mathcal{O}}}( {\mathcal{G}},\ \underline{for}(\cdot)\ )\ ;
\label{adjoint1}\\ Hom_{{\mathcal{A}}}({\mathcal{A}}\otimes{\mathcal{F}},\ \cdot\ )&=Hom_{{\mathcal{O}}}( {\mathcal{F}},\
\underline{for}(\cdot)\ )\ ,
\label{adjoint2}\end{align} where $\underline{for}$ is the forgetful functor.
\begin{Thm}\label{tensor}
\begin{equation}RS_2({\mathcal{D}}\overset{L}{\underset{=}{\otimes}}(\cdot))=
{\mathcal{A}}\overset{L}{\underset{=}{\otimes}}R{\mathcal{S}}_2(\cdot)\ .
\end{equation}
\begin{equation}RS_1({\mathcal{A}}\overset{L}{\underset{=}{\otimes}}(\cdot))=
{\mathcal{D}}\overset{L}{\underset{=}{\otimes}}R{\mathcal{S}}_1(\cdot)\ .
\end{equation}
\end{Thm}
\begin{pf} By theorem \ref{equiv2}, it suffices to prove the first
equality. Let ${\mathcal{F}}_1$ be an object in $D^b\text{Mod}({\mathcal{A}})$ and let
${\mathcal{F}}_2$ be an object in $D^b\text{Mod}({\mathcal{O}}_X)$. Using \eqref{adjoint1},
\eqref{adjoint2}, Mukai's theorem and theorem \ref{equiv},
\begin{align} Hom({\mathcal{A}}&\overset{L}{\underset{=}{\otimes}}{\mathcal{F}}_2,{\mathcal{F}}_1)\notag\\
&=Hom({\mathcal{F}}_2,\underline{for}({\mathcal{F}}_1))\notag\\ &=Hom(R\cal
S_1({\mathcal{F}}_2),\underline{for}(RS_1({\mathcal{F}}_1)))\notag\\
&=Hom({\mathcal{D}}\overset{L}{\underset{=}{\otimes}}R{\mathcal{S}}_1({\mathcal{F}}_2),RS_1({\mathcal{F}}_1))\notag\\
&=Hom(RS_2({\mathcal{D}}\overset{L}{\underset{=}{\otimes}}R\cal
S_1({\mathcal{F}}_2)),T^{-g}({\mathcal{F}}_1))\notag\\
&=Hom(RS_2({\mathcal{D}}\overset{L}{\underset{=}{\otimes}}T^gR{\mathcal{S}}_1({\mathcal{F}}_2)),{\mathcal{F}}_1)\ .
\end{align} Writing ${\mathcal{F}}_2$ as $R{\mathcal{S}}_2({\mathcal{G}})$ for some ${\mathcal{G}}$, we get
\begin{equation} Hom({\mathcal{A}}\overset{L}{\underset{=}{\otimes}}R{\mathcal{S}}_2({\mathcal{G}}),{\mathcal{F}}_1)=
Hom(RS_2({\mathcal{D}}\overset{L}{\underset{=}{\otimes}}{\mathcal{G}}),{\mathcal{F}}_1)\ .
\end{equation} The theorem is proved.
\end{pf}
\noindent{\bf Remark}\ This result appears to conflict with the interchange
of tensor and pontrjagin product stated in theorem Mukai2. Note, however,
that in the definition of ${\mathcal{D}}\otimes_{{\mathcal{O}}} {\mathcal{G}}$, one is using
the right ${\mathcal{O}}$-module structure on ${\mathcal{D}}$ to define the tensor product and
the left ${\mathcal{O}}$-module structure on the resulting sheaf.
\vskip 12pt
We digress to give a
\begin{pf*}{Short proof of Proposition \ref{fm of B}} By theorem
\ref{tensor},
\begin{align} RS_2({\mathcal{D}}_{\{0\}\rightarrow Y}) &=RS_2({\mathcal{D}}\otimes k(0))\notag\\
&={\mathcal{A}}\otimes R{\mathcal{S}}_2(k(0))\notag\\ &={\mathcal{A}}\otimes {\mathcal{O}}_X={\mathcal{A}}\ .
\end{align}
\end{pf*}
A more important corollary is the following. Let
$D^b_{coh}\text{Mod}({\mathcal{A}})$
(resp. $D^b_{coh}\text{Mod}({\mathcal{D}}_Y)$) denote the subcategory of complexes with
${\mathcal{A}}$-coherent (resp. ${\mathcal{D}}$-coherent) cohomology.
\begin{Thm} The functors $RS_1$ and $RS_2$ restrict to equivalences
between the categories $D^b_{coh}\text{Mod}({\mathcal{A}})$ and
$D^b_{coh}\text{Mod}({\mathcal{D}}_Y)$.
\end{Thm}
\begin{pf} We give the proof in one direction, the other direction being
the same. The category $D^b_{coh}\text{Mod}({\mathcal{A}})$ of complexes with
coherent cohomology is generated by sheaves of the form ${\mathcal{A}}\otimes{\mathcal{F}}$,
where ${\mathcal{F}}$ is a coherent ${\mathcal{O}}_X$-module. It is well-known that $R\cal
S_1({\mathcal{F}})$ the belongs to $D^b_{coh}\text{Mod}({\mathcal{O}}_Y)$. By theorem
\ref{tensor},
\begin{equation}
RS_1({\mathcal{A}}\otimes{\mathcal{F}})=
{\mathcal{D}}\overset{L}{\underset{=}{\otimes}}R{\mathcal{S}}_1({\mathcal{F}})\ .
\end{equation}
This completes the proof, for
${\mathcal{D}}\overset{L}{\underset{=}{\otimes}}(\cdot)$\ maps
$D^b_{coh}\text{Mod}({\mathcal{O}}_Y)$ to $D^b_{coh}\text{Mod}({\mathcal{D}}_Y)$.
\end{pf}
\subsection{} Given a complex ${\mathcal{F}} \in \text{Ob}\ D^b_{coh}\text{Mod}({\mathcal{D}}_Y)$,
its dual complex is
\begin{equation}
\Delta^{{\mathcal{D}}_Y}({\mathcal{F}}) = R {\mathcal{H}}om_{{\mathcal{D}}_Y}({\mathcal{F}}, T^g({\mathcal{D}}_Y))\ .
\end{equation} Note that $\Delta^{{\mathcal{D}}_Y}({\mathcal{F}})$ is naturally a complex of
right
${\mathcal{D}}$-modules, but we regard it as a complex of left ${\mathcal{D}}$-modules, using the
antiinvolution
\begin{align}
{\mathcal{D}} &\longrightarrow {\mathcal{D}}\notag\\ L = \sum f_I \xi^I &\mapsto \sum(-1)^{|I|}
\xi^I f_I
\ \ .
\end{align}
Given ${\mathcal{F}} \in \text{Ob}\ D^b\text{Mod}({\mathcal{A}})$, define
$\Delta^{{\mathcal{A}}}{\mathcal{F}} = R {\mathcal{H}}om({\mathcal{F}}, T^g({\mathcal{A}}))$.
In particular,
\begin{align}
\Delta^{\cal
A}({\mathcal{A}}\overset{L}{\underset{=}{\otimes}}{\mathcal{F}})
&={\mathcal{A}}\overset{L}{\underset{=}{\otimes}}\Delta^{{\mathcal{O}}_X}({\mathcal{F}})\ ;\\
\Delta^{{\mathcal{D}}}({\mathcal{D}}\overset{L}{\underset{=}{\otimes}}{\mathcal{F}})
&={\mathcal{D}}\overset{L}{\underset{=}{\otimes}}\Delta^{{\mathcal{O}}_Y}({\mathcal{F}})\ .
\end{align}
\begin{Prop}\label{grothendieck}
\begin{align} RS_2\Delta^{{\mathcal{D}}_Y} &= T^{-g}\Delta^{{\mathcal{A}}}RS_2\ ,\\
\Delta^{{\mathcal{D}}_Y} RS_1 &= RS_1\Delta^{{\mathcal{A}}}T^{-g}\ .
\end{align}
\end{Prop}
\begin{pf} By theorem \ref{equiv2} it suffices to prove the first equality.
It also suffices to consider the case where ${\mathcal{F}}$ is an object in
$D^b\text{Mod}({\mathcal{D}}_Y)$ of the form ${\mathcal{F}} = {\mathcal{D}} \otimes_{{\mathcal{O}}}{\mathcal{G}}$, for some
object ${\mathcal{G}} \in
\text{Ob}\ D^b\text{Mod}({\mathcal{O}}_Y)$. Then
$\Delta^{{\mathcal{D}}}({\mathcal{F}}) = {\mathcal{D}} \otimes \Delta^{{\mathcal{O}}_Y}({\mathcal{G}})$. By \cite[p. 161]{Muk}
\begin{equation} R{\mathcal{S}}_2 \Delta^{{\mathcal{O}}_Y} = T^{-g}\Delta^{{\mathcal{O}}_X}R{\mathcal{S}}_2\ .
\end{equation} Then
\begin{align} &RS_2\Delta^{{\mathcal{D}}_Y}({\mathcal{F}}) = RS_2({\mathcal{D}} \otimes \Delta^{{\mathcal{O}}_Y}({\mathcal{G}}))\\
&= {\mathcal{A}} \otimes R {\mathcal{S}}_2\Delta^{{\mathcal{O}}_Y}({\mathcal{G}})\ \ \ \ \text{(by theorem
\ref{tensor})}\notag\\ &= {\mathcal{A}} \otimes T^{-g} \Delta^{{\mathcal{O}}_X} R\cal
S_2({\mathcal{G}})\notag\\ &= {\mathcal{A}} \otimes T^{-g}R {\mathcal{H}}om_{{\mathcal{O}}}(R{\mathcal{S}}_2({\mathcal{G}}),
T^g({\mathcal{O}}_X))\notag\\ &= T^{-g}R {\mathcal{H}}om_{{\mathcal{A}}}({\mathcal{A}} \otimes R{\mathcal{S}}_2({\mathcal{G}}),
T^g({\mathcal{A}}))\notag\\ &= T^{-g} R {\mathcal{H}}om_{{\mathcal{A}}}(RS_2({\mathcal{F}}), T^g({\mathcal{A}})) =
T^{-g}\Delta^{{\mathcal{A}}} RS_2({\mathcal{F}})\ .
\end{align}
\end{pf}
\subsection{}
For the rest of this section we assume $char(k)=0$. Recall that a ${\mathcal{D}}$-module
${\mathcal{F}}$ is said to be holonomic if its characteristic variety has the least
possible dimension, namely $g$.
\begin{Prop}\label{Borel} Let ${\mathcal{F}}$ be a
coherent
${\mathcal{D}}$-module. Then ${\mathcal{F}}$ is holonomic if and only if $H^i(\Delta^{{\mathcal{D}}}({\mathcal{F}})) =
0$ for $i \ne 0$.
\end{Prop}
\begin{pf} \cite[p. 230]{Bo}\end{pf}
Since holonomicity of a ${\mathcal{D}}$-module is a local condition,
one expects it to be encoded globally when one takes the Fourier
transform.
\begin{Thm} Let ${\mathcal{F}} \in \text{Ob}\ D^b\text{Mod}({\mathcal{A}})$ be
a complex such that the cohomology of $RS_1({\mathcal{F}})$ is concentrated in
a single degree $i$ and $R^iS_1({\mathcal{F}})$ is ${\mathcal{D}}$-coherent.
Then $R^iS_1({\mathcal{F}})$ is holomonic if and only if the cohomology of
$RS_1\Delta^{{\mathcal{A}}}({\mathcal{F}})$ is concentrated in degree $g-i$.
\end{Thm}
\begin{pf} Let $\hat{{\mathcal{F}}}=R^iS_1({\mathcal{F}})$. Regarding $\hat{{\mathcal{F}}}$ as a
complex in degree $0$,\hfill\break
$H^j(\Delta^{{\mathcal{D}}}(\hat{{\mathcal{F}}}))=H^j(\Delta^{{\mathcal{D}}}T^iRS_1({\mathcal{F}}))$.
Then by proposition \ref{Borel}, $\hat{{\mathcal{F}}}$
is holonomic if and only if
$H^j(\Delta^{{\mathcal{D}}}T^iRS_1({\mathcal{F}})) = 0$ for $j \ne 0$.
By proposition
\ref{grothendieck},
\begin{align}\label{holonomic}
&H^j(\Delta^{{\mathcal{D}}}T^iRS_1({\mathcal{F}})) = H^j(\Delta^{{\mathcal{D}}}RS_1 T^i({\mathcal{F}}))\notag\\ &=
H^j(RS_1\Delta^{{\mathcal{A}}}T^{i-g}({\mathcal{F}}))= H^j(RS_1T^{g-i}\Delta^{\cal
A}({\mathcal{F}}))\ .
\end{align}
This vanishes for $j\ne 0$ if and only if
$R^lS_1\Delta^{{\mathcal{A}}}({\mathcal{F}})$ vanishes for $l\ne g-i$.
\end{pf}
We leave the detailed study of this condition to a future work.
\section{Characteristic Variety}\label{char var}
In many important examples, it is possible to be quite explicit about the
characteristic variety of the transform of a coherent ${\mathcal{A}}$-module.
Let $\{{\mathcal{A}}(m)\}$ be the filtration in example \ref{B itself}. If ${\mathcal{F}}$
is a coherent ${\mathcal{A}}$-module, then one has the usual notion of good filtration
with respect to $\{{\mathcal{A}}(m)\}$. As the Fourier-Mukai transform\ exchanges local data for global
data, it is worth noting that good filtrations exist globally.
\begin{Prop} Let $Z$ be a projective scheme of finite type over $k$, and
let ${\mathcal{A}}$ be a sheaf of ${\mathcal{O}}_Z$-algebras. Regarding ${\mathcal{A}}$ as an
${\mathcal{O}}$-module by letting ${\mathcal{O}}$ act on the right, assume ${\mathcal{A}}$ is
quasicoherent. If ${\mathcal{L}}$ is an ample line bundle on $Z$ and ${\mathcal{M}}$ is a
sheaf of coherent left ${\mathcal{A}}$-modules, then there is presentation of
${\mathcal{M}}$ of the form
\begin{equation}\label{presentation} ({\mathcal{A}}\otimes {\mathcal{L}}^{n_2})^{r_2}\longrightarrow
({\mathcal{A}}\otimes {\mathcal{L}}^{n_1})^{r_1}\longrightarrow{\mathcal{M}}\longrightarrow 0\ .
\end{equation}
\end{Prop}
\begin{pf} The proof is the same as for the case ${\mathcal{A}}={\mathcal{O}}$.\ (cf.
\cite[p. 122]{H} )
\end{pf} In particular,
\begin{Cor} If ${\mathcal{F}}$ is a sheaf of coherent ${\mathcal{A}}$ (resp. ${\mathcal{D}}_Y$)
modules, then ${\mathcal{M}}$ has a global good filtration by coherent ${\mathcal{O}}_X$
(resp. ${\mathcal{O}}_Y$) modules.
\end{Cor}
Let ${\mathcal{F}}$ be a coherent ${\mathcal{A}}$-module, and let $\{{\mathcal{F}}_m\}$ be a good
filtration. It follows from \eqref{filtration} that
\begin{equation}
Gr{\mathcal{A}}=Sym({\mathfrak{g}})\otimes{\mathcal{O}}_X\ .
\end{equation}
Thus, for each $\xi\in{\mathfrak{g}}$ we have a homogeneous map of degree $1$
\begin{equation}
Gr{\mathcal{F}}\overset{\dot\xi}\longrightarrow Gr{\mathcal{F}}\ .
\end{equation}
Identifying ${\mathcal{A}}$ with ${\mathcal{O}}[x(1),...,x(g)]$ on a sufficiently small
open set, we get a commutative diagram
\begin{equation}\label{comm}
\begin{CD}
{\mathcal{F}}_m @>{x(i)}>> {\mathcal{F}}_{m+1} \\ @VVV @VVV \\ Gr_m{\mathcal{F}} @>{\dot{\xi}_i}>>
Gr_{m+1}{\mathcal{F}}
\end{CD} \ \ .
\end{equation} Set $R^jS_1({\mathcal{F}},\psi)=(R^j{\mathcal{S}}_1({\mathcal{F}}),\nabla^j)$, and set
$\nabla^j=\sum \omega^i\nabla^j(i)$. It follows from the explicit formula
for $S_1$ that for all $j$, the diagram
\begin{equation}\label{comm transform}
\begin{CD} R^j{\mathcal{S}}_1({\mathcal{F}}_m) @>{\nabla^j(i)} >> R^j{\mathcal{S}}_1({\mathcal{F}}_{m+1}) \\
@VVV @VVV \\ R^j{\mathcal{S}}_1(Gr_m{\mathcal{F}}) @> {R^j{\mathcal{S}}_1(\dot{\xi}_i)}>> R^j\cal
S_1(Gr_{m+1}{\mathcal{F}})
\end{CD}
\end{equation} commutes.
Let ${\mathcal{K}}_m$ denote the kernel of the natural map
\begin{equation}
{\mathfrak{g}}\otimes Gr_m{\mathcal{F}}\longrightarrow Gr_{m+1}{\mathcal{F}}\ .
\end{equation}
Let us say that the filtered ${\mathcal{A}}$-module ${\mathcal{F}}$ satisfies {\it filtered W.I.T
with index $i$} (cf. \cite[p.156]{Muk}) if
$R^j{\mathcal{S}}_1({\mathcal{F}})=0$ for $j\ne i$ and the same is also true of ${\mathcal{F}}_m$ and
${\mathcal{K}}_m$ for $m$ sufficiently large. Following Mukai, we denote the
surviving cohomology sheaf of $RS_1({\mathcal{F}})$ by
$\hat{{\mathcal{F}}}$. Then we have
\begin{equation}
0\longrightarrow \hat{{\mathcal{K}}}_m\longrightarrow {\mathfrak{g}}\otimes \widehat{Gr_m{\mathcal{F}}}\longrightarrow
\widehat{Gr_{m+1}{\mathcal{F}}}\longrightarrow 0
\end{equation} for $m$ sufficiently large.
By the preceding remarks, we therefore have
\begin{Prop}\label{good filtration} Let ${\mathcal{F}}$ be a filtered
${\mathcal{A}}$-module satisfying filtered W.I.T.. Then
1. $\{\hat{{\mathcal{F}}_m}\}$ is a good ${\mathcal{D}}$-filtration on $\hat{{\mathcal{F}}}$.
2. $\widehat{Gr_m{\mathcal{F}}}=Gr_m\hat{{\mathcal{F}}}$ for $m$ sufficiently large.
\end{Prop}
Assume ${\mathcal{F}}$ satisfies filtered W.I.T. with index $i$, and let
${{\mathcal{I}}}(\hat{{\mathcal{F}}})\subset Sym({\mathfrak{g}})\otimes{\mathcal{O}}_Y$ denote the characteristic
ideal sheaf of
$\hat{{\mathcal{F}}}$. Fix an affine open subset ${\mathcal{U}}\subset Y$, let $A=\Gamma(\cal
U,{\mathcal{O}}_Y)$ and let $I=
\Gamma({\mathcal{U}},{\mathcal{I}}(\hat{{\mathcal{F}}}))$. This ideal may be described as follows.
We have a map
\begin{equation}
\text{Sym}({\mathfrak{g}}) \overset{gr\psi}{\longrightarrow} H^0(X,{\mathcal{E}}nd (Gr\cal
F))
\end{equation}
coming from the $Gr{\mathcal{A}}$-module structure on $Gr{\mathcal{F}}$. Redefining the
filtration if necessary, we may assume that ${\mathcal{K}}_m$ and ${\mathcal{F}}_m$ satisfy
W.I.T. for all $m$. Since ${\mathcal{U}}$ is affine,
proposition \ref{good filtration} gives
\begin{align}
\Gamma({\mathcal{U}},Gr\hat{{\mathcal{F}}})&=H^i(X\times\cal
U,{\mathcal{P}}\otimes\alpha_1^*(Gr{\mathcal{F}}))\notag\\ &=H^i(X,\alpha_{1*}({\mathcal{P}}|_{X\times\cal
U})\otimes(Gr{\mathcal{F}}))\ .
\end{align}
It is clear from the construction that the
$\text{Sym}({\mathfrak{g}})$-module structure on
$\Gamma({\mathcal{U}},Gr\hat{{\mathcal{F}}})$ is given by the composition
\begin{align}
\text{Sym}({\mathfrak{g}})\overset{gr\psi}{\longrightarrow}
H^0(X,{\mathcal{E}}nd (Gr{\mathcal{F}}))\rightarrow H^0(X,&\cal
End(Gr{\mathcal{F}} \otimes
\alpha_{1*}({\mathcal{P}}|_{X\times{\mathcal{U}}})))\notag\\ &\rightarrow
\text{End}(H^i(X, Gr{\mathcal{F}} \otimes \alpha_{1*}({\mathcal{P}}|_{X\times{\mathcal{U}}})))\ .
\end{align} Putting this together with the $A$-module structure,
we have a map
\begin{equation} A\otimes \text{Sym}({\mathfrak{g}})
\overset{\lambda_{{\mathcal{U}}}}{\longrightarrow}
\text{End}(\Gamma({\mathcal{U}},Gr\hat{{\mathcal{F}}}))\ .
\end{equation} Thus we have
\begin{Thm} \label{char}
\begin{equation}
\Gamma({\mathcal{U}},{\mathcal{I}}(\hat{{\mathcal{F}}}))=\sqrt{\text{ker}(\lambda_{{\mathcal{U}}})}\ .
\end{equation}
\end{Thm}
We will make use of this result in section \ref{nak}.
\section{The Krichever Construction}\label{krich}
Let us briefly explain how the Krichever construction fits into the present
framework. Assume now that $X=Y=Jac(C)$, where $C$ is a smooth curve of
positive genus. Pick a base point $P\in C$ and let $C\overset a\longrightarrow X$ be
the associated abel map. Taking ${\mathcal{F}}=a_*({\mathcal{O}}_C(*P))$, it is easy to see
that ${\mathcal{F}}$ admits a ${\mathcal{A}}$-module structure. It suffices to take
representative cocycles $\{c_{nm}(i)\}$ for a basis of $H^1(C,{\mathcal{O}})$ with
respect to an open cover $\{{\mathcal{U}}_n\}$, and solve the equations
\begin{equation}
c_{nm}(i)=f_n(i)-f_m(i)
\end{equation}
with $f_n(i)\in\Gamma({\mathcal{U}}_n,{\mathcal{O}}(*P))$. (Note that we are identifying
${\mathcal{O}}(*P)$ with its endomorphism sheaf.) An important ingredient here is
the {\it flag} on $H^1(C,{\mathcal{O}})$ coming from the natural maps
\begin{equation}
H^0(C,{\mathcal{O}}(nP)/{\mathcal{O}})\longrightarrow H^1(C,{\mathcal{O}})\ .
\end{equation}
Denoting the images of these maps by $V_n$, we have a sequence of
subsheaves of ${\mathcal{D}}$,
\begin{equation}
{\mathcal{O}}_Y={\mathcal{D}}_0\subset{\mathcal{D}}_1\subset ...\ ,
\end{equation}
where ${\mathcal{D}}_n$ is the ${\mathcal{O}}$-algebra generated by the vector fields belonging
to $V_n$. In particular,
\begin{equation}
{\mathcal{D}}_1={\mathcal{O}}[\xi]\ ,
\end{equation}
where $\xi$ is a basis of the one-dimensional space $V_1$. Now we may also
filter ${\mathcal{A}}$ by subalgebras
\begin{equation}
{\mathcal{O}}_X={\mathcal{A}}_0\subset{\mathcal{A}}_1\subset ...\
\end{equation}
in the same way, so that if we set
\begin{equation}
{\mathcal{A}}_i(m)={\mathcal{A}}_i\cap{\mathcal{A}}(m)\ ,
\end{equation}
then
\begin{equation}
Gr_i{\mathcal{A}}=Sym(V_i)\otimes{\mathcal{O}}\ .
\end{equation}
Theorem \ref{equiv2} extends to this more general situation:
$$\text{The Fourier-Mukai transform\ gives an equivalence of categories}$$
$$D^b\text{Mod}({\mathcal{A}}_i) \leftrightarrow D^b\text{Mod}({\mathcal{D}}_i)\ .$$
Now there is an essentially canonical ${\mathcal{A}}_1$-module structure on ${\mathcal{O}}(*P)$.
Indeed, let $z$ be a local parameter at $P$. Then $a^*({\mathcal{A}}_1)$ is the
subsheaf of ${\mathcal{O}}(*P)[x]$ characterized by the following growth condition:
If ${\mathcal{U}}$ is a neighborhood of $P$, then
\begin{equation}
\Gamma({\mathcal{U}},a^*({\mathcal{A}}_1))=\{ \sum f_ix^i \in
\Gamma({\mathcal{U}},{\mathcal{O}}(*P)[x])\ |
\ \text{for all}\ j, \sum\limits_{i\ge j} \binom ij \frac{f_i}{z^{i-j}}\in
\Gamma({\mathcal{U}},{\mathcal{O}})\ \}\ .\end{equation}
Then ${\mathcal{O}}(*P)$ is in fact a sheaf of ${\mathcal{A}}_1|_C$-algebras under
\begin{align}
{\mathcal{A}}_1|_C&\longrightarrow {\mathcal{O}}(*P)\notag\\
f(x)&\mapsto f(x=0)\ .
\end{align}
The key to the Krichever construction is
\begin{Prop} Let ${\mathcal{G}}=\widehat{{\mathcal{O}}(*P)}$ regarded as ${\mathcal{D}}_1$-module. Then
${\mathcal{G}}|_{X-\Theta}$ is {\it canonically} isomorphic to ${\mathcal{D}}_1$.
\end{Prop}
This proposition is simply a translation into the language of this paper of
well-known results. The canonical generator is the Baker-Akhiezer
function. The discussion given here is similar to that of \cite{R}. The
important point is that
\begin{equation}
H^0(C,{\mathcal{O}}(*P))=End_{{\mathcal{A}}_1}({\mathcal{O}}(*P))=End_{{\mathcal{D}}_1}({\mathcal{G}})\ .
\end{equation}
Thus, if we let $\chi$ denote the canonical generator of ${\mathcal{G}}|_{X-\Theta}$,
then for all $f\in H^0(C,{\mathcal{O}}(*P))$, there exists a unique $L_f\in
H^0(X-\Theta,{\mathcal{D}}_1)$ such that
\begin{equation}
f\chi=L_f\chi\ .
\end{equation}
This is the Burchnall-Chaundy representation of $H^0(C,{\mathcal{O}}(*P))$, done for
all line bundles at once. The famous result is that the $L_f$'s satisfy
the KP-hierarchy.
\section{Further Examples}\label{nak}
Nakayashiki has studied generalizations of the Krichever construction in
which the curve and point are replaced by an arbitrary variety together
with an ample hypersurface. In this section we will illustrate the results
of section \ref{char var} using a somewhat more general version of his
examples. In particular we will obtain some refinements of his
results about characteristic varieties.
Let $Z$ be a smooth projective variety, and take $X$ to be its albanese
variety. Assume for simplicity that the albanese map
\begin{equation}
Z\overset a\longrightarrow X
\end{equation}
is an imbedding. Let ${\mathcal{A}}_Z=a^*({\mathcal{A}})$. Then an
${\mathcal{A}}_Z$-module is the same as an ${\mathcal{A}}$-module supported on $Z$, so we have a
functor
$${\mathcal{A}}_Z\text{-modules} \longrightarrow {\mathcal{D}}_Y\text{-modules}\ .$$
Let $D\subset Z$
be an ample hypersurface.
As in the Krichever construction, ${\mathcal{O}}(*D)$ may be given the structure of an
algebra over ${\mathcal{A}}_Z$, making $\widehat{{\mathcal{O}}(*D)}$ a ${\mathcal{D}}$-module. (Note that
${\mathcal{O}}(*D)$ is W.I.T. of index 0.) Nakayashiki refers to such ${\mathcal{D}}$-modules as
BA-modules.
In general, when one has a splitting on a sheaf ${\mathcal{R}}$, there is an
induced map
\begin{equation}
{\mathfrak{g}} \longrightarrow H^0(X,{\mathcal{E}}nd({\mathcal{R}})/{\mathcal{O}})
\end{equation} by virtue of \eqref{cobdry}. In the present example, this is
a map
\begin{equation}
{\mathfrak{g}} \overset{\psi'}{\longrightarrow} H^0(Z, {\mathcal{O}}(*D)/{\mathcal{O}})
\end{equation} splitting the natural map
\begin{equation} H^0(Z, {\mathcal{O}}(*D)/{\mathcal{O}}) \longrightarrow H^1(Z, {\mathcal{O}}) = {\mathfrak{g}}\ .
\end{equation}
Conversely, let ${\mathfrak{g}} \overset{T}{\rightarrow} H^0(Z, {\mathcal{O}}(rD)/{\mathcal{O}})$ be
any left inverse of the natural map $H^0(Z, {\mathcal{O}}(rD)/{\mathcal{O}})
\rightarrow H^1(Z, {\mathcal{O}}) = {\mathfrak{g}}$. Then there is a splitting $\psi$ on
${\mathcal{O}}(*D)$ such that $\psi' = T$. We say that the splitting is {\it
associated} to
$T$. Equivalently,
let $D_r$ denote the scheme $(D, {\mathcal{O}}/{\mathcal{O}}(-rD))$ and let ${\mathcal{M}} = \cal
O(rD)/{\mathcal{O}})$, which we regard as a line bundle on $D_r$. Then we think
of $\psi$ as being associated to the rational map
\begin{equation}
D_r\longrightarrow \Bbb P({\mathfrak{g}}^*)
\end{equation}
induced by the linear system $T({\mathfrak{g}})$.
Now let ${\mathcal{O}}(*D)$ be endowed with a ${\mathcal{A}}_Z$-module structure
associated to a fixed map $T$. From $T$ we get a map
\begin{equation}
{\mathfrak{g}} \otimes {\mathcal{O}}_{D_r} \overset{t}{\longrightarrow} {\mathcal{M}}\ ,
\end{equation} from which we may construct a Koszul complex (where ${\mathcal{O}} =
{\mathcal{O}}_{D_r})$
\begin{equation} 0 \rightarrow \wedge^g {\mathfrak{g}} \otimes {\mathcal{M}}^{-g + 1}
\cdots \overset{\wedge_3 t}{\longrightarrow} \wedge^2 {\mathfrak{g}}
\otimes {\mathcal{M}}^{-1} \overset{\wedge_2 t}{\longrightarrow} {\mathfrak{g}}
\otimes {\mathcal{O}} \overset{t}{\longrightarrow} {\mathcal{M}} \rightarrow 0.
\end{equation}
Now let ${\mathcal{H}}$ be a sheaf of coherent ${\mathcal{O}}_Z$-modules, and set
\begin{equation}
{\mathcal{F}} = {\mathcal{H}} \otimes {\mathcal{O}}(*D)\ ,
\end{equation} regarded as a ${\mathcal{A}}_Z$-module. We make the simplifying
hypothesis that
the injections
\begin{equation}
{\mathcal{O}}(krD) \longrightarrow {\mathcal{O}}((k+1)rD)
\end{equation} induce injections
\begin{equation}
{\mathcal{F}}_k \longrightarrow {\mathcal{F}}_{k+1}\ ,
\end{equation}
where ${\mathcal{F}}_k=
{\mathcal{H}} \otimes {\mathcal{O}}(krD)$.
This is equivalent to
\vskip 14pt
\noindent{\bf Assumption}
\begin{equation}
{\mathcal{T}}\kern -.2em or^1({\mathcal{H}},{\mathcal{O}}_D)=0\ .
\end{equation} This gives us a filtration on ${\mathcal{F}}$, with associated graded
sheaf
\begin{equation}
Gr{\mathcal{F}}_k={\mathcal{H}}|_{D_r}\otimes{\mathcal{M}}^k\ ,
\end{equation}
upon which ${\mathfrak{g}}$ acts through the map $T$.
We want this to be a good filtration.
\begin{Lem}\label{base locus} The following are equivalent.
\begin{enumerate}
\item ${\mathfrak{g}} \otimes {\mathcal{H}} \rightarrow {\mathcal{M}} \otimes {\mathcal{H}}
\rightarrow 0$ is exact.
\item The baselocus of $T({\mathfrak{g}})$ does not meet the support of
${\mathcal{H}}$.
\item The complex $(\wedge^i{\mathfrak{g}} \otimes {\mathcal{M}}^{1-i}) \otimes
{\mathcal{H}}$ is exact
\item If we set ${\mathcal{R}}_j = \text{ker}(\wedge_j t)$, then
\begin{equation} 0 \rightarrow {\mathcal{R}}_j \otimes {\mathcal{H}} \rightarrow \wedge^j{\mathfrak{g}}
\otimes {\mathcal{M}}^{1-j} \otimes {\mathcal{H}} \rightarrow {\mathcal{R}}_{j-1}\otimes
{\mathcal{H}} \rightarrow 0\notag
\end{equation} is exact for all $j$.
\item The filtration $\{{\mathcal{F}}_k\}$ is a good ${\mathcal{A}}_Z$-filtration.
\end{enumerate}
\end{Lem}
\begin{pf} The equivalence of 1 and 2 follows from Nakayama's lemma. If 2
holds, then the Koszul complex is exact at points in the support of
${\mathcal{H}}$, from which 3 follows. Since 1 is part of 3, the first three
statements are equivalent. Then 4 follows because $\wedge^j
{\mathfrak{g}} \otimes {\mathcal{M}}^{1-j} \rightarrow {\mathcal{R}}_{j-1} \rightarrow 0$ is exact at
all points in $\text{supp}({\mathcal{H}})$, and the
${\mathcal{R}}_j$'s are all projective. But 1 is part of 4, so 1 through 4 are
equivalent. Now the filtration ${\mathcal{F}}_k$ is good if and only if
\begin{equation}\label{also action}
{\mathfrak{g}} \otimes Gr_k{\mathcal{F}} \rightarrow Gr_{k+1}{\mathcal{F}}\longrightarrow 0
\end{equation} is exact for large $k$. But $Gr_k{\mathcal{F}} = {\mathcal{H}}|_{{\mathcal{D}}_r}
\otimes {\mathcal{M}}^k$, so 1 and 5 are equivalent.
\end{pf}
Assume now that $\{ {\mathcal{F}}_k\}$ is a good $\cal
A_Z$-filtration.
\begin{Lem} ${\mathcal{F}}$ satisfies filtered W.I.T. with index $0$.
\end{Lem}
\begin{pf} It is clear that ${\mathcal{F}}_k$ satisfies W.I.T. with
index $0$ if $k$ is sufficiently large. Let
${\mathcal{K}}_k$ denote the kernel of \eqref{also action}. Since ${\mathcal{M}}$ is
invertible,
\begin{equation}
{\mathcal{K}}_k = {\mathcal{K}}_1 \otimes {\mathcal{M}}^k\ .
\end{equation} Since ${\mathcal{M}}$ is ample on $D_r$, there exists $k$ such that
\begin{equation} H^i(D_r, {\mathcal{K}}_j \otimes {\mathcal{L}}) = 0
\end{equation} for any line bundle ${\mathcal{L}}$ which is the pullback of a
degree-0 line bundle on $X$, any $j \ge k$ and all $i > 0$.
Thus for large $k$, ${\mathcal{K}}_k$ also satisfies W.I.T. with index $0$.
\end{pf}
This puts us in
the position to apply theorem \ref{char}.
Let ${\mathcal{U}}\subset Y$ be affine open and let $A=\Gamma(\cal
U,{\mathcal{O}}_Y)$. Let
$\epsilon_1$ and $\epsilon_2$ denote the projections on $D_r\times Y$. We
must study the graded
$A\otimes
\text{Sym}({\mathfrak{g}})$-module
\begin{equation} H\overset{\text{def}}=
\bigoplus_{k=0}^{\infty}H^0(D_r,{\mathcal{H}}|_{D_r}\otimes
\epsilon_{1*}({\mathcal{P}}|_{D_r\times{\mathcal{U}}})\otimes{\mathcal{M}}^k)\ .
\end{equation} Let
\begin{equation} S=H^0(D_r,\bigoplus_{k=0}^{\infty} {\mathcal{M}}^k)\ .
\end{equation} Then $H$ is a graded
$S$-module. Moreover, we have the map
\begin{equation}\label{linear map}
{\mathfrak{g}} \overset{T}{\longrightarrow} H^0(Z, {\mathcal{O}}(rD)/{\mathcal{O}})=
H^0(D_r, {\mathcal{M}})\ ,
\end{equation}
which induces
\begin{equation}\label{induced}
Sym({\mathfrak{g}}) \longrightarrow Sym( H^0(D_r, {\mathcal{M}}))\ .
\end{equation}
Then $Sym({\mathfrak{g}})$ acts on
$H$ through the composition of \eqref{induced} with the natural
homomorphism
\begin{equation}
Sym( H^0(D_r, {\mathcal{M}}))\longrightarrow S\ .
\end{equation}
If we apply Proj to the composite map $Sym({\mathfrak{g}})\longrightarrow S$, we recover the
rational map
\begin{equation} D_r\overset{\Psi}{--\rightarrow}\Bbb P({\mathfrak{g}}^*)\ .
\end{equation}
associated to the linear map $T$, \eqref{linear map}. Moreover, applying Proj
to the graded $A\otimes S$-module $H$, we get the sheaf
\begin{equation}
\tilde{H}=(\alpha_1^*({\mathcal{H}})\otimes{\mathcal{P}})|_{D_r\times{\mathcal{U}}}
\end{equation}
on $D_r\times {\mathcal{U}}$. Thus, the consideration of $H$ as an $A\otimes
Sym({\mathfrak{g}})$-module may be viewed on the sheaf level as taking the direct
image of $(\alpha_1^*({\mathcal{H}})\otimes{\mathcal{P}})|_{D_r\times{\mathcal{U}}}$ under the
map
$\Psi\times 1$. (Recall that by lemma
\ref{base locus}, $\Psi$ is defined on the support of ${\mathcal{H}}|_{D_r}$.)
The discussion above gives us the main result of this section.
\begin{Thm}\label{charac var} Let ${\mathcal{G}}=\hat{{\mathcal{F}}}$, where ${\mathcal{F}}=\cal
H\otimes{\mathcal{O}}_Z(*D)$, ${\mathcal{T}}or^1({\mathcal{H}},{\mathcal{O}}_D)=0$, and the ${\mathcal{A}}_Z$-module
structure on ${\mathcal{O}}(*D)$ is associated to $\Psi:D_r--\to \Bbb P({\mathfrak{g}}^*)$. Then
the characteristic variety of
${\mathcal{G}}$, viewed as a subvariety of $\Bbb P({\mathfrak{g}}^*)\times Y$, is the support of
the sheaf
\begin{equation} Gr{\mathcal{G}}=(\Psi\times 1)_*
((\alpha_1^*(\cal
H)\otimes{\mathcal{P}})|_{D_r\times Y})\ .
\end{equation}
\end{Thm}
In \cite{N2} the case $Z=X$ is considered, and it is proved that the
codimension of the characteristic variety is $dim(X)-dim Supp ({\mathcal{H}})$.
The more general theorem \ref{charac var} yields quite detailed information
in this case, and in particular has Nakayashiki's result as a corollary.
We are now dealing with a smooth, ample hypersurface $D\subset X$. Then
there is a natural class of ${\mathcal{A}}$-module structures on ${\mathcal{O}}(*D)$. These are
described in \cite{N2} in terms of factors of automorphy, but may also be
seen somewhat more geometrically. Let ${\mathfrak{h}}$ be the space of vector
fields on $X$. We have the {\it Gauss map}
\begin{equation}
D\overset\Psi\longrightarrow \Bbb P({\mathfrak{h}}^*)\ .
\end{equation}
The normal bundle to $D$ is $\Psi^*({\mathcal{O}}(1))$, and thus we have a linear map
\begin{equation}
{\mathfrak{h}} \overset{\lambda}{\longrightarrow} H^0(D, {\mathcal{N}})\ .
\end{equation}
The composition of $\lambda$ with the canonical map $H^0(D, {\mathcal{N}})\to{\mathfrak{g}}$ is
an isomorphism.
Thus one may consider ${\mathcal{A}}$-module structures on ${\mathcal{O}}(*D)$
associated to the Gauss map. That is, we can choose the ${\mathcal{A}}$-module
structure in such a way that the induced map
\begin{equation}
{\mathfrak{g}} \overset{\psi'}{\longrightarrow} H^0(X, {\mathcal{O}}(*D)/{\mathcal{O}})
\end{equation}
is precisely the composition
\begin{equation}
{\mathfrak{g}}\simeq{\mathfrak{h}}\overset{\lambda}{\longrightarrow} H^0(D,{\mathcal{N}})\subset
H^0(X, {\mathcal{O}}(*D)/{\mathcal{O}})\ .
\end{equation}
We will call such an ${\mathcal{A}}$-module structure {\it canonical}. In the
language of this paper, Nakayashiki's ${\mathcal{D}}$-modules are obtained as the Fourier-Mukai transform\
of ${\mathcal{A}}$-modules of the form ${\mathcal{H}}\otimes{\mathcal{O}}(*D)$, where ${\mathcal{O}}(*D)$ has
a canonical ${\mathcal{A}}$-module structure. However, one may as well consider
a more general class of ${\mathcal{A}}$-module structures, namely those
associated to any $g$-dimensional linear system $V\subset H^0(D,{\mathcal{N}})$
which is basepoint-free and maps isomorphically onto ${\mathfrak{g}}$. The example
of the Gauss map shows that this is the generic situation. As a
corollary of theorem \ref{charac var}, we have
\begin{Thm} Let $D\subset X$ be a
smooth ample hypersurface. Let
$V\subset H^0(D,{\mathcal{N}})$ be a $g$-dimensional basepoint-free linear
system mapping isomorphically onto ${\mathfrak{g}}$, and let
$\Psi:D\to \Bbb P({\mathfrak{g}}^*)$ be the corresponding morphism. Let ${\mathcal{H}}$ be a
coherent
${\mathcal{O}}_X$-module such that
${\mathcal{T}}\kern -.2em or^1({\mathcal{H}},{\mathcal{O}}_D)=0$\ , and set ${\mathcal{G}}=\widehat{\cal
H\otimes{\mathcal{O}}(*D)}$. Give ${\mathcal{G}}$ the ${\mathcal{D}}$-module structure induced by a
${\mathcal{A}}$-module structure on ${\mathcal{O}}(*D)$ associated to $V$. Then the
characterisic
variety of ${\mathcal{G}}$ is
\begin{equation} ss({\mathcal{G}})=\Psi(Supp({\mathcal{H}}|_D))\times Y
\ .\end{equation}
(In particular, $ codim(ss({\mathcal{G}})=dim(X)-dim Supp ({\mathcal{H}})$\ .)
\end{Thm}
\begin{pf} Because ${\mathcal{N}}$ is ample, the morphism $\Psi$ is
finite. Therefore, tensoring
${\mathcal{H}}|_D$ with a line bundle has no effect on the support of its direct
image. The result then follows from theorem \ref{charac var}.\end{pf}
\section{Some remarks on commuting rings of matrix partial differential
operators}\label{PDOs}
In some sense, the origins of the present subject date to work of Burchnall
and Chaundy on commuting rings of ordinary differential operators
\cite{BC}. Such rings are always of dimension one. As Nakayashiki has
observed in \cite{N2}, the Fourier-Mukai transform\ allows one to represent the ring
$H^0(X,{\mathcal{O}}(*D))$, $X$ an abelian variety and $D$ a smooth ample hypersurface,
by matrix valued partial differential operators in $g$ variables, the size
of the matrix being the $g$-fold self-intersection number, $D^g$. We want
to offer some further observations on this question.
Consider again the data $(Z,D,{\mathcal{H}})$ in section \ref{nak}. Fix an integer
$r$ and a subspace $V\subset H^0(Z,{\mathcal{O}}(rD)/{\mathcal{O}})$ mapping isomorphically onto
its image under the natural map
\begin{equation} H^0(Z,{\mathcal{O}}(rD)/{\mathcal{O}})\longrightarrow {\mathfrak{g}}\ .
\end{equation} As in section \ref{krich}, we get a subsheaf ${\mathcal{A}}_1\subset
{\mathcal{A}}$ by imitating the construction of ${\mathcal{A}}$, replacing ${\mathfrak{g}}$ by $V$
throughout. Similarly, we have a subsheaf ${\mathcal{D}}_1\subset {\mathcal{D}}_Y$ generated over
${\mathcal{O}}_Y$ by the vector fields belonging to the image of $V$. As in
section
\ref{nak}, we have a rational map
\begin{equation} D_r\overset{\Psi}{--\rightarrow}\Bbb P(V^*)\ .
\end{equation} Assuming now that $supp({\mathcal{H}})$ does not meet the baselocus
of $\Psi$, we can introduce a coherent ${\mathcal{A}}_1$-module structure on
${\mathcal{F}}={\mathcal{H}} \otimes
{\mathcal{O}}(*D)$, filtered by the submodules
${\mathcal{H}} \otimes
{\mathcal{O}}(krD)$ exactly as before. We now have
\begin{align} Gr_1{\mathcal{A}} &=Sym(V)\otimes{\mathcal{O}}_Z\\ D_r&=Spec({\mathcal{O}}/{\mathcal{O}}(-rD))\\
{\mathcal{M}}&={\mathcal{O}}(rD)/{\mathcal{O}}\ \text{(thought of as a line bundle on $D_r$)}\\ Gr{\mathcal{F}}&=
\bigoplus_{l=0}^{\infty}{\mathcal{H}}|_{D_r}\otimes{\mathcal{M}}^l\ ,
\end{align} where the $Sym(V)$-module structure on $Gr{\mathcal{F}}$ is defined by
the inclusion
\begin{equation}\label{inclusion} V\longrightarrow H^0(D_r,{\mathcal{M}})\ .
\end{equation} Set
$
{\mathcal{G}}=\hat{{\mathcal{F}}}$, regarded as a sheaf of ${\mathcal{D}}_1$-modules. We will examine
conditions under which ${\mathcal{G}}$ is free in a neighborhood of a given line
bundle ${\mathcal{L}}\in Pic^0(Z)$. In order to have the equality
\begin{equation}
\widehat{Gr{\mathcal{F}}}=Gr\hat{{\mathcal{F}}}\
\end{equation} in a neighborhood of ${\mathcal{L}}$, we make the assumptions
\begin{align} &\text{For all}\ k\ge 0\ , i>0\ ,\ H^i(Z,{\mathcal{L}}\otimes{\mathcal{F}}_k)=0\
.\\ &\text{There exists}\ j\ge 2\ \text{such that for all}\ i\ne j\ ,
\ H^i(Z,{\mathcal{L}}\otimes{\mathcal{F}}_{-1})=0\ .
\end{align} For ${\mathcal{G}}$ to be free over ${\mathcal{D}}_1$ in a neighborhood of $\cal
L$, it is sufficient that the fiber of $Gr{\mathcal{G}}$ at ${\mathcal{L}}$ be free over
$Sym(V)$. We therefore have
\begin{Prop}\label{generically free} For ${\mathcal{G}}$ to be free over ${\mathcal{D}}_1$
in a neighborhood of ${\mathcal{L}}$, it is sufficient that
$$
\bigoplus_{l=0}^{\infty}H^0(D_r,({\mathcal{L}}\otimes{\mathcal{H}})|_{D_r}\otimes{\mathcal{M}}^l)
$$ be freely generated as a module over $Sym(V)$.
\end{Prop} Regarding the hypothesis of this proposition, we have the
following purely algebraic lemma. Let $M$ be an arbitrary finitely
generated graded
$Sym(V)$-module, with $M_j=0$ for $j<0$. For all $j$ we have a complex
\begin{equation}\label{complex}
\wedge^2(V)\otimes M_{j-1}\overset{\beta_j}\longrightarrow V\otimes
M_j\overset{\alpha_j}\longrightarrow M_{j+1}\ ,
\end{equation} where $\alpha_j$ is the action of $V$, and $\beta_j$ is
defined by
\begin{equation} v\wedge w\otimes m\mapsto v\otimes wm-w\otimes vm\ .
\end{equation}
\begin{Lem}\label{free} $M$ is free over $\text{Sym}(V)$ if and only if the
sequence
\eqref{complex} is exact for all $j$.
\end{Lem}
\begin{pf} If $M = \overset{\ell}{\underset{i=1}{\oplus}}
\text{Sym}(V)[n_j]$, where
\begin{equation}
\text{Sym}(V)[n_i]_{j} = \text{Sym}^{j + n_i}(V)\ ,
\end{equation} then the exactness of the complexes \eqref{complex} follows
from the well-known fact that the natural complex
\begin{equation}
\label{always exact}
\wedge^2(V) \otimes \text{Sym}^k(V) \rightarrow V \otimes
\text{Sym}^{k+1}(V) \rightarrow \text{Sym}^{k+2}(V)
\end{equation} is always exact.
Conversely, suppose that \eqref{complex} is always exact. For all
$j$, choose a subspace $U_j \subset M_j$ complementary to the image of
$\alpha_{j-1}$. What we must show is that for all $j$, the natural map
\begin{equation}
\overset{j}{\underset{i=1}{\oplus}} \text{Sym}^i(V) \otimes U_{j-i}
\overset{\gamma_j}{\longrightarrow} M_j
\end{equation} is injective, for then $M$ is isomorphic to
\begin{equation}
\oplus U_j \otimes_k \text{Sym}(V)[-j]\ .
\end{equation} Given $\ell$, assume $\gamma_j$ is injective for $j < \ell$.
Then $M_{\ell-1} \approx \overset{\ell - 1}{\underset{i=0}{\oplus}}
\text{Sym}^i(V) \otimes U_{\ell - 1 - i}$ and we have a commutative diagram
\begin{equation}
\begin{matrix} V \otimes \left(\operatornamewithlimits{\oplus}\limits^{\ell -
1}_{i=0}
\text{Sym}^i (V)
\otimes U_{\ell-1-i}\right)\\
\qquad \qquad \qquad \downarrow \delta \qquad\qquad \searrow
\alpha_{\ell - 1}\\
\operatornamewithlimits{\oplus}\limits^{\ell}_{i=1} \text{Sym}^i (V)
\otimes U_{\ell - i}
\overset{\gamma_{\ell}}{\longrightarrow}M_{\ell}
\end{matrix}
\end{equation} Since
$M_{\ell - 2} \approx \overset{\ell - 2}{\underset{i=0}{\oplus}}
\text{Sym}^i(V) \otimes U_{\ell - 2 - i}$, the exactness of \eqref{always
exact} implies that
$\text{Ker}(\delta) = \text{Im}(\beta_{\ell - 1})$. Thus
$\text{ker}(\delta) = \text{Ker}(\alpha_{\ell - 1})$, which says that
$\gamma_{\ell}$ is injective.
\end{pf}
The geometric interpretation of this lemma is the following. Let $S$ be a
scheme over $k$, $V$ a finite dimensional vector space over
$k$,
${\mathcal{M}}$ a line bundle on $S$, and $V
\overset{T}{\longrightarrow} H^0(S, {\mathcal{M}})$ a linear map. As in section
\ref{nak}, we associate to $T$ a Koszul complex
\begin{equation}
\dots \rightarrow \wedge^3 V \otimes {\mathcal{M}}^{-2} \overset{\wedge_3
t}{\longrightarrow} \wedge^2 V \otimes {\mathcal{M}}^{-1}
\overset{\wedge_2 t}{\longrightarrow} V \otimes {\mathcal{O}}
\overset{t}{\longrightarrow} {\mathcal{M}} \rightarrow 0\ .
\end{equation} Let ${\mathcal{R}}_i = \text{ker}(\wedge_i t)$. Let ${\mathcal{H}}$ be a
sheaf of ${\mathcal{O}}_S$-modules. Then we have complexes
\begin{equation}
\label{exact1} 0 \rightarrow {\mathcal{R}}_1 \otimes {\mathcal{H}} \rightarrow V \otimes
{\mathcal{H}}
\rightarrow {\mathcal{M}} \otimes {\mathcal{H}} \rightarrow 0
\end{equation}
\begin{equation}
\label{exact2} 0 \rightarrow {\mathcal{R}}_2 \otimes {\mathcal{H}} \rightarrow \wedge^2 V
\otimes {\mathcal{M}}^{-1} \otimes {\mathcal{H}} \rightarrow {\mathcal{R}}_1 \otimes
{\mathcal{H}} \rightarrow 0.
\end{equation} As in lemma \ref{base locus}, these sequences are exact if
and only if
\begin{equation}
\label{exact0} V \otimes {\mathcal{H}} \rightarrow {\mathcal{M}} \otimes {\mathcal{H}} \rightarrow
0
\end{equation} is exact. We therefore have
\begin{Thm} Assume \eqref{exact0} is exact. Consider the graded
$\text{Sym}(V)$-module
\begin{equation} M = \underset{j\ge 0}{\oplus} H^0(S,{\mathcal{H}} \otimes \cal
M^j)\ .
\end{equation} Then $M$ is free over $\text{Sym}(V)$ if and only if
\begin{align} 0 \rightarrow H^1(S, {\mathcal{R}}_2 \otimes {\mathcal{H}} \otimes \cal
M^j)&
\rightarrow \wedge^2 V \otimes H^1(S, {\mathcal{H}} \otimes {\mathcal{M}}^{j-1})
\ \text{is exact for all}\ j\ge 1\ ,\text{and}\\ H^0(S,{\mathcal{R}}_1\otimes\cal
H)=0\ .\end{align}
\end{Thm}
\begin{pf} The maps $\alpha_j$ and $\beta_j$ in \eqref{complex} are given in
this case by tensoring \eqref{exact1} and
\eqref{exact2} with ${\mathcal{M}}^i$ and taking cohomology:
\begin{equation}
\begin{matrix} 0 \rightarrow H^0(S, {\mathcal{R}}_1 \otimes {\mathcal{H}}\otimes {\mathcal{M}}^j)
\rightarrow V \otimes M_j \overset{\alpha_j}{\longrightarrow} M_{j+1}\\
\qquad \quad \uparrow \qquad \nearrow
\beta_j\\
\wedge^2(V) \otimes M_{j-1}
\end{matrix}
\end{equation} Therefore, \eqref{complex} is exact for $j\ge 1$ precisely
when
\begin{equation} H^1(S, {\mathcal{R}}_2 \otimes {\mathcal{H}} \otimes {\mathcal{M}}^j) \rightarrow
\wedge^2V \otimes H^1(S, {\mathcal{H}} \otimes {\mathcal{M}}^{j-1})
\end{equation} is injective. The second condition appears because we have
defined $M$ as a sum over nonnegative $j$, but have not assumed that
$H^0(S,{\mathcal{H}} \otimes {\mathcal{M}}^{-1})=0$. The theorem then follows from lemma
\ref{free}.
\end{pf}
Let us put all these ingredients together. We have a map
\begin{equation} V\overset{T}{\longrightarrow} H^0(D_r, {\mathcal{M}})\ , \ \ \cal
M = {\mathcal{O}}(rD)/{\mathcal{O}}\ ,
\end{equation}
giving us subsheaves ${\mathcal{A}}_1\subset {\mathcal{A}}$, ${\mathcal{D}}_1\subset {\mathcal{D}}$, and a
${\mathcal{A}}_1$-module structure on ${\mathcal{O}}(*D)$. We have a sheaf ${\mathcal{H}}$ such that
${\mathcal{T}}or^1({\mathcal{H}}, {\mathcal{O}}_D) = 0$ and
$\text{supp}({\mathcal{H}})$ does not meet the baselocus of
$\text{Im}(T)$. We have ${\mathcal{G}}=\widehat{{\mathcal{H}}\otimes{\mathcal{O}}(*D)}$ and
${\mathcal{F}}_k={\mathcal{H}}\otimes{\mathcal{O}}(krD)$. The
Koszul complex associated to
$T$ is therefore exact on the support of ${\mathcal{H}}|_{D_r}$, and we can apply
the previous theorem in combination with theorem \ref{generically free}.
\begin{Thm}\label{cohomological} For ${\mathcal{G}}$ to be free over ${\mathcal{D}}_1$ in
a neighborhood of
${\mathcal{L}}
\in \text{Pic}^0(Z)$, the following conditions are sufficient:
\begin{enumerate}
\item For all $k\ge 0$,\ $i>0$,\ $H^i(Z,{\mathcal{L}}\otimes{\mathcal{F}}_k)=0$.
\item There exists $j\ge 2$ such that for all $i\ne j$\ ,
$H^i(Z,{\mathcal{L}}\otimes{\mathcal{F}}_{-1})=0$.
\item $0 \rightarrow H^1(D_r, {\mathcal{R}}_2
\otimes{\mathcal{L}}\otimes {\mathcal{H}} \otimes {\mathcal{M}}^j)
\rightarrow \wedge^2 V \otimes H^1(D_r, {\mathcal{L}}\otimes{\mathcal{H}} \otimes \cal
M^{j-1})$\ is exact for all $j\ge 1$
\item $H^0(D_r,{\mathcal{R}}_1\otimes\cal
L\otimes
{\mathcal{H}})=0$.\end{enumerate}
\end{Thm}
Nakayashiki's embedding may now be recovered. As in section \ref{nak}, we
take $Z$ to be $X$ itself, and we assume $D$ is smooth. We take an $\cal
A$-module structure on ${\mathcal{O}}(*D)$ associated to a $g$-dimensional
basepoint-free linear system $V\subset H^0(D,{\mathcal{N}})$
mapping isomorphically onto ${\mathfrak{g}}$.
We take ${\mathcal{H}}={\mathcal{O}}(D)$. This affects only the
filtration, not the ${\mathcal{A}}$-module structure. Thus
\begin{equation}
{\mathcal{F}}_k={\mathcal{O}}((k+1)D)\ .
\end{equation}
\begin{Thm} Set ${\mathcal{G}}=\widehat{{\mathcal{O}}(*D)}$. Then ${\mathcal{G}}$ is a
free
${\mathcal{D}}$-module in a neighborhood of any ${\mathcal{L}} \ne {\mathcal{O}}$.
\end{Thm}
\begin{pf} By theorem
\ref{cohomological}, it suffices to verify the following:
\begin{enumerate}
\item For all $i>0$, $k>0$, $H^i(X,{\mathcal{L}}(kD))=0$.
\item There exists $j\ge 2$ such that for all $i\ne j$,
$H^i(X,{\mathcal{L}})=0$.
\item For all $j\ge 1$,\ $0\to
H^1(D,{\mathcal{R}}_2\otimes {\mathcal{L}}\otimes {\mathcal{N}}^{j+1})\to
\wedge^2(V)\otimes H^1(D,{\mathcal{L}}\otimes {\mathcal{N}}^j)$\ is exact, where $\cal
R_j$ are the kernel sheaves of the Koszul complex $\wedge^j(V)\otimes\cal
N^{1-j}$.
\item $H^0(D, {\mathcal{R}}_1\otimes{\mathcal{L}}\otimes{\mathcal{N}})=0$.
\end{enumerate}
Items 1 and 2 are well-known. See, for example, \cite[sec. 13, sec.
16]{Mum}. It then follows from the exact sequence
\begin{equation} 0 \rightarrow {\mathcal{L}}((k-1)D) \rightarrow {\mathcal{L}}(kD)
\rightarrow
{\mathcal{L}} \otimes {\mathcal{N}}^k \rightarrow 0
\end{equation} that for all $k$ and all $0<i<g-1$,\ $H^i(D, {\mathcal{L}} \otimes
{\mathcal{N}}^k) = 0$. Since ${\mathcal{R}}_{g-1}={\mathcal{N}}^{1-g}$, it follows by
descending induction that
\begin{equation}\label{descent} H^i(D, {\mathcal{R}}_j \otimes {\mathcal{L}} \otimes \cal
N^k) = 0
\text{ for $0<i<j$ and all $k$.}
\end{equation}
In particular, 3
holds. Taking $k=1$ we get a stronger statement, also
by descending induction:
\begin{equation} H^i(D, {\mathcal{R}}_j \otimes {\mathcal{L}} \otimes \cal
N) = 0
\text{ for $i<j$.}
\end{equation}
Thus 4 holds.
\end{pf}
{}From the standpoint of integrable systems, the relevant feature of a $\cal
D$-module is its endomorphism ring. As we saw in section
\ref{krich}, if $C$ is a curve embedded in its Jacobian,
$\widehat{{\mathcal{O}}(*P)}$ is a ${\mathcal{D}}_1$-module, where ${\mathcal{D}}_1 =
{\mathcal{O}}[\xi]$. Extending this ${\mathcal{D}}_1$-module structure to a
${\mathcal{D}}$-module structure, we have
\begin{equation} H^0(C,{\mathcal{O}}(*P)) = \text{End}_{{\mathcal{A}}}({\mathcal{O}}_C(*P)) =
\text{End}_{{\mathcal{D}}}(\widehat{{\mathcal{O}}_C(*P)})\ .
\end{equation} It is the analysis of this endomorphism ring which leads to
the $KP$-hierarchy. Indeed, having trivialized $\widehat{{\mathcal{O}}_C(*P)}$ as a
${\mathcal{D}}_1$-module, its ${\mathcal{D}}$-endomorphisms are then differential operators
in one variable with $g-1$ parameters, satisfying certain nonlinear
differential equations.
More generally, one may hope to associate dynamics to the endomorphism ring
of a
${\mathcal{D}}$-module coherent over a proper subalgebra, ${\mathcal{O}}[\xi_1,
\dots,
\xi_n]
\subset {\mathcal{D}}$, $n < g$. Those modules which are free over the smaller
algebra provide a natural starting point for such an investigation. Note
that the presence of any nontrivial endomorphisms in such a setting is
already a strong condition on the ${\mathcal{D}}$-module, but one which can easily be
satisfied by the methods presented here. Such examples will be the object of
study in the sequel.
Finally, by way of advertisement, we mention
\vskip 12pt
\noindent{\bf Example: The Fano Surface.}
\vskip 12pt If $Z \subset \Bbb C \Bbb P^4$ is a smooth cubic hypersurface,
then its family of lines\hfill\break
\hbox{$S = \{\ell \in \text{Gr}(2, 5)\ |\ \ell
\subset Z\}$} is a smooth surface \cite{CG}. For generic $s \in S$
corresponding to a line $\ell_s$, the set \hbox{$\{t \in S\ |\ \ell_t
\cap
\ell_s = \{pt\}\}$} is a smooth ample hypersurface $D_s$, the incidence
divisor. We have isomorphisms
\begin{equation}
\text{Alb}(S) \approx \text{Pic}^0(S) \approx J(Z)\ ,
\end{equation} where $J(Z)$ is the intermediate Jacobian of $Z$. The
albanese map is identified with the assignment $s \rightsquigarrow D_s$,
which gives an embedding
$S
\subset \text{Pic}^0(S)$. The dimension of $\text{Pic}^0(S)$ is 5.
Setting ${\mathcal{N}} =$ normal bundle of $D_s$ in $S$ and $T_s(S) =$ tangent space
to $S$ at $s$, we have an isomorphism $T_s(S) \approx H^0(D_s, {\mathcal{N}})$. The
image of the natural map $H^0(D_s, {\mathcal{N}})
\rightarrow H^1(S, {\mathcal{O}})$, gives a subspace
\begin{equation}
\Bbb C^2 \approx V \subset H^1(S, {\mathcal{O}}) \approx \Bbb C^5.
\end{equation}
Set ${\mathcal{D}}_1={\mathcal{O}}[\xi_1,\xi_2]\subset{\mathcal{D}}$, where $\xi_1$ and $\xi_2$ are a basis
for $V$. Then the conditions of theorem
\ref{cohomological} are fulfilled with ${\mathcal{H}} = {\mathcal{O}}(2D)$. Thus
$\widehat{{\mathcal{O}}(*D)}$ is a ${\mathcal{D}}$-module, locally free as a $\cal
D_1$-module at a generic point. The rank of $\widehat{{\mathcal{O}}(*D)}$ as a
${\mathcal{D}}_1$-module is the degree of the map $D_s \rightarrow \Bbb P^1$
corresponding to the linear system $H^0(D_s, {\mathcal{N}})$. This degree is also
5. Thus we have a representation of $H^0(S, {\mathcal{O}}(*D))$ as
$5
\times 5$ matrix partial differential operators in two variables, with
$3 (=
\dim(H^1(S, {\mathcal{O}})) - \dim(V))$ parameters.
|
1996-02-19T06:20:16 | 9602 | alg-geom/9602013 | en | https://arxiv.org/abs/alg-geom/9602013 | [
"alg-geom",
"math.AG"
] | alg-geom/9602013 | V. Batyrev | Victor V. Batyrev and Yuri Tschinkel | Rational points on some Fano cubic bundles | 8 pages., LaTeX | null | null | null | null | We consider some families of smooth Fano hypersurfaces $X_{n+2}$ in ${\bf
P}^{n+2} \times {\bf P}^3$ given by a homogeneous polynomial of bidegree
$(1,3)$. For these varieties we obtain lower bounds for the number of
$F$-rational points of bounded anticanonical height in arbitrary nonempty
Zariski open subset $U \subset X_{n+2}$. These bounds contradict previous
expectations about the distribution of $F$-rational points of bounded height on
Fano varieties.
| [
{
"version": "v1",
"created": "Sat, 17 Feb 1996 19:42:55 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Batyrev",
"Victor V.",
""
],
[
"Tschinkel",
"Yuri",
""
]
] | alg-geom | \section{Cubic bundles}
Let $X_{n+2}$ be a hypersurface in ${\bf P }^n \times {\bf P }^3$ $(n \geq
1)$ defined by the equation
$$ P({\bf x},{\bf y}) = \sum_{i =0}^3 l_i({\bf x})y_i^3 = 0$$
where
$$P({\bf x},{\bf y}) \in {\bf Q } [ x_0, \ldots, x_n,
y_0,\ldots, y_3 ]$$
and $l_0({\bf x}), \ldots, l_3({\bf x})$ are
homogeneous linear forms in $x_0, \ldots, x_n$.
Put $k = \max ( n+1, 4)$.
We shall always assume that any $k$ forms among
$ l_0({\bf x}), \ldots, l_3({\bf x})$ are linearly independent.
It is elementary to
check the following statements:
\begin{prop}
The hypersurface $X_{n+2}$ is a smooth Fano variety
containing a Zariski open subset
$U_{n+2}$ which is isomorphic to ${\bf A}^{n+2}$.
\end{prop}
\begin{prop}
Let $U_P \subset {\bf P }^n$ be the Zariski open subset
defined by the condition
\[ \prod_{i =0}^3 l_0({\bf x}) \neq 0. \]
Then the fibers of the natural projection $\pi \,:\, X_{n+2} \rightarrow {\bf P }^n$
over closed points of $U_P$ are
smooth diagonal cubic surfaces in ${\bf P }^3$.
\end{prop}
\noindent
By Lefschetz theorem, we have:
\begin{prop}
The Picard group of $X_{n+2}$ over an arbitrary field containing ${\bf Q }$ is
isomorphic to ${{\bf Z }}\oplus {\bf Z }$.
\label{pic}
\end{prop}
\section{Heights on cubic surfaces}
Let $F$ be a number field, ${\rm Val}(F)$ the set of all
valuations of $F$, $W$ a projective algebraic variety over $F$, $W(F)$
the set of $F$-rational points of $W$,
$D$ a very ample divisor on $W$, and ${ \gamma} =
\{ s_0, \ldots, s_m\}$ a basis over $F$
of the space of global sections $ \Gamma (W, {\cal O}(D))$.
The {\em height function associated with $D$ and
${ \gamma}$}
$$H(W, D,{ \gamma},x)\; : \; W(F) \rightarrow {\bf R }_{>0} $$
is given by the formula
\[ H(W, D,{ \gamma},x) = \prod_{v \in {\rm Val}(F)}
\max_{i=0, \ldots, m} | s_i(x) |_v, \]
where $|\cdot |_v \,:\, F_v \rightarrow {{\bf R }}_{>0}$ is the multiplier
of a Haar measure on the additive group of the
$v$-adic completion of $F$.
\begin{dfn}
{\rm Let $Z \subset W$ be a locally closed algebraic
subset of $W$, $B$ a positive
real number. Define
\[ N(Z, D,{ \gamma},B) := {\rm Card}\{ x \in W(F) \cap Z \mid
H(W, D,{ \gamma},x) \leq B \}.\]}
\end{dfn}
\noindent
The following classical statement is due to A. Weil:
\begin{theo}
Let ${ \gamma'} = \{ s_0', \ldots, s_m' \}$ be another basis in
$ \Gamma (W, {\cal O}(D))$.
Then there exist two positive constants $c_1,c_2$ such that
\[ c_1 \leq \frac{H(W, D,{ \gamma},x)}{H(W, D,{ \gamma'},x)} \leq c_2 \]
for all $x \in W(F)$.
\label{weil}
\end{theo}
For a smooth projective variety $W$ we denote by $-K_W$
the anticanonical divisor on $W$.
\begin{theo}
Let $Y \subset {\bf P}^3$ be a smooth cubic surface defined over $F$,
${ \gamma} = \{ s_0, s_1, s_2, s_3 \}$ the basis of
global sections of ${\cal O}(-K_Y)$ corresponding to the standard
homogeneous coordinates on ${{\bf P }}^3$.
Assume that $Y$ can be obtained by blowing up of $6$ $F$-rational
points in ${\bf P }^2$. Then for any nonempty
Zariski open subset $U \subset Y$ one has
\[ N(U, -K_Y, { \gamma}, B) \geq c B (\log B)^3 \]
for all $B > 0$ and for some positive constant $c$.
\label{cub}
\end{theo}
\noindent
{\em Proof.} Assume that $Y$ is obtained by blowing up of
$p_1, \ldots, p_6 \in {\bf P }^2(F)$. By \ref{weil},
we can assume without loss
of generality that $p_1 = (1:0:0)$, $p_2 = (0:1:0)$ and
$p_3 = (0:0:1)$. Denote by
$Y_0$ the Del Pezzo surface obtained by blowing up $p_1, p_2, p_3$.
Let $f \,: \, Y \rightarrow Y_0$ be the contraction
of exceptional curves $C_4, C_5, C_6 \subset Y$ lying over $p_4,p_5,p_6$.
Let $V$ be the $10$-dimensional space over $F$ of all homogeneous
polynomials of degree
$3$ in variables $z_0, z_1, z_2$. We identify
$ \Gamma (Y, {\cal O}(-K_Y))$ with the subspace in $V$ consisting
of all polynomials vanishing in $p_1, \ldots, p_6$.
Analogously, we identify
$ \Gamma (Y_0, {\cal O}(-K_{Y_0}))$ with the subspace in $V$ consisting
of all polynomials vanishing in $p_1,p_2,p_3$. Let
$ \gamma_0 = \{ s_0, \ldots, s_6 \} \subset V$ the extension of the basis
${ \gamma}$ to a basis of the subspace $ \Gamma (Y_0, {\cal O}(-K_{Y_0})) \subset V$.
The surface $Y_0$ is an smooth equivariant compactification of the split
$2$-dimensional algebraic torus over $F$
$$({\bf G}_{m})^2 = {\bf P }^2 \setminus
\{ l_{12}, l_{13}, l_{23} \}$$
where $l_{ij}$ denotes the projective line in ${\bf P }^2$ through $p_i$ and $p_j$.
Since $Y_0$ is a smooth toric variety,
the main theorem in \cite{BaTschi1} shows that
the following asymptotic formula holds:
\begin{equation}
N(({\bf G}_{m})^2, -K_{Y_0}, \gamma_0', B) =
c_0 B (\log B)^3(1 + o(1)),\;\;
B \rightarrow \infty,
\label{f0}
\end{equation}
where $c_0$ is some positive constant and
\[ \gamma_0' = \{ z_0z_1z_2, z_1^2z_2, z_1 z_2^2, z_2^2z_0, z_2z_0^2,
z_0^2z_1, z_0 z_1^2 \}. \]
Let $U$ be any nonempty Zariski open subset in $Y$. We denote by
$U_0$ a nonempty open subset in $U$ such that the restriction
of $f$ on $U_0$ is an isomorphism and $f(U_0)$ is contained in
$({\bf G}_{m})^2 \subset Y_0$. Since
\[ \prod_{v \in {\rm Val}(F)}
\max_{i=0, \ldots, 3} | s_i(x) |_v \leq
\prod_{v \in {\rm Val}(F)}
\max_{i=0, \ldots, 6} | s_i(x) |_v \]
holds for every $F$-rational point $x \in U_0$, we
obtain
\begin{equation}
N(U_0, -K_Y, \gamma, B) \geq N(U_0, -K_{Y_0}, \gamma_0, B)
\label{f1}
\end{equation}
for any $B >0$. By \ref{weil}, there exists a positive constant
$c_3$ such that
\begin{equation}
N(U_0, -K_{Y_0}, \gamma_0, B) \geq c_3 N(U_0, -K_{Y_0}, \gamma_0',
B).
\label{f2}
\end{equation}
On the other hand,
\begin{equation}
N(U_0, -K_{Y_0}, \gamma_0',
B) = N(({\bf G}_{m})^2, -K_{Y_0}, \gamma_0', B) - N(Z, -K_{Y_0}, \gamma_0',
B),
\label{f3}
\end{equation}
where $Z = ({\bf G}_{m})^2 \setminus U_0$. Let $Z_1, \ldots,
Z_l$ be the irreducible components of $Z$, $\overline{Z}_i$
the closure of $Z_i$ in $Y_0$ $(i =1, \ldots, l)$. It is known
that
\begin{equation}
N(Z_i, -K_{Y_0}, \gamma_0',B) \leq c_4 B^{2/({\rm deg}\,
\overline{Z}_i)}
\label{f4}
\end{equation}
holds for some positive constant $c_4$,
where ${\rm deg}\,\overline{Z}_i$ denotes the degree of
$\overline{Z}_i$ with respect to the anticanonical divisor
$-K_{Y_0}$. Since every irreducible curve $C \subset Y_0$ with
${\rm deg}\, C = 1$ is a component of $Y_0 \setminus ({\bf G}_m)^2$, we
have ${\rm deg}\,\overline{Z}_i \geq 2$; i.e.,
\begin{equation}
N(Z_i, -K_{Y_0}, \gamma_0',B) \leq c_4 B
\label{f5}
\end{equation}
holds for all $i = 1, \ldots, l$.
It follows from the asymptotic formula (\ref{f0}) combined with
(\ref{f1}), (\ref{f2}) and (\ref{f3})
that there exists a positive constant $c$ such that
\[ N(U_0, -K_Y, { \gamma}, B) \geq c B (\log B)^3 \]
holds for all $B > 0$. This yields the statement, since $U_0$ is
contained in $U$.
\hfill $\Box$
\begin{coro}
Let
$Y$ be a smooth diagonal
cubic surface in ${\bf P }^3$ defined by the equation
\[ a_0 y_0^3 + a_1 y_1^3 + a_2 y_2^3 + a_3 y_3^3 = 0 \]
with coefficients $a_0, \ldots, a_3$ in a number field
$F$ which contains ${{\bf Q }}(\sqrt{-3})$. Assume that
there exist numbers $b_0, \ldots, b_3 \in F^*$ such that
$a_i = b_i^3$ $(i =0, \ldots, 3)$. Then for any nonempty
Zariski open subset $U \subset Y$ one has
\[ N(U, -K_Y, \gamma, B) \geq c B (\log B)^3 \]
for all $B > 0$ and some positive constant $c$.
\label{cubic1}
\end{coro}
\noindent
{\em Proof.} It follows from our assumptions on the coefficients
$a_0, \ldots, a_3$ and on the field $F$ that
all $27$ lines on $Y$ are defined over $F$. Hence,
$Y$ can be obtained from ${{\bf P }}^2$ by blowing up of $6$
$F$-rational points. Now the statement follows from
\ref{cub}. \hfill $\Box$
\section{Rational points on $X_{n+2}$}
We have the natural isomorphism
\[ \Gamma (X_{n+2}, {\cal O}(-K_{X_{n+2}}) ) \cong
\Gamma ({\bf P }^3, {\cal O}(1)) \otimes \Gamma ({\bf P }^n, {\cal O}(n)).\]
Let $\{ t_0, \ldots, t_m \}$ be a basis
in $ \Gamma ({\bf P }^n, {\cal O}(n))$ and $\{ s_0, s_1, s_2, s_3 \}$ the
standard basis in $ \Gamma ({\bf P }^3, {\cal O}(1))$. Denote by
$ \gamma$ the basis of $ \Gamma (X_{n+2}, {\cal O}(-K_{X_{n+2}}) )$
consisting of $s_i \otimes t_j$ $(i=0,\ldots, 3; \; j =
0, \ldots, m)$.
\begin{theo} Let $n \geq 3$. Then
for any nonempty Zariski open subset $U \subset X_{n+2}$ and for
any field $F$ containing ${{\bf Q }}(\sqrt{-3})$, one has
\[ N(U, -K_{X_{n+2}}, \gamma, B) \geq c B (\log B)^3 \]
for all $B > 0$ and some positive constant $c$.
\label{t1}
\end{theo}
\noindent
{\em Proof.} Consider the projection $\pi\,:\,U\rightarrow {\bf P }^n$.
Let $U'$ be a Zariski open subset in $\pi (U) \cap U_P$ such that
the image of $U'$ under the dominant (rational) mapping
\[ \psi \;:\; {\bf P }^n - - \rightarrow {\bf P }^3 \]
\[ \psi(x_0 : \ldots : x_n) =
(l_0({\bf x}): \ldots : l_3({\bf x})) \]
is Zariski open in ${\bf P }^3$.
Denote by
$ \varphi $ the finite morphism
\[ \varphi \; : \; {\bf P }^3 \rightarrow {\bf P }^3, \]
\[ \varphi (z_0 : \ldots : z_3) = (z_0^3 : \ldots : z_3^3). \]
Since ${\bf P }^3(F)$ is Zariski dense in ${\bf P }^3$,
there is a point $p \in {\bf P }^3(F) \cap \varphi ^{-1}(\psi(U'))$.
Since $U'(F) \cap \psi^{-1}( \varphi (p))$
is Zariski dense in $\psi^{-1}( \varphi (p))$,
there exists $q \in U'(F) \cap
\psi^{-1}( \varphi (p))$. Therefore, the fiber of $\pi $ over $q$
is a diagonal cubic surface $Y_q$ such that and $U \cap Y_q \subset Y_q$ is
a nonempty Zariski open subset. It remains to apply \ref{cubic1}.
\hfill $\Box$
\begin{theo}
Let $n =2$. Then there exists a
number field $F_0$ depending only on $X_{n+2}$ such that
for any nonempty Zariski open subset $U \subset X_{n+2}$ for any field
$F$ containing $F_0$ one has
\[ N(U, -K_{X_{n+2}}, \gamma, B) \geq c B (\log B)^3 \]
for all $B > 0$ and some positive constant $c$.
\label{t2}
\end{theo}
\noindent
{\em Proof.}
Let $U'$ be a Zariski open subset in $\pi (U) \cap U_P$. We have the
linear embedding
\[ \psi \;:\; {\bf P }^2 \hookrightarrow {\bf P }^3 \]
defined by $l_0({\bf x}), \ldots, l_3({\bf x})$.
Then $ \varphi ^{-1}(\psi({\bf P }^2))$ is a smooth diagonal cubic surface
$S \subset {\bf P }^3$ defined over ${\bf Q }$. Let $F_0$ be a finite
extension of ${\bf Q }(\sqrt{-3})$ such that $S(F_0)$ is Zariski dense in $S$.
Then there exists a point $p \in S(F_0)$ such that $q = \varphi (p)$ is contained
in $U'$. Therefore,
the fiber of $\pi$ over $q$
is a diagonal cubic surface $Y_q$, and $U \cap Y_q \subset Y_q$ is
a nonempty Zariski open subset. It remains to apply
\ref{cubic1}.
\hfill $\Box$
\begin{theo}
Let $n = 1$. Then for
any nonempty Zariski open subset $U \subset X_{n+2}$, there exists a
number field $F_0$ $($which depends on $U$$)$ such that for any field
$F$ containing $F_0$, one has
\[ N(U, -K_{X_{n+2}}, \gamma, B) \geq c B (\log B)^3 \]
for all $B > 0$ and some positive constant $c$.
\label{t3}
\end{theo}
\noindent
{\em Proof.}
Let $U'$ be a Zariski open subset in $\pi (U) \cap U_P$. We have the
linear embedding
\[ \psi \;:\; {\bf P }^1 \hookrightarrow {\bf P }^3 \]
defined by $l_0({\bf x}), \ldots, l_3({\bf x})$.
Then $ \varphi ^{-1}(\psi({\bf P }^1))$ is an algebraic curve
$C \subset {\bf P }^3$ which is a complete intesection of two
diagonal cubic surfaces defined over ${\bf Q }$.
Let $F_0$ be a finite extension of ${\bf Q }(\sqrt{-3})$ such that
there exists an $F_0$-rational point $p \in C(F_0) \cap
\varphi ^{-1}(U')$.
Then the fiber of $\pi$ over $q = \varphi (p)$
is a diagonal cubic surface $Y_q$ and $U \cap Y_q \subset Y_q$ is
a nonempty Zariski open subset. It remains to apply
\ref{cubic1}.
\hfill $\Box$
\section{Conclusions}
The following statement, inspired by the Linear Growth conjecture
of Manin (\cite{manin1}) and by extrapolation
of known results (circle method, flag varieties,
toric varieties), has been expected to be true
\cite{bat.man,franke-manin-tschinkel}:
\begin{conj}
Let $X$ be a smooth Fano variety over a number field $E$.
Then there exist
a Zariski open subset $U \subset X$ and a finite
extension $F_0$ of $E$ such that for all number fields $F$
containing $F_0$ the following
asymptotic formula holds
\[ N(U, -K_X, \gamma, B) = c B (\log B)^{t-1} (1 + o(1)), \;\;
B \rightarrow \infty ,\]
where $t$ is the rank of the Picard group of
$X$ over $F$.
\label{conjecture}
\end{conj}
Some lower and upper bounds for $N(U, -K_X, \gamma, B)$ for Del Pezzo
surfaces and Fano threefolds have been obtained in
\cite{manin1,MaTschi}.
The conjecture \ref{conjecture} was refined by E. Peyre who proposed
an adelic interpretation for the constant $c$ introducing Tamagawa numbers
of Fano varieties \cite{peyre}. This refined version of
the conjecture has been proved for toric varieties in
\cite{BaTschi,BaTschi1}.
The statements in Theorems \ref{t1}, \ref{t2}, \ref{t3} and the
property \ref{pic} show
that Conjecture \ref{conjecture} is not true for Fano
cubic bundles $X_{n+2}$ $(n \geq 1)$.
|
1996-02-28T06:21:02 | 9602 | alg-geom/9602022 | en | https://arxiv.org/abs/alg-geom/9602022 | [
"alg-geom",
"math.AG"
] | alg-geom/9602022 | Chikashi Miyazaki | Chikashi Miyazaki and Wolfgang Vogel | Towards a theory of arithmetic degrees | LaTeX, 14 pages | null | null | null | null | The aim of this paper is to start a systematic investigation of the
arithmetic degree of projective schemes as introduced by D. Bayer and D.
Mumford. One main theme concerns itself with the behaviour of this arithmetic
degree under hypersurface sections. The notion of arithmetic degree involves
the new concept of length-multiplicity of embedded primary ideals. Therefore it
is much harder to control the arithmetic degree under a hypersurface section
than in the case for the classical degree theory. Nevertheless it has important
and interesting applications. We describe such applications to the
Castelnuovo-Mumford regularity and to Bezout-type theorems.
| [
{
"version": "v1",
"created": "Wed, 28 Feb 1996 01:14:46 GMT"
},
{
"version": "v2",
"created": "Wed, 28 Feb 1996 03:53:31 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Miyazaki",
"Chikashi",
""
],
[
"Vogel",
"Wolfgang",
""
]
] | alg-geom | \section{Introduction}
The aim of this paper is to start a systematic investigation of the
arithmetic degree of projective
schemes as introduced in \cite{BM}. One main theme concerns itself
with the
behaviour of
this arithmetic degree under hypersurface sections, see Theorem 2.1.
The classical intersection theory only considers the top-dimensional
(or
isolated) primary
components. However, the notion of arithmetic degree involves the
new concept of
length-multiplicity of embedded primary ideals as considered in \cite{BM},
\cite{EH},
\cite{H}, \cite{K}, \cite{STV}. Therefore it is much harder to control
the
arithmetic
degree under a hypersurface section than in the case for the classical
degree theory.
We describe in \S 3 an upper bound for the arithmetic degree in terms of
the Castelnuovo-Mumford regularity, see Theorem 3.1. In addition, we
generalize Bezout's theorem via iterated hypersurface sections, see
Theorem 4.1. We conclude in \S 5 by studying two examples.
\section{Arithmetic degree and hypersurface sections}
Before stating our main result of this section we need to introduce
the concept
of the length-multiplicity of (embedded) primary components.
Let $K$ be an arbitrary field and $S$ the polynomial ring
$K[x_0,\cdots,x_n]$.
Let ${\sf m}=(x_0,\cdots,x_n)$ be the homogeneous maximal ideal
of $S$.
Let $I$ be a
homogeneous ideal of $S$.
\vspace{5mm}
\noindent{\bf Definition (\cite{BM}):}
Let $\sf p$ be a homogeneous prime ideal belonging to $I$. For
a primary
decomposition
$I=\cap {\sf q}$ we take the primary ideal $\sf q$ with
$\sqrt{{\sf q}}={\sf p}$.
Let $J$ be the intersection of all primary components of $I$ with
associated prime ideals
${\sf p}_1$ such that
${\sf p}_1\rixrel{\neq}{\subset}{\sf p}$. If
the prime ideal
$\sf p$ is an isolated component of $I$, we set $J=S$. We define
the
length-multiplicity
of $\sf q$ denoted by $\mbox{mult}_I({\sf p})$, to be the length
$\ell$
of a maximal
strictly increasing chain of ideals
$${\sf q}\cap J =: J_\ell\subset J_{\ell-1}\subset\cdots\subset
J_1\subset J_0 := J$$
where $J_k$, $1\leq k\leq \ell-1$, equals ${\sf q}_k\cap J$ for
some
$\sf p$-primary
ideal ${\sf q}_k$.
Despite the non-uniqueness of embedded components, the number
$\ell=\mbox{mult}_I({\sf p})$ is
well-defined.
\vspace{5mm}
For a finitely generated graded $S$-module $M$, let $H(M,\ell)$
be the
Hilbert function of $M$ for all
integers $\ell$, that is, $H(M,\ell)$ is the dimension of the vector
space
$[M]_\ell$
over $K$. It is well-known that the Hilbert function of $M$ is a
polynomial
in $\ell$
for $\ell$ large enough. We denote this polynomial by $P(M,\ell)$.
We set $\Delta(H(M,\ell))=H(M,\ell)-H(M,\ell-1)$, $\Delta^0
H(M,\ell)=H(M,\ell)$,
and $\Delta^r(H(M,\ell))=
\Delta^{r-1}(\Delta H(M,\ell))$ for all integers $r\geq 2$. Moreover,
we set
$$\Delta_\tau(H(M,\ell))=H(M,\ell)-H(M,\ell-\tau)$$
for all integers $\tau\geq 1$. Further, the Hilbert polynomial $P(M,\ell)$
is written as
$$P(M,\ell) = \frac{e}{d!}\ell^d + (\mbox{lower order terms}),
\quad e\neq 0.$$
Then we define $h$-dim $M=d$ (homogeneous dimension) and degree
of $M$ by
$\deg M := e$. Also, we write, for any ideal $I$ of $S$, $\dim
I$ and $\deg I$
for $h$-dim $S/I$ and $\deg S/I$ respectively. In case $P(M,\ell)=0$,
we
define $h$-dim $M=-1$ and
$\deg M=\sum_{\ell\in{\bf Z}} H(M,\ell)$. In particular, $\dim
{\sf m}=-1$ and $\deg {\sf m}=1$.
\vspace{5mm}
\noindent{\bf Definition (\cite{BM}):}
For an integer $r\geq -1$, we define
\begin{eqnarray*}
\mbox{arith-deg}_r (I) & = &
\sum_{\stackrel{{\sf p} \mbox{\ \scriptsize is a prime
ideal}}{\mbox{\scriptsize such that}
\dim{\sf p}=r}} \mbox{mult}_I({\sf p})\cdot\deg {\sf p}\\
& = & \sum_{\stackrel{{\sf p}\in \mbox{\scriptsize Ass }
S/I}{\mbox{\scriptsize such that }
\dim{\sf p}=r}}
\mbox{mult}_I(\sf p)\cdot\deg{\sf p}
\end{eqnarray*}
\vspace{3mm}
\noindent{\bf Definition (\cite{H}):}
Let $I$ be a homogeneous ideal of $S$. Let $r$ be an integer with
$r\geq -1$. We
define the ideal $I_{\geq r}$ as the intersection of all primary
components
$\sf q$
of $I$ with $\dim{\sf q}\geq r$.
\vspace{5mm}
\noindent{\bf Remark :}
Although primary decomposition is not uniquely determined, the
ideal $I_{\ge r}$ does not depend on the choice of
primary decomposition of $I$ and is again a homogeneous ideal.
\vspace{5mm}
The aim of this section is to prove the following theorem.
\vspace{5mm}
\noindent{\bf Theorem 2.1:}{\em\ Let $r$ be an integer with $r\geq
0$. Let
$I$ be
a homogeneous ideal of $S$. Let $F$ be a homogeneous polynomial of
$S$ with
$\mbox{degree }(F)=\tau\geq 1$. Assume that $F$ does not belong
to any
associated
prime ideal $\sf p$ of $I$ with $\dim {\sf p}\geq r$. Then we
have
$$\mbox{arith-deg}_{r-1}(I,F) - \mbox{arith-deg}_{r-1}(I_{\geq
r+1},F)\geq\tau\cdot
\mbox{arith-deg}_r (I)$$
and the equality holds if and only if $F$ does not belong to any
associated
prime
ideal $\sf p$ of $I$ with $\dim{\sf p}=r-1$.}
\vspace{5mm}
\noindent{\bf Corollary 2.2:}{\em\ Under the above condition,
$$\mbox{arith-deg}_{r-1}(I,F)\geq\tau\cdot\mbox{arith-deg}_r(I)$$
and the equality holds if and only if $F$ does not belong to any
associated
prime
ideal $\sf p$ of $I$ with $\dim{\sf p}=r-1$ and the ideal $(I_{\geq
r+1},F)$ has
no associated prime ideals of dimension $(r-1)$.}
\vspace{5mm}
\noindent{\em Proof.} Corollary 2.2 follows immediately from Theorem
2.1.
\vspace{5mm}
We note that Corollary 2.2 and Lemma 3 of \cite{KS} yield Theorem 2.3 of
\cite{STV}.
\vspace{5mm}
We want to consider generic hyperplane sections.
The following useful lemma is obtained from \cite{BF}, (4.2)
and \cite{F}, (5.2), which was pointed out to us by H. Flenner.
\vspace{5mm}
\noindent{\bf Lemma 2.3:}{\em\
Let $I$ be a homogeneous ideal of $S$. We set $A=S/I$. Let $h=0$
be the
defining
equation of a generic hyperplane of ${\bf P}_K^n$ ($K$: infinite
field).
Then we have
$$\mbox{Ass}(A/h)\verb+\+\{{\sf m}\}\rixrel{=}{\subset}
\bigcup_{{\sf p}\in
\mbox{\scriptsize Ass } A}
\mbox{Min} (A/({\sf p},h)),$$
where $\mbox{Min}(A/({\sf p},h))$ is the set of minimal primes
belonging to
$({\sf p},h)$.}
\vspace{5mm}
\noindent{\bf Corollary 2.4:}{\em\
Let $r$ be an integer $\geq 1$. Let $H$ be a generic hyperplane in
${\bf
P}_K^n$, given
by $h=0$. Then we have:
$$\mbox{arith-deg}_r (I) = \mbox{arith-deg}_{r-1} (I,h).$$
}
\vspace{5mm}
\noindent{\em Proof.} Corollary 2.4 follows immediately from Theorem
2.1 and Lemma 2.3.
\vspace{5mm}
We note that Corollary 2.4 is stated in \cite{BM}, page 33 without
proof.
\vspace{5mm}
Before we turn to the proof of Theorem 2.1, two technical results
are
needed. First we
state a more or less known result describing
a
different
characterization of the arithmetic degree, which is purely algebraic
and,
in fact,
serves as the definition in \cite{H}.
\vspace{5mm}
\noindent{\bf Lemma 2.5:}{\em\
Let $r$ be a non-negative integer. Let $I$ be a homogeneous ideal
of $S$.
Then we have
$$\mbox{arith-deg}_r (I) = \Delta^r (P(S/I,\ell) - P(S/I_{\geq
r+1},\ell))$$
for all integers $\ell$.}
\vspace{5mm}
\noindent{\bf Lemma 2.6:}{\em\
Let $r$ be an integer with $r\geq 1$. Let $I$ be a homogeneous ideal
of $S$
and $F$
a homogeneous polynomial of $S$ with $\deg(F)=\tau\geq 1$. Assume
that $F$
does not
belong to any associated prime ideal ${\sf p}$ of $I$ with $\dim
{\sf p}\geq r+1$. Then we have
\begin{eqnarray*}
\lefteqn{\mbox{arith-deg}_{r-1}(I,F)-\Delta^{r-1}P(S/(I_{\geq r+1},F),\ell)
+
\Delta^{r-1}P(S/(I,F)_{\geq r},\ell)} \\
& & \qquad\qquad= \tau\cdot\mbox{arith-deg}_r (I) + \Delta^{r-1}
P([0:F]_{S/I},\ell-\tau)
\end{eqnarray*}
for all integers $\ell$.}
\vspace{5mm}
\noindent{\em Proof.}
From the exact sequences
$$0\rightarrow [0:F]_{S/I}(-\tau)\rightarrow
S/I(-\tau)\stackrel{F}{\rightarrow}S/I
\rightarrow S/(I,F)\rightarrow 0$$
and
$$0\rightarrow S/I_{\geq r+1}(-\tau)\stackrel{F}{\rightarrow}S/I_{\geq
r+1}\rightarrow
S/(I_{\geq r+1},F)\rightarrow 0,$$
we have
$$\Delta_\tau P(S/I,\ell)=P(S/(I,F),\ell)-P([0:F]_{S/I},\ell-\tau)$$
and
$$\Delta_\tau P(S/I_{\geq r+1},\ell)=P(S/(I_{\geq r+1},F),\ell)$$
for all integers $\ell$.
Note that
$P(S/I,\ell) - P(S/I_{\ge r+1},\ell)$
is a numerical polynomial of degree $r$, see (2.5).
Thus we have
\begin{eqnarray*}
\lefteqn{\mbox{arith-deg}_{r-1}(I,F)-\Delta^{r-1}P(S/(I_{\geq r+1},F),\ell)
+
\Delta^{r-1}P(S/(I,F)_{\geq r},\ell)}\\[2mm]
& \qquad\qquad= & \Delta^{r-1}(P(S/(I,F),\ell)-P(S/(I,F)_{\geq
r},\ell))\\
& & -\Delta^{r-1}P(S/(I_{\geq r+1},F),\ell)+\Delta^{r-1}P(S/(I,F)_{\geq
r},\ell)\\[2mm]
& \qquad\qquad= & \Delta^{r-1}(\Delta_\tau
P(S/I,\ell)+P([0:F]_{S/I},\ell-\tau))\\
& & -\Delta^{r-1}\Delta_\tau P(S/I_{\geq r+1},\ell)\\[2mm]
& \qquad\qquad= & \Delta_\tau (\Delta^{r-1}(P(S/I,\ell)-P(S/I_{\geq
r+1},\ell)))\\
& & +\Delta^{r-1}P([0:F]_{S/I},\ell-\tau)\\[2mm]
& \qquad\qquad= & \tau\cdot\Delta^r(P(S/I,\ell)-P(S/I_{\geq r+1},\ell))
+
\Delta^{r-1}P([0:F]_{S/I},\ell-\tau)\\[2mm]
& \qquad\qquad= & \tau\cdot
\mbox{arith-deg}_r(I)+\Delta^{r-1}P([0:F]_{S/I},\ell-\tau),
\end{eqnarray*}
by Lemma 2.5. Hence the assertion is proved.
\vspace{5mm}
The following lemma is used in the proof of Theorem 2.1 and Lemma
4.3.
\vspace{5mm}
\noindent{\bf Lemma 2.7:}{\em\
Let $I$ be a homogeneous ideal of $S$. Let $r$ be an integer. Let
$F$ be a
homogeneous
polynomial of $S$ with degree $(F)\geq 1$ such that $F$ does not
belong to
any associated
prime ideal $\sf p$ of $I$ with $\dim{\sf p}\geq r$. Then we
have
$${(I_{\geq u},F)}_{\geq r} = (I,F)_{\geq r}$$
for all integers $u=-1,0,\cdots,r+1$.}
\vspace{5mm}
\noindent{\em Proof.}
We want to prove that
$${(I_{\geq u},F)}_{\sf p} = (I,F)_{\sf p}$$
for all prime ideals $\sf p$ with $\dim{\sf p}\geq r$. We may
assume
that $F\in{\sf p}$.
If $\sf p$ does not contain any primary component $\sf q$ of
$I$ with
$\dim({\sf q})
<u$, the proof is done. Now assume that there is a primary component
$\sf q$ of $I$ with
$\dim({\sf q})<u$ such that ${\sf p}\rixrel{=}{\supset}{\sf q}$.
Since $r\leq\dim({\sf p})
\leq\dim({\sf q})<u$, we see that $u=r+1$ and ${\sf p}=\sqrt{\sf q}$.
Thus we
have that $\sf p$ is an associated prime ideal of $I$ with $\dim{\sf p}=r$ and
$F\in{\sf p}$, which contradicts the hypothesis.
\vspace{5mm}
\noindent{\em Proof of Theorem 2.1.}
First we prove the case $r\geq 1$. Applying (2.5) and (2.6), we
have
\begin{eqnarray*}
\lefteqn{\mbox{arith-deg}_{r-1}(I,F)-\mbox{arith-deg}_{r-1}(I_{\geq
r+1},F)}\\
& \quad= & \tau\cdot\mbox{arith-deg}_r(I) + \Delta^{r-1}P(S/(I_{\geq
r+1},F),\ell)\\
& & -\Delta^{r-1}P(S/(I,F)_{\geq r},\ell) +
\Delta^{r-1}P([0:F]_{S/I},\ell-\tau)\\
& & -\Delta^{r-1}(P(S/(I_{\geq
r+1},F),\ell)-P(S/(I_{\geq r+1},F)_{\geq r},\ell))\\
& \quad= & \tau\cdot\mbox{arith-deg}_r(I) + \Delta^{r-1}[P(S/(I_{\geq
r+1},F)_{\geq r},\ell)
-P(S/(I,F)_{\geq r},\ell)]\\
& & + \Delta^{r-1}P([0:F]_{S/I},\ell-\tau).
\end{eqnarray*}
By the assumption, $[0:F]_{S/I}$ has at most $(r-1)$-dimensional
support,
which means
$\Delta^{r-1}P([0:F]_{S/I},\ell-\tau)\geq 0$. Further,
$\Delta^{r-1}P([0:F]_{S/I},\ell
-\tau)=0$ if and only if $F$ does not belong to any associated
prime ideal
$\sf p$ with
$\dim{\sf p}=r-1$. On the other hand, $S/(I_{\geq r+1},F)_{\geq
r}=S/(I,F)_{\geq r}$
by Lemma 2.7. Therefore we have
$$\mbox{arith-deg}_{r-1}(I,F) - \mbox{arith-deg}_{r-1}(I_{\geq
r+1},F)\geq\tau\cdot
\mbox{arith-deg}_r(I)$$
and the equality holds if and only if $F$ does not belong to any
associated
prime ideal
$\sf p$ with $\dim{\sf p}=r-1$.
Next we prove the case $r=0$. Now we see that
\begin{eqnarray*}
\mbox{arith-deg}_{-1}(I,F) & = & \mbox{length}_S(I,F)_{\geq 0}/(I,F)\\
& = & \sum_{\ell=0}^{N}(\dim_K{[S/(I,F)]}_\ell - \dim_K{[S/(I,F)_{\geq
0}]}_\ell)\\
& = & \sum_{\ell=0}^N(\dim_K{[S/I]}_\ell-\dim_K{[S/I]}_{\ell-\tau}\\
& & + \dim_K
{\left[[0:F]_{S/I}\right]}_{\ell-\tau}
- \dim_K{[S/(I,F)_{\geq 0}]}_\ell)
\end{eqnarray*}
for large $N$. Similarly, we see that
\begin{eqnarray*}
\mbox{arith-deg}_{-1}(I_{\geq 1},F) & = &
\sum_{\ell=0}^N\left(\dim_K{[S/I_{\geq 1}]}_\ell
- \dim_K[S/I_{\geq 1}]_{\ell-\tau}\right.\\
& & \left. - \dim_K{[{S/(I_{\geq 1},F)}_{\geq 0}]}_\ell\right)
\end{eqnarray*}
for large $N$. Hence we have
\begin{eqnarray*}
\lefteqn{\mbox{arith-deg}_{-1}(I,F) - \mbox{arith-deg}_{-1}(I_{\geq
1},F)}\\
& \qquad= & \sum_{\ell=0}^N(\dim_K{[I_{\geq 1}/I]}_\ell - \dim_K
{[I_{\geq
1}/I]}_{\ell-\tau})\\
& & - \sum_{\ell=0}^N\dim_K {[{(I_{\geq 1},F)}_{\geq 0}/{(I,F)}_{\geq
0}]}_\ell
+ \sum_{\ell=0}^N\dim_K{[{[0:F]}_{S/I}]}_{\ell-\tau}\\
& \qquad= & \sum_{\ell=N-\tau+1}^N\dim_K{[I_{\geq 1}/I]}_\ell
-
\sum_{\ell=0}^N\dim_K
{[{(I_{\geq 1},F)}_{\geq 0}/{(I,F)}_{\geq 0}]}_\ell\\
& & + \sum_{\ell=0}^N\dim_K{[{[0:F]}_{S/I}]}_{\ell-\tau}
\end{eqnarray*}
for large $N$. By the assumption,
$\sum_{\ell=0}^N\dim_K{[{[0:F]}_{S/I}]}_{\ell-\tau}
=\mbox{length}_S({[0:F]}_{S/I}) \geq 0$ for large $N$, and ${[0:F]}_{S/I}
=0$
if and only if
$F$ is a non-zero-divisor of $S/I$. On the other hand, we see
$\dim_K{[I_{\geq 1}/I]}_\ell
=P(I_{\geq 1}/I,\ell)=\mbox{arith-deg}_0 (I)$ for large $\ell$.
Further, we have
${(I_{\geq 0},F)}_{\sf p} = (I,F)_{\sf p}$ for all prime ideals
$\sf p$ with
$\dim{\sf p}=0$ by Lemma 2.7. Hence we have
$$\mbox{arith-deg}_{-1}(I,F) -
\mbox{arith-deg}_{-1}(I_{\geq 1},F)\geq\tau\cdot
\mbox{arith-deg}_0 (I)$$
and the equality holds if and only if $F$ is a non-zero-divisor of
$S/I$. This
completes the proof of Theorem 2.1.
\section{Castelnuovo-Mumford regularity}
Bayer and Mumford \cite{BM} give a bound for the arithmetic degree
in terms of the Castelnuovo-Mumford regularity. The aim of this section
is to describe improved bound on this degree.
Let $m = m(I)$ be the Castelnuovo-Mumford regularity (see, e.g.,
\cite{BM}, \cite{EG}, \cite{M}) for a homogeneous ideal $I$ of the
polynomial ring $S = K[x_0,\cdots,x_n]$. Then our main result is
the following theorem.
\vspace{5mm}
\noindent{\bf Theorem 3.1:}{\em\
Let $I$ be a homogeneous ideal of $S$.
Let $m = m(I)$ be the Castelnuovo-Mumford regularity of $I$.
Then we have, for any integer $r\geq 0$
$$\mbox{arith-deg}_r (I)\leq \Delta^r P(S/I,\ell)$$
for all integers $\ell\geq m-1$.}
\vspace{5mm}
We want to give two corollaries. The first one shows
that
(3.1) improves
the bound given in \cite{BM}, Proposition 3.6.
\vspace{5mm}
\noindent{\bf Corollary 3.2:}{\em\
For all $r \geq 0$, we have
\begin{eqnarray*}
\mbox{arith-deg}_r (I)
& \leq & \Delta^r P(S/I,m-1)\\
& \leq & \left(\begin{array}{c}m+n-r-1\\n-r\end{array}\right)\\
& \leq & m^{n-r}
\end{eqnarray*}}
\noindent{\bf Corollary 3.3:}{\em\
Let $t=\mbox{depth } S/I$. Then we have, for an integer
$r\geq 0$,
$$\mbox{arith-deg}_r (I)\leq\Delta^r H(S/I,\ell)$$
for all $\ell\geq m+r-t-1$ if $r-t$ is even, and for all $\ell\geq
m+r-t$
if $r-t$ is
odd.}
\vspace{5mm}
Before embarking on the proof of Theorem 3.1 and the corollaries
we need
two
lemmas. The first one follows from \cite{S} Nr.79 (see also \cite{SV},
Proof of Lemma
I.4.3).
\vspace{5mm}
\noindent{\bf Lemma 3.4:}{\em\
Let $I$ be a homogeneous ideal of $S$ and
$t=\mbox{depth } S/I$.
Then we have
\renewcommand{\labelenumi}{(\theenumi)}
\renewcommand{\theenumi}{\alph{enumi}}
\begin{enumerate}
\item $P(S/I,\ell) = H(S/I,\ell) - \sum_{i=0}^d (-1)^i\dim_K
{[H_{\sf m}^i(S/I)]}_\ell$
for all integers $\ell$.
\item $P(S/I,\ell) = H(S/I,\ell)$ for all $\ell\geq m-t$.
\end{enumerate}
}
\vspace{5mm}
\noindent{\bf Lemma 3.5:}{\em\
Let $I$ be a homogeneous ideal of $S$.
Then we have
$$\Delta^r P(I,\ell)\geq 0$$
\vspace{-10mm}
\noindent for all $\ell\geq m-1$ and $r\geq 0$.
}
\vspace{5mm}
\noindent{\em Proof.}
For a generic hyperplane $H$ given by $h=0$, we can take an exact
sequence
$$0\rightarrow I(-1)\stackrel{h}{\rightarrow}
I\rightarrow I_H\rightarrow 0 ,$$
where $I_H=(I,h)/h$.
From the exact sequence, we have
$$\Delta P(I,\ell) = P(I_H,\ell)$$
for all $\ell$ and $I_H$ is $m$-regular. Repeating this step, we
see that
$$\Delta^r P(I,\ell) = P(I_{H_1\cap\cdots\cap H_r},\ell)$$
for all $\ell$ and for generic hyperplanes $H_1,\cdots,H_r$ defined
by
$h_1=0,\cdots,h_r=0$, resp., where
$$I_{H_1\cap\cdots\cap H_r} = (I,h_1,\cdots,h_r)/(h_1,\cdots,h_r),$$
and that
$I_{H_1\cap\cdots\cap H_r}$ is $m$-regular. So ${(I_{H_1\cap\cdots\cap
H_r})}_{\geq 0}$
is also $m$-regular and \linebreak $\mbox{depth}_S S/{(I_{H_1\cap\cdots\cap
H_r})}_{\geq 0}\geq 1$.
Therefore we have
\begin{eqnarray*}
\Delta^r P(I,\ell) & = & P(I_{H_1\cap\cdots\cap H_r},\ell)\\
& = & P(S,\ell) - P(S/I_{H_1\cap\cdots\cap H_r},\ell)\\
& = & P(S,\ell) - P(S/{(I_{H_1\cap\cdots\cap H_r})}_{\geq 0},\ell)\\
& = & H(S,\ell) - H(S/{(I_{H_1\cap\cdots\cap H_r})}_{\geq 0},\ell)\\
& = & H({(I_{H_1\cap\cdots\cap H_r})}_{\geq 0},\ell)\geq 0
\end{eqnarray*}
for $\ell\geq m-1$, by (3.4), (b).
\vspace{5mm}
\noindent{\em Proof of Theorem 3.1.}
Without the loss of generality, we may assume\linebreak that $I$
is a saturated
ideal. First we
prove the case $r=0$. By Lemma 2.5, we have
$$(\ast)\qquad\qquad\mbox{arith-deg}_0(I)=P(S/I,\ell) - P(S/I_{\geq
1},\ell).$$
Now we want to show that $I_{\geq 1}$ is $m$-regular. From the short
exact
sequence
$$0\rightarrow I_{\geq 1}/I\rightarrow S/I\rightarrow S/I_{\geq
1}\rightarrow 0$$
and the fact that $I_{\geq 1}/I$ has at most 1-dimensional support and
by Grothendieck's vanishing theorem $H_{\sf m}^i(I_{\ge 1}/I) = 0$
for $i \ge 2$,
we have
$$0\rightarrow H_{\sf m}^1 (I_{\geq 1}/I)\rightarrow
H_{\sf m}^1(S/I)\rightarrow H^1_{\sf m}
(S/I_{\geq 1})\rightarrow 0$$
and $H_{\sf m}^i(S/I)\cong H_{\sf m}^i (S/I_{\geq 1})$ for $i\geq
2$.
Thus we have $I_{\geq 1}$
is $m$-regular. Hence
$$P(S/I_{\geq 1},\ell) = H(S/I_{\geq 1},\ell)\geq 0$$
for all $\ell\geq m-1$, by (3.4), (b). Therefore we have from $(\ast)$
$$\mbox{arith-deg}_0 (I)\leq P(S/I,\ell)$$
for all $\ell\geq m-1$.
Now let us assume $r>0$. By Corollary 2.4 we see
$$\mbox{arith-deg}_r (I) = \mbox{arith-deg}_0 (I,h_1,\cdots,h_r)$$
for generic hyperplanes $h_1,\cdots,h_r$. Thus we have
$$\mbox{arith-deg}_r (I)\leq P(S/(I,h_1,\cdots,h_r),\ell)$$
for all $\ell\geq m-1$. On the other hand, we see
\begin{eqnarray*}
P(S/(I,h_1,\cdots,h_r),\ell) & = & \Delta P(S/(I,h_1,\cdots,h_{r-1}),\ell)
\\
& & \qquad \vdots\\
& = & \Delta^r P(S/I,\ell)
\end{eqnarray*}
for all $\ell$. Hence the assertion is proved.
\vspace{5mm}
\noindent{\em Proof of Corollary 3.2.}
By Lemma 3.5, we have
$$\Delta^r P(S/I,\ell)\leq\Delta^r P(S,\ell)$$
for all $\ell\geq m-1$. On the other hand,
$\Delta^r P(S,\ell) =
\left(\begin{array}{c}
n+\ell-r\\n-r\end{array}\right)$. Hence the assertion
follows from Theorem 3.1.
\vspace{5mm}
\noindent{\em Proof of Corollary 3.3.}
By Lemma 3.4, we see that
$$\Delta^r P(S/I,\ell) = \Delta^r H(S/I,\ell)$$
for all $\ell\geq m+r-t$, and that
$$\Delta^r P(S/I,m+r-t-1)=\Delta^r H(S/I,m+r-t-1)-(-1)^{r-t}\dim_K
{[H_{\sf m}^t(S/I)]}_{m-t-1}$$
because
$$P(S/I,m-t-1) = H(S/I,m-t-1) - (-1)^t[H_{\sf m}^t(S/I)]_{m-t-1}.$$
Hence the assertion follows from Theorem 3.1.
\section{Bezout-type results}
The aim of this section is to state properties of arithmetic degree
under
iterated
hyperplane sections, and Bezout-type results. Our Theorem 4.1 describes
a Bezout's theorem in terms of the arithmetic degree.
\vspace{5mm}
\noindent{\bf Theorem 4.1:}{\em\
Let $I$ be a homogeneous ideal of $S := K[x_0,x_1,\cdots,x_n]$.
Let $r\geq
0$ and
$s\geq 1$ be integers with $r+1\geq s$. Let $F_1,\cdots,F_s$ be homogeneous
polynomials
of $S$ such that $F_i$ does not belong to any associated prime ideal
${\sf p}$ of
$(I,F_1,\cdots,F_{i-1})$ with $\dim {\sf p}\geq r-i+1$, for all
$i=1,\cdots,s$. Then we have
\renewcommand{\labelenumi}{(\theenumi)}
\renewcommand{\theenumi}{\roman{enumi}}
\begin{enumerate}
\item $\mbox{arith-deg}_{r-s} (I,F_1,\cdots,F_s)\geq\left[\prod_{i=1}^s
\mbox{degree} (F_i)\right]\cdot\mbox{arith-deg}_r (I)$;
\item We have equality in (i) if and only if
$({(I,F_1,\cdots,F_{i-1})}_{\geq r-i+2},F_i)$
has no $(r-i)$-dimensional primes and $F_i$ does not belong to any
associated prime ideal
${\sf p}$ of $(I,F_1,\cdots,F_{i-1})$ with $\dim {\sf p}=r-i$,
for all
$i=1,\cdots,s$;
\item Assume that there is an integer $t$ with $-1\leq t\leq r+1$
such that
$F_i$ does
not belong to any associated prime ideal ${\sf p}$ of $(I_{\geq
t},F_1,\cdots,F_{i-1})$ with
$\dim {\sf p}\geq r-i+1$, for all $i=1,\cdots,s$, and $(I_{\geq
t},F_1,\cdots,F_s)$ has no
$(r-s)$-dimensional associated prime ideals. Then we have equality
in (i)
if and only if
$F_i$ does not belong to any associated prime ideal ${\sf p}$ of
$(I,F_1,\cdots,F_{i-1})$
with $\dim {\sf p}=r-i$, for all $i=1,\cdots,s$.
\end{enumerate}}
\vspace{5mm}
\noindent{\em Proof.}
(i) and (ii) follow from (2.2).
In order to prove (iii) we need Lemma 4.2 and Lemma 4.3 below. First
we
replace the ideal
$I$ of (4.2) by the ideal $I_{\geq t}$ of (iii). Then Lemma 4.3 shows
that
we can apply
(ii) of (4.1). This provides our result (iii).
\vspace{5mm}
We note that special cases of (4.1) describe
generalizations of
classical results in the degree theory (see, e.g., \cite{FV}, \cite{V}).
\vspace{5mm}
We prove the two lemmas.
\vspace{5mm}
\noindent{\bf Lemma 4.2:}{\em\
Let $I$ be a homogeneous ideal of $S$. Let $r$ and $s$ be integers
with
$1 \leq s \leq r+1$. Let $F_1, \cdots, F_s$ be homogeneous polynomials
of $S$
with\linebreak
degree $(F_i) \geq 1$, $i=1,\cdots,s$, such that $F_i$ does not
belong to
any associated
prime ideal $\sf p$ of $(I,F_1,\cdots,F_{i-1})$ with $\dim{\sf p}\geq
r-i+1$, for all
$i=1,\cdots,s$. If the ideal $(I,F_1,\cdots,F_s)$ has no
$(r-s)$-dimensional associated
prime ideals, then the ideal $({(I,F_1,\cdots,F_{i-1})}_{\geq
r-i+2},F_i)$ has no
$(r-i)$-dimensional associated prime ideals, for all $i=1,\cdots,s$.}
\vspace{5mm}
\noindent{\em Proof.}
It is easy to see that the ideal $(I,F_1,\cdots,
F_i)$
has no
$(r-i)$-dimensional associated prime ideals. By Theorem 2.1, we have
$$\mbox{arith-deg}_{r-i} {(I,F_1,\cdots,F_i)}_{\geq r-i} -
\mbox{arith-deg}_{r-i}
({(I,F_1,\cdots,F_{i-1})}_{\geq r-i+2},F_i)\geq 0.$$
This shows that $\mbox{arith-deg}_{r-i}({(I,F_1,\cdots,F_{i-1})}_{\geq
r-i+2},F_i)=0$.
Thus the assertion is proved.
\vspace{5mm}
\noindent{\bf Lemma 4.3:}{\em\
Let $I$ be a homogeneous ideal of $S$. Let $r$ and $s$ be integers
with
$1\leq s\leq r+1$.
Let $t$ be an integer with $-1\leq t\leq r+1$. Let $F_1,\cdots,F_s$
be
homogeneous
polynomials of $S$ with degree $(F_i)\geq 1$, $i=1,\cdots,s$, such
that
$F_i$ does not
belong to any associated prime ideal $\sf p$ of $(I,F_1,\cdots,F_{i-1})$
with
$\dim{\sf p}\geq r-i+1$, for all $i=1,\cdots,s$. Then we have
$${(I_{\geq t},F_1,\cdots,F_{i-1})}_{\geq r-i+2}=
{(I,F_1,\cdots,F_{i-1})}_{\geq r-i+2}$$
for $i=1,\cdots,s$.}
\vspace{5mm}
\noindent{\em Proof.}
By Lemma 2.7, we have
\begin{eqnarray*}
{(I,F_1,\cdots,F_{i-1})}_{\geq r-i+2} & = &
{({(I,F_1,\cdots,F_{i-2})}_{\geq r-i+3},
F_{i-1})}_{\geq r-i+2}\\
& = & \cdots=\ {(\cdots{({(I,F_1)}_{\geq r},F_2)}_{\geq
r-1},\cdots,F_{i-1})}_{\geq r-i+2}\\
& = & {(\cdots{({(I_{\geq t},F_1)}_{\geq r},F_2)}_{\geq
r-1},\cdots,F_{i-1})}_{\geq r-i+2}.
\end{eqnarray*}
On the other hand, we see
$${(I_{\geq t},F_1,F_2,\cdots,F_{i-1})}_{\geq r-i+2}\subset
{(\cdots{({(I_{\geq t},F_1)}_{\geq r},F_2)}_{\geq
r-1},\cdots,F_{i-1})}_{\geq r-i+2}$$
and
$${(I_{\geq t},F_1,F_2,\cdots,F_{i-1})}_{\geq r-i+2}\supset
{(I,F_1,\cdots,F_{i-2})}_{\geq r-i+2}.$$
Therefore we have ${(I_{\geq t},F_1,\cdots,F_{i-1})}_{\geq r-i+2}
=
{(I,F_1,\cdots,F_{i-1})}_{\geq r-i+2}$ for all\linebreak $i=1,\cdots,s$.
\vspace{5mm}
\section{Some examples}
The first example sheds some light on Theorem 2.1 and Corollary 2.2
in case that
$F$ has degree one and is a non-zero-divisor on $S/I$. It shows that
we have no
equality in Corollary 2.2 even under these assumptions.
\vspace{5mm}
\noindent{\em Example 1:}
Let $S=K[x_0,x_1,x_2,x_3,y_1,y_2,\cdots,y_r]$ be a polynomial ring,
where
$r$ is a
non-negative integer. Take
${\sf q}=(x_0x_3-x_1x_2,x_0^2,x_1^2,x_0x_1)\subset S$, which
is a primary ideal belonging to $(x_0,x_1)$ (cf. \cite{SV}, Claim
1 on page
182).
We set $I={\sf q}\cap (x_0^2,x_1,x_2)$ and
$F(x_0,x_1,x_2,x_3)=x_3+G(x_0,x_1,x_2)$, where $G(x_0,x_1,x_2)$
is a linear form. Then we have by (2.1)
$$\mbox{arith-deg}_{r-1}(I,F) - \mbox{arith-deg}_{r-1}(I_{\geq r+1},F)
=\mbox{arith-deg}_r (I).$$
Now we will show that
$$\mbox{arith-deg}_{r-1}(I,F)>\mbox{arith-deg}_r (I).$$
For simplicity we assume that $G=0$, that is, $F=x_3$. Clearly,
$\mbox{arith-deg}_r (I)\linebreak=1$.
On the other hand,
\begin{eqnarray*}
(I,x_3) & = & (x_0^2,x_1^2,x_0x_1,x_0x_2x_3-x_1x_2^2) + (x_3) \\
& = & (x_0^2,x_1,x_3)\cap (x_0^2,x_1^2,x_2^2,x_0x_1,x_3).
\end{eqnarray*}
Hence $\mbox{arith-deg}_{r-1}(I,x_3)=2$. Also, we have
\begin{eqnarray*}
(I_{\geq r+1},x_3) & = & ({\sf q},x_3)\\
& = & (x_0^2,x_1^2,x_0x_1,x_1x_2,x_3)\\
& = & (x_0^2,x_1,x_3)\cap (x_0^2,x_1^2,x_2,x_3,x_0x_1).
\end{eqnarray*}
Hence $\mbox{arith-deg}_{r-1}(I_{\geq r+1},x_3)=1$.
We note that
$$\mbox{arith-deg}_{r-1}(I,x_3)>\mbox{arith-deg}_r (I)$$
even in the case that $x_3$ is a non-zero-divisor on $S/I$.
\vspace{5mm}
The second example shows that the bound of Theorem 3.1 is sharp and
improves the
result of \cite{BM}, Proposition 3.6 (see Corollary 3.2).
\vspace{5mm}
\noindent{\em Example 2:}
Take
$I=(x_0^2x_1,x_0x_2^2,x_1^2,x_2,x_2^3)
=(x_0^2,x_2)\cap(x_1,x_2^2)\cap(x_0^2,
x_1^2,
x_0x_2^2,x_2^3)\linebreak\subset S := K[x_0,x_1,x_2]$.
We get $m=5$,
$P(S/I,\ell)=4$ for all $\ell\geq 4$. We consider the case $r=0$
in (3.1)
and (3.2).
Then we have
$$4 = \deg I = \mbox{arith-deg}_0 (I)\leq P(S/I,m-1)=4
<\left(\begin{array}{c}
5+2-0-1\\2-0\end{array}\right)=15.$$
\vspace{5mm}
|
1996-09-28T19:01:13 | 9602 | alg-geom/9602004 | en | https://arxiv.org/abs/alg-geom/9602004 | [
"alg-geom",
"math.AG"
] | alg-geom/9602004 | null | Eriko Hironaka | Alexander Stratifications of Character Varieties | 30 pages with 2 figures. (Revised Sept. 25, 1996) LaTeX2e | null | null | null | null | There is a natural stratification of the character variety of a finitely
presented group coming from the jumping loci of the first cohomology of
one-dimensional representations. Equations defining the jumping loci can be
effectively computed using Fox calculus. In this paper, we give an exposition
of Fox calculus in the language of group cohomology and in the language of
finite abelian coverings of CW complexes. Work of Simpson, Arapura and others
show that if $\Gamma$ is the fundamental group of a compact K\"ahler manifold,
then the strata are finite unions of translated affine subtori. If follows that
for K\"ahler groups the jumping loci must be defined by binomial ideals. We
discuss properties of the jumping loci of general finitely presented groups and
apply the ``binomial criterion" to obtain new obstructions for one-relator
groups to be K\"ahler.
| [
{
"version": "v1",
"created": "Fri, 2 Feb 1996 23:03:19 GMT"
},
{
"version": "v2",
"created": "Sat, 28 Sep 1996 16:47:58 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Hironaka",
"Eriko",
""
]
] | alg-geom | \section{Introduction}
Let $X$ be homotopy equivalent to a finite CW complex
and let $\Gamma$ be the fundamental group of $X$. One
would like to derive geometric properties of
$X$ from a finite presentation
$$
\langle\ x_1,\dots,x_r\ : \ R_1,\dots,R_s\ \rangle
$$
of $\Gamma$. Although the isomorphism problem is
unsolvable for finite presentations, Fox calculus
can be used to effectively compute invariants of
$\Gamma$, up to second commutator, from the
presentation. In this paper, we study a natural
stratification of the character variety $\chargp{\Gamma}$
of $\Gamma$, associated to Alexander invariants,
which we will call the {\it Alexander stratification.}
We relate properties of the stratification to
properties of unbranched coverings of $X$ and to the
existence of irrational pencils on $X$ when
$X$ is a compact K\"ahler manifold. Furthermore,
we obtain obstructions for a group $\Gamma$ to be
the fundamental group of a compact K\"ahler manifold.
This paper is organized as follows. In section 2, we give
properties of the Alexander stratification as an invariant
of arbitrary finitely presented groups. We begin with
some notation and basic definitions of Fox calculus in
section 2.1. In section 2.2, we relate the Alexander stratification
to jumping loci for group cohomology
and in section 2.3 we translate the definitions
to the language of coherent sheaves.
This allows one to look at Fox calculus as
a natural way to get from a presentation of a group
to a presentation of a canonically associated coherent sheaf, as we show in section 2.4.
Another way to view the Fox calculus is geometrically,
by looking at the CW complex associated to a finitely
generated group. We show how the first Betti number of
finite abelian coverings can be computed in terms
of the Alexander strata in section 2.5.
In section 3, we relate group theoretic
properties to properties of the Alexander stratification.
Of special interest to us in this paper are torsion
translates of connected algebraic subgroups of
$\chargp{\Gamma}$, we will call them {\it rational planes},
which sit inside the Alexander strata. In section 4, we show how
these rational planes relate to geometric properties of $X$.
For example, in 4.1 we show that the first Betti number
of finite abelian coverings of $X$ depends only on a
finite number of rational planes in the Alexander
strata. This follows from a theorem of Laurent on
the location of torsion points on an algebraic subset
of an affine torus. When $X$ is a compact K\"ahler
manifold, we relate the rational planes to the existence
of irrational pencils on $X$ or on a finite unbranched
covering of $X$. This gives a much weaker, but simpler
version of a result proved by Beauville \cite{Beau:Ann}
and Arapura \cite{Ar:Higgs}
which asserts that when $X$ is a compact
K\"ahler manifold the first
Alexander stratum is a finite union of rational
planes associated to the irrational pencils of $X$ and
of its finite coverings (see 4.2).
Simpson in \cite{Sim:Subs} shows that if $X$ is a compact
K\"ahler manifold, then the Alexander strata for
$\pi_1(X)$ are all finite unions of rational planes.
Since the ideals defining
the Alexander strata of a finitely presented group
are computable and rational planes are zero sets of
binomial ideals, one can test whether a
group could not be the fundamental group of K\"ahler
manifold in a practical way: by computing ideals defining
the Alexander strata and showing that their radicals are
not binomial ideals. In section 4.3 we use the above
line of reasoning to obtain an
obstruction for a finitely presented group of a
certain form to be K\"ahler.
It gives me pleasure
to thank G\'erard Gonzalez-Sprinberg and
the Institut Fourier for their hospitality during
June 1995 when I began work on this paper.
I would also like to thank the referee for helpful remarks, including
a suggestion for improving the example at the end of section 4.3.
\vspace{12pt}
\section{Fox Calculus and Alexander Invariants}
\subsection{Notation.}
For any group $\Gamma$, we denote by $\mathrm{ab}(\Gamma)$
the abelianization of $\Gamma$ and
$$
\mathrm{ab} : \Gamma \rightarrow \mathrm{ab}(\Gamma)
$$
the abelianization map. By $F_r$, we mean the free group
on $r$ generators $x_1,\dots,x_r$. For any ring $R$,
we let $\Lambda_r(R)$ be the ring of Laurent polynomials
$R[t_1^{\pm 1},\dots,t_s^{\pm 1}]$. When the ring $R$ is
understood,
we will write $\Lambda_r$ for $\Lambda_r(R)$.
Note that $\Lambda_r(R)$ is canonically isomorphic
to the group ring $R[\mathrm{ab}(F_r)]$ by the map
$t_i \mapsto \mathrm{ab}(x_i)$. Let $\mathrm{ab}$ also denote the map
$$
\mathrm{ab} : F_r \rightarrow \Lambda_r(R)
$$
given by composing the abelianization map with the injection
$$
\mathrm{ab}(F_r) \rightarrow R[\mathrm{ab}(F_r)] \cong \Lambda_r(R).
$$
A finite presentation of a group $\Gamma$ can be
written in two ways. One is by
$$
\langle \ F_r\ : \ {\cal R} \ \rangle,
$$
where ${\cal R} \subset F_r$ is a finite subset.
Then $\Gamma$ is isomorphic to the quotient
group
$$
\Gamma = {F_r}/{N({\cal R})},
$$
where $N({\cal R})$ is the normal subgroup of $F_r$
generated by ${\cal R}$. The other is by
a sequence of homomorphisms
$$
F_s \mapright{\psi} F_r \mapright{q} \Gamma,
$$
where $q$ is onto and the normalization of the
image of $\psi$ is the kernel of $q$.
Let $\chargp{\Gamma}$ be the group of characters
of $\Gamma$. Then $\chargp{\Gamma}$ has the structure
of an algebraic group with coordinate ring
${\Bbb C}[\mathrm{ab}(\Gamma)]$. (One can verify this by noting
that that the closed
points in $\mathrm{Spec}({\Bbb C}[\mathrm{ab}(\Gamma)])$ correspond
to homomorphisms from $\mathrm{ab}(\Gamma)$ to ${\Bbb C}^*$.)
A presentation $\langle F_r \ : \ {\cal R} \rangle$
of $\Gamma$ gives an embedding of $\chargp{\Gamma}$
in $\chargp{F_r}$. The latter can be
canonically identified with
the affine torus $({\Bbb C}^*)^r$ as follows. To a
character $\rho \in \chargp{F_r}$ we identify the point
$(\rho(x_1),\dots,\rho(x_r)) \in ({\Bbb C}^*)^r$. The image
of $\chargp{\Gamma}$ in $({\Bbb C}^*)^r$ is the zero set of
the subset of $\Lambda_r({\Bbb C})$ defined by
$$
\{\ \mathrm{ab}(R) - 1\ :\ R \in {\cal R} \ \} \subset {\Bbb C}[\mathrm{ab}(F_r)] \cong \Lambda_r({\Bbb C}).
$$
Given any homomorphism, $\alpha : \Gamma' \rightarrow \Gamma$
between two finitely presented groups, let
$\chargp{\alpha} : \chargp{\Gamma} \rightarrow
\chargp{\Gamma'}$ be the map given by composition.
Let $\alpha_{\mathrm{ab}} : \mathrm{ab}(\Gamma) \rightarrow \mathrm{ab}(\Gamma')$ be
the map canonically induced by $\alpha$ and
let $\chargp{\alpha}^* : {\Bbb C}[\mathrm{ab}(\Gamma')] \rightarrow
{\Bbb C}[\mathrm{ab}(\Gamma)]$ be the linear extension of
$\alpha_{\mathrm{ab}}$. Then it is easy to verify that $\chargp{\alpha}$ is
an algebraic morphism and $\chargp{\alpha}^*$ is the
corresponding map on coordinate rings:
$\chargp{\alpha}^*(f) (\rho) = f(\chargp{\alpha}(\rho)),$ for
$\rho \in \Gamma$ and $f \in {\Bbb C}[\mathrm{ab}(\Gamma')]$.
In \cite{Fox:CalcI}, Fox develops a calculus to
compute invariants, originally discovered by
Alexander, of finitely presented groups. The
calculus can be defined as follows: fix $r$ and,
for $i=1,\dots,r$,
let
$$
D_i : F_r \rightarrow \Lambda_r({\Bbb Z})
$$
be the map given by
\begin{eqnarray*}
D_i(x_j) &=& \delta_{i,j}, \mbox{ and}\cr
D_i(fg) &=& D_i(f) + \mathrm{ab}(f) D_i(g).
\end{eqnarray*}
The map
$$
D = (D_1,\dots,D_r) : F_r \rightarrow \Lambda_r({\Bbb Z})^r
$$
is called the {\it Fox derivative} and the
$D_i$ are called the $i$th partials. Now let $\Gamma$
be a group with finite presentation
$$
\langle F_r\ : \ {\cal R} \rangle
$$
and let $q : F_r \rightarrow \Gamma$ be the quotient
map. The {\it Alexander matrix} of $\Gamma$ is the
$r \times s$ matrix of partials
$$
M(F_r,{\cal R})
= \left [\ (\chargp{q})^* D_i(R_j)\ \right ].
$$
For any $\rho \in \chargp{\Gamma}$, let
$M(F_r,{\cal R})(\rho)$
be the $r \times s$ complex matrix given by evaluation on $\rho$
and define
$$
V_i(\Gamma) = \{\ \rho \in \chargp{\Gamma}\ | \
\mathrm{rank}\ M(F_r,{\cal R})(\rho) < r-i\ \}.
$$
These are subvarieties of $\chargp{\Gamma}$ defined by the ideals of
$(r-i) \times (r-i)$ minors of $M(F_r, {\cal R})$.
We will call the nested sequence of algebraic subsets
$$
\chargp{\Gamma} \supset V_1(\Gamma) \supset \dots \supset V_r(\Gamma)
$$
the {\it Alexander stratification} of $\Gamma$.
One can check that the Tietze transformations on group presentations
give different Alexander matrices, but
don't effect the $V_i(\Gamma)$. Hence the Alexander stratification
is independent of the presentation. Later in section 2.4 (Corollary 2.4.3)
we will prove the independence by other methods.
\subsection{Jumping loci for group cohomology.}
For any group $\Gamma$,
let $C^1(\Gamma,\rho)$ be the set of {\it crossed
homomorphisms} $f : \Gamma \rightarrow {\Bbb C}$
satisfying
$$
f(g_1g_2) = f(g_1) + \rho(g_1)
f(g_2).
$$
Then $C^1(\Gamma,\rho)$ is a vector space over ${\Bbb C}$.
Note that for any $f \in
C^1(\Gamma,\rho)$, $f(1)=0$.
Here are two elementary lemmas, which will be useful throughout
the paper.
\begin{lemma} Let $\alpha : \Gamma' \rightarrow \Gamma$ be a
homomorphism of groups and let $\rho \in \chargp{\Gamma}$. Then
right composition by $\alpha$ defines a vector space homomorphism
$$
T_\alpha : C^1(\Gamma,\rho) \rightarrow C^1(\Gamma',\chargp{\alpha}(\rho)).
$$
\end{lemma}
\proof Take any $f \in C^1(\Gamma,\rho)$. Then, for
$g_1,g_2 \in \chargp{\Gamma'}$,
\begin{eqnarray*}
T_\alpha(f)(g_1g_2) &=& f(\alpha(g_1g_2))\cr
&=&f(\alpha(g_1)\alpha(g_2))\cr
&=&f(\alpha(g_1))+\rho(\alpha(g_1))f(\alpha(g_2)) \cr
&=&T_\alpha(f)(g_1) + \chargp{\alpha}(\rho)(g_1)(T_\alpha(f))(g_2).
\end{eqnarray*}
Thus, $T_\alpha(f)$ is in $C^1(\Gamma',\chargp{\alpha}(\rho))$.
\qed
\begin{lemma} Let $g,x \in \Gamma$ and let $f \in C^1(\Gamma,\rho)$,
for any $\rho \in \chargp{\Gamma}$. Then
$$
f(gxg^{-1}) = f(g)(1 - \rho(x)) + \rho(g) f(x).
$$
\end{lemma}
\proof This statement is easy to check by expanding the left hand
side and noting that
$$
f(g^{-1}) = - \rho(g)^{-1} f(g),
$$
for any $g \in \Gamma$.\qed
Let
$$
U_i(\Gamma) = \{\ \rho \in \chargp{\Gamma} \ |
\ \dim C^1(\Gamma,\rho) > i\ \}.
$$
This defines a nested sequence
$$
\chargp{\Gamma} \supset U_0(\Gamma) \supset
U_1(\Gamma) \supset \dots.
$$
In section 2.4 (Corollary 2.4.3), we will show that $U_i(\Gamma) = V_i(\Gamma)$, for all
$i \in {\Bbb N}$.
Define, for $\rho \in \chargp{\Gamma}$,
$$
B_1(\Gamma,\rho) = \{\ f : \Gamma \rightarrow
{\Bbb C}
\ | \ f(g) = (\rho(g) - 1)c \ \mbox{for some
constant $c \in {\Bbb C}$} \ \}.
$$
Then $B_1(\Gamma,\rho)$ is a subspace of
$C^1(\Gamma,\rho)$.
Define
$$
\mathrm{H}^1(\Gamma,\rho) =
{C^1(\Gamma,\rho)}/{B^1(\Gamma,\rho)}.
$$
This is the {\it first cohomology group of
$\Gamma$ with respect to the representation $\rho$}.
Let
$$
W_i(\Gamma) = \{\ \rho\in \chargp{\Gamma}\ |\
\dim \mathrm{H}^1(\Gamma,\rho) \ge i\ \},
$$
for $i\in {\Bbb Z}_+$. We will call the $W_i(\Gamma)$
the {\it jumping loci} for the first cohomology
of $\Gamma$.
This defines a nested sequence
$$
\chargp{\Gamma} = W_0(\Gamma) \supset
W_1(\Gamma) \supset \dots.
$$
If $\rho = \chargp{1}$ is
the identity character in $\chargp{\Gamma}$,
then $\rho(g) = 1$, for all $g \in \Gamma$.
Thus, $B^1(\Gamma,\rho) = \{0\}$.
Also, $C^1(\Gamma,\chargp{1})$ is the set
of all homomorphisms from $\Gamma$ to ${\Bbb C}$
and is isomorphic to the abelianization of
$\Gamma$ tensored with ${\Bbb C}$. Thus,
$$
\dim \mathrm{H}^1(\Gamma,\chargp{1}) =
\dim C^1(\Gamma,\chargp{1}) = d,
$$
where
$d$ is the rank of the abelianization of
$\Gamma$.
If $\rho \neq \chargp{1}$, then $B^1(\Gamma,\rho)$ is
isomorphic to the field of constants ${\Bbb C}$, so
$$
\dim C^1(\Gamma,\rho) = \dim \mathrm{H}^1(\Gamma,\rho)
+ 1.
$$
We have thus shown the following.
\begin{lemma} The jumping loci $W_i(\Gamma)$ and the
nested sequence $U_i(\Gamma)$ are related as follows:
\begin{eqnarray*}
W_i(\Gamma) = U_i(\Gamma) &\qquad& \mbox{for
$i\neq d$}\\
W_i(\Gamma) = U_i(\Gamma) \cup \{\chargp{1}\}
&\qquad&\mbox{for $i = d$.}
\end{eqnarray*}
\end{lemma}
\heading{Remark.} The jumping loci could also have been defined
using the cohomology of local systems. Let $X$ be a topological space
homotopy equivalent to a finite CW complex with $\pi_1(X) = \Gamma$.
Let $\widetilde{X} \rightarrow X$ be the universal cover of $X$. Then
for each $\rho \in \chargp{\Gamma}$, each $g \in \Gamma$ acts on
$\widetilde{X} \times {\Bbb C}$ by its action as covering automorphism on
$\widetilde{X}$ and by multiplication by $\rho(g)$ on ${\Bbb C}$. This
defines a local system ${\Bbb C}_\rho \rightarrow X$ over $X$. Then
$W_i(\Gamma)$ is the jumping loci for the rank of the cohomology group
$\mathrm{H}^1(X,{\Bbb C}_\rho)$ with coefficients in the local system ${\Bbb C}_\rho$.
\subsection{Coherent sheaves over the character variety.}
Let $\Gamma$ be a finitely presented group
and let
$\dual{C^1(\Gamma,\rho)}$ be the dual space of
$C^1(\Gamma,\rho)$. We will construct
sheaves ${\cal C}^1(\Gamma)$ and
$\dual{{\cal C}^1(\Gamma)}$ over
$\chargp{\Gamma}$ whose stalks are
$C^1(\Gamma,\rho)$ and
$\dual{C^1(\Gamma,\rho)}$, respectively.
Then, the jumping loci $U_i(\Gamma)$ defined
in the previous section, are just the
jumping loci for
the dimensions of stalks of ${\cal C}^1(\Gamma)$
and $\dual{{\cal C}^1(\Gamma)}$.
This just gives a translation of the previous section
into the language of sheaves, but using this language
we will show that a presentation for $\Gamma$
induces a presentation of $\dual{{\cal C}^1(\Gamma)}$
as a coherent sheaf such that the presentation map on
sheaves is essentially the Alexander matrix.
We start by constructing ${\cal C}^1(F_r)$ for free groups.
\begin{lemma} For any $r$ and $\rho \in \chargp{F_r}$,
$C^1(F_r,\rho)$ is isomorphic to ${\Bbb C}^r$,
and has a basis given by $\langle x_i \rangle_\rho$,
where
$$
\langle x_i \rangle_\rho (x_j) = \delta_{i,j}.
$$
\end{lemma}
\proof By the product rule, elements of $C^1(F_r,\rho)$
only depend on what happens to the generators of $F_r$.
Since there are no relations on $F_r$, any choice
of values on the basis elements determines an element
of $C^1(F_r,\rho)$.
\qed
Let
$$
E_r = \bigcup_{\rho \in \chargp{F_r}} C^1(F_r,\rho)
$$
be the trivial ${\Bbb C}^r$-vector bundle over $\chargp{F_r}$ whose
fiber over $\rho \in \chargp{F_r}$ is $C^1(F_r,\rho)$.
For each generator $x_i$ of $F_r$, define
$$
\langle x_i \rangle : \chargp{F_r}
\rightarrow \bigcup_{\rho \in \chargp{F_r}}
C^1(F_r,\rho),
$$
by $\langle x_i \rangle (\rho) = \langle x_i \rangle_\rho$.
The maps $\langle x_1 \rangle, \dots, \langle x_r \rangle$
are global sections of $E_r$
over $\chargp{F_r}$.
Let ${\cal C}^1(F_r)$ be the corresponding sheaf of sections of the
bundle $E_r \rightarrow \chargp{F_r}$.
The module $M_r$ of global sections of ${\cal C}^1(F_r)$ is a free
$\Lambda_r$-module of rank $r$,
generated by $\langle x_1 \rangle, \dots, \langle x_r \rangle$,
and ${\cal C}^1(F_r)$ is the sheaf associated to $M_r$ (in the sense
of \cite{Hart:AG}, p.110).
Fix a presentation
$$
F_s \mapright{\psi} F_r \mapright{q} \Gamma,
$$
of $\Gamma$. This induces maps on character varieties
$$
\cd
{
&\Gamma&\mapright{\chargp{q}}
&\chargp{F_r}&\mapright{\chargp{\psi}}
&\chargp{F_s}\cr
&&&\Vert&&\Vert\cr
&&&({\Bbb C}^*)^r&&({\Bbb C}^*)^s.
}
$$
Let ${\cal C}^1(F_r)_\Gamma$ and ${\cal C}^1(F_s)_\Gamma$ be the pullbacks
of ${\cal C}^1(F_r)$ and ${\cal C}^1(F_s)$ over $\chargp{\Gamma}$.
These are the sheafs associated to the modules:
$$
M_r(\Gamma) = M_r \otimes_{{\Bbb C}[\mathrm{ab}(F_r)]} {\Bbb C}[\mathrm{ab}(\Gamma)]
\cong {\Bbb C}[\mathrm{ab}(\Gamma)]^r
$$
and
$$
M_s(\Gamma) = M_s \otimes_{{\Bbb C}[\mathrm{ab}(F_s)]} {\Bbb C}[\mathrm{ab}(\Gamma)]
\cong {\Bbb C}[\mathrm{ab}(\Gamma)]^s,
$$
respectively.
Let
$$
{\cal T}_\psi : {\cal C}^1(F_r)_\Gamma \rightarrow {\cal C}^1(F_s)_\Gamma
$$
be the homomorphism of sheaves defined by composing sections by $\psi$.
For any $\rho \in \chargp{\Gamma}$,
the stalk of ${\cal C}^1(F_r)_\Gamma$ over $\rho$ is given by
$C^1(F_r,\chargp{q}(\rho))$. Since $q \circ \psi$
is the trivial map, the stalk of ${\cal C}^1(F_s)_\Gamma$ over
$\rho$ is given by $C^1(F_s,\chargp{1})$.
For any $\rho \in \chargp{\Gamma}$, the map on stalks determined by ${\cal T}_\psi$
is the map
$$
({\cal T}_\psi)_\rho : C^1(F_r,\chargp{q}(\rho)) \rightarrow C^1(F_s,\chargp{1})
$$
defined by $({\cal T}_\psi)_\rho(f) = f\circ\psi$.
Let $M_\Gamma(F_r,{\cal R})$ be the sub ${\Bbb C}[\mathrm{ab}(\Gamma)]$-module
of $M_r(\Gamma)$ given by the kernel of the map
\begin{eqnarray*}
M_r(\Gamma) &\rightarrow& M_s(\Gamma)\\
f\otimes g &\mapsto& (f \circ \psi) \otimes g
\end{eqnarray*}
Let ${\cal C}^1(\Gamma)$ be the kernel of ${\cal T}_\psi$. That is,
${\cal C}^1(\Gamma)$ is the sheaf associated to $M_{\Gamma}(F_r,{\cal R})$.
\begin{lemma} The stalk of ${\cal C}^1(\Gamma)$ over
$\rho \in \chargp{\Gamma}$ is isomorphic to $C^1(\Gamma,\rho)$.
\end{lemma}
\proof We need to show that the kernel of $({\cal T}_\psi)_\rho$ is isomorphic to
$C^1(\Gamma,\rho)$. Let
$$
(T_q)_\rho : C^1(\Gamma,\rho) \rightarrow C^1(F_r,\chargp{q}(\rho))
$$
be the homomorphism given by composing with $q$ as in Lemma 2.2.1.
Since $q$ is surjective, it follows that $(T_q)_\rho$ is injective.
The composition $T_\psi \circ (T_q)_\rho$ is right composition by
$\psi \circ q$, which is trivial, so the image of $(T_q)_\rho$ lies
in the kernel of $\Psi$. Now suppose, $f \in C^1(F_r,\chargp{q}(\rho))$
is in the kernel of $\Psi$. Then $f$ is trivial on $\psi(F_s)$.
Since $\chargp{q}(\rho)$ is trivial on $\psi(F_s)$, Lemma 2.2.2 implies
that $f$ is trivial on the normalization of $\psi(F_s)$ in
$F_r$. Thus, $f$ induces a map from $\Gamma$ to ${\Bbb C}$ which is twisted
by $\rho$.\qed
\begin{lemma}
Let $\alpha : \Gamma' \rightarrow \Gamma$ be a homomorphism of groups
and let $\chargp{\alpha} : \chargp{\Gamma} \rightarrow \chargp{\Gamma'}$
be the corresponding morphism on character varieties. Let
${\cal C}(\Gamma)$ and ${\cal C}(\Gamma')$ be the sheaves associated to $\Gamma$
and $\Gamma'$ and let ${\cal C}(\Gamma')_\Gamma$ be the pullback of
${\cal C}(\Gamma')$ over $\chargp\Gamma$. Then the
map ${\cal T}_\alpha : {\cal C}(\Gamma) \rightarrow {\cal C}(\Gamma')$ defined by composing
sections by $\alpha$ is a homomorphism of sheaves.
\end{lemma}
\proof The statement follows from Lemma 2.2.1.\qed
\begin{corollary} There are exact sequences of sheaves
$$
0 \rightarrow {\cal C}^1(\Gamma)\ \mapright{}\ {\cal C}^1(F_r)_\Gamma
\ \mapright{{\cal T}_\psi}\ {\cal C}^1(F_s)_\Gamma
$$
and
$$
\dual{{\cal C}^1(F_s)}_\Gamma\ \mapright{\dual{{\cal T}_\psi}}\
\dual{{\cal C}^1(F_r)}_\Gamma
\rightarrow \dual{{\cal C}^1(\Gamma)} \rightarrow 0.
$$
\end{corollary}
We have seen that the modules of holomorphic sections of
${{\cal C}^1(F_r)}$ and ${\cal C}^1(F_s)$ are freely generated
over ${\cal C}[\mathrm{ab}(\Gamma)]$ of ranks $r$ and $s$, respectively.
Similarly, the dual sheaves $\dual{{\cal C}^1(F_r)}$ and $\dual{{\cal C}^1(F_s)}$
are freely generated.
This gives $\dual{{\cal C}^1(\Gamma)}$ the structure of a coherent sheaf.
In section 2.4 we will show that the Alexander Matrix gives
a presentation for global sections of $\dual{{\cal C}^1(\Gamma)}$.
\subsection{Jumping loci and the Alexander stratification.}
In this section, we show that for a given group $\Gamma$, the jumping
loci $U_i(\Gamma)$ defined in 2.2 is the same as the
Alexander stratification $V_i(\Gamma)$.
For any group $\Gamma$, there is an exact bilinear pairing
$$
({\Bbb C}\Gamma)_\rho \times C^1(\Gamma,\rho) \rightarrow {\Bbb C}
$$
and the pairing is given by
$$
({\Bbb C}\Gamma)_\rho =
{\Bbb C}\Gamma/{\{g_1g_2 - g_1 - \rho(g_1)g_2 | g_1,g_2 \in\Gamma\}},
$$
where
$$
[g,f] = f(g).
$$
The pairing determines a ${\Bbb C}$-linear map
$$
\Phi[\Gamma]_\rho : ({\Bbb C}\Gamma)_\rho \rightarrow \dual{C^1(\Gamma,\rho)},
$$
where, for $g \in ({\Bbb C}\Gamma)_\rho$ and $f \in C^1(\Gamma,\rho)$,
$\Phi[\Gamma]_\rho(f) (g) = [g,f] = f(g)$.
\begin{lemma} Let $\alpha : \Gamma' \rightarrow \Gamma$ be a group
homomorphism.
For each $\rho \in \chargp{\Gamma}$, we have a commutative
diagram
$$
\cd
{
&({\Bbb C}\Gamma')_{\chargp{\alpha}(\rho)}
&\mapright{\Phi[\Gamma']_{\chargp{\alpha}(\rho)}}
&\dual{C^1(\Gamma',\chargp{\alpha}(\rho))}\cr
&\mapdown{\alpha}&&\mapdown{\dual T_\alpha}\cr
&({\Bbb C}\Gamma)_\rho &\mapright{\Phi[\Gamma]_\rho}&\dual{C^1(\Gamma,\rho)}
}
$$
where $\dual T_\alpha$
is the dual map to $T_\alpha : C^1(\Gamma,\rho) \rightarrow C^1(\Gamma',
\chargp{\alpha}(\rho))$.
\end{lemma}
\proof
For $g \in ({\Bbb C}\Gamma')_{\chargp{\alpha}}(\rho)$ and
$f \in C^1(\Gamma,\rho)$, the pairing $[,]$ gives
$$
[g,T_\alpha(f)] = {T_\alpha}(f)(g) = f(\alpha(g)) = [\alpha(g),f].
$$
\qed
Let $\dual{M_r}$ be the global holomorphic sections of $\dual{{\cal C}^1(F_r)}$.
Define
$$
\Phi : {\Bbb C} F_r \rightarrow \dual{M_r}
$$
by
\begin{eqnarray*}
\Phi(x_i) &=& \dual{\langle x_i \rangle}\cr
\Phi(g_1g_2) &=& \Phi(g_1) +
\mathrm{ab}(g_1)\Phi(g_2)\qquad \mbox{for}\ g_1,g_2 \in F_r,
\end{eqnarray*}
where
$$
\dual{\langle x_i \rangle}_\rho : C^1(F_r,\rho) \rightarrow {\Bbb C}
$$
is given by
$$
\dual{\langle x_i \rangle}_\rho(\langle x_j \rangle_\rho) = \delta_{i,j}.
$$
Define, for any $\rho \in \chargp{F_r}$ and $g \in {\Bbb C} F_r$, with
image $g_\rho$ in $({\Bbb C} F_r)_\rho$, $\Phi_\rho(g_\rho) =
\Phi_\rho(g)(\rho) \in \dual{C^1(F_r,\rho)}$, where
$$
\Phi_\rho(g)(\rho)(f) = f(g)
$$
for all $f \in C^1(F_r,\rho)$. Then $\Psi_\rho = \Psi[F_r]_{\rho}$.
Since $\dual{M_r}$ is generated freely by the global sections
$$
\dual{\langle x_1 \rangle},\dots,\dual{\langle x_r \rangle}
$$
as a $\Lambda_r({\Bbb C})$-module, we can identify $\dual{M_r}$ with
$\Lambda_r({\Bbb C})^r$.
Thus, the map $\Phi$ is the extension of the Fox derivative
$$
D : F_r \rightarrow \Lambda_r({\Bbb Z})^r
$$
in the obvious way to ${\Bbb C} [F_r] \rightarrow \Lambda_r({\Bbb C})^r$.
Let ${\cal D}_r({\cal R})$ be the sub $\Lambda_r$-module of $\Lambda_r({\Bbb C})^r$ spanned by
$\Phi({\cal R})$. For $\rho \in \chargp{\Gamma}$, let ${\cal D}_r({\cal R})(\rho)$
be the subspace of ${\Bbb C}^r$ spanned by the vectors obtained by
evaluating the $r$-tuples of functions in $\Phi({\cal R})$ at $\rho$.
\begin{lemma} Let $\langle F_r : {\cal R} \rangle$ be a presentation for $\Gamma$.
For each $\rho \in \chargp{\Gamma}$, the dimension of
$C^1(\Gamma,\rho)$ is given by
$$
r - \dim({\cal D}_r({\cal R})(\rho)).
$$
\end{lemma}
\proof Let
$$
F_s \mapright{\psi} F_r \mapright{q} \Gamma
$$
be the sequence of maps determined by the presentation. Then,
for each $\rho \in \chargp{\Gamma}$, by Corollary 2.3.4,
there is an exact sequence
$$
\dual{C^1(F_s,\chargp{1})}\ \mapright{\dual{{\cal T}_\psi}}
\
\dual{C^1(F_r,\chargp{q}(\rho))} \ \mapright{\dual{{\cal T}_q}}
\
\dual{C^1(\Gamma,\rho)} \mapright{} 0.
$$
By Lemma 2.4.1, the following diagram commutes:
$$
\cd{
&({\Bbb C} F_s)_{\chargp{1}} &\mapright{\Phi[F_s]_{\chargp{1}}} &\dual{C^1(F_s,\chargp{1})}\cr
&\mapdown{\psi} &&\mapdown{\dual{T_\psi}}\cr
&({\Bbb C} F_r)_{\chargp{q}(\rho)} &\mapright{\Phi[F_r]_{\chargp{q}(\rho)}}
&\dual{C^1(F_r,\chargp{q}(\rho))}\cr
&\mapdown{q} &&\mapdown{\dual{T_q}}\cr
&({\Bbb C}\Gamma) &\mapright{\Phi[\Gamma]_\rho} &\dual{C^1(\Gamma,\rho)}
}
$$
Thus,
\begin{eqnarray*}
\dim C^1(\Gamma,\rho) &=& \dim C^1(F_r,\chargp{q}(\rho)) -
\dim (\mbox{image} (\dual{{\cal T}_\psi})).
\end{eqnarray*}
Since $\Phi[F_s]_{\chargp{1}}$ is onto
\begin{eqnarray*}
\mbox{image}(\dual{{\cal T}_\psi}) &=& \mbox{image}(\Phi[F_s]_{\chargp{1}}
\circ \dual{{\cal T}_\psi})\cr
&=& \mbox{image}(\Phi[F_r]_{\chargp{q}(\rho)} \circ \psi)
\end{eqnarray*}
For any $\rho$, $C^1(F_r,\chargp{q}(\rho))$ is isomorphic to ${\Bbb C}^r$.
Putting this together, we have
\begin{eqnarray*}
\dim C^1(\Gamma,\rho) &=& r - \dim \Phi[F_r]_{\chargp{q}(\rho)}({\cal R})\cr
&=& r - \dim {\cal D}_r({\cal R})(\rho).
\end{eqnarray*}
\qed
\begin{corollary} For any finitely presented group $\Gamma$, the jumping
loci $U_i (\Gamma)$ for the cohomology of $\Gamma$ is the
same as the Alexander stratification $V_i(\Gamma)$.
\end{corollary}
\subsection{Abelian coverings of finite CW complexes.}
In this section we explain the Fox calculus and Alexander
stratification in terms of finite abelian coverings of a finite CW
complex. The relations between homology of coverings of a
$K(\Gamma,1)$ and the group cohomology of $\Gamma$
are well known (see, for
example, \cite{Bro:Coh}). The results of this section
come from looking at Fox calculus from this point of view.
Let $X$ be a finite CW complex and let
$\Gamma = \pi_1(X)$. Suppose $\Gamma$ has presentation
given by $\langle x_1,\dots,x_r :
R_1,\dots,R_s \rangle$.
Then $X$ is homotopy equivalent to a CW complex with cell
decomposition whose tail end is given by
$$
\dots \supset \Sigma_2 \supset \Sigma_1 \supset
\Sigma_0,
$$
where $\Sigma_0$ consists of a point $P$,
$\Sigma_1$ is a bouquet of $r$ oriented circles
$S^1$ joined at $P$. Identify $F$ with
$\pi_1(\Sigma_1)$ so that each $x_i$ is the
positively oriented loop around the $i$-th
circle. Each
$R_i$ defines a homotopy class of map from $S^1$
to
$\Sigma_1$. The 2-skeleton
$\Sigma_2$ is the union of $s$ disks attached
along their boundaries to $\Sigma_1$ by
maps in the homotopy class defined by
$R_1,\dots,R_s$.
Let $\alpha : \Gamma \rightarrow G$ be any epimorphism
of $\Gamma$ to a finite abelian group $G$. Let
$\tau_\alpha : X_\alpha \rightarrow X$ be the regular
unbranched covering determined by $\alpha$ with $G$ acting as group of
covering automorphisms. Our aim is to show how Fox calculus can
be used to compute the first Betti number of $X_\alpha$. Choose
a basepoint $1P \in \tau_\alpha^{-1}(P)$. For each $i$-chain
$\sigma \in \Sigma_i$ and $g\in G$, let $g
\sigma$ be the the component of its preimage
which passes through $gP$. For each generating
$i$-cell in
$\Sigma_i$, there are exactly $G$ copies
of isomorphic cells in its preimage. Thus
$X_\alpha$ has a cell decomposition
$$
\dots \supset \Sigma_{2,\alpha} \supset
\Sigma_{1,\alpha} \supset \Sigma_{0,\alpha},
$$
where the $i$-cells in $\Sigma_{i,\alpha}$ are
given by the set $\{g \sigma \ :\ g\in G,\sigma
\mbox{ an $i$-cell in $\Sigma_i$}\}$. With this
notation if $\sigma$
attaches to $\Sigma_{i-1,\alpha}$ according
to the homotopy class of mapping $f: \partial\sigma
\rightarrow \Sigma_{i-1}$, where $\partial\sigma$
is the boundary of $\sigma$, then $g\sigma$
attaches to $\Sigma_{i-1,\alpha}$ by the
map $f' : \partial g\sigma
\rightarrow \Sigma_{i-1,\alpha}$ lifting $f$
at the
basepoint
$gP$.
Let $C_i$ be the $i$-chains on $X$ and
let $C_{i,\alpha}$ be the $i$-chains on
$X_\alpha$. Then there is a commutative
diagram for the chain complexes for $X$ and
$X_\alpha$:
$$
\cd
{
&\dots&\mapright{}&C_{2,\alpha}
&\mapright{\delta_{2,\alpha}}&C_{1,\alpha}
&\mapright{\delta_{1,\alpha}}&C_{0,\alpha}&\mapright{\epsilon} &{\Bbb Z}\cr
&&&\mapdown{\tau_\alpha}&&\mapdown{\rho_\alpha}
&&\mapdown{\tau_\alpha}&&\cr
&\dots&\mapright{}&C_2&\mapright{\delta_2}&C_1
&\mapright{\delta_1}&C_0,&&
}
$$
where the map $\epsilon$ is the augmentation map
$$
\epsilon(\sum_{g \in G} (a_g
g)) =
\sum_{g\in G} a_g.
$$
Let
$\langle x_1 \rangle_\alpha,\dots,
\langle x_r \rangle_\alpha$ be the elements
of $C_{1,\alpha}$ given by
lifting
$x_1,\dots,x_r$, considered as loops on
$\Sigma_1$, to 1-chains on $\Sigma_{1,\alpha}$ with
basepoint
$1P$.
Then
$C_{1,\alpha}$ can be identified with
${\Bbb C}[G]^r$, with basis $\langle x_1\rangle, \dots,
\langle x_r \rangle$
and $C_{0,\alpha}$ can be identified with
${\Bbb C}[G]$, where each $g \in
G$ corresponds to $gP$.
The above commutative diagram can be rewritten
as
\begin{eqnarray}
\cd
{
&\dots&\mapright{}& {\Bbb Z}[G]^s
&\mapright{\delta_{2,\alpha}}&{\Bbb Z}[G]^r
&\mapright{\delta_{1,\alpha}}&{\Bbb Z}[G]
&\mapright{\epsilon}&{{\Bbb Z}} \cr
&&&\mapdown{\tau_\alpha}&&\mapdown{\rho_\alpha}
&&\mapdown{\tau_\alpha}&& \cr
&\dots&\mapright{}&{{\Bbb Z}}^s&\mapright{\delta_2}
&{{\Bbb Z}}^r
&\mapright{\delta_1}&{{\Bbb Z}.}&&
}
\end{eqnarray}
For any finite set $S$, let $|S|$ denote its
order. The map $\epsilon$ is surjective, so we
have the formula
\begin{equation}
\begin{array}{rl}
b_1(X_\alpha) &= \mathrm{nullity} (\delta_{1,\alpha})
-
\mathrm{rank} (\delta_{2,\alpha})\\
&= (r-1)|G| + 1 - \mathrm{rank}(\delta_{2,\alpha}),
\end{array}
\end{equation}
where $b_1(X_\alpha)$ is the rank of ${\ker{\delta_{1,\alpha}}}/
{\mathrm{image} (\delta_{2,\alpha})}$ and is the rank of $H_1(X_\alpha;{\Bbb Z})$.
We will rewrite this
formula in terms of the Alexander
stratification.
\begin{lemma} The map $\delta_{1,\alpha}$
is given by
$$
\delta_{1,\alpha} (\sum_{i=1}^r f_i \langle x_i
\rangle_\alpha) = \sum_{i=1}^r
f_i\chargp{q_\alpha}^*{(t_i-1)}.
$$
\end{lemma}
\proof
It's enough to notice that
the lift of $x_i$ to $C_{1,\alpha}$ at the
basepoint $1P$ has end point $\chargp{q_\alpha}^*(t_i)P$.
\qed
We will now relate the map $\delta_{2,\alpha}$
with the Fox derivative.
Recall that $\Sigma_1$ equals a bouquet of $r$
circles $\wedge_r S^1$. Let $\tau : {\cal L}_r
\rightarrow \wedge_r S^1$ be the universal
abelian covering. Then ${\cal L}_r$ is a lattice
on
$r$ generators with
$\mathrm{ab}(F_r)$ acting as covering automorphisms.
The vertices of the lattice can be identified
with $\mathrm{ab}(F_r)$. Let
$K_\alpha = \ker(\alpha \circ q) \subset F_r$
and let $\widetilde{K_\alpha}$ be its image in
$\mathrm{ab}(F_r)$. Then $\Sigma_{1,\alpha} =
{{\cal L}_r}/{\widetilde{K_\alpha}}$ and we have a
commutative diagram
$$
\cd
{
&{\cal L}_r &\mapright{\eta_\alpha}&\Sigma_{1,\alpha}\cr
&\mapdown{\tau}&&\mapdown{\tau_\alpha}\cr
&\wedge_r S^1&=&\Sigma_1
}
$$
where $\eta_\alpha : {\cal L}_r \rightarrow
\Sigma_{1,\alpha}$ is the quotient map. Let
$(\eta_\alpha)_* : C_1({\cal L}_r) \rightarrow
C_1(\Sigma_{1,\alpha})$ be the induced map on one
chains. Then identifying $C_1({\cal L}_r)$ with
${\Bbb Z}[\mathrm{ab}(F_r)]^r$ and $C_1(\Sigma_{1,\alpha})$ with
${\Bbb Z}[G]^r$, we have $(\eta_\alpha)_* =
{(\chargp{q_\alpha}^*)}^r$.
Choose $1\tilde P \in \tau^{-1}(P)$.
Let $C_1({\cal L}_r)$ be the 1-chains on ${\cal L}_r$.
Let $\langle x_1
\rangle,\dots,
\langle x_r \rangle$ be the lifts of
$x_1,\dots,x_r$ to $C_1({\cal L}_r)$ at the base
point $1\tilde P$.
This determines an identification of
$C_1({\cal L}_r)$ with $\Lambda({\Bbb Z})^r$ and determines
a choice of homotopy lifting map $
\ell : \pi_1(\Sigma_1)
\rightarrow C_1({\cal L}_r)$.
\begin{lemma}
The identifications $F_r = \pi_1(\Sigma_1)$
and $\Lambda_r({\Bbb Z}) = C_1({\cal L}_r)$, make the
following diagram commute
$$
\cd
{
&\pi_1(\Sigma_1) &\mapright{\ell} &C_1({\cal L}_r)\cr
&\Vert &&\Vert\cr
&F_r &\mapright{D} &\Lambda_r({\Bbb Z}).
}
$$
\end{lemma}
\proof
By definition, both maps $\ell$ and $D$
send $x_i$ to $\langle x_i \rangle$, for
$i=1,\dots,r$. We have left to check
products. Let $f, g \in F_r$, be thought of
as loops on $\wedge_r S^1$. Then the lift
of $f$ has endpoint $\mathrm{ab}(f)$.
Therefore, $\ell(fg) = \ell(f) +
\mathrm{ab}(f)\ell(g)$. Since these rules are the
same as those for the Fox derivative map, the
maps must be the same.
\qed
\begin{corollary} Let $\Gamma$ be
a finitely presented group with presentation
$\langle F_r : {\cal R} \rangle$.
Let $\alpha : \Gamma \rightarrow G$ be an
epimorphism to a finite abelian group $G$.
Let $M(F_r,{\cal R})_\alpha$ be the matrix $M(F_r,{\cal R})$
with $\chargp{q}_\alpha^*$ applied to all the entries.
Then
$$
\cd
{
&C_{2,\alpha} &\mapright{\delta_{2,\alpha}}
&C_{1,\alpha} \cr
&\Vert &&\Vert\cr
&{\Bbb Z}[G]^s &\mapright{M(F_r,{\cal R})_\alpha} &{\Bbb Z}[G]^r.
}
$$
\end{corollary}
\proof
Let $\sigma_1,\dots,\sigma_s$ be the $s$ disks
generating the $2$-cells $C_2$. For
each $i=1,\dots,s$ and $g \in G$, let
$g\sigma_i$ denote the lift of
$\sigma_i$
at $gP$. Let $R_1,\dots,R_s$ be the elements of
${\cal R}$. By Lemma 2.5.2, the boundary
$\partial\sigma_i $ maps to
$D(R_i)$ in $C_1({\cal L}_r)$. Thus, the boundary
of
$g
\sigma_i$ equals $g D(R_i)$,
and for
$g_1,\dots,g_s \in {\Bbb Z}[G]$,
$$
\delta_{\alpha,2}(\sum_{i=1}^s g_i
\sigma_i) = \sum_{i=1}^s g_i D(R_i).
$$
This is the same as the application of
$M(F_r,{\cal R})_\alpha$ on the
$s$-tuple
$(g_1,\dots,g_s)$.
\qed
We now give a
formula for the first Betti number $b_1(X_\alpha)$ in terms of the
Alexander stratification in the case where $G$
is finite.
Tensor the top row in diagram (1) by ${\Bbb C}$. Then
the action of $G$ on ${\Bbb C}[G]$ diagonalizes to get
$$
{\Bbb C}[G] \cong \bigoplus_{\rho \in \chargp{G}}
{\Bbb C}[G]_\rho,
$$
where ${\Bbb C}[G]_\rho$ is a one-dimensional subspace
of ${\Bbb C}[G]$ and $g \in G$ acts on ${\Bbb C}[G]_\rho$
by multiplication by $\rho(g)$.
The top row of diagram (1) becomes
\begin{eqnarray*}
\bigoplus_{\rho\in\chargp{G}}
{\Bbb C}[G]_\rho^s
\ \mapright{\delta_{\alpha,2}} \ \bigoplus_{\rho \in
\chargp{G}}
{\Bbb C}[G]_\rho^r
\ \mapright{\delta_{\alpha,1}} \ \bigoplus_{\rho \in
\chargp{G}}
{\Bbb C}[G]_\rho
\ \mapright{\epsilon} \ {\Bbb C}.
\end{eqnarray*}
The map $\delta_{\alpha,2}$ considered
as a matrix $M(F_r,{\cal R})_\alpha$, as in
Lemma 2.5.3, decomposes into blocks
$$
M(F_r,{\cal R})_\alpha = \bigoplus_{\rho \in \chargp{G}}
M(F_r,{\cal R})_\alpha(\rho),
$$
where, if $M(F_r,{\cal R})_\alpha = [f_{i,j}]$, then
$M(F_r,{\cal R})_\alpha(\rho) = [f_{i,j}(\rho)]$. We thus
have the following formula for the rank
of $M(F_r,{\cal R})_\alpha$:
\begin{eqnarray}
\mathrm{rank}
(M(F_r,{\cal R})_\alpha) &=&
\sum_{\rho\in\chargp{G}} \mathrm{rank}(M(F_r,{\cal R})_\alpha(\rho)).
\end{eqnarray}
Recall that the Alexander stratification $V_i(\Gamma)$ was
defined to be the zero set
in
$\chargp{\Gamma}$ of the $(r-i) \times (r-i)$
ideals of $M(F_r,{\cal R})$. For any $\rho \in \chargp{G}$,
$M(F_r,{\cal R})_\alpha(\rho) =
M(F_r,{\cal R})(\chargp{\alpha}(\rho)) =
M(F_r,{\cal R})(\chargp{q_\alpha}(\rho))$, since
$\widetilde{\alpha}(f) (\rho) =
f(\chargp{\alpha}(\rho))$ and
$\widetilde{q_\alpha}(f)(\rho) =
f(\chargp{q_\alpha}(\rho))$.
We thus have the following Lemma.
\begin{lemma} For $\rho \in \chargp{G}$,
$\chargp{\alpha}(\rho) \in V_i(\Gamma)$ if
and only if $\mathrm{rank}
(M(F_r,{\cal R})_{\alpha}(\rho)) < r-i$.
\end{lemma}
For each $i=0,\dots,r-1$, let
$\chi_{V_i(\Gamma)}$ be the indicator function
for $V_i(\Gamma)$. Then, for $\rho \in \chargp{G}$, we have
\begin{eqnarray}
\mathrm{rank} (M(F_r,{\cal R})_{\alpha}(\rho)) = r -
\sum_{i=0}^{r-1}
\chi_{V_i(\chargp{\Gamma})}(\chargp{\alpha}(\rho)).
\end{eqnarray}
\begin{lemma}
For the special character $\widehat{1}$,
$$
\mathrm{rank} (M(F_r,{\cal R})_\alpha(\chargp{1})) = r - b_1(X)
$$
and $\mathrm{rank} (M(F_r,{\cal R})_\alpha(\chargp{1})) = r$
if and only if $\chargp{\Gamma} = \{\chargp{1}\}$ and
$\Gamma$ has no nontrivial abelian quotients.
\end{lemma}
\proof
The group $G$ acts trivially on $\Lambda_{\alpha,
\widehat{1}}$. Thus, in the commutative diagram
$$
\cd
{
&\Lambda_{\alpha,\widehat{1}}^s
&\mapright{M(F_r,{\cal R})_\alpha(\chargp{1})}
&\Lambda_{\alpha,\chargp{1}}^r
&\mapright{\delta_{\alpha}(\chargp{1})}
&\Lambda_{\alpha,\chargp{1}}\cr
&\mapdown{}&&\mapdown{}
&&\mapdown{}\cr
&({\Bbb C})^s&\mapright{\delta_2}&({\Bbb C})^r
&\mapright{\delta_1}&{{\Bbb C}}
}
$$
the vertical arrows are isomorphisms.
We thus have
\begin{eqnarray*}
\mathrm{rank} (M(F_r,{\cal R})_{\alpha}(\chargp{1})) &=& \mathrm{rank}
(\delta_2)\\
&=& r- b_1(X).
\end{eqnarray*}
\qed
\begin{proposition} Let $\Gamma$ be a finitely
presented group and let
$\alpha :
\Gamma
\rightarrow G$ be an epimorphism where $G$ is a
finite abelian group. Let $\chargp{\alpha} :
\chargp{G} \hookrightarrow \chargp{\Gamma}$ be
the inclusion map induced by $\alpha$. Then
$$
b_1(X_\alpha) = b_1(X) + \sum_{i=1}^{r-1} | V_i(\Gamma)
\cap \widehat{\alpha} (\widehat{G} \setminus
\widehat{1}) |.
$$
\end{proposition}
\proof
Starting with formula (2) and Corollary 2.5.3, we have
\begin{eqnarray*}
b_1(X_\alpha) &=&(r-1)|G| + 1 - \mathrm{rank}
(M(F_r,{\cal R})_\alpha)\\
&=& r- \mathrm{rank} (M(F_r,{\cal R})_\alpha(\chargp{1})) +
\sum_{\rho\in \chargp{G} \setminus \chargp{1}}
(r-1) - \mathrm{rank} (M(F_r,{\cal R})_\alpha(\rho)).
\end{eqnarray*}
By Lemma 2.5.5, the left hand summand equals $b_1(X)$ and
by (4) the right hand side can be written in terms of the
indicator functions:
$$
b_1(X_\alpha) = b_1(X) + \sum_{\rho \in \chargp{G}
\setminus
\chargp{1}} \sum_{i=1}^{r-1}
\chi_{V_i(\chargp{\Gamma})}(\chargp{\alpha}(\rho))
$$
and the claim follows.
\qed
\begin{corollary} Let $\Gamma = \pi_1(X)$ be a finitely
presented group and $\alpha : \Gamma \rightarrow G$ an
epimorphism to a finite abelian group $G$, as above.
Then
$$
b_1(X_\alpha) =
\sum_{i=1}^r |W_i(\Gamma) \cap \chargp{\alpha}
(\chargp{G})|.
$$
\end{corollary}
\heading{Example.} We illustrate the above
exposition
using the well known case of the trefoil knot in the
three sphere $S^3$:
$$
\epsffile{trefoil}
$$
One presentation of the fundamental group of the complement is
$\Gamma =
\langle x,y : xyx y^{-1}x^{-1}y^{-1} \rangle$.
Then $\Sigma_1$ is a bouquet of two circles
and $F = \pi_1(\Sigma_1)$ has two generators
$x,y$ one for each positive loop around the
circles. The maximal abelian covering of
$\Sigma_1$ is the lattice ${\cal L}_2$. Now
take the relation $R = xyxy^{-1}x^{-1}y^{-1}
\in F$. The lift of $R$
at the origin of the
lattice is drawn in the figure below.
$$
\epsffile{fig1}
$$
Note that the order in which the path segments
are taken does not matter in computing the
1-chain. One can
verify that $D(R)$ is
the 1-chain defined by
$$
(1-t_x+t_xt_y)\langle x \rangle
+(-t_xt_y^{-1}
+ t_x -
t_x^2)\langle y \rangle.
$$
Thus, the Alexander matrix for the relation
$R$ is
$$
M(F_r,{\cal R}) = \left[\begin{array}{c}1-t
+ t^2\cr -1 + t - t^2\end{array}\right].
$$
Here $t_x$ and $t_y$ both map to the generator
$t$ of ${\Bbb Z}$ under the abelianization of $\Gamma$.
The Alexander stratification of $\Gamma$ is thus given by
\begin{eqnarray*}
V_0(\Gamma) &=& \chargp{\Gamma} = {\Bbb C}^*;\cr
V_1(\Gamma) &=& V(1-t+t^2);\cr
V_i(\Gamma) &=& \emptyset \qquad \mbox{for $i \ge 2$}.
\end{eqnarray*}
Note that the torsion points on $V_1(\Gamma)$ are the two primitive
$6$th roots of unity $\exp{(\pm{2\pi}/6)}$.
Now let $\alpha : \Gamma \rightarrow G$ be any epimorphism onto
an abelian group. Then since $\mathrm{ab}(\Gamma) \cong {\Bbb Z}$, $G$ must
be a cyclic group of order $n$ for some $n$. This means the
image of
$\chargp{\alpha} : \chargp{G} \rightarrow {\Bbb C}^*$ is the set of
$n$-th roots of unity in ${\Bbb C}^*$. Let $X_n$ be the $n$-cyclic
unbranched covering of the complement of the trefoil corresponding to
the map
$\alpha = \alpha_n$. By Proposition 2.5.6,
$$
b_1(X_n) = \left \{ \begin{array}{ll}3 & \mbox{if $6 | n$}\cr
1 & \mbox{otherwise.}\end{array}\right.
$$
\section{Group theoretic constructions and Alexander invariants.}
\subsection {Group homomorphisms.}
Let $\Gamma$ and $\Gamma'$ be finitely presented groups
and let $\alpha : \Gamma' \rightarrow \Gamma$ be a
group homomorphism. In this section, we look at what
can be said about the Alexander strata of the groups
$\Gamma$ and $\Gamma'$ in terms of $\alpha$.
\begin{lemma} The homomorphism
$$
T_\alpha : C^1(\Gamma,\rho) \rightarrow C^1(\Gamma',
\chargp{\alpha}(\rho))
$$
given by composition with $\alpha$
induces a homomorphism
$$
\widetilde{T_\alpha} : H^1(\Gamma,\rho)
\rightarrow H^1(\Gamma',\chargp{\alpha}(\rho)).
$$
\end{lemma}
\proof It suffices to show that if
$f$ is an element of $B^1(\Gamma,\rho)$, then
$T_\alpha(f)$ is an element of $B^1(\Gamma',\chargp{\alpha}(\rho))$.
For any $f \in B^1(\Gamma,\rho)$, there is a constant
$c \in {\Bbb C}$ such that for all $g \in \Gamma$,
$$
f(g) = (1-\rho(g))c.
$$
Then, for any $g' \in \Gamma'$,
\begin{eqnarray*}
T_\alpha(f) (g') &=& (1 - \rho(\alpha(g'))c\\
&=& (1 - \chargp{\alpha}(\rho)(g'))c.
\end{eqnarray*}
Thus, $T_\alpha(f)$ is in $B^1(\Gamma',\chargp{\alpha}(\rho))$. \qed
The following lemma follows easily from the definitions.
\begin{lemma} If $\alpha : \Gamma' \rightarrow \Gamma$
is a group homomorphism, then
(1) implies (2) and (2) implies (3),
where (1), (2), and (3) are the
following statements.
\begin{description}
\item{(1)} $\widetilde{T_\alpha} : H^1(\Gamma,\rho)
\rightarrow H^1(\Gamma',\chargp{\alpha}(\rho))$ is
injective;
\item{(2)} $\dim H^1(\Gamma,\rho) \leq \dim H^1(\Gamma',\chargp{\alpha}(\rho)$, for all $\rho \in \chargp{\Gamma}$; and
\item{(3)}
$\chargp{\alpha}(W_i(\Gamma)) \subset W_i(\Gamma')$.
\end{description}
\end{lemma}
\begin{proposition} If $\alpha : \Gamma' \rightarrow
\Gamma$ is an epmiorphism, then
$$
\widetilde{T_\alpha} : H^1(\Gamma,\rho)
\rightarrow H^1(\Gamma',\rho)
$$
is injective.
Furthermore,
$$
\chargp{\alpha}(V_i(\Gamma)) \subset V_i(\Gamma').
$$
\end{proposition}
\proof
To show the first statement we need to show that if
$T_\alpha(f) \in B^1(\Gamma',\chargp{\alpha}(\rho))$
for some $\rho \in \chargp{\Gamma}$, then
$f \in B^1(\Gamma,\rho)$.
If $f \in C^1(\Gamma,\rho)$ and $T_\alpha(f)
\in B^1(\Gamma',\chargp{\alpha}(\rho))$, then for
some $c \in {\Bbb C}$ and all $g' \in \Gamma'$ we have
$$
T_\alpha(f) = (1 - \chargp{\alpha}(\rho)(g'))c.
$$
Take $g \in \Gamma$. Since $\alpha$ is surjective,
there is a $g' \in \Gamma'$ so that $\alpha(g') = g$.
Thus,
\begin{eqnarray*}
f(g) &=& f(\alpha(g'))\\
&=& T_\alpha(f)(g')\\
&=& (1- \chargp{\alpha}(\rho)(g'))c\\
&=& (1 - \rho(\alpha(g'))c\\
&=& (1 - \rho(g))c.
\end{eqnarray*}
Since this holds for all $g \in \Gamma$, $f$ is in
$B^1(\Gamma,\rho)$.
The second statement follows
from Lemma 3.1.2, Lemma 2.2.3 and
Corollary 2.4.3, since
$\chargp{\alpha}$ is injective and
sends the trivial character to the trivial character.
\qed
\begin{proposition} If $\alpha : \Gamma' \rightarrow
\Gamma$ is a monomorphism whose image has finite
index in $\Gamma$, then, for any $\rho \in \chargp{\Gamma}$,
$$
\widetilde{T_\alpha} :
H^1(\Gamma,\rho) \rightarrow H^1(\Gamma',\chargp{\alpha}(\rho))
$$
is injective.
\end{proposition}
\proof We can assume that $\Gamma'$ is a subgroup of
$\Gamma$. Take any $\rho \in \chargp{\Gamma}$.
We can think of $\chargp{\alpha}(\rho)$ as the restriction of the
representation $\rho$ on $\Gamma$ to the subgroup
$\Gamma'$.
The map $\widetilde{T_\alpha}$ is then the restriction
map
$$
\mathrm{res}^\Gamma_{\Gamma'} : H^1(\Gamma,\rho) \rightarrow
H^1(\Gamma',\chargp{\alpha}(\rho))
$$
in the notation of Brown (\cite{Bro:Coh}, III.9).
Furthermore, one can define a {\it transfer map}
$$
\mathrm{cor}^\Gamma_{\Gamma'} : H^1(\Gamma',\chargp{\alpha}
(\rho)) \rightarrow H^1(\Gamma,\rho)
$$
with the property that
$$
\mathrm{cor}^\Gamma_{\Gamma'}\circ \mathrm{res}^\Gamma_{\Gamma'} :
H^1(\Gamma,\rho) \rightarrow H^1(\Gamma,\rho)
$$
is multiplication by the index $[\Gamma:\Gamma']$
of $\Gamma'$ in $\Gamma$ (see \cite{Bro:Coh},
Proposition 9.5).
This implies that $\mathrm{res}^\Gamma_{\Gamma'}$ is
injective.
\qed
Note that Proposition 3.1.4 does not hold
if $\alpha(\Gamma)$ does not have finite
index. For example, let $\alpha :
F_{1} (= {\Bbb Z}) \hookrightarrow
F_{2}$ be the inclusion of the free group on
one generator into that free group on two
generators, sending the generator of $F_{1}$
to the first generator of $F_{2}$.
Then for any $\rho \in \chargp{F_{2}}$,
$$
\dim \ H^1(F_{2},\rho) = 2 > 1 =
\dim \ H^1(F_{1},\chargp{\alpha}(\rho)).
$$
\subsection{Free products.}
In this section, we treat free products of finitely
presented groups. The easiest case is a free group.
Since there are no relations, it is easy to see that
$$
V_i(F_r) = \chargp{F_r} = ({\Bbb C}^*)^r
$$
for $i=1,\dots,r-1$ and is empty for $i \ge r$.
Thus,
$$
W_i(F_r) = \left\{\begin{array}{ll} ({\Bbb C}^*)^r
\qquad &\mbox{if $i=1,\dots,r-1$,}\\
\{\chargp{1}\} &\mbox{if $i = r$}
\end{array}\right.
$$
and is empty for $i > r$.
\begin{proposition} If $\ \Gamma = \Gamma_1 * \dots * \Gamma_k$
is a free product of $k$ finitely presented groups,
then
$$
V_i(\Gamma) = \sum_{i_1 + \dots + i_k} V_{i_1}(\Gamma_1) \oplus
\dots \oplus V_{i_k}(\Gamma_k).
$$
\end{proposition}
\proof We first do the case $k=2$. Suppose $\Gamma$ is isomorphic to the
free product $\Gamma_1 *
\Gamma_2$, where $\Gamma_1$ and $\Gamma_2$ are
finitely presented groups with
presentations $\langle F_{r_1},{\cal R}_1 \rangle$ and
$\langle F_{r_2},{\cal R}_2 \rangle$, respectively. Suppose
${\cal R}_1 = \{R_1,\dots,R_{s_1}\}$
and
${\cal R}_2 = \{S_1,\dots,S_{s_2}\}$. Then, setting $r = r_1 + r_2$
and noting the isomorphism $F_r \cong F_{r_1} *
F_{r_2}$, $\Gamma$ has the finite presentation
$\langle F_r,{\cal R} \rangle$ where ${\cal R} = \{
R_1,\dots,R_{s_1},S_1,\dots,S_{s_2}\}$.
The character group
$\chargp{F_r}$ splits
into the product $\chargp{F_r} =
\chargp{F_{r_1}} \times \chargp{F_{r_2}}$. Thus, each
$\rho \in \chargp{\Gamma}$ can be written as $\rho =
(\rho_1,\rho_2)$, where $\rho_1 \in \chargp{F_{r_1}}$ and
$\rho_2 \in \chargp{F_{r_2}}$.
The vector
space ${\cal D}_r({{\cal R}})(\rho)$ splits into a direct sum
${\cal D}({{\cal R}})(\rho) = {\cal D}({{\cal R}_1}) (\rho_1) \oplus
{\cal D}({{\cal R}_2})(\rho_2)$ so we have
$$
\dim {\cal D}({{\cal R}})(\rho) = \dim {\cal D}({{\cal R}_1})(\rho_1)+ \dim {\cal D}({{\cal R}_2})(\rho_2).
$$
The rest follows by induction.
\qed
\subsection{Direct products.}
In this section we deal with groups $\Gamma$
which are finite products of finitely
presented groups.
\begin{lemma}
Let $\Gamma$ be the direct product
of free groups $F_{r_1}\times\dots\times F_{r_k}$.
Let $q_i : \Gamma \rightarrow F_{r_i}$ be the
projections. Let $r = r_1 + \dots + r_k$
and let $m = \max\{r_1,\dots,r_k\}$. Then
$$
V_i(\Gamma) =
\left\{\begin{array}{ll}
\bigcup_{i<r_j}\chargp{q_j} (\chargp{F_{r_j}})
&\qquad\mbox{if $1 \leq i < m$;}\\
\{ \chargp 1 \} &\qquad\mbox{if $m \leq i <
r$;}\\
\emptyset &\qquad\mbox{if $i \ge r$.}
\end{array}\right.
$$
\end{lemma}
\proof
We know from section 3.2 that
$$
V_i(F_{r_j}) = \left\{
\begin{array}{ll}
\chargp{F_{r_j}}&\qquad\mbox{for $i < r_j$;}\\
\emptyset&\qquad\mbox{for $i \ge r_j$.}
\end{array}
\right.
$$
By Proposition 3.1.3, the epimorphisms
$q_j : \Gamma \rightarrow F_{r_j}$ give inclusions
$$
\chargp{q_j}(\chargp{F_{r_j}}) \subset V_i(\Gamma)
$$
for all $j$ such that $i < r_j$.
This gives the inclusion
$$
\bigcup_{i<r_j} \chargp{q_j}(\chargp{F_{r_j}})
\subset V_i(\Gamma)
$$
for all $i < m$.
Let $x_{i,1},\dots,x_{i,r_i}$ be the generators
for $F_{r_i}$, for $i=1,\dots,k$.
Let $F_r = F_{r_1}*\dots*F_{r_k}$. For $i,j=1,\dots,k$,
$i < j$, $\ell = 1,\dots,r_i$ and $m =
1,\dots,r_j$, let
$R_{i,\ell,j,m} = [x_{i,\ell},x_{j,m}]$.
Let
$$
{\cal R} = \{\ R_{i,\ell,j,m} \ : \
i \neq j\ \}.
$$
Then $\langle F_r,{\cal R} \rangle$ is a presentation for
$\Gamma$. Let $\Lambda_r$ be the Laurent
polynomials in the generators $t_{i,\ell}$,
$i=1,\dots,k$, $\ell = 1,\dots,r_i$ and
associate this to the ring of functions on
$\chargp{F_r} = \chargp{\Gamma}$ by sending
$x_{i,\ell}$ to $t_{i,\ell}$.
We have
$$
D(R_{i,\ell,j,m}) = (1- t_{j,m})
\langle x_{i,\ell} \rangle +
(t_{i,\ell} - 1) \langle x_{j,m}\rangle.
$$
It immediately follows that $M(F_r,{\cal R})(\chargp{1})$
is the zero matrix, so $\chargp{1} \in
V_i(\Gamma)$ for $i<r$ and $\chargp{1} \not\in
V_i(\Gamma)$ for $i \ge r$.
Now consider $\rho \in \chargp{F_r} =
\chargp{\Gamma}$ with $\rho \neq \chargp{1}$. We
will show that if $\rho \in \chargp{q_i}(F_{r_i})$
then $\rho \in V_n(\Gamma)$ for $n < r_i$ and $\rho
\not\in V_n(\Gamma)$ for $n \ge r_i$.
If $\rho \not\in \chargp{q_i}(F_{r_i})$ for any
$i$, then we will show that $\rho \not\in
V_1(\Gamma)$.
Let
$\rho_{i,\ell}$,
$i=1,\dots,k$ and $\ell=1,\dots,r_i$, be
the component of $\rho$ corresponding to
the generator $t_{i,\ell}$ in $\Lambda_r$.
For each $i=1,\dots,k$, let $s_i = r_1 + \dots
+ \widehat{r_i} + \dots + r_k$.
Take $\rho \in \chargp{q_i}(F_{r_i})$. We know
from Proposition 3.1.3
that $\rho \in V_n(\Gamma)$ for $n < r_i$. Also,
$\rho_{j,m} = 1$, for all $j=1,\dots,\hat
i,\dots,k$. Since $\rho \neq \chargp{1}$,
$\rho_{i,\ell} \neq 1$ for
some $\ell$. Consider the $s_i \times s_i$
minor of $M(F_r,{\cal R})(\rho)$ with rows
corresponding to the generators $\langle
x_{j,m}\rangle$
and columns
corresponding to generators
$R_{i,\ell,j,m}$, where
$j=1,\dots,\hat{i},\dots,k$ and $m=1,\dots,r_j$.
This is the $s_i \times s_i$ matrix
$$
(1-\rho_{i,\ell})I_{s_i}
$$
where $I_{s_i}$ is the $s_i \times s_i$
identity matrix. Thus, rank $M(F_r,{\cal R})(\rho) \ge s_i$.
This means that $\rho \not\in V_n(\Gamma)$ for $n
\ge (r-s_i) = r_i$.
Now take $\rho \not\in \chargp{q_i}(F_{r_i})$
for any $i$. Then, for some $i$ and $j$
with $i\neq j$, and some $\ell$
and $m$, we have $\rho_{i,\ell} \neq 1$
and $\rho_{j,m} \neq 1$. Consider the minor
of $M(F_r,{\cal R})(\rho)$ with columns corresponding to all
generators except $x_{i,\ell}$,
and rows corresponding to relations
$R_{i,\ell,j',m'}$, where $j'=1,\dots,\hat
i,\dots,k$ and $m'=1,\dots,r_{j'}$, and
$R_{i,\ell',j,m}$, where $\ell' = 1,\dots,\hat
\ell,\dots,r_i$.
This is the $r-1 \times r-1$ matrix
$$
\left [
\begin{array}{ll}
\pm(1-\rho_{i,\ell})I_{s_i} & 0\\
0 & \pm(1-\rho_{j,m})I_{r_i-1}
\end{array}
\right ]
$$
which has rank $r-1$.
Thus, $\rho$ is not in $V_1(\Gamma)$.
\qed
\begin{corollary} Let $\Gamma$ be the direct
product of finitely presented groups
$$
\Gamma = \Gamma_1 \times \dots \times \Gamma_k
$$
with $r_1,\dots,r_k$ generators, respectively.
Let
$$
P = F_{r_1} \times \dots \times F_{r_k}.
$$
Then
$$
V_i(\Gamma) \subset V_i(P)
$$
for each $i$ and, in particular,
$$
V_i(\Gamma) \subset\{\chargp{1}\}
$$
if $\max\{r_1,\dots,r_k\} \le i$.
\end{corollary}
\proof This follows from Lemma 3.1.2 and Proposition 3.1.3.
\qed
In particular, if $\Gamma$ is abelian, we have the following
result.
\begin{corollary} If $\Gamma$ is an abelian group, then
$$
V_i(\Gamma) = \left \{
\begin{array}{ll}
\{\chargp{1}\}&\qquad\mathrm{if} \ 1 \leq i < \mathrm{rank} (\Gamma)\cr
\emptyset&\qquad\mathrm{otherwise.}
\end{array}
\right .
$$
Here $\mathrm{rank}(\Gamma)$ means the rank of the abelianization of $\Gamma$.
\end{corollary}
\section{Applications.}
Let $X$ be any topological space homotopy equivalent to a finite CW
complex with fundamental group $\Gamma$.
In this section, we will study the role that rational
planes in the Alexander
strata $V_i(\Gamma)$ and the jumping loci $W_i(\Gamma)$
relate to the the geometry of $X$.
\subsection{Betti numbers of abelian coverings.}
Let $X$ be homotopy equivalent to a finite CW complex.
Let $\Gamma = \pi_1(X)$. We will relate the first
Betti number of finite abelian coverings of $X$ to
rational planes in the jumping loci $W_i(\Gamma)$.
Let $\alpha : \Gamma \rightarrow G$ be an epimorphism
onto a finite abelian group $G$. Assume that $\Gamma$
is generated by $r$ elements. Then
by Corollary 2.5.7, we have
$$
b_1 (X_\alpha) = \sum_{i=1}^r |W_i(\Gamma) \cap \chargp{\alpha}(\chargp{G})|.
$$
Since $G$ is finite, all points in $\alpha(\chargp{G})$
have finite order. Thus, to compute $b_1(X_\alpha)$ for
finite abelian coverings $X_\alpha$, we need only
know about the torsion points on $W_i(\Gamma)$.
The position of torsion points $\mathrm{Tor}(V)$ for any
algebraic subset $V \subset ({\Bbb C}^*)^r$ is described by
the following result due to Laurent \cite{Laur:Equ}.
\begin{theorem}(Laurent) If $V \subset ({\Bbb C}^*)^r$ is
any algebraic subset, then there exist rational
planes
$P_1,\dots,P_k$ in $({\Bbb C}^*)^r$ such that $P_i
\subset V$ for each $i = 1,\dots,k$ and
$$
\mathrm{Tor}(V) = \bigcup_{i=1}^k \mathrm{Tor}( P_i).
$$
\end{theorem}
\noindent
From this theorem it follows that, to any finitely
presented group $\Gamma$, we can associate
a collection of finite sets of rational planes ${\cal P}_i$, such that
$$
\mathrm{Tor}(V_i(\Gamma)) = \bigcup_{P \in {\cal P}_i} \mathrm{Tor}(P).
$$
We thus have the following.
\begin{corollary} The rank of co-abelian, finite index subgroups of
a finitely presented group $\Gamma$ depends only on the rational planes
contained in the Alexander strata $V_i(\Gamma)$.
\end{corollary}
\subsection{Existence of irrational pencils.}
Let $X$ be a compact K\"ahler manifold.
An {\it irrational pencil on $X$} is a surjective
morphism
$$
X \rightarrow C_g,
$$
where $C_g$ is a Riemann surface
of genus $g \ge 2$. In this section, we will discuss
the relation between properties of the Alexander
stratification for $\Gamma = \pi_1(X)$ and
the existence of irrational pencils on $X$.
Let $\Gamma_g$ be the fundamental group of
$C_g$.
Then $\Gamma_g$ has presentation $\langle F_{2g},
R_g \rangle$, where $R_g$ is the single element
$$
[x_1,x_{g+1}][x_2,x_{g+2}]\dots[x_g,x_{2g}].
$$
The Fox derivative of $R_g$ is given by
$$
D(R_g) = \sum_{i=1}^g (t_i - 1)\langle x_i
\rangle + \sum_{i=g+1}^{2g} (1-t_i)\langle x_i
\rangle.
$$
Thus, we have
$$
V_i(\Gamma_g) = \left\{
\begin{array}{ll}
\chargp{\Gamma_g}\cong({\Bbb C}^*)^{2g} &\qquad\mbox{if
$1 \leq i < 2g-1$;}\cr
\{\chargp 1\} &\qquad\mbox{if $i=2g-1$;}\cr
\emptyset &\qquad\mbox{if $i>2g-1$.}
\end{array}
\right.
$$
and for the jumping loci
$$
W_i(\Gamma_g) = \left\{
\begin{array}{ll}
\chargp{\Gamma_g}\cong ({\Bbb C}^*)^{2g} &\qquad\mbox{if $1\leq i<2g-1$;}\cr
\{\chargp{1}\} &\qquad\mbox{if $2g-1 \le i \le
2g$;}\cr
\emptyset &\qquad\mbox{if $i > 2g$.}
\end{array}
\right.
$$
Given an irrational pencil $X \rightarrow C_g$, the
Stein factorization gives a map
$$
X \rightarrow C_h \rightarrow C_g,
$$
where the map from $C_h$ to $C_g$ is a finite
surjective morphism and $X$ has connected fibers.
Then $h \ge g$ and there is a surjective group
homomorphism
$$
\pi_1(X) \rightarrow \Gamma_h.
$$
By Proposition 3.1.3, this implies that there is
an inclusion
$$
W_i(\Gamma_h) \rightarrow W_i(\pi_1(X)),
$$
for all $i$.
We can thus conclude the following.
\begin{proposition} If $X$ has an irrational pencil
of genus $g$, then for some $h \ge g$,
$W_i(\pi_1(X))$ contains an affine subtorus of dimension
$2h$, for $i=1,\dots,2h-2$.
\end{proposition}
The question arises, do the maximal affine
subtori in
$W_i(\pi_1(X))$ all come from irrational pencils?
This was answered in the affirmative by Beauville
\cite{Beau:Ann} for $W_1(\pi_1(X))$
(see also \cite{G-L:HighOb},\cite{Ar:Higgs} and \cite{Cat:Mod}.)
This shows that the irrational pencils on
$X$ only depend on the topological type of $X$
(see also \cite{Siu:Strong}).
Now suppose $V \subset W_i(\Gamma)$ is a translate
of an affine subtorus by a character $\rho \in
\chargp{\Gamma}$ of finite order.
Then, since $\Gamma$ is finitely generated, the image
of $\rho$ is finite in ${\Bbb C}^*$.
Let $\widetilde X \rightarrow X$ be the finite
abelian unbranched covering associated to this map.
Then the corresponding map on fundamental groups
$$
\alpha : \pi_1(\widetilde X) \rightarrow \pi_1(X)
$$
has image equal to the kernel of $\rho$.
Thus, $\chargp{\alpha}(\rho)$ is the trivial
character in $\chargp{\pi_1(\widetilde X)}$ and
$\chargp{\alpha}(V)$ is a connected subgroup,
i.e., an affine subtorus of
$\chargp{\pi_1(\widetilde X)}$.
As we discuss in the next section, a theorem
of Simpson shows that all the jumping loci
$W_i(\pi_1(X))$ are finite unions of rational
planes.
This leads us to the following question:
\heading{Question.} Can all the rational planes
in the jumping loci
$W_i(\pi_1(X))$ be explained by irrational
pencils on $X$ or on finite abelian coverings
of $X$?
\vspace{12pt}
\subsection{Binomial criterion for K\"ahler groups.}
If $\Gamma$ is a group such that there is an isomorphism
$\Gamma \cong \pi_1(X)$ for some compact K\"ahler manifold $X$,
we will say that $\Gamma$ is {\it K\"ahler}.
A {\it binomial ideal} in $\Lambda_r({\Bbb C})$ is
an ideal generated by {\it binomial} elements of the form
$$
t^\lambda - u
$$
where $\lambda \in {\Bbb Z}^r$, $t^\lambda =
t_1^{\lambda_1}\dots t_r^{\lambda_r}$ and
$u \in {\Bbb C}$ is a unit.
The following is straightforward.
\begin{lemma}
If $V \subset ({\Bbb C}^*)^r$ is a rational plane
then $V$ is defined by a binomial ideal where
the units $u$ are roots of unity.
\end{lemma}
In (\cite{Ar:Higgs}, Theorem 1), Arapura shows that $W_i(\Gamma)$ is
a finite union of unitary translates of affine tori.
Simpson (\cite{Sim:Subs}, Theorem 4.2) extends Arapura's result, showing
that the $W_i(\Gamma)$ are actually translates of rational tori.
\begin{theorem} (Simpson)
If $\Gamma$ is K\"ahler, then $W_i(\Gamma)$
is a finite union of rational planes for all $i$.
\end{theorem}
\begin{corollary}
If $\Gamma$ is K\"ahler, then any irreducible component
of $V_i(\Gamma)$ is defined by a binomial ideal.
\end{corollary}
\proof By Lemma 2.2.3, $V_i(\Gamma)$ equals $W_i(\Gamma)$
except when $i$ equals the rank of the abelianization of $\Gamma$.
Suppose the latter holds. Then, again by Lemma 2.2.3, $V_i(\Gamma)$
is $W_i(\Gamma)$ minus the identity character $\chargp{1}$. But
$V_i(\Gamma)$ is a closed algebraic set, so $\chargp{1}$ is an
isolated component of $W_i(\Gamma)$. Thus, since $W_i(\Gamma)$
is a finite union of rational planes, so is $V_i(\Gamma)$.
The rest follows from Lemma 4.3.1.
\qed
\heading{Remark.} Stated in terms of the ideals of
minors (also known
as Alexander ideals or fitting ideals) of an
Alexander matrix, Simpson's theorem implies
a property of the radical of these ideals
for K\"ahler groups.
Subtler and interesting questions can be asked about the
fitting ideals themselves. We leave this as a topic
for further research.
\vspace{12pt}
Let $R_g$ be the standard relation for $\pi_1(C_g)$, where
$C_g$ is a Riemann surface of genus $g$.
It is possible from Corollary 4.3.3 to make
many examples of nonK\"ahler finitely presented groups.
For example, we have the following Proposition
(cf. \cite{Ar:SurFun}, \cite{Gro:Sur}, \cite{Sim:Subs}).
\begin{proposition} Let $g \ge 2$ and let
$$
\Gamma = \langle x_1,\dots,x_{2g} :
S_1,\dots,S_s \rangle,
$$
where
$$
S_i = u_{i,1}R_g u_{i,1}^{-1}
\dots u_{i,k_i}R_g u_{i,k_i}^{-1},
$$
for $i=1,\dots,s$.
Let
$$
p_i = \mathrm{ab}(u_{i,1}) + \dots + \mathrm{ab}(u_{i,k_i})
$$
considered as a polynomial in $\Lambda_r$.
Then, if $\Gamma$ is K\"ahler,
the set of common zeros $V(p_1,\dots,p_s)$
must be defined by binomial ideals.
\end{proposition}
\proof
The Fox derivative $D: F_{2g} \rightarrow
{\Bbb Z}[\mathrm{ab}(F_{2g})]^{2g}$ takes each $S_i$ to
$$
D(S_i) = (\mathrm{ab}(u_{i,1}) + \dots + \mathrm{ab}(u_{i,k_i}))
D(R_g).
$$
Thus, the $i$th row of the Alexander matrix $M(F_{2g},{\cal R})$
equals $M(F_{2g},R_g)$, considered as row vector,
multiplied by $p_i$.
It follows that the rank of $M(F_{2g},{\cal R})$ is at most
1 and equals 1 outside of the set of common zeros of $p_1,\dots,p_s$
and the point $(1,\dots,1)$.
The rest is a consequence of Corollary 4.3.3.
\qed
\heading{Example.} Fix $g \ge 3$, and let $\Gamma$ be given
by
$$
\Gamma = \langle x_1,\dots,x_{2g} : S_1,S_2 \rangle,
$$
where
$$
S_1 = x_1 R_{2g} x_1^{-1} \dots x_g R_{2g} x_g^{-1}
$$
and
$$
S_2 = x_{g+1} R_{2g} x_{g+1}^{-1} \dots x_{2g} R_{2g} x_{2g}^{-1}.
$$
Then
\begin{eqnarray*}
D(S_1) &=& (t_1 + \dots + t_g) D(R_g)\\
D(S_2) &=& (t_{g+1} + \dots + t_{2g}) D(R_g)
\end{eqnarray*}
which implies that $V_1(\Gamma)$ contains $\chargp{1}$ and
the points in
$$
V(t_1 + \dots + t_g) \cap V(t_{g+1} + \dots + t_{2g}).
$$
This is isomorphic to the product of the hypersurface
in $({\Bbb C}^*)^g$ defined by $V = V(t_1 + \dots + t_g)$ with itself. Since
$g \ge 3$, this hypersurface is not defined by a binomial ideal.
Thus, $\Gamma$ is not K\"ahler.
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